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We introduce a novel technique for verification and model synthesis of
sequential programs.
Our technique is based on learning a regular model of the set of feasible paths
in a program, and testing whether this model contains an incorrect behavior.
Exact learning algorithms require checking equivalence between the model and
the program, which is a difficult problem, in general undecidable.
Our learning procedure is therefore based on the framework of probably
approximately correct (PAC) learning, which uses sampling instead and provides
correctness guarantees expressed using the terms error probability and
Besides the verification result, our procedure also outputs the model with the
said correctness guarantees.
Obtained preliminary experiments show encouraging results, in some cases
even outperforming mature software verifiers.
§ INTRODUCTION
Formal verification of software aims to prove software properties
through rigorous mathematical reasoning. Consider, for example,
the C statement specifying that the value of the variable
must be positive. If the assertion is formally verified,
it cannot be violated in any possible execution during runtime.
Formal verification techniques are, however, often computationally expensive.
Although sophisticated heuristics have been developed to improve
scalability of the techniques, formally verifying real-world software is still
considered to be impractical.
The complexity of current programs and the associated state space explosion
make the programs hard to be analyzed precisely, and over-approximating analyses
in abstract domains often flood users with false warnings.
Unsound analyses, on the other hand, show good scalability while
keeping the number of false warnings low <cit.>.
However, in case unsound analyses do not find and error, they mostly do not
provide any statistical guarantees of correctness of the program.
Statistical guarantees may be needed in the development of safety-critical
systems, for example during certification of software used in the aviation
industry <cit.>.
A common technique to ensure quality in industry is software
Errors in software can be detected by exploring different software behaviors
via injecting various testing vectors.
Testing cannot, however,
guarantee software is free from errors. Consider again the assertion
Unless all system behaviors are
explored by testing vectors, it is unsound to conclude that the
value of is always positive. Various techniques have been
proposed to improve its coverage, but it is an inherent feature of software
testing that it cannot establish program properties conclusively.
In this paper, we propose a novel learning-based approach that aims to balance
scalability and coverage of existing software engineering
techniques. In order to be scalable, as for software testing, our
new technique explores only a subset of all program behaviors.
Moreover, we apply
machine learning to generalize observed program behaviors for better
semantic coverage.
Our technique allows software engineers to combine scalable testing with
high-coverage formal analyses and improve the quality assurance process.
We hope that this work reduces the dichotomy between
formal and practical software engineering techniques.
In our technical setting, we assume programs are annotated with
program assertions. A program assertion is a Boolean expression
intended to be true every time it is encountered during program execution.
Given a
program with assertions, our task is to check whether all
assertions evaluate to true on all possible executions. In principle,
the problem can be solved by examining all program executions. It is,
however, prohibitive to inspect all executions exhaustively since there may
be infinitely many of them.
One way to simplify the analysis is to group the set of program executions to paths of a control flow graph.
A control flow graph (CFG) is derived from the syntactic structure of a
program source code. Each execution of a program corresponds to a path
in its control flow graph. One can therefore measure the completeness
of software testing on CFGs. Line coverage, for instance,
gives the ratio of explored edges in the CFG of the tested
program, while branch coverage is the ratio of explored
branches of this CFG. Note that such syntactic measures of code
coverage approximate program executions only very roughly. Executions that
differ in the
number of iterations in a simple program loop have the same
line and branch coverages, although their computation may be
drastically different. A full syntactic code coverage does not
necessarily mean all executions have been explored by software testing.
Observe that program executions traversing the same path in a CFG perform the
same sequence of operations (although maybe with different values).
Consider a path corresponding to a program execution in a CFG.
Such a path can be characterized by the sequence of
decisions that the execution took when traversing conditional
statements in the CFG. We call a sequence of such
decisions a decision vector. A decision vector is feasible if it
represents one or more (possibly infinitely many) program executions, and
infeasible if it represents a sequence of branching choices than can never
occur in an execution of the program.
To check whether all assertions evaluate to $\mathit{true}$ on all executions, it
suffices to examine all feasible decision vectors and check they do not
represent any assertion-violating program execution.
Although feasibility of a decision vector can be determined by using
an off-the-shelf Satisfiability Modulo Theories (SMT) solver,
the set of feasible decision vectors is in general difficult to compute
Therefore, we apply algorithmic learning, in particular the framework of
probably approximately correct learning, to construct a regular approximation
of this set.
The task of program verification can be described as determining whether there
exists a feasible decision vector such that a path characterized by the vector
leads to an error node in the control flow graph of the program.
In general, existence of such a vector is undecidable; moreover, the set of
feasible decision vectors is difficult to compute, in general, it may even not
be recursively enumerable.
OL: now talk about inspiration in natural language processing?
We can determine feasibility of a decision vector by constructing a path formula
of the path corresponding to the decision vector, and then discharge the
formula using an off-the-shelf SMT solver.
Within the framework of probably approximately correct (PAC) learning with queries,
learning algorithms query about target concepts to construct
hypotheses. The constructed hypotheses are then validated by
sampling. If a hypothesis is invalidated by witnessing a counterexample, learning
algorithms refine the invalidated hypothesis by the witness and more
queries. If, on the other hand, a hypothesis conforms to all
samples, PAC learning algorithms return the inferred hypothesis with
statistical guarantees. In our approach, we adopt a PAC learning
algorithm with queries to infer a regular language approximation to the set of feasible decision vectors of a program.
In each query, the PAC learning algorithm asks whether a posed
decision vector is feasible. The technique proposed in this paper answers the
query by determining feasibility of the decision vector with an SMT
solver. After a number of queries, the PAC learning algorithm constructs
a regular set of decision vectors as a hypothesis. Our technique
validates the hypothesis by software testing in the following way.
We collect a number of
program executions and check if the hypothesis contains all (feasible)
decision vectors of the sample executions. If any feasible decision
vector does not belong to the hypothesis, the PAC learning algorithm
refines the hypothesis with the witnessed counterexample and more queries.
Otherwise, we obtain a regular set
approximating the set of feasible decision vectors of the program with high
OL: the last sentence is vague... I don't like the “with high probability” part.
In our approach, we use a learning algorithm to infer a regular language that
approximates the set of feasible decision vectors of a program's control flow
The learning algorithm proceeds in iterations.
During an iteration, the algorithm selects decision vectors, tests their
feasibility, and builds a hypothesis—a finite automaton approximating the
set of feasible decision vectors of the program.
At the end of an iteration, the algorithm checks the quality of the presented
hypothesis by testing whether a number of decision vectors, taken randomly from
the language of the hypothesis, are feasible in the control flow graph of the
If the quality test fails, the algorithm proceeds with refining the hypothesis.
To grasp the statistical guarantees provided by PAC learning, consider the
task of checking defects in a large shipment using uniform
sampling. Because of the size of the shipment, it is impractical to check
every item. We instead want to know, with a given confidence $\delta$, if the defect probability is at
most $\epsilon$. This can be done by
selecting $r$ (to be determined later) randomly chosen items. If all
chosen items are good, the method reports that the defect probability
is at most $\epsilon$. We argue the simple method can err with
probability at most $1 - \delta$. Suppose $r$ randomly chosen items
are tested without any defect, but the defect probability is, in fact,
more than $\epsilon$.
Under this thesis, the probability that
$r$ random items are all good is lower than $(1 - \epsilon)^r$. That is,
the method is incorrect with probability lower than
$(1 - \epsilon)^r$. Take $r$ such that
$(1 - \epsilon)^r < 1 - \delta$. The simple method
reports incorrect results with probability at most $1 -
\delta$; we equivalently say the result of the method is $\pacofed$-correct.
Suppose we test items chosen randomly and fail to find any defect after testing
$r$ items.
What can we say about the probability that an item randomly chosen from the
shipment will be a defect?
Assume we want to claim “the probability that a chosen item is a defect is at
most $\epsilon$ under uniform distribution.”
(Or, equivalently, that the probability that the item is good is at least $1-
\epsilon$.)
How confident can we be about our claim?
If our claim is false, the probability of selecting $r$ good items is at most
$(1 - \epsilon)^r < 1$.
For any $0 < \delta < 1$, there is an $r$ such that $1 - (1 - \epsilon)^{r} >
\delta$.
This means that the probability of finding a defect in $r$ random items is at
least $\delta$.
If none from $r$ random items is a defect, we say the shipment is good with
defect probability at most $\epsilon$ and confidence at
least $\delta$.
Observe that if we want to increase the confidence in our claim, we either need
to increase the claimed error probability or the number of tested items.
On the other hand, if we want to decrease the error probability, we either need
to decrease our confidence or increase the number of tested items.
Using a similar argument, it can be shown that our PAC learning algorithm
returns a regular set approximating the set of feasible decision
vectors of the program with the error probability $\epsilon$ and confidence $\delta$
of our choice.
If the inferred set contains no decision vector representing an assertion-violating program
execution, our technique concludes the verification with statistical guarantees about correctness.
In the other case, when the quality test succeeds, the algorithm
intersects the hypothesis with a finite automaton obtained from the control flow
graph of the program, and tests whether there is an error node reachable in the
In case there is such an error node, because the hypothesis
is approximate, the algorithm checks whether the decision vector leading to the
error node is indeed feasible.
In case it is, we report an error.
If not, we continue in the next iteration of the learning algorithm, using the
infeasible decision vector as a counterexample for refinement of the hypothesis.
If the intersection does not contain any error node, we can conclude that the
verified program is probably correct, with some statistical guarantees.
Our learning-based approach finds a balance between formal analysis
and testing. Rather than exploring program behaviors exhaustively, our
technique infers an approximation of the set of feasible decision vectors by
queries and sampling.
Although the set of feasible decision vectors is
in general not computable, PAC learning with queries may still return a
regular set approximation of it with a quantified guarantee.
Such an approximate model
with statistical guarantees can be useful for program
With an approximate model that is $\pacofed$-correct and proved to be free from assertion violation, one can conclude that the program is also $\pacofed$-correct.
The statistical guarantees are different from syntactic code
coverages in software testing.
Recall that our application of PAC learning works over decision vectors.
Decision vectors in turn represent
program executions. When our technique does not find any assertion
violation, the statistical guarantees give software engineers a
semantic coverage about program executions. Along with conventional
syntactic coverages, such information may help software engineers
estimate the quality of software.
We implement a prototype, named (PAC learning-based Model synthesizer
and ANalyzer), of our procedure based on program verifiers , ,
and the concolic tester .
We evaluate the prototype on the benchmarks from the recursive category of SV-COMP 2015 <cit.>.
The results are encouraging—we can find all errors that can be found by .
We also provide quantified guarantee accompanied by a faithful approximate
model for several examples that are challenging for program verifiers and
concolic testers.
This approximate model can later be reused, e.g., for verifying the same program with a different set of program assertions.
Our contributions are summarized in the following:
* We show the PAC learning algorithm can be applied to synthesize
a faithful approximate model of the set of feasible decision vectors of
a program.
Such a model can be useful in many different aspect of program verification
(cf. Section <ref> for details).
We believe it is not hard to adopt our approach to handle different type of
systems (e.g., black box systems) and to obtain approximate models on
a different level of abstraction (e.g., on a function call graph).
* We develop a verification procedure based on the approximate model obtained from PAC learning. The procedure integrates the advantages of both testing and verification. It uses testing techniques to collect samples and catch bugs. The PAC learning algorithm generalizes the samples to obtain an approximate model that can then be analyzed by verification techniques for statistical guarantees.
We note that exact learning cannot be applied to the task of learning a regular
set of feasible vectors.
The reason for this is that the set of feasible decision vectors is in general
indeed not regular, so if the learning algorithm keeps iterating, one cannot know
whether it just needs to refine a few more times, or whether the algorithm never
terminates, due to the set of feasible vectors being not regular.
When compared with statistical model checking, our technique is different,
although both make statistical claims.
Statistical model checking tests whether a property holds in a given model, and
makes statistical claims about the property, not generating models.
Our technique generates models and makes statistical claims about
A program assertion is a Boolean expression intended to be true
when it is evaluated during execution. Given a program annotated with
assertions, we would like to check whether all assertions evaluate to
true on all possible executions. In principle, the problem can be
solved by examining all program executions. It is however prohibitive
to inspect all executions exactly. Control flow graphs are used to
approximate program executions in practice. The syntax-based
approximation makes most program analysis techniques inherently
intentional. Researches are invested to recover real program
behaviors from the syntactic abstraction.
OL: I don't understand the 3 last sentences
Rather than control flow graphs, we propose to represent program
executions by decision vectors. A decision vector characterizes
a program execution by recording conditional decisions. For each
conditional statement during an execution, a decision is made and
directs the execution. The sequence of conditional decisions during a
program execution is the decision vector of the execution. Using an
Satisfiability Modulo Theories (SMT) solver, one can simultaneously
examine all program executions with the same decision vector.
An arbitrary sequence of conditional decisions may not correspond to a
program execution. We say a decision vector is feasible if it
corresponds to a program execution. Computing the set of feasible
decision vectors nonetheless is not easy. One certainly can get the
decision vector from any program execution. Yet it is not clear how to
compute all feasible decision vectors. Similar problems arise
in natural language processing.
English corpora are abundant. Can English grammar be reconstructed
automatically? As in natural language processing, we apply machine
learning to find feasible decision vectors.
Consider the set of feasible decision vectors as an unknown
formal language. We use a learning algorithm to infer an approximate regular
language. The learning algorithm represents inferred regular languages
by finite automata. In order to construct an automaton, the learning
algorithm needs to know whether a decision vector is feasible or
not. When an automaton is inferred, the learning algorithm uses a
number of random decision vectors to test the quality of the
automaton. If the inferred automaton passes the tests, we use it to
represent feasible decision vector and hence executions of a given
After an approximation to feasible decision vectors is derived, we
check if any decision vector leads to assertion violation. We
compute error decision vectors by transforming the control flow
graph into a finite automaton accepting decision vectors leading to
assertion violation. If the intersection of feasible and error
decision vectors is empty, there is no assertion violation in the
program. Otherwise, recall that feasible decision vectors are
approximate. We check if the error decision vector leads to
assertion violation by an SMT solver before reporting to the user.
Similar to natural language processing, we do not know the set of
feasible decision vectors exactly and never will. Subsequently,
there is no way to know if the inferred regular language captures
feasible decision vectors precisely. The theory of Probably
Approximately Correct (PAC) learning nonetheless allows us to quantify
the quality of approximations to feasible decision vectors.
To give a glimpse, consider checking defects in a (very large)
shipment by sampling uniformly. Suppose we fail to find any defect
after testing $r$ random items. What can we say about the
probability of defects in the shipment? How confident are we
about our claim? Assume we want to claim “the probability of defects
is at most $\epsilon$ under the uniform distribution.” If our
claim is false, the probability of selecting $r$ good items is at most
$(1 - \epsilon)^r < 1$. For any $0 < \delta < 1$, there is an $r$ that
$1 - (1 - \epsilon)^{r} > \delta$. Then the probability of finding a
defect in $r$ random items is at least $\delta$. If $r$ random items
fail to defects, we say the shipment is good with at most $\epsilon$
of error and at least $\delta$ of confidence.
Based on a similar argument, the learning algorithm guarantees
to generate a regular approximation to feasible decision vectors
with error and confidence of our choice. Even though our technique
cannot verify a program conclusively, it for instance can show that a
program is correct with at most $1 \%$ of error and at least $99 \%$
of confidence. Such statistical statements can be informative when
formal verification fails to perform in circumstances.
We note that exact learning is not applicable to finding feasible
decision vectors. Since feasible decision vectors are unknown,
one cannot decide whether it is correctly inferred. Exact learning
subsequently does not know when to finish. Also, our technique is
different from statistical model checking although both make
statistical claims. In statistical model checking, a property is
verified against a given model. It makes statistical claims about the
property and does not generates models. Our technique generates models
and makes statistical claims about models.
§ PRELIMINARIES
Let $\mathcal{X}$ be the set of program variables and $\mathcal{F}$
the set of function and predicate symbols.
We use $\mathcal{X'}$ for the set $\{x' \mid x \in
\mathcal{X} \}$. The set $\mathcal{T}[\mathcal{X}, \mathcal{F}]$ of
transition formulae consists of well-formed first-order logic
formulae over $\mathcal{X}, \mathcal{X'}$, and $\mathcal{F}$.
For a transition formula $f \in \mathcal{T}[\mathcal{X}, \mathcal{F}]$
and $n \in \bbfN$, we use $\lift{f}{n}$ to denote
the formula obtained from $f$ by replacing all free variables $x \in
\mathcal{X}$ and $x' \in \mathcal{X'}$ with $\lift{x}{n}$ and
$\lift{x}{n+1}$ respectively.
We represent a program with a single procedure using a control flow
(Section <ref> extends the notion to programs with
multiple procedures and procedure calls.)
A control flow graph (CFG) is a graph $G=(V, E, v_i, v_r, V_e, \fp)$
where $V = V_b \cup V_s$ is a finite set of nodes consisting of
disjoint sets of branching nodes $V_b$ and sequential
nodes $V_s$, $v_i
\in V$ is the initial node, $v_r \in V_s$ is the return node, $V_e \subseteq V$ is the set of
error nodes, $\fp\subseteq \mathcal{X}$ is the set of formal parameters, and $E$ is a finite set of edges such that $E \subseteq V \times
\mathcal{T}[\mathcal{X}, \mathcal{F}] \times V$ and the following conditions hold:
* for any branching node $v_b \in V_b$, there are exactly two nodes $v'_0,
v'_1 \in V$ with $(v_b, f_0, v'_0), (v_b, f_1, v'_1) \in E$, where
$f_0, f_1 \in \mathcal{T}[\mathcal{X}, \mathcal{F}]$ are transition
* for any non-return sequential node $v_s \in V_s \setminus \{v_r\}$, there is exactly one node
$v' \in V$ with $(v_s, f, v') \in E$; and
* for the return node $v_r \in V_s$, there is no $v' \in V$ such
that $(v_r, f, v') \in E$ for any $f \in \mathcal{T}[\mathcal{X},
\mathcal{F}]$.
A control flow graph (CFG) of a program is a tuple $G=(V_b \cup V_s, E, v_i, V_e, \fp)$
where $V_b \cup V_s$ is the (finite) set of nodes consisting of
disjoint sets of branching nodes $V_b$ and sequential nodes $V_s$, the node $v_i
\in V_s$ is the initial node, the set $V_e \subseteq V_b \cup V_s$ is the set of
error nodes, $\fp\subseteq \mathcal{X}$ is the set of formal parameters, and the set $E \subseteq (V_b \cup V_s) \times
\mathcal{T}[\mathcal{X}, \mathcal{F}] \times (V_b \cup V_s)$ is the set of
edges, where
* for any branching node $v_b \in V_b$, there are exactly two nodes $v'_0,
v'_1 \in V_b \cup V_s$ with $(v_b, f_0, v'_0), (v_b, f_1, v'_1) \in E$, where
$f_0, f_1 \in \mathcal{T}[\mathcal{X}, \mathcal{F}]$ are transition
formulae such that $f_0 \lor f_1$ is a valid formula; and
* for any sequential node $v_s \in V_s$, there is at most one node
$v' \in V_b \cup V_s$ with $(v_s, f, v') \in E$ for $f \in
\mathcal{T}[\mathcal{X}, \mathcal{F}]$.
We say $v'$ is a successor of $v$ if $(v, f, v') \in E$.
Assume, moreover, that the two successors $v'_0$ and $v'_1$ of the
branching node $v$ are ordered.
Intuitively, the `1' corresponds to the if branch and the `0'
corresponds to the else branch.
We call $v'_0$ and $v'_1$ the
$0$-successor and $1$-successor of $v$ respectively.
Similarly, $f_0$ and $f_1$ are called the $0$-transition and
$1$-transition formulae of $v$.
Note that the definition of a CFG allows us to describe nondeterministic
choice, which is commonly used to model the environment.
To be more specific, a nondeterministic choice from a branching node $v$ can be
represented by defining both the $0$-transition and $1$-transition formulae of
$v$ as $\bigwedge_{x \in
\mathcal{X}} x=x'$.
A path in the CFG $G$ is a sequence $\pi = \pthof{v_0, f_1, v_1, f_2, v_2,
\ldots, f_m, v_m}$ such that $v_0 = v_i$ and $(v_j, f_{j+1}, v_{j+1}) \in E$ for
every $0 \leq j < m$.
The path $\pi$ is feasible if the path formula
$\bigwedge_{k=1}^m \lift{f_k}{k}$ is satisfiable.
It is an error
path if $v_j \in V_e$ for some $0 \leq j \leq m$.
The task of our analysis is to check whether $G$ contains
a feasible error path.
A sequence $w = a_1 a_2
\cdots a_n$ with $a_j \in \{0,1\}$ for $1 \leq j \leq n$ is called a
word over $\{0,1\}$. The length of $w$ is $|w| = n$. The
word of length 0 is the empty word $\lambda$. We also use
$w[j]$ to denote the $j$-th symbol $a_j$.
If $u, w$ are words over $\{0,1\}$, $u \cdot w$ denotes the
concatenation of $u$ and $w$.
A language $L$ over $\{0,1\}$ is a set of words over $\{0,1\}$.
$L^{<k} = \{ w \in L \mid | w | < k\}$.
A language over $\{0,1\}$ is a set of words over $\{0,1\}$.
Let $L$ be a language over
$\{0,1\}$. We define $L^{\leq k} = \{ w \in L \mid | w | \leq k\}$ and,
similarly, $L^{=k} = \{ w \in L : | w | = k\}$.
$L^{<k} = \{ w \in L \mid | w | < k\}$.
For a word $d = d_1 d_2 \dots d_n$, we use $d[j]$ to denote the symbol $d_j$ if
$0 \leq j \leq n$; otherwise $d[j]$ is undefined.
We introduce the function $\DEC$ that maps a path $\pi$ of $G$ to a sequence of
decisions made in the branching nodes traversed by $\pi$.
Formally, $\DEC$ is a function from paths to words over $\{0,1\}$ defined recursively as follows:
$\DEC(\pthof{v_0, f_1, v_1, f_2, v_2, \ldots, f_m, v_m}) = \DEC(\pthof{v_0, f_1}) \cdot\DEC(\pthof{v_1, f_2, v_2, \ldots, f_m, v_m})$ such that
\begin{eqnarray*}
\DEC(\pthof v) \!\!\!&=& \lambda \\
\DEC(\pthof{v, f}) \!\!\!&=&
\!\!\!\left\{
\begin{array}{ll}
\lambda & \textmd{if } v \in V_s,\\
0 & \textmd{if } v \in V_b \textmd{ and } \\
& f \textmd{ is the
$0$-transition formula of } v,\\
1 & \textmd{if } v \in V_b \textmd{ and }\\
& f \textmd{ is the
$1$-transition formula of } v.\\
\end{array}
\right.
\end{eqnarray*}
For a path $\pi$, $\DEC (\pi)$ is the decision vector of $\pi$.
We lift $\DEC$ to a set of paths $\Pi$ and define decision vectors of
$\Pi$ as $\DEC (\Pi) = \{ \DEC (\pi) \mid \pi \in \Pi \}$.
A finite automaton (with $\lambda$-moves) $A$ is a
tuple $A = (\Sigma, Q, q_i, \Delta,
F)$ consisting of a finite alphabet $\Sigma$, a finite set of
states $Q$, an
initial state $q_i \in Q$, a transition relation $\Delta
\subseteq Q \times (\Sigma \cup \{ \lambda \}) \times Q$, and a set of
accepting states $F \subseteq Q$.
A transition $(q, \lambda,
q') \in \Delta$ is called a $\lambda$-transition.
A word $w$ over $\Sigma$ is
accepted by $A$ if there are
states $q_0, \dots, q_m \in Q$ and symbols (or
$\lambda$'s) $a_1, \dots, a_m \in (\Sigma \cup \{ \lambda \})$,
such that $w = a_1 \cdots a_m$,
for every $0 \leq j < m$ there is a transition $(q_j, a_{j+1}, q_{j+1}) \in \Delta$,
and further $q_0 = q_i$ and $q_{m} \in F$.
The language of $A$ is defined as $L (A) =
\{ w \mid w \textmd{ is accepted by } A \}$. A language $R$ is regular if
$R = L (A)$ for some finite automaton $A$. The finite automaton $A$ is
deterministic if its transition relation is a function from $Q
\times \Sigma$ to $Q$. For any finite automaton $A$, there exists
a deterministic finite automaton (DFA) $B$ such that $L (A) = L (B)$.
A finite automaton with $\epsilon$-transitions (FA-$\epsilon$) is
a tuple $A_{\epsilon} = (\Sigma, Q, q_i, \Delta_{\epsilon}, F)$ where $\Sigma$,
$Q$, $q_i$, and $F$ have the same meaning as in the definition of an FA (with
the additional constraint that $\epsilon \not\in \Sigma$), but the transition
$\Delta_{\epsilon} \subseteq Q \times (\Sigma \cup \{\epsilon\}) \times Q$ can
also contain the so-called $\epsilon$-transitions.
Note that any FA-$\epsilon$ $A_{\epsilon}$ can be transformed into an FA $A$
using the standard textbook algorithm for removing $\epsilon$-transitions.
We define the language of $A_{\epsilon}$ to coincide with the language of $A$.
A pushdown automaton (PDA) is a tuple $P = (\Sigma, Q, \Gamma, q_i,
\Delta, F)$ where $\Sigma$ is a finite input alphabet, $Q$ is a finite
set of states, $\Gamma$ is a finite stack alphabet, $q_i \in Q$
is the initial state, $F \subseteq Q$ is the set of final states,
and $\Delta \subseteq Q \times (\Sigma \cup \{\lambda\}) \times (\Gamma \cup
\{\lambda\})\times (\Gamma \cup \{\lambda\}) \times Q$ is a transition
We use $\pdatrans q a b c {q'}$ to denote the transition $(q, a, b, c, q')$,
and we sometimes simplify $\pdatrans q a \lambda \lambda {q'}$ to
$(q, a, q')$.
We define a configuration of $P$ as a pair $(q, \gamma) \in Q \times
\Gamma^*$.
A word $w$ over $\Sigma$ is accepted by $P$ if there exists a sequence of
configurations $(q_0, \gamma_0), \ldots, (q_m, \gamma_m) \in Q \times \Gamma^*$
and a sequence of symbols (or $\lambda$'s) $a_1, \ldots, a_m \in (\Sigma \cup
\{\lambda\})$, such that $w = a_1 \cdots a_m$, $q_0 = q_i$, $\gamma_0 =
\epsilon$, $q_m \in F$, and for every $0 \leq j < m$ it holds that
there are some $b_j, b_{j+1} \in (\Gamma \cup \{\lambda\})$ and $\gamma'_j,
\gamma'_{j+1} \in \Gamma^*$ such that $\gamma_j = b_j \gamma'_j, \gamma_{j+1} =
b_{j+1} \gamma'_{j+1}$, and there is $\pdatrans {q_j} {a_{j+1}} {b_j}
{b_{j+1}} {q_{j+1}} \in \Delta$.
The language of $P$ is defined as $\langof P = \{w \mid w \text{~is accepted
by~} P\}$.
§ OVERVIEW
In this section, we give an overview of our verification procedure.
Let $G$ be a CFG of a program. Our goal is to check whether there is a
feasible error path in $G$.
More concretely, consider the set $\Pi$ of feasible paths
in $G$ and the set $\badpaths$ of error paths in
We call the languages $\DEC (\Pi)$ and $\DEC (\badpaths)$ over the alphabet
$\{ 0, 1 \}$ as feasible decision
vectors and error decision vectors respectively.
The program is correct if the intersection $\DEC
(\Pi) \cap \DEC (\badpaths)$ is empty, i.e., if $G$ contains no feasible error path.
We therefore would like to check if $\DEC (\badpaths) \cap \DEC (\Pi)$
is empty.
Components of our verification procedure
Representation of the language $\DEC (\Pi)$ of all feasible
decision vectors in $G$ is not so easy.
In general, this language may not be regular or even computable.
In our procedure, we construct a candidate finite automaton $C$ that
approximates $\DEC(\Pi)$,
the set of feasible decision vectors of $G$.
We infer $C$ using a probably approximately correct (PAC) online
automata learning algorithm <cit.>.
The use of PAC learning provides us with statistical guarantees about the
correctness of $C$—we can claim that $C$ is $\pacofed$-correct, i.e., with
confidence $\delta$, the deviation of $\langof C$ from $\DEC(\Pi)$
is less than $\epsilon$ (we give a proper explanation of the terms in
Section <ref>).
On the other hand, it is straightforward to convert $G$ to a finite automaton $B$
accepting the set of all error decision vectors $\DEC (\badpaths)$.
Intuitively, states of $B$ correspond to nodes of $G$, the initial state of $B$
corresponds to the initial node of $G$, and accepting states of $B$ correspond
to $G$'s error nodes.
An edge from a sequential node is translated to a
$\lambda$-transition. For a branching node, the edges to its $0$- and
$1$-successors are translated to transitions over symbols $0$ and
$1$ respectively (cf. Section <ref>).
A high-level overview or our learning procedure is given in
Figure <ref> (the procedure is similar in structure to the
one of <cit.>).
It consists of two main components:
The learning algorithm asks the teacher two kinds of questions:
membership (“Is a given decision vector feasible?”) and
equivalence (“Is a given candidate finite automaton
$\pacofed$-correct?”) queries.
The teacher resolves the queries, at the same time observing whether some of
the tested decision vectors corresponds to a feasible error path.
By posing these queries, either the learning algorithm iteratively constructs
a $\pacofed$-correct approximation of $\DEC(\Pi)$ or our procedure finds
a feasible error decision vector.
However, in our setting, we do not have a unique target; any language
satisfying the two conditions is qualified. This is very similar to the setting
of learning-based compositional
verification <cit.>, where the learning algorithm is
modified to return a counterexample in case the system is buggy.
We modified the learning algorithm in a similar way. At the bottom of Figure <ref>, we add two tests to handle the case when the system is buggy, i.e., $L(C)\cap \DEC(\Pi) \neq \emptyset$.
OL: I don't understand the last sentence, YFC: try again, OL: I still don't understand the last sentence, YFC:yet another try.
As with other online learning-based techniques, we need to devise
a mechanical teacher that answers queries from the learning
Checking membership queries (i.e., membership in the set $\DEC(\Pi)$ of
feasible decision vectors) is relatively
easy—for example, given a decision vector $d$, we obtain its corresponding
path $\pi$ by unfolding the CFG $G$ according to $d$, and use
an off-the-shelf solver
to decide whether $\pi$ is feasible or not
(cf. Section <ref>).
When the automata learning algorithm infers a candidate finite automaton $C$,
we need to check whether $\langof C$ approximates $\DEC (\Pi)$, i.e., whether $C$ is
Since we cannot compare $\DEC(\Pi)$ with
$\langof C$ directly, we employ the sampling-based approximate equivalence technique of
PAC learning.
While generally unsound, the technique still provides statistical guarantee
about the correctness of the inferred model (details are given in
Section <ref>).
OL: OLD
When the automata learning algorithm infers a candidate finite automaton $C$, since
$\DEC (\Pi)$ is unknown to us, checking $L (C) = \DEC
(\Pi)$ is not straightforward.
Instead of deciding whether $L (C)=\DEC (\Pi)$, we aim to infer a
probable approximation to $\DEC (\Pi)$ using
the technique of PAC learning.
Intuitively, we want to infer a finite automaton that accepts “nearly all words in $\DEC(\Pi)$” and can accept any word that is not in $\DEC(\Pi)$.
Now the meaning of “nearly all words in $\DEC(\Pi)$” is still vague.
It will become clear after Section <ref>, where we will introduce how we extend the PAC learning theory for exact learning to our setting.
We will use the PAC learning theory to substitute the equivalence checking with
random sampling and get a statistical guarantee of correctness.
When an automaton approximation $C$ of $\DEC(\Pi)$ such that $L(C) \cap L(B) = \emptyset$ is found, we will conclude that the program is “probably approximately correct”. Otherwise, a feasible decision vector may be returned when $\DEC(\Pi) \cap L(B) \neq \emptyset$. In the next three sections, we will introduce further details of the three components.
OL: OLD END
To be more specific, with a sampling mechanism required to support PAC guarantee, we can answer whether $L (C) \supseteq \DEC(\Pi)$ on a candidate finite automaton $C$ probably approximately by the following steps (the top-right of Figure <ref>):
* We compute the number of samples required for obtaining the PAC guarantee
and pick a set $S \subseteq \DEC(\Pi)$ of decision vectors using the sampling
* We check if any of the decision vectors in $S$ is not in $L(C)$.
For the case that all of them are in $L(C)$, we conclude that the system is
“probably approximately correct”.
* Otherwise, there must be a decision vector $s\in S$, such that $s$ is not in $L(C)$.
By checking if $s \in \DEC(\Pi)$, we either conclude that $s$ is a feasible
error decision vector or return it to the learning algorithm to refine the
next conjecture.
In Section <ref>, we summarize the algorithm and show the theoretical guarantees of our algorithm.
More precisely, we will select a
number of decision vectors from $L (S_G)$
where $S_G$ is a finite automaton derived from the CFG $G$. One
must be careful in designing the sample language which decision
vectors are selected from.
OL: say sth more
The naïve universal sample language will not work because the
difference between the empty language and $\DEC (\Pi)$ is
insignificantly small.
OL: what do you mean by “insignificant”
Section <ref> explains
how to construct the finite automaton $S_G$ from $G$.
After the sample language is determined, we design a sampling
algorithm to choose decision vectors randomly from $L (S_G)$.
Since different decision vectors can be selected with different
probabilities, our
sampling algorithm decides OL: I don't understand a distribution on $L (S_G)$. The
distribution in turn determines the quality of the inferred finite
automaton (see <ref>).
For instance, suppose a
sampling algorithm only selects decision vectors of length $1$. Then
PAC learning does not have any guarantee on the difference of $L (C)$
and $\DEC (\Pi)$ for decision vectors of length greater than $1$. In
order to attain better statistical claim, it is necessary to sample
decision vectors of various lengths from $L (S_G)$ fairly
(Section <ref>).
Using the sampling algorithm, we test the conformance of $L (C)$ and
$\DEC (\Pi)$ on a number of random decision vectors. If they
agree on these decision vectors, we have a finite
automaton $C$ satisfying <ref> for the considered
error and confidence parameters.
The approximation $L (C)$ is then intersected with the error decision
vectors $L (B)$. If the result is empty, an approximation to CFG $G$
has no feasible error path. The program is therefore probably approximately
correct. Otherwise, let $\be \in L (C) \cap L (B)$ be an error
decision vector. If it is a feasible decision vector, a
witness to an error in the CFG $G$ is found. Otherwise,
we ask the learning algorithm to modify $C$ by the
counterexample $\be$ (Section <ref>).
§ PAC AUTOMATA LEARNING
Here we explain the PAC automata learning algorithm that we use to find an approximation to $\DEC(\Pi)$. Classical PAC automata learning algorithm cannot be used directly for the purpose of program verification. It has to be modified to handle the case when the program contains an error.
The classical PAC automata learning algorithm was obtained from modifying the requirement of the exact automata learning algorithm <cit.>. Our modification follows the same route.
In this section, we first describe the classical “exact” automata learning algorithm of regular languages and then describe how to modify it for verification.
Then we explain how to relax the requirement of an exact automata learning algorithm to infer an approximation to $\DEC (\Pi)$.
§.§ Exact Learning of Regular Languages
Suppose $R$ is a target regular language such that its description is
not directly accessible.
Automaton learning algorithms <cit.> infer automatically a finite automaton $A_R$ recognizing $R$.
The setting of an online learning algorithm
assumes a teacher who has access to $R$ and can answer the
following two types of queries:
* Membership query $\MEM(w)$: is the word $w$ a member of $R$, i.e., $w \in R$?
* Equivalence query $\EQ(C)$: is the language of the finite automaton $C$ equal to
$R$, i.e., $L (C) = R$?
If not, what is a counterexample to this equality
(a word in the symmetric difference of $L(C)$ and $R$)?
The learning algorithm will then construct a finite automaton $A_R$ such
that $L (A_R) = R$ by interacting with the teacher.
Such an algorithm works iteratively: In each iteration, it
performs membership queries to get information about $R$ from the teacher.
Using the results of the queries, it proceeds by
constructing a candidate automaton $C$ and, finally, makes an
equivalence query $\EQ (C)$. If $L (C) = R$, the algorithm terminates
with $C$ as the resulting finite automaton $A_R$.
Otherwise, the teacher returns a word $w$ distinguishing $L (C)$ from the
target language $R$. The learning algorithm uses $w$ to
modify the conjecture for the next iteration.
The mentioned learning algorithms are guaranteed to find a finite automaton $A_R$
recognizing $R$ using a number of queries polynomial to the
number of states of the minimal DFA recognizing $R$. In the rest of the paper, we denote “online automata learning” simply as “automata learning”.
§.§ Learning for Program Verification
Under the context of program verification, it may be the case that $\DEC(\Pi)\cap L(B)\neq \emptyset$; in such a case, our procedure should return a feasible error path in the program. This is very similar to the setting
of learning-based
verification <cit.>, where the learning algorithm is
modified to return a counterexample in case the system contains an error.
We modified the used learning algorithm in a similar way.
To be more specific, when the classical learning algorithm poses an equivalence query $\EQ(C)$, we first check whether there exists a decision vector $c$ such that $c\in L(C)\cap L (B)$ and then test if $c\in \DEC(\Pi)$.
* In case that the two tests identified a decision vector $c$ such that $c
\in L(C)\cap L(B)$ and $c \not \in \DEC(\Pi)$, then $c$ is in the difference
of $L(C)$ and $\DEC(\Pi)$ and hence a valid counterexample for the classical
learning algorithm to refine the next conjecture automaton $C$.
* In case that the two tests identified a decision vector $c$ such that $c
\in L(C)\cap L(B)$ and $c \in \DEC(\Pi)$, then $c$ is a feasible error
decision vector and we report $c$ to the user.
* In case that $L(C) \cap L(B) =\emptyset$, the modified learning
algorithm poses an equivalence query $\EQ(C)$ to the teacher.
Given a teacher answers membership and equivalence queries about $\DEC(\Pi)$, the modified automata learning algorithm has the following properties.
Assume $\DEC(\Pi)$ is a regular set. The modified automata learning algorithm eventually finds a counterexample $c \in L(B)\cap \DEC(\Pi)$ when $L(B)\cap \DEC(\Pi)\neq \emptyset$. It eventually finds a finite automaton recognizing $\DEC(\Pi)$ when $L(B)\cap \DEC(\Pi)= \emptyset$.
Observe that when the program does not contain any error, the behavior of the modified learning algorithm is identical to the classical one and hence is still an exact automata learning algorithm. Next we explain how to relax the requirements of the exact automata learning algorithm to obtain a PAC automata learning algorithm that is suitable for program verification.
§.§ Probably Approximately Correct Learning
The techniques for learning automata we just discussed in the previous section assume a teacher who has the ability to answer equivalence queries.
Such an assumption is, however, invalid in our procedure.
Checking $\DEC(\Pi) = \langof C$ can be undecidable.
Consider, for instance, a finite state machine in a black-box.
The teacher is in this case unable
to answer equivalence queries since the structure of the
machine is hidden from him. Even in the case the teacher has access to the description
of the target language,
the complexity of answering equivalence queries can be prohibitive
(e.g., in the case of nondeterministic finite automata, it is
a PSPACE-complete problem).
Assuming such a teacher can be unrealistic sometimes.
Compared to equivalence queries, membership queries can be answered
rather easily even for black-box representations of the target language.
The teacher simply needs to send the inputs to the black-box and observe the output.
It is therefore natural to attempt to substitute equivalence testing by
observing results of membership queries of a set of sampled words in the two
regular languages.
This approach is clearly unsound—it is in general impossible to check
equivalence of a pair of infinite sets by sampling a finite number of elements.
On the other hand,
Angluin showed in <cit.>
that if we substitute equivalence queries with sampling, we can still make
statistical claims about the difference of the inferred set and the target set.
Assume that we are given a probability distribution $D$ over the elements of
a universe $\mathcal{U}$, and a hypothesis in the form
$$\probof{w \in
\mathcal{U}|_D}{\neg\varphi(w)} \leq \epsilon.$$
In the hypothesis, the term $\probof{w \in \mathcal{U}|_D}{\neg\varphi(w)}$ denotes the
probability that the formula $\varphi(w)$ is invalid for $w$ chosen randomly from
$\mathcal{U}$ according to the distribution $D$.
We call $\epsilon$ the error parameter
and use the term confidence to denote the
least probability that the hypothesis is correct.
We say that $\varphi(w)$ is $\pacofed$-valid if $\probof{w \in
\mathcal{U}|_D}{\neg\varphi(w)} \leq \epsilon$ with confidence $\delta$.
In the setting of automata learning, the considered universe is $\Sigma^*$ and the target regular
language is $R \subseteq \Sigma^*$.
The task of an equivalence query $\EQ(C)$ is changed from checking
exact equivalence, which we can express as checking that $\forall w \in \Sigma^*:
w \notin R \ominus \langof C$ (we use $\ominus$ to denote the symmetric difference operator), to
checking approximate equivalence, i.e., checking whether the formula
$\varphi(w) = w \notin R \ominus \langof C$ is $\pacofed$-valid.
In other words, we check whether $\probof{w \in \Sigma^*|_D}{w \in R \ominus \langof C}
\leq \epsilon$ with confidence $\delta$.
For a fixed $R$ and a candidate $C$, we say that $C$ is
$\pacofed$-correct if $w \notin R \ominus \langof C$ is $\pacofed$-valid.
The teacher checks the $\pacofed$-correctness of $C$ by picking $r$ samples according
to $D$ and testing if all of them are not in $R \ominus \langof C$.
For the $i$-th equivalence query of the learning algorithm, the number of samples $q_i$ needed to establish that $C$ is
$\pacofed$-correct is given by Angluin in <cit.> as
\begin{equation}
\label{eq:num-samples}
q_i = \left\lceil \frac{1}{\epsilon} \left(\ln \frac{1}{1-\delta}+i \ln 2 \right)\right\rceil .
\end{equation}
Since the inferred set $C$ is guaranteed to be $\pacofed$-correct, this approach is termed probably approximately correct
(PAC) learning <cit.>.
OL: OLD FOLLOWS
Suppose $D$ is a probability distribution over $\Sigma^*$ and suppose that we can pick
a word from $\Sigma^*$ according to this distribution.
Let $R$ be a target set, $\epsilon$ the error, and $\delta$ the confidence
Given an equivalence query $\EQ (C)$, the
response of the teacher is now of the form “with probability at least $\delta$, the sets $C$ and $R$ are identical with error at most $\epsilon$.”
\begin{equation}
\tag{Claim 1}
\mathit{Prob}_{D} {[}R \ominus C{]} \leq \epsilon \textmd{ with
confidence } \delta
\label{eqn:claim}
\end{equation}
where $R \ominus C$ is the symmetric difference of $R$ and $C$ and
$\mathit{Prob}_{D} {[}E{]}$ is the probability that a sample word picked
from $\Sigma^*$ according to the distribution $D$ is in $E$.
The teacher first determines the number $r$ of sample words needed to be tested
so that the claim holds with the given error and confidence parameters.
According to <cit.>, $r$ can be computed from the error and confidence parameters and the number of equivalence queries that have been posed.
If $C$ and $R$ agree on $r$ samples chosen according to the distribution $D$,
the teacher answers the equivalence query positively.
Otherwise, the teacher answers negatively and provides a counterexample, which
can be used by the learning algorithm for inferring a more precise candidate.
The number of random samples
needed for the $j$-th candidate $C_j$ is <cit.>:
\begin{equation*}
\lceil \frac{1}{\epsilon}(\ln \frac{1}{1-\delta} + j \ln 2) \rceil
\end{equation*}
To give some ideas about the number of samples, suppose we would like
to find a finite automaton with error $\epsilon = 0.1$ and confidence
$\delta = 0.01$ over a distribution $D$. If the first candidate
conforms to $8$ random samples on the distribution $D$, the first
candidate is statistically correct with respect to the parameters
$\epsilon$ and $\delta$. Otherwise, the second candidate need
conform to $14$ random sample for the same statistical correctness.
The third, fourth, and fifth candidates require $21$, $28$, and $35$
random samples respectively. It does not take too many random samples
to attain $90\%$ of error with $99\%$ of confidence over the
distribution $D$. Random sampling can be a practical alternative when
answering equivalence queries is impossible.
way to substitute equivalence queries with a number of membership
queries and get a probably approximately correct (PAC)
guarantee. (Give an example of $D$) Assume a distribution $D$ of
strings in $R$, for the $i$-th equivalence query on a conjecture
automaton $C_i$, his algorithm samples $q_i=\lceil
\frac{1}{\epsilon}(\ln \frac{1}{1-\delta} + i \ln 2) \rceil$ strings
from $R$ according to the given distribution $D$, where $\delta$ and
$\epsilon$ are two user specified numbers for the confidence level and
error rate, respectively.
For example, when $\delta=0.01$ and $\epsilon=0.1$, the number of membership queries needed for the $50$-th iteration $q_50 \approx 393$.
The algorithm then checks if any sampled string are in the difference of $R$ and $L(C_i)$ by membership queries to the oracle and membership tests to the automaton $C_i$. If a string $s$ is found in the difference of $R$ and $L(C_i)$, it will be returned to the learning algorithm as a counterexample required by the equivalence query.
Otherwise, the learning algorithm stops and conclude confidence $1-\delta$ that the difference of $L(C_i)$ and $R$ is smaller than $\epsilon$ according to the distribution $D$.
When $\delta=0.01$ and $\epsilon=0.1$, if the algorithm did not find any inconsistency with the $q_i$ queries, then we can conclude that with $99 \%$ confidence, difference between $L(C_i)$ and $R$ is smaller than $90 \%$.
Below we explain why PAC equivalence query can provide such a guarantee.
We use $\mathbf{P}(L(C_i),R)_D$ to denote the difference between $L(C_i)$ and $R$ according to the distribution $D$.
The possibility that the equivalence query makes a mistake at the $i$-th iteration, that is $\mathbf{P}(L(C_i),R)_D > \epsilon$ but is not detected by the $q_i$ membership queries, is $(1-\mathbf{P}(L(C_i),R)_D)^{q_i}$, which is smaller than $(1-\epsilon) ^{q_i}$.
The possibility that the equivalence query makes a mistake at any iteration is bounded by $\delta$.
The possibility that the equivalence query makes a mistake at any iteration is $\Sigma_{i=1}^{\infty}(1-\epsilon) ^{q_i}$.
(1) $\Sigma_{i=1}^{\infty}(1-\epsilon) ^{q_i} \leq \Sigma_{i=1}^{\infty} e^{-\epsilon q_i}$: When $\epsilon$ is small, $e^{- \epsilon} = 1 +
\frac{(- \epsilon)^1}{1!} + \frac{(- \epsilon)^2}{2!}
+ \frac{(- \epsilon)^3}{3!}+ \frac{(- \epsilon)^4}{4!}+ \frac{(- \epsilon)^5}{5!}+\ldots$
(Taylor Series). The number $\frac{(- \epsilon)^{2j}}{(2j)!}
+ \frac{(- \epsilon)^{2j+1}}{(2j+1)!}$ is always positive for all $j\leq 1$.
Hence we have $(1-\epsilon) \leq e^{-\epsilon}$.
(2) $\Sigma_{i=1}^{\infty}e^{-\epsilon q_i} \leq \Sigma_{i=1}^{\infty} \frac{\delta}{2^i}$: $e^{-\epsilon q_i} = e^{-\epsilon \lceil \frac{1}{\epsilon}(\ln \frac{1}{\delta} + i \ln 2) \rceil} \leq e^{-\epsilon \frac{1}{\epsilon}(\ln \frac{1}{\delta} + i \ln 2) } =
e^{ -\ln \frac{2^i}{\delta}}=
e^{ \ln \frac{\delta}{2^i}}= \frac{\delta}{2^i}$.
(3) $\Sigma_{i=1}^{\infty} \frac{\delta}{2^i} \leq \delta$.
More precisely, we will select a
number of decision vectors from $L (S_G)$
where $S_G$ is a finite automaton derived from the CFG $G$. One
must be careful in designing the sample language which decision
vectors are selected from.
OL: say sth more
The naïve universal sample language will not work because the
difference between the empty language and $\DEC (\Pi)$ is
insignificantly small.
OL: what do you mean by “insignificant”
Section <ref> explains
how to construct the finite automaton $S_G$ from $G$.
After the sample language is determined, we design a sampling
algorithm to choose decision vectors randomly from $L (S_G)$.
Since different decision vectors can be selected with different
probabilities, our
sampling algorithm decides OL: I don't understand a distribution on $L (S_G)$. The
distribution in turn determines the quality of the inferred finite
automaton (see <ref>).
For instance, suppose a
sampling algorithm only selects decision vectors of length $1$. Then
PAC learning does not have any guarantee on the difference of $L (C)$
and $\DEC (\Pi)$ for decision vectors of length greater than $1$. In
order to attain better statistical claim, it is necessary to sample
decision vectors of various lengths from $L (S_G)$ fairly
(Section <ref>).
Using the sampling algorithm, we test the conformance of $L (C)$ and
$\DEC (\Pi)$ on a number of random decision vectors. If they
agree on these decision vectors, we have a finite
automaton $C$ satisfying <ref> for the considered
error and confidence parameters.
The approximation $L (C)$ is then intersected with the error decision
vectors $L (B)$. If the result is empty, an approximation to CFG $G$
has no feasible error path. The program is therefore probably approximately
correct. Otherwise, let $\be \in L (C) \cap L (B)$ be an error
decision vector. If it is a feasible decision vector, a
witness to an error in the CFG $G$ is found. Otherwise,
we ask the $L^*$ algorithm to modify $C$ by the
counterexample $\be$ (Section <ref>).
§ RESOLVING EQUIVALENCE QUERIES BY SAMPLING
The current section discusses how to design a mechanism that the
teacher can use for equivalence queries to provide the $\pacofed$-correctness
guarantee, as defined in Section <ref>.
Given a probability distribution $D$ over the set of feasible decision vectors
$\DEC(\Pi)$, we can use $D$ to give a formal definition of the quality of
a candidate automaton $C$.
In particular, we use as a measure the probability with which a decision vector
chosen randomly from $\DEC(\Pi)$ (according to the distribution $D$) is
contained in $C$.
A sampling mechanism offering such a distribution must satisfy the following conditions:
* Only decision vectors in $\DEC(\Pi)$ are sampled.
* The samples are independent and identically distributed (IID),
i.e., the distribution is fixed and the probability of sampling a particular
element does not depend on the previously picked samples.
In this paragraph, we introduce the random input sampling mechanism.
We treat all nondeterministic choices and formal parameters of the program as input variables
and assume that all input variables are over finite domains.
Each set of initial values of input variables yields a path in the CFG
of the program.
Based on this observation, random input sampling works by
(1) picking uniformly at random a set of initial values for input variables of
the program and then
(2) obtaining the corresponding decision vector by traversing the CFG of the
program using the picked values.
The sampling mechanism forms a distribution such that the probability of
a decision vector $d$ being chosen is proportional to the number of program paths
corresponding to $d$.
The issue of random input sampling is that it suffers from the well-known fact
that coverage of input values is not a good approximation of program path
Depending on the sizes of input domains of program variables, some paths
might have only a negligible probability of being selected—for instance,
given two 64-bit integers and , the probability of taking
the true branch in the test is equal to $2^{-128}$.
The situation gets even worse for input variables over unbounded domains.
Even with an extremely high coverage rate of input variables' values,
many paths may still not be explored, while other are explored
In order to get a sampling mechanism with a better distribution over program
paths, we developed a technique that randomly explores program's paths
using a concolic
tester <cit.>, which is
an efficient means for exploring decision vectors corresponding to rare paths.
We describe the technique and prove its properties in the rest of this section.
§.§ Concolic Testing
Concolic testing is a testing approach that explores paths in the CFG of
a program while searching for bugs.
The algorithm begins with a decision vector generated by randomly picked input values.
Then, it finds the next decision vector by flipping some decision made in the
chosen path and obtains new input values that lead the program execution
according to the new path.
This mechanism gives rare paths a much greater chance to be explored.
The selection of which decision should be flipped depends on the used search strategy of the tester.
In our procedure, we use the concept of a batched sample.
A batched sample is defined as a set of decision vectors of the size $k$
(where $k$ is a given parameter) obtained from a concolic tester by exploring
$k$ paths using its search strategy.
We denote $D_k$ the distribution over elements of $(\Sigma^*)^k$ obtained in
this way.
Our procedure restarts the concolic tester after taking every batched sample.
The previous point gives us the guarantee that the probability of taking each
batched sample remains the same during the execution our procedure (we assume
that the concolic tester does not keep state information between its restarts),
and that the distribution is IID and, therefore, meets condition 2 defined above.
The principal functioning of concolic testers guarantees that condition 1 is
also met.
§.§ Generalized Stochastic Equivalence
In this section, we show that our sampling mechanism using batched samples has
the property required for the $\pacofed$-correctness guarantee of the
learning algorithm given in Section <ref>.
Recall that for the set of feasible decision vectors of a program $\DEC(\Pi)$
and a candidate automaton $C$ inferred by the learning algorithm using some
distribution $D$ over $\Sigma^*$, if the teacher gives the answer yes
for the equivalence query $\EQ(C)$, it guarantees with confidence $\delta$ that
\begin{equation}
\vspace{-0.2mm}
\probof{w\in \Sigma^*|_{D}}{w \in \DEC(\Pi) \ominus \langof{C}} \leq \epsilon .
\end{equation}
Since our sampling technique uses batched samples from the universe
$\mathcal{U}_k = (\Sigma^*)^k$ w.r.t. the distribution $D_k$ instead of
elements of $\Sigma^*$ and distribution $D$, we need to change the provided
guarantee in our modification of the learning algorithm.
If our algorithm answers yes, it guarantees that
\begin{equation}
\label{eq:mod-guarantee}
\vspace{-0.2mm}
\probof{S\in \mathcal{U}_k|_{D_k}}{\exists w\in S: w \in \DEC(\Pi) \ominus \langof{C}} \leq \epsilon
\end{equation}
with confidence $\delta$ (we hereafter use the term $\pacofed$-correct to
denote this form of guarantee).
When a teacher receives an equivalence query $\EQ(C)$, it uses a concolic tester to obtain $q_i$ (given in (<ref>)) batch samples.
For each batch sample $S$, the teacher checks if there exists a decision vector
$w\in S$ such that $w \notin \langof{C}$ (by definition $w\in \DEC(\Pi)$).
The teacher answers yes if there is no such $w$.
Otherwise, the teacher checks if $w$ is an error decision vector and either
reports $w$ as a feasible error decision vector or returns $w$ to the learning
algorithm to refine the next conjecture.
The following lemma shows that if we use the number $q_i$ batched samples for testing the equivalence, we
obtain the modified $\pacofed$-correctness guarantee from (<ref>).
Let $\epsilon$ and $\delta$ be the error and confidence parameters, and $R$ be
the target language. If no decision vector $w \notin \langof C$ is found in the $q_i$ batched samples, then it holds that $C$ is $\pacofed$-correct.
Based on the fact that $\langof C \cap \langof B =\emptyset$ (the property of the modified learning algorithm in Section <ref>) and the lemma above, we obtain the following corollary.
Let $\epsilon$ and $\delta$ be the error and confidence parameters, and $R$ be
the target language. If no decision vector $w \notin \langof C$ is found in $q_i$ batched samples, then it holds that the program is $\pacofed$-correct.
Note that in the algorithm above, the probability of a decision vector being picked for the first call is proportional to the number of paths it corresponds to. However, the probability of the 2nd decision vector being picked is actually depending on the 1st decision vector. Sampling using concolic testing violates the i.i.d. requirement (Section <ref>) for a sampling mechanism and hence cannot be used directly.
Given the sample language $L (S_G)$ for a CFG $G$, we give a sampling
mechanism which offers certain statistical bounded path coverage
guarantee with our technique. Let $k$ be a bound of maximal sample
length. Our sampling mechanism consists of the following steps:
* a number $r$ between $0$ and $k$ is
chosen uniformly.
* return a decision vector chosen uniformly from $L (S_G)^{=r}$ if
it is not empty; otherwise, go to step <ref>.
When $L$ is a regular language, uniformly choosing a word from
$L^{=r}$ can be done efficiently <cit.>. With our
sampling algorithm, a decision vector $\bd$ with length at most $k$ is
chosen with probability $\frac{1}{(k+1)|L (S_G)^{=|\bd|}|}$. Let us
denote the distribution on $L (S_G)$ by $D (S_G)$.
Let $\Pi$ be the set of feasible paths of the CFG $G$ and $C$ be the
conjecture automata from an equivalence query. With our sampling
mechanism, if an automaton candidate $C$ passes the random test, one
conclude that the difference between feasible decision vectors $\DEC
(\Pi)$ and $L (C)$ is at most $\epsilon$ of error with at least
$\delta$ of confidence on the distribution $D (S_G)$.
Formally, $\mathit{Prob}_{D (S_G)} {[}\DEC (\Pi) \ominus L (C){]} \leq
\epsilon \textmd{ with confidence at least } \delta $.
The sampling mechanism defines a probability distribution that each
non-empty length below the bound $k$ in $L (S_G)$ has an equal chance to
be picked. This has many advantages over other alternatives.
For example, we could select a decision vector in $L (S_G)^{\leq k}$
uniformly. Although it looks appealing, uniform selection on
$L (S_G)^{\leq k}$ is problematic. Let us assume $L (S_G)^{\leq k}$
contains all decision vectors of length at most $k$ for argument's
sake. Observe that a half of the decision vectors in $L
(S_G)^{\leq k}$ are of length $k$; a quarter of $L (S_G)^{\leq k}$ are
of length $k-1$.
Over $95\%$ of the words in
$\Phi^{\leq k}$ are of length bigger than $k-5$.
If a decision vector from $L (S_G)^{\leq k}$ is uniformly selected,
it is more likely than not to have length close to $k$. Such a sampling
mechanism would not be representative for our purposes.
In order to offer the kind of guarantee mentioned above, we need to create a
mechanism that samples uniformly at random words in $\DEC(\Pi)^{<k}$.
A naive approach is the following: we can sample uniformly at random words in
$(1+2)^{<k}$ by generating a random number $r$ between 1 and $|(1+2)^{<k}|$,
picking the $r$-th word in $(1+2)^{<k}$ (according to a predefined order), and
performing a membership query on the word to decide if the word is in
If the word is not in $\DEC(\Pi)^{<k}$, the sampling mechanism drops it and
randomly picks another one.
The procedure is repeated until a word in $\DEC(\Pi)^{<k}$ is found.
However, this procedure might be inefficient.
Consider a program that executes a loop 10 times at the beginning and then
Let $\Pi$ be the feasible paths of the program and pick $k=12$. Then only words
in $\DEC(\Pi)$ all prefixes of $1^{10}2$.
If we sample uniformly at random words in $(1+2)^{<12}$, a huge portion of them
are not in $\DEC(\Pi)$, i.e., only $12$ (the number of prefixes of $1^{10}2$)
out of the $2^{11}$ (all words of length smaller than $12$) words are correct
Hence some heuristics to alleviate the problem is needed.
Our heuristics make use of the fact that $\DEC(\Pi)$ is prefix closed, i.e., if a word $w$ is in $\DEC(\Pi)$, all prefixes of $w$ are also in $\DEC(\Pi)$. Each time when we find a sample word $w$ that is not in $\DEC(\Pi)$, we perform a binary search to find the shortest prefix of $w$ that is not in $\DEC(\Pi)$, and store it in a set $N$. For the next sampling, we exclude all words and their suffixes in $N$ and hence reduce the chance of doing some redundant computation.
§.§ Efficient Procedure and Data Structure for Sampling
To concretise the idea, below we describe a data structure to efficiently store
the set $N$ and use it for sampling.
The structure is a binary tree with numbers labeled on its nodes.
The label number denotes the total number of possible feasible branching
decisions below the node.
For each tree node, the edge to its left child is labeled $1$ and the one to
its right child is labeled $2$.
Initially, we have only the root node labeled with $2^k-1$.
Each time we find a shortest prefix $p$ that is not in $\DEC(\Pi)$ with length
$s$, we update the tree in three steps: (1) Add a branch to $p$ and all
corresponding new nodes to the tree and label all new added nodes with the
maximal number of possible feasible branching decisions below the node.
For example, a node of level $i$ (i.e., the vertical distance to the root is
$i$) has initially $2^{k-i}-1$ possible feasible branching decisions below it.
Note that we do not change the labels of all existing nodes in this step.
(2) For all nodes in this branch to $p$ (including the root node), we subtract
from their labeled number $2^{k-s}-1$, the number of words in $(1+2)^{<k}$ with
prefix $p$.
We do this because all words with prefix $p$ should be excluded from next
After this step, the label of the node for $p$ is $0$, which means no feasible
branching decision with such a prefix.
(3) For all nodes with only one child, we also add the missing child and label
it with $2^{k-i}-1$, where $i$ is the level of the newly added child.
This step can simplify the procedure of sampling that we will present in the
next paragraph.
Each time when we want to sample a new word, we first generate a random number between $1$ and the label of the root, store the number in a variable $n$, and then start the sampling procedure from the root node. We refer to the node we working currently working at the current node and use a variable $s$ to store the sample word. Initially, the current node is the root and $s$ is $\epsilon$.
* If the current node does not have any children, then we convert $n$ to a binary word $w$ (begin with the high digit), append the suffix of $w$ after the first occurrence of $1$ to $s$, and return $s$ as the sample word. For instance, if $n=10$, then $w=1010$ and the suffix of $w$ to be appended is $010$.
* If $n$ is smaller than the label of the left child node, then we change the current node to the left child node and append $0$ to $s$. Otherwise, we subtract the label of the left child node from $n$, change the current node to the right child, and append $1$ to $s$.
§ RESOLVING MEMBERSHIP QUERIES
In this section, we describe how a membership query $\MEM(\bd)$
in the algorithm in Figure <ref>
is discharged by the teacher.
Let $\Pi$ be the set of feasible paths of a CFG $G$. When the learning
algorithm asks a membership query
$\MEM (\bd)$, the teacher needs to check whether the decision vector $\bd$ is
in the set of feasible decision vectors $\DEC (\Pi)$.
To answer the query, the teacher first constructs a path $\pi=\pthof{v_0, f_1, v_1,
f_2, v_2, \ldots,v_{m-1}, f_m, v_m}$ in $G$ such that
* there are exactly $|\bd|$ occurrences of branching nodes in
the prefix $\pthof{v_0, f_1, v_1, f_2, v_2, \ldots,v_{m-1}}$ of $\pi$,
* if $v_k$ is the $j$-th branching node in $\pi$, it holds that
$\DEC(v_k,f_{k+1}) = \bd[j]$, and
* $v_{m-1}$ is a branching node.
Recall that $\pi$ is a feasible path if and only if $\varphi = \bigwedge_{j=1}^m
\lift{f_j}{j}$ is satisfiable.
Therefore, the teacher can simply construct the formula $\varphi$ from the path
$\pi$ and check its satisfiability using an off-the-shelf constraint solver.
Alternatively, the teacher can check feasibility by translating the path into a
sequence of program statements (with conditions removed and substituted by
assertions on the values of the conditions) and asking a symbolic executor or software model checker whether the final line of the constructed program is reachable.
The alternative option is easier to implement but usually suffers from some performance penalty.
Note that other paths can also be mapped to the same branching decision as $\pi$.
One may wonder that why it is sufficient to check just the path $\pi$. The main reason is that all such paths are with the prefix $\pi$ and also with $v_{m-1}$ as the last branching node. Since all edges from sequential nodes are non-blocking, the edge $(v_{m-1},f_m, v_m)$ is the last possibility to make those paths infeasible. Therefore if $\pi$ is feasible, then all such paths are also feasible.
§ ERROR DECISION VECTORS
Let $\badpaths$ be the set of (not necessarily feasible) error paths in
a CFG.
In this section, we show how we construct a finite automaton accepting the
set of all error decision vectors $\DEC (\badpaths)$ of the given CFG.
This automaton will later be intersected with the automaton representing the
set of feasible paths to determine whether the CFG contains a feasible error
Let $G = (V_b \cup V_s, E, v_i, v_r, V_e, \fp)$ be a CFG.
We define the error trace automaton for $G$ as the finite automaton $B =
(\{0, 1\}, V_b \cup V_s, v_i, \Delta_E, V_e)$ where $\Delta_E$ is defined as follows:
* $(v, 0, v'_0) \in \Delta_E$ if $v \in V_b\setminus V_e$, $(v, f_0, v'_0) \in E$,
and $v'_0$ is the $0$-successor of $v$;
* $(v, 1, v'_1) \in \Delta_E$ if $v \in V_b\setminus V_e$, $(v, f_1, v'_1) \in E$,
and $v'_1$ is the $1$-successor of $v$;
* $(v, \lambda, v') \in \Delta_E$ if $v \in V_s\setminus V_e$ and $(v, f, v') \in
E$; and
* $(v, 0, v), (v, 1, v) \in \Delta_E$ if $v \in V_e$.
Informally, $B$ contains a state for every node and a transition for every
edge of $G$.
It reads a symbol in each state corresponding to a branching node and performs
$\lambda$-transitions for states corresponding to sequential
nodes. For every error node, $B$ reads all remaining symbols
and accepts the input word. It is straightforward to see that
$B$ accepts exactly the set of decision vectors corresponding to
error paths in $G$.
Let $G = (V, E, v_i, v_r, V_e, \fp)$ be a CFG and $\badpaths$ the set
of error paths in $G$. Let $B$ be the error trace automaton for $G$. It holds that $L (B) = \DEC (\badpaths)$.
In Section <ref>, we describe an extension of our procedure to
programs with procedure calls.
Because representing the set of error decision vectors using a finite automaton
is in this setting imprecise, the section also discusses an extension that
represents the set of error paths in a program with procedure calls using
pushdown automata.
§ THE MAIN PROCEDURE
A detailed flow chart of our verification procedure
We summarize our procedure in this section.
Let $G$ be the CFG of the verified program, $k$ be the size of a batched
sample, $\epsilon$ be the error parameter, and $\delta$ be the confidence
The goal of our procedure is to
either find a feasible error decision vector of $G$ or show that $G$ is
In the latter case, we also
accompany our answer with a $\pacofed$-correct regular representation
of the set of feasible decision vectors of $G$.
Let $\Pi$ be the set of feasible paths of $G$, $D_k$ be the distribution defined
by our sampling mechanism
(cf. Section <ref>), and $\langof B$ be the
set of error decision vectors of $G$
(cf. Section <ref>).
A detailed
flow chart of our procedure can be found in Figure <ref>.
First, the bottom part of the figure describes our learning algorithm.
We extend the online automata learning algorithm with two additional
tests for verification, as described in Section <ref>.
In particular, when the automata learning algorithm outputs a candidate $C$,
before sending teacher the equivalence query $\EQ(C)$,
we first test whether $\langof C$ contains a feasible error decision vector $c$.
In case it does, we report $c$ as an error.
Otherwise, in the case $c$ is both in $\langof C$ and $\langof B$ but is not feasible,
we return $c$ to the learning algorithm to further refine the conjecture.
The top part of the figure describes our design of a mechanical teacher.
The task of the teacher is to answer queries from the learning algorithm.
Membership queries of the form $\MEM(w)$ can be answered by constructing the
path corresponding to the decision vector $w$ and the associated path formula,
which is then solved using a constraint solver
(cf. Section <ref>).
Equivalence queries, on the other hand, are discharged using a concolic tester
by checking whether there is a decision vector $s$ in the set of batched
samples $S$ such that it does not belong to the language of $C$
(cf. Section <ref>).
If no such a decision vector exists, we conclude that the program is
Otherwise, we test whether $s \in \langof B$; if this holds, we report
that we have found a feasible error decision vector.
In the case $s \notin \langof B$, it holds that $s$ is a feasible decision
vector in $\DEC(\Pi)$ but not in the language of the current conjecture
$\langof C$.
If this happens, we return $s$ to the automata learning algorithm to refine the
conjecture and continue with the next iteration of the learning loop.
In general, our procedure is not guaranteed to terminate.
When the procedure terminates and reports an error (either by the teacher or the learning algorithm), a feasible
error decision vector is found and the program is reported to be incorrect.
If the teacher approves an approximate finite automaton $C$, our procedure
reports that $C$ is an approximate
model of $\DEC(\Pi)$ w.r.t. the $\pacofed$-correctness guarantee, which, in turn, implies that the program is $\pacofed$-correct.
From Lemma <ref>, we have the following theorem
Let $\epsilon$ and $\delta$ be the error and confidence parameters respectively
and $\Pi$ be the set of feasible paths of a given CFG $G$.
If our procedure terminates with an approximate finite automaton $C$, the
program is $\pacofed$-correct.
Moreover, we obtain the following corollary.
Suppose our procedure reports a program $P$ is $\pacofed$-correct.
If we run the concolic tester with the same search strategy and batch size used
in our procedure on $P$, with confidence $\delta$, the concolic tester will
find an error with a probability less than $\epsilon$.
Thanks to the properties of the modified automata learning algorithm, when $\DEC(\Pi)$ is a regular set, our algorithm is guaranteed to terminate and either (1) return a counterexample $c \in L(B)\cap \DEC(\Pi)$ or (2) find an approximate model of $\DEC(\Pi)$ that is disjoint with $L(B)$.
We design a mechanical teacher to answer queries from the learning
algorithm. If the mechanical teacher reports an error, a feasible
error decision vector is found. The CFG is incorrect. If the teacher
finds an approximate finite automaton $C$, the CFG
is correct with respect to the parameters $\epsilon$ and $\delta$.
$G$ : a CFG; $0 < \epsilon < 1$ : error; $0 < \delta < 1$ :
run the $L^*$ algorithm with equivalence queries resolved by
Algorithm <ref>
Algorithm <ref> reports ERROR
$G$ is incorrect
$G$ is correct
The $L^*$ algorithm assumes a teacher to answer membership and
equivalence queries. For an membership query $\MEM (\bd)$
with a decision vector $\bd$, we use the algorithm in
Section <ref> to resolve the query. For an
equivalence query $\EQ (C)$, we test the candidate finite automaton
$C$ on a number of decision vectors chosen from $L (S_G)$ by
our sampling algorithm (Section <ref>
and <ref>). If there is any discrepancy between $\DEC
(\Pi)$ and $L (C)$, the sample is returned as a counterexample to the
learning algorithm. Otherwise, $C$ is probably approximate $\DEC
(\Pi)$. We then check if $L (C) \cap L (E_G) = \emptyset$. If so, the
CFG $G$ is probably
approximately correct. Otherwise, note that $L (C)$ is but an
approximation to $\DEC (\Pi)$. We check if
$\be \in L (C) \cap L$ corresponds to a feasible path by a membership
query. If so, $G$ has an error. Otherwise, $\be$ is returned as a
counterexample (Algorithm <ref>).
$\EQ (C_i)$ : the $i^{\textmd{th}}$ equivalence query with a
finite automaton $C_i$; $k$ : batch size for concolic tester
$\mathit{YES}$ if $G$ is probably approximately correct;
$\mathit{CE} (\be)$ if there is a counterexample $\be$
$L (C_i) \cap L (E_G) \neq \emptyset$
pick $\be \in L (C_i) \cap L (E_G)$
$\MEM (\be)$ is $\mathit{YES}$
report ERROR
return $\mathit{CE} (e)$
$r_i \leftarrow \lceil \frac{1}{\epsilon}(\ln \frac{1}{1-\delta} + i
\ln 2) \rceil$
$j \leftarrow 1$ $r_i$
obtain a set of decision vectors $S$ of size $k$ from a concolic tester
$\bd\in S$
$\MEM (\bd)$ is $\mathit{NO}$ but $\bd \in L (C)$, or
$\MEM (\bd)$ is $\mathit{YES}$ but $\bd \not\in L (C)$
$\mathit{CE} (\bd)$
Resolving Equivalence Queries
Note that the number of samples from $L (S_G)$ varies on
different equivalence queries. For the $i^{\textmd{th}}$ candidate
finite automaton $C_i$ with $\mathit{Prob}_{D (S_G)} [\DEC (\Pi) \ominus
L (C_i)] > \epsilon$, the probability of $C_i$ passing the $r_i$
random tests is less than
\begin{eqnarray*}
(1 - \epsilon)^{r_i}
& < & \sum\limits^{\infty}_{i = 1} (1 - \epsilon)^{r_i}
< \sum\limits^{\infty}_{i=1} e^{-\epsilon r_i}
\leq \sum\limits^{\infty}_{i=1} e^{-(\ln \frac{1}{(1-\delta)} + i\ln 2)}\\
& = & \sum\limits^{\infty}_{i=1} e^{\ln {(1-\delta)} + \ln 2^{-i}}
= (1 - \delta) \cdot \sum\limits^{\infty}_{i = 1} 2^{-i}
= 1 - \delta.
\end{eqnarray*}
In other words, if the teacher returns $\mathit{YES}$ on the
$i^{\textmd{th}}$ equivalence query with a candidate finite automaton
$C_i$, we have at most $1 - \delta$ of chance to have $L (C_i)$
significantly different from $\DEC (\Pi)$. Equivalently, we have at
least $\delta$ of confidence to claim that $L (C_i)$ approximates
$\DEC (\Pi)$ within $\epsilon$ of error.
Let $G$ be a CFG, $0 < \epsilon, \delta < 1$, and $D$ the
distribution on $L (S_G)$ determined in
Section <ref>. If Algorithm <ref>
reports “$G$ is correct,” then $G$ is correct with at most
$\epsilon$ of error and at least $\delta$ of confidence on $D (S_G)$.
If one can construct in a similar manner an NFA $F$ for feasible paths in $G$, then the verification problem is reduced to the emptiness problem of $L(F)\cap L(B)$. However, we did not find a suitable way to construct $F$. However, with PAC learning algorithm for automata, we should be able to find an automaton that is very “similar” to $L(F)$ and use it for verification.
With the two mechanism provided, we can invoke PAC-based automata learning algorithm to find a candidate automaton for $L(F)$. As mentioned, the learning algorithm works iteratively. In each iteration, the algorithm asks some number of membership queries that are answered using the mechanism mentioned above. At the end of an iteration, the algorithm proposes a candidate automaton, which is then checked by a PAC-based equivalence query (described in Section 3.1). Assume that we use a confidence level $90\%$ and allow an error rate $5\%$, at the $30$-th iteration of the learning with a candidate automaton $C_{30}$, we need to sample using the above mentioned mechanism $q_{30}=\lceil \frac{1}{0.05}(\ln \frac{1}{1-0.9} + 30 \ln 2) \rceil=462$ words of length smaller than $30$ and check if they are in the difference of $L(C_{30})$ and $L(F)$. Since we have the automaton $C_{30}$, we can do a membership test of automata and know whether a word is in its language. We can check if a word is in $L(F)$ by posing a membership query on the word. Once we find a word in the difference of $L(C_{30})$ and $L(F)$, the word will be returned to the learning algorithm to refine the next conjecture.
If no difference is found, we can conclude that with a confidence level $90\%$, we are sure that $95\%$ of the traces in the program with less than $30$ decisions are correct.
Notice that the confidence level is obtained by summing up the chance of the learning makes an incorrect (namely the difference to $L(F)$ is larger than $5\%$ is this case) conjecture automaton in any iteration. The number still holds even if we apply such an refinement procedure.
§ HANDLING PROCEDURE CALLS
In this section, we extend our formalism of CFGs to handle programs with
multiple procedures and procedure calls.
Moreover, we also refine our representation of error traces.
The issue of using finite automata to represent error decision vectors in the
said setting is that when returning from a procedure call, a finite automaton
cannot remember an (unbounded) number of return points (in the case of
recursive procedures).
Therefore, an overapproximation, such as a nondeterministic jump to any possible
return point, needs to be used.
The said overapproximation is, however, too imprecise and yields numerous
spurious errors.
To address this issue, we also describe our extension of the representation of
error decision vectors using pushdown automaton.
This representation of the set of error decision vectors in programs with
procedure calls is precise.
In this section, we extend our formalism of CFGs to handle programs with
multiple procedures.
We use a PDA to represent error decision vectors in this setting.
The issue of using finite automata to represent error decision vectors in the
said setting is that when returning from a procedure call, a finite automaton
cannot remember an unbounded number of return points (in the case of
recursive procedures).
Therefore, an overapproximation, such as a nondeterministic jump to any possible
return point, needs to be used.
The said overapproximation is, however, too imprecise and yields numerous
spurious errors.
In contrast, PDAs can represent the set of error decision vectors precisely.
On the other hand, we still use a finite automaton to represent the
approximation of the set of feasible decision vectors $\DEC(\Pi)$.
As a consequence, except that we need to use PDA operations instead of FA operations and
handle procedure calls in the membership queries, all other components remain unchanged for the setting of multiple procedures.
§.§ Extending CFGs with Procedure Calls
Consider a set of procedure names $\procs$
and a set of procedure parameter names $\params$, disjoint from
We use $\proccalls{\procs}[\mathcal{X}, \mathcal{F}]$ to denote the set of all
pairs $(p, g)$ where $p \in \procs$ is a procedure name and $g$ is a well-formed
first-order logic formula over $\mathcal{X}, \params$, and $\mathcal{F}$.
Assume the set of procedure names $\procs$.
A CFG with calls (CFGC) is defined as a graph $G = (V, E, v_i, v_r, V_e,
\fp)$ where $V, v_i, v_r, V_e,$ and $\fp$ are defined in the same way as for
a CFG, and
$E \subseteq (V \times \mathcal{T}[\mathcal{X}, \mathcal{F}] \times V) \cup
(V \times (\procs \times \mathcal{T}[\mathcal{X}, \mathcal{F}] \times \mathcal{T}[\mathcal{X}, \mathcal{F}]) \times V)$
is an extended set of edges that apart from local CFG edges $(v, f, v')$ for $f \in
\mathcal{T}[\mathcal{X}, \mathcal{F}]$ also contains procedure
call edges $e = (v, (p, g_{\mathit{in}}, g_{\mathit{out}}), v')$ for $(p,
g_{\mathit{in}}, g_{\mathit{out}}) \in \procs \times \mathcal{T}[\mathcal{X},
\mathcal{F}] \times \mathcal{T}[\mathcal{X}, \mathcal{F}]$ and sequential nodes $v$.
The $g_{\mathit{in}}$ and $g_{\mathit{out}}$ components of $e$ correspond to
formulae for passing actual values to formal parameters of
$p$ (formula $g_{\mathit{in}}$) and passing the return value of $p$ back to the
caller procedure (formula $g_{\mathit{out}}$).
In this extension, we define a program as a set of CFGCs $\prog = \{G_1,
\ldots, G_n\}$ together with a (bijective) mapping $\cfg_{\prog}: \procs \to \prog$ that
assigns procedure names to CFGCs.
We abuse notation and use $\prog$ to denote $\cfg_{\prog}$, i.e.,
$\cfgof{p}$ denotes the CFGC of a procedure $p$ in a program $\prog$.
We assume that all CFGCs in $\prog$ have pairwise disjoint sets of nodes, and
further assume an entry point $\fun{main} \in \procs$.
In this paragraph, we give an informal description of how we extend the
definition of a path from a program consisting of single CFG to a program consisting of a set of CFGCs and a dedicated entry point.
Given a procedure call edge $e$ in a CFGC $G$, we call the inlining of
$e$ in $G$ the CFGC $G'$ obtained from $G$ by substituting $e$ with the CFGC of
the called procedure.
We use $\semof{\prog}$ to denote the set of CFGs obtained from
$\cfgof{\fun{main}}$ by performing all possible (even recursively called)
sequences of inlinings, and removing any left procedure call edges from the
output CFGCs.
A path in $\prog$ is then a sequence $\pi =
\pthof{v_0, f_1, v_1, f_2, v_2, \ldots, f_m, v_m}$ such that
there exists a CFG $G' \in \semof{\prog}$ for which it holds that $\pi \in
§.§ Encoding Error Decision Vectors with Pushdown Automata
In this section, we describe how we construct the PDA encoding the set of error
traces in the considered extension.
The general idea is the same as the one for the use of finite automata
(described in Section <ref>).
The main difference is that we add jumps between CFGCs (corresponding to
procedure call edges), which use the stack to remember which state the PDA
should return to after the procedure call terminates.
In the following, given a CFGC $G = (V, E, v_i, v_r, V_e, \fp)$, we use $V(G),
E(G), \ldots, \fp(G)$ to denote the corresponding components of $G$, and,
moreover, we use $V_s(G)$ and $V_b(G)$ to denote the set of sequential and
branching nodes of $G$ respectively.
Consider a program $\prog = \{G_1, \ldots, G_n\}$.
We construct the error path
automaton as the PDA $B_P = (\{0, 1\}, Q, Q, q_i, \Delta, F)$ in the
following way:
* $Q = V(G_1) \cup {} \cdots {} \cup V(G_n)$,
* $q_i = v_i(G_k)$ such that $\cfgof{\fun{main}} = G_k$,
* $F = V_e(G_1) \cup {} \cdots {}\cup V_e(G_n)$,
* $\Delta = \Delta_1 \cup {} \cdots {} \cup \Delta_n$ where every $\Delta_j$ is
defined as follows:
* $(v, 0, v'_0) \in \Delta_j$ if $v \in V_b(G_j)\setminus V_e(G_j)$, $(v, f_0, v'_0) \in E(G_j)$,
and $v'_0$ is the $0$-successor of $v$;
* $(v, 1, v'_1) \in \Delta_j$ if $v \in V_b(G_j)\setminus V_e(G_j)$, $(v, f_1, v'_1) \in E(G_j)$,
and $v'_1$ is the $1$-successor of $v$;
* $(v, \lambda, v') \in \Delta_j$ if $v \in V_s(G_j)\setminus
V_e(G_j)$ and $(v, f, v') \in E(G_j)$;
* $(v, 0, v), (v, 1, v) \in \Delta_j$ if $v \in V_e(G_j)$; and
* $(v, [\lambda;\lambda/v'], v_i(G_k)), (v_r(G_k), [\lambda;v'/\lambda], v') \in \Delta_j$ if $v \in
V_s(G_j)\setminus V_e(G_j)$, $(v, (p, g_{\mathit{in}}, g_{\mathit{out}}),
v') \in E(G_j)$, and $G_k = \cfgof{p}$.
Let $\prog$ be a program, $\badpaths$ the set of error paths of $\prog$, and
$B_P$ be the error path PDA for $\prog$. Then it holds that $\langof{B_P} =
\DEC (\badpaths)$.
§ IMPLEMENTATION
We created a prototype tool that implements the verification procedure
described in this paper.
The tool uses several third-party libraries and tools.
First, it uses CIL (C Intermediate Language) <cit.> to
convert the verified C program to a set of CFGCs, from
which we construct the error trace pushdown automaton $B_P$.
Further, we use the library <cit.> to perform operations of
automata, such as testing their membership and emptiness, or computing their
For learning automata, we use the implementation of various learning algorithms
within the library <cit.>.
Membership queries are discharged using a concolic tester, mentioned as an
alternative option in Section <ref>.
Given a decision vector, our tool uses the CFG of the program to generate
a path corresponding to the decision vector.
The path is passed in the form of a sequence of program statements to
the software model checker <cit.>, which
checks its feasibility.
It is possible to switch the model checker with other checkers, such as
To deal with equivalence queries, we modified the concolic tester
<cit.> to generate a batch of $k$ decision vectors, as described in Section <ref>.
As may fail to generate the decision vector of a program
execution when the execution terminates abnormally, we modified to take a finite prefix of the execution in this case.
One issue of that we encountered is that when it processes a condition
composed using Boolean connectives, it expands the condition into a cascade of
statements corresponding to the Boolean expression, making the
program longer and harder to learn.
We addressed this by modifying so that it can process conditions with
Boolean connectives inside without expanding them, and in this way we increased
the performance and precision of the analysis.
We also implemented the following three optimizations to improve the
performance of the prototype.
Intersection with Bad Automaton
Recall that our modified learning algorithm (described in
Section <ref>) first checks whether the intersection of
the language of the conjecture $L(C)$ and the bad language $L(B_P)$ is empty.
Checking emptiness of a PDA is, however, more difficult than that of
a finite automaton.
To speed up the procedure, we build a finite automaton $B_O$ that
over-approximates the error language and always first checks whether $L(C)\cap
L(B_O)=\emptyset$, which is an emptiness test for finite automata.
We check $L(C)\cap L(B_P)=\emptyset$ only for the cases that the
previous test fails.
Counterexample from the Learning Algorithm
When an equivalence query returns a counterexample $c$, automata
learning algorithms usually do not guarantee that $c$ is not a valid
counterexample in the next conjecture automaton.
In our preliminary experiments, we found out that it happens very often that the
mechanical teacher returns the same counterexample in several consecutive
Therefore, we decided to check whether $c$ is still a valid counterexample (by
a membership query) for the learning algorithm before proceeding to the
emptiness test.
In the case $c$ is valid, it will be immediately returned to the learning
algorithm to refine the conjecture.
Handling Membership Queries
The main bottleneck of our approach is the time spent for membership
In our implementation, the software model checker
is used to check whether a path is feasible.
For each membership query, if we invoke with a system
call, a Java virtual machine will be created and the components of need to be loaded, which is very time consuming.
To make membership queries more efficient, we modified to run in a server mode so that it can check more than a single path without
being re-invoked.
The module can then construct an automaton $E_G$ accepting
the decision vectors of error paths in $G$.
With a specified decision vector, can produce the path
induced by the decision vector in $G$ as a C program.
The module can answer membership and equivalence queries.
For the membership query of a decision vector, first
obtains the path induced by the decision vector from .
then asks the external verifier
<cit.> to check if the path is
feasible or not, that is, if the end of the path is reachable or not.
For the equivalence checking of a conjecture $C$, can
uniformly select samples from the decision vectors of feasible paths
in $G$ and then perform membership queries of the selected samples.
Again, during the selection, whether a decision vector is feasible or
not is checked by .
To check if $L(C) \cap L(E_G) = \emptyset$ in the equivalence
checking, relies on the automata library
The module implements Algorithm <ref> by
connecting the module with the external library
<cit.>, which is a learning framework with
different kinds of learning algorithms implemented in.
If the result of an equivalence query of $C$ is respectively
$\mathit{YES}$ or $\mathit{ERROR}$, reports that $G$ is
correct or that $G$ is incorrect.
Otherwise, a counterexample is found and asks to
refine the conjecture $C$.
A main bottleneck of our approach is membership query where in our
implementation, is used to check if a path is feasible.
Based on the original design of , for each membership
query, we have to invoke once.
This is very inefficient because spends much time in
loading its components in an invocation.
To make membership queries more efficient, we modified such that more than one path can be checked in a single process.
Another choice for performance improvement is to find a suitable tool
that can translate a path to an SMT formula, which can be solved by
SMT solvers to see if the path is feasible.
§ EXPERIMENTS
This section presents our experimental results to justify the
claims made in this paper.
We evaluated the performance of our prototype using the recursive
category of SV-COMP 2015 <cit.> as the benchmark.
The recursive category consists of 24 non-trivial examples such as
Ackermann, McCarthy 91, and Euclidean algorithms. Among eight
participating tools, only two can solve 20 or more examples correctly.
and is one
of the most difficult categories in the competition [The winner
of the category received 27 out of 40 points.].
Among the 24 examples, 8 of them contain an error.
We performed our experiments with the error parameter $\epsilon = 0.1$, confidence
$\delta = 0.9$, and size of batched samples $k = 10$.
We ran our prototype on each example three times in all experiments.
The provided statistical data were calculated based on the average of the
three runs unless explicitly stated otherwise.
We set the timeout to 900 s to match the rules of SV-COMP 2015.
§.§ Comparison of Learning Algorithms
We evaluated our approach with different automata learning algorithms
implemented within the library.
There are five active online automata learning algorithms implemented in
: Angluin's $L^*$ <cit.>, $L^*$-columns,
Kearns/Vazirani (KV) <cit.>, Rivest/Schapire
(RS) <cit.>, and
$\mathit{NL}^*$ <cit.>.
Among the search strategies provided by , we chose the
random branch strategy.
The experimental results are in Table <ref>.
Comparison of learning algorithms
1c| 5c|Algorithms
1c| KV $L^*$ $L^*$-col. RS $NL^*$
Verified 15 9.67 10 11.33 8.33
Bug found 6 6.33 6 6.33 6
by Bad 4 4.67 4 3 4.33
by 2 1.67 2 3.33 1.67
False positives 0 0 1 0.33 1
False negatives 2 1.67 1.33 1.67 1
Timeouts 1 6.33 5.67 4.33 7.67
# of $\MEM$ queries 2896 8898 10071 15377 14463
# of $\EQ$ queries 548 78 77 367 67
Total time [s] 2406 6668 6565 5786 7972
$\MEM$ queries time 30 % 59 % 58 % 63 % 70 %
The results show that KV is the algorithm with the
best performance—it solved 21 out of the 24 examples.
Our technique solves more than any participant in the recursive
category of SV-COMP 2015 but the winner.
The main reason for the performance difference is that KV uses a
tree-based data structure to store query results.
Compared to other learning algorithms that use table-based structures,
KV requires much less number of membership queries to maintain the
consistency of the tree-based structure.
For all learning algorithms except RS, the number of error paths found
by the emptiness test of the intersection of the conjecture and the bad
automaton is more than that found by .
In our experiments, the time spent for membership queries is usually
the performance bottleneck.
Table <ref> shows that membership queries took
30 % of the total execution time for KV and at least 58 % for
other algorithms.
§.§ Comparison of Search Strategies
We also evaluated the performance of our algorithm against different search strategies.
According to <cit.>, the most
efficient ones are random branch strategy (RBS for short) and
control-flow directed strategy (CDS for short).
Therefore, we tested the performance of our prototype using these two
We selected KV as the learning algorithm in this experiment.
The results are shown in Table <ref>.
Comparison of search strategies of
1c| 2c|Search Strategy
1c| RBS CDS
Verified 15 15
Bug found 6 6
False positives 0 0
False negatives 2 2
Timeouts 1 1
# of $\MEM$ queries 2896 2362
# of $\EQ$ queries 548 463
Total time [s] 2406 2013
Time for one sample [s] 0.75 0.82
Table <ref> shows that although the average
time for taking one sample with CDS is more than with RBS, the total
time is less.
The main reason is that CDS explores untouched branching points more
aggressively than RBS but requires more overhead.
Our experiments conform the results
in <cit.>.
The result is a bit surprising.
Before this experiment, we expect that the CFG directed strategy will
lead to a better performance, because it explores untouched branching
points more aggressively than the random branching strategy.
Moreover, in the results of <cit.>, the
performance of CFG directed strategy is slightly better than the
random branching strategy.
However, it turns out this is not the case in our application.
We solved more examples using the the random branching strategy than the CFG directed strategy. The reason we found is that it took more time to make one sample using the CFG directed strategy than the random branching strategy.
In our experiment, sampling takes XX% of the total execution time with the random branching strategy and XX% for the CFG directed strategy.
Although the random branching strategy has a better performance in terms of the number of solved examples, one might have doubt that the CFG directed strategy might generate a more precise over-approximation than the random branching strategy when the example is bug-free. To evaluate the quality of the learned automata, we run to get for each strategy and for each automaton from the last learning iteration 100 samples. Each sample contains 10 decisions. Then we check how many of the samples are contained in the languages of the automata learned. The result is shown in Table <ref>. From the table, surprisingly, we found the quality of the automata learned with the two strategies is almost the same. Also, observe that the guarantee of our procedure is that the sample coverage rate is higher than 90% in the worst case. Our experimental results show that the quality of the automata produced by our procedure matches the expectation of from the theory.
§.§ Evaluation of CREST with Restarts
Evaluation of with and without restart. For each example, the
number of batches and the number of iterations used to find bugs are obtained
respectively from the worst run in scenario (1) and from the best run in
scenario (2).
1c| 2c|Settings
Examples Batch Size 10 Never Restart
Ackermann02 batch 3 iteration 14
Addition02 batch 1 iteration 2
Addition03 Timeout Timeout
BallRajamani-SPIN2000 batch 1 iteration 1
EvenOdd03 batch 1 iteration 2
Fibonacci04 batch 4 iteration 2
Fibonacci05 Timeout Timeout
McCarthy91 batch 1 iteration 2
To justify our modification to the $\pacofed$-correctness guarantee given in Section <ref>,
we show in the experiment below that running in batches does
not decrease its bug-hunting capabilities.
We compared the performance of with two different scenarios:
(1) restart after each 10 decision vectors and
(2) never restart.
We performed the experiment on the 8 buggy examples in the recursive
category and calculated in how many examples found a bug for
within the timeout period.
In Table <ref>, we chose RBS as the search strategy.
We also tried the experiments with the CFG strategy and got a similar result.
We list the worst result for
scenario (1) and the best result for scenario (2) that we received in our three
We found out that the worst runs in scenario (1) can still find
with a little overhead all bugs found by the best runs in scenario (2).
§.§ Evaluating Quality of Learned Automata
Besides the performance in terms of the running time, we also compared the
quality of the learned automata produced by our prototype using the
two strategies for the 15 successfully verified bug-free examples.
To evaluate the quality of the learned automata, for each example, we
ran with the given search strategy to get 100 batched samples and
tested how many of the them are all accepted by the learned
The average values of the runs are shown in
Table <ref> where evaluation strategies are
strategies used to generate the testing batched samples.
The table shows that the quality of the automata
learned with the two strategies is almost the same.
Also, observe that the guarantee of our procedure is that the sample
coverage is higher than 90 %.
Our experimental results show that the quality of the automata
produced by our procedure matches the theoretical expectations.
Comparison of search strategies in terms of the quality of
the learned automata. The total number of tested batches is 1500.
2c| 2c|$\EQ$ query strategy
2c| RBS CDS
Evaluation Accepted 1487 1473
(RBS) Ratio 99.13 % 98.2 %
Evaluation Accepted 1489 1500
(CDS) Ratio 99.27 % 100 %
Finally, we tested how many words generated by are not covered in the automata learned with the KV algorithm and RBS.
Again, we ran with RBS in two scenarios:
(1) restart after each 10 decision vectors and
(2) never restart.
For each learned automaton (there were 15) and each scenario, we generated 1000
decision vectors and checked how many of them are accepted by the
In total, for scenario (1), we observed 1487 accepted batches of size 10 (for
the total of 15 000 tested vectors), yielding the correctness 99.13 %.
For scenario (2), we observed 14 977 accepted vectors, for the correctness
99.86 %.
We notice that no matter which strategy we use, the learned automaton accepts
over 99 % of the decision vectors produced by .
§ DISCUSSION
There are several advantages of having a program model with statistical guarantees.
For instance, the model can be reused for verifying a different set of properties of the program.
Assume that the new property to be verified is described as an error path
automaton $B'$ and $C$ is the learned automaton.
If $\langof {B'} \cap \langof C =\emptyset$, we verified the program with the
new property and the same $\pacofed$-correctness guarantee.
For the case that there exists a decision vector $w\in \langof {B'} \cap
\langof C$, we test whether $w$ is feasible and either report that $w$ is
a feasible error decision vector w.r.t. $B'$ or continue the learning algorithm
with $w$ as a counterexample for refining the next conjecture.
In this paper we focus on checking validity of program assertions.
The verification step is handled by making an intersection of the conjecture
automaton $C$ and the error path automaton $B$ and testing its emptiness.
This procedure can be generalized to more sophisticated safety properties by
replacing the tests $\langof C \cap \langof B =\emptyset$ and $s \in \langof B$
with other tests.
For example, we can check the property “the program contains at most 10
consecutive 1-decisions on any path” with a statistical guarantee of the
correctness of the received answer.
By extending the alphabet $\{0,1\}$ with program labels, one can also check temporal
properties related to those labels, e.g., “label $A$ should be reached within
10 decisions after label $B$ is reached.”
One possible extension of our work is to learn sequences of feasible function
calls instead of decision vectors.
This might lead to a more compact model in contrast to the current approach.
However, in this case the alphabet of the model to be learned will be all
function names in the program, which is usually much larger than 2, the size of
the alphabet in our work.
Moreover, in this setting, it is much harder to answer membership queries;
a program path composed of function calls
might perform a complex traversal through loops and branches in between the
calls, making the problem of checking feasibility of a program path already
One benefit of our approach is that, in principle, it can be extended to black
box system verification and model synthesis.
By observing the behavior of the environment, we may find some pattern (e.g., some
statistical distribution) of the inputs and then, based on that, design
a sampling mechanism.
Under the assumption that the behavior of the environment remains unchanged, we can verify or
synthesize the model of the system w.r.t. the given sample distribution.
The design of the sampling mechanism is one of the most difficult part in the entire procedure. Our first proposal was to give all decision vectors with length smaller than $k$ the sample probability to be chosen, where $k$ is a big integer number. However, we found that most of the decision vectors are infeasible. In our experience, in it very often that we can claim an automaton with empty language is 99% similar to $\DEC(\Pi)$ using this sampling mechanism.
OL: remove the previous paragraph? it doesn't suit me there ...
§ RELATED WORKS
Exact automata learning algorithm was first proposed by
Angluin <cit.> and later improved by many
people <cit.>.
The concept of probably approximately correct (PAC) learning was first proposed
by Valiant in his seminal work <cit.>.
The idea of turning an exact learning algorithm to a PAC learning algorithm can
be found in Section 1.2 of <cit.>.
Applying PAC learning to testing has been considered before <cit.>.
The work in <cit.> considers a program that manipulates
graphs and check if the output graph of the program has properties such as
being bipartite, $k$-colorable, etc.
Our work considers assertion checking, which is more general than the
specialized properties.
The work <cit.> considers more
theoretical aspects of the problem.
The author estimates the maximal number of queries required to infer a model of
a black box machine.
The context is quite different, e.g., the work does not discuss how to sample
according to some distribution efficiently to produce the desired guarantee
(bounded path coverage) as we do in this paper.
The $L^*$ algorithm has been used to infer the model of error traces of a program.
In <cit.>, instead of decision vectors, the authors try to
learn the sequences of function calls leading to an error.
Their teacher is implemented using a bounded model checker and hence can only guarantee correctness up to a given bound.
The authors do not make use of the PAC learning technique as we did in this work.
Both our approach and statistical model
checking <cit.>
provide statistical guarantees.
As mentioned in the introduction, statistical model checking assumes a given
model while our technique generates models of programs with statistical guarantee. Those
models can be analyzed using various techniques and reused for
verifying different properties.
§ AN INLINING IN A CFGC
Consider a procedure call edge $e = (v, (p, g_{\mathit{in}},
g_\mathit{out}), v')$ of
$G_j = (V_j, E_j, v_{ij}, v_{rj}, V_{ej}, \fp{}_j)$ for $1 \leq j \leq n$.
Further assume that
$\cfgof{p} = G_k = (V_k, E_k, v_{ik}, v_{rk}, V_{ek}, \fp{}_k)$ for $1 \leq k
\leq n$.
The inlining of $e$ in $\prog$ is the program $\prog^{\#} = \{G_1, \ldots,
G_{j-1}, G^{\#}_j, G_{j+1}, \ldots, G_n\}$ such that
$G^{\#}_j = (V_j \cup V^{\#}_k, E^{\#}, v_{ij},
v_{rj}, V_{ej} \cup V^{\#}_{ek}, \fp{}_j)$
where $V^{\#}_k$ is a set of fresh nodes (i.e., $V^{\#}_k$ is disjoint from the
set of
nodes of any CFGC in $\prog$) and $V^{\#}_{ek} \subseteq V^{\#}_k$.
Moreover, there exist bijections $\subst: V^{\#}_k \to V_k$ and $\subst_e:
\to V_{ek}$ such that $\subst_e \subseteq \subst$.
The set of edges $E^{\#}$ is defined as
$E^{\#} = (E_j \setminus \{e\}) \cup E^{\#}_k \cup \{(v, g_{\mathit{in}},
v^{\#}_{ik}), (v^{\#}_{rk}, g_{\mathit{out}}, v')\}$
where $E^{\#}_k = \{(v^{\#}_k, f, v^{\#\prime}_k) \mid (\substof{v^{\#}_k}, f,
\substof{v^{\#\prime}_k}) \in E_k\}$.
Let $\inlngsof{\prog}$ be the smallest (potentially infinite) set that contains
$\prog$ and is closed w.r.t.inlinings in $\fun{main}$, i.e., if $\inlngsof{\prog}$ contains a program
$\prog^{\#}$ with a procedure call edge $e$ in the CFGC
$\cfgprimeof{\fun{main}}$, it also contains the inlining of $e$ in $\prog^{\#}$.
We abuse notation and use $\subst$ uniformly to denote for all programs
$\prog^{\#} \in \inlngsof{\prog}$ the mapping of the nodes of the CFGCs
in $\prog^{\#}$ to their original nodes in $\prog$ (for nodes $V$ of $\prog$, we
assume that $\subst|_{V} = \mathrm{id}$, i.e., that the restriction of $\subst$
to $V$ is the identity relation).
We denote as $\semof{\prog}$ the set of CFGs (without procedure call edges)
obtained by starting from the set $\inlngsof{\prog}$,
collecting the CFGCs of $\fun{main}$ functions of all inlinings of $\prog$
into the set $M = \{\cfgprimeof{\fun{main}} \mid \prog^{\#} \in
\inlngsof{\prog}\}$, and, finally, transforming the CFGCs of $M$ into
CFGs of $\semof{\prog}$ by substituting each procedure call edge $(v^{\#}, (p^{\#},
g^{\#}_{\mathit{in}}, g^{\#}_{\mathit{out}}), v^{\#\prime})$ with the edge
$(v^{\#}, \FF, v^{\#\prime})$.
We extend the definition of a path as follows:
A path in $\prog$ is a sequence $\pi =
\pthof{v_0, f_1, v_1, f_2, v_2, \ldots, f_m, v_m}$ such that
there exists a CFG $G' \in \semof{\prog}$ and a CFG path
$\pi' = \pthof{v'_0, f_1, v'_1, f_2, v'_2, \ldots, f_m, v'_m}$ in $G'$ such that
$\forall 0 \leq j \leq m: v_j = \substof{v'_j}$.
|
1511.00759
|
Lifespan of Solutions to the Euler-Poisson System]On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary
We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid motion is described by the Euler-Poission system in moving bounded simply connected domains.
A family of equilibrium solutions of the system are the perfect balls moving at constant velocity.
We show that for smooth data which are small perturbations of size $\epsilon$ of these static states, measured in appropriate Sobolev spaces, the solution exists and remains of size $\epsilon$ on a time interval of length at least $c\epsilon^{-2},$ where $c$ is a constant independent of $\epsilon.$ This should be compared with the lifespan $O(\epsilon^{-1})$ provided by local well-posdness. The key ingredient of our proof is finding a nonlinear
transformation which removes quadratic terms from the nonlinearity. An important difference with the related gravity water waves problem is that unlike the constant gravity for water waves, the self-gravity in the Euler-Poisson system is nonlinear. As a first step in our analysis we also show that the Taylor sign condition always holds and establish local well-posedness for this system.
Support of the National Science Foundation grants DMS-1253149 for the first and second, NSF-1045119 for the third, and DMS-1361791 for the fourth authors is gratefully acknowledged. The third author was also supported by the NSF under Grant No.0932078000 while in residence at the MSRI in Berkeley, CA during Fall 2015.
§ INTRODUCTION
We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force in two dimensional spaces. We assume that the fluid domain is bounded and simply connected and the surface tension is zero. Denoting the fluid domain by $\Omega(t)\subset\R^2,$ the fluid velocity by $\bfv$, and the pressure by $P,$ the evolution is described by the system
\begin{align}\label{main eq}
\begin{cases}
\bfv_{t}+(\bfv\cdot\nabla)\bfv=-\nabla P-\nabla\phi\quad&\mbox{in}\quad \Omega(t),t\geq0,\\
\textrm{div}\,\bfv=0,\quad\textrm{curl}\,\bfv=0\quad &\mbox{in}\quad \Omega(t), t\geq0,\\
P=0\quad &\textrm{on}\quad \partial\Omega(t),
%\\(1,\bfv)\in T\left((t,\partial\Omega(t)\right)\quad &t\geq0,
\end{cases}
\end{align}
where the self-gravity Newtonian potential $\phi$ satisfies
This system is commonly referred to as the incompressible and irrotational Euler-Poisson system.
In the equilibrium case where the total force from the pressure and self-gravity are balanced, a ball in $\R^2,$ possibly moving with constant velocity, gives a static solution of the system (<ref>).
An important stability condition for this problem is the Taylor sign condition
$$\frac{\partial P}{\partial \bfn}<0,$$
where $\bfn$ is the unit outward pointing normal to the boundary of the fluid region.
In the three dimensional version of this problem Nordgren <cit.> proved local well-posedness without the irrotationality assumption, but assuming that initially the Taylor sign condition holds. A priori estimates were consequently obtained by Lindblad and Nordgren <cit.> in the two dimensional case, but well-posedness was not proved. In our case where the fluid is incompressible and irrotational, the Taylor sign condition holds automatically.
Indeed by taking divergence of the first equation in (<ref>) and using the fact that $\Delta\phi=2\pi$ in $\Omega(t)$ we see that in $\Omega(t)$
$$\Delta P=-\Delta \phi-|\nabla v|^2 =-2\pi-|\nabla v|^2 <0,$$
so by the Hopf's Maximum principle
$${\frac{\partial P}{\partial \bfn}<0}.$$
The objective of this paper is to show that if $\epsilon\ll1$ is the size of the difference of the smooth initial data from one of the equilibrium states above, measured in various Sobolev spaces, a unique solution exists and its lifespan has a lower bound of order $O(\epsilon^{-2})$. This should be compared with the $O(\epsilon^{-1})$ estimate from local well-posedness. As a first step in the proof of this result we
establish local well-posedness for data of arbitrary size. The key to obtaining our long-time $O(\epsilon^{-2})$ estimate is a fully nonlinear `normal form' transformation which removes quadratic terms from the nonlinearity in the equation. More precisely we find a new unknown and a coordinate change such that in the new coordinates the new unknown satisfies an equation with only cubic and higher order nonlinearity. The use of normal form transformations for evolution PDEs goes back to <cit.> where bilinear transformations of the unknown are used to study nonlinear Klein-Gordon equations. For the gravity water wave problem a fully nonlinear transformation was discovered by the last author in <cit.>. An important difference of the Euler-Poisson system with the gravity water wave problem is that in the Euler-Poisson system the contribution of gravity is nonlinear. Indeed in the two dimensional gravity water wave equation the gravity is given by the constant vector $\pmat{0\\-1}.$ Finally note that since the fluid domain $\Omega(t)$ is bounded, dispersive tools are not available to prove global well-posedness at this point. The precise statement of our result is given in Theorem <ref> below.
To state our result we first discuss the reduction of the system (<ref>) to a system on the boundary $\partial\Omega(t).$ We occasionally use the notation $\Omega_t:=\Omega(t)$. When there is no risk of confusion we simply write $\Omega$; similarly we occasionally write the parametrization of $\partial\Omega:=\partial\Omega(t)$ as $z=z(\cdot)$ instead of $z=z(t,\cdot)$. Moreover, we use the usual identification $\pmat{x\\y}\mapsto z=x+iy$ of $\R^2$ with $\bbC$ to identify $\Omega$ with a domain in the complex plane.
Let $z(t,\alpha),~\alpha\in\R$ be a counterclockwise and $2\pi-$periodic Lagrangian parametrization of $\partial\Omega.$ By this we mean
so in particular
$$z_{tt}(t,\alpha)=\bfv_t(t,z(t,\alpha))+(\bfv\cdot \nabla\bfv)(t,z(t,\alpha))$$
is the acceleration. The conditions $\textrm{div}\,\bfv=0,~\textrm{curl}\,\bfv=0$ now imply that $\overline{\bfv}$ is anti-holomorphic in $\Omega$ and therefore $\zbar_t$ is the boundary value of a holomorphic function in $\Omega.$ It then follows, cf. Proposition <ref> in Appendix <ref>, that
where $H$ denotes the Hilbert transform associated to $\Omega$ defined by
$$Hf(z_0):=\frac{\pv}{\pi i}\int_{\partial\Omega}\frac{f(z)}{z-z_0}dz:=\frac{\pv}{\pi i}\int_0^{2\pi}\frac{f(z(t,\beta))}{z(t,\beta)-z(t,\alpha)}z_\beta(t,\beta)d\beta$$
for $z_0=z(t,\alpha)\in\partial\Omega.$ Since $z$ is a counterclockwise parametrization of $\partial\Omega$ the unit exterior normal of this boundary is given by ${\bold n}:=\frac{-iz_\alpha}{|z_\alpha|},$ and since $P$ is constant on $\partial\Omega$ we can write $\nabla P(t,z)=iaz_\alpha$ for a real-valued function
It follows from (<ref>), our identification of $\R^2$ with $\bbC,$ and these observations that $z$ satisfies the fully nonlinear system
or equivalently
The remainder of this paper is devoted to the study of this equation. Note that once a solution $z$ to (<ref>) is found, one can recover $\bfv$ by solving the Dirichlet problem
Δ=0, in Ω
=z_t, on ∂Ω.
We can now state the main result of this paper. See also Theorems <ref> and <ref> for more quantitative formulations.
Let $\Omega_0$ be a bounded simply-connected domain in $\bbC$ with smooth boundary $\partial\Omega_0$ satisfying $\abs{\Omega_0}=\pi,$ and denote the associated Hilbert transform by $H_0.$ Suppose $z_0(\alpha)=e^{i\alpha}+\epsilon f(\alpha)$ is a parametrization of $\partial\Omega_0$ and $z_1(\alpha)=v_0+\epsilon g(\alpha)$ where $f$ and $g$ are smooth and $g$ satisfies $H_0\overline{g}=\overline{g},$ and $v_0\in\bbC$ is a constant. Then there is $T>0$ and a unique classical solution $z(t,\alpha)$ of (<ref>) on $[0,T)$ satisfying $(z(0,\alpha),z_t(0,\alpha))=(z_0(\alpha),z_1(\alpha)).$ Moreover if $\epsilon>0$ is sufficiently small the solution can be extended at least to $T^*=c\epsilon^{-2}$ where $c$ is a constant independent of $\epsilon.$
The normalization $\abs{\Omega_0}=\pi$ is made only for notational convenience. By the incompressibility of the flow the area of $\Omega(t)$ remains constant during the evolution, and our proof goes through without this assumption by renormalizing the transformations in Section <ref>.
The constant $v_0\in\bbC$ corresponds to the fact that we consider the stability of the equilibrium solution $e^{i\alpha}+v_0t.$ In practice we work in the center of mass coordinates (see Section <ref>) to reduce the analysis to the case $v_0=0.$
We now continue with a brief historical survey of developments related to equation (<ref>) followed by a discussion of the main difficulties in the proof of Theorem <ref> and the ideas for resolving them. The mathematical study of the closely related water wave problem goes back to <cit.>. Numerous studies on local well-posedness for the water wave problem with or without surface tension, bottom, and/or vorticity can be found in <cit.>, and works on water waves with angled crests can be found in <cit.>. Also as mentioned above, in the presence of self-gravity local well-posedness in dimension three and a-priori estimates in dimension two were obtained in <cit.>. For the gravity water wave problem, first Wu obtained almost global well-posedness in dimension two in <cit.>. Then, global well-posedness in three dimensions was solved independently by Wu in <cit.> and by Germain, Masmoudi, and Shatah in <cit.>. The $2$d result was later extended to global well-posedness in <cit.>. See also <cit.> for other related developments. As the literature on this subject is vast, we refer the reader to these articles and the references therein as well as the recent survey article <cit.> for more comprehensive accounts. We mention that for the almost global and global existence results in <cit.>, besides the normal form transformations, the proofs rely crucially on the dispersive properties of the localized solutions.
We now turn to the discussion of the proof of Theorem <ref>. As mentioned earlier, our main idea is to find a new unknown and a coordinate change such that the new unknown satisfies an equation with only cubic and higher order nonlinearity in the new coordinates. To understand what we mean by cubic we have to specify what kinds of terms are considered to be small. Recall that we are studying the stability of the static solution[More precisely we consider the stability of the solutions $z(t,\alpha)=e^{i\alpha}+v_0t$ where $v_0\in\bbC$ is a constant initial velocity. However, by working in the center of mass frame we are able to reduce to the case $v_0=0.$ See Section <ref> for more details.] $z(t,\alpha)\equiv e^{i\alpha},~z_t(t,\alpha)\equiv0.$ It therefore makes sense to consider a quantity depending on $z$ to be small if it is zero when $z$ is the static solution. For instance the quantities $|z|^2-1$ and $z_t$ are considered small, and to say that the nonlinearity is “cubic and higher order" means that every term in the nonlinearity is the product of at least three small terms. Let $\ep:=|z|^2-1$ and denote by $\delta$ the projection of $\ep$ onto the space of functions which are holomorphic outside $\Omega$ (see Appendix <ref>), that is,
\delta:=(I-H)\ep
where $H$ is the Hilbert transform. Our first step is to prove that $\delta$ satisfies an equation of the form
Note that to prove energy estimates for this equation we need to have control on the size of the coefficient $a$, and that the dependency of $a$ on $z$ is nonlinear. A careful computation then shows that the contribution of the term $ia\pa\delta$ to the nonlinearity is only quadratic. We remedy this problem by working in a different set of coordinates by introducing an appropriate coordinate transformation $k:\R\to\R.$ Given such $k$ and with the notation
\chi:=\delta\circ k^{-1},\quad A:=(ak_\alpha)\circ k^{-1}, \quad b=k_t\circ k^{-1}
we see that $\chi$ satisfies
\left((\pt+b\pa)^2+iA\pb-\pi\right)\chi(t,\beta)=\mathrm{cubic}.
The idea now is to choose $k$ in such a way that $b$ and $A-\pi$ are quadratic. Here in the static case the transformation $k$ is simply the identity and $A$ is the constant $\pi.$ We will show in Section <ref> that these conditions will be satisfied if we choose $k$ such that $(I-H)(\log\zbar+ik)=0$ and $k$ satisfies an additional normalization. See Section <ref> for more details. We refer the reader to <cit.> for the related transformation in the case of the water wave problem where the origin of these ideas can be found, and where a discussion on the relation with the bilinear normal form transformations of <cit.> is provided. We also emphasize that unlike the gravity water wave problem, the contribution of the self-gravity term on the right hand side of (<ref>) is nonlinear. With this choice of $k$ we have obtained our cubic equation, and we focus on the energy estimates for the equation
\left((\pt+b\pa)^2+iA\pb-\pi\right)\Theta=\mathrm{cubic}
where $\Theta=(I-H)f$ for some $f.$ Unfortunately the operator $iA\pb-\pi$ is not positive even when restricted to the class of functions satisfying $\Theta=(I-H)f.$ On the other hand, we observe that if $z(t,e^{i\alpha})\equiv e^{i\alpha},$ so that $A=\pi,$ then a Fourier expansion shows that $i\pb-1$ is indeed positive on the class of functions with only negative frequencies, i.e. $\Theta=(I-\bbH)f$ where now $\bbH$ is the Hilbert transform associated to the unit circle. This suggests that the negative part of $iA\pb-\pi$ should be higher order with respect to our energy, and in Section <ref> we will show that this is indeed the case. The most natural way to see this structure will be to work with the quantity $(z\circ k^{-1})\Theta$ instead of $\Theta$ (see Lemma <ref>), however to be able to control the negative part of the energy without loss of derivatives, a very careful choice of the energies will be needed. The detailed execution of these ideas is contained in Section <ref>.
Finally we mention that our proof of long-time existence relies on the existence of a local-in-time solution. We have provided a proof of local well-posedness for the system (<ref>) in Section <ref>. Here the Riemann mapping is used to transfer the analysis from the evolving domain $\Omega$ to the unit disk $\bbD\subset\bbC.$ A key step in this analysis is obtaining a lower bound on $a,$ which implies the Taylor sign condition. We refer the reader to <cit.> for more details on the significance of this condition.
§.§ Organization of the paper
The rest of this paper is organized as follows. In Section <ref> we collect some analytic tools which are used in the rest of the paper. The proof of the long-time existence statement of Theorem <ref> is the content of Sections <ref>–<ref>. The proof relies on the existence of a local-in-time solution, but as local well-posedness is not the primary focus of this paper the proof of local well-posedness is postponed to Section <ref>, where Riemann mapping coordinates are introduced and the quasilinear structure of the equation revealed. In Section <ref> we introduce the normal form transformation and obtain the desired cubic equation discussed above. In Section <ref> we investigate the relation between the original and transformed quantities, and how estimates on one set of quantities translate to estimates for the other set. In section <ref> we introduce the energies and carry out the energy estimates, and finally in Section <ref> we combine the results from the previous three sections to conclude the proof of long-time existence. Appendix <ref> contains a review of facts that are used about the Hilbert transform in this paper, and for the convenience of the reader we have provided a list of notations we use, before the references.
§ ANALYSIS TOOLS
In this section we collect a number general estimates which will be used in the rest of the paper. Most notably we will provide classical estimates on certain singular integral operators adapted to our case. Throughout this section we let
$$\zed:[0,2\pi]\to\partial \Omega\subseteq \C$$
be a parametrization of the (closed) boundary of a domain $\Omega$ in $\C.$ We require $\zed$ to be at least $C^1,$ but most of the results in this section hold under the weaker assumption that $\zed$ is Lipchitz. By abuse of notation, for a function $A:\partial\Omega\to\C,$ we write $ A(\alpha)$ instead of $A(\zed(\alpha)).$ In this context $A^\prime(\alpha)$ means $\pa A(\zed(\alpha)),$ so for instance if $A=|\zed|^2-1$ then $A^\prime=2\Re \zedbar \zed_\alpha.$ In the proof of local well-posedness $\zed$ will usually be chosen as $\zed(\alpha)=e^{i\alpha}$ or $\zed(t,\alpha)=Z(t,\alpha).$ For the long-time existence we will often consider $\zed(t,\alpha)=\zeta(t,\alpha)$ or $\zed(t,\alpha)=z(t,\alpha)$ (the definitions for $Z$ and $\zeta$ will be given in later sections).
Even though the functions in this section depend only on $\alpha$ and not on $t,$ we use the notations $L^p_\alpha$ and $H^s_\alpha$ for the Lebesgue and Sobolev spaces in the variable $\alpha$ to be consistent with the rest of the paper. The following standard Sobolev estimate will be used throughout this work, often without reference.
[Sobolev] There is a constant $C$ such that for all $f$ in the Sobolev space $H_\alpha^1$
f_^∞≤C( f_^2+f_^2).
We now turn to the main estimates of this section. We are interested in bounding operators of the forms
C_1(A,f)(α):=∫_0^2π∏_i≤m(A_i(α)-A_i(β))/((α)-(β))^m+1-k((α)-(β))^kf(β)dβ, k≤m+1,
C_2(A,f)(α):=∫_0^2π∏_i≤m(A_i(α)-A_i(β))/((α)-(β))^m-k((α)-(β))^k∂_βf(β)dβ, k≤m.
The two propositions below are due, in their original forms, to Calderon <cit.>, Coifman, McIntosh, Meyer <cit.>, Coifman, David, and Meyer <cit.>, and here we only provide the straightforward modifications necessary for their application in our periodic setting. See also Wu <cit.> for the proof of the second part of this proposition using these results and the Tb Theorem.
Suppose $\zed$ satisfies
for some constant $c_0.$ Then there is a constant $C=C(c_0)$ such that the following statements hold.
* For any $f\in \La^{2}, A'_{i}\in \La^{\infty}, 1\leq i\leq m$,
\begin{align*}
\|C_{1}(A,f)\|_{\La^{2}}\leq C\|A'_{1}\|_{\La^{\infty}}...\|A'_{m}\|_{\La^{\infty}}\|f\|_{\La^{2}}.
\end{align*}
* For any $f\in \La^{\infty}, A'_{i}\in \La^{\infty}, 2\leq i\leq m, A'_{1}\in \La^{2}$,
\begin{align*}
\|C_{1}(A,f)\|_{\La^{2}}\leq C\|A'_{1}\|_{\La^{2}}\|A'_{2}\|_{\La^{\infty}}...\|A'_{m}\|_{\La^{\infty}}\|f\|_{\La^{\infty}}.
\end{align*}
Propositions <ref> is a consequence of Propositions 3.2 in <cit.>. Here we describe the modifications necessary to apply this result to our setting. We restrict attention to the case $m=1,~ k=0,$ and write $A$ instead of $A_1.$ The general case can be handled in a similar way. With $\chi$ denoting the characteristic function of the interval $[0,2\pi]$ we have
\begin{align}\label{L2 C1 explicit}
\end{align}
Since $A$ appears only as $A(\alpha)-A(\beta)$ in this expression, we may assume without loss of generality that $A(0)=A(2\pi)=0.$ We introduce some more notation. First let $\chi_j,~j=1,2,3$ be the characteristic function of the interval $[\frac{2(j-1)\pi}{3},\frac{2j\pi}{3}].$ Next define $\tA$ by $\tA(\alpha)=A(\alpha)$ if $\alpha\in[-4\pi,4\pi]$ and $\tA(\alpha)=0$ if $\alpha\notin[-4\pi,4\pi].$
\begin{align*}
K:=\{w\in\bbC\,|\,w=\dfrac{\zed(\alpha')-\zed(\beta')}{\alpha'-\beta'}\quad \textrm{for some}\quad |\alphap-\betap|\leq \frac{5\pi}{3}\}\subseteq \C.
\end{align*}
From the assumptions of Proposition <ref> it follows that $K$ does not contain the origin $w=0$ in $\C.$ Let $K^\prime\supseteq K$ be a compact set containing $K$ such that $0\notin K^\prime,$ and let $\phi_K$ be a cut-off function supported in $K^\prime$ and equal to one on $K.$ If follows that the function
is smooth. With these definitions we have
dβ)^2dα=:∑_i,j=1^3 L_i,j
and we will estimate these integrals separately for different values of $i,j.$ First we treat the case $i=j,$ and for simplicity of notation we assume $i=j=1:$
L_1,1 =∫_χ(α)χ_1(α)(∫_F((α)-(β)/α-β)(α)-(β)/(α-β)^2χ(β)χ_1(β)f(β)dβ)^2dα
≲__C^1_α^'_^∞^2χχ_1f_^2([0,2π])^2≲A^'_^∞^2 f_^2([0,2π])^2,
where we have used Propositions 3.2 in <cit.> to pass to the last line. The case where $j={i+1}$ is similar, and we now treat the case $i=1,~j=3.$ Using again Propositions 3.2 in <cit.> and the periodicity of $A,~f,$ and $\zed$ we have
L_1,3 =∫_χ(α)χ_1(α)(∫_A(α)-A(β-2π)/((α)-(β-2π))^2χ(β)χ_3(β)
≲__C^1_α A^'_^∞^2 f_^2([0,2π])^2.
The remaining cases can be handled using similar arguments.
A similar argument as in the proof of Proposition <ref> allows us to deduce the following result from Proposition 3.3 in <cit.>. We omit the proof.
Suppose $\zed$ satisfies
sup_ α≠β |e^iα-e^iβ/(α)-(β)|≤c_0
for some constant $c_0.$ Then there is a constant $C=C(c_0)$ such that the following statements hold.
* For any $f\in \La^{2}, A'_{i}\in \La^{\infty}, 1\leq i\leq m$,
\begin{align*}
\|C_{2}(A,f)\|_{\La^{2}}\leq C\|A'_{1}\|_{\La^{\infty}}...\|A'_{m}\|_{\La^{\infty}}\|f\|_{L^{2}}.
\end{align*}
* For any $f\in \La^{\infty}, A'_{i}\in \La^{\infty}, 2\leq i\leq m, A'_{1}\in \La^{2}$,
\begin{align*}
\|C_{2}(A,f)\|_{\La^{2}}\leq C\|A'_{1}\|_{\La^{2}}\|A'_{2}\|_{\La^{\infty}}...\|A'_{m}\|_{\La^{\infty}}\|f\|_{\La^{\infty}}.
\end{align*}
The next lemma is a simple computation which is used in estimating derivatives of expressions such as $C_1(A,f)$ and $C_2(A,f).$
𝐊 f (α)=∫_0^2πK(α,β)f(β)dβ
where $K(\alpha,\beta)$ or $e^{i(\alpha}-e^{i\beta})K(\alpha,\beta)$ are continuous and $K$ is $C^1$ away from the diagonal in $[0,2\pi]\times[0,2\pi].$ Then
∂_α𝐊 f(α)=𝐊 f_α(α)+∫_0^2π(∂_α+∂_β)K(α,β)f(β)dβ.
This follows from integration by parts.
The following two lemmas are important corollaries of Propositions <ref> and <ref> and Lemma <ref>. Recall that for a $C^1$ parametrization $\zeta:[0,2\pi]\to\partial\Omega$ of the boundary of $\Omega$ the Hilbert transform is given by
Suppose $\zeta:[0,2\pi]\to\partial\Omega\subseteq \C$ satisfies $\sum_{i=1}^{\ell+1}\|\pa^j\zeta\|_{\La^2}\leq c$ for some nonzero constant $c,$ where $\ell\geq 4$ is a fixed integer, and
sup_ α≠β |e^iα-e^iβ/(α)-(β)|≤c_0
Then there is a constant $C=C(j,c,c_0)$ such that for $4\leq j \leq \ell$
In particular
The second estimate follows from the first by writing
To prove the first estimate we use Lemma <ref> to distribute the derivative on $f$ and $g.$ In the case where all derivatives fall on $f$ Proposition <ref> gives
∫_0^2πg(β)-g(α)/ζ(β)-ζ(α)^jf(β)dβ_^2 ≲g_^∞^j-1f_^2.
When all derivatives fall on $g$ we use the boundedness of the Hilbert transform and Proposition <ref> to estimate
The case when $j-1$ derivatives fall on $g$ and one derivative on $f$ can be estimated directly by Proposition <ref>. When $j-1$ derivatives fall on $g$ and none on $f$ we have
∫_0^2π(^j-1g(β)-^j-1g(α))(ζ_β(β)-ζ_α(α))/(ζ(β)-ζ(α))^2f(β)dβ_^2≲ ∫_0^2π^j-1g(β)-^j-1g(α)/(ζ(β)-ζ(α))^2f(β)ζ_β(β)dβ_^2
which can be estimated using Proposition <ref>. All other cases can simply be estimated by bounding the contributions of both $f$ and $g$ in $\La^\infty$ and using the embedding $\La^\infty\hookrightarrow\La^2.$
Under the assumptions of Lemma <ref> for any $\ell\geq4$
∑_j≤ℓ^j(I-)f_^2≤C ∑_j≤ℓ^jf_^2,
where $C$ depends on the $H_\al^\ell$ norm of $\zeta.$
This follows from Lemma <ref> and Propositions <ref> and <ref> by writing
where $\eta:=\zeta_\alpha-i\zeta,$ Here to compute the commutator $[\pa,\CH]=[\zeta_\alpha,\CH]\frac{\pa}{\zeta_\alpha}$ we have used Lemma <ref>.
As another corollary of Proposition <ref> and Lemma <ref> we get the following $\La^\infty$ estimate for $C_1(A,f),$ which is similar to Proposition 3.4 in <cit.>.
Suppose $\zed$ satisfies
sup_ α≠β |e^iα-e^iβ/(α)-(β)|≤c_0
for some constant $c_0,$ and $A_i^\prime,$ $A_i^{\prime\prime},$ $f,$ and $f^\prime$ are in $\La^\infty.$ Assume further that $\|\zed\|_{H_\alpha^2}\leq M.$ Then there exists a constant $C=C(c_0)$ such that
\begin{align*}
\|C_{1}(A,f)\|_{\La^{\infty}}\leq C(1+M)\prod_{i\leq m}\left(\|A''_{i}\|_{\La^{\infty}}+\|A'_i\|_{\La^\infty}\right)\left(\|f\|_{\La^{\infty}}+\|f'\|_{\La^\infty}\right).
\end{align*}
This follows from applying the Sobolev inequality to $C(A,f)$ and using Lemma <ref> and Proposition <ref>. Note that in view of the embedding $\La^\infty\hookrightarrow\La^2$ we may replace the $\La^2$ norms appearing on the right hand side of the statement of Proposition <ref> by $\La^\infty$ norms.
We close this section by stating the following estimates from <cit.> (see also <cit.> Lemma 5.2) which are proved using Fourier analysis. The adaptations from the case of the real line to the circle are straightforward and omitted. Here
$\H$ is the Hilbert transform on the circle.
\begin{lemma}\label{lem: Yosihara} Let $r\geq0,$ $q>1/2,$ and $s\geq 1.$ Then for any smooth functions $a$ and $u$
\Aligns{
\|[a,\H]u\|_{H_\alpha^r}\lesssim \|a\|_{H_\alpha^{r+p}}\|u\|_{H_\alpha^{q-p}},\quad p\geq0.
\end{lemma}
\section{The Normal Form Transformation}\label{sec: normal form}
In this section we begin the study of the Cauchy problem of the system \eqref{z eq temp} with small initial data. We start with the following important representation formula for the boundary contribution of the gravity term.
\begin{lemma}\label{lem: gravity}
$-2\partial_\zbar\phi=-\frac{\pi}{2}(I-\Hbar)z=-\pi z+\frac{\pi}{2}(I+\Hbar)z.$
\end{lemma}
\begin{proof}
With $\bfx=z(\alpha)$ and by the dominated convergence theorem
\Aligns{
\nabla\phi(\bfx)=-\int\int_{\Omega}\nabla_\bfy\log(|\bfx-\bfy|)dy=-\lim_{\epsilon\to0}\int\int_{\Omega\minus B_\epsilon}\nabla_\bfy\log(|\bfx-\bfy|)d\bfy
where $B_\epsilon$ is a ball of radius $\epsilon$ centered at $\bfx.$ We now identify $\R^2$ with $\C$ in the usual way and abuse notation to write for instance $\nabla\phi=\partial_{x}\phi+i\partial_{y}\phi.$ Defining the vector fields
\Aligns{
X=(\log(|\bfx-\bfy|),0),\qquad Y=(0,\log(|\bfx-\bfy|)),
we have
\Aligns{
\nabla\phi(\bfx)=-\lim_{\epsilon\to0}\int\int_{\Omega\minus B_\epsilon}(\div X+i\,\div Y)d\bfy=-\lim_{\epsilon\to0}\int_{\partial(\Omega\minus B_\epsilon)}(X+iY)\cdot N d\sigma(\bfy)
where $d\sigma$ is the line element of the boundary and $N$ the outward pointing normal vector. The boundary has two parts: $C_\epsilon$ corresponding to $\partial B_\epsilon$ and $\Gamma_\epsilon$ corresponding to $\partial \Omega_{\epsilon}.$ We can find $\delta_1(\epsilon)$ and $\delta_2(\epsilon)$ which are $O(\epsilon)$ and such that $\Gamma_\epsilon$ is parametrized by $z(\cdot):[0,2\pi]\minus[\alpha-\delta_1,\alpha+\delta_2]\to\Gamma_\epsilon.$ The outward pointing normal vector is therefore given by $-iz_\alpha/|z_\alpha|$ in complex notation or $\frac{1}{|z_\alpha|}(\Im z_\alpha,-\Re z_\alpha)$ in real notation. Similarly there are numbers $\eta_1(\epsilon)<\eta_2(\epsilon)$ in $(0,2\pi)$ such that in complex notation $C_\epsilon$ is parametrized by $\theta\in(\eta_1,\eta_2)\mapsto \bfx+\epsilon e^{i\theta}.$ It follows form the computation above and the $2\pi$-periodicity of $z(\cdot)$ that
\begin{align*}%\label{gradient phi}
\nabla\phi(\bfx)=&i\lim_{\epsilon\to0}\int_{\alpha+\delta_1}^{2\pi+\alpha-\delta_2}\log(|z(\alpha)-z(\beta)|)z_\beta(\beta)d\beta\\
&-\lim_{\epsilon\to0}\epsilon\log|\epsilon|\int_{\eta_1}^{\eta_2}(1,i)\cdot N_{C_\epsilon}d\theta\\
\end{align*}
\end{proof}
In view of Lemma \ref{lem: gravity} we can replace \eqref{z eq temp} by
\begin{align}\label{z eq temp 1}
\begin{cases}
\end{cases}
\end{align}
The norms in which the data are assumed to be small will be made precise below. Our main objective here to transform equation \eqref{z eq temp 1} to an equation for which the nonlinearity is small of cubic order. Again the exact meaning of the term ``cubic" will be clarified below, but roughly speaking we consider a quantity to be `small' if the corresponding quantity in the case of the static solution $z(t,α)≡e^i, z_t(t,α)≡0$ is zero. This implies, for instance, that the quantity $(z_α-1)(|z|^2-1)z_t$ is thought of as cubic. However, before we can investigate the structure of \eqref{z eq temp 1} we need to know the existence of a solution, at least locally in time. Theorem \ref{thm: lwp} on local well-posedness for \eqref{z eq temp 1} is therefore the first stepping stone in our analysis. Since local well-posedness is not the focus of this work, we postpone the proof of Theorem \ref{thm: lwp} to Section \ref{sec: RM} and until then we treat it as a black box.
\begin{theorem}\label{thm: lwp}
Let $s\geq 5$. Assume that $z_{0}\in H^{s+\frac{1}{2}}_{\alpha}$ $z_{1}\in H^{s+\frac{1}{2}}_{\alpha}$ and $|z_{0}(\alpha)-z_{0}(\beta)|\geq c'_{0}|e^{i\alpha}-e^{i\beta}|$ for some constant $c'_{0}>0$. Then there is $T>0$, depending on the norm of the initial data, so that \eqref{z eq temp 1} with initial data $(z,z_t)\vert_{t=0}=(z_0,z_1)$ has a unique solution $z=z(t,\alpha)$ for $t\in[0,T)$ satisfying for all $j\leq s$,
\begin{align*}
&\pa^{j}z, \pa^{j}z_{t}\in C\left([0,T],H^{\frac{1}{2}}_{\alpha}\right),\\
&\pa^{j}z_{tt}\in C\left([0,T],L^{2}_{\alpha}\right),
\end{align*}
and $|z(t,\alpha)-z(t,\beta)|\geq \frac{c'_{0}}{2}|e^{i\alpha}-e^{i\beta}|$ for all $\alpha\neq\beta.$ Moreover, if $T^{*}$ is the supremum over all such time $T$, then either $T^{*}=\infty$, or
\begin{align*}
\sup_{t< T^*}\left(\|z_{tt}\|_{H_\alpha^{4}}+\|z_t\|_{H_\alpha^\frac{9}{2}}\right)+\sup_{\substack{t< T^*\\ \alpha\neq\beta}}\left|\frac{e^{i\alpha}-e^{i\beta}}{z(t,\alpha)-z(t,\beta)}\right|=\infty.
\end{align*}
\end{theorem}
\begin{remark}
Note that to prove local well-posedness we differentiate equation \eqref{z eq temp} with respect to time (cf. eq \eqref{zt eq}) to reveal the quasilinear structure, and treat the resulting equation as a second order equation for $z_t.$ The original unknown $z$ is then obtained from $z_t$ by integration, which explains the choice of regularity for the initial data. See Section \ref{sec: RM} for more details.
\end{remark}
In what follows we will use the notation
\Aligns{
for the anti-holomorphic part of the contribution of the gravity and
\Aligns{
for the holomorphic part of the conjugate of the gravity term. We will show later that $g^a$ and $g^h$ are small in an appropriate sense. With this notation we rewrite the equations for $z$ and $$ as
\Align{\label{z eq}
z_{tt}+iaz_\alpha=-\pi z+g^a
\Align{\label{zbar eq}
\zbar_{tt}-ia\zbar_\alpha=-\pi\zbar+g^h.
For future reference we also record the time-differentiated versions of equation \eqref{z eq} and \eqref{zbar eq}. Differentiation of \eqref{z eq} and use of anti-holomorphicity of $z_t$ give
\Align{\label{zt eq}
Similarly, differentiating \eqref{zbar eq} we get
\Align{\label{zbart eq}
\zbar_{ttt}-ia\zbar_{t\alpha}=ia_t\zbar_\alpha+\frac{\pi}{2}[z_t,H]\frac{\zbar_\alpha}{z_\alpha}.
Since $|z|$ is not expected to be small, we want to linearize these equations about the static solution $z_0(α):=e^iα$ in some sense, to exploit the smallness of the initial data. This will be achieved in Subsection \ref{subsec: delta eq}, but before that we will need to establish some basic identities involving $H$ and $.$ This will be the content of Subsection \ref{subsec: basic identities}. A final point to keep in mind when thinking about the smallness of the solution is that if we start with the static solution $z_0(α) := e^iα$ but with arbitrary constant initial velocity, then the domain will move
in the direction of the initial velocity without changing its geometry. Therefore to properly interpret small quantities as those which are small when the static solution is the unit disk centered at zero, we need to appropriately renormalize the solution to account for this motion with constant velocity. It turns out that this issue can be resolved simply by choosing coordinates in which the center of mass is static. We begin the analysis in this section by clarifying this point in Subsection \ref{subsec: center of mass}.
\subsection{Center of Mass}\label{subsec: center of mass}
In this subsection we first show that the center of mass $C=C_Ω(t)$ moves along a straight line with constant speed, that is, $C_tt=0,$ which is consistent with the fact that no external force acts on the system. Then we derive a formula for the center of mass only involving quantities defined on the boundary $∂Ω(t)$, which will be useful later. We begin by recalling the definition of the center of mass
\Align{\label{center of mass}
C_\Omega(t):=\frac{1}{\pi}\iint\limits_{\Omega}\bfx \,dxdy.
\begin{proposition}\label{prop: center of mass}
The center of mass $C:=C_\Omega$ satisfies
\Aligns{
\frac{d^2C}{dt^2}=0.
\end{proposition}
\begin{proof}
We prove that
\Align{\label{com 1}
\pi\frac{d^2C}{dt^2}=-\int_{\partial\Omega}\phi \,\vec{n}\,dS,
where $\phi$ is the gravity potential, ${\bfn=-\frac{iz_\alpha}{|z_\alpha|}}$ is the exterior unit normal of $\partial\Omega,$ and $dS=|z_\alpha|d\alpha$ is the line element of the boundary. We assume \eqref{com 1} for the moment and prove that the integral on the right hand side vanishes. Recall that $\phi$ satisfies $\partial_z\partial_\zbar\phi=\frac{\pi}{2}$ inside $\Omega.$ Integration in $z$ gives $\partial_\zbar\phi=\frac{\pi}{2}z+A(\zbar)$ where $A$ is an anti-holomorphic function inside $\Omega,$ and another integration in $\zbar$ gives $\phi=\frac{\pi}{2}z\zbar+A(\zbar)+B(z)$ for some holomorphic function $B.$ Moreover, from Lemma \ref{lem: gravity} we know that for points on the boundary $\partial_\zbar\phi=\partial_\zbar(\phi-B)=\frac{\pi}{4}(I-\Hbar)z.$ With this notation we rewrite \eqref{com 1} as
\Align{\label{com 3}
\pi\frac{d^2C}{dt^2}=&i\int_0^{2\pi}\left(\frac{\pi}{2}z\zbar+A(\zbar)\right)z_\alpha d\alpha+i\int_{\partial\Omega} B(z)dz\\
=&-\frac{\pi i}{2}\int_0^{2\pi}\zbar\, z\, z_\alpha d\alpha-\frac{\pi i}{4}\int_0^{2\pi}((I-\Hbar)z) \,z\,\zbar_\alpha d\alpha\\
=&\frac{\pi i}{4}\int_0^{2\pi}z(\Hbar z) \,\zbar_\alpha d\alpha=\frac{\pi i}{4}\int_{\partial\Omega}z\Hbar z d\zbar.
Now recall that the (conjugate) Hilbert transform is defined as
\Aligns{
\Hbar f(\zbar):=-\frac{\pv}{\pi i}\int_{\partial\Omega}\frac{f(w)}{\wbar-\zbar}d\wbar:=-\frac{1}{\pi i }\lim_{\epsilon\to0}\int_{\partial\Omega\backslash B_\epsilon(\zbar)}\frac{f(w)}{\wbar-\zbar}d\wbar,}
where the last limit converges in the $L^2$ sense. In particular if $f,g\in L^2$ then
\Aligns{
-\pi i{\int_{\partial\Omega}}g(z)\Hbar f(z)d\zbar=&{\int_{\partial\Omega}}g(z)\lim_{\epsilon\to0}\int_{\partial\Omega\backslash B_\epsilon(z)}\frac{f(w)}{\wbar-\zbar}d\wbar d\zbar=\lim_{\epsilon\to0}\int_{\partial\Omega}\int_{\partial\Omega-B_\epsilon(z)}\frac{g(z)f(w)}{\wbar-\zbar}d\wbar d\zbar\\
=&\lim_{\epsilon\to0}\int_{\partial\Omega}\int_{\partial\Omega-B_\epsilon(w)}\frac{g(z)f(w)}{\wbar-\zbar}d\zbar d\wbar=\lim_{\epsilon\to0}\int_{\partial\Omega}\int_{\partial\Omega-B_\epsilon(z)}\frac{g(w)f(z)}{\zbar-\wbar}d\wbar d\zbar\\
\int_{\partial\Omega-B_{\epsilon}(z)}\frac{g(w)}{\zbar-\wbar}d\wbar d\zbar=\pi i\int_{\partial\Omega}f(\zbar)\Hbar g(\zbar)d\zbar.
Applying this observation to $f(z)=g(z)=z$ we see that $\int_{\partial\Omega}z\Hbar z d\zbar=-\int_{\partial\Omega}z\Hbar zd\zbar$ and therefore in view of \eqref{com 3} we get $\frac{d^2C}{dt^2}=0.$ Finally we establish \eqref{com 1} by direct differentiation. For this we denote the flow map by $X,$ that is, $$X(t,\cdot):\Omega(0)\to\Omega(t)$$ satisfies $\frac{dX(t,\bfx)}{dt}=V(t,X(t,\bfx)),~\frac{d^2X(t,\bfx)}{dt^2}=-\nabla P(t,X(t,\bfx))-\nabla\phi(t,X(t,\bfx)).$ Then since the flow is incompressible we have
\Aligns{
and hence
\Aligns{
\pi\frac{d^2C}{dt^2}=&-\iint\limits_{\Omega(0)}\nabla \left(P(t,X(t,\bfx^\prime))+\nabla\phi(t,X(t,\bfx^\prime))\right)d\bfx^\prime
=-\iint\limits_{\Omega(t)}(\nabla P(t,\bfx)+\nabla\phi(t,\bfx))d\bfx=-\int_{\partial\Omega}\phi\,\vec{n}\,dS,
as desired.
\end{proof}
The formula \eqref{center of mass} is in terms of the domain $Ω(t)$, but since we work with the boundary equation \eqref{z eq temp 1}, it is more convenient to derive a formula for center of mass only involving quantities defined on the boundary. This is achieved in the following proposition.
\begin{proposition}\label{prop: bdry formula CM}
Let us denote
\begin{align}\label{def: ep delta}
\ep:=|z|^{2}-1,\quad \delta:=(I-H)\ep.
\end{align}
Then the center of mass \eqref{center of mass} as a complex number can be written as
\begin{align}\label{bdry formula CM}
\end{align}
where $z(t,\alpha)$ is the parametrization of $\partial\Omega(t)$.
\end{proposition}
\begin{proof}
We can write the center of mass as $\frac{1}{\pi}\iint_{\Omega(t)}(x+iy)dxdy$. Using the divergence theorem, we have
\begin{align*}
\iint_{\Omega(t)}xdxdy=\iint_{\Omega(t)}\textrm{div}\left(\frac{x^{2}}{2},0\right)dxdy=&\int_{\partial\Omega(t)}\left(\frac{x^{2}}{2},0\right)\cdot \left(\frac{y_{\alpha}}{|z_{\alpha}|},-\frac{x_{\alpha}}{|z_{\alpha}|}\right)ds\\
\end{align*}
\begin{align*}
i\iint_{\Omega(t)}ydxdy=i\iint_{\Omega(t)}\textrm{div}\left(0,\frac{y^{2}}{2}\right)dxdy=&i\int_{\partial\Omega(t)}\left(0,\frac{y^{2}}{2}\right)\cdot \left(\frac{y_{\alpha}}{|z_{\alpha}|},-\frac{x_{\alpha}}{|z_{\alpha}|}\right)ds\\
\end{align*}
Therefore we have
\begin{align*}
\iint_{\Omega(t)}(x+iy)dxdy=-\frac{i}{2}\int_{0}^{2\pi}|z|^{2}z_{\alpha}d\alpha=&-\frac{i}{2}\int_{0}^{2\pi}\ep z_{\alpha}d\alpha\\
=&-\frac{i}{2}\int_{0}^{2\pi}\left(\frac{I-H}{2}\ep\right) z_{\alpha}d\alpha=-\frac{i}{4}\int_{0}^{2\pi}\delta z_{\alpha}d\alpha.
\end{align*}
This completes the proof.
\end{proof}
We have the following corollary.
\begin{corollary}\label{prop: trans CM}
Let $v^{0}_{c}$ and $c^{0}$ be the initial velocity and position of the center of mass respectively. If $z=z(t,\alpha)$ is a solution to \eqref{z eq temp 1} then $z(t,\alpha)-c^{0}-v^{0}_{c}t$ is also a solution to \eqref{z eq temp 1}. Moreover, $z-c^{0}-v^{0}_{c}t$ parametrizes the boundary of a domain whose center of mass is always at the origin.
\end{corollary}
\begin{proof}
This follows from Proposition \ref{prop: center of mass} and the fact that $z_{tt}, z_{\alpha}$ and $H$ are invariant under the transformation
\begin{align}\label{trans CM}
z(\alpha,t)\mapsto z(\alpha,t)-c^{0}-v^{0}_{c}t.
\end{align}
\end{proof}
In what follows, we will only consider this normalized solution to \eqref{z eq temp 1}, that is we assume that the center of mass is always at the origin, and this assumption is justified by Corollary~\ref{prop: trans CM}. Therefore in view of Proposition \ref{prop: bdry formula CM} and Corollary \ref{prop: trans CM} we always have
\begin{align*}
\int_{0}^{2\pi}\ep z_{\alpha}d\alpha=\int_{0}^{2\pi}\delta z_{\alpha}d\alpha=0.
\end{align*}
\subsection{Basic Identities}\label{subsec: basic identities}
In this subsection we record some basic identities which will be used in the remainder of this work. A few more standard properties of the Hilbert transform are recalled in Appendix \ref{app: Hilbert transform}. In the remainder of this section we assume that the parametrization $z$ of $∂Ω(t)$ has regularity $C^2_t,α.$
\subsubsection{Commutation Relations}
We compute the commutators of various operators with the Hilbert transform.
\begin{lemma}\label{lem: operator H commutator}
For any $2\pi-$periodic function $f$ in $C^2_{t,\alpha}$
\begin{enumerate}[(i)]
\item $[\partial_t,H]f=[z_t,H]\frac{f_\alpha}{z_\alpha},$
\item $[\partial_t^2,H]f=2[z_t,H]\frac{f_{t\alpha}}{z_\alpha}+[z_{tt},H]\frac{f_\alpha}{z_\alpha}+\frac{1}{\pi i}\int_0^{2\pi}\left(\frac{z_t(\beta)-z_t(\alpha)}{z(\beta)-z(\alpha)}\right)^2f_\beta(\beta)d\beta,$
\item$\partial_\alpha Hf=z_\alpha H\frac{f_\alpha}{z_\alpha},$
\item$[a\partial_\alpha,H]f=[az_\alpha,H]\frac{f_\alpha}{z_\alpha},$
\item$[\partial_t^2+ia\partial_\alpha,H]f=-\frac{\pi}{2}[(I-\Hbar)z,H]\frac{f_\alpha}{z_\alpha}+2[z_t,H]\frac{f_{t\alpha}}{z_\alpha}+\frac{1}{\pi i}\int_0^{2\pi}\left(\frac{z_t(\beta)-z_t(\alpha)}{z(\beta)-z(\alpha)}\right)^2f_\beta(\beta)d\beta.$
\end{enumerate}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem: operator H commutator}]
\begin{enumerate}[(i)]
\item \Aligns{
[\partial_t,H]f&=\frac{\pv}{\pi i}\int_0^{2\pi}\left(\frac{f(\beta)z_{t\beta}(\beta)}{z(\beta)-z(\alpha)}-\frac{(z_t(\beta)-z_t(\alpha))f(\beta)z_\beta(\beta)}{(z(\beta)-z(\alpha))^2}\right)d\beta\\
&=-\frac{\pv}{\pi i}\int_0^{2\pi}\frac{(z_t(\beta)-z_t(\alpha))f_\beta(\beta)}{(z(\beta)-z(\alpha))z_\beta(\beta)}z_\beta(\beta)d\beta=[z_t,H]\frac{f_\alpha}{z_\alpha}.
\item \Aligns{
[\partial_t^2,H]f=&\partial_t([z_t,H]\frac{f_\alpha}{z_\alpha})+\partial_t H\partial_t f=H\partial_t^2f\\
=&[z_{tt},H]\frac{f_\alpha}{z_\alpha}+2[z_t,H]\frac{f_{t\alpha}}{z_\alpha}+\frac{1}{\pi i}\int_{0}^{2\pi}\left(\frac{z_t(\beta)-z_t(\alpha)}{z(\beta)-z(\alpha)}\right)^2f_\beta(\beta)d\beta.
\item \Aligns{
\partial_\alpha H f&=\frac{\pv}{\pi i}\int_0^{2\pi}\frac{f(\beta)}{(z(\beta)-z(\alpha))^2}z_\alpha(\alpha)z_\beta(\beta)d\beta\\
&=z_\alpha(\alpha)\frac{\pv}{\pi i}\int_0^{2\pi}\frac{f_\beta(\beta)}{z\beta)-z(\alpha)}d\beta=z_\alpha H\frac{f_\alpha}{z_\alpha}.
\item \Aligns{
[a\partial_\alpha,H]f=az_\alpha H\frac{f_\alpha}{z_\alpha}-H(af_\alpha)=[az_\alpha,H]\frac{f_\alpha}{z_\alpha}.
\item This part is a corollary of the previous parts combined with equation \eqref{z eq temp} and Lemma \ref{lem: gravity}.
\end{enumerate}
\end{proof}
\begin{lemma}\label{lem: partialt f H g}
For any $2\pi-$periodic function $f$ and $g$ in $C^2_{t,\alpha}$
\Aligns{
\partial_t[f,H]g=[f_t,H]g+[f,H]g_t+f[z_t,H]\frac{f_\alpha}{z_\alpha}-[z_t,H]\frac{\partial_\alpha(fg)}{z_\alpha}.
\end{lemma}
\begin{proof}
Using part $(i)$ of Lemma \ref{lem: operator H commutator} we get
\Aligns{
\partial_t[f,H]g=\partial_t(fHg)-\partial_tH(fg)&=f_tHg+fHg_t+f[z_t,H]\frac{g_\alpha}{z_\alpha}-H(f_tg)-H(fg_t)-[z_t,H]\frac{\partial_\alpha(fg)}{z_\alpha}\\
\end{proof}
Next we recored the following important computation relating $[,]z$ and the area of $Ω.$
\begin{lemma}\label{lem: zbar Hbar commutator}
If $z:[0,2\pi]\to\partial\Omega$ is a counterclockwise parametrization then
\Aligns{
\end{lemma}
\begin{proof}
Since the parametrization is counterclockwise the exterior normal $\bfn$ is given by $$\bfn=-\frac{iz_\alpha}{|z_\alpha|}=\frac{y_\alpha-ix_\alpha}{|z_\alpha|}$$ in complex notation. It follows that with $z=x+iy$
\Aligns{
[\zbar,\Hbar]z&=\frac{1}{\pi i}\int_0^{2\pi }z(\beta)\zbar_\beta(\beta)d\beta=\frac{1}{2\pi i}\int_0^{2\pi}(z(\beta)\zbar_\beta(\beta)-\zbar(\beta)z_\beta(\beta))d\beta\\
&=\frac{1}{\pi}\int_0^{2\pi}\Im(z(\beta)\zbar_\beta(\beta))d\beta=\frac{1}{\pi}\int_0^{2\pi}(x_\beta(\beta)y(\beta)-y_\beta(\beta) x(\beta))d\beta\\
&=-\frac{1}{\pi}\int_{\partial\Omega}\pmat{x\\y}\cdot \bfn\,|z_\beta(\beta)|d\beta=-\frac{1}{\pi}\iint_{\Omega}\div\pmat{x\\y}dxdy=-\frac{2|\Omega|}{\pi}.
\end{proof}
\begin{lemma}\label{lem: z H}
For any $2\pi-$periodic function $f$ in $C^2_{t,\alpha}$
\Aligns{
\end{lemma}
\begin{proof}
This is an immediate consequence of the definition of the Hilbert transform and the periodicity of $f.$
\end{proof}
\begin{lemma}\label{lem: fgh commutator}
For any $2\pi-$periodic function $f$, $g,$ and $h$ in $C^2_{t,\alpha}$
\Aligns{
\end{lemma}
\begin{proof}
\Aligns{
\end{proof}
\begin{lemma}\label{lem: H plus Hbar}
Suppose $f$ and $g$ are $2\pi-$periodic functions in $C^{2}_{t,\alpha}$ which are anti-holomorphic inside $\Omega.$ Then with the notation $\ep=|z|^{2}-1$
\Aligns{
[f,H\frac{1}{z_\alpha}+\Hbar\frac{1}{\zbar_\alpha}]g_\alpha=-\frac{1}{\pi i}\int_0^{2\pi}\frac{(f(\alpha)-f(\beta))g_\beta(\beta)\zbar(\alpha)\zbar(\beta)\left(\frac{\ep(\alpha)}{\zbar(\alpha)}-\frac{\ep(\beta)}{\zbar(\beta)}\right)}{|z(\beta)-z(\alpha)|^2}d\beta.
\end{lemma}
\begin{proof}
\Aligns{
[f,H\frac{1}{z_\alpha}+\Hbar\frac{1}{\zbar_\alpha}]g_\alpha=&\frac{\pv}{\pi i}\int_0^{2\pi}\left(\frac{1}{z(\beta)-z(\alpha)}-\frac{1}{\zbar(\beta)-\zbar(\alpha)}\right)(f(\alpha)-f(\beta))g_\beta(\beta)d\beta\\
=&-[f,\Hbar]\frac{g_\alpha}{\zbar_\alpha}+\zbar[f,\Hbar]\frac{\zbar g_\alpha}{\zbar_\alpha}-\frac{1}{\pi i}\int_0^{2\pi}\frac{(f(\alpha)-f(\beta))g_\beta(\beta)\zbar(\alpha)\zbar(\beta)\left(\frac{\ep(\alpha)}{\zbar(\alpha)}-\frac{\ep(\beta)}{\zbar(\beta)}\right)}{|z(\beta)-z(\alpha)|^2}d\beta.
Since $\frac{g_\alpha}{\zbar_\alpha}$ is anti-holomorphic in side $\Omega,$ the first two terms on the last line above are zero, and this proves the lemma.
\end{proof}
\subsubsection{The Relation between $H$ and $\Hbar$}
In the static case where the boundary of the domain $Ω$ is exactly the unit circle, the corresponding Hilbert transform $$̋ satisfies $\overline{\H}=-\H+2\Av$ where $\Av (f):=\frac{1}{2\pi}\int_0^{2\pi}f(\alpha)d\alpha$. Here we prove an important lemma which quantifies the failure of this identity when $\Omega$ is a small perturbation of the unit disc.
For any $2\pi-$periodic function $f$ in $C^2_{t,\alpha}$
f =-zHf/z+z[ε,H]f_α/z_α+E(f)
where $\ep:=|z|^2-1$ and $E(f)=E_1(f)+E_2(f)+E_3(f)$ with
E_1(f):= -1/πi∫_0^2πf(β)(ϵ(α)/z(α)-ϵ(β)/z(β))(z(α)z(β))^2/(z(α)-z(β))^2∂_β(ϵ(β)/z(β))dβ
E_2(f):= -1/πi∫_0^2πf(β)(ϵ(β)/z(β)-ϵ(α)/z(α))^2(z(α)z(β))^2/(z(β)-z(α))|z(β)-z(α)|^2∂_β(ϵ(β)/z(β))dβ
E_3(f):= -1/πi∫_0^2πf(β)(ϵ(β)/z(β)-ϵ(α)/z(α))^2(z(α)z(β))^2/(z(β)-z(α))|z(β)-z(α)|^2∂_β(1/z(β))dβ.
Recall the following relations:
\begin{align*}
\zbar=\frac{1+\epsilon}{z},\quad \zbar_{\beta}(\beta)=\frac{\epsilon_{\beta}(\beta)z(\beta)-z_{\beta}(\beta)(1+\epsilon(\beta))}{\left(z(\beta)\right)^{2}}.
\end{align*}
We have
\begin{align}\label{Hbar pre}
\begin{split}
(\overline{H}f)(\alpha)=&-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)\zbar_{\beta}(\beta)}{\zbar(\beta)-\zbar(\alpha)}d\beta\\
=&-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)}{\frac{1+\epsilon(\beta)}{z(\beta)}-\frac{1+\epsilon(\alpha)}{z(\alpha)}}\left(\left(\frac{\epsilon(\beta)}{z(\beta)}\right)_{\beta}-\frac{z_{\beta}(\beta)}{(z(\beta))^{2}}\right)d\beta\\
=&-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)}{\frac{1}{z(\beta)}-\frac{1}{z(\alpha)}}\left(\left(\frac{\epsilon(\beta)}{z(\beta)}\right)_{\beta}-\frac{z_{\beta}(\beta)}{(z(\beta))^{2}}\right)d\beta\\
&-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)\left(\frac{\epsilon(\alpha)}{z(\alpha)}-\frac{\epsilon(\beta)}{z(\beta)}\right)}{\left(\frac{1}{z(\beta)}-\frac{1}{z(\alpha)}\right)\left(\frac{1}{z(\beta)}-\frac{1}{z(\alpha)}+\frac{\epsilon(\beta)}{z(\beta)}-\frac{\epsilon(\alpha)}{z(\alpha)}\right)}\left(\left(\frac{\epsilon(\beta)}{z(\beta)}\right)_{\beta}-\frac{z_{\beta}(\beta)}{(z(\beta))^{2}}\right)d\beta\\
=&-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)}{\frac{1}{z(\beta)}-\frac{1}{z(\alpha)}}\left(\left(\frac{\epsilon(\beta)}{z(\beta)}\right)_{\beta}-\frac{z_{\beta}(\beta)}{(z(\beta))^{2}}\right)d\beta\\
&-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)\left(\frac{\epsilon(\alpha)}{z(\alpha)}-\frac{\epsilon(\beta)}{z(\beta)}\right)}{\left(\frac{1}{z(\beta)}-\frac{1}{z(\alpha)}\right)^{2}}\left(\left(\frac{\epsilon(\beta)}{z(\beta)}\right)_{\beta}-\frac{z_{\beta}(\beta)}{(z(\beta))^{2}}\right)d\beta\\
&-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)\left(\frac{\epsilon(\alpha)}{z(\alpha)}-\frac{\epsilon(\beta)}{z(\beta)}\right)^{2}}{\left(\frac{1}{z(\beta)}-\frac{1}{z(\alpha)}\right)^{2}\left(\frac{1}{z(\beta)}-\frac{1}{z(\alpha)}+\frac{\epsilon(\beta)}{z(\beta)}-\frac{\epsilon(\alpha)}{z(\alpha)}\right)}\left(\left(\frac{\epsilon(\beta)}{z(\beta)}\right)_{\beta}-\frac{z_{\beta}(\beta)}{(z(\beta))^{2}}\right)d\beta.
\end{split}
\end{align}
The `constant term' above (the second term in the first line) is
\begin{align}\label{Hbar const}
\begin{split}
&\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)z(\alpha)z(\beta)}{z(\alpha)-z(\beta)}\frac{z_{\beta}(\beta)}{(z(\beta))^{2}}d\beta\\
=&\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)z(\alpha)z_{\beta}(\beta)}{\left(z(\alpha)-z(\beta)\right)z(\beta)}d\beta=-zH\left(\frac{f}{z}\right).
\end{split}
\end{align}
The `linear terms' above (the first term in the first line and the second term in the second line) are given by
\begin{align}\label{Hbar linear 1}
-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)z(\alpha)z(\beta)}{z(\alpha)-z(\beta)}\left(\frac{\epsilon(\beta)}{z(\beta)}\right)_{\beta}d\beta.
\end{align}
\begin{align}\label{Hbar linear 2}
\begin{split}
&\frac{1}{\pi i}\int_{0}^{2\pi}f(\beta)\left(\frac{\epsilon(\alpha)}{z(\alpha)}-\frac{\epsilon(\beta)}{z(\beta)}\right)\left(\frac{1}{\frac{1}{z(\beta)}-\frac{1}{z(\alpha)}}\right)_{\beta}d\beta\\
=&-\frac{1}{\pi i}\int_{0}^{2\pi}f_{\beta}(\beta)\left(\frac{\epsilon(\alpha)}{z(\alpha)}-\frac{\epsilon(\beta)}{z(\beta)}\right)\left(\frac{1}{\frac{1}{z(\beta)}-\frac{1}{z(\alpha)}}\right)d\beta\\
&+\frac{1}{\pi i}\int_{0}^{2\pi}f(\beta)\left(\frac{\epsilon(\beta)}{z(\beta)}\right)_{\beta}\left(\frac{1}{\frac{1}{z(\beta)}-\frac{1}{z(\alpha)}}\right)d\beta.
\end{split}
\end{align}
The last term in (<ref>) cancels with (<ref>). Therefore the `linear term' in $\overline{H}f$ is given by
\begin{align}\label{Hbar linear}
-\frac{1}{\pi i}\int_{0}^{2\pi}f_{\beta}(\beta)\frac{\epsilon(\alpha)z(\beta)}{z(\alpha)-z(\beta)}d\beta+\frac{1}{\pi i}\int_{0}^{2\pi}f_{\beta}(\beta)\frac{\epsilon(\beta)z(\alpha)}{z(\alpha)-z(\beta)}d\beta
\end{align}
where in the last step we used the fact that $f(0)=f(2\pi)$. The remaining terms in $\overline{H}f$ are the first term in the second line and the two terms in the third line of (<ref>). The first term in the second line can be written as
\begin{align}\label{Hbar R 1}
\begin{split}
E_{1}(f):=-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)\left(\frac{\epsilon(\alpha)}{z(\alpha)}-\frac{\epsilon(\beta)}{z(\beta)}\right)(z(\alpha)z(\beta))^{2}}{(z(\alpha)-z(\beta))^{2}}\partial_{\beta}\left(\frac{\epsilon(\beta)}{z(\beta)}\right)d\beta.
\end{split}
\end{align}
The first term in the third line of (<ref>) can be written as
\begin{align}\label{Hbar R 2}
E_{2}(f):=-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)\left(\frac{\epsilon(\beta)}{z(\beta)}-\frac{\epsilon(\alpha)}{z(\alpha)}\right)^{2}(z(\alpha)z(\beta))^{2}}{(z(\beta)-z(\alpha))|z(\beta)-z(\alpha)|^{2}}\partial_{\beta}\left(\frac{\epsilon(\beta)}{z(\beta)}\right)d\beta.
\end{align}
The second term in the third line of (<ref>) can be written as
\begin{align}\label{Hbar R 3}
E_{3}(f):=-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)\left(\frac{\epsilon(\beta)}{z(\beta)}-\frac{\epsilon(\alpha)}{z(\alpha)}\right)^{2}(z(\alpha)z(\beta))^{2}}{(z(\beta)-z(\alpha))|z(\beta)-z(\alpha)|^{2}}\partial_{\beta}\left(\frac{1}{z(\beta)}\right)d\beta.
\end{align}
Note that if we measure smallness of quantities by comparison with the static case $z\equiv e^{i\alpha},$ then by Lemma <ref>, $E(f)$ is order of $\ep^2$ smaller than $f.$ This observation will be made precise when we carry out the estimates in Sections <ref> and <ref>.
§.§ The $\delta$ Equation
In this section we derive an equation for the small quantity
Note that in view of our small data assumptions we expect the quantities $\varepsilon$ and $\delta$ to be (linearly) small. Our main goal here is to show that $\delta$ satisfies a constant-coefficient PDE with cubic nonlinearity. This will be accomplished in two steps. In the first step we show that the nonlinear part of $(\partial_t^2+ia\partial_\alpha)\delta$ is cubic. If we then replace the operator $\partial_t^2+ia\partial_\alpha$ by $\partial_t^2+i\pi\partial_\alpha,$ corresponding to the value of $a$ in the static case, we will notice that the resulting error is only quadratic. For this reason, in the second step we perform a change of variables $\beta(t,\alpha)=k^{-1}(t,\alpha)$ such that the nonlinearity in the equation for $(\partial_t^2+i\pi\partial_\beta)\delta$ has no quadratic part. The first step is achieved in the following proposition.
The quantities $\delta=(I-H)\varepsilon$ and $\delta_t=\partial_t\delta$ satisfy
(∂_t^2+ia∂_α-π)δ=_1:= π/2[E(z),H]_α/z_α+π/2(I-H)E()
(∂_t^2+i∂_α-π)δ_t= _2:= -ia_t∂_αδ+π/2((I-H)E()-[z_t,H]E()/z_α)
where $E(f)$ is as in Lemma <ref>. Moreover, we can write
We want to apply the last part of Lemma <ref>. To this end we first compute $(\partial_t^2+ia\partial_\alpha)\ep.$
(∂_t^2+ia∂_α)= (z_tt+iaz_α)+(_tt+ia_α)z+2z_t_t
= -π/2((I-)z-z(I-H))+2∂_t(z_t),
and since $z\zbar_t$ is holomorphic
Applying Lemma <ref> we get
(∂_t^2+ia∂_α)δ= π/2(I-H)(z(I-H)-(I-)z)+π/2[(I-)z,H]_α/z_α
= π/2(I-H)(z(I-H)-(I-)z)+π/2[(I-)z,H]_α/z_α
The last two terms already have the right form so we concentrate on the first two. Using Lemma <ref> we write
and hence
where to pass to the last equality we have used the fact that $\left(\frac{1}{2}(I-H)\right)^2=\frac{1}{2}(I-H).$ To understand the contributions of $\deltabar$ and $(I-\Hbar)z$ we use Lemma <ref> to replace $\Hbar$ by $H.$ For $\deltabar,$ noting that $H\frac{1}{z}=-\frac{1}{z}$ we get
= +zH(-1/z)-z[,H]_α/z_α-E()
= z(I+H)-z[,H]_α/z_α-E()
= z(I+H)-z(I+H)_α/z_α+z(I+H)_α/z_α-E(),
which implies
-π/2(I-H)= π/2(I-H)(z(I+H)(_α/z_α))+π/2(I-H)E()
= π/4(I-H)(zδ(I+H)(_α/z_α))+π/2(I-H)E().
Similarly for $(I-\Hbar)z$ we have
It follows from this and Lemma <ref> that
π/2[(I-)z,H]_α/z_α= -π/2[zδ,H]_α/z_α+π/2[E(z),H]_α/z_α
= -π/4[zδ,H](I+H)ϵ_α/z_α-π/4[zδ,H](I-H)ϵ_α/z_α+π/2[E(z),H]_α/z_α.
By Lemma <ref>, the second term in (<ref>) can be written as
\begin{align}\label{qua vanish temp 3}
\end{align}
The first term in (<ref>) can be written as
\begin{align*}
\end{align*}
By $(iii)$ in Lemma <ref>, the second term in (<ref>) can be written as
\begin{align*}
\end{align*}
Combining these observations with the fact that
we get
Equation (<ref>) now follows from combining this identity with (<ref>) and (<ref>).
Finally, equations (<ref>) and (<ref>) are direct consequences of Lemmas <ref>, <ref>, and <ref> and equation (<ref>).
By comparing the terms on the right hand sides of the equations (<ref>) and (<ref>) with their corresponding values in the static case, one can see that the nonlinearity is cubic. This is least clear for the first term involving $a_t$ in the equation for $\delta_t$ so in the following lemma we present a formula for $a_t$ which sheds some light the structure of this term.
Let $K^*$ denote the formal adjoint of $K:=\Re H=\frac{1}{2}(H+\Hbar)$, i.e.,
\begin{align*}
K^*g(\alpha)=-\Re\frac{\pv}{\pi i}\int_0^{2\pi}\frac{z_\alpha(\alpha)}{|z_\alpha(\alpha)|}\frac{|z_\beta(\beta)|}{z(\beta)-z(\alpha)}g(\beta)d\beta=-\Re\left\{\frac{z_\alpha}{|z_\alpha|}H\frac{|z_\beta|g}{z_\beta}\right\}.
\end{align*}
(I+K^*)(a_t|z_α|)=[-iz_α/|z_α|{ 2[z_t,H]_ttα/z_α+2[z_tt,H]_tα/z_α-[g^a,H]_tα/z_α
Using equation (<ref>), (<ref>) and Lemma <ref> we have
(I-H)(ia_t_α)= (I-H)(_ttt-ia_tα-π/2[z_t,H]_α/z_α)
= [∂_t^2-ia∂_α,H]_t-π/2(I-H)([z_t,H]_α/z_α)
= 2[z_t,H]_ttα/z_α+[z_tt,H]_tα/z_α-[iaz_α,H]_tα/z_α
= 2[z_t,H]_ttα/z_α+2[z_tt,H]_tα/z_α-[g^a,H]_tα/z_α
The lemma now follows by multiplying the two sides of this equation by $\frac{-iz_\alpha}{|z_\alpha|}$ and taking real parts and also observing that
\begin{align*}
\frac{\pi}{2}(I-H)\left([z_{t},H]\frac{\zbar_{\alpha}}{z_{\alpha}}\right)=-\frac{\pi}{2}(I-H)\left(\pt(I-H)\zbar\right)=\frac{\pi}{2}[z_{t},H]\frac{\pa g^{h}}{z_{\alpha}}.
\end{align*}
For future reference we also record the following representation for $K^*$ which is more amenable to estimates.
For any real valued $2\pi-$periodic function $f$
Using the definition $K^*f=-\Re\{\frac{z_\alpha}{|z_\alpha|}H\frac{|z_\beta|}{z_\beta}f\}$ of $K^*$ we have
-2K^*g= z_α/|z_α|H|z_β|f/z_β+_α/|z_α||z_α|f/_α
= 2/|z_α|{[z_α,H]|z_β|f/z_β}+1/|z_α|(H+)(|z_β|f)
= 2{1/|z_α|[z_α-iz,H]|z_β|f/z_β}+1/|z_α|(H+)(|z_α|f)+2{i/|z_α|[z,H]|z_β|f/z_β}.
The lemma now follows by noting that
We now turn to the left hand side of (<ref>). As mentioned above, the nonlinear contribution of $a$ to (<ref>) can be seen to be only quadratic, and therefore a change of variables is necessary to retain the cubic structure. More precisely suppose
is an increasing function such that $k-\alpha$ is $2\pi$ periodic and $k$ is differentiable on $(0,2\pi)$. Let us define
ζ(t,):=z∘k^-1(t,), χ(t,):=δ∘k^-1(t,).
Then introducing
b:=k_t∘k^-1, A= (ak_α)∘k^-1
we have
z_t∘k^-1=(∂_t+b∂_)ζ, (az_α)∘k^-1=Aζ_.
In particular,
We wish to choose the change of variables $k$ in such a way that $b$ consists of quadratic and higher order terms, and $A$ has no linear terms. This is achieved in the following three propositions. First in Propositions <ref> and <ref> we derive the desired representations for $b$ and $A$ under various assumptions on $k.$ Then in Remark <ref> we explain how to construct $k$ satisfying these assumptions.
Suppose that $z(t,\cdot)$ is a simple closed curve containing origin in its interior for each $t$, and that $k$ is increasing and such that $k-\alpha$ is $2\pi$ periodic and $(I-H)(\zbar e^{ik})=(I-H)(\log \zbar+ik)=0.$ Then
\begin{align*}
&(I-H)k_{t}=-i(I-H)\frac{\zbar_{t}\ep}{\zbar }-i[z_{t},H]\frac{\left(\log(\zbar e^{ik})\right)_{\alpha}}{z_{\alpha}}.\\
&\qquad\qquad\qquad\quad-(I-H)\frac{\zbar_{tt}\ep}{\zbar}+(I-H)\frac{g^{h}\ep}{\zbar}+[z_{tt}-g^{a},H]\frac{\left(\log(\zbar e^{ik})\right)_{\alpha}}{z_{\alpha}}.
\end{align*}
The conditions on $k$ in the proposition can be understood in the following way. First note that if we fix a value of $\arg(z(t,0))$ (uniquely determined up to an integer multiple of $2\pi$) then $\log \zbar(t,\cdot)$ is an unambiguously defined continuous function of the real variable $\alpha$ for each fixed $t.$ Moreover, if $z(t,\cdot)$ is a simple closed curve surrounding the origin, then by the periodicity assumption on $k,$ the curve $\zbar e^{ik}$ does not contain the origin in its interior. Therefore $\log (\zbar e^{ik})$ is defined unambiguously as a complex logarithm, its value agrees with $\log \zbar +ik,$ and for any other choice of $\arg(z(t,0))$ it differs from this by an additive constant, so in particular the condition $(I-H)(\log \zbar +ik)=0$ is independent of this choice. The conditions on $k$ can now be understood as requiring that $\zbar e^{ik}$ be the boundary value of a holomorphic function $F,$ such that $0\notin\{F(z)\vert z\in\Omega\}$ and therefore $\log F$ is also well-defined and holomorphic.
Differentiating $(I-H)\left(\log(\zbar e^{ik})\right)=0$ on both sides with respect to $t$, we have:
\begin{align*}
0=(I-H)\left(\frac{\zbar_{t}}{\zbar}+ik_{t}\right)-[z_{t},H]\frac{\left(\log(\zbar e^{ik})\right)_{\alpha}}{z_{\alpha}},
\end{align*}
which implies
\begin{align*}
(I-H)k_{t}=i(I-H)\left(\frac{\zbar_{t}}{\zbar}\right)-i[z_{t},H]\frac{\left(\log(\zbar e^{ik})\right)_{\alpha}}{z_{\alpha}}.
\end{align*}
In view of the fact $(I-H)(\zbar_{t}z)=0$, the first term on the right hand side above can be written as
\begin{align*}
-i(I-H)\frac{\zbar_{t}\ep}{\zbar },
\end{align*}
which gives the first formula in the proposition. For the second formula, we apply the operator $ia\partial_{\alpha}$ to the equation $(I-H)\left(\log(\zbar e^{ik})\right)=0$, to arrive at
\begin{align*}
0=(I-H)\left(\frac{ia\zbar_{\alpha}}{\zbar}-ak_{\alpha}\right)-i[az_{\alpha},H]\frac{\left(\log(\zbar e^{ik})\right)_{\alpha}}{z_{\alpha}}.
\end{align*}
Using (<ref>) we get
\begin{align*}
(I-H)(ak_{\alpha})=&(I-H)\left(\frac{\zbar_{tt}}{\zbar}+\pi-\frac{g^{h}}{\zbar}\right)+[z_{tt}+\pi z-g^{a},H]\frac{\left(\log(\zbar e^{ik})\right)_{\alpha}}{z_{\alpha}}\\
=&(I-H)\left(\frac{\zbar_{tt}z}{|z|^2}-\frac{zg^{h}}{|z|^2}\right)+[z_{tt}-g^{a},H]\frac{\left(\log(\zbar e^{ik})\right)_{\alpha}}{z_{\alpha}}\\
=&(I-H)(\zbar_{tt}z)-(I-H)\frac{\zbar_{tt}\ep}{\zbar}+(I-H)\frac{g^{h}\ep}{\zbar}+[z_{tt}-g^{a},H]\frac{\left(\log(\zbar e^{ik})\right)_{\alpha}}{z_{\alpha}}.
\end{align*}
Here we used the fact that $(I-H)(zg^{h})=0$. The first term above can be written as
\begin{align*}
\end{align*}
and this completes the proof.
Now suppose we define $k$ in a way that $(I-H)\zbar e^{ik}=0.$ Then in view of Proposition <ref>, to prove that $b$ is quadratic and $ak_\alpha$ contains no linear terms we need to understand the invertibility properties of $\Re (I-H)$ (note that $ak_\alpha$ and $k_t$ are real). In fact, a proper understanding of this is necessary also for controlling various other quantities, such as $\ep$ from our control of $\delta.$ A rigorous quantitative treatment of this in the context of small data problem will be given when we carry out the estimates in Sections <ref> and <ref>, but for now we note that if $f$ is real valued, then regarding the last two terms on the second line of (<ref>) in Lemma <ref> as $O(\ep),$[Note that in the static case $z(\alpha)=e^{i\alpha}$ the `average' $[z,H]\frac{f}{z}$ is real if $f$ is real-valued. We may therefore treat the imaginary part of this average as perturbative.]
Therefore, roughly speaking, if $\ep$ is small, then we expect $\Re(I-H)$ to be invertible on the space of functions in $\La^{2}$ which satisfy $\Re\AV(f)=0,$ where [Note that the sign convention is such that $\AV(1)=-1$.]
With this observation in mind we compute $\AV$ for $\ep,~b,$ and $ak_\alpha$ in the following proposition.
Suppose that $z(t,\cdot),~t\in I,$ for some interval $I\subseteq \R,$ is a simple closed curve containing the origin in its interior for each $t,$ $|\Omega|=\pi,$ and that $\zbar e^{ik}$ is the boundary value of a holomorphic function $F(t,z)$ such that $\log F(t,z)$ is also holomorphic and $F(t,0)\in\R_{+},~\forall t\in I.$ Then
(a k_α)=-π+1/2πi∫_0^2π_tz_tβdβ+1/2πi∫_0^2π(_tt-g^h)z_β/|z|^2dβ-1/2πi∫_0^2π(z_tt-g^h/z)∂_βlogFdβ,
(k_t)=/2π∫_0^2π_t/|z|^2z_βdβ-/2π∫_0^2πlogF (zz_tβ-z_tz_β/z^2)dβ.
For $\ep$ we have
by Lemma <ref>. To compute $\AV(ak_\alpha)$ we write $\zbar e^{ik}=F(z)$ where $F$ is as in the statement of the lemma. Differentiating with respect to $\alpha$ and multiplying by $ia$ we get
Using equations (<ref>) and (<ref>) and the relation $\frac{1}{|z|^2}=1-\frac{\ep}{|z|^2}$ we get
ak_α/z= π/z+_tt-g^h/|z|^2+π∂_zlogF+(z_tt-g^a/z)∂_zlogF
= π/z+_tt-g^h-(_tt-g^h)/|z|^2+π∂_zlogF+(z_tt-g^a/z)∂_zlogF.
It follows that
(ak_α)= -1/2πi∫_0^2πak_βz_β/zdβ
= -π+1/2πi∫_0^2π_tz_tβdβ+1/2πi∫_0^2π(_tt-g^h)z_β/|z|^2dβ-1/2πi∫_0^2π(z_tt-g^a/z)∂_βlogFdβ.
The computation for $\AV(k_t)$ is similar. We differentiate the equation $\zbar e^{ik}=F(t,z)$ with respect to time to get
_t e^ik+ik_t F=∂_t (F)
k_t=i_t/-i∂_t (logF).
It follows that
k_t/zz_α=i_tz_α-i_tz_α/|z|^2-i∂_t( z_αlogF/z)+ilogF (zz_tα-z_tz_α/z^2).
Therefore, since $\zbar_t$ and $\log F$ are holomorphic,
(k_t)=1/2π∫_0^2π_t/|z|^2z_βdβ-1/2π∫_0^2πlogF (zz_tβ-z_tz_β/z^2)dβ+∂_t(ilogF(0,t)).
The last equality in the lemma now follows by taking real parts of this expression and noting that $\log F(0,t)\in\R,~\forall t.$
It follows from the previous two propositions that if $k$ satisfies the conditions in these propositions then $A$ and $b$ have the desired smallness properties. In the following remark we explain how to construct $k$ satisfying the hypotheses of these propositions. Note that the fact that $k$ is increasing will follow from the definition of $k$ below and the smallness assumptions in our problem. See Proposition <ref> and the proof of Theorem <ref> for more details.
Suppose $z(t,\cdot)$ is a simple closed curve containing the origin in its simply connected interior for each $t\in I,$ where $I$ is some time interval. We explain how to construct a function $k:I\times\R\to\R$ such that $k-\alpha$ is periodic and $\zbar e^{ik}$ is the boundary value of a function $F$ such that $F$ and $\log F$ are holomorphic inside $\Omega,$ so in particular $(I-H)(\log \zbar+ik)=0.$ Moreover, we normalize $k$ such that $\log F(t,0)\in \R$ for all $t\in I.$
We fix a choice of the logarithm so that $\log z -i\alpha$ is continuous and $2\pi$ periodic. We let $u$ be the solution of the Dirichlet problem in $\Omega$ with boundary value $\log|z|$ and let $v$ be the harmonic conjugate of $u$ which exists because the domain is simply connected. It is then easy to see that if $k:=v\vert_{\partial\Omega}+\arg z$ then $\zbar e^{ik}$ is the boundary value of a holomorphic function $F$ such that $\log F$ is also holomorphic and $k-\alpha$ is $2\pi$ periodic. It remains to show that $k$ may be chosen such that $\log F(t,0)\in\R.$ For this note that $0\notin \{F(t,z)~|~z\in\Omega\}$ and the function
$$G(t,z):=F(t,z)e^{-i\arg F(t,0)}$$
is also holomorphic and $0\notin \{G(t,z)~|~z\in\Omega\},$ so $\log G$ is also holomorphic. Moreover, $G$ now satisfies $G(t,0)=|F(t,0)|\in\R$ and the boundary value of $G$ is $\zbar e^{ip}$ where $p(t,\alpha)=k(t,\alpha)-\arg F(t,0).$ In other words, we have found $p$ such that $(I-H)(\zbar e^{ip})=(I-H)(\log \zbar+ip)=0$ and $\zbar e^{ip}$ is the boundary value of a holomorphic $G$ such that $G(t,0)\in \R$ and $\log G$ is holomorphic.
Suppose $k$ is defined as in Remark <ref>. Assume also that $k$ is increasing. Then $\chi:=(I-H)\delta\circ k^{-1}$ and $v:=\delta_t\circ k^{-1}$ satisfy the equations
\begin{equation}\label{delta eq}
\end{equation}
\begin{equation}\label{deltat eq}
(\pt+b\pap)^2v+iA\pap v-\pi v=N_2
\end{equation}
where $N_j:=\calN_j\circ k^{-1}$ and $\calN_1$ and $\calN_2$ are as defined in equations (<ref>) and (<ref>) respectively.
§ RELATIONS BETWEEN ORIGINAL AND TRANSFORMED QUANTITIES
In the previous section we derived an equation for the transformed quantities $\delta=(I-H)\ep$ and $\chi=\delta\circ k^{-1},$ defined as in (<ref>)–(<ref>), where $k$ was chosen according to Remark <ref>. In order to prove energy estimates for this equation it will be important to be able to transfer estimates on $\delta$ to estimates on $\ep$ and conversely. More precisely we define the following quantities
\begin{align}\label{quantities}
\begin{split}
&\zeta=z\circ k^{-1},\quad u=z_{t}\circ k^{-1}, \quad w=z_{tt}\circ k^{-1}\\
&\chi=\delta\circ k^{-1},\quad v=(\partial_{t}\delta)\circ k^{-1}=(\partial_{t}+b\partial_{\alphap})\chi,\\
&\mu=\ep\circ k^{-1},\quad \eta=\zeta_\alpha-i\zeta,\quad \ep=|z|^2-1.
\end{split}
\end{align}
We will use $\CH$ for the Hilbert transform in the variable $\zeta$ and $H$ for the Hilbert transform in $z.$ Our goal in this section will be to obtain algebraic and analytic relations between the `transformed' quantities
χ, v, (+b)v
and the `original' quantities
ζ, u, w, η, μ.
Note that by comparison with the static case (where $z\equiv e^{i\alpha}$ and $k\equiv \alpha$) we expect the `small quantities' to be
\begin{align}
&\mathrm{original:}\quad \eta,~\mu,~u,~w,\label{osqs}\\
&\mathrm{transformed:}\quad \chi,~v,~(\pt+b\pa)v.\label{tsqs}
\end{align}
The analytic relations in this section will be derived under the following bootstrap assumption, where $\ell\geq 5$ is a fixed integer and $M \leq M_0 <\infty$ are small numbers to be fixed:
∑_k≤ℓ(∂_^kw_L_^2+∂_^ku_L_^2+∂_^k η_L_^2)≤M<M_0,
for $t\in I$ where $I$ is some interval containing $0$.
We start with the following estimates for $\zeta$ which will be used in many other computations.
There exists $\alphap_0=\alphap_0(t)$ such that
ζ(·)-e^i(_0+·)_L^∞_∩L^2_≤Cη_^2≤C M,
μ_L^∞_∩L^2_≤Cη_^2≤C M.
* If $M_0$ in (<ref>) is sufficiently small then there exist non-zero constants $c$ and $C$ such that for all $j\leq \ell$ and $k\leq \ell-1$
c≤∂_^j ζ__L_^2,∂_^k ζ__L_^∞≤C.
* If $M_0$ in (<ref>) is sufficiently small then for all $k\leq \ell$
\begin{align*}
\sum_{j\leq k}\|\partial_\alphap^j\mu_\alphap\|_{L_\alphap^2}\leq C\sum_{j\leq k}\|\partial_\alphap^j\eta\|_{L_\alphap^2},\qquad \sum_{j\leq k-1}\|\partial_\alphap^j\mu_\alphap\|_{L_\alphap^{\infty}}\leq C\sum_{j\leq k-1}\|\partial_\alphap^j\eta\|_{L_\alphap^{\infty}}.
\end{align*}
* Note that since $0\in\Omega(0)$ and $|\zeta(t,\alphap)|\geq\frac{1}{2}$ for all $\alphap,$ $0\in\Omega(t)$ as long as the bootstrap assumptions hold. Direct differentiation implies that $f(\alpha):=e^{-i\alpha}\zeta(\alpha)$ satisfies $\|f_\alphap\|_{L_\alphap^2}\leq \|\eta\|_{\Lap^2}.$ Moreover since the area of $\Omega$ is a preserved by the flow, there exists $\gamma\in [0,2\pi]$ such that $|f(\gamma)|=1,$ or equivalently $f(\gamma)=e^{i\alphap_0}$ for some $\alphap_0.$ Now for any other $\alphap\in[0,2\pi]$ we have
proving the first inequality. The second inequality is a direct consequence of the first and the definition of $\mu.$
* From the definition of $\eta$ we have
The desired estimates now follow from the previous part by induction on $j$ and use of the Sobolev inequality $\|\partial_\alphap^k\eta\|_{L_\alphap^\infty}\leq C(\|\partial_\alphap^k\eta\|_{L_\alphap^2}+\|\partial_\alphap^{k+1}\eta\|_{L_\alphap^2}).$
* This estimate is a direct consequence of the previous part and the relation $\mu_\alphap=\zetabar\eta+\zeta\etabar.$
A corollary of Proposition <ref> is the following result which allows us to use the tools from Section <ref> in the remainder of this section and in the next section.
Under the bootstrap assumption (<ref>), and if $M_0$ is sufficiently small,
\begin{align*}
|\zeta(\alphap)-\zeta(\betap)|\geq \frac{1}{10}|e^{i\alphap}-e^{i\betap}|.
\end{align*}
Since $\zeta(\alpha\pm2\pi)=\zeta(\alpha)$ and $e^{i(\alpha\pm2\pi)}=e^{i\alpha}$ it suffices to prove the corollary for $\alpha$ and $\beta$ such that $|\alpha-\beta|\leq\frac{3\pi}{2}.$ Since for this range of $\alpha$ and $\beta$ we have $|e^{i\alpha}-e^{i\beta}|\gtrsim |\alpha-\beta|,$
\begin{align*}
\abs{\zeta(\alphap)-\zeta(\betap})&=\abs{\int_{\betap}^{\alphap}\zeta_{\alphap'}(\alpha'')d\alpha''}=\abs{i\int_{\betap}^{\alphap}\zeta(\alpha'')d\alpha''
\left(e^{i\alphap}-e^{i\betap}\right)}-\abs{\int_{\betap}^{\alphap}O(M)d\alpha''}\\
&\geq \frac{1}{10}|e^{i\alpha}-e^{i\beta}|.
\end{align*}
if $M$ is sufficiently small.
As immediate important consequences of Proposition <ref> and Corollary <ref> we record the following two corollaries.
If $M_0$ in (<ref>) is sufficiently small, then for any $2\leq k\leq \ell,$ any $2\pi-$periodic function $f,$ and with $E$ defined as in Lemma <ref>
∑_j≤k^j E(f)_^2≤C(∑_j≤k^jμ_^2^2)(∑_j≤k^jf_^2)≤C M^2 ∑_j≤k^jf_^2.
From the definition of $E$ this is a direct corollary of Proposition <ref> and Lemma <ref>. Notice that by Proposition <ref> $\|\pap^{\ell+1}\mu\|_{\Lap^2}\leq CM$ and $1\lesssim\|\pap^{\ell+1}\zeta\|_{\Lap^2}\lesssim1$ under the bootstrap assumptions (<ref>).
If $M_0$ in (<ref>) is sufficiently small, then for any $2\leq k\leq \ell,$
Since $\chi=(I-\CH)\mu,$ this follows from Lemma <ref> and Proposition <ref>.
Next we record the following algebraic relations.
With the same notation as (<ref>)
\begin{align}
&\partial_{\alphap}\chi=\left(I-\zeta_{\alphap}\calH\frac{1}{\zeta_{\alphap}}\right)\mu_\alphap\label{chi mu}\\
+\overline{\calH}\right)u\zetab-[u,\calH]\frac{\mu_\alphap}{\zeta_{\alphap}},\label{u v}\\
&\quad\quad\quad\quad\quad\quad-\frac{1}{\pi i}\int_0^{2\pi}\left(\frac{u(t,\alphap)-u(t,\betap)}{\zeta(t,\betap)-\zeta(t,\alphap)}\right)^2\mu_\betap(t,\betap)d\betap.\label{vt w}
\end{align}
\begin{align*}
\partial_{\alphap}\chi=&\left(\zeta\zetab-1\right)_{\alphap}
\end{align*}
Composing with $k^{-1}$ we get the first identity. Similarly
\begin{align*}
\partial_{t}\delta=&\partial_{t}(I-H)(z\zbar-1)=(z\zbar-1)_{t}-\partial_{t}\left(H(z\zbar-1)\right)\\
\end{align*}
To derive the third formula, we need to compute a time derivative as follows:
\begin{align*}
\partial_{t}\left([z_{t},H]\frac{f_{\alpha}}{z_{\alpha}}\right)=[z_{tt},H]\left(\frac{f_{\alpha}}{z_{\alpha}}\right)+[z_{t},H]\frac{f_{t\alpha}}{z_{\alpha}}+\frac{1}{\pi i}\int_{0}^{2\pi}\left(\frac{z_{t}(\beta)-z_{t}(\alpha)}{z(\beta)-z(\alpha)}\right)^{2}f_{\beta}(\beta)d\beta.
\end{align*}
\begin{align*}
\partial^{2}_{t}\delta=&2z_{tt}\zbar+2z_{t}\zbar_{t}-(H+\Hbar)(z_{tt}\zbar+z_{t}\zbar_{t})\\
&-\frac{1}{\pi i}\int_{0}^{2\pi}\left(\frac{z_{t}(\beta)-z_{t}(\alpha)}{z(\beta)-z(\alpha)}\right)^{2}(z\zbar)_{\beta}d\beta.
\end{align*}
The third formula follows by precomposing with $k^{-1}$.
The estimates for $u$ and $w$ are given in the following proposition.
If $M_0$ in (<ref>) is sufficiently small, then there are non-zero constants $c$ and $C$ such that for any $2\leq k \leq \ell$
\begin{align}
&c \sum_{j\leq k}\|\partial_\alphap^j v\|_{L^{2}_\alphap}^2 \leq \sum_{j\leq k}\|\partial_\alphap^j u\|_{L^{2}_\alphap}^2\leq C \sum_{j\leq k}\|\partial_\alphap^j v\|_{L^{2}_\alphap}^2, \label{u v 1}\\
&\left|\sum_{j_\leq k}\|\pap^j(\partial_t+b\partial_\alphap)v\|_{\Lap^2}-\sum_{j\leq k}\|\pa^jw\|_{\Lap^2}\right|\leq C\sum_{j\leq k}\|\pap^ju\|_{\Lap^2}^2.\label{u v 2}
\end{align}
In particular
First we prove (<ref>). We begin by rewriting (<ref>) as
2u= v/ +ζ/[μ,](u)/ζ_+1/[u,]μ_/ζ_+1/E(u)-2/(u)
Now we estimate the terms above in $H_\alphap^k.$ First note that by Proposition <ref> and Sobolev
∑_j≤k^jv/_^2≲∑_j≤k^jv_^2, ∑_j≤k^j(u)_^2≲∑_j≤k^ju_^2,
so it suffices to bound the contribution of all other terms on the right hand sides by $M\sum_{j\leq k}\|\pap^ju\|_{\Lap^2}.$ The contribution of $E(u\zetabar)$ is already handled in Corollary <ref>. For $\AV(u\zetabar)$ note that since $u$ is anti-holomorphic inside $\Omega$
which is bounded by $M\|u\|_{L^\infty}$ (note that $\AV(u\zetabar)$ is a constant as a function of $\alpha$). The contribution of the other terms is handled by Lemma <ref>.
For (<ref>) we use (<ref>) and a similar argument as for the proof of (<ref>) to bound the contributions of the last integral in (<ref>), $|u|^2,$ $[u,\CH]\frac{(u\zetabar)_\alpha}{\zeta_\alpha},$ and $[\ubar,\CHbar]\frac{(u\zetabar)_\alpha}{\zetabar_\alpha}$ by $\sum_{j\leq k}\|\pa^ju\|_{\La^2}^2.$ The contribution of $[w,\CH]\frac{\mu_\alpha}{\zeta_\alpha}$ is bounded by $M\sum_{j\leq k}\|\pa^jw\|_{\La^2},$ by Lemma <ref>. Finally, applying the identity
to $f=w\zetabar$ and $f=|u|^2$ and using similar arguments as above we can estimate the contribution of $(\CH+\CHbar)(w\zetabar+|u|^2)$ by
M∑_j≤k ^jw_^2+∑_j≤k^j u_^2^2.
Here to estimate $\AV(w\zetabar)$ we have noted that
∫_0^2πwζ_/ζd=∫_0^2πwμ_/ζd-∫_0^2πw_d=∫_0^2πwμ_/ζd-∫_0^2πu_ d,
where for the last equality we have written $z_t=F(t,\zbar)$ for some anti-holomorphic function $F$ to get $z_{tt}=F_t+F_\zbar\zbar_t=F_t+\frac{z_{t\alpha}\zbar_t}{\zbar_\alpha}.$ The desired estimates now follow from the bootstrap assumptions (<ref>) if $M_0$ is sufficiently small. Note that the term $\int_{0}^{2\pi}u_{\alphap}\ubar d\alphap$ is bounded by $\sum_{j\leq k}\|\pap^{j}u\|_{\Lap^{2}}^{2}$ because it does not depend on $\alphap$. Therefore it vanishes when the spatial derivatives hit it.
Our next goal is to estimate $\eta$ and its higher derivatives. To this end we rearrange equation (<ref>) to get
πη=iw-(A-π)ζ_-i g^a∘k^-1.
To use this equation we first need to estimate $A-\pi.$ This is accomplished in the next proposition.
If $M_0$ in (<ref>) is sufficiently small then for any $2\leq k\leq \ell$
∑_j≤k^j(A-π)_^2≤C( ∑_j≤k^j u_^2^2+∑_j≤k^jη_^2^2+∑_j≤k^jw_^2∑_j≤k^jη_^2).
Using Propositions <ref> and <ref> and the fact that $\partial_\alphap \log(\zetabar e^{i\alphap})=\frac{\etabar}{\zetabar}$ we write
Now since $A$ is real
A-π =(I-)(A-π)+1/2(+)(A-π)
Moreover, by Lemma <ref> we can write
Notice that by Corollary <ref> and Proposition <ref>
and by Proposition <ref> if $M_0$ in (<ref>) is sufficiently small
∑_j≤k^j(g^a∘k^-1)_^2≤C ∑_j≤k^jη_^2.
It follows from this, Proposition <ref>, Lemma <ref>, and (<ref>) that
∑_j≤k^j(I-)(A-π)_^2 +(A-π)_^2
≤C(∑_j≤k^j u_^2^2+∑_j≤k^jη_^2^2+∑_j≤k^jw_^2∑_j≤k^jη_^2).
Similarly the bootstrap assumptions give
∑_j≤k(^j(ζ[μ,]A_/ζ_)_^2+^j E(A-π)_^2)≤CM ∑_j≤k^j(A-π)_^2.
Combining these estimates with (<ref>) we arrive at the desired conclusion if $M$ is sufficiently small.
We now go back to the analysis of equation (<ref>). As observed in the proof of Proposition <ref> we can write $g^a\circ k^{-1}=\frac{\pi}{2}(\zeta\chi+E(\zeta)).$ This shows that equation (<ref>) by itself is not enough to obtain estimates on $\eta$ and its higher derivatives in terms of $(\pt+b\pap)\chi$ and $(\pt+b\pap)v$ and their higher derivatives. To get such estimates we will also need to use the original equation (<ref>), which in turn requires estimates on the right hand side of (<ref>). These estimates are also of independent interest in proving energy estimates, so before stating the final estimates for $\eta$ we state the following estimates on the right hand sides of the equations (<ref>) and (<ref>).
Let $N_1$ and $N_2$ be as in Corollary <ref>. Then if $M_0$ in (<ref>) is sufficiently small, for any $3\leq k \leq \ell$
∑_j≤k( ^j N_1_^2+^j N_2_^2 ) ≲∑_j≤k ( ^jη_^2 + ^j u_^2 + ^j w_^2)^3
We begin with $N_1.$ Using Lemmas <ref> and <ref> and Corollary <ref> we can bound the contributions of the first two terms on the right hand side of (<ref>) by the right hand side of (<ref>). Similarly, in view of equation (<ref>), the contributions of the last two terms on the right hand side of (<ref>) can be bounded by the right hand side of (<ref>) by using Lemma <ref> and Propositions <ref> and <ref>. This completes the estimates for $N_1.$ The contribution of $N_2$ can be treated in a similar way. Indeed except for the first term on the right hand side of (<ref>) all other terms can be estimated by similar arguments as above using Propositions <ref> and <ref> and Lemmas <ref>, <ref>, and <ref>. Here we will also use the observations that
and $\pt\ep=z_t\zbar+z\zbar_t.$ We omit the details. Finally the term $a_t$ is treated independently in the proof of Lemma <ref> below.[We note that the treatment in Lemma <ref> does not rely on the validity of Proposition <ref>. In fact we only use the estimates for $N_1$ in the proof of Proposition <ref> below and the proof of the estimates for $a_t$ in Lemma <ref> are even independent of this proposition.]
Using equation (<ref>), we can now combine Propositions <ref>, <ref>, and <ref> to prove the following proposition.
If $M_0$ in (<ref>) is sufficiently small then for any $3\leq k\leq\ell,$
∑_j_≤k (^jw_^2+^ju_^2+^jη_^2)≤C∑_j≤k(^j(∂_t+b)v_^2+^j(∂_t+b)χ_^2).
In view of Proposition <ref> we only need to prove this estimate for $\eta.$ From Proposition <ref> we know that $(A-\pi)$ is quadratic. In equation (<ref>), using Lemma <ref> we can write the term $g^{a}\circ k^{-1}$ as
\begin{align*}
g^{a}\circ k^{-1}=\frac{\pi}{2}(I+\overline{\calH})\zeta=\frac{\pi}{2}(\calH+\overline{\calH})\zeta=\frac{\pi}{2}\zeta\chi+E(\zeta)
\end{align*}
and equation (<ref>) can be written as
\begin{align}\label{eta eq rough}
\pi\eta=iw-\frac{\pi i}{2}\zeta\chi-\frac{\pi i}{2}E(\zeta)-(A-\pi)\zeta_{\alphap}
\end{align}
The arguments for estimating $\eta$ itself and its derivatives are different. For $\eta$ we use equation (<ref>) and the definition of $v:=(\pt+b\pap) \chi$ to get
For higher derivatives of $\eta$ we instead use the following system which is obtained by differentiating (<ref>) and the second equation in (<ref>)
((A-π)ζ_), ℓ≥1
We start with the estimates for $\eta$ itself. In view of equation (<ref>), we need to obtain an estimate for $\zeta\chi$. On the other hand, by Propositions <ref>, <ref>, <ref> and Corollary <ref>, the second equation in (<ref>) gives us an estimate for $\pap(\zeta\chi)$:
\begin{align}\label{esti: deri zetachi final}
\|\pap(\zeta\chi)\|_{\Lap^{2}}\lesssim\|(\pt+b\pap)v\|_{\Lap^{2}}+M\|\eta\|_{\Lap^{2}}+M^{2}\sum_{j\leq 3}\left(\|\pap^ju\|_{\Lap^2}+\|\pap^jw\|_{\Lap^2}+\|\pap^j\eta\|_{\Lap^2}\right).
\end{align}
In order to obtain the $L^{2}$-estimate for $\zeta\chi$, we still need to
know the value of $\zeta\chi$ at least at one point. Note that by Proposition <ref>
\begin{align*}
\int_{0}^{2\pi}\zeta\chi\cdot \frac{\zeta_{\alphap}}{\zeta}d\alphap=0.
\end{align*}
\begin{align*}
\int_{0}^{2\pi}\zeta\chi d\alphap=i\int_{0}^{2\pi}\chi\eta d\alphap,
\end{align*}
from which we have
\begin{align*}
\left|\int_{0}^{2\pi}\Re(\zeta\chi)d\alphap\right|,\quad \left|\int_{0}^{2\pi}\Im(\zeta\chi)d\alphap\right|\lesssim M\|\eta\|_{\Lap^{2}}.
\end{align*}
These together with (<ref>) imply that
\begin{align}\label{esti: zetachi}
\|\zeta\chi\|_{\Lap^{2}}\lesssim\|(\pt+b\pap)v\|_{\Lap^{2}}+M\|\eta\|_{\Lap^{2}}+M^{2}\sum_{j\leq 3}\left(\|\pa^ju\|_{\Lap^2}+\|\pap^jw\|_{\La^2}+\|\pap^j\eta\|_{\Lap^2}\right).
\end{align}
Substituting this into (<ref>), using Corollary <ref> and Proposition <ref>, and taking $M>0$ sufficiently small we get
η_^2≲ (+b)v_^2+(+b)χ_^2+M^2∑_j≤3^jη_^2
Finally applying Propositions <ref> and <ref> and Corollary <ref> to equation (<ref>), for and $3\leq k\leq \ell,$ we get the bound
Combining this with (<ref>) and choosing $M$ sufficiently small gives
which completes the proof of the proposition.
The next step in our analysis is to obtain estimates for quantities of the form $\|\pap^j(\partial_t+b\pap)f\|_{\Lap^2}$ in terms of $\|(\partial_t+b\pap)\pa^jf\|_{\Lap^2},$ which in turn will be bounded by the higher order energies to be defined in the next section. For this we first obtain estimates on $b$ and its derivatives.
If $M_0$ in (<ref>) is sufficiently small then for $2\leq k \leq \ell$
The proof is similar to that of Proposition <ref>. Recall from Propositions <ref> and <ref> that
The Proposition now follows from similar arguments as those in the proof of Proposition <ref>.
An important corollary of Propositions <ref> and <ref> is the following result.
If $M_0$ in (<ref>) is sufficiently small then for $2\leq k \leq \ell,$
∑_j_≤k (^jw_^2+^ju_^2+^jη_^2)≤C∑_j≤k((∂_t+b)^jv_^2+(∂_t+b)^jχ_^2).
We first note that for any function $f$
^j(∂_t+b)f=(∂_t+b)^jf+∑_1≤i≤j ji ^ib ^j+1-if,
and therefore by Sobolev
^j(∂_t+b)f_^2≤(∂_t+b)^jf_^2+c∑_i≤max{2,j} ^ib_^2∑_i≤j^if_^2.
Summing this estimate over $j\leq k$ for $f=\chi$ and $f=v$ and using Propositions <ref>, <ref> and <ref>, Corollary <ref>, and the bootstrap assumption (<ref>) we get the desired result.
§ ENERGY ESTIMATES
In this section we define the energy and prove energy estimates for equations (<ref>) and (<ref>). The main energy estimates are stated in Proposition <ref> below. We consider an equation of the form
In most applications $\Theta$ will be the boundary value of a holomorphic function outside $\Omega$ decaying to zero as $|z|\to\infty,$ that is, $\Theta=(I-\CH)f$ for some $f.$ More precisely, the relevant choices of $\Theta$ are $\chi=(I-\CH)\mu$ and $v=(\pt+b\pap)\chi$. Since $v$ cannot be written as $(I-\CH)f$ we define the new unknown
\begin{align}\label{v tilde}
\tilde{v}:=(I-\CH)v.
\end{align}
Associated to (<ref>) we define the following basic energy
and for the choices of $\Theta$ above we let
We will show below that if $\Theta=(I-\CH)f$ for some $f$ then $i\int_0^{2\pi}\Theta_\alpha\Thetabar d\alphap$ is non-negative. It is not, however, in general true that $\F_0^\Theta$ is non-negative even if $\Theta=(I-\CH)f,$ but this is the case if $\partial\Omega$ is a an exact circle. This can be seen by noting that in this case the Fourier expansion of $\Theta$ contains only negative frequencies if $\Theta=(I-\CH)f$, and then carrying out the integration on the frequency side after an application of Plancherel. Therefore, we expect that for small data where $\partial\Omega$ is nearly a circle, $\F_0^\Theta$ can be written as a positive term plus `higher order terms.' This can be achieved for instance by writing $\CH$ as the Hilbert transform on the circle plus an error. While this intuition is helpful, we will not use this argument in our applications, but instead explicitly decompose $\F_0^\Theta$ as the sum of a positive term and a `higher order' difference in terms of known quantities for choices of $\Theta$ that interest us. We will postpone this computation to after defining the higher order energies and now only prove the following general estimate.
The integral $i\int_0^{2\pi}\Theta_\alpha\Thetabar d\alphap$ is real and if $\Theta=(I-\CH)f$ for some $2\pi-$periodic function $f,$ then
Note that this lemma does not apply to the choice $\Theta=v,$ which is why we have replaced $v$ by $\tv$ in the definition of $E_0(v).$
Integration by parts shows that the integral $i\int_0^{2\pi}\Theta_\alphap\Thetabar d\alphap$ is equal to its conjugate and is therefore real, and hence
i∫_0^2πΘ_d= {i∫_0^2πΘ_/|ζ_||ζ_|d}.
Now note that if $\Theta=(I-\CH)f$ then by Proposition <ref> we can write $\Theta$ as the boundary value of a function $F$ which is holomorphic in $\Omega^c$ and decays as $|\zeta|^{-1}$ when $|\zeta|\to\infty.$ A simple computation using the holomorphicity of $F$ in $\Omega^c$ and the Cauchy-Riemann equations gives
{iΘ_/|ζ_|}=-⟨F, ∂F/∂n ⟩
where $\bfn:=-\frac{iz_\alpha}{|z_\alpha|}$ is the exterior normal of $\Omega$ and $\langle F , G \rangle:= f_1 g_1 +f_2 g_2$ for complex numbers $F=f_1+if_2$ and $G=g_1+ig_2.$ From Green's formula and with $ds$ denoting the arc-length measure we get
i∫_0^2πΘ_d=-∫_∂Ω^c⟨F, ∂F/∂n ⟩ds=∬_Ω^c|∇F|^2 dxdy ≥0,
where we have used the decay properties of $F$ stated above to justify the use of Green's formula.
With this basic positivity estimate in place we turn to the following energy identity for $E_0^\Theta.$
Suppose $\Theta$ satisfies equation (<ref>). Then
Note that if $\Theta=\chi$ or $v,$ then by the results of Section <ref> the first integral on the right hand side above is of `order four'. However, in the definition of $E_0(v)$ we have used $\Theta=\tv,$ so to have this smallness we still need to show that $\tv$ satisfies a `cubic' equation. This will be accomplished in Proposition <ref> below.
Precomposing with $k$ we can rewrite (<ref>) as
(^2+ia-π)θ=g, θ:=Θ∘k, g=G∘k.
It follows that
d/dt E_0^Θ= ∫_0^2π2/a{^2θ} dα-∫_0^2πa_t/a^2|θ|^2dα
= 2∫_0^2π1/a{g}dα-∫_0^2πa_t/a^2(|θ|^2-π|θ|^2)dα.
Composing back with $k^{-1}$ we get the desired identity.
We now turn to the higher energy estimates for (<ref>). For simplicity of notation we define
and note that
Applying this identity to (<ref>) we get
The $j$th order energy is now defined as
and in analogy with the undifferentiated case we let
The following lemma follows from a similar argument to the proof of Lemma <ref>.
$\pt E_j^\theta=R_j(t)$ where
Supposing for the moment that we know how to deal with the non-positive part $\F_j^\Theta$ of the energy, we can use Corollary <ref> and Proposition <ref> to estimate the quantities appearing in the bootstrap assumption (<ref>) in terms of the positive parts of the energy $\E_j(\chi)$ and $\E_j(v).$ The only difficulty with this is that in the definition of $\E_j(v)$ we have replaced $v$ by $\tv,$ so in the next proposition we show that the conclusions of Corollary <ref> hold with $v$ replaced by $\tv.$
If $M_0$ in (<ref>) is sufficiently small then for $2\leq k \leq \ell,$
∑_j_≤k (^jw_^2^2+^ju_^2^2+^jη_^2^2)≤C∑_j≤k(_j^χ+_j^).
In view of Corollary <ref> and Proposition <ref> we only need to show that under the assumptions of the proposition
To see this we first write
(+b)^j[u,]χ_/ζ_= ^j(+b)[u,]χ_/ζ_-∑_i=1^j ji ^ib ^j-i[u,]χ_/ζ_.
By Corollary <ref>, Proposition <ref>, and Lemma <ref> the contribution of the last term above can be bounded as
^ib ^j-i[u,]χ_/ζ__^2^2≲M ∑_i≤j((+b)^iv_^2+(+b)^iχ_^2).
To estimate the first term on the right hand side of (<ref>) we first note that
[z_t,H]δ_α/z_α=[z_tt,H]δ_α/z_α+[z_t,H]δ_t/z_α+1/πi∫_0^2π(z_t(α)-z_t(β)/z(β)-z(α))^2δ_β(β) dβ,
(+b)[u,]χ_/ζ_=[w,]χ_/ζ_+[u,]v/ζ_+1/πi∫_0^2π(u()-u()/ζ()-ζ())^2χ_() d.
It follows from this, Corollary <ref>, Proposition <ref>, Proposition <ref>, and Lemma <ref> that
^j(+b)[u,]χ_/ζ__^2^2≲M ∑_i≤j((+b)^iv_^2+(+b)^iχ_^2).
Combining (<ref>)–(<ref>) we get (<ref>).
We now turn to the issue of non-positivity of $\F_j^\Theta.$ Note that even if $\Theta$ can be written as $(I-\CH)f$ this will not in general imply that $\pap^j\Theta$ is the boundary value of a function holomorphic outside of $\Omega,$ so even the first integral in the definition of $\F_j^\Theta$ above may not be non-negative for $j\geq1.$ Nevertheless, as for $\F_0^\Theta,$ we are able to show that the negative part of $\F_j^\Theta$ is of higher order for the choices of $\Theta$ we need in the energy estimates. The following simple observation is the main step in this direction.
Suppose $\Theta:=(I-\CH)f$ for some $2\pi-$periodic function $f.$ Then with $g=\zeta\Theta$
i∫_0^2πΘ_d-∫_0^2π|Θ|^2d =∫_0^2πg_d-∫_0^2π(iζ_α+1)|Θ|^2d-∫_0^2πiμΘ_d
The first equality follows from
To get the inequality it suffices to show that $i\int_0^{2\pi}g_\alphap\gbar d\alphap\geq0.$ For this note that
and that $[\zeta,\CH]f$ is independent of $\alphap.$ It follows that
$$i\int_0^{2\pi}g_\alphap\gbar d\alphap=i\int_0^{2\pi}\pap[(I-\CH)(\zeta f)]\overline{[(I-\CH)(\zeta f)]}d\alphap$$
which is non-negative by Lemma <ref>.
Lemma <ref> shows that the difference between the energy and a positive term is of higher order. Note however, that the lower order term involves an extra derivative of $\Theta.$ This causes a problem only when we consider $\pap^\ell v,$ where $\ell$ is the maximum number of derivatives we commute. But in this case we can write
and the main term here is already bounded by the energy of $\chi,$ that is,
Since $\|(\pt+b\pap)\pap^\ell\chi\|_{\Lap^2}^2$ is precisely the negative term in the energy of $\pap^\ell v$ this idea can be used to resolve the issue in the case where we commute the maximum number of derivatives. We will now make this argument more precise, starting with a few important identities stated only for the choices of $\Theta$ which will be used in the energy estimates, namely $\Theta=\chi$ and $\tv.$
If $\chi$ and $v$ are as in (<ref>) then
The first identity follows from commuting $\pap^j$ with $\CH$ in the definition $\chi=(I-\CH)\mu$ of $\chi$ and noting that
The proof of the second identity is similar where we use the definition $\tv=(I-H)v.$
We can now prove the following positivity estimate.
* If $M_0$ in (<ref>) is sufficiently small then for $\ell \geq 2$
* If $M_0$ in (<ref>) is sufficiently small then for $\ell \geq 2$
_ℓ^≥-C∑_i=0^ℓ(_i^χ+_i^)^2 - C∑_i=0^ℓ(_i^χ+_i^)^3/2-C_ℓ^χ.
* We assume $M_0$ is small enough that Corollary <ref> holds. We start with the estimate for $\chi.$ By Lemma <ref>
It follows that
_i^χ= i∫_0^2πf_id-∫_0^2π|f_i|^2d
To estimate the first line above we apply Lemma <ref> with $\Theta=(I-\CH)\pap^j\mu=f_i$ to get (for $i\leq \ell$)
i∫_0^2πf_id-∫_0^2π|f_i|^2d≥ -(ζ_^∞η_^∞+μ_^∞)f_i_^2^2-μ_^∞f_i_^2f_i_^2
≥ -C∑_j=0^i(_j^χ+_j^)^3/2,
by Corollary <ref> and Proposition <ref>. Here to estimate $\|\pap f_\ell\|_{\Lap^2}$ we have noted that
To estimate the second line in (<ref>) it suffices to show that for $i\leq \ell$
But (<ref>) is a direct consequence of Corollary <ref> and Lemma <ref>. Combining (<ref>), (<ref>), and (<ref>) we get the estimate for $\chi.$
The estimate for $\tv$ is similar. Using Lemma <ref> we write
^i =ϕ_i+ψ_i
ϕ_i:=(I-)^iv, ψ_i=-∑_m=1^i^i-m[η,]^m v/ζ_α.
The argument is now the same as for $\chi$ where we replace $g_i$ by $\psi_i$ and $f_i$ by $\phi_i$ everywhere. The only difference is that (<ref>) is now replaced by
which is responsible for the loss of one derivative.
* Note that with $c_i:={{\ell}\choose{i}}$
^ℓv =(+b)^ℓχ+∑_i=1^ℓc_i ^ib ^ℓ+1-iχ.
It follows from this, Corollary <ref>, Proposition <ref>, and Proposition <ref> that
^ℓv_^2^2≤C∑_i=0^ℓ(_i^χ+_i^)^2+C _ℓ^χ,
and from the second identity in Lemma <ref> that
\begin{align*}
\|\pap^\ell \tv\|_{\Lap^2}^2\leq C\sum_{i=0}^{\ell}(\E_i^\chi+\E_i^{\tv})^2+C \E_\ell^\chi.
\end{align*}
From the definition of $\F_\ell^{\tv}$ and in view of Proposition <ref> if $M_{0}$ is sufficiently small it follows that
_ℓ^≥i∫_0^2π^j+1^jd-C∑_i=0^ℓ(_i^χ+_i^)^2- C _ℓ^χ,
so it suffice to show
For this we use Lemma <ref> to write
f=(I-)^ℓv, g=-∑_i=1^ℓ^ℓ-i[η,]^i v/ζ_.
arguing as in (<ref>) and (<ref>) we just need to show that
But this is again a consequence of Corollary <ref>, Proposition <ref>, and Lemma <ref>. This now proves (<ref>) which concludes the proof of the Lemma.
Combining Lemmas <ref> and <ref> we see that if $M_0$ in (<ref>) is sufficiently small we can find constants $c_{1}$, $C_1$ and $C_2$ such that with $R_k$ as in Lemma <ref>
∑_k≤ℓ_k^χ(t)+∑_k≤ℓ-1_k^(t)≤∑_k≤ℓ(E_k^χ(0)+E_k^(0))+C_1∑_k≤ℓ(_k^χ(t)+_k^(t))^3/2+∑_k≤ℓ∫_0^t |R_k(t)|dt
_ℓ^(t)-c_1_ℓ^χ(t)≤ ∑_k≤ℓ(E_k^χ(0)+E_k^(0))+C_2∑_k≤ℓ(_k^χ(t)+_k^(t))^3/2+C_2∑_k≤ℓ(_k^χ(t)+_k^(t))^2
+∑_k≤ℓ∫_0^t |R_k(t)|dt.
Adding an appropriate multiple of the second estimate to the first we get the following energy estimate.
If $M_0$ in (<ref>) is sufficiently small then with $N_k$ as in Lemma <ref>
∑_k≤ℓ(_k^χ(t)+_k^(t))≤ C∑_k≤ℓ(E_k^χ(0)+E_k^(0))+ C∑_k≤ℓ(_k^χ(t)+_k^(t))^3/2+ C∑_k≤ℓ(_k^χ(t)+_k^(t))^2
We now turn to the estimates for $R_k.$ For notational convenience we define
Our first step will be to compute the equation for $\tv.$
$\tv=(I-H)v$ satisfies
(^2+ia-π)(∘k)= (I-H)(^2+ia-π)δ_t+2[z_t,H](^2+ia-π)δ/z_α
From Lemma <ref> we have
(^2+ia-π)(∘k)= (^2+ia-π)(I-H)δ_t=(I-H)(^2+ia-π)δ_t-[^2+ia,H]δ_t
= (I-H)(^2+ia-π)δ_t-π/2[(I-)z,H]δ_tα/z_α+2[z_t,H]δ_tt/z_α
= -π/2[(I-)z,H]δ_tα/z_α-2[z_t,H](iaδ-πδ)/z_α
The last three terms above already have the right form, so we only need to consider
Note that if $g$ is the boundary value of a decaying holomorphic function $F$ outside of $\Omega,$ i.e., $g=(I-H)f_1$ then $\frac{g_\alpha}{z_\alpha}$ is the boundary value of $F_z$ so $\frac{g_\alpha}{z_\alpha}=(I-H)f_2$ for some $f_2.$ We will use this observation repeatedly in the rest of this proof. Applying this observation to $III$ we see that since $\delta=(I-H)\ep$
For $II$ we use (<ref>) to write
The first two terms have the right form and we can rewrite the last term as
2π[z_t,H]/z_α(zδ_α/z_α)= π[(I+H)z_t,H]δ_α/z_α+π[z_t,H]z(/z_α)^2δ
= π[z[,H]z_tα/z_α,H]δ_α/z_α+π[E(z_t),H]δ_α/z_α
The first term in the last line above can be written as
\begin{align*}
&\pi z[(I+H)z_{t},H]\left(\frac{\pa}{z_{\alpha}}\right)^{2}\delta=\pi z[(H+\Hbar)z_{t},H]\left(\frac{\pa}{z_{\alpha}}\right)^{2}\delta\\
=&\pi z\left[z[\ep,H]\frac{z_{t\alpha}}{z_\alpha},H\right]\left(\frac{\partial_{\alpha}}{z_{\alpha}}\right)^{2}\delta+\pi[E(z_t),H]\left(\frac{\pa}{z_{\alpha}}\right)^{2}\delta.
\end{align*}
For the second term $2\pi [[z_{t},H],z]\left(\frac{\pa}{z_{\alpha}}\right)^{2}\delta$, we use Jacobi identity to write this as
\begin{align*}
\end{align*}
By Lemma <ref>, we have
\begin{align*}
\end{align*}
Therefore the contribution we need to consider is
\begin{align*}
\int_{0}^{2\pi}z_{t}(\beta)\partial_{\beta}\left((I+\Hbar)\frac{\ep_{\beta}}{z_{\beta}}\right)(\beta)d\beta=\int_{0}^{2\pi}z_{t}(\beta)G_{\zbar}(\zbar(\beta))\zbar_{\beta}(\beta)d\beta=0.
\end{align*}
Here $G_{\zbar}(\zbar(\beta))=\frac{\partial_{\beta}}{\zbar_{\beta}}\left((I+\Hbar)\frac{\ep_{\beta}}{z_{\beta}}\right)$ is the boundary value of an anti-holomorphic function $G_{\zbar}(\zbar)$ in $\Omega$. We also used the fact that $z_{t}(\beta)$ is the boundary value of an anti-holomorphic function in $\Omega$. Finally for $I$ we compute
I= -π/2[(I-)z,H]δ_tα/z_α
= π/2[(I+)z,H]δ_tα/z_α
= π/2[zδ,H]δ_tα/z_α+π/2[E(z),H]δ_tα/z_α
= π/2[zδ,H]/z_α(I-H)(z_t)-π/2[zδ,H]/z_α[z_t,H]_α/z_α+π/2[E(z),H]δ_tα/z_α.
Again the last two terms have the right form and for the first we use Lemma <ref> with $f=z,$ $g=\delta$ and $h=\frac{\pa}{z_\alpha}(I-H)(z_t\zbar)$ and the fact that for and $f_1$ and $f_2$
to write
Here for the last step we have used the fact that since $\delta$ and $(I-H)(z_t\zbar)$ are boundary values of holomorphic functions $F_1$ and $F_2,$ respectively, in $\Omega^c$ going to zero as $|z|\to\infty,$
The following estimate is used for estimating the second integral in the definition of $N_k.$
If $M_0$ in (<ref>) is sufficiently small
We use Lemmas <ref> and <ref> and the Sobolev embedding $H_\alphap^1\hookrightarrow \Lap^\infty.$ Recalling that $A=(ak_\alpha)\circ k^{-1},$ from Lemma <ref> and precomposition with $k^{-1}$ we get
(I+^*)(a_t/a∘k^-1 A|ζ_|)={ -iζ_/|ζ_|{2[u,]_/ζ_+[2w-g^a∘k^-1,]_/ζ_
Recalling that $g^1\circ k^{-1}=\frac{\pi}{2}\zeta\chi+E(\zeta),$ it follows from this, Lemmas <ref> and <ref>, and Corollary <ref> that
∑_i=0^1^i((I+^*)(a_t/a∘k^-1 A|ζ_|))_^2≤C.
On the other hand,
1/Aa_t/a∘k^-1=1/A^2|ζ_|(I+^*)(a_t/a∘k^-1 A|ζ_|)-1/A^2|ζ_|^*(a_t/a∘k^-1 A|ζ_|)=:I-1/A^2|ζ_|II.
By (<ref>) and Propositions <ref> and <ref>
and therefore in view of Propositions <ref> and <ref> to complete the proof of the lemma it suffice to show that
For this we use Lemma <ref>. Note that since $K^*f=-\Re\{\frac{z_\alpha}{|z_\alpha|}H\frac{|z_\alpha|f}{z_\alpha}\}$ we may replace $z$ by $\zeta$ and $z_\alpha$ by $\zeta_\alphap$ everywhere in formula derived in Lemma <ref> to get a representation for $\CK^*.$ Using this observation and Lemma <ref>, <ref> we get with $f=\frac{a_t}{a}\circ k^{-1}A|\zeta_\alphap|$
II= 1/π|ζ_|∫_0^2πf()|ζ_()|d+(f|ζ_|)/|ζ_|
The contribution of the second line above can be bounded by the right hand side of (<ref>) using Lemma <ref>, Corollary <ref>, and Proposition <ref>. To estimate the contribution of the first line of (<ref>) we go back to equation (<ref>) which we rewrite as
f|ζ_|=a_t/a∘k^-1 A|ζ_|^2=i_(+b)w-πu__-(A-π)u__-πi/2[,1/ζ_+1/_α]ζ_.
Moreover, we can write
(g)=∫_0^2πηg/ζ d+i∫_0^2πg d
so to prove (<ref>) for the first line of (<ref>), it suffices to bound $\int_0^{2\pi} gd\alphap$ by the right hand side of (<ref>) with $g$ replaced by each of the terms on the right hand side of (<ref>). For the last two terms of (<ref>) the contributions are of the right form in view of Lemma <ref> and Proposition <ref>. For the second term of (<ref>) it suffices to note that since $u$ is anti-holomorphic inside $\Omega$
∫_0^2πu__d=∫_0^2πu_ d+i∫_0^2πu_d=∫_0^2πu_ d
which can be bounded by the right hand side of (<ref>). Finally for the first term of (<ref>) we write $z_t=F(t,\zbar)$ for an anti-holomorphic function to get
z_tt=F_t+F__t=F_t+z_tα_t/_α, z_ttt=F_tt+z_ttα/_α_t+(_ttα_t+z_tα_tt)_α-_tαz_tα_t/_α^2.
Since $F_{tt}$ is anti-holomorphic, it follows that
∫_0^2π(+b)w_d=∫_0^2π w_ d+∫_0^2π(_+u_)
_-_u_u/_ d
which can be bounded by the right hand side of (<ref>). This completes the proof of (<ref>) and hence of the lemma.
If $M_0$ in (<ref>) is sufficiently small then for all $j\leq \ell$ and with $\Theta=\chi$ or $v$
This is a direct corollary of the definition of $\E,$ Lemma <ref>, Proposition <ref>, and Corollaries <ref> and <ref>.
The last step before stating the main result of this section is to obtain an expression for the time derivative of $b$ and then estimates for it.
Suppose that $k$ is given as in Remark <ref> and that it is increasing. Then $k_{tt}=(\pt+b\pa)b\circ k$ satisfies
\begin{align*}
(I-H)k_{tt}=&-i(I-H)\frac{\zbar_{tt}\ep+\zbar_{t}\ep_{t}}{\zbar}+i(I-H)\frac{\zbar_{t}^{2}\ep}{\zbar^{2}}-i[z_{t},H]\frac{(\log(\zbar e^{ik}))_{t\alpha}+ik_{t\alpha}}{z_{\alpha}}+i[z_{t},H]\frac{1}{z_{\alpha}}\pa\left(\frac{\zbar_{t}\ep}{\zbar}\right)\\
&-i[z_{tt},H]\frac{(\log(\zbar e^{ik}))_{\alpha}}{z_{\alpha}}-\frac{1}{\pi}\int_{0}^{2\pi}\left(\frac{z_{t}(\beta)-z_{t}(\alpha)}{z(\beta)-z(\alpha)}\right)^{2}(\log(\zbar e^{ik}))_{\beta}d\beta
\end{align*}
\begin{align*}
\Re\AV(\pt k_t)=\frac{\Im}{2\pi}\int_0^{2\pi}\left(\frac{z_{t\beta}z-z_tz_\beta}{z^2}\right)k_td\beta+\frac{\Re}{2\pi}\pt\int_0^{2\pi}\frac{\zbar_t\ep}{|z|^2}z_\beta d\beta+\frac{\Re}{2\pi}\pt \int_0^{2\pi}(\log (\zbar e^{ik}))_\beta \frac{z_t}{z}d\beta.
\end{align*}
Differentiating the first formula in Proposition <ref> with respect to time, we obtain
\begin{align*}
&=-i\pt(I-H)\frac{\zbar_{t}\ep}{\zbar}-i\pt[z_{t},H]\frac{\left(\log(\zbar e^{ik})\right)_{\alpha}}{z_{\alpha}}+[z_{t},H]\frac{k_{t\alpha}}{z_{\alpha}}\\
\end{align*}
Direct computations imply that
\begin{align*}
II=&-i[z_{t},H]\frac{\left(\log(\zbar e^{ik})\right)_{t\alpha}}{z_{\alpha}}-i[z_{tt},H]\frac{\left(\log(\zbar e^{ik})\right)_{\alpha}}{z_{\alpha}}-\frac{1}{\pi}\int_{0}^{2\pi}\left(\frac{z_{t}(\beta)-z_{t}(\alpha)}{z(\beta)-z(\alpha)}\right)^{2}\left(\log(\zbar e^{ik})\right)_{\beta}d\beta.
\end{align*}
Putting all these together, the first formula in the proposition follows. The second formula follows from differentiating the last formula in Proposition <ref> with respect to time.
We are finally ready to prove the main result of this section.
If $M_0$ in (<ref>) is sufficiently small then with $R_k$ as in Lemma <ref>
∑_k≤ℓ(_k^χ(t)+_k^(t))≲ ∑_k≤ℓ(_k^χ(0)+_k^(0))+ ∑_k≤ℓ(_k^χ(t)+_k^(t))^3/2+ ∑_k≤ℓ(_k^χ(t)+_k^(t))^2
+∑_k≤ℓ∫_0^t (_k^χ(s)+_k^(s))^2 ds.
By Corollary <ref> we only need to estimate the nonlinear term $R_k.$ Here $R_k$ is defined in Lemma <ref> and $G_j$ is given in (<ref>) as
It follows from Corollary <ref> that we only need to consider the first integral in the expression for $R_k$ in Lemma <ref>. In particular we need to show that
We begin with the contribution of $\pap^jG.$ When $\Theta=\chi$ this is already dealt with in Propositions <ref> and <ref> and Corollaries <ref> and <ref>. When $\Theta=\tv$ we use the equation derived for $\tv$ in Proposition <ref>. But then in view of Proposition <ref>, the contribution of $\pap^jG$ when $\Theta=\tv$ is also handled by simlar arguments as before using Lemmas <ref>, <ref>, <ref>, Propositions<ref>, <ref>, <ref>, <ref>, and Corollaries <ref>, <ref>. We omit the details. To estimate the contribution of the second term on the right hand side of (<ref>) we note that
and use Proposition <ref> to express $(\pt+b\pap)b$ in terms of quantities we can already control. Here we also use the observation that
and that $\pt \eta=u_{\alphap}-iu.$ The proof of the proposition can now be completed by appealing to Propositions <ref>, <ref>, <ref> and Corollary <ref>.
§ LONG TIME WELL-POSEDNESS
In this final section we prove long-time existence for solutions of the system
\begin{equation}\label{z final eq}
\begin{cases}
&z_{tt}+iaz_\alpha=-\frac{\pi}{2}(I-\Hbar)z,\quad \zbar_t=H\zbar_t\\
&z(0,\alpha)=z_0(\alpha), \quad z_t(0,\alpha)=z_1(\alpha)
\end{cases},
\end{equation}
with small initial data. More precisely we will complete the proof of Theorem <ref>. This section is divided into two parts. To use the energy estimates from the previous section we need to transfer the smallness of the data for equation (<ref>) to the initial smallness of the quantities appearing in the bootstrap assumption (<ref>) and the initial energy defined in the previous section. This will be accomplished in Subsection <ref>. Then in Subsection <ref> we will establish Theorem <ref>, by showing long-time existence of solutions to (<ref>) assuming that initially the bootstrap assumptions (<ref>) hold and that the energy defined in the previous section is sufficiently small.
§.§ A discussion for initial data
We consider initial data $z_0(\alpha)=e^{i\alpha}+\epsilon f(\alpha)$ and $z_1(\alpha)=\epsilon g(\alpha)$ for the system (<ref>) such that $z_0$ is a simple closed curve containing the origin in the interior, parametrized counterclockwisely, and such that $(f,g)\in H_\alpha^s\times H^{s}_\alpha,~s\geq15.$ Furthermore, we assume
for some $\lambda >0.$ We let $H_0$ be the Hilbert transform associated to the initial domain $\Omega(0)$ bounded by $z_0$ and $k_0(\alpha)=k(0,\alpha)$ be defined according to Remark <ref>. Using equation (<ref>) we can now uniquely determine initial values $z_2$ and $a_0$ for $z_{tt}$ and $a$ respectively. Here to get the initial value for $a$ one can for instance use the Riemann mapping formulation of the problem as discussed in Section <ref>. Alternatively one could use the double-layered potential as in <cit.>, see also <cit.>, <cit.> and <cit.>. More precisely, let us write (<ref>) as
\begin{align}\label{a initial temp 0}
\end{align}
Applying $(I-H)$ on both sides we obtain
\begin{align}\label{a initial temp 1}
i(I-H)\left((a-\pi)\zbar_{\alpha}\right)=(I-H)\zbar_{tt}-i\pi (I-H)(\zbar_{\alpha}+i\zbar).
\end{align}
Using the holomorphicity of $\zbar_{t}$ and multiplying both sides of (<ref>) by $\frac{-iz_{\alpha}}{|z_{\alpha}|}$ then taking the real part, we get
\begin{align}\label{a initial temp 2}
(I+K^{*})\left((a-\pi)|z_{\alpha}|\right)=-\Re\left\{\frac{iz_{\alpha}}{|z_{\alpha}|}\left([z_{t},H]\frac{\zbar_{t\alpha}}{z_{\alpha}}-\pi i(I-H)(\zbar_{\alpha}+i\zbar)\right)\right\}.
\end{align}
Note that $z_{\alpha}=ie^{i\alpha}+\epsilon f_{\alpha}(\alpha)$. An argument similar to the proof of Lemma <ref> using (<ref>) and (<ref>) implies that
\begin{align}\label{a L2}
\|a-\pi\|_{\La^{2}}\lesssim\|z_{\alpha}-iz\|_{\La^{2}}+\epsilon\|z_{t}\|_{\La^{2}}
\end{align}
if $\epsilon$ is small enough. The $H^{s}_{\alpha}$ estimate for $(a-\pi)$ can be derived similarly:
\begin{align}\label{a Hs}
\|(a-\pi)|z_{\alpha}|\|_{H^{s}_{\alpha}}\lesssim\|z_{\alpha}-iz\|_{H^{s}_{\alpha}}+\epsilon\|z_{t}\|_{H^{s}_{\alpha}}.
\end{align}
As for $a$ the initial value for $z_{tt}$ can be determined and estimated using the equation (<ref>)
\begin{align}\label{ztt initial}
\zbar_{tt}=i(a-\pi)\zbar_{\alpha}+\pi i(\zbar_{\alpha}+i\zbar)+\frac{\pi}{2}(I+H)\zbar.
\end{align}
Finally we let $k_1(\alpha)=\pt k (\alpha,0),$ where $k$ is extended using Theorem <ref> and Remark <ref>.
Our goal in this subsection is to prove the following proposition.
Let $z_0,~z_1,~f,~g,~z_2,~k_0,~k_1,~a_0,$ and $H_0$ be defined as above and let $M_0>0$ and $\ell\in \N,~\ell\leq s-2$ be fixed constants. Then there exists $\epsilon_0>0,$ depending only on $\|f\|_{H_\alpha^s}$ and $\|g\|_{H_\alpha^{s}}$, such that if $\epsilon <\epsilon_0$ then $k_0$ is a diffeomorphism and
k_0,α-1_^∞≤1/2, k_α-1_H^s-1_α≲z_α-iz_H^s-1_α.
Moreover, if $\epsilon<\epsilon_0$ and we define
ζ_0:=z_0∘k_0^-1, η_0:=ζ_0-iζ_0, u_0:=z_1∘k_0^-1, w_0:=z_2∘k_0^-1,
∑_j≤ℓ(^jη_0_^2+^ju_0_^2+^j w_0_^2)≤M_0/2, |ζ_0|^2≥1/2.
Finally if we extend $z_0,~z_1$ to a local-in-time solution $(z,z_t)$ of (<ref>), with the corresponding Hilbert transform $H,$ and we define $b_0:=k_1\circ k^{-1},~A_0=:(a_0\pa k_0)\circ k_0^{-1},$ and
:=|z|^2-1, δ:=(I-H)_0, χ:=δ∘k^-1, v=δ_t∘k^-1, =(I-H)v,
then if $\epsilon <\epsilon_0$
for a fixed $R_0>0$ independent of $\epsilon.$
Let the $F(\cdot)$ be the holomorphic function with the boundary value $\zbar_{0}e^{ik_{0}}$. Differentiating the equation $(I-H_{0})(\zbar_{0}e^{ik_{0}})=0$ with respect to $\alpha$ we get
\begin{align}\label{kalpha initial temp 1}
(I-H_{0})k_{0,\alpha}=i(I-H_{0})\frac{\zbar_{0,\alpha}+i\zbar_{0}}{\zbar_{0}}-i[z_{0,\alpha}-iz_{0},H_{0}]\frac{\pa(\log F)}{z_{0,\alpha}}.
\end{align}
On the other hand, for the initial data we have
\begin{align}\label{small initial energies}
\|z_{0,\alpha}-iz_{0}\|_{H^{s}_{\alpha}},\quad \|z_{1}\|_{H^{s}_{\alpha}},\quad \|z_{2}\|_{H^{s}_{\alpha}}\leq C_{0}\epsilon.
\end{align}
In fact, the first two estimates are straightforward from the construction of $z_{0}$ and $z_{1}$ and the last one follows from (<ref>) and (<ref>). Equation (<ref>) together with
the relation
\begin{align}
k_{\alpha}-1=\frac{i\zbar_{\alpha}-\zbar}{\zbar}-i\pa\left(\log F\right)
\end{align}
implies that
\begin{align}\label{ave kalpha}
\AV(k_{\alpha}-1)\lesssim\|z_{\alpha}-iz\|_{H^{1}_{\alpha}}.
\end{align}
Here we used the fact that $\|\log F\|_{L^{\infty}_{\alpha}}$ is bounded by an absolute constant, which follows from the definition of $F$. Therefore writing $k_{0,\alpha}$ in terms of $\Re(I-H_{0})k_{0,\alpha}$ gives the desired estimate (<ref>) for $k_{\alpha}$. The other statements of the proposition follow from (<ref>), the relation
\begin{align}\label{chain rule}
\pa\left(f\circ k^{-1}\right)=\frac{f_{\alpha}\circ k^{-1}}{k_{\alpha}\circ k^{-1}},
\end{align}
and arguments similar to those in Section <ref>
§.§ Completion of the proof
In view of Proposition <ref> the proof of long-time well-posedness will be complete once we prove the following theorem.
Let $z_0,~z_1$ be as in Proposition <ref> and denote by $z(t,\alpha)$ the local-in-time solution of (<ref>). Then there exist constant $M_0,~c,$ and $\epsilon_1$ such that if (<ref>) and (<ref>) hold with $\epsilon <\epsilon_1$ then (<ref>) has a unique classical solution in $[0,\frac{c}{\epsilon^2}].$
Let $T^*>0$ be the maximal time of existence guaranteed by Theorem <ref>. We want to show that $T^*\geq \frac{c}{\epsilon^2}$ for some $c$ independent of $\epsilon.$ Let $T\leq T^*$ be defined as
T:=sup{t∈[0,T^*) | k_α(t,α)>1/100, ∀α∈[0,2π]}.
In particular $k$ is a diffeomorphism and continuous in time
for all $t\leq T.$ Moreover, the energy $\E(t)$ defined in (<ref>) is continuous in $[0,T].$ Next, define $T_{M_{0}}\leq T$ as
T_M_0:=sup{ t≤T | ∑_j≤ℓ(^jη_^2+^ju_^2+^jw_^2)≤M_0},
and $T_\epsilon \leq T$ as
T_ϵ:=sup{ t≤T | ^1/2(t)≤2 CR_0ϵ},
where $C$ is the constant in Proposition <ref>.
§.§.§ Step 1
We show that $T_\epsilon\leq T_{M_0},$ provided $\epsilon_1$ is sufficiently small. Indeed, if this is not the case then by Corollary <ref> for all $t\in[0,T_{M_0}]$
and choosing $\epsilon_1\leq \frac{M_0}{2C_1(M_0)R_0}$ we get a contradiction with the maximality of $T_{M_0}.$
§.§.§ Step 2
We show that there exists a constant $c_1=c_1(M_0,R_0)$ such that if $\epsilon_1$ is sufficiently small and $T\leq T_0:=\frac{c_1}{\epsilon^2}$ then $T_\epsilon=T,$ and hence by the previous step $T_{M_0}=T_\epsilon=T.$ To see this, assume the contrary and first let $\epsilon_1$ be so small that the conclusion of the previous step holds. Then we can apply Proposition <ref> with $t=\frac{c_1}{\epsilon^2}\leq T_\epsilon,$ and conclude that if $\epsilon_1$ and $c_1$ are sufficiently small then $\E^{\frac{1}{2}}(t)\leq 2CR_0\epsilon$ proving the claim by contradiction. Here note that since $t\leq T_\epsilon$ the last integral in the statement of Proposition <ref> can be bounded by $16c_1R_0^4C^4\epsilon^2<4C^2R_0^2\epsilon^2$ if $c_1$ is sufficiently small.
§.§.§ Step 3
We show that there exists $c_2=c_2(M_0,R_0)$ such that if $\epsilon_1$ is sufficiently small and $T_1:=\frac{c_2}{\epsilon^2}\leq T_0$ then $k_\alpha\geq\frac{1}{100}$ for all $t\in[0,\min\{T^*,T_1\}).$ Suppose $\epsilon_1$ is small enough that the conclusions of the previous two steps hold. From the definition of $b$
k_α=(b_∘k) k_α,
and hence
But then by Proposition <ref> and Corollary <ref> if $t\leq \min\{T_1,T^*\}$ and $c_2$ is sufficiently small it follows that
§.§.§ Step 4
Finally we show that $T^*\geq \frac{c}{\epsilon^2}$ for a sufficiently small constant $c.$ By Theorem <ref> it suffices to show that if $T^*<\frac{c}{\epsilon^2}$ the $H_\alpha^{10}$ norms of $z_t$ and $z_{tt}$ remain bounded for $t<T^*$ and
Let $\epsilon_1$ be small enough that the conclusions of the previous steps hold, and let $c=c_2$ be as in Step 3. Then if $T^*<\frac{c_2}{\epsilon}, $ it follows from the previous three steps that $T_\epsilon=T_{M_0}=T=T^*.$ By Corollary <ref>, $|\zeta_\alphap(t,\alpha)|\geq \frac{1}{2}$ for all $t\leq T^*$ and all $\alpha\in[0,2\pi],$ and therefore combining with the fact that $k_\alpha\geq \frac{1}{100}$ we get (<ref>). Moreover, from the definition of $T_{M_0}$ the $H_\alphap^{10}$ norms of $u$ and $w$ are bounded up to $T^*,$ so by the chain rule, we only need to prove that the derivatives of $k$ up to order $10$ remain bounded for $t\in[0,T^*).$ But this follows from Proposition <ref> and successive differentiation of the first indentity in (<ref>).
§ RIEMANN MAPPING COORDINATES AND LOCAL WELL-POSEDNESS
In this section we outline the proof Theorem <ref> by investigating the quasilinear structure of the equation
More precisely, we find a quasilinear equation whose well-posedness implies that of equation (<ref>). This is achieved by differentiating (<ref>) with respect to time and exploiting the holomorphicity of various quantities. Once the equivalent quasilinear system is found, the proof of well-posedness is standard and follows for instance from the vanishing-viscosity method in <cit.>. To avoid repetition we only prove the equivalence of (<ref>) with a quasilinear equation and refer the reader to <cit.> for the details of the vanishing-viscosity method.
To get a quasilinear equation we differentiate (<ref>) with respect to time, noting that $(I+H)\zbar$ is the boundary value of a holomorphic function in $\Omega(t)$, to get
_ttt-ia_tα=ia_t_α+π/2[z_t,H]_α/z_α, H_t=_t.
Even though the proof of local existence for (<ref>) can be carried out in these coordinates, the structure of the equation will be more clear in Riemann mapping coordinates which we now introduce. Since we are interested in local existence, we fix a point $\bfx_0\in\Omega(0)$ and define the Riemann mapping for $t$ such that $\bfx_0\in\Omega(t).$ For such $t$ we define the Riemann mapping $\Phi(t.\cdot):\Omega(t)\to \D$ using the normalization $\Phi(t,\bfx_0)=0$ and $\Phi_z(t,\bfx_0)>0$ (in particular $\Phi_z(t,\bfx_0)$ is real). To $\Phi$ we associate the coordinate change $h:\R\to\R$ defined by $e^{ih(t,\alpha)}=\Phi(t,z(t,\alpha)).$ Alternatively let $\chi_1(\cdot)=z(0,\cdot):[0,2\pi]\to\partial\Omega(0)$ be the parametrization of the initial boundary and extend the definition of $\chi_{1}(\cdot)$ to $\bbR$ periodically. Similarly let $X(t,\cdot):\Omega(0)\to\Omega(t)$ denote the flow of the velocity vector field, that is, $\dot X(t,\cdot)=\bfv(t,X(t,\cdot))$. Finally let $\chi_2(\cdot):=-i\log(\cdot):\partial\D\to \bbR$ be the inverse parametrization of the boundary of the unit disc. In this notation $h$ is the composition change of variables $h:=\chi_2\circ\Phi\circ X\circ\chi_1,$
[bend right]rrrrrr[swap]h(t,·)
∂Ω(0)⊆Ω(0) rX(t,·) Ω(t) rΦ(t,·)
and the new unknowns in Riemann mapping coordinates are
Z(t,):=z(t,h^-1(t,)), Z_t(t,):=z_t(t,h^-1(t,)),
Z_tt(t,)=z_tt(t,h^-1()), Z_ttt(t,):=z_ttt(t,h^-1(t,)).
To avoid confusion we separate the subscripts corresponding to partial differentiation by a comma, so for instance $Z_{,\alphap}(t,\alphap)=\pap Z(t,\alphap).$ We denote by $\H$ the Hilbert transform on the circle which can be written as
\Align{\label{H circle}
\H f(\alphap):=\frac{\pv}{\pi i}\int_0^{2\pi}\frac{f(\betap)}{e^{i\betap}-e^{i\alphap}}ie^{i\betap}d\betap=&\frac{\pv}{2\pi i}\int_0^{2\pi}f(\betap)\cot\left(\frac{\betap-\alphap}{2}\right)d\betap+\frac{1}{2\pi}\int_0^{2\pi}f(\betap)d\betap\\
=&\,\tH f(\alphap)+\Av(f),
\Aligns{
&\tH f(\alphap):=\frac{\pv}{2\pi i}\int_0^{2\pi}f(\betap)\cot\left(\frac{\betap-\alphap}{2}\right)d\betap,\quad \Av(f):=\frac{1}{2\pi}\int_0^{2\pi}f(\betap)d\betap.
For notational convenience we also introduce the following new variables and operators in Riemann mapping coordinates:
\Aligns{
\A:=(a h_\alpha)\circ h^{-1},\quad G:=\frac{\pi}{2}((I+H)\zbar)\circ h^{-1}
\Aligns{
\CH f(\alphap):=\frac{\pv}{\pi i}\int_0^{2\pi}\frac{f(\betap)}{Z(t,\betap)-Z(t,\alphap)}Z_{,\betap}(t,\betap)d\betap.
With this notation, precomposing with $h^-1(t,·)$ we can rewrite equations \eqref{z eq RM} and \eqref{zt eq RM} as
\Align{\label{Z eq RM}
\Zbar_{tt}-i\A{\Zbar_{,\alphap}}=-\pi \Zbar+G
\begin{align}\label{Zt eq RM}
\Zbar_{ttt}-i\A{\Zbar_{t,\alphap}}=i\frac{a_t}{a}\circ h^{-1}\A{\Zbar_{,\alphap}}+\frac{\pi}{2}[Z_t,\CH]{\frac{\Zbar_{,\alphap}}{Z_{,\alphap}}}, \quad \H\Zbar_t=\Zbar_t.
\end{align}
Note that if we let
\Aligns{
B:=h_t\circ h^{-1}
we can rewrite \eqref{Zt eq RM} as
\Align{\label{Quasi temp}
\left((\pt+B\pap)^2-i\A\pap\right)\Zbar_t=i\frac{a_t}{a}\circ h^{-1}\A{{\Zbar_{,\alphap}}}+\frac{\pi}{2}[Z_t,\CH]{\frac{\Zbar_{,\alphap}}{Z_{,\alphap}}}.
To understand the quasilinear structure of this equation we need to compute $,$ $B,$ and $a_t/a∘h^-1$ in terms of the unknowns. We begin with $,$ where in addition we verify that $$ is in fact a positive quantity so that the Taylor sign condition holds.
\begin{proposition}\label{prop: A1} $\A_1:=\A|{Z_{,\alphap}}|^2$ is positive and is given by
\Aligns{
\A_1=&\Im[Z_t,\tH]{\Zbar_{t,\alphap}}+\pi\Im[\Zbar,\tH]{Z_{,\alphap}}\\
\end{proposition}
\begin{proof}
Multiplying \eqref{Z eq RM} by ${Z_{,\alphap}}$ we get
\Align{\label{A temp 1 RM}
i\A_1=i\A|{Z_{,\alphap}}|^2=\Zbar_{tt}{Z_{,\alphap}}+\pi\Zbar {Z_{,\alphap}}- G{Z_{,\alphap}}.
Note that since $\Phi(t,Z(t,\alphap))=e^{i\alphap}$ and $\Phi_{z}$ is non-vanishing,
\Aligns{
is holomorphic inside $\D.$ Moreover writing $\zbar_t=F(t,z)$ where $F$ is holomorphic inside $\Omega(t)$ we get
\Aligns{
\zbar_{tt}=F_t(t,z)+F_z(t,z)z_t=F_t(t,z) + \frac{\zbar_{t\alpha}z_t}{z_\alpha},
and hence
\Aligns{
\Zbar_{tt}=F_t(t,Z)+\frac{{\Zbar_{t,\alphap}}Z_t}{{Z_{,\alphap}}}.
Therefore we can apply $(I-\H)$ to \eqref{A temp 1 RM} to get
\Aligns{
i(I-\H)\A_1=(I-\H)({\Zbar_{t,\alphap}}Z_t)+\pi(I-\H)(\Zbar {Z_{,\alphap}}).
Taking imaginary parts of the two sides, keeping in mind that $\A_{1}$ is real, yields
\Align{\label{A temp 2 RM}
\A_1-\Av(\A_1)=\Im[Z_t,\H]{\Zbar_{t,\alphap}}+\pi\Im[\Zbar,\H]{Z_{,\alphap}}.
Note that from \eqref{A temp 1 RM}
\Align{\label{A temp 2 p RM}
\Av(\A_1)=-i \Av({\Zbar_{t,\alphap}}Z_t)-\pi i\Av(\Zbar {Z_{,\alphap}}).
\Aligns{
[\Zbar,\H]{Z_{,\alphap}}=&\Zbar\tH {Z_{,\alphap}}+\Zbar \Av({Z_{,\alphap}})-\tH(\Zbar {Z_{,\alphap}})-\Av(\Zbar {Z_{,\alphap}})\\
=&[\Zbar,\tH]{Z_{,\alphap}}-\Av(\Zbar {Z_{,\alphap}})
\Aligns{
Using the fact that $\Av(\Zbar {Z_{,\alphap}})$ and $\Av({\Zbar_{t,\alphap}}Z_t)$ are purely imaginary, these computations and \eqref{A temp 2 RM} combine to give
\Aligns{
\A_1=\Im[Z_t,\tH]{\Zbar_{t,\alphap}}+\pi\Im[\Zbar,\tH]{Z_{,\alphap}}.
To see that the right hand side above is positive note that
\begin{align*}
\Im([\overline{Z},\tH]{Z_{,\alphap}})&=-\frac{1}{2\pi}\Re\int_{0}^{2\pi}\left(\overline{Z}(t,\alpha')-\overline{Z}(t,\beta')\right)\cot\left(\frac{\betap-\alphap}{2}\right)\partial_{\beta'}\left(Z(t,\beta')-Z(t,\alpha')\right)d\beta'\\
\end{align*}
\begin{align*}
\begin{split}
\Im[Z_t,\tH]{\Zbar_{t,\alphap}}&=-\frac{1}{2\pi}\Re\int_{0}^{2\pi}\left(Z_{t}(t,\alpha')-Z_{t}(t,\beta')\right)\cot\left(\frac{\betap-\alphap}{2}\right)\partial_{\beta'}\left(\overline{Z}_{t}(t,\beta')-\overline{Z}_{t}(t,\alpha')\right)d\beta'\\
\end{split}
\end{align*}
\end{proof}
The computation for $a_t/a∘h^-1$ is more involved. In order to state the result we introduce the notation
\Align{\label{D notation}
D_\alpha:=\frac{1}{z_\alpha}\partial_\alpha,\qquad D_{\alphap}:=\frac{1}{{Z_{,\alphap}}}\partial_{\alphap}
We also define
\begin{align*}
\end{align*}
With this notation we state our next proposition.
\begin{proposition}\label{prop: at over a}
\Aligns{
\frac{a_t}{a}\circ h^{-1}-\frac{1}{\A_1}\Av\left(\A_1\frac{a_t}{a}\circ h^{-1}\right)=\frac{1}{\A_1}\Im\Bigg\{&-\frac{\pi}{2}\left[[Z_{t},\cH]\frac{{\Zbar_{,\alphap}}}{{Z_{,\alphap}}},\H\right]{Z_{,\alphap}}-\frac{\pi}{2}[(I+\CHbar)Z,\H]{\Zbar_{t,\alphap}}\\
&+2[Z_{t},\H]{\Zbar_{tt,\alphap}}+2[Z_{tt},\H]{\Zbar_{t,\alphap}}-[Z_t,Z_t;D_\alphap \Zbar_t]\Bigg\},
\begin{align*}
\Av\left(\A_{1}\frac{a_{t}}{a}\circ h^{-1}\right)=&-2i\Av\left(Z_{t}\partial_{\alphap}(\Zbar_{tt}-(D_{\alphap}\Zbar_{t})Z_{t})\right)-i\Av\left(Z_{tt}\pap\Zbar_t\right)-i\Av\left(Z_t^2\pap D_\alphap \Zbar_t\right)\\
&-i\Av\left({\Zbar_{t,\alphap}}Z_{tt}\right)+\frac{\pi i}{2}\Av\left({\Zbar_{t,\alphap}}(I+\CHbar)Z\right)+\frac{\pi i}{2}\Av\left({Z_{,\alphap}} [Z_t,\CH]\frac{{\Zbar_{,\alphap}}}{{Z_{,\alphap}}}\right).
\end{align*}
\end{proposition}
\begin{proof}
Multiplying \eqref{Zt eq RM} by ${Z_{,\alphap}}$ gives
\Align{\label{A1 temp 1}
{Z_{,\alphap}}\left(\Zbar_{ttt}-i\A{\Zbar_{t,\alphap}}-\frac{\pi}{2}[Z_t,\CH]\frac{{\Zbar_{,\alphap}}}{{Z_{,\alphap}}}\right)=i\A_1\frac{a_{t}}{a}\circ h^{-1}.
In order to understand the holomorphicity properties of $\Zbar_{ttt}$ we recall that
\Align{\label{holomorphicity zbart}
With the notation introduced in \eqref{D notation},
\Aligns{
F_t\circ z=\zbar_{tt}-(D_\alpha\zbar_t)z_t, \quad F_z\circ z=D_\alpha \zbar_t,\quad {F_{zz}\circ z=D_\alpha^2\zbar_t,}\quad F_{tz}\circ z =D_\alpha(\zbar_{tt}-(D_\alpha \zbar_t)z_t)
where the lasts identity follows form differentiating the first with respect to $\alpha.$
Substituting back into the equation for $\zbar_{ttt}$ we get
\Aligns{
\zbar_{ttt}=F_{tt}\circ z+2z_tD_\alpha(\zbar_{tt}-(D_\alpha\zbar_t)z_t)+z_{tt}D_\alpha \zbar_t+z_t^2D_\alpha^2\zbar_t.
We now precompose with $h^{-1}$ to get
\Align{\label{A1 temp 2}
\Zbar_{ttt}=F_{tt}\circ Z+2Z_tD_\alphap(\Zbar_{tt}-(D_\alphap\Zbar_t)Z_t)+Z_{tt}D_\alphap \Zbar_t+Z_t^2D_\alphap^2\Zbar_t.
We will substitute this into \eqref{A1 temp 1} and apply $(I-\H).$ To this end we first note that if $f$ is holomorphic then since $Z_{,\alphap}$ is also holomorphic, $(I-\Hp)(Z_{,\alphap} f)=0,$ which allows us to compute
\begin{align*}
&(I-\Hp)({Z_{,\alphap}}F_{tt}\circ Z)=0,\quad (I-\Hp)({Z_{,\alphap}}\Zbar_{t})=0,\\
\end{align*}
\Aligns{
In view of \eqref{Z eq RM} and holomorphicity of $Z\Zbar_{t,\alphap},$
\begin{align*}
-i(I-\H)(\A{\Zbar_{t,\alphap}}{Z_{,\alphap}})=&(I-\H)\left(\Zbar_{t,\alphap}(Z_{tt}+\pi Z-\Gbar)\right)\\
\end{align*}
Moreover using Lemma \ref{lem: operator H commutator},
\begin{align*}
\end{align*}
Putting these together and using the notation introduced before the proposition we get
\begin{align*}
i(I-\H)(\A_{1}\frac{a_{t}}{a}\circ h^{-1})=&-\frac{\pi}{2}\left[[Z_{t},\cH]\frac{{\Zbar_{,\alphap}}}{{Z_{,\alphap}}},\H\right]{Z_{,\alphap}}-\frac{\pi}{2}[(I+\CHbar)Z,\H]{\Zbar_{t,\alphap}}\\
&+2[Z_{t},\H]{\Zbar_{tt,\alphap}}+2[Z_{tt},\H]{\Zbar_{t,\alphap}}-[Z_t,Z_t;D_\alphap \Zbar_t].
\end{align*}
The first statement of the proposition now follows by taking imaginary parts on both sides of this equation. The second statement follows from taking the averages of the two sides of \eqref{A1 temp 1} and using \eqref{A1 temp 2} and \eqref{Z eq RM} as well as the facts that by the holomorphicity of $F$ and $F_{tt}$
$$\Av({Z_{,\alphap}} F_{tt}\circ Z )==\frac{1}{2\pi}\int_{\partial\Omega}F_{tt} dz=0$$
$$\Av({\Zbar_{t,\alphap}}Z)=-\frac{1}{2\pi}\int_0^{2\pi}\Zbar_t Z_{,\alphap}d\alphap=\frac{1}{2\pi}\int_{\partial\Gamma}F dz=0.$$
\end{proof}
Finally we turn to $B:=h_t∘h^-1.$
\begin{proposition}\label{prop: B}
Suppose the Riemann mapping $\Phi$ satisfies $\Phi(t,\bfx_0)=0,$ $\Phi_z(t,\bfx_0)>0$ for all $t\leq T,$ where $T$ is such that $\bfx_0\in\Omega(t)$ for $t\leq T.$ Then $B$ satisfies
\Aligns{
B-\Av(B)= \Re\left(\left[\frac{Z_{t}}{e^{i\alphap}},\H\right]\frac{e^{i\alphap}}{{Z_{,\alphap}}}\right),
\Aligns{
\Av(B)=\frac{1}{2\pi}\Re\int_{0}^{2\pi}\frac{Z_{t}}{{Z_{,\alphap}}}d\alphap.
\end{proposition}
\begin{proof}
Differentiating the equation $\Phi(t,{Z(t,\alphap)})=e^{i\alphap}$ with respect to $t$ gives
\Aligns{
0=\Phi_t\circ Z +\Phi_z\circ Z (Z_t-B{Z_{,\alphap}})=\Phi_t\circ Z+\frac{ie^{i\alphap}(Z_t-B{Z_{,\alphap}})}{{Z_{,\alphap}}}
which can be rearranged as
\Aligns{
B=\frac{\Phi_t\circ Z}{ie^{i\alphap}}+\frac{Z_t}{{Z_{,\alphap}}}.
Since $\Phi(t,\bfx_{0})=0$ for all $t\in[0,T]$, applying $(I-\H)$ gives
\Aligns{
Taking the real parts on both sides of above gives
\Aligns{
Note that
\Aligns{
\frac{1}{2\pi i}\int_{0}^{2\pi}\frac{\Phi_{t}(t,\Phi^{-1}(t,e^{i\alphap}))}{ie^{2i\alphap}}ie^{i\alphap}d\alphap=\frac{1}{i}\Phi_{tz}(t,\bfx_{0}){(\Phi^{-1})_{w}(t,0)}
is purely imaginary by our choice of normalization for the Riemann mapping. Since $B$ is real, it follows that
\Aligns{
\Av(B)=\frac{1}{2\pi}\Re\int_{0}^{2\pi}\frac{Z_{t}}{{Z_{,\alphap}}}d\alphap.
\end{proof}
Summarizing the computations above, we get the following corollary of \eqref{Z eq RM}, \eqref{Quasi temp}, and Propositions \ref{prop: A1}, \ref{prop: at over a}, and \ref{prop: B}.
\begin{corollary}\label{cor: Quasi}
If $z$ is a solution to \eqref{z eq RM} and the Riemann mapping $\Phi$ is defined according to the normalization above, then $Z:=z\circ h^{-1}$ satisfies
\begin{align}\label{Quasi 1}
\begin{cases}
&(\partial_{t}+B\partial_{\alphap})^{2}\Zbar_{t}-i\A\partial_{\alphap}\Zbar_{t}=i\dfrac{a_t}{a}\circ h^{-1}\dfrac{\A_{1}}{{Z_{,\alphap}}}+\dfrac{\pi}{2}[Z_t,\CH]\dfrac{{\Zbar_{,\alphap}}}{{Z_{,\alphap}}}:=g\\
\end{cases},
\end{align}
\begin{align}\label{Quasi 2}
\begin{cases}
&\A=\dfrac{|\Zbar_{tt}+\dfrac{\pi}{2}(I-\cH)\Zbar|^{2}}{\A_{1}},\quad \Zbar_{tt}=(\partial_{t}+B\partial_{\alphap})\Zbar_{t}\\
\end{cases},
\end{align}
and $\A_{1}, ~\dfrac{a_{t}}{a}\circ h^{-1},$ and $B$ are given in Propositions \ref{prop: A1}, \ref{prop: at over a}, and \ref{prop: B} respectively.
\end{corollary}
\begin{remark}
The significance of \eqref{Quasi 2} is that in proving local well-posedness for \eqref{Quasi 1} we will use \eqref{Quasi 2} as the \emph{definition} of $\A$ and $\frac{1}{{Z_{,\alphap}}}.$ As we will discuss below, we will separately show that the resulting solution is a solution of the original system \eqref{Z eq RM}.
\end{remark}
We have now seen how to go from the original system to
\begin{align}\label{Actual eq 1}
\begin{cases}
&(\partial_{t}+B\partial_{\alphap})^{2}V+\A|D|V=\dfrac{a_t}{a}\circ h^{-1}L+\dfrac{\pi}{2}[\Vbar,\CH]\dfrac{\Wbar_{\alphap}}{W_\alphap}=:g\\
\end{cases},
\end{align}
\begin{align}\label{Actual eq 2}
\begin{cases}
&\Av(B)=\frac{1}{2\pi}\Re\int_{0}^{2\pi}\frac{\Vbar L}{i\A_{1}}d\alphap,\\
&\A=\dfrac{|(\pt+B\pap)V+\dfrac{\pi}{2}(I-\cH)\Wbar|^{2}}{\A_{1}},\quad \\
\end{cases}
\end{align}
and $_1$ and $a_t/a∘h^-1$ are defined in Propositions \ref{prop: A1} and \ref{prop: at over a} with $Z, _t,$ and $_tt$ replaced by $W, V,$ and $(+B)V$ respectively.
Here $W=Z, V=_t$, $|D|=√(-^2)$ and we have used the fact that if $u$ is the boundary value of a holomorphic function in the disc, then $|D|u=-iű=-iu$. We next discuss how to go back to the original water wave system from \eqref{Actual eq 1}--\eqref{Actual eq 2}.
\begin{proposition}\label{prop: going back}
Suppose $(W,V)$ is a solution to \eqref{Actual eq 1} and \eqref{Actual eq 2} on some time interval $J$ extending from $t=0$ such that $\bfx_0\in\Omega(t)$ for all $t\in J.$ Then the following statements hold.
\begin{enumerate}
\item $W$ and $V$ are boundary values of holomorphic functions and $L-\frac{i\A_{1}}{W_{\alphap}}=0$, if initially $W$ and $V$ are boundary values of holomorphic functions and $L-\frac{i\A_{1}}{W_{\alphap}}=0$.
\item If $h$ is the solution to
\begin{align*}
\frac{dh}{dt}=B(h,t),\quad h(0,\alpha)=\alpha,
\end{align*}
then $z:=W\circ h$ satisfies \eqref{z eq RM}.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{enumerate}
\item
We will derive a linear differential system for the quantities $(I-\H)V$, $(I-\H)W$ and $\frac{i\A_{1}}{W_{\alphap}}-L$ for which uniqueness of solutions holds. Since these quantities are zero initially, they must be zero during the evolution. In this process, we will use $\calR$ to denote linear terms in these quantities, whose exact definition may change from line to line. If we want to make the dependence precise, we use expressions such as $\calR((I-\H)V, (I-\H)W,...)$. We start with the equation for $W$. Applying $(I-\H)$ on both sides we get
\begin{align}\label{homo 1}
\begin{split}
\partial_{t}\left((I-\H)W\right)+B\partial_{\alphap}\left((I-\H)W\right)
=&(I-\H)\Vbar-\left[\frac{I-\H}{2}\Re[\Vbar e^{-i\alphap},\H]\frac{e^{i\alphap}}{W_{\alphap}},\H\right]\frac{I+\H}{2}W_{\alphap}+\calR_{1}+\calR_{2}\\
=&(I-\H)\Vbar-\left[\frac{I-\H}{2}\Re[\Vbar e^{-i\alphap},\H]\frac{e^{i\alphap}}{\frac{I+\H}{2}W_{\alphap}},\H\right]\frac{I+\H}{2}W_{\alphap}+\calR_{1}+\calR_{2}+\calR_{3}\\
\end{split}
\end{align}
\begin{align}\label{W errors}
\begin{split}
&\calR_{2}=-\left[\frac{I-\H}{2}\Re[\Vbar e^{-i\alphap},\H]e^{i\alphap}\left(\frac{L}{i\A_{1}}-\frac{1}{W_{\alphap}}\right),\H\right]\frac{I+\H}{2}W_{\alphap},\\
&\calR_{3}=-\left[\frac{I-\H}{2}\Re[\Vbar e^{-i\alphap},\H]\left(\frac{e^{i\alphap}}{W_{\alphap}}-\frac{e^{i\alphap}}{\frac{I+\H}{2}W_{\alphap}}\right),\H\right]\frac{I+\H}{2}W_{\alphap}.
\end{split}
\end{align}
Note that in view of Lemma \ref{lem: Yosihara},
\begin{align}\label{esti R1-3}
\|\calR_{j}\|_{H^{s}_{\alphap}}\leq C\left(\left\|L-\frac{i\A_{1}}{W_{\alphap}}\right\|_{L^{2}_{\alphap}}+\|(I-\H)W\|_{L^{2}_{\alphap}}\right).
\end{align}
To derive an equation for $(I-\H)V$, we introduce the notation $\calP:=(\pt+B\pap)^{2}+\A|D|$. Then the first equation in \eqref{Actual eq 1} can be written as
\begin{align}\label{homo 2 pre}
\calP\left(\frac{I-\H}{2}V\right)=-\calP\left(\frac{I+\H}{2}V\right)+e^{-i\alphap}\frac{ie^{i\alphap}}{\frac{I+\H}{2}W_{\alphap}}\A_{1}\frac{a_{t}}{a}\circ h^{-1}+\frac{\pi}{2}[\Vbar,\calH]\frac{\Wbar_{\alphap}}{W_{\alphap}}+\calR_{4}
\end{align}
\begin{align}\label{V error}
\begin{split}
\calR_{4}=&\left((\pt+B\pap)\left(\frac{I-\H}{2}V\right)+\pi\frac{I-\overline{\bbH}}{2}\Wbar+\frac{\pi}{2}(\tcalH-\calH)\Wbar\right)\frac{a_{t}}{a}\circ h^{-1}\\
&+e^{-i\alphap}\left(\frac{\tL e^{i\alphap}}{i\A_{1}}-\frac{e^{i\alphap}}{\frac{I+\H}{2}W_{\alphap}}\right)i\A_{1}\left(\frac{a_{t}}{a}\circ h^{-1}\right).
\end{split}
\end{align}
\begin{align*}
&(\tcalH f)(\alphap):=\frac{\pv}{\pi i}\int_0^{2\pi}\frac{f(\betap)}{\PH W(\betap)-\PH W(\alphap)}\PH W_\betap(\betap)d\betap.
\end{align*}
Note that
\begin{align*}
&(\tcalH-\calH)f=\calR((I-\H)W, (I-\H)W_{\alphap}),\\
&L-\tL=\calR((I-\H)V, (I-\H)W, (I-\H)W_{\alphap}, (\pt+B\pap)(I-\H)V).
\end{align*}
\begin{claim}\label{claim: gain regularity}
Given any $f\in H^{s}_{\alphap}$, there is a constant $C=C\left(\|f\|_{H^{s}_{\alphap}}\right)$, such that
\begin{align*}
&\left\|(I-\H)f\left(\frac{\tL e^{i\alphap}}{i\A_{1}}-\frac{e^{i\alphap}}{\frac{I+\H}{2}W_{\alphap}}\right)\right\|_{H^{s}_{\alphap}}\\
\leq &C\left(\left\|L-i\A\Wbar_{\alphap}\right\|_{L^{2}_{\alphap}}+\|(I-\H)V\|_{H^{s}_{\alphap}}+\|(\pt+B\pap)(I-\H)V\|_{L^{2}_{\alphap}}+\|(I-\H)W\|_{H^{s}_{\alphap}}\right).
\end{align*}
\end{claim}
\begin{proof}[Proof of Claim \ref{claim: gain regularity}]
First we compute
\begin{align*}
\A_{1}(I-\H)\left(\frac{\tL e^{i\alphap}}{\A_{1}}\right)=&(I-\H)(e^{i\alphap}\tL)+[\H,\A_{1}]\left(\frac{\tL e^{i\alphap}}{\A_{1}}\right)\\
=&\left[\frac{I-\H}{2}B,\H\right]\left(\frac{I+\H}{2}V_{\alphap}\cdot e^{i\alphap}\right)+\pi(I-\H)e^{i\alphap}\frac{I+\overline{\H}}{2}\Wbar\\
&-\frac{1}{2}\left[(I-\H)\PHbar\Vbar\PH V_{\alphap},\H\right]\left(\frac{e^{i\alphap}}{\frac{I+\H}{2}W_{\alphap}}\right)\\
&-\frac{\pi}{2}\left[(I-\H)\PHbar\Wbar\PH W_{\alphap},\H\right]\left(\frac{e^{i\alphap}}{\frac{I+\H}{2}W_{\alphap}}\right)+\calR\\
=&\left[\frac{\PHbar\Vbar}{\PH W_{\alphap}},\H\right]\left(e^{i\alphap}\left(\frac{I+\H}{2}\right)V_{\alphap}\right)\\
&-\left[\PHbar\Vbar\PH V_{\alphap},\H\right]\left(\frac{e^{i\alphap}}{\frac{I+\H}{2}W_{\alphap}}\right)+\calR=\calR.
\end{align*}
Therefore since $\A_{1}$ is bounded away from zero
\begin{align*}
(I-\H)f\left(\frac{\tL e^{i\alphap}}{i\A_{1}}-\frac{e^{i\alphap}}{\frac{I+\H}{2}W_{\alphap}}\right)=&[f,\H]\PH\left(\frac{\tL e^{i\alphap}}{i\A_{1}}-\frac{e^{i\alphap}}{\frac{I+\H}{2}W_{\alphap}}\right)+\calR\\
\end{align*}
and the claim follows from Lemma \ref{lem: Yosihara}.
\end{proof}
Applying $(I-\H)$ to both sides of \eqref{homo 2 pre} and with
\begin{align*}
\calS:=i\A_{1}\frac{a_{t}}{a}\circ h^{-1}-\left(\frac{I+\H}{2}\right)W_{\alphap}\left(\calP\left(\frac{I+\H}{2}\right)V-\frac{\pi}{2}[\Vbar,\calH]\frac{\Wbar_{\alphap}}{W_{\alphap}}\right)
\end{align*}
we obtain
\begin{align}\label{homo V temp 1}
\begin{split}
\end{split}
\end{align}
To see that the first two terms in the last line are linear in $(I-\H)V, (I-\H)W,$ and $\frac{i\A_{1}}{W_{\alphap}}-L$, we want to mimic the proof of Proposition \ref{prop: at over a}, for which we need to introduce the Riemann mapping. First let $h$ be the function on $\bbR$ defined by
\begin{align}\label{h construct}
\frac{dh}{dt}=B(h,t),\quad h(\alpha,0)=\alpha.
\end{align}
Since $h$ is a diffeomorphism at $t=0$ and $h_{\alphap}$ satisfies the linear ODE
\begin{align*}
\frac{dh_{\alpha}}{dt}=B_{\alphap}h_{\alpha},
\end{align*}
$h$ is a diffeomorphism at least for a short time
and $\partial_{t}(f\circ h)=(\partial_{t}+B\pap)f\circ h$ for all time.
Let $\TPhi^{-1}(t,\cdot)$ be the holomorphic function with boundary value $\TPhi^{-1}(t,e^{i\alphap})=\left(\frac{I+\H}{2}W\right)(t,\alphap)$. Since $\TPhi^{-1}_{w}(0,\cdot)$ is never zero on the disc $\D$, the same is true for $\TPhi^{-1}_{w}(t,\cdot)$ for small $t$ by the Cauchy integral formula for the derivative of a holomorphic function. Therefore $\TPhi^{-1}(t,\cdot)$ has an inverse, which we denote by $\TPhi(t,\cdot): \TPhi^{-1}(t,\D)\to \D$. Note that with this definition $\TPhi(t,\frac{I+\H}{2}W(t,\alphap))=e^{i\alphap}$. It follows that if $f$ is the boundary value of a holomorphic function on $\D$, i.e., $f(\alphap)=F(e^{i\alphap})$ for a holomorphic function $F$ on $\D$, then $f\circ h$ is the boundary value of the holomorphic function $G=F\circ \TPhi$ on $\TPhi^{-1}(t,\D)$. Introducing the variable
\begin{align*}
\tz:=\frac{I+\H}{2}W\circ h
\end{align*}
we can write
\begin{align*}
\frac{I+\H}{2}V\circ h=\widetilde{F}(t,\tz).
\end{align*}
Now the same argument as in the proof of Proposition \ref{prop: at over a} implies that $(I-\H)\calS$ and $[e^{-i\alphap},\H]\calS=2e^{-i\alphap}\Av(\calS)$ are linear in $(I-\H)V, (I-\H)W,$ and $\frac{i\A_{1}}{W_{\alphap}}-L$.
Next we compute the left hand side of \eqref{homo V temp 1}.
\begin{align*}
\end{align*}
\begin{align*}
\end{align*}
Combining these observations with \eqref{homo V temp 1}, we get
\begin{align}\label{homo 2}
\calP\left(\frac{I-\H}{2}V\right)=\calR.
\end{align}
Note that by Claim \ref{claim: gain regularity} and \eqref{esti R1-3}, to bound, say, the $H^{2}_{\alphap}$ norms of $(I-\H)V$ and $(I-\H)W$, we only need to use the $L^{2}_{\alphap}$ norm of $L-i\frac{\A_{1}}{W_{\alphap}}$. Therefore to derive the equation for $L-i\frac{\A_{1}}{W_{\alphap}}$, we can write terms involving derivatives of $(I-\H)V$ and $(I-\H)W$ as $\calR$. To derive this equation for $L-\frac{i\A_1}{W_\alphap}.$ we first note that
\Aligns{
where for the last equality we have used the fact that $\CH f-\tcalH f=\calR$.This computation and the fact that $|D|V=-i\pap V+|D|(I-\H)V$ allow us to write the first equation in \eqref{Actual eq 1} as
\Align{\label{homo 3 temp 1}
(\pt+B\pap)\left(L-i\A\pap\Wbar\right)=\frac{i}{W_\alphap}\left(\A_1\frac{a_{t}}{a}\circ h^{-1}-\A_1\frac{\bfa_{t}}{\bfa}\circ h^{-1}\right)-\A|D|(I-\H)V+\calR.
Here we have used the notation
\Aligns{
\bfa:=\frac{\A\circ h}{h_{\alpha}}
so in particular since $\A=\frac{L\Lbar}{\A_1}$
\Align{\label{homo 3 temp 2}
\A_1\frac{\bfa_{t}}{\bfa}\circ h^{-1}=\frac{\Lbar(\pa+B\pap)L}{\A}+\frac{L(\pt+B\pap)\Lbar}{\A}-(\pt+B\pap)\A_1-\A_1B_\alphap.
To write \eqref{homo 3 temp 1} as a homogeneous linear equation in $(I-\H)V, (I-\H)W,$ and $L-\frac{i\A_1}{W_\alphap}$ we need to study the right hand side of \eqref{homo 3 temp 2} more carefully. Since by \eqref{Actual eq 1} and with the notation $L_t:=(\pt+B\pap)L$ the quantity $\Lbar(L_t+\A|D|V)-\frac{\pi}{2}\Lbar(\tcalH-\CH)V$ is purely real,
\Aligns{
\frac{\Lbar L_t}{\A}+\frac{L\Lbar_t}{\A}=&2\frac{\Lbar(L_t+\A|D|V)}{\A}-\Lbar |D|V-L|D|\Vbar+\calR\\
=&2\frac{\Lbar}{\A}(L_t+\A|D|V)-L|D|\Vbar-\Lbar|D| V+\calR\\
=&2\A_1\frac{a_t}{a}\circ h^{-1}-L|D|\Vbar-\Lbar|D| V+\calR,
which means
\begin{align*}
\A_{1}\frac{\bfa_{t}}{\bfa}\circ h^{-1}=2\A_{1}\frac{a_{t}}{a}\circ h^{-1}-L|D|\Vbar-\Lbar|D|V-(\pt+B\pap)\A_{1}-\A_{1}B_{\alphap}.
\end{align*}
Together with \eqref{homo 3 temp 1} and \eqref{homo 3 temp 2} this gives
\Align{\label{homo 3 temp 3}
(\pt+B\pap)\left(L-i\A\pap\Wbar\right)=&\frac{i}{W_\alphap}\left(-\A_1\frac{a_t}{a}\circ h^{-1}+\Lbar|D|V+L|D|\Vbar+(\pt+B\pap)\A_1+\A_1B_\alphap\right)+\calR\\
Since $\calT$ is purely real,
\begin{align*}
\calT=\Im(i\calT)=\Im\left((I-\H)i\calT\right)+\Av(\calT).
\end{align*}
First we compute
\begin{align*}
L|D|\Vbar+\Lbar|D|V=&L|D|\Vbar-i\A W_{\alphap}|D|\left(\frac{I-\H}{2}\right)V-i\A W_{\alphap}|D|\left(\frac{I+\H}{2}\right)V+\calR\\
&-i\A W_{\alphap}|D|\left(\frac{I+\H}{2}\right)V+\calR\\
=&\left((\pt+B\pap)\frac{I+\H}{2}V+\frac{\pi}{2}(I-\cH)\Wbar\right)|D|\Vbar-i\A W_{\alphap}|D|\left(\frac{I+\H}{2}\right)V+\calR.
\end{align*}
\begin{align}\label{homo 3 temp 6}
\begin{split}
\frac{i}{W_{\alphap}}(I-\H)\left(L|D|\Vbar+\Lbar|D|V\right)
&\qquad\qquad\qquad\qquad\left.-i\A \left(\frac{I+\H}{2}\right)W_{\alphap}|D|\left(\frac{I+\H}{2}\right)V\right)+\calR.
\end{split}
\end{align}
With the notation $\calU:=(\pt+B\pap)\frac{I+\H}{2}V+\frac{\pi}{2}(I-\cH)\Wbar$
\begin{align}\label{homo 3 temp 7}
\begin{split}
\calU|D|\Vbar=&\calU|D|\PHbar\Vbar+\calU|D|\OPHbar\Vbar\\
=&\calU i\pap\frac{I-\H}{2}\Vbar+\calR=i\calU \pap\left(\frac{I-\H}{2}(\pt+B\pap)W\right)+\calR\\
=&i\calU B_{\alphap}\left(\frac{I+\H}{2}\right)W_{\alphap}+i(\pt+B\pap)\left(\calU\left(\frac{I+\H}{2}\right)W_{\alphap}\right)-i\left(\frac{I+\H}{2}\right)W_{\alphap}(\pt+B\pap)\calU+\calR.
\end{split}
\end{align}
Combining \eqref{homo 3 temp 3}--\eqref{homo 3 temp 7} we get
\begin{align*}
(I-\H)(i\calT)=&i(I-\H)\left(-\A_{1}\frac{a_{t}}{a}\circ h^{-1}+L|D|\Vbar+\Lbar|D|V+(\pt+B\pap)\A_{1}+\A_{1}B_{\alphap}\right)\\
=&-(I-\H)\left(\calU B_{\alphap}\left(\frac{I+\H}{2}\right)W_{\alphap}\right)-(I-\H)\left((\pt+B\pap)
\left(\calU\left(\frac{I+\H}{2}\right)W_{\alphap}\right)\right)\\
&+(I-\H)\left(\left(\frac{I+\H}{2}\right)W_{\alphap}\left((\pt+B\pap)\calU+\A|D|\left(\frac{I+\H}{2}\right)V\right)-i\A_{1}\frac{a_{t}}{a}\circ h^{-1}\right)\\
=&-(I-\H)\left(\calU B_{\alphap}\left(\frac{I+\H}{2}\right)W_{\alphap}\right)-(I-\H)\left((\pt+B\pap)
\left(\calU\left(\frac{I+\H}{2}\right)W_{\alphap}\right)\right)\\
\end{align*}
To compute the second term, we first note that
\begin{align*}
\calU=(\pt+B\pap)\PH V+\pi\PHbar \Wbar-\frac{\pi}{2}(I+\tcalH)\Wbar+\calR:=\tcalU+\calR.
\end{align*}
By a computation similar to Proposition \ref{prop: A1} it follows that
\begin{align*}
=&-(\pt+B\pap)\left(\left[\Vbar,\H\right]\PH V_{\alphap}+\pi\left[\Wbar,\H\right]\PH W_{\alphap}\right)\\
\end{align*}
\begin{align}\label{homo 3 temp 10}
\begin{split}
\Im((I-\H)i\calT)=&-(\pt+B\pap)\Im \left(\left[\Vbar,\H\right]\PH V_{\alphap}+\pi\left[\Wbar,\H\right]\PH W_{\alphap}\right)\\
&-\Im [B,\H]\partial_{\alphap}\left(\calU\left(\frac{I+\H}{2}\right)W_{\alphap}\right)+(\pt+B\pap)\A_{1}-\pt\Av(\A_{1})
&-\Im(I-\H)\left(\calU B_{\alphap}\left(\frac{I+\H}{2}\right)W_{\alphap}\right)\\
=&-\Im [B,\H]\partial_{\alphap}\left(\calU\left(\frac{I+\H}{2}\right)W_{\alphap}\right)+\A_{1}B_{\alphap}-\Im(I-\H)\left(\calU B_{\alphap}\left(\frac{I+\H}{2}\right)W_{\alphap}\right)+\calR\\
=&-\Im [B,\H]\partial_{\alphap}\left(\tcalU\left(\frac{I+\H}{2}\right)W_{\alphap}\right)+\A_{1}B_{\alphap}-\Im(I-\H)\left(\tcalU B_{\alphap}\left(\frac{I+\H}{2}\right)W_{\alphap}\right)+\calR.
\end{split}
\end{align}
We compute
\begin{align*}
&-[B,\H]\partial_{\alphap}\left(\tcalU\left(\frac{I+\H}{2}\right)W_{\alphap}\right)-(I-\H)\left(\tcalU B_{\alphap}\left(\frac{I+\H}{2}\right)W_{\alphap}\right)\\
=&-B_{\alphap}(I-\H)\left(\tcalU\PH W_{\alphap}\right)-\partial_{\alphap}[B,\H]\left(\tcalU\PH W_{\alphap}\right).
\end{align*}
Note that
\begin{align*}
-\Im\left(B_{\alphap}(I-\H)\left(\tcalU\PH W_{\alphap}\right)\right)=-\A_{1} B_{\alphap}+\Av(\A_{1})B_{\alphap}+\calR
\end{align*}
\begin{align*}
-\Im\left(\partial_{\alphap}[B,\H]\left(\tcalU\PH W_{\alphap}\right)\right)=&-\Im\left(\partial_{\alphap}[B,\H](i\A_{1})\right)+\calR\\
=&-\Im\left(\partial_{\alphap}[B,\Av](i\A_{1})\right)+\calR=-\Av(\A_{1}) B_{\alphap}+\calR.
\end{align*}
Combining these observations with \eqref{homo 3 temp 10} we obtain
\begin{align*}
\end{align*}
\item This is a direct consequence of the fact that $L=\frac{i\A_{1}}{W_{\alphap}}$ and the definition of $h$.
\end{enumerate}
\end{proof}
The proof of Theorem \ref{thm: lwp} now follows from local the well-posedness of \eqref{Actual eq 1}--\eqref{Actual eq 2}:
\begin{proof}[Proof of Theorem \ref{thm: lwp}]
By Proposition \ref{prop: going back} it suffices to show local well-posedness of \eqref{Actual eq 1}--\eqref{Actual eq 2}. The proof of local well-posedness for the system \eqref{Actual eq 1}--\eqref{Actual eq 2} is almost identically the same as the proof of Theorem 5.10 in \cite{Wu1} where the vanishing viscosity method us used. In fact the only difference is that unlike in \cite{Wu1}, here we also need to control $W=Z.$ But by \eqref{Actual eq 1} $W$ satisfies a transport equation and therefore control of $W$ follows from control of $V$ by integration. We refer the reader to \cite{Wu1} Section 5, and leave the necessary routine modifications to the reader.
\end{proof}
\appendix
%\section{Proof of Lemmas \ref{lem: gravity} and \ref{lem: operator H commutator}}\label{app: proofs}
\section{The Hilbert Transform}\label{app: Hilbert transform}
In this appendix we recall some facts about the Hilbert transform. If $Ω$ is a bounded domain in $$ with $C^2_t,α$ boundary and $f$ is a function defined on $∂Ω$ then the Hilbert transform $Hf$ of $f$ with respect to $Ω$ is defined as
\Aligns{
Hf(z_0):=\lim_{\epsilon\to 0^{+}}\frac{1}{\pi i}\int_{\gamma_\epsilon}\frac{f(w)}{w-z_0}dw,
where $γ_ϵ$ is the portion of $∂Ω$ obtained by removing a segment of $∂Ω$ which lies within a circle of radius $ϵ$ centered at $z_0∈∂Ω.$ Given a $C^2_t,α$ parametrization $z:[0,2π]→∂Ω$ of $∂Ω$ we identify $2π-$periodic functions on $$ with functions on $∂Ω,$ and for any such function $f$ we write
\Aligns{
H f(\alpha):=\frac{\pv}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)}{z(\beta)-z(\alpha)}z_{\beta}(\beta)d\beta.
The relevant results from this appendix are summarized in the following proposition.
\begin{proposition}\label{prop: hilbert}
Suppose that $\Omega$ is a bounded domain in $\bbC$ with $C^{2}$ boundary $\partial\Omega$. Let $f$ be a Lipschitz continuous function on $\partial\Omega$ and $H f$ be its Hilbert transform. Then $H f=f$ if and only if $f$ is the boundary value of a holomorphic function in $\Omega$ and $H f=-f$ if and only if $f$ is the boundary value of a holomorphic function $F$ in $\Omega^{c}$ satisfying $F(z)\to0$ as $|z|\rightarrow\infty$.
\end{proposition}
\begin{proof}
Suppose that $\Omega$ is a bounded domain in $\bbC$ and $\gamma:=\partial\Omega$ has $C^{2}_{t,\alpha}$. Let $f$ be a continuous function defined on $\partial\Omega$. The following Cauchy integral
\begin{align}\label{Cauchy integral}
C_f(z):=\frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{w-z}dw
\end{align}
defines a holomorphic function when $z\slashed{\in}\gamma$. In this subsection, we will introduce the Hilbert transforms associated to $\Omega$ and $\Omega^{c}$ by considering the limit of $C_f(z)$ as $z$ approaches $z_{0}$ from $\Omega$ and $\Omega^{c}$ where $z_0$ is a point on $\partial\Omega.$ Here all integrals are understood as counterclockwise, unless otherwise stated. Let us first consider the limit from the inside.
\begin{align}\label{hilbert temp 1}
\lim_{z\rightarrow z_{0}}\frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{w-z}dw=\lim_{z\rightarrow z_{0}}\frac{1}{2\pi i}\int_{\gamma_{\epsilon}+\xi_{\epsilon}}\frac{f(w)}{w-z}dw=\lim_{\epsilon\rightarrow0^{+}}\lim_{z\rightarrow z_{0}}\frac{1}{2\pi i}\int_{\gamma_{\epsilon}+\xi_{\epsilon}}\frac{f(w)}{w-z}dw,
\end{align}
where $\gamma_{\epsilon}$ is the portion of $\gamma$ obtained by subtracting the segment $\xi_{\epsilon}$ about $z_{0}$ which lies within the circle of radius $\epsilon$ centered at $z_0.$ We recognize the limit over $\gamma_\epsilon$ as one half of the Hilbert transform of $f$ associated to $\Omega$:
\begin{align}\label{hilbert temp 2}
\frac{1}{2}H f(z_{0})=\lim_{\epsilon\rightarrow 0^{+}}\frac{1}{2\pi i}\int_{\gamma_{\epsilon}}\frac{f(w)}{w-z_{0}}dw=\lim_{\epsilon\rightarrow0^{+}}\lim_{z\rightarrow z_{0}}\frac{1}{2\pi i}\frac{f(w)}{w-z}dw.
\end{align}
On the other hand,
\Aligns{
\lim_{\epsilon\rightarrow0^{+}}\lim_{z\to z_0}\int_{\xi_\epsilon}\frac{f(w)}{w-z}dw&=\lim_{\epsilon\rightarrow0^{+}}\lim_{z\to z_0}\left(\int_{\xi_\epsilon}\frac{f(w)-f(z_0)}{w-z}dw+\int_{\xi_\epsilon}\frac{f(z_0)}{w-z}dw\right)\\
&=\lim_{\epsilon\rightarrow0^{+}}\lim_{z\to z_0}\int_{\xi_0}\frac{f(z_0)}{w-z}dw.
Now with $C_\epsilon$ denoting the part of the circle of radius $\epsilon$ centered at $z_0$ which lies within $\Omega$ we have
\Align{\label{hilbert temp 3}
\lim_{\epsilon\rightarrow0^{+}}\lim_{z\to z_0}\int_{\xi_\ep}\frac{dw}{w-z}&=\lim_{\epsilon\rightarrow0^{+}}\lim_{\epsilon\rightarrow0^{+}}\int_{\xi_\epsilon+C_\epsilon}\frac{dw}{w-z}-\lim_{\epsilon\rightarrow0^{+}}\int_{C_\epsilon}\frac{dw}{w-z_0}\\
&=2\pi i-\lim_{\epsilon\rightarrow0^{+}}\int_{\pi+O(\epsilon)}^{2\pi+O(\epsilon)}\frac{i\epsilon e^{i\theta}}{\epsilon e^{i\theta}}=\pi i.
Combining this with \eqref{hilbert temp 1} and \eqref{hilbert temp 2} we get
\Aligns{
H f(z_0)=2\lim_{z\to z_0}C_f(z)-f(z_0).
Since $C_f$ is a holomorphic function inside $\Omega,$ and $\lim_{z\to z_0}C_f(z)=f(z_0)$ if $f$ can be extended to a holomorphic function inside $\Omega,$ we conclude that $f$ is the boundary value of a holomorphic function inside $\Omega$ if and only if $Hf(z_0)=f(z_0)$ for all $z_0\in\partial\Omega.$
The computation is similar for the case where $z\to z_0$ from the outside (i.e. $z\in \Omega^c$). In this case in \eqref{hilbert temp 3} we define $C_\epsilon$ to be the part of the circle of radius $\epsilon$ centered at $z_0,$ parametrized clockwisely, which lies in $\Omega^c.$ It then follows that
\Aligns{
\int_{\xi_\epsilon+C_\epsilon}\frac{dw}{w-z}=-2\pi i,
and hence
\Aligns{
Hf(z_0)=2\lim_{z\to z_0}C_f(z)-3f(z_0),
where now the limit is understood to be from the outside. Now notice that from the definition \eqref{Cauchy integral} of the Cauchy integral that $C_f$ is holomorphic in $\Omega^c$ and decays like $\frac{1}{|z|}$ as $|z|\to\infty.$ Therefore if $f$ if $Hf=-f$ then $f$ is the boundary value of a holomorphic function in $\Omega^c$ decaying like $\frac{1}{|z|}$ as $|z|\to\infty.$ Conversely, if $f$ is the boundary value of such a holomorphic function, then defining $U=\{\frac{1}{z}\mathrm{~s.t.~}z\in\Omega^c\}\subseteq \C,$ we have
\Aligns{
\lim_{\stackrel{z\to z_0}{z\in\Omega^c}} C_f(z)=\lim_{\stackrel{z\to z_0}{z\in\Omega^c}}\int_{\partial\Omega}\frac{f(w)}{w-z}dw=\lim_{\stackrel{z\to 1/z_0}{z\in U}}\frac{1}{z}\int_{\partial U}\frac{\frac{f(1/u)}{u}}{u-\frac{1}{z}}du=f(z_0),
and therefore $Hf(z_0)=-f(z_0).$
\end{proof}
\section*{Notations}
For the reader's convenience we give the definitions of some of the symbols used commonly in this work.
\Aligns{
&H f(t,\alpha)=\frac{\pv}{\pi i}\int_0^{2\pi}\frac{f(t,\beta)}{z(t,\beta)-z(t,\alpha)}z_\beta(\beta)d\beta.\\
&\CH f(t,\alpha)=\frac{\pv}{\pi i}\int_0^{2\pi}\frac{f(t,\beta)}{\zed(t,\beta)-\zed(t,\alpha)}\zed_\beta(\beta)d\beta,\quad \zed(t,\cdot)=z(t,j(t,\cdot)),\quad j(t,\cdot):[0,2\pi]\to[0,2\pi] \mathrm{~ a ~ diffeomorphism}.\\
&\H f(t,\alphap)=\frac{\pv}{\pi i}\int_0^{2\pi}\frac{f(t,\betap)}{e^{i\betap}-e^{i\alphap}}ie^{i\betap}d\betap, \qquad \tH f(t,\alpha)=\frac{\pv}{2\pi i}\int_0^{2\pi} f(t,\beta)\cot\left(\frac{\beta-\alpha}{2}\right)d\beta.\\
&\AV(f):=\frac{1}{2\pi i}\int_0^{2\pi}\frac{f(\alpha)}{z(t,\alpha)}z_\alpha(t,\alpha)d\alpha, \qquad \Av(f)=\frac{1}{2\pi}\int_0^{2\pi}f(\alpha)d\alpha.\\
&Kf=\Re H=\frac{1}{2}(H+\Hbar)f,\quad \CK f=\Re \CH f=\frac{1}{2}(\CH+\CHbar)f,\quad f\mathrm{~real~valued}.\\
&K^*f=-\Re\left\{\frac{z_\alpha}{|z_\alpha|}H\frac{|z_\alpha|}{z_\alpha}f\right\},\quad \CK^*f=-\Re\left\{\frac{\zed_\alpha}{|\zed_\alpha|}\CH\frac{|\zed_\alpha|}{\zed_\alpha}f\right\}, \quad f\mathrm{~ real ~ valued}.\\
&a=-\frac{1}{|z_\alpha|}\frac{\partial P}{\partial n},\quad {\bfn \mathrm{~unit~exterior~ normal}}.
Let $h$ be as defined in Figure \ref{fig: h} and $k$ as defined in Remark~\ref{prop: k existence}.
\Aligns{\allowdisplaybreaks
&Z(t,\alphap)=z(t,h^{-1}(t,\alphap)), \quad \zeta(t,\alpha)=z(t,k^{-1}(t,\alpha)).\\
&Z_t(t,\alphap)=z_t(t,h^{-1}(t,\alphap)), \quad Z_{tt}(t,\alphap)=z_{tt}(t,h^{-1}(t,\alphap)),\quad Z_{ttt}(t,\alphap)=z_{ttt}(t,h^{-1}(t,\alphap)).\\
%&\zeta(t,\alpha)=z(t,k^{-1}(t,\alpha)), \quad \zeta_t(t,\alphap)=z_t(t,k^{-1}(t,\alpha)).\\
%& \zeta_{tt}(t,\alpha)=z_{tt}(t,k^{-1}(t,\alpha)),\quad \zeta_{ttt}(t,\alpha)=z_{ttt}(t,k^{-1}(t,\alpha)).\\
&B=h_t\circ h^{-1},\qquad b=k_t\circ k^{-1}.\\
&\A=(ah_\alpha)\circ h^{-1},\quad \A_1=\A|{Z_{,\alphap}}|^2,\quad A=(ak_\alphap)\circ k^{-1}.\\
&D_\alpha=\frac{1}{|z_\alpha|}\pa,\quad D_\alphap=\frac{1}{|{Z_{,\alphap}}|}\pap.\\
&[Z_{t},Z_{t};D_{\alphap}\Zbar_{t}]=\frac{ie^{i\alphap}}{\pi i}\int_{0}^{2\pi}\left(\frac{Z_t(t,\betap)-Z_t(t,\alphap)}{e^{i\betap}-e^{i\alphap}}\right)^2\frac{e^{i\betap}}{Z_\betap(t,\betap)}{\Zbar_{t,\betap}}(t,\betap)d\beta'.\\
&\ep=|z|^2-1,\quad \mu=\ep\circ k^{-1},\quad \delta=(I-H)\ep,\quad \chi=\delta\circ k^{-1},\quad \eta=\zeta_\alpha-i\zeta.\\
&u=z_t\circ k^{-1},\quad w=z_{tt}\circ k^{-1},\quad v=\delta_{t}\circ k^{-1}.
\bibliographystyle{plain}
\bibliography{twobodybib}
\bigskip
\centerline{\scshape Lydia Bieri, Shuang Miao, Sohrab Shahshahani, Sijue Wu}
\medskip
% please put the address of the first author
\centerline{Department of Mathematics, The University of Michigan}
\centerline{2074 East Hall, 530 Church Street
Ann Arbor, MI 48109-1043, U.S.A.}
\centerline{\email{[email protected], [email protected], [email protected], [email protected]}}
\end{document}
|
1511.00769
|
ufca]J. E. G. Silva
pici]R. V. Maluf
pici]C. A. S. Almeida
[ufca]Universidade Federal do Cariri (UFCA), Instituto de formação de professores, Rua Olegário Emídio de Araújo, Brejo Santo-Ce ,63.260.000-Brazil
[pici]Universidade Federal do Ceará (UFC), Departamento de Física, Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil
Local Lorentz Violating Gravity Finsler Gravity Randers spacetime
We investigate the properties of a real scalar field in the Finslerian Randers spacetime, where
the local Lorentz violation is driven by a geometrical background vector. We propose a dynamics for the scalar field by a minimal coupling of the scalar field and the Finsler metric. The coupling is intrinsically defined on the Randers spacetime, and it leads to a non-canonical kinetic term for the scalar field. The nonlinear dynamics can be split into a linear and nonlinear regimes, which depend perturbatively on the even and odd powers of the Lorentz-violating parameter, respectively. We analyze the plane-waves solutions and the modified dispersion relations, and it turns out that the spectrum is free of tachyons up to second-order.
§ INTRODUCTION
Despite the current lack of a complete theory of quantum gravity, several candidate theories assume that some symmetries present at low energy regimes might no longer be valid at Planck scale. For instance, the string theory <cit.>, spacetime noncommutativity <cit.>, Horava-Lifshitz gravity <cit.>, loop quantum gravity (LQG) <cit.>, Doubly Special Relativity (DSR) <cit.> and the Very Special Relativity (VSR) <cit.> admit the possibility of absence of Lorentz symmetry for the spacetime. The violation of Lorentz symmetry may be the result of a spontaneous breaking of tensor fields which acquire nonvanishing vacuum expectation values <cit.> or the condensation of a ghost (scalar) field leading to modifications of the dispersion relations <cit.>. A field theoretical framework to test the Lorentz symmetry is provided by the
Standard Model Extension (SME) <cit.>. For a comprehensive review of tests on Lorentz and CPT violation, we indicate the Ref.<cit.>.
The violation of the local Lorentz symmetry can be extended to curved spacetimes by means of the so-called Finsler geometry <cit.> .
In this anisotropic geometry, the intervals are evaluated by a non-quadratic function, called the Finsler function <cit.>. The lack of quadratic restriction provides modified dispersion relation for the fields, a hallmark of
Lorentz violation <cit.>. Applications of Finsler geometry can also be found in optics <cit.> and condensed matter physics <cit.>.
One of the most important Finsler spacetimes is the Randers spacetime where the anisotropy is driven by a background vector field $a_\mu$ which changes the length of intervals of the spacetime <cit.>. The cosmological and astrophysical
effects of the Randers spacetime were analyzed in Refs. <cit.>. In the context of the SME, the Randers spacetime arises as a kind of bipartite-Finsler space, in the classical point particle Lagrangian for the CPT-Odd fermionic sector <cit.>. Other SME-based Finsler spaces can be found in Refs. <cit.>.
In this work, we propose a minimal coupling of a real scalar field and the Finsler metric in the Randers spacetime. Unlike the
tangent bundle theories <cit.>, whose dynamics lies on $TTM$, we propose a position dependent field and Lagrangian. Our dynamics also differs from the osculating method, where the direction-dependent is worked out as a constraint <cit.>. For a Finslerian action, we employ an extension of the known as the Shen functional, where the square of the components of the gradient of the field is evaluated with the Finsler metric. As a Finsler volume, we choose the Busemann-Hausdorff volume which provides an anisotropic factor, such as in Bogoslovsky space <cit.>.
The work is organized as follows. In section <ref> we review the basic definitions and properties of the Randers spacetime. In section <ref> we propose the Finsler action, obtain the Finslerian equation of motion and
analyze the important regimes. The modified dispersion relation and stability are studied in section <ref> and section <ref>, for linear and nonlinear regimes, respectively.
Final remarks, conclusions and perspectives are outlined in section <ref>.
§ RANDERS SPACETIME
In Randers spacetime, given a four-velocity $\dot{x}^{\mu}=dx^{\mu}/dt$, the infinitesimal interval of a worldline is defined by $ds=F_R(x,\dot{x})dt$ where <cit.>
\begin{eqnarray}
\label{randersfunction}
F_{R}(x,\dot{x}) & := & \alpha(x,\dot{x})+\beta(x,\dot{x})\nonumber\\
& = & \sqrt{-g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}} + \zeta a_{\mu}(x)\dot{x}^{\mu},
\end{eqnarray}
and $\zeta $ is a real parameter controlling the local Lorentz-violation.
The norm of the background Randers covector $a_{\mu}$ is evaluated with the Lorentzian metric $g$, $a^{2}(x):=g_{\mu\nu}(x)a^{\mu}a^{\nu}$. In this work we adopt the mostly-plus metric convention $(-,+,+,+)$. The perturbative character of the Lorentz violation is encoded in the small value of the linear term, constrained to $0\leq \zeta a <1$.
We assume that $\zeta$ is a constant, bigger than Planck length and with a dimension of length $L$. The Randers background vector is assumed to have mass dimension one, as expected for a background vector field arising from reminiscent quantum gravity effects in four dimensions <cit.>.
The Randers function can be written as $F_{R}=\sqrt{-g^{R}_{\mu\nu}(x,\dot{x})\dot{x}^\mu \dot{x}^\nu}$, where the
anisotropic timelike Randers metric $g^{R}_{\mu\nu}(x,\dot{x})$ is defined as <cit.>
\begin{eqnarray}
\label{hessianoffinslerfunction}
g^{R}_{\mu\nu}(x,\dot{x})&:=&-\frac{1}{2}\frac{\partial^{2}F_{R}^{2}(x,y)}{\partial \dot{x}^{\mu}\partial \dot{x}^{\nu}}\nonumber\\
&=& \frac{F_R}{\alpha}g_{\mu\nu} + \frac{\beta}{\alpha}\tilde{l}_{\mu}\tilde{l}_{\nu} - \zeta(\tilde{l}_{\mu}a_{\nu}+\tilde{l}_{\nu}a_{\mu}) - \zeta^{2}a_{\mu}a_{\nu},
\end{eqnarray}
where $\tilde{l}_{\mu}:=\partial\alpha / \partial\dot{x}^\mu = -g_{\mu\nu}(x)\dot{x}^{\nu}/\alpha$, such that $\tilde{l}_{\mu}\dot{x}^\mu = \alpha$.
We define the action for a free massive particle in the Randers-Finsler spacetime as <cit.>
\begin{equation}
\label{finsleraction}
S^{R}=-m\int_{I}dt F_{R}(x,\dot{x})=-m\int_{I} dt \sqrt{-g_{\mu\nu}^{R}(x,\dot{x})\dot{x}^{\mu}\dot{x}^{\nu}}.
\end{equation}
The free particle action (<ref>) is analogous to the Lagrangian of a charged particle in a Lorentzian spacetime with an electromagnetic background vector $a_\mu$.
The canonical momentum is $P_{\mu}^{R}=P_{\mu} - m\zeta a_{\mu},$
where $P_{\mu}:=mg_{\mu\nu}(x)U^{\nu}$ is the Lorentzian conjugate momentum and $U^{\mu}:=\dot{x}^{\mu}/\alpha(\dot{x})$ is
the Lorentzian unitary 4-velocity.
The Finsler metric $g^{R}_{\mu\nu}(x,\dot{x})$ provides a nonlinear duality between the covariant $P^{R}_{\mu}$ and the contravariant $P^{R\mu}$, given by $P^{R}_{\mu}=g_{\mu\nu}^{R}(x,P^{R})P^{R\nu}$ <cit.>.
Thus, the contravariant components of the momentum are given by $P^{R\mu}=m\dot{x}^{\mu}/F(x,\dot{x})$ <cit.>.
The Finsler function for the covariant vector $P^{R}_\mu $ is
=\sqrt{-g^{*\mu\nu}(x)P^{R}_\mu P^{R}_\nu} + a^{*\mu}(x)P^{R}_\mu,$$
$$a^{*\mu}(x) = - \zeta\frac{a^{\mu}}{(1\mp \zeta^{2} a^{2})},\ \ g^{*\mu\nu}(x) = \frac{( 1\mp \zeta^{2} a^{2})g^{\mu\nu} - \zeta^{2} a^{\mu}a^{\nu}}{(1\mp\zeta^{2} a^{2})^{2}}.$$
The signs $-,+$ stand for a timelike and spacelike background vector $a^\mu$ <cit.>. The dual Finsler metric $g^{*F}(x,P_{\mu}^{R})$ is defined by <cit.>
$$g^{*F \mu\nu}(x,P_{\mu}^{R})=-\frac{1}{2}\frac{\partial^{2}F^{*2}(x,P{\mu}^{R})}{\partial P_{\mu}^{R}\partial P_{\mu}^{R}}
The Finsler metric also provides a deformation of the mass shell, given by <cit.>
\begin{equation}
\label{finslermassshell}
g^{*R \mu\nu}(x,P_{\mu}^{R})P^{R}_{\mu}P^{R}_{\nu}=-m^{2}.
\end{equation}
In Randers spacetime, the modified dispersion relation (MDR) is $g^{\mu\nu}P_{\mu}^{R}P^{R}_{\nu} - 2m\zeta (a\cdot P^{R}) = - (1\mp \zeta^{2} a^{2})m^{2}$,
which is an elliptical hyperboloid of two sheets <cit.>. Considering the momentum 4-vector $P^{R\mu}=(E,\vec{p})$, the dual mass-shell $g^{R}_{\mu\nu}(x,P^{R})P^{R\mu}P^{R\nu}=-m^2$ yields to $(g_{\mu\nu}+\zeta^2 a_\mu a_\nu)P^{R\mu}P^{R\nu}-2\zeta m a_\mu P^{R\mu}=-m^2$. In a flat spacetime $g_{\mu\nu}=\eta_{\mu\nu}$ and for a constant Randers background vector $a^\mu =(-a_0, \vec{a})$, the modified mass-shell is $(1-\zeta^2 a_{0}^2)E^2 + 2\zeta a_0 (m-\zeta (\vec{a}\cdot \vec{p}))E-|\vec{p}|^2 +\zeta (\vec{a}\cdot \vec{p})[2m-\zeta (\vec{a}\cdot \vec{p})]=-m^2$. For a timelike background vector $a^\mu =(-a_0, \vec{0})$, the modified mass-shell lies inside the Lorentz-invariant lightcone, similar to the dispersion relations analysed in the Ref.<cit.>. Further, the asymptote of the deformed mass-shell corresponds to the Randers lightcone. Then, though the particle reaches Lorentzian superluminal velocities, its speed does not exceed the speed of light in the Randers spacetime.
§ SCALAR FIELD DYNAMICS
After analyzing the dynamics of a point particle in the anisotropic Randers spacetime, let us now consider the real scalar field dynamics in this local Lorentz violating spacetime. We assume that the real scalar field is a function of the position $x$ only, i.e., $\Phi=\Phi(x)$.
Consider the action functional defined by a minimal coupling between the scalar field and the dual Finsler metric, namely
\begin{equation}
\label{shenaction2}
S_{\Phi}=-\frac{1}{2}\int d^{4}x \sqrt{-g(x)} \left(1\mp\zeta^{2}a^2(x)\right)^{\frac{5}{2}}\left[g^{F*\mu\nu}(x,d\Phi)\partial_{\mu}\Phi\partial_{\nu}\Phi + V(\Phi)\right],
\end{equation}
where $V(\Phi)=m^2 \Phi^2$ for the free scalar field and the signs $-,+$ stand for a timelike and spacelike background vector $a$ respectively. The anisotropic volume form $d^{4}x\sqrt{-g(x)}(1\mp\zeta^{2}a^2(x))^{5/2}$, is an extension of the Busemann-Haussdorf volume for the Randers spacetime <cit.>.
It is worthwhile to mention that the Finslerian action in Eq.(<ref>) bears some resemblance to the so-called k-essence models <cit.>.
The nonquadratic Lagrangian density defined by Eq.(<ref>) can be split into two terms
\begin{eqnarray}
\mathcal{L}_{\Phi}=\mathcal{L}^{b}_{\Phi}+\mathcal{L}^{NQ}_{\Phi},
\end{eqnarray}
where $\mathcal{L}^{b}_{\Phi}$ is the bilinear Lagrangian density constructed with the quadratic terms field derivatives $\partial\Phi$ and with the potential term, given by
\begin{eqnarray}
\label{bilinearlagrangian}
\mathcal{L}^{b}_{\Phi} & = &-\frac{1}{2}\Bigg\{\Big[(1\mp\zeta^{2}a^{2})g^{\mu\nu}(x)-2\zeta^{2}a^{\mu}(x)a^{\nu}(x)\Big]\partial_{\mu}\Phi\partial_{\nu}\Phi + \nonumber\\
& & + V(\Phi)(1\mp\zeta^2 a^2)^2\Bigg\}\sqrt{-(1\mp\zeta^2 a^2)g},
\end{eqnarray}
and $\mathcal{L}^{NQ}_{\Phi}$ corresponds to the nonquadratic terms in the Lagrangian $\mathcal{L}_{\Phi}$, whose expression is given by
\begin{equation}
\label{nonlinearlagrangian}
\mathcal{L}^{NQ}_{\Phi}=-\zeta (a\cdot \partial\Phi)\sqrt{[-(1\mp\zeta^{2}a^{2})(\partial\Phi)^{2}+\zeta^{2}(a\cdot\partial\Phi)^{2}]}\sqrt{-(1\mp\zeta^2 a^2)g},
\end{equation}
where, $(\partial\Phi)^2:=g^{\mu\nu}(x)\partial_{\mu}\Phi\partial_{\nu}\Phi$ and $a\cdot\partial\Phi:=a^{\mu}\partial_{\mu}\Phi$. Note that for a Lorentzian spacetime, i.e. $\zeta=0$, the nonquadratic Lagrangian density $\mathcal{L}^{NQ}_{\Phi}$
vanishes and the bilinear Lagrangian reduces to the Lorentz-invariant $\mathcal{L}_{\Phi}=-\frac{1}{2}\Big[g^{\mu\nu}(x)\partial_{\mu}\Phi\partial_{\nu}\Phi + 2 V(\Phi)\Big]\sqrt{-g}$.
The bilinear Lagrangian $\mathcal{L}_{\Phi}^{b}$ in (<ref>) can be expanded in powers of $\zeta$ as
\begin{eqnarray}
\mathcal{L}_{\Phi}^{b} & = & \Bigg\{-\frac{1}{2}\Big[(\partial\Phi)^{2}+V(\Phi)\Big]-\frac{\zeta^{2}}{2}\Bigg[\mp\frac{3}{2}a^2(\partial\Phi)^2 - 2(a\cdot \partial \Phi)^2 \mp \frac{5}{2}a^{2}V(\Phi)\Bigg]\nonumber\\
& & + \mathcal{O}(\zeta^4)\Bigg\}\sqrt{-g}.
\end{eqnarray}
Therefore, the Local Lorentz violating effects arise in second order in $\zeta$ parameter. The LV terms have a similar form of those proposed in the context of the Standard Model Extension (SME) <cit.>. Indeed, defining the symmetric and dimensionless tensor $k^{\mu\nu}_{\Phi\Phi}$ as $k^{\mu\nu}_{\Phi\Phi}:=\zeta^2 a^{\mu}a^{\nu}$, the bilinear term can be regarded as a CPT even Lorentz-violating lagrangian of the Higgs sector in the minimal SME <cit.>.
Extremizing the Finslerian action (<ref>), the Euler-Lagrange equation yields the equation of motion for the field $\Phi$
\begin{equation}
\frac{1}{(1\mp\zeta^{2}a^{2})^{\frac{5}{2}}\sqrt{-g}}\partial_{\mu}\Big[(1\mp\zeta^{2}a^{2})^{\frac{5}{2}}\sqrt{-g}g^{F*\mu\nu}(x,d\Phi)\partial_{\nu}\Phi\Big]=\frac{\partial V(\Phi)}{\partial \Phi}.
\label{shenkleingordonequation}
\end{equation}
The Eq.(<ref>) is a nonlinear Finslerian extension of the Klein-Gordon equation. By defining the nonlinear D'Alembertian operator
\begin{equation}
\Box^{F}\Phi:=\frac{1}{(1\mp\zeta^{2}a^{2})^{\frac{5}{2}}\sqrt{-g}}\partial_{\mu}\Big[(1\mp\zeta^{2}a^{2})^{\frac{5}{2}}\sqrt{-g}g^{F*\mu\nu}(x,d\Phi)\partial_{\nu}\Phi\Big],
\end{equation}
the nonlinear Finslerian Klein-Gordon equation can be rewritten as
\begin{equation}
\Box^{F}\Phi=\frac{\partial V(\Phi)}{\partial \Phi}.
\end{equation}
The nonlinear Finslerian D'Alembertian operator $\Box^{F}\Phi$ is an extension of the so-called Shen Laplacian <cit.>, defined on Riemann-Finsler spaces, to Pseudo-Finsler Spacetimes.
The free nonlinear Finslerian Klein-Gordon equation, where $V(\Phi)=m^2 \Phi^2$, can be rewritten as
\begin{eqnarray}
\label{nonlinearfinsleriankleingordonequation}
g^{F*\mu\nu}(x,d\Phi)\partial_{\mu}\partial_{\nu}\Phi + \partial_{\mu}\Bigg[\log \Big[(1\mp\zeta^2 a^2)^{\frac{5}{2}}\sqrt{-g}\Big]g^{F*\mu\nu}(x,d\Phi)\Bigg]\partial_{\nu}\Phi = m^{2}\Phi.
\end{eqnarray}
For a flat spacetime with a constant background field $a$, the anisotropic volume factor $(1\mp\zeta^2 a^2)^{5/2}$ can be absorbed in a change of coordinates and the Klein-Gordon equation (<ref>) yields to
\begin{equation}
g^{F*\mu\nu}(d\Phi)\partial_{\mu}\partial_{\nu}\Phi + \partial_{\mu}(g^{F*\mu\nu}(d\Phi))\partial_{\nu}\Phi = m^{2}\Phi.
\end{equation}
Consider the ray approximation, where the wave length $\lambda$ is much smaller then the geometrical characteristic length $L$, i.e., $\lambda<<L$ <cit.>.
A ray ansatz for the scalar field has the form <cit.>
\begin{equation}
\label{rayansatz}
\Phi(x):=Re\Big[(a_1 + \epsilon a_2 + \cdots)e^{-i\frac{\psi(x)}{\epsilon}}\Big],
\end{equation}
where $\epsilon:=\lambda/L$ and the phase function $\psi(x)$ is called the eikonal. The differential of the field $\Phi$ (<ref>) is given by
\begin{equation}
d\Phi=\frac{1}{\epsilon}A(x)k_{1} + \theta_{1},
\end{equation}
where, $A(x):=Re\Big[-i(a_1 + \epsilon a_2 + \cdots)e^{-i\frac{\psi(x)}{\epsilon}}\Big]$, $k_{1} :=k_{\mu}dx^{\mu}$,
$k_{\mu}:=\partial_{\mu}\psi$ and $\theta_{1}:=\theta_{\mu}dx^{\mu}$, where $\theta_{\mu}:=Re\Big[(\partial_{\mu}(a_1 + \epsilon a_2 + \cdots)e^{-i\frac{\psi(x)}{\epsilon}})\Big]$. At leading order, $g^{F*\mu\nu}(x,d\Phi)=g^{F*\mu\nu}(x,A k)=g^{F*\mu\nu}(x,k)$.
Therefore, the nonlinear Finslerian Klein-Gordon equation (<ref>) yields the modified dispersion relation
\begin{equation}
\label{leadingordermdr}
\end{equation}
Then, at leading order, the wave 1-form $k$ modified dispersion relation in (<ref>) satisfies the point particle
modified mass shell (<ref>).
§ LINEAR DYNAMICS SECTOR
The bilinear Lagrangian density $\mathcal{L}_{\Phi}^{b}$ in (<ref>) already exhibits interesting anisotropic modifications in its own. The Euler-Lagrange equations from the bilinear Lagrangian $\mathcal{L}_{\Phi}^{b}$ yields the equation
\begin{eqnarray}
\label{linearkleingordonequation}
&& \frac{\partial_{\mu}\partial^{\mu}\Phi}{(1\mp\zeta^{2}a^{2})}- \frac{2\zeta^{2}}{(1\mp\zeta^{2}a^{2})^{2}}\Big[a^{\mu}a^{\nu}\partial_{\mu}\partial_{\nu}\Phi+ (a\cdot\partial\Phi)\partial_{\mu}a^{\mu}+ a^{\mu}\partial_{\mu}a^{\nu}\partial_{\nu}\Phi \nonumber\\ && + a(\partial_{\mu}a)\partial^{\mu}\Phi - \frac{2\zeta^2 a^{\mu}\partial_{\mu}a(a\cdot\partial\Phi)}{(1\mp\zeta^2 a^2)}\Big]
+ \frac{\partial_{\mu}\Big[\log{(\sqrt{-g}(1\mp\zeta^2 a^2)^{\frac{5}{2}})}\Big]}{(1+\zeta^2 a^2)^2}\times \nonumber\\ &&
\Big[(1\mp\zeta^{2}a^{2})\partial^{\mu}\Phi - 2\zeta^{2}(a\cdot\partial\Phi)a^{\mu}\Big]
= \frac{\partial V(\Phi)}{\partial\Phi}.
\end{eqnarray}
In a flat spacetime $(g^{\mu\nu}=\eta^{\mu\nu})$ for a constant background vector $a^{\mu}$, after a rescaling of the coordinates and field, the linear anisotropic Klein-Gordon equation yields to
\begin{equation}
\label{rescaledlinearfinslerkleingordonequation}
\partial_{\mu}\partial^{\mu}\Phi-\zeta^2 (a^{\mu}a^{\nu}\partial_{\mu}\partial_{\nu}\Phi)=\frac{\partial V(\Phi)}{\partial\Phi}.
\end{equation}
Considering a free massive scalar field, where $V(\Phi)=m^2 \Phi^2$, the Fourier transform ansatz for $\Phi$
\begin{equation}
\Phi(x):=\int\frac{d^{4}k}{(2\pi)^{4}}\tilde{\Phi}(k)e^{-i k\cdot x},
\end{equation}
in the Klein-Gordon equation Eq.(<ref>) yields the modified dispersion relation
\begin{equation}
\label{scalarfieldmodifieddispersionrelation}
k^{2}-\zeta^2 (a\cdot k)^2=-m^{2},
\end{equation}
where $k^2 :=\eta^{\mu\nu}k_{\mu}k_{\nu}$, $a^2 := \eta^{\mu\nu}a_{\mu}a_{\nu}$ and $a\cdot k:=\eta^{\mu\nu}a_{\mu}k_{\nu}$.
The modified dispersion relation (MDR) in (<ref>) has a form similar to the modified particle mass shell. The MDR in the linear regime (<ref>) resembles the MDR for the graviton in the so-called Bumblebee model <cit.>.
For $k_{\mu}=(k_0,\vec{k})$, the modified dispersion relation (<ref>) takes the form
\begin{equation}
\label{modifieddispersionrelationlinearregime}
(1+\zeta^2 a^2_0)k_0^{2} - 2\zeta^{2}a_{0}k_0(\vec{a}\cdot\vec{k})
- \left[|\vec{k}|^2 + m^{2} - \zeta^2 (\vec{a}\cdot\vec{k})^2\right] = 0.
\end{equation}
From (<ref>), the relation between the frequency and wave vector is given by
\begin{equation}
\label{frequencyfunction}
k_0(\vec{k}) =\frac{1}{(1+\zeta^2 a^2_0)}\left[\zeta^2 a_{0}(\vec{a}\cdot\vec{k})\pm\sqrt{(|\vec{k}|^2 + m^2)(1+\zeta^2 a_0^2)-\zeta^2 (\vec{a}\cdot\vec{k})^2}\right].
\end{equation}
One should notice that for an arbitrary configuration of the background vector, the dispersion relations assume the form $k_{0}=E_{\pm}$ with $|E_{+}|\neq|E_{-}|$. This difference between the absolute values of the positive and negative energy states impairs the usual quantum description for particles and antiparticles, leading to problems with respect to the locality of the quantum theory, as discussed in <cit.>. Nevertheless, it is interesting to open up the discussion of the spectral consistency of the model for some particular configurations of $a^\mu$.
For $\zeta=0$ we recover the Lorentz-invariant relation $k_{0}=\pm\sqrt{|\vec{k}|^{2}+m^{2}}$. Taking now a timelike background vector $a_{\mu}=(a_0,\vec{0})$, the corresponding dispersion relation is
\begin{equation}
\end{equation}
which is like $k_{0}=\pm E$ and, the interpretation of the negative-energy states can be consistently carry out. The group velocity is
\begin{equation}
\end{equation}
which is smaller than 1, assuring causality for this mode.
For a spacelike vector, viz., $a_{\mu}=(0,\vec{a})$, the modified dispersion relation can be written as
\begin{equation}
k_0(\vec{k}) =\pm\sqrt{|\vec{k}|^2\left(1-\zeta^{2}|\vec{a}|^{2}\cos^{2}\theta\right) + m^2},
\end{equation}
where $\theta$ is the angle between $\vec{a}$ and $\vec{k}$. Again we find a physically acceptable dispersion relation, related with the group velocity
\begin{equation}
v_{g}= \frac{|\vec{k}|(1-\zeta^{2}|\vec{a}|^{2}\cos^{2}\theta)}{\sqrt{m^{2}+|\vec{k}|^{2}(1-\zeta^{2}|\vec{a}|^{2}\cos^{2}\theta)}},
\end{equation}
which becomes smaller than 1 for $\zeta^{2}|\vec{a}|^{2}<1$. So, no superluminal signals are created for both configurations.
Finally, we conclude that a lightlike Randers background vector $a^{\mu}$ is inconsistent with the usual quantum interpretation of the negative-energy states as antiparticles. The above results are in accordance with previous studies in the literature <cit.>.
§ NONLINEAR REGIME
Now let us consider the effects of both linear and nonlinear regimes by expanding the equation of motion (<ref>) up to second-order in $\zeta$ in the flat spacetime with constant $a^{\mu}$.
The resulting equation is given by
\begin{eqnarray}
\label{flatkleingordonequationuptosecondorder}
&&\partial_{\mu}\partial^{\mu}\Phi + \frac{\zeta}{\sqrt{-(\partial\Phi)^2}}\Bigg\{(a\cdot\partial\Phi)\partial_{\mu}\partial^{\mu}\Phi +2a^{\mu}(\partial_{\mu}\partial_{\nu}\Phi)\partial^{\nu}\Phi
\nonumber\\ && + \frac{(a\cdot\partial\Phi)\partial^{\mu}\Phi\eta^{\rho\sigma}(\partial_{\mu}\partial_{\rho}\Phi)\partial_{\sigma}\Phi}{(-\partial\Phi)^2}\Bigg\} -\zeta^2 (a^{\mu}a^{\nu}\partial_{\mu}\partial_{\nu}\Phi)=m^2 \Phi.
\end{eqnarray}
Considering a single plane wave solution of form $\Phi(x):=Re(e^{-i k\cdot x})
$, the Eq.(<ref>) has the modified quartic dispersion relation
\begin{equation}
k^4+2\zeta^2 k^2 (a\cdot k)^2 + 2m^2 k^2 - 2\zeta^2 m^2 (a\cdot k)^2 + \zeta^4 (a\cdot k)^4=0.
\end{equation}
Quartic dispersion relation can also be found in the nonminimal SME <cit.>.
For a timelike Randers vector $a_{\mu}=(a_0,\vec{0})$, we obtain
\begin{equation}
\begin{split}
&(1-\zeta^2 a_0^2)^{2}k_{0}^4 - 2\Big[(|\vec{k}|^2 + m^2) - \zeta^2 a_0^2 |\vec{k}|^2\Big]k_0^2 +(|\vec{k}|^2 + m^2)^2=0.
\end{split}
\label{mdrupsecondordertimelike}
\end{equation}
Once again there are two positive $(k_{0;1}^{+},k_{0;2}^{+})$ and two negatives $(k_{0;1}^{-},k_{0;2}^{-})$ roots, where
\begin{equation*}
k_{0;1,2}^{+}= \sqrt{\frac{(1-\zeta^2 a_0^2)|\vec{k}|^2 + m^2 \pm \zeta a_0 m\sqrt{2(1-\zeta^2 a_0^2) |\vec{k}|^2 + (2-\zeta^2 a_0^2)m^2}}{(1-\zeta^2 a_0^2)^{2}}},
\end{equation*}
\begin{equation}
\label{nonlinearfrequency}
k_{0;1,2}^{-}=- \sqrt{\frac{(1-\zeta^2 a_0^2)|\vec{k}|^2 + m^2 \pm \zeta a_0 m\sqrt{2(1-\zeta^2 a_0^2) |\vec{k}|^2 + (2-\zeta^2 a_0^2)m^2}}{(1-\zeta^2 a_0^2)^{2}}}.
\end{equation}
The behaviour of the frequencies $k_{0;1,2}^{+}$ is shown in the Figure <ref> for $m=1$ and $\zeta a_0 =0.5$. $k_{0;1}^{+}$ (dashed line) and $k_{0;2}^{+}$ (dotted line) are positive what leads to the positivity of the energy.
The absence of complex frequencies provides stability to the states of the scalar field, which means that no exponential decreasing or increasing factor due to the imaginary part appears in the plane-wave, as discussed in details in the Ref.<cit.>.
The perturbed mass-shells lie inside the Lorentz-invariant lightcone (thick line). The asymptotes represent the deformed Randers lightcone. Further, the difference of the frequencies to the usual frequency (thin line) is proportional to the product $\zeta a_0$ at leading order.
The group velocity has the form
\begin{equation}
\label{groupvelocitiesupsecondorder}
v_{g;1,2}^{+} = \frac{(1-\zeta^2 a_0^2)|\vec{k}|\left[1\pm\zeta a_0 m/\sqrt{(1-\zeta^2 a_0^2)(m^{2}+2|\vec{k}|^2) + m^2}\right]}{\sqrt{(1-\zeta^2 a_0^2 )^{2}|\vec{k}|^2 + m^2 \pm \zeta a_0 m \sqrt{(1-\zeta^2 a_0^2)(m^{2}+2|\vec{k}|^2) +m^2}}},
\end{equation}
\begin{equation}
v_{g;1,2}^{-}=-\frac{(1-\zeta^2 a_0^2)|\vec{k}|\left[1\pm\zeta a_0 m/\sqrt{(1-\zeta^2 a_0^2)(m^{2}+2|\vec{k}|^2) + m^2}\right]}{\sqrt{(1-\zeta^2 a_0^2 )^{2}|\vec{k}|^2 + m^2 \pm \zeta a_0 m \sqrt{(1-\zeta^2 a_0^2)(m^{2}+2|\vec{k}|^2) +m^2}}}.
\end{equation}
[Positive frequencies as functions of the 3-momentum $\vec{k}$, the perturbed second-order Klein-Gordon equation and a timelike Randers vector. Both the biggest frequency (dashed line) and the smallest frequency (dotted line) behave as the Lorentz-invariant limit (thin line).]
[Group velocity for the perturbed second-order Klein-Gordon equation and a timelike Randers vector. The group velocities for the biggest frequency (dashed line) and the smallest frequency (dotted line) have superluminal behaviour above a $\zeta a_0$-dependent energy scale Lorentz-invariant one.]
Frequencies and group velocity
We plotted the group velocities $v_{g;1,2}^{+}$ in Fig.<ref>. We set $m=1$
and $\zeta a_0=0.5$ (dotted line) and $\zeta a_0=0.7$ (dashed line). The group velocities exhibit superluminal behaviour above an energy scale, in a same way as analysed by Kostelecky and Lehnert in the fermionic sector of the SME <cit.>.
§ CONCLUSIONS AND PERSPECTIVES
In this work, we proposed a dynamics for a scalar field intrinsically defined on the local Lorentz-violating Finsler-Randers spacetime.
The action functional proposed is defined through a minimal coupling between the scalar field and the Finsler-Randers metric. We also assume that the anisotropy deforms the volume by means of an extension of the so-called Busemann-Hausdorff Finslerian volume.
The resulting action exhibits a non-canonical kinetic term, as in k-essence models. By expanding the Finslerian action in powers of the background vector,
we obtain Lorentz violating terms similar to those of the Standard Model Extension (SME). The Finslerian equation of motion is an extension of the Klein-Gordon equation by using the so-called Shen D'Alembertian.
The analysis of the perturbed modified dispersion relations (MDR) for the free field revealed that tachyonic modes are absent in the bilinear sector, regardless of the causal nature of the background Randers vector. For the nonlinear regime perturbed to second-order, the fourth-order MDR has a positive energy (stable) spectrum whose group velocities exhibit superluminal effects above an energy scale that depends on $\zeta a_0$. The UV causality issues are similar to those
studies in the fermionic sector of the SME <cit.>. However, since the Randers vector also deforms the lightcone, the particles do not exceed the deformed Randers speed of light.
As perspectives, we point out the analysis of the effects of this nonlinear dynamics in cosmological scenarios. A relevant extension of this work is the analysis of the characteristic surface which could resolve some causality issues at the lightcone. For a complete analysis of the modified dispersion relation, the nonlinear Klein-Gordon equation demands
the use of nonlocal and fractional operators that we leave as a perspective. The quantization of this classical theory and the analysis of some process in order to find upper bounds to the background Randers vector are in progress.
§ ACKNOWLEDGEMENTS
This work was partially supported by the Brazilian agencies Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) (grant no. 99999.
006822/2014-02) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (grant numbers 305766/2012-0 and 305678/2015-9). J.E.G. Silva acknowledges the Indiana University Center for Spacetime Symmetries (IUCSS) for the kind hospitality.
KS V. A. Kostelecký and S. Samuel, Phys. Rev. D 39,
683 (1989); Phys. Rev. Lett. 63, 224 (1989); Phys. Rev. D 40, 1886 (1989).
E. Witten, Nucl. Phys. B 268, 253 (1986). S. M. Carroll, J. A. Harvey, V. A. Kostelecký, C. D. Lane, and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001).
horava P. Hořava, Phys. Rev. D 79, 084008 (2009);
G. Calcagni, JHEP 0909, 112 (2009); M. Visser, Phys. Rev. D 80, 025011 (2009).
J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia, Phys. Rev. Lett. 84, 2318 (2000);
Phys. Rev. D 65, 103509 (2002).
J. Magueijo and L. Smolin,
Phys. Rev. Lett. 88, 190403 (2002).
A. G. Cohen and S. L. Glashow,
Phys. Rev. Lett. 97, 021601 (2006).
espontaneous lorentz violation string V. A. Kostelecký and S. Samuel, Phys. Rev. D 42, 1289 (1990); V. A. Kostelecký and R. Potting, Nucl. Phys. B 359,
545 (1991).
N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Mukohyama,
JHEP 0405, 074 (2004). S. Mukohyama,
JCAP 0610, 011 (2006).
D. Colladay and V. A. Kostelecký,
Phys. Rev. D 55, 6760 (1997);
Phys. Rev. D 58, 116002 (1998).
V. A. Kostelecký,
Phys. Rev. D 69, 105009 (2004).
rmp V. A. Kostelecký and N. Russell, Rev. Mod. Phys. 83, 11 (2011).
V. A. Kostelecký,
Phys. Lett. B 701, 137 (2011).
G. Amelino-Camelia, L. Barcaroli, G. Gubitosi, S. Liberati and N. Loret,
Phys. Rev. D 90, no. 12, 125030 (2014).
G. W. Gibbons, J. Gomis and C. N. Pope,
Phys. Rev. D 76, 081701 (2007).
D. Bao, S. Chern, Z. Shen, An introduction to Riemann-Finsler geometry,
Springer, New York, 1991.
Z. Shen, Lectures on Finsler geometry, World Scientific, Singapore, 2001.
V. A. Kostelecký and N. Russell,
Phys. Lett. B 693, 443 (2010).
F. Girelli, S. Liberati and L. Sindoni,
Phys. Rev. D 75, 064015 (2007).
J. Skakala and M. Visser,
Int. J. Mod. Phys. D 19, 1119 (2010).
A. Joets and R. Ribotta, Opt. Commun., 107, (1994).
M. Cvetic and G. W. Gibbons,
Annals Phys. 327, 2617 (2012).
G. Randers,
Phys. Rev. 59, 195 (1941).
R. Miron, D. Hrimiuc, H. Shimada, S. Sabau, The geometry of Hamilton and Lagrange spaces, (Kluwer
Academic Publs., Dordrecht, 2001).
Z. Chang and X. Li,
Phys. Lett. B 668, 453 (2008); Phys. Lett. B 676, 173 (2009).
Z. Chang and X. Li,
Phys. Lett. B 663, 103 (2008).
V. A. Kostelecký, N. Russell and R. Tso,
Phys. Lett. B 716, 470 (2012).
J. E. G. Silva and C. A. S. Almeida,
Phys. Lett. B 731, 74 (2014).
D. Colladay and P. McDonald,
Phys. Rev. D 85, 044042 (2012).
N. Russell,
Phys. Rev. D 91, no. 4, 045008 (2015).
M. Schreck,
Phys. Rev. D 91, no. 10, 105001 (2015).
G. S. Asanov,
Rept. Math. Phys. 13, 13 (1978).
C. Pfeifer and M. N. R. Wohlfarth,
Phys. Rev. D 84, 044039 (2011); Phys. Rev. D 85, 064009 (2012).
P. C. Stavrinos, A. P. Kouretsis and M. Stathakopoulos,
Gen. Rel. Grav. 40, 1403 (2008). P. C. Stavrinos,
Gen. Rel. Grav. 44, 3029 (2012).
G. Y. Bogoslovsky and H. F. Goenner,
Gen. Rel. Grav. 31, 1565 (1999).
S. I. Vacaru,
Class. Quant. Grav. 28, 215001 (2011).
JHEP 9809, 011 (1998).
V. A. Kostelecký and R. Lehnert,
Phys. Rev. D 63, 065008 (2001).
H. Busemann,
Ann. of Math., 48, 234 (1947).
Comment. Math. Helvet. 24, 156 (1950).
C. Armendariz-Picon, T. Damour and V. F. Mukhanov, Phys. Lett. B 458, 209 (1999).
R. Bluhm and V. A. Kostelecký, Phys. Rev. D
71, 065008 (2005).
maluf R. V. Maluf, C. A. S. Almeida, R. Casana and M. M. Ferreira, Jr., Phys. Rev. D 90, 025007 (2014); R. V. Maluf, J. E. G. Silva and C. A. S. Almeida,
Phys. Lett. B 749, 304 (2015).
greenbergO. W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002).
pereira B. Pereira-Dias, C. A. Hernaski, and J. A. Helayël-Neto, Phys. Rev D 83, 084011 (2011).
V. A. Kostelecký and M. Mewes,
Phys. Rev. D 80, 015020 (2009).
T. Padmanabhan,
Cambridge, UK: Cambridge Univ. Pr. (2010).
|
1511.01050
|
Three science and engineering
problems of recent interests—index coding, locally recoverable distributed storage, and guessing games
on graphs—are discussed and the connection between their optimal solutions is elucidated.
By generalizing recent results by Shanmugam and Dimakis and by Mazumdar on the complementarity between the optimal
broadcast rate of an index coding problem on a directed graph
and the normalized rate of a locally recoverable distributed storage problem
on the same graph, it is shown that
the capacity region and the optimal rate region of these two problems are complementary.
The main ingredients in establishing this result are the notion of confusion graph introduced by Alon et al. (2008),
the vertex transitivity of a confusion graph, the characterization of the index coding capacity region
via the fractional chromatic number of confusion graphs, and the characterization of the optimal rate region of the locally recoverable distributed storage
via the independence number of confusion graphs.
As the third and final facet of the complementarity,
guessing games on graphs by Riis are discussed as special cases of the locally recoverable distributed storage problem,
and it is shown that the winning probability of the optimal strategy for a guessing game and
the ratio between the winning probabilities of the optimal strategy and a random guess can be characterized, respectively,
by the capacity region for index coding and the optimal rate region for distributed storage.
§ INTRODUCTION
This paper discusses index coding, locally recoverable distributed storage, and guessing games on directed graphs,
with the goal of elucidating
the relationship between the optimal solutions of these three problems.
Index coding.
A server has multiple messages and wishes to send them to their corresponding receivers.
Each receiver has some side information about a subset of messages (not including the desired message).
The goal is to exploit the side information of the receivers to minimize the number of required transmissions.
This problem arises in many contexts such as intra-cell communication, content broadcasting and coded caching <cit.>,
as well as satellite communication as originally introduced by Birk and Kol <cit.>.
It is easy to see that index coding is a special case of network coding which is a key open problem in network information theory.
Effros, El Rouayheb, and Langberg <cit.> showed that
for any network coding problem, there exists an equivalent index coding problem.
This equivalence makes index coding even more intriguing.
So far the index coding problem has been attacked using tools from various disciplines such as graph theory <cit.>, algebra <cit.>, interference alignment <cit.>, source coding <cit.>, and random coding <cit.>.
Locally recoverable distributed storage. A set of servers collectively store data such that
if a server fails, its contents can be efficiently reconstructed from the contents of the other servers (among many others, see <cit.>).
The goal is to design a distributed storage code that maximizes the amount of data that can be stored while satisfying the single-failure recovery constraint.
In the mentioned references, it is assumed that each server is able to connect to all the other servers of the system.
However, in a real system, due to some constraints, a server may only have access to a subset of the other servers.
In <cit.>, Mazumdar took this point into consideration by assuming that the topology of the system is given by a directed graph and established a duality between the normalized rate of such a distributed storage system and the broadcast rate of the index coding problem.
In particular, Mazumdar showed that for any directed graph on $n$ nodes, the broadcast rate of the index coding problem and the normalized rate of the locally recoverable distributed storage problem sum up to $n$.
In an independent concurrent work, Shanmugam and Dimakis <cit.> established a similar result for vector linear codes and showed that the dual code of a linear index code is a linear code for the locally recoverable distributed storage problem and vice versa.
Guessing games on graphs.
Consider a cooperative game among multiple players, in which a value (hat color) is assigned to each player independently.
The value of each player is not revealed to her, but she knows the values assigned to a subset of the other players she can observe.
The players should simultaneously guess their own values and they win if all of them guess correctly.
After the values are assigned, the players are not allowed to communicate, but they can come up with a strategy beforehand to maximize their probability of winning. What is the best strategy and how much improvement over random guesses
can we make by employing the optimal strategy?
This interesting puzzle, which apparently seems to be in the realm of recreational mathematics,
was introduced originally by Riss <cit.> for studying multiple unicast network coding problems.
Its relationship to index coding was observed firstly by Yi, Sun, Jafar, and Gesbert <cit.>.
In this paper, we define the $\lambdav$-directed capacity $C(\lambdav)$ of the index coding problem and the $\lambdav$-directed optimal rate $R(\lambdav)$ of the distributed storage problem and rewrite the capacity region of the index coding problem and the optimal rate region of the distributed storage problem in terms of the set of $\lambdav$-directed capacities, and the set of $\lambdav$-directed optimal rates respectively.
Denoting the all-ones vector by $\ov$, $C(\ov)$ is equal to the reciprocal of the broadcast rate of the index coding problem and $R(\ov)$ equals the reciprocal of the normalized rate of the distributed storage.
The complementarity between the capacity region and the optimal rate region of the two problems is demonstrated by generalizing the complementarity result by Mazumdar <cit.>, and by Shanmugam and Dimakis <cit.>, to a complementarity relationship between $C(\lambdav)$ and $R(\lambdav)$ for any nonnegative real $n$-tuple $\lambdav$.
The same technique is used to make a complementary connection between the sum-capacity of the index coding problem and the optimal sum-rate of the distributed storage.
Next, we will discuss that the optimal guessing number of guessing games on graphs is inversely related to the optimal sum-rate of the distributed storage problem which together with the complementary connection between the sum-capacity and the optimal sum-rate makes the inverse relationship between the optimal complementary guessing number and the sum-capacity of the index coding problem clear (see Fig. <ref>).
As a result of the clear connection between the optimal solutions of these three problems, the known results for one of these problems can be easily extended to the other problems. We will conclude the paper by one such example.
ic[c]Index coding
ds[c]Distributed storage
gg[c]Optimal guessing number
gg2[c]guessing number
ggc[c]Optimal complementary
The relationship between index coding, distributed storage, optimal guessing number, and optimal complementary guessing number.
The rest of the paper is organized as follows.
We first review some mathematical preliminaries in Section <ref>.
In Section <ref>, formal definitions of the problems are presented.
In Section <ref>, we overview the characterization of the capacity region of the index coding problem, the optimal rate region of the distributed storage problem, and the guessing number of the guessing games via the notion of confusion graph.
Section <ref> is the main part of the paper in which the complementarity between the index coding problem and the distributed storage problem is presented and the guessing game on a graph is shown to be a special case of the distributed storage problem on the same graph.
Finally, in Section <ref>, we conclude by mentioning some results for the guessing games that due to the established connections, also hold for the index coding problem.
§ MATHEMATICAL PRELIMINARIES
Throughout the paper, $\ov$ denotes the $n \times 1$ vector of all ones.
The notations $V(\G)$ and $E(\G)$ mean the vertex set and the edge set of a graph $\G$ respectively.
For $v_1, v_2 \in V(\G)$, $v_1 \sim v_2$ denotes that there exists an edge between $v_1$ and $v_2$.
The set of positive integers that are less than or equal to $n$, $\{1,2,\ldots, n\}$, is denoted by $[n]$.
§.§ Graph Coloring
A (vertex) coloring of an undirected graph $\G$
is a mapping that assigns a color to each vertex
such that no two adjacent vertices share the same color.
The chromatic number $\chi(\G)$ is the minimum number of colors such that a coloring of the graph exists.
More generally, a $b$-fold coloring assigns a set of $b$ colors to each vertex
such that no two adjacent vertices share the same color.
The $b$-fold chromatic number $\chi^{(b)}(\G)$ is the minimum number of colors such that a $b$-fold coloring
The fractional chromatic number of the graph is defined as
\[
\chi_f(\G) = \lim_{b \rightarrow \infty} \frac{\chi^{(b)}(\G)}{b} = \inf_b \frac{\chi^{(b)}(\G)}{b},
\]
where the limit exists since $\chi^{(b)}(\G)$ is subadditive.
$\chi_f(\G) \le \chi(\G)$.
§.§ Vertex Transitive Graphs
An automorphism of a graph $\G$ is a
bijective function $\sigma: V(\G) \to V(\G)$, such that the pair of vertices $(\sigma(u),\sigma(v))$ form an edge iff (if and only if) the pair $(u,v)$ form an edge.
A graph $\G$ is said to be vertex transitive if for any two vertices $u$ and $v$ of $\G$, there exists some automorphism $\sigma: V(\G) \to V(\G)$ such that $\sigma(u) = v$.
The independence number of the undirected graph $\G$ denoted $\alpha(\G)$ is the cardinality of the largest independent set of the graph.
For any vertex transitive graph $\G$ we have
\[
\chi_f(\G) = |V(\G)|/\alpha(\G).
\]
§.§ Disjunctive Graph Product
Given two (undirected) graphs $\G_1$ and $\G_2$, the disjunctive product $\G = \G_1*\G_2$ produces a graph $\G$ on
the Cartesian product of the original vertex sets, i.e., $V(\G) = V(\G_1) \times V(\G_2)$ and the edge set is constructed from
the original edge sets according to the following rule.
$(u_1,u_2) \sim (v_1,v_2)$ iff
$u_1 \sim v_1$ or $u_2 \sim v_2$.
The fractional chromatic number of the disjunctive product is multiplicative.
\[
\chi_f(\G_1*\G_2) = \chi_f(\G_1)\chi_f(\G_2).
\]
§.§ Confusion Graphs
The notion of confusion graph is originally introduced by Alon, Hassidim, Lubetzky, Stav, and Weinstein <cit.>.
Consider a directed graph $G=(V,E)$ with $V = [n]$. Let $A_j = \{i \in V \suchthat (i,j) \in E\}$, $j \in [n]$ and let
$\tv = (t_1, \ldots, t_n)$ be a length-$n$ integer tuple.
Two binary $n$-tuples $x^n, z^n \in \prod_{i=1}^n \{0,1\}^{t_i}$ are said to be confusable
at node $j \in [n]$ if $x_j \ne z_j$ and $x_i = z_i$ for all $i \in A_j$.
We say $x^n$ and $z^n$ are confusable if they are confusable at some node $j$.
Given a directed graph $G$ and
a length-$n$ integer tuple $\tv = (t_1, \ldots, t_n)$, the confusion graph
$\G_{\tv}(G)$ is an undirected graph with $\prod_{i=1}^n 2^{t_i}$ vertices such that
every vertex corresponds to a binary tuple $x^n$ and
two vertices are connected iff the corresponding
binary tuples are confusable.
As an example, the confusion graph of the directed graph in Fig. <ref>(a) corresponding to $(t_1, t_2, t_3) = (1,1,1)$ is depicted in Fig. <ref>(b).
(a) A 3-node directed graph.
(b) The confusion graph corresponding to the integer tuple $(t_1, t_2, t_3) = (1,1,1)$.
Every confusion graph is vertex transitive.
The following lemma shows that given any directed graph $G$,
confusion graph corresponding to a larger integer tuple has a larger fractional chromatic number.
The proof is presented in Appendix <ref>.
Let $\sv = (s_1, \ldots, s_n)$ and $\tv = (t_1, \ldots, t_n)$ be two integer tuples such that $\sv \leq \tv$.
Then for any directed graph $G$ with $n$ vertices, we have
\begin{align}
\label{eq:sleqt}
\chi_f(\G_\sv(G)) \leq \chi_f(\G_\tv(G)).
\end{align}
§ PROBLEM DEFINITIONS
§.§ Index Coding
In the index coding problem, a sender wishes to communicate a tuple of $n$ messages, $x^n = (x_1, \ldots, x_n)$, $x_j \in \{0,1\}^{t_j}$, to their corresponding receivers using a shared noiseless channel.
Receiver $j \in [n]$ has prior knowledge of a subset $x(A_j) := (x_i \suchthat i \in A_j)$, $A_j \subseteq [n] \setminus \{j\}$, of the messages and wishes to recover $x_j$.
It is assumed that the sender is aware of $A_1, \ldots, A_n$.
The goal is to minimize the amount of information that should be broadcast from the sender to the receivers so that every receiver can recover its desired message.
Any instance of this problem, referred to collectively as the index coding problem,
is fully specified by the side information sets $A_1, \ldots, A_n$. Equivalently, it can be specified
by a side information graph $G$ with $n$ nodes, in which
a directed edge $i \to j$ represents that receiver $j$ has message $i$ as side information ($i \in A_j$).
For instance, Fig. <ref>(a) shows the directed graph representing the index coding problem with $A_1 = \{2,3\}$, $A_2 = \{1\}$, and $A_3=\{1,2\}$.
We often identify an index coding problem with its side information graph and simply write “index coding problem $G$.”
A $(t_1, \ldots, t_n, r)$ index code is defined by
* an encoder $\phi: \prod_{i=1}^n \{0,1\}^{t_i} \to \{0,1\}^r$ that maps $n$-tuple of messages $x^n$
to an $r$-bit index and
* $n$ decoders $\psi_j: \{0,1\}^r \times \prod_{k \in A_j} \{0,1\}^{t_k} \to \{0,1\}^{t_j}$ that maps the received index $\phi(x^n)$ and the side information $x(A_j)$ back to $x_j$ for $j \in [n]$.
Thus, for every $x^n \in \prod_{i=1}^n \{0,1\}^{t_i}$,
\[
\psi_j(\phi(x^n), x(A_j)) = x_j, \quad j \in [n].
\]
A rate tuple $(R_1,\ldots,R_n)$ is said to be achievable for the index coding problem $G$
if there exists a $(t_1, \ldots, t_n, r)$ index code such that
\[
R_j \leq \frac{t_j}{r}, \quad j \in [n].
\]
The capacity region $\Cr$
of the index coding problem is defined as the closure of the set of achievable rate tuples.
Let $\lambdav = (\lambda_1, \ldots, \lambda_n)$ be a non-negative real tuple.
Define the $\lambdav$-directed capacity $C(\lambdav)$ of the index coding problem $G$ as
\begin{align}
\label{eq:clambda}
C(\lambdav) = \max \{R \suchthat R \lambdav \in \Cr\}.
\end{align}
The capacity region can be written in terms of $\lambdav$-directed capacities.
\begin{equation}
\label{eq:lambdacapacity}
\Cr = \bigcup_{\lambdav \in \mathbb{R}_{\ge 0}^n \suchthat \sum_{i=1}^n \lambda_i = n} \left\lbrace\Rv \in \mathbb{R}_{\ge 0}^n \suchthat \Rv \leq C(\lambdav) \lambdav\right\rbrace.
\end{equation}
If $\lambdav_1 = c \lambdav_2$ for some constant $c$, then $C(\lambdav_1) \lambdav_1 = C(\lambdav_2) \lambdav_2$ and thus in (<ref>), it suffices to do the union over vectors $\lambdav$ such that $\sum_{i=1}^n \lambda_i = n$.
The $\ov$-directed capacity of the index coding problem $G$ is referred to as the symmetric capacity,
\[
\csym = C(\ov) = \max \{R \suchthat (R, \ldots, R) \in \Cr\}.
\]
The reciprocal of the symmetric capacity is referred to as the broadcast rate $\beta$ of the index coding problem $G$.
For any nonnegative real vector $\muv = (\mu_1, \ldots, \mu_n)$, the $\muv$-weighted sum-capacity $\bar{C}(\muv)$ of the index coding problem $G$ is defined as
\[
\bar{C}(\muv) = \max\left\lbrace\sum_{i=1}^n \mu_i R_i \suchthat (R_1, \ldots, R_n) \in \Cr\right\rbrace.
\]
The $\ov$-weighted sum-capacity is simply referred to as sum-capacity,
\[
\csum = \bar{C}(\ov) = \max\left\lbrace\sum_{i=1}^n R_i \suchthat (R_1, \ldots, R_n) \in \Cr\right\rbrace.
\]
§.§ Locally Recoverable Distributed Storage
In the locally recoverable distributed storage problem, data is to be stored in a network of $n$ servers such that by having access to all of the servers the stored data can be exactly recovered.
Let $x_j \in \{0,1\}^{t_j}$ denote the content of server $j$, $j \in [n]$.
It is assumed that each server has access to the contents of a subset of the other servers $x(A_j)$, where $A_j \subseteq [n] \setminus \{j\}$ and is referred to as the recoverability set of node $j$.
The goal is to find the maximum amount of data that can be stored in the network such that
if any of the servers fails, its content can be recovered from the contents of its recoverability set.
Any instance of the distributed storage problem is fully represented by the storage recovery graph $G=(V,E)$ in which each node represents a server and there exists a directed edge $i \to j$ iff server $i$ is in the recoverability set of server $j$.
A $(t_1, \ldots, t_n, r)$ distributed storage code is defined by
* a message set $[2^r]$,
* a one-to-one encoding function $x^n: [2^r] \to \prod_{i=1}^n \{0,1\}^{t_i}$ that assigns a codeword $x^n(m)$ to each message $m \in [2^r]$, (the set $\mathcal{C} = \{x^n(1), \ldots, x^n(2^r)\}$ is referred to as the codebook), and
* $n$ recovery functions $f_j: \prod_{k \in A_j} \{0,1\}^{t_k} \to \{0,1\}^{t_j}$ that maps the contents of the recoverability set $x(A_j)$ to $x_j$ for $j \in [n]$.
Thus, for every $x^n \in \mathcal{C}$,
\[
f(x(A_j)) = x_j, \quad j \in [n].
\]
A rate tuple $(R'_1,\ldots,R'_n)$ is said to be achievable for the distributed storage problem $G$
if there exists a $(t_1, \ldots, t_n, r)$ distributed storage code such that
\[
R'_j \geq \frac{t_j}{r}, \quad j \in [n].
\]
The optimal rate region $\Rr$
of the distributed storage problem is defined as the closure of the set of achievable rate tuples.
For any non-negative real tuple $\lambdav = (\lambda_1, \ldots, \lambda_n)$, the $\lambdav$-directed optimal rate $R(\lambdav)$ of the distributed storage problem $G$ is defined as
\begin{align}
\label{eq:clambdap}
R(\lambdav) = \min \{R' \suchthat R' \lambdav \in \Rr\}.
\end{align}
The optimal rate region can be written in terms of $\lambdav$-directed optimal rates.
\begin{equation}
\label{eq:lambdacapacity-storage}
\Rr = \bigcup_{\lambdav \in \mathbb{R}_{\ge 0}^n \suchthat \sum_{i=1}^n \lambda_i = n} \left\lbrace \Rv' \in \mathbb{R}^n \suchthat \Rv' \geq R(\lambdav) \lambdav \right\rbrace.
\end{equation}
The $\ov$-directed optimal rate of the distributed storage problem $G$ is referred to as the symmetric coding rate,
\begin{align*}
\rsym = R(\ov) = \min \{R' \suchthat (R', \ldots, R') \in \Rr \}.
\end{align*}
The reciprocal of the symmetric coding rate is sometimes referred to as the normalized rate.
For any nonnegative real vector $\muv = (\mu_1, \ldots, \mu_n)$, the $\muv$-weighted optimal sum-rate $\bar{R}(\muv)$ of the distributed storage problem $G$ is defined as
\[
\bar{R}(\muv) = \min \left\lbrace\sum_{i=1}^n \mu_i R'_i \suchthat (R'_1, \ldots, R'_n) \in \Rr\right\rbrace.
\]
The $\ov$-weighted optimal sum-rate $\bar{R}(\ov)$ is simply referred to as optimal sum-rate
\begin{align*}
\rsum = \bar{R}(\ov) = \min \left\lbrace\sum_{i=1}^n R'_i \suchthat (R'_1, \ldots, R'_n) \in \Rr\right\rbrace.
\end{align*}
§.§ Guessing Games on Directed Graphs
Given a directed graph $G=(V,E)$, $V = [n]$, consider the following cooperative game among $n$ players.
Player $j \in [n]$ is associated to node $j$ and is assigned a value $x_j \in \{0,1\}^{t_j}$ independently from the other players.
It is assumed that player $j$ can observe the values assigned to a subset of the other players $x(A_j):= (x_i \suchthat i \in A_j)$, $A_j \subseteq [n] \setminus \{j\}$.
$A_j$ is referred to as the set of neighbors of player (node) $j$.
The players should simultaneously guess their own value and the goal is to maximize the probability that all players guess their value correctly.
No communication is allowed between the players, but they can agree on a strategy beforehand.
Note that the setting that we presented here is slightly different from the original one defined by Riis <cit.> in that the range of the values assigned to the players can be different.
A $(t_1, \ldots, t_n, W)$ guessing strategy is defined by
* $n$ functions $h_j: \prod_{k \in A_j} \{0,1\}^{t_k} \to \{0,1\}^{t_j}$ that maps the values of the neighbors $x(A_j)$ to $x_j$, for $j \in [n]$, and
* a set $W$ of $n$-tuples that can be guessed correctly using these functions,
\[
W := \bigcap_{j=1}^n \left\lbrace x^n \in \prod_{i=1}^n \{0,1\}^{t_i} \suchthat h_j(x(A_j)) = x_j\right\rbrace.
\]
Let $P_\textrm{win}$ be the probability of winning, namely, the probability that everyone guesses her value correctly.
If the players adopt a $(t_1, \ldots, t_n, W)$ strategy, i.e., player $j$, $j \in [n]$ uses function $h_j$ to guess her value based on the values of her neighbors, then
\begin{align*}
P_\textrm{win} = |W|/\prod_{i=1}^n 2^{t_i}.
\end{align*}
Let $P_\textrm{rand}$ be the probability of winning if
every player guesses her value randomly.
As player $j$, $j \in [n]$, is correct with probability $\frac{1}{2^{t_j}}$ independent of others, we have
\[
P_\textrm{rand} = 1/\prod_{i=1}^n 2^{t_i}.
\]
To measure the performance of adopting a strategy, in <cit.> guessing number is defined as the logarithm of the ratio between the winning probabilities of an strategy and a random guess.
In the following, we define the guessing number for the general case where the range of the values assigned to the players are different.
Given a directed graph $G$, the guessing number of a $(t_1, \ldots, t_n, W)$ guessing strategy is defined as
\begin{align}
\label{eq:guessing-number}
k(G,\tv) = \log_s \left( P_\textrm{win}/P_\textrm{rand} \right),
\end{align}
where $s = 2^{\frac{1}{n}\sum_{i=1}^n t_i}$,
and $P_\textrm{win}$ is the probability that the players win if they adopt that strategy.
Note that for the case where $t_j = t, \forall j$, we have
$k(G, t) = \log(|W|)/t$.
The optimal guessing number $k(G)$ of a directed graph $G$ is defined as
\begin{align}
\label{eq:guessing-number-optimal}
k(G) = \sup_\tv \sup k(G,\tv),
\end{align}
where the second supremum is over all $(t_1, \ldots, t_n, W)$ strategies.
The following is an alternative way to measure the performance of adopting a strategy.
Given a directed graph $G$, the complementary guessing number of a $(t_1, \ldots, t_n, W)$ guessing strategy is defined as
\begin{align}
\label{eq:guessing-number-comp}
k'(G,\tv) = \log_s \left( 1/P_\textrm{win} \right),
\end{align}
where $s = 2^{\frac{1}{n}\sum_{i=1}^n t_i}$ and $P_\textrm{win}$ is the probability that the players win if they adopt that strategy.
The optimal complementary guessing number is defined in a similar way.
\begin{align}
\label{eq:guessing-number-optimal-comp}
k'(G) = \inf_\tv \inf k'(G,\tv),
\end{align}
where the second infimum is over all $(t_1, \ldots, t_n, W)$ guessing strategies.
For any $(t_1, \ldots, t_n, W)$ guessing strategy, we have
$k'(G,\tv) = n - k(G,\tv)$,
and thus
\begin{align}
\label{eq:k-k'}
k'(G) = n - k(G).
\end{align}
Based on the definitions, it is easy to see the following relationship between the guessing games and the distributed storage problem.
Given any directed graph $G$, a $(t_1, \ldots, t_n, r)$ distributed storage code exists iff a $(t_1, \ldots, t_n, W)$ guessing strategy with $|W| \geq 2^r$ exists.
We will return to the relationship between these two problems at the and of Section <ref>.
§ CAPACITY REGION, OPTIMAL RATE REGION, AND GUESSING NUMBER VIA CONFUSION GRAPH
In order to shed light on how these three problems are related,
we will review the characterization of the capacity region of the index coding problem, the optimal rate region of the distributed storage problem, and the guessing number of guessing games on graphs via the notion of confusion graph.
§.§ Index Coding
In <cit.>, the capacity region is characterized through the fractional chromatic number of the confusion graph.
The capacity region $\Cr$ of the index coding problem $G$ is the closure
of all rate tuples $(R_1, \ldots, R_n)$ such that
\begin{equation}
\label{eq:graphregion3}
R_j \leq \frac{t_j}{\log(\chi_f(\G_{\tv}(G)))}, \quad j \in [n],
\end{equation}
for some $\tv = (t_1, \ldots, t_n)$.
§.§ Distributed Storage and Guessing Game
In <cit.>, Mazumdar characterized the normalized rate of the distributed storage problem through the independence number of the confusion graph. The following is a simple generalization of Mazumdar's result.
A rate tuple $(R'_1, \ldots, R'_n)$ is achievable for the distributed storage problem $G$ iff there exists an integer tuple $\tv = (t_1, \ldots, t_n)$ such that
\begin{align*}
R'_j \geq \frac{t_j}{\floor{\log(\alpha(\G_\tv(G)))}}, \quad j \in [n].
\end{align*}
Sufficiency (achievability).
For a given tuple $\tv = (t_1, \ldots, t_n)$, consider a maximal independent set of the confusion graph
$\G = \G_{\tv}(G)$.
By the definition of the confusion graph, no two $n$-tuples in this independent set are confusable
and therefore for these $\alpha(\G)$ $n$-tuples, contents of each server is a function of the contents of its recoverability set.
Therefore, it is possible to use these $\alpha(\G)$ $n$-tuples to store $r = \floor{\log(\alpha(\G))}$ bits in the distributed network.
This proves the existence of a $(t_1, \ldots, t_n, \floor{\log(\alpha(\G_\tv(G)))})$ distributed storage code.
Necessity (converse).
Consider any $(t_1, \ldots, t_n, r)$ distributed storage code, which has at least $2^r$ distinct $n$-tuples that satisfy the required function relationship.
By definition, these $n$-tuples form an independent set of the
confusion graph $\G = \G_\tv(G)$.
Thus, $\alpha(\G) \ge 2^r$, or equivalently,
$r \le \floor{\log(\alpha(\G))}$.
Therefore, any achievable $(R'_1, \ldots, R'_n)$ must satisfy
\[
R'_j \ge \frac{t_j}{\floor{\log(\alpha(\G_\tv(G)))}}, \quad j \in [n],
\]
for some $\tv = (t_1,\ldots,t_n)$.
\begin{align*}
\frac{kt_j}{\floor{k\log(\alpha(\G_\tv(G)))}} \le \frac{kt_j}{k\log(\alpha(\G_\tv(G)))-1},
\end{align*}
letting $k \to \infty$ establishes the following.
The optimal rate region $\Rr$ of the distributed storage problem $G$ is the closure of all rate tuples $(R'_1, \ldots, R'_n)$ such that
\begin{align*}
R'_j \ge \frac{t_j}{\log(\alpha(\G_\tv))}, \quad j \in [n],
\end{align*}
for some $\tv = (t_1, \ldots, t_n)$.
Using an argument similar to the proof of Proposition <ref>, the optimal guessing number of the guessing game on directed graph $G$ can be characterized via the independence number of confusion graphs.
For the guessing game on directed graph $G$ on $n$ nodes we have
\begin{align*}
k(G) = \sup_{\tv \in \mathbb{Z}_{\geq 0}^n} ~\log(\alpha(\G_\tv))/(\frac{1}{n}\sum_{i=1}^n t_i).
\end{align*}
§ RELATIONSHIPS
In this section, we first discuss the complementarity between the index coding and the distributed storage and then clarify the relationship between guessing games and the distributed storage problem.
§.§ Index Coding Versus Distributed Storage
For any length-$n$ integer tuple $\tv$, the confusion graph $\G_\tv$ is vertex transitive. Therefore, Lemma <ref> yields
\begin{align}
\label{eq:chivsalpha}
\log(\chi_f(\G_\tv)) = \sum_{i=1}^n t_i - \log(\alpha(\G_\tv)),
\end{align}
which is the key to establishing the complementarity between the index coding problem and the distributed storage problem.
We start by stating the following proposition.
The proof of the proposition is relegated to Appendix <ref>.
For any directed graph $G$ on $n$ nodes and any $\lambdav \in \mathbb{Q}_{\geq 0}^n$,
\begin{align}
\label{eq:Clambda}
C(\lambdav) &= \sup_{r \suchthat r\lambdav \in \mathbb{Z}_{\geq 0}^n} \frac{r}{\log(\chi_f(\G_{r\lambdav}(G)))},\\
\label{eq:Cplambda}
R(\lambdav) &= \inf_{r \suchthat r\lambdav \in \mathbb{Z}_{\geq 0}^n} \frac{r}{\log(\alpha(\G_{r\lambdav}(G)))}.
\end{align}
The following theorem is the main result of this paper which establishes the complementarity between the $\lambdav$-directed capacity $C(\lambdav)$ and the $\lambdav$-directed optimal rate $R(\lambdav)$, for any nonnegative real tuple $\lambdav$.
For any directed graph $G$ on $n$ nodes and any $\lambdav \in \mathbb{R}_{\geq 0}^n$, we have
\begin{align}
\label{eq:duality}
\frac{1}{C(\lambdav)} = \sum_{i=1}^n \lambda_i - \frac{1}{R(\lambdav)}.
\end{align}
For $\lambdav \in \mathbb{Q}_{\geq 0}^n$, we have
\begin{align}
\label{eq:duality1}
C(\lambdav) &= \sup_{r \suchthat r\lambdav \in \mathbb{Z}_{\geq 0}^n} \frac{r}{\log(\chi_f(\G_{r\lambdav}(G)))} \\
\label{eq:duality2}
&= \sup_{r \suchthat r\lambdav \in \mathbb{Z}_{\geq 0}^n} \frac{r}{r\sum_{i=1}^n \lambda_i - \log(\alpha(\G_{r\lambdav}(G)))} \\
%&= \frac{1}{\sum_{i=1}^n \lambda_i - \sup_{r \suchthat r\lambdav \in \mathbb{Z}_{\geq 0}^n} \frac{\log(\alpha(\G_{r\lambdav}(G)))}{r}} \\
%&= \frac{1}{\sum_{i=1}^n \lambda_i - \frac{1}{\inf_{r \suchthat r\lambdav \in \mathbb{Z}_{\geq 0}^n} \frac{r}{\log(\alpha(\G_{r\lambdav}(G)))}}} \\
\label{eq:duality3}
&= \frac{1}{\sum_{i=1}^n \lambda_i - \frac{1}{R(\lambdav)}},
\end{align}
where (<ref>) and (<ref>) follow from Proposition <ref>, and (<ref>) follows from (<ref>).
For $\lambdav \not \in \mathbb{Q}^n$, (<ref>) holds due to the continuity of the functions $C(\lambdav)$ and $R(\lambdav)$ and $\mathbb{Q}$ being dense in $\mathbb{R}$.
Due to equations (<ref>) and (<ref>),
the above theorem establishes the complementarity between the two problems in the strong sense that given the capacity region of the index coding problem $G$ (more precisely, given the boundary points of the capacity region), Theorem <ref> completely determines the optimal rate region for the distributed storage problem $G$ and vice versa.
This includes as an special case the complementarity between the symmetric capacity of the index coding problem and the symmetric coding rate of the distributed storage established by Mazumdar <cit.>, and by Shanmugam and Dimakis <cit.>.
Setting $\lambdav = \ov$ in Theorem <ref> yields
\begin{align}
\frac{1}{\csym} = n - \frac{1}{\rsym}.
\end{align}
Equation (<ref>) can also be used to show how the sum-capacity of the index coding problem is related to the optimal sum-rate of the distributed storage problem.
The proof is straightforward and is omitted.
\begin{align}
\frac{1}{\csum} = 1 - \frac{1}{\rsum}.
\end{align}
§.§ Guessing Games Versus Distributed Storage
The following proposition shows that for any directed graph $G$, the optimal guessing number of the guessing game on $G$ is inversely related to the optimal sum-rate of the distributed storage problem $G$.
For any directed graph $G$ on $n$ nodes
\begin{align*}
k(G) = \frac{n}{\rsum}.
\end{align*}
For any tuple $\tv$, let $s(\tv) = 2^{\frac{1}{n}\sum_{i=1}^n t_i}$, then
\begin{align}
\rsum &= \min_{R' \in \Rr} \sum_{i=1}^n R'_i \nonumber \\
\label{eq:ggvsds1}
&= \inf_{\tv} \frac{\sum_{i=1}^n t_i}{\log(\alpha(\G_\tv))} \\
%&= \inf_{\tv} \frac{n \log(s(\tv))}{\log(\alpha(\G_\tv))} \nonumber \\
&= \frac{n}{\sup_{\tv} \frac{\log(\alpha(\G_\tv))}{\log(s(\tv))}}
= \frac{n}{k(G)},
\label{eq:ggvsds2}
\end{align}
where (<ref>) follows from Proposition <ref>, and (<ref>) follows from Proposition <ref>.
Combining Theorems <ref> and <ref>, and (<ref>) yields the inverse relationship between the optimal complementary guessing number and the index coding sum-capacity.
For any directed graph $G$ on $n$ nodes
\begin{align*}
k'(G) = \frac{n}{\csum}.
\end{align*}
§ CONCLUDING REMARKS
In this paper, we surveyed three problems and elaborated on the connections between their optimal solutions.
This can be used in directly translating any achievability scheme or converse result for one problem to the other problems.
For example, in <cit.>, it is shown that even for undirected graphs, the guessing strategy based on the fractional clique covering <cit.> is not optimal.
It is also shown via an example that non-Shannon inequalities provide a better bound on the guessing number of undirected graphs than Shannon inequalities.
The established connections between the problems answer the corresponding open problems in the context of index coding, namely, even when restricted to undirected graphs, neither the fractional clique covering inner bound nor the polymatroidal outer bound is tight for the index coding problem.
§ ACKNOWLEDGMENTS
This work was supported by the National Science Foundation under Grant CCF-1320895
and the Korean Ministry of Science, ICT and Future Planning under the
Institute for Information and Communications Technology Promotion Grant B0132-15-1005
(Development of Wired-Wireless Converged 5G Core Technologies).
§ PROOF OF LEMMA <REF>
First assume that $s_j + k = t_j$ for some $j \in [n]$ and some positive integer $k$ and $s_i = t_i, \forall i \not = j$.
In this case, we will prove the lemma by contradiction.
Assume that (<ref>) does not hold.
Then as any confusion graph is vertex transitive, by Lemma <ref>, we have
\begin{align}
\label{eq:cont-assumption}
\alpha(\G_\tv(G)) > 2^k \alpha(\G_\sv(G)).
\end{align}
Each vertex of $\G_\tv$ is associated with an $n$-tuple that has $t_i$ bits for user $i \in [n]$.
Consider the $\alpha(\G_\tv)$ $n$-tuples in a maximal independent set of $\G_\tv$ and partition them into (at most $2^k$) subsets based on the first $k$ bits of user $j$.
As these $k$ bits are the same for all the members of each partition, after removing these $k$ bits from all the $n$-tuples, each partition will correspond to an independent set of $\G_\sv$.
However, there are at most $2^k$ partitions and hence if (<ref>) holds, due to the pigeonhole principle, there exists a partition with more than $\alpha(\G_\sv)$ members, i.e., there exists an independent set of size more than $\alpha(\G_\sv)$ in $\G_\sv$, which contradicts the definition of the independence number of a graph. Therefore, (<ref>) holds if the two integer tuples differ only at one element. Applying this (at most $n$ times) to length-$n$ tuples that differ only at one element, completes the proof of the lemma.
§ PROOF OF PROPOSITION <REF>
Due to Proposition <ref>,
\Cr = cl(\Cr^{\degree})
\begin{equation*}
\Cr^{\degree} = \{\Rv \in \mathbb{R}^n_{\geq 0} \suchthat \Rv \leq \frac{\tv}{\log(\chi_f(\G_\tv))} ~\text{for some}~ \tv \in \mathbb{Z}^n_{\geq 0} \}.
\end{equation*}
The following lemma shows that the $\lambdav$-directed capacity defined in (<ref>), can also be defined in terms of $\Cr^{\degree}$.
For any non-negative real tuple $\lambdav$,
\begin{align}
C(\lambdav) = \sup \{R \suchthat R \lambdav \in \Crn\}.
\end{align}
Let $R^* = \sup \{R \suchthat R \lambdav \in \Crn\}$, then $R^* \lambdav \in \Cr$ and by the definition of $C(\lambdav)$ we have
$R^* \leq C(\lambdav)$.
Assume that $R^* < C(\lambdav)$.
\[ \epsilon = \frac{1}{2} (C(\lambdav)-R^*) \min_{i \suchthat \lambda_i > 0} \lambda_i,\]
and define the $\epsilon$-neighborhood $N_\epsilon(C(\lambdav) \lambdav)$ as
\[
N_\epsilon(C(\lambdav) \lambdav) = \bigcap_{i \suchthat \lambda_i > 0} \left\lbrace\Rv \in \mathbb{R}^n \suchthat e_i^T (C(\lambdav) \lambdav - \Rv) < \epsilon\right\rbrace,
\]
where all of the components of the $n \times 1$ vector $e_i$ are zero, except the $i$-th component, which is one.
If $N_\epsilon(C(\lambdav) \lambdav) \cap \Crn = \emptyset$, then it contradicts the fact that $C(\lambdav) \lambdav$ belongs to $\Cr$.
Alternatively, if $N_\epsilon(C(\lambdav) \lambdav) \cap \Crn \not = \emptyset$, then there exists $R > R^*$ such that $R\lambdav \in \Crn$, which contradicts the definition of $R^*$.
Therefore, $R^* = C(\lambdav)$ and the proof is complete.
[Proof of Proposition <ref>]
Let $\lambda = (\frac{a_1}{b}, \ldots, \frac{a_n}{b})^T$, $b \in \mathbb{N}$, and $a_1, \ldots, a_n \in \mathbb{Z}_{\geq 0}$.
If $r\lambdav \in \mathbb{Z}^n_{\geq 0}$, then due to Proposition <ref>, we have
\[
\frac{r \lambdav}{\log(\chi_f(\G_{r\lambdav}(G)))} \in \Cr.
\]
Therefore, we have
\[
C(\lambdav) \geq \sup_{r \suchthat r\lambdav \in \mathbb{Z}^n_{\geq 0}} \frac{r}{\log(\chi_f(\G_{r\lambdav}(G)))}.
\]
Next, let $R$ be any real number such that $R\lambda \in \Cr^0$.
Then, there exists integer tuple $\tv$ such that
$R\lambdav \leq \tv/\log(\chi_f(\G_\tv))$,
and hence $R \leq \frac{t_i}{\lambda_i}/\log(\chi_f(\G_\tv))$, for all $i$ such that $\lambda_i > 0$.
\begin{align}
\label{eq:argmin}
j = \arg\min_{i \suchthat \lambda_i > 0} \frac{t_i}{\lambda_i},
\end{align}
then we have
\begin{align}
\label{eq:appendix1}
R \leq \frac{q}{a_j \log(\chi_f(\G_\tv))},
\end{align}
where $q = t_jb$.
Due to (<ref>), $a_j \tv \geq q\lambdav = t_j(a_1, \ldots, a_n)^T \in \mathbb{Z}_{\geq 0}^n$ and we have
\begin{align}
\label{eq:ineq1}
\log(\chi_f(\G_{q \lambda})) &\leq \log(\chi_f(\G_{a_j \tv})) \\
\label{eq:ineq2}
&\leq \log(\chi_f(\G_\tv^{a_j})) \\
\label{eq:ineq3}
&= a_j \log(\chi_f(\G_\tv)),
\end{align}
where (<ref>) follows from Lemma <ref>, (<ref>) follows from the fact that the set of edges of $\G_{a_j \tv}$ is a subset of the set of edges of $\G_\tv^{a_j}$, and (<ref>) follows from Lemma <ref>.
Combining (<ref>) and (<ref>), we have
\[
R \leq \frac{q}{\log(\chi_f(\G_{q \lambda}))} \leq \sup_{r \suchthat r\lambdav \in \mathbb{Z}_{\geq 0}^n} \frac{r}{\log(\chi_f(\G_{r\lambdav}(G)))},
\]
which together with Lemma <ref> yields
\begin{align*}
C(\lambdav) \leq \sup_{r \suchthat r\lambdav \in \mathbb{Z}_{\geq 0}^n} \frac{r}{\log(\chi_f(\G_{r\lambdav}(G)))},
\end{align*}
and hence (<ref>) holds.
Following similar steps as above, one can show that (<ref>) also holds.
|
1511.00991
|
We give a simple example of spacetime metric, illustrating that homogeneity
and isotropy of space slices at all moments of time is not obligatory lifted
to a full system of six Killing vector fields in spacetime, thus it cannot be
interpreted as a symmetry of a four dimensional metric. The metric depends
on two arbitrary and independent functions of time. One of these functions is
the usual scale factor. The second function cannot be removed by coordinate
transformations. We prove that it must be equal to zero, if the metric satisfies
Einstein's equations and the matter energy momentum tensor is homogeneous
and isotropic. A new, equivalent, definition of homogeneous and isotropic
spacetime is given.
Cosmological models constitute a classic part of general relativity and attract
now great interest because of large amount of observational data. Most of the
cosmological models are based on cosmological principle which requires our
universe to be homogeneous and isotropic [1–5].
In particular, all space slices corresponding to constant time must be
homogeneous and isotropic.
We consider spacetime $\MM$ with coordinates $\lbrace t,x^\mu\rbrace$,
$\mu=1,2,3$, and metric of Lorentzian signature $(+---)$. We assume that $t$ is
the time coordinate and every section $t=\const$ is spacelike. According to
cosmological principle every section of constant time must be a
three-dimensional space of constant curvature which is homogeneous and
isotropic. Then the usual ansatz for the metric is
\begin{equation} \label{qnxbgk}
ds^2=dt^2+a^2\overset{\circ}g_{\mu\nu}(x)dx^\mu dx^\nu.
\end{equation}
where $\overset{\circ}g_{\mu\nu}(x)$ is a negative definite metric on
three-dimensional space slices $\MS$ of constant curvature which does not depend
on time. This metric contains only one arbitrary function $a(t)>0$ which is
called the scale factor. This is the most general form of the homogeneous and
isotropic metric in a suitable coordinate system (see, for example,
The form of the metric depends on the coordinates chosen on space slices. We
choose stereographic coordinates. Then the metric takes the diagonal form
\begin{equation} \label{emfree}
g=\begin{pmatrix} 1 & 0 \\
0 & \frac{\displaystyle a^2\eta_{\mu\nu}}{\displaystyle \big(1+b_0x^2\big)^2}
\end{pmatrix},
\end{equation}
where $b_0=-1,0,1$, $\eta_{\mu\nu}:=\diag(---)$ is the negative definite
Euclidean metric, and $x^2:=\eta_{\mu\nu}x^\mu x^\nu\le0$. Since we have chosen
negative definite metric on space slices, the values $b_0=-1,0,1$ describe
constant curvature spaces of positive, zero, and negative curvature,
respectively. For positive and zero curvature spaces, stereographic coordinates
are defined on the whole Euclidean space $x\in\MR^3$. For negative curvature
spaces stereographic coordinates are defined inside the ball $|x^2|<1/b_0$.
Let us make the coordinate transformation $x^\mu\mapsto x^\mu/a$. Then metric
becomes non-diagonal, and the scale factor disappears:
\begin{equation} \label{emfrei}
1+{\displaystyle\frac{\dot b^2x^2}{4b^2\big(1+bx^2\big)^2}} &
{\displaystyle\frac{\dot bx_\nu}{2b\big(1+bx^2\big)^2}} \\[2mm]
{\displaystyle\frac{\dot bx_\mu}{2b\big(1+bx^2\big)^2}} &
{\displaystyle\frac{\eta_{\mu\nu}}{\big(1+bx^2\big)^2}} \end{pmatrix},
\end{equation}
\begin{equation} \label{qbgsty}
\end{equation}
and dot denotes derivative with respect to time.
We see that the metric of a homogeneous and isotropic universe can be
non-diagonal, and may do not contain the scale factor. Moreover, the scalar
curvature of space slices which is proportional to $b(t)$ explicitly depends
on time.
Let us now simply drop non-diagonal terms, add the scale factor, and put
$g_{00}=1$. Then the metric is
\begin{equation} \label{emfrep}
g=\begin{pmatrix} 1 & 0 \\
0 & {\displaystyle\frac{a^2\eta_{\mu\nu}}{\big(1+bx^2\big)^2}} \end{pmatrix}.
\end{equation}
This metric contains two independent arbitrary functions of time: $a(t)>0$
and $b(t)$. It is nondegenerate for all values of $b$ including zero. It is
interesting because allows one to analyse, in a general case, solutions which go
through zeroes $b=0$. If such solutions exist, then the universe can change the
sign of its space slices curvature during evolution.
Metric on space slices $\MS$ of constant time $t=\const$ is
\begin{equation} \label{qbavsj}
\end{equation}
depending on time $t$ as a parameter. Straightforward calculations show
that three-dimensional curvature tensor for metric (<ref>) has the form
\begin{equation*}
\hat R_{\mu\nu\rho\s}=-\frac{4b}{a^2}
\end{equation*}
Three-dimensional Ricci tensor and scalar curvature are
\begin{equation*}
\hat R_{\mu\nu}=-\frac{8b}{a^2}h_{\mu\nu},\qquad \hat R=-\frac{24b}{a^2}.
\end{equation*}
We see that that all space slices are spaces of constant curvature. Therefore
there are six independent Killing vector fields $\hat K_i=\hat K_i^\mu\pl_\mu$,
$i=1,\dotsc,6$, which act on space slices. Symmetry transformations on space
slices must be prolongated on the whole spacetime. The usual assumption is that
they act trivially on time coordinate <cit.>. That is Killing vector
fields on spacetime have zero time component:
$K_i:=\lbrace 0,\hat K_i^\mu\rbrace$. By construction, any linear combination of
Killing vector fields $K$ satisfies the Killing equation
\begin{equation} \label{qbsvwg}
\nb_\al K_\bt+\nb_\bt K_\al=0,\qquad\al,\bt=0,1,2,3,
\end{equation}
because they do not have time component and time enters space components as a
parameter. The Killing equation can be rewritten for the contravariant
\begin{equation} \label{ekilco}
g_{\al\g}\pl_\bt K^\g+g_{\bt\g}\pl_\al K^\g+K^\g\pl_\g g_{\al\bt}=0.
\end{equation}
This equation is fulfilled for all moments of time.
The $(\al,\bt)=(0,0)$ component of Killing equation (<ref>) for diagonal
metric (<ref>) is identically satisfied. The $(\al,\bt)=(0,\mu)$
components reduce to equation
\begin{equation} \label{qnbcgf}
\pl_0\hat K^\mu=0.
\end{equation}
The space components $(\al,\bt)=(\mu,\nu)$ of the Killing equation decouple
\begin{equation*}
h_{\mu\rho}(t,x)\pl_\nu\hat K^\rho+h_{\nu\rho}(t,x)\pl_\mu\hat K^\rho
+\hat K^\rho\pl_\rho h_{\mu\nu}(t,x)=0.
\end{equation*}
These equations are identically satisfied for all moments of time by
There is an interesting situation. On one hand, all space slices of metric
(<ref>) are homogeneous and isotropic. On the other hand, any
and isotropic metric must have form (<ref>). The answer is the following.
The whole four-dimensional metric (<ref>) is not homogeneous and
isotropic in a sense that Killing equations (<ref>) are not fulfilled.
Indeed, the six independent Killing vector fields on space slices are
\begin{equation} \label{qbsghu}
\begin{split}
\hat K_{0\mu}&=(1+bx^2)\pl_\mu-\frac2bx_\mu x^\nu\pl_\nu,
\\
\hat K_{\mu\nu}&=x_\mu\pl_\nu-x_\nu\pl_\mu,\qquad \mu,\nu=1,2,3.
\end{split}
\end{equation}
The first three Killing vectors generate translations at the origin of the
coordinate system $x^2=0$, and the last three Killing vectors generate
rotations. We see that the first three Killing vectors explicitly depend on time
through $b(t)$, and Eq.(<ref>) is not fulfilled. This example shows that
homogeneity and isotropy of space slices does not provide sufficient condition
for the whole four-dimensional metric to be homogeneous and isotropic. The
equivalent definition is the following.
The spacetime $\MM$ is called homogeneous and isotropic if: 1) All sections $\MS$ of constant time $t=\const$ are three-dimensional spaces
of constant curvature;2) Extrinsic curvature of the embedding $\MS\hookrightarrow\MM$ is homogeneous
and isotropic for all $t$.
Indeed, the first requirement provides the existence of such coordinate system
where metric is block diagonal <cit.>
\begin{equation*}
g=\begin{pmatrix} 1 & 0 \\ 0 & h_{\mu\nu}(t,x) \end{pmatrix},
\end{equation*}
where $h_{\mu\nu}(t,x)$ is a constant curvature metric for all $t$. For this
metric the extrinsic curvature is <cit.>
\begin{equation*}
K_{\mu\nu}=-\frac12\dot h_{\mu\nu}.
\end{equation*}
If it is homogeneous and isotropic, then it must be proportional to the metric,
and we get the differential equation
\begin{equation} \label{qbncgt}
\dot h_{\mu\nu}=f h_{\mu\nu},
\end{equation}
where $f(t)$ is an arbitrary sufficiently smooth function of time.
If $f=0$, then nothing should be proved, and the metric has form (<ref>)
for $a=\const$.
Let $f\ne0$. Then we introduce new time coordinate $t\mapsto t'$ defined by the
differential equation
\begin{equation*}
\end{equation*}
Afterwards equation (<ref>) becomes
\begin{equation*}
\frac{dh_{\mu\nu}}{dt'}=h_{\mu\nu}.
\end{equation*}
Its general solution is
\begin{equation*}
h_{\mu\nu}(t',x)=C\ex^{t'}\overset\circ g_{\mu\nu}(x),\qquad C=\const\ne0,
\end{equation*}
where $\overset\circ g_{\mu\nu}(x)$ is a constant curvature metric on $\MS$
which do not depend on time. It implies Eq. (<ref>).
In general relativity, we assume that metric satisfies Einstein's equations
\begin{equation} \label{eincos}
\end{equation}
where $R_{\al\bt}$ is the Ricci tensor, $R$ is the scalar curvature, and
$\Lm$ is the cosmological constant. The matter energy-momentum tensor is denoted
by $T_{\al\bt}$.
The cosmological principle requires not only the metric but also the
energy-momentum tensor to be homogeneous and isotropic. The most general form
of the homogeneous and isotropic energy-momentum tensor in the coordinate
system defined by Eq.(<ref>) is <cit.>
\begin{equation} \label{engmom}
T_{\al\bt}=\begin{pmatrix}{\cal E} & 0 \\0& -{\cal P}h_{\mu\nu} \end{pmatrix},
\end{equation}
where $\CE(t)$ and $\CP(t)$ are the matter energy density and pressure.
One can easily calculate the Einstein tensor for metric (<ref>). The
off-diagonal component is
\begin{equation*}
R_{0\mu}=-\frac{4\dot b x_\mu}{(1+bx^2)^2}.
\end{equation*}
To satisfy Einstein's equations (<ref>) we must put
\begin{equation*}
\dot b\quad\Leftrightarrow\quad b=\const
\end{equation*}
because all other terms are diagonal. Thus for homogeneous and isotropic matter
we return to the original metric (<ref>) on the equations of motion.
Metric (<ref>) describes spacetime which has homogeneous and isotropic
space slices. We have shown that this important property is not sufficient for
describing homogeneous and isotropic universe. The reason is that three of
the six Killing vectors on space slices do depend on time, and their lift to the
whole spacetime does not satisfy the four-dimensional Killing equations. The
sufficient condition for the metric to be homogeneous and isotropic is (i) all
space slices must be spaces of constant curvature and (ii) time derivative of
the spacial part of the metric must be homogeneous and isotropic (in the
coordinate system described above). Fortunately, metric of type (<ref>)
seems to be excluded by Einstein's equations, thus there is no reason to worry
too much about this dilemma.
This work is supported by the Russian Science Foundation under grant
A. Friedmann.
Über die Krümmung des Raumes.
Zs. Phys., 10:377–386, 1922.
A. Friedmann.
Über die Möglichkeit einer Welt mit konstanter negativer
Krümmung des Raumes.
Zs. Phys., 21:326–332, 1924.
G. Lemaitre.
LUnivers en expansion.
Ann. Soc. Sci. (Bruxelles), A53:51–85, 1933.
H. P. Robertson.
Kinematics and world structure.
Ap. J., 82:284–301, 1935.
A. G. Walker.
On Milne's theory of world-structure.
Proc. London Math. Soc. Ser.2, 42(1):90–127, 1937.
S. Weinberg.
Gravitation and Cosmology.
Wiley, New York, 1972.
L. P. Eisenhart.
Continuous Groups of Transformations.
Dover Publ., New York, 1961.
R. M. Wald.
General Relativity.
The University of Chicago Press, Chicago, 1984.
|
1511.00721
|
This paper develops a convex approach for sparse one-dimensional deconvolution
that improves upon L1-norm regularization, the standard convex approach.
We propose a sparsity-inducing non-separable non-convex bivariate penalty function
for this purpose.
It is designed to enable the convex formulation of ill-conditioned linear inverse problems with quadratic data fidelity terms.
The new penalty overcomes limitations of separable regularization.
We show
how the penalty parameters should be set to ensure that the objective function is convex,
provide an explicit condition to verify the optimality of a prospective solution.
We present an algorithm (an instance of forward-backward splitting)
for sparse deconvolution using the new penalty.
§ INTRODUCTION
Methods for sparse regularization
can be broadly categorized as convex or non-convex.
In the standard convex approach, the regularization terms (penalty functions) are convex; and the objective function,
consisting of both data fidelity and regularization terms, is convex <cit.>.
The convex approach has several benefits:
the objective function is free of extraneous local minima,
globally convergent optimization algorithms can be leveraged <cit.>.
Despite the attractive properties of convex regularization,
non-convex regularization often performs better <cit.>.
Classical and recent examples are in edge preserving tomography
and compressed sensing <cit.>,
In the non-convex approach, penalty functions are non-convex
as they can be designed to induce sparsity more effectively than convex ones.
However, the convexity of the objective function is generally sacrificed.
Consequently, non-convex regularization is hampered by complications:
the objective function
will generally possess many sub-optimal local minima in which optimization algorithms can become entrapped.
It turns out, non-convex penalties can be utilized
without giving up the convexity of the objective function and corresponding benefits.
This is achieved by carefully specifying the penalty in accordance with the data fidelity term,
as described by Blake, Zimmerman, and Nikolova
In recent work, a class of sparsity-inducing non-convex penalties has been developed
to formulate convex objective functions
and applied to several signal estimation problems <cit.>.
This approach maintains the benefits of the convex framework (absence of spurious local minima, etc.),
yet estimates sparse signals more accurately than convex regularization (e.g., the norm)
due to the sparsity-inducing properties of non-convex regularization.
However, this previous work considers only separable (additive) penalties,
which have fundamental limitations.
In this paper, we introduce a parameterized sparsity-inducing non-separable non-convex bivariate penalty function.
The penalty is designed to enable the convex formulation of ill-conditioned linear inverse problems with quadratic data fidelity terms.
The new penalty overcomes limitations of separable non-convex regularization.
We show
how the penalty parameters should be set to ensure the objective function is convex.
We also show
how this bivariate penalty can be incorporated into linear inverse problems of $ N $ variables ($ N > 2 $),
and we provide an explicit condition to verify the optimality of a prospective solution.
We derive two iterative algorithms for optimization using the new penalty,
and demonstrate its effectiveness for one-dimensional sparse deconvolution.
§.§ Basic problem statement
We consider the problem of bivariate sparse regularization (BISR)
with a quadratic data fidelity term:
\begin{equation}
\label{eq:biprob}
\hat x =
\arg \min_{x \in \RR^2} \bigg\{ f(x) = \half \norm{ y - H x }_2^2 + \lam \psi(x) \bigg\}
\end{equation}
where $ \lam > 0 $,
$ H $ is a $ 2 \times 2 $ matrix,
and $ \psi \colon \RR^2 \to \RR $ is a bivariate penalty.
[In Sec. <ref>, it will be shown how to
extend this bivariate problem to an $ N $-point linear inverse problem.]
In this paper, we suppose $ H\tr H $ is Toeplitz,
as this naturally arises in deconvolution problems.
we write $ H\tr H = K(\gamma) $ where $ \gamma = (\gamma_1, \gamma_2) \in \RR^2 $ and
\begin{equation}
\label{eq:defK}
% H\tr H =
\half
\begin{bmatrix}
\gamma_1 + \gamma_2 & \gamma_1 - \gamma_2
\\
\gamma_1 - \gamma_2 & \gamma_1 + \gamma_2
\end{bmatrix}
= Q \, \Gamma \, Q\tr
\end{equation}
\begin{equation}
\label{eq:defQG}
Q =
\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1
\\
1 & -1
\end{bmatrix},
\quad
\Gamma =
\begin{bmatrix}
\gamma_1 & 0
\\
0 & \gamma_2
\end{bmatrix}.
\end{equation}
This is an eigenvalue decomposition of $ H\tr H $.
The parameters $ \gamma_1 $ and $ \gamma_2 $ are the eigenvalues
of the positive semidefinite matrix $ H\tr H $;
hence, they are nonnegative.
First, suppose $ \psi $ is
a separable convex penalty, e.g., $ \psi(x) = \abs{x_1} + \abs{x_2} $
corresponding to the norm (Fig. <ref>(a)).
Then the objective function $ f $ in (<ref>) is convex,
but it does not induce sparsity as effectively as non-convex penalties can.
In particular, $ \ell_1 $ norm regularization tends to underestimate the true signal values.
Second, suppose $ \psi $ is a separable non-convex penalty,
i.e., $ \psi(x) = \phi(x_1) + \phi(x_2) $, as illustrated in Fig. <ref>(b).
Then the objective function $ f $ is convex only if $ \phi $ is suitably chosen.
In particular,
for the class of penalties we consider,
the objective function is convex only if $ \phi''(t) \ge -\min\{\gamma_1, \gamma_2\}/\lam $,
where $ \gamma_1 $ and $ \gamma_2 $ are the eigenvalues of $ H\tr H $.
(See Lemma <ref>, <cit.>, and <cit.>.)
When $ H $ is singular, the minimum eigenvalue of $ H\tr H $ is zero
and $ \phi $ must be convex
(i.e., it induces sparsity relatively weakly).
Hence, if $ H $ is singular and we restrict the penalty $ \psi $ to be separable,
then we can not use sparsity-inducing non-convex regularization without
sacrificing the convexity of the objective function $ f $.
Indeed, when separable non-convex penalties are utilized for their strong sparsity-inducing properties,
the convexity of $ f $ is generally sacrificed.
Our aim is to prescribe a non-separable penalty $ \psi $ so that
the objective function is guaranteed to be convex even though the penalty $ \psi $ itself is not.
Such a penalty will be given in Sec. <ref>.
It turns out, when we utilize a non-separable non-convex penalty to strongly induce sparsity,
we need not sacrifice the convexity of the objective function $ f $,
even when $ H $ is singular.
Separable bivariate penalties, convex and non-convex.
§.§ Related work
The design of non-convex regularizers ensuring convexity of an objective function
proposed as part of the Graduated Non-Convexity (GNC) approach <cit.>
and for binary image estimation <cit.>.
Most methods for non-convex sparse regularization do not aim to maintain convexity of the objective function.
The $ \ell_p $ pseudo-norm ($ 0 \le p < 1 $) is widely used,
but other regularizers have also been advocated
Several algorithms have been developed specifically for the $ \ell_0 $ pseudo-norm:
matching pursuit <cit.>,
greedy $\ell_1$ <cit.>,
iterative hard thresholding and its variations <cit.>,
smoothed $\ell_0$, <cit.>,
and single best replacement <cit.>.
The penalty developed here is different from regularizers described in the literature,
most of which are separable or are compositions of separable regularizers and linear operators.
Non-separable penalties
(e.g., mixed norms <cit.>)
are generally used to capture statistical relationships among signal values
(possibly linearly transformed
[Isotropic two-dimensional total variation,
which induces joint sparsity of horizontal and vertical gradients,
exemplifies this type of regularization.])
or to induce structured sparsity;
they are not intended for
unstructured, uncorrelated, `pure' sparsity.
Moreover, most sparsity-inducing penalties
are neither
parameterized nor utilized in the way undertaken here: namely, to induce sparsity subject to the constraint that the objective function is convex.
As the proposed method leads to a convex problem,
convex sparse optimization methods may be used or adapted for its solution.
Representative algorithms are
the iterative shrinkage/thresholding algorithm (ISTA/FISTA)
proximal methods
alternating direction method of multipliers (ADMM)
majorization-minimization (MM) <cit.>.
Several algorithms are suitable for general non-convex sparse regularization problems,
such as
iteratively reweighted least squares (IRLS) <cit.>,
iteratively reweighted (IRL1)
related algorithms
non-convex MM
of GNC
and other methods
Non-convex regularization has also been used for blind deconvolution <cit.>
and low-rank plus sparse matrix decompositions <cit.>.
Convergence of these algorithms are generally to local optima only.
conditions for convergence to a global minimizer,
or to guarantee that all local minimizers are near a global minimizer,
have been recently reported <cit.>.
Whereas Refs. <cit.> focuses on convergence guarantees for given penalties,
we focus on the design of penalties.
§.§ Notation
We write the vector $ x \in \RR^N $ as $ x = (x_1, \ x_2, \ \dots, \ x_{N}) $.
Given $ x \in \RR^N $, we define $ x_n = 0 $ for $ n \notin \{ 1 , 2 , \dots, N \} $.
(This simplifies expressions involving summations over $ n $.)
The norm of $ x \in \RR^N $ is defined as $ \norm{ x }_1 = \sum_n \abs{ x_n } $.
If the matrix $ A $ is positive semidefinite, we write $ A \mge 0 $.
If the matrix $ A - B $ is positive semidefinite, we write $ A \mge B $.
\displaystyle
\phi(t; a) =
\frac { \abs{ t } } { 1 + a \abs{ t } / 2 } ,
\quad a \ge 0
\phi(t; a) =
\begin{cases}
\frac{1}{a} \, \log(1 + a \abs{t}), \ & a > 0
\\
\abs{t}, & a = 0
\end{cases}
\phi(t; a) =
\begin{cases}
\frac{2}{a\sqrt{3}}
\Bigl(
\tan\inv \Bigl( \frac{1+2 a \abs{t}}{\sqrt{3}} \Bigr) - \frac{\pi}{6}
\Bigr),
\ & a > 0
\\
\abs{t}, & a = 0
\end{cases}
§ UNIVARIATE PENALTIES
The bivariate penalty to be given in Sec. <ref> will be
based on a parameterized non-convex univariate penalty function
$ \phi(\, \cdot \, ; a) \colon \RR \to \RR $
with parameter $ a \ge 0 $.
We shall assume $ \phi $ has the following properties:
* $ \phi( \, \cdot \,; a) $ is continuous on $ \RR $
* $ \phi(\, \cdot \,; a) $ is twice continuously differentiable, increasing, and concave on $ \RR_+ $
* $ \phi(0; a) = 0 $
* $ \phi(t; 0) = \abs{t} $
* $ \phi(-t; a) = \phi(t; a) $
* $ \phi'(0^+; a) = 1 $
* $ \phi''(0^+; a) = -a $
* $ \phi''(t; a) \geq -a $ for all $ t \neq 0 $
* $ \phi(t; a) $ is decreasing and convex in $ a $.
* $ \phi(t; a) = (b/a) \, \phi\big(at / b; b\big) $ for $ a, b > 0 $
It follows from symmetry that $ \phi'(-t) = -\phi'(t) $ and $ \phi''(-t) = \phi''(t) $.
The scaling property of $ \phi $ also induces a scaling property of $ \phi' $ and $ \phi'' $.
\begin{align}
\label{eq:phidscale}
\phi'(t; a) & = \phi'(a t / b; b),
\\
\label{eq:phiddscale}
\phi''( t ; a)
& =
(a/b) \, \phi''( a t / b ; b).
\end{align}
Table <ref> lists several penalty functions that satisfy the above properties.
For example,
the rational <cit.>,
logarithmic, and arctangent functions
(when suitably normalized).
The arctangent penalty is illustrated in Fig. <ref> for several values of $ a $.
For larger $ a $, the penalty functions increase more slowly, are more concave on the positive real line,
and induce sparsity more strongly (i.e., by mildly penalizing large values).
A comparison of the three penalties listed in Table <ref> is illustrated in
Figure 1 of Ref. <cit.> for a fixed value of $ a $.
Of the three penalties,
the arctangent penalty increases the slowest for a fixed value of $ a $.
We mention that we do not use the simpler form of the arctangent penalty,
$ \phi( t ; a ) = ( 1 / a ) \tan\inv( a \abs{t} ) $,
as it does not satisfy $ \phi''(0^+, a) = -a $ which is property P7 listed above.
Penalty $ \phi( t ; a) $ for several values of $ a $.
A univariate penalty $ \phi $, its corresponding function $ s $,
and the first and second-order derivatives of $ s $.
The function $ s $ is twice continuously differentiable and concave.
Corresponding to a penalty $ \phi $ having the above properties, we define a smooth concave function.
Let $ \phi \colon\RR \to \RR $ be a penalty function satisfying the properties listed above.
For $ a \ge 0 $,
we define $ s \colon \RR \to \RR $,
\begin{equation}
\label{eq:defs}
s( t ; a ) = \phi( t ; a ) - \abs{ t }.
\end{equation}
Figure <ref> illustrates the function $ s $ corresponding to the arctangent penalty for $ a = 0.3 $.
The following proposition follows straightforwardly <cit.>.
Let $ a \ge 0 $.
Let $ \phi \colon\RR \to \RR $ be a penalty function satisfying the properties listed above.
The function $ s \colon\RR \to \RR $ in Definition <ref>
is twice continuously differentiable, concave, and satisfies
\begin{equation}
\label{eq:sbound}
-a \le s''( t ; a) \le 0.
\end{equation}
This property will be of particular importance.
Note that the value $ \phi''(0) $ is not defined since $ \phi $ is not differentiable at zero.
But the value $ s''(0) $ is defined (and is equal to $ -a $).
\begin{equation}
\label{eq:s2a}
s''( t ; a ) =
\begin{cases}
-a, \ \ & t = 0
\\
\phi''( t ; a ) & t \neq 0.
\end{cases}
\end{equation}
Also, although $ \phi'(0) $ is not defined [because $ \phi'(0^+) = 1 $ and $ \phi'(0^-) = -1 $],
the value $ s'(0) $ is defined [$ s'(0) = 0 $].
Many of the properties listed above for $ \phi $ are inherited by $ s $,
such as the symmetry and scaling properties:
\begin{align}
s(-t ; a) & = s(t ; a)
\\
\label{eq:scaling_s}
s(t; a) & = (b/a) \, s\big(at / b; b\big)
\\
\label{eq:ssp1}
s'(t; a) & = s'\big(at / b; b\big)
\\
\label{eq:scaling_sd2}
s''(t; a) & = (a/b) \, s''\big(at / b; b\big)
\end{align}
The following proposition is proven in Appendix <ref>.
Let $ \phi \colon\RR \to \RR $ satisfy the properties listed above.
Let $ s \colon\RR \to \RR $ be given by Definition <ref>.
Let $ \lam > 0 $.
If $ 0 \le a \le 1/\lam $,
then the functions
$ g \colon\RR \to \RR $
\begin{equation}
\label{eq:defgprop}
g( t ) =
\half t ^2 + \lam s( t ; a)
\end{equation}
$ f \colon\RR \to \RR $
\begin{equation}
f( t ) =
\half t ^2 + \lam \phi( t ; a)
\end{equation}
are convex functions.
Regions $ \SP_1 $ through $ \SP_4 $ in Definition <ref>.
§ BIVARIATE CONCAVE FUNCTION
The bivariate penalty to be given in Sec. <ref> will be defined in terms of a concave bivariate function $ S $.
The role of $ S $, in describing the bivariate penalty, will be analogous to
the role of $ s $ in describing the univariate penalty $ \phi $.
Accordingly, the properties of $ S $ will be important for the properties of the bivariate penalty.
Let $ a = (a_1, a_2) $ with $ a_i \ge 0 $.
Let $ \phi \colon\RR \to \RR $ be a univariate penalty function having the properties listed in Sec. <ref>.
Let $ s \colon\RR \to \RR $ be given by Definition <ref>.
If at least one of $ \{ a_1, a_2 \} $ is non-zero,
we define the function
$ S \colon \RR^2 \to \RR $ as
\[
% \label{eq:defS}
S( x ; a ) =
\begin{cases}
s( x_1 + r x_2 ; \alpha ) + (1 - r) \, s( x_2 ; a_1 ) , & x \in \SP_1
\\
s( r x_1 + x_2 ; \alpha ) + (1 - r) \, s( x_1 ; a_1 ) , & x \in \SP_2
\\
s( r x_1 + x_2 ; \alpha ) + (1 + r) \, s( x_1 ; a_2 ) , & x \in \SP_3
\\
s( x_1 + r x_2 ; \alpha ) + (1 + r) \, s( x_2 ; a_2 ) , & x \in \SP_4
\end{cases}
\]
\begin{equation}
\label{eq:ralpha}
\alpha = \frac{ a_1 + a_2 }{2},
\qquad
r = \frac{ a_1 - a_2 }{ a_1 + a_2 }
\end{equation}
and sets $ \SP_i \subset \RR^2 $ are
\begin{align}
\SP_1 & = \{ x \in \RR^2 \mid x_2 ( x_1 - x_2 ) \ge 0 \}
\\
\SP_2 & = \{ x \in \RR^2 \mid x_1 (x_1 - x_2) \le 0 \}
\\
\SP_3 & = \{ x \in \RR^2 \mid x_1 (x_1 + x_2) \le 0 \}
\\
\SP_4 & = \{ x \in \RR^2 \mid x_2 (x_1 + x_2) \le 0 \}
\end{align}
as shown in Fig. <ref>.
If both $ a_i = 0 $, we define $ S(x ; 0 ) = 0 $.
The non-negative parameters $ a_i $ characterize
how strongly concave $ S $ is.
Figure <ref>
illustrates $ S $ for the parameter values $ a_1 = 1.5 $ and $ a_2 = 0.3 $.
The level sets of $ S $ are not ellipses, even though they appear ellipsoidal.
The function $ S $ has three symmetries:
\begin{align}
\label{eq:sym1}
S( x_1, x_2 ; a )
& =
S( x_2, x_1 ; a )
\\
\label{eq:sym2}
% S( x_1, x_2 ; a )
& =
S( -x_1, -x_2 ; a )
\\
\label{eq:sym3}
% S( x_1, x_2 ; a )
& =
S( -x_2, -x_1 ; a )
\end{align}
(any two of which imply the remaining one).
$ S $ is symmetric with respect to the origin and the two lines $ x_1 = x_2 $ and $ x_1 = -x_2 $.
These symmetries follow directly from Definition <ref>
and from the symmetry of the univariate function $ s $.
Non-separable concave function $ S $ in Definition <ref>.
The following lemmas are proven in the Appendix.
It will be useful in the proofs to note some identities.
First, note that $ S(0; a) = 0 $.
From the definitions of $ \alpha $ and $ r $ in (<ref>), we have:
r ≤1,
(1 + r) α = a_1
r α = (a_1 - a_2) / 2,
(1 - r) α = a_2.
The bivariate function $ S \colon \RR^2 \to \RR $
in Definition <ref>
is twice continuously differentiable and concave on $ \RR^2 $.
Let $ a = (a_1, a_2) $ with $ a_i \ge 0 $.
The Hessian of the bivariate function $ S $ in Definition <ref>
\begin{equation}
\label{eq:ellipse}
- \half
\begin{bmatrix}
a_1 + a_2 & a_1 - a_2
\\
a_1 - a_2 & a_1 + a_2
\end{bmatrix}
\mle
\nabla^2 S(x ; a )
\mle
\quad
\text{for all} \
x \in \RR^2.
\end{equation}
\begin{equation}
\label{eq:ellipseB}
\mle
\nabla^2 S(x ; a )
\mle
\quad
\text{for all} \
x \in \RR^2
\end{equation}
where $ K(a) $ is defined by its eigenvalue decomposition
\begin{equation}
\begin{bmatrix}
a_1 & 0
\\
0 & a_2
\end{bmatrix}
\half
\begin{bmatrix}
a_1 + a_2 & a_1 - a_2
\\
a_1 - a_2 & a_1 + a_2
\end{bmatrix}
\end{equation}
where $ Q $ is the orthonormal matrix
defined in (<ref>).
Furthermore, the lower bound is attained at $ x = 0 $, i.e.,
\begin{equation}
\label{eq:SKZ}
\nabla^2 S( 0 ; a ) = -K(a).
\end{equation}
Lemma <ref> states that
$ S $ is maximally concave at the origin.
The lemma also gives the Hessian at the origin in terms of the parameters $ a_i $.
Lemma <ref> is a key result for the subsequent results.
Lemma <ref> can be illustrated in terms of ellipses.
If $ M $ is a positive semidefinite matrix, then the set $ \ellip[M] = \{ x : x\tr M\inv x \le 1 \} $ is an ellipsoid <cit.>.
In addition, $ M_1 \mle M_2 $ if and only if $ \ellip[M_1] \subseteq \ellip[M_2] $.
In Fig. <ref>,
we set $ a_1 = 1.5 $ and $ a_2 = 0.3 $.
The ellipses corresponding to $ K(a) $ and $ -\nabla^2 S( x ; a ) $
are shown in gray and black, respectively.
For each $ x $, the black ellipse is contained within the gray ellipse, illustrating (<ref>).
For large $ x $,
the black ellipse shrinks,
indicating that the function $ S $ becomes less concave away from the origin.
The shrinkage behavior of the ellipse for large $ x_i $ is different in different quadrants:
it shrinks faster in quadrants 1 and 3 than in quadrants 2 and 4.
At $ x = 0 $, the black and gray ellipses coincide,
reflecting the fact that the Hessian of $ S $ is equal to $ -K(a) $ at the origin (<ref>).
Illustration of Lemma <ref>.
The black ellipses are inside the gray ellipses,
indicating that
$ 0.5 x\tr K x + S(x; a )$ is convex.
Let $ a = (a_1, a_2) $ with $ a_i \ge 0 $.
Let $ S \colon \RR^2 \to \RR $ be the function
in Definition <ref>.
$ K(\gamma) = Q \, \Gamma \, Q\tr $
with eigenvalues $ \gamma_i \ge 0 $
be the positive semidefinite matrix
defined in (<ref>).
The function $ g \colon \RR^2 \to \RR $,
\begin{equation}
\label{eq:defg}
g(x; a, \gamma) = \half x\tr K(\gamma) \, x + \lam \, S(x; a),
\
\lam > 0,
\end{equation}
is convex
\begin{equation}
\label{eq:aicond}
0 \le a_1 \le \gamma_1 / \lam,
\qquad
0 \le a_2 \le \gamma_2 / \lam.
\end{equation}
Since $ g $ is twice continuously differentiable, it is sufficient to show
the Hessian of $ g $ is positive semidefinite.
The Hessian of $ g $ is given by
\begin{align}
\label{eq:deffH}
\nabla^2 g(x; a , \gamma )
& = K(\gamma) + \lam [ \nabla^2 S(x; a) ]
\\
& = Q\tr \Gamma \, Q + \lam [ \nabla^2 S(x; a) ].
\end{align}
From Lemma <ref> it follows that
\begin{equation}
\nabla^2 g(x; a, \gamma )
\mge
\begin{bmatrix}
\gamma_1 - \lam a_1 & 0
\\
0 & \gamma_2 - \lam a_2
\end{bmatrix}
\end{equation}
$ g(x; a, \gamma ) $ is convex if $ \gamma_i - \lam a_i \ge 0 $ for $ i = 1, 2 $.
This proves the result.
Theorem <ref> is illustrated in Fig. <ref>.
In this example, we set
$ \gamma_1 = 1.5 $, $ \gamma_2 = 0.3 $, and $ \lambda = 15.0 $.
Hence, the critical parameters are $ a_1^{*} = \gamma_1 / \lam = 0.1 $ and $ a_2^{*} = \gamma_2 / \lam = 0.02 $.
If either $ a_1 $ or $ a_2 $ is greater than the
respective critical value, then the function $ g $ in (<ref>) will be non-convex.
In Fig. <ref>(a),
we set
$ a_i = 0.9 \gamma_i / \lam < a_i^{*} $
to satisfy condition (<ref>),
hence $ g $ is convex.
In contrast,
in Fig. <ref>(b),
we set
$ a_i = 1.1 \gamma_i / \lam > a_i^{*} $
violating condition (<ref>),
$ g $ is non-convex.
The lack of convexity can be recognized in both the surface and contour plots.
Illustration of convexity condition (<ref>).
Function $ g $ is convex as $ a_i $ satisfy (<ref>).
Function $ g $ is not convex as $ a_i $ violate (<ref>).
§ BIVARIATE PENALTIES
In this section, we define a non-convex non-separable bivariate penalty.
Our intention is to strongly induce sparsity in
solutions of problem (<ref>) while maintaining the convexity of the problem.
The penalty is parameterized by two non-negative parameters $ a_1 $ and $ a_2 $,
which we restrict so as to ensure convexity of the objective function.
Let $ a = (a_1, a_2) $ with $ a_i \ge 0 $.
Let $ \phi \colon\RR \to \RR $ be a univariate penalty function having the properties listed in Sec. <ref>.
Let $ S \colon\RR^2 \to \RR $ be
the corresponding function in Definition <ref>.
We define the bivariate penalty function
$ \psi \colon \RR^2 \to \RR $ as
\begin{equation}
\label{eq:defpsi}
\psi( x ; a )
= S( x ; a ) + \norm{ x }_1.
\end{equation}
If $ a_1 \neq a_2 $, then the penalty $ \psi $ is non-separable.
Figure <ref>
illustrates $ \psi $ for the parameter values $ a_1 = 1.5 $ and $ a_2 = 0.3 $.
The degree of non-convexity differs in different quadrants.
Note in Fig. <ref> that
the contours of $ \psi $ resemble those of the separable non-convex penalty in Fig. <ref>,
but the curvature is more pronounced in quadrants 2 and 4
and less pronounced in quadrants 1 and 3.
The parameters $ a_1 $ and $ a_2 $ determine the precise behavior of the penalty.
Non-separable non-convex penalty $ \psi $ in Definition <ref>.
It is informative to consider special cases of the bivariate penalty.
If $ a_1 > 0 $ and $ a_2 = 0 $,
then $ \psi $ simplifies to
\begin{equation}
\label{eq:pensing}
\psi(x) = \abs{x_1} + \abs{x_2} + \phi( x_1 + x_2 ; a_1/2) - \abs{ x_1 + x_2 }.
\end{equation}
If $ a_1 = a_2 $, then the penalty reduces to a separable function,
$ \psi(x) = \phi(x_1; a_1) + \phi(x_2; a_1) $
(see Fig. <ref>(b)).
If $ a_1 = a_2 = 0 $,
then it further reduces to the norm,
$ \psi(x) = \abs{x_1} + \abs{x_2} $
(see Fig. <ref>(a)).
In any case,
if either $ a_1 $ or $ a_2 $ is positive, then $ \psi $ is non-convex.
The following theorem, based on Theorem <ref>,
states how to restrict the parameters $ a_i $
to ensure problem (<ref>) is convex.
Let $ \psi \colon \RR^2 \to \RR $ be the bivariate penalty in Definition <ref>.
Suppose $ H\tr H = Q\tr \Gamma Q $
where $ Q $ and $ \Gamma $ are given by (<ref>).
If $ a = (a_1, a_2) $
$ 0 \le a_i \le \gamma_i / \lam $,
then the bivariate objective function $ f := \RR^2 \to \RR $,
\begin{equation}
\label{eq:biofun}
f(x ; a ) = \half \norm{ y - H x }_2^2 + \lam \, \psi(x; a),
\end{equation}
is convex.
We write
\begin{equation}
f(x ; a ) = g(x ; a ) + \half y\tr y - y\tr H x + \lam \, \norm{ x }_1
\end{equation}
where $ g $ is given by (<ref>) with $ K = H\tr H $ therein.
From Theorem <ref>, $ g $ is convex.
Hence, $ f $ is convex
because it is the sum of convex functions.
Theorem <ref> gives a range for parameters $ a_1 $ and $ a_2 $
to ensure the objective function $ f $ is convex.
Precisely, the parameters should be bounded, respectively, by the eigenvalues of $ (1/\lam) H\tr H $.
To maximally induce sparsity, the parameters should be set to the maximal (critical) values,
$ a_i = \gamma_i/\lam $.
Note that, even when the matrix $ H $ is singular (i.e., $ \gamma_1 = 0 $ or $ \gamma_2 = 0 $),
the bivariate penalty can be non-convex without spoiling the convexity
of the objective function $ f $
(if at least one of $ \gamma_i $ is positive).
In other words,
we need not sacrifice the convexity of the objective function $ f $
in order to use sparsity-inducing non-convex penalties,
even when $ H $ is singular.
This is an impossibility when the penalty $ \psi $ is a separable function.
Other bivariate penalties can be defined that ensure the objective function is convex;
however, the one defined here
satisfies a further property we think should be required of a bivariate penalty.
Namely, the proposed bivariate penalty lies between
the two separable penalties corresponding to the
minimum and maximum parameters $ a_i $.
Let $ a = (a_1, a_2) $ with $ a_i \ge 0 $.
$ a_{\min} = \min\{ a_1, a_ 2 \} $
$ a_{\max} = \max\{ a_1, a_ 2 \} $.
The bivariate penalty $ \psi $ in Definition <ref> satisfies
\begin{equation}
\label{eq:bounds}
\phi(x_1; a_{\max}) + \phi(x_2; a_{\max})
\le \psi(x; a) \le
\phi(x_1; a_{\min}) + \phi(x_2; a_{\min}).
\end{equation}
The theorem is proven in Appendix <ref>.
(The supplemental material has an animated illustration of these bounds.)
Illustration of Theorem <ref>.
The non-separable penalty $ \psi $ lies between two separable penalties.
The inequality in Theorem <ref> is tight:
the lower and upper bounds are individually satisfied with equality on the lines $ \{ x = (t, t) \} $
and $ \{ x = (t, -t) \} $,
as illustrated in Fig. <ref>.
Note that when $ a_1 = a_2 $ (i.e., $ a_{\min} = a_{\max} $),
the theorem requires the penalty $ \psi $ to be separable.
Indeed, the penalty (<ref>) is separable when $ a_1 = a_2 $.
We think a bivariate penalty should satisfy the inequality of Theorem <ref> for the following reason.
In this work, we aim to induce pure sparsity (i.e., not structured sparsity, etc.).
Therefore, when $ H\tr H $ is a diagonal matrix
we should use a separable penalty.
(A separable penalty best reflects an iid prior.)
It follows that when $ H\tr H = \gamma_1 I $,
the most suitable penalty (maintaining convexity of $ f $)
is the separable one:
$ \phi(x_1, \gamma_1/\lam) + \phi(x_2, \gamma_1/\lam) $.
A parameterized bivariate penalty should recover this
separable penalty as a special case.
if $ H\tr H = \gamma_2 I $ with $ \gamma_2 < \gamma_1 $, then
the most suitable penalty is again a separable one:
$ \phi(x_1, \gamma_2/\lam) + \phi(x_2, \gamma_2/\lam) $.
But $ \gamma_2 < \gamma_1 $
means the corresponding data fidelity term is less strongly convex
and thus the penalty term must be less strongly non-convex.
Consequently, we must have
\[
\phi(x_1, \gamma_1/\lam) + \phi(x_2, \gamma_1/\lam)
\phi(x_1, \gamma_2/\lam) + \phi(x_2, \gamma_2/\lam).
\]
When $ H\tr H $
has distinct eigenvalues $ \gamma_1 $ and $ \gamma_2 $,
the most suitable bivariate penalty should lie between these two separable penalties.
Theorem <ref> assures this.
If not suitably designed and utilized, it is conceivable that a non-separable penalty may
lead to correlation or structure in the estimated signal that is not present in original sparse signal.
To avoid unintentionally inducing correlation in the estimated signal,
it seems reasonable that
the bivariate penalty should
exhibit some similarity to the corresponding separable penalties
(that reflect unstructured sparsity).
Theorem <ref> indicates the bivariate penalty (<ref>)
conforms to the relevant separable penalties.
It is still possible that some erroneous correlation might be introduced,
but such correlation is not evident in the experimental results.
We attribute this to Theorem <ref>.
One may question the legitimacy of a method wherein penalty parameters
are set according to the data fidelity term.
Conventionally, the penalty term should reflect prior knowledge
of the signal to be estimated;
it should not depend on $ H $, which represents the observation model.
(This is formalized in the Bayesian perspective
where the objective function corresponds to a likelihood function
and the penalty term corresponds to a prior).
The approach taken here, wherein the parameters
of the penalty term are based on properties of $ H $,
appears to violate this principle.
the common practice of
restricting the penalty to be convex also violates this principle.
Probability densities
generalized Gaussian <cit.>,
mixture models <cit.>,
Bessel-K <cit.>, and $\alpha$-stable <cit.>),
that accurately model sparsity,
correspond to non-convex penalties.
Using the norm as a penalty corresponds to the Laplace distribution (a relatively weak sparsity model).
The proposed bivariate sparse regularization (BISR) approach
is simply intended to follow a sparsity prior more closely.
§.§ Separable penalties
To clarify the value of non-separable regularization,
we note a limitation of separable penalties.
Let the univariate penalty $ \phi \colon\RR \to \RR $ satisfy the properties of Sec. <ref>.
The objective function $ f := \RR^2 \to \RR $,
\begin{equation}
f(x ; a ) = \half \norm{ y - H x }_2^2
+ \lam \, \phi(x_1; a)
+ \lam \, \phi(x_2; a),
\end{equation}
with $ \lam > 0 $ and $ a \ge 0 $,
is convex only if
\begin{equation}
\phi''(0^+) \ge -(1/\lam) \min\{ \gamma_1, \gamma_2 \},
\end{equation}
or equivalently,
$ 0 \le a \le \min \{ \gamma_1, \gamma_2 \} / \lam $,
where $ \gamma_i $ are the eigenvalues of $ H\tr H $.
Let $ u \in \RR^2 $ be an eigenvector of $ H\tr H $ corresponding to
its minimum eigenvalue $ \gamma_{\min} $, i.e.,
\begin{equation}
H\tr H u = \gamma_{\min} u.
\end{equation}
Consider $ f $ on a line in the direction of $ u $.
define $ g \colon \RR \to \RR $ as
\begin{align}
g(t) & = f( t u; a )
\\
& =
\half \norm{ y - t H u }_2^2
+ \lam \, \phi( t u_1 ; a)
+ \lam \, \phi( t u_2 ; a).
\end{align}
We will show that $ g $ is not convex when $ a > \gamma_{\min} $.
It will follow that $ f $ is not convex,
because the restriction of a multivariate convex function to any line must also be convex.
By the properties of $ \phi $,
the function $ g $ is twice continuously differentiable on $ \RR_+ $
and its second derivative is given by
\begin{align}
& =
u H\tr H u
+ \lam u_1^2 \, \phi''( t u_1 ; a)
+ \lam u_2^2 \, \phi''( t u_2 ; a)
\\
& =
\gamma_{\min} u\tr u
+ \lam u_1^2 \, \phi''( t u_1 ; a)
+ \lam u_2^2 \, \phi''( t u_2 ; a).
\end{align}
Since $ \phi''( 0^+ ; a ) = -a $ is a defining property of $ \phi $,
we have
\begin{align}
g''(0^+) & =
\gamma_{\min} u\tr u
- \lam u_1^2 \, a
- \lam u_2^2 \, a
\\
& =
(\gamma_{\min} - \lam a) \, u\tr u.
\end{align}
Hence, convexity of $ g $ requires $ \gamma_{\min} - \lam a \ge 0 $;
i.e., $ a \le \gamma_{\min} / \lam $.
According to Lemma <ref>, a separable non-convex penalty,
that ensures convexity of the objective function, is limited by the minimum eigenvalue of $ H\tr H $.
It cannot exploit the greater eigenvalue.
This is unfavorable, because when one of the eigenvalues is close to zero,
a separable penalty can be only mildly non-convex
and provides negligible improvement relative to the norm;
when $ H\tr H $ is singular,
we recover the norm.
In contrast,
a non-separable penalty can exploit both eigenvalues independently.
non-separable penalties are most advantageous when the eigenvalues of $ H\tr H $ are quite different in value.
§ SPARSE RECONSTRUCTION
Practical problems in signal processing involve far more than two variables.
Therefore, the proposed bivariate penalty (<ref>) and convexity condition (<ref>)
are of little practical use on their own.
In this section we show how they can be used to solve an $ N $-point
linear inverse problem (with $ N > 2 $).
We consider the problem of estimating a signal $ x \in \RR^N $ given $ y $,
\begin{equation}
\label{eq:obsmod}
y = H x + w
\end{equation}
$ H $ is a known linear operator,
$ x $ is known to be sparse, and
$ w $ is additive white Gaussian noise (AWGN).
We formulate the estimation of $ x $ as an optimization problem
with bivariate sparse regularization (BISR),
\begin{equation}
% \label{eq:spdeconv}
\label{eq:defFdeconv}
\hat x = \arg \min_{x \in \RR^N } \;
\biggl\{
F(x) =
\half \norm{ y - H x }_2^2 + \frac{\lam}{2} \sum_n \psi( (x_{n-1}, x_n) ; a)
\biggr\},
\quad
\lam > 0
\end{equation}
$ a = (a_1, a_2) $
and $ \psi \colon \RR^2 \to \RR $ is the proposed bivariate penalty (<ref>).
In the penalty term,
the first and last signal values pairs, $ ( x_0 , x_1 ) $ and $ ( x_{N}, x_{N+1} ) $, straddle
the end-points of $ x $.
As noted in Sec. <ref>,
we define $ x_n = 0 $ for $ n \notin \{ 1 , 2 , \dots, N \} $,
which simplifies subsequent notation.
If $ a_1 = a_2 $,
then the bivariate penalty is separable,
$ \psi(u; a) = \phi(u_1; a_1) + \phi(u_2; a_1) $,
and the $ N $-point penalty term in (<ref>)
reduces to $ \lam \sum_n \phi(x_n, a_1) $.
we recover the standard (separable) formulation of sparse regularization.
In particular,
if $ a_1 = a_2 = 0 $,
$ \psi( u ; 0 ) = \abs{ u_1 } + \abs{ u_2 } $
the $ N $-point penalty term reduces to $ \lam \norm{ x }_1 $,
the classical sparsity-inducing convex penalty.
In order to induce sparsity more effectively,
we allow $ \psi $ to be non-separable; i.e., $ a_1 \neq a_2 $.
To that end, the following section addresses the problem of how to set $ a_1 $ and $ a_2 $
in the bivariate penalty $ \psi $ to ensure convexity of the $ N $-variate objective function $ F $ in (<ref>).
§.§ Convexity condition
Let $ F \colon \RR^N \to \RR $ be defined in (<ref>)
$ \psi \colon \RR^2 \to \RR $ is a parameterized bivariate penalty as defined in Definition <ref>.
Let $ P $ be a positive semidefinite symmetric tridiagonal Toeplitz matrix,
\begin{equation}
\label{eq:defP}
P =
\begin{bmatrix}
p_0 & p_1 & & & \\
p_1 & p_0 & p_1 & & \\
% & p_1 & p_0 & p_1 & \\
& \ddots & \ddots & \ddots \\
& & p_1 & p_0 & p_1 \\
& & & p_1 & p_0
\end{bmatrix},
\end{equation}
such that
$ 0 \mle P \mle H\tr H $.
If the bivariate function $ f \colon \RR^2 \to \RR $
defined as
\begin{equation}
\label{eq:defp}
f(u) = \half u\tr
\begin{bmatrix}
p_0 & 2 p_1 \\
2 p_1 & p_0
\end{bmatrix}
u + \lam \psi( u; a )
\end{equation}
is convex,
then $ F $ is convex.
The lemma is proven in Appendix <ref>.
According to the lemma,
it is sufficient to restrict $ \psi $ so as to ensure convexity of the bivariate function $ f $ in (<ref>).
the allowed penalty parameters $ a_i $
can be determined from the tridiagonal matrix $ P $.
Using Theorem <ref> and Lemma <ref>, we obtain Theorem <ref>.
Let $ F \colon \RR^N \to \RR $ be defined in (<ref>)
$ \psi \colon \RR^2 \to \RR $ is a parameterized bivariate penalty as defined in Definition <ref>.
Let $ P $ be a symmetric tridiagonal Toeplitz matrix (<ref>)
$ 0 \mle P \mle H\tr H $.
\begin{equation}
\label{eq:aipconds}
0 \le a_1 \le (p_0 + 2 p_1)/\lam,
\quad
0 \le a_2 \le (p_0 - 2 p_1)/\lam,
\end{equation}
then $ F $ in (<ref>) is convex.
Note the eigenvalue value decomposition
\begin{equation}
\begin{bmatrix}
p_0 & 2 p_1 \\
2 p_1 & p_0
\end{bmatrix}
\begin{bmatrix}
p_0 + 2 p_1 & 0
\\
0 & p_0 - 2 p_1
\end{bmatrix}
\end{equation}
where $ Q $ is the orthonormal matrix (<ref>).
Hence, $ f $ in (<ref>) can be written as
\begin{equation}
f(u) =
\frac{1}{2}
\begin{bmatrix}
p_0 + 2 p_1 & 0
\\
0 & p_0 - 2 p_1
\end{bmatrix}
\lam \,
\psi( u; a ).
\end{equation}
By Theorem <ref>,
if $ 0 \le a_1 \le (p_0 + 2 p_1)/\lam $
and $ 0 \le a_2 \le (p_0 - 2 p_1)/\lam $,
then $ f $ is convex.
It follows from Lemma <ref> that $ F $ in (<ref>) is convex.
§.§ Optimality condition
In this section, we derive an explicit condition to
verify the optimality of a prospective minimizer of
the objective function $ F $ in (<ref>).
The optimality condition is also useful for monitoring
the convergence of an optimization algorithm
(see the animation in the supplemental material).
The general condition to characterize minimizers of a convex
function is expressed in terms of the subdifferential.
If $ F $ is convex, then $ x\opt \in \RR^N $ is a minimizer if and only if
$ 0 \in \partial F( x \opt ) $
where $ \partial F $ is the subdifferential of $ F $.
We seek an expression for the subdifferential of the objective function $ F $.
The function $ F $ in (<ref>)
has a regularization term that is non-differentiable, non-convex, and non-separable.
But with the aid of (<ref>),
we may write the regularization term as:
\begin{align}
\half \sum_{n} \psi( ( x_{n-1} , x_{n} ) ; a )
& = \half \sum_{n} \Bigl[ S( ( x_{n-1} , x_{n} ) ; a )
+ \norm{ ( x_{n-1} , x_{n} ) }_1 \Bigr]
\\
& = \half \sum_{n} \Bigl[ S( ( x_{n-1} , x_{n} ) ; a )
+ \abs{ x_{n-1} } + \abs{ x_{n} } \Bigr]
\\
\label{eq:RS}
& = \norm{ x }_1
+ \half \sum_{n} S( ( x_{n-1} , x_{n} ) ; a )
\end{align}
where $ x_n = 0 $ for $ n \notin \{ 1 , 2 , \dots, N \} $.
We define
$ \SN \colon \RR^N \to \RR $
\begin{equation}
\label{eq:defSN}
\SN( x ; a ) = \half \sum_{n} S( ( x_{n-1} , x_{n} ) ; a ) .
\end{equation}
The function $ \SN $ is differentiable, it being the sum of differentiable functions.
Using (<ref>), we may express the objective function $ F $ in (<ref>) as
\begin{equation}
\label{eq:Fsimp}
F(x) =
\half \norm{ y - H x }_2^2
\lam \SN( x ; a )
\lam \norm{ x }_1.
\end{equation}
The benefit of (<ref>)
compared to (<ref>)
is that the regularization term (which is non-differentiable, non-convex, and non-separable)
is separated into
a differentiable part and a convex separable part.
The $ \SN $ term is differentiable and its gradient is easily evaluated.
The norm is separable and convex and its subdifferential is easily evaluated.
The gradient of $ \SN $ is given by
\begin{equation}
\label{eq:SNgrad}
[ \nabla \SN( x ; a ) ]_n
\half
S_1( (x_{n}, x_{n+1}) ; a)
\half
S_2( (x_{n-1}, x_n) ; a)
\end{equation}
where $ S_i $ is the partial derivative of $ S((x_1, x_2)) $ with respect to $ x_i $.
They are tabulated in (<ref>) and (<ref>).
The subdifferential of the norm is separable <cit.>,
\begin{equation}
\partial \norm{ x }_1 = \sign( x_1 ) \times \cdots \times \sign ( x_N )
\end{equation}
where $ \sign $ is the set-valued signum function
\begin{equation}
\sign( t ) :=
\begin{cases}
\{ 1 \}, & t > 0
\\
[-1, 1], \ & t = 0
\\
\{ -1 \}, & t < 0.
\end{cases}
\end{equation}
Since the first two terms of (<ref>) are differentiable, the subdifferential of $ F $ is
\begin{equation}
\partial F(x) =
H\tr ( H x - y )
\lam \nabla \SN( x ; a )
\lam \partial \norm{ x }_1.
\end{equation}
Hence the condition $ 0 \in \partial F(x\opt) $
can be expressed as
\begin{equation}
% \frac{ 1 }{ \lam }
( 1 / \lam )
H\tr ( y - H x\opt )
\nabla \SN( x\opt ; a )
\in
\partial \norm{ x\opt }_1.
\end{equation}
Expressing this condition component-wise,
we have the following result.
If $ a = (a_1, a_2) $
is chosen so that
the objective function $ F $ in (<ref>)
is convex,
then $ x\opt $ minimizes $ F $
if and only if
\begin{equation}
\label{eq:optcond}
\frac{ 1 }{ \lam }
[ H\tr ( y - H x\opt ) ]_n
[ \nabla \SN( x\opt ; a ) ]_n
\in
\sign ( x\opt_n ),
\ \
n = 1, \dots, N.
\end{equation}
This condition can be depicted using a scatter plot
as in Fig. <ref> below.
The points in the scatter plot
show the left-hand-side of (<ref>) versus
$ x_n $ for $ n = 1, \dots, N $.
A signal $ x\opt $ is a minimizer of $ F $ if and only if
the points in the scatter plot lie on the graph of the set-valued signum function
(e.g., Fig. <ref>).
§.§ Sparse Deconvolution
We apply Theorem <ref> to the sparse deconvolution problem.
In this case, the linear operator $ H $ represents convolution,
\begin{equation}
[H x]_n = \sum_k h_{n - k} \, x_{ k }.
\end{equation}
That is, $ H $ is a Toeplitz matrix.
It represents a linear time-invariant (LTI) system
with frequency response given by
the Fourier transform of $ h $,
\begin{equation}
\label{eq:Hom}
H( \om ) = \sum_n h_n \, \myE^{ - \myJ \om n}.
\end{equation}
the matrix $ P $ in (<ref>) represents
an LTI system with a real-valued frequency response,
\begin{align}
P( \om )
& = p_1 \myE^{-\myJ \om} + p_0 + p_1 \myE^{\myJ \om}
\\
\label{eq:Pcos}
& = p_0 + 2 p_1 \cos(\om).
\end{align}
Specializing Theorem <ref> to the problem of deconvolution,
we have the following result.
Let $ H $ in (<ref>) represent convolution.
Let $ P $ in (<ref>) satisfy
\begin{equation}
\label{eq:ZPH}
0 \le P( \om) \le \abs{ H( \om ) }^2,
\ \
\forall \om,
\end{equation}
where $ H(\om) $ is given by (<ref>).
If $ 0 \le a_1 \le P(0)/\lam $
and $ 0 \le a_2 \le P(\pi)/\lam $,
then $ F $ in (<ref>) is convex.
Using the convolution property of the discrete-time Fourier transform, condition (<ref>) is equivalent to $ 0 \mle P_{\infty} \mle H_{\infty}\tr H_{\infty} $,
where these are doubly-infinite Toeplitz matrices corresponding to discrete-time signals defined on $ \ZZ $.
Since any principal sub-matrix of a positive semidefinite matrix is also positive semidefinite,
the inequality is also true for finite matrices $ P $ and $ H\tr H $,
which can be recognized as principal sub-matrices of the corresponding doubly-infinite matrices.
Therefore, $ 0 \mle P \mle H\tr H $,
and by Theorem <ref>,
the objective function $ F $ is convex
if $ a_i $ satisfy (<ref>).
Noting that $ P(0) = p_0 + 2 p_1 $ and $ P(\pi) = p_0 - 2 p_1 $
yields the result.
To induce sparsity as strongly as possible,
$ P( \om ) $ should be as close as possible to the upper bound $ \abs{ H( \om ) }^2 $.
The determination of $ P( \om ) $ satisfying such constraints can
be efficiently and exactly performed using semi-definite programming (SDP)
as described by Dumitrescu <cit.>.
Since $ P $ is low-order here, the SDP computation is negligible.
An example is illustrated in Fig. <ref>.
The impulse response $ h $
is shown in Fig. <ref>(a).
The square magnitude of the frequency response $ \abs{ H( \om ) }^2 $
is shown in Fig. <ref>(b).
The frequency response
P ( \om ) = 0.4 + 0.2 \cos(\om)
is real-valued, non-negative,
and approximates $ \abs{ H( \om ) }^2 $ from below.
According to Theorem <ref>, the objective function $ F $ is
convex if
$ 0 \le a_1 \le 0.6/ \lam $
$ 0 \le a_2 \le 0.2/ \lam $.
This filter will be used in Example 1 below (Sec. <ref>).
Filters $ H(\om) $ and $ P(\om) $ for Example 1.
Filters $ H(\om) $ and $ P(\om) $ for Example 2.
Another example is illustrated in Fig. <ref>.
The frequency response $ H( \om ) $ has a null at $ \om = \pi $.
Hence, the system $ H $ is not invertible.
Since $ H(\pi) = 0 $,
any $ P $ satisfying (<ref>) also has $ P(\pi) = 0 $.
We find $ P( \om ) = 0.38 ( 1 + \cos \om ) $ satisfies (<ref>);
see Fig. <ref>(b).
according to Theorem <ref>, the objective function $ F $ is
convex if $ 0 \le a_1 \le 0.76/ \lam $ and $ a_2 = 0 $.
For $ \{ a_1 > 0, a_2 = 0 \} $, the multivariate penalty is non-convex and non-separable.
For $ \{ a_1 = a_2 = 0 \} $, the penalty is simply the norm
(convex and separable).
This filter will be used in Example 2 below (Sec. <ref>).
In reference to the filters $ H $ illustrated in Figs. <ref> and <ref>,
it is informative to consider the case of a separable
If $ \psi $ is a separable penalty (i.e., $ a_1 = a_2 $),
then the objective function $ F $ is
convex only if $ 0 \le a_1 = a_2 \le \min_{\om} \abs{ H(\om) }^2 $.
For the filter of Fig. <ref> this leads to the constraint
$ 0 \le a_1 = a_2 \le 0.26 $,
meaning that the separable penalty may be non-convex (i.e., $ a_1 = a_2 > 0 $ is allowed).
On the other hand,
for the filter of Fig. <ref>
this leads to the constraint $ a_1 = a_2 = 0 $,
i.e., $ \psi(x, a) = \abs{ x_1 } + \abs{ x_2 } $.
When the filter $ H $ is not invertible,
the only non-convex penalties maintaining convexity of the objective function $ F $
are non-separable penalties.
Since inverse problems often involve singular or nearly singular operators $ H $,
this motivates the development of non-separable penalties as herein.
§ ALGORITHMS
§.§ Iterative L1 minimization
We present an iterative L1 norm minimization algorithm
to solve (<ref>).
The derivation is based on majorization-minimization (MM)
The MM principle consists of the iteration
\begin{equation}
\label{eq:defMM}
x\iter{k+1} \in \arg\min_{x} F\maj (x; x\iter{k})
\end{equation}
$ k $ is the iteration index and
$ F\maj $ denotes a majorizer of the objective function $ F $, i.e.,
\begin{align}
F\maj (x; v) & \ge F(x), \ \ \text{for all $x$, $v$}
\\
F\maj (v; v) & = F(v), \ \ \text{for all $v$.}
\end{align}
Majorization of concave function $ S $ by linear function $ S\maj $.
The function $ S \colon \RR^2 \to \RR $ defined in Lemma <ref>
is twice continuously differentiable and concave.
Therefore, a majorizer of $ S $
is given by
\begin{equation}
S\maj (x; v) = S(v) + [\nabla S(v)]\tr (x - v)
\end{equation}
where $ \nabla S $ is the gradient of $ S $.
See Fig. <ref>.
The majorizer $ S\maj (x; v) $ is linear in $ x $; hence convex in $ x $.
A majorizer of the bivariate penalty $ \psi $
is then obtained using (<ref>),
\begin{align}
\psi\maj (x; v) & = S\maj(x; v) + \norm{ x }_1
\\
& = S(v) + [\nabla S(v)]\tr (x - v) + \norm{ x }_1.
\end{align}
The majorizer $ \psi\maj (x; v) $ is convex in $ x $, it being a sum of convex functions.
Recall the objective function $ F $ in (<ref>)
can be expressed as (<ref>).
Therefore, a majorizer of the objective function $ F $ in (<ref>) is given by
\begin{equation}
F\maj(x; v) =
\half \norm{ y - H x }_2^2
\lam \SN\maj( x ; v ; a )
\lam \norm{ x }_1
\end{equation}
$ \SN\maj( x ; v ; a ) $ is a majorizer of $ \SN( x ; a ) $.
We recall the bivariate function $ S $
is concave and twice continuously differentiable.
Therefore, the $ N $-variate function $ \SN $
is concave as it is a sum of concave functions.
Likewise, $ \SN $ is twice continuously differentiable.
Therefore, a majorizer of $ \SN $ is given by
\begin{equation}
\SN\maj (x; v; a ) = \SN(v ; a) + [\nabla \SN(v ; a )]\tr (x - v)
\end{equation}
where $ \nabla \SN $ is given by (<ref>).
Hence, $ F\maj $ is given by
\begin{equation}
\label{eq:defFM}
F\maj(x; v)
\half \norm{ y - H x }_2^2
\lam
[\nabla \SN(v ; a )]\tr x
\lam \norm{ x }_1
+ C(v)
\end{equation}
where $ C(v) $ does not depend on $ x $.
The MM iteration (<ref>) is then given by
\begin{equation}
\label{eq:defMMF}
x\iter{k+1} \in \arg\min_{ x \in \RR^N }
\Bigl\{
\half \norm{ y - H x }_2^2
\lam
[\nabla \SN( x\iter{k} ; a )]\tr x
\lam \norm{ x }_1
\Bigr\}.
\end{equation}
This is a standard norm optimization problem which can be solved
by several methods (e.g. proximal methods, ADMM).
Hence, the solution to (<ref>) can be
obtained by iterative norm minimization.
Based on the theory of MM algorithms,
it is guaranteed that $ x\iter{k} $ converges to the minimizer of $ F $
when $ F $ is strictly convex
If $ H $ is not invertible, then $ F $ may be convex without being strictly convex.
In this case:
(a) the function value $ F(x\iter{k}) $ converges to the minimum value of $ F $,
(b) if $ F $ has a unique minimizer, then $ x\iter{k} $ converges to it.
See Theorems 4.1 and 4.4 of
§.§ Iterative thresholding
We present an iterative thresholding algorithm to solve (<ref>).
The algorithm is an immediate application of forward-backward splitting (FBS) <cit.>.
The FBS algorithm minimizes a function of the form $ f_1 + f_2 $ where
both $ f_1 $ and $ f_2 $ are convex and additionally
$ \nabla f_1 $ is Lipschitz continuous.
To apply the FBS algorithm to problem (<ref>),
we express $ F $ using (<ref>).
The first two terms of (<ref>) constitute the smooth convex function $ f_1 $.
We remark that since $ \Theta $ is concave, the Lipschitz constant of $ \nabla f_1 $ is bounded
by $ \rho $ where $ \rho $ is the maximum eigenvalue of $ H\tr H $.
The $ \ell_1 $ norm term in (<ref>) constitutes the (non-smooth) convex function $ f_2 $.
An FBS algorithm to solve problem (<ref>) is then given by
\begin{align}
z\iter{k} & =
x\iter{k} + \mu \Bigl[ H\tr (y - H x\iter{k}) - \lam \nabla \SN(x\iter{k} ; a ) \Bigr]
\\
x\iter{k+1} & = \soft( z\iter{k} , \mu \lam )
\end{align}
$ 0 < \mu < 2 / \rho $
where $ \rho $ is the maximum eigenvalue of $ H\tr H $.
The parameter $ \mu $ can be viewed as a step-size.
The soft thresholding function
\begin{equation}
\soft(t , T)
\begin{cases}
t - T, \ \ & t \ge T
\\
0, & \abs{ t } \le T
\\
t + T, & t \le -T
\end{cases}
\end{equation}
is applied element-wise to vector $ z\iter{k} $.
As an FBS algorithm, it is guaranteed that $ x\iter{k} $ converges to a minimizer of $ F $.
This algorithm resembles the classical
iterative shrinkage/thresholding algorithm (ISTA) <cit.>.
Note that ISTA was derived in <cit.> using
MM and was shown to converge for $ 0 < \mu < 1/\rho $.
the same algorithm derived using FBS is known to converge for twice this step size.
This is a practical advantage because the larger step-size generally yields
faster convergence of the algorithm.
We implement the algorithm with $ \mu = 1.9 / \rho $.
We further note that the iterative thresholding algorithm (<ref>)
has the property that $ F( x\iter{k} ) $ monotonically decreases.
$ 0 < \mu < 1/\rho $, this monotonic decreasing property follows from the MM-based derivation.
For $ 0 < \mu < 2/\rho $,
the proximal theory of FBS <cit.>
ensures convergence but not the monotonic decreasing property.
However, for this larger range of $ \mu $
the algorithm does in fact have
the monotonic decreasing property <cit.>.
§ NUMERICAL EXAMPLES
§.§ Example 1
Example 1 of sparse deconvolution
using bivariate sparse regularization (BISR).
The 100-point sparse signal illustrated in Fig. <ref>(a)
is convolved with the impulse response $ h $ shown in Fig. <ref>.
The convolved signal is corrupted by additive white Gaussian noise (AWGN)
with standard deviation $ \sigma = 4 $.
The corrupted signal is shown in Fig. <ref>(b).
To perform sparse deconvolution using BISR,
we need to define the univariate penalty $ \phi $ and
set the regularization parameters $ \lam $ and $ a = (a_1, a_2) $
in the objective function (<ref>).
For $ \phi $,
we use the arctangent penalty in Table <ref>.
We set $ \lam $ straightforwardly as
\begin{equation}
\label{eq:lamrule}
\lam = \beta \sigma \norm{ h }_2
% = \beta \sigma \sqrt{ \sum_n \abs{h_n}^2 }
\end{equation}
which follows from an analysis of
sparse optimality conditions
We set $ \beta = 2.5 $, similar to the `three sigma' rule.
This choice of $ \beta $ is not intended to minimize the mean square error,
but rather to inhibit false impulses appearing in the estimated sparse signal.
To set $ a $, we use the non-negative function $ P(\om) $
shown in Fig. <ref>.
This filter has $ P(0) = 0.6 $ and $ P(\pi) = 0.2 $.
Therefore, according to Theorem <ref>,
the objective function is convex if
$ 0 \le a_1 \le 0.6/\lam $
$ 0 \le a_2 \le 0.2/\lam $.
To maximally induce sparsity, we set $ a_i $ to their respective maximal values.
To perform deconvolution using BISR (i.e., to minimize the objective function $ F $)
we use the iterative thresholding algorithm (Sec. <ref>)
with a step-size of $ \mu = 1.9 / \rho $.
We run the algorithm until a stopping condition is satisfied.
As a stopping condition, we use
\norm{ x\iter{k+1} - x\iter{k} }_\infty
\le 10^{-4}
\times
\norm{ x\iter{k} }_\infty
where $ k $ is the iteration index.
The run-time (averaged over 50 realizations) is about 8.8 milliseconds
using a 2013 MacBook Pro (2.5 GHz Intel Core i5) running Matlab R2011a.
The BISR solution is shown in Fig. <ref>(c).
It has a root-mean-square-error (RMSE) of 2.7,
about 56% that of the norm solution,
shown in Fig. <ref>(d) for comparison.
Figure <ref>(e) shows the reconstruction error of both solutions.
The relative accuracy of the BISR solution is further illustrated in the scatter plot of Fig. <ref>(f),
which shows the error of the two solutions plotted against each other.
Most of the points lie below the diagonal line, meaning that the
BISR solution has less error than the norm solution.
Optimality condition for Example 1.
We verify the optimality of the obtained BISR solution using (<ref>)
as illustrated by the scatter plot in Fig. <ref>.
The obtained solution is confirmed to be a global minimizer of the objective function
the points in the scatter plot lie on the graph of the signum function.
We compare the proposed BISR method with several other algorithms
in Table <ref> and Fig. <ref>.
For the comparison,
we vary the noise standard deviation
$ \sigma $ and generate 200 sparse signals and noise realizations
for each value of $ \sigma $.
Each sparse signal consists of 10 randomly located impulses
with amplitudes uniformly distributed between $-$100 and 100.
Table <ref> gives the average RMSE and run-time
(with all algorithms implemented in Matlab on the same computer).
Average RMSE and run-time for Example 1
Algorithm $ \sigma\!=\!1 $ $ \sigma\!=\!2 $ $ \sigma\!=\!4 $ $ \sigma\!=\!8 $ $ \sigma\!=\!16 $ (msec)
L1 1.17 2.32 4.43 8.19 13.47 3.5
L1+debiasing 0.62 1.26 2.57 5.46 11.92 3.6
Lp (p = 0.5) 0.66 1.19 2.39 5.12 12.05 4.3
SBR (L0) 0.65 1.15 2.38 5.33 15.35 2.1
IMSC 0.50 1.00 2.23 5.00 11.02 323.9
IPS 0.51 1.02 2.23 4.89 10.96 5.1
BISR (log) 0.52 1.11 2.53 5.62 11.58 8.2
BISR (rat) 0.51 1.06 2.41 5.42 11.46 8.6
BISR (atan) 0.50 1.03 2.30 5.13 11.22 8.8
Average RMSE for Example 1.
Included in the comparison are the following methods:
$ \ell_1 $ norm regularization,
$ \ell_1 $ norm regularization with debiasing,
$ \ell_p $ pseudo-norm regularization with $ p $ of $ 0.5 $,
the single best replacement (SBR) algorithm <cit.>,
iterative maximally sparse convex (IMSC) regularization <cit.>,
the iterative p-shrinkage (IPS) algorithm <cit.>.
Each method provides an improvement over $ \ell_1 $ norm regularization.
The first approach is $ \ell_1 $ norm regularization with debiasing.
Debiasing is a post-processing step that applies unbiased least squares approximation
to re-estimate the non-zero amplitudes <cit.>.
This method solves the systematic underestimation of non-zero amplitudes
from which $ \ell_1 $ norm regularization suffers,
but it is still influenced by noise in the observed data.
As shown in Table <ref>, several other methods
generally perform better than $ \ell_1 $ regularization with debiasing.
Non-convex regularization using the $ \ell_p $ pseudo-norm with
$ 0 < p < 1 $ is also a common approach for improving upon $ \ell_1 $ norm regularization.
We use $ p = 0.5 $ with $ \lambda $ set according to (<ref>)
with $ \beta = 8 $, a value we found worked well on average for this example.
This method performed similarly as $ \ell_1 $ norm regularization with debiasing.
For non-convex regularization using the $ \ell_0 $ pseudo-norm
we use the SBR algorithm <cit.>.
A comparison of several methods for $ \ell_0 $ pseudo-norm regularization
showed the SBR algorithm to be state-of-the-art <cit.>.
We use SBR with $ \lam $ set according to (<ref>) with $ \beta = 70 $, which we found
worked well here except for high noise levels.
We were unable to find a single $ \beta $ value that worked well
over the range of $ \sigma $ considered here.
The SBR algorithm performed similarly to $ \ell_p $ pseudo-norm regularization,
except for the highest noise level of $ \sigma = 8 $.
The IMSC algorithm for sparse deconvolution proceeds by solving
a sequence of convex sub-problems <cit.>.
The regularization term of each sub-problem is individually designed to
be maximally non-convex (i.e., sparsity-inducing).
In contrast to the proposed BISR approach,
IMSC does not solve a prescribed optimization problem — each iteration yields a new convex problem to be solved.
The formulation of each problem in IMSC requires the solution to a semidefinite problem (SDP);
therefore, the IMSC approach is very slow.
(IMSC is up to 100 times slower than other algorithms; see Table <ref>.)
In IMSC, the parameter $ \lam $ can be set the same as for $ \ell_1 $ deconvolution;
hence, we again set $ \lam $ using (<ref>) with $ \beta = 2.5 $.
From Table <ref>, IMSC performs better than the preceding methods.
The IPS algorithm <cit.>
generalizes the classic iterative shrinkage-thresholding algorithm (ISTA) <cit.>.
The IPS algorithm replaces soft-thresholding in ISTA by a
threshold function that does not underestimate large-amplitude signal values.
Notably, IPS can be understood as a method that seeks to minimize a
prescribed (albeit implicit) non-convex objective function;
furthermore, each iteration of IPS can be understood as the exact minimization
of a convex problem.
as IPS is an iterative thresholding algorithm, it is computationally very efficient.
For this example, IPS gives excellent results when
the parameter $ \lam $ is set using (<ref>) with $ \beta = 2.5 $.
It performs about the same as IMSC but runs much faster.
To complete the comparison, we compare the proposed BISR method
using the three penalties listed in Table <ref>.
(The BISR method can be used with any penalty function satisfying the properties
listed at the beginning of Sec. <ref>.)
Referring to Table <ref>,
the three penalties give about the same result at low noise levels;
at higher noise levels, the arctangent penalty tends to perform better than the other two penalties.
Compared to the other algorithms,
BISR with the arctangent penalty does about as well as IPS and IMSC
at low noise levels, but not quite as well at higher noise levels.
Note that the proposed BISR approach is the only algorithm
minimizing a prescribed convex objective function.
§.§ Example 2
This is like Example 1, except we use the convolution filter $ H $ shown in Fig. <ref>.
This filter is not invertible.
The sparse signal (Fig. <ref>(a))
is convolved with the impulse response $ h $
and corrupted by AWGN ($ \sigma = 4 $).
We set $ \lam $ using (<ref>) with $ \beta = 2.5 $ as in Example 1.
Since the filter $ P $ has $ P(0) = 0.76 $ and $ P(\pi) = 0 $,
Theorem <ref> states that
the objective function (<ref>) is convex if
$ 0 \le a_1 \le 0.76/\lam $
$ a_2 = 0 $.
We run the algorithm of Sec. <ref>
with the same stopping condition
to obtain the BISR solution (Fig. <ref>(c)).
The BISR solution has an RMSE of 1.8, which is about 50% that of the
norm solution (Fig. <ref>(d)).
As in Example 1, the optimality of the obtained BISR solution can be readily verified.
Example 2 of sparse deconvolution
using bivariate sparse regularization (BISR).
Average RMSE and run-time for Example 2
Algorithm $ \sigma\!=\!1 $ $ \sigma\!=\!2 $ $ \sigma\!=\!4 $ $ \sigma\!=\!8 $ $ \sigma\!=\!16 $ (msec)
L1 1.37 2.70 5.01 8.96 14.01 5.1
L1+debiasing 0.76 1.55 3.14 6.56 13.31 5.2
Lp (p = 0.5) 1.07 1.57 2.89 6.14 13.38 5.7
SBR (L0) 0.73 1.44 2.73 6.59 17.64 3.4
IMSC 0.54 1.18 2.69 6.26 12.38 305.9
IPS 0.59 1.20 2.66 5.88 12.41 6.9
BISR (log) 0.75 1.60 3.19 6.65 12.45 13.9
BISR (rat) 0.73 1.56 3.08 6.49 12.37 14.6
BISR (atan) 0.75 1.56 3.00 6.29 12.25 15.4
It is noteworthy in this example that
we obtain a strictly non-convex penalty ensuring convexity of the objective function $ F $,
because in this example the convolution filter $ H $ is not invertible.
It is possible only because the utilized penalty is non-separable.
In Table <ref> we compare the same algorithms as in Example 1.
The regularization parameter $ \lam $ for each algorithm was chosen as in Example 1.
The remarks made in Example 1 mostly apply here, but with a few exceptions as follows.
In contrast to the generally tendency, the $ \ell_p $ pseudo-norm
performed worse than the debiased $ \ell_1 $ norm solution for the lowest noise level ($ \sigma = 1 $).
It appears that the performance of the $ \ell_p $ pseudo-norm and SBR methods
may degrade for low or high noise levels when
one attempts to set $ \lam $ proportional to $ \sigma $.
In addition, in contrast to Example 1,
the BISR approach did not outperform
$ \ell_p $ pseudo-norm or SBR for several noise levels.
IPS performs very well at most noise levels.
§ LIMITATIONS OF BIVARIATE PENALTIES
The examples in Sec. <ref> demonstrate the improvement
attainable by non-separable bivariate penalties in comparison with separable penalties;
however, the degree of improvement depends on the linear operator $ H $ in the
data fidelity term.
In particular, for some $ H $, a bivariate penalty
will offer little or no improvement in comparison with a separable penalty.
In reference to Figs. <ref> and <ref>,
the effectiveness of a bivariate penalty for sparse deconvolution depends on how well
$ \abs{ H(\om) }^2 $ can be approximated by a function $ P(\om) $
of the form (<ref>) satisfying condition (<ref>).
Some filters $ H $ can not be well approximated by such a filter $ P $.
For example, if $ H(\om) $ is Gaussian or has a sharp peak,
then it can not be well approximated by such a filter $ P $.
In this case, a bivariate penalty
does not offer significant improvement in comparison with a separable penalty.
For a second example,
if the frequency response $ H(\om) $
has multiple nulls (e.g., a $ K $-point moving average filter with $ K > 2 $),
then the only filter $ P $ of the form (<ref>)
satisfying condition (<ref>) is identically zero, $ P(\om) = 0 $.
In this case, the bivariate penalty reduces to the norm and
the proposed BISR approach offers no improvement.
In both situations,
a higher-order filter $ P $ is needed to accurately approximate $ H $,
and a higher-order non-separable penalty is needed to strongly induce sparsity.
Hence, to extend the applicability of non-separable sparse regularization,
it will be necessary generalize the bivariate penalty to
non-separable $ K $-variate penalties ($ K > 2 $).
Provided such an extension can be constructed,
we expect the proposed BISR approach will be applicable to more general problems.
§ CONCLUSION
This paper aims to develop a convex approach for sparse signal estimation that improves upon $ \ell_1 $ norm regularization, the standard convex approach.
We consider ill-conditioned linear inverse problems with a quadratic data fidelity term.
We focus in particular on the deconvolution problem.
The proposed method is based on a non-convex penalty designed to ensure that the objective function is convex.
Our previous work <cit.> using this idea considered only separable (additive) penalties;
this is a fundamental limitation when the observation matrix is singular or near singular.
The non-separable bivariate penalty introduced in this paper overcomes this limitation.
The proposed bivariate sparse regularization (BISR) approach
provides a mechanism by which to improve upon $ \ell_1 $ norm regularization
while adhering to a convex framework.
The greater generality of non-separable regularization, as compared with separable regularization,
allows for the design of regularizers that more effectively induce sparsity.
Both BISR and norm regularization lead to convex optimization problems which can be solved
by similar optimization techniques.
While we have focused on deconvolution,
we note that several filtering methods can be formulated as sparse deconvolution problems,
e.g., peak detection <cit.>
and denoising <cit.>.
Hence, the proposed approach (and extensions thereof) is not limited to deconvolution.
§ ACKNOWLEDGMENT
The authors gratefully acknowledge constructive comments from
the anonymous reviewers.
§ PROOFS
§.§ Proof of Proposition <ref>
Let $ 0 \le a \le 1/\lam $.
By Proposition <ref>, $ s $ is twice continuously differentiable.
Hence, so is $ g $.
to show $ g $ is convex, we show its second derivative is non-negative.
From (<ref>), we have
\begin{equation}
s''( t ; a ) \ge -a.
\end{equation}
Since $ a \le 1/\lam $, we have
\begin{equation}
s''( t ; a ) \ge - \frac{ 1 }{ \lam }.
\end{equation}
\begin{equation}
g''( t ) = 1 + \lam s''( t ; a ) \ge 0.
\end{equation}
Therefore, $ g $ is convex.
From (<ref>),
we have $ f( t ) = g( t ) + \lam \abs{ t } $.
Since $ f $ is the sum of convex functions, $ f $ is convex.
§.§ Partial derivatives
Some following proofs will require the partial derivatives of $ S(x; a) $.
From Definition <ref>, they are as follows.
\begin{align}
\label{eq:defS1}
\frac{ \partial S }{ \partial x_1} ( x ; a ) & =
\begin{cases}
s'( x_1 + r x_2 ; \alpha ) , & x \in \SP_1
\\
r s'( r x_1 + x_2 ; \alpha ) + (1 - r) \, s'( x_1 ; a_1 ) , & x \in \SP_2
\\
r s'( r x_1 + x_2 ; \alpha ) + (1 + r) \, s'( x_1 ; a_2 ) , & x \in \SP_3
\\
s'( x_1 + r x_2 ; \alpha ) , & x \in \SP_4
\end{cases}
\end{align}
\begin{align}
\label{eq:defS2}
\frac{ \partial S }{ \partial x_2} ( x ; a ) & =
\begin{cases}
r s'( x_1 + r x_2 ; \alpha ) + (1 - r) \, s'( x_2 ; a_1 ) , & x \in \SP_1
\\
s'( r x_1 + x_2 ; \alpha ) , & x \in \SP_2
\\
s'( r x_1 + x_2 ; \alpha ) , & x \in \SP_3
\\
r s'( x_1 + r x_2 ; \alpha ) + (1 + r) \, s'( x_2 ; a_2 ) , & x \in \SP_4
\end{cases}
\end{align}
\begin{align}
\label{eq:defS11}
\frac{ \partial^2 S }{ \partial^2 x_1} ( x ; a ) & =
\begin{cases}
s''( x_1 + r x_2 ; \alpha ) , & x \in \SP_1
\\
r^2 s''( r x_1 + x_2 ; \alpha ) + (1 - r) \, s''( x_1 ; a_1 ) , & x \in \SP_2
\\
r^2 s''( r x_1 + x_2 ; \alpha ) + (1 + r) \, s''( x_1 ; a_2 ) , & x \in \SP_3
\\
s''( x_1 + r x_2 ; \alpha ) , & x \in \SP_4
\end{cases}
\end{align}
\begin{align}
\label{eq:defS22}
\frac{ \partial^2 S }{ \partial^2 x_2} ( x ; a ) & =
\begin{cases}
r^2 s''( x_1 + r x_2 ; \alpha ) + (1 - r) \, s''( x_2 ; a_1 ) , & x \in \SP_1
\\
s''( r x_1 + x_2 ; \alpha ) , & x \in \SP_2
\\
s''( r x_1 + x_2 ; \alpha ) , & x \in \SP_3
\\
r^2 s''( x_1 + r x_2 ; \alpha ) + (1 + r) \, s''( x_2 ; a_2 ) , & x \in \SP_4
\end{cases}
\end{align}
\begin{align}
\label{eq:defS12}
\frac{ \partial^2 S }{ \partial x_1 \partial x_2} ( x ; a ) & =
\begin{cases}
r s''( x_1 + r x_2 ; \alpha ) , & x \in \SP_1
\\
r s''( r x_1 + x_2 ; \alpha ) , & x \in \SP_2
\\
r s''( r x_1 + x_2 ; \alpha ) , & x \in \SP_3
\\
r s''( x_1 + r x_2 ; \alpha ) , & x \in \SP_4
\end{cases}
\end{align}
§.§ Scaling identities
Some following proofs will use scaling identities.
From the scaling property in Sec. <ref>,
it follows that
\begin{align}
s( t, \alpha )
& = \frac{ a_1 }{ \alpha } \, s\Bigl( \frac{ \alpha }{ a_1 } t ; a_1 \Bigr)
\\
\label{eq:stalpha}
& = ( 1 + r ) \, s\Bigl( \frac{ t }{ 1 + r } ; a_1 \Bigr)
% \forall t
\end{align}
and similarly
\begin{align}
s( t, a_1 )
& = \frac{ 1 }{ 1 + r } \, s\bigl( ( 1 + r ) t ; \alpha \bigr)
\\
s'( t, a_1 )
& = s'\bigl( ( 1 + r ) t ; \alpha \bigr)
\\
\label{eq:s2da1}
s''( t, a_1 )
& = (1 + r) s''\bigl( ( 1 + r ) t ; \alpha \bigr).
% \forall t.
\end{align}
\begin{align}
s( t, \alpha )
& = \frac{ a_2 }{ \alpha } \, s\Bigl( \frac{ \alpha }{ a_2 } t ; a_2 \Bigr)
\\
\label{eq:sta2}
& = ( 1 - r ) \, s\Bigl( \frac{ t }{ 1 - r } ; a_2 \Bigr)
\end{align}
\begin{align}
s( t, a_2 )
& = \frac{ 1 }{ 1 - r } \, s\bigl( ( 1 - r ) t ; \alpha \bigr)
\\
s'( t, a_2 )
& = s'\bigl( ( 1 - r ) t ; \alpha \bigr)
\\
\label{eq:s2da2}
s''( t, a_2 )
& = (1 - r) s''\bigl( ( 1 - r ) t ; \alpha \bigr).
\end{align}
§.§ Proof of of Lemma <ref>
We first show that $ S $ in Definition <ref> is consistent on the common boundaries of the sets $ A_i $.
Consider the common boundary of $ A_1 $ and $ A_4 $, i.e., the line $ \{ (x_1, 0) : x_1 \in \RR \} $.
From the $ A_1 $ side,
\begin{align}
S(x ; a ) \Big|_{\substack{x_2 = 0 \\ x \in A_1}}
& = s( x_1 ; \alpha ) + (1 - r ) s( 0 ; a_1 )
\\
& = s( x_1 ; \alpha)
\end{align}
where we used $ s( 0 ; \cdot ) = 0 $.
From the $ A_4 $ side,
\begin{align}
S(x ; a ) \Big|_{\substack{x_2 = 0 \\ x \in A_4}}
& = s( x_1 ; \alpha ) + (1 + r) \, s( 0 ; a_2 )
\\
& = s( x_1 ; \alpha).
\end{align}
Hence, $ S $ is consistently defined on the common boundary of $ A_1 $ and $ A_4 $.
Consider the common boundary of $ A_1 $ and $ A_2 $, i.e., the line $ \{ (x_1, x_1) : x_1 \in \RR \} $.
From the $ A_1 $ side,
\begin{equation}
S(x ; a ) \Big|_{\substack{x_2 = x_1 \\ x \in A_1}}
= s( (1 + r) x_1 ; \alpha ) + (1 - r ) s( x_1 ; a_1 ).
\end{equation}
From the $ A_2 $ side,
\begin{equation}
S(x ; a ) \Big|_{\substack{x_2 = x_1 \\ x \in A_2}}
= s( (1+r) x_1 ; \alpha ) + (1 - r) \, s( x_1 ; a_2 ).
\end{equation}
Hence, $ S $ is consistently defined on the common boundary of $ A_1 $ and $ A_2 $.
Similarly, it can be shown that $ S $ is consistently defined on the other
common boundaries of the sets $ A_i $.
Hence $ S $ is continuous because $ s $ is continuous.
We now show $ S $ is differentiable on $ \RR^2 $.
We need only show $ S $ is differentiable on the common boundaries of the sets $ A_i $
because $ s $ is differentiable.
Consider the common boundary of $ A_1 $ and $ A_4 $.
From the $ A_1 $ side,
\begin{equation}
\frac{ \partial S }{ \partial x_1} ( x ; a )
\Big|_{\substack{x_2 \to 0 \\ x \in A_1}}
= s'( x_1 ; \alpha )
\end{equation}
using (<ref>),
\begin{align}
\frac{ \partial S }{ \partial x_2} ( x ; a )
\Big|_{\substack{x_2 \to 0 \\ x \in A_1}}
& = r s'( x_1 ; \alpha ) + (1 - r) s'( 0 ; a_1 )
\\
& = r s'( x_1 ; \alpha )
\end{align}
using (<ref>)
and $ s'(0, \cdot) = 0 $.
From the $ A_4 $ side,
\begin{equation}
\frac{ \partial S }{ \partial x_1} ( x ; a )
\Big|_{\substack{x_2 \to 0 \\ x \in A_4}}
= s'( x_1 ; \alpha )
\end{equation}
\begin{align}
\frac{ \partial S }{ \partial x_2} ( x ; a )
\Big|_{\substack{x_2 \to 0 \\ x \in A_4}}
& = r s'( x_1 ; \alpha ) + (1 + r) s'( 0 ; a_2 )
\\
& = r s'( x_1 ; \alpha ).
\end{align}
Hence, the partial derivatives of $ S $ are continuous on the common of boundary of $ A_1 $ and $ A_4 $.
Consider the common boundary of $ A_1 $ and $ A_2 $.
From the $ A_1 $ side,
\begin{align}
\frac{ \partial S }{ \partial x_1} ( x ; a )
\Big|_{\substack{x_2 \to x_1 \\ x \in A_1}}
& = s'( (1 + r ) x_1 ; \alpha )
\\
& = s'( x_1 ; a_1 )
\end{align}
where we used the scaling property (<ref>) and identity (<ref>).
Also, using the same properties,
\begin{align}
\frac{ \partial S }{ \partial x_2} ( x ; a )
\Big|_{\substack{x_2 \to x_1 \\ x \in A_1}}
& = r s'( (1 + r ) x_1 ; \alpha ) + (1 - r) s'( x_1 ; a_1 )
\\
& = s'( x_1 ; a_1 ).
\end{align}
From the $ A_2 $ side, we similarly have
\begin{align}
\frac{ \partial S }{ \partial x_1} ( x ; a )
\Big|_{\substack{x_2 \to x_1 \\ x \in A_2}}
& = r s'( (1 + r ) x_1 ; \alpha ) + (1 - r) s'( x_1 ; a_1 )
\\
& = s'( x_1 ; a_1 )
\end{align}
\begin{align}
\frac{ \partial S }{ \partial x_2} ( x ; a )
\Big|_{\substack{x_2 \to x_1 \\ x \in A_2}}
& = s'( (1 + r) x_1 ; \alpha ).
\\
& = s'( x_1 ; a_1 ).
\end{align}
Hence, the partial derivatives of $ S $ are continuous on the common of boundary of $ A_1 $ and $ A_2 $.
Similarly, it can be shown that the partial derivatives of $ S $
are continuous on the other common boundaries of the sets $ A_i $.
Hence $ S $ is differentiable on $ \RR^ 2 $.
We now show $ S $ is twice differentiable.
We need only show the second-order partial derivatives of $ S $
are continuous on the common boundaries of the sets $ A_i $
because $ s $ is twice differentiable.
Consider the common boundary of $ A_1 $ and $ A_4 $.
Using (<ref>), we obtain
\begin{equation}
\frac{ \partial^2 S }{ \partial^2 x_1} ( x ; a )
\Big|_{\substack{x_2 \to 0 \\ x \in A_1}}
= s''( x_1 ; \alpha )
\end{equation}
\begin{equation}
\frac{ \partial^2 S }{ \partial^2 x_1} ( x ; a )
\Big|_{\substack{x_2 \to 0 \\ x \in A_4}}
= s''( x_1 ; \alpha ).
\end{equation}
Hence $ \partial^2 S / \partial^2 x_1 $ is continuous on the common boundary of $ A_1 $ and $ A_4 $.
Using (<ref>), we obtain
\begin{align}
\frac{ \partial^2 S }{ \partial^2 x_2} ( x ; a )
\Big|_{\substack{x_2 \to 0 \\ x \in A_1}}
& = r^2 s''( x_1 ; \alpha ) + ( 1 - r ) s''(0 ; a_1)
\\
& = r^2 s''( x_1 ; \alpha ) - ( 1 - r ) a_1
\end{align}
where we used $ s''(0; a_1) = -a_1 $,
\begin{align}
\frac{ \partial^2 S }{ \partial^2 x_2} ( x ; a )
\Big|_{\substack{x_2 \to 0 \\ x \in A_4}}
& = r^2 s''( x_1 ; \alpha ) + ( 1 + r ) s''(0 ; a_2)
\\
& = r^2 s''( x_1 ; \alpha ) - ( 1 + r ) a_2
\end{align}
where we used $ s''(0; a_2) = -a_2 $.
Using (<ref>), we have
\begin{equation}
(1 - r) a_1 = (1 + r) a_2 = \frac{ 2 a_1 a_2}{a_1 + a_2},
\end{equation}
hence $ \partial^2 S / \partial^2 x_2 $ is continuous on the common boundary of $ A_1 $ and $ A_4 $.
Consider the common boundary of $ A_1 $ and $ A_2 $.
Using (<ref>), we obtain
\begin{equation}
\frac{ \partial^2 S }{ \partial^2 x_1} ( x ; a )
\Big|_{\substack{x_2 \to x_1 \\ x \in A_1}}
= s''( (1 + r) x_1 ; \alpha )
\end{equation}
\begin{align}
\frac{ \partial^2 S }{ \partial^2 x_1} ( x ; a )
\Big|_{\substack{x_2 \to x_1 \\ x \in A_2}}
& = r^2 s''( (1 + r) x_1 ; \alpha ) + (1 - r) s''(x_1 ; a_1)
\\
& = r^2 s''( (1 + r) x_1 ; \alpha ) + (1 - r) (1+r) s''( (1+r) x_1 ; \alpha)
\\
& = s''( (1 + r) x_1 ; \alpha )
\end{align}
where we used (<ref>).
Hence $ \partial^2 S / \partial^2 x_2 $ is continuous on the common boundary of $ A_1 $ and $ A_2 $.
Similarly, it can be shown that the remaining partial derivatives are continuous on
the other common boundaries of the sets $ A_i $.
Hence $ S $ is twice continuously differentiable on $ \RR^2 $.
We now show $ S $ is concave
by showing that $ \nabla^2 S(x) $ is negative semidefinite for all $ x \in \RR^2 $.
we show that $ -[\nabla^2 S(x)] $ is positive semidefinite
by showing that its principal minors are non-negative (Sylvester's criterion).
The determinant of the Hessian of $ S $
\begin{equation}
\det [\nabla^2 S(x)] =
\frac{ \partial^2 S }{ \partial^2 x_1 }
\frac{ \partial^2 S }{ \partial^2 x_2 }
\left( \frac{ \partial^2 S }{ \partial x_1 \partial x_2} \right)^2
\end{equation}
is given by
\begin{align}
\det [\nabla^2 S(x; a)] =
\\
\nonumber
\qquad
\begin{cases}
(1 - r) \, s''( x_2 ; a_1 ) \, s''( x_1 + r x_2 ; \alpha ), & x \in \SP_1
\\
(1 - r) \, s''( x_1 ; a_1 ) \, s''( r x_1 + x_2 ; \alpha ), & x \in \SP_2
\\
(1 + r) \, s''( x_1 ; a_2 ) \, s''( r x_1 + x_2 ; \alpha ), & x \in \SP_3
\\
(1 + r) \, s''( x_2 ; a_2 ) \, s''( x_1 + r x_2 ; \alpha ), & x \in \SP_4.
\end{cases}
\end{align}
Since $ s''(t; \cdot ) $ is negative for all $ t $ and $ \abs{ r } $ is bounded by 1,
it follows that $ \det [\nabla^2 S(x)] $ is non-negative for all $ x \in \RR^2 $.
Hence, the determinant of $ -[\nabla^2 S(x)] $ is non-negative.
Moreover, $ \partial^2 S / \partial^2 x_i ( x ; a ) \le 0 $ for all $ x \in \RR^2 $
for $ i = 1, 2 $.
Hence, the principal minors of $ -[\nabla^2 S(x)] $ are non-negative.
This proves $ S $ is concave on $ \RR^2 $.
§.§ Proof of Lemma <ref>
Using (<ref>)-(<ref>), the Hessian of $ S $ is given by
\begin{equation}
\nabla^2 S( x ; a )
\begin{cases}
\begin{bmatrix}
s''( x_1 + r x_2 ; \alpha )
r s''( x_1 + r x_2 ; \alpha )
\\
r s''( x_1 + r x_2 ; \alpha )
r^2 s''( x_1 + r x_2 ; \alpha ) + (1 - r) \, s''( x_2 ; a_1 )
\end{bmatrix},
\ \
x \in \SP_1
\\[2em]
\begin{bmatrix}
r^2 s''( r x_1 + x_2 ; \alpha ) + (1 - r) \, s''( x_1 ; a_1 )
r s''( r x_1 + x_2 ; \alpha )
\\
r s''( r x_1 + x_2 ; \alpha )
s''( r x_1 + x_2 ; \alpha )
\end{bmatrix},
x \in \SP_2
\\[2em]
\begin{bmatrix}
r^2 s''( r x_1 + x_2 ; \alpha ) + (1 + r) \, s''( x_1 ; a_2 )
r s''( r x_1 + x_2 ; \alpha )
\\
r s''( r x_1 + x_2 ; \alpha )
s''( r x_1 + x_2 ; \alpha )
\end{bmatrix},
x \in \SP_3
\\[2em]
\begin{bmatrix}
s''( x_1 + r x_2 ; \alpha )
r s''( x_1 + r x_2 ; \alpha )
\\
r s''( x_1 + r x_2 ; \alpha )
r^2 s''( x_1 + r x_2 ; \alpha ) + (1 + r) \, s''( x_2 ; a_2 )
\end{bmatrix},
x \in \SP_4.
\end{cases}
\end{equation}
We write this as
\begin{equation}
\label{eq:ShessSA}
\nabla^2 S( x ; a )
\begin{cases}
s''(x_1 + r x_2; \alpha)
\begin{bmatrix}
1 & r
\\
r & r^2
\end{bmatrix}
s''(x_2; a_1)
\begin{bmatrix}
0 & 0
\\
0 & 1 - r
\end{bmatrix},
\ \
x \in \SP_1
\\[2em]
s''( r x_1 + x_2 ; \alpha )
\begin{bmatrix}
r^2 & r
\\
r & 1
\end{bmatrix}
s''(x_1 ; a_1)
\begin{bmatrix}
1 - r & 0
\\
0 & 0
\end{bmatrix},
x \in \SP_2
\\[2em]
s''( r x_1 + x_2 ; \alpha )
\begin{bmatrix}
r^2 & r
\\
r & 1
\end{bmatrix}
s''(x_1; a_2 )
\begin{bmatrix}
1 + r & 0
\\
0 & 0
\end{bmatrix},
x \in \SP_3
\\[2em]
s''( x_1 + r x_2 ; \alpha )
\begin{bmatrix}
1 & r
\\
r & r^2
\end{bmatrix}
s''(x_2 ; a_2 )
\begin{bmatrix}
0 & 0
\\
0 & 1 + r
\end{bmatrix},
x \in \SP_4.
\end{cases}
\end{equation}
Using (<ref>) and (<ref>), we write (<ref>) as
\begin{equation}
\label{eq:ShessS}
\nabla^2 S( x ; a )
\begin{cases}
s''(x_1 + r x_2; \alpha)
\begin{bmatrix}
1 & r
\\
r & r^2
\end{bmatrix}
s''( (1 + r) x_2; \alpha)
\begin{bmatrix}
0 & 0
\\
0 & 1-r^2
\end{bmatrix},
\ \
x \in \SP_1
\\[2em]
s''( r x_1 + x_2 ; \alpha )
\begin{bmatrix}
r^2 & r
\\
r & 1
\end{bmatrix}
s''( (1 + r) x_1; \alpha)
\begin{bmatrix}
1-r^2 & 0
\\
0 & 0
\end{bmatrix},
x \in \SP_2
\\[2em]
s''( r x_1 + x_2 ; \alpha )
\begin{bmatrix}
r^2 & r
\\
r & 1
\end{bmatrix}
s''( (1 - r) x_1; \alpha)
\begin{bmatrix}
1-r^2 & 0
\\
0 & 0
\end{bmatrix},
x \in \SP_3
\\[2em]
s''( x_1 + r x_2 ; \alpha )
\begin{bmatrix}
1 & r
\\
r & r^2
\end{bmatrix}
s''( (1 - r) x_2; \alpha)
\begin{bmatrix}
0 & 0
\\
0 & 1-r^2
\end{bmatrix},
x \in \SP_4.
\end{cases}
\end{equation}
Note that the matrices in (<ref>) are positive semidefinite
because $ \abs{ r } \le 1 $.
We also have $ -\alpha \le s''(t; \alpha) < 0 $ for all $ t $.
\begin{equation}
\label{eq:nablaSgte}
\nabla^2 S( x ; a )
\mge
\begin{bmatrix}
1 & r
\\
r & 1
\end{bmatrix},
\ \ x \in \RR^2.
\end{equation}
Using (<ref>) and (<ref>),
we write (<ref>) as
\begin{equation}
\nabla^2 S( x ; a )
\mge
- \half
\begin{bmatrix}
a_1 + a_2 & a_1 - a_2
\\
a_1 - a_2 & a_1 + a_2
\end{bmatrix}.
\end{equation}
Since $ s''(0; \alpha) = -\alpha $ [see (<ref>)],
we similarly have
\begin{equation}
\nabla^2 S( 0 ; a )
- \half
\begin{bmatrix}
a_1 + a_2 & a_1 - a_2
\\
a_1 - a_2 & a_1 + a_2
\end{bmatrix}.
\end{equation}
§.§ Proof of Theorem <ref>
Without loss of generality, assume $ a_1 > a_2 $.
From (<ref>) and (<ref>),
the inequality (<ref>) can be written as
\begin{equation}
\label{eq:Sbounds}
s(x_1; a_1) + s(x_2; a_1)
\le S(x; a) \le
s(x_1; a_2) + s(x_2; a_2)
\end{equation}
where $ S $ is given in Definition <ref>.
First we prove (<ref>) for $ x \in \SP_1 $.
Second, we prove it for $ x \in \SP_3 $.
The proofs for $ x \in \SP_2 $ and $ x \in \SP_4 $ are essentially identical by symmetries.
Note that since $ a_1 \ge a_2 \ge 0 $, we have
$ a_1 \ge \alpha \ge a_2 $
$ 0 \le r \le 1 $.
Let $ x \in \SP_1 $. We seek to prove
\begin{equation}
\label{eq:A1L}
% S( x ; a ) =
s( x_1 + r x_2 ; \alpha ) + (1 - r) \, s( x_2 ; a_1 )
\ge
s(x_1; a_1) + s(x_2; a_1).
\end{equation}
Using (<ref>), we have
\begin{align}
s( x_1 + r x_2 ; \alpha )
& =
(1 + r ) \, s\Bigl( \frac{ x_1 + r x_2 }{ 1 + r } ; a_1 \Bigr)
\\
& \ge
( 1 + r ) \, \Bigl[ \frac{ 1 }{ 1 + r } s( x_1 ; a_1 ) + \frac{ r }{ 1 + r } s( x_2 ; a_1 ) \Bigr]
% \ \ \text{ (as $ s $ is concave) }
\\
\label{eq:sx1rx2}
& =
s( x_1 ; a_1 ) + r \, s( x_2 ; a_1 )
\end{align}
where the inequality is due to $ s $ being a concave function.
Adding $ (1 - r) \, s( x_2 ; a_1 ) $ to both sides of (<ref>) gives (<ref>).
Let $ x \in \SP_1 $.
(That is, $ 0 \le x_2 \le x_1 $
$ x_1 \le x_2 \le 0 $.)
We seek to prove
\begin{equation}
\label{eq:A1U}
s(x_1; a_2) + s(x_2; a_2)
\ge
s( x_1 + r x_2 ; \alpha ) + (1 - r) \, s( x_2 ; a_1 ).
% = S( x ; a ).
\end{equation}
From (<ref>) we have
\begin{equation}
\label{eq:sx1a1}
s( x_2 ; a_1 ) = \frac{ 1 }{ 1 + r } \, s( (1+r) x_2 ; \alpha ).
\end{equation}
Suppose $ 0 \le x_2 \le x_1 $.
As $ s $ is concave and $ s(0 ; \cdot \,) = 0 $, we have
\begin{equation}
\label{eq:sx1alpha}
s( x_1 ; \alpha ) \ge \frac{ x_1 }{ x_1 + r x_2 } \, s( x_1 + r x_2 ; \alpha )
\end{equation}
because $ 0 \le x_1 \le x_1 + r x_2 $.
\begin{equation}
\label{eq:s1rx2}
s( (1+r) x_2 ; \alpha ) \ge
\frac{ (1 + r) x_2 }{ x_1 + r x_2 } \, s( x_1 + r x_2 ; \alpha )
\end{equation}
because $ 0 \le (1 + r ) x_2 \le x_1 + r x_2 $.
From (<ref>) and (<ref>)
it follows that
\begin{equation}
\label{eq:rsx2a1}
r s( x_2 ; a_1 ) \ge
\frac{ r x_2 }{ x_1 + r x_2 } \, s( x_1 + r x_2 ; \alpha ).
\end{equation}
Adding (<ref>) and (<ref>) gives
\begin{equation}
\label{eq:sx1rsx2sx1}
s( x_1 ; \alpha ) + r s( x_2 ; a_1 ) \ge s( x_1 + r x_2 ; \alpha ).
\end{equation}
Adding $ (1 - r ) \, s( x_2 ; a_1 ) $ to both sides of (<ref>) gives
\begin{equation}
s( x_1 ; \alpha ) + s( x_2 ; a_1 ) \ge s( x_1 + r x_2 ; \alpha ) + (1 - r ) \, s( x_2 ; a_1 ).
\end{equation}
Since $ \alpha \ge a_2 $, we have $ s( x_1 ; a_2 ) \ge s( x_1 ; \alpha ) $.
since $ a_1 \ge a_2 $, we have $ s( x_2 ; a_2 ) \ge s( x_2 ; a_1 ) $.
It follows that (<ref>) holds.
For $ x_1 \le x_2 \le 0 $ the proof is similar.
Let $ x \in \SP_3 $.
We seek to prove
\begin{equation}
\label{eq:A3L}
s( r x_1 + x_2 ; \alpha ) + (1 + r) \, s( x_1 ; a_2 )
\ge
s(x_1; a_1) + s(x_2; a_1).
\end{equation}
Suppose $ x_2 \ge -x_1 \ge 0 $.
Since $ 0 \le r \le 1 $ it follows that $ 0 \le r x_1 + x_2 \le x_ 2 $.
As $ s $ is concave and $ s(0 ; \cdot \,) = 0 $, we have
\begin{equation}
\label{eq:sx1aB}
s( r x_1 + x_2 ; \alpha ) \ge \frac{ r x_1 + x_2 }{ x_2 } \, s( x_2 ; \alpha )
\end{equation}
\begin{equation}
\label{eq:sx1mx1}
s( x_1 ; \alpha ) = s( -x_1 ; \alpha) \ge -\frac{ x_1 }{ x_2 } s( x_2 ; \alpha )
\end{equation}
where we use the fact that $ s $ is symmetric.
Multiplying (<ref>) by $ r $ and adding (<ref>), we have
\begin{equation}
\label{eq:srxg}
s( r x_1 + x_2 ; \alpha ) + r s( x_1 ; \alpha ) \ge s( x_2 ; \alpha )
\end{equation}
Since $ a_2 \le \alpha $, we have
$ s( x_1 ; a_2 ) \ge s( x_1 ; \alpha) $.
Multiplying by $ (1 + r) $ and rearranging, we have
\begin{equation}
\label{eq:srxf}
(1 + r ) s( x_1 ; a_2 ) - r s( x_1 ; \alpha ) \ge s( x_1 ; \alpha ).
\end{equation}
Adding (<ref>) and (<ref>), we have
\begin{equation}
s( r x_1 + x_2 ; \alpha ) + (1 + r) \, s( x_1 ; a_2 )
\ge
s(x_1; \alpha ) + s(x_2; \alpha ).
\end{equation}
Since $ \alpha \le a_1 $, we have $ s( t ; \alpha) \ge s(t ; a_1) $ for all $ t $.
It follows that (<ref>) holds.
For $ x_2 \le -x_1 \le 0 $ the proof is similar.
Let $ x \in \SP_3 $.
We seek to prove
\begin{equation}
\label{eq:A3U}
s( r x_1 + x_2 ; \alpha ) + (1 + r) \, s( x_1 ; a_2 )
\le
s(x_1; a_2) + s(x_2; a_2).
\end{equation}
From (<ref>) we have
\begin{equation}
s( r x_1 + x_2 ; \alpha ) = ( 1 - r ) \, s\Bigl( \frac{ r x_1 + x_2 }{ 1 - r } ; a_2 \Bigr).
\end{equation}
Since $ s $ is symmetric, we have $ s( x_1 ; a_2 ) = s( -x_1 ; a_2 ) $.
\begin{align}
s( r x_1 + x_2 ; \alpha ) + r s( x_1 ; a_2 )
& =
( 1 - r ) \, s\Bigl( \frac{ r x_1 + x_2 }{ 1 - r } ; a_2 \Bigr)
+ r s( -x_1 ; a_2 )
\\
& \le
s\Bigl( ( 1 - r ) \frac{ r x_1 + x_2 }{ 1 - r } - r x_1 ; a_2 \Bigr)
\\
\label{eq:sx2a2p}
& =
s( x_2 ; a_2 )
\end{align}
where the inequality is due to $ s $ being a concave function.
Adding $ s( x_1 ; a_2 ) $ to both sides of (<ref>) gives the
desired inequality (<ref>).
§.§ Proof of Lemma <ref>
We express $ F $ in (<ref>) as
\begin{align}
F(x) & = \half \norm{ y - H x }_2^2 + \frac{\lam}{2} \sum_n \psi( (x_{n-1}, x_n) ; a)
\\
& =
\half y\tr y - y\tr H x + \half x\tr \, ( H\tr H - P) \, x
g( x )
% \half x\tr P x + \frac{\lam}{2} \sum_n \psi( (x_{n-1}, x_n) ; a),
\end{align}
where $ g \colon \RR^N \to \RR $ is defined as
\begin{equation}
g(x) =
\half x\tr P x + \frac{\lam}{2} \sum_n \psi( (x_{n-1}, x_n) ; a).
\end{equation}
The first three terms are convex.
Hence $ F $ is convex if $ g $ is convex.
We express $ P $ as a sum of matrices, each comprising a single $ 2 \times 2 $ block.
\begin{equation}
P =
\cdots
% +
% \begin{bmatrix}
% \half p_0 & p_1 & & & \\
% p_1 & \half p_0 & & & \\
% & & 0 & & \\
% & & & 0 & \\
% & & & & \ddots
% \end{bmatrix}
% \hspace{6em}
% \hspace{6em}
\begin{bmatrix}
0 & & & & \\
& \half p_0 & p_1 & & \\
& p_1 & \half p_0 & & \\
& & & 0 & \\
& & & & 0
\end{bmatrix}
\begin{bmatrix}
0 & & & & \\
& 0 & & & \\
& & \half p_0 & p_1 & \\
& & p_1 & \half p_0 & \\
& & & & 0
\end{bmatrix}
\ \cdots \
\end{equation}
\begin{equation}
P =
\begin{bmatrix}
\half p_0 & & & & \\
& 0 & & & \\
& & 0 & & \\
& & & 0 & \\
& & & & \half p_0 \\
\end{bmatrix}
\begin{bmatrix}
\half p_0 & p_1 & & & \\
p_1 & \half p_0 & & & \\
& & 0 & & \\
& & & 0 & \\
& & & & 0 \\
\end{bmatrix}
\begin{bmatrix}
0 & & & & \\
& \half p_0 & p_1 & & \\
& p_1 & \half p_0 & & \\
& & & 0 & \\
& & & & 0 \\
\end{bmatrix}
\end{equation}
\[
\begin{bmatrix}
0 & & & & \\
& 0 & & & \\
& & \half p_0 & p_1 & \\
& & p_1 & \half p_0 & \\
& & & & 0 \\
\end{bmatrix}
% +
% \begin{bmatrix}
% 0 & & & & \\
% & 0 & & & \\
% & & 0 & & \\
% & & & 3 & 1 \\
% & & & 1 & 3 \\
% \end{bmatrix}
\ \cdots \
\begin{bmatrix}
0 & & & & \\
& 0 & & & \\
& & 0 & & \\
& & & \half p_0 & p_1 \\
& & & p_1 & \half p_0 \\
\end{bmatrix}.
\]
Hence $ g $ may be expressed as
\begin{align}
g(x) & =
% \frac{p_0}{4} (x_1^2 + x_N^2)
% +
\half \sum_n
\left(
\big( x_{n-1}, \, x_n \big)
\begin{bmatrix}
\half p_0 & p_1 \\
p_1 & \half p_0
\end{bmatrix}
\begin{pmatrix} x_{n-1} \\ x_n \end{pmatrix}
\lam \psi( (x_{n-1}, x_n) ; a)
\right)
\\
& =
% \frac{p_0}{4} (x_1^2 + x_N^2)
% +
\half \sum_n f( (x_{n-1}, x_n) )
\end{align}
where $ f \colon \RR^2 \to \RR $ is given by (<ref>).
If $ f $ is convex, then $ g $ is the sum of convex functions and is thus convex itself.
A. Achim and E. E. Kuruoğlu.
Image denoising using bivariate $\alpha$-stable distributions in the
complex wavelet domain.
IEEE Signal Processing Letters, 12(1):17–20, January 2005.
M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo.
Fast image recovery using variable splitting and constrained
IEEE Trans. Image Process., 19(9):2345–2356, September 2010.
M. S. Asif and J. Romberg.
Fast and accurate algorithms for re-weighted $l_1$-norm minimization.
IEEE Trans. Signal Process., 61(23):5905–5916, December 2013.
F. Bach, R. Jenatton, J. Mairal, and G. Obozinski.
Optimization with sparsity-inducing penalties.
Foundations and Trends in Machine Learning, 4(1):1–106, 2012.
İ. Bayram.
On the convergence of the iterative shrinkage/thresholding algorithm
with a weakly convex penalty.
http://arxiv.org/abs/1510.07821, October 2015.
İ. Bayram.
Penalty functions derived from monotone mappings.
IEEE Signal Processing Letters, 22(3):265–269, March 2015.
I. Bayram, P.-Y. Chen, and I. Selesnick.
Fused lasso with a non-convex sparsity inducing penalty.
In Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing
(ICASSP), May 2014.
A. Beck and M. Teboulle.
A fast iterative shrinkage-thresholding algorithm for linear inverse
SIAM J. Imag. Sci., 2(1):183–202, 2009.
A. Blake and A. Zisserman.
Visual Reconstruction.
MIT Press, 1987.
A. Blumensath.
Accelerated iterative hard thresholding.
Signal Processing, 92(3):752–756, 2012.
S. Boyd and L. Vandenberghe.
Convex Optimization.
Cambridge University Press, 2004.
A. Bruckstein, D. Donoho, and M. Elad.
From sparse solutions of systems of equations to sparse modeling of
signals and images.
SIAM Review, 51(1):34–81, 2009.
C. L. Byrne.
Iterative Optimization in Inverse Problems.
CRC Press, 2014.
E. J. Candès, M. B. Wakin, and S. Boyd.
Enhancing sparsity by reweighted l1 minimization.
J. Fourier Anal. Appl., 14(5):877–905, December 2008.
P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud.
Deterministic edge-preserving regularization in computed imaging.
IEEE Trans. Image Process., 6(2):298–311, February 1997.
R. Chartrand.
Fast algorithms for nonconvex compressive sensing: MRI
reconstruction from very few data.
In IEEE Int. Symp. Biomed. Imag. (ISBI), pages 262–265, July
R. Chartrand.
Nonconvex splitting for regularized low-rank + sparse decomposition.
IEEE Trans. Signal Process., 60(11):5810–5819, November 2012.
R. Chartrand.
Shrinkage mappings and their induced penalty functions.
In Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing
(ICASSP), pages 1026–1029, May 2014.
R. Chartrand, E. Y. Sidky, and P. Xiaochuan.
Nonconvex compressive sensing for X-ray CT: an algorithm
In Asilomar Conf. on Signals, Systems and Computers, pages
665–669, November 2013.
L. Chen and Y. Gu.
The convergence guarantees of a non-convex approach for sparse
IEEE Trans. Signal Process., 62(15):3754–3767, August 2014.
P.-Y. Chen and I. W. Selesnick.
Group-sparse signal denoising: Non-convex regularization, convex
IEEE Trans. Signal Process., 62(13):3464–3478, July 2014.
H. Chipman, E. Kolaczyk, and R. McCulloch.
Adaptive Bayesian wavelet shrinkage.
J. Am. Stat. Assoc., 92(440):1413–1421, December 1997.
E. Chouzenoux, A. Jezierska, J. Pesquet, and H. Talbot.
A majorize-minimize subspace approach for $\ell_2-\ell_0$ image
SIAM J. Imag. Sci., 6(1):563–591, 2013.
P. L. Combettes and J.-C. Pesquet.
Proximal thresholding algorithm for minimization over orthonormal
SIAM J. Optim., 18(4):1351–1376, 2008.
P. L. Combettes and J.-C. Pesquet.
Proximal splitting methods in signal processing.
In H. H. Bauschke et al., editors, Fixed-Point Algorithms for
Inverse Problems in Science and Engineering, pages 185–212.
Springer-Verlag, 2011.
P. L. Combettes and V. R. Wajs.
Signal recovery by proximal forward-backward splitting.
Multiscale Modeling & Simulation, 4(4):1168–1200, 2005.
I. Daubechies, M. Defriese, and C. De Mol.
An iterative thresholding algorithm for linear inverse problems with
a sparsity constraint.
Commun. Pure Appl. Math, LVII:1413–1457, 2004.
I. Daubechies, R. DeVore, M. Fornasier, and C. Gunturk.
Iteratively reweighted least squares minimization for sparse
Comm. Pure App. Math., 63(1):1–38, January 2010.
Y. Ding and I. W. Selesnick.
Artifact-free wavelet denoising: Non-convex sparse regularization,
convex optimization.
IEEE Signal Processing Letters, 22(9):1364–1368, September
B. Dumitrescu.
Positive trigonometric polynomials and signal processing
Springer, 2007.
J. M. Fadili and L. Boubchir.
Analytical form for a Bayesian wavelet estimator of images using
the Bessel K form densities.
IEEE Trans. Image Process., 14(2):231–240, February 2005.
M. Figueiredo, J. Bioucas-Dias, and R. Nowak.
Majorization-minimization algorithms for wavelet-based image
IEEE Trans. Image Process., 16(12):2980–2991, December 2007.
M. Figueiredo and R. Nowak.
An EM algorithm for wavelet-based image restoration.
IEEE Trans. Image Process., 12(8):906–916, August 2003.
M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright.
Gradient projection for sparse reconstruction: Application to
compressed sensing and other inverse problems.
IEEE J. Sel. Top. Signal Process., 1(4):586–598, December
S. Foucart.
Hard thresholding pursuit: an algorithm for compressive sensing.
SIAM J. Numer. Anal., 49(6):2543–2563, 2010.
J.-J. Fuchs.
Convergence of a sparse representations algorithm applicable to real
or complex data.
IEEE. J. Sel. Top. Signal Processing, 1(4):598–605, December
J. J. Fuchs.
Identification of real sinusoids in noise, the global matched filter
In 15th IFAC Symp. on System Identification, pages 1127–1132,
Saint-Malo, France, July 2009.
G. Gasso, A. Rakotomamonjy, and S. Canu.
Recovering sparse signals with a certain family of nonconvex
penalties and DC programming.
IEEE Trans. Signal Process., 57(12):4686–4698, December 2009.
D. Geman and G. Reynolds.
Constrained restoration and the recovery of discontinuities.
IEEE Trans. Pattern Anal. and Machine Intel., 14(3):367–383,
March 1992.
A. Gholami and S. M. Hosseini.
A general framework for sparsity-based denoising and inversion.
IEEE Trans. Signal Process., 59(11):5202–5211, November 2011.
T. Goldstein and S. Osher.
The split Bregman method for L1-regularized problems.
SIAM J. Imag. Sci., 2(2):323–343, 2009.
G. Harikumar and Y. Bresler.
A new algorithm for computing sparse solutions to linear inverse
In Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing
(ICASSP), volume 3, pages 1331–1334, May 1996.
M. W. Jacobson and J. A. Fessler.
An expanded theoretical treatment of iteration-dependent
majorize-minimize algorithms.
IEEE Trans. Image Process., 16(10):2411–2422, October 2007.
N. Kingsbury and T. Reeves.
Redundant representation with complex wavelets: how to achieve
In Proc. IEEE Int. Conf. Image Processing (ICIP), volume 1,
pages 45–48, 2003.
I. Kozlov and A. Petukhov.
Sparse solutions of underdetermined linear systems.
In W. Freeden et al., editor, Handbook of Geomathematics.
Springer, 2010.
K. Lange, E. C. Chi, and H. Zhou.
A brief survey of modern optimization for statisticians.
Int. Stat. Rev., 82(1):46–70, 2014.
A. Lanza, S. Morigi, and F. Sgallari.
Convex image denoising via non-convex regularization.
In J.-F. Aujol, M. Nikolova, and N. Papadakis, editors, Scale
Space and Variational Methods in Computer Vision, volume 9087 of Lecture Notes in Computer Science, pages 666–677. Springer, 2015.
P.-L. Loh and M. J. Wainwright.
Regularized M-estimators with nonconvexity: Statistical and
algorithmic theory for local optima.
J. Machine Learning Research, 16:559–616, 2015.
D. A. Lorenz.
Non-convex variational denoising of images: Interpolation between
hard and soft wavelet shrinkage.
Current Development in Theory and Application of Wavelets,
1(1):31–56, 2007.
D. Malioutov and A. Aravkin.
Iterative log thresholding.
In Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing
(ICASSP), pages 7198–7202, May 2014.
S. Mallat.
A wavelet tour of signal processing.
Academic Press, 1998.
Y. Marnissi, A. Benazza-Benyahia, E. Chouzenoux, and J.-C. Pesquet.
Generalized multivariate exponential power prior for wavelet-based
multichannel image restoration.
In Proc. IEEE Int. Conf. Image Processing (ICIP), pages
2402–2406, September 2013.
H. Mohimani, M. Babaie-Zadeh, and C. Jutten.
A fast approach for overcomplete sparse decomposition based on
smoothed l0 norm.
IEEE Trans. Signal Process., 57(1):289–301, January 2009.
L. B. Montefusco, D. Lazzaro, and S. Papi.
A fast algorithm for nonconvex approaches to sparse recovery
Signal Processing, 93(9):2636–2647, 2013.
N. Mourad and J. P. Reilly.
Minimizing nonconvex functions for sparse vector reconstruction.
IEEE Trans. Signal Process., 58(7):3485–3496, July 2010.
M. Nikolova.
Estimation of binary images by minimizing convex criteria.
In Proc. IEEE Int. Conf. Image Processing (ICIP), pages
108–112 vol. 2, 1998.
M. Nikolova.
Markovian reconstruction using a GNC approach.
IEEE Trans. Image Process., 8(9):1204–1220, 1999.
M. Nikolova.
Energy minimization methods.
In O. Scherzer, editor, Handbook of Mathematical Methods in
Imaging, chapter 5, pages 138–186. Springer, 2011.
M. Nikolova, M. K. Ng, and C.-P. Tam.
Fast nonconvex nonsmooth minimization methods for image restoration
and reconstruction.
IEEE Trans. Image Process., 19(12):3073–3088, December 2010.
X. Ning, I. W. Selesnick, and L. Duval.
Chromatogram baseline estimation and denoising using sparsity
Chemometrics and Intelligent Laboratory Systems, 139:156–167,
December 2014.
D. P. Palomar and Y. C. Eldar, editors.
Convex Optimization in Signal Processing and Communications.
Cambridge University Press, 2010.
A. Parekh and I. W. Selesnick.
Convex denoising using non-convex tight frame regularization.
IEEE Signal Processing Letters, 22(10):1786–1790, October
A. Parekh and I. W. Selesnick.
Convex fused lasso denoising with non-convex regularization and its
use for pulse detection.
http://arxiv.org/abs/1509.02811, October 2015.
J. Portilla and L. Mancera.
L0-based sparse approximation: two alternative methods and some
In Proceedings of SPIE, volume 6701 (Wavelets XII), San Diego,
CA, USA, 2007.
J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli.
Image denoising using scale mixtures of Gaussians in the wavelet
IEEE Trans. Image Process., 12(11):1338–1351, November 2003.
K. Qiu and A. Dogandzic.
Sparse signal reconstruction via ECME hard thresholding.
IEEE Trans. Signal Process., 60(9):4551–4569, September 2012.
B. D. Rao, K. Engan, S. F. Cotter, J. Palmer, and K. Kreutz-Delgado.
Subset selection in noise based on diversity measure minimization.
IEEE Trans. Signal Process., 51(3):760–770, March 2003.
A. Repetti, M. Q. Pham, L. Duval, E. Chouzenoux, and J.-C. Pesquet.
Euclid in a taxicab: Sparse blind deconvolution with smoothed l1/l2
IEEE Signal Processing Letters, 22(5):539–543, May 2015.
I. W. Selesnick.
Sparsity-assisted signal smoothing.
In R. Balan et al., editors, Excursions in Harmonic Analysis,
Volume 4. Birkhäuser Basel, 2015.
I. W. Selesnick and I. Bayram.
Sparse signal estimation by maximally sparse convex optimization.
IEEE Trans. Signal Process., 62(5):1078–1092, March 2014.
I. W. Selesnick, A. Parekh, and I. Bayram.
Convex 1-D total variation denoising with non-convex
IEEE Signal Processing Letters, 22(2):141–144, February 2015.
L. Sendur and I. W. Selesnick.
Bivariate shrinkage functions for wavelet-based denoising exploiting
interscale dependency.
IEEE Trans. Signal Process., 50(11):2744–2756, November 2002.
Y. She.
Thresholding-based iterative selection procedures for model selection
and shrinkage.
Electronic Journal of Statistics, 3:384–415, 2009.
C. Soussen, J. Idier, D. Brie, and J. Duan.
From Bernoulli-Gaussian deconvolution to sparse signal
IEEE Trans. Signal Process., 59(10):4572–4584, October 2011.
X. Tan, W. Roberts, J. Li, and P. Stoica.
Sparse learning via iterative minimization with application to MIMO
radar imaging.
IEEE Trans. Signal Process., 59(3):1088–1101, March 2011.
J. Trzasko and A. Manduca.
Highly undersampled magnetic resonance image reconstruction via
homotopic L0-minimization.
IEEE Trans. Medical Imaging, 28(1):106–121, January 2009.
S. Voronin and R. Chartrand.
A new generalized thresholding algorithm for inverse problems with
sparsity constraints.
In Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing
(ICASSP), pages 1636–1640, May 2013.
Y. Wang and W. Yin.
Sparse signal reconstruction via iterative support detection.
SIAM J. Imag. Sci., 3(3):462–491, 2010.
D. Wipf and S. Nagarajan.
Iterative reweighted $\ell_1$ and $\ell_2$ methods for finding sparse
IEEE. J. Sel. Top. Signal Processing, 4(2):317–329, April
J. Woodworth and R. Chartrand.
Compressed sensing recovery via nonconvex shrinkage penalties.
http://arxiv.org/abs/1504.02923, April 2015.
H. Zou and R. Li.
One-step sparse estimates in nonconcave penalized likelihood models.
Ann. Statist., 36(4):1509–1533, 2008.
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1511.01008
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Threshold graphs are a prevalent and widely studied class of simple graphs. They have several equivalent definitions which makes them a go-to class for finding examples and counter examples when testing and learning. This versatility has led to many results about threshold graphs and similar structures. We look to generalize this class of graphs to oriented graphs (directed simple graphs.) We give generalizations to four of the most versatile definitions and show their equivalence in the oriented case. We finish with a proof enumerating the number of these oriented threshold graphs which relates to the Fibonacci numbers.
§ INTRODUCTION
§.§ History
Threshold graphs were first seen in several publications in the mid 1970s. Papers in a variety of areas independently developed basic definitions for a class of graphs which gets it's name from a 1973 paper titled Set-packing Problems and Threshold Graphs by Chvátal and Hammer <cit.>. These graphs have been found in numerous applications since their introduction. These cover a wide range of subjects including applications in set-packing, parallel processing, resource allocation, scheduling, and psychology. There is a great introduction to threshold graphs and their applications in the book Threshold Graphs and Related Topics by Mahadev and Peled, <cit.>.
In recent years, the limit points of threshold graphs (as graphons) have been studied in a paper by Diaconis, Holmes, and Jansen, <cit.>. This gives an interesting result that their limits can be realized as $\lb 0, 1\rb-$valued increasing functions on the unit square.
Another recent result is by Cloteaux, LaMar, Moseman, and Shook, <cit.>. Their generalization to directed graphs is focused on degree sequences and unique realizations. This work is extended by Reilly, Scheinerman, and Zhang, <cit.>. These extensions generalize definitions of simple threshold graphs into directed graphs and demonstrate their equivalence with the definitions of Cloteaux, et al. These definitions deal predominately with directed graphs in which 2-cycles (multiedges in the underlying graph) are permitted in order to obtain unique realizability.
In this paper we'll look at oriented simple graphs where we prohibit such 2-cycles and see some surprisingly lovely results.
§.§ Background
Mahadev and Peled in <cit.> give a thorough treatment of the class of threshold graphs. Here we give the basic definition and some equivalences.
TG Let $G$ be a graph. We say that $G$ is agraph, threshold threshold graph if there exists a threshold $t\in \R$ and a vertex weight function $w:V(G)\to \R$ such that $e=(x,y)\in E$ if and only if $w(x)+w(y) > t$.
Though this is a fairly simple definition to work with, there are several equivalences that will be worth considering. To understand them we need two definitions:
split graph, splitA graph, $G=(V,E)$, is said to be split if the vertex set $V$ can be partitioned into two classes $K$ and $I$ such that $K$ induces a clique in $G$, and $I$ is an independent set in $G$.
A graph, $G=(V,E)$, on four vertices is a switch switchif there is an ordering of the vertices, $a,b,c,d$ such that $ab,cd\in E$ and $ad,bc\notin E$.
We say a graph is switch-free if it contains no induced switches.
The definition of a switch describes several different graphs. For example, $C_4$, 2 copies of $K_2$, and $P_3$ the path with three edges are all switches.
We can now state 4 characterizations of threshold graphs.
TGequiv <cit.> The following are equivalent:
* $G$ is a threshold graph.
* $G$ is a split graph and the vertex neighborhoods are nested.
* $G$ is switch-free.
* The graph $G$ can be constructed by starting with a single vertex and sequentially adding either a dominating vertex or an isolated vertex at each step.
Equivalence (iv) of Theorem <ref> allows a very nice constructive bijection between binary sequences of length $n-1$ and threshold graphs on $n$ vertices. We define a threshold graph by creating a binary sequence $\bar s =(s_i)_{i=1}^n$ where $s_i=1$ if the vertex added is dominating, or $s_i=0$ if it is isolated. Given such a sequence we define $T(\bar s)$ to be the threshold graph associated with it. In this construction the very first vertex is both isolated and dominating; we therefore classifying it as a 0 or 1 is somewhat misleading. We will always classify the first vertex as $\star$ when giving a threshold graph in its sequential form. We use the convention that the sequence is constructed right to left, thinking of the first vertices added as least significant, as in least significant digits in a number.
§ ORIENTED THRESHOLD GRAPHS
There are several definitions for a threshold graph in the undirected case, Theorem <ref>. We begin by developing an analogous vocabulary for oriented graphs and then state a theorem presenting several equivalent definitions of an oriented threshold graph.
dtgdef digraph, thresholdAn oriented graph, $G=(V,E)$, is said to be threshold if there exists a weight function on the vertices $w:V\to \R$ and a threshold value $t\in\R$ such that $\overrightarrow{xy}\in E$ if and only if $|w(x)|+|w(y)| \ge t$ and $w(x)>w(y)$.
Since we insist on strict inequality, $w(x)>w(y),$ so that our oriented threshold graphs contain no loops. This also lets us think of edges running `downhill.'
§.§ Background and Oriented Threshold Equivalence
Before we state our main theorem which is directly analogous to Theorem <ref>, we need to develop vocabulary to state corresponding statements in the oriented case.
The first generalization we'll explore is nested neighborhoods. These next definitions will help us generalize the concept of nested neighborhoods to oriented graphs where we have not just a total neighborhood, but have in and out neighborhoods as well.
displit digraph, splitdisplitA directed graph is said to be displit if the vertex set can be partitioned into three classes, $V=B\cup I\cup T$ (Bottom, Independent, and Top), with the properties:
* $I$ is an independent set,
* the graph induced by $B\cup T$ is a tournament,
* all edges between $T$ and $B\cup I$ are directed from $T$,
* all edges between $B$ and $T\cup I$ are directed into $B$.
A small example of a displit graph may be helpful in understanding this definition.
A small displit graphdisplitexa
[scale=.75, transform shape, >=triangle 45]
[place] (2) at (-2,2)T;
[place] (1) at (-1,1)T;
[place] (0) at (0,0)I;
[place] (-1) at (-1,-1)B;
[place] (-2) at (-2,-2)B;
[place] (i) at (1.4,0)I;
[place] (ii) at (2.8,0)I;
EdgeStyle=[->,bend right]
EdgeStyle=[->, bend left]
LaMar prior to working on the unigraphic sequence problem for digraphs gave a definition for split digraphs in <cit.>, the definition we just gave fits within his, but is slightly more restrictive. With this stronger definition, we end up with a smaller class of graphs, but we are able to say much more about the structure of our class, both by giving an inductive construction from a ternary sequence, and by defining what it means to have nested neighborhoods in the sense of directed threshold graphs.
doms in-dominatedout-dominatingA vertex, $v$, in a digraph $D$ is called an out-dominating (in-dominated) if it is adjacent to every other vertex in $D$ and is a source (resp. sink).
Let $\sigma:V\to 2^V$ be a function from a set to its power set. We say the function $\sigma$ is nestedfunction, nestednested, on $S\subset V$ if for every $x,y\in S$ we have $\sigma(x)\subseteq \sigma(y)\cup \lb y\rb$ or $\sigma(y)\subseteq \sigma(x)\cup\lb x\rb$. We denote $\sigma(x)\subseteq \sigma(y)\cup\lb y\rb$ by $x\unlhd^\sigma y$.
We say the function $\sigma$ is strictly nested on $S\subset V$, if for every $x,y\in S$ we have either $\sigma(x)\subseteq \sigma(y)$ or $\sigma(y)\subseteq \sigma(x)$. For $x,y\in V$ we denote $\sigma(x)\subsetneq \sigma(y)$ by $x\lhd^\sigma y$.
If $\sigma$ is nested (resp. strictly nested) on all of $V$, we say that $\sigma$ is nested (resp. strictly nested.)
Recall that in Theorem <ref> one of the conditions that makes a graph threshold is that it is split and has nested neighborhoods. That is, the neighborhood function of a threshold graph is a nested function. With this in mind, the proper way to view nested and strictly nested functions in terms of directed threshold graphs is via the following definition.
pnn Let $D$ be a displit graph with clique $K=T\cup B$ and independent set $I$. We say $D$ has properly nested neighborhoods properly nested neighborhoods if the following hold for $N,N^+, N^-: V\to P(V)$ the neighborhood, out-neighborhood and in-neighborhood functions:
* $N$ is nested,
* $N^+$ and $N^-$ are nested on $I$ and for $x,y\in I$ we have if $x\unlhd^N y$ then $x\unlhd^- y$ and $x\unlhd^+ y$ (I've suppressed the Ns in the inequalities to make the notation less cumbersome.)
* $N^+$ and $N^-$ are strictly nested on $K$ and for $x,y\in K$ we have $x\lhd^+ y$ if and only if $y\lhd^- x$.
The properly nested neighborhoods condition states that the total neighborhoods are nested. For vertices in $I$, the size of in and out neighborhoods are directly correlated, whereas in $B\cup T$ the in and out neighborhoods are inversely correlated. Figure <ref> gives an example of a graph which has properly nested neighborhoods.
Now, we can generalize Theorem <ref> to directed threshold graphs.
equivdef The following are equivalent for a graph $G=(V,E)$:
* $G$ is an oriented threshold graph.
* $G$ is a transitive orientation of a threshold graph.
* $G$ is a displit graph and has properly nested neighborhoods.
* $G$ can be constructed from the one vertex empty graph by successively adding a independent vertex, an out-dominating vertex or an in-dominated vertex.
($a\implies b$): Let $w:V(G)\to\R$ be the weight function and $t\in \R$ the threshold value for $G$. To show the underlying graph is threshold, use the weight function $|w(v)|$ and the same threshold $t$. For transitivity, suppose $x\rightarrow y$ and $y\rightarrow z$. By definition then, we know that $|w(x)|+|w(y)|\ge t$ and $|w(y)|+|w(z)|\ge t$ and that $w(x)>w(y)>w(z)$. We need to consider 2 cases to show that $x\to z$.
Case 1) $w(y)\ge 0$: Then $w(x)>0$ so $|w(x)|\ge |w(y)|$ therefore $|w(x)|+|w(z)|\ge |w(y)|+|w(z)|\ge t.$ So $x\rightarrow z$.
Case 2) $w(y)\le 0$: Then $w(z)<0$ so $|w(z)|\ge |w(y)|$ therefore $|w(x)|+|w(z)|\ge |w(x)|+|w(y)|\ge t.$ So again $x\rightarrow z$.
This shows that $G$ is transitive, and thus a transitive orientation of a threshold graph.
($b\implies c$): Let $G$ be the underlying threshold graph associated with the sequence $\bar s$ (Theorem <ref>.) The initial vertex in the sequential construction is given the label $\star$. Now, the collection of $\star$ and the $0$'s form an independent set; call this set of vertices $I$. The collection of $1$s form a clique, call this set $K$. Set $T = N^-(\star)$ and $B= N^+(\star)$. Since every $1$ was adjacent to $\star$, this partitions $K$. By the transitivity of the ordering, if $t\in T$ and $b\in B$, then $t\to \star$ and $\star\to b$ so $t\to b$. This gives us the partition $T\cup I\cup B.$ We have that all edges between $T$ and $B$ are oriented correctly.
We still need to show that edges between $K$ and $I$ are oriented correctly. That is, we need to show the edges are directed from $T$ to $I$, and the edges are directed to $B$ from $I.$ To do this we first show that $N^+$ and $N^-$ are nested on $I$. Since the underlying graph is threshold, the neighborhood function is nested (this is the nested neighborhoods condition of $(ii)$ in Theorem <ref>.) So, let $i,j\in I$ with $i\unlhd^N j$. Suppose there is $x\in N^+(i)\wout N^+(j)$. Then $x\in N^-(j)$ since $i\unlhd^N j$. But this means $i\to x \to j$ and transitivity gives $i\to j$, however, that is impossible as $i,j\in I$. So we must have $N^+(i)\subseteq N^+(j)$. A similar argument gives $N^-(i)\subseteq N^-(j)$. This shows property $ii$) of the properly nested condition, Definition <ref>.
Using this, and noting that for all $i\in I$ we have $i\unlhd^N \star$, we get that $N^+(i)\subseteq N^+(\star)=B$ and $N^-(i)\subseteq N^-(\star)=T$, showing that the graph is displit.
To show the third condition of properly nested neighborhoods, let $x,y\in K$ with $x\to y$. Then by transitivity $N^+(y)\subsetneq N^+(x)$ (the inclusion is strict because $y\notin N^+(y)$) and $N^-(x)\subsetneq N^-(y)$ (again because $x\notin N^-(x)$) which completes all conditions.
($c\implies d$): Let $i\in I$ be minimal in $I$ with respect to total neighborhoods. If $N(i)=\emptyset$ then it is an isolate. If not, we have that either its in-neighborhood is non-empty or its out-neighborhood is non-empty. Say $x\in N^+(i)$. Then, $x\in N^+(j)$ for all $j\in I$ since the neighborhood function is nested on $I$. Let $y$ be the maximum element in the order given by $N^-$ being strictly nested on $K$. Then since $x\in N^+(j)$ for all $j\in I$, we get that $j\in N^-(x)\subsetneq N^-(y)$ for all $j\in I$. This show $y$ is dominated by $I$. To show $y$ is dominated by $K$, suppose to get a contradiction, $y\to z$ for some $z\in K$. Then $y\in N^-(z)\subsetneq N^-(y)$ which is impossible. This means $y$ is an in-dominated vertex. A similar argument gives that if $x\in N^-(i)$ then the maximal vertex with respect to the strictly nested order on $N^+$ is an out-dominating vertex.
In order to inductively choose an independent, out-dominating, or in-dominated vertex we must show that the removal of such a vertex leaves us with a transitive digraph with properly nested neighborhoods. If the vertex is isolated, its removal has no effect on the neighborhoods or the transitivity of the graph. If the vertex is dominating or dominated, then its removal decreases every neighborhood in exactly the same way, leaving comparability conditions intact. The transitivity also remains as removal of a vertex in any transitive graph leaves the graph transitive.
This gives us an inductive construction of the directed threshold graph as a sequence of independent, in-dominated, and out dominating vertices as required.
($d\implies a$): The assumption gives a sequence of zeros, ones, and negative ones, say $(s_i)_{i=1}^n$. If we forget (temporarily) about the sign on the ones, we have a sequence of zeros and ones corresponding to removing the direction on the edges. This underlying graph is constructed by adding isolates or dominating vertices. This means it is an unoriented threshold graph. There is an injective weight function and threshold for this underlying graph <cit.>, say $(w_i)_{i=1}^n$ and $t$. It is enough to show then, that the weight function
$$\vec w_i=\begin{cases} s_iw_i, & \mathrm{if} s_i\ne 0 \cr w_i, & \mathrm{if} s_i=0.\end{cases}$$
gives the correct orientation of the edges.
Let's make an observation about the weights of the vertices. Notice that the weight of a vertex is directly correlated to the size of its neighborhood. This means the later in the sequence a $1$ happens, the higher the weight of the vertex associated to it. Conversely, the later in the sequence a $0$ happens, the lower the weight of the associated vertex.
With this, we see that the above weight function satisfies $|\vec w_i|>|\vec w_j|$ whenever $i>j$ and $|s_i|=1$. This means that the orientation of the graph given by the above weight function is the same as the orientation given by the sequential construction.
dtgseq This last equivalence gives a ternary sequence which can be translated into an oriented threshold graph. We call the graph a sequence $s$ produces, $\dtg(s)$
Let $\bar s = (1,-1,0,-1,\star)$. The sequential construction yields the following graph.
(Left) $\dtg(1,-1,0,-1,\star)$ and (right) the directed threshold graph corresponding to the weight function $(15,-12, 3, -9, 6)$ and threshold $15$.dtg1
[scale=.75, transform shape,>=triangle 45]
[place] (5) at (-12.5,0)1;
[place] (-4) at (-11.5,-.8)-1;
[place] (1) at (-10,-1.1)0;
[place] (-3) at (-8.5,-.8)-1;
[place] (2) at (-7.5,0)$\star$;
[place] (50) at (-2.5,0)15;
[place] (-40) at (-1.5,-.8)-12;
[place] (10) at (0,-1.1)3;
[place] (-30) at (1.5,-.8)-9;
[place] (20) at (2.5,0)6;
EdgeStyle=[->,bend left]
EdgeStyle=[<-, bend left]
EdgeStyle=[->,bend left]
EdgeStyle=[<-, bend left]
The vertex weights shown in the figure, $(15, -12, 3, -9, 6)$ with threshold $t=15$ also give the same graph.
§ SEQUENTIAL FORM AND ENUMERATION
A few things before we go further: to draw and think about these oriented threshold graphs, the sequential definition is quite a bit more malleable; we work with it. Recall that in the undirected case, the first vertex drawn is always an independent vertex and we denoted it by $\star$. We'll use this convention with oriented threshold graph sequences as well. Another simplification, instead of $+1$ and $-1$ we simply write $+$ and $-$ (resp.)
Looking more closely at these sequences, things get a little messy. In the undirected case, it is easy to just count the sequences, $\lb 0,1\rb^{n-1}$ ($n-1$ as the first vertex drawn does not matter.) Things are a little more subtle in the case of directed threshold graphs. Notice (in figure <ref>) that the sequence $(+-0\,\star)$ gives the same graph as the sequence $(-+0\,\star)$. The isomorphism switches the last two vertices, as shown in the following figure.
(Left) $\dtg(-1,1,0,\star)$, and (right) $\dtg(1,-1,0,\star)$mppm
[scale=.75, transform shape,>=triangle 45]
[place] (+) at (-1.6,0)+;
[place] (-) at (1.6,0)-;
[place] (0) at (0,-1.1)0;
[place] (s) at (0,-2.9)$\star$;
[place] (-) at (-10.6,0)-;
[place] (+) at (-7.4,0)+;
[place] (0) at (-9,-1.1)0;
[place] (s) at (-9,-2.9)$\star$;
pmcomm Given a sequence $\bar s:=(s_i)_{i=1}^n$, if there is a $k\in [n]$ such that $|s(k)|=|s(k-1)|$ then the sequence $s' = (s_1,s_2, \cdots, s_{k-2}, s_k, s_{k-1},s_{k+1},\cdots, s_n)$ produces a digraph isomorphic to the one produced by $s$.
Clearly if $s_k=s_{k-1}$ we're fine. So without loss of generality assume $s_k=+$ and $s_{k-1}=-$. So there is an edge $\overrightarrow{k(k-1)}$. Now, just note the neighborhoods $N^+(k), N^-(k), N^+(k-1), N^-(k-1)$ do not change when we swap the order of $k$ and $k-1$, as the only edge affected is the one between them, and its order is switched as was needed.
endplus As the $\star$ at the beginning of any sequence can be thought of as a $+, -,$ or $0$, we can always think of $-$s adjacent to $\star$ as $+$'s.
Using the previous lemma and remark we obtain a `canonical' representation for any isomorphism class, namely,
$$(+^{p_l},-^{m_l}, 0^{z_l}, +^{p_{l-1}}, -^{m_{l-1}}, 0^{z_{l-1}}, \cdots, +^{p_1}, -^{m_1}, 0^{z_1}, +^{p_0} \star)$$
where $z_i\ne0$ for all $i$. The notation $+^{p_i}$, $-^{m_i}$, and $0^{z_i}$ simply mean $p_i$ $+$s, $m_i$ $-$s, and $z_i$ $0$s (resp.)
canon digraph, threshold canonical formThere is a bijection between isomorphism classes of directed threshold graphs and sequences of the form
$$(+^{p_l},-^{m_l}, 0^{z_l}, +^{p_{l-1}}, -^{m_{l-1}}, 0^{z_{l-1}}, \cdots, +^{p_1}, -^{m_1}, 0^{z_1}, +^{p_0}\star)$$
where the $z_i$ are positive integers and $p_i$ and $m_i$ are non-negative integers. We call a sequence of this form canonical.
Let $G$ be a directed threshold graph. Then by Theorem <ref> (d) there is a sequence of the characters $+,-$ and $0$ corresponding to $G$. By Lemma <ref> any grouping of $+$s and $-$s can be rearranged so that all $+$s are to the left of all $-$s in the grouping without changing the isomorphism class. Also, by Remark <ref>, if there is a grouping of $+$s and $-$s by $\star$ we can consider them as all $+$s. Each of these groupings is separated by a grouping of $0$s. This gives us a canonical sequence for the graph $G$.
Now, every ternary sequence gives a unique representation of the form
$$(+^{p_l},-^{m_l}, 0^{z_l}, +^{p_{l-1}}, -^{m_{l-1}}, 0^{z_{l-1}}, \cdots, +^{p_1}, -^{m_1}, 0^{z_1}, \star). $$
Since each graph gives a ternary sequence, if we can show that two sequences that have different canonical forms give non-isomorphic graphs, we're done.
So let $s$ and $t$ be two different ternary sequences in canonical form. Let's get through a few trivial cases first. If we take the underlying undirected graphs of $s$ and $t$, and they are non-isomorphic, then $s$ and $t$ themselves cannot be isomorphic. To get to the undirected underlying graphs, we just look at the binary sequences where $+$s and $-$s are mapped to $1$, and $0$s are mapped to $0$. If these aren't the same, we're done. So assume, $s$ and $t$ have the same length and the same number of $0$s moreover the indices of the $0$s are the same. Let $i$ be the leftmost index in which $s$ and $t$ differ, without loss of generality, say $s_i=+$ and $t_i=-$. Let $k$ be the next index ($k<i$) where $s_k=t_k=0$ (this exists because $s$ and $t$ are in canonical form, and if it didn't, then $s$ and $t$ would be a sequence of $+$s.) Then the grouping of $0$s that include $s_k$ and $t_k$ form an independent set, say $I$. Now, all vertices in $I$ have the same degree, $n-k$.
From here there are two cases: there is is another $+$ or $-$ after the grouping of $0$s, or $\star$ is the next vertex after the grouping of zeros in which index $k$ lies.
In the first case, there are no other vertices of degree $n-k$ besides those in $I$. In the sequence $s$, the number of vertices in the in-neighborhood of the vertices in $I$ is greater than they are in $t$. This makes the two graphs non-isomorphic.
In the other case, the number of $+$ and $-$ are different between the two sequences, meaning the displit partition of the vertices is different showing that the sequences represent non-isomorphic graphs.
Having a ternary canonical representation for oriented threshold graphs gives us an easy way to count the number of isomorphism classes of directed threshold graphs on $n$ vertices.
number The number of isomorphism classes of directed threshold graphs on $n$ vertices is $F_{2n}$ the $2n$ Fibonacci number (where $F_0=0, F_1=1$.)
We find a recursion relation on the classes by looking at the sequences in canonical form. We can always create a new sequence in canonical from one in canonical form by augmenting it with a $0$ or $+$, but only sequences that have a $0$ or $-$ can be augmented with a $-$ to form a new sequence in canonical form. Let $T(n)$ be the number of sequences in canonical form, and $P(n)$ be the number of sequences in canonical form starting with a $+$. Because a sequence in canonical form cannot have $-$ before a $+$, we get that $T(n) = T(n-1)+T(n-1)+(T(n-1)-P(n-1))$ where the first two terms come from augmenting a $0$ or $+$ to an old sequence, and the last term from augmenting a $-$. Since we could always have augmented a sequence with a $+$, we get that $P(n)=T(n-1)$. This gives us the recurrence $T(n)=3T(n-1)-T(n-2)$. The initial conditions are that $T(1)=1$ (being the sequence $+$) and $T(2)=2$ (from the sequences $+\star$ and $0\star$.)
Let's look at the Fibonacci sequence for a second. Specifically, let's look at $F_{2n}$.
\begin{eqnarray} F_{2n} & = F_{2n-1}+F_{2n-2}\cr & = 2F_{2n-2}+F_{2n-3} \cr & = 3F_{2n-2}-F_{2n-4} \cr & = 3F_{2(n-1)} - F_{2(n-2)} \end{eqnarray}
Also, notice that $F_0=1$ and $F_2=2$. Therefore we have the same recursion and starting values as the even Fibonacci numbers, and we're done.
Putting this together with the characterization of directed threshold graphs, <ref>, we see that the number of isomorphism classes of orientation of threshold graphs which are transitive is $F_{2n}$. This follows directly from statement $b)$ from <ref> states that every transitive ordering of a threshold graph is directed threshold. This means that the number of transitive orientations come from tertiary sequences and the number of isomorphism classes we've just shown is $F_{2n}$.
Canonical representation also allows us to count the number of non-isomorphic transitive orientations of a specific threshold graph.
ntransorient Let $G$ be a threshold graph given by the sequence $(+^{p_l}, 0^{z_l}, \dots, +^{p_1}, 0^{p_1}, +^{p_0}, \star).$ The number of non-isomorphic transitive orientations of $G$ is
$$\prod_{i=1}^l (p_l+1).$$
An orientation of $G$ is given by turning some of the $+$s into $-$s. Canonical form states that we get the same graph if we put $-$s at the end of the string of $+$/$-$s. So, for each block of $+$s we simply have a choice of where to start putting $-$s. There are $p_i+1$ choices for each block. The last block doesn't get any $-$s. The product of these choices is the total number of orientations which are transitive.
|
1511.00654
|
[mycorrespondingauthor]Corresponding author
Tufts University,
Department of Mathematics, 503 Boston Avenue, Medford, MA 02155, USA.
This paper presents a new type of Gronwall-Bellman inequality, which arises from a class of integral equations with a mixture of nonsingular and singular integrals. The new idea is to use a binomial function to combine the known Gronwall-Bellman inequalities for integral equations having nonsingular integrals with those having singular integrals. Based on this new type of Gronwall-Bellman inequality, we investigate the existence and uniqueness of the solution to a fractional stochastic differential equation (SDE) with fractional order $0 < \alpha < 1$. This result generalizes the existence and uniqueness theorem related to fractional order $\frac{1}{2} < \alpha < 1$ appearing in <cit.>. Finally, the fractional type Fokker-Planck-Kolmogorov equation associated to the solution of the fractional SDE is derived using Itô's formula.
Gronwall-Bellman inequality; Fractional stochastic differential equations (SDEs); Existence and uniqueness; fractional Fokker-Planck equation.
§ INTRODUCTION
It is well known that integral inequalities are instrumental in studying the qualitative analysis of solutions to differential and integral equations <cit.>. Among these inequalities, the distinguished Gronwall-Bellman type inequality from <cit.>, and its associated extensions, <cit.>, are capable of affording explicit bounds on solutions of a class of linear differential equations with integer order. The following lemma concerns a standard Gronwall-Bellman inequality in <cit.> for a differential equation with order one or equivalently an integral equation with nonsingular integrals.
Suppose $h(t)$, $k(t)$ and $x(t)$ are continuous functions on $t_{0} \leq t < T, 0 < T\leq \infty,$ with $k(t) \geq 0$. If $x(t)$ satisfies
\begin{eqnarray*}
x(t) \leq h(t) + \int_{t_{0}}^{t}k(s)x(s)\mathrm{d}s,
\end{eqnarray*}
\begin{eqnarray*}
x(t) \leq h(t) + \int_{t_{0}}^{t}h(s)k(s)\exp\Bigg [\int_{s}^{t}k(u)\mathrm{d}u\Bigg ]\mathrm{d}s.
\end{eqnarray*}
Moreover, if $h(t)$ is nondecreasing, then
\begin{eqnarray*}
x(t) \leq h(t)\exp\Bigg [\int_{t_{0}}^{t}k(s)\mathrm{d}s\Bigg ].
\end{eqnarray*}
In order to investigate the qualitative properties of solutions to differential equations of fractional order, there are several generalizations of Gronwall-Bellman inequalities developed by many researchers <cit.>. Let's recall the following generalized Gronwall-Bellman inequality proposed in <cit.> for a fractional differential equation with order $\beta > 0$ or equivalently an integral equation with singular integrals.
Suppose $\beta > 0$, $a(t)$ is a nonnegative function which is locally integrable on $0 \leq t < T, 0 < T\leq \infty$, and $g(t)$ is a nonnegative, nondecreasing continuous function defined on $0 \leq t < T$ with $g(t) \leq M~(constant)$. If $u(t)$ is nonnegative and locally integrable on $0 \leq t < T $ with
\begin{eqnarray*}
u(t) \leq a(t) + g(t)\int_{0}^{t}(t -s)^{\beta - 1}u(s)\mathrm{d}s
\end{eqnarray*}
on this interval, then
\begin{eqnarray*}
u(t) \leq a(t) + \int_{0}^{t}\Bigg [ \sum_{n = 1}^{\infty} \frac{(g(t)\Gamma(\beta))^{n}}{\Gamma(n\beta)}(t - s)^{\{n\beta - 1\}}a(s)\Bigg]\mathrm{d}s,
\end{eqnarray*}
where $\Gamma(t)$ is the gamma function. Furthermore, if $a(t)$ is nondecreasing on $0 \leq t < T$, then
\begin{eqnarray*}
u(t) \leq a(t)E_{\beta}(g(t)\Gamma(\beta)t^{\beta}),
\end{eqnarray*}
where $E_{\beta}(z)$ is the Mittag-Leffler function defined by $E_{\beta}(z) = \sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(k\beta +1)}$ for $z > 0$.
From many real applications, such as in physics, theoretical biology, and mathematical finance, there is substantial interest in a class of fractional SDEs <cit.>. The fractional SDEs take the form
\begin{eqnarray}\label{fraceq1}
\mathrm{d}x(t) = b(t,x(t))\mathrm{d}t + \sigma_{1}(t, x(t))\mathrm{d}t^{\alpha} + \sigma_{2}(t, x(t))\mathrm{d}B_{t},
\end{eqnarray}
where the initial value is $x(0) = x_{0}$, $0 < \alpha < 1$, and $B_{t}$ is the standard Brownian motion. According to <cit.>, the integral equation corresponding to Eq. (<ref>) is
\begin{eqnarray}\label{fracinteg}
x(t) = x_{0} + \int_{0}^{t}b(s, x(s))\mathrm{d}s + \alpha\int_{0}^{t}(t - s)^{\alpha - 1}\sigma_{1}(s, x(s))\mathrm{d}s + \int_{0}^{t}\sigma_{2}(s, x(s))\mathrm{d}B_{s}.
\end{eqnarray}
Since $0 < \alpha < 1$, there are nonsingular and singular integrals in the integral equation Eq. (<ref>). However, the above mentioned types of Gronwall-Bellman inequalities, such as Lemmas <ref> and <ref>, are not applicable to studying the qualitative properties of the solution to Eq. (<ref>) or Eq.(<ref>).
The first goal of this paper, presented in Section <ref>, is to derive a new type of Gronwall-Bellman inequality which is applicable to study the qualitative behaviors of the solution to the fractional SDE Eq. (<ref>) or the stochastic integral equation Eq. (<ref>). The second goal, accomplished in Section <ref>, is to apply the results from Section <ref> to investigate existence and uniqueness of the solution to the fractional SDE Eq. (<ref>) of order $0 < \alpha < 1$. Finally, in Section <ref>, a fractional type Fokker-Planck-Kolmogorov equation associated to the solution of the fractional SDE Eq. (<ref>) is derived.
§ GENERALIZATION OF THE GRONWALL-BELLMAN INEQUALITY
In this section, we develop a new integral inequality, Eq. (<ref>) below, by verifying three claims. The first claim is established by using the method of induction and taking advantage of the binomial function; the second claim is verified by taking advantage of properties of the Gamma function; the third claim is verified by employing Gamma functions, Mittag-Leffler functions and exponential functions. The established integral inequality is applicable to the fractional SDE Eq.(<ref>) or the stochastic integral equation Eq. (<ref>). Also this new integral inequality can be considered as a generalization of the integral inequalities in Lemmas <ref> and <ref>.
Let $0 < \alpha < 1$ and consider the time interval $I = [0, T)$, where $T\leq~\infty$. Suppose $a(t)$ is a nonnegative function, which is locally integrable on $I$ and $b(t)$ and $g(t)$ are nonnegative, nondecreasing continuous function defined on $I$, with both bounded by a positive constant, $M$. If $u(t)$ is nonnegative, and locally integrable on $I$ and satisfies
\begin{eqnarray}\label{condforineq}
u(t) \leq a(t) + b(t)\int_{0}^{t}u(s)\mathrm{d}s + g(t)\int_{0}^{t}(t -s)^{\alpha - 1}u(s)\mathrm{d}s,
\end{eqnarray}
\begin{eqnarray}\label{fracgrowineq}
u(t) \leq a(t) + \sum_{n = 1}^{\infty}\sum_{i=0}^{n} \dbinom{n}{i}b^{n - i}(t)g^{i}(t)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha + n - i)}\int_{0}^{t}(t - s)^{\{i\alpha - (i+1-n)\}}a(s)\mathrm{d}s.
\end{eqnarray}
Let $\phi$ be a locally integrable function and define an operator $B$ on $\phi$ as follows
\begin{eqnarray*}
B\phi(t) := b(t)\int_{0}^{t}\phi(s)\mathrm{d}s + g(t)\int_{0}^{t}(t - s)^{\alpha - 1}\phi(s)\mathrm{d}s, ~~~t\geq 0.
\end{eqnarray*}
From the inequality, Eq. (<ref>),
\begin{eqnarray*}
u(t) \leq a(t) + Bu(t).
\end{eqnarray*}
This implies
\begin{eqnarray}\label{iterineq}
u(t) \leq \sum_{k = 0}^{n - 1}B^{k}a(t) + B^{n}u(t).
\end{eqnarray}
In order to get the desired inequality, Eq. (<ref>), from Eq. (<ref>), there are three claims to be verified.
The first claim provides a general bound on $B^{n}(t)$:
\begin{eqnarray}\label{guessiter}
B^{n}u(t) \leq \sum_{i = 0}^{n}\dbinom{n}{i}b^{n - i}(t)g^{i}(t)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha + n - i)}\int_{0}^{t}(t - s)^{\{i\alpha - (i+1-n)\}}u(s)\mathrm{d}s.
\end{eqnarray}
The method of induction will be used to verify the inequality in Eq. (<ref>). First let $n = 1$. Then the inequality in Eq. (<ref>) is true. Now, suppose that the inequality, Eq. (<ref>), holds for $n = k$, and then compute $B^{n}$ when $n = k+1$,
\begin{eqnarray*}
\begin{aligned}
B^{k+1}u(t) &= B(B^{k}u(t))\leq b(t)\int_{0}^{t}\sum_{i = 0}^{k}\dbinom{k}{i}b^{k-i}(s)g^{i}(s)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha + k - i)}\int_{0}^{s}(s - \tau)^{\{i\alpha - (i+1-k)\}}u(\tau)\mathrm{d}\tau\mathrm{d}s\\
&+ g(t)\int_{0}^{t}(t - s)^{\alpha - 1}\sum_{i = 0}^{k}\dbinom{k}{i}b^{k-i}(s)g^{i}(s)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i)}\int_{0}^{s}(s - \tau)^{\{i\alpha - (i+1-k)\}}u(\tau)\mathrm{d}\tau\mathrm{d}s.\\
%& := C(t) + G(t).
\end{aligned}
\end{eqnarray*}
\begin{eqnarray*}
C(t) := b(t)\int_{0}^{t}\sum_{i = 0}^{k}\dbinom{k}{i}b^{k-i}(s)g^{i}(s)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha + k - i)}\int_{0}^{s}(s - \tau)^{\{i\alpha - (i+1-k)\}}u(\tau)\mathrm{d}\tau\mathrm{d}s,
\end{eqnarray*}
\begin{eqnarray*}
G(t) := g(t)\int_{0}^{t}(t - s)^{\alpha - 1}\sum_{i = 0}^{k}\dbinom{k}{i}b^{k-i}(s)g^{i}(s)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i)}\int_{0}^{s}(s - \tau)^{\{i\alpha - (i+1-k)\}}u(\tau)\mathrm{d}\tau\mathrm{d}s.
\end{eqnarray*}
Then, compute $C(t)$ and $G(t)$ term by term to reach the desired inequality Eq. (<ref>). Since $b(t)$ and $g(t)$ are nonnegative and nondecreasing functions,
\begin{eqnarray}\label{computationofC(t)}
\begin{aligned}
C(t) &\leq \sum_{i = 0}^{k}b^{k-i+1}(t)g^{i}(t)\dbinom{k}{i}\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i)}\int_{0}^{t}\int_{0}^{s}(s - \tau)^{\{i\alpha -(i+1-k)\}}u(\tau)\mathrm{d}\tau\mathrm{d}s\\
&= \sum_{i = 0}^{k}b^{k-i+1}(t)g^{i}(t)\dbinom{k}{i}\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i)}\int_{0}^{t}\int_{\tau}^{t}(s - \tau)^{\{i\alpha -(i+1-k)\}}u(\tau)\mathrm{d}s\mathrm{d}\tau\\
&= \sum_{i = 0}^{k}b^{k-i+1}(t)g^{i}(t)\dbinom{k}{i}\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i+1)}\int_{0}^{t}(t - \tau)^{\{i\alpha -(i-k)\}}u(\tau)\mathrm{d}\tau\\
&= b^{k+1}(t)\dbinom{k}{0}\frac{1}{\Gamma(k+1)}\int_{0}^{t}(t - \tau)^{k}u(\tau)\mathrm{d}\tau\\
&+ b(t)\sum_{i = 1}^{k}b^{k-i}(t)g^{i}(t)\dbinom{k}{i}\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i+1)}\int_{0}^{t}(t - \tau)^{\{i\alpha -(i-k)\}}u(\tau)\mathrm{d}\tau.\\
\end{aligned}
\end{eqnarray}
Similarly, compute $G(t)$
\begin{eqnarray}\label{computationofG(t)}
\begin{aligned}
G(t) &\leq \sum_{i = 0}^{k}b^{k-i}(t)g^{i+1}(t)\dbinom{k}{i}\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i)}\int_{0}^{t}(t - s)^{\alpha - 1}\int_{0}^{s}(s - \tau)^{\{i\alpha -(i+1-k)\}}u(\tau)\mathrm{d}\tau\mathrm{d}s\\
&= \sum_{i = 0}^{k}b^{k-i}(t)g^{i+1}(t)\dbinom{k}{i}\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i)}\int_{0}^{t}\int_{\tau}^{t}(t - s)^{\alpha - 1}(s - \tau)^{\{i\alpha -(i+1-k)\}}u(\tau)\mathrm{d}s\mathrm{d}\tau\\
&= \sum_{i = 0}^{k}b^{k-i}(t)g^{i+1}(t)\dbinom{k}{i}\frac{[\Gamma(\alpha)]^{i+1}}{\Gamma((i+1)\alpha+k-i)}\int_{0}^{t}(t - \tau)^{\{(i+1)\alpha -(i+1-k)\}}u(\tau)\mathrm{d}\tau\\
&= g^{k+1}(t)\dbinom{k}{k}\frac{[\Gamma(\alpha)]^{k+1}}{\Gamma((k+1)\alpha)}\int_{0}^{t}(t - \tau)^{\{(k+1)\alpha - 1\}}\mathrm{d}\tau\\
&+ b(t)\sum_{i = 1}^{k}b^{k-i}(t)g^{i}(t)\dbinom{k}{i-1}\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i+1)}\int_{0}^{t}(t - \tau)^{\{i\alpha -(i-k)\}}u(\tau)\mathrm{d}\tau.\\
\end{aligned}
\end{eqnarray}
Combining Eq. (<ref>) and Eq. (<ref>) yields
\begin{eqnarray*}
\begin{aligned}
B^{k+1}u(t) &= C(t) + G(t)\leq b^{k+1}(t)\dbinom{k}{0}\frac{1}{\Gamma(k+1)}\int_{0}^{t}(t - \tau)^{k}u(\tau)\mathrm{d}\tau\\
&+ b(t)\sum_{i = 1}^{k}[\dbinom{k}{i-1} + \dbinom{k}{i}]b^{k-i}(t)g^{i}(t)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i+1)}\int_{0}^{t}(t - \tau)^{i\alpha - (i - k)}u(\tau)\mathrm{d}\tau\\
&+ g^{k+1}(t)\dbinom{k}{k}\frac{[\Gamma(\alpha)]^{k+1}}{\Gamma((k+1)\alpha)}\int_{0}^{t}(t - \tau)^{(k+1)\alpha - 1}\mathrm{d}\tau\\
&= \sum_{i=0}^{k+1}\dbinom{k+1}{i}b^{k+1-i}(t)g^{i}\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+k-i+1)}\int_{0}^{t}(t - \tau)^{i\alpha-(i-k)}u(\tau)\mathrm{d}\tau.
\end{aligned}
\end{eqnarray*}
This implies that for any $n\in\mathbb{N}^{+}$, the first claim, Eq. (<ref>), holds.
The second claim shows that $B^{n}u(t)$ vanishes as $n$ increases. For each $t$ in $[0, T)$,
\begin{eqnarray}\label{guess_second}
B^{n}u(t) \to 0, ~as~n\to\infty.
\end{eqnarray}
For the purpose of notation simplification during the proof of the second claim, define
\begin{eqnarray*}
H_{n}(t) := \sum_{i = 0}^{n}\dbinom{n}{i}b^{n - i}(t)g^{i}(t)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha + n - i)}\int_{0}^{t}(t - s)^{\{i\alpha - (i+1-n)\}}u(s)\mathrm{d}s.
\end{eqnarray*}
Note that $\Gamma(x)$ is positive and decreasing on $(0, 1]$ but positive and increasing on $[2, \infty)$. Let $x_{i} = i\alpha + n - i$. Then, the sequence $x_{i}$ is decreasing over $[0, n]$ since $x_{i+1} - x_{i} = \alpha - 1 < 0$ when $i$ is an integer and $i\in [0, n]$. This means $x^{\min}_{i} = n\alpha$ and $x^{\max}_{i} = n$. Furthermore, for a fixed $\alpha$, there exists a large enough $n_{0}$ such that for any $n > n_{0}$, there is $n\geq\frac{2}{\alpha}$. So the sequence satisfies $x_{i} \geq 2$ for any integer $i\in[0, n]$ if $n$ is large enough. Thus, for any $i\in[0, n]$, $\Gamma(x^{\min}_{i}) < \Gamma(x_{i})$ and
\begin{eqnarray*}
H_{n}(t) \leq \frac{1}{\Gamma(n\alpha)}\sum_{i = 0}^{n}\dbinom{n}{i}b^{n - i}(t)g^{i}(t)[\Gamma(\alpha)]^{i}\int_{0}^{t}(t - s)^{\{i\alpha - (i+1-n)\}}u(s)\mathrm{d}s, ~n>n_{0}.
\end{eqnarray*}
Also for $\alpha \in (0, 1)$, $\Gamma(\alpha) > 1$. Therefore,
\begin{eqnarray*}
H_{n}(t) \leq \frac{[\Gamma(\alpha)]^{n}}{\Gamma(n\alpha)}\sum_{i = 0}^{n}\dbinom{n}{i}b^{n - i}(t)g^{i}(t)\int_{0}^{t}(t - s)^{\{i\alpha - (i+1-n)\}}u(s)\mathrm{d}s.
\end{eqnarray*}
Let $y_{i} = i\alpha + n - i - 1$. Similar to the sequence $x_{i}$, there is $y^{\min}_{i} = n\alpha - 1 \geq 1$ for a large enough $n$ and $y^{\max}_{i} = n - 1$. Since $t\in[0, T)$, split the interval $[0, T)$ into two subintervals $[0, 1]$ and $[1, T)$. For $t\in[0, 1]$, $(t - s)^{y_{i}} \leq t^{y_{min}} = t^{n\alpha - 1}$ while if $t\in[1, T)$, $(t - s)^{y_{i}}\leq t^{y^{\max}_{i}} = t^{n}$. Thus,
\begin{eqnarray}\label{boundofoperator}
\begin{aligned}
H_{n}(t) &\leq \frac{[\Gamma(\alpha)]^{n}\max\{t^{n\alpha - 1}, t^{n}\}}{\Gamma(n\alpha)}\sum_{i = 0}^{n}\dbinom{n}{i}b^{n - i}(t)g^{i}(t)\int_{0}^{t}u(s)\mathrm{d}s\\
&= \frac{[\Gamma(\alpha)]^{n}\max\{t^{n\alpha - 1}, t^{n}\}}{\Gamma(n\alpha)}(b(t) + g(t))^{n}\int_{0}^{t}u(s)\mathrm{d}s.\\
\end{aligned}
\end{eqnarray}
Notice that $b(t)$ and $g(t)$ are both bounded by a positive constant $M$, i.e., $b(t)\leq M$ and $g(t) \leq M$, and $u(s)$ is locally integrable over $0\leq t < T$. This means that from Eq. (<ref>), $H_{n}(t) \to 0$ as $n\to\infty$ because the Gamma function, $\Gamma(n\alpha)$, is growing faster than a power function. Therefore, the second claim, Eq. (<ref>), is verified since $B^{n}u(t) \leq H_{n}(t)$ for any $n\in\mathbb{N}^{+}$.
The third claim establishes that the right hand side (RHS) of Eq. (<ref>) exists on $0 \leq t <T$. In order to show this statement, we first prove that for $0 \leq t < T$, the following infinite sum of sequences denoted by $L(t; \tau)$ is convergent.
\begin{eqnarray}\label{mixfunc}
\begin{aligned}
L(t; \tau) &:= \sum_{n=0}^{\infty}\sum_{i=0}^{n}\dbinom{n}{i}b^{n-i}(t)g^{i}(t)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha+n-i+1)}\tau^{i\alpha+n-i}\\
&= \sum_{i=0}^{\infty}g^{i}(t)[\Gamma(\alpha)]^{i}\tau^{i\alpha}\sum_{n=i}^{\infty}\dbinom{n}{i}b^{n-i}(t)\frac{1}{\Gamma(i\alpha+n-i+1)}\tau^{n-i}\\
&= \sum_{i=0}^{\infty}g^{i}(t)[\Gamma(\alpha)]^{i}\tau^{i\alpha}\frac{1}{\Gamma(i\alpha+1)}\sum_{n=i}^{\infty}\dbinom{n}{i}b^{n-i}(t)\frac{1}{(i\alpha+n-i)\cdots(i\alpha+1)}\tau^{n-i},\\
\end{aligned}
\end{eqnarray}
where $(i\alpha+n-i)\cdots(i\alpha+1)$ is a product and it takes one if $(i\alpha+n-i) < i\alpha+1$. Let $k=n-i$, then compute
\begin{eqnarray}\label{computationofproduct}
\begin{aligned}
\dbinom{n}{i}\frac{1}{(i\alpha + n - i)\cdots(i\alpha + 1)} &= \frac{(k+i)!}{i!k!}\frac{1}{(i\alpha + k)\cdots(i\alpha + 1)}
= \frac{1}{k!}\frac{(k+i)\cdots(i+1)}{(i\alpha + k)\cdots(i\alpha + 1)}\\
&=\frac{1}{\alpha^{k}}\frac{1}{k!}\frac{(k+1)\cdots(i+1)}{(k/\alpha + i)\cdots(i + 1/\alpha)}\leq \frac{1}{\alpha^{k}}\frac{1}{k!}.
\end{aligned}
\end{eqnarray}
Substituting $k = n - i$ and Eq. (<ref>) into Eq. (<ref>) gives
\begin{eqnarray*}
L(t; \tau) \leq \sum_{i=0}^{\infty}\frac{g^{i}(t)[\Gamma(\alpha)]^{i}\tau^{i\alpha}}{\Gamma(i\alpha+1)}\sum_{k=0}^{\infty}\frac{1}{\alpha^{k}}\frac{b^{k}(t)\tau^{k}}{k!} = E_{\alpha}(g(t)\Gamma(\alpha)\tau^{\alpha})\exp(\frac{1}{\alpha}b(t)\tau),
\end{eqnarray*}
which is finite for $0\leq t < T$. Furthermore, since $b(t) \leq M$ and $g(t)\leq M$, define
\begin{eqnarray*}
L(M;\tau) := \sum_{i=0}^{\infty}\frac{M^{i}[\Gamma(\alpha)]^{i}\tau^{i\alpha}}{\Gamma(i\alpha+1)}\sum_{k=0}^{\infty}\frac{1}{\alpha^{k}}\frac{M^{k}\tau^{k}}{k!} = E_{\alpha}(M\Gamma(\alpha)\tau^{\alpha})\exp(\frac{1}{\alpha}M\tau),
\end{eqnarray*}
which means $L(M; \tau)$ is finite and $L(t; \tau)\leq L(M; \tau)$.
Then, compute the RHS of Eq. (<ref>)
\begin{eqnarray*}
\begin{aligned}
RHS &= a(t) + \sum_{n = 1}^{\infty}\sum_{i=0}^{n} \dbinom{n}{i}b^{n - i}(t)g^{i}(t)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha + n - i + 1)}\int_{0}^{t}\frac{\mathrm{d}}{\mathrm{d}t}(t - s)^{\{i\alpha + n - i\}}a(s)\mathrm{d}s \\
& \leq a(t) + \sum_{n = 1}^{\infty}\sum_{i=0}^{n} \dbinom{n}{i}M^{n - i}M^{i}\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha + n - i + 1)}\int_{0}^{t}\frac{\mathrm{d}}{\mathrm{d}t}(t - s)^{\{i\alpha + n - i\}}a(s)\mathrm{d}s \\
&= a(t) + \int_{0}^{t}\frac{\mathrm{d}L(M; t-s)}{\mathrm{d}t}a(s)\mathrm{d}s.
\end{aligned}
\end{eqnarray*}
Since the Mittag-Leffler function $E_{\alpha}(t^{\alpha})$ is an entire function in $t^{\alpha}$, see <cit.>, the exponential function $\exp(t)$ is uniformly continuous in $t$, and both $t^{\alpha - 1}$ and $a(t)$ are locally integrable over $0 \leq t < T$, the integral $\int_{0}^{t}\frac{\mathrm{d}L(M; t-s)}{\mathrm{d}t}a(s)\mathrm{d}s$ is finite. This implies that the $RHS$ of Eq. (<ref>) is finite. So the last claim is also verified, thereby completing the proof.
Suppose the conditions in Theorem <ref> are satisfied and furthermore, $a(t)$ is nondecreasing on $0 \leq t < T$. Then
\begin{eqnarray*}
u(t) \leq a(t)E_{\alpha}(g(t)\Gamma(\alpha)t^{\alpha})\exp(\frac{1}{\alpha}b(t)t).
\end{eqnarray*}
From the proof of Theorem <ref>,
\begin{eqnarray*}
\begin{aligned}
u(t) &\leq a(t) + \sum_{n = 1}^{\infty}\sum_{i=0}^{n} \dbinom{n}{i}b^{n - i}(t)g^{i}(t)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha + n - i)}\int_{0}^{t}(t - s)^{\{i\alpha - (i+1-n)\}}a(s)\mathrm{d}s.
\end{aligned}
\end{eqnarray*}
Since $a(t)$ is nondecreasing,
\begin{eqnarray*}
\begin{aligned}
u(t) &\leq a(t)\sum_{n = 0}^{\infty}\sum_{i=0}^{n} \dbinom{n}{i}b^{n - i}(t)g^{i}(t)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha + n - i)}\int_{0}^{t}(t - s)^{\{i\alpha - (i+1-n)\}}\mathrm{d}s\\
&\leq a(t)\sum_{n = 0}^{\infty}\sum_{i=0}^{n} \dbinom{n}{i}b^{n - i}(t)g^{i}(t)\frac{[\Gamma(\alpha)]^{i}}{\Gamma(i\alpha + n - i+1)}t^{i\alpha + n - i}\\
&\leq a(t)E_{\alpha}(g(t)\Gamma(\alpha)t^{\alpha})\exp(\frac{1}{\alpha}b(t)t).
\end{aligned}
\end{eqnarray*}
This completes the proof.
From Theorem <ref> and Corollary <ref>, we see that if $\alpha = 1$, Theorem <ref> and Corollary <ref> are the same as Lemma <ref>; while if $b(t)\equiv 0$, Theorem <ref> and Corollary <ref> become Lemma <ref>.
§ EXISTENCE AND UNIQUENESS OF THE SOLUTION TO FRACTIONAL SDES
In this section, using the main results from Section <ref>, we investigate the existence and uniqueness of the solution to the fractional SDE Eq. (<ref>) with fractional order $0 < \alpha < 1$. By application of the classical Picard-Lindelöf successive approximation scheme and the standard Gronwall-Bellman inequality, existence and uniqueness of the solution to Eq. (<ref>) with fractional order $\frac{1}{2} < \alpha < 1$ is discussed in <cit.>. However, the case with $0 < \alpha \leq \frac{1}{2}$ remains to be investigated. We can apply the generalized Gronwall-Bellman inequality developed in Section <ref> to derive existence and uniqueness of the solution to Eq. (<ref>) when $0 < \alpha < 1$.
Let $0 < \alpha < 1, T > 0$, and $B_{t}$ be a $m-$dimensional Brownian motion on a complete probability space $\Omega\equiv (\Omega, \mathcal{F}, \mathbb{P})$. Assume that $b(\cdot, \cdot), \sigma_{1}(\cdot, \cdot): [0, T]\times\mathbb{R}^{n}\to\mathbb{R}^{n}, \sigma_{2}(\cdot, \cdot): [0, T]\times\mathbb{R}^{n}\to\mathbb{R}^{n\times m}$ are measurable functions satisfying the linear growth condition,
\begin{eqnarray}\label{linearcond}
|b(t, x)|^{2} + |\sigma_{1}(t, x)|^{2} + |\sigma_{2}(t, x)|^{2} \leq K^{2}(1 + |x|^{2}),
\end{eqnarray}
for some constant $K > 0$ and the Lipschitz condition,
\begin{eqnarray}\label{lipschitzcond}
|b(t, x) - b(t, y)| + |\sigma_{1}(t, x) - \sigma_{1}(t, y)| + |\sigma_{2}(t, x) - \sigma_{2}(t, y)| \leq L|x - y|,
\end{eqnarray}
for some constant $L > 0$. Let $x_{0}$ be a random variable, which is independent of the $\sigma-$algebra $\mathcal{F}_{t}\subset\mathcal{F}_{\infty}$ generated by $\{B_{t}, t\geq 0\}$ and satisfies $\mathbb{E}|x_{0}|^{2} < \infty$. Then, the fractional stochastic differential equation Eq. (<ref>) has a unique $t-$continuous solution $x(t, \omega)$ with the property that $x(t, \omega)$ is adapted to the filtration $\mathcal{F}_{t}^{x_{0}}$ generated by $x_{0}$ and $\{B_{t}, t\geq 0\}$, and
\begin{eqnarray*}
\mathbb{E}\Bigg[\int_{0}^{T}|x(t)|^{2}\mathrm{d}t\Bigg] < \infty.
\end{eqnarray*}
(Existence) From Eq. (<ref>), the corresponding equivalent stochastic integral equation of the fractional stochastic differential equation Eq. (<ref>) is rewritten as
\begin{eqnarray*}
x(t) = x_{0} + \int_{0}^{t}b(s, x(s))\mathrm{d}s + \alpha\int_{0}^{t}(t - s)^{\alpha - 1}\sigma_{1}(s, x(s))\mathrm{d}s + \int_{0}^{t}\sigma_{2}(s, x(s))\mathrm{d}B_{s},
\end{eqnarray*}
where $0 \leq t < T$ and $0 < \alpha < 1$. For more details about this equivalence between Eq. (<ref>) and Eq. (<ref>), we refer to <cit.>. By the method of Picard-Lindelöf successive approximations, define $x^{0}(t) = x_{0}$ and $x^{k}(t) = x^{k}(t, \omega)$ inductively as follows
\begin{eqnarray}\label{picarditeration}
x^{k+1}(t) = x_{0} + \int_{0}^{t}b(s, x^{k}(s))\mathrm{d}s + \alpha\int_{0}^{t}(t - s)^{\alpha - 1}\sigma_{1}(s, x^{k}(s))\mathrm{d}s + \int_{0}^{t}\sigma_{2}(s, x^{k}(s))\mathrm{d}B_{s}.
\end{eqnarray}
Applying the inequality $|x + y + z|^{2} \leq 3|x|^{2} + 3|y|^{2} + 3|z|^{2}$ leads to
\begin{eqnarray*}\label{triangularineq}
\begin{aligned}
\mathbb{E}|x^{k+1}(t) - x^{k}(t)|^{2} &\leq 3\mathbb{E}\bigg|\int_{0}^{t}\bigg(b(s, x^{k}(s)) - b(s, x^{k-1}(s))\bigg)\mathrm{d}s\bigg|^{2}\\
&+ 3\mathbb{E}\bigg|\alpha\int_{0}^{t}(t - s)^{\alpha - 1}\bigg(\sigma_{1}(s, x^{k}(s)) - \sigma_{1}(s, x^{k-1}(s))\bigg)\mathrm{d}s\bigg|^{2}\\
&+ 3\mathbb{E}\bigg|\int_{0}^{t}\bigg(\sigma_{2}(s, x^{k}(s)) - \sigma_{2}(s, x^{k-1}(s))\bigg)\mathrm{d}s\bigg|^{2}\\
&:= I_{1} + I_{2} + I_{3}.
\end{aligned}
\end{eqnarray*}
Using the Cauchy-Schwartz inequality on the first two terms, $I_{1}$ and $I_{2}$, plus Itô's Isometry, see in <cit.>, in the third term, $I_{3}$, produces
\begin{eqnarray*}\label{triangularineq}
\begin{aligned}
\mathbb{E}|x^{k+1}(t) - x^{k}(t)|^{2} &\leq 3T\mathbb{E}\int_{0}^{t}\bigg(b(s, x^{k}(s)) - b(s, x^{k-1}(s))\bigg)^{2}\mathrm{d}s\\
&+ 3\alpha^{2}\int_{0}^{t}(t - s)^{\alpha - 1}\mathrm{d}s\mathbb{E}\int_{0}^{t}(t - s)^{\alpha - 1}\bigg(\sigma_{1}(s, x^{k}(s)) - \sigma_{1}(s, x^{k-1}(s))\bigg)^{2}\mathrm{d}s\\
&+ 3
\mathbb{E}\int_{0}^{t}\bigg(\sigma_{2}(s, x^{k}(s)) - \sigma_{2}(s, x^{k-1}(s))\bigg)^{2}\mathrm{d}s\\
&:= J_{1} + J_{2} + J_{3}.
\end{aligned}
\end{eqnarray*}
Finally, using the Lipschitz condition Eq. (<ref>) on all terms, $J_{1}, J_{2}, J_{3}$, evaluating the first integral in the second term, $J_{2}$, and combining the first and third terms, $J_{1}$ and $J_{3}$, yields
\begin{eqnarray}\label{iterineqofappl}
\begin{aligned}
\mathbb{E}|x^{k+1}(t) - x^{k}(t)|^{2} &\leq 3L^{2}(1+T)\int_{0}^{t}\mathbb{E}|x^{k}(s) - x^{k-1}(s)|^{2}\mathrm{d}s\\
&+ 3L^{2}(1+T)\int_{0}^{t}(t - s)^{\alpha - 1}\mathbb{E}|x^{k}(s) - x^{k-1}(s)|^{2}\mathrm{d}s.
\end{aligned}
\end{eqnarray}
Thus, for locally integrable function $\phi(t)$, define an operator $B$ as follows
\begin{eqnarray*}
B\phi(t) := 3L^{2}(1+T)\Bigg\{\int_{0}^{t}\phi(s)\mathrm{d}s + \int_{0}^{t}(t - s)^{\alpha - 1}\phi(s)\mathrm{d}s\Bigg\}.
\end{eqnarray*}
Then, iterating Eq. (<ref>) yields
\begin{eqnarray*}
\mathbb{E}|x^{k+1}(t) - x^{k}(t)|^{2} \leq B(\mathbb{E}|x^{k}(t) - x^{k-1}(t)|^{2}) \leq \cdots \leq B^{k}(\mathbb{E}|x^{1}(t) - x^{0}(t)|^{2}).
\end{eqnarray*}
Since $0 < \alpha < 1$ and $\mathbb{E}|x^{1}(t) - x^{0}(t)|^{2}$ is nonnegative and locally integrable, from the first claim, Eq. (<ref>), and the Eq. (<ref>) in the proof of the second claim in Section <ref>, we know that
\begin{eqnarray*}
\begin{aligned}
\mathbb{E}|x^{k+1}(t) - x^{k}(t)|^{2} &\leq B^{k}(\mathbb{E}|x^{1}(t) - x^{0}(t)|^{2})\\ &\leq \frac{[\Gamma(\alpha)]^{k}\max\{t^{k\alpha - 1}, t^{k}\}}{\Gamma(k\alpha)}[6L^{2}(1+T)]^{k}\int_{0}^{t}\mathbb{E}|x^{1}(s) - x^{0}(s)|^{2}\mathrm{d}s.
\end{aligned}
\end{eqnarray*}
Similarly, apply the Cauchy-Schwartz inequality, the Itô's Isometry, and the linear growth condition, Eq. (<ref>), instead of Lipschitz condition, Eq. (<ref>), to compute
\begin{eqnarray*}
\mathbb{E}|x^{1}(t) - x^{0}(t)|^{2} \leq 3(1+T)K^{2}(1 + \mathbb{E}|x_{0}|^{2})(t + t^{\alpha}).
\end{eqnarray*}
This implies
\begin{eqnarray}\label{supoutside}
\sup_{0 \leq t \leq T}\mathbb{E}|x^{k+1}(t) - x^{k}(t)|^{2} \leq M_{0} \frac{[\Gamma(\alpha)]^{k}\max\{T^{k\alpha - 1}, T^{k}\}}{\Gamma(k\alpha)}[6L^{2}(1+T)]^{k},
\end{eqnarray}
where $M_{0} = 3(1+T)K^{2}(1 + \mathbb{E}|x_{0}|^{2})\bigg(\frac{T^{2}}{2} + \frac{T^{\alpha + 1}}{\alpha + 1}\bigg)$ is independent of $k$ and $t$. Thus, for any $m > n > 0$,
\begin{eqnarray*}
\begin{aligned}
\|x^{m}(t) - x^{n}(t)\|^{2}_{L^{2}(\mathbb{P})} &\leq \sum_{k=n}^{m}\|x^{k+1}(t) - x^{k}(t)\|^{2}_{L^{2}(\mathbb{P})} = \sum_{k=n}^{m}\int_{0}^{T}\mathbb{E}|x^{k+1}(t) - x^{k}(t)|^{2}\mathrm{d}t\\
& \leq M_{1}\sum_{k=n}^{m}\frac{[\Gamma(\alpha)]^{k}\max\{T^{k\alpha - 1}, T^{k}\}}{\Gamma(k\alpha)}[6L^{2}(1+T)]^{k}\\
& \to 0, ~as~m,~n\to\infty,
\end{aligned}
\end{eqnarray*}
where $M_{1} = 3(1+T)K^{2}(1 + \mathbb{E}|x_{0}|^{2})\bigg(\frac{T^{3}}{2} + \frac{T^{\alpha + 2}}{\alpha + 1}\bigg)$ is independent of $k$ and $t$.
This means the successive approximations $(x^{k}(t))$ are mean-square convergent uniformly on $[0, T]$.
It remains now to show that the sequence of successive approximations $(x^{k}(t))$ is almost surely convergent. First, apply Chebyshev's inequality to yield
\begin{eqnarray*}
\begin{aligned}
\sum_{k = 1}^{\infty}\mathbb{P}\bigg\{\sup_{0 \leq t \leq T}|x^{k+1}(t) - x^{k}(t)| > \frac{1}{k^{2}}\bigg\} &\leq \sum_{k=1}^{\infty}k^{4}\mathbb{E}\bigg(\sup_{0\leq t\leq T}|x^{k+1}(t) - x^{k}(t)|\bigg)^{2}\\
&= \sum_{k=1}^{\infty}k^{4}\mathbb{E}\bigg(\sup_{0\leq t\leq T}|x^{k+1}(t) - x^{k}(t)|^{2}\bigg).
\end{aligned}
\end{eqnarray*}
By computations similar to those leading to Eq. (<ref>) and Doob's Maximal Inequality for martingales,
\begin{eqnarray*}
\sum_{k = 1}^{\infty}\mathbb{P}\bigg\{\sup_{0 \leq t \leq T}|x^{k+1}(t) - x^{k}(t)| > \frac{1}{k^{2}}\bigg\} \leq M_{0}\sum_{k=1}^{\infty}\frac{[\Gamma(\alpha)]^{k}\max\{T^{k\alpha - 1}, T^{k}\}}{\Gamma(k\alpha)}[6L^{2}(1+T)]^{k}k^{4},
\end{eqnarray*}
which is finite. Then, applying the Borel-Cantelli lemma yields,
\begin{eqnarray*}
\mathbb{P}\bigg\{\sup_{0 \leq t \leq T}|x^{k+1}(t) - x^{k}(t)| > \frac{1}{k^{2}} ~\textrm{for infinitely many}~k\bigg\} = 0.
\end{eqnarray*}
So there exists a random variable $x(t)$, which is the limit of the following sequence
\begin{eqnarray*}
x^{k}(t) = x^{0}(t) + \sum_{n = 0}^{k-1}(x^{n+1}(t) - x^{n}(t))\to x(t) ~a.s.,
\end{eqnarray*}
uniformly on $[0, T]$. Also $x(t)$ is $t-$continuous since $x^{k}(t)$ is $t-$continuous for all $k$. Therefore, taking the limit on both sides of Eq. (<ref>) as $k\to\infty$, there is a stochastic process $x(t)$ satisfying Eq. (<ref>).
(Uniqueness) The uniqueness is due to the Itô Isometry and the Lipschitz condition, Eq. (<ref>). Let $x_{1}(t) = x_{1}(t,\omega)$ and $x_{2}(t) = x_{2}(t, \omega)$ be solutions of Eq. (<ref>), which have the initial values, $x_{1}(0) = y_{1}$ and $x_{2}(0) = y_{2}$, respectively. Similarly, apply the Cauchy-Schwartz inequality, the Itô Isometry, and the Lipschitz condition Eq. (<ref>) to compute
\begin{eqnarray*}
\begin{aligned}
\mathbb{E}|x_{1}(t) - x_{2}(t)|^{2} &\leq 4\mathbb{E}|y_{1} - y_{2}|^{2} + 4L^{2}(1 + T)\int_{0}^{t}\mathbb{E}|x_{1}(s) - x_{2}(s)|^{2}\mathrm{d}s\\
&+ 4\alpha L^{2}T^{\alpha}\int_{0}^{t}(t - s)^{\alpha - 1}\mathbb{E}|x_{1}(s) - x_{2}(s)|^{2}\mathrm{d}s. \end{aligned}
\end{eqnarray*}
By application of the generalized Gronwall-Bellman inequality in Corollary <ref>, we have
\begin{eqnarray*}
\mathbb{E}|x_{1}(t) - x_{2}(t)|^{2} \leq 4\mathbb{E}|y_{1} - y_{2}|^{2}E_{\alpha}(4\alpha L^{2}T^{\alpha}\Gamma(\alpha)t^{\alpha})\exp(\frac{1}{\alpha}4L^{2}(1 + T)t).
\end{eqnarray*}
Since $x_{1}(t)$ and $x_{2}(t)$ both satisfy the stochastic integral equation Eq. (<ref>), the initial values $y_{1}$ and $y_{2}$ are both equal to $x_{0}$. This means $\mathbb{E}|x_{1}(t) - x_{2}(t)|^{2} = 0$ for all $t > 0$. Furthermore,
\begin{eqnarray*}
\mathbb{P}\bigg\{|x_{1}(t) - x_{2}(t)| = 0,~\textrm{for all}~0 \leq t \leq T\bigg\} = 1.
\end{eqnarray*}
Therefore, the uniqueness of the solution to Eq. (<ref>) is proved.
§ FRACTIONAL FOKKER-PLANCK-KOLMOGOROV EQUATION
Based on the existence and uniqueness Theorem <ref> developed in Section <ref>, we derive the fractional Fokker-Planck-Kolmogorov equation associated to the unique solution of the fractional SDE, Eq. (<ref>). Before deriving the fractional Fokker-Planck-Kolmogorov equation, we first introduce an Itô formula from <cit.> to the following Itô process
\begin{eqnarray}\label{fraceq14}
x(t) = x_{0} + \int_{0}^{t}b(s,x(s))\mathrm{d}s + \int_{0}^{t}\sigma_{1}(s, x(s))\mathrm{d}s^{\alpha} + \int_{0}^{t}\sigma_{2}(s, x(s))\mathrm{d}B_{s},
\end{eqnarray}
where $0 < \alpha < 1$, $B_{t}$ is the $m-$dimensional standard Brownian motion, and functions $b, \sigma_{1}, \sigma_{2}$ satisfy the conditions in Theorem <ref>.
Let $X(t)$ satisfy the Eq. (<ref>) and furthermore, let $V\in C[R^{+}\times R^{n}, R^{m}]$, and assume that $V_{t}$, $V_{x}$, $V_{xx}$ exist and continuous for $(t, x)\in R^{+}\times R^{n}$, where $V_{x}(t, x)$ is an $m\times n$ Jacobian matrix of $V(t, x)$ and $V_{xx}(t, x)$ is an $m\times n$ Hessian matrix. Then,
\begin{eqnarray*}
\mathrm{d}V(t, X(t)) = L_{1}V(t, X(t))\mathrm{d}t + L_{2}V(t, X(t))\mathrm{d}t^{\alpha} + L_{3}V(t, X(t))\mathrm{d}B_{t},
\end{eqnarray*}
\begin{eqnarray*}
L_{1}V(t, x) = V_{t}(t, x) + V_{x}(t, x)b(t, x) + \frac{1}{2}\sigma_{2}(t, x)^{T}V_{xx}(t, x)\sigma_{2}(t, x)
\end{eqnarray*}
\begin{eqnarray*}
L_{2}V(t, x) = V_{x}(t, x)\sigma_{1}(t, x), ~~~L_{3}V(t, x) = V_{x}(t, x)\sigma_{2}(t, x).
\end{eqnarray*}
By applying the existence and uniqueness Theorem <ref> and Itô's formula, Lemma <ref>, the following fractional Fokker-Planck-Kolmogorov equation is established.
Let $B(t)$ be the $m-$dimensional standard Brownian motion. Suppose that $X(t)$ is the solution to the fractional SDE Eq. (<ref>) whose coefficient functions $b, \sigma_{1}$ and $\sigma_{2}$ satisfy the conditions in Theorem <ref>. Then the transition probabilities $P^{X}(t, x) = P^{X}(t, x|0, x_{0})$ of $X(t)$ satisfy the following fractional type differential equation
\begin{eqnarray}\label{fracfokkplanck}
\mathrm{d}P^{X}(t, x) = A_{x}^{*}P^{X}(t, x)\mathrm{d}t + B_{x}^{*}P^{X}(t, x)\mathrm{d}t^{\alpha}
\end{eqnarray}
with initial condition $P^{X}(0, x) = \delta_{x_{0}}(x)$, the Dirac delta function with mass on $x_{0}$, and $A^{*}_{x}, B^{*}_{x}$ are spatial operators defined respectively by
\begin{eqnarray*}
A^{*}_{x}h(x) = -\sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}[b_{i}(t, x)h(x)] + \frac{1}{2}\sum_{i=1}^{j=1}\sum_{i=1}^{j=1}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}[\sum_{k=1}^{m}\delta_{2}^{ik}\delta_{2}^{jk}(t, x)h(x)]
\end{eqnarray*}
\begin{eqnarray*}
B_{x}^{*} = -\sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}[\delta_{1}^{i}h(x)],
\end{eqnarray*}
where $b = (b_{1}, \cdots, b_{n})^{T}, \delta_{1} = (\delta_{1}^{1}, \cdots, \delta_{1}^{n})^{T}$ and $\delta_{2}$ is an $n\times m$ matrix with elements $[\delta_{2}]_{ij} = \delta_{2}^{ij}$.
Let $f\in C_{c}^{\infty}(R^{n})$, i.e. $f$ is an infinitely differential function on $R^{n}$ with compact support. Since $X(t)$ is the solution of the stochastic fractional differential equation Eq. (<ref>), this means $X(t)$ satisfies the stochastic integral equation Eq. (<ref>). So apply Itô formula Lemma <ref> on $f(X(t))$ to yield
\begin{eqnarray}\label{itoequation}
\begin{aligned}
f(X(t)) - f(x_{0}) &= \int_{0}^{t}\bigg(f_{x}(X(s))b(s, X(s)) + \frac{1}{2}\sigma_{2}^{T}(s, X(s))f_{xx}(X(s))\sigma_{2}(s, X(s))\bigg)\mathrm{d}s\\
&+ \int_{0}^{t}f_{x}(X(s))\sigma_{1}(s, X(s))\mathrm{d}s^{\alpha} + \int_{0}^{t}f_{x}(X(s))\sigma_{2}(s, X(s))\mathrm{d}B_{s}.
\end{aligned}
\end{eqnarray}
Notice the fact that
\begin{eqnarray*}
\int_{0}^{t}f_{x}(X(s))\sigma_{1}(s, X(s))\mathrm{d}s^{\alpha} = \alpha\int_{0}^{t}(t - s)^{\alpha - 1}f_{x}(X(s))\sigma_{1}(s, X(s))\mathrm{d}s,
\end{eqnarray*}
and more details on this equality can be found in <cit.>. Thus Eq. (<ref>) can be written as
\begin{eqnarray}\label{itoequation1}
\begin{aligned}
f(X(t)) - f(x_{0}) &= \int_{0}^{t}\bigg(f_{x}(X(s))b(s, X(s)) + \frac{1}{2}\sigma_{2}^{T}(s, X(s))f_{xx}(X(s))\sigma_{2}(s, X(s))\bigg)\mathrm{d}s\\
&+ \alpha\int_{0}^{t}(t - s)^{\alpha - 1}f_{x}(X(s))\sigma_{1}(s, X(s))\mathrm{d}s + \int_{0}^{t}f_{x}(X(s))\sigma_{2}(s, X(s))\mathrm{d}B_{s}.
\end{aligned}
\end{eqnarray}
Since the integral $\int_{0}^{t}f_{x}(X(s))\sigma_{2}(s, X(s))\mathrm{d}B_{s}$ is a martingale with respect to the filtration $\mathcal{F}_{t}$, take conditional expectations on both sides of Eq. (<ref>) to obtain
\begin{eqnarray}\label{conditionalexpectation}
\begin{aligned}
\mathbb{E}[f(X(t))|X(0) = x_{0}] - f(x_{0}) &= \mathbb{E}\bigg[\int_{0}^{t}f_{x}(X(s))b(s, X(s))\mathrm{d}s\bigg|X(0) = x_{0}\bigg]\\ &+ \mathbb{E}\bigg[\frac{1}{2}\int_{0}^{t}\sigma_{2}^{T}(s, X(s))f_{xx}(X(s))\sigma_{2}(s, X(s))\mathrm{d}s\bigg|X(0) = x_{0}\bigg]\\
&+ \mathbb{E}\bigg[\alpha\int_{0}^{t}(t - s)^{\alpha - 1}f_{x}(X(s))\sigma_{1}(s, X(s))\mathrm{d}s\bigg|X(0) = x_{0}\bigg].
\end{aligned}
\end{eqnarray}
By Fubini's Theorem and integration by parts, the above Eq. (<ref>) can be rewritten as
\begin{eqnarray*}
\begin{aligned}
\int_{R^{n}}f(x)P^{X}(t, x)\mathrm{d}x - f(x_{0})
&= \int_{0}^{t}\int_{R^{n}}f_{x}(x)b(s, x)P^{X}(s, x)\mathrm{d}x\mathrm{d}s\\
&+ \frac{1}{2}\int_{0}^{t}\int_{R^{n}}\sigma_{2}^{T}(s, x)f_{xx}(x)\sigma_{2}(s, x)P^{X}(s, x)\mathrm{d}x\mathrm{d}s\\
&+ \alpha\int_{0}^{t}\int_{R^{n}}(t - s)^{\alpha - 1}f_{x}(x)\sigma_{1}(s, x)P^{X}(s, x)\mathrm{d}x\mathrm{d}s\\
&= \int_{R^{n}}f(x)\int_{0}^{t}A^{*}_{1}P^{X}(s, x)\mathrm{d}s\mathrm{d}x\\
&+ \int_{R^{n}}f(x)\alpha\int_{0}^{t}(t - s)^{\alpha - 1}A^{*}_{2}P^{X}(s, x)\mathrm{d}s\mathrm{d}x.
\end{aligned}
\end{eqnarray*}
Since $f\in C_{c}^{\infty}(R^{n})$ is arbitrary and $ C_{c}^{\infty}(R^{n})$ is dense in $L^{2}(R^{n})$,
\begin{eqnarray}\label{integralequationoffokkerplanckeq}
P^{X}(t, x) - \delta_{x_{0}}(x) = \int_{0}^{t}A^{*}_{1}P^{X}(s, x)\mathrm{d}s + \alpha\int_{0}^{t}(t - s)^{\alpha - 1}A^{*}_{2}P^{X}(s, x)\mathrm{d}s,
\end{eqnarray}
where $\delta_{x_{0}}(x)$ is a generalized function taking value $\delta_{x_{0}}(x) = P^{X}(0,x_{0})$. Finally, take the derivative with respect to time $t$ on both sides of Eq. (<ref>) to yield the desired result Eq. (<ref>).
§ CONCLUSION
In this paper, a new type of Gronwall-Bellman inequality is established for a class of integral equations with a mixture of nonsingular and singular integrals. This new type of Gronwall-Bellman inequality can be considered as a generalization of known Gronwall-Bellman inequalities dealing with an integral equation having nonsingular or singular integrals, separately. With this new type of Gronwall-Bellman inequality, existence and uniqueness of the solution to a fractional SDE with fractional oder $0 < \alpha < 1$ is investigated. Furthermore, based on the existence and uniqueness result, a fractional type Fokker-Planck-Kolmogorov equation associated to the solution of a fractional SDE is derived.
§ ACKNOWLEDGMENTS
The author wishes to thank Dr. Marjorie Hahn for her advice, fruitful discussion, encouragement and patience with my research, and Dr. Xiaozhe Hu for his helpful discussion.
§ REFERENCES
|
1511.00907
|
$^1$ Division of Theoretical Physics, Department of Physics,
Yaroslavl State P G Demidov University, Sovietskaya 14,
150000 Yaroslavl, Russia
$^2$ Division of Physics, Yaroslavl Higher Military School of Air Defence, Moskovskiy Prosp. 28,
150001 Yaroslavl, Russia
[email protected], $^{\ast}[email protected], $^{\ast \ast}[email protected]
The processes of neutrino production of electron–positron pairs, $\nu \bar\nu \to e^- e^+$ and $\nu \to \nu e^- e^+$, in a magnetic field of arbitrary strength, where electrons and positrons can be created in the states corresponding to excited Landau levels, are analysed. The results can be applied for calculating the efficiency of the electron–positron plasma production by neutrinos in the conditions of the Kerr black hole accretion disc considered by experts as the most possible source of a short cosmological gamma burst.
§ INTRODUCTION
An intense electromagnetic field makes possible the processes which are forbidden in a vacuum such as
the neutrino production of an electron–positron pair $\nu \to \nu e^- e^+$.
The list of papers devoted to an analysis of this process and the collection of the results obtained
could be found e.g. in <cit.>.
In most cases, calculations of this kind were made either in the crossed field approximation,
or in the limit of a superstrong field much greater than the critical value of
$B_e = m_e^2/e \simeq 4.41\times 10^{13}$ G (we use natural units
$c = \hbar = k_{\rm{B}} = 1$),
when the electrons and positrons are born
in states corresponding to the ground Landau level.
However, there exist physical situations of the so-called moderately strong magnetic
$p_\perp^2 \ge e B \gg m_e^2$, when electrons and positrons mainly occupy the ground
Landau level, however, a noticeable fraction may be produced at the next levels.
The indicated hierarchy of physical parameters corresponds to the conditions of the Kerr black hole accretion disk, regarded by experts as the most likely source of a short cosmological gamma-ray burst.
The disc is a source of copious neutrinos and anti-neutrinos, which partially annihilate above the disc
and turn into $e^{\mp}$ pairs, $\nu \bar\nu \to e^- e^+$. This process was proposed and investigated in many details
(for the list of references see e.g. <cit.>)
as a possible mechanism for creating relativistic, $e^{\mp}$-dominated jets that could power observed
gamma-ray bursts.
In <cit.>, in addition to $\nu \bar\nu$ annihilation, the contribution
of the magnetic field-induced process $\nu \to \nu e^- e^+$ to the neutrino energy deposition rate around
the black hole was also included for the first time.
The authors <cit.> concluded in part, that the process $\nu \to \nu e^- e^+$ could dominate over
the basic process $\nu \bar\nu \to e^- e^+$.
They used the result for the energy deposition rate in the process $\nu \to \nu e^- e^+$
obtained in <cit.> in the crossed field limit, while in those physical conditions ($B$ to 180 $B_e$, $E_\nu$ to 25 MeV) the approximation of a crossed field is poorly applicable
(as well as the approximation of a superstrong field when $e^-e^+$ are created in the ground Landau level).
The next Landau levels can be also excited, as we have shown in our paper <cit.>.
Furthermore, the authors <cit.> considered the process $\nu \bar\nu \to e^- e^+$
without taking account of the magnetic field influence.
Thus, the aim of this paper is the study of the processes $\nu \bar\nu \to e^- e^+$ and $\nu \to \nu e^- e^+$
in the physical conditions of the moderately strong
magnetic field, where the electrons and positrons would be born in the states corresponding to the excited
Landau levels.
Possible astrophysical applications are discussed.
§ NEUTRINO PROCESS $\NU \TO \NU E^- E^+$ IN A STRONG MAGNETIC FIELD
The total probability of the process $\nu \to \nu e^-_{(n)} e^+_{(\ell)}$, when the electron and the positron are created
in the $n$th and $\ell$th Landau levels,
is, in a general case, the sum of the probabilities of the four polarization channels:
\begin{equation}
\label{eq:Wtot}
W_{n \ell} = W^{--}_{n \ell} + W^{-+}_{n \ell} + W^{+-}_{n \ell} + W^{++}_{n \ell} \, .
\end{equation}
For each of the channels, the differential probability over the final neutrino momentum
per unit time, after integration over the momenta of the electron and positron, is reduced to one
nontrivial integral:
\begin{eqnarray}
\D W^{s s'}_{n \ell} =
\frac{\beta \, \D^3 P'}{(2 \pi)^4 16 E E'} \,
\int \,
\frac{\D p_z}{\varepsilon_n \, \varepsilon'_{\ell}} \, \delta(\varepsilon_n + \varepsilon'_{\ell} - q_0) \,
|{\cal M}_{n \ell}^{s s'}|^2 \, ,
\label{eq:dw2}
\end{eqnarray}
where $\varepsilon_n = \sqrt{M_n^2 + p_z^2}$, $M_n = \sqrt{m_e^2 + 2 \beta n}$, $\beta = e B$.
The energy of the initial
neutrino should exceed a certain threshold value.
In the reference frame,
where the momentum of the initial neutrino directed at an angle
$\theta$ to the magnetic field, the threshold energy is given by:
\begin{equation}
E \, \sin \theta \ge M_{n} + M_{\ell} \, .
\label{eq:condE}
\end{equation}
Some details of calculations can be found in our paper <cit.>.
The probability of the $\nu \to \nu e^- e^+$ process defines
its partial contribution into the neutrino opacity of the medium.
The estimation of the neutrino mean free
path with respect to this process gives the result which is too large <cit.>
compared with the typical size of any compact astrophysical object,
where a strong magnetic field could exist.
However, a mean free path does not exhaust the neutrino physics in
a medium. In astrophysical applications, we could consider
the values that probably are more essential, namely, the mean values
of the neutrino energy and momentum losses, caused by the influence of an external magnetic field.
These values can be described by the four-vector of losses $Q^{\alpha}$,
\begin{equation}
Q^\alpha \, = \, E \int q^\alpha \, \D W = - E \, ({\cal I}, {\bf F}) \,.
\label{eq:Q0}
\end{equation}
where $q$ is the difference of the momenta of the initial and final neutrinos,
$q = P - P'$, $\D W$ is the total differential probability of the process.
The zeroth component of $Q^{\alpha}$ is connected with the mean energy lost
by a neutrino per unit time due to the process considered,
${\cal I} = \D E/\D t$.
The space components of the four-vector (<ref>) are similarly
connected with the mean neutrino momentum loss per unit time,
${\bf F} = \D {\bf P}/\D t$.
It should be noted that the four-vector of losses $Q^{\alpha}$ can be used for evaluating
the integral effect of neutrinos on plasma in the conditions of not very dense plasma,
where an one-interaction approximation of a neutrino with plasma is valid.
In <cit.>, the formula for the energy deposition rate was taken, which
was calculated in the crossed field limit <cit.>.
However, in the region of the physical parameters
used in <cit.> ($B$ to 180 $B_e$, $E_\nu$ to 25 MeV), the approximation of a crossed field is poorly applicable,
as well as the approximation of a superstrong field when $e^- e^+$ are created in the ground Landau level.
The contribution of the next Landau levels which can be also excited, should be taken into account.
In <cit.>, the results
are presented of our calculation of the mean neutrino energy losses caused by
the process $\nu \to \nu e^- e^+$ in a moderately strong magnetic field, i.e. in the conditions
of the Kerr black hole accretion disk.
It was shown that the crossed field limit gives the overstated result which is in orders of magnitude greater
than the sum of the contributions of lower excited Landau levels.
On the other hand, the results with $e^- e^+$ created at the ground Landau level give the main contribution to
the energy deposition rate, and almost exhaust it at $B = 180 B_e$.
This would mean that the conclusion <cit.> that the contribution of the process
$\nu \to \nu e^- e^+$ to the efficiency of the electron-positron plasma production by neutrino exceeds
the contribution of the annihilation channel $\nu \bar\nu \to e^- e^+$, and that the first process
dominates the energy deposition rate, does not have a sufficient basis.
A new analysis of the efficiency of energy deposition by neutrinos through both processes, $\nu \bar\nu \to e^- e^+$
and $\nu \to \nu e^- e^+$, in a hyper-accretion disc around a black hole should be performed, with taking
account of our results <cit.> for the process $\nu \to \nu e^- e^+$.
§ THE STRONG MAGNETIC FIELD INFLUENCE ON THE PROCESS $\NU \BAR\NU \TO E^- E^+$
The local energy-momentum
deposition rate due to the process $\nu \bar\nu \to e^- e^+$ is defined by the equation <cit.>:
\begin{equation}
Q^\alpha_{\nu \bar\nu} =
\int \frac{\D^3 p}{(2 \pi)^3} \, f_{\nu} (p) \int \frac{\D^3 p'}{(2 \pi)^3} \, f_{\bar\nu} (p') \,
\left( p^\alpha + p'^\alpha \right) \frac{(p p')}{E E'} \,
\sigma (\nu \bar\nu \to e^- e^+) \,,
\label{eq:Q1}
\end{equation}
where $\sigma$ is the cross-section of the process,
$p$ and $p'$ are the four-momenta of the neutrino and antineutrino,
$f_{\nu} (p)$ and $f_{\bar\nu} (p')$ are the local distribution functions depending
on the distribution functions at the surface of
the black hole accretion disc, and on the details of propagation.
In a strong magnetic field, the cross-section takes the form:
\begin{equation}
\sigma (\nu \bar\nu \to e^- e^+) = \sum\limits_{f=e,\mu,\tau} \, \sum\limits_{n, \ell} \,
\sigma (\nu_f {\bar\nu}_f \to e^-_{(n)} e^+_{(\ell)}) \,,
\label{eq:sigmaB}
\end{equation}
where the upper limit of summation over $n, \ell$ is defined by the condition
$(M_n + M_{\ell})^2 \le ( p + p' )_{\mprl}^2$, ($q_{\mprl}^2=q_0^2-q_z^2$, if $z$ is along $\bf B$).
Unlike the cross-section in vacuum where it depends on the Mandelstam parameter $S$ only,
the cross-section in a magnetic field depends on the set of kinematic variables, e.g.: energies $E, E'$, two polar angles and one azimuth angle. In figure <ref>, we take for the sake of illustration the case $E=E'$, and take certain angles. The dependence is presented of $\sigma$ (solid line) and $\sigma_{\rm vac}$ (dashed line) on $E$.
The cross-section has a peculiar “sawtooth” profile due to the square-root singularities <cit.>,
which is similar
to the profile of the process $\gamma^\ast \to e^- e^+$ width in a strong field <cit.>.
After averaging over small intervals $E \pm \Delta E$, the dependence becomes smoother.
It can be seen that in calculations of the energy-momentum deposition rate by integration
over the neutrino and antineutrino momenta, the field influence appears to be inessential.
Cross-section of the process $\nu \bar\nu \to e^- e^+$ in the case $E=E'$ and for fixed angles:
the energy dependence in the field (solid line) and in vacuum (dashed line);
$\sigma_0 = {4 \, G_{\mathrm{F}}^2 \, m_e^2}/{\pi}$ is the so-called typical weak cross-section.
§ CONCLUSIONS
* The processes $\nu \to \nu e^- e^+$ and $\nu \bar\nu \to e^- e^+$ are investigated in the magnetic field of an arbitrary strength, when $e^- e^+$
can be produced in the excited Landau levels.
* The neutrino energy losses due to the process $\nu \to \nu e^- e^+$ are calculated. The results should be used for calculations of the efficiency of the $e^- e^+$ plasma production by neutrinos in
the conditions of the Kerr black hole accretion disk. In these conditions, the crossed field limit gives the overstated result which is in orders of magnitude greater
than the sum over the lower Landau levels.
* The cross-section of the process $\nu \bar\nu \to e^- e^+$ in a strong field, has a peculiar “sawtooth” profile, which is close to the vacuum cross-section after averaging. In calculations of the energy-momentum deposition rate by integration over the neutrino and antineutrino momenta, the field influence appears to be inessential.
The study was performed with the support by the Project No. 92 within the base part of the State Assignment
for the Yaroslavl University Scientific Research, and was supported in part by the
Russian Foundation for Basic Research (Project No. ).
§ REFERENCES
Kuznetsov A and Mikheev N 2013 Electroweak Processes in External Active Media
(Berlin, Heidelberg: Springer-Verlag)
Zalamea I and Beloborodov A M 2011 Mon. Not. R. Astron. Soc. 410 2302
Kuznetsov A V, Rumyantsev D A and Savin V N 2014
Int. J. Mod. Phys. A 29 1450136
Kuznetsov A V and Mikheev N V 1997 Phys. Lett. B 394 123
Kuznetsov A V and Mikheev N V 1997
Phys. At. Nucl. 60 1865 (Original Russian text: Yad. Fiz. 60 2038)
Birkl R, Aloy M A, Janka H-Th and Müller E 2007
Astron. Astrophys. 463 51
Klepikov N P 1954 (In Russian) Zh. Eksp. Teor. Fiz. 26 19
Daugherty J K and Harding A K 1983
Astrophys. J. 273 761
Baier V N and Katkov V M 2007 Phys. Rev. D 75 073009
The function $f (E/m_e)$ for $B=180 \,B_e$ obtained in the crossed field limit (dotted line),
with $e^- e^+$ created at the ground (0,0) Landau level (dashed line), and for the sum of all lower
Landau levels which are excited in this energy interval according
to the condition (<ref>)(solid line).
§ PREPARING YOUR PAPER
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* Strite S and Morkoc H 1992 J. Vac. Sci. Technol. B 10 1237
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J. Appl. Phys. 87 965
* Nakamura S, Senoh M, Nagahama S, Iwase N, Yamada T, Matsushita T, Kiyoku H
and Sugimoto Y 1996 Japan. J. Appl. Phys. 35 L74
* Akasaki I, Sota S, Sakai H, Tanaka T, Koike M and Amano H 1996
Electron. Lett. 32 1105
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* Cheng G X 2001 Raman and Brillouin Scattering-Principles and Applications (Beijing: Scientific)
* Szytula A and Leciejewicz J 1989 Handbook on the Physics and Chemistry of Rare Earths vol 12, ed K A Gschneidner Jr and L Erwin (Amsterdam: Elsevier) p 133
* Kuhn T 1998 Density matrix theory of coherent ultrafast dynamics Theory of Transport Properties of Semiconductor Nanostructures (Electronic Materials vol 4) ed E Schöll (London: Chapman and Hall) chapter 6 pp 173–214
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1511.00637
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Realistic $f(T)$ model describing the de Sitter epoch of the dark energy dominated universe
S. B. Nassur$^{(a)}$[e-mail:[email protected]], A. V. Kpadonou$^{(a,c)}$[e-mail: [email protected]], M. E. Rodrigues$^{(b)}$[e-mail: [email protected]],
M. J. S. Houndjo$^{(a,d)}$[e-mail:
[email protected]] and J. Tossa$^{(a)}$[e-mail: [email protected]]
$^a$ Institut de Mathématiques et de Sciences Physiques (IMSP)
01 BP 613, Porto-Novo, Bénin
$^{b}$ Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará - Campus Universitário de
Abaetetuba, CEP 68440-000, Abaetetuba, Pará, Brazil
$^{c}$ Ecole Normale Supérieure de Natitingou - Université de Parakou - Bénin
$^{d}$ Faculté des Sciences et Techniques de Natitingou - Université de Parakou - Bénin
We consider an exponential model within the so-called $f(T)$ theory of gravity, where $T$ denotes the torsion scalar. We focus our work on a cosmological feature of such a model, checking whether it may describe the de Sitter stage of the current universe through the analysis of the redshift $z$. Our results shows that the model reproduces the de Sitter stage only for low-redshifts, where the perturbation function goes toward zero as the low values of the redshift are reached, whereas the effective parameter of equation of state goes to $-1$, which is the expected behavior for any model able to reproduce de Sitter stage.
Pacs numbers: 98.80.-k, 04.50.Kd, 95.36.+x
§ INTRODUCTION
According to the recent astronomical data from Type Ia supernovae <cit.> as well as from the CMB spectrum <cit.>, it is well known that our universe is experiencing an accelerating expanded phase. In order to comply with this feature, dark energy content for the universe is assumed <cit.> within the framework of the General Relativity (GR), having undesired properties such as the violation of some energy conditions. Nowadays, it is a well-known fact that modifying the law of gravity leads to possible explanations for the acceleration mechanism of the universe <cit.>. Various modified theory are proposed, one assumed as an algebraic function of the curvature $R$, the so-called $f(R)$ theory of gravity <cit.>, other as an algebraic function for both the curvature and the Gauss-Bonnet invariant $G$ <cit.>.The recent and interesting kind of modified theory of gravity is the one base on the torsion scalar, as algebraic function of this later, denoted by $f(T)$ <cit.>.The modified theory of gravity undertaken in this paper, called $f(T)$, is a modified version of the teleparallel (TT) essential based on the torsion scalar $T$. In this way, the torsion, as the TT term, is replaced by an algebraic function of the torsion, and this, always in the optic to comply with the cosmological data. In this paper, we are interested in the cosmological behavior of realistic models of modified gravity describing the de Sitter epoch of the current universe. As a check, we will study homogeneous perturbations around the de Sitter solution of the dark energy density, to see whether we are able to regain the well known results. Note that the analysis is possible only the explicit form of the algebraic function is known. We therefore consider an exponential form of $f(T)$. Remember that this kind of model is well-known is the framework of $f(R)$ <cit.>, where the cosmological evolution has been explored. More precisely in the framework of $f(T)$ theory of gravity, Bamba and collaborators considered an exponential model and studied cosmological evolutions of the equation of state for dark energy. They result shows that the crossing of the phantom divide line of $\omega_{DE}=-1$ may be observed when a logarithmic correction is performed to the exponential model.
In the paper <cit.>, it is shown that for $\omega$ is pure-imaginary, the model is unstable. Also, it is shown that the model defined by the equation (39) is stable if $)<q<1$. On the other hand, in a work developed by Bamba and collaborators, it has been shown that the model (39) of our paper should not produce “finite time singularities" <cit.>. The well known of finite time singularities is the Big-Rip. Note that the Big-Rip appears only in the phantom phase. Therefore, the model under consideration here may just reproduce a quintessence universe, meaning that the greater value of the effective parameter of equation of state is $-1$. In this paper, we still obey these features of the considered model, considering the suitable values of the parameter $q$ as pointed out in <cit.>, and try to check whether such a model should describe the de Sitter stage of our current universe. This paper can be view as an extension of these cited paper, still in the optic to point out the how much this the exponential $f(T)$ model can be assumed as a viable one. To do so, we perform a perturbation around the de Sitter point and such for the behaviour of the perturbation function $y(z)$. We see that as the low red-shifts are being reached, the perturbation parameter goes to zero. This means that the exponential $f(T)$ can describe the de Sitter stage only for low red-shift. It is why for our study in this paper, we limit the values of $z$ to the lowest ones. It is can be view from our results that the behaviour of the parameter of the equation of state related to the dark energy is the same as in the Bamba's paper <cit.>, despite of using different values of the parameter $q$ belonging to the interval imposed by <cit.>.
Also, in this paper, within suitable values of the input parameters
we plot the cosmological parameters $\omega_{DE}$ and $\omega_{eff}$ and also the behaviour of the perturbation function $y(z)$. Our results perfectly fit with the cosmological observational data and more precisely, for the low-redshift, the function $y(z)$ goes towards zero whereas $\omega_{eff}$ tends to $-1$, showing the stability of the model under consideration around the de Sitter solution.The paper is organized as follows: In $\ref{sec2}$ we present the generality on $f(T)$, putting up the general equation of motion and its associated generalized Friedmann equations from which all the analysis are realized. The $\ref{sec3}$ is devoted for obtaining the general scale factor according to the model. The conclusion and perspective are presented in the last section.
§ GENERALITY ON $F(T)$ THEORY
The modified theory of gravity based on the torsion scalar is the one for which the geometric part of the action is an algebraic function depending on the torsion. In the same way as in the Teleparallel gravity, the geometric elements are described using orthonormal tetrads components defined in the tangent space at each point of the manifold. In general the line element can be written as
\begin{eqnarray}
ds^2=g_{\mu\nu}dx^\mu dx^\nu=\eta_{ij}\theta^i\theta^j\,,
\end{eqnarray}
where we define the following elements
\begin{eqnarray}
dx^\mu=e_{i}^{\;\;\mu}\theta^{i}\,\quad \theta^{i}=e^{i}_{\;\;\mu}dx^{\mu}.
\end{eqnarray}
Note that $\eta_{ij}=diag(1,-1,-1,-1)$ is the metric related to the Minkowskian spacetime and the $\{e^{i}_{\;\mu}\}$ are the components of the tetrad which satisfy the following identity
\begin{eqnarray}
e^{\;\;\mu}_{i}e^{i}_{\;\;\nu}=\delta^{\mu}_{\nu},\quad e^{\;\;i}_{\mu}e^{\mu}_{\;\;j}=\delta^{i}_{j}.
\end{eqnarray}
The connection in use in this theory is the one of Weizenbock's, defined by
\begin{eqnarray}
\Gamma^{\lambda}_{\mu\nu}=e^{\;\;\lambda}_{i}\partial_{\mu}e^{i}_{\;\;\nu}=-e^{i}_{\;\;\mu}\partial_\nu e_{i}^{\;\;\lambda}.
\end{eqnarray}
Once the previous connection is assumed, one can then expression the main geometric objects; the torsion tensor's components as
\begin{eqnarray}
T^{\lambda}_{\;\;\;\mu\nu}= \Gamma^{\lambda}_{\mu\nu}-\Gamma^{\lambda}_{\nu\mu},
\end{eqnarray}
which is used in the definition of the contorsion tensor as
\begin{eqnarray}
\end{eqnarray}
The above objects (torsion and contorsion) are used to define a new tensor $S_{\lambda}^{\;\;\mu\nu}$ as
\begin{eqnarray}
\end{eqnarray}
The torsion scalar is defined from the previous tensor and the torsion tensor as
\begin{eqnarray}
\end{eqnarray}
Let us write the action for the modified $f(T)$ theory with matter as follows
\begin{equation}
\label{1}
S=\int d^{4}x e\left[\frac{f(T)}{2\kappa^{2}}+\mathcal{L}_{(matter)}\right],
\end{equation}
where $e\equiv\det[e^{i}\,_{\mu}]=\sqrt{-g}$ denotes the determinant of the tetrad, and $g$ the determinant of the space-time metric and the algebraic function $f(T)$ is assumed as the sum of the Teleparallel gravity term ( the torsion scalar $T$), and an additive algebraic function $F(T)$ as
\begin{equation}
\label{2}
\end{equation}
But for the instance we will continue working in the general scheme using the algebraic function $f(T)$, having in view that the additive function $F(T)$ has to be used later.By varying Eq. $\ref{1}$ with respect to the tetrad, one gets the following general equation of motion <cit.>,
\begin{eqnarray}
S^{\;\;\; \nu \rho}_{\mu} \partial_{\rho} T f_{TT} +
[e^{-1} e^{i}_{\;\; \mu}\partial_{\rho}(e e^{\;\; \mu}_{i}S^{\;\;\; \nu\lambda}_{\alpha} )
+T^{\alpha}_{\;\;\; \lambda \mu} S^{\;\;\; \nu \lambda}_{\alpha} ]f_{T}+
\frac{1}{4}\delta^{\nu}_{\mu}f=\frac{\kappa^{2}}{2} \mathcal{T}^{\nu}_{\mu}, \label{eq10}
\end{eqnarray}
where $\mathcal{T}^{\nu}_{\mu}$ denotes the stress tensor such that
\begin{eqnarray}
\mathcal{T}^{\nu}_{\mu}=\left(\rho+p\right)u_{\mu}u^{\nu}-p\delta^{\nu}_{\mu},
\end{eqnarray}
$\rho$ and $p$ being the energy density and the pressure of ordinary content of the universe respectively and $u^{\mu}$, the four-velocity such that $u^{\mu}u_{\mu}=1$. Still with the considered signature (+,-,-,-), we assume the line element for the flat Robertson-Walker universe as
\begin{equation}
\label{4}
dS^{2}=dt^{2}-a^{2}(t)d{\bf x}^{2},
\end{equation}
where $a(t)$ is the universe scalar factor. Therefore, the torsion scalar is performed giving
\begin{equation}
\label{5}
\end{equation}
From Eq.$(\ref{eq10})$, and using (<ref>), one gets the following gravitational field equations:
\begin{equation}
\label{6}
\end{equation}
\begin{equation}
\label{7}
2{\dot T}Hf_{TT}(T)+2({\dot H}+3H^{2})f_{T}(T)+\frac{1}{2}f(T)=-\kappa^{2}p.
\end{equation}
Here the “dot" denotes the derivative with respect to the cosmic time $t$, $H={\dot a}(t)/a(t)$, the Hubble parameter and
$\rho$ and $p$ , the matter energy-density and pressure, respectively. The matter conservation equation reads
\begin{equation}
\label{8}
\dot\rho+3H(\rho+p)=0.
\end{equation}
By assuming that the matter content is a perfect fluid, the pressure and the energy density are related by the barotropic equation of state (EoS)
\begin{equation}
\label{9}
\end{equation}
where $\omega$ is the EoS-parameter for matter. For standard matter (non-relativist matter), $\omega_{m}=0$ and $\rho_{m}=\rho^{(0)}_{m}a(t)^{-3}$, whereas
for radiation (relativist matter), $\omega_{r}=1/3$ and $\rho_{r}=\rho^{(0)}_{r}a(t)^{-4}$.
Now, making use of the relation $f(T)=T+F(T)$, one can write the Eqs.$(\ref{6})-(\ref{7})$ as functions of the effective energy density, $\rho_{eff}$, and effective pressure,
$p_{eff}$ as
\begin{eqnarray}
\label{10}
\rho_{eff}& = &\frac{3}{\kappa^{2}}H^{2},\\
\label{11}
p_{eff}& = &-\frac{1}{\kappa^{2}}\left( 2{\dot H}+3H^{2} \right),
\end{eqnarray}
\begin{eqnarray}
\label{12}
\rho_{eff}& = &\rho-\frac{1}{2\kappa^{2}}\left[F(T)-2TF_{T}(T)\right],\\
\label{13}
p_{eff}& = &p+\frac{1}{\kappa^{2}}\left[F(T)+4\left({\dot H}+3H^{2}\right)F_{T}(T)+4H{\dot T}F_{TT}(T)\right].
\end{eqnarray}
Let us define the dark energy density $\rho_{DE}$ as $\rho_{DE}=\rho_{eff}-\rho$ and introduce the variable
\begin{equation}
\label{14}
\end{equation}
where, $\rho^{(0)}_{m}$ is the energy density of matter at present time, $\bar{m}^{2}$ being the mass scale
\begin{equation}
\label{15}
\bar{m}^{2}\equiv\frac{\kappa^{2}\rho^{0}_{m}}{3}\simeq1.5\times10^{-67}eV^{2},
\end{equation}
and $\chi$ defined by
\begin{equation}
\label{16}
\chi\equiv\frac{\rho^{0}_{r}}{\rho^{0}_{m}},
\end{equation}
where $\rho^{(0)}_{r}$ is the current radiation density, $z$ the redshift parameter, $z=1/a(t)-1$, and $y_{H}(z)$, written as a function of $z$.
The EoS parameter for the dark energy, $\omega_{DE}$, is written as
\begin{equation}
\label{17}
\omega_{DE}=-1+\frac{1}{3}(z+1)\frac{1}{y_{H}(z)}\frac{dy_{H}(z)}{dz}.
\end{equation}
By combining Eq.$(\ref{11})$ with Eq.$(\ref{5})$ and using Eq.$(\ref{14})$, one gets
\begin{equation}
\label{18}
\frac{dy_{H}(z)}{dz}+J_{1}y_{H}(z)+J_{2}=0,
\end{equation}
\begin{equation}
\label{19}
\end{equation}
\begin{eqnarray}
\label{20}
J_{2}& = &\frac{1}{(z+1)\left(1+F_{T}(T)-12\bar{m}^{2}\left(y_{H}(z)+(z+1)^{3}+\chi(z+1)^{4}\right)F_{TT}(T)\right)}\times\nonumber\\
\end{eqnarray}
The torsion scalar can be written in function of the red-shift as
\begin{equation}
\label{22}
\end{equation}
where we used $d/dt=-(z+1)H(z)d/dz=H(t)d/d(\ln a(t))$. By taking the trace form of Eq. $\ref{eq10}$ in the vacuum, one gets the following equation
\begin{equation}
\label{23}
3{\dot H}\left(2Tf_{TT}(T)+f_{T}(T)\right)-2Tf_{T}(T)+f(T)=0.
\end{equation}
Since we are considering the de Sitter universe, the torsion has to be taken as the de Sitter one, known as a constant, because of the constance of the Hubble parameter, such that the trace equation becomes
\begin{equation}
\label{24}
\end{equation}
where $T_{dS}=constant$ is the de Sitter torsion scalar.Now, in order to check how much our model deviates from the de Sitter one, say, checking it convergence to the de Sitter model, we study perturbations around the de Sitter solution of the dark energy density. Then, we write the following equation
\begin{equation}
\label{25}
y_{H}(z)\simeq y_{0}+y(z),
\end{equation}
where $y_{0}=-T_{dS}/6\bar{m}^{2}$ is a constant and the stability requires $|y(z)|\ll1$. Eq.$(\ref{22})$ leads to
\begin{equation}
\label{26}
\end{equation}
In this case, by neglecting the contribution of radiation and taking into account only the non-relativist matter contribution, but assumed to be much smaller
than $y_{0}$, Eq.$(\ref{18})$ becomes
\begin{equation}
\label{27}
\frac{dy}{dz}+\frac{\sigma}{(z+1)}y=\beta(z+1)^{2},
\end{equation}
\begin{equation}
\label{28}
\sigma=-\frac{3\left(2f_{T}(T_{dS})-1\right)}{f_{T}+2T_{dS}f_{TT}(T_{dS})},
\end{equation}
\begin{equation}
\label{29}
\beta=\frac{3\left(f_{T}(T_{dS})-1\right)}{f_{T}+2T_{dS}f_{TT}(T_{dS})}
\end{equation}
Here we have performed the variation of Eq.$(\ref{26})$ with respect to $T$, and also have used Eq.$(\ref{24})$.
The solution of Eq.$(\ref{27})$ is
\begin{equation}
\label{31}
\end{equation}
where $C_{0}$ is an integration constant.As a check for our approach, let us now consider an exponential model, to see the behaviour of the dark energy. In this paper, we assume the following exponential model; $f(T)=T+\alpha T\left(1-e^{qT_0/T}\right)$. Note that this model has been early used by Bamba and collaborators in <cit.> where they studied the cosmological evolutions of the equation of state for dark energy. They showed that the crossing of the phantom divide line of $\omega_{DE}=-1$ can be realized by combining to the exponential model a logarithmic one. In this section, our goal is just to check the convergence of the exponential model to the de Sitter one. In this way, we have
\begin{equation}
\label{48}
F(T)=\alpha T\left(1-e^{qT_0/T}\right),
\end{equation}
where $T_{0}=-6H_0^2$ is the scalar tensor at present time, with $H_0=74.2\pm 3.6 Km s^{-1} Mpc^{-1}$, and $\alpha=-(1-\Omega_m^{(0)})/\left(1-\left(1-2q\right)e^q\right)$. By considering $ T_{dS}/T_{0}\ll1$, and using Eq (<ref>) within the expansion of the exponential function, one gets the following expression for the de Sitter torsion
\begin{eqnarray}
T_{dS}=-\frac{qT_0}{2}\left(\alpha-\sqrt{\alpha^2 -8\alpha}\right)
\end{eqnarray}
In order to reach to an interesting physical result, among other conditions, we assume the case where $q>0$ and $\alpha<0$. Therefore, one gets
\begin{eqnarray}
\sigma=\frac{-3\left[2+\frac{1}{2}(\alpha-\sqrt{\alpha^{2}-8\alpha}-4)\right]}
{\frac{1}{2}(\alpha-\sqrt{\alpha^{2}-8\alpha}-4)+\frac{4}{\alpha-\sqrt{\alpha^{2}-8\alpha}}-1}, \\
\beta=\frac{3}{\frac{1}{2}(\alpha-\sqrt{\alpha^{2}-8\alpha}-4)+\frac{4}{\alpha-\sqrt{\alpha^{2}-8\alpha}}-1},
\end{eqnarray}
and the perturbation function $y(z)$ takes the following form
\begin{eqnarray}
\end{eqnarray}
and whose the evolution versus redshift is realized at the Fig. <ref>
From the equation (<ref>), we obtain the corresponding expression of $y_H(z)$ as
\begin{eqnarray}
\end{eqnarray}
The graphs representing the evolution of this function are plotted at the Fig. <ref>. Here, we just try to show what happens at the low-redshift level. It can be observed that as the lowest redshift are being reached, i.e, $z\rightarrow -1^{+}$, the curves at the right hand side seem to go toward zero. This is just the effect for having used a very large interval, i.e, $]-1,+\infty[$ because, indeed, at the level of low-redshift the curved tend to some values different from zero. Within the above expression, we perform the ones of the parameters of equation of state of the dark energy $\omega_{DE}$ and the effective $\omega_{eff}$ as
\begin{eqnarray}
\omega_{DE}&=&-1-\frac{\sigma\left(3+\sigma\right)\left(z+1\right)^{-\sigma}-3\beta\left(z+1\right)^3}{3\left(3+\sigma\right)y_H(z)},\\
\omega_{eff}&=&-1-\frac{\sigma\left(3+\sigma\right)\left(z+1\right)^{-\sigma}-3\left(3+\sigma+\beta\right)\left(z+1\right)^3}{3\left(3+\sigma\right)\left[y_H(z)+\left(z+1\right)^3\right]},
\end{eqnarray}
whose the graphs representatives are plotted respectively at Fig. <ref> and Fig. <ref> .
We see from the Fig. <ref> that as the lowest redshift are being reached, the parameter $\omega_{DE}$ tends to $-1$ for the values of the input parameters $q$ and $\alpha$. The same is almost obtained for the effective parameter of EoS $\omega_{eff}$ showing for any of the considered values of the input parameters $q$, the curves go toward $-1$. This is an interesting result. Note that the effective parameter $\omega_{eff}$ is the one that characterizes the whole content of the universe. Also, it is well known that the de Sitter universe is the one for which one has $\omega_{eff}=-1$. Therefore, due to fact that $\omega_{eff}\rightarrow -1$ for $z\rightarrow -1^{+}$, one can conclude that within the exponential $f(T)$ model, the de Sitter stage of the universe is realized for the lowest values of the redshift. This result completely agrees with the one obtained by Bamba and collaborators <cit.> where they showed within others values of the input parameters that the parameter $\omega_{DE}$ goes toward $-1$ as the lowest redshift are being reached $z\rightarrow -1^{+}$ with an exponential $f(T)$ model. In our case, due to the fact that we approximate the dark energy to the effective content of the universe, it is obvious that our result is in agreement with the one of <cit.>. We point out that our study is just about low red-shift. Note that as the title mentions, we are interested to see whether the exponential model should describe the de Sitter stage of our current universe. Therefore, we just have to limit our analysis to the low. This king of study is in the same way as the well know $\Lambda CDM$ model, which works only for the current times. It is well known that for early times, $\Lambda CDM$ model does not work. The spirit is just the same here. We just our study to the low red-shift because knowing that this cannot lead any interesting result for high red-shift.
§ TIME EVOLUTION
Our goal in this section in to obtain the expression of the scale factor for with, at the level of low-redshift, the standard de Sitter expression may be recovered. To do this, let us consider the Eqs.$(\ref{25})$ and $(\ref{31})$ in the case where $-\sigma=\gamma$ is a positive real number:
\begin{equation}
\label{42}
\end{equation}
where $y_{0}=H^{2}_{dS}/{\bar{m}}^{2}=-T_{dS}/6{\bar{m}}^{2}$ and $C_{0}$ is a constant. We will assume $C_{0}>0$.
From the first equation of motion $(\ref{10})$, one gets
\begin{equation}
\label{43}
\frac{H^{2}}{{\bar{m}}^{2}}=y_{H}(z)+(z+1)^{2}\equiv y_{0}+C_{0}(z+1)^{\gamma}+\frac{1}{1+\frac{1}{2T_{dS}f_{TT}(T_{dS})}(1-f_{T}(T_{dS}))}(z+1)^{3},
\end{equation}
and the explicit expression of $H$ depending on cosmic time $t$ may be obtained. By substituting $(z+1)$ by $1/a(t)$, one gets
\begin{equation}
\label{44}
\left(\frac{{\dot a}(t)}{a(t)}\right)=H^{2}_{dS}+(C_{0}{\bar{m}}^{2})\left(\frac{1}{a(t)}\right)^{\gamma}.
\end{equation}
In the previous expression we have neglected the matter contribution. Now by considering $t>0$, the general solution for the expanding universe is
\begin{equation}
\label{45}
a(t)=\left(\frac{C_{0}{\bar{m}}^{2}}{H^{2}_{dS}}\right)^{\frac{1}{\gamma}}\left[\sinh\left(\frac{H_{dS}}{2}\gamma t+\phi\right)\right]^{\frac{2}{\gamma}},
\end{equation}
where $\phi$ is a positive constant. The above expression can be rewritten, given
\begin{equation}
\label{46}
a(t)=a_{0}e^{H_{dS}t}\left[1-e^{-(H_{dS}\gamma t+2\phi)}\right]^{\frac{2}{\gamma}},
\end{equation}
where $a_{0}$ is the value of the scalar factor $a(t)$ at $t=0$ and $a_{0}=\frac{1}{2}\left(\frac{C_{0}\bar{m}^{2}e^{2\phi}}{H_{dS}}\right)^{\frac{1}{\gamma}}$. Therefore, the Hubble parameter gives
\begin{equation}
\label{47}
H=H_{dS}\coth\left(\frac{H_{dS}}{2}\gamma t+\phi\right).
\end{equation}
The interesting feature here is to observe that, as $\gamma>0$, for the the redshifts reaching the low possible ones, i.e, $z\rightarrow -1^{+}$, the scale factor becomes
$a(t)\simeq a_{0}e^{H_{dS}t}$, and
for $t>0$, $H\simeq H_{dS}$. These results, once again confirm that the de Sitter stage of the current dark energy dominated universe may be described for the lowest values for the redshift.
The curves plotted present the evolution of the perturbation function $y$ versus $z$. The Blue one is plotted for $q=0.03$, the Red for $q=0.04$, the Magenta for $q=0.05$ and the Green one for $q=0.06$. The parameters $C_0$ and $\Omega_{m}^{(0)}$ are set to $0.5$ and $0.26$, respectively.
The graph presents the evolution of $y_H$ versus $z$ for a wide interval of the redshift. The Blue one is plotted for $q=0.03$, the Red for $q=0.04$, the Magenta for $q=0.05$ and the Green one for $q=0.06$. All of them are plotted for $C_0=0.5$, $\Omega_{m}^{(0)}=0.26$.
The graph presents the evolutions of the parameter of the equation of state related to the dark energy according to the physical evolution of the redshift. The Blue one is plotted for $q=0.03$, the Red for $q=0.04$, the Magenta for $q=0.05$ and the Green one for $q=0.06$. The parameters $C_0$ and $\Omega_{m}^{(0)}$ are set to $0.5$ and $0.26$, respectively.
This graph presents the evolutions of the parameter of the effective equation of state versus $z$, the blue for $q=0.03$, the Red for $q=0.04$, the Magenta for $q=0.05$ and the Green one for $q=0.06$. Also here $C_0$ and $\Omega_{m}^{(0)}$ are set to $0.5$ and $0.26$, respectively.
§ CONCLUSION
In this paper, we consider the modified theory of gravity based on the torsion scalar, as the generalized version of the TT, where we assume an exponential model for the action. We then search whether the model may be used to describe the de Sitter stage of the current dark energy. To do so, the function $y_H(z)$, as the ratio of the energy density of the dark energy over the current ordinary matter, and perturb it around its de Sitter value. What has to be checked is whether within the exponential model, the perturbation $y(z)$ function may vanish or converge toward a value whose the magnitude should be very less than 1.According to our results, the perturbation function $y(z)$ converges and goes toward zero as the lowest values ($z=-1^{+}$) are approached. As it is well known, for the de Sitter universe, the parameter of the effective equation of motion is $-1$. Beside to the interesting convergence of the perturbation function, we see that for the same values of the redshift, the parameter of the effective equation of state $\omega_{eff}$ goes toward $-1$, which is its known value of the de Sitter universe. At the same time, the parameter $y_H(z)$ reaches the corresponding de Sitter values $y_0$, as explicitly presented at the left hand side of the Fig. <ref>. In order to check the validity of the fact to get stability only for low-redshifts, we first try to perform the time evolution expression for both the scale factor and the Hubble parameter. The interesting feature here is that for the lowest values of $z$ and within the cosmological considerations, we see that the de Sitter expressions are recovered for the scale factor and the Hubble factor, confirming once again that the model can describe the de Sitter stage of the current accelerating expanded universe only for lowest values of $z$.
From ours results, we conclude that the exponential $f(T)$ model, for suitable values of the input parameters may perfectly describe the de Sitter stage of the current dark energy dominated universe.
Acknowledgement: B. S. B. Nassur Maeva thanks a lot DAAD for financial support.
1de1207.1646 T. M. Davis et al., Astrophys. J. 666, 716 (2007) [arXiv: astro-ph/0701510].
2de1207.1646 Dunkley et al [WMAP Collaboration], collaborations: arXiv: 0803.0586 [astro-ph], E. Komatsu et al. [WMAP Collaboration], arXiv: 0803.0547 [astro-ph].
3de1207.1646 T. Padmanabhan, AIP Conf. Proc. 861, 179 (2006) arXiv: astro-ph/0603114.
1dediego A. G. Riess et al. [Supernova Search Team Collaboration], Astron. J. 116: 1009 (1998);
2dediego Edmund J. Compeland, M. Sami and Shinji Tsujikawa, Int. J. Mod. Phys. D. 15 (2006) 1753-1936 [arXiv: hep-th/0603057v3].
ddd T. Padmanabhan, AIP Conf. Proc. 843, 111 (2006) [astro-ph/0602117].
ddd J. Frieman, M. Turner and D. Huterer, Ann. Rev. Astron. Astrophys. 46, 385 (2008) [arXiv: 0803.0982[astro-ph]].
P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003) [astro-ph/0207347].
8a14deines1 S. Nojiri and S. D. Odintsov, ECONF C 0602061, 06 (2006); Int. J. Geom. Meth. Mod. Phys. 4, 115-146 (2007) [arXiv: hep-th/0601213]; Phys. Rept. 505, 59-144 (2011) [arXiv: 1011.0544].
dd T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, Phys. Rev. D 84 (2011) 024020 [arXiv: 1104.2669[gr-qc]].
M. J. S. Houndjo, Int. J. Mod. Phys. D. 21, 1250003 (2012). arXiv: 1107.3887 [astro-ph.CO].
dd M. J. S. Houndjo and O. F. Piattella, Int. J. Mod. Phys. D. 21 1250024 (2012) [arXiv: 1111.4275 [gr-qc]].
dd D. Momeni, M. Jamil and R. Myrzakulov, Euro. Phys. J. C. 72, [arXiv: 1107.5807 [Physics.gen-ph]].
M. J. S. Houndjo, C. E. M. Batista, J. P. Campos and O. F. Piattella, Can. J. Phys. 91, 548-553 (2013), [arXiv: 1203.6084[gr-qc]].
dines F. G. Alvarenga, M. J. S. Houndjo, A. V. Monwanou and Jean B. Chabi-Orou, Journal of Modern Physics 4, 130-139 (2013), [arXiv: 1205.4678[gr-qc]].
15a19deines1 S. Nojiri and S. D. Odintsov, Phys. Lett. B. 631, 1 (2005), [hep-th/0508049].
dd S. Nojiri, S. D. Odintsov, A. Toporensky and P. Tretyakov, [arXiv: 0912.2488].
dd K. Bamba, S. D. Odintsov, L. Sebastiani and S. Zerbini, [arXiv: 0911.4390].
dd K. Bamba, C-Q. Geng, S. Nojiri and S. D. Odintsov, [arXiv: 0909.4397].
M. E. Rodrigues, M. J. S. Houndjo, D. Momeni and R. Myrzakulov, Can. J. Phys, 92, 173-176 (2014), [arXiv: 1212.4488].
dines2 M. J. S. Houndjo, M. E. Rodrigues, D. Momeni and R. Myrzakulov, Accepted for publication in Can. J. Phys, [arXiv: 1301.4642[gr-qc]].
20a57deines1 J. Amorós, J. de Haro and S. D. Odintsov, Physical Review D 87, 104037 (2013) [arXiv:1305.2344[gr-qc]].
dd K. Bamba, J. de Haro and S. D. Odintsov, JCAP 1302 (2013) 008 [arXiv:1211.2968[gr-qc]].
dd K. Bamba, S. Nojiri and S. D. Odintsov, [arXiv: 1304.6191[gr-qc]].
dd G. R. Bangochea, R. Ferraro, Phys. Rev. D. 79, 124019 (2009) [arXiv: 0812.1205[gr-qc]].
dd E. V. Linder, Phys. Rev. D 81 127301 (2010) [Erratum-ibid D 82, 109902 (2010)][arXiv: 1005.3039 [astro-ph.CO]].
M. Jamil, D. Momeni and R. Myrzakulov, Eur.Phys. J. C. 72 (2012) 2267 [arXiv: 1212.6017[gr-qc]].
dd R. Myrzakulov, Entropy 14 (2012) 1627 [arXiv: 1212.2155[gr-qc]].
M. R. Setare and N. Mohammadipour, JCAP 1211 (2012) 030 [arXiv: 1211.1375[gr-qc]]; JCAP 01 (2013) 015 [arXiv: 1301.4891[gr-qc]].
dd M. Jamil, D. Momeni, R. Myrzakulov and P. Rudra, J. Phys. Soc. Jap. 81 (2012) 114004 [arXiv: 1211.0018[Physics.gen-ph]].
dd M. E. Rodrigues, M. J. S. Houndjo, D. Saez-Gomez and F. Rahaman, Phys. Rev. D. 86 (2012) 104059 [arXiv: 1209.4859[gr-qc]].
M. Jamil, D. Momeni and R. Myrzakulov, Eur. Phys. J. C 72 (2012) 2122 [arXiv: 1209.1298[gr-qr]].
dd R. Myrzakulov, Eur.Phys. J. C 72 (2012) 2203 [arXiv: 1207.1039[gr-qc]].
dd M. J. S Houndjo, D. Momeni and R. Myrzakulov, Int. J. Mod. Phys. D 21 (2012) 1250093 [arXiv: 1206.3938[Physics.gen-ph]].
dd M. E. Rodrigues, M. H. Daouda and M. J. S. Houndjo, [arXiv: 1205.0565[gr-qc]].
dd M. R. Setare and M. J. S. Houndjo, Can. J. Phys. 91 (2013) 260-267,[arXiv: 1203.1315[gr-qc]].
dd S. Nesseris, S. Basilakos, E. N. Saridakis, L. Perivolaropoulos, Phys. Rev. D 88, 103010 (2013); arXiv:1308.6142 [astro-ph.CO].
dd A. Paliathanasis, S. Basilakos, E.N. Saridakis, U. Valparaiso, S. Capozziello, K. Atazadeh, F. Darabi and M. Tsamparlis, arXiv:1402.5935 [gr-qc].
dd S. Capozziello, P. A. Gonzalez, Emmanuel N. Saridakis and Yerko Vasquez, JHEP 1302, (2013) 039; arXiv:1210.1098 [hep-th].
dd [pdf, ps, other]
Solar system constraints on f(T) gravity
Lorenzo Iorio, Emmanuel N. Saridakis, Mon. Not. Roy. Astron. Soc. 427 (2012) 1555; arXiv:1203.5781 [gr-qc].
dd Yi-Fu Cai, Shih-Hung Chen, J. B. Dent, S. Dutta, E. N. Saridakis, Quantum Grav. 28, (2011) 215011; arXiv:1104.4349 [astro-ph.CO].
dd J. B. Dent, S. Dutta, E. N. Saridakis, JCAP 1101, 009 (2011); arXiv:1010.2215 [astro-ph.CO].
dd Shih-Hung Chen, J. B. Dent, S. Dutta, E. N. Saridakis, Phys. Rev. D 83 , 023508 (2011), arXiv:1008.1250 [astro-ph.CO].
dd K. Bamba, M. Jamil, D. Momeni and R. Myrzakulov, [arXiv: 1202.6114[Physics.gen-ph]].
dd K. Bamba, S. 'i. Nojiri and S. D. Odintsov, Phys. Rev. D 85 (2012) 104036 [arXiv: 1202.4057[gr-qc]].
dd M. Jamil, D. Momeni and R. Myrzakulov, Eur. Phys. J. C 72 (2012) 2267 [arXiv: 1212.6017[gr-qc]]; Gen. Rel. Grav. 45 (2013) 263 [arXiv: 1211.3740[physics.gen-ph]]; Eur. Phys. J. C 72 (2012) 2122 [arXiv: 1209.1298[gr-qc]]; Eur. Phys. J. C 72 (2012) 2075 [arXiv: 1208.0025[gr-qc]].
dd M. Jamil, K. Yesmakhanova, D. Momeni and R. Myrzakulov, Central Eur. J. Phys. 10 (2012) 1065 [arXiv: 1207.2735[gr-qc]].
dd M. J. S. Houndjo, D. Momeni and R. Myrzakulov, Int. J. Mod. Phys. D 21 (2012) 1250093 [arXiv: 1206.3938[physics.gen-ph]].
M. Jamil, D. Momeni and R. Myrzakulov, Eur. Phys. J. C 72 (2012) 1959 [arXiv: 1202.4926[physics.gen-ph]].
dd M. H. Daouda, M. E. Rodrigues and M. J. S. Houndjo, Phys. Lett. B 715 (2012) 241 [arXiv: 1202.1147[gr-qc]].
dd M. Jamil, S. Ali, D. Momeni and R. Myrzakulov, Eur. Phys. J. C 72 (2012) 1998 [arXiv: 1201.0895[physics.gen-ph]].
dd M. Jamil, D. Momeni N. S. Serikbayev and R. Myrzakulov, Astrophys. Space Sci. 339, 37, (2012) [arXiv: 1112.4472[physics.gen-ph].
M. Jamil, D. Momeni and M. A. Rachid, Eur. Phys. J. C 71, 1711 (2011) [arXiv: 1107.1558[physics.gen-ph]].
dd M. Hamani Daouda, M. E. Rodrigues and M. J. S. Houndjo, Eur. Phys. J. C 72 (2012) 1893 [arXiv: 1111.6575[gr-qc]]; Eur. Phys. J. C 72 (2012) 1890 [arXiv: 1109.0528[physics.gen-ph]]; Eur. Phys. J. C 71 (2011) 1817 [arXiv: 1108.2920[astro-ph.CO]].
dd R. Myrzakulov, Gen. Rev. Grav. 44 (2012) 3059 [arXiv: 1008.4486[physics.gen-ph]].
K. K. Yerzhanov, S. R. Myrzakul, I. I. Kulnazarov and R. Myrzakulov, [arXiv: 1006.3879[gr-qc]].
dd R. Myrzakulov, Eur. Phys. J. C 71 (2011) 1752 [arXiv: 1006.1120[gr-qc]].
dd M. E. Rodrigues, M. J. S. Houndjo, D. Momeni and R. Myrzakulov, [arXiv: 1302.43.72[physics.gen-ph]].
dd J. M. Bardeen, B. Carter and S. W. Hawking, Math. Phys. 31 (1973) 161-170.
N. Tamanini and C. G. Boehmer, Pyhs. Rev. D 86 044009 (2012) [arXiv: 1204.4593[gr-qc]].
ddbarrow Baojiu Li, T. P. Sotiriou and J. D. Barrow, Phys. Rev. D 83, 064035 (2011); Phys. Rev. D 83, 104030 (2011).
M. J. S. Houndjo, D. Momeni, R. Myrzakulov and M. E. Rodrigues, [arXiv: 1304.1147 [physics.gen-ph]].
dines3 C. Deliduman and B. Yapiskan, [arXiv: 1103.2225v2[gr-qc]].
12de1108.6184v2 K. Bamba, C. Q. Geng and C. C. Lee, J CAP 1008, 021 (2010); [arXiv: 1005.4572[astro-ph.CO]].
bamba1 K. Bamba, C. Q. Geng, C. C. Lee and L. L. Luo, JCAP 1101, 021 (2011); [arXiv: 1011. 0508v2[astro-ph.CO]].
1205.3421 Kazuharu Bamba, Salvatore Capozziello, Shin'ichi Nojiri and Sergei D. Odintsov;[arXiv: 1205. 3421v3[gr-qc]]
perturbation3 Shinh-Hung Chen, James B. Dent, Sourish Dutta, and Emmanuel N. Saridakis;[arXiv:1008. 1250v2[astro-ph.CO]]
bamba K. Bamba, C. Q. Geng, C. C. Lee and L. L. Luo, JCAP 1101, 021 (2011); [arXiv: 1011. 0508v2[astro-ph.CO]]
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1511.00702
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Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley, California 94720, USA.
Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley, California 94720, USA.
Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley, California 94720, USA.
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA.
Laboratoire de Physique Théorique, École Normale Supérieure, CNRS, Paris, France.
Instituut voor Theoretische Fysica, KU Leuven, Belgium.
Department of Physics, New York City College of Technology, The City University of New York, Brooklyn, NY 11201, USA
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA.
Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley, California 94720, USA.
Bath engineering, which utilizes coupling to lossy modes in a quantum system to generate non-trivial steady states, is a tantalizing alternative to gate- and measurement-based quantum science. Here, we demonstrate dissipative stabilization of entanglement between two superconducting transmon qubits in a symmetry-selective manner. We utilize the engineered symmetries of the dissipative environment to stabilize a target Bell state; we further demonstrate suppression of the Bell state of opposite symmetry due to parity selection rules. This implementation is resource-efficient, achieves a steady-state fidelity $\mathcal{F} = 0.70$, and is scalable to multiple qubits.
Advances in quantum circuit engineering <cit.> have enabled coherent control of multiple long-lived qubits based on superconducting Josephson junctions <cit.>. Conventional approaches for further boosting coherence involve minimizing coupling to lossy environmental modes, but this poses an increasingly non-trivial challenge as chip designs scale and increase in complexity. An alternate approach, quantum bath engineering <cit.>, explicitly utilizes this coupling in conjunction with microwave drives, to modify the dissipative environment and dynamically cool to a desired quantum state. Bath engineering in superconducting qubits has resulted in the stabilization of a single qubit on the Bloch sphere <cit.>, a Bell-state of two qubits housed in the same cavity <cit.>, many-body states <cit.>, and a variety of non-classical resonator states <cit.>. Additionally, theoretical proposals have been put forward for dissipative error correction <cit.> and ultimately a universal quantum computation <cit.>.
These approaches require careful selection of the bath modes, and typically many drives to excite these modes so as to produce a non-trivial ground state. Bath engineering schemes have typically focused on sculpting a density of states conducive to cooling, relying on the conservation of energy between drive, qubit, and resonator modes in multi-photon processes. In this Letter, we harness an additional degree of freedom: the spatial symmetry of the bath, which mandates conservation of parity. We combine both spectral and symmetry selectivity of the bath to provide a scalable protocol for generating on-demand entanglement using only a single microwave drive with a controllable spatial profile. As a demonstration of this scheme, we generate and stabilize a two-qubit entangled state of choice in the single-excitation subspace using two tunable 3D transmon qubits <cit.> in independent microwave cavities. Our results demonstrate the viability of this protocol for stabilizing many-body entangled states with high fidelity in extended arrays.
Cavity-mediated qubit coupling. a: schematic of aperture-coupled cavities, with weakly-coupled input ports $\kappa^{in}_i$, strongly-coupled output port $\kappa^{out}$, and inter-cavity coupling $J$. b: Transmission spectrum of the coupled cavity modes, showing the symmetric (blue) and antisymmetric (red) peaks. c: Pump-probe spectroscopy of the coupled qubit modes, exhibiting an avoided crossing. Cavity $B$ is driven at the symmetric cavity resonance conditioned on the qubit state $|gg\rangle$, and cavity $A$ is driven at a swept Rabi pump frequency $\omega^R$. A dip in transmission (blue) indicates that $\omega^R$ is resonant with a qubit mode. Vertical stripes are an artifact of data acquisition. The dashed line is a fit of the spectral data.
The experiments are implemented (Figure <ref>a) using two aperture-coupled copper waveguide cavities (indexed as $A$ and $B$ throughout this Letter), with an independent flux-tunable transmon embedded in each cavity. The cavities are fabricated with near-identical resonance frequencies $\omega^c_{A,B} \equiv \omega^c = 2\pi \times 7.114 \text{ GHz}$; the qubits are flux-tuned to resonance at $\omega^q_{A,B} \equiv \omega^q = 2\pi \times 6.200 \text{ GHz}$. The full set of qubit and cavity parameters are tabulated in the Supplemental Material [See supplementary information.]. The cavities are individually addressable via a weakly-coupled port ($\kappa^{in}_i$) through which we apply qubit pulses and bath drives; cavity $A$ has an additional strongly coupled port for readout.
The unitary dynamics of the system are described by a Hamiltonian that can be subdivided into qubit, cavity, and drive components:
\begin{equation}
\hat{\mathcal{H}} = \hat{\mathcal{H}}_q + \hat{\mathcal{H}}_a + \hat{\mathcal{H}}_d
\end{equation}
where, in the rotating wave approximation,
\begin{equation}
\begin{array}{lcl}
\hat{\mathcal{H}}_q & = & \dsum\limits_{i=A,B}\left[\dfrac{\omega^q}{2}\hat{\sigma}^z_i + g_i\left(\hat{\sigma}^+_i \hat{a}_i + \hat{\sigma}^-_i \hat{a}^\dagger_i\right) \right]\\
\hat{\mathcal{H}}_a & = & \dsum\limits_{i=A,B} \left[ \omega^c\hat{a}^\dagger_i \hat{a}_i + J\left(\hat{a}_A \hat{a}_B^\dagger + \hat{a}_A^\dagger \hat{a}_B\right) \right] \\
\hat{\mathcal{H}}_d & = & \dsum \limits_{i=A,B} \epsilon^d_i \left[ \hat{a}_i^\dagger e^{-i(\omega^d t +\phi_i)} +\hat{a}_i e^{i(\omega^d t +\phi_i)}\right] \\
\end{array}
\end{equation}
Here, $\hat{\sigma}_i$ are Pauli operators on the qubits; $\hat{a}_i^\dagger$ are creation operators in the cavity modes; $\epsilon^d_i$ are Rabi drives applied at a single frequency $\omega^d$ to the respective cavities with a tunable phase $\phi_i$; and $g_i$ are the qubit-cavity couplings. Decay mechanisms not accounted for in these unitary dynamics include qubit energy relaxation ($\Gamma_1$) and dephasing ($\Gamma_\phi$), and cavity photon leakage ($\kappa$).
The effects of the coupling terms $g$ and $J$ manifest in both the qubit and cavity sectors. The central cavity resonances hybridize into symmetric and antisymmetric modes, with the former having a lower frequency (Figure <ref>b). We define these modes as $\omega^c_\pm = \omega^c \mp J$. In the dispersive limit where the qubit-cavity detuning $\Delta_\pm \equiv \omega^q -\omega^c_\pm$ is large in comparison to $g$, the qubit-cavity coupling creates a photon-mediated $XY$ interaction between the qubits, lifting the degeneracy in the single-excitation subspace <cit.>. Defining $\delta = J\dfrac{g_Ag_B}{\Delta_+\Delta_-}$, the coupled eigenstates and eigenenergies are given by the following:
\begin{equation}
\begin{array}{lclclcl}
\ket{T_-} &\equiv& \ket{gg}&;& \omega_{\ket{T_-}} &=& 0 \\
\ket{T_0} &\equiv &\dfrac{\ket{ge} + \ket{eg}}{\sqrt{2}} & ; & \omega_{\ket{T_0}}& =& \omega^q - \delta \\
\ket{S} &\equiv& \dfrac{\ket{ge} - \ket{eg}}{\sqrt{2}} & ; & \omega_{\ket{S}} &= &\omega^q + \delta \\
\ket{T_+} &\equiv& \ket{ee}&;& \omega_{\ket{T_+}} &=& 2\omega^q \\
\end{array}
\end{equation}
Note that the single-excitation states $\ket{S}$ and $\ket{T_0}$ are maximally-entangled Bell states. We can then define full basis states of the system including the cavity modes, as
\begin{equation}
\ket{i,j,k} = \ket{n_+} \otimes \ket{n_-} \otimes \ket{\psi_q}
\end{equation}
where $n_\pm$ indexes the Fock state of the respective hybridized cavity modes and $\ket{\psi_q}$ is a coupled qubit state $\ket{\psi_q} \in \{\ket{S}, \ket{T_{0,\pm}} \}$. Figure <ref>c shows the qubit-sector avoided crossing of width $2\delta = 2\pi \times 2.7$ MHz, in quantitative agreement with independently-characterized system parameters.
Protocol for cooling to $\ket{0,0,T_0}$ via $\omega^c_-$ (left) and $\omega^c_+$ (right). Each set of levels outlined in grey dashed lines represents a rung on the Jaynes-Cummings ladder; the states $\ket{\psi}$ are the coupled qubit states. Not drawn to scale. The illustrated drives (arrows) represent $\omega^d_{\ket{T_0}}(\pm)$ from Equation <ref>. Symmetry selection rules require that if cooling via $\omega^c_+$, the drive must be overall symmetric (indicated by blue lines), with $\phi = \{0, \pi\}$; if cooling via $\omega^c_-$, the drive must comprise one antisymmetric (red) photon for each symmetric photon ($\overline{n}_+ = \overline{n}_-$). If this condition is met, stochastic leakage of cavity photons (purple, $\kappa$) brings the system to the entangled state $\ket{0,0,T_0}$. Leakage from the entangled state is dominated by qubit decay (green, $\Gamma_1$); leakage rates not shown include transitions between $\ket{S}$ and $\ket{T_0}$, and off-resonant pumping to $\ket{T_+}$.
We now aim to stabilize the entangled state of choice ($\ket{S}$ or $\ket{T_0}$) by taking advantage of the distinct symmetries of the bath modes at $\omega^c_+$ and $\omega^c_-$ [Since the target entangled states are in fact eigenstates of the coupled Hamiltonian, it is in principle possible to coherently pulse to these states. However, because the splitting is small, a coherent pulse with narrow enough bandwidth to drive selectively to one of these states would need to be several microseconds long, and therefore would be spoiled by qubit decay.].
We do this by simultaneously applying a two-photon at the individual cavity ports while varying the relative phase between the cavities (Figure <ref>). This work represents a generalization to arbitrary drive phase of the proposal in <cit.>; the full theoretical treatment of this generalization (including dynamics) is presented in the Supplement <cit.>.
Our cooling protocol relies on transitions between two neighboring rungs of the Jaynes-Cummings ladder (typically the $n_\pm = \left\{0,1\right\}$ subspaces). The appropriate drive frequencies are given by
\begin{equation}
\begin{array}{rcl}
\omega^d_{\ket{T_0}}(\pm) & = & \frac{1}{2} \left\{ \omega^c_\pm + \left[ \tilde{\omega}^q + 2\chi_\pm \right] -\delta \right\} \\
\omega^d_{\ket{S}}(\pm) & = & \frac{1}{2} \left\{ \omega^c_\pm + \left[ \tilde{\omega}^q + 2\chi_\pm \right] +\delta \right\} \\
\end{array}\
\label{eq:freqs}
\end{equation}
where $\chi_\pm$ is a cross-Kerr term leading to a $n_\pm$-dependent shift in the effective qubit frequency and $\tilde{\omega}^q$ represents the dressed qubit frequency, which has a power-dependent red shift due to the off-resonant displacement of the cavity field by the drive [Because the qubit-cavity couplings $g_i$ differ, the qubit frequencies shift by different amounts when exposed to the same intra-cavity field. To correct for this, we place the bare qubit frequencies slightly off of resonance such that the dressed qubit frequencies $\tilde{\omega}^q_i$ are identical. This adjustment is power-dependent, but is on the order of 1 MHz.].
When a microwave drive is applied at one of these frequencies, a two-photon transition is created between the un-driven ground state $\ket{0,0, T_-}$ and the resonant partner state $\ket{\psi} \in \{\ket{1,0,S}, \ket{1,0,T_0}, \ket{0,1,S}, \ket{0,1,T_0}\}$. However, when $n_\pm > 0$ the cavities decay stochastically and irreversibly at a rate $\kappa_+ (\kappa_-) = 2\pi \times$ 650 (820) kHz to $\ket{0,0, T_0}$ or $\ket{0,0,S}$, where the final qubit state is the same as that of the state targeted by the pump. There are no transitions from this state that are resonant with the drive. In the case of a $T_1$ decay, the drive rapidly repumps the qubits, thus creating a stabilized entangled state. A weak off-resonant pumping into $\ket{T_+}$, which is depleted by $T_1$ rather than by active cooling, sets an upper limit on the cooling rate.
In Figure <ref>, we implement this protocol by applying simultaneous, amplitude-balanced drives with a relative phase $\phi \equiv \phi_B-\phi_A$ to the input of the cavities. Panel (a) shows the sequence of pulses: we apply the bath drive for a fixed interval of $\tau = 10$ $\mu s$, and sweep the drive frequency ($\omega^d$, $y$-axis) and relative phase ($\phi$, $x$-axis). We then reconstruct the joint qubit density matrix $\rho$ using tomographic reconstruction techniques <cit.> based on high-power readout <cit.>. Figure <ref>(b) shows the fidelity to $\ket{S}$ (red) and to $\ket{T_0}$ (blue), where the fidelity to a target state $\ket{\psi}$ is given by $\mathcal{F} = \langle \psi| \rho | \psi \rangle$.
The symmetry-selective aspect of the protocol manifests at three symmetry points. In particular, there are four bands in which the protocol achieves entanglement, corresponding to the frequencies in Equation <ref>: entanglement via $\omega^c_+$ $(\omega^c_-)$ occurs at $\omega^d = 2\pi \times 6.572 \text{ }(6.713) \pm 0.0013$ GHz. However, at $\phi = 0, \phi = \pi$, and $\phi \sim 180^\circ \pm 67^\circ$, the resonant transitions are selectively suppressed for one of the target states, and the suppressed states are reversed between the $\omega^c_+$ and $\omega^c_-$ cooling bands.
The phase-dependent suppression can be understood as a parity selection rule that is dynamically generated by altering the drive profile across the coupled cavities. The starting permutation-exchange parity is comprised of the initial qubit state ($\ket{T_-}$, a symmetric state) and the two photons used to generate the drive (which vary from symmetric to antisymmetric with $\phi$); the output parity is comprised of the qubit state symmetry and the dissipated photon. Conservation of parity requires that the net parity of the output state respect that of the input state - remembering that the net exchange symmetry of two antisymmetric components is overall even. By varying the relative phase of the drives, we vary the input symmetry and therefore control the parity selection rules.
Under an even-parity drive, when the cooling drive is comprised of two symmetric or two antisymmetric photons (i.e. $\phi = 0$ or $\pi$), we can only cool to the qubit state whose parity is the same as the cavity output photon. Indeed, population in the antisymmetric $\ket{S}$ is fully suppressed in the lower (symmetric) band at $\phi = \{0,\pi\}$, and $\ket{T_0}$ is similarly suppressed in the upper band (where the scattered photon is antisymmetric). There also exits a relative phase at which the drive is comprised equally of symmetric and antisymmetric photons, leading to an overall odd-parity drive. This phase is $\phi \approx 180^\circ \pm 67^\circ$ in these experiments, and differs from $\pi \pm \pi/2$ because of the detuning $\omega^c_- \neq \omega^c_+$ <cit.>. At these phases, the parity of the target qubit state must be opposite that of the cavity output photon. Cooling to $\ket{T_0}$ occurs only via the anti-symmetric cavity in this case, and cooling to $\ket{S}$ occurs via the $\omega^c_+$ mode. These symmetry restrictions are lifted for generic $\phi$, in which case both cavity modes can be equivalently used to target $\ket{T_0}$ or $\ket{S}$, and only energy conservation of input and output photons is required. Thus, simply by tuning a readily-adjustable parameter in our driving profile, we can turn a given entangled state from a forbidden into a symmetry-protected state.
Symmetry- and frequency- selective bath engineering. (a) The sequence of drive, qubit, and cavity pulses used in the experiment. We apply the bath drive for time $\tau$, then perform one of a set of tomographic rotations followed by a projective readout. (b) Symmetry and frequency dependence of the cooling drive. We plot $\mathcal{F}_{\ket{S}}-\mathcal{F}_{\ket{T_0}}$ such that $\ket{S}$ is red and $\ket{T_0}$ is blue. At the symmetry points $\phi=0, \phi=\pi$, and $\overline{n}_+ = \overline{n}_-$ <cit.>, the drive is both frequency- and symmetry- selective. The $\ket{\psi_1} \leftrightarrow \ket{\psi_2}$ notation indicates the transition with which the drive is resonant for the labelled band. Transitions between higher cavity occupation states are red-detuned by $\chi_+ = 2\pi \times$ 2.5 MHz for the lower-frequency bands, and $\chi_- = 2\pi \times$ 1.4 MHz for the higher-frequency bands.
The undulation in the cooling bands is an effect of the phase-dependence of $\tilde{\omega}^q$, due to the detuning between $\omega^c_+$ and $\omega^c_-$: a drive of fixed amplitude is closer in frequency and therefore coupled more strongly to the lower-frequency symmetric mode, resulting in a stronger AC Stark shift at $\phi \approx 0$. The broadening of the cooling spectrum at $\phi = 0$ represents the same phenomenon, this time manifesting as power-broadening. The faint red-shifted cooling bands, detuned by $\chi_\pm$, represent cooling between higher photon-number subspaces, as labeled.
Cooling dynamics. (a) We prepare $\ket{S}$ using $\phi=\pi$ by cooling via the antisymmetric cavity mode (inset). (b) Similarly, we prepare $\ket{T_0}$ using $\phi = \pi$ via the symmetric cavity mode. In both cases, we fix $\phi$ and $\omega^d$; apply the drive for time $\tau$; and then tomographically reconstruct the resultant joint qubit state. The experimental data are represented as dots; solid lines are fits to a coupled rate equation with rates as noted.
By moving to the time-domain (Figure <ref>), we can resolve the effects of the several dynamical rates that govern the non-equilibrium steady state. For each experiment we fix $\omega^d$ and $\phi$, and apply the bath drive for a variable time $\tau$, again finally tomographically reconstructing the joint qubit state. We utilize $\phi = \pi$ such that parity rules require cooling to $\ket{S}$ $\left(\ket{T_0}\right)$ via $\omega^c_-$ $\left(\omega^c_+\right)$. The dominant rates in the system are $\Gamma_p$, the pumping rate from $\ket{T_-}$ to the target state; $\Gamma_p'$, a weak off-resonant pumping to $\ket{T_+}$; $\Gamma_1$, the spontaneous decay rates of the qubits; and $\Gamma_{\phi}$, the effective dephasing rate between $\ket{S}$ and $\ket{T_0}$. Provided that we meet the inequality $\Gamma_\phi, \Gamma_p' <\Gamma_1 <\Gamma_p$, we expect the steady-state population to be entangled. We fit the data to a coupled rate equation and extract the pumping and decay rates. The quality of the fit to a simple exponential indicates that the dynamics of this system are dominated by incoherent processes, which is consistent with $\kappa_\pm \gg \Gamma_p$: in this regime, photons stochastically leak from the cavity much more quickly than the drive is able to repopulate them.
The steady state saturates to the entangled state of choice after a transient ring-up (dominated by $\Gamma_p$) and a small-overshoot (related to $\Gamma_\phi$). The steady-state fidelities are $\mathcal{F}(\ket{T_0}) = 0.70$ and $\mathcal{F}(\ket{S}) = 0.71$, well beyond the threshold $F=0.5$ indicative of quantum entanglement. The fidelity loss is dominated by residual $\ket{T_-}$ population and by transitions to the entangled state of opposite symmetry $\ket{T_0} \leftrightarrow \ket{S}$. Increasing $\Gamma_p$ in principle helps to depopulate the inital $\ket{T_-}$ state; however, increasing the pump leads to power-broadening of both the desired transition and of the off-resonant pumping to $\ket{T_+}$. Since $\ket{T_+}$ decays equally to $\ket{S}$ and to $\ket{T_0}$, this creates a drive-dependent dephasing that sets an upper limit on the pumping rate.
In a future on-chip implementation with currently-accessible qubit coherence times, this protocol can be expected to produce on-demand entanglement with fidelity in excess of 0.90.
In this work, we have demonstrated symmetry-selective bath engineering, harnessing both the spatial symmetry and the density of states of the dissipative environment to achieve and preserve on-demand entanglement. The engineered symmetries in our system distinguish it from the two-qubit bath engineering experiment in Ref. <cit.>, where cooling to $\ket{S}$ is achieved by utilizing far-detuned qubits in a single cavity; stabilizing entanglement in this system required six microwave drives, and only $\ket{S}$ was accessible. In our implementation, the resonant construction of the photonic lattice imprints itself onto the effective qubit Hamiltonian and lifts the degeneracy in the single-excitation subspace. The lifting of this degeneracy allows us to reduce the number of required drives from six to one, and the use of separate cavities allows us to easily modify the spatial profile of this drive in order to capitalize on the permutation symmetries of the coupled cavity resonances.
Our work demonstrates that engineering symmetries of a dissipative environment provides a powerful route to quantum control. Because we achieve entanglement in a resource-efficient manner, our protocol is amenable to scaling to multiple qubits and cavities. In a multi-qubit implementation, the even and odd permutation symmetries generalize to a quasi-momentum across the lattice - and critically, adjusting the profile of a driving tone across the lattice still provides symmetry selectivity. Furthermore, this architecture is well-suited to the preparation of many-body $W$-states <cit.>, the protection of high-symmetry states (i.e. quadrupoles), and the study of driven-dissipative dynamics in quantum lattices. The ease of access to single-qubit manipulation and readout makes this experimental geometry a promising testbed for transport and studies of long-range entanglement in these systems.
We acknowledge useful discussions with Vinay Ramasesh and Shyam Shankar. This research is based on work supported in part by the U.S. Army Research Office (ARO) under grant no. W911NF-14-1-0078. MES acknowledges support from the Fannie and John Hertz Foundation; LM acknowledges support from the Berkeley Fellowship and the National Science Foundation (NSF) Graduate Research Fellowship. HET acknowledges support from ARO Grant No. W911NF-15-1-0299 and NSF Grant No. DMR-1151810. CA acknowledges support from NSF Grant No. DMR-1151810 and the Eric and Wendy Schmidt Transformative Technology Fund. MK gratefully acknowledges support from the Professional Staff Congress of the City University of New York award No. 68193-0046.
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1511.00976
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Department of Optics, Palacký University, 17. listopadu 1192/12, CZ-771 46 Olomouc, Czech Republic
RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 84511 Bratislava, Slovakia
RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 84511 Bratislava, Slovakia
Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 84511 Bratislava, Slovakia
Faculty of Informatics, Masaryk University, Botanická 68a, 60200 Brno, Czech Republic
03.65.Ta, 03.67.-a, 03.65.-w, 03.65.Aa
The existence of incompatible measurements, epitomized by Heisenberg's uncertainty principle, is one of the distinctive features of quantum theory. So far, quantum incompatibility has been studied for measurements that test the preparation of physical systems. Here we extend the notion to measurements that test dynamical processes, possibly consisting of multiple time steps.
Such measurements are known as testers and are
implemented by interacting with the tested process through a sequence of state preparations, interactions, and measurements.
Our first result is a characterization of the incompatibility of quantum testers, for which we provide necessary and sufficient conditions.
Then, we propose a quantitative measure of incompatibility. We call this measure the robustness of incompatibility and define it as the minimum amount of noise that has to be added to a set of testers in order to make them compatible. We show that (i) the robustness is lower bounded by the distinguishability of the sequence of interactions used by the tester and (ii) maximum robustness is attained when the interactions are perfectly distinguishable.
The general results are illustrated in the concrete example of binary testers probing the time-evolution of a single-photon polarization.
§ INTRODUCTION
Quantum theory challenges our intuition in many ways,
a prominent example being the existence of incompatible measurements <cit.>.
Observable quantities, such as the position and the velocity of a particle, can be incompatible, in the sense that it is impossible to measure them in a single experiment unless some amount of noise is added <cit.>.
The existence of incompatible measurements is at the root of the wave-particle duality <cit.> and, more generally, of many striking distinctions between quantum and classical physics.
For example, the existence of incompatible measurements implies the no cloning theorem <cit.>: by contradiction, if we could make two perfect copies of an arbitrary quantum state, we could perform one measurement on one copy and another measurement on the other, so that no pair of measurements would be incompatible. This argument applies also to approximate universal cloning <cit.>, whose optimal performance is limited by the incompatibility of quantum measurements <cit.>. Moreover, if all quantum measurements were compatible, then quantum states could be represented as probability distributions over a classical phase space, whose points would be labeled by outcomes of all possible quantum measurements. For composite systems, such a classical description would prevent the violation of Bell inequalities <cit.>, inhibiting important applications such as device-independent cryptography <cit.>.
More recently, the existence of incompatible measurements has been recognized as equivalent to the existence of Einstein-Podolski-Rosen (EPR) steering, a weaker form of non-locality whereby the choice of measurement on one system determines the ensemble decomposition of the state of another system. In short, the argument <cit.> is as follows: if all quantum measurements were compatible, then one could explain the phenomenon of steering in terms of a hidden state model <cit.>, wherein the state of the steered system is defined before the measurement. Vice-versa, if every instance of EPR steering could be explained by a hidden state model, then all quantum measurements should be compatible. Based on this argument, one can establish a quantitative connection between incompatibility and EPR steering, which has been explored extensively in Refs. <cit.>.
Due to its fundamental implications, quantum incompatibility has been the object of intense research <cit.> (see Ref. <cit.> for a recent review).
So far, all investigations have focused on the standard scenario where the goal of measurements is to
test properties of the system's preparation <cit.>.
However, one can consider more general scenarios, where the goal of the measurement is to test a property of a dynamical process <cit.>. For example, imagine that we are given an optical device that transforms the polarization of photons in some unknown way. To gain some knowledge of the device, we can ask how well it preserves the vertical polarization. This property can be tested by preparing a vertically polarized photon, sending it through the device, and finally performing a polarization measurement with a vertically aligned polarizer. Similarly, we may ask how well the device preserves the horizontal, or the diagonal polarization (defined as the polarization aligned by +45 degrees with respect to vertical polarization), or whether the device transforms the vertical polarization into the diagonal one. All these questions correspond to different experimental setups, that one can use to test different properties of the unknown process. These properties are often complementary <cit.> in the sense that they cannot be tested in a single experiment — i.e. they correspond to incompatible measurement setups.
The aim of this paper is to provide a precise characterization of what setups are incompatible when we test quantum processes. Our prime motivation is fundamental: we want to explore the new forms of complementarity arising from the study of quantum dynamics. On a more practical side, we expect that our generalized notion of quantum incompatibility will have new applications in quantum information, in the same way as the early studies of quantum complementarity and incompatibility led to the discovery of new quantum protocols. In our investigation we will start from the simplest case of processes that consist of a single time step. To test such processes, we consider setups that consist of (i) preparing the input of the process in a known state (possibly a joint state of the input with an ancillary system), (ii) letting the state evolve through the process, and (iii) performing a measurement on the output, as in Figure <ref>.
Pictorial representation of a setup testing a property of a
quantum process $\mathcal E$. The input and output of the process are labeled as $0$ and $1$, respectively. The setup consists of the preparation of the input (and possibly an ancilla) into state $\Psi$ and in the execution of a measurement (POVM) $\st P$ on the output.
Any such setup can be described by a quantum tester, a suitable generalization of the notion of positive operator-valued measure.
A measurement setup as in Figure <ref> can be represented in a compact way using the notion of quantum tester <cit.>, a generalization of the notion of positive operator-valued measure (POVM) <cit.>. More specifically, a tester is a collection of operators that can be used to compute the outcome probabilities for the setup under consideration.
Specifically, each outcome $j\in [n]:=\{1,\dots,n\}$ is associated with a positive operator $T_j$ acting on the tensor product of Hilbert spaces of input and output systems. Consequently the tester consists of collection of such positive operators $\{T_j, j\in[n]\}$.
The probability of recording the outcome $j$ is given by the generalized Born rule
\begin{align}\label{genborn}
p_j = \Tr [ T_j E] \, ,
\end{align}
where $E$ is the Choi operator <cit.> of the tested process $\mathcal{E}$ (see Section <ref> for the precise definitions and for more details).
It is important to stress that, like the notion of POVM, the notion of tester involves a certain level of abstraction: since the tester describes only the probabilities of the outcomes, different experimental implementations giving rise to the same statistics are identified. For example, suppose that we want to know whether a process is unital, i.e. whether it preserves the maximally mixed state. A natural way to test unitality is to prepare the maximally mixed state and to perform a state tomography on the output. However, the maximally mixed state can be prepared in many different ways: for example, one could set up a stochastic mechanism that, with equal probabilities, prepares a photon with vertical or horizontal polarization. Alternatively, the mechanism could prepare a photon with diagonal (+45 degrees) and antidiagonal (-45 degrees) polarizations. Or one could prepare two photons in a maximally entangled state, so that the reduced state of each photon is maximally mixed. Despite being physically different, all these procedures will eventually lead to the same statistics, and, therefore, to the same tester.
The statistical point of view will be crucial for our notion of incompatibility. We will regard two testers as compatible if their outcomes can be generated in a single experiment and the corresponding probability distribution has marginals coinciding with the probability distributions predicted by the original testers. This is a purely information-theoretic notion of compatibility: it states that the statistics of two testers can be merged into the statistic of a third tester. It is worth stressing, however, that the physical implementation of the third tester may be very different from the physical implementations of the original testers.
A key point of our work will be to identify the sources of incompatibility that affect the tests of dynamical processes. A well-known source of incompatibility is the incompatibility of the measurements performed at the output: a tester that prepares a vertically polarized photon and measures the output with a vertically oriented polarizer is incompatible with a tester that prepares a vertically polarized photon and measures the output with a diagonally oriented polarizer. However, in the case of processes there is another source of incompatibility, namely the incompatibility of the inputs: for example, one can test the action of a process on vertically polarized photons, or one can test it on horizontally polarized photons, but there is no joint setup that performs both tests at the same time. Note that taking a superposition of horizontal and vertical polarizations would not work, because the action of the process on the superposition does not give enough information about the action of the
process on the individual states that are being superposed.
One of the first results in our paper is a necessary and sufficient condition for the statistical compatibility of two (or more) testers.
Afterwards, we provide a quantitative measure of incompatibility, based on the amount of noise needed to make two (or more) testers compatible. This notion, called robustness of incompatibility, will allow us to give an interesting lower bound, where the amount of incompatibility of two setups is lower bounded by the distinguishability of the input states used to probe the unknown process. As a result of this bound, we find that only setups with the same local input states can be compatible.
A complete analysis of the compatibility conditions is presented in the case of two-qubit, two-outcome testers.
All our results can be generalized to the case of processes consisting of multiple time steps <cit.>, each step transforming an input into an output.
Such multi-time processes can be tested by preparing an input for the first step and applying a sequence of operations, as in Figure <ref>.
Pictorial representation of a measurement setup testing a property of multi-time
quantum processes.
The paper is structured as follows:
In Section <ref> we introduce the mathematical framework
of process POVMs suited for the analysis of tester
incompatibility questions. In Section <ref> we define incompatibility of testers.
The Section <ref> introduces a measure
of incompatibility of testers that is evaluated in Sections <ref> and <ref> in cases
when the incompatibility is rooted in the incompatibility of the input states and final
measurements, respectively. In Section <ref> we investigate in details the
incompatibility of two-outcome testers, especially, we focus
on factorized qubit case. In Section <ref> we generalize the
incompatibility consideration for general
quantum networks and
we point out that the introduced incompatibility measure
is bounded from below by success probability characterizing the
minimum-error discrimination of corresponding quantum devices.
In Section <ref> we summarize our findings. Technical results are gathered in the Appendix.
§ BACKGROUND ON QUANTUM TESTERS
§.§ Testing quantum processes
Quantum testers <cit.> provide a compact way of representing experimental setups designed to test unknown quantum processes. Let us start from the simplest case, where the tested process consists of a single time step.
A setup testing such processes consists of
* the joint preparation of an input system and an ancilla,
* the application of the tested process on the input, and
* the execution of a joint measurement on the output and the ancilla, as in Figure <ref>.
In the following we will label the input, output, and ancilla system as $0$, $1$, and $\rm anc$, respectively. We denote by $\spc H_0$, $\spc H_1$, and $\spc H_{\rm anc}$ the corresponding Hilbert spaces and by $d_0$, $d_1$, and $d_{\rm anc}$, the corresponding dimensions, respectively. Moreover, we denote by $\set S$ the set of possible outcomes of the final measurement.
Mathematically, the above setup is specified by a triple $\map T =(\Hs_{\rm anc}, \Psi, {\st P} )$, where
* $\Hs_{\rm anc}$ is the Hilbert space of the ancilla used in the experiment,
* $\Psi$ is a density operator, acting on the tensor product Hilbert space $\Hs_0\otimes\Hs_{\rm anc}$ and representing a joint preparation of the input system and the ancilla, and
* ${\st P} = \{P_j,j \in \set S\}$ is a positive operator-valued measure (POVM) on $\Hs_1\otimes\Hs_{\rm anc}$, representing a joint measurement on the output system and the ancilla.
The tested process is described by a completely positive trace-nonincreasing linear map $\map E$, transforming operators on the input Hilbert space $\Hs_0$ into operators on the output Hilbert space $\Hs_1$. For deterministic processes (also known as quantum channels) the map $\map E$ is trace-preserving, see e.g. <cit.>.
When the process $\map E$ is tested with the setup $\map T$, the probability that the measurement produces the outcome $j$ is given by
\begin{align}\label{probabilities}
p_j(\map T, \mathcal{E})=\tr{[ P_j \, (\mathcal{E}\otimes\mathcal{I}_{\rm anc})(\Psi) ]},
\end{align}
where $\mathcal{I}_{\rm anc}$ is the identity mapping on the ancilla.
The probabilities defined in this way are non-negative and sum up to one if the tested process is deterministic. A remarkable property of quantum theory is that, under minimal requirements, every admissible map sending quantum processes to probability distributions can be physically implemented via some
setup $\map T$, meaning that one can always find an ancillary system, an input state, and a measurement that give rise to the desired mapping $\map E \mapsto p_j (\map E)$ <cit.>.
The probabilities in Eq. (<ref>) can be written down in a compact way using the Choi isomorphism <cit.>, whereby the process $ \map E$ is represented by the positive (semidefinite) operator $E$ defined by
\begin{align}\label{Choi}
E : = \left( \map E \otimes \map I \right) (|\Omega\>\<\Omega|)\, ,
\end{align}
where $|\Omega \> \in \spc H_0 \otimes \spc H_0$ is the unnormalized maximally entangled state
\begin{align}\label{omega}
|\Omega\> : = \sum_{m=1}^{d_0} \, |m\>|m\> \, ,
\end{align}
$\{\ket{m} \}_{m=1}^{d_0}$ being an orthonormal basis for $\Hs_0$.
In terms of the Choi operator, the outcome probabilities
can be rewritten as <cit.>
\begin{align}\label{testerprob}
p_j (\map T, \map E) = \Tr \left[ T_j \, E \right] \, ,
\end{align}
where $T_j$ is the operator on $\spc H_1\otimes \spc H_0$ defined by <cit.>
\begin{align}\label{Tj}
T_j: = \Tr_{\rm anc} [ ( P_j \otimes I_0 ) \, ( I_1 \otimes {\tt SWAP} \, \Psi^{T_0} \, {\tt SWAP} \, )].
\end{align}
Here $\Psi^{T_0}$ denotes the partial transpose of $\Psi$ on the Hilbert space $\spc H_0$, and $\tt SWAP$ is the unitary operator that swaps the Hilbert spaces $\spc H_0$ and $\spc H_{\rm anc}$ in order to have them consistently ordered.
It is easy to see that the operators $\{ T_j \, , j\in\set S\}$ satisfy the conditions
\begin{align}
\label{positivity} {\rm positivity:} \quad & T_j \ge 0 \, , \quad \forall j\in \set S \\
\label{normalization} {\rm normalization:} \quad & \sum_{j\in\set S} \, T_j = I_1\otimes \rho \, ,
\end{align}
where $\rho$ is a density operator on $\spc H_0$. Physically, equations (<ref>) and (<ref>) guarantee the positivity and normalization of the outcome probabilities.
The above observations lead to the definition of quantum tester:
Let ${\st T} = \{ T_j,j\in \set S\}$ be a collection of operators on $\spc H_1 \otimes \spc H_0$. We say that $\st T$ is a quantum tester if it satisfies the conditions (<ref>) and (<ref>), for some suitable density operator $\rho$ on $\spc H_0$. We call the operator $\rho$ the normalization state of the tester $\st T$.
Quantum testers have also been called process POVMs in Ref. <cit.> and measuring co-strategies in Ref. <cit.>.
When there is no ambiguity, we will omit the explicit specification of the outcome set. For example, we will write $\st T = \{ T_j\}$ instead of $\st T = \{ T_j\, , j\in\set S\}$ and
\[ \sum_j T_j \quad {\rm instead~of}\quad \sum_{j\in\set S} T_j \, . \]
§.§ Physical implementation of quantum testers
We have seen that every experimental setup testing quantum processes can be described by a tester. The converse is also true: for every tester, one can find a setup that generates the corresponding statistics.
We say that a setup $ \map T = (\spc H_{\rm anc}, \Psi, {\st P} )$ is a physical implementation of the tester $\st T$ if it satisfies
the condition
\begin{align}
p_j( \map T, \map E) = \Tr \left[ T_j \, E \right] \, ,
\end{align}
for every outcome $j $ and for every process $\map E$.
A canonical way to construct physical implementations is provided by the following:
For a given tester ${\st T} = \{ T_j \}$, Let
* $\rho$ be the normalization state in Eq. (<ref>),
* $\spc H_\rho$ be the support of $\rho$,
* $|\Psi_\rho\>\in \spc H_0\otimes \spc H_\rho$ be the unit vector defined by
\begin{equation}
|\Psi_\rho\> := \left(I_0 \otimes \rho^{\frac 12}\right)\, | \Omega \> ,
\end{equation}
* $\Psi_\rho$ be the density operator $ \Psi_\rho : = |\Psi_\rho\>\<\Psi_\rho| $, and
* $\st P = \{ P_j \}$ be the POVM defined by
\begin{align}\label{canonicalPOVM}
P_j : = \left(I_0 \otimes \rho^{-\frac 12}\right) \, T_j \, \left(I_0 \otimes \rho^{-\frac 12}\right) \, ,
\end{align}
where $\rho^{-\frac 12}$ is the inverse of $\rho^{\frac 12}$ on its support.
Then, the triple $\map T = ( \spc H_\rho, \, \Psi_\rho \, , {\st P})$ is a physical implementation of the tester $\st T$.
We call the POVM ${\st P}$ defined in Eq. (<ref>) the canonical POVM associated with the tester $\st T$. The implementation defined in Proposition <ref> will be called the canonical implementation of the tester $\st T$.
Proposition <ref> tells us that every tester can be implemented with an ancilla of the size of the support of $\rho$, the normalization state associated with the tester. In general, one can construct other implementations where the size of the ancilla is larger, or even smaller, as we will see later in an example. Nevertheless, all the physical implementations of a given tester must satisfy a common property, highlighted by the following proposition:
Let $\st T$ be a quantum tester and let $\map T = ( \spc H_{\rm anc}, \Psi, {\st P})$ be a physical implementation of $\st T$. Then, the input state $\Psi$ must satisfy the condition
\begin{align}
\Tr_{\rm anc} [ \Psi ] = \rho^{T} \, ,
\end{align}
where $\rho$ is the normalization state defined in Eq. (<ref>) and the transposition is defined with respect to the basis used in Eq. (<ref>).
A simple proof can be found in Appendix <ref>. In words, proposition <ref> identifies the normalization state with (the transpose of) the local state on the input system. Moreover, it implies that all the physical implementations of the same tester must have the same marginal state on the input system. This property will play a crucial role in deciding the compatibility of testers.
§.§ Ancilla-free testers
The simplest example of testers are those that can be implemented without ancillas, as in Figure <ref>.
Diagrammatic representation of an ancilla-free quantum tester.
Precisely, we adopt the following definition.
A tester $\st T$ is ancilla-free if it admits an implementation $\map T = (\spc H_{\rm anc}, \Psi, {\st P})$ where the ancilla Hilbert space is trivial, namely $\spc H_{\rm anc} = \mathbb C$. When this is the case, we say that the implementation $\map T$ is ancilla-free.
Ancilla-free testers have a very simple characterization.
A tester ${\st T} =\{ T_j\, , j \in\set S \}$ is ancilla-free if and only if there exists a POVM ${\st P} = \{ P_j\, , j\in\set S \}$ and a density operator $\rho$ such that one has
\begin{align}\label{ancillafree}
T_j = P_j\otimes \rho \, , \qquad \forall j \in \set S \, .
\end{align}
Every ancilla-free implementation $\map T = (\mathbb C, \Psi, {\st P} )$ has $\Psi = \rho^T$.
A proof can be found in Appendix <ref>.
An example of ancilla-free tester is a tester designed to probe how an optical device preserves the vertical polarization of a single photon.
In this case, implementation consists in preparing photon in the vertical polarization state $\Psi = |V\>\< V|$, feeding it in the input port of the device, and performing the projective measurement $ {\st P} = \{ |V\>\<V| \, , |H\>\< H| \}$ on the output.
The corresponding tester is
\begin{equation}
{\st T}_V = \{ |V\>\<V| \otimes |V\>\<V| \, , |H\>\<H| \otimes |V\>\<V| \} .
\end{equation}
Another example is tester designed to probe how the device preserves the horizontal polarization. In this case the tester is
\begin{equation}
{\st T}_H = \{ |H\>\<H| \otimes |H\>\<H| \, , |V\>\<V| \otimes |H\>\<H| \} .
\end{equation}
Yet another example is a tester designed to probe the unitality of a process. For a process acting on the polarization of a photon, the test of unitality can be performed by preparing the input in the maximally mixed state $\Psi = I/2$ and by checking whether the output state $\map E ( I/2)$ is still maximally mixed. To check, one can perform a tomographically complete POVM $\st P$ on the output and establish whether the statistics of the outcomes is compatible with the maximally mixed state. For example, one can choose at random among three polarization measurements, performing the six-outcome POVM ${\st P} = \{ P_1,P_2, \dots, P_6\}$ with
\begin{align}\label{tomopovm}
\begin{array}{llll}
P_1 & = \frac 13 \, |V\>\< V | \, , \qquad & P_2 & = \frac 13 \, |H\>\< H | \\ \\
P_3 &= \frac 13 \, |D\>\< D | \, , & P_4 &= \frac 13 \, |A\>\< A |\\ \\
P_5 & = \frac 13 \, |R\>\< R | \, , & P_6 & = \frac 13 \, |L\>\< L | \, ,
\end{array}
\end{align}
where $ |D\> \, , |A \> \,, |R\> \, , |L\>$ denote the states of diagonal, antidiagonal, right-handed, and left-handed polarization, respectively. If the process $\map E$ is unital, the six possible outcomes should occur with equal probabilities. The above setup corresponds to the tester ${\st T} = \{ T_1, T_2, \dots, T_6\}$ with
\begin{equation}
T_j := P_j\otimes \frac I2 \, , \qquad \forall j\in \{ 1,2, \dots, 6\} .
\end{equation}
Note that an alternative way to implement the same tester is to prepare two-photon maximally entangled state
\[ |\Phi^+ \> : = \frac { | H\> |H\> + |V\>|V\>}{\sqrt 2}\]
and to perform POVM ${\st P'} = \{ P_1', P_2' , \dots , P_6'\}$ with
\[ P_j' : = P_j \otimes I \, . \]
This implementation is canonical (according to Definition <ref>), but not ancilla-free: it uses the second photon as ancilla. Besides the ancilla-free testers, other examples of testers are provided in Appendix <ref>.
§ CHARACTERIZING COMPATIBILITY
Before analyzing the case of testers, we briefly define the notion of compatibility for POVMs. We refer the reader to Ref. <cit.> for a more in-depth presentation.
§.§ Compatibility of POVMs
POVMs provide statistical description of experiments designed to probe the preparation of quantum systems. Each POVM represents, so to speak, a different “experimental question" that we can ask the system.
Given two such questions, it is important to know whether the system can answer both questions at the same time. More precisely, it is important to know whether there exists a third experiment that produces the same outcomes as
the two original experiments, with the same probability distributions.
Concretely, suppose that the system is prepared in the state $\rho$ and measured either with the POVM ${\st P} = \{ P_j \}$ or with the POVM $ {\st Q} =\{ Q_k \}$. In one case, the measurement will produce the outcome $j$ with probability
\begin{align}\label{p}
p_j = \Tr [ P_j \rho ] \, ,
\end{align}
while, in the other case the measurement will produce the outcome $k$ with probability
\begin{align}\label{q}
\qquad \, q_k = \Tr [ Q_k \rho] \, .
\end{align}
Now, we would like to find an experiment that provides both outcomes $j$ and $k$, with a joint probability distribution $r_{jk}$ that reproduces the statistics of the original measurements, when one takes the marginals:
\begin{align}\label{marginal} \sum_{k } r_{jk} = p_j \qquad {\rm and} \qquad \sum_{j } r_{jk} = q_k
\end{align}
for every $j$ and every $k$. The statistics of the third experiment will be determined by a (joint) POVM $ {\st R} = \{ R_{jk} \}$, so that one has
\begin{align}\label{r}
r_{jk} = \Tr [ R_{jk} \rho] \, ,
\end{align}
for every $j$ and $k$.
If the state $\rho$ is known, the condition (<ref>) can be trivially satisfied by choosing $R_{jk}= p_j q_k \, I$, meaning that one can draw $j$ and $k$ at random according to the (known) probability distributions in Eqs. (<ref>) and (<ref>). The interesting scenario is when the state $\rho$ is not known to the experimenter. Here we require the condition (<ref>) to hold for every quantum state $\rho$.
Under this requirement, Eq. (<ref>) becomes equivalent to the condition
\begin{align}\label{compatiblePOVM}
\sum_{k} \, R_{jk} = P_j \qquad {\rm and} \qquad
\sum_{j} \, R_{jk} = Q_k
\end{align}
for every $j$ and every $k$. The above discussion motivates the following definition.
Two POVMs $\st P= \{ P_j\}$ and $\st Q=\{ Q_k\}$ are compatible if there exists a third POVM ${\st R}= \{ R_{jk}\}$ such that Eq. (<ref>) is satisfied for every $j$ and $k$. If no such POVM exists, we say that $\st P$ and $\st Q$ are incompatible.
An example of compatible POVMs is given by commuting POVMs, i.e. POVMs $\st P$ and $\st Q$ satisfying the condition $[P_j, Q_k] = 0$ for every $j$ and $k$. In this case, one can define the joint POVM ${\st R}= \{ R_{jk}\}$ with operators
\begin{equation}
R_{jk} := P_j Q_k ,
\end{equation}
whose positivity is guaranteed by the commutation of $P_j$ and $Q_k$.
On the other hand, there are examples of compatible POVMs that
are not commuting. A nice counterexample was introduced by Busch <cit.>, who considered two unsharp measurements of horizontal/vertical and diagonal/antidiagonal polarizations, ${\st P}=\{P_1,I-P_1\}$ and
${\st Q}=\{Q_1,I-Q_1\}$, with
\begin{align}
P_1 &=\frac{1+p}{2} \, |V\>\< V| + \frac{1-p}{2} \, |H\>\< H| , \\
Q_1 &= \frac{1+q}{2} \, |D\>\< D| + \frac{1-q}{2} \, |A\>\< A| ,
\end{align}
and showed that these are compatible whenever $p^2+q^2\leq 1$. Clearly, the operators $P_1$ and $Q_1$ do not commute, except in the trivial case when $p$ or $q$ is zero. In summary, we obtained the following observation.
For POVMs, commutativity implies compatibility,
but not vice-versa.
§.§ Compatibility of quantum testers
In analogy with the POVM case, we can regard quantum testers as different “experimental questions" that we can ask about physical processes. The only difference is that now the questions we can ask are of a more dynamical nature — essentially, they are questions about how processes transform different inputs.
Now, we want to know whether two experimental questions can be answered at the same time. Let us represent the two questions with two corresponding testers ${\st A} = \{ A_j\} $ and ${\st B} = \{ B_k\}$, respectively, and let us represent the tested process $\mathcal E$ by its Choi operator $E$.
When the first, resp. the second, question is asked, the probability to obtain the outcome $j$, resp. $k$ is given by
\begin{align}\label{pqproc} p_j = \Tr [ A_j \, E ] \qquad\text{resp.}\qquad q_k = \Tr [ B_k\, E ] \, ,
\end{align}
cf. Eq. (<ref>). Now, we are looking for an experimental setup that can answer both questions, by providing the outcomes $j$ and $k$ with the right probabilities. In other words, we want to find a tester ${\st C} = \{ C_{jk}\}$, such that the probability distribution $\{ r_{jk} \}$ defined by
\begin{align}
r_{jk} := \Tr [ C_{jk}\, E ]
\end{align}
reproduces the statistics of the original testers, namely
\begin{align} \sum_{k} r_{jk} = p_j \qquad {\rm and} \qquad \sum_{j} r_{jk} = q_k \, .
\end{align}
Requiring this condition to hold for every possible process (including both deterministic and non-deterministic processes) leads to the following definition.
Two quantum testers ${\st A}=\{A_j\}$ and ${\st B}=\{B_k\}$ are
compatible if there exists a (joint) tester ${\st C}=\{C_{jk}\}$
such that
\begin{align}
\label{compatibilitytester}
\sum_k C_{jk}=A_j \qquad {\rm and} \qquad
\sum_j C_{jk}=B_k
\end{align}
for every $j$ and $k$. If no such tester exists, we say that ${\st A}$ and ${\st B}$ are incompatible.
§.§ Necessary and sufficient conditions for compatibility
In this part we will analyze the immediate implications of our definition of compatibility. First of all, it is easy to see that compatible testers must have the same normalization:
Let $\st A$ and $\st B$ be two testers and let $\rho$ and $\sigma$ be the corresponding normalization states, as in Eq. (<ref>). If $\st A$ and $\st B$ are compatible, then one must have
\begin{align}\label{joint_conditions}
\rho = \sigma \, .
\end{align}
Let $ \st C$ be the joint tester that reproduces the statistics of $\st A$ and $\st B$. Then, one has
\begin{align}
I_1\otimes\rho &= \sum_j A_j = \sum_j \left( \sum_k C_{jk}\right)\notag\\
& =\sum_k \left( \sum_j C_{jk}\right) = \sum_k B_k =I_1\otimes\sigma.
\end{align}
Hence, one must have $\rho = \sigma$.
As an application, Proposition <ref> shows that the tester
\begin{align}\label{TV} {\st T}_V = \{ |V\>\<V| \otimes |V\>\<V| \, , |H\>\<H| \otimes |V\>\<V| \} \, ,
\end{align}
designed to probe the preservation of the vertical polarization, is incompatible with the tester
\begin{align}\label{TH} {\st T}_H = \{ |H\>\<H| \otimes |H\>\<H| \, , |V\>\<V| \otimes |H\>\<H| \} \, ,\end{align}
designed to probe the preservation of the horizontal polarization. Indeed, ${\st T}_V$ has the normalization state $\rho = |V\>\< V|$, while ${\st T}_H$ has the normalization state $\sigma = |H\>\< H|$. Interestingly, all the operators in the testers ${\st T}_V$ and ${\st T}_H$ commute. Summarizing, we have the following observation.
For testers, commutativity does not imply compatibility.
As the above counterexample shows, the reason why two commuting testers may fail to be compatible is that their normalization states do not coincide. Interestingly, once this obstacle is removed, commuting testers become compatible:
Two commuting testers $\st A$ and $\st B$ are compatible if and only if they have the same normalization states.
The “only if" part follows immediately from proposition <ref>. Let us show the “if" part. Denote by $\rho$ the normalization state. Since the testers commute, the normalization state commutes with all the operators: indeed, one has
\begin{align}
[I_1\otimes \rho \, , B_k] & = \sum_{j} \, [ A_j , B_k] =0, \\
[A_j, I_1\otimes \rho] & = \sum_{k} \, [ A_j , B_k] =0 ,
\end{align}
for every $j$ and $k$.
As a consequence, also the inverse $(I_1\otimes \rho)^{-1}$ (on the support of $\rho$) commutes with $A_j$ and $B_k$ for every $j$ and $k$. Using this fact, we define the tester ${\st C}= \{ C_{jk}\}$ with operators
\begin{equation}
C_{jk} = A_j B_k (I_1 \otimes \rho)^{-1},
\end{equation}
whose positivity follows from the commutation of $A_j$, $B_k$ and $(I_1\otimes \rho)^{-1}$. It is immediate to verify the normalization condition (<ref>) and the compatibility conditions (<ref>).
More generally, we now provide a necessary and sufficient condition for the compatibility of two arbitrary (possibly non-commuting) testers. The condition is expressed in terms of the canonical implementation of the testers introduced in Proposition <ref>.
Two quantum testers $\st A$ and $\st B$ are compatible if and only if
* the normalization states associated with $\st A$ and $\st B$ in Eq. (<ref>) coincide, and
* the canonical POVMs associated with $\st A$ and $\st B$ are compatible.
The proof can be found in Appendix <ref>. The physical meaning of Theorem <ref> is that two testers $\st A$ and $\st B$ are compatible if and only if they can be implemented with two experimental setups $\map A = ( \spc H_{\rm anc}, \Psi, {\st P})$ and $\map B = ( \spc H_{\rm anc} \, , \Psi \, , {\st Q})$ using the same ancilla, preparing the same input state, and measuring compatible POVMs $\st P$ and $\st Q$.
Note that thanks to Theorem <ref>, we do not need to search over all possible physical implementations of the two testers. All the information regarding the compatibility of the testers can be read out from their canonical implementation.
§ QUANTIFYING INCOMPATIBILITY
The incompatibility of POVMs is a resource for various quantum information tasks, such as steering <cit.> or device independent cryptography <cit.>. It plays a strong role also in approximate cloning <cit.>.
It is then natural to expect that also the incompatibility of testers will serve as a resource for information processing, at a higher level where the information is encoded into processes, rather than states. From this point of view, it is natural to look for suitable measures that quantify the amount of incompatibility of two or more testers. In the following we will provide one such measure, which we call the robustness of incompatibility. Our constriction is inspired by a convex method previously used for POVMs <cit.>.
§.§ Convexity of the set of testers
Testers with the same set of outcomes form a convex set. Precisely, we have the following
Let ${\st A} = \{ A_j \, , j\in\set S \}$ and ${\st B} = \{ B_j \, , j\in\set S\}$
be two sets of operators, acting on the same Hilbert space and indexed by the same index set $\set S$. The convex combination of $\st A$ and $\st B$ with weight $\lambda \in [0,1]$ is the set of operators $\st C = \{ C_j \, , j\in\set S\}$ defined by
\begin{equation}\label{eq:convextester}
C_j : = (1-\lambda) \, A_j + \lambda\, B_j \, .
\end{equation}
We will denote the convex combination as
\begin{align}\label{cc}
{\st C}:= (1-\lambda) \, {\st A} + \lambda\, {\st B} \, . \end{align}
Denoting by $ \set T ( \spc H_0, \spc H_1,\set S)$ the set of all testers with input space $\spc H_0$, output space $\spc H_1$, and outcome space $\set S$, we have the following
The set $ \set T ( \spc H_0, \spc H_1,\set S) $ is convex.
It is easy to see that the set of operators $\st C$ defined in Eq. (<ref>) is indeed a tester: the operators $C_j$ are obviously positive and one has the normalization condition
$\sum_j \, C_j = I_1 \otimes \tau$ with $\tau = (1- \lambda) \rho + \lambda \, \sigma$ ,
where $\rho$ and $\sigma$ are the normalization states of $\st A$ and $\st B$, respectively. Finally, convexity of the set of density matrices implies that the operator $\tau$ is a density matrix. Hence, the set of operators $\st C$ is properly normalized.
§.§ The robustness of incompatibility
Operationally, convex combinations of testers corresponds to the randomization of different experimental setups. In particular, we can consider the case where an ideal setup, designed to measure a quantity of interest, is randomly mixed with another setup, regarded as noise.
Now, suppose that the ideal tester $\st A$ is mixed with the noise ${\st N}^{(\st A)}$ and that the ideal tester $\st B$ is mixed with the noise ${\st N}^{(\st B)}$. By adding enough noise, we can make the resulting testers compatible. This idea motivates the following
Two testers $\st A$ and $\st B$ are $\lambda$-compatible if there exist two testers ${\st N}^{(\st A)}$ and $\st N^{(\st B)}$ such that the randomized testers
$ (1-\lambda) {\st A} + \lambda {\st N}^{(\st A)}$ and $ (1-\lambda) {\st B} + \lambda {\st N}^{(\st B)}$
are compatible.
In this setting the probability $\lambda$ can be interpreted as a measure of the level of added noise. Intuitively, the amount of noise needed to break the incompatibility of two testers can be regarded a measure of the degree of incompatibility: the higher the noise, the higher the incompatibility. Based on this idea we define a quantitative measure of incompatibility, in terms of the minimum amount of added noise required to make two testers compatible:
The robustness of incompatibility $\Rt({\st A},{\st B})$ is the minimum $\lambda$ such that the testers $\st A$ and $\st B$ are
In the rest of the paper we will analyze the properties of the
robustness of incompatibility.
§.§ Why different noises?
Our definition of the robustness of incompatibility is similar to the definition of robustness used in the literature on POVMs <cit.>. Except in one detail: for POVMs, one usually assumes that the added noises coincide, i.e. ${\st N}^{(\st A)}=\st N^{(\st B)}$. We do not make this assumption here, because assuming equal noises in the case of testers would result in a notion of robustness with undesired physical properties.
This point is clarified by the following proposition, proven in Appendix <ref>.
Let $\st A$ and $\st B$ be two testers with distinct normalization states and let $\st N$ be an arbitrary tester, representing the noise. Then, the testers $ (1-\lambda) {\st A} + \lambda {\st N}$ and $ (1-\lambda) {\st B} + \lambda {\st N}$ are compatible only if $\lambda = 1$.
In other words, if two testers have different normalization states, then adding the same noise on both will not make them compatible, except in the trivial case when the noise completely replaces the original testers. As a result, even two testers that are arbitrarily close to each other would be maximally incompatible, just because they have slightly different normalization states. A tester could be maximally incompatible with a noisy version of itself, even if the noise is arbitrarily small. For these reasons, we regard the definition of robustness with equal added noises as not physically interesting in the case of testers.
§.§ Bounds on the robustness of incompatibility
The robustness of incompatibility satisfies some obvious bounds: first one has the lower bound $\Rt({\st A},{\st B})\geq 0$.
An upper bound is given
by the following
Every pair of testers $\st A$ and $\st B$ is $\lambda$-compatible
with $\lambda=1/2$.
It is enough to set ${\st N}^{(\st A)} := \st B$ and ${\st N}^{(\st B)} := \st A$. With this choice, the randomized testers $1/2\, {\st A} + 1/2\, {\st N}^{(\st A)}$ and $ 1/2\, {\st B} + 1/2\, {\st N}^{(\st B)}$ coincide and therefore are trivially compatible.
In summary, the robustness of incompatibility for a pair of testers has values in the interval
\begin{align}\label{bounds}
0\leq \Rt({\st A},{\st B})\leq \frac 12 \, .
\end{align}
The lower bound is attained if and only if the testers $\st A$ and $\st B$ are compatible. In the following we will see that the upper bound is also attainable.
§ INCOMPATIBILITY DUE TO THE INPUT STATES
The incompatibility between two testers can have two different origins: it can come from the incompatibility of the input states, or from the incompatibility of the measurements on the output. In this section we quantify the contribution of the input states.
§.§ The robustness of state incompatibility
We know that, in order to be compatible, two testers must have the same normalization state. When this is not the case, it is possible to give a lower bound on the incompatibility of the testers in terms of the normalization states.
Let $\st A$ and $\st B$ be two testers and let $\rho$ and $\sigma$ be their normalization states, respectively. If $\st A$ and $\st B$ are $\lambda$-compatible, then there must exist two states
$\widetilde{\rho}$ and $ \widetilde{\sigma}$ such that
\begin{align}
\label{eq:ncnorm}
(1-\lambda)\rho+\lambda \widetilde{\rho}=(1-\lambda)\sigma+\lambda \widetilde{\sigma}\,.
\end{align}
Hence, defining the robustness of state incompatibility
\begin{align}
\Rs (\rho, \sigma ) : = \min \{ \lambda ~|~ \exists \widetilde \rho \, , \widetilde \sigma \, : {\rm Eq.}~(\ref{eq:ncnorm})~{\rm holds} \}
\end{align}
we have the lower bound
\begin{align}\label{lowerb}
\Rt({\st A},{\st B}) \ge \Rs(\rho, \sigma ) \, .
\end{align}
Suppose that the testers $ (1-\lambda) {\st A} + \lambda {\st N}^{(\st A)}$ and $ (1-\lambda) {\st B} + \lambda {\st N}^{(\st B)}$ are compatible. Then, they must have the same normalization states (see Proposition <ref>). On the other hand, the normalization states are $(1-\lambda)\rho+\lambda \widetilde{\rho}$ and $(1-\lambda)\sigma+\lambda \widetilde{\sigma}$, where $\widetilde \rho$ and $\widetilde \sigma$ are the normalization states of ${\st N}^{(\st A)}$ and ${\st N}^{(\st B)}$, respectively. Hence, Eq. (<ref>) should hold. Minimizing over $\lambda$ one finally gets the lower bound (<ref>).
§.§ Maximally incompatible testers
Using the lower bound (<ref>), it is easy to construct quantum testers that achieve the maximum value of the robustness of incompatibility. We call these testers maximally incompatible:
We say that two testers $\st A$ and $\st B$ are maximally incompatible if $\Rt({\st A},{\st B}) = 1/2$.
An example of maximally incompatible tester is as follows: consider two testers $\st A$ and $\st B$ that probe qubit processes and suppose that $\st A$ and $\st B$ have normalization states $\rho = |0\>\<0|$ and $\sigma = |1\>\<1|$, respectively.
Then, Eq. (<ref>) implies the inequality
\begin{align}
\nonumber \lambda & \ge \lambda \, \< 1| \widetilde \rho |1\> \\
\nonumber & = \< 1| \, \left[ (1-\lambda) \, |0\>\< 0| + \lambda \, \widetilde \rho \right] \, |1\>\\
\nonumber & = \< 1| \, \left[ (1-\lambda) \, |1\>\<1| + \lambda \, \widetilde \sigma \right] \, |1\>\\
\label{derivationbound} & \ge 1-\lambda,
\end{align}
which in turn implies $\lambda \ge 1/2$. Since the inequality must hold for every $\lambda$, the robustness of state incompatibility should also satisfy the inequality $\Rs (\rho, \sigma ) \ge \frac 12$. Hence, Eq. (<ref>) yields the lower bound
\begin{equation}
\Rt({\st A},{\st B}) \ge \Rs (\rho, \sigma ) \ge \frac 12 .
\end{equation}
Combining it with the upper bound (<ref>) we obtain the equality
\begin{equation}
\Rt({\st A},{\st B}) = \frac 12.
\end{equation}
Concrete example of maximally incompatible qubit testers are the optical testers ${\st T}_V$ and ${\st T}_H$ designed to test the preservation of the vertical polarization and horizontal polarization, cf. Eqs. (<ref>) and (<ref>) for the explicit expression. Note that the testers ${\st T}_V$ and ${\st T}_H$ are commuting, and are still, maximally incompatible.
For testers, not only commutativity does not imply compatibility, but also there exist commuting testers that are maximally incompatible!
Remark (testers vs POVMs). The attainability of the upper bound $\Rt({\st A},{\st B}) = 1/2$ highlights an important difference between the incompatibility of testers and the incompatibility of POVMs. For POVMs, it is known that the upper bound is not achievable for finite dimensional systems <cit.>
(see Section <ref> for qualitative details). In contrast, the robustness of tester incompatibility can reach the maximum value $\Rt({\st A},{\st B}) = 1/2$ even for two-dimensional systems.
For POVMs, this maximum value can only be attained for infinite dimensional
systems <cit.>.
§.§ Orthogonal testers are maximally incompatible
Adapting the inequalities used in Eq. (<ref>), one can see that the robustness of incompatibility attains its maximum value $ \Rt({\st A},{\st B}) = 1/2$ whenever the testers $\st A$ and $\st B$ are orthogonal, in the following sense:
Two testers $\st A$ and $\st B$ are orthogonal if one has $A_jB_k= 0$ for every pair of outcomes $j$ and $k$.
It is easy to see that the orthogonality of two testers is equivalent to the orthogonality of their normalization states.
Let $\st A$ and $\st B$ be two testers and let $\rho$ and $\sigma$ their normalization states, respectively.
Then, the following are equivalent
* $\st A$ and $\st B$ are orthogonal
* $\rho$ and $\sigma$ are orthogonal, namely $ \rho \, \sigma = 0$.
By definition, one has $ I_1 \otimes \rho\,\sigma = \sum_{j,k} A_j B_k$. Hence, condition 1 implies condition 2. On the other hand, taking the trace on both sides one has
\begin{equation}
d_1 \,\Tr [ \rho \, \sigma ] = \sum_{j,k} \Tr [A_j B_k] \ge \Tr [A_j B_k] \ge 0 \, ,
\end{equation}
having used the fact that $A_j$ and $B_k$ are positive.
Hence, the condition 2 implies $\Tr [A_jB_k] = 0$, and, since $A_j$ and $B_k$ are positive, $A_jB_k= 0$.
Note that orthogonal testers are commuting, because the condition $A_j B_k= 0$ trivially implies $[A_j, B_k] =0$.
In summary, we have seen that orthogonal testers are always maximally incompatible. It is an open problem whether maximally incompatible testers must be orthogonal.
§.§ The case of jointly diagonal testers
In some cases, the robustness of tester incompatibility coincides with the robustness of state incompatibility, meaning that the lower bound (<ref>) is attained. One such case involves pairs of jointly diagonal testers, defined as follows.
Two testers ${\st A}=\{A_j\}$ and ${\st B}=\{ B_k\}$ are jointly diagonal if all the operators $A_j$ and $B_k$ are diagonal in the same basis.
Note that jointly diagonal testers are a strict subset of the set of commuting testers: while commuting testers satisfy the relation $[A_j, B_k] = 0$ for every $j$ and $k$, jointly diagonal testers have also to satisfy the relations $[A_j, A_k]=0$ and $[B_j, B_k]=0$, for every $j$ and $k$. For jointly diagonal testers we have the following.
Let $\st A$ and $\st B$ be two jointly diagonal testers and let $\rho$ and $\sigma$ be the corresponding normalization states.
Then, one has
\begin{equation}
\Rt({\st A},{\st B}) = \Rs (\rho, \sigma ).
\end{equation}
The proof can be found in Appendix <ref>.
An obvious example of jointly diagonal testers is the example of the testers ${\st T}_V$ and ${\st T}_H$, designed to test the preservation of the vertical and horizontal polarizations, respectively.
This kind of incompatibility originates in the mutually exclusive choices of inputs needed to test the properties represented by ${\st T}_V$ and ${\st T}_H$.
Again, we stress that the incompatibility of inputs is sufficient to make two testers maximally incompatible.
§.§ Computing the robustness of state incompatibility
The robustness of state incompatibility has a direct interpretation in terms of distinguishability. Specifically, we have the following
For every pair of density operators $\rho$ and $\sigma$, the robustness of state incompatibility is given by
\begin{align}
\label{eq:lambda_mingen}
\Rs (\rho, \sigma)=\frac{\|\rho-\sigma\|}{\|\rho-\sigma\|+2}\, \, ,
\end{align}
where $\| \cdot \| = \Tr |\cdot |$ denotes the trace norm.
The above result can be concisely derived from the semidefinite programming approach to minimum error state discrimination, due to Yuen, Kennedy, and Lax <cit.>, combined with the operational interpretation of the trace distance, following from Helstrom's theorem <cit.>. A more explicit derivation with a nice geometric interpretation can be found in Appendix <ref>.
In the special case of two-dimensional systems (qubits), the robustness of state incompatibility has a simple expression in terms of the Bloch vectors of the states $\rho$ and $\sigma$, i.e. the vectors $\st r = (r_x,r_y,r_z)$ and $\st s = (s_x,s_y,s_z)$ in the expressions
\begin{equation}
\rho=\frac{1}{2}(I+{\st r}\cdot\boldsymbol{\sigma}) \quad \text{and} \quad\sigma=\frac{1}{2}(I+{\st s}\cdot\boldsymbol{\sigma}) ,
\end{equation}
where $\boldsymbol{\sigma} = (\sigma_x,\sigma_y,\sigma_z)$
is the vector of the Pauli matrices. Indeed, using the expression
\begin{equation}
\| \rho-\sigma\| = \| {\st r} -{\st s} \| := \sqrt{ \sum_{i=x,y,z} ( r_i - s_i )^2},
\end{equation}
one obtains the following corollary.
For every pair of qubit density operators $\rho$ and $\sigma$, the robustness of state incompatibility is given by
\begin{align}
\label{eq:lambda_min}
\Rs (\rho,\sigma)=\frac{\|\bf{r}-\bf{s}\|}{\| {\st r}-{\st s}\|+2}\,.
\end{align}
§ INCOMPATIBILITY DUE TO THE OUTPUT MEASUREMENTS
In the previous section we quantified how the normalization states affect the incompatibility of two testers. In this section we carry out a similar analysis for the measurements. We focus our attention on testers that have the same normalization states — the rationale being that in such scenario the incompatibility arises purely from the measurements.
§.§ Upper bound on the measurement-induced incompatibility
Two POVMs $\st P$ and $\st Q$, representing two measurements on the same quantum system, are said to be $\lambda$-compatible <cit.> if there exist two POVMs $ {\st J}^{(\st P)}$ and ${\st J}^{(\st Q)}$ such that the POVMs
\begin{equation}
(1-\lambda) \, { \st P} + \lambda \, {\st J}^{(\st P)} \quad\text {and} \quad (1-\lambda) \, { \st Q} + \lambda\, {\st J}^{(\st Q)}
\end{equation}
are compatible. In this context, $ {\st J}^{(\st P)}$ and $ {\st J}^{(\st Q)}$ are regarded as introducing “noise” (or “junk") in the statistics of the desired measurements $\st P$ and $\st Q$.
The robustness of incompatibility is then defined in the natural way.
The robustness of incompatibility of two POVMs $\st P$ and $\st Q$, denoted by $\Rm( \st P,\st Q) $, is the minimum $\lambda$ such that $\st P$ and $\st Q$ are $\lambda$-compatible.
Now, when two testers have the same normalization state, one can upper bound their incompatibility in terms of the incompatibility of the canonical POVMs introduced in definition (<ref>). The upper bound is as follows:
Let $\st A$ and $\st B$ be two testers with the same normalization state $\rho$ and let $\st P$ and $\st Q$ be the canonical POVMs associated with $\st A$ and $\st B$, respectively. Then, one has
\begin{align}\label{measureinc}
\Rt({\st A}, {\st B}) \le \Rm( \st P, \st Q ) \, .
\end{align}
The proof is straightforward: one way to make $\st A$ and $\st B$ compatible is to take their canonical implementations and add enough noise to the canonical POVMs in order to make them compatible. In other words, if $\st P$ and $\st Q$ are $\lambda$-compatible, then also $\st A$ and $\st B$ are $\lambda$-compatible. Taking the minimum over $\lambda$, one obtains the desired upper bound.
§.§ Reachability of maximal incompatibility
As we mentioned earlier, the robustness of incompatibility of two POVMs $\st P$ and $\st Q$ in finite dimensions is always smaller than $1/2$ <cit.>. Physically, this can be seen by considering the process of optimal universal cloning <cit.>, which outputs approximate copies of the original state, mixed with a suitable amount of white noise. By using the optimal cloner, one can make every two POVMs compatible by measuring different copies, at the price of a noisy statistics <cit.>. For a quantum system of dimension $d$, the above procedure gives the bound
\begin{equation}
\Rm(\st P,\st Q)\leq\frac{1}{2}\left(1-\frac{1}{1+d}\right)\,.
\end{equation}
Consequently, two testers with the same normalization states cannot reach the maximum value of the robustness $ \Rt (\st A, \st B)= 1/2 $, unless the dimension of the output system is infinite. For systems of finite dimensions, it is natural to ask what the maximum amount of measurement-induced incompatibility is. We conjecture that the maximum amount is achieved by setups that measure two mutually unbiased bases on the output; further discussion on this point can be found in Appendix <ref>.
§.§ The case of testers with pure normalization state
The upper bound $\Rt({\st A}, {\st B}) \le \Rm({\st P}, {\st Q})$ comes from using the canonical implementation of the testers $\st A$ and $\st B$. In principle, however, the bound may not be saturated, because some non-canonical implementation may have “more compatible" POVMs than the canonical implementation. The bound is saturated, however, in the case of testers with pure normalization state:
Let $\st A$ and $\st B$ be two testers and let $\st P$ and $\st Q$ be their canonical POVMs. If the testers $\st A$ and $\st B$ have the same pure normalization state, then the equality
\begin{equation}
\Rt({\st A}, {\st B}) = \Rm( \st P, \st Q )
\end{equation}
The proof can be found in Appendix <ref>. Note that all testers with pure normalization state are ancilla-free, that is, they can be implemented by preparing the (transpose of) the normalization state, applying the tested process, and measuring the output. It is an open question whether the equality $\Rt({\st A}, {\st B}) = \Rm( \st P, \st Q )$ holds for all ancilla-free testers.
§ INCOMPATIBILITY OF TWO-OUTCOME TESTERS
Here we consider the case of two-outcome testers ${\st A} = \{A_1,A_2\} $ and ${\st B} = \{B_1,B_2\} $. Even in this simple case, we will see that the incompatibility of testers is a subtler issue than the incompatibility of POVMs. We will first provide general results, later moving to a concrete example in the qubit case.
§.§ Characterization of compatibility
In the case of twe-outcome testers, compatibility has a simple algebraic characterization:
A pair of two-outcome quantum testers ${\st A} = \{A_1,A_2\} $ and ${\st B} = \{B_1,B_2\} $
are compatible if and only if
* $\st A$ and $\st B$ have the same normalization state, denoted by $\rho$
* there exists a positive operator $C$ such that
\begin{align}
\nonumber C &\leq A_1, \\
\nonumber C & \leq B_1, \\
I_1\otimes \rho + C &\ge A_1 + B_1\, .
\label{twooutcomecomp}
\end{align}
By definition, $\st A$ and $\st B$ are compatible if and only if there exists a tester ${\st C}$ such that
\begin{align}
A_1 &= C_{11} + C_{12},\nonumber\\
B_1 &= C_{11} + C_{21},\nonumber\\
A_2 &= C_{21} + C_{22},\nonumber\\
B_2 & = C_{12} + C_{22} .
\end{align}
Setting $C:=C_{11}$, the above equations imply
\begin{align}
C_{12}&=A_1-C, \nonumber \\
C_{21}&=B_1-C, \nonumber \\
C_{22}&=I_1\otimes \rho- C_{11} - C_{12} - C_{21}\nonumber\\
& = I_1\otimes \rho -A_1-B_1 + C\, .
\end{align}
The positivity of the above operators is equivalent to the conditions in Eq. (<ref>).
We stress that the equality of the normalization states is essential for compatibility. Even in the extreme case $A_1=B_1$ the testers $\st A$ and $\st B$ may not be compatible, due to the fact that they have different normalization states. In the following we will see that, when $A_1= B_1$, the incompatibility of the testers $\st A$ and $\st B$ is equal to the robustness of normalization.
§.§ The case of comparable testers
Here we consider the case of comparable two-outcome testers, defined as testers satisfying one of the relations
\begin{equation}
A_j \le B_k \qquad {\rm or} \qquad B_k \le A_j
\end{equation}
for at least one pair of outcomes $(j,k) \in \{1,2\}\times \{1,2\}$. As a special case, testers with $A_1 =B_1$ are comparable.
Unlike in the case of POVMs, where comparability implies compatibility <cit.>, comparable testers may not be compatible.
However, they have a remarkable property, expressed by the following proposition.
For comparable two-outcome testers $\st A=\{A_1,A_2\}$ and $\st B=\{B_1,B_2\}$ with normalizations $\rho$ and $\sigma$ the robustness of incompatibility is equal to the robustness of normalizations,
\begin{equation}
\Rt(\st A,\st B) = \Rs(\rho,\sigma) \, .
\end{equation}
The proof can be found in Appendix <ref>.
§.§ Example: testing linear polarizations
In this paragraph we analyze the incompatibility of setups consisting in the preparation of a pure qubit state and in the measurement of a sharp observable on the output.
For concreteness, we refer to the case of polarization qubits and we analyze setups that probe the action of
an unknown process on photons with a given linear polarization.
We consider two setups. In one setup, the input photon has linear polarization at $\theta/4$ degrees relative to the vertical axis and the output photon is measured with a polarizing filter at $\varphi/4$ degrees. In the other setup, the input photon has linear polarization at $-\theta/4$ and the output photon is measured with a filter at $-\varphi/4$ degrees. The factor 4 is just for later convenience. The two setups are described by the testers
$\st A=\{A_1,A_2\}$, $\st B=\{B_1,B_2\}$, defined by
\begin{align}
\label{eq:defABq}
A_1&=P_{-\varphi/2}\otimes P_{-\theta/2}, \quad &B_1&=P_{\varphi/2}\otimes P_{\theta/2}, \nonumber \\
A_2&=P_{\pi-\varphi/2}\otimes P_{-\theta/2}, \quad &B_2&=P_{\varphi/2-\pi}\otimes P_{\theta/2} \,
\end{align}
where we used the notation
\[ P_\alpha:=\frac{1}{2}(I+\sin\alpha\,\sigma_x+\cos\alpha\,\sigma_z) \,. \]
Illustration of the optimal choice of the normalizations $\widetilde\rho$, $\widetilde\sigma$ of the admixed testers $\st N^{(\st A)}$, $\st N^{(\st B)}$ for the incompatibility of testers defined in Eq. (<ref>)
This figure illustrates the splitting of the parameter space
of two-outcome factorized rank-one testers into two regions $\mathsf M$ and $\mathsf D$
investigated separately. The robustness for testers is given
by robustness of normalization states in the (shaded blue) region $\mathsf M$
(Proposition <ref>). In the remaining
region ${\mathsf D}$ this is no longer the case as it is illustrated
in Fig. <ref>.
Physically, $\theta/2$ and $\varphi/2$ represent the angles between the polarization filters preparing the input photon and measuring the output photon, respectively.
By varying the angles $\theta$ and $\varphi$ we obtain different pairs of testers, which can be represented in a square, as in Fig. <ref>.
Note that, unless $\theta = 0$, the two testers $\st A$ and $\st B$ are incompatible, because they have different normalization states.
Moreover, even testers with $\theta= 0$ may be incompatible, due to the incompatibility of the output measurements: for $\theta = 0$, we have the equality
\[ \Rt(\st A, \st B)=\Rm(\st P, \st Q) \, , \]
showing that the incompatibility of the testers is quantified by the incompatibility of the canonical POVMs associated with them. Note that the canonical POVMs are compatible only if $\varphi$ is equal to 0 or to $\pi$.
To start our analysis, we evaluate the lower bound arising from the incompatibility of the input states. Combining Proposition <ref> with
Equation (<ref>) we obtain
\begin{align}
\label{eq:minlforAB}
\Rt(\st A, \st B)
\geq \Rs(\rho, \sigma )
=\frac{\sin \frac{\theta}{2}}{1+\sin \frac{\theta}{2}}\, ,
\end{align}
where $\rho$ and $\sigma$ are the normalization states of $\st A$ and $\st B$, respectively.
Then, it is interesting to characterize the cases where the bound is saturated.
To this purpose, we define a suitable parameter region, which we call “region $\mathsf M$":
\begin{equation}
\mathsf M\equiv \Big\{(\theta,\varphi)\;|\;\theta,\varphi\in[0,\pi]\, ,\, \sin \frac{\theta}{2}\geq \frac{\sin \varphi }{2+\sin \varphi}\Big\} \, .
\end{equation}
Note that the region $\mathsf M$ contains all the testers
testing how well a given polarization is preserved, that is, all testers with $\varphi = \theta$.
More generally, the points in region $\mathsf M$ are illustrated in Fig. <ref>.
With a slight abuse of terminology, we say that two testers $\st A$ and $\st B$ belong to region $\mathsf M$ if the corresponding parameters belong to region $\mathsf M$. We then have the following proposition.
If a pair of testers $\st A,\st B$ belongs to region $\mathsf M$, then their incompatibility is quantified by the incompatibility of the input states. In formula:
\begin{equation}
\label{eq:achieveinM}
\Rt(\st A,\st B)= \Rs (\rho, \sigma) = \frac{\sin \frac{\theta}{2}}{1+\sin \frac{\theta}{2}} \, .
\end{equation}
The proof of this proposition is in Appendix <ref>. A plot of the robustness of incompatibility is provided in Fig. <ref>.
Outside region $\mathsf M$ the situation is much trickier. Here we do not have close formulas for the robustness of incompatibility and we had to resorted to numerical evaluation via semidefinite programming (SDP),
as outlined in Appendix <ref>. The result of the evaluation is plotted in Fig. <ref>. Conceptually, our main findings are
* outside region $\mathsf M$, the robustness of tester incompatibility is strictly larger than the robustness of state incompatibility
* for every fixed value of $\theta$, the maximum of the robustness is attained when the angle $\varphi$ is equal to $\pi/2$, corresponding to mutually unbiased measurements on the output, or equivalently, to polarizing filters with a relative angle of 45 degrees.
Robustness of incompatibility of two qubit testers as defined in Eq. (<ref>). Upper figure shows dependence on $\theta$ and situations for various choices of $\varphi$. Solid line depicts bound of Eq. (<ref>). Bottom figure shows dependence on $\varphi$ for various choices of $\theta$. Dotted lines represent Eq. (<ref>) which coincides with $\Rt(\st A,\st B)$ for $\theta \geq 0.6797$.
Before concluding, we note that our results can be extended from testers involving sharp measurements to testers involving arbitrary two-outcome POVM, provided that a suitable technical condition is satisfied.
Consider a pair of two-outcome testers $\st A$ and $\st B$ with operators
\begin{align}
\label{eq:defABqPOVM}
A_1&=E_1\otimes P_{-\theta/2}, &B_1&=F_1\otimes P_{\theta/2} , \nonumber \\
A_2&=E_2\otimes P_{-\theta/2}, &B_2&=F_2\otimes P_{\theta/2},
\end{align}
where $\{E_1,E_2\}$, $\{F_1,F_2\}$ are arbitrary qubit POVMs. Then, one has the equality
\begin{equation}
\Rt(\st A,\st B)= \Rs(\rho, \sigma )=\frac{\sin \frac{\theta}{2}}{1+\sin \frac{\theta}{2}} \, ,
\end{equation}
whenever the angle $\theta$ satisfies the condition $\theta \geq 2 \arcsin (1/3)$.
The proof is presented in Appendix <ref>.
§ EXTENSION TO MEASUREMENT SETUPS WITH MULTIPLE TIME STEPS
So far we have discussed the compatibility of pairs of experiments designed to test quantum channels, however, the definitions and most of the results hold in a more general setting: we can consider
* the compatibility of more than two quantum testers
* the case of testers that probe quantum processes with multiple time steps.
A quantum process with multiple time steps can be viewed as the quantum version of a causal network, consisting of an ordered sequence of processes correlated by the presence of a quantum memory (see Figure <ref> down).
Illustration of quantum causal network, or $N$-comb (lower orange) that can be “measured” by a quantum $N$-tester (upper blue).
Here $\spc H_i, \, i\in \{0,1,\dots,2N-1\}$ are the Hilbert spaces describing the inputs and outputs of the network (and outputs and inputs of the $N$-tester), $\spc L_j , j\in \{0,1,\dots,N-1\}$ are the Hilbert spaces of the internal memories of the network, $\spc K_l \, , l\in\{0,1,\dots,N-1\}$ are the Hilbert spaces of the internal memories of the $N$-tester.
$\Psi$ is an input state to the $N$-tester (which can be chosen to be pure without loss of generality), $\mathcal U^{(l)}$, $l\in\{1,2,\dots,N-1\}$ are its quantum channels (which can be chosen to be unitary without loss of generality), and $\st P:=\{ P_j\}_{j\in J}$ is a POVM, representing a measurement on the final output systems of the network of tested processes $\mathcal E^{(j)},j\in\{1,2,\dots,N\}$.
We call a quantum causal network of such form a quantum comb <cit.> (also known as quantum strategy in the context of quantum games <cit.>).
More specifically, we call quantum $N$-comb a causal network consisting of $N$ steps.
Two $N$-combs are of the same type if they have the same sequence of input/output Hilbert spaces. Like quantum channels, quantum $N$-combs can be represented by Choi operators. An operator $\RC^{(N)}$ is a quantum N-comb (illustrated in Figure <ref> down) if and only if it is positive and satisfies the equations
\begin{align}
\nonumber \Tr_{2N-1}\left[ \RC^{(N)}\right] & = I_{2N-2} \otimes \RC^{(N-1)},\\
\nonumber \Tr_{2N-3} \left[ \RC^{(N-1)} \right] & = I_{2N-4} \otimes \RC^{(N-2)}, \\
\nonumber & ~\, \vdots \\
\Tr_1 \left[ \RC^{(1)} \right] & = I_{0},
\label{normalizationNcomb}
\end{align}
where $\Tr_n$ denotes the partial trace over the Hilbert space $\spc H_n$,
$I_n$ denotes the identity operator on $\spc H_n$ and $ \RC^{(n)} $
is a positive operator on $\bigotimes_{i=0}^{2n-1} \spc H_{i}$.
In order to extract information about the quantum network we need to
perform an experiment that provides inputs to the network and processes the outputs received at all time steps. The most general way to interact with a quantum causal network is by connecting it with another causal network, as illustrated in Figure <ref>.
Again, we will call the testing network a tester <cit.> — more specifically, we will call $N$-tester a network designed to test processes of $N$ time steps. Quantum testers have been recently considered in the tomographic characterization of non-Markovian evolutions <cit.>, a scenario where probing multitime quantum processes becomes highly relevant.
Two $N$-testers are of the same type if they have the same sequence of input/output Hilbert spaces. An $N$-tester illustrated in Figure <ref> (up) is described by a set of positive operators $\st T = \{ T_j\}_{j\in \set S}$, where $\set S$ is the set of possible outcomes and each operator $T_j$ acts on the tensor product Hilbert space $\spc H_{\rm network} = \bigotimes_{i=0}^{2N-1} \spc H_i$. The normalization condition for a tester is given by the following set of equations
\begin{align}
\nonumber \sum_{j\in \set S } T_j & = I_{2N-1} \otimes \Theta^{(N)},\\
\nonumber \Tr_{2N-2} \left[ \Theta^{(N)} \right] & = I_{2N-3} \otimes \Theta^{(N-1)}, \\
\nonumber & ~\, \vdots \\
\nonumber \Tr_2 \left[ \Theta^{(2)} \right] & = I_{1}\otimes \Theta^{(1)} ,\\
\label{normalizationNtester}
\Tr \left[\Theta^{(1)}\right] & =1 \, ,
\end{align}
where $\Theta^{(n)}$ is a positive operator on
$\bigotimes_{i=0}^{ 2n-2} \spc H_i$. We call such positive operator
$\Theta^{(N)}\equiv\Theta$ the normalization of the tester $\st T$.
When quantum $N$-comb and quantum $N$-tester are combined the probability
of the outcome $j\in \set S$ is given by the generalized Born rule
\begin{equation}
p_j=\tr{\left[ T_j \, \RC^{(N)} \right]}\,.
\end{equation}
Different $N$-testers represent different (and possibly complementary) ways to extract information about a quantum $N$-comb. In the following we will formulate the elementary properties of compatibility for two or more $N$-testers. Let us stress that previous sections treat
the case of 1-testers, which we for simplicity denoted until now as testers and 1-combs traditionally called channels.
Let $ \{\st T^{(x)} \,, \, x\in \set X\}$ be a set of testers of the same type, the $x$-th tester with outcomes in the set $\set S_x$ and the normalization $\Theta_x$. The testers are compatible if there exists a joint tester ${\st C} = \{ C_{\bf{k}}\}_{\bf{k}\in \set S_1\times\dots\times \set S_{|X|}}$ such that for all $x\in X$ and for every $j_x\in \set S_x$
\begin{align}\label{compatibilityKtesters}
T^{(x)}_{j_x} = \sum_{{\bf k}: k_x= j_x} C_{\mathbf{k}}\,,
\end{align}
where ${k}_x$ is the $x$-th component of vector $\bf{k}$.
As in the case of compatibility of pairs of 1-testers, one can show that a set of testers $\{\st T^{(x)}\}_{x\in X}$ are compatible only if they have the same normalizations. Moreover, Theorem <ref> generalizes to the following
For each $N$-tester $\st T^{(x)}$ with normalization $\Theta_x$ define the canonical POVM $\st P^{(x)}=\{ P^{(x)}_{j_x}\}$
\begin{equation}
P^{(x)}_{j_x} = \left(I_{2N-1} \otimes \Theta_x^{-\frac 12}\right) \, T^{(x)}_{j_x} \, \left(I_{2N-1} \otimes \Theta_x^{-\frac 12}\right) .
\end{equation}
The testers $\{ \st T^{(x)} \, , x\in \set X\}$ are compatible if and only if
* their normalizations coincide ($\Theta_x\equiv\Theta$ for all $x$)
* the canonical POVMs $\{ \st P^{(x)} \, , x\in \set X\}$ are compatible.
The proof is a direct generalization of the proof of Theorem
The incompatibility of multitime testers can be quantified in the same way as we did in the $N=1$ case. Again, the idea is to measure the incompatibility of a set of testers based on the amount of “noise” that one has to add in order to make them compatible.
The testers $\{\st T^{(x)},x\in \set X\}$ are $\lambda$-compatible if
for any $x$ there exists a tester $\widetilde{\st T}^{(x)}$ with the
outcome set $\set S_x$ such that the testers $\{(1-\lambda ) \st T^{(x)} +
\lambda \widetilde{\st T}^{(x)}, x\in \set X\}$ are compatible.
The robustness of incompatibility of a set of testers $\mathsf T:=\{\st T^{(x)}\}_{x\in \set X}$, denoted by $\Rt(\mathsf T)$, is the minimal $\lambda$ such that the testers in the set $\mathsf T$ are $\lambda$-compatible.
Note that every set of testers is $\lambda$-compatible with $\lambda = 1-|\set X|^{-1}$, as one can see from a simple adaptation of Proposition <ref>: essentially, one can always make the testers compatible by uniformly mixing them. In other words, we have the bounds
\[0\leq \Rt (\set T)\leq 1-|\set X|^{-1} \, , \]
valid for every set $\set T$ of testers containing $|\set X| $ elements.
We now show that the upper bound can be saturated. To this purpose, we formulate a lower bound in terms of normalization operators $\Theta_x$. The crucial observation is that the normalization operator $\Theta_x$ is an $N$-comb—that is, it satisfies the conditions in Eq. (<ref>). Physically, this means that $\Theta_x$ represents a quantum causal network, consisting of a sequence of $N$ time steps. The distinguishability of the causal networks associated with the original testers gives a lower bound to the robustness of incompatibility:
Let $ \set T= \{\st T^{(x)} \,, \, x\in X\}$ be a set of testers of the same type
with normalizations $\Theta_x$, respectively. Then the
robustness of incompatibility of this set is lower bounded by
\begin{equation}
\Rt (\set T)\ge 1- \frac 1{ |X| \, p_{\rm succ}},
\end{equation}
where $p_{\rm succ}$ is the maximum probability of success in distinguishing among the quantum causal networks associated with operators $ \left\{ \Theta_x , \, x\in X\right\}$. In particular, the
bound is saturated whenever the networks are perfectly distinguishable.
Assume the testers are $\lambda$-compatible for a certain $\lambda$ and let $\st C=\{C_{\bf{j}}\}$ be the joint tester that guarantees the compatibility. Let us denote by $\Theta$ the normalization of the joint tester. Then necessarily for all $x\in X$
\begin{align}\label{dual}
\Theta \ge (1-\lambda) \, \Theta_x \, .
\end{align}
In order to compute the robustness, we have to minimize $\lambda$ over all operators $\Theta$ subject to the constraint that $\Theta$ is the normalization of a joint tester satisfying the compatibility condition. We now relax this constraint and assume only that $\Theta$ satisfies the normalization conditions in Eq. (<ref>).
Defining $\mu := (1-\lambda)^{-1} |X|^{-1}$, we have that minimizing $\lambda$ under the condition (<ref>) is equivalent to minimizing $\mu$ under the condition
\begin{align}
\mu \, \Theta \ge \frac 1{|X|} \, \Theta_x, \qquad \forall x\in X \, .
\end{align}
Now, the minimization of $\mu$ under the condition that $\Theta$ is a comb is a semidefinite program. This semidefinite program was recognized in Ref. <cit.> as the dual to the maximization of the success probability in the discrimination of the networks $\{ \Theta_x \, , x \in\set X\}$. More precisely, Theorem 1 of Ref. <cit.> guarantees that the minimum of $\mu$ is equal to the maximum probability of success $p_{\rm succ}$. Hence, we must have $\mu \ge p_{\rm succ}$, or equivalently
\[ \lambda \ge 1 - \frac 1 { |X| p_{\rm succ}} \, . \]
Since the inequality holds for every $\lambda$, it must hold also for the minimum $\lambda$, leading to the bound $\Rt (\set T)\ge 1 - 1/( |X| p_{\rm succ})$. If the quantum networks $\{ \Theta_x \, ,x\in X\}$ are perfectly distinguishable, one has $p_{\rm succ} =1$ and, therefore, $\Rt (\set T)\geq 1- 1/|X|$. On the other hand, we already mentioned that every set of testers is $\lambda$-compatible with $\lambda = 1-1/|X|$, which concludes the proof.
In the case of two testers ($|\set X|=2$), the above bound has a nice expression in terms of the operational distance between two quantum causal networks <cit.>. Specifically, one has
\begin{equation}
p_{\rm succ} = \frac 12 \left( 1 + \frac 12 \left\| \Theta_1 - \Theta_2 \right\|_{\rm op} \right) ,
\end{equation}
where $\| \cdot \|_{\rm op}$ is the operational norm <cit.>. Inserting this expression in the lower bound we then obtain
\begin{equation}
\label{eq:generalbound}
\Rt \ge \frac{\left\| \Theta_1 - \Theta_2 \right\|_{\rm op} }{2+ \left\| \Theta_1 - \Theta_2 \right\|_{\rm op} } \, .
\end{equation}
Let us note that in the $N=1$ case the operational norm coincides with the trace norm, which implies that Eqs. (<ref>) and (<ref>) match.
§ CONCLUSIONS
In this paper we have introduced the notion of compatibility for measurement setups designed to probe quantum dynamical processes. We highlighted how the time structure, with its division into inputs and outputs, affects the notion of compatibility. In particular, we highlighted the differences between the compatibility of ordinary quantum measurements, described by POVMs, and more general setups with multiple time steps, described by quantum testers.
When testing processes consisting of a single time step, the key differences between POVMs and testers are the following:
* For POVMs, commutativity implies compatibility. For testers, the implication is false, whenever the testers entail preparations of distinct input states. In this way, even testers consisting of mutually commuting projectors can turn out to be incompatible.
* For POVMs, the maximum amount of incompatibility that can be found in a finite dimensional system increases with the dimension, reaching the largest value only for infinite dimensional systems <cit.>. For testers, the maximum amount of incompatibility is the same for all testers with non-trivial input: two testers are maximally incompatible whenever they entail the preparation of orthogonal input states.
* For two-outcome POVMs, the ability to compare one element of a POVM with one element of the other POVM implies compatibility. For testers, the implication is false: two comparable testers may not be compatible. Also in this case, the incompatibility originates in the incompatibility of input states.
Physically, the differences arise from the fact that the incompatibility of testers can arise from a different source than the incompatibility of POVMs. Such a different source is the incompatibility of the input states: essentially, probing a process on a certain input precludes the possibility of probing the process on another input.
As a result, the incompatibility of two testers can arise from two contributions: the incompatibility of the input states sent to the process and the incompatibility of the measurements performed on the output. To quantify these contributions, we provide lower and upper bounds on the tester incompatibility in terms of the state and measurement incompatibilities, respectively.
For simplicity, we illustrated most of our results in the case of testers probing processes consisting of a single time step.
However, all results can be generalized to testers that probe quantum processes consisting of multiple time steps.
In particular, we showed that two general testers can be incompatible, because they use two distinct sequences of interactions in order to probe an unknown multiple time step process. The distinguishability of the sequences of interactions provides a lower bound to the incompatibility of the resulting testers and, again, maximum incompatibility is obtained when the sequences are perfectly distinguishable.
Since the incompatibility of ordinary measurements is a resource in several applications (steering, device independent cryptography, etc.), we believe that the research program on the incompatibility of testers, initiated in this paper, will have an impact on the design of new quantum protocols. At a more fundamental level, the study of the dynamical properties of quantum causal networks is expected to shed light on foundational questions about time and causal structure in quantum theory.
§ ACKNOWLEDGMENTS
We thank the anonymous referees for stimulating us to make the paper more self contained and more readable for a wider group of readers. We thank also Anna Jenčová for stimulating discussions. We acknowledge support from the SRDA grant APVV-0808-12 (QETWORK),
VEGA Grant No. 2/0125/13 (QUICOST), from the the Foundational Questions Institute (FQXi-RFP3-1325), from the National Natural Science Foundation of China through Grants 11450110096 and 11350110207, from the 1000 Youth Fellowship Program of China, and from the HKU Seed Funding for Basic Research.
DR was supported via SASPRO Program No. 0055/01/01 (QWIN) and
MS acknowledges support by the European Social Fund and the state budget of the Czech Republic under Operational Program Education for Competitiveness (Project
No. CZ.1.07/2.3.00/30.0004) and by the Development Project
of Faculty of Science, Palacky University. MZ acknowledges
the support of Czech Science Foundation (GA ČR) project
No. GA16-22211S.
§ PROOF OF PROPOSITION <REF>
Given a physical implementation $\map T = (\spc H_{\rm anc}, \Psi, {\st P})$, the corresponding tester $\st T$ is given by Eq. (<ref>). Using this expression, we obtain
\begin{align}
\nonumber \sum_j T_j & = \sum_j \Tr_{\rm anc} [ ( P_j \otimes I_0 ) \, ( I_1 \otimes {\tt SWAP} \Psi^{T_0} {\tt SWAP} )] \\
& = \Tr_{\rm anc} [ I_1 \otimes \Psi^{T_0} ] = I_1 \otimes \rho ,
\end{align}
where $\rho: = \Tr_{\rm anc} [ \Psi^{T_0} ]$.
§ PROOF OF PROPOSITION <REF>
Let $ \map T = (\mathbb C, \Psi, {\st P})$ be an ancilla-free implementation. Then, Eq. (<ref>) gives
\begin{equation}
T_j = ( P_j \otimes I_0 ) \, ( I_1 \otimes \Psi^{T} )= P_j\otimes \Psi^T \, ,\quad \forall j .
\end{equation}
The normalization of the tester reads
\begin{equation} \sum_{j} T_j = \left( \sum_j P_j \right) \otimes \Psi^T = I_1 \otimes \Psi^T \, ,
\end{equation}
which, compared with the normalization condition (<ref>), implies $\rho = \Psi^T$.
Conversely, suppose that the tester operators are of the form $ T_j = P_j\otimes \rho$.
Setting $\Psi := \rho^T$, it is immediate to check that $ \map T = (\mathbb C, \Psi, {\st P})$ is an ancilla-free implementation.
§ FURTHER EXAMPLES OF TESTERS
§.§.§ Testers with classical ancilla
In the case of ancilla-free testers, there are no correlations between the state sent as input to the unknown process and the measurement performed on its output. For some applications, like quantum process tomography <cit.>, it is important to test the action of the process on multiple input states, keeping track of which state has been used as a probe.
This task can be accomplished by using a classical system as ancilla.
Schematically, we can consider a setup as in Figure <ref>, where the input state is prepared in a state $\rho_k$, correlated with a classical random variable, which assumes the value $k$ with probability $q_k$. Then, the output is measured with a POVM ${\st P}^{(k)} = \left\{ P^{(k)}_{j_k} \, , j_k \in \set S_k \right\}$, whose outcome set $\set S_k$ possibly depends on the index $k$. This dependence allows to keep track of the value of $k$, as different values of $k$ can correspond to disjoint sets of outcomes.
Diagrammatic representation of a tester with classical ancilla.
Note that the joint state of the input system and the classical random variable $k$ can be represented as the quantum-classical state
\begin{align}\label{qcstate}
\Psi = \sum_k \, q_k \, \rho_k \otimes |k\>\< k| \, ,
\end{align}
where $ \{ |k\> \}$ is an orthonormal set of states of a suitable ancilla. In this picture, a measurement that depends on $k$ is just a joint measurement on the system and the ancilla.
In fact, Eq. (<ref>) is not needed: if the state $\Psi$ is quantum-classical, then every measurement performed on the output will give the same statistics as some suitable quantum-classical POVM.
This observation motivates the following
We say that a tester $\st T$ can be implemented with a classical ancilla if there exists an implementation $\map T = (\spc H_{\rm anc} \, , \Psi, {\st P})$ such that the state $\Psi$ is quantum-classical.
Testers that can be implemented with a classical ancilla have the form ${\st T} = \{ T_{ j_k} \}$, with
\begin{align}\label{classicalanc}
T_{j_k} = \sum_k q_k\, P^{(k)}_{j_k} \otimes \rho_k \, .
\end{align}
Mathematically, the set of testers with classical ancilla is nothing but the convex hull of the set of ancilla-free testers.
This fact can be easily seen by comparing Eq. (<ref>) with Eq. (<ref>).
§.§.§ Testers with genuinely quantum ancilla
Some testers cannot be implemented with a classical ancilla. This is the case, for example, of testers containing non-separable operators — i.e. operators that cannot be written in the form
\[ T_j = \sum_k \, A_{jk} \otimes B_{jk} \]
with positive $A_{jk}$ and $B_{jk} $.
A concrete example arises when testing how well a process preserves the maximally entangled state $|\Phi_+\>$. To this purpose, we can prepare the input and the ancilla in the state $\Psi = |\Phi_+\>\< \Phi_+| $, apply the process $\map E$ on the first system, and then measure the output and the ancilla with two-outcome POVM $ {\st P} =\{ P_1,P_2\}$ with
\begin{equation}
P_1 = |\Phi_+\>\<\Phi_+| \, , \quad P_2 = I\otimes I -|\Phi_+\>\<\Phi_+| .
\end{equation}
This setup corresponds to the two-outcome tester ${\st T} = \{T_1, T_2\}$ with
\begin{equation}
T_1 = \frac 1 d \, |\Phi_+\>\<\Phi_+| \, , \quad T_2 = \frac 1 d \, \left( I\otimes I - |\Phi_+\>\<\Phi_+| \right) .
\end{equation}
This tester cannot be realized with a classical ancilla, because the operator $T_1$ does not have the separable form of Eq. (<ref>).
§ PROOF OF THEOREM <REF>
Suppose that ${\st A} = \{ A_j\} $ and ${\st B}= \{ B_k\}$ are compatible. Then, there exists a tester ${\st C}= \{ C_{jk}\}$ such that the compatibility condition (<ref>) holds. Note that Eq. (<ref>) implies that $\st A$, $\st B$, and $\st C$ have the same normalization state, call it $\rho$. Now, define the canonical POVMs associated with $\st A$, $\st B$, and $\st C$, namely the POVMs ${\st P} = \{ P_j\}$, ${\st Q} = \{ Q_k\}$, and ${\st R} = \{ R_{jk}\}$ with operators
\begin{align}
P_{j} &= \left(I\otimes \rho^{-\frac12}\right)\,A_{j}\, \left(I\otimes \rho^{-\frac12}\right), \nonumber\\
Q_{k}&= \left(I\otimes \rho^{-\frac12}\right)\,B_{k}\, \left(I\otimes \rho^{-\frac12}\right), \\
R_{jk}&= \left(I\otimes \rho^{-\frac12}\right)\,C_{jk}\, \left(I\otimes \rho^{-\frac12}\right) .\nonumber
\end{align}
Then, it is immediate to obtain the compatibility conditions for the canonical POVMs $\st P$ and $\st Q$. Indeed, one has
\begin{align}
\sum_{k} R_{jk}&= \left(I\otimes \rho^{-\frac12}\right) \left(\sum_k C_{jk}\right) \left(I\otimes \rho^{-\frac12}\right)\nonumber\\
& = \left(I\otimes \rho^{-\frac12}\right) A_j \left(I\otimes \rho^{-\frac12}\right) \\
& \equiv P_j \, ,\nonumber
\end{align}
and similarly,
\begin{align}
\sum_{j} R_{jk}&= \left(I\otimes \rho^{-\frac12}\right) \left(\sum_j C_{jk}\right) \left(I\otimes \rho^{-\frac12}\right)\nonumber\\
& = \left(I\otimes \rho^{-\frac12}\right) B_k \left(I\otimes \rho^{-\frac12}\right) \\
& \equiv Q_k \, . \nonumber
\end{align}
Conversely, if the normalization states are the same and the canonical POVMs $\st P$ and $\st Q$ are compatible, then one can use the joint POVM ${\st R}$ to define the tester ${\st C}$ with operators
\begin{equation}
C_{jk} := \left(I\otimes \rho^{\frac12}\right) R_{jk} \left(I\otimes \rho^{\frac12}\right) .
\end{equation}
By construction, $\st C$ is a joint tester for $\st A$ and $\st B$: indeed, one has
\begin{align}
\sum_{k} C_{jk}&= \left(I\otimes \rho^{\frac12}\right) \left( \sum_k R_{jk} \right)\left(I\otimes \rho^{\frac12}\right) \nonumber\\
& = \left(I\otimes \rho^{\frac12}\right) P_j \left(I\otimes \rho^{\frac12}\right)\\
&= A_j \, ,\nonumber
\end{align}
having used the fact that $\rho^{\frac 12}$ is invertible on its support. Similarly, one has
\begin{align}
\sum_{j} C_{jk}&= \left(I\otimes \rho^{\frac12}\right) \left( \sum_j R_{jk} \right)\left(I\otimes \rho^{\frac12}\right) \nonumber\\
& = \left(I\otimes \rho^{\frac12}\right) Q_k \left(I\otimes \rho^{\frac12}\right)\\
&= B_k \, .\nonumber
\end{align}
This concludes the proof.
§ PROOF OF PROPOSITION <REF>
Let $\rho$, $\sigma$, and $\tau$ be the normalization states of $ \st A$, $\st B$, and $\st N$, respectively. Then, the compatibility condition implies the relation
\[ (1-\lambda) \, \rho + \lambda \, \tau = (1-\lambda) \, \sigma + \lambda\, \tau \, , \]
(cf. Proposition <ref>). This relation can be satisfied only if $\lambda = 1$ .
§ PROOF OF PROPOSITION <REF>
Since $\st A$ and $\st B$ are diagonal in the same basis, also the operators
$I_1\otimes \rho$ and $I_1\otimes \sigma$ are diagonal in the same basis. As a result, also the canonical POVMs $\st P$ and $\st Q$, defined by
\begin{align}
P_j&= \left(I_1 \otimes \rho^{-\frac 12} \right) \, A_j \, \left(I_1 \otimes \rho^{-\frac 12} \right),\nonumber \\
Q_k &= \left(I_1 \otimes \sigma^{-\frac 12} \right) \, B_k \, \left(I_1 \otimes \sigma^{-\frac 12} \right) \, .
\end{align}
are diagonal in the same basis.
Now, let $(\lambda \, , \widetilde \rho \, ,\widetilde \sigma)$ be a triple satisfying Eq. (<ref>) and let $\omega$ be the state
\begin{align}
\label{eq:appnormt}
\omega & = (1-\lambda) \, \rho+\lambda \, \widetilde{\rho}\nonumber\\
&\equiv(1-\lambda) \, \sigma+\lambda \, \widetilde{\sigma} \, .
\end{align}
Without loss of generality, we can choose the operators $I_1 \otimes \widetilde \rho $, $I_1\otimes \widetilde \sigma$, and $I_1 \otimes \omega$ to be diagonal in the same basis,
since $\widetilde \rho $, $\widetilde \sigma$ can be made diagonal by taking only their diagonal elements, which preserves validity of Eq. (<ref>) and does not change $\lambda$.
Using this fact, we define the testers ${\st N}^{(\st A)}$ and ${ \st N}^{(\st B)}$ as
\begin{align}
N^{(\st A)}_j &: = \left(I_1 \otimes \widetilde \rho^{\frac 12} \right) \, \tilde P_j \, \, \left(I_1 \otimes \widetilde \rho^{\frac 12} \right),\nonumber \\
N^{(\st B)}_k &: = \left(I_1 \otimes \widetilde \sigma^{\frac 12} \right) \, \tilde Q_k \, \, \left(I_1 \otimes \widetilde \sigma^{\frac 12} \right) ,
\end{align}
where $\{\tilde P_j\}$, $\{\tilde Q_k\}$ are arbitrary POVMs diagonal in the common basis.
Note that also ${\st N}^{(\st A)}$ and ${ \st N}^{(\st B)}$ are diagonal in the same basis as $\st A$ and $\st B$. Now, by construction the testers $ (1-\lambda) {\st A} + \lambda {\st N}^{(\st A)}$ and $ (1-\lambda) {\st B} + \lambda {\st N}^{(\st B)}$ have the same normalization state, namely $\omega$. Moreover, their canonical POVMs $\overline P$ and $\overline Q$, defined by
\begin{align}
\overline P_j &: = \left(I_1 \otimes \omega^{-\frac 12} \right) \, \left[ (1-\lambda) {A}_j + \lambda {N}^{(\st A)}_j \right] \, \left(I_1 \otimes \omega^{-\frac 12} \right),\nonumber \\
\overline Q_k &: = \left(I_1 \otimes \omega^{-\frac 12} \right) \, \left[ (1-\lambda) {B}_k + \lambda {N}^{(\st B)}_k \right] \, \left(I_1 \otimes \omega^{-\frac 12} \right),
\end{align}
are also diagonal in the same basis. Hence, they can be jointly measured. Using Theorem <ref>, we conclude that the testers $\st A$ and $\st B$ are $\lambda$-compatible, whenever their normalization states $\rho$ and $\sigma$ are $\lambda$-compatible. Taking the minimum over $\lambda$ we finally obtain the desired result.
§ PROOF OF PROPOSITION <REF>
Eq. (<ref>) can be rewritten as
\begin{equation}
\widetilde\rho-\widetilde\sigma = \left(\frac1\lambda -1\right)\, ( \sigma - \rho),
\end{equation}
which implies that the operators $\widetilde \rho - \widetilde \sigma$ and $\sigma-\rho$ must be proportional to one another, with the proportionality constant
\begin{equation}
\frac1\lambda - 1 = \frac{\| \widetilde \sigma -\widetilde \rho \|}{\| \sigma - \rho\|}.
\end{equation}
Clearly, the minimum value of $\lambda$ is attained when the norm $\| \widetilde \rho -\widetilde \sigma \|$ is maximal, compatibly with the requirement that $\widetilde \rho - \widetilde \sigma$ and $\rho-\sigma$ be proportional.
We now show that one can always choose $\widetilde \rho$ and $\widetilde \sigma$ so that the norm has the maximum possible value, namely $\| \widetilde \rho -\widetilde \sigma\| = 2$.
To this purpose, we define the operator
\begin{equation}
\Delta:=\frac{\sigma - \rho}{\|\sigma - \rho\|} .
\end{equation}
Since $\Delta$ is self-adjoint, it can be decomposed as
\begin{equation}
\Delta=\Delta_+ - \Delta_- ,
\end{equation}
with $\Delta_+\ge 0$, $\Delta_- \ge 0$, and $\Delta_+ \Delta_- = 0$.
Moreover, $\Delta$ satisfies $\Tr[\Delta] = 0$ and $\| \Delta \| = 1$, which imply
\begin{align}
\tr [ \Delta_+ - \Delta_-] = 0 \qquad {\rm and} \qquad
\tr [ \Delta_+ + \Delta_-] = 1 \, ,
\end{align}
or equivalently, $\tr [\Delta_+]=\tr[ \Delta_-]=1/2$.
Hence, we can define the density operators
\begin{align}
\widetilde\rho:=2 \Delta_+ \qquad {\rm and} \qquad \widetilde\sigma:=2 \Delta_- ,
\end{align}
which satisfy
\begin{equation}
\widetilde \rho - \widetilde \sigma = 2 \Delta = \frac { 2 (\sigma - \rho) }{\| \sigma - \rho\|}.
\end{equation}
In other words, $ \widetilde \rho - \widetilde \sigma$ and $\sigma - \rho$ are proportional and the proportionality constant is
\begin{equation}
\frac2{\| \rho-\sigma\| } = \frac 1 {\Rs (\rho,\sigma)} -1.
\end{equation}
In conclusion, we obtained
\begin{equation}\Rs(\rho,\sigma) = \frac{ \| \rho-\sigma\|}{ \|\rho - \sigma\| + 2 }.
\end{equation}
The above proof has a nice geometric interpretation, highlighted in Figure <ref>.
To obtain common normalization $I\otimes \omega$ for the same $\lambda$, lines connecting $\rho$ with $\sigma$ and
$\widetilde\rho$ with $\widetilde\sigma$ must be parallel.
Geometrically, the convex combination $\omega = (1-\lambda)\rho+\lambda \widetilde{\rho}$ represents a point in the segment joining $\rho$ and $\widetilde \rho$. Measuring the length of segments with the trace distance
\begin{equation}
d(\rho, \sigma) = \| \rho - \sigma\|,
\end{equation}
one has
\begin{align}
\nonumber \lambda &= \frac{ d ( \rho, \omega)}{d (\rho , \widetilde \rho)} \\
\nonumber & = \frac{ d ( \rho, \omega)}{d (\rho , \omega ) + d (\omega, \widetilde \rho) } \\
\label{lambda} & = \frac{ 1 }{1 + \frac{d (\omega, \widetilde \rho)}{d(\omega,\rho) } } \, .
\end{align}
Now, the relation
\begin{equation}
(1-\lambda)\rho+\lambda \widetilde{\rho}=(1-\lambda)\sigma+\lambda \widetilde{\sigma}
\end{equation}
implies that
* the points $\rho,\sigma,\widetilde{\rho}$, and $\widetilde{\sigma}$ belong to the same plane ,
* the point $\omega$
is the intersection of the segment joining $\rho$ and $\widetilde \rho$ with the segment joining $\sigma$ and $\widetilde \sigma$, and
* the triangles with vertices $(\rho, \sigma, \omega)$ and $(\widetilde \rho, \widetilde \sigma, \omega)$ are similar.
Using the similarity of the triangles, Eq. (<ref>) becomes
\begin{align}
\lambda = \frac{ 1 }{1 + \frac{d (\widetilde \rho, \widetilde \sigma)}{d(\rho,\sigma) } } .
\end{align}
Now, the distance $d (\widetilde \rho, \widetilde \sigma)$ is maximized by choosing $\widetilde \rho$ and $\widetilde \sigma$ to be as far as possible,
but compatibly with the condition that $\widetilde \rho$ and $\widetilde \sigma$ must be states.
Hence, we have the bound
\begin{align}
\Rs (\rho,\sigma) \ge \frac{ 1 }{1 + \frac{2}{d(\rho,\sigma) } } \ge \frac{ \| \rho -\sigma \| }{\|\rho-\sigma\| +2 } ,
\end{align}
with the equality if and only if there exist states $\widetilde \rho$ and $\widetilde \sigma$ at distance $d(\widetilde \rho, \widetilde \sigma)=2$.
In qubit case the common normalization given by Bloch vector $\st x$ points at the intersection of lines connecting Bloch vectors $\st r$ with $\st{\widetilde r}$ and $\st s$ with $\st{\widetilde s}$, while the vectors $\st{r}-\st{s}$ and $\st{\widetilde r}-\st{\widetilde s}$ must be parallel.
In the qubit case, these states can be easily found by exploiting the geometry of the Bloch sphere.
Indeed, the similarity of the triangles with vertices $(\rho, \sigma, \omega)$ and $(\widetilde \rho, \widetilde \sigma, \omega)$ implies that the segment joining $\widetilde \rho$ with $\widetilde \sigma$ should be parallel to the segment joining $\rho$ with $\sigma$.
Hence, we can maximize the length $d (\widetilde \rho,\widetilde \sigma)$ by choosing $\widetilde \rho$ and $\widetilde \sigma$ to be the extreme points of the diameter parallel to the segment joining $\rho$ with $\sigma$, as in Figure <ref>.
In terms of the Bloch vectors $\st r, \st s, \st {\widetilde r}, \st {\widetilde s} $ associated with the states $\rho, \sigma, \widetilde \rho, \widetilde \sigma$, the condition that the segments are parallel reads
\begin{equation}
\frac{ \st {\widetilde r} - \st {\widetilde s} }{\| \st {\widetilde r} - \st {\widetilde s} \|} = \frac{ \st s-\st r}{\| \st r - \st s\|}.
\end{equation}
Choosing $ \st {\widetilde r}$ and $ \st {\widetilde s}$ on the diameter, the above condition yields
\begin{equation}
\st {\widetilde r} = \frac{ \st s-\st r}{\| \st r - \st s\|} \qquad {\rm and} \qquad \st {\widetilde s} = \frac{ \st r-\st s}{\| \st r - \st s\|}.
\end{equation}
For quantum systems of higher dimensions, the existence of orthogonal states $\widetilde \rho$ and $\widetilde \sigma$ does not follow directly from the geometric picture, but can be shown from the spectral properties of the density matrices, as we did in the proof at the beginning of this Appendix.
§ CONJECTURE ON THE MAXIMUM AMOUNT OF MEASUREMENT-INDUCED INCOMPATIBILITY
Currently, it is not known which pair of POVMs are the most incompatible, although a natural conjecture is that the maximum amount of incompatibility is attained by projective measurements on two mutually unbiased bases <cit.>. For example, one can pick the computational basis
\begin{align}\label{computational}
\set B_C = \{ |j\> \, , j = 0,\dots, d-1 \}
\end{align}
and the Fourier basis
\begin{align}\label{fourier}
\set B_F =\{ |e_k\> \, , k = 0, \dots, d-1\} \, ,
\end{align}
defined by
\begin{equation}
|e_k\> = \frac 1 {\sqrt d} \, \sum_{j=0}^{d-1} \, e^{\frac{ 2\pi i jk}d} \, |j\> \, .
\end{equation}
For the corresponding pair of projective POVMs, Haapasalo <cit.> has shown that
\begin{align}
\label{eq:defmu}
\Rm({\st P}, {\st Q}) = \frac{1}{2}\left(1-\frac{1}{\sqrt d }\right).
\end{align}
In the case of testers, we conjecture that the maximum of the measurement-induced incompatibility is reached when the canonical POVMs are measurements on two mutually unbiased bases for the input-output Hilbert space $\spc H = \spc H_1\otimes \spc H_0$. For example, we can choose the computational and Fourier bases in Eqs. (<ref>) and (<ref>), with $d := d_1 d_0$, and define the testers $\st A$ and $\st B$ with
\begin{align}
\label{eq:defPQpair}
A_j=\frac{1}{d_0} |j\>\< j| \qquad {\rm and} \qquad B_k=\frac{1}{d_0} |e_k\>\< e_k|\,,
\end{align}
and $j,k$ ranging from $0$ to $d-1$.
Both testers $\st A$ and $\st B$ have the normalization state $\rho = I_0/d_0$. Their canonical POVMs $\st P$ and $\st Q$ are given by
\begin{equation}
P_j = |j\>\< j| \qquad {\rm and} \qquad Q_k = |e_k\>\< e_k|,
\end{equation}
respectively. Combining Eqs. (<ref>) and (<ref>), we obtain the bound
\begin{equation}
\Rt({\st A}, {\st B}) \le \Rm( {\st P}, {\st Q}) = \frac{1}{2}\left(1-\frac{1}{\sqrt {d_0 d_1}}\right).
\end{equation}
We conjecture that the r.h.s. is the maximum amount of measurement-induced incompatibility that can be observed for a process with $d_0$-dimensional input space and $d_1$-dimensional output space.
§ PROOF OF PROPOSITION <REF>.
The proof consists of three steps:
Step 1. We show that, for pure normalization states, the testers $\st A$ and $\st B$ coincide with their canonical POVMs $\st P$ and $\st Q$, respectively. Denote the normalization state by $\rho = |\psi\>\<\psi|$.
By definition, one has
\begin{equation}
\sum_j A_j=\sum_k B_k=I_1\otimes |\psi\>\<\psi|.
\end{equation}
This condition implies that $A_j$ and $B_k$ have the product form
$A_j=a_j\otimes |\psi\>\<\psi|$ and $B_k=b_k \otimes |\psi\>\<\psi| $, where ${\st a} = \{a_j\}$ and ${\st b} = \{b_k\}$ are POVMs. Note that the canonical POVMs act on the Hilbert space $\spc H_0\otimes \Supp ( |\psi\>\<\psi|) \simeq \spc H_0$ and satisfy $\st P = \st A \simeq \st a$ and $\st Q = \st B \simeq \st b$.
Step 2. We show that, for the evaluation of the robustness of incompatibility, it is enough to restrict the attention to noise testers ${\st N}^{(\st A)}$ and ${\st N}^{(\st B)}$ with the same normalization state $\rho = |\psi\>\<\psi|$. Indeed, assume that the mixed testers
\begin{equation}
(1-\lambda)\, {\st A } + \lambda\, {\st N}^{(\st A)} \quad {\rm and} \quad (1-\lambda)\, {\st B } + \lambda\, {\st N}^{(\st B)}
\end{equation}
are compatible. Compatibility means that there exists a tester $\st C$ such that
\begin{align}
\nonumber \sum_k C_{jk} &=(1-\lambda) \, A_j+\lambda \, N^{(\st A)}_j,\\
\label{sandwich}
\sum_j C_{jk} &=(1-\lambda) \, B_k+\lambda \, N^{(\st B)}_k \, ,
\end{align}
for all $j$ and $k$.
Denoting by $\widetilde \rho$, $\widetilde \sigma$, and $\omega$ the normalization states of ${\st N}^{(\st A)}$, ${\st N}^{(\st B)}$, and $\st C$, respectively, the above compatibility relations imply
\begin{align} \omega & = (1-\lambda) \, |\psi\>\<\psi| + \lambda\, \widetilde \rho \nonumber\\
& = (1-\lambda) \, |\psi\>\<\psi| + \lambda\, \widetilde \sigma \, ,
\end{align}
and, of course,
\begin{equation}
\widetilde \rho = \widetilde \sigma.
\end{equation}
Now, define the testers $ \widetilde {\st N}^{(\st A)}$, $ \widetilde {\st N}^{(\st B)}$, and $\widetilde {\st C}$ with operators
\begin{align}
\widetilde N_j^{(\st A)} &:= \frac{(I_1\otimes |\psi\>\<\psi| ) \, N_j^{(\st A)} \, (I_1\otimes |\psi\>\<\psi| ) }{ \<\psi| \widetilde \rho| \psi\> } , \nonumber\\ \nonumber \\
\widetilde N_k^{(\st B)} & : = \frac{(I_1\otimes |\psi\>\<\psi| ) \, N_k^{(\st B)} \, (I_1\otimes |\psi\>\<\psi| ) }{ \<\psi| \widetilde \rho| \psi\> } ,
\\ \nonumber \\
\widetilde C_{jk} &: = \frac{(I_1\otimes |\psi\>\<\psi| ) \, C_{jk}\, (I_1\otimes |\psi\>\<\psi| ) }{ \<\psi| \omega| \psi\> } .\nonumber
\end{align}
By definition, all these testers have the same normalization state, equal to $\rho = |\psi\>\<\psi|$.
Moreover, pinching both sides of Eq. (<ref>) with the projector $(I_1\otimes |\psi\>\<\psi|)$ we obtain the relation
\begin{align}
\nonumber \sum_k \widetilde C_{jk} &= (1-\widetilde \lambda) \, A_j+ \widetilde \lambda \, \widetilde N^{(\st A)}_j,\\
\label{compatibility2}
\sum_j \widetilde C_{jk} &= (1-\widetilde \lambda) \, B_k+\widetilde \lambda \, \widetilde N^{(\st B)}_k ,
\end{align}
valid for every $j$ and $k$ with
\begin{align}
\widetilde \lambda &= \frac{ \<\psi | \widetilde\rho |\psi\>}{ \< \psi| \omega |\psi\> }\, \lambda .
\end{align}
Using the relation
\begin{align*}
\frac{ \<\psi | \widetilde\rho |\psi\>}{ \< \psi| \omega |\psi\> } = \frac{ \<\psi | \widetilde\rho |\psi\>}{ (1-\lambda) + \lambda \, \<\psi| \widetilde \rho |\psi\> }
\le 1,
\end{align*}
we then obtain
\begin{equation}
\widetilde \lambda \le \lambda.
\end{equation}
In conclusion, the search for the minimum $\lambda$ can be restricted without loss of generality to noise testers ${\st N}^{(\st A)}$ and ${\st N}^{(\st B)}$ of the form
\begin{align}
N_j^{(\st A)} &= n_j^{(\st A)} \otimes |\psi\>\<\psi| \nonumber \\
N_j^{(\st B)} &= n_j^{(\st B)} \otimes |\psi\>\<\psi|\,,
\end{align}
where ${\st n}^{(\st A)} = \left\{n_j^{(\st A)} \right \}$ and ${\st n}^{(\st B)} = \left\{n_j^{(\st B)} \right \}$ are suitable POVMs.
Step 3. Note that the $\lambda$-compatibility conditions (<ref>) are equivalent to the $\lambda$-compatibility of the canonical POVMs. Minimizing over $\lambda$, we then obtain the lower bound $\Rt({\st A}, {\st B})\ge \Rm({\st P},{\st Q})$. Combining this lower bound with the upper bound of Eq. (<ref>), we obtain the equality $\Rt({\st A}, {\st B}) = \Rm ({\st P},{\st Q})$.
§ PROOF OF PROPOSITION <REF>
Without loss of generality, let us assume $A_1\leq B_1$. Let $ \rho $ and $\sigma$ be the normalization states of the testers $\st A$ and $\st B$, respectively, and suppose that the relation
\begin{equation}
(1-\lambda) \, \rho + \lambda\, \widetilde \rho = (1-\lambda) \, \sigma + \lambda\, \widetilde \sigma,
\end{equation}
holds for suitable density operators $\widetilde \rho$ and $\widetilde \sigma$. Then
define the testers ${\st N}^{(\st A)}, {\st N}^{(\st B)}$ and $\st C$ with operators
\begin{align}
N_1^{(\st A)} & = 0, &
N_2^{(\st A)} &= I_1\otimes \widetilde \rho, \nonumber\\
N_1^{(\st B)} &= 0, &
N_2^{(\st B)} & = I_1\otimes \widetilde \sigma \, ,
\end{align}
\begin{align}
C_{12}&=0,\nonumber \\
C_{21}&=(1-\lambda)(B_1-A_1),\nonumber \\
C_{22}&=(1-\lambda)(I\otimes\sigma-B_1)+\lambda I\otimes\widetilde{\sigma} .
\end{align}
With the above definitions, one has
\begin{align}
\sum_{k} C_{jk}&= (1-\lambda) \, A_j+\lambda \,N^{(\st A)}_j\nonumber\,,\\
\sum_{j} C_{jk}&= (1-\lambda) \,B_k+\lambda \, N^{(\st B)}_k\,.
\end{align}
Hence, $\st A$ and $\st B$ are $\lambda$-compatible. Minimizing over $\lambda$, we obtain the upper bound $\Rt({\st A}, {\st B}) \le \Rs(\rho,\sigma)$. Combining this bound with the lower bound of Eq. (<ref>), we then have the equality $\Rt({\st A}, {\st B}) = \Rs(\rho,\sigma)$.
§ PROOF OF PROPOSITION <REF>
We will prove the Proposition <ref> by demonstrating a particular choice of testers ${\st N}^{(\st A)}, {\st N}^{(\st B)}$ and a joint tester $\st C$ satisfying all the requirements of the robustness of incompatibility for $\lambda=\Rt(\st A,\st B)=\Rs(P_{-\theta/2},P_{\theta/2})$ as specified in Eq. (<ref>). We further set
\begin{align}
\label{eq:defABtildaq}
N_1^{(\st A)}&=\Big[\frac{1-\delta}{2} P_{(\varphi+\pi)/2}+\frac{1+\delta}{2} P_{(\varphi-\pi)/2}\Big] \otimes P_{\pi/2}, \nonumber \\
N_2^{(\st A)}&=I \otimes \widetilde \rho - N_1^{(\st A)}, \nonumber \\
N_1^{(\st B)}&=\Big[\frac{1-\delta}{2} P_{-(\varphi+\pi)/2}+\frac{1+\delta}{2} P_{-(\varphi-\pi)/2}\Big] \otimes P_{-\pi/2}, \nonumber \\
N_2^{(\st B)}&=I \otimes \widetilde\sigma - N_1^{(\st B)},
\end{align}
\begin{align}
\label{eq:defdelta}
\delta&=-\,\frac{\sin \varphi}{2}\,\frac{1- \sin \frac{\theta}{2}}{ \sin \frac{\theta}{2}}\,.
\end{align}
Let us stress that the associated states
$\widetilde \rho=P_{\pi/2}$ and $\widetilde\sigma=P_{-\pi/2}$
are orthogonal as it is required in order to saturate the bound (<ref>).
By definition we can express the mixed state $\omega$ as
\begin{align}
\label{eq:sformomega}
\omega&=\frac{1}{2}\left[(1-\lambda)(\rho+\sigma)+\lambda (\widetilde{\rho}+\widetilde{\sigma})\right] \nonumber \\
&=\frac{1-\lambda}{2}(P_{-\theta/2}+P_{\theta/2})+\frac{\lambda}{2} I,
\end{align}
which will be convenient in subsequent calculations.
Using $\st{\bar A}$ and $\st{\bar B}$ for the mixed testers
we define the joint tester $\st C=\{C_{11},C_{12},C_{21},C_{22}\}$ as follows
\begin{align}
C_{11}&= C, & C_{12}&=\bar{A}_1-C, \\
C_{21}&= \bar{B}_1-C, & C_{22}&=I\otimes \omega +C -\bar{A}_1-\bar{B}_1, \nonumber
\end{align}
\begin{align}
\label{eq:defG}
C=&\frac{1-\lambda}{2} (\cos^2 \frac{\varphi}{2}+\sin^2 \frac{\varphi}{2}\sin \frac{\theta}{2})\Big[\ket{v_1}\bra{v_1}+\ket{v_2}\bra{v_2}\Big] \nonumber \\
&+\frac{1-\lambda}{2} \,\cos \frac{\varphi}{2}\cos \frac{\theta}{2}\,\Big[\ket{v_1}\bra{v_2}+\ket{v_2}\bra{v_1}\Big]
\end{align}
\begin{align}
\ket{v_1}&=\ket{\frac{\varphi}{2}}\ket{\frac{\pi}{2}}, \quad \quad \ket{v_2}=\ket{-\frac{\varphi}{2}}\ket{-\frac{\pi}{2}}, \nonumber\\
\ket{\beta}&=\cos \frac{\beta}{2}\ket{0} + \sin \frac{\beta}{2}\ket{1}.
\end{align}
To demonstrate $\lambda$-compatibility of $\st A$, $\st B$ it suffices to show (see Proposition <ref>) that
\begin{align}
\label{eq:ineqtoshow}
0\leq C &\leq \bar{A}_1, \bar{B}_1,\\
\bar{A}_1+\bar{B}_1 &\leq C+I\otimes\omega
\label{eq:ineqDG}
\end{align}
Since $\<v_1\ket{v_2}=0$, the nonzero eigenvalues of $C$ are the same as for the matrix
\begin{align}
\label{eq:matg}
\frac{1-\lambda}{2}
\left(\begin{array}{cc}
a & b \\
b & a
\end{array}
\right),
\end{align}
where $a=\cos^2 \frac{\varphi}{2}+\sin^2 \frac{\varphi}{2}\sin \frac{\theta}{2}$, $b= \cos \frac{\varphi}{2}\cos \frac{\theta}{2}$.
After some algebra the requirement of non-negativity of the eigenvalues leads to the definition of the region $\mathsf M$. Thus, in region $\mathsf M$ we proved $C\geq 0$.
Let us define
\begin{align}
\label{eq:defD}
&D\equiv \bar{A}_1+\bar{B}_1-I\otimes\omega, \nonumber\\
&Q\equiv P_{\varphi/2}\otimes P_{\pi/2}+P_{-\varphi/2}\otimes P_{-\pi/2}, \nonumber\\
&Q^\perp=I-Q, \nonumber\\
&S\equiv \sigma_X \otimes \sigma_Z,
\end{align}
where $\sigma_X,\sigma_Z$ are the Pauli matrices.
Then Eq. (<ref>) can be rewritten as $D\leq C$.
In the Appendix <ref> we prove
\begin{align}
\label{eq:D}
D&=QDQ+Q^\perp D Q^\perp=C-S C S,
\end{align}
which implies $D\leq C$, because $C-D=S C S\geq 0$ due to preservation of eigenvalues of $C\geq 0$ by unitary rotation $S$.
Thus, we proved Eq. (<ref>).
Next, we show that due to the symmetry of the problem
\begin{align}
\label{eq:GAGB}
C &\leq \bar{A}_1 \Leftrightarrow C \leq \bar{B}_1
\end{align}
For this purpose we define hermitian and unitary operator $T\equiv \sigma_Z \otimes \sigma_Z$, for which $T^2=I$.
It is easy to verify by direct calculation from Eqs. (<ref>), (<ref>), (<ref>) that
\begin{align}
\label{eq:GAGBproof1}
TCT=C \quad T \bar{A}_1 T= \bar{B}_1.
\end{align}
Since conjugation with $T$ is reversible and preserves eigenvalues we get
\begin{align}
\label{eq:GAGBproof2}
C\leq\bar{A}_1\;\Leftrightarrow\; TCT\leq T \bar{A}_1 T \;\Leftrightarrow \;C \leq \bar{B}_1,
\end{align}
where we used Eq. (<ref>).
In the following we prove $C\leq\bar{A}_1$ by demonstrating positivity of the matrix of $\bar{A}_1-C$ in the basis $\{\ket{v_1},\ket{v_2},\ket{v_3},\ket{v_4}\}$, where
\begin{align}
\ket{v_3}&=S\ket{v_2}=\ket{\frac{\varphi}{2}-\pi}\ket{\frac{\pi}{2}}, \nonumber \\
\ket{v_4}&=S\ket{v_1}=\ket{\pi-\frac{\varphi}{2}}\ket{-\frac{\pi}{2}}.
\end{align}
Direct calculation of the matrix elements yields
\begin{align}
\label{eq:A1Gmat}
\bar{A}_1-C&=
\left(\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & x & y & 0 \\
0 & y & z & 0 \\
0 & 0 & 0 & 0
\end{array}
\right),
\end{align}
\begin{align}
\label{eq:defxyz}
x&=\frac{1}{2}\left( 1- \frac{\cos^2 \frac{\varphi}{2}+\sin^2 \frac{\varphi}{2}\sin \frac{\theta}{2}}{1+\sin \frac{\theta}{2}}\right), \nonumber \\
y&= \frac{\sin \frac{\varphi}{2}\cos \frac{\theta}{2}}{2(1+\sin \frac{\theta}{2})}, \\
z&= \frac{1}{2}\left( 1- \frac{\sin^2 \frac{\varphi}{2}\sin \frac{\theta}{2}}{1+\sin \frac{\theta}{2}}\right) . \nonumber
\end{align}
Thus, it suffice to examine eigenvalues of matrix
\begin{align}
\label{eq:A1Gmsmall}
\left(\begin{array}{cc}
x & y \\
y & z
\end{array}
\right),
\end{align}
which can be analytically shown to be non-negative for all $\theta,\varphi\in[0,\pi]$.
In conclusion we proved validity of Eqs. (<ref>), (<ref>) for all $(\theta,\varphi)\in\mathsf M$ and thus demonstrated the existence of the joint tester $\st C$ needed for proving $\Rt(\st A,\st B)=\Rs(P_{-\theta/2},P_{\theta/2})$ claimed in the proposition.
§ DIAGONAL FORM OF THE OPERATOR $D$
Let us first explicitly write operator $D$
\begin{align}
\label{eq:Dexplicit}
D=&\frac{1-\lambda}{2}\Big[(P_{-\varphi/2}-P_{\pi-\varphi/2}) \otimes P_{-\theta/2} \nonumber \\
& \quad \quad\;\; +(P_{\varphi/2}-P_{\varphi/2-\pi})\otimes P_{\theta/2}\Big] \nonumber \\
&+\frac{\lambda\delta}{2}\Big[ (P_{(\pi-\varphi)/2}-P_{-(\pi+\varphi)/2}) \otimes P_{-\pi/2} \nonumber \\
& \quad \quad\;\; +(P_{(\varphi-\pi)/2}-P_{(\pi+\varphi)/2}) \otimes P_{\pi/2} \Big]\, ,
\end{align}
where we used (<ref>), (<ref>). It can be written more compactly as
\begin{align}
\label{eq:Dshort}
D=\frac{1-\lambda}{2}(H+THT)+ \frac{\lambda\delta}{2}(K+TKT),
\end{align}
\begin{align}
\label{eq:defHK}
H&=(P_{-\varphi/2}-P_{\pi-\varphi/2}) \otimes P_{-\theta/2}, \nonumber \\
K&=(P_{(\pi-\varphi)/2}-P_{-(\pi+\varphi)/2}) \otimes P_{-\pi/2},
\end{align}
and $T\equiv \sigma_Z \otimes \sigma_Z$ is a tensor product of Pauli matrices.
Our aim is to show that operator $D$ does not mix subspaces defined by projectors $Q,Q^\perp$, i.e.
\begin{align}
\label{eg:QDQp}
QDQ^\perp=Q^\perp DQ=0.
\end{align}
Thanks to hermicity of operator $D$ it suffices to show $QDQ^\perp=0$.
We observe that $TQT=Q$ and consequently $[Q,T]=0$. Similarly, $TQ^\perp T=Q^\perp$ implies $[Q^\perp,T]=0$. This means it is crucial to calculate operators $QHQ^\perp$, $QKQ^\perp$ and the remaining terms of $QDQ^\perp$ can be obtained by conjugation with $T$. For such calculation the following formula is useful
\begin{align}
\label{eg:Pproducts}
P_{\alpha} P_{\beta} P_{\gamma} = \ket{\alpha}\bra{\gamma}\;\cos\frac{\alpha-\beta}{2}\cos\frac{\beta-\gamma}{2}.
\end{align}
After a longer, but straightforward calculation one obtains
\begin{align}
\label{eq:HKresult1}
QHQ^\perp=&\frac{\sin \frac{\varphi}{2} \cos\frac{\theta}{2} }{2} \Big[\ket{-\frac{\varphi}{2}}\bra{\frac{\varphi}{2}-\pi}\otimes\ket{-\frac{\pi}{2}}\bra{\frac{\pi}{2}} \nonumber \\
&\quad\quad\quad\quad\quad\;\; - \ket{\frac{\varphi}{2}}\bra{\pi-\frac{\varphi}{2}}\otimes\ket{\frac{\pi}{2}}\bra{-\frac{\pi}{2}} \Big] \nonumber \\
&+\sin \varphi\frac{1-\sin\frac{\theta}{2}}{2} \ket{\frac{\varphi}{2}}\bra{\frac{\varphi}{2}-\pi}\otimes P_{\pi/2}, \nonumber \\
QKQ^\perp=&\ket{-\frac{\varphi}{2}}\bra{\pi-\frac{\varphi}{2}} \otimes P_{-\pi/2}.
\end{align}
Thanks to Eq. (<ref>) it is easy to evaluate
\begin{align}
\label{eq:HKresult2}
&QHQ^\perp+TQHQ^\perp T=\sin \varphi \; \frac{1-\sin\frac{\theta}{2}}{2}\; \times \nonumber \\
&\quad\;\;\times \Big[ \ket{\frac{\varphi}{2}}\bra{\frac{\varphi}{2}-\pi}\otimes P_{\pi/2} + \ket{-\frac{\varphi}{2}}\bra{\pi-\frac{\varphi}{2}}\otimes P_{-\pi/2}\Big], \nonumber \\
&QKQ^\perp+T QKQ^\perp T= \nonumber \\
&\quad\;\;= \Big[ \ket{\frac{\varphi}{2}}\bra{\frac{\varphi}{2}-\pi}\otimes P_{\pi/2} + \ket{-\frac{\varphi}{2}}\bra{\pi-\frac{\varphi}{2}}\otimes P_{-\pi/2}\Big],
\end{align}
where the terms with $\cos\frac{\theta}{2}$ effectively disappeared due to conjugation.
Finally, using Eqs. (<ref>), (<ref>) and definitions (<ref>), (<ref>) we get $QDQ^\perp=0$, because
\begin{align}
\frac{1-\lambda}{2}\sin \varphi \; \frac{1-\sin\frac{\theta}{2}}{2}\; +\frac{\lambda\delta}{2}=0.
\end{align}
This allows us to write
\begin{align}
\label{eq:fineqHK}
D=(Q+Q^\perp)D(Q+Q^\perp)=QDQ+ Q^\perp D Q^\perp.
\end{align}
Let us calculate $QDQ$.
Direct calculation using (<ref>) shows that
\begin{align}
\label{eq:QKQ}
QKQ&=\Big[P_{-\varphi/2}\left(P_{(\pi-\varphi)/2}-P_{-(\pi+\varphi)/2}\right)P_{-\varphi/2}\Big]\otimes P_{-\pi/2} \nonumber \\
&= \Big[\frac{1}{2} P_{-\varphi/2} - \frac{1}{2} P_{-\varphi/2} \Big]\otimes P_{-\pi/2}=0
\end{align}
\begin{align}
\label{eq:QHQ}
QHQ=&\frac{1-\sin\frac{\theta}{2}}{2}\cos \varphi \ket{v_1}\bra{v_1} +
\frac{1+\sin\frac{\theta}{2}}{2} \ket{v_2}\bra{v_2} \nonumber \\
&+\frac{1}{2}\cos\frac{\theta}{2}\cos\frac{\varphi}{2} \left(\ket{v_1}\bra{v_2}+\ket{v_2}\bra{v_1}\right).
\end{align}
Thanks to Eqs. (<ref>), (<ref>), (<ref>) and the fact that $T\ket{v_1}=\ket{v_2}$ we obtain
\begin{align}
\label{eq:QDQfin}
QDQ=\frac{1-\lambda}{2}(QHQ+TQHQT)=C\, .
\end{align}
The last thing we have to show is $Q^\perp D Q^\perp=-SCS$.
Let us note the following identities
\begin{align}
\label{eq:QpDQpstart}
SHS&=-THT, \qquad ST=-TS, \nonumber \\
SKS&=-TKT, \qquad SQS=Q^\perp ,
\end{align}
which from Eqs. (<ref>), (<ref>) imply $SDS=-D$ and
\begin{align}
\label{eq:QpDQp}
&Q^\perp D Q^\perp=SQSDSQS=-SQDQS=-SCS .
\end{align}
Combining equations (<ref>), (<ref>), (<ref>) we obtain Eq. (<ref>) we wanted to prove.
§ PROOF OF PROPOSITION <REF>
Since $E_2=I-E_1$ and $F_2=I-F_1$, we can parametrize both POVMs by spectral decompositions of the effects $E_1$, $F_1$
\begin{align}
\label{eq:defEFPOVM}
E_1&=e_1 \ket{u_1}\bra{u_1}+e_2 \ket{u_2}\bra{u_2}\nonumber \\
F_1&=f_1 \ket{w_1}\bra{w_1}+f_2 \ket{w_2}\bra{w_2},
\end{align}
where $e_i,f_j\in[0,1]$ and $\{\ket{u_1},\ket{u_2}\}$, $\{\ket{w_1},\ket{w_2}\}$ are two orthonormal qubit bases.
Effects $E_1$, $F_1$ as well as the corresponding POVMs can be convexly decomposed into four projective measurements (extremal POVMs),
\begin{align}
\label{eq:defEFconv}
E_1&=\sum_{a=1}^4 c_a E^a_1, \quad \quad F_1=\sum_{b=1}^4 d_b F^b_1,
\end{align}
\begin{align}
\label{eq:defEiFj}
E^1_1&=0, \quad &F^1_1&=0, \nonumber \\
E^2_1&=\ket{u_1}\bra{u_1}, \quad &F^2_1&=\ket{w_1}\bra{w_1}, \nonumber \\
E^3_1&=\ket{u_2}\bra{u_2}, \quad &F^3_1&=\ket{w_2}\bra{w_2}, \\
E^4_1&=I, \quad &F^4_1&=I. \nonumber
\end{align}
The decomposition in Eq. (<ref>) is unique and such that
\begin{align}
\label{eq:sumconvcoef}
\sum_{a=1}^4c_a&=1, \quad &\sum_{b=1}^4d_b&=1.
\end{align}
The two outcome POVMs defined by effects $E^1_1$, $E^4_1$, $F^1_1$, $F^4_1$ are trivial, i.e. their outcomes can be generated without actually measuring the quantum state.
The first pair and the second pair of POVMs defined by effects $E^2_1$, $E^3_1$, $F^2_1$, $F^3_1$ are related by relabeling of outcomes (e.g. $E^2_1=E^3_2$, $E^2_2=E^3_1$).
We define $1$-testers
\begin{align}
{\st A}^a&=\{A^a_1,A^a_2\},&\st B^b&=\{B^b_1, B^b_2\}, \nonumber \\
A^a_k&=E^a_k\otimes P_{-\theta/2}, &B^b_k&=F^b_k\otimes P_{\theta/2} ,
\end{align}
where $k=1,2$ and $a,b=1,2,3,4$. Showing that all pairs of testers $\st A^a$ and $\st B^b$ (for all $a$, $b$) are $\lambda$-compatible will be later used to show compatibility of 1-testers $\st A$ and $\st B$.
Firstly, the outcomes of trivial POVMs can be generated without measuring the state, and so it is clear that those $1$-testers defined above that contain trivial POVM will be $\lambda$-compatible with $\lambda$ obeying Proposition <ref> with any other product $1$-tester $\st B=\{F_1\otimes \sigma,F_2\otimes \sigma\}$.
For example, for $A^4_1=I\otimes \rho$
it would suffices to choose the joint and the admixed testers as
$\widetilde{A}^4_1=I\otimes\widetilde\rho$, $\widetilde{B}_1=F_1\otimes \widetilde\sigma$ and
\begin{align}
C_{11}&=(1-\lambda)B_1 +\lambda \widetilde{B}_1, &C_{12}&=(1-\lambda)B_2 +\lambda \widetilde{B}_2, \nonumber \\
C_{21}&=0, &C_{22}&=0 .
\end{align}
$C_{11}+C_{21}=\bar{B}_1$, $C_{12}+C_{22}=\bar{B}_2$ and
\begin{align}
C_{21}+C_{22}&=0=\bar{A}^4_2, \nonumber \\
C_{11}+C_{12}&=(1-\lambda)I\otimes\sigma + \lambda I\otimes \widetilde{\sigma} \nonumber \\
&=(1-\lambda)I\otimes\rho + \lambda I\otimes \widetilde{\rho}= \bar{A}^4_1.
\end{align}
Thus, for $\theta \geq 2 \arcsin (1/3)\approx 0.6797$ $1$-testers $\st A^a$, $\st B^b$ are $\lambda$-compatible, because either one of them contains trivial POVM or the pair is unitarily equivalent to $1$-testers in Proposition <ref>.
Now it suffices to show that $\lambda$-compatibility of $1$-testers $\st A^a$, $\st B^b$ for all $a,b$ implies $\lambda$-compatibility of $1$-testers $\st A$, $\st B$ from the proposition <ref>.
This can be done as follows.
The fact that for $\theta \geq 2 \arcsin (1/3)$ $1$-testers $\tA^a$, $\tB^b$ are $\lambda$-compatible can be expressed using Proposition <ref> by existence of operators $C^{ab}$ satisfying
\begin{align}
\label{eq:ineqGABab}
0\leq C^{ab} &\leq \bar{A}^{a}_1, \bar{B}^b_1,\\
\label{eq:ineqGABab2}
\bar{A}^{a}_1+\bar{B}^b_1 &\leq C^{ab}+I\otimes\omega.
\end{align}
Let us note that $\forall a,b$ $A^{a}_1+A^{a}_2=I\otimes P_{-\theta/2}$, $B^{b}_1+B^{b}_2=I\otimes P_{\theta/2}$ and since $\lambda$ is given by the lower bound (<ref>) also
$\forall a,b$ $\widetilde A^{a}_1+\widetilde A^{a}_2=I\otimes P_{\pi/2}$, $B^{b}_1+B^{b}_2=I\otimes P_{-\pi/2}$ and as a consequence the normalization of the joint tester $\omega$ is the same $\forall a,b$.
To prove $\lambda$-compatibility of $\st A$, $\st B$ we define the admixed $1$-testers $\st{\widetilde A}$, $\st{\widetilde B}$ and the joint $1$-tester $\st{C}$
\begin{align}
\widetilde A_1&=\sum_{a=1}^4 c_a \widetilde A^{a}_1, &\widetilde B_1&=\sum_{b=1}^4 d_b \widetilde B^{b}_1, \nonumber\\
C&=\sum_{a,b=1}^4 c_a d_b C^{ab}.
\end{align}
Let us remind that $A_1=\sum_{a=1}^4 c_a A^{a}_1$, $B_1=\sum_{b=1}^4 d_b B^{b}_1$, which implies
\begin{align}
\bar{A}_1=\sum_{a=1}^4 c_a \bar{A}^{a}_1, \quad &\bar{B}_1=\sum_{b=1}^4 d_b \bar{B}^{b}_1.
\end{align}
Due to $C^{ab}\geq 0$, $c_a,d_b\geq0$ we conclude $C\geq 0$, because $C$ is a nonnegative sum of positive-semidefinite operators.
We also easily get
\begin{align}
C&=\sum_{a,b=1}^4 c_a d_b C^{ab}\leq \sum_{a,b=1}^4 c_a d_b \bar{A}^{a}_1= \bar{A}_1, \nonumber \\
C&=\sum_{a,b=1}^4 c_a d_b C^{ab}\leq \sum_{a,b=1}^4 c_a d_b \bar{B}^{b}_1= \bar{B}_1,
\end{align}
where we used (<ref>) and (<ref>).
Finally, we use Eq. (<ref>) to write
\begin{align}
\bar{A}_1+\bar{B}_1&=\sum_{a,b=1}^4 c_a d_b ( \bar{A}^{a}_1+\bar{B}^{b}_1 )\leq I\otimes \omega + C ,
\end{align}
which concludes the proof.
§ SDP FOR $\LAMBDA$-COMPATIBILITY
Proposition <ref> can be used to construct SDP for solving $\lambda$-compatibility of two two-outcome testers $\st A=\{A_1,A_2\}$ and $\st B=\{B_1,B_2\}$ such that $A_1+A_2=I\otimes\rho$ and $B_1+B_2=I\otimes\sigma$. According to Definition <ref> in $\lambda$-compatibility we search for the smallest $\lambda$ such that the testers $(1-\lambda)\st A+\lambda\st{\widetilde A}$ and $(1-\lambda)\st B+\lambda\st{\widetilde B}$ are compatible for some $\st{\widetilde A}$ and $\st{\widetilde B}$.
First of all, the necessary condition $I \otimes \bar \rho = I\otimes \bar \sigma$ from Eq. (<ref>) for the normalizations $\bar \rho=(1-\lambda)\rho+\lambda \widetilde\rho$ and $\bar \sigma = (1-\lambda)\sigma + \lambda\widetilde\sigma$ needs to be satisfied. Thus,
in addition to search over operators $C$, we have to expand the search also over the mixed-in elements $\widetilde A_1$, $\widetilde B_1$ and their normalizations $\widetilde{\rho}$, $\widetilde{\sigma}$. Defining
\begin{align}
\bar A_i &=(1-\lambda)A_i+\lambda \widetilde A_i,\nonumber\\
\bar B_j &=(1-\lambda)A_j+\lambda \widetilde B_j,\\
\omega &=(1-\lambda)\rho+\lambda\widetilde\rho,\nonumber\\
&= (1-\lambda)\sigma+\lambda\widetilde\sigma,\nonumber
\end{align}
the problem can be recast as the following bilinear SDP
\begin{align}\begin{array}{ll}
\text{Find} & \lambda_0:=\inf\lambda\\
\text{subject to } &0 \leq \widetilde A_1, \widetilde B_1, \widetilde\rho, \widetilde\sigma, C\\
&C \leq \bar A_1,\bar B_1\\
&\bar A_1+\bar B_1\leq C+I\otimes\omega,\\
&\widetilde A_1\leq I\otimes\widetilde\rho,\widetilde B_1\leq I\otimes\widetilde\sigma,\\
\end{array}
\end{align}
where the last condition comes from the common normalization to $\omega$. We can linearize the program by rescaling relevant operators by $1/\lambda$. Using the definition of $\omega$ and setting $\mu=(1-\lambda)/\lambda$, $H=\frac{1}{\lambda}C$ the SDP program can be equivalently stated as
\begin{align}
\begin{array}{ll}
\text{Find} &\mu_0:=\sup\mu\\
\text{subject to}
&0\leq\widetilde A_1, \widetilde B_1, \widetilde\rho, \widetilde\sigma, H,\\
&H\leq\mu A_1+\widetilde A_1,\\
&H\leq\mu B_1+\widetilde B_1,\\
&\mu(A_1+B_1-I\otimes\sigma)+\widetilde A_1+\widetilde B_1\leq H+I\otimes\widetilde\sigma,\\
&\widetilde A_1\leq I\otimes\widetilde\rho,\widetilde B_1\leq I\otimes\widetilde\sigma,\\
\end{array}
\end{align}
where the unknown objects are $\mu, H, \widetilde A_1, \widetilde B_1, \widetilde\rho, \widetilde\sigma$.
Then the minimal $\lambda$ is determined as
\begin{equation}
\lambda_0=\frac{1}{1+\mu_0}.
\end{equation}
W. Heisenberg,
Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,
Zeitschrift für Physik 43, 172–-198 (1927).
E. Arthurs, J.L. Kelly,
On the Simultaneous Measurement of a Pair of Conjugate Observables,
Bell Syst. Tech. J. 44, 725 (1965).
P.J. Coles, J. Kaniewski, S. Wehner,
Equivalence of wave–particle duality to entropic uncertainty,
Nature Communications 5, 581 (2014).
B.-G. Englert,
Fringe Visibility and Which-Way Information: An Inequality,
Phys. Rev. Lett. 77, 2154 (1996).
W. Wootters, W. Zurek,
A Single Quantum Cannot be Cloned,
Nature 299, 802–803 (1982).
D. Dieks,
Communication by EPR devices,
Physics Letters A 92, 271-272 (1982).
V. Bužek, M. Hillery,
Quantum Copying: Beyond the No-Cloning Theorem,
Phys. Rev. A 54, 1844 (1996).
N. Gisin, S. Massar,
Optimal quantum cloning machines,
Phys. Rev. Lett. 79, 2153 (1997).
R. Werner,
Optimal cloning of pure states,
Phys. Rev. A 58, 1827 (1998).
H.F. Hofmann,
How weak values emerge in joint measurements on cloned quantum systems,
Phys. Rev. Lett. 109, 020408 (2012).
M.M. Wolf, D. Perez-Garcia, C. Fernandez,
Measurements incompatible in Quantum Theory cannot be measured jointly in any other local theory,
Phys. Rev. Lett. 103, 230402 (2009).
J. Barrett, L. Hardy, A. Kent,
No signalling and quantum key distribution,
Phys. Rev. Lett. 95, 010503 (2005).
A. Acín, N. Gisin, L. Masanes,
From Bell’s Theorem to Secure Quantum Key Distribution,
Phys. Rev. Lett. 97, 120405 (2006).
M.T. Quintino, T. Vértesi, N. Brunner,
Joint Measurability, Einstein-Podolsky-Rosen Steering, and Bell Nonlocality,
Phys. Rev. Lett. 113, 160402 (2014).
R. Uola, T. Moroder, O. Gühne,
Joint measurability of generalized measurements implies classicality,
Phys. Rev. Lett. 113, 160403 (2014).
S.J. Jones, H.M. Wiseman, A.C. Doherty,
Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering,
Phys. Rev. A 76, 052116 (2007).
V. Handchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R.F. Werner, R. Schnabel,
Observation of one-way Einstein-Podolsky–Rosen steering,
Nat. Photonics 6, 596-–9 (2012).
P. Skrzypczyk, M. Navascués, D. Cavalcanti,
Quantifying Einstein–Podolsky–Rosen steering,
Phys. Rev. Lett. 112, 180404 (2014).
P. Skrzypczyk, D. Cavalcanti,
Loss-tolerant Einstein-Podolsky–Rosen steering for arbitrary-dimensional states: Joint measurability and unbounded violations under losses,
Phys. Rev. A 92, 022354 (2015).
M. Piani, J. Watrous,
Necessary and sufficient quantum information characterization of Einstein–Podolsky–Rosen steering,
Phys. Rev. Lett. 114, 060404 (2015).
G. Ludwig,
Foundations of Quantum Mechanics I
(Berlin: Springer, 1983).
P. Busch,
Unsharp reality and joint measurements for spin observables,
Phys. Rev. D 33, 2253-–2261 (1986).
P. Lahti, S. Pulmannová,
Coexistent observables and effects in quantum mechanics,
Rep. Math. Phys. 39, 339-–351 (1997).
P. Lahti,
Coexistence and joint measurability in quantum mechanics,
Int. J. Theor. Phys. 42, 893-–906 (2003).
P. Stano, D. Reitzner, T. Heinosaari,
Coexistence of qubit effects,
Phys. Rev. A 78, 012315 (2008).
P. Busch, H.-J. Schmidt,
Coexistence of qubit effects,
Quant. Info. Proc. 9, 143 (2010).
S. Yu, N.-L. Liu, L. Li, C.H. Oh,
Joint measurement of two unsharp observables of a qubit,
Phys. Rev. A 81, 062116 (2010).
T. Heinosaari, J. Kiukas, D. Reitzner,
Coexistence of effects from an algebra of two projections,
J. Phys. A: Math. Theor. 47, 225301 (2014).
R. Kunjwal, C. Heunen, T. Fritz,
All joint measurability structures are quantum realizable,
Phys. Rev. A 89, 052126 (2014).
T. Heinosaari, J. Kiukas, D. Reitzner,
Noise robustness of the incompatibility of quantum measurements,
Phys. Rev. A 92, 022115 (2015).
T. Heinosaari, T. Miyadera, M. Ziman,
An Invitation to Quantum Incompatibility,
Journal of Physics A 49, 123001 (2016).
G. Chiribella, G.M. D'Ariano, P. Perinotti,
Memory Effects in Quantum Channel Discrimination,
Phys. Rev. Lett. 101, 180501 (2008).
M. Ziman,
Process POVM: A mathematical framework for the description of process tomography experiments,
Phys. Rev. A 77, 062112 (2008).
Y. Aharonov, S. Popescu, J. Tollaksen, L. Vaidman,
Multiple-time states and multiple-time measurements in quantum mechanics,
Phys. Rev. A 79, 052110 (2009).
G. Chiribella, G.M. D'Ariano, P. Perinotti,
Theoretical framework for quantum networks,
Phys. Rev. A 80, 022339 (2009).
N. Bohr,
The Quantum Postulate and the Recent Development of Atomic Theory,
Nature 121, 580-–590 (1928).
G. Gutoski, J. Watrous,
Towards a general theory of quantum games,
in Proceedings of the thirty-ninth annual ACM Symposium on Theory of computing, pp. 565–574 (2007).
A.S. Holevo,
Probabilistic and statistical aspects of quantum theory
(North-Holland Series in Statistics and Probability, Vol. 1, North-Holland Publishing Company, Amsterdam, 1982)
T. Heinosaari, M. Ziman,
The mathematical language of quantum theory: From uncertainty to entanglement
(Cambridge University Press, Cambridge, UK, 2012).
G. Chiribella, G.M. D'Ariano, P. Perinotti,
Transforming quantum operations: quantum supermaps,
Europhysics Letters 83, 30004 (2008).
M.-D. Choi, Linear Algebra Appl. 10, 285 (1975).
P. Busch, T. Heinosaari, J. Schultz, N. Stevens,
Comparing the degrees of incompatibility inherent in probabilistic physical theories,
Europhysics Letters 103, 10002 (2013).
E. Haapasalo,
Robustness of incompatibility for quantum devices,
J. Phys. A 48, 255303 (2015).
T. Heinosaari, J. Schultz, A. Toigo, M. Ziman,
Maximally incompatible quantum observables,
Phys. Lett. A 378, 1695–1699 (2014).
H. Yuen, R. Kennedy, M. Lax,
Optimum testing of multiple hypotheses in quantum detection theory,
IEEE Transactions on Information Theory 21, 125 (1975).
C.W. Helstrom,
Quantum Detection and Estimation Theory
(Academic Press, 1976).
T. Heinosaari,
A simple sufficient condition for the coexistence of quantum effects,
J. Phys. A: Math. Theor. 46, 152002 (2013).
G. Chiribella, G.M. D'Ariano, P. Perinotti,
Quantum Circuits Architecture,
Phys, Rev. Lett. 101, 180501 (2008).
RiWoMo2015 M. Ringbauer, C. J. Wood, K. Modi, A. Gilchrist, A.G. White, A. Fedrizzi,
Characterizing Quantum Dynamics with Initial System-Environment Correlations,
Phys. Rev. Lett. 114, 090402 (2015).
G. Chiribella,
Optimal networks for quantum metrology: semidefinite programs and product rules,
New Journal of Physics 14, 125008 (2012).
G. Gutoski,
On a measure of distance for quantum strategies,
J. Math. Phys. 53, 032202 (2012).
M. A. Nielsen and I. L. Chuang,
Quantum Computation and Quantum Information
(Cambridge University Press, Cambridge, UK, 2000).
T. Durt, B.-G. Englert, I. Bengtsson, K. Zyczkowski,
On mutually unbiased bases,
Int. J. Quantum Information 8, 535–640 (2010).
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firstpage–lastpage 2015
The Local Group of galaxies offer some of the most discriminating
tests of models of cosmic structure formation. For example,
observations of the Milky Way (MW) and Andromeda satellite
populations appear to be in disagreement with N-body simulations of
the “Lambda Cold Dark Matter” ($\Lambda$CDM) model: there are far
fewer satellite galaxies than substructures in cold dark matter
halos (the “missing satellites” problem); dwarf galaxies seem to
avoid the most massive substructures (the “too-big-to-fail”
problem); and the brightest satellites appear to orbit their host
galaxies on a thin plane (the “planes of satellites”
problem). Here we present results from Apostle (A Project Of
Simulating The Local Environment), a suite of cosmological
hydrodynamic simulations of twelve volumes selected to match the
kinematics of the Local Group (LG) members. Applying the code
to the LG environment, we find that our simulations match the
observed abundance of LG galaxies, including the satellite galaxies
of the MW and Andromeda. Due to changes to the structure of halos
and the evolution in the LG environment, the simulations reproduce
the observed relation between stellar mass and velocity dispersion
of individual dwarf spheroidal galaxies without necessitating the
formation of cores in their dark matter profiles. Satellite systems
form with a range of spatial anisotropies, including one similar to
that of the MW, confirming that such a configuration is not
unexpected in $\Lambda$CDM. Finally, based on the observed velocity
dispersion, size, and stellar mass, we provide new estimates of the
maximum circular velocity for the halos of nine MW dwarf
cosmology: theory – galaxies: formation – galaxies: evolution –
galaxies: mass functions – methods: N-body simulations
§ INTRODUCTION
The ability of the
cold dark matter (CDM) model to predict observables on different
scales and at different epochs lies at the root of its remarkable
success. Predictions for the anisotropy of the microwave background
radiation <cit.>, and the large scale distribution of
galaxies <cit.>, were made soon after the model was
formulated, and have since been spectacularly validated by
observations <cit.>. However, observations on scales
currently testable only within the Local Group (LG) have yielded
results that appear to be in conflict with CDM predictions.
The “missing satellites” problem <cit.>
refers to the apparent paucity of luminous satellite galaxies compared
to the large number of dark matter substructures predicted in
$\Lambda$CDM[Throughout this paper, we use $\Lambda$CDM to
refer to the $\Lambda$ Cold Dark Matter model, where $\Lambda$
refers to the cosmological constant model for dark energy, which is
constrained primarily by observations outside the Local Group. We
also assume that the dark matter is collisionless.] . That the
number of observed dwarf galaxies does not directly mirror the number
of substructures is perhaps no surprise: it has long been predicted
that processes such as supernova feedback <cit.>,
and cosmic reionisation <cit.> should reduce
the star formation efficiency in low mass halos, and even prevent the
smallest ones from forming stars altogether.
The potential of these processes to bring the stellar mass function in
$\Lambda$CDM into agreement with observations has already been
demonstrated using semi-analytical models <cit.>. However, these do not
investigate whether the resulting satellites would also match the
observed kinematics, or indeed predict ratios between stellar mass and
maximum circular velocity, $\vm$, that are far lower than
observed. Extending the success of $\Lambda$CDM from the overall
galaxy population down to the number of observed Local Group dwarf
galaxies through direct simulations has remained a challenge.
Furthermore, the apparent excess of substructures in $\Lambda$CDM is
not just limited to the lowest mass scales: simulations also predict
the presence of subhalos so massive that they should not be affected
by reionisation (and hence deemed “too big to fail”,
<cit.>), but whose internal structure seems
incompatible with that of the brightest observed satellites. While the
two main galaxies within the LG both have several satellite galaxies
whose rotation curves are consistent with massive subhalos, their
number falls short of the number of such substructures predicted to
surround Milky Way (MW) or M31 mass halos in $\Lambda$CDM.
Finally, it has long been known that most of the eleven brightest, so
called “classical” MW satellites, appear to orbit the Galaxy on a
thin plane, and a similar (but distinct) plane has subsequently also
been identified among satellites of Andromeda. While $\Lambda$CDM
satellite systems are known to be anisotropic, the satellite systems
of the MW and Andromeda are purported to be extremely unusual
The problems, at times reported as fatal for $\Lambda$CDM, arise when
observations are confronted with predictions from dark matter only
(DMO) simulations that treat the cosmic matter content as a single
collisionless fluid, a poor approximation on scales where baryonic
processes are important. It has, of course, long been recognised that
the distribution of light is not a precise tracer of dark matter, but
simple models for populating dark matter structures with galaxies do
not capture the complexity of galaxy formation physics. On the other
hand, hydrodynamic simulations have confirmed the importance of
baryonic effects, but have either focussed on individual dwarf
galaxies, ignoring the LG environment <cit.>, or have not yet reproduced the observed LG galaxy
population <cit.>. The lack of a single
model able to reconcile all of the LG observations with $\Lambda$CDM
predictions has led to renewed interest in alternatives to CDM, such
as warm <cit.> or self-interacting dark matter
Here we test the $\Lambda$CDM model with a new suite of cosmological
hydrodynamic simulations, with initial conditions tailored to match
the LG environment. In particular, we focus on pairs of halos that
match the separation, approach velocity, and relative tangential
velocity of the Milky Way (MW) and Andromeda (M31). From a large
cosmological simulation, we have selected twelve pairs of halos with
combined virial masses of $\sim2.3\pm0.6\times 10^{12} \Ms$,
compatible with the most recent dynamical constraints
<cit.>. The selection and set-up of
our initial conditions are described in more detail by
We have re-simulated each LG volume at several resolutions, both as
dark matter only (DMO) simulations, and as hydrodynamic simulations,
with the code developed for the Evolution and Assembly of
GaLaxies and their Environments project (Eagle,
In this paper, we give an introduction to the simulations, and compare
our results to the observed LG galaxy population. We show that the
abundance of galaxies in the LG can be reproduced within $\Lambda$CDM
using a galaxy formation model calibrated on much larger scales. By
comparing the DMO and hydrodynamic simulations, we show that the
apparent discrepancies between observations and $\Lambda$CDM
predictions can be resolved once baryonic effects are included.
This paper is organised as follows. In Section <ref>, we
give a brief overview of the code
(Section <ref>), followed by a description of the
Apostle volumes (Section <ref>). In
Section <ref>, we present our key results, and
implications for the questions outlined above: the stellar mass
function (Section <ref>), and the satellites “too big
to fail” (Section <ref>). We demonstrate that our
simulations also produce the right satellite galaxies in the right
satellite halos (Section <ref>), and provide estimates for
$\vm$ of observed satellites based on the measured velocity dispersion
and stellar mass (Section <ref>). We revisit the apparent
kinematical and spatial alignment of satellites in
Section <ref> and conclude with a summary in
Section <ref>.
§ METHODS
§.§ The EAGLE Galaxy Formation Model
The simulations presented in this paper were performed with the code
developed for the Evolution and Assembly of GaLaxies and their
Environments (Eagle, <cit.>)
project. The code is a substantially modified version of P-Gadget-3, which itself is an improved version of the publicly
available Gadget-2 code <cit.>. Gravitational
accelerations are computed using the standard Tree-PM scheme of P-Gadget-3, while hydrodynamic forces are computed in the smoothed
particle hydrodynamics (SPH) scheme of Anarchy described in
Dalla-Vecchia et al. (in prep.) and <cit.> , which uses
the pressure-entropy formalism introduced by <cit.>.
is an evolution of the models used in the Gimic
<cit.> and Owls <cit.> projects and has
been calibrated to reproduce the $z=0.1$ stellar mass function and
galaxy sizes from $10^8\Ms$ to $10^{11}\Ms$ in a cosmological volume
of $100^3$ Mpc$^3$. In addition, the simulations also
successfully reproduce many other properties and scaling laws of
observed galaxy populations, including the evolution of the stellar
mass function <cit.>, and the luminosities and colours
of galaxies <cit.>.
The subgrid physics model of is described in detail by
<cit.>. It includes radiative cooling, star formation,
stellar evolution and stellar mass loss, and thermal feedback that
captures the collective effects of stellar winds, radiation pressure
and supernova explosions. It also includes black hole growth fuelled
by gas accretion and mergers, and feedback from active galactic nuclei
(AGN) <cit.>. Within the Local Group,
AGN feedback is negligible, and the main processes regulating the
formation of galaxies are gas cooling and heating by the UV
background, star formation, and supernova feedback, which are
described in more detail below.
Following <cit.>, net cooling rates are computed
separately for 11 elements, assuming ionisation equilibrium in the
presence of uniform UV and X-ray backgrounds from quasars and galaxies
<cit.>, and the cosmic microwave background
(CMB). Hydrogen is assumed to reionise instantaneously at $z=11.5$,
which is implemented by turning on the ionising background. At higher
redshifts the background is truncated at 1 Ryd, limiting its effect to
preventing the formation of molecular hydrogen. During reionization an
extra 2 eV per proton mass are injected to account for the increase in
the photoheating rates of optically thick gas over the optically thin
rates that are used otherwise. For hydrogen this is done at $z=11.5$,
ensuring that the gas is quickly heated to $10^4$ K, but for HeII the
extra heat is distributed in time following a Gaussian centered at
$z=3.5$ with $\sigma(z) = 0.5$, which reproduces the observed thermal
history <cit.>. In order to prevent
artificial fragmentation of the ISM, a temperature floor is imposed on
the gas through a polytropic equation of state with index $\gamma =
4/3$, normalised to $T = 8 \times10^3$K at a gas threshold density of
$n_\mathrm{H} = 10^{-1}$cm$^{-3}$ <cit.>.
The star formation rate is assumed to be pressure-dependent
<cit.> and follows the observed Kennicutt-Schmidt star
formation law with a metallicity-dependent density threshold
<cit.>. Energy feedback from star formation is
implemented using the stochastic, thermal prescription of
<cit.>. The expectation value for the energy
injected per unit stellar mass formed decreases with the metallicity
of the gas and increases with the gas density to account for
unresolved radiative losses and to help prevent spurious numerical
losses. The injected energy is calibrated to reproduce the observed,
$z=0$ galaxy stellar mass function and sizes <cit.>. On
average it is close to the energy available from core collapse
supernovae alone <cit.>. Galactic winds develop
naturally, without imposing mass loading factors, velocities or
In our highest resolution simulations, each of the main galaxies
contains more than $20$ million particles, comparable to the best
simulations of individual MW sized galaxies published to
date. Nevertheless, they still barely resolve the scale height of the
MW thin disk, and the effective resolution is also limited by the
equation of state imposed on the gas. Furthermore, the resolution and
the physics included in the code do not resolve individual
star forming regions or supernova feedback events. We rely instead on
a well-calibrated subgrid physics model to parametrise the star
formation and feedback processes. For our study, we have used the
same parameter values that were used in the $100^3$Mpc$^3$ L100N1504
reference simulation <cit.> independently of
While there is clearly scope for future improvements, the relevant
properties discussed in this paper, such as the stellar mass function
and the circular velocity function of substructures are well
converged, as demonstrated in the Appendix. This indicates that our
numerical resolution is sufficient to capture the physical mechanisms
of structure formation, gas accretion and outflows in the model.
§.§ The APOSTLE simulations
twelve Local Group regions are zoom simulations selected from a DMO
simulation of $100^3$Mpc$^3$ with $1620^3$ particles in WMAP-7
cosmology <cit.>. The resimulation volumes were selected
to match the available dynamical constraints of the Local Group. Each
volume contains a pair of halos in the virial mass[For halo
masses, we generally quote $m_{200}$, the mass enclosed in a
spherical volume whose mean overdensity is 200$\times$ the critical
density.] range $5\times10^{11} \Ms$ to $2.5\times10^{12} \Ms$, with
median values of $1.4 \times10^{12}\Ms$ for the primary (more massive)
halo and $0.9 \times10^{12}\Ms$ for the secondary (less massive) halo
of each pair. The combined masses of the primary and secondary range
from $1.6\times10^{12}\Ms$ to $3.6\times10^{12}\Ms$ with a median mass
of $2.3\times10^{12}\Ms$, consistent with recent estimates of
$2.40_{-1.05}^{+1.95}\times10^{12}\Ms$ from dynamical arguments and
CDM simulations <cit.>, or $2.3 \pm 0.7 \times 10^{12}$
based on equations of motions that take into account the observed
velocities of galaxies in the local volume <cit.>.
[height=4.35cm]gas_4_3-1.jpg (4,68)z=12 (3,3)0.223077cm1.5pt
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(4,68)z=0.5 (3,3)1.933 cm1.5pt
[height=4.35cm]gas_4_3-8.jpg (4,68)z=0.25 (3,3)2.32 cm1.5pt
[height=4.35cm]gas_4_3-9.jpg (4,68)z=0 (3,3)2.9cm1.5pt
[height=4.35cm, trim=35 0 35 0, clip]wide.jpg (1.5,1.5)z=0
Top three rows: evolution of gas density (red colours) and
stellar density (blue colours) in a comoving volume of side length
$150 \times 200$ h$^{-1}$ kpc, centred on one of the main LG
galaxies and its progenitors at resolution L1. For scale, a bar of
length $100 h^{-1}$ kpc is shown on each panel. At $z=12$, before
reionisation, stars have already formed in some of the highest
density regions, and feedback from supernovae has begun to blow
bubbles into the gas. Over time stars form in many more halos, and
star forming regions merge to form larger galaxies. By $z=1$, the
main galaxy has formed, and continues to accrete both gas and
satellites, many of which lose their gas on infall and are also
tidally disrupted. At $z=0$, the central galaxy is surrounded by
many satellite galaxies, and a complex stellar halo with visible
shells and streams. Bottom row: slice through the entire LG in the
same simulation.
(3,3)Dark matter
Top left: projected dark matter density in one of our
resimulations at resolution level L2 in a cube of side length
4 Mpc. Circles indicate the locations of the eleven brightest
satellites of one of the main galaxies, whose spatial distribution
is as anisotropic as that of the eleven brightest Milky Way
satellites, and which align with the filament that contains most of
the halos and galaxies in the region. The main panels contrast the
vast number of dark matter substructures (left) with the stellar
light distribution (right) in the 2 Mpc cube indicated by the square
in the top left panel. The small panels in the top row are of side
length 125 kpc and reveal in more detail the stellar component of
some of the different types of galaxies formed in this simulation;
central galaxies (first and second columns) and satellite galaxies
(third and fourth columns), which have realistic sizes, colours, and
morphologies. Dark matter substructures are abundant in the Apostle simulations, but due to the complexity of galaxy
formation, starlight paints a very different picture.
We further require that the two halos be separated by $800 \pm 200$
kpc, approaching with radial velocity of $\left(0-250\right)$ km/s and
with tangential velocity below $100$ km/s; to have no additional halo
larger than the smaller of the pair within 2.5 Mpc from the midpoint
of the pair, and to be in environments with a relatively unperturbed
Hubble flow out to 4 Mpc. More details about the selection criteria,
and implications of the different dynamical constraints on the total
mass of the LG may be found in <cit.>.
The high resolution initial conditions were created using second-order
Lagrangian perturbation theory <cit.>. The cosmological
parameters and the linear phases of the parent volume, which are based
on the public multi-scale Gaussian white noise field Panphasia,
are given in Tables 1 and 6 of <cit.>, who also describes
the method used to make the Local Group zoom initial conditions.
Each region sampled with baryons and at the highest resolution
comprises a sphere of at least 2.5 Mpc radius from the LG barycentre
at $z=0$. Outside of these regions, dark matter particles of
increasing mass are used to sample the large scale environment of the
$100^3$ Mpc$^3$ parent simulation. To investigate the impact of
baryons, we also repeated all our simulations as dark matter only
(DMO), where the dark matter particle masses in the high resolution
region are larger by a factor of
$(\Omega_\mathrm{b}+\Omega_{\mathrm{DM}})/\Omega_{\mathrm{DM}}$ than
in the corresponding hydrodynamic simulations.
The three different resolution levels of the Apostle simulations
labelled “L1”, “L2” and “L3” have primordial gas (DM) particle
masses of approximately $1.0 (5.0) \times 10^4 \Ms$, $1.2 (5.9)
\times10^5 \Ms $ and $1.5 (7.5) \times10^6 \Ms$, respectively, and
maximum gravitational softening lengths of 134 pc, 307 pc and 711
pc. L3 is close to the resolution of the L100N1504
simulation. While the simulations use the Planck-1 cosmology
<cit.>, Apostle was performed in the slightly
different WMAP-7 cosmology <cit.>, with density
parameters at $z=0$ for matter, baryons and dark energy of
$\Omega_\mathrm{M}=0.272$, $\Omega_\mathrm{b}=0.0455$ and
$\Omega_\mathrm{\lambda}=0.728$, respectively, a Hubble parameter of
H$_0=70.4$ km/s Mpc$^{-1}$, a power spectrum of (linear) amplitude on
the scale of 8$h^{-1}$Mpc of $\sigma_8=0.81$ and a power-law spectral
index $n_s=0.967$. On LG scales, we expect the effect of cosmological
parameters to be minimal.
Stellar mass functions from 12 Apostle simulations at
resolution L2 compared to observations. In the left and centre,
shaded regions show the mass functions of satellites within 300 kpc
of each of the primary (left) and secondary (centre) of the two main
Local Group galaxies from each simulation volume, while lines show
the observed stellar mass function within 300 kpc of M31 (left) and
the MW (centre). In the right, the shaded region shows all galaxies
within $2$ Mpc of the Local Group barycentre in the simulations,
while the line is the stellar mass function of all known galaxies
within the same region. On each panel, the dark colour-shaded areas
bound the 16th and 84th percentiles; light shaded areas indicate the
full range among our twelve Local Group realisations. For
comparison, the grey area on each panel corresponds to the mass
function of all dark matter halos. All observational data are taken
from the latest compilation by <cit.>. Note that while the M31 satellite count is
likely to be complete to $10^5\Ms$, the count of satellites of the
MW and the total count within $2$ Mpc should be considered as lower
limits to the true numbers due to the limited sky coverage of local
galaxy surveys and the low surface brightness of dwarf galaxies. See
Fig. <ref> for numerical
§ RESULTS
§.§ Formation of LG galaxies
While the Local Group observations are all made at $z=0$, and the
focus of our paper is on the relation between the observable stellar
component and the underlying dark matter model, our simulations allow
us to follow its evolution from before the formation of the first
stars to the present day. Fig. <ref> illustrates the
evolution of gas and stars in a comoving region of side length $200
\times 150$ h$^{-1}$ kpc from $z=12$ to $0$, centred on the particles
that become one of the central galaxies in one of the Apostle
simulations at resolution level L1. At early times, the gas traces the
filamentary structure, and stars begin to form in the highest density
regions, often found in the nodes at the intersection of filaments. In
this simulation, the first stars begin to form at $z\sim17$, in the
progenitors of what will become the pair of main LG galaxies,
analogues to the Milky Way and M31. Immediately after the first stars
have formed, feedback associated with star formation begins to blow
out gas from the then very low mass dark matter and gas halos. At
$z=11.5$, reionization heats the intergalactic gas and rarefies gas
already collapsed in halos, quenching further gas cooling and
accretion into small halos. As a result, the formation of new galaxies
is disrupted, until sufficiently massive halos begin to form. Over
time, star formation begins anew in more and more halos, while
individual star-forming regions merge to assemble larger galaxies.
Shortly after $z\sim3$, the proto-galaxy undergoes a final major
merger, with minor mergers continuing up to $z=0$. The progenitor
continues to accrete new satellites that mostly lose their gas on
infall due to ram-pressure stripping. A stellar halo also builds up,
with shell-like and stream-like substructures originating from tidally
disrupted satellite galaxies. By $z=0$, a pair of large disk galaxies
have formed, both surrounded by shells and streams, along with many
dwarf galaxy satellites.
§.§ Galaxies that only scratch the
Fig. <ref> illustrates the dark matter and starlight in
another of our resimulations at redshift $z=0$. The top left panel
shows the dark matter distribution in a cube of side length 4 Mpc,
encompassing the spherical volume commonly considered as the Local
Group. It reveals a cosmic filament that envelopes the two principal
halos and most of the galaxies in the region. The bottom row zooms in
on a region of side length 2 Mpc around the simulated LG barycentre,
contrasting the distribution of dark matter (left panel) and star
light (right panel). While the simulations contains tens of thousands
of dark matter substructures, galaxies appear as highly biased tracers
of the dark matter, forming almost exclusively in the most massive
Also highlighted in the top left panel are the positions of the
satellite halos that host the eleven brightest satellites of one of
the central galaxies. The alignment of the satellites is indicative of
a thin plane seen in projection, that is also aligned with the
orientation of the filament.
The small insets in Fig. <ref> show the stellar structure
of some of the many galaxies formed in this simulation. The images use
multi-band colours rendered using a spectrophotometric model
<cit.>. A variety of disk and spheroid morphologies,
luminosities, colours, and sizes are clearly visible, reminiscent of
the diversity of observed LG galaxies.
§.§ No missing satellites
Fig. <ref> shows the galaxy stellar mass functions in the
simulations, using data from all twelve of the Apostle volumes
at resolution L2. Results are plotted both within $300$ kpc from each
of the two main galaxies per volume (labelled “primary” and
“secondary” in order of halo mass), as well as within $2$ Mpc from
the LG barycentre, which includes both central and satellite galaxies.
The primary and secondary galaxies have $\tol{20}{10}{6}$ and
$\tol{18}{8}{5}$ satellites more massive than $M_*=10^5\Ms$ inside
$300$ kpc, respectively, where the errors indicate the scatter
equivalent to $1\sigma$ about the median values. This is in good
agreement with the observed number of MW and M31 satellites. Within
$2$ Mpc of the LG barycentre, there are $\sim60$ galaxies with
$M_*>10^5\Ms$ presently known; our simulations produce
$\tol{90}{20}{15}$. The modest number of luminous galaxies is in stark
contrast to the very large number of dark matter halos found within
the same volume, indicated by the grey shaded area in
Fig. <ref>. While feedback from supernovae and stellar winds
regulates star formation in those halos where a dwarf galaxy has
formed, reionisation has left most of the low mass halos completely
dark. The observed stellar mass function of the LG and those of the MW
and M31 satellites are within the 1 $\sigma$ scatter of the average
stellar mass function in our resimulations over most of the stellar
mass range. The relative scatter is larger for the satellite galaxies,
reflecting the larger relative sampling error, and the fact that the
relative variation in single-halo mass among the different Apostle volumes is larger than that of the total LG mass.
Excluding substructures, the stellar masses of the Milky Way and M31
analogues in our simulations lie in the range $1.5 - 5.5 \times
10^{10} \Ms$, on the low end compared to the observational estimates
for the Milky Way ($5 \times 10^{10} \Ms$ <cit.>) but lower than those for M31 ($10^{11} \Ms$
<cit.>). As noted by <cit.>, the subgrid
physics used in the Reference model of the code, which we have
adopted in this work, generally results in slightly low stellar masses
in halos of around $10^{12}\Ms$ compared to abundance matching
expectations <cit.>, while the Milky Way and M31 both
appear to lie above the average stellar-to-halo mass relation. While
the predicted abundance of satellites and dwarf galaxies within the
Local Group depends on its total mass, as discussed in
Section <ref>, and in more detail by
<cit.>, we have selected our Local Group analogues based
on their dynamical properties in a pure dark matter simulation, and
independently of the stellar mass in the primaries, which may be
affected by the limitations of subgrid physics model.
That the simulations reproduce the stellar mass function of galaxies
and satellites in the LG over all resolved mass scales is remarkable,
given that these simulations use the very same model that
matches the shape and normalisation of the galaxy stellar mass
function in large cosmological volumes. Not only are our simulations
free of the “missing satellites” problem, but they indicate that the
observed stellar mass functions of the LG volume and of the MW and M31
satellites are entirely consistent with $\Lambda$CDM.
Cumulative number of halos as a function of maximum circular
velocity, $\vm$, averaged over 12 Apostle volumes at
resolution level L2. The four bottom curves correspond to satellite
halos within $300$ kpc of each of the two main galaxies; the top
two curves to all systems within $2$ Mpc from the LG barycentre.
Grey/black curves are from dark matter only (DMO) simulations.
Coloured curves are for systems that contain luminous galaxies in
the hydrodynamic runs. Red circles show measurements of the most
massive MW satellites by <cit.>. For
guidance, the dashed line denotes a $\vm$ value of
$30$km/s. The abundance of satellites with $\vm >30$km/s is
halved in the hydrodynamic simulations and matches the Milky Way
observations. At lower values of $\vm$, the drop in the
abundance relative to the DMO case increases as fewer and fewer
subhalos host an observable galaxy. See Fig. <ref>
for numerical convergence.
§.§ The Baryon Bailout
We next consider the “too-big-to-fail” problem
<cit.>. As demonstrated by
<cit.> using the Aquarius dark matter only (DMO)
simulations <cit.>, a Milky Way mass halo in
$\Lambda$CDM typically contains at least one satellite substructure
that matches the velocity dispersion profiles measured for each of the
five Milky Way dwarf spheroidal satellites for which high-quality
kinematic data are available. However, that work addressed neither the
question of whether those halos which match the kinematics of a
particular satellite would actually host a comparable galaxy, nor
whether an observed satellite galaxy can be found to match each of the
many predicted satellite halos. Indeed, the identification in the same
simulations of an excess of massive substructures with no observable
counterparts, and the implication that the brightest satellites of the
Milky Way appear to shun the most massive CDM substructures,
constitutes the “too big-to-fail” problem
<cit.>. A simple characterisation of the
too-big-to-fail problem is given by the number of satellite halos with
maximum circular velocities, $\vm = \max \left(\sqrt{G M(<r) / r}
\right)$, above $\sim 30$ km/s, where all satellite halos are expected
to be luminous <cit.>. Only three MW
satellites are consistent with halos more massive than this limit (the
two Magellanic Clouds and the Sagittarius dwarf), whereas dark
matter only (DMO) $\Lambda$CDM simulations of MW-sized halos
produce two to three times this number.
Ratio between the maximum circular velocity, $\vm$, of
individual, isolated halos in the hydrodynamic simulation and the
DMO simulation of the same volume, as a function of $\vm$, at
resolution L1. The red line shows the binned median ratio. The loss
of baryons and the truncated growth leads to a reduction in $\vm$
for halos below $\vm \sim 100$km/s.
Indeed, as shown in Fig. <ref>, when we consider the DMO
counterparts of our LG simulations, the MW and M31 halos each contain
an average of $7-8$ satellites with $\vm >30$ km/s inside 300 kpc,
more than twice the observed number of luminous satellites. This is
despite the fact that, in order to match the most recent dynamical
constraints <cit.>, the average halo
masses of M31 and the MW in the Apostle simulations are lower
than those in which the problem was first identified
Left: circular velocity profiles of all satellites with $\vm
> 12$ km/s and within 300 kpc from the halo centre, in four
individual LG halos from two Apostle simulations at resolution
L1. Right: as above, for the satellite galaxies in the corresponding
hydrodynamic simulations. Overplotted in red on each panel are the
half-mass radii and corresponding circular velocities of nine
observed MW dwarf spheroidal satellites adopted from <cit.>. While three of the four halos in the DMO
simulations have multiple massive subhalos without observable
counterparts, the discrepancy is resolved in the hydrodynamic
simulations. As the LMC, SMC and the Sagittarius dSph, all
consistent with $\vm > 30$ km/s, are excluded from this sample, we
have also removed the three satellites with the highest $\vm$ values
from each panel.
The situation changes, however, when we consider the hydrodynamic simulations: each main galaxy in our hydrodynamic
simulation has on average only $3-4$ luminous satellites with $\vm
>30$ km/s. Furthermore, the average velocity function of the most
massive substructures across the Apostle simulations appears to
be in excellent agreement with the MW estimates quoted by
<cit.> and overplotted as red circles in
Fig. <ref>. It should be noted that the true $\vm$ values
of dwarf spheroidal galaxies cannot easily be measured, and their
estimates rely on additional assumptions. For Fig. <ref>,
we have chosen to use measurements obtained independently from our own
simulations, but we revisit this topic in Section <ref> where
we provide ranges of the likely $\vm$ values of nine MW dwarf
spheroidals with measured stellar masses and velocity dispersions.
Structural parameters and $\vm$ estimates for Milky-Way
satellite galaxies with M$_* > 10^5 \Ms$
$M_*$ $\rh$ $\sqrt{\left<\sigma^2_{los}(\rh)
\right>} $ $\vm$ $\vm$ $\vm$ $\vm$ (DMO) $\vm$ (hydro)
$ [\Ms]$ [pc] [km/s] [km/s] [km/s] [km/s] [km/s]
Notes and References (1) (2) (2) (3) (4) (5) this work
(6) this work (7)
Carina $4.3^{+1.1}_{-0.9}\times10^5$ $334 \pm 37$ $6.4 \pm0.2$ 16 17 $11.4 ^{+1.1}_{-1.0}$ $13.7^{+4.8}_{-2.2} $ $14.6 ^{+6.4}_{-2.9}$
Draco $2.2^{+0.7}_{-0.6} \times 10^5$ 291 $\pm$ 14 10.1 $\pm$0.5 – 35 $20.5 ^{+4.8}_{-3.9}$ $27.6^{+12}_{-4.1}$ $23.7^*$
Fornax $1.7^{+0.5}_{-0.4} \times 10^7$ 944 $\pm$ 53 10.7$\pm$0.2 21 21 $17.8 ^{+0.7}_{-0.7}$ $19.6^{+1.4}_{-1.1}$ $20.4^*$
Leo I $5.0^{+1.8}_{-1.3} \times 10^6$ 388 $\pm$ 64 9.0 $\pm$0.4 22 30 $16.4 ^{+2.3}_{-2.0}$ $ 18.5^{+9.0}_{-3.4}$ $18.8^{+12.3}_{-4.0}$
Leo II $7.8^{+2.5}_{-1.9} \times 10^5$ 233 $\pm$ 17 6.6 $\pm$ 0.5 – 18 $12.8 ^{+2.2}_{-1.9}$ $14.2^{+6.4}_{-2.6}$ $17.7^{+6.0}_{-3.2} $
Sculptor $2.5^{+0.9}_{-0.7} \times 10^6$ 375 $\pm$ 54 9.0 $\pm$ 0.2 26 27 $17.3 ^{+2.2}_{-2.0}$ $20.9^{+9.6}_{-4.0}$ $20.1^{+7.2}_{-1.6} $
Sextans $5.9^{+2.0}_{-1.4} \times 10^5$ 768 $\pm$ 47 7.1 $\pm$ 0.3 12 11 $11.8 ^{+1.0}_{-0.9}$ $12.4^{+1.6}_{-1.0}$ $13.2^{+2.5}_{-1.5}$
Ursa Minor $3.9^{+1.7}_{-1.3} \times 10^5$ 588 $\pm$ 58 11.5 $\pm$ 0.6 – 29 $20.0 ^{+2.4}_{-2.2}$ $22.6^{+6.6}_{-4.0}$ $21.5^{+2.8}_{-0.4} $
CanVen I $2.3^{+0.4}_{-0.3} \times 10^5$ 750 $\pm$ 48 7.6 $\pm$ 0.5 – – $11.8^{1.3}_{-1.2}$ $12.8^{+2.8}_{-1.6}$ $13.1^{+3.4}_{-1.4}$
(1): <cit.>, (2): <cit.>,
(3): <cit.>, (4) <cit.>, (5) <cit.>, (6) satellite halos from DMO APOSTLE
simulations that match the observed $\vh$ at the observed
$\rh$, (7) satellite halos from hydrodynamic Apostle
simulations that match the observed $\vh$ at the observed
$\rh$, and that host galaxies that match the observed stellar
mass. $^*$For Fornax and Draco, there are too few simulated counterparts to
estimate the range reliably.
Circular velocity profiles for satellites from a total of
four halos from two of our LG simulations at resolution L1, which
have been matched to 9 observed dSph galaxies. On the top left,
black lines show satellites from the DMO simulations whose
circular velocities are within 3$\sigma$ of the value corresponding
to the observed velocity dispersion at the observed half-light
radius. On the top right, grey lines show satellites from the
corresponding hydrodynamic simulations selected with the same
criteria. On the bottom left, cyan lines are for satellites from
the hydrodynamic simulations, selected by stellar mass to be within
a factor of 2 of the observed value, and on the bottom right, dark blue
lines are for satellites from the hydrodynamic simulation that
satisfy both the stellar mass and circular velocity criteria. On all panels, red symbols show measurements
for observed dwarf spheroidals at the projected 3D half-light
radius, adopted from <cit.>.
Several factors contribute to the reduction in the measured satellite
$\vm$ function in our hydrodynamic simulations compared to DMO
simulations: (i) a reduction in the mass of each subhalo due to
baryonic effects as discussed below, (ii) the failure of a fraction of
subhalos of $\vm < 30$ km/s to form any stars, and (iii) those halos
of $\vm < 30$ km/s today that actually contain observable dwarf
galaxies having been affected by tidal stripping, even more strongly
than typical satellite halos of the same mass today.
In Fig. <ref>, we compare the maximum circular velocity
of individual isolated halos matched in our hydrodynamic and DMO
simulations. In agreement with <cit.> and
<cit.>, we find that while the more massive halos of
$\vm > 100$ km/s such as those that host the MW and M31 are not
significantly affected by baryonic effects, the halos of dwarf
galaxies end up being less massive than their DMO counterparts,
because the loss of baryons due to reionization and supernova
feedback, results in a reduced halo growth rate and leads to a $\sim
15\%$ reduction in $\vm$. The average reduction in mass is similar for
the halos of satellite and isolated galaxies prior to infall, but the
more massive satellites in the hydrodynamic simulations experience a
further mass loss relative to their DMO counterparts due to the
ram-pressure stripping of the remaining gas.
For halos below 30 km/s, the intrinsic reduction in $\vm$ due
to baryonic effects is compounded by the fact that not all low-mass
halos host galaxies: at $10$ km/s the fraction of luminous systems is
well below $10\%$ and decreases even further towards lower masses.
Fig. <ref> shows a comparison of
individual circular velocity curves of the most massive satellites
within 300 kpc of four of the main LG galaxies, in the DMO simulations
(black, left column) and in the corresponding hydrodynamic (blue,
right column) simulations. While there is considerable scatter due to
the fact that individual satellites can evolve differently in the two
simulations, particularly after infall, it can be seen that the
satellites in the hydrodynamic simulations have systematically lower
circular velocity curves compared to the DMO counterparts. While three
of the four halos in the DMO simulations contain a number of
satellites whose circular velocity curves cannot be matched by any of
the observed satellite galaxies shown, the velocity curves of
satellites in the hydrodynamic simulations are consistent with the
observed stellar kinematics.
§.§ $\mathrm{V}_\mathrm{max}$ estimates for dwarf spheroidal
Because the visible stellar components of dwarf
spheroidal galaxies probe only the innermost part of their dark matter
halos, connecting the measured line-of-sight velocity dispersion,
$\sigma_{los}$, and the size of a satellite to the maximum circular
velocity, $\vm$, of its halo is not straightforward, and relies on
several assumptions.
<cit.> estimated $\vm$ values of individual Milky Way
satellite galaxies, assuming that their stellar and dark matter
components follow King profiles and NFW profiles,
respectively. <cit.> also gave values for $\vm$
for nine Milky-Way satellites with stellar masses above
$10^5\Ms$. Using the result of <cit.> that the uncertainty
on the enclosed mass for an observed line-of-sight velocity dispersion
is minimal at the stellar half-light radius, $\rh$, they used the
circular velocity profiles of satellite halos from the Aquarius
DMO simulations to determined the most likely $\vm$ value of satellite
halos in CDM that match the measured values of $\rh$ and
$\sigma_{los}(\rh)$. Independently, using the same simulations,
<cit.> also determined the $\vm$ values of five dwarf
spheroidals with resolved kinematics. Instead of relying on the
velocity dispersion at the half-light radius, they determine the most
likely value of $\vm$ for a given observed satellite from the
best-fitting velocity dispersion profile in the simulated halos. The
observed structural parameters of nine individual Milky Way satellite
galaxies compiled by <cit.>, and the $\vm$ estimates of
<cit.>, <cit.> and
<cit.> are reproduced in Table <ref>.
(43,45)Leo I
(30,31)Leo II
(52,25)U Mi
(24,16)CV I
(43,45)Leo I
(30,31)Leo II
(52,25)U Mi
(24,16)CV I
Stellar mass as a function of maximum circular velocity,
$\vm$, for isolated galaxies (blue circles, left) and satellite
galaxies (red circles, right) from two LG simulations at resolution
L1. Also shown are measurements for the nine MW satellites listed in
Table 1, with stellar masses adopted from <cit.>
and the range of possible $\vm$ values computed from the observed
$\rh$ and $\sigma_{los}(\rh)$ as discussed in
Section <ref>, based on halos from the DMO simulations
(grey) and based on satellite galaxies from the hydrodynamic
simulations (black). While isolated galaxies in the simulation fall
below the MW satellites, the reduction in $\vm$ particularly of
low-mass satellites by tidal stripping, brings the simulated
satellite galaxies into good agreement with observations.
For those galaxies where multiple estimates are available, the $\vm$
values of <cit.> tend to be lower than those
obtained by the two other authors. Indeed, as pointed out by
<cit.>, these low values would still lead to an
overprediction of the MW satellite $\vm$ function in our simulations,
even though the extrapolation down to 10 km/s is somewhat
misleading, because it only includes $\vm$ values for a small fraction
of the known MW satellites.
However, the most likely subhalo in a DMO simulation like Aquarius whose enclosed mass inside $\rh$ corresponds to the
observed $\sigma_{los}(\rh)$ may give a low estimate of the
satellite's true $\vm$. Not only do baryons change the dark matter
subhalos, as discussed in Section <ref>; but not all
subhalos are expected to host satellite galaxies, and the probability
for a low mass subhalo to host a satellite galaxy depends on its
$\vm$, and the typical stellar mass of a satellite galaxy also depends
on $\vm$. Hence, not all satellite halos have the same probability of
matching an observed satellite galaxy of a known stellar mass.
Indeed, the sample of satellite galaxies analysed by
<cit.> contains nine of the twelve most luminous
known MW satellites, which are likely to be amongst the satellites
whose halos have the highest $\vm$ values. $\Lambda$CDM predicts
many more low $\vm$ halos than high $\vm$ halos. While
point out that this does not affect
their analysis, it may amplify any potential bias caused by the
implicit assumption that all halos have an equal chance of hosting a
satellite of a given stellar mass.
In Fig. <ref>, we examine the circular velocity
profiles of individual satellite halos and galaxies in Apostle,
and compare them to individual observed MW satellites. On each panel,
we show the circular velocity curves of satellites located within 300
kpc from the four central galaxies in two LG simulations at resolution
L1, and compare them to the circular velocities inferred from
$\sigma_{los}(\rh)$ for nine observed MW dwarf spheroidal galaxies.
On the top left panel of Fig. <ref> (black
curves), we have selected satellite subhalos from the DMO simulation,
requiring that the circular velocity $\vc$ be within three times the
quoted observational uncertainty of $\sigma_{los}(\rh)$ for a given
satellite at its observed half-light radius, assuming the relation
between $\vc$ and $\sigma_{los}(\rh)$ of <cit.>.
For the remaining three panels of Fig. <ref>, we
have selected satellite subhalos from the corresponding hydrodynamic
simulations. In the top right (grey curves), we use the same selection
criterion as for the DMO case. For the bottom left panel (cyan
curves), we have not applied any velocity criterion, but require the
stellar mass in the simulation to be within $50\%$ of the observed
stellar mass of the individual satellite. Finally, in the bottom right
panel (dark blue curves), we combine the two previous criteria, and select
satellites from the hydrodynamic simulation whose stellar mass and
measured circular velocity are both compatible with those of the
observed satellite.
Due to the range in concentration of CDM halos, both the DMO and
hydrodynamic simulations allow a large range of $\vm$ values for most
observed satellites, in particular for those where the half-light
radius, $\rh$, is small compared to the radius where the circular
velocity is maximal, $\rmax$. It is also worth noting that when
satellites are selected purely by stellar mass, we find that many of
the simulated galaxies live in subhalos that are consistent with the
observed kinematics. As discussed in Section <ref>, a
notable exception is the Draco dSph, which appears to have an
unusually high halo mass for its stellar mass.
Combining the velocity and stellar mass criteria generally reduces the
range of compatible $\vm$ values for a particular satellite. We list
the most likely $\vm$ value for each of the nine observed satellites
according to our simulations in the two rightmost columns of Table
<ref>, for both the DMO simulation using only the velocity
criterion, and for the hydrodynamic simulation, using both the
velocity and stellar mass constraints.
It should be noted that while the ranges quoted for the values of
<cit.> denote the uncertainty of the most likely
value, for our own values we give the 16th and 84th percentiles of
$\vm$ for the compatible satellite halos (equivalent to $\pm 1 \sigma$
for a normal distribution). The latter can be much larger, and reflect
the diversity of halos in $\Lambda$CDM and in our simulations. It is
also worth pointing out again that any of the quoted $\vm$ estimates
from observed velocity dispersions are only valid in the $\Lambda$CDM
context and, for the values that also use the stellar mass, under the
additional assumptions made in our simulations.
§.§ The right satellites in the right halos
We have shown in Figs. <ref> and <ref> that the
Apostle simulations reproduce the number of satellites as a
function of stellar mass and of $\vm$, as inferred from the stellar
velocity dispersion and sizes. However, this success does not
necessarily imply that the simulations reproduce the stellar
mass–$\vm$ relation for individual satellites. Furthermore, the
stellar-to-total mass ratios of individual dwarf spheroidals show a
surprising amount of scatter: Fornax is roughly 100 times brighter
than Draco, but appears to inhabit a halo of similar mass.
In our simulations, the key to understanding this puzzling result lies
in the fact that, in the $\vm$ range below 30 km/s, where some halos
remain dark, those satellite halos that host galaxies similar to the
Milky Way dwarf spheroidals have typically experienced much more
severe tidal stripping, with a resulting reduction in mass or $\vm$
greater than expected for typical halos of the same mass
<cit.>. This phenomenon is illustrated in
Fig. <ref>, which compares the stellar mass–$\vm$
relation of both central and satellite galaxies in our simulations
with data for the actual satellites of the Milky Way. While isolated
halos in our simulations fall below the stellar mass–$\vm$ relation
of observed satellites, the strong tidal stripping experienced by the
satellite galaxies and their halos brings them into good agreement
with the observed MW dwarf spheroidal data. The effect of stripping
can also account for the large scatter in stellar mass of dwarf
galaxies in a very narrow $\vm$ range. In this scenario, the high
stellar mass–$\vm$ ratio of satellites like Fornax and Sextans would
be explained in part by tidal stripping. By contrast, the low
stellar-to-halo mass ratio of Draco is more typical of isolated dwarf
galaxies in our simulations, suggesting that Draco has not yet
experienced strong tidal effects.
Angular distribution and kinematics of the eleven brightest satellites for a
Milky-Way like system in our simulation (red), and for the observed
Milky Way satellites (blue). Circles show the
location on the sky, while triangles
indicate the orientation of the corresponding angular momentum
vector. For both systems, the equator is chosen
to align with the respective plane of satellites. The eleven brightest satellites in the simulated system are
distributed on a plane just as flat as those of the Milky Way and
several of them have a coherent rotation. Selecting the brightest satellites, systems as
anisotropic as the MW's can be formed in $\Lambda$CDM, although
they are not typical.
In summary, not only do our hydrodynamic simulations reproduce both
the observed satellite stellar mass function and satellite circular
velocity function; but satellite galaxies of stellar masses comparable to
observed dwarf spheroidals also live in halos with compatible velocity
profiles. The reduction in subhalo mass due to baryonic effects, and
the strong stripping of halos that host the luminous satellites,
combine to give a satellite population that not only matches the Milky
Way and M31 satellite luminosity function, but also the total velocity
function of the observed satellite population.
§.§ Unsurprisingly aligned
Finally, the anisotropy, first noticed by <cit.>, and the
apparent orbital alignment of the 11 brightest, so-called
“classical” MW satellites, have been regarded as highly improbable
in $\Lambda$CDM <cit.>. Fig. <ref> compares
the observed angular distribution and orbital kinematics of the 11
brightest Milky Way satellites to the 11 brightest satellites around
one of our simulated LG galaxies at $z=0$, as identified in
Fig. <ref>. Both the simulated and observed satellite
populations show highly anisotropic distributions.
To characterise the anisotropy of each satellite system, we compute
the ratio of the minimal to maximal eigenvalues, $c$ and $a$, of the
reduced inertia tensor <cit.> defined by the 11
brightest satellites, ${I_{\alpha,\beta} =
\sum_{i=1}^{11}{r_{i,\alpha}r_{i,\beta}} / r_i^2}$. From a total of
24 MW or M31-like halos in our 12 LG simulations at resolution L2, we
find values of $\sqrt{c/a}$ in the range 0.34 – 0.67, compared to
0.36 for the MW and 0.53 for M31.
Clearly, $\Lambda$CDM can produce satellite systems with a range of
anisotropies, and consistent with measurements for both the MW and M31.
However, considering only the 11 brightest satellites, the satellite
distribution of the MW appears to be more anisotropic than all but one
of our 24 systems, which still leaves open the question of its
statistical significance. It should be noted that, by the same
criterion, the satellite system of M31 is much less anisotropic than
that of the MW, and indeed quite typical of our simulated LG
galaxies. Given that the MW and M31 clearly formed within the same
cosmology with different satellite anisotropies, it may be premature
to consider the anisotropy of one a failure of the cosmological model.
The inclusion of baryonic physics in the Apostle simulations
allow us to directly identify the brightest satellite galaxies out of
a much larger number of satellite halos. However, their spatial
anisotropy is clearly not caused by baryonic effects; it is intrinsic
to the assembly of $\Lambda$CDM satellite systems. As
<cit.> have recently demonstrated, even when subsets of
observed satellites are carefully chosen to exhibit maximal anisotropy
<cit.>, rigorous statistical analysis shows that
the observed spatial and kinematic anisotropies are not inconsistent
with $\Lambda$CDM and that such apparently “unusual” systems are in
fact quite ubiquitous.
That satellite alignments are consistent with, and indeed expected to
exist in $\Lambda$CDM does not render them uninteresting. Each
individual satellite system can still contain information about the
assembly history of the halo and its relation to the large scale
structure. Across our simulations, the satellite plane most like the
MW's is aligned with a cosmic filament that envelopes the Local
Group. This supports the scenario proposed by <cit.>,
whereby the accretion of satellites from the “cosmic web” imparts a
degree of coherence to the timing and direction of satellite
§ SUMMARY
Apostle is a series of high-resolution zoom simulations of
the Local Group within the $\LCDM$ cosmology, performed both as dark
matter only, and including baryonic processes with the code. We
find that our simulations accurately reproduce the observed stellar
mass function of the Local Group, and also result in satellite
populations that are in good agreement with the observed relation
between dwarf galaxies and their dark matter halos.
We conclude that the “problems” often cited as challenges to
$\Lambda$CDM are resolved in simulations that reproduce the dynamical
constraints of the Local Group environment and include a realistic
galaxy formation model. Reionisation and supernova feedback allow
galaxy formation to proceed only in a small subset of dark matter
halos, eliminating the “missing satellites” problem. It is notable
that the very same galaxy formation model calibrated to reproduce the
galaxy population in cosmologically representative volumes naturally
produces a LG galaxy population in volumes consistent with the LG
kinematics in $\Lambda$CDM.
The loss of baryons due to reionisation and star formation feedback,
and from satellites through ram pressure stripping, affects the growth
of low-mass halos, leading to a reduction in their maximum circular
velocity compared to a DMO simulation. Combined with the effect of
tidal stripping, strongly enhanced for luminous, low mass halos, this
not only resolves the “too-big-to-fail” problem, but it also leads
to a satellite circular velocity function that matches observations,
at the same time as it matches the observed stellar mass
function. Furthermore, kinematics of individual simulated satellites
are consistent with the kinematics of observed satellites matched by
stellar mass. We also find that individual galaxies formed in our
simulations follow the observed relation between $\vm$ and stellar
mass, and use our DMO and hydrodynamic simulations to estimate the
range of likely $\vm$ values for nine dwarf spheroidals with measured
stellar mass and $\sigma_{los}(\rh)$ in the $\Lambda$CDM cosmology.
Finally, we find a diversity of satellite systems with spatial
anisotropy similar to those of the M31, and one system that is similar
to the “plane of satellites” around the MW. We conclude that the
observed anisotropies of these satellite systems do not falsify
$\Lambda$CDM, but may reflect their assembly histories within the
$\Lambda$CDM paradigm.
Another often-cited difficulty for $\Lambda$CDM is the inference of
constant density cores in dark matter halos
<cit.>, whereas N-body simulations predict
cusps. While the observed kinematics of LG dwarf spheroidals are, in
fact, consistent with either cores or cusps <cit.>, it
has also been argued that cores are required to solve the
too-big-to-fail problem <cit.>. This is not the case: the
star formation and feedback model in our simulations and the effect of
tidal stripping give rise to a realistic LG galaxy population without
cores. Hence, we conclude that cores are not necessary to solve the
perceived small scale problems of $\Lambda$CDM.
Our simulations predict that the relation between stellar mass and
$\vm$ should differ between satellites and isolated dwarf galaxies, as
the observed satellites with high stellar mass–$\vm$ ratios live in
halos that experienced particularly strong tidal stripping. This
prediction is testable: alternative scenarios in which the dark matter
halos are modified independently of environment should give much more
similar relations between the two.
Our simulations are still limited by resolution and the shortcomings
of the subgrid physics model. Nevertheless, they result in galaxy
populations compatible with many of the observations of the LG galaxy
population and do not suffer from any of the problems often
interpreted as a failure of $\Lambda$CDM. Using a model already shown
to reproduce the galaxy population on much larger scales, they suggest
that the success and predictive power of the $\Lambda$CDM cosmology
extend far into the LG regime, once the effects of galaxy formation
and of the particular LG environment are taken into account.
§ ACKNOWLEDGEMENTS
We are indebted to Dr. Lydia Heck who looks after the supercomputers
at the ICC. This work was supported by the Science and Technology
Facilities Council [grant number ST/F001166/1 and RF040218], the
European Research Council under the European Union's Seventh Framework
Programme (FP7/2007-2013) / ERC Grant agreement 278594
'GasAroundGalaxies', the National Science Foundation under Grant
No. PHYS-1066293, the Interuniversity Attraction Poles Programme of
the Belgian Science Policy Office [AP P7/08 CHARM]. T. S. acknowledges
the Marie-Curie ITN CosmoComp and the support of the Academy of
Finland grant 1274931. C. S. F. acknowledges ERC Advanced Grant 267291
'COSMIWAY' and S. W. acknowledges ERC Advanced Grant 246797
'GALFORMOD'. R. A. C. is a Royal Society University Research
Fellow. This work used the DiRAC Data Centric system at Durham
University, operated by the Institute for Computational Cosmology on
behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk), and resources
provided by WestGrid (www.westgrid.ca) and Compute Canada / Calcul
Canada (www.computecanada.ca). The DiRAC system is funded by BIS
National E-infrastructure capital grant ST/K00042X/1, STFC capital
grant ST/H008519/1, STFC DiRAC Operations grant ST/K003267/1, and
Durham University. DiRAC is part of the National E-Infrastructure.
§ NUMERICAL CONVERGENCE
Stellar and halo mass functions (left panel) and maximum
circular velocity functions (right panel) within 2 Mpc from the
barycentre of one of the Apostle volumes at three different
resolutions, for galaxies in the hydrodynamic simulations, and halos
in the corresponding DMO simulations. The stellar and halo mass
functions, and the velocity functions of galaxies and halos are well
converged at resolution L2.
As discussed in Section <ref>, we have performed our
simulations at 3 resolution levels, varying by a factor of $\sim144$
in particle mass. Fig. <ref> illustrates the numerical
convergence of the mass functions (left panel) and velocity function
(right panel) for one Apostle volume at the three resolution
levels. For both the halo mass function and halo velocity function,
results are shown from the DMO simulation. Here, convergence is
excellent. For the galaxy mass function, and the velocity function of
the halos containing galaxies (coloured curves in both figures),
results are shown from the hydrodynamic simulation.
Convergence of the cumulative stellar mass function remains very
good. By comparison to Fig. <ref>, which includes results from
all twelve Apostle volumes at resolution L2, it is clear that
the difference in the stellar mass function due to numerical
resolution is much smaller than the variation between the different
Apostle volumes.
The velocity function of galaxies, which falls below the corresponding
total velocity function of halos in the DMO simulation due the effect
of baryons on halos at $\vm \sim 60$ km/s, and due to the appearance
of “dark” halos at t $\vm \sim 30$ km/s, is equally well
converged. In particular, both the average reduction in $\vm$ of
subhalos, and the fraction of dark halos, are independent of
|
1511.00782
|
2010 Mathematics Subject Classification. 47A13, 32A50, 30H20, 32D15, 47B35
Key words and phrases. Geometric Arveson-Douglas Conjecture, Complex harmonic analysis, Extension of holomorphic functions, Bergman spaces
The second author would like to thank for the hospitality of Taxes A&M University during her visit here. She also want to thank for the financial support from the program of China Scholarships Council(CSC).
In this paper we introduce techniques from complex harmonic analysis to prove a weaker version of the Geometric Arveson-Douglas Conjecture for complex analytic subsets that is smooth on the boundary of the unit ball and intersects transversally with it. In fact, we prove that the projection operator onto the corresponding quotient module is in the Toeplitz algebra $\toe$, which implies the essential normality of the quotient module. Combining some other techniques we actually obtain the $p$-essential normality for $p>2d$, where $d$ is the complex dimension of the analytic subset. Finally, we show that our results apply for the closure of a radical polynomial ideal $I$ whose zero variety satisfies the above conditions. A key technique is defining a right inverse operator of the restriction map from the unit ball to the analytic subset generalizing the result of Beatrous's paper <cit.>.
§ INTRODUCTION
A complex Hilbert space $\mathcal{H}$ is called a Hilbert module (over the polynomial ring $\mathbb{C}[z_1,\cdots,z_n]$) if for every $p\in\mathbb{C}[z_1,\cdots,z_n]$ there is a bounded linear operator $M_p$ on $\mathcal{H}$ and the map $\mathbb{C}[z_1,\cdots,z_n]\to B(\mathcal{H}),~p\to M_p$ is an algebra homomorphism. Examples include the Bergman module, Hardy module and the Drury-Arveson module over $\bn$, the unit ball of $\mathbb{C}^n$. A submodule $P\subset\mathcal{H}$ is a Hilbert subspace of $\mathcal{H}$ that is closed under the module multiplications $M_p$. Let $Q$ be the orthogonal complement of $P$ in $\mathcal{H}$. Then $Q$ is the quotient Hilbert module with the homomorphism taking $z_i$ to
the compression of $M_{z_i}$ to $Q$. A Hilbert module $\mathcal{H}$ is said to be essentially normal ($p$-essentially normal ) if the commutators $[M_{z_i}, M_{z_j}^*]$ belong to the compact operators $\mathcal{K}(\mathcal{H})$ ( Schatten $p$ class $\mathcal{S}^p$), for any $1\leq i, j\leq n$.
Hilbert modules play an important role in multivariate operator theory.
In his paper <cit.>, Arveson made a conjecture, which was refined by the first author in <cit.> to the following form:
Let $\mathcal{H}$ be one of $\ber$, $H^2(\bn)$ and $H^2_n(\bn)$. Assume $I$ is a homogeneous ideal of $\mathbb{C}[z_1,\cdots,z_n]$
and $P$ is the closure of $I$ in $\mathcal{H}$. Then for all $p>\dim Z(I)$, the quotient module $Q=P^{\perp}$ is $p$-essentially normal. Here $Z(I)$ is the zero set of $I$ and $\dim Z(I)$ denotes the complex dimension of $Z(I)$.
The Arveson-Douglas Conjecture has been proved under various conditions. Arveson <cit.> provd the case when $I$ is generated by monomials; Guo <cit.> proved the case when $n=2$; Guo and Wang <cit.> proved the case when $I$ is a principal homogenous ideal, when the complex dimension of $Z(I)\leq1$, and for any homogenous ideal $I$ when the dimension $n\leq 3$; they <cit.> also proved the case when $I$ is a quasi-homogeneous ideal and $n=2$; the first author and Wang <cit.> proved the case for $\ber$ when $I$ is any principal ideal; in <cit.>, the first author and Sarkar reduced the quasi-homogenous case to the homogenous ones; Shalit <cit.> proved the case when the submodule possesses the stable division property.
In many cases( cf. <cit.>), especially when $I$ is radical, one can prove that $[I]$, the closure of the ideal $I$, consists of all the holomorphic functions in $\mathcal{H}$ that vanish on $V(I)$. Here
V(I)=\{z\in\bn: p(z)=0, \forall p\in I\}.
In these cases, we can relate the submodule $[I]$ to its zero variety. The following conjecture first appeared in <cit.> and is a reformulation of the Arveson-Douglas Conjecture in these special cases.
Let $M$ be a homogeneous variety in $\bn$. Let
P=\{f\in\mathcal{H}: f|_M=0\}
and $Q=P^{\perp}$. Then the quotient module $Q$ is $p$-essentially normal for every $p>\dim M$. Here $\mathcal{H}$ is an analytic Hilbert module(cf. <cit.>), such as the Bergman module $\ber$, the Hardy module $H^2(\bn)$ or the Drury-Arveson module $H^2_n(\bn)$.
There is also a weaker form of the two conjectures: to replace $p$-essential normality with essential normality.
Shalit and Kennedy <cit.> proved the case when $M$ can be decomposed into varieties having “good” properties; Engliš and Eschmeier <cit.> proved the case when $M$ is a homogeneous subvariety that is smooth away from the origin and the case when $M$ is a (not necessarily homogeneous) smooth submanifold that intersects $\pbn$ transversally; the first author, Tang and Yu <cit.> proved the weaker form under the assumption that $M$ is a (not necessarily homogenous) complete intersection space that intersects $\pbn$ transversally and has no singular point on $\pbn$.
In this paper we mainly consider the weaker form of the Geometric Arveson-Douglas Conjecture on a not necessarily homogeneous variety. The methods of this paper differ from those in <cit.> and <cit.>. We obtain essential normality by proving that the projection operator onto the submodule is in the Toeplitz algebra. This approach is new and allows us to analyse the conjecture using tools from complex harmonic analysis. Part of our ideas come from Suárez's paper <cit.> and the first author, Tang and Yu's paper <cit.>.
In section <ref>, we use some ideas in Suárez's paper <cit.> to show that the existence of a positive measure on $M$ that defines an equivalent norm on the quotient module will imply the essential normality of the quotient module.
(Theorem <ref>)
If there exists a positive measure $\mu$ on $M$ such that the $L_{\mu}^2$ norm and Bergman norm are equivalent on the quotient module $Q$, i.e., $\exists C, c>0$ such that $\forall f\in Q$,
$$c\|f\|^2\leq\int_M|f(w)|^2d\mu(w)\leq C\|f\|^2,$$
then the quotient module $Q$ is essentially normal.
Equivalently, we show that the existence of an extension map which is a right inverse of the restriction map from $\bn$ to the zero variety(cf. <cit.> and section 4.2 in <cit.>) implies the essential normality of the quotient module.(See the remark in the end of section <ref>). Moreover, as a consequence of the proof, we have the following interesting corollary:
(Corollary <ref>)
Assume the same hypotheses as in Theorem <ref>, then the projection operators $P$ and $Q$ are in the Toeplitz algevra $\toe$.
In section <ref>, we combine techniques from complex harmonic analysis with those from classical operator theory to prove that such a measure always exists under some conditions. More precisely, we have the following theorem:
(Theorem <ref>)
Suppose $\tilde{M}$ is a complex analytic subset of an open neighborhood of $\clb$ satisfying the following conditions:
(1) $\tilde{M}$ intersects $\pbn$ transversally.
(2) $\tilde{M}$ has no singular point on $\pbn$.
Let $M=\tilde{M}\cap\bn$ and let $P=\{f\in\ber: f|_M=0\}$. Then the submodule $P$ is essentially normal.
Applying the previous corollary to this case, we show that the corresponding projection operator is in the Toeplitz $C^*$-algebra.
(Corollary <ref>)
Suppose $M$ satisfies the properties of Theorem <ref>, then the projection operator onto $P$ is in the Toeplitz algebra $\toe$.
In connection with the theory of holomorphic extension, our approach proves the existence of an extension operator from the analytic subset to the unit ball that is bounded in the $L^2$ norm. This result extends the result in <cit.>. We conclude section <ref> with a remark related to this topic, including the following corollary.
(Corollary <ref>)
Suppose $\tilde{M}$ is a $d$-dimensional complex analytic subset of an open neighborhood of $\clb$ which intersects $\pbn$ transversally and has no singular point on $\pbn$. Let $M=\tilde{M}\cap\bn$, $\mu=(1-|w|^2)^{n-d}dv_d$ on $M$ and $\mathcal{P}$ be the closed subspace of $L^2(\mu)$ generated by analytic polynomials. Then $\mathcal{P}$ is exactly the range of the restriction operator
R: \ber\to L^2(\mu), f\mapsto f|_M.
Equivalently, there is an extension operator
E: \mathcal{P}\to\ber,
which is a right inverse of $R$.
Section <ref> is an attempt towards the full Geometric Arveson-Douglas Conjecture. We combine our methods and the methods in <cit.> to show:
(Theorem <ref>)
Under the assumptions of Theorem <ref>, the quotient module $Q$ is $p$-essentially normal for all $p>2d$.
In fact, we prove that the range space $\mathcal{P}$ is $p$-essentially normal for $p>d$. Although we didn't get the $p$($p>d$)-essential normality of $Q$, we still think it's possible to prove it using similar methods.
In section <ref>, we relate the Geometric Arveson-Douglas Conjecture with the Arveson-Douglas Conjecture. We prove the following theorem by extending the proof in the appendix of <cit.> to our case.
(Theorem <ref>)
Suppose $I\subset\mathbb{C}[z_1,\cdots,z_n]$ is a radical polynomial ideal. If $\tilde{M}:=Z(I)$ has no singular point on $\pbn$ and intersects $\pbn$ transversally, then $[I]=\{f\in\ber: f|_M=0\}$. Here $M=\tilde{M}\cap\bn$. As a consequence, the quotient module $Q=[I]^{\perp}$ is $p$-essentially normal, $p>2d$. Moreover, the projection operator onto $Q$ is in the Toeplitz algebra $\toe$.
Our result generalizes the essential normality results of those in <cit.> and <cit.>. Moreover, it was not shown in the previous two papers that the projection operators are in the Toeplitz algebra. This could be useful, for example, in the study of joint invertibility and joint Fredholmness of Toeplitz operators on the submodules and quotient modules.
This new method is simple and calculable. We believe that there is more to be discovered. We hope to obtain generalizations to $p$-essential normality( $p>d$) and varieties satisfying other assumptions in the future.
§ PRELIMINARIES
Let $\bn$ be the unit ball of $\cn$. The Bergman space $\ber$ is defined as all holomorphic functions on $\bn$ that is square integrable.
\ber=\{f:\bn\to\mathbb{C}|~f\mbox{ is holomorphic and }\int_{\bn}|f(w)|^2dv_n(w)<\infty\}.
Here $v_n$ is the normalized volume measure on $\bn$. For $g\in L^{\infty}(\bn)$, the Toeplitz operator is defined by
T_g:\ber\to\ber, f\mapsto P_{\ber}(gf)
where $P_{\ber}$ is the projection operator from $L^2(\bn)$ to $\ber$. The Toeplitz algebra $\toe$ is the $C^*$ subalgebra of $B(\ber)$ generated by $T_g$, $g\in L^{\infty}(\bn)$. The space $\ber$ is a Hilbert module with module multiplications $T_p, p\in \mathbb{C}[z_1,\cdots,z_n]$. By definition, a submodule of $\ber$ is a closed subspace that is invariant under $T_{z_i}, i=1,\cdots,n$.
In the rest of this paper, $M$ will always mean a subset of $\bn$, $P$ means the submodule of $\ber$ consisting of all functions that vanish on $M$ and $Q=P^{\perp}$. We also use $P$, $Q$ to denote the corresponding projection operators to each space. Note that under this setting,
$$Q=\overline{span}\{k_z: z\in M\}.$$
In order to prove the essential normality of the submodule $P$, we need the following lemma, which is well known.
Suppose $P$ is a submodule of the Bergman module $\ber$, then the following are equivalent:
(1) The submodule $P$ is essential normal.
(2) The quotient module $Q$ is essential normal.
(3) $[P, M_{z_i}]\in \K$, $i=1,\cdots,n$.
(4) $[Q, M_{z_i}]\in \K$, $i=1,\cdots,n$.
Next we introduce some elementary tools in complex harmonic analysis, which will be used frequently in this paper.
For $z\in\bn$, write $P_z$ for the orthogonal projection onto the complex line $\mathbb{C}z$ and $Q_z=I-P_z$. The function
$$\varphi_z(w)=\frac{z-P_z(w)-(1-|z|^2)^{1/2}Q_z(w)}{1-\langle w,z\rangle}$$
is the (unique) automorphism of $\bn$ that satisfies $\varphi_z\circ\varphi_z=id$ and $\varphi_z(0)=z$. The following Lemma is in Chapter 2 of <cit.>.
Suppose $a$, $z$, $w\in\bn$, then
$$1-\langle\varphi_a(z),\varphi_a(w)\rangle=\frac{(1-\langle a,a\rangle)(1-\langle z,w\rangle)}{(1-\langle z,a\rangle)(1-\langle a,w\rangle)}.$$
(2) As a consequence of (1),
$$1-|\varphi_a(z)|^2=\frac{(1-|a|^2)(1-|z|^2)}{|1-\langle z,a\rangle|^2}.$$
(3) The Jacobian of the automorphism $\varphi_z$ is
$$(J\varphi_z(w))=\frac{(1-|z|^2)^{n+1}}{|1-\langle w,z\rangle|^{2(n+1)}}.$$
The pseudo-hyperbolic metric on $\bn$ is defined by
And the hyperbolic metric is defined by
Thus $\rho(z,w)=\tanh\beta(z,w)$. It's well known that the two metrics are invariant under actions of $Aut(\bn)$, the group of holomorphic automorphisms of $\bn$. That is, given $\psi\in Aut(\bn)$,
for all $z$, $w\in\bn$. For $r>0$, $z\in\bn$, write
$$D(z,r)=\{w\in\bn: \beta(w,z)< r\}=\{w\in\bn: \rho(w,z)< s_r\},$$
where $s_r=\tanh r$. In this paper, we use $D(z,r)$ to denote the hyperbolic ball in $\bn$ and use $D_d(z',r)$ to denote the hyperbolic ball in $\mathbb{B}_d$. The notation $B(z,\delta)$ is used to denote the Euclidian ball with center $z$ and radius $\delta$.
For $z\in\bn$, $r>0$, the hyperbolic ball $D(z,r)$ consists of all $w$ that satisfy:
where $P=P_z$, $Q=Q_z$, and
Thus $D(z,r)$ is an ellipsoid with center $c$, radius of $s_r\rho$ in the $z$ direction and $s_r\sqrt{\rho}$ in the directions perpendicular to $z$. Therefore the Lebesgue measure of $D(z,r)$ is
where $C>0$ is a constant depending only on $n$.
Note that when we fix $r$, $\rho$ is comparable with $1-|z|^2$. Hence $v(D(z,r))$ is comparable with $(1-|z|^2)^{n+1}$.
Suppose $\nu$ is a positive, finite, regular, Borel measure. The operator
$$T_{\nu}f(z)=\int_{\bn}\frac{f(w)}{(1-\langle z,w\rangle)^{n+1}}d\nu(w)$$
defines an analytic function for every $f\in H^{\infty}$. The following lemma can be seen from the proof of Lemma 2.1 in <cit.>.
Let $\nu$ be a positive, finite, regular, Borel measure on $\bn$ and $r>0$. Then the following are equivalent.
When one of these conditions holds, $\nu$ is called a Carleson measure ( for $\ber$).
(1)$\sup_{z\in\bn}\int_{\bn}\frac{(1-|z|^2)^{n+1}}{|1-\langle w, z\rangle|^{2(n+1)}}d\nu(w)<\infty,$
(2)$\exists C>0:\int|f|^2d\nu\leq C\int|f|^2dv~\mbox{for all}~f\in\ber,$
(4)$T_{\nu}$ extends to a bounded linear operator on $\ber$.
Suppose $\nu$ is a Carleson measure, by Fubini's Theorem, we have:
$$\langle Tf,g\rangle=\int_{\bn}f(w)\overline{g(w)}d\nu(w),~~~\forall f, g\in\ber.$$
The operator $T_{\nu}$ plays an important role in this paper. As a corollary of Theorem 7.3 in <cit.>, we have:
Suppose $\nu$ is a Carleson measure, then the operator $T_{\nu}$ belongs to the Toeplitz algebra of $L^{\infty}$ symbols $\mathcal{T}(L^{\infty})$.
The following lemma is crucial to our proof of essential normality. One can find a proof in <cit.>.
If $f\in C(\clb)$, then $T_f$ essentially commutes with every operator in the Toeplitz algebra $\toe$.
We will use the following lemma frequently in calculation. One can find a proof in <cit.>.
For $z\in\bn$, $c$ real, $t>-1$, define
I_c(z)=\int_S\frac{d\sigma(\zeta)}{|1-\langle z,\zeta\rangle|^{n+c}}
J_{c,t}(z)=\int_{\bn}\frac{(1-|w|^2)^tdv(w)}{|1-\langle z,w\rangle|^{n+1+t+c}}.
When $c<0$, then $I_c$ and $J_{c,t}$ are bounded in $\bn$. When $c>0$, then
I_c(z)\approx(1-|z|^2)^{-c}\approx J_{c,t}(z).
I_0(z)\approx \log\frac{1}{1-|z|^2}\approx J_{0,t}(z).
The notation $a(z)\approx b(z)$ means that the ratio $a(z)/b(z)$ has a positive finite limit as $|z|\to1$.
§ CARLESON MEASURE AND HOLOMORPHIC EXTENSION
In this section we reveal some connection between the Geometric Arveson-Douglas Conjecture and the holomorphic extension theory. Our idea come from Suárez's paper <cit.>.The following theorem is the main theorem of this section. It's connection with holomorphic extension theory is discussed in Remark <ref> in the end of this section. Also, a new result extending that of <cit.> is stated in Remark <ref> in the next section.
If there exists a positive, finite, regular, Borel measure $\mu$ on $M$ such that the $L_{\mu}^2$ norm and Bergman norm are equivalent on $Q$, i.e., $\exists C, c>0$ such that $\forall f\in Q$,
$$c\|f\|^2\leq\int_M|f(w)|^2d\mu(w)\leq C\|f\|^2,$$
then the submodule $P$ is essentially normal.
First, we prove that the measure $\mu$ is a Carleson measure. From the assumption, we have for any $z\in\bn$,
\begin{eqnarray*}
\int_{\bn}\frac{(1-|z|^2)^{n+1}}{|1-\langle w,~z\rangle|^{2(n+1)}}d\mu(w)&=&\int_{M}|k_z(w)|^2d\mu(w)\\
\end{eqnarray*}
The second equality is because $k_z-Qk_z\in P$, therefore $k_z(w)=Qk_z(w), \forall w\in M$.
By Lemma <ref>, $\mu$ is a Carleson measure.
Next we show that the projection $P$ is a continuous function calculous of $T_{\mu}$ and therefore is in the Toeplitz algebra. From the equation
$$\langle T_{\mu}f,~f\rangle=\int_M|f(w)|^2d\mu(w),~~~\forall f\in\ber$$
we see that $T_{\mu}$ is positive and vanishes on $P$. Also, the equivalence of $L^2(\mu)$-norm and Bergman norm on $Q$ implies that $T_{\mu}$ is bounded below on $Q$. Therefore $0$ is isolated in $\sigma(T_{\mu})$ and $P=\ker T_{\mu}$. Take any continuous function $f$ on $\mathbb{R}$ that vanishes at $0$ and equals $1$ on the rest of the spectrum, then $Q=f(T_{\mu})$.
Finally, by Lemma <ref>, $Q$ is in the Toeplitz algebra. By Lemma <ref> and Lemma <ref>, the quotient module $Q$ is essentially normal and so is the submodule $P$. This completes the proof.
As a consequence of the proof of Theorem <ref>, we have the following:
Assume the same as Theorem <ref>, then the projection operators $P$ and $Q$ are in the Toeplitz algevra $\toe$.
The following example is the starting point of our research.
Suppose $M$ is the intersection of a $d$-dimension hyperplane and $\bn$, where $d<n$. We can identify $M$ with the unit ball in $\mathbb{C}^d$. Let $P$ be the submodule of $\ber$ consisting of all the functions that vanish on $M$ and $Q$ be the orthogonal complement of $P$.
Let $\rho$ be the weighted bergman measure on $M$: $d\rho=c(1-|z|^2)^{n-d}dm_d$, where $m_d$ is the Lebesgue measure on $M$ and $c>0$ is chosen such that $\rho(M)=1$ . It is well-known that the weighted Bergman space on $M$ defined by
$$L_{a,n-d}^2(M)=\{f~\mbox{is an analytic function on}~M:~\int_M|f(w)|^2d\rho(w)<\infty\}$$
is a reproducing kernel Hilbert space on $M$ with reproducing kernels
$$\{K_z^{n-d}(w)=\frac{1}{(1-\langle w,~z\rangle)^{n+1}}:~z\in M\}.$$
Since $Q$ is also the reproducing kernel Hilbert space with the same reproducing kernels, one can identify the two spaces(cf.<cit.>).
It's also clear that $\rho$ is a Carleson measure for the $n$-dimensional Bergman space $\ber$. Then the equation
$$\langle T_{\rho}f,~g\rangle=\int_Mf(w)\overline{g(w)}d\rho(w),~~~\forall f,~g\in\ber$$
shows that $T_{\rho}=Q$.
From Example <ref>, one might suspect that there is always a measure on $M$ that defines the same norm on $Q$. We show that this is not the case, even when $M$ consists of finite points.
Suppose $M=\{a_1,\cdots,a_m\}\subset\bn$, then there exists a positive measure $\mu$ on $M$ such that $\forall f\in Q$,
$$\|f\|^2=\int_M |f|^2d\mu$$
if and only if $m=1$.
The “if” part is obvious, we prove the “only if” part.
Suppose $\mu$ is supported on $M$ such that the equation holds, then $\mu=\sum\limits_{i=1}^mc_i\delta_i$, where $c_i\geq0$ and $\delta_i$ are the point masses at $a_i$, $i=1,\cdots,m$. For any tuple $x_1,\cdots,x_m\in\mathbb{C}$, let $x=\sum\limits_{i=1}^mx_ik_{a_i}\in Q$. Then
\|x\|^2=\sum_{i,j}x_i\overline{x_j}\langle k_{a_i}, k_{a_j}\rangle.
On the other hand,
\begin{eqnarray*}
\int_M|x(w)|^2d\mu(w)&=&\sum_{i=1}^m c_i|x(a_i)|^2\\
&=&\sum_{i=1}^mc_i(1-|a_i|^2)^{-(n+1)}|\sum_{j=1}^mx_j\langle k_{a_j}, k_{a_i}\rangle|^2.
\end{eqnarray*}
Let $G$ be the $m\times m$ matrix $(\langle k_{a_i}, k_{a_j}\rangle)_{ij}$, then
\|x\|^2=\begin{pmatrix}x_1& \dots & x_m\end{pmatrix}G\begin{pmatrix}\overline{x_1}\\ \vdots\\ \overline{x_m}\end{pmatrix}
\int_M|x(w)|^2d\mu(w)=\begin{pmatrix}x_1&\dots&x_m\end{pmatrix}G\begin{pmatrix}d_1&\dots&0\\ \vdots&\ddots&\vdots\\0&\dots&d_m\end{pmatrix}G^*\begin{pmatrix}\overline{x_1}\\ \vdots\\ \overline{x_m}\end{pmatrix}
where $d_i=c_i(1-|a_i|^2)^{-(n+1)}$. Since $x_i$ are arbitrary, we have
G=G\begin{pmatrix}d_1&\dots&0\\ \vdots&\ddots&\vdots\\0&\dots&d_m\end{pmatrix}G^*.
This only holds when $G$ is diagonal, which implies $m=1$.
Given a positive Carleson measure $\mu$ on $M$, the restriction map
R: \ber\to L^2(\mu), f\mapsto f|_M
is bounded. Assume that $\ker R=P$, then by the open mapping theorem, the hypotheses in Theorem <ref> is equivalent to that $R$ has closed range. If $L$ is a closed subspace of $L^2(\mu)$ containing $Range R$ and there is a bounded linear operator
$E: L\to \ber$ such that $RE=Id_{L}$, then $Range R=L$, therefore is closed. These seems redundant but will be very useful in practice. For example, given Theorem 4.3 in <cit.>, one easily sees that the weighted measure in <cit.> satisfies Theorem <ref>. So the essential normality of the corresponding submodule follows. On the other hand, suppose $\mu$ satisfies the assumption of Theorem <ref>, the map
E: Range R\to Q\subset\ber, Rf\mapsto f\in Q
is an extension operator. So this also provides a way to solve the holomorphic extension problem related to the topic in <cit.>.
Theorem <ref> also relates to the reverse Carleson inequality(cf. <cit.>). One can think of the requirements in Theorem <ref> as a reverse Carleson inequality on the zero variety.
Suppose $M$ is an interpolating sequence, then the corresponding projection $P$ is in the Toeplitz algebra and the submodule $P$ is essentially normal.
§ GEOMETRIC ARVESON-DOUGLAS CONJECTURE
In this section, we construct a measure satisfying the hypotheses of Theorem <ref> for any complex analytic subset that intersects $\pbn$ transversally and has no singular point on $\pbn$.
Let $\Omega$ be a complex manifold. A set $A\subset\Omega$ is called a (complex) analytic subset of $\Omega$ if for each point $a\in\Omega$ there are a neighborhood $U\ni a$ and functions $f_1,\cdots,f_N$ holomorphic in this neighborhood such that
A\cap U=\{z\in U: f_1(z)=\cdots=f_N(z)=0\}.
A point $a\in A$ is called regular if there is a neighborhood $U\ni a$ in $\Omega$ such that $A\cap U$ is a complex submanifold of $\Omega$. A point $a\in A$ is called a singular point of $A$ if it's not regular.
Let $Y$ be a manifold and let $X, Z$ be two submanifolds of $Y$. We say that the submanifolds $X$ and $Z$ are transversal if $\forall x\in X\cap Z$, $T_x(X)+T_x(Z)=T_x(Y)$.
Suppose $\tilde{M}$ is a complex analytic subset of an open neighborhood of $\clb$ satisfying the following conditions:
(1) $\tilde{M}$ intersects $\pbn$ transversally.
(2) $\tilde{M}$ has no singular point on $\pbn$.
Let $M=\tilde{M}\cap\bn$ and let $P=\{f\in\ber: f|_M=0\}$. Then the submodule $P$ is essentially normal.
Note that in this case, condition (1) is equivalent to that $\tilde{M}$ is not tangent with $\pbn$ at every point of $\tilde{M}\cap\pbn$. Condition (2) implies that $\tilde{M}$ has only finite singular points inside $\bn$.
Under the assumption of Theorem <ref>, the projection operator $P$ is in the Toeplitz algebra $\toe$.
In order to prove Theorem <ref>, we need to establish a few lemmas.
Let $\alpha$ be the intersection of a $d$-dimensional affine space and $\bn$. Then $\alpha$ is a $d$-dimensional ball. Let $r$ be the radius of $\alpha$ and $v$ be the volume measure on $\alpha$. Then for any function $f$ holomorphic on $\alpha$ and any $R>0$, $z\in\alpha$,
$$\int_{\alpha\cap D(z,R)}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z,w\rangle)^{n+1}}dv(w)=r^{-2}C_Rf(z).$$
Here $D_d(0,R)$ means the hyperbolic ball in $\mathbb{B}_d$ centered at $0$ with radius $R$ and $\nu$ is the volume measure on $\mathbb{B}_d$.
Let $z_0$ be the center of $\alpha$ and let
The affine space $\beta$ is the intersection of a hyperplane and $\bn$, therefore can be identified with $\mathbb{B}_d$. Clearly $\phi$ is biholomorphic. For $z\in\alpha$, consider the map
By <cit.>, $\gamma$ is an affine space containing $0$. Hence $\gamma$ can also be identified with $\mathbb{B}_d$. So $\varphi_z\phi^{-1}$ is an automorphism of $\mathbb{B}_d$ and therefore preserves the hyperbolic metric. We get
\begin{eqnarray*}
\phi(D(z,R)\cap\alpha)
\end{eqnarray*}
\begin{eqnarray*}
&&\int_{\alpha\cap D(z,R)}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z,w\rangle)^{n+1}}dv(w)\\
&=&\int_{D_d(\phi(z),R)}f\phi^{-1}(\eta)\frac{(1-|\phi^{-1}(\eta)|^2)^{n-d}}{(1-\langle z,\phi^{-1}(\eta)\rangle)^{n+1}}d\nu(\phi^{-1}(\eta))\\
\end{eqnarray*}
The last equation comes from the following argument.
In general, if $g$ is holomorphic on $\mathbb{B}_d$, for $R>0$ and $\xi\in\mathbb{B}_d$,
\begin{eqnarray*}
&=&\int_{D_d(0,R)}g\varphi_{\xi}(w)\frac{(1-|\varphi_{\xi}(w)|^2)^{n-d}}{|1-\langle\xi,\varphi_{\xi}(w)\rangle)^{n+1}}\frac{(1-|\xi|^2)^{d+1}}{(1-\langle w,\xi\rangle|^{2(d+1)}}d\nu(w)\\
&=&\int_{D_d(0,R)}g\varphi_{\xi}(w)\frac{(1-\langle\xi,w\rangle)^{n+1}(1-|\xi|^2)^{n-d}(1-|w|^2)^{n-d}(1-|\xi|^2)^{d+1}}{(1-|\xi|^2)^{n+1}|1-\langle\xi,w\rangle|^{2(n-d)}|1-\langle w,\xi\rangle|^{2(d+1)}}d\nu(w)\\
&=&\int_{D_d(0,R)}g\varphi_{\xi}(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle w,\xi\rangle)^{n+1}}d\nu(w)\\
\end{eqnarray*}
This completes the proof.
For $t>0$, we have
$$\lim_{r\to1-}\sup_{z\in\mathbb{B}_d}\int_{w\in\mathbb{B}_d:r<|w|<1}\frac{(1-|w|^2)^t}{|1-\langle z,w\rangle|^{d+1}}d\nu(w)=0.$$
$$I(z)=\int_S\frac{1}{|1-\langle z,\zeta\rangle|^{d+1}}d\sigma(\zeta).$$
Where $S$ is the unit sphere in $\mathbb{C}^d$ and $\sigma$ is the volume measure on $S$.
By Lemma <ref>, there exists $C>0$ such that
$$I(z)\leq C(1-|z|^2)^{-1}.$$
\begin{eqnarray*}
\int_{r<|w|<1}\frac{(1-|w|^2)^t}{|1-\langle z,w\rangle|^{d+1}}d\nu(w)&=&\int_r^1\int_S\frac{(1-s^2)^t}{|1-\langle z,s\zeta\rangle|^{d+1}}s^{2d-1}d\sigma(\zeta)ds\\
&\leq&C\int_r^1(1-s^2)^{t-1}ds\to 0.~~~~~~(r\to1-)
\end{eqnarray*}
This completes the proof.
Suppose $\tilde{M}$ is as in Theorem <ref>. We first assume that $\tilde{M}$ is connected. For $0\leq s<t\leq 1$, define
$$M_s^t=\{z\in M|~s\leq|z|<t\}.$$
Write $M_s=M_s^1, M^t=M_0^t$.
Since $\tilde{M}$ has no singular point on $\pbn$, we can cover $\pbn\cap\tilde{M}$ with finite open sets $\{U_i\}$, $U_i\subset\tilde{M}$ such that:
(1) For each $i$, we can find $n-1$ of the canonical basis of $\mathbb{C}^n$, denoted $e_{i_1},\cdots,e_{i_{n-1}}$ such that for any $z\in U_i$, the $n$ vectors $\{z, e_{i_1},\cdots, e_{i_{n-1}}\}$ spans $\mathbb{C}^n$.
(2) $\tilde{M}$ has local coordinates on each $U_i$, i.e., there exists open set $\Omega_i\subset\mathbb{C}^d$ and $\varphi_i:\Omega_i\to U_i$ which is one to one and holomorphic.
Fix $z\in U_i$, apply the Gram-Schimidt process to $\{z,e_{i_1},\cdots,e_{i_{n-1}}\}$ to obtain a new basis $\{f_1^z,\cdots,f_n^z\}$, then $z=(z_1,0,\cdots,0)$ under this basis. Let $G^z:\Omega_i\to U_i$, $G^z=(g_1^z,\cdots,g_n^z)$ be the expression of $\varphi_i$ under the new basis. Note that the new basis and expression depend continuously on $z$.
Since $\tilde{M}$ intersects $\pbn$ transversally, by possibly refining the cover $\{U_i\}$ we can assume that for each $U_i$, $\forall z\in U_i$, $(\frac{\partial g_1^z}{\partial z_1},\cdots,\frac{\partial g^z_1}{\partial z_d})$ is non-zero at $z$. Since the matrix $[\frac{\partial g_i^z}{\partial z_j}(z)]_{1\leq i\leq n,1\leq j\leq d}$ has rank $d$, by possibly refining $\{U_i\}$ again we could get $2\leq k_1,\cdots,k_{d-1}\leq n$ for each $U_i$, such that the determinant $\frac{\partial(g_1^z,g_{k_1}^z,\cdots,g_{k_{d-1}}^z)}{\partial(z_1,\cdots,z_d)}|_z\neq0$, $\forall z\in U_i$. Let $\epsilon$ be the Lebesgue number of the cover $\{U_i\}$ and let $V_i=\{z\in U_i|~d(z,\partial U_i)>\frac{1}{2}\epsilon\}$, then $\pbn\cap\tilde{M}\subset\cup V_i$. The function $\frac{\partial(g_1^z,g_{k_1}^z,\cdots,g_{k_{d-1}}^z)}{\partial(z_1,\cdots,z_d)}(w)$ is uniformly continuous on $\{(z,w)|z\in\bar{V_i},w\in U_i\}$. Therefore $\exists\delta>0$ such that $\forall z\in V_i$, $\forall w\in B(z,\delta)$, $\frac{\partial(g_1^z,g_{k_1}^z,\cdots,g_{k_{d-1}}^z)}{\partial(z_1,\cdots,z_d)}(w)\neq0$. By the implicit function theorem, we have:
There exists a finite open cover $\{V_i\}$ of $\pbn\cap\tilde{M}$ and $\delta>0$ such that for any fixed $V_i$, we can pick $d-1$ numbers out of $\{2,\cdots,n\}$, assume they are $\{2,\cdots,d\}$ without loss of generality, such that $\forall z\in\bar{V_i}$ and $\forall w\in B(z, \delta)$,
under the basis $\{f_1^z,\cdots,f_n^z\}$, where $w'=(w_1,\cdots,w_d)$. The functions $F_i^z(w')$ are holomorphic on $w'$ and depends continuously on $z$.
In the later discussion, whenever we fix a $z\in V_i$, we will discuss under the new basis $\{f_i^z\}_{i=1}^n$ and the new expression $(w',F_{d+1}^z,\cdots,F_n^z)$ and we will omit the superscript “$z$” for convenience. Moreover, we will denote any constant that depends only on $M$ by $C$ as long as it doesn't cause confusion. So $C$ may refer to different constant in different places.
By Proposition 1 in <cit.>, the assumptions in Theorem <ref> implies $\tilde{M}$ has only finite singular points in $\bn$. Let $\Sigma=\{z_1,\cdots,z_m\}$ be the set of all singular points of $\tilde{M}$ inside $\bn$.
Take $0<s_1<1$ such that $\Sigma\cap M_{s_1}=\emptyset$. Then the volume measure $v_d$ is well-defined on $M_{s_1}$. In local coordinates, $v_d$ corresponds to the volume form $E(w)dx_1\wedge dy_1\wedge\cdots\wedge dy_d$, where $E(w)$ is the square root of the absolute value of the determinant of the matrix representation of the metric tensor on $M_{s_1}$. Note that $E(w)$ is uniformly continuous on $z$ and $w$. Let
where $\delta_{z_i}$ is the point mass at $z_i$. For $s_1<s<1$, let
We will prove that for $s$ sufficiently close to $1$, $\mu_s$ satisfies the assumption of Theorem <ref>, therefore Theorem <ref> holds.
Fix $z\in V_i$(and the basis depending on $z$), define a map
$$p_z:\tilde{M}\cap B(z,\delta)\to T\tilde{M}|_z$$
$$(w',F_{d+1}(w'),\cdots,F_n(w'))\mapsto (w',\sum_{i=1}^d\frac{\partial F_{d+1}}{\partial w_i}(z')(w_i-z_i),\cdots,\sum_{i=1}^d\frac{\partial F_n}{\partial w_i}(z')(w_i-z_i))$$
Here $T\tilde{M}|_z$ is the tangent space of $\tilde{M}$ at $z$. Note that by construction, $F_i(z')=0, i=d+1,\cdots,n$.
Clearly, $p_z$ is one to one and holomorphic, $p_z(w)-w\perp z$ and
Fix $R>0$, then there exists $1>s_2>s_1$, such that
(1) $\forall z\in M_{s_2}$, $D(z,R)\subset B(z,\delta)$.
(2) $\forall z\in M_{s_2}$, $\forall w\in D(z,R)$, $p_z(w)\in\bn$.
(3) $\sup\limits_{w\in D(z,R)}|\frac{1-|p_z(w)|^2}{1-|w|^2}-1|\to0,~~~|z|\to1$.
(4) $\sup\limits_{w\in D(z,R)}\beta(p_z(w),w)\to 0$, $|z|\to1$.
By Lemma <ref>, it's easy to see that (1) holds as long as we take $s_2$ sufficiently close to $1$.
To prove (2), we notice first that Lemma <ref> also implies
$$\sup_{w\in D(z,R)}|w-z|=O((1-|z|^2)^{\frac{1}{2}}).$$
$$\sup_{w\in D(z,R)}|p_z(w)-w|=O(1-|z|^2).$$
Since $\langle z,p_z(w)\rangle=\langle z,w\rangle\neq0$, $\varphi_z(p_z(w))$ is well defined. It's easy to verify that
$$\varphi_z(\xi)\in\bn \mbox{ if and only if }\xi\in\bn.$$
So we only need to make sure that $\varphi_z(p_z(w))\in\bn$. Since
$$|\varphi_z(p_z(w))-\varphi_z(w)|=\frac{(1-|z|^2)^{\frac{1}{2}}}{|1-\langle w,z\rangle|}O(1-|z|^2)=O((1-|z|^2)^{\frac{1}{2}})$$
$$|\varphi_z(w)|\leq s_{\sss R},$$
when we take $s_2$ sufficiently close to $1$, we have $|\varphi_z(p_z(w))|<1$. Therefore (2) is proved.
We prove (4) first. Take $s_2$ so close to $1$ that $\forall z\in M_{s_2}$, $\forall w\in D(z,R)$, $\varphi_z(p_z(w))\in D(0,2R)$.
On $D(0,2R)$, the hyperbolic distance and Euclidian distance are equivalent. Hence
$$\beta(p_z(w),w)=\beta(\varphi_z(p_z(w)),\varphi_z(w))\leq C|\varphi_z(p_z(w))-\varphi_z(w)|\to0,$$
as $|z|\to1$.
Finally, since $p_z(w)=\varphi_w\varphi_w(p_z(w))$ and $|\varphi_w(p_z(w))|\to0$,
apply Lemma <ref> (2), we have
Notice that $|\varphi_w(p_z(w))|=\rho(w,p_z(w))$ tends to $0$ uniformly. We have (3). This completes the proof.
For $1>s>s_1$, the measure
is a Carleson measure.
Fix $R>0$, by Lemma <ref>, we only need to prove that
$$\int_{D(z,R)\cap M_s}(1-|w|^2)^{n-d}dv_d(w)\leq C(1-|z|^2)^{n+1}$$
for some constant $C>0$. Since
$$\frac{1-|w|^2}{1-|z|^2}=\frac{1-|\varphi_z(w)|^2}{|1-\langle\varphi_z(w),z\rangle|^2}\leq C,$$
it suffices to show
$$v_d(D(z,R))\leq C(1-|z|^2)^{d+1}.$$
$$v_d(D(z,R))=\int_{\{w':w\in D(z,R)\}}E(w')dv(w')\leq C\int_{\{w':w\in D(z,R)\}}dv(w').$$
By definition, $|\varphi_{z'}(w')|\leq|\varphi_z(w)|$, so
$$\{w':w\in D(z,R)\}\subset D_d(z',R).$$
$$v_d(D(z,R))\leq C(1-|z|^2)^{d+1}.$$
This completes the proof.
There exists a constant $C>0$ such that
The lemma follows from the fact that $T_{\delta}$ has closed range and is positive.
For any $\epsilon>0$, there exists $1>s_3>s_1$ and $R>0$ such that,
$$\sup_{z\in M_{s_3}}\int_{M_{s_3}}\frac{(1-|z|^2)^{\frac{n-d}{2}}(1-|w|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)<\infty.$$
(2) $\forall z\in M_{s_3}$,
$$\int_{M_{s_3}\backslash D(z,R)}\frac{(1-|z|^2)^{\frac{n-d}{2}}(1-|w|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)<\epsilon.$$
We prove (1) and (2) together. For $z\in\bar{V_i}\cap M_{s_1}$ and $R>0$,
\begin{eqnarray*}
& &\int_{M_{s_3}}\frac{(1-|z|^2)^{\frac{n-d}{2}}(1-|w|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)\\
&\leq&\int_{B(z,\delta)\cap M_{s_3}}\frac{(1-|z|^2)^{\frac{n-d}{2}}(1-|w|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)\\
&+&\int_{M_{s_3}\backslash B(z,\delta)}\frac{(1-|z|^2)^{\frac{n-d}{2}}(1-|w|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)
\end{eqnarray*}
For the second part, the integrand is smaller than $C\delta^{-2(n+1)}(1-s_3^2)^{n-d}$ because
$$|1-\langle z,w\rangle|\geq(1-Re\langle z,w\rangle)\geq\frac{1}{2}(|z|^2+|w|^2-2Re\langle z,w\rangle)=\frac{1}{2}|z-w|^2.$$
So when $s_3$ is close to $1$, the second part will be smaller than $\frac{1}{2}\epsilon$.
For the first part,
\begin{eqnarray*}
&&\int_{B(z,\delta)\cap M_{s_3}}\frac{(1-|z|^2)^{\frac{n-d}{2}}(1-|w|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)\\
&=&\int_{\{w':w\in B(z,\delta)\cap M_{s_3}\}}\frac{(1-|z'|^2)^{\frac{n-d}{2}}(1-|w|^2)^{\frac{n-d}{2}}}{|1-\langle z',w'\rangle|^{n+1}}E(w')dv(w')\\
&\leq&C\int_{\mathbb{B}_d}\frac{(1-|z'|^2)^{\frac{n-d}{2}}(1-|w'|^2)^{\frac{n-d}{2}}}{|1-\langle z',w'\rangle|^{n+1}}dv(w')\\
&=&C\int_{\mathbb{B}_d}\frac{(1-|z'|^2)^{\frac{n-d}{2}}(1-|\varphi_{z'}(\eta')|^2)^{\frac{n-d}{2}}}{|1-\langle z',\varphi_{z'}(\eta')\rangle|^{n+1}}\frac{(1-|z'|^2)^{d+1}}{|1-\langle z',\eta'\rangle|^{2(d+1)}}dv(\eta')\\
&=&C\int_{\mathbb{B}_d}\frac{(1-|\eta'|^2)^{\frac{n-d}{2}}}{|1-\langle z',\eta'\rangle|^{d+1}}dv(\eta').
\end{eqnarray*}
Where the second equality from the bottom is by change of variable $w'=\varphi_{z'}(\eta')$. By the proof of Lemma <ref>, the integral above is uniformly bounded, this proves (1).
The above argument also gives
\begin{eqnarray*}
&&\int_{M_{s_3}\cap B(z,\delta)\backslash D(z,R)}\frac{(1-|z|^2)^{\frac{n-d}{2}}(1-|w|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)\\
&\leq&\int_{\{\varphi_{z'}(w'):w\in B(z,\delta)\cap M_{s_3}\backslash D(z,R)\}}\frac{(1-|\eta'|^2)^{\frac{n-d}{2}}}{|1-\langle z',\eta'\rangle|^{d+1}}dv(\eta').
\end{eqnarray*}
Claim: There exists $c>0$ such that for any $R>0$,
$$\{\varphi_{z'}(w'):w\in B(z,\delta)\cap M_{s_3}\backslash D(z,R)\}\cap cs_R\mathbb{B}_d=\emptyset.$$
Assume the claim, then (2) follows from Lemma <ref>.
Now we prove the claim. For $z\in M_{s_1}$, $w\in B(z,\delta)$, let $\eta=\varphi_z(w)$, $\eta'$ be the first $d$ entries of $\eta$. Then $\eta'=\varphi_{z'}(w')$.
\begin{eqnarray*}
|\eta|^2-|\eta'|^2&=&\frac{1-|z|^2}{|1-\langle w,z\rangle|^2}\sum_{i=d+1}^n|f_i^z(w')|^2\\
&\leq& C\frac{1-|z'|^2}{|1-\langle w',z'\rangle|^2}|w'-z'|^2\\
&\leq& C\bigg(\frac{1}{|1-\langle w',z'\rangle|^2}|z_1-w_1|^2+\sum_{i=2}^d\frac{1-|z'|^2}{|1-\langle w',z'\rangle|^2}|w_i|^2\bigg)\\
\end{eqnarray*}
If $w\notin D(z,R)$, then $|\eta|=|\varphi_z(w)|\geq s_{\sss R}$. Therefore $|\eta'|\geq\frac{1}{\sqrt{C+1}}s_{\sss R}$. Take $c=\frac{1}{\sqrt{C+1}}$ and the proof is complete.
proof of Theorem <ref>
First, we prove the theorem under the assumption that $M$ is connected. Then the dimension of $M$ at every regular point is the same. Let $0<d<n$ be the dimension.
Let $\epsilon>0$ be determined later. Let $R>0$ and $s_3$, $s_2$ be as in Lemma <ref> and Lemma <ref>. Let $s=\max\{s_2, s_3\}$ and $1>s'>s$ be such that $\forall z\in M_{s'}$, $D(z,2R)\cap M\subset M_s$. We may enlarge $s$ (and the associated $s'$ )in the proof and still denote it by $s$ ( $s'$ ).
We will prove that $T_{\mu_s}^3\geq cT_{\mu_s}$ for some $c>0$. Since $T_{\mu_s}$ is self-adjoint and $\ker T_{\mu_s}=P$. $T_{\mu_s}$ is bounded below on $Q$. This will give us the desired result, by Theorem <ref>.
Denote $\mathcal{P}(\mu_s)$ to be the closure of the restriction of all analytic polynomials to $M$ in $L^2(\mu_s)$ . Clearly $Range R\subset\mathcal{P}(\mu_s)$. Suppose $s'<t<1$, for every $z\in M_{s'}^t$, there is an open neighborhood $U\ni z$ contained in $M_s$ and doesn't touch $\pbn$ such that $M$ has local coordinates on $U$. It's easy to prove that for a compact set $V\subset U$, there is a constant $C>0$ such that $\forall p\in \mathbb{C}[z_1,\cdots,z_n]$, $\forall z\in V$,
$$|p(z)|^2\leq C\int_U|p(w)|^2dv_d(w)\leq C'\int_M|p(w)|^2d\mu_s(w).$$
Clearly the same is true for $z=z_i, i=1,\cdots,m$.
Suppose $K\subset M$ is compact, then $K$ is contained in $M^t$ for some $t<1$. Assume $t>s'$, we can cover $\overline{M_{s'}^t}$ with finite compact neighborhoods $V_i$ as above. So there is a constant $C>0$ such that for any analytic polynomial $p$, $\forall z\in M_{s'}^t\cup \Sigma$,
|p(z)|^2\leq C\int_M|p(w)|^2d\mu_s(w).
For $z\in M^{s'}\backslash\Sigma$, using the maximum modulus principle, we have
|p(z)|^2\leq\sup_{w\in M_{s'}^t\cup\Sigma}|p(w)|^2\leq C\int_M|p(w)|^2d\mu_s(w).
This means the evaluation at every point in $M$ is bounded on $\mathcal{P}(\mu_s)$. Therefore we can think of $f\in\mathcal{P}(\mu_s)$ as a pointwisely defined function on $M$(instead of an equivalence class in $L^2(\mu_s)$). Also, it's easy to prove that under this definition, $\forall f\in\ber, \forall z\in M, Rf(z)=f(z)$.
In conclusion, the space $\mathcal{P}(\mu_s)$ is a reproducing kernel Hilbert space on $M$, and the reproducing kernels on any compact subset are uniformly bounded.
Consider the operator
$$T:\mathcal{P}(\mu_s)\to L^2(\mu_s),~Tf=f\chi_{\sss M_s^{s'}}.$$
Then $T$ is compact: suppose $\{f_k\}\subset \mathcal{P}(\mu_s)$ and $f_k$ weakly converges to $0$. Then $f_k$ converges to $0$ pointwisely and are uniformly bounded on $M_s^{s'}$. By the above argument and the dominance convergence theorem,
\|Tf_k\|^2=\int_{M_s^{s'}}|f(w)|^2d\mu_s(w)\to0.
So $T$ is compact, therefore $|T|$ is compact.
Since $\|Tf\|=\||T|f\|$, $\forall f\in\mathcal{P}(\mu_s)$. Using the spectral decomposition of $|T|$, we see that for any $0<a<1$, there exists a finite codimensional subspace $L\subset \mathcal{P}(\mu_s)$, such that $\forall f\in L$,
$$\int_M|f|^2d\mu_{s'}\geq a\int_M|f|^2d\mu_s.$$
We will use this in the last part of our proof.
Define the operator
$$\tTmus: \Ps\to\Ps$$
$$\tTmus f(z)=\int_Mf(w)\frac{1}{(1-\langle z,w\rangle)^{n+1}}d\mu_s(w),~\forall z\in M$$
By definition, $\forall F\in\ber$, $\tTmus Rf=R\Tmus F$. Since
\|R\Tmus F\|_{\mu_s}^2=\langle \Tmus^3 F, F\rangle\leq\|\Tmus\|^2\langle\Tmus F, F\rangle=\|RF\|_{\mu_s}^2
$\tTmus$ is bounded on $\Ps$. We will show that $\tTmus$ is bounded below.
For $z\in M_{s'}$, $f=RF\in\Ps$, $F\in\ber$,
$$\tTmus f(z)=\int_{\Sigma}f(w)K_w(z)d\mu_s(w)+\int_{M_s}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z,w\rangle)^{n+1}}dv_d(w).$$
Consider the map $p_z: D(z,2R)\cap M\to TM|_z$ defined before Lemma <ref>, by (4) of Lemma <ref>, by enlarging $s$, we could assume $\beta(p_z(w),w)<\frac{1}{2}R$, $\forall w\in D(z,2R)$.
$$p_z(D(z,2R)\cap M)\supset D(z,\frac{3}{2}R)\cap TM|_z$$
$$p_z^{-1}(D(z,\frac{3}{2}R)\cap TM|_z)\supset D(z,R)\cap M.$$
\begin{cases}
\int_{\Sigma}f(w)K_w(z)d\mu_s(w) & z\in\Sigma\\
\int_{p_z^{-1}(D(z,\frac{3}{2}R)\cap TM|_z)}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z,w\rangle)^{n+1}}dv_d(w) & z\in M_{s'}
\end{cases}
II(z)=\tTmus f(z)-I(z)
%\int_{M_{s}}f(w)\frac{(1-|w|)^{n-d}}{(1-\langle z,w\rangle)^{n+1}}dv_d(w) & z=z_i\\
%\int_{\Sigma}f(w)K_w(z)d\mu_s(w)+\int_{M_s\backslash p_z^{-1}(D(z,\frac{3}{2}R)\cap TM|_z)}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z,w\rangle)^{n+1}}dv_d(w) & z\in M_{s'}.
Then $I(z)+II(z)=\tTmus f(z)$, $\forall z\in M_{s'}\cup\Sigma$.
For $I(z)$,
$$\int_{M}|I(z)|^2d\mu_{s'}=\langle T_{\delta}^3F,F\rangle+\int_{M_{s'}}|I(z)|^2(1-|z|^2)^{n-d}dv_d(z).$$
By Lemma <ref>, the first part is greater than $c\langle T_{\delta}F,F\rangle=c\int_M|f(w)|^2d\delta$.
If $z\in M_{s'}$,
\begin{eqnarray*}
I(z)&=&\int_{p_z^{-1}(D(z,\frac{3}{2}R)\cap TM|_z)}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z,w\rangle)^{n+1}}dv_d(w)\\
&=&\int_{D(z,\frac{3}{2}R)\cap TM|_z}fp_z^{-1}(\eta)\frac{(1-|p_z^{-1}(\eta)|^2)^{n-d}}{(1-\langle z,\eta\rangle)^{n+1}}\frac{E(w')}{E(z')}dv(\eta)\\
&=&\int_{D(z,\frac{3}{2}R)\cap TM|_z}fp_z^{-1}(\eta)\frac{(1-|\eta|^2)^{n-d}}{(1-\langle z,\eta\rangle)^{n+1}}g(\eta)dv(\eta)
\end{eqnarray*}
By Lemma <ref> (3) and the absolute continuity of $E$, we could enlarge $s$ (so that the Euclidian size of $D(z,2R)$ is small enough) such that $g(\eta)$ is sufficiently close to $1$ and
$$|g(\eta)-1|\leq\epsilon g(\eta).$$
By Lemma <ref>,
$$\int_{D(z,\frac{3}{2}R)\cap TM|_z}fp_z^{-1}(\eta)\frac{(1-|\eta|^2)^{n-d}}{(1-\langle z,\eta\rangle)^{n+1}}dv(\eta)=C_zf(z),$$
where $C_z\geq C_{\frac{3}{2}R}$. And
\begin{eqnarray*}
&&|\int_{D(z,\frac{3}{2}R)\cap TM|z}fp_z^{-1}(\eta)\frac{(1-|\eta|^2)^{n-d}}{(1-\langle z,\eta\rangle)^{n+1}}(g(\eta)-1)dv(\eta)|\\
&\leq&\epsilon\int_{D(z,\frac{3}{2}R)\cap TM|_z}|fp_z^{-1}(\eta)|\frac{(1-|\eta|^2)^{n-d}}{|1-\langle z,\eta\rangle|^{n+1}}g(\eta)dv(\eta)\\
&\leq&\epsilon\int_{M_{s}}|f(w)|\frac{(1-|w|^2)^{n-d}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)
\end{eqnarray*}
\begin{eqnarray*}
& &-\epsilon^2\int_{M_{s'}}\bigg(\int_{M_s}|f(w)|\frac{(1-|w|^2)^{n-d}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)\bigg)^2(1-|z|^2)^{n-d}dv_d(z)
\end{eqnarray*}
Using Holder's inequality and Lemma <ref> (1), the second part is smaller than
\begin{eqnarray*}
&&\epsilon^2\int_{M_{s'}}\bigg(\int_{M_s}\frac{(1-|w|^2)^{\frac{n-d}{2}}(1-|z|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)\bigg)\\
& &~~\cdot\bigg(\int_{M_s}|f(w)|^2\frac{(1-|w|^2)^{\frac{3(n-d)}{2}}(1-|z|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)\bigg)dv_d(z)\\
&\leq&C\epsilon^2\int_{M_s}\int_{M_{s'}}\frac{(1-|w|^2)^{\frac{n-d}{2}}(1-|z|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(z)\\
\end{eqnarray*}
The above estimation is inspired from <cit.>. We will use the same kind of argument in the estimation of $II(z)$.
Combining the above, we have
$$\int_M|I(z)|^2d\mu_{s'}\geq C_1\int_M|f|^2d\mu_{s'}-C_2\epsilon^2\int_M|f|^2d\mu_s.$$
Next we estimate $II(z)$.
\begin{eqnarray*}
&=&\sum_{i=1}^m\bigg|\int_{M_s}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z_i,w\rangle)^{n+1}}dv_d(w)\bigg|^2(1-|z_i|^2)^{n+1}+\int_{M_{s'}}\bigg|T_{\delta}F(z)+\\
& &\int_{M_s\backslash p_z^{-1}(D(z,\frac{3}{2}R)\cap TM|_z)}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z,w\rangle)^{n+1}}dv_d(w)\bigg|^2(1-|z|^2)^{n-d}dv_d(z)\\
\end{eqnarray*}
$$A=\sum_{i=1}^m\bigg|\int_{M_s}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z_i,w\rangle)^{n+1}}dv_d(w)\bigg|^2(1-|z_i|^2)^{n+1},$$
$$C=\int_{M_{s'}}\bigg|\int_{M_s\backslash p_z^{-1}(D(z,\frac{3}{2}R)\cap TM|_z)}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z,w\rangle)^{n+1}}dv_d(w)\bigg|^2(1-|z|^2)^{n-d}dv_d(z).$$
Let $a=d(\Sigma,M_s)$, we have
\begin{eqnarray*}
\end{eqnarray*}
where the first inequality is because
|1-\langle z_i,w\rangle|\geq 1-Re\langle z_i,w\rangle\geq1/2(|z_i|^2+|w|^2-Re\langle z_i,w\rangle)=1/2|z_i-w|^2
and the second inequality is by Holder's inequality.
By taking $s$ closer to $1$, we could make
Similar argument will give us
Now we estimate $C$.
\begin{eqnarray*}
C&\leq&\int_{M_{s'}}\bigg|\int_{M_s\backslash D(z,R)}f(w)\frac{(1-|w|^2)^{n-d}}{(1-\langle z,w\rangle)^{n+1}}dv_d(w)\bigg|^2(1-|z|^2)^{n-d}dv_d(z)\\
&\leq&\int_{M_{s'}}\bigg(\int_{M_s\backslash D(z,R)}\frac{(1-|w|^2)^{\frac{n-d}{2}}(1-|z|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)\bigg)\\
& &\cdot\bigg(\int_{M_s\backslash D(z,R)}|f(w)|^2\frac{(1-|w|^2)^{\frac{3(n-d)}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(w)\bigg)(1-|z|^2)^{\frac{n-d}{2}}dv_d(z)\\
&\leq&\epsilon\int_{M_s}\bigg(\int_{M_{s'}\backslash D(w,R)}\frac{(1-|w|^2)^{\frac{n-d}{2}}(1-|z|^2)^{\frac{n-d}{2}}}{|1-\langle z,w\rangle|^{n+1}}dv_d(z)\bigg)\\
& &\cdot|f(w)|^2(1-|w|^2)^{n-d}dv_d(w)\\
\end{eqnarray*}
Combining the three inequalities, we get
Finally, we have
\begin{eqnarray*}
\int_M|\tTmus f(z)|^2d\mu_s(z)&\geq&\int_M|\tTmus f(z)|^2d\mu_{s'}(z)\\
\end{eqnarray*}
This holds for all $f\in\Ps$. From the argument in the beginning, we can find a finite codimensional space $L\subset \mathcal{P}(\mu_s)$ such that $\forall f\in L$,
Therefore $\forall f\in L$,
\int_M|\tTmus f(z)|^2d\mu_s(z)\geq(\frac{1}{2}C-C'\epsilon^2)\int_M|f|^2d\mu_s
Take $\epsilon>0$ such that $\alpha=\frac{1}{2}C-C'\epsilon^2>0$. Then
\|\tTmus f\|_{\mu_s}^2\geq\alpha\|f\|_{\mu_s}^2,~~~\forall f\in L.
Next we show that $\ker\tTmus=\{0\}$. Consider the commuting diagram
\xymatrix{
\Ps\ar[r]^{\tTmus}&\Ps
Since $\tTmus$ is positive, it suffices to show that $Range\tTmus$ is dense in $\Ps$. We already know that $Range\Tmus$ is dense in $Q$( since $\ker\Tmus=\{0\}$). Therefore $R\Tmus(Q)$ is dense in $R(Q)$, which is dense in $\Ps$. So $Range\tTmus\supset R\Tmus(Q)$ is dense in $\Ps$. Hence $\ker\tTmus=\{0\}$.
Now suppose $\tTmus$ is not bounded below, then there exists a pairwise orthogonal sequence $\{f_n\}\subset\Ps$, $\|f_n\|_{\mu_s}=1$ such that $\|\tTmus(f_n)\|_{\mu_s}\to0$, $n\to\infty$. Since $L$ is finite codimensional,
\|f_n-Lf_n\|_{\mu_s}\to0, n\to\infty.
\|\tTmus Lf_n\|_{\mu_s}\leq\|\tTmus f_n\|_{\mu_s}+\|\tTmus(f_n-Lf_n)\|_{\mu_s}\to0,~~~n\to\infty,
a contradiction. So $\tTmus$ is bounded below.
$$\|\tTmus f\|_{\mu_s}^2\geq c\|f\|_{\mu_s}^2,~~~\forall f\in\Ps,$$
then $\forall F\in\ber$,
\langle\Tmus^3 F, F\rangle=\|\tTmus RF\|_{\mu_s}^2\geq c^2\|RF\|_{\mu_s}^2=c^2\langle\Tmus F, F\rangle.
This means $\Tmus^3\geq c^2\Tmus$, which implies that $\Tmus$ is bounded below on $Q$. Therefore $\|\dot\|_{\mu_s}$ and $\|\dot\|$ are equivalent on $Q$. This completes the proof when $\tilde{M}$ is connected.
If $\tilde{M}$ is not connected, then by the theorem in <cit.>, after restricting it to a smaller neighborhood of $\clb$, we can divide $\tilde{M}$ into finitely many connected components, each two having positive Euclidian distance(although they may have different dimensions). Then we divide $II(z)$ into more parts, the rest of the proof remains unchanged.
Suppose $M$ satisfies the properties of Theorem <ref>, then the projection operator onto $P$ is in the Toeplitz algebra $\toe$.
(1) Set $d\nu=(1-|w|^2)^{n-d}dv_d|_{M_s}$, then $T_{\nu}$ is a finite rank perturbation of $T_{\mu_s}$ and $\ker T_{\nu}=P$. Restricting to $Q$, $T_{\mu_s}$ is invertible and $T_{\nu}$ is a finite rank perturbation. So $T_{\nu}$ has index $0$. Since $T_{\nu}$ doesn't vanish on $Q$, it is invertible. Therefore $0$ is also isolated in $\sigma(T_{\nu})$, i.e., the measure $\nu$ also satisfies the hypotheses of Theorem <ref>. But how to prove it directly is still a problem. Also, for any positive measure $\mu$ on $M$ that is greater than $\nu$, since $T_{\mu}\geq T_{\nu}$ and they have the same kernel, $\mu$ also satisfies the assumptions in Theorem <ref>. In particular, we can choose $\mu$ to be the measure that defines the weighted Bergman space on $M$(cf. <cit.>). For measures $\mu$ satisfying the hypotheses of Theorem <ref>, the spaces $\mathcal{P}(\mu)$ are the same(with equivalent norms). Sometimes we write $\mathcal{P}$ for simplicity.
(2) By Proposition 4.4 in <cit.>, the extensions associated to the quotient module and the module $\mathcal{P}$ are unitarily equivalent. If we can prove that the analytic polynomials are dense in the weighted Bergman space $L_{a,n-d}^2(M)$ defined in <cit.>, i.e., $\mathcal{P}=L_{a,n-d}^2(M)$, then the quotient module also defines a K-homology class of the boundary $\tilde{M}\cap\partial\bn$, which we expect to be the fundamental class of $\tilde{M}\cap\partial\bn$ defined by the CR-structure on it.
(3) Now that we already know the projection $Q$ is in the Toeplitz algebra, there is an easier way to check if a measure $\mu$ satisfy the hypotheses of Theorem <ref>. For example, suppose we know that
$$\|{T_{\mu}}_x-Q_x\|\leq a<1,~~\forall x\in M_{\mathcal{A}}/\bn,$$
where $S_x$ for an operator $S$ is defined in <cit.>, then from Theorem 10.1 in <cit.>, it's easy to prove that $T_{\mu}$ is Fredholm as an operator on $Q$. Therefore $T_{\mu}$ is invertible whenever $T_{\mu}$ doesn't vanish on $Q$.
By the previous remark (1), the proof of Theorem <ref> gives an extension operator from $\mathcal{P}(\mu_s)$ to $\ber$. This is a generalization of Theorem 4.3 in <cit.>. We state it as a corollary.
Suppose $\tilde{M}$ is a $d$-dimensional complex analytic subset of an open neighborhood of $\clb$ which intersects $\pbn$ transversally and has no singular point on $\pbn$. Let $M=\tilde{M}\cap\bn$, $\mu=(1-|w|^2)^{n-d}dv_d$ on $M$ and $\mathcal{P}$ be the closed subspace of $L^2(\mu)$ generated by analytic polynomials. Then $\mathcal{P}$ is exactly the range of the restriction operator
R: \ber\to L^2(\mu), f\mapsto f|_M.
Equivalently, there is an extension operator
E: \mathcal{P}\to\ber,
which is a right inverse of $R$.
In the case of <cit.> and <cit.>, $\mathcal{P}(\mu_s)$ coincides with the weighted Bergman space on $M$. We don't know whether the two spaces are the same in general.
The papers <cit.> and <cit.> prove the existence of a extension map by giving an integral formula directly and proving the boundedness of it. In fact, our proof also shows the existence of an integral formula defining the extension operator $E$. $\forall f\in\mathcal{P}$, $\forall z\in\bn$,
Ef(z)=\langle Ef, K_z\rangle=\langle f, E^*K_z\rangle_{\mathcal{P}}=\int_M f(w)\overline{E^*K_z(w)}d\mu(w).
This is a new result in the theory of holomorphic extension.
§ $P$-ESSENTIAL NORMALITY FOR $P>2D$
This section is an attempt to solving the unweakened Geometric Arveson-Douglas Conjecture. Throughout this section, we assume $M$ satisfies the hypothesis of Theorem <ref>. Let $\mu=(1-|w|^2)^{n-d}dv_d$, where $d$ is the complex dimension of $M$. Then from Remark <ref> above, $\mu$ satisfies the hypothesis of Theorem <ref>. The restriction operator $R$ is one-to-one when restricted to the quotient space $Q$. In this section, we will show that the range space $\mathcal{P}\subset L^2(\mu)$ is $p$-essentially normal( $p>d$) as a Hilbert module. Moreover, we will show that for $p>2d$, the two modules are “equivalent” modulo $\mathcal{S}^p$, in particular, $Q$ is $p$-essentially normal for $p>2d$.
The following lemma is a generalization of Lemma 7 in <cit.>.
Suppose $2\leq p<+\infty$ and $G(z,w)$ is $\mu$-measurable in both variables. Let $A_G$ be the integral operator on $L^2(\mu)$ defined by
A_Gf(z)=\int_M G(z,w)K_w(z)f(w)d\mu(w).
\int_M\int_M|G(z,w)|^p|K_w(z)|^2d\mu(z)d\mu(w)<+\infty,
then $A_G$ is in the Schatten $p$ class $\mathcal{S}^p$.
When $p=2$ the result is well-known. When $|G(z,w)|$ is bounded, let $h(z)=(1-|z|^2)^{-1/4}$, then
\begin{eqnarray*}
\int_M|G(z,w)K_w(z)|h(z)d\mu(z)&\leq&C\int_M\frac{1}{|1-\langle z,w\rangle|^{n+1}(1-|z|^2)^{1/4}}d\mu(z)\\
&\leq&C\int_{\bn}\frac{1}{|1-\langle z,w\rangle|^{n+1}(1-|z|^2)^{1/4}}dv(z)\\
\end{eqnarray*}
\int_M|G(z,w)K_w(z)|h(w)d\mu(w)\leq Ch(z).
By Schur's test, $A_G$ defines a bounded operator. Now let $d\nu$ be the measure $|K_w(z)|^2d\mu(z)d\mu(w)$ on $M\times M$ and consider the map
L^2(\nu)+L^{\infty}(\nu)\to\mathcal{B}(L^2(\mu)),~G(z,w)\to A_G.
Then by non-commutative interpolation(cf. <cit.> ), the lemma is true.
\hat{T}_{\mu}: L^2(\mu)\to L^2(\mu),~\hat{T}_{\mu}f(z)=\int_M f(w)K_w(z)d\mu(w),
\hat{M}_{z_i}: L^2(\mu)\to L^2(\mu),~f\mapsto z_if.
Then the commutator $[\hat{M}_{z_i},\hat{T}_{\mu}]\in\mathcal{S}^p$, $\forall p>2d$.
It's easy to check that
[\hat{T}_{\mu},\hat{M}_{z_i}]f(z)=\int_M(w_i-z_i)K_w(z)f(w)d\mu(w),~\forall z\in M,~\forall f\in L^2(\mu).
Let $G(z,w)=w_i-z_i$ in the last lemma, then it suffices to prove that
\int_M\int_M|z-w|^p|K_w(z)|^2d\mu(w)d\mu(z)<+\infty,~\forall p>2d.
Now when $|z-w|>\delta$ the intergrade is bounded. On the other hand, using the same technique in the last section,
\begin{eqnarray*}
&&\int_{B(z,\delta)\cap M}|z-w|^p|K_w(z)|^2d\mu(w)\\
&\leq& C\int_{\mathbb{B}_d}|z'-w'|^p\frac{(1-|w'|^2)^{n-d}}{|1-\langle z',w'\rangle|^{2(n+1)}}dv_d(w')\\
&\leq&C\int_{\mathbb{B}_d}\frac{(1-|w'|^2)^{n-d}}{|1-\langle z',w'\rangle|^{2(n+1)-p/2}}\\
\end{eqnarray*}
\begin{eqnarray*}
\end{eqnarray*}
Using local coordinates, the last integral is less than
which is bounded when $p>2d$. This completes the prove.
The module $\mathcal{P}$ is $p$-essentially normal for all $p>d$. That is $[M_{z_i},M_{z_j}^*]\in\mathcal{S}^p$, $p>d$, where $M_{z_i}$ are the multiplication operators on $\mathcal{P}$.
Since the module action on $L^2(\mu)$ is normal, from Proposition 4.1 in <cit.>, it suffices to show that $[\mathcal{P}, \hat{M}_{z_i}]\in\mathcal{S}^p$, $p>2d$. Clearly $\hat{T}_{\mu}$ is self-adjoint and $\mathcal{P}$ is a $C^{\infty}$-functional calculous of $\hat{T}_{\mu}$. Combining lemma <ref> and Proposition 5 in Appendix I of <cit.> one gets the desired result.
Now we are ready to prove the main theorem of this section.
Under the assumptions of Theorem <ref>, the quotient module $Q$ is $p$-essentially normal for all $p>2d$.
Consider the following commuting graph:
\xymatrix{
\mathcal{P}\ar[r]^{M_{z_i}}&\mathcal{P}
Then $S_{z_i}=R^{-1}M_{z_i}R$. Therefore
On the other hand, for any $f$, $g\in Q$,
\langle Rf,Rg\rangle=\int_Mf(z)\overline{g(z)}d\mu(z)=\langle T_{\mu}f,g\rangle.
Hence $R^*R=T_{\mu}$. Also, from the following commuting graph,
\xymatrix{
\mathcal{P}\ar[r]^{\tilde{T}_{\mu}}&\mathcal{P}
$\tilde{T}_{\mu}=RT_{\mu}R^{-1}=RR^*RR^{-1}=RR^*$. From lemma <ref>, $[\tilde{T}_{\mu}, M_{z_i}]=[\hat{T}_{\mu},\hat{M}_{z_i}]|_{\mathcal{P}}\in\mathcal{S}^p$, $p>2d$. Therefore
\begin{eqnarray*}
&=&R^*M_{z_i}M_{z_j}^*R^{*-1}-R^*M_{z_j}^*M_{z_i}R^{*-1}~~~~(\mbox{modulo }\mathcal{S}^p)\\
\end{eqnarray*}
for $p>2d$. This completes the proof.
Although our proof doesn't give $p$-essential normality for $p>d$, we still think it is true in our case. In the case of a hyperplane, the operators $[\hat{T}_{\mu}, \hat{M}_{z_i}]|_{\mathcal{P}}$ are $0$, therefore there are no obstructions in this case. It's possible that after modifying the measure and using more detailed estimation, one can show that the above operators are in $\mathcal{S}^p$, $p>d$, which would imply the unweakened Geometric Arveson-Douglas Conjecture.
§ ARVESON-DOUGLAS CONJECTURE
Suppose $I\subset\mathbb{C}[z_1,\cdots,z_n]$ is a polynomial ideal. Let $\tilde{M}=Z(I)$, the zero variety of $I$. If $\tilde{M}$ satisfies the hypotheses of Theorem <ref>, then we know that the quotient module $Q$ is $p$-essential normal, $\forall p>2d$. In this section, we show that when $I$ is radical, $[I]=P$. Here $[I]$ is the closure of $I$ in $\ber$. Therefore we prove the weak Arveson-Douglas Conjecture for such ideals.
Suppose $I\subset\mathbb{C}[z_1,\cdots,z_n]$ is a radical polynomial ideal. If $\tilde{M}:=Z(I)$ has no singular point on $\pbn$ and intersects $\pbn$ transversally, then $[I]=\{f\in\ber: f|_M=0\}$. Here $M=\tilde{M}\cap\bn$. As a consequence, the quotient module $Q=[I]^{\perp}$ is $p$-essentially normal, $p>2d$. Moreover, the projection operator onto $Q$ is in the Toeplitz algebra $\toe$.
To prove the above theorem, we will show that the extension operators corresponding to the subvariety $1/t\tilde{M}$ for $t$ in a small interval $[1,t_0]$ are uniformly bounded.
In general, suppose $\tilde{M}$ satisfies the assumptions of Theorem <ref>. Then for $t$ close enough to $1$, $1/t\tilde{M}$ also satisfies the hypotheses of Theorem <ref>. Equivalently, there is an extension operator from $M^t$ to $t\bn$. In the proof of Theorem <ref>, the number $\epsilon$ depends on $\tilde{M}$, $R$ and $s$ depend on $\epsilon$ and $\tilde{M}$ and $s'$ depends on $s$ and $R$. By taking a larger $R$, we can find $0<s<s'<1$ and $t_1>1$ such that $\forall 1\leq t\leq t_1$, the measure
is suitable for the proof of Theorem <ref>. That means, suppose $f\in L_a^2(t\bn)$,
$$\langle T_{\mu_s^t}^3f,f\rangle\geq\alpha\langle T_{\mu_s^t}f,f\rangle$$
for some $\alpha>0$ that doesn't depend on $t$.
We remind the reader that there is in fact a normalizing constant between the spaces corresponding $1/t\tilde{M}$ and $\tilde{M}$, but since it tends to $1$ as $t$ tends to $1$, we omit the difference.
Let $\mathcal{P}_t$, $Q_t$, $T_{\mu_s^t}$ and $R_t$ be the obvious ones. Then $\mathcal{P}_t=Range R_t$. Define
E_t: \mathcal{P}_t\to Q_t, f|_{M^t}\mapsto f\in Q_t,
\tilde{T}_{\mu_s^t}:\mathcal{P}_t\to\mathcal{P}_t, f\mapsto R_tT_{\mu_s^t}E_tf.
Then $\tilde{T}_{\mu_s^t}$ are uniformly bounded for all $t$.
Let $L_1\subset\mathcal{P}_1$ be the finite co-dimensional subspace that $\forall f\in L_1$,
\int_{M_s^{s'}}|f|^2d\mu_s^1\leq1/3\int_M^1|f|^2d\mu_s^1.
Then $\forall f\in R_1^{-1}L_1\subset Q_1$,
\langle T_{\mu_s^1}^3f,f\rangle\geq\alpha\langle T_{\mu_s^1}f,f\rangle\geq\frac{\alpha}{\|T_{\mu_s^1}\|}\|T_{\mu_s^1}f\|^2.
\int_{M^1}|T_{\mu_s^1}f|^2d\mu_s^1\geq c\|T_{\mu_s^1}f\|^2.
This means for every function $g\in\mathcal{L}_1:=\tilde{T}_{\mu_s^1}L_1$,
\|E_1g\|\leq C\|g\|_{\mu_s^1}.
There is a natural inclusion $\mathcal{P}_t\subset\mathcal{P}_{t'}, t'\leq t$, given by restriction. In particular, $\mathcal{P}_t\subset\mathcal{P}_1$, $\forall t\geq1$. Let $L_t=L_1\cap\mathcal{P}_t$. Then for $t$ close enough to $1$, $\forall f\in L_t$,
\int_{M_s^{s'}}|f|^2d\mu_s^t\leq1/2\int_{M^t}|f|^2d\mu_s^t.
So $\forall g\in\mathcal{L}_t:=\tilde{T}_{\mu_s^t}L_t$,
\|E_tg\|\leq C\|g\|_{\mu_s^t}.
The spaces $\mathcal{L}_t\subset\mathcal{P}_t$ have the same co-dimension: since the polynomials are dense, we can find finite dimensional space $N$ consisting of(restriction of) polynomials such that $N+L_1=\mathcal{P}_1$, $N\cap L_1=\{0\}$. Then it's easy to prove that $N+L_t=\mathcal{P}_t$, $N\cap L_t=\emptyset$. Since the operators $\tilde{T}_{\mu_s^t}$ are one to one and surjective, codim$\mathcal{L}_t=$ codim$L_t=\dim N$.
We denote the norms $\|\cdot\|_{\mu_s^t}$ by $\|\cdot\|_t$ from this point.
Let $\eta>0$ be determined later. We can take finite dimensional space $\mathcal{N}$ consisting of polynomials such that $\mathcal{N}+\mathcal{L}_1=\mathcal{P}_1$ and $\forall f\in\mathcal{N}, \forall g\in\mathcal{L}_1$,
\frac{|\langle f,g\rangle_1|}{\|f\|_1\|g\|_1}<\eta.
Claim: $\exists t_0>1$, such that $\forall 1\leq t\leq t_0$, $\forall f\in\mathcal{N}$, $\forall g\in\mathcal{L}_t$,
\frac{|\langle f,g\rangle_t|}{\|f\|_t\|g\|_t}<1/2.
Suppose the claim is not true, then there exists a sequence $t_n\to1$, $f_n\in\mathcal{N}, g_n\in\mathcal{L}_{t_n}$, $\|f_n\|_{t_n}=\|g_n\|_{t_n}=1$, such that
\langle f_n,g_n\rangle_{t_n}\geq1/2.
Then $g_n=\tilde{T}_{\mu_s^{t_n}}h_n$, $h_n\in L_{t_n}$, by previous discussion,
\|h_n\|_{t_n}\leq C\|g_n\|_{t_n}=C
for some $C>0$. It's easy to show that the norms $\|\cdot\|_t$ are uniformly equivalent on the space $\mathcal{N}$. So $f_n$(has a subsequence) tends to some $f\in\mathcal{N}$ uniformly on all $\|\cdot\|_{t_n}$ norms. Hence for sufficiently large $n$ we have
|\langle f,g_n\rangle_{t_n}|\geq1/3.
We also have $\|f\|_1=1$. So
|\langle \tilde{T}_{\mu_s^{t_n}}f,h_n\rangle_{t_n}|=|\langle f,g_n\rangle_{t_n}|\geq1/3.
Finally, we prove that
|\langle\tilde{T}_{\mu_s^{t_n}}f,h_n\rangle_{t_n}-\langle\tilde{T}_{\mu_s^1}f,h_n\rangle_1|\to0, n\to\infty.
$$|\langle\tilde{T}_{\mu_s^1}f,h_n\rangle_1|=|\langle f,\tilde{T}_{\mu_s^1}h_n\rangle_1|<\eta\|f\|_1\|\tilde{T}_{\mu_s^1}h_n\|_1\leq C\eta,$$
Take $\eta$ such that $C\eta\leq1/4$, this proves the claim
\begin{eqnarray*}
|\tilde{T}_{\mu_s^{t_n}}f(z)|&\leq&|\int_{M_s^{t_n}}f(w)\frac{(t_n^2-|w|^2)^{n-d}}{(t_n^2-\langle z,w\rangle)^{n+1}}dv_d(w)|\\
& &+|\sum_{i=1}^mf(z_i)\frac{(t_n^2-|z_i|^2)^{n+1}}{(t_n^2-\langle z,z_i\rangle)^{n+1}}|\\
&\leq&C\int_{M_s^{t_n}}\frac{(t_n^2-|w|^2)^{n-d}}{|t_n^2-\langle z,w\rangle|^{n+1}}dv_d(w)+C\\
&\leq&C+C\int_{M_s^{t_n}\cap B(z,\delta)}\frac{(t_n^2-|w|^2)^{n-d}}{|t_n^2-\langle z,w\rangle|^{n+1}}dv_d(w)\\
&\leq&C+C\int_{t_n\mathbb{B}_d}\frac{(t_n^2-|w'|^2)^{n-d}}{|t_n^2-\langle z',w'\rangle|^{n+1}}dv(w')\\
\end{eqnarray*}
The above estimation is similar with those used in the last section, we omit the details. The third inequality from the bottom is from Lemma <ref>.
\begin{eqnarray*}
& &+\int_{M_s^1}\bigg(\tilde{T}_{\mu_s^{t_n}}f(z)-\tilde{T}_{\mu_s^1}f(z)\frac{(1-|z|^2)^{n-d}}{(t_n^2-|z|^2)^{n-d}}\bigg)\overline{h_n(z)}d\mu_s^{t_n}(z)\\
& &+\sum_{i=1}^m\bigg(\tilde{T}_{\mu_s^{t_n}}f(z_i)(t_n^2-|z_i|^2)^{n+1}-\tilde{T}_{\mu_s^1}f(z_i)(1-|z_i|^2)^{n+1}\bigg)\overline{h_n(z_i)}\\
\end{eqnarray*}
Clearly $III_n\to0, n\to\infty$. For the first part,
\begin{eqnarray*}
\end{eqnarray*}
For the second part, since
\begin{eqnarray*}
& &+2(C+C\log\frac{1}{1-|z|^2})^2(1-|z|^2)^{n-d}\frac{(1-|z|^2)^{n-d}}{(t_n^2-|z|^2)^{n-d}}\\
\end{eqnarray*}
By the dominance convergence theorem,
\int_{M_s^1}|\tilde{T}_{\mu_s^{t_n}}f(z)-\tilde{T}_{\mu_s^1}f(z)\frac{(1-|z|^2)^{n-d}}{(t_n^2-|z|^2)^{n-d}}|^2(t_n^2-|z|^2)^{n-d}dv_d(z)\to0.
Using the holder's inequality, we see that $II_n\to0$. So the proof of claim is complete.
For $\forall f\in\mathcal{N}, \forall g\in\mathcal{L}_t$,
\begin{eqnarray*}
&&\|f+g\|_t^2=\|f\|_t^2+\|g\|_t^2+2Re\langle f,g\rangle_t\\
\end{eqnarray*}
By continuity, there is a constant $C'>0$ such that $\forall 1\leq t\leq t_0$, $\forall f\in\mathcal{N}$, $\|E_tf\|\leq C'\|f\|_t$. So for any $h\in\mathcal{P}_t$, $h=f+g$, $f\in\mathcal{N}$, $g\in\mathcal{L}_t$,
\|E_t(f+g)\|^2\leq\|E_tf\|^2+\|E_tg\|^2\leq C(\|f\|_t^2+\|g\|_t^2)\leq2C\|f+g\|_t^2.
So the extension operators $E_t$ are uniformly bounded.
Knowing the above result, the proof of Theorem <ref> is exactly the same as in the appendix of <cit.>.
§ SUMMARY
This paper combines ideas from various subject to solve the Geometric Arveson-Douglas Conjecture. First, we view the quotient module as a reproducing kernel Hilbert module on the variety(cf. <cit.>) and seek an equivalent norm given by a measure. The fact that operators of the form $T_{\mu}$ are in the Toeplitz algebra(cf. <cit.>) is crucial to our proof. Second, we use the idea that Toeplitz operators are “localized” (cf. <cit.><cit.>) and apply it on the variety, instead of the whole unit ball. Finally, we observe that the fact that the size of the hyperbolic balls tend to $0$ uniformly as the centers tend to the boundary forces the Bergman reproducing kernels $K_z(w)$ to act “almost” like reproducing kernels on the quotient space. Using Theorem 4.7 in <cit.>, we see that these kind of argument also work for the weighted Bergman spaces.
The techniques from complex harmonic analysis reveals some connection between the Geometric Arveson-Douglas Conjecture and the extension of holomorphic functions(cf. <cit.>). This idea first appeared in <cit.> and has inspired us to seek connections from different angles.
Under the hypotheses of this paper, quotient modules are closely related to weighted Bergman spaces on the variety. Therefore whatever is true for the Bergman space may also be true on the quotient space. Moreover, the idea of looking at the operator $T_{\mu}$ “locally” on $M$ offers a way to study the Geometric Arveson-Douglas Conjecture for more general varieties. These relationships will be the future focuses of our research.
Another direction we plan to consider is extending the index result in <cit.>. Recall in <cit.>, a generalization of the Boutet de Monvel result is obtained using the methods in <cit.>. It seems likely we can extend the proofs to cover our case.
We would like to thank Xiang Tang for discussing with us since the early stage of our research, for reading the drafts of this paper and for the valuable suggestions he gave us. We also would like to thank Kai Wang for the valuable discussions over Wechat. The second author would like to thank Kunyu Guo, her advisor in Fudan University, for inviting her to the world of mathematical research and the advice he gave over emails. She also want to thank her fellows in Fudan University, especially Zipeng Wang, for drawing her attention to complex harmonic analysis before her visit to Texas A&M University.
AronszajnN. Aronszajn, Theory of Reproducing Kernels. Trans. Amer. Math. Soc. 68 (1950). 337-404. MR0051437
Arv III W. Arveson, Subalgebras of $C^*$-algebras III: Multivariate operator theory. Acta Math. 181 (1998), no. 2, 159-228. MR1668582
Arv Dirac W. Arveson, The Dirac Operator of a Commuting d-Tuple. (English summary) J. Funct. Anal. 189 (2002), no. 1, 53-79. MR1887629
Arv p summable W. Arveson, $p$-Summable commutators in dimension $d$. J. Operator Theory 54 (2005), no. 1, 101-117. MR2168861
BDT P. Baum, R. G. Douglas and M. Taylor, Cycles and relative cycles in analytic K-homology. J. Differential Geom. 30(1989), no.3, 761-804
Beatrous F. Beatrous, $L^p$-estimates for extensions of holomorphic functions. Michigan Math. J. 32 (1985), no. 3, 361-380. MR0803838
Guo Chen X. Chen; K. Guo, Analytic Hilbert modules. Chapman & Hall/CRC Research Notes in Mathematics, 433. Chapman & Hall/CRC, Boca Raton, FL, 2003. MR1988884
Complex analytic sets E. M. Chirka, Complex Analytic Sets. Translated from the Russian by R. A. M. Hoksbergen. Mathematics and its Applications (Soviet Series), 46 Kluwer Academic Publishers Group, Dordrecht, 1989 MR1111477
connes A. Connes, Non-commutative differential geometry. Publications Mathematiques de l'IHES 62.1 (1985): 41-144.
Dou index R. G. Douglas, A New Kind of Index Theorem. Analysis, geometry and topology of elliptic operators, 369-382, World Sci. Publ., Hackensack, NJ, 2006. MR2246775
Douglas Sarkar R. G. Douglas and J. Sarkar, Essentially reductive weighted shift Hilbert modules. (English summary)
J. Operator Theory 65 (2011), no. 2, 379-401. MR2785850
Douglas Wang R. G. Douglas and K. Wang, A harmonic analysis approach to essential normality of principal submodules. J. Funct. Anal. 261 (2011), no. 11, 3155-3180. MR2835994
DYTR. G. Douglas, X. Tang and G. Yu, An Analytic Grothendieck Riemann Roch Theorem. arXiv:1404.4396
Englis M. Engliš and J. Eschmeier, Geometric Arveson-Douglas Conjecture. Adv. Math. 274 (2015), 606-630. MR3318162
Guo K. Guo, Defect Operators for Submodules of $H_d^2$, J. fur Reine und Angew. Math. 573 (2004), 181-209. MR2084587
Guo Wang K. Guo and K. Wang, Essentially Normal Hilbert modules and K-homology. Math. Ann. 340 (2008), no. 4, 907-934. MR2372744
Guo Wang quosi K. Guo and K. Wang, Essentially Normal Hilbert Modules and K-homology. II. Quasi-homogeneous Hilbert Modules Over the Two Dimensional Unit Ball. J. Ramanujan Math. Soc. 22 (2007), no. 3, 259-281. MR2356345
localized operator J. Isralowitz, M. Mitkovski and B. Wick, Localization and Compactness in Bergman and Fock Spaces, to appear in Indiana Univ. Math. J., arXiv:1306.0316
JMT M. Jevtić, X. Massaneda and P. J. Thomas, Interpolating sequences for weighted Bergman spaces of the ball.
Michigan Math. J. 43 (1996), no. 3, 495-517. MR1420589
Sha Ken M. Kennedy and O. Shalit, Essential Normality and the Decomposability of Algebraic Varieties. New York J. Math. 18 (2012), 877-890. MR2991427
Luecking D. H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives. Amer. J. Math. 107 (1985), no. 1, 85-111. MR0778090
Mcdonald G. McDonald, Fredholm Properties of a Class of Toeplitz Operators on the Ball, Indiana Univ. Math. J. 26 (1977), 567-576
Suarez Wick M. Mitkovski, D. Suárez and B. D. Wick, The essential norm of operators on $A_{\alpha}^p(\bn)$. (English summary)
Integral Equations Operator Theory 75 (2013), no. 2, 197-233. MR3008923
interpolation J. Peetre and G. Sparr, Interpolation and non-commutative integration. Ann. Mat. Pura Appl. (4) 104 (1975), 187-207.
Rudin W. Rudin, Function Theory in the Unit Ball of $\mathbb{C}^n$. Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2008. MR2446682
Shalit O. Shalit, Stable Polynomial Division and Essential Normality of Graded Hilbert Modules, J. Lond. Math. Soc. (2) 83 (2011), no. 2, 273-289. MR2776637
Suarez07 D. Suárez, The Essential Norm of Operators in the Toeplitz Algebra on $A^p(\bn)$. (English summary)
Indiana Univ. Math. J. 56 (2007), no. 5, 2185-2232. MR2360608
hankel K. Zhu, Schatten class Hankel operators on the Bergman space of the unit ball. Amer. J. Math. 113 (1991), no. 1, 147-167. MR1087805
Ronald G. Douglas
Texas A&M University, College Station, TX, 77843
E-mail address: [email protected]
Yi Wang
Fudan University, Shanghai, China, 200433
E-mail address: [email protected]
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1511.00748
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Character formulas and BGG resolutions]Character formulas and Bernstein-Gelfand-Gelfand resolutions for Cherednik algebra modules
We study blocks of category $\OO$ for the Cherednik algebra having the property that every irreducible module in the block admits a BGG resolution, and as a consequence prove a character formula conjectured by Oblomkov-Yun.
Stephen Griffeth
Instituto de Matemática y Física
Universidad de Talca
Emily Norton
Department of Mathematics
Kansas State University
§ INTRODUCTION
A fundamental problem in representation theory is the calculation of characters of irreducible objects. For category $\OO$ of the rational Cherednik algebra the complete solution is equivalent to calculating the decomposition matrix $[\Delta(\lambda):L(\mu)]$. For the cyclotomic rational Cherednik algebras, the entries of the decomposition matrix are values of affine parabolic Kazhdan-Lusztig polynomials <cit.>, <cit.>, <cit.>; this amounts, in principle, to a recursive algorithm for calculating characters. Special classes of modules exist, however, for which we can write closed character formulas. Our work in the present paper was motivated by the desire to understand the reasons for the existence of such formulas in a particular class of examples, those appearing in Conjecture 9.10.1 of Oblomkov-Yun <cit.>:
$$\sum_{n>0} \mathrm{dim}_\CC(L_{1/2n}(D_{2n}))x^{n-1}=(1-4x)^{-3/2} $$
$$\sum_{n>0} \mathrm{dim}_\CC(L_{1/2n}(C_{2n}))x^n=(1-4x)^{-3/2}(1+\sqrt{1-4x})^2/4.$$
In these formulas, $L_c(W)$ denotes the $H_c(W)$ module $L_c(\Triv)$. This conjecture is a consequence of the following graded dimension formula (see Corollary <ref>):
\begin{equation} \label{graded dim}
\mathrm{dim}_\CC(L^d)=\dim_\CC(L^{N-d})=\sum_{\substack{0 \leq \ell \leq d \\ d-\ell \in 2 \ZZ}} {2n \choose \ell}
\end{equation}
if $0 \leq d \leq n$, $N=2n$, and $W=W(C_{2n})$; or if $0 \leq d \leq n-1$, $N=2n-2$, and $W=W(D_{2n})$; and in which $L^d$ denotes the degree $d$ part of $L=L_{1/2n}(\Triv)$.
In most branches of Lie theory, those irreducible modules for which character formulas as simple as this exist tend to admit BGG resolutions, that is, resolutions by standard (also known as Verma) modules. Not only do the modules involved in (<ref>) admit BGG resolutions, but so do all the simples in the same block.
Our first theorem characterizes this situation and gives a practical method for proving that BGG resolutions exist (see Theorem <ref> for a more general result which would also apply to the module category of a cellular algebra):
Let $C$ be a highest weight category over a field $F$, with simple objects indexed by a finite poset $\Lambda$. The following conditions are equivalent:
(a) Every simple object in $C$ admits a resolution by standard objects.
(b) The radical of every standard object in $C$ is generated by homomorphic images of standard objects.
(c) $[\Delta(\lambda):L(\mu)]=\dim\Hom(\Delta(\mu),\Delta(\lambda))$ for all $\lambda,\mu\in\Lambda$.
Earlier known examples of blocks of category $\OO_c$ satisfying these conditions were closely related to Haiman's work on diagonal coinvariant rings; see <cit.>, <cit.>, <cit.>.
The proof we give of (<ref>) utilizes the tools developed in <cit.>, which completely describe the submodule structure of the standard module $\Delta_c(\Triv)$ when the parameter $c_0$ associated with the conjugacy class of reflections containing the transpositions is not a positive rational number of denominator less than $n$. The most interesting case is when $c_0=\ell/n$ for a positive integer $\ell$ coprime to $n$. In this case, we can give a very complete description of the structure of the standard modules in the principal block of category $\OO_c$ for the rational Cherednik algebra of the group $G(r,1,n)$, and in particular we can compute the graded dimensions of their simple heads. Our next theorem specifies the set of parameters $c$ for which the conditions in Theorem <ref> hold for the principal block of category $\OO_c$. We order pairs of integers lexicographically: $(i_1,k_1) > (i_2,k_2)$ if $k_1>k_2$ or $k_1=k_2$ and $i_1>i_2$. We mark the boxes of the trivial partition $(n)$ as follows: for a box $b$, if there exist integers $0 \leq i \leq r-1$ and $k$ so that
$$(d_i-i)/r=d_0/r+\mathrm{ct}(b) c_0+k,$$
then we place a label $(i,k)$ in the box $b$ (so a box may receive more than one label, but a given $i$ appears at most once). An example of such a labeling when $n=8$ and $r=6$ is
mathmode, boxsize=3.5em
(1,-2) (3,-1) (5,3)
Using this notation we give the following characterization of when the conditions from Theorem <ref> hold for the principal block (see Theorem <ref> for the proof):
Suppose $c_0=\ell/n$ for a positive integer $\ell$ coprime to $n$. Then every simple module in the principal block has a BGG resolution if and only if when read left to right, and assuming the labels appearing in a given box are ordered in decreasing fashion, we have
Thus in the example preceding the theorem there are simples in the principal block not admitting BGG resolutions. On the other hand, it is straightforward to check that for the parameters appearing in (<ref>), the condition of Theorem <ref> is satisfied: for the type $D_{2n}$ version, take $c_0=1/2n$ and $d_0=d_1=0$, in which case the labels are $(0,0)$ in the first box, and $(1,-1)$ in the box with content $n$; for the type $C_{2n}$ version, take $c_0=d_0=-d_1=1/2n$ so that the labels are $(0,0)$ appearing in the first box and $(1,-1)$ appearing in the box with content $n-1$.
When the principal block $B_0$ of $\OO_c$ satisfies the conditions in Theorems <ref> and <ref>, we can conjecturally calculate BGG resolutions of simples using a graph $\Gamma$ depicting the submodule structure of standard modules. The graph $\Gamma$ has a vertex for each $\lambda\in B_0$, and an arrow $\mu\rightarrow \lambda$ for each $\mu$ which is the highest weight of a submodule of $\Del(\lambda)$ of the type defined in Theorem <ref>. These generate the radical of $\Del(\lambda)$ (but not always minimally). We conjecture (<ref>) that the standards in the BGG resolution of $L(\lambda)$ may be found by, first, taking the subgraph $\Gamma_\lambda$ spanned by all $\mu$ such that there is a path from $\mu$ to $\lambda$ in $\Gamma$; and then, deleting all subgraphs $\Gamma_\nu$ whenever a single arrow $\nu\rightarrow\mu$ can be factored as $\nu\rightarrow\rho\rightarrow\mu$ for vertices $\nu,\mu,\rho\in\Gamma_\lambda$. We expect that the vertices $\mu$ which remain after this procedure are those such that $\Del(\mu)$ appears in the minimal BGG resolution of $L(\lambda)$; moreover, $\Delta(\mu)$ occurs with multiplicity $1$ in homological degree $d(\mu,\lambda)$, where $d(\mu,\lambda)$ is the length of the longest chain of directed arrows from $\mu$ to $\lambda$ in $\Gamma$. For $r=2$ and equal parameters $1/2n$, we can write down all maps between standard modules and we verify (<ref>) that this algorithm is indeed correct. We also have calculated some examples for $G(r,1,rn)$ at equal parameters $1/rn$; the conjectural algorithm produces the correct graded characters for simple modules in the blocks in our examples, see Section <ref>.
For the Cherednik algebra, BGG resolutions that are minimal in a certain sense are minimal resolutions in the sense of commutative algebra; <cit.> used this observation to conjecture a combinatorial formula for the Betti numbers of the ideals of certain subspace arrangements. In this direction, here we develop new tools for establishing the existence of BGG resolutions, thus reducing the calculation of Betti numbers of a number of subspace arrangements to the calculation of character formulas for Cherednik algebra modules. We expect that in our categories of interest satisfying the conditions of Theorem <ref>, the character of a simple module should determine the terms in its minimal BGG resolution. For example, we have:
Suppose $W=B_{2n}=G(2,1,2n)$ and the parameters $c=1/2n$ are constant.
The module $L_c(\mathrm{triv})=L((2n),\emp)$ is the coordinate ring of the scheme-theoretic intersection of
$$\{x_1^2=x_2^2=\cdots=x_{2n}^2 \}$$
and the set of points where at least $n$ coordinates are equal to $0$. For $0\leq i \leq n$, the coefficient of $v^i$ in the graded character of $L_c(\Triv)$ is the sum $(-1)^i\Del_i$ where
$$\Del_i=\Del((2n-i,1^i),\emp)\oplus\bigoplus_{1 \leq a \leq i}\Del((n-a,1^{i-a}),(n+a+1-i,1^{a-1}))$$
and the $(2n-i)$'th term is the sum $(-1)^{2n-i}\Del_{2n-i}$ where
$$\Del_{2n-i}=\Del(\emp,(2n-i,1^i)^t)\oplus\bigoplus_{1 \leq a \leq i}\Del((n+a+1-i,1^{a-1})^t,(n-a,1^{i-a})^t)$$
For $0\leq i \leq n$, the $i$'th term of the minimal BGG resolution of $L_c((2n),\emp)$ is $\Del_i$ and the $(2n-i)$'th term is $\Del_{2n-i}$.
Acknowledgments. We acknowledge Fondecyt proyecto 1110072 for financing E.N.'s visit to Chile in March 2015 where this project started, and S.G. acknowledges Fondecyt proyecto 1151275 for continuing financial support. E.N. is grateful to O. Dudas and H. Miyachi for enlightening discussions. We thank S. Arkhipov, R. Bezrukavnikov, V. Ginzburg, and I. Loseu for helpful remarks about graded BGG reciprocity, and C. Berkesch-Zamaere and S. Sam for sharing their expertise in commutative algebra.
§ BGG RESOLUTIONS
In this section we give definitions, fix notation, and prove some very general results on BGG resolutions in a class of categories containing highest weight categories with finitely many isoclasses of simple objects.
§.§ Axiomatics
Let $F$ be a field and let $C$ be an artinian $F$-linear abelian category. Assume $\Lambda$ is a finite poset such that for each $\lambda \in \Lambda$, there is an object $\Delta(\lambda)\in C$ satisfying:
(a) The objects $L(\lambda)=\Del(\lambda)/\Rad(\Delta(\lambda))$ are a complete set of non-isomorphic simple objects of $C$,
(b) we have $\End(\Del(\lambda))=F=\End(L(\lambda))$, and
(c) if $[\Rad(\Del(\lambda)):L(\mu)] \neq 0$ then $\mu < \lambda$.
For instance, if $C$ is a highest weight category in the sense of <cit.> with finitely many isoclasses of simple objects, then it satisfies these axioms. We will refer to the objects $\Delta(\lambda)$ as standard objects.
§.§ Tight multiplicities and singular vectors
Let $C$ be a category as in <ref>.
By right exactness of Hom we have
\begin{equation} \label{hom inequality}
\dim\Hom(\Del(\lambda),M) \leq \dim\Hom(\Del(\lambda),N)+\dim\Hom(\Del(\lambda),M/N)
\end{equation}
for each object $M$ in $C$ and every subobject $N$ of $M$. Thus by induction on the length of $M$ we have
$$\dim\Hom(\Del(\lambda),M)\leq [M:L(\lambda)].$$
An object $M$ in $C$ has tight multiplicities if for all $\lambda \in \Lambda$ we have
Thus if $M$ has tight multiplicities, all occurrences of an irreducible factor $L(\lambda)$ in a composition series for $M$ are accounted for by homomorphisms from $\Del(\lambda)$ to $M$.
If $M$ has tight multiplicities, so does every subobject and every quotient object. If $M$ and $N$ have tight multiplicities, so does $M \oplus N$.
It is obvious that direct sums of objects with tight multiplicities have tight multiplicities. The proofs for quotient objects and subobjects are dual to one another; we prove that every subobject inherits tight multiplicities.
Suppose $M$ has tight multiplicities and that $N \subseteq M$ is a subobject. We prove that $N$ has tight multiplicities by induction on the length of $M/N$, which we may therefore assume to be $1$. If $M/N \cong L(\lambda)$, then using (<ref>) we obtain
$$\dim\Hom(\Del_c(\mu),N) \geq \dim\Hom(\Del_c(\lambda),M)-\delta_{\lambda \mu}$$
where $\delta_{\lambda \mu}$ is $0$ unless $\lambda=\mu$ in which case it is $1$. This proves the result.
An object $M$ in $C$ is generated by singular vectors if it is equal to the largest subobject $N$ of $M$ containing the images of all homomorphisms from standard objects $\Delta(\lambda)$ to $M$ (one could also write generated by highest weight vectors, the more common terminology in classical Lie theory).
Suppose $M$ is an object in $C$. Then $M$ has tight multiplicities if and only if every submodule of $M$ is generated by singular vectors.
First suppose $M$ has tight multiplicities. Then by Lemma <ref> the same is true of every subobject, so it suffices to prove that if $M$ has tight multiplicities then it is generated by singular vectors. Let $N$ be the subobject of $M$ generated by the images of homomorphisms from standard objects to $M$. For each $\lambda \in \Irr W$ we have
\begin{align*}
[M:L(\lambda)]&=\dim\Hom(\Del(\lambda),M)=\dim\Hom(\Del(\lambda),N) \\ &\leq [N:L(\lambda)] \leq [M:L(\lambda)], \end{align*}
implying equality throughout and hence $N=M$.
Conversely, assume that every subobject of $M$ is generated by singular vectors. We prove by induction on the length of $M$ that $M$ has tight multiplicities. This is clear if the length of $M$ is $1$. Let
$$0=M_0 \subseteq M_1 \subseteq \cdots \subseteq M_\ell=M$$
be a composition series. By assumption every subobject of $M_{\ell-1}$ is generated by singular vectors; by induction, $M_{\ell-1}$ has tight multiplicities. This implies that
$$\sum_\lambda \dim\Hom(\Del(\lambda),M)) \geq \ell-1.$$
But if equality holds then the subobject $N$ of $M$ generated by homomorphisms from standard objects is contained in $M_{\ell-1}$, contradicting that $M$ is generated by singular vectors.
If the radical of each $\Delta(\lambda)$ is generated by singular vectors, then every subobject of every standard object is generated by singular vectors.
We proceed by induction on the poset $\Lambda$. If $\lambda$ is minimal then $L(\lambda)=\Del(\lambda)$ is simple and hence every subobject is generated by singular vectors. In general, we have
with $\Rad \Del(\lambda)$ generated by singular vectors by hypothesis, so there is a sum $\Del_1$ of standard objects surjecting onto $\Rad \Del(\lambda)$. Using the axioms <ref> we may assume that the standard summands in $\Del_1$ are all smaller than $\lambda$. So the inductive hypothesis and Lemmas <ref> and <ref> together imply that every subobject of $\Del_1$ is generated by singular vectors, and hence the same holds for its quotient $\Rad\Del(\lambda)$. But every proper subobject of $\Del(\lambda)$ is a subobject of $\Rad\Del(\lambda)$, so every subobject of $\Del(\lambda)$ is generated by singular vectors.
§.§ BGG resolutions
A Bernstein-Gelfand-Gelfand resolution (or briefly, BGG resolution) of an object $M$ of $C$ is an exact sequence
$$\cdots \longrightarrow \Del_m \lra \Del_{m-1} \lra \cdots \lra \Del_1 \lra \Del_0\lra M \lra 0$$
in which each $\Del_i$ is a (possibly empty) finite direct sum of standard objects. It is minimal if the differential maps $\Delta_i$ into $\mathrm{Rad}(\Delta_{i-1})$. We observe that a minimal BGG resolution necessarily has $\Del_i=0$ for $i$ sufficiently large: since the radical of $\Del_i$ is the sum of the radicals of its standard summands, our axioms <ref> imply that for each standard summand $\Del(\lambda)$ of $\Del_i$ there is a standard summand $\Del(\mu)$ of $\Del_{i-1}$ with $\lambda < \mu$.
If $\Del_1 \rightarrow \Delta_0 \rightarrow M \rightarrow 0$ is exact, where $\Del_1=\bigoplus_{i \in I} \Delta(\lambda_i)$ and $\Del_0=\bigoplus_{j \in J} \Delta(\mu_j)$ are finite sums of standard objects, then there are subsets $I_0 \subseteq I$ and $J_0 \subseteq J$ and an exact sequence
$$\bigoplus_{i \in I_0} \Delta(\lambda_i) \rightarrow \bigoplus_{j \in J_0} \Delta(\mu_j) \rightarrow M \rightarrow 0$$
so that the image of $\bigoplus_{i \in I_0} \Delta(\lambda_i)$ is contained in $\bigoplus_{j \in J_0} \mathrm{Rad}(\Delta(\mu_j))$.
If the image of $\Delta(\lambda_{i_0})$ is not contained in $\bigoplus_{j \in J} \mathrm{Rad}(\Delta(\mu_j))$ then there is some index $j_0 \in J$ so that the composite $\Delta(\lambda_{i_0}) \rightarrow \Delta(\mu_{j_0}) \rightarrow L(\mu_{j_0})$ is non-zero. Since the top of $\Delta(\lambda_{i_0})$ is $L(\lambda_{i_0})$, it follows that $\lambda_{i_0}=\mu_{j_0}$ and hence the map $\Delta(\lambda_{i_0}) \rightarrow \Delta(\mu_{j_0})=\Delta(\lambda_{i_0})$ is a scalar multiple of the identity. In particular the map $\Delta(\lambda_{i_0}) \rightarrow \bigoplus \Delta(\mu_j)$ is injective. Hence the sequence
$$\bigoplus_{i \neq i_0} \Delta(\lambda_i) \rightarrow \left(\bigoplus_{j \in J} \Delta(\mu_j) \right)/\mathrm{Im}(\Delta(\lambda_{i_0})) \rightarrow M \rightarrow 0$$
is exact. Since
$$\mathrm{Im}(\Delta(\lambda_{i_0})) \cap \bigoplus_{j \neq j_0} \Delta(\mu_j)=0$$
the map
$$\bigoplus_{j \neq j_0} \Delta(\mu_j) \rightarrow \left(\bigoplus_{j \in J} \Delta(\mu_j) \right)/\mathrm{Im}(\Delta(\lambda_{i_0}))$$
is injective; it is therefore an isomorphism because both sides have the same length. We may thus eliminate $i_0$ and $j_0$ from our index sets. Repeating this procedure proves the lemma.
If a BGG resolution exists, then $M$ is generated by singular vectors. If an irreducible object $L(\lambda)$ has a BGG resolution, then the first map has domain $\Del_0$ a sum of standard objects; since the only standard object with a non-zero map to $L(\lambda)$ is $\Del(\lambda)$, and since there is a unique such map up to scalars, we may discard all the summands of $\Del_0$ and assume the first step in the resolution is the quotient map $\Del(\lambda) \rightarrow L(\lambda)$. Therefore if $L(\lambda)$ has a BGG resolution then $\Rad \Del(\lambda)$ is generated by singular vectors.
If every submodule of every standard module in $C$ is generated by singular vectors then every quotient of a sum of standard modules has a minimal BGG resolution.
First suppose that every subobject of every standard object in $C$ is generated by singular vectors. By Lemmas <ref> and <ref>, every subobject of every sum of standard objects in $C$ is generated by singular vectors. Let $\Del_0$ be a sum of standard modules and let $M=\Del_0/N$ be a quotient of $\Del_0$. By hypothesis $N$ is generated by singular vectors, so it is a quotient of a finite sum $\Del_1$ of standard objects. We have obtained an exact sequence
$$ \Delta_1 \rightarrow \Del_0 \rightarrow M \rightarrow 0,$$
and by lemma <ref> we may assume that the image of $\Del_1$ is contained in the radical of $\Del_0$. Iterating produces a minimal BGG resolution.
The category $C$ has tight multiplicities if every standard object in $C$ does. Examples include defect one blocks in Cherednik algebra category $\OO$ (see 5.2.4 of <cit.>)).
We now restate and prove Theorem <ref> using these definitions.
Assume $C$ is a category as in <ref>. The following are equivalent:
* $C$ has tight multiplicities.
* Every simple object in $C$ has a BGG resolution.
* The radical of every standard object in $C$ is generated by singular vectors.
(a) implies (b) follows from Lemma <ref> and Lemma <ref>. (b) implies (c) is the discussion following the definition of BGG resolutions. (c) implies (a) follows from Lemma <ref> and Lemma <ref>.
For finite dimensional complex semisimple Lie algebras, there is another way of stating this criterion.
Let $C$ be a block of category $\OO$ for a finite dimensional semisimple Lie algebra over $\CC$. Then every simple in $C$ has a BGG resolution if and only if every Verma module is multiplicity free.
It is a consequence of classical Lie theory that
$$[\Delta(\lambda):L(\mu)] \neq 0 \ \iff \mathrm{Hom}(\Delta(\mu),\Delta(\lambda)) \neq 0$$
and that in this case $\dim\Hom(\Delta(\mu),\Delta(\lambda))=1$. It follows that $C$ has tight multiplicities if and only if every Verma module is multiplicity free.
We remark that König <cit.> proves this corollary by a different method; we do not know if the standing hypotheses of his paper hold in the cases of interest to us.
§.§ BGG resolutions and character formulas
Suppose $M$ is a module in $C$ that admits a BGG resolution
$$0 \rightarrow \Delta_\ell \rightarrow \cdots \rightarrow \Delta_0 \rightarrow M$$
In the Grothendieck group of $C$ this gives
$$[M]=\sum (-1)^i [\Delta_i]=\sum (-1)^i c_{\lambda,i} [\Delta_c(\lambda)],$$
$$\Delta_i \cong \bigoplus_{\lambda} \Delta_c(\lambda)^{\oplus c_{\lambda,i}}.$$
§.§ Wreath products
In <cit.>, Chuang and Tan study filtrations of modules of wreath product algebras and the corresponding graded decomposition numbers. Here we combine their results with ours to show that the conditions of Theorem <ref> are inherited by wreath products as well.
Let $A$ be a finite-dimensional $F$-algebra, let $I$ be a finite partially ordered set, and for each $i \in I$ let $\Delta(i)$ be an $A$-module. With $C=A \mathrm{-mod}$ we assume that the conditions from <ref> hold. Let $n$ be a positive integer and write
$$A_n=A^{\otimes n} \rtimes S_n$$
for the wreath product of $A$ with $S_n$. A collection $\lambda=(\lambda^i)_{i \in I}$ of partitions indexed by $i \in I$ with $\sum_{i,j} \lambda^i_j=n$ is an $I$-partition of $n$. Given an $I$-partition $\lambda$ of $n$ and a collection of $A$-modules $M(i)$, $i \in I$, we write $A_\lambda=\bigotimes_{i \in I} A_{|\lambda^i|}$ and define
$$M(\lambda)=\mathrm{Ind}_{A_\lambda}^{A_n} \bigotimes_{i \in I} M(i)^{\otimes |\lambda^i|} \otimes S^{\lambda_i}.$$
Given two $I$-partitions $\lambda$ and $\mu$ of $n$, we write $\lambda \leq \mu$ if $\lambda=\mu$ or for each $i \in I$,
$$\sum_{j \geq i} |\lambda^j| \leq \sum_{j \geq i} |\mu^j|,$$
with a strict inequality for some $i \in I$. This defines a partial order on the set of $I$-partitions of $n$.
If $A$-mod, with standard objects $\Delta(i)$, satisfies the conditions from <ref>, then the category $A_n$-mod satisfies the conditions from <ref>, with standard objects $\Delta(\lambda)$ indexed by the poset of $I$-partitions $\lambda$ of $n$.
Proposition 3.7 of <cit.> implies that axioms (a) and (b) of <ref> hold for $A_n$, while Proposition 4.7 of <cit.> implies that (c) holds.
Let $i$ and $j$ be positive integers with $i+j=n$, let $\mu$ be an $I$-partition of $i$ and let $\lambda$ be an $I$-partition of $j$. Then the module $\mathrm{Ind}_{A_i \otimes A_j}^{A_n} \Delta(\mu) \otimes \Delta(\lambda)$ is a direct sum of standard modules for $A_n$.
This follows from the definition of standard module together with part (1) of Lemma 3.3 of <cit.>.
If $A$ is a finite dimensional $F$-algebra, $I$ is a finite poset, and $\Delta(i)$, for $i \in I$, are $A$-modules so that the conditions of <ref> and the conclusions of Theorem <ref> hold, then they hold for $A_n$ with standard modules $\Delta(\lambda)$, where $\lambda$ ranges over all $I$-partitions of $n$.
It suffices to prove that the radical of each $\Delta(\lambda)$ is a homomorphic image of a sum of $\Delta(\mu)$'s. We first treat the special case $\lambda^i=\emptyset$ for all but one $i \in I$. In this case $\Delta(\lambda)=\Delta(i)^{\otimes n} \otimes S^\lambda$, where $\lambda=\lambda^i$ is the only non-empty component. Then by <cit.> Lemma 3.5 part (3),
$$\mathrm{rad}(\Delta(\lambda))=\sum_{j=1}^n \Delta(i)^{\otimes j-1} \otimes \mathrm{Rad} \Delta(i) \otimes \Delta(i)^{\otimes n-j} \otimes S^\lambda.$$
Now if the radical of $\Delta(i)$ is a quotient of a direct sum of standard modules $M$, then there is a surjection of $A_{n-1} \otimes A_1$-modules
$$\Delta(i)^{n-1} \otimes M \otimes S^\lambda \rightarrow \Delta(i)^{n-1} \otimes \mathrm{Rad} \Delta(i) \otimes S^\lambda$$
which upon inducing and using the fact that as a $A_n$-module, $\mathrm{Rad}(\Delta(\lambda))$ is generated by the target, gives a surjection of $A_n$ modules
$$\mathrm{Ind}_{A_{n-1} \otimes A_1}^{A_n} \Delta(i)^{n-1} \otimes M \otimes S^\lambda \rightarrow \mathrm{Rad} \Delta(\lambda).$$
The domain here is a sum of standard modules by Lemma <ref>, proving this special case.
In general by using parts (3) and (4) of Lemma 3.5 from <cit.> we have
$$\mathrm{Rad}(\Delta(\lambda))=\mathrm{Ind}_{A_\lambda}^{A_n} \mathrm{Rad}( \bigotimes_{i \in I} \Delta(\lambda_i))=\mathrm{Ind}_{A_\lambda}^{A_n} (\sum_{i \in I} \bigotimes_{j \neq i} \Delta(\lambda_j) \otimes \mathrm{Rad}(\Delta(\lambda_i))).$$
Combining this equation with the special case already proved and Lemma <ref> finishes the proof.
In the category $\OO$ for the Cherednik algebra of a symmetric group, there are certain combinatorially defined blocks called RoCK blocks which have the structure of a wreath product (see <cit.>, Definition 52).
If $B_{w,\rho}\subset\OO_c(S_n)$ is a RoCK block of weight $w$ and $e$-core $\rho$ then $B_{w,\rho}$ has tight multiplicities.
$\OO_c(S_n)$ is equivalent as a highest weight category to $\mathcal{S}_q(n)-\mathrm{mod}$, the category of finite-dimensional modules over the $q$-Schur algebra <cit.>. Here $c=1/e$ with $2\leq e\leq n$, and $q$ is a primitive $e$'th root of $1$. Also, $\mathcal{S}_q(n)-\mathrm{mod}$ is highest-weight equivalent to the unipotent block of $GL(n,q)$ over a field of characteristic $\ell$ where $\ell>>0$ has order $e$ mod $q$ <cit.>. Combining these facts with theorems proved by Miyachi and Turner, we have:
If $B_{w,\rho}$ is a RoCK block in $\OO_{1/e}(S_n)$ of weight $w$ and $e$-core $\rho$ then $B_{w,\rho}\simeq B^e_{1,\emp}\wr S_w$.
$B^e_{1,\emp}$ denotes a weight $1$ block, equivalent to the principal block of $\OO_{1/e}(S_e)$. It follows from the analysis in <cit.> that any weight $1$ block has tight multiplicities. Now the conclusion follows from Corollary <ref>.
Call a block $B$ multiplicity-free if $[\Del(\lambda):L(\mu)]\in\{0,1\}$ for all $\lambda,\mu\in B$. A block with tight multiplicities need not be multiplicity-free. The RoCK blocks provide an example of this: <cit.> and <cit.> discovered a formula for decomposition numbers for RoCK blocks in terms of Littlewood-Richardson coefficients, and any Littlewood-Richardson coefficient appears as a decomposition number in some RocK block, so RoCK blocks in general are not multiplicity-free.
Does the Scopes equivalence class of RoCK blocks of weight $w$ exhaust all blocks of weight $w$ in $\bigoplus_n\OO_c(S_n)$ which have tight multiplicities? In other words, for type $A$ Cherednik algebras, is every block that has tight multiplicities highest weight equivalent to a RoCK block?
Outside type $A$, the principal block $B^0_{1/4}(E_6)$ of $\OO_{1/4}(E_6)$ is an example of a wreath product block which has tight multiplicities: $B^0_{1/4}(E_6)\simeq B^0_{1/4}(D_4)\wr S_3$ (where $B^0_{1/4}(D_4)$ denotes the principal block of $\OO_{1/4}(D_4)$) by results of Miyachi <cit.>. As mentioned in the introduction, for $D_4$ one should take $c_0=1/4$ and $d_0=d_1=0$, in which case the labels are $(0,0)$ in the first box, and $(1,-1)$ in the box with content $2$. Then $B^0_{1/4}(D_4)$ is $\ttt$-diagonalizable by Lemma <ref> and has tight multiplicities by Theorem <ref>. Therefore by Corollary <ref>, $B^0_{1/4}(E_6)$ has tight multiplicities.
§.§ BGG algebras and the $\Ext^1$ quiver
For the rest of this section, let $C$ be a highest weight category with finite poset $\Lambda$ indexing the simple and standard objects.
We always have that $\dim\Ext^1(L(\lambda),L(\mu))$ is equal to the number of times $L(\mu)$ appears in the first radical layer of $P(\lambda)$, defined as $\rad_1 P(\lambda)=\Rad P(\lambda)/\Rad(\Rad P(\lambda))$. Suppose there is a contravariant duality functor $\delta:C\rightarrow C$, $\delta^2\simeq\mathrm{Id}$, such that $\delta(L)=L$ for any simple $L\in C$. Then we say that the highest weight category $C$ is BGG <cit.>. In this case, $\dim\Ext^1(L(\mu),L(\lambda))=\dim\Ext^1(L(\lambda),L(\mu))$ for any $\lambda,\mu\in\Lambda$, as can be seen by taking a nonsplit short exact sequence $0\rightarrow L(\mu)\rightarrow E \rightarrow L(\lambda) \rightarrow 0$ and applying $\delta$ to it.
The $\Ext^1$ quiver of $C$ is defined to be the quiver with vertices $\lambda\in\Lambda$ and $\dim\Ext^1(L(\lambda),L(\mu))=:a_{\lambda,\mu}$ arrows $\lambda\rightarrow\mu$. Let us denote the $\Ext^1$ quiver for $C$ as $Q_{\Ext^1}(C)$. From the remarks above, if $C$ is BGG then $Q_{\Ext^1}(C)$ is a double quiver, that is, $a_{\lambda,\mu}=a_{\mu,\lambda}$.
§.§ Primitive homomorphisms between standard modules
Let $\lambda,\mu\in\Lambda.$ Call a homomorphism $\phi:\Del(\mu)\rightarrow\Del(\lambda)$ primitive if it does not factor through any highest weight submodule $M$ of $\Del(\lambda)$ with highest weight $\nu$, $\mu<\nu<\lambda$. Call a homomorphism $\phi:\Del(\mu)\rightarrow\Del(\lambda)$ imprimitive if there exists a highest weight submodule $M$ of $\Del(\lambda)$ of highest weight $\nu$, $\mu<\nu<\lambda$, such that a singular vector $v_\mu$ generating the image of $\phi$ belongs to $M$. Define $\Hom_{\mathrm{imprim}}(\Del(\mu),\Del(\lambda)$ to be the subspace of $\Hom(\Del(\mu),\Del(\lambda))$ generated by the imprimitive homomorphisms from $\Del(\mu)$ to $\Del(\lambda)$. Now define $\Hom_{\mathrm{prim}}(\Del(\mu),\Del(\lambda))$ to be the quotient of $\Hom(\Del(\mu),\Del(\lambda))$ by $\Hom_{\mathrm{imprim}}(\Del(\mu),\Del(\lambda))$. Note that if $\dim\Hom(\Del(\mu),\Del(\lambda))=1$ then $\Hom(\Del(\mu),\Del(\lambda))$ either consists entirely of primitive homomorphisms or imprimitive homomorphisms. If $\dim\Hom(\Del(\mu),\Del(\lambda))\in\{0,1\}$ for all $\mu,\lambda\in\Lambda$, then $\Hom_{\mathrm{prim}}(\Del(\mu),\Del(\lambda))$ or $\Hom_{\mathrm{imprim}}(\Del(\mu),\Del(\lambda))$ is either $0$ or simply the Hom space between the standards, any Hom space being the one or the other.
We may define another quiver $Q_{\mathrm{prim}}(B)$ for the block $B$: it has vertices $\lambda\in\Lambda$ and $b_{\mu,\lambda}$ arrows $\mu\rightarrow\lambda$ if $\dim\Hom_{\mathrm{prim}}(\Del(\mu),\Del(\lambda))=b_{\mu,\lambda}$.
§ THE CHEREDNIK ALGEBRA
Let $V$ be a finite dimensional $\CC$-vector space and $W \subseteq \mathrm{GL}(V)$ a finite subgroup of the group of linear transformations of $V$. A reflection is an element $r \in W$ such that the codimension of the fix space of $r$ in $V$ is $1$. Let $R$ be the set of reflections in $W$, and for each $r \in R$ let $c_r \in \CC$ be a number such that $c_r=c_{w r w^{-1}}$ for all $r \in R$ and $w \in W$. We also fix a linear form $\alpha_r \in V^*$ with
$$\mathrm{fix}(r)=\{\alpha_r=0 \}.$$
Nothing will depend upon our choice of $\alpha_r$, but the choice of $c_r$ is very important. We will write $c=(c_r)_{r \in R}$ for the collection of $c_r$'s.
Given a vector $y \in V$, the corresponding Dunkl operator on $\CC[V]$ is given by the formula
$$y(f)=\partial_y(f)-\sum_{r \in R} c_r \la \alpha_r,y \ra \frac{f-r(f)}{\alpha_r} \quad \hbox{for $f \in \CC[V]$.}$$
The rational Cherednik algebra is the subalgebra $H_c=H_c(W,V)$ of $\mathrm{End}_\CC(\CC[V])$ generated by $W$, $\CC[V]$, and the Dunkl operators $y$ for all $y \in V$.
§.§ Category $\OO_c$
Category $\OO_c$ is the category of finitely generated $H_c$-modules on which each Dunkl operator $y \in V$ acts locally nilpotently. The objects of $\OO_c$ that can be constructed most directly are certain induced modules. Specifically, given a $\CC W$-module $U$ we inflate it to a $\CC[V^*] \rtimes W$-module by letting each $y \in V$ act by $0$. Then we put
$$\Delta_c(U)=\mathrm{Ind}_{\CC[V^*] \rtimes W}^{H_c} U.$$
If $\Lambda$ is an index set for the irreducible $\CC W$-modules, $\lambda \in \Lambda$, and $S^\lambda$ is the corresponding irreducible, then we will also write
These $\Delta_c(\lambda)$ are the standard modules for $H_c$. Each carries a contravariant form $\la \cdot,\cdot \ra_c$ whose radical is the radical of the module $\Delta_c(\lambda)$. The quotient module
$$L_c(\lambda)=\Delta_c(\lambda)/\mathrm{Rad}(\la \cdot,\cdot \ra_c)$$
is irreducible, and as $\lambda$ runs over $\Lambda$ these give a complete set of non-isomorphic irreducible objects of $\OO_c$.
§.§ Partial orders on $\Irr W$
There is a partial order $<_c$ on $\Irr W$ called the $c$-order with respect to which $\OO_c(W)$ is a highest weight category <cit.>. This partial order is defined using the action of the Euler element $h$, see section 3.1 of <cit.>: $h$ acts by a scalar $h_c(\lambda)$ on the highest weight $\lambda$ of any standard module $\Del(\lambda)$ in $\OO_c(W)$. We say that $\lambda>_c\mu$ if $h_c(\mu)-h_c(\lambda)\in\ZZ_{\geq 0}$. For the groups of type $G(r,1,n)$, we may define $h_c(\lambda)$ in terms of charged contents. Let $\tilde{c}(b)=rc_0\ct(b)+d_{\beta(b)}$ (see Section <ref>) and $\hat{c}(\lambda)=\sum_{b\in\lambda}\tilde{c}(b)$. Then
On the other hand, there is the natural partial order $<$ on $\Irr W$ given by the transitive closure of the relation: $\mu<\lambda$ if there is a nonzero homomorphism $\Del(\mu)\rightarrow\Del(\lambda)$.
Note that Cherednik category $\OO_c(W)$ always satisfies BGG reciprocity <cit.>:
\begin{equation}\label{BGG}
\end{equation}
It is a consequence of BGG reciprocity that $\OO_c(W)$ has a unique coarsest partial order. This minimal partial order is given by $<$:
(<cit.>, Lemma 4.5) If $\prec$ coarsens the $c$-order $<_c$ and $\prec$ gives a highest weight structure on $\OO_c(W)$, then $\prec$ refines $<$.
§.§ BGG properties of $\OO_c(W)$
Cherednik Category $\OO_c(W)$ for $W$ a real reflection group has a duality functor $\delta$ fixing simples, as described in <cit.>, Proposition 4.7 and Remark 4.9. Thus $\OO_c(W)$ is BGG when $W$ is real. Furthermore, if the parameter $c$ is real and $W$ is a complex reflection group, then $\OO_c(W)$ also has a duality functor, defined as follows. Fix a $W$-equivariant complex-antilinear isomorphism $\phi:\hh\rightarrow\hh^*$. If $c$ is real, then $\phi$ induces a complex-antilinear ring anti-involution $\bar{}:H_c(W)\rightarrow H_c(W)$ such that $\bar{x}=\phi^{-1}(x)$, $\bar{y}=\phi(y)$, and $\bar{w}=w^{-1}$ for all $x\in\hh^*$, $y\in\hh$, $w\in W$. Given a module $M$ in $\OO_c(W)$, let $DM$ be its graded complex-antilinear dual, with action given by $(hf)(m)=f(\bar{h}m)$ for $h\in H_c(W)$, $f\in DM$, $m\in M$. This gives a complex-antilinear additive exact functor $\delta:\OO_c(W)\rightarrow\OO_c(W)$, $\delta(M)=DM$. Then $\delta$ fixes simples. Thus $\OO_c(W)$ is also BGG so long as $c$ is real, which is the case in all examples considered in this paper.
§ ORTHOGONAL FUNCTIONS AND TABLEAU TECHNIQUES FOR CHEREDNIK ALGEBRA REPRESENTATION THEORY
§.§ The group $G(r,p,n)$.
Let $n$ and $r$ be positive integers. The group $G(r,1,n)$ consists of all $n$ by $n$ matrices with exactly one non-zero entry in each row and column, and such that the non-zero entries are all $r$'th roots of $1$. It acts on $V=\CC^n$ as a reflection group. Write $s_{ij}$ for the matrix interchanging the $i$'th and $j$'th coordinates and leaving the remaining coordinates fixed, let $\zeta=e^{2 \pi i /r}$ be a fixed primitive $r$'th root of $1$, and write $\zeta_i$ for the diagonal matrix with $1$'s on the diagonal except in position $i$ and with $i$'th diagonal entry equal to $\zeta$.
Let $p$ be a positive integer dividing $r$. The group $G(r,p,n)$ is the subgroup of $G(r,1,n)$ consisting of matrices so that the product of the non-zero entries is an $r/p$'th root of $1$. The set of reflections in $G(r,p,n)$ is
$$R=\{\zeta_i^\ell s_{ij} \zeta_i^{-\ell} \ | \ 1 \leq i< j \leq n, \ 0 \leq \ell \leq r-1 \} \cup \{\zeta_i^{p\ell} \ | \ 1 \leq i \leq n, \ 1 \leq \ell \leq r/p-1 \}.$$
§.§ The rational Cherednik algebra for $G(r,p,n)$.
We put $W=G(r,1,n)$. It is convenient to reparametrize $c$ in this case. We do so following the conventions in <cit.>, (4.2) and (4.3). This parametrization may be specified by giving the defining relations for $H_c=H_c(W,V)$: it is the quotient of the algebra $T(V \oplus V^*) \rtimes W$ by the relations
$$[x_i,x_j]=0=[y_i,y_j] \ \hbox{for all $1 \leq i,j \leq n$,}$$
$$y_i x_j=x_j y_i+c_0 \sum_{0 \leq \ell \leq r-1} \zeta^{-\ell} \zeta_i^\ell s_{ij} \zeta_i^{-\ell} \quad \hbox{for $1 \leq i \neq j \leq n$,}$$
$$y_i x_i=x_i y_i + 1-\sum_{0 \leq j \leq r-1} (d_j-d_{j-1}) e_{ij} - c_0 \sum_{\substack{j \neq i \\ 0 \leq \ell \leq r-1}} \zeta_i^\ell s_{ij} \zeta_i^{-\ell} \quad \hbox{for $1 \leq i \leq n$,}$$
$$e_{ij}=\frac{1}{r} \sum_{0 \leq \ell \leq r-1} \zeta^{-\ell j} \zeta_i^\ell.$$
The rational Cherednik algebra for $G(r,p,n)$ may be realized as the subalgebra of $H_c$ generated by $x_1,\dots,x_n,y_1,\dots,y_n$ and $G(r,p,n)$.
§.§ A third commutative subalgebra of $H_c$.
The rational Cherednik algebra contains subalgebras $\CC[x_1,\dots,x_n]$ and $\CC[y_1,\dots,y_n]$ isomorphic to polynomial rings; the variables $x_i$ and $y_i$ act nilpotently on any finite dimensional representation. There is, however, a third polynomial subalgebra that often acts with simple spectrum. This is generated by the version of the Cherednik-Dunkl operators, also known as trigonometric Dunkl operators, introduced by Dunkl and Opdam in <cit.>. These are given by the formula
$$z_i=y_i x_i+c_0 \phi_i \quad \text{where} \quad \phi_i=\sum_{\substack{1 \leq j < i \\ 0 \leq \ell \leq r-1}} \zeta_i^\ell s_{ij} \zeta_i^{-\ell}.$$
We will write $\ttt=\CC[z_1,\dots,z_n,\zeta_1,\dots,\zeta_n]$ for the (commutative, though this is not obvious) subalgebra of $H_c$ generated by the Cherednik-Dunkl operators $z_i$ and the matrices $\zeta_i$.
§.§ $r$-partitions
We will use the following conventions for multi-partitions. A partition is a weakly decreasing sequence $\lambda=(\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k)$ of non-negative integers. Given positive integers $r$ and $n$, an $r$-partition of $n$ is a sequence $\lambda=(\lambda^0,\lambda^1,\dots,\lambda^{r-1})$ of partitions $\lambda^i$ such that $n=\sum \lambda^i_j$. We will frequently visualize partitions via Young diagrams. For instance, the partition $(3,3,2)$ may be visualized as
As is usual, we will identify two partitions that differ by a string of $0$'s, so that for instance $(3,3,2)=(3,3,2,0,0)$. The content of a box in (the diagram of) a partition $\lambda$ is
where $\mathrm{col}(b)$ is $i$ if $b$ is in the $i$th column (counting from the left) and $\mathrm{row}(b)$ is $j$ if $b$ is in the $j$th row (counting from the top). Thus the list of contents of the boxes of $(3,3,2)$, read as in English from left to right and then top to bottom, is $(0,1,2,-1,0,1,-2,-1)$. For an $r$-partition $\lambda$ and a box $b \in \lambda$ we define
$$\beta(b)=i \quad \text{if} \quad b \in \lambda^i.$$
§.§ When are two standard modules in the same block?
The charged content $c(b)$ of a box $b$ in an $r$-partition $\lambda$ is given by the formula
\begin{equation} \label{charged content}
c(b)=(d_{\beta(b)}-\beta(b))/r+ \mathrm{ct}(b) c_0
\end{equation}
where if is a box of $\lambda^\ell$, $\mathrm{ct}(b)$ means the content of $b$ as a box of the ordinary partition $\lambda^\ell$. We define the $c$-weight of a box $b$ to be the element of $\CC / \ZZ$ given by
$$w_c(b)=c(b) \ \text{mod} \ \ZZ,$$
and the $c$-weight of $\lambda$ is the multiset of the $c$-weights of its boxes,
$$w_c(\lambda)=\{w_c(b)\;|\;b \in \lambda\}$$
Suppose the parameters $c$ are nonzero. By Theorem 2.11 of <cit.> and the Double Centralizer Theorem <cit.>,
$\Delta_c(\lambda)$ and $\Delta_c(\mu)$ belong to the same block if and only if $w_c(\lambda)=w_c(\mu)$.
The translation between Lyle-Mathas' criterion and ours is given by $e^{-2\pi iw_c(b)}=\mathrm{res}(b)$.
In the cases we study in this paper, when the denominator of $c_0$ is exactly $n$, we could use <cit.> to characterize the $r$-partitions indexing standard modules in the principal block directly.
§.§ Tableaux and representations
Let $\lambda$ be an $r$-partition of $n$. A standard Young tableau on $\lambda$ is a filling $T$ of the boxes of $\lambda$ by the integers $1,2,\dots,n$ in such a way that the entries are increasing left to right and top to bottom within each component partition $\lambda^\ell$. We will denote the set of all standard Young tableaux on $\lambda$ by $\mathrm{SYT}(\lambda)$. When we write $T^{-1}(i)$ we will mean the box in $\lambda$ labeled by $i$.
The set of irreducible representations of $\CC G(r,1,n)$ is in bijection with the set of $r$-partitions of $n$ in such a way that if $S^\lambda$ is the irreducible $\CC G(r,1,n)$-module indexed by the $r$-partition $\lambda$, then there is a basis $v_T$ of $S^\lambda$ indexed by $T \in \mathrm{SYT}(\lambda)$ with
$$\zeta_i v_T=\zeta^{\beta(T^{-1}(i))} v_T \quad \text{and} \quad \phi_i v_T=r \mathrm{ct}(T^{-1}(i)) v_T \quad \hbox{for $1 \leq i \leq n$.} $$
§.§ The $\ttt$-action on standard modules.
We reintroduce some of the notation from Section 2 of <cit.>. Let $\alpha=(\alpha_1,\alpha_2,...,\alpha_n)\in\ZZ^n_{\geq 0}$. For $w\in S_n$, define a left action on $\ZZ^n_{\geq 0}$ by
Let $w_\alpha\in S_n$ be the longest element (in Bruhat order) of all $w$ such that $w\cdot\alpha$ is a nondecreasing sequence. Finally, let $x^\alpha=x_1^{\alpha_1}x_2^{\alpha_2}...x_n^{\alpha_n}$.
The standard module $\Del_c(\lambda)$ is isomorphic, as a vector space, to $\CC[\hh]\otimes \lambda$; as a basis of $\Del_c(\lambda)$ we may take
$$\{x^\alpha w_\alpha^{-1} v_T\;|\;\alpha\in\ZZ^n_{\geq0},\;T\in\mathrm{SYT}(\lambda)\}$$
According to Theorem 5.1 of <cit.> the action of $\ttt$ on this basis is given, for $1 \leq i \leq n$, by
$$\zeta_i x^\alpha w_\alpha^{-1} v_T=\zeta^{\beta(T^{-1} w_\alpha(i))-\alpha_i} x^\alpha w_\alpha^{-1} v_T$$
\begin{align*}
z_i x^\alpha w_\alpha^{-1} v_T&=\left(\alpha_i+1-(d_{\beta(T^{-1} w_\alpha(i))}-d_{\beta(T^{-1} w_\alpha(i))-\alpha_i-1})-r \mathrm{ct}(T^{-1} w_\alpha(i))c_0 \right) x^\alpha w_\alpha^{-1} v_T \\ &+\text{lower terms}.
\end{align*}
§.§ $\ttt$-semisimplicity of $\Delta_c(\lambda)$.
The standard module $\Delta_c(\lambda)$ for $H_c$ is, for generic $c$, diagonalizable with respect to $\ttt$. The precise result is as follows (see Lemma 7.1 from <cit.>): for each component $\lambda^i$ of $\lambda$, define its diameter $\mathrm{diam}(\lambda^i)$ to be the maximum difference between the contents of two of its boxes. Thus the diameter of $(3,3,2)$ is $4$. Assume $c_0 > 0$ (up to some obvious symmetries, this is the only really interesting case). Define $m$ to be the maximum over all components $\lambda^i$ of $\lambda$ that are not single columns of the diameter $\mathrm{diam}(\lambda^i)$. Then $\Delta_c(\lambda)$ is $\ttt$-diagonalizable if and only if $c_0$ is not a rational number with denominator at most $m$, and furthermore no equation of the form
$$k=c(b_1)-c(b_2) \quad \hbox{for $b_1 \in \lambda^i$, $b_2 \in \lambda^j$, $i \neq j$ and $k \in \ZZ$}$$
holds, where $c(b)$ denotes the charged content of $b$ as in (<ref>). In this situation, for each pair $(\alpha,T) \in \ZZ_{\geq 0}^n \times \mathrm{SYT}(\lambda)$ there is a unique $f_{\alpha,T} \in \Delta_c(\lambda)$ with $f_{\alpha,T}=x^\alpha w_\alpha^{-1} v_T+\text{lower terms}$ and
\begin{align*}
z_i f_{\alpha,T}&=\left(\alpha_i+1-(d_{\beta(T^{-1} w_\alpha(i))}-d_{\beta(T^{-1} w_\alpha(i))-\alpha_i-1})-r \mathrm{ct}(T^{-1} w_\alpha(i))c_0 \right)f_{\alpha,T}\\ \zeta_i f_{\alpha,T}&=\zeta^{\beta( T^{-1} w_\alpha(i))} f_{\alpha,T}
\end{align*}
for $1 \leq i \leq n$.
The above results imply the following lemma, giving a sufficient condition for every standard module in the principal block to be $\ttt$-diagonalizable.
If $w_c(b) \neq w_c(b')$ for all $b \neq b'$ then every standard module in the same block as $\Delta_c(\lambda)$ is $\ttt$-diagonalizable. In particular, if $c_0$ is not a rational number of denominator at most $n-1$, then every standard module in the same block as $\Delta_c(\mathrm{triv})$ is $\ttt$-diagonalizable.
§.§ The submodule structure of $\Delta_c(\lambda)$ and tight multiplicities.
We will repeatedly make use Theorem 7.5 from <cit.>. For the reader's convenience we include a statement of this result here.
Assume $\Delta_c(\lambda)$ is $\ttt$-diagonalizable, so that the basis elements $f_{\alpha,T}$ are all well-defined. Given a box $b \in \lambda$ and a positive integer $k$, define a $\CC$-subspace $M_{b,k}$ by
$$M_{b,k}=\CC \{ f_{\alpha,T} \ | \ \alpha^-_{T(b)} \geq k \}.$$
Given two boxes $b_1,b_2 \in \lambda$ and a positive integer $k$, let
$$M_{b_1,b_2,k}=\CC \{ f_{\alpha,T} \ | \ \hbox{ $ \alpha^-_{T^{-1}(b_1)} - \alpha^-_{T^{-1}(b_2)} \geq k$, with equality implying $w_\alpha^{-1}(T(b_1)) < w_\alpha^{-1}(T(b_2))$} \}.$$
Now we state our main tool, Theorem 7.5 from <cit.> (which is a generalization of Theorem ??? from <cit.>).
Suppose $\Delta_c(\lambda)$ is $\ttt$-diagonalizable. Then the lattice of submodules of $\Delta_c(\lambda)$ is generated by those of following two forms:
(1) $M_{b,k}$, for a box $b \in \lambda$ and a positive integer $k$ such that $k=r c(b)+\beta(b)-d_{\beta(b)-k}$;
(2) $M_{b_1,b_2,k}$, for boxes $b_1,b_2 \in \lambda$ and a positive integer $k$ with $k=r c(b_1)+\beta(b_1)-r c(b_2)-\beta(b_2) \pm r c_0 $ and $k=\beta(b_1)-\beta(b_2) \ \mathrm{mod} \ r$.
We will refer to the submodules $M_{b,k}$ and $M_{b_1,b_2,k}$ appearing in Theorem <ref> as fundamental submodules.
The lowest degree subspace of a given submodule consists of singular vectors. Thus for each fundamental submodule we obtain a map from a standard module to $\Delta_c(\lambda)$. We can specify which standard modules map into $\Delta_c(\lambda)$ this way by examining the $\ttt$-eigenvalues of the $f_{\alpha,T}$'s spanning the lowest degree part.
Suppose $\Delta_c(\lambda)$ is diagonalizable and that $c_0>0$.
(a) In case (1) from Theorem <ref>, the isotype of the lowest degree subspace of $M_{b,k}$ is the module $S^\mu$ where $\mu$ is obtained from $\lambda$ by removing the subdiagram of $\lambda^{\beta(b)}$ consisting of $b$ and all boxes (weakly) below and to the right of it, and setting $\mu^{\beta(b)-k}$ equal to this subdiagram, leaving all other components of $\lambda$ unchanged.
(b) In case (2) from Theorem <ref>, when the plus sign holds in the term $+rc_0$ (resp., the minus sign holds in the term $-rc_0$), the isotype of the lowest degree space of $M_{b_1,b_2,k}$ is the module $S^\mu$, where $\mu$ is obtained from $\lambda$ by removing the subdiagram of $\lambda^{\beta(b_1)}$ consisting of $b_1$ and all boxes below it (resp., to its right), and attaching it to $\lambda^{\beta(b_2)}$ in such a way that $b_1$ is the box directly below $b_2$ (resp., to its right).
The next corollary is a sufficient condition for the principal block of $\OO_c$ to have tight multiplicities that is practical to verify in examples.
Suppose that every standard module in the principal block is diagonalizable and that the following condition holds: for every standard module and every submodule $M$ of the form $M_{b,k}$ or $M_{b_1,b_2,k}$ as in the previous theorem, if the lowest degree subspace of $M$ is contained in a module of the form (2), then $M$ is as well. Then the principal block has tight multiplicities.
If $M$ and $N$ are submodules that are both generated by singular vectors, then so is their sum $M+N$. So it suffices to prove that submodule of the forms (1) and (2) is generated by singular vectors. Let $M$ be such a submodule and let $N \subseteq M$ be the submodule generated by the singular vectors in $M$. By Theorem <ref> $N$ is in the lattice of submodules, so equal to a sum of intersections of modules of the forms (1) and (2). Evidently $N$ contains the lowest degree piece $S$ of $M$. By our hypothesis $S$ is only contained in a submodule of the form (2) if $M$ is. On the other hand, it follows from the definitions that if $S$ is contained in a submodule of the form (1) then so is $M$. Therefore every submodule of the form (1) or (2) that contains $S$ contains all of $M$, and hence $N=M$ as desired.
§.§ Graded dimension of $L_c(\Triv)$
Suppose $c_0=\ell/n$ for a positive integer $\ell$ coprime to $n$. Then the set of $f_\alpha$ such that
(a) we have $\alpha_n^--\alpha_1^- \leq \ell r$ with equality implying $w_\alpha^{-1}(n) > w_\alpha^{-1}(1)$, and
(b) for each pair of integers $k$ and $m$ with $0 \leq m \leq n-1$ and $k >0$ such that
$$d_0-d_{-k}+rm \ell/n=k,$$
we have $\alpha_{m+1}^- < k$
is a homogeneous basis of $L_c(\Triv)$. In particular the graded dimension of $L=L_c(\Triv)$ for $G(2,2,2n)$ with $c=1/2n$ is
$$\sum \mathrm{dim}_\CC(L^d)q^d= \sum_{\substack{0 \leq \ell \leq n-1 \\ n-1-\ell \in 2 \ZZ}} {2n \choose \ell} q^{n-1} + \sum_{d=0}^{n-2} \left(\sum_{\substack{0 \leq \ell \leq d \\ d-\ell \in 2 \ZZ}} {2n \choose \ell} \right) (q^d+q^{2(n-1)-d}),$$
and the graded dimension of $L=L_c(\Triv)$ for $G(2,1,2n)$ with $c=1/2n$ is
$$ \sum \mathrm{dim}_\CC(L^d)q^d= \sum_{\substack{0 \leq \ell \leq n \\ n-\ell \in 2 \ZZ}} {2n \choose \ell} q^n + \sum_{d=0}^{n-1} \left(\sum_{\substack{0 \leq \ell \leq d \\ d-\ell \in 2 \ZZ}} {2n \choose \ell} \right) (q^d+q^{2n-d}).$$
That the given set of $f_\alpha$'s give a basis of $L_c(\Triv)$ follows immediately from Theorem <ref>. In the case of $G(2,1,n)$ with the parameters $c_0=1/2n$ and $d_0=0=d_1$, this specializes to show that
\begin{equation} \label{Dtriv basis} \{f_\alpha \ | \ \alpha^-_{n+1}=0 \ \text{and} \ \alpha^-_{2n}-\alpha^-_1 \leq 2 \ \hbox{with equality implying $w_\alpha^{-1}(2n) > w_\alpha^{-1}(1)$}\}\end{equation}
is a basis of $L=L_c(\Triv)$. Moreover, since $d_0=0$, the Cherednik algebra of type $G(2,2,n)$ is a subalgebra of that of type $G(2,1,n)$ and the module $L$ restricts to this subalgebra to give the spherical irreducible.
The condition $\alpha^-_{n+1}=0$ means that any $\alpha$ indexing a basis element of $L$ must have at least $n+1$ $0$'s, or equivalently, at most $n-1$ non-zero entries. In the presence of this constraint, the condition $\alpha^-_{2n}-\alpha^-_1 \leq 2$ implies that the largest entry in $\alpha$ is at most $2$, and finally the condition $w_\alpha^{-1}(2n) > w_\alpha^{-1}(1)$ if some $2$ appears means that all the zeros appear to the left of all the twos. Thus the top degree piece occurs in dimension $2(n-1)$, in accordance with the claimed graded dimension formula. By symmetry it suffices to prove that the dimension of each graded piece in degree at most $n-1$ is as claimed.
The set of possible $\alpha^-$'s that can arise is
$$\{(0^k, 1^\ell ,2^m \ | \ k \geq n+1 \ \text{and} \ k+\ell+m=2n \}.$$
Given such a $\alpha^-$, the conditions in (<ref>) imply that choosing a $\alpha$ rearranging to $\alpha^-$ and corresponding to a basis element of $L$ is equivalent to choosing the position of the $\ell$ $1$'s, which may be done in exactly ${2n \choose \ell}$ ways. It follows that, for $d \leq n-1$, the dimension of the polynomial degree $d$ piece of $L$ is
$$\mathrm{dim}_\CC(L^d)=\sum_{\substack{0 \leq \ell \leq d \\ d-\ell \in 2 \ZZ}} {2n \choose \ell}$$
in which the $\ell$'th summand comes from $\alpha$'s with $\ell$ $1$'s and $(d-\ell)/2$ $2$'s. This proves the type $G(2,2,2n)$ formula, and the type $G(2,1,2n)$ case is entirely analogous.
These formulas together with some elementary manipulatorics prove the conjectures of Oblomkov-Yun mentioned in the introduction.
The Oblomkov-Yun conjecture for $\dim L(B_{2n})$ concerns the Cherednik algebra of type $G(2,1,2n)$ at equal parameters $1/2n$; generalizing to the Cherednik algebra of $G(r,1,rn)$ at equal parameters $1/rn$, it turns out that for $r>1$ there is a natural dimension formula for $L(\Triv)$ that specializes to the dimension formula for $L(B_{2n})$ when $r=2$. Take equal parameters $c_0=c_1=...=c_{r-1}=1/rn$ for $H_c(G(r,1,rn))$. Let $\zeta$ be a primitive $r$'th root of $1$. Then for each $0\leq k\leq r-1$,
$$d_k=\sum_{1\leq \ell\leq r-1}\zeta^{k\ell} c_\ell=\begin{cases}(r-1)/rn & \hbox{ if }k=0\\ -1/rn & \hbox{ if }1\leq k \leq r-1\end{cases}$$
We have $L(\Triv)=L((rn),\emp,...,\emp)$. The $r$-version of the dimension formula for $L(B_{2n})$ is straightforward to prove, using Theorem <ref> and generalizing the argument in Corollary <ref>:
* The radical of $\Del(\Triv)$ is the sum of the following $r$ submodules:
* For each $k=1,...,r-1$, there is a submodule $M_{b,k}$ where $b$ is the $kn$'th box of $\Triv$.
* There is a single submodule of the form $M_{b_1,b_2,r}$ by taking $b_1$ and $b_2$ to be the rightmost and leftmost boxes, respectively, of $\Triv$.
* $L(\Triv)$ has the basis
$$\{ f_\alpha \ | \ \alpha_{kn}^-\leq k-1 \ \hbox{for each} \ 1\leq k\leq r-1,\ \alpha_{rn}^-\leq r \ \hbox{with equality implying} \ w_\alpha^{-1}(rn)>w_\alpha^{-1}(1)\}$$
\begin{align*}
\dim L(\Triv)=&\sum_{j_r=0}^n\sum_{j_{r-1}=0}^{n-j_r}{rn\choose j_{r-1}}\sum_{j_{r-2}=0}^{2n-j_r-j_{r-1}}{rn-j_{r-1}\choose j_{r-2}}\sum_{j_{r-3}=0}^{3n-j_r-j_{r-1}-j_{r-2}}{rn-j_{r-1}-j_{r-2}\choose j_{r-3}}...\\
&\qquad\qquad\qquad\qquad...\sum_{j_1}^{(r-1)n-j_r-...-j_2}{rn-j_{r-1}-...-j_2\choose j_1}
\end{align*}
§.§ Coordinate rings of subspace arrangements
In many cases of interest, a quotient of the polynomial represnetation $\Delta_c(\Triv)$ may be identified with the coordinate ring of a subscheme of $V$, and information about the subscheme obtained from representation theory. For instance, the coordinate ring of the set of points in $\CC^n$ having at least $k$ coordinates equal to one another is the quotient of the polynomial representation of the type $A_n$ Cherednik algebra at parameter $c=1/k$ by its socle, which is a unitary representation. A conjectural BGG resolution (see <cit.>) of this unitary representation is then a minimal resolution of the coordinate ring of the $k$-equals arrangement. The general principle allows the calculation of data of interest in commutative algebra in terms of the combinatorics used by representation theorists.
We can combine Theorem <ref> with Lemma 2.9 from <cit.> to give a similar description of $L_c(\Triv)$ for types $G(2,1,2n)$ and $G(2,2,2n)$.
The module $L_c(\Triv)$ for $G(2,2,2n)$ at parameter $c=1/2n$ is the coordinate ring of the scheme theoretic intersection of the set of points in $\CC^{2n}$ with $x_1^2=x_2^2=\cdots=x_{2n}^2$ with the set of points having at least $n+1$ coordinates equal to zero. Likewise, the module $L_c(\Triv)$ for $G(2,1,2n)$ at parameter $c=1/2n$ is the coordinate ring of the scheme theoretic intersection of the set of points in $\CC^{2n}$ with $x_1^2=x_2^2=\cdots=x_{2n}^2$ with the set of points having at least $n$ coordinates equal to zero.
In the $G(2,2,2n)$ case, Lemma 2.9 from <cit.> implies that the two ideal of $x_1^2=\cdots=x_{2n}^2$ and the ideal of the set of points having at least $n+1$ coordinates equal to zero are both submodules. Now Theorem 4.4 implies that these must be equal to the submodules $M_{b_1,b_2,2}$ (where $b_1$ is the box of content $2n-1$ and $b_2$ is the box of content zero in $\Triv$) and $M_{b,1}$ (where $b$ is the box of content n), respsectively. Therefore the radical of $\Delta_c(\Triv)$ is the sum of these two ideals and the result follows. The case $G(2,1,2n)$ is analogous.
§.§ Jantzen filtration
In this subsection we explain how to explicitly describe the Jantzen filtration on $\Delta_c(\lambda)$ in case it is $\ttt$-diagonalizable.
We will write $H(G,V)$ and $\Delta(\lambda)$ for the Cherednik algebra and its standard modules assuming the parameters $c$ are polynomial variables. The contravariant form $\la \cdot,\cdot \ra$ on $\Delta(\lambda)$ then takes value in the ring $A=\CC[c_r]_{r \in R}$ of polynomials in the parameters. Fixing a numerical parameter $c$, we write $m_c \subseteq A$ for the corresponding maximal ideal, so that we have quotient maps $H(G,V) \rightarrow H_c(G,V)=H(G,V)/m_cH(G,V)$ and $\pi:\Delta(\lambda) \rightarrow \Delta_c(\lambda)=\Delta(\lambda)/m_c \Delta(\lambda)$.
Define a filtration on $\Delta(\lambda)$ by
$$\Delta(\lambda)^{\geq d}=\{f \in \Delta(\lambda) \ | \ \la f,g \ra \in I_c^d \ \hbox{for all $g \in \Delta(\lambda)$.} \}.$$
The Jantzen filtration on $\Delta_c(\lambda)$ is the image of this filtration by the quotient map,
$$\Delta_c(\lambda)^{\geq d}=\pi(\Delta(\lambda)^{\geq d}).$$
We note that we may also compute the Jantzen filtration by first localizing $A$, replacing it in the definitions by the local ring $A_c$ at $m_c$ before specializing to $A/m_c$. If $c$ is a parameter such that the polynomials $f_{\alpha,T}$ are well-defined at $c$, these may also be regarded as elements of the standard module localized at $m_c$.
The next lemma describes an explicit basis of the $d$th submodule in the Jantzen filtration in terms of the fundamental submodules defined following Theorem <ref>.
Assume that $\Delta_c(\lambda)$ is $\ttt$-diagonalizable. We have
$$\Delta_c(\lambda)^{\geq d}=\CC \{f_{\alpha,T} \ | \ \hbox{$f_{\alpha,T}$ belongs to at least $d$ fundamental submodules.} \}$$
Each $f_{\alpha,T} \in A_c \otimes_A \Delta(\lambda)$ may be constructed by applying the intertwining operators $\sigma_i$ and $\Phi$ in some sequence (depending on $(\alpha,T))$ to a certain basis element $w_0 v_T \in S^\lambda$, in such a way that once we enter a fundamental submodule we never leave it. The norm $\la f_{\alpha,T},f_{\alpha,T} \ra$ is a rational function of $c$ which may be written as a product of linear factors divided by another product of linear factors using the recursions in the proof of Theorem 6.1 from <cit.>. Examining these recursions shows that the denominators are never zero at $c$, and the numerator acquires precisely one zero every time we enter some fundamental submodule.
§.§ Tableaux indexation for $f_{\alpha,T}$
Given $T \in \mathrm{SYT}(\lambda)$ and $\alpha \in \ZZ_{\geq 0}^n$, define fillings $P$ and $Q$ of the boxes of $\lambda$ by the rules
$$P(b)=w_\alpha^{-1} (T(b)) \quad \text{and} \quad Q(b)=\alpha_{P(b)} \quad \hbox{for all $b \in \lambda$.}$$
We may recover $\alpha$ and $T$ from $P$ and $Q$ by the rules
$$\alpha_i=Q(P^{-1}(i)) \quad \text{and} \quad T(b)=w_\alpha(P(b)).$$
These mappings define inverse bijections between $\ZZ_{\geq 0}^n \times \mathrm{SYT}(\lambda)$ and the set of pairs $(P,Q)$, where $P$ is a bijection from the boxes of $\lambda$ to the set $\{1,2,\dots,n\}$, $Q$ is a filling of the boxes of $\lambda$ by non-negative integers, weakly increasing left to right and top to bottom, and we have
$$P(b) > P(b') \quad \hbox{whenever $b < b'$ and $Q(b)=Q(b')$.}$$
Here we have defined an ordering on the boxes by $b< b'$ if $T(b) < T(b')$ for all standard Young tableaux $T$ (in other words, if $b$ is up and to the left of $b'$ in the same component of $\lambda$). We will write $\gimel(\lambda)$ for this set of pairs $(P,Q)$; it is identified via the above bijections with the set of vertices of the calibration graph defined in <cit.>.
§.§ Indexation of the irreducibles in the principal block
Given $0 \leq i \leq r-1$ and a box $b \in (n)$ for which the equation
$$(d_i-i)/r=d_0/r+\mathrm{ct}(b) c_0+k_i$$
holds for some integer $k_i$, we put $b_i=b$. Note that since the denominator of $c_0$ is $n$, the box $b$ and the integer $k_i$ are uniquely determined by $i$ if they exist. This produces a list of boxes $b_i$, one for each $0\leq i \leq r-1$ for which the equation has a solution, each such box being accompanied by the pair of integers $(i,k_i)$. Notice that we may well have $b_i=b_j$ for some $i \neq j$. We can record these data as in the introduction, by placing a label $(i,k_i)$ in the box $b_i$. We repeat the example from the introduction here:
\ytableausetup{mathmode, boxsize=3.5em}
\begin{ytableau}
\begin{matrix} (0,0) \\ (1,-2) \end{matrix} & (3,-1) & & & & \begin{matrix} (5,3) \\ (2,-2) \end{matrix} & &
\end{ytableau}
Given a multipartition $\lambda$ with $w_c(\lambda)=w_c(\Triv)$ we define a pair of tableaux on $\Triv$: first, we observe that since the denominator of $c_0$ is exactly $n$, there is a unique box $\overline{b}$ of $\lambda$ for each box $b$ of $\Triv$ satisfying
$$c(\overline{b})=c(b) \ \mathrm{mod} \ \ZZ.$$
We then define two fillings of the boxes of $(n)$ as follows: for the first, we put $S(b)=\beta(\overline{b})$ in the box $b$, and for the second, we place the integer $T(b)$ defined by the equation
in the box $b$. We write $S(\lambda)$ and $T(\lambda)$ for these tableaux. It is not difficult to see that $T(\lambda)$ is uniquely determined from $S(\lambda)$ and the list $(b_i,k_i)$ unless $S(\lambda)$ contains only one integer $i$ (that is, unless $\lambda^i=\emptyset$ for all but one index $i$).
This defines a bijection from the set of $r$-partitions $\lambda$ with $w_c(\lambda)=w_c(\Triv)$ onto the set of pairs $(S,T)$ of tableaux on $(n)$ with the following properties. First, we identify the right-hand border of the last box of $(n)$ with the left-hand border of the first box of $(n)$ so that the set of boxes becomes circular. Then the set of boxes $b$ with $S(b)=i$ is an interval (in this circular version of $(n)$) containing the box $b_i$ labeled $(i,k_i)$ (and empty if there is no box $b_i$), we have $T(b_i)=k_i$, and $T(b)=k_i$ or $T(b)=k_i \pm \ell$ for all boxes $b$ with $S(b)=i$, with $T(b)=k_i+\ell$ if when traveling left to right from $b_i$ to $b$ we cross from the $n$th box to the $1$st.
For the example of the block labeling given above and in the introduction and with $c_0=3/8$, here are possible pairs $(S,T)$:
S=\ytableausetup{mathmode, boxsize=2em}
\begin{ytableau}
0 & 0 & 5 & 5 & 5 & 5 & 5 & 0
\end{ytableau}
T=\ytableausetup{mathmode, boxsize=2em}
\begin{ytableau}
0 & 0 & 3 & 3 & 3 & 3 & 3 & -3
\end{ytableau}
corresponds to
\lambda=\left(\ytableausetup{mathmode, boxsize=1em}
\begin{ytableau} \hfil & \hfil \\ \hfil \end{ytableau}, \emptyset,\emptyset,\emptyset,\emptyset, \begin{ytableau} \hfil & \hfil \\ \hfil \\ \hfil \\ \hfil \end{ytableau}\right)
S=\ytableausetup{mathmode, boxsize=2em}
\begin{ytableau}
2 & 3 & 3 & 3 & 2 & 2 & 2 & 2
\end{ytableau}
T=\ytableausetup{mathmode, boxsize=2em}
\begin{ytableau}
1 & -1 & -1 & -1 & -2 & -2 & -2 & -2
\end{ytableau}
corresponds to
\lambda=\left(\ytableausetup{mathmode, boxsize=1em}
\emptyset,\emptyset,\begin{ytableau} \hfil & \hfil & \hfil &Â \hfil \\ \hfil \end{ytableau}, \begin{ytableau} \hfil & \hfil & \hfil \end{ytableau} ,\emptyset,\emptyset \right)
§.§ Submodules and maps
In terms of the previous parametrization, the submodule structure is determined using Theorem <ref> as follows: there is a submodule of the form $M_{b,k}$ whenever $b$ is labeled by $(i,k_i)$ in the block tableau and $S(b)=j \neq i$ while $(j,T(b))>(i,k_i)$, with $k=r(T(b)-k_i)+j-i$. There is a submodule of the form $M_{b_1,b_2,k}$ for each pair of adjacent boxes $b_1,b_2$ with $(S(b_1),T(b_1))>(S(b_2),T(b_2))$ where $k=r(T(b_1)-T(b_2))+S(b_1)-S(b_2)$. In addition, if $b_1$ is the last box of $(n)$ and $b_2$ is the first box, and we have $(S(b_1),T(b_1)) > (S(b_2),T(b_2)-\ell)$ then $M_{b_1,b_2,k}$ is a submodule with $k=r(T(b_1)-T(b_2)+\ell)+S(b_1)-S(b_2)$.
Thus for the first pair $(S,T)$ above, and labeling the boxes as follows
\ytableausetup{mathmode, boxsize=2em}
\begin{ytableau}
b_3 & b_2 & b_1 & \hfil & \hfil & b_6 & b_4 & b_5
\end{ytableau}
we have submodules $M_{b_1,b_2,23}$, $M_{b_3,11}$, $M_{b_4,b_5,41}$, and $M_{b_6,33}$.
§.§ Tight multiplicities
Given pairs of integers $(i_1,k_1)$ and $(i_2,k_2)$ we write $(i_1,k_1)>(i_2,k_2)$ if $k_1 > k_2$ or $k_1=k_2$ and $i_1>i_2$.
Suppose $c_0=\ell/n$ for a positive integer $\ell$ coprime to $n$. Then the principal block has tight multiplicities if and only if the following condition holds: read from left to right, and breaking ties between labels in the same box by reading $(i,k)$ before $(j,k')$ if $(i,k)>(j,k')$, in the block tableau, if the labels are $((i_1,k_1),\dots,(i_s,k_s)$ then we have
$$(i_1,k_1) > (i_2,k_2)> \cdots > (i_s,k_s) > (i_1,k_1-\ell).$$
Suppose first that the condition
$$(i_1,k_1) > (i_2,k_2)> \cdots > (i_s,k_s) > (i_1,k_1-\ell)$$
on the labels read left to right holds. It then follows from the translation of Theorem <ref> in <ref> that:
(1) Whenever submodules $M_{b,k}$ and $M_{b_1,b_2,k'}$ both appear with $\beta(b)=\beta(b')$, then we have $k<k'$ and hence the lowest degree space of $M_{b,k}$ is not contained in $M_{b_1,b_2,k'}$.
(2) Whenever submodules $M_{b_1,b_2,k}$ and $M_{b_1',b_2',k'}$ appear, we have $\beta(b_1) \neq \beta(b_1')$.
Now these conditions imply that the hypotheses of Corollary <ref> are satisfied, so the radical of $\Delta_c(\lambda)$ is generated by singular vectors.
The case in which $S(b)=i$ for all $b$ is similar but easier, as only one submodule of the form $M_{b_1,b_2,k}$ can appear. This proves the sufficiency of our condition.
If a standard module $\Delta_c(\lambda)$ has tight multiplicities, then its composition length is equal to the number of irreducible $G(r,1,n)$-modules, counted with multiplicities, appearing in the space of singular vectors. On the other hand, this composition length is, by Theorem 7.1 of <cit.>, equal to the number of distinct intersections of the submodules of types (1) and (2) appearing in Theorem <ref>. Now we have a map from irreducible $G(r,1,n)$-submodules of the space of singular vectors to submodules of $\Delta_c(\lambda)$ given by $S \mapsto \CC[V] S$, and evidently $\CC[V] S$ has lowest degree subspace $S$ so this map is an injection. Again using Theorem <ref>, by irreducibility of $S$ the submodule $\CC[V] S$ of $\Delta_c(\lambda)$ must be equal to an intersection of submodules of the forms (1) and (2). It now follows by comparing cardinalities that if $\Delta_c(\lambda)$ has tight multiplicities, then every such intersection is of the form $\CC[V] S$ for some irreducible $G(r,1,n)$-submodule $S$ of the space of singular vectors, and that distinct intersections have different lowest degree parts $S$.
We first suppose that some label $(i,k)$ appearing in the block tableau has $(i,k)<(i_1,k_1-\ell)$. If the box $b$ has the label $(i,k)$, then setting $\lambda$ equal to the multiparition with $\lambda^i=(n)$ and all other $\lambda^j$ empty, the standard module $\Delta_c(\lambda)$ has a submodule $M_{b,r(k_1-k)+(i_1-i)}$ with
$$r(k_1-k)+(i_1-i) > r \ell.$$
If $b_1$ is the last box of $(n)$ and $b_1$ is the first, then $\Delta_c(\lambda)$ also has the submodule $M_{b_1,b_2,r \ell}$. But the lowest degree piece of $M_{b,r(k_1-k)+(i_1-i)}$ is contained in $M_{b_1,b_2,r \ell}$ without having a containment of
$M_{b,r(k_1-k)+(i_1-i)}$ in $M_{b_1,b_2,r \ell}$, so the distinct intersections $M_{b,r(k_1-k)+(i_1-i)}$ and $M_{b,r(k_1-k)+(i_1-i)} \cap M_{b_1,b_2,r \ell}$ have the same lowest degree space. By the argument of the previous paragraph $\Delta_c(\lambda)$ does not have tight multiplicities.
Now suppose that there are labels $(i,k)$ and $(j,m)$ with $(i,k) < (j,m)$ and $(i,k)$ appearing strictly to the left of $(j,m)$ in the block tableau. Let $\lambda$ be the multipartition with $\lambda^j=(n)$ and all other components empty. Let $b$ be the box of $\lambda^j$ determined by $c(b)=c(b')$ mod $\ZZ$, where $b'$ is the box of $(n)$ labeled by $(i,k)$, let $b_1$ be the rightmost box of $\lambda^j$ and let $b_2$ be the leftmost box. We then have submodules
$$M=M_{b_1,b_2,r \ell} \quad \text{and} \quad N=M_{b,r(m-k+\ell)+j-i}$$
with the lowest degree piece of $N$ contained in $M$ but $N$ not contained in $M$. We conclude as in the previous paragraph.
§.§ The quiver of primitive homs and tight multiplicities
We have the quiver $Q_{\mathrm{prim}}(B_0)$ for the principal block $B_0$ of $\OO_c(G(r,1,n))$ at $c_0=\ell/n$: it has vertices $\lambda\in\Lambda$ and an arrow $\mu\rightarrow\lambda$ if $\dim\Hom_{\mathrm{prim}}(\Del(\mu),\Del(\lambda))=1$. (Recall that all Hom spaces are $0$ or $1$-dimensional in $B_0$). We will state a criterion in terms of quivers for detecting whether $B_0$ has tight multiplicities.
We will need a graded version of BGG reciprocity to hold for $B_0$.
Let $\lambda,\mu\in\OO_c(G(r,1,n))$. Then:
There is an equivalence of $\OO_c(G(r,1,n))$ with a block of type $A$ affine parabolic category $\OO$ <cit.>. A graded version of the latter category is Koszul <cit.>. The argument of Beilinson-Bernstein <cit.> implies that the graded lift of a standard object coincides with the Jantzen filtration on that standard. Proposition 1.8 of <cit.> implies that the Jantzen filtration on a standard in $\OO_c(G(r,1,n))$ goes to the Jantzen filtration on a standard in the type $A$ affine parabolic category $\OO$ under the equivalence of <cit.>. This implies the statement.
Suppose $B$ is a block of a highest weight category with finitely many simple objects, which is BGG and satisfies a graded BGG reciprocity rule as in Lemma <ref>. Then $B$ has tight multiplicities if and only if the $\Ext^1$ quiver $Q_{\Ext^1}(B)$ is the double of the quiver $Q_{\mathrm{prim}}(B)$.
Since $B$ is BGG, $Q_{\Ext^1}(B)$ is guaranteed to be a double quiver, and it suffices to consider the number of arrows $\mu\rightarrow\lambda$ in $Q_{\Ext^1}(B)$ for $\mu<\lambda$. First, let us establish that the information we need about $\Ext^1$ is contained in the structure of $\Del(\lambda)$. We know that $\dim\Ext^1(L(\lambda),L(\mu))=[\rad_1 P(\lambda):L(\mu)]$, and the latter is equal to the coefficient of $v$ in the graded decomposition number $[P(\lambda):L(\mu)](v)$. Now we use the graded BGG reciprocity rule as in Lemma <ref>. If $\sigma\neq\lambda,\mu$ then $[\Del(\sigma):L(\lambda)](v)$ and $[\Del(\sigma):L(\mu)](v)$ both belong to $v\NN[v]$; also, $[\Del(\mu):L(\lambda)](v)=0$ because $\mu<\lambda$. Thus a single term in the sum, when $\sigma=\lambda$, produces the coefficient of $v$ in $[P(\lambda):L(\mu)](v)$; and $\left([\Del(\lambda):L(\lambda)](v)\right)\left([\Del(\lambda):L(\mu)](v)\right)=[\Del(\lambda):L(\mu)](v)$. Therefore the coefficient of $v$ in $[\Del(\lambda):L(\mu)](v)$ is equal to $\dim\Ext^1(L(\lambda),L(\mu))$.
Now, for the first direction of the proof, assume that $B$ has tight multiplicities. Let $\mu<\lambda$. Then
and the latter is $\dim\Ext^1(L(\lambda),L(\mu))$ plus the sum of the coefficients of $v^k$, $k>1$. On the other hand, we have
If $\phi:\Del(\mu)\rightarrow\Del(\lambda)$ factors through a highest weight module $M(\nu)$, $\mu<\nu<\lambda$, then $\im\phi$ is a submodule of a submodule in $\Del(\lambda)$, so belongs to the radical of the radical of $\Del(\lambda)$; thus it doesn't give rise to an extension between $L(\lambda)$ and $L(\mu)$ and so it doesn't contribute to the coefficient of $v$ in $[\Del(\lambda):L(\mu)](v)$. Any primitive hom $\phi:\Del(\mu)\rightarrow\Del(\lambda)$ gives rise to an extension $0\rightarrow L(\mu)\rightarrow E\rightarrow L(\lambda)\rightarrow 0$ since the head of $\im(\phi)$ must belong to $\rad_1\Del(\lambda)\subset\rad_1 P(\lambda)$ if $\phi$ does not factor through any other standard module. So $\dim\Hom_{\mathrm{prim}}(\Del(\mu),\Del(\lambda))\leq\dim\Ext^1(L(\lambda),L(\mu))$ in general, with equality in the situation of tight multiplicities.
For the converse direction of the proof, assume $Q_{\Ext^1}(B)$ is the double of $Q_{\mathrm{prim}}(B)$. Take $\Del(\lambda)\in B$. We will show that $\Rad\Del(\lambda)$ is generated by singular vectors. Let $\mu_1,...,\mu_k$ be a minimal set of $W$-irreps generating $\Rad\Del(\lambda)$. Suppose that for some $\mu_i$ there is no map $\Del(\mu_i)\rightarrow\Del(\lambda)$. Since $\mu_i$ is part of a minimal generating set for $\Rad\Del(\lambda)$, $H_c\cdot\mu_i$ does not belong to $\Rad(\Rad\Del(\lambda))$. Note that $\rad_1\Del(\lambda)=\Rad\Del(\lambda)/\Rad(\Rad\Del(\lambda))$ is semisimple, so the head of $H_c\cdot\mu_i$, which is $L(\mu_i)$, belongs to $\rad_1 \Del(\lambda)\subset\rad_1 P(\lambda)$. But this implies that $\dim\Ext^1(L(\lambda),L(\mu_i))>\dim\Hom_{\mathrm{prim}}(\Del(\mu_i),\Del(\lambda))$, contradicting the assumption.
Now Lemma <ref> and Theorem <ref> imply:
The principal block $B_0$ of $\OO_c(G(r,1,n))$ at $c_0=\ell/n$, $\mathrm{gcd}(\ell,n)=1$, has tight multiplicities if and only if the quivers $Q_{\Ext^1}(B_0)$ and $Q_{\mathrm{prim}}(B_0)$ are the same.
§.§ Graded decomposition numbers
Let $B_0$ be the principal block of $\OO_c(G(r,1,n))$ in the case that $B_0$ is $\ttt$-diagonalizable and has tight multiplicities. Define a graph $\Gamma$ for $B_0$ as follows. The vertices of $\Gamma$ are all $\lambda\in B_0$. There is an arrow $\mu\rightarrow\lambda$ if and only if $\mu$ is the lowest degree subspace of a fundamental submodule of $\Del(\lambda)$. The graph $\Gamma$ encodes much of the structure of the block.
Let $P_\lambda$ be the subgraph of $\Gamma$ coinciding with the lattice of intersections of the fundamental submodules of $\Del(\lambda)$ defined in Theorem <ref>. Let $d(\mu,\lambda)$ be the length of the longest chain of arrows from $\mu$ to $\lambda$ in $\Gamma$, if such a chain exists.
The graded decomposition numbers are given by
$$[\Delta(\lambda):L(\mu)](v)=v^{d(\mu,\lambda)}\hbox{ if $\mu$ is a vertex on }P_\lambda$$
Otherwise, $[\Delta(\lambda):L(\mu)](v)=0$.
We know that $[\Del(\lambda):L(\mu)]=\dim\Hom(\Del(\mu),\Del(\lambda))$ is $0$ if $\mu$ is not the highest weight of the intersection of fundamental submodules of $\Del(\lambda)$, and $1$ if it is, because any nonzero homomorphism $\Del(\mu)\rightarrow\Del(\lambda)$ is the inclusion of an intersection of fundamental submodules by the remarks in the proof of Theorem <ref>, and $B_0$ has tight multiplicities. Also, we know by Lemma <ref> that the power of $v$ in the graded decomposition number is equal to the number of fundamental submodules of $\Del(\lambda)$ containing the homomorphic image of $\Del(\mu)$. If the longest length of a chain of arrows $\mu\rightarrow\lambda$ is $1$ then $\mu$ labels a fundamental submodule of $\Del(\lambda)$ which is not contained in any other fundamental submodule. Suppose by induction that all the vertices $\nu$ corresponding to intersections of $k$ fundamental submodules have distance $k$ from $\lambda$, for any $k\leq d$, where $d$ is less than the number of fundamental submodules of $\Del(\lambda)$. Suppose $\mu$ is the highest weight of the intersection of exactly $d+1$ fundamental submodules. Then the image of $\Del(\mu)$ is a proper submodule of the intersection $I$ of some $d$ fundamental submodules. Say the highest weight of $I$ is $\nu$. Then the longest path $\nu\rightarrow\lambda$ has length $d$, so there is a path of length at least $d+1$ from $\mu$ to $\lambda$. Suppose there is a path of length greater than $d+1$, whose first arrow is $\mu\rightarrow\xi$. Then $\xi$ must be the intersection of $\ell\geq d+1$ fundamental submodules since any intersection of a smaller collection of fundamental submodules satisfies the induction hypothesis. But then the submodule with highest weight $\mu$ is a proper submodule of the submodule with highest weight $\xi$, so it must be the intersection of more than $d+1$ fundamental submodules. Therefore $d+1$ is the longest length of a path $\mu\rightarrow\lambda$, which completes the induction.
The $|\Lambda|\times|\Lambda|$ matrix with entries $[\Del(\lambda):L(\mu)](v)$ is the graded decomposition matrix. If rows and columns are arranged so that the partial order $>$ on $\Lambda$ is nonincreasing across rows and down columns then the matrix is upper-triangular with $1$'s on the diagonal.
§.§ The inverse of the graded decomposition matrix
We now describe a formula for the inverse of the graded decomposition matrix whose entries were given by Theorem <ref>. This formula should really hold in any tight block with simple multiplicities equipped with a minimal partial order. Of course, such a formula is given by abstract Kazhdan-Lusztig theory in terms of the Poincaré polynomial $p_{\mu,\lambda}$ evaluated at $-1$, c.f. Proposition (3.2) in <cit.>, if we can calculate $\dim\Hom^n(L(\lambda),\nabla(\mu))$. In a tight block, however, everything is controlled by homomorphisms between standards, and we may define a naive polynomial which does the same job as the Poincaré polynomial when evaluated at $-1$.
Let $\Gamma(\Hom)$ be the graph with vertices $\lambda\in\Lambda$ and an arrow $\mu\rightarrow\lambda$ if there is a nonzero homomorphism $\Del(\mu)\rightarrow\Del(\lambda)$. For $\mu\leq\lambda$, let $b_n(\mu,\lambda)$ be the number of chains of $n$ arrows from $\mu$ to $\lambda$ in $\Gamma(\Hom)$.
Now we define a polynomial in $\NN[v]$ which keeps track of all chains of arrows from $\mu$ to $\lambda$ in $\Gamma(\Hom)$:
$$B_{\mu,\lambda}=\sum_{n=0}^{d(\mu,\lambda)} b_n(\mu,\lambda) v^n$$
By Theorem <ref>, the number of chains of homs of length $n$ from $\mu$ to $\lambda$ is the number of paths in the $v$-dec matrix starting at $(\mu,\mu)$, ending at $(\lambda,\lambda)$, traveling up and left and bouncing off exactly $n$ nonzero $v$-dec numbers and $n-1$ times off the diagonal. The product of the entries on the bounce points of such a path is $v^{d(\mu,\lambda)}$.
$$[L(\lambda)](v)=\sum_{\mu\leq \lambda} B_{\mu,\lambda}(-1)v^{d(\mu,\lambda)}[\Del(\mu)]$$
Take $\lambda\in\Lambda$. To find $[L(\lambda):\Del(\mu)](v)$, i.e. the coefficient of $[\Del(\mu)]$ in the formula, we will induct on $d(\mu,\lambda)$. For $\mu=\lambda$, $B_{\lambda,\lambda}=1$ and $d(\lambda,\lambda)=0$, and we have $[L(\lambda):\Del(\lambda)](v)=1$ which is true. Assume by induction that if $d(\sigma,\lambda)<n$ that $[L(\lambda):\Del(\sigma)](v)=B_{\sigma,\lambda}(-1)v^{d(\sigma,\lambda)}$. Take $\mu$ such that $d(\mu,\lambda)=n$.
Consider the first arrow in a chain from $\mu$ to $\lambda$ in $\Gamma(\Hom)$, the arrow originating at $\mu$. If $\sigma$ is the target of an arrow originating at $\mu$, then the contribution to $B_{\mu,\lambda}$ from all those chains $\mu$ to $\lambda$ in $\Gamma(\Hom)$ which pass through $\sigma$ is: $vB_{\sigma,\lambda}$. Therefore:
$$B_{\mu,\lambda}=v\left(\sum_{\substack{\mu<\sigma\leq\lambda \\ \dim\Hom(\Del(\mu),\Del(\sigma))\neq0}}B_{\sigma,\lambda}\right)$$
But if $\dim\Hom(\Del(\mu),\Del(\sigma))\neq0$, then $[\Del(\sigma):L(\mu)](v)=v^{d(\mu,\sigma)}$ by Theorem <ref>, and if $\sigma>\mu$ then by definition $d(\mu,\sigma)\geq1$. Then $d(\sigma,\lambda)<n$ since $d(\mu,\lambda)=d(\mu,\sigma)+d(\sigma,\lambda)$. Applying induction, $[L(\lambda):\Del(\sigma)](v)=B_{\sigma,\lambda}(-1)v^{d(\sigma,\lambda)}$. Now observe that the dot product of the row vector $[L(\lambda):\Del(-)](v)$ with the column vector $[\Del(-):L(\mu)](v)$ is $0$, since the $|\Lambda|\times|\Lambda|$ matrix with entries $[L(\sigma):\Del(\tau)](v)$ is the inverse of the matrix with entries $[\Del(\sigma):L(\tau)](v)$. Then we have:
$$[L(\lambda):\Del(\mu)](v)=-\left(\sum_{\substack{\mu<\sigma\leq\lambda\\ \dim\Hom(\Del(\mu),\Del(\sigma)\neq0}} B_{\sigma,\lambda}(-1)v^{d(\sigma,\lambda)}v^{d(\mu,\sigma)}\right)=B_{\mu,\sigma}(-1)v^{d(\mu,\lambda)}$$
§.§ BGG resolutions
We describe an algorithm which computes the standards appearing in a minimal BGG resolution of any $L(\lambda)$ in $B_0$. Let $\Lambda_{\leq\lambda}=\{\mu\leq\lambda\}$ be the poset ideal of $\Lambda$ generated by $\lambda$. Define the subgraph $\Gamma_\lambda$ of $\Gamma$ to be that containing all $\mu\leq\lambda$ and their arrows.
Next, any time there's an arrow $\nu\rightarrow\mu$ with $\nu\in\Lambda_\lambda^i$ and $\mu\in\Lambda_\lambda^j$ and $i-j>1$, delete that arrow and remove $\Gamma_\nu$ from $\Gamma_\lambda$. The graph $\bar{\Gamma}_\lambda$ which remains is, essentially, the complex of standard modules resolving $L(\lambda)$: conjecturally, the standards in the minimal BGG resolution $\Del_\bullet\rightarrow L(\lambda)\rightarrow 0$ are
This algorithm constructs the inverse of the $v$-decomposition matrix of the principal block $B_0$ described in Theorem <ref>: $[L(\lambda):\Del(\mu)](v)=(-1)^{d(\mu,\lambda)}v^{d(\mu,\lambda)}$ if $\mu\in\bar{\Gamma}_\lambda$, and $0$ otherwise.
Row $\lambda$ of the inverse of the graded decomposition matrix is given by the formula <ref>. Thus Conjecture <ref> is equivalent to the claim that if $\mu\leq\lambda$, then,
* $B_{\mu,\lambda}(-1)=(-1)^{d(\mu,\lambda)}$ if every path from $\mu$ to $\lambda$ in $\Gamma$ has the same length $d(\mu,\lambda)$
* $B_{\mu,\lambda}(-1)=0$ if there are paths from $\mu$ to $\lambda$ of different lengths in $\Gamma$.
Let us prove part (<ref>) by induction:
(<ref>) Suppose every path $\mu$ to $\lambda$ in $\Gamma$ has length $d(\mu,\lambda)$. Then all vertices $\nu$ lying on a path from $\mu$ to $\lambda$ in $\Gamma$ also have this property. Consider all nonzero homs out of $\Del(\mu)$ which lie on a path of homs to $\lambda$: the subgraph $C$ of these lying in $\Gamma_\lambda$ is the edge graph of an $N$-cube, where $N$ is the number of outgoing arrows from $\mu$ in $\Gamma_\lambda$. There is a hom from $\Del(\mu)$ to $\Del(\sigma)$ for each vertex $\sigma$ of $C$ by <ref>: $C$ is the lattice of intersections of some collection of fundamental submodules of a standard module which have pairwise intersections not equal to any fundamental submodule, plus the apex of $C$ which is the standard module itself. Then we have:
$$B_{\mu,\lambda}=v\sum_{j=1}^N\sum_{\substack{\sigma\in C \\ d(\mu,\sigma)=j}}p_{\sigma,\lambda}$$
Evaluating at $v=-1$ and using induction gives:
$$B_{\mu,\lambda}(-1)=(-1)\sum_{j=1}^N{N\choose j}(-1)^{d(\mu,\lambda)-j}=(-1)\left(\sum_{j=0}^N{N\choose j}(-1)^{d(\mu,\lambda)-j}-(-1)^{d(\mu,\lambda)}\right)=(-1)^{d(\mu,\lambda)}$$
§ EXAMPLES
We illustrate the material of the preceding sections with pictures of diagonalizable principal blocks with tight multiplicities and of BGG resolutions of finite-dimensional modules. The graphs $\Gamma$ below are built up inductively, starting by putting vertices for those $\lambda$ in the principal block $B_0$ such that there cannot be any $\nu\neq\lambda$ in $B_0$ with a nonzero map $\Del(\lambda)\rightarrow\Del(\nu)$. Next, for a given vertex $\lambda$ in $\Gamma$, there is a vertex for each submodule of the form $M=M_{b,?}$ or $M_{b_1,b_2,?}$ of $\Del(\lambda)$, together with an arrow $\mu\rightarrow\lambda$. Each such vertex for a submodule $M\subset \Del(\lambda)$ is labeled by the irreducible representation $\mu$ of $\CC G(r,1,rn)$ that occupies the lowest degree subspace of $M$. $M\subset\Del(\lambda)$ then determines a map $\Del(\mu)\rightarrow\Del(\lambda)$, since the lowest degree subspace always consists of singular vectors.
The arrows in shades of red and orange denote the primitive homs between standards, while the green arrows are nonzero compositions of homs. To obtain the $\Ext^1$ quiver for simples, delete the green arrows and double the orange arrows.
§.§ BGG resolutions and character formulas
According to <ref>, to find the standards in a minimal BGG resolution of some $L(\lambda)$, one should take the subgraph $\Gamma_\lambda$ consisting of partitions with paths to $\lambda$, and delete from it anything that lies on a path through a green arrow to $\lambda$. The result is the graph called $\bar{\Gamma}_\lambda$ which conjecturally “is" the BGG resolution. We have included several examples of $\bar{\Gamma}_\chi$ for finite-dimensional modules $L(\chi)$, $\chi$ a twist of the trivial rep. $\Del_i$, the $i$'th term in the resolution $\Del_\bullet\rightarrow L(\Triv)\rightarrow 0$, is the direct sum of the $\Del(\lambda)$ where $\lambda$ occurs on the $i$'th level from the top of $\bar{\Gamma}_\chi$ (the character $\chi$ is at level $0$). We find that for $G(3,1,3)$, $G(3,1,6)$, and $G(4,1,4)$ at equal parameters $1/rn$, this produces the right dimensions of finite-dimensional reps according to Corollary <ref>. Furthermore, we have checked that Conjecture <ref> actually holds for the whole principal block in the cases of $G(3,1,3)$ at $c=1/3$, and $G(4,1,4)$ at $c=1/4$, by computing the $v$-decomposition matrix using <ref> and finding its inverse and checking this matches the answer given by our graphical algorithm.
We may then calculate the graded dimension formula for any $L(\lambda)$ in a tight, diagonalizable principal block of $\OO_c$ using the BGG resolution of $L(\lambda)$ as follows: for $\mu$ in the graph $\bar{\Gamma}_\lambda$ , let $\tilde{c}(b)=rc_0\ct(b)+d_{\beta(b)}$ for each box $b$ in $\mu$ and let
\begin{equation}\label{charge}
\tilde{c}_\lambda(\mu)=\left(\sum_{b\in\lambda}\tilde{c}(b)\right)-\sum_{b\in\mu}\tilde{c}(b)
\end{equation}
Note that $\tilde{c}_\lambda$ is the same function as that given by the Euler element $h_c$, except shifted so that $\Del(\lambda)$ is $\ZZ_{\geq0}$-graded with $\lambda$ in degree $0$. When $\lambda=\Triv$ we will write $\tilde{c}$ for $\tilde{c}_\Triv$. Let $\hd(\mu)=i$ be the homological degree of $\mu$, i.e. the unique $i$ such that $\Del(\mu)$ is a direct summand of $\Del_i$. Thus $\hd(\mu)=i$ if $\mu$ appears on the $i$'th level from the top of the graph $\bar{\Gamma}_\lambda$, starting from level $0$ for $\lambda$, i.e $\hd(\mu)=d(\mu,\lambda)$. The graded dimension of $L(\lambda)$ is given by the formula:
\begin{equation}\label{gr char}
\sum_{t\geq 0}\dim_\CC (L(\lambda)^i) t^i=\frac{\sum_{\mu\in\bar{\Gamma}_\lambda}(-1)^{\hd(\mu)}\dim_\CC(\mu) t^{\tilde{c}_\lambda(\mu)}}{(1-t)^n}
\end{equation}
§.§ The principal block of $\OO_{1/rn}(G(r,1,rn))$
In the examples below, we write $c=1/rn$ as shorthand for $c_0=c_1=...=c_{r-1}=1/rn$. We have $d_0=(r-1)/rn$ and $d_i=-1/rn$ if $1\leq i \leq r-1$. Let $B_0$ be the principal block of $\OO_c(G(r,1,rn))$. Every standard module in $B_0$ is $\ttt$-diagonalizable by Lemma <ref>. Following Section <ref> we have a labeling of $\Triv=(rn)$ with a distinct box $b_i$ for each $0\leq i\leq r-1$. The box $b_0$ is the box with content $0$, so it is the first box, and $(0,k_0)=(0,0)$. For $0<i\leq r-1$, $b_i$ is the $(r-i)n$'th box, and $(i,k_i)=(i,-1)$. Read from left to right, we have
so by Theorem <ref>, the principal block $B_0$ has tight multiplicities.
The partitions in $B_0$ may be found as follows (see <ref>). Label the Young diagram of $(rn)$ with the numbers $1$ through $rn$ from left to right, then connect up the two ends so that the numbers $1$ through $rn$ are arranged increasing clockwise around a circle. Make $k$ cuts to make $k$ connected subsets $Y_1,...,Y_k$, $k \leq r$, such that each $Y_a$ for $a>1$ contains a multiple of $n$ (if $Y_a$, $a>1$, does not contain a box labeled by a multiple of $n$ then the collection $Y_1,...,Y_k$ cannot give rise to a collection of partitions in the block). For each $1\leq b \leq r-1$, $bn\in Y_a$ for some $a$. Then $Y_a$ may be turned into a (labeled) hook $\lambda^{r-b}$ and put in component $r-b$, such that $\lambda^{r-b}$ has $bn$ in the upper left corner and all numbers increasing (mod $rn$) from bottom to top, left to right. Additionally, $Y_1$ may be placed in the $0$'th component if it is bent into the hook $\lambda^0$ having $1$ in the upper left corner, with numbers increasing left to right across the arm, and bottom to top up the leg, except that $rn$ may occur in the box below $1$. $\lambda$ is the $r$-partition of $rn$ produced by bending some such collection of $Y_a$ at multiples of $n$ or at $1$ and placing them in the appropriate components.
In the case $r=2$, i.e. for $B_{2n}$, this amounts to the following characterization:
The bipartitions $(\lambda,\mu)$ in the principal block of $\OO_{1/2n}(B_{2n})$ consist of:
* $(\lambda,\emptyset)$, $\lambda\vdash 2n$ a hook
* $(\emptyset,\mu)$, $\mu\vdash 2n$ a hook
* $((a,1^k),(n+1-k,1^{n-1-a}))$, $1\leq a\leq n-1$, $0\leq k\leq n$.
The principal block for $B_{2n}$ at equal parameters $c=1/2n$ has weight $2$ by <cit.>, Proposition 1.10. Weight $2$ blocks for Iwahori-Hecke algebras of type $B_n$ were studied by Fayers in <cit.>. The image of $B_0$ under $\KZ$ functor coincides with the “prototype" of a Type I block in Fayers' classification (<cit.>, Propositions 2.1, 2.2).
In the case of the principal block for $\OO_{1/2n}(B_{2n})$, we may explicitly describe all maps between standards in the block in terms of bipartitions. The calculation and case by case description is lengthy; we summarize and refer the reader to the examples for $B_{2n}$, $n=1,2,3,4$ below (<ref>,<ref>). The graph $\Gamma$ admits a planar embedding such that given a vertex $\lambda\in\Gamma$, the subgraph $P_\lambda$ spanned by those $\mu$ such that there is a nonzero map $\Del(\mu)\rightarrow\Del(\lambda)$ is either a point, a directed line segment connecting two points or, in the case of $\lambda=((1^{n+1},1^{n-1})$, two directed line segments to $\lambda$, or, encloses a two-dimensional polygon. The polygons which occur in $\Gamma$ are either triangles or diamonds. In fact, there are only two triangles which arise in $\Gamma$, and they are:
[scale=1.5,font=,every text node part/.style=align=center]
(top1) at (-3,0) $(n,1^n),\;\emp$;
(mid1) at (-1.5,-1) $(n-1,1^n),\;(1)$;
(bot1) at (-3,-2) $(n-1,1^{n+1}),\;\emp$;
(top2) at (3,0) $\emp,\;(n+2,1^{n-2})$;
(mid2) at (1.5,-1) $(1),\;(n+1,1^{n-2})$;
(bot2) at (3,-2) $\emp,\;(n+1,1^{n-1})$;
(mid1) edge node (top1)
(bot1) edge node (mid1)
(bot1) edge node (top1)
(mid2) edge node (top2)
(bot2) edge node (mid2)
(bot2) edge node (top2);
To read the graded decomposition numbers off of $\Gamma$, look at the vertex $\lambda$ in the graph, and look at the polygon whose edges flow into vertex $\lambda$: then $[\Del(\lambda):L(\mu)](v)=0$ unless $\mu$ is a vertex in that polygon, in which case $[\Del(\lambda):L(\mu)](v)=1$ if $\lambda=\mu$, $v$ if there is a single path $\mu\rightarrow\lambda$, and $v^2$ if the path $\mu\rightarrow\lambda$ traverses one side of a single triangle or diamond.
Theorem <ref> now follows from the discussion above and <ref>.
Let $\Del_i$ denote the coefficient of $v^i$, up to signs, in the graded character of $L_{1/2n}(\Triv)$ in $\OO_{1/2n}(B_{2n})$. The $\mu$ appearing as highest weights of summands of $\Del_i$ are those obtained from $((2n),\emp)$ by $i$ single-chunk-of-boxes moves of the forms described in Corollary <ref>, and which can't be so obtained by a smaller number of such moves. This proves Theorem <ref>. Thus every $\mu\in\Lambda$ appears in the character except for those $\mu=(\mu^0,\mu^1)$ such that either $\mu^0=\emp$ and $\mu^1$ has more than $n+1$ boxes in the first row, or $\mu^1=\emp$ and $\mu^0$ has more than $n+1$ boxes in the first column.
If $\mu=(\emp,\mu^1)$ and $\mu^1$ has more than $n+1$ boxes in the first row, then $\mu^1$ is not comparable to $\Triv$ in the partial order, so does not appear in the character formula. If $\mu=(\mu^0,\emp)$ and $\mu^0$ has more than $n+1$ boxes in the first column, then we may calculate $B_{\mu,\Triv}$, and thus $[L(\lambda)](v)$ by <ref>, as follows. Let $\mu^0=(b,1^{2n-b})$ with $a:=2n-b\geq n+1$. There is a single chain of homs $\mu\rightarrow\Triv$ which passes through the green arrow and this path has length $a$. Any other chain of homs $\mu\rightarrow\Triv$ passes through one of the vertices $((n-1,1^n),(1))$, $((n-2,1^n),(1^2))$,...,$((b,1^n),(1^{n-b}))$. Given one of these vertices $\nu$, any path from $\nu$ to $\Triv$ in $\Gamma$ has length $d(\nu,\Triv)$. For such a $\nu$, there is a single path from $\mu$ to $\nu$ along arrows from $\Gamma$ which does not go through the other vertices in this list lying below $\nu$ in the partial order. Then there is a single path of homs from $\mu$ to $\nu$, unless $\nu$ is $((b,1^n),(1^{n-b}))$, that goes across the diagonal of a diamond with top vertex $\nu$. We then have
$$B_{\mu,\Triv}=v^a+vB_{((b,1^n),(1^{n-b})),\Triv}+\sum_{k=1}^{n-b-1} (v^k+v^{k+1})B_{((b+k,1^n),(1^{n-b-k})),\Triv}$$
and so
Then for these $\mu$, $[L(\Triv):\Del(\mu)](v)=0$ by <ref>.
Next, by the part of Conjecture <ref> which we proved (<ref>), if every path $\mu$ to $\Triv$ in $\Gamma$ has length $d(\mu,\Triv)$ then $[L(\Triv):\Del(\mu)]=(-1)^{d(\mu,\Triv)}v^{d(\mu,\Triv)}$. This implies the formula.
The same argument works for any $\lambda$ in place of $\Triv$ with $\lambda\geq((n,1^n),\emp)$. A similar argument works for $\lambda\geq(\emp,(n+2,1^{n-2}))$. Any $\lambda\in B_0$ is comparable to at most one of $((n,1^n),\emp)$, $(\emp,(n+2,1^{n-2}))$. Thus Conjecture <ref> holds for $L(\lambda)$ in either case. If $\lambda$ is not comparable to either of $((n,1^n),\emp)$, $(\emp,(n+2,1^{n-2}))$, then any $\mu\leq\lambda$ satisfies the conditions of (<ref>) and so the character formula of Conjecture <ref> holds by the proof of (<ref>). This verifies Conjecture <ref> for the principal block of $B_{2n}$ at equal parameters $c=1/2n$, the case concerning the Oblomkov-Yun dimension formula proved in <ref>.
§.§ Graded dimensions
The pictures below contain all information necessary to calculate graded dimensions of simple modules. In $\Gamma$, vertical spacing between partitions is proportionate to difference between their charged contents, and the minimum of such in $\Gamma$ is $1$. In BGG resolutions, latitudes are homological degrees. We then obtain graded dimensions and dimensions over $\CC$ of the finite-dimensional representations, which agree, in the case of $L(\Triv)$, with Corollary <ref>:
In <ref>, let $\chi_1=(\emp,(3),\emp)$ and $\chi_2=(\emp,\emp,(3))$. We have $\tilde{c}_{\chi_i}(\lambda)=2-\sum_{b\in\lambda}\tilde{c}(b)$, see <ref>, while $\tilde{c}(\lambda)=5-\sum_{b\in\lambda}\tilde{c}(b)$. Applying formula <ref>:
\begin{align*}
\sum_{t\geq 0}\dim_\CC(L(\Triv)^i) t^i &= 1+3t+6t^2+7t^3+3t^4\\
\sum_{t\geq 0}\dim_\CC (L(\chi_1)^i) t^i%&=\frac{1-3t+3t^2-t^3}{(1-t)^3}\\
\sum_{t\geq 0}\dim_\CC (L(\chi_2)^i) t^i%&=\frac{1-3t^2+3t^4-t^6}{(1-t)^3}\\
\end{align*}
Evaluating at $t=1$, we get $\dim L(\Triv)=20$, $\dim L(\emp,(3),\emp)=1$, and $\dim L(\emp,\emp,(3))=8$.
In <ref>,<ref> let $\Triv=((6),\emp,\emp)$, $\chi_1=(\emp,(6),\emp)$, and $\chi_2=(\emp,\emp,(6))$. Then $\tilde{c}(\lambda)=9.5-\sum_{b\in\lambda}\tilde{c}(b)$, and $\tilde{c}_{\chi_i}(\lambda)=6.5-\sum_{b\in\lambda}\tilde{c}(b)$, see <ref>. Applying formula <ref>:
\begin{align*}
\sum_{t\geq 0}\dim_\CC (L(\Triv)^i) t^i%&=\frac{1-5t^3-6t^5-5t^6+60t^7+69t^8-330t^9+270t^{10}-15t^{11}-30t^{12}-24t^{13}+15t^{14}}{(1-t)^6}\\
\sum_{t\geq 0}\dim_\CC (L(\chi_1)^i) t^i%&=\frac{1-25t^3+30t^4+93t^5-250t^6+225t^7-75t^8-5t^9+6t^{10}}{(1-t)^6}\\
\sum_{t\geq 0}\dim_\CC (L(\chi_2)^i) t^i%&=\frac{1-6t^2+84t^5-210t^6+216t^7-105t^8+20t^9}{(1-t)^6}\\
\end{align*}
Evaluating at $t=1$, we get $\dim L(\Triv)=538$, $\dim L(\emp,(6),\emp)=65$, and $\dim L(\emp,\emp,(6))=42$.
In <ref>, we have $\tilde{c}(\lambda)=9-\sum_{b\in\lambda}\tilde{c}(b)$, while $\tilde{c}_\chi(\lambda)=5-sum_{b\in\lambda}\tilde{c}(b)$ for $\chi\neq\Triv$ a character with $(4)$ in a single component, see <ref>. Applying formula <ref>:
\begin{align*}
\sum_{t\geq 0}\dim_\CC (L(\Triv)^i) t^i
\sum_{t\geq 0}\dim_\CC (L(\emp,(4),\emp,\emp)^i) t^i&=1+4t+4t^2\\
\sum_{t\geq 0}\dim_\CC (L(\emp,\emp,(4),\emp)^i) t^i&=1+4t+6t^2\\
\sum_{t\geq 0}\dim_\CC (L(\emp,\emp,\emp,(4))^i) t^i&=1+4t+10t^2+16t^3+13t^4+4t^5\\
\end{align*}
Evaluating at $t=1$, we get $\dim L(\Triv)=152$, $\dim L(\emp,(4),\emp,\emp)=9$, $\dim L(\emp,\emp,(4),\emp))=11$, and $\dim L(\emp,\emp,\emp,(4))=48$.
§.§ The principal blocks of $B_2$ at $c=1/2$, $B_4$ at $c=1/4$, and $B_6$ at $c=1/6$.
56[scale=1.4,every text node part/.style=align=center]
(E11) at (-3,4) $\yng(2),\;\emp$;
(E21) at (-2,3) $\emp,\;\yng(2)$;
(E22) at (-4,3) $\yng(1,1),\;\emp$;
(E31) at (-3,2) $\emp,\;\yng(1,1)$;
(f11) at (1,4) $\yng(4),\;\emp$;
(f21) at (1,2) $\yng(3,1),\;\emp$;
(f22) at (5,2) $\emp,\;\yng(4)$;
(f31) at (3,1) $\yng(1),\;\yng(3)$;
(f41) at (1,0) $\yng(2,1,1),\;\emp$;
(f42) at (3,0) $\yng(1,1),\;\yng(2)$;
(f43) at (5,0) $\emp,\;\yng(3,1)$;
(f51) at (3,-1) $\yng(1,1,1),\;\yng(1)$;
(f61) at (1,-2) $\yng(1,1,1,1),\;\emp$;
(f62) at (5,-2) $\emp,\;\yng(2,1,1)$;
(f71) at (5,-4) $\emp,\;\yng(1,1,1,1)$;
(g11) at (-3.5,-1) $\yng(6),\;\emp$;
(g21) at (-3.5,-3) $\yng(5,1),\;\emp$;
(g22) at (2.5,-3) $\emp,\;\yng(6)$;
(g31) at (-3.5,-5) $\yng(4,1,1),\;\emp$;
(g32) at (-1.5,-5) $\yng(2),\;\yng(4)$;
(g33) at (2.5,-5) $\emp,\;\yng(5,1)$;
(g41) at (-1.5,-6) $\yng(2,1),\;\yng(3)$;
(g42) at (0.5,-6) $\yng(1),\;\yng(4,1)$;
(g51) at (-3.5,-7) $\yng(3,1,1,1),\;\emp$;
(g52) at (-1.5,-7) $\yng(2,1,1),\;\yng(2)$;
(g53) at (0.5,-7) $\yng(1,1),\;\yng(3,1)$;
(g54) at (2.5,-7) $\emp,\;\yng(4,1,1)$;
(g61) at (-1.5,-8) $\yng(2,1,1,1),\;\yng(1)$;
(g62) at (0.5,-8) $\yng(1,1,1),\;\yng(2,1)$;
(g71) at (-3.5,-9) $\yng(2,1,1,1,1),\;\emp$;
(g72) at (0.5,-9) $\yng(1,1,1,1),\;\yng(1,1)$;
(g73) at (2.5,-9) $\emp,\;\yng(3,1,1,1)$;
(g81) at (-3.5,-11) $\yng(1,1,1,1,1,1),\;\emp$;
(g82) at (2.5,-11) $\emp,\;\yng(2,1,1,1,1)$;
(g91) at (2.5,-13) $\emp,\;\yng(1,1,1,1,1,1)$;
(E21) edge node [left] (E11)
(E22) edge node [right] (E11)
(E31) edge node [left] (E21)
(E31) edge node [right](E22);
(f21) edge node (f11)
(f31) edge node (f11)
(f31) edge node (f22)
(f41) edge node (f21)
(f42) edge node (f21)
(f42) edge node (f31)
(f43) edge node (f31)
(f51) edge node (f41)
(f51) edge node (f42)
(f61) edge node (f51)
(f62) edge node (f42)
(f62) edge node (f43)
(f71) edge node (f51)
(f71) edge node (f62);
(f43) edge node (f22)
(f61) edge node (f41);
(g21) edge node (g11)
(g31) edge node (g21)
(g32) edge node (g11)
(g32) edge node (g22)
(g33) edge node (g22)
(g41) edge node (g21)
(g41) edge node (g32)
(g42) edge node (g32)
(g42) edge node (g33)
(g51) edge node (g31)
(g52) edge node (g31)
(g52) edge node (g41)
(g53) edge node (g41)
(g53) edge node (g42)
(g54) edge node (g42)
(g61) edge node (g51)
(g61) edge node (g52)
(g62) edge node (g52)
(g62) edge node (g53)
(g71) edge node (g61)
(g72) edge node (g61)
(g72) edge node (g62)
(g73) edge node (g53)
(g73) edge node (g54)
(g81) edge node (g71)
(g81) edge node (g72)
(g82) edge node (g62)
(g82) edge node (g73)
(g91) edge node (g72)
(g91) edge node (g82);
(g54) edge node (g33)
(g71) edge node (g51);
§.§ The principal block of $\OO_{1/8}(B_8)$
56[scale=1.35,every text node part/.style=align=center]
(01) at (-5,0)$\yng(8),\;\emp$;
(11) at (-5,-2)$\yng(7,1),\;\emp$;
(12) at (5,-2)$\emp,\;\yng(8)$;
(21) at (-5,-4)$\yng(6,1,1),\;\emp$;
(22) at (5,-4)$\emp,\;\yng(7,1)$;
(31) at (-2.5,-5)$\yng(3),\;\yng(5)$;
(41) at (-5,-6)$\yng(5,1,1,1),\;\emp$;
(42) at (-2.5,-6)$\yng(3,1),\;\yng(4)$;
(43) at (0,-6)$\yng(2),\;\yng(5,1)$;
(44) at (5,-6)$\emp,\;\yng(6,1,1)$;
(51) at (-2.5,-7)$\yng(3,1,1),\;\yng(3)$;
(52) at (0,-7)$\yng(2,1),\;\yng(4,1)$;
(53) at (2.5,-7)$\yng(1),\;\yng(5,1,1)$;
(61) at (-5,-8)$\yng(4,1,1,1,1),\;\emp$;
(62) at (-2.5,-8)$\yng(3,1,1,1),\;\yng(2)$;
(63) at (0,-8)$\yng(2,1,1),\;\yng(3,1)$;
(64) at (2.5,-8)$\yng(1,1),\;\yng(4,1,1)$;
(65) at (5,-8)$\emp,\;\yng(5,1,1,1)$;
(71) at (-2.5,-9)$\yng(3,1,1,1,1),\;\yng(1)$;
(72) at (0,-9)$\yng(2,1,1,1),\;\yng(2,1)$;
(73) at (2.5,-9)$\yng(1,1,1),\;\yng(3,1,1)$;
(81) at (-5,-10)$\yng(3,1,1,1,1,1),\emp$;
(82) at (0,-10)$\yng(2,1,1,1,1),\;\yng(1,1)$;
(83) at (2.5,-10)$\yng(1,1,1,1),\;\yng(2,1,1)$;
(84) at (5,-10)$\emp,\;\yng(4,1,1,1,1)$;
(91) at (2.5,-11)$\yng(1,1,1,1,1),\;\yng(1,1,1)$;
(101) at (-5,-12)$\yng(2,1,1,1,1,1,1),\;\emp$;
(102) at (5,-12)$\emp,\;\yng(3,1,1,1,1,1)$;
(111) at (-5,-14)$\yng(1,1,1,1,1,1,1,1),\;\emp$;
(112) at (5,-14)$\emp,\;\yng(2,1,1,1,1,1,1)$;
(121) at (5,-16)$\emp,\;\yng(1,1,1,1,1,1,1,1)$;
(11) edge node (01)
(21) edge node (11)
(22) edge node (12)
(41) edge node (21)
(31) edge node (01)
(31) edge node (12)
(42) edge node (11)
(42) edge node (31)
(43) edge node (31)
(43) edge node (22)
(44) edge node (22)
(51) edge node (21)
(51) edge node (42)
(52) edge node (42)
(52) edge node (43)
(53) edge node (43)
(53) edge node (44)
(61) edge node (41)
(62) edge node (41)
(62) edge node (51)
(63) edge node (51)
(63) edge node (52)
(64) edge node (52)
(64) edge node (53)
(65) edge node (53)
(71) edge node (61)
(71) edge node (62)
(72) edge node (62)
(72) edge node (63)
(73) edge node (63)
(73) edge node (64)
(81) edge node (71)
(82) edge node (71)
(82) edge node (72)
(83) edge node (72)
(83) edge node (73)
(84) edge node (64)
(84) edge node (65)
(91) edge node (82)
(91) edge node (83)
(101) edge node (81)
(101) edge node (82)
(102) edge node (73)
(102) edge node (84)
(111) edge node (101)
(111) edge node (91)
(112) edge node (83)
(112) edge node (102)
(121) edge node (112)
(121) edge node (91);
(65) edge node (44)
(81) edge node (61);
§.§ The principal block of $\OO_{1/3}(G(3,1,3))$
56[scale=1.5,every text node part/.style=align=center]
(N11) at (-3,5)$\yng(3),\;\emp,\;\emp$;
(N21) at (-3,2)$\yng(2,1),\;\emp,\;\emp$;
(N22) at (0,2)$\emp,\;\yng(3),\;\emp$;
(N23) at (3,2)$\emp,\;\emp,\;\yng(3)$;
(N31) at (-2,1)$\yng(1),\;\yng(2),\;\emp$;
(N41) at (-2.5,0)$\yng(1,1),\;\yng(1),\;\emp$;
(N42) at (2.2,0)$\emp,\;\yng(2),\;\yng(1)$;
(N51) at (-3,-1)$\yng(1,1,1),\;\emp,\;\emp$;
(N52) at (0,-1)$\emp,\;\yng(2,1),\;\emp$;
(N53) at (3,-1)$\emp,\;\emp,\;\yng(2,1)$;
(N61) at (1,-2)$\emp,\;\yng(1),\;\yng(1,1)$;
(N71) at (0,-4)$\emp,\;\yng(1,1,1),\;\emp$;
(N72) at (3,-4)$\emp,\;\emp,\;\yng(1,1,1)$;
(N21) edge node (N11)
(N23) edge node (N11)
(N31) edge node (N11)
(N31) edge node (N22)
(N41) edge node (N21)
(N41) edge node (N31)
(N42) edge node (N31)
(N42) edge node (N23)
(N51) edge node (N41)
(N52) edge node (N42)
(N53) edge node (N21)
(N53) edge node (N23)
(N61) edge node (N41)
(N61) edge node (N42)
(N61) edge node (N53)
(N71) edge node (N52)
(N71) edge node (N61)
(N72) edge node (N51)
(N72) edge node (N61);
(N42) edge node (N22)
(N51) edge node (N21)
(N52) edge node (N22)
(N52) edge node (N31)
(N71) edge node (N41)
(N72) edge node (N53);
§.§ The (conjectural) BGG resolutions of $L(\Triv)$, $L(\emp,\emp,(3))$, and $L(\emp,(3),\emp)$ for $G(3,1,3)$ when $c=1/3$.
56[scale=1.5,every text node part/.style=align=center]
(N0) at (0,0)$\yng(3),\;\emp,\;\emp$;
(N11) at (-2,-1.5)$\yng(2,1),\;\emp,\;\emp$;
(N12) at (0,-1.5)$\yng(1),\;\yng(2),\;\emp$;
(N13) at (2,-1.5)$\emp,\;\emp,\;\yng(3)$;
(N21) at (-2,-3)$\yng(1,1),\;\yng(1),\;\emp$;
(N22) at (2,-3)$\emp,\;\yng(2),\;\yng(1)$;
(N23) at (0,-3)$\emp,\;\emp,\;\yng(2,1)$;
(N3) at (0,-4.5)$\emp,\;\yng(1),\;\yng(1,1)$;
(N11) edge node (N0)
(N12) edge node (N0)
(N13) edge node (N0)
(N21) edge node (N11)
(N21) edge node (N12)
(N22) edge node (N12)
(N22) edge node (N13)
(N23) edge node (N11)
(N23) edge node (N13)
(N3) edge node (N21)
(N3) edge node (N22)
(N3) edge node (N23);
56[scale=1.5,every text node part/.style=align=center]
(x0) at (-1,0)$\emp,\;\yng(3),\;\emp$;
(x1) at (-2,-1)$\yng(1),\;\yng(2),\;\emp$;
(x2) at (-3,-2)$\yng(1,1),\;\yng(1),\;\emp$;
(x3) at (-4,-3)$\yng(1,1,1),\;\emp,\;\emp$;
(y01) at (3,0)$\emp,\;\emp,\;\yng(3)$;
(y11) at (2,-1)$\emp,\;\yng(2),\;\yng(1)$;
(y12) at (4,-1)$\emp,\;\emp,\;\yng(2,1)$;
(y21) at (1,-2)$\emp,\;\yng(2,1),\;\emp$;
(y22) at (3,-2)$\emp,\;\yng(1),\;\yng(1,1)$;
(y31) at (2,-3)$\emp,\;\yng(1,1,1),\;\emp$;
(x3) edge node (x2)
(x2) edge node (x1)
(x1) edge node (x0)
(y31) edge node (y21)
(y31) edge node (y22)
(y21) edge node (y11)
(y22) edge node (y11)
(y22) edge node (y12)
(y11) edge node (y01)
(y12) edge node (y01);
§.§ The (conjectural) BGG resolution of $L(\Triv)$ for $G(3,1,6)$ when $c=1/6$.
The graph $\bar{\Gamma}_{\Triv}$ encoding the resolution consists of eight cubes glued into a $2 \times 2 \times 2$ cube.
[scale=1.5,every text node part/.style=align=center]
(01) at (0,0)$\yng(6),\;\emp,\;\emp$;
(11) at (-2.5,-1.5)$\yng(5,1),\;\emp,\;\emp$;
(12) at (0,-1.5)$\yng(3),\;\yng(3),\;\emp$;
(13) at (2.5,-1.5)$\yng(1),\;\emp,\;\yng(5)$;
(21) at (-5,-3)$\yng(4,1,1),\;\emp,\;\emp$;
(22) at (1,-3)$\yng(1,1),\;\emp,\;\yng(4)$;
(23) at (-3,-3)$\yng(3,1),\;\yng(2),\;\emp$;
(24) at (-1,-3)$\yng(2),\;\yng(3,1),\;\emp$;
(25) at (3,-3)$\yng(1),\;\yng(3),\;\yng(2)$;
(26) at (5,-3)$\emp,\;\emp,\;\yng(5,1)$;
(31) at (-1.7,-4.5)$\yng(1,1,1),\;\emp,\yng(3)$;
(32) at (0,-4.5)$\yng(1,1),\;\yng(2),\;\yng(2)$;
(33) at (3.3,-4.5)$\emp,\;\emp,\;\yng(4,1,1)$;
(34) at (-3.3,-4.5)$\yng(2,1),\;\yng(2,1),\;\emp$;
(35) at (5,-4.5)$\emp,\;\yng(3),\;\yng(2,1)$;
(36) at (1.7,-4.5)$\yng(1),\;\yng(3,1),\;\yng(1)$;
(37) at (-5,-4.5)$\yng(3,1,1),\;\yng(1),\;\emp$;
(41) at (1,-6)$\emp,\;\emp,\;\yng(3,1,1,1)$;
(42) at (3,-6)$\emp,\;\yng(2),\;\yng(2,1,1)$;
(43) at (-3,-6)$\yng(1,1,1),\;\yng(1),\;\yng(2)$;
(44) at (-5,-6)$\yng(2,1,1),\;\yng(1,1),\;\emp$;
(45) at (5,-6)$\emp,\;\yng(3,1),\;\yng(1,1)$;
(46) at (-1,-6)$\yng(1,1),\;\yng(2,1),\;\yng(1)$;
(51) at (0,-7.5)$\emp,\;\yng(1),\;\yng(2,1,1,1)$;
(52) at (2.5,-7.5)$\emp,\;\yng(2,1),\;\yng(1,1,1)$;
(53) at (-2.5,-7.5)$\yng(1,1,1),\;\yng(1,1),\;\yng(1)$;
(61) at (0,-9)$\emp,\;\yng(1,1),\;\yng(1,1,1,1)$;
(11) edge node (01)
(12) edge node (01)
(13) edge node (01)
(22) edge node (11)
(23) edge node (11)
(23) edge node (12)
(25) edge node (12)
(22) edge node (13)
(25) edge node (13);
(24) edge node (12)
(36) edge node (25)
(34) edge node (24)
(36) edge node (24);
(21) edge node (11)
(26) edge node (13)
(31) edge node (21)
(37) edge node (21)
(37) edge node (23)
(32) edge node (25)
(35) edge node (25)
(33) edge node (26)
(35) edge node (26)
(42) edge node (32)
(46) edge node (32)
(42) edge node (35);
(45) edge node (36)
(46) edge node (36)
(45) edge node (35)
(52) edge node (45);
(32) edge node (23)
(34) edge node (23)
(46) edge node (34)
(43) edge node (32)
(43) edge node (37)
(44) edge node (37)
(44) edge node (34)
(53) edge node (44);
(31) edge node (22)
(32) edge node (22)
(33) edge node (22)
(41) edge node (31)
(41) edge node (33)
(42) edge node (33)
(43) edge node (31)
(51) edge node (41);
(51) edge node (43)
(53) edge node (43)
(52) edge node (46)
(53) edge node (46)
(51) edge node (42)
(52) edge node (42)
(61) edge node (51)
(61) edge node (52)
(61) edge node (53);
§.§ The (conjectural) BGG resolutions of $L(\emp,(6),\emp)$ and $L(\emp,(6),\emp)$ for $G(3,1,6)$ at $c=1/6$. The principal block of $G(4,1,4)$ at $c=1/4$.
56[scale=1.5,every text node part/.style=align=center]
(11) at (2,0)$\emp,\;\yng(6),\;\emp$;
(21) at (.5,-1.5)$\yng(3),\;\yng(3),\;\emp$;
(22) at (2,-1.5)$\emp,\;\yng(5,1),\;\emp$;
(23) at (3.5,-1.5)$\emp,\;\yng(4),\;\yng(2)$;
(31) at (-1,-3)$\yng(3,1),\;\yng(2),\;\emp$;
(32) at (.5,-3)$\yng(2),\;\yng(3,1),\;\emp$;
(33) at (2,-3)$\yng(1),\;\yng(3),\;\yng(2)$;
(34) at (3.5,-3)$\emp,\yng(4,1),\;\yng(1)$;
(41) at (-2.5,-4.5)$\yng(3,1,1),\;\yng(1),\;\emp$;
(42) at (-1,-4.5)$\yng(2,1),\;\yng(2,1),\;\emp$;
(43) at (.5,-4.5)$\yng(1,1),\;\yng(2),\;\yng(2)$;
(44) at (2,-4.5)$\yng(1),\;\yng(3,1),\;\yng(1)$;
(51) at (-4,-6)$\yng(3,1,1,1),\;\emp,\;\emp$;
(52) at (-2.5,-6)$\yng(2,1,1),\;\yng(1,1),\;\emp$;
(53) at (-1,-6)$\yng(1,1,1),\;\yng(1),\;\yng(2)$;
(54) at (.5,-6)$\yng(1,1),\;\yng(2,1),\;\yng(1)$;
(61) at (-4,-7.5)$\yng(2,1,1,1,1),\;\emp,\;\emp$;
(62) at (-2.5,-7.5)$\yng(1,1,1,1),\;\emp,\yng(2)$;
(63) at (-1,-7.5)$\yng(1,1,1),\;\yng(1,1),\;\yng(1)$;
(71) at (-2.5,-9)$\yng(1,1,1,1,1),\;\emp,\;\yng(1)$;
(x01) at (0,-10)$\emp,\;\emp,\;\yng(6)$;
(x11) at (-1,-11)$\yng(1),\;\emp,\;\yng(5)$;
(x12) at (1,-11)$\emp,\;\yng(4),\;\yng(2)$;
(x21) at (-2,-12)$\yng(1,1),\;\emp,\yng(4)$;
(x22) at (0,-12)$\yng(1),\;\yng(3),\;\yng(2)$;
(x23) at (2,-12)$\emp,\;\yng(4,1),\;\yng(1)$;
(x31) at (-3,-13)$\yng(1,1,1),\;\emp,\;\yng(3)$;
(x32) at (-1,-13)$\yng(1,1),\;\yng(2),\;\yng(2)$;
(x33) at (1,-13)$\yng(1),\;\yng(3,1),\;\yng(1)$;
(x34) at (3,-13)$\emp,\;\yng(4,1,1),\;\emp$;
(x41) at (-2,-14)$\yng(1,1,1),\;\yng(1),\;\yng(2)$;
(x42) at (0,-14)$\yng(1,1),\;\yng(2,1),\;\yng(1)$;
(x43) at (2,-14)$\yng(1),\;\yng(3,1,1),\;\emp$;
(x51) at (-1,-15)$\yng(1,1,1),\;\yng(1,1),\;\yng(1)$;
(x52) at (1,-15)$\yng(1,1),\;\yng(2,1,1),\;\emp$;
(x61) at (0,-16)$\yng(1,1,1),\;\yng(1,1,1),\;\emp$;
(22) edge node (11)
(23) edge node (11)
(34) edge node (22)
(34) edge node (23);
(21) edge node (11)
(32) edge node (21)
(32) edge node (22)
(33) edge node (21)
(33) edge node (23)
(44) edge node (32)
(44) edge node (33)
(44) edge node (34);
(31) edge node (21)
(42) edge node (31)
(42) edge node (32)
(43) edge node (31)
(43) edge node (33)
(54) edge node (43)
(54) edge node (42)
(54) edge node (44);
(41) edge node (31)
(52) edge node (41)
(52) edge node (42)
(53) edge node (41)
(53) edge node (43)
(63) edge node (52)
(63) edge node (53)
(63) edge node (54);
(51) edge node (41)
(61) edge node (52)
(61) edge node (51)
(62) edge node (51)
(62) edge node (53)
(71) edge node (61)
(71) edge node (62)
(71) edge node (63);
(x61) edge node (x51)
(x61) edge node (x52)
(x51) edge node (x41)
(x51) edge node (x42)
(x52) edge node (x42)
(x52) edge node (x43)
(x41) edge node (x31)
(x41) edge node (x32)
(x42) edge node (x32)
(x42) edge node (x33)
(x43) edge node (x33)
(x43) edge node (x34)
(x31) edge node (x21)
(x32) edge node (x21)
(x32) edge node (x22)
(x33) edge node (x22)
(x33) edge node (x23)
(x34) edge node (x23)
(x21) edge node (x11)
(x22) edge node (x11)
(x22) edge node (x12)
(x23) edge node (x12)
(x11) edge node (x01)
(x12) edge node (x01);
[scale=1.5,every text node part/.style=align=center]
(0) at (-1,3)$\yng(4),\;\emp,\;\emp,\;\emp$;
(11) at (-7,-1)$\yng(3,1),\;\emp,\;\emp,\;\emp$;
(12) at (-2,-1)$\emp,\;\yng(4),\;\emp,\;\emp$;
(13) at (1,-1)$\emp,\;\emp,\;\yng(4),\;\emp$;
(14) at (4,-1)$\emp,\;\emp,\;\emp,\;\yng(4)$;
(21) at (-5,-3)$\yng(2),\;\yng(2),\;\emp,\;\emp$;
(22) at (1,-3)$\yng(1),\;\emp,\;\yng(3),\;\emp$;
(31) at (-4.5,-4)$\yng(2,1),\;\yng(1),\;\emp,\;\emp$;
(32) at (-1,-4)$\emp,\;\yng(3),\;\yng(1),\;\emp$;
(33) at (3,-4)$\emp,\;\emp,\;\yng(3),\;\yng(1)$;
(41) at (-7,-5)$\yng(2,1,1),\;\emp,\;\emp,\;\emp$;
(42) at (-5.2,-5) $\yng(1,1),\;\emp,\;\yng(2),\;\emp$;
(43) at (-3.4,-5) $\emp,\;\yng(3,1),\;\emp,\;\emp$;
(44) at (-1.6,-5) $\yng(1),\;\yng(2),\;\yng(1),\;\emp$;
(45) at (.2,-5) $\emp,\;\emp,\;\yng(3,1),\;\emp$;
(46) at (2,-5) $\emp,\;\yng(2),\;\emp,\;\yng(2)$;
(47) at (4,-5) $\emp,\;\emp,\;\emp,\;\yng(3,1)$;
(51) at (-4,-6)$\yng(1,1),\;\yng(1),\;\yng(1),\;\emp$;
(52) at (-1.5,-6)$\yng(1),\;\yng(2,1),\;\emp,\;\emp$;
(53) at (0,-6)$\emp,\;\yng(2),\;\yng(1),\;\yng(1)$;
(61) at (-5.5,-7)$\yng(1,1,1),\;\emp,\;\yng(1),\;\emp$;
(62) at (-1,-7)$\emp,\;\yng(2),\;\yng(1,1),\;\emp$;
(63) at (-3,-7)$\yng(1,1),\;\yng(1,1),\;\emp,\;\emp$;
(64) at (.5,-7)$\emp,\;\emp,\;\yng(2),\;\yng(1,1)$;
(65) at (2,-7)$\emp,\;\yng(2,1),\;\emp,\;\yng(1)$;
(66) at (3.5,-7)$\emp,\;\yng(1),\;\emp,\;\yng(2,1)$;
(71) at (-.5,-8)$\emp,\;\yng(1),\;\yng(1),\;\yng(1,1)$;
(81) at (-7,-9)$\yng(1,1,1,1),\;\emp,\;\emp,\;\emp$;
(82) at (-4,-9)$\emp,\;\yng(2,1,1),\;\emp,\;\emp$;
(83) at (-1,-9) $\emp,\;\yng(1,1),\;\emp,\;\yng(1,1)$;
(84) at (2,-9)$\emp,\;\emp,\;\yng(2,1,1),\;\emp$;
(85) at (4,-9)$\emp,\;\emp,\;\emp,\;\yng(2,1,1)$;
(91) at (-2.5,-10)$\emp,\;\yng(1),\;\yng(1,1,1),\;\emp$;
(92) at (.5,-10)$\emp,\;\emp,\;\yng(1),\;\yng(1,1,1)$;
(10a) at (-4,-13)$\emp,\;\yng(1,1,1,1),\;\emp,\;\emp$;
(10b) at (-1,-13)$\emp,\;\emp,\;\yng(1,1,1,1),\;\emp$;
(10c) at (2,-13)$\emp,\;\emp,\;\emp,\;\yng(1,1,1,1)$;
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§.§ The (conjectural) BGG resolution of $L(\Triv)$ for $G(4,1,4)$ when $c=1/4$.
The graph $\bar{\Gamma}_{\Triv}$ encoding the resolution is a hypercube.
56[scale=1.5,every text node part/.style=align=center]
(N0) at (0,0)$\yng(4),\;\emp,\;\emp,\;\emp$;
(N11) at (-3,-1.5)$\yng(3,1),\;\emp,\;\emp,\;\emp$;
(N12) at (-1,-1.5)$\yng(2),\;\yng(2),\;\emp,\;\emp$;
(N13) at (1,-1.5)$\yng(1),\;\emp,\;\yng(3),\;\emp$;
(N14) at (3,-1.5)$\emp,\;\emp,\;\emp,\yng(4)$;
(N21) at (-5,-3)$\yng(2,1),\;\yng(1),\;\emp,\;\emp$;
(N22) at (-3,-3)$\yng(1,1),\;\emp,\;\yng(2),\;\emp$;
(N23) at (-1,-3)$\yng(1),\;\yng(2),\;\yng(1),\;\emp$;
(N24) at (1,-3)$\emp,\;\yng(2),\;\emp,\;\yng(2)$;
(N25) at (3,-3)$\emp,\;\emp,\yng(3),\;\yng(1)$;
(N26) at (5,-3)$\emp,\;\emp,\;\emp,\;\yng(3,1)$;
(N31) at (-3,-4.5)$\yng(1,1),\;\yng(1),\;\yng(1),\;\emp$;
(N32) at (-1,-4.5)$\emp,\;\yng(2),\;\yng(1),\;\yng(1)$;
(N33) at (1,-4.5)$\emp,\;\yng(1),\;\emp,\;\yng(2,1)$;
(N34) at (3,-4.5)$\emp,\;\emp,\;\yng(2),\;\yng(1,1)$;
(N4) at (0,-6)$\emp,\;\yng(1),\;\yng(1),\;\yng(1,1)$;
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§.§ An informative (non)-example.
Consider $\OO_c(B_2)$ when $c_0=1/2$, $d_0=1$. There are five simples labeled by the five bipartitions of $2$ and they all belong to the same block. By Lemma <ref> every $\Delta(\lambda)$ is $\ttt$-diagonalizable. $L_c(\Triv)$ is finite-dimensional and has a basis of Jack polynomials by <cit.>.
On the left, the graph $\Gamma$; on the right, the $\Ext^1$ quiver of the block, $Q_{\Ext^1}$:
56[scale=1,every text node part/.style=align=center]
(N1) at (-4,0)$\yng(2),\;\emp$;
(N2) at (-2,-1)$\yng(1,1),\;\emp$;
(N3) at (-4,-2)$\yng(1),\;\yng(1)$;
(N4) at (-2,-3)$\emp,\;\yng(2)$;
(N5) at (-4,-4)$\emp,\;\yng(1,1)$;
(Q1) at (2,0)$\yng(2),\;\emp$;
(Q2) at (4,-1)$\yng(1,1),\;\emp$;
(Q3) at (2,-2)$\yng(1),\;\yng(1)$;
(Q4) at (4,-3)$\emp,\;\yng(2)$;
(Q5) at (2,-4)$\emp,\;\yng(1,1)$;
(N5) edge node (N4)
(N4) edge node (N3)
(N3) edge node (N2)
(N2) edge node (N1);
(N5) edge node (N3)
(N3) edge node (N1);
(Q5) edge node (Q4)
(Q4) edge node (Q3)
(Q3) edge node (Q2)
(Q2) edge node (Q1)
(Q4) edge node (Q1);
We see that $Q_{\Ext^1}$ does not coincide with the double of $Q_{\mathrm{prim}}$, which is the subquiver of $\Gamma$ consisting of the orange arrows. Therefore $\OO_c(B_2)$ cannot have tight multiplicities by Corollary <ref>. However, the Serre subcategory of $\OO_c(B_2)$ spanned by $\{L_c(\tau)\;|\;\tau\leq((1^2),\;\emp)\}$ does have tight multiplicities since the subquivers of $Q_{\Ext^1}$ and $Q_{\mathrm{prim}}$ coincide there, by Corollary <ref> again. It follows that the only simple in $\OO_c(B_2)$ which does not have a BGG resolution is $L_c(\Triv)$.
BB Beilinson, A.; Bernstein, J. A proof of Jantzen conjectures. Adv. Soviet Math. 16, Part 1, p.1-50, AMS, 1993.
BEG Berest, Y.; Etingof, P.; Ginzburg, V. Finite-dimensional representations of rational Cherednik algebras. Int. Math. Res. Not. 2003, no. 19, 1053-1088.
BGS Berkesch-Zamaere, C.; Griffeth, S.; Sam, S. Jack polynomials as fractional quantum Hall states and the Betti numbers of the $(k+1)$-equals ideal, Comm. Math. Phys. 330 (2014), no. 1, 415-434.
ChTa Chuang, J.; Tan, K.M. Representations of wreath products of algebras. Mathematical proceedings of the Cambridge Philosophical Society, 135(3) (2003), 295-411.
CPS Cline, E; Parshall, B.; Scott, L. Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391 (1988), 85-99.
CPS2 Cline, E; Parshall, B.; Scott, L. Abstract Kazhdan-Lusztig theories. Tôhoku Math. J. 45 (1993), 511-534.
DuOp Dunkl, C.; Opdam, E. Dunkl operators for complex reflection groups. Proc. London Math. Soc. (3) 86 (2003), no. 1, 70-108.
Fa Fayers, M. Weight two blocks of Iwahori-Hecke algebras of type $B$. J. Algebra 303 (2006), no. 1, 154-201.
GGOR Ginzburg, V.; Guay, N.; Opdam, E.; Rouquier, R. On the category $\mathcal{O}$ for rational Cherednik algebras. Invent. Math. 154 (2003), no. 3, 617-651.
Gor Gordon, I. On the quotient ring by diagonal invariants. Invent. Math. 153 (2003), no. 3, 503-518.
GoGr Gordon, I.; Griffeth, S. Catalan numbers for complex reflection groups. Amer. J. Math. 134 (2012), no. 6, 1491-1502.
Gri Griffeth, S. Orthogonal functions generalizing Jack polynomials. Trans. Amer. Math. Soc. 362 (2010), no. 11, 6131-6157.
GGJL Griffeth, S.; Gusenbauer, A.; Juteau, D.; Lanini, M. Parabolic degeneration of rational Cherednik algebras. arXiv:1502.08025.
Ir Irving, R. $BGG$ algebras and the $BGG$ reciprocity principle. J. Algebra 135 (1990), 363-380.
Koe König, S. Cartan decompositions and BGG resolutions. Manuscripta Math. 86 (1995), 103-111.
LeMi Leclerc, B.; Miyachi, H. Some closed formulas for canonical bases of Fock spaces. Represent. Theory 6 (2002), 290-312.
Lo Losev, I. Proof of Varagnolo-Vasserot conjecture on cyclotomic categories O. arXiv:1305.4894.
LyMa Lyle, S.; Mathas, A. Blocks of cyclotomic Hecke algebras. Adv. Math. 216 (2007), no. 2, 854-878.
Mi1 Miyachi, H. Unipotent blocks of finite general linear groups in non-defining characteristic. PhD thesis, Chiba University, Chiba, March 2001.
Mi2 Miyachi, H. Rouquier blocks in Chevalley groups of type $E$. Adv. Math. 217 (2008), no. 6, 2841-2871.
ObYu Oblomkov, A.; Yun, Z. Geometric representations of graded and rational Cherednik algebras. arXiv:1407.5685
Rouq Rouquier, R. $q$-Schur algebras and complex reflection groups. Mosc. Math. J. 8 (2008), no. 1, 119-158, 184.
RSVV Rouquier, R.; Shan, P.; Varagnolo, M.; Vasserot, É. Categorifications and cyclotomic rational double affine Hecke algebras. Invent. Math. (to appear), 1-116.
Shan Shan,P. Graded decomposition matrices of $v$-Schur algebras via Jantzen filtration. Represent. Theory 16(2012), 212-269.
Ta Takeuchi, M. The group ring of $GL_n(q)$ and the $q$-Schur algebra. J. Math. Soc. Japan 48 (1996), no. 2, 259-274.
Tu2 Turner, W. Equivalent blocks of finite general linear groups. J. Algebra 247 (2002), No. 1, 244-267.
Tu1 Turner, W. RoCK blocks. Memoirs of the American Mathematical Society vol. 202, no. 947 (2009).
We Webster, B. Rouquier's conjecture and diagrammatic algebra. arXiv:1306.0074.
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1511.00972
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The updated ATLAS Jet Trigger for the LHC Run II
Sebastien Prince
on behalf of the ATLAS collaboration
After the current shutdown, the LHC is about to resume operation for a new data-taking period, when it will operate with increased luminosity, event rate and center of mass energy. The new conditions will impose more demanding constraints on the ATLAS online trigger reconstruction and selection system. To cope with such increased constraints, the ATLAS High-Level Trigger, placed after a first hardware-based Level 1 trigger, has been redesigned by merging two previously separated software-based processing levels. In the new joint processing level, the algorithms run in the same computing nodes, thus sharing resources, minimizing the data transfer from the detector buffers and increasing the algorithm flexibility.
The jet trigger software selects events containing high transverse momentum hadronic jets. It needs optimal jet energy resolution to help rejecting an overwhelming background while retaining good efficiency for interesting jets. In particular, this requires the CPU-intensive reconstruction of tridimensional energy deposits in the ATLAS calorimeter to be used as the basic input to the jet finding algorithms. To allow this costly reconstruction step, a partial detector readout scheme was developed, that effectively suppresses the low activity regions of the calorimeter and significantly reduces the needed resources. In this paper we describe the overall jet trigger software and its physics performance. We then focus on detailed studies of the algorithm timing and the performance impact of the full and partial calorimeter readout schemes. We conclude with an outlook of the jet trigger plans for the next LHC data-taking period.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
The CERN Large Hadron Collider (LHC) <cit.> was in shutdown in 2013 and 2014. Such a downtime period, separating the data-taking periods known as Run I and Run II, was required to perform the maintenance and upgrade work necessary to increase the collision energy, luminosity and bunch frequency that the LHC can achieve.
For the LHC detectors, such as ATLAS <cit.>, the increase in energy and luminosity means that there is be an increase by approximately a factor five of interesting collisions. One of the challenges for the ATLAS detector is to be able to record this higher production of interesting events while maintaining a good efficiency at low energy. An additional challenge comes from the expected higher number of interactions per bunch crossing, called pileup. This higher pileup increases the detector occupancy and thus the time required to reconstruct an event. The challenge is, in other words, to control the increased trigger rate with a limited latency.
The ATLAS trigger <cit.> is the system that decides, in real-time, whether to record or not an event. Its main purpose is to discard the least interesting events to reduce its input frequency of 20 MHz (40 MHz), the bunch crossing frequency of the Run I (Run II) LHC, to an achievable recording rate of the order of 400 Hz (1 kHz). To accomplish this, the ATLAS trigger system is divided into consecutive levels: Level 1, implemented in hardware, and a High-Level Trigger (HLT), implemented in software <cit.>. Each level allows the subsequent ones to have a longer latency, and thus more refined selection algorithms.
As the LHC is a hadronic collider, colored particles are the most prevalent high-energy particles produced. Due to color confinement, these particles shower and hadronize to form collimated sprays of particles, called jets. These particle jets are detected as energy deposits in the electromagnetic and hadronic calorimeters. However, the ATLAS jets are reconstructed from all calorimetric energy deposits, including those associated to non colored particles. The main increase in the jet rate comes from low-energy objects, since their high cross-section increases further with the higher collision energy. Also, the rate of jets not coming from the hard-scattering process, considered as background, increases as the pileup does. Therefore, the trigger system specialized in selecting jets, the jet trigger, had to undergo major changes to maintain a good selection performance in Run II.
§ JET TRIGGER PERFORMANCE IN RUN I
In Run I, the HLT was divided into two software levels: the Level 2 and the Event Filter. The algorithms used by the jet trigger to reconstruct jets were different across the three trigger levels. They were chosen such that they maximize the performance within the available latency. At all levels, the primary selection criterion of the jet triggers is on the transverse energy of the jets built by the algorithms.
At Level 1, the algorithm was a sliding window <cit.> with trigger towers as input, built across the whole calorimeters. Trigger towers are the sum of cells along the depth of the calorimeters, at a specific angular[ATLAS uses as angular variables the azimuthal angle $\phi$ and the pseudorapidity $\eta=-\ln\tan\frac{\theta}{2}$, where $\theta$ is the polar angle.] position. The position of the sliding window at a local energy maximum is called a region of interest (RoI) <cit.>.
At Level 2, two algorithms were used, depending on whether cells or trigger towers were used as input. If the input were calorimeter cells, the algorithm was a three-iteration cone algorithm <cit.> with a starting axis pointing at the RoI. Only cells around the RoI were taken as input. Instead, if the input were trigger towers, the algorithm was instead the same collinear- and infrared-safe algorithm that is used for offline jet reconstruction: anti-$k_t$ <cit.>. The input trigger towers were not restricted to be near the RoI. Since the trigger towers are less granular than cells, the timing of the algorithm still satisfied the Level 2 latency requirement even without any angular distance restriction.
At the Event Filter, anti-$k_t$ was also used but now taking as input topological clusters over the whole calorimeters, as in the offline jet reconstruction. Topological clusters are noise-suppressed objects formed by merging in three dimensions neighboring calorimeter cells that are above some noise thresholds <cit.>.
The performance of the Event Filter decision can be investigated from the trigger efficiency, shown in Fig. *fig:EF. The trigger efficiency is the ratio of the number of jets reconstructed online with respect to the number of offline-reconstructed jets. The plateau region reaches 100% efficiency and the data agree with the Monte Carlo simulation in that region, showing that the trigger had a good performance. As can be seen, the turn-on region is not centered on the threshold value: in the case of EF_j75 (an Event Filter trigger requiring a jet with at least 75 GeV of transverse energy), it is around 110 GeV. Figure *fig:Offset shows that relative difference between the energy calculated online and offline, across the full $p_\text{T}$ range. This discrepancy is due to the usage of different jet energy scales. Indeed, the offline reconstruction permits the application of a more precise calibration <cit.>, while the short latency of the trigger system didn't allow for such detailed corrections.
For data and Monte Carlo simulations, (a) the Event Filter trigger efficiency curves and (b) their offset relative to the offline $E_\text{T}$ <cit.>.
§ JET TRIGGER IMPROVEMENTS FOR RUN II
To be able to record the higher rate of interesting events in Run II, the ATLAS trigger system received a multitude of upgrades <cit.> during the long shutdown. An important change relevant to the jet trigger is the merging of the Level 2 and the Event Filter, such that the HLT became only one trigger level, effectively pooling the computing resources of the two levels and minimizing data transfers. The merged HLT allows the CPU-intensive step of building topological clusters for all events passing the Level 1. As the topological clusters are inherently noise-suppressed, they provide higher resolution than other types of input. Another important upgrade to the trigger system relevant to the jet trigger is the new faster readout boards that allow accessing the cell information across the whole calorimeters more frequently. In practice, this means being able to build topoclusters over the whole calorimeters for all events passing first trigger level, implying that the Level 2 algorithms can be skipped altogether. Therefore, in Run II, the jet trigger algorithms are a sliding window over trigger towers at Level 1 and anti-$k_T$ over topological clusters at the HLT, both types of input being taken over the whole calorimeters.
Although simulations showed that this new design would respect the latency constraints of the online environment, a fallback plan was still conceived: a partial scan of the calorimeters <cit.>. The partial scan is an improvement over the algorithms limited to the information near the RoIs as it accesses all the RoIs simultaneously. This avoids processing multiple times energy deposits in regions covered by more than one RoI, while still suppressing the low activity regions of the calorimeters.
Figure *fig:PSTime shows that, in simulation, the partial scan indeed reduces the timing of building topological clusters on average from 6% to 10% of that of the full calorimeter scan, depending on the size of the RoI. Where the partial scan falls short however is in the jet energy measurement. Figure *fig:PSDiff shows the relative measured energy difference between the partial scan and the full scan techniques. Although at higher $E_\text{T}$ the difference is small, it is non negligible at lower $E_\text{T}$, where the production rate of jets is higher. This is mitigated by using larger RoIs, but a difference still remains. For this reason, as long as the trigger decision can be achieved within the HLT latency when using a full scan of the calorimeters, such a technique is to be preferred.
Comparison in simulation of the performance of partial scan, for two sizes of the region of interest, against full scan (a) for the topological clustering timing and (b) for the jet transverse energy measurement <cit.>.
§ JET TRIGGER PERFORMANCE IN RUN II
To implement the new design of the jet trigger within the merged HLT, the jet trigger software had to be completely rewritten. This was taken as an opportunity to streamline the software, which would facilitate implementing further refinements. Indeed, a streamlined code allowed to implement some of the offline jet energy corrections in the trigger: the pileup subtraction and the jet energy scale calibration <cit.>.
To assess the online behavior of the new software and the performance of the new jet energy corrections, the early data-taking was crucial. Shown in Fig. <ref> are the efficiency curves of some HLT jet triggers, for a wide range of threshold values, using the first week of stable beam data of Run II <cit.>. The plateau regions for the data all reach 100% efficiency and agree with those of the Monte Carlo simulations, showing that the new code kept a good performance in that respect. Also, an inspection of the figure reveals that the turn-on regions are nearly centered on their threshold value. A more quantitative description would require further work, but already this qualitative observation gives credence that the new jet energy calibrations are performing well.
Comparison of the HLT efficiency curves between the Run II first week of data and the Monte Carlo simulation <cit.>.
§ CONCLUSION
The Run II jet trigger code had to be completely rewritten to adapt to the new running conditions of the LHC. This new software relies on the improvement of the trigger, both in design and in hardware. To improve upon the Run I performance, the new code can now apply some of the jet energy corrections as they are applied offline. All of these updates show great performance.
In the future, to further the improvements to the jet trigger, adding supplementary offline-inspired jet energy corrections is planned. These new corrections would mainly improve the resolution of the jet energy measured online.
L. Evans and P. Bryant,
JINST 3, S08001 (2008).
ATLAS Collaboration,
JINST 3, S08003 (2008).
ATLAS Collaboration,
Eur. Phys. J. C 72, 1849 (2012).
ATLAS Collaboration,
ATLAS-TDR-016 (2003).
W. Lampl et al.,
ATL-LARG-PUB-2008-002 (2008).
R. Blair et al.,
JINST 3, P04001 (2008).
ATLAS Collaboration,
CERN-OPEN-2008-020 (2008).
M. Cacciari, G. P. Salam and G. Soyez,
JHEP 0804, 063 (2008).
ATLAS Collaboration,
Eur. Phys. J. C 73, 2304 (2013).
ATLAS Collaboration,
Eur. Phys. J. C 75, 17 (2015).
ATLAS Collaboration,
ATLAS-TDR-023 (2013).
A. Tavares Delgado,
ATL-DAQ-PROC-2015-019 (2015).
All the ATLAS jet trigger public plots are available at
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1511.00795
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We study the probability that all eigenvalues of the Laguerre unitary ensemble of $n$ by $n$ matrices are in $(0,t)$, i.e., the largest eigenvalue distribution.
Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely $\alpha_n(t)$ and $\beta_n(t),$
as well as three auxiliary quantities, denoted by $r_n(t),~R_n(t)$ and $\sigma_n(t).$
We establish the second order differential equations for both $\beta_n(t)$ and $r_n(t).$
By investigating the soft edge scaling limit when $\alpha=O(n)$ as $n\rightarrow\infty$ or $\alpha$ is finite,
we derive a $P_{II},$ the $\sigma$-form, and the asymptotic solution of the probability.
In addition, we develop differential equations for orthogonal polynomials $P_{n}(z)$
corresponding to the largest eigenvalue distribution of LUE and GUE with $n$ finite or large.
For large $n,$ asymptotic formulas are given near the singular points of the ODE.
Moreover, we are able to deduce a particular case of Chazy's equation for $\varrho(t)=\Xi'(t)$
with $\Xi(t)$ satisfying the $\sigma$-form of $P_{IV}$ or $P_V.$
§ INTRODUCTION
A unitary ensemble is well defined for Hermitian matrices $M=(M_{ij})_{n\times n}$ with probability density
p(M)dM∝e^- v(M)(dM), (dM)=∏_i=1^ndM_ii∏_1≤j<k≤nd(ReM_jk) d (Im M_jk).
Here $v(M)$ is a matrix function <cit.> defined via Jordan canonical form and $\vol(dM)$ is called the volume element <cit.>.
The joint probability density function of the eigenvalues $\{x_j\}_{j=1}^n$ of this unitary ensemble is given in <cit.> by
\begin{equation}
\frac{1}{D_n(a,b)}\;\frac{1}{n!}{\prod\limits_{1\leq j<k\leq n}|x_{k}-x_{j}|}^{2}\prod\limits^{n}_{j=1}w(x_j),
\end{equation}
where $D_n(a,b)$ is the normalization constant which reads
\begin{equation}\label{DnLUE}
D_n(a,b)=\frac{1}{n!}\int_{[a,b]^{n}}{\prod\limits_{1\leq j<k\leq n}|x_{k}-x_{j}|}^{2}\prod\limits^{n}_{j=1}w(x_j)dx_{j},
\end{equation}
and $w(x)=e^{-v(x)}$
is a positive weight function supported on $[a,b]$ with finite moments
$$\mu_k:=\int_a^b x^{k}w(x)dx,\qquad k=0,1,2,\cdots.$$
It is shown, in <cit.>, that $D_n(a,b)$ can be evaluated as the determinant of the Hankel (or moment) matrix, that is,
A unitary ensemble is called the Laguerre unitary ensemble (LUE) if in (<ref>)
$$v(x)=x-\alpha\ln x,$$
or, what amounts to the same thing, in (<ref>)
$$w(x)=x^\alpha e^{-x},\qquad x\in[0,\infty),\quad\alpha>0.$$
A special case of LUE is $M=XX^*$ and $\alpha=p-n,$
where $X=X_1+iX_2$ is an $n\times p$ $ (n\leq p)$ random matrix with each element of $X_1$ and $X_2$ chosen independently
as a Gaussian random variable, see <cit.>.
Denote by $\mathbb{P}(n,t)$ the probability that the largest eigenvalue in LUE is not larger than $t,$ then
ℙ(n,t) = D_n(t)/D_n(0,∞),where $D_n(t):=D_n(0,t).$
Tracy and Widom <cit.> have obtained the Jimbo-Miwa-Okamoto (J-M-O) $\sigma$-form <cit.> of $P_V$ for
$$\sigma_n(t):=t\frac{d}{d t}\ln\mathbb{P}(n,t)$$
by making use of the Fredholm determinant. Basor and Chen <cit.> have derived the same $\sigma$-form by studying the Hankel determinant $D_n(t)$
with the help of the ladder operators related to orthogonal polynomials.
In their work, another four quantities associated with $\mathbb{P}(n,t)$ are considered,
i.e. $\alpha_n(t),~\beta_n(t),~r_n(t)$ and $R_n(t),$ and the relationships between them are established.
In addition, a $P_V$ is derived for $R_n(t)$ (or $\alpha_n(t)$).
Based on these results, we obtain in this paper the second order differential equation for $\beta_n(t)$ as well as $r_n(t).$
The soft edge scaling limit of the smallest eigenvlue distribution on $(t,\infty)$ in LUE with $\alpha=\mu n=O(n)$ and
$t=\left(\sqrt{\mu +1}-1\right)^2 n-\frac{\left(\sqrt{\mu +1}-1\right)^{4/3}}{(\mu +1)^{1/6}}n^{1/3}s$
is analyzed in <cit.>.
Concerning the largest eigenvalue distribution, we show that for $\alpha=O(n)$ or finite,
t=c_1 n+c_2 n^{1/3}s,\qquad\sigma(s):=\frac{c_2}{c_1}\lim\limits_{n\to\infty}n^{-2/3}\sigma_{n}(t)$$
$$c_1=\left(\sqrt{\mu +1}+1\right)^2,\qquad c_2=\frac{\left(\sqrt{\mu +1}+1\right)^{4/3}}{(\mu +1)^{1/6}},
\qquad\mu=\begin{cases}
\frac{\alpha}{n}, & \quad \alpha=O(n)\\
0, & \quad \alpha\;\text{is finite}\\
\end{cases},$$
the aforementioned $\sigma$-form of $P_V$ reduces down to the same $\sigma$-form of $P_{II}$ as presented in <cit.>.
The $P_{V},$ the ODEs for $\beta_n(t)$ and $r_n(t)$ can likewise be reduced to a $P_{II}.$
According to the ODE for $\sigma(s),$ we are able to provide the behavior of $\mathbb{P}(n,t)$
for large $n$ when $s\rightarrow\infty$ or $s\rightarrow-\infty.$
By means of the ladder operators valid for the orthogonal polynomials $P_{n}(z)$ associated with the general weight function $w(x)=e^{-v(x)},$
we deal with our problem and the largest eigenvalue distribution of GUE, and show that
the corresponding $\phi_n(z):=e^{-v(z)/2}P_n(z)$ satisfy different second order ODEs for finite $n$ but the same one for large $n.$
In the case of large $n,$ we develop the asymptotic behavior and Taylor expansion of $\phi_n(z)$ near the singular points of the corresponding ODE.
Moreover, a Chazy's equation <cit.> is derived for $\varrho(t):=\Xi'(t)$ with $\Xi(t)$ satisfying the $\sigma$-form of $P_{IV}$ or $P_V,$
and this result is applied to different ensembles including the largest eigenvalue distribution of LUE and GUE.
This paper is built up as follows. In Section 2, we introduce the ladder operator technique and restate the results of <cit.> which are used throughout this paper for further derivation.
We produce the ODE for $\beta_n(t)$ and establish a mapping for $r_n(t)$ and $R_n(t).$
The soft edge scaling limit is studied in Section 3.
The limiting behavior of $\phi_n(z)$ in the neighbourhood of the singular points is then presented in Section 4.
Finally, Section 5 is devoted to a derivation of Chazy's equations.
§ PRELIMINARIES
Monic polynomials $\{P_{n}(x)\}$ orthogonal with respect to a generic weight $w(x)$ on [a,b] is defined by the relations
∫_a^bP_m(x)P_n(x)w(x)dx= h_nδ_mn, m≥0, n≥0,
where $h_n$ is the square of the $L^2$ norm of the polynomial $P_n(x)$ and
P_n(x) =x^n+ p_1(n)x^n-1+⋯+P_n(0).
An immediate consequence of the orthogonality relation is the three-term recurrence relation <cit.>
xP_n(x)= P_n+1(x)+α_nP_n(x)+β_nP_n-1(x), n≥0
with initial conditions
Substituting (<ref>) into this relation gives rise to
\alpha_{n}=p_1(n)-p_1(n+1),\qquad n\geq0$$
with $p_1(0):=0,$ which immediately yields
From the recurrence relation (<ref>) and the orthogonality relation (<ref>), we get
The lowering and raising ladder operators (see e.g. <cit.>, <cit.> for a precise statement) are
\begin{equation}\label{ladderoperator}
\begin{aligned}
\left(\frac{d}{dz}+B_n(z)\right)P_n(z)&=\beta_n A_n(z)P_{n-1}(z),\\
\left(\frac{d}{dz}-B_n(z)-v'(z)\right)P_{n-1}(z)&=-A_{n-1}(z)P_n(z),
\end{aligned}
\end{equation}
\begin{equation}\label{AnBn}
\begin{aligned}
A_n(z)&=\left .\frac{P_n^2(y)w(y)}{h_n(y-z)}\right |_{y=a}^{y=b}+\frac{1}{h_n}\int_{a}^{b}\frac{v'(z)-v'(y)}{z-y} P_{n}^2(y)w(y)dy,\\
B_n(z)&=\left .\frac{P_n(y)P_{n-1}(y)w(y)}{h_{n-1}(y-z)}\right |_{y=a}^{y=b}+\frac{1}{h_{n-1}}\int_{a}^{b}\frac{v'(z)-v'(y)}{z-y} P_{n}(y)P_{n-1}(y)w(y)dy,
\end{aligned}
\end{equation}
and $v(z):=-\ln w(z).$ The compatibility conditions ($S_1$), ($S_2$) and ($S_2'$) for the ladder operators, see <cit.>, are given by
\begin{align}
1+(z-\alpha_n)\left(B_{n+1}(z)-B_n(z)\right)&=\beta_{n+1}A_{n+1}(z)-\beta_n A_{n-1}(z),\tag{$S_2$}\\
\end{align}
The discontinuous Laguerre weight
\begin{align}
w(x)=(A+B\theta(x-t))x^\alpha e^{-x},\qquad A\geq0,\quad A+B\geq0,
\end{align}
\begin{equation}
\begin{aligned}
\begin{cases}
A+B, & \quad \text{if } x>t\\
A, & \quad \text{if } x\leq t\\
\end{cases}
\end{aligned}
\end{equation}
is investigated in <cit.>, and the case where $A=0$ and $B=1$ leads to the smallest eigenvalue distribution of LUE on $(t,\infty).$
For our problem at hand, which is the case where $A=1$ and $B=-1,$ it is shown that
\begin{equation}\label{AnBnLUE}
\begin{aligned}
\end{aligned}
\end{equation}
R_n(t):=-P^2_n(t,t)/h_n(t)t^αe^-t, r_n(t):=-P_n(t,t)P_n-1(t,t)/h_n-1(t)t^αe^-t,where $P_j(t,t):=P_j(z,t)\mid_{z=t}.$ It should be noted that the $t$ dependence through the weight induces $t$ dependence of $P_j(x),$ $h_j$ and their allied quantities. For the sake of brevity, we shall not display the independence on $t$ for latter discussion unless we have to.
By using the compatibility conditions and taking the derivative of the orthogonality relation (<ref>)
with respect to $t,$ in addition to a $P_V$ and the $\sigma$-form,
Basor and Chen show that $r_n$ and $R_n$ which are closely related to $\alpha_n$ and $\beta_n$
satisfy a couple of difference equations.
We recall a number of results from <cit.> and make some remarks for our new observations.
The relations between $\sigma_n(t):=t\frac{d}{dt}\ln\mathbb{P}(n,t)$ and other quantities are
\begin{align}
\sigma_n&=t\frac{d}{dt}\ln D_{n}=-t\sum\limits^{n-1}_{j=0}R_{j}\non\\
\sigma_n'&=r_{n},\qquad t\sigma''_n=tr_n'=\beta_{n}'.\non
\end{align}
The quantities $r_n$ and $R_n$ satisfy the following coupled Riccati equations:
\begin{align}
t r_{n}'&=\left(\frac{1}{R_{n}}+\frac{1}{R_{n}-1}\right)r_{n}^{2}+(2n+\alpha)\frac{R_{n}}{R_{n}-1}r_{n}+n(n+\alpha)\frac{R_{n}}{R_{n}-1},\label{Riccattirn}\\
t R_{n}'&=t R_{n}^2+(2n+\alpha-t)R_{n}+2r_n.\label{RiccatiRn}
\end{align}
* The difference equations for $r_n(t)$ and $R_n(t)$ read
r_n+1+r_n=(t-2n-1-α-t R_n)R_n,
The equations in $(b)$ can be rewritten as
x_n+1x_n = y^2_n-(2n+α)y_n+n(n+α)/y_n^2,
y_n+y_n-1 = -(-t+2n-1+α)x_n-(2n-1+α)/x^2_n-2x_n+1,where
$$x_n:= 1-\frac{1}{R_{n-1}},\qquad y_n:=-r_n.$$
This mapping is very similar to (25) in <cit.> which leads to discrete Painlev$\acute{e}$ equations.
* The recurrence coefficients $\alpha_n$ and $\beta_n$ are expressed in terms of $r_n$ and $R_n$ as follows:
α_n = 2n+1+α+t R_n,
β_n = 1/1-R_n((2n+α)r_n+n(n+α)+r^2_n/R_n).
* The following equation holds
\begin{align}\label{odebetanrn}
\left((2n+\alpha)^2 -4\beta_n \right)r_n^{2}+2(2n+\alpha)(n(n+\alpha)-\beta_n)r_n+(n(n+\alpha)-\beta_n)^{2} -(\beta_n')^{2}=0.
\end{align}
Equation (<ref>) is given at the end of the proof of Thereom 6 in <cit.>.
Solving for $r_n$ from it, differentiating both sides of the resulting equation with respect to t
and noting that $\beta_{n}'=t r_{n}',$
we establish the differential equation for $\beta_n=\beta_n(t):$
t ^2 (2 n^2 (α+n)^2 (α+2
n)^2-8 n (α+n) (α^2+3 n^2+3 αn) β_n+6 (α+2 n)^2 β_n^2.
. -8 β_n^3+2 ((α+2 n)^2-4 β_n) β_n'^2+((α+2 n)^2-4 β_n)^2 β_n”)^2
=((α+2 n)^4-α^2 (α+2 n)t-8
(α+2 n)^2 β_n+16 β_n^2)^2
·(4 β_n(n (α+n)-β_n)^2+((α+2 n)^2-4 β_n) β_n'^2).
* The quantity
satisfies the following equation
S”_n = (1/2S_n+1/S_n-1)(S'_n)^2-1/tS'_n-(S_n-1)^2/t^2(α^2/2·1/S_n)
which is a $P_V$ <cit.> with
* The differential equation for $\sigma_{n}$ reads
\begin{align}\label{JMOP5}
(t \sigma_{n}'')^{2}=(\sigma_{n}-(t-2n-\alpha) \sigma_{n}')^{2}+4\sigma_{n}'^{2}(\sigma_{n}-t \sigma_{n}'-n(n+\alpha)),
\end{align}
which is the Jimbo-Miwa-Okamoto $\sigma$-form <cit.> of $P_V$ (see (<ref>) below) with
\begin{equation}\label{paralargest}
\begin{aligned}
\nu_1=0,\qquad\nu_{2}=n,\qquad\nu_{3}=n+\alpha.
\end{aligned}
\end{equation}
Combining (<ref>), (<ref>) and (<ref>) gives
\begin{align}
\sigma_n&=\frac{\alpha^2}{4}\cdot\frac{R_n}{1-R_n}-\frac{1}{4}(4n+2\alpha-t) tR_n-\frac{1}{4}t^2 R_n^2-\frac{1}{4}\frac{t^2(R_n')^2}{R_n(1-R_n)}\label{sigmanRn}\\
&=-\frac{\alpha^2}{4}\cdot\frac{1}{S_n}+\frac{t }{4}\cdot\frac{4n+2\alpha-t}{S_n-1}-\frac{1}{4}\cdot\frac{t^2}{(S_n-1)^2}+\frac{1}{4}\cdot\frac{t^2 (S_n')^2}{(S_n-1)^2S_n}.\label{sigmanSn}
\end{align}
This is the desired relation for $\sigma_n$ and $S_n$ as it was demonstrated in <cit.>.
§ SOFT EDGE SCALING LIMIT: $P_{II},$ THE $\SIGMA$-FORM OF $P_{II}$ AND THE TAIL BEHAVIOR OF $\MATHBB{P}(N,T)$
For large $n,$ we define
t=c_1 n+c_2 n^{1/3} s,
c_1=\left(\sqrt{\mu +1}+1\right)^2,\qquad
c_2=\frac{\left(\sqrt{\mu +1}+1\right)^{4/3}}{(\mu +1)^{1/6}},\qquad\mu=\begin{cases}
\frac{\alpha}{n}, & \quad \alpha=O(n)\\
0, & \quad \alpha\;\text{is finite}\\
\end{cases}.
In the case of finite $\alpha,$ we have
$$c_1=4,\qquad c_2=4^{2/3},\qquad t=4n+4^{2/3}n^{1/3}s.$$
We also use the following symbols:
y(s) := c_2/c_1lim_n→∞S_n(t)/n^2/3,
σ(s) := c_2/c_1lim_n→∞σ_n(t)/n^2/3,
r(s) := c_2^2/c_1lim_n→∞r_n(t)/n^1/3.
Combining $\eqref{sigmarnbetan}$ with $\sigma_n'(t)=r_{n}(t),$ and (<ref>) with (<ref>), we obtain
\begin{align}
\end{align}
Based on the results for finite $n$ presented in the previous sections, we are able to find the relations and ODEs for the above four quantities.
The equation for $\sigma(s)$ indicates the behavior of $\mathbb{P}(n,t)$ when both $n$ and $|s|$ large.
A $P_{II}$ and the $\sigma$-form of $P_{II}$ are established in <cit.>
for the smallest eigenvalue distribution of LUE with $\alpha=O(n).$
The analysis there is done directly on the associated $P_V$ equation.
For our problem, we shall now present a further and more detailed investigation of $\sigma(s).$
* The variables $\sigma(s),~y(s),$ and $r(s)$ are connected by the relations
\begin{align}
\sigma(s)&=-\frac{s}{y(s)}-\frac{1}{y^2(s)}+\frac{ \left(y'(s)\right)^2}{4y^3(s)}\label{sigmay}\\
&=-\frac{r'(s)^2}{4 r(s)}-r(s)^2+s r(s),\label{sigmar}\\
\end{align}
* The following equation is valid
y”(s)=3/2·y'(s)^2/y(s)-2s y(s)-4,
in which we introduce
so that $w(s)$ satisfies the $P_{II}$ with $\alpha=0,$ namely
w''(s)=2w^{3}(s)+s w(s).
* The equation for $\sigma(s)$ reads
which can be brought into the $\sigma$-form of $P_{II}$ with $\theta=0$ by making the replacements
$s\rightarrow-2^{-1/3}s$ and $\sigma (s)\rightarrow-2^{1/3}\sigma (s).$ Moreover, for $r(s),$ we have
(<ref>) can be verified by (<ref>).
To be specific, substituting $\sigma_n(t)$ by $\frac{c_1}{c_2}n^{2/3}\sigma(s),$
and $S_n(t)$ by $\frac{c_1}{c_2}n^{2/3}y(s)$ in (<ref>),
we obtain (<ref>) as the only retained leading order term when $n\rightarrow\infty.$
(<ref>) is a direct consequence of the first expression in (<ref>), which
combined with (<ref>) results in (<ref>).
Finally, equation (<ref>) arises from (<ref>), (<ref>) from (<ref>),
and (<ref>) from (<ref>) (as well as from (<ref>), see below).
In case (<ref>) we take into account (<ref>).
We can prove (<ref>)-(<ref>) in another way, by use of (<ref>)-(<ref>) and $\sigma'(s)=r(s).$
Indeed, differentiating both sides of (<ref>), in view of (<ref>), we get (<ref>).
Replacing $r(s)$ by $\sigma'(s)$ and $r'(s)$ by $\sigma''(s)$ in (<ref>) yields (<ref>).
Solve for $\sigma(s)$ from (<ref>), then differentiation gives (<ref>).
Now we go ahead with the evaluation of $\mathbb{P}(n,t).$ The definitions of $\sigma_n(t)$ and $\sigma(s)$ imply
\begin{align}
\sigma(s)=\frac{d}{ds}\ln\mathbb{\hat{P}}(s),\nonumber
\end{align}
where $\mathbb{\hat{P}}(s):=\lim\limits_{n\to\infty}\mathbb{P}(n,c_1 n+c_2 n^{1/3} s).$
To obtain the expansion of $\sigma(s)$ as $s\rightarrow-\infty,$ we assume
Substituting this into (<ref>) yields
σ(s)=s^2/4-1/8 s+9/64
s^4-189/128 s^7+21663/512 s^10+O(1/s^13),
from which follows
ℙ̂(s) = ι_1exp(s^3/12)/(-s)^1/8exp(-3/64 s^3+63/256
s^6-2407/512 s^9+O(1/s^12))
= ι_1exp(s^3/12)/(-s)^1/8(1-3/2^6 s^3+2025/2^13 s^6-2470825/2^19
s^9+26389914075/2^27 s^12+O(1/s^12)),
where the second equality results from the Taylor series for the exponential function.
Here $\iota_1$ is the normalization constant and is given in <cit.> by
where $\varsigma'(-1)$ is the derivative of the Riemann zeta function evaluated at -1. The above result agrees with (1.19) in <cit.>, since $w(z)$ plays the same role as $q(s;\lambda)$ there.
In fact, because of (<ref>) and $y(s)=w^{-2}(s),$ we have
\sigma'(s)=-w^{2}(s).
As $s\rightarrow\infty,$ to continue, we write
$$\mathbb{\hat{P}}(s)=1-\varepsilon f(s),$$
where $\varepsilon>0$ is sufficiently small and $f(s)>0,$ then
$$\sigma(s)=\frac{\varepsilon f'(s)}{1-\varepsilon f(s)}.$$
Hence from the left hand side of (<ref>) there follows a quadratic polynomial in $\varepsilon$
and the constant term of which has to be 0, namely
f^{(3)}(s)^2-4 s f''(s)^2+4 f'(s) f''(s)=0.
If we define
$$h(s):=\frac{f'(s)}{f(s)}=\frac{d}{ds}\ln f(s),$$
(h”(s)+3 h(s)
h'(s)+h^3(s))^2-4 s (h'(s)+h^2(s))^2+4 h(s)
$$h(s)=\lambda_0 s^{1/2}+\sum\limits^{\infty}_{k=0}\frac{\nu_k}{s^{k/2}},$$
then the coefficients can be determined by use of the preceding equation for $h(s)$ and we obtain
which implies
Referring to <cit.>, we see that $\iota_2=\frac{1}{16\pi}.$
§ THE BEHAVIOR OF $Z^{\ALPHA/2}E^{-Z/2}P_N(Z)$ ON $(0,T)$ AND OF $E^{-Z^2/2}P_N(Z)$ ON $(-\INFTY,T)$ FOR FINITE $N$ AND LARGE $N$
We now turn our attention to the ladder operators given by (<ref>)
to develop the differential equation for orthogonal polynomials $P_n(z)$ defined by (<ref>) and (<ref>).
Eliminating $P_{n-1}(z)$ from the lowering and raising operators, incorporated with $(S_2'),$ produces
v')P_n'+(B_n'- A_n' /A_nB_n+∑^n-1_j=0A_j)P_n=0,
which is presented in <cit.> and <cit.>.
To continue, we set, with suitable continuation in $z,$
$$P_n(z)=e^{v(z)/2}\phi_n(z),\qquad v(z)=-\ln w(z),$$
and, in light of (<ref>), establish
Below we will apply the above equation to the largest eigenvalue distribution of LUE on $(0,t)$ and of GUE on $(-\infty,t).$
For the Laguerre case, that is,
using the same notations as in Section <ref> for finite $n$ and as in Section <ref> for large $n$,
we have the following results :
* For finite $n$
\begin{equation}\label{finitephin}
\begin{aligned}
\phi_{n}''(z)&+\left(\frac{1}{z}+\frac{1}{z-t}-\frac{1}{z-t+t R_n(t)}\right)\phi_{n}'(z)\\
&+\left(\frac{-\frac{1}{4}\alpha^2}{z^2}+\frac{\frac{1}{2}(2n+\alpha+1)-\kappa_1(t)-\kappa_2(t)}{z}+\frac{\kappa_1(t)}{z-t}+\frac{\kappa_2(t)}{z-t+t R_n(t)}-\frac{1}{4}\right)\phi_{n}(z)=0,
\end{aligned}
\end{equation}
\begin{equation}\label{finitepara}
\begin{aligned}
\kappa_1(t)=\frac{1}{2}R_n(t)-\frac{R_n'(t)}{2R_n(t)}-\frac{\sigma
\kappa_2(t)=\frac{R_n'(t)}{2 R_n(t)}+\frac{R_n'(t)}{2(1-R_n(t))},
\end{aligned}
\end{equation}
and $\sigma_n(t)$ can be expressed in terms of $R_n(t)$ by using (<ref>).
* For $z=c_1 n+c_2 n^{1/3}z^*,$ and $\phi(z^*):=\lim\limits_{n\to\infty}\phi_n(z)$
\begin{equation}\label{inftyphi}
\begin{aligned}
\phi''(z^*)&+\left(\frac{1}{z^*-s}-\frac{1}{z^*-s+r(s)}\right)\phi'(z^*)\\
&+\left(\frac{\frac{r'(s)^2}{4 r(s)}-\frac{r'(s)}{2 r(s)}-s r(s)+r^2(s)}{z^*-s}+\frac{\frac{r'(s)}{2 r(s)}}{z^*-s+r(s)}-z^*\right)\phi(z^*)=0.
\end{aligned}
\end{equation}
Substituting (<ref>) and $v(z)=z-\alpha\ln z$ in (<ref>) furnishes
\begin{equation}
\begin{aligned}
\phi_{n}''(z)&+\left(\frac{1}{z}+\frac{1}{z-t}-\frac{1}{z-t+t R_n(t)}\right)\phi_{n}'(z)\non\\
&+\left(\frac{-\frac{1}{4}\alpha^2}{z^2}+\frac{\frac{1}{2}(2n+\alpha+1)-\kappa_1(t)-\kappa_2(t)}{z}+\frac{\kappa_1(t)}{z-t}+\frac{\kappa_2(t)}{z-t+t R_n(t)}-\frac{1}{4}\right)\phi_{n}(z)=0,
\end{aligned}
\end{equation}
κ_1(t) = -1/t(r_n(t)/R_n(t)+α/2+n)+∑^n-1_j=0R_j(t)+1/2,
κ_2(t) = 1/t-t R_n(t)(r_n(t)/R_n(t)+α/2+n)-1/2.
Applying (<ref>) to cancel $r_n(t)$ in $\kappa_1(t)$ and $\kappa_2(t),$
and taking into consideration $\sum\limits^{n-1}_{j=0}R_{j}(t)=-\frac{\sigma_n(t)}{t},$
we get (<ref>) and hence (<ref>).
Using the first equality of (<ref>) and the definition of $\sigma(s),$ we can show that
$$c_2\lim\limits_{n\to\infty}n^{1/3}\kappa_1(t)=-\frac{r'(s)}{2r(s)}-\sigma(s),\qquad c_2\lim\limits_{n\to\infty}n^{1/3}\kappa_2(t)=\frac{r'(s)}{2r(s)}.$$
Denote the coefficient of $\phi_n(z)$ in (<ref>) by $C_L,$ that is,
$$C_L=\frac{-\frac{1}{4}\alpha^2}{z^2}+\frac{\frac{1}{2}(2n+\alpha+1)}{z}-\frac{1}{4}-\frac{\kappa_1(t)+\kappa_2(t)}{z}+\frac{\kappa_1(t)}{z-t}+\frac{\kappa_2(t)}{z-t+t R_n(t)}.$$
Then, for $z=c_1 n+c_2 n^{1/3}z^*$ and $t=c_1 n+c_2 n^{1/3}s,$ we have
c_2^2lim_n→∞n^2/3C_L = c_2^2lim_n→∞n^2/3(-1/4α^2/z^2+1/2(2n+α+1)/z-1/4)
+c_2^2lim_n→∞n^2/3κ_2(t)/z-t+t R_n(t)
= -z^*-0+-r'(s)/2 r(s)-σ(s)/z^*-s+r'(s)/2 r(s)/z^*-s+r(s)
= -r'(s)/2 r(s)-σ(s)/z^*-s+r'(s)/2 r(s)/z^*-s+r(s)-z^*.
Noting that, for $\phi(z^*):=\lim\limits_{n\to\infty}\phi_n(z),$
$$\phi'(z^*)=c_2\lim\limits_{n\to\infty} n^{1/3}\phi_{n}'(z),
\qquad\phi''(z^*)=c_2^2\lim\limits_{n\to\infty} n^{2/3}\phi_{n}''(z),$$
according to (<ref>) multiplied by $c_2^2n^{2/3},$ as $n\rightarrow\infty,$
we conclude that (<ref>) is valid.
Now we consider the largest eigenvalue distribution of GUE on $(-\infty,t).$
Let $P_n(x)$ be monic polynomials orthogonal with respect to the deformed Hermite weight with one jump
0, & \quad \text{if } x>t\\
2e^{-x^2}, & \quad \text{if } x\leq t\\
\end{cases},$$
\int_{-\infty}^{\infty}P_m(x)P_{n}(x)w(x)dx= 2h_n(t)\delta_{mn},
or, what amounts to the same thing,
\int_{-\infty}^{t}P_m(x)P_{n}(x)e^{-x^2}dx= h_n(t)\delta_{mn}.$$
Consequently, the results in <cit.> with $\beta=-2$ there in the weight function are valid
for our $P_n(x)$ which corresponds to the largest eigenvalue distribution of GUE on $(-\infty,t).$ To begin with,
A_n(z)=2(α_n(t)/z-t+1), B_n(z)=r_n(t)/z-t,
where $\alpha_n(t)$ is the recurrence coefficient, that is,
and $r_n(t)$ is defined by
Based on the results in <cit.>, for
we obtain the following analogue of the preceding theorem:
* For finite $n$
\begin{equation}\label{GUEphin}
\begin{aligned}
\phi_{n}''(z)&+\left(\frac{1}{z-t}-\frac{1}{z-t+\alpha_n(t)}\right)\phi_{n}'(z)\\
\end{aligned}
\end{equation}
\begin{equation}\label{finiteGUEpara}
\begin{aligned}
\kappa_3(t)&=\frac{\left(\alpha_n'(t)\right)^2}{4\alpha_n(t)}-\frac{\alpha_n'(t)}{2\alpha_n(t)}-\alpha_n^3(t)+2t\alpha_n^2(t)+(-t^2+2n+1)\alpha_n(t),\qquad \kappa_4(t)=\frac{\alpha_n'(t)}{2\alpha_n(t)},
\end{aligned}
\end{equation}
and $\alpha_n(t)$ satisfies
For $z=\sqrt{2n}+\frac{z^*}{\sqrt{2}n^{1/6}},~t=\sqrt{2n}+\frac{s}{\sqrt{2}n^{1/6}},$
and $\phi(z^*):=\lim\limits_{n\to\infty}\phi_{n}(z)$
\begin{equation}\label{inftyphiGUE}
\begin{aligned}
\phi''(z^*)&+\left(\frac{1}{z^*-s}-\frac{1}{z^*-s+u(s)}\right)\phi'(z^*)\\
\end{aligned}
\end{equation}
where $u(s):=\lim\limits_{n\to\infty}\sqrt{2}n^{1/6}\alpha_{n}(t)$ is a solution of (<ref>).
Substitution of (<ref>) in (<ref>) leads to
\begin{equation}\label{GUEphin1}
\begin{aligned}
\phi_{n}''(z)&+\left(\frac{1}{z-t}-\frac{1}{z-t+\alpha_n(t)}\right)\phi_{n}'(z)\\
\end{aligned}
\end{equation}
The following results are established in <cit.> (see (22), (24), (26), (28) and (31)),
\begin{align}
\end{align}
To remove $\alpha_{n-1}(t)$ from (<ref>), we add (<ref>) to it and get
so that on account of (<ref>), equation (<ref>) follows from (<ref>).
Equation (<ref>) is true due to (<ref>)-(<ref>).
In fact, from (<ref>) and (<ref>), there follows
which combined with (<ref>) implies (<ref>).
For $t=\sqrt{2n}+\frac{s}{\sqrt{2}n^{1/6}}$ and $u(s)=\lim\limits_{n\to\infty}\sqrt{2}n^{1/6}\alpha_{n}(t),$
we get
\qquad u''(s)=\lim\limits_{n\to\infty}\frac{\alpha_{n}''(t)}{\sqrt{2}n^{1/6}}.$$
If we divide both sides of (<ref>) by $\sqrt{2}n^{1/6}$ and taking $n\rightarrow\infty,$
then we obtain
In addition, according to (<ref>),
For $z=\sqrt{2n}+\frac{z^*}{\sqrt{2}n^{1/6}}$ and $\phi(z^*)=\lim\limits_{n\to\infty}\phi_{n}(z),$ we have
Dividing both sides of (<ref>) by $2n^{1/3},$
as $n\rightarrow\infty,$ (<ref>) is seen to be true.
Note that (<ref>) is identical with (<ref>). This result is mainly due to the relation between Hermite and Laguerre polynomials.
Indeed, monic Hermite polynomials $\{H_n(z)\}$ can be reduced to monic Laguerre polynomials $\{L_n^\alpha(z)\}$ by
$$H_{2n}(z)=L_n^{\left(-\frac{1}{2}\right)}(z^2),\qquad H_{2n+1}(z)=L_n^{\left(\frac{1}{2}\right)}(z^2).$$
Observe that $\alpha=\pm\frac{1}{2}$ corresponds to $\mu=0$ (see Section <ref>). Write
z_L := 4n+4^2/3n^1/3z^*, t_L:=4n+4^2/3n^1/3s,
z := √(2n)+z^*/√(2)n^1/6, t:=√(2n)+s/√(2)n^1/6.
Replacing $n$ by $2n$ in $z$ and $t,$ we find
z^2 = 4n+4^2/3n^1/3z^*+(z^*)^2/4^2/3n^1/3∼z_L,
t^2 = 4n+4^2/3n^1/3s+s^2/4^2/3n^1/3∼t_L,
Here the symbol $\sim$ refers to the limiting procedure $n\rightarrow\infty.$
The above analysis suggests that (<ref>) and (<ref>) are obtained by using the same scaling method.
Introducing into (<ref>) the new variable $x$ defined by $x=-\frac{z^*-s}{r(s)}$
and the new function $f(x)=\phi(z^*),$ we obtain an equation in the form
p(x):=1/x-1/x-1, q(x):=a_0/x+a_1/x-1+a_2+a_3x,
a_1=-1/2r'(s), a_2=-sr^2(s), a_3=r^3(s).
Note that the dependence on $s$ of $\{a_i\}$ is not displayed for ease of notations,
in addition, the Painlev$\acute{e}$ equation
For any interval $[c_0,x]$ excluding $0,1,$ and $\infty,$ by writing
$$f(x)=F(x)\exp\left(-\frac{1}{2}\int_{c_0}^x p(z)dz\right),$$
it follows from (<ref>) that
\begin{equation}
\begin{aligned}
\begin{cases}
\frac{1}{4x^2}+\frac{a_0-\frac{1}{2}}{x}+a_2-a_1-\frac{5}{4}, & \qquad\text{as} \quad x\rightarrow0,\\
-\frac{3}{4(x-1)^2}+\frac{a_1+\frac{1}{2}}{x-1}-a_1^2-a_1-\frac{1}{4}, & \qquad \text{as } \quad x\rightarrow1,\\
a_2+a_3 x,& \qquad \text{as } \quad x\rightarrow\infty,
\end{cases}
\end{aligned}
\end{equation}
according to (<ref>), we have the following asymptotic formulas
\begin{equation}
\begin{aligned}
\begin{cases}
C_1\frac{\sqrt{2\lambda x}}{{\exp\left(\lambda x\right)}}
{{\rm M}\left(\frac{1}{2}-{\frac {a_0-\frac{1}{2}}{2\lambda }},1,2\lambda x\right)}+C_2\frac{\sqrt{2\lambda x}}{{\exp\left(\lambda x\right)}}
{{\rm U}\left(\frac{1}{2}-{\frac {a_0-\frac{1}{2}}{2\lambda }},1,2\lambda x\right)}, & \qquad\text{as} \quad x\rightarrow0,\bigskip\\
C_3{\frac {{\exp\left(\left(a_{{1}}+\frac{1}{2}\right)x\right)}}{
\sqrt {x-1}}}+C_4{\frac {\left(\left(a_{{1}}+\frac{1}{2}\right)x-a_1\right) \exp\left(-\left(a_{{1}}+\frac{1}{2}\right)x\right)}{\sqrt {x-1}}}, & \qquad \text{as} \quad x\rightarrow1,\bigskip\\
C_5{{\rm Ai}\left(-\frac{a_2}{\sqrt[3]{a_3^2}}-\sqrt[3]{a_3}x\right)}+C_{6}
{{\rm Bi}\left(-\frac{a_2}{\sqrt[3]{a_3^2}}-\sqrt[3]{a_3}x\right)},& \qquad \text{as} \quad x\rightarrow\infty,\non
\end{cases}
\end{aligned}
\end{equation}
where $\lambda :=\frac{1}{2}\sqrt{-4a_{{2}}+4a_{{1}}+5},$
${\rm M}(\mu,\nu,z)$ and ${\rm U}(\mu,\nu,z)$ are Kummer's confluent hypergeometric functions M and U respectively,
${\rm Ai}(z)$ and ${\rm Bi}(z)$ are the Airy functions of the first and second kind respectively, and $C_1$-$C_6$ are arbitrary constants.
We readily see that
$x=0$ is a regular singular point <cit.> for (<ref>) since $xp(x)$ and $x^2q(x)$ are analytic at 0.
From (<ref>) it follows that
D(f):=(x^2-x)f”(x)-f'(x)+(a_3x^3 +b_2x^2+ b_1x-a_0)f(x)=0,
b_1=-\frac{1}{4}r'(s)^2+2sr^2(s)-r^3(s),\qquad b_2=-sr^2(s)-r^3(s).
Note that
$$Y_0=\sum\limits^{\infty}_{n=0}c_{0,n} x^{\tau+n},\qquad c_{0,0}=1,
provided that for $n\geq0$
(τ+n+1)^2 c_0,n+1-((τ+n-1)(τ+n)-a_0)c_0,n-b_1 c_0,n-1-b_2 c_0,n-2-a_3 c_0,n-3=0. Here and in what follows $c_{0,-3}=c_{0,-2}=c_{0,-1}:=0.$
The two solutions of (<ref>) are given by
$$\left.Y_0\right|_{\tau=0}=\sum\limits^{\infty}_{n=0}c_{0,n} x^{n},
\qquad \left.\frac{\partial Y_0}{\partial \tau}\right|_{\tau=0}=\sum\limits^{\infty}_{n=1}d_{0,n} x^{n}+\ln x \cdot \left.Y_0\right|_{\tau=0}$$
$$d_{0,n}=\left.\frac{\partial c_{0,n}}{\partial \tau}\right|_{\tau=0}.$$
See <cit.>.
Using relation (<ref>), we find $\{c_{0,n}\}$ with $n$ large appear in the following form
c_0,n=∑^∞_ℓ=3θ_0,ℓ/n^ℓ, θ_0,3:=1,
where the coefficients $\{\theta_{0,\ell}\}_{\ell\geq1}$ are determined by
\begin{equation}\label{largencntau}
\begin{aligned}
\ell\theta_{0,\ell+3}=&\left(-\tau(1+2\ell)+\frac{1}{2}(\ell+1)\ell-a_1\right)\theta_{0,\ell+2}\\
\end{aligned}
\end{equation}
For $\ell=1$ the sum term is to be replaced by 0, so that
In case $\tau=0,$ we obtain from (<ref>)
(n+1)^2 c_0,n+1-(n(n-1)-a_0)c_0,n-b_1 c_0,n-1-b_2 c_0,n-2-a_3 c_0,n-3=0, n≥0,
and from (<ref>)
\begin{equation}
\begin{aligned}
\ell\theta_{0,\ell+3}=&\left(\frac{1}{2}(\ell+1)\ell-a_1\right)\theta_{0,\ell+2}\\
&+\sum\limits^{\ell+1}_{k=3}\left\{(-1)^{\ell-k}\binom{\ell+1}{k-3}-\left(b_1+2^{\ell-k+2}b_2+3^{\ell-k+2}a_3\right)\binom{\ell+1}{k-1}\right\}\theta_{0,k},\qquad \ell\geq1.
\end{aligned}
\end{equation}
Differentiation of (<ref>) and (<ref>) yields, respectively,
-∂c_0,n/∂τ((τ+n-1)(τ+n)-a_0)-∂c_0,n-1/ ∂τb_1-∂c_0,n-2/ ∂τb_2-∂c_0,n-3/ ∂τa_3=0, n≥0,
ℓ∂θ_0,ℓ+3/∂τ = -(1+2ℓ)θ_0,ℓ+2+∑^ℓ+1_k=3(-1)^ℓ-k[-2ℓ+1k-2+2τℓ+1k-1]θ_0,k
. -(b_1+2^ℓ-k+2b_2+3^ℓ-k+2a_3)ℓ+1k-1}∂θ_0,k/∂τ, ℓ≥1.
Setting $\tau=0$ in the above two formulas, we find
(n+1)^2 d_0,n+1 + 2(n+1)c_0,n+1-(2n-1)c_0,n
- (n(n-1)-a_0)d_0,n-b_1 d_0,n-1-b_2 d_0,n-2-a_3 d_0,n-3=0, n≥0,
and, by denoting $\nu_{0,j}=\left.\frac{\partial\theta_{0,j}}{\partial \tau}\right|_{\tau=0},$
\begin{equation}\label{largecn'tau}
\begin{aligned}
\ell\nu_{0,\ell+3}=&-(1+2\ell)\theta_{0,\ell+2}-2\sum\limits^{\ell+1}_{k=3}(-1)^{\ell-k}\binom{\ell+1}{k-2}\theta_{0,k}+\left(\frac{1}{2}(\ell+1)\ell-a_1\right)\nu_{0,\ell+2},\\
&+\sum\limits^{\ell+1}_{k=3}\left\{(-1)^{\ell-k}\binom{\ell+1}{k-3}-\left(b_1+2^{\ell-k+2}b_2+3^{\ell-k+2}a_3\right)\binom{\ell+1}{k-1}\right\}\nu_{0,k},\qquad \ell\geq1.
\end{aligned}
\end{equation}
Observe that for large $n$
d_0,n=∑^∞_ℓ=4ν_0,ℓ/n^ℓ, ν_0,4=-3.
Now we proceed to find out the solutions of (<ref>) in the form of Taylor series near $x=1$
which is a regular singular point since $(x-1)p(x)$ and $(x-1)^2q(x)$ are analytic at 0.
Equation (<ref>) can be rewritten as
\begin{equation}\label{x=1equation}
\begin{aligned}
\end{aligned}
\end{equation}
Y_1(x)=∑^∞_n=0c_1,n (x-1)^λ+n,
providing that
\begin{equation}\label{z=1c1c2c3}
\begin{aligned}
\left(\lambda^2-1\right)c_{1,1}+(a_1+\lambda(\lambda-1))c_{1,0}=0&,\\
\lambda(\lambda+2)c_{1,2}+(a_1+\lambda(\lambda+1))c_{1,1}-a_1^2c_{1,0}=0&,\\
\end{aligned}
\end{equation}
and for $n\geq3$
Taking $c_{1,0}=\frac{1}{2}\lambda,$ we have
so that there are three solutions of (<ref>) given by
\qquad \left.Y_1\right|_{\lambda=0},\qquad\left.\frac{\partial Y_1}{\partial \lambda}\right|_{\lambda=0}.$$
Given that $c_{1,0}=\frac{1}{2}\lambda,$ we obtain from (<ref>)
.c_1,n|_λ=0=0, n=0,1,2,3,
so that by mathematical induction it follows from (<ref>) that
$$\left.c_{1,n}\right|_{\lambda=0}=0,\qquad n\geq0,$$
which implies
\left.Y_1\right|_{\lambda=0}=0,\qquad
\left.\frac{\partial Y_1}{\partial \lambda}\right|_{\lambda=0}
=\sum\limits^{\infty}_{n=0}\left.\frac{\partial c_{1,n}}{\partial \lambda}\right|_{\lambda=0} (x-1)^{n}.
Consequently, there are two solutions of (<ref>) given by
$$\left.Y_1\right|_{\lambda=2}=\sum\limits^{\infty}_{n=0}\left(\left.c_{1,n}\right|_{\lambda=2}\right) (x-1)^{n+2},
\qquad 2\left.\frac{\partial Y_1}{\partial \lambda}\right|_{\lambda=0}=\sum\limits^{\infty}_{n=0}d_{1,n} (x-1)^{n},$$
$$d_{1,n}=2\left.\frac{\partial c_{1,n}}{\partial \lambda}\right|_{\lambda=0}.$$
Differentiation of (<ref>) gives
\begin{equation}\label{z=1cn'}
\begin{aligned}
&2(\lambda+n)c_{1,n+1}+(2(\lambda+n)-1)c_{1,n}+\left((\lambda+n)^2-1\right)\frac{\partial c_{1,n+1}}{\partial\lambda}\\
&+(a_1+(\lambda+n-1)(\lambda+n))\frac{\partial c_{1,n}}{\partial \lambda}-a_1^2\frac{\partial c_{1,n-1}}{\partial\lambda}
+A\frac{\partial c_{1,n-2}}{\partial\lambda}+a_3\frac{\partial c_{1,n-3}}{\partial\lambda}=0.
\end{aligned}
\end{equation}
In particular, for $\lambda=0:$
Choosing $\lambda=2$ in (<ref>) leads to
When $n$ is large, we find from (<ref>)
c_1,n = (-1)^n∑^∞_ℓ=1θ_1,ℓ/n^ℓ,
θ_1,1 := λ/2,
θ_1,2 = (a_0-λ(λ-1))θ_1,1,
2θ_1,3 = (a_0-3λ+2)θ_1,2-(a_1^2+a_2+a_3)θ_1,1,
3θ_1,4 = (a_0+5(1-λ))θ_1,3-2(a_1^2+a_2+a_3+λ^2-3λ+1)θ_1,2-(a_1^2+4a_2-a_3)θ_1,1,
ℓθ_1,ℓ+1 = (a_0+ℓ(ℓ+1)/2-(2ℓ-1)λ-1)θ_1,ℓ
. -ℓ-1k+1(a_1^2+2^ℓ-k-2A-3^ℓ-k-2a_3)}θ_1,k+2
-(ℓ-1)(a_1^2+2^ℓ-2A-3^ℓ-2a_3)θ_1,2-(a_1^2+2^ℓ-1A-3^ℓ-1a_3)θ_1,1, ℓ≥4.
Recalling $d_{1,n}=2\left.\frac{\partial c_{1,n}}{\partial \lambda}\right|_{\lambda=0}$
and bearing in mind $\left.c_{1,n}\right|_{\lambda=0}=0$ which suggests $\left.\theta_{1,\ell}\right|_{\lambda=0}=0,$
we obtain for large $n,$
d_1,n = (-1)^n ∑^∞_ℓ=1ν_1,ℓ/n^ℓ,
\nu_{1,\ell}=2\left.\frac{\partial \theta_{1,\ell}}{\partial \lambda}\right|_{\lambda=0}
and satisfy
ν_1,1 = 1,
ν_1,2 = a_0,
2ν_1,3 = (a_0+2)ν_1,2-(a_1^2+a_2+a_3),
3ν_1,4 = (a_0+5)ν_1,3-2(a_1^2+a_2+a_3+1)ν_1,2-(a_1^2+4a_2-a_3),
ℓν_1,ℓ+1 = (a_0+(ℓ-1)(ℓ+2)/2)ν_1,ℓ
-(ℓ-1)(a_1^2+2^ℓ-2A-3^ℓ-2a_3)ν_1,2-(a_1^2+2^ℓ-1A-3^ℓ-1a_3), ℓ≥4.
In addition, for $\lambda=2,$
θ_1,1 = 1,
θ_1,2 = a_0-2,
2θ_1,3 = (a_0-4)θ_1,2-(a_1^2+a_2+a_3)
3θ_1,4 = (a_0-5)θ_1,3-2(a_1^2+a_2+a_3-1)θ_1,2-(a_1^2+4a_2-a_3),
ℓθ_1,ℓ+1 = (a_0+ℓ(ℓ+1)/2-4ℓ+1)θ_1,ℓ
-(ℓ-1)(a_1^2+2^ℓ-2A-3^ℓ-2a_3)θ_1,2-(a_1^2+2^ℓ-1A-3^ℓ-1a_3), ℓ≥4.
§ CHAZY'S EQUATIONS
Noticing that $r_n(t)=\sigma_n'(t)$ is valid for both our problem and the largest eigenvalue distribution of GUE (see <cit.>),
we can find the ODE for $r_n(t)$ from the $\sigma$-form of a particular $P_V$ satisfied by $\sigma_n(t).$
In general, we can develop the ODE for $\varrho(t):=\Xi'(t)$
with $\Xi(t)$ satisfying the $\sigma$-form of a general $P_{V}.$
The extension of these considerations to $P_{IV}$ is not difficult.
Chazy's equation for the $\sigma$-form of $P_V$
Recall the J-M-O $\sigma$-form of $P_V$ <cit.>
(tΞ”)^2=(Ξ-t Ξ'+2 (Ξ')^2+(ν_1+ν_2+ν_3) Ξ')^2-4 Ξ' (Ξ'+ν_1) (Ξ'+ν_2) (Ξ'+ν_3),
where $\Xi=\Xi(t)$ and $\nu_1,~\nu_2,~\nu_3$ are constants. Here we use the new notation $\Xi$ instead of $\sigma.$
To get the ODE for $\varrho(t):=\Xi'(t),$ we need to eliminate $\Xi$ in (<ref>).
Differentiating both sides of (<ref>),
solving for $\Xi(t)$ and plugging it back into (<ref>), we obtain
(t (ϱ'+t ϱ”)+8
ϱ^3+6 (ν_1+ν_2+ν_3) ϱ^2+4 (ν_1ν_2+ν_1ν_3+ν_2 ν_3) ϱ+2 ν_1 ν_2 ν_3)^2
= (4 ϱ+ν_1+ν_2+ν_3-t)^2
·(t^2 (ϱ')^2+4 ϱ^4+4 (ν_1+ν_2+ν_3)
ϱ^3+4 (ν_1ν_2+ν_1ν_3+ν_2 ν_3) ϱ^2+4 ν_1 ν_2 ν_3ϱ).
With the change of variables,
t = 2 i e^z,\qquad
\vartheta(z)=-2i\varrho(t)-\frac{i}{2} \left(\nu_1+\nu _2+\nu_3\right),\nonumber
where $i^{2}=-1,$ we see that $\vartheta(z)$ satisfies
\begin{align}\label{ChazyII2equation}
\left(\frac{d^2 \vartheta}{d z^2}-2 \vartheta^3-\alpha_{1} \vartheta-\beta_{1}\right)^2=-4
\left(\vartheta-e^z\right)^2 \left(\left(\frac{d \vartheta}{dz}\right)^2-\vartheta^4-\alpha_{1} \vartheta^2-2\beta_{1}\vartheta-\gamma_{1}\right),
\end{align}
recognized to be the second member of the Chazy II system <cit.> with
\begin{equation}\label{ChazyII2parameters}
\begin{aligned}
\alpha_1&=\frac{1}{2} \left(3 \nu _1^2+3 \nu _2^2+3 \nu _3^2-2 \nu _1\nu _2-2 \nu _1\nu _3-2 \nu _2 \nu _3\right),\\
\beta_1&=\frac{i}{2} \left(\nu _1-\nu _2-\nu _3\right) \left(\nu _1+\nu _2-\nu _3\right) \left(\nu _1-\nu _2+\nu
\gamma_1&=-\frac{1}{16} \left(\nu _1+\nu _2-3 \nu _3\right) \left(3 \nu _1-\nu _2-\nu _3\right) \left(\nu _1-3 \nu
_2+\nu _3\right) \left(\nu _1+\nu _2+\nu _3\right).
\end{aligned}
\end{equation}
Now we proceed to apply our results to different ensembles.
For our problem, we have
Substituting (<ref>) for $\nu_1,~\nu_2$ and $\nu_3$ in (<ref>) and (<ref>)
gives rise to, respectively, the ODE for $r_n(t)$
(t (r_n'+t r_n”)+8
r_n^3+6 (2n+α) r_n^2+4n(n+α) r_n)^2
=(4 r_n+2n+α-t)^2(t^2 (r_n')^2+4 r_n^4+4 (2n+α)
r_n^3+4 n(n+α) r_n^2),
and the Chazy equation (<ref>) for
$$\vartheta(z)=-2i\left.r_n(t)\right|_{t = 2 i e^z}-\frac{i}{2} \left(2n+\alpha\right)$$
\begin{equation}\label{ourparameters}
\begin{aligned}
\alpha_{1}=\frac{1}{2} \left(3 \alpha ^2+4 n^2+4 \alpha n\right),
\qquad\beta_{1}=\frac{i}{2} \alpha ^2 (\alpha +2 n),
\qquad\gamma_{1}=\frac{1}{16} (2n-\alpha) (\alpha +2 n)^2 (3\alpha +2 n).\non
\end{aligned}
\end{equation}
By means of the Riccati equations for $R_n$ and $r_n,$ we can derive the ODE for $r_n$ in an alternative way which is straightforward but tedious.
Solving for $R_{n}$ from (<ref>) yields
R_n=t r_n'+2
r_n^2±√(Δ_n)/2 (t
r_n'-(2n+α) r_n -n(n+α)),
with $\Delta_n:=4 n (\alpha +n) r_n^2+4 (\alpha +2 n) r_n^3+4 r_n^4+(t r_n')^2.$
Choosing either sign, substituting the resulting $R_n$ into (<ref>)
and clearing the square root, we obtain (<ref>) again.
The outage probability of a single-user MIMO communication system can be calculated via
the moment generating function $\mathcal{M}(\lambda)$ of the mutual information.
Under some assumptions, see <cit.> and <cit.>, $\mathcal{M}(\lambda)$ is shown to be
D_n(t,λ) = (∫_0^∞x^i+jw_dlag(x,t)dx)_i,j=0^n-1,
H_n(t) := td/dtlnD_n(t,λ),and
$$w_{\rm dlag}(x,t)=(x+t)^\lambda x^\alpha e^{-x}.$$
Moreover, the $\sigma$-form of $P_V$ with
ν_1=λ, ν_2=-n, ν_3=-n-αis established for
and the following is established
$$r_n(t):=\frac{\lambda}{h_{n-1}}\int_0^\infty \frac{P_n(x)P_{n-1}(x)}{x+t}w_{\rm dlag}(x,t)dx=-\Xi'(t),$$
∫_0^∞P_m(x)P_n(x)w_dlag(x,t)dx= h_nδ_mn, m≥0, n≥0.Plugging (<ref>) into (<ref>) leads to the ODE for
from which follows the Chazy's equation (<ref>) for
$$\vartheta(z)=2i\left.r_n(t)\right|_{t = 2 i e^z}+\frac{i}{2} \left(2n+\alpha-\lambda\right)$$
\begin{equation}
\begin{split}
\alpha _1&=\frac{1}{2} \left(4 n^2+4 n \alpha+3 \alpha ^2+4n\lambda+2 \alpha \lambda +3\lambda ^2\right),\notag\\
\beta _1&=-\frac{i}{2} (\alpha-\lambda )(\alpha+\lambda) (2 n+\alpha+\lambda ),\notag\\
\gamma _1&=\frac{1}{16} (2n+\alpha -\lambda) (2n-\alpha +\lambda) (2n+3\alpha +\lambda) (2n+\alpha+3 \lambda).\notag
\end{split}
\end{equation}
Denote by $D_n(t)$ the determinant of the Hankel matrix generated from the moments of the time-dependent Jacobi weight
$$w(x,t)=(1-x)^\alpha(1+x)^\beta e^{-tx},\qquad -1\leq x\leq1,\quad t\in\mathbb{R},$$
$$D_n(t)=\det\left(\int_{-1}^1 x^{i+j}w(x,t)dx\right)_{i,j=0}^{n-1}.$$
It is proved in <cit.> that
$$\Xi(t):=t\frac{d}{dt}\ln D_n(t/2)-\frac{nt}{2}+n(n+\beta)$$
satisfies the $\sigma$-form of $P_V$ with
ν_1=-α, ν_2=n, ν_3=n+β.
where $r_n(t)$ is defined by
$$r_n(t):=\frac{\alpha}{h_{n-1}}\int_{-1}^1 \frac{P_n(x)P_{n-1}(x)}{1-x}w(x,t)dx,$$
∫_-1^1P_m(x)P_n(x)w(x,t)dx= h_nδ_mn, m≥0, n≥0.Substitution of (<ref>) in (<ref>) gives the ODE for
from which we obtain (<ref>) for
$$\vartheta(z)=2i\left.r_n(t)\right|_{t = i e^z}-\frac{i}{2} \left(2n-\alpha+\beta\right)$$
\begin{equation}
\begin{split}
\alpha _1&=\frac{1}{2} \left(4 n^2+4 n \alpha+3 \alpha ^2+4n\beta+2 \alpha\beta +3 \beta ^2\right),\\
\beta _1&=-\frac{i}{2} (\alpha -\beta ) (\alpha+\beta ) (2n+\alpha +\beta),\\
\gamma _1&=\frac{1}{16} (2n+\alpha -\beta) (2n-\alpha+\beta) (2n+3 \alpha +\beta)(2n+\alpha +3 \beta).\nonumber
\end{split}
\end{equation}
The weight
$$w(x,t)=e^{-t/x}x^\alpha(1-x)^\beta,\qquad 0\leq x\leq1,\quad t\geq0$$
is studied in <cit.> and the J-M-O $\sigma$-form of $P_V$ is found for
$$\Xi(t):=t\frac{d}{dt}\ln D_n(t)-n(n+\alpha+\beta)$$
ν_1=-(n+α+β), ν_2=n, ν_3=-β.
Here again $D_n(t)$ is the Hankel determinant
$$D_n(t)=\det\left(\int_{0}^1 x^{i+j}w(x,t)dx\right)_{i,j=0}^{n-1}.$$
Take into account the following result in <cit.>
\int_{0}^{1}P_m(x)P_{n}(x)w(x,t)dx= h_n\delta_{mn},\qquad m\geq0,\quad n\geq0,
then, with the aid of (<ref>), we find the ODE from (<ref>) for
so that (<ref>) holds for
$$\vartheta(z)=-2i\left.\varrho(t)\right|_{t = 2 i e^z}+\frac{i}{2} \left(\alpha+2\beta\right)$$
\begin{equation}
\begin{split}
\alpha _1&=\frac{1}{2} \left(8 n^2+8 n \alpha+3 \alpha ^2+8 n \beta +4 \alpha \beta +4\beta ^2\right),\notag\\
\beta _1&=-\frac{i}{2} \alpha (2n+\alpha) (2n+\alpha +2 \beta),\notag\\
\gamma _1&=-\frac{1}{16} (\alpha -2 \beta )(\alpha +2 \beta ) (4n+\alpha +2\beta) (4 n +3 \alpha +2\beta).\notag
\end{split}
\end{equation}
Chazy's equation for the $\sigma$-form of $P_{IV}$
We go ahead to deal with the $\sigma$-form of $P_{IV}$ in the spirit of the previous section.
The J-M-O $\sigma$-form of $P_{IV}$ <cit.> is given by
(Ξ”)^2 =4(t Ξ'-Ξ)^2 -4 Ξ' (Ξ'+ν_1)(Ξ'+ν_2),where $\Xi=\Xi(t)$ and $\nu_1,~\nu_2$ are constants. From this equation follows the ODE for $\varrho(t):=\Xi'(t),$
(ϱ”+6 ϱ^2+4(ν_1+ν_2)ϱ+2ν_1 ν_2)^2 =4 t^2 ((ϱ')^2+4ϱ^3+4(ν_1+ν_2)ϱ^2+ 4ν_1ν_2ϱ).
\vartheta(z)=-\frac{1}{2}\left.\varrho(t)\right|_{t=\frac{z}{\sqrt{2}}}-\frac{1}{6} \left(\nu _1+\nu_2\right),$$
we find
\begin{equation}\label{generalChazyII1}
\left(\vartheta''(z)-6 \vartheta(z)^2-\alpha_1\right){}^2=z^2 \left( \vartheta'(z)^2-4 \vartheta(z)^3-2\alpha_1 \vartheta(z)-\beta_1\right),
\end{equation}
which is the first member of the Chazy II system <cit.> with
\begin{equation}\label{ourparameters1}
\begin{aligned}
\alpha_1=\frac{1}{6}\left(-\nu _1^2+\nu_1\nu _2-\nu _2^2\right),\qquad\beta _1= -\frac{1}{54}\left(\nu _1-2 \nu _2\right)\left(2 \nu _1-\nu _2\right)\left(\nu _1+\nu _2\right).
\end{aligned}
\end{equation}
Now we intend to show the application of the above result.
Regarding the deformed Hermite weight with one jump,
\left(1+\frac{\beta}{2}\right)e^{-x^2}, & \quad \text{if } x>t\\
\left(1-\frac{\beta}{2}\right)e^{-x^2}, & \quad \text{if } x\leq t\\
\end{cases},$$
the following formulas are established in <cit.> (The $\tilde{x}$ there has been repalced here with $t$),
\begin{equation}\label{formulasknown}
\begin{aligned}
\frac{d}{dt}\ln D_n(t)&=2tr_n(t)-2(n+r_n(t))(\alpha_n+\alpha_{n-1}),\\
\frac{d^2}{dt^2}\ln D_n(t)&=2r_n(t).
\end{aligned}
\end{equation}
Recall here that
r_n(t) = βP_n(t,t)P_n-1(t,t)/h_n-1e^-t^2,
D_n(t) = (∫_-∞^∞x^i+jw(x;t)dx)_i,j=0^n-1,
h_nδ_mn = ∫_-∞^∞P_m(x)P_n(x)w(x;t)dx,
zP_n(z) = P_n+1(z)+α_nP_n(z)+β_nP_n-1(z).
From (<ref>), we can derive the $\sigma$-form of $P_{IV}$
$$ \Xi(t):=\frac{d}{dt}\ln D_n(t)$$
ν_1=0, ν_2=2n.
Hence the the ODE for
follows immediately from (<ref>)
and, as a consequence, the Chazy's equation (<ref>) holds for
α_1= -2/3n^2, β_1=-8/27n^3.
Denote by $\mathbb{P}_{\max}(n,t)$
the largest eigenvalue distribution of GUE on $(0,t)$ and by $\mathbb{P}_{\min}(n,t)$ the smallest one on $(t,\infty).$
Then we readily see in this last example that
according as
$$\beta=-2\qquad \text{or}\qquad2.$$
Indeed, for instance, for $\beta=-2,$ we have
D̅_n(t) := (∫_-∞^t x^i+je^-x^2dx)_i,j=0^n-1=D_n(t)/2^n,
D̅_n(t)/D̅_n(∞) = ℙ_max(n,t),
so that
\frac{d}{dt}\ln\mathbb{P}_{\max}(n,t)=\frac{d}{dt}\ln \bar{D}_n(t)=\frac{d}{dt}\ln D_n(t)=\Xi(t).
In addition, we notice that
\int_{-\infty}^{t}P_m(x)P_{n}(x)e^{-x^2}dx=\frac{1}{2}h_n\delta_{mn}=:\hbar_n\delta_{mn}.$$
The $\sigma$-form of $P_{IV}$ with parameters given by (<ref>) is also satified by
where $\mathbb{P}(n,t)$ is the probability that $(-t,t)$ is free of eigenvalues in GUE.
See <cit.> and <cit.>.
equation (<ref>) is valid for
with parameters given by (<ref>).
This result is in agreement with the one in <cit.> found by using the Riccati equations for $r_n$ and $R_n.$
The financial support of the Macau Science and Technology Development Fund under grant number FDCT 077/2012/A3, FDCT 130/2014/A3
is gratefully acknowledged. We also like to thank the University of Macau for generous support: MYRG 2014–00011 FST, MYRG 2014–00004 FST.
E. Basor, Y. Chen,
Painlev$\acute{e}$ V and the distribution function of a discontinuous linear statistic in the Laguerre unitary ensembles,
J. Phys. A: Math. Theor., 42 (2009), 035203 (18pp).
E. Basor, Y. Chen,
Perturbed Laguerre unitary ensembles, Hankel determinants, and information theory,
Math. Meth. Appl. Sci., DOI: 10.1002/mma.3399, 2014.
E. Basor, Y. Chen, T. Ehrhardt,
Painlev$\acute{e}$ V and time-dependent Jacobi polynomials,
J. Phys. A: Math. Theor., 43 (2010), 015204 (25pp).
M. Cao, Y. Chen, J. Griffin,
Continuous and discrete Painlev$\acute{e}$ equations arising from the gap probability distribution of the finite n Gaussian Unitary Ensembles,
J. Stat. Phys., 157 (2014), 363–375.
Y. Chen, D. Dai,
Painlev$\acute{e}$ V and a Pollaczek-Jacobi type orthogonal polynomials,
J. Approx. Theory, 162 (2010), 2149–2167.
Y. Chen, M. Ismail,
Ladder operators and differential equations for orthogonal polynomials,
J. Phys. A: Math. Gen., 30 (1997), 7817–7829.
Y. Chen, M. Ismail,
Jacobi polynomials from compatibility conditions,
Proc. Amer. Math. Soc., 133 (2) (2004), 465–472.
Y. Chen, A. Its,
Painlev$\acute{e}$ III and a singular linear statistics in Hermitian random matrix ensembles, I,
J. Approx. Theory, 162 (2010), 270–297.
Y. Chen, M. R. McKay,
Coulumb fluid, Painlev$\acute{e}$ transcendents, and the information theory of MIMO systems,
IEEE Trans. Inf. Theory, 58 (7) (2012), 4594–4634.
Y. Chen, G. Pruessner,
Orthogonal polynomials with discontinuous weights,
J. Phys. A: Math. Gen., 38 (2005), 191–198.
C. M. Cosgrove,
Chazy's second-degree Painlev$\acute{e}$ equations,
J. Phys. A: Math. Gen., 39 (2006), 11955–11971.
A. R. Forsyth,
A treatise on differential equations, sixth edition,
Macmillan Co. Ltd., 1956.
P. J. Forrester,
The spectrum edge of random matrix ensembles,
Nucl. Phys. B, 402 (1993), 709–728.
P. J. Forrester, N. S. Witte,
The distribution of the first eigenvalue spacing at the hard edge of the Laguerre unitary ensemble,
Kyushu. J. Math., 61 (2007), 457–526.
N. R. Goodman,
Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction),
Ann. Math. Statist., 34 (1963), 152–177.
B. Grammaticos, A. Ramani,
Discrete Painlev$\acute{e}$ equations: an integrability paradigm,
Phys. Scr., 89 (2014), 038002 (13pp).
N. J. Higham,
Functions of Matrices: Theory and Computation,
SIAM, 2008.
L. K. Hua,
Harmonic analysis of functions of several complex variables in the classical domains,
AMS, Providence, RI, 1963.
A. T. James,
Distributions of matrix variates and latent roots derived from normal samples,
Ann. Math. Statist., 35 (1964), 475–501.
M. Jimbo, T. Miwa,
Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II,
Physica D, 2 (1981), 407–448.
A. P. Magnus,
Painlev$\acute{e}$-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials,
J. Comput. Appl. Math., 57 (1995), 215–237.
M. L. Mehta,
Random Matrices. 3rd ed., Pure and Applied Mathematics (Amsterdam), vol 142, Elsevier/Academic Press, Amsterdam, 2004.
K. Okamoto,
On the $\tau$-function of the Painlev$\acute{e}$ equations,
Physica D, 2 (1981), 525–535.
A. Perret, G. Schehr,
Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices,
accepted for publication in Random Matrices: Theory Appl., 2015 (28pp).
G. Szeg$\ddot{o}$,
Orthogonal Polynomials (American Mathematical Society Colloquium Publications),
vol 23, AMS, New York, 1939.
C. A. Tracy, H. Widom,
Level-spacing distributions and the Airy kernel,
Commun. Math. Phys., 159 (1994), 151–174.
C. A. Tracy, H. Widom,
Fredholm determinants, diffrential equations and matrix models,
Commun. Math. Phys., 163 (1994), 33–72.
C. A. Tracy, H. Widom,
The distributions of random matrix theory and their applications,
in New Trends in Mathematical Physics, V. Sidoravi$\breve{c}$ius ed.,
Springer Netherlands, (2009), 753–765.
E. T. Whittaker, G. N. Watson,
A course of modern analysis. 4th ed.,
Cambridge university press, 1927.
|
1511.00915
|
Web and Media group,
VU University Amsterdam, The Netherlands,
Department of Philosophy, Linguistics and Theory of Science,
University of Gothenburg, Sweden
Dipartimento di Matematica e Informatica,
University of Ferrara, Italy
Recently, we see a new type of interfaces for programmers based on web
technology. For example, JSFiddle, IPython Notebook and R-studio. Web
technology enables cloud-based solutions, embedding in tutorial web
pages, attractive rendering of results, web-scale cooperative
development, etc. This article describes SWISH, a web front-end for
Prolog. A public website exposes SWI-Prolog using SWISH, which is used
to run small Prolog programs for demonstration, experimentation and
education. We connected SWISH to the ClioPatria semantic web toolkit,
where it allows for collaborative development of programs and queries
related to a dataset as well as performing maintenance tasks on the
running server and we embedded SWISH in the Learn Prolog Now! online
Prolog book.
§ INTRODUCTION
Web technology has emerged to a state where it becomes useable for
implementing programming development environments. All major modern
browsers now implement HTML5 and JavaScript and there are mature
components available such as the
CodeMirror[<https://codemirror.net/>] and
ACE[<http://ace.c9.io>] code editors, the
Bootstrap[<http://getbootstrap.com/>] framework for styling
and UI widgets and vizualization libraries such as
D3.js.[<http://d3js.org/>] Using web technology rather than
traditional GUI based technology such as
Eclipse,[<https://eclipse.org/>] Microsoft Visual
XEmacs,[<http://www.xemacs.org/>] etc. has various
advantages. Being network transparent, it allows for controlling cloud
hosted applications as well as Prolog processes running on headless
devices. Web technology provides a great infrastructure for
mashups, pages that integrate material from several sources.
For example, embedding Prolog in tutorial pages or embedding Prolog
queries that can be modified and re-evaluated in documents that describe
data collections.
With SWISH (SWI-Prolog for Sharing), we provide this
technology for (SWI-)Prolog. SWISH consists of JavaScript client
(browser) code and a number of Prolog libraries that realise the server
as a Prolog application. The client code consists of a series of
jQuery[<https://jquery.com/>] plugins that deal with
editing source code, managing a shared source repository, entering
queries and rendering answers produced by Prolog. The server-side
libraries serve the overall web application, implement the source store
and support the editor with predicate documentation, templates,
cross-reference results, etc. For executing Prolog queries, SWISH relies
on Pengines (Prolog engines,
<cit.>). A pengine is a Prolog engine that
can be controlled similarly to Prolog running in a terminal using HTTP
requests. The SWISH infrastructure was originally developed as a Prolog
version of JSFiddle. It was later reimplemented as a modular jQuery
based infrastructure aiming at collaborative exploration of data hosted
on a SQL or SPARQL server. This use-case is described in
This article is organised as follows. related describes related
work, which in our case are the systems that have inspired us.
application describes the architecture and components of SWISH.
In applications we describe four applications of the current
system. We conclude with future work and conclusions.
§ RELATED WORK
We are not aware of other initiatives that aim at developing a rich
web-based development environment for Prolog. We do not compare SWISH
with traditional editor or GUI based development environments for Prolog
because web-based environments provide new opportunities and pose new
challenges. Instead, we discuss three applications that have served as
inspiration for SWISH: JSFiddle,[<https://jsfiddle.net/>]
R-Studio[<http://www.rstudio.com/>] and IPython
* As stated, the initial inspiration for SWISH was JSFiddle. Unlike
JSFiddle though, Prolog is executed on the server rather than in the
* R-Studio <cit.> is an interface to the R
statistical package. Although not a web application, it is based on the
Qt webkit framework and uses web based technology in the background.
R-Studio came into the picture when the COMMIT/ project provided a grant
for developing SWISH as a toolkit for analysis of relational (SQL) data.
The R-studio interface has a similar layout as SWISH, providing a source
window, a console and an output plane that typically shows results in
tables or charts.
* IPython Notebook <cit.> allows mixing markdown or
HTML text with Python sources. The rendered Notebook shows the text,
sources and possible results in the form of numbers, tables or charts.
SWISH embodies most of the ideas behind JSFiddle and R-Studio. Embedding
of SWISH in documents is demonstrated in lpn. Interactive
editing of documents that embed SWISH is discussed in future work
Both R-Studio and IPython Notebook work on the basis of
authentication (either to the OS or application), after which
any command may be executed. SWISH can operate both as a public service
granting access to non-intrusive queries and as an authenticated service
to run arbitrary queries, for example for maintenance purposes. See
§ THE SWISH APPLICATION
SWISH consists of two parts. The client side, running in a
browser, is implemented as a series of jQuery plugins, using Bootstrap
for styling and RequireJS[<http://requirejs.org/>] for
package management. The server side is completely implemented in
SWI-Prolog <cit.>. It builds on top of
the SWI-Prolog HTTP server libraries, the Pengines library and the IDE
support libraries that provide data for auto completion, documentation
and highlighting.
In the following sections we describe SWISH in terms of interface
components, where we discuss the requirements, the user aspects, the
client code and supporting server functionality for each component.
First, we provide a screendump that illustrates the main components
in swish.
Screendump of SWISH. The left pane shows the source code,
while the top-right pane shows a query runner that
exploits the current selected answer renderer and
buttons on how to continue after the first answer. The
bottom-right pane provides the query editor with access to
example queries stored in the source, query history, apply
solution modifiers, result presentation and a
button to start the query.
§.§ The code editor
A proper editor is the most important component of a usable programming
environment. The editor must support the language, including syntax
highlighting, auto indentation, code completion based on templates and
already existing code and highlighting of errors and warning from the
compiler. The editor is used both for editing the source code and
editing queries.
Prolog is a difficult language to support in code editors due to the
lack of reserved keywords, e.g., the word if in C starts an
if-statement if not embedded in comment or a string, but the word
is in Prolog can refer to the built-in predicate is/2is2, but also
some predicate with a different arity, just a constant, etc. Another
example is X-Y which can both be an arithmetic expression or a
pair as used with e.g., keysort/2keysort2. Next to the lack of keywords
the ability to extend the syntax using new operators complicates the
implementation of syntax support while editing. SWI-Prolog's built-in
Emacs oriented editor resolves this problem by closely integrating
Prolog with the editor. While typing, the current term (clause or
directive) is parsed and analysed in the context of the current file and
the file's imports after each keystroke. If the term has valid syntax,
all tokens are coloured according to their syntactic role as well their
relation to the remainder of the program. For example, a call to a
non-existing predicate is coloured red, a call to a built-in or imported
predicate is blue and a call to a locally defined predicate is black.
The libraries that implement this analysis have been decoupled from the
built-in editor, both to support source colouring for the SWI-Prolog
documentation system PlDoc <cit.> and
ProDT[<http://prodevtools.sourceforge.net>, these libraries
are not yet used by ProDT.]
There are two dominant open source and actively maintained in-browser
code editors available: ACE and CodeMirror. When we started SWISH, ACE
had a very basic Prolog mode and CodeMirror had none. We nevertheless
opted for CodeMirror because its highlighting is based on raw JavaScript
code rather than a regular expression based template language as used
for ACE. The low level implementation allows for a novel highlighting
implementation. The highlighter consists of a JavaScript implemented
Prolog tokeniser. Tokenizing Prolog is sufficient to colour
comments, quoted material (strings, quoted atoms), variables and
constants (atoms and numbers). It is also sufficient to support smart
indentation. As discussed above, it is not sufficient for highlighting
the role played by atoms and compound terms.[An additional
complication is formed by CodeMirror's token-based highlighting which
does not support look-ahead. As a consequence, we must decide on the
colour of e.g., asserta( while we do not know the arity of the
The semantic highlighter classifies, in addition to
the syntactic category such as comment or
variable, terms that define or call predicates
based on cross-referencing the source code.
We provide semantic highlighting as illustrated in
semhighlight by (1) forwarding the changes to the content of
the editor to the server server which maintains a mirror of the editor
content and (2) asking the server to produce a list of semantically
enriched tokens for the source. The tokens are returned as a
list-of-lists, where each inner list represents the tokens for a source
term (clause or directive). Grouping the tokens per source term allows
for incremental update (not yet implemented) as well as
re-synchronisation (see below). For example, a fragment of an enriched
token list may look like this: [ functor, (undefined_goal),
variable (singleton), …]. The JavaScript tokeniser
matches its tokens with this list. If the basic type (e.g., `functor' or
`variable') matches, it uses the enrichment information (e.g.,
`singleton') to decide on the style. If the basic token type does not
match, it highlights the token using the basic syntactical
category and schedules a request to the server for a new list of
enriched tokens. This request is sent if the user pauses typing for 2
seconds. The request is accompanied by the full source if this is small
or the list of changes since the last request if the source is large.
While waiting for up-to-date enriched tokens, the JavaScript
highlighting code heuristically tries to re-synchronise and either uses
the uncertain results or falls back to the plain tokens.
Re-synchronisation checks whether a single added, deleted or modified
token gets the token stream in-sync. If this fails it tries to
re-synchronise on a full-stop with the next clause or directive.
A CodeMirror hover plugin is used to show basic information
about tokens if the pointer hovers over it. For goals, this includes
where the goal is defined (ISO, SWI-Prolog built-in, a library, locally)
and the documentation summary information if available. This information
is requested from the server.
A CodeMirror template plugin is configured from templates
(e.g., atom_length(+Atom, -Length)) extracted from the SWI-Prolog
manual and PlDoc documentation of imported libraries. This plugin shows
a menu of applicable predicates with their templates on
Finally, if the user uses the button to execute a query,
the program is compiled. If the compiler generates errors or warnings,
these are inserted as notes in the source code.
§.§ Source code and query management
As JSFiddle formed the initial inspiration for SWISH, SWISH has a
facility to save the program. The current version of SWISH explicitly
targets the cooperative development of Prolog programs and queries
related to a dataset (see cliopatria). This triggered the
implementation of a more organised storage facility. The server-side
storage module is implemented in Prolog and inspired by GIT. Each file
is versioned independently rather than maintaining the version of a
hierarchy of files. Files can be referenced by content using their GIT
compatible SHA1 hash or by name. The name can be considered a
version head and refers to the latest version with that name.
The file save and load interface provides the following operations:
* Saving a file anonymously, which produces a randomly
generated URL similar to JSFiddle.
* Saving a file by name.
* Saving a new version. The interface shows the available
versions and the modifications.
* Forking a file under a new name. The history remains
linked to the original.
Prolog source files can include other sources on the same
server using :- include(filename)., including the latest
version or :- include(hash). to include a specific version.
Prolog source files can embed example queries using structured
comments, where each sequence from to the matching full stop
token is added to the menu of the query panel (see
lpn). The comment below illustrates a call to append/3append3 embedded
in the source window.
/** <examples>
?- append([one], [two,three], List).
§.§ The query editor
The query editor is based on the same jQuery plugin that
realises the code editor and thus profits from the syntax
highlighting, template insertion and hovering plugins. In addition,
it provides three popup menus:
Examples This menu is filled from the structured comments
described above. The examples menu is shown in lpn.
History This menu provides earlier executed queries.
Solutions This menu embeds an existing query in a meta-call to
alter the result. Currently provided operations are Aggregate
(count all), Order by, Distinct, Limit,
Time and Debug (trace). aggregate shows
how the menu is used to count the solutions of a goal.
The menu can be used to count results,
order them, filter duplicates, etc. The upper runner shows
answers to the query as a table.
§.§ Running a query: runners in the answer pane
The answer pane is a placeholder for runners, where each runner
represents a query. The answer pane provides a menu for operations on
all runners inside it. Provided actions are ,
, and . The query may
be completed, running or waiting for user input. SWISH can manage
multiple active queries at the same time, up to an application defined
maximum (default 3).
Each runner provides its own set of commands to control the specific
query. During execution a runner provides an button.
After query evaluation completes with an answer and more answers may be
available the runner allows for asking the next 1, 10, 100 or 1,000
results or to the query. In addition, the runner shows a
text input field when the application wants to read input and
may show debugger interaction buttons if the tracer is being used (see
A runner can render answers in two modes, the classical Prolog mode or
as a table, similar to what many database interfaces provide. The
`table' mode is particularly appealing for querying datasets (see
aggregate), while the former is more suitable for rendering
small amounts of complex answers such as the chessboard position in
swish. By default, Prolog terms are rendered as structured HTML
objects, where the rendered text is the same as Prolog's writeq/1writeq1
The server can provide rendering libraries that render Prolog
terms using dedicated HTML. In swish, the `chess' renderer is
loaded due to the :- use_rendering(chess) directive. The `chess'
renderer translates a list of length $N$ holding integers in the range
$1..N$ as a chessboard with queens. In addition to the chess rendering
library, SWISH provides rendering libraries for sudoku puzzles, parse
trees and tables. The ClioPatria version adds a renderer for RDF
resources that renders the resource more compactly and provides a
hyperlink for obtaining details. If a term can be rendered in multiple
ways, the interface provides a hover menu to select between the
alternatives. render illustrates this functionality. The render
facility is similar to the Prolog portray/1portray1 hook that allows changing
the result printed for terms with a specific shape. However, it can
exploit the full potential of HTML (or SVG) and the interface allow for
switching the selected rendering.
A rendering library is a module that must define a non-terminal (grammar
rule) term_rendering//3term_rendering3, calling
to produce HTML from the Prolog input term, a list of variable bindings
(Name = Variable) and user provided options. In the current version, new
rendering modules must be loaded into the SWISH server and cannot be
created by the SWISH user.
With the `chess' render library, a list of integers is
interpreted as queens on a chessboard. The user can
select rendering as a `Prolog term' to see the actual
§.§.§ Server side execution of the query
Server-side execution of a query is supported by the Pengines
<cit.> library. The Pengines library allows
for creating a Prolog engine represented by a Prolog thread. Optionally,
the pengine is handed a Prolog program that is loaded into the pengine's
workspace (program space). The workspace is a temporary module that is
disposed of after the pengine terminates. The pengine may be asked
questions through HTTP queries, similar to a traditional Prolog user
interacting with Prolog running in a
If the SWISH user hits the button, the content of the
source pane is used to create a new pengine. Subsequently, the content
of the query pane is sent as the one and only query that will be
executed by the pengine.[Pengines can execute multiple queries.
We do not use this facility because a fresh pengine starts in a
predictable state (standard operators, empty dynamic database).] Before
execution, the query is verified to be safe, unless sandboxing is
disabled (see cliopatria). The sandbox component is discussed
The pengine's default working module may be pre-loaded with code. SWISH
uses this facility to redefine the Prolog I/O predicates such as read/1read1,
write/1write1, format1,2,3, etc. The ClioPatria version
(cliopatria) also preloads the RDF libraries, so users can run
queries on the RDF database without explicitly importing the required
§.§.§ Sandboxing queries
A Prolog environment contains global state in the form of loaded
modules, defined operators, dynamic predicates, etc. Prolog exposes a
rich and potentially dangerous interface to the operating system. For an
anonymous services, we want each query to start in a well defined state
and we must ensure that execution of the query does not make unwanted
changes to the hosting computer or leaks sensitive information.
Both for education purposes and data analysis one can write meaningful
programs without making permanent changes to the server or the server's
filesystem. That is where the sandbox library comes in. The sandbox
library is active while loading the source, where it refuses to add
clauses to other modules than the pengine's workspace and where it only
accepts a restricted set of directives, also aimed at keeping
all changes local to the workspace. Prior to execution, the sandbox
unfolds the query and compares all reachable goals with a whitelist. The
whitelist contains all side-effect free built-in Prolog predicates, safe
meta-predicates (e.g., findall/3findall3) and allows for using the dynamic
database, provided that the head of the affected predicate is not
module-qualified (and thus the referenced predicate lives in the
temporary program space of the Pengine) and the body is safe. It does
not allow for cross-module calls (Module:Goal) to
private predicates and does not provide access to
object-enumeration predicates such as current_atom/1current_atom1,
current_predicate/1current_predicate1, etc., both to avoid leaking sensitive information.
The sandbox test fails under one of these conditions:
* It discovers a (meta-) goal for which it cannot deduce the
called code. The traditional example is read(X), call(X).
If such a goal is encountered, it signals an instantiation
error, together with a trace that explains how the insufficiently
instantiated goal can be reached. Note that it can deal with normal
high-order predicates if the meta-argument is specified. For
example, the following goal is accepted as safe.
?- maplist(plus(1), [1,2,3])
* It discovers a goal that is not whitelisted. In this case
it signals a permission error, again accompanied with
a trace that explains how the goal can be reached. Note that
pure Prolog predicates are unfolded, also if it concerns predicates
from the libraries or belonging to the set of built in predicates.
* It discovers a cross-module (M:Goal) call to a predicate
that is not public. Normally, SWI-Prolog, in the tradition of
Quintus Prolog, allows for this. Allowing it in SWISH would imply
that no data can be kept secret. With this limitation, libraries can
keep data in local dynamic predicates that remain invisible to
non-authorised users.
§.§.§ Debugging
The SWISH debugger is based on the traditional 4-port debugging model
for Prolog. tracer shows the tracer in action on sublist/2sublist2 from
the Lists example source. The debugger was triggered by a
break-point on line 10 set by clicking on the line-number in the code
editor. The debugging interaction is deliberately kept simple and
similar to traditional programming environments. A retry button
is added to the commonly seen `step into', `step over' and `step out'
for highlighting the unique feature of Prolog to re-evaluate a goal.
Debugging applications in SWISH
§ PORTABILITY
The SWISH client libraries are based on mature and well maintained
JavaScript libraries. The client runs all modern major browsers with
HTML5, CSS and JavaScript support. It is frequently tested on FireFox,
Chrome, Safari and Internet Explorer 11.
The server code is basically non-portable. Many of the required
libraries and features are shared with at least one other Prolog
implementation, but none is capable to support the full range. Below we
summarise the main problems.
* The scale of the involved Prolog libraries demands for
a closely compatible Prolog module system. Probably only
SICStus and YAP can be used without significant rewrites.
* The HTTP server libraries are heavily based on C code
that interacts with the SWI-Prolog foreign language interface
to Prolog streams. YAP has copied the low-level libraries and is
capable to run (an old version of) these libraries.
* The Pengines library depends on the HTTP library and
the multi-thread interface. The SWI-Prolog thread API is
also provided by YAP and XSB.
* The sandbox library (sandbox) assumes that
whitelisted predicates are indeed safe. This requires robust
handling of invalid calls and resource overflows. Few Prolog
systems can satisfy this requirement. SICStus Prolog would
be a candidate, but SICStus does not support multi-threading.
* The semantic syntax highlighting depends on detailed
source layout information provided by read_term/3read_term3. SWI-Prolog's
support for term layout is an extended version of the Quintus
Prolog term layout functionality.
* Significant parts of the code rely on SWI-Prolog version 7
extensions, notably the dict and string types
that facilitate a natural mapping between Prolog and JSON data.
From the above list it should be clear that a fully functional port of
SWISH to another Prolog system is not immediately feasible. YAP probably
comes closest but still requires a significant amount of work.
There is a more realistic scenario though. In this setup, SWI-Prolog
provides the web interface and most of the development tools and another
language, not even necessarily Prolog, provides the query solving. The
interface between the two can be based on interprocess communication or,
if the target system is robust, safe and capable of supporting
threads, by linking the target system into the process and using the
C interface for communication.
§ APPLICATIONS
In this section we describe and evaluate four publicly available SWISH
applications. All these services are regularly updated to run the latest
version of SWISH and SWI-Prolog.
§.§ SWISH
SWISH[<http://swish.swi-prolog.org>] runs a plain publicly
accessible copy of SWISH that allows running sandboxed (see
sandbox) Prolog programs. The server has collected 10,800
programs between September 29, 2014 and June 2, 2015. Over the month May
2015, it has executed 258,809 Prolog queries. The web site is regularly
used by users of the ##prolog IRC channel to discuss programming
solutions and is in active use for education.[Steve Matuszek,
UMBC (via e-mail: “Thank you very much for this fantastic resource! I
used it while teaching Prolog this semester, and it really helped
tighten the loop for my students. We spent zero time on tool
installation and other overhead, and all the time on understanding the
concepts. I even had them turn their assignments in via SWISH, with
their test queries in the examples block.”]
§.§ ClioPatria
ClioPatria is a semantic web (RDF) framework for SWI-Prolog. It consists
of an RDF triple store, a SPARQL server and a web frontend to manage the
server and explore the data in the RDF store. ClioPatria can be extended
using cpacks (ClioPatria pack or plugin). SWISH is available
as a ClioPatria
and makes the Prolog shell available for querying as
well as maintenance tasks. Without login, user can run side-effect free
queries over the RDF data stored in ClioPatria's RDF database. After
login with administrative rights, the sandbox limitations are lifted and
the Prolog shell can be used to perform maintenance tasks on the RDF
data such as data transformation, cleanup, etc., as well as program
maintenance tasks such as reloading modified source files.
SWISH has been used in the Talk Of Europe creative
camp[<http://www.talkofeurope.eu/>] to explore data on the
speeches in the European
parliament.[<http://purl.org/linkedpolitics>] Although
still immature, users appreciated the ability to define more efficient
and expressive queries than provided by the SPARQL query interface.
Above all, the ability to save and share programs that perform
interesting tasks on the data was frequently used, in particular to seek
help fixing queries.
§.§ Learn Prolog Now!
Learn Prolog Now![<http://www.learnprolognow.org>] is an
online version of a Prolog book by Patrick Blackburn, Johan Bos, and
Kristina Striegnitz <cit.>. We established a proof of
concept that embeds SWISH in the online course
material.[<http://lpn.swi-prolog.org>] It is realised as a
Prolog hosted proxy that fetches the pages from the main site
and serves the enhanced pages to the user. The proxy identifies and
classifies the code fragments in terms of `source code' `queries' and
dependencies. Next, it adds a button to the source fragments that, when
pressed, replaces the HTML element with in
running SWISH filled with the source while the example queries are added
to the menu (lpn). The queries are executed
by <http://swish.swi-prolog.org>. Program and queries are
transferred using the following HTTP parameters: code (the
source code), background (source code that is loaded into the
pengine but not visible in the editor), examples (queries that
appear in the menu) and q (the initial query).
The proxy server served 19,700 pages during May 2015.
Screendump of Learn Prolog Now with opened SWISH instance
that shows the collected source as well as example queries
from the subsequent text that are classified as relating to
this source. The embedded SWISH provides all functionality
available in the stand-alone SWISH. If the user presses the
close button, SWISH will be removed and the original code
§.§ on SWISH
on SWISH[<http://cplint.lamping.unife.it/>]
is a web application based on SWISH for reasoning with probabilistic logic programs under the distribution semantics. The Prolog source window
is used to write a logic program with annotated disjunction. A query in the form of a ground atom is answered by returning its probability of being true in the program. The computation of the probability is done with the system <cit.> in the server using Pengines. The input program is translated into an internal representation using source to source transformation.
SWISH was modified only in the JavaScript code for the runner. The source code is prepended code for loading the library and enabling the source to source transformation while the query is wrapped into a call of the inference predicate. This call has a variable argument which will hold the probability and will be shown to the user in the answer pane.
§.§ TRILL on SWISH
TRILL on SWISH[<http://trill.lamping.unife.it/>] is a
probabilistic OWL reasoner that reuses SWISH. As SWISH for ClioPatria,
described in cliopatria, it is a ClioPatria cpack. The
Prolog source window is replaced by an RDF/XML editor window that can be
used to upload an OWL ontology while the query editor can be used to
pose Prolog queries against the OWL ontology. The probability of queries
is computed using TRILL <cit.>, a reasoner in Prolog
that adopts the distribution semantics for probabilistic description
Also for TRILL on SWISH we had only to modify the JavaScript code for the runner. The source code sent to the Pengine is obtained by adding Prolog code for parsing an RDF/XML string, by calling the parsing predicate and by wrapping the query in a meta-call that performs syntactic checks for misspellings.
§ FUTURE WORK
Although definitely usable in its current state, SWISH is work in
progress. We are confident that the basic component selection and
organisation of the server and client code are stable. More work is
needed to improve the current system. Notably the semantic highlighting
is not yet perfect and often fails to degrade gradually if the server
side annotation does not match the client tokens perfectly. The
Pengine's sandbox protection is often too restrictive, while several
security flaws have been reported and fixed already. It is, and probably
always will be, advised to run public SWISH-enabled services in an
operating system sandbox. The current server suffers from memory leaks
and stability problems. Although the situation has improved
significantly, the main demo server needs to be restarted about once a
week.[A restart of the server has only small consequences to
active users. Open queries are killed. The source code mirror is lost,
but automatically recovered if the client asks for a new set of
semantically enriched tokens.]
We foresee several extensions to SWISH that will improve current
applications and enable new opportunities for deploying SWISH.
* ClioPatria's authorised usage of SWISH shows some of
the potential for controlling servers or embedded Prolog engines.
In addition to small temporary Prolog programs, we would like to
be able to edit existing and create new Prolog modules as well as
pages in other languages, such as JavaScript, HTML and CSS. Full
editing capabilities would allow for shared development of server
software without shell access to the server on which SWISH enabled
software is running.
* Multi-document editing can enhance the sandboxed SWISH
application by providing input and output documents. Compare this
to TRILL (trill) using an RDF/XML document as input.
* We plan to provide a markdown-based format specifically
for writing tutorials and well as dataset analysis documents. The first
will look like the Learn Prolog Now! example discussed in lpn.
For the second, we envision a document with embedded code and query
fragments, where the query fragments produce tables or charts. This
is similar to IPython Notebook.
* Turn <http://swish.swi-prolog.org> into a reliable
and scalable resource. Examine the possibility for schools to
instantiate a private version that is preloaded with course material and
§ CONCLUSION
This article presented SWISH, SWI-Prolog for Sharing.
SWISH provides a web-enabled interface to Prolog that is based on ideas
from JSFiddle, R-Studio and IPython Notebook. It consists of a
JavaScript client side, while the server side is based on SWI-Prolog's
HTTP and Pengines (Prolog engines) libraries. SWISH can be deployed in
many settings, such as education, data analysis and server development
and maintenance. SWISH as a whole is tied to SWI-Prolog, but other
languages, not even limited to Prolog, could be controlled from
SWI-Prolog. SWISH is made available as open source and can be downloaded
from github.[<https://github.com/SWI-Prolog/swish>]
§.§ Acknowledgements
The development of SWISH was supported by the Dutch national program
|
1511.01072
|
Department of Chemistry and Biochemistry, Queens College, City University of New York, 65-30 Kissena Boulevard, Queens, New York 11367[mailing address] & PhD programs in Chemistry and Physics, and Initiative for the Theoretical Sciences, Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY 10016
Department of Chemistry, James Franck Institute, Institute for Biophysical Dynamics and Computation Institute, University of Chicago, 5735 S. Ellis Avenue, Chicago, Illinois 60637
The definition of the classical transition state theory (TST) as a $t\rightarrow 0_+$ limit of the flux-side time correlation function relies on the assumption that simultaneous measurement of population and flux is a well defined physical process. However, the noncommutativity of the two measurements in quantum mechanics makes the extension of such a concept to the quantum regime impossible. For this reason, quantum TST (QTST) has been generally accepted as any kind of quantum rate theory reproducing the TST in the classical limit, and there has been a broad consensus that no unique QTST retaining all the properties of TST can be defined. Contrary to this widely held view, Hele and Althorpe (HA) [J. Chem. Phys. 138, 084108 (2013)] recently suggested that a true QTST can be defined as the exact $t\rightarrow 0_+$ limit of a certain kind of quantum flux-side time correlation function and that it is equivalent to the ring polymer molecular dynamics (RPMD) TST. This work seeks to question and clarify certain assumptions underlying these suggestions and their implications. First, the time correlation function used by HA as a starting expression is not related to the kinetic rate constant by virtue of linear response theory, which is the first important step in relating a $t=0_+$ limit to a physically measurable rate. Second, a theoretical analysis calls into question a key step in HA's proof which appears not to rely on an exact quantum mechanical identity. The correction of this makes the true $t=0_+$ limit of HA's QTST different from the RPMD-TST rate expression, but rather equal to the well-known path integral quantum transition state theory rate expression for the case of centroid dividing surface. An alternative quantum rate expression is then formulated starting from the linear response theory and by applying a recently developed formalism of real time dynamics of imaginary time path integrals [S. Jang, A. V. Sinitskiy, and G. A. Voth, J. Chem. Phys. 140, 154103 (2014)]. It is shown that the $t\rightarrow 0_+ $ limit of the new rate expression vanishes in the exact quantum limit.
§ INTRODUCTION
How to extend the transition state theory (TST)<cit.> to the quantum regime has been a long standing theoretical challenge.<cit.> Within the classical mechanics, the TST is a well defined theory with firmly established computational methods and simulation protocols. However, the very concept of measuring flux at a localized dividing surface is at odds with the quantum mechanical uncertainty principle. Moreover, the notion of sorting out “trajectories" that do not recross, which allows the TST to be the upper bound of a true barrier crossing rate, is difficult to envision quantum mechanically. In fact, it does not seem clear whether a practical quantum TST (QTST) that translates all the assumptions of the TST to quantum regime can be developed at all.
While genuine QTSTs serving as a rigorous upper bound for quantum barrier crossing rates have been developed,<cit.> they are either difficult to implement or lack quantitative accuracy. Thus, here we adopt a loose definition of QTST as a quantum barrier crossing rate theory approaching the TST in the classical limit. Various QTSTs that allow practical calculations have been developed based on a wide range of theoretical approaches.
One approach is to invoke approximate time dependent quantum dynamics<cit.> near the barrier top, which results in time dependent rates, and to define the QTST as the steady state limit of the time dependent rate expression. Another approach is to abandon explicit consideration of the dynamics from the outset and to extract the rate from the quantum partition function representing metastable reactant states.<cit.> Justification for this latter approach can be made through analytic continuation of the partition function to the complex time domain or by an argument of detailed balance. Theoretical formulations unifying the two approaches have also been developed.<cit.>
Recently, Hele and Althorpe (HA) proposed a new formulation of QTST,<cit.> hereafter termed as HA-QTST, and suggested that they have proven its equivalence to the ring polymer molecular dynamics TST (RPMD-TST).<cit.> They argued that HA-QTST is a true QTST in the sense that it corresponds to an exact $t\rightarrow 0_+$ limit of a quantum flux-side correlation function. A follow-up work<cit.> presented another formal analysis suggesting again that the HA-QTST is exact under a condition of no recrossing, which however was defined in a formal way without a physical definition of quantum mechanical recrossing. A careful theoretical examination of this overall formalism issue is therefore important for our general understanding of QTST.
A distinctive aspect of HA-QTST is that the formulation starts from a special kind of quantum time correlation function which is apparently constructed based on a mathematical consideration of its $t\rightarrow 0_+$ limit, the physical basis of which is not clear. In addition, as will be analyzed in detail in Sec. III, HA's proof for the equivalence between the $t\rightarrow 0_+$ limit of their quantum correlation function and the RPMD-TST<cit.> rate can evidently only be understood through an incorrect or at best approximate application of a quantum mechanical identity. When this is corrected, we find that the true limit of HA-QTST for the case of centroid dividing surface becomes identical to the well-known path integral quantum transition theory (PI-QTST) expression above the crossover temperature,<cit.> which can be understood in a much simpler manner<cit.> and amounts to classical approximation for the dynamical factor. Thus, as will be explained in Sec. III, the implications<cit.> of HA-QTST need to be reassessed.
As a formalism alternative to HA-QTST, we then present a new path integral based quantum rate expression, starting from the well established Yamamoto expression<cit.> for the exact quantum rate based on the linear response theory,<cit.> and employing a recently developed quantum dynamics formalism,<cit.> hereafter called as real time dynamics of imaginary time path integral (RDIP). In the classical limit, the $t=0_+$ limit of our rate expression becomes equal to classical TST. However, if the quantum limit is taken first, the $t=0_+$ limit of the rate expression can be shown to be zero due to quantum delocalization. This is consistent with other work,<cit.> and confirms the assessment<cit.> that constructing a rigorous QTST as the $t=0_+$ limit of a quantum dynamics is likely to be impossible.
The remainder of this paper is organized as follows. Section II provides a review of the linear response theory as a preliminary step. Section III presents a detailed analysis of HA-QTST.
Then, in Sec. IV, we develop a new formulation of QTST employing the RDIP approach. Sec. V provides concluding remarks.
§ TIME DEPENDENT RATE EXPRESSION BASED ON QUANTUM LINEAR RESPONSE THEORY
A schematic of free energy profile along the reaction coordinate, $q$. The reactant is denoted as $a$ and the product is denoted as $b$. $q=d$ is the dividing surface between the two.
As a preliminary step, we here provide a short review of the exact quantum rate expression<cit.> based on the linear response theory.<cit.>This will serve as the basis for the discussion in Sec. III and the formulation developed in Sec. IV.
Let us consider an activated rate process from a reactant ($a$) to a product ($b$). Figure 1 shows a schematic of the free energy profile. Here, we assume that the reaction occurs in a one dimensional coordinate $q$, but the formalism can be readily extended to multidimensional situation. The population functions of $a$ and $b$, denoted here as $h_a(q)$ and $h_b(q)$, respectively, satisfy the condition of $h_{a} (q) +h_{b} (q)=1$.
We here make the simplest and well known choice of $h_b(q)=\Theta (q-d)$, where $\Theta$ is the step function and $d$ is the dividing surface (point for one dimensional case) between $a$ and $b$. The quantum mechanical population operators are then defined as $\hat h_a=h_a(\hat q)$ and $\hat h_b= h_b(\hat q)$, and the flux operator from $a$ to $b$ can be expressed as
F̂=i/ħ [Ĥ,h_b (q̂)]=p̂/2m h_b'(q̂)+h_b'(q̂)p̂/2m ,
where $\hat p$ is the momentum operator conjugate to $\hat q$, $m$ is the mass of the particle, and $h_b'(q)=\delta (q-d)$. Starting from the linear response theory<cit.> and the assumption of rate behavior in which the longest time scale of the system is the rate of interconversion from wells $a$ to $b$, as detailed in Appendix A, on can obtain the following expression for the time dependent forward rate constant from $a$ to $b$:
k(t)=1/Z_a 1/β ∫_0^βdλ Tr { e^-(β-λ)Ĥ ĥ_a e^-λĤ F̂(t)} ,
where $\beta=1/(k_BT)$ and $Z_a=Tr\left \{e^{-\beta \hat H}\hat h_a \right\}$, the reactant partition function. The operator $\hat F(t)$ in the above expression is the time dependent flux operator defined as
F̂(t) = e^iĤt/ħF̂ e^-iĤt/ħ
= p̂(t)/2m h_b'(q̂(t))+h_b'(q̂(t))p̂(t)/2m ,
where $\hat p(t)=e^{i\hat Ht/\hbar}\hat p e^{-i\hat H t/\hbar}$ and $\hat q(t)=e^{i\hat Ht/\hbar} \hat q e^{-i\hat H t/\hbar}$. Utilizing the fact that $\hat F(t)=i[\hat H, \hat h_b(t)]/\hbar$ and the cyclic invariance of the trace operation,
we can rewrite Eq. (<ref>) as follows:
k(t) = -1/Z_a 1/β ∫_0^βdλTr { e^-(β-λ)Ĥ i/ħ[Ĥ,ĥ_a] e^-λĤ ĥ_b(q̂(t))}
= 1/Z_a 1/β ∫_0^βdλTr { e^-(β-λ)Ĥ F̂ e^-λĤ ĥ_b(q̂(t))} ,
where, in the second equality, the fact that $\hat F=-i[\hat H,\hat h_a]/\hbar$ has been used.
For direct quantum dynamics simulation, Eq. (<ref>) may be more advantageous than Eq. (<ref>) because it allows sampling of trajectories starting only from the barrier region. However, the evaluation of the exact quantum flux operator still remains a challenging task.
It can be shown that the exact $t=0_+$ limit of Eq. (<ref>) or (<ref>) is zero, which is a manifestation of fundamental quantum principles. Since the population and flux operators do not commute, measurement of the flux right after the preparation of the system, i.e., population measurement, results in both positive and negative fluxes with an equal probability. Thus, the structure of the linear response theory dictates that the exact $t=0_+$ limit of the reactive flux is zero in the quantum regime. Therefore, any quantum rate formulation based on the exact $t=0_+$ limit should address how this fundamental quantum mechanical property can be somehow avoided.
§ QTST BY HELE AND ALTHORPE
One feature that sets the HA-QTST<cit.> apart from other earlier theories is that, instead of Eq. (<ref>) or (<ref>), the rate formulation starts from a quantum time correlation function they have termed as generalized Kubo transform of the side-side correlation function (GKSCF).
The apparent motivation<cit.> for introducing GKSCF is to identify a time correlation function with nonzero $t=0_+$ limit, and HA has constructed it starting from a complex time path integral representation of the quantum partition function $Z=Tr\{e^{-\beta \hat H}\}$. In our notation, as detailed in Appendix B, the GKSCF can be expressed as
C̃_ss(t) = ∫dq__1⋯∫dq__P∫dη__1⋯∫dη__P
×∫dz_1⋯∫dz__P ρ(q,)𝒢(q, ,z;t) h_b(f(q)) h_b (f(z)) ,
where ${\bf q}$ and $\boldeta$ are path coordinates defined by Eqs. (<ref>) and (<ref>), $f({\bf q})$ is a symmetric function of $q_k$'s defined in the space of imaginary time paths. The density $\rho({\bf q},\boldeta)$ is defined by Eq. (<ref>) and represents the canonical density operator. The term ${\mathcal G}({\bf q},\boldeta,{\bf z};t)$ is defined by Eq. (<ref>) and retains all the dynamical information. Note that we have incorporated the position of the dividing surface $d$ into the definition of the population function, unlike the original formulation by HA.<cit.> Thus, $f({\bf q})$ in Eq. (<ref>) is independent of $d$.
Although mathematically well defined, the physical meaning of GKSCF is less clear. HA rendered the $t=0_+$ limit of the time derivative of the correlation function to be non-zero by making the flux and side dividing surfaces equal.<cit.> However, this procedure is not possible if one adheres to the rules of quantum mechanics and the requirement to express the rate in an exact and general form, e.g., from the linear response theory [Eqs. (<ref>)-(<ref>)]. This is also illustrated by the fact that Fig. 2 of Ref. hele-jcp138 depicts the manipulation of the Feynman diagrams for the correlation function, which appear to be disallowed due to the non-commutation of the operators.
It is possible to further analyze the GKSCF as follows.
From the definition of ${\mathcal G}({\bf q}, \boldeta,{\bf z};t)$ given by Eq. (<ref>), it is clear that
𝒢(q, ,z;0)=δ(q-z)δ() .
Thus, for $t=0$, Eq. (<ref>) becomes
C̃_ss(0) = ∫dq__1⋯∫dq__P ρ(q,0) h_b(f(q)) ,
where $\rho({\bf q},0)=\prod_{k=1}^P \langle q_k|e^{-\epsilon \hat H}|q_{k+1}\rangle$, with $\epsilon=\beta/P$ and the cyclic boundary condition of $q_{_{P+1}}=q_1$.
Note that the population function $h_b$ is a nonlinear function. As a result, Eq. (<ref>) does not have any corresponding quantum mechanical operator expression. This is because the imaginary time in the Feynman path integral is a fictitious time, each labeling a different realization of the system in the quantum canonical ensemble. Any function that depends nonlinearly on the collection of the imaginary time path coordinates mixes up different realizations of the ensemble in a nonlinear manner. Such a function has temperature as an implicit variable, and does not have any analogue for the case of a pure quantum state. Thus, $h_b\left (f({\bf q})\right)$ cannot be related to any genuine physical observable except in the classical limit or zero temperature limit. This is true even for the case where $f({\bf q})=q_0=(q_1+\cdots+q_{_p})/P$, namely the imaginary time path centroid. Therefore, it is not possible to establish unambiguous relationship between $\tilde C_{ss}(0)$ (and thus $\tilde C_{ss}(t)$) with physically measurable quantities. This is in strong contrast with the physical basis for Eq. (<ref>) or (<ref>), as demonstrated in detail in Appendix A. Thus, it appears that the starting expression used by HA<cit.> lacks a clear physical justification unlike the linear response rate expression.<cit.> This does not necessarily mean that the time derivative of $\tilde C_{ss}(t)$ cannot be used as a quantum rate. However, no fundamental physical basis appears to exist which suggests that such a limit should be considered as a true QTST.<cit.> With this issue clarified, we can now provide a further analysis of HA-QTST to discuss another important issue.
As the next step, HA calculates the time derivative of Eq. (<ref>) as follows:
= -∫dq__1⋯∫dq__P∫dη__1⋯∫dη__Pρ(q,)
×∫dz_1⋯∫dz__P h_b(f(q))h_b(f(z)) d/dt𝒢(q, ,z;t) ,
where the detailed expression for $d{\mathcal G}({\bf q}, \boldeta, {\bf z};t)/dt$ is given by Eq. (<ref>) in Appendix C.
Through partial integration of the resulting expression for Eq. (<ref>), as detailed in Appendix C and using the definition that $h_b$ is a step function, we obtain the following expression
C̃_fs(t) = ∫dq__1⋯∫dq__P∫dη__1⋯∫dη__P
×∫dz_1⋯∫dz__P ρ(q,) 𝒢(q, ,z;t)
×Θ(f(z)-d) δ(f(q)-d)
×1/2m∑_k=1^P ∂f(q)/∂q_k { p̅_+(q_k-1-η_k-1/2,q_k+η_k/2;ϵ) .
. +p̅_-(q_k-η_k/2,q_k+1+η_k+1/2;ϵ) } ,
where we have introduced new imaginary time momentum averages as follows:
p̅_+(x',x”;ϵ)= ⟨x'|e^-ϵĤp̂|x”⟩/⟨x'|e^-ϵĤ|x”⟩ ,
p̅_-(x',x”;ϵ)=⟨x'|p̂ e^-ϵĤ |x”⟩/⟨x'|e^-ϵĤ |x”⟩ .
Equation (<ref>) can be shown to be equivalent to Eq. (31) of Ref. hele-jcp138. At $t=0$, it becomes
C̃_fs(0) = ∫dq__1⋯∫dq__P ρ(q,0) Θ(f(q)-d) δ(f(q)-d)
×1/2m∑_k=1^P ∂f(q)/∂q_k { p̅_+(q_k-1,q_k;ϵ) +p̅_-(q_k,q_k+1;ϵ) } ,
where the identity of Eq. (<ref>) has been used. This expression can be shown to be zero, as detailed in Appendix D, due to the cyclic symmetry of the imaginary time path integral.
Invoking an analogy to the classical TST, HA defines their QTST as the $t=0_+$ limit of Eq. (<ref>). For this, they use a short time approximation for real time propagators, $e^{\pm it\hat H/\hbar}$'s, within the definition of ${\mathcal G}({\bf q}, \boldeta, {\bf z};t)$, Eq. (<ref>). This procedure leads to classical-like real time momenta $p_k$'s satisfying the following relation:[However, it should be made clear that $p_k$ is simply an integrand in the unit of momentum and that no apparent theoretical justification is available for the suggestion that it is a dynamical variable following the RPMD equation of motion as in the RPMD-TST.<cit.>]
z_k=q_k+p_k/mt .
HA then replaces the integrands $z_k$'s in Eq. (<ref>) with $p_k$'s, which is mathematically valid as long as $t>0$. Taking the limit of $t=0_+$, the resulting expression can be shown to be
= 1/(2πħ)^P∫dq__1⋯∫dq__P∫d η__1⋯∫d η__P
×∫dp_1⋯∫dp__P ρ(q,) exp(ip ·/ħ)
×Θ(f(q+p/m0_+)-d) δ(f(q)-d)
×1/2m∑_k=1^P ∂f(q)/∂q_k { p̅_+(q_k-1-η_k-1/2,q_k+η_k/2;ϵ) .
. +p̅_-(q_k-η_k/2,q_k+1+η_k+1/2;ϵ) } .
Based on the following expansion
f(q+p/m0_+)=f(q)+p/m·∇f(q)0_+ ,
and employing the constraint of $f({\bf q})=d$, one can make the following replacement:
Θ(f(q+p/m0_+)-d) =Θ(p·∇f(q) ) .
Thus, Eq. (<ref>) can be expressed as
k_HAZ_a= 1/(2πħ)^P∫dq__1⋯∫dq__P∫d η__1⋯∫d η__P
×∫dp_1⋯∫dp__P ρ(q,) exp(ip ·/ħ)
×Θ(p·∇f(q) ) δ(f (q)-d)
×1/2m∑_k=1^P ∂f(q)/∂q_k { p̅_+(q_k-1-η_k-1/2,q_k+η_k/2;ϵ) .
. +p̅_-(q_k-η_k/2,q_k+1+η_k+1/2;ϵ) } .
The above expression, or Eq. (<ref>), is similar to Eq. (38) of Ref. hele-jcp138, which serves as the key step in HA's suggestion<cit.> that their result constitutes a derivation of RPMD-TST<cit.> However, there is an important difference, which is clarified below. The difference lies in the fact that Eq. (38) of Ref. hele-jcp138 amounts to replacing $\bar p_+$/$\bar p_-$ in Eq. (<ref>) with $p_k$. In other words, HA replace<cit.> the average imaginary time momenta defined along the imaginary time paths at $t=0$ with the real time momenta defined by Eq. (<ref>), which results from the short time classical-like approximation for the real time propagator. However, the equations of motion for the real time momenta, $p_k$'s, are as yet undefined. In fact, these simply serve as dummy integrands. Thus, $\bar p_+$/$\bar p_-$ and $p_k$'s correspond to two distinctively different sets of variables, and the interchange between them is not justified unless stated explicitly as being an approximation, which HA did not appear to do.
The procedure noted above appears to have resulted from a mixing of the path integral and operator formulations<cit.> of quantum mechanics. For example, the ring polymer flux operator $\hat F$ defined by Eq. (32) of Ref. hele-jcp138 has its meaning only within the specific convention prescribed by HA in their work.<cit.> Adhering to such a convention<cit.> makes their Eq. (31) equivalent to the $t=0_+$ limit of Eq. (<ref>) of this work. However, at the next stage of formulation, the momentum operator being applied to each position state is replaced with a real time value of momentum, which is not allowed quantum mechanically. Thus, the follow-up steps of using $S({\bf q},{\bf p})$ in Eq. (33) or (38) of Ref. hele-jcp138 cannot be justified quantum mechanically. This is tantamount to assuming that position states are eigenstates of the momentum operators in the ring polymer flux operator, which is not true.[Similar incorrect application can also be identified in going from Eq. (17) to Eq. (21) of Ref. hele-jcp138.] Although it is possible to derive an expression in which the real time momenta appear explicitly through an alternative procedure,[S. C. Althorpe, Private Communication] as detailed in Appendix C, the resulting expression is still different from Eq. (38) of Ref. hele-jcp138.
Despite the concerns over HA's formulation as noted above, it might still be possible for the correctly calculated $t=0_+$ limit of the derivative of GKSCF, which is Eq. (<ref>), to be equivalent to the RPMD-TST rate expression. In order to check this, Eq. (<ref>) can be evaluated further employing a normal mode transformation used by HA.<cit.> Let us introduce $\tilde p_l({\bf q})$ and $\tilde \eta_l ({\bf q})$ such that
p̃_l(q)=∑_k=1^P p_k T_k,l(q) ,
η̃_l(q)=∑_k=1^P η_k T_k,l (q) ,
where $T_{k,0}({\bf q})=B({\bf q})^{-1/2}\partial f({\bf q})/\partial q_k$ with $B({\bf q})=\sum_{k=1}^P (\partial f({\bf q})/\partial q_k)^2$. Other components of $T_{k,l}({\bf q})$ can be determined such that $\sum_{k} T_{k,l}({\bf q}) T_{k,l'}({\bf q})=\delta_{ll'}$. Then, Eq. (<ref>) can be expressed as
k_HAZ_a= 1/(2πħ)^P∫dq__1⋯∫dq__P
×∫d η̃__0(q)⋯∫d η̃__P-1 (q) ∫d̃p_0 (q)⋯∫dp̃__P-1 (q)
×ρ(q,) exp(ip̃ ·/ħ) Θ(p·∇f(q) ) δ(f (q)-d)
×1/2m∑_k=1^P ∂f(q)/∂q_k { p̅_+(q_k-1-η_k-1/2,q_k+η_k/2;ϵ) .
. +p̅_-(q_k-η_k/2,q_k+1+η_k+1/2;ϵ) } .
Integration over $\tilde \eta_k({\bf q})$ for $k\neq 0$ in the above expression can be performed easily because the integral over $\tilde p_k ({\bf q})$ amounts to a Fourier integral expression for the delta function. Thus,
k_HAZ_a= 1/2πħ∫dq__1⋯∫dq__P
×∫d η̃__0(q)∫d̃p_0(q) ρ(q,_0) exp(i p̃_0(q) η̃_0 (q) /ħ)
×Θ(p̃_0 (q)) δ(f (q)-d)
×1/2m∑_k=1^P ∂f(q)/∂q_k { p̅_+(q_k-1-η_0,k-1/2,q_k+η_0,k/2;ϵ) .
. +p̅_-(q_k-η_0,k/2,q_k+1+η_0,k+1/2;ϵ) } ,
where $\boldeta_0=(T_{1,0},\cdots, T_{P,0})\tilde \eta_0({\bf q})$.
As detailed in Appendix D, $\bar p_+$ and $\bar p_-$ defined by Eqs. (<ref>) and (<ref>) can be calculated explicitly, resulting in Eqs. (<ref>) and (<ref>). Employing these expressions and also performing explicit integration over $\tilde p_0({\bf q})$, under the assumption that the integrand vanishes in the limit of $\tilde p_0({\bf q})\rightarrow \infty$, we obtain the following expression:
k_HAZ_a= 1/4πħϵ∫dq__1⋯∫dq__P
×∫d η̃__0(q)ρ(q,_0)δ(f (q)-d)
×1/η̃_0(q)∑_k=1^P ∂f(q)/∂q_k { η_0,k-1(q)+2η_0,k(q)+η_0,k+1(q)/2
-q_k-1+q_k+1 } ,
where $\eta_{0,k}({\bf q})=T_{k,0}\tilde \eta_0({\bf q})$. Taking the average of the above integral with an equivalent one resulting from the following variable transformation $(q_1,\cdots,q_{_P})\rightarrow (q_{_P},\cdots,q_1)$ and $\tilde \eta_0 ({\bf q}) \rightarrow -\tilde \eta_0 ({\bf q})$, we then obtain
k_HAZ_a= 1/4πħϵ∫dq__1⋯∫dq__P
×∫d η̃__0(q)ρ(q,_0)δ(f (q)-d)
×∑_k=1^P ∂f(q)/∂q_k T_k-1,0+2T_k,0+T_k+1,0/2 .
The integration over $\tilde \eta_0({\bf q})$ in the above expression can be done explicitly following the same procedure used by HA.<cit.> Although the resulting expression is finite, which confirms the mathematical motivation behind defining the dividing surfaces of flux and side functions to coincide, the resulting expression is clearly different from the RPMD-TST rate expression.<cit.> This also can be seen easily from the fact that Eq. (<ref>) does not involve any real time momentum variables.
In order to demonstrate the physical implication of the corrected HA-QTST rate expression derived above, let us consider the following case of centroid dividing surface:
f(q)=1/P∑_k=1^Pq_k .For this case, $\partial f/\partial q_k=1/P$, $B=1/P$, $T_{k,0}=1/\sqrt{P}$, and
∑_k=1^P ∂f(q)/∂q_k T_k-1,0+2T_k,0+T_k+1,0/2=2/√(P) .
It is also straightforward to show that
ρ(q,η_0)≈exp{ - mP/2ħ^2 η̃_0^2 } ρ(q,0) .
Inserting the above expressions into Eq. (<ref>) and performing Gaussian integration over $\tilde \eta_0$, we find that
k_HA Z_a≈1/2πħβ ρ_c(d) ,
ρ_c(d)=√(2πβħ^2/m) ∫dq_1 ⋯∫dq__P ρ(q,0) δ(q_0-d) .
It is interesting to note that the above rate expression is exactly equal to the PI-QTST rate expression above the crossover temperature.<cit.> However, this is not surprising considering that HA's formulation relies on a purely classical nature of the short real time dynamics, only within which the definition of $t=0_+$ limit makes sense. Their definition of free energy space of the reactant, for the case of centroid dividing surface, is also the same as that for PI-QTST.<cit.> Thus, our finding above regarding the correct $t=0_+$ limit of the time derivative of the GKSCF implies that the net effect of having the flux and side dividing surfaces coincide and taking the exact $t=0_+$ limit is making classical approximation for the dynamical factor as implied in the above crossover temperature PI-QTST rate expression.<cit.>
In a follow-up paper,<cit.> Althorpe and Hele also suggest that HA-QTST is exact in a certain limit of no-recrossing. However, their conclusion is based on the fact that any effect of a non-ideal initial condition disappears in the steady state limit. In addition, their no-recrossing condition, which is defined as the limit where their generalized flux-side time correlation function becomes independent of time, does not appear to naturally arise from quantum dynamical considerations. Thus, the physical basis for the condition of no-recrossing seems unclear and difficult to verify by independent means.
§ REAL TIME DYNAMICS OF THE IMAGINARY TIME PATH INTEGRAL
Recently, we developed a formalism of RDIP,<cit.> which allows calculation of a general class of nonlinear Kubo-transformed time correlation functions and thus can be used to evaluate the rigorous rate expression based on the linear response theory, Eq. (<ref>).
With the path integral representation of the canonical density operator given by Eq. (<ref>), it is straightforward to show that Eq. (<ref>) can be expressed as
k(t)= 1/Z_a∫dq∫dp U(q,p)h_a,0 (q) F_0(t;q,p) ,
where $U({\bf q},{\bf p})$ is defined by Eq. (<ref>) and
h_a,0(q)=1/P∑_k=1^P h_a (q_k) ,
F_0(t;q, p) =Tr{Ŝ(t;q, p̅) F̂} ,
with $\hat S(t;{\bf q},\bar {\bf p})\equiv e^{-i\hat Ht/\hbar}\hat S({\bf q},\bar {\bf p})e^{i\hat H t/\hbar}$, the time dependent version of $\hat S({\bf q},\bar {\bf p})$ defined by Eq. (<ref>). Note that $\bar {\bf p}$ is the midpoint momentum vector as defined below Eq. (<ref>).
At time $t=0$,
F_0(0;q, p)=F_0(q,p)=Tr { Ŝ (q, p̅) F̂ }
=1/P∑_k=1^P ∫dηJ(q_k, p̅_k;η) ⟨q_k-η/2|F̂|q_k+η/2⟩ .
As detailed in Appendix E, the above expression is equivalent to the following classical-like expression:
F_0(q,p)=1/P ∑_k=1^P p̅_k/m h_b'(q_k) =1/P∑_k=1^P p̅_k/m δ(q_k-d) ,
where the second equality results from our starting assumption that $h_b(q_k)=\Theta(q_k-d)$.
Now, let us introduce ${\bf q}_{cl}(t)$ and ${\bf p}_{cl}(t)$, the path vectors consisting of $p_{k,cl}(t)$'s and $q_{k,cl}(t)$'s, each evolving classically from $\bar p_k$ and $q_k$.
Then, Eq. (<ref>) can be expressed as
F_0(q,p)=. d/dt h_b,0(q_cl(t))|_t=0 .
This motivates the following approximation:
F_0(t;q, p) ≈ F_0(q_cl(t),p_cl(t))
= d/dt h_b,0(q_cl(t)) .
With the above approximation, the rate expression of Eq. (<ref>) can be expressed as
k(t) ≈ 1/Z_a∫dq∫dp U(q,p)h_a,0 (q) F_0(q_cl(t),p_cl(t))
= 1/Z_a∫dq∫dp U(q,p)h_a,0 (q) d/dt h_b,0(q_cl(t)) .
Evaluation of the above expression does not require quantum dynamical time evolution, and is thus feasible.
For the case where $P=1$, the $t=0_+$ limit of this expression is equivalent to the classical TST. However, for the quantum case, it is possible to show that the $t=0_+$ limit of the above expression is of order $1/P$, which vanishes in the exact path integral limit. Detailed proofs are provided in Appendix F. Thus, Eq. (<ref>) satisfies the exact property that the $t=0_+$ limit vanishes, while possibly being amenable for practical calculations. This points to the possibility of developing a path integral expression for the QTST based on the linear response theory which also reproduces the correct behavior in the $t=0_+$ limit. Previous tests<cit.> of the RDIP method for harmonic oscillator was confirmed to be exact even for nonlinear operators. Therefore, the result of the above rate expression is expected to be exact for the case of a quadratic barrier.
§ CONCLUDING REMARKS
The very concept of the TST relies on the principles of classical mechanics. It either assumes that instantaneous back-to-back measurements of population and flux are possible<cit.> or that completely deterministic trajectories<cit.> can be defined in phase space. In the quantum regime, these assumptions become invalid. Therefore, it seems likely that no quantum rate theory retaining all the properties of classical TST can be found. Even with this fundamental limitation, it may still be possible to define a quantum rate as long as the measurement of rate is defined so as to be consistent with the history<cit.> of quantum preparation. Thus, a formulation of quantum rate theory approaching TST in the classical limit, which serves as the definition of QTST here, is suggested to be possible, although it does not necessarily have to be unique.
Two popular frameworks for defining a quantum rate have been the scattering formulation<cit.> and the linear response theory,<cit.> with the former favored in the field of gas phase dynamics and the latter in the field of condensed phase dynamics. In both cases, a physically meaningful QTST can be defined in the steady state limit where the reactant and product population states have completely decohered due to spatial separation or action of environments. Only in the classical limit where the decoherence time becomes effectively zero, can such a steady state limit be replaced with the $t=0_+$ limit. In this sense, the suggestion by HA that a true QTST can be defined as the exact $t=0_+$ limit of a quantum time correlation function and that it also reproduces the rate expression of RPMD-TST, which is approximate in nature even in the $t=0_+$ limit, is unclear. Such a suggestion also conflicts with a significant body of earlier works on how QTST might be defined, and has therefore motivated us to carry out a detailed analysis of HA-QTST.
In this paper, we have identified two significant concerns with the development and assumptions of HA-QTST. First, we note that the GKSCF introduced by HA as the starting point of their QTST is not related to a time correlation function for the observable rate constant<cit.> from the linear response theory.<cit.> This is an important issue because the rate is defined as the $t=0_+$ limit of its time derivative, which should be affected significantly by the nature of the initial expression and the initial physical conditions. Second, we identified an incorrect application of a quantum mechanical identity, namely the replacement of average imaginary time momenta along the imaginary time paths at $t=0$ with the real time momenta variables under the classical approximation at $t=0_+$. The two are independent variables, and such a replacement is not strictly allowed quantum mechanically. When the $t=0_+$ limit is evaluated employing a corrected expression, the resulting $t=0_+$ limit turns out to be the same as the PI-QTST rate expression above the crossover temperature<cit.> for the case of centroid dividing surface. This confirms the equivalence of assumptions of the two approaches. In other words, the HA's formulation amounts to making classical approximation for the quantum dynamical factor and does not provide an exact quantum mechanical derivation of the RPMD-TST rate expression. This conclusion is also consistent with our previous analysis based on the RDIP formulation<cit.> showing that RPMD differs from the exact quantum dynamics even for harmonic oscillators at zero time.
Finally, in Sec. IV, we have presented an alternative path integral approach for evaluating the quantum rate expression based on the exact linear response theory quantum rate expression<cit.> and by employing the formalism of RDIP.<cit.> We have shown that the $t=0_+$ limit of this rate expression becomes zero in the quantum regime, which reproduces the known quantum mechanical behavior in the $t=0_+$ limit. Application of this formalism to the quadratic barrier model is expected to be exact considering its result for harmonic oscillator system.<cit.> Whether this alternative approach will also be amenable to practical calculations of general anharmonic systems will be a focus of future research.
SJ acknowledges the support for this research from the National Science Foundation (CHE-1362926), the Office of Basic Energy Sciences, Department of Energy (DE-SC0001393), and the Camille Dreyfus Teacher Scholar Award. GAV acknowledges the support of the National Science Foundation (NSF) through grant CHE-1465248.
§ REVIEW OF THE LINEAR RESPONSE THEORY FORMULATION OF QUANTUM RATE THEORY
The rate expression based on the quantum linear response theory<cit.> is well established,<cit.> and we here provide a brief review of the derivation for the sake of completeness.
Consider a perturbed Hamiltonian, $\hat H_\gamma =\hat H - \gamma \hat h_a$, where $\gamma$ is a small parameter. Given that this perturbation has been imposed for a long enough time (for $t<0$), the ensemble can be represented by the following (unnormalized) canonical density operator:
ρ̂_γ=exp{-β(Ĥ-γĥ_a)} .
Up to the first order of $\gamma$, this can be approximated as
ρ̂_γ≈e^-βĤ+γ∫_0^βdλ e^-(β-λ) Ĥ ĥ_a e^-λĤ .
Assume that the perturbation of $-\gamma \hat h_a$ disappears for $t\geq0$. Then, the population of the product ($b$) within the linear response theory can be approximated as
p_b(t)≈1/Z+γZ_a (Z_b
+γ∫_0^βdλTr{ e^-(β-λ)Ĥ ĥ_a e^-λĤ e^iĤ t/ħĥ_b e^-iĤ t/ħ} ) ,
where $Z_a=Tr\{e^{-\beta \hat H} \hat h_a\}$ and $Z_b=Tr\{e^{-\beta \hat H} \hat h_b\}$. Expanding the denominator up to the first order and introducing the thermal equilibrium (with respect to the unperturbed Hamiltonian), $\langle \hat h_b\rangle = Z_b/Z=p_{b,eq}$,
p_b(t)≈Z_b/Z(1+γ/Z_b∫_0^βdλ Tr { e^-(β-λ) Ĥĥ_a e^-λĤ δĥ_b(t)}) ,
δĥ_b(t)=e^iĤ t/ħ (ĥ_b-⟨ĥ_b⟩)e^-iĤt/ħ .
=γ/Z∫_0^βdλTr{e^-(β-λ) Ĥ h_a e^-λĤ δĥ_b(t)} .
Taking the time derivative of this,
d/dt δp_b(t)=γ/Z∫_0^βdλ Tr{e^-(β-λ) Ĥĥ_a e^-λĤ F̂(t)} ,
where $\hat F(t)$ is defined by Eq. (<ref>).
On the other hand, given that population changes follow rate behavior near the chemical equilibrium,
d/dtδp_b(t) = k_f p_a(t)-k_b p_b(t)
= k_f(δp_a(t)+p_a,eq)-k_b (-δp_a(t)+p_b,eq)
= (k_f+k_b)δp_a(t)=k_fZ/Z_b δp_a(t) ,
where $k_f$ is the forward rate and $k_b$ is the backward rate, $p_{a,eq}=Z_a/Z$, and the detailed balance condition of $k_f p_{a,eq}=k_b p_{b,eq}$ has been used.
Under the assumption that there exists a time $t_p$ in the plateau region longer than the transient relaxation time but much smaller than the reaction time,
. d/dtδp_b(t)|_t_p≈k_fZ/Z_bδp_a(0) .
Equating Eqs. (<ref>) and (<ref>), we obtain
k_f≈γZ_b/δp_a(0) Z^2 ∫_0^βdλ Tr{e^-(β-λ) Ĥĥ_a e^-λĤ F̂(t_p)} .
While $\delta p_a(0)$ can be evaluated directly from Eq. (<ref>), for the case where the barrier is high enough and the population is dominated by those near the bottoms of the reactant and product wells, $\delta p_a(0)$ can be determined by using the detailed balance condition for the perturbed Hamiltonian.<cit.> In other words, for the perturbed Hamiltonian, $Z_a^\gamma \approx Z_a(1+\beta\gamma)$ whereas $Z_b^\gamma\approx Z_b$. Therefore,
p_a(0)≈Z_a(1+βγ)/Z+Z_aβγ .
δp_a(0)=p_a(0)-p_a,eq= Z_a(1+βγ)/Z+Z_aβγ -Z_a/Z≈βγZ_aZ_b/Z^2 ,
where approximation up to the first order of $\gamma$ has been made in obtaining the second equality. Plugging this expression into Eq. (<ref>), we obtain
k_f≈1/Z_a 1/β ∫_0^βdλ Tr { e^-(β-λ)Ĥ ĥ_a e^-λĤ F̂(t_p)} .
Equation (<ref>) is a general time dependent version of this expression, from which the plateau behavior can be examined directly.
§ PATH INTEGRAL REPRESENTATION IN THE COMPLEX TIME DOMAIN
Consider the following expression for the quantum partition function defined in the complex time domain:
Z=Tr{e^-βĤ} = Tr { e^-ϵĤ e^it Ĥ/ħe^-itĤ/ħ ⋯.
. ×e^-ϵĤ e^it Ĥ/ħe^-itĤ/ħ} .
In the above expression, inserting $\hat 1=\int dq_i' |q_i'\rangle \langle q_i'|$ after each $e^{-\epsilon \hat H}$, $\hat 1=\int dz_i |z_i\rangle \langle z_i|$ after each $e^{it\hat H/\hbar}$, and $\hat 1=\int dq_i''|q_i''\rangle \langle q_i''|$ after each $e^{-it\hat H/\hbar}$, we obtain
× ⋯
×⟨q__P-1”|e^-ϵĤ|q__P'⟩⟨q__P'|e^itĤ/ħ|z__P⟩⟨z__P|e^-itĤ/ħ|q__P”⟩ .
Let us introduce
q_i=q_i'+q_i”/2 ,
η_i=(q_i'-q_i”) .
Then, Eq. (<ref>) can be expressed as follows:
×∫dz_1⋯∫dz__P ρ(q,)𝒢(q, ,z;t)
ρ(q,)=∏_k=1^P⟨q__k-η__k/2|e^-ϵĤ|q__k+1+η__k+1/2 ⟩ ,
𝒢(q, ,z;t)=
∏_k=1^P ⟨q__k+η_k/2|e^itĤ/ħ|z_k⟩⟨z_k|e^-itĤ/ħ|q_k-η_k/2⟩ .
Note that $\int dz_1\cdots \int dz_{_P} {\mathcal G}({\bf q}, \boldeta,{\bf z};t) =\delta (\eta_1)\cdots \delta (\eta_{_P})$.
§ EVALUATION OF EQ. (<REF>)
The time derivative of ${\mathcal G}({\bf q}, \boldeta, {\bf z};t)$ in Eq. (<ref>) can be expressed as
d/dt𝒢(q, ,z;t)
=∑_k=1^Pi/ħ{⟨q_k+η_k/2|Ĥ e^itĤ/ħ|z_k⟩⟨z_k|e^-itĤ/ħ|q_k-η_k/2⟩
- ⟨q_k+η_k/2|e^itĤ/ħ|z_k⟩⟨z_k|e^-itĤ/ħĤ |q_k-η_k/2⟩}
= ∑_k=1^P i/ħ{ (-ħ^2/2m (∂^2/∂x_k^2 - ∂^2/∂y_k^2 ) +V(x_k)-V(y_k))
×∏_j≠k^P⟨x_j|e^itĤ/ħ|z_j⟩⟨z_j|e^-itĤ/ħ|y_j⟩ ,
where $x_k=q_k+\eta_k/2$ and $y_k=q_k-\eta_k/2$.
Then, Eq. (<ref>) can be calculated by employing Eq. (<ref>), performing partial integration for the terms involving $\partial^2/\partial x_k^2$ and $\partial^2/\partial y_k^2$, and utilizing the fact that
×⟨y_1|e^-ϵĤ|x_2⟩⋯⟨y__P-1|e^-ϵĤ|x__P⟩ ,
h_b(f(q))=h_b(f(x+y/2)) .
For example,
∫dq ∫d∫dz ρ(q,) h_b(f(q))h_b(f(z))
×(∂^2/∂x_k^2 ⟨x_k|e^itĤ/ħ|z_k⟩⋯)
=∫dq ∫d∫dz 𝒢(q, ,z;t) h_b(f(z))
×∂^2/∂x_k^2 (ρ(x+y/2,x-y ) h_b(f(x+y/2))) .
In the integrand of the above expression,
∂^2/∂x_k^2 (ρ(x+y/2,x-y ) h_b(f(x+y/2)))
=(∂^2/∂x_k^2 ρ(x+y/2,x-y ) ) h_b(f(x+y/2))
+2(∂/∂x_k ρ(x+y/2,x-y ) ) (∂/∂x_k h_b(f(x+y/2)) )
+ ρ(x+y/2,x-y )(∂^2/∂x_k^2 h_b(f(x+y/2)) ) .
A similar expression can be obtained for the partial derivatives with respect to $y_k$. Note the following identities.
(-ħ^2/2m ∂^2/∂x_k^2 +V(x_k)) ρ(x+y/2,x-y )
= ⟨y__p|e^-ϵĤ|x_1⟩⋯⟨y_k-1|e^-ϵĤ Ĥ|x_k⟩⋯⟨y__P-1|e^-ϵĤ|x__P⟩ .
(-ħ^2/2m ∂^2/∂y_k^2 +V(y_k)) ρ(x+y/2,x-y )
= ⟨y__p|e^-ϵĤ|x_1⟩⋯⟨y_k|Ĥ e^-ϵĤ |x_k+1⟩⋯⟨y__P-1|e^-ϵĤ|x__P⟩ .
When summed over all $k$, the contribution of the above two terms to the integral, Eq. (<ref>), cancel out because $e^{-\epsilon \hat H}$ commutes with $\hat H$. In addition, the term involving the second derivative of $h_b$ with respect to $x_k$ in Eq. (<ref>) is equal to an analogous term involving the second derivative of $h_b$ with respect to $y_k$. Therefore, the contribution of these terms to the integral, Eq. (<ref>) vanishes as well.
As a result,
d/dtC̃_ss(t) = ∫dq__1⋯∫dq__P∫dη__1⋯∫dη__P
×∫dz_1⋯∫dz__P 𝒢(q, ,z;t) h_b(f(z))h_b'( f(x+y/2))
×∑_k=1^P {(ħ/im∂/∂x_k ρ(x+y/2,x-y ) ) (∂/∂x_k f(x+y/2) ) .
-. (ħ/im∂/∂y_k ρ(x+y/2,x-y ) ) (∂/∂y_k f(x+y/2) ) } ,
ħ/im∂/∂x_k ρ(x+y/2,x-y)=
-1/m⟨y_k-1|e^-ϵĤ p̂ |x_k⟩∏_l≠k ⟨y_l-1|e^-ϵĤ|x_l⟩ ,
ħ/im∂/∂y_k ρ(x+y/2,x-y)=
1/m⟨y_k|p̂ e^-ϵĤ |x_k+1⟩∏_l≠k ⟨y_l|e^-ϵĤ|x_l+1⟩ .
Inserting these expressions into Eq. (<ref>), we obtain Eq. (<ref>).
For the $t=0_+$ limit of Eq. (<ref>), an alternative expression can be found through direct evaluation of $d {\mathcal G} ({\bf q}, {\bf \eta}, {\bf z};t)/dt$ in that limit. For short enough time $t$,
⟨q_k+η_k/2|e^itĤ/ħ |z_k⟩⟨z_k|e^-itĤ/ħ |q_k-η_k/2⟩
≈m/2πħt exp{-im/ħt (q_k-z_k)η_k} .
Inserting the above approximation into Eq. (<ref>) and using the resulting expression, Eq. (<ref>) can be expressed as
C̃_ss(t) = ∫dq__1⋯∫dq__P∫dη__1⋯∫dη__P
×(∏_k=1^P m/2πħt exp{-im/ħt (q_k-z_k)η_k} )
×h_b(f(q)) h_b (f(z)) .
Now, utilizing the variable transformation of Eq. (<ref>), which is valid as long as $t\neq 0$, this can be expressed as
C̃_ss(t) = ∫dq__1⋯∫dq__P∫dη__1⋯∫dη__P
×∫dp_1⋯∫dp__P ρ(q,)
×(∏_k=1^P 1/2πħ exp{im/ħ p_kη_k} )
×h_b(f(q)) h_b (f(q+t/mp)) .
Taking time derivative of the above expression directly, $\tilde C_{fs}(t)=-d\tilde C_{ss}(t)/dt$ in the short time limit turns out to have the following expression:
C̃_fs(t)= -∫dq__1⋯∫dq__P∫dη__1⋯∫dη__P
×∫dp_1⋯∫dp__P ρ(q,)
×(∏_k=1^P 1/2πħ exp{im/ħ p_kη_k} ) Θ(f(q)-d)
×δ(f(q+t/mp)-d)p/m·∇f(q) .
where the fact that $h_b(x)=\Theta (x-d)$ has been used.
In the limit of $t=0_+$ this becomes
C̃_fs(0_+)= -∫dq__1⋯∫dq__P∫dη__1⋯∫dη__P
×∫dp_1⋯∫dp__P ρ(q,)
×(∏_k=1^P 1/2πħ exp{im/ħ p_kη_k} )
×Θ(-p/m·∇f(q)0_+ )
×δ(f(q)-d+p/m·∇f(q)0_+)p/m·∇f(q) .
Despite some similarity, the above expression is different from Eq. (38) of Ref. hele-jcp138 and there appears to be no obvious way to convert one to the other. This is in contrast to the classical case where variable transformation from $q$ and $p$ to $q'(0_+)$ and $p'(0_+)$ can be made through Liouville's theorem as detailed in Appendix F.
§ EVALUATION OF EQ. (<REF>)
For the evaluation of Eq. (<ref>), first consider the numerator, which can be approximated as
⟨x'|e^-ϵĤ p̂ |x” ⟩≈⟨x'|e^-ϵV̂ e^-ϵT̂ p̂|x”⟩
=e^-ϵV(x')1/2πħ∫_-∞^∞ dp p e^-ϵp^2/(2m) +i(x'-x”)p/ħ
≈im(x'-x”)/ħϵ ⟨x'|e^-ϵĤ |x” ⟩ .
p̅_+(x',x”;ϵ)≈im(x'-x”)/ħϵ .
Similarly, one can show that
p̅_-(x',x”;ϵ)≈im(x'-x”)/ħϵ .
Thus, to the leading order, $p_+(x',x'',\epsilon)=p_-(x',x'',\epsilon)$.
Inserting Eqs. (<ref>) and (<ref>) into Eq. (<ref>), we obtain
C̃_fs(0) ≈∫dq__1⋯∫dq__P ρ(q,0)Θ(f(q)-d) δ(f(q)-d)
×i/2ħϵ ∑_k=1^P (q_k-1-q_k+1)∂f(q)/∂q_k .
In the above integration, $q_k$'s are dummy integrands. Therefore, the result should be invariant with respect to the following replacement: $(q_1,\cdots,q_{_P})\rightarrow (q_{_P},\cdots,q_1)$. Upon this replacement, the summation can be rearranged such that $\sum_k (q_{k-1}-q_{k+1})\partial f/\partial q_k \rightarrow -\sum_k (q_{k-1}-q_{k+1})\partial f/\partial q_k$. Since the two terms cancel out, this proves that $\tilde C_{fs}(0)=0$ to its leading order. The next term is of order $O(1/\sqrt{P})$, which disappears in the $P\rightarrow \infty$ limit.
§ REAL TIME DYNAMICS FORMULATION OF IMAGINARY TIME PATH INTEGRAL
In a recent work,<cit.> we have shown that the canonical density operator for standard Hamiltonian, $\hat H=\hat p^2/(2m)+V(\hat q)$, can be expressed as
e^-βĤ= ∫dq ∫dp U(q,p) Ŝ(q, p̅) ,
where ${\bf q}\equiv (q_1,\cdots,q_{_P})$, ${\bf p}\equiv(p_1,\cdots,p_{_P})$, and $\bar {\bf p}\equiv (\bar p_1,\cdots, \bar p_{_P})$ with the definition of $\bar p_k=(p_k+p_{k-1})/2$ and the cyclic boundary condition $p_0=p_{_P}$. In the above expression, $U({\bf q}, {\bf p})$ is defined by
U(q, p)=(1/2πħ)^P ∏_k=1^P{e^-βV( q_k)/Pe^-βp_k^2/(2mP) .
×. e^ip_k(q_k-q_k+1)/ħ } ,
with $q_{_{P+1}}=q_1$, and $\hat S({\bf q},{\bf p})$ is defined by
Ŝ(q,p̅)=1/P∑_k=1^P ∫d η J(q_k,p̅_k;η)|q_k+η/2⟩⟨q_k-η/2| ,
J(q,p̅_k;η)= e^-βD_V(q;η)/P e^iη/ħ p̅_k ,
where $ D_V (q;\eta)\equiv (V(q+\eta/2)+V(q-\eta/2))/2-V(q)$.
For the flux operator defined by Eq. (<ref>),
= ∫dp' p'/2m { ⟨q_k-η/2|p'⟩⟨p'|q_k+η/2⟩h_b'(q_k+η/2).
. +h_b'(q_k-η/2)⟨q_k-η/2|p'⟩⟨p'|q_k+η/2⟩}
={h_b'(q_k+η/2)+h_b'(q_k-η/2)} ∫dp' p'/2m 1/2πħ e^-ip'η/ħ
={h_b'(q_k+η/2)+h_b'(q_k-η/2)} iħ/2m∂/∂η δ(η) .
Inserting this into Eq. (<ref>) and conducting partial integration, we find that
F_0(q,p)=1/P ∑_k=1^P { . -iħ/m ∂/∂η J(q_k,p̅_k; η)|_η=0 h_b'(q_k) .
-iħ/2m J(q_k,p̅_k;0)
×. . (∂/∂η h_b'(q_k+η/2)+∂/∂η h_b'(q_k-η/2) )|_η=0 } .
In the above expression, the two terms within the parenthesis of the last line, which involves derivatives of $h_b'(q_k\pm \eta/2)$, cancels out.
Employing the expression for Eq. (<ref>), it is easy to show that the derivative of $J(q_k,\bar p_k;\eta)$ with respect to $\eta$ at $\eta=0$ results in $i\bar p_k/\hbar$.
As a result, Eq. (<ref>) simplifies to Eq. (<ref>).
§ $T=0_+$ LIMIT OF EQ. (<REF>)
In the limit of $t\rightarrow 0_+$, Eq. (<ref>) approaches
k(0_+) = 1/Z_A∫dq∫dp U(q,p) (1/P∑_k=1^P h_a(q_k))
×(1/P∑_k=1^P p̅_k/m h_b'(q_k,cl(0_+))) .
The classical analog of this expression corresponds to the case where $P=1$ and $U({\bf q},{\bf p})$ is replaced with $e^{-\beta H}/(2\pi \hbar)$.
k_cl(0_+)= 1/Z_a∫∫dq d p/2πħ e^-βH(q,p) h_a(q)p/m h_b'(q(0_+)) .
Let us consider the simple case where $h_b(q)=\Theta (q-d)$. First, due to the Liouville theorem, $dqdp\ e^{-\beta H(q,p)}$ in the integrand can be replaced with $dq'dp'\ e^{-\beta H(q',p')}$ where $q'= q(0_+)$ and $p'= p(0+)$. Then, $q=q'(0_-)$. Dropping primes, we then obtain the following expression:
k_cl(0_+) = 1/Z_a∫∫dq d p/2πħ e^-βH(q,p) h_a(q(0_-))p/m δ(q-d)
= 1/Z_a∫∫dq d p/2πħ e^-βH(q,p) Θ(p)p/m δ(q-d)
= k_cl^TST ,
where the following relation has been used.
h_a(q(0_-))δ(q-d)=lim_ϵ→0 (1-Θ(-ϵp) )δ(q-d)
=Θ(p)δ(q-d) .
Equation (<ref>) is the well-known classical TST rate expression.
In contrast to the above classical case, the zero time limit for the quantum case, Eq. (<ref>) can be shown to be zero.
In order to show this, the double summation in Eq. (<ref>) can be divided into two terms as follows:
k(0_+)=1/Z_a∫dq∫dp U(q,p)
×1/P^2(∑_k=1^P h_a(q_k)p̅_k/m h_b'(q_k,cl(0_+) ).
+. ∑_k=1^P∑_l≠k^P h_a(q_k)p̅_l/m h_b'(q_l,cl(0_+) ) ) .
For the case where $h_b(q)=\Theta(q-d)$, going through the same procedure of variable transformation and time translation as in deriving Eq. (<ref>), we obtain
k(0_+)=1/Z_a∫dq∫dp U(q,p)
×1/P^2(∑_k=1^P Θ(p̅_k)p̅_k/m δ(q_k-d) .
+. ∑_k=1^P∑_l≠k^P h_a (q_k)p̅_l/m δ(q_l-d ) ) .
In the above expression, the first term involving single sum over $k$ is nonzero as in the classical case but its contribution is of order $(1/P)$ and vanishes in the limit of $P\rightarrow\infty$.
The second term involving double summation can be evaluated explicitly and shown to be zero due to symmetry as follows:
∫dq∫dp U(q,p) 1/P^2∑_k=1^P∑_l≠k^P h_a (q_k)p̅_l/m δ(q_l-d )
=i/2βħP (2πm P/β)^P/2∫dq e^-βV_RP (q)/P
×∑_k=1^P∑_l≠k^P h_a(q_k) (q_l-q_l+1)(δ(q_l-d )+δ(q_l+1-d))
=i/4βħP (2πm P/β)^P/2∫dq e^-βV_RP (q)/P ∑_k=1^P∑_l≠k^P h_a (q_k)
×(q_l-q_l+1 +q_l+1-q_l) (δ(q_l-d )+δ(q_l+1-d))
=0 .
V_RP(q)=∑_k=1^P{ V(q_k)+mP^2/2β^2ħ^2(q_k-q_k+1)^2} .
In Eq. (<ref>), the first equality is obtained by explicit integration over $p_k$'s, and the second equality is obtained using the fact that $V_{RP}({\bf q})$ is invariant with respect to exchange of $q_l \rightarrow q_{l+1}$ and the reversal of the ordering of indices.
P. Pechukas, in Dynamics of Molecular Collisions, Part B, edited by W. H.
Miller (Plenum, New York, 1976).
J. T. Hynes, in Theory of Chemical Reaction Dynamics, edited by M. Baer
(CRC Press, Boca Raton, 1985).
D. Chandler, J. Stat. Phys. 42, 49 (1986).
P. Ha̋nggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251
E. Pollak, in Dynamics of Molecules and Chemical Reactions, edited by R.
E. Wyatt and J. Z. H. Zhang (Marcel Dekker, New York, 1996).
F. J. McLafferty and P. Pechukas, Chem. Phys. Lett. 27, 511 (1974).
W. H. Miller, Acc. Chem. Res. 26, 174 (1993).
E. Pollak, J. Chem. Phys. 74, 6765 (1981).
E. Pollak, J. Chem. Phys. 107, 64 (1997).
G. A. Voth, D. Chandler, and W. H. Miller, J. Phys. Chem. 93, 7009
G. A. Voth, D. Chandler, and W. H. Miller, J. Chem. Phys. 91, 7749
G. A. Voth, J. Phys. Chem. 97, 8365 (1993).
N. F. Hansen and H. C. Andersen, J. Chem. Phys. 101, 6032 (1994).
N. F. Hansen and H. C. Andersen, J. Phys. Chem. 100, 1137 (1996).
E. Pollak and J.-L. Liao, J. Chem. Phys. 108, 2733 (1998).
S. Jang and G. A. Voth, J. Chem. Phys. 112, 8747 (2000).
I. Affleck, Phys. Rev. Lett 46, 388 (1981).
M. J. Gillan, J. Phys. C 20, 3621 (1987).
M. Messina, G. K. Schenter, and B. C. Garrett, J. Chem. Phys. 98, 8525
M. Messina, G. K. Schenter, and B. C. Garrett, J. Chem. Phys. 99, 8644
G. Mills, G. K. Schenter, D. E. Makarov, and H. Jónsson, Chem. Phys. Lett
278, 91 (1998).
J. Cao and G. A. Voth, J. Chem. Phys. 105, 6856 (1996).
T. J. H. Hele and S. C. Althorpe, J. Chem. Phys. 138, 084108 (2013).
I. R. Craig and D. E. Manolopoulos, J. Chem. Phys. 122, 084106 (2005).
I. R. Craig and D. E. Manolopoulos, J. Chem. Phys. 123, 034102 (2005).
S. C. Althorpe and T. J. Hele, J. Chem. Phys. 139, 084115 (2013).
E. Geva, S. Jang, and G. A. Voth, in Encyclopedia of Materials Modeling:
Vol. I, Fundamental Models and Methods, S. Yip, Editor (Springer-Verlag,
T. Yamamoto, J. Chem. Phys. 33, 281 (1960).
A. A. Stuchebrukhov, J. Chem. Phys. 95, 4258 (1991).
R. Kubo, J. Phy. Soc. Japan 12, 570 (1957).
R. Kubo, M. Yokota, and S. Nakajima, J. Phy. Soc. Japan 12, 1203
S. Jang, arXiv: 1308.3805 (2013).
S. Jang, A. V. Sinitskiy, and G. A. Voth, J. Chem. Phys. 140, 154103
J. Costley and P. Pechukas, Chem. Phys. Lett. 83, 139 (1981).
P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University
Press, Oxford, UK, 1967).
J. J. Sakurai, Modern Quantum Mechanics (The Benjamin/Cummings Publishing
Company, Inc., Menlo Park, CA, 1985).
R. Omnès, Rev. Mod. Phys. 64, 339 (1992).
W. H. Miller, S. D. Schwartz, and J. W. Tromp, J. Chem. Phys. 79, 4889
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§ INTRODUCTION
A Berry phase <cit.> is a shift in the phase of a quantum-mechanical wavefunction due to adiabatic evolution of parameters in the Hamiltonian. Berry phases are encoded in a geometric object: a connection on a vector bundle over the parameter space. A well-known system that exhibits a Berry phase is the spin-half particle in an external magnetic field; here, the Berry connection is simply the Dirac monopole.
In <cit.>, Sonner and Tong found a similar setup in which the Berry connection that arises is a single 't Hooft-Polyakov monopole <cit.>. The construction of Sonner and Tong uses $\mathcal N = 4$ supersymmetric quantum mechanics on a $\mathbb{CP}^1$ target space, where the background parameters being varied are twisted masses associated with an isometry of $\mathbb{CP}^1$. Later, in <cit.>, Sonner and Tong showed that the fact that their Berry connection obeys the Bogomolny-Prasad-Sommerfield (BPS) monopole equations <cit.> is a consequence of supersymmetry, via an argument similar to Cecotti and Vafa's derivation of the tt* equations <cit.>, or the calculation of the effective action on the Coulomb branch for gauged linear models <cit.>.
The purpose of this article is to point out that every multi-monopole solution of the BPS equations can be realised in a similar way, as a Berry phase in $\mathcal N = 4$ quantum mechanics. The key is to replace the twisted mass deformation with a new deformation, which, as far as the author is aware, has not previously appeared in the literature. This new deformation is constructed from matrix-valued functions of the moment map for a holomorphic isometry of $\mathbb{CP}^1$, and it preserves supersymmetry under the condition that these matrices obey a certain set of first-order equations.
In fact, these equations are nothing other than the Nahm equations. The computation of the Berry phase in the quantum-mechanical model with this deformation turns out to be identical to the Nahm construction of BPS monopoles. The Nahm equations <cit.> have already been studied in connection with many aspects of mathematical physics, including spectral curves <cit.>, D-branes <cit.>, hyperkähler geometry <cit.>, geometric Langlands <cit.> and the vacuum geometry of theories in three or four dimensions <cit.>. There is also a one-dimensional supersymmetric model <cit.> in which a version of the Nahm equations emerges as Dirac brackets in the quantum theory.
The present work offers a different perspective on the Nahm equations, showing that they are intimately related to Kähler geometry. While it may not immediately seem natural to view Nahm matrices as being defined over $\mathbb {CP}^1$ (as opposed to a real interval), this approach enables us to use the $\mathcal N = 4$ supersymmetry of quantum mechanics with Kähler target space in an elementary way to show that the gauge fields obtained from the Nahm construction must satisfy the BPS monopole equations.
Furthermore, our supersymmetric deformation generalises for any Kähler manifold with a holomorphic isometry. This leads to a procedure by which solutions of the BPS equations can be obtained from the zero modes of a certain deformation of the Dolbeault operators on any such Kähler manifold.
§ $\MATHCAL N = 4$ QUANTUM MECHANICS AND THE NAHM EQUATIONS
We begin by describing a class of supersymmetric deformations of quantum mechanics on Kähler target spaces with holomorphic isometry, which, as we will see, are closely related to the Nahm equations.
Let $\mathcal{M}$ be a complex manifold with a hermitian metric $g_{i\bar j} = ( g_{ \bar i j})^\star $ such that $\omega = i g_{i \bar j} dz^i \wedge d\bar z^{\bar j} $ obeys the Kähler condition,
\begin{eqnarray}
d \omega = 0 \nonumber
\end{eqnarray}
It is a standard fact that there exists a quantum-mechanical theory with target space $\mathcal{M}$ that is invariant under $\mathcal N = 4$ supersymmetry <cit.>. The fields in the theory are a collection of chiral multiplets $(\phi^i, \psi^i)$, where $\phi^i$ are the complex coordinates on $\mathcal{M}$ and $\psi^i$ are two-component complex spinors in the holomorphic tangent bundle of $\mathcal{M}$.
Now suppose that $\mathcal{M}$ admits a holomorphic isometry, that is, there exists a global vector field $K = k^i \partial_i + \bar{k}^{\bar i} \partial_{\bar i}$, with $(k^i)^\star = \bar k^{\bar i}$, obeying the holomorphicity condition,
\begin{eqnarray}
\mathcal L_K J = 0 \nonumber
\end{eqnarray}
and the Killing equation,
\begin{eqnarray}
\mathcal L_K \omega = 0 \nonumber
\end{eqnarray}
where $J = idz^i \otimes \partial_i - i d\bar z^{\bar i} \otimes \partial_{\bar i}$ is the complex structure. Furthermore, suppose that there exists a global moment map $\mu$ for this isometry, obeying
\begin{eqnarray}
d\mu = \iota_K \omega \nonumber
\end{eqnarray}
It was shown in <cit.> that one can gauge the isometry in a way that is invariant under $\mathcal N = 4$ supersymmetry. One introduces a vector multiplet $(u_t, x^a, \lambda, D)$, where $u_t$ is a $U(1)$ gauge field, $x^1, x^2, x^3$ are real scalars, $\lambda$ is a two-component complex spinor and $D$ is a real auxiliary scalar field. The Lagrangian is
\begin{eqnarray}
L = && \frac 1 {2e^2} \left( \dot x^a \dot x^a + i \bar \lambda \dot \lambda + D^2 \right) + g_{\bar i j} D_t \bar\phi^{\bar i} D_t \phi^j - g_{\bar i j} \bar k^{\bar i} k^j x^a x^a - \mu D \nonumber \\
&& +i g_{\bar i j} \bar {\psi}^{\bar i}D_t \psi^j - i \nabla_j k_{\bar i} x^a \bar{\psi}^{\bar i} \sigma^a \psi^j
- \frac 1 4 R_{i\bar k j \bar l} (\psi^{i T} \varepsilon \psi^j ) (\bar{\psi}^{\bar k} \varepsilon \bar{\psi}^{\bar l T}) - \bar k_i \psi^{i T} \varepsilon \lambda + k_{\bar i} \bar{\psi}^{\bar i} \varepsilon \bar \lambda^T \nonumber
\end{eqnarray}
Here, $\sigma^a$ are the Pauli matrices and $\varepsilon_{21} = - \varepsilon_{12} = +1$ is the alternating symbol. The covariant derivatives are defined as
\begin{eqnarray}
D_t \phi^i = \dot \phi^i + u_t k^i, \ \ \ \ \ \ \ \ \ \ \ \ D_t \psi^i = \dot \psi^i + \Gamma^i_{jk} \dot \phi^j \psi^k + u_t \nabla_j k^i \psi^j \nonumber
\end{eqnarray}
and $\Gamma^j_{kl} = g^{j\bar m} \partial_k g_{\bar m l}$ and $R_{i \bar k j \bar l} = \partial_j \partial_{\bar l} g_{i \bar k} - g^{m \bar n} \partial_j g_{\bar n i} \partial_{\bar l} g_{m \bar k}$ are the standard expressions for the connection and curvature on a Kähler manifold. One can verify that this Lagrangian is invariant under the supersymmetry transformations
\begin{eqnarray}
\delta \phi^i & = & - \psi^{i T} \varepsilon \xi \nonumber \\
\delta \psi^i & = & \left( -i D_t \phi^i - i k^i x^a \sigma^a \right) \varepsilon \bar \xi^T - \frac 1 2 \Gamma^i_{jk} ( \psi^{j T} \varepsilon \psi^k) \xi \nonumber \\
\delta u_t & = & \frac i 2 \bar \lambda \xi - \frac i 2 \bar \xi \lambda \nonumber \\
\delta x^a & = & \frac i 2 \bar \lambda \sigma^a \xi - \frac i 2 \bar \xi \sigma^a \lambda \nonumber \\
\delta \lambda & = & \dot x^a \sigma^a \xi - i D \xi \nonumber \\
\delta D & = & \frac 1 2 \bar \xi \dot \lambda + \frac 1 2 \dot{\bar \lambda } \xi
\label{susytrans}
\end{eqnarray}
It is well-known that one can introduce an $\mathcal N = 4$ invariant mass term for the chiral multiplet fields. The construction is simple: we replace $x^a$ by $ x^a + m^a$, where $m^1, m^2, m^3$ are constants. The three mass parameters transform as a triplet under the $SU(2)_R$ symmetry. We refer to this deformation as a twisted mass, by analogy with terminology used for two-dimensional sigma models. Of course, $x^a \mapsto x^a + m^a$ is a mere redefinition of variables, so as things stand, this deformation has no effect on the physics of the theory. However, if we send the gauge coupling $e$ to zero so that fluctuations of the fields in the vector multiplet cost infinite energy, and fix a particular zero-energy configuration for these fields, say $x^a = \lambda = D = 0$, then the parameters $m^a$ become genuine mass parameters.
One may ask whether it is possible to introduce a deformation similar to the twisted mass, but with the parameters $m^a$ replaced by functions that depend on the fields $\phi^i$ and $\bar \phi^{\bar i}$. It turns out that such a construction exists if one is prepared to replace the parameters $m^a$ with a triplet of hermitian $k \times k$ matrices, which we denote as $T^a$. To put matrices into a scalar-valued Lagrangian, we borrow a technique from <cit.>: we introduce a “spin” variable, a $k$-component bosonic vector $\varphi$ whose indices can be contracted with the indices on the matrices $T^a$. We give $\varphi$ a first-order kinetic term, and we gauge the symmetry $\varphi \mapsto e^{i \theta} \varphi$ by introducing a gauge field $\alpha_t$ that acts as a Lagrange multiplier imposing the constraint $\varphi^\dagger \varphi = 1$. Some experimentation reveals that a possible supersymmetric deformation of the theory is
\begin{eqnarray}
L = && i \varphi^\dagger \dot \varphi + \alpha_t (\varphi^\dagger \varphi - 1) + \frac 1 {2e^2} \left( \dot x^a \dot x^a + i \bar \lambda \dot \lambda + D^2 \right) \nonumber \\
&& + g_{\bar i j} D_t \bar\phi^{\bar i} D_t \phi^j - g_{\bar i j} \bar k^{\bar i} k^j ( x^a + \varphi^\dagger T^a \varphi)( x^a + \varphi^\dagger T^a \varphi) - \mu D \nonumber \\
&& +i g_{\bar i j} \bar {\psi}^{\bar i}D_t \psi^j - i \nabla_j k_{\bar i} ( x^a + \varphi^\dagger T^a \varphi )\bar{\psi}^{\bar i} \sigma^a \psi^j - i k_{\bar i} \bar k_{j} \varepsilon^{abc} (\varphi^\dagger [T^b , T^c] \varphi) \bar{\psi}^{\bar i } \sigma^a \psi^j \nonumber \\
&& - \frac 1 4 R_{i\bar k j \bar l} (\psi^{i T} \varepsilon \psi^j ) (\bar{\psi}^{\bar k} \varepsilon \bar{\psi}^{\bar l T}) - \bar k_i \psi^{i T} \varepsilon \lambda + k_{\bar i} \bar{\psi}^{\bar i} \varepsilon \bar \lambda^T \label{perturbedlagrangian}
\end{eqnarray}
where the hermitian matrices $T^a$ are functions of the moment map $\mu(\phi^i, \bar \phi^{\bar i})$, and obey the Nahm equation,
\begin{eqnarray}
\frac d {d\mu} T^a (\mu) = \frac i 2 \varepsilon^{abc} [T^b (\mu) , T^c (\mu) ] \label{nahm}
\end{eqnarray}
Indeed, whenever equation (<ref>) is satisfied, the variation of (<ref>) under the supersymmetry transformations (<ref>) reduces to a total derivative upon imposing the equations of motion for $u_t$, $\varphi$ and $\psi^i$. By the standard Noether procedure, one obtains an expression for the conserved supercharges,
\begin{eqnarray}
Q = i g_{\bar i j} D_t \bar\phi^{\bar i} \psi^j + i \bar k_i (x^a +\varphi^\dagger T^a \varphi) \sigma^a \psi^i - \frac i 2 \mu \varepsilon \bar\lambda^T + \frac 1 {2e^2} \left( \dot x^a \sigma^a \varepsilon \bar\lambda^T + i D \varepsilon \bar \lambda^T \right) \nonumber
\end{eqnarray}
There are two key differences between the twisted mass deformation and the new deformation described. Firstly, as mentioned already, the twisted mass only affects the physics of the theory in the limit $e \to 0$ in which the vector multiplet fields are frozen to a chosen supersymmetric configuration. The new deformation is different: it is also physically relevant for non-zero gauge coupling $e$.
Secondly, in the limit $e \to 0$, the twisted mass deformation remains supersymmetric if we do not include the gauge field $u_t$ in our theory; indeed, this is what is usually meant by a twisted mass deformation in the literature. (For instance, in the background $x^a = \lambda = D = 0$, the supersymmetry transformation for the fermion is modified to $\delta \psi^i = ( -i \dot \phi^i - i k^i m^a \sigma^a ) \varepsilon \bar \xi^T - \frac 1 2 \Gamma^i_{jk} ( \psi^{j T} \varepsilon \psi^k) \xi$ and the supersymmetry algebra has a central charge.) Our new deformation, however, is only invariant under supersymmetry when the gauge field is present, even in the limit $e \to 0$, because the equation of motion for $u_t$ is necessary in order to ensure that the variation of the Lagrangian is a total derivative on-shell. We will return to this point in the next section, where we quantise the theory.
§ SUPERSYMMETRIC GROUND STATES AND THE ASSOCIATED WEYL EQUATION
In the Nahm construction of BPS monopoles, after a solution to the Nahm equations is obtained, the next step is to find zero modes of a certain Weyl operator built from this Nahm data. The purpose of this section is to explain how this step of the Nahm construction is related to a physical problem in our quantum-mechanical model: determining the supersymmetric ground states.
Throughout this section, we take the limit $e \to 0$. (In later sections, we will discuss what happens when we relax this condition.) As we have already discussed, this means that any fluctuations of the vector multiplet fields cost infinite energy, so the vector multiplet fields are frozen and take the form
\begin{eqnarray}
x^a = {\rm constant}, \ \ \ \ \lambda = 0, \ \ \ \ D = 0 \label{background}
\end{eqnarray}
We can think of $(x^1, x^2, x^3)$ as background parameters in the theory, while the chiral multiplet fields and the $k$-component field $\varphi$ remain dynamical. To quantise this theory, one must therefore quantise these dynamical fields in the supersymmetric background (<ref>).
Following Witten <cit.>, canonical quantisation promotes the chiral multiplet fields to linear operators acting on differential forms on the Kähler manifold $\mathcal M$,
\begin{eqnarray}
\phi^i \mapsto z^i \times \ \ \ \ \ \ \ \ \ & & \ \ \ \ \ \ \ \ \bar{\phi}^{\bar i} \mapsto \bar {z}^{\bar i} \times \nonumber
\\
g_{i \bar j} \dot{\bar{\phi}^{\bar j}} \mapsto - i \nabla_i \ \ \ \ \ & & \ \ \ \ \ \ g_{\bar i j} \dot\phi^j \mapsto - i \nabla_{\bar i} \nonumber
\\
\psi^i \mapsto \left( \begin{array}{c} dz^i \wedge \\ i g^{i \bar j} \iota_{\partial_{\bar j}} \end{array} \right) \ \ \ & & \ \ \ \bar \psi^{\bar i} \mapsto \left( \begin{array}{cc} g^{\bar i j} \iota_{\partial_j} & - i d\bar z^{\bar i} \wedge \end{array} \right) \label{dictionary}
\end{eqnarray}
Meanwhile, the canonical commutation relations for the $k$-component bosonic variable $\varphi_p$ are
\begin{eqnarray}
[\varphi_p, \varphi^\dagger_q] = \delta_{pq} \nonumber
\end{eqnarray}
We define a state $| 0 \rangle $ such that $ \varphi_p | 0 \rangle = 0 $ for $p = 1, ... , k$. Observe that the equation of motion obtained by varying the Lagrange multiplier field $\alpha_t$ imposes the constraint $ \sum_{p = 1}^k \varphi_p^\dagger \varphi_p = 1 $. Therefore, the allowed states in the theory are of the form
\begin{eqnarray}
\sum_{p = 1}^k \eta_p \varphi^\dagger_p |0\rangle \nonumber
\end{eqnarray}
where $\eta_1, ... , \eta_k$ are differential forms on $\mathcal M$. Collecting these differential forms into a $k$-component column vector $\eta = (\eta_1, ... , \eta_k )^T$, we see that the operator $\varphi^\dagger T^a \varphi$ acts on $\eta$ as multiplication by the matrix $T^a(z^i, \bar z^{\bar i})$.
Since the theory is a $U(1)$ gauge theory, we must fix a gauge when canonically quantising. We pick the $u_t = 0$ gauge. In this gauge the equation of motion for $u_t$ reads as $i\bar k_i \dot \phi^i + i k_{\bar i} \dot{\bar \phi^{\bar i}} - \nabla_j k_{\bar i} \bar{\psi}^{\bar i} \psi^j = 0$, and this equation must be imposed as a constraint on the Hilbert space. By the dictionary (<ref>), the physical states in the Hilbert space are those differential forms that are invariant under the action of the isometry, that is, those that are annihilated by the Lie derivative with respect to the Killing vector field $K = k^i \partial_i + \bar k^{\bar i} \partial_{\bar i}$,
\begin{eqnarray}
\mathcal L_K \eta = 0 \label{physicalstates}
\end{eqnarray}
To complete the quantisation procedure, we must specify the inner product on the states. This is given by the usual expression,
\begin{eqnarray}
\langle \eta ' | \eta \rangle = \sum_{p = 1}^k \int \eta_p \wedge \star \eta_p' \nonumber
\end{eqnarray}
We are now almost ready to write down the supercharges as differential operators. Before doing so, it helps to organise the various Dolbeault operators and contraction operators as doublets of the Lefshetz $SU(2)$ action on the Kähler manifold $\mathcal M$, which descends from the $SU(2)_R$ action in the quantum mechanics. We define
\begin{eqnarray}
d_\alpha = \left( \begin{array}{c} \partial \\ - i \bar \partial^\dagger \end{array} \right) \ \ \ \ \bar d_\alpha = \left( \begin{array}{cc} \partial^\dagger & i \bar \partial \end{array} \right) \nonumber
\end{eqnarray}
\begin{eqnarray}
\iota_\alpha = \left( \begin{array}{c} i \iota_{(k^i \partial_i)}^\dagger \\ - \iota_{(\bar k^{\bar i} \partial_{\bar i}) }\end{array} \right) \ \ \ \ \bar \iota_\alpha = \left( \begin{array}{cc} -i \iota_{(k^i \partial_i)} & \ -\iota_{(\bar k^{\bar i} \partial_{\bar i} )}^\dagger \end{array} \right) \nonumber
\end{eqnarray}
The Kähler condition, Killing's equation, holomorphicity of the Killing field and the Nahm equation (<ref>) can be summarised as a list of (anti)commutation relations,
\begin{eqnarray}
\{ d_\alpha, d_\beta \} = \{ \bar d_\alpha, \bar d_\beta \} = 0, \ \ \ \ \{d_\alpha, \bar d_\beta \} = \Delta \delta_{\alpha \beta} = & \frac 1 2 (d d^\dagger + d^\dagger d) \delta_{\alpha \beta} \nonumber
\end{eqnarray}
\begin{eqnarray}
\{ \iota_\alpha, \iota_\beta \} = \{ \bar \iota_\alpha , \bar \iota_\beta \} = 0 , \ \ \ \ \{ \iota_\alpha , \bar \iota_\beta \} = & \frac 1 2 \sum_\gamma \{ \iota_\gamma, \bar \iota_\gamma \} \delta_{\alpha \beta} \nonumber
\end{eqnarray}
\begin{eqnarray}
\{ d_\alpha , \iota_\beta \} = \{ \bar d_\alpha, \bar \iota_\beta \} = 0 , \ \ \ \ \ \ \{ d_\alpha, \bar \iota_\beta \} - \{ \iota_\alpha , \bar d _\beta \} = & \frac 1 2 \sum_{\gamma} \left( \{ d_\gamma , \bar \iota _\gamma \} - \{ \iota_\gamma , \bar d_\gamma \} \right) \delta_{\alpha \beta} \nonumber
\end{eqnarray}
\begin{eqnarray}
\mathcal L_K = & \frac i 2 \sum_{\gamma} \left( \{ d_\gamma , \bar \iota_\gamma \} + \{ \iota_\gamma , \bar d_\gamma \} \right) \nonumber
\end{eqnarray}
\begin{eqnarray}
[d_\alpha , T^a ] = - & \frac { i }{2} \varepsilon^{abc} [T^b, T^c] \iota_\alpha, \ \ \ \ [\bar d_\alpha , T^a ] = + \frac i 2 \varepsilon^{abc} [T^b, T^c ] \bar \iota_\alpha \nonumber
\end{eqnarray}
\begin{eqnarray}
[\iota_\alpha, T^a ] = [\bar \iota_\alpha, T^a ] = 0 \label{relations}
\end{eqnarray}
Having established this notation, we find the following expressions for the supercharges,
\begin{eqnarray}
Q_\alpha = d_\alpha +(x^a + T^a) \sigma^a_{\alpha \beta } \iota_\beta, \ \ \ \ \bar Q_\alpha = \bar d_\alpha + (x^a + T^a ) \bar \iota_\beta \sigma_{\beta \alpha} \label{supercharges}
\end{eqnarray}
The Hamiltonian is
\begin{eqnarray}
& H = \Delta + \frac 1 2 (x^a + T^a)(x^a + T^a ) \{ \iota_\gamma , \bar \iota_\gamma \} + (x^a + T^a) \sigma_{\gamma \delta}^a \{ \iota_\delta, \bar d_\gamma\} - i \varepsilon^{abc} [T^b, T^c ] \sigma^a_{\gamma \delta } \iota_\delta \bar \iota_\gamma \nonumber
\end{eqnarray}
One can check that the list of relations (<ref>) ensures that the supercharges and Hamiltonian preserve the space of physical states (<ref>), and, when acting on this space, obey the supersymmetry algebra
\begin{eqnarray}
\{ Q_\alpha, Q_\beta \} = \{ \bar Q_\alpha , \bar Q_\beta \} = [H, Q_\alpha] = [H, \bar Q_\alpha ] = 0, \ \ \ \ \ \{ Q_\alpha , \bar Q_\beta \} = H \delta_{\alpha \beta} \nonumber
\end{eqnarray}
Note that the gauge fixing constraint $\mathcal L_K \eta = 0$ is essential to obtain the commutator $ [H, Q_\alpha] = [H, \bar Q_\alpha ] = 0$. (By contrast, when one uses constant, scalar-valued twisted masses $(m^1, m^2, m^3)$, the commutation relations $ [H, Q_\alpha] = [H, \bar Q_\alpha ] = 0$ hold regardless of the constraint, a sign that the theory remains supersymmetric even without gauging the isometry; in this case the $Q_\alpha$ and $\bar Q_\beta$ anticommutator becomes $\{ Q_\alpha , \bar Q_\beta \} = H \delta_{\alpha \beta} - i \mathcal L_K m^a \sigma^a_{\alpha \beta}$, and $\mathcal L_K$ is interpreted as a central charge. This is the setup considered in <cit.>.)
One may notice a similarity between the supercharges $Q_\alpha$ and $\bar Q_\alpha$ in (<ref>) and the Weyl operators used in the Nahm construction of monopoles. Zero modes of these supercharges have a physical interpretation: they represent the ground states of energy $H=0$.
In one special example, the supercharges reduce precisely to the Weyl operators familiar from the Nahm construction. This is when $\mathcal {M}$ is the Riemann sphere $\mathbb{CP}^1$. We use a stereographic coordinate $z$ on $ \mathbb{CP}^1 \backslash \{ \infty \} $, chosen so that the isometry is the rotation $z \mapsto e^{-i\theta} z$. Then the metric is of the form $g_{z \bar z} = g\left( |z |^2 \right)$ for some positive function $g$, and the Killing vector field associated to the rotation is $-iz \partial_z + i \bar z \partial_{\bar z}$. The moment map for the isometry is $\mu = \mu \left(| z |^2\right)$, where $\mu' = g$; it is a monotonically increasing function of $| z|$, taking values in the interval $[ \mu(0), \mu(\infty)]$.
To see that the supercharges on $\mathcal M = \mathbb {CP}^1$ really do reduce to the familiar Weyl operators, we first observe that the operators $ \sigma^a_{\gamma \delta } \{ \iota_\delta, \bar d_\gamma\} $ and $ \sigma^a_{\gamma \delta }\iota_\delta \bar \iota_\gamma $ that appear in the Hamiltonian vanish on (1,0)- and (0,1)-forms, and what remains of the Hamiltonian is positive definite because $ h^{1,0} (\mathbb {CP}^1) = h^{0,1} (\mathbb {CP}^1) = 0$ while $\frac 1 2 (x^a + T^a) (x^a + T^a ) \{ \iota_\gamma , \bar \iota_\gamma \} = (x^a + T^a) (x^a + T^a ) |z|^2 g\left(| z |^2 \right) $ is the square of a hermitian matrix. So for the purposes of finding supersymmetric ground states, we may restrict our attention to linear combinations of (0,0)- and (1,1)-forms. Let us write such a state as
\begin{eqnarray}
\eta = \frac 1 {\sqrt{2\pi} } \left( f_1 + f_2 \omega \right) \nonumber
\end{eqnarray}
where $f_1$ and $f_2$ are $k$-component vector-valued functions on $\mathbb{CP}^1$ and $\omega = i g_{z\bar z} dz \wedge d \bar z$ is the Kähler form. The gauge fixing constraint (<ref>) implies that $f_1, f_2$ are functions of the moment map $\mu$. The condition $Q_\alpha \eta = \bar Q_\alpha \eta = 0 $ defining supersymmetric ground states is equivalent to
\begin{eqnarray}
\left( \frac d {d\mu} - \left( x^a 1_{k\times k}+ T^a(\mu) \right) \otimes \sigma^a \right) \left( \begin{array}{c} f_1 (\mu) \\ f_2 (\mu) \end{array} \right) = 0 \label{weyl}
\end{eqnarray}
which is precisely the Weyl equation associated with the Nahm data $T^a(\mu)$.
§ BERRY'S CONNECTION AND THE INVERSE NAHM TRANSFORM
In the previous section, we described the supersymmetric ground states in our theory in the limit $e \to 0$ in which the vector multiplets are frozen. We now consider what happens when the gauge coupling $e$ is small but non-zero, so that $x^a $ may vary slowly with time. We shall see that the quantum dynamics is governed by a Berry connection, and the construction of this Berry connection is related to the last step of the Nahm construction in which solutions of the BPS monopole equations are finally obtained.
We will continue to think of $x^a$ as a background parameter, but now as one that varies slowly with time. The supersymmetric ground states are still represented by differential forms annihilated by the supercharge operators given in (<ref>), but since $x^a$ appears as a parameter in these supercharge operators, the states will change adiabatically as $x^a$ varies. If $\mathcal M$ is compact, the spectrum of the Hamiltonian is discrete, and as $x^a$ varies, the number of supersymmetric ground states can only ever change by an even number. This is because, for every excited state, there is at least one supercharge, say $Q_\alpha$, that fails to annihilate that state. So the excited states in the theory come in pairs, $\{|\eta\rangle , Q_\alpha | \eta \rangle \}$. Thus the Witten index $I = {\rm Tr} (-1)^F$ is preserved as $x^a$ varies, and the number of supersymmetric ground states always remains equal to the Witten index modulo 2.
Let us make the assumption that, in the particular quantum-mechanical model we are considering, the number of supersymmetric ground states is exactly equal to $| I |$ for all $x^a$. (We will soon comment on what happens if this condition is violated.) We pick a basis of ground states, $\{ |m (\vec x) \rangle\} $, with $m= 1, ... , |I|$, obeying the orthonormality condition
\begin{eqnarray}
\langle m (\vec x) | n (\vec x) \rangle= \delta_{mn} \nonumber
\end{eqnarray}
Writing a general state as a linear combination,
\begin{eqnarray}
\sum_m \chi_m (t) | m (\vec x (t)) \rangle \nonumber
\end{eqnarray}
where $\chi_m(t)$ are complex coefficients with $\sum_m \chi^\star_m \chi_m = 1$, it is easy to see that the Schrödinger equation,
\begin{eqnarray}
i \partial_t \left( \sum_m \chi_m (t) | m (\vec x(t)) \rangle \right) = \hat H \left( \sum_m \chi_m (t) | m (\vec x(t)) \rangle \right) = 0 \nonumber
\end{eqnarray}
implies that the coefficients $\chi_m(t)$ satisfy the first-order equation of motion,
\begin{eqnarray}
\dot \chi_m = i \dot x^a \sum_{n} A_a (\vec x)_{mn} \chi_n \label{timeevolution}
\end{eqnarray}
\begin{eqnarray}
A_a(\vec x)_{mn} = i \langle m(\vec x) | \frac \partial {\partial x^a } | n(\vec x) \rangle \label{gaugefield}
\end{eqnarray}
The hermitian matrix $A_a(\vec x)_{mn}$ is the Berry connection for our model. Geometrically, we can think of $(x^1, x^2, x^3)$ as coordinates on a parameter space $\mathbb R^3$, and we can view $\chi_m$ as coordinates on the fibres of a complex vector bundle of rank $|I|$ over this $\mathbb R^3$. This bundle is known as the vacuum bundle. Equation (<ref>) is the condition of parallel transport with respect to the connection $A_a$. The Berry connection $A_a(\vec x)$ is a smooth connection; however, if we relax the condition that the number of supersymmetric ground states equals $| I|$ for all $(x^1,x^2,x^3)$, that is, if extra pairs of ground states appear at certain points in the parameter space $\mathbb R^3$, then the vacuum bundle is ill-defined at those points in the parameter space and the Berry connection develops singularities at those points.
Let us return to our special example where $\mathcal M $ is a rotationally invariant $\mathbb{CP}^1$. Here, the definition of the Berry connection (<ref>) is identical to the standard construction of the Yang-Mills fields in a BPS monopole from Nahm data,
\begin{eqnarray}
A_a (\vec x)_{mn} = i \int_{\mu(0)}^{\mu(\infty)} d\mu \left( f_{m,1}^\dagger ( \vec x, \mu) \frac \partial {\partial x^a } f_{n,1} (\vec x, \mu)+ f_{m,2}^\dagger (\vec x , \mu) \frac \partial {\partial x^a } f_{n,2} (\vec x, \mu) \right) \nonumber
\end{eqnarray}
The rank of the vacuum bundle is equal to the number of supersymmetric ground states, and this depends on the precise form of the Nahm data. In <cit.>, it was shown by an index theorem <cit.> that if the matrices $T^a$ have simple poles at $\mu = \mu(0)$ and $\mu(\infty)$, with residues defining the $k$-dimensional irreducible representation of $su(2)$, then there are two solutions to the Weyl equation (<ref>) with finite $L^2$-norm, that is, there are two supersymmetric ground states. Thus, the Berry connection (<ref>) is a $U(2)$ connection. If we rescale $\mu$ by an additive constant so that $\mu(0) = - \mu(\infty) $, and impose the condition $T^t(-\mu) = T(\mu)$, the Berry connection is also an $SU(2)$ connection <cit.>.
So far, we have only identified the Yang-Mills fields in the BPS monopole solution. It still remains to identify the Higgs field. This is done by defining an endomorphism $\Phi_{mn} (\vec x)$ on the vacuum bundle to be the matrix elements of the moment map,
\begin{eqnarray}
\Phi(\vec x)_{mn} = - \langle m(\vec x) | \mu (\phi^i, \bar \phi^{\bar i}) | n(\vec x) \rangle \label{higgsfield}
\end{eqnarray}
We postpone the physical interpretation of $\Phi_{mn}$ for the next section.
In the example where $\mathcal M = \mathbb {CP}^1$, this coincides with the usual construction of the Higgs field from Nahm data,
\begin{eqnarray}
\Phi (\vec x)_{mn} = - \int_{\mu(0)}^{\mu(\infty)} d\mu \left( \mu f_{m,1}^\dagger ( \vec x, \mu) f_{n,1} (\vec x, \mu)+ \mu f_{m,2}^\dagger (\vec x , \mu) f_{n,2} (\vec x, \mu) \right) \nonumber
\end{eqnarray}
It is a standard result that the $SU(2)$ connection $A_a$ and endomorphism $\Phi$ obtained in this way obey the BPS monopole equation $D_a \Phi = \frac 1 2 \varepsilon^{abc} F_{bc}$. In fact, there is a one-one correspondence between Nahm data and BPS monopole solutions <cit.>. The situation where the Nahm matrices are $1 \times1$ matrices (i.e. constant twisted masses) and the resulting BPS monopole solution has unit magnetic charge was studied by Sonner and Tong in <cit.>. What we have done is generalise the procedure of Sonner and Tong for Nahm matrices of arbitrary size, showing that the construction produces all multi-monopole solutions of the BPS equations.
Note that the role of the pole condition described above is only to ensure that there are two supersymmetric ground states for all values of $x^a$, so the vacuum bundle is of rank two. The BPS equation $D_a \Phi = \frac 1 2 \varepsilon^{abc} F_{bc}$, however, holds regardless of the residues of the poles of $T^a$ at $\mu = \mu(0)$ and $\mu(\infty)$. For example, if we set $T^a$ to be $2\times 2$ zero matrices on $\mathbb {CP}^1$, so the residues of the poles of $T^a$ at $\mu = \mu(0)$ and $ \mu(\infty)$ are direct sums of two copies of the 1-dimensional trivial representation of $su(2)$, then the system has four supersymmetric ground states, and $A_a$ and $\Phi$ are an $SU(2) \times SU(2)$ connection and endomorphism on a rank four complex vector bundle over $\mathbb R^3$, equal to the direct sum of two $SU(2)$ monopoles of unit magnetic charge.
The BPS equation also holds regardless of the choice of the Kähler manifold $\mathcal M$ (although for general $\mathcal M$ there is no guarantee that $A_a$ and $\Phi$ will be non-singular); demonstrating this will be the goal of the next section.
§ THE EFFECTIVE ACTION AND THE BPS MONOPOLE EQUATIONS
In this final section, we integrate out the fast degrees of freedom in our model, that is, we integrate out the fields in the chiral multiplets, leaving an effective action for the fields in the vector multiplet. We will see that both the Berry connection $A_a(\vec x)_{mn}$ and the endomorphism $\Phi(\vec x)_{mn}$ defined in the previous section will appear in this effective action, and the requirement that this effective action is invariant under $\mathcal N = 4$ supersymmetry forces $A_a$ and $\Phi$ to obey the BPS monopole equations. Our presentation here roughly follows <cit.>, though the idea of using supersymmetry to constrain connection terms in an effective action can be traced back to <cit.>. (In fact, the result in <cit.> is a special case of the result presented here, with $\mathcal M = \mathbb C$ and $T^a$ equal to the $1\times 1$ zero matrices; in this setup, the vacuum bundle is a rank one vector bundle, singular at the origin in $\mathbb R^3$, and the connection $A_a$ is the Dirac monopole, while the endomorphism $\Phi$ obeys $\partial_a \Phi = \frac 1 2 \varepsilon^{abc} F_{bc}$.)
A subtlety in the computation of this effective action is that the system has multiple, degenerate, ground states. Therefore, the effective action depends not only on the vector multiplet fields, but also on the variable $\chi_m(t)$ that indicates which ground state the system is in at any moment in time. (Alternatively, one can present the effective action as a matrix-valued weight in the path integral; this equivalent approach was taken in <cit.>.)
One can check by dimensional analysis <cit.> that the effective action contains two types of bosonic terms at lowest order in the derivative expansion: a term proportional to $\dot x^a$, and another proportional to $D$. The first of these terms can be deduced from the discussion in the previous section. We already know that, as $x^a$ varies over time, $\chi_m$ obeys the equation of motion (<ref>). The action that gives rise to such an equation of motion is
\begin{eqnarray}
L_{\dot{\vec x}} = i \chi^\dagger \dot \chi +\beta_t (\chi^\dagger \chi - 1 ) + \dot x^a \left( \chi^\dagger A_a(\vec x) \chi \right) \nonumber
\end{eqnarray}
and this provides the term in the effective action proportional to $\dot x^a$. (Note that we have introduced an additional variable $\beta_t$, which acts as a Lagrange multiplier imposing the normalisation $\chi^\dagger \chi = 1$; we used a similar trick with $\varphi$ and $\alpha_t$ when we constructed the original Lagrangian, equation (<ref>).)
Next, rather than allowing $x^a$ to vary, we consider instead what happens if we allow $D$ to acquire a non-zero, but still time-independent, vacuum expectation value. By examining the original Lagrangian, we learn that turning on a vacuum expectation value for $D$ shifts the energy of the ground states by
\begin{eqnarray}
\Delta H = D \langle \mu \rangle \nonumber
\end{eqnarray}
This energy shift can be written in terms of the endomorphism $\Phi_{mn}(\vec x)$ defined in (<ref>) as
\begin{eqnarray}
\Delta H = - D \left(\chi^\dagger \Phi(\vec x) \chi \right) \nonumber
\end{eqnarray}
It follows that the second of the terms in the effective action, the term depending on $D$, is
\begin{eqnarray}
L_{D} = + D \left( \chi^\dagger \Phi(\vec x) \chi \right) \nonumber
\end{eqnarray}
All that remains is to find the fermionic terms in the effective action. These are constrained by the requirement that the effective action must be invariant under the supersymmetry transformations for $x^a$, $\lambda$ and $D$ in (<ref>). As shown in <cit.>, a supersymmetric completion of $L_{\dot{\vec x}} + L_D$ only exists if $A_a$ and $\Phi$ obey
\begin{eqnarray}
D_a \Phi = \frac 1 2 \varepsilon^{abc} F_{bc} \label{BPS}
\end{eqnarray}
\begin{eqnarray}
D_a \Phi = \frac \partial {\partial x^a} \Phi - i [A_a, \Phi ], \ \ \ \ \ \ F_{bc} = \frac \partial {\partial x^b} A_c - \frac \partial {\partial x^c} A_b - i [A_b , A_c] \nonumber
\end{eqnarray}
and, provided equation (<ref>) is obeyed, the unique possibility for the effective action is
\begin{eqnarray}
L = && \frac 1 {2e^2} \left( \dot x^a \dot x^a + i \bar \lambda \dot \lambda + D^2 \right)
+ i \chi^\dagger \dot \chi +\beta_t (\chi^\dagger \chi - 1 ) \nonumber \\ && + \dot x^a \left( \chi^\dagger A_a \chi \right)+ \frac 1 2 (\bar \lambda \sigma^a \lambda ) \left( \chi^\dagger D_a \Phi \chi \right) + D \left(\chi^\dagger \Phi \chi \right) \nonumber
\end{eqnarray}
Of course, equation (<ref>) is the BPS monopole equation.
It is pleasing that both the Nahm equation and the BPS equation drop out of our quantum-mechanical model as conditions for $\mathcal N = 4$ supersymmetry: the Nahm equation ensures that our matrix-valued version of the twisted mass deformation is supersymmetric, while the BPS equation does the same for the effective action for the vector multiplet. To impose $\mathcal N = 4$ supersymmetry, it is necessary to think of the Nahm matrices as being defined over a Kähler manifold $\mathcal M$ with a holomorphic isometry, rather than on a real interval; the Nahm matrices then form a triplet under the Lefschetz $SU(2)_R$ action. When $\mathcal M$ is $\mathbb {CP}^1$, the steps towards the computation of the effective action are in correspondence with the steps in the standard Nahm construction of BPS monopoles. But the arguments presented here imply that the procedure is more general: starting from any arbitrary Kähler manifold with a holomorphic isometry, and from any solution of the Nahm equations, there is a canonical way to construct a (possibly singular) connection and endomorphism on a complex vector bundle over $\mathbb R^3$ obeying the BPS monopole equations.
§ ACKNOWLEDGEMENTS
The author would like to thank David Tong for a careful reading of this article. The author is supported by Gonville and Caius College and the ERC Grant agreement STG 279943.
berry M. V. Berry, “Quantal Phase Factors Accompanying Adiabatic Changes,” Proc. R. Soc. A 392 (1984) 45.
sonner1 J. Sonner and D. Tong, “Non-Abelian Berry Phases and BPS Monopoles,” Phys. Rev. Lett. 102 (2009) 191801, arXiv:0809.3783.
thooft G. 't Hooft, “Magnetic Monopoles in Unified Gauge Theories,” Nucl. Phys. B 79 (1974) 276.
polyakov A. M. Polyakov, “Particle Spectrum in Quantum Field Theory,” JETP 20 (1974) 194.
sonner2 J. Sonner and D. Tong, “Berry Phase and Supersymmetry,” JHEP 0901 (2009) 063, arXiv:0810.1280.
bogomolny E. B. Bogomolny, “The Stability of Classical Solutions,” Sov. J. Nucl. Phys. 24 (1976) 49.
ps M. K. Prasad and C. M. Sommerfield, “An Exact Classical Solution for the 't Hooft Monopole and the Julia-Zee Dyon,” Phys. Rev. Lett. 35 (1975) 760.
cecotti S. Cecotti and C. Vafa, “Topological-Antitopological Fusion,” Nucl. Phys. B 367 (1991) 359.
smilga A. Smilga, “Vacuum Structure in the Chiral Supersymmetric Quantum Electrodynamics,” JETP 64 (1986) 8.
denef F. Denef, “Quantum Quivers and Hall/Hole Halos,” JHEP 0210 (2002) 023,
nahm W. Nahm, “A Simple Formalism For The BPS Monopole,” Phys. Lett. B 90 (1980) 413.
corrigan E. Corrigan and P. Goddard, “Construction of Instanton and Monopole Solutions and Reciprocity,” Ann. Phys. 154 (1984) 253.
hitchin N. Hitchin, “On the Construction of Monopoles,” Commun. Math. Phys. 89 (1983) 145.
nakajima H. Nakajima, “Monopoles and Nahm's Equations,” in Einstein Metrics and Yang-Mills Connections, ed. T. Mabuchi and S. Mukai (1990) 193.
donaldson S. K. Donaldson, “Nahm's Equations and the Classification of Monopoles,” Commun. Math. Phys. 96 (1984) 387.
hitchinspec N. Hitchin, “Monopoles and Geodesics,” Commun. Math. Phys. 83 (1982) 579.
diaconescu D.-E. Diaconescu, “D-branes, Monopoles and Nahm Equations,” Nucl. Phys. B 503 (1997) 220, arXiv:hep-th/9608163.
hashimoto K. Hashimoto and S. Terashima, “Stringy Derivation of Nahm Construction of Monopoles,” JHEP 0509 (2005) 055, arXiv:hep-th/0507078.
dancer A. Dancer, “Nahm's Equations and Hyperkähler Geometry,” Commun. Math. Phys. 158 (1993) 545.
witten E. Witten, “Geometric Langlands and the Equations of Nahm and Bogomolny,” arXiv:0905.4795.
gaiotto S. Cecotti, D. Gaiotto and C. Vafa, “tt$^\star$ Geometry in 3 and 4 Dimensions,” JHEP 1405 (2014) 055, arXiv:1312.1008.
fil S. Fedoruk, E. Ivanov and O. Lechtenfeld, “Nahm Equations in Supersymmetric Mechanics,” JHEP 1206 (2012) 147, arXiv:1204.4474.
wittensusy E. Witten, “Constraints on Supersymmetry Breaking,” Nucl. Phys. B 202 (1982) 253.
bagger J. Bagger and E. Witten, “The Gauge Invariant Supersymmetric Nonlinear Sigma Model,” Phys. Lett. B 118 (1982) 103.
gomis J. Gomis and F. Passerini, “Wilson Loops as D3-Branes,” JHEP 0701 (2007) 097, arXiv:hep-th/0612022.
me1 D. Tong and K. Wong, “Monopoles and Wilson Lines,” JHEP 1406 (2014) 048, arXiv:1401.6167.
me2 D. Tong and K. Wong, “ADHM Revisited: Instantons and Wilson Lines,” Phys. Rev. D 91 (2015) 026007, arXiv:1410.8523.
ivanov E. Ivanov, M. Konyushikhin and A. Smilga, “SQM with Non-Abelian Self-Dual Fields: Harmonic Superspace Description,” JHEP 1005 (2010) 033, arXiv:0912.3289.
ivanov2 E. Ivanov and M. Konyushikhin, “N = 4, 3D Supersymmetric Quantum Mechanics in Non-Abelian Monopole Background,” Phys. Rev. D 82 (2010) 085014, arXiv:1004.4597.
callias C. Callias, “Index Theorems on Open Spaces,” Commun. Math. Phys. 62 (1978) 213.
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1511.01020
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Galaxy morphologies and star-formation
rates depend on environment. Galaxies in
underdense regions are generally star-forming and disky whereas
galaxies in overdense regions tend to be early-type and not actively
forming stars. The mechanism(s) responsible for star-formation
quenching and morphological transformation remain unclear,
although many processes have been proposed. We study the dependence
of star-formation and morphology on X-ray luminosity for galaxies in
Sloan Digital Sky Survey Data Release 7 (SDSS-DR7) groups and
clusters. While controlling for stellar and halo mass
dependencies, we find that galaxies in X-ray strong groups and
clusters have preferentially low star-forming and disk
fractions – with the differences being
strongest at low stellar masses. The trends that we observe do not
change when considering only galaxies found within or outside of the
X-ray radius of the host group. When considering central
and satellite galaxies separately we find that this dependence on
X-ray luminosity is
only present for satellites, and we show that our
results are consistent with “galaxy stangulation” as a mechanism
for quenching these satellites. We investigate the dynamics of the
groups and clusters in the sample, and find that
the velocity distributions of
galaxies beyond the virial radius in low X-ray luminosity
halos tend
to be less Gaussian in
nature than the rest of the data set. This may be indicative of low
X-ray luminosity groups and clusters having
enhanced populations of star-forming
and disk galaxies as a result of recent accretion.
galaxies: clusters: general – galaxies: evolution – galaxies:
groups: – galaxies: statistics
§ INTRODUCTION
Numerous studies have shown a strong environmental dependence on the
star-forming and
morphological properties of galaxies <cit.>.
Low density
regimes tend to be dominated by star-forming, late-type galaxies
whereas high density areas, such as galaxy clusters, tend to be
primarily populated by quiescent, early-type galaxies. Within
individual clusters, galaxy morphologies tend to distribute as a
function of local density (or equivalently cluster-centric radius),
with high fractions of late-type galaxies being found at large radii
and the regions near the cluster core being dominated by early-types
<cit.>. This
effect has become known as the morphology-density relation.
While galaxies tend to distribute based on their star-forming and
morphological properties, the mechanism(s) responsible for the quenching of
star-formation and morphological transformations in galaxies are not
well constrained – although many
have been proposed. Both mergers and impulsive galaxy-galaxy
interactions (“harassment”) <cit.> can induce
star-burst events in galaxies leading to rapid consumption of gas
reserves and star-formation quenching. Within the virial radius of a
group or cluster the stripping of gas from galaxies becomes
efficient. Both the stripping of hot halo gas (“strangulation”)
<cit.> and cold gas stripping due to a dense
intra-cluster medium (“ram-pressure”) <cit.> can
quench star-formation. As well, tidal interactions can affect
gas reservoirs by transporting
gas from the galactic halo outwards which subsequently allows it to
more easily be stripped from the galaxy <cit.>.
On top of these environmental quenching mechanisms, previous authors
have found that secular processes, which depend on galaxy mass,
appear to play a significant role in star-formation quenching
<cit.>. The emergent picture for star
-formation quenching
appears to be some combination of environmental quenching mechanisms
and internal, secular processes. In particular, <cit.>
suggests that in the low redshift Universe, environmental quenching is
dominant for
galaxies with $M_\star \la 10^{10.5}\,\Msun$, whereas for galaxies
with $M_\star \ga 10^{10.5}\,\Msun$ mass quenching plays the more
important role.
While environmental and mass quenching within individual halos are
seemingly strong effects, it is important to realize that groups and
clusters are not isolated structures. In particular, galaxies can be
pre-quenched in group halos prior to infall into a
larger cluster. This “pre-processing” suggests that many galaxies
may already be quenched upon cluster
infall. Simulations have shown that between $\sim 25$ and $45$ per
cent of infalling cluster galaxies may have been pre-processed <cit.>. Observationally, <cit.> find that $\sim 25$
per cent of the infall population reside in subhalos for massive
clusters ($M_H \ga 10^{14.5}\,\Msun$). This pre-quenching of galaxies
in groups could potentially be driven by galaxy interactions and
mergers which are favoured in the group regime as a result of lower
relative velocities between member galaxies <cit.>.
An important method for studying the quenching mechanisms in groups
and clusters is to study the dependence of the star-formation and
morphological properties of galaxies on the conditions of their host
halo (e.g. halo mass, X-ray luminosity, etc.). In particular, if
quenching mechanisms depend on the density of the intra-group/cluster medium
(IGM/ICM) – for example, ram-pressure stipping of cold gas – then one
would expect to see galaxy populations which are preferentially
passive in halos with high X-ray
luminosities. Such correlations have been looked for in previous
studies, primarily within cluster environments.
In particular, <cit.> finds no
positive correlation between the fraction of old galaxies and X-ray
gas density. <cit.> conclude that the level of
star-formation found in their “low-$L_X$” sample is consistent with
levels seen in their CNOC1 sample consisting of higher mass clusters.
<cit.> and <cit.> both study the fractions of blue
galaxies at intermediate redshifts and find no discernible trend
between blue fraction and X-ray luminosity. Using multivariate
regression <cit.> find that cluster star-formation
depends on cluster richness but find no additional dependence on X-ray
luminosity. In addition, they find no significant correlation between
star-forming fraction and any global cluster property ($M_{200}$,
$\sigma_v$, $N_{\mathrm{gal}}$, and $L_X$). <cit.> find no
dependence of blue fraction on X-ray
luminosity and the only slight dependence they find between disk fraction and
X-ray luminosity is within the central and most dense regions.
Conversely, <cit.> find that galaxies in their “low-$L_X$” sample
have preferentially high disk fractions compared to galaxies in their
“high-$L_X$” sample. <cit.> find that the
bulge-dominated fraction for galaxies in high X-ray luminosity
clusters is higher than for those in low X-ray luminosity clusters.
In contrast with their star-formation results, <cit.> do
find a significant anti-correlation
between blue fraction and X-ray luminosity.
Finally, <cit.> find an anti-correlation between blue
fraction and X-ray temperature for galaxies in intermediate redshift
In this paper we re-visit the dependence of galaxy star-formation and
morphological properties on the X-ray luminosity of the host halo.
Specifically, as a result of the large SDSS X-ray sample presented in
<cit.>, we are able to control for stellar mass, halo
mass, and radial dependencies through fine-binning of the data set.
This allows us
to more directly investigate the effect of X-ray
luminosity on galaxies in different environments.
The results of this study are presented as follows. In
<ref> we briefly describe the SDSS group catalogues
utilized in this work, as well as the star-formation and
morphology catalogues which we match to the group data set. In
<ref> we present the primary results
of this paper, specifically, the differences between star-forming and
morphological trends in environments with different X-ray luminosities. In
<ref> we provide a discussion of the results
presented in this paper. Finally, in
<ref> we provide a summary of the key results and make
concluding statements.
In this paper we assume a $\Lambda$ cold dark matter cosmology with
$\Omega_M=0.3$, $\Omega_\Lambda=0.7$, and
§ DATA
§.§ Yang group catalogue
This work relies heavily on the group catalog of
<cit.>. The Yang group catalogue is constructed by applying
the iterative halo-based group finder of <cit.> to the New
York University Value-Added Galaxy Catalogue (NYU-VAGC;
), which is based on the Sloan Digital Sky Survey
Data Release 7 (SDSS-DR7; ). The Yang group
catalogue has a wide range of halo masses, spanning from
$\sim10^{12}\,\Msun$ to $\sim10^{15}\,\Msun$. The catalogue
contains both objects which would be classified as groups
($10^{12} \la M_H \la 10^{14}\,\Msun$) and as
clusters ($M_H \ga 10^{14}\,\Msun$), however for brevity we will
refer to all systems as groups regardless of mass.
Groups are initially
populated using the traditional friends-of-friends (FOF) algorithm
<cit.>, as well
as assigning galaxies not yet linked to FOF groups as the centres of
potential groups. Next, the characteristic luminosity, $L_{19.5}$,
defined as the combined luminosity of all group members with
$^{0.1}M_r-5\log h \le -19.5$, is calculated for each group. Using the
value of $L_{19.5}$ along with an assumption for the group
mass-to-light ratio, $M_H/L_{19.5}$, a tentative halo mass is assigned
on a group-by-group basis. The tentative halo mass is used to
calculate a virial radius and velocity dispersion for each group,
which are then used to add or remove galaxies from the system.
Galaxies are assigned to groups under the
assumption that the distribution of galaxies in phase space follows
that of dark matter particles – the distribution of which is assumed
to follow a spherical NFW profile <cit.>. This process is
iterated until the group memberships no longer change.
Final halo masses given in the Yang group catalogue
are determined using the ranking of the characteristic stellar mass,
$M_{\star,\mathrm{grp}}$, and assuming a relationship between
$M_H$ and $M_{\star,\mathrm{grp}}$ <cit.>.
$M_{\star,\mathrm{grp}}$ is defined by Yang et al. as
\begin{equation}
M_{\star,\mathrm{grp}} = \frac{1}{g(L_{19.5},\,L_{\mathrm{lim}})}\sum_i
\frac{M_{\star,i}}{C_i},
\end{equation}
where $M_{\star,i}$ is the stellar mass of the $i$th member
galaxy, $C_i$ is the completeness of the survey at the position of
that galaxy, and $g(L_{19.5},\,L_{\mathrm{lim}})$ is a correction
factor which accounts for galaxies missed due to the magnitude limit
of the survey. The statistical error in $M_H$ is on the order of
$0.3\,\mathrm{dex}$ and mostly independent of halo mass <cit.>.
§.§ SDSS X-ray catalogue
To study the X-ray properties of the group sample, we utilize the SDSS
X-ray catalogue of <cit.>.
Wang et al. use ROSAT All Sky
Survey (RASS) X-ray images in conjuction with optical groups
identified from SDSS-DR7 <cit.> to estimate X-ray
luminosities around $\sim65\,000$ spectroscopic groups..
To identify X-ray luminosities for individual groups, the algorithm of
<cit.> is employed. Beginning from an optical group, the
most massive galaxies (MMGs) of that group are identified – up to
four MMGs are kept. The RASS fields in which the MMGs reside are then
identified, and an X-ray source catalogue is generated in the
$0.5-2.0\,\mathrm{keV}$ band. The maximum X-ray emission density
point is used to identify the X-ray centre of the group, and any X-ray
sources not associated with the group (e.g. point source quasars or
stellar objects cross-matched from RASS and SDSS-DR7), within one virial radius,
are masked out. Values for the X-ray background, centred on the X-ray
centre, are determined and subtracted off and the X-ray luminosity, $L_X$, is
calculated by integrating the source count profile within the X-ray
Determining X-ray luminosities in this manner is succeptible to
“source confusion”. Due to
projection it is possible for more than one group to contribute to the
X-ray emission within the X-ray
radius, leading to an overestimation of the X-ray luminosity for a
given group. To account for this effect Wang et al. calculate the
“expected” average X-ray flux, $F_{X,i}$, for each group using the average
$L_X - M_H$ relation taken from <cit.>. They then
calculate the sum of the expected fluxes from each group for
multi-group systems and
determine the contribution fraction, $f_{\mathrm{mult},i}$, for each
group defined as,
\begin{equation}
f_{\mathrm{mult},i} = F_{X,i}/\Sigma_i F_{X,i}.
\end{equation}
The contribution factor will approximate the fraction of the observed
X-ray luminosity intrinsic to the individual group in question,
therefore applying this fraction to each group will act to debias
the measured X-ray luminosity from source confusion contamination.
Within the Wang catalogue 817 groups have $S/N > 3$, compared to the
total of 34522 groups with positive detections (positive count rates
after background subtraction) and $S/N > 0$. We run our analysis for
groups with $S/N >3$ as well as groups with $S/N > 0$ and find that
our choice of signal-to-noise cut does not change the trends that we
observe, therefore we focus on the total sample
($S/N > 0$) to ensure a sample size which is large enough to finely
bin the data in various properties simultaneously.
§.§ Final data set
Density contours for log X-ray luminosity versus log halo
mass. Dashed line
corresponds to the linear least-squares best-fit relationship.
Smoothed distributions for halo mass and X-ray
luminosity within the sample. Distributions
are shown for both the X-ray strong (red, dashed) and the X-ray weak
(blue, solid) samples.
Shaded regions correspond to $2\sigma$ confidence intervals
obtained from random bootstrap resampling.
Left: star-forming fraction versus stellar mass for the four
X-ray luminosity quartiles of the data sample. Right:
disk fraction versus stellar mass for the four
X-ray luminosity quartiles of the sample. Error bars
correspond to 1$\sigma$ Bayesian binomial confidence
intervals given in <cit.>.
To obtain the final data set, we match the Wang
SDSS X-ray catalogue to the Yang SDSS group catalogue, giving us
both optical and X-ray group properties for the sample. To
obtain individual galaxy properties we further match the data set to
various public SDSS catalogues, as follows.
We utilize stellar masses given
in the NYU-VAGC, which are computed following the methodology of
To obtain star-formation rates (SFRs) and specific star formation
rates ($\mathrm{SSFR} = \mathrm{SFR}/M_{\star}$) we match the catalogue of
<cit.> to the sample. SFRs given by
Brinchmann et al. are determined using emission line fluxes whenever
possible, however in the case of no clear emission lines or
contamination from active galactic nuclei, SFRs are determined using
the strength of the 4000 Å break ($D_n4000$) <cit.>.
We obtain galaxy morphologies from the catalogue of
<cit.>. Simard et al. perform two-dimensional bulge + disk
decompositions for over one million galaxies from the Legacy area of
the SDSS-DR7, using three different fitting models: a pure Sérsic
model, a bulge + disk model with de Vaucouleurs ($n_b = 4$) bulge, and
a bulge + disk model with a free $n_b$. To distinguish between disky
and elliptical galaxies we utilize the galaxy Sérsic index, $n_g$,
from the pure Sérsic decomposition. We also use the $V_{\mathrm{max}}$
weights given by Simard et al. to correct for the incompleteness of
our sample.
We calculate group-centric distances for each galaxy using the
redshift of the group and the angular seperation between the galaxy and the
luminosity-weighted centre of its host group. We normalize all of the
galaxy radii by the virial radius of the host group, $R_{180}$, which
we calculate as in <cit.>
\begin{equation}
R_{180} =
+ z_g)^{-1},
\end{equation}
where $z_g$ is the redshift of the group center.
Star-forming (solid lines) and disk (dashed lines) fractions
versus stellar mass, for different halo mass bins and the XRW
(blue) and XRS (red) samples. Error bars
correspond to 1$\sigma$ Bayesian binomial confidence
intervals given in <cit.>.
The final data set includes groups with halo masses ranging between
$10^{13} - 10^{15}\,\Msun$, and galaxies with stellar masses ranging from
$10^9 - 10^{11.3}\,\Msun$. Group X-ray luminosities in the data set are
between $10^{39.6} - 10^{46.4}\,\mathrm{erg}\,\mathrm{s^{-1}}$, with a median value
of $10^{43.9}\,\mathrm{erg}\,\mathrm{s^{-1}}$, and are strongly
correlated with halo
mass (see Fig. <ref>). We do not make an explicit radial
cut, however over 99 per cent of member galaxies fall within 1.5
virial radii. Our final sample contains 3902 low redshift ($z<0.1$)
groups housing a collective 41173 galaxies. The
catalogue of
<cit.> contains $\sim 35000$ groups, the fact that the
final sample in this work is significantly smaller is for two
reasons. Firstly,
we restrict our sample to redshifts smaller than 0.1 which reduces
the number of groups from $\sim 35000$ at $z<0.2$ to $\sim 18000$ at
$z<0.1$. The second important cut is that we require $10^{13} < M_H < 10^{15}\,\Msun$,
a number of groups in the Wang et al. catalogue have halo masses, $M_H
< 10^{13}\,\Msun$ (where halo masses have been obtained from the catalogue of
), this cut reduces the remaining number of groups
from $\sim 18000$ to $\sim 3900$. It should be noted that the
majority of the $M_H < 10^{13}\,\Msun$ groups removed from the data
set are groups with very low membership.
To determine the effect of X-ray luminosity on star-formation and
morphology we consider two X-ray luminosity samples for the majority
of our analysis, which we refer to as the X-ray weak (XRW) and X-ray
strong (XRS) samples. Similar to Wang et al., we define the XRS sample
to consist of
all galaxies found above the best-fit $\log M_H - \log L_X$ line (see
Fig. <ref>), and correspondingly the XRW sample consists of
all galaxies found below the $\log M_H - \log L_X$ trend line. This leads to an
approximately equal number of galaxies within the XRW and XRS
samples. We also performed
our analysis with a cut between the two X-ray samples
at the median X-ray luminosity of the data
set, as well as
defining the two samples using the first and fourth quartiles, however
these alternative definitions of the two X-ray samples do not
change the trends that we observe.
Smoothed distributions for halo mass and X-ray luminosity are shown in
Fig. <ref> for both X-ray luminosity samples. Density
distributions are calculated using the
function in the statistical computing language
<cit.>[http://www.R-project.org/] using a
Gaussian kernel.
We study the dependence of star-formation rates and morphology on
stellar mass by binning the data by stellar mass and calculating the
disk and star-forming fractions for each bin. Binning by stellar mass
is important to account for the systematic dependence of star-formation and
morphology on stellar mass
<cit.>. Additionally, as the relative
balance between environmental and mass quenching is not well
understood, it is important to investigate the effects of environment
at a given stellar mass.
We define the star-forming fraction,
$f_{SF}$, as the fraction of galaxies in
each bin with $\log \mathrm{SSFR} > -11$. <cit.> show that at
low redshift the division between the red sequence and the blue cloud
is found at $\log \mathrm{SSFR} \simeq -11$ across a wide range of halo masses.
For each stellar mass bin the star-forming fraction is given by
\begin{equation}
f_{SF} =
\frac{V_{\mathrm{max}}\;\text{weighted}\;\#\;\text{galaxies}\;\text{with}\;\log SSFR>-11}{V_{\mathrm{max}}\;\text{weighted}\;\text{total}\;\#\;\text{galaxies}}.
\end{equation}
Similarly we define the disk fraction, $f_D$, as the fraction of
galaxies in each bin with Sérsic index, $n < 1.5$. For each
stellar mass bin this is given by
\begin{equation}
f_D =
\frac{V_{\mathrm{max}}\;\text{weighted}\;\#\;\text{galaxies}\;\text{with}\;n<-1.5}{V_{\mathrm{max}}\;\text{weighted}\;\text{total}\;\#\;\text{galaxies}}.
\end{equation}
We also ran our analysis using a dividing cut at Sérsic indices of
$n=1.0$ and
$n=2.0$ to define a disk galaxy, however using
these alternative definitions for a disk galaxy does not alter the
trends that we observe.
§ RESULTS
Star-forming (solid lines) and disk (dashed lines)
fractions versus stellar mass, for galaxies outside of their host
X-ray radius and for different halo mass bins and
the two $L_X$ samples. Error bars
correspond to 1$\sigma$ Bayesian binomial confidence
intervals given in <cit.>.
Same as Fig. <ref> for galaxies inside of
their host X-ray radius.
SF and disk excess versus stellar mass for both galaxies
within (purple, solid) and
outside (green, dashed) of the X-ray radius. Panels a-d show SF excess
for four halo mass bins and panels e-h show disk excess for
four halo mass bins. Shaded regions represent
$1\sigma$ confidence intervals.
§.§ Star-forming and morphology trends in strong and weak $L_X$
To investigate the effect of X-ray luminosity on galaxy properties, in
Fig. <ref> we show star-forming and disk fractions, as a function
of stellar mass, for subsamples corresponding to the four X-ray
luminosity quartiles of the data set.
Examination of Fig. <ref>a and <ref>b show
that star-forming and disk fractions follow a consistent marching order
with respect to X-ray luminosity. The disk and star-forming fractions
decrease as X-ray luminosity increases.
We note that the results in Fig. <ref> consider all halo
masses in the sample, however it has been found that galaxy morphology and
star-formation depend on local density and halo mass
(however also see: ). As shown in
Fig. <ref> the data show a strong correlation between X-ray
luminosity and halo mass, therefore we must determine if differences
shown in Fig. <ref> are simply a result of galaxies in
higher $L_X$ environments being housed
in preferentially high-mass halos.
To control for any potential halo mass effect, we further bin the data into
narrow halo mass bins and re-examine the dependence of galaxy
properties on X-ray
luminosity, considering now the XRW and XRS
samples from Fig. <ref>. Fig. <ref> shows
star-forming (solid) and disk (dashed) fractions as a function of
stellar mass for four different halo
mass bins – ranging from $10^{13} - 10^{15}\,\Msun$ with bin widths
of $0.5\,\mathrm{dex}$. Data are binned according to stellar mass and
markers are plotted at the median bin values. For each halo mass bin
we show star-forming and disk
fractions from the X-ray strong and X-ray weak samples.
For both star-forming and disk fractions we continue to see a residual trend
with X-ray luminosity, even after controlling for any halo mass
dependence: star-forming and disk fractions are systematically higher
in the XRW sample. We see the strongest trends in the intermediate
and high mass
halos. The difference between the
strong (red) and the weak (blue) X-ray luminosity samples is clearest at low stellar
mass, and in all halos the two samples converge at
moderate to high stellar mass.
§.§ Radial dependence of star-forming and morphology trends
Within host groups X-ray emission is concentrated at relatively small
group-centric radii, with X-ray emission generally extending out to
half a virial radius <cit.>. If the
trends we are observing are a result of increased gas density, we
would expect to see enhanced trends (i.e. a larger difference between
the XRS and XRW
samples) at small group-centric radii and suppressed trends at
large radii. To test this we further divide the data into subsets
corresponding to those galaxies that lie within the X-ray emission
radius (using the X-ray radius, $R_{Xray}$, given in ) and
those galaxies that lie outside of the X-ray radius. We again plot
star-forming/disk fraction versus stellar mass, in narrow halo mass bins,
for the large and small radius subsamples. The results of this analysis
are shown in Fig. <ref> & <ref>, where
the two figures correspond to disk fraction and star-forming fraction trends
for the large and small radius subsamples, respectively.
Examination of Fig. <ref> & <ref>
shows that for both galaxies found within their host halo's X-ray
radius, and those found outside, we still see an increase in
star-forming and disk fractions in the XRW sample – as before this
effect is strongest in the intermediate to high mass halos and at low
stellar mass. Also the disk and star-forming
fractions tend to be higher at large radii, which is consistent with
the morphology-density relation.
To further investigate if the increase
in star-forming and disk fractions in the XRW sample compared to the
XRS sample – which we will
refer to as the “SF excess” and the “disk excess” – depends on
whether you consider galaxies within or outside the X-ray radius, we
show SF and disk excess versus stellar mass in
Fig. <ref>. We quantitatively define
SF and disk excess as
\begin{align}
& \mathrm{SF}\;\mathrm{excess} = f_{SF}(XRW) -
f_{SF}(XRS) \label{eq:sf_excess} \\
& \mathrm{Disk}\;\mathrm{excess} = f_{D}(XRW) -
f_{D}(XRS) \label{eq:d_excess},
\end{align}
where $f_{SF}(XRW)$ and $f_{SF}(XRS)$ are the star-forming fractions in
the XRW and XRS samples respectively, and analogously for $f_{D}(XRW)$
and $f_{D}(XRS)$.
We find no radial dependence for SF and disk excess as the two radial
subsamples in Fig. <ref> show
overlap for all halo and stellar masses. With the exception in
Fig <ref>c where the SF
excess, for low-mass galaxies, is stronger for galaxies within
the X-ray radius.
§ DISCUSSION
We find that star-forming and disk fractions are systematically lower in
the XRS sample than galaxies in XRW
environments. This trend persists even upon controlling for any halo
mass dependence, however the observed difference between the XRS
and the XRW sample is not enhanced when considering
only those galaxies within the X-ray radius of the host halo.
There are two major observed effects which have been found to impact
the distributions of
early-type and late-type galaxies within cluster environments. The so
called “Butcher-Oemler” (BO) effect is the observational trend that
the blue fraction of cluster galaxies are positively correlated with redshift
<cit.>. However, it
should be noted that there is still debate when it comes to the
physical nature of the BO effect (for example, see:
). Since we are only
considering low-redshift $(z < 0.1)$ galaxies the BO effect should be
Smoothed radial distributions of galaxies in the XRW (blue,
solid) and XRS (red, dashed) samples.
Shaded regions correspond to $2\sigma$ confidence intervals
obtained from random bootstrap resampling.
The second major effect is the previously mentioned morphology-density
relationship. In order to determine if the morphology-density
relation is affecting the trends we observe, we must check if there are
significant differences
in the radial distributions of the XRS and the XRW
samples. For
instance, if the XRW sample is found at systematically high
group-centric radii compared to the XRS sample, then the
morphology-density relation could explain why we find systematically larger
star-forming and disk fractions in the XRW sample. In
Fig. <ref> we plot the smoothed radial
distributions for both the XRS and the XRW samples. We
see no systematic difference between the two distributions, in fact
they are nearly indistinguishable from one another. We
conclude that the two X-ray samples have radial distributions
which do not differ substantially from one another, and therefore any
observed differences
between the XRS and XRW samples are not being driven by
differing radial distributions.
§.§ AGN contamination
BPT identified AGN fraction versus stellar mass for our XRW and XRS
samples. Error bars correspond to $1\sigma$
Bayesian binomial confidence intervals given in <cit.>.
When considering X-ray properties of galaxy groups it is important to
ensure that the observed X-ray emission is due to the hot IGM and
not due to contamination from AGN or other X-ray sources. In
<cit.> bright point sources, such as stars and quasars, are
masked out, however it is still important to ensure that
our results are not being contaminated by galaxies housing non point
source AGN.
In Fig. <ref> we plot AGN fraction versus stellar mass for
the XRW and XRS samples. We use AGN classified by
<cit.>, which are identified using the location of
galaxies on the BPT diagram <cit.>. It should be noted
that <cit.> show that between 20 and 50 per cent
(depending on the dividing line between AGN and star-formings galaxies used) of
X-ray identified AGN fail to be classified as AGN on the BPT diagram.
We see that the AGN fraction tends to be
larger within the XRS sample, however at all stellar masses the number of AGN
galaxies is a modest fraction (less than five per cent) of the total
sample, for both XRS and XRW galaxies. Most relevant is the fact that
at low stellar mass the AGN
fraction is consistently below one per cent, for both the XRW and XRS
samples, and that the trends that we observe with X-ray luminosity are
exclusively seen at low stellar mass (e.g. Fig <ref>). We
examined disk and
star-forming fractions for a subsample of the data with galaxies
identified as AGN removed and
found that removing AGN galaxies from the sample does not change the
observed trends. Furthermore, we examined trends
after removing all groups that
house galaxies identified as AGN and again found no change in
the observed trends. Therefore we conclude that AGN are
not a significant
contributor to the observed trends in star-forming and disk fractions.
§.§ Implications for star-formation quenching
The relative importance of various galaxy quenching mechanisms is an
important, open question. Galaxy populations in groups
can be classified as either “central” (located at the centre
of the group dark matter halo) or “satellite”
galaxies. These two populations are
expected to evolve differently <cit.>, and
therefore when
attempting to elucidate information on the quenching of galaxies it is
important to consider centrals and satellites as distinct populations.
In Fig. <ref> we
plot SF and disk excess (Equations <ref> &
<ref>) versus stellar mass, considering
separately central and satellite galaxies. Central galaxies are
defined as the most massive group
galaxies and satellite galaxies are defined as all group galaxies which have
not been classified as centrals.
SF and disk excess versus stellar mass for both centrals
(gold, solid) and satellites (gray, dashed). Shaded regions
correspond to $1\sigma$ confidence
When considering satellite galaxies in Fig. <ref>a we find
that galaxies within the XRW sample have consistently larger
star-forming fractions at low stellar mass (SF excess $> 0$), while at
large stellar mass the
XRW and XRS samples are indistinguishable. When considering only
central galaxies we find that there is no
difference between the XRW and XRS samples (SF excess $\approx 0$) when
considering star-forming fraction. We observe qualitatively similar
trends for disk excess in Fig. <ref>b. This implies that
whatever effect
X-ray luminosity has on star-forming and morphlogical properties it
only affects satellite galaxies, central galaxies are insensitive to
the group X-ray properties. This is not surprising given that central
galaxies are massive, and we see no difference between the XRS and XRW
at large stellar mass.
One interpretation of the differences we observe between the XRW and
XRS samples would be to invoke the ram-pressure stripping of satellite
galaxies. The rate at which galaxies will lose gas through
ram-pressure stripping will increase in proportion to $L_X$
<cit.>. Therefore, if ram-pressure is an important
mechanism when it comes to the quenching of galaxies, a decrease
in star-forming fraction should be observed with increasing X-ray
luminosity. It be noted that
although we observe very similar trends for star-forming and disk
fractions, it is not clear whether ram-pressure stripping can
efficiently drive galaxy morphology transformations
from late to early type <cit.>. Prior studies
<cit.> have found
evidence of ram-pressure
stripping. We note as well that other studies <cit.> have found no clear trend between
star-forming or blue fractions and X-ray luminosity. At first glance
the results shown in Fig. <ref> are consistent with ram
pressure stripping; at low stellar masses there are lower star-forming
fractions in the XRS sample. One difference
between the results we observe and previous studies is that we narrowly bin our
data in stellar mass.
Since star-forming and morphological properties depend strongly on
stellar mass, any residual dependence on X-ray luminosity may be lost
without controlling for stellar mass. In addition our sample size is
significantly larger than most previous studies, so it may be that
trends with X-ray luminosity are subtle enough to be missed without
large statistics.
If the trends we detect are driven by ram-pressure we would expect a
radial dependence of our trends with X-ray
luminosity. The efficiency of ram-pressure stripping
is proportional to $\rho v^2$ <cit.>, where $\rho$ is the IGM density and $v$ is
the speed of the member galaxies. Since the IGM density is highest at
small group-centric radii, the efficiency of ram-pressure stripping
should increase towards small radii. In Fig. <ref> we
showed that the observed SF excess does not strongly depend on radius.
We conclude that this lack of radial dependence is
inconsistent with the ram-pressure stripping scenario.
Mean stellar metallicity versus stellar mass for
star-forming (blue, dashed) and passive (red, solid) galaxies,
divided by galaxies in
the XRS (top) and XRW (bottom) samples. Shaded regions correspond
to $1\sigma$ confidence intervals obtained from random bootstrap resampling.
Another often envoked mechanism for regulating star-formation is
“galaxy strangulation” <cit.>. Strangulation is a mechanism in
which the replenishment of cold gas onto galaxies is halted, which in
turn leads to galaxy quenching once the galaxy has exhausted its
existing cold gas reservoirs. The timescales over which a galaxy will
be quenched by strangulation are longer than the quenching times
associated with the direct stripping of cold gas reserves
(ram-pressure). Recently, <cit.> have argued that it
is possible to differentiate between strangulation and direct
stripping using metallicity differences between star-forming and
quiescent galaxy populations. We direct the reader to
<cit.> for a more complete discussion, however the main idea
is that quenching by strangulation will result in higher
metallicities for passive galaxies compared to star-forming galaxies.
This is a result of star-formation continuing even after the gas
supply has been halted which will increase stellar metallicity until
the cold gas reserves have been exhausted and the galaxy has therefore
been quenched. This trend in metallicity is not expected from direct
stripping, where star-formation shuts off quickly after the removal of
cold gas.
To investigate the effect of strangulation on the galaxy sample in
this study we follow Peng et al. and calculate mean stellar
versus stellar mass considering star-forming and passive galaxies
separately, for galaxies within our XRW sample as well as our XRS
sample. Metallicities are matched to our sample from the catalogue of
<cit.>, and mean metallicities are plotted in stellar
mass bins with widths of $0.15\,\mathrm{dex}$. Not all of the
galaxies within this sample have measured metallicities, therefore for
this aspect of the analysis our XRW and XRS samples are reduced to
10939 (52 per cent of total sample) and 8851 (44 per cent of total
sample) member galaxies, respectively.
In Fig. <ref> we see higher stellar
metallicities for passive galaxies compared to star-formers, which
we interpret as evidence for strangulation playing a significant role
in star-formation quenching. Of particular interest for this work is
the behaviour at low stellar mass which is where the dependence of
star-formation and morphology on X-ray luminosity is observed (see
Fig. <ref>). We see
a somewhat stronger strangulation signal (ie. difference between passive and
star-former metallicity) for galaxies in the XRS sample compared to
the XRW sample, at low stellar mass.
In light of this observed difference, it important
to note that compiling this subsample of
galaxies with measured metallicities does not affect all stellar
masses equally. Specifically, low-mass galaxies are
preferentially removed from the sample when matching to the
metallicity catalogue. In particular, 69 per cent of low-mass
($M_\star < 10^{9.5}\,\Msun$) galaxies in the XRS sample do not have
measured metallicities, whereas in the XRW sample 75 per cent of
low-mass galaxies do not have measured metallicities. Not only
are low-mass galaxies being preferentially lost, but the fraction
of low-mass galaxies being lost is slightly different between the
two X-ray samples. Therefore, although the results in
Fig. <ref> are consistent with strangulation – and more
specifically, somewhat stronger strangulation at the low-mass end of
the XRS sample – we suggest that this trend be interpretted with
caution as completeness differences could be playing some role.
§.§ Group evolutionary/dynamical state
The dynamical state of galaxy groups is an important evolutionary
indicator and can potentially have an impact on galaxy
properties. Trends with X-ray luminosity may reflect that the XRW and
XRS samples have different dynamical properties as it is expected that
more evolved groups with relaxed dynamics would be more X-ray luminous
Theoretically the velocity distribution of galaxies
within a group in dynamical equilibrium should have a characteristic
Gaussian shape. Groups lacking this Gaussian distribution can
therefore be considered as being unevolved, dynamically young
systems. To investigate the dependence on the dynamical state of the
groups in our data set we follow the procedure of
<cit.> and apply the Anderson-Darling normality (ADN) test to
the velocity distributions of the galaxies in the group sample. The ADN test
is a non-parametric test which compares the cumulative
distribution function (CDF) of the data to the CDF of a normal
distribution to determine the probability (p-value) that the
difference between the distribution of the data and that of a
Gaussian is as large as observed (or larger), under the assumption that the data
is in fact normally distributed. For our dynamical analysis we use a
subset of the data
consisting of only those groups with eight or more members (31820
galaxies in 1456 groups), in order to
ensure reasonable statistics when applying the ADN test.
To obtain values for the ADN statistic for each
of our groups we employ the
function in the statistical computing language
<cit.> – large values of the
ADN statistic are indicative of less Gaussian distributions.
Initially, we examine the dynamical states of galaxies within the XRW
and XRS samples globally (i.e. no radial cuts) and we find no
systematic differences between the dynamical states of XRW and XRS
galaxies. <cit.> study the difference between X-ray underluminous
Abell (AXU) clusters and normal Abell clusters. They find that while
both AXU and normal Abell clusters show Gaussian velocity
distributions within the virialized region $(R < 1.5\,R_{200})$,
within the exterior regions $(1.5\,R_{200} \le R \le 3.5\,R_{200})$
the AXU clusters show sharply peaked, non-Gaussian velocity
distributions. The authors interpret these leptokurtic velocity
distributions in the outer cluster regions as evidence that AXU
clusters have experienced recent
accretion/merging. If the XRW groups have experienced more recent
accretion of galaxies from
the field and smaller groups than the XRS groups, then this could
contribute to the dependence we
observe between star-forming and disk fractions on X-ray luminosity.
Galaxies in underdense regions (the field, low-mass groups) have been
found to be preferentially star-forming with late-type morphologies.
Accordingly, groups experiencing recent accretion may contain more
star-forming, late-type, galaxies when compared to groups which are
dynamically older.
To investigate this possibility we study the dynamical states of
groups in both the XRW and XRS samples, and divide member galaxies
into two radial subsamples: those found in the inner regions $(R <
R_{180})$ of their host group, and those found in the outer regions
$(R \ge R_{180})$ of their host group. This is similar to the
analysis performed by <cit.>. Instead of making an
arbitrary, discrete cut to define Gaussian and
non-Gaussian groups we treat the AD statistic
values as continuous and compare
the distributions of ADN statistics from the
four subsamples (XRW inner, XRW outer, XRS inner, XRS outer) to
determine whether there are any significant differences in
dynamical states.
To quantitatively compare the distributions we utilize the two-sample
Anderson Darling (AD2) test. The AD2 test is similar to the ADN test,
however instead of comparing observed data to the
normal distribution, it compares the CDFs of two data samples to
determine whether they are drawn from the same underlying
distribution. We apply the AD2 test to the distributions of ADN
statistic values for the XRW and XRS samples to
determine if the dynamical
states vary between the inner and outer regions. To perform the AD2
test between the subsamples we use the
function in the statistical computing language
We find
no evidence $(\text{p-value} = 0.38)$ for different dynamical
states in the inner and outer regions of the XRS sample, however
for the XRW sample we find strong evidence $(\text{p-value} = 3
\times 10^{-7})$ that the dynamical state of galaxies in the outer
region is different from those in the inner region. When we examine
the distributions of ADN statistics for the four
subsamples we find that
the ADN statistic values for the XRW outer
subsample are systematically higher
than for the other three subsamples. This suggests that the
velocity distributions for galaxies outside of the virial radius in the XRW
sample are less Gaussian than the rest of the data set.
This result is consistent with <cit.>, who find
non-Gaussian velocity distributions for galaxies in the outer regions
of X-ray underluminous Abell clusters. This result supports the
notion that the increased number
of star-forming and late-type galaxies we observe in the XRW
sample can potentially be explained by underluminous X-ray groups experiencing
recent accretion of field galaxies and small galaxy groups, as
this recent accretion can give rise to less Gaussian velocity
distributions in the exteriors of these groups.
We do note that it remains difficult to simultaneously explain the
dynamical results together with the fact that we observe no dependence of
SF and disk excess on radius (Fig. <ref>).
§ SUMMARY & CONCLUSIONS
We have used a sample of galaxies taken from X-ray
emitting groups and clusters in the SDSS to study the effect of X-ray
luminosity on galaxy star-formation and morphological properties. Using a data
set spanning a large range in stellar mass ($10^9 -
10^{11.3}\,\mathrm{M_\odot}$), halo mass ($10^{13} -
10^{15}\,\mathrm{M_\odot}$), and X-ray luminosity ($10^{39.6} -
10^{46.4}\,\mathrm{erg}\,\mathrm{s^{-1}}$) we have investigated the
differences between disk and star-forming fractions within different X-ray
environments. The main results of this paper are as follows:
* Star-forming and disk fractions are preferentially lower
within the X-ray strong sample when
compared to galaxies within the X-ray weak sample – this trend
remains after controlling for any halo mass dependence.
* This difference between the X-ray strong and X-ray weak samples
is most apparent at intermediate to high halo mass
and at low stellar mass.
* The differences we observe between the X-ray weak and X-ray
strong samples do not depend on whether we consider galaxies
inside of, or outside their host halo's X-ray radius.
* The enhancement of star-forming and disk fractions we observe
in the X-ray weak sample is present for satellites but not central
galaxies, which is not surprising given that the difference
between X-ray samples is only seen at low stellar mass.
* Our results are consistent with quenching by
strangulation, in particular we see a somewhat stronger
strangulation signal at low stellar mass within the XRS sample.
* We find that in the X-ray weak sample the velocity
distributions of galaxies outside of the virial radius are less
Gaussian than galaxies within the virial radius. We find no
differences between the dynamical states of inner and outer
galaxies within the X-ray strong sample.
With the large
sample of SDSS X-ray and spectroscopic groups we are able to study
star-forming and disk fractions while simultaneously controlling for stellar
mass, halo mass, and radial depedencies, thereby allowing a robust
analysis of the effects of X-ray luminosity on star formation and
We find that galaxies outside the virial radius of X-ray underluminous
groups have dynamics which are less Gaussian than the other groups in
the sample. This may indicate that recent accretion onto low X-ray
luminosity groups contributes to an excess of star-forming, late-type
galaxies. The fact that the X-ray weak sample
shows weaker stangulation could simply be due to the lower $L_X$
environment reducing the efficiency of strangulation, or it could
be a result of recently accreted galaxies having had less time to
be quenched by environmental quenching mechanisms like
stangulation. Naively, one would
expect to observe a corresponding enhancement of star-forming,
late-type galaxies in the exteriors of low X-ray luminosity groups
compared to X-ray strong groups; however this is not observed. The
results presented in this work therefore require a detailed theoretical
treatment to fully explain the trends observed.
§ ACKNOWLEDGMENTS
We thank the anonymous referee for their various helpful comments and
suggestions which have improved the paper. IDR and LCP thank the National
Science and Engineering Research
Council of Canada for funding. The authors thank
F. Evans for matching together the various SDSS catalogues used in
this research. We thank X. Yang et al. for
making their
SDSS DR7 group catalogue publicly available, L. Simard et al. for the
publication of their SDSS DR7 morphology catalogue, J. Brinchmann et al. for
publication of their SDSS SFRs, G. Kauffmann et al. for the publication of
SDSS AGN galaxies, the NYU-VAGC
team for the
publication of their SDSS DR7 catalogue, and A. Gallazzi et al. for making
publicly available their SDSS metallicities. This research would not have
been possible without access to these public catalogues.
Funding for the SDSS has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science
Foundation, the U.S. Department of Energy, the National Aeronautics
and Space Administration, the Japanese Monbukagakusho, the Max Planck
Society, and the Higher Education Funding Council for England. The
SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions are the
American Museum of Natural History, Astrophysical Institute Potsdam,
University of Basel, University of Cambridge, Case Western Reserve
University, University of Chicago, Drexel University, Fermilab, the
Institute for Advanced Study, the Japan Participation Group, Johns
Hopkins University, the Joint Institute for Nuclear Astrophysics, the
Kavli Institute for Particle Astrophysics and Cosmology, the Korean
Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos
National Laboratory, the Max-Planck-Institute for Astronomy (MPIA),
the Max-Planck-Institute for Astrophysics (MPA), New Mexico State
University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States
Naval Observatory, and the University of Washington.
|
1511.00682
|
We consider the inclusion of the most important vibrational modes in the quantisation of the dodecahedral $B=7$ Skyrmion. In contrast to a rigid body quantisation, this formalism allows a spin $\frac{3}{2}$ state to lie below the spin $\frac{7}{2}$ state, in agreement with experimental data. There is also a low lying spin $\frac{1}{2}$ state and two spin $\frac{5}{2} $ states. We find that the excited spin $\frac{7}{2}$ state has a smaller root mean square charge radius than the other states. This prediction is an important signature of the Skyrme model, in conflict with more conventional nuclear models.
§ INTRODUCTION
The Skyrme model is a non-linear field theory of pions which admits soliton solutions called Skyrmions <cit.>. These are classically stable due to the topology of the system and each Skyrmion has a conserved topological charge, $B$. After quantisation Skyrmions are identified as nuclei with topological charge equal to baryon number.
The theory is non-renormalisable and so a first principles quantisation is beyond current methods. Instead, one must reduce the degrees of freedom in the problem to a finite number and quantise these. Each charge $B$ Skyrmion may be separated into $B$ charge one Skyrmions. These have six zero modes, three rotations and three translations. Thus to calculate quantities such as the binding energy of a nucleus one should take account of at least $6B$ degrees of freedom. Unfortunately this means quantising on a $6B$ dimensional space and little progress has been made, even for $B=2$ <cit.>. Instead, one must select a subset of modes.
The simplest idea is to only include the zero modes of the Skyrmion, those transformations which leave the static energy unchanged. These are the rotations and isorotations (we stay in the centre of mass frame, allowing us to ignore translations). This procedure ignores vibrational modes, dynamical oscillations around the Skyrmion. Zero mode quantisation has had some success, such as reproducing the energy spectra of some light nuclei <cit.> and a natural description of the Hoyle state <cit.>. However, there are also some failures. For example, the binding energies are all much too large. This is to be expected when we truncate the degrees of freedom from $6B$ to $6$.
Another failure of zero mode quantisation is the prediction of a spin $\frac{7}{2}$ ground state for the $^7$Be/$^7$Li isodoublet. The dodecahedral symmetry of the $B=7$ Skyrmion rules out low energy states with spin $\frac{1}{2}$, $\frac{3}{2}$ and $\frac{5}{2}$. In reality, experimental data show that all these states exist and the ground state has spin $\frac{3}{2}$. The first excited state of $^7$Li has spin $\frac{1}{2}$ and lies $0.5$ MeV above the ground state while the spin $\frac{7}{2}$ state is the second excited state lying $4.6$ MeV above. In this paper we shall see that the inclusion of vibrational modes in the quantisation procedure resolves this problem.
The $^7$Li and $^7$Be nuclei are special. Among all nuclei with $B<30$ they are the only ones that have an observed spin $\frac{7}{2}$ state lying below the lowest spin $\frac{5}{2}$ state. The $B=7$ Skyrmion is also special. It has the largest finite symmetry group of any known Skyrmion with non-zero pion mass. We shall see that this large symmetry group is the reason why the spin $\frac{7}{2}$ state has abnormally low energy.
The $^7$Li nucleus is usually described using a cluster model <cit.> which asserts that the nucleus is made of two interacting clusters. These are an alpha particle and a tritium nucleus. This model successfully reproduces the energy spectrum and some electrostatic properties of the nucleus. We shall see that the inclusion of vibrational modes in Skyrmion quantisation highlights a connection between the Skyrme model and the ideas of clustering.
This paper is organised as follows. In section $2$ we review the Skyrme model and the structure of the $B=7$ vibrational space. We discuss how one should include vibrations in the quantisation procedure and the effects of the Finkelstein-Rubinstein constraints in section $3$. Details of the quantisation are laid out in section $4$, alongside the results of our calculations and a comparison with the experimental data.
§ THE $B=7$ SKYRMION AND ITS VIBRATIONAL SPACE
§.§ The Skyrme Model
The Skyrme model can be defined in terms of the three pion fields, $ \boldsymbol\pi(t,\boldsymbol{x})$. These are combined into an $SU(2)$-valued field
\begin{equation}
U(t,\boldsymbol{x}) = \sigma(t,\boldsymbol{x}) + i \boldsymbol{\pi}(t,\boldsymbol{x})\cdot \boldsymbol\tau \, ,
\end{equation}
where $\boldsymbol\tau$ are the Pauli matrices and $\sigma$ is an auxiliary field which satisfies $\sigma^2 + \boldsymbol\pi\cdot\boldsymbol\pi = 1$. This ensures that $U \in SU(2)$. Many quantities are most easily expressed in terms of the right current $R_\mu = (\partial_\mu U)U^\dagger$. The Lagrange density is given by
\begin{equation}
\mathcal{L} = -\frac{F_\pi^2}{16}\text{Tr}\left( R_\mu R^\mu\right) + \frac{1}{32e^2}\text{Tr}\left([R_\mu,R_\nu][R^\mu,R^\nu]\right) + \frac{1}{8}m_\pi^2F_\pi^2\text{ Tr}(U-\boldsymbol{1}_2)
\end{equation}
where $F_\pi$ is the pion decay constant, $e$ is a dimensionless parameter and $m_\pi$ is the pion mass. It is more natural to work in Skyrme units. In these, the energy and length units are $F_\pi / 4e$ and $2/eF_\pi$ respectively. The Lagrangian becomes
\begin{equation}
L = \int -\frac{1}{2}\text{Tr}\left( R_\mu R^\mu\right) + \frac{1}{16}\text{Tr}\left([R_\mu,R_\nu][R^\mu,R^\nu]\right) + m^2\text{ Tr}(U-\boldsymbol{1}_2)\, d^3x
\end{equation}
where $m=2m_\pi/e F_\pi$ is the dimensionless pion mass.
A Skyrmion is a solution of the field equations which minimises the static energy. This is interpreted as the classical mass of the Skyrmion and is given by
\begin{equation} \label{massf}
\mathcal{M}_B = \int -\frac{1}{2}\text{Tr}\left( R_i R_i\right) - \frac{1}{16}\text{Tr}\left([R_i,R_j][R_i,R_j]\right) - m^2\text{ Tr}(U-\boldsymbol{1}_2)\, d^3x \, .
\end{equation}
For this to be finite the Skyrme field must take a constant value, $U = \boldsymbol{1}_2$, at spatial infinity. This one point compactification of space means that $U$ is a map from $\mathbb{R}^3 \cup \{\infty\} \cong S^3$ to $SU(2)$, which is topologically equivalent to $S^3$. These maps are labelled by an integer as $\pi_3(S^3) = \mathbb{Z}$. The integer is identified with the baryon number, $B$, and can be calculated explicitly from the Skyrme field,
\begin{equation}
B = \int \mathcal{B}(x)\, d^3x=-\frac{1}{24\pi^2}\int \epsilon_{ijk}\text{Tr}(R_i R_j R_k)\, d^3x \,
\end{equation}
where $\mathcal{B}$ is the baryon density.
To visualise a Skyrmion we plot a surface of constant baryon density. This is then coloured to express the direction of the pion field, $\hat{\boldsymbol\pi}$, as it varies over the surface. We use the same colouring scheme as in <cit.>. The Skyrmion is coloured white/black when $\hat{\pi}_3$ equals $\pm 1$ and red, green and blue when $\hat{\pi}_1+i\hat{\pi}_2$ is equal to $1$, $\exp(2\pi i /3)$ and $\exp(4\pi i /3)$ respectively.
§.§ The vibrational space of the $B=7$ Skyrmion
The $B=7$ Skyrmion has dodecahedral symmetry as seen in figure <ref>. There is $D_5$ symmetry around each face of the Skyrmion and $D_3$ symmetry around each vertex. These, alongside the additional reflection symmetry, generate the full symmetry group of the Skyrmion $Y_h$.
A surface of constant baryon density for the $B=7$ Skyrmion.
The vibrational space was numerically generated and studied in <cit.> by considering small perturbations around the $B=7$ Skyrmion. Two low frequency modes were found, one of which had a clear physical interpretation and a clean peak in the power spectrum. We will assume that this is the lowest energy vibrational mode and exclude all others from our analysis. Each point in the vibrational space corresponds to a deformed Skyrme configuration. Our aim is to understand the structure of the space and to find subspaces where the Skyrme configurations have enhanced symmetry.
The vibrational mode we consider has five fold degeneracy and so spans a $5$-dimensional vibrational space which we denote $\mathcal{V}_5$. Each point $\boldsymbol v \in \mathcal{V}_5$ corresponds to a quadrupole deformation tensor of the Skyrmion, $Q(\boldsymbol{v})$. There is a natural mapping from a hyperplane in $\mathbb{R}^6$ (isomorphic to $\mathcal{V}_5$) to the space of quadrupole tensors. It is
\begin{equation} \label{r6toquad}
(v_1,v_2,v_3,v_4,v_5,v_6) \mapsto \begin{pmatrix}
v_1 & 2^{-\frac{1}{2}} v_6 & 2^{-\frac{1}{2}} v_5 \\
2^{-\frac{1}{2}} v_6 & v_2 & 2^{-\frac{1}{2}} v_4 \\
2^{-\frac{1}{2}} v_5 & 2^{-\frac{1}{2}} v_4 & v_3
\end{pmatrix} \, ,
\end{equation}
where $\boldsymbol v$ satisfies $(1,1,1,0,0,0)\cdot \boldsymbol v = 0$ to ensure the quadrupole tensor is traceless. We can add vectors on the hyperplane; this is equivalent to adding the quadrupole tensors in $\mathbb{R}^3$. We choose the normalisation so that a unit vector $\boldsymbol{\hat{v}}$ maps to a quadrupole which satisfies $Q_{ij}Q_{ij} = 1$. Each quadrupole tensor, $Q(\boldsymbol v)$, has an associated symmetry group which acts on $\mathbb{R}^3$. Any symmetry shared by the quadrupole tensor and the $B=7$ Skyrmion is a symmetry of the Skyrme configuration at the point $\boldsymbol v$.
In <cit.> it was found that the vibration we consider preserves the Skyrmion's $D_5$ symmetry along certain lines in $\mathcal{V}_5$. Physically, this vibration pulls on two opposite faces of the dodecahedron and breaks the Skyrmion into three clusters: a $B=3$ torus sandwiched between two $B=2$ tori. This can happen in six ways as there are six pairs of faces on the Skyrmion. Hence there are six special lines in $\mathcal{V}_5$ which preserve $D_5$ symmetry. They are evenly spaced and are aligned with the vertices of a regular $5$-simplex. We must position the $5$-simplex in $\mathcal{V}_5$ so that each vertex, $\boldsymbol{v}_a$, maps to a quadrupole tensor which is circle invariant around the axis passing through the Skyrmion faces that are being pulled upon. This ensures that the Skyrme configuration at $\boldsymbol{v}_a$ preserves $D_5$ symmetry. We use the Veronese mapping to help us. This is a map from $\mathbb{R}P^2$ to a $2$-dimensional subspace of $\mathcal{V}_5$. Explicitly it takes
\begin{equation}
(x_1,x_2,x_3) \mapsto \left(x_1^2 - \frac{1}{3}r^2,x_2^2 - \frac{1}{3}r^2,x_3^2 - \small\frac{1}{3}r^2,x_2 x_3,x_1x_3,x_1 x_2\right)\, .
\end{equation}
This then maps to a quadrupole via (<ref>) which is circle invariant around $(x_1,x_2,x_3)$. For example, the Skyrmion has $D_5$ symmetry around the axis $\boldsymbol{x}_1=(0,0,1)$. This goes, via the Veronese mapping, to the $6$-vector
\begin{equation}
\boldsymbol{v}_1 = (-6^{-\frac{1}{2}},-6^{-\frac{1}{2}},(2/3)^\frac{1}{2},0,0,0)
\end{equation}
which maps to the quadrupole
\begin{equation}
Q_1 = \begin{pmatrix}
-6^{-\frac{1}{2}} & 0 & 0 \\
0 & -6^{-\frac{1}{2}} & 0 \\
0 & 0 & (2/3)^\frac{1}{2}
\end{pmatrix} \, .
\end{equation}
This is circle invariant around $\boldsymbol{x}_1$ as desired. Repeating this process, we may generate the vertices of the $5$-simplex in $\mathcal{V}_5$ from the lines which pass through the faces of dodecahedron. This procedure has the corollary that all six vertices of the $5$-simplex lie on the $2$-dimensional Veronese surface. We denote the $5$-simplex vertices as $\boldsymbol{v}_a \in \mathcal{V}_5$ and the corresponding quadrupole tensors $Q_a$; these are circle invariant around $\boldsymbol{x}_a$. Any configuration which lies on the line $\lambda \boldsymbol{v}_a \in \mathcal{V}_5,\, \lambda \in \mathbb{R}$ has $D_5$ symmetry. The parameter $\lambda$ is the amplitude of the vibration. For $\lambda > 0$ the Skyrmion deforms as described above: a pair of opposite faces are pulled upon. When $\lambda < 0$ the faces are pushed together and the Skyrmion flattens out. The full vibration is displayed in figure <ref>.
A vibration in $\mathcal{V}_5$ which preserves $D_5$ symmetry. The parameter $\lambda$ measures the amplitude of the vibration. This figure was generated using the gradient flow approximation to dynamics. The minimum energy Skyrmion is at $\lambda=0$. This deforms into a ring-like configuration for $\lambda<0$ and three clusters for $\lambda>0$.
We may use the geometry of the $5$-simplex to find additional symmetric subspaces in $\mathcal{V}_5$. The planes passing through an edge of the simplex can be written as
\begin{equation}
\mu \boldsymbol{v}_a + \nu \boldsymbol{v}_b\, ,
\end{equation}
where $a \neq b$ and $\mu, \nu \in \mathbb{R}$. The corresponding quadrupole has a $C_2$ symmetry, shared with the $B=7$ Skyrmion, about the axis $\boldsymbol{x}_a \times \boldsymbol{x}_b$. This is enhanced to a $D_2$ symmetry when $\mu=\nu$.
The $5$-simplex has $20$ triangular faces. A line passing through the centre of a face takes the form
\begin{equation} \label{3lines}
\lambda(\boldsymbol{v}_a+\boldsymbol{v}_b+\boldsymbol{v}_c)\, ,
\end{equation}
where $a \neq b \neq c$. In fact, this line passes through two triangular faces which are dual to each other. Thus there are only ten distinct lines. The quadrupole tensor derived from (<ref>) has only two distinct eigenvalues. Thus it is circle invariant around the eigenvector of the non-degenerate eigenvalue. This eigenvector passes through a vertex of the $B=7$ Skyrmion which has $D_3$ symmetry. Thus the Skyrme configurations on these $10$ lines in $\mathcal{V}_5$ retain $D_3$ symmetry. Note that, since these quadrupoles are circle invariant, these points in $\mathcal{V}_5$ also lie on the Veronese surface discussed earlier. It is instructive to view the physical picture. When $\lambda > 0$ the three component quadrupole tensors pull on three pairs of opposite faces. Three faces always surround a vertex of the Skyrmion, as do the opposite faces; the remaining three pairs form a ring around its centre. The quadrupole tensors around the vertex sum to give a quadrupole which pulls in the direction of the surrounded vertex. This is seen in Figure <ref>. When large, this vibration breaks the Skyrmion into two $B=3$ Skyrmions sandwiching a $B=1$ Skyrmion. When $\lambda < 0$ the faces surrounding the vertex are pushed upon and the $B=7$ Skyrmion breaks into $7$ individual $B=1$ Skyrmions.
When three faces of the Skyrmion are pulled equally, a $D_3$ symmetry remains. The sum of the quadrupoles which pull on the faces of the Skyrmion give a quadrupole which is circle invariant about the red axis which passes through a vertex as shown.
A vibration in $\mathcal{V}_5$ which preserves $C_3$ symmetry. The parameter $\lambda$ measures the amplitude of the vibration. The minimum energy Skyrmion is at $\lambda=0$. This deforms into seven individual distorted Skyrmions for $\lambda<0$ and two clusters for $\lambda>0$.
The analysis so far is based on small perturbations around the Skyrmion. We believe that there will be a bifurcation where the exact symmetries discussed above will break. For example, the $D_3$ symmetry will break to a $C_3$ symmetry. This allows the asymptotic configuration in $\mathcal{V}_5$ to be a two-cluster system consisting of a $B=3$ and $B=4$ Skyrmion. This has lower energy than the three-cluster system described in the previous paragraph. These are hard to distinguish near the origin of $\mathcal{V}_5$ and so the difference will not be apparent in our analysis. Thus we shall assume that the Skyrme configuration along this vibration, at large amplitudes, will be the $C_3$ symmetric $3+4$ cluster configuration instead of the $D_3$ symmetric $3+1+3$ configuration. The entire vibration is displayed in figure <ref>.
§ QUANTISATION ON THE VIBRATIONAL SPACE
Our aim is to quantise the $B=7$ Skyrmion taking the lowest energy vibrational mode into account. The manifold we quantise must contain all configurations in $\mathcal{V}_5$ including those generated by rotations and isorotations. Explicitly the manifold is
\begin{equation}
\mathcal{N} = \frac{\mathcal{V}_5 \times SU(2) \times SU(2)}{D}
\end{equation}
where $D$ is a finite group encoding the dodecahedral symmetry of the Skyrmion. We can think of this manifold as a family of Skyrme configurations parametrised by vibrational ($\boldsymbol s$), rotational ($\phi,\theta,\psi$) and isorotational ($\alpha, \beta, \gamma$) coordinates. The angular coordinates are two sets of Euler angles. To quantise we promote all these parameters to dynamical degrees of freedom by allowing them to depend on time. This ansatz allows us to define angular velocities $\boldsymbol{b}$, isoangular velocities $\boldsymbol{a}$ and find the kinetic energy
\begin{equation}
T = \frac{1}{2}(\dot{\boldsymbol s},\boldsymbol{a}, \boldsymbol{b}).g(\boldsymbol s).(\dot{\boldsymbol s}, \boldsymbol{a}, \boldsymbol{b})^T\, ,
\end{equation}
where $g(\boldsymbol s)$ is the metric on $\mathcal{N}$, which depends on the Skyrme configuration at $\boldsymbol s$.
With the kinetic energy written in this way, the quantum kinetic operator is well known <cit.>. It is proportional to the Laplace-Beltrami operator, $\Delta$. Explicitly
\begin{equation}
\Delta = \frac{1}{\sqrt{|g|}}\partial_i\left(\sqrt{|g|}g^{ij}\partial_j\right) \, ,
\end{equation}
where $|g|$ is the determinant of the metric. This preserves the classical symmetries of the kinetic energy after quantisation.
The potential energy, $V(\boldsymbol s)$, is the mass (<ref>) of the configuration at $\boldsymbol s$. The Hamiltonian on $\mathcal{N}$ is
\begin{equation} \label{ham}
\mathcal{H} = -\frac{\hbar^2}{2} \Delta + V(\boldsymbol s) \, .
\end{equation}
To find bound states of definite energy we solve the stationary Schrödinger equation arising from this Hamiltonian,
\begin{equation} \label{Schro}
\mathcal{H}\Psi = E \Psi \, .
\end{equation}
Formally, the wavefunction $\Psi$ is a section of a complex line bundle over $\mathcal{N}$.
There are constraints on $\Psi$ which encode the fact that nucleons are fermions. These are the Finkelstein-Rubinstein (FR) constraints <cit.>. They can be written in terms of the classical symmetries of the Skyrmion. For example, the $B=7$ Skyrmion is invariant under a $2 \pi / 5$ rotation around the $3$-axis followed by a $-4 \pi / 5$ isorotation around the $3$-axis in isospace. In operator form, this $C_5$ symmetry puts the following constraint on the wavefunction
\begin{equation} \label{FR1}
e^{\frac{2\pi i}{5}\hat{L}_3}e^{-\frac{4\pi i}{3}\hat{K}_3} \Psi = -\Psi \, ,
\end{equation}
where $\hat L_3$ and $\hat K_3$ are the body fixed angular momentum operators defined in the usual way. Similarly, the $C_3$ symmetry gives the constraint
\begin{equation} \label{FR2}
e^{2\pi i /3\, \boldsymbol{n}_1 \cdot \boldsymbol{\hat L}}e^{i\xi \,\boldsymbol{n}_2 \cdot \boldsymbol{\hat K}}\Psi=\Psi \, .
\end{equation}
where $\boldsymbol{n}_1=(-\sqrt{\frac{2}{15}(5-\sqrt{5})},0,\sqrt{\frac{1}{15}(5+2\sqrt{5})})$ is a vector which passes through one of the dodecahedron's vertices while $\xi$ and $\boldsymbol{n}_2$ define the isorotation required to return the Skyrmion to its original colouring. The FR signs can be calculated using the rational map ansatz <cit.>. The procedure is set out in <cit.>.
The constraints (<ref>) and (<ref>) both apply when the Skyrmion has dodecahedral symmetry. This occurs at the origin of $\mathcal{V}_5$. For a generic point $\boldsymbol s$ there is no symmetry and thus no constraints. In the previous Section we found lines in $\mathcal{V}_5$ which had enhanced symmetry. One set of these preserved $D_5$ symmetry. Thus, on these lines, only a constraint such as (<ref>) applies, as well as an additional constraint which enhances the $C_5$ symmetry to $D_5$. Another set of lines preserved $C_3$ symmetry, meaning the wavefunction must satisfy a constraint such as (<ref>) on these.
Now the problem is formulated. To include the lowest vibrational mode when studying the states of $^7$Li/$^7$Be we must solve (<ref>), an $11$-dimensional Schrödinger equation, subject to (<ref>) and (<ref>) at $\boldsymbol s=0$, just a constraint such as (<ref>) on six lines in the vibrational space (representing the $D_5$ preserving directions in $\mathcal{V}_5$) and a constraint such as (<ref>) on ten lines. There are further constraints on the edges of the $5$-simplex. To set up and solve this problem rigorously is too hard and so we will make some simplifying assumptions below.
In certain cases the metric, $g$, will simplify due to the symmetries of the system. In particular the kinetic operator can separate into a part which only acts via the rotational coordinates and a part which only acts via the vibrational ones. We denote this as
\begin{equation}
\Delta = \Delta_s + \nabla^2 \, .
\end{equation}
We can then solve the Schrödinger equation using separation of variables
\begin{equation} \label{SepWv}
\Psi = u(\boldsymbol s)\Theta(\phi,\theta,\psi,\alpha,\beta,\gamma) \, ,
\end{equation}
where we call $u$ the vibrational wavefunction and $\Theta$ the rotational wavefunction. The latter is a solution of the rigid body Schrödinger equation. This problem has been studied extensively, see <cit.> for details. The solutions are tensor products of Wigner D-functions and each solution has six conserved quantities: total spin ($J^2$), total isospin ($I^2$), body fixed spin ($L_3$) and isospin ($K_3$) projections and space fixed spin ($J_3$) and isospin ($I_3$) projections. The space fixed projections do not affect the energy spectrum and as such they are often suppressed in the bra-ket notation where we denote the state $ D^J_{L_3 J_3}(\phi,\theta,\psi) \otimes D^I_{K_3 I_3}(\alpha,\beta,\gamma)$ as $\ket{J \, L_3} \ket{I\,K_3}$.
One may satisfy the FR constraints using the rotational wavefunction by taking appropriate linear combinations of the Wigner functions. Often the constraints rule out certain spin states. The dodecahedral symmetry of the $B=7$ Skyrmion rules out states with $J=\frac{1}{2}, \frac{3}{2} \text{ and } \frac{5}{2}$ and $I=\frac{1}{2}$. However, the constraints apply to the entire wavefunction $\Psi$, not just the rotational part $\Theta$. We may alternatively satisfy the FR constraints at the origin of $\mathcal{V}_5$ by insisting that $\Psi$ vanishes there. So there are two ways to satisfy the FR constraints at $\boldsymbol s=0$:
* The rotational wavefunction, $\Theta$, is permitted by both the FR constraints. There are no restrictions on the vibrational wavefunction, $u(\boldsymbol s)$.
* The vibrational wavefunction is zero at the origin, $u(\boldsymbol 0) = 0$. There are fewer restrictions on $\Theta$.
These two options also apply on any subspace of vibrational space with enhanced symmetry.
Let us fix $I = \frac{1}{2}$ and look for low energy states ($J \leq \frac{7}{2}$). We will now review which spin states are allowed by each symmetry we've encountered, many of these were discovered and discussed in <cit.>.
The dodecahedron at the origin of $\mathcal{V}_5$. The only allowed state has $J=\frac{7}{2}$.
The $D_5$ symmetric lines on the vertices of the $5$-simplex. No spin $\frac{1}{2}$ states are allowed. One spin $\frac{3}{2}$ state is allowed and one spin $\frac{5}{2}$ states is allowed. There are two permitted $\frac{7}{2}$ states: the dodecahedral state and an orthogonal one.
The $C_2$ symmetry present on the edges of the $5$-simplex. This is not a very restrictive symmetry. There are two, four, six and eight states allowed with spin $\frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$ and $\frac{7}{2}$ respectively.
The $D_2$ symmetry present at the centre of the edges of the $5$-simplex. There are one, two, three and four states allowed with spin $\frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$ and $\frac{7}{2}$ respectively.
The $C_3$ symmetric lines going through the triangular faces of the $5$-simplex. Many states are allowed here. Two spin $\frac{1}{2}$ states, two spin $\frac{3}{2}$ states, three spin $\frac{5}{2}$ states and three spin $\frac{7}{2}$ states.
We now look for the low energy states. A rotational wavefunction with spin $\frac{7}{2}$ is allowed everywhere in vibrational space. Thus the corresponding wavefunction, $\Psi_{J=\frac{7}{2}}$, can be of type (a). There are no spin $\frac{3}{2}$ states allowed at the origin of $\mathcal{V}_5$. Thus the spin $\frac{3}{2}$ wavefunction is of type (b) and $u_{J=\frac{3}{2}}$ must vanish at $\boldsymbol s=0$. We can schematically calculate the energy difference of these states using a harmonic approximation. The spin $\frac{7}{2}$ vibrational wavefunction is in the ground state of $\mathcal{V}_5$ while the spin $\frac{3}{2}$ wavefunction must be excited in one direction so that it vanishes at the origin. So the spin $\frac{3}{2}$ state has one unit of vibrational energy more than the spin $\frac{7}{2}$ state. However it will have less rotational energy as the spin is smaller. The ordering of these states depends on the relative energy contributions from vibrations and rotations. We make the approximation that all other vibrations contribute equally to the states. Thus to compare these low lying states we only need to account for the vibration in one direction in $\mathcal{V}_5$, the smallest energy direction.
In the harmonic approximation the direction of the vibration does not matter as the potential on $\mathcal{V}_5$ is isotropic. However in the full model the direction will be important. A generic direction in $\mathcal{V}_5$ will break the $B=7$ Skyrmion into seven $B=1$ Skyrmions. This has high potential energy compared to the break up into clusters we saw in Section $2$ where the Skyrmion could break into fewer, higher-charge Skyrmions. Thus we believe that the smallest vibration will not be in a generic direction. Instead it will be along one of the high symmetry directions previously discussed.
§ QUANTISING THE $B=7$ SKYRMION
We will now quantise the $B=7$ Skyrmion taking a single vibrational mode into account. To do this we must decide on the direction of the vibration in $\mathcal{V}_5$. We shall assume the lowest energy direction is along either a $C_3$ preserving line or a $D_5$ preserving line since these have low energy configurations asymptotically.
The symmetry present on these lines restricts the form of the metric which is 7-dimensional. It is standard convention to split the metric into submatrices. We follow <cit.> and write
\begin{equation}
g = \begin{pmatrix}
\Lambda & \multicolumn{2}{c}{$0$} \\
\multirow{2}{*}{$0$} & U & -W \\
& -W^T & V
\end{pmatrix}
\end{equation}
where $U$, $W$ and $V$ are $3\times3$ matrices and $\Lambda$ is a scalar. The kinetic energy is invariant under the action of the symmetry group of the vibration. This restriction means that, along the symmetric lines in $\mathcal{V}_5$, the cross terms in the metric vanish and the kinetic energy is separable in the sense described in the previous Section. As such the wavefunction takes the form (<ref>) on these lines, with the vibrational parameter $s$ now 1-dimensional.
Consider a rotational state with spin $J$ and denote the rotational energy contribution $E_J(s)$ so that
\begin{equation}
\nabla^2 \Theta_J = E_J(s)\Theta_J \, .
\end{equation}
Note that the rotational energy contribution is a function of $s$ through its dependence on the moments of inertia which vary as the Skyrmion deforms. Then the Schrödinger equation (<ref>) reduces to the $1$-dimensional equation
\begin{equation} \label{Schro2}
\left( - \frac{\hbar^2}{2\sqrt{|g|}}\partial_s\left( \frac{\sqrt{|g|}}{\Lambda}\partial_s\right) +V(s)+E_J(s)\right)u(s) = Eu(s) \, .
\end{equation}
To solve this we must first generate $g(s)$, $V(s)$ and $E_J(s)$. We will do this using gradient flow.
Gradient flow generates a path of steepest descent in field space. We use the separated Skyrmion clusters as initial configurations which are then evolved in gradient flow time $\tau$ according to
\begin{equation} \label{flow}
\frac{d \boldsymbol{\pi} }{d \tau} = - \frac{\delta \mathcal{M}_7}{\delta \boldsymbol{\pi}} \, ,
\end{equation}
where $\boldsymbol{\pi}$ are the pion fields and $\mathcal{M}_7$ is the potential energy (<ref>). This flow reduces the potential energy of the system and ends at a stationary point of field space. The fields $\boldsymbol{\pi}(\tau)$ approximate the Skyrme configurations along a half line in $\mathcal{V}_5$. The solution of (<ref>) is beyond analytic calculation and so we must use a numerical code to calculate the flow. The energy $V(\tau)$ and the metric $g(\tau)$ are calculated at numerous points during the process.
The metric at time $\tau$ can be expressed in terms of the currents $R_i = (\partial_i U)U^{-1}$ and $T_i = \frac{i}{2}[\tau_i,U]U^{-1}$. The moments of inertia and $\Lambda$ are given by
\begin{align}
&\Lambda = -\int \text{Tr}\left( R_\tau R_\tau + [R_\tau,R_i][R_\tau,R_i]\right) d^3 x \\
& U_{ij} = -\int \text{Tr}\left(T_iT_j + \frac{1}{4}[R_k,T_i][R_k,T_j]\right) d^3 x \\
& W_{ij} = \int \epsilon_{jlm}x_l\text{Tr}\left(T_iR_m + \frac{1}{4}[R_k,T_i][R_k,R_m]\right)d^3 x \\
& V_{ij} = -\int \epsilon_{ilm}\epsilon_{jnp}x_lx_n\text{Tr}\left( R_mR_p + \frac{1}{4}[R_k,R_m][R_k,R_p]\right)d^3 x \, \text{.}
\end{align}
Gradient flow time is an unnatural parameter when the Skyrmion clusters are widely separated and near the dodecahedral configuration. Thus, once we have found our quantities numerically we change variables to the geodesic distance, $s$ <cit.>. This can be defined in terms of the vibrational kinetic energy by demanding
\begin{equation}
T_\text{vib} = \frac{1}{2}\dot s^2 = \frac{1}{2}\Lambda(\tau)\dot{\tau}^2
\end{equation}
which means that
\begin{equation}
s(\tau) = \int^\tau \sqrt{\Lambda(\tau')} \, d\tau' \, .
\end{equation}
There are several advantages to this new coordinate. First, the geodesic distance is related to the cluster separation, $r$, asymptotically. We can calculate how the moments of inertia vary with $r$ and this gives an asymptotic check of the numerics. Additionally we are able to add an analytic tail to the numerically derived potential and moments of inertia. Further, we may now calculate the harmonic frequency near the origin of $\mathcal{V}_5$ and compare it to what was calculated in <cit.>. We find the frequency to be $0.34$ compared with $0.302$ as found in <cit.>. These are approximately the same, showing the methods are consistent. The small difference is likely due to the different pion masses used. Finally, the new coordinate simplifies the Schrödinger equation (<ref>). It now reads
\begin{equation} \label{Schro3}
\left( - \frac{\hbar^2}{2} \frac{d^2}{ds^2}-\frac{\hbar^2}{4}\partial_s\log(|g|)\frac{d}{ds} +V(s)+E_J(s)\right)u(s) = Eu(s)
\end{equation}
From now on, $s$ will refer exclusively to the geodesic distance.
§.§ The $C_3$ direction
The initial configuration for the $C_3$ direction is constructed using a symmetrised product ansatz of a $B=3$ Skyrmion with a $B=4$ Skyrmion. These are orientated as in figure <ref>. The $C_3$ symmetry constrains the form of the metric. We find that $U$, $V$ and $W$ are all diagonal. Further
\begin{equation}
U_{11} = U_{22}, \, V_{11} = V_{22}, \, \text{and } \, W_{11} = W_{22} \, .
\end{equation}
We have set $\Lambda = 1$ by choosing our parameter to be the geodesic distance.
We now look at specific rotational wavefunctions. Although this direction in $\mathcal{V}_5$ only has $C_3$ symmetry, it has approximate $D_3$ symmetry near the origin. This means that a wavefunction disallowed by $D_3$ symmetry would have extra constraints imposed on it nearby in the full vibrational space. This would increase the energy of the state. Thus we focus on states which are allowed by $D_3$ symmetry. The rotational wavefunctions we consider are presented in table <ref>.
Spin FR-allowed states
$J=\frac{1}{2}$ $\ket{\Theta}_{\frac{1}{2}} = \ket{\frac{1}{2},\frac{1}{2}}\ket{\frac{1}{2},\frac{1}{2}} - \ket{\frac{1}{2}, -\frac{1}{2}}\ket{\frac{1}{2}, -\frac{1}{2}}$
$J=\frac{3}{2}$ $\ket{\Theta}_{\frac{3}{2}} =\ket{\frac{3}{2},\frac{1}{2}}\ket{\frac{1}{2},\frac{1}{2}}+\ket{\frac{3}{2},-\frac{1}{2}}\ket{\frac{1}{2},-\frac{1}{2}}$
$J=\frac{5}{2}$ $\ket{\Theta}^{(1)}_{\frac{5}{2}} =\ket{\frac{5}{2},\frac{1}{2}}\ket{\frac{1}{2},\frac{1}{2}}+\ket{\frac{5}{2},-\frac{1}{2}}\ket{\frac{1}{2},-\frac{1}{2}}$
$J=\frac{7}{2}$ $\ket{\Theta}^{(1)}_{\frac{7}{2}} =\ket{\frac{7}{2},\frac{1}{2}}\ket{\frac{1}{2},\frac{1}{2}}+\ket{\frac{7}{2},-\frac{1}{2}}\ket{\frac{1}{2},-\frac{1}{2}}$
The low energy states allowed by $D_3$ symmetry.
Consider the spin $\frac{3}{2}$ state. The full wavefunction is of the form
\begin{equation} \label{32state}
\Ket{\Psi}_\frac{3}{2} = u_\frac{3}{2}(s)\left( \, \Ket{\frac{3}{2},\frac{1}{2}}\Ket{\frac{1}{2},\frac{1}{2}}+\Ket{\frac{3}{2},-\frac{1}{2}}\Ket{\frac{1}{2},-\frac{1}{2}} \right) \, .
\end{equation}
The vibrational wavefunction must satisfy $u_\frac{3}{2}(0)=0$ so that the full wavefunction $\Psi_\frac{3}{2}$ is consistent with the additional FR constraint (<ref>) at $s=0$. The derivative must be non-zero here or the vibrational wavefunction will be trivial everywhere. Inserting (<ref>) into the Schrödinger equation (<ref>) we find that $u_\frac{3}{2}$ satisfies
\begin{align}
\Bigg( \frac{\hbar^2}{2}\Big( \frac{V_{11}}{2\left(U_{11}V_{11}-W_{11}^2\right)}+\frac{3U_{11}}{2\left(V_{11}U_{11}-W_{11}\right)} + \frac{1}{U_{33}V_{33} - W_{33}^2} \Big(\frac{9}{4}U_{33} + \frac{1}{4}V_{33} - &\frac{3}{2}W_{33}\Big) \Big) \nonumber \\
-\frac{\hbar^2}{2}\frac{d^2}{ds^2}-\frac{\hbar^2}{4}\partial_s\log(|g|)\frac{d}{ds}+ V(s)\Bigg) u_\frac{3}{2}(s) = E u_\frac{3}{2}(s)& \, .
\end{align}
We would like to understand the contributions from rotations and vibrations separately. There is no unique way to split the energy; we choose to define the rotational energy contribution as the rigid body energy of the undeformed Skyrmion, $E_J(0)$. We may then split the energy $E$ into three parts: the classical mass of the Skyrmion $\mathcal{M}_7 = V(0)$, the contribution from the rigid rotation $E_J(0)$, and the energy contribution from the vibration $\epsilon_\text{vib}$. We write $E = \mathcal{M}_7 + E_J(0) + \epsilon_\text{vib}$ and the Schrödinger equation becomes
\begin{equation} \label{Schro4}
\left( -\frac{\hbar^2}{2}\frac{d^2}{ds^2}-\frac{\hbar^2}{4}\partial_s\log(|g|)\frac{d}{ds} + V_\text{eff}(s)\right)u_\frac{3}{2}(s) = \epsilon_\text{vib} u_\frac{3}{2}(s)
\end{equation}
where $V_\text{eff}(s) = V(s)-\mathcal{M}_7 + E_J(s)-E_J(0)$. Note that $V_\text{eff}$(0) = 0. Equation (<ref>) is then solved numerically using a shooting technique.
There are two spin $\frac{5}{2}$ states. These have different values of $L_3$ and thus each state has a different effective potential. Each full wavefunction is of the form
\begin{equation}
\Ket{\Psi}_\frac{5}{2} = u_\frac{5}{2}(s)\Ket{\Theta}_\frac{5}{2}
\end{equation}
with each vibrational wavefunction being zero at $s=0$, just like the spin $\frac{3}{2}$ case.
There are three spin $\frac{7}{2}$ states. We focus on the lowest energy state. This is a linear combination of the three states,
\begin{equation}
\Ket{\Psi}_\frac{7}{2} = u(s) \ket{\Theta}^{(1)}_{\frac{7}{2}} + v(s) \ket{\Theta}^{(2)}_{\frac{7}{2}} + w(s) \ket{\Theta}^{(3)}_{\frac{7}{2}}
\end{equation}
\begin{equation} \label{sevenhalfscond}
u(0) = v(0) = (7/18)^{1/2} \text{ and } w(0) = \sqrt{2}/3
\end{equation}
to ensure that $\Psi_\frac{7}{2}$ satisfies the additional FR constraint (<ref>) at $s=0$. This gives three uncoupled Schrödinger equations for $u$, $v$ and $w$. Generally these three independent equations will not produce a shared energy eigenstate as the effective potential is different for each component rotational wavefunction. However, we can obtain a shared eigenvalue by enforcing an additional boundary condition that the probability distribution is maximal at the origin. This gives
\begin{equation}
0 = \frac{1}{2}\frac{d}{ds}\biggr(|\Psi|^2\biggr)\biggr\rvert_{s=0} = \left(u\dot u+ v\dot v+ w\dot w\right)\rvert_{s=0} \, .
\end{equation}
This condition, alongside (<ref>), produces a discrete eigenvalue spectrum.
The spin $\frac{1}{2}$ state is similar to the spin $\frac{3}{2}$ state and takes a form analogous to (<ref>) with the same conditions on the vibrational wavefunction. However, it has additional constraints in the full vibrational space. It must vanish on the $D_5$ preserving lines in $\mathcal{V}_5$ due to the FR constraints. The wavefunction we construct is concentrated along a $C_3$ direction. This direction is maximally far away from the $D_5$ lines. This can be seen geometrically: the $C_3$ lines go through the centre of the $5$-simplex faces while the $D_5$ lines pass through the vertices. Thus the wavefunction we construct should already be small on the $D_5$ lines. A modification is required to make the wavefunction vanish on the $D_5$ lines which will cost energy. Thus, we expect the true spin $\frac{1}{2}$ state to have higher energy than what is calculated here.
§.§.§ Calibration of the model
Before comparing our results to experimental data we must calibrate the model. All previous calibrations are based on zero mode quantisation and as such we don't necessarily expect our choice of parameters to match previous studies. The vibrational energy contribution is of order $\hbar$ while the rotational energy contribution is of order $\hbar^2$. Thus the relative energies of the states will be sensitive to the value of $\hbar$.
In figure <ref> the quantum energy of each state is plotted (in Skyrme units) for various values of $\hbar$. The most important feature of the plot is that the spin $\frac{7}{2}$ state increases in energy, relative to the other states, as $\hbar$ increases. This is because the spin $\frac{7}{2}$ state has the largest rotational energy and the smallest vibrational energy; rotational effects dominate for large $\hbar$ while vibrational effects dominate for small $\hbar$. To match experimental data the spin $\frac{7}{2}$ state must lie between the spin $\frac{3}{2}$ state and the first spin $\frac{5}{2}$ state. This occurs when
\begin{equation}
55 < \hbar < 65 \, ,
\end{equation}
and as such we demand that $\hbar$ lies in this interval. For illustrative purposes we fix $\hbar=60$.
The quantum energy of each state (in Skyrme units) as a function of $\hbar$.
We are left to choose the value of $F_\pi$, as $\hbar$ fixes the dimensionless constant $e$ through the identity $\hbar = 2e^2$. We will consider two alternative calibrations:
(i) $F_\pi = 60$ MeV
(ii) $F_\pi = 139$ MeV.
Parameter choice (i) gives a good fit to the size of the gaps in the energy spectrum but underestimates the total mass of the Skyrmion compared to the total mass of $^7$Li. Choice (ii) gives a reasonable value of the total mass but overestimates the gaps in the spectrum. We have fixed the dimensionless pion mass $m$ to $1$ throughout.
§.§.§ Results
We solved the Schrödinger equation (<ref>) for all states in discussed in section <ref>. The numerically generated vibrational wavefunction $u(s)$, potential $V(s)-\mathcal{M}_7$ and effective potential $V_\text{eff}(s)$ for each state is plotted in table <ref>. We also note the classical mass of the Skyrmion $\mathcal{M}_7$, the energy contribution from rotations $E_J(0)$ and the contribution from vibrations $\epsilon_\text{vib}$, as well as the total energy of each state $E$. Our results are then compared to experimental data in table <ref> for each calibration (i) and (ii).
State Vibrational wavefunction and potentials $\mathcal{M}_7 + E_J(0) + \epsilon_\text{vib}$ $E$
$\ket{\Psi}_\frac{1}{2}$ 985.13 + 5.95 + 18.24 1009.32
$\ket{\Psi}_\frac{3}{2}$ 985.13 + 8.81 + 17.31 1011.25
$\ket{\Psi}_\frac{5}{2}^{(1)}$ 985.13 + 13.56 + 15.55 1014.24
$\ket{\Psi}_\frac{5}{2}^{(2)}$ 985.13 + 13.56 + 19.42 1018.11
$\ket{\Psi}_\frac{7}{2}$ 985.13 + 20.24 + 7.39 1012.76
The numerical results for quantisation along the $C_3$ direction. We display the vibrational wavefunction, potential and effective potential for each spin state from table <ref>. All results are in Skyrme units with $\hbar=60$.
3c|Energy relative to ground state (MeV)
State Experiment Calibration (i) Calibration (ii)
$\Ket{\Psi}_\frac{1}{2}$ $0.48$ $-5.29$ $-12.25$
$\Ket{\Psi}_\frac{3}{2}$ $0$ $0$ $0$
$\Ket{\Psi}_\frac{7}{2}$ $4.63$ $4.14$ $9.58$
$\Ket{\Psi}_\frac{5}{2}^{(1)}$ $6.68$ $8.19$ $18.97$
$\Ket{\Psi}_\frac{5}{2}^{(2)}$ $7.46$ $18.79$ $43.53$
A comparison of the experimentally obtained energy spectrum of $^7$Li (column 1) with the results from our calculation using Calibration (i) (column 2) and Calibration (ii) (column 3). The experimental data is from <cit.>.
The results are promising. All of the states considered are seen experimentally. The ordering is correct apart from the spin $\frac{1}{2}$ and $\frac{3}{2}$ states. We argued earlier that the spin $\frac{1}{2}$ state has higher energy than our calculation suggests as it must vanish in a subspace of $\mathcal{V}_5$. This may remedy the ordering issue. Most importantly, the spin $\frac{7}{2}$ state lies between the spin $\frac{3}{2}$ and $\frac{5}{2}$ states. The size of the gaps in the energy spectrum are reasonable for calibration (i) and much too large for calibration (ii). The ratios of the energy gaps between states are independent of this choice, though do depend on $\hbar$. We find that
\begin{equation}
\frac{E_{J=\frac{5}{2}} - E_{J=\frac{3}{2}}}{E_{J=\frac{7}{2}}-E_{J=\frac{3}{2}}}=1.98
\end{equation}
which is reasonably close to the experimental result, $1.44$.
The second spin $\frac{5}{2}$ state has very high energy. This can be understood by considering the body-fixed spin classically. The highly excited state has $|L_3| = \frac{5}{2}$. This means that the spin is around the $3$-axis. This gives a large energy contribution since the Skyrmion is prolate in this direction. The lower energy spin $\frac{5}{2}$ state has $|L_3| = \frac{1}{2}$ which allows it to rotate about an axis orthogonal to the prolate one. The state we have found probably does not correspond to the experimental state we have compared it to in table <ref>. In the cluster model <cit.> this state has a different structure than the others. It is described by a neutron orbiting a $^6$Li nucleus. Thus, it could be that we only see this spin state if we include a vibration which can split the $B=7$ Skyrmion into these clusters.
The next three experimental states of $^7$Li have spin $\frac{7}{2}$, $\frac{3}{2}$ and $\frac{3}{2}$. These have natural descriptions in this model. The $\frac{7}{2}$ state is orthogonal to the one allowed by the dodecahedron and has a single excited vibration in $\mathcal{V}_5$. The excited spin $\frac{3}{2}$ states will have isospin $\frac{3}{2}$, a possibility we neglected for simplicity. This would also describe the ground states of $^7$B and $^7$He which have spin $\frac{3}{2}$.
The mass of the $^7$Li nucleus is $6535$ MeV. Calibration (ii) gives the total mass of the ground state to be $6404$ MeV which is very close to the experimental value. Calibration (i) gives a much smaller value, only $2764$ MeV. There are several ways this could be remedied. First, we have only taken one of the Skyrmion's vibrational modes into account. There are approximately $6B$ modes, all of which will contribute to the energy. The Casimir energy contribution is also large, a $40\%$ correction in the $B=1$ sector <cit.>. Finally, the Lagrangian may be altered to include a $6$th order term which can be chosen to contribute positively to the mass. When this term is the same order as the other terms in the Lagrangian, the Skyrmion solutions do not change significantly <cit.>. Thus the calculation in this paper would not vary greatly except for the total energy. These three factors could combine to give a reasonable value for the total mass. They also highlight the uncertainty in calculations of total mass in the Skyrme model.
Inclusion of the $C_3$ vibration has given us a good model of the spin $\frac{3}{2}$, $\frac{7}{2}$ and $\frac{5}{2}$ states of the $^7$Li/$^7$Be isodoublet. Further, it brings us closer to the cluster model of nuclei. This is apparent when we plot the classical baryon density at the maximum of the vibrational wavefunctions which shows us a classical approximation of the quantum state, before rotational averaging. These are plotted in figure <ref>. We see that the spin $\frac{3}{2}$ state exhibits clustering while the spin $\frac{7}{2}$ state does not. This goes against conventional wisdom in the cluster model where the ground state is generally the most isotropic.
Plots of the baryon density at the maximum value of the vibrational wavefunctions. The spin $\frac{3}{2}$ state is on the left while the spin $\frac{7}{2}$ state is on the right.
§.§ The $D_5$ direction
To simplify this calculation we orient the Skyrmion as it is in figure <ref>. The $D_5$ symmetry constrains $U$, $V$ and $W$ to be diagonal with
\begin{equation}
U_{11} = U_{22}, \, V_{11} = V_{22}, \, \text{and} \, W_{11} = W_{22} =0 \, .
\end{equation}
The spin states allowed by $D_5$ symmetry are presented in table <ref>. The spin $\frac{7}{2}$ state which is allowed by the dodecahedron is given by
\begin{equation}
\tilde{\ket{\Theta}}_{\frac{7}{2}}=\sqrt{\frac{7}{10}}\tilde{\ket{\Theta}}^{(1)}_{\frac{7}{2}} - \sqrt{\frac{3}{10}}\tilde{\ket{\Theta}}^{(2)}_{\frac{7}{2}} \, .
\end{equation}
Spin FR-allowed states
$J=\frac{3}{2}$ $\tilde{\ket{\Theta}}_{\frac{3}{2}} =\ket{\frac{3}{2},\frac{3}{2}}\ket{\frac{1}{2},-\frac{1}{2}}+\ket{\frac{3}{2},-\frac{3}{2}}\ket{\frac{1}{2},\frac{1}{2}}$
$J=\frac{5}{2}$ $\tilde{\ket{\Theta}}_{\frac{5}{2}} =\ket{\frac{5}{2},\frac{3}{2}}\ket{\frac{1}{2},-\frac{1}{2}}-\ket{\frac{5}{2},-\frac{3}{2}}\ket{\frac{1}{2},\frac{1}{2}}$
$J=\frac{7}{2}$ $\tilde{\ket{\Theta}}^{(1)}_{\frac{7}{2}} =\ket{\frac{7}{2},\frac{3}{2}}\ket{\frac{1}{2},-\frac{1}{2}}+\ket{\frac{7}{2},-\frac{3}{2}}\ket{\frac{1}{2},\frac{1}{2}}$
The low energy states allowed by $D_5$ symmetry.
State Vibrational wavefunction and potentials $\mathcal{M}_7 + E_J(0) + \epsilon_\text{vib}$ $E$
$\tilde{\ket{\Psi}}_\frac{3}{2}$ 985.13 + 8.84 + 26.47 1020.45
$\tilde{\ket{\Psi}}_\frac{5}{2}$ 985.13 + 13.54 + 25.23 1023.91
$\tilde{\ket{\Psi}}_\frac{7}{2}$ 985.13 + 20.28 + 10.09 1015.50
The numerical results for quantisation along the $D_5$ direction. We display the vibrational wavefunction, potential and effective potential for each spin state from table <ref>. All results are in Skyrme units, with $\hbar=60$.
We solve the Schrödinger equation and present the numerically generated vibrational wavefunction $u(s)$, potential $V(s)-\mathcal{M}_7$ and effective potential $V_\text{eff}(s)$ for each state in table <ref>. We also note the classical mass of the Skyrmion $\mathcal{M}_7$, the energy contribution from rotations $E_J(0)$ and the contribution from vibrations $\epsilon_\text{vib}$, as well as the total energy of each state $E$. We see that the $C_3$ direction produces lower energy states and should be considered the lower energy direction in $\mathcal{V}_5$. The states arising from the $D_5$ direction have higher energy than any experimentally discovered state and so are not relevant to the known energy spectrum of the $^7$Li/$^7$Be isodoublet. Earlier it was noted that the direction of the vibration in $\mathcal{V}_5$ is important. These results highlight this fact.
§.§ The root mean square matter radius
We saw in figure <ref> that the different states appear to have different sizes. The simplest quantitative measure of the size of a nucleus is the root mean square (rms) matter radius, $\langle r_m\rangle$. We can calculate this for each value of $s$ by taking the square root of
\begin{equation}
r_m^2(s) = \frac{\int |\boldsymbol x|^2 \rho(\boldsymbol x,s) \, d^3 x}{\int \rho(\boldsymbol x,s) \, d^3 x}
\end{equation}
where $\rho(\boldsymbol x,s)$ is the energy density of the Skyrme configuration at $s$. For a given state, the rms matter radius is then
\begin{equation}
\langle r_m\rangle = \bra{\Psi_J}r_m(s)\ket{\Psi_J} = \int r_m(s) u_J^2(s) \sqrt{|g|}\, ds
\end{equation}
where we have taken the vibrational wavefunctions to be normalised. We find that the matter radius of the spin $\frac{3}{2}$ state, in Skyrme units, is
\begin{equation} \label{rms}
\langle r_m\rangle_{\frac{3}{2}} = \, 1.85 .
\end{equation}
Experiments are unable to measure the matter radius directly. However in most nuclei the matter and charge radii are very similar. Thus we compare (<ref>) to the experimentally determined charge radius, $2.444$ fm. Our result depends on our choice of $F_\pi$. Calibration (i) gives a matter radius of $2.22$ fm, close to the experimental value. However calibration (ii) gives a very small radius, $0.96$ fm. Earlier we found that calibration (i) gave a better match to the energy spectrum. This result adds weight to the idea that it is the better choice. Regardless, ratios of lengths are independent of $F_\pi$. As such we can compare the matter radii for the spin $\frac{7}{2}$ and spin $\frac{3}{2}$ states and have more trust in the result. We find that
\begin{equation}
\frac{\langle r_m\rangle_{\frac{3}{2}}}{\langle r_m\rangle_{\frac{7}{2}}} = 1.07 \, .
\end{equation}
Thus we predict that the ground state of $^7$Li is $7 \%$ larger than the second excited state, which has spin $\frac{7}{2}$. The rms charge radius of an excited state is difficult to measure experimentally. As such there is no data to confirm our prediction. This is an important signature for the Skyrme model as this prediction is in conflict with the standard cluster model and shell model predictions.
§ CONCLUSION AND OUTLOOK
In this paper we have considered the inclusion of vibrational modes in the quantisation of the $B=7$ Skyrmion. We argued that to understand the low lying states of the $^7$Li/$^7$Be isodoublet one can truncate to quantisation along a $1$-dimensional line in vibrational space, $\mathcal{V}_5$. The space has a rich structure best understood using the geometry of a $5$-simplex. Using this, we picked special directions in the space to quantise along. The calculation gives a reasonable energy spectrum, much closer to the experimental data than had previously been found using zero mode quantisation. Most importantly, the spectrum includes all experimentally seen states and has the spin $\frac{7}{2}$ state lying above the spin $\frac{3}{2}$ state.
During the quantisation procedure some cluster structure emerged. We found that the Skyrmion picks out the $C_3$ direction as the lowest energy direction. This is remarkable since this is the vibration used in the basic $4+3$ cluster model. This brings the Skyrme model closer to the cluster models which are used widely in nuclear physics. The advantage of the Skyrme model is that the dynamics of the clusters are fully determined by the Lagrangian. They can merge smoothly into the $B=7$ Skyrmion or be infinitely separated; our formalism takes account of all configurations in between.
We predict that the excited spin $\frac{7}{2}$ state of $^7$Li is smaller than the spin $\frac{3}{2}$ ground state. The result depends crucially on the dodecahedral symmetry of the $B=7$ Skyrmion. This symmetry appears to persist in modified Skyrme models except in extreme BPS models <cit.> <cit.>. Thus this prediction is an important signature for soliton models of finite nuclei.
Vibrational modes have the capacity to fix many issues in the Skyrme model including the high binding energies and small radii found using zero mode quantisation. They also have a fascinating and rich geometric structure. For these reasons alone, more work should be done to understand the vibrational spaces of Skyrmions. It is somewhat surprising that their inclusion leads to a resolution of problems in the $B=7$ sector. Hopefully a similar analysis in other sectors can produce more surprises.
§.§ Acknowledgments
I would like to thank Professor Nick Manton for numerous useful discussions and for comments on the manuscript of this paper. I am grateful to Chris King for discussions regarding the structure of vibrational spaces. The work is supported by an STFC studentship.
Sk T.H.R. Skyrme,
A nonlinear field theory.
Proc. Roy. Soc. A260 (1961) 127.
AttDeu R. Leese, N. S. Manton and B. Schroers,
Attractive Channel Skyrmions and the Deuteron.
Nucl. Phys. B442 (1995) 228.
Light O. Manko, N. S. Manton and S. Wood,
Light nuclei as quantized Skyrmions.
Phys. Rev. C76 (2007) 055203.
Hoyle P. H. C. Lau and N. S. Manton,
States of Carbon-12 in the Skyrme Model.
Phys. Rev. Lett. 113 (2014) 232503.
Tang Y. C. Tang, K. Wildermuth and L. D. Pearlstein,
Cluster Model Calculation on the Energy Levels of the Lithium Isotopes.
Phys. Rev. 123 (1961) 548.
108 D. T. J. Feist, P. H. C. Lau and N. S. Manton,
Skyrmions up to Baryon Number 108.
Phys. Rev. D87 (2013) 085034.
Kim W. K. Baskerville,
Vibrational spectrum of the $B=7$ Skyrme soliton.
hep-th/9906063 (1999).
LB G. W. Gibbons and N. S. Manton,
Classical and quantum dyanmics of BPS monopoles.
Nucl. Phys. B274 (1986) 183.
FRc D. Finkelstein and J. Rubinstein,
Connection between Spin, Statistics, and Kinks.
J. Math. Phys. 9 (1968) 1762.
RMan C. J. Houghton, N. S. Manton and P. M. Sutcliffe,
Rational maps, monopoles and Skyrmions.
Nucl. Phys. B510 (1998) 507.
Steffan S. Krusch,
Homotopy of Rational Maps and the Quantization of Skyrmions.
Ann. Phys. 304 (2003) 103.
Olga7 O. V. Manko and N. S. Manton,
On the spin of the $B=7$ Skyrmion.
J. Phys. A40 (2007) 3683.
Duet E. Braaten and L. Carson,
Deuteron as a Soliton in the Skyrme Model.
Phys. Rev. Lett. 56 (1986) 3525.
book N. Manton and P. Sutcliffe,
Topological Solitons.
Cambridge University Press: Cambridge (2004).
data D. R. Tilley, C. M. Cheves, J. L. Godwin, G. M. Hale, H. M. Hofmann, J. H. Kelley, C. G. Sheu and H. R. Weller,
Energy levels of light nuclei $A=5, 6, 7$.
Nucl. Phys. A708 (2002) 3.
Casimir F. Meier and H. Walliser,
Quantum Corrections to Baryon Properties in Chiral Soliton Models.
Phys. Rept. 289 (1997) 383.
6thTheses I. Floratos,
Multi-skyrmion solutions of a sixth order skyrme model.
Durham theses. http://etheses.dur.ac.uk/3988 (2001).
lowbinding M. Gillard, D. Harland and J. M. Speight,
Skyrmions with low binding energies.
Nucl. Phys. B895 (2015) 272.
BPSm C. Adam, J. Sánchez-Guillén and A. Wereszczyński,
A Skyrme-type proposal for baryonic matter.
Phys. Lett. B691 (2010) 105.
|
1511.01033
|
Chemistry in star forming filaments
1. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany
We present simulations of star forming filaments incorporating – to our the largest chemical network used to date on-the-fly in a 3D-MHD simulation. The network contains 37 chemical species and about 300 selected reaction rates. For this we use the newly developed package KROME <cit.>. Our results demonstrate the feasibility of using such a complex chemical network in 3D-MHD simulations on modern supercomputers. We perform simulations with different strengths of the interstellar radiation field and the cosmic ray ionisation rate and find chemical and physical results in accordance with observations and other recent numerical work.
§ INTRODUCTION
Modelling the chemical evolution of the gas during the process of star formation on various scales is a numerically and theoretically challenging task. Several authors have incorporated reduced chemical networks in their 3D, magneto-hydrodynamical (MHD) simulations <cit.>. The newly designed chemistry package KROME <cit.> is a versatile package which allows the user to incorporate chemistry in MHD simulations in a very efficient manner and simultaneously guarantees the freedom to choose any desired network as well as its associated cooling and heating processes. Furthermore, recently the importance of filamentary structures in star forming clouds has been re-emphasized <cit.>, which make them an optimal target to test the feasibility of using a detailed chemical network in fully self-consistent 3D-MHD simulations.
§ INITIAL CONDITIONS, FLASH SOLVER, AND CHEMICAL NETWORK
The simulations presented here use the same initial conditions as those presented in <cit.>. The simulated filaments have an initial width of about 0.1 pc, a length of 1.6 pc, and a mass per unit length of 75 M$_{\odot}$/pc. The initial magnetic field strength is 40 $\mu$G, in agreement with recent observations <cit.>. The field is oriented either perpendicular or parallel to the filament. We perform several simulations with varying strengths of the interstellar radiation field (ISRF) and the cosmic ray ionisation rate (CRIR).
In order to model the chemical evolution of the gas, we use the KROME package <cit.>. The network used contains 37 species and 287 reactions and is designed to model the formation of CO and H$_2$ in great detail (also including the formation of H$_2$ on dust). We calculate the attenuation of the ISRF as well as the self-shielding factors for H$_2$ and CO formation in the FLASH code with the TreeCol algorithm <cit.>. We take into account all relevant cooling and heating mechanism present in the interstellar medium, which also includes the cooling via line emission of CO and is isotopologues $^{13}$CO and C$^{18}$O as well as a detailed calculation of the dust temperature, $T_\text{dust}$. A more detailed description of the chemistry network can be found in <cit.>.
§ RESULTS
§.§ Time evolution of a fiducial run
In the following, we present the time evolution of the run with a parallel magnetic field, G$_0$ = 1.7, and a CRIR of 1.3 $\times$ 10$^{-17}$ s$^{-1}$. In the left panel of Fig. <ref> we plot the spatial distribution of H, H$_2$, C, and CO at the end of the simulation, which reveals some differences in the radial distribution. Whereas H$_2$ and CO are concentrated towards the centre of the filament, H and C are more extended.
Left: Spatial distribution of H, H$_2$, C, and CO at the end of the simulation with a parallel magnetic field, G$_0$ = 1.7, and a CRIR of 1.3 $\times$ 10$^{-17}$ s$^{-1}$ along a slice through the centre of the filament. Right: From left to right: Radial dependence of the density of H$_2$, H, H$^+$, and CO, C, and C$^+$ at after the start of this simulation. The black line in the upper left panel shows the total hydrogen density divided by two.
The radial density profiles of H$_2$, H, H$^+$, CO, C, and C$^+$ at 4 different times are shown in the right panel of Fig. <ref>. The constant increase of the central number densities of H, H$_2$, C, and CO over time is caused by the contraction of the filament along the radial direction. On the other hand, $n_\text{H$^+$}$ and $n_\text{C$^+$}$ remain almost unchanged over time as they recombine when the filament becomes denser. In the upper left panel we also show the number density of all H atoms (divided by two in order to be comparable to $n_\text{H$_2$}$). In the centre of the filament most of the hydrogen is bound in H$_2$, only outside $\sim$ 0.1 pc, where the black and purple line start to differ, hydrogen mainly occurs in atomic or ionised form. Hence, there is a gradual conversion of H$^+$ over H to H$_2$ towards the centre of the filament as well as of C$^+$ over C to CO, with $n_\text{C}$ exceeding $n_\text{CO}$ at $\sim$ 0.1 pc as well.
§.§ Impact of the ISRF and the CRIR
Next, we study the influence of the ISRF and CRIR on the properties of the filaments (right panel of Fig. <ref>). Increasing the CRIR naturally results in a higher ionisation fraction of gas. This is reflected by the abundances of H$^+$ and C$^+$, which are 1 - 2 orders of magnitude higher in runs with . Also $T_\text{gas}$ shows a slight increase with increasing CRIR, which is due to the larger amount of energy released by various dissociation reactions caused by cosmic rays. Increasing the strength of the ISRF only marginally affects the chemical composition of the gas. However, $T_\text{gas}$ and $T_\text{dust}$ are increased by a few Kelvin, which is most likely due to the increased photoelectric heating. Interestingly, we find that for all runs $T_\text{dust}$ decreases towards the centre of the filaments. We attribute this to the progressive attenuation of the ISRF – mainly responsible for dust heating – towards the centre of the filament <cit.>.
§ DISCUSSION AND CONCLUSIONS
§.§ Physical interpretation
Assuming a polytropic relation $T_\text{dust} \propto \rho^{\gamma - 1}$, we find $\gamma$ to be in a range from 0.9 to 0.95, which is in good agreement with HERSCHEL observations by <cit.>. We emphasize that $T_\text{gas}$ and $T_\text{dust}$ are markedly different, which requires independent measurements for both quantities in observations. We attribute this to the fact that the collisional coupling between gas and dust at is not yet strong enough to assure similar temperatures.
It can also be seen that the ratio $R = \frac{n_{\text{CO}}}{n_\text{H$_2$}}$ is not constant along the radial direction. There is a strong decline of $R$ with increasing radius of about 2 orders of magnitude, which is due to the fact that the formation of CO happens at somewhat smaller radii (i.e. higher gas column densities) than that of H$_2$. Our work therefore suggests that caution is recommended when using CO line intensities (i.e. the X-factor) to obtain total gas masses. In the centre of the filament, however, $R$ is comparable for all our runs with a value around $1.5 \times 10^{-4}$, and thus in good agreement with observations.
§.§ Numerical cost/Performance
The simulations were carried out on computing nodes with 2 - 5 GB memory per CPU, which corresponds to state-of-the-art computing nodes at modern supercomputing facilities. We compare our runs with an isothermal ($T$ = 15 K) simulation without a chemical network. We find that the inclusion of the network increases the computation costs per timestep by roughly a factor of 5 – 7.
§.§ Conclusions and outlook
We present first results of simulations of star forming filaments using – to our knowledge – the largest chemical network ever applied in 3D-MHD simulations <cit.>, combining the versatile chemistry package KROME <cit.> with the TreeCol algorithm <cit.>. We show that in terms of memory consumption such simulations are feasible on modern supercomputers. The results appear to be promising and in good agreement with observational results.
This work paves the way for many future applications: The detailed chemical modelling will allow us to produce synthetic observations of several molecular lines as well as of continuum emission. Furthermore, the simulations can improve our understanding of the X-factor required for the mass determination of gaseous objects. Moreover, the network will allow us to study in detail the evolution of other molecules like H$_2$O or the cosmic ray tracer H$_3^+$.
[André et al.(2010)André, Men'shchikov, Bontemps,
Könyves, Motte, Schneider, Didelon, Minier, Saraceno,
Ward-Thompson, di Francesco, White, Molinari, Testi, Abergel,
Griffin, Henning, Royer, Merín, Vavrek, Attard,
Arzoumanian, Wilson, Ade, Aussel, Baluteau, Benedettini,
Bernard, Blommaert, Cambrésy, Cox, di Giorgio, Hargrave,
Hennemann, Huang, Kirk, Krause, Launhardt, Leeks, Le Pennec,
Li, Martin, Maury, Olofsson, Omont, Peretto, Pezzuto, Prusti,
Roussel, Russeil, Sauvage, Sibthorpe, Sicilia-Aguilar, Spinoglio,
Waelkens, Woodcraft, & Zavagno]Andre10
André, P., Men'shchikov, A., Bontemps, S., et al. 2010, A&A, 518,
[Clark et al.(2012)Clark, Glover, & Klessen]Clark12
Clark, P. C., Glover, S. C. O., & Klessen, R. S. 2012, MNRAS, 420, 745
[Clark et al.(2013)Clark, Glover, Ragan, Shetty, &
Clark, P. C., Glover, S. C. O., Ragan, S. E., Shetty, R., & Klessen,
R. S. 2013, ApJL, 768, L34
[Grassi et al.(2014)Grassi, Bovino, Schleicher, Prieto,
Seifried, Simoncini, & Gianturco]Grassi14
Grassi, T., Bovino, S., Schleicher, D. R. G., et al. 2014, MNRAS, 439,
[Palmeirim et al.(2013)Palmeirim, André, Kirk,
Ward-Thompson, Arzoumanian, Könyves, Didelon, Schneider,
Benedettini, Bontemps, Di Francesco, Elia, Griffin, Hennemann,
Hill, Martin, Men'shchikov, Molinari, Motte, Nguyen Luong,
Nutter, Peretto, Pezzuto, Roy, Rygl, Spinoglio, &
Palmeirim, P., André, P., Kirk, J., et al. 2013, A&A, 550, A38
[Seifried & Walch(2015a)]Seifried15b
Seifried, D. & Walch, S. 2015a, ArXiv:1510.06544
[Seifried & Walch(2015b)]Seifried15
Seifried, D. & Walch, S. 2015b, MNRAS, 452, 2410
[Sugitani et al.(2011)Sugitani, Nakamura, Watanabe,
Tamura, Nishiyama, Nagayama, Kandori, Nagata, Sato, Gutermuth,
Wilson, & Kawabe]Sugitani11
Sugitani, K., Nakamura, F., Watanabe, M., et al. 2011, ApJ, 734, 63
[Walch et al.(2015)Walch, Girichidis, Naab, Gatto,
Glover, Wünsch, Klessen, Clark, Peters, Derigs, &
Walch, S., Girichidis, P., Naab, T., et al. 2015, MNRAS, 454, 238
[Wünsch et al.(2015, in prep.)Wünsch, Walch, Whitworth,
& Dinnbier]Wunsch15
Wünsch, R., Walch, S., Whitworth, A., & Dinnbier, F. 2015, in prep.
|
1511.00941
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ArgoNeuT Collaboration
$^{1}$Fermi National Accelerator Laboratory, Batavia, IL 60510
$^{2}$Yale University, New Haven, CT 06520
$^{3}$Syracuse University, Syracuse, NY 13244
$^{4}$Kansas State University, Manhattan, KS 66506
$^{5}$Michigan State University, East Lansing, MI 48824
$^{6}$Universita dell'Aquila e INFN, L'Aquila, Italy
$^{7}$Pacific Northwest National Laboratory, Richland, WA 99354
$^{8}$University of Bern, Bern, Switzerland
$^{9}$The University of Texas at Austin, Austin, TX 78712
$^{10}$INFN - Laboratori Nazionali del Gran Sasso, Assergi, Italy
The ArgoNeuT collaboration reports the first measurement of neutral current $\pi^{0}$ production in $\nu_{\mu}$-argon and $\bar{\nu}_{\mu}$-argon scattering. This measurement was performed using the ArgoNeuT liquid argon time projection chamber deployed at Fermilab's NuMI neutrino beam with an exposure corresponding to 1.2$\times 10^{20}$ protons-on-target from the Fermilab Main Injector and a mean energy for $\nu_{\mu}$ of 9.6 GeV and for $\bar{\nu}_{\mu}$ of 3.6 GeV. We compare the measured cross section and kinematic distributions to predictions from the GENIE and NuWro neutrino interaction event generators.
Valid PACS appear here
§ INTRODUCTION
Interest in the precise measurement of neutrino-nucleus cross-sections has grown in recent years due to their effect on the interpretation of neutrino oscillation data. Few precise measurements exist for $\nu$ and $\bar{\nu}$ neutral current (NC) neutral pion ($\pi^{0}$) production <cit.>. Cross-section uncertainties for neutral current neutrino scattering become important as precision oscillation measurements attempt to measure charge-parity violation in the neutrino sector ($\delta_{CP}$ mixing parameter)<cit.> and disentangle the neutrino mass hierarchy question. The NC$\pi^{0}$ channel is of particular importance to neutrino oscillation experiments as it can be experimentally misidentified as $\nu_{e}$ or $\bar{\nu}_{e}$ charged current production. This misidentification complicates the interpretation of $\nu_{\mu} \rightarrow \nu_{e}$ appearance oscillation measurements, which are required for the detection of neutrino CP violation.
Future short-baseline oscillation experiments, such as MicroBooNE <cit.>, SBND <cit.>, and ICARUS <cit.> as well as long baseline experiments, such as DUNE <cit.>, plan to utilize large scale liquid argon time projection chambers (LArTPCs)<cit.> to detect neutrino interactions. This detector technology offers exemplary electromagnetic shower reconstruction capabilities as well as electron/photon discrimination ability, as recently demonstrated by the ICARUS and ArgoNeuT collaboration <cit.>. While a previous measurement of the energy reconstruction of $\pi^{0}$ mesons from cosmic ray production has been performed for LArTPC's <cit.>, prior to the measurement presented in this paper, no direct measurement of the neutral current neutrino-argon interaction rate has been performed. The difficulty of identifying NC$\pi^{0}$ interactions in neutrino experiments has lead to few measurements of this process and thus the uncertainty in the cross-section is often a large systematic in $\nu_{e}$ appearance oscillation measurements. The characterization of the NC$\pi^{0}$ production in a LArTPC has increased importance as these LArTPC experiments attempt to disentangle possible hints of new physics from $\nu_{e}$ appearance as reported by the LSND collaboration <cit.> and the MiniBooNE collaboration <cit.>.
The unique electron/photon discrimination power offered by LArTPCs will allow future experiments, such as the forthcoming MicroBooNE experiment, to either confirm or rule out any excess seen in electron like events thought to originate from $\nu_{\mu} \rightarrow \nu_{e}$ oscillations. Moreover this $e/\gamma$ discrimination allows LArTPCs the ability to better characterize the dominant background, namely mis-identified $\pi^{0} \rightarrow \gamma \gamma$. In order to accomplish this, precise characterization of NC$\pi^{0}$ production is of the utmost importance.
Fig. <ref> shows the predicted production cross-section for semi-inclusive NC$\pi^{0}$ utilizing both the GENIE <cit.> and NuWro <cit.> neutrino event generators. Both generators predict similar cross-sections in the lower energy region ($<$10 GeV), however the NuWro event generator only includes $\Delta$ resonant production of the $\pi^{0}$ meson, and is thus known to become deficient at higher energies. This known difference does not impact this analysis since the neutrino energies we are interested in are below 10 GeV. Accurate modelling of this production requires knowledge of both the underlying neutrino-nucleon interactions and of final state interactions.
Semi-inclusive neutral current $\pi^{0}$ production as a function of neutrino energy on an argon target as predicted by the GENIE and NuWro event generators.
In this paper we present the first measurement of neutrino induced NC$\pi^{0}$ production on an argon target. The interaction final state utilized in this analysis is defined as:
\begin{equation}
\nu_{\mu} + \mbox{Ar} \rightarrow \nu_{\mu} + \pi^{0} + \mbox{X},
\end{equation}
\begin{equation}
\bar{\nu}_{\mu} + \mbox{Ar} \rightarrow \bar{\nu}_{\mu} + \pi^{0} + \mbox{X}
\end{equation}
where NC$\pi^{0}$ is defined as an event topology where there is no electron or muon in the final state, at least one $\pi^{0}$ meson that decays to two photons, and any other number of final state nucleons or mesons (X) are present. In the instance where multiple photons are observed in the final state, all possible combinations of photon pairs are considered when attempting to reconstruct the $\pi^{0}$ meson from where the photons originated. This definition differs slightly from much of the historical neutral current $\pi^{0}$ data <cit.> which typically require one and only one $\pi^{0}$ meson and little other activity in the detector (typically a single proton). This definition is used in this analysis to help mitigate the low statistics of the data sample. This difference in final state definition, in addition to the neutrino scattering occurring off a much higher Z nuclei such as argon, complicates any direct comparison to historic data. However, where possible comparisons are made to previous measurements.
§ OVERVIEW OF THE ANALYSIS
Fig. <ref> shows a simulated Monte Carlo (MC) NC$\pi^{0}$ event inside ArgoNeuT. This particular event demonstrates the semi-inclusive topology in which the neutrino interacts with the nucleus and causes the ejection of a single $\pi^{0}$ meson and a large number of other final state particles. This $\pi^{0}$ then immediately decays into a pair of photons that convert to electron / positron ($e^{+} e^{-}$) pairs a distance characteristic of the 14 cm radiation length of liquid argon from the neutrino interaction point (referred to as the event vertex). The ionization caused by the $e^{+} e^{-}$ pair thus registers on the read-out wires as two clusters of charge (“showers”) pointing back to a common vertex.
Event display for a Monte Carlo neutral current $\pi^{0}$ event simulated in the ArgoNeuT detector.
In order to identify and reconstruct these NC$\pi^{0}$ events, the analysis proceeds in four parts.
*Event Selection
First, candidate NC$\pi^{0}$ events are identified utilizing a series of selection criteria that are chosen to reject charge current (CC) interactions and search for topologies consistent with $\pi^{0} \rightarrow \gamma\gamma$ decays inside the ArgoNeuT detector. The selection criteria utilized is outline in Section <ref>.
*Energy Corrections
ArgoNeuT's small volume causes many of the photons resulting from a $\pi^{0} \rightarrow \gamma\gamma$ decay to not be fully contained within the TPC. From MC studies, 60$\%$ of the electromagnetic showers coming from $\pi^{0} \rightarrow \gamma\gamma$ decay have less than 50$\%$ of their energy contained. Moreover, 40$\%$ of events have both photon showers with less than 50$\%$ containment. By using the prior that the pair of photons observed in the event come from a decay of a $\pi^{0}$ meson, it is possible to correct back the missing energy due to loss from poor containment based on the opening angles of the photon pair and the topological location of the shower inside the detector. The templates used for the energy corrections are based on the simulation of these events inside the detector as well as the topology of the reconstructed event. MC description of relevant event observables is provided in Section <ref> with the full procedure for the energy corrections and its results given in Section <ref>.
*Reconstructed $\pi^{0}$ Kinematics
Following the application of the energy corrections, the data is presented as a function of $\pi^{0}$ kinematic variables and the reconstructed photon energy and momentum.
The momentum of the two photon pair defined as
\begin{equation}\label{eqn:EnergyMomentum}
P_{\gamma\gamma} = \sqrt{E_{1}^{2} + E_{2}^{2}+2E_{1}E_{2} \cos(\theta_{\gamma\gamma})},
\end{equation}
where $E_{i}$ is the energy of the photon (ordered by their energy) and $\theta_{\gamma\gamma}$ is the opening angle between the photon pair. Another kinematic observable of the pair of photons is the cosine of the angle of the photon pair ($\gamma\gamma$) with respect to the beam as defined by
\begin{equation}\label{eqn:CosPi0}
\cos(\gamma\gamma) = \frac{\vec{P}_{z}^{\gamma^{1}} + \vec{P}_{z}^{\gamma^{2}}}{P_{\gamma\gamma}} ,
\end{equation}
where $\vec{P}_{z}^{\gamma^{i}}$ is the z component of the momentum of the photon in the pair and $P_{\gamma\gamma}$ is the reconstructed momentum of the pair of photons (which is the momentum of $\pi^{0}$ mesons in the lab frame if they were correctly identified). Distributions of these quantities may be found in Section <ref>.
*Ratio of NC($\pi^{0}$) to CC
One way to interpret the data beyond the reconstructed kinematics of the $\pi^{0}$ and allow a comparison with results obtained from other experiments and theory is to convert the observed event rate into a ratio of efficiency corrected NC$\pi^{0}$ production to efficiency corrected inclusive charged current (CC) production. This ratio can be written as
\begin{equation}\label{eqn:RatioDefn}
Ratio(NC/CC) = \frac{\sigma(NC\pi^{0})}{\sigma(CC)} = \frac{\sum\limits_{NC} \frac{S_{i}^{NC} - B_{i}^{NC}}{\epsilon_{i}^{NC}\Phi_{\nu}N_{Targets}}}{\sum\limits_{CC} \frac{S_{i}^{CC} - B_{i}^{CC}}{\epsilon_{i}^{CC}\Phi N_{Targets}}},
\end{equation}
where $S_{i}^{NC/CC}$ is the number of the signal events in the particular bin from data from NC$\pi^{0}$/CC events, $B_{i}^{NC/CC}$ is the predicted background coming from MC scaled to the appropriate protons-on-target (P.O.T), and $\epsilon_{i}^{NC/CC}$ is the efficiency for NC$\pi^{0}$ or CC events taken from simulation. For identical flux and number of targets Eq. <ref> simplifies to
\begin{equation}\label{eqn:RatioSimp}
\frac{\sum\limits_{NC} \frac{S_{i} - B_{i}}{\epsilon_{i}}}{\sum\limits_{CC} \frac{S_{i} - B_{i}}{\epsilon_{i}}} = \frac{N(NC\pi^{0})}{N(CC)}.
\end{equation}
The numerator represents all events with no muon or electron and at least one $\pi^{0}$ observed in the final state exiting the target nucleus. This interaction can be accompanied by any number of other nucleons or other mesons. The denominator represents events with an identified muon in the final state coming from the target nucleus and any other number of other nucleons or final state mesons. One complication in making this simplification arises because the anti-neutrino beam is actually a mixture of a neutrinos and anti-neutrinos. In order to address this, the sample is broken into two components
\begin{equation}\label{eqn:nuRatioOverview}
\frac{N_{\nu}(NC\pi^{0})}{N_{\nu}(CC)} = \frac{\mbox{Number of }\nu \mbox{ induced NC }\pi^{0} \mbox{ Events}}{\mbox{Number of } \nu \mbox{ induced CC Events}},
\end{equation}
\begin{equation}\label{eqn:antinuRatioOverview}
\frac{N_{\bar{\nu}}(NC\pi^{0})}{N_{\bar{\nu}}(CC)} = \frac{\mbox{Number of }\bar{\nu} \mbox{ induced NC }\pi^{0} \mbox{ Events}}{\mbox{Number of } \bar{\nu} \mbox{ induced CC Events}},
\end{equation}
ensuring that the simplification made to obtain Eq. <ref> takes into account the flux in the anti-neutrino beam due to neutrinos and anti-neutrinos for both the CC and NC sample. In Eq. <ref> and Eq. <ref>, the denominator is taken directly from an analysis of charged current interactions measured in the ArgoNeuT detector <cit.>. This analysis utilized the MINOS Near Detector (MINOS-ND), a 0.98 kton magnetized steel-scintillator calorimeter <cit.>, for non-contained muons exiting in the forward direction to determine the sign of the outgoing lepton and thus could distinguish the species of neutrino interaction as well as the momentum for that lepton. This previous measurement also takes into account the different acceptances for neutrinos and anti-neutrinos.
For the numerator in Eq. <ref> and Eq. <ref>, the neutral current channel, the technique of utilizing the MINOS-ND to distinguish the species of neutrino is not possible. Instead, a MC based estimate on the NuMI beam composition is used to estimate the fraction of NC$\pi^{0}$ events that come from $\nu$ and $\bar{\nu}$ interactions. This means that the number extracted for $N_{\nu}(NC\pi^{0})$ is anti-correlated with $N_{\bar{\nu}}(NC\pi^{0})$ and thus the ratio extracted is also anti-correlated. This separation technique, however, allows for comparisons to other experimental results and thus is a useful tool to interpret the ArgoNeuT data. Details of the procedure to extract this measurement are presented in Section <ref>.
*Flux Averaged NC($\pi^{0}$) Cross-Section
Furthermore, we give a measurement of the flux-averaged absolute cross-section for NC$\pi^{0}$ production on an argon nucleus. A similar procedure as described above for extracting the component of the cross-section due to $\nu$ and $\bar{\nu}$ interactions is followed and results are compared to the GENIE and NuWro neutrino event generator in Section <ref>.
§ EVENT SELECTION
The ArgoNeuT detector <cit.> is a 47.5 $\times$ 40 $\times$ 90 cm$^{3}$($x$-$y$-$z$) active volume LArTPC with the longest dimension ($z$) situated along the beam axis and two wire planes positioned on beam right. ArgoNeuT ran in the NuMI-LE (Neutrinos at the Main Injector, Low Energy option) beam at Fermi National Accelerator Laboratory <cit.> and collected 0.085$\times 10^{20}$ protons-on-target (POT) in neutrino mode and 1.20$\times 10^{20}$ POT in anti-neutrino mode. Tab. <ref>, taken from Ref. <cit.>, provides the total flux for the anti-neutrino mode beam used in this analysis.
3cTotal anti-neutrino mode fluxes
$E_{\nu}$ GeV $\nu$ Flux $\bar{\nu}$ Flux
0 - 50 3.9 $\pm 0.4 \times 10^{7}$ 2.4 $\pm 0.3 \times 10^{5}$
The neutrino and antineutrino fluxes for the anti-neutrino mode beam, taken from Ref. <cit.>, used in this analysis. The flux unit is $\nu_{\mu}$/GeV/m$^{2}$/$10^{9}$ POT
A 481 V/cm electric field is imposed that allows ionization trails created by charged particles traversing the argon medium to be drifted toward the sensing wires. The signal from the wire planes, oriented 60$^{\circ}$ with respect to one another, is combined to provide 3D reconstruction of the neutrino interaction along with particle identification and calorimetric information.
ArgoNeuT utilizes the LArSoft software package <cit.> that provides a full rendering of charged particles interacting inside the ArgoNeuT detector. LArSoft provides a full simulation of the experiment and electronics response as well as a simulation of neutrino interactions utilizing the GENIE neutrino event generator and GEANT4 <cit.> for the propagation of particles inside the detector. The propagation of particles into the MINOS-ND is done using GEANT3 <cit.> and a standalone version of the MINOS-ND simulation is employed to characterize the matching of tracks passing from ArgoNeuT to MINOS. Monte Carlo events are treated in the reconstruction package identically as data events.
Three volumes inside the ArgoNeuT detector are defined and used throughout this analysis. All three volumes employ a right-handed coordinate system with positive $Y$ vertical, positive $Z$ parallel to the neutrino beam axis, and the origin placed at the upstream end of the ArgoNeuT TPC, with the detector centered on $Y=0$, and $X=0$ at the TPC sense-wire plane. We also use the conventional polar angles $\theta$ and $\phi$ to denote vector directions, as well as $\theta_x$ and $\theta_y$, the angles with respect to the $X$ and $Y$ axes, respectively.
* Active Volume:
Is the volume of the entire ArgoNeuT TPC defined as: 0 cm$<$ $X$ $<$47.5 cm, - 20 cm$<$ $Y$ $<$20 cm, 0 cm$<$ $Z$ $<$90 cm.
* Fiducial Volume:
A volume definded to allow for a small volume of argon between any interaction and the active boundary. This distance is chosen to mirror that used in the inclusive charged current analysis. This allows for a ratio between the measured neutral current and charge current rate to be easily compared. The fiducial volume is thus defined as: 3 cm$<$ $X$ $<$44.5 cm, - 16 cm$<$ $Y$ $<$16 cm, 6 cm$<$ $Z$ $<$86 cm.
* Photon Conversion Volume (PCV):
A volume defined such that a volume of argon exists between the point where the photon converts to an $e^{+} e^{-}$ pair (defined as the photon vertex) and the boundary of the detector. This volume allows for a photon that converts near the boundary to still be identified via a dE/dX measurement. The photon conversion volume is defined as: 5 cm$<$ $X$ $<$42.5 cm, - 15 cm$<$ $Y$ $<$15 cm, 5 cm$<$ $Z$ $<$85 cm.
The signal events for this analysis are characterized by a neutrino interaction occurring inside the active volume of the detector that produces at least one $\pi^{0}$ via a neutral current interaction and subsequently decays to a pair of photons. In order to be considered a neutral current interaction no track may be reconstructed and identified as a muon or electron of either sign. Photons from the decay of the $\pi^{0}$ must convert to $e^{+} e^{-}$ pairs inside the PCV in order to be considered in our selection.
Background events for this analysis are categorized inclusively as any event that is not already identified as NC$\pi^{0}$ event. Four selection requirements are used to identify candidate NC$\pi^{0}$ events. These selection requirements are chosen to reject events that appear to come from a charged current interaction, and thus produce a charged lepton in the detector volume, and to identify events that have a topology consistent with the presence of a $\pi^{0} \rightarrow \gamma\gamma$ decay.
We first reject events in which a muon track found in MINOS-ND that is matched to a track in the ArgoNeuT detector. The front face of MINOS-ND is approximately 1.5 m downstream of ArgoNeuT, and the center of ArgoNeuT is located 20 cm below the center of the MINOS fiducial volume. An ArgoNeuT-MINOS-ND track match is defined by the following criteria: (i) the track has a MINOS-ND hit within 20 cm of the front face of the MINOS-ND detector; (ii) the MINOS-ND track must start within 35 cm of the projected ArgoNeuT track location in the $y-z$ plane; (iii) the ArgoNeuT and MINOS-ND track direction cosine differences must satisfy the requirements $|\delta\cos(\theta_{x})|<1.0$, $|\delta\cos(\theta_{y})|<1.0$, and $|\delta\cos(\theta_{z})|<0.5$. These selection requirements are similar (although much more inclusive) to those used in previous ArgoNeuT CC analyses (<cit.>) and have been shown to be efficient at identifying charged current interactions within the ArgoNeuT detector. If such a track exists, the event is rejected as likely coming from a charged current interaction. The dominant inefficiency for this selection comes from the incorrect matching of non-related tracks in the MINOS-ND to charged pion tracks present in the NC$\pi^{0}$ interaction.
The next selection requirement applied is designed to reject charged current events missed by the anti-matching to the MINOS-ND. These events can fail the anti-matching because the muon produced in a charged current interaction does not exit in the direction of the MINOS-ND or because the track is poorly reconstructed and thus does not match back to the track found in MINOS-ND. To reject these events we utilize the reconstruction information from inside the ArgoNeuT TPC and veto an event that has a topology consistent with having a muon originating from a neutrino interaction vertex, as expected in a charged current interaction. These events must have at least two tracks that are identified as either a muon/proton, muon/pion or muon/muon pair emanating from a common vertex. This selection was chosen over simply removing any event with a minimum ionizing track (MIP) present in the event in order to preserve the statistics of the selected sample and avoid removing neutral current events with charged pions in the final state that are incorrectly identified. The dominant inefficiency comes from multiple misidentification of charged pions as muons thus causing NC$\pi^{0}$ events to be incorrectly excluded from the sample. However, this is a relatively small inefficiency compared to other selection criteria. The choice to not reject any event with a minimum ionizing track was made to help increase the statistics of the sample of candidate NC$\pi^{0}$ events where, in addition to the neutral pion, charged pions and high momentum protons which exit the chamber are present.
The next event selection identifies $\pi^{0} \rightarrow \gamma\gamma$ decays by leveraging the powerful track reconstruction techniques available in LArTPC's. Correlated groups of short-length tracks consistent with electrons or positrons produced in electromagnetic showers can be identified and reliably separated from tracks produced by pions, muons, and protons. The selection accomplishes this identification by analyzing the components of an electromagnetic shower as if it is made of many small tracks and attempting to identify the starting “trunk” of the shower. The electromagnetic nature of short tracks can be verified by a particle identification (PID) procedure that correlates energy loss in the TPC, $dE/dX$, with the range of the tracks in liquid argon, as well as taking into account the topological reconstruction of the small tracks. This also allows for a determination of the energy of each track. Fig. <ref> shows a simulated event where the electromagnetic showers have been broken into smaller components based on their track-like structure. Importantly, the beginning of the shower is reliably reconstructed as a small track component of the shower and can be identified by analyzing the reconstructed components.
A simulated neutral current $\pi^{0}$ event reconstructed using the small track algorithm in order to break up an electromagnetic shower into smaller track like segments for analysis and identification. Inside the photon showers, highlighted in the image, the unique colors indicate the individual small tracks that have been reconstructed.
The procedure for selecting $\pi^{0} \rightarrow \gamma\gamma$ topologies using these small tracks is to first require that at least two such small tracks are found in the event. If during the application of any subsequent requirement the number of small tracks present is less than two, the event is removed from consideration. Short tracks with a PID assignment consistent with that of a muon, pion, or proton are removed from consideration. Next, pairs of small tracks are kept for consideration if their start points are separated by at least 4.0 cm. This requirement identifies pairs of small tracks coming from a pair of photons that are separated in space removing highly ionizing parts of a single shower that have been broken into many small tracks while preserving the unique starting portion of the distinct shower pairs. Next we require greater than 75 $\%$ of the first 4 cm of the track to have a $dE/dX$ value $\geq$ 3.5 MeV/cm. If the track is shorter than 4 cm in length than the requirement is that the majority ($\geq50\%$) track have a $dE/dX$ $\geq$ 3.5 MeV/cm. This selection is designed to only keep highly ionizing short tracks, such as those coming from a photon conversion into an $e^{+} e^{-}$ pairs, and reject those likely due to a minimum ionizing particle. Finally, to separate distinct photon conversions from the background of electron and single photon, we cluster the hits in the event using a density based spatial clustering algorithm (DBSCAN) and require that any small track share less than 85$\%$ of its hits with any other small track from within the same DBSCAN based cluster. This DBSCAN algorithm groups together nearby distributions of charge and associates them into one object. The result of this requirement is that two spatially close small tracks that belong to the same DBSCAN cluster (sharing the majority of their hits) will be removed from consideration.
Finally, events are visually examined (“hand-scanned”) by physicists (“scanners”) to identify two electromagnetic showers originating from a NC$\pi^{0}$ interaction. The hand-scan procedure both identifies events consistent with a $\pi^{0} \rightarrow \gamma\gamma$ decay and rejects background events that have passed prior selection requirements. In general, the hand-scan of events takes place in three parts. Step one selects events with associated photon showers resulting from a NC $\pi^{0}$ interaction. Here the topology of the event is taken into consideration and the scanner looks for two showers pointing back to a common vertex point. Part two has the scanner select the clusters in each view by defining the shower's start point and axis. The shower axis represents a cylinder around which hits associated with the shower will be selected. When the start point and axis are identified for both planes of wires, the preliminary 3D angle shower object is created. In step three the scanner includes or excludes any hits not initially associated to the shower and builds the 3D shower object and evaluates the quality of the reconstructed shower.
The scanner has the ability to review the shower selection by reconstructing the 3D shower's vertex location, angles ($\theta , \phi$), reconstructed energy, and $dE/dX$ over the first 2.4 cm of the shower. This allows the scanner to check that the shower is consistent with a pair converting photon (dE/dX values $\geq$ 2.5 MeV/cm) and begins within the appropriate boundary (PCV). If the shower is deemed to have been correctly identified, it is then processed into an offline file for further analysis. Once fully reconstructed offline, showers whose $dE/dX$ profile in the first 2.4 cm of the shower is greater than 2.5 MeV/cm, to distinguish photon from electron induced showers, are kept for analysis.
§ MATCHING ELECTROMAGNETIC SHOWER ANGLES
Following the identification of candidate NC$\pi^{0}$ events, the reconstruction of the selected electromagnetic showers becomes necessary in order to further identify events consistent with coming from $\pi^{0} \rightarrow \gamma\gamma$ decay. In order to do this, the energy of the shower must be obtained. However, as will be discussed further in Section <ref>, the majority of photon showers coming from $\pi^{0}$ decays are not contained within the ArgoNeuT detector. Fig. <ref> demonstrates the problem by plotting the reconstructed energy of the candidate data events compared to the true MC Energy for NC$\pi^{0}$ events and the simulated deposited energy inside the active volume of the detector.
The reconstructed energy of candidate NC$\pi^{0}$ events compared to the simulated deposited energy from NC$\pi^{0}$ interactions inside the active volume (blue) and the MC true energy of the photons (red). The distributions have been area normalized.
However, utilizing the fine grain tracking detection of LArTPC's, it is possible to reconstruct the angle of the photons given the visible portion of the shower in the detector. Fig. <ref> shows the distribution for the candidate NC$\pi^{0}$ data events compared to the simulated deposited angle calculated using the charge weighting of the visible shower and the MC true angle. Using the reconstructed start point and angle of the shower, made possible by the tracking capabilities of the detector, the data closely tracks the true angles.
The reconstructed angles theta ($\theta$) and phi ($\phi$) for the photon showers of candidate NC$\pi^{0}$ events compared to the simulated deposited angle from NC$\pi^{0}$ interactions inside the active volume (blue) and the Monte Carlo true angles of the photons (red). The deposited angles are calculated using the start point of the shower and a charge weighted sum of the visible shower.
The well-reconstructed shower angles permit extraction of information about the energy and momentum of the event by utilizing the assumption that the two electromagnetic showers originate from the decay of $\pi^{0} \rightarrow \gamma\gamma$ and constructing the opening angle between the two photons ($\theta_{\gamma\gamma}$). Fig. <ref> shows the $\theta_{\gamma\gamma}$ distribution for the candidate NC$\pi^{0}$ data events compared to the simulated deposited angle that would be calculated using only a charge weighting of the visible shower and the simulated true angle. The distribution tracks well with the true information up to small opening angle, where the reconstruction has difficulty disentangling the two showers. This provides confidence to the hypothesis that these events do in fact come from a $\pi^{0} \rightarrow \gamma\gamma$ and we will utilize this ability to reconstruct angles well within a LArTPC in the next section to attempt to correct back the missing energy.
The reconstructed opening angle between two electromagnetic showers for candidate NC$\pi^{0}$ events compared to the simulated opening angle from NC$\pi^{0}$ interactions inside the active volume (blue) and the simulated true opening angle of the photons (red). The distributions have been area normalized.
§ ENERGY CORRECTIONS
Having reconstructed the portion of the shower that is contained inside ArgoNeuT, corrections must be developed to model the portion of the shower that is not contained within the active volume. The mean momentum for $\pi^{0}$'s created in a neutrino-argon interaction in the ArgoNeuT detector is 730 MeV. At these energies, the photon showers have a radiation length of 14 cm and photons typically convert to electron positron pairs within three radiation lengths, which is the same order as the size of the volume of the ArgoNeuT TPC. Furthermore, these $\pi^{0}$'s can be created anywhere within the fiducial volume, thus the probability of the energy containment of both photons is low. Fig. <ref> shows the simulated energy containment from photons coming from $\pi^{0}$'s with a similar momentum and position distribution as the NC$\pi^{0}$ sample. The fraction of the energy contained is defined as
\begin{equation}\label{eqn:EnergyDep}
1 - \frac{|E_{Deposited} - E_{Total}|}{E_{Total}},
\end{equation}
where $E_{Deposited}$ is the total energy deposited by the electromagnetic shower from the photon inside the active volume. $E_{Total}$ is the true energy of the photon that caused the electromagnetic shower. This definition is chosen such that the energy containment of the shower is between zero (poor containment) and one (full containment)
The fraction of energy from simulated $\pi^{0} \rightarrow \gamma\gamma$ events contained in the ArgoNeuT volume. These events have a mean momentum of 0.5 GeV and are simulated uniformly in their initial ($x$, $y$, $z$) location in the TPC volume.
The dominant cause of energy loss is the shower exiting the boundary of the detector. This happens when a photon converts near the boundary and is directed toward one of the TPC walls, thus resulting in the majority of its energy escaping. Given the radiation length of a photon in argon, the photon would have to convert very far away from any boundary of the ArgoNeuT TPC in order to have near full containment.
A series of corrections based on the topological reconstruction of the visible component of the electromagnetic showers are applied. These corrections aim at adding back the energy loss due to poor containment of the electromagnetic shower inside the active volume.
The details of the derivation of the energy corrections are left for discussion in Appendix <ref>, but broadly speaking the approach adopted for this analysis corrects back this energy loss taking as a prior that two identified photon showers from the event selection described in Section <ref> come from the decay of a $\pi^{0}$. Utilizing this assumption, a relationship between the opening angle of the two photons, $\theta_{\gamma\gamma}$, and the momentum of the $\pi^{0}$, $P_{\pi^{0}}$, is derived. This relationship provides the basis for the application of the subsequent energy corrections. Specifically, any energy correction that is applied to a photon based on its distance to the nearest boundary is not allowed to cause the calculated $P_{\gamma\gamma}$ from Eq. <ref> to exceed the inferred momentum from $\theta_{\gamma\gamma}$. The amount of containment of the electromagnetic shower within the active volume of the detector is determined by simulating a large number of $\pi^{0} \rightarrow \gamma\gamma$ events inside the ArgoNeuT TPC and building templates for the characteristics of the energy loss based on the location of the photon's conversion. The procedure for constructing the templates is broken into three steps described in greater detail next.
§.§ Angle hypothesis of the $\pi^{0}$ momentum
An event that is identified as having two or more reconstructed showers will have a hypothesis formed assuming that two of the showers come from the decay of a $\pi^{0}$. Using this hypothesis, an initial estimate of the momentum of the $\pi^{0}$ ($P_{\gamma\gamma}$) responsible for these two electromagnetic showers is calculated based on the opening angle between the showers. When the $\pi^{0}$ decays the photons have an angle $\theta_{\gamma\gamma}$ $\leq$180$^{\circ}$ due to the boost from the rest frame to the lab frame. The greater the momentum of the $\pi^{0}$ in the lab frame the smaller the opening angle $\theta_{\gamma\gamma}$ will be, as shown in Fig. <ref>. This correlation breaks down for angles less than 10$^{\circ}$ due to the difficulty of disentangling overlapping showers. To derive the correlation between $\theta_{\gamma\gamma}$ and $P_{\pi^{0}}$ (between 10$^{\circ}$ and 180$^{\circ}$) a polynomial fit is used to provide an analytical expression
\begin{equation}\label{eqn:AngleMomentum}
P_{\gamma\gamma} = C_{0} + \sum_{i = 1}^{6} C_{i}\theta_{\gamma\gamma}^{i},
\end{equation}
with the constants $C_{i}$ given in Appendix <ref>.
The projection of the $\pi^{0}$ momentum versus the mean opening angle between the two decay photons. The full 2-d relationship is given in Fig. <ref> in Appendix <ref>. The fitted function provides a bound on the momentum of the hypothesized $\pi^{0}$ system, which any subsequent energy correction to the photons is not allowed to violate.
Comparing the $P_{\pi^{0}}$ calculated using the energy of the photon showers deposited in the active volume given in Eq. <ref> to the polynomial fit to the hypothesized $P_{\pi^{0}}$ obtained from the opening angle distribution allows a bound to any subsequently applied energy correction. Namely, when considering the application of a correction to one of the two photons ($\gamma_{1}, \gamma_{2}$), due to its containment, the subsequently calculated $P_{\pi^{0}}$ using Eq. <ref> must not exceed the $P_{\pi^{0}}$ calculated using Eq. <ref>. The opening angle sets a minimum value on $P_{\pi^{0}}$ from the kinematics. However, the distribution in opening angle becomes rather sharply peaked about the minimum as $P_{\pi^{0}}$ increases. So, while not excluded from kinematics, a value higher than $P_{\pi^{0}}$ from opening angle is unlikely, and becomes less likely as the opening angle decreases.
Thus with each energy correction applied to one of the photons, one can evaluate if this correction would exceed a reasonable hypothesis for the momentum of the $\pi^{0} \rightarrow \gamma\gamma$ system. If it would, the correction is not applied to the photon. If it does not, the correction is applied and the next energy correction is attempted. The detailed procedure for applying this hypothesis is provided in Appendix <ref>.
§.§ Energy Corrections
With the momentum hypothesis, $P_{\gamma \gamma}$, formed a series of energy corrections are attempted to correct back the energy loss due to poor containment within the ArgoNeuT TPC. The details of the corrections are described in the Appendix in sections <ref> and <ref>, an overview of which we cover here.
Photons of from higher momentum $\pi^{0}$'s are less well contained within the TPC and thus are subject to having their energy missed due to containment. Using Eq. <ref> to estimate the initial momentum of the $\pi^{0}$ system an energy correction is applied to $\gamma_{1}$ and $\gamma_{2}$ (where $\gamma_{1}$ is ordered such that it is the most energetic of the photon pair). $P_{\pi^{0}}$ is then calculated using Eq. <ref> and if the correction is found to violate the initial hypothesis formed using the opening angle, the correction is not applied to $\gamma_{1}$ but instead applied to $\gamma_{2}$ and $P_{\pi^{0}}$ is recalculated. If after this, the correction is still found to violate the initial hypothesis the correction for $\gamma_{1}$ is swapped for the correction to $\gamma_{2}$ and the procedure is repeated. The energy correction can be applied to both, either, or none of the photons for any given event.
Next, a set of corrections are applied based on where the shower vertex is located inside the detector and the direction the shower is pointing. The amount of energy that is deposited inside the detector is strongly correlated with where the photon first converts inside the TPC and how much argon there is between the vertex and the nearest TPC boundary. The basis of these energy corrections depends on the well known electromagnetic shower profile within liquid argon <cit.>. We calculate the distance to the nearest wall in $x$, $y$, $z$ space using the “straight line” distance between the photon shower vertex and the nearest boundary which the shower is pointed towards. To correct back the energy loss due to this topological location of the electromagnetic shower, we plot the fraction of energy contained (as defined in Eq. <ref>) versus the distance to the boundary. A polynomial fit provides a functional form for the energy correction based on its distance to that given boundary. Similar to before, each of these corrections must not violate the initial $P_{\pi^{0}}$ hypothesis formed using Eq. <ref>, and a correction is kept for any individual photon only if these criteria are met.
§.§ Results of Energy Corrections
The complete set of results utilizing the application of the template based energy corrections and the procedure described briefly above is given in Appendix <ref>. From MC studies, less than 10$\%$ of events receive energy corrections that would move the observed shower energy above the true value. Fig. <ref> shows the outcome of the full suite of energy corrections to the reconstructed $\gamma\gamma$ invariant mass, defined as $M_{\gamma\gamma} = \sqrt{4 E_{1}^{\gamma}E_{2}^{\gamma} \sin^{2}(\frac{\theta_{\gamma\gamma}}{2})}.$
The invariant mass of the $\gamma\gamma$ system after the application of the template based energy corrections for a MC sample of $\pi^{0}$ decays inside the ArgoNeuT detector. The distribution is fit with a Gaussian plus polynomial between 0 MeV$\leq M_{\gamma\gamma}\leq$240 MeV. This fit returns a mean value of 140.2$\pm$0.8 MeV with a RMS of 37.6$\pm$0.61 MeV.
The distribution is fit to a Gaussian plus a polynomial (to model the low energy mis-reconstruction) between 0 MeV$\leq M_{\gamma\gamma}\leq$240 MeV. The result of the fit returns a mean value for the Gaussian of 140.2$\pm$0.8 MeV with a width of 37.5$\pm$0.61 MeV (to be compared to $M_{\pi^{0}} = 135$ MeV). This demonstrates that the template based energy corrections do adjust back the photon's energy closer to its true energy and thus give us another tool to identify candidate NC$\pi^{0}$ events.
§ RECONSTRUCTED $\PI^{0}$ KINEMATICS
After applying all corrections to the energy of the showers in our data sample, we require the reconstructed invariant mass, $ M_{\gamma\gamma}$, to lie within the range 60 MeV$\leq M_{\gamma\gamma}\leq$240 MeV. This requirement selects events that reconstruct inside $\pm$2$\sigma$ of the invariant mass of the $\pi^{0}$. The size of the window is selected based on the reconstructed mass peak RMS from the sample of events used to calibrate the energy of the photons.
Fig. <ref> shows the reconstructed invariant mass distribution of anti-neutrino mode data events before the requirement that all events fall between 60 MeV$\leq M_{\gamma\gamma}\leq$240 MeV. A Gaussian plus linear function is fitted to the data, yielding an invariant mass of $131.1 \pm 8.4$ MeV with a width of $81.4 \pm 11.1$ MeV. After requiring events with an invariant mass between 60 MeV$\leq M_{\gamma\gamma}\leq$240 MeV, we reconstruct the invariant mass with a peak of $138.4\pm7.0$ MeV and a width of $54.7\pm7.2$ MeV. The fitted mean is consistent with the 135 MeV $\pi^{0}$ mass taking into account statistical uncertainties and the systematic energy scale error associated with the energy correction scheme.
(Left) $\pi^{0}$ invariant mass plot for all NC$\pi^{0}$ candidate events before requiring that all events fall between 60 MeV$\leq M_{\gamma\gamma}\leq$240 MeV. A clear peak near the invariant mass of the $\pi^{0}$ meson can be seen. (Right) $\pi^{0}$ invariant mass plot zoomed into the invariant mass peak between 60 MeV$\leq M_{\gamma\gamma}\leq$240 MeV. The mean of the fitted Gaussian shifts after the selection of events between 60 MeV$\leq M_{\gamma\gamma}\leq$240 MeV because the low invariant data points have been excluded from the fit. These events include interactions from both $\nu$ and $\bar{\nu}$ scattering events.
Tab. <ref> summarizes the effects of all selection criteria and the application of the energy correction procedure used for this analysis. In total, 123 data events survive all the cuts, consistent with the expected 159 events (101 NC$\pi^{0}$ events and 58 background events) predicted from the GENIE MC. As evident from Fig. <ref>, the data are consistent with a model where $\pi^{0}$ production fully accounts for the two photon mass distribution in the range 60-240 MeV. This observation is consistent with predictions from the MC using the GENIE production model defined in Appendix B.
Number of MC Events
Event Selection Scaled to 1.20$\times10^{20}$ POT Signal Acceptance Background Rejection
(Signal / Background) $\%$ $\%$
Total Number of Events 615 / 10,019 - -
Anti-MINOS Matching 494 / 2,475 80$\%$ 75$\%$
Charged Current Veto 365 / 1,664 74$\%$ 33$\%$
Small Track Reconstruction 285 / 792 78$\%$ 52$\%$
Shower Reconstruction 188 / 126 66$\%$ 84$\%$
$<$dE/dX$>$ $\geq$ 2.5 MeV/cm 158 / 107 84$\%$ 15$\%$
60 MeV$\leq M_{\gamma\gamma}\leq$240 MeV 101 / 58 64$\%$ 46$\%$
Data Passing all event selection 123 2c|
Summary of NC $\pi^{0}$ event selection cuts applied to ArgoNeuT Monte Carlo and data.
Of the background events remaining after all the cuts are applied, 95.7$\%$ are charged current events with a $\pi^{0}$ produced in the neutrino interaction. These are events where the muon was not reconstructed well enough to be matched to the MINOS-ND nor be identified within the TPC. The remaining 4.3$\%$ results from either a mis-identified particle as a photon shower or a $\pi^{+}$ created in the neutrino interaction that underwent charge exchange producing a $\pi^{0}$ that is mis-identified as coming from the primary interaction point.
Fig. <ref> shows the momentum of the $\pi^{0}$, from Eq. <ref> and the cosine of the angle of the $\pi^{0}$ with respect to the beam as defined by Eq. <ref>. Despite the low statistics, both of these distributions have general agreement with the MC prediction in shape while the MC over predicts the peak.
$\pi^{0}$ Momentum and cosine of the angle of the $\pi^{0}$ for the set of anti-neutrino data events with MC backgrounds scaled to 1.2$\times 10^{20}$ POT. These events include interactions from both $\nu$ and $\bar{\nu}$ scattering events.
The photon energy for both photons in the event, the photon z momentum ($P_{z}$) for each photon ($\gamma_{1}, \gamma_{2}$, where the highest energy photon is listed first), and the sum of the z momentum for our data events with MC backgrounds scaled to 1.2$\times 10^{20}$ POT. These events include interactions from both $\nu$ and $\bar{\nu}$ scattering events.
The data presented in Fig. <ref> and Fig. <ref> represent the observed number of events when compared to MC prediction as well as the kinematic distributions that go into these comparisons.
§ RATIO OF NC($\PI^{0}$) TO CC
Normalizing the NC$\pi^{0}$ production rate to the inclusive charged current rate measured by ArgoNeuT over the same running condition reduces many systematic uncertainties, particularly those associate with flux. This procedure also facilitates comparison with previous measurements of this ratio reported by other neutrino experiments <cit.>.
In order to construct the ratio of NC$\pi^{0}$ to CC separately for neutrinos and anti-neutrinos, we divide the anti-neutrino beam into its components. For the charged current sample, <cit.>, the species of neutrino is determined by the MINOS-ND measuring the sign of the charged muon. In the case of NC$\pi^{0}$ production, no such data driven sign determination is possible. Instead we utilize MC to estimate the fraction of events coming from $\nu$ and $\bar{\nu}$ interactions. Our assumption that $f^{\nu}=74 \pm 15\%$ and $f^{\bar{\nu}}=26 \mp 15\%$ of NC$\pi^{0}$ originate from $\nu$ and $\bar{\nu}$ interactions, respectively, consistent with predictions from GENIE. GENIE also predicts that these fractions are independent of the $\nu$ and $\bar{\nu}$ energy. The uncertainty on this fraction is taken as a systematic and is described in Section <ref>. This technique of separating the data sample utilizing the MC fraction of the beam means that the resulting fractions will be anti-correlated with one another through the systematic uncertainty of the beam content.
Taking neutrino interactions as described in Eq. <ref>, to illustrate the procedure, we define the numerator as:
\begin{equation} \label{eqn:NumeratorOfRatio}
N_{Events}(NC\pi^{0}) = \sum\limits_{i = bin} \frac{S_{i}^{\nu} - B_{i}^{\nu}}{\epsilon_{i}^{\nu}},
\end{equation}
where $S_{i}^{\nu}$ is the number of the signal events in a given bin from data that originates from neutrino ($\nu$) interactions and can be written as
\begin{equation}
S_{i}^{\nu} = f^{\nu} D_{i},
\end{equation}
where $f^{\nu}$ is the fraction of events coming from neutrino interactions (in this case approximately 74$\%$) and $D_{i}$ is the number of data events in that particular bin. In Eq. <ref>, $B_{i}^{\nu}$ is the predicted background coming from $\nu$ interactions scaled to the appropriate protons on target (P.O.T). Finally, the term $\epsilon_{i}^{\nu}$ is the efficiency for neutrino induced NC$\pi^{0}$ events taken from MC is given by
\begin{equation}
\epsilon_{i}^{\nu} = \frac{\nu\mbox{ induced NC}\pi^{0}\mbox{'s passing all cuts in the i'th bin}}{\nu\mbox{ induced NC}\pi^{0}\mbox{'s generated in the i'th bin}}
\end{equation}
and is estimated to be $20.3\%$($15.4 \%$) for $\nu$ ($\bar{\nu}$) and is flat as a function of $P_{\pi^{0}}$ and $\cos(\pi^{0})$.
An analogous procedure is followed for the anti-neutrino component of these interactions. Tab. <ref> summarizes the results of the calculation from Eq. <ref> for both the neutrino and anti-neutrino components of the anti-neutrino mode beam. The two sets of numbers for each type of interaction represent the parent distribution the data and background MC were drawn from (either $P_{\pi^{0}}$ (left hand side of <ref>) or cosine of the $\pi^{0}$ (right hand side of Fig. <ref>)).
3|c|Efficiency Corrected NC$\pi^{0}$ Production
Parent Distribution Species Events $\pm$ Stat. Error
$\pi^{0}$ Momentum Neutrino 311.4 $\pm$ 75.0
$\pi^{0}$ Momentum Anti-Neutrino 97.5 $\pm$ 51.7
Cosine $\pi^{0}$ Neutrino 328.5 $\pm$ 74.7
Cosine $\pi^{0}$ Anti-Neutrino 104.2 $\pm$ 51.6
Summary of the efficiency corrected semi-inclusive NC$\pi^{0}$ production on an argon target for the ArgoNeuT anti-neutrino data sample)
A similar procedure is followed for the charged current sample from Ref. <cit.> using Eq. <ref> but this time for CC inclusive. Tab. <ref> summarizes the results of the calculation of the MC corrected charged current production for both the neutrino and anti-neutrino components of the anti-neutrino mode beam. The corrections applied to the CC inclusive sample include taking into account the various acceptances due to the neutrino and anti-neutrino components. The two sets of numbers for each type of interaction represent the parent distribution the data and background MC where drawn from (either the momentum of the lepton ($P_{\mu}$) or angle of the lepton ($\theta_{\mu}$)).
3|c|MC Corrected CC Production
Parent Distribution Species Events $\pm$ Stat. Error
Lepton Momentum Neutrino 3425.9 $\pm$ 9.9
Lepton Momentum Anti-Neutrino 2470.3 $\pm$ 4.4
Lepton Angle Neutrino 3385.8 $\pm$ 14.4
Lepton Angle Anti-Neutrino 2353.2 $\pm$ 9.3
Summary of the MC corrected charged current production on an argon target for the ArgoNeuT anti-neutrino data sample)
§.§ Systematic Error
* MC estimation of the neutral current beam composition:
As was stated earlier, we take from MC the fraction of the anti-neutrino beam that produces NC$\pi^{0}$ from neutrino or anti-neutrino interactions to be $f^{\nu}=74 \pm 15\%$ and $f^{\bar{\nu}}=26 \mp 15\%$. This error on the fraction is based on the difference in the fraction of the beam as estimated from the charged current inclusive sample and the fraction calculated from NC$\pi^{0}$ MC. Taking the numbers from Tab. <ref>, the fraction of the beam from $\nu$ and $\bar{\nu}$ is measured as
\begin{equation}
\frac{\nu\mbox{-CC}}{\mbox{Total CC}} = 59\% \mbox{ and } \frac{\bar{\nu}\mbox{-CC}}{\mbox{Total CC}} = 41\%
\end{equation}
Since there is no direct analogue to measure for NC$\pi^{0}$ production, the $\pm 15\%$ systematic covers this difference in the fraction calculated from the CC-inclusive data and those calculated from the NC$\pi^{0}$ MC. This conservatively assumes the maximum error in the beam composition is solely due to modelling of the beam.
* Ratio extraction from different parent histograms:
The MC corrected ratio of NC$\pi^{0}$ to charged current production can be taken from either the momentum of the NC$\pi^{0}$ (CC-lepton) or from the cosine of the NC$\pi^{0}$ ($\theta$ of the CC-lepton). These two MC corrected histograms give slightly different results for the ratio, as summarized in Tab. <ref>. The final answer is taken as the mean of the two results and the variation on the mean is taken as a systematic on the final ratio.
3|c|NC$\pi^{0}$/CC Production Ratio
Parent Distribution Neutrino Ratio Anti-Neutrino Ratio
$\pm$ (stat. only) $\pm$ (stat. only)
Momentum 0.091 $\pm$ 0.022 0.039 $\pm$ 0.021
Angle 0.097 $\pm$ 0.022 0.044 $\pm$ 0.022
Summary of the ratios computed from either the momentum or angle distributions from Tab. <ref> and Tab. <ref>
* Energy Correction Templates:
The energy correction templates are allowed to vary between those derived for $\pi^{0} \rightarrow \gamma\gamma$ decays, shown in Fig. <ref>, Fig. <ref>, and Fig. <ref> in Appendix <ref> and the templates derived for single electrons. The variation in templates allows events to shift slightly bin-to-bin as well as in and out of the sample via the $M_{\gamma\gamma}$ cut defined in Section <ref>. The variation in the ratio due to the differences in the energy templates is taken as a systematic on the final answer.
Tab. <ref> summarizes the systematic errors and their relative magnitude on the final ratio as well as the total systematic taken on the computed ratio for neutrinos and anti-neutrinos.
3|c|Systematic Errors
Source of the Error Error on Error on
Neutrino Ratio Anti-Neutrino Ratio
Beam Composition $\pm$ 0.014 $\pm$ 0.006
Parent Histogram $\pm$ 0.006 $\pm$ 0.005
Energy Templates $\pm$ 0.002 $\pm$ 0.003
Total Systematic Error $\pm$ 0.015 $\pm$ 0.008
Summary of the systematic errors on the ratio of NC production to CC production)
§.§ Results
Taking the mean of the two parent distributions and adding the systematics the final ratios for both neutrino and anti-neutrino interactions are
\begin{equation}
\frac{\sigma_{\nu}(NC\pi^{0})}{\sigma_{\nu}(CC)} = 0.094 \pm 0.022\mbox{(stat.)} \pm 0.015 \mbox{(sys.)}
\end{equation}
\begin{equation}
\frac{\sigma_{\bar{\nu}}(NC\pi^{0})}{\sigma_{\bar{\nu}}(CC)} = 0.042 \pm 0.022\mbox{(stat.)} \pm 0.008 \mbox{(sys)}
\end{equation}
for neutrinos with a mean energy of 9.6 GeV and anti-neutrinos with a mean energy of 3.6 GeV. The total inclusive ratio calculated from the sum of the neutrino and anti-neutrino component is
\begin{equation}
\frac{\sigma(NC\pi^{0})}{\sigma(CC)} = 0.136 \pm 0.031\mbox{(stat.)} \pm 0.017 \mbox{(sys)}.
\end{equation}
The result is plotted on Fig. <ref> showing the computed ratio of NC$\pi^{0}$ production to inclusive CC scattering as computed using GENIE and NuWro neutrino simulations on an argon target. To compute the ratio from GENIE and NuWro we take the total neutral current production cross-section and scale it by the fraction of those events that produce $\geq 1 \pi^{0}$ in the event and divide by the charged current inclusive cross-section.
Ratio of the NC$\pi^{0}$ production to inclusive CC scattering cross-sections for both neutrino (red) and anti-neutrino (blue) scattering as measured by ArgoNeuT and as computed using the GENIE and NuWro neutrino simulation on an argon target. For reference, the results obtained by the SciBooNE collaboration for a neutrino beam with a mean energy of 1.1 GeV on a polystyrene target (C8H8) as well as the results from the K2K collaboration for a neutrino beam with a mean energy of 1.3 GeV on a water target are shown. The ArgoNeuT measurement is shown with statistical and total error bars.
Comparing the ArgoNeuT measured values for the ratio of NC$\pi^{0}$/CC to those reported by the SciBooNE collaboration for a neutrino beam with a mean energy of 1.1 GeV on a polystyrene target (C8H8) (0.077 $\pm 0.5$(stat.) $\pm 0.5$ (sys.)) <cit.> and with the K2K collaboration for a neutrino beam with a mean energy of 1.3 GeV on a water target (0.064 $\pm$ 0.001 (stat.) $\pm$ 0.007 (sys.))<cit.> we find that the ArgoNeuT ratio is slightly higher for the higher energy beam.
§ FLUX AVERAGED NC$\PI^{0}$ CROSS-SECTION
In addition to calculating the ratio of NC$\pi^{0}$ to CC, one can also calculate the flux averaged absolute cross-section for the NC$\pi^{0}$ production. This is defined as
\begin{equation} \label{eqn:NuCrossSection}
\sigma_{\nu}(NC\pi^{0}) = \sum\limits_{i = bin} \frac{S_{i}^{\nu} - B_{i}^{\nu}}{\epsilon_{i}^{\nu}\Phi_{\nu}N_{Targets}}
\end{equation}
\begin{equation}
\sigma_{\bar{\nu}}(NC\pi^{0}) = \sum\limits_{i = bin} \frac{S_{i}^{\bar{\nu}} - B_{i}^{\bar{\nu}}}{\epsilon_{i}^{\bar{\nu}}\Phi_{\bar{\nu}}N_{Targets}},
\end{equation}
where $S_{i}^{\nu / \bar{\nu}}$, $B_{i}^{\nu / \bar{\nu}}$ and $\epsilon_{i}^{\nu / \bar{\nu}}$ are defined just as before and $N_{Targets}$ represents the number of argon nuclei in the fiducial volume and $\Phi_{\nu / \bar{\nu}}$ is the neutrino/anti-neutrino flux exposure given in Tab. <ref>.
Similar to Section <ref>, the component of the sample coming from $\nu$ and $\bar{\nu}$ is derived using the MC. The total integrated flux for the $\nu$ and $\bar{\nu}$ components of the anti-neutrino beam is taken from Ref. <cit.>.
§.§ Cross-Section Systematic Error
In addition to the systematics described before, three new systematics are present when the result is interpreted as an integrated cross-section. The integrated flux is assigned a flat 11$\%$ uncertainty that accounts for the uncertainty in hadron production and beam line modelling and has been used in previous ArgoNeuT analyses <cit.>. Uncertainties in the number of targets and P.O.T. are taken into account as well, although these contribute only at the few percent level. The full description of the systematics applied to the integrated cross-section are given in Tab. <ref>.
3|c|Systematic Errors
Source of the Error $\%$ Error on $\sigma(\nu)$ $\%$ Error on $\sigma(\bar{\nu})$
Beam Composition $\pm$ 15$\%$ $\pm$ 15$\%$
Flux Normalization $\pm$ 11$\%$ $\pm$ 11$\%$
Parent Histogram $\pm$ 4.3$\%$ $\pm$ 8.1$\%$
Number of Targets $\pm$ 2$\%$ $\pm$ 2$\%$
Energy Templates $\pm$ 1$\%$ $\pm$ 1$\%$
P.O.T. $\pm$ 1$\%$ $\pm$ 1$\%$
Total Systematic Error $\pm$ 18.7$\%$ $\pm$ 20.4$\%$
Summary of the systematic errors on the ratio of NC production to CC production)
§.§ Cross-Section Results
Taking the mean of the two parent distributions and adding the full systematics, the cross-section for both neutrino and anti-neutrino interactions is measured to be
\begin{equation}
\sigma_{\nu}(NC\pi^{0}) = (7.1 \pm 1.7\mbox{(stat.)} \pm 1.3 \mbox{(sys.)})\times10^{-40}\mbox{cm}^{2}
\end{equation}
\begin{equation}
\sigma_{\bar{\nu}}(NC\pi^{0}) = (0.5 \pm 0.2\mbox{(stat.)} \pm 0.1\mbox{(sys.)})\times10^{-40}\mbox{cm}^{2}
\end{equation}
per argon nucleon with anti-neutrinos at a mean energy of 3.6 GeV and neutrinos at a mean energy of 9.6 GeV. These results are plotted in Fig. <ref> as well as the inclusive result taken by adding together the contributions due to the two components.
\begin{equation}
\sigma(NC\pi^{0}) = (7.6 \pm 1.7\mbox{(stat.)} \pm 1.4\mbox{(sys.)})\times10^{-40}\mbox{cm}^{2}
\end{equation}
These results are shown with a comparison to the predictions from the GENIE and NuWro event generators. The predictions of GENIE and NuWro are consistent with the ArgoNeuT measurements.
The NC$\pi^{0}$ production cross-section for both neutrino (red) and anti-neutrino (blue) scattering on argon as measured in ArgoNeuT and as predicted by the GENIE (solid) and NuWro (dashed) generators. The ArgoNeuT measurement has been flux averaged and is shown with both statistical and total uncertainties.
Since this is the first time this process has been measured on an argon nuclei, comparison with previous NC$\pi^{0}$ data proves difficult. For example, MiniBooNE (<cit.>) reports an absolute cross-section for single $\pi^{0}$ neutral current production of (4.76$\pm$0.05 (stat)$\pm$ 0.76 (sys))$\times$10$^{-40}$ cm$^2$/nucleon at a mean energy for neutrinos of 808 MeV and (1.48$\pm$0.05 (stat)$\pm$0.23 (sys))$\times$10$^{-40}$ cm$^2$/nucleon at a mean energy for anti-neutrinos of 664 MeV. While both these results are of the same order as the ArgoNeuT reported cross-section, the MiniBooNE's target nuclei was CH$_{2}$, and the scaling to the more dense nuclei of argon is not well understood.
§ DISCUSSION
In order to perform this measurement many novel techniques for identifying and reconstructing electromagnetic showers in a small volume LArTPC were employed and are presented here along side the measurement of neutral current $\pi^{0}$ production. One such technique presented here is a method to account for the missing energy of the photon shower when it is loss outside the fiducial boundary. By utilizing the well known electromagnetic shower development in liquid argon with the visible portion of the shower in the detector it is possible to correct back the loss energy. Improvements on this technique will enable future larger LArTPC experiments, such as MicroBooNE, SBND, and ICARUS, to increase their acceptance of NC$\pi^{0}$ production within their detectors.
The $\pi^{0}$ kinematic distributions presented in Section <ref> agree within the experimental uncertainties in both rate the shape with the predicted simulations. Future larger LArTPC's will have the capability to improve greatly on these measurements and probe with even better resolution these distributions.
The interpretation of these kinematic distributions as the ratio of the efficiency corrected neutral current $\pi^{0}$ production to total inclusive charged current cross-section on argon as well as flux averaged absolute cross-section are the first of their kind done on an argon target. Both the ratio and the flux averaged absolute cross-section are found to be consistent with predictions from simulation as well as previous data measurements. The ArgoNeuT measurements provide information on the $A$ dependence of neutrino cross sections that may prove helpful when attempting to estimate future cross-section uncertainties for LArTPC's.
§ CONCLUSIONS
In conclusion, the ArgoNeuT Collaboration reports the first measurement of neutrino and anti-neutrino semi-inclusive neutral current $\pi^{0}$-production on an argon target. We present: 1) kinematic distribution of the $\pi^{0}$ mesons produced in the neutrino-argon interaction, 2) the ratio of the GENIE and NuWro event generators corrected neutral current $\pi^{0}$ production to the total inclusive charged current production and 3) the flux averaged cross-section for neutrinos with a mean energy of 9.6 GeV and for anti-neutrinos with a mean energy of 3.6 GeV. Both the ratio and the cross-section are broken into a contribution from the neutrino and the anti-neutrino profile based on a Monte Carlo estimate of the beam composition. Both of the values obtained are consistent with predictions from the GENIE and NuWro neutrino generator Monte Carlo.
ArgoNeuT gratefully acknowledges the cooperation of the MINOS collaboration in providing data for the use in this analysis. We would also like to acknowledge the support of Fermilab (Operated by Fermi Research Alliance, LLC under Contract No. De‐AC02‐07CH11359 with the United States Department of Energy), the Department of Energy, and the National Science Foundation in the construction, operation and data analysis of ArgoNeuT.
§ ELECTROMAGNETIC ENERGY CORRECTIONS INSIDE THE ARGONEUT TPC
In this appendix we provide a detailed overview of the procedures used to generate the energy correction templates used in this analysis and initially described in Section <ref>. We begin with a discussion of the angle reconstruction seen for single particle $\pi^{0}$ MC that becomes the basis of the template approach to applying energy corrections. This is followed by a detailing of the various energy correction procedures and templates that are applied. Finally, a summary of the cross-checks that were performed showing the robustness of the energy corrections is presented.
§.§ Momentum Hypothesis from reconstructed angles
Utilizing one of the great strength's of the LArTPC technology, its fine grain tracking information, allows for reconstruction of the angle of an electromagnetic shower despite very little of the shower being contained. This was demonstrated in Section <ref> and Fig. <ref> shows the performance of the shower reconstruction on simulated $\pi^{0}\rightarrow\gamma\gamma$ events using the hand-scan and automated shower reconstruction tools described in Section <ref>. These tools faithfully reconstruct the true angle of the photon shower and thus the angular reconstruction can be trusted to represent the underlying true photon angles. This information becomes very useful when attempting to correct back the energy loss of a shower due to poor containment. Specifically, it provides another method to calculate a hypothesis for the initial momentum the $\pi^{0}$ based on the opening angle between the pair of photons in the event ($\theta_{\gamma\gamma}$).
Single particle Monte Carlo angles that have been fully reconstructed using the shower hand-scan + automated shower tools.
As mentioned previously, Fig. <ref> shows the strong correlation between the initial momentum of the $\pi^{0}$ and the opening angle $\theta_{\gamma\gamma}$. Fig. <ref> provides the two dimensional (2D) distribution from which the fitted function, Eq. <ref>, that defines the $\pi^{0}$ momentum as a function of the opening angle between the photons. The constants from the fit are $C_{0}$ = 2202.3, $C_{1}$ = -94.9, $C_{2}$ = 2.1, $C_{3}$ = -0.025, $C_{4}$ = 0.00017, $C_{5}$ = -6.0$\times 10^{-7}$, $C_{6}$ = 8.5$\times 10^{-10}$.
Momentum versus opening angle shown as a 2-d distribution. The profile fit with a function shown in Fig. <ref> that allows a hypothesis for the upper bound on the momentum of the $\pi^{0}$ system.
Comparing the difference between the momentum of the $\pi^{0}$ calculated utilizing Eq. <ref> and the momentum utilizing Eq. <ref>, as is done in Figure <ref>, shows that the angle method reconstructs the momentum to within $\pm$30$\%$ with a slight bias towards overestimating the momentum. While this accuracy is insufficient to directly extract the physics of the $\pi^{0}$ system, using this as a starting point for our energy correction calculation is sufficient. By being able to bound the hypothesis for the momentum of the $\pi^{0}\rightarrow\gamma\gamma$ we can then go about deriving energy corrections using topological information.
Difference between $\pi^{0}$ momentum estimated from $\gamma\gamma$ opening angle and true $\pi^{0}$ momentum, from MC. This method defined in Eq. <ref> reconstructs the momentum of the $\pi^{0}$ system to within $\pm$30$\%$.
§.§ Linear Correction
Fig. <ref> shows the fraction of the contained energy (as defined in Eq. <ref>) as a function of the initial true momentum of the $\pi^{0}$.
Template for the linear energy correction applied to the photons based on the $P_{\pi^{0}}$.
Utilizing the initial momentum hypothesis formed using Eq. <ref> for the $\pi^{0}$ ($P_{\pi^{0}}^{I}$) the linear correction that is attempted to be applied to the photons has the form
\begin{equation}
E^{Flat Corr}_{\gamma^{i}} = (P_{\pi^{0}}^{I} \times C_{0}^{Linear}) + C_{1}^{Linear},
\end{equation}
where the constants $C_{i}$ are obtained from the fit of Fig. <ref> and found to be $C_{0} = 0.51$ and $C_{1} = $-1.4e$\times 10^{-4}$.
Thus the first correction that is tried is a simple linear correction applied to the photon energy as defined in Eq. <ref>.
\begin{equation}\label{eqn:FlatCorr}
E_{\gamma^{i}} = E^{0}_{\gamma^{i}} + E^{0}_{\gamma^{i}} \times E^{Flat Corr}_{\gamma^{i}},
\end{equation}
where $E^{0}_{\gamma^{i}}$ is the original energy of the photon (uncorrected). If this correction for any of the photon pairs does not cause the momentum of the $\pi^{0}$ system as calculated using Eq. <ref> to exceed the hypothesis for the momentum as calculated using Eq. <ref>, then the correction is kept and stored. Otherwise the correction becomes
\begin{equation}
E^{Flat Corr}_{\gamma^{i}} = 0.0.
\end{equation}
§.§ X, Y, Z Topology Corrections
The next set of corrections that are applied are based on where the shower vertex is found inside the detector. The amount of energy that is deposited inside the detector is strongly correlated with where the photon first converts in X,Y,Z. Fig. <ref> is a schematic demonstrating how the geometry and topological distribution of the shower, where the shower is created and what direction the shower is pointing, will determine how much of the shower is contained. For example, as is illustrated on the left of Fig. <ref>, a shower that is created at the front (small Z) of the detector but is pointed down (negative Y) will have worse shower containment when compared to a shower that is created at the same point but is pointed up (positive Y).
Illustration of the topological layout of an electromagnetic shower (represented as a yellow cone) and the direction the shower is propagating. The left hand side represents two showers created at the same Z location but with different Y initial positions. The right hand side shows three showers all created with the same Y-Z location, but with different initial X positions. Each of these cases demonstrates the topology of the shower matters when calculating the shower containment. This analysis uses the “straight line” distance to the wall in X, Y, and Z independently to calculate an energy correction.
Moreover, as is illustrated on the right side of Fig. <ref>, three different showers all created at the same Z location and pointing upward ($\phi\sim 90$) can have vastly different containment based on where they are in X (drift direction). For this reason, when we calculate the distance to the nearest wall in X, Y, and Z we use the “straight line” distance (illustrated as dashed line in Fig. <ref>) instead of the “pointing line” (illustrated with the arrow and referred to at the $R$ distance). By taking into account all three spatial variables separately and in turn there is a greater chance of correcting the component of the energy loss that is due to the spatial variable that matters, instead of rolling all the information into one variable ($R$).
Taking the origin of the detector to be on the beam right, center of the upstream face and defining the shower vertex relative to this position we are able to calculate the distance of the “closest” boundary in $X$, $Y$, and $Z$ using the following relations:
$Z_{Boundary} = 90$ if $-90 \leq \phi \leq 90$
$Z_{Boundary} = 0$ if $90 \leq \phi \leq 180$ or $-180 \leq \phi \leq -90$
$Y_{Boundary} = 20$ if $\phi \geq 0$
$Y_{Boundary} = -20$ if $\phi \leq 0$
$X_{Boundary} = 0.0$ if $\theta \geq 0$
$X_{Boundary} = 50.0$ if $\theta \leq 0$
To correct back the energy loss due to the topological location of the electromagnetic shower, we plot the fraction of energy contained (as defined in Eq. <ref>) versus the distance to the X,Y, and Z boundary. We then fit the projection of these distributions with a polynomial that minimizes the $\chi^{2}$/NDF to give us a functional form for the energy correction. For each photon we calculate its distance to any boundary and apply back an energy correction based on this function of $x$, $y$, and $z$ distance.
Z Distance to a Boundary Template
Fig. <ref> shows the polynomial fit to the plot of the average fraction of the energy contained as a function of the distance to the Z boundary. The best fit returns an eighth degree polynomial based on the distance of the shower vertex to the nearest Z boundary defined by the Equation below
Templates of the fraction of the energy contained as a function of the shower's distance Z boundarywith the constants in the polynomial fit given by $C_{0}^{z}$ = 0.32, $C^{z}_{1}$ = 0.028, $C^{z}_{2}$ = -0.0014, $C^{z}_{3}$ = 2.5e-05, $C^{z}_{4}$ = -2.8e-08, $C^{z}_{5}$ = -2.4e-09, $C^{z}_{6}$ = -1.6e-12, $C^{z}_{7}$ = 3.3e-13, $C_{z}^{8}$ = -1.8e-15
\begin{equation}
E^{Z Dist Corr}_{\gamma^{i}} = \sum_{i=0}^{8} C_{i}^{z} Z_{Dist}^{i},
\end{equation}
where the constants are read from the polynomial fit for the energy of a given photon becomes
\begin{equation}
E_{\gamma^{i}} = E^{0}_{\gamma^{i}} + E^{Flat Corr}_{\gamma^{i}} + (E^{0}_{\gamma^{i}} \times E^{Z Dist Corr}_{\gamma^{i}}),
\end{equation}
where $E^{0}_{\gamma^{i}}$ is the original energy of the photon (uncorrected) and $E^{Flat Corr}_{\gamma^{i}}$ is defined in Eq. <ref>. If this correction for any of the photon pairs does not cause the momentum of the $\pi^{0}$ system as calculated using Eq. <ref> to exceed the hypothesis for the momentum as calculated using Eq. <ref>, then the correction is kept and stored as
\begin{equation}\label{eqn:ZCorr}
E^{Z Corr}_{\gamma^{i}} = E^{0}_{\gamma^{i}} \times E^{Z Dist Corr}_{\gamma^{i}}.
\end{equation}
If it does exceed the momentum hypothesis then it is stored as
\begin{equation}
E^{Z Corr}_{\gamma^{i}} = 0.0.
\end{equation}
Y Distance to a Boundary Template
Fig. <ref> shows the polynomial fit to the plot of the average fraction of the energy contained as a function of the distance to the Y boundary. The best fit returns a seventh degree polynomial based on the distance of the shower vertex to the nearest Y boundary defined by the Equation below
Templates of the fraction of the energy contained as a function of the shower's distance Y boundary with the constants in the polynomial fit given by $C_{0}^{y}$ = 0.41, $C_{1}^{y}$ = -0.054, $C_{2}^{y}$ = 0.014, $C_{3}^{y}$ = -0.0010, $C_{4}^{y}$ = 2.2$\times 10^{-5}$, $C_{5}^{y}$ = 6.3$\times 10^{-7}$, $C_{6}^{y}$ = -3.3$\times 10^{-8}$, and $C_{7}^{y}$ = 3.7$\times 10^{-10}$
\begin{equation}
E^{Y Dist Corr}_{\gamma^{i}} = \sum_{i=0}^{7} C_{i}^{y} Y_{Dist}^{i},
\end{equation}
where the constants are read from the polynomial fit. Now the calculation for the energy of a given photon becomes
\begin{equation}
E_{\gamma^{i}} = E^{0}_{\gamma^{i}} + E^{Flat Corr}_{\gamma^{i}} + E^{Z Corr}_{\gamma^{i}} + (E^{0}_{\gamma^{i}} \times E^{Y Dist Corr}_{\gamma^{i}}),
\end{equation}
where $E^{0}_{\gamma^{i}}$ is the original energy of the photon (uncorrected) and $E^{Flat Corr}_{\gamma^{i}}$ is defined in Eq. <ref> and $E^{Z Corr}_{\gamma^{i}}$ is defined in Eq. <ref>. If this correction for any of the photon pairs does not cause the momentum of the $\pi^{0}$ system as calculated using Eq. <ref> to exceed the hypothesis for the momentum as calculated using Eq. <ref>, then the correction is kept and stored as
\begin{equation}\label{eqn:YCorr}
E^{Y Corr}_{\gamma^{i}} = E^{0}_{\gamma^{i}} \times E^{Y Dist Corr}_{\gamma^{i}}.
\end{equation}
If it does exceed the momentum hypothesis then it is stored as:
\begin{equation}
E^{Y Corr}_{\gamma^{i}} = 0.0.
\end{equation}
X Distance to a Boundary Template
Fig. <ref> shows the polynomial fit to the plot of the average fraction of the energy contained as a function of the distance to the X boundary. The best fit returns a fifth degree polynomial based on the distance of the shower vertex to the nearest X boundary defined by the Equation below
Templates of the fraction of the energy contained as a function of the shower's distance X boundary with the constants in the polynomial fit given by $C_{0}^{x}$ = 0.25 , $C_{1}^{x}$ = 0.088, $C_{2}^{x}$ = -0.0078, $C_{3}^{x}$ = 0.00031, $C_{4}^{x}$ = -5.6$\times 10^{-6}$, and $C_{5}^{x}$ = 3.7$\times 10^{-8}$
\begin{equation}
E^{X Dist Corr}_{\gamma^{i}} = \sum_{i=0}^{5} C_{i}^{x} X_{Dist}^{i},
\end{equation}
where the constants are read from the polynomial fit. Now the calculation for the energy of a given photon becomes
\begin{eqnarray*}
E_{\gamma^{i}} & = & E^{0}_{\gamma^{i}} + E^{Flat Corr}_{\gamma^{i}} + E^{Z Corr}_{\gamma^{i}} \\
& & + E^{Y Corr}_{\gamma^{i}} + (E^{0}_{\gamma^{i}} \times E^{X Dist Corr}_{\gamma^{i}}),
\end{eqnarray*}
where $E^{0}_{\gamma^{i}}$ is the original energy of the photon (uncorrected) and $E^{Flat Corr}_{\gamma^{i}}$ is defined in Eq. <ref> and $E^{Z Corr}_{\gamma^{i}}$ is defined in Eq. <ref> and $E^{Y Corr}_{\gamma^{i}}$ is defined in Eq. <ref>. If this correction for any of the photon pairs does not cause the momentum of the $\pi^{0}$ system as calculated using Eq. <ref> to exceed the hypothesis for the momentum as calculated using Eq. <ref>, then the correction is kept and stored as
\begin{equation}\label{eqn:XCorr}
E^{X Corr}_{\gamma^{i}} = E^{0}_{\gamma^{i}} \times E^{X Dist Corr}_{\gamma^{i}}.
\end{equation}
If it does exceed the momentum hypothesis than it is stored as
\begin{equation}
E^{X Corr}_{\gamma^{i}} = 0.0.
\end{equation}
Then end result of all these corrections is for any single photon to have its energy calculated as:
\begin{equation}\label{eqn:finalPhotonE}
E_{\gamma^{i}} = E^{0}_{\gamma^{i}} + E^{Flat Corr}_{\gamma^{i}} + E^{Z Corr}_{\gamma^{i}} + E^{Y Corr}_{\gamma^{i}} + E^{X Corr}_{\gamma^{i}}.
\end{equation}
The final results of these corrections appear to be insensitive the order they are applied as long as the constraint based on the $\pi^{0}$ momentum is applied. In this analysis, the corrections are applied as they are laid out here.
§.§ Energy Correction Results
The results of the application of the template based energy corrections and procedure described above shown in Fig. <ref>. With each subsequent correction, more events move towards full containment (closer to 1) while only a relatively small fraction of the photons ever have their energy corrected above the MC-truth value (these photons appear with values less than 0). The order of the corrections applied here start with the “flat correction” shown in pink (described in Section <ref>), then the “Z distance” shown in red, “Y distance” shown in green, and “X distance” shown in black.
The fraction of energy from simulated $\pi^{0} \rightarrow \gamma\gamma$ events that after the application of the template based energy corrections. (Top) The blue distribution is the initial fraction of deposited energy inside the TPC and black is the result after the application of all the corrections. (Bottom) Shows the same fraction of deposited energy as the flat energy correction (pink), Z distance (red), Y distance (green), and X distance (black) is applied.
In order to ensure none of these templates would sculpt a result based on the $\pi^{0} \rightarrow \gamma\gamma$ system, the behavior of the templates themselves were studied for a sample of single particle photons and single particle electrons. These events were simulated with the same initial momentum and position distribution as the $\pi^{0}$ system and the topological templates were derived for the electron sample. Comparisons to the fits from the electron and photon templates against those derived for the $\pi^{0} \rightarrow \gamma\gamma$ sample showed very little difference in either their shape of the magnitude of the correction. This is as expected since the development of the shower inside the detector has little to do with how the shower was created and more to do with the physical process of the electromagnetic properties of the argon.
In addition to checking the templates against single electrons and photon MC, a check that the order in which the templates are applied/derived was also performed. In the analysis the corrections are applied in the order described in Section <ref>. The results were also computed by changing the order of the topological templates (from Z, Y, X distances to the boundary to all possible unique rearrangements of these three variables). The flat momentum based correction was always applied first, however the resulting templates were shown to be insensitive (up to a scaling) to which order they are applied.
Fig. <ref> shows the impact of the template based energy correction procedure to the invariant mass of the $\pi^{0}$, the energy of the photons, as well as the momentum of the $\pi^{0}$. In black is shown the true distribution, in blue the information from only the deposited energy within the volume of the TPC, and then the application of each of the subsequent energy corrections.
The invariant mass of the $\pi^{0} \rightarrow \gamma\gamma$ events fit with a Gaussian function (after applying all energy corrections) between 60 MeV$\leq M_{\gamma\gamma}\leq$240 MeV, the energy of the photons as well as $P_{\pi^{0}}$, calculated using Eq. <ref>, for the $\pi^{0} \rightarrow \gamma\gamma$ events that after the application of the template based energy corrections.
authorJ. Beringer et al. (Particle Data Group),
journalPhys. Rev. D volume86
pages010001 (year2012).
authorS. J. Barish et al.,
journalPhys. Rev. Lett. volume33
pages448 (year1974).
authorM. Derrick et al.,
journalPhys. Rev. D volume23
pages569 (year1981).
authorW. Y. Lee et al.,
journalPhys. Rev. Lett. volume38
pages202 (year1977).
authorW. Krenz et al. [Gargamelle Neutrino Propane Col-
laboration and Aachen-Brussels-CERN-Ecole Po],
journalNucl. Phys. B volume135
pages45 (year1978).
authorS. Nakayama et al. [K2K Collaboration],
journalPhys. Lett. B volume619
pages255 (year2005) [arXiv:hep-ex/0408134].
authorA. A. Aguilar-Arevalo et al. [MiniBooNE Collaboration],
journalPhys. Lett. B volume664
pages41 (year2008) [arXiv:hep-ex/0803.3423].
authorA. A. Aguilar-Arevalo et al. [MiniBooNE Collaboration],
journalPhys. Rev. D volume83
pages052009 (year2011) [arXiv:hep-ex/1010.3264].
authorA. A. Aguilar-Arevalo et al. [MiniBooNE Collaboration],
journalPhys. Rev. D volume81
pages013005 (year2010) [arXiv:hep-ex/0911.2063].
authorY. Kurimoto et al. [SciBooNE Collaboration],
journalPhys. Rev. D volume81
pages033004 (year2010) [arXiv:hep-ex/0910.5768].
authorHasegawa, M. et al. [K2K Collaboration],
journalPhys. Lett. B volume619
pages255-262 (year2005) [arXiv:hep-ex/0408134].
authorM. Freund,
journalPhys. Rev. D volume64
pages053003 (year2001) [arXiv:hep-ph/0103300].
authorThe MicroBooNE Technical Design Report[MicroBooNE Collaboration],
journal<http://www-microboone.fnal.gov/publications/TDRCD3.pdf> volume
pages (year2012).
authorC. Adams et al. [LAr1-ND Collaboration],
journalarXiv:hep-ex/1309.7987 volume
pages (year2013).
authorR. Acciarri et al. [SBN Program],
journalarXiv:hep-ex/1503.01520 volume
pages (year2015).
authorC. Adams et al. [LBNE Collaboration],
journalarXiv:hep-ex/1307.7335 volume
pages (year2013).
authorC. Rubbia,
journalCERN-EP/77-08 volume
pages (year1977).
authorICARUS Collaboration,
journalJournal of Instrumentation volume6 P07011
pages (year2011).
authorA. Szelc,
journal“Results from and Status of ArgoNeuT and MicroBooNE” presentation at Neutrino 2014 volume
pages<https://indico.fnal.gov/getFile.py/access?contribId=294 sessionId=25 resId=0 materialId=slides confId=8022> (year2014).
authorICARUS Collaboration,
journalActa Phys. Polon volumeB41
pages103-125 (year2010).
authorC. Athanassopoulos et al.,
journalPhys. Rev. Lett. volume75
pages2650 (year1995);
journal volume77
pages3082 (year1996);
journal volume81
pages1774 (year1998);
journalPhys. Rev. C volume58
pages2489 (year1998);
author A. Aguilar et al.,
journalPhys. Rev. D volume64
pages112007 (year2001).
authorA. A. Aguilar-Arevalo et al. [MiniBooNE Collaboration],
journalPhys. Rev. Lett. volume110
pages161801 (year2013).
author C. Andreopoulos et al.,
journalNucl. Instr. $\&$ Meth. A volume506
pages250 (year2003) Version 2.8.0 was used for this analysis .
authorT. Golan, C. Juszczak and J. T. Sobczy,
journalPhys. Rev. C volume86
pages015505 (year2012) We use version 11m for this analysis.
authorC. Anderson et al,
journalJINST Vol volume7
pagesP10019 (year2012).
author K. Anderson et al.,
journalFERMILAB-DESIGN-1998-01 volume
pages (year1998).
authorR. Acciarri et al,
journalPhys. Rev. D volume89
pages112003 (year2014).
author D.G. Michael et al. [MINOS Collaboration],
journalNucl. Instr. $\&$ Meth. A volume596
pages190 (year2008).
authorE. Church,
journalarXiv:1311.6774 volume
pages (year2013).
author S. Agostinelli et al.,
journalNucl. Instr. $\&$ Meth. A volume506
pages250 (year2003).
author Application Software Group,
journalCERN Program Library Long Writeup volume
pagesW5013,CERN (year1994).
authorR. Acciarri et al,
journalPhys. Rev. Lett. volume113
pages261801 (year2014).
authorHitlin, D. et al.,
journalNucl. Instrum. Meth.volume137
pages225 (year1976).
authorNelson, C. et al.,
journalNucl. Instrum. Meth. volume216
pages381 (year1983).
|
1511.01058
|
§ INTRODUCTION
Scattering amplitude is a leading research area which has various applications in phenomenology and attracts attention in a wide range of formal theories <cit.>. The major objects we focus on in the field of scattering amplitude are polynomial functions and rational functions. In modern mathematics, an efficient tool of dealing with rational functions stems from the theoretical structure of algebraic geometry <cit.>. Thus, the study of scattering amplitudes may bring us a fascinating connection between mathematics and physics. Many concepts and methods in algebraic geometry play a significant role in both the calculation and the theoretical analysis in scattering amplitude. One of the most widely used concepts is syzygy <cit.>, which is the relation set of an m-tuple polynomial function.
Up to now, syzygies have appeared in lots of research topics in scattering amplitude, such as the IBP relation <cit.>, which is used to determine irreducible loop integrals. It can also be used to fix the ambiguity of the integrands of the non-planar amplitudes. Another application is to simplify the Grassmannian integral form of the $\mathcal{N}=4$ Super Yang-Mills non-planar amplitude <cit.>. Furthermore, syzygies can potentially be used to construct the loop-level scattering amplitudes from unitarity cuts <cit.> and to probe the amplitude relations beyond the KK-relation <cit.> and the BCJ-relation <cit.> in Yang-Mills theory.
We usually need a highly efficient algorithm to obtain syzygies in most applications in physics. To give compact expressions for the quantities of interest, such as the IBP relations of the loop integrals, we also need to get the locally minimal generating set for the syzygies. In this paper,we develop an effective algorithm to obtain the irreducible basis of the syzygies of an ideal. Meanwhile, the Gröbner basis of the ideal is also obtained. Current algorithms, such as MMT <cit.> and F5 <cit.>, are very efficient to get the Gröbner basis <cit.> of syzygies or ideals. However, an algorithm to reduce the number of necessary syzygies is still needed. In <cit.>, Gluza et al also considered reducing a new syzygy obtained by the known syzygies, recursively, in their algorithm. The major advantage of our method is that we deal with more syzygies effectively and efficient feed-back is used to verify the irreducibility of each new syzygy added. This leads to that all the irreducible syzygies can be obtained even before the end of the main loop. This algorithm, with a promising significance in both mathematics and theoretical physics, can also be used in other areas that need to classify irreducible algebraic relations. The major difference between this algorithm and MMT <cit.> is that we only use the leading terms of polynomials to justify the permitted critical pairs and reduce the generated syzygies in each step. At each step, we can feedback to up-levels to guarantee that our justification is correct. This difference also makes our algorithm faster than MMT in practice.
This paper is organized as following. In Section <ref> we will first give a warming-up example, and then illustrate our new method for syzygies of ideals in detail with the generalization to module cases. Section <ref> will focus on the application of this algorithm in the IBP relations of a specific two-loop diagram. In Section <ref>, we will give some conclusions and remarks on this new algorithm.
§ A DIRECT METHOD TO COMPUTE SYZYGIES
A syzygy of a polynomial m-tuple $\mathbf{f}=(f_1, f_2\cdots f_m)$ is a m-tuple $\mathbf{S}=(a_1,\cdots,a_m)$ such that $\mathbf{S}\cdot\mathbf{f}=0$. The traditional method to get the syzygies of an ideal is based on the Gröebner basis techniques. The locally minimal generating set of the obtained syzygies can only be obtained by the syzygies of the syzygy module. This algorithm is usually not so efficient. Another method is based on the linear algebra techniques <cit.> which does not rely on the Gröebner basis. This method does not guarantee the completeness of the syzygies. In this section, we propose a new method to formulate all the syzygies completely. Meanwhile our method guarantees that the generating set of the syzygy module is locally minimal. The Gröebner basis is obtained automatically in the process.
The general strategy to get the locally minimal generating set of syzygies is to decompose these syzygies by the syzygy of the leading monomials. This is realized by computing the S-polynomial from a chosen critical pair step by step. In each step, this generates an S-polynomial $S_{i,j}={lcm(i,j)\over LT(f_i)} f_i-{lcm(i,j)\over LT(f_j)} f_j$ with new leading terms. Finally if there exists a vanishing $S_{i,j}$, we can get the syzygy of the original m-tuple by combining all the two-pairs inversely as shown in Fig. <ref>.
Critical pair decomposition.
We give some definitions first:
* $f$: A polynomial in the polynomial ring $R=K[x_1\cdots x_n]$, where $K$ is an algebraic closed field.
* $>$: A monomial order on the polynomial ring $R$. In this paper we usually choose the Degree Reverse Lexicographic order. Let $x^\alpha$ and $x^\beta$ be monomials in $R$, where $\alpha(\beta)$ is the exponent vector $\alpha(\beta)\in \mathbb{Z}^n$. We say $x^\alpha>x^\beta$ if $\sum_{i=1}^n\alpha_i>\sum_{i=1}^n\beta_i$ or if $\sum_{i=1}^n\alpha_i=\sum_{i=1}^n\beta_i$, and in the difference $\alpha-\beta$, the rightmost nonzero entry is negative.
* $\succ$: For a polynomial m-tuple, we define an additional order $\vec{e}_1\succ\vec{e}_2\cdots\succ\vec{e}_m$. The monomial order for this m-tuple is a POT extension of $>$: We say $x^\alpha e_i\succ x^\beta e_j$ if $i<j$ or if $i=j$ and $x^\alpha>x^\beta$
* $I=\langle f_1\cdots f_m\rangle$: An ideal generated by $f_1 \cdots f_m$ in $R$, where $m\in \mathbb{Z}_{>0}$.
* $\langle g_1\cdots g_m\rangle$: A Gröebner basis for an ideal in $R$, where $m\in \mathbb{Z}_{>0}$.
* $LT(f)$: The leading term of $f$ with respect to the order $>$.
* $LCM(i,j)$: The lowest common multiple of the two polynomial $f_i$ and $f_j$.
* $lcm(i,j)$: The lowest common multiple of the leading term for two polynomial $f_i$ and $f_j$.
§.§ A Warming-up Example
Now let us consider an ideal $\langle f_1=xy^3+z, f_2=x^2y^2+z^4,f_3=z^5\rangle$. There are three principle syzygies $(f_2, -f_1,0), (f_3,0,-f_1),(0,f_3,-f_2)$. A general syzygy has the form of $(f_{c1}, f_{c2}, f_{c3})$. The eliminating between pairs can be done in a decreasing monomial order. The highest critical pair cancelation happens between $\{f_1, f_2\}, \{f_1, f_3\}, \{f_2, f_3\}$.We use $\{f_i,f_j\}$ ($\{i,j\}$ for simplicity) to label the critical pair. For $\{f_1, f_2\}$, the cancelation between the highest terms leads to $LT(f_{c1})=m_1 x, LT(f_{c2})=m_1 (-y)$, where $m_1$ is an monomial in the polynomial ring $K[x,y,z]$. For the pairs $\{f_1, f_3\}$ and $\{f_2, f_3\}$, the leading terms are $LT(f_{c1})=m_1 z^5, LT(f_{c3})=m_1 (-x y^3)$ and $LT(f_{c2})=m_1 z^5, LT(f_{c2})=m_1 (-x^2 y^2)$. For the last two cases, the leading terms of the syzygies can be simplified by the principle syzygy $(f_3,0,-f_1),(0,f_3,-f_2)$. Hence such pair $\{f_1, f_3\}$ and $\{f_2, f_3\}$ can be reduced by other syzygies. We will call a syzygy's leading monomial of the first term a barrier. A new syzygy can be added to the syzygy set if and only if its product factor can not be divided by any existing barrier.
We denote the syzygies and the barriers as
\bordermatrix{
\text{index}&\text{Effect}&f_1&f_2&f_3\cr
\mathbf{S}_1&\text{T}&\underline{x^2 y^2}+z^4 & -x y^3-z & 0 \cr
\mathbf{S}_2&\text{T}& \underline{z^5} & 0 & -x y^3-z \cr
\mathbf{S}_3&\text{T} &0 & \underline{z^5} & -x^2 y^2-z^4 \cr
where the barriers are underlined. We use $\text{T}$ to mark an irreducible syzygy for the original m-tuple $f_i$ at this step. The syzygy set of m-tuple $f_i$ denoted as $\mathcal{S}=\{\mathbf{S}_1,\mathbf{S}_2,\mathbf{S}_3\}$ is called the Top-syzygy set .
Now we can proceed to the next step. Following the rules described above, the only allowed pairs are those with product factors that can not be divided by the barriers. Here only the pair $\{f_1, f_2\}$ with the coefficients $(x, -y)$ is allowed.
After processing this pair, the ideal can be written as
$$\langle xy^3+z, x^2y^2+z^4,z^5, f_4=-yz^4+xz\rangle.$$
and there is one more syzygy $\mathbf{S}_{1,2}=(x,-y,0,-1)$ to be added to complete the syzygy set. This extra relation is an inheritance of the cancelation that takes place in the last step. Then the syzygies become:
Illustrations on the warming-up example.
\bordermatrix{
\text{Index}&\text{Effect}&f_1&f_2&f_3&f_4\cr
\mathbf{S}_1&\text{T}& \underline{x^2 y^2}+z^4 & -x y^3-z & 0 & 0 \cr
\mathbf{S}_2&\text{T}& \underline{z^5} & 0 & -x y^3-z & 0 \cr
\mathbf{S}_3&\text{T}& 0 & \underline{z^5} & -x^2 y^2-z^4 & 0 \cr
\mathbf{S}_{1,2}&&\underline{x} & -y & 0 & -1 \cr
%\text{True}& 0 & 0 & \underline{-y z^3}+x & -z^4 \cr
At this moment, we should make sure that the barriers can't be divide by each other by performing linear transformations. First, add the product of $\mathbf{S}_{1,2}$ and $-xy^2$ to $\mathbf{S}_1$ ($\mathbf{S}_{1,2}$ and $\mathbf{S}_1$ represent the fourth and the first syzygy in the syzygy matrix, i.e. the fourth and the first row in the syzygy matrix, respectively). Now we have
\left(
\begin{array}{cccc}
\underline{z^4} & -z & 0 & xy^2 \\
\underline{z^5} & 0 & -x y^3-z & 0 \\
0 & \underline{z^5} & -x^2 y^2-z^4 & 0 \\
\underline{x} & -y & 0 & -1 \\
%0 & 0 & \underline{-y z^3}+x & -z^4 \\
\end{array}
\right).
Next, we see that the first term in $\mathbf{S}_2$ can be divided by the first term in $\mathbf{S}_1$. So we multiply $\mathbf{S}_1$ with $-z$ and add it to $\mathbf{S}_2$; and the syzygy matrix becomes
\left(
\begin{array}{cccc}
\underline{z^4} & -z & 0 & xy^2 \\
0 & \underline{z^2} & -x y^3-z & -xy^2z \\
0 & \underline{z^5} & -x^2 y^2-z^4 & 0 \\
\underline{x} & -y & 0 & -1 \\
%0 & 0 & \underline{-y z^3}+x & -z^4 \\
\end{array}
\right).
Likewise, we further simplify $\mathbf{S}_3$ using $\mathbf{S}_2$. The matrix then reads,
\left(
\begin{array}{cccc}
\underline{z^4} & -z & 0 & xy^2 \\
0 & \underline{z^2} & -x y^3-z & -xy^2z \\
0 & 0 & \underline{xy^3z^3}-x^2 y^2 & xy^2z^4 \\
\underline{x} & -y & 0 & -1 \\
% 0 & 0 & \underline{-y z^3+x} & -z^4 \\
\end{array}
\right).
Now there are three new two-pair syzygies, i.e. those containing the new polynomial $-yz^4+xz$. Again by the rules above, the only permitted pair is $\{f_3, f_4\}$. Here the relation is $(0,0,f_4/z, -z^4)$. From these two-pair relations the syzygies with barriers are updated as
\left(
\begin{array}{cccc}
\underline{z^4} & -z & 0 & xy^2 \\
0 & \underline{z^2} & -x y^3-z & -xy^2z \\
0 & 0 & \underline{xy^3z^3}-x^2 y^2 & xy^2z^4 \\
\underline{x} & -y & 0 & -1 \\
0 & 0 & \underline{-y z^3}+x & -z^4 \\
\end{array}
\right).
We observe that $\mathbf{S}_3$ can be further reduced by $\mathbf{S}_4$ to zero. Thus $\mathbf{S}_3$ should be removed from the syzygy matrix and the Top-syzygy set. The syzygy matrix becomes
\bordermatrix{
\text{Index}&\text{Effect}&f_1&f_2&f_3&f_4\cr
\mathbf{S}_1&\text{T}& \underline{z^4} & -z & 0 & xy^2 \cr
\mathbf{S}_2& \text{T }&0 & \underline{z^2} & -x y^3-z & -xy^2z \cr
\mathbf{S}_{1,2} & & \underline{x} & -y & 0 & -1 \cr
\mathbf{S}_4&\text{T}& 0 & 0 & \underline{-y z^3}+x & -z^4 \cr
and the Top-syzygy set becomes $\mathcal{S}=\{\mathbf{S}_1, \mathbf{S}_2, \mathbf{S}_4\}$.
We can continue to add a new polynomial to the ideal, namely the polynomial generated by the pair $\{f_3, f_4\}$ with the coefficients $(y, z)$, and the ideal is updated to
$$\langle xy^3+z, x^2y^2+z^4,z^5, -yz^4+xz,f_5=xz^2\rangle.$$
We need a new syzygy that generates $f_5$ and the non-principle syzygy $\{f_4, f_5\}$. The updated syzygy matrix becomes:
\bordermatrix{
\text{Index}&\text{Effect}&f_1&f_2&f_3&f_4&f_5\cr
\mathbf{S}_1&\text{T}& \underline{z^4} & -z & 0 & xy^2 &0\cr
\mathbf{S}_2& \text{T}&0 & \underline{z^2} & -z & 0 & -x y^2 \cr
\mathbf{S}_{1,2}&& \underline{x} & -y & 0 & -1 &0\cr
\mathbf{S}_4&\text{T}& 0 & 0 & \underline{x} & 0 & -z^3 \cr
\mathbf{S}_{3,4}&& 0 & 0 & \underline{y} & z&-1 \cr
\mathbf{S}_5&\text{F}& 0 & 0 &0& \underline{x z} & y z^3-x \cr
Here the new rank-2 syzygy $\mathbf{S}_5$ denoted by $F$ can generate a new barrier which can not be reduced to zero. However if we keep track of it back to the level above, we find this syzygy can be reduced to zero. Hence it can not contribute a new irreducible syzygy to the Top-syzygy set $\mathcal{S}$. To keep the Top-syzygies irreducibility, we introduce a rule that if a T-type syzygy is reduced to zero only by the F-type syzygies, we still keep this T-type syzygy in the Top-syzygy set $\mathcal{S}$.
The next permitted pair is $\{f_1, f_5\}$ with the coefficients $(z^2, -y^3)$, and it generates a new polynomial $f_6=z^3$. The syzygy $\mathbf{S}_{1,5}$, together with non-principle syzygy $\mathbf{S}_6$–$\{f_3, f_6\}$ with the coefficients $(1, -z^2)$ and $\mathbf{S}_7$–$\{f_5, f_6\}$ with the coefficients $(z, -x)$, update the syzygy matrix. The syzygy $\mathbf{S}_6$ is T-type while $\mathbf{S}_7$ is F-type. Thus we can add $\mathbf{S}_6$ to the Top-syzygy set. After rewriting, the syzygy matrix refreshes to
\bordermatrix{
\text{Index}&\text{Effect}&f_1&f_2&f_3&f_4&f_5&f_6\cr
\mathbf{S}_1&\text{T}& 0 & \underline{-z} & 0 & xy^2 & 0 & z^2+xy^3z \cr
\mathbf{S}_{1,2}& & \underline{x} & -y & 0 & -1 & 0 & 0 \cr
%\mathbf{S}_4&\text{T}& 0 & 0 & 0 & 0 & \underline{-z^3} & xz^2 \cr
\mathbf{S}_{3,4}& & 0 & 0 & 0 & \underline{z} &-1 & yz^2 \cr
\mathbf{S}_{1,5}&& \underline{z^2} & 0 & 0 & 0 & -y^3 & -1 \cr
\mathbf{S}_6&\text{T}& 0 & 0 & \underline{1} & 0 & 0 & -z^2 \cr
\mathbf{S}_4&\text{T}& 0 & 0 & 0 & 0 & \underline{z} & -x \cr
$\mathbf{S}_2$ is reduced to zero by $\mathbf{S}_6$ and some other non F-type syzygies. Hence it should be removed. $\mathbf{S}_4$ is reduced to zero only by F-type syzygies. This indicates that $\mathbf{S}_4$ and $\mathbf{S}_7$ is equivalent. We replace the elements in $\mathbf{S}_4$ by those in $\mathbf{S}_7$ and leave the index untouched. Hence the Top-syzygy set is $\mathcal{S}=\{\mathbf{S}_1, \mathbf{S}_4, \mathbf{S}_6\}$ at this step.
The next permitted pair is $\{f_4, f_6\}$ with the coefficients $(1, yz)$, and it generates a new polynomial $f_7 = xz$. After rewriting, the syzygy matrix refreshes to
\bordermatrix{
\text{Index}&\text{Effect}&f_1&f_2&f_3&f_4&f_5&f_6& f_7\cr
\mathbf{S}_1&\text{T}& 0 & \underline{-z} & 0 & 0 & 0 & z^2 &xy^2 \cr
\mathbf{S}_{1,2}& & \underline{x} & -y & 0 & 0 & 0 & yz & -1 \cr
\mathbf{S}_{3,4}&& 0 & 0 & 0 & 0 & \underline{-1} & 0 & z \cr
\mathbf{S}_{1,5}&& \underline{z^2} & 0 & 0 & 0 & 0 & -1&-y^3z \cr
\mathbf{S}_6&\text{T}& 0 & 0 & \underline{1} & 0 & 0 & -z^2 & 0 \cr
\mathbf{S}_4&\text{T}& 0 & 0 & 0 & 0 & 0 & \underline{x} & -z^2 \cr
\mathbf{S}_{4,6}& & 0 & 0 & 0 & \underline{1} & 0 & yz & -1 \cr
The next permitted pair is $\{f_1, f_7\}$ with the coefficients $(z, -y^3)$, and it generates a new polynomial $f_8 = z^2$. Taken into consideration the non-principle syzygy $\{f_7, f_8\}$ with the coefficients $(z, -x)$, the syzygy matrix can be reduced as
\bordermatrix{
\text{Index}&\text{Effect}&f_1&f_2&f_3&f_4&f_5&f_6& f_7&f_8\cr
\mathbf{S}_1&\text{T}& 0 & \underline{-z} & 0 &0 & 0 & 0&xy^2 &z^3\cr
\mathbf{S}_{1,2}& & \underline{x} & -y & 0 & 0 & 0 & 0 & -1&yz^2 \cr
\mathbf{S}_{3,4}&& 0 & 0 & 0 & 0 & \underline{-1} & 0 & z &0\cr
\mathbf{S}_{1,5}&& 0& 0 & 0 & 0 &0 &\underline{-1}&0&z \cr
\mathbf{S}_6&\text{T}& 0 & 0 & \underline{1} & 0 & 0 & 0 & 0 &-z^3\cr
%\mathbf{S}_7&\text{F}& 0 & 0 & 0 & 0 & 0 & 0 & -z^2 &xz\cr
\mathbf{S}_{4,6}& & 0 & 0 & 0 & \underline{1} & 0 & 0 & -1&yz^2 \cr
\mathbf{S}_{1,7}& & \underline{z} & 0 & 0 & 0 & 0 & 0 & -y^3 & -1 \cr
\mathbf{S}_{4}&\text{T}& 0 & 0 & 0 & 0 & 0 & 0 & \underline{z} & -x \cr
There is no more permitted pair, so the program halts. The final ideal is
$$\langle xy^3+z, x^2y^2+z^4,z^5, -yz^4+xz,f_5=xz^2,f_6=z^3,f_7=xz,f_8=z^2 \rangle.$$
In processing the last two critical pair, the Top-syzygy set is invariant. Now we obtain the syzygy $\mathcal{S}=\{\mathbf{S}_1,\mathbf{S}_4,\mathbf{S}_6\}$ of the original 3-tuple ideal generators,
\left(
\begin{array}{ccc}
z^4+x^2 y^2 & -x y^3-z & 0 \\
x z^4 & -y z^4 & y z^3-x \\
x y^3 z^3-z^4 & -y^4 z^3 &y^4 z^2+1
\end{array}
\right).
The simplest form of the syzygies is obtained by pairing the elements in red in Fig. <ref>. The syzygy module obtained here is the same as that obtained by Singular <cit.>. But the number of the generators of the module is smaller than that in Singular. According to Fig. <ref> and the analysis in <cit.>, the polynomials in the final step whose barriers are not $1$ just form the Gröbner basis of the ideal.
§.§ Syzygies Of General Ideals
In this section we give a general description of our method. We arrange the generators of the ideal in an order $LT(f_i)<LT(f_j)$ for $i<j$. We also assume all the monomials in $f_j$ can not be divided by the $LT(f_i)$ for all $i<j$. This is easy to be realized by dividing each $f_j$ by all $f_i$ with $i<j$, which is called Top-reduction. Our method is to generate the syzygies inductively from a series of critical pairs, as shown in Fig. <ref>.
Performing syzygies by a series of critical pairs.
We divide the generators into two levels, namely, before and after dealing with the critical pair, as shown in the first two rows in Fig. <ref>. In the up-row, there are already some obvious syzygies. If a pair induces a syzygy which can be simplified by the already known syzygies, then this pair is not permitted. This condition can be used to set up some barriers for the critical pairs. Among the allowed pairs, we prefer to deal with the one that generates the S-polynomial with the lowest leading terms in the order $>$. To summarize, in each step, we need to implement the following procedures
* Get all the trivial syzygies
* Reduce the syzygies
* Build up the barriers
* Find a critical pair and generate an S-polynomial
* Top-reduction on the S-polynomial
Get all the trivial syzygies Now we set up the known syzygies on level $t$ as shown in Fig. <ref>. There are four kinds of syzygies on level $t$ which can be directly read off
* Syzygies inherited from the syzygies on level $t-1$
* Syzygy induced by the critical pair on level $t-1$, $(\cdots { lcm_{i,j}\over LT(f_j)}\cdots{lcm_{i,j}\over LT(f_i)}\cdots0, -1)$. We denote such syzygies as $\mathbf{S}_{i,j}$
* Principle syzygies $\mathbf{S}^p$ with $f_{m+t}$
* Non-principle rank-2 syzygies with $f_{m+t}$. On level $t$, they are just
$(0,\cdots, {f_{m+t}\over c_{i,m+t}},\cdots, {-f_i\over c_{i,m+t} } )$, where $c_{i,m+t}$ is the maximal common factor of $f_i$ and $f_{m+t}$. We use $\mathbf{S}^e_{i,m+t}$ to denote such a syzygy
For the last kind of syzygies, they are permitted only when they are not blocked by the former syzygies. For setting up the barriers, the principle syzygies with $f_{m+t}$ are not necessary since they always generate some abandoned syzygies. To see this, we consider the principle syzygy between $f_1$ and $f_{m+t}$ without loss of generality
$$\mathbf{S}^p_{1,m+t}=\left(f_{m+t},\cdots, 0_i,\cdots, 0_j,\cdots, -f_1\right).$$
The syzygies inherited from the level $(t-1)$ contain
$$\mathbf{S}^p_{1,i}=\left(f_i,\cdots, -f_1,\cdots, 0_j,\cdots, 0_{m+r}\right)$$
$$\mathbf{S}^p_{1,j}=\left(f_j,\cdots, 0_i,\cdots, -f_1,\cdots, 0_{m+r}\right).$$
According to the syzygy induced by the critical pair
$$\mathbf{S}_{i,j}=(0_1,\cdots ,{ lcm_{i,j}\over LT(f_j)},\cdots, -{lcm_{i,j}\over LT(f_i)}, \cdots, -1),$$
it is easy to see that $f_{m+t}={ lcm_{i,j}\over LT(f_j)} f_i- {lcm_{i,j}\over LT(f_i)} f_j$ and
$$\mathbf{S}^p_{1,m+t}={lcm_{i,j}\over LT(f_j)}\mathbf{S}^p_{1,i}- {lcm_{i,j}\over LT(f_i)}\mathbf{S}^p_{1,j}+f_1 \mathbf{S}_{i,j}.$$
Back to the level $(t-1)$, $\mathbf{S}_{i,j}$ is vanishing and the corresponding syzygy of $\mathbf{S}^p_{1,m+t}$ on level $(t-1)$ is
$$\mathbf{S}^{p,t-1}_{1,m+t}={lcm_{i,j}\over LT(f_j)}\mathbf{S}^p_{1,i}-{lcm_{i,j}\over LT(f_i)}\mathbf{S}^p_{1,j}.$$
Hence the syzygy $\mathbf{S}^p_{1,m+t}$ is reduced to zero directly on level $t-1$ by $\mathbf{S}^p_{1,i}$ and $\mathbf{S}^p_{1,j}$. Obviously $\mathbf{S}^p_{1,m+t}$ can be deleted without affecting the completeness of the syzygy module. Furthermore, for the justification of irreducibility, if a syzygy $\mathbf{S}$ is reduced to zero by $\mathbf{S}^p_{1,m+t}$ on level $t$, it is also reduced to zero by $\mathbf{S}^p_{1,i}$ and $\mathbf{S}^p_{1,j}$ on level $(t-1)$. Hence we do not include any principle syzygy with $f_{m+t}$.
Syzygies, Rewriting and Barriers
The known syzygies can usually be reduced with each other and this process provides more barriers for upcoming critical pairs. We first reduce each syzygy using $\mathbf{S}_{i,j}$. Then we set up the initial barrier $\langle B^0_{f_i}\rangle$ for each $f_i$. If an pair is blocked by in $\langle B^0_{f_i}\rangle$, then the pair multiplying an monomial is also blocked. The barrier $\langle B^0_{f_i}\rangle$ is an ideal in the monomial ring. For each syzygy other than $\mathbf{S}_{i,j}$, we select the leading term of the first non-vanishing polynomial in the syzygy in order $\vec{e}_1\succ\vec{e}_2\succ\cdots\succ\vec{e}_{m+t}$. Then we add this leading term into the generating set for $\langle B^0_{f_i}\rangle$. For each critical pair, it induces a syzygy $\mathbf{S}_{i,j}$ which forbids a particular critical pair $\{i, j\}$ for upcoming syzygies. It also adds a new generator ${LCM(i,j)\over LT(f_i)}$ to the ideal $\langle B^0_{f_i}\rangle$ if $\mathbf{S}_{i,j}[k]=0$ for all $k<i$. We use $\mathbf{S}_{i,j}[k]$ to denote the k-column value of $\mathbf{S}_{i,j}$.
For the syzygies which can create generators for $\langle B^0_{f_i}\rangle$, they usually can be reduced using each other. To see how this happens, we choose two syzygies $\mathbf{S}_1, \mathbf{S}_2$ which create the generators $m_1, m_2$ for $\langle B^0_{f_i}\rangle$. Then the form of the two syzygies are
\begin{eqnarray*}
\mathbf{S}_1&=&\{0_1,\cdots,0_{i-1}, m_1+h_1, \cdots\}\\
\mathbf{S}_2&=&\{0_1,\cdots,0_{i-1}, m_2+h_2, \cdots\},
\end{eqnarray*}
where $m_1, m_2$ are monomials and $h_1,h_2$ are polynomials.
If $m_1=t \times m_2$, where $t$ is an element in the monomial ring, then $\mathbf{S}_1$ can be reduced by $\mathbf{S}_2$ as
$$\mathbf{S}'_1=\mathbf{S}_1-t \mathbf{S}_2=\{0_1,\cdots,0_{i-1}, h_1- t h_2, \cdots\}.$$
The syzygy $\mathbf{S}'_1$ will create a generator $m_3=LT(h_1- t h_2)$ for the barrier ideal of $f_i$. Then the barrier ideal is $\langle B^1_{f_i}\rangle=\langle m_3, m_1, m_2,\cdots\rangle=\langle m_3, m_2,\cdots\rangle$, where $\cdots$ denote the other generators of $\langle B^0_{f_i}\rangle$ from other syzygies. The involution of the barrier ideal under rewriting have $\langle B^0_{f_i}\rangle \subseteq \langle B^1_{f_i}\rangle$. When $\langle B^0_{f_i}\rangle= \langle B^1_{f_i}\rangle$, this indicates that $\mathbf{S}'_1$ can also be reduced by other syzygies. The rewriting will stop only when $\langle B^0_{f_i}\rangle \subset \langle B^1_{f_i}\rangle$ or $m_3$ is reduced to zero. If $m_3$ is reduced to zero, the final $\mathbf{S}'_1$ after several rewriting steps will add a generator to the barrier ideal $\langle B^0_{f_{i+1}}\rangle$. The generator-creating rewriting process of syzygies can be done recursively. Hence we conclude that the rewriting can enlarge the barrier ideal or leave the ideal unchanged. And after the steps of rewriting for all columns $f_{i}$, we get the maximal barrier ideal $\langle B_{f_{i}}\rangle$ for each $f_{i}$.
If there are some $\mathbf{S}_{i,j}[k]\neq 0$ for $k<i$, this kind of syzygies do not get reduced in the steps above. They only forbid a particular critical pair $\{i, j\}$ to upcoming syzygies. To reduce them, we can use the syzygy $\mathbf{S}^e$ when $\mathbf{S}[i]$ can be divided by $LT(\mathbf{S}^e[i])$ and $\mathbf{S}^e[k<i]=0$, or when $\mathbf{S}[j]$ can be divided by $LT(\mathbf{S}^e[j])$ and $\mathbf{S}^e[k<j]=0$. In either case, the critical pair $\{i, j\}$ has been blocked by the barrier ideal generator $LT(\mathbf{S}^e[i])$ or $LT(\mathbf{S}^e[j])$. After rewriting, $\mathbf{S}_{i,j}$ becomes $\mathbf{S}'_{i,j}=\mathbf{S}_{i,j}-m_1 \mathbf{S}^e$, where $m_1={\mathbf{S}[i]\over LT(\mathbf{S}^e[i])}, \text{or} {\mathbf{S}[j]\over LT(\mathbf{S}^e[j])}$. Furthermore $\mathbf{S}'_{i,j}$ does not contain the critical pair. $\mathbf{S}'_{i,j}$ also creates a generator for the barrier ideal $\langle B_{f_{i}}\rangle$ and return to the rewriting steps for the involution of barrier ideals.
Irreducibility among critical pairs
In addition to the rewriting of the syzygies, to maintain the irreducibility of the syzygies, it needs to be guaranteed that the critical pairs are irreducible among themselves. To justify the irreducibility of the critical pairs, only the leading terms of said pairs are relevant. The irreducibility is easy to be observed from the cell complex for the monomial ideas <cit.>.
The cell complex can be set up on the present level $t$ or on level $(t-1)$ in advance. As an example, we consider three leading terms $\{x^2y,xz^2, yz^3\}$ on level $t$. We first define the vertices of the simplex $\bigtriangleup$ to be the monomial elements as shown in Fig. <ref>.
Example for Cell complex.
The edge of $\bigtriangleup$ is labelled by the least common multiple of the monomial element. And the face of $\bigtriangleup$ which the three edges form is labelled by the least common multiple of the labels of the edges. The edges $e_{12}, e_{23}, e_{13}$ imply three syzygies $(z^2, -xy,0)$, $(0, yz,x)$, $(z^3, 0,x^2)$. The least common multiple of $e_{13}$ is equal to the least common multiple of the face. Hence the syzygy corresponding to $e_{13}$ is reducible. This rule holds generally and is very convenient for determining whether the syzygies generated by the critical pairs are reducible. In fact if the edges form a face, and the least common multiple of the face is equal to the least common multiple of a edge, the syzygy corresponding to the edge is reducible.
The cell complex on level $(t-1)$ is shown in the following example. We set that the ideal generators include $f_{i}=x_4, f_{j}=x_2x_5x_6, f_{k}=x_1x_2x_3+x_1$ on level $(t-1)$. Then we perform the critical pair for $\{f_j, f_k\}$. A new generator $f_{k+1}=-x_1x_5x_6$ of the ideal is obtained on level $t$ as shown in Fig. <ref>.
Performing a critical pair.
On level $(t-1)$, according to the cell complex, the three edges are denoted as $e_{ij}, e_{ik}, e_{jk}$. These edges are characterized by the least common multiples $x_2x_4x_5x_6, x_1x_2x_3x_4, x_1x_2x_3x_5x_6$ for leading terms of $(f_i, f_j), (f_i, f_k), (f_j, f_k)$ respectively. The face $s_{ijk}$ formed by the three edges is characterized by the least common multiple $m_s=x_1x_2x_3x_4x_5x_6$ for the leading terms of $(f_i, f_j,f_k)$. None of the edge common multiple is equal to $m_s$. Hence the three critical pairs are irreducible among each other. On level $t$, according to the cell complex, the irreducible edges are shown in Fig. <ref>.
Cell Complex for the critical pairs, the symbol such as $1234$ denote the monomial $x_1x_2x_3x_4$ for convenience.
Now we discuss the critical pair for edge $e_{i,k+1}$. The syzygy corresponding to $e_{i,k+1}$ is $\mathbf{S}_{i,k+1}=\{x_1x_5x_6,0,0,x_4\}$ on level $t$. On level $(t-1)$, $\mathbf{S}_{i,k+1}$ become $\mathbf{S}'_{i,k+1}=\{x_1x_5x_6, x_1x_3x_4,-x_4x_5x_6\}$. This syzygy is the product of $x_4$ and the syzygy corresponding to $ e_{jk}$
$$x_4 (x_1 x_3 f_j - x_5 x_6 f_k)=-x_1 x_5 x_6 f_i. $$
This relation is reducible as the product of $x_4$ and the least common multiple $x_1x_2x_3x_5x_6$ for the edge $ e_{jk}$ divide the $m_s$ for the face $s_{ijk}$. Hence according to the cell complex, such syzygy can be reduced to
\begin{eqnarray*}
\mathbf{S}_{i,j}&=&\{x_2x_5x_6,-x_4,0\}\\
\mathbf{S}_{i,k}&=&\{x_1x_2x_3+x_1,0,-x_4\}.
\end{eqnarray*}
This is also direct to check
$$\mathbf{S}'_{i,k+1}+{x_1x_3} \mathbf{S}_{i,j} -{x_5 x_6} \mathbf{S}_{i,k}=0.$$
Hence, $\mathbf{S}_{i,k+1}$ is reducible on $\mathbf{S}_{i,j}$, $\mathbf{S}_{i,k}$, $\mathbf{S}_{j,k}$. We just delete $\mathbf{S}_{i,k+1}$. In practice, the rules for forbidding such reducible critical pairs are summarized as following: When performing a critical pair $\{i,j\}$ on level $(t-1)$ and generating new ideal generator $f_{m+i+1}$, choose generator $f_k$ on level $(t-1)$, such that $k\neq i, k\neq j$ and the corresponding S-polynomial $S_{k,i}, S_{k,j}$ can be reduced to zero, then $\mathbf{S}_{m+t+1,k}$ is reducible by the syzygy module on level $(t-1)$ when ${lcm(LT(f_{m+t+1}), LT(f_k))\over LT(f_{m+t+1})}lcm(LT(f_i),LT(f_j))$ divide $lcm(LT(f_i),LT(f_j),LT(f_k))$. The proof of this statement is direct by the cell complex as shown in the above example.
Get the permitted critical pair
After updating the barriers, it is easy to observe which critical pair is permitted. If the monomials from a critical pair belong to $\langle B_{f_i}\rangle$, such pair is not permitted. And all the used critical pairs are not permitted in the following steps. Among all the allowed critical pairs, we choose a pair such that the leading term of S-polynomial is the lowest in the $>$ order. When performing a critical pair, a new polynomial is generated. In the following, we always assume a full top-reduction is performed on new polynomials.
For an ideal $\langle f_1,f_2\cdots f_i\cdots f_j\cdots f_m\rangle$, if a critical pair $\{i,j\}$ generates a polynomial $f_{m+1}$ with leading term being able to be divided by the leading terms of the generators $f_s$, then the $f_{m+1}$ is reduced to a polynomial $f'_{m+1}$ with the lower order leading term. All the irreducible syzygies will not refer to $f_{m+1}$, which means that we can replace $f_{m+1}$ with $f'_{m+1}$.
Without loss of generality, we suppose that $LT(f_{m+1})$ is divided by $LT(f_m)$, then any critical pair $\{i,m+1\}$ can be taken as the composition of $\{i,m\}$ and $\{m,m+1\}$, which is easy to see from the triangle diagram in Fig. <ref>. The critical pairs, denoted by edges in Fig. <ref>, are characterized by the minimal common factor of the leading terms of $f$. Three critical pair edges form a triangle. This triangle characterize the syzygy of the syzygy module of the three elements. If the face common factor equals an edge common factor, then according to <cit.> the syzygy corresponding to the edge is reducible. Since all syzygies are decomposed as the compositions of the critical pairs, any irreducible syzygy does not refer to $f_{m+1}$. We replace $f_{m+1}$ with $f'_{m+1}$, which is named as Top-reduction. The Top-reduction is done recursively and finally we have a fully reduced $f'_{m+1}$. If the Top-reduction gives us $f'_{m+1}=0$, this will induce a syzygy. This syzygy is a high rank syzygy. After adding this to the syzygy module, the critical pair can never be used in further steps.
If a S-polynomial is reduce to zero under Top-reduction, then there is a higher rank trivial syzygy
& f_1&\cdots & f_{i-1}& f_i & \cdots &f_{j-1}& f_j &\cdots &f_m \cr
& a_1&\cdots &a_{i-1} &{LCMij\over LT(f_i)}+a_i &\cdots&a_{j-1} & {LCMij\over LT(f_j)}+a_j&\cdots &a_m
\end{array}\right)$$
where $a_i$ is the coefficient of $f_i$ under Top-reduction for $S_{i,j}$. This only happens in the beginning of the critical pair decomposition. Our algorithm is able to forbid such redundant pairs.
Triangle diagram for our theorem.
End criterion
The former procedures are proceeded recursively and stop if all the critical pairs are not permitted. This means that all other syzygies can be reduce to zero under the existing syzygies, and we finally obtain the all the irreducible syzygies. These syzygies form a basis of the syzygy module. Moreover, we also get the Gröbner basis of the ideal. The Gröbner basis is formed by the polynomials in final step whose barriers are not equal to $\langle 1\rangle$. This procedure is shown in flow chart Fig. <ref>.
End criterion.
Independence criterion for the syzygy
In performing each critical pair, we can keep the new added syzygy from being reduced by the old ones. This is only part of the story for the irreducibility among syzygies. To remove all the reducible syzygies and obtain the locally minimal generating set of the syzygies, we need to deal with much more in each step. Let's first discuss relations between syzygies on level $(t-1)$ and those on level $t$.
* $\{\text{Syzygies on level $t-1$}\}\subset\{\text{Syzygies on level $t$}\}$.
* A permitted rank-2 syzygy on level $t$ combining with the critical pair induced syzygy will generate a higher rank syzygy on level $t-1$.
When we use the syzygies on level $t$ to set up the barriers, new permitted rank-2 syzygies can not be reduced on level $t$. If it is reduced to zero at on level $t-1$, we denote this syzygy as $F$. It means that this syzygy generates a new barrier to the following steps, but a reducible syzygy on level $t-1$. In turn, this also generates a linearly reducible syzygy for the original m-tuple polynomial and we do not need to add it to the Top-syzygy set. If it can not be reduced to zero on level $t-1$, then we denote it a $T$ and add it to the Top-syzygy set.
We suppose that the rewriting on level $t-1$ is performed and a syzygy is reduced to zero. If it is of $F$-type, then we just remove it. For $T$-type syzygy, we need to be more careful. If it is reduced to zero all by $T$-type and critical pair syzygies, then we remove it. If it is reduced to zero by a $F$-type syzygy and some other type of syzygies, then we replace it by this $F$-type syzygy in the Top-syzygy set. This is because the two syzygies are equivalent. In order to continue the following steps, this $F$-type syzygy is used. However for the sake of obtaining the right and simple Top-syzygies, we use the $T$-type syzygy. Hence the replacing $F$-type syzygy also come from the eliminating pair of the $T$-type syzygy. Remark that if the original $T$-type syzygy is reduced to zero by several $F$-type syzygies, we should replace it syzygy by the last adding $F$-type syzygy. The irreducibility of these syzygies is justified by rewriting of the syzygy module as is shown in Fig. <ref>.
Top-reduction of the syzygy module.
The above procedure is performed recursively. In principle, the reducibility among the syzygies can not be fully determined at each level. However, in performing the critical pairs one by one, if we keep to reduce all the syzygies at each level, and delete all the abandoned syzygies one by one, we can finally get the irreducible syzygies for zero level generators. This is easy to see. For convenience, we define the leading critical pair for each syzygy as following: A pair with coefficients $(C^{S}_{i}, C^{S}_{j})$ of column $i$ and column $j$ in $\mathbf{S}$ is called the leading critical pair if $C^{S}_{i}LT(f_i)+C^{S}_{j}LT(f_j)=0$ and $C^{S}_{i}LT(f_i)$ is of the largest monomial order comparing with other pairs, where $C^{S}$ is the leading term of $\mathbf{S}$. We only need to verify that at the end of our procedure if $\mathbf{S}_t=c_1\mathbf{S}_1+c_2\mathbf{S}_2\cdots+c_K\mathbf{S}_K$, then $\mathbf{S}_t$ will be reduced to zero. According to the equation, the leading critical pair of $\mathbf{S}_t$ should be reducible by the leading critical pair of $\mathbf{S}_k$, $k\in{1\cdots K}$. The leading critical pair in each $\mathbf{S}_k$ contains the index of the first non-zero column except for those critical pair induced syzygies. Otherwise there will be more allowed critical pairs and the main loop can not break, thus the leading critical pair of $\mathbf{S}_t$ is able to be reduced by the syzygies $\mathbf{S}_k$, $k\in{1\cdots K}$.
Our algorithm generates the complete syzygy module. In fact, we include all the possible syzygy in performing each critical pair. We only abandon those reducible syzygies. After performing all the irreducible critical pairs, there is no other syzygy not in the module. To prove this, we only need to verify that any syzygy $\mathbf{S}$ can be decomposed by $\mathbf{S}_1, \mathbf{S}_2\cdots \mathbf{S}_K$, where $\mathbf{S}_{i\in [1, K]}$ is the syzygy module we obtained. For each syzygy of the ideal generators, the leading critical pair should be a summation of the multiples of some critical pairs $\{i_1,j_1\},\cdots, \{i_{n_1},j_{n_2}\}$. If the critical pair does not generate a new ideal generator, this critical pair will generate a syzygy $\mathbf{S}_i$ which is in our syzygy module. Then we have $\mathbf{S}=\mathbf{S}'+\mathbf{S}_i$. For $\mathbf{S}'$, the leading pair is of small order. If the critical pair generates a new ideal generator, the $\mathbf{S}$ become a syzygy $\mathbf{S}^1$ in next level. After transforming the leading critical pair, we find that the syzygy $\mathbf{S}$ either decreases the order or becomes syzygy on next level. Then we transform the leading critical pair step by step. Finally, the steps stop when $\mathbf{S}$ become either zero or a syzygy on the final level. The ideal generators on the final level form a Gröbner basis. For the first case, it just means $\mathbf{S}$ lies in our syzygy module. For the last case, $\mathbf{S}$ is able to be decomposed as the summation of the syzygies induced by the critical pairs. At finial step, each critical pair is blocked by the barriers. Then the corresponding syzygies are all in the syzygy module. Hence $\mathbf{S}$ belong to the syzygy module.
Now, we get the full description of this new algorithm, we name it as $\text{C2Z}$ algorithm. In the following, we apply our algorithm to several examples.
A short cut for regular sequence We first discuss two-tuple regular sequence $(f_1, f_2)$. The principle syzygy is $(f_2, -f_1)$. In our algorithm, this can generate a barrier ideal $\langle LT(f_2)\rangle$ on $f_1$. Then the critical pair $\{1,2\}$ is permitted when $LT(f_1)$ and $LT(f_2)$ have a proper greatest common factor. Our algorithm will stop when we get a Gröbner basis. The syzygy module of $(f_1, f_2)$ is induced by all the syzygies in the following levels. If a new syzygy on the following level is of T-type, then the corresponding Top-syzygy $(h_1, h_2)$ of $(f_1, f_2)$ is not blocked by the barrier ideal $\langle LT(f_2)\rangle$. Here $h_1, h_2$ are polynomials. Hence $LT(f_2)$ is not a divisor of $LT(h_1)$. But $(f_1, f_2)$ is a regular sequence. Then $f_2$ should be a divisor of $h_1$. In turn, $LT(f_2)$ is a divisor of $LT(h_1)$. Then we get the T-type syzygy does not exist. The syzygy module is generated by the principle syzygy for regular sequence $(f_1, f_2)$.
Likely, for $m$-tuple regular sequence $(f_1, f_2,\cdots, f_m)$, a syzygy arising on a following level $t$ generate a Top-syzygy. If it is of T-type, the corresponding Top-syzygy is of form $\mathbf{S}_t=(0_1, 0_2,\cdots, 0_i, h_{i+1}, h_{i+2}, \cdots, h_{m})$ in general. For regular sequence, we have $h_{i+1}\in \langle f_{i+2}, f_{i+3},\cdots, f_{m}\rangle$. Since we finally get the Gröbner basis for the ideal. $\mathbf{S}_t$ will be reduced by the principle syzygies on a level below $t$. No new syzygy is added to the syzygy module for the $m$-tuple regular sequence $(f_1, f_2,\cdots, f_m)$. Furthermore, the principle syzygies are possible to be reduced among each other. This is easy to be realized by the cell complex composed by the leading terms of entry in $(f_1, f_2,\cdots, f_m)$. Such kind of reduction is also included in our algorithm.
To test the program, we need to compare with the popular software Singular <cit.>. By Singular, the local minimal generating set of the syzygy module is obtained according to the Gröbner basis of the syzygies of the syzygy module. In the Gröbner basis, a syzygy of the syzygy module has
$$(h_1,\cdots, h_{i-1}, h_{i}, h_{i+1}, \cdots, h_K)\cdot (\mathbf{S}_1,\cdots,\mathbf{S}_{i-1},\mathbf{S}_i,\mathbf{S}_{i+1}, \cdots,\mathbf{S}_K)=0,$$
where $K$ is an integral number denoting the generator number of the syzygy module. If one of the entry of the syzygy $h_i\in \mathbb{Z}$, then $\mathbf{S}_i$ is reducible. Let's consider the following ideals with integral coefficients
\begin{eqnarray*}
I_1&=&\left\langle x_2 + x_2^2 + x_1 x_3,\ x_1 + 3 x_1^2,\ x_1^2, x_1 x_2 + x_2^2 + x_3 + x_3^2,\
x_2^2 + x_1 x_3 + x_3^2\right\rangle,\\
I_2&=&\left\langle x_3 + x_1 x_3 + x_2 x_3,\ x_1 + x_1 x_2 + 2 x_3^2,\ x_1 x_2 + x_3 + x_1 x_3 + x_2 x_3,\
x_1 x_2 + x_2^2 + x_3 + x_2 x_3,\right.\\
&&\left.x_1 x_2 + x_2^2 + x_1 x_3 + x_2 x_3, x_2^2 + 2 x_1 x_3 \right\rangle,\\
I_3&=&\left\langle x_3 + x_4 + x_1 x_4 + x_3 x_4,\ x_1 + x_1 x_4,\ x_2^2 + x_3 + x_3^2 \right\rangle,\\
I_4&=&\left\langle x_1 + 2 x_2 + x_1 x_4,\ x_1 x_2 + x_1 x_3 + x_4 + x_2 x_4,\
x_2 x_3 + x_3^2,\ x_1 + x_2 + x_4 + x_1 x_4,\ x_1 x_2 \right\rangle,\\
I_5&=&\left\langle x_1 x_2 x_3 + x_2 x_4^2,\
x_1 x_4 + x_4^2,\ x_4^3,\ x_4 + x_1 x_2 x_4 + x_3^2 x_4, x_3^3 + x_1 x_2 x_4,\ x_2 x_4^2 \right\rangle,\\
I_{6}&=&\left\langle x_1^2 x_3,x_1 x_2^2 x_3^2+x_1 x_2 x_3^3,x_1 x_2 x_3^4+x_3\right\rangle,\\
I_{7}&=&\left\langle x_1^6+x_1 x_2^3 x_3+x_1 x_2 x_3^4+x_2,x_1^4 x_3^2+x_1^3 x_2^2+x_1^2 x_2^3 x_3,x_1+x_2^3 x_3^2,x_1 x_2^2 x_3^3,x_3^6,x_1 x_2 x_3^2+x_3^3\right\rangle,\\
I_{8}&=&\left\langle x_1^3 x_2 x_3^3+x_1^3+x_1^2 x_2^2,x_1^6 x_3+x_1 x_2^2 x_3+x_1 x_2 x_3^2+x_1,x_3^3,x_1^3 x_3^2+x_1^3+x_3,x_1 x_3^4+x_2^2 x_3^3\right\rangle,\\
I_{9}&=&\left\langle x_1^4 x_2 x_3^3+x_1^2 x_2 x_3^5+x_1 x_2^3 x_3^3,x_1^2 x_2^4 x_3+x_2^6,x_1^3 x_2^3 x_3^2+x_1^2 x_2+x_2 x_3^2,x_1^3 x_2^2 x_3^2,x_1^2 x_2^2+x_1\right\rangle.\\
\end{eqnarray*}
We compare the our results and the timing of this algorithm with Singular on a computer with CPU 2.4GHz and RAM 8G as shown in following
\begin{array}{|c|c|c|c|c|c|c|}\hline
& T_1/s & n_1 & T_2/s & n_2&T_3/s& n_3\\ \hline
I_1& 0.04 & 5 & 0.03& 6 & 0.29&5\\ \hline
I_2& 0.05 & 6 &0.05& 7 & 0.57&6\\ \hline
I_3& 0.09 & 3 & 0.23 & 4 &0.38&3\\ \hline
I_4& 0.27 & 7 & 0.14 & 10 & 1.2&7\\ \hline
I_5& 0.32 & 6 & 0.19 & 9 &1.0 &6\\ \hline
I_{6}& 0.09 & 2 &0.14& 3 & 0.28&2\\ \hline
I_{7}& 0.88 & 8 & 0.15 & 12 & ?&?\\ \hline
I_{8}& 1.59 & 11 & 0.26 & 17 & 1.61&11\\ \hline
I_{9}& 0.53 & 6 & 0.15 & 15 & ? &?\\ \hline
\end{array},
where the $T_1, n_1$ is the timing and number of syzygies of our program, $T_2, n_2$ is the timing and number of syzygies in Singular, $T_3, n_3$ is the timing and number for getting the local minimal generating set of syzygies in Singular, the symbol “$?$" denotes that the time is longer than half hour in our computer and we do not have the output of $n_3$. We also test for the ideals contain an uncertain constant $a$.
\begin{eqnarray*}
I^a_1&=&\left\langle x_2^2 x_3,x_1 x_4+x_2 a^2,x_1+x_3^2 a,x_1 x_2+x_2^2 x_3 x_4+x_2 x_4 a\right\rangle,\\
I^a_2&=&\left\langle x_2 x_3^2+x_3^3+x_3 a,x_1^2 a,x_1^3+x_3^2,x_2+x_3,x_1^2 x_3+x_1 x_3^2+a^3,x_1^2 x_2+x_1 x_3 a+x_1 a+x_2^3,x_2\right\rangle,\\
I^a_3&=&\left\langle x_1^2 x_2+x_1 x_3+x_1 a+x_2^2,x_1^2+x_1 a^2+x_1,x_1^2+a,x_3,x_1^2 x_2+x_3^2 a\right\rangle,\\
I^a_4&=&\left\langle x_1 x_2^2+x_1 x_2 a+x_1 x_3 a+x_3^2 a,x_1^2 x_3+x_1 x_3 a,x_2 x_3 a\right\rangle,\\
I^a_{5}&=&\left\langle x_1^2 x_3 a+x_1 x_2^2+x_2^3 x_3,x_1^3 x_2+x_2+x_3 a^2,x_1^3 x_3+x_1^3 a,x_1^2 x_3+x_1 x_2 x_3 a+x_3^3 a+x_3^2 a^2,x_1 x_3 a^2,x_1^2 x_2^2\right\rangle,\\
I^a_{6}&=&\left\langle x_1 x_2 x_3+x_3 a^2,x_1 x_2 x_3^2+x_2^2 a,x_2,x_2^2,x_3^2 a^2\right\rangle.
\end{eqnarray*}
\begin{array}{|c|c|c|c|c|c|c|}\hline
& T_1/s & n_1 & T_2/s & n_2&T_3/s& n_3\\ \hline
I^a_1& 0.32 & 6 & 0.14& 8 & 0.41& 6\\ \hline
I^a_2& 0.09 & 7 &0.01& 9 & 0.38& 7\\ \hline
I^a_3& 0.03 & 4 & 0.24 & 5 &0.40& 4\\ \hline
I^a_4& 0.09 & 3 & 0.14 & 4 & 0.29& 3\\ \hline
I^a_{5}& 0.11 & 8 & 0.24 & 9 &? & ?\\ \hline
I^a_{6}& 0.02 & 4 & 0.14& 5 & 0.29& 4\\ \hline
\end{array},
Our program was written in Mathematica. So there is still of much space to improve our program in $C^{++}$.
§.§ Remarks On Syzygies Of Modules
From the algorithm of syzygies for an ideal, it is straightforward to extend this algorithm to calculate the syzygies of any module. In fact, we calculate the syzygies of each column of the module. Then we combine them together to get the syzygies of the module. First, taking the the elements of the first column as an ideal, we calculate the syzygy directly by $\text{C2Z}$ . We denote this syzygy set by $\mathbf{B}^1$. Then we perform dot product for all the syzygies in $\mathbf{B}^1$ with the second column. The obtained result generates a new ideal. The syzygy set of this new ideal is denoted as $\mathbf{C}^2$. Then, the syzygy module of the first two columns is $\mathbf{S}^e_j=\mathbf{C}^2_{j,i}\cdot \mathbf{S}^1_i$, where the $j$ denotes the syzygy index and $i$ is the component index for each syzygy. Repeating these operations to the last column, we obtain the final syzygies of the module. In each step, our algorithm can guarantee the minimal generating set of $\mathbf{C}^2$. There are extra reductions after taking the dot product $\mathbf{C}^2_{j,i}\cdot \mathbf{S}^1_i$. An obvious example is a syzygy in $\mathbf{C}^2$ was just a syzygy of $\mathbf{S}^1$, dot product of them get a row of zero entry, which is obvious redundant. If two syzygies in $\mathbf{C}^2$ differ by a syzygy of $\mathbf{S}^1$, one of them is reduced by the other one in the final syzygy module. But in our method, this redundancy still remains for the syzygy of a module. To remove this redundancy, we need to get the syzygies $S_S$ of $\mathbf{S}^1_i$. In fact what we need is the $\mathbf{C}^2_{j,i} \mod S_S$. To get an efficient algorithm to eliminate the redundancy is beyond the scope of this paper. We leave this to future works.
§ REVISITING THE IBP RELATIONS
One important application of using syzygies is to obtain the IBP relations in generalized unitarity cuts<cit.>. The IBP relations are from the fact that any total partial derivative of rational functions of loop momentum is vanishing under loop integration
\begin{align}
\int dl_1^d\cdots dl_L^d \partial_{l_I^\mu} v_J^\mu{C_{IJ}\over D_1 D_2\cdots D_m}=0~,
\end{align}
where $v^\mu_J\in\{l^\mu_1,l^\mu_2, \cdots, l^\mu_L, p^\mu_1,\cdots ,p^\mu_\beta\}$ for general $n$ point amplitude ($n$ point amplitudes have $\beta$ irreducible external-leg momentums, then we have $\beta \le 4$). We define the $O_{IJ}\equiv \partial_{l_I^\mu} v_J^\mu$ as the generators of the IBP relations. In Lorentz invariant parameters, $s_a\in\{l_{I_1}\cdot l_{I_2}, l_{I_1}\cdot p_J\}$ and $D_a$, the IBP generators are defined in Baikov's method <cit.> as
\begin{align}
{{O}_{IJ}}=d {{\delta }_{IJ}}+\frac{\partial {{s}_{b}}}{\partial l_{I}^{\mu }}v_{J}^{\mu }\frac{\partial {{D}_{a}}}{\partial {{s}_{b}}} {\partial\over\partial D_a}~,
\end{align}
where $I\in\{1,2,\cdots,L\}$, $J\in\{1,2,\cdots,L+\beta\}$, $a,b\in\{1,2,\cdots, m\}$ and $d$ is the dimension of spacetime. When acting IBP generators on the dominators, it is useful to keep the dominators free of double poles according to the combination of the generators
\begin{align}
\sum_{IJ}c_{IJ}\frac{\partial {{s}_{b}}}{\partial l_{I}^{\mu }}v_{J}^{\mu }\frac{\partial {{D}_{a}}}{\partial {{s}_{b}}}+c_a D_a=0~.
\end{align}
We need to find all the solutions of these equations for $c_{IJ}, c_a$. With the definition
\begin{align}
{{Q}_{IJ}^a}=\frac{\partial {{s}_{b}}}{\partial l_{I}^{\mu }}v_{J}^{\mu }\frac{\partial {{D}_{a}}}{\partial {{s}_{b}}}~,
\end{align}
it is equivalent to find syzygies for the module
\begin{align}
\cdots &\cdots&\cdots&\cdots \\
\cdots &\cdots&\cdots&\cdots \\
\end{array}\right)~.
\end{align}
As a result, our analysis on syzygies comes to the story. In order to illustrate this application, we take a characteristic two-loop diagram as an example in Fig. <ref>. We will show that such diagram can reduce to zero under the generalized unitarity cut. Hence in physics, the four-point massless diagram is not a master integral.
An example two-loop diagram.
The loop integration of this diagram is
\begin{align}
L=\int{{{d}^{4}}{{l}_{1}}}{{{d}^{4}}{{l}_{2}}}\frac{{{C }}}{D_1D_2D_3D_4D_5D_6}~,
\end{align}
\begin{align}
\end{align}
Extra $D_7$, $D_8$, $D_9$ are fake propagators in order to keep the one-to-one correspondence between $D$ and $s$, where projections $s$ are defined as
\begin{align}
&{{s}_{1}}=l_{1}^{2}~,~~{{s}_{2}}={{l}_{1}}\cdot {{l}_{2}}~,~~{{s}_{3}}={{p}_{1}}\cdot {{l}_{1}}~,\nonumber\\
&{{s}_{4}}={{p}_{2}}\cdot {{l}_{1}}~,~~{{s}_{5}}={{p}_{4}}\cdot {{l}_{1}}~,~~{{s}_{6}}=l_{2}^{2}~,\nonumber\\
&{{s}_{7}}={{p}_{1}}\cdot {{l}_{2}}~,~~{{s}_{8}}={{p}_{2}}\cdot {{l}_{2}}~,~~{{s}_{9}}={{p}_{4}}\cdot {{l}_{2}}~.
\end{align}
$D_7$ is used to keep the one-to-one relation between $D_a$ and $s_a$. If we use symbols ${{v}_{J}}=\{{{l}_{1}},{{l}_{2}},{{p}_{1}},{{p}_{2}}\}$, where $J=1,2,3,4$, and ${{l}_{I}}=\{{{l}_{1}},{{l}_{2}}\}$, where $I=1,2$, then the effects of IBP operators $O_{IJ}$ on the propagators $D_a$ are given by
\begin{align}
{{O}_{IJ}}{{D}_{a}}=d{{\delta }_{IJ}}{{D}_{a}}+v_{J}^{\mu }\frac{\partial {{s}_{b}}}{\partial l_{I}^{\mu }}\frac{\partial {{D}_{a}}}{\partial {{s}_{b}}}~,
\end{align}
where $d$ is the dimension of spacetime. As a result, the module $M$ is
\begin{align}
\left(
\begin{array}{ccl}
Q_{11}&\rightarrow&\{2D_{1} , D_{1}{\tiny\text{+}}D_{2} , D_{1}{\tiny\text{+}}D_{3}\text{-}2 k_{12} , 0 , 0 , D_{1}\text{-}D_{4}{\tiny\text{+}}D_{6} , 0 , D_{1}{\tiny\text{+}}D_{8} , 0\}\\
Q_{12}&\rightarrow&\{\text{-}D_{1}\text{-}D_{4}{\tiny\text{+}}D_{6} , \text{-}D_{1}{\tiny\text{+}}D_{6}\text{-}D_{7} , \text{-}D_{1}{\tiny\text{+}}D_{4}{\tiny\text{+}}D_{6}\text{-}D_{7}\text{-} D_{9} , 0 , 0 , \text{-}D_{1}{\tiny\text{+}}D_{4}{\tiny\text{+}}D_{6} , 0 , \text{-}D_{1}\text{-}2 D_{4}{\tiny\text{+}}D_{5}{\tiny\text{+}}D_{6} , 0 \}\\
Q_{13}&\rightarrow&\{D_{1}\text{-}D_{2} , D_{1}\text{-}D_{2} , D_{1}\text{-}D_{2}\text{-}2 k_{12} , 0 , 0 , D_{1}\text{-}D_{2}\text{-}D_{4}{\tiny\text{+}}D_{7} , 0 , D_{1}\text{-}D_{2}{\tiny\text{+}}2 k_{14} , 0 \}\\
Q_{14}&\rightarrow&\{ D_{2}\text{-}D_{3}{\tiny\text{+}}2 k_{12} , D_{2}\text{-}D_{3} , D_{2}\text{-}D_{3} , 0 , 0 , D_{2}\text{-}D_{3}\text{-}D_{4}{\tiny\text{+}}D_{9}{\tiny\text{+}}2 k_{12} , 0 , D_{2}\text{-}D_{3}\text{-}2 k_{14} , 0 \}\\
Q_{15}&\rightarrow&\{ \text{-}D_{1}{\tiny\text{+}}D_{8} , \text{-}D_{1}{\tiny\text{+}}D_{8}\text{-}2 k_{14} , \text{-}D_{1}{\tiny\text{+}}D_{8}{\tiny\text{+}}2 k_{12} , 0 , 0 , \text{-}D_{1}\text{-}D_{4}{\tiny\text{+}}D_{5}{\tiny\text{+}}D_{8} , 0 , \text{-}D_{1}{\tiny\text{+}}D_{8} , 0 \}\\
Q_{22}&\rightarrow& \{0 , 0 , 0 , 2 D_{4} , D_{4}{\tiny\text{+}}D_{5} , \text{-}D_{1}{\tiny\text{+}}D_{4}{\tiny\text{+}}D_{6} , D_{4}{\tiny\text{+}}D_{7} , 0 , D_{4}{\tiny\text{+}}D_{9}\} \\
Q_{23}&\rightarrow& \{0 , 0 , 0 , \text{-}D_{4}{\tiny\text{+}}D_{7} , \text{-}D_{4}{\tiny\text{+}}D_{7}{\tiny\text{+}}2 k_{14} , D_{1}\text{-}D_{2}\text{-}D_{4}{\tiny\text{+}}D_{7} , \text{-}D_{4}{\tiny\text{+}}D_{7} , 0 , \text{-}D_{4}{\tiny\text{+}}D_{7}{\tiny\text{+}}2 k_{12}\} \\
Q_{24}&\rightarrow&\{0 , 0 , 0 , \text{-}D_{4}{\tiny\text{+}}D_{9} , \text{-}D_{4}{\tiny\text{+}}D_{9}\text{-}2 (k_{12}{\tiny\text{+}}k_{14}) , D_{2}\text{-}D_{3}\text{-}D_{4}{\tiny\text{+}}D_{9}{\tiny\text{+}}2 k_{12} , \text{-}D_{4}{\tiny\text{+}}D_{9}{\tiny\text{+}}2 k_{12} , 0 , \text{-}D_{4}{\tiny\text{+}}D_{9} \}\\
Q_{25}&\rightarrow&\{0 , 0 , 0 , \text{-}D_{4}{\tiny\text{+}}D_{5} , \text{-}D_{4}{\tiny\text{+}}D_{5} , \text{-}D_{1}\text{-}D_{4}{\tiny\text{+}}D_{5}{\tiny\text{+}}D_{8} , \text{-}D_{4}{\tiny\text{+}}D_{5}{\tiny\text{+}}2 k_{14} , 0 , \text{-}D_{4}{\tiny\text{+}}D_{5}\text{-}2 (k_{12}{\tiny\text{+}}k_{14})\}\\
Q_1&\rightarrow&\{D_1 , 0 , 0 , 0, 0 , 0, 0 , 0 , 0\}\\
Q_6&\rightarrow&\{0 , 0 , 0 , 0, 0 , D_6, 0 , 0 , 0\}\nonumber
\end{array}
\right)~,
\end{align}
where $k_{12}=p_1\cdot p_2$ and $k_{14}=p_1\cdot p_4$.
Finally, we found 64 syzygies. We have verified that the syzygy module is same as the output of Singular with 87 syzygies. We first use one of them which is of first order
\begin{align}
\textbf{C}^{(1)}&=\left(0, 0, 0, 0, 0, \text{-}D_4 {\tiny{\text{+}}} D_5, \text{-}D_1 \text{-} 2 D_4 {\tiny{\text{+}} } 2 D_5 {\tiny{\text{+}}} D_8, 0, 0, D_1 \text{-} D_4 \text{-} D_6, 0\right), \nb\\
\textbf{c}^{(1)}&=\left(0, 0, 2 D_1 {\tiny{\text{+}} } 2 D_4 \text{-} 2 D_5 \text{-} 2 D_8, 2 D_1 {\tiny{\text{+}} }2 D_4 \text{-} 2 D_5 \text{-} 2 D_8, 2 D_4 \text{-} 2 D_5\right).\nb
\end{align}
According to the integrand reduction <cit.>, the naive power counting of renormalizable condition constrains the irreducible integrals as following, $\frac{D_7^{n_7} D_8^{n_8} D_9^{n_9}}{D_1 D_2\cdots D_6}$ with constraints $n_7\leq 4$, $n_8+n_9\leq 2$, and $n_7+n_8+n_9\leq 4$. An IBP relation induce a integral equation among the irreducible integrand
\begin{align}
0=\int \sum_{IJ}\frac{O_{IJ}c_{IJ} }{D_1\cdots D_6}+\sum_{a=1}^6{c_a\over D_1\cdots D_6},
\end{align}
where $c_{IJ}$ and $c_a$ are the components of $\textbf{C}^{(1)}$ and $\textbf{c}^{(1)}$ respectively.
We shall show how IBP relations work by demonstrating the IBP relation generated by the first syzygy which we pick. First, we calculate the action of operator $O_{IJ}$ on vector $C_{IJ}^{(1)}$ and the result is simple: $-d D_1-2 d D_4+2 d D_5+d D_8$. Then we add it with $\sum _{i=1}^6 c_i$, and we have an IBP relation
$$\int \frac{(4-d) D_1+(6-2 d) D_4+(2 d-6) D_5+(d-4) D_8}{D_1 D_2 \ldots D_6 }=0.$$
Similarly, IBP relations in diagrams with less propagators can reduce $\int\frac{D5}{D_1 D_2 \ldots D_6 }$ to
$$-\frac {(3 d - 10) (3 d - 8) } {4 (d - 4)^2 k_{12}^2}\int\frac{1}{D_3 D_4 D_6}.$$
Finally, the IBP relation is:
$$\int\frac{(d-4) D_8}{D_1 D_2 \ldots D_6 }= \int\frac{(4-d)} {D_2 D_3 D_4 D_5 D_6 }+\frac{(6-2 d)}{D_1 D_2 D_3 D_5 D_6}+\frac {(2 d-6)(3 d - 10) (3 d - 8) } {4 (d - 4)^2 k_{12}^2D_3 D_4 D_6}.$$
This result is consistent with the output of FIRE5 <cit.>. Our program spent 42 seconds while FIRE5 spent 36 seconds in getting this result.
To reduce the integral further, we need to use more IBP relations. For the general $r$-order syzygies, we use $m_{i_1\cdots i_r}$ denotes the product of $\{D_1, D_2,\cdots, D_9, k_{12}, k_{14}\}$, where the subindex $i\in \{1\cdots 9\}$ represent a $D_i$ factor in the product, $i\in {a, b}$ denotes a factor $k_{12}$ and $k_{14}$ in the product respectively. For example $m_{1a}=D_1 k_{12}, m_{12}=D_1 D_2$.We choose the following syzygies
\begin{align}
\textbf{C}^{(2)}=&(-m_{1b}+m_{2a}+m_{3b}-m_{8a},0,m_{1b}+m_{8a},m_{1b},m_{2a},0,\nb\\&-2 m_{4a}-2 m_{4b}+m_{5a}+m_{7a}+m_{7b}+m_{9b}+2 m_{ab},-2 m_{4a}-m_{4b}+m_{5a},-m_{4b},-m_{7a})\nb\\
\textbf{c}^{(2)}=&(m_{1b}-m_{2a}-m_{3b}+m_{8a}-2 m_{ab},m_{1b}-m_{2a}-m_{3b}+m_{8a}+2 m_{ab},\nb\\ &m_{1b}-m_{2a}-m_{3b}+m_{8a}+2 m_{ab}, 2 m_{4a}+2 m_{4b}-m_{5a}-m_{7a}-m_{7b}-m_{9b}-4 m_{ab},\nb\\ &2 m_{4a}+2 m_{4b}-m_{5a}-m_{7a}-m_{7b}-m_{9b}-4 m_{ab},\nb\\ &m_{1b}-m_{2a}-m_{3b}+2 m_{4a}+2 m_{4b}-m_{5a}-m_{7a}-m_{7b}+m_{8a}-m_{9b}-2 m_{ab}),\nb
\end{align}
\begin{align}
\textbf{C}^{(3)}=&(-2 m_{14}-m_{17}+4 m_{24}-m_{27}-m_{29}-2 m_{34}+m_{37}+4 m_{4a}+m_{78}+m_{89},0,\nb\\ &2 m_{14}+m_{17}-2 m_{24}+2 m_{34}-4 m_{4a}-m_{78}-m_{89},m_{17}-2 m_{24},-m_{27}-m_{29},0,\nb\\ &2 m_{47}-m_{57}-m_{59}+2 m_{7a},m_{47}-m_{57}-m_{59},m_{47},m_{77}+m_{79})\nb\\
\textbf{c}^{(3)}=&(2 m_{14}+m_{17}-4 m_{24}+m_{27}+m_{29}+2 m_{34}-m_{37}-4 m_{4a}-m_{78}-2 m_{7a}-m_{89},\nb\\ &2 m_{14}+m_{17}-4 m_{24}+m_{27}+m_{29}+2 m_{34}-m_{37}-8 m_{4a}-m_{78}-2 m_{7b}-m_{89}-2 m_{9b},\nb\\ &2 m_{14}+m_{17}-4 m_{24}+m_{27}+m_{29}+2 m_{34}-m_{37}-4 m_{4a}-m_{78}+2 m_{7a}-m_{89},\nb\\ &-2 m_{47}+m_{57}+m_{59}-4 m_{7a},-2 m_{47}+m_{57}+m_{59}-2 m_{7a}+2 m_{7b}+2 m_{9b},\nb\\ &2 m_{14}+m_{17}-4 m_{24}+m_{27}+m_{29}+2 m_{34}-m_{37}-2 m_{47}-4 m_{4a}+m_{57}+m_{59}\nb\\ &-m_{78}-2 m_{7a}-m_{89}).\nb
\end{align}
\begin{align}
\textbf{C}^{(4)}=&(-4 m_{119}-2 m_{124}-2 m_{127}+4 m_{129}+4 m_{134}+2 m_{137}-4 m_{139}+2 m_{145}+2 m_{148}\nb\\ &-8 m_{14a}-4 m_{17a}+2 m_{189}+8 m_{19a}+8 m_{224}-2 m_{227}-2 m_{229}-12 m_{234}+4 m_{237}\nb\\ &+2 m_{239}-3 m_{245}-3 m_{248}+20 m_{24a}+2 m_{278}-4 m_{27a}+2 m_{289}+4 m_{334}-2 m_{337}\nb\\ &+2 m_{345}-16 m_{34a}-2 m_{378}+4 m_{37a}-m_{458}-4 m_{45a}-m_{488}+16 m_{4aa}+4 m_{78a},\nb\\ &0,4 m_{119}+2 m_{124}+2 m_{127}-2 m_{129}-4 m_{134}-2 m_{137}+4 m_{139}-2 m_{145}-2 m_{148}\nb\\ &+8 m_{14a}+4 m_{17a}-2 m_{189}-8 m_{19a}-4 m_{224}+8 m_{234}+2 m_{245}+2 m_{248}-16 m_{24a}\nb\\ &-2 m_{278}-2 m_{289}-4 m_{334}-2 m_{345}+16 m_{34a}+2 m_{378}+m_{458}+4 m_{45a}+m_{488}\nb\\ &-16 m_{4aa}-4 m_{78a},-2 m_{124}+2 m_{127}-2 m_{129}-2 m_{137}+4 m_{17a}-4 m_{224}+4 m_{234}\nb\\ &+2 m_{245}+2 m_{248}-8 m_{24a},-2 m_{129}-2 m_{227}-2 m_{229}+2 m_{234}+2 m_{237}+m_{245}\nb\\ &+m_{248}-4 m_{24a}-4 m_{27a},0,-2 m_{159}+2 m_{179}-2 m_{244}+4 m_{247}+4 m_{249}-4 m_{24a}\nb\\ &-2 m_{257}-2 m_{259}-2 m_{279}+4 m_{27a}-2 m_{299}+4 m_{29a}+2 m_{345}-4 m_{347}+2 m_{357}\nb\\ &+2 m_{379}-4 m_{37a}+m_{455}-m_{457}+m_{458}-4 m_{45a}-m_{478}+8 m_{47a}-4 m_{57a}-4 m_{79a}+8 m_{7aa},\nb\\ &-2 m_{159}-2 m_{244}+2 m_{247}+2 m_{249}-2 m_{257}-2 m_{259}+2 m_{345}-2 m_{347}+2 m_{357}\nb\\ &+m_{455}+m_{458}-4 m_{45a}+4 m_{47a}-4 m_{57a},-2 m_{244}+2 m_{247}+2 m_{249}-2 m_{347}+4 m_{47a},\nb\\ &2 m_{179}+2 m_{244}-2 m_{247}-2 m_{249}+2 m_{277}+2 m_{279}-2 m_{377}-m_{457}-m_{478}+4 m_{77a}),\nb\\
\textbf{c}^{(4)}=&(4 m_{119}+2 m_{124}+2 m_{127}-4 m_{129}-4 m_{134}-2 m_{137}+4 m_{139}-2 m_{145}-2 m_{148}\nb\\ &+8 m_{14a}+4 m_{17a}-2 m_{189}-8 m_{19a}-8 m_{224}+2 m_{227}+2 m_{229}+12 m_{234}-4 m_{237}\nb\\ &-2 m_{239}+3 m_{245}+3 m_{248}-16 m_{24a}-2 m_{278}-2 m_{289}-4 m_{29a}-4 m_{334}+2 m_{337}\nb\\ &-2 m_{345}+16 m_{34a}+2 m_{378}+m_{458}+4 m_{45a}+m_{488}-16 m_{4aa}-4 m_{78a}-8 m_{7aa},\nb\\ &4 m_{119}+2 m_{124}+2 m_{127}-4 m_{129}-4 m_{134}-2 m_{137}+4 m_{139}-2 m_{145}-2 m_{148}\nb\\ &+8 m_{14a}+4 m_{17a}-2 m_{189}-16 m_{19a}-4 m_{19b}-8 m_{224}+2 m_{227}+2 m_{229}+12 m_{234}\nb\\ &-4 m_{237}-2 m_{239}+3 m_{245}+3 m_{248}-28 m_{24a}-2 m_{278}+4 m_{27a}-4 m_{27b}-2 m_{289}\nb\\ &-4 m_{29b}-4 m_{334}+2 m_{337}-2 m_{345}+24 m_{34a}+4 m_{34b}+2 m_{378}-4 m_{37a}+4 m_{37b}\nb\\ &+m_{458}+8 m_{45a}+2 m_{45b}+m_{488}+4 m_{48a}+2 m_{48b}-32 m_{4aa}-8 m_{4ab}-4 m_{78a}-8 m_{7ab},\nb\\ &4 m_{119}+2 m_{124}+2 m_{127}-4 m_{129}-4 m_{134}-2 m_{137}+4 m_{139}-2 m_{145}-2 m_{148}\nb\\ &+8 m_{14a}+4 m_{17a}-2 m_{189}-8 m_{19a}-8 m_{224}+2 m_{227}+2 m_{229}+12 m_{234}-4 m_{237}\nb\\ &-2 m_{239}+3 m_{245}+3 m_{248}-24 m_{24a}-2 m_{278}+8 m_{27a}-2 m_{289}+4 m_{29a}-4 m_{334}+2 m_{337}\nb\\ &-2 m_{345}+16 m_{34a}+2 m_{378}-8 m_{37a}+m_{458}+4 m_{45a}+m_{488}-16 m_{4aa}-4 m_{78a}+8 m_{7aa},\nb\\ &2 m_{159}-2 m_{179}+2 m_{244}-2 m_{245}-4 m_{247}-4 m_{249}+8 m_{24a}+4 m_{257}+4 m_{259}+2 m_{279}\nb\\ &-8 m_{27a}+2 m_{299}-8 m_{29a}-2 m_{345}+4 m_{347}-4 m_{357}-2 m_{379}+8 m_{37a}-m_{455}+m_{457}\nb\\ &-m_{458}+4 m_{45a}+m_{478}-8 m_{47a}+8 m_{57a}+4 m_{79a}-16 m_{7aa},2 m_{159}-2 m_{179}+4 m_{19b}\nb\\ &-2 m_{247}-2 m_{249}+4 m_{24a}+2 m_{257}+2 m_{259}+2 m_{279}-4 m_{27a}+4 m_{27b}+2 m_{299}-4 m_{29a}\nb\\ &+4 m_{29b}-2 m_{345}+2 m_{347}-4 m_{34b}-2 m_{357}-2 m_{379}+4 m_{37a}-4 m_{37b}-m_{455}+m_{457}-m_{458}\nb\\ &+4 m_{45a}-2 m_{45b}+m_{478}-4 m_{47a}-2 m_{48b}+8 m_{4ab}+4 m_{57a}+4 m_{79a}-8 m_{7aa}+8 m_{7ab},\nb\\ &4 m_{119}+2 m_{124}+2 m_{127}-4 m_{129}-4 m_{134}-2 m_{137}+4 m_{139}-2 m_{145}-2 m_{148}+8 m_{14a}+2 m_{159}\nb\\ &-2 m_{179}+4 m_{17a}-2 m_{189}-8 m_{19a}-8 m_{224}+2 m_{227}+2 m_{229}+12 m_{234}-4 m_{237}-2 m_{239}\nb\\ &+2 m_{244}+3 m_{245}-4 m_{247}+3 m_{248}-4 m_{249}-16 m_{24a}+2 m_{257}+2 m_{259}-2 m_{278}+2 m_{279}\nb\\ &-2 m_{289}+2 m_{299}-4 m_{29a}-4 m_{334}+2 m_{337}-4 m_{345}+4 m_{347}+16 m_{34a}-2 m_{357}+2 m_{378}\nb\\ &-2 m_{379}-m_{455}+m_{457}+8 m_{45a}+m_{478}-8 m_{47a}+m_{488}-16 m_{4aa}+4 m_{57a}-4 m_{78a}\nb\\ &+4 m_{79a}-8 m_{7aa}).\nb
\end{align}
\begin{align}
\textbf{C}^{(5)}=&(22 m_{11}-34 m_{12}+14 m_{13}+32 m_{14}-6 m_{15}+46 m_{17}-11 m_{18}-20 m_{19}-76 m_{1a}+8 m_{1b}\nb\\ &-84 m_{24}+24 m_{25}+6 m_{27}+16 m_{28}+6 m_{29}-50 m_{2a}+72 m_{34}-18 m_{35}-26 m_{37}\nb\\ &-7 m_{38}-88 m_{3b}-20 m_{48}-104 m_{4a}+24 m_{5a}+40 m_{6a}+80 m_{6b}-26 m_{78}-40 m_{7a}\nb\\ &-40 m_{7b}+14 m_{89}+66 m_{8a}-40 m_{9b},-40 m_{1b}-40 m_{3b}-40 m_{8a}+80 m_{ab},\nb\\ & -22 m_{11}+17 m_{12}-18 m_{13}-12 m_{14}+6 m_{15}-46 m_{17}+11 m_{18}+36 m_{1a}-48 m_{1b}+52 m_{24}\nb\\ &-12 m_{25}-8 m_{28}-52 m_{34}+12 m_{35}+9 m_{38}+104 m_{4a}+40 m_{4b}-24 m_{5a}-40 m_{6b}+26 m_{78}\nb\\ &+6 m_{89}-66 m_{8a},-4 m_{11}+17 m_{12}+40 m_{14}-6 m_{15}-46 m_{17}+2 m_{18}-48 m_{1b}+52 m_{24}\nb\\ &-12 m_{25}-8 m_{28}-20 m_{48}+40 m_{4b}-40 m_{6b}+20 m_{78},m_{12}+20 m_{14}-20 m_{19}-40 m_{1a}\nb\\ &+m_{23}+20 m_{24}+6 m_{27}+6 m_{29}-50 m_{2a}-20 m_{34}+20 m_{37}+40 m_{6a}-40 m_{7a}, \nb\\ &16 m_{7a}+16 m_{7b}+16 m_{9b},4 m_{14}+5 m_{15}-17 m_{17}+8 m_{19}+8 m_{1a}+16 m_{1b}-18 m_{24}\nb\\ &+9 m_{27}+9 m_{29}-18 m_{2a}+12 m_{34}-3 m_{35}-9 m_{37}-16 m_{3b}-80 m_{44}+32 m_{45}+8 m_{47}+4 m_{48}\nb\\ &+80 m_{49}+136 m_{4a}+64 m_{4b}+32 m_{57}-32 m_{59}-94 m_{5a}-20 m_{77}+6 m_{78}-20 m_{79}-90 m_{7a}\nb\\ &-56 m_{7b}-10 m_{89}+4 m_{8a}-56 m_{9b}-144 m_{ab},\nb\\ &4 m_{14}+5 m_{15}+8 m_{19}+16 m_{1a}+16 m_{1b}-9 m_{24}+8 m_{34}+m_{35}-40 m_{44}+6 m_{45}\nb\\ &-8 m_{46}-6 m_{47}+2 m_{48}+40 m_{49}+64 m_{4a}-8 m_{4b}+8 m_{56}+26 m_{57}-6 m_{59}-50 m_{5a}-16 m_{6a}\nb\\ &-16 m_{6b}-8 m_{89},-4 m_{14}+4 m_{15}-8 m_{17}+16 m_{1b}-9 m_{24}-14 m_{45}+8 m_{46}-46 m_{47}+2 m_{48}\nb\\ &-8 m_{4b}-8 m_{56}+32 m_{57}-16 m_{6b}+8 m_{78},\nb\\ &4 m_{14}-5 m_{17}-4 m_{19}+4 m_{34}-m_{37}-40 m_{44}+20 m_{47}+40 m_{49}+72 m_{4a}-8 m_{67}+8 m_{69}\nb\\ &-26 m_{77}-26 m_{79}+50 m_{7a}),\nb\\
\textbf{c}^{(5)}=&(-22 m_{11}+34 m_{12}-14 m_{13}-32 m_{14}+6 m_{15}-46 m_{17}+11 m_{18}+20 m_{19}+84 m_{1a}-8 m_{1b}\nb\\ &+84 m_{24}-24 m_{25}-6 m_{27}-16 m_{28}-6 m_{29}+52 m_{2a}-72 m_{34}+18 m_{35}+26 m_{37}+7 m_{38}\nb\\ &+88 m_{3b}+20 m_{48}+24 m_{4a}-80 m_{4b}-12 m_{5a}-40 m_{6a}-80 m_{6b}+26 m_{78}+132 m_{7a}+80 m_{7b}\nb\\ &-14 m_{89}-70 m_{8a}+80 m_{9b}+176 m_{ab},-22 m_{11}+34 m_{12}-14 m_{13}-32 m_{14}+6 m_{15}-46 m_{17}\nb\\ &+11 m_{18}+20 m_{19}+112 m_{1a}-6 m_{1b}+84 m_{24}-24 m_{25}-6 m_{27}-16 m_{28}-6 m_{29}+50 m_{2a}\nb\\ &-72 m_{34}+18 m_{35}+26 m_{37}+7 m_{38}+90 m_{3b}+20 m_{48}+208 m_{4a}+40 m_{4b}-48 m_{5a}\nb\\ &-40 m_{6a}-80 m_{6b}+26 m_{78}+40 m_{7a}+52 m_{7b}-14 m_{89}-82 m_{8a}+52 m_{9b}-100 m_{ab},\nb\\ &-22 m_{11}+34 m_{12}-14 m_{13}-32 m_{14}+6 m_{15}-46 m_{17}+11 m_{18}+20 m_{19}+68 m_{1a}-8 m_{1b}\nb\\ &+84 m_{24}-24 m_{25}-6 m_{27}-16 m_{28}-6 m_{29}+48 m_{2a}-72 m_{34}+18 m_{35}+26 m_{37}+7 m_{38}\nb\\ &+88 m_{3b}+20 m_{48}+184 m_{4a}+80 m_{4b}-36 m_{5a}-40 m_{6a}-80 m_{6b}+26 m_{78}-52 m_{7a}-14 m_{89}\nb\\ &-62 m_{8a}-176 m_{ab},-4 m_{14}-5 m_{15}+17 m_{17}-8 m_{19}+18 m_{24}-9 m_{27}-9 m_{29}+36 m_{2a}\nb\\ &-12 m_{34}+3 m_{35}+9 m_{37}+32 m_{3b}+80 m_{44}-32 m_{45}-8 m_{47}-4 m_{48}-80 m_{49}-136 m_{4a}\nb\\ &-144 m_{4b}-32 m_{57}+32 m_{59}+66 m_{5a}-16 m_{6a}-32 m_{6b}+20 m_{77}-6 m_{78}+20 m_{79}+182 m_{7a}\nb\\ &+136 m_{7b}+10 m_{89}-8 m_{8a}+136 m_{9b}+288 m_{ab},-4 m_{14}-5 m_{15}+17 m_{17}-8 m_{19}-18 m_{1b}\nb\\ &+18 m_{24}-9 m_{27}-9 m_{29}+18 m_{2a}-12 m_{34}+3 m_{35}+9 m_{37}+14 m_{3b}+80 m_{44}-32 m_{45}-8 m_{47}\nb\\ &-4 m_{48}-80 m_{49}-192 m_{4a}-104 m_{4b}-32 m_{57}+32 m_{59}+94 m_{5a}-16 m_{6a}-32 m_{6b}+20 m_{77}\nb\\ &-6 m_{78}+20 m_{79}+154 m_{7a}+68 m_{7b}+10 m_{89}-4 m_{8a}+68 m_{9b}+244 m_{ab},-22 m_{11}+34 m_{12}\nb\\ &-14 m_{13}-28 m_{14}-7 m_{15}-37 m_{17}+11 m_{18}+20 m_{19}+84 m_{1a}-8 m_{1b}+86 m_{24}-8 m_{25}\nb\\ &-15 m_{27}-16 m_{28}-15 m_{29}+52 m_{2a}-76 m_{34}+13 m_{35}+35 m_{37}+7 m_{38}+88 m_{3b}+80 m_{44}\nb\\ &-32 m_{45}-8 m_{47}+16 m_{48}-80 m_{49}+16 m_{4a}-96 m_{4b}-32 m_{57}+32 m_{59}+46 m_{5a}-40 m_{6a}\nb\\ &-80 m_{6b}+20 m_{77}+28 m_{78}+20 m_{79}+130 m_{7a}+136 m_{7b}-12 m_{89}-70 m_{8a}+136 m_{9b}+176 m_{ab})
\end{align}
For convenience, we let $k_{12}=-1/2, k_{14}=-1/2, k_{24}=1$.
According to these chosen syzygies, we can obtain the IBP relations,
\begin{align*}
&\int\frac{4 D_7 (-4 + d)}{D_1 D_2 \ldots D_6 }=\int\frac{1}{D_1 D_2 \ldots D_6 }\times\\
&\left(2 (d-3) m_{12}+(7 d-36) m_{14}+\frac{1}{2} (d-6) m_{15} \right.+(d-3) m_{16}+(d-7) m_{17}-(d-2) m_{19}-(d-3) m_{23} \\
&+(65-16 d) m_{24}+2 (d-3) m_{26} +4 (d-4) m_{27} +4 (d-4) m_{29}+(8 d-30) m_{34}+\frac{1}{2} (d+2) m_{35} \\
&+(d-3) m_{36}+(19-5 d) m_{37}-(d-3) m_{38}+9 (d-2) m_{45} -14 (d-2) m_{47}-4 (d-2) m_{49}\\
&+\frac{7}{2} (d-2) m_{57} +\frac{3}{2} (d-2) m_{59}-4 (d-3) m_{68}+2 (d-2) m_{69} -(d-7) m_{78}-2 (d-5) m_{89}-(d-5) m_1\\
&+(8 d-31) m_4 \left.+\frac{1}{2} (9 d-35) m_5+(d-5) m_6-m_{25}+5 m_{48} \right)\\
&\int\frac{D_9 (-3 + d)}{D_1 D_2 \ldots D_6 }=\int\frac{1}{D_1 D_2 \ldots D_6 }\times\\
&\left(-(d-4) m_1+(d-4) m_2+13 (d-4) m_4+2 (2 d-9) m_5+\frac{1}{2} (7 d-33) m_7-2 (d-4) m_8+2 (d-3) m_{11}\right.\\
&-3 (d-3) m_{12}+2 (d-3) m_{13}+4 (3 d-13) m_{14}+\left(d-\frac{13}{2}\right) m_{15}+\frac{1}{2} (7 d-33) m_{17}-(d-3) m_{18}\\
&-(d-3) m_{19}+(98-25 d) m_{24}-\frac{1}{2} (d-4) m_5 m_{24}+4 (d-3) m_{26}+\frac{1}{2} (13 d-51) m_{27}+2 (d-3) m_{28}\\
&+\frac{1}{2} (13 d-51) m_{29}+2 (6 d-23) m_{34}-\frac{1}{2} (13 d-51) m_{37}-2 (d-3) m_{38}+(d-1) m_5 m_{44}\\
&+\left(9 d-\frac{35}{2}\right) m_{45}-\frac{1}{2} (d-1) m_7 m_{45}+\frac{1}{2} (d-4) m_8 m_{45}-14 (d-2) m_{47}\\
&+d m_{48}-4 (d-2) m_{49}-\frac{1}{2} (d-1) m_4 m_{55}+\frac{3}{2} (d-2) m_{57}+\frac{7}{2} (d-2) m_{59}\\
&\left.-4 (d-3) m_{68}+2 (d-2) m_{69}-\frac{3}{2} (3 d-13) m_{78}-\frac{1}{2} (11 d-45) m_{89}+m_{25}+\frac{5 m_{35}}{2}\right)\\
& \int\frac{1/2 (4 - d)}{D_1 D_2 \ldots D_6 }=\int\frac{1}{D_1 D_2 \ldots D_6 }\times\\
&\left(-\frac{1}{2} (2 d+1) m_1+\frac{1}{2} (d-4) m_2+\frac{3}{2} (d-4) m_3+\left(12-\frac{5 d}{2}\right) m_4 \right.\\
&+\left(\frac{29}{2}-4 d\right) m_5+\left(\frac{41}{2}-6 d\right) m_7-\frac{5}{4} (d-4) m_8-\frac{3}{2} (d-3) m_9\\
&+\frac{3}{2} (d-3) m_{11}-3 (d-3) m_{12}+\frac{1}{2} (d-3) m_{13}-\frac{3}{2} (3 d-17) m_{14}\\
&+\frac{1}{2} (d-1) m_{15}+\left(7-\frac{3 d}{2}\right) m_{17}-\frac{3}{4} (d-3) m_{18}+3 (d-3) m_{19}+\left(9 d-\frac{75}{2}\right) m_{24}\\
&+\left(\frac{17}{2}-2 d\right) m_{27}+2 (d-3) m_{28}+\left(\frac{17}{2}-2 d\right) m_{29}-\frac{1}{2} (7 d-27) m_{34}\\
&-\frac{3}{2} (d-2) m_{35}+\left(2 d-\frac{17}{2}\right) m_{37}-\frac{1}{4} (d-3) m_{38}-7 (d-2) m_{45}+10 (d-2) m_{47}\\
&+\frac{1}{2} (d-12) m_{48}+4 (d-2) m_{49}-2 (d-2) m_{57}-(d-2) m_{59}-2 (d-2) m_{69}+\frac{1}{4} (3 d-19) m_{78}\\
&\left.-\frac{1}{4} (3 d+1) m_{89}+\frac{m_6}{2}+\frac{m_{25}}{2}\right).
\end{align*}
According to former IBP relations and IBP relations of diagrams with less propagators, the result can be further reduced as following:
\begin{align*}
\int\frac{D_7}{D_1 D_2 \ldots D_6 }&=\frac{(d-4 )}{(d-3 ) } I_5^b-2I_5^a+ \frac{6d-20 }{d-4} I_4^a - \frac{(9d-30)(3 d-8 ) }{2 (d-4 )^2 }I_3^b + \frac{(6d-20 ) (3 d-8)}{(d-4)^2}I_3^a\\
\int\frac{D_9}{D_1 D_2 \ldots D_6 }&=\frac{3 (3 d-10) (3 d-8)}{(d-4)^2 }I_3^b-\frac{5 (3 d-10) (3 d-8) }{(d-4)^2}I_3^a+\frac{2 (3 d-10) (3 d-8) }{(d-4)^2}I_3^c\\
&-\frac{6 (d-4)}{(d-3)}I_5^b-\frac{2 (3 d-10) }{(d-4)}I_4^a+4I_5^a\\
\int\frac{1}{D_1 D_2 \ldots D_6 }&=-\frac{6 (d-3)}{(d-4) }I_5^a-6I_5^b+\frac{(9 d-30) (d-3)}{(d-4)^2 }I_4^a\\
&-\frac{3 (3 d-10) (3 d-8) (d-3)}{(d-4)^3 }I_3^a+\frac{3 (3 d-10) (3 d-8) (d-3)}{(d-4)^3 }I_3^c,
\end{align*}
\begin{eqnarray*}
I_5^a&=&\int {1\over D_1 D_2 D_3 D_5 D_6}, I_5^b=\int {1\over D_2 D_3 D_4 D_5 D_6},
I_4^a=\int {1\over D_1 D_3 D_5 D_6}, \\
I_3^a&=&\int {1\over D_3 D_4 D_6},~~~~~~ ~~I_3^b=\int {1\over D_3 D_5 D_6}, ~~~~~~~I_3^c=\int {1\over D_2 D_5 D_6}.
\end{eqnarray*}
This results are same as the output from FIRE5. Other higher order integrands can also be reduced under these IBP relations. Our program spent 125 seconds to reduce each integral into the irreducible basis while FIRE5 spent 76 seconds.
§ CONCLUSION AND OUTLOOK
In this paper, we have shown a new effective method on calculating syzygies. With a significant feedback in each step, our method can safely protect the irreducibility of our basis, which can not be given by other methods in current study. Evidence is given in the paper to show that this algorithm is effective for general ideals, and hints are obtained on the generalizations to modules (although currently we cannot guarantee the full rigorous irreducibility of syzygies for modules). Through this effective method, the idea that mathematical structures of scattering amplitudes are featured with different syzygies can come to practical use via some simple steps of programming.
As a result, physics can be read off through this useful method. As an application, this paper mainly describes s specific context, namely, the IBP relations of a specific two-loop diagram in a Yang-Mills field theory. However, via the general illustrations on the method given before, one can easily simplify all possible IBP relations for a general diagram. Thus, we gain the general and systematic approach on how to simplify the IBP relations and determine irreducible integrals.
This method is fundamental enough that can be generalized to a wide range of applications in the study of scattering amplitudes. First, one can use this method to investigate some other theories and investigate the irreducible integrals of diagrams. Second, tree-level amplitudes or the integrands of loop-level amplitudes can be taken as generators for an ideal, so all possible irreducible relations beyond degree-zero (KK relations) and degree-one (BCJ relations) may be found. One can also use this method to explore simplifications of Grassmannian integral form, and construction of loop amplitudes from unitarity cuts. Furthermore, this algorithm and its ideas, can be used in all possible areas in mathematics and physics that need to simplify complicated algebraic relations (namely, syzygies) of several polynomial (rational) functions. We leave them to future works.
§ ACKNOWLEDGMENTS
GC, RX and HZ thank K. Larsen and Y. Zhang for useful comments and kind suggestions. Useful discussions with Y. E. Cheung, Y. Wang and S. Zhou are gratefully acknowledged. GC, RX and HZ have been supported by the Fundamental Research Funds for the Central Universities under contract 020414340080, NSF of China Grant under contract 11405084, the Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China (No.Y5KF171CJ1) and the Jiangsu Ministry of Science and Technology under contract BK20131264. JL and YZ will thank the department of physics in Nanjing University for hosting. We also thank Y. Gao, T. Han for hospitality and Key Laboratory of Theoretical Physics for hosting. We thank Tianheng Wang for lots of help in revising our manuscript.
R. Britto, F. Cachazo, B. Feng and E. Witten,
Phys. Rev. Lett. 94, 181602 (2005)
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov and J. Trnka,
arXiv:1212.5605 [hep-th].
Bayer, Dave, and David Mumford. What can be computed in algebraic geometry. Computational algebraic geometry and commutative algebra. Vol. 34. Symposia Matematica, 1993.
W. Adams and P. Loustaunau, An Introduction to Gröbner Bases. American Mathematical Society, Graduate Studies in Mathematics, Volume 3, 1994.
Postnikov, Alexander, and Boris Shapiro. Trees, parking functions, syzygies, and deformations of monomial ideals. Transactions of the American Mathematical Society 356.8 (2004): 3109-3142.
Buchberger B. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Innsbruck, 1965
Buchberger B. Gröbner Bases : an Algorithmic Method in Polynomial Ideal Theory. In Recent trends in multidimensional system theory, Reidel, Ed. Bose, 1985.
J. Gluza, K. Kajda and D. A. Kosower,
Phys. Rev. D 83, 045012 (2011)
[arXiv:1009.0472 [hep-th]].
R. M. Schabinger,
JHEP 1201, 077 (2012)
[arXiv:1111.4220 [hep-ph]].
Y. Zhang,
arXiv:1408.4004 [hep-th].
A. G. Grozin,
Int. J. Mod. Phys. A 26, 2807 (2011)
[arXiv:1104.3993 [hep-ph]].
D. A. Kosower and K. J. Larsen,
Phys. Rev. D 85, 045017 (2012)
[arXiv:1108.1180 [hep-th]].
H. Ita,
arXiv:1510.05626 [hep-th].
K. J. Larsen and Y. Zhang,
arXiv:1511.01071 [hep-th].
B. Chen, G. Chen, Y. K. E. Cheung, Y. Li, R. Xie and Y. Xin,
arXiv:1411.3889 [hep-th].
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. Postnikov and J. Trnka,
JHEP 1506, 179 (2015)
[arXiv:1412.8475 [hep-th]].
S. Franco, D. Galloni, B. Penante and C. Wen,
JHEP 1506, 199 (2015)
[arXiv:1502.02034 [hep-th]].
B. Chen, G. Chen, Y. K. E. Cheung, R. Xie and Y. Xin,
arXiv:1507.03214 [hep-th].
P. Benincasa,
arXiv:1510.03642 [hep-th].
R. Frassek, D. Meidinger, D. Nandan and M. Wilhelm,
arXiv:1506.08192 [hep-th].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka,
JHEP 1506, 202 (2015)
[arXiv:1412.8584 [hep-th]].
Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower,
Nucl. Phys. B 425, 217 (1994)
P. Du, G. Chen and Y. K. E. Cheung,
JHEP 1409, 115 (2014)
[arXiv:1401.6610 [hep-th]].
R. Kleiss and H. Kuijf,
Nucl. Phys. B 312, 616 (1989).
Z. Bern, J. J. M. Carrasco and H. Johansson,
Phys. Rev. D 78, 085011 (2008)
[arXiv:0805.3993 [hep-ph]].
Sun, Yao. Signature-Based Gröbner Basis Algorithms—Extended MMM Algorithm for computing Gröbner bases." arXiv:1308.2371 (2013).
Faugere, Jean-Charles. A new efficient algorithm for computing Gröbner basis without reduction to zero (F5)(15/6/2004).
Eder, Christian, and John Perry. F5C: a variant of FaugereˇŻs F5 algorithm with reduced Gröbner bases. Journal of Symbolic Computation 45.12 (2010): 1442-1458.
Roune, Bjarke Hammersholt, and Michael Stillman. Practical Gröbner basis computation. Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. ACM, 2012.
Gräbe, Hans-Gert. Minimal primary decomposition and factorized Gröbner basis. Applicable Algebra in Engineering, Communication and Computing 8.4 (1997): 265-278.
Hodges, Tim. Computing Syzygies of Homogeneous Polynomials using Linear Algebra. Diss. Colorado State University, 2014.
W. Decker et al,
Singular, A computer algebra system for polynomial computations (http://www.singular.uni-kl.de, 2015).
Möller, H. Michael, Teo Mora, and Carlo Traverso. Gröbner bases computation using syzygies. Papers from the international symposium on Symbolic and algebraic computation. ACM, 1992.
Eisenbud, David. The geometry of syzygies. Graduate Texts in Mathematics (2003).
S. Caron-Huot and K. J. Larsen,
JHEP 1210, 026 (2012)
[arXiv:1205.0801 [hep-ph]].
P. A. Baikov, Nucl. Instrum. Meth. A 389, (1997) 347
[arXiv: hep-ph/9611449].
P. A. Baikov, Phys. Lett. B 389, (1996) 347
[arXiv: hep-ph/9603267].
A. G. Grozin, [arXiv: 1104.3993 [hep-ph]].
Y. Zhang,
JHEP 1209, 042 (2012)
[arXiv:1205.5707 [hep-ph]].
G. Ossola, C. G. Papadopoulos and R. Pittau,
JHEP 0803, 042 (2008)
[arXiv:0711.3596 [hep-ph]].
R. K. Ellis, W. T. Giele and Z. Kunszt,
JHEP 0803, 003 (2008)
[arXiv:0708.2398 [hep-ph]].
P. Mastrolia, G. Ossola, T. Reiter and F. Tramontano,
JHEP 1008, 080 (2010)
[arXiv:1006.0710 [hep-ph]].
S. Badger, B. Biedermann and P. Uwer,
Comput. Phys. Commun. 182, 1674 (2011)
[arXiv:1011.2900 [hep-ph]].
G. Ossola, C. G. Papadopoulos and R. Pittau,
Nucl. Phys. B 763, 147 (2007)
V. Hirschi, R. Frederix, S. Frixione, M. V. Garzelli, F. Maltoni and R. Pittau,
JHEP 1105, 044 (2011)
[arXiv:1103.0621 [hep-ph]].
G. Bevilacqua, M. Czakon, M. V. Garzelli, A. van Hameren, A. Kardos, C. G. Papadopoulos, R. Pittau and M. Worek,
Comput. Phys. Commun. 184, 986 (2013)
[arXiv:1110.1499 [hep-ph]].
W. T. Giele, Z. Kunszt and K. Melnikov,
JHEP 0804, 049 (2008)
[arXiv:0801.2237 [hep-ph]].
R. K. Ellis, W. T. Giele, Z. Kunszt and K. Melnikov,
Nucl. Phys. B 822, 270 (2009)
[arXiv:0806.3467 [hep-ph]].
P. Mastrolia, E. Mirabella and T. Peraro,
JHEP 1206, 095 (2012)
[JHEP 1211, 128 (2012)]
[arXiv:1203.0291 [hep-ph]].
A. V. Smirnov,
Comput. Phys. Commun. 189, 182 (2015)
[arXiv:1408.2372 [hep-ph]].
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1511.00639
|
We explore some integrals associated with the Riesz function and establish relations to other functions from number theory that have appeared in the literature. We also comment on properties of these functions.
2010 Mathematics Subject Classification 11M26, 11M06.
§ INTRODUCTION
A now well-known criterion for the Riemann hypothesis was offered by Riesz [10] (see also [12, pg.382] and [14]), who stated that, a necessary and sufficient condition for the Riemann hypothesis is
\begin{equation}x\sum_{n\ge1}\frac{\mu(n)}{n^2}e^{-x/n^2}=O(x^{\frac{1}{4}+\epsilon}).\end{equation}
Throughout this paper $\mu(n)$ will denote the M$\ddot{o}$bius function [12], and $\dot{R}(x)$ will denote the Riesz series on the left side of (1.1).
In [2, 3] Bartz gave an explicit formula for Merten's function [12, pg.372, Theorem 14.27] $\sum_{n\le x}\mu(n)$ without the assumption of the Riemann hypothesis, and subsequently described the analytic character of two functions defined when there are no multiple zeros (non-trivial)
of the Riemann zeta function. One of these functions is defined by
$$m(z)=\lim_{n\rightarrow\infty}\sum_{0<\Im\rho<T_n}\frac{e^{\rho z}}{\zeta'(\rho)}.$$
Bartz further offers that [2, Theorem 2], this function can be continued analytically to a meromorphic function on the whole
complex plane, and satisfies the functional equation
(Here the bar designates the complex conjugate.) In the next section we will offer a criteria for the Riemann hypothesis for a
Laplace transform involving
§ MAIN INTEGRALS
An explicit formula (which is known [11]) for the Riesz function that will become central in this section is given by $x>0$
\begin{equation}\dot{R}(x)=\sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}-\sum_{n\ge1}\frac{n!x^{-n}}{\zeta'(-2n)}.\end{equation}
To see this formula, one may apply the Residue theorem (using a positively oriented circle of radius $\frac{1}{2}+M$ centered at the origin) to
after noting simple poles at the complex zeros of the Riemann zeta function, $\rho=\frac{1}{2}+i\gamma,$ $\gamma\in\mathbb{R},$ the trivial zeros at $z=-2j,$ $j\in\mathbb{N},$ and the
zeros of the gamma function at $z=2i+2,$ $i\in\mathbb{N}_0.$ Further, using the formula [1]
for positive integers $n,$
we may write the series on the far right hand side of (2.1) as
\begin{equation}\sum_{n\ge1}\frac{n!x^{-n}}{\zeta'(-2n)}=\frac{1}{2}\sum_{n\ge1}\frac{(-1)^nn!}{\zeta(2n+1)(2n)!}\left(\frac{2\pi}{\sqrt{x}}\right)^{2n}.\end{equation}
The series (2.2) might be realized as a Fourier cosine integral in the following way. First, note that
and that
\begin{equation}\int_{0}^{\infty}e^{-ax}\cos(\sqrt{x}b)dx=\frac{1}{a}\sum_{n\ge0}\frac{n!}{(2n)!}\left(-\frac{b^2}{a}\right)^n.\end{equation}
Consequently, by uniform convergence, we may write the series in (2.2) as
\begin{equation}\int_{0}^{\infty}t\left(x\sum_{n\ge1}n\mu(n)e^{-x(nt)^2}\right)\cos(t2\pi)dt.\end{equation}
We may also use (2.3) to write this as
\begin{equation}\frac{x}{(2\pi)^2}\int_{0}^{\infty}e^{-xt/(2\pi)^2}\left(\sum_{n\ge1}\frac{\mu(n)}{n}\cos(\frac{\sqrt{t}}{n})\right)dt,\end{equation}
which involves the function of Bartz [2, eq.(2.8)].
A necessary and sufficient condition for the Riemann hypothesis, is (for all $\epsilon>0$)
\begin{equation}\sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}-\frac{x}{(2\pi)^2}\int_{0}^{\infty}e^{-xt/(2\pi)^2}\left(\sum_{n\ge1}\frac{\mu(n)}{n}\cos(\frac{\sqrt{t}}{n})\right)dt=O(x^{\frac{1}{4}+\epsilon}).\end{equation}
We now turn our attention to a useful study on Fourier integrals employed by Csordas [4]. The following definition, along with applicable theorems, was used there (see also that papers' references) to determine the nature of the zeros and other properties of the function represented by the Fourier cosine integral
These type of results had its beginnings with that of Pólya [9].
Definition 2.2 A function $k:\mathbb{R}\rightarrow\mathbb{R}$ is said to be an 'admissible kernel,' if it
satisfies (i) $k(t)\in C^{\infty}(\mathbb{R}),$ (ii) $k(t)>0$ for $t\in\mathbb{R},$ (iii) $k(t)=k(-t)$ for $t\in\mathbb{R},$ (iv) $\frac{d}{dt}k(t)<0$ for $t>0,$ and (v) for some $\epsilon>0,$
as $t\rightarrow\infty.$
Some interesting properties are known about $f(x)$ when $k(t)$ satisfies Definition 2.2 [4]. Namely, by the Riemann-Lebesgue Lemma,
$f(x)\rightarrow0$ as $|x|\rightarrow\infty.$ Additionally, $f(x)$ is then an entire function of order $\frac{2+\epsilon}{1+\epsilon}<2.$
We may observe that we may make the the change of variables in (2.4) with $t$ replaced by $t/\sqrt{x}$
to get that
\begin{equation}\int_{0}^{\infty}t\left(\sum_{n\ge1}n\mu(n)e^{-(nt)^2}\right)\cos(t2\pi/\sqrt{x})dt,\end{equation}
still represents the series in (2.2). However, our $k(t)$ function here is odd and subsequently it is not the case that the function $f(x)$ represented by our integral can have only real zeros. Our $k(t)$ implies that our $f(x)$ has infinitely many non-real zeros, and finitely many real zeros. To see this, we need only observe that if we let $s(t)=\sum_{n\ge1}n\mu(n)e^{-(nt)^2},$ then $s(t)>0$ when $t>0,$ $s(t)=s(-t),$ and $s'(t)<0$ when $t>0.$ Comparing these properties with the work of Csordas [4] gives our claim. In fact, it was already shown in Rieszs' study [10] that $\dot{R}(x)$ has infinitely many imaginary zeros using a different approach. See also [13] for more properties on the zeros of the Riesz function.
§ REMARKS ON A RECENT GENERALIZATION
Recently, Dixit et. al. [5, 6] studied a more general series than a function considered by Hardy and Littlewood, with a similar condition to (1.1). Their function is given by
\begin{equation} P_{z}(y)=\sum_{n\ge1}\frac{\mu(n)}{n}e^{-y/n^2}\cosh(\frac{\sqrt{y}z}{n}).\end{equation}
offer the condition that the Riemann hypothesis implies $P_{z}(y)=O_{z,\epsilon}(y^{-\frac{1}{4}+\epsilon})$ as $y\rightarrow\infty$ for
all $\epsilon>0.$ We offer some more comments on $P_{z}(y)$ herein. It is well-known [7] that (for $y>0$ and $\Re(\beta)>0$)
\begin{equation} \int_{0}^{\infty}e^{-t^2/{4\beta}}\cosh(\alpha t)\cos(yt)dt=\sqrt{\pi/\beta}e^{\alpha^2\beta}e^{-\beta y^2}\cos(2\alpha\beta y).\end{equation}
Or equivalently,
\begin{equation} e^{-t^2/{4\beta}}\cosh(\alpha t)=\sqrt{\pi/\beta}e^{\alpha^2\beta}\int_{0}^{\infty}e^{-\beta y^2}\cos(2\alpha\beta y)\cos(yt)dy.\end{equation}
Put $\beta=n^2,$ and $\alpha=1/n,$ where $n\in\mathbb{N}.$ Then sum over $\mu(n)/n$ to get, by uniform convergence,
\begin{equation} \sum_{n\ge1}\frac{\mu(n)}{n}e^{-t^2/{4n^2}}\cosh(\frac{t}{n})=\sqrt{\pi}e\int_{0}^{\infty}\sum_{n\ge1}\frac{\mu(n)}{n^2}e^{-n^2 y^2}\cos(2n y)\cos(yt)dy.\end{equation}
We need to check that the function
$$\bar{k}(t)=\sum_{n\ge1}\frac{\mu(n)}{n^2}e^{-n^2 t^2}\cos(2n t),$$
is an admissible kernel according to Definition 2.2. The condition (ii) that $\bar{k}(t)>0$ for $t\in\mathbb{R},$ can not be met for all $t\in\mathbb{R}.$
For example, in considering $\bar{k}(t)$ partial sums, we see that $t=\pi/2$ gives
\begin{equation}\sum_{1\le n \le 3}\frac{\mu(n)e^{-n^2\pi^2/4}\cos(n\pi)}{n^2}=-\frac{1}{36}e^{-\frac{9\pi^2}{4}}\left(-4+9e^{5\pi^2/4}+36e^{2\pi^2}\right), \end{equation}
which is $<0.$ Subsequently (ii) only holds for a subset of $\mathbb{R}.$ If instead we choose $\alpha=i/n,$ initially in our computations from (3.3), we find that applying our same analysis leads to an admissible kernel for the function $P_{iz'}(y),$ $z'\in\mathbb{R},$ which implies it would have only real zeros if $z'\in\mathbb{R}.$
Riesz [10] noted that $|\dot{R}(x)|<|x|e^{|x|},$ since $x\in\mathbb{R},$ and hence $\dot{R}(x)$ has order one. Recall the Hermite polynomials are generated by
and the Hermite numbers are $H_n:=H_n(0).$
The function $\dot{R}(x^2)/x^2$ may take the form
where $H_n$ is the $n$th Hermite number, and therefore is an entire function of order $\lambda,$ given by
§ MORE ON THE RIESZ CRITERION AND OTHER POSSIBLE DIRECTIONS
We make some further comments on Theorem 2.1 and offer some possible further directions. First if we first note that
the series on the far right hand side of (2.1) is $O(\frac{1}{x}),$ we may write
\begin{equation}\dot{R}(x)=\sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}+O(\frac{1}{x}).\end{equation}
\begin{equation}\sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}=O(w(x)).\end{equation}
\begin{equation}\dot{R}(x)=O(\max\{w(x),\frac{1}{x}\}).\end{equation}
This, together with Riesz criterion (1.1), tells us that proving $w(x)=x^{\frac{1}{4}+\epsilon},$ $\epsilon>0,$ would imply the Riemann Hypothesis. This is equivalent to the observation that (4.1) says $\dot{R}(x)\sim \sum_{\rho}\frac{x^{\rho/2}\Gamma(1-\frac{\rho}{2})}{\zeta'(\rho)}.$
A possible direction for further research would be to consider the following integral, which we produce
from some observations. We start with Riesz's integral [10]
\begin{equation}\frac{\Gamma(1-\frac{s}{2})}{\zeta(s)}=\int_{0}^{\infty}x^{-(\frac{s}{2}+1)}\dot{R}(x)dx.\end{equation}
Now if we assume a suitable test function $T(x)$ has a Mellin transform $\bar{T}(s),$ which exists in the region $\frac{1}{2}+\eta\le\Re(s)\le 2-\eta,$ ($\eta>0$), then
\begin{equation}\int_{0}^{\infty}x^{s-1}\left(\int_{0}^{\infty}T(x\sqrt{t})\frac{\dot{R}(t)}{t}dt\right)dx=\frac{\bar{T}(s)\Gamma(1-\frac{s}{2})}{\zeta(s)}.\end{equation}
So we may write
\begin{equation}\int_{0}^{\infty}T(x\sqrt{t})\frac{\dot{R}(t)}{t}dt=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\bar{T}(s)\Gamma(1-\frac{s}{2})}{\zeta(s)}x^{-s}ds.\end{equation}
If we choose our test function to be the dirac delta function $T(t)=\delta(t-a)$ and choosing $c=\frac{1}{2}+\epsilon,$ we can obtain the Riesz condition after noting that Littlewood [8, pg.161] showed that a criterion for the Riemann Hypothesis is that $\sum_{n\ge1}\mu(n)n^{-\frac{1}{2}-\epsilon}$ converges for every $\epsilon>0.$ It would be interesting to see some further examples by choosing other test functions $T(x).$ For example, choosing a function $T_{\epsilon_1}(t)$ whose limit is $\lim_{\epsilon_1\rightarrow0}T_{\epsilon_1}(t)=T(t)=\delta(t-a),$ and estimating the integral in (4.6) involving $T_{\epsilon_1}(t)$ may lead to improvements or new criteria.
Acknowledgement. We thank Professor Dixit and Professor Wolf for helpful comments.
G. Andrews, R. Askey, and R. Roy. Special Functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, New York, 1999.
K. M. Bartz, On some complex explicit formulae connected with the M$\ddot{o}$bius function, I, Acta Arithmetica 57, p.283–293 (1991).
K. M. Bartz, On some complex explicit formulae connected with the M$\ddot{o}$bius function, II, Acta Arithmetica 57, p.295–305 (1991).
G. Csordas, Fourier Transforms of Positive Definite Kernels and the Riemann $\xi$-function, Computational Methods and Function Theory, Volume 15, Issue 3, pp 373–391 (2015).
A. Dixit, Analogues of the general theta transformation formula, Proc. Roy. Soc. Edinburgh, Sect. A, 143 (2013), 371–399
A. Dixit, A. Roy and A. Zaharescu, Riesz-type criteria and theta transformation analogues, J. Number Theory 160, p. 385–408 (2016).
I. S. Gradshteyn and I. M. Ryzhik, eds., Table of Integrals, Series, and Products, 7th ed., Academic Press,
San Diego, 2007.
G. H. Hardy and J. E. Littlewood, Contributions to the Theory of the Riemann Zeta-Function and the
Theory of the Distribution of Primes, Acta Math., 41 (1916), 119–196.
G. Pólya, Uber trigonometrische Integrale mit nur reellen Nullstellen, J. Reine Angew. Math. 158 (1927), 6–18.
M. Riesz, Sur l'hypoth'ese de Riemann, Acta Math., 40 (1916), 185–190.
G.W. Smith, On a function of Marcel Riesz, http://arxiv.org/abs/1209.5652, September 2012.
E. C. Titchmarsh, The Theory of the Riemann Zeta Function, Clarendon Press, Oxford, 1986.
H.Wilf, On the zeros of Riesz' function in the analytic theory of numbers, Illinois J. Math., 8 (1964), pp. 639–641
M. Wolf, Evidence in favor of the Baez-Duarte criterion for the Riemann Hypothesis,
Computational Methods in Science and Technology, v.14 (2008) pp.47–54
1390 Bumps River Rd.
Centerville, MA
E-mail: [email protected]
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1511.00743
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Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA
Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 Canada.
Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 Canada.
MAL gratefully acknowledges a Canada Research Chair and NSERC Discovery grants. MF and HW gratefully acknowledge NSERC Discovery grants.
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1511.00701
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Department of Mathematics, University of York,
Heslington, York YO10 5DD, UK
$^1$E-mail: [email protected]
School of Mathematical Sciences, University of Nottingham,
Nottingham NG7 2RD, UK
$^2$E-mail: [email protected]
$^3$E-mail: [email protected]
Thermal phenomena in quantum field theory can be detected with the aid of particle detectors coupled to quantum fields along stationary worldlines, by testing whether the response of such a detector satisfies the detailed balance version of the KMS condition at a constant temperature. This relation holds when the interaction between the field and the detector has infinite time duration. Operationally, however, detectors interact with fields for a finite amount of time, controlled by a switching function of compact support, and the KMS detailed balance condition cannot hold exactly for finite time interactions at arbitrarily large detector energy gap. In this large energy gap regime, we show that, for an adiabatically switched Rindler detector, the Unruh temperature emerges asymptotically after the detector and the field have interacted for a time that is polynomially long in the large energy. We comment on the significance of the adiabaticity assumption in this result.
§ PARTICLE DETECTORS AND THE DETAILED BALANCE KMS CONDITION
The concept of a particle is ambiguous in quantum field theory in generic, curved spacetimes due to the lack of global symmetries. Particle detectors<cit.> resolve this ambiguity by making the notion of a particle operational in the sense that, for a particle detector interacting with a field, there will be absorption or emission of quanta which corresponds to particle exchange. This is the content of the dictum “a particle is what a particle detector detects"<cit.>.
The discovery of celebrated phenomena such as the Hawking<cit.> and Unruh<cit.> effects indicates that there exists an intimate relationship between general relativity, quantum physics and thermal physics. Localised observers in spacetime, equipped with ensembles of particle detectors coupled to quantum fields, can register thermal effects, which occur due to the detector-field interactions<cit.>, by studying the response of their detectors. More precisely, the detailed balance version of the KMS condition<cit.> can be used to relate the energy gap of a detector transition, the temperature registered by the observer and the ratio between the absorption and emission responses of an ensemble of detectors<cit.> by
\begin{equation}
\frac{1}{T} = \frac{1}{E} \ln \left(\frac{\mathcal{F}(-E)}{\mathcal{F}(E)} \right),
\label{dbKMS}
\end{equation}
where $E$ is the energy gap of a detector state transition, $T$ is the state temperature and $\mathcal{F}(E)$ is the response of the detector. For example, a linearly uniformly accelerated detector, which interacts with a field in the Minkowski vacuum state, satisfies the detailed balance condition at the Unruh temperature, $T= a/(2\pi)$, in the conventional approximation of weak coupling and long interaction time.
If we focus our attention to weakly-coupled, point-like particle detectors, condition (<ref>) holds whenever a detector follows an orbit that is stationary with respect to the field KMS state and, moreover, interacts with the field for an infinite amount of (detector proper) time. If eq. (<ref>) holds in an asymptotic regime, we say that the response is asymptotically thermal. For example, a detector which falls geodesically from infinity into a Schwarzschild black hole, as it interacts with a field in the Hartle-Hawking-Israel state, will satisfy the KMS condition at the asymptotic past infinity, where the geodesic is stationary with respect to the radiation, but condition (<ref>) will not be satisfied at any finite distance, as the detector falls into the black hole. The $(1+1)$-dimensional case has been studied by a combination of numerical and analytic techniques<cit.>. In this case, the detailed balance condition is satisfied asymptotically, but not exactly, in the sense that the interaction is asymptotically stationary.
From an operational point of view, the interaction of a detector with the field is controlled by a smooth switching function of compact support in the detector proper time, $\chi \in C_0^\infty(\mathbb{R})$, and not for an infinite amount of time. It is possible to show that eq. (<ref>) cannot hold exactly whenever the interaction is timelike compact<cit.> and the detector energy gap, $E$, becomes large. Nevertheless, one may ask whether this condition holds asymptotically, as the interaction time becomes long. Moreover, one may ask the question, how long one needs to wait in order for the detailed balance condition to up to energy $E$, as $E \rightarrow \infty$?
The purpose of this contribution is to answer this question in the case of a two-state, pointlike Unruh-DeWitt detector coupled to a scalar field in the Minkowski vacuum state though a smoothly-switched and timelike compact interaction Hamiltonian, along an orbit of uniform linear acceleration. To this end, we shall introduce the model in section <ref> and make precise the notion of asymptotic thermality in section <ref>. Our main result, namely, that the response of a uniformly accelerated detector is asymptotically thermal at the Unruh temperature in a time scale that is polynomial in the energy in the large energy gap regime, is presented in section <ref>. This polynomial time can be achieved under technical assumptions concerning the switching of the detector. We discuss this issue in section <ref>.
§ DETECTOR-FIELD MODEL
We consider a detector-field system which consists of a Klein-Gordon field, $\Phi$, weakly coupled to a two-level, point-like particle detector in 4-dimensional Minkowski space.
The kinematical Hilbert space of the system is given by the tensor product $\mathscr{F}_\text{s}(\mathcal{H}_{\Phi}) \otimes \mathcal{H}_\text{D}$, where $\mathscr{F}_\text{s}(\mathcal{H}_{\Phi}) = \mathbb{C}\otimes_s\oplus_{n=1}^\infty \mathcal{H}_{\Phi}^{\otimes_sn}$ is the bosonic Fock space of the Klein-Gordon field and $\mathcal{H}_\text{D}$ is the detector two-dimensional Hilbert space.
The dynamics are implemented by the Hamiltonian $H = H_{\Phi} \otimes I_{\text{D}} + I_{\Phi} \otimes H_{\text{D}} + H_{\text{int}}$, where $H_{\Phi}$ is the Hamiltonian operator of the free scalar field, $H_{\text{D}}$ is the detector Hamiltonian, and $H_{\text{int}}$ is the interaction Hamiltonian. The detector Hamiltonian can be written in terms of creation and annihilation operators as $H_{\text{D}} = E d^\dagger d$ and acts on the basis energy eigenstates of the Hilbert space as $H_{\text{D}} |0\rangle = 0 |0\rangle$ and $H_{\text{D}} |1 \rangle = E |1 \rangle$. We consider $E>0$, so that $|0\rangle$ is the ground state of the detector. The interaction Hamiltonian is
\begin{equation}
H_{\text{int}}(\tau) = c \chi(\tau) \Phi(\textsf{x}(\tau)) \otimes \mu(\tau),
\label{Hint}
\end{equation}
where $c \in \mathbb{R}$ is a small coupling constant, $\mu: \mathcal{H}_{\text{D}} \rightarrow \mathcal{H}_{\text{D}}$ is the monopole moment operator of the detector, $\tau$ is the detector's proper time and $\chi \in C_0^\infty(\mathbb{R})$ is a smooth switching function of compact support that controls the interaction of the field and the detector along the worldline of the detector.
If one supposes that the initial state of the system is given by $|\phi \rangle \otimes |0 \rangle$, where $|\phi \rangle$ is the Minkowski vacuum state, one can switch on the interaction and let the system evolve. After the interaction has been switched off, the probability of a detector transition to state $|1 \rangle$ is proportional to the response function, which is given, to leading order in $c$, by
\begin{equation}
\mathcal{F}(E) = \int_{-\infty}^\infty \! d\tau' \, \int_{-\infty}^\infty \! d\tau'' \, \chi(\tau') \chi(\tau'') \, \ee^{-\ii E (\tau'-\tau'')} \mathcal{W}\left(\mathsf{x}\left(\tau'\right), \mathsf{x}\left(\tau''\right)\right),
\label{ResponseFn}
\end{equation}
where $\mathcal{W}$ is the pullback of the Wightman two-point function to the detector's worldline in the field state $|\phi \rangle$.
The function defined by eq. (<ref>) does not satisfy the detailed balance condition whenever $\chi$ is a function of compact support <cit.>. Our objective is now to see under what conditions $\mathcal{F}$ satisfies condition (<ref>) asymptotically for a detector following a Rindler trajectory, for a detector coupled to the field for a finite amount of time. We make precise the definition of asymptotic thermality in section <ref>.
§ ASYMPTOTIC THERMALITY
We introduce a time scale in the problem by performing a rescaling of the proper time of the detector. We consider an adiabatic scaling $\lambda$ which produces a long and slow switching $\chi(\tau) \rightarrow \chi(\tau/\lambda)$ as $\lambda \rightarrow \infty$. Concordantly, the response function (<ref>) is rescaled by the time scale $\lambda$ as
\begin{equation}
\mathcal{F}_\lambda(E)/\lambda = \frac{1}{(2 \pi)^2} \int_{-\infty}^{\infty} \! d \omega \, \left|\hat{\chi}(\omega)\right|^2 \left( \frac{E+\omega/\lambda}{\ee^{2 \pi(E + \omega/\lambda)/a} - 1}\right),
\label{Flambda}
\end{equation}
where we have specialised eq. (<ref>) to the case of the Minkowski vacuum 2-point function along the Rindler trajectory, and used the stationarity of the state to obtain formula (<ref>). When the scale $\lambda$ is related to the energy, we establish an energy-dependent timescale.
The pointwise limit as $\lambda \rightarrow \infty$ of eq. (<ref>) defines a function that satisfies the detailed balance condition at any fixed value of $E$ at the Unruh temperature. This means that after an infinite time has elapsed, the detector's transition probability is a Planckian spectrum at $T = a/(2\pi)$. This pointwise limit is a property of all stationary KMS states.
We consider $\lambda = \lambda(E)$, as the energy $E \rightarrow \infty$, to be a positive, strictly increasing function of $E$. In other words, we consider the time scale to be a function of the large energy gap. We define<cit.> the response to be asymptotically thermal at temperature $T$ if there exist positive functions of $E$, $\mathcal{B}^+$ and $\mathcal{B}^-$, such that
\begin{equation}
\frac{1}{T} - \mathcal{B}^{-}(E) \leq \frac{1}{E}\ln \left( \frac{\mathcal{F}_{\lambda(E)}(-E)}{\mathcal{F}_{\lambda(E)}(E)}\right) \leq \frac{1}{T} + \mathcal{B}^{+}(E)
\label{AsympDBC}
\end{equation}
and $\mathcal{B}^{\pm} \to 0$, perhaps at some prescribed rate, as $E \rightarrow \infty$. Moreover, if such $\mathcal{B}^{\pm}$ functions exist and $\lambda$ is a polynomial function of $E$, we say that the response of the detector is polynomially asymptotically thermal.
Polynomially asymptotic thermality is relevant in the sense that if an observer, equipped with an ensemble of detectors, and following a Rindler trajectory, wishes to detect the Unruh temperature up to the large energy scale $E$, the amount of time that the observer needs to wait after carefully switching on the interaction of the detectors is polynomial in the large energy scale.
§ ASYMPTOTIC UNRUH EFFECT
The question of whether or not a detector satisfies the detailed balance condition asymptotically depends on the detailed form of the switching, $\chi \in C_0^\infty$. In the case of a uniformly accelerated detector, there exists a class of switching functions for which it is possible to find polynomially suppressed functions $\mathcal{B}^\pm$, which allow for the asymptotic detailed balance condition to hold for the rescaled response function, $\mathcal{F}_\lambda$, as $E \rightarrow \infty$. This class consists of the set of functions $\chi \in C_0^\infty$ whose Fourier transforms, $\hat{\chi}$, become suppressed sufficiently fast. The technical conditions that define this set of switching functions can be formulated precisely<cit.>. However, we feel that, for the purposes of this contribution, it is more enlightening to present an example of a switching function that belongs to this class. This example was introduced in the context of quantum stress energy tensor smearing functions<cit.>. Consider the $C^\infty(\mathbb{R})$ function
\begin{align}
f(\tau) =
\begin{cases}
(\kappa \tau)^{-3/2}e^{-1/(4 {\kappa \tau})}, & \tau>0\\
0, & \tau \leq 0.
\end{cases}
\end{align}
Here $\kappa$ is a constant introduced for dimensional reasons, $[\kappa] = [1/\tau]$. A smooth switching function of compact support is defined by $\chi(\tau) = f(\tau) f\left(\kappa^{-1} - \tau \right)$. The Fourier transform, $\hat{\chi}$, exists by the convolution theorem and the rapid decay of $f$ at large positive argument, and this further allows one to write $\hat{f}$ as a boundary value of the Laplace transform of $f$. In this way, controlling the decay properties of $\hat{f}$ is sufficient to control the decay properties of $\hat{\chi}$. It suffices for us to say that $\hat{\chi}$ has sufficiently strong decay for our purposes and refer the reader to our references<cit.> for details.
The bottom line is that, for a switching function with sufficiently strong decay, such as the function $\chi$ provided in our example, the functions $\mathcal{B}^\pm$ can be estimated and one obtains that the asymptotic detailed balance condition (<ref>) is satisfied at the Unruh temperature, $T = a/(2\pi)$, with polynomially suppressed $\mathcal{B}^\pm$. With the specific switching function described above, choosing $\lambda = \alpha (2\pi E/a)^{1+p}$, where $\alpha > \pi \kappa /(2a)$ and $p > 1$, we obtain the bounds
\begin{align}
\mathcal{B}^{-}(E) & = \frac{4}{(2\pi)^5 a} \left( \frac{2 \pi E}{a} \right)^{-(1+p)/2} + \mathcal{O}\left( \left( \frac{2 \pi E}{a} \right)^{-p} \right), \label{B-thm}\\
\mathcal{B}^{+}(E) & = \frac{\left|\left| \omega \hat{\chi} \right|\right|^2_{L^2}}{\left|\left| \hat{\chi} \right|\right|^2_{L^2}}\frac{ \pi^2}{3 a^2} \left(\frac{2 \pi E}{a}\right)^{-(4+2p)} + \mathcal{O}\left( \left( \frac{2 \pi E}{a} \right)^{-2(3+2p)-1} \right). \label{B+thm}
\end{align}
§ FINAL REMARKS
We would like to stress that the question of whether or not a detector satisfies the detailed balance depends on the detailed form of the switching. To emphasise this further, one may consider the following situation: Suppose that one smoothly switches on a detector, and that this switching procedure takes a fixed detector proper time interval $\Delta \tau_s$. Once this is done, one lets the detector and the field interact at a constant interaction strength during a time $\Delta \tau$ and, finally, one switches off the interaction smoothly during a time $\Delta \tau_s$. Such a switching function can be constructed, for example, by integrating bump functions of compact support in the detector proper time. Consider now a time scale $\lambda$, that rescales the constant interaction time $\Delta \tau \rightarrow \lambda \Delta \tau$, but leaves the switching tails fixed. As $E \rightarrow \infty$, the asymptotic detailed balance condition will not be satisfied for any polynomial time scale, $\lambda = P(E)$, where $P$ is a positive, polynomially increasing function of the energy <cit.>. This means that the careful and slow switching of the interaction is necessary to detect a temperature, up to a large energy scales, at late a time which is polynomial in the large energy.
§ ACKNOWLEDGMENTS
C.J.F. thanks Prof R. Longo and the organisers of the session on Quantum Field Theory for arranging financial support under the ERC Advanced Grant “Operator Algebras and Conformal Field Theory". B.A.J.-A. thanks the organisers of the Fourteenth Marcel Grossmann Meeting for their kind hospitality at Sapienza - Università di Roma, and acknowledges the financial support of Consejo Nacional de Ciencia y Tecnología (CONACYT), México, REF 216072/311506 and of the School of Mathematical Sciences at the University of Nottingham. J.L. was supported in part by STFC (Theory Consolidated Grant ST/J000388/1).
W. G. Unruh,
“Notes on black hole evaporation”,
Phys. Rev. D 14 (1976) 870.
B. S. DeWitt,
“Quantum Gravity: The New Synthesis”, in General Relativity: and Einstein centenary survey, edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979) 680.
W. G. Unruh and R. M. Wald,
“What happens when an accelerating observer detects a Rindler particle”,
Phys. Rev. D 29 (1984) 1047.
W. G. Unruh, public communication.
S. W. Hawking,
“Particle Creation by Black Holes”,
Commun. Math. Phys. 43 (1975) 199
[Commun. Math. Phys. 46 (1976) 206].
D. Buchholz and R. Verch,
“Unruh versus Tolman: On the heat of acceleration”,
arXiv:1505.01686 [gr-qc].
R. Haag, N. M. Hugenholtz and M. Winnink,
“On the Equilibrium states in quantum statistical mechanics”,
Commun. Math. Phys. 5 (1967) 215.
R. Kubo,
“Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in magnetic and conduction problems”,
J. Phys. Soc. Jap. 12 (1957) 570.
P. C. Martin and J. S. Schwinger,
“Theory of many particle systems. 1.”,
Phys. Rev. 115 (1959) 1342.
S. Takagi,
“Vacuum noise and stress induced by uniform accelerator: Hawking-Unruh effect in Rindler manifold of arbitrary dimensions”,
Prog. Theor. Phys. Suppl. 88 (1986) 1.
B. A. Juárez-Aubry and J. Louko,
“Onset and decay of the 1 + 1 Hawking-Unruh effect: what the derivative-coupling detector saw”,
Class. Quant. Grav. 31 (2014) 24, 245007
[arXiv:1406.2574 [gr-qc]].
C. J. Fewster, B. A. Juárez-Aubry and J. Louko,
in preparation (2015).
C. J. Fewster and L. H. Ford,
“Probability Distributions for Quantum Stress Tensors Measured in a Finite Time Interval,”
arXiv:1508.02359 [hep-th].
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1511.00968
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Advances in High Energy Physics
[Milano]Dipartimento di Fisica, Università di Milano-Bicocca, Milano I-20126 - Italy
[INFNMiB]INFN - Sezione di Milano Bicocca, Milano I-20126 - Italy
Milano,INFNMiB]A. Nucciotti
Recent years have witnessed many exciting breakthroughs in neutrino physics.
The detection of neutrino oscillations has proved that neutrinos are massive particles but the assessment of their absolute
mass scale is still an outstanding challenge in today particle physics and cosmology.
Since low temperature detectors were first proposed for neutrino physics experiments in 1984,
there have been tremendous technical progresses: today this technique offers the high energy resolution and scalability required
to perform competitive experiments challenging the lowest electron neutrino masses.
This paper reviews the thirty-year effort aimed at realizing a calorimetric measurements with sub-eV neutrino mass sensitivity using low temperature detectors.
§ INTRODUCTION
Almost two decades ago, the discovery of neutrino flavor oscillations firmly demonstrated that neutrinos are massive particles <cit.>. This was a crucial breach in the Standard Model of fundamental interactions which assumed massless neutrinos.
Flavor oscillations show that the three active neutrino flavor states ($\nu_e$, $\nu_\mu$, and $\nu_\tau$) are a superposition of three mass states ($\nu_1$, $\nu_2$, and $\nu_3$), and allow to measure the difference between the squared mass of the neutrino mass states; but they are not at all sensitive
to the absolute masses of the neutrinos.
Today, assessing the neutrino mass scale is still an outstanding task for particle physics,
as the absolute value of the neutrino mass would provide an important parameter to extend the Standard Model of particle physics and understand the origin of fermion masses beyond the Higgs mechanism.
Furthermore, due to their abundance as big-bang relics, neutrinos
strongly affect the large-scale structure and dynamics of the universe by means of their gravitational interactions, which hinder
the structure clustering with an effect that is dependent on their mass <cit.>. In the framework of $\Lambda$CDM cosmology (the model with Cold Dark Matter and a cosmological constant $\Lambda$), the scale dependence of clustering observed in the Universe can indeed be used to set an upper limit on the neutrino mass sum $m_\Sigma=\sum_i m_i$, where $m_i$ is the mass of the $\nu_i$ state.
Depending on the model complexity and the input data used, this limit spans in the range between about 0.3 and 1.3 eV <cit.>; more recently, by combining cosmic microwave background data with galaxy surveys and data on baryon acoustic oscillations a significantly lower bound on the neutrino mass sum of 0.23 eV has been published <cit.>,
although this value is strongly model-dependent.
The oscillation discovery and the accurate cosmological observations revived and boosted the interest in neutrino physics[This is also confirmed by the Nobel Prizes in Physics awarded in the years 2002, 2008 and, very recently, 2015.], with the start of many ambitious experiments for different high precision measurements and the rate of publishing papers increased by almost an order of magnitude; but in spite of the enhanced experimental efforts very little is known about neutrinos and their properties.
Several crucial pieces are still missing, in particular: the absolute neutrino mass scale, the neutrino mass ordering (the so-called mass hierarchy), the neutrino nature (Dirac or Majorana fermion), the magnitude of the CP (charge and parity) violation phases, and the possible existence of sterile neutrinos.
This paper is devoted to the assessment of the absolute neutrino mass scale, and in particular to the direct measurement of the electron neutrino mass via calorimetric experiments.
After a brief overview of
our present picture for massive neutrinos, I will introduce both the theoretical and the experimental issues involved in the direct determination of the neutrino mass, and discuss past and current calorimetric experiments, with a focus on experiments with low temperature detectors.
§ THE NEUTRINO MASS PATTERN AND MIXING MATRIX
Most of the existing experimental data on neutrino oscillations can be explained by assuming a three-neutrino framework, where
any flavor state $\nu_l$ ($l=e, \mu, \tau$) is described as a superposition of mass states $\nu_i$ ($i=1, 2, 3$), or
\begin{equation}
|\nu_l \rangle = \sum_i U_{li} |\nu_i \rangle
\end{equation}
where $U_{li}$ is the $3\times3$ Pontecorvo-Maki-Nakagawa-Sakata unitary
mixing matrix (see e.g. <cit.>). As a consequence, the neutrino flavor is no longer a conserved quantity and for neutrinos propagating in vacuum the amplitude of the process $\nu _l \to \nu_{l'}$ is not vanishing.
The $U_{li}$ mixing matrix
is parametrized by three angles, conventionally denoted as $\Theta _{12}$, $\Theta _{13}$ and $\Theta _{23}$, one CP violation phase $\delta$, and two Majorana phases $\alpha _1$, $\alpha _2$ – these two have physical consequences only if neutrinos are Majorana particles, i.e. identical to their antiparticles, but do not affect neutrino oscillations.
To these six parameters – three angles and three phases – the three mass values $m_i$ must be added also, for a total of nine unknowns altogether.
In the years, oscillation experiments measuring the flux of solar, atmospheric, reactor, and accelerator neutrinos
have contributed to precisely determine many of these unknowns.
In a three-neutrino scheme, the oscillation parameters measured by the various experiments paint two possible scenarios regarding the neutrino mass ordering:
Normal Hierarchy (left) and Inverted Hierarchy (right). The absolute scale is not accessible by presently available data.
The oscillation probabilities depend, in general, on the neutrino energy, on the source-detector distance, on the elements of the mixing matrix, and on the neutrino mass squared differences $\Delta m^2_{ij} \equiv m^2_{j} - m^2_{i}$.
At present, the three mixing angles and the two mass splittings, conventionally $\Delta m^2_{21}$ (from solar neutrino oscillations) and $\Delta m^2_{31}$ (from atmospheric neutrino oscillations), have been determined with reasonable accuracy <cit.>.
However, the available data are not yet able to discriminate the neutrino mass ordering. It proves convenient to assume $m_1<m_2$, so $\Delta m^2_{21}>0$; with these choices, $\Delta m^2_{21} \ll |\Delta m^2_{31}| \cong |\Delta m^2_{32}|$ and we are left with two possibilities: either $m_1<m_2<m_3$ (normal ordering, i.e. $\Delta m^2_{31}>0$) or $m_3<m_1<m_2$ (inverted ordering, i.e. $\Delta m^2_{31}<0$)[Compare also with Fig. <ref>.]. In both schemes, there is a Quasi-Degeneracy (QD) of the three neutrino masses when $m_1 \simeq m_2 \simeq m_3 $, with $m_{i} \gg \sqrt{\Delta m^2_{31}} \simeq 5\times 10^{-2}$ eV.
Depending on the value of the lightest of the mass values, the neutrino mass ordering can also follow a Normal Hierarchy (NH), with $m_1 \ll m_2 \ll m_3$ (in which $m_2 \simeq \sqrt{\Delta m^2_{21}}$ and $m_3 \simeq \sqrt{\Delta m^2_{31}}$), or an Inverse Hierarchy (IH), with $m_3 \ll m_1 < m_2$ (in which $m_1$ and $m_2$ are quasi-degenerate) – see Fig. <ref>. As a final remark, as shown in Fig. <ref>, independently of the mass scheme, oscillation results state that at least two neutrinos are massive, with masses larger than $ \sqrt{\Delta m^2_{21}} \simeq 8.7 \times 10^{-3}$ eV.
Most of the oscillation data are well described by the three-neutrino schemes. However, there are a few anomalous indications (the so-called reactor neutrino anomaly) <cit.> that cannot be accommodated within this picture. If confirmed, they would indicate the existence of additional neutrino families, the sterile neutrinos. These neutrinos do not directly participate in the standard weak interactions and would manifest themselves only when mixing with the familiar active neutrinos. Future reactor experiments will test this fascinating possibility.
Assessing the
neutrino mass ordering, i.e. the sign of $|\Delta m^2_{31}|$, is of fundamental importance not only because it would address the correct theoretical extension of the Standard Model, but also because it can impact on many important processes in particle physics (like neutrinoless double beta decay).
In addition, the phase $\delta$ governing CP violation in the flavor oscillation experiments remains unknown, and a topic of considerable interest <cit.>.
A worldwide research program is underway to address these important open issues in the near future by precise study of the various oscillation patterns.
The oscillation experiments, however,
are not able to access the remaining unknown quantities, i.e. the absolute mass scale and the two Majorana phases.
Their determination is the ultimate goal of nuclear beta decay end-point experiments and neutrinoless double beta decay searches.
§ WEAK NUCLEAR DECAYS AND NEUTRINO MASS SCALE
Fundamental neutrino properties, in particular its absolute mass and its nature, can be investigated by means of suitable weak decays, where flavor state neutrinos are emitted along with charged leptons and/or pions. There are two complementary approaches for the measurement of the neutrino mass in laboratory experiments: the precise spectroscopy of beta decay at its kinematical end-point, and the search for neutrinoless double beta decay. Though the expected effective mass sensitivity for neutrinoless double beta decay search is higher, this process implies a strong model-dependence since it requires the neutrino to be a Majorana particle.
Direct neutrino mass measurement, by analyzing the kinematics of electrons emitted in a beta decay, is the most sensitive model independent method to assess the neutrino mass absolute value[Analogue measurements involving pion or tau decays give much weaker limits on $m_{\nu_{\mu}}$ or $m_{\nu_{\tau}}$.].
The beta decay is a nuclear transition involving two nuclides $(A,Z-1)$ and $(A,Z)$
\begin{equation}
(A,Z-1) \rightarrow (A,Z) + e^- + \overline\nu_e
\end{equation}
where $A$ and $Z$ are, respectively, the mass and atomic numbers of the involved nuclei. Neglecting the nuclear recoil, the kinetic energy $E_0$ available to the electron and anti-neutrino in the final state is given by
\begin{equation}
E_0 = E_\beta + E_{\overline\nu} = M(A,Z-1) - M(A,Z) = Q
\end{equation}
where $M$ indicates the mass of the atoms in the initial and final state.
In practice, this method exploits only momentum and energy conservation: it measures the minimum energy carried away by the neutrino – i.e. its rest mass – by observing the highest energy electrons emitted in this three body decay.
To balance the energy required to create the emitted neutrinos, the highest possible kinetic energy $E_\beta$ of the electrons is slightly reduced. This energy deficit may be noticeable when measuring with high precision the higher energy end (the so-called end-point) of the emitted electron kinetic energy distribution $N_\beta(E_\beta,m_{\nu_e})$. If one neglects the nucleus recoil energy, $N_\beta(E_\beta,m_{\nu_e})$ is described in the most general form by
\begin{equation}
N_\beta(E_\beta,m_{\nu_e}) = p_\beta E_\beta (E_0 - E_\beta)\sqrt{(E_0 - E_\beta)^2 - m_{\nu_e}^2} F(Z,E_\beta) S(E_\beta) [1+\delta_R(Z,E_\beta)]\theta(E_0 - E_\beta -m_{\nu_e})
\label{eq:SPEBETA}
\end{equation}
where $F(Z,E_\beta)$ is the Coulomb correction (or Fermi function) which accounts for the effect of the nuclear charge on the wave function of the emitted electron, $S(E_\beta)$ is the form factor which contains the nuclear matrix element $\cal{M}(E_\beta)$ of the electroweak interaction and can be calculated using the V-A theory, and $\delta_R(Z,E_\beta)$ is the radiative electromagnetic correction, usually neglected due to its exiguity.
$\theta$ is the Heaviside step function, which confines the spectrum in the physical region $(E_0 - E_\beta -m_{\nu_e})>0$.
The term $p_\beta E_\beta (E_0 - E_\beta)\sqrt{(E_0 - E_\beta)^2 - m_{\nu_e}^2}\ $
is the phase space term in a three-body decay, for which the nuclear recoil has been neglected; $p_\beta$ is the electron momentum.
For the sake of completeness, it is worth noting that the particle emitted in the experiments considered here is the electron anti-neutrino $\overline{\nu}_e$. Since the CPT theorem assures that particle and antiparticle have the same rest mass, from now on I will speak simply of “neutrino mass” both for $\nu_e$ and $\overline{\nu}_e$.
Moreover, it must be stressed that since the effect of the neutrino mass in nuclear beta decay is due purely to kinematics, this measurement does not give any information on the Dirac or Majorana origin of the neutrino mass.
From oscillation experiments, we know that any neutrino flavor state is a superposition of mass states. Therefore, (<ref>) can be generalized as
\begin{equation}
N_\beta(E_\beta,m_{\nu_e}) = R(E_\beta) \sum_{i=1}^3 |U_{ei}|^2 \sqrt{(E_0 - E_\beta)^2 - m_i^2}\theta(E_0 - E_\beta -m_i)
\label{eq:SPEBETA-mixing}
\end{equation}
$R(E_\beta)$ is a term which groups all terms in (<ref>) which do not depend on the neutrino mass, $|U_{ei}|$ is the electron row of the neutrino mixing matrix, and $m_i$ are the masses of the neutrino mass states. The square root term is the part of the phase space factor
sensitive to the neutrino masses. An example of the resulting spectrum is shown in Fig. <ref>.
Expected electron spectrum following $\beta$ decay. Blue curve: expected spectrum in the case $m _{\nu _e}=0$. Red curve: expected spectrum in the case $m _{\nu _e}=200$ meV. The red dashed curves show the contributions to the total spectrum from the three mass states, with the mass differences driven by present values of $ \sqrt{\Delta m^2_{21}}$ and $ \sqrt{\Delta m^2_{31}}$ <cit.>.
Since the individual neutrino masses are too close to each other to be resolved experimentally, the measured spectra can still be analyzed with (<ref>), but the quantity
\begin{equation}
m _{\nu _e} \equiv m_\beta = \sqrt {\sum _{i=1} ^3 | U _{ei} | ^2 m_i^2}.
\label{eq:effective-mnue}
\end{equation}
should now be interpreted as an effective electron neutrino mass, where the sum is over all mass values $m _i$. Therefore, a limit on $m_{\nu_e}$ implies trivially an upper limit on the minimum value $m_{min}$ of all $m_i$, independent of the mixing parameters $U_{ei}$: $m_{min} \le m_{\nu_e}$, i.e. the lightest neutrino cannot be heavier than $m_{\nu_e}$.
By using currently available information from oscillation data <cit.>, it is possible to formulate the values of the neutrino masses (and of as well) as a function of the lightest mass, i.e. $m_1$ in the Normal Hierarchy (NH) and $m_3$ in the Inverted one (IH). This is done in Fig. <ref>, which shows that in the case of NH the main contribution to is mainly due to $m_2$.
In the case of IH, has practically the same value of $m_1$ and $m_2$. Finally, in the case of QD spectrum $m_{\nu_e} \simeq m_1 \simeq m_2 \simeq m_3 $ in both schemes. From the figure it is also clear that the allowed values for in the two mass schemes are quite different: in the case of IH there is a lower limit for of about 0.04 eV, while in the NH this limit is of about 0.01 eV. Therefore, if a future experiment will determine an upper bound for smaller than 0.04 eV, this would be a clear indication in favor of the NH mass pattern. Finally, Fig. <ref> shows that the ultimate sensitivity needed for a direct neutrino mass measurement is set at about 0.01 eV, the lower bound in case of NH.
Effective electron neutrino mass $m_{\beta}$ as a function of the lightest mass in both NH and IH mass schemes. The values of all three mass states are also plotted for comparison.
However, if experiments on neutrino oscillations provide us with the values of all neutrino mass-squared differences $\Delta m_{ij}^2$ (including their signs) and the mixing parameters $|U_{ei}|^2$, and the value of $m_{\nu_e}^2$ has been determined in a future search, then the individual neutrino mass squares can be determined
\begin{equation}
m_j ^2 = m_{\nu_e} ^2 - \sum _{i=1} ^3 | U_{ei} | ^2 \Delta m_{ij}^2 \qquad (\Delta m_{ij} ^2 = m_i ^2 - m_j^2)
\end{equation}
On the other hand, if only the absolute values $| \Delta m_{ij} ^2 |$ are known (but all of them), a limit on $m_{\nu_e}^2$ from beta decay may be used to define an upper limit on the maximum value $m_{max}$ of $m _i$
\begin{equation}
m_{max}^2 \le m_{\nu_e} ^2 + \sum _{i < j} | \Delta m_{ij}^2 |
\end{equation}
In other words, knowing $ | \Delta m _{ij} ^2 | $ one can use a limit on $m_{\nu _e}$ to constrain the heaviest active neutrino.
At present, the most stringent experimental constraint on $m _{\nu _e}$ is the one obtained by the Troitzk <cit.> and the Mainz <cit.> neutrino mass experiments, $m _{\nu _e} < 2.05$ eV at 95% C.L.: this falls in the QD region for both mass schemes.
Another type of weak process sensitive to the neutrino mass scale is the neutrinoless double beta decay (), a second order weak decay that violates the total lepton number conservation by two units, and whose existence is predicted for many even-even nuclei
\begin{equation}
(A,Z) \to (A,Z + 2) + e^-_1 + e^-_2
\end{equation}
The search for is the only available experimental tool to demonstrate the Majorana character of the neutrino (i.e. $\nu\equiv\bar{\nu}$). In fact, the observation of always requires and implies that neutrinos are massive Majorana particles <cit.>. However, there are many proposed mechanisms
which could contribute to the transition amplitude, and only when the is mediated by a light mass Majorana neutrino the observed decay is useful for determining the neutrino mass. In this case the measured decay rate is
given by
\begin{equation}
\frac{1}{\tau_{1/2}^{0\nu}}= \frac{m_{\beta\beta}^2}{m_e^2} F_N
\end{equation}
where $\tau_{1/2}^{0\nu}$ is the decay half-life, $m_e$ is electron mass, and $m_{\beta\beta}$ is the effective Majorana mass, defined below.
The nuclear structure factor $F_N$ is given by
\begin{equation}
F_N = G^{0\nu}(Q_{\beta\beta},Z) |M^{0\nu}|^2
\end{equation}
where $G^{0\nu}$ is the accurately calculable phase space integral, and $M^{0\nu}$ is the nuclear matrix element which is subject to uncertainty <cit.>.
At present, the discrepancies among different nuclear model calculations of $M^{0\nu}$ amount to a factor of about 2 to 3.
These reflect on $F_N$ and are an unavoidable source of systematic uncertainties
in the determination of $m_{\beta\beta}$ from the experimental data.
Measuring the lifetime of different isotopes would allow to disentangle the model dependency linked to the exact mechanism causing the
and to reduce the systematic uncertainties on $m_{\beta\beta}$.
If the decay is observed, and the nuclear matrix elements are known, one can deduce the corresponding $m_{\beta\beta}$ value, which in turn is related to the oscillation parameters through
\begin{equation}
m_{\beta\beta} = \left | \sum _{i=1} ^3 |U_{ei}|^2 m_i e^{i \alpha _i} \right |
\label{eq:effective-majorana-mass}
\end{equation}
Due to the presence of the unknown Majorana phases $\alpha _i$, cancellation
of terms in (<ref>) is possible, and $m_{\beta\beta}$ could be smaller than any of the $m_i$.
Therefore, unlike the direct neutrino mass measurement, a limit on $m_{\beta\beta}$ does not allow to constrain the individual mass values $m_i$ even when the mass differences $\Delta m_{ij}^2$ are known.
On the other hand, the observation of the decay and the accurate determination of the $m_{\beta\beta}$ value, would not only establish that neutrinos are massive Majorana particles, but would contribute considerably to the determination of the absolute
neutrino mass scale. Moreover, if the neutrino mass scale would be known from independent measurements, one could possibly obtain from the measured $m_{\beta\beta}$ also some information about the CP violating Majorana phases <cit.>.
Relationship between the effective Majorana mass $m_{\beta\beta}$ and the effective electron neutrino mass $m_{\beta}$ for both IH (blue) and NH (red) mass schemes. The width of the bands is caused by the unknown Majorana phases.
Given the present knowledge of the neutrino oscillation parameters, it is possible to derive the relation between the effective Majorana mass and the lightest neutrino mass in the different neutrino mass schemes. This is done in a number of papers (see e.g. <cit.>). Fig. <ref> shows the effective Majorana mass as a function of the effective electron neutrino mass in both the NH and IH mass schemes, demonstrating the complementarity of the two methods.
As a final remark, and $\beta$ decays both depend on different combinations of the neutrino mass values and oscillation parameters. The decay rate is proportional to the square of a coherent sum of the Majorana neutrino
masses because the process originates from exchange of a virtual neutrino. On the other hand, in beta decay one can determine an incoherent sum because a real neutrino is emitted. That shows clearly that a complete neutrino physics program cannot renounce either of these two experimental approaches. The various methods that constrain the neutrino absolute mass scale are not redundant but rather complementary. If, ideally, a positive measurement is reached in all of them ( decay, $\beta$ decay, cosmology) one can test the results for consistency and with a bit of luck determine the Majorana phases.
§ THE DIRECT NEUTRINO MASS MEASUREMENT VIA SINGLE NUCLEAR BETA DECAY
As already pointed out, the most useful tool to constrain kinematically the neutrino mass is the study of the “visible” energy in single beta decay.
The experimental beta spectra are normally analyzed by means of a transformation which produces a quantity generally linear with the kinetic energy $E_\beta$ of the emitted electron
\begin{equation}
K(E_\beta)\equiv \sqrt{\frac{N_\beta(E_\beta,m_{\nu_e})}{p_\beta E_\beta F(Z,E_\beta) S(E_\beta) [1+\delta_R(Z,E_\beta)]}}
=(E_0-E_\beta)\left( 1 - \frac{m_{\nu_e}^2}{(E_0-E_\beta)^2}\right)^{1/4}
\end{equation}
The graph of this quantity as a function of $E_\beta$ is named Kurie plot. In a Kurie plot, each bin has the same error bar and therefore the same statistical weight.
Assuming massless neutrinos and infinite energy resolution, the Kurie plot is a straight line intersecting the X-axis at the transition energy $E_0$. In case of massive neutrino, the Kurie plot is distorted close to the end-point and intersects with vertical tangent the X-axis at the energy $E_0-m_{\nu_e}$. The two situations are depicted in Fig. <ref>.
$^3$H Kurie plot close to the end-point, computed for neutrino masses equal to 0 (blue) and 1 eV (red), and an energy resolution $\sigma_E$ of 0 (full line) and 0.5 eV (dashed line).
Most of the information on the neutrino mass is therefore contained in the final part of the Kurie plot, which is the region where the counting rate is lower. In particular, the relevant energy interval is and the fraction of events occurring here is
\begin{equation}
F(\delta \! E) =\int^{E_0}_{E_0-\delta \! E} N_\beta(E,m_{\nu_e} \! = \! 0) d \! E \approx
C\left( \frac {\delta \! E} {E_0} \right)^3
\end{equation}
where $C$ is a constant of order unity which depends on the details of the beta transition.
From this it is apparent that kinematical mass measurements require beta decaying isotopes with the lowest end-point energy.
Tritium is one of the best and most used isotopes thanks to its very low transition energy, $E_0 = 18.6$ keV; nonetheless, the fraction of events falling in the last 5 eV of the tritium spectrum is only $4\times 10^{-11}$.
Every instrumental effect such as energy resolution or background will tend to hinder or even wash out this tiny signal. In Fig. <ref> the effect on the spectral end-point of an energy resolution of 0.5 eV is shown.
This distorts the Kurie plot in the opposite way with respect to the neutrino mass effect. It is therefore mandatory to evaluate and/or measure the detector response function, which includes the energy resolution but is not entirely determined by it. Finally, the analysis of the final part of the Kurie plot is complicated by the background due to cosmic rays and environmental radioactivity. Because of the low beta counting rate in the interesting region, spurious background counts may affect the neutrino mass determination.
The possibility to use beta decay to directly measure the neutrino mass was first suggested by Fermi <cit.> in 1934, but the first sensitive experiments where performed only in the '70s. The first experiments were the one of Bergkvist <cit.>
and the one of the ITEP group <cit.>, both of which used magnetic spectrometers to analyze the electrons emitted by tritium sources. This experimental approach has clear advantages such as 1) the high specific activity of tritium, 2) the high energy resolution and luminosity of spectrometers, and 3) the possibility to select and analyze only the electrons with kinetic energies close to $E_0$.
In the '80s and through the '90s, experiments with spectrometers using tritium were reporting largely negative <cit.> (see Fig. <ref>) or even an unlikely finite value of about 35 eV <cit.>.
These were all signs of under- or over- corrected instrumental effects which were causing systematic shifts <cit.>.
In fact, despite the relative conceptual simplicity of the kinematic direct determination of the neutrino mass, it has been soon recognized that there are many subtle effects which threaten the accuracy of these measurements.
Some are related to beta decay itself, since the atom or the molecule containing the decaying nucleus can be left in an excited state, leading even in this case to dangerous distortions of the Kurie plot (see <ref>).
Other are due to the scattering and absorption of the electrons in the source itself. And last but not least, systematic effects are also caused
by the imperfect characterization of the detector response.
In the past 30 years many experiments using tritium were performed. Starting from the '90s, magnetic spectrometers were gradually abandoned for electrostatic retarding spectrometers with adiabatic magnetic collimation <cit.>. Many improvements in the detectors, in the tritium source, and in the data analysis and processing allowed to constantly improve the statistical sensitivity and to minimize the systematic uncertainties, as it is shown in Fig <ref>.
Today, owing to a continuous and strenuous investigation of all experimental effects and systematic uncertainties,
the measurements reported by the two most sensitive experiments <cit.> are compatible with a zero mass, with the systematic errors reduced to the same level of statistical ones.
Historical trend of the measured with spectrometer using tritium (taken from <cit.>).
Nevertheless, today direct neutrino mass measurements remain affected by an intrinsic potential bias. As it already happened in the past, in a sensitive experiment small miscorrections of instrumental effects may again either mimic or cancel the traces of a small positive neutrino mass. A weak unexpected effect not included in the data analysis may compensate and hide the signal of a small mass within the statistical sensitivity of an experiment, which would therefore report a nicely compatible with the null hypothesis and thus quote just an upper limit. On the other hand, in a future experiment with a statistical sensitivity approaching the range predicted by oscillation parameters a slightly excessive correction for an expected effect could mimic the signal for a tiny mass which would not contradict the community expectations. For these reasons, direct neutrino mass measurements call for a continuous crosscheck from different independent experiments to confirm both positive and negative findings.
Already in the '80s when the negative squared masses and the positive claim from ITEP were puzzling the neutrino community, A. De Rujula proposed the use of other beta decaying isotopes with low decay energy. In <cit.> it was noticed the has an endpoint around 2 keV, much more favorable than the one of tritium. This isotope was at that time discarded because of its long half life around $10^9$ years.
The focus of <cit.> was therefore on the isotope , which decays by Electron Capture (EC) with a very low transition energy.
In the EC process <cit.>
\begin{equation}
(A,Z) + e^- \rightarrow (A,Z-1) + \nu_e
\end{equation}
the available decay energy
\begin{equation}
Q = M(A,Z) - M(A,Z-1)
\end{equation}
where $M$ indicates the mass of the atoms in the initial and final state.
Neglecting the nuclear recoil, the energy $Q$ is shared between the neutrino and the radiation emitted in the de-excitation of the daughter atom
\begin{equation}
\label{eq:q_ec}
Q= E_\nu + E_X
\end{equation}
Here $E_X$ includes the energy of X-rays, Inner Bremsstrahlung photons, and Auger and Coster-Kronig electrons emitted in the atomic de-excitation
of the daughter atom and adds up the binding energy of the captured electron, allowing for a small indetermination due to the natural width of the atomic energy levels.
Because of energy conservation, the end-points of the spectra of these electrons or photons – where the massive neutrino emitted in the EC is at rest – is sensitive to the neutrino mass.
It is worth noting here that the kinematics of EC decay probes the mass the neutrino $\nu_e$ whereas the regular beta decays which probes the mass of the anti-neutrino $\overline{\nu}_e$ but, as already recalled above, the CPT theorem warrants the equality of these quantities.
In particular, two measurements were discussed in 1981 for the isotope: the end-point of the IBEC (Inner Bremsstrahlung in EC) spectrum <cit.>
\begin{equation}
\label{eq:ibec}
^{163}\mathrm{Ho} \rightarrow (^{163}\mathrm{Dy^{H^*}} + \nu_e) \rightarrow ^{163}\mathrm{Dy^{H^\prime}} + \gamma(k) + \nu_e
\end{equation}
and the end-point of SEEEC (Single Electron Ejection in EC) spectrum <cit.>
\begin{equation}
\label{eq:seeec}
^{163}\mathrm{Ho} \rightarrow (^{163}\mathrm{Dy^{H^*}} + \nu_e) \rightarrow ^{163}\mathrm{Dy^{H_1H_2}} + e^- + \nu_e
\end{equation}
Even if at that time the $Q$ of EC was largely unknown, this decay was already considered very promising for a sensitive neutrino mass measurement, since it was clear that the $Q$ value is one of the lowest available.
Both processes (<ref>) and (<ref>) start with a first intermediate atomic vacancy H$^*$ caused by the EC, where the $^*$ reminds that the state is not necessarily on-shell. The energy of the vacant state has its own natural width.
Because of the low $Q$ value, this first vacancy H can be created only in one of the M1, M2, N1, N2, O1, O2, or P1 shells of the Dy daughter atom.
In the IBEC process (<ref>) a photon is emitted during the virtual transition of an electron from H$^\prime$ to the intermediate state H$^*$, from which the electron was captured.
For each possible final vacancy H$^\prime$, and for $m_{\nu_e}=0$, the spectrum of the emitted photons is not made of mono-energetic lines at $k=E(\mathrm{H^\prime})-E(\mathrm{H^*})$, where $E(\mathrm{H})$ is the ionization energy of the H shell in Dy, but is a continuum with a kinematic limit $k\le k^{\mathrm{H^\prime}}_{max} = Q - E(\mathrm{H^\prime})$, and the total photon spectrum is therefore a superposition of several spectra with different end-points.
The spectral end-points follow the three-body statistical shape
\begin{equation}
N^{\mathrm{H^\prime}}(k) \propto k (k^{\mathrm{H^\prime}}_{max}-k)\sqrt{(k^{\mathrm{H^\prime}}_{max}-k) - m_{\nu_e}^2}
\end{equation}
In general, since the IBEC is a second-order effect, its intensity is very low. However, the photon emission may experience large resonant enhancements for photons with energies $k_{res}$ equal to the ones of the characteristic X-ray transitions of the daughter atom.
In particular, A. De Rujula has shown for that when H$^\prime$ is one of the N1, N2, O1, O2 shells then the dominant resonance close to the end-point is with the X-ray transitions H$^\prime\rightarrow$M1, that is when the intermediate vacancy of the virtual transition H corresponds to the M1 shell. In this case the distance between the resonance and the end-point is $k^{\mathrm{H^\prime}}_{max}-k_{res}=Q-E(\mathrm{M1})$, which for is equal to a few hundreds electronvolts.
Unfortunately, calculations <cit.> showed that with a $Q$ at around 2.8 keV, an IBEC measurement with is not going to be statistically competitive with the tritium experiments, also because of complex destructive interference patterns.
The SEEEC process (<ref>) is the analogous of the IBEC with the role of the IB photon played by an Auger (or Coster-Kronig) electron.
The spectrum of the ejected electron is a continuum with an end-point at $E^{\mathrm{H1H2}}_{max} = Q - E(\mathrm{H1})-E(\mathrm{H2})$, for $m_{\nu_e}=0$.
Also in this case, the kinematics of a 3-body decay process applies, and a phase space term $p_\nu E_\nu$ appears in the spectral shape of ejected electrons.
The continuous spectra show many resonances for different combinations of H,H1 and H2, but close to the end-point the dominant ones result from the M1 capture and are at $E_{res}=E(\mathrm{M1})-E(\mathrm{H1})-E(\mathrm{H2})$. These resonances provide an enhancement of the spectrum close to end-point, thereby increasing the statistical sensitivity to $m_{\nu_e}$. The inclusive spectrum of all the ejected electrons is quite complicate because of the many possible end-points $E^{\mathrm{H1H2}}_{max}$ and resonance peaks: nevertheless, the authors in <cit.> argue that the end-point region of this spectrum is unaffected by all the atomic details, since it is dominated by the upper tails of few resonances and maintains its usable sensitivity to $m_{\nu_e}$, although the estimated $F(\Delta E)$, depending on the $Q$ value, may be substantially lower than for tritium.
One stressed advantage of IBEC and SEEEC measurements is that, unlike what happens in tritium beta decay, the probability of atomic excitations in the final state – such as shake-up or shake-off processes – is strongly suppressed and estimated to be $<1/Z^2$ (see also <ref>).
More than 30 years later, none of the above suggestions has been successfully exploited to perform an experiment with a competitive sensitivity on $m_{\nu_e}$.
Of the various attempts to perform an IB end-point measurements <cit.>, only the one of P.T. Springer <cit.> reported a limit on of about 225 eV obtained by fitting the end-point of the X-ray spectrum.
Most of the measurements performed on to directly measure the neutrino mass followed instead another proposal from C.L. Bennett et al. <cit.> in 1981.
In <cit.> it is suggested that and the transition energy $Q$ can be determined or constrained by measuring the ratios of absolute capture rates[A better treatment includes $C_i$ factors for the nuclear shape factor.]
\begin{equation}
\label{eq:ratios}
\lambda_i/\lambda_j=(n_i p_{\nu,i} E_{\nu,i} \beta^2_i B_i)/(n_j p_{\nu,j} E_{\nu,j} \beta^2_j B_j)
\end{equation}
where neutrino momentum $p_{\nu,i}$ is given by
\begin{equation}
p^2_{\nu,i} = (Q - E_i)^2 - m_{\nu_e}^2= E^2_{\nu,i} - m^2_{\nu_e}
\end{equation}
$n_i$ is the fraction of occupancy of the $i$-th atomic shell,
$\beta_i$ is the Coulomb amplitude of the electron radial wave function
(essentially, the modulus of the wave function at the origin), and $B_i$
is an atomic correction for electron exchange and overlap.
Following this idea, practically all the experimental researches on EC of so far focused on the atomic
emissions - photons and electrons contributing to $E_X$ in (<ref>) - following the EC and used the capture ratios to determine the $Q$ value <cit.>.
Unfortunately the accuracy achieved for and $Q$ with this method is adversely affected by the limited knowledge of the atomic parameters
in (<ref>).
As repeatedly underlined by A. De Rujula and M. Lusignoli <cit.>, there is one experimental approach to the measurement of the neutrino mass from the EC which overcomes all the difficulties above: the calorimetric measurement of all the energy released in the EC ($E_X$ in (<ref>)) except for the energy of the neutrino. This will be discussed in <ref>.
Today all expectations for a new direct measurement of the neutrino mass with a substantially improved statistical sensitivity
are directed to the KATRIN experiment <cit.>. KATRIN uses a large electrostatic spectrometer which will analyze the tritium beta decay end-point with an energy resolution of about 1 eV and with an expected statistical sensitivity of about 0.2 eV. KATRIN reaches the maximum size and complexity practically achievable for an experiment of its type and no further improved project can be presently envisaged.
As an alternative for the study of tritium end-point, Project8 proposes a new experimental approach based on the
detection of the relativistic cyclotron radiation detection emitted by the beta electrons <cit.>, which
is presently under development <cit.>.
§ CALORIMETRIC MEASUREMENTS
§.§ General considerations
In the global effort to cure the weaknesses of direct neutrino mass measurements with spectrometers yielding negative which started to show up since the '80, J.J. Simpson first proposed the calorimetric approach <cit.>.
In an ideal calorimetric experiment, the source is embedded in the detector and therefore only the neutrino energy escapes to detection. The part of the energy spent for the excitation of atomic or molecular levels is measured through the de-excitation of these states, provided that their lifetime be negligible with respect to the detector time response. In other terms, the kinematical parameter which is effectively measured is the neutrino energy $E_\nu$ (or $E_0-E_\nu$), in the form of a missing energy, a common situation in experimental particle physics.
The advantages of a calorimetric measurement are 1) the measurement of all the energy temporarily stored in excited states, 2) the absence
of source effects – such as self-absorption, and 3) the lack of backscattering from the detector.
The effect of final states on the tritium beta spectrum was discussed throughly in many works <cit.>. In the following for simplicity we consider the so-called sudden approximation or first order perturbation of an atomic tritium beta decay, neglecting the sum over the mass eigenstates $m_i$.
Due to the excited final states, the measured beta spectrum is a combination of different spectra characterized by different transition energies $E_0 - V_i$, where $V_i$ is the energy of the $i$-th excited state
\begin{equation}
\label{eq:speb2}
N_\beta(E_\beta,m_{\nu_e}) \approx
\sum_i{w_i p_\beta E_\beta
(E_0 - E_\beta - V_i)^2 \left( 1 - \frac{m_{\nu_e}^2 }{(E_0 - E_\beta -
V_i)^2}\right)^{1/2}F(Z,E_\beta) S(E_\beta) }
\end{equation}
with $w_i$ describing the transition probability to the final $i$-th excited state. The spectral shape induced by the excited states is misleading when one tries to extract the value of the neutrino mass. Assuming that the neutrino mass is null and summing up over all the final states, from equation (<ref>) one obtains
\begin{equation}
\label{eq:speb2m0}
N_\beta(E_\beta,0) \approx
p_\beta E_\beta (E_0 - E_\beta - \langle V_i \rangle)^2
\left( 1 + \frac {\langle V_i^2 \rangle - \langle V_i \rangle ^2} {(E_0 - E_\beta - \langle V_i \rangle)^2}\right)
F(Z,E_\beta) S(E_\beta)
\end{equation}
which approximates the single beta spectrum (<ref>) with a negative squared neutrino mass equal to , where $\sigma$ is the variance of the final state spectrum given by (Fig. <ref>), and
with an end-point shifted by $\langle V_i \rangle$.
These plots compare the effect of a final excited state ($V_1=1$ eV and $\omega_1=0.1$ eV) on the beta spectrum as measured with a calorimeter (blue) and with a spectrometer (red), and the effect of $=1$ eV (green). The fractional spectrum deviation is the quantity $1-N^*(E_\beta)/N(E_\beta)$, where $N^*(E_\beta)$ and $N(E_\beta)$ are respectively the observed beta spectra with and without excited final state.
In case of tritium atom, the Shrödinger equation can be solved analytically and one gets $\sigma^2=$740.5 eV$^2$. In all the other cases, the final state distributions are estimated numerically.
The situation changes completely in the calorimetric approach. Even in this case the observed spectrum is a combination of different spectra. It can be obtained by operating the following replacements
\begin{equation}
\label{eq:eshift}
\begin{array}{c}
E_\beta \rightarrow E_\beta^\prime = E_\beta - V_i \\ [5.0mm]
p_\beta = (E_\beta^2 - m_e^2)^{1/2} \rightarrow p_\beta^\prime = \left((E_\beta - V_i)^2 - m_e^2 \right)^{1/2}
\end{array}
\end{equation}
motivated by the distinguishing feature of the calorimeters to measure simultaneously the beta electron energy and the de-excitation energy $V_i$ of the final state.
By combining (<ref>) and (<ref>) one gets
\begin{eqnarray*}
\label{eq:spebetacalo0}
& \approx & (E_0 - E_\beta)^2 \left(\displaystyle 1 - \frac{m_{\nu_e}^2}{(E_0 -E_\beta)^2}\right)^{1/2} \\
& & \sum_i{w_i (E_\beta - V_i) \left((E_\beta - V_i)^2 - m_e^2\right)^{1/2} \,\,
F(Z,E_\beta -V_i) S(E_\beta-V_i) }
\end{eqnarray*}
Observing that $F(Z,E_\beta -V_i) S(E_\beta-V_i)\approx F(Z,E_\beta) S(E_\beta)$ and expanding in a series of powers of $V_i/E_\beta$, one obtains
\begin{eqnarray}
\label{eq:spebetacalo1}
& \approx & p_\beta E_\beta (E_0 - E_\beta)^2 \left(\displaystyle 1 - \frac{m_{\nu_e}^2}{(E_0 -E_\beta)^2}\right)^{1/2}
F(Z,E_\beta) S(E_\beta) \nonumber\\
& & \sum_i{w_i \left( 1 - \frac{V_i}{E_\beta} - \frac{V_i E_\beta}{E_\beta^2 - m_e^2} +
\frac{V_i^2}{2(E_\beta^2 - m_e^2)}\right) }
\end{eqnarray}
Apart from the sum term, for a null neutrino mass the equation (<ref>)
describes a beta spectrum with a linear Kurie plot in the final region ($E_\beta \gg V_i$);
Fig. <ref> shows as the influence of the excited final states on the calorimetric beta spectrum is confined at low energy.
Therefore, a calorimeter provides a faithful reconstruction of the beta spectral shape over large energy range below the end-point . This is not true for spectrometers for which the measured spectrum at the end-point presents a deviation of the same size of that caused by a finite neutrino mass. Furthermore it is apparent from Fig. <ref> as the presence of an excited state causes the spectrum of a spectrometer to mimic a lower $E_0$ along with a negative .
The possibility to observe a substantial undistorted fraction of the spectrum is very useful to check systematic effects and to prove the general reliability of a calorimetric experiment.
As a general drawback, calorimeters present a major inconvenience which may be a serious limitation for the approach.
In a calorimeter, the whole beta spectrum is acquired and the detector technology poses important restrains to the source strength.
This in turns limits the statistics that can be accumulated. The consequences on the achievable statistical sensitivity are discussed in the next section.
First of all the counting rate must be controlled to avoid distortions of the spectral shape due to pile-up pulses.
Then the concentration of the decaying isotope maybe not freely adjustable.
For example at the time of J.J. Simpson experiments, the only way to make a sensitive calorimetric measurement was to ion implant tritium
in semiconductor ionization detectors such as Si(Li) or High Purity Ge. There is however a trade off between the required tritium implantation dose
– i.e. the tritium concentration – and the acceptable radiation damage. The tritium activity is then limited by the detector size in relation to
its energy resolution.
This first generation of calorimetric experiments exploited Si(Li) or Ge detectors with implanted tritium, but suffered for their intrinsic energy resolution which is limited to about 200 eV at 20 keV. With these experiments a limit on of about 65 eV was set <cit.>.
At the same time, these experiments showed that the calorimetric approach does not cancel all the systematic uncertainties. As it was already recognized by J.J. Simpson in <cit.>, one source of systematic uncertainty relates to the precise evaluation of the resolution function of these solid states detectors. The resolution function is obtained through X-ray irradiation from an external source. The response of the detector may be different for X-rays entering the detector from one direction and the betas emitted isotropically within the detector volume. Moreover, the beta emission is localized in the deep region of the detector where an incompletely recovered irradiation damage may lead to incomplete charge collection, while X-ray interactions are distributed in the whole detector volume.
Soon it became clear that calorimeters may also be affected by solid states effect. The “17 keV neutrino saga” <cit.> started off from an unexpected feature observed first by J.J. Simpson in the low energy part of the tritium spectrum measured with the implanted Si(Li) detectors <cit.>.
While a neutrino with a mass of 17 keV was finally deemed inexistent and the observed kink ascribed to a combination of various overlooked instrumental effects in spectrometric experiment <cit.>, the evidence in calorimetric measurements remained unexplained.
The invoked explanations include environmental effects in silicon and germanium and remain of interest for future calorimetric experiments.
One of these solid state effects was first described by S.E. Koonin in 1991 <cit.>: it is a solid state effect known as Beta Environmental Fine Structure (BEFS), which introduces oscillatory patterns in the energy distribution of the electrons emitted by a beta isotope in a lattice.
It is an effect analogous to the Extended X-ray Absorption Fine Structure (EXAFS) and it will be addressed in more detail in <ref>.
So far only tritium beta decay was considered, but all the arguments above apply to other isotopes undergoing nuclear beta decay.
In particular, as it will be shown quantitatively in next section, isotopes with a transition energy lower than that of tritium are better suited
for a calorimetric experiment. The rest of the present work will focus on two such isotopes: and , which have a $Q$ around 2.5 keV.
In fact, already in the '80s many authors realized that low temperature detectors could offer a solution for making calorimetric measurements with high energy resolution and could be used either for tritium or, better, the lower $Q$ beta emitters and ( <ref>).
A final remark from the discussion above is that the spectrometer and the calorimeter methods have both complicated but totally different systematic effects. Therefore, once that it is demonstrated that the achievable sensitivities are of the same order of magnitude in the two cases, it is scientifically very sound to develop complementary experiments exploiting these two techniques.
§.§ Sensitivity of calorimeters: analytical evaluation
It is useful to derive an approximate analytic expression for the statistical sensitivity of a calorimetric neutrino mass experiment (see for example <cit.>).
The primary effect of a finite mass $m_{\nu_e}$ on the beta spectrum is to cause the
spectrum to turn more sharply down to zero at a distance $m_{\nu_e}$ below the endpoint
$E_0$ (higher panel of Fig. <ref>).
To rule out such a mass, an experiment must be sensitive to the number of counts expected in this interval.
The fraction of the total spectrum within $\Delta E$ of the endpoint $E_0$ is given by
\begin{equation}\label{}
F_{\Delta \! E}(m_{\nu_e}) =\int^{E_0}_{E_0-\Delta \! E} N_\beta(E,m_{\nu}) d \! E
\end{equation}
For $m_{\nu_e}=0$ this is approximately
\begin{equation}
F_{\Delta \! E}(0) \approx \left( \frac {\Delta \! E} {E_0} \right)^3
\end{equation}
For a finite mass it is found also
\begin{equation}
F_{\Delta \! E}(m_{\nu_e}) \approx F_{\Delta \! E}(0) \left(1 - \frac {3 m_{\nu}^2} {2 \Delta \! E^2} \right)
\end{equation}
In addition to the counting statistics, the effect must be detected in the presence of an external background,
and of the background due to undetected pile-up of two events.
Decays which occur within a definite time interval cannot be resolved by a calorimetric detector, giving rise to the phenomenon of pile-up.
This implies that a certain fraction of the detected events is the sum of two or more single events. In particular, two low energy events can sum up and contribute with a count in the region close to the transition energy, contaminating the spectral shape in the most critical interval.
In a first approximation the external background can be neglected.
The pile-up spectrum can then be approximated by
assuming a constant pulse-pair resolving time, $\tau_R$, such that events with greater separation are
always detected as being doubles, while those at smaller separations are always interpreted as
singles with an apparent energy equal to the sum of the two events. In reality, the resolving time
will depend on the amplitude of both events, and the sum amplitude will depend on the
separation time and the filter used, so a proper calculation would have to be done through a Monte
Carlo applying the actual filters and pulse-pair detection algorithm being used in the experiment. However, this
approximation is good enough to get the correct scaling and an approximate answer.
In practice, $\tau_R$ depends on the high frequency signal-to-noise ratio but it is of the order of the detector rise time.
Effects of pile-up on the experimental energy spectrum of beta decay. Higher panel: beta spectrum compared with pile-up spectrum. Lower panel: zoom around the end-point, with a comparison between 0 and finite neutrino mass beta spectra
With these assumptions, for a pulse-pair resolving time of the detector $\tau_R$, the fraction of events which suffer with not-identified pile-up of two events for a Poisson time distribution is
\begin{equation}
\label{eq:PPU}
P(\Delta t < \tau_R) = 1 - e^{-A_\beta \tau_R} \approx A_\beta \tau_R
\end{equation}
where $A_\beta$ is the source activity in the detector and $\Delta t$ is the time separation between two events.
The beta spectrum of the unresolved pile-up events is given by the convolution product
\begin{equation}
\label{eq:spebpu}
N_{pp}(E) = (1 - e^{-A_\beta \tau_R}) \int_0^{E_0} N_\beta(E^\prime,0)N_\beta(E-E^\prime,0)d \! E^\prime = (1 - e^{-A_\beta \tau_R}) N_\beta(E,0) \otimes N_\beta(E,0)
\end{equation}
The coincidence probability, in a first approximation, is given by $\tau_R A_\beta$.
As shown in the lower part of Fig. <ref>, a fraction $F^{pp}_{\Delta E}$ these events will fall in the region
within $\Delta E$ of the endpoint $E_0$, and can be approximated by
\begin{equation}
F^{pp}_{\Delta E} = \int^{E_0}_{E_0-\Delta} N_{pp}(E) d \! E \approx \tau_R A_\beta \int^{E_0}_{E_0-\Delta \! E} N_\beta(E,0) \otimes N_\beta(E,0)d \! E
\end{equation}
Measuring for a length of time $t_M$, the signal-to-background ratio in the region
within $\Delta E$ of the endpoint $E_0$ can be expressed as
\begin{equation}\label{eq:S2B}
\frac{signal}{background} = \frac{A_\beta N_{det} t_M |F_{\Delta \! E}(m_{\nu_e})-F_{\Delta \! E}(0)|}{\sqrt{A_\beta N_{det} t_M (F_{\Delta \! E}(0)+F^{pp}_{\Delta E})}} = \sqrt{A_\beta T}\frac{|F_{\Delta \! E}(m_{\nu_e})-F_{\Delta \! E}(0)|}{\sqrt{(F_{\Delta \! E}(0)+F^{pp}_{\Delta E})}}
\end{equation}
where $N_{det}$ is the number of detectors and $T=N_{det} t_M$ is the exposure.
This ratio must be about 1.7 for a 90% confidence limit. Therefore, in absence
of background, an approximated expression for the 90% C.L. limit on – $\Sigma(m_{\nu_e})_{90}$ – can be written as <cit.>
\begin{equation}
\label{eq:sensitivity}
\Sigma_{90}(m_\nu) = 1.13 \frac{E_0}{\sqrt[4]{t_{M}A_\beta N_{det}}} \left[ \frac{\Delta E}{E_0} + \frac{3}{10}\frac{E_0}{\Delta E} \tau_R A_\beta \right]^{\frac{1}{4}}
\end{equation}
The two terms in (<ref>) arise from the statistical fluctuations of, respectively, the beta and pile-up
spectrum in (<ref>). Equation (<ref>) shows the importance of improving the detector energy resolution and of minimizing the pile-up by reducing the detector rise time. On the other hand it shows also that the largest reduction on the limit can only come by
substantially increasing the total statistics $N_{ev} = t_{M}A_\beta N_{det}$.
Statistical neutrino mass sensitivity for calorimetric measurements of the beta decay. Lines are obtained from (<ref>) according to <cit.>. Symbols are obtained with Monte Carlo simulations ( <ref>).
If the pile-up is negligible, i.e. when the following condition is met
\begin{equation}\label{eq:condizione}
\tau_R A_\beta \ll \frac{10}{3} \frac{\Delta E^2}{E_0^2}
\end{equation}
from (<ref>) one can write the 90% confidence limit sensitivity as
\begin{equation}\label{eq:S90}
\Sigma_{90}(m_{\nu_e}) \approx1.13 \sqrt[4]{\frac{E_0^3 \Delta E}{ N_{ev} }}
\end{equation}
where energy interval $\Delta E$ in (<ref>) cannot be taken smaller than about 2 times the detector energy resolution
$\Delta E_{\rm FWHM}$.
It is then apparent that to increase the sensitivity one has both to improve the energy resolution and to augment the statistics; however, there is a technological limit to the resolution improvements, thus the statistics $N_{ev} = t_{M}A_\beta N_{det}$ is in fact the most important factor in (<ref>).
For a more complete treatment, also in presence of a not negligible pile-up, refer to
A similar approach for assessing the statistical sensitivity of EC decay cannot be pursued with the same simplicity because of the more complex spectrum (see <ref>).
Nevertheless it is worth anticipating that with some approximations – discussed in <ref> – one can at least
easily show that
\begin{equation}\label{eq:sensho}
\Sigma_\mathrm{EC}(m_{\nu_e}) \propto E_0 - E_{i_{max}}
\end{equation}
where $E_{i_{max}}$ is the energy of the Lorentzian peak whose high energy tail dominates the end-point region, i.e. the M1 peak in (<ref>).
Equation <ref> is to be compared to equation <ref>, which gives
\begin{equation}\label{eq:sensre}
\Sigma_\mathrm{\beta}(m_{\nu_e}) \propto E_0^{4/3}
\end{equation}
From (<ref>) it is apparent that for EC experiments in general – and for in particular – it is not only winning to have the lowest possible $Q$, but the end-point energy must be as close as possible to the binding energy of the deeper shell accessible to the EC.
§.§ LTD for calorimetric neutrino mass measurements
In 1981 A. De Rujula was already discussing with E. Fiorini about the possibility of performing a calorimetric measurement of the electron capture process in , apparently without any useful conclusion. It was only 3 years later – in 1984 – that
two independent seminal papers proposed for the first time the use of
phonon-mediated detectors operated at low temperatures (simply called here low temperature detectors, LTDs)
for single particle detection with high energy resolution. E. Fiorini and T. Niinikoski <cit.> proposed to apply these new detectors to various rare events searches in a calorimetric configuration, while D. McCammon et al. <cit.> initiated the application to X-ray detection. It was immediately clear to D. McCammon et al. that this could be extended to the spectroscopy of an internal beta source by realizing high energy resolution calorimeters with implanted tritium <cit.>.
In 1985, few years after A. De Rujula suggested the use of and for a sensitive neutrino mass measurement in <cit.>,
S. Vitale et al. came up with the first operative proposal for an experiment using LTDs to measure the spectrum calorimetrically <cit.>.
The same year, also N. Coron et al. started a research program aiming at exploiting LTDs to perform the calorimetry of EC decay <cit.>, which was soon discontinued for what concerns .
In the following years, the Genova group pioneered the development of LTDs aimed at a direct neutrino mass measurement using the beta decay.
The experiment was later called MANU and produced its first result in 1992.
Some years later, in 1993, the Milano group, mostly focused on carrying out a search with LTDs, also opened a research line to develop high energy resolution LTDs for a calorimetric measurement of beta decay. This project was named MIBETA and came to the first measurement in 1999.
In 2005, the MANU and MIBETA experiments merged in the international project MARE.
In parallel to the work on , starting from 1995 the Genova group was also carrying on a research for a calorimetric measurement of the EC decay.
This activity, later on, was first absorbed in MARE and then transferred into the HOLMES project.
In 2012, the Heidelberg group, former member of the MARE collaboration, presented its own R&D program for a calorimetric experiment, ECHo.
Recently, also the Los Alamos group started a preliminary work for a experiment, with a project named NuMECS.
All these experiments and projects will be discussed in the next two sections.
It is worth mentioning two other groups that in these three decades participated to the efforts to develop LTDs for neutrino mass measurements.
The Oxford group developed arrays of indium based Superconducting Tunnel Junctions (STJ) to search for the 17 keV neutrino in the $^{63}$Ni beta decay <cit.> and to measure precisely the exchange effect in the low energy part of the spectrum of the same decay <cit.>.
The Duke University group developed Transition Edge Sensors (TESs) based detectors for measuring calorimetrically the tritium decay <cit.>,
but this project was abandoned before obtaining a statistically meaningful sample.
All these activities were triggered in the early '80s by the lucky coincidence to have the need for a tool to perform calorimetric measurements of new low $Q$ beta isotopes just at the time when a new promising particle detection technology was appearing on the scene.
It took more than 20 years to the LTD technology to actually be mature enough to sustain the ambitions of calorimetric neutrino mass experiments
Nowadays, LTDs can indeed deliver to this science case what they have been developed for. In particular, LTDs provide better
energy resolution and wider material choice than conventional detectors.
The energy resolution of few electronvolts is comparable to that of spectrometers and the restrictions caused by the full spectrum
detection are lifted by the parallelization of the measurement with large arrays of detectors.
Still, the detectors time constants of the order of microseconds and, correspondingly, the read-out bandwidth remain the most serious
technical constraint to the full exploitation of LTDs in this field.
§.§.§ LTD basic principles
A complete overview of LTDs can be found in <cit.>, while
the status-of-the-art is well summarized in the proceedings of the bi-yearly international workshop on Low Temperature Detectors <cit.>.
LTDs were initially proposed as perfect calorimeters, i.e. as devices able to thermalize thoroughly the energy released by the impinging particle.
In this approach, the energy deposited by a single quantum of radiation into an energy absorber (weakly connected to a heat sink) determines an increase of its temperature $T$. This temperature variation corresponds simply to the ratio between the energy $E$ released by the impinging particle and the heat capacity $C$ of the absorber, i.e. is given by $\Delta T = E/C$. The only requirements are therefore to operate the device at low temperatures (usually $< 0.1$ K) in order to make the heat capacity of the device low enough, and to have a sensitive enough thermometer coupled to the energy absorber.
Often LTDs with a total mass not exceeding 1 mg and few hundreds micron linear dimensions are called low temperature (LT) microcalorimeters.
In the above linear approximation, using simple statistical mechanics arguments, it can be shown that the internal energy of an LTD weakly linked to
a heat sink fluctuates according to
\begin{equation}
\label{eq:thermodinamicallimit}
\Delta E_{rms}^2 = k_B T^2 C
\end{equation}
where $T$ is the equilibrium operating temperature, $k_B$ is the Boltzmann constant, and independent of the weak link thermal conductance $G$.
Eq. <ref> is often referred to as the thermodynamical limit to the LTD sensitivity and the internal energy fluctuations as Thermodynamic Fluctuation Noise (TFN). Although, strictly speaking, (<ref>) is not the best energy resolution achievable by an LTD,
it turns out that when a sensitive enough thermometer is considered and all sources of broadband noise are included in the calculation, the
real thermodynamical limit of the energy resolution of an LTD can be expressed as <cit.>
\begin{equation}
\label{eq:TLriso}
\Delta E_{rms}^2 = \xi^2 k_B T_b^2 C_0
\end{equation}
where now $T_b$ is the heat sink temperature, $C_0$ is the heat capacity at $T_b$, and $\xi$ is a numerical parameter of order one which is derived
from the LTD thermal details and for the optimal operating temperature. A detailed analysis of the optimal energy resolution for various thermometers
can be found in <cit.>.
From the above and (<ref>) it is evident that the LTD absorber with its $C$ together with the thermometer with its sensitivity are the crucial
ingredients for obtaining high energy resolution detectors.
A sensitive thermometer is the one which allows to transduce the temperature fluctuations of the TFN to a signal larger than the other noise sources
intrinsic to the thermometer itself and to the signal read-out chain.
Today, this condition has been met – and (<ref>) is achieved – for LT microcalorimeters using at least three types of optimized thermometers: semiconductor thermistors, transition edge sensors (TES), and Au:Er metallic magnetic sensors.
The thermal sensor of a LTD does not only affect the achievable energy resolution, but also determines the speed of the detector, i.e. it determines the time scale of the signal formation with the details of the thermal mechanisms entering in the temperature transduction.
Although the detector speed is a crucial parameter in calorimetric neutrino mass experiments, a complete technical treatment for the three sensor technologies is out of the scope of present work. Here it is enough to say that the three technologies above are sorted from the slowest to the fastest: the numerical values for the achievable speeds (from hundreds of nanoseconds to hundreds of microseconds) will be given in the following sections.
Each sensor technology has its pros and cons which have driven the choice for the various neutrino mass experiments. The traded off parameters, which include the achievable performances, the ease of fabrication, and the read-out technology, will be discussed in <ref>.
Next section is dedicated to the other critical component, i.e. the absorber.
§.§.§ Energy absorber and thermalization process
Under many respects, the absorber of LTDs plays the most crucial role in calorimetric experiments.
First of all (<ref>) shows that the absorber heat capacity $C$ sets the achievable energy resolution.
When designing LTDs, usually the absorber is chosen to be made out of a dielectric and diamagnetic material, so that $C$ is described
only by the Debye term, which is proportional to $(T / \Theta_D)^3$ at low temperatures, and can be extremely small for a good material with large
Debye temperature $\Theta_D$. Insulators and semiconductors are often good examples of suitable dielectric and diamagnetic materials.
Metals are instead discarded because of the electron heat capacity, which is proportional to $T$ and remains large also at very low temperatures, thereby dominating the total $C$ of the absorber.
Superconductors are in principle also suitable, since the electronic contribution to the specific heat vanishes exponentially below the critical temperature $T_c$, and only the Debye term remains.
For microcalorimeters, the situation is different because their reduced size allows to tolerate also the heat capacity of a metal so that other considerations may be adopted to select the absorber material.
Microcalorimeters for calorimetric measurements of the tritium, , or , decay spectra must contain the unstable isotope in their absorbers.
As it will be discussed in more details in the following, while tritium and can be included by various means in materials with no special relation with hydrogen or holmium, is naturally found in physical and chemical forms suitable for making LTDs, i.e. superconducting metal and dielectric compounds.
In addition to the electronic and phononic heat capacities considered above, other contributions caused by nuclear heat capacity or by impurities may become important in certain conditions <cit.>. As it will be discussed later, this can be the case for metallic rhenium and embedded .
The above leads to the conclusion that there is large a flexibility in the choice of the material for the absorber of microcalorimeters for calorimetric neutrino mass experiments: dielectrics and normal or superconducting metals have all indeed been used.
In spite of this apparent flexibility, in such experiments it turned out that the ideal energy resolution (<ref>) is quite hard to achieve
because of the details of the chain of physical processes which transform the energy deposited as ionization into the altered equilibrium thermal distribution of phonons – i.e. the $\Delta T$ above – sensed by the thermometers. This chain – also called the thermalization process – is responsible for the introduction of a fluctuation in the deposited energy $E$ which is finally converted in the measured $\Delta T$: the so-called thermalization noise.
The chain starts with the hot electron-hole pairs created by the primary ionizing interaction: on a time scale of 10$^{-10}$ s, this energy is degraded and partitioned between colder electronic and phononic excitations by means of electron-electron and electron-phonon scattering.
The chain then proceeds with the conversion of the electronic excitations into phonons accompanied by a global cooling of all excitations, and it ends
with the new thermal distribution of phonons which corresponds to a temperature increase $\Delta T$ above the equilibrium operating temperature.
The total time scale of this latter process and its details strongly depend on the material.
Only when the time elapsed between the primary interaction and the signal formation is long enough to allow the phonon system to relax to the new quasi-equilibrium distribution, the detector works really as a calorimeter. In commonly used thermal sensors, the measured physical quantity is sensitive
to the temperature of sensor electrons: therefore, at the end of the thermalization there must be a last heat flow from the absorber phonons to the sensor electrons through a link which ultimately acts as a throttle for the signal rise.
The extra noise shows up every time the deposited energy is not fully converted into heat <cit.>, i.e. into the new quasi-equilibrium thermal distribution, and gets trapped in long – compared to thermalization and signal formation time scales – living excitations.
In a simplified picture, if $\eta$ is the fraction of deposited energy $E$ which actually goes into heat, the achievable energy resolution may be written as
\begin{equation}
\label{eq:thermnoise}
\Delta E_{rms}^2= \xi^2 k_BT_c^2C_0 + (1-\eta)^2 EF\omega
\end{equation}
where $F$ is the Fano factor and $\omega$ is the average excitation energy of the long living states. The second term is given by the statistical fluctuation
of the number of long living states.
The parameters $\eta$ and $\omega$ are peculiar of each type of material. The parameter $\eta$ may depend on the operating temperature and on the signal
time scale: often, the thermalization slows down at low temperatures and the signal time scale must be adapted accordingly.
Under all these respects, metals are the ideal material because they show fast and complete thermalization at every temperature – i.e. $\eta=1$ is achieved on time scales of the order of nanoseconds or less – thanks to the strong interactions between electrons and phonons.
Microcalorimeters with metallic absorbers in electrical contact with the sensor are often called hot-electron microcalorimeters <cit.>. In hot-electron microcalorimeters the thermalization ultimately warms up the absorber electronic system and the hot absorber electrons can directly warm up the sensor electrons without throttling, therefore showing a very fast response time.
On the contrary, dielectrics often suffer of a large thermalization noise translating in a degraded energy resolution which increases with the deposited energy. In dielectrics, impurities and defects can act as traps which lie energetically inside the forbidden band-gap. Following the primary
ionization created by the incident particle, electrons and holes can get trapped before their recombination to phonons. Experimentally it is found that
$\omega$ can be as large as few tens of electronvolts, so that the second term (<ref>) may easily dominate the energy resolution also
for values $\eta$ approaching unity.
Semiconductors may be better than dielectrics, owing to their smaller band-gap.
But only metals, semi-metals (such as bismuth) and zero-gap semiconductors (such as HgTe) have been successfully employed in microcalorimeters showing energy resolutions close to the thermodynamical limit (<ref>) <cit.>.
In principle, superconductors should provide a further improvement thanks to their band-gap of few millielectronvolts: unfortunately, the thermalization in superconductors is a complex process in which $\eta$ can be very small.
In superconductors the electronic excitations produced in the thermalization process described above are broken Cooper pairs, also called
quasi-particles <cit.>.
Microscopic calculations from the Bardeen-Cooper-Schrieffer theory predict that, indeed, a large part of the energy released inside the
absorber can be trapped in quasi-particles states which can live for many seconds at temperatures below 0.1 K.
The energy release inside a superconductor leads to a long living state far from equilibrium in
which many Cooper pairs are continuously broken by phonons produced when quasi-particles recombine.
A model describing this situation was proposed by S.B. Kaplan <cit.>,
who found that the time the quasi-particles need to recombine ($\tau_{qp}$) would be somewhere between 1 and 10 seconds. Analogous results were
obtained by the analysis of A.G. Kozorezov <cit.>.
The global result of these models is that in superconductors $\eta$ is expected to be very small on a time scale useful for a LTD.
Despite of these theoretical considerations, it is an experimental fact that some superconducting materials perform well as absorbers in cryogenic detectors.
Indeed, deviations from the predicted temperature dependence of quasi-particle lifetime $\tau_{qp}$ have been reported: for example,
tin has been used for making LTDs with an energy resolution approaching the thermodynamical limit, thanks to a fast and complete energy thermalization. This is apparently one characteristic shared also by other soft superconductors such as lead an indium.
So far, no generally accepted explanation has been given for these apparent discrepancies between experimental results and theory, and the topic
of quasi-particles recombination in LTDs remains an active field of research.
There are two other important sources of energy resolution degradation which are often observed in LTDs <cit.>.
The first is the escape from the absorber of high energy phonons during the first stages of the thermalization process, which adds another fluctuation component to the finally thermalized energy.
The second is the accidental direct detection of high energy phonons by the thermal sensors, which determines an excess systematic broadening of the energy resolution because its probability varies with the interaction position.
§.§.§ Temperature sensors, read-out, and signal processing
The LTDs used for neutrino mass calorimetric measurements fall in the category of the low temperature microcalorimeters and are designed to
provide energy resolutions better than about 10 eV, possibly approaching the thermodynamical limit.
As shown in <ref>, the detector speed – i.e. the detector signal bandwidth, or its rise time $\tau_R$ – is another
parameter guiding the design.
Furthermore, neutrino mass experiments with LTDs need to use large arrays of detectors. This calls for ease of both fabrication and signal read-out.
Along with the selection of the absorber material containing the source, the above are the main guidelines for the design of an LTD based neutrino mass experiment. The choice of the sensor technology is one of the first steps in the design.
To date, only three technologies have been exploited. These are the semiconductor thermistors, the transition edge sensors, and the magnetic metallic sensors, and they will be briefly discussed here (more details can be found in <cit.>). The possibility of employing other technologies, such as the one of superconducting microwave microresonators, is also investigated but its perspectives are not clear yet <cit.>.
The application of LTDs to the spectroscopy of and decays fully overlaps the range of use of microcalorimeters developed for soft X-ray
spectroscopy; therefore, in the following the discussion will be restricted to thermal sensors for X-ray detection.
Constant current biasing of thermistors (left). Temperature dependence of the thermistor resistivity for a $T_0$ of 3 K (blue) and 10 K (red) (right).
Semiconductor thermistors
These sensors are resistive elements with a heavy dependence of the resistance on the temperature.
Usually, they consist of small crystals of germanium or silicon with a dopant concentration slightly below the metal-to-insulator transition <cit.>.
The sensor low temperature resistivity is governed by variable range hopping (VRH) conduction and it is often well described by the
expression $\rho(T)=\rho_0 \exp(T_0/T)^{1/2}$, where $T_0$ and $\rho_0$ are parameters controlled by the doping level <cit.> (Fig. <ref>).
Semiconductor thermistors are high impedance devices – 1-100 M$\Omega$ – and are usually parameterized by the sensitivity $A$, defined as $-d \log R / d \log T$, which typically ranges from 1 to 10. Semiconductor thermistors can be realized also in amorphous film form, like NbSi.
Silicon thermistors are fabricated using multiple ion implantation in high purity silicon wafers to introduce the dopants in a thin box-like volume defined by photolithographic techniques. Germanium thermistors are fabricated starting from bulk high purity germanium crystals doped by means of neutron irradiation (nuclear transmutation doping, NTD) <cit.>. Single NTD germanium sensors are obtained by dicing and further
processing using a combination of craftsmanship and thin film techniques.
In early times, the weak coupling to the heat sink was provided by the electrical leads used for the read-out; nowadays, microelectronic planar technologies and silicon micro-machining are used to suspend the sensors on thin silicon nitride membranes or thin silicon beams.
Thermistors are read-out in a constant current biasing configuration which allows to convert the thermal signal $\Delta T$ in a voltage signal $\Delta V$ (Fig. <ref>). Because of their high impedance, thermistors are best matched to JFETs.
Semiconductor thermistor present few drawbacks. First of all their high impedance requires the JFET front end to be placed as close as possible – centimeters – to the devices to minimize microphonic noise, and bandwidth limitations due to signal integration on parasitic electrical capacitance. Since commonly used silicon JFET must operate at temperatures not lower than about 110 K, this becomes quickly a technical challenge when increasing the number of detectors. Secondly, it has been experimentally observed that conductivity of semiconductor thermistors deviates from linearity at low temperatures <cit.>. The deviation is understood in terms of a finite thermal coupling between electrons and phonons, whose
side effect is to intrinsically limit the signal rise times to hundreds of microseconds for temperatures below 0.1 K.
Semiconductors are now an established and robust technology, and arrays of microcalorimeters based on these devices have been widely used for X-ray spectroscopy <cit.> achieving energy resolutions lower than 5 eV with tin or HgTe absorbers.
Constant voltage biasing of a TES (left). Temperature dependence of the TES resistivity at $T_c$ (right)
Superconducting transition edge sensors (TESs)
TES are also resistive devices made out of thin films of superconducting materials whose resistivity changes sharply from 0 to a finite value in a very narrow temperature interval around the critical temperature $T_c$ (Fig. <ref>).
The superconducting material can be an elemental superconductor (such as tungsten or iridium), although it is more often a bilayer made of a normal metal and a superconductor.
With bilayers, the $T_c$ of the superconductor is reduced by the proximity effect and can be controlled by adjusting the relative thicknesses of the two layers.
Common material combinations used to fabricate TES bilayer with a $T_c$ between 0.05 and 0.1 K are Mo/Au, Mo/Cu, Ti/Au or Ir/Au.
TES fabrication exploits standard thin film deposition techniques, photolithographic patterning, and micro-machining. Sensors can be designed to have, at the operating point, a sensitivity $A$ as high as 1000 and a resistance usually less than 1 $\Omega$. The most common ways to isolate TES microcalorimeters
from the heat sink are the use of thin silicon nitride membranes or thin silicon beams.
TES are read-out at a constant voltage and their low impedance is ideal to use SQUIDs to amplify the current signal induced by a particle interaction (<ref>). The constant voltage biasing provides the condition to achieve the extreme electro-thermal feedback (ETF) regime <cit.> which leads to substantial improvements in resolution, linearity, response speed, and dynamic range. This regime also eases the operation of large pixel count arrays because ETF produces a self-biasing effect that causes the temperature of the film to remain in stationary equilibrium within its transition region.
With respect to semiconductor thermistors, TESs offer many advantages: 1) large arrays can be fully fabricated with standard micro-fabrication processes, 2) the larger electron-phonon coupling allows signal rising as fast as few microseconds, and 3) the low impedance reduces the sensitivity to environmental mechanical noise.
The main drawbacks of TESs are the limited dynamic range, the adverse sensitivity to magnetic fields of TES and SQUID, and the not fully understood physics of superconducting transitions and excess noise sources <cit.>.
TES microcalorimeter arrays are being actively developed as X-ray spectrometers for many applications, which include material analysis and X-ray astrophysics <cit.>. TES sensors are particularly well suited to be coupled to metallic (gold) or semi-metallic (bismuth) absorbers, providing fast response and energy resolutions lower than few electronvolts (Fig. <ref>).
Read-out of a paramagnetic sensor (left). Temperature dependence of the sensor magnetization (right)
Magnetic metallic sensors
These sensors are quite different from the previous two, and their successful development is more recent <cit.>.
They are paramagnetic sensors exposed to a small magnetic field. The temperature rise $\Delta T$ causes a change in the sensor magnetization,
which is sensed by a SQUID magnetometer. The non dissipative read-out scheme avoids the noise sources typical of dissipative systems, such as
the Johnson noise of semiconductor thermistors and of TESs.
State of the art sensors use Er$^+$ paramagnetic ions localized in a Au metallic host (Au:Er sensors). The use of a metallic host
ensures a very fast sensor response time, since the spin-electron relaxation time for Au:Er is around 0.1 at about 0.05 K.
Microcalorimeters with magnetic metallic sensors (Magnetic Metallic Calorimeters, MMCs) are usually fully made out of gold to obtain both a fast and
efficient energy thermalization to the absorber electronic system and a quick equilibration with the sensor electrons. Despite its high sensitivity, the paramagnetic sensor has an intrinsically large heat capacity: therefore the gold absorbers may be relatively large without adversely affecting the MMC performance.
These microcalorimeters, in general, do not need special measures for thermally isolating the devices from the heat sink because the signal is predominantly
developed in the electronic system and the electron-phonon coupling is rather weak and slower at low temperatures.
An interesting feature of MMCs is the availability of a complete and successful modeling, which allow a precise design tailored to each specific application.
The micro-fabrication of MMCs is somewhat more cumbersome than for TES microcalorimeters but, for the large part, can be carried out with a standard micro-fabrication process <cit.>.
Presently, the most used design for arrays of MMCs has planar sensors on meander shaped pickup coils and achieves record energy resolutions of few electronvolts for soft X-rays (Fig. <ref>) accompanied by large dynamic range and good linearity.
Energy resolutions achieved with a TES ($\Delta E_\mathrm{FWHM}=1.5$ eV, courtesy of NIST) (left) and with an MMC <cit.> (right).
Signal read-out
Neutrino mass experiments are necessarily carried out using LTD arrays with a large pixel count, and this calls for the implementation of an efficient multiplexing system for reading out many sensors with the smallest possible number of amplifiers. This, in turn, reduces the number of read-out leads from
room temperature to the array and the power dissipation at low temperature.
Therefore, in order to be of some use for future experiments, a sensor technology must be compatible with some sort of multiplexed read-out,
not causing restrictions on the available signal bandwidth and degradation of the resolving power. This makes the semiconductor thermistors
not appealing for sensitive neutrino mass experiments. The opposite is true for the other two technologies owing to the use of a SQUID read-out.
TES arrays with SQUID read-out can be multiplexed according to three schemes <cit.>: Time Division (TDM) <cit.>, Frequency Division (FDM) <cit.> and Code Division (CDM) <cit.>. The three schemes differ by the set of orthogonal modulation functions used to encode the signals. TDM and FDM (in the MHz band) are the most mature ones, and they have been already applied to the read-out of many multi-pixel scientific instruments. The more recently developed CDM combines the best features of TDM and FDM, and is useful for applications demanding fast response and high resolution.
Recent advancements on microwave multiplexing ($\mu$MUX) suggest that this is the most suitable system for neutrino mass experiments, since it provides a larger bandwidth for the same multiplexing factor (number of multiplexed detector signals).
It is based on the use of rf-SQUIDs as input devices, with flux ramp modulation <cit.> (Fig. <ref>). The modulated rf-SQUID signals are read-out by coupling the rf-SQUID to superconducting LC resonators in the GHz range and using the homodyne detection technique. By tuning the LC resonators at different frequencies it is straightforward to multiplex many RF carriers.
The feasibility of this approach has been demonstrated in <cit.> only with two channels, but it is making quick progresses
as shown with the multiplexed arrays of TES bolometers for millimeter astronomy of MUSTANG2 <cit.>.
The $\mu$MUX is suitable for a fully digital approach based on the Software Defined Radio (SDR) technique <cit.>. The comb of frequency carriers is generated by digital synthesis in the MHz range and up-converted to the GHz range by $IQ$-mixing. The GHz range comb is sent to the cold $\mu$MUX chips coupled to the TES array through one semi-rigid cryogenic coax cable, amplified by a cryogenic low noise High Electron Mobility Transistor (HEMT) amplifier, and sent back to room temperature through another coax cable.
The output signal is down-converted by $IQ$-mixing, sampled with a fast analog-to-digital converter, and digital mixing techniques are used to recover the signals of each TES in the array (channelization).
Because of their excellent energy resolution combined with a very fast response time, the multiplexed read-out of MMCs is more demanding than for TESs. To date, although some results have been obtained also with TDM, $\mu$MUX is the most promising approach for multiplexing MMCs <cit.>, even if its development for these devices is still in progress.
Circuit schematic for a two channel microwave multiplexed read out of a TES array (From <cit.>).
Signal processing
One of the conditions to obtain a thermodynamically limited energy resolution is to process the microcalorimeter signals with the Optimal Filter (OF) <cit.>. For this purpose the signal waveforms must be fully digitized and saved to disk for further
off-line processing. This approach allows also to apply various specialized signal filters to the same waveform, with the aim of improving time resolution – thereby reducing $f_{pp}$ –, rejecting spurious events, and gating background induced events with a coincidence analysis.
The storage of the raw data needed for off-line signal processing and for building the energy spectrum to be analyzed sets a practical limit to the lower energy limit of the final energy spectrum.
While there is no issue for LTD experiments, such as dark matter or searches, to save digitized waveforms for later off-line analysis,
this becomes quickly unpractical for sub-eV neutrino mass experiments. The pulses collected over the whole spectrum would amount to about $10^{14}$, and digitizing pulses
with 512 samples with 16 bit depth would translate in a storage of about 100 PB (un-compressed), way more than LHC data.
The only viable strategy is then to save only the relevant event parameters calculated by the pulse processing software. These parameters
must include at least the energy, the arrival time, and a couple of other useful shape parameters.
Assuming one can limit the useful parameters to 5 and that each is saved as a 4 bytes number, then the required storage reduces to about 1 PB,
which is still quite a large but manageable number.
It looks therefore likely that only fractions of the spectrum of the order of 10% will be available for a neutrino mass analysis.
For and , this means that the analysis will be forcibly limited to an energy interval which extends, respectively, about 1200 eV and 750 eV – just right of the main peak of the spectrum ( <ref>) – below the spectrum end-point.
§.§ Additional direct neutrino measurements with LTDs
Thanks to the acquisition of the full energy spectrum, LTD calorimeters offer the opportunity to perform other interesting investigations on the data collected for the neutrino mass measurement: these include the searches for massive sterile neutrinos and for the cosmic relic neutrinos
(Cosmic Neutrino Background, C$\nu$B). These by-products of the neutrino mass measurements are largely out of the scope of the present work
and are discussed here briefly only for the sake of completeness.
The finite neutrino mass manifesting in neutrino oscillations is already an important breach in the Standard Model of fundamental interactions, but the neutrino sector could hold more surprises. In fact, recent re-analysis of existing
data from reactor oscillation experiments together with some anomalies observed in short baseline accelerator oscillation experiments (LSND, MiniBOONE) and in solar experiment calibration with neutrino sources (GALLEX), point to the existence of at least a fourth generation of neutrinos <cit.>.
These hypothetical neutrinos would be sterile in the sense that they would feel only gravitational interactions, along with those induced by mixing with the other ordinary neutrinos.
Combined analysis of the available data from various sources leads to an additional mass splitting of $\Delta m_{sterile}^2 \approx 1$ eV, with
a mixing parameter of about $\sin^2 (2\theta) \approx 0.1$.
Sterile Right Handed neutrinos are indeed introduced naturally when one tries to extend the Standard Model to include the mass of active neutrinos ($\nu$MSM) <cit.>. Moreover sterile neutrinos in the keV mass range are perfect candidate as Warm Dark Matter (WDM) particles <cit.>.
The calorimetric spectra of or are suitable to investigate the emission of heavy sterile neutrinos with a mixing angle $\theta$. Assuming the electron neutrino $\nu_e$ is a mixture of two mass eigenstates $\nu_H$ and $\nu_L$, with masses $m_H \gg m_L$, then $\nu_e = \nu_L \cos \theta + \nu_H \sin \theta$ and the measured energy spectrum is $N(E,m_L,m_H,\theta) = N(E,m_L) \cos^2 \theta + N(E,m_H) \sin^2 \theta$. The emission of heavy neutrinos would manifest as a kink in the spectrum at an energy of $Q - m_H$ for heavy neutrinos $\nu_H$ with masses between about 0 and $Q-E_{th}\lesssim2.5$ keV, where $E_{th}$ is the experimental low energy threshold. It is worth noting that the strategy is of course the very analogous of the one adopted by J.J. Simpson <cit.> and which started off the already mentioned saga of the 17 keV neutrino. Moreover, such search may be affected by systematic uncertainties due to the background and to the ripple observed in the spectrum and caused by the BEFS ( <ref>).
An alternative and possibly more robust approach to the search of sterile neutrino emission in has been proposed in <cit.>.
Cosmology predicts that there are about 55 neutrinos/cm$^3$ in the universe as left-overs of the Big Bang. Their average temperature today is about 1.95 K and therefore their observation is extremely difficult. It has been proposed that the C$\nu$B could be detected via the induced beta decay on beta decaying isotopes: for example $\nu_e +^3$H$ \rightarrow^3$He$ + $e$^-$.
This reaction could be detected as a peak at an energy of $Q+m_{\nu_e}$ in beta decay spectra. The expected rate can be calculated starting from the beta decay lifetime,
and for 100 g of tritium it would be of about 10 counts/year <cit.>. Unfortunately 100 g is 10$^6$ times the amount of tritium contained in KATRIN, and the situation is not more favorable for other isotopes like or .
The reactions $\nu_e +^{187}$Re$ \rightarrow^{187}$Os$ + $e$^-$ and $\nu_e + $e$^- +^{163}$Ho$ \rightarrow^{163}$Dy$^*$ are expected to give
yearly about $10^{-10}$ <cit.> and $10^{-5}$ events per gram of target isotope <cit.>, respectively.
Recently, a dedicated experiment called PTOLEMY has been proposed <cit.>: it combines a large area surface-deposition tritium target, the KATRIN magnetic/electrostatic filtering, LTDs, RF tracking and time-of-flight systems.
The possibility to detect heavy sterile neutrino dark matter (WDM) via the above induced beta decays in and has also been investigated but, again, the expected rates are hopelessly low <cit.>.
§ PAST EXPERIMENTS
§.§ Rhenium experiments with LTDs
was mentioned in <cit.> as interesting alternative to tritium because its transition energy of about 2.5 keV is one of the lowest known. Thanks to this characteristic, the useful fraction of events close to the end-point is $\sim$350 times higher for than for tritium.
In addition, the half lifetime of about $4\times 10^{10}$ years together with
the large natural isotopic abundance (62.8 %) of $^{187}$Re allows to get useful beta sources without any isotopic separation process. The beta decay rate in natural rhenium is of the order of $\sim$1 Bq/mg, almost ideally suited to calorimetric detection with LTDs.
As soon as the idea of developing LTDs for X-ray spectroscopy with energy resolutions caught, metallic rhenium became one of the
most appealing materials also for making the X-ray absorbers.
First of all metallic rhenium is a superconductor with a critical temperature $T_c$ of about 1.69 K therefore, ideally, it is a good candidate for photon detection free of thermalization noise.
Then the combination of high $Z$, high density ($\rho=21.02$ g/cm$^3$), and high Debye temperature ($\Theta_D\approx 417$ K) makes metallic rhenium a unique material for designing X-ray detectors with low heat capacity $C$ – i.e. high sensitivity – and high photon stopping power.
Unfortunately, it became soon clear that metallic rhenium absorbers do not behave as expected and metallic rhenium was abandoned in favor of other more friendly materials as absorber for X-ray microcalorimeters.
The result of early efforts on rhenium absorbers for X-ray microcalorimeters are reported in <cit.>.
The long time constants (up to 100 ms) and a significant deficit in the signal amplitude are the distinguishing features of microcalorimeters
with metallic rhenium absorbers. In the same years the Genova group was finding similar results, as discussed below.
Although the explanation of the observed behavior most probably resides in the superconductivity of rhenium, also the heat capacity
may contribute to poor and inconsistent performance.
In fact, according to <cit.> the specific heat of rhenium in the normal state is given by
\begin{equation}
c(T) = 40.6\,T^{-2} + 0.034\,T^{-3} + 2290\,T + 27 \,T^3 \hbox{$\mu$J/(mol K)}
\end{equation}
where the last two terms are the contributions from normal conduction electrons and phonons, respectively.
The first two terms are due to the nuclear heat capacity, which arises from the interaction between
the large nuclear quadrupole moment of the two natural isotopes of rhenium – both have nuclear spin 5/2 – and
the electrical field gradient at the nucleus in the non-cubic rhenium lattice.
When rhenium is in the superconducting state the nuclear heat capacity term should vanish
since the slow spin-lattice relaxation thermally isolates the nuclear spin system.
In the superconducting state a non-reproducible small fraction of the
normal-state nuclear heat capacity may still be observed if trapped magnetic flux causes regions in the specimen to remain normal.
In spite of these difficulties, research on LTDs with metallic rhenium went on for the purpose of making detectors for calorimetric neutrino mass experiments, although other dielectric materials were also tested (see <ref>).
§.§ The beta decay spectrum
The beta decay
\begin{equation}
^{187}\mathrm{Re} (5/2^+) \rightarrow ^{187}\mathrm{Os} (1/2^-) + e^- + \overline\nu_e \\
\end{equation}
is a unique first forbidden transition. Unlike non-unique transitions, the nuclear matrix element is computable, even if the calculation is not straightforward as in the case of tritium. In the literature, it is possible to find detailed calculations of both the matrix element and the Fermi function for this process <cit.>. The electron and the neutrino are emitted in the $p_{3/2}$ and the $s_{1/2}$ states, respectively, or vice versa. Higher partial waves are strongly suppressed because of the low transition energy.
The distribution of the kinetic energies $E$ of the emitted electrons, calculated neglecting the neutrino mixing is (according to <cit.>)
\begin{eqnarray}
\label{eq:reniospe}
N(E,m_{\nu_e})&=& N^{p_{3/2}}(E,m_{\nu_e})+N^{s_{1/2}}(E,m_{\nu_e})\nonumber\\
&=& C p E (E_0 -E)^2 [ F_1(Z,E)p^2+F_0(Z,E)p_\nu^2] \sqrt{1 - \frac{m_{\nu_e}^2}{(E_0-E)^2}}
\end{eqnarray}
where $p$ is the electron momentum, $p_\nu^2=((E_0-E)^2-m_{\nu_e}^2)$ is the neutrino momentum, and $F_0(Z,E)$ and $F_1(Z,E)$ are the relativistic Coulomb factors, which take into account the distortion of the electron wave function due to the electromagnetic
interaction of the emitted electron in $s_{1/2}$ and $p_{3/2}$ states with the atomic nucleus.
In general, the Coulomb factor takes the form
\begin{eqnarray}
F_{k-1}(Z,E) = \left( \frac{\Gamma (2k+1)}{\Gamma(k) \Gamma(1+2\gamma_k)} \right)^2 (2pR )^{2(\gamma_k -k)} |\Gamma (\gamma_k +iz)|^2 e^{\pi z}
\end{eqnarray}
\begin{eqnarray}
\gamma_k &=& \sqrt{k^2-(\alpha Z)^2} \nonumber \\
z &=& \alpha Z \frac{E}{p}.
\end{eqnarray}
where $\Gamma$ is the gamma function, $\alpha$ is the fine structure constant, and $R$ is the nuclear radius.
It can be found numerically that the $p_{3/2}$ component of the spectrum is dominant, i.e. <cit.>
\begin{equation}
\frac{I_{s_{1/2}}}{I_{p_{3/2}}}=\frac{\int_0^{E_0}N^{s_{1/2}}(E,m_{\nu_e})d \! E}{\int_0^{E_0}N^{p_{3/2}}(E,m_{\nu_e})d \!E} \approx 10^{-4}
\end{equation}
This has been confirmed experimentally in <cit.> (see <ref>).
It can also be shown that (<ref>) can be approximated by the expression
\begin{equation}
N(E,m_{\nu_e})= C^\prime (1 + f_{^{187}Re}(E))(E_0-E)^2 \sqrt{1- \frac{m_{\nu_e}}{(E_0-E)^2}}
\end{equation}
where the correction factor $f_{^{187}Re}(E)$ is shown in Fig. <ref>.
Theoretical shape of the beta decay spectrum (left). Deviation of the spectrum from a simple quadratic form (right).
§.§ Statistical sensitivity and systematics
An accurate assessment of calorimetric neutrino mass experimental sensitivity requires the use of Monte Carlo frequentist approach <cit.>.
The parameters describing the experimental configuration are the total number of decays $N_{ev}$,
the FWHM of the Gaussian energy resolution , the fraction of unresolved pile-up events $f_{pp}$, and the radioactive background $B(E)$.
The total number of events is given by , where $ N_{det}$, $A_\beta$ and $t_M$ are the total number of detectors, the beta decay rate in each detector, and the measuring time, respectively. As discussed in <ref>, $f_{pp}=\tau_R A_\beta$, where $\tau_R$ is the time resolution of the detectors.
The $B(E)$ function is usually taken as a constant, $B(E)=bT$, where $b$ is the average background count rate for unit energy
and for a single detector, and $T=N_{det}\times t_M$ is the experimental exposure.
A set of experimental spectra are simulated and fitted with $m^2_\nu$, $E_0$, $N_{ev}$, $f_{pp}$ and $b$ as
free parameters. The 90% C.L. $m_{\nu_e}$ statistical sensitivity $\Sigma_{90}(m_{\nu_e})$ of the simulated experimental
configuration can be obtained from the distribution of the found by fitting the spectra.
The statistical sensitivity is then given by $\Sigma_{90}(m_{\nu_e}) = \sqrt{1.7 \sigma_{m_{\nu_e}^2}}$, where $\sigma_{m_{\nu_e}^2}$ is the standard deviation of the distribution.
Experimental exposure required for the target statistical sensitivity in the second column with $b=0$.
isotope sensitivity $A_\beta$ $\tau$ $N_{ev}$ exposure $T$
[eV] [Hz] [$\mu$s] [eV] [counts] [detector$\times$year]
0.2 10 3 3 $1.3\times10^{14}$ $4.1\times10^{5}$
0.1 10 1 1 $10.3\times10^{14}$ $3.3\times10^{6}$
The symbols in Fig. <ref> are the results of Monte Carlo simulations for various experimental parameters and for $b=0$ compared to the analytic estimate.
As an example, Tab. <ref> reports two experimental configurations which could allow to achieve statistical sensitivities of about 0.2 and 0.1 eV, respectively. The two sensitivities could be attained measuring for 10 years, respectively, $4\times 10^4$ and $3.2\times 10^5$ detectors, while the total mass of natural metallic rhenium in the two cases would be about 400 g and 3.2 kg, respectively.
A flat background remains almost negligible as long as it is much lower than the pile-up contribution at the end-point, i.e.
$b \ll \approx A_\beta f_{pp} / (2 E_0)$. For the two experiments in Tab. <ref> this translates in a constant background,
lower than about $1\times10^{-2}$ and $4\times10^{-3}$ counts/day/eV, respectively, which should be achievable without operating the arrays in the extreme low background conditions of an underground laboratory.
Given the strong dependence of the sensitivity on the total statistics, for a fixed experimental exposure $T$ – that is, for a fixed measuring time and a fixed experiment size – and for fixed detector performance – and $\tau_R$ –, it always pays out to increase the single detector activity $A_\beta$ as high as technically feasible, even at the expenses of an increasing pile-up level.
Of course, since the rhenium specific activity is practically fixed, the ultimate limit to $A_\beta$ is set by the
tolerable heat capacity of the absorber.
With the same Monte Carlo approach it is also possible to investigate the source of
systematic uncertainties peculiar to the calorimetric technique.
As shown in <cit.>, it appears that the most crucial and worrisome
sources of systematics are the uncertainties related to the Beta Environmental Fine Structure (BEFS), the theoretical spectral shape of the beta decay, the energy response function, and the radioactive background. These sources are briefly discussed in the following.
Monte Carlo computed statistical sensitivity of a decay calorimetric measurement for different experimental configurations and for $N_{ev}=10^{14}$ (left). BEFS ripple prediction for metallic rhenium and for for an infinite instrumental resolution (right).
The BEFS is a modulation of the beta emission probability due to the atomic and molecular surrounding of the decaying nuclei <cit.>, which is
explained by the electron wave structure in terms of reflection and interference.
The BEFS oscillations depend on the inter-atomic distance, while their amplitude is tied to the electron-atom scattering cross-section: although the phenomenon is completely understood, its description is quite complex and the parameters involved are not all known a priori.
So far it could be detected only in the low energy region – $E\lesssim1.5$ keV – of the spectra, where both the beta rate and the BEFS are larger, but
as far as the effect on the neutrino mass determination is concerned, its effect extends way up to the end-point (Fig. <ref>).
For a safe extrapolation up to the end-point and to minimize the systematic uncertainties, the
BEFS must be characterized using much higher statistics beta spectra and independent EXAFS analysis of the material containing rhenium.
The theoretical description of the decay spectrum given in <ref> is slightly contradicted by experimental observation, since
available high statistics spectra are in fact better interpolated as $N(E)=(E_0-E)^2$, i.e. with $f_{^{187}Re}(E)\approx 1$. This deviation from theory
has not found a plausible explanation yet[Fedor Šimkovic, private communication], and it will become troublesome when larger
statistics experiments will call for more accurate description of the spectrum.
The detector response function is probed by means of X-ray sources which are not exactly monochromatic and which do not replicate
the same type of interactions in the absorber as the beta decay. In fact, the X-ray interactions happen at a shallow depth, whereas the beta decays are uniformly distributed in the volume; moreover, in the case of X-rays the energy is deposited by a primary photo-electron followed by a cascade of secondary X-rays and Auger electrons, whereas in beta decay all the energy is deposited along one single track.
It is therefore extremely important – yet challenging – to fully understand the measured response function in order to disentangle the contributions to its shape caused by the external X-rays.
In calorimetric experiments, since the beta source cannot be switched off, the environmental and cosmic background in the energy
range of the beta spectrum cannot be directly assessed. Therefore a constant background is usually included in the fit model as the safest hypothesis. This hypothesis may happen to be not accurate enough for future high statistics measurements.
§.§ MANU
The research program in Genova which lead to the MANU experiment started in 1985 <cit.> with a focus on the
use of metallic rhenium absorbers. At that time there was absolutely no knowledge about the behavior of this material as
absorber for LTDs. Therefore, the first years were devoted to study the heat capacity and the thermalization efficiency $\eta$ of
metallic rhenium.
The outcomes of the preliminary phase are summarized in <cit.>. The thermalization efficiency $\eta$ was
studied for many superconductors in form of small
single crystals (cubic millimeters) of Al, Pb, In, Ti, Nb, Va, Zn and Re, and a quasi universal dependence on the
ratio $T/\Theta_D$ was found, with $\eta$ dropping sharply for $T$ lower than about $2\times 10^{-4}\Theta_D$.
In particular rhenium thermalization was investigated for single- and poly-crystals
between 50 mK and 200 mK. The rise-time was limited to about 200 $\mu$s, preventing to assess the thermalization efficiency at shorter
time scales. Also, for rhenium it was found that full thermalization is attained for an operating temperature above about 83 mK.
The effect of magnetic fields was also investigated, and an unexpected and unexplained reduction of $\eta$ for magnetic fields increasing
up to 20 Gauss was found.
The MANU experiment detector (from <cit.>).
The MANU experiment final spectrum (left) and its fitting residuals showing the BEFS ripples (right) (from <cit.>).
Heavy sterile mass exclusion plot of MANU <cit.> (left). BEFS in the MANU spectrum <cit.> (right).
The first observation of spectrum was reported in <cit.>.
After this, a period was spent to optimize the microcalorimeter performance, also exploiting the gained understanding of
metallic rhenium absorbers: energy resolutions as good as about 30 eV were demonstrated with small – about $50$ $\mu$g – rhenium absorbers.
In 2001 the results of the high statistics measurement of MANU were published <cit.>.
The MANU experiment's microcalorimeter was a NTD germanium thermistor coupled with epoxy resin to a
1.572 mg rhenium single crystal. Two ultrasonically bonded aluminum wires provided both the path for the electrical signal and the thermal contact to the heat sink at 60 mK (Fig. <ref>).
The detector had a thin shield against environmental radiation made out of ancient Roman lead.
A weak $^{55}$Fe source ($10^{-3}$ counts/s) allowed to monitor the gain stability during the measurement, while the energy calibration
was established through a removable fluorescence source emitting the K lines of Cl, Ca, and K.
Signals were read-out by a cold stage with unitary gain using a JFET at about 150 K, digitized at 12 bit in 1024 long records, and processed with an optimal filter. Further processing was used to detect pile-up events <cit.>.
The high statistics measurement lasted for about 3 months and the detector performance are listed in Tab. <ref> <cit.>.
The $^{55}$Fe K$_\alpha$ line had a perfectly Gaussian shape with tails lower than 0.1%. The calibration with the fluorescence source showed
that the energy resolution is practically constant with energy, and the deviation from linearity of the energy response was of about 0.16% at
the spectrum end-point.
The fit of the spectrum (Fig. <ref>) gave a squared
neutrino mass of $m_{\nu_e}^2=-462^{+579}_{-679}$ eV$^2$, which translated in an upper limit $m_{\nu_e} \le26$ eV at 95% C.L., or 19 eV at 90% C.L. <cit.>.
The results reported in <cit.> were the most precise measurements of the transition energy $E_0$ and of the half-life
at the time of publishing; the half-life in particular is of great interest for geochronology for determining the age of minerals and meteorites.
This high statistics measurement allowed also to observe for the first time the BEFS <cit.> and to set a limit for the emission of sterile neutrinos with masses below 1 keV <cit.> (Fig. <ref>).
§.§ MIBETA
The Milano program for a neutrino mass measurement with started in 1992 with an
R&D to fabricate silicon implanted thermistors in collaboration with FBK <cit.>. The final objective was to make large arrays
of high resolution microcalorimeters using micro-machining <cit.>.
NTD germanium based microcalorimeters were also tested.
In the light of the encouraging results obtained at Genova, at first the program concentrated on metallic rhenium absorbers.
Many single- and poly-crystalline samples were tested with disappointing results: small signals, long time constants, and inconsistently varying pulse shapes. A possible correlation with the sample purity and with residual magnetic fields was individuated, but this was not enough to
The research program therefore moved onto the systematic testing
of dielectric rhenium compounds as microcalorimeter absorbers.
From the beginning, the most suitable compounds looked like those based on the ReO$_{4}^{-}$ (perrhenate) anion. A non exhaustive list of tested compounds includes: Re$_{2}$(CO)$_{10}$, K$_{2}$ReCl$_{6}$, (NH$_{4}$)ReO$_{4}$, KReO$_{4}$, AgReO$_{4}$. The tests with Re$_{2}$(CO)$_{10}$ failed since this compound sublimates in vacuum at room temperature. The second-to-fourth materials, despite the good theoretical expectations and the large signal-to-noise ratio, showed a quite poor energy resolution – exceeding 100 eV at 6 keV –
which could be explained as due to a large thermalization noise.
Silver perrhenate (AgReO$_{4}$), on the other hand, immediately exhibited good properties with limited thermalization noise.
The calibration peaks were sufficiently symmetric, and energies resolutions as good as 18 eV FWHM at 6 keV were achieved. AgReO$_{4}$ crystals are transparent, crumbly, and slightly hygroscopic, with a specific activity of about $5.4 \times 10^{-4}$Hz/ <cit.>.
One of the MIBETA microcalorimeters with .
The MIBETA experiment ran between 2002 and 2003 an array of 10 microcalorimeters for a high statistics measurement, which was preceded by a campaign of measurements dedicated to tuning the set-up and to reducing the background <cit.>.
Final MIBETA calibration spectrum (left). Kurie plot of the final MIBETA data (right).
The array was made of crystals with masses ranging from 250 to 300 $\mu$g to
limit event pile-up, for a total mass of 2.683 mg. The crystals were attached to silicon implanted thermistors with epoxy resin, and four ultrasonically bonded aluminum wires were used both as signal leads and heat links to the heat bath, stabilized at 25 mK.
The 10 microcalorimeters were enclosed in two copper holders without lead shieldings, to avoid the background caused by lead fluorescence at 88 keV, which in turn provokes escape peaks in very close to the beta end-point.
The stability and performance of all detectors were monitored with
a movable multi-line fluorescence source at 2 K, which was activated for 25 min every 2 h to emit the
K lines of Al, Cl, Ca, Ti, and Mn. When not used for the calibration, the primary source was pulled
inside a massive shield of ancient Roman lead <cit.>,
in order to minimize the contribution to the radioactive background caused by the IB of .
The data acquisition program controlled the movements of the source and tagged the events collected
during the calibrations.
The first stage of the electronic chain used 10 JFETs cooled to about 120 K and placed few centimeters
below the detectors. A 16-bit data acquisition system digitalized and saved to disk the signals for an Optimal Filter based off-line
The high statistics measurement of MIBETA lasted for about 7 months. In the final analysis, the data from
two detectors, with poorer energy resolution, were not included. The total active mass was therefore 2.174 mg,
for a activity of 1.17 Bq. The final beta spectrum obtained from the sum of the 8 working
detectors corresponds to about 8745 hours$\times$mg <cit.>.
The performance of the detectors were quite stable during the run and are reported in Tab. <ref>.
All X-ray peaks in the calibration spectrum showed tails on the low energy side, and the thermalization noise of caused
their width to increase with the energy (Fig. <ref>).
The fit of spectrum (Fig. <ref>) gave a squared
neutrino mass of $m_{\nu_e}^2=-112\pm 207_{stat} \pm 90_{sys}$, which translates in an upper limit $m_{\nu_e} \le15$ eV at 90% C.L.
The systematic error was dominated by the uncertainties on the energy resolution function, on the background, and on the theoretical
shape of the spectrum.
Escape peaks due to the exposure of microcalorimeters to a $^{44}$Ti source <cit.> (left) BEFS in the MIBETA final spectrum <cit.> (right).
Additional lower statistics measurements with the same set-up were carried out to study and reduce the background, and to investigate the energy response function. In particular, using the escape peaks caused at about 17 keV by the irradiation with a $^{44}$Ti gamma source (see Fig. <ref>) as comparison, it was possible to partly understand the complex shape of the X-ray calibration peaks and to establish that at least the longest of the observed tails were due to surface effects <cit.>.
Although BEFS (see <ref> and Fig. <ref>) is almost one order of magnitude fainter in AgReO$_{4}$
than in metallic rhenium, it was observed also in the high statistics spectra of MIBETA <cit.> (Fig. <ref>). In particular, the BEFS ripple interpolation allowed to determine that $p_{3/2}$ to $s_{1/2}$ branching ratio in the beta emission is $0.84\pm0.30$, which is compatible with the expected prevalent $p_{3/2}$ emission (see <ref>).
Comparison of the MANU and MIBETA experiments.
source / absorber
metallic Re
Si implanted thermistor
number of detectors
total mass [$\mu$g]
1572 (Re)
2174 ()
measuring time $t_M$ [h]
$\approx 2800$
$\approx 8700$
total activity $A$ [Bq]
1.1 (above 350 eV)
1.17 (above 700 eV)
energy resolution [eV]
96 (at 5.9 keV)
28.5 (average at end-point)
rise time [$\mu$s]
$\approx 1000$
492 (average 10-90%)
statistics $N_{ev}$
$6\times 10^{6}$ (above 420 eV)
$6.2\times 10^{6}$ (above 700 eV)
pile-up fraction $f_{pp}$
$2.3\times 10^{-4}$
$E_0$ [eV]
$2470\pm 1_{stat} \pm 4_{sys}$
$2465.3\pm 0.5_{stat} \pm 1.6_{sys}$
$\tau_{1/2}$ [$10^{10}$ y]
$4.12\pm 0.02_{stat} \pm 0.11_{sys}$
$4.32\pm 0.02_{stat} \pm 0.01_{sys}$
$m_{\nu_e}^2$ [eV$^2$]
$-112\pm 207_{stat} \pm 90_{sys}$
90% [eV]
background $b$ [c/eV/day]
$3\times 10^{-4}$
$1.7\times 10^{-4}$
§.§ MARE
The MANU and MIBETA results, together with the constant advance in the LTD technology,
made it reasonable to propose a larger scale project: the Microcalorimeter Arrays for a Neutrino Mass Experiment (MARE).
The ambition of MARE was to establish a sub-eV neutrino mass sensitivity through a gradual deployment approach.
The project was started in 2005 by a large international collaboration <cit.> and it was organized
in two phases.
The final objective of a sub-eV statistical sensitivity on the electron neutrino mass was the goal of the second phase. To accomplish this, the program was to gradually deploy several large arrays – about $10^4$ elements each – of detectors, with energy and time resolutions of the order of 1 eV and 1 , respectively. Each pixel was planned to have a
source activity of about few counts per second in order to collect a total
statistics of about $10^{14}$ beta decays in up to ten years of measurement time (see Fig. <ref>) <cit.>. Fig. <ref> shows also the MARE sensitivity to the emission of heavy sterile neutrinos with masses below 2 keV ( <ref>).
Statistical sensitivity of MARE final experiment. Curves are calculated for the deployment of 10000 pixels per year for the first 5 years (left). Statistical sensitivity to the emission of heavy neutrinos with mass $m_H$. From lower to upper curve: Monte Carlo simulation with $N_{ev}=10^{14}$, $f_{pp}=10^{-6}$ and $\Delta E = 1.5$ eV; Monte Carlo simulation with $N_{ev}=8\times10^{9}$, $f_{pp}=10^{-5}$ and $\Delta E = 15$ eV; MIBETA experiment (unpublished); MANU experiment <cit.> (right).
Phase 1 – also called MARE-1 – had the task to ascertain the most suitable technical approach for the final experimental phase, also with the help of smaller scale experiments.
An R&D program was started with the aim to improve the understanding of the superconducting rhenium absorbers and of their
optimal coupling to sensors, and to develop the appropriate array technology and multiplexed read-out scheme <cit.>.
At the same time, two intermediate size experiments carried out with the available technologies aimed to reach a
neutrino mass sensitivity of the order of 1 eV, and to improve the understanding of all the systematics peculiar of the calorimetric approach with .
Furthermore, MARE-1 started to explore the alternative use of for a calorimetric measurement of the neutrino mass.
Given the unavoidable competition with the KATRIN experiment, the time schedule for MARE was quite tight.
The physics of metallic rhenium as absorber for the MARE detectors was the focus of the Genova and the Heidelberg groups.
The best technologies available for the MARE-2 arrays were: 1) the Transition Edge
Sensors (TES) with Frequency Division Multiplexing, investigated by the Genova group and Physikalisch-Technische Bundesanstalt (PTB, Berlin, Germany); 2) the Metallic Magnetic Calorimeters (MMC) with Microwave Squid Multiplexing, developed by the Heidelberg group; and 3) Microwave Kinetic Inductance Detectors (MKID) with Microwave Multiplexing, explored by the Milano group.
The Genova group, in collaboration with Miami and Lisbon, planned an experiment consisting of an array of 300 TES detectors, with a total mass of about 1 mg of rhenium single crystals <cit.>.
With energy and time resolutions of about 10 eV and 10 , respectively, the sensitivity attainable in 3
years of measuring time was estimated to be around 1.8 eV at 90% C.L., for a statistics of about $3\times 10^{10}$ decays.
The Milano group, together with the NASA/GSFC and Wisconsin groups, deployed an array of silicon
implanted thermistors coupled to absorbers.
The experiment used 8 of the 36 pixel arrays that NASA/GSFC had developed for the XRS2 instrument <cit.>.
With 288 pixels attached to about 500 crystals, and with energy and time resolutions of about 25 eV and 250 ,
respectively, a sensitivity around 3.3 eV at 90% C.L. was expected in 3
years of measuring time, with a statistics of about $7 \times 10^{9}$ decays.
Unfortunately, the MARE-1 outcomes were quite disappointing, and MARE-2 ended up canceled before taking off.
The lack of success of the MARE initiative was mostly the consequence of the final acknowledgment of the impossibility
to fabricate rhenium microcalorimeters matching the specifications set by the aimed sub-eV sensitivity.
The systematic investigations carried out at Heidelberg with rhenium absorbers coupled to MMC, despite some noteworthy
progress, arrived to conclusions similar to that of past works: rhenium absorbers behave inconsistently, showing
a large deficit in the energy thermalization accompanied by long time constants <cit.>.
Therefore the challenging idea of improving and scaling up the pioneer experiments
using metallic rhenium absorbers turned out to be a dead end road.
Indeed, also the other experimental efforts of MARE-1 encountered several difficulties <cit.>.
For example, the setting up of arrays of crystals turned out to be more troublesome than expected.
The freshly polished surfaces of crystals shaped to cuboids resulted to be incompatible with the sensor coupling methods used successfully in MIBETA with as-grown small crystals. Despite the use of a micro-machined array of silicon implanted thermistors, the performance of the pixels were irreproducible and, while gradually populating and testing the XRS2 array with crystals, the performance of the instrumented pixels started to degrade.
This made the array finally unusable. Given the sorts of the MARE project, also this branch of the MARE-1 program was thus dropped in 2013.
31 crystal glued on the first XRS2 array of MARE-1 (left).
The 16 usable pixel give $\approx 47$ eV at 2.6 keV, $\tau_R \approx 1$ ms. Spectrum measured with the best pixel (right).
§.§ Future of rhenium experiments
From the MARE experience, it is clear that a large scale neutrino mass experiment based on beta decay is not foreseeable in the
near future. It would require a major step forward in the understanding of the superconductivity of rhenium but, after more than 20
years of efforts, this is not anymore in the priorities of the LTD scientific community.
Besides the intrinsic problems of metallic rhenium, there are other considerations which make rhenium microcalorimeters not quite an appealing choice for high statistics measurements.
Because of the large half life of , the specific activity of metallic rhenium is too low to design pixels with both high performance and
high intensity beta sources. The activity required by a high statistics experiment must be therefore distributed over a large number of pixels – of the order of $10^5$ – while the difficulties inherent with the production of high quality metallic rhenium absorbers contrast with the full micro-fabrication of the arrays. MARE-1 also demonstrated that is not a viable alternative to metallic rhenium.
For these reasons the new hope for a calorimetric neutrino mass experiment with LTDs is .
§ CURRENT EXPERIMENTS
§.§ Calorimetric absorption spectrum of EC
A. De Rujula introduced the idea of a calorimetric measurement of EC decay already in 1981 <cit.>, but
it was only one year later that this idea was fully exploited in the paper written with M. Lusignoli <cit.>.
The EC decay
\begin{equation}
^{163}\mathrm{Ho} + e^- \rightarrow ^{163}\mathrm{Dy} + \nu_e
\end{equation}
has the lowest known $Q$ value, around 2.5 keV, and its half-life of about 4750 years is much shorter than the $^{187}$Re one.
In <cit.> the authors compute the calorimetric spectrum and give also an estimate of the statistical sensitivity to the neutrino mass at the spectrum end-point, including the presence of the pile-up background.
Unfortunately, at that time the experimental measurements of the $Q$ value were scattered between
2.3 keV to 2.8 keV causing a large uncertainties on the achievable statistical sensitivity.
Calculated EC calorimetric spectrum for $Q=2.3$ keV (blue) and $Q=2.8$ keV (red), and for $\Delta E=2$ eV and $N_ev=10^{14}$.
A calorimetric EC experiment records all the de-excitation energy and therefore it measures the escaping neutrino energy $E_\nu$ – see (<ref>). The de-excitation energy $E_c$ is the energy released by all the atomic radiation emitted in the process of filling the vacancy left by the EC decay, mostly electrons with energies up to about 2 keV
(the fluorescence yield is less than $10^{-3}$) <cit.>. The calorimetric spectrum has lines at the ionization energies $E_i$ of the captured electrons. These lines have a natural width $\Gamma_i$ of a few eV, therefore the actual spectrum is a continuum with marked peaks with Breit-Wigner shapes (Figure <ref>).
The spectral end-point is shaped by the same neutrino phase space factor $(Q-E)\sqrt{(Q-E)^2-m_{\nu_e}^2}$ that appears in a beta decay spectrum, with the total de-excitation energy $E_c$ replacing the electron kinetic energy $E$.
For a non-zero $m_{\nu_e}$, the de-excitation (calorimetric) energy $E_c$ distribution is expected to be
\begin {eqnarray}
\label{eq:E_c-distr}
N_{EC}(E_c,m_{\nu_e}) = {G_{\beta}^2 \over {4 \pi^2}}(Q-E_c) \sqrt{(Q-E_c)^2-m_{\nu}^2} \sum_i n_i C_i \beta_i^2 B_i {\Gamma_i \over 2\,\pi}{1 \over (E_c-E_i)^2+\Gamma_i^2/4}
\end{eqnarray}
where $G_{\beta} = G_F \cos \theta_C$ (with the Fermi constant $G_F$ and the Cabibbo angle $\theta_C$), $E_i$ is the binding energy of the $i$-th atomic shell, $\Gamma_i$ is the natural width, $n_i$ is the fraction of occupancy, $C_i $ is the nuclear shape factor,
$\beta_i$ is the Coulomb amplitude of the electron radial wave function (essentially, the modulus of the wave function at the origin) and $B_i$
is an atomic correction for electron exchange and overlap.
The sum in (<ref>) runs over the Dy shells which are accessible to the EC with the available $Q$ (M1, M2, N1, N2, O1, O2, and P1).
The expression (<ref>) is derived in <cit.>, where numerical checks to test the validity of the approximations made are also presented.
Until about 2010 only three calorimetric absorption measurements were reported in the literature:
* the ISOLDE collaboration used a Si(Li) detector with an implanted source <cit.>;
* Hartman and Naumann used a high temperature proportional counter with organometallic gas <cit.>;
* Gatti et al. used a cryogenic calorimeter with a sandwiched source <cit.>.
However, none of these experiments had the sensitivity required for an
end-point measurement, therefore they all gave results in terms of capture rate ratios.
The most evident limitations of these experiments were statistics and energy resolution.
One further serious trouble for the Si(Li) and the cryogenic detectors was the incomplete energy detection caused by implant damages and weak
thermal coupling of the source, respectively.
Recently a new generation of calorimetric holmium experiments has been stimulated by the MARE project.
In fact, despite the shortcomings of previous
calorimetric experiments, and theoretical and experimental uncertainties, a calorimetric absorption
experiment seems the only way to achieve sub-eV sensitivity for the neutrino mass.
Moreover, low temperature X-ray microcalorimeters have reached the necessary maturity to
be used in a large scale experiment with good energy and time resolution, hence they are the detectors of choice for a
sub-eV holmium experiment.
Thanks to the short lifetime, the limited number of nuclei needed for a neutrino mass experiment – $10^{11}$ nuclei for 1 decay/s –
can be introduced in the energy absorber of a low temperature microcalorimeter.
Therefore, holmium experiments can leverage the microcalorimeters development for high energy resolution soft X-ray spectroscopy, whereas rhenium experiments would need a dedicated development of detectors with metallic rhenium absorbers.
Small footprint kilo-pixel arrays can be fully fabricated with well established micro-fabrication techniques.
Indeed, in microcalorimeters with metallic absorbers such as gold, the relatively high concentration of holmium (J = 7/2) could cause an excess heat capacity due to hyperfine level splitting in the metallic host <cit.> and thereby degrade the microcalorimeter performance. Low temperature measurements have been already carried out in the framework of the MARE project to assess the gold absorber heat capacity (at temperatures $<150$ mK), both with holmium and erbium implanted ions <cit.>. Those tests did not show any excess heat capacity, but more sensitive investigations need to be carried out.
The Genova group pioneered the application of LTDs to the measurement of the calorimetric spectrum <cit.>
and continued this research until it converged in the MARE project <cit.>. For long, the focus has been on the production of the isotope, on the chemistry of metallic holmium, and on the techniques to embed the isotope in the detector absorbers.
The new experiments, now ready to start the production of high resolution detectors for the high statistics
calorimetric measurement of the EC decay spectrum, will be the subject of next sections.
§.§ The $Q$ value of decay
Until very recently, the question of the exact $Q$ value of the EC decay was not settled.
Although the results showed a general tendency to accumulate around 2.8 keV, especially restricting to the calorimetric measurements <cit.>, the reliability of the capture ratios as tool for determining $Q$ remained questionable.
Indeed, the $Q$ has never been measured directly from the end-point of the EC spectrum, but only from the capture ratios $\lambda_i/\lambda_j$, whose accuracy is limited ( <ref>). The currently recommended value of $Q$ is 2.555$\pm$0.016 keV <cit.>, but it is deduced from a limited set of data.
The statistical sensitivity of a experiment depends strongly on how close the end-point and the M2 capture peak are.
To a good degree of approximation, the Lorentzian tail of the M1 peak centered at $E_{M1}=B(\mathrm{M1})$ dominates the end-point, and for equal to zero one has
\begin{equation}
N_{EC}(E_c,m_{\nu_e}=0) \propto \frac{(Q-E_c)^2}{(E_{M1}-E_c)^2} = \frac{(Q^\prime-E^\prime)^2}{E^{\prime\,2}}
\end{equation}
where $E^\prime = E_c-E_{M1}$ and $Q^\prime = Q - E_{M1}$. It can be shown that, in these conditions, the neutrino mass sensitivity is $\Sigma_{EC}(m_{\nu_e}) \propto Q^\prime$.
The uncertainty on $Q$, therefore, turns into the difficulty to design a experiment and to predict its sensitivity reach (Fig. <ref>).
Indeed, the shift of attention from to has been eased by the reasonable hope that a very low $Q$ could greatly enhance the achievable sensitivity of an experiment.
Monte Carlo computed statistical sensitivity as function of $Q$ for $= 3\times10^{13}$ and with $=3\times10^{-4}$, $= 1$ eV, and no background.
Very recently, the $Q$ value was determined from a measurement of - mass difference using the Penning trap mass spectrometer SHIPTRAP <cit.>. The measured value $\Delta m = 2833\pm 30_{stat}\pm15_{sys}$ confirms the most recurrent $Q$ measured in recent calorimetric experiments, although chemical shifts may still be expected for embedded in the LTD absorbers.
The knowledge of the $Q$ value is indeed a crucial ingredient for the optimal design of an experiment, while its limited precision and accuracy
prevent from using it as fixed parameter when the experimental data are interpolated to assess the neutrino mass <cit.>.
Nevertheless, a comparison of the $Q$ from the interpolation with a value obtained with an independent measurement – such as the - mass difference – is a powerful tool to pinpoint systematic effects.
In any case, the direct assessment of $Q$ from the end-point of the calorimetric spectrum, remains the first important goal of
upcoming high statistics measurements.
Calculated experimental EC calorimetric spectrum for $Q=2.8$ keV, $\Delta E=2$ eV, $f_{pp}=10^{-4}$, and $N_ev=10^{14}$ (blue). The pile-up spectrum is the red curve (left). Monte Carlo estimated statistical sensitivity for $\Delta E=1$ eV, $\tau_R=1$ $\mu$s, and for $f_{pp}=10^{-3}$, $10^{-4}$, $10^{-5}$, and $10^{-6}$ (from top to bottom). To two dashed lines correspond to (top)
$A_{EC}=1000$ Bq and $N_{dey}\times t_M=10^7$ detector$\times$year and (bottom) $A_{EC}=1$ Bq and $N_{dey}\times t_M= 3\times10^9$ detector$\times$year
Calculated experimental EC calorimetric spectrum for $Q=2.8$ keV, $\Delta E=1$ eV, for a constant exposure of $10^5$ detector$\times$year, and for (top to bottom) $\tau_R=10$ $\mu$s, $1$ $\mu$s, and $0.1$ $\mu$s (left). Sensitivity to heavy sterile neutrinos detected from kinks in a calorimetric spectrum with $Q=2.8$ keV, $N_{ev}=3\times10^{13}$, $\Delta E=1$ eV, and $f_{pp}=3\times10^{-4}$.
§.§ Statistical sensitivity
While the complexity of both the EC and the pile-up spectra make an analytical estimate of the statistical sensitivity an impossible task, a
Monte Carlo approach analogous to the one described in <ref> can give useful results <cit.>.
Most of the considerations made for are also valid in the case . The general conclusion about the importance of the total statistics
is well exemplified by Fig. <ref> for the now established $Q$ value of 2800 eV: indeed, the high $Q$ value raises the stakes of the experimental challenge and the prospects for a sub-eV sensitivity are scaled down.
Table <ref> shows the exposures required for two possible experiments aiming at a sensitivity of 0.2 and 0.1 eV, respectively (see also Fig. <ref>).
Although it may be possible to design microcalorimeters with a high activity, sub-eV sensitivity will likely require arrays with total
number of channels of the order of $10^6$. Indeed, there are several limitations to the possible activity , such as, for example, the effect of the nuclei on the detector performance or detector cross-talk and dead time considerations.
As shown in <ref>, a high activity causes an increase of $f_{pp}$ and thereby a reduced sensitivity to radioactive
background. This, along with the relative thinner aspect-ratio of microcalorimeters with ,
makes it likely that it is not strictly necessary to operate arrays in an underground laboratory <cit.>.
No high statistics measurement faced so far with the task of a careful estimation of systematic uncertainties. Nevertheless it is fair to say that there are some substantial differences between the systematic uncertainties expected for and experiments, which are worth a mention.
To avoid spectral distortions due to the escape of radiation, the absorbers must provide a $4\pi$ encapsulation with a minimum thickness of few microns. For gold absorbers, Monte Carlo simulations indicate a thickness of 2 $\mu$m for a 99.99998% (99.927%) absorption of 2 keV electrons (photons).
Furthermore, the M1 and M2 peaks in the calorimetric spectrum provide a useful tool to evaluate the detector response function overcoming the problems related to the use of an external X-ray source ( <ref>). The same peak can also be exploited for energy calibration, for tracking and correcting gain drifts, and for easing the summation of the spectra measured with the many pixels of the arrays.
Experimental exposure required for various target statistical sensitivities, with no background and two
different sets of detector parameters.
$Q$ target sensitivity $A_{EC}$ $\Delta E$ $\tau_R$ $N_{ev}$ exposure $T$
[eV] [eV] [counts/s] [eV] [ $\mu s$ ] counts [detector$\times$year]
2800 0.2 100 1 0.1 $9.8\times10^{15}$ $3.1\times10^6$
2800 0.1 100 0.3 0.1 $1.9\times10^{17}$ $5.9\times10^7$
§.§ A more precise description of the EC spectrum
While the question of the actual $Q$ value of the EC transition is now settled, many authors are still debating about the
precise shape of the calorimetric spectrum.
Indeed, (<ref>) is only an approximation. Already in the original work <cit.>, it was demonstrated the applicability of two approximations: the neglection of possible interference between the capture from different levels,
and the inclusion of transitions with off-shell intermediate states such as K and L.
Riisager in 1988 <cit.> discussed the distortions of the Lorentzian peak shape
expected when considering
that, in the atomic radiation cascade, the atomic phase space available at each step is altered by the natural width of previous transitions.
More recently, beginning with Robertson papers <cit.>, some authors started to recognize that the sum in (<ref>) must be
extended to more transitions which initially were deemed as negligible <cit.>.
This is caused by the incomplete overlap between the Ho and Dy atomic wave functions.
Recalling that calorimeters measure the neutrino energy $E_\nu$, while writing (<ref>) it was assumed
\begin{equation}
\end{equation}
where $B(\mathrm{H})$ is the binding energy of shell H in a Dy atom, i.e. the energy to fill the hole H in the Dy$^+$ atom.
But this is not correct. The hole H is in a neutral Dy atom with an extra (the eleventh) 4f electron, because the parent Ho atom
has an electronic configuration which differs from the one of Dy in the number of $4f$ electrons (11 vs. 10) (see also <cit.>).
Following <cit.> this can be expressed as
\begin{equation}
\end{equation}
where $E_R$ is a correction which accounts for the imperfect atomic wave functions overlap. So the capture peaks in the calorimetric
spectrum are expected to be shifted by a small amount which Roberston <cit.> calculated as $E_R\approx B(4f)_\mathrm{Ho}$, where $B(4f)_\mathrm{Ho}$ is the binding energy of the $4f$ electron in the Ho atom.
The atomic wave function mismatch goes along with the possibility of shake-up and shake-off processes, adding more
final states to the EC transition and, therefore, more terms in the sum in (<ref>).
These processes are the ones responsible for the presence in the final states of two (or more) vacancies created in the Dy atom, along with the extra $4f$ electron.
The second vacancy is left by an atomic electron which has been shaken by the sudden change in the wave functions to an higher bound unoccupied state (shake-up) or to the continuum (shake-off).
In the case of shake-up processes the neutrino energy is given by <cit.>
\begin{equation}
E_\nu = Q-B(\mathrm{H1})-B(\mathrm{H2}) - E_R
\end{equation}
and the contribution to (<ref>) it is just another Lorentzian peak term with $E_i = B(\mathrm{H1})+B(\mathrm{H2}) +E_R$.
The case of the shake-off process is more complex because it is a three-body process
\begin{equation}
\label{eq:shakeoff}
^{163}\mathrm{Ho} \rightarrow ^{163}\mathrm{Dy^{H_1H_2}} + e^- + \nu_e
\end{equation}
\begin{equation}
E_\nu + E_e = Q-B(\mathrm{H1})-B(\mathrm{H2}) - E_R
\end{equation}
The corresponding contribution to (<ref>) is not a narrow line since $E_e$ adds up to the observable atomic dis-excitation $B(\mathrm{H1})-B(\mathrm{H2}) - E_R$. The actual shape of the energy spectrum of the shaken-off electrons can be calculated as shown in <cit.>.
In general, the probability for the multi-hole processes is small and it can be estimated to be of the order of $10^{-5}$ <cit.>.
The precise calculation of the 2- or 3-hole processes probability is treated in many recent papers <cit.>, with the purpose
to improve past results from <cit.>, although, so far, all calculations apparently consider only shake-up processes.
In Figure <ref> the dashed line is the EC calorimetric spectrum calculated including 2-holes excitations and using the parameters
calculated in <cit.>.
Predicted single and double holes calorimetric spectrum of the EC decay of $^{163}$Ho with $Q=2.8$ keV (left).
The experimental spectrum analogous to the one in Fig. <ref> with double hole transitions included (right)..
The awareness of the above corrections to (<ref>) triggered some skepticism about the actual feasibility of a neutrino mass measurement from the end-point
of the calorimetric EC spectrum. The main argument is that, since the neutrino mass is searched as the difference between
the observed experimental spectrum and the theoretical one for $m_{\nu_e}=0$, the a priori knowledge of the latter one
is an absolute condition.
However, it can be argued that all the above corrections are immaterial for the neutrino mass measurement since, at the end-point, they
have no other effect than altering the overall rate, i.e. the spectrum normalization, while the shape remains determined by the phase space factor.
The change in the rate may be even in the direction of a favorable increase.
This is indeed the beauty of the calorimetric experimental approach.
For what concerns the additional 2- or 3-holes transitions, a more subtle threat to the neutrino mass measurement with is brought
by the underlying pile-up spectrum: as it can be seen in Fig. <ref>, these higher order transitions cause additional peaks to appear in the end-point vicinity.
§.§ $^{163}$Ho production
is not a naturally occurring isotope: it was discovered at Princeton in a sample of that was neutron irradiated in a nuclear reactor <cit.>.
To carry out neutrino mass experiments with , the isotope must be produced in fairly large amounts. Upcoming medium size experiments will have to contain about $10^{16}$ nuclei of – i.e. about 3 $\mu$g – for a total activity of the order of $10^5$ Bq. The isotope production and separation are critical steps in every plan for an ambitious holmium neutrino mass experiment.
There are many nuclear reactions which can be exploited to produce . A comprehensive critical evaluation of all possible production routes is presented in <cit.> although, presently, not all the cross-sections of the considered processes are experimentally known.
In general, the production process starts with a nuclear reaction, which can be either direct – such as $^{nat}\mathrm{Dy}(p,xn)^{163}\mathrm{Ho}$ – or indirect – such as $^{162}\mathrm{Er}(n,\gamma)^{163}\mathrm{Er}\rightarrow^{163}\mathrm{Ho}$.
These reactions unavoidably co-produce other long-living radioactive species – also owing to the presence of un-necessary isotopes in the target material – which need to be removed to prevent interferences to the neutrino mass measurement. Chemical separation of holmium can remove most of them, with the notable exception of the beta decaying isomer $^{166m}$Ho ($\tau_{1/2}=1200$ years, $E_0=1854$ keV).
Geant4 Monte Carlo simulations performed for gold absorbers show that each Bq of $^{166m}$Ho can contribute about 1 count/eV/day to the background level in the end-point region of the spectrum <cit.>. Therefore, $^{166m}$Ho must be removed by means of a further isotope mass separation step.
The key parameters of the entire process are the isotope production rate, the $^{166m}$Ho/ ratio, and the efficiencies of chemical and mass separations. They determine the amount of starting material that is required to have the target number of nuclei to be embedded in the detector absorbers. Of course, also the final embedding process causes further isotope losses which must be considered, although in some approaches the embedding is part of the production process – e.g. when the embedding is achieved by means of the same accelerator used for mass separation.
When all efficiencies entering in the process are considered, the needed for the next high statistic measurements is likely to increment to tens or hundreds of MBq.
Early experiments used the same process with which the isotope was discovered, i.e. neutron irradiation of . Another route used for past experiments is based on proton spallation with Ta targets.
The experiments presented in the following use either neutron irradiation of
enriched targets or proton irradiation of natural Dy targets.
Neutron irradiation of an enriched sample is a very efficient route. The starting material is usually enriched which is available as by-product of the production of isotopes for medical applications. The large thermal neutron cross-section $\sigma(\mathrm{n},\gamma)\approx 20$ barns together with the availability of high thermal neutron flux nuclear reactors – as the one of the Institut Laue-Langevin (ILL, Grenoble, France) with a thermal neutron flux of about $1.3\times10^{15}$ n/s/cm$^2$ <cit.> – give an estimated production rate of about 50 kBq()/week/mg(), for enriched at 30% in .
This rate may be reduced by the yet unknown cross-section of the burn-up process $^{163}\mathrm{Ho}(n,\gamma)^{164}\mathrm{Ho}$.
Neutron irradiation causes also the production of $^{166m}$Ho owing to the presence of impurities such as $^{164}$Er and $^{165}$Ho in the enriched target. If the $^{164}$Er production route prevails, for a 10% isotopic abundance in the target a co-production of about $A(^{166m}\mathrm{Ho})/A(^{163}\mathrm{Ho})\approx 0.001$ can be expected.
One drawback of this route is the cost for the enriched procurement.
The production via proton irradiation of natural Dy target depends on the proton energy and has a production rate which is not competitive with high-flux reactors, especially for large amounts.
In <cit.> the production rate as a function of the total cumulative charge is estimated to be about few Bq($^{163}$Ho)/$\mu$Ah/g($^{nat}$Dy) for 24 MeV protons.
$^{166m}$Ho is produced by the neutrons from the reaction $^{164}\mathrm{Dy}(p,n)$ in $(n,\gamma)$ captures on $^{164}\mathrm{Dy}$ or on $^{165}\mathrm{Ho}$ contaminations. Monte Carlo simulations give a co-production lower than $A(^{166m}\mathrm{Ho})/A(^{163}\mathrm{Ho})\approx 10^{-6}$.
In spite of its low efficiency, the use of a natural target and the limited $^{166m}$Ho co-production make this route appealing for small scale experiments.
Latest measured spectrum from ECHo, see text (from <cit.>).
§.§ ECHo
ECHo is a project carried out by the Heidelberg group in collaboration with many other European and Indian groups <cit.>. The mid term goal of this project – ECHo-1k – is a medium scale experiment with an array of 1000 MMCs, each implanted with 1 Bq of <cit.>.
With energy and time resolutions of at least 5 eV and 1 $\mu$s, respectively, a statistical sensitivity of about 20 eV at 90% C.L. is expected after one year of measurement (Tab. <ref>). The microcalorimeters are derived from the gold detectors with Au:Er sensors designed and fabricated by the Heidelberg group for soft X-rays spectroscopy.
So far, the results of two prototypes with in the absorbers have been presented.
For the first prototype the isotope was implanted at the isotope separation on-line facility ISOLDE (CERN). Here the ,
produced by spallation with protons on Ta, was accelerated, mass separated, and implanted in the absorbers of four detectors. A total activity of $10^{-2}$ Bq was enclosed between two gold films with dimensions $190\times190\times5$ $^3$.
The results of the characterization of these detectors are reported in <cit.>, and include
an energy resolution of about 8 eV and a remarkable rise time of about 130 ns.
In the high statistics spectrum, the peaks due to a contamination of co-produced $^{144}$Pm are visible, although decaying with time.
In addition there are structures on the high energy side of the N1 peak which are tentatively interpreted as due to higher order EC transitions.
From this measurement, the intensities of the N1 and M1 lines give $Q=2.800\pm0.080_{stat}$ keV <cit.>.
For the second prototype the isotope is produced at the ILL high flux nuclear reactor by neutron
irradiating an enriched target. The sample is chemically purified at Mainz both before and after irradiation.
The in the target is then mass separated and implanted off-line at ISOLDE in the absorbers of two maXs-20 chips.
The maXs-20 chips are arrays of 16 MMCs designed and optimized for soft X-ray spectroscopy <cit.>.
About 0.2 Bq are encapsulated between two gold layers with dimensions $250\times240\times5$ $^3$.
Preliminary measurements (see Fig. <ref>) show an energy resolution of about 12 eV and a strong reduction of the background, and confirm the structures
on the right side of the N1 <cit.>. The persistence of these structures, in spite of the improvements in the background and in the instrumental line shape, supports their interpretation as due to processes related to the EC decay.
Another preliminary analysis discussed in <cit.> interprets these as the broad structures expected for shake-off transitions.
Present ECHo activities are aimed at running ECHo-1k in the next years (2016–2018) and include the development of the microwave multiplexed read-out of the MMCs <cit.>, the optimization of MMCs design, and the production of 10 MBq of high purity .
HOLMES TES pixel
§.§ HOLMES
HOLMES is an experiment carried out by the Genoa and Milano groups in collaboration with NIST, ILL, PSI, and Lisbon <cit.>.
The baseline program is to deploy an array of about 1000 TES based microcalorimeters each with about 300 Bq of fully embedded in the absorber, with the goal of energy and time resolutions as close as possible to 1 eV and 1 , respectively (Tab. <ref>).
In this configuration, HOLMES can collect about $3\times10^{13}$ decays in 3 years of measuring time and the expected statistical sensitivity is about 1.5 eV at 90% C.L.
The choice of this configuration is driven by the aim to collect the highest possible statistics with a reasonable exposure. Despite the
high pile-up level and the technical challenge that derives from it, this provides a net improvement on the achievable sensitivity and a lower impact of the radioactive background.
The amount of needed for the experiment is estimated to be about 100 MBq and it is being produced at ILL by neutron irradiation of an enriched target, subjected to chemical pre-purification and post-separation at PSI (Villigen, Switzerland).
A custom ion implanter is being set-up in Genova to embed the isotope in the detector absorbers. It consists of a Penning sputter ion source, a magnetic/electrostatic mass analyzer, an acceleration section, and an electrostatic scanning stage. The full system is being designed to achieve an optimal mass separation of vs. $^{166m}$Ho. The implanter will be integrated with a vacuum chamber for the simultaneous evaporation of gold, first to control the concentration, and then to deposit a final Au layer to prevent the from oxidizing.
The cathode of the ion source will be made of high purity metallic holmium to avoid end-point deformations due to the different $Q$ shifts in diverse chemical species. The metallic holmium will be obtained by thermal reduction at about 2000 K, using the reaction Ho$_2$O$_3$+2Y(met)$\rightarrow$2Ho(met)+ Y$_2$O$_3$ <cit.>.
HOLMES uses TES microcalorimeter arrays with $\mu$MUX read-out, both fabricated at NIST (Boulder, USA). The DAQ exploits the
Reconfigurable Open Architecture Computing Hardware (ROACH2) board equipped with a FPGA Xilinx Virtex6 <cit.>, which has been developed in the framework of CASPER (Collaboration for Astronomy Signal Processing and Electronic Research).
Presently, the collaboration is working on the optimization of the isotope production processes. Two samples have been irradiated at ILL and processed at PSI. ICP-MS is used to assess the amount of produced and the efficiency of the chemical separation. From preliminary assessments, the total available activity is about 50-100 MBq.
The optimization of the pixel design is also in progress <cit.> and fig. <ref> shows the design that best matches HOLMES specifications. The absorber is made of gold and, to avoid interference to the superconducting transition, it is placed side-by-side with the Mo/Cu sensor on a silicon nitride membrane. The design also includes features to control the microcalorimeter speed. Energy and time resolutions are
within a factor 2-3 of the target ones, owing also to new algorithms for pile-identification <cit.>.
HOLMES is expected to start data taking in 2018, but a smaller scale experiment with a limited number of pixels will run in 2017, with the aim to collect a statistics of about $10^{10}$ decays in a few months.
NUMECs TES detectors (left) and the latest spectrum measured from NuMECS (right), see text (from <cit.>).
§.§ NuMECS
NuMECS is a collaboration of several US institutions (LANL, NIST, NSCL, CMU) with the goal to critically assess the
potential of holmium calorimetric neutrino mass measurements <cit.>. The NuMECS program includes the validation of the
isotope production, purification, and sensor incorporation techniques, the scalability to high resolution LTD arrays, and the
understanding of underlying nuclear and atomic physics.
Recent work has successfully tested the production via proton irradiation of a natural dysprosium target.
About 3 MBq of have been produced by irradiating about 13 g of high purity natural dysprosium with $3.4\times10^4$ $\mu$Ah of 25 MeV protons at the Los Alamos National Laboratory Isotope Production Facility (IPF). At the same time, a cation-exchange high performance liquid chromatography (HPLC) procedure for the chemical separation of holmium has been developed and a separation efficiency of about 70% has been measured.
For the present testing phase, NuMECS uses TES microcalorimeters fabricated by NIST and specially designed to be mechanically robust.
The TES sensor is at the end of a silicon beam, close to a pad used for testing the attachment of a wide range of absorbers (Fig. <ref>).
To incorporate the isotope in the microcalorimeter absorber, NuMECS exploits the drying of solutions containing the isotope
onto thin gold foils. After testing many procedures, the best results were recently obtained by deposition of an aqueous solution on nanoporous gold on a regular gold foil, followed by annealing in dilute H$_2$ atmosphere at $800^\circ$C. The microcalorimeter absorber is made by folding and pressing a small piece of the gold foil.
Figure <ref> shows a spectrum collected in 40 hours <cit.>. The activity is about 0.1 Bq.
Peaks have a low energy tail and show an excess broadening, explained as caused by thermalization noise.
A fit of the M1 peak gives a of about 43 eV, inclusive of the peak natural width.
All peaks are found within 1% of the tabulated positions. Remarkably, the spectrum does not show any of the satellite peaks predicted in <cit.>, although the statistics is still limited. There is instead an unexplained shoulder on the high energy side of the N1 peak, which
resemble the structure observed by ECHo and interpreted as shake-off transition in <cit.>.
NuMECS future plans include the deployment of four 1024 pixel arrays, aiming to a statistical sensitivity of about 1 eV.
Comparison of the ECHo, HOLMES, and NuMECS projects.
× 1cECHo 1cHOLMES 1cNUMECS
production 1c$^{162}Er(n,\gamma)$ 1c$^{162}Er(n,\gamma)$ 1cDy(p,)
absorber 1cgold 1cgold 1cnanoporous gold
sensor 1cAu:Er magnetic 1cTES Mo/Cu 1cTES Mo/Cu
4cpresent status
$\Delta E$ at M1 peak [eV] 1c12 1c– 1c43 (incl. $\Gamma_\mathrm{M1}$)
$\tau_{rise}$ [] 1c– 1c– 1c–
$A_{EC}$ [Bq] 1c0.2 1c– 1c0.1
4cprojected ($E_0=2800$ eV)
$N_{det}$ 1c1000 1c1000 1c4096
$\Delta E$ [eV] 1c$<5$ 1c1 1c–
$\tau_{rise}$ [] 1c$<1$ 1c1 1c–
$A_{EC}$ [Bq/detector] 1c1 1c300 1c100
$f_{pp}$ 1c$10^{-6}$ 1c$3\times 10^{-4}$ 1c–
$t_M$ [y] 1c1 1c3 1c1
$\Sigma_{90}(m_{\nu_e})$ [eV] 1c20 1c2 1c1
§ SUMMARY AND OUTLOOK
The use of and as an alternative to tritium for the direct measurement of the neutrino mass was
proposed in the same years when the low temperature detector technology was moving the first steps.
The idea of making low temperature detectors with rhenium absorbers immediately caught up, both because
it appeared to be of almost immediate realization and because of its appealing impact on X-ray spectroscopy.
Unfortunately, on the long run the technological difficulties inherent to the use of superconducting rhenium caused
the interest of the low temperature detector community to fade away, and the neutrino mass projects to have the
same fate as the X-ray applications of rhenium detectors.
measurements took more time to take off, as if they were awaiting for the readiness of the technology of microcalorimeter arrays applied to
high resolution spectroscopy of soft X-rays.
Now, neutrino mass experiments are ready to leverage this mature technology, and the interest of the
low temperature detector community is high, as demonstrated by the number of parallel efforts.
Despite the unluckily high $Q$ value, the good prospects to perform high statistics neutrino mass measurements in the
next couple of years are also attracting the attention of the neutrino physics community as a valid complementary alternative to KATRIN.
§ ACKNOWLEDGMENTS
I would like to thank Andrea Giachero, Marco Faverzani, Elena Ferri, Andrei Puiu, and Monica Sisti, who in various ways supported me
with the writing of this paper; Maurizio Lusignoli and Alvaro De Rujula, for the many useful discussions; and Loredana Gastaldo, Michael Rabin and Mark Philip Croce, for providing me with updated informations on their experiments.
|
1511.00817
|
Low Temperature Laboratory, Department of Applied Physics, P.O. Box 15100, FI-00076 Aalto University, Finland
University of Jyvaskyla, Department of Physics and Nanoscience Center, P.O. Box 35, 40014 University of Jyväskylä, FINLAND
University of Jyvaskyla, Department of Physics and Nanoscience Center, P.O. Box 35, 40014 University of Jyväskylä, FINLAND
Centro de Física de Materiales (CFM-MPC), Centro
Mixto CSIC-UPV/EHU, Manuel de Lardizabal 5, E-20018 San
Sebastián, Spain
Donostia International Physics Center (DIPC), Manuel
de Lardizabal 5, E-20018 San Sebastián, Spain
In superconductors spin-split by an exchange field, thermal effects
are coupled to spin transport. We show how an oscillating
electromagnetic field in such systems creates spin imbalance, that
can be detected with a spin-polarized probe. The sign and
magnitude of the probe signal result from a competition between processes converting
field induced spin energy imbalance to spin imbalance, dominant at
low frequencies, and
microwave-driven pair breaking at high frequencies. In the presence of spin-flip scattering,
we show that ac excitation also leads to multistabilities in
the superconducting state.
§ INTRODUCTION
Long-lived spin excitations are interesting for spintronics
applications, and the spin transport in superconductors has
recently attracted renewed attention in this context.
Spin accumulation
in a superconductor can be generated by injecting current from a
spin-polarized electrode, for example a ferromagnet. A second approach
for spin injection studied in a number of recent experiments
is to use a magnetic field or proximity to ferromagnetic insulators to
Zeeman split the density of states of the superconductor
<cit.> (cf. Fig. <ref>),
so that injection of current from an unpolarized probe also generates
observable spin accumulation. The spin-splitting also changes the
quasiparticle physics so that one component of the imbalance only
relaxes via inelastic scattering,
leading to long observed spin lifetimes and relaxation lengths.
The physics of the long-ranged quasiparticle spin accumulation in
spin-split superconductors is closely connected to their
thermoelectric properties. Magnetic interactions in superconductors
break the spin-resolved electron-hole symmetry, enabling large
thermoelectric responses. This is predicted to occur due to magnetic
impurities <cit.>, spin-active interfaces,
<cit.> and in
superconductor–ferromagnet systems in the presence of exchange
fields. <cit.> The large thermoelectric effect in spin-split superconductor/ferromagnet
tunnel junction has been observed very recently (see
Ref. kolenda2015-otc).
A superconductor absorbs microwave electromagnetic radiation, in the
presence of an internal exchange field $\vec{h}$. In the steady
state, this generates a
spin imbalance $\delta\mu_s$, an excess of quasiparticles
whose spins are either aligned ($\delta\mu_s>0$) or anti-aligned
($\delta\mu_s<0$) with the axis of the exchange field.
The quasiparticle spectrum is spin-split by the exchange field.
Coupling to microwaves generates spin-conserving quasiparticle
transitions that change energy by $\pm\omega$
and perturb the electron distribution (dotted).
Elastic spin-flip scattering transforms quasiparticles
to the opposite spin species, converting energy imbalance
to spin imbalance.
The thermoelectric mechanisms are also connected to photoelectric
effects in superconductors <cit.>,
where absorbed radiation is converted to a dc voltage observed in a
probe electrode. Based on the above discussion, a photo-spin-electric effect should be present also in spin-split
superconductors — the absorbed radiation generates spin imbalance,
which relaxes slowly via inelastic scattering. This is interesting to
consider e.g. in the context of measurements that use microwave signals to
probe spin resonances of the quasiparticles. <cit.>
In this work, we discuss how an electric ac field in diffusive spin-split
superconductors produces spin imbalance [see
Fig. <ref>(a)]. We find that the ac driving generates
spin imbalance that is either parallel or antiparallel to the exchange
field, depending on the drive frequency. Nonequilibrium states
generated by ac fields in conventional superconductors have long been
studied, <cit.> and we extend the picture to
include spin splitting. Although interaction with the fields conserves
spin, combining it with elastic spin-flip scattering from
e.g. magnetic impurities results to a nonequilibrium steady state
with nonzero spin imbalance [see Fig. <ref>(b)]. We
discuss how the effect can be detected via ferromagnetic probes
[Eq. (<ref>)]. Similar photoelectric effects are known
to occur also in the absence of spin splitting,
<cit.> but they require weak elastic scattering. We also predict that the spin imbalance
results to an instability in the superconducting order parameter,
permitting multiple non-zero values for it in a temperature range
around $T_c$, leading to hysteresis and
providing a second characteristic signature of the effect.
The manuscript is organized as follows. We outline the model in
Sec. <ref> and discuss observables accessible with
spin-polarized electrical probes in Sec. <ref>.
Modification of the superconducting order parameter is discussed in
Sec. <ref>, and we conclude in Sec. <ref>.
§ KINETIC EQUATIONS
We consider a diffusive superconductor film with a Zeeman field induced either by an external magnetic
field <cit.> or for example proximity to a magnetic
insulator.<cit.> In order to describe a nonequilibrium situation
we apply the quasiclassical Keldysh-Green function
formulation, <cit.>
and write the Usadel equation for the spin-split superconductor (here and below, we set $\hbar=e=k_B=1$):
\begin{gather}
\label{eq:usadel}
[i \epsilon\hat{\tau}_3 - i (\vec{h}\cdot\vec{S})\hat{\tau}_3 - \hat{\Delta} - i\check{\sigma}, \check{g}]
\,.
\end{gather}
The function
$\check{g}(t,t')$ is a matrix which in the Keldysh-Nambu-spin space has the form
\[
\check{g}=\left( \begin{array}{cc}
\hat g^R & \hat g^K\\
0 & \hat g^A
\end{array} \right)\; ,
\]
where $\hat g^{R,A,K}$ are the retarded, advanced and Keldysh 2$\times$2
matrices in the Nambu ($\tau_j$) and spin ($s_j$) spaces and $\vec{S}=(s_1,s_2,s_3)$.
The exchange field $\vec{h}$
is induced by an external magnetic field <cit.> or for
example proximity to a magnetic insulator.
<cit.> Here, $D$ is the diffusion
constant of the superconductor, $\Delta$ is the order parameter, and
$\check{\sigma}$ a self-energy corresponding to spin-flip and
inelastic scattering (electron-phonon or electron-electron). We use a
gauge where the electric potential is $\varphi=0$, and coupling to
electromagnetic fields is via a vector potential $\vec{A}$ appearing
in the covariant derivative
$\hat{\nabla}X = \nabla X - [i \vec{A}\tau_3, X]$.
We assume that the superconducting film is in a uniform time-dependent
electric field ${\cal E}(t)=A_0\omega_0\sin(\omega_0 t)$. We follow a
similar approximation procedure as in
Ref. eliashberg1970-fss. In a spatially uniform
situation, assuming the film is thinner than the skin depth, the
vector potential enters equivalently as a self-energy
$\check{\sigma}_A(t,t')=-i D {\vec A}(t)\cdot \hat{\tau}_3
\check{g}(t,t') {\vec A}(t') \hat{\tau}_3$ which
after time averaging is given by
\[
\check{\sigma}_A(E)=-i\frac{DA_0^2}{4}\tau_3[\check{g}(E+\omega_0)+\check{g}(E-\omega_0)]\tau_3\; .
\]
Considering time dynamics implied by Eq. (<ref>), this expression describes the effect of the ac field
in the leading order in the small parameter $DA_0^2\ll\omega_0$.
The self-energy term in Eq. (<ref>) also takes into account
a number of relaxation processes present in real superconductors. This includes spin-flip scattering
<cit.> due to magnetic
$\check{\sigma}_{\rm sf}=-\frac{i}{8\tau_{\rm
spin-orbit scattering
$\check{\sigma}_{\rm so}=-\frac{i}{8\tau_{\rm
and phonon scattering $\check{\sigma}_{\rm ph}$ (see
Appendix <ref>). Below, we parameterize
$\tau_{\rm sf/so}^{-1} = \frac{1 \pm \beta}{2}\tau_{sn}^{-1}$, where
the parameter $-1\le\beta\le1$ describes which of the spin-flip and
spin-orbit scattering mechanisms is stronger.
For example, the scattering
rates in Al wires were found to be $\tau_{\rm sn}\approx\unit[100]{ps}$ and $\beta\approx0.5$
in Ref. poli2008-sir, so that $T_c\tau_{\rm sn}\sim20$.
We also include orbital dephasing,
$\check{\sigma}_{\rm orb}=-\frac{i}{2\tau_{\rm
which is relevant if an external magnetic field is used to generate
the exchange field $h=g\mu_BB$.
The scattering rate associated with the orbital effect is <cit.>
$\tau_{\rm orb}^{-1}=\frac{T_{c0}\alpha_{\rm orb}}{2}(h/T_{c0})^2$, where
$T_{c0}\approx0.567\Delta_0$ is the BCS critical temperature, and
the parameter $\alpha_{\rm orb}=T_{c0}DW^2/(12g^2\mu_B^2)$ depends on
the film thickness $W$.
The Keldysh component of $\check g$ can be expressed in terms of the
retarded and advanced matrices and distribution function matrix $\hat{f}$, $\hat{g}^K = \hat{g}^R \hat{f} - \hat{f}\hat{g}^A$. In particular $\hat{f}$ parameterizes the quasiparticle nonequilibrium modes.
Below, we choose the $z$-axis parallel to the Zeeman field.
In this case, the retarded function is spin-diagonal and we
write it in the form
$\hat{g}^R = \sum_{\sigma=\uparrow/\downarrow}s_{\sigma}[g_{\sigma,1}\tau_1+g_{\sigma,3}\tau_3]$,
where $s_{\uparrow/\downarrow}=[1\pm{}s_z]/2$.
Similarly, we write the distribution function as
$\hat{f} = \sum_{\sigma=\uparrow/\downarrow}s_{\sigma}[f_{L\sigma} + \hat{\tau}_3f_{T\sigma}]$.
The distribution functions $f_{T\sigma}$ characterize the charge imbalance
and $f_{L\sigma}$ the energy imbalance in the two spin bands. An
alternative representation
$f_{L3/T3}=(f_{T/L,\uparrow}-f_{T/L,\downarrow})/2$ was used in
Ref. silaev2015-lrs.
In terms of these functions, Eq. (<ref>) results in kinetic
equations for the components of $\hat{f}$. In the steady state they
are rate equations expressing a balance of excitation and relaxation
\begin{align}
\label{eq:kinetic-eq}
\hat{\cal I}_{A}[\hat{f}]
+ \hat{\cal I}_{\rm sf+so}[\hat{f}]
+ \hat{\cal I}_{\Delta}[\hat{f}]
+ \hat{\cal I}_{\rm relax}[\hat{f}]
= 0
\,,
\end{align}
where the collision integrals $\hat{\cal I}$ are related to the
corresponding self-energies and $\hat{\Delta}$ via
$\hat{\cal I} = (\hat{g}^R \hat{Z} - \hat{Z}\hat{g}^A)/8$,
$\hat{Z}=i\hat{\sigma}^R\hat{f} - i\hat{f}\hat{\sigma}^A -
Below, we find that $f_{T\sigma}=0$ for our problem, so that
${\cal I}_\Delta=0$.
For the electromagnetic collision integral we get
\begin{align}
\hat{Z}_A
\frac{DA_0^2}{4}
\sum_\pm
\tau_3[g^R_\pm(\hat{f}-\hat{f}_\pm) - (\hat{f}-\hat{f}_\pm)g^A_\pm]\tau_3
\,,
\end{align}
where $\hat{f}_\pm(E) = \hat{f}(E\pm\omega_0)$. The
$s_{\uparrow/\downarrow}$ components read
\begin{align}
\label{IA}
{\cal I}_{A,\sigma}(E)
\Tr{}s_\sigma\hat{\cal I}
\frac{DA_0^2}{4}
\sum_\pm
R_{L,\sigma}(E, E \pm \omega)
\\\notag
[f_{L,\sigma}(E) - f_{L,\sigma}(E\pm\omega)]
\,.
\end{align}
The $\hat{\tau}_3s_\sigma$ components vanish, reflecting charge conservation.
The $\pm\omega$ terms indicate driven quasiparticle transitions up/down in energy.
In terms of the Fermi distribution function $f_\sigma=\frac{1-f_{L\sigma}}{2}$,
the second line acquires the typical structure for fermion transitions,
The kernel $R$ is
\begin{align}
R_{L,\sigma}(E, E')
\Im g_{\sigma,1,E}\Im g_{\sigma,1,E'}
\,,
\end{align}
where $N_\sigma(E)=\Re g_{\sigma,3,E}$ is the spin-dependent density
of states. The result Eq. (<ref>) is equivalent to a standard photoabsorption
collision integral for each spin.
For the elastic spin-flip and spin-orbit scattering, we have
\begin{align}
{\cal I}_{{\rm sn},\sigma}
\frac{S_{\uparrow\downarrow}}{4\tau_{\rm sn}}(f_{L,\sigma}-f_{L,-\sigma})
\,,
\\
S_{\uparrow\downarrow} &= N_{\uparrow}N_{\downarrow} + \beta\Im g_{\uparrow,1}\Im g_{\downarrow,1}
\,.
\end{align}
To find analytical results we describe inelastic relaxation
within a relaxation-time approximation, for which we
\begin{align}
{\cal I}_{\mathrm{in},\sigma} = \frac{N_\sigma}{2\tau_{\mathrm{in}}}(f_{L\sigma}-f_L^{(0)})
\,.
\end{align}
We also obtain numerical results with a more detailed model for
electron-phonon scattering (see Appendix <ref>).
Based on the above equations, we can first solve the components of
$\hat{g}^R$ from Eq. (<ref>), which can be done analytically
in some cases, or in general numerically. This provides the coefficients in the
kinetic equations (<ref>). Alternatively, we also solve
Eq. (<ref>) directly numerically (see
Appendix <ref>), which ensures self-consistency of the
spectral functions.
§ SPIN IMBALANCE
Schematic cross-section of a superconductor
(S) / ferromagnetic insulator (FI) hybrid thin-film structure of
thickness $d\ll{}L$, driven by an oscillating ac electric field
corresponding to voltage $V$ at frequency $\omega$.
The S/FI layer is coupled to a ferromagnetic detector probe (F)
via a tunnel junction, where the dc current $I_{\rm det}$
is measured.
Experimentally, spin imbalance in the superconductor can be probed by electrical
measurements that use spin-filtering probe junctions, for example
ferromagnets. The dc current-voltage relation will in these cases
contain a component that depends on the polarization of the probe and
the spin imbalance in the sample.
The current measured by a spin-filtering tunnel probe
[see Fig. <ref>] coupled to the superconductor and biased
at $V=0$ is given by<cit.>
\begin{align}
\label{eq:Idet}
\mu
\\
\mu &=
\frac{1}{4}\int_{-\infty}^\infty\dd{\epsilon}[N_\uparrow f_{T\uparrow} + N_\downarrow f_{T\downarrow}]
\,,
\\
\mu_z &=
\frac{1}{4}\int_{-\infty}^\infty\dd{\epsilon}\{
N_\uparrow [f_{L\uparrow} - f_{L,d}]
N_\downarrow [f_{L\downarrow} - f_{L,d}]
\}
\,,
\end{align}
where $P_{det}$ is the detector polarization in $z$-direction,
$R_{det}$ the junction resistance, and
$f_{L,d}=\tanh\frac{E}{2T_{\rm det}}$ the equilibrium distribution
in the detector. The current in the detector is a measure of the charge ($\mu$) and spin ($\mu_z$) imbalances.
The ac drive only excites the modes $f_{L,\uparrow/\downarrow}$ which
carry no charge imbalance, so that on this level of analysis, no
photoelectric effect is present ($\mu=0$). Moreover,
Eq. (<ref>) together with a spin-independent relaxation
time yields no spin imbalance ($\mu_z=0$). This follows
from $R_{L,\sigma}(E,E')=R_{L,\sigma}(E',E)$ and
I}_{A,\sigma}(E)=0$, reflecting conservation of spin.
In practice, however, elastic spin-flip scattering cannot be ignored,
and the associated scattering times can be short compared to the
inelastic collisions, $\tau_{\rm sn}\ll{}\tau_{\mathrm{in}}$. Under
such conditions, ac drive can result to nonzero spin imbalance
$\mu_z\ne0$ in the steady state. Away from the strict diffusive limit,
magnetic impurities result to a photoelectric effect of order
$\ell_{\rm el}^2A_0^2$ also in the absence of the Zeeman
splitting. <cit.> Here, we
concentrate only on the diffusive limit and hence the Zeeman splitting is crucial.
Writing the solution of the kinetic equations
Eqs. (<ref>),(<ref>) using the relaxation time
approximation for inelastic processes, and considering the limit of
weak spin-flip scattering
$\tau_{\mathrm{in}}\ll\tau_{sn}$, $DA_0^2\tau_{\mathrm{in}}\ll1$, we find
\begin{align}
\simeq
\frac{2{\cal I}_{A,\sigma}^{(0)}\tau_{\mathrm{in}}}{N_\sigma}
\sigma \frac{S_{\uparrow\downarrow}\tau_{\mathrm{in}}^2}{2\tau_{sn}}
\frac{N_\uparrow{\cal I}_{A\downarrow}^{(0)}-N_\downarrow{\cal I}_{A\uparrow}^{(0)}}{N_\uparrow{}N_\downarrow}
\,.
\end{align}
where ${\cal I}_{A\sigma}^{(0)}={\cal I}_{A\sigma}[f_L^{(0)}]$, and
$f_L^{(0)}=\tanh\frac{E}{2T}$ is the equilibrium distribution. As
noted above, the first term does not contribute to spin imbalance, but
the second term does. The result is illustrated in
Fig. <ref>(b): the transitions driven by the ac field
generate an imbalance of quasiparticles inside both spin bands, which
the spin-flip scattering converts to a spin imbalance at energies
Current at the detector probe under ac excitation,
for $\tau_{sn}T_{c0}=12.5$, $\beta=0.5$, $h/\Delta_0=0.17$, $DA_0^2/\Delta_0=4\times10^{-4}$,
$\alpha_{\rm orb}=0.1$.
Solid lines correspond to a numerical solution of Eq. (<ref>) with
the phonon model with $\tau_{eph,0}T_{c0}=100$,
and dotted lines to Eq. (<ref>) with $\tau_{\mathrm{in}}=(T_{c0}/T)^3\tau_{eph,0}$.
In a typical situation, however, we expect that spin-flip scattering is fast
compared to inelastic relaxation ($\tau_{sn}\ll\tau_{\mathrm{in}}$). In this limit
we have for the detector current,
\begin{align}
\label{eq:Idetapprox}
I_{\rm det}
-\frac{P_{\rm det}DA_0^2\tau_{\mathrm{in}}}{4R_{\rm det}}
\int_{-\infty}^\infty\dd{E}
\sum_\pm
[f_{L}^{(0)}(E) - f_{L}^{(0)}(E\pm\omega)]
\\\notag
\frac{
N_{\uparrow}(E)R_{L\downarrow}(E,E\pm\omega) - N_\downarrow(E) R_{L\uparrow}(E,E\pm\omega)
\end{align}
The result is shown in Fig. <ref> for representative
parameters. The figure also shows results computed numerically using a
phonon model (see Appendix <ref>). The two are qualitatively
similar, up to differences largely originating from temperature and
energy dependence in the relaxation rates. As the frequency increases,
the amplitude of the signal also increases up to the point
$\omega\approx{}2h$, where the process depicted in
Fig. <ref>(b) saturates. At large frequencies
$\omega\gtrsim{}2\Delta$ the detector current changes sign, as
microwave-driven pair breaking starts to contribute. In this regime,
the resulting signal can be understood as a thermoelectric current
<cit.> driven by a
temperature difference $T_S>T_F$ caused by the heating of the
superconductor by the drive.
In addition to the nonequilibrium-generated signal, oscillating
electric fields can introduce a voltage drop $V_{ac}$ across the
detector tunnel barrier.
This results to an additional signal from photoassisted tunnelling.
Within the above approach, the total current is given by the Tien-Gordon
result, <cit.>
\begin{align}
I = \sum_{n=-\infty}^\infty J_n^2\left(\frac{eV_{ac}}{\hbar\omega_{ac}}\right)
I_{\rm det}(V_{dc} + n\hbar\omega_{ac}/e)
\,,
\end{align}
where $I_{\rm det}(V)$ is the current-voltage relation in the absence of
$V_{ac}$, and $J_n$ are Bessel functions. In the leading order in
driving amplitude ($V_{\rm ac}\to0$) and without dc bias ($V=0$),
\begin{align}
I_{\rm det}(0)
\\
I_{0,T} &= \frac{V_{\rm ac}^2}{4\omega^2}[I_{\rm eq}(\omega) + I_{\rm eq}(-\omega)]
\,,
\end{align}
where $I_{\rm det}(0)$ is the nonequilibrium current (<ref>),
$I_{0,T}$ the photoassisted current, and $I_{\rm eq}(V)$ is the
IV-relation of the FIS junction when the superconductor is at
equilibrium. We have
\begin{align}
\frac{P_{\rm det}V_{\rm ac}^2}{2R_{\rm det}\omega^2}
\int_{-\infty}^\infty\dd{E}
\Bigl[
\\\notag
f_L^{(0)}(E+h) - \frac{f_L^{(0)}(E-\omega+h) + f_L^{(0)}(E+\omega+h)}{2}
\\\notag
f_L^{(0)}(E-h) + \frac{f_L^{(0)}(E-\omega-h) + f_L^{(0)}(E+\omega-h)}{2}
\Bigr]
\,,
\end{align}
where $N_0=\Re\frac{|E|}{\sqrt{E^2-\Delta^2}}$ is the BCS density of
states. Consider now $\omega,h\lesssim{}T$. We find
\begin{align}
-\frac{P_{\rm det}V_{ac}^2h}{2R_{\rm det}\Delta_0^2} \eta(T/\Delta)
\,,
\end{align}
where $\eta\sim{}1$. The photoassisted tunneling current has the same sign
$I<0$ as a thermoelectric current generated by heating the
ferromagnet, $T_S<T_F$, and therefore also the same sign as the
nonequilibrium effect (<ref>) at the low
frequencies. The ratio of the photoassisted tunneling to the
nonequilibrium one is
$I_{\rm det}(0)/I_{0,T}\propto(A_0\omega_0\ell_{\mathrm{in}}/V_{ac})^2$ where
In principle, $V_{ac}$ can be suppressed by suitable circuit design.
An approach for suppressing the photoassisted tunnelling used in a
previous experiment <cit.> measuring the gap
enhancement due to microwave drive was to use a large-area lateral
tunnel junction with a high capacitance. Assuming the microwave
currents in the S-film and through the detector junction are of
similar magnitude (cf. Ref. mooij1983-nem), one has
where $Z_{S,\ell_{\mathrm{in}}}$ is the impedance of the $S$ film of length
$\ell_{\mathrm{in}}$ and $Z_T$ is the junction impedance. Moreover, the presence
of photoassisted tunneling can in principle be recognized from the
appearance of frequency replicas in the tunneling I-V relation, which
should not be present in the nonequilibrium signal.
§ GAP INSTABILITY
The nonequilibrium spin accumulation affects the magnitude of the
superconducting order parameter, potentially leading to large changes
and collapse of superconductivity for large driving amplitudes.
Let us consider the effect of the driving on the
superconducting order parameter,
\begin{align}
\label{eq:gapeqn}
\Delta
\frac{\lambda}{8}
\int_{-E_0}^{E_0}\dd{E}
\tr\frac{\tau_1+i\tau_2}{2}\hat{g}^K(E)
\,.
\end{align}
Here, we assume singlet pairing, and $\lambda$ is the corresponding
coupling constant and $E_0$ the BCS cutoff. The simplest situation is
obtained by neglecting spin-flip and spin-orbit scattering. In this
case, we can observe that the only spin structure in the equations
arises from the Zeeman term, $\vec{h}\cdot\vec{S}$. Treating inelastic
collisions within a relaxation time approximation, we find in leading
\begin{align}
\frac{N_\sigma\delta f_{L\sigma}}{\tau_{\mathrm{in}}}
\frac{DA_0^2}{2}\sum_{\pm}R_{L\sigma}(E,E\pm\omega) [f_L^{(0)}(E) - f_L^{(0)}(E \pm \omega)]
\,.
\end{align}
The nonequilibrium part of the gap equation now reads
\begin{align}
\label{eq:gapeqn-nosf}
\delta \Delta
\frac{\lambda}{4}
\int_{-\infty}^\infty dE
\sum_{\sigma=\pm}\sum_{\pm}\frac{R_{\pm}^{(0)}(E)F_0(E)}{N_0(E)}
\\\notag&\qquad
\times[f_0(E+\sigma h) - f_0(E+\sigma h \pm \omega)]
\\
\frac{\lambda}{4}
\frac{DA_0^2\tau_{\mathrm{in}}\omega}{2T}
\int_{-\infty}^\infty dE
\sum_{\pm}\frac{\pm R_{\pm}^{(0)}(E)F_0(E)}{N_0(E)}
\end{align}
where on the second line we expand around $T\gg{}h,\Delta$, and
$F_0(E) = \frac{1}{4}\Im \tr \tau_1 \hat{g}^R(E)\rvert_{h=0}$. The result is the
same as for zero Zeeman field. Without spin-flip scattering, the
exchange field does not have a significant effect at high temperatures,
and the result coincides with known results in
Ref. eliashberg1970-fss: the superconducting gap is
enhanced by the driving, and the superconducting branch extends to
$T>T_c$. Numerical calculations also indicate that the exchange field
does not cause significant qualitative changes at lower temperatures
either (see below).
The spin imbalance generated by the spin-flip scattering
however modifies the above conclusion, provided these processes are
not slow compared to energy relaxation. As above, let us now assume
$\tau_{\rm sn}\ll{}\tau_{\mathrm{in}}$. In this case we find
\begin{align}
\delta\Delta
\frac{\lambda}{4}
\int_{-\infty}^\infty\dd{\epsilon}
\frac{
\\\notag
\sum_\pm
R_{L}(E,E\pm\omega)[f_L^{(0)}(E) - f_{L}^{(0)}(E\pm\omega)]
\,,
\end{align}
where $R_{L}=\frac{1}{2}\sum_\sigma{}R_{L\sigma}$, and
$F_\sigma=\Re{}g_{\sigma,1}$ is the coherence function. The
difference to Eq. (<ref>) is in that the elastic
spin-flip scattering forces the quasiparticle distributions for both
spins to be the same, rather than being copies of a single
distribution shifted by the exchange field.
In order to obtain analytical results, let us consider a situation in
which the effect of scattering on the spectral functions is small,
$1/\Delta\ll{}\tau_{\rm sf},\tau_{\rm so},\tau_{\rm orb}$. Then,
$N_\sigma(E)=N(E-\sigma h)$, $F_\sigma(E)=F(E-\sigma h)$,
where $N(E) = N(-E) = \Re[E/\sqrt{E^2-\Delta^2}]$ is the BCS density
of states, and $F(E) = -F(-E) = \Re[\Delta / \sqrt{E^2 - \Delta^2}]$
the BCS coherence function. Close to the critical temperature
$T\approx{}T_c(h)$ and neglecting the orbital effect, the gap equation
can be expanded to the form <cit.>
\begin{align}
\label{eq:gl}
\ln\frac{T_{c}}{T}
\frac{7\zeta(3) - 186\zeta(5)\frac{h^2}{4\pi^2T^2}}{8\pi^2T^2}\Delta^2
\frac{DA_0^2\tau_{\mathrm{in}}\omega}{4T} P(\frac{\omega}{\Delta}, \frac{h}{\Delta})
\,.
\end{align}
The nonequlibrium part $\delta\Delta$ results to an extra term,
<cit.> whose parameter dependence is
given by the dimensionless function
\begin{align}
\int_{1}^\infty\dd{x}
\frac{
[Z(x,y) - Z(x+w,y)]
[x(x+w) + 1]
\sqrt{x^2 - 1} \sqrt{(x+w)^2 - 1}
\\\notag
\theta(w-2)
\int_1^{w-1}\dd{x}
\frac{
[x(x-w) + 1]
\sqrt{x^2 - 1} \sqrt{(x-w)^2 - 1}
\end{align}
\begin{align}
\frac{1}{2}
\sum_{\alpha=\pm} \frac{
\sum_{\gamma=\pm}F[x\Delta + (\alpha + \gamma) y\Delta]
\sum_{\gamma=\pm}N[x\Delta + (\alpha + \gamma) y\Delta]
\,.
\end{align}
In the absence of spin splitting, $Z(x,y=0)=1/x$.
The function $P$ is plotted in Fig. <ref>.
We can also find its asymptotic behavior for $w\to0$,
\begin{align}
\label{eq:P-lowfreq}
\simeq
\begin{cases}
\sqrt{\frac{2yw}{1+y}}
\frac{w\ln\frac{8}{w}}{2 + 2y}
\,,
& w\ll{}y<1\,,
\\
\sqrt{8w} - (\ln\frac{8}{w} - 1)w - \frac{w^{3/2}}{\sqrt{2}}
\,,
& w\ll{}1\ll{}y\,,
\\
(\ln(8/w) - 1)w,
\,.
\end{cases}
\end{align}
For $w\gg1$, on the other hand,
\begin{align}
\label{eq:P-highfreq}
\simeq
\begin{cases}
\pi/w\,,\qquad y\ll{}1\ll{}w
\,,
\\
2/w\,,\qquad{} 1\ll{}w,y ; \; |y-w|\gg{}1
\,.
\end{cases}
\end{align}
These limits do not include the feature at $\Delta=h$.
Function $P$ for different exchange fields $h$.
The resonant dips are located at $\Delta = h$.
As follows from the gap equation (<ref>), close to $T=T_c$ the
$\Delta(T)$ relation is determined by
$T/T_c-1 \propto{} P(\Delta/\omega,h/\omega)$. The order parameter
$\Delta(T)$ is then given by the curves in
Fig. <ref>, with $y$-axis $\propto{}T-T_c$.
Additional features in the $\Delta(T)$ relation may appear
at $\Delta(T)=h$. In particular, the relation is multivalued around
this point, indicating that we can expect discontinuous transitions as
a function of temperature. This is related to changes in possible
relaxation channels around $\Delta\sim{}h$: for $\Delta(T)>h$ the
spin-averaged density of states is gapped at $E<|\Delta|-|h|$, but for
$\Delta(T)<h$ the averaged DOS is gapless, as the spin splitting is
large enough to separate the energy gaps of the two spin species. The
gap enhancement by microwave driving however continues to increase in the $\Delta(T)<h$
regime as long as $\omega<2\Delta(T)$, and is larger at some
frequencies than without spin splitting.
Order parameter $\Delta$ for
(a) different exchange fields for $DA_0^2/\omega=0.032$, and
(b) different drive amplitudes for $h/\Delta_0=0.4$.
Dotted lines correspond $DA_0^2=0$.
Spin-flip scattering time is $\tau_{sn}T_{c0}=12.5$,
$\beta=0.5$ and $\alpha_{\rm orb}=0.01$,
$\omega/\Delta_0=0.2$. Inelastic relaxation is described by
a phonon model with $\tau_{\text{e-ph},0}T_{c0}=100$.
Order parameter $\Delta$ for different spin-flip scattering times
$\tau_{sn}$. Here, $h/\Delta_0=0.4$, and other parameters are as in
Fig. <ref>(a).
The inelastic relaxation is modeled either via phonon model (solid lines)
or relaxation time approximation (dashed). Results for $h=0$ are also
shown (dotted).
To obtain a more accurate picture, we can solve
Eqs. (<ref>) and (<ref>)
Such results are shown in Figs. <ref>
and <ref>. Figure <ref>(a) indicates
the appearance of a multivalued gap, when $h\gtrsim\omega$ and
$\Delta\approx{}h$, as found analytically. From
Fig. <ref> we can note that the feature occurs for a
wide range of scattering times. The feature is absent if there is no
exchange field, and also if there is no spin-flip scattering, so that
the qualitative conclusions based on Eq. (<ref>) also
apply at lower temperatures.
§ DISCUSSION
Microwave electric field in a spin-split superconductor drives the
quasiparticles lying above the superconducting gap into a
nonequilibrium state. In the absence of spin-flipping processes, it
however cannot generate charge or spin imbalance in the diffusive
limit. Presence of spin-flip scattering enables generation of spin
imbalance, and results to a photoelectric signal observable with
ferromagnetic probes. The effect is closely related to the
thermoelectric effects in magnetic superconductors. In addition, the
excitation causes an instability in the superconducting order
parameter when the energy gap becomes comparable to the exchange
splitting of the spectrum.
To conclude, we describe production of spin imbalance and a
photo-spin-electric effect in spin-split superconductors, and present
calculations in relevant parameter ranges. The effects are
experimentally accessible with state-of-the art methods. A
number of recent experiments in similar systems exist,
<cit.> including also
microwave excitation in the GHz frequency range. <cit.>
In addition to using ferromagnetic probes for detecting the photo-spin
signal, the effect can also be seen indirectly by
observing the discontinuous transitions in the superconducting order
parameter. These are experimentally accessible via measurements of the
tunnelling DOS in the superconductor, or for example via measurement
of the supercurrent.
P.V. and T.T.H acknowledge the Academy of Finland for financial
support. T.T.H. acnowledges funding from the European Research
Council (Grant No. 240362-Heattronics). The work of F.S.B.
was supported by Spanish Ministerio de Economía y Competitividad (MINECO)
through the Project No. FIS2014-55987-P and Grupos Consolidados UPV/EHU
del Gobierno Vasco (Grant No. IT-756-13).
§ ELECTRON-PHONON INTERACTION
We use a simplified model for inelastic relaxation due to
electron-phonon interaction, <cit.>
\begin{align}
\hat{\sigma}^R(t)
\tilde{D}^R(t) \hat{g}^K(t)
+ \tilde{D}^K(t) \hat{g}^R(t)
\Bigr\}
\\
\hat{\sigma}^K(t)
\tilde{D}^R(t) - \tilde{D}^A(t)] [\hat{g}^R(t) - \hat{g}^A(t)]
\\\notag
+ \tilde{D}^K(t) \hat{g}^K(t)
\Bigr\}
\,,
\end{align}
where the Fourier transformed functions
\begin{align}
\tilde{D}^{R/A}(\omega) &= \pm i\omega^2 \sgn\omega
\,,
\\
\tilde{D}^{K}(\omega) &= [\tilde{D}^{R}(\omega) - \tilde{D}^{A}(\omega)]\coth\frac{\omega}{2T_{\rm ph}}
\,,
\end{align}
arise from weighed Fermi surface averages of the phonon Green
functions. Parts that do not contribute to the collision integral have
been subtracted. The collision integral assumes a standard form,
\begin{align}
{\cal I}_{L\sigma}
\int_{-\infty}^\infty\frac{\dd{\omega}}{8\pi}
\omega|\omega|\tr[\hat{g}^{RA}(E)\hat{g}^{RA}(E-\omega)s_\sigma]
\\\notag&\times
\{
f_{L\sigma,E}f_{L\sigma,E-\omega} - 1 + [f_{L\sigma,E}-f_{L\sigma,E-\omega}]\coth\frac{\omega}{2T_{\rm ph}}
\}
\end{align}
where $\hat{g}^{RA}=\hat{g}^R - \hat{g}^A$. The prefactor $g_{eph}$
can be defined in terms of a relaxation rate
\begin{align}
\tau_{eph,0}^{-1} = 4\frac{7\zeta(3)g_{eph}}{\pi} T_{c,0}^3
\,,
\end{align}
at temperature $T_{c,0}$ in the normal state at
Fermi surface, where $\hat{\sigma}^R=-i\hat{\tau}_3/(2\tau_{\rm eph,0})$.
§ NUMERICAL DETAILS
We solve Eq. (<ref>) numerically via a Jacobian-free
Newton-GMRES method. <cit.> To obtain a preconditioner,
we use automatic differentiation to compute the Jacobian of the
energy-local terms, excluding the self-energy parts
$[\check{\sigma}_{\rm eph} + \check{\sigma}_A,\check{g}]$. The energy
convolutions in the self-energies, after energy discretization, are
computed via fast Fourier transforms.
|
1511.00768
|
$^1$Department of Physics and Institute of Astronomy,
National Tsing-Hua University, Hsin-Chu, Taiwan
$^2$Crimean Astrophysical Observatory, 298409, Nauchny, Crimea
$^3$National Astronomical Research Institute of Thailand (NARIT),
Siripanich Building, 191 Huaykaew Road, Muang District, Chiangmai, Thailand
$^4$Special Astrophysical Observatory, Russian Academy of Sciences,
Nizhnii Arkhyz, Russia
$^5$Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel
$^6$ Department of Applied Mathematics,
National Hsinchu University of Education, Hsin-Chu, Taiwan
Motivated by the previously reported high orbital decay rate of the
planet WASP-43b,
eight newly transit light curves are obtained and presented.
Together with other data in literature, we perform
a self-consistent timing analysis with data
covering a timescale of 1849 epochs.
The results give an orbital decay rate
$dP/dt$ = $-0.02890795\pm 0.00772547$ sec/year, which is
one order smaller than previous values.
This slow decay rate corresponds to a normally assumed theoretical value
of stellar tidal dissipation factor.
In addition, through the frequency analysis, the
transit timing variations presented here are unlikely to be periodic, but
could be signals of a slow orbital decay.
§ INTRODUCTION
The discoveries of new extra-solar planets continue to be exciting
and have revealed many new implications about the formation and evolution of
planetary systems.
Though the majority of them was detected by the method of Doppler shift
(Marcy & Butler 1998),
other methods such as transit, direct imaging etc. also made
impressed contributions.
Because the orbital configurations of extra-solar planetary systems
are generally quite different from our Solar system, many investigations
on their dynamical properties and evolution have been done
(see Jiang & Ip 2001, Ji et al. 2002,
Jiang et al. 2003, Jiang & Yeh 2004a,
Jiang & Yeh 2004b, Jiang & Yeh 2007,
Mordasini et al. 2009).
In addition, the further observational effort also produces
new fruitful results continuously. For example,
Kepler space telescope has discovered many multiple planetary systems
through the transit method (Lissauer et al. 2011).
Maciejewski et al. (2010) and Jiang et al. (2013)
discovered possible transit timing variations (TTVs) which could
imply the existence of additional bodies in these planetary systems.
Lee et al. (2014) also found planetary companions around evolved stars
through the method of radial velocities by Doppler shift.
Among these,
those extreme systems with very short orbital period
have particularly raised many interesting questions
such as where they could have
formed, how they would have migrated to current positions, and how
stable their current orbits are etc.
Their physical properties have also been seriously investigated with
great effort. For example, WASP-12 planetary system,
discovered by Hebb et al. (2009), was one of the well known
extreme systems that attracted much attention.
The planet was argued to be losing mass by exceeding its Roche lobe.
Due to the falling of planetary gas towards the host star through the
first Lagrange point, it is likely to form an accretion disk
(Li et al. 2010).
This might lead to the transfer of metals and thus enhance the stellar
Maciejewski et al. (2011) employed a high-precision photometric
monitoring to study this system and greatly improved the determination
of WASP-12b planetary properties.
On the other hand, the WASP-43 planetary system, first discovered by
Hellier et al. (2011), is another case with
an even smaller orbit.
The planet is moving around a low mass K star with
an orbital period only about 0.8 days.
With a mass of 1.8 Jupiter Mass,
it is one of the most massive exoplanets carrying
an extremely short orbital period.
The existence of WASP-43 system has therefore triggered
the study of thermal radiation from exoplanets.
For example, Wang et al. (2013) confirmed the thermal emission
from the planet WASP-43b.
Chen et al.(2014) observed one transit and one occultation
event in many bands simultaneously. They detected
the day-side thermal emission in the $K$-band.
Moreover, Kreidberg et al. (2014)
determined the water abundance in the atmosphere of WASP-43b
based on the observations through Hubble Space Telescope.
As discussed in Jiang et al. (2003),
a system with a close-in planet would experience an orbital decay through
star-planet tidal interactions.
Indeed, through the XMM-Newton observations,
Czesla et al. (2013) showed an X-ray detection and
claimed that WASP-43 is an active K-star, which could be related with
tidal interactions.
In order to obtain more precise measurements of the
characteristics of this system, Gillon et al. (2012)
performed an intense photometric monitoring by ground-based
telescopes. The physical parameters have been measured
with much higher precision. Employing their data, the
atmosphere of WASP-43b was modeled. However,
they concluded that their transit data presented
no sign of transit timing variations.
Later, through a timing analysis on the transits of
WASP-43b, Blecic et al. (2014)
proposed an orbital period decreasing rate
about 0.095 second per year. With the data from
Gran Telescopio Canarias (GTC),
Murgas et al. (2014) also claimed an orbital decay
with period decreasing rate about 0.15 second per year
and suggested that a further timing analysis over future years
would be important.
Motivated by the above interesting results,
we employ two telescopes to monitor the WASP-43b transit events
and obtain eight new transit light curves.
Combining our own data
with available published
photometric transit data of WASP-43b, we investigate
the possible timing variations or orbital decay here.
Since these data cover more than 1800 epochs of the orbital evolution,
our results shall serve as the most updated reference for this
system. Our observational data are described in Section 2,
the analysis of light curves is in Section 3, the results of
transit timing variations are presented in Section 4,
and finally the concluding remarks
are provided in Section 5.
§ OBSERVATIONAL DATA
§.§ Observations and Data Reduction
In this project,
two telescopes were employed to observe
the transits of
WASP-43b. One is the 1.25-meter telescope (AZT-11)
at the Crimean Astrophysical Observatory (CrAO) in Nauchny, Crimea,
and another is the 60-inch telescope (P60) at Palomar Observatory
in California, USA.
We successfully performed one complete transit observation
with AZT-11 in 2012 and seven with P60 in 2014 and 2015.
A summary of the above observations is presented in Table 1.
After some standard procedures such as flat-field corrections etc.,
we first use the IRAF task, $daofind$, to pick bright
stars and then the task, $phot$, to measure these stars' fluxes
in each image. The light curves of these bright stars
are thus determined (Jiang et al. 2013, Sun et al. 2015).
In order to decide which stars could
be comparison stars, we first choose those with higher
brightness consistency as candidates, i.e. candidate stars.
Therefore, we calculate the
Pearson's correlation coefficient between
any two of these light curves and use 0.9 as the criterion.
The candidate stars are a set of stars in which
the correlation coefficient between any pairs of their light curves
must be more than 0.9, in order to ensure the brightness
consistency and that none of the candidate stars
are variable objects.
Finally, the flux of WASP-43 is divided by any possible combination
of the fluxes from candidate stars.
For example, when there are three candidate stars,
the flux of WASP-43 is divided by the summation
of all three candidate stars, the summation of any possible pairs
from these candidate stars, and also the flux of individual
candidate stars.
Each of the above operations leads to one calibrated light curve,
and the one with the smallest out-of-transit root-mean-square
deviation becomes the light curve of WASP-43.
Note that the out-of-transit root-mean-square deviation
is determined after the normalization process,
which would be described in Section 2.3 later.
The comparison stars are those involved in the determination of
the light curve of WASP-43.
The number of bright stars, candidate stars, and comparison stars
are listed in Table 2.
§.§ Other Observational Data from Literature
In addition to our own light curves, it will be very helpful
to employ those publicly available transit data
in previous work.
With both our own and other transit light curves,
we could therefore cover a large number of
epochs for the investigation on possible transit timing variations.
We review all WASP-43b papers and find that
there are five papers in which the electronic files
of transit light curves are provided.
Gillon et al. (2012) employed the 60cm telescope, TRAPPIST
(TRAnsiting Planets and PlanetesImals Small Telescope),
in the Astrodon $I+z$ filter to obtain 20 light curves
and the 1.2m Euler Swiss telescope in
the Gunn-$r'$ filter to obtain three light curves.
Two of the above light curves are actually
for the same transit event. That is, Epoch 38
is observed by both telescopes. Note that the epochs are given
an identification number following the convention that the transit observed
in Hellier et al. (2011) is Epoch 0.
Chen et al. (2014) observed the transit event of Epoch 499 with the
GROND instrument mounted on 2.2m MPG/ESO telescope
in seven bands: Sloan $g', r', i', z'$ and NIR $J, H, K$.
We take the light curve in $J$ band,
because only $J, H$, and $K$ bands have the information
of seeing and the wavelength of $J$ band is the closest to $R$ band,
the one we used for our own observations.
Maciejewski et al. (2013) provided the light curves of Epoch 543
and Epoch 1032.
Murgas et al. (2014) used GTC (Gran Telescopio Canarias)
instrument OSIRIS to obtain long-slit spectra. We choose the
white light-curve data to do the analysis in this paper.
In addition, there are
seven light curves available from Ricci et al.(2015).
Therefore, we take 23 light curves from Gillon et al. (2012).
In addition, we get one light curve from Chen et al. (2014),
two light curves from Maciejewski et al. (2013),
one from Murgas et al. (2014), and seven from
Ricci et al.(2015).
In total, we have 34 light curves
taken from published papers.
We do not simply use the mid-transit times written in these papers,
but re-analyze all the photometric data with the same procedure
and software to perform
parameter fitting in a consistent way.
Because all these data go
through the same transit modeling and fitting procedure
as our own data,
it can ensure that the results are reliable.
§.§ The Normalization and Time Stamp of Light Curves
For all the previously mentioned light curves,
including eight from our work and 34 from the
published papers, we further consider the airmass and
seeing effects here.
As the procedure in Murgas et al. (2014),
a 3rd-degree polynomial is used to model the airmass effect,
and a linear function is employed to model
the seeing effect.
The original light curve, $F_{o}(t)$,
can be expressed as:
\begin{equation}
F_{o}(t) = F(t) \mathcal{P}(t) \mathcal{Q}(s),
\end{equation}
where $F(t)$ is the corrected light curve,
$\mathcal{P}(t) = a_0 + a_1 t + a_2 t^2+ a_3 t^3$,
and $\mathcal{Q}(s) = 1 + c_0 s$,
where $s$ is the seeing of each image.
(Note that, in Maciejewski et al. 2013 and Ricci et al. 2015,
the seeing is not known and no seeing correction can be done.
We thus set $\mathcal{Q}(s)=1$ for light curves from these two papers.)
We numerically search the best values of five parameter
$a_0, a_1, a_2, a_3, c_0$ to make out-of-transit part
of $F(t)$ close to unity with smallest standard deviations,
and thus normalize all the light curves.
$F(t)$ would be simply called the observational
light curves and used in any further analysis
for the rest of this paper.
On the other hand, the time stamp we use is the Barycentric Julian Date
in the Barycentric Dynamical Time ($BJD_{TDB}$).
We compute the UT time of mid exposure from the recorded header
and convert the time stamp to
$BJD_{TDB}$ as in Eastman et al. (2010).
§ THE ANALYSIS OF LIGHT CURVES
The Transit Analysis Package (TAP) presented by
Gazak et al. (2012) is used to obtain transit models and
corresponding parameters from all the above 42 light curves.
TAP employs the light-curve models of Mandel & Agol (2002),
the wavelet-based likelihood function developed by Carter & Winn (2009),
and Markov Chain Monte Carlo (MCMC) technique to
determine the parameters.
All 42 light curves are loaded into TAP
and analyzed simultaneously.
For each light curve, five MCMC chains of length 500,000 are computed.
To start an MCMC chain in TAP, we need to set the initial values
of the following parameters:
orbital period $P$, orbital inclination $i$, semi-major axis $a$
(in the unit of stellar radius $R_{\ast}$), the planet's radius $R_{\rm p}$
(in the unit of stellar radius),
the mid-transit time $T_{\rm m}$, the linear limb darkening
coefficient $u_1$, the quadratic limb darkening
coefficient $u_2$, orbital eccentricity $e$ and
the longitude of periastron $\omega$. Once the initial
values are set, one could choose any one of the above to be:
(1) completely fixed (2)
completely free to vary or (3) varying following a Gaussian function,
i.e., Gaussian prior, during the MCMC chain in TAP.
Moreover, any of the above parameters which is not completely fixed
can be linked among different light curves.
The orbital period is treated as
a fixed parameter $P$ = 0.81347753, which is taken from Table 5 of
Gillon et al. (2012). The initial values of inclination $i$,
semi-major axis, and planet's radius are all from Gillon et al.(2012),
i.e. $i$=82.33, $a/R_{\ast}$=4.918, and $R_{\rm p}/R_{\ast}$=0.15945.
They are completely free to vary and linked among all light curves.
We leave the mid-transit times
$T_{\rm m}$ to be completely free during TAP runs and
it is only linked among those light curves in the same transit events.
Two light curves from Gillon et al. (2012) are for
the same transit event, i.e. epoch 38,
and another two from Ricci et al. (2015) are for epoch 1469.
A Gaussian prior centered on the values
of quadratic limb darkening coefficients
with certain $\sigma$ are set for TAP runs.
The quadratic limb darkening coefficients and $\sigma$
for $I+z$ and Gunn-$r'$ filters
are set as the values in Gillon et al. (2012),
and the one for white light curve follows the values used
in Murgas et al. (2014).
For $i, I, J, R$, and $V$ filters,
we linearly interpolate
from Claret (2000, 2004) to the values effective temperature
$T_{\rm eff}$ = 4400 K, log$g$ = 4.5 ${\rm cm/s^{2} }$,
metallicity [Fe/H] = 0, and micro-turbulent velocity
$V_{\rm t}$ = 0.5 ${\rm km/s}$ (Hellier et al. 2011).
In order to consider the possible small differences mentioned in
Southworth (2008), a Gaussian prior centered
on the theoretical values with $\sigma$ = 0.05 is set
for our limb darkening coefficients $u_1$ and $u_2$ during
TAP runs.
The details of parameter setting for TAP runs are listed in
Table 3 and Table 4.
There are five chains in each of our TAP runs, and all of the chains
are added together into the final results. The 15.9, 50.0 and
84.1 percentile levels are recorded. The 50.0 percentile, i.e., median level,
is used as the best value, and the other two percentile levels
give the error bar.
The mid-transit time for the corresponding epoch
of each transit event is obtained.
In order to examine whether there is any outlier, these mid-transit times are
fitted by a linear function. It is found that the mid-transit time of
epoch 1469 has the largest deviation and is more than 3$\sigma$ away
from the linear function. We thus remove two light curves of
epoch 1469 from our data set and re-run TAP through the same procedure.
We finally obtain the mid-transit time for the corresponding epoch
of each transit event, as those presented in Table 5.
They will be used to establish a new ephemeris and study
the transit timing variations in next section.
The results of inclination,
semi-major axis, and planet's radius are listed in Table 6.
These values are consistent with all those published in previous work.
For example, comparing with the results in Gillon et al.(2012)
or Ricci et al (2015), our results
are all extremely close to theirs, if error bars are considered.
Our error bars are actually smaller than theirs.
This shows that our analysis with more light curves gives
stronger observational constraints.
Moreover, the observational light curves and
best fitting models of our own data are presented in Figure 1,
where the points are observational data and solid curves are
the best fitting models.
These eight light curves of our own work
are available in a machine-readable form in
the electronic version of Table 7.
§ TRANSIT TIMING VARIATIONS
§.§ A New Ephemeris
When all mid-transit times of 39 epochs in Table 5
are considered, we obtain a new ephemeris
by minimizing $\chi^2$ through
fitting a linear function as
T^C_m (E) = T_0 + P E,
where $T_0$ is a reference time, $E$ is an epoch
(The transit observed in Hellier et al. 2011 is defined to be
epoch $E=0$, and other transits' epochs are defined accordingly.),
$P$ is the orbital period, and $T^C_{\rm m}(E)$
is the calculated mid-transit time at a given epoch $E$.
We find that
$T_{0} = 2455528.86860518\pm 0.00003632$ ($BJD_{TDB}$),
$P = 0.81347392\pm 0.00000004$ (day).
The corresponding $\chi^2$ = 266.2076.
Because the degree of freedom is 37, the reduced $\chi^2$,
$\chi^2_{red}(37)$ = 7.1948.
Using this new ephemeris,
the $O-C$ diagram is presented as the data points in Figure 2.
The large value of reduced $\chi^2$ of the linear fitting presented here
indicates that a certain level of TTVs does exist.
§.§ A Model of Orbital Decay
Through the transit timing analysis, Blecic et al. (2014)
and Murgas et al. (2014) proposed a possible orbital decay for
the planet WASP-43b. However, their transit data were up to about
epoch 1000 only. It would be very interesting to see
whether our newly observed data gives the transit timing
with a trend of orbital decay.
Assume the orbital period is $P_q$, and the predicted
mid-transit time at epoch $E$ is $T_{S}(E)$.
For convenience, the mid-transit time of epoch 0, $T_S(0)$, is set to be zero,
so the mid-transit time of epoch 1
is $T_S(1)=P_q$, and the elapsed time
$\delta t_1= P_q$. If there is a small amount of
period changing $\delta P$ from time $t= T_S(1)$ to
$t=T_S(2)$, the elapsed time is $\delta t_2= P_q + \delta P$.
Suppose there is a further period changing with $\delta P$
from time $t= T_S(2)$ to $t= T_S(3)$,
so the elapsed time $\delta t_3= P_q + 2\delta P$.
Following this continuous period decay, we
have $\delta t_i= P_q + (i-1)\delta P$,
where $i=1, 2, ...,(E-1), E$.
Summing up all the above $\delta t_i$, we obtain
$T_{S}(E) = E P_q + [E(E-1)/2]\delta P $.
as in Blecic et al. (2014), a model of orbital decay can be obtained
by minimizing $\chi^2$ through
fitting a function as
\begin{equation}
T_{S}(E) = T_{q0} + P_q E + \delta P \frac{E(E-1)}{2}
\end{equation}
where $T_{q0}$ is a reference time, $E$ is an epoch,
$P_{q}$ is the orbital period, $\delta P$
is the amount of period changing between each mid-transit time
starting from $t=T_S(1)$.
When only the data of earlier work
with transits before epoch 1100, i.e.
Gillon et al. (2012), Chen et al. (2014),
Maciejewski et al. (2013), and Murgas et al. (2014),
are considered, we have
$T_{q0} = 2455528.86809115\pm 0.00006471$,
$P_q = 0.81347925\pm 0.00000055$,
$\delta P = -1.03181346\times 10^{-8}\pm 0.10711789\times 10^{-8}$.
The corresponding $\chi^2$= 131.7672, and
$\chi^2_{red}(24)= 5.4903$.
Using the above best-fitted parameters for
$T_{S}(E)$ and the new ephemeris for $T^C_{\rm m}(E)$,
the $ T_{S}(E)- T^C_{\rm m}(E)$
as a function of epoch $E$ is plotted as the dashed curve
in the $O-C$ diagram, together with
data points as shown in Figure 2.
Both the units of $P_q$ and $\delta P$ are days,
and we obtain
$dP/dt=\delta P/P_q$ = $-0.40027520\pm 0.04155436 $ sec/year.
We find that this result is consistent
with the orbital decay rate stated in previous works.
When all the data in Table 5 are considered, we obtain
$T_{q0} = 2455528.86851783\pm 0.00004318$,
$P_q=0.81347448\pm 0.00000016$,
$\delta P = -7.45173434\times 10^{-10}\pm 1.98109164\times 10^{-10}$.
The corresponding $\chi^2_{red}(36)=7.0057$.
The larger value of reduced $\chi^2$ is due to the
larger number of data points in this case.
Using the above best-fitted parameters for
$T_{S}(E)$ and the new ephemeris for $T^C_{\rm m}(E)$,
the $ T_{S}(E)- T^C_{\rm m}(E)$
as a function of epoch $E$ is plotted as the solid curve
in the $O-C$ diagram, together with
data points as shown in Figure 2.
Comparing the solid curve with the dashed curve in Figure 2,
it is obvious that the
data points around epoch 1500 and epoch 1900 do not follow the
dashed curve. That is, the newly obtained transits do not
follow the predicted transit timings in previous models.
On the other hand, for the solid curve in Figure 2, the overall
orbital decay rate is
$dP/dt=\delta P/P_q$ = $-0.02890795\pm 0.00772547$ sec/year, which is
one order smaller than the values in previous work.
Therefore, with our newly observed transits,
we obtain a very different orbital decay rate.
These results indicate that,
if there is any orbital decay, the decay rate shall be much smaller
than those values proposed in previous works.
This slower orbital decay rate leads to a new estimate of
the stellar tidal dissipation factor $Q_{\ast}$.
Following the equation in Blecic et al. (2014),
we obtain a value of $Q_{\ast}$ about the order of $10^5$,
which is within the range of normally assumed theoretical value
from $10^5$ to $10^{10}$.
§.§ The Frequency Analysis
In order to search for possible periodicities of transit timing
variations from the timing residuals,
Lomb-Scargle normalized periodogram (Press & Rybicki 1989)
is used. Figure 3 shows the resulting spectral
power as a function of frequencies.
The false-alarm probability of the largest power
of frequencies is 0.20, which is very far from the usual
thresholds 0.05 or 0.01 for a confirmed frequency.
Therefore, our results show that there is no evidence
for periodic TTVs.
§ CONCLUDING REMARKS
Employing telescopes at two observatories,
we monitor the transits of exoplanet WASP-43b
and obtain eight new transit light curves.
Together with the light curves from published papers,
they are all further analyzed through the same procedure.
The transit timings are obtained,
and a new ephemeris is established.
The newly determined inclination $i=82.149^{+0.084}_{-0.086}$,
semi-major axis $a/R_{\ast}=4.837^{+0.021}_{-0.022}$,
and planet's radius $R_{\rm p}/R_{\ast}=0.15929^{+0.00045}_{-0.00045}$
are all consistent with previous work.
Our results reconfirm that a certain level of
TTVs does exist, which is the same as what was claimed
in Blecic et al.(2014) and Murgas et al.(2014) previously.
However, the results here show that the transit timings of new data
do not follow the fast trend of the orbital decay suggested
in Blecic et al. (2014) and Murgas et al. (2014).
Our results lead to an orbital decay rate
$dP/dt$ = $ -0.02890795\pm 0.00772547$ sec/year, which is
one order smaller than the previous values.
This slower rate corresponds to a larger
stellar tidal dissipation factor $Q_{\ast}$
in the range of normally assumed theoretical value.
On the other hand,
the false-alarm probabilities in the frequency analysis indicate
that these TTVs are unlikely to be periodic.
The TTVs we present here could be signals of a slow orbital decay.
We conclude that, in order to further investigate and understand
this interesting system, both realistic theoretical modeling
and much more high-precision observations
are desired in the future.
§ ACKNOWLEDGMENT
We thank the anonymous referee for good suggestions
which greatly improved this paper. We also thank the
helpful communications with
Gillon, M., Gazak, J. Z., Maciejewski, G., and Ngeow, C.-C..
This work is supported in part
by the Ministry of Science and Technology, Taiwan,
under MOST 103-2112-M-007-020-MY3 and
NSC 100-2112-M-007-003-MY3.
Bl Blecic, J. et al. 2014, ApJ, 781, 116
Ca Carter, J. A. & Winn, J. N. 2009, ApJ, 704, 51
Che Chen, G. et al. 2014, A&A, 563, A40
Cl2000 Claret, A. 2000, A&A, 363, 1081
Cl2004 Claret, A. 2004, A&A, 428, 1001
Czesla, S., Salz, M., Schneider, P. C., Schmitt, J. H. M. M. 2013,
A&A, 560, A17
East Eastman, J., Siverd, R., & Gaudi, B. S. 2010, PASP, 122, 935
Ga Gazak, J. Z. et al. 2012, Advances in Astronomy, 2012, 697967
Gi Gillon, M. et al. 2012, A&A, 542, A4
Hebb Hebb, L. et al. 2009, ApJ, 693, 1920
He Hellier, C. et al. 2011, A&A, 535, L7
Ji1 Ji, J., Li, G., Liu, L. 2002, ApJ, 572, 1041
Jiang, I.-G., Ip, W.-H. 2001, A&A, 367, 943
Jiang, I.-G., Ip, W.-H., Yeh, L.-C. 2003, ApJ, 582, 449
JiangY1 Jiang, I.-G., Yeh, L.-C. 2004a, AJ, 128, 923
JiangY2 Jiang, I.-G., Yeh, L.-C. 2004b,
Int. J. Bifurcation and Chaos, 14, 3153
Jiang, I.-G., Yeh, L.-C. 2007, ApJ, 656, 534
Jiang, I.-G. et al. 2013, AJ, 145, 68
K1 Kreidberg, L. et al. 2014, ApJ, 793, L27
Lee, B.-C., Han, I., Park, M.-G., Mkrtichian, D. E., Hatzes, A. P.,
Kim, K.-M. 2014, A&A, 566, A67
Li, S.-L., Miller, N., Lin, D. N. C., Fortney, J. J. 2010,
Nature, 463, 1054
Liss Lissauer, J. J. et al. 2011, ApJS, 197, 8
Mac Maciejewski, G. et al. 2010, MNRAS, 407, 2625
Maci Maciejewski, G. et al. 2011, A&A, 528, A65
Maciejewski, G. et al. 2013, Information Bulletin on Variable Stars, 6082, 1
Marcy Marcy, G. W., Butler, R. P. 1998,
ARA&A, 36, 57
Man Mandel, K. & Agol, E. 2002, ApJ, 580, L171
Mor Mordasini, C., Alibert, Y., Benz, W. 2009,
A&A, 501, 1139
Mu Murgas, F. et al. 2014, A&A, 563, A41
Pr Press, W. H. & Rybicki, G. B. 1989, ApJ, 388, 277
Ricci Ricci, D. et al. 2015, PASP, 127, 143
Sun Sun, L.-L. et al. 2015,
Research in Astronomy and Astrophysics, 15, 117
Sou Southworth, J. 2008, MNRAS, 386, 1644
Wan Wang, W., van Boekel, R., Madhusudhan, N., Chen, G.,
Zhao, G., Henning, Th. 2013, ApJ, 770, 70
Table 1: The log of observations of this work
Run UT Date Instrument Filter Interval (JD-2450000)
Exposure No. of
1 2012 Mar 24 AZT-11 R 6011.212 - 6011.302 30
2 2014 Mar 12 P60 R 6728.698 - 6728.789 10
3 2014 Mar 16 P60 R 6732.767 - 6732.856 10
4 2014 Apr 07 P60 R 6754.731 - 6754.820 12
5 2014 Dec 24 P60 R 7015.864 - 7015.940 12
6 2015 Jan 06 P60 R 7028.864 - 7028.955 12
7 2015 Jan 15 P60 R 7037.815 - 7037.905 12
8 2015 Jan 19 P60 R 7041.882 - 7041.979 12
[Table 1]The log of observations of this work.
For each run, the UT date,
instrument, filter, observational interval (JD-2450000),
exposure time (second), and the number of images
are listed.
Table 2: The numbers of stars
Run No. of Brighter Stars No. of Candidates No. of Comparisons oot rms
1 5 3 1 0.0045
2 18 3 3 0.0028
3 12 4 2 0.0041
4 24 5 2 0.0020
5 5 2 2 0.0029
6 11 2 2 0.0035
7 16 3 3 0.0042
8 13 10 6 0.0027
[Table 2]The number of stars in the process for choosing comparison
stars. The out-of-transit
root-mean-square of light curves are also listed.
Table 3: The parameter setting
Parameter Initial Value During MCMC Chains
$P$(day) 0.81347753 fixed
$i$(deg) 82.33 free, linked among all
$a$/$R_{\ast}$ 4.918 free, linked among all
$R_{\rm p}$/$R_{\ast}$ 0.15945 free, linked among all
$T_{\rm m }$ set-by-eye free, only linked if same transit events
$u_1$ Claret (2000,2004) a Gaussian prior, not linked
$u_2$ Claret (2000,2004) a Gaussian prior, not linked
$e$ 0.0 fixed
${\omega}$ 0.0 fixed
[Table 3]
The parameter setting. The initial values of $P, i, a/R_{\ast},
R_{\rm p}/R_{\ast}$ are adopted from Table 5 of Gillon et al.(2012).
Table 4: The quadratic limb darkening coefficients
filter $u_1$ $u_2$
$^{a}I+z$ $0.440\pm 0.035$ $0.180\pm 0.025$
$^a$Gunn-$r'$ $0.625\pm 0.015$ $0.115\pm 0.010$
$^{b}$white $0.394\pm 0.087$ $0.289\pm 0.119$
$^{c}i$ $0.4767\pm 0.05$ $0.2067\pm 0.05$
$^{c}I$ $0.4401\pm 0.05$ $0.2200\pm 0.05$
$^{c}J$ $0.2560\pm 0.05$ $0.2959\pm 0.05$
$^{c}R$ $0.6012\pm 0.05$ $0.1492\pm 0.05$
$^{c}V$ $0.7598\pm 0.05$ $0.0427\pm 0.05$
$^{d}$clear $0.6805\pm 0.05$ $0.0960\pm 0.05$
[Table 4]The quadratic limb darkening coefficients.
$^a$set as the values in Gillon et al. (2012)
$^b$set as the values in Murgas et al. (2014)
$^c$calculated for $T_{\rm eff} = 4400$ K,
log$g$ = 4.5 ${\rm cm/s^{2} }$, [Fe/H] = 0,
and $V_{\rm t}$ = 0.5 ${\rm km/s}$.
$^d$calculated as the average of those for $V$ and $R$ bands.
Table 5: The results of light-curve analysis for
the mid-transit time
Epoch Data Source $T_m$($BJD_{TDB}-2450000$)
11 (a) 5537.81659 $^{+0.00045}_{-0.00048}$
22 (a) 5546.76493 $^{+0.00020}_{-0.00021}$
27 (a) 5550.83218 $^{+0.00019}_{-0.00018}$
38 (a) 5559.78048 $^{+0.00012}_{-0.00012}$
43 (a) 5563.84773 $^{+0.00021}_{-0.00020}$
49 (a) 5568.72833 $^{+0.00012}_{-0.00012}$
59 (a) 5576.86368 $^{+0.00015}_{-0.00015}$
65 (a) 5581.74392 $^{+0.00011}_{-0.00011}$
70 (a) 5585.81297 $^{+0.00029}_{-0.00029}$
76 (a) 5590.69256 $^{+0.00019}_{-0.00018}$
87 (a) 5599.64047 $^{+0.00024}_{-0.00024}$
97 (a) 5607.77505 $^{+0.00012}_{-0.00012}$
124 (a) 5629.73981 $^{+0.00011}_{-0.00010}$
140 (a) 5642.75474 $^{+0.00013}_{-0.00013}$
141 (a) 5643.56875 $^{+0.00023}_{-0.00022}$
152 (a) 5652.51574 $^{+0.00038}_{-0.00038}$
168 (a) 5665.53206 $^{+0.00026}_{-0.00026}$
173 (a) 5669.59939 $^{+0.00017}_{-0.00018}$
189 (a) 5682.61543 $^{+0.00018}_{-0.00018}$
200 (a) 5691.56374 $^{+0.00013}_{-0.00013}$
211 (a) 5700.51237 $^{+0.00018}_{-0.00018}$
243 (a) 5726.54407 $^{+0.00018}_{-0.00018}$
499 (b) 5934.79276 $^{+0.00019}_{-0.00019}$
543 (c) 5970.58598 $^{+0.00025}_{-0.00027}$
593 (f) 6011.25910 $^{+0.00056}_{-0.00054}$
950 (d) 6301.66872 $^{+0.00006}_{-0.00005}$
1032 (c) 6368.37476 $^{+0.00060}_{-0.00068}$
1442 (e) 6701.89857 $^{+0.00016}_{-0.00017}$
1475 (f) 6728.74255 $^{+0.00023}_{-0.00024}$
1480 (f) 6732.80936 $^{+0.00049}_{-0.00047}$
1485 (e) 6736.87743 $^{+0.00046}_{-0.00048}$
1486 (e) 6737.69125 $^{+0.00022}_{-0.00022}$
1496 (e) 6745.82472 $^{+0.00034}_{-0.00035}$
1507 (f) 6754.77378 $^{+0.00016}_{-0.00016}$
1550 (e) 6789.75311 $^{+0.00033}_{-0.00033}$
1828 (f) 7015.89837 $^{+0.00025}_{-0.00024}$
1844 (f) 7028.91466 $^{+0.00024}_{-0.00023}$
1855 (f) 7037.86178 $^{+0.00029}_{-0.00028}$
1860 (f) 7041.92985 $^{+0.00020}_{-0.00020}$
[Table 5]The results of light-curve analysis for
the mid-transit time $T_m$.
The epoch is the number of transit calculated from the first transit
presented in Hellier et al. (2011). Data sources: (a) Gillon et al.(2012),
(b) Chen et al.(2014), (c) Maciejewski et al.(2013), (d) Murgas et al.(2014), (e) Ricci et al.(2015), and (f) this work.
Table 6: The results of light-curve analysis for
the inclination, semi-major axis, and planet's radius
Parameter Value
$i$ 82.149 $^{+0.084}_{-0.086}$
$a$/$R_{\ast}$ 4.837 $^{+0.021}_{-0.022}$
$R_{\rm p}$/$R_{\ast}$ 0.15929 $^{+0.00045}_{-0.00045}$
[Table 6]The results of light-curve analysis for the
inclination $i$, semi-major axis $a$/$R_{\ast}$,
and planet's radius $R_{\rm p}$/$R_{\ast}$.
Table 7: The photometric light-curve data of this work
Run Epoch TDB-based BJD Relative Flux
1 593 2456011.21824144 0.999658
2456011.21859823 0.997367
2456011.21895503 0.997222
2 1475 2456728.70459992 1.005345
2456728.70501139 1.002147
2456728.70542342 1.001576
3 1480 2456732.77312609 0.999979
2456732.77353873 1.004087
2456732.77395244 1.002193
[Table 7]The photometric light-curve data of this work.
This table is available in its entirety in the on-line journal.
A portion is shown here for guidance.
The normalized relative flux as a function of
the time (the offset from mid-transit time and in TDB-based BJD)
of eight transit light curves of this work: points are the data and
curves are models. The corresponding residuals are shown at the
bottom of light curves.
The $O-C$ diagram.
The filled circles are for our work. The triangles are
for the data from Gillon et al. (2012), the square is for
the data from Chen et al. (2014),
the diamonds are for the data from Maciejewski et al. (2013),
the cross is for the
data from Murgas et al. (2014), and the open circles are
for the data from Ricci et al. (2015).
The dashed curve is the model determined by
fitting with those data before epoch 1100 only.
The solid curve is the model determined by
fitting with all data.
The spectral power as a function of frequencies for the data points
shown in Figure 2.
The false-alarm probability of the largest power
of frequencies is 0.20 and shown
as the bottom dotted line. The middle dotted line shows 0.05,
and the top dotted line shows 0.01 false-alarm probability.
|
1511.00938
|
We consider query answering using views on graph databases, i.e. databases
structured as edge-labeled graphs. We mainly consider views and queries
specified by Regular Path Queries (). These are queries selecting pairs of nodes
in a graph database that are connected via a path whose sequence of edge labels
belongs to some regular language. We say that a view determines a query if for all graph databases , the view image $\V(\D)$ always contains enough
information to answer on . In other words, there is a well defined
function from $\V(\D)$ to $\Q(\D)$.
Our main result shows that when this function is monotone, there exists a
rewriting of as a query over the view instance $\V(\D)$. In
particular the rewriting query can be evaluated in time polynomial in the size
of $\V(\D)$. Moreover this implies that it is decidable whether an query
can be rewritten in using views.
§ INTRODUCTION
We consider the problem of answering queries using views on graph
databases. Graph databases are relational databases where all relation symbols
are binary. In other words a graph database can be viewed as an edge-labeled
directed graph.
Graph-structured data can be found in many important scenarios. Typical examples
are the semantic Web via the format RDF and social
networks. Graph-structured data differs conceptually from relational databases
in that the topology of the underlying graph is as important as the
data it contains. Usual queries will thus test whether two nodes are
connected and how they are connected <cit.>.
In many contexts it is useful to know whether a given set of queries can be used
to answer another query. A typical example is the data integration setting where
data sources are described by views
of a virtual global database. Queries over the global
database are then rewritten as queries over the views. Another example is
caching: answers to some set of queries against a data source are cached, and
one wishes to know if a newly arrived query can be answered using the cached
information, without accessing the source. This problem also finds application in the
context of security and privacy. Suppose access to some of the information in a
database is provided by a set of public views, but answers to other queries are
to be kept secret. This requires verifying that the disclosed views do not
provide enough information to answer the secret queries.
All these problems can be phrased in terms of views and query rewriting using
views, which is a typical database problem, not specific to graph databases,
that has received considerable attention (see <cit.> among others).
When graph databases are concerned, the
difference lies only in the kind of queries under consideration <cit.>.
Over graph databases, typical queries have at least the expressive power of
Regular Path Queries (), defined in <cit.> (see
also the survey <cit.>). An selects pairs of nodes
connected by a path whose sequence of edge labels satisfies a given regular
expression. A view, denoted by , is then specified using a finite set of
s. When evaluated over a graph database , the view yields a new
graph database $\V(\D)$ where each $V_i\in\V$ is a new edge relation symbol.
We are interested in knowing whether the view always provides enough
information to answer another query , i.e. whether $\Q(\D)$
can be computed from $\V(\D)$ for all databases . When this is the case we say
that determines , and we look for a
rewriting of using , i.e. a new query, in some query
language, that expresses $\Q$ in terms of $\V$.
We are then interested in finding an algorithm for evaluating the rewriting, i.e. an algorithm computing $\Q(\D)$ from $\V(\D)$.
These two related questions, determinacy and query rewriting, have been studied
for relational databases and graph databases. Over relational databases,
determinacy is undecidable already if the queries and views are defined by
union of conjunctive queries, and its decidability status is open for views and
queries specified by conjunctive queries () <cit.>. Over graph
databases and queries and views, the decidability status of determinacy is
also open <cit.>. Determinacy has been shown to be
decidable in a scenario where views and queries can only test whether there is
a path of distance $k$ between the two nodes, for some given
$k$ <cit.>.
This scenario lies at the intersection of and and contains already non
trivial examples. For instance the view $\path_3$ and $\path_4$, giving
respectively the pairs of nodes connected by a path of length 3 and 4,
determines the query $\path_5$ asking for the pairs of nodes connected by a
path of length 5 <cit.> (see also Example <ref> in
Section <ref>).
Clearly when can be rewritten in terms of , the rewriting witnesses that
determines . On the other hand determinacy does not say that one can find
a rewriting definable in a particular language, nor with particular computational
It is then natural to ask which rewriting language $\cLR$ is sufficiently
powerful so that determinacy is equivalent to the existence of a rewriting
definable in $\cLR$. This clearly depends on the language used for defining the
query and the view.
Consider again the case of $\path_5$ that is determined by $\path_3$ and
$\path_4$. A rewriting $R(x,y)$ of $\path_5$ in terms of $\path_3$ and $\path_4$ is
defined by:
\begin{equation*}
\exists u ~ ( \path_4(x,u) \land \forall v ~(\path_3(v,u) \rightarrow
\path_4(v,y)) )
\end{equation*}
and it can be shown that there is no rewriting definable in , nor in (cf. Example <ref>). In the case of views and queries
defined by s it is still an open problem to know whether first-order logic
is a sufficiently powerful rewriting language. Even worse, it is not even
known whether there always exists a rewriting that can be evaluated in time
polynomial in the size of the view instance <cit.>, ie. polynomial data
complexity. A similar situation arises over graph-databases and views
and queries <cit.>.
It can be checked that in the example above there exists no monotone rewriting
of $\path_5$ (see again Example <ref>). In particular, as s define only monotone
queries, no rewriting is definable in . Monotone query languages such as
, , and their extensions are of crucial importance in many
database applications. The possibility of expressing rewritings in these
languages is subject to a monotonicity restriction.
This is why in this paper we are considering a stronger notion of determinacy,
referred to as monotone determinacy, by further requiring that the
mapping from view instances to query results is monotone.
In the case when views and queries are defined by s, monotone determinacy
can be shown to be equivalent to the existence of a rewriting in
<cit.>. As this latter problem is decidable <cit.>,
monotone determinacy for s is decidable.
We consider here monotone determinacy for graph databases and views and queries
defined by s.
We first observe that monotone determinacy corresponds to the notion called
losslessness under the sound view assumption
in <cit.>, where it was shown to be decidable. We then
concentrate on the rewriting problem.
We know that there exist cases of monotone rewritings that are not expressible
in <cit.> (see also Example <ref> in
Section <ref>). We thus need a more powerful language in order
to express all monotone rewritings.
It is not too hard to show that if determines then there exists a
rewriting with data complexity, as well as a rewriting
with data complexity. Our main result shows that if moreover determines in a monotone way, there exists a rewriting definable in , which therefore can
be evaluated in polynomial time.
Our proofs are constructive, hence the rewriting can be computed from
and .
As a corollary this implies that it is decidable whether a query has a
rewriting definable in using a view , where both and are
defined using s. This comes from the fact that our main result implies
that the existence of a rewriting in is equivalent to monotone
determinacy, a decidable property as mentioned above.
Related work
The work which is most closely related to ours is that of the “Four
Italians”. In particular, the notion of losslessness under the exact view
assumption introduced in <cit.> corresponds to what we
call determinacy; similarly the notion of losslessness under the sound view
assumption corresponds to what we call monotone determinacy. Monotone determinacy is
also mentioned in the thesis <cit.> under the name of “strong
determinacy”. It is shown there that it corresponds to the existence of a
monotone rewriting.
A lot of attention has been devoted to the problem of computing the set of
certain answers to a query w.r.t a set of views, under the sound view
assumption (see the precise definition of certain answers in
Section <ref>). For views and queries, the problem is
shown to be equivalent to testing whether the given instance homomorphically
embeds into a structure $\tempQV$ computed from the view and the query
<cit.>. In general this shows that the data complexity of
computing the certain answers is -complete. Building on results on
Constraint Satisfaction Problems <cit.>, it was also
shown in <cit.> that for an view , an query and for each $l,k$, with $l \leq k$, there is a program $\Qlk$ which
is contained in the certain answers to given and is, in a sense,
maximally contained: i.e. $\Qlk$ contains all programs which are
contained in the certain answers and use at most $l$ head variables and at most
$k$ variables in each rule.
If we assume that determines in a monotone way, it is easy to see that
the query computing the certain answers under the sound view assumption is a
rewriting of using (i.e the certain answers of a view instance
$\V(\D)$ are precisely the query result $\Q(\D)$).
However there are possibly other rewritings (they only need to agree on
instances of the form $\V(\D)$, but may possibly differ on instances not in the
image of .) While computing the certain answers is -hard, our
main result shows that there exists another rewriting which is expressible in
, and has therefore polynomial time data complexity.
Nevertheless our proof makes use of the structure $\tempQV$ mentioned above, and
our rewriting turns out to be the query $\Qlk$ associated with and
for some suitable values of $l$ and $k$.
§ PRELIMINARIES
Graph databases and paths
A binary schema is a finite set of relation symbols of arity 2. All the schemas
used in this paper are binary. A graph database is a finite
relational structure over a (binary) schema $\sigma$. We will also say a
$\sigma$-structure. Alternatively can be viewed as a directed edge-labeled
graph with labels from the alphabet $\sigma$. The elements of the domain of are referred to as nodes. The number of elements in is denoted by
$\size{\D}$. If $A$ is a set of elements of , we denote by $\D[A]$ the
substructure of induced by $A$.
Given a graph database , a path $\pi$ in from $x_0$ to $x_m$ is a
finite sequence $\pi = x_0a_0x_1\ldots x_{m-1}a_{m-1}x_m$, where each $x_i$ is
a node of , each $a_i$ is in $\sigma$, and $a_i(x_i,x_{i+1})$ holds in for each $i$.
A simple path is a path
such that no node occurs twice in the sequence.
The label of $\pi$, denoted by $\lambda(\pi)$, is the word $a_0a_1\ldots
a_{m-1} \in \sigma^*$. By abuse of notation, we sometimes view a path $\pi$ as a
graph database, which contains only the nodes and edges that occur in the
Queries and query languages
A query over a schema $\sigma$ is a mapping
associating to each graph database over $\sigma$ a finite relation
$\Q(\D)$ over the domain of . We will only consider binary queries, that is queries
that return binary relations, and work with the following query languages.
A Regular Path Query (often abbreviated as ) over $\sigma$ is given by a regular
expression over the alphabet $\sigma$. If is an , we denote by $L(\Q)$ the language
corresponding to its regular expression. On a graph database, such a query
selects all the pairs $(x, y)$ of nodes such that there exists a path $\pi$
from $x$ to $y$ with $\lambda(\pi) \in L(\Q)$.
For instance the query $\path_3$ of the introduction is an corresponding
to the regular expression $\sigma\sigma\sigma$ (also denoted
$\sigma^3$). Another example is the $(\sigma\sigma)^*$ that select pairs
of nodes connected via a path of even length.
A Context-Free Path Query
over $\sigma$
is defined similarly but using a context-free grammar instead of a regular
A Conjunctive Regular Path Query (sometimes abbreviated ) over $\sigma$ is a conjunctive
query whose atoms are specified using s over $\sigma$.
For instance the query
\begin{equation*}
\exists z~ Q_1(x,z) \land Q_2(z,y) \land Q_3(z,y)
\end{equation*}
where $Q_1 = a^+$, $Q_2 = b$ and $Q_3 = c$ selects pairs of nodes $(x,y)$ which
are connected via a path labeled $a^+b$ and another path labeled $a^+c$ sharing
their $a^+$ part. This cannot be expressed by an .
A query over schema $\sigma$ is defined by a finite set of rules of the form
\begin{equation*}
I(\bar x) :\!\!-~ I_1(\bar x_1) \land \cdots \land I_m(\bar x_m)
\end{equation*}
where each $I_i$ is a relational symbol, either a symbol from $\sigma$, or an
internal symbol. $I(\bar x)$ is called the head of the rule and $I$ must
be an internal symbol. The variables $\bar x$ are among $\bar x_1 \dots \bar x_m$ and
the variables of $\bar x_i$ not occurring in $\bar x$
should be understood as existentially quantified. One of the internal symbols,
referred to as the goal,
is binary and is designated as being the output of the query. The evaluation of
a query computes the internal relations incrementally
starting from the empty ones by applying greedily the rules (see <cit.>).
It is easy to see that any Regular or Context-Free Path Query, and
therefore any Conjunctive Regular Path Query, can be expressed in . Hence
is the most expressive of the query languages presented above. It is
also well known that each query can be evaluated in polynomial time,
data complexity, using the procedure briefly sketched above.
We will consider restrictions of limiting the maximal arity of
the internal symbols and the number of variables in each rule. This is
classical in the context of Constraint Satisfaction Problems
() <cit.> that we will use in
Section <ref>. In the context of , programs are boolean (i.e. the goal has arity 0) and
denotes the fragment allowing at most $k$ variables in each
rule and internal symbols of arity at most $l$.
Here we are dealing with binary
programs. In order to stay close to the
notations and results coming from , we generalize this definition and let
denote the programs having at most $k+r$ variables in each
rule and internal symbols of arity at most $l+r$, where $r$ is the arity
of the goal, in our case $r=2$.
If $\sigma$ and $\tau$ are (binary) schemas, a view from $\sigma$ to
$\tau$ is a set consisting of one binary query over $\sigma$ for each symbol in
$\tau$. If $\V$ consists of the queries $\{V_1,\ldots,V_n\}$, with a little
abuse of notation, we let each $V_i$ also denote the corresponding symbol in
$\tau$. For a graph database over $\sigma$, we denote by $\V(\D)$ the graph
database over $\tau$ where each binary symbol $V_i$ is instantiated as
$V_i(\D)$. We say that a view consisting of the queries $\{V_1,\ldots,V_n\}$
is an view if each $V_i$ is an . We define similarly Context-Free Path
Query views and Conjunctive Regular Path Query views.
In what follows whenever we refer to a view and a query , unless
otherwise specified, we always assume that is over the schema $\sigma$ and is a view from $\sigma$ to $\tau$. A view instance $\S$ is a
$\tau$-structure such that $\S=\V(\D)$ for some database .
Determinacy and rewriting
The notion of determinacy specifies when a query can be answered completely
from the available view. The following definitions are taken from <cit.>.
We say that a view determines a query if :
\begin{equation*}
\forall \D,\D', \quad \V(\D) = \V(\D') \ \Rightarrow \ \Q(\D) = \Q(\D')
\end{equation*}
In other words, $\Q(\D)$ only depends on the view instance $\V(\D)$ and not on
the particular database yielding the view. Observe that determinacy says
that there exists a function defined on view images such that
$\Q(\D)=\f(\V(\D))$ for each database . We call the function induced
by $\Q$ using $\V$.
A rewriting of using is a query over the schema $\tau$ such that
$\R(\V(\D))=\Q(\D)$ for all $\D$.
Notice that there can be possibly many rewritings, while the
function induced by $\Q$ using $\V$ is unique.
In fact the domain of is defined to be
the set of view images, that is, all the $\tau$-structures $\S$ such that there
exists a database $\D$ with $\V(\D) = \S$. Thus, is fully defined by the
identity $\Q(\D)=\f(\V(\D))$, and is therefore unique. On the contrary,
rewritings are defined as queries over $\tau$, which means that they are mappings defined
over all $\tau$-structures $\S$, even those which are not of the form
$\S = \V(\D)$. In particular, this means that the condition
$\Q(\D) = \R(\V(\D))$ is not sufficient to fully define $\R$, as it
can take arbitrary values on $\tau$-structures that are not of the form
$\V(\D)$. Of course any rewriting coincides with the function when restricted to view images.
Consider again the view defined by the two s $V_1=\sigma^3$ and
$V_2=\sigma^4$ testing for the existence of a path of length 3 and 4, respectively.
Let $\Q=\sigma^5$ be the testing for the existence of a path of
length 5.
It turns out that determines <cit.>. This is not immediate to
see but, as mentioned in Section <ref>, one can verify that a rewriting of
using can be expressed in first-order by the following query:
\begin{equation*}
\exists u ~ ( V_2(x,u) \land \forall v ~(V_1(v,u)
\Rightarrow V_2(v,y)))
\end{equation*}
As shown in Figure <ref>, the function induced by using
is not monotone. This implies that no monotone query can be a
rewriting, in particular there exists no nor rewriting.
Consider now the $\Q'=\sigma^2$. One can verify that does not
determine $\Q'$. Indeed the database consisting of a single node with no
edge, and the database consisting of a single path of length 2, have
the same empty view but disagree on $\Q'$.
[shorten >=1pt,->]
vertex=[circle,fill=black,minimum size=3pt,inner sep=0pt]
/in 0/0, 1/1, 2/2, 3/3, 4/4, 5/5 [vertex] () at (,5) ;
/→in 0/1, 1/2, 2/3, 3/4, 4/5 () – (→);
/in 0/0.25, 1/1.25, 2/2.25, 3/3.25, 4/4.25, 5/5.25 (lab) at (, 4.75) $x_\name$;
(D) at (-0.5,5) $\D :$;
/in 0/0, 1/1, 2/2, 3/3, 4/4 [vertex] () at (,0) ;
/→in 0/1, 1/2, 2/3, 3/4 () – (→);
/in 01/3, 11/2, 21/1 [vertex] () at (4,) ;
/→in 01/11, 11/21, 21/4 () – (→);
/in 02/3, 12/2, 13/1 [vertex] () at (,2) ;
/→in 11/02, 02/12, 12/13 () – (→);
//in 0/0.25/-0.25, 1/4.25/2.75, 2/4.25/1.75, 3/3.25/-0.25, 4/4.25/-0.25, 5/1.25/1.75
(lab) at (, ) $x_\name$;
(D') at (-0.5,2.5) $\D' :$;
Illustration for Example <ref>: $\D$ and
$\D'$ are such that $\V(\D) \subseteq \V(\D')$, but $(x_0,x_5) \in
\Q(\D)$, whereas $(x_0,x_5) \notin \Q(\D')$. Hence the function
induced by using is not monotone.
It is important at this point to understand the difference between determinacy
and rewriting. If determines then there exists a rewriting of using .
However there are possibly many rewritings of using . Each of them agrees on
the function induced by using when restricted to view images, but
can take arbitrary values on structures that are not in the image of the view.
Consider for instance the view and the query of
Example <ref>. The query:
\begin{equation*}
\exists u,u' ~ V_2(x,u) \land V_1(x,u')\land \forall v ~(V_1(v,u)
\Rightarrow V_2(v,y))
\end{equation*}
is also a rewriting of using . It is equivalent to the rewriting of
Example <ref> on $\tau$-instances $\S$ such that $\S=\V(\D)$
for some . Indeed whenever $V_2(x,u)$ holds in $\V(\D)$, the database $\D$ contains a path
of length 4 from $x$ to $u$, hence if $u'$ is the node at distance 3
from $x$ in this path, $V_1(x,u')$ also holds in $\V(\D)$. However the two rewritings may
differ on instances which are not in the view image, such as an instance consisting of a single
$V_2$-labeled edge.
The determinacy problem for a query language $\cL$ is the problem of
deciding, given an input view defined in $\cL$ and a query of $\cL$, whether determines .
Determinacy does not say whether there exists a rewriting definable in a
particular query language, or computable with a particular data complexity. This clearly
depends on the language used for specifying the views and queries.
The rewriting problem for a query language $\cL$ is the problem of
finding a rewriting for a query of $\cL$ using a view defined in $\cL$
whenever determines .
These two problems have been thoroughly investigated in the case that $\cL$ is
<cit.>. However the determinacy problem for remains wide open and it is not clear what would be a good (low data
complexity) rewriting language for . Note that a similar situation arises
in the case that $\cL$ is <cit.>.
§ DETERMINACY PROBLEM
We have already mentioned above that the determinacy problem for is open.
For Context-Free Path Queries and for Conjunctive Regular Path Queries,
determinacy is undecidable. Actually the problem is already undecidable when
the query is an . These undecidability results are formalized in the two
following propositions.
Given a Context-Free Path Query view $\V$ and a Regular Path Query $\Q$,
it is undecidable whether $\V$ determines $\Q$.
We prove this by reduction from the universality problem for context-free
languages. Let $L$ be a context-free language over some alphabet
$\sigma$. Let $\$$ be a fresh symbol that does not appear in $\sigma$. Let
$\V = \set{V}$ where $V$ is defined by $L(V) = \$\cdot L \cdot \$$. Let $\Q$
be defined by $L(\Q) = \$\cdot \sigma^* \cdot \$$. Then $\V$ determines $\Q$
if and only if $L$ is universal over $\sigma$.
* Assume that $L$ is universal. Then $\Q = V$ and it is easy to
check that $R = V$ is a rewriting of $\Q$ using $\V$.
* Conversely, assume that $L$ is not universal. Then there exists
$w \in \sigma^*$ such that $w \notin L$. Consider the database $\D$
consisting of a simple path labeled by $\$\cdot w \cdot\$$, and the
empty database $\D'$. Then
$\V(\D) = \emptyset = \V(\D')$, but $\Q(\D)$ contains the first and
last node of the path, whereas $\Q(\D')$ is empty.
Hence, $\V$ does not determine $\Q$.
Given a Conjunctive Regular Path Query view $\V$ and a Regular Path Query
$\Q$, it is undecidable whether $\V$ determines $\Q$.
We prove this by reduction from the word problem for graph databases.
\begin{array}{ll}
\textsc{Problem} : & \textsc{Word problem for graph databases} \\
\textsc{Input} : & \text{A list of pairs } (u_i,v_i)_{0 < i \leq
n}, \text{a pair } (u,v),
\text{ where } u, v, u_i, v_i, \text{ for every } i \\
& \text{are words over } \sigma, \text{ viewed as \RPQ{}s } \\
\textsc{Question} : & \text{Is the following statement true?} \\
& \text{For every graph database \D over } \sigma, \text{if } \forall i, u_i(\D) = v_i(\D), \text{ then } u(\D) = v(\D)
\end{array}
A straightforward reduction from the word problem for finite semigroups shows:
The word problem for graph databases is undecidable.
We prove this by reduction from the word problem for finite semigroups. This
problem has the same input as the word problem for graph databases but asks
whether for all semigroup $S$ and all homomorphism $h$ from $\sigma^*$ to $S$
such that $h(u_i)=h(v_i)$ for all $i$, it is the case that $h(u)=h(v)$.
We now prove that any input is accepting for the word problem for finite
semigroups if and only if it is accepting for the word problem for graph databases.
* Assume that the input is accepting for the word problem for finite
semigroups. Let be a graph database such that for all $i$, $u_i(\D) =
v_i(\D)$. From $\D$, we compute the semigroup $S_\D$ and the homomorphism $h :
\sigma^* \rightarrow S_\D$ as follows:
* The elements of $S_\D$ are the set of pairs $w(\D)$ for all $w\in
\sigma^*$. As is finite $S_\D$ is finite.
* Let $x$ and $y$ be two elements of $S_\D$. Let $u,v \in \sigma^*$ such
that $x = u(\D)$ and $y = v(\D)$. Then $x\cdot y$ is defined as $u\cdot
v(\D)$. It is easy to check that this operation is associative and well
defined (i.e. does not depend on the specific choice of $u$ and $v$).
* For all $\alpha \in \sigma$ we set $h(\alpha) = \alpha(\D)$. Hence
for all $u\in\sigma^*$ we have $h(u)=u(D)$.
By construction we therefore have for all $i$, $h(u_i) =
h(v_i)$. Hence, $h(u) = h(v)$, which implies that $u(\D) = v(\D)$.
* Assume that the input is accepting for the word problem for
graph databases. Let $S$ be a finite semigroup, and $h$ an homomorphism from
$\sigma^*$ to $S$, such that, for all $i$, $h(u_i) = h(v_i)$. From $S$ and
$h$, we define the graph database $\D_h$ as follows:
* The sets of nodes of $\D_h$ is $h(\sigma^+) \cup
\set{\varepsilon}$. This set is finite since $h(\sigma^+)$ is a subset of
* Let $x$ and $y$ be two nodes of $\D_h$. Then there is an edge $\alpha$
from $x$ to $y$ if either $x = \varepsilon$ and $y = h(\alpha)$ or $x
\neq \varepsilon$ and $x \cdot h(\alpha) = y$.
Assume that $(x,y) \in u_i(\D_h)$. Then either $x = \varepsilon$, hence $y
= h(u_i) = h(v_i)$ and $(x,y)\in v_i(\D_h)$, or $x \cdot h(u_i) = y$, which
implies that $x \cdot h(v_i) = y$ and $(x,y)\in v_i(\D_h)$. Hence,
$u_i(\D_h)=v_i(\D_h)$ for all $i$ and therefore $u(\D_h) = v(\D_h)$. Hence,
$(\varepsilon,h(u)) \in v(\D_h)$, which implies that there is a path $v$ from
$\varepsilon$ to $h(u)$ and thus that $h(u) = h(v)$.
$(u_i,v_i)_{0 < i \leq n}$ and $(u,v)$ be an input
for the word problem. Let $\sigma'$ be a copy of $\sigma$ using only fresh
symbols. For each $\alpha\in\sigma$, we use $\alpha'$ to denote the
corresponding symbol in $\sigma'$. We define the following query and view:
* $\Q$ is the defined by $L(\Q) = \set{u,v'}$ where $v'$ is a copy
of $v$ using symbols of $\sigma'$.
* For all $\alpha \in \sigma$, $V_\alpha$ is a query of the view defined
by the $L_\alpha = \set{\alpha,\alpha'}$.
* For all $i$, $V_i$ is also a query of the view defined by the $L_i = \set{u_i,v'_i}$, where
$v'_i$ is a copy of $v_i$ using symbols of $\sigma'$.
* For all $\alpha,\beta \in \sigma$, $T_{\alpha,\beta}$ is a query of the view defined
by the : $\alpha(x,y) \wedge \exists z,t ~\beta'(z,t)$.
* For all $\alpha,\beta \in \sigma$, $T'_{\alpha,\beta}$ is a query of the view defined
by the : $\alpha'(x,y) \wedge \exists z,t ~ \beta(z,t)$.
We now prove that $\V = \set{V_\alpha, V_i, T_{\alpha,\beta}, T'_{\alpha,\beta} \ | \
\alpha,\beta\in\sigma, 0< i \leq n}$ determines $\Q$ if and only if the input is
accepting for the word problem for graph databases.
* Assume that the input is accepting for the word problem for graph
databases. Let $\D$ and $\D'$ be two graph databases such that $\V(\D) =
\V(\D')$. Consider first the case where $\D$ uses symbols from both
$\sigma$ and $\sigma'$, then $T_{\alpha,\beta}$ and $T'_{\alpha,\beta}$
reveal $\D$ entirely, which implies that $\D = \D'$, and thus $\Q(\D) =
\Q(\D')$. Similarly, if both $\D$ and $\D'$ use only symbols from $\sigma$,
then $V_\alpha$ reveals $\D$ entirely ensuring that $\D = \D'$. It remains
to consider the case where $\D$ only uses symbols from $\sigma$ and $\D'$
only uses symbols from $\sigma'$. Notice that, since $V_\alpha(\D) = V_\alpha(\D')$, then $\D$ and $\D'$
are isomorphic (by renaming each $\alpha$ to $\alpha'$).
Let $(x,y) \in u_i(\D)$. Hence, $(x,y) \in V_i(\D)$, which implies that
$(x,y)\in V_i(\D')$, and finally that $(x,y)\in v'_i(\D')$. By isomorphism
$(x,y)\in v_i(\D)$. Similarly, we can show that $(x,y)\in v_i(\D)$ implies
$(x,y) \in u_i(\D)$. Hence, $u(\D) = v(\D)$. Let $(x,y) \in \Q(\D)$. Then,
$(x,y) \in u(\D)$, which implies that $(x,y) \in v'(\D')$, and thus that
$(x,y) \in \Q(\D')$. A similar reasoning also gives the converse, and we
can conclude that $\V$ determines $\Q$.
* Assume that $\V$ determines $\Q$. Let $\D$ be a graph database over
$\sigma$ that satisfies the condition for the word problem. Let $\D'$ be
the copy of $\D$ given by renaming the symbols in $\sigma$ by the
corresponding symbols in $\sigma'$. Remark now that $\V(\D) =
\V(\D')$. Indeed, $V_\alpha(\D) = V_\alpha(\D')$ is given by the fact that
$\D'$ is a copy of $\D$ over $\sigma'$. $V_i(\D) = V_i(\D')$ is given by
the fact that $\D$ satisfies the condition for the word problem. Finally,
$T_{\alpha,\beta}(\D) = T_{\alpha,\beta}(\D') = T'_{\alpha,\beta}(\D) = T'_{\alpha,\beta}(\D') = \emptyset$
comes from the fact that $\D$ (resp. $\D'$) uses only symbols from $\sigma$
(resp. $\sigma'$).
Since $\V$ determines $\Q$, this implies that $\Q(\D) = \Q(\D')$. Let
$(x,y) \in u(\D)$. Then $(x,y) \in \Q(\D)$, which implies that $(x,y) \in
\Q(\D')$. Hence, $(x,y) \in v'(\D')$, and since $\D'$ is a copy of $\D$,
this yields $(x,y) \in v(\D)$. A similar reasoning also gives the converse,
and we can conclude that the input is accepting for the word problem for
graph databases.
§ VIEWS AND REWRITING
We have seen in the previous section that knowing whether a given view determines a given query is often computationally a difficult task. In this
section we assume that determines and we investigate how can be computed
from the given view instance.
A possibility is to use the following generic algorithm :
Given a $\tau$-structure , compute a $\sigma$-structure such that
$\V(\D)=\S$ (reject if no such exists) and return $\Q(\D)$.
As we know that
determines this procedure always returns the correct answers on view images. Therefore the query over $\tau$ defined by this algorithm is a rewriting of using .
For all the query languages considered in this paper, computing $\V(\D)$ and
$\Q(\D)$ can be done in time polynomial in . Hence it remains to be
able to test whether there exists a such that $\V(\D)=\S$ and, if yes,
compute such a .
The first issue, testing whether a $\tau$-instance is in the image of the view, is already
a challenging task and will be investigated in the next section. We start with
the second problem, i.e. computing a such that $\V(\D)=\S$, if it exists.
§.§ Looking for a view preimage
We assume in this section that is a view from $\sigma$ to $\tau$ and that
we are given a $\tau$-structure that is in the image of . We are now
looking for a such that $\V(\D)=\S$, knowing that one such exists.
Our first result below shows that for views, if such a exists then
there is one whose size is polynomial in . It is essentially a pumping
Let be an view from $\sigma$ to $\tau$. Let be a
$\tau$-structure. If $\S=\V(\D)$ for some then $\S=\V(\D')$, for some $\D'$ of size quadratic in .
Let and be as in the statement of the lemma. We show that if there
exists $\D$ such that $\S=\V(\D)$ then there exists a new database $\D'$ of
size $O(\size{\S}^2)$ such that $\V(\D') = \V(\D)$. $\D'$ is obtained from
$\D$ in several steps. First $\D$ is “normalized", without altering its
view, so that nodes not occurring in appear in only one path linking two
nodes of . The normalized $\D$ turns out to consist of a constant number
of disjoint paths between each pair of nodes of $\S$ (where the constant only
depends on the size of the view automaton). Then a Ramsey argument is used
to show that these paths can be “cut" without changing the view. The
resulting database $\D'$ thus consists of a constant number of paths of
constant length between each pair of nodes of $\S$. The size of $\D'$ is
therefore $O(\size{\S}^2)$. We now formalize this argument.
Assume that there exists a database such that $\S=\V(\D)$. We prove the
lemma by constructing a new database $\D'$ such that $\V(\D') = \V(\D)$, with
$\size{\D'} = O(\size{\S}^2)$.
Let $A=\langle S_\V,\delta_\V,q^0_\V,F_\V \rangle$ be the product automaton
of all the deterministic minimal automata of all the regular expressions of
the s in . Let $N(\V)$ be the number of states of $A$, i.e $|S_\V|$.
In what follows, for $w \in \sigma^*$, $\delta_\V(\cdot,w)$ denotes the
function from $S_\V$ to $S_\V$ sending $q$ to $p$ such that there is a run of
$A$ on $w$ starting in state $q$ and arriving in state $p$.
We say that a path $\pi$ from $u$ to $v$ in a database $\D'$ is $\V$-minimal
if $u,v$ are elements of $\V(\D')$ and no other nodes of $\pi$ are in the domain of $\V(\D')$.
We first build a database $\D_1$ such that :
* $\V(\D_1)=\V(\D)$;
* each node of $\D_1$ is in a $\V$-minimal path and no two $\V$-minimal
paths in $\D_1$ intersect;
* the number of $\V$-minimal paths in $\D_1$ is bounded by
$\size{\V(\D)}^2 \cdot N(\V)^{N(\V)}$.
$\D_1$ is constructed as follows: All elements of $\V(\D)$ are elements of
$\D_1$. Moreover, for each function $f : S_\V \rightarrow S_\V$ and each pair
$(x,y)$ of elements of $\V(\D)$, if there exists a $\V$-minimal path $\pi$
from $x$ to $y$ in and such that $f = \delta_\V(\cdot,\lambda(\pi))$, then
we add to $\D_1$ a copy of $\pi$ that uses only fresh, non-repeating nodes,
except for $x$ and $y$. Figure <ref> illustrates the
main idea of this construction.
It is now easy to check that $\D_1$ has the desired properties. The second
bullet holds by construction. Clearly the number of $f: S_\V \rightarrow
S_\V$ is bounded by $N(\V)^{N(\V)}$ hence the third bullet holds. It remains
to check that $\V(\D_1) = \V(\D)$. There is an obvious canonical homomorphism
sending $\D_1$ to $\D$. Hence $\V(\D_1) \subseteq \V(\D)$. For the converse
direction, consider a path $\pi$ witnessing the fact that
$(u,v)\in\V(\D)$. Decompose $\pi$ into $\V$-minimal paths. By construction,
each of these $\V$-minimal paths can be simulated in $\D_1$. Hence $(u,v) \in
\V(\D_1)$.
From $\D_1$ we construct the desired $\D'$ by replacing each
$\V$-minimal path of $\D_1$ by another one whose length is
bounded by a constant $r$ and without affecting the view image. Altogether
$\D'$ will have a size bounded by $r \cdot \size{\V(\D)}^2 \cdot N(\V)^{N(\V)}$,
hence polynomial in $|\V(\D)|$ as desired.
Let $r$ be the Ramsey's number that guarantees the existence of a
monochromatic 3-clique in an $r$-clique using $N(\V)^{N(\V)}\cdot
2^{N(V)^{N(V)}}$ colors.
Consider a $\V$-minimal path $\pi = xa_0x_1a_1\ldots x_{m}a_{m}y$ in
$\D_1$ such that $m > r$. For $1\leq s<t\leq m$ we denote by $\pi_{s \rightarrow t}$
the subpath of $\pi$ that starts at position $s$ and ends at position $t$, that is
$\pi_{s \rightarrow t} = x_sa_sx_{s+1}a_{s+1}\ldots a_{t-1}x_t$.
To each pair of nodes $(x_i,x_j)$ in $\pi$ with $i<j$, we attribute the color
$(f_{ij},\Delta_{ij})$ where:
\begin{align*}
f_{ij} &= \delta_\V(\cdot,\lambda(\pi_{i \rightarrow j}))\\
\Delta_{ij} &= \set{f : S_\V \rightarrow S_\V \ | \ \exists \alpha, &&i<\alpha<j
\textrm{ and }\\
&&& f = \delta_\V(\cdot,\lambda(\pi_{i \rightarrow
\alpha}))}.
\end{align*}
Then, by our choice of $r$, we know that there exist $i<j<k$ such that
$f_{ij}=f_{jk}=f_{ik}$ and $\Delta_{ij}=\Delta_{jk}=\Delta_{ik}$. Let $\pi'$ be
the path constructed from $\pi$ by replacing the
subpath $\pi_{i \rightarrow k}$ by $\pi_{j \rightarrow k}$.
Let $\D_2$ be the database constructed from $\D_1$ by replacing $\pi$ by
$\pi'$. We now prove that $\V(\D_2) = \V(\D_1)$. As $\D_2$ still has all the
properties of $\D_1$ listed above, by repeating this operation until all
$\V$-minimal paths have length less than $r$ we eventually get the desired
database $\D'$.
Let $(u,v) \in \V(\D_1)$ as witnessed by a path $\mu$ in $\D_1$. Then $\mu$
neither starts nor ends in an internal node of $\pi$ as internal nodes do not appear in
$\V(\D_1)$. Hence either $\mu$ does not use $\pi$ or it uses all of it. In the
former case, $\mu$ witnesses the fact that $(u,v)\in\V(\D_2)$. In the latter,
notice that $f_{ik}=f_{jk}$ implies that $\lambda_\V(\cdot,\lambda(\pi)) =
\lambda_\V(\cdot,\lambda(\pi'))$, hence replacing $\pi$ by
$\pi'$ in $\mu$ witnesses the fact that $(u,v)\in\V(\D_2)$. Altogether we have
shown that $\V(\D_1) \subseteq \V(\D_2)$.
Suppose now that $(u,v)\in\V(\D_2)$ as witnessed by a path $\mu$ in $\D_2$. If
$\mu$ does not go through $x_j$ (i.e. $x_j$ is not an internal node of $\mu$),
it is also a path in $\D_1$ and
$(u,v)\in\V(\D_1)$. If $\mu$ goes through $x_j$ but does not end between $x_j$
and $x_k$ we can also conclude that $(u,v)\in\V(\D_1)$ using the fact that
$f_{ik}=f_{jk}$. It remains to consider the case when $\mu$ ends with $x_j a_j
\ldots a_{\beta-1} x_\beta$ for some $\beta$ with $j < \beta < k$ (in
particular $v=x_\beta$). As $\Delta_{ij}=\Delta_{jk}$ there exists $\alpha$
with $i < \alpha < j$ such that $\delta_\V(\cdot,\lambda(\pi_{i\rightarrow
\alpha})) = \delta_\V(\cdot,\lambda(\pi_{j\rightarrow \beta}))$. From this we
can construct a path $\mu'$ in $\D_1$ replacing in $\mu$ the segment $x_j a_j
\ldots a_{\beta-1} x_\beta$ by $x_i a_i \ldots a_{\alpha-1} x_\alpha$,
witnessing the fact that $(u,x_\alpha) \in \V(\D_1)$, a contradiction as
$x_\alpha$ is not an element of $\V(\D_1)$. Altogether we have proved that
$\V(\D_2) \subseteq \V(\D_1)$. Hence, $\V(\D_2) = \V(\D_1)= \V(\D)$.
[shorten >=1pt,->]
vertex=[circle,fill=black,minimum size=3pt,inner sep=0pt]
viewnode=[draw, circle, minimum size=5pt, inner sep = 0pt]
//in 1/0/2, 2/3/2, 3/0/0, 4/3/0 [viewnode] () at (,) ;
[vertex] (5) at (1.5,1) ;
(1) edge[out=15,in=165] (2) ;
(1) edge[out=255,in=105] (3) ;
(1) edge[out=-30,in=150] (5) ;
(3) edge[out=30,in=210] (5) ;
(5) edge[out=30,in=210] (2) ;
(5) edge[out=-30,in=150] (4) ;
//in 1/1.5/2.40, 2/0.1/1, 3/1/1.5, 4/2.5/1.4, 5/1/0.4, 6/2.5/0.5 () at (,) $\pi_\name$;
//in 1/-0.25/2.25, 2/-0.35/0.25, 3/3.25/2.25, 4/3.25/0.25 () at (,) $x_\name$;
() at (-1.5,2) $\D :$;
[shorten >=1pt,->]
vertex=[circle,fill=black,minimum size=3pt,inner sep=0pt]
viewnode=[draw, circle, minimum size=5pt, inner sep = 0pt]
//in 1/0/2, 2/3/2, 3/0/0, 4/3/0 [viewnode] () at (,) ;
[vertex] (5) at (1.5,0.8) ;
[vertex] (5b) at (1.5,-0.3) ;
[vertex] (5c) at (1.5,1.7) ;
[vertex] (5d) at (-1,-0.5) ;
(1) edge[out=15,in=165] (2) ;
(1) edge[out=255,in=105] (3) ;
(3) edge[out=30,in=195] (5) ;
(5) edge[out=15,in=260] (2) ;
(3) edge[out=-15,in=180] (5b) ;
(3) edge[out=-15,in=180] (5b) ;
(5b) edge[out=0,in=195] (4) ;
(1) edge[out=-15,in=180] (5c) ;
(5c) edge[out=0,in=195] (2) ;
(1) edge[out=225,in=90] (5d) ;
(5d) edge[out=0,in=-115] (4) ;
//in 1/1.5/2.40, 2/0.1/1, 3/0.8/1.6, 3/-1/1, 4/2.5/0.95, 4/2.2/1.6, 5/1/0.4, 5/0.8/-0.4, 6/2.2/-0.4, 6/0/-0.8
() at (,) $\pi_\name$;
//in 1/-0.25/2.25, 2/-0.35/0.25, 3/3.25/2.25, 4/3.25/0.25 () at (,) $x_\name$;
() at (-1.5,2) $\D_1 :$;
Illustration of the transformation from $\D$ to $\D_1$ in Lemma <ref>.
Nodes are colored white or black depending on whether they appear in $\V(\D)$ or not.
In view of Lemma <ref>, we know that if determines then there exists a rewriting with data complexity. Indeed is the
query computed by the following non-deterministic polynomial time algorithm: on
an input $\tau$-structure , guess from a database of polynomial size,
check that $\V(\D)=\S$ and then evaluate on . There also exists a rewriting
with data complexity, by considering all databases of
polynomial size such that $\V(\D)=\S$. Altogether we get:
Let and be such that determines . Then there exists a
rewriting of using with data complexity, and another one with
data complexity.
It is not known whether, for views and queries, determinacy implies the existence of
a rewriting with polynomial time data complexity. The complexity bounds of
Corollary <ref> are the current best known bounds. We will see in
the next sections that if we further assume that the function induced by using is monotone then there exists a rewriting of using definable in and therefore computable in polynomial time.
Using a more intricate pumping argument it is possible to show that for any
Conjunctive Regular Path Query view , the fact that a view instance is
in the image of can also be witnessed by a database of polynomial size. Hence
Corollary <ref> extends to Conjunctive Regular Path Queries.
However we will see that for Context-Free Path Query views there is no
recursive bound on the size of a database yielding a given view instance. This
will follow from Lemma <ref> showing that,
for Context-Free Path Query views, checking whether a view
instance is in the image of the view is undecidable.
§.§ Testing for view images
We now consider the following problem. We are given a view $\V$ from $\sigma$ to $\tau$
and a $\tau$-structure $\S$ and we are asking whether there exists a
$\sigma$-structure such that $\V(\D)=\S$.
Note that this problem is related to the previous one. In view of
Lemma <ref> we immediately get an algorithm for testing
membership in the image of an view : on input $\S$ guess a database $\D$
of size polynomial in $\S$ and check $\V(\D)=\S$. We will see that testing for
view images is -hard for Regular Path Query views and undecidable for
Context-Free Path Query views.
Moreover one can show that if testing for view images can be done in then, for and such that determines , then there exists a rewriting
of using with polynomial time data complexity. The polynomial time
algorithm works as follows. On a view instance , it first tests whether there
exists a database such that $\S=\V(\D)$. If not it rejects. If yes, consider
the schema adding two new letters $a$ and $b$ and consider the query $Q_{a,b}$
asking for a path in the language $a\cdot L(Q) \cdot b$. Define $\V'$ as $\V
\cup \set{\Q_{a,b},V_a,V_b}$ where $V_a$ and $V_b$ return all pairs of nodes
linked by $a$ and $b$ respectively. For each pair $(x,y)$ of nodes of $\S$, let
$\S'$ be expanded with the empty relation for $\Q_{a,b}$, a single pair
$(u,x)$ for $V_a$ and a single pair $(y,v)$ for $V_b$ where $u$ and $v$ are two
new nodes. We then test whether $\S'$ is a view image. A simple argument shows
that the test says yes iff $(x,y) \not\in\Q(\D)$ and the algorithm works in
time polynomial in the size of .
Unfortunately, as already mentioned, the test for view images is -hard
already for views.
There is an view from $\sigma$ to $\tau$ such that given a
$\tau$-structure it is -hard to test whether there exists a
$\sigma$-structure such that $\V(\D)=\S$.
We reduce 3-Colorability to our problem. The proof is a simple
variation of the reduction found in <cit.> to prove
that computing certain answers under the sound view assumption is -hard
in data complexity.
Let $\sigma = \{rg,gr,bg,gb,rb,br\}$ and $\tau=\{V_1, V_2\}$. By abuse of notation, we will
refer to an element of $\sigma$ as $\alpha\beta$, with $\alpha$ and
$\beta$ two symbols in $\{r,g,b\}$, and $\alpha \neq \beta$. Let be
the following view from $\sigma$ to $\tau$:
* $\V = \set{V_1,V_2}$
* $L(V_1) =
\set{rg,gr,bg,gb,rb,br}$
* $L(V_2) = \set{\alpha_1\beta_1\cdot\alpha_2\beta_2 \ | \ \beta_1 \neq
\alpha_2}$.
Let $G = (U,W)$ be a connected graph. From $G$ we define a
$\tau$-structure $\S_G$, in which the interpretation of $V_1$ is:
$$\set{(x,y) \ | \ (x,y) \in W \textrm{ or } (y,x) \in W}$$
and the interpretation of $V_2$ is the empty relation.
We show that $G$ is 3-colorable iff there exists $\D$ such that $\V(\D)=E_G$.
Intuitively, the idea is that $\sigma$ describes the colors of the edges of G,
that is the color of the two end points of each edge. For instance, if $x$ and
$y$ are linked by $rg$, then it should be understood that $x$ is red and $y$ is
green. $V_1$ checks that each pair of nodes that are connected in $G$ are
colored with (at least) two different colors, and $V_2$ checks if there is any
error, that is, if a node is required to have more than one color. Since $V_2$
is empty, any graph database such that $\V(\D) = \S$ cannot have any
such error, and would thus be 3-colorable.
More precisely, assume that $G$ is 3-colorable. Then there exists a coloring
function $c : U \rightarrow \{r,g,b\}$ such that $c(x)
\neq c(y)$ for all $(x,y) \in W$. We define as the $\sigma$-structure such that,
for each $\alpha\beta \in \sigma$, the interpretation of $\alpha\beta$ in $\D$
\begin{align*}
\set{(x,y) \ | \ & (x,y) \in W \textrm{ or } (y,x) \in W,\\
&\textrm{ and } c(x) = \alpha, c(y) = \beta }.
\end{align*}
It is then easy to check that $\V(\D) = \S_G$. Indeed, for all $x,y,z \in \D$,
if $\alpha_1\beta_1(x,y)$ and $\alpha_2\beta_2(y,z)$ hold in $\D$, then
$\beta_1 = c(y) = \alpha_2$, hence $(x,z) \notin V_2(\D)$, so $V_2(\D)$ is
Conversely, assume that there exists a graph database such that $\V(\D) =
\S_G$. Consider the coloring function $c : U \rightarrow \{r,g,b\}$ defined as:
$c(x) = \alpha$ if there exists $y$ such that $\alpha\beta(x,y)$ holds in .
Since $V_2(\D)$ is empty, it is immediate to check that $c(x)$ is uniquely
defined and that $c$ is a proper $3$-coloring of $G$.
If we go from regular languages to context-free ones, then the problem becomes
Let $\V$ be a Context-Free Path Query view from $\sigma$ to $\tau$. Let $\S$ be a
$\tau$-instance. Then it is undecidable whether there exists a
$\sigma$-structure $\D$ such that $\V(\D) = \S$.
We prove this by reduction from the universality problem for context-free
languages. Let $L$ be a context-free language over some alphabet
$\sigma$. Let $\$$ be a fresh symbol that does not appear in $\sigma$. Let
$\V = \set{V_1, V_2}$, where $V_1$ is defined by $L(V_1) = \$\cdot L \cdot
\$$ and $V_2$ is defined by $L(V_2) = \$\cdot \sigma^* \cdot \$$. Finally,
let $\S$ be the view instance that contains a single pair $(x,y)$ in $V_2$
and no pair in $V_1$. Then there exists $\D$ such that $\V(\D) = \S$ if and
only if $L$ is not universal over $\sigma$.
* Assume that there exists a database $\D$ such that $\V(\D) = \S$. Then
there exists a path $\pi$ from $x$ to $y$ such that $\lambda(\pi) \in
L(V_2)$. Hence there exists $w\in \sigma^*$ such that $\lambda(\pi) =
\$\cdot w\cdot\$$. However, $\lambda(\pi) \notin L(V_1)$. Hence $w \notin
L$ and $L$ is not universal.
* Conversely, assume that $L$ is not universal. Then there exists $w \in
\sigma^*$ such that $w\notin L$. Then it is easy to check that the database
$\D$ consisting of a simple path labeled by $\$\cdot w \cdot\$$ satisfies
$\V(\D) = \S$.
A more intricate argument shows that undecidability already holds for a fixed view definition .
There exists a fixed Context-Free Path Query view from $\sigma$ to $\tau$ such that, given a
$\tau$-structure , it is undecidable whether there exists a $\sigma$-structure such that $\V(\D) = \S$.
Let $\sigma = \set{(,;,),a,b,\$,1}$. Let $\boldsymbol{\sigma}$ be a copy of
$\sigma$ with fresh symbols. For $\alpha \in \sigma$, we denote by
$\boldsymbol{\alpha}$ the corresponding symbol in $\boldsymbol{\sigma}$. For
$w$ a word, $\tilde{w}$ denote the word corresponding to $w$ read from right
to left. $\V$ consists of views that reveal each symbol in $\sigma$, that
is, for all $\alpha \in \sigma$, $\V$ contains a view $V_\alpha$ defined by
$L(V_\alpha) = \set{\alpha}$. Additionally, $\V$ contains the queries
$V_u$, $V_v$, $V'_u$, $V'_v$, $V_g$ and $V_c$ defined by the following
L(V_u) = \left \{
\begin{array}{cc}
\boldsymbol{\$} \cdot w \cdot \boldsymbol{\$} \cdot \po i_1 \pv v_1 \pv u_1
\pf \ldots \po i_n \pv v_n \pv u_n \pf \cdot \boldsymbol{\$} \ |\\
w,u_k,v_k \in
\set{\boldsymbol a,\boldsymbol b}^*, i_k \in \boldsymbol 1^*,
u_1\cdot\ldots\cdot u_n = \tilde w
\end{array}
\right \}
L(V_v) = \left \{
\begin{array}{cc}
\boldsymbol{\$} \cdot w \cdot \boldsymbol{\$} \cdot \po i_1 \pv v_1 \pv u_1
\pf \ldots \po i_n \pv v_n \pv u_n \pf \cdot \boldsymbol{\$} \ |\\
w,u_k,v_k \in
\set{\boldsymbol a,\boldsymbol b}^*, i_k \in \boldsymbol 1^*,
v_1\cdot\ldots\cdot v_n = \tilde w
\end{array}
\right \}
L(V'_u) = \left \{
\begin{array}{cc}
\boldsymbol{\$} \cdot w \cdot \boldsymbol{\$} \cdot \po i_1 \pv v_1 \pv
u_1 \pf \ldots \po i_n \pv v_n \pv u_n \pf \cdot \boldsymbol{\$} \ |\\
w,u_k,v_k \in
\set{\boldsymbol a,\boldsymbol b}^*, i_k \in \boldsymbol 1^*,
u_1\cdot\ldots\cdot u_n \neq \tilde w
\end{array}
\right \}
L(V'_u) = \left \{
\begin{array}{cc}
\boldsymbol{\$} \cdot w \cdot \boldsymbol{\$} \cdot \po i_1 \pv v_1 \pv
u_1 \pf \ldots \po i_n \pv v_n \pv u_n \pf \cdot \boldsymbol{\$} \ |\\
w,u_k,v_k \in
\set{\boldsymbol a,\boldsymbol b}^*, i_k \in \boldsymbol 1^*,
v_1\cdot\ldots\cdot v_n \neq \tilde w
\end{array}
\right \}
L(V_g) = \left \{
\begin{array}{cc}
\$ \cdot (u_1;v_1;i_1) \cdot \ldots \cdot
(u_n;v_n;i_n) \cdot \boldsymbol{\$} \cdot \boldsymbol{\sigma}^* \cdot \boldsymbol{\$} \cdot
\boldsymbol{\sigma}^* \cdot \po i' \pv v' \pv u' \pf \ |\\
u_k,v_k \in
\set{a,b}^*, i_k\in 1^*, u',v' \in \set{\boldsymbol a, \boldsymbol b}^*,
i' \in \boldsymbol 1^*, i' > i_n
\end{array}
\right \}
L(V_c) = \left \{
\begin{array}{ccc}
\$ \cdot (u_1;v_1;i_1) \cdot \ldots \cdot
(u_n;v_n;i_n) \cdot \boldsymbol{\$} \cdot \boldsymbol{\sigma}^* \cdot \boldsymbol{\$} \cdot
\boldsymbol{\sigma}^* \cdot \po i' \pv v' \pv u' \pf \ |\\
u_k,v_k \in
\set{a,b}^*, i_k\in 1^*, u',v' \in \set{\boldsymbol a, \boldsymbol b}^*,
i' \in \boldsymbol 1^*,\\
\exists k, i_k = \varphi(i'), u_k \neq
\varphi(\tilde u') \text{ or } v_k \neq \varphi(\tilde v')
\end{array}
\right \}
where $\varphi$ is the function that maps each symbol in $\boldsymbol \sigma$
to the corresponding symbol in $\sigma$.
One can check that all these languages are actually context-free languages.
We now prove that, given a view instance $\S$ for this specific view $\V$, it
is undecidable whether there exists a database $\D$ such that $\V(\D) =
\S$. We prove this by reduction from the Post Correspondence Problem
(PCP). Let $(u_i,v_i,i)_{0 < i \leq n}$ be an instance of PCP over
$\set{a,b}$, where the third argument explicitly gives the index of each
pair. We build the following view instance $\S$:
[shorten >=1pt, ->]
vertex=[circle,fill=black,minimum size=3pt,inner sep=0pt]
[vertex,label=above:$x_0$] (1) at (0,0) ;
[vertex,label=above:$x_1$] (2) at (1,0) ;
(1) edge node[below] $V_\$$ (2);
[vertex] (3) at (2,0) ;
(2) edge node[below] $V_($ (3);
[vertex] (4) at (3,0) ;
(3) edge[dashed] node[below] $``V_{u_1}"$ (4);
[vertex] (5) at (4,0) ;
(4) edge node[below] $V_;$ (5);
[vertex] (6) at (5,0) ;
(5) edge[dashed] node[below] $``V_{v_1}"$ (6);
[vertex] (7) at (6,0) ;
(6) edge node[below] $V_;$ (7);
[vertex] (8) at (7,0) ;
(7) edge node[below] $V_1$ (8);
[vertex, label=above:$x_2$] (9) at (8,0) ;
(8) edge node[below] $V_)$ (9);
[vertex, label=above:$x_n$] (11) at (1,-2) ;
(9) edge[dotted, thick] (11);
[vertex] (12) at (2,-2) ;
(11) edge node[below] $V_($ (12);
[vertex] (13) at (3,-2) ;
(12) edge[dashed] node[below] $``V_{u_n}"$ (13);
[vertex] (14) at (4,-2) ;
(13) edge node[below] $V_;$ (14);
[vertex] (15) at (5,-2) ;
(14) edge[dashed] node[below] $``V_{v_n}"$ (15);
[vertex] (16) at (6,-2) ;
(15) edge node[below] $V_;$ (16);
[vertex] (17) at (7,-2) ;
(16) edge[dashed] node[below] $V_1^n$ (17);
[vertex, label=above:$x_{n+1}$] (18) at (8,-2) ;
(17) edge node[below] $V_)$ (18);
[vertex, label=below right:$x_{end}$] (22) at (9,-4) ;
(18) edge node[right] $V_u$, $V_v$ (22) ;
(10,-1.95) – (10,0) node [black,midway,xshift=1.7cm] PCP encoding;
(10,-4) – (10,-2.05) node [black,midway,xshift=1.95 cm] solution encoding;
We now show that there exists $\D$ such that $\V(\D) = \S$ if and only
if the PCP instance is satisfiable. Intuitively, $\S$ consists of two
parts. The first part, from $x_0$ to $x_{n+1}$ is the encoding of the
PCP instance. It uses letters from $\sigma$ that are all revealed by
the view. All tuples are simply enumerated in the natural order, where
the $i$th tuple is encoded between $x_i$ and $x_{i+1}$. The dashed
arrows $V_{u_i}$ and $V_{v_i}$ represent the correct succession of
$V_a$ and $V_b$ that naturally encode $u_i$ and $v_i$, whereas the
$V_1^i$ part is the unary encoding of $i$, the index of the tuple. The
second part of the instance states the existence of a solution for this
instance, and uses “hidden" letters from $\boldsymbol \sigma$. $V_u$
and $V_v$ states that there exists a solution, and the fact that all
other views are empty checks that this solution is correct.
* Assume that there exists a database $\D$ such that $\V(\D) =
\S$. Then there exists a path $\pi$ from $x_{n+1}$ to $x_{end}$ such
that $\lambda(\pi) \in L(V_u)$. Hence, this path is of the form $\$
\cdot w \cdot \$ \cdot \po i_{1} \pv v'_{1} \pv u'_{1} \pf \ldots \po
i_{m} \pv v'_{m} \pv u'_{m} \pf \cdot \$$, where $w$ is a word in
$\boldsymbol \sigma^*$ and $u'_{1} \ldots u'_{m} = \tilde w$. Remark
that is also holds that $v'_{1} \ldots v'_{m} = \tilde w$, otherwise
$\lambda(\pi) \in V'_v$, which would imply that $(x_{n+1},x_{end})\in
V'_v(\D)$, and lead to a contradiction.
Hence, $u'_1\ldots u'_m = v'_1 \ldots v'_m$. It remains to show that
each $\po i_{i} \pv v'_{i} \pv u'_{i} \pf$ is an encoding of the
mirror of some tuple in the PCP instance, which would imply a
solution as $\tilde u'_m\ldots \tilde u'_1 = \tilde v'_m \ldots
\tilde v'_1$. In other words, $u_{\size{i_m}}\ldots u_{\size{i_1}} =
v_{\size{i_m}}\ldots v_{\size{i_1}}$.
Assume that one of the $\po i_{i} \pv v'_{i} \pv u'_{i} \pf$ is not
the mirror of some tuple encoded in the first half of the
instance. Remark that $\size{i_i} \leq n$. Otherwise, there exists a
path whose label is in $L(V_g)$, which leads to a
contradiction. Hence, either $u'_i \neq \tilde u_{\size{i_i}}$ or
$v'_i \neq \tilde v_{\size{i_i}}$. Both cases lead to the existence
of a path whose label is
in $L(V_c)$, and thus to a contradiction.
* Assume that there exists a solution $i_1\ldots i_m$ to the PCP
instance. Then the database $\D$ that consists of the following
simple path is such that $\V(\D) = \S$:
$$\$(u_1;v_1;1)\ldots(u_n;v_n;1^n)\boldsymbol{\$}\bold{u_{i_1}\ldots u_{i_m}}\boldsymbol{\$}
\po \bold 1^{i_m} \pv \bold{\tilde v_{i_m}} \pv \bold{\tilde u_{i_m}} \pf
\ldots \po \bold 1^{i_1} \pv \bold{\tilde v_{i_1}} \pv \bold{\tilde u_{i_1}}
\pf \boldsymbol{\$}$$
where $\bold{u_i}$ and $\bold{v_i}$ simply represent the corresponding
$u_i$ and $v_i$ written using $\boldsymbol a$ and $\boldsymbol b$ instead of
$a$ and $b$.
Note that in the proof of Lemma <ref> the view instance is a
coding of a PCP instance and the corresponding database a coding of a
solution. As there is no recursive bound on the size of a solution of a PCP
instance, for Context-Free Path Query views, there are no recursive bound on
the size of a database that yields a given view instance. This is to be
compared with the polynomial bound for views shown in
Lemma <ref>.
§ MONOTONE DETERMINACY AND REWRITING
As Example <ref> shows, there is an view and an query such that determines but the function induced by using is not
monotone, therefore having no rewriting. It is natural to wonder
whether the monotonicity of the function induced by the query is the only limit
for the existence of an rewriting. Recall from the introduction that if and are defined using s and determines , then the function induced
by using is monotone iff there exists a rewriting. In the case of
views and queries the analog does not hold. We will see that, even if we
assume monotonicity, an rewriting need not exist; however in the next
section we will show that a rewriting definable in always exists. We
start by formalizing the notion of monotone determinacy.
[Monotone determinacy]
We say that a view determines a query in a monotone way if determines and the function induced by using is monotone.
It is rather immediate to see that monotone determinacy is equivalent to the
following property for and :
\begin{equation*}
\forall \D,\D', \quad \V(\D) \subseteq \V(\D') \ \Rightarrow \ \Q(\D)
\subseteq \Q(\D')
\end{equation*}
This turns out to coincide with the notion of losslessness under the
sound view assumption defined in <cit.>, that was
shown to be decidable, actually -complete, for s.
The monotone determinacy problem for s is -complete.
Note that in the proof of Proposition <ref>, the
rewriting is always monotone when the view determines the query. Therefore, for
Context-Free Path Query views and queries, monotone determinacy is
Recall from Example <ref> that there exist a view and a query
such that the view determines the query but not in a monotone way.
We now assume given an view and an query such that determines in a monotone way. It was observed in <cit.>
that even in this case there might be no rewriting definable in .
In fact, given and defined using , it is decidable whether an rewriting exists and the problem is
-complete <cit.>. As testing monotone determinacy
is -complete, a simple complexity argument shows that an rewriting is not guaranteed to exist under monotone determinacy.
Here is a concrete example witnessing this fact.[A similar example was
claimed in <cit.> but it seems that in this
example and are such that does not determine .]
Let $\sigma=\set{a,b,c}$. Let and be defined as follows:
* $\Q = ab^*a \ | \ ac^*a$
* $\V=\set{V_1,V_2,V_3}$ with
* $V_1 = ab^*$
* $V_2 = ac^*$
* $V_3 = b^*a \ | \ c^*a$
One can verify that determines as witnessed by the following
rewriting $\R(x,y)$:
\begin{equation*}
\exists z ~ V_1(x,z) \wedge V_2(x,z) \wedge V_3(z,y)
\end{equation*}
That is a rewriting is illustrated in Figure <ref>. Consider
the database of Figure <ref> which is a typical database such
that $(x,y) \in \Q(\D)$. The choice of $z$ witnessing $(x,y) \in
R(\V(\D))$ is then immediate. Conversely, consider the database $\D'$ of
Figure <ref>. It is a typical database such that $(x,y) \in
R(\V(\D))$. The top path shows that $(x,y)~\in~\Q(\D)$.
[shorten >=1pt,->]
vertex=[circle,fill=black,minimum size=3pt,inner sep=0pt]
//in 1/0/0, 2/1/0, 3/2/0, 4/3/0, 5/4/0, 6/5/0, 7/6/0 [vertex] () at (,) ;
/→in 1/2, 2/3, 3/4, 4/5, 5/6, 6/7 () – (→);
() at (0.5,0.2) $a$ ; () at (5.5,0.2) $a$ ;
in 1.5, 2.5, 3.5, 4.5 () at (,0.25) $b$ ;
/in x/0, z/1, y/6 () at (,-0.25) $\name$;
(D) at (-0.5,0) $\D :$;
[shorten >=1pt,->]
vertex=[circle,fill=black,minimum size=3pt,inner sep=0pt]
//in 1/0/0, 2a/1/1, 2b/1/-1, 3a/2/1, 3b/3/-1, 4a/3/1, 5/4/0, 6/5/0, 7/6/0
[vertex] () at (,) ;
/→in 1/2b, 2b/3b, 3b/5, 5/6, 6/7 () – (→);
/→in 1/2a, 2a/3a, 3a/4a, 4a/5 () – (→);
() at (0.25,0.65) $a$ ; () at (0.25,-0.65) $a$ ;
in 1.5, 2.5 () at (,1.25) $b$ ;
() at (2,-0.8) $c$ ;
() at (3.7,0.7) $b$ ;
() at (3.65,-0.65) $c$ ;
() at (4.5,0.25) $b$ ; () at (5.5,0.20) $a$ ;
/in x/-0.2, z/4.1, y/6.1 () at (,-0.25) $\name$;
() at (-0.5,1) $\D' :$ ;
Databases $\D$ and $\D'$ for Example <ref>.
Since is monotone, determines in a monotone way. It can also be shown
(for instance using the decision procedure provided in <cit.>)
that no rewriting exists.
In the previous example we have exhibited a Conjunctive Regular Path Query
rewriting. However the following example suggests that Conjunctive Regular Path Query is not expressive
enough as a rewriting language.
Let $\sigma=\set{a}$. Let and be defined as follows:
* $\Q = a(a^6)^* \ | \ aa(a^6)^*$
(words of length 1 or 2 modulo 6)
* $\V=\set{V_1,V_2}$ with
* $V_1 = a \ | \ aa$(words of length 1 or 2)
* $V_2 = aa \ | \ aaa$(words of length 2 or 3)
It can be verified that determines in a monotone way as witnessed by the following
rewriting $\R(x,y)$:
\begin{equation*}
\exists z ~ V_1(x,z) \wedge T^*(z,y)
\end{equation*}
where $T(x,y)$ is defined as:
\begin{align*}
\exists z_1,z_2 ~ & V_1(x,z_1) \wedge V_2(x,z_1) \wedge V_1(z_1,z_2) \wedge\\
& V_2(z_1,z_2) \wedge V_1(z_2,y) \wedge V_2(z_2,y)
\end{align*}
The query $T$ is such that if $T(x,y)$ holds in $\V(\D)$, then in $\D$ the nodes $x$ and $y$ are either
linked by a path of length $6$ or by both a path of length $5$ and a path of
length $7$. This fact can be checked by a simple case analysis. One such case is
illustrated in Figure <ref>. In this case there is no path of
length $6$ in $\D$, but the top path has length $5$, and the path starting with the
bottom segment and then the last two top segments has length $7$.
From this, a simple induction shows that if $T^*(x,y)$ holds in $\V(\D)$, then in $\D$
the nodes $x$ and $y$
are either linked by a path of length $0$ modulo $6$, or by both a path of length
$1$ modulo $6$ and a path of length $5$ modulo $6$.
Assume now that $\R(x,y)$ holds in $\V(\D)$. Then in $\D$ there exists a $z$ such that $x$ is at
distance $1$ or $2$ from $z$, and such that $T^*(z,y)$ holds in $\V(\D)$. Assume first that
$z$ and $y$ are at distance $0$ modulo $6$ in $\D$. In this case, regardless of the
distance between $x$ and $z$, $\Q(x,y)$ holds in $\D$. Otherwise, in $\D$ there exist both a
path of length $1$ modulo $6$ and a path of length $5$ modulo $6$ from $z$ to
$y$. Therefore, if $x$ and $z$ are at distance $1$, the first path from $z$ to
$y$ yields a path of length $2$ modulo $6$ and, if $x$ and $z$ are at
distance $2$, the second path from $z$ to $y$ yields a path of length $1$
modulo $6$, see Figure <ref>.
Conversely, it is easy to check that $\R(x,y)$ holds in $\V(\D)$ whenever $\Q(x,y)$ holds in $\D$. This
follows from the fact that $T(x,y)$ holds in $\V(\D)$ for all $x$ and $y$ that are at distance $6$ in $\D$.
Notice that is monotone. A tedious combinatorial argument can show that cannot be expressed as a Conjunctive Regular Path Query.
[shorten >=1pt,->]
vertex=[circle,fill=black,minimum size=3pt,inner sep=0pt]
/in 1/0, 2/2, 3/4, 4/6 [vertex] () at (,0) ;
/→in 1/2, 2/3, 3/4 () edge[out=60,in=120] (→);
/→in 1/2, 2/3, 3/4 () edge[out=-60,in=-120] (→);
/in x/-0.25, y/6.25 () at (,-0.25) $\name$;
/in 1/2, 2/4 () at (,-0.5) $z_\name$;
() at (1,0.8) $V_1:a$;
() at (3,0.8) $V_1:aa$;
() at (5,0.8) $V_1:aa$;
() at (1,-0.8) $V_2:aaa$;
() at (3,-0.8) $V_2:aa$;
() at (5,-0.8) $V_2:aa$;
Example <ref>: An arbitrary database $\D$ whose view satisfies $T(x,y)$. Each arrow of the
form $V_i: w$ from a node $u$ to a node $v$ should be understood as a path
from $u$ to $v$ whose label is $w$ which witnesses $(u, v) \in V_i(D)$.
[shorten >=1pt,->]
vertex=[circle,fill=black,minimum size=3pt,inner sep=0pt]
/in 1/0, 2/2, 3/6 [vertex] () at (,0) ;
(1) – (2);
(2) – (3);
/in x/-0.25, y/6.25 () at (,-0.25) $\name$;
() at (2,-0.25) $z$;
() at (1,0.25) $V_1:a \textrm{ or } a^2$;
() at (4,0.25) $T^*:(a^6)^*$;
[shorten >=1pt,->]
vertex=[circle,fill=black,minimum size=3pt,inner sep=0pt]
/in 1/0, 2/2, 3/6 [vertex] () at (,0) ;
(1) – (2);
(2) edge[out=60,in=120] (3);
(2) edge[out=-60,in=-120] (3);
/in x/-0.25, y/6.25 () at (,-0.25) $\name$;
() at (2,-0.25) $z$;
() at (1,0.25) $V_1:a$;
() at (4,1.25) $T^*:a(a^6)^*$;
[color = gray] () at (4,-1.30) $T^*:a^5(a^6)^*$;
[shorten >=1pt,->]
vertex=[circle,fill=black,minimum size=3pt,inner sep=0pt]
/in 1/0, 2/2, 3/6 [vertex] () at (,0) ;
(1) – (2);
(2) edge[out=60,in=120] (3);
(2) edge[out=-60,in=-120] (3);
/in x/-0.25, y/6.25 () at (,-0.25) $\name$;
() at (2,-0.25) $z$;
() at (1,0.25) $V_1:aa$;
[color = gray] () at (4,1.25) $T^*:a(a^6)^*$;
() at (4,-1.30) $T^*:a^5(a^6)^*$;
The three cases of Example <ref>. The parts that
are not used for are shaded out.
The careful reader has probably noticed that in both examples above a
rewriting can be expressed in . As we will see later, it easily follows
from the results of <cit.> that this is always true in
general: if and are defined by s and determines in a
monotone way, then there exists a rewriting of using definable in (actually universal ).
§ DATALOG REWRITING
In this section we prove our main result, namely:
If $\V$ and $\Q$ are and $\V$ determines $\Q$ in a monotone way then
there exists a rewriting of $\Q$ using $\V$.
Theorem <ref> also implies that the monotone determinacy
problem for coincides with the problem of the existence of a rewriting. The latter is therefore decidable by Corollary <ref>:
Let $\V$ and $\Q$ be . It is decidable,
-complete, whether there exists a rewriting of $\Q$ using
Our proof being constructive, the rewriting can be computed from and .
Main idea and sketch of the proof
The starting point is the relationship between rewriting and certain answers
under monotone determinacy. One can easily show that if the view determines the
query in a monotone way then the certain answers query is a rewriting. However
certain answers for views and queries are -hard to
compute <cit.>. Here we show that there exists
another rewriting (which of course coincides with certain answers on view
images) that is expressible in . This other rewriting is suggested by
the relationship between certain answers and Constraint Satisfaction
Problems (). Following <cit.> we adopt here the
homomorphism point of view for s: Each is defined by a
structure, called the template, and its solutions are all the structures
mapping homomorphically into the template.
Indeed <cit.> showed that, for $\V$ and $\Q$, certain
answers can be expressed as a whose template depends only on $\V$ and
$\Q$. It is known from <cit.> that for every $l$ and $k$
with $l \leq k$, and every template, there exists a query
approximating the defined by this template. Even if its “approximation” does not compute precisely the associated to and ,
if it is exact on view images, then it is a rewriting.
We show that if the view determines the
query in a monotone way then there is an $l$ and a $k$, depending only on and , such that the approximation is exact on
view images. This proves the existence of a rewriting.
This is done in two steps. We first show that there exists a approximation which is exact on view images of simple path databases. Then
we show how to lift this result on all view images. The first step is proved by
a careful analysis of the properties of view images of simple path
databases. The second steps exploits monotonicity.
We now provide more details.
§.§ Monotone rewritings, certain answers and
Let be a view from $\sigma$ to $\tau$ and be a query on
$\sigma$-structures. The certain answers of $\Q$ on a $\tau$-structure $\S$
w.r.t. are defined as
\begin{equation*}
\cert_{\Q, \V}(\S)=\bigcap_{\D \ | \
\S \subseteq \V(\D)} \Q(\D)
\end{equation*}
This notion is usually referred to as certain answers under the sound
view assumption or open world assumption in the
literature <cit.>. It is straightforward to check
that if determines in a monotone way, the query $\cert_{\Q, \V}$ is a
rewriting of using , i.e. $\cert_{\Q, \V}(\V(\D))=\Q(\D)$ for each
$\sigma$-structure .
Therefore any language known to express certain answers is a suitable rewriting language under monotone determinacy.
The following proposition, proved in <cit.>, shows that, for
views and queries, certain answers (and therefore rewritings) can be
expressed as (the negation of) a .
Let be an view from $\sigma$ to $\tau$ and be an query over
$\sigma$. There exists a $\tau$-structure $\tempQV$ having a
set of distinguished source nodes and a set of distinguished
target nodes such that, if $\V$ determines $\Q$ in a monotone way, the
following are equivalent, for each $\sigma$-structure and each
pair of nodes $u, v$ of $\D$:
* $(u,v) \in \Q(\D)$
$(u, v) \in \cert_{\Q,\V}(\V(\D))$
$\V(\D)$ has no homomorphism to $\tempQV$ sending $u$ to a source node and $v$ to a target node.[More precisely <cit.> further proved that <ref>. and <ref>. are
equivalent not only for $\V(\D)$ but for all $\tau$-structures, and even without the assumption that determines in a monotone way.]
In the sequel, by $\neg\CSP(\tempQV)$ (resp. $\CSP(\tempQV)$) we refer to the
set of all triplets $(\S,u,v)$ such that is a $\tau$-structure, $u,v$ are
nodes of and, there is no homomorphism (resp. there is a homomorphism) from
to $\tempQV$ sending $u$ to a source node and $v$ to a target
node[are usually defined as boolean problems, i.e. without the
nodes $u,v$. As queries are binary, these parameters are necessary for
our presentation.
The problem $\CSP(\tempQV)$, as defined here, can be viewed as a classical $\CSP$
problem by extending the signature with two unary predicates, interpreted as the source and
the target nodes, as done in <cit.>.].
In view of Proposition <ref>, if
determines in a monotone way, $(\V(\D),u,v) \in \neg\CSP(\tempQV)$ iff
$(u,v) \in \Q(\D)$.
Observe that $\neg\CSP(\tempQV)$ naturally defines a binary query associating with each
$\tau$-structure $\S$ the set of all pairs $(u, v)$ of nodes of $\S$ such that
$(\S,u,v) \in \neg\CSP(\tempQV)$.
By abuse of notation, when clear from the context, we will let $\neg\CSP(\tempQV)$ also denote this binary query.
The structure $\tempQV$ of Proposition <ref> can be effectively computed
from $\Q$ and $\V$. Moreover observe that $\CSP(\tempQV)$ can be expressed in existential
. This shows, as mentioned in Remark <ref>, that if $\V$ and $\Q$ are and
$\V$ determines $\Q$ in monotone way, then there always exists a rewriting of using definable in (universal) ; moreover this rewriting can be effectively computed
from $\Q$ and $\V$.
It is well known that the certain answers query is a rewriting that can be
computed in (this follows for instance from
Proposition <ref>). Assuming is not , $\cert_{\Q,
\V}$ cannot always be computed in polynomial time, not even under the
assumption that $\V$ determines $\Q$ in a monotone way. Indeed it has been
shown <cit.> that there exists $\Q$ and $\V$ defined by
s such that $\cert_{\Q, \V}$ has -hard data complexity. An easy
reduction from this problem shows that the lower bound remains valid if we
further assume that determines in a monotone way:
There exist an view and an query such that determines in a monotone way and it is -hard to decide – given a $\tau$-structure
and nodes $(u, v)$ of – whether $(u,v) \in
\cert_{\Q,\V}(\S)$.
We show in the next section that when determines in a monotone way there
is another rewriting expressible in , hence computable in polynomial
time. Before we do this we remark that the complexity of $\cert_{\Q,\V}$
can be extended to Context-Free Path Query views and queries.
Let $\V$ be a Context-Free Path Query view and $\Q$ be a . Then
$\cert_{\Q,\V}$ can be evaluated with data complexity.
Let $\V$ be a Context-Free Path Query view, and $\Q$ be a over some
schema $\sigma$. We prove that $\cert_{\Q,\V}$ can be evaluated with
data complexity by reducing it to the case of regular path
views. Let $A = \langle S,\delta,q_0,F \rangle$ be a deterministic minimal
automaton for $L(\Q)$. In what follows, $\delta(.,w)$ denotes the function
from $S$ to $S$ associating to a state $p$ the state reached by $A$ when
reading $w$ starting from $p$. For all $V \in \V$, we define the language $L_V$
$$L_V = \set{w \in \sigma^* \
| \ \exists w' \in L(V) ~~\delta(\cdot,w) = \delta(\cdot,w')}.$$
We claim that $L_V$ is a regular language. To see this recall that for
each function $f$ from $S$ to $S$ the language $L_f$ defined as
$L_f=\set{w \in \sigma^* \ | \ \delta(\cdot,w)=f}$ is regular and notice
that $L_V$ is a union of such languages. We remark here for later that $L_V$ is
constructible as soon that it is decidable whether $L(V) \cap L_f$ is non
empty. This is in particular the case when $L(V)$ is context-free.
We now define a new view $\tilde \V$ defined as the view:
$$\tilde \V = \set{\tilde V \ | \ V\in \V \text{ and } L(\tilde V)=L_V}$$
Let $\S$ be a view instance
for $\V$. We define $\tilde \S$ as a copy of $\S$ where each $V$ relation is
replaced by $\tilde V$. Hence, $\tilde \S$ is a view instance for $\tilde
\V$. We now show that:
$$\cert_{\Q,\V}(\S) = \cert_{\Q,\tilde \V}(\tilde \S)$$
and thus $\cert_{\Q,\V}(\S)$ can be evaluated in in the size of
$\tilde \S$, which is also the size of $\S$.
* Assume that $(u,v) \in \cert_{\Q,\tilde \V}(\tilde \S)$. Hence,
for all $\D$ such that $\tilde \V(\D) \supseteq \tilde \S$, there
exists a path $\pi$ from $u$ to $v$ such that $\lambda(\pi) \in L(\Q)$.
Let $\D$ be a database such that $\V(\D) \supseteq \S$. Remark that, for
all $V \in \V$, $L(V) \subseteq L(\tilde V)$. Hence, $\tilde \V(\D)
\supseteq \tilde \S$. Hence, there exists
a path $\pi$ in $\D$ from $u$ to $v$ such that $\lambda(\pi) \in
L(\Q)$, which means that $(u,v) \in \cert_{\Q,\V}(\S)$.
* Conversely, assume that $(u,v) \notin
\cert_{\Q,\tilde \V}(\tilde \S)$. Hence, there exists a database
$\D$ such that $\tilde \V(\D) \supseteq \tilde \S$, but no path from $u$
to $v$ in $\D$ satisfies $\Q$. From $\D$, we build a database $\D'$ as
* Start with $\D'$ as a copy of $\D$.
* For all $V \in \V$, for all $(x,y) \in \S$, if $(x,y) \in V$,
then $(x,y) \in \tilde V$ in $\tilde \S$. We pick a path $\pi$ in
$\D'$ from $x$ to $y$ of label $w'$ such that $w' \in
L(\tilde V)$. Hence, there exists $w\in L(V)$ such that
$\delta(\cdot,w') = \delta(\cdot,w)$. Then, we add in $\D'$ a
simple path from $x$ to $y$ using only fresh nodes of label
$w$. Hence $(x,y) \in V(\D')$.
Remark then that $\V(\D') \supseteq \S$. Let $\pi'$ be a path from
$u$ to $v$ in $\D'$. Then $\pi'$ is of the form $\pi' =
\pi_1\mu_1\pi_2\ldots\pi_{n-1}\mu_{n-1}\pi_n$, where each $\pi_i$ is
a path that was originally in $\D$ and each $\mu_i$ is a new path
using only fresh nodes. Then, for each $\mu_i$, there exists a path
$\rho_i$ in $\D$ with the same starting and ending nodes and such
that $\delta(\cdot,\lambda(\mu_i)) =
\delta(\cdot,\lambda(\rho_i))$. Hence, we can define a path $\pi$ of
$\D$ as $\pi =
\pi_1\rho_1\pi_2\ldots\pi_{n-1}\rho_{n-1}\pi_n$. Hence,
$\delta(\cdot,\lambda(\pi')) = \delta(\cdot,\lambda(\pi))$.
Since $(u,v) \notin \cert_{\Q,\tilde \V}(\tilde \S)$, then
$\delta(q_0,\lambda(\pi))\notin F$. Hence,
$\delta(q_0,\lambda(\pi'))\notin F$, which proves that $(u,v) \notin
\cert_{\Q,\V}(\S)$.
The proposition has the following consequence:
Let $\Q$ be a and $\V$ be a Context-Free Path Query view such that $\V$
determines $\Q$ in a monotone way. Then there exists a rewriting of $\Q$
using $\V$ that can be evaluated with data complexity.
Notice that the proof of Proposition <ref> and
therefore also Corollary <ref> do not assume that the
language defining the views are context-free and work with any
language. However, in order to effectively construct the rewriting, it is
necessary that the formalism used to define the views has a decidable emptiness
test for the intersection with a regular language.
§.§ Existence of a rewriting
We now show that for each query and each view such that determines in a monotone way, there exists a rewriting.
The existence of such a rewriting stems from links between s and
. Recall from Proposition <ref> that if determines in a monotone way, $\neg\CSP(\tempQV)$, viewed as a binary query, is a rewriting of
using . It
is known that to each problem (i.e. arbitrary template), one can associate
a canonical program, for each $l, k$, with $l \leq k$. This program
can equivalently be described in terms of a two-player game, and can be thought
of as a maximal “approximation” of the complement of a problem, in a
precise sense (the interested reader is referred
to <cit.> for more details). Our main contribution
consists in proving that, for some explicit values of $l$ and $k$ (depending on
and ), this approximation is “exact” when restricted to view
images (i.e. computes precisely $\neg\CSP(\tempQV)$), and is therefore a
rewriting over such instances.
We now present the $(l,k)$-two-player game of <cit.>,
and its correspondence with .
[$(l,k)$-two-player game]
Let $l,k$ be two integers, with $l\leq k$, let be a $\tau$-structure and
$u,v$ be two nodes of . The $(l,k)$-game on $(\S,\tempQV,u,v)$ is played by two
players as follows:
* The game begins with $A_0 = \emptyset$ and $h_0$ being the empty
function over $A_0$.
For $i \geq 0$, round $i+1$ is defined as follows:
* Player 1 selects a set $A_{i+1}$ of nodes
of , with
$|A_{i+1}| \leq k$ and $|A_i \cap A_{i+1}| \leq l$.
* Player 2 responds by giving a homomorphism
$h_{i+1} : \S[A_{i+1}] \rightarrow \tempQV$ that coincides
with $h_i$ on $A_i \cap A_{i+1}$ and such that
$h_{i+1}(u)$ is a source node and $h_{i+1}(v)$ is a
target node whenever $u$ or $v$ are in $A_{i+1}$.
Player 1 wins if at any point Player 2 has no possible move. Player 2
wins if she can play forever.
The existence of a winning strategy for Player 1 is expressible in :
Let $l,k$ be two integers, with $l \leq k$, and and be an query and an view. Then there exists a program $\datlk(x,y)$ in
such that for every graph database , $\datlk(\S)$ is the set
of pairs $(u,v)$ such that Player 1 has a winning strategy for the
$(l,k)$-two-player game on $(\S,\tempQV,u,v)$.
Moreover the program in the above lemma can be effectively constructed from $\tempQV$, and therefore from and . It will be simply denoted by $\datlk$ when and are clear from the
We are now ready to state the main technical result of our paper.
Let and be an view and an query such that determines in a
monotone way. There exists $l$ such that $\datll$ is a rewriting of using .
Theorem <ref> is an immediate consequence of this proposition. The
rest of this section is devoted to proving Proposition <ref>.
This is done in two steps. We first prove that there exists $l$ such that
$\datll$ is a rewriting of using , when restricted to view images of
simple path graph databases. We then show that this suffices for $\datll$ to be
a rewriting of using .
Observe that if there is a homomorphism from a $\tau$-structure $\S$ to
$\tempQV$ sending $u$ to a source node and $v$ to a target node, then Player 2
has a winning strategy for the $(l,k)$-two-player game on $(\S,\tempQV,u,v)$. This
strategy consists in
always playing the restriction of the homomorphism on the set selected by
Player 1. In this sense the program $\datlk$ is a under-approximation of the $\neg\CSP(\tempQV)$ problem: if $(u,v)\in\datlk(\S)$ then
$(\S,u,v) \in \neg\CSP(\tempQV)$. If moreover $\S=\V(\D)$ for some
$\sigma$-structure then, by Proposition <ref>,
$(u,v)\in\datlk(\V(\D))$ implies $(u,v)\in\Q(\D)$. We will refer to this
property by saying that $\datlk$ is always sound.
The converse inclusion does not necessarily hold. If $(u, v) \notin
\datlk(\S)$ then Player 2 has a winning strategy, but this only means that
she can always exhibit partial homomorphisms from to $\tempQV$ (sometimes called
local consistency checking); this is in general not sufficient to
guarantee the existence of a suitable global homomorphism.
However here we are not interested in arbitrary $\tau$-structures, but only
structures of the form $\V(\D)$ for some simple path graph database . We now
show that, thanks to the particular properties of these structures, local
consistency checking is sufficient to obtain a global homomorphism, for some
suitable $l$ and $k=l+1$. In other words, the program $\datll$ computes
precisely $\neg\CSP(\tempQV)$ on views of simple path graph databases.
The case of simple path graph databases
Let and be an view and an query. There exists $l$ such that for every
simple path database from $u$ to $v$,
\begin{equation*}
(u,v) \in \datll(\V(\D)) \text{ iff } (\V(\D),u,v) \in \neg\CSP(\tempQV).
\end{equation*}
In particular if determines in a monotone way,
\begin{equation*}
(u,v) \in \datll(\V(\D)) \text{ iff } (u,v) \in \Q(\D).
\end{equation*}
Let and be an view and an query, and let $\D$ be a graph database
consisting of a simple path from node $u$ to node $v$. Assume $u, v \in \V(\D)$.
We will show, in Lemma <ref> below, that for large enough $l$, if
Player 2 has a winning strategy on the game on $(\V(\D),\tempQV, u,v)$ then we
can exhibit a homomorphism witnessing the fact that $(\V(\D), u, v) \in
\CSP(\tempQV)$. Before that we prove crucial properties of
$\V(\D)$ which will be exploited in the sequel. For that
we need the following simple definitions and claims.
Let $\D$ consist of the simple path $\pi=x_0a_1x_1\ldots x_{m-1}a_mx_{m}$,
with $x_0=u$ and $x_{m}=v$. Moreover let
$\S=\V(\D)$ and let
$A=\langle S_\V,\delta_\V,q^0_\V,F_\V \rangle$ be the product automaton of
all the deterministic minimal automata of all the regular expressions of the
s in . Let $N(\V)$ be the number of states of $A$, i.e. $|S_\V|$.
In what follows, for $q\in S_\V$ and $w \in \sigma^*$, $\delta_\V(q,w)$
denotes the state $p \in S_\V$ such that there is a run of $A$ on $w$
starting in state $q$ and arriving in state $p$.
For every $k \leq m+1$, and every $i,j \leq k$, we say that $x_i \sim_k x_j$ in $\V(\D)$
if, for all $V \in \V$, for all $r \geq k$,
\begin{equation*}
(x_i,x_r) \in V(\D) \quad \Leftrightarrow \quad (x_j,x_r) \in V(\D)
\end{equation*}
For all $k$, the relation $\sim_k$ is an equivalence relation over $\{x_i \ | \ i \leq k
\}$. We now prove the main property of $\V(\D)$, namely that the index of all
$\sim_k$ is bounded by the size of $\V$.
For all $k \leq m+1$:
$$\Big |\{x_i \ | \ i \leq k \} / \sim_k \Big |
\leq N(\V)$$
To each node $x_i$ in $\pi$ with $i \leq k$, we associate a state
$\varphi(x_i) \in S_\V$ defined as :
$$\varphi(x_i) =
\delta_\V(q_\V^0,\lambda(\pi_{i \rightarrow k}))$$
$\pi_{s \rightarrow t}$ is defined as the subpath of $\pi$ that starts at
position $s$ and ends at position $t$, that is $\pi_{s \rightarrow t} = x_{s}
a_{s} x_{s+1} a_{s+1}\ldots a_{t-1} x_t$.
Assume that there exist two nodes $x_i$ and $x_j$, with $i,j \leq k$, that
have the same image in $\varphi$. It follows that:
$$\delta_\V(q_\V^0,\lambda(\pi_{i \rightarrow k})) =
\delta_\V(q_\V^0,\lambda(\pi_{j \rightarrow k})) $$
Let us prove that
$x_i \sim_k x_j$. Assume that there exist $r \geq k$ and $V\in
\V$ such that $(x_i,x_r)\in V(\D)$. Then
$\delta_\V(q_\V^0,\lambda(\pi_{i \rightarrow r}))$ is final for $V$.
Remark that $\lambda(\pi_{i \rightarrow r}) = \lambda(\pi_{i
\rightarrow k})\lambda(\pi_{k \rightarrow r}$), from which we can
deduce that :
$$\delta_\V(q_\V^0,\lambda(\pi_{i \rightarrow r})) =
\delta_\V(\varphi(x_i),\lambda(\pi_{k \rightarrow r})) $$
$$\delta_\V(q_\V^0,\lambda(\pi_{i \rightarrow r})) =
\delta_\V(\varphi(x_j),\lambda(\pi_{k \rightarrow r})) $$
We can now
conclude that $\delta_\V(q_\V^0,\lambda(\pi_{j \rightarrow r}))$ is
final for $V$, which means that $(x_j,x_r) \in V(\D)$. A symmetric argument
easily proves the other direction of the equivalence. Hence,
$x_i \sim_k x_j$, and we can finally conclude that there cannot be more
that $N(\V)$ distinct equivalence classes of $\sim_k$ over the
nodes $\{x_i \ | \ i \leq k \}$ of $\pi$.
The following easily verified property of the equivalence relations $\sim_k$ will
also be useful:
Let $k_1,k_2 \leq m+1$, with $k_1 \leq k_2$. Let $x$ and $y$ be two elements
of $\pi$ that occur before $x_{k_1}$. Then $x \sim_{k_1} y$ implies $x
\sim_{k_2} y$.
We are now ready to prove the statement of the Proposition.
Let $l = |\tempQV| \cdot N(\V)$.
We prove that $(u,v) \in \datll(\S)$ iff
$(\S, u, v) \in \neg\CSP(\tempQV)$.
In view of the fact that $\datll$ encodes the $(l, l+1)$-two-player game in the sense of Lemma <ref>, it is enough to prove the following:
Player 2 has a winning strategy for the $\,(l,l+1)\,$- two-player game on
$\,(\S,\tempQV,u,v)$ iff there is an homomorphism from to
$\tempQV$ sending $u$ to a source node and $v$ to a target node.
The right-left direction is obvious. If there is a suitable homomorphism $h : \S
\rightarrow \tempQV$, then Player 2 has a winning strategy which consists in
playing according to $h$.
Conversely, assume that Player 2 has a winning strategy for the
$(l,l+1)$-two-player game on $(\S,\tempQV, u, v)$. Let $\{s_1,s_2,\ldots,s_r\}$ be an
ordering of the elements of , according to the order on $\pi$, that is, in
such a way that $\forall j\leq k$, $s_j$ occurs before $s_k$ in $\pi$.
Clearly $s_1 =u$ and $s_r=v$. If $r\leq l +1$, Player 1 can select all elements of in a single round, and then
Player 2 has to provide a full homomorphism from to $\tempQV$,
which concludes the proof.
Assume $r > l+1$. For ease of notations, we will number rounds starting from
$l+1$. This can be seen just as a technicality, or equivalently as Player 1
selecting the empty set for the first $l$ rounds. Since Player 2 has a winning
strategy, she has, in particular, a winning response against the following play
of Player 1 :
* On round $l+1$, Player 1 plays $A_{l+1} =
\{s_1,\ldots,s_{l+1}\}$. Player 2 has to respond with a partial
homomorphism $h_{l+1}$, which she can do, since she has a winning
* Assume that, on round $i$, $A_i$ is of size $l+1$ and its element
of biggest index is $s_{i}$ (as it is the case on round $l+1$).
Given the choice of $l$, the set $A_i$ is sufficiently “big", that is
by Claim <ref>, there exist two elements $s_j,s_k \in
A_i$ such that $s_j \sim_{i} s_k$, and $h_i(s_j) = h_i(s_k)$. On
round $i+1$, Player 1 picks $A_{i+1} = (A_{i} - \{s_j\}) \cup
\{s_{i+1}\}$. This choice maintains that $A_{i+1}$ is of size
$l+1$ and that its element of
biggest index is $s_{i+1}$. Once again, Player 2 has to respond with
a partial homomorphism $h_{i+1}$, which she can do.
* Following this play, on round $r$, $A_{r}$ contains $s_r$, the element of biggest index in .
From now on, we no longer care about Player 1's move, that is, we arbitrarily set $A_i = \emptyset$ for all
$i > r$.
We can now define $h$ as follows :
$$h(s_i) =
\left\{
\begin{array}{ll}
h_{l+1}(s_i) & \mbox{if } i \leq l+1 \\
h_{i}(s_i) & \mbox{if } l+1 < i \leq r
\end{array}
\right.
Observe that, by definition, the mapping $h$ sends $u$ to a source node and $v$ to a target node
(since so do all the $h_i$'s used in the game). It remains to prove that
$h$ is an homomorphism from to $\tempQV$. We prove by
induction on $i \geq l+1$ that :
$(H_1)$ $h$ is a homomorphism from $\S[\{s_1,\ldots,s_i\}]$
to $\tempQV$.
$(H_2)$ $h$ coincides with $h_i$ on $A_i$.
$(H_3)$ for all $j \leq i$, there exists $s \in A_i$ such that
$s_j \sim_i s$ and $h(s_j) = h(s)$.
Base case : For $i = l+1$, the mapping $h$ coincides by definition with
$h_{l+1}$ on $\{s_1,\ldots,s_{l+1}\}$. Hence, $(H_1)$ and $(H_3)$
follow easily.
Inductive case : Assume that there exists $i$ with $l+1 \leq i < r$
such that $(H_1)$,$(H_2)$ and $(H_3)$ holds for $i$; we prove them for
$(H_2)$Let $s \in A_{i+1}$. If $s = s_{i+1}$, then, by
definition, $h(s_{i+1}) = h_{i+1}(s_{i+1})$. Otherwise, $s\in A_i
\cap A_{i+1}$. $(H_2)$ for $i$ implies that $h(s) = h_i(s)$, and the
definition of $h_{i+1}$ thus yields $h_{i+1}(s) = h_i(s) =
h(s)$. Hence, $(H_2)$ holds for $i+1$.
$(H_3)$Let $j \leq i+1$. If $j = i+1$, then $s_j \in A_{i+1}$,
and the result is obvious. Otherwise, $(H_3)$ for $i$ implies that
there exists $s \in A_i$ such that $s_j \sim_i s$ and $h(s_j) =
h(s)$. From Claim <ref>, we deduce that $s_j \sim_{i+1}
s$. If $s\in A_{i+1}$, there is nothing more to prove. Otherwise, it
means that $s$ is exactly the element that was removed from $A_i$ on
round $i+1$, which means that there exists another element $s' \in
A_{i}\cap A_{i+1}$ such that $s \sim_{i} s'$ and $h_i(s) = h_i(s')$.
Then Claim <ref> and $(H_2)$ imply that $s_j \sim_{i+1} s'$
and $h(s_j) = h(s')$. Hence $(H_3)$ holds for $i+1$.
$(H_1)$By definition, $h$ already preserves any self-loop.
Moreover, $(H_1)$ for $i$ implies that $h$
is a homomorphism from $\S[\{s_1,\ldots,s_i\}]$ to
$\tempQV$. Hence, any edge between two elements of
$\{s_1,\ldots,s_i\}$ in $\S$ is already preserved by $h$. Let $s_j \in
\{s_1,\ldots,s_i\}$. Remark that, since $\pi$ is a simple path,
there are no edges from
$s_{i+1}$ to $s_j$ in $\S$. Thus, we just have to prove that all edges from
$s_j$ to $s_{i+1}$ are preserved by $h$.
$(H_3)$ for $i+1$ implies that there exists an element $s\in A_{i+1}$
such that $s_j \sim_{i+1} s$ and $h(s_j) = h(s)$. Since $h_{i+1}$ is
a homomorphism on $\S[A_{i+1}]$, it preserves all edges
from $s$ to $s_{i+1}$. Moreover, $(H_2)$ for $i+1$ implies that $h$
and $h_{i+1}$ coincide on $A_{i+1}$, which means that $h$ preserves
all edges from $s$ to $s_{i+1}$. Finally, the definition of
$\sim_{i+1}$ implies that $s_j$ and $s$ have the same edges to
$s_{i+1}$. Hence, $h$ preserves all edges from $s_j$ to $s_{i+1}$.
Finally, $(H_1)$ applied for $r$ proves that $h$ is indeed a
homomorphism from to $\tempQV$.
This completes the proof of Lemma <ref>.
Now assume determines in a monotone way, then from Proposition <ref>
it immediately follows that $(u,v) \in \datll(\V(\D))$ iff $(u,v) \in \Q(\D)$.
This completes the proof of Proposition <ref>.
From simple paths to arbitrary graph databases
Proposition <ref> shows that if determines in a monotone
way then $\datll$ is a rewriting of using , when restricted to simple path
databases. It remains to lift this result to arbitrary graph databases. In
a sense, the following result shows that the general case can always be
reduced to the simple path case.
Let and be an view and an query such that determines in a
monotone way. Assume $\mathcal{P}$ is a query of schema $\tau$ such that:
* $\mathcal{P}$ is closed under homomorphisms: for all
databases $\S,\S'$, and all pair of elements $(u,v)$ of
, if $(u,v) \in \mathcal{P}(\S)$ and there exists a homomorphism
$h:\S \rightarrow \S'$ then $(h(u),h(v)) \in \mathcal{P}(\S')$.
* $\mathcal{P}$ is sound and complete for all simple path
for all simple path databases $\D$ from $u$ to $v$ such that $u$ and $v$ are in the
domain of $\V(\D)$, we have $(u,v) \in \mathcal{P}(\V(\D))$ iff
$(u,v)\in \Q(\D)$.
* $\mathcal{P}$ is always sound: for all graph databases and
elements $u$ and $v$ of $\V(\D)$, if $(u,v) \in \mathcal{P}(\V(\D))$
then $(u,v) \in \Q(\D)$.
Then $\mathcal{P}$ is a rewriting of using .
Let be a database, and $(u,v)$ be a pair of elements of $\V(\D)$,
such that $(u,v) \in \Q(\D)$. Then there exists in a path $\pi_0$ from $u$
to $v$, such that $\lambda(\pi_0) \in L(\Q)$.
Consider the simple path $\pi = x_0 a_0 x_1 \ldots x_m a_m x_{m+1}$ defined
such that $\lambda(\pi) = \lambda(\pi_0)$. Since $\V$ determines $\Q$ in a
monotone way and $\lambda(\pi) \in L(\Q)$, then $x_0$ and $x_{m+1}$ are in
the domain of $\V(\pi)$, and $(x_0,x_{m+1})\in \Q(\pi)$.
Hence, (<ref>) implies that $(x_0,x_{m+1})\in
\mathcal{P}(\V(\pi))$.
Additionally, it is clear that there exists a homomorphism $h$ from $\pi$ to
with $h(x_0) = u$ and $h(x_{m+1}) = v$. Observe that $h$ extends to the views of
$\pi$ and $\D$, that is $h$ is an homomorphism from $\V(\pi)$ to $\V(\D)$,
and (<ref>) thus implies that $(u,v)\in \mathcal{P}(\V(\D))$.
The other direction is immediately given by (<ref>).
We now have all the elements to prove Proposition <ref>.
Let and be an view and an query such that determines in a
monotone way.
By Proposition <ref> there exists $l$ such that $\datll$ is sound and complete
over simple path databases.
Moreover each query is preserved under homomorphisms, and we have already observed that all $\datlk$ are always sound.
It then follows from Proposition <ref> that there exists $l$ such that $\datll$ is a rewriting of using .
This proves Proposition <ref> and therefore Theorem <ref>.
§ CONCLUSIONS
We have seen that if an view determines an query in a monotone way
then a rewriting can be computed from and .
As a corollary it is decidable whether there exists a rewriting to
an query using views.
These results extends to 2-way-. A 2-way-is defined using a regular
expression over the alphabet $\sigma\cup\bar\sigma$. It asks for pairs of nodes
linked by a 2-way-path using the symbol $a$ for traversing an edge of label $a$
in the direction of the arrow, and the symbol $\bar a$ for backward traversing an edge of
label $a$. This query language has been
studied in <cit.>. In particular <cit.> gives an extension of
Corollary <ref> and of Proposition <ref> for
2-way-. Building from these two results it is possible to extend the
results of Section <ref> to 2-way-s. The details are more
complicated and omitted here, but the general idea is the same.
We may wonder whether a simpler query language than could suffice to express monotone rewritings of queries using views.
For instance all examples we are aware of use only the transitive closure of binary
Conjunctive Regular Path Queries. It is then natural to ask whether linear (where at most
one internal predicate may occur in the body of each rule), using internal
predicates of arity at most 2, can express all monotone rewritings. We leave
this interesting question for future work.
Finally we conclude by mentioning that we don't know yet whether the monotone
determinacy problem for Conjunctive Regular Path Query is decidable. Likewise,
deciding whether an view determines an query, without the
monotonicity assumption, is still an open problem.
|
1511.01068
| |
1511.00632
|
[ca]Corresponding author
]Ivan Mizera
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
]the Alzheimer's Disease Neuroimaging Initiativelabel1
We propose a prediction procedure for the functional linear quantile
regression model by using partial quantile covariance techniques and
develop a simple partial quantile
regression (SIMPQR) algorithm to efficiently extract partial quantile
regression (PQR) basis for estimating functional coefficients. We
further extend our partial quantile covariance techniques to
functional composite quantile regression (CQR) defining partial
composite quantile covariance. There are three major contributions.
(1) We define partial quantile covariance between two scalar variables
through linear quantile regression. We compute PQR basis by
sequentially maximizing the partial quantile covariance between the
response and projections of functional covariates. (2) In order to
efficiently extract PQR basis, we develop a SIMPQR algorithm analogous to simple partial least
squares (SIMPLS). (3) Under the homoscedasticity assumption, we extend
our techniques to partial composite quantile covariance and use it to
find the partial composite quantile regression (PCQR) basis. The
SIMPQR algorithm is then modified to obtain the SIMPCQR algorithm. Two
simulation studies show the superiority of our proposed methods. Two
real data from ADHD-200 sample and ADNI are analyzed using our
proposed methods.
Functional linear quantile regression; Partial quantile covariance; PQR basis; SIMPQR; Partial composite quantile covariance; CPQR basis; ADHD; ADNI
[label1]Part of the data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging
Initiative (ADNI) database (adni.loni.ucla.edu). As such, the investigators within the ADNI contributed to
the design and implementation of ADNI and/or provided data but did not participate in analysis or writing
of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.ucla.edu/wp-content/
§ INTRODUCTION
Nowadays, there is great need in the analysis of complex neuroimaging
data obtained from various cross-sectional and clustered neuroimaging
studies. These neuroimaging studies are essential to advancing our
understanding of the neural development of neuropsychiatric and
neurodegenerative disorders, substance use disorders, the normal brain
and the interactive effects of environmental and genetic factors on
brain structure and function. Such large imaging studies include the
ADNI (Alzheimer's Disease Neuroimaging Initiative), the longitudinal
magnetic resonance imaging (MRI) study of schizophrenia, autism, and
attention deficit hyperactivity disorder (ADHD), the NIH human
connectome project, among many others. Neuroimaging studies usually
collect structural, neurochemical, and functional images over both
time and space
These structural, neurochemical, and functional imaging modalities
include computed axial tomography (CT), diffusion tensor imaging
(DTI), functional magnetic resonance imaging (fMRI), magnetic
resonance imaging (MRI), magnetic resonance spectroscopy (MRS),
positron emission tomography (PET), single photon emission tomography
(SPECT), electroencephalography (EEG), and magnetoencephalography
(MEG), among many others. For instance, by using anatomical MRI,
various measures of the morphology of the cortical and subcortical
structures (e.g., hippocampus) are extracted to understand
neuroanatomical differences in brain structure across different
populations <cit.>. In DTI,
various diffusion properties and fiber tracts are extracted for
quantitative assessment of anatomical connectivity across different
Functional images, such as resting-state functional MRI (rsfMRI), have
been widely used in behavioral and cognitive neuroscience to
understand functional segregation and integration of different brain
regions across different populations
A common feature of many imaging techniques is that massive
functional data are observed/calculated at the same design points, such as time for functional images (e.g., PET and fMRI) and arclength for structure imaging (e.g. DTI).
As an illustration, we present two smoothed functional data that we encounter in neuroimaging studies.
First, we consider the BOLD rsfMRI signal, which is based on hemodynamic responses secondary to resting-state.
We plot the estimated hemodynamic response functions (HRF) with 172 time courses from 20 randomly selected children at a selected region of interest (ROI) of
Anatomical Automatic Labeling (AAL) atlas <cit.> from the New York University (NYU) Child Study Center from the ADHD-200 Sample Initiative Project. Although the canonical form of the HRF is often used, when applying rsfMRI in a clinical
population with possibly altered hemodynamic responses (Figure <ref> (a)), using the subject's own HRF in
rsfMRI data analysis may be advantageous because HRF variability is greater across subjects
than across brain regions within a subject <cit.>. We
are particularly interested in delineating the structure of the
variability of the HRF and their capacity of predicting ADHD index with a set of covariates of interest, such
as diagnostic group <cit.>. Secondly, we plot one diffusion property, called fractional
anisotropy (FA), measured at 83 grid points along the midsagittal corpus callosum (CC) skeleton
(Figure <ref> (b)) from 30 randomly selected
infants from the NIH Alzheimer's Disease Neuroimaging Initiative (ADNI) study. The corpus callosum (CC) is the largest
fiber tract in the human brain and is a topographically organized structure. It is responsible for much of
the communication between the two hemispheres and connects homologous areas in the two
cerebral hemispheres. Scientists are
particularly interested in delineating the structure of the
variability of these functional FA data and their prediction ability on mini-mental state examination (MMSE) with a set of covariates of interest, such
as genetic information. MMSE is
one of the most widely used screening tests on Alzheimer's Disease to provide brief and objective
measures of cognitive functioning <cit.>. We will systematically investigate
these two prediction problems using functional imaging data over time or space in Section <ref> after we develop our methodology.
Representative functional neuroimaging data: (Left) the estimated hemodynamic response functions (HRF) corresponding to resting-state from 20 children at NYU from the ADHD-200 Sample Initiative Project and (Right) fractional anisotropy (FA) along the midsagittal corpus callosum (CC) skeleton from 30 randomly selected subjects from the NIH Alzheimer's Disease Neuroimaging Initiative (ADNI) study.
A functional linear regression model, where the responses such as the
neurological or clinical outcomes (e.g. ADHD index or MMSE) are
modeled by a set of scalar covariates and functional covariates of
interest (e.g. HRF along time courses or FA along arclength), is a
powerful statistical tool for addressing these scientific questions
In particular, denoting the neurological or clinical outcome of the
$i$-th subject by $y_i$, $i=1,\dots,n$, the functional linear
regression model is of the form
\begin{equation}\label{pfqreq1}
y_i=\alpha+\ve x_i^T\ve \beta + \int_0^1 \ve z_i^T(t) \ve \gamma(t)dt +\epsilon_i,
\end{equation}
where $\alpha$ is the intercept, $\ve \beta=(\beta_1, \cdots,
\beta_p)^T$ is a $p\times 1$ vector of coefficients, ${\ve
x}_i=(x_{i1},\cdots,x_{ip})^T$ is a $p\times 1$ vector of scalar
covariates of interest, $\ve \gamma(t)=(\gamma_1(t), \cdots,
\gamma_q(t))^T$ is a $q\times 1$ vector of coefficient functions of
$t$, ${\ve z}_i(t) = (z_{i1}(t),\cdots,z_{iq}(t))^T$ is a $q\times 1$
vector of functional covariates, and $\epsilon_{i}$ is a random error.
It is usually assumed that $\epsilon_i$ is independent and identical
copy of normal distribution with zero mean and variance $\sigma^2$.
For simplicity, we let $t \in [0,1]$. Model (<ref>) is a
generalization of the classical linear regression model corresponding
to the case $\ve \gamma(t)$ is a constant. If it is not constant, the
contributions of $ \ve z_i(t)$ characterized by $\ve \gamma(t)$ change
in terms of $t$. The model has been well studied and applied in many
fields including neuroimaging data analysis
To facilitate the estimation of $\ve \gamma(t)$, we usually require
that it satisfies certain smoothness conditions and restrict it onto a
functional space. For example, we may require that it its second
derivative exists and that the square of $\gamma(t)$ is integrable,
that is, $\ve \gamma(t) \in L_2[0,1]$. Even in such a case, the
estimation is still an infinite-dimensional problem.
The common practice is to project $\ve \gamma(t)$ into a functional space with a finite functional basis. There are three major methods to choose
the functional basis: general basis, functional principal component basis (fPC), and partial least square basis (PLS). There are various options on the selection
of general basis, for example B-spline basis <cit.>, wavelet basis <cit.> and so on. In order to provide a good approximation of
the functional coefficients, a large number of basis should be chosen. However, this may cause overfitting of the model and to remedy that various penalty methods have been proposed
<cit.>. The fPC method has been extensively studied <cit.> where the fPC of ${\ve z}_i(t)$
serve as the basis. Its generalization to the reproducing kernel Hilbert space (RKHS) was proposed by Cai and Yuan <cit.> who also studied its minimax rates. Although fPC basis are more data-adapted than the general basis as they use the information of functional covariates and the formed space
can explain most of the variation of ${\ve z}_i(t)$, it is not necessary all the fPC basis will contribute to the variation of the responses.
Therefore, another appealing choice is the PLS basis which use both the information of functional covariates and the responses. The PLS basis use the linear projects of ${\ve z}_i(t)$ which best predict the responses <cit.>.
An alternative to model (<ref>) is the functional linear
quantile regression where the conditional quantiles of the responses
are modeled by a set of scalar covariates and functional covariates.
There are at least three advantages to use conditional quantiles
instead of conditional means. First, quantile regression, in
particular median regression, provides an alternative and complement
to mean regression while being resistant to outliers in responses. It
is more efficient than mean regression when the errors follow a
distribution with heavy tails. Second, quantile regression is capable
of dealing with heteroscedasticity, the situation when variances
depend on some covariates. More importantly, quantile regression can
give a more complete picture on how the responses are affected by
covariates: for example, some tail behaviors of the responses
conditional on covariates. For more background on quantile regression,
see the monograph of Koenker <cit.>. In our case,
we consider functional linear quantile regression: for given $\tau \in
\begin{equation}\label{pfqreq2}
Q_{\tau}\left(y_i|\ve x_i,\ve z_i(t)\right)=\alpha_\tau+\ve x_i^T\ve \beta_\tau + \int_0^1 \ve z_i^T(t) \ve \gamma_\tau(t)dt,
\end{equation}
where $Q_{\tau}\left(y_i|\ve x_i,\ve z_i(t)\right)$ is the $\tau$-th
conditional quantile of $y_i$ given covariates $\ve x_i$ and ${\ve
z}_i(t)$, $\alpha_\tau$ is the intercept, $\ve
\beta_\tau=(\beta_{1\tau}, \cdots, \beta_{p\tau})^T$ is a $p\times 1$
vector of coefficients and $\ve \gamma_\tau(t)=(\gamma_{1\tau}(t),
\cdots, \gamma_{q\tau}(t))^T$ is a $q\times 1$ vector of coefficient
functions. In the existing literature, model (<ref>) has been
well studied and various methods have been proposed. As in functional
linear regression, to estimate functional coefficients $\ve
\gamma_\tau(t)$ it is convenient to restrict it in a functional space
with a finite basis. Similarly, general basis like B-spline can be
used to approximate the coefficient functions
<cit.>. fPC basis have also
been throughly investigated with and without scalar covariate $\ve
x_i$ while only one functional covariate presents
<cit.>. However,
there is no analogue to the PLS basis method in functional linear
regression model. Therefore, none of the existing methods for model
(<ref>) is able to provide more efficient prediction by
extracting information from the responses.
In this paper, we propose a prediction procedure for the functional
linear quantile regression model (<ref>) by using partial
quantile covariance techniques and develop an algorithm inspired by
simple partial linear regression, SIMPLS <cit.>, to
efficiently extract partial quantile regression (PQR) basis for
estimating functional coefficients. We further extend our partial
quantile covariance techniques to functional composite quantile
regression (CQR) <cit.> by defining partial composite
quantile covariance. The major contributions of this paper can be
summarized as follows. We first define partial quantile covariance
between two scalar variables through linear quantile regression.
Motivated by extracting PLS basis in functional linear regression, we
found PQR basis by sequentially maximizing the partial quantile
covariance between the response and projections of functional
covariates. In order to efficiently extract PQR basis, we develop a
simple partial quantile regression (SIMPQR) algorithm analogue to
SIMPLS. Under the homoscedasticity assumption, we extend our
techniques to partial composite quantile covariance and use it to find
the partial composite quantile regression (PCQR) basis. The SIMPQR
algorithm is then modified to obtain the SIMPCQR algorithm.
The rest of this paper is organized as follows. In Section 2, we
define partial quantile covariance and describe how to use it to extract PQR basis in functional linear quantile regression model.
In Section 3, we develop the SIMPQR algorithm and discuss its properties. We discuss how to calculate the PCQR basis by using
partial composite quantile covariance and propose the SIMPCQR algorithm in Section 4.
Two sets of simulation studies are presented in Section 5 with the known ground truth to
examine the finite sample performance of our proposed methodology. In Section 6, we use PQR and PCQR to predict ADHD index and
MMSE using data from NYU site from ADHD-200 sample and ADNI, respectively. Some discussions and future research directions are given in Section 7.
§ PARTIAL FUNCTIONAL LINEAR QUANTILE REGRESSION
In model (<ref>), we assume without loss of generality that $t
\in [0,1]$ and restrict the functional coefficients $\ve\gamma_\tau(t)
\in L_2[0,1]$.
For simplicity, we assume $q=1$, that is, we only
consider one functional covariate. The extension of our methodology to
more functional covariates is straightforward.
The estimation
$\ve\gamma_\tau(t)$ is in general a difficult question as it lies in
an infinite-dimensional space. However, if it can be well approximated
in a finite element space, say, $H[0,1]$, the solution for the model
(<ref>) can be found. Let $b_{k\tau}(t)$, $k=1,\dots,K$ be a
basis of $H[0,1]$ and $\ve\gamma_\tau(t) =
\sum_{k=1}^K\ve\gamma_{k\tau}b_{k\tau}(t)$. Model (<ref>)
can be then rewritten as
\begin{equation}\label{pfqreq3}
Q_{\tau}\left(y_i|\ve x_i,\ve z_i(t)\right)=\alpha_\tau+\ve x_i^T\ve \beta_\tau + \sum_{k=1}^K \ve z_{ki} \ve \gamma_{k\tau},
\end{equation}
where $\ve z_{ki}= \int_{0}^1\ve z_i(t)b_{k\tau}(t) dt$. Model
(<ref>) is simply a linear quantile regression problem, which
is essentially a linear programing problem; its solutions can be
obtained by many algorithms—for example, the simplex method
<cit.>, the interior point method
<cit.>, the MM algorithm <cit.>
and many others, already implemented in various statistical softwares
like in <cit.>.
In the literature, there are many methods devoted to find the crucial
basis functions in model (<ref>). The general basis B-spline
was proposed and studied by Cardot and others
<cit.>. In various models,
fPC basis has also been studied <cit.>. However, neither basis does use
information of the responses and hence they are less efficient to do
prediction. In this session, as the motivation of our proposal, we
first review the PLS basis in model (<ref>) where both
information of the functional covariates and the responses are used to
choose the basis functions. Then we propose our developed methodology
to choose basis for model (<ref>), namely, partial quantile
regression (PQR) basis.
In functional linear regression model (<ref>), the first PLS basis is chosen to be
\begin{equation}\label{pfqreq4}
b(t) = \arg_{b(t)}\min_{\alpha,\beta,b(t)}\sum_{i=1}^n\left(y_i-\alpha-\ve x_i^T\ve \beta - \int_0^1 \ve z_i(t)b(t)dt\right)^2,
\end{equation}
which is the analogue to the partial least square regression in
multivariate analysis. The subsequent basis is chosen by iteratively
using (<ref>) after taking account of and subtracting the
information from previous basis. For more details, see Delaigle and
Hall <cit.>. The essential idea of criteria
(<ref>) is to find a direction $b(t)$ so that the projection
of $\ve Z(t)$ on it explains as much as possible the variation of the
response after adjusting some covariates. Therefore, as shown in
<cit.>, it is equivalent to find a basis
$b(t)$ such that the partial covariance
\begin{equation}\label{pfqreq5}
COV\left(Y-\alpha-\ve X\ve \beta, \int_0^1 \ve Z(t)b(t)dt\right)
\end{equation}
is maximized, where $Y = (y_1,\dots,y_n)^T$, $\ve X = (\ve x_1,\dots,\ve x_n)^T$ and $\ve Z(t) = (\ve z_1(t),\dots,\ve z_n(t))^T$. Based on this equation, Delaigle and Hall <cit.> found an equivalent space with the same dimension as
the PLS space and proved the estimation and precision consistency.
The parameters in model (<ref>) are estimated by solving
\begin{equation}\label{pfqreq6}
\min_{\alpha,\beta,b(t)}\sum_{i=1}^n\rho_\tau\left(y_i-\alpha-\ve x_i^T\ve \beta - \int_0^1 \ve z_i (t)b(t)dt\right),
\end{equation}
where $\rho_\tau(u) =u(\tau-I(u<0))$ is the quantile loss function
<cit.> with $I$ as the indicator function. When
$\tau=0.5$, the loss is $\rho_\tau(u) =|u|/2$ and the results is then
the median, or least absolute deviation (LAD) regression. To adapt to
the idea of PLS basis, that is, to find a direction $b_\tau(t)$ so
that the projection of $\ve z(t)$ on it contributes as much as
possible to predict the quantile of the response after adjusting some
covariates, we first propose the concepts of quantile covariance (QC)
and partial quantile covariance (PQC). For given $\tau \in (0,1)$ and
a random variable $X$, the partial quantile covariance $COV_{qr}(Y,Z)
$ between two random variables $Y$ and $Z$ is of the form
\begin{equation}\label{pfqreq7}
COV_{qr}(Y,Z) =\arg_{\gamma_\tau}\inf_{\alpha,\beta_\tau,\gamma_\tau}E\left(\rho_\tau\left(Y-\alpha-\beta_\tau X-\gamma_\tau Z \right)\right),
\end{equation}
where we first normalize $Z$ to have mean zero and variance one. If
there is no $X$, then $COV_{qr}(Y,Z) $ is quantile covariance between
$Y$ and $Z$. The quantile covariance measures the contribution of $Z$
to the $\tau$-th quantile of $Y$. It was first proposed and studied by
Dodge and Whittaker <cit.> in the context of partial
quantile regression. Li, Li and Tsai <cit.> proposed a
similar concept of quantile correlation and used it to study quantile
autoregressive model.
To find the partial quantile regression basis (PQR), similar to that of PLS to maximize the covariance we propose to compute the $b_\tau(t)$ by maximizing
\begin{equation}\label{pfqreq8}
COV_{qr}\left(Y-\alpha_\tau-\ve X\ve \beta_\tau, \int_0^1 \ve Z(t)b_\tau(t)dt\right).
\end{equation}
The subsequent basis is computed by iteratively maximizing
(<ref>) after taking account of and subtracting the
information of the previous basis. Let $\ve Z_k = \int_0^1\ve
Z(t)b_{k\tau}(t)dt$, where $b_{k\tau}(t)$ is the $k$-th PQR basis.
Denote $\ve Z_{(k+1)}(t)$ as $\ve Z(t)$ after subtracting the
information from the first $k$ basis. Then the $(k+1)$-th basis
$b_{(k+1)\tau}(t)$ is obtained by maximizing the partial quantile
\begin{equation}\label{pfqreq9}
COV_{qr}\left(Y-\alpha_\tau-\ve X\ve \beta_\tau-\sum_{j=1}^k\ve Z_{j} \ve \gamma_{j\tau}, \int_0^1 \ve Z_{(k+1)}(t)b_\tau(t)dt\right).
\end{equation}
We will discuss detailed algorithms in the next section. Once we find
an adequate number, $K$ of functional basis elements, we have the
approximation model (<ref>), where the parameters are obtained
by minimizing
\begin{equation}\label{pfqreq10}
\sum_{i=1}^n\rho_\tau\left(y_i-\alpha_\tau-\ve x_i^T\ve \beta_\tau - \sum_{k=1}^K \ve z_{ki} \ve \gamma_{k\tau}\right).
\end{equation}
The number of PQR basis can be chosen use BIC or cross validation (CV) as in choosing the number of fPC basis adapted by Kato and other authors
§ SIMPQR ALGORITHM
In this section, we propose a simple partial quantile regression
(SIMPQR) algorithm to iteratively extract the PQR basis from the
functional covariates $\ve z(t)$. Similar algorithm has been studied
by Dodge and Whittaker <cit.> in partial quantile
regression with multiple covariates. It is parallel to the SIMPLS for
partial least square regression <cit.>. The motivation is
to subsequently maximize (<ref>) after accounting and
subtracting the information of the previous basis. To simplify the
description of the SIMPQR algorithm, we will drop the scalar
covariates $\ve x$ in model (<ref>) in this section. Let
$0<t_1<\dots<t_m<1$ denote the discretized sample points for the
functional covariates and we assume they are equally spaced. Recall
that we set $q=1$ and we focus on only one functional covariate $\ve
z(t)$. The SIMPQR algorithm is described as follows.
* Step 1, Initialization: Normalize $\ve z_i(t_j)$ for each $j$ so that it has mean zero and variance one.
* Step 2, Repeat:
* Compute a functional basis $b_\tau(t_j) = COV_{qr}(Y,\ve Z(t_j))$ for $j=1,\dots,m$ and rescale it to have $\sum b^2_\tau(t_j)=1$.
* Project $\ve z_i(t_j)$ onto the basis $(b_\tau(t_1),\dots,b_\tau(t_m))^T$ to obtain $\ve z_{i} = \sum \ve z_i(t_j) b_\tau(t_j)$. Denote $\ve Z = (\ve z_1,\dots,\ve z_n)^T$ as the projections for each subjects.
* Predict $\ve z_i(j)$ by using simple linear regression with the projection $\ve z_i$ as the covariate, denoting the result by $\hat{\ve z}_i(j)$.
* Subtract the information from the projection $\ve z_i$ by replacing $\ve z_i(j)$ by their residuals
$\ve z_i(j)-\hat{\ve z}_i(j)$.
* Step 3, Stop: Check stopping criterion and retain the projections $\ve Z_1,\dots,\ve Z_K$.
* Step 4, Model: Fit the model (<ref>) by minimizing equation (<ref>).
The SIMPQR algorithm follows the same line of SIMPLS with the
covariance being replaced by quantile covariance. The nature of our
proposed quantile covariance implies that it is not necessary to
adjust the response $Y$ each time after a new basis is obtained. The
resulting functional basis is orthogonal to each other due to the
prediction in step 2.3. However, it is worth noting that due to the
nonadditivity of conditional quantiles, we need to fit model
(<ref>) after all the basis elements are picked out, instead
of estimating the coefficients of each basis projections once they are
§ PARTIAL FUNCTIONAL LINEAR COMPOSITE QUANTILE REGRESSION
Despite the success of quantile regression (QR), its relative efficiency to the least square regression can be arbitrarily small <cit.>.
Composite quantile regression (CQR) proposed by Zou and Yuan <cit.> inherits some good properties of QR and is capable of providing
more efficient estimators under certain conditions. Given two random variables $X$ and $Y$ and quantile level set $0<\tau_1<\dots<\tau_L<1$, the CQR parameters $(\alpha_{\tau_1},\dots,\alpha_{\tau_L},\beta)$ are defined as
\begin{equation}\label{pfqreq11}
\inf_{\alpha_{\tau_1},\dots,\alpha_{\tau_L},\beta}E\sum_{l=1}^L\left(\rho_{\tau_l}\left(Y-\alpha_{\tau_l}-\beta X \right)\right),
\end{equation}
where $\rho_{\tau_l}$ is the $\tau_l$-th quantile loss function. Under the homoscedasticity assumption, that is, the model errors do not depend on
covariates, all conditional regression quantiles are parallel and they have the same slope $\beta$ but different intercepts. The CQR is equivalent to
fit QR at different quantile levels. However, CQR estimators are more efficient.
For given $0<\tau_1<\dots<\tau_L<1$ and a random variable $X$ similar
to (<ref>), the partial composite quantile covariance (PCQC)
$COV_{cqr}(Y,Z) $ between two random variable $Y$ and $Z$ is of the
\begin{equation}\label{pfqreq12}
COV_{cqr}(Y,Z) =\arg_{\gamma}\inf_{\alpha_{\tau_1},\dots,\alpha_{\tau_L},\beta,\gamma}E\sum_{l=1}^L\left(\rho_{\tau_l}\left(Y-\alpha_{\tau_l}-\beta X-\gamma Z \right)\right),
\end{equation}
where we first normalize $Z$ to have mean zero and variance one. If
there is no $X$, then $COV_{cqr}(Y,Z) $ is composite quantile
covariance (CQC) between $Y$ and $Z$. The composite quantile
covariance measures the contribution of $Z$ to the quantiles of $Y$ at
levels $0<\tau_1<\dots<\tau_L<1$. There are some connections between
composite quantile covariance and covariance; however, these are beyond
the scope of this paper and we plan to discuss them elsewhere.
With the definition of PCQC, we can obtain the PCQR basis for functional linear composite quantile regression by maximizing
\begin{equation}\label{pfqreq13}
COV_{cqr}\left(Y-\ve X\ve \beta, \int_0^1 \ve Z(t)b(t)dt\right),
\end{equation}
for a given quantile level set $0<\tau_1<\dots<\tau_L<1$. The
subsequent basis is computed by iteratively maximizing
(<ref>) after accounting and subtracting the information of
the previous basis. Once the PCQR basis is found, the functional
linear composite quantile regression can be easily fitted by a linear
program, for example in
<cit.>. The algorithm to compute the PCQR basis follows
the same line of SIMPQR in the last section; we only need to replace
$COV_{qr}$ by $COV_{cqr}$ in step 2.1 and keep the rest unchanged.
§ SIMULATION STUDIES
In this section, we investigate the finite sample performance of our
proposed prediction methods, namely partial quantile regression (PQR)
basis and partial composite quantile regression (PCQR) basis methods.
We compare them with the fPC basis method in functional linear
quantile regression (QRfPC) and functional linear composite quantile
regression (CQRfPC) models. In addition, we compare them with PLS
basis and fPC basis methods in functional linear regression model. We
conduct our simulations in two settings where the first one is in
favor of the fPC basis and the second one is a more general case. Both
simulations show superior or comparable performance of our proposed
Simulation I. In this simulation, we adapt the setup in Kato <cit.>. In particular, the model is of the form
\begin{eqnarray*}
Y &=& \int_{0}^{1} \gamma(t) Z(t) dt + \varepsilon,\\
\gamma(t)&=& \sum_{j=1}^{50} \gamma_j \phi_j(t); ~~\gamma_1=0.5, \gamma_j=\frac{20}{3}(-1)^{j+1} j^{-2}, j \geq 2, \phi_j(t)=2^{1/2}\cos(j \pi t),\\
Z(t) &=& \sum_{j=1}^{50} v_j U_j \phi_j(t);~~ v_j = (-1)^{j+1} j^{-1.1/2},U_j \sim U[-3^{1/2},3^{1/2}].
\end{eqnarray*}
Each $X_i(t)$ was observed at $m=201$ equally spaced grid points on $[0,1]$. We choose the sample size $n$ to be $100$, $200$, and $500$. The error
$\varepsilon$ follows either Gaussian with mean zero and variance one or Cauchy distribution.
In this design we have
\begin{equation*}
Q_\tau(Y|X) = F_{\varepsilon}^{-1}(\tau) + \int_0^1 \gamma(t) Z(t) dt,
\end{equation*}
where $F_{\varepsilon}$ is the cumulative distribution function of $\varepsilon$. It should be pointed out that
the simulation set up is in favor of fPC basis methods as the functional coefficients lie on the same fPC space of
functional covariates. It is expected that fPC basis methods may be superior to other methods.
To facilitate the comparison, we set $\tau=0.5T$ for QR methods and $\tau_l=l/(1+L)$ with $L=9$ for CQR methods. One criteria we use is
the mean integrated errors (MISE) of the functional coefficients,
\begin{eqnarray*}
\mbox{MISE} =\frac{1}{S}\sum_{s=1}^S
\sum_{j=1}^m\left(\hat\gamma_s(t_j)-\gamma(t_j)\right)^2
= \mbox{Bias}^2+\mbox{Var},
\end{eqnarray*}
\begin{equation*}
\mbox{Bias}^2 =
\sum_j\left(\frac{1}{S}\sum_s\hat\gamma_s(t_j)-\gamma(t_j)\right)^2
\end{equation*}
\begin{equation*}
\mbox{Var}=\frac{1}{S}\sum_s
\sum_j\left(\hat\gamma_s(t_j)-\frac{1}{S}\sum_s\hat\gamma_s(t_j)\right)^2.
\end{equation*}
In the simulation, we set the total number of replication $S=100$. For the first three cutoff levels,
Table <ref> gives us a summary of the different configurations of parameters for the six methods.
Although the simulation design is in favour of fPC based methods,
for the small number of cutoff levels, the PLS, PQR and PCQR methods perform better regarding the performance measurements of $\mbox{Bias}^2$ and MISE.
Due to the natures of sensitivity against skewness of errors, Figure
<ref> shows that the performances of PLS and fPC are much worse
in general compared with the other four methods when the errors follow
the Cauchy distribution.
On the other hand, when the Gaussian errors are employed, for the lower cutoff levels, the PLS, PQR and PCQR methods are very similar.
And when the number of cutoff levels becomes larger, the PCQR performs slightly better than the PLS while PQR performs much better than the PCQR.
fPC based methods are similar to each other crossing all cutoff levels.
The averaged mean squared error (MSE) of the responses is another
prediction performance criteria we consider. Figure <ref>
indicates that the prediction errors are much lower for PLS, PQR, PCQR
methods compared with those for fPC based methods, due to the fact
that fPC based methods are only data driven while the other three
methods are both data and response adapted. For the Gaussian errors,
although with regard to the functional coefficients estimation PQR is
better than both PLS and PCQR methods, taking into consider of the
prediction errors, the PLS and PCQR methods perform better than PQR.
For the Cauchy errors, PQR performs the best out of the PLS, PQR, PCQR
methods which indicates that PQR is more robust against the skewness
of error distribution.
Simulation II. In this simulation, we take the $Z_i(t)$s from a real data study, and generate the $Y_i$s according to the linear model of
$$Y = \int_{0}^{1} \gamma(t) Z(t) dt + \varepsilon,$$
where the error $\varepsilon$ is taken as Gaussian and Cauchy.
The centres of errors are taken as zero while the scales are taken as the empirical standard deviation of the true responses multiplied by $\sqrt{5}$.
The $Z_i$s are taken from a benchmark Phoneme dataset, which can be downloaded from http://statweb.stanford.edu
/ tibs/ElemStatLearn/.
In these data, $Z_i(t)$ represents log-periodgrams constructed from recordings of different phonemes.
The periodgrams are available at $256$ equally-spaced frequencies $t$,
which for simplicity we denote by $0=t_1<t_1<\ldots<t_{m}=1$, where $m=256$ <cit.>.
We used $n=1717$ data curves $Z_i(t)$ that correspond to the phonemes “aa” as in “dark” and “ao” as in “water”.
This example can also be found in <cit.>.
Computing the first $J=20$ empirical fPCbasis functions $\hat{\phi}_1(t),\ldots,\hat{\phi}_{J}(t)$, we consider four different curves $\gamma(t)$ by taking
$\gamma(t)=\sum_{j=1}^J a_j \hat{\phi}_i(t)$ for four different sequence of $a_j$s: (i) $a_j=(-1)^j \cdot \mathbf{1}\{0\leq j\leq 5\}$;
(ii) $a_j=(-1)^j \cdot \mathbf{1}\{6\leq j\leq 10\}$; (iii) $a_j=(-1)^j \cdot \mathbf{1}\{11\leq j\leq 15\}$; (iv) $a_j=(-1)^j \cdot \mathbf{1}\{16\leq j\leq 20\}$.
Going through case (i) to (iv), the models become less favorable for fPC, while we will see the PLS, PQR and PCQR methods manage to capture the interaction
between $Z$ and $Y$ using only a few terms.
We take $\tau = 0.5$ and compare the six methods by looking at the $\mbox{MISE}$, $\mbox{Bias}^2$ and $\mbox{Var}$. As shown in Figure <ref>,
from case (i) to (iv), PLS, PQR and PCQR methods perform better and better compared with the fPC based methods.
In fact, all the PLS, PQR and PCQR methods manage to obtain a very good fitting using only a much small number of components no matter how the errors are distributed.
This shows great superiority of our proposed methods when the functional coefficients do not lie on the fPC space.
Figure <ref> displays the prediction errors MSE when the errors follow Gaussian (left panels) and Cauchy (right panels) distributions.
The PLS, PQR and PCQR methods predict better in general compared with fPC based methods.
Except for the PLS of Cauchy errors, the MSEs of PQR and PCQR methods decrease immediately with the increase of cutoff levels,
while the fPC based methods performed differently under each case.
From case (i) to (iv), the MSEs of fPC based methods begin to drop significantly after a larger and larger cutoff level.
And for the same cutoff levels, the differences of the prediction errors between the PLS, PQR and PCQR methods and the fPC based methods become more and more signifiant from case (i) to (iv).
One interesting phenomenon here is that although the PCQR method outperforms the PQR method when the errors are Gaussian distributed,
the PQR method regains its superiority when the errors are Cauchy distributed.
Compared with what we have observed from simulation I,
it may indicate that the PCQR method is only a slightly less favourable alternative to the PLS method when the errors are symmetric.
On the other hand, when the errors are distributed in an extremely skewed manner, the PCQR method could not out-perform the PQR method.
That is exactly the same situation as the fPC based methods when the CQR method is implemented.
§ REAL DATA ANALYSIS
Real Data Analysis I: ADHD-200 fMRI Data. We apply our proposed
method to a dataset on attention deficit hyperactivity disorder (ADHD)
from the ADHD-200 Sample Initiative Project. ADHD is the most commonly
diagnosed behavioral disorder of childhood, and can continue through
adolescence and adulthood. The symptoms include lack of attention,
hyperactivity, and impulsive behavior. The dataset we use is the
filtered preprocessed resting state data from New York University
(NYU) Child Study Center using the Anatomical Automatic Labeling (AAL)
<cit.> atlas. AAL contains 116 Regions of
Interests (ROI) fractionated into functional space using
nearest-neighbor interpolation. After cleaning the raw data that
failed in quality control or has missing data, we include 120
individuals in the analysis.
The response of interest is the ADHD index, Conners' parent rating scale-revised, long version (CPRS-LV), a continuous behavior score reflecting the severity of the ADHD disease. In the AAL atalas data, the mean of the grey scale in each region is calculated for 172 equally spaced time points. We choose six parts of the brain which contain at least 4 ROIs, namely cerebelum, temporal, vermis, parietal, occipital, and frontal. The six functional predictors for each candidate part are computed by taking the average grey scale of the ROIs corresponding to each part, see Figure <ref> (Left) for some selected subjects at cerebellum. The scalar covariates of primary interest include gender (female/male), age, handedness (continuous between -1 and 1, where -1 denotes totally left-handed and 1 denotes totally right-handed), diagnosis status (categorical with 3 levels: ADHD-combined, ADHD-inattentative and Control as baseline), medication status (yes/no), Verbal IQ, Performance IQ and Full4 IQ. We build model to predict ADHD index adjusting these 9 scalar covariates (coded with dummy variables) using each of the six functional predictors. We consider the models for each individual functional covariates adjusting for the 9 scalar covariates.
Figure <ref> displays the changes of MSEs for all six methods,
with the increase of the number of cutoff level $L$ for different brain regions.
Here the quantile level $\tau$ is chosen to be fixed as $0.5$. As shown in the figure, PLS, PQR and PCQR methods perform much better than the fPC based methods while PCQR shows a significant superiority. In general, for each method only a few basis functions is capable of predicting the response well and additional basis functions
do not decrease MSE much. This is more obvious for PLS, PQR and PCQR methods as they consider information from the response while choose basis functions.
Real Data Analysis II: ADNI DTI Data. We use our model methods to analyze a real DTI data set with $n = 214$ subjects collected
from NIH Alzheimer's Disease Neuroimaging Initiative (ADNI) study. Data used in the preparation of this article were obtained from the Alzheimer's Disease Neuroimaging
Initiative (ADNI) database (adni.loni.ucla.edu). The ADNI was launched in 2003 by the National Institute on
Aging (NIA), the National Institute of Biomedical Imaging and Bioengineering (NIBIB), the Food and Drug
Administration (FDA), private pharmaceutical companies and non-profit organizations, as a $60 million,
5-year public private partnership. The primary goal of ADNI has been to test whether serial magnetic
resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and
neuropsychological assessment can be combined to measure the progression of mild cognitive impairment
(MCI) and early Alzheimer's disease (AD). Determination of sensitive and specific markers of very early
AD progression is intended to aid researchers and clinicians to develop new treatments and monitor their
effectiveness, as well as lessen the time and cost of clinical trials. The Principal Investigator of this initiative
is Michael W. Weiner, MD, VA Medical Center and University of California, San Francisco. ADNI is the
result of efforts of many coinvestigators from a broad range of academic institutions and private corporations,
and subjects have been recruited from over 50 sites across the U.S. and Canada. The initial goal of ADNI was
to recruit 800 subjects but ADNI has been followed by ADNI-GO and ADNI-2. To date these three protocols
have recruited over 1500 adults, ages 55 to 90, to participate in the research, consisting of cognitively normal
older individuals, people with early or late MCI, and people with early AD. The follow up duration of
each group is specified in the protocols for ADNI-1, ADNI-2 and ADNI-GO. Subjects originally recruited
for ADNI-1 and ADNI-GO had the option to be followed in ADNI-2. For up-to-date information, see www.adni-info.org.
The significance level is an ongoing public-private partnership to test whether genetic, structural and functional
neuroimaging, and clinical data can be integrated to assess the progression of mild cognitive
impairment (MCI) and early Alzheimer's disease (AD). The structural brain MRI data and
corresponding clinical and genetic data from baseline and follow-up were downloaded from
the ADNI publicly available database (https://ida/loni/usc/edu).
The DTI data were processed by two key steps including a weighted least squares estimation
method Basser et al. <cit.>; Zhu et al. <cit.> to construct the diffusion tensors
and a FSL TBSS pipeline Smith et al. <cit.> to register DTIs from multiple subjects to
create a mean image and a mean skeleton. Speciffically, maps of fractional anisotropy (FA)
were computed for all subjects from the DTI after eddy current correction and automatic
brain extraction using FMRIB software library. FA maps were then fed into the TBSS tool,
which is also part of the FSL. In the TBSS analysis, the FA data of all the subjects were
aligned into a common space by non-linear registration and the mean FA image were created
and thinned to obtain a mean FA skeleton, which represents the centers of all WM tracts
common to the group. Subsequently, each subjects aligned FA data were projected onto this
skeleton. We focus on the midsagittal corpus callosum skeleton and associated FA curves from all
subjects, see Figure <ref> (Right) for some selected subjects. The corpus callosum (CC) is the largest fiber
tract in the human brain and is a topographically organized structure, see Figure <ref> (Left). It is responsible for much of
the communication between the two hemispheres and connects homologous areas in the two
cerebral hemispheres. It is important in the transfer of visual, motoric, somatosensory, and auditory information.
We are interested in predicting mini-mental state examination (MMSE) scores, one of the most widely used screening tests, which are used to provide brief, objective measures of cognitive functioning for almost fifty years.
The MMSE scores has been seen as a reliable and valid clinical measure quantitatively assessing the severity of cognitive impairment.
It was believed that the MMSE scores to be affected by demographic features
such as age, education and cultural background, but not gender <cit.>. After quality control and excluding the missing data,
we include 200 subjects from the total 217 subjects.
The functional covariate is fractional anisotropy (FA) values along the corpus callosum (CC) fiber tract with 83 equally spaced grid points,
which can be treated as a function of arc-length. The scale covariates are the
gender variable (coded by a dummy variable indicating for male), the age of the subject (years), the education level (years), an indicator for Alzheimer's disease
(AD) status (19.6%) and an indicator for mild cognitive impairment (MCI) status (55.1%), and genotypes for apolipoprotein E $\epsilon$-$4$ (coded by three indicator variables for four levels).
The MSEs are shown in figure <ref>. In general, PLS, PQR and PCQR methods present consistently better than fPC based methods while
PCQR outperforms PQR and PCQR methods.
The phenomenon has been observed from the previous read data analysis, which indicates that for brain imaging data
PCQR method has a improved prediction accuracy compared with PQR and PCQR methods. With the number of functional basis increases,
the MSEs do not decreases much for fPC based methods while constantly decrease for PLS, PQR and PCQR methods. This indicates
that the fPC basis is not suitable to do prediction though they may account a large portion of the variations of functional covariates. The PLS, PQR and PCQR methods
is capable of explaining a large percentage variation of the response and reducing the MSEs by proving appropriate basis functions. Our proposed methods show
great superiority to the fPC based methods and the PLS methods and provide a powerful tool to do prediction in practice.
§ DISCUSSION
In this paper, we first define the concept partial quantile covariance (PQC) to measure the contribution of one covariate to the response.
We then propose the partial functional linear quantile regression method to use partial quantile regression (PQR) to extract PQR basis to
effectively predict the response. This is motivated by the success of the partial least square (PLS) basis in functional linear regression model.
The key idea is to use both information from the functional covariates and the response and therefore both PQR basis and PLS basis can be treated
as supervised learning while fPC based methods are semi-supervised learning as they only use information from the functional covariates. The
algorithm SIMPQR we developed is analogue to that of SIMPLS. We extend PQC to partial composite quartile covariance (PCQC) and
propose the PCQR basis and its SIMPCQR algorithm under the homoscedasticity condition.
The simulations show that PLS, PQR and PCQR in general perform better than the fPC based methods. However, PQR method is more robust against skewness of error distribution while the PLS and PCQR methods act similarly to each other and perform better than PQR method when the error distribution is symmetric.
This advantage from PQR method can be explained by the general nature of quantile method which obtains its robustness by sacrificing certain efficiency.
By assuming homoscedasticity, the PCQR method acts similarly to the PLS method when the error distributions are symmetric
but retains its robustness when the error distributions are extremely skewed.
Our proposed methods, PQR and PCQR methods, significantly outperform other methods, especially those fPC based methods in both ADHD-200 fMRI
data analysis and ADNI DTI data analysis. In ADHD-200 fMRI data analysis, our methods are capable of reducing much more MSEs by using only a few basis while
fPC based method are not even by adding more basis. In
ADNI DTI data analysis, both PQR and PCQR methods reduce significant amount MSEs with more and more basis. On the other hand, fPC based
methods perform poor even with more basis. Overall in the two neuroimaging data analysis, PCQR
performs slightly better than PQR though.
The consistency of the PLS methods was proved by Delaigle and Hall <cit.> where they found an equivalent space with explicit expressed basis functions
to the PLS basis space. For PQR and PCQR methods, it is difficult to find such equivalent space and therefore their consistency may not be easy to show. The difficulty
of the problem lies on the iterative nature of PQR and PCQR methods where basis is sequentially extracted. One way to overcome that is to find preselected number of basis
simultaneously <cit.>. Another direction is to impose certain structure
on the selected basis, for example, sparsity and smoothness in PLS methods <cit.>. This can be done for simultaneous basis selection as well <cit.>.
In both simulation studies and real data analysis, only univariate functional covariate case is considered.
However, the extension of PQR and PCQR methods to multivariate functional covariates is straightforward.
The computation becomes more complex and intensive due to the iterative basis extraction nature.
Such complexity is expected to be significantly reduced
by applying simultaneous basis selection or imposing certain structure on the selected basis. Further details are out of the scope of this manuscript
and will be pursuit in the future research.
§ ACKNOWLEDGMENTS
Dengdeng Yu and Dr. Linglong Kong's research were supported by the startup from the University of Alberta and grants from the
Natural Sciences and Engineering Research Council of Canada. Dr. Ivan Mizera's research was supported by grants from the
Natural Sciences and Engineering Research Council of Canada. Dr. Linglong Kong also wants to thank the support of the Program on Low-dimensional
Structure in High-dimensional Systems (LDHD) at the Statistical and Applied Mathematical Sciences Institute (SAMSI) during his visit in 2014.
Part of data collection and sharing for this project was funded by the Alzheimer's Disease
Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904). ADNI
is funded by the National Institute on Aging, the National Institute of Biomedical Imaging
and Bioengineering, and through generous contributions from the following: Alzheimer's
Association; Alzheimer's Drug Discovery Foundation; BioClinica, Inc.; Biogen Idec Inc.;
Bristol-Myers Squibb Company; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company;
F. Hoffmann-La Roche Ltd and its affilated company Genentech, Inc.; GE Healthcare;
Innogenetics, N.V.; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development,
LLC.; Johnson & Johnson Pharmaceutical Research & Development LLC.; Medpace,
Inc.; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Novartis Pharmaceuticals
Corporation; Pfizer Inc.; Piramal Imaging; Servier; Synarc Inc.; and Takeda
Pharmaceutical Company. The Canadian Institutes of Health Research is providing funds
to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the
Foundation for the National Institutes of Health (www.fnih.org). The grantee organization
is the Northern California Institute for Research and Education, and the study is coordinated by the
Alzheimer's Disease Cooperative Study at the University of California, San
Diego. ADNI data are disseminated by the Laboratory for Neuro Imaging at the University
of California, Los Angeles. This research was also supported by NIH grants P30 AG010129
and K01 AG030514.
§ REFERENCES
3||c|| 3|c|fPC 3|c|QRfPC 3|c|CQRfPC
Error L $n$ $\rm{Bias}^2$ Var MISE $\rm{Bias}^2$ Var MISE $\rm{Bias}^2$ Var MISE
9*Gaussian 3*1 100 3.63 0.07* 3.70 3.63 0.09 3.72 3.63 0.07* 3.71
200 3.64 0.03* 3.67 3.63 0.04 3.68 3.63 0.04 3.67
500 3.69 0.02* 3.70 3.68 0.02* 3.71 3.69 0.02* 3.70
3*2 100 0.78 0.36* 1.14* 0.77 0.39 1.16 0.78 0.36* 1.14*
200 0.86 0.17* 1.03 0.86 0.18 1.04 0.86 0.17* 1.03
500 0.86 0.09* 0.95 0.86 0.10 0.96 0.86 0.09* 0.95
3*3 100 0.32 0.34* 0.67* 0.33 0.38 0.70 0.32 0.35 0.67*
200 0.28 0.19* 0.47* 0.28 0.22 0.50 0.28 0.20 0.48
500 0.29 0.08* 0.38 0.29 0.09 0.38 0.29 0.08* 0.38
9*Cauchy 3*1 100 7.32 $>$100 $>$100 3.65 0.12 3.76 3.66 0.09* 3.75
200 4.82 54.42 59.24 3.63 0.10 3.73 3.64 0.08* 3.72
500 43.78 $>$100 $>$100 3.71 0.03 3.74 3.72 0.02* 3.74
3*2 100 7.56 $>$100 $>$100 0.77 0.40* 1.17* 0.76 0.42 1.18
200 2.14 $>$100 $>$100 0.78 0.21* 0.99 0.77 0.22 1.00
500 $>$100 $>$100 $>$100 0.81 0.10* 0.91 0.81 0.10* 0.91
3*3 100 6.03 $>$100 $>$100 0.28 0.48* 0.76* 0.27 0.53 0.80
200 3.67 $>$100 $>$100 0.31 0.24* 0.55* 0.31 0.27 0.58
500 $>$100 $>$100 $>$100 0.31 0.11* 0.41* 0.31 0.12 0.42
3||c|| 3|c|PLS 3|c|PQR 3|c|PCQR
Error L $n$ $\rm{Bias}^2$ Var MISE $\rm{Bias}^2$ Var MISE $\rm{Bias}^2$ Var MISE
9*Gaussian 3*1 100 0.54 0.82 1.36* 0.50* 0.91 1.41 0.52 0.84 1.36*
200 0.63 0.20 0.83 0.57* 0.26 0.83 0.60 0.21 0.81*
500 0.59 0.07 0.66 0.52* 0.10 0.62* 0.56 0.08 0.64
3*2 100 0.11* 1.07 1.18 0.15 1.18 1.33 0.12 1.07 1.19
200 0.12* 0.29 0.41* 0.16 0.36 0.52 0.13 0.29 0.43
500 0.11* 0.10 0.21* 0.14 0.13 0.27 0.12 0.10 0.23
3*3 100 0.08 2.58 2.66 0.07 2.28 2.36 0.06* 2.65 2.71
200 0.04* 0.87 0.91 0.04* 1.10 1.13 0.04* 0.96 0.99
500 0.02* 0.26 0.28* 0.02* 0.43 0.45 0.02* 0.30 0.32
9*Cauchy 3*1 100 71.08 $>$100 $>$100 0.49 1.23 1.72* 0.48* 1.39 1.87
200 41.24 $>$100 $>$100 0.48 0.43 0.91* 0.47* 0.47 0.94
500 $>$100 $>$100 $>$100 0.46* 0.16 0.62* 0.48 0.17 0.64
3*2 100 $>$100 $>$100 $>$100 0.16 1.96 2.12 0.12* 3.01 3.13
200 $>$100 $>$100 $>$100 0.15 0.67 0.82* 0.12* 0.98 1.09
500 $>$100 $>$100 $>$100 0.14 0.23 0.37* 0.11* 0.29 0.41
3*3 100 $>$100 $>$100 $>$100 0.13* 5.84 5.97 0.20 14.28 14.48
200 $>$100 $>$100 $>$100 0.06* 2.59 2.65 0.11 5.45 5.56
500 $>$100 $>$100 $>$100 0.02* 1.01 1.02 0.04 1.67 1.71
Simulation I: $*$ flags the minimum values of the six methods in each measurement category of $\rm{Bias}^2$, Var and MISE.
Simulation I: the MISEs with Gaussian (left) and Cauchy (right) errors, sample size $n=100$, $200$, and $500$ from up to down.
Simulation I: the averaged MSEs with Gaussian (left) and Cauchy (right) errors, sample size $n=100$, $200$, and $500$ from up to down.
Simulation II: the MISEs with Gaussian (left) and Cauchy (right) errors, case I, II, II, and IV from up to down.
Simulation II: the averaged MSEs with Gaussian (left) and Cauchy (right) errors, case I, II, II, and IV from up to down.
Real Data Analysis I: ADHD-200 fMRI data, From up to down there are cerebelum, vermis, and occipital on the left panels and
temporal, parietal, and frontal on the right panel.
Real Data Analysis II: (Left) The midsagittal corpus callosum (CC) skeleton overlaid with fractional anisotropy (FA) from one randomly selected subject and (Right) the MSE of mini-mental state examination (MMSE) at different cut-off levels.
|
1511.00611
|
[email protected] Department of Physics,
University of Trieste, Trieste, Italy.
[email protected], [email protected]
Pailan College of Management and Technology, Bengal
Pailan Park, Kolkata-700 104, India.
Eurasian International Center for Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan.
Abstract: In this paper, we study the Power Law Entropy Corrected Holographic Dark Energy (PLECHDE) model in the framework of a non-flat Universe and of Hořava-Lifshitz cosmology with infrared cut-off given by recently proposed Granda-Oliveros cut-off, which contains one term proportional to the Hubble parameter squared $H^2$ and one term proportional to the first time derivative of the Hubble parameter $\dot{H}$. Moreover, this cut-off is characterized by two constant parameters, $\alpha$ and $\beta$. For the two cases corresponding to non-interacting and interacting DE and Dark Matter (DM), we derive the evolutionary form of the energy density of DE $\Omega_D'$, the Equation of State (EoS) parameter of DE $\omega_D$ and the deceleration parameter $q$. Using the parametrization of the EoS parameter $\omega_D\left(z\right)=\omega_0+\omega_1 z$, we obtain the expressions of the two parameters $\omega_0$ and $\omega_1$. We also study the statefinder parameters $\left\{ r,s \right\}$, the snap and lerk cosmographic parameters and the squared speed of the sound $v_s^2$. We also calculate the values of the quantities we study for different values of the running parameter $\lambda$ and for different set of values of $\alpha$ and $\beta$.
Keywords: Dark Energy; Hořava-Lifshitz; Granda-Oliveros cut-off.
§ INTRODUCTION
Recent cosmological and astrophysical data obtained with observations done thanks to the Supernova Cosmology Project, the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck satellites, the Sloan Digital Sky Survey (SDSS) and X-ray experiments <cit.> give clear indications that the observable present day Universe is experiencing a phase of expansion with accelerated rate, which is practically the expansion with accelerated rate which the Universe undergoes, with the first happened during the inflationary period. The present day cosmic acceleration is one of the biggest challenges in the understanding of the standard models of gravity and particle physics.
Three main different classes of models have been suggested and well studied till now with the aim to give a proper explanation to the accelerated expansion of the present day observable Universe:
* the Cosmological Constant (CC) $\Lambda_{CC}$ model;
* Dark Energy (DE) models;
* Theories of Modified Gravity models.
The first and also the simplest candidate introduced with the aim to explain the present day observed accelerated expansion of the Universe is the Cosmological Constant $\Lambda_{CC}$, which can be considered as an extra term added to Einstein's equations. One of the main features of the Cosmological Constant $\Lambda_{CC}$ is that it has an Equation of State (EoS) parameter $\omega$ exactly equal to $ -1$, i.e. $\omega_{\Lambda_{CC}} \equiv -1$. According to what we know thanks to the Quantum Field Theory (QFT), a cut-off at the Planck (or at the electro-weak scale) leads to the production of a Cosmological Constant $\Lambda_{CC}$ which is of the order of $10^{123}$ (or $10^{55}$), respectively, times bigger than the value we are able to observe. The fact that we still do not have a fundamental symmetry which is able to put the precise value of the Cosmological Constant $\Lambda_{CC}$ to exactly zero (i.e., $\Lambda_{CC}\equiv 0$) or, instead, to a very small value (i.e., $\Lambda_{CC} \approx 0$) produces the so-called Cosmological Constant problem, also known as fine tuning problem.
Moreover, it is also well-known that the Cosmological Constat $\Lambda_{CC}$ model is affected by another problem, which is the Cosmic Coincidence problem <cit.>. The Cosmic Coincidence problem states that the DM and the vacuum energy are almost equal at the present epoch of the Universe even if they had an independent evolution and they had an evolution starting from different mass scales. Many proposals have been suggested till now with the purpose and the hope to obtain an explanation to the Cosmic Coincidence problem <cit.>.
The second class of models which are suggested and widely studied with the aim to give a plausible explanation to the present day accelerated expansion of the Universe considers Dark Energy (DE) models.
The observational evidences of the cosmic accelerated expansion imply that, if, on cosmological scale, the theory of Einstein's General Relativity (GR) is valid, we must have that the present day observable Universe must have as dominant component an unknown missing energy component which has some particular features, in particular: 1) its pressure $p$ must be sufficiently negative if it wants to be able to produce the rate of accelerated expansion of the Universe we are able to observe and 2) it must not be clustered on cosmological scales (i.e. on large scale length). The present day observed cosmic accelerated expansion of the Universe can be described, in relativistic cosmology, introducing a perfect fluid with energy density $\rho $ and pressure $p$ satisfying the following condition: $\rho + 3p < 0$ (which implies that the pressure $p$ must be negative in order the condition is satisfied). This kind of fluid with a sufficient negative pressure $p$ in order to satisfy the condition $\rho+3p<0$ is referred as Dark Energy (DE). The fact that the relation $\rho + 3p < 0$ must be satisfied leads to the fact that the EoS parameter $\omega $ (defined as the ratio of the pressure $p$ and the energy density $\rho$) satisfies the following condition: $\omega \equiv \frac{p}{\rho} <-1/3$. Instead, from an observational point of view, it is still a challenging task to constrain its exact value. The fundamental theory which can explain the microscopic physics of DE is still not known up to now, for this reason scientists continue to reconstruct and suggest different possible models which are mainly based on its macroscopic behavior.
Furthermore, recent cosmological experiments and observations have clearly indicated that the largest part of the total energy density $\rho_{tot}$ of the present Universe is contained in the two Dark Sectors <cit.>, i.e. DE and DM, which represent, respectively, the 68.3$\%$ and the 26.8$\%$ of the total energy density of the present day observable Universe. We also know that the Baryonic Matter (BM) we are able to observe with our scientific instruments contributes only for approximately the $ 4.9 \%$ of the total energy density $\rho_{tot}$ of the present day Universe. Moreover, we have that the contribution produced by the radiation term to the total cosmic energy density $\rho_{tot}$ can be safely considered practically negligible, i.e we have that $\rho_{rad} \approx 0$.
Other different candidates introduced and suggested for the DE problem are given by the dynamical DE scenarios with a time dependent EoS parameter $\omega$, then not anymore constant. According to analysis of the available SNe Ia observational data, it has been derived that time-varying DE models lead to a better fit compared with a model with Cosmological Constant $\Lambda_{CC}$. There are two main different categories suggested for dynamical DE scenarios: (i) scalar fields models, which include k-essence <cit.>, quintessence <cit.>, tachyon <cit.>, phantom <cit.>, dilaton <cit.> and quintom field <cit.>, (ii) interacting DE models, which include for example Chaplygin gas <cit.> and Agegraphic DE (ADE) models <cit.>.
The complete description of the DE features and nature must come from a consistent theory of Quantum Gravity (QG).
Unfortunately, we still do not have a complete and widely accepted theory of Quantum Gravity (QG) and then some approximations for this theory can be made: some examples are given by the Loop Quantum Gravity (LQG) and String Theory.
The third and last class of models proposed in order to give an explanation to the present day accelerated expansion of Universe involves extended theories of gravity, which correspond to a modification of the action of the gravitational fields. Some of the most famous and studied models of Modified Gravity are the $f\left(T\right)$ modified gravity model (with $T$ being the torsion scalar), braneworld models, the $f \left(G\right)$ modified gravity model (with $G$ being the Gauss-Bonnet invariant defined as $G=R^2-4R_{\mu \nu}R^{\mu \nu} + R_{\mu \nu \lambda \sigma}R^{\mu \nu \lambda \sigma}$, $R_{\mu \nu}$ being the Ricci curvature tensor and $R_{\mu \nu \lambda \sigma}$ being the Riemann curvature tensor), the $f \left(R\right)$ modified gravity model (with $R$ being the Ricci scalar curvature), the Dvali-Gabadadze-Porrati (DGP) model, the $f \left(R,T\right)$ modified gravity model, the Dirac-Born-Infeld (DBI) model and the Brans-Dicke model <cit.>.
Using the holographic principle which was recently introduced by Fischler $\&$ Susskind <cit.>, a model dubbed as Holographic DE (HDE) model has been recently proposed in the paper of Li <cit.>. The HDE model is one of the most famous and studied candidate of DE <cit.>.
It is well-known that the holographic principle assumes a fundamental role in both black hole and string
theories. It was recently demonstrated in the work of Cohen et al. <cit.> that, in the framework of the QFT, the UV cut-off, which is indicated with $\Lambda_{UV}$, is related to the IR cut-off, which is given by $L$, due to the limitations produced by the formation of a black hole. If the vacuum energy $\rho_D$ density produced by the UV cut-off is given by the relation $\rho_D = \Lambda_{UV}^4$, then we have that the total energy density of a given size $L$ must be less or at least equal to the mass corresponding to the system-size black hole, i.e. we must have that:
\begin{eqnarray}
E_D \leq E_{BH} , \label{1}
\end{eqnarray}
which implies that:
\begin{eqnarray}
L^3 \rho_D \leq M_p^2 L, \label{1cinzia}
\end{eqnarray}
where $M_p = \left( 8\pi G_N \right)^{-1/2} \approx 10^{18}\, GeV$ represents the reduced Planck mass and $G_N=6.67\cdot10^{-11}Nm^2kg^{-2}$ represents the Newton's gravitational constant. If the largest possible cut-off $L$ of the system is that one which is able to saturate the inequality given in Eq. (<ref>), we derive the following expression for the energy density $\rho_D$ of the HDE model:
\begin{eqnarray}
\rho_D = 3n^2 M_p^2 L^{-2}, \label{2}
\end{eqnarray}
where $n$ represents a dimensionless constant parameter. It has been obtained that, in the case of a Universe that is not flat (i.e. for a value of the curvature parameter which is different from zero) the value of such constant is given by $n = 0.815^{+0.179}_{-0.139}$ while for a flat Universe (i.e. when the curvature parameter is equal to zero), we have that the value of $n$ is given by $n=0.818_{-0.097}^{+0.113}$ <cit.>.
The expression of the energy density $\rho$ of the HDE model can be also obtain using a different approach <cit.>. It must be here underlined that the black hole entropy $S$ has an important role in the derivation of the HDE energy density $\rho_D$. In fact, we know that the derivation of the HDE energy density strongly depends on the entropy-area relation given, in Einstein's gravity, by the relation $S \approx A \approx L^2$ (where $A$ gives the area of the black hole horizon). According to the laws of the thermodynamics of black holes <cit.>, a maximum value of the entropy in a box with a dimension of $L$ (which is also referred as Bekenstein-Hawking entropy bound), is given by the relation $S_{BH} \approx M_p^2 L^2$, which goes as the area $A$ of the box (given approximatively by the expression $A \approx L^2$) rather than the volume $V$ of the box (which is given by $V \approx L^3$). Moreover, for macroscopic systems having some self-gravitation effects which cannot be ignored, we have that the expression of the Bekenstein entropy bound (which is indicated with $S_B$) can be obtained multiplying the energy $E$, given by the relation $E \approx \rho_DL^3$, and the linear size $L$ of the system. If we impose that the Bekenstein entropy bound must be smaller than the Bekenstein-Hawking entropy (i.e., if we impose that $S_B \leq S_{BH}$, which implies that $E\cdot L \leq M_p^2 L^2$), we obtain the same result obtained from energy bound arguments, i.e. we obtain that $\rho_D \leq M_p^2L^{-2}$.
Using the holographic principle, Cohen et al. <cit.> recently proposed that the vacuum energy density must be proportional to the Hubble parameter $H$. In this particular model, both the fine-tuning and coincidence problems can be solved, but it is still not possible to give a reasonable explanation to the present day cosmic accelerated expansion of the Universe since the effective Equation of State (EoS) parameter $\omega_{eff}$ for such vacuum energy is equal to zero, then it is different from what it is requested for the HDE model. In a recent paper, Li <cit.> suggested that the future event horizon of the Universe can be used as possible IR cut-off. This DE model not only has a reasonable value for the DE energy density but it also leads to an accelerated solution for the cosmological expansion.
Jamil et al. <cit.> studied the EoS parameter $\omega_D$ of the HDE model choosing a Newton's gravitational constant $G_N$ which is not constant but it is time dependent, i.e. we have $G_N\left(t\right)$; furthermore, they obtained that the EoS parameter $\omega_D$ can be significantly modified when the low-redshift $z$ limit is considered.
Chen et al. <cit.> studied the HDE model in order to obtain an inflationary epoch in the early evolutionary stages of our Universe. The HDE model was recently considered in other works with different IR cut-offs, for example the Hubble horizon, the particle horizon and the future event horizon <cit.>. Moreover, correspondences between some scalar field models and the HDE model have been recently proposed <cit.>, while in other works, the HDE model was accurately studied in different modified gravity theories, like for example scalar-tensor gravity, $f\left(R\right)$, DGP model, braneworld and Brans-Dicke cosmology <cit.>.
Different HDE models have also been constrained and tested by using different astronomical and cosmological observations <cit.> and also thanks to the anthropic principle <cit.>. It is also known that the HDE model fits well cosmological data obtained using the data obtained from observations of SNeIa and CMB radiation anisotropies <cit.>.
The definition of the entropy-area relation can be modified considering quantum effects which are motivated from the Loop Quantum Gravity (LQG). The relation entropy-area $S\left(A\right)$ has an interesting modification (correction), i.e. the power-law correction <cit.> which arises in dealing with the entanglement of quantum fields in and out the horizon.
The power-law corrected entropy-area relation $S\left( A \right)$ has the following specific form <cit.>:
\begin{eqnarray}
S\left( A \right)=c_0 \left( \frac{A}{a_1^2} \right)\left[ 1+c_1f\left( A \right) \right],\label{powerlawentropyold}
\end{eqnarray}
where the term $ f\left( A \right) $ is given by the following power-law relation:
\begin{eqnarray}
f\left( A \right) = \left( \frac{A}{a_1^2} \right)^{-\nu},\label{powerlawentropyold}
\end{eqnarray}
with $c_0$ and $c_1$ indicating two constant parameters, $a_1$ being the UV cut-off at the horizon and $\nu$ being a fractional power which depends on the amount of mixing of ground and excited states. For a large horizon area (i.e. for $A>>a_1^2$), the contribution given by the term $f \left(A\right)$ to the entropy $S\left( A \right)$ can be considered practically negligible and, therefore, the mixed state entanglement entropy asymptotically approaches the ground state (Bekenstein-Hawking) entropy.
Another useful way to write the expression of the entropy area relation $S\left( A \right)$ for the power-law corrected entropy is given by the following relation:
\begin{eqnarray}
S\left( A \right)= \frac{A}{4G}\left( 1-K_{\alpha}A^{1-\alpha /2} \right), \label{1--}
\end{eqnarray}
with $\alpha$ representing a dimensionless constant parameter and the term $K_{\alpha}$ is a constant which is defined as follows:
\begin{eqnarray}
K_{\alpha} = \frac{\alpha \left( 4\pi \right)^{\alpha /2 -1}}{\left( 4-\alpha \right)r_c^{2-\alpha}}, \label{2}
\end{eqnarray}
where the term $r_c$ indicates the cross-over scale. Moreover, we have that the quantity $A=4\pi R_h^2$ gives the area of the horizon (with the term $R_h$ indicating the radius of the horizon). The second term present in Eq. (<ref>) gives the power-law correction to the entropy-area law. In order the entropy is a well-defined quantity, we need to have that the parameter $\alpha$ is positive defined, i.e. we must have that the condition $\alpha>0$ must be satisfied. Motivated by the relation defined in Eq. (<ref>), a new version of HDE (also known with the name of Power-Law Entropy-Corrected HDE (PLECHDE) model) was recently introduced as follows:
\begin{eqnarray}
\rho_D = 3n^2M_p^2L^{-2} - \varepsilon M_p^2 L^{-\delta}, \label{lgo2}
\end{eqnarray}
with $\varepsilon$ being a positive dimensionless parameter and $\delta$ begin a positive exponent.
In the limiting case of $ \varepsilon =0$ (or, equivalently, for $\delta \rightarrow \infty$), Eq. (<ref>) reduces to the well-known expression of the HDE energy density. The correction term present in Eq. (<ref>) is of the same order of to the first one only when $L$ assumes a very small value. Then, at the very early evolutionary phases and stages of our Universe history (i.e., when the Universe underwent the inflationary phase), the contribution of the correction term in the PLECHDE energy density can be safely considered relevant but, when the Universe became larger, the PLECHDE energy density reduced to the ordinary HDE energy density. Therefore, PLECHDE model can be also considered as a model of entropic cosmology which is able to unify the early-time inflation and late-time cosmic acceleration of the Universe.
In some recent works, Hořava <cit.> recently introduced a new theory of gravity which is renormalizable with higher spatial derivatives in four dimensions. This theory leads to the Einstein's gravity (i.e.
to GR) with non-vanishing value of the Cosmological Constant $\Lambda_{CC}$ in the infrared (IR) limit and it also have some improved behaviors and features in the ultraviolet (UV) regime. Hořava gravity can be also considered similar to a scalar field theory previously proposed by Lifshitz <cit.> in which we have that the temporal dimension $t$ has a weight equal to three if the space dimension has a weight of one. For this reason, the gravity theory proposed by Hořava is also known with the name of Hořava-Lifshitz gravity. The Hořava-Lifshitz gravity has been extended and studied in detail in some papers, like for example <cit.>, and it has been applied as a possible cosmological framework of our Universe <cit.>.
Furthermore, Hořava-Lifshitz theory is not Lorentz invariant (with the exception of the IR limit), test particles do not follow geodesics, it is non-relativistic and we also have that the speed of light $c$ diverges in the UV limit. We have four different versions of Hořava-Lifshitz theory of gravity:
(i) with projectability condition, (ii) without projectability condition, (iii) with detailed balance and (iv) without detailed balance. Having a first look, it seems that this model of Quantum Gravity (QG) has a well defined IR limit and it also reduces to General Relativity, but as it was obtained by Mukohyama <cit.>, Hořava-Lifshitz gravity behaves like General Relativity plus DM. For some relevant works on the scenario where the cosmological evolution is ruled by Hořava-Lifshitz gravity see <cit.>.
Because of these characteristics, a great effort in extending, examining and improving the physical features and properties of the theory itself have been done <cit.>. Furthermore, applications of Hořava-Lifshitz gravity as a cosmological context produces the Hořava-Lifshitz cosmology, which has some interesting features. For example, it is possible to examine the perturbation spectrum <cit.>, some particular solution subclasses <cit.>, the matter bounce <cit.>, the production of gravitational waves <cit.>, the phenomenology of DE <cit.>, the properties of black hole <cit.> and the astrophysical phenomenology <cit.>. Hořava-Lifhsitz cosmology has been recently investigated raking into account and choosing different infrared cut (IR)-offs and different approaches.
Setare $\&$ Jamil <cit.> considered the HDE model with a varying Newton's gravitational constant $G$ in the framework of Hořava-Lifshitz cosmology.
Jamil et al. <cit.> studied the behavior of the Generalized Second Law of Thermodynamics (GSLT) in the context of Hořava-Lifshitz cosmology using the dynamical apparent horizon as infrared (IR) cut-off of the system.
Karami et al. <cit.> studied the Logarithmic Entropy Corrected New Agegraphic DE (LECNADE) model in the context of Hořava-Lifshitz cosmology.
Jamil et al. <cit.> considered the NADE model in the context of the Hořava-Lifshitz cosmology.
Karami et al. <cit.> studied the Power-Law Entropy Corrected NADE (PLECNADE) model in the framework of Hořava-Lifshitz cosmology.
Pasqua et al. <cit.> studied the Power Law and the Logarithmic Ricci Dark Energy Models in the framework of Hořava-Lifshitz Cosmology.
Jawad et al. <cit.> studied the power-law solution of the new agegraphic modified $f\left(R\right)$ Hořava-Lifshitz gravity.
Chattopadhyay and Pasqua <cit.> studied the modified holographic Ricci DE (RDE) model in the framework of modified $f\left( R\right)$ Hořava-Lifshitz gravity.
Jawad et al. <cit.> obtained a holographic reconstruction of the modified $f\left( R\right)$ Hořava-Lifshitz gravity with the scale factor $a\left(t\right)$ given in the in power-law form.
Anyway, even if this extended research is available, a lot of ambiguities are still presents about the fact that Hořava-Lifshitz gravity can be considered a reliable theory and it is able to accurately describe the cosmological behavior of our Universe.
This work differs from the ones cited above and other available in literature since we are considering an IR cut-off, known as Granda-Oliveros cut-off, based on purely dimensional grounds which Granda $\&$ Oliveros recently proposed. We must also underline here that DE models with Granda-Oliveros cut-off belong to generalized Nojiri-Odintsov HDEs classes <cit.>. It also differs from the work of Pasqua $\&$ Chattopadhyay <cit.> since the Authors considered the Logarithmic Entropy-Corrected Holographic Dark Energy (LECHDE) model in the framework of Hořava-Lifshitz cosmology with Granda-Oliveros cut-off while we are considering here the power law correction to the entropy. This new cut-off contains a term proportional to the time derivative of the Hubble parameter $\dot{H}$ and one term proportional to the squared Hubble parameter $H^2$ and it is indicated with $L_{GO}$. The final expression of $L_{GO}$ is given by <cit.>:
\begin{equation}
L_{GO}=\left( \alpha H^{2}+\beta \dot{H}\right) ^{-1/2}. \label{lgo5}
\end{equation}
$\alpha $ and $\beta $ indicate two constant dimensionless parameters. In the limiting case corresponding to $ \alpha = 2$ and $\beta = 1$, we obtain that the expression of $L_{GO}$ defined in Eq. (<ref>) becomes proportional to the average radius of the Ricci scalar curvature $R^{-1/2}$ when the curvature parameter $k$ assumes the values of zero. In a recent paper, Wang $\&$ Xu <cit.> have constrained the HDE model with GO cut-off for a non-flat Universe using observational data. The best fit values of the pair $\left(\alpha, \beta \right)$ with their confidence level they found are given by $\alpha = 0.8824^{+0.2180}_{-0.1163}(1\sigma)\,^{+0.2213}_{-0.1378}(2\sigma)$ and $\beta = 0.5016^{+0.0973}_{-0.0871}(1\sigma)\,^{+0.1247}_{-0.1102}(2\sigma)$ for non flat Universe (i.e. for $k\neq 0$), while for a flat Universe (i.e. for $k=0$) they found that are $\alpha = 0.8502^{+0.0984}_{-0.0875}(1\sigma)\,^{+0.1299}_{-0.1064}(2\sigma)$ and $\beta = 0.4817^{+0.0842}_{-0.0773}(1\sigma)\,^{+0.1176}_{-0.0955}(2\sigma)$.
We decided to consider the GO scale $L_{GO}$ defined in Eq. (<ref>) as IR cut-off for some specific reasons. If the IR cut-off chosen is given by the particle horizon, the HDE model cannot produce an accelerated expansion of the Universe <cit.>. If we consider as cut-off of the system the future event horizon, the HDE model has a causality problem. The DE models which consider the GO scale depend only on local quantities, then it is possible to avoid the causality problem, moreover it is also possible to obtain the accelerated phase of the Universe.
Replacing $L$ with $L_{GO}$ in the expression of the energy density of DE $\rho_D$ given in Eq. (<ref>), we get the energy density of the PLECHDE model $\rho_D$ as follows:
\begin{eqnarray}
\rho_D=\frac{3n^2 M_p^2}{L_{GO}^2} - \frac{\varepsilon M_p^2}{L_{GO}^{\delta}}. \label{8}
\end{eqnarray}
In the following Sections we will study the main properties and features of the cosmological parameters obtained using the energy density $\rho_D$ given in Eq. (<ref>) of the PELCHDE model with GO cut-off we are studying.
This paper is organized in the following way. In Section 2, we describe the most important features of Hořava-Lifshitz cosmology. In Section 3, we study the PLECHDE model with Granda-Oliveros cut-off in the context of Hořava-Lifshitz cosmology. Moreover, we derive the evolutionary form of the energy density of DE, the Equation of State (EoS) parameter, the evolutionary form of the fractional energy density and the deceleration parameter for both non interacting and interacting Dark Sectors. In Section 4, we study the low redshift limit of the EoS parameter, which is parametrized as $\omega_D = \omega_0 + \omega_1 z$, obtaining the expressions of $\omega_0$ and $\omega_1$ for both cases corresponding to non interacting and interacting Dark Sectors. In Section 5, we study the statefinder pair $\left\{ r,s \right\}$ for the model we are studying. In Section 6, we derive and study the expressions of the snap and of the lerk cosmographic parameters for the model taken into account in this paper. In Section 7, we study the squared speed of the sound $v_s^2$ for the model considered in order to check its stability. Finally, in Section 8, we write the Conclusions of this work.
§ HOŘAVA-LIFSHITZ GRAVITY
In this Section, we introduce the main physical and cosmological characteristics of Hořava-Lifshitz gravity. These information will be useful in order to obtain the cosmological information we want to derive for the model we are studying.
Considering the projectability condition, we have that the metric in the (3+1)-dimensional Arnowitt-Deser-Misner formalism can be written as follows <cit.>:
\begin{eqnarray}
ds^2 = -N^2dt^2+g_{ij}\left(dx^i+N^i dt \right)\left(dx^j+N^jdt \right), \label{9}
\end{eqnarray}
where $t$ indicates the cosmic time while the dynamical variables $g_{ij}$, $N$ and $N^i$ indicate, respectively, the 3-dimensional metric tensor, the lapse function and the shift vector. The projectability condition leads to the fact that the lapse function $N$ is space-independent, instead the 3-dimensional metric $g_{ij}$ and the shift vector $N^i$ still depend on both space and time. Moreover, we have that the indices are raised and lowered thanks to the metric tensor $g_{ij}$. The scaling transformations of the coordinates $x^i$ and $t$ are given by the following relations:
\begin{eqnarray}
x^i &&\rightarrow l x^i, \\
t&&\rightarrow l^z t, \label{tra}
\end{eqnarray}
where the quantities $t$, $z$, $\l$ and $x^i$ represent, respectively, the temporal coordinate, the dynamical critical exponent, the scaling factor and the spatial coordinates.
In this paper, we have that $z=3$, then the scaling transformation of the temporal coordinate defined in Eq. (<ref>) can be rewritten as follows:
\begin{eqnarray}
t\rightarrow l^3 t.
\end{eqnarray}
The gravitational action of Hořava-Lifshitz cosmology, indicated with $S_g$, can be decomposed into two different parts, i.e. a kinetic part, given by $L_K$, and a potential part, given by $L_V$, and it is given by the following relation:
\begin{eqnarray}
S_g = \int dt \, d^3x \sqrt{g} N \left( L_K + L_V \right),
\end{eqnarray}
with $g$ indicating the determinant of the metric tensor $g^{\mu \nu}$.
The assumption of detailed balance <cit.> allows to reduce the number of possible terms in
the expression of the gravitational action $S_g$. Moreover, it also permit a quantum inheritance principle, because the $(D + 1)$-dimensional
theory takes the renormalization properties of the $D$-dimensional theory. Considering the detailed balance condition, the gravitational action of the Hořava-Lifshitz gravity $S_g$ is given by the following expression <cit.>:
\begin{eqnarray}
S_g &=& \displaystyle \int{d^3x \, dt \, N \sqrt{g}} \left\{\frac{2 \left(K_{ij}K^{ij}-\lambda K^2\right) }{\kappa^2} \right.\nonumber\\
&& + \left(\frac{\kappa^2}{2\omega^4}\right)C_{ij}C^{ij} -\left(\frac{\kappa^2\mu}{2\omega^2}\right)\left( \frac{\eta^{ijk}}{\sqrt{g}}\right)R_{il} \bigtriangledown_j R^l_k \nonumber\\
&& +\left(\frac{\kappa^2\mu^2}{8}\right)R_{ij}R^{ij} \nonumber\\
&& + \left. \frac{\kappa^2\mu^2}{8\left(3\lambda - 1\right)}\left[\frac{\left(1-4\lambda\right)R^2}{4}+\Lambda R - 3 \Lambda^2\right]\right\}, \label{10}
\end{eqnarray}
where the terms $C_{ij}$ and $K_{ij}$ represent, respectively, the Cotton tensor and the extrinsic curvature which are defined in the following way:
\begin{eqnarray}
C_{ij} &=& \frac{e^{ijk}}{\sqrt{g}} \bigtriangledown_k \left(R^j_i - \frac{R \delta^j_i}{4}\right), \label{12}\\
K_{ij} &=& \frac{1}{2N}\left(\dot{g}_{ij}- \bigtriangledown_iN_j - \bigtriangledown_jN_i\right). \label{11}
\end{eqnarray}
Furthermore, the quantity $\Lambda$ represents a positive dimensionless constant which is related to the cosmological constant in the infrared (IR) limit, the quantity $\eta^{ijk}$ indicates the totally antisymmetric unit tensor and $\lambda$ represents a dimensionless constant (also known as running parameter). More information about the running parameter $\lambda$ will be given later on.
The three parameters $\mu$, $\kappa$ and $\omega$ represents three constants which has mass dimension, respectively, of 1, -1 and 0.
If we want to include the matter component in a Universe ruled by Hořava-Lifshitz gravity, we have that there are two options which can be taken into account. In the first option, we include a scalar field $\phi$ with action $S_{\phi}$ which is given by the following relation <cit.>:
\begin{eqnarray}
S_m \equiv S_{\phi} &=& \int dtd^3x N\sqrt{g} \left[ \left(\frac{3\lambda -1}{4}\right)\frac{\dot{\phi}^2}{N^2} +m_1m_2 \phi \nabla ^2 \phi - \frac{1}{2}m_2^2\phi \nabla ^4 \phi \right. \nonumber \\
&&\left. + \frac{m_3^2\phi \nabla ^6 \phi}{2} - V\left( \phi \right) \right],
\end{eqnarray}
where the three quantities $m_1$, $m_2$ and $m_3$ represent three constant parameters while the term indicated with $V\left( \phi \right) $ represents the potential term. Furthermore, we have that the equation of motion for the field $\phi$ can be written as follows:
\begin{eqnarray}
\ddot{\phi} + 3H\dot{\phi}+ \left(\frac{2}{3\lambda -1}\right) \frac{dV\left( \phi \right)}{d \phi} = 0,
\end{eqnarray}
with the condition $3\lambda - 1 \neq 0$, i.e. $\lambda \neq 1/3$ in order to avoid singularities. Moreover, an overdot indicates a derivative with respect to the cosmic time $t$.
The second option we have in order to insert the matter component is obtained taking into account a hydrodynamical approximation adding a cosmological stress-energy tensor to the gravitational field equations; we must also consider the condition that the formalism of the General Relativity must be obtained when the low-energy limit is considered <cit.>. In this case, the energy density $\rho_m$ and the pressure $p_m$ of DM satisfy the following continuity equation:
\begin{eqnarray}
\dot{\rho}_m +3H\left(\rho_m+p_m\right) = 0. \label{13}
\end{eqnarray}
In this paper, we have decided to consider the hydrodynamical approximation.
Eq. (<ref>), as it is well-known, has several problems, like strong coupling problems, instability and inconsistency <cit.>. It is possible overcome these problems invoking the Vainshtein mechanism, as it was already done in the paper of Mukohyama in the case of spherical space-times <cit.> and in the paper of Wang $\&$ Wu in the cosmological setting <cit.>. These considerations were also carried out by considering the gradient expansion method <cit.>. Another possible approach which can be taken into account is given by the introduction of an extra $U\left(1\right)$ symmetry: this kind of approach was considered for the first time by Hořava $\&$ Melby-Thompson <cit.> in the limiting case corresponding to $\lambda =1$, and subsequently generalized to the case with any possible value of $\lambda$ in the paper of da Silva <cit.>. These works were also extended to the case with absence of the projectability condition <cit.>. In both cases, i.e. with and without the projectability condition, the spin-0 gravitons are eliminated because of the $U\left(1\right)$ symmetry, therefore all the problems related to them are then solved.
In the cosmological framework, we consider a FLRW metric which is recovered for the following values of $N$, $g_{ij}$ and $N^i$:
\begin{eqnarray}
N &=& 1, \label{14} \\
g_{ij}&=&a^2\left(t\right)\gamma_{ij},\label{15} \\
N^i&=&0, \label{16}
\end{eqnarray}
where $\gamma_{ij}$ is given be the following relation:
\begin{eqnarray}
\gamma_{ij}dx^idx^j=\frac{dr^2}{1-kr^2}+r^2d\Omega^2_2, \label{17}
\end{eqnarray}
with the term $d\Omega^2_2$ representing the angular part of the metric.
Taking the variation of the action $S_g$ obtained in Eq. (<ref>) with respect to the metric components $N$ and $g_{ij}$, we obtain the modified Friedmann equations in the framework of Hořava-Lifshitz cosmology as follows:
\begin{eqnarray}
H^2 &=& \left[\frac{\kappa^2}{6\left(3\lambda-1\right)}\right]\rho_m + \frac{\kappa^2}{6\left(3\lambda-1\right)}\left[\frac{3\kappa^2\mu^2k^2}{8\left(3\lambda-1\right)a^4} \right. \nonumber\\
&& + \left. \frac{3\kappa^2\mu^2\Lambda^2}{8\left(3\lambda-1\right)}\right] - \frac{\kappa^4\mu^2\Lambda k}{8\left(3\lambda - 1\right)^2a^2}\label{18} , \\
\dot{H}+\frac{3}{2}H^2 &=& - \left[\frac{\kappa^2}{4\left(3\lambda - 1\right)}\right]p_m \nonumber\\
&& - \frac{\kappa^2}{4\left(3\lambda - 1\right)}\left[\frac{\kappa^2\mu^2k^2}{8\left(3\lambda-1\right)a^4} - \frac{3\kappa^2\mu^2\Lambda^2}{8\left(3\lambda-1\right)} \right] \nonumber\\
&& - \frac{\kappa^4\mu^2\Lambda k}{16\left(3\lambda - 1\right)^2a^2}. \label{19}
\end{eqnarray}
In the limiting case of a flat Universe, i.e. for $k = 0$, the higher order derivative terms do not produce contributions to the action. Instead, for a non flat universe, i.e. for $k \neq 0$, the higher derivative terms give a relevant contribution for small volumes, i.e. for small values of $a$,
while this contribution becomes practically negligible when $a$ assumes large values (in this case we recover a good agreement with the results of General Relativity).
Considering the Friedmann equations given in Eqs. (<ref>) and (<ref>), we define the energy density $\rho_D$ and the pressure $p_D$ of DE as follows:
\begin{eqnarray}
\rho_D &\equiv& \frac{3\kappa^2\mu^2k^2}{8\left(3\lambda-1\right)a^4} + \frac{3\kappa^2\mu^2\Lambda^2}{8\left(3\lambda-1\right)}, \label{20} \\
p_D &\equiv& \frac{\kappa^2\mu^2k^2}{8\left(3\lambda-1\right)a^4} - \frac{3\kappa^2\mu^2\Lambda^2}{8\left(3\lambda-1\right)}. \label{21}
\end{eqnarray}
The first term of the right hand side of both Eqs. (<ref>) and (<ref>) (which scales as $a^{-4}$) indicates effectively the dark radiation term which is present in Hořava-Lifshitz cosmology, instead the second term (which is constant) has a cosmological constant term-like behavior.
Furthermore, Eqs. (<ref>) and (<ref>) obey the following continuity equation:
\begin{eqnarray}
\dot{\rho}_D +3H\left(\rho_D+p_D\right) = 0. \label{22}
\end{eqnarray}
We also have that Eqs. (<ref>) and (<ref>) lead to the standard Friedmann equations if we take into account the following considerations:
\begin{eqnarray}
G_{cosmo} = \frac{\kappa^2}{16\pi\left(3\lambda-1\right)}, \label{23} \\
\frac{\kappa^4\mu^2\Lambda}{8\left(3\lambda-1\right)^2} = 1. \label{24}
\end{eqnarray}
The term $G_{cosmo}$ indicates the Newton's cosmological constant. We must underline that, in gravitational theories which lead to a violation of the Lorentz invariance (which happens in theories like Hořava-Lifshitz gravity), the Newton's gravitational constant $G_{grav}$ (which is one of the terms present in the gravitational action) is different from the Newton's cosmological constant $G_{cosmo}$ (which is one of the terms present in Friedmann equations). We have that $G_{cosmo}$ and $G_{grav}$ are equal if Lorentz invariance is recovered.
For completeness, we give the definition of $G_{grav}$, which can we written as follows:
\begin{eqnarray}
G_{grav} = \frac{\kappa^2}{32\pi}, \label{25}
\end{eqnarray}
as we easily derive using the results of Eq. (<ref>). Moreover, we observe that, in the IR limit (corresponding to the limiting case of $\lambda = 1$), which also implies that the Lorentz invariance is restored, $G_{cosmo}$ and $G_{grav}$ assume an equivalent form. Then, we can also state that the running parameter $\lambda$ gives information about
possibility of breaking the Lorentz invariance. In fact, a value of $\lambda = 1$ indicates validity of
Lorentz invariance, while $\lambda \neq 1$ indicates that Lorentz invariance has been broken.
In a recent work of Dutta $\&$ Saridakis <cit.>, authors concluded that $|\lambda - 1 | \leq 0.02$ with $1\sigma$ confidence level while its best fit value is $|\lambda_{b.f.} - 1 | \approx 0.02$.
Moreover, using Eqs. (<ref>), (<ref>), (<ref>) and (<ref>), it is possible to rewrite the modified Friedmann equations given in Eqs. (<ref>) and (<ref>) in the usual forms as follows:
\begin{eqnarray}
H^2+\frac{k}{a^2} &=& \frac{8\pi G_{cosmo}}{3}\left(\rho_m+\rho_D \right), \label{26}\\
\dot{H}+ \frac{3}{2}H^2+\frac{k}{2a^2} &=& -4\pi G_{cosmo} \left(p_m+p_D \right). \label{27}
\end{eqnarray}
In the following Section, we will derive some important cosmological quantities for the model considered, in particular the Equation of State (EoS) parameter $\omega_D$, the evolutionary form of the energy density of DE $\Omega_D'$ and the deceleration parameter $q$.
§ PLECHDE MODEL WITH GO CUT-OFF IN HOŘAVA-LIFSHITZ COSMOLOGY
We now discuss the main features and properties of the PLECHDE model with Granda-Oliveros cut-off in the context of Hořava-Lifhsitz cosmology. We must underline that we consider a spatially non-flat FLRW Universe which is filled by both Dark Sectors, i.e. DE and DM.
We start remembering that the DE energy density $\rho_D$ with GO cut-off can be written as follows:
\begin{eqnarray}
\rho_D=\frac{3n^2 M_p^2}{L_{GO}^2} - \frac{\varepsilon M_p^2}{L_{GO}^{\delta}}. \label{8-1-1}
\end{eqnarray}
Since we have that the Planck mass can be expressed as $M_p^2 = \left( 8\pi G_{grav} \right)^{-1}$, we can rewrite the expression of the energy density $\rho_D$ defined in Eq. (<ref>) as follows:
\begin{eqnarray}
\rho_D=\frac{3n^2 }{8\pi G_{grav} L_{GO}^2} - \frac{\varepsilon }{8\pi G_{grav} L_{GO}^{\delta}}. \label{8-1-2}
\end{eqnarray}
We now introduce the expressions of the dimensionless fractional energy densities for DM, DE and also for the curvature parameter $k$ which are defined, respectively, in the following way:
\begin{eqnarray}
\Omega_m &=& \frac{\rho_m}{\rho_{cr}}=\left(\frac{8\pi G_{cosmo}}{3H^2}\right)\rho_m, \label{28} \\
\Omega_D &=& \frac{\rho_D}{\rho_{cr}}=\left(\frac{8\pi G_{cosmo}}{3H^2}\right)\rho_D\nonumber \\
&=&\left(\frac{n^2}{H^2L_{GO}^2}\right)\gamma_n, \label{lgo3} \\
\Omega_k &=& -\frac{k}{a^2H^2}, \label{30}
\end{eqnarray}
where $\rho_{cr}$ indicates the critical energy density (i.e. the energy density necessary to obtain the flatness of the Universe) which is defined as follows:
\begin{eqnarray}
\rho_{cr} = \frac{3H^2}{8\pi G_{cosmo}}, \label{31}
\end{eqnarray}
and the term $\gamma_n$ is given by the following relation:
\begin{eqnarray}
\gamma_n = \frac{G_{cosmo}}{G_{grav}}\left(1 - \frac{\varepsilon}{3n^2L_{GO}^{\delta-2}} \right). \label{lgo4}
\end{eqnarray}
Using the expressions of the fractional energy densities defined in Eqs. (<ref>), (<ref>) and (<ref>), the first Friedmann equation defined in Eq. (<ref>) can be written in an equivalent way as follows:
\begin{eqnarray}
1 - \Omega_k = \Omega_D + \Omega_m . \label{32}
\end{eqnarray}
Eq. (<ref>) has the property that it relates all the fractional energy densities considered in this work.
§.§ Non Interacting Case
We start considering the case corresponding to a FLRW Universe containing DE and pressureless DM (i.e., we have $p_m =0$) and in absence of interaction between DE and DM. Moreover, we consider that the Dark Sectors evolve according to conservation laws which expressions are given by the following continuity equations:
\begin{eqnarray}
\dot{\rho}_D +3H\left(1+\omega_D\right)\rho_D = 0, \label{38} \\
\dot{\rho}_m +3H\rho_m = 0, \label{39}
\end{eqnarray}
which are equivalent to the expressions:
\begin{eqnarray}
\rho'_D +3\left(1+\omega_D\right)\rho_D = 0, \label{38prime} \\
\rho'_m +3\rho_m = 0. \label{39prime}
\end{eqnarray}
We must also underline that the prime $'$ and the overdot are related by the following relation:
\begin{eqnarray}
\frac{d}{dt} = H \frac{d}{dx},
\end{eqnarray}
where we have that the parameter $x$ is defined as $x=\ln a$.
As described in the Introduction, we decided to choose as IR cut-off the GO scale $L_{GO}$, which has been previously defined in Eq. (<ref>).
The first time derivative of the expression of $L_{GO}$ given in Eq. (<ref>) is given by the following expression:
\begin{eqnarray}
\dot{L}_{GO}=-H^{3}L_{GO}^{3}\left[ \alpha \left(\frac{\dot{H}}{H^{2}}\right)+\beta \left(\frac{\ddot{H}}{2H^{3}}\right)\right]. \label{lgo7}
\end{eqnarray}
Instead, the first derivative with respect to the cosmic time $t$ of the energy density of DE $\rho_D$ given in Eq. (<ref>) is given by:
\begin{eqnarray}
\dot{\rho}_D = 6H^3 \left[ \alpha \left(\frac{\dot{H}}{H^{2}}\right)+\beta \left(\frac{\ddot{H}}{2H^{3}}\right)\right] \left( \frac{n^2}{8\pi G_{grav}} -\frac{\varepsilon \delta}{48\pi G_{grav}L_{GO}^{\delta -2}} \right). \label{lgo8}
\end{eqnarray}
where we have used the result of Eq. (<ref>).
Differentiating the Friedmann equation given in Eq. (<ref>) with respect to the cosmic time $t$ and using Eq. (<ref>), we obtain that the term $\left[ \alpha \left(\frac{\dot{H}}{H^{2}}\right)+\beta \left(\frac{\ddot{H}}{2H^{3}}\right)\right]$ can be rewritten as follows:
\begin{eqnarray}
\left[ \alpha \left(\frac{\dot{H}}{H^{2}}\right)+\beta \left(\frac{\ddot{H}}{2H^{3}}\right)\right]= \frac{1+\frac{\dot{H}}{H^2}+\Omega_D \left(\frac{u}{2} -1 \right) }{8\pi G_{cosmo} \left( \frac{n^2}{8\pi G_{grav}} -\frac{\varepsilon \delta}{48\pi G_{grav}L_{GO}^{\delta -2}} \right)}, \label{lgo9}
\end{eqnarray}
where the dimensionless parameter $u$ is defined as follows:
\begin{eqnarray}
u= \frac{\Omega_m}{\Omega_D} = \frac{1-\Omega_k}{\Omega_D}-1. \label{lgu}
\end{eqnarray}
In Eq. (<ref>), we used the result of Eq. (<ref>) in order to find a result for $\Omega_m$.
Using the definition of $L_{GO}$ given in Eq. (<ref>) and the expression of the fractional energy density of DE $\Omega_D$ given in Eq. (<ref>), after some algebraic calculations, we obtain that the term $\frac{\dot{H}}{H^2}$ can be written as follows:
\begin{eqnarray}
\frac{\dot{H}}{H^2} =\frac{1}{\beta} \left(\frac{\Omega_D}{n^2\gamma_n} -\alpha \right) . \label{lgo6}
\end{eqnarray}
Inserting the result of Eq. (<ref>) in Eq. (<ref>) and later on Eq. (<ref>) into Eq. (<ref>), we obtain the following expression for $\dot{\rho}_D$:
\begin{eqnarray}
\dot{\rho}_D = \frac{6H^3}{8\pi G_{cosmo} \beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n}\right) -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) \right], \label{lgo10prim}
\end{eqnarray}
which leads to the following expression of the evolutionary form of the DE energy density $\rho'_D$:
\begin{eqnarray}
\rho'_D = \frac{\dot{\rho}_D}{H} =\frac{6H^2}{8\pi G_{cosmo} \beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n}\right) -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) \right]. \label{lgo10evo}
\end{eqnarray}
Using the definition of the fractional energy density of DE given in Eq. (<ref>), we can rewrite the result of Eq. (<ref>) as follows:
\begin{eqnarray}
\rho'_D = \frac{2\rho}{\Omega_D \beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n}\right) -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) \right]. \label{lgo10evoother}
\end{eqnarray}
From the continuity equations for DE given in Eqs. (<ref>) and (<ref>), we can derive the following expression for the EoS parameter of DE $\omega_D$:
\begin{eqnarray}
\omega_D &=& -1 -\frac{\dot{\rho}_D}{3H\rho_D} \nonumber \\
&=& -1 -\frac{\rho'_D}{3\rho_D}. \label{omeganino}
\end{eqnarray}
Using in Eq. (<ref>) the expression of $\dot{\rho}_D$ obtained in Eq. (<ref>) or equivalently the expression of $\rho'_D$ obtained in Eq. (<ref>), we can write $\omega_D$ as follows:
\begin{eqnarray}
\omega_D = -1 - \frac{2}{3\beta \Omega_D}\left[ \left(\frac{\Omega_D}{n^2 \gamma_n}\right) -\alpha + \beta + \beta \Omega_D \left(\frac{u}{2} -1 \right) \right]. \label{}
\end{eqnarray}
We now derive an expression for the evolutionary form of the fractional energy density of DE $\Omega_D'$.
Differentiating the expression of $\Omega_D$ given in Eq. (<ref>) with respect to the variable $x$, we obtain the following expression:
\begin{eqnarray}
\Omega_D' = \Omega_D \left[ \frac{\rho_D'}{\rho_D} - 2\left(\frac{\dot{H}}{H^2}\right) \right] \label{maybritt}.
\end{eqnarray}
Using in Eq. (<ref>) the expressions of $\left(\frac{\dot{H}}{H^2}\right)$ and $\rho'_D$ given, respectively, in Eqs. (<ref>) and (<ref>), we obtain the evolutionary form of the fractional energy density of DE as follows:
\begin{eqnarray}
\Omega_D'= \frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} \right]. \label{lgo11}
\end{eqnarray}
§.§ Interacting Case
We now extend the calculations we have made in the previous subsection obtaining the same cosmological quantities but in the case of presence of a kind of interaction between the two Dark Sectors, i.e. DM and DE.
The presence of an interaction between the two Dark Sectors implies that the energy conservation laws are not held separately, therefore we have the following expressions:
\begin{eqnarray}
\dot{\rho}_D +3H\left(1+\omega_D\right)\rho_D &=& -Q, \label{72} \\
\dot{\rho}_m +3H\rho_m &=& Q, \label{73}
\end{eqnarray}
which are equivalent to the following relations:
\begin{eqnarray}
\rho'_D +3\left(1+\omega_D\right)\rho_D &=& -\frac{Q}{H}, \label{72-1} \\
\rho'_m +3\rho_m &=& \frac{Q}{H}. \label{73-1}
\end{eqnarray}
The quantity $Q$ in the continuity equations given in Eqs. (<ref>), (<ref>), (<ref>) and (<ref>) indicates the interaction term which is in general a function of other cosmological parameters, like for example energy densities of DE and DM $\rho_D$ and $\rho_m$, the Hubble parameter $H$ and the deceleration parameter $q$. Many candidates have been suggested in order to describe the behavior of the interaction term $Q$, we decided to consider in this paper one interaction term which is proportional to the energy density of DE $\rho_D$ and to the Hubble parameter $H$ as follows <cit.>:
\begin{eqnarray}
Q = 3b^2H \rho_D,\label{74}
\end{eqnarray}
where the quantity $b^2$ indicates a coupling parameter (also known with the name of transfer strength) between the Dark Sectors <cit.>. The limiting case corresponding to $b^2 = 0$ leads to the non-interacting FLRW Universe, which has been studied in the previous subsection.
The presence of interaction between DE and DM can be detected during the formation of the Large Scale Structures (LSS). It is suggested that the dynamical equilibrium of some collapsed structures like for example the clusters of galaxies (the cluster Abell A586 is one good example) results to be modified because of the coupling between the two Dark Sectors <cit.>. The main concept is that the virial theorem is affected by the exchange of energy between the two Dark Sectors which leads to a bias in the estimation made for the virial masses of clusters when the usual virial conditions are taken into account. This fact gives a probe in the near Universe of the coupling of the Dark components. Some other observational signatures on the dark sectors mutual interaction can be also found in the probes of the cosmic expansion history by using the results we know about Supernovae Ia (SNe Ia), Baryonic Acoustic Oscillations (BAO) and CMBR shift data <cit.>. Thanks to observational data of Gold SNe Ia samples, CMBR anisotropies and the Baryonic Acoustic Oscillations (BAO), it was possible to estimate that the coupling parameter between DM and DE must assume a small positive value. This condition satisfies the requirement for solving the cosmic coincidence problem and also constraints given by the second law of thermodynamics <cit.>.
Observations of the CMBR and of clusters of galaxies indicate a value of the coupling parameter $b^2$ which is $b^2 < 0.025$. <cit.>. Negative values of $b^2$ are not taken into account since they would lead to the violation of laws of thermodynamics. We also need to emphasize that other interaction terms can be also taken into account <cit.>.
Following the same procedure made for the non interacting case, we obtain the following expression for $\dot{\rho}_D$:
\begin{eqnarray}
\dot{\rho}_D &=& \frac{6H^3}{8\pi G_{cosmo} \beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n}\right) -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) - \left(\frac{8\pi G_{cosmo}}{6H^3}\right)\beta Q \right], \label{lgo12!}
\end{eqnarray}
which leads to the following evolutionary form of the DE energy density:
\begin{eqnarray}
\rho'_D &=& \frac{6H^2}{8\pi G_{cosmo} \beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n}\right) -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) -\left( \frac{8\pi G_{cosmo}}{6H^3}\right)\beta Q \right]. \label{lgo12}
\end{eqnarray}
Using in Eq. (<ref>) the expression of $Q$ we have considered in Eq. (<ref>), we can write $\rho'_D$ as follows:
\begin{eqnarray}
\rho'_D &=& \frac{6H^2}{8\pi G_{cosmo} \beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n}\right) -\alpha + \beta + \beta \Omega_D \left( \frac{u-2-3b^2}{2} \right) \right]. \label{lgo12evoint}
\end{eqnarray}
Using the definition of fractional energy density of DE given in Eq. (<ref>), we can rewrite Eq. (<ref>) as follows:
\begin{eqnarray}
\rho'_D &=& \frac{2\rho_D}{\Omega_D \beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n}\right) -\alpha + \beta + \beta \Omega_D \left( \frac{u-2-3b^2}{2} \right) \right]. \label{lgo12evoint2}
\end{eqnarray}
From the continuity equations for DE given in Eqs. (<ref>) and (<ref>), we can derive the following expression for the EoS parameter $\omega_D$:
\begin{eqnarray}
\omega_D &=& -1 -\frac{\dot{\rho}_D}{3H\rho_D} - \frac{Q}{3H\rho_D} \nonumber \\
&=& -1 -\frac{\rho'_D}{3\rho_D} - \frac{Q}{3H\rho_D}.\label{omeganinoint}
\end{eqnarray}
Using in Eq. (<ref>) the expression of $\dot{\rho}_D$ obtained in Eq. (<ref>) or equivalently the expression of $\rho'_D$ obtained in Eq. (<ref>) along with the definition of $Q$ given in Eq. (<ref>), we can write $\omega_D$ as follows:
\begin{eqnarray}
\omega_D = -1 - \frac{2}{3\beta \Omega_D}\left[ \frac{\Omega_D}{n^2 \gamma_n} -\alpha + \beta + \beta \Omega_D \left(\frac{u}{2} -1 \right) \right],\label{mona1}
\end{eqnarray}
which is the same result of the non interacting case.
We now want to find the expression for $\Omega_D'$ for the interacting case.
Using the general expression of $\Omega_D'$ given in Eq. (<ref>) along with the expressions of $\frac{\dot{H}}{H^2}$ and $\rho_D'$ given, respectively, in Eqs. (<ref>) and (<ref>), we obtain the evolutionary form of the fractional energy density of DE as follows:
\begin{eqnarray}
\Omega_D' &=& \frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right)\left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} - \frac{3}{2}\Omega_D \beta b^2 \right]. \label{lgo13-1}
\end{eqnarray}
For completeness, we here obtain also the expression of the deceleration parameter $q$, which is generally defined as follows:
\begin{eqnarray}
q&=&-\frac{\ddot{a}a}{\dot{a}^2} \nonumber \\
&=& -\frac{\ddot{a}}{aH^2} \nonumber \\
&=&-1-\frac{\dot{H}}{H^2}. \label{89}
\end{eqnarray}
The deceleration parameter $q$ can be used in order to quantify the status of the acceleration of the Universe <cit.>. In particular, a negative value of the present day value of $q$ indicates an accelerating Universe, whereas a positive value of the present day value of $q$ indicates a Universe which is either decelerating or expanding at the coasting <cit.>. In order to have a negative value of the deceleration parameter, we must have $\ddot{a}>0$.
Using the expression of $\frac{\dot{H}}{H^2}$ given in Eq. (<ref>), we can write the deceleration parameter $q$ as follows:
\begin{eqnarray}
q&=&-1- \frac{1}{\beta}\left( \frac{\Omega_D}{n^2\gamma_n} -\alpha \right)\nonumber \\
&=&-1 + \frac{\alpha}{\beta} -\frac{1}{\beta}\left( \frac{\Omega_D}{n^2\gamma_n} \right)\nonumber \\
&=& \frac{\left( \alpha - \beta \right)n^2 \gamma_n - \Omega_D}{\beta n^2 \gamma_n} . \label{89-finale}
\end{eqnarray}
We can easily derive that the expression of $\omega_D$ given in Eq. (<ref>) and the expression of $q$ given in Eq. (<ref>) are related through the following relation:
\begin{eqnarray}
\omega_D = \frac{2q}{3\Omega_D} - \frac{1+u}{3}.
\end{eqnarray}
We must also underline that, in the limiting case of $b^2=0$ (i.e. in absence of interaction), we recover the same results of the non interacting case obtained in the previous subsection.
§ LOW REDSHIFT EXPANSION
In the previous Section, we have derived the general expression of the EoS parameter of DE $\omega_D$ as function of the other cosmological parameters.
We now consider a particular parametrization of the EoS parameter of DE $\omega_D$ as function of the redshift $z$, which is given by $\omega_D\left(z\right) = \omega_0 + \omega_1 z$, and we will calculate the expressions of the two parameters $\omega_0$ and $\omega_1$ for both non interacting and interacting Dark Sectors.
§.§ Non Interacting Case
We start studying the case corresponding to absence of interaction between the two Dark Sectors.
As previously stated, the EoS parameter of DE $\omega_D$ written in a parameterized way as function of the redshift $z$ is given by the following relation <cit.>:
\begin{eqnarray}
\omega_D\left(z\right) = \omega_0 + \omega_1 z, \label{49}
\end{eqnarray}
therefore we obtain an expression of the EoS parameter of DE $\omega_D$ which is function of the redshift $z$ too while the dependence on the other cosmological parameters will appear in the final expressions of the two parameters $\omega_0$ and $\omega_1$.
We must also remember here that the relation between the scale factor $a$ and the redshift $z$ is given by the following expression:
\begin{eqnarray}
a = \frac{1}{1+z} =\left( 1+z \right)^{-1} , \label{50}
\end{eqnarray}
which leads to the following expression for the redshift $z$:
\begin{eqnarray}
z= a^{-1}-1. \label{50a}
\end{eqnarray}
Using the continuity equation for DE obtained in Eq. (<ref>) in the definition of $\omega_D$ given in (<ref>), we derive that the energy density of DE $\rho_D$ evolves according to the following relation <cit.>:
\begin{eqnarray}
\frac{\rho_D}{\rho_{D_0}} = a^{-3 \left( 1+ \omega_0 - \omega_1 \right)} e^{3\omega_1z}, \label{51}
\end{eqnarray}
where $\rho_{D_0}$ indicates the present day value of the energy density of DE $\rho_D$.
The Taylor expansion of the energy density of DE $\rho_D$ around the point $a_0 = 1$ yields:
\begin{eqnarray}
\ln{\rho_D} = \ln{\rho_{D_0}} + \left. \frac{d\ln{ \rho_D}}{d\ln{ a}}\right|_0\ln{a} + \left. \frac{1}{2} \frac{d^2\ln{ \rho_D}}{d\left(\ln{ a}\right)^2}\right|_0\left(\ln{a}\right)^2 + ... , \label{52}
\end{eqnarray}
where $a_0$ indicates the present value of the scale factor $a$. Using the expression of $a$ given in Eq. (<ref>) in the Taylor expansion given in Eq. (<ref>), we obtain, for small redshifts, the following relation:
\begin{eqnarray}
\ln a = -\ln \left( 1+z \right) \simeq -z + \frac{z^2}{2}, \label{53}
\end{eqnarray}
where we used the property of logarithm $\ln x^n = n \ln x$ (with $n$ real number) and the Taylor expansion of the logarithm given by $\ln \left(1 + x \right) = \sum _{n=1}^{\infty} \left( -1 \right)^{n+1}\frac{x^n}{n} $.
Therefore, Eqs. (<ref>) and (<ref>) lead, respectively, to the following two relations:
\begin{eqnarray}
\frac{\ln{\left(\rho_D/\rho_{D_0}\right)}}{\ln{a}} &=& -3\left(1+\omega_0\right) - \frac{3}{2}\omega_1z, \label{54} \\
\frac{\ln{\left(\rho_D/\rho_{D_0}\right)}}{\ln{a}} &=& \left. \frac{d\ln{ \rho_D}}{d\ln{ a}}\right|_0 - \left. \frac{1}{2} \frac{d^2\ln{ \rho_D}}{d\left(\ln{ a}\right)^2}\right|_0 z. \label{55}
\end{eqnarray}
Making a comparison of the results obtained in Eqs. (<ref>) and (<ref>), we easily derive the following relations for the two parameters $\omega_0$ and $\omega_1$:
\begin{eqnarray}
\omega_0 &=& \left. -\frac{1}{3}\frac{d\ln{ \rho_D}}{d\ln{ a}}\right|_0 - 1, \label{56} \\
\omega_1 &=& \left. \frac{1}{3}\frac{d^2\ln{ \rho_D}}{d\left(\ln{ a}\right)^2}\right|_0. \label{57}
\end{eqnarray}
Using the definition of DM and DE given, respectively, in Eqs. (<ref>) and (<ref>), we can obtain the following relation between the critical energy density $\rho_{cr}$ and the energy densities of DM and DE $\rho_m$ and $\rho_D$:
\begin{eqnarray}
\rho_{cr} = \frac{\rho_m}{\Omega_m} = \frac{\rho_D}{\Omega_D}, \label{58a}
\end{eqnarray}
which leads to the following result for $\rho_D$:
\begin{eqnarray}
\rho_D = \left(\frac{\rho_m}{\Omega_m}\right)\Omega_D. \label{58b}
\end{eqnarray}
From the continuity equation for DM obtained in Eq. (<ref>), we derive that the energy density $\rho_m$ of DM evolves according to the relation $\rho_m = \rho_{m_0}a^{-3}$, where the constant $\rho_{m_0}$ indicates the present day value of $\rho_m$. Using the expression of $\rho_m$ we have above obtained along with the relation between all the fractional energy densities given in Eq. (<ref>), we can rewrite Eq. (<ref>) as follows:
\begin{eqnarray}
\rho_D = \left[\frac{\rho_{m_0}a^{-3}}{\left(1 - \Omega_k - \Omega_D\right)}\right]\Omega_D. \label{58}
\end{eqnarray}
Substituting Eq. (<ref>) into Eq. (<ref>), we derive the following general expression for the parameter $\omega_0$:
\begin{eqnarray}
\omega_0 = -\frac{1}{3} \left[\frac{\Omega'_D}{\Omega_D} + \frac{\Omega'_D + \Omega'_k}{\left(1 - \Omega_k - \Omega_D\right)} \right]_0. \label{59}
\end{eqnarray}
Moreover, inserting the result of Eq. (<ref>) in Eq. (<ref>), we easily derive the following general expression for the parameter $\omega_1$:
\begin{eqnarray}
\omega_1= \frac{1}{3} \left[\frac{\Omega''_D}{\Omega_D} - \frac{\Omega'^2_D}{\Omega^2_D} + \frac{\Omega''_D + \Omega''_k}{\left(1 - \Omega_k - \Omega_D\right)} + \frac{ \left( \Omega'_D + \Omega'_k \right)^2}{\left(1 - \Omega_k - \Omega_D\right)^2} \right]_0.
\label{60}
\end{eqnarray}
We now want to obtain the final expression for both $\omega_0$ and $\omega_1$, therefore we must obtain the quantities involved in the relations we have obtained in Eqs. (<ref>) and (<ref>).
We have already obtained the evolutionary form of the fractional energy density of DE in Eq. (<ref>), which is given by:
\begin{eqnarray}
\Omega_D'= \frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} \right]. \label{lgo11new}
\end{eqnarray}
We now need to find the expressions of the quantities $\Omega_D''$, $\Omega_k'$ and $\Omega_k''$ in order to have the expressions of all the quantities necessary in order to calculate the final expressions of both $\omega_0$ and $\omega_1$.
Differentiating the expression of the fractional energy density of the curvature parameter $\Omega_k$ given in Eq. (<ref>) with respect to the variable $x$, we obtain the following relation:
\begin{eqnarray}
\Omega'_k = -2 \Omega_k \left( 1+ \frac{\dot{H}}{H^2} \right). \label{lgo13}
\end{eqnarray}
Inserting in Eq. (<ref>) the expression of $ \left(\frac{\dot{H}}{H^2}\right) $ derived in Eq. (<ref>), we obtain the following final expression for the evolutionary form of the fractional energy density of curvature $\Omega_k'$:
\begin{eqnarray}
\Omega_k'=-\frac{2\Omega_k}{\beta}\left( \frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta \right). \label{lgo15}
\end{eqnarray}
Differentiating the expression of $\Omega_D'$ given in Eq. (<ref>) with respect to the variable $x$, we derive that the second derivative of $\Omega_D$ with respect to the variable $x$ is given by:
\begin{eqnarray}
\Omega_D'' = \frac{2}{\beta}\left\{ \Omega_D'\left[\frac{1-2\Omega_D}{n^2\gamma_n} +\alpha + \beta\left(\frac{ u}{2}-1\right) \right] +\Omega_D \left[ \frac{\beta u'}{2}
- \frac{\gamma_n'\left( 1-\Omega_D \right)}{n^2 \gamma_n^2} \right] \right\}. \label{lgo19}
\end{eqnarray}
Moreover, differentiating the expression of $\Omega_k'$ given in Eq. (<ref>) with respect to the variable $x$, we derive that the second derivative of $\Omega_k$ with respect to the variable $x$ is given by:
\begin{eqnarray}
\Omega_k'' = -\frac{2}{\beta}\left[ \Omega'_k \left(\frac{\Omega_D}{n^2 \gamma_n} +\beta - \alpha \right)
+\frac{\Omega_k}{n^2 \gamma_n}\left( \Omega'_D - \frac{\Omega_D \gamma'_n}{\gamma_n} \right)\right]. \label{lgo16}
\end{eqnarray}
We can now make some consideration about the quantities $\gamma_n$ and $u$ and also their derivatives and their present day values.
Using the definitions of $\gamma_n$ and $u$ given, respectively, in Eqs. (<ref>) and (<ref>), we can easily derive their derivatives with respect to the variable $x$, which are given by the following relations:
\begin{eqnarray}
u'&=&\frac{\dot{u}}{H} = -\frac{\Omega_k'}{\Omega_D} - \left( 1 - \Omega_k\right)\frac{\Omega_D'}{\Omega_D^2}\nonumber \\
&=&-\frac{\Omega_k'}{\Omega_D} - \left( u +1 \right) \frac{\Omega_D'}{\Omega_D}, \label{lgo17old} \\
\gamma_n' &=& \frac{H^2 \varepsilon \left( 2-\delta \right)}{3 \beta n^2 L^{\delta-4}\left\{n^2 -\frac{\varepsilon \delta}{8L^{\delta -2}} \right\}} \left\{ \beta +\frac{\Omega_D}{n^2 \gamma_n} -\alpha +\beta \Omega_D \left(\frac{u}{2} -1 \right) \right\}. \label{lgo18}
\end{eqnarray}
Using in Eq. (<ref>) the expressions of $\Omega_D'$ and $\Omega_k'$ derived, respectively, in Eqs. (<ref>) and (<ref>), we obtain the following expression for $u'$:
\begin{eqnarray}
u'&=&\frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -u\left( u+1 \right). \label{lgo17}
\end{eqnarray}
In order to study the behavior of the quantities $\gamma_{n}$ and $\gamma_{n}'$, we need to make some preliminary considerations. We know that the power law correction to the entropy area law gives a reasonable contribution only at early stages of the Universe while with the passing of the time its contribution becomes less important. For this reason, we can tell that, for $\gamma_{n}$ and $\gamma_{n}'$, the power law corrections can be considered practically negligible. We then have:
\begin{eqnarray}
\gamma_{n} &=& \frac{G_{cosmo}}{G_{grav}} = \frac{2}{3\lambda -1}, \label{lim1}\\
\gamma_{n}' &=& 0. \label{lim2}
\end{eqnarray}
Taking into account the above reasons, in the following calculations, we will neglect the derivatives of $\gamma_{n}$ and we will consider as expression for $\gamma_{n}$ the one obtained in Eq. (<ref>).
Using the general definition of $u$ given in Eq. (<ref>), we have that $u_0$ is given by the following quantity:
\begin{eqnarray}
u_0= \frac{1-\Omega_{k_0}}{\Omega_{D_0}} -1, \label{u0}
\end{eqnarray}
while, using the result of Eq. (<ref>), we have that $u_0'$ can be written as follows:
\begin{eqnarray}
u_0'&=& \frac{2}{\beta}\left( \frac{1}{n^2\gamma_{n_0}} +\frac{\beta - \alpha}{\Omega_{D_0}} \right) \left[ \Omega_{k_0} - \left(u_0+1 \right)\left( 1- \Omega_{D_0} \right) \right] -u_0\left(u_0+1 \right) .\label{u0prime}
\end{eqnarray}
Moreover, we have that the present day value of $\gamma_{n}$ is given by:
\begin{eqnarray}
\gamma_{n_0} &=& \frac{G_{cosmo}}{G_{grav}} = \frac{2}{3\lambda -1}, \label{lim1-1}
\end{eqnarray}
i.e. $\gamma_{n_0}$ is equivalent to the expression of $\gamma_{n}$, $\gamma_{n_0} = \gamma_{n} $.
Therefore, using the expression of $\gamma_{n_0} $ obtained in Eq. (<ref>), we can write $u_0'$ as folloes:
\begin{eqnarray}
u_0'&=& \frac{2}{\beta}\left( \frac{3\lambda-1}{2n^2} +\frac{\beta - \alpha}{\Omega_{D_0}} \right) \left[ \Omega_{k_0} - \left(u_0+1 \right)\left( 1- \Omega_{D_0} \right) \right] -u_0\left(u_0+1 \right) .\label{u0prime}
\end{eqnarray}
Inserting the results of Eqs. (<ref>) and (<ref>) in the general expression of the parameter $\omega_0$ given in Eq. (<ref>), we derive the following expression for $\omega_0$:
\begin{eqnarray}
\omega_0 &=& -\frac{2}{3\beta}\left[\left(\frac{1}{n^2\gamma_{n_0}} + \frac{\beta - \alpha}{\Omega_{D_0}}\right) + \frac{\beta u_0}{2}\left( \frac{1-\Omega_{k_0}}{1-\Omega_{D_0} - \Omega_{k_0}} \right)\right]. \label{59new}
\end{eqnarray}
We now want to calculate the terms involved in the expression of the parameter $\omega_1$.
Using the definition of the evolutionary form of the fractional energy density of DE $\Omega_D'$ given in Eq. (<ref>), we have that:
\begin{eqnarray}
\frac{\Omega_D'}{\Omega_D}= \frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \frac{\left( 1-\Omega_D \right)}{\Omega_D} + \frac{ \beta u}{2} \right]. \label{lgo11newNAN}
\end{eqnarray}
Using the expression of $\Omega_D''$ obtained in Eq. (<ref>), we have that the term $\left(\frac{\Omega_D''}{\Omega_D}\right)$ is equal to:
\begin{eqnarray}
\frac{\Omega_D''}{\Omega_D} = \frac{2}{\beta \Omega_D}\left\{ \Omega_D'\left[\frac{1-2\Omega_D}{n^2\gamma_n} +\alpha + \beta\left(\frac{ u}{2}-1\right) \right] \right\} + u'. \label{iele1NON}
\end{eqnarray}
Adding the expressions of $\Omega_D''$ and $\Omega_k^{''}$ given, respectively, in Eqs. (<ref>) and (<ref>), we obtain:
\begin{eqnarray}
\frac{\Omega_D^{''}+\Omega_k^{''}}{1-\Omega_D-\Omega_k} &=& \frac{2}{\beta}\left\{\Omega_D' \left[\frac{1}{n^2\gamma_n} + \frac{\beta u}{2\left( 1-\Omega_D-\Omega_k \right)} \right] \right\} \nonumber \\
&& -\frac{2}{\beta}\frac{\Omega_D' + \Omega_k'}{1-\Omega_D-\Omega_k}\left( \frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta \right) + \frac{ u'\Omega_D}{1-\Omega_D-\Omega_k}. \label{iele2}
\end{eqnarray}
Therefore, adding Eqs. (<ref>) and (<ref>), we can write:
\begin{eqnarray}
\frac{\Omega_D''}{\Omega_D} + \frac{\Omega_D^{''}+\Omega_k^{''}}{1-\Omega_D-\Omega_k} &=& \frac{2}{\beta \Omega_D}\left\{ \Omega_D'\left[\frac{1-2\Omega_D}{n^2\gamma_n} +\alpha + \beta\left(\frac{ u}{2}-1\right) \right] \right\} \nonumber \\
&&+\frac{2}{\beta}\left\{\Omega_D' \left[\frac{1}{n^2\gamma_n} + \frac{\beta u}{2\left( 1-\Omega_D-\Omega_k \right)} \right] \right\} \nonumber \\
&&-\frac{2}{\beta}\left(\frac{\Omega_D' + \Omega_k'}{1-\Omega_D-\Omega_k}\right)\left( \frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta \right) + \frac{\left(1-\Omega_k\right)u'}{1-\Omega_D-\Omega_k}. \label{ieleNA2}
\end{eqnarray}
Finally, adding the expressions of the evolutionary form of the fractional energy density of DE $\Omega_D'$ and $\Omega_k^{'}$ given, respectively, in Eqs. (<ref>) and (<ref>), we obtain:
\begin{eqnarray}
\frac{\Omega_D'+\Omega_k'}{1-\Omega_D-\Omega_k} = \frac{2}{\beta}\left[\left( \frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta \right) +\frac{\beta u \Omega_D}{2\left( 1-\Omega_D-\Omega_k\right)} \right].\label{iele3non}
\end{eqnarray}
Therefore, considering the results of above equations in Eq. (<ref>), the final expression of the parameter $\omega_1$ can be written as follows:
\begin{eqnarray}
\omega_1&=& \frac{4}{3\beta^2 \Omega_{D_0}}\left\{\left[\left(\frac{\Omega_{D_0}}{n^2\gamma_{n_0}} - \alpha + \beta\right) \left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} \right]\left[\frac{1-2\Omega_{D_0}}{n^2\gamma_{n_0}} +\alpha + \beta\left(\frac{ u_0}{2}-1\right) \right] \right\} \nonumber \\
&&+\frac{4}{3\beta^2}\left\{\left[\left(\frac{\Omega_{D_0}}{n^2\gamma_{n_0}} - \alpha + \beta\right) \left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} \right] \left[\frac{1}{n^2\gamma_{n_0}} + \frac{\beta u_0}{2\left( 1-\Omega_{D_0}-\Omega_{k_0} \right)} \right] \right\} \nonumber \\
&&-\frac{4}{3\beta^2}\left( \frac{\Omega_{D_0}}{n^2\gamma_{n_0}} -\alpha + \beta \right)\left[\left( \frac{\Omega_{D_0}}{n^2\gamma_{n_0}} -\alpha + \beta \right) +\frac{\beta u_0 \Omega_{D_0}}{2\left( 1-\Omega_{D_0}-\Omega_{k_0}\right)} \right] \nonumber \\
&&+ \left\{ \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_{n_0}} + \frac{\beta - \alpha}{\Omega_{D_0}} \right) \left[\Omega_{k_0} - \left(u_0+1 \right) \left(1-\Omega_{D_0} \right) \right] -u_0\left( u_0+1 \right) \right\}\frac{1-\Omega_{k_0}}{3\left(1-\Omega_{D_0}-\Omega_{k_0}\right)}\nonumber \\
&&-\frac{4}{3\beta^2}\left[\left(\frac{\Omega_{D_0}}{n^2\gamma_{n_0}} - \alpha + \beta\right) \frac{\left( 1-\Omega_{D_0} \right)}{\Omega_{D_0}} + \frac{ \beta u_0}{2} \right]^2 \nonumber \\
&&+\frac{4}{3\beta^2}\left[\left( \frac{\Omega_{D_0}}{n^2\gamma_{n_0}} -\alpha + \beta \right) +\frac{\beta u_0 \Omega_{D_0}}{2\left( 1-\Omega_{D_0}-\Omega_{k_0}\right)}\right]^2 . \label{71}
\end{eqnarray}
We must emphasize here that $\Omega_{D_0}$ and $\Omega_{k_0}$ represent, respectively, the present day values of the fractional energy densities for DE and curvature parameter $k$.
Using the expression of $\gamma_{n_0}$ obtained in Eq. (<ref>), we can finally write $\omega_0$ and $\omega_1$ as follows:
\begin{eqnarray}
\omega_0 &=& -\frac{2}{3\beta}\left[\left(\frac{3\lambda -1}{2n^2} + \frac{\beta - \alpha}{\Omega_{D_0}}\right) + \frac{\beta u_0}{2}\left( \frac{1-\Omega_{k_0}}{1-\Omega_{D_0} - \Omega_{k_0}} \right)\right], \label{59newfine}\\
\omega_1&=& \frac{4}{3\beta^2 \Omega_{D_0}}\left\{\left[\left(\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2} - \alpha + \beta\right) \left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} \right]\times \right. \nonumber \\
&&\left.\left[\frac{\left(1-2\Omega_{D_0}\right)\left(3\lambda -1 \right)}{2n^2} +\alpha + \beta\left(\frac{ u_0}{2}-1\right) \right] \right\} \nonumber \\
&&+\frac{4}{3\beta^2}\left\{\left[\left(\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2} - \alpha + \beta\right) \left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} \right] \times \right. \nonumber \\
&&\left. \left[\frac{\left(3\lambda -1 \right)}{2n^2 } + \frac{\beta u_0}{2\left( 1-\Omega_{D_0}-\Omega_{k_0} \right)} \right] \right\} \nonumber \\
&&-\frac{4}{3\beta^2}\left[ \frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } -\alpha + \beta \right]\left\{\left[ \frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } -\alpha + \beta \right] +\frac{\beta u_0 \Omega_{D_0}}{2\left( 1-\Omega_{D_0}-\Omega_{k_0}\right)} \right\} \nonumber \\
&&+ \left\{ \frac{2}{\beta}\left[ \frac{\left(3\lambda -1 \right)}{2n^2 } + \frac{\beta - \alpha}{\Omega_{D_0}} \right] \left[\Omega_{k_0} - \left(u_0+1 \right) \left(1-\Omega_{D_0} \right) \right] -u_0\left( u_0+1 \right) \right\}\left(\frac{1-\Omega_{k_0}}{3\left(1-\Omega_{D_0}-\Omega_{k_0}\right)}\right)\nonumber \\
&&-\frac{4}{3\beta^2}\left\{\left[\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } - \alpha + \beta\right] \frac{\left( 1-\Omega_{D_0} \right)}{\Omega_{D_0}} + \frac{ \beta u_0}{2} \right\}^2 \nonumber \\
&&+\frac{4}{3\beta^2}\left\{\left[ \frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } -\alpha + \beta \right] +\frac{\beta u_0 \Omega_{D_0}}{2\left( 1-\Omega_{D_0}-\Omega_{k_0}\right)}\right\}^2 . \label{71nino}
\end{eqnarray}
We now want to calculate the values of $\omega_0$ and $\omega_1$ for three different values of the running parameter $\lambda$, in particular $\lambda =1.02$ and
$\lambda = 0.98$ (which are values suggested by the work of Dutta and Saridakis) and $\lambda =1$ which leads to the Lorentz invariance case. Moreover, we will make the same examples also for the limiting case corresponding to the Ricci scalar curvature, i.e. when $\alpha =2$ and $\beta =1$.
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$:
\begin{eqnarray}
\omega_0 &\approx & -1.81193, \\
\omega_1 &\approx & 0.416014,
\end{eqnarray}
therefore we obtain the following equation for the EoS parameter of DE $\omega_D$:
\begin{eqnarray}
\omega_D &\approx& -1.81193 + 0.416014z. \label{ceizze2}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 1.02$.
At present time, i.e. for $z=0$, Eq. (<ref>) leads to $\omega_D \approx -1.81193 $ while the value $\omega_D = -1$ is obtained for a redshift of $z\approx 1.95$.
Moreover, inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 0.98$:
\begin{eqnarray}
\omega_0 &=& -1.69188, \\
\omega_1 &=& 0.388546 ,
\end{eqnarray}
therefore we derive the following equation for the EoS parameter of DE $\omega_D$:
\begin{eqnarray}
\omega_D &\approx & -1.69188 + 0.388546 z. \label{ceizze1}
\end{eqnarray}
In Figure <ref>, we plot the behavior of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 0.98$.
At present time, i.e. for $z=0$, Eq. (<ref>) leads to $\omega_D \approx -1.69188$ while the value $\omega_D = -1$ is obtained for a redshift of $z\approx 1.78$.
Finally, inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1$:
\begin{eqnarray}
\omega_0 &=& -1.75191, \\
\omega_1 &=& 0.40228,
\end{eqnarray}
therefore we derive the following equation for the EoS parameter of DE $\omega_D$:
\begin{eqnarray}
\omega_D &\approx & -1.75191 + 0.40228 z. \label{ceizze3}
\end{eqnarray}
In Figure <ref>, we plot the behavior of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 1.00$.
At present time, i.e. for $z=0$, Eq. (<ref>) leads to $\omega_D \approx -1.75191$ while the value $\omega_D = -1$ is obtained for a redshift of $z\approx 1.86$.
We now want to obtain the expressions for $\omega_D$ for the Ricci scale, which is recovered for $\alpha =2$ and $\beta=1$.
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$:
\begin{eqnarray}
\omega_0 &\approx & -0.552428, \\
\omega_1 &\approx & 0.247249,
\end{eqnarray}
therefore we obtain the following equation for the EoS parameter of DE $\omega_D$:
\begin{eqnarray}
\omega_D &\approx& -0.552428 + 0.247249 z. \label{ceizzericci1}
\end{eqnarray}
In Figure <ref>, we plot the behavior of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 1.02$ in the limiting case of Ricci scale.
At present time, i.e. for $z=0$, Eq. (<ref>) leads to $\omega_D \approx -0.552428$ while the value $\omega_D = -1$ is obtained for a redshift of $z\approx -1.81$.
Moreover, inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 0.98$:
\begin{eqnarray}
\omega_0 &=& -0.492207, \\
\omega_1 &=&0.220506,
\end{eqnarray}
therefore we derive the following equation for the EoS parameter of DE $\omega_D$:
\begin{eqnarray}
\omega_D &\approx & -0.492207 + 0.220506 z. \label{ceizzericci2}
\end{eqnarray}
In Figure <ref>, we plot the behavior of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 0.98$ in the limiting case of Ricci scale.
At present time, i.e. for $z=0$, Eq. (<ref>) leads to $\omega_D \approx -0.492207$ while the value $\omega_D = -1$ is obtained for a redshift of $z\approx -2.30$.
Finally, inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1$:
\begin{eqnarray}
\omega_0 &=& -0.522318, \\
\omega_1 &=& 0.233877,
\end{eqnarray}
therefore we derive the following equation for the EoS parameter of DE $\omega_D$:
\begin{eqnarray}
\omega_D &\approx & -0.522318 + 0.233877 z. \label{ceizzericci3}
\end{eqnarray}
In Figure <ref>, we plot the behavior of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 1.00$ in the limiting case of Ricci scale.
At present time, i.e. for $z=0$, Eq. (<ref>) leads to $\omega_D \approx -0.522318$ while the value $\omega_D = -1$ is obtained for a redshift of $z\approx -1.84$.
We now want ot obtain the present day values of the deceleration parameter $q$ for the three different values of the running parameter $\lambda$ we are considering and for both sets of values of $\alpha$ and $\beta$ we are studying.
Using the general expression of $q$ obtained in Eq. (<ref>) along with the considerations done in this Section, we have that the present day value of the deceleration parameter $q$ (indicated with $q_0)$ is given by:
\begin{eqnarray}
q_0 &=& \frac{\left( \alpha - \beta \right)n^2 \gamma_{n_0} - \Omega_{D_0}}{\beta n^2 \gamma_{n_0}} . \label{89-finale-2old}
\end{eqnarray}
Using in Eq. (<ref>) the expression of $\gamma_{n0}$ given in Eq. (<ref>), we can write $q_0$ as follows:
\begin{eqnarray}
q_0 &=& -1 + \frac{\alpha}{\beta} - \frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2\beta n^2} . \label{89-finale-2}
\end{eqnarray}
We can now calculate the value of $q_0$ according to the values of the parameters involved.
We first consider the case with $\alpha =0.8824$ and $\beta =0.5016$.
We have that, for $\lambda = 1.02$, $q_0$ assumes the value of $-1.37734$, for $\lambda = 0.98$, it assumes the value of $-1.25288$ while for $\lambda = 1.00$ it assumes the value of $-1.31511$.
We now consider the limiting case of Ricci scale, which is recovered for $\alpha =2$ and $\beta =1$.
We obtain that, for $\lambda = 1.02$, $q_0$ assumes the value of $-0.0716745$, for $\lambda = 0.98$, it assumes the value of $-0.00924687$ while for $\lambda = 1.00$ it assumes the value of $-0.0404607$.
We can see, then, that for all the values of the running parameter $\lambda$ considered, the present day value of the deceleration parameter $q_0$ assumes a negative value for both set of values of $\alpha$ and $\beta$ we have chosen, which indicates an accelerated expansion of the Universe, result which is in agreement with the most recent cosmological observations. We also observe that for the Ricci scale case, we obtain values of $q_0$ which are closer to $q=0$ (which indicates the transition from decelerated to accelerated Universe) with respect to the case with $\alpha =0.8824$ and $\beta =0.5016$.
§.§ Interacting Case
We now consider the interacting case, obtaining the same quantities of the previous subsection but with the contribution produced by the interaction between the two Dark Sectors.
Using the result of Eq. (<ref>) along with the relation between all the fractional energy densities derived in Eq. (<ref>), we can write the following relation for $\rho_D$:
\begin{eqnarray}
\rho_D = \left( \frac{\rho_m}{\Omega_m}\right)\Omega_D = \left[\frac{\rho_m}{\left(1 - \Omega_k - \Omega_D\right)}\right]\Omega_D. \label{76}
\end{eqnarray}
From the result of Eq. (<ref>), we can easily derive the following expression:
\begin{eqnarray}
\frac{d\ln{\rho_D}}{d\ln{a}} = \frac{\rho'_m}{\rho_m} - \frac{\Omega'_m}{\Omega_m} + \frac{\Omega'_D}{\Omega_D}. \label{77}
\end{eqnarray}
We also obtain, using the results of Eqs. (<ref>) and (<ref>), that the energy density of DE $\rho_D$ evolves according to the following relation:
\begin{eqnarray}
\frac{\rho_D}{\rho_{D_0}} = a^{-3\left(1+\omega_0 - \omega_1+b^2\right)}e^{3\omega_1 z}.
\label{78}
\end{eqnarray}
Using the Taylor expansion given in Eq. (<ref>) for small redshifts, Eq. (<ref>) leads to the following relation:
\begin{eqnarray}
\frac{\ln{\left(\rho_D / \rho_{D_0}\right)}}{\ln{a}} = -3\left(1+\omega_0+b^2\right) - \frac{3}{2} \omega_1 z. \label{79}
\end{eqnarray}
Comparing the results of Eqs. (<ref>) and (<ref>), we obtain that the parameters $\omega_0$ and $\omega_1$ for the interacting DE and DM can be written as follows:
\begin{eqnarray}
\omega_0 &=& \left. -\frac{1}{3}\frac{d\ln{\rho_D}}{d\ln{a}}\right|_0 - 1 - b^2, \label{80} \\
\omega_1 &=& \left. \frac{1}{3}\frac{d^2\ln{\rho_D}}{d \left(\ln{a}\right)^2}\right|_0. \label{81}
\end{eqnarray}
Inserting the result of Eq. (<ref>) in Eq. (<ref>) and considering the result of Eq. (<ref>), it is possible to write the parameter $\omega_0$ as follows:
\begin{eqnarray}
\omega_0 = -\frac{1}{3}\left[\frac{\Omega'_D}{\Omega_D}+\frac{\Omega'_D + \Omega'_k}{\left(1 - \Omega_k - \Omega_D\right)} \right]_0-b^2\left(\frac{1-\Omega_k}{1 - \Omega_k - \Omega_D}\right)_0. \label{82}
\end{eqnarray}
Moreover, we also obtain the following relation for the parameter $\omega_1$:
\begin{eqnarray}
\omega_1 &=& \frac{1}{3}\left[\frac{3b^2\Omega'_D}{1 - \Omega_k - \Omega_D} + \frac{3b^2\Omega_D\left(\Omega'_D+\Omega'_k\right)}{\left(1 - \Omega_k - \Omega_D\right)^2}+\frac{\Omega''_D}{\Omega_D} \right. \nonumber\\
&-& \left. \frac{\Omega'^2_D}{\Omega^2_D}+\frac{\Omega''_D+\Omega''_k}{1 - \Omega_k - \Omega_D}+\frac{\left(\Omega'_D+\Omega'_k\right)^2}{\left(1 - \Omega_k - \Omega_D\right)^2}\right]_0. \label{83}
\end{eqnarray}
We now derive the explicit forms of the parameter $\omega_0$ and $\omega_1$ for the interacting case using the results of Eqs. (<ref>) and (<ref>). Therefore, we need to calculate the expressions of the quantities involved.
We have already obtained the expression of the evolutionary form of the fractional energy density of DE $\Omega_D'$ for the interacting case in Eq. (<ref>) and it is given by the following relation:
\begin{eqnarray}
\Omega_D' &=& \frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right)\left( 1-\Omega_D \right) + \frac{\Omega_D \beta \left(u-3b^2\right)}{2} \right]. \label{paolone1}
\end{eqnarray}
Differentiating the result of Eq. (<ref>) with respect to the variable $x$ leads to the following expression for $\Omega_D''$:
\begin{eqnarray}
\Omega_D'' = \frac{2}{\beta}\left\{ \Omega_D'\left[\frac{1-2\Omega_D}{n^2\gamma_n} +\alpha + \beta\left(\frac{u -2 -3b^2}{2}\right) \right]
+ \frac{\Omega_D \beta u'}{2} \right\}. \label{paolone2}
\end{eqnarray}
The expression of $\Omega_k'$ and $\Omega_k''$ are the same as those obtained for the non interacting case and given, respectively, in Eqs. (<ref>) and (<ref>).
Also in the interacting case, the expressions of $\gamma_{n}$ and $\gamma_{n}'$ are the same as the non-interacting case, given respectively in Eqs. (<ref>) and (<ref>). Instead, the expression of $u'$ is given by:
\begin{eqnarray}
u' &=& \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -\left(u -3b^2 \right)\left( u+1 \right), \label{lgo17int}
\end{eqnarray}
which implies that the final expression $u_0'$ is given by the following relation:
\begin{eqnarray}
u_0'&=& \frac{2}{\beta}\left( \frac{1}{n^2\gamma_{n_0}} +\frac{\beta - \alpha}{\Omega_{D_0}} \right) \left[ \Omega_{k_0} - \left(u_0+1 \right)\left( 1- \Omega_{D_0} \right) \right] -\left( u_0 -3b^2\right)\left(u_0+1 \right). \label{lgo17-2-2old}
\end{eqnarray}
Using the expression of $\gamma_{n_0}$ obtained in Eq. (<ref>), we can write Eq. (<ref>) as follows:
\begin{eqnarray}
u_0'&=& \frac{2}{\beta}\left( \frac{3\lambda-1}{2n^2} +\frac{\beta - \alpha}{\Omega_{D_0}} \right) \left[ \Omega_{k_0} - \left(u_0+1 \right)\left( 1- \Omega_{D_0} \right) \right] -\left( u_0 -3b^2\right)\left(u_0+1 \right). \label{lgo17-2-2}
\end{eqnarray}
Inserting in Eq. (<ref>) the results obtained in Eqs. (<ref>) and (<ref>), we obtain the following expression for $\omega_0$:
\begin{eqnarray}
\omega_0 &=& -\frac{2}{3\beta}\left[\left(\frac{1}{n^2\gamma_{n_0}} + \frac{\beta - \alpha}{\Omega_{D_0}}\right) + \frac{\beta u_0}{2}\left( \frac{1-\Omega_{k_0}}{1-\Omega_{D_0} - \Omega_{k_0}} \right)\right] , \label{omega0}
\end{eqnarray}
which is the same result of the non interacting case.
In order to find the expression of the parameter $\omega_1$, we follow the same procedure of the non interacting case.
Using the definition of the evolutionary form of the fractional energy density of DE $\Omega_D'$ given in Eq. (<ref>), we obtain that:
\begin{eqnarray}
\frac{\Omega_D'}{\Omega_D}= \frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \frac{\left( 1-\Omega_D \right)}{\Omega_D} + \frac{ \beta \left(u -3b^2\right)}{2} \right]. \label{lgo11new}
\end{eqnarray}
Moreover, using the expression of $\Omega_D''$ given in Eq. (<ref>), we obtain that the term $\left(\frac{\Omega_D''}{\Omega_D}\right)$ is equal to:
\begin{eqnarray}
\frac{\Omega_D''}{\Omega_D} = \frac{2}{\beta \Omega_D}\left\{ \Omega_D'\left[\frac{1-2\Omega_D}{n^2\gamma_n} +\alpha + \beta\left(\frac{ u}{2}-1\right) -\frac{3}{2}\beta b^2 \right] \right\} + u'. \label{iele1dar}
\end{eqnarray}
Adding the expressions of $\Omega_k''$ and $\Omega_D^{''}$ given, respectively, in Eqs. (<ref>) and (<ref>), we obtain:
\begin{eqnarray}
\frac{\Omega_D^{''}+\Omega_k^{''}}{1-\Omega_D-\Omega_k} &=& \frac{2}{\beta}\left\{\Omega_D' \left[\frac{1}{n^2\gamma_n} + \frac{\beta \left(u-3b^2\right)}{2\left( 1-\Omega_D-\Omega_k \right)} \right] \right\} \nonumber \\
&& -\frac{2}{\beta}\frac{\Omega_D' + \Omega_k'}{1-\Omega_D-\Omega_k}\left( \frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta \right) \frac{\Omega_D u'}{1-\Omega_D-\Omega_k}. \label{iele2dar}
\end{eqnarray}
Therefore, adding Eqs. (<ref>) and (<ref>), we can write:
\begin{eqnarray}
\frac{\Omega_D''}{\Omega_D} + \frac{\Omega_D^{''}+\Omega_k^{''}}{1-\Omega_D-\Omega_k} &=& \frac{2}{\beta \Omega_D}\left\{ \Omega_D'\left[\frac{1-2\Omega_D}{n^2\gamma_n} +\alpha + \beta\left(\frac{ u}{2}-1\right) -\frac{3}{2}\beta b^2 \right] \right\} \nonumber \\
&&+\frac{2}{\beta}\left\{\Omega_D' \left[\frac{1}{n^2\gamma_n} + \frac{\beta \left(u -3b^2\right)}{2\left( 1-\Omega_D-\Omega_k \right)} \right] \right\} \nonumber \\
&&-\frac{2}{\beta}\frac{\Omega_D' + \Omega_k'}{1-\Omega_D-\Omega_k}\left( \frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta \right) + \frac{u'\left(1-\Omega_k\right)}{1-\Omega_D-\Omega_k}\label{iele2-2}.
\end{eqnarray}
Finally, adding the expressions of $\Omega_k'$ and $\Omega_D^{'}$, given, respectively, in Eqs. (<ref>) and (<ref>), we obtain:
\begin{eqnarray}
\frac{\Omega_D'+\Omega_k'}{1-\Omega_D-\Omega_k} = \frac{2}{\beta}\left[\left( \frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta \right) +\frac{\beta \left(u-3b^2\right) \Omega_D}{2\left( 1-\Omega_D-\Omega_k\right)} \right].\label{iele3}
\end{eqnarray}
Moreover, inserting the results of above equations in Eq. (<ref>), we obtain the following expression for $\omega_1$:
\begin{eqnarray}
\omega_1 &=& \frac{2b^2}{\beta\left(1 - \Omega_{k_0} - \Omega_{D_0}\right)}\left[\left(\frac{\Omega_{D_0}}{n^2\gamma_{n_0}} - \alpha + \beta\right)\left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} - \frac{3}{2}\Omega_{D_0} \beta b^2 \right]\nonumber \\
&& +\frac{2b^2\Omega_{D_0}}{\beta\left(1 - \Omega_{k_0} - \Omega_{D_0}\right)}\left[\left( \frac{\Omega_{D_0}}{n^2\gamma_{n_0}} -\alpha + \beta \right) +\frac{\beta \left(u_0-3b^2\right) \Omega_{D_0}}{2\left( 1-\Omega_{D_0}-\Omega_{k_0}\right)} \right] \nonumber \\
&& -\frac{4}{3\beta^2}\left[\left(\frac{\Omega_{D_0}}{n^2\gamma_{n_0}} - \alpha + \beta\right) \frac{\left( 1-\Omega_{D_0} \right)}{\Omega_{D_0}} + \frac{ \beta \left(u_0 -3b^2\right)}{2} \right]^2 \nonumber \\
&& +\frac{4}{3\beta^2 \Omega_{D_0}}\left\{ \left[\left(\frac{\Omega_{D_0}}{n^2\gamma_{n_0}} - \alpha + \beta\right)\left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} - \frac{3}{2}\Omega_{D_0} \beta b^2 \right]\times \right. \nonumber \\
&&\left. \left[\frac{1-2\Omega_{D_0}}{n^2\gamma_{n_0}} +\alpha + \beta\left(\frac{ u_0}{2}-1\right) -\frac{3}{2}\beta b^2 \right] \right\} \nonumber \\
&&+\frac{4}{3\beta^2}\left\{\left[\left(\frac{\Omega_{D_0}}{n^2\gamma_{n_0}} - \alpha + \beta\right)\left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} - \frac{3}{2}\Omega_{D_0} \beta b^2 \right] \times \right. \nonumber \\
&&\left. \left[\frac{1}{n^2\gamma_{n_0}} + \frac{\beta \left(u_0 -3b^2\right)}{2\left( 1-\Omega_{D_0}-\Omega_{k_0} \right)} \right] \right\} \nonumber \\
&&-\frac{4}{3\beta^2}\left[\left( \frac{\Omega_{D_0}}{n^2\gamma_{n_0}} -\alpha + \beta \right) +\frac{\beta \left(u_0-3b^2\right) \Omega_{D_0}}{2\left( 1-\Omega_{D_0}-\Omega_{k_0}\right)} \right]\left( \frac{\Omega_{D_0}}{n^2\gamma_{n_0}} -\alpha + \beta \right) \nonumber \\
&&+\left\{ \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_{n_0}} + \frac{\beta - \alpha}{\Omega_{D_0}} \right) \left[\Omega_{k_0} - \left(u_0+1 \right) \left(1-\Omega_{D_0} \right) \right] -\left(u_0 -3b^2 \right)\left( u_0+1 \right) \right\}\times \nonumber \\
&&\frac{1-\Omega_{k_0}}{3\left(1-\Omega_{D_0}-\Omega_{k_0}\right)} \nonumber \\
&& +\frac{4}{3\beta^2}\left[\left( \frac{\Omega_{D_0}}{n^2\gamma_{n_0}} -\alpha + \beta \right) +\frac{\beta \left(u_0-3b^2\right) \Omega_{D_0}}{2\left( 1-\Omega_{D_0}-\Omega_{k_0}\right)} \right]^2\label{omega1nino}.
\end{eqnarray}
Considering the expression of $\gamma_{n_0}$ given in Eq. (<ref>), we can rewrite Eqs. (<ref>) and (<ref>) as follows:
\begin{eqnarray}
\omega_0 &=& -\frac{2}{3\beta}\left\{\left[\frac{\left(3\lambda -1 \right)}{2n^2 } + \frac{\beta - \alpha}{\Omega_{D_0}}\right] + \frac{\beta u_0}{2}\left( \frac{1-\Omega_{k_0}}{1-\Omega_{D_0} - \Omega_{k_0}} \right)\right\} , \label{omega0-1}
\end{eqnarray}
\begin{eqnarray}
\omega_1 &=& \frac{2b^2}{\beta\left(1 - \Omega_{k_0} - \Omega_{D_0}\right)}\left\{\left[\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } - \alpha + \beta\right]\left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} - \frac{3}{2}\Omega_{D_0} \beta b^2 \right\}\nonumber \\
&& +\frac{2b^2\Omega_{D_0}}{\beta\left(1 - \Omega_{k_0} - \Omega_{D_0}\right)}\left\{\left[ \frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } -\alpha + \beta \right] +\frac{\beta \left(u_0-3b^2\right) \Omega_{D_0}}{2\left( 1-\Omega_{D_0}-\Omega_{k_0}\right)} \right\} \nonumber \\
&& -\frac{4}{3\beta^2}\left\{\left[\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } - \alpha + \beta\right] \frac{\left( 1-\Omega_{D_0} \right)}{\Omega_{D_0}} + \frac{ \beta \left(u_0 -3b^2\right)}{2} \right\}^2 \nonumber \\
&& +\frac{4}{3\beta^2 \Omega_{D_0}}\left\{ \left[\left(\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } - \alpha + \beta\right)\left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} - \frac{3}{2}\Omega_{D_0} \beta b^2 \right]\times \right. \nonumber \\
&&\left. \left[\frac{\left(1-2\Omega_{D_0}\right)\left(3\lambda -1 \right)}{2n^2 } +\alpha + \beta\left(\frac{ u_0}{2}-1\right) -\frac{3}{2}\beta b^2 \right] \right\} \nonumber \\
&&+\frac{4}{3\beta^2}\left\{\left[\left(\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } - \alpha + \beta\right)\left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} - \frac{3}{2}\Omega_{D_0} \beta b^2 \right] \times \right. \nonumber \\
&&\left. \left[\frac{\left(3\lambda -1 \right)}{2n^2 } + \frac{\beta \left(u_0 -3b^2\right)}{2\left( 1-\Omega_{D_0}-\Omega_{k_0} \right)} \right] \right\} \nonumber \\
&&-\frac{4}{3\beta^2}\left\{\left[ \frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } -\alpha + \beta \right] +\frac{\beta \left(u_0-3b^2\right) \Omega_{D_0}}{2\left( 1-\Omega_{D_0}-\Omega_{k_0}\right)} \right\}\left[ \frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } -\alpha + \beta \right] \nonumber \\
&&+\left\{ \frac{2}{\beta}\left[ \frac{\left(3\lambda -1 \right)}{2n^2 } + \frac{\beta - \alpha}{\Omega_{D_0}} \right] \left[\Omega_{k_0} - \left(u_0+1 \right) \left(1-\Omega_{D_0} \right) \right] -\left(u_0 -3b^2 \right)\left( u_0+1 \right) \right\}\times \nonumber\\
&&\frac{1-\Omega_{k_0}}{3\left(1-\Omega_{D_0}-\Omega_{k_0}\right)} \nonumber \\
&& +\frac{4}{3\beta^2}\left\{\left[ \frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } -\alpha + \beta \right] +\frac{\beta \left(u_0-3b^2\right) \Omega_{D_0}}{2\left( 1-\Omega_{D_0}-\Omega_{k_0}\right)} \right\}^2. \label{omega1nino-1}
\end{eqnarray}
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$:
\begin{eqnarray}
\omega_0 &\approx& -1.81193, \\
\omega_1 &\approx&\left(0.416014 -0.74713 b^2\right),
\end{eqnarray}
therefore we obtain the following equation for the EoS parameter of DE $\omega_D$ as function of the redshift $z$:
\begin{eqnarray}
\omega_D &\approx& -1.81193 + \left(0.416014 -0.74713 b^2\right) z\label{ceizze2int}.
\end{eqnarray}
In Figure <ref>, we plot the behavior of $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 1.02$.
For $z=0$, Eq. (<ref>) leads to $\omega_D \approx -1.81193 $ while the value $\omega_D = -1$ is obtained for $z \approx \frac{0.81193}{\left(0.416014 -0.74713 b^2\right)}$.
For $b^2=0.025$, we obtain:
\begin{eqnarray}
\omega_D &\approx& -1.81193 + 0.39733 z\label{ceizze2int2},
\end{eqnarray}
while $\omega_D = -1$ is obtained for $z \approx 2.04$.
Moreover, for $\lambda = 0.98$, we obtain:
\begin{eqnarray}
\omega_0 &\approx& -1.69188, \\
\omega_1 &\approx& \left(0.388546 -0.74713 b^2\right),
\end{eqnarray}
therefore we obtain the following equation for the EoS parameter of DE $\omega_D$ as function of the redshift $z$:
\begin{eqnarray}
\omega_D &\approx& -1.69188 + \left(0.388546 -0.74713 b^2\right) z.\label{ceizze1intlalla}
\end{eqnarray}
In Figure <ref>, we plot the behavior of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 0.98$.
For $z=0$, Eq. (<ref>) leads to $\omega_D \approx 1.69188 $ while the value $\omega_D = -1$ is obtained for $z\approx \frac{0.69188}{\left(0.388546 -0.74713 b^2\right) }$.
For $b^2=0.025$, we obtain:
\begin{eqnarray}
\omega_D &\approx& -1.69188 + 0.36987 z,\label{ceizze1int}
\end{eqnarray}
while $\omega_D = -1$ is obtained for $z \approx 1.87$.
Finally, for $\lambda = 1.00$, we obtain the following values for $\omega_0$ and $\omega_1$:
\begin{eqnarray}
\omega_0 &\approx& -1.75191, \\
\omega_1 &\approx& \left(0.40228 -0.74713 b^2\right),
\end{eqnarray}
therefore we obtain the following equation for the EoS parameter of DE $\omega_D$ as function of the redshift $z$:
\begin{eqnarray}
\omega_D &\approx& -1.75191 +\left(0.40228 -0.74713 b^2\right) z.\label{ceizzeintint}
\end{eqnarray}
In Figure <ref>, we plot the behavior of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 1.00$.
For $z=0$, Eq. (<ref>) leads to $\omega_D \approx -1.75191 $ while the value $\omega_D = -1$ is obtained for $z \approx \frac{-1.75191}{\left(0.40228 -0.74713 b^2\right)}$.
For $b^2=0.025$, we obtain:
\begin{eqnarray}
\omega_D &\approx& -1.75191 +0.38360 z,\label{ceizzeintint2}
\end{eqnarray}
while $\omega_D = -1$ is obtained for for $z \approx 1.96$.
We now consider the limiting case corresponding to the Ricci scale, which is recovered for $\alpha =2$ and $\beta =1$.
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$:
\begin{eqnarray}
\omega_0 &\approx& -0.552428, \\
\omega_1 &\approx& \left(0.247249 - 1.44407 b^2\right) z,
\end{eqnarray}
therefore we obtain the following equation for the EoS parameter of DE $\omega_D$ as function of the redshift $z$:
\begin{eqnarray}
\omega_D &\approx& -0.552428 + \left(0.247249 - 1.44407 b^2\right) z\label{ceizze2intricci}.
\end{eqnarray}
In Figure <ref>, we plot the behavior of $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 1.02$ for the Ricci scale case.
For $z=0$, Eq. (<ref>) leads to $\omega_D \approx -0.552428 $ while the value $\omega_D = -1$ is obtained for $z \approx \frac{-0.447572 }{\left(0.247249 - 1.44407 b^2\right) }$.
For $b^2=0.025$, we obtain:
\begin{eqnarray}
\omega_D &\approx& -0.552428 + 0.211115 z\label{ceizze2intricci2},
\end{eqnarray}
while $\omega_D = -1$ is obtained for $z \approx -2.11$.
Moreover, for $\lambda = 0.98$, we obtain:
\begin{eqnarray}
\omega_0 &\approx& -0.492207, \\
\omega_1 &\approx& \left(0.220506 - 1.44407 b^2\right) z,
\end{eqnarray}
therefore we obtain the following equation for the EoS parameter of DE $\omega_D$ as function of the redshift $z$:
\begin{eqnarray}
\omega_D &\approx& -0.492207 + \left(0.220506 - 1.44407 b^2\right) z.\label{ceizze1intricci}
\end{eqnarray}
In Figure <ref>, we plot the behavior of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 0.98$ for the Ricci scale case.
For $z=0$, Eq. (<ref>) leads to $\omega_D \approx -0.492207 $ while the value $\omega_D = -1$ is obtained for $z\approx \frac{-0.507793}{\left(0.220506 - 1.44407 b^2\right)} $.
For $b^2=0.025$, we obtain:
\begin{eqnarray}
\omega_D &\approx& -0.492207 + 0.188958z.\label{ceizze1intricci2}
\end{eqnarray}
while $\omega_D = -1$ is obtained for $z \approx -2.69 $.
Finally, for $\lambda = 1.00$, we obtain the following values for $\omega_0$ and $\omega_1$:
\begin{eqnarray}
\omega_0 &\approx& -0.522318, \\
\omega_1 &\approx& \left(0.233877 - 1.44407 b^2\right) z ,
\end{eqnarray}
therefore we obtain the following equation for the EoS parameter of DE $\omega_D$ as function of the redshift $z$:
\begin{eqnarray}
\omega_D &\approx& -0.522318 + \left(0.233877 - 1.44407 b^2\right) z.\label{ceizzeintintricci}
\end{eqnarray}
In Figure <ref>, we plot the behavior of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>).
Plot of the EoS parameter of DE $\omega_D$ given in Eq. (<ref>) for $\lambda = 1.00$ for the Ricci scale case.
For $z=0$, Eq. (<ref>) leads to $\omega_D \approx -0.522318 $ while the value $\omega_D = -1$ is obtained for $z \approx \frac{-0.477682}{\left(0.233877 - 1.44407 b^2\right)}$.
For $b^2=0.025$, we obtain:
\begin{eqnarray}
\omega_D &\approx& -0.522318 + 0.197775 z.\label{ceizzeintintricci2}
\end{eqnarray}
while $\omega_D = -1$ is obtained for $z \approx -2.42$.
We can clearly see in Eq. (<ref>) that the interaction parameter $b^2$ plays an important role in the final expression of the parameter $\omega_1$ for the interacting case. We can also observe that,
in the limiting case corresponding to $b^2=0$, the expression of the parameter $\omega_1$ given in Eq. (<ref>) leads to the expression for non-interacting case, given in Eq. (<ref>).
§ STATEFINDER DIAGNOSTIC
The study and the investigation of cosmological quantities like for example the Hubble parameter $H$, the EoS parameter of DE $\omega_D$ and the deceleration parameter $q$ have attracted a lot of
attention in present day cosmology. Since the different DE models usually lead to a positive Hubble parameter $H$ and a negative deceleration parameter $q$, i.e. to to $H > 0$ and $q < 0$, at the present epoch, the Hubble and the deceleration parameters $H$ and $q$ can not effectively discriminate
between the various DE models. We have, therefore, that a higher order of time derivative of the scale factor $a\left( t \right)$ is required in order to have a better and deeper comprehension and understanding of the DE model taken into account. Sahni et al. <cit.>
and Alam et al. <cit.>, using the third time derivative of the scale factor $a(t)$, introduced the statefinder pair $\left\{r,s\right\}$ with the aim to remove the problems related to the values assumed by the Hubble parameter $H$ and the deceleration parameter $q$ at the present epoch. The general expressions of the statefinder parameters $r$ and $s$ are given, respectively, by the following relations:
\begin{eqnarray}
r &=& \frac{{ {...} \atop a}}{aH^3}, \label{r1}\\
s &=& \frac{r -1}{3\left(q-1/2\right)}, \label{s1}
\end{eqnarray}
The statefinder parameters can be also written as functions of the total energy density $\rho$ and the total pressure $p$ of the model considered, defined as $\rho = \rho_D + \rho_m$ and $p = p_D$, as follows:
\begin{eqnarray}
r &=& 1 + \frac{9}{2}\left(\frac{\rho + p}{\rho}\right)\frac{\dot{p}}{\dot{\rho}} \nonumber \\
&=& 1 + \frac{9}{2}\left(\frac{\rho_m + \rho_D + p_D}{\rho_m + \rho_D}\right)\frac{\dot{p}_D}{\dot{\rho}_m+\dot{\rho}_D} , \label{34p2}\\
s&=& \left(\frac{\rho + p}{p}\right)\frac{\dot{p}}{\dot{\rho}} \nonumber \\
&=& \left(\frac{\rho_m +\rho_D + p_D}{p_D}\right)\frac{\dot{p}_D}{\dot{\rho}_m+\dot{\rho}_D}, \label{35p2}
\end{eqnarray}
which can be also written as follows:
\begin{eqnarray}
r &=& 1 + \frac{9}{2}\left(\frac{\rho + p}{\rho}\right)\frac{p'}{\rho'}\nonumber \\
&=& 1 + \frac{9}{2}\left(\frac{\rho_m + \rho_D + p_D}{\rho_m + \rho_D}\right)\frac{p_D'}{\rho_m' + \rho_D'}, \label{34p2}\\
s&=& \left(\frac{\rho + p}{p}\right)\frac{p'}{\rho'} \nonumber \\
&=& \left(\frac{\rho_m + \rho_D + p_D}{p_D}\right)\frac{p_D'}{\rho_m' + \rho_D'}. \label{35p2}
\end{eqnarray}
An alternative way to write the statefinder parameters $r$ and $s$ involves the Hubble parameter $H$ and its higher time derivatives as follows:
\begin{eqnarray}
r&=& 1 + 3\left(\frac{\dot{H}}{H^2}\right)+ \frac{\ddot{H}}{H^3}, \label{r2}\\
s&=& -\frac{3H\dot{H}+\ddot{H}}{3H\left( 2\dot{H}+3H^2 \right)}\nonumber \\
&=& -\frac{3\dot{H}+\ddot{H}/H}{3\left( 2\dot{H}+3H^2 \right)}. \label{s2}
\end{eqnarray}
The most important features of the statefinder parameters which must be taken into account when a particular DE model is studied is that the point with coordinate corresponding to $\left\{r, s\right\} = \left\{1, 0\right\}$ in the $\left\{r, s\right\}$ plane indicates the fixed point corresponding to the flat $\Lambda$CDM model. Departures of given DE models from this fixed point are good ways to establish the distance of these models from the flat $\Lambda$CDM model.
Moreover, we must underline here that, in the $\left \{r, s\right \}$ plane, the sector of positive $s$, i.e. $s > 0$, corresponds to quintessence-like models of DE while the sector of negative $s$, i.e. $s < 0$, corresponds to phantom-like models of DE. Furthermore, an evolution from phantom to quintessence or from quintessence to phantom is given by crossing of the fixed point $\left\{r, s\right\} = \left\{1, 0\right\}$ corresponding to the $\Lambda$CDM in the $\left \{r, s \right \}$ plane <cit.>.
Braneworld, Cosmological Constant $\Lambda_{CC}$, Chaplygin gas and quintessence models were investigated by Alam et al. <cit.> using the statefinder diagnostic: they observed that the statefinder pair could differentiate between these different proposed models. An investigation on statefinder parameters for differentiating between DE and modified gravity was carried out in the paper of Wang et al. <cit.>. Statefinder diagnostics for the $f\left(T\right)$ modified gravity model has been studied in the paper of Wu $\&$ Yu <cit.>.
Other authors have been studied the properties of various DE models from the viewpoint of statefinder diagnostic <cit.>.
We now want to study the statefinder pair for the model considered in this paper, for this reason we need to derive the quantities useful in order to obtain the final expression of the pair $\left\{ r,s \right\}$. We underline here that for the statefinder parameter $r$ we will use the expression given in Eq. (<ref>) since it will be easier to calculate the terms involved, in fact we have already obtained the expression of $\left(\frac{\dot{H}}{H^2}\right)$.
§.§ Non Interacting Case
We start studying the behavior of the statefinder parameters for the case corresponding to the non interacting Dark Sectors in order to find if the model we are considering leads to a point close to the one of the $\Lambda$CDM model or it has a departure from it.
We have already derived the expression of $\left(\frac{\dot{H}}{H^2}\right)$ in Eq. (<ref>). Differentiating the expression of $\left(\frac{\dot{H}}{H^2}\right)$ given in Eq. (<ref>) with respect to the cosmic time $t$, we obtain:
\begin{eqnarray}
\frac{d}{dt} \left( \frac{\dot{H}}{H^2} \right)= \frac{\ddot{H}}{H^2} -2\left(\frac{\dot{H}^2}{H^3}\right). \label{jorge}
\end{eqnarray}
Dividing the result of Eq. (<ref>) by the Hubble parameter $H$, after some algebraic calculations, we can easily find the following expression for the term $\left(\frac{\ddot{H}}{H^3}\right)$:
\begin{eqnarray}
\frac{\ddot{H}}{H^3} &=& 2\left( \frac{\dot{H}}{H^2} \right)^2 + \left( \frac{\dot{H}}{H^2} \right)'. \label{}
\end{eqnarray}
Then, we have that the final expression of the statefinder parameter $r$ can be written as follows:
\begin{eqnarray}
r&=& 1 + 3\left(\frac{\dot{H}}{H^2}\right) + 2\left( \frac{\dot{H}}{H^2} \right)^2 + \left( \frac{\dot{H}}{H^2} \right)' . \label{}
\end{eqnarray}
Differentiating the expression of $\left( \frac{\dot{H}}{H^2} \right)$ obtained in Eq. (<ref>) with respect to the variable $x$, we find the following expression for $\left( \frac{\dot{H}}{H^2} \right)'$:
\begin{eqnarray}
\left( \frac{\dot{H}}{H^2} \right)' = \frac{1}{\beta}\left(\frac{\Omega'_D}{n^2 \gamma_n} \right). \label{}
\end{eqnarray}
Therefore, we conclude that $r$ can be written as follows:
\begin{eqnarray}
r&=& 1 + \frac{3}{\beta} \left(\frac{\Omega_D}{n^2\gamma_n} -\alpha \right) \nonumber \\
&& +\frac{2}{\beta^2} \left(\frac{\Omega_D}{n^2\gamma_n} -\alpha \right)^2 + \frac{1}{\beta}\left(\frac{\Omega'_D}{n^2 \gamma_n} \right). \label{rfinalenino}
\end{eqnarray}
Inserting in Eq. (<ref>) the expression of the evolutionary form of the fractional energy density of DE $\Omega_D'$ obtained in Eq. (<ref>), we obtain the following expression for the statefinder parameter $r$:
\begin{eqnarray}
r&=& 1 + \frac{3}{\beta} \left(\frac{\Omega_D}{n^2\gamma_n} -\alpha \right) +\frac{2}{\beta^2} \left(\frac{\Omega_D}{n^2\gamma_n} -\alpha \right)^2 \nonumber \\
&& + \frac{1}{\beta n^2 \gamma_n}\left\{\frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} \right] \right\}. \label{nataliar}
\end{eqnarray}
Considering the present day values of the quantities involved in the final expression of the statefinder parameter $r$ given in Eq. (<ref>) and using the expression of $\gamma_{n_0}$ obtained in Eq. (<ref>), we have that the expression of the present day value of the statefinder parameter $r$ can be also written as follows:
\begin{eqnarray}
r_0&=& 1 + \frac{3}{\beta} \left[\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2} -\alpha \right] +\frac{2}{\beta^2} \left[\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2} -\alpha \right]^2\nonumber \\
&& + \frac{\left(3\lambda -1\right)}{\beta^2n^2}\left\{\left[\frac{\left( 3\lambda -1 \right)\Omega_{D_0}}{2n^2} - \alpha + \beta\right] \left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} \right\} . \label{rzerofinalenino}
\end{eqnarray}
The present day value of the statefinder parameter $s$ can be obtained from Eq. (<ref>) and it is given by:
\begin{eqnarray}
s_0 = \frac{r_0 -1}{3\left(q_0-1/2\right)}, \label{s11}
\end{eqnarray}
where $r_0$ is given in Eq. (<ref>) while $q_0$ is given in Eq. (<ref>).
Inserting in Eq. (<ref>) the values of the parameters involved, we find that $r_0\approx 6.00853$ for $\lambda =1.02$, $r_0 \approx 5.0452$ for $\lambda =0.98$ and $r_0 \approx 5.51566$ for $\lambda =1.00$.
Using the values of $r_0$ derived above along with the values of $q_0$ obtained in previous Section, we can easily obtain that $s_0 \approx -0.889295$ for $\lambda =1.02$, $s_0\approx -0.769247$ for $\lambda =0.98$ and $s_0 \approx -0.829271$ for $\lambda =1.00$.
Therefore we obtain the following pairs of values: $\left\{ r_0,s_0 \right\} \approx \left\{ 6.00853, -0.889295\right\}$ for $\lambda =1.02$, $\left\{ r_0,s_0 \right\} \approx \left\{ 5.0452,-0.769247 \right\}$ for $\lambda =0.98$ and $\left\{ r_0,s_0 \right\} \approx \left\{ 5.51566, -0.829271\right\}$ for $\lambda =1.00$.
We can observe that the values of the statefinder pair $\left\{ r,s \right\}$ for all cases of the running parameter $\lambda$ taken into account considerably differs from the values corresponding to the $\Lambda$CDM model. Moreover, since we have $s<0$ for all cases considered, we can conclude that we are dealing with a phantom-like model.
We now consider the limiting case corresponding to the Ricci scale, which is recovered for $\alpha =2$ and $\beta =1$.
Inserting in Eq. (<ref>) the values of the parameters involved, we find that $r_0\approx 0.489371$ for $\lambda =1.02$, $r_0 \approx 0.453289$ for $\lambda =0.98$ and $r_0 \approx 0.46851 $ for $\lambda =1.00$.
Using the values of $r_0$ derived above along with the values of $q_0$ obtained in previous Section, we can easily obtain that $s_0 \approx 0.297739$ for $\lambda =1.02$, $s_0\approx 0.357856$ for $\lambda =0.98$ and $s_0 \approx 0.3278$ for $\lambda =1.00$.
Therefore we obtain the following pairs of values: $\left\{ r_0,s_0 \right\} \approx \left\{ 0.489371,0.297739 \right\}$ for $\lambda =1.02$, $\left\{ r_0,s_0 \right\} \approx \left\{ 0.453289, 0.357856\right\}$ for $\lambda =0.98$ and $\left\{ r_0,s_0 \right\} \approx \left\{ 0.46851, 0.3278\right\}$ for $\lambda =1.00$.
We can clearly observe that, in this case, we obtain values which are closer to the point corresponding to the $\Lambda$CDM model if compared with the result of the other set of values of $\alpha$ and $\beta$. Moreover, since we obtained $s>0$ for all cases considered, we have that for the limiting case of Ricci scale we deal with a quintessence-like model.
§.§ Interacting Case
We now consider the case corresponding to presence of interaction between the Dark Sectors.
Following the same procedure of the non interacting case, we have that the expression of the statefinder parameter $r$ for the interacting case can be written as follows:
\begin{eqnarray}
r_0&=& 1 + \frac{3}{\beta} \left[\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2} -\alpha \right] \nonumber \\
&& +\frac{2}{\beta^2} \left[\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2} -\alpha \right]^2\nonumber \\
&& + \frac{\left(3\lambda -1\right)}{\beta^2n^2}\left\{\left[\frac{\left( 3\lambda -1 \right)\Omega_{D_0}}{2n^2} - \alpha + \beta\right] \left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2}\left( u_0 -3b^2 \right)\right\}. \label{rzerofinaleninoint}
\end{eqnarray}
The present day value of the statefinder parameter $s$ can be obtained from Eq. (<ref>) and it is given by:
\begin{eqnarray}
s_0 = \frac{r_0 -1}{3\left(q_0-1/2\right)}, \label{s11int}
\end{eqnarray}
where $r_0$ has been obtained in Eq. (<ref>) while $q_0$ has been obtained in Eq. (<ref>).
Inserting in Eq. (<ref>) the values of the parameters involved, we find that $r_0\approx 6.00853 - 6.40954 b^2$ for $\lambda =1.02$, $r_0\approx 5.0452 - 6.03617 b^2$ for $\lambda =0.98$ and $r_0 \approx 5.51566 - 6.22285 b^2$ for $\lambda = 1.00$.
Using the expression of $r_0$ derived above along with the expression of $q_0$ obtained in previous Section, we can easily obtain that $s_0 \approx -0.889295 + 1.13805 b^2$ for $\lambda =1.02$, $s_0\approx -0.769247 + 1.14785 b^2$ for $\lambda =0.98$ while $s_0\approx -0.829271 + 1.14278 b^2$ for $\lambda =1.00$.
Therefore, we obtain $\left\{ r_0,s_0 \right\} \approx \left\{ 6.00853 - 6.40954 b^2, -0.889295 + 1.13805 b^2 \right\}$ for $\lambda = 1.02$, $\left\{ r_0,s_0 \right\} \approx \left\{ 5.0452 - 6.03617 b^2, -0.769247 + 1.14785 b^2\right\}$ for $\lambda = 0.98$ and $\left\{ r_0,s_0 \right\} \approx \left\{ 5.51566 - 6.22285 b^2,-0.829271 + 1.14278 b^2 \right\}$ for $\lambda = 1.00$.
In the limiting case of $b^2 = 0.025$, we obtain $\left\{ r_0,s_0 \right\} \approx \left\{ 5.84829, -0.86084 \right\}$ for $\lambda = 1.02$, $\left\{ r_0,s_0 \right\} \approx \left\{ 4.89429, -0.74055 \right\}$ for $\lambda = 0.98$ and $\left\{ r_0,s_0 \right\} \approx \left\{ 5.36009,-0.80070\right\}$ for $\lambda = 1.00$. Moreover, in the limiting case of $b^2=0$, we recover the same results of the non interacting case.
We obtain, then, that also in the interacting case, the values of the pair $\left\{ r,s \right\}$ differ (even if they are closer with respect to the non interacting case) from the $\Lambda$CDM model for all the cases of the running parameter $\lambda$ considered. Moreover, since we obtain $s<0$ for all the cases of the running parameter $\lambda$ considered, we conclude that we deal with a phantom-like model for the set of values of the parameters considered.
We now consider the limiting case corresponding to the Ricci scale, then for $\alpha =2$ and $\beta=1$.
Inserting in Eq. (<ref>) the values of the parameters involved, we find that $r_0\approx 0.489371 - 3.21502 b^2$ for $\lambda =1.02$, $r_0\approx 0.453289 - 3.02774 b^2$ for $\lambda =0.98$ while $r_0\approx 0.46851 - 3.12138 b^2$ for $\lambda =1.00$.
Using the expressions of $r_0$ derived above along with the expressions of $q_0$ obtained in previous Section, we can easily obtain that $s_0 \approx 0.297739 + 1.87462 b^2$ for $\lambda =1.02$, $s_0\approx 0.357856 + 1.98184 b^2 b^2$ for $\lambda =0.98$ while $s_0\approx 0.3278 + 1.92514 b^2$ for $\lambda =1.00$.
Therefore we obtain the following pairs of values: $\left\{ r_0,s_0 \right\} \approx \left\{ 0.489371 - 3.21502 b^2,0.297739 + 1.87462 b^2 \right\}$ for $\lambda =1.02$, $\left\{ r_0,s_0 \right\} \approx \left\{ 0.453289 - 3.02774 b^2,0.357856 + 1.98184 b^2 \right\}$ for $\lambda =0.98$ and $\left\{ r_0,s_0 \right\} \approx \left\{ 0.46851 - 3.12138 b^2,0.3278 + 1.92514 b^2 \right\}$ for $\lambda =1.00$.
In the limiting case of $b^2 = 0.025$, we obtain $\left\{ r_0,s_0 \right\} \approx \left\{0.40900 ,0.34460 \right\}$ for $\lambda = 1.02$, $\left\{ r_0,s_0 \right\} \approx \left\{ 0.37760,0.40740 \right\}$ for $\lambda = 0.98$ and $\left\{ r_0,s_0 \right\} \approx \left\{0.390476, 0.37593\right\}$ for $\lambda = 1.00$. Therefore, for $b^2=0.025$, the results obtained lead to a values of the statefinder parameters which are more distant with respect to the point $\left\{ r_0,s_0 \right\} = \left\{1,0 \right\}$ corresponding to the $\Lambda$CDM model. Moreover, since we obtain that $s>0$ for all the cases of the running parameter $\lambda$ considered, we conclude we deal with a quintessence-like model for the set of values of the parameters taken into account. Furthermore, in the limiting case of $b^2=0$, we recover the same results of the non interacting case.
§ COSMOGRAPHIC PARAMETERS
In this Section, we want to obtain some important cosmological information about the PLECHDE model with Granda-Oliveros cut-off we are considering using the properties of the cosmographic parameters.
Standard candles (like SNe Ia) represent powerful instruments in present day cosmology since they can be used in order to reconstruct the Hubble diagram, i.e. the redshift-distance relation up to high redshifts $z$. It is quite common to constrain a parameterized model against available cosmological data in order to check the validity of the model considered and in order to constraint the free parameters of the model. However, it is known that this type of approach is highly model-dependent, for this reason there are still doubts in scientific community on the validity and reliability of the constraints on the derived cosmological quantities obtained with this method.
In order to avoid this kind of problem, it is possible to consider the cosmography, i.e. it is possible to expand the scale factor $a\left( t \right)$ in Taylor series with respect to the cosmic time $t$. This type of expansion produces a distance-redshift relation which is based only on the assumption of the FLRW metric, therefore it is fully model independent because it is independent on the particular form of the solution of the cosmic equations. Cosmography can be considered as a milestone in the study of the main properties of Universe dynamics, which any theoretical model studied and considered has to take into account and also to satisfy.
It is useful to introduce the following four quantities <cit.>:
\begin{eqnarray}
q &=& -\left(\frac{\ddot{a}}{a}\right)H^{-2}= -\frac{\ddot{a}a}{\dot{a}^2} = -\frac{a^{\left(2\right)}a}{\dot{a}^2}, \label{par1} \\
j &=& \left(\frac{1}{a}\frac{d^3a}{dt^3}\right)H^{-3} = \frac{a^{\left(3\right)}a^2}{\dot{a}^3}, \label{par2} \\
s &=& \left(\frac{1}{a}\frac{d^4a}{dt^4}\right)H^{-4}= -\frac{a^{\left(4\right)}a^3}{\dot{a}^4}, \label{par3} \\
l &=& \left(\frac{1}{a}\frac{d^5a}{dt^5}\right)H^{-5}= \frac{a^{\left(5\right)}a^4}{\dot{a}^5}, \label{par4}
\end{eqnarray}
where the number in parenthesis indicates the order of the derivative with respect to the cosmic time $t$ while the numbers without parenthesis indicates the power of the relevant quantity.
In general, we have that the $i$-th parameter $x^i$ can be obtained thanks to the following general expression:
\begin{eqnarray}
x^{i} &=& \left( -1 \right)^{i+1}\left(\frac{1}{H^{i}}\right)\frac{a^{\left(i\right)}}{a}\nonumber \\
&=& \left( -1 \right)^{i+1} \frac{a^{\left(i\right)}a^{i-1}}{\dot{a}^{i+1}}.
\end{eqnarray}
The quantities given in Eqs. (<ref>), (<ref>), (<ref>) and (<ref>) are known, respectively, as deceleration, jerk, snap and lerk parameters. We must underline that we have already derived the expressions of the deceleration parameter $q$.
In order to avoid confusion with the statefinder parameter $s$, we will denote since now the snap parameter with $s_{cosmo}$.
Making some algebraic calculations, we can obtain the following useful relations between the time derivatives of the Hubble parameter and the cosmographic parameters $q$, $j$, $s_{cosmo}$ and $l$:
\begin{eqnarray}
\frac{d H}{dt}&=&\dot{H} = -H^2 \left(1 + q \right), \label{ddd1} \\
\frac{d^2H}{dt^2}&=&\ddot{H}= H^3 \left(j+3q+2 \right), \label{ddd2} \\
\frac{d^3H}{dt^3}&=& \dot{\ddot{H}} =H^4 \left[s_{cosmo} - 4j -3q\left(q+4\right) -6 \right], \label{ddd3} \\
\frac{d^4H}{dt^4}&=& \ddot{\ddot{H}} =H^5 \left[l-5s_{cosmo} +10\left( q+2 \right)j +30\left( q+2 \right)q +24 \right]. \label{ddd4}
\end{eqnarray}
The present-day values of these cosmographic parameters, denoted with the subscript 0 (which indicates the value of the parameter for $z=0$ or equivalently for $t=0$) can be used in order to characterize the evolutionary status of the present day Universe. For example, a negative value of $q_0$ indicates an accelerated expansion of the Universe (as it is suggested by recent cosmological measurements), while the value of $j_0$ allows to discriminate among different accelerating models.
More information on the calculations made in order to obtain the previous equations can be found in the paper of Capozziello et al. <cit.>.
Some constraints about the present day values of the snap parameter $s_{cosmo}$ and of the lerk parameter $l$ have been recently obtained. For example, Capozziello $\&$ Izzo <cit.> have found that $s_{cosmo,0} = 8.32 \pm 12.16$, while John <cit.> has derived that $s_{cosmo,0} = 36.5 \pm 52.9$ and $l_0= 142.7 \pm 320$. As we can clearly see, the errors associated with the values derived in these two works for the snap and lerk cosmographic parameters $s_{cosmo}$ and $l$ are of the order of 200$\%$, for future more precise comparisons between cosmological constraints of $s_{cosmo}$ and $l$ and the values obtained from theoretical models, it will be useful to have better constraints with more accurate errors.
Using the definitions of the cosmographic parameters given in Eqs. (<ref>), (<ref>), (<ref>) and (<ref>), we can easily obtain the fifth order Taylor expansion of the scale factor $a\left( t \right)$ as follows:
\begin{eqnarray}
\frac{a\left(t\right)}{a\left(t_0\right)} &=& 1+H_0 \left( t -t_0 \right) - \left(\frac{q_0}{2}\right)H_0^2 \left( t -t_0 \right)^2 + \left(\frac{j_0}{3!}\right)H_0^3 \left( t -t_0 \right)^3 \nonumber \\
&&+\left(\frac{s_0}{4!}\right)H_0^4 \left( t -t_0 \right)^4 + \left(\frac{l_0}{5!}\right)H_0^5 \left( t -t_0 \right)^5 + O \left[ \left( t-t_0 \right)^6 \right], \label{exp}
\end{eqnarray}
where $t_0$ represents the present day age of the Universe, which is given by $t\approx 1/H_0$. It must be here underlined that Eq. (<ref>) is also the fifth order expansion of $\left(1 + z\right)^{-1}$, since, from the definition of redshift, we obtain:
\begin{eqnarray}
z = \frac{a\left( t_0 \right)}{a\left( t \right)} -1. \label{}
\end{eqnarray}
The jerk parameter $j$ is also another name of the statefinder parameter $r$ we have studied in the previous Section and it represents a natural next step beyond the Hubble parameter $H$ and the deceleration parameter $q$.
The snap parameter $s_{cosmo}$, which involves the fourth time derivative of the scale factor $a\left(t \right)$, is also
sometimes called kerk parameter and it has been well discussed in the works of Dabrowski <cit.>, Dunajski $\&$ Gibbons <cit.> and Arabsalmania $\&$ Sahni <cit.>.
Another useful relation which can be used in order to find the expression of $s_{cosmo}$ involves the deceleration and the jerk parameters and it is given by:
\begin{eqnarray}
s_{cosmo} &=& \frac{\dot{j}}{H}-j\left( 2+3q \right)\nonumber \\
& =& j' -j\left( 2+3q \right). \label{scosmogeneral}
\end{eqnarray}
The lerk parameter $l$ involves the fifth time derivative of scale factor $a\left( t \right)$. More information about the lerk parameter can be found in the paper of Dabrowski <cit.>. An useful relation between the lerk parameter $l$ and the deceleration and snap parameters is given by:
\begin{eqnarray}
l= s_{cosmo}' -\left( 3+4q \right)s_{cosmo}. \label{lcosmo}
\end{eqnarray}
Using the definition of the snap parameter $s_{cosmo}$ given in Eq. (<ref>), we can rewrite Eq. (<ref>) as follows:
\begin{eqnarray}
l= s_{cosmo}'-\left( 3+4q \right)\left[ j'-j\left( 2+3q \right) \right].\label{lcosmo2}
\end{eqnarray}
Moreover, from the definition of the snap parameter $s_{cosmo}$ given in Eq. (<ref>), we obtain the following expression for the derivative of the snap parameter $s_{cosmo}$ with respect to the variable $x$:
\begin{eqnarray}
s_{cosmo}' = j'' - j'\left( 2+3q \right) -3jq'.\label{sprimo}
\end{eqnarray}
Therefore, $l$ can be also written as follows:
\begin{eqnarray}
l&=& j'' - j' \left( 5+7q \right) -j\left( 3q' - 3q -2 \right). \label{lcosmo3}
\end{eqnarray}
§.§ Non Interacting Case
We now want to derive the final expressions for $s_{cosmo}$ and $l$ for the non interacting case.
We start calculating the final expression of the snap parameter $s_{cosmo}$.
Differentiating with respect to the variable $x$ the expression of the statefinder parameter $r$ given in Eq. (<ref>), we find that:
\begin{eqnarray}
j' &\equiv& r' = \left(\frac{\Omega'_{D}}{n^2\gamma_{n}} \right) \left[\frac{3}{\beta} +\frac{4}{\beta^2} \left(\frac{\Omega_{D}}{n^2\gamma_{n}} -\alpha \right) \right] + \frac{1}{\beta}\left(\frac{\Omega''_{D}}{n^2 \gamma_{n}} \right) . \label{jprime}
\end{eqnarray}
Using the expression of $\Omega_D''$ for the non interacting case, we can write $j'$ as follows:
\begin{eqnarray}
j' &=& \frac{1}{\beta n^2\gamma_n} \left[ \Omega_D' \left( \frac{2}{\beta n^2 \gamma_n} - \frac{2\alpha}{\beta} + u -2 \right) + \Omega_D u'\right]. \label{jprime2}
\end{eqnarray}
Using in Eq. (<ref>) the expressions of the evolutionary form of the fractional energy density of DE $\Omega_D'$ and $u'$ for the non interacting case, we obtain:
\begin{eqnarray}
j' &=& \frac{1}{\beta n^2\gamma_n} \left\{ \frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} \right] \left( \frac{2}{\beta n^2 \gamma_n} - \frac{2\alpha}{\beta} + u -2 \right) \right. \nonumber \\
&&\left.+ \frac{2}{\beta}\left( \frac{\Omega_D}{n^2 \gamma_n} + \beta - \alpha \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -u\Omega_D\left( u+1 \right)\right\}. \label{jprime3}
\end{eqnarray}
Therefore, we can conclude that the cosmographic parameter $s_{cosmo,non}$ for the non interacting case can be written as follows:
\begin{eqnarray}
s_{cosmo,non} &=& \frac{1}{\beta n^2\gamma_n} \left\{ \frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} \right]\times \right. \nonumber \\
&&\left. \left( \frac{2}{\beta n^2 \gamma_n} - \frac{2\alpha}{\beta} + u -2 \right) \right. \nonumber \\
&&\left.+ \frac{2}{\beta}\left( \frac{\Omega_D}{n^2 \gamma_n} + \beta - \alpha \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -u\Omega_D\left( u+1 \right)\right\}\nonumber \\
&& -j\left( 2+3q \right). \label{scosmogeneralfinalnon}
\end{eqnarray}
considering the present day values of the parameter involved along with the expression of $\gamma_{n0}$ given in Eq. (<ref>), we can write:
\begin{eqnarray}
s_{cosmo,non0} &=& \left(\frac{3\lambda -1}{2\beta n^2}\right) \left\{ \frac{2}{\beta}\left[\left(\frac{\left( 3\lambda -1 \right)\Omega_{D_0}}{2n^2} - \alpha + \beta\right) \left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} \right] \times \right. \nonumber \\
&&\left. \left( \frac{3\lambda -1}{\beta n^2 } - \frac{2\alpha}{\beta} + u_0 -2 \right) \right. \nonumber \\
&&\left.+ \frac{2}{\beta}\left[ \frac{\left( 3\lambda -1 \right)\Omega_{D_0}}{2n^2 } + \beta - \alpha \right] \times \right. \nonumber \\
&& \left.\left[\Omega_{k_0} - \left(u_0+1 \right) \left(1-\Omega_{D_0} \right) \right] -u_0\Omega_{D_0}\left( u_0+1 \right)\right\}\nonumber \\
&& -j_0\left( 2+3q_0 \right). \label{scosmogeneralfinalnon0}
\end{eqnarray}
Inserting in Eq. (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$, that $s_{cosmo,non0} \approx 17.1515$, for $\lambda = 0.98$ we obtain $s_{cosmo,non0}\approx 11.9$ while for $\lambda = 1.00$ we obtain $s_{cosmo,non0}\approx 14.3768$. We obtain, therefore, values of $s_{cosmo,non0}$ which are between the errors obtained in the works of Capozziello $\&$ Izzo <cit.> and John <cit.>.
We now consider the case corresponding to the Ricci scale, which is recovered in the limiting case of $\alpha =2$ and $\beta =1$.
Inserting in Eq. (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$, that $s_{cosmo,non0} \approx -0.230849$, for $\lambda = 0.98$ we obtain $s_{cosmo,non0}\approx -0.15236$ while for $\lambda = 1.00$ we obtain $s_{cosmo,non0}\approx -0.18851$. Also for the limiting case of the Ricci scale, we obtain values of $s_{cosmo,non0}$ which are between the errors obtained in the works of Capozziello $\&$ Izzo <cit.> and John <cit.>.
We now want to obtain the final expression for the lerk parameter $l$, therefore, in order to use the general expression given in Eq. (<ref>), we need to calculate the expressions of $j''$ and $q'$.
Considering the expression of $r'$ given in Eq. (<ref>), we obtain the following expression for $j''$:
\begin{eqnarray}
j'' &=& r''= \frac{1}{\beta n^2\gamma_n} \left[ \Omega_D'' \left( \frac{2}{\beta n^2 \gamma_n} - \frac{2\alpha}{\beta} + u -2 \right)+2 \Omega_D' u' + \Omega_D u'' \right] . \label{jsecond}
\end{eqnarray}
We have already obtained the expressions of the evolutionary form of the fractional energy density of DE $\Omega_D'$, $\Omega_D''$ and $u'$ for the non interacting case. We now need to find the expression of $u''$ for the non interacting case.
Using the general expression of $u'$, we find the following relation for $u''$:
\begin{eqnarray}
u'' &=& \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega'_k - u' \left(1-\Omega_D \right) + \Omega_D'\left(1+u \right) \right] -u\left( u+1 \right) \nonumber \\
&&-\frac{2}{\beta}\left[ \frac{\left(\beta - \alpha\right)\Omega_D'}{\Omega_D^2} \right] \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] \nonumber \\
&&-u'\left( 2u+1 \right). \label{usecond}
\end{eqnarray}
Inserting the expressions of $\Omega_k'$, $\Omega_D'$ and $u'$ in Eq. (<ref>), we obtain:
\begin{eqnarray}
u'' &=& -\frac{4\Omega_k}{\beta^2}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right)\left( \frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta \right) \nonumber \\
&&-\frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left(1-\Omega_D \right) \times \nonumber \\
&&\left\{ \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -u\left( u+1 \right) \right\}\nonumber \\
&&+\frac{4}{\beta^2}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left(1+u \right) \left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} \right]\nonumber \\
&&-\frac{4}{\beta^2}\left( \frac{\beta - \alpha}{\Omega_D^2} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right]\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} \right] \nonumber \\
&&-\left\{ \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -u\left( u+1 \right)\right\}\left( 2u+1 \right). \label{usecond2}
\end{eqnarray}
The final expression of $j''$ can be obtained inserting in Eq. (<ref>) the expressions of $u'$, $\Omega_D'$ and $u''$ for the non interacting case.
Finally, differentiating the expression of $q$ given in Eq. (<ref>) with respect to the variable $x$, we have that $q'$ is given by:
\begin{eqnarray}
q'= - \frac{1}{\beta}\left( \frac{\Omega_D'}{n^2\gamma_n} \right). \label{qfinale}
\end{eqnarray}
Therefore, for the non interacting case, using in Eq. (<ref>) the expression of the evolutionary form of the fractional energy density of DE $\Omega_D'$ for the non interacting case, we obtain the following expression for $q'$:
\begin{eqnarray}
q'= - \frac{2}{\beta^2}\left( \frac{1}{n^2\gamma_n} \right)\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} \right]. \label{}
\end{eqnarray}
We can now calculate the present day values of the lerk parameter $l$ taking into account all the results of the above equations.
For $\lambda = 1.02$, we obtain that $l_{non,0}\approx 80.5574$, for $\lambda = 0.98$ we derive that $l_{non,0}\approx 57.667$ while for $\lambda = 1.00$ we obtain $l_{non,0} \approx 68.3621$. We obtain, therefore, values of $l_{non0}$ which are between the errors obtained in the work of John <cit.>.
We now consider the case corresponding to the Ricci scale, which is recovered for $\alpha =2$ and $\beta =1$.
Inserting in Eq. (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$, that $l_{non,0} \approx -10.0602$, for $\lambda = 0.98$ we obtain $l_{non,0}\approx -11.2141$ while for $\lambda = 1.00$ we obtain $l_{non,0}\approx -10.6489$. Also for the limiting case of the Ricci scale, we obtain, values of $l_{non0}$ which are between the errors obtained in the work of John <cit.>.
§.§ Interacting Case
We now consider the interacting case, following the same procedure of the non interacting case in order to find the final expressions of $s_{cosmo}$ and $l$.
We start calculating $s_{cosmo}$. We still start from the following equation for $j'$:
\begin{eqnarray}
j' &=& \frac{1}{\beta n^2\gamma_n} \left[ \Omega_D' \left( \frac{2}{\beta n^2 \gamma_n} - \frac{2\alpha}{\beta} + u -2 \right) + \Omega_D u'\right]. \label{jprime2int}
\end{eqnarray}
Instead, using in Eq. (<ref>) the expressions of the evolutionary form of the fractional energy density of DE $\Omega_D'$ and $u'$ for the interacting case, we obtain:
\begin{eqnarray}
j' &=& \frac{1}{\beta n^2\gamma_n} \left\{\frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right)\left( 1-\Omega_D \right) + \frac{\Omega_D \beta }{2}\left( u-3b^2 \right) \right] \times \right. \nonumber \\
&&\left. \left( \frac{2}{\beta n^2 \gamma_n} - \frac{2\alpha}{\beta} + u -2 \right) \right. \nonumber \\
&&\left.+ \frac{2}{\beta}\left( \frac{\Omega_D}{n^2 \gamma_n} + \beta - \alpha \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -\Omega_D\left(u-3b^2\right)\left( u+1 \right)\right\}. \label{}
\end{eqnarray}
Therefore, we can conclude that the cosmographic parameter $s_{cosmo,int}$ for the interacting case can be written as follows:
\begin{eqnarray}
s_{cosmo,int} &=&\frac{1}{\beta n^2\gamma_n} \left\{\frac{2}{\beta}\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right)\left( 1-\Omega_D \right) + \frac{\Omega_D \beta }{2}\left( u-3b^2 \right) \right] \times \right. \nonumber \\
&&\left. \left( \frac{2}{\beta n^2 \gamma_n} - \frac{2\alpha}{\beta} + u -2 \right) \right. \nonumber \\
&&\left.+ \frac{2}{\beta}\left( \frac{\Omega_D}{n^2 \gamma_n} + \beta - \alpha \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -\Omega_D\left(u-3b^2\right)\left( u+1 \right)\right\}\nonumber \\
&& -j\left( 2+3q \right). \label{scosmogeneralfinalint}
\end{eqnarray}
Considering the present day values of the quantities involved, we can write the present day value of $s_{cosmo,int}$ as follows:
\begin{eqnarray}
s_{cosmo,int0} &=&\left(\frac{3\lambda-1}{2\beta n^2}\right) \left\{\frac{2}{\beta}\left[\left(\frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2} - \alpha + \beta\right)\left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta }{2}\left( u_0-3b^2 \right) \right]\times \right. \nonumber \\
&&\left. \left( \frac{3\lambda -1}{\beta n^2 } - \frac{2\alpha}{\beta} + u_0 -2 \right) \right. \nonumber \\
&&\left.+ \frac{2}{\beta}\left[ \frac{\left(3\lambda -1 \right)\Omega_{D_0}}{2n^2 } + \beta - \alpha \right]\times \right. \nonumber \\
&&\left. \left[\Omega_{k_0} - \left(u_0+1 \right) \left(1-\Omega_{D_0} \right) \right] -\Omega_{D_0}\left(u_0-3b^2\right)\left( u_0+1 \right)\right\}\nonumber \\
&& -j_0\left( 2+3q_0 \right). \label{scosmogeneralfinalint0}
\end{eqnarray}
Inserting in Eq. (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$, that:
\begin{eqnarray}
s_{cosmo,int0} \approx 17.1515 - 11.5155 b^2. \label{debbymary1}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $s_{cosmo,int0} $ for the case with $\lambda = 1.02$.
Plot of $s_{cosmo,int0} $ obtained in Eq. (<ref>) for $\lambda = 1.02$.
Instead, for $\lambda = 0.98$ we obtain:
\begin{eqnarray}
s_{cosmo,int0} \approx 11.9 - 6.41689 b^2.\label{debbymary2}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $s_{cosmo,int0} $ for the case with $\lambda = 0.98$.
Plot of $s_{cosmo,int0} $ obtained in Eq. (<ref>) for $\lambda = 0.98$.
Finally, for $\lambda = 1.00$, we obtain:
\begin{eqnarray}
s_{cosmo,int0} \approx 14.3768 - 8.89771 b^2.\label{debbymary3}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $s_{cosmo,int0} $ for the case with $\lambda = 1.00$.
Plot of $s_{cosmo,int0} $ obtained in Eq. (<ref>) for $\lambda = 1.00$.
In the limiting case of $b^2=0$, the results of the interacting case lead to the same results of the non interacting case. Instead, for $b^2=0.025$, the results of the interacting case reduce to $s_{cosmo,int0}\approx 16.86366$ for $\lambda = 1.02$, $s_{cosmo,int0}\approx 11.7396$ for $\lambda = 0.98$ and $s_{cosmo,int0}\approx 14.1544 $ for $\lambda = 1.00$.
We now consider the limiting case corresponding to the Ricci scale, i.e. for $\alpha =2$ and $\beta =1$.
Inserting in Eq. (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$:
\begin{eqnarray}
s_{cosmo,int0} \approx -0.230849 + 18.273 b^2.\label{debbymary4}
\end{eqnarray}
In Figure (<ref>), we plot the behavior of $s_{cosmo,int0} $ for the case with $\lambda = 1.02$.
Plot of $s_{cosmo,int0} $ obtained in Eq. (<ref>) for $\lambda = 1.02$.
Instead, for $\lambda = 0.98$, we obtain:
\begin{eqnarray}
s_{cosmo,int0} \approx -0.15236 + 18.3226 b^2.\label{debbymary5}
\end{eqnarray}
In Figure (<ref>), we plot the behavior of $s_{cosmo,int0} $ for the case with $\lambda = 0.98$.
Plot of $s_{cosmo,int0} $ obtained in Eq. (<ref>) for $\lambda = 0.98$.
Finally, for $\lambda = 1.00$, we obtain:
\begin{eqnarray}
s_{cosmo,int0} \approx -0.18851 + 18.315 b^2. \label{debbymary6}
\end{eqnarray}
In Figure (<ref>), we plot the behavior of $s_{cosmo,int0} $ for the case with $\lambda = 1.00$.
Plot of $s_{cosmo,int0} $ obtained in Eq. (<ref>) for $\lambda = 1.00$.
In the limiting case of $b^2=0$, the results of the interacting case lead to the same results of the non interacting case. Instead, for $b^2=0.025$, the results of the interacting case reduce to $s_{cosmo,int0}\approx 0.22598$ for $\lambda = 1.02$, $s_{cosmo,int0}\approx 0.305705 $ for $\lambda = 0.98$ and $s_{cosmo,int0}\approx 0.269365$ for $\lambda = 1.00$.
We now want to find the final expression of the lerk parameter $l$ for the interacting case.
We follow also in this case the same procedure of the non interacting case.
For the interacting case, we have that $u''$ is given by the following relation:
\begin{eqnarray}
u'' &=& \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega'_k - u' \left(1-\Omega_D \right) + \Omega_D'\left(1+u \right) \right] \nonumber \\
&&-\frac{2}{\beta}\left[ \frac{\left(\beta - \alpha\right)\Omega_D'}{\Omega_D^2} \right] \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] \nonumber \\
&&-u'\left( 2u-3b^2+1 \right). \label{usecondint}
\end{eqnarray}
Inserting the expressions of $\Omega_k'$, $\Omega_D'$ and $u'$ in Eq. (<ref>), we obtain:
\begin{eqnarray}
u'' &=& -\frac{4\Omega_k}{\beta^2}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left( \frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta \right) \nonumber \\
&&- \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left(1-\Omega_D \right) \times \nonumber \\
&&\left\{ \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -\left(u-3b^2\right)\left( u+1 \right) \right\} \nonumber \\
&&+ \frac{4}{\beta^2}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left(1+u \right) \left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right)\left( 1-\Omega_D \right) + \frac{\Omega_D \beta }{2}\left( u-3b^2 \right) \right] \nonumber \\
&&-\frac{4}{\beta^2}\left( \frac{\beta - \alpha}{\Omega_D^2} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right]\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right)\left( 1-\Omega_D \right) + \frac{\Omega_D \beta }{2}\left( u-3b^2 \right) \right] \nonumber \\
&&-\left( 2u-3b^2+1 \right)\left\{ \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] \right. \nonumber \\
&&\left.-\left(u-3b^2\right)\left( u+1 \right) \right\} \label{usecondint2}.
\end{eqnarray}
The final expression of $j''$ can be obtained inserting in Eq. (<ref>) the expressions of $u'$, $\Omega_D'$ and $u''$ for the interacting case.
For the interacting case, using in Eq. (<ref>) the expression of the evolutionary form of the fractional energy density of DE $\Omega_D'$ for the interacting case, we obtain the following expression for $q'$:
\begin{eqnarray}
q'= - \frac{2}{\beta^2}\left( \frac{1}{n^2\gamma_n} \right)\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right)\left( 1-\Omega_D \right) + \frac{\Omega_D \beta }{2}\left( u-3b^2 \right) \right]. \label{}
\end{eqnarray}
We can now calculate the present day values of $l$ taking into account all the results of the above equations.
For the interacting case, we obtain, for $\lambda = 1.02$, that the present day value of the lerk parameter is given by the following relation:
\begin{eqnarray}
l_{int,0} \approx 80.5574 + \left(-174.752 + 88.9182 b^2\right)b^2. \label{driutti1}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $l_{int,0}$ obtained in Eq. (<ref>).
Plot of $l_{int,0}$ given in Eq. (<ref>) for $\lambda = 1.02$.
For $\lambda = 0.98$ we derive that the present day value of the lerk parameter is given by the following relation:
\begin{eqnarray}
l_{int,0} \approx 57.667 + \left(-136.334 + 70.4551b^2 \right)b^2. \label{driutti2}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $l_{int,0}$ obtained in Eq. (<ref>).
Plot of $l_{int,0}$ given in Eq. (<ref>) for $\lambda=0.98$.
Finally, for $\lambda =1.00$, we obtain:
\begin{eqnarray}
l_{int,0} \approx 68.3621 + \left(-154.359 + 79.4812 b^2\right)b^2. \label{driutti3}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $l_{int,0}$ obtained in Eq. (<ref>).
Plot of $l_{int,0}$ given in Eq. (<ref>) for $\lambda =1.00$.
In the limiting case of $b^2=0$, the results of the interacting case lead to the same results of the non interacting case. Instead, for $b^2=00.25$, we obtain, for $\lambda = 1.02$, that $l_{int,0} \approx 76.2442$, for $\lambda = 0.98$, we obtain $l_{int,0} \approx 54.3027$ while for $\lambda = 1.00$ we obtain $l_{int,0} \approx 64.5520 $.
We now consider the limiting case of Ricci scale, which is obtained in the limiting case of $\alpha =2$ and $\beta =1$.
For the interacting case, we obtain, for $\lambda = 1.02$, that the present day value of the lerk parameter is given by the following relation:
\begin{eqnarray}
l_{int,0} \approx -10.0602 - \left(80.2631 + 20.5776 b^2\right) b^2 . \label{driutti4}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $l_{int,0}$ obtained in Eq. (<ref>).
Plot of $l_{int,0}$ given in Eq. (<ref>) for $\lambda = 1.02$ for the limiting case corresponding to the Ricci scale.
For $\lambda = 0.98$ we derive that the present day value of the lerk parameter is given by the following relation:
\begin{eqnarray}
l_{int,0}\approx -11.2141 - \left(82.7856 + 22.721 b^2\right) b^2. \label{driutti5}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $l_{int,0}$ obtained in Eq. (<ref>).
Plot of $l_{int,0}$ given in Eq. (<ref>) for $\lambda = 0.98$ for the limiting case corresponding to the Ricci scale.
Finally, for $\lambda =1.00$, we obtain:
\begin{eqnarray}
l_{int,0}\approx -10.6489 - \left(81.5413 + 21.701 b^2\right) b^2. \label{driutti6}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $l_{int,0}$ obtained in Eq. (<ref>).
Plot of $l_{int,0}$ given in Eq. (<ref>) for $\lambda = 1.00$ for the limiting case corresponding to the Ricci scale.
In the limiting case of $b^2=0$, the results of the interacting case lead to the same results of the non interacting case. Instead, for $b^2=00.25$, we obtain, for $\lambda = 1.02$, that $l_{int,0} \approx -12.0796$, for $\lambda = 0.98$ we obtain $l_{int,0} \approx -13.2979$ while for $\lambda = 1.00$ we obtain $l_{int,0} \approx -12.7010$.
We obtain, therefore, values of $s_{cosmo,int0}$ and $l_{int0}$ which are between the errors obtained in the works of Capozziello $\&$ Izzo <cit.> and John <cit.> for both sets of values of $\alpha$ and $\beta$ we considered and for all the values of $\lambda$ taken into account.
§ SQUARED SPEED OF THE SOUND $V_S^2$
We now consider an important quantity which is used in order check the stability of any DE model studied, named as squared speed of sound $v_s^2$, which is generally defined as follows <cit.>:
\begin{eqnarray}
v_s^2 = \frac{\dot{p}}{\dot{\rho}} = \frac{p'}{\rho'} , \label{genvs}
\end{eqnarray}
where, as before, $p= p_D$ and $\rho = \rho_D + \rho_m$ are, respectively, the total pressure and the total energy density of the DE model taken into account. We must also remember that we are considering pressureless DM. Inserting in Eq. (<ref>) the expressions of $p$ and $\rho$, we can write:
\begin{eqnarray}
v_s^2 &=& \frac{\dot{p}_D}{\dot{\rho}_D + \dot{\rho}_m}\nonumber \\
&=& \frac{p'_D}{\rho'_D + \rho'_m}. \label{genvs2}
\end{eqnarray}
The sign of the squared speed of the sound $v_{s}^{2}$ has a fundamental role when the stability of a background
evolution is wanted to be studied since it discriminates between instable and stable models. A negative value of $v_s^2$ indicates a classical instability of a given perturbation in General Relativity (GR) <cit.> while a positive value of $v_s^2$ indicates a stable model. Myung <cit.> observed that $v_{s}^{2}$ for the HDE model is always negative if the future event horizon is taken as IR cut-off of the system, while for Chaplygin gas and tachyon models it is observed to be non negative. Kim et al. <cit.> observed that the squared speed of the sound $v_{s}^{2}$ for the Agegraphic DE (ADE) model is always negative leading to an instability of the perfect fluid for this particular model. In a recent paper, Sharif $\&$ Jawad <cit.> have obtained that the interacting new HDE model leads to a negative value of $v_{s}^{2}$. Jawad et al. <cit.> observed that $v_s^2$ remains negative for the HDE model based reconstructed $f\left(G\right)$ model with the choice of the scale factor in the power law form. In the recent work of Pasqua et al. <cit.>, the Authors observed that the interacting Modified Holographic Ricci DE (MHRDE) model in the framework of $f\left(R,T\right) = \mu R + \nu T$ modified gravity model (with $\mu$ and $\nu$ being two constant parameters) is classically stable.
Pasqua et al. <cit.> showed that the New ADE (NADE) model based on the Generalized Uncertainty Principle (GUP) with power-law form of the scale factor $a\left(t\right)$ is classically instable.
§.§ Non Interacting Case
We can start studying the behavior of the squared speed of the sound $v_s^2$ for the non interacting case.
We have already obtained the expression of $\rho_D'$ in Eq. (<ref>), we now need to find the expressions of $p_D'$ and $\rho_m'$.
From the continuity equation for DE given in Eq. (<ref>), using the general definition of the EoS parameter of DE $\omega_D$, we can write the following expression of the pressure of DE $p_D$:
\begin{eqnarray}
p_D = -\rho_D - \frac{\rho_D'}{3} . \label{pressurevs}
\end{eqnarray}
Therefore, we obtain that the derivative of the pressure of DE $p_D$ with respect to the variable $x$ is given by:
\begin{eqnarray}
p'_D = -\rho'_D - \frac{\rho_D''}{3} . \label{pressurevs}
\end{eqnarray}
We must now calculate the expression of $\rho_D''$. We have that the expression of $\rho'_D$ obtained in Eq. (<ref>) is given by:
\begin{eqnarray}
\rho'_D = \frac{2\rho_D}{ \beta \Omega_D}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) \right]. \label{lgo10evootherderi}
\end{eqnarray}
Differentiating the expression of $\rho_D'$ given in Eq. (<ref>) with respect to the variable $x$, we obtain that:
\begin{eqnarray}
\rho''_D &=& =\frac{2\rho_D'}{ \beta \Omega_D}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) \right] \nonumber \\
&&+\frac{2\rho_D\Omega_D'}{ \beta \Omega_D^2}\left(\beta - \alpha \right) +\rho_D u'. \label{rhosecond}
\end{eqnarray}
The expressions of the EoS parameter of DE $\omega_D$ and $u'$ have been derived in Eqs. (<ref>) and (<ref>). Using the result of Eq. (<ref>) in Eq. (<ref>), we can write:
\begin{eqnarray}
p'_D &=& - \frac{2\rho_D'}{ 3\beta \Omega_D}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) + \frac{3\beta \Omega_D}{2}\right] \nonumber \\
&&+\frac{2\rho_D\Omega_D'}{ 3\beta \Omega_D^2}\left( \alpha - \beta \right) -\frac{\rho_D u'}{3} . \label{pressurevs2pao}
\end{eqnarray}
Inserting in Eq. (<ref>) the expressions of the evolutionary form of the fractional energy density of DE $\Omega_D'$, $\rho_D'$ and $u'$ for the non interacting case, we obtain:
\begin{eqnarray}
p'_D &=& - \frac{4\rho_D}{ 3\beta^2 \Omega_D^2}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) + \frac{3\beta \Omega_D}{2}\right]\times \nonumber \\
&&\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) \right] \nonumber \\
&& +\frac{4\rho_D}{ 3\beta^2 \Omega_D^2}\left(\alpha - \beta \right)\times \nonumber \\
&&\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} \right] \nonumber \\
&&-\frac{\rho_D}{3}\left\{ \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -u\left( u+1 \right) \right\} . \label{pressurevs3}
\end{eqnarray}
We must now find an expression for $\rho_m'$.
From the continuity equation for DM given in Eq. (<ref>), we obtain:
\begin{eqnarray}
\rho_m'= -3\rho_m= -3u\rho_D , \label{rhomprime}
\end{eqnarray}
where we used the general definition of $u$.
Therefore, we have that:
\begin{eqnarray}
\rho_D' +\rho_m'= \rho_D' -3u\rho_D . \label{rhosum}
\end{eqnarray}
Inserting in Eq. (<ref>) the expression of $\rho_D'$ given in Eq. (<ref>), we obtain:
\begin{eqnarray}
\rho_D' +\rho_m'= \frac{2\rho_D}{ \beta \Omega_D}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) \right] -3u\rho_D . \label{rhosumnew}
\end{eqnarray}
Finally, we can write $v_s^2$ as follows:
\begin{eqnarray}
v_s^2 = \frac{A_1}{B_1}, \label{}
\end{eqnarray}
where $A_1$ and $B_1$ are defined as follows:
\begin{eqnarray}
A_1 &=& - \frac{4}{ 3\beta^2 \Omega_D^2}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) + \frac{3\beta \Omega_D}{2}\right]\times \nonumber \\
&&\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) \right] \nonumber \\
&&+\frac{4}{ 3\beta^2 \Omega_D^2}\left(\beta - \alpha \right)\times \nonumber \\
&&\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right) \left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} \right] \nonumber \\
&&-\frac{1}{3}\left\{ \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -u\left( u+1 \right) \right\}, \label{a1}\\
B_1 &=& \frac{2}{ \beta \Omega_D}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) -\frac{3\beta \Omega_D u}{2} \right]. \label{b1}
\end{eqnarray}
Using the present day values of the parameters involved along with the expression of $\gamma_{n_0}$ given in Eq. (<ref>), we can write the terms $A_1$ and $B_1$ as follows:
\begin{eqnarray}
A_1 &=& - \frac{4}{ 3\beta^2 \Omega_{D_0}^2}\left[\frac{\Omega_{D_0}\left( 3\lambda -1 \right)}{2n^2} -\alpha + \beta + \beta \Omega_{D_0} \left( \frac{u_0-2}{2} \right) + \frac{3\beta \Omega_{D_0}}{2}\right]\times \nonumber \\
&&\left[\frac{\Omega_{D_0}\left( 3\lambda -1 \right)}{2n^2} -\alpha + \beta + \beta \Omega_{D_0} \left( \frac{u_0-2}{2} \right) \right] \nonumber \\
&&+ \frac{4\left(\beta - \alpha \right)}{ 3\beta^2 \Omega_{D_0}^2}\times \nonumber \\
&&\left\{\left[\frac{\Omega_{D_0}\left( 3\lambda -1 \right)}{2n^2} - \alpha + \beta\right] \left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} \right\} \nonumber \\
&&-\frac{1}{3}\left\{ \frac{2}{\beta}\left[ \frac{\left( 3\lambda -1 \right)}{2n^2 } + \frac{\beta - \alpha}{\Omega_{D_0}} \right] \left[\Omega_{k_0} - \left(u_0+1 \right) \left(1-\Omega_{D_0} \right) \right] -u_0\left( u_0+1 \right) \right\}, \label{a11}\\
B_1 &=& \frac{2}{ \beta \Omega_{D_0}}\left[\frac{\Omega_{D_0}\left( 3\lambda -1 \right)}{2n^2} -\alpha + \beta + \beta \Omega_{D_0} \left( \frac{u_0-2}{2} \right) -\frac{3\beta \Omega_{D_0} u_0}{2} \right]. \label{b11}
\end{eqnarray}
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$:
\begin{eqnarray}
A_1 &\approx& -6.96235,\\
B_1 &\approx& 1.08621,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& -6.40974. \label{vs1}
\end{eqnarray}
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 0.98$:
\begin{eqnarray}
A_1 &\approx& -5.87013,\\
B_1 &\approx& 0.726044,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& -8.08509. \label{vs1}
\end{eqnarray}
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.00$:
\begin{eqnarray}
A_1 &\approx& -6.40543,\\
B_1 &\approx& 0.906129,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& -7.069. \label{vs1}
\end{eqnarray}
The negative values obtained for the squared speed of the sound $v_s^2$ for all cases of the running parameter $\lambda$ considered indicate that the model we are studying is unstable for the set of values considered.
We now consider the limiting case corresponding to the Ricci scale, i.e. for $\alpha =2$ and $\beta =1$.
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$:
\begin{eqnarray}
A_1 &\approx& -0.0630672,\\
B_1 &\approx& -2.6923,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& 0.023425. \label{vs1}
\end{eqnarray}
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 0.98$:
\begin{eqnarray}
A_1 &\approx& 0.0794059,\\
B_1 &\approx& -2.87297,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& -0.027639. \label{vs1}
\end{eqnarray}
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.00$:
\begin{eqnarray}
A_1 &\approx& 0.0108893,\\
B_1 &\approx& -2.78263,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& -0.00391329. \label{vs1}
\end{eqnarray}
We observe that the squared soeed of the sound $v_s^2$ can assume both positive and negative values according to the value of $\lambda$ considered.
Therefore, we can conclude that we can obtain both a stable or an unstable model according to the value of the running parameter $\lambda$ considered.
§.§ Interacting Case
We now want to study the behavior of the squared speed of the sound $v_s^2$ in the case of interacting Dark Sectors.
From the continuity equation for DE given in Eq. (<ref>), using the general definition of the EoS parameter of DE $\omega_D$, we can write the following expression of the pressure of DE $p_D$:
\begin{eqnarray}
p_D &=& -\rho_D - \frac{\rho_D'}{3} -\frac{Q}{3H} \nonumber \\
&=& -\rho_D - \frac{\rho_D'}{3} -b^2 \rho_D\nonumber \\
&=& -\rho_D\left( 1+b^2 \right) - \frac{\rho_D'}{3}. \label{pressurevsint}
\end{eqnarray}
Differentiating Eq. (<ref>) with respect to the variable $x$, we obtain the following expression for $p'_D$:
\begin{eqnarray}
p'_D = -\rho'_D\left( 1+b^2 \right) - \frac{\rho_D''}{3}. \label{pressurevsintnew}
\end{eqnarray}
We have already obtained the expression of $\rho'_D$ in Eq. (<ref>) and it is given by:
\begin{eqnarray}
\rho'_D =\frac{2\rho_D}{ \beta\Omega_D}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2-3b^2}{2} \right) \right]. \label{lgo12evovs2}
\end{eqnarray}
We must now calculate the expression of $\rho_D''$.
Differentiating the expression of $\rho_D'$ given in Eq. (<ref>) with respect to the variable $x$,we obtain that:
\begin{eqnarray}
\rho''_D &=& =\frac{2\rho_D'}{ \beta \Omega_D}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2-3b^2}{2} \right) \right] \nonumber \\
&&+\frac{2\rho_D\Omega_D'}{ \beta \Omega_D^2}\left(\beta-\alpha \right) +\rho_D u'. \label{rhosecond}
\end{eqnarray}
Using the result of Eq. (<ref>) in Eq. (<ref>), we can write:
\begin{eqnarray}
p'_D &=& - \frac{2\rho_D'}{ 3\beta \Omega_D}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) + \frac{3\beta \Omega_D}{2}\right] \nonumber \\
&&+\frac{2\rho_D\Omega_D'}{ 3\beta \Omega_D^2}\left(\alpha - \beta \right) -\frac{\rho_D u'}{3} , \label{pressurevs2}
\end{eqnarray}
which is the same general expression obtained for the non interacting case.
Inserting in Eq. (<ref>) the expressions of $\Omega_D'$, $\rho_D'$ and $u'$ obtained for the interacting case, we obtain:
\begin{eqnarray}
p'_D &=& - \frac{4\rho_D}{ 3\beta^2 \Omega_D^2}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) + \frac{3\beta \Omega_D}{2}\right]\times \nonumber \\
&&\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2-3b^2}{2} \right) \right] \nonumber \\
&&+\frac{4\rho_D}{ 3\beta^2 \Omega_D^2}\left(\alpha - \beta \right) \times \nonumber \\
&&\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right)\left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} - \frac{3}{2}\Omega_D \beta b^2 \right] \nonumber \\
&&-\frac{\rho_D}{3}\left\{ \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -\left(u-3b^2\right)\left( u+1 \right) \right\} . \label{pressurevs3}
\end{eqnarray}
We must now find an expression for $\rho_m'$.
From the continuity equation for DM given in Eq. (<ref>), we obtain:
\begin{eqnarray}
\rho_m'&=& -3\rho_m + \frac{Q}{H}\nonumber \\
&=& -3u\rho_D +3b^2\rho_D\nonumber \\
&=& 3\rho_D \left( b^2 - u \right), \label{rhomprime}
\end{eqnarray}
where we used the general definition of $u$.
Therefore, we have that:
\begin{eqnarray}
\rho_D' +\rho_m'= \rho_D' + 3\rho_D \left( b^2 - u \right) . \label{rhosum2}
\end{eqnarray}
Inserting in Eq. (<ref>) the expression of $\rho_D'$ given in Eq. (<ref>), we obtain:
\begin{eqnarray}
\rho_D' +\rho_m'= \frac{2\rho_D}{ \beta\Omega_D}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2-3b^2}{2} \right) \right] + 3\rho_D \left( b^2 - u \right) . \label{}
\end{eqnarray}
Finally, we can write $v_s^2$ as follows:
\begin{eqnarray}
v_s^2 = \frac{A_2}{B_2}, \label{}
\end{eqnarray}
where $A_2$ and $B_2$ are defined as follows:
\begin{eqnarray}
A_2 &=&- \frac{4}{ 3\beta^2 \Omega_D^2}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2}{2} \right) + \frac{3\beta \Omega_D}{2}\right]\times \nonumber \\
&&\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2-3b^2}{2} \right) \right] \nonumber \\
&&+\frac{4\left( \alpha - \beta \right)}{ 3\beta^2 \Omega_D^2} \times \nonumber \\
&&\left[\left(\frac{\Omega_D}{n^2\gamma_n} - \alpha + \beta\right)\left( 1-\Omega_D \right) + \frac{\Omega_D \beta u}{2} - \frac{3}{2}\Omega_D \beta b^2 \right] \nonumber \\
&&-\frac{1}{3}\left\{ \frac{2}{\beta}\left( \frac{1}{n^2 \gamma_n} + \frac{\beta - \alpha}{\Omega_D} \right) \left[\Omega_k - \left(u+1 \right) \left(1-\Omega_D \right) \right] -\left(u-3b^2\right)\left( u+1 \right) \right\}, \label{a2} \\
B_2 &=& \frac{2}{ \beta\Omega_D}\left[\frac{\Omega_D}{n^2\gamma_n} -\alpha + \beta + \beta \Omega_D \left( \frac{u-2-3b^2}{2} \right) \right] + 3 \left( b^2 - u \right).\label{b2}
\end{eqnarray}
Using the present day values of the parameters involved along with the expression of $\gamma_{n_0}$ given in Eq. (<ref>), we can write $A_2$ and $B_2$ as follows:
\begin{eqnarray}
A_2 &=&- \frac{4}{ 3\beta^2 \Omega_{D_0}^2}\left[\frac{\Omega_{D_0}\left( 3\lambda -1 \right)}{2n^2} -\alpha + \beta + \beta \Omega_{D_0} \left( \frac{u_0-2}{2} \right) + \frac{3\beta \Omega_{D_0}}{2}\right]\times \nonumber \\
&&\left[\frac{\Omega_{D_0}\left( 3\lambda -1 \right)}{2n^2} -\alpha + \beta + \beta \Omega_{D_0} \left( \frac{u_0-2-3b^2}{2} \right) \right] \nonumber \\
&&+\frac{4\left( \alpha - \beta \right)}{ 3\beta^2 \Omega_{D_0}^2} \times \nonumber \\
&&\left\{\left[\frac{\Omega_{D_0}\left( 3\lambda -1 \right)}{2n^2} - \alpha + \beta\right]\left( 1-\Omega_{D_0} \right) + \frac{\Omega_{D_0} \beta u_0}{2} - \frac{3}{2}\Omega_{D_0} \beta b^2 \right\} \nonumber \\
&&-\frac{1}{3}\left\{ \frac{2}{\beta}\left[ \frac{\left( 3\lambda -1 \right)}{2n^2 } + \frac{\beta - \alpha}{\Omega_{D_0}} \right]\times \right. \nonumber \\
&&\left.\left[\Omega_{k_0} - \left(u_0+1 \right) \left(1-\Omega_{D_0} \right) \right] -\left(u_0-3b^2\right)\left( u_0+1 \right) \right\}, \label{a22} \\
B_2 &=& \frac{2}{ \beta\Omega_{D_0}}\left[\frac{\Omega_{D_0}\left( 3\lambda -1 \right)}{2n^2} -\alpha + \beta + \beta \Omega_{D_0} \left( \frac{u_0-2-3b^2}{2} \right) \right] + 3 \left( b^2 - u_0 \right)\label{b22}.
\end{eqnarray}
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$:
\begin{eqnarray}
A_2 &\approx& -6.96235 + 10.5769 b^2,\\
B_2 &\approx&1.08621,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& -6.40974 + 9.73741 b^2. \label{schi1}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $v_s^2 $ obtained in Eq. (<ref>).
Plot of the squared speed of the sound $v_s^2$ given in Eq. (<ref>) for $\lambda = 1.02$.
In the limiting case of $b^2 =0$ we recover the same result of the non interacting case, while for $b^2=0.025$ we obtain $A_2 \approx -6.69793$ and $v_s^2 \approx -6.1663$.
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 0.98$:
\begin{eqnarray}
A_2 &\approx& -5.87013 + 10.2167 b^2,\\
B_2&\approx& 0.726044,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& -8.08509 + 14.0718 b^2. \label{schi2}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $v_s^2$ obtained in Eq. (<ref>).
Plot of the squared speed of the sound $v_s^2$ given in Eq. (<ref>) for $\lambda = 9.98$.
In the limiting case of $b^2 =0$ we recover the same result of the non interacting case, while for $b^2=0.025$ we obtain $A_2 \approx -5.6147$ and $v_s^2 \approx -7.7339$.
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.00$:
\begin{eqnarray}
A_2 &\approx& -6.40543 + 10.3968 b^2,\\
B_2 &\approx& 0.906129,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& -7.069 + 11.4739 b^2. \label{schi3}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $v_s^2 $ obtained in Eq. (<ref>).
Plot of the squared speed of the sound $v_s^2$ given in Eq. (<ref>) for $\lambda = 1.00$.
In the limiting case of $b^2 =0$ we recover the same result of the non interacting case, while for $b^2=0.025$ we obtain $A_2 \approx -6.14551$ and $v_s^2 \approx -6.7821$.
We can observe that the squared speed of the sound $v_s^2$ assumes a negative value for all the cases of the running parameter $\lambda$ considered for $b^2$ in the range $\left[0, 0.025 \right]$, therefore we deal with a model which is unstable for the set of values considered.
We now consider the limiting case corresponding to the Ricci scale, i.e. in the limiting case of $\alpha =2$ and $\beta =1$.
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.02$:
\begin{eqnarray}
A_2 &\approx& -0.0630672 + 8.88923 b^2,\\
B_2 &\approx& -2.6923,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& 0.023425 - 3.30172 b^2. \label{schi4}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $v_s^2 $ obtained in Eq. (<ref>).
Plot of the squared speed of the sound $v_s^2$ given in Eq. (<ref>) for $\lambda = 1.02$ for the limiting case of Ricci scale.
In the limiting case of $b^2 =0$ we recover the same result of the non interacting case, while for $b^2=0.025$ we obtain $A_2 \approx 0.159164$ and $v_s^2 \approx -0.0591181$.
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 0.98$:
\begin{eqnarray}
A_2 &\approx& 0.0794059 + 8.70857 b^2,\\
B_2 &\approx& -2.87297,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& -0.027639 - 3.03121 b^2. \label{schi5}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $v_s^2$ obtained in Eq. (<ref>).
Plot of the squared speed of the sound $v_s^2$ given in Eq. (<ref>) for $\lambda = 0.98$ for the limiting case of Ricci scale.
In the limiting case of $b^2 =0$ we recover the same result of the non interacting case, while for $b^2=0.025$ we obtain $A_2\approx 0.29712$ and $v_s^2 \approx -0.103419$.
Inserting in Eqs. (<ref>) and (<ref>) the values of the parameters involved, we obtain, for $\lambda = 1.00$:
\begin{eqnarray}
A_2 &\approx& 0.0108893 + 8.7989 b^2,\\
B_2 &\approx& -2.78263,
\end{eqnarray}
which lead to the following value of $v_s^2$:
\begin{eqnarray}
v_s^2 &\approx& -0.00391329 - 3.16208 b^2. \label{schi6}
\end{eqnarray}
In Figure <ref>, we plot the behavior of $v_s^2 $ obtained in Eq. (<ref>).
Plot of the squared speed of the sound $v_s^2$ given in Eq. (<ref>) for $\lambda = 1.00$ for the limiting case of Ricci scale.
In the limiting case of $b^2 =0$ we recover the same result of the non interacting case, while for $b^2=0.025$ we obtain $A_2 \approx 0.230862$ and $v_s^2 \approx -0.0829653$.
We can conclude that, for $\lambda = 1.02$, we can obtain a model which can be stable or unstable according to the value of $b^2$. Instead, for $\lambda =1.00$ and $\lambda = 0.98$, we obtain an unstable model for all the range of values of $b^2$.
§ CONCLUSIONS
In this work, we studied the Power Law Entropy Corrected versions of the HDE (PLECHDE) model with infrared (IR) cut-off the one recently suggested and studied by Granda and Oliveros, which contains two terms, one proportional to the Hubble parameter squared $H^2$ and one proportional to the first derivative with respect of the cosmic time $t$ of the Hubble parameter $H$, i.e. $\dot{H}$. Moreover, this model is characterized by two constant parameters indicated with $\alpha$ and $\beta$. In the limiting case of $\alpha=2$ and $\beta=1$, we obtain that the Granda-Oliveros cut-off becomes proportional to the average radius of the Ricci scalar curvature. We have investigated this model in a FLRW Universe in the framework of Hořava-Lifshitz gravity for both non-interacting and interacting DE and DM. We must underline that we have considered three different values of the running parameter $\lambda$ (which is one of the parameter characterizing the Hořava-Lifshitz gravity): in particular, following the results of Dutta $\&$ Saridakis <cit.>, we have considered $\lambda =1.02$ and $\lambda = 0.98$. We must also underline that Lorentz invariance is restored for $\lambda =1$. We also considered the case with $\lambda =1$ in order to obtain results when the Lorentz invariance is restored. Moreover, we also studied the limiting case corresponding to the Ricci scale for all the values of the running parameter $\lambda$ taken into account.
Using a low redshift expansion of the EoS parameter of DE as $\omega_D \left( z \right) = \omega_0 + \omega_1 z$, we calculated the expressions of the parameters $\omega_0$ and $\omega_1$ as functions of the DE and curvature fractional energy density parameters and of the interaction parameter $b^2$ for the interacting case. We found that the parameter $\omega_0$ assumes the same expression for both non interacting and interacting Dark Sectors. Instead, $\omega_1$ has a clear dependence on $b^2$ for the interacting case. We also calculated the value of the redshift which lead to $\omega_D = -1$ and the present day values of the EoS parameter of DE $\omega_D$.
We also derived an expression for deceleration parameter $q$, which was found to be function of the fractional energy density of DE $\Omega_D$ and of the parameter $n$, the running parameter $\lambda$ and the two constants $\alpha$ and $\beta$ characterizing the GO cut-off. The values obtained for the deceleration parameter $q$ for the present day values of the parameters involved indicate that the model considered leads to an accelerated Universe since $q$ assumes negative values, which is in agreement with the most recent cosmological observations.
Studying the statefinder parameters $r$ and $s$, for $\alpha = 8824$ and $\beta = 0.5016$, we found that the PLECHDE model with GO cut-off considered in this work leads to points that are far from the point corresponding to the $\Lambda$CDM model for both non interacting and interacting Dark Sectors; moreover, since we obtained $s<0$, we derive that we deal with a phantom-like model. Instead, for the limiting case corresponding to the Ricci scale (recovered for $\alpha=2$ and $\beta=1$), we obtain points with are closer to the point corresponding to the $\Lambda$CDM model (with a departure which is a bit more evident for the interacting case); moreover, since we obtained $s>0$, we derive that we deal with a quintessence-like model for this case.
We have also derived the expressions of the cosmographic parameters $s_{cosmo}$ and $l$, also known as snap and lerk parameters. We must remember that the cosmographic parameters are useful in order to characterize the main properties of a particular DE model. Being dependent on the higher time derivatives of the scale factor $a\left(r\right)$, the cosmographic parameter $s_{cosmo}$ and $l$ can give more cosmological details respect to the Hubble and and deceleration parameters $H$ and $q$. We have derived that the present day values of $s_{cosmo}$ and $l$ are between the errors of the values found in some recent papers for both non interacting and interacting DE and DM and for both sets of values of $\alpha$ and $\beta$ considered in this paper.
We also studied the behavior of the squared speed of the sound $v_s^2$ in order to check the stability of the model we are studying. We obtained that, for both non interacting and interacting Dark Sectors, the model we are considering in unstable since we obtained a negative value of $v_s^2$ for the case with $\alpha = 8824$ and $\beta = 0.5016$. Instead, for the limiting case corresponding to the Ricci scale, we can obtain a model which is stalbe or unstable depending on the value of the running parameter $\lambda$, on the absence or presence of interaction and (in the case an interaction between the Dark Sectors really exists) on the strength of the interaction.
Some of the parameters involved in the equations used and derived in this work have not a precise value yet or they have a considerable errors (like it happens for the snap and lerk cosmographic parameters), so it is difficult to obtain exact constraints and comparisons using these parameters. For this reason, future precision cosmological missions could help to obtain better constraints in order to also allow the accurate determination of the values of the parameter derived in this work.
[Abazajian et al., (2004)]sds2 K. Abazajian, et al., Astron. J. 128, 502 (2004)
[Ade et al., (2013)]planck Planck Collaboration, P.A.R. Ade et al., arXiv,1303.5076 (2013)
[Allen et al., (2004)]xray S.W. Allen et al., Mon. Not. Roy. Astron. Soc. 353, 457 (2004)
[Astier et al., (2006)]sn3 P. Astier et al., Astron. Astrophys. 447, 31 (2006)
[Bennett et al., (2003)]cmb1 C.L. Bennett et al., Astrophys. J. 148, 1 (2003)
[de Bernardis et al., (2000)]sn4 P. de Bernardis et al., Nature 404, 955 (2000)
[Riess et al., (1998)]sn1 A.G. Riess et al., Astron. J. 116, 1009 (1998)
[Seljak et al., (2005)]1c U. Seljak et al., Phys. Rev. D 71, 103515 (2005)
[Spergel et al., (2003)]cmb2 D.N. Spergel et al., Astrophys. J. Suppl. Ser. 148, 175 (2003)
[Tegmark et al., (2004)]sds1 M. Tegmark et al., Phys. Rev. D 69, 103501 (2004)
[Copeland et al., (2006)]16 E.J. Copeland, M. Sami, S. Tsujikawa, International Journal of Modern Physics D 15, 1753 (2006)
[Berger & Shojae, (2006)]delcampoc M.S. Berger, H. Shojae, Phys. Rev. D 73, 083528 (2006)
[del Campo et al., (2009)]delcampo S. del Campo, R. Herrera, D. Pavon, J. Cosmol. Astropart. Phys. 0901, 020 (2009)
[Griest, (2002)]delcampoe K. Griest, Phys. Rev. D 66, 123501 (2002)
[Jamil & Rahaman, (2009)]delcampof M. Jamil, F. Rahaman, Eur. Phys. J. C 64, 97 (2009)
[Jamil et al., (2010)]delcampol M. Jamil, A. Sheykhi, M.U. Farooq, Int. J. Mod. Phys. D 19, 1831 (2010)
[Jimenez, & Maroto, (2009)]delcampob J.B. Jimenez, A.L. Maroto, AIP Conf. Proc. 1122, 107 (2009)
[Leon & Saridakis, (2010)]delcampoa G. Leon, E N. Saridakis, Phys. Lett. B 693, 1 (2010)
[Zhang, (2005)]delcampod X. Zhang, Mod. Phys. Lett. A 20, 2575 (2005)
[Peiris, (2003)]twothirds H.V. Peiris et al., Astrophys. J. Suppl. Ser. 148, 213 (2003)
[Armendariz-Picon et al., (1999) ]kess4 C. Armendariz-Picon, T. Damour, V. Mukhanov, Phys. Lett. B 458, 209 (1999)
[Armendariz-Picon et al., (2000)]kess2 C. Armendariz-Picon, V. Mukhanov, P.J. Steinhardt, Phys. Rev. Lett. 85, 4438 (2000)
[Chiba et al., (2000)]kess3 T. Chiba, T. Okabe, M. Yamaguchi, Phys. Rev. D 62, 023511 (2000)
[Ratra & Peebles, (1988)]quint1 B. Ratra, P.J.E. Peebles, Phys. Rev. D 37, 3406 (1988)
[Wetterich, (1988)]quint2 C. Wetterich, Nuclear Physics B 302, 668 (1988)
[Zlatev, (1999)]quint3 I. Zlatev, L. Wang, P.J. Steinhardt, Physical Review Letters 82, 896 (1999)
[Padmanabhan, (2002)]tac2 T. Padmanabhan, Phys. Rev. D 66, 021301 (2002)
[Padmanabhan & Choudhury, (2002)]tac3 T. Padmanabhan, T.R. Choudhury, Phys. Rev. D 66, 081301 (2002)
[Sen, (2002)]tac1 A. Sen, Journal of High Energy Physics 4, 48 (2002)
[Boisseau et al., (2000)]pha6 B. Boisseau, G. Esposito-Farese, D. Polarski, A.A. Starobinsky, Phys. Rev. Lett. 85, 2236 (2000)
[Caldwell, (2002)]pha1 R. Caldwell, Phys. Lett. B 545, 23 (2002)
[Chimento & Lazkoz, (2003)]pha5 L.P. Chimento. R. Lazkoz, Phys. Rev. Lett. 91, 211301 (2003)
[Nojiri & Odintsov, (2003a)]pha2 S. Nojiri, S.D. Odintsov, Phys. Lett. B 565, 1 (2003)
[Arkani-Hamed et al., (2004a) ]dil1 N. Arkani-Hamed, P. Creminelli, S. Mukohyama, M. Zaldarriaga, J. Cosmol. Astropart. Phys. 4, 1 (2004)
[Gaperini et al., (2002)]dil2 M. Gasperini, F. Piazza, G. Veneziano, Phys. Rev. D 65, 023508 (2002)
[Piazza & Tsujikawa, (2004)]dil3 F. Piazza, S. Tsujikawa, J. Cosmol. Astropart. Phys. 7, 4 (2004)
[Anisimov, (2005)]qui1 A. Anisimov, J. Cosmol. Astropart. Phys. 6, 6 (2005)
[Cai et al., (2007)]qui6 Y.F. Cai, M.Z. Li, J.X. Lu, Y.S. Piao, T.T. Qiu, X.M. Zhang, Phys. Lett. B 651, 1 (2007)
[Elizalde, et al., (2004)]qui2 E. Elizalde, S. Nojiri, S.D. Odintsov, Phys. Rev. D 70, 043539 (2004)
[Zhao & Zhang, (2006)]qui8 W. Zhao, Y. Zhang, Phys. Rev. D 73, 123509 (2006)
[Bento et al., (2002)]cgas2 M.C. Bento, O. Bertolami, A.A. Sen, Phys. Rev. D 66, 043507 (2002)
[Kamenshchik et al., (2001)]cgas1 A. Kamenshchik, U. Moschella, V. Pasquier, Physics Letters B 511 265 (2001)
[Setare, (2007c)]cgas3 M.R. Setare, European Physical Journal C 52, 689 (2007)
[Wei & Cai, (2008)]ade1 H. Wei, R.G. Cai, Physics Letters B 660, 113 (2008)
[Cai, (2007)]ade2 R.G. Cai, Phys. Lett. B 657, 228 (2007)
[Abdalla & Odintsov, (2005)]fr11 M.C.B. Abdalla, S.D. Odintsov, Class. Qunt. Grav. 22, L35 (2005)
[Aghmohammadi et al., (2010)]fr15 A. Aghmohammadi, K. Saaidi M.R. Abolhassani, Int. J. Theor Phys. 49, 709 (2010)
[Alvarenga et al., (2013)]frt5 F.G. Alvarenga, A. de la Cruz-Dombriz, M.J.S. Houndjo, M.E. Rodrigues, D. Saez-Gomez, Phys. Rev. D 87, 103526 (2013)
[Appleby & Battye, (2007)]fr13 S.A. Appleby, R.A. Battye, Phy. Lett. B 654, 7 (2007)
[Arkani-Hamed et al., (2004b)]fr8 N. Arkani-Hamed, H.C. Cheng, M.A. Luty, S. Mukohyama, JHEP 05, 074 (2004)
[Bamba et al., (2011)]ft8 K. Bamba, C.Q. Geng, C.C. Lee, L.W. Luo, J. Cosmol. Astropart. Phys. 1101, 021 (2011)
[Bengochea & Ferraro, (2009)]ft1 G.R. Bengochea, R. Ferraro, Phys. Rev. D 79, 124019 (2009)
[Capozziello, (2002)]fr1 S. Capozziello, Int. J. Mod. Phys. D 11, 483 (2002)
[Capozziello et al., (2006)]fr12 S. Capozziello, S. Nojiri, S.D. Odintsov, A. Troisi, Phys. Lett. B 639, 135 (2006)
[Deffayet et al., (2002)]bra1 C. Deffayet, G. Dvali, G. Gabadadze, Phys. Rev. D 65, 044023 (2002)
[Dvali et al., (2000)]dgp1 G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485, 208 (2000)
[Easson, (2004)]fr14 D.A. Easson, Int. Mod. Phys. A 19, 5343 (2004)
[Freese & Lewis, (2002)]fr6 K. Freese, M. Lewis, Phys. Lett. B 540, 1 (2002)
[Jawad et al., (2013b)]miofr A. Jawad, S. Chattopadhyay, A. Pasqua, Astrophys. Space Sci. 346, 273 (2013)
[Jawad et al., (2013c)]miofg1 A. Jawad, S. Chattopadhyay, A. Pasqua, European Physical Journal Plus 128, 88 (2013)
[Karami & Abdolmaleki, (2012)]ft6 K. Karami, A. Abdolmaleki, J. Cosmol. Astropart. Phys. 1204, 007 (2012)
[Li et al., (2011)]ft3 B. Li, T.P. Sotiriou, J.D. Barrow, Phys. Rev. D 83, 064035 (2011)
[Myrzakulov, (2012)]frt2 R. Myrzakulov, European Physical Journal C 72, 2203 (2012)
[Nojiri & Odintov, (2003b)]fr10 S. Nojiri, S.D. Odintsov, Phys. Rev. D 68, 123512 (2003)
[Ovalle et al., (2013)]miors J. Ovalle, F. Linares, A. Pasqua, A. Sotomayor, Classical and Quantum Gravity 30, 175019 (2013)
[Pasqua & Chattopadhyay, (2013b)]mioft1 A. Pasqua, S. Chattopadhyay, Canadian Journal of Physics 91, 351 (2013)
[Sahni & Shtanov, (2003)]bra2 V. Sahni, Y. Shtanov, J. Cosmol. Astropart. Phys. 11, 14 (2003)
[Starobinsky, (1980)]fr3 A.A. Starobinsky, Phys. Lett. B 91, 99 (1980)
[Fischler & Susskind, (1998)]holo1 W. Fischler, L. Susskind, arXiv,hep-th/9806039 (1998)
[Hsu, (2004)]holo4 S.D.H. Hsu, Physics Letters B 594, 13 (2004)
[Huang & Li, (2004)]holo5 Q.G. Huang, M. Li, J. Cosmol. Astropart. Phys. 8, 13 (2004)
[Susskind, (1995)]holo2 L. Susskind, J. Math. Phys. 36, 6377 (1995)
[Li, (2004)]li M. Li, Phys. Lett. B 603, 1 (2004)
[Elizalde et al., (2005)]hde33 E. Elizalde, S. Nojiri, S.D. Odintsov, P. Wang, Phys. Rev. D 71, 103504 (2005)
[Gong, (2004)]hde13 Y. Gong, Phys. Rev. D 70, 064029 (2004)
[Guberina et al., (2005)]hde12 B. Guberina, R. Horvat, H. Stefancic, J. Cosmol. Astropart. Phys. 5, 1 (2005)
[Hořava & Minic, (2000)]hde1 P. Hořava, D. Minic, Physical Review Letters 85, 1610 (2000)
[Karami & Fehri, (2010a)]15 K. Karami, J. Fehri, Physics Letters B 684, 61 (2010)
[Kinney & Tzirakis, (2008)]fis8 W.H. Kinney, K. Tzirakis, Phys. Rev. D 77, 103517 (2008)
[Lidsey, & Huston, (2007)]fis7 J.E. Lidsey, I. Huston, J. Cosmol. Astropart. Phys. 7, 2 (2007)
[Myung & Seo, (2009)]fis3 Y.S. Myung, M.G. Seo, Physics Letters B 671, 435 (2009)
[Saridakis, (2008a)]saridakis2 E N. Saridakis, J. Cosmol. Astropart. Phys. 0804 020 (2008)
[Saridakis, (2008b)]saridakis3 E N. Saridakis, Phys. Lett. B 661 335 (2008)
[Setare, (2007b)]hde5 M.R. Setare, Phys. Lett. B 648, 329 (2007)
[Setare & Saridakis, (2009)]saridakis11 M.R. Setare, E N. Saridakis, Phys. Lett. B 671 331 (2009)
[Sheykhi, (2009)]hde26 A. Sheykhi, Physics Letters B 681, 205 (2009)
[Sheykhi, (2010)]30 A. Sheykhi, Classical & Quantum Gravity 27, 025007 (2010)
[Sheykhi, & Jamil (2011)]hde20 A. Sheykhi, M. Jamil, General Relativity & Gravitation 43, 2661 (2011)
['t Hooft, (2006)]fis5 G. 't Hooft, International Journal of Modern Physics D 15, 1587 (2006)
[Wang et al., (2005)]hde10 B. Wang, Y. Gong, E. Abdalla, Astrophys. J. 624, 141 (2005)
[Wang et al., (2008)]33 B. Wang, C.Y. Lin, D. Pavón, E. Abdalla, Physics Letters B 662, 1 (2008)
[Zhang, (2006)]hde35 X. Zhang, Phys. Rev. D 74, 103505 (2006)
[Cohen et al., (1999)]15cohen A.G. Cohen, D.B. Kaplan, A.E. Nelson, Physical Review Letters 82, 4971 (1999)
[Li et al., (2009)]n2primo M. Li, X.D. Li, S. Wang, Y. Wang X. Zhang, J. Cosmol. Astropart. Phys. 0912, 014 (2009)
[Guberina et al., (2007)]8!!! B. Guberina, R. Horvat, H. Nikolic, J. Cosmol. Astropart. Phys. 1, 12 (2007)
[Hawking, (1976)]9a!!! S.W. Hawking, Phys. Rev. D 13, 191 (1976)
[Bekenstein, (1973)]9!!! J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973)
[Jamil et al., (2009b)]11 M. Jamil, E N. Saridakis, M.R. Setare, Physics Letters B 679, 172 (2009)
[Chen et al., (2009)]A73 B. Chen, S. Pi, J.Z. Tang, J. Cosmol. Astropart. Phys. 8, 7 (2009)
[Jamil et al., (2009a)]12e M. Jamil, M.U. Farooq, M.A. Rashid, European Physical Journal C 61, 471 (2009)
[Jamil et al., (2011)]12f M. Jamil, K. Karami, A. Sheykhi, International Journal of Theoretical Physics 50, 3069 (2011)
[Jamil & Sheykhi, (2011)]12g M. Jamil, A. Sheykhi, International Journal of Theoretical Physics 50, 625 (2011)
[Karami & Fehri, (2010b)]12c K. Karami, J. Fehri, International Journal of Theoretical Physics 49, 1118 (2010)
[Sadjadi & Jamil, (2011)]12a H.M. Sadjadi, M. Jamil, General Relativity and Gravitation 43, 1759 (2011)
[Wang et al., (2006)]13a B. Wang, C.Y. Lin, E. Abdalla, Physics Letters B 637, 357 (2006)
[Chattopadhyay & Debnath, (2009)]14 S. Chattopadhyay, U., Debnath, Astrophys. Space Sci. 319, 183 (2009)
[Karami et al., (2011)]14b K. Karami, M.S. Khaledian, M. Jamil, Phys. Scr. 83, 025901 (2011)
[Bisabr, (2009)]15c Y. Bisabr, General Relativity and Gravitation 41, 305 (2009)
[Feng & Zhang, (2009)]15a C.J. Feng, X. Zhang, Physics Letters B 680, 399 (2009)
[Karami & Khaledian, (2011)]15g K. Karami, M.S. Khaledian, Journal of High Energy Physics 3, 86 (2011)
[Nozari & Rashidi, (2009)]15e K. Nozari, N. Rashidi, International Journal of Theoretical Physics 48, 2800 (2009)
[Nozari & Rashidi, (2010)]15d K. Nozari, N. Rashidi, International Journal of Modern Physics D 19, 219 (2010)
[Setare, (2007a)]15h M.R. Setare, Phys. Lett. B 644, 99 (2007)
[Setare & Jamil, (2010c)]15i M.R. Setare, M. Jamil, EPL (Europhysics Letters) 92, 49003 (2010)
[Wei, (2009)]15b H. Wei, Nuclear Physics B 819, 210 (2009)
[Eqnvist et al., (2005)]cons1 K. Enqvist, S. Hannestad, M.S. Sloth, J. Cosmol. Astropart. Phys. 2, 4 (2005)
[Feng et al., (2005)]cons8 B. Feng, X. Wang, X. Zhang, Physics Letters B 607, 35 (2005)
[Kao et al., (2005)]cons7 H.C. Kao, W.L. Lee, F.L. Lin, Phys. Rev. D 71, 123518 (2005)
[Micheletti, (2010)]cons6 S. Micheletti, J. Cosmol. Astropart. Phys. 4, 9 (2010)
[Shen at al., (2005)]cons9 J. Shen, B. Wang, E. Abdalla, R.K. Su, Physics Letters B 609, 200 (2005)
[Zhang, (2009)]cons4 X. Zhang, Phys. Rev. D 79, 103509 (2009)
[Huang & Li, (2005)]ant1 Q.G. Huang, M. Li, J. Cosmol. Astropart. Phys. 3, 1 (2005)
[Lu et al., (2010)]16d J. Lu, E N. Saridakis, M.R. Setare, L. Xu, J. Cosmol. Astropart. Phys. 3, 31 (2010)
[Wang et al., (2007)]16a B. Wang, J. Zang, C.Y. Lin, E. Abdalla, S. Micheletti, Nuclear Physics B 778, 69 (2007)
[Wu et al., (2008)]16b Q. Wu, Y. Gong, A. Wang, J.S. Alcaniz, Phys. Lett. B 659, 34 (2008)
[Zhang, (2010)]16e X. Zhang, Physics Letters B 683, 81 (2010)
[Das et al., (2008)]das18 S. Das, S. Shankaranarayanan, S. Sur, Phys. Rev. D 77, 064013 (2008)
[Radicella & Pavon, (2010)]das18a N. Radicella, D. Pavon, Phys. Lett. B 691, 121 (2010)
[Hořava, (2009b)]23 P. Hořava, Phys. Rev. D 79, 084008 (2009)
[Hořava, (2009c)]A176 P. Hořava, Physical Review Letters 102, 161301 (2009)
[Hořava, (2010)]A22 P. Hořava, Physics Letters B 694, 172 (2010)
[Lifshitz, (1949)]lif1 E.M. Lifshitz, Zh. Eksp. Teor. Fiz. 11, 255 (1949)
[Afshordi, (2009)]A140 N. Afshordi, Phys. Rev. D, 80, 081502 (2009)
[Alexandre et al., (2004)]A141 J. Alexandre, K. Farakos, P. Pasipoularides, A. Tsapalis, Phys. Rev. D 81, 045002 (2010)
[Blas et al., (2010)]A145 D. Blas, O. Pujolàs, S. Sibiryakov, Physical Review Letters 104, 181302 (2010)
[Bogdanos, (2010)]A146 C. Bogdanos, E.N. Saridakis, Classical and Quantum Gravity 27, 075005 (2010)
[Cai et al., (2009)]A148 R.G. Cai, L.M. Cao, N. Ohta, Phys. Rev. D 80, 024003 (2009)
[Germani et al., (2009)]A171 C. Germani, A. Kehagias, K. Sfetsos, Journal of High Energy Physics 9, 60 (2009)
[Klusoň, (2009)]A192 J. Klusoň, Journal of High Energy Physics 11, 78 (2009)
[Myung, (2009)]A202 Y.S. Myung, Phys. Lett. B 679, 491 (2009)
[Bakas et al., (2010)]A143 I. Bakas, F. Bourliot, D. Lüst, M. Petropoulos, Classical and Quantum Gravity 27, 045013 (2010)
[Cai & Zhang, (2009)]A151 Y.F. Cai, X. Zhang, Phys. Rev. D 80, 043520 (2009)
[Cai et al., (2009)]A7 R.G. Cai, B. Hu, H.B. Zhang, Phys. Rev. D 80, 041501 (2009)
[Carloni et al., (2010)]A156 S. Carloni, E. Elizalde, P.J. Silva, Classical and Quantum Gravity 27, 045004 (2010)
[Dutta & Saridakis, (2010)]A163 S. Dutta, E.N. Saridakis, J. Cosmol. Astropart. Phys. 1, 13 (2010)
[Gao et al., (2010)]A170 X. Gao, Y. Wang, W. Xue, R. Brandenberger, J. Cosmol. Astropart. Phys. 2, 20 (2010)
[Greenwald et al., (2010)]A172 J. Greenwald, A. Papazoglou, A. Wang, Phys. Rev. D 81, 084046 (2010)
[Kim et al., (2009)]A188 S.S. Kim, T. Kim, Y. Kim, Phys. Rev. D 80, 124002 (2009)
[Kiritsis & Kofinas, (2009)]46 E. Kiritsis, G. Kofinas, Nuclear Physics B 821, 467 (2009)
[Kiritsis & Kofinas, (2010)]A190 E. Kiritsis, G. Kofinas, Journal of High Energy Physics 1, 122 (2010)
[Lü et al., (2009)]A36 H. Lü, J. Mei, C.N. Pope, Physical Review Letters 103, 091301 (2009)
[Mukohyama et al., (2009)]A201 S. Mukohyama, K. Nakayama, F. Takahashi, S. Yokoyama, Physics Letters B 679, 6 (2009)
[Sotiriou et al., (2009)]A126 T.P. Sotiriou, M. Visser, S. Weinfurtner, Journal of High Energy Physics 10, 33 (2009)
[Wang & Wu, (2009)]76 A. Wang, Y. Wu, J. Cosmol. Astropart. Phys. 7, 12 (2009)
[Mukohyama, (2009)]58 S. Mukohyama, Phys. Rev. D 80, 064005 (2009)
[Mukohyama, (2010)]57 S. Mukohyama, Classical and Quantum Gravity 27, 223101 (2010)
[Majhi & Samanta, (2010)]53 B.R. Majhi, S. Samanta, Annals of Physics 325, 2410 (2010)
[Neupane, (2009)]59 I.P. Neupane, Phys. Lett. B 673, 111 (2009)
[Saridakis, (2010)]63 E. N. Saridakis, European Physical Journal C 67, 229 (2010)
[Wang, (2011)]74 A. Wang, Modern Physics Letters A 26, 387 (2011)
[Nishioka, (2009)]A112 T. Nishioka, Classical and Quantum Gravity 26, 242001 (2009)
[Orlando & Reffert, (2009)]A113 D. Orlando, S. Reffert, Classical and Quantum Gravity 26, 155021 (2009)
[Visser, (2009)]73 M. Visser, Phys. Rev. D 80, 025011 (2009)
[Volovik, (2009)]A135 G.E. Volovik, Soviet Journal of Experimental and Theoretical Physics Letters 89, 525 (2009)
[Wang & Maartens, (2010)]A221 A. Wang, R. Maartens, Phys. Rev. D 81, 024009 (2010)
[Minamitsuji, (2010)]mina M. Minamitsuji, Physics Letters B 684, 194 (2010)
[Brandenberger, (2009)]A69 R. Brandenberger, Phys. Rev. D 80, 023535 (2009)
[Takahashi, (2009)]A131 T. Takahashi, J. Soda, Physical Review Letters 102, 231301 (2009)
[Appignani, (2010)]app C. Appignani, R. Casadio, S. Shankaranarayanan, J. Cosmol. Astropart. Phys. 4, 6 (2010)
[Bertoldi et al., (2009)]A65 G. Bertoldi, B.A. Burrington, A. Peet, Phys. Rev. D 80, 126003 (2009)
[Danielsoon, (2009)]A78 U.H. Danielsson, L. Thorlacius, Journal of High Energy Physics, 3, 70 (2009)
[Mann, (2009)]A110 R.B. Mann, Journal of High Energy Physics 6, 75 (2009)
[Lin et al., (2012)]kai K. Lin, S. Mukohyama, A. Wang, Phys. Rev. D 86, 104024 (2012)
[Setare & Jamil, (2010a)]suracon5 M.R. Setare, M. Jamil, J. Cosmol. Astropart. Phys. 2, 10 (2010)
[Jamil et al., (2010)]suracon3 M. Jamil, E.N. Saridakis, M.R. Setare, J. Cosmol. Astropart. Phys. 11, 32 (2010)
[Karami et al., (2012)]suracon1 K. Karami, M. Jamil, M. Roos, S. Ghaffari, A. Abdolmaleki, Astrophys. Space Sci. 340, 175 (2012)
[Jamil & Saridakis, (2010)]suracon4 M. Jamil, E.N. Saridakis, J. Cosmol. Astropart. Phys. 7, 28 (2010)
[Karami et al., (2012)]suracon2 K. Karami, A. Sheykhi, M. Jamil, et al., Canadian Journal of Physics 90, 473 (2012)
[Pasqua et al.(2015)]miohl1 A. Pasqua et al., International Journal of
Theoretical Physics 54, 972 (2015)
[Jawad et al.(2014)]miohl2 A. Jawad, S. Chattopadhyay, A. Pasqua, European Physical Journal Plus 129, 51 (2014)
[Chattopadhyay & Pasqua(2014)]miohl3 S. Chattopadhyay, A. Pasqua, Canadian Journal of Physics 92, 200 (2014)
[Jawad et al.(2013)]miohl4 A. Jawad, S. Chattopadhyay, A. Pasqua, Astrophysics and Space Science 346, 273 (2013)
[Nojiri & Odintsov, (2006)]valerossi S. Nojiri, S.D. Odintsov, Gen. Rel. Grav. 38, 1285 (2006)
[Pasqua & Chattopadhyay, (2013a)]suracon6 A. Pasqua, S. Chattopadhyay, Astrphys. Space Sci. 348, 541 (2013)
[Granda & Oliveros, (2008)]grandaoliverosa L.N. Granda, A. Oliveros, Physics Letters B 669, 275 (2008)
[Granda & Oliveros, (2009)]grandaoliveros L.N. Granda, A. Oliveros, Physics Letters B 671, 199 (2009)
[Wang & Xu, (2010)]wangalfa Y. Wang, L. Xu, Phys. Rev. D, 81, 083523 (2010)
[Hsu, (2008)]hsunuovo S.D.H. Hsu, Phys. Lett. B 669, 275 (2008)
[Arnowitt, (2004)]arno R.L. Arnowitt, S. Deser, C.W. Misner, Gravitation, an introduction to current research, Louis Witten ed. (Wiley 1962), chapter 7, pp 227-265 (2004)
[Hořava, (2009a)]A177 P. Hořava, Journal of High Energy Physics 3, 20 (2009)
[Calcagni, (2009)]12calcagni G. Calcagni, Journal of High Energy Physics 9, 112 (2009)
[Chaichian et al., (2010)]14cha M. Chaichian, S. Nojiri, S.D. Odintsov, M. Oksanen, A. Tureanu, Classical and Quantum Gravity 27, 185021 (2010)
[Wang & Wu, (2011)]75 A. Wang, Q. Wu, Phys. Rev. D 83, 044025 (2011)
[Izumi & Mukohyama, (2011)]29 K. Izumi, S. Mukohyama, Phys. Rev. D 84, 064025 (2011)
[Hořava & Melby-Thompson, (2010)]24 P. Hořava, C.M. Melby-Thompson, Phys. Rev. D 82, 064027 (2010)
[da Silva, (2011)]17 A.M. da Silva, Classical and Quantum Gravity 28, 055011 (2011)
[Zhu et al., (2011)]89 T. Zhu, Q. Wu, A. Wang, F.W. Shu, Phys. Rev. D 84, 101502 (2011)
[Dutta & Saridakis, (2013)]duttasari S. Dutta, E.N. Saridakis, J. Cosmol. Astropart. Phys. 5, 13 (2013)
[Perivolaropoulos, (2005)]A209 L. Perivolaropoulos, J. Cosmol. Astropart. Phys. 10, 1 (2005)
[Amendola & Tocchini-Valentini, (2001)]q1 L. Amendola, D. Tocchini-Valentini, Phys. Rev. D 64, 043509 (2001)
[Farooq et al., (2010)]q1-5 M.U. Farooq, M. Jamil, M.A. Rashid, Int. J. Theor. Phys 49, 2278 (2010)
[Jamil & Farooq, (2010b)]q1-10 M. Jamil, M.U. Farooq, J. Cosmol. Astropart. Phys. 03, 001 (2010)
[Setare & Jamil, (2010b)]q1-3 M.R. Setare, M. Jamil, Phys. Lett. B 690, 1 (2010)
[Zimdhal, (2001)]q1-7 W. Zimdahl, D. Pavon, Phys. Lett. B 521, 133 (2001)
[Abdalla et al., (2009)]1 E. Abdalla, L.R. Abramo, L. Sodré, B. Wang, Physics Letters B 673, 107 (2009)
[Bertolami et al., (2007)]9 O. Bertolami, F. Gil Pedro, M. Le Delliou, Physics Letters B 654, 165 (2007)
[Bertolami et al., (2009)]10O. Bertolami, F. Gil Pedro, M. Le Delliou, General Relativity and Gravitation, 41, 2839 (2009)
[Guo, (2007)]A173 Z.K. Guo, N. Ohta, S. Tsujikawa, Phys. Rev. D 76, 023508 (2007)
[He et al., (2009)]22 J.H. He, B. Wang, P. Zhang, Phys. Rev. D 80, 063530 (2009)
[Feng et al., (2008)]feng08 C. Feng et. al., Phys. Lett. B 665, 111 (2008)
[Ichiki et al., (2008)]q4 K. Ichiki et. al., J. Cosmol. Astropart. Phys. 06, 005 (2008)
[Jamil & Rashid, (2008)]jamil-08-2008 M. Jamil, M.A. Rashid, Eur. Phys. J. C. 56, 429 (2008)
[Da̧browski, (2005)]dabro M.P. Da̧browski, Physics Letters B 625, 184 (2005)
[Alam et al., (2003)]alam U. Alam, V. Sahni, T. Deep Saini, A.A. Starobinsky, Mon. Not. R. Astron. Soc. 344, 1057 (2003)
[Huterer & Turner, (1999)]28 D. Huterer, M.S. Turner, Phys. Rev. D 60, 081301 (1999)
[Weller & Albrecht, (2001)]84 J. Weller, A. Albrecht, Physical Review Letters 86, 1939 (2001)
[Sahni et al., (2003)]sahni V. Sahni, T.D. Saini, A.A. Starobinsky, U. Alam, Soviet Journal of Experimental and Theoretical Physics Letters 77, 201 (2003)
[Wu & Yu, (2010)]wu1 P. Wu, H. Yu, Phys. Lett. B 693, 415 (2010)
[Wang et al., (2009)]wang F.Y. Wang, Z.G. Dai, S. Qi, Astron. Astrophys. 507, 53 (2009)
[Khodam-Mohammadi & Malekjani, (2011)]state1 A. Khodam-Mohammadi, M. Malekjani, Astrophys. Space Sci. 331, 265 (2011)
[Malekjani& Khodam-Mohammadi, (2010)]state2 M. Malekjani, A. Khodam-Mohammadi, Int. J. Mod. Phys. D 19, 1857 (2010)
[Malekjani& Khodam-Mohammadi, (2011)]state3 M. Malekjani, A. Khodam-Mohammadi, N. Nazari-pooya, Astrophys. Space Sci. 332, 515 (2011)
[Malekjani & Khodam-Mohammadi, (2013)]state6 M. Malekjani, A. Khodam-Mohammadi, Astrophys. Space Sci. 343, 451461 (2013)
[Visser, (2004)]cosmo2 M. Visser, Class. Quant. Grav. 21, 2603 (2004)
[Weinberg, (1972)]cosmo1 S. Weinberg, Gravitation and cosmology, Wiley, New York 1972.
[Capozziello et al., (2011)]capozziello S. Capozziello, R. Lazkoz, V. Salzano, Phys. Rev. D 84, 124061 (2011)
[Capozziello & Izzo, (2008)]values0 S. Capozziello, L. Izzo, Astronomy & Astrophysics 490, 31 (2008)
[John, (2004)]values0-2 M.V. John, The Astrophysical Journal 614, 1, (2004)
[John, (2008)]values0-1 M.V. John, The Astrophysical Journal 630, 667 (2008)
[Dunajski, & Gibbons, (2008)]duna M. Dunajski, G. Gibbons, Classical and Quantum Gravity 25, 235012 (2008)
[Arabsalmani & Sahni, (2011)]arab M. Arabsalmani, V. Sahni, Phys. Rev. D 83, 043501 (2011)
[Myung, (2007)]myung Y.S. Myung, Phys. Lett. B 652 223 (2007)
[Kim et al., (2008)]kim K.Y. Kim, H.W. Lee, Y.S. Myung, Phys. Lett. B 660, 118 (2008)
[Sharif & Jawad, (2005)]sharif M. Sharif, A. Jawad, Eur. Phys. C 72 2097 (2012)
[Jawad et al., (2013a)]jawad A. Jawad, A. Pasqua, S. Chattopadhyay, Astrophys. Space Sci. 344 489 (2013)
[Pasqua et al., (2013b)]miofrt A. Pasqua, S. Chattopadhyay, I. Khomenko, Canad. J. Phys. 91, 632 (2013)
[Pasqua et al., (2013a)]miovs2 A. Pasqua, S. Chattopadhyay, I. Khomenko, International Journal of Theoretical Physics 52, 2496 (2013)
|
1511.00624
|
$^1$ Institut für Quantenoptik und Quanteninformation
der Österreichischen Akademie der Wissenschaften,
Technikerstrasse 21a,
A-6020 Innsbruck, Austria
$^2$ Institute of Physics and Astronomy,
University of Potsdam,
Karl-Liebknecht-Str. 24/25,
D-14476 Potsdam, Germany
$^3$ Institute of Atomic and Subatomic Physics,
TU Wien,
Stadionallee 2,
A-1020 Vienna, Austria
$^4$ The University of Hong Kong,
Hong Kong
$^5$ Institut für Experimentalphysik,
Universität Innsbruck,
Technikerstrasse 25,
A-6020 Innsbruck, Austria
The electric-field noise above a layered structure composed of a planar metal electrode covered by a thin dielectric is evaluated and it is found that the dielectric film considerably increases the noise level, in proportion to its thickness. Importantly, even a thin (mono) layer of a low-loss dielectric can enhance the noise level by several orders of magnitude compared to the noise above a bare metal. Close to this layered surface, the power spectral density of the electric field varies with the inverse fourth power of the distance to the surface, rather than with the inverse square, as it would above a bare metal surface. Furthermore, compared to a clean metal, where the noise spectrum does not vary with frequency (in the radio-wave and microwave bands), the dielectric layer can generate electric-field noise which scales in inverse proportion to the frequency. For various realistic scenarios, the noise levels predicted from this model are comparable to those observed in trapped-ion experiments. Thus, these findings are of particular importance for the understanding and mitigation of unwanted heating and decoherence in miniaturized ion traps.
§ INTRODUCTION
Electric-field fluctuations above metal surfaces are a common problem in many areas of physics and a severe limitation to precision measurements as diverse as space-based gravitational-wave detectors <cit.>, nano-cantilevers probing dispersion forces <cit.>, and the shielding of particle beams <cit.>. In trapped-ion systems, electric-field noise at around 1 MHz and at distances of a few tens or hundreds of µm from metallic electrodes significantly heats the ions <cit.>. This sets a limit on the coherence times that can be achieved in miniaturized trap designs which are currently developed for scalable quantum information processing. Ever since the observation of unexpectedly high heating rates <cit.> which could not be explained by the noise of the trapping circuitry, the role of the electric noise from surfaces in ion traps has attracted much experimental and theoretical attention. While a perfect conductor would not generate electric noise beyond the very low level of blackbody radiation, larger fluctuating electric fields are in principle expected above real conductors made of metals with non-vanishing resistive losses. However, early investigations <cit.> showed that the noise levels expected from the metal's resistance are generally still far too low to account for the experimentally observed heating rates in ion traps. There are experimental indications that in some instances the high heating rates observed are related to conditions on the electrodes' surfaces. Various mechanisms have been proposed, including models based on fluctuating patch-potentials <cit.>, adatom dipoles, two-level fluctuators <cit.>, or diffusing adatoms and charges <cit.>. Finding exactly which of these effects is significant in any given experiment, and whether other effects also play a role, constitutes an active area of experimental and theoretical research <cit.>.
In this work the electric-field noise generated by a thin layer of a dielectric on top of a flat metal electrode is investigated. This scenario mimics surface conditions that are typically encountered in trapped-ion experiments: the surface of the metal electrodes, having been exposed to air and humidity, will usually be covered by a non-metallic layer such as native oxides or hydrocarbon compounds. Recent experiments with trapped ions have indeed observed a considerable reduction of the electric-field noise after in-situ cleaning of the electrode surface with lasers <cit.>, ion-beam milling <cit.> or plasma cleaning <cit.>. In this paper the contamination layer is modeled as a thin film with dielectric losses. By this means, analytic results for the spectral power of electric-field fluctuations $S_{E}$ at a distance, $d$, above the surface are calculated.
The analysis presented here shows that the presence of even a very thin dielectric (mono-layer) can increase the absolute level of electric-field noise by several orders of magnitude compared to a bare metal surface. It also shows that, for moderate distances from the surface $d > \delta$, where $\delta$ is the metal's skin depth, the distance dependence of the noise spectrum changes from a $d^{-2}$ to a $d^{-4}$ scaling. Such a behavior is often attributed to localized surface potentials of microscopic origin, but arises here from a purely macroscopic description <cit.>. For many dielectric materials covering the electrodes, the permittivity $\epsilon$ and loss tangent $\tan\theta$ can be considered constant over a range of frequencies $\omega$ <cit.>, so that the power spectrum of the electric-field fluctuations decreases as $1/\omega$ with increasing frequency. The dielectric thickness and electrical properties could be measured independently with microwave loss spectroscopy <cit.> or surface scanning probes, providing a more detailed test of this model.
The rest of this paper is organized as follows. The fluctuation–dissipation theorem is described in section <ref>, and a qualitative estimate is given using the method of image charges for the noise expected above a clean metal surface and above a metal covered with a dielectric. As a cross-check, in section <ref> the noise spectrum is more rigorously computed for both a bare metal and a covered metal using methods of fluctuation electrodynamics. The absolute levels of electric-field noise are given for common metals under realistic surface conditions in section <ref>. Section <ref> discusses the relevance of the results and the outlook for experimentation in light of them.
§ ELECTRIC-FIELD FLUCTUATIONS FOR A CHARGED PARTICLE
This paper considers a single charged particle (or ion) interacting with its surroundings as shown in figure <ref>. The point-like ion is suspended in vacuum a distance, $d$, above a conducting plane. At the surface of the plane, there is a material of thickness $t_{\rm d}\ll d$ characterized by a (real) permittivity $\epsilon$ and loss tangent $\tan\theta$. The materials composing the structure are at temperature $T$.
A point-like particle is suspended in vacuum a distance $d$ above a conducting plane. The plane is covered with material of thickness $t_{\rm d}\ll d$ characterized by a permittivity $\epsilon$ and loss tangent $\tan\theta$. The materials composing the structure are at temperature $T$.
§.§ Fluctuation–Dissipation Theorem
Consider a single particle above a plane composed of some materials as in figure <ref>. When the nearby materials have some non-zero temperature, they will transmit energy to the particle through fluctuating forces. The motion of particle can also be damped by the surrounding materials via dissipative forces. The fluctuation–dissipation theorem states that for a system composed of a single particle at equilibrium with its surroundings, at a temperature $T$, the energy that is transmitted to the particle by the surrounding material's fluctuating forces must be equal to the energy lost through dissipative forces to the environment <cit.>. The fluctuations from the surrounding materials are a property of the material's temperature $T$ and will affect the particle, even if the particle is no longer in equilibrium. Using the methods and notation outlined by Kubo <cit.>, the electric-field fluctuations above a metal surface, with and without a dielectric layer, are computed as follows.
Consider a point-like particle moving in one dimension, where the dissipative force $F_\mathrm{d}$ is proportional to the speed of the particle, $u$, so that
\begin{equation}
F_\mathrm{d}= m\gamma u,
\end{equation}
where $m\gamma$ is the damping coefficient. More generally, this kind of formula applies in Laplace-Fourier space, with frequency-dependent $\gamma[\omega]$. This damping rate $\gamma[\omega]$ can be found by giving the particle an oscillatory motion at frequency $\omega$ and calculating the dissipated power due to this motion. In addition to friction, the particle is subject to a random force of thermally activated origin. Of interest for us is the power spectrum of the force fluctuations, $S_\mathrm{F}( \omega )$. The convention used here is that of a single-sided power spectral density (PSD) (units of ${\rm N^2/Hz}$) which is given by
\begin{equation}
S_\mathrm{F}(\omega)= 2 \int_{-\infty}^\infty {\rm d}\tau \langle \delta F(\tau) \delta F(0)\rangle {\rm e}^{-{\rm i}\omega \tau},
\label{eq:def-power-spectrum}
\end{equation}
where $\delta F(\tau)$ is the time-dependent variation of the force, $F$, from its long-term mean value. The fluctuation–dissipation theorem links the fluctuating force to the dissipative damping, such that
\begin{equation}
S_\mathrm{F}(\omega)=4 k_\mathrm{B}T m\operatorname{Re} \gamma[\omega],
\label{eq:flucDiss}
\end{equation}
where $k_\mathrm{B}$ is Boltzmann's constant, and $\operatorname{Re} \gamma[\omega]$ is the real part of the damping rate. If the particle has a charge $q$, then the PSD of the fluctuating force is related to the power spectral density, $S_{E}$, of the electric-field fluctuations at the location of the particle by
\begin{equation}
\label{eq:SFtoSE}
\end{equation}
The problem of computing the fluctuating electric field can thus be cast as a problem of calculating the dissipated power due to a forced motion of the charged particle. In order to calculate the dissipation, the form of the electric field due to the charge above the surface is found. The losses due to this electric field can then be computed. This is done for a clean metal surface in section <ref>, and for a system in which a thin dielectric covers the metal in section <ref>.
§.§ Ohmic Losses in the Metal
The static electric field due to a charged particle above an ideal conductor is half of a dipole pattern. This is the same pattern as would arise in the upper half-space if two particles of charge $+q$ and $-q$ were separated by a distance $2d$, as shown in figure <ref>. This method of electric images <cit.> allows the electric field at the conductor surface due to a charged particle a distance $d$ above the surface to be easily calculated. This is done by summing the fields of the real charge and its mirror charge. In this case the electric field at the surface of the conductor only has a non-zero component of the electric field normal to the surface given by
\begin{equation}
E_z=-\frac{q d}{2\pi\epsilon_0 R^3},
\label{eq:surfaceField}
\end{equation}
where $\epsilon_0$ is the permittivity of free space and $R$ is the distance from the charged particle to the location on the surface. The coordinates used here assume that the origin is located on the surface of the metal directly under the unperturbed charged particle, so that the $z$-axis goes through the particle.
For a single charged particle above the metal surface, the electric field above a metal surface forms a dipole pattern. The method of electric images allows the $z$ component of the electric field at the metal surface due to the charged particle a distance $d$ above the surface to be easily calculated.
The surface charge, $\sigma_\mathrm{s}$, present on an ideal conductor to produce the electric field at its surface is given by
\begin{equation}
\sigma_\mathrm{s}=\epsilon_0 E_z.
\end{equation}
If the charged particle is given a sinusoidal motion at frequency $\omega$, with a velocity-amplitude $\bm{u}$, the surface charge will be time-dependent and will produce a surface current, with amplitude $\bm{J}_\mathrm{s}$. For motion normal to the surface of the conductor, only radial surface currents, with amplitude $J_{\mathrm{s}r}$, will be produced. Utilizing the continuity equation, these can be shown to be
\begin{equation}
J_{\mathrm{s}r}(r)=-\frac{1}{r}\int^r_0 {\rm d}r' r' \frac{\partial \sigma_\mathrm{s}}{\partial t}
\,,
\end{equation}
where $r$ is the radial distance from the $z$ axis (i.e. $r^2=x^2+y^2$). The term $\partial \sigma_\mathrm{s} / \partial t$ can be expressed as the time derivative of the electric field. For a charged particle moving normal to the surface of the metal, with a small velocity amplitude, $|u_z| \ll d\omega$, this is
\begin{equation}
\frac{\partial \sigma_\mathrm{s}}{\partial t} = u_z \frac{q \left(2 d^2-r^2\right)}{2\pi R^5}.
\end{equation}
For an ideal conductor, the surface charges would respond instantly to the motion of the charge and reproduce the dipole pattern of figure <ref> at each moment in time. However, for materials with a non-zero resistivity, the induced surface currents produce an electric field parallel to the surface. For metals commonly used to fabricate ion traps, such as copper, gold, and aluminum, and considering the case where the oscillating charge is about 100 µm from the electrode, the resistance is so small that the field lines are not qualitatively different from figure <ref> up to frequencies in the THz band. At higher frequencies the ideal conductor approximation breaks down and it becomes necessary to treat the metal more generally with a complex permittivity. The analysis in this section is restricted to estimating the electric-field noise up to GHz frequencies with a distance between charged particle and surface greater than 100 nm. This is the regime in which trapped-ion experiments operate, and is also relevant for many other experimental systems.
When the oscillating charge is much further away from the metal than the metal's skin depth, $\delta$, the current density in the conductor falls off exponentially with the distance from the surface <cit.>. In this sub-surface region, the amplitude of the radial current density, $j_r$, can then be approximated by a constant effective current density within the skin depth (i.e. $j_r=J_{\mathrm{s}r}/\delta$) and 0 elsewhere, so that
\begin{align}
j_r(z) & \approx \frac{q r u_z}{2 \pi R^3 \delta } & & -\delta<z \le 0\\\notag
j_r(z) & \approx 0 & & z \le -\delta,
\end{align}
where $u_z$ is the amplitude of the $z$-component of the velocity of the particle. Within a metal of resistivity $\rho$, the cycle-averaged power-loss density $\langle p_\mathrm{loss} \rangle$, is then
\begin{equation}
\langle p_\mathrm{loss} \rangle=\frac{1}{2}\rho j^2_r,
\end{equation}
where $\rho$ is the resistivity of the conductor. By integrating the power-loss density over the volume of the whole conductor, this provides the total average dissipated power $\overline{P}_\mathrm{loss}$ in the conductor as a function of the amplitude of the $z$-component of the velocity $u_z$:
\begin{equation}
\overline{P}_\mathrm{loss}\approx\frac{q^2 \rho u^2_z}{16 \pi \, d^2 \delta }= \frac{1}{2}m u^2_z \operatorname{Re} \gamma[\omega] .
\label{eq:power-loss-and-Re-gamma}
\end{equation}
From this, the real part of the damping rate can be obtained. Using the fluctuation–dissipation theorem (see equation (<ref>)), the electric-field spectrum is found. Far from the surface ($d>\delta$) this is
\begin{equation}
\label{eq:metalNoiseFar}
S^\mathrm{F}_{E,\perp}\approx\frac{k_\mathrm{B}T\rho}{2\pi \, d^2 \delta}.
\end{equation}
For currents flowing within a thin film of metal for which the thickness is less than the skin depth ($t_\mathrm{m} < \delta$), the current is confined to a smaller region than it would be in a bulk metal. This increases the losses, and the resulting electric-field fluctuations above such thin films are
\begin{equation}
\label{eq:metalNoiseThinFilm}
S^\mathrm{TF}_{E,\perp}\approx\frac{k_\mathrm{B}T\rho}{2\pi \, d^2 t_\mathrm{m}}.
\end{equation}
If the ion-electrode distance is smaller than both the skin depth and the metal's thickness ($d < \delta, t_\mathrm{m}$), then the currents (and electric fields) are confined even closer to the surface: to within a depth $\simeq d$ <cit.>. The electric-field noise is then approximately
\begin{equation}
\label{eq:metalNoiseNear}
S^\mathrm{N}_{E,\perp}\approx\frac{k_\mathrm{B}T\rho}{2\pi \, d^3}.
\end{equation}
These results, obtained here by applying the fluctuation–dissipation theorem to a charge–image charge pair, reproduce essentially the same results derived independently by applying the fluctuation–dissipation theorem to a Green's function formalism of electrodynamics <cit.>, which is further discussed in section <ref>.
The electric-field fluctuations above a metal due to resistive losses in the metal share many characteristics with fluctuations due to Johnson-Nyquist voltage noise of the electrodes and the connected circuitry. The power spectrum is proportional to the resistivity of the electrical components and for ion-electrode separations greater than the skin depth, the power spectrum scales as $1/d^2$ <cit.>. However, one difference from voltage noise is that as the ion approaches the electrode to distances $d$ less than the skin depth $\delta$ (provided the electrode thickness, $t_{\rm m}$ is greater than $d$), the power spectrum scales as $1/d^3$.
§.§ Losses in a Thin Dielectric Layer
In this section, the electric-field noise above a metal electrode covered with a thin layer of an isotropic dielectric with a thickness $t_{\rm d}\ll d$ is estimated. The dielectric is characterized by a complex permittivity, $\varepsilon=\epsilon(1 + \rm{i}\tan\theta)$, a real permittivity $\epsilon$, and loss tangent $\tan\theta$. It is further assumed here that the dielectric's loss tangent is not large ($\tan\theta < 1$) so that the electric-field pattern above the surface is still well approximated by a dipole pattern (see figure <ref>).
The static energy density $w_0$ in the thin dielectric layer can be written as a function of the (real) static electric field $\bm{E}_0$ as,
\begin{equation}
w_0=\frac{1}{2}\epsilon \bm{E}_0 \cdot \bm{E}_0 = \frac{1}{2}\bm{D}_0\cdot \bm{E}_0,
\end{equation}
where $\bm{E}_0$ is the electric field in the dielectric layer due to the charged particle,
and $\bm{D}_0=\epsilon\bm{E}_0$ is the (real) static electric displacement.
If there is a time-dependent change in the electric field, $\bm{E}(t)$, so that the total electric field is then $\bm{E}_\mathrm{total}(t)=\bm{E}_0+\bm{E}(t)$, the time dependent energy density, $w(t)$, is given by
\begin{equation}
\label{eq:ModEDensity}
w(t)=\frac{1}{2}\bm{D}_0\cdot \bm{E}_0 + \bm{D}_0 \cdot \bm{E}(t) + \frac{1}{2}\bm{D}(t) \cdot \bm{E}(t),
\end{equation}
where $\bm{D}(t)$ is the change in the electric displacement from its static value $\bm{D}_0$.
Consider that the charged particle at a distance, $d$, above the surface undergoes a small-amplitude motion, $\delta\bm{r}(t)=\delta\bm{r}\cos(\omega t)$, at frequency $\omega$ and with amplitude $\delta\bm{r}$ ($|\delta\bm{r}|\ll d$), which produces a change in the electric field, $\bm{E}(t)$. The first two terms on the right hand side of equation (<ref>)
therefore cycle-average to a constant or zero. Consequently, only the third term will
contribute to the energy lost during a cycle of motion.
The cycle-averaged rate of change of the energy density with time $\langle \partial w/\partial t \rangle$ is the time-averaged power loss density $\langle p_\mathrm{loss} \rangle$ in the dielectric,
\begin{equation}
\langle p_\mathrm{loss} \rangle = \langle \bm{E}\cdot \frac{\partial \bm{D}}{\partial t} \rangle.
\end{equation}
Using the complex formalism for the electric field, the power-loss density can be written as
\begin{equation}
\langle p_\mathrm{loss} \rangle = \frac{1}{2}\operatorname{Re}\left [\bm{\hat{E}}^*\cdot \frac{\partial}{\partial t}\bm{\hat{D}}\right ].
\end{equation}
where $\bm{\hat{E}}$ and $\bm{\hat{D}}$ are the complex amplitudes of the electric and displacement fields. The complex amplitudes are defined by their relation to the time varying fields as,
\begin{align}
\bm{E}(t) &= \operatorname{Re}\left[ \bm{\hat{E}}\,{\rm e}^{-{\rm i}\omega t} \right]\\ \notag
\bm{D}(t) &= \operatorname{Re}\left[ \bm{\hat{D}}\,{\rm e}^{-{\rm i}\omega t} \right],
\end{align}
where $\omega$ is the frequency of the oscillations in the electric field and the complex amplitude of the displacement field is $\bm{\hat{D}=\varepsilon \bm{\hat{E}}}$. The cycle-averaged power loss density is then,
\begin{equation}
\label{eq:powerLossCycle}
\langle p_\mathrm{loss} \rangle=\frac{1}{2}\operatorname{Re}\left[-{\rm i}\omega \bm{\hat{E}}^*\cdot\bm{\hat{D}} \right]=\frac{\omega}{2}\epsilon\tan\theta|\bm{\hat{E}}|^2.
\end{equation}
If the motion of the particle is parallel to the surface of the metal in the $x$-direction with a small amplitude ($\delta x \ll d$), then the complex amplitude of the $z$-component of the electric field $\hat{E}_z$ at the surface can be expanded in $\delta x$ using equation (<ref>) as
\begin{equation}
\hat{E}_z= \frac{-3 q d}{2 \pi \varepsilon R^3}\frac{x \delta x}{R^2},
\label{eq:Ez-oscillating-charge}
\end{equation}
where $x$ is the co-ordinate of the location on the layered surface below the charged particle and the factor $1/\varepsilon$ describes the dielectric screening in the material. This approximation is equivalent to considering the oscillating charge as a dipole in the low-frequency limit.
The power density can then be computed as a function of $\delta x$. By integrating over the volume of the thin dielectric and averaging over a cycle (see eq. <ref>), the cycle-averaged power lost in the dielectric is found. The average power dissipated in the dielectric $\overline{P}_\mathrm{d}$ as a function of the amplitude of the oscillatory motion $\delta x$ is
\begin{equation}
\overline{P}_\mathrm{d}=\frac{3}{64\pi}
\frac{ \tan\theta }{ \epsilon (1 + \tan^2\theta) }
\frac{q^2 t_{\rm d} \omega (\delta x)^2 }{ d^4 } ,
\end{equation}
Using again the second equality in equation (<ref>) and knowing the amplitude $u_x = \omega \delta x$ of the particle velocity, the damping rate $m \operatorname{Re} \gamma[\omega]$ is found. Using equations (<ref>, <ref>), the spectrum of electric-field fluctuations parallel to the surface above the dielectric layer is
\begin{equation}
S^\mathrm{D}_{E,\parallel} = \frac{3}{8\pi}
\frac{ \tan\theta }{ \epsilon (1 + \tan^2\theta) }
\frac{k_\mathrm{B}T t_{\rm d}}{ \omega d^4 }.
\label{eq:result-noise-parallel-1}
\end{equation}
This analysis can also be done for the dissipation of motion and electric-field fluctuations normal to the surface, for which the power spectrum due to the dielectric covering is
\begin{equation}
S^\mathrm{D}_{E,\perp} = \frac{3}{4 \pi }
\frac{ \tan\theta }{ \epsilon (1 + \tan^2\theta) } \frac{k_\mathrm{B}T t_{\rm d}}{\omega d^4}.
\label{eq:result-noise-perpendicular-1}
\end{equation}
The noise due to the dielectric thin film occurs in addition to any noise due to the finite resistance of the metal plate itself (see equations <ref>-<ref>). However, as shown in section <ref>, for typical experimental parameter regimes, the noise from even very thin dielectric coatings (mono-layers) exceeds the noise due to resistive losses of the metal by several orders of magnitude, and so the noise from the metal can generally be neglected.
§ CALCULATION FROM FLUCTUATION ELECTRODYNAMICS
In this section, the spectrum of the electric-field noise is calculated with the help of fluctuation electrodynamics in thermal equilibrium, using the fluctuation–dissipation theorem <cit.>:
\begin{equation}
S_{E,ij}( \bm{r}, \omega ) =
\frac{ 4 k_\mathrm{B} T }{ \omega }
%2 \hbar
%\coth\frac{ \hbar \omega }{ 2 k_\mathrm{B} T }
\mathop{\rm Im} G_{ij}( \bm{r}, \bm{r}; \omega ).
\label{eq:FDT-for-E}
\end{equation}
This classical approximation is valid because of the low-frequencies under consideration, $\hbar\omega \ll k_{\rm B} T$.
The spectrum, $S_{E,ij}$, gives the spectral expansion of the cross-correlation function,
$\langle E_{i} E_{j} \rangle$, see equation (<ref>).
The Green tensor, $G_{ij}( \bm{r}, \bm{r}'; \omega )$, allows the electric field at the position of the trap centre, $\bm{r}$, radiated by a point dipole with complex amplitude $\hat{\bm{d}}$, located at $\bm{r}'$ and oscillating at a frequency $\omega$ to be calculated:
\begin{equation}
E_i( \bm{r}, t ) =
\mathop{\rm Re} \Big[
\sum_{j}
G_{ij}( \bm{r}, \bm{r}'; \omega )
\hat d_j\, {\rm e}^{ - {\rm i} \omega t }
% + \mathrm{c.c.}
\Big].
\label{eq:def-E-Green-App}
\end{equation}
When evaluated at the metal plate [$\bm{r} = (x, y, z = 0)$] this recovers the field given in equation (<ref>). This field can be split into a free-space contribution and the reflection from the surface. Evaluating this in a frequency range where the distance to the surface is much shorter than the wavelength of the electric field, retardation can be neglected and the situation can be evaluated using electrostatics. The imaginary part of the reflection corresponds to the losses in the metal, relevant in equation (<ref>), and this yields, for fields parallel to the surface <cit.>,
\begin{equation}
G_\mathrm{pp}( \bm{r}, \bm{r}; \omega ) \approx \frac{ 1 }{ 8\pi\epsilon_0 }
\int\limits_{0}^{\infty}\!{\rm d}k \,k^2\, R_\mathrm{r}( \omega, k )
\, {\rm e}^{ - 2 k d }.
\label{eq:parallel-Green}
\end{equation}
Here, $R_\mathrm{r}( \omega, k )$ is the surface's electrostatic reflection coefficient. If the latter is independent of $k$, then the integral with respect to $k$ in equation (<ref>) can be simply performed
\[
\int\limits_{0}^{\infty}\!{\rm d}k \,k^2
\, {\rm e}^{ - 2 k d } =
\frac{ 1 }{ 4 d^3 }
\]
and yields the field generated by an image dipole with amplitude $R_\mathrm{r}( \omega )$.
According to the fluctuation–dissipation theorem expressed in equation (<ref>), the imaginary part of this image-dipole amplitude (related to dissipation in the surface) determines the electric-field noise (its fluctuation strength).
§.§ Bare Metal
For a clean surface without contaminants, the reflection coefficient is $k$-independent <cit.>, provided spatial dispersion (i.e. the anomalous skin effect) can be neglected. At distances greater than a few nanometers above the surface this is indeed the case and the method of image dipoles can be applied.
\begin{equation}
\mbox{bare metal:} \qquad
R_\mathrm{r}( \omega, k ) =
R_{\rm m}( \omega )
\,,\qquad
R_{\rm m}( \omega )
\frac{ \varepsilon_{\rm m}( \omega ) -
\epsilon_0 }{ \varepsilon_{\rm m}( \omega ) +
\epsilon_0 }.
\label{eq:bare-metal-R}
\end{equation}
This holds because, at low frequencies and for a good conductor, the complex dielectric function, $\varepsilon_{\rm m}$, is dominated by the conductivity $1/(\rho\omega)$, which is large compared to $\epsilon_0$. For example, the DC resistivity of gold typically exhibits $1/(\epsilon_0 \rho) \sim 10^{18}\,{\rm s}^{-1}$. Consequently,
\begin{equation}
\varepsilon_{\rm m}( \omega ) \approx
\frac{ {\rm i} }{ \rho \omega } + \ldots .
\label{eq:eps-and-sigma}
\end{equation}
and to
a good approximation the dissipative part of the image dipole is
\begin{equation}
\mathop{\rm Im} R_\mathrm{r}( \omega, k ) \approx
2 \mathop{\rm Im}\left( - \frac{ \epsilon_0 }{ \varepsilon_{\rm m}( \omega ) } \right)
% \approx \frac{ 2 \epsilon_0 \omega }{ \sigma }
\approx 2 \epsilon_0 \rho \omega
\ll 1
\label{eq:B}
\end{equation}
which is small, as expected for a good conductor. The field spectrum from equation (<ref>) becomes
\begin{equation}
\mbox{metal: } \qquad
S_{E,\parallel}( d, \omega ) \approx
\frac{ k_\mathrm{B} T \rho }{ 4 \pi\, d^3 }
\label{eq:result-metal}
\end{equation}
which is white. For the noise normal to the surface, a similar calculation <cit.> leads to a spectrum which is twice as large: $S_{E,\perp}( d, \omega ) = 2 S_{E,\parallel}( d, \omega )$, as also found in equation (<ref>).
Note that the approximations used here do not reproduce a perfect conductor since they vanish in the limit $\rho \to 0$. For this case, retardation must be taken into account to capture the noise in the leading order. Explicit formulas can be found in Ref. <cit.>. It should also be noted that the short-distance approximation breaks down when $d$ becomes comparable to the skin depth in the metal: $d \sim \delta = [2\rho / (\mu \omega) ]^{1/2} \approx 75$ µm for gold at $1\,{\rm MHz}$ with a resistivity of 22.1 nΩ$\cdot$m at a temperature of 293 K <cit.>, where gold's permeability $\mu=\mu_0$ is the vacuum permeability $\mu_0$. The $1/d^3$ scaling of equation (<ref>) applies provided $d \ll \delta$. In the opposite limit, $d \gg \delta$, equation (<ref>) must be multiplied by $2 d / \delta$, meaning that the noise exhibits a scaling of $\sim 1/d^2$ <cit.>. This produces the same results obtained in section <ref>. For gold at $d = 100$ µm the noise level expected from equation (<ref>) is $S_E \approx 10^{-17}$ , much smaller than what is observed experimentally in ion traps <cit.>. Much larger noise levels can arise from covering layers as follows.
§.§ Dielectric Covering Layer
For a metal covered with a dielectric layer (thickness $t_{\rm d}$,
complex permittivity $\varepsilon$), the (electrostatic) reflection coefficient is <cit.>
\begin{equation}
R_\mathrm{d}( \omega, k ) =
\frac{ R_{\varepsilon} + R_{\mathrm{m}\varepsilon}\, {\rm e}^{ - 2 k t_{\rm d} } }{
1 + R_{\varepsilon} R_{\mathrm{m}\varepsilon}\, {\rm e}^{ - 2 k t_{\rm d} } }
\,,
\label{eq:layer-R}
\end{equation}
where $R_{\varepsilon}$ and $R_{\mathrm{m}\varepsilon}$ are the reflection coefficients of the interfaces vacuum-dielectric and dielectric-metal respectively:
\begin{align}
R_{\varepsilon} &= \frac{ \varepsilon - \epsilon_0 }{
\varepsilon + \epsilon_0 }
\,, \\
R_{\mathrm{m}\varepsilon} &=
\frac{ \varepsilon_{\rm m}( \omega ) - \varepsilon }
{\varepsilon_{\rm m}( \omega ) + \varepsilon }.
\end{align}
The complex permittivity, $\varepsilon$, involves the loss tangent in its imaginary part, $\varepsilon = \epsilon ( 1 + \rm{i} \tan \theta)$.
Equation (<ref>) can be approximated for the purposes of this analysis: from the integral in equation (<ref>) it can be seen that the main $k$-vectors are $k = {\cal O}( 1 / d )$, so $k t_{\rm d} \ll 1$ for a thin layer. Combined with the assumption $|\varepsilon| =
{\cal O}( \epsilon_0 ) \ll |\varepsilon_{\rm m}|$, which is valid for a low-loss dielectric coating above a metal [see discussion above equation (<ref>)],
a series expansion can be performed for the two small parameters $k t_{\rm d}$ and $\epsilon_0 / \varepsilon_{\rm m}$ to give
\begin{equation}
R_\mathrm{d}( \omega, k ) \approx
- 2 k t_{\rm d} \frac{ \epsilon_0 }{ \varepsilon }
- 2 \frac{ \epsilon_0 }{ \varepsilon_{\rm m} }.
\label{eq:expanded-R-layer}
\end{equation}
Note the factor $k$ in the second term which, following integration with respect to $k$, leads to a different scaling with respect to distance, $d$.
\[
\int\limits_{0}^{\infty}\!{\rm d}k \,k^3
\, {\rm e}^{ - 2 k d } =
\frac{ 3 }{ 8 d^4 }
\]
For a highly conductive substrate, this is also the dominating term in equation (<ref>). The noise above a metal covered in a dielectric layer is thus equal to the sum of the noise from the dielectric layer and of the noise from the metal [given by equation (<ref>)]
\begin{equation}
\mbox{layer:}\qquad
S_{E,\parallel}^\mathrm{d}( d, \omega ) \approx
\frac{ 3 k_\mathrm{B} T t_{\rm d} }{ 8 \pi \epsilon_0 \omega \, d^4 }
% \mathop{\rm Im}\left( - \epsilon_0 / \varepsilon \right)
\mathop{\rm Im}\left( - \frac{ \epsilon_0 }{ \varepsilon } \right)
% + S_{E,\parallel}^{\rm met}( d )
+ \frac{ k_\mathrm{B} T \rho }{ 4 \pi\, d^3 }.
\label{eq:spectrum-layer}
\end{equation}
This can be rewritten in terms of the loss tangent and the DC permittivity, given that
\begin{equation}
\mathop{\rm Im}\left( - \frac{ \epsilon_0 }{ \varepsilon } \right)
=\frac{ \epsilon_0 \tan\theta }{ \epsilon (1 + \tan^2\theta) }.
\end{equation}
This method therefore independently reproduces the result of equation (<ref>) which was derived by the methods of image charges. For some technical details and the extension of this calculation beyond electrostatics, see
the Appendix.
§ RESULTS FOR COMMON ELECTRODE MATERIALS
The model presented in section <ref> is quite general. It can be used to consider electrodes for which the dielectric covering is an intrinsic dielectric layer, such as a native oxide, as well as ones which are contaminated by some other non-conductive material. A thin dielectric layer covering ion trap electrodes has been measured on electrodes, which have significant electric-field noise with a level of approximately $10^{-11} \ldots 10^{-9}$ at an ion-electrode separation $d\approx 50 \ldots 100$ µm <cit.>. The model presented here predicts comparable levels of noise for both contaminated gold electrodes and metals which form a native oxide such as copper.
Noble metals, such as gold, do not form oxides. Nonetheless, following exposure to air – and particularly following the vacuum-bake process typically used in preparing trapped-ion systems – the metal surface is typically covered with a few mono-layers of a dielectric substance such as hydrocarbons <cit.>. The level of noise expected above a gold surface using the model presented here is estimated in section <ref>. Many metals develop a native oxide upon exposure to air and this native oxide can be a dielectric. This is the case for aluminium <cit.>, niobium <cit.> and copper <cit.>, all of which are standard materials for electrodes in ion traps. The level of noise expected above these metals (and their native oxides) is calculated in section <ref>.
The levels of noise above metal electrodes with various dielectric coverings, calculated in sections <ref> and <ref>, can be compared to the level of noise above a bare metal. While the properties of the dielectric layers can vary significantly, the relevant properties of good metals are such that expected level of noise from the bare metal is relatively consistent between materials. In typical miniaturized surface ion-trap experiments the ion-electrode separation ($50\,$µ${\rm m} < d$) is about equal to the skin depth of the electrode material ($\delta \sim$ 50 µm) which is much greater than the range of electrode thicknesses used in miniaturized ion traps ($100\,\mathrm{nm}< t_{\rm m} <$ 10 µm). Using equation <ref> the electric-field noise expected 50 µm above a bare metal at 1 MHz is approximately $10^{-16} \ldots 10^{-14}$ .
§.§ Contamination
Metal electrodes can be contaminated with dielectric substances upon exposure to air. For instance, a pure gold surface will be contaminated with at least 0.4 nm of hydrocarbons (a mono-layer) within minutes of exposure to air <cit.>. While the contamination on gold films exposed to air has been characterized to be largely hydrocarbon in nature with an approximate thickness of 0.4-2 nm, the exact chemical structure and the radio-frequency electrical characteristics of these surface contaminants are not currently known.
Consider, therefore, a gold electrode at room temperature with a 0.4 nm thick hydrocarbon film on the surface having the electrical characteristics of a known hydrocarbon compound (pentane) <cit.>. This contamination would have a relative permittivity $\epsilon/\epsilon_0 \simeq 2$, with a loss tangent $\tan\theta \simeq 0.01$. Using equation <ref>, the power of the electric-field fluctuations 50 µm above the surface at 1 MHz would be of order $10^{-11}$ .
§.§ Native Oxides
Many metals develop an oxide layer, called a native oxide, on any surface exposed to air and humidity. These oxides can form a dielectric a few nanometers thick and this is the case for metals commonly used in miniaturized ion traps, such as copper, aluminum and niobium. The exact details of their thickness, chemical and and electrical properties can depend upon environmental conditions, as well as on the underlying metal. In some instances it is possible to reduce the electric-field noise experienced by trapped atomic ions above metallic electrodes by modifying the surface of electrodes which have a native-oxide layer <cit.>.
In this section native oxides covering their associated metals are considered. For each native oxide, the relative permittivity $\epsilon/\epsilon_0$, loss tangent $\tan\theta$ and thickness $t_\mathrm{d}$ is estimated. And from these parameters, the corresponding power spectrum of electric field is provided using equations <ref> and <ref>. In each instance the noise at a distance of 50 µm above a planar surface at 300 K is calculated.
The alumina layer that forms as a native oxide on the surface of aluminum typically has a thickness, $t_{\rm d} \approx 4$ nm <cit.>, a relative permittivity, $\epsilon/\epsilon_0 \simeq 8.5$ <cit.>, and a loss tangent, $\tan\theta \simeq 0.001$ <cit.>. From equations <ref> and <ref>, the expected electric-field noise 50 µm above an aluminum surface with a native oxide at 1 MHz is approximately $0.5\times10^{-12}$ parallel to the surface and $1\times10^{-12}$ normal to the surface.
Niobium oxides have widely varying properties depending upon the exact stoichiometric ratio, crystal structure and test conditions <cit.>. For illustration, a 5 nm thick layer of Ni2O5 with a relative permittivity, $\epsilon/\epsilon_0\simeq41$, and a room-temperature loss tangent, $\tan\theta\simeq0.01$ is considered here <cit.>. Again from equations <ref> and <ref> the expected power spectral density (PSD) 50µm above the surface at 1 MHz is around $1.5\times10^{-12}$ parallel to the surface and $3\times10^{-12}$ normal to the surface.
Copper oxides also have widely varying properties depending on exactly how they are produced. They tend to have large relative permittivities and high losses <cit.>. Their thickness grows over time on exposure to the humidity in air without limit. Because of these wide variations it is hard to give a general level of expected noise. However, for illustration, a 5 nm thick layer of CuO with a relative permittivity, $\epsilon/\epsilon_0\simeq20$, and loss tangent, $\tan\theta\simeq0.5$, is considered here. From equation (<ref>) the expected electric-field noise 50µm above the surface at 1 MHz is of order $10^{-10}$ .
§.§ Distance and Frequency Scaling of Common Materials
Our analysis presented in sections <ref> and <ref> has shown that a thin dielectric layer can significantly modify the electric field noise spectrum and change its scaling with distance and frequency. For a situation where the ion-surface separation $d$ is much larger than the dielectric layer, $t_{\rm d}$ as well as the thickness of the current layer in the metal (either $t_{\rm m}$ or $\delta$), the distance scaling changes from $d^{-2}$ to $d^{-4}$. From the simple model described in section <ref>, this result follows from the fact that in a dielectric layer local losses scale as $|E_z(t)|^2$, which falls much faster, with increasing $d$, than the radial current density squared, $j_r^2(t)$, responsible for resistive losses in the metal (see sec. <ref>). Compared to the bare metal, the result for $S_E(\omega)$ for a dielectric layer contains another factor of $\omega^{-1}$, but in general also the frequency and temperature dependence of the dielectric loss must be taken into account and
\begin{equation}
S_E(\omega) \sim \frac{ T \, \mathop{\rm Im} \varepsilon(\omega; T)}{\omega}.
\end{equation}
For a simple Debye model for the dielectric constant, $\varepsilon(\omega)\approx \epsilon/(1+i\omega \tau)$, where $\tau$ is a characteristic damping time, one would obtain $S_E(\omega)=const.$ for $\omega\ll \tau^{-1}$ and $S_E(\omega)\sim 1/\omega^2$ for $\omega\gg \tau^{-1}$. However, it is know that most real materials have a much weaker frequency dependence in the RF to microwave frequency regime <cit.> and therefore, depending in detail on the dielectric material, a scaling $S_E(\omega)\sim \omega^{-\alpha}$, with $\alpha\sim 1$ is expected.
The temperature dependence of the complex permittivity of materials varies widely. However, for materials whose permittivity does not change substantially with temperature, the noise would scale linearly with temperature $T$. For the native oxides of aluminum and niobium, the loss tangent tends to decrease with temperature <cit.>. In general, microwave and radio-frequency spectroscopy with conventional tools, or using a trapped ion as a probe could be used to infer the temperature dependence of the complex permittivity.
Figure <ref> shows the normalized electric-field-noise levels vs. distance for a bare gold or copper electrode, a gold electrode with 0.4 nm hydrocarbon (HC) contamination and a 5 nm film of copper oxide on a copper electrode. For thick metal electrodes ($t_\mathrm{m} > d$), the noise above a bare metal scales as $1/d^2$ [as $1/d^3$] when the distance $d$ between the charged particle and the metal surface is larger [smaller] than the skin depth $\delta$, respectively. Even very thin layers of common dielectric materials covering the metal electrodes will produce an electric field noise above the surface, which is orders of magnitude above that produced from a metal and scales as $1/d^4$. Assuming the loss tangent $\tan\theta$ and permittivity $\epsilon$ are essentially constant <cit.> the expected power spectrum is inversely proportional to the frequency $\omega$.
The normalized electric-field-noise-levels and distance scaling for a bare gold or copper electrode, a gold electrode with 0.4 nm hydrocarbon (H.C.) contamination and a 5 nm film of copper oxide on a copper electrode. Dashed lines: fields parallel to the surface; solid lines: fields normal to surface. The noise above a bare metal scales as $1/d^2$ when the distance $d$ between the charged particle and the metal surface is larger than the skin depth $\delta$. When $d$ is smaller than the skin depth, the scaling changes to $1/d^3$. Even very thin layers of dielectric materials covering the metal electrodes will produce an electric field noise above the surface, which is orders of magnitude above that produced from a metal and scales as $1/d^4$. Reference noise level and skin depth are $S_E=1\times10^{-16}$ , $\delta = 75$ µm.
§ OUTLOOK AND SUMMARY
Numerous mechanisms have previously been put forward to account for the electric-field noise observed in miniaturized ion traps above conductors. The challenge is to match the observed levels of noise which are well above those predicted for bare metals, and their scaling with relevant parameters like distance, frequency, and temperature. Numerous experiments have been performed to characterise the noise, often with apparently-conflicting results. Taken together the results seem to point to the fact that different experiments are limited by different, possibly multiple, sources of noise <cit.>.
The mechanism considered in this paper is by no means a panacea to explain all experimental observations. Rather it is to be added to the list of noise sources which must be considered (and if necessary eliminated) in any given experiment. Thin dielectric coatings that cover a metallic electrode have been analyzed here and it is found that electric-field fluctuations many orders of magnitude stronger than above a clean metal surface are to be expected. This is consistent with a number of experimental results, which reduced the electric-field fluctuations by modifying the surface. For instance, it has been shown that treatments which altered the native oxide of superconducting cavities were able to improve the quality factors of such cavities <cit.>. In ion traps, laser-ablation cleaning has been seen to cause a slight reduction in the electric-field noise above aluminum electrodes <cit.>, and plasma cleaning has been used to reduce the electric-field noise above niobium electrodes <cit.> and copper/aluminum electrodes <cit.>.
For metals such as gold, which does not support a native oxide, the analysis presented here shows that even mono-layers of dielectrics which adhere to a non-passivated gold surface exposed to air will produce substantial electric-field noise. This is consistent with experimental results which show that argon-ion cleaning of gold electrodes can significantly reduce the electric-field noise above such surfaces <cit.>. The model presented here could be tested in detail with setups <cit.> where a controlled surface coating is deposited on the trap electrodes. One would expect a difference between islands and continuous films, amorphous or annealed. Alternatively, the electric properties of surface layers may be tested with microwaves whose fields are confined to the sub-surface region by the skin effect. Similar techniques have been applied for superconducting cavities <cit.>. More generally, the crucial role of electrode coatings put forward here may help to understand why some traps develop increased anomalous heating over time (“aging"), while others perform well over periods of months.
Noise of the type modelled here can be distinguished from other noise sources. For instance, in trapped-ion systems, if Johnson-Nyquist noise is the dominant source of fluctuating electric fields, this will predominantly originate in the attendant electronics in the system, rather than the ion-trap electrodes themselves. Consequently the noise level varies as a function of the temperature of the electronics. In contrast, noise due to dielectric coverings on the electrodes varies as a function of the electrode temperature, which can be controlled independently of the attendant electronics.
In addition to highlighting a possible source of noise in trapped-ion experiments, the analysis of this paper suggests a novel method of reducing the electric-field noise in experiments. If there is an existing dielectric layer on the electrodes, it could be modified to increase its (real) permittivity $\epsilon$ or reduce its loss tangent $\tan\theta$. This would reduce the electric-field noise (see equation <ref>). For example, copper electrodes exposed to air will invariably have a layer of copper oxide CuO on them. Copper oxide can transition, by means of a temperature treatment, to a giant permittivity material with a relative permittivity $\epsilon/\epsilon_0 \simeq 10^4$ <cit.>. For metals such as gold, which are easily contaminated upon exposure to air, it may be possible to mitigate contamination through passivation of the bare metal by a thin film of a substance with a large permittivity during fabrication. For instance, a film of a ceramic such as SrTiO3 with a relative permittivity $\epsilon/\epsilon_0 \simeq 10^4$ may provide a suitably high dielectric screening and passivation <cit.>.
The simple model presented here of an infinite sheet of conductor with a uniform layer of a dielectric coating could be extended to include other situations. For instance, the expected electric-field noise for three-dimesional electrodes or non-uniform patches of various materials could be calculated with the same basic theory. It is expected that the distance scaling would depend upon the geometry of the electrodes <cit.> and patches of high-loss materials would increase the electric-field noise locally <cit.>. Such customization of the theory presented here would allow for the model to be applied to more specific experimental situations.
In summary, this paper describes how the expected thermal noise above metal electrodes coated with various dielectric materials can be calculated using a simple macroscopic model. It is shown that native oxides of common metals and mono-layers of hydrocarbon contamination can produce levels of electric-field noise which could be of concern to a number of experiments.
§ ACKNOWLEDGEMENTS
The authors thank Nikos Daniilidis, Yves Colombe, Philipp Schindler, Ron Folman, Baruch Horovitz, and Ferdinand Schmidt-Kaler for discussions. This work was supported by the Austrian Science Fund (FWF) via the project Q-SAIL, the SFB FOQUS, the START Grant Y 591-N16 and the Institut für Quanteninformation GmbH. C.H. acknowledges support from the DFG through the DIP program (FO 703/2-1).
§ APPENDIX: EFFICIENT CALCULATION OF ELECTRIC NOISE SPECTRA
§.§ Introduction
The experimental conditions typical for ion traps can
be summarized as
* low frequencies
$\hbar \omega \ll k_{\rm B} T$
* good metallic conductors
$\varepsilon_m( \omega ) \approx {\rm i} / (\rho \omega)$
* trap–surface distance small compared to wavelength
$d \ll c / \omega$
* but comparable to skin depth
$\delta = [2 \rho / (\mu_0 \omega)]^{1/2} \sim d$
* thin dielectric coating with thickness
$t \ll d, \delta$
Under these conditions, the fluctuation–dissipation theorem
can be simplified because of the low frequencies involved.
The spectral correlation function of the electric
field is used here for positive frequencies only.
Attention: this convention for the noise spectrum is a
factor $2$ larger than in other papers <cit.>
\begin{equation}
S_{E,ij}( {\bf r}, \omega ) =
\frac{ 4 k_\mathrm{B} T }{ \omega }
\mathop{\rm Im} G_{ij}( {\bf r}, {\bf r}; \omega ).
\label{eq:FDT-for-E-App}
\end{equation}
The Green tensor is defined (and normalized) according
to electric dipole radiation by equation (<ref>)
\begin{equation}
% E_i( {\bf r}, t ) =
% \sum_{j}
% G_{ij}( {\bf r}, {\bf r}'; \omega )
% d_j\, {\rm e}^{ - {\rm i} \omega t } + \mathrm{c.c.},
E_i( \bm{r}, t ) =
\mathop{\rm Re} \Big[
\sum_{j}
G_{ij}( \bm{r}, \bm{r}'; \omega )
\hat d_j\, {\rm e}^{ - {\rm i} \omega t }
% + \mathrm{c.c.}
\Big].
\label{eq:def-E-Green}
\end{equation}
for a point dipole with complex amplitude $\hat{\bf d}$, frequency $\omega$,
located at position ${\bf r}'$.
§.§ Green tensor
We work in a planar geometry: expansion in plane waves with wave vector
${\bf k}$ parallel to surface <cit.>.
For ${\bf r} = {\bf r}'$,
only the diagonal components of the Green tensor
are nonzero. It splits naturally in two contributions:
free space radiation and reflection from the surface.
The first part is the same as if the surface were at infinite distance; it gives an imaginary part of
[adapted from Eqs.(11, 12) in <cit.>; attention:
the correlation function there is not symmetrized]:
\begin{equation}
\mathop{\rm Im} G_{ij}^{\infty}( {\bf r}, {\bf r}; \omega ) =
\delta_{ij} \frac{ \omega^3 }{ 6 \pi \varepsilon_0 c^3 }.
\end{equation}
The reflection from the surface gives the distance ($d$) dependent
parts for fields parallel ($p$) and normal ($n$) to the surface
[from Eq.(14) in <cit.>]:
\begin{eqnarray}
\mathop{\rm Im} G_{pp}( z, \omega ) &=&
\frac{ 1 }{ 8 \pi \varepsilon_0 }
\int\limits_{0}^\infty \!
{\rm d}k \, {\rm e}^{ - 2 k d }
\left(
\frac{ \omega^2 }{ c^2 }
\mathop{\rm Im} r_s( k )
k^2 \mathop{\rm Im} r_p( k )
\right)
\nonumber
\\
\mathop{\rm Im} G_{nn}( z, \omega ) &=&
\frac{ 1 }{ 4 \pi \varepsilon_0 }
\int\limits_{0}^\infty \!
{\rm d}k \, {\rm e}^{ - 2 k d }
\mathop{\rm Im}
r_p( k ).
\label{eq:expansion-G}
\end{eqnarray}
We have used here
the non-retarded approximation, applying the general rules: relevant
$k$-vectors much larger than $\omega/c$. This is because their
typical size is (from the exponential) $k \sim 1/d \gg \omega/c$.
This applies to fields above the
surface (outside the dielectric): they correspond
to electrostatic fields with $\nabla^2 {\bf E} = 0$, hence
the normal wavevector $k_z = ( \omega^2 / c^2 - k^2)^{1/2}$
is approximately purely imaginary, $k_z = {\rm i} k$.
We see that if the s-polarization does not contribute, the
normal field spectrum is a factor $2$ above that for parallel fields.
§.§ Reflection coefficients
These are given by the Fresnel formulas. For a simple interface $12$
(field incident from
medium $1$), they depend on normal wave vectors $q_a$ in the media
($a = 1,2$)
\begin{equation}
q_{a} = ( \varepsilon_a \mu_0 \omega^2 - k^2 )^{1/2},
\label{eq:def-medium-q}
\end{equation}
where $\varepsilon_a$ is the dielectric function ($\varepsilon_0$
for vacuum). The two principal polarizations (also denoted
TM, TE instead of $p$, $s$) yield:
\begin{eqnarray}
r_p^{12} &=&
\frac{ \varepsilon_2 q_{1} - \varepsilon_1 q_{2} }%
{ \varepsilon_2 q_{2} + \varepsilon_1 q_{2} }
\\
r_s^{12} &=&
\frac{ q_{1} - q_{2} }%
{ q_{1} + q_{2} }.
\end{eqnarray}
For a layered system (media $012$ from top to bottom, incidence
from `above') <cit.>
\begin{equation}
r^{012} =
\frac{ r^{01} + r^{12}\, {\rm e}^{ 2 {\rm i} q_1 t } }{
1 + r^{01} r^{12} \, {\rm e}^{ 2 {\rm i} q_1 t } },
\end{equation}
where $t$ is the thickness of the layer (medium $1$). This formula
has the same structure for both polarizations (the planar geometry
preserves the two principal polarizations).
Our goal is now: simplify these expressions
without compromising too much accuracy as
appropriate for our parameters. Assumption: dielectric
function of layer is comparable to vacuum, while
below there is a good conductor. This gives
$\varepsilon_1 \sim \varepsilon_0 \ll \varepsilon_2$
(to be understood in absolute values, all medium dielectric
functions are in general complex).
Using the non-retarded approximation, we expand the square roots
in Eq.(<ref>) and get the approximate medium wave
\begin{eqnarray}
q_a \approx {\rm i} k
- \frac{ {\rm i} \varepsilon_a \mu_0 \omega^2 }{ 2 k }
\,, a = 0, 1\,, \qquad
q_2 \approx
( 2 {\rm i} / \delta^2 - k^2 )^{1/2}
\,,
\end{eqnarray}
where the dielectric function of the conducting medium 2
has been re-written with the skin depth $\delta$:
\begin{equation}
\varepsilon_2( \omega ) \approx \frac{ {\rm i} }{ \omega \rho }
= \frac{ 2 {\rm i} }{ \mu_0 \omega^2 \delta^2 }.
\end{equation}
Since we allow for $d \sim \delta$, we have to deal with
$k$-vectors $k \sim 1/\delta$ and we do not expand $q_2$.
§.§.§ s-Polarization
This polarization is often ignored in the non-retarded limit.
We start with it because the discussion is somewhat simpler.
In the leading order for the vacuum–dielectric interface
\begin{equation}
r^{01}_s \approx
\frac{ (\varepsilon_1 - \varepsilon_0) \mu_0
\omega^2 }{ 4 k^2 }
\left( \frac{ \varepsilon_1 }{ \varepsilon_0 } - 1 \right)
\frac{ \omega^2 }{ 4 k^2 c^2 }
\ll 1
\end{equation}
while for the dielectric-metal interface, we have
${\cal O}(1)$ reflection:
\begin{equation}
r^{a2}_s \approx
\frac{ {\rm i} k - q_2 }{ {\rm i} k + q_2 }
\,,\qquad
a = 0, 1
\,.
\end{equation}
This is not unitary since $q_2$ is in general complex.
For the layered geometry, we neglect the 01 reflection compared
to 12 and get the following approximation
\begin{equation}
r_s^{012} \approx
\frac{ {\rm i} k \delta - (2{\rm i} - k^2\delta^2)^{1/2} }{
{\rm i} k \delta + (2 {\rm i} - k^2\delta^2)^{1/2} }
\, {\rm e}^{ - 2 k t },
\end{equation}
which is just a factor ${\rm e}^{ - 2 k t }$ smaller than
the reflection from the bare metal surface. (The exponential takes
into account that the `strong reflection' happens at the lower
interface, whose distance from the charge is $d + t$.) The
dielectric losses in the thin layer do not appear in this
order. (This is OK as long as its dielectric function
$\varepsilon_1$ is much smaller than that of the metal.)
Dielectric losses of covered metal, expressed
as imaginary part of reflection coefficients. The thick curves
give the exact calculation (including retardation), the thin
(colored) ones the approximations discussed in the text.
(left) s-Polarization: bare and covered metals give
the same result.
p-Polarization: upper lines with layer, lower lines bare metal.
$\omega/2\pi = 1\,{\rm MHz}$.
Parameters for copper: conductivity from wikipedia
($\delta = 65\,{\rm \mu m}$),
oxide layer with $t = 5\,{\rm nm}$, dielectric constant $20$,
loss tangent $0.5$.
For aluminium:
conductivity from wikipedia
($\delta = 75\,{\rm \mu m}$),
oxide layer with $t = 2\,{\rm nm}$, dielectric constant $8.5$,
loss tangent $10^{-3}$.
The skin depth $\delta$ is used to scale the $k$-vector.
With this scaling, the s-polarization gives a `universal
curve' that is independent of the material. Note that
in p-polarization, the losses scale with the thickness and
the loss tangent.
Fig.<ref>(left) shows that these approximations work
very well for a broad range of $k$-vectors. As apparent from
Eqs.(<ref>), the imaginary parts of the reflection
coefficients determine the noise spectrum; this is shown in the
§.§.§ p-Polarization
The situation is somewhat reversed: at the vacuum-dielectric
interface, we have ${\cal O}( 1 )$ reflection:
\begin{equation}
r^{01}_p \approx \frac{ \varepsilon_1 - \varepsilon_0 }
{ \varepsilon_1 + \varepsilon_0 },
\end{equation}
while at the dielectric-metal interface there is a (nearly) perfect
\begin{equation}
r^{a2}_p \approx 1 + 2 \frac{ (2{\rm i} - k^2\delta^2)^{1/2} }{ k \delta }
\varepsilon_a \omega \rho
\,,\qquad
a = 0,1
\,,
\end{equation}
whose imaginary part is small because
$\varepsilon_a \omega \rho = {\cal O}( 10^{-11} )$
for good conductors and MHz frequencies
(this is simply the ratio
$\varepsilon_a / \varepsilon_2( \omega )$).
This small deviation
from perfect reflection is at the end responsible for the imaginary
Expansion of the layered reflection coefficient for
a thin layer $q_1 t \approx {\rm i} k t \ll 1$ gives
\begin{equation}
r^{012}_p \approx
1 + 2 \frac{ (2{\rm i} - k^2\delta^2)^{1/2} }{ k \delta }
\varepsilon_0 \omega \rho
- 2 \frac{ \varepsilon_0 }{ \varepsilon_1 } k t.
\label{eq:rp-approx}
\end{equation}
Here, we have also expanded the exponential
${\rm e}^{ 2 {\rm i} q_1 t }$.
Fig.<ref>(right) shows that these approximations work
very well for a broad range of $k$-vectors.
§.§ Spectra
Use scale factor $1/\delta$ to introduce a dimensionless
$k$-vector (see the products $k\delta$ in the formulas above).
Then by combining the FDT and the expansion of the Green function,
we find
\begin{eqnarray}
S_{E,p}(d, \omega ) &=&
\frac{ k_{B} T }{ 2\pi \varepsilon_0 \omega \, \delta^3 }
\left[
s_p( d / \delta; \omega )
\frac{ \omega^2 \delta^2 }{ c^2 }
s_s( d / \delta )
\right]
\\
S_{E,n}( d, \omega ) &=&
\frac{ k_{B} T }{ \pi \varepsilon_0 \omega \, \delta^3 }
s_p( d / \delta; \omega ),
\end{eqnarray}
where the dimensionless functions $s_s$ and $s_p$ are given
in Eqs.(<ref>, <ref>,
<ref>) below.
Observe the factor 2 between parallel and normal noise spectra,
apart from the contribution of the s-polarization. That one
comes, however, with the small coefficient
\begin{equation}
\frac{ \omega^2 \delta^2 }{ c^2 }
2 \rho \omega \varepsilon_0
\sim 2 \times 10^{-12} \ll 1.
\label{eq:small-ratio-epsilons}
\end{equation}
This turns the prefactor into the convenient reference level
\begin{equation}
S^{\rm ref}_E =
\frac{ k_{B} T \rho }{ 2\pi \, \delta^3 }
\sim 4 \times 10^{-17} \frac{ {\rm V^2/m^2} }{ {\rm Hz} }
\label{eq:spec-normalization}
\end{equation}
for good conductors (room temperature, 1 MHz secular frequency).
§.§.§ s-Polarization
Negligible as mentioned above, discussed only for completeness.
The integral
\begin{equation}
s_s( d / \delta ) \approx
\int\limits_0^{\infty}\!{\rm d}x\,
\mathop{\rm Im}
\frac{ {\rm i} x - (2{\rm i} - x^2)^{1/2} }{
{\rm i} x + (2{\rm i} - x^2)^{1/2} }
\, {\rm e}^{ - 2 x (d + t)/\delta }
\label{eq:s-integral-approx}
\end{equation}
goes for short distances,
$d' = d + t \ll \delta$, to a constant because for $d' = 0$,
it converges to $2 / 3$.
For large distances, $d' = d + t \gg \delta$, the domain $x \ll 1$
gives the dominant contribution. Expanding the integrand,
we get
\begin{equation}
s_s \approx
% expand root around q = sqrt( 2i ), lowest order sufficient to get
% part linear in x
% ( i x - q ) / ( i x + q )
% approx - 1 (1 - 2 i x / q) = - 1 + (2i/q) x
% Im gives x
% old version: q = sqrt(i), Im gives sqrt(2) x (OK)
\int\limits_{0}^{\infty}\!{\rm d}x\,
% \sqrt{2} \,x % old
x % new
\,{\rm e}^{ - 2 x d'/\delta }
% \frac{ 1 }{ \sqrt{ 8 } \, (z' / \delta) ^2 }% old
\frac{ 1 }{ 4\, (d' / \delta) ^2 }.% new
\label{eq:large-distance-s}
\end{equation}
The two asymptotes compare well with the numerical integration
(which is immediate to compute as well), as Fig.<ref>
Dimensionless noise spectrum for electric fields, bare metal
(left) s-Polarization. Magenta: constant $2 / 3$;
blue: Eq.(<ref>) $\sim 1/d^2$.
(right) p-Polarization (without the prefactor
$\varepsilon_0 \omega \rho$). Magenta:
Eq.(<ref>) $\sim 1/d^3$;
red: Eq.(<ref>) $\sim 1/d^2$.
§.§.§ p-Polarization
Using the approximate form (<ref>) for the reflection coefficient, there are two contributions. One for the bare
metal surface (the expansion assumes that $d' = d + t \approx d$,
superscript m for `metal'):
\begin{equation}
s_p^{\rm m}( d / \delta; \omega ) =
2 \varepsilon_0 \omega \rho
\int\limits_0^\infty\!{\rm d}x\, x
\mathop{\rm Im} ( 2 {\rm i} - x^2 )^{1/2}\,
{\rm e}^{ - 2 x d / \delta}.
\label{eq:p-integral-approx-m}
\end{equation}
For short distance, perform a large-$x$ expansion (exponential kept for
convergence) and obtain:
\begin{equation}
d \ll \delta: \qquad
s_p^{\rm m}( d / \delta; \omega ) \approx
2 \varepsilon_0 \omega \rho
\int\limits_0^\infty\!{\rm d}x\, x^2 \,
{\rm e}^{ - 2 x d / \delta}
\frac{ \varepsilon_0 \omega \rho }{ 2 (d/\delta)^3 }.
\label{eq:short-p}
\end{equation}
And for large distance, small $x$ one obtains:
\begin{equation}
d \gg \delta: \qquad
s_p^{\rm m}( d / \delta; \omega ) \approx
% \sqrt{2} \, \varepsilon_0 \omega \rho% old
2 \varepsilon_0 \omega \rho % new
\int\limits_0^\infty\!{\rm d}x\, x \,
{\rm e}^{ - 2 x d / \delta}
% \frac{ \varepsilon_0 \omega \rho }{ 2\sqrt{2} (z/\delta)^2 }% old
\frac{ \varepsilon_0 \omega \rho }{ 2 (d/\delta)^2 }.
\label{eq:large-p}
\end{equation}
See Fig.<ref>(right) how accurate these asymptotes are.
Note that the p-polarization involves, at large distance, the same (small) prefactor as
the s-polarized contribution, since $(\omega\delta/c)^2 =
2 \varepsilon_0 \omega \rho$ [Eq.(<ref>)].
In particular, at large distances, both polarizations are comparable so
that parallel and normal fields have the same noise spectrum (we use $d' \approx d$
for consistency with the expansion of $r_p^{012}$):
\begin{equation}
d \gg \delta: \qquad
S^{\rm m}_{E,p}( d, \omega ) =
S^{\rm m}_{E,n}( d, \omega )
\approx
\frac{ k_{\rm B} T \rho }{ 2\pi \varepsilon_0 \delta^3 }
% \frac{ 1 }{ \sqrt{ 2 } \, (z / \delta) ^2 }% old
\frac{ 1 }{ (d / \delta) ^2 }.% new
\label{eq:large-distance-bare}
\end{equation}
The other contribution depends on the properties of the
dielectric layer. Here the integral is simpler (superscript
d for `dielectric layer'):
\begin{equation}
s_p^{\rm d}( d / \delta; \omega ) =
- 2 \mathop{\rm Im} \frac{ \varepsilon_0 }{ \varepsilon_1 }
\frac{ t }{ \delta }
\int\limits_0^\infty\!{\rm d}x\, x^3
{\rm e}^{ - 2 x d / \delta}
- \mathop{\rm Im} \frac{ \varepsilon_0 }{ \varepsilon_1 }
\frac{ t }{ \delta }
\frac{ 3 }{ 4 (d/\delta)^4 }.
\label{eq:p-integral-approx-d}
\end{equation}
Even a layer of a few nanometers makes this the dominant
contribution, as can be seen in the plots [Fig.<ref>].
We recover again the factor $1/2$ between parallel
and normal spectra
\begin{equation}
S_{E,p}( d, \omega ) =
\frac12 S_{E,n}( d, \omega ) \approx
\frac{ k_{\rm B} T }{ 2 \pi \varepsilon_0 \omega \delta^3 }
\mathop{\rm Im}
\left( \frac{ - \varepsilon_0 }{ \varepsilon_1 } \right)
\frac{ t }{ \delta }
\frac{ 3 }{ 4 (d/\delta)^4 },
\label{eq:layer-spec}
\end{equation}
This expression is independent of the properties of the metallic substrate
because the skin depth $\delta$ drops out.
§.§ Normalized Power Spectra for Common Materials
A convenient reference level of the noise is:
\begin{equation}
S_E^{\rm ref}
= \frac{ k_{\rm B} T \times \rho \varepsilon_0 \omega
}{ 2\pi \varepsilon_0 \omega\,\delta^3 }
= \frac{ k_{\rm B} T \rho }{ 2\pi \,\delta^3 },
\label{eq:reference-noise-level}
\end{equation}
which is typical for a bare metal at distance $z \approx \delta$
(see caption Fig.<ref> for values).
The noise spectra parallel and perpendicular to the surface are
shown in
normalized to Eq.(<ref>).
The thick lines give the results of
numerical integrations, combining the contributions from
the metal and the dielectric layer and summing
p- and s-polarization.
The blue line, on top of the dashed black one,
is given by the asymptote (<ref>) and depends
on the layer properties only.
One sees that the oxide coating (or the hydrocarbon contamination
layer) dominates the noise spectrum by more than 6 orders of
magnitude. For the bare metal, both p- and s-polarizations
contribute equally when the distance $z$ is larger than the skin depth
$\delta$ – this doubling of the noise would not have been
found from strict electrostatics. Red line: large-distance
asymptote (<ref>) for the bare metal.
Electric noise spectra vs. distance $d$ from
bare and covered surfaces. Dashed: fields parallel to the surface;
solid: fields normal to surface.
Thin red line: asymptote
$\delta^2 / d^2$
thin blue line (coincident with parallel noise):
asymptote $\sim 1/d^4$ of Eq.(<ref>).
Top left: copper with conductivity from wikipedia
(skin depth $\delta = 65\,{\rm \mu m}$),
oxide layer with $t = 5\,{\rm nm}$, dielectric constant $20$,
loss tangent $0.5$. Reference noise level
[Eq.(<ref>)] is
$S_E^{\rm ref} \approx 4\times 10^{-17}\,{\rm (V/m)^2 / Hz}$.
Top right: gold with wikipedia conductivity
(skin depth $\delta = 75\,{\rm \mu m}$),
contamination layer (HC for hydrocarbon compounts)
with $t = 0.4\,{\rm nm}$,
dielectric constant $2$,
loss tangent $0.01$. Reference level
$S_E^{\rm ref} \approx 3.5\times 10^{-17}\,{\rm (V/m)^2 / Hz}$.
Bottom: comparison of both materials, identical to Fig.<ref>.
§ REFERENCES
|
1511.00932
|
The X-ray rings of LMXB ]The X-ray dust scattered rings of the black hole low mass binary
G. Vasilopoulos & M. Petropoulou]G. Vasilopoulos$^1$Email: [email protected] & M. Petropoulou$^{2}$Einstein Postdoctoral Fellow
$^{1}$Max-Planck-Institut für extraterrestrische Physik,Giessenbachstraße, 85748 Garching, Germany
$^2$ Department of Physics and Astronomy, Purdue University, 525 Northwestern
Avenue, West Lafayette, IN 47907, USA
|
1511.00752
|
\begin{eqnarray}}
\def\ee{\end{eqnarray}}
\def\p{\partial}
\def\no{\nonumber}
\def\e{\epsilon}
\def\de{\delta}
\def\De{\Delta}
\def\om{\omega}
\def\Om{\Omega}
\def\f{\frac}
\def\th{\theta}
\def\la{\lambda}
\def\lab{\label}
\def\b{\bigg}
\def\var{\varphi}
\def\na{\nabla}
\def\ka{\kappa}
\def\al{\alpha}
\def\La{\Lambda}
\def\ga{\gamma}
\def\Ga{\Gamma}
\def\ti{\tilde}
\def\wti{\widetilde}
\def\wh{\widehat}
\def\ol{\overline}
\def\ul{\underline}
\def\Th{\Theta}
\def\si{\sigma}
\def\Si{\Sigma}
\def\oo{\infty}
\def\q{\quad}
\def\z{\zeta}
\def\co{\coloneqq}
\def\eqq{\eqqcolon}
\def\di{\displaystyle}
\def\bt{\begin{theorem}}
\def\et{\end{theorem}}
\def\bc{\begin{corollary}}
\def\ec{\end{corollary}}
\def\bl{\begin{lemma}}
\def\el{\end{lemma}}
\def\bp{\begin{proposition}}
\def\ep{\end{proposition}}
\def\br{\begin{remark}}
\def\er{\end{remark}}
\def\bd{\begin{definition}}
\def\ed{\end{definition}}
\def\bpf{\begin{proof}}
\def\epf{\end{proof}}
\def\bex{\begin{example}}
\def\eex{\end{example}}
\def\bq{\begin{question}}
\def\eq{\end{question}}
\def\bas{\begin{assumption}}
\def\eas{\end{assumption}}
\def\ber{\begin{exercise}}
\def\eer{\end{exercise}}
\def\mb{\mathbb}
\def\mbR{\mb{R}}
\def\mbZ{\mb{Z}}
\def\mc{\mathcal}
\def\mcS{\mc{S}}
\def\ms{\mathscr}
\def\lan{\langle}
\def\ran{\rangle}
\def\lb{\llbracket}
\def\rb{\rrbracket}
\title{Decay properties of axially symmetric D-solutions to the steady Navier-Stokes equations}
\author{{Shangkun Weng\thanks{School of mathematics and statistics, Wuhan University, Wuhan, Hubei Province, China, 430072. Email: [email protected]}}}
\pagestyle{myheadings} \markboth{Decay properties of the axially symmetric D-solutions to steady Navier-Stokes equations}{}\maketitle
\begin{abstract}
We investigate the decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations. The achievements of this paper are two folds. One is improved decay rates of $u_{\th}$ and $\na {\bf u}$, especially we show that $|u_{\th}(r,z)|\leq c\left(\f{\log r}{r}\right)^{\f 12}$ for any smooth axially symmetric D-solutions to the Navier-Stokes equations. These improvement are based on improved weighted estimates of $\om_{\th}$ and $A_p$ weight for singular integral operators, which yields good decay estimates for $(\na u_r, \na u_z)$ and $(\om_r, \om_{z})$, where $\bm{\om}=\textit{curl }{\bf u}= \om_r {\bf e}_r + \om_{\th} {\bf e}_{\th}+ \om_z {\bf e}_z$. Another is the first decay rate estimates in the $Oz$-direction for smooth axially symmetric flows without swirl. We do not need any small assumptions on the forcing term.
\end{abstract}
\begin{center}
\begin{minipage}{5.5in}
Mathematics Subject Classifications 2010: Primary 76D05; Secondary 35Q35.\\
Key words: Navier-Stokes, axially symmetric, decay, $A_p$ weight.
\end{minipage}
\end{center}
\section{Introduction and main results}
In this paper, we will investigate the decay properties of the smooth axially symmetric solutions to the steady Navier-Stokes equations
\be\lab{sns} \begin{cases}
({\bf u}\cdot\nabla) {\bf u} +\nabla p -\Delta {\bf u} ={\bf f},\q \q \forall {\bf x}\in \mbR^3\\
\text{div }{\bf u}=0,\\
\displaystyle \lim_{|x|\to \oo} {\bf u}({\bf x}) ={\bf u}_{\oo}=0
\end{cases}
\ee
with finite Dirichlet integral
\be\lab{ns-dirichlet}
\int_{\mbR^3} |\nabla {\bf u}({\bf x})|^2 d{\bf x}<+\oo.
\ee
Here ${\bf u}, p$ and ${\bf f}$ denote the fluid velocity, the pressure and the body force. ${\bf u}_{\oo}$ is a constant vector and we also assume the viscosity to be $1$ for simplicity. One can also consider the same problem in the exterior domains $\Om \subset\mbR^3$ with the no-slip boundary condition on $\p\Om$, where the complement of $\Om$ is a compact axially symmetric domain. For simplicity, we only consider the whole space case and ${\bf u}_{\oo}=0$ in this paper, some of our results can also be extended to the exterior domain case.
The fundamental contribution to the existence of weak solutions to the stationary Navier-Stokes equations is due to Leray \cite{leray33}, where he constructed the weak solution of (\ref{sns}) with no-slip boundary conditions and constant velocity at infinity. Leray's solution has finite Dirichlet integral, and is usually refereed as $D$-solution. Ladyzhenskaya \cite{lady69} and Fujita \cite{fujita} also considered the nonhomogeneous boundary conditions case. It was easy to show that the $D$-solutions are smooth provided that data are smooth. In \cite{finn59,finn61}, Finn showed that any $D$-solution in three dimensional exterior domain converged uniformly pointwise to the prescribed vector ${\bf u}_{\oo}$ at infinity and, moreover, in the case ${\bf u}_{\oo}\neq 0$, he showed that if $|{\bf u}({\bf x})- {\bf u}_{\oo}|\leq C |{\bf x}|^{-\al}$ for some $\al>\f 12$ as ${\bf x}\to \oo$, then $|{\bf u}({\bf x})-{\bf u}_{\oo}|\leq \bar{C} |{\bf x}|^{-1}$ as ${\bf x}\to \oo$. Finn also suggested a calss of physical reasonable (PR) solutions to (\ref{sns}) in the three-dimensional exterior domain satisfying ${\bf u}({\bf x})= O(|{\bf x}|^{-1})$ if ${\bf u}_{\oo}=0$ or ${\bf u}({\bf x})-{\bf u}_{\oo}=O(|{\bf x}|^{-\f 12-\e})$ for some $\e>0$, if ${\bf u}_{\oo}\neq 0$. Finn \cite{finn65} then established the existence and uniqueness of a physically reasonable solution in a three dimensional exterior domain when the data are small enough. It is easy to show a PR-solution is a D-solution. However, whether the converse implication holds true has remained open for long times. In the case of ${\bf u}_{\oo}\neq 0$, Babenko \cite{babenko73} showed that every D-solution is a PR-solution if the force is of bounded support. Galdi \cite{galdi92} also proved the same result for ${\bf u}_{\oo}=0$, under the assumption that ${\bf u}$ obeys the ``energy inequality" and the viscosity is sufficiently large. In \cite{np95}, the authors established the existence and uniqueness of solution to (\ref{sns}) with the same decay rate as that of the fundamental solution of the Stokes problem, under some smallness assumptions on the data. For the investigation of the asymptotic profile of (\ref{sns}) with ${\bf u}_{\oo}=0$, one can refer to \cite{dg00,ks11,np00}. For more information about the recent results about these problems, one can refer to \cite{farwig98,fs98,galdi11}.
%It is well-known that a generalized solution to (\ref{sns}) belonging to $W^{1,2}_{loc}(\mbR^3)$ implie that ${\bf u}$ is smooth. Here we include the result stated in Galdi's book \cite{galdi11}.
%\bt\lab{galdi}(Theorem X.5.1 in \cite{galdi11}).
%Let ${\bf u}({\bf x})$ be a generalized solution of (\ref{sns}) satisfying (\ref{ns-dirichlet}) and $p({\bf x})$ be the associated pressure, then there exists $p_1\in \mbR$ such that
%\lim_{|{\bf x}|\to \oo} |D^{\al} {\bf u}({\bf x})| + \lim_{|{\bf x}|\to \oo} |D^{\al} (p({\bf x})-p_1)|=0
%uniformly for all multi-index $\al=(\al_1,\al_2,\al_3)\in [\mb{N}\cup \{0\}]^3$.
From the above introduction, we see that the theory about the decay properties of D-solutions to (\ref{sns}) is quite incomplete in the case of large forcing term. In this article, we will investigate the D-solution ${\bf u}$ with additional axially symmetric property to simplify this problem. On the other hand, this paper can be regarded as a continuation of my previous study \cite{cw15} with Prof. Chae, where we established some interesting Liouville type theorems for smooth axially symmetric D-solutions. From \cite{cw15}, we can see that there is a close relation between the decay properties of D-solution to (\ref{sns}) with the famous open problem of the triviality of D-solution to (\ref{sns}) with ${\bf f}\equiv 0$ and ${\bf u}_{\oo}=0$. One can also refer to \cite{chae14,cy13,cw15,galdi11,kpr15} for more recent results about this triviality problem. Now we introduce the mathematical setup of our problem. More precisely, we introduce the cylindrical coordinate
\be\no
r=\sqrt{x_1^2+x_2^2},\q \theta= \arctan\f{x_2}{x_1},\q z=x_3.
\ee
We denote ${\bf e}_r, {\bf e}_{\th}, {\bf e}_z$ the standard basis vectors in the cylindrical coordinate:
\be\no
{\bf e}_r= (\cos\th,\sin\th, 0),\q {\bf e}_{\th}= (-\sin\th,\cos\th,0),\q {\bf e}_z= (0,0,1).
\ee
A function $f$ is said to be {\it axially symmetric} if it does not depend on $\th$. A vector-valued function ${\bf u}= (u_r, u_{\th}, u_z)$ is called {\it axially symmetric} if $u_r, u_{\th}$ and $u_z$ do not depend on $\th$. A vector-valued function ${\bf u}= (u_r, u_{\th}, u_{z})$ is called {\it axially symmetric with no swirl} if $u_{\th}=0$ while $u_r$ and $u_z$ do not depend on $\th$.
Assume that ${\bf u}({\bf x})= u_r(r,z) {\bf e}_r+ u_{\th}(r,z) {\bf e}_{\th} + u_z(r,z){\bf e}_z$ is a smooth D-solution to (\ref{sns}). The corresponding asymmetric steady Navier-Stokes equations read as follows.
\be\lab{asym-sns}
\begin{cases}
(u_r\p_r+ u_z\p_z) u_r-\f{u_{\th}^2}{r} + \p_r p =\left(\p_r^2+\f{1}{r}\p_r+\p_z^2-\f{1}{r^2}\right) u_r+f_r, \\
(u_r\p_r+ u_z\p_z) u_{\th} +\f{u_r u_{\th}}{r}=\left(\p_r^2+\f{1}{r}\p_r+\p_z^2-\f{1}{r^2}\right)u_{\th}+f_{\th},\\
(u_r\p_r+ u_z\p_z) u_z + \p_z p=\left(\p_r^2+\f{1}{r}\p_r+\p_z^2\right) u_z+ f_z,\\
\p_r u_r +\f{u_r}{r} +\p_z u_z=0.
\end{cases}
\ee
Define the vorticity $\bm{\om}({\bf x})=\text{curl }{\bf u}({\bf x})= \om_r(r,z) {\bf e}_r + \om_{\th}(r,z) {\bf e}_{\th}+ \om_z(r,z) {\bf e}_z$, where
\be\no
\om_r= -\p_z u_{\th},\q \om_{\th}= \p_z u_r -\p_r u_z,\q \om_z= \f 1r \p_r(r u_{\th}).
\ee
The equations satisfied by $\om_r,\om_{\th}$ and $\om_z$ are listed as follows.
\be\lab{vorticity1}
&&(u_r\p_r+ u_z\p_z) \om_r - (\om_r\p_r +\om_z\p_z) u_r =\left(\p_r^2+\f{1}{r}\p_r+\p_z^2-\f{1}{r^2}\right)\om_r- \p_z f_{\th},\\\lab{vorticity2}
&&(u_r\p_r+ u_z\p_z) \om_{\th} - \f{u_r\om_{\th}}{r}+\f{1}{r}\p_z (u_{\th}^2) =\left(\p_r^2+\f{1}{r}\p_r+\p_z^2-\f{1}{r^2}\right)\om_{\th}+\p_z f_r- \p_r f_z,\\\lab{vorticity3}
&&(u_r\p_r+ u_z\p_z) \om_z - (\om_r\p_r +\om_z\p_z) u_z =\left(\p_r^2+\f{1}{r}\p_r+\p_z^2\right)\om_z+\f{1}{r}\p_r(r f_{\th}).
\ee
For the investigation of the decay properties of D-solutions to the exterior stationary Navier-Stokes equations (\ref{sns}), one should trace back to the important papers by Gilbarg and Weinberger \cite{gw74,gw78}. For the two dimensional exterior domain, Gilbarg and Weinberger \cite{gw74} showed that the weak solution constructed by Leray was bounded and converged to a limit ${\bf u}_0$ in a mean square sense, while the pressure converged pointwise. In \cite{gw78}, they further investigated any weak solutions to (\ref{sns}) with finite Dirichlet integral, and found that the weak solution ${\bf u}$ may not be bounded, but it grew more slowly than $(\log r)^{\f 12}$. The pressure has a finite limit at infinity and the velocity either had a limit in the mean or $\int_0^{2\pi} |{\bf u}(r,\th)|^2 d\th \to \oo$ as $r\to \oo$. The vorticity $\om= \p_{x_2} u_1-\p_{x_1} u_2$ decayed more rapidly than $r^{-\f34}(\log r)^{\f 18}$ and the first derivatives of the velocity decayed more rapidly than $r^{-\f34}(\log r)^{\f 98}$ at infinity.
Inspired by \cite{gw74} and \cite{gw78}, Choe and Jin \cite{cj09} first obtained the following decay rates for smooth axially symmetric solutions to (\ref{sns}): Let $\Om$ be an exterior domain, suppose ${\bf f}\in H^1(\Om)$ be an axially symmetric vector field with
\be\lab{cj-force}
\|{\bf f}\|_{H^{-1}(\Om)}+\|{\bf f}\|_{H^{1}(\Om)}+ \b\|\b(\f{r}{\log r}\b)^{1/2} f_{\th}\b\|_{L^2(\Om)}+\|r\p_z f_r\|_{L^2(\Om)}+ \|r\p_r f_z\|_{L^2(\Om)} \leq M
\ee
for some constant $M$. Then the axially symmetric solution $({\bf u}, p)$ with finite Dirichlet integral satisfied
\be\lab{cj1}
|u_r(r,z)|+ |u_z(r,z)|&\leq& c(M)\b(\f{\log r}{r}\b)^{\f 12},\\\lab{cj2}
|u_{\th}(r,z)| &\leq& c(M)\f{(\log r)^{1/8}}{r^{3/8}},\\\lab{cj3}
|\om_{\th}(r,z)|&\leq& \f{c(M)}{r^{\f78}}
\ee
for large $r$.
Based on these results, we can obtain better improved decay rate estimates for $u_{\th}, \bm{\om}$ and $\na {\bf u}$. We first obtain some weighted and decay estimates of $\na u_r$ and $\na u_z$ by $A_p$ weight method, since $\na u_r$ and $\na u_z$ can be expressed as singular integral operators of $\om_{\th}$. These yield weighted energy and decay estimates of $\om_r$ and $\om_z$ by using the equations of $\om_r$ and $\om_z$. Finally, we use the integral formula of $u_{\th}$ in terms of $\om_r$ and $\om_z$ to improve the decay rates. Our first main result is stated as follows.
\bt\lab{main1}{\it
Suppose ${\bf f}\in H^1(\mbR^3)$ be an axially symmetric vector field with
\be\lab{force}
\|{\bf f}\|_{H^{-1}(\mbR^3)}+\|{\bf f}\|_{H^{1}(\mbR^3)}+ \b\|\b(\f{r}{\log r}\b)^{1/2} f_{\th}\b\|_{L^2(\mbR^3)}+\|r^2 \text{curl }{\bf f}\|_{L^2(\mbR^3)}\leq M
\ee
for some constant $M$. Then the axially symmetric solution $({\bf u}, p)$ to (\ref{sns}) with finite Dirichlet integral satisfied
\be\lab{main11}
|u_{\th}(r,z)| &\leq& c(M)\b(\f{\log r}{r}\b)^{\f 12}.
\ee
for large $r$. Moreover, we have the following estimates for $\na {\bf u}$:
\be\lab{main12}
|\om_{\th}(r,z)| &\leq& c(M) r^{-(\f{19}{16})^-},\\\lab{main13}
|\na u_r(r,z)|+|\na u_z(r,z)|&\leq & c(M) r^{-(\f{9}{8})^-},\\\lab{main14}
|\om_r(r,z)|+|\om_z(r,z)|&\leq& c(M) r^{-(\f{67}{64})^-},\\\lab{main15}
|\na u_{\th}(r,z)|&\leq& c(M) r^{-(\f{67}{64})^-},
\ee
where we denote $a^-$ to be any constant less than $a$.
\et
Another main result in this paper is the first decay rates estimate in the $Oz$-direction for the D-solution of (\ref{sns}) without swirl. We realize that it is possible to derive the weighted energy estimates of $\Om\co \f{\om_{\th}}{r}$ in the $Oz$ direction. Together with previous weighted energy estimates on $\om_{\th}$, we can derive the decay rate of $\om_{\th}$ with respect to $\rho=\sqrt{r^2+z^2}$, which also yields the decay rate of ${\bf u}$, since ${\bf u}$ has an integral representation formula in terms of $\om_{\th}$.
\bt\lab{main2}{\it
Suppose ${\bf f}\in H^1(\Om)$ be an axially symmetric vector field without swirl, and satisfying (\ref{force}) and
\be\lab{force2}
\b\|\rho\f{(\p_z f_r-\p_r f_z)}{r}\b\|_{L^2(\mbR^3)}\leq M.
\ee
%%\|{\Bf F}\|_{H^{-1}(\Om)}+ \B\|\B(\F{R}{\Log R}\B)^{1/2} F_{\Th}\B\|_{L^2(\Om)}+\|R\P_Z F_R\|_{L^2(\Om)}+ \|R\P_R F_Z\|_{L^2(\Om)} \Leq M
%For Some Constant $M$.
Then the axially symmetric $D$-solution $({\bf u}, p)$ to (\ref{sns}) without swirl satisfied
\be\lab{main21}
|{\bf u}(r,z)| &\leq& \f{c(M)}{(1+\rho)^{(\f{1}{8})^-}},\\\lab{main22}
|\bm{\om}(r,z)|&\leq& \f{c(M)}{(1+\rho)^{(\f{3}{8})^-}}
\ee
where $\rho=\sqrt{r^2+z^2}$.
\et
%There is a close relation between the decay properties of D-solution to (\ref{sns}) with the famous open problem of the triviality of D-solution to (\ref{sns}) with ${\bf f}\equiv 0$ and ${\bf u}_{\oo}=0$. One can refer to \cite{cw15,galdi11,kpr15} for more information.
After this introduction section, some preliminary tools including a decay lemma and $A_p$ weight for singular integral operators will be introduced. Then we prove Theorem \ref{main1} in Section \ref{mainsection1}. We first improve the weighted energy estimates and the decay rates of $\om_{\th}$ by a bootstrap argument, these yields some good weighted estimates and decay rates of $\na u_r$ and $\na u_z$ by employing the $A_p$ weight method and the decay lemma. Based on these and the equations of $\om_r$ and $\om_z$, one can obtain some decay rates of $\om_r$ and $\om_z$, which yields better decay rates of $u_{\th}$. In Section \ref{mainsection2}, we first realizes it is possible to obtain some weighted energy estimates of $\Om$ in the $Oz$ direction, which enables us to get some decay properties of ${\bf u}$ in the $Oz$ direction.
\section{Preliminary}
\subsection{A decay lemma}
The following decay lemma is proved by the techniques developed in \cite{gw74}, \cite{gw78} and \cite{cj09}.
\bl\lab{rdecay}
{\it Suppose an smooth axially symmetric function $f(x)$ satisfies the following weighted energy estimates
\be\lab{rdecay1}
&&\int_{\mbR^3} r^{e_1} |f(r,z)|^2 dx \leq C,\\\lab{rdecay2}
&&\int_{\mbR^3} r^{e_2} |\na f(r,z)|^2 dx \leq C,\\\lab{rdecay3}
&&\int_{\mbR^3} r^{e_3} |\nabla\p_z f(r,z)|^2dx \leq C
\ee
with nonnegative constants $e_1, e_2,e_3$. Then for any $r>0$, we have
\be\lab{rdecay4}
&&\int_{-\oo}^{\oo} |f(r,z)|^2 dz \leq C r^{-\f 12(e_1+e_2)-1},\\\lab{rdecay5}
&&\int_{-\oo}^{\oo} |\p_z f(r,z)|^2 dz \leq C r^{-\f 12(e_2+e_3)-1},\\\lab{rdecay6}
&&|f(r,z)|^2\leq C r^{-\f14(e_1+2e_2+e_3)-1},\q \forall z\in \mbR^3.
\ee
%|f(r,z)|^2 \leq C r^{-\f 14(e_1+2e_2+e_3)-1},\q \forall z\in\mbR^3.
\el
\bpf
For any integer $n\geq 0$, from (\ref{rdecay1}), we have
\be\no
\int_{2^n}^{2^{n+1}} \int_{-\oo}^{\oo} r^{e_1} |f(r,z)|^2 rdr dz \leq C.
\ee
By the intermediate value theorem, there exists $r_n\in [2^n, 2^{n+1}]$ such that
\be\no
\int_{-\oo}^{\oo} |f(r_n,z)|^2 dz \leq C r_n^{-e_1-2}.
\ee
For $\forall r>0$, choose $r_n>r$ and then
\be\no
\int_{-\oo}^{\oo} |f(r,z)|^2dz&=& \int_{-\oo}^{\oo} |f(r_n, z)|^2dz- 2\int_{r}^{r_n} \int_{-\oo}^{\oo} f(s,z)\p_s f(s,z) ds dz \co I_1 +I_2,\\\no
|I_2|&\leq& C\b(\int_{r}^{r_n} \int_{-\oo}^{\oo} \f{|f(s,z)|^2}{s^2} sds dz\b)^{\f12} \b(\int_{r}^{r_n} \int_{-\oo}^{\oo} |\p_s f(s,z)|^2 s ds dz\b)^{\f 12} \\\no
&\leq& \f{C}{r^{\f12(e_1+e_2+2)}}\b(\int_{r}^{\oo} \int_{-\oo}^{\oo} s^{e_1} |f(s,z)|^2 sds dz\b)^{\f12} \b(\int_{r}^{\oo} \int_{-\oo}^{\oo}s^{e_2}|\p_s f(s,z)|^2 s ds dz\b)^{\f 12}\\\no &\leq& O(r^{-\f 12(e_1+e_2)-1}).
\ee
Sending $n\to \oo$, $I_1\to 0$. Hence we arrive at (\ref{rdecay4}). Similarly, we also have (\ref{rdecay5}).
To prove (\ref{rdecay6}), for $|z|\leq z_1$ for some $z_1\geq 1$, we use
\be\no
|f(r,z)|^2&=&\f{1}{2z_1} \int_{-z_1}^{z_1} |f(r,t)|^2 dt+ \b(|f(r,z)|^2-\f{1}{2z_1} \int_{-z_1}^{z_1} |f(r,t)|^2 dt\b)= J_1 +J_2.
\ee
By the mean value theorem and (\ref{rdecay4})-(\ref{rdecay5}), it follows
\be\no
|J_2|&=& (|f(r,z)|^2- |f(r,z_*)|^2) \leq \int_{-z_1}^{z_1} \b|\f{\p}{\p t}|f(r,t)|^2\b| dt \\\no
&\leq& c\b(\int_{-\oo}^{\oo}|f(r,t)|^2dt\b)^{\f 12}\b(\int_{-\oo}^{\oo}|\p_t f(r,t)|^2 dt\b)^{\f 12}\\\no
&\leq& C r^{-\f14(e_1+2e_2+e_3)-1},\\\no
J_1&\leq& \f{C}{z_1 r^{\f 12(e_1+e_2)+2}}.
\ee
Then letting $z_1$ goes to $\oo$, we obtain (\ref{rdecay6}).
\epf
%{\it If an axially symmetric function $f(x)$ satisfies the following weighted energy estimates
%&&\int_{\mbR^3} (z^2+1)^{\f{e_1}{2}} f^2(x) dx \leq C(M),\\\lab{zdecay2}
%&&\int_{\mbR^3} (z^2+1)^{\f{e_2}{2}} |\na f(x)|^2 dx \leq C(M),\\\lab{zdecay3}
%&&\int_{\mbR^3} (z^2+1)^{\f{e_3}{2}}|\nabla^2 f(x)|^2dx \leq C(M)
%with nonnegative constants $e_1, e_2,e_3$. Then we have
%&&\int_{0}^{\oo} |f(r,z)|^2 rdr = O(|z|^{-\f 12(e_1+e_2)}),\\\lab{zdecay5}
%&&\int_{0}^{\oo} |\nabla f(r,z)|^2 rdr = O(|z|^{-\f 12(e_2+e_3)}),\\\lab{zdecay6}
%&&|f(r,z)|^2= C |z|^{-\f14(e_1+2e_2+e_3)} r^{-1}.
%For any integer $n\geq 0$, from (\ref{rdecay1}), we have
%\int_{2^n}^{2^{n+1}} \int_{0}^{\oo} |z|^{e_1} f^2(r,z) rdr dz \leq C(M).
%By the intermediate value theorem, there exists $z_n\in [2^n, 2^{n+1}]$ such that
%\int_{0}^{\oo} |f^2(r,z_n)| rdr \leq C(M) |z_n|^{-e_1-1}.
%For $\forall z>0$, choose $z_n>z$ and
%\int_{0}^{\oo} f^2(r,z) rdr= \int_{0}^{\oo} f^2(r, z_n) rdr- 2\int_{z}^{z_n} \int_{0}^{\oo} f(r,t)\p_t f(r,t) sds dz \co I_1 +I_2.
%|I_2|&\leq& C\b(\int_{z}^{z_n} \int_{0}^{\oo} f^2(r,t) rdr dt\b)^{\f12} \b(\int_{z}^{z_n} \int_{0}^{\oo} |\p_t f(r,t)|^2 r dr dt\b)^{\f 12} \\\no
%&\leq& \f{C}{|z|^{\f12(e_1+e_2)}}\b(\int_{z}^{z_n} \int_{0}^{\oo} |t|^{e_1}f^2(r,t) rdr dt\b)^{\f12} \b(\int_{z}^{z_n} \int_{0}^{\oo}|t|^{e_2}|\p_t f(r,t)|^2 r dr dt\b)^{\f 12}\\\no
%&\leq& O(|z|^{-\f 12(e_1+e_2)}).
%Sending $n\to \oo$, $I_1\to 0$. Hence we arrive at (\ref{zdecay4}). Similarly, we have (\ref{zdecay5}).
%To prove (\ref{zdecay6}), Choose some $r_1> r$, we have
%|f(r,z)|^2&=&\f{1}{r_1-r} \int_{r}^{r_1} |f(s,z)|^2 ds+ \b(|f(r,z)|^2-\f{1}{r_1-r} \int_{r}^{r_1} |f(s,z)|^2 ds\b)= J_1 +J_2.
%By the mean value theorem, it follows
%|J_2|&=& (|f(r,z)|^2- |f(r_*,z)|^2) \leq \int_{r}^{r_1} \b|\f{\p}{\p s}|f(s,z)|^2\b| ds \\\no
%&\leq& cr^{-1}\b(\int_{r}^{\oo}|f(s,z)|^2 sds\b)^{\f 12}\b(\int_{r}^{\oo}|\nabla f(s,z)|^2 sds\b)^{\f 12}\\\no
%&\leq& C r^{-1} |z|^{-\f14(e_1+2e_2+e_3)},\\\no
%J_1&\leq& \f{C}{(r_1-r)r |z|^{\f 12(e_1+e_2)}}.
%Then let $r_1$ goes to $\oo$, we obtain (\ref{zdecay6}).
\subsection{$A_p$ weight and singular integral operator}
We first give the classical definition of $A_p$ weight.
\begin{definition}\label{sio1}
Let $p\in (1,\infty)$. A real valued function $w(x)$ is said to be in $A_p$ class if it satisfies
\begin{equation}\label{sio2}
\sup_{B\subset R^3}\bigg(\frac{1}{|B|}\int_Bw(x)dx\bigg)\bigg(\frac{1}{|B|}\int_B w(x)^{-\frac{p'}{p}} dx\bigg)^{\frac{p}{p'}}<\infty,
\end{equation}
where the supremum is taken over all balls $B$ in $R^3$. Here $p'$ is the H$\ddot{o}$lder conjugate of $p$, i.e. $\frac{1}{p}+\frac{1}{p'}=1$
\end{definition}
For function $w(x)\in A_p$, we can extend the Calderon-Zygmund inequality for the singular integral operator with the integral having weight function $w(x)$.
\bt\label{sio3} (\cite{stein93} page 205.)
{\it Let $p\in (1,\infty)$. Suppose $T$ is a singular integral operator of the convolution type, and $w(x)\in A^p$. Then for $f\in L^p(R^3)$,
\begin{equation}\label{sio4}
\int_{R^3}|T f(x)|^p w(x)dx\leq C \int_{R^3}|f(x)|^p w(x)dx.
\end{equation}
\et
\begin{lemma}\label{sio5}
For any $p\in (1,\infty)$, the function $w(x)=r^{\al}$, where $r=\sqrt{x_1^2+x_2^2}$ and $\al\in (-2, 2(p-1))$, is in $A_p$ class.
\end{lemma}
\bpf
Similar results has been proved in \cite{cl02}. Let $x_0\in \mbR^3$ be given. Set $B=B_s(x_0)$ and $d=\sqrt{x_{01}^2 + x_{02}^2}$. If $d\geq 2s$, then $d-s\leq \sqrt{x_1^2+x_2^2}\leq d+s$ for all $x\in B$. Thus if $\al\geq 0$
\begin{eqnarray}\nonumber
&\quad&\bigg(\frac{1}{|B|}\int_B w(x)dx\bigg)\bigg(\frac{1}{|B|}\int_B w(x)^{-\frac{p'}{p}} dx\bigg)^{\frac{p}{p'}}\\\nonumber
&\leq&\bigg(\frac{3}{4\pi s^3}\int_B (d+s)^{\al}dx\bigg)\bigg(\frac{3}{4\pi s^3}\int_B (d-s)^{-\frac{\al}{p-1}}dx\bigg)^{(p-1)}\\\nonumber
&=&C_2\frac{(d+s)^{\al}}{(d-s)^{\al}}\leq C_3.
\end{eqnarray}
For $\al<0$, one can also obtain a similar thing. If $d<2s$, then the cylinder $\{(x_1,x_2,x_3)\in \mbR^3|x_1^2+x_2^2<(d+s)^2, |x_3-x_3^0|<s\}$ contains the ball $B$. Thus if $\al\in (-2, 2(p-1))$,
\begin{eqnarray}\nonumber
&\quad \bigg(\frac{1}{|B|}\int_B w(x)dx\bigg)\bigg(\frac{1}{|B|}\int_B w(x)^{-\frac{p'}{p}} dx\bigg)^{\frac{p}{p'}}\\\nonumber
&\leq & \bigg(\frac{3}{4\pi s^3}2\pi \int_{x_3^0-s}^{x_3^0+s}\int_0^{d+s}\rho^{\al}\rho d\rho dx_3\bigg)\times \bigg(\frac{3}{4\pi s^3}2\pi \int_{x_3^0-s}^{x_3^0+s}\int_0^{d+s}\rho^{-\frac{\al}{p-1}+1} d\rho dx_3\bigg)^{(p-1)}\\\nonumber
&\leq & \b(\frac{3}{2 s^3}\f{2}{2+\al}s(d+s)^{2+\al}\b)\bigg(\frac{3}{2 s^3}\f{2}{2-\f{\al}{p-2}}s (d+s)^{2-\frac{\al}{p-1}}\bigg)^{(p-1)}\\\nonumber
&\leq & C(\al,p)<\oo.
\end{eqnarray}
Since the ball $B$ is arbitrary, the proof is completed.
\epf
\section{Proof of Theorem \ref{main1}.}\lab{mainsection1}
\subsection{Improved estimates for $\om_{\th}$.}
Since ${\bf f}\in H^{-1}(\mbR^3)$, the standard existence theory tells us that there exists a weak solution ${\bf u}$ to (\ref{sns}) with finite Dirichlet integral. If ${\bf f}\in H^1(\mbR^3)$, by the $L^q$ estimates of the Stokes system, then $\na^2 {\bf u}\in L^2(\mbR^3)$ and ${\bf u}\in L^{\oo}(\mbR^3)$.We first give the following basic decay estimates for $D$-solutions to (\ref{sns}).
\bl\lab{first}
{\it Let $({\bf u}, p)$ be an axially symmetric smooth D-solutions to (\ref{sns}) with the forcing term ${\bf f}$ satisfying (\ref{force}). Then there is a constant $c(M)$ such that
\be\lab{first1}
\int_{-\oo}^{\oo} |u_r(r,z)|^2 + |u_{\th}(r,z)|^2 dz &\leq& c(M),\\\lab{first2}
%\int_{-\oo}^{\oo} |u_z(r,z)|^4 dz &\leq& \f{c(M)}{r},\\\lab{first3}
\int_{-\oo}^{\oo} |\om_{\th}(r,z)|^2 dz &\leq& \f{c(M)}{r}.
\ee
\el
\bpf
The inequality (\ref{first1}) was proven in Lemma 3.2 of \cite{cj09}. %In their proof, they had to used that $(\f{u_r}{r},\f{u_{\th}}{r})\in L^2(\mbR^3)$. However, since $u_z(0,z)$ may not be zero, $\f{u_z(r,z)}{r}$ may not belong to $L^2(\mbR^3)$. For (\ref{first2}), we see
%&\q&\b|\int_{-\oo}^{\oo} |u_z(r,z)|^{4} dz \b|= \b|-4\int_{-\oo}^{\oo} \int_r^{\oo}|u_z(s,z)|^2 u_z(s,z)\p_s u_z(s,z) ds dz\b|\\\no
%&\leq &\f{4}{r}\int_{-\oo}^{\oo} \int_r^{\oo} |u_z(s,z)|^3 |\p_s u_z(s,z)| s ds dz\\\no
%&\leq&\f{c}{r} \b(\int_{-\oo}^{\oo} \int_0^{\oo} |u_z(s,z)|^6 s ds dz\b)^{1/2}\b(\int_{-\oo}^{\oo} \int_0^{\oo} |\p_s u_z(s,z)|^2 s ds dz\b)^{1/2}\\\no
%&\leq& \f{c(M)}{r}.
The second estimate (\ref{first2}) follows from (\ref{rdecay4}) in Lemma \ref{rdecay}, since $\om_{\th}$ and $\na\om_{\th}$ belong to $L^2(\mbR^3)$.
\epf
\bl\lab{omega-theta}
{\it Let $({\bf u}, p)$ be an axially symmetric smooth D-solutions to (\ref{sns}) with the forcing term ${\bf f}$ satisfying (\ref{force}). Suppose that
\be\lab{omega-theta1}
|u_r(r,z)|+ |u_{\th}(r,z)|+ |u_z(r,z)|\leq C (1+r)^{-\de},\q \forall z\in \mbR
\ee
holds for some $\de\in [0,1]$. Then the following estimates holds
\be\lab{omega-theta2}
\int_{\mbR^3}|\om_{\th}|^2 dx &\leq& c(M), \\\lab{omega-theta3}
\int_{\mbR^3}r^{1+\de}|\na\om_{\th}|^2 dx &\leq& c(M),\\\lab{omega-theta4}
\int_{\mbR^3}r^{1+3\de}|\p_z\na\om_{\th}|^2 dx &\leq& c(M).
\ee
In particular, by Lemma \ref{rdecay}, we obtain the following decay rate for $\om_{\th}$:
\be\lab{omega-theta8}
|\om_{\th}(r,z)| \leq C(M) (1+r)^{-\f 78-\f 58\de}.
\ee
\el
\bpf
Since $\na {\bf u}\in L^2(\mbR^3)$, (\ref{omega-theta2}) holds. Take a cut-off function $\eta\in C_0^{\oo}(\mbR^+), \eta=\eta(r)$, satisfying $0\leq \eta\leq 1$, $\eta=0$ on $r\leq r_0$ or $r\geq 2 r_1$, $\eta=1$ on $2 r_0<r<r_1$, so that $|\na\eta|\leq \f{c}{r_0}$ on $r_0\leq r \leq 2 r_0$, $|\na \eta|\leq \f{C}{r_1}$ on $r_1<r<2 r_1$, and $\na \eta=0$ elsewhere. Here $r_0$ is a fixed positive constant. We will let $r_1$ tends to $\oo$ at the end.
%|\na\eta| \begin{cases}
%\leq \f{c}{\rho_0}\q &\text{on}\q \rho_0<\rho< 2\rho_0,\\
Multiplying (\ref{vorticity2}) by $\eta^2 r^{a_1} \om_{\th}$ and integrating over $\mbR^3$, after some computations we obtain
%LHS%&=&\int_{\mbR^3} r^{a_1} \om_{\th} \eta^2 \b[(u_r\p_r+ u_z\p_z)\om_{\th}-\f{u_r}{r} \om_{\th} -\f{1}{r}\p_z({u_{\th}^2})\b] dx \\\no
%&=&-\pi\int_{-\oo}^{\oo}\int_0^{\oo} r^{a_1} 2\eta \eta' \f{r u_r+ z u_z}{\sqrt{r^2+z^2}} \om_{\th}^2 r dr dz - \pi\int_{-\oo}^{\oo}\int_0^{\oo} a_1 r^{a_1-1} \eta^2 \om_{\th}^2 u_r r drdz\\\no
%&\quad&-\int_{\mbR^3} r^{a_1-1} \eta^2 u_r \om_{\th}^2 dx- 2\int_{\mbR^3} r^{a_1-2}\eta^2 u_{\th} \p_z u_{\th}\om_{\th} dx\co \sum_{k=1}^4 I_k,\\\no
%RHS %&=& \int_{\mbR^3} r^{a_1} \eta^2\om_{\th}(\p_r^2+\f{1}{r}\p_r+\p_z^2-\f 1{r^2})\om_{\th} dx +\int_{\mbR^3} r^{a_1}\eta^2 \om_{\th}(\p_z f_r-\p_r f_{z}) dx\\\no
%&=&- 2\pi \int_{-\oo}^{\oo}\int_0^{\oo} r^{a_1} |\na \om_{\th}|^2 \eta^2 rdr dz- 2\pi\int_{-\oo}^{\oo}\int_0^{\oo} 2r^{a_1}\om_{\th} \eta\eta' \f{r\p_r\om_{\th}+ z\p_z\om_{\th}}{\sqrt{r^2+z^2}} rdr dz\\\no
%&\quad&-a_1\int_{\mbR^3} r^{a_1-1} \eta^2\om_{\th} \p_r \om_{\th} r dr dz-\int_{\mbR^3} r^{a_1-2}\eta^2 \om_{\th}^2 dx+\int_{\mbR^3} r^{a_1}\eta^2 \om_{\th}(\p_z f_r-\p_r f_{z}) dx\\\no
%&\co& \sum_{k=1}^5 J_k.
\be\no
0&=& \int_{\mbR^3}\eta^2 r^{a_1}|\na \om_{\th}|^2 dx+ \int_{\mbR^3}\eta^2 r^{a_2} \b|\f{\om_{\th}}{r}\b|^2 dx -\f12\int_{\mbR^3} \p_r^2(\eta^2 r^{a_1})|\om_{\th}|^2 dx\\\no
&\q&- \f12 \int_{\mbR^3} r^{-1}\p_r(\eta^2 r^{a_1}) |\om_{\th}|^2 dx- \int_{\mbR^3} \eta^2 r^{a_2}\om_{\th}(\p_z f_r -\p_r f_z) dx -\f12\int_{\mbR^3} \p_r(\eta^2 r^{a_1}) u_r |\om_{\th}|^2 dx\\\no
&\q&- \int_{\mbR^3} \eta^2 r^{a_1-1} u_r \om_{\th}^2 dx - 2\int_{\mbR^3} \eta^2 r^{a_1-1}\om_{\th} u_{\th} \p_z u_{\th} dx \\\no
&\co& \int_{\mbR^3}\eta^2 r^{a_1}|\na \om_{\th}|^2 dx+ \int_{\mbR^3}\eta^2 r^{a_2} \b|\f{\om_{\th}}{r}\b|^2 dx + \sum_{k=1}^3 J_k + \sum_{k=1}^3 I_k.
\ee
Then for $a_1=1+\de$, we have the following estimates
\be\no
\sum_{j=1}^2|I_j|&\leq&C\int_{\mbR^3} |u_r| r^{a_1-1} \om_{\th}^2 dx \leq C\|r^{a_1-1}u_r\|_{L^{\oo}} \|\om_{\th}\|_{L^2}^2\leq C\|\om_{\th}\|_{L^2}^2,\\\no
|I_3|&\leq&C\|r^{a_1-1}u_{\th}\|_{L^{\oo}} \|\p_z u_{\th}\|_{L^2} \|\om_{\th}\|_{L^2}\leq C\|\p_z u_{\th}\|_{L^2} \|\om_{\th}\|_{L^2},\\\no
\sum_{k=1}^2 |J_k|&\leq& \int_{r\geq r_0} r^{a_1-2} |\om_{\th}|^2 dx,\q \textit{since }\q 0\leq a_1\leq 2, \\\no
%&\leq& \e \|\eta r^{\f{a_1}{2}}\na \om_{\th}\|_{L^2}^2 + C(\e)\|\f{\om_{\th}}{r}\|_{L^2}^{2-a_1}\|\om_{\th}\|_{L^2}^{a_1},\\\no
%|J_4|&\leq&\int_{\mbR^3}\b|\f{\om_{\th}}{r}\b|^{2-a_1} |\om_{\th}|^{a_1} dx \leq \|\f{\om_{\th}}{r}\|_{L^2}^{2-a_1}\|\om_{\th}\|_{L^2}^{a_1},\q \textit{if}\q 0\leq a_1\leq 2,\\\no
|J_3|&\leq&C\|\om_{\th}\|_{L^2(\mbR^3)}\|r^2(\p_z f_r-\p_r f_z)\|_{L^2},\q\textit{since }\q 0\leq a_1\leq 2.
\ee
Assuming (\ref{omega-theta1}), letting $r_1\to \oo$, then one obtains (\ref{omega-theta3}).
Furthermore, we can obtain (\ref{omega-theta4}) by using the equation for $\p_z\om_{\th}$:
\be\lab{omega-theta-partialz}
\p_z\b((u_r\p_r+ u_z\p_z)\om_{\th}-\f{u_r}{r} \om_{\th} -\f{1}{r}\p_z({u_{\th}^2})\b)=\b(\p_r^2+\f{1}{r}\p_r+\p_z^2-\f 1{r^2}\b)\p_z\om_{\th}+\p_z(\p_z f_r-\p_r f_z).
\ee
Multiplying (\ref{omega-theta-partialz}) by $\eta^2 r^{b_1} \p_z\om_{\th}$ and integrating over $\mbR^3$, we get an integral identity with left and right hand sides as
%LHS%&=&-\int_{\mbR^3}\b[(u_r\p_r+ u_z\p_z)\om_{\th}-\f{u_r}{r} \om_{\th} -\f{1}{r}\p_z({u_{\th}^2})\b)\b]\p_z (\eta^2 r^b\p_z \om_{\th}) dx \\\no
%&=&-\int_{\mbR^3}(u_r\p_r+ u_z\p_z)\om_{\th} \eta^2 r^{b_1} \p_z^2\om_{\th} dx-2\int_{\mbR^3} (u_r\p_r+ u_z\p_z)\om_{\th} r^{b_1}\eta \eta' \f{z\p_z\om_{\th}}{\sqrt{r^2+z^2}} dx\\\no
%&\quad&+ \int_{\mbR^3} r^{b_1-1} u_r \eta^2\om_{\th} \p_z^2\om_{\th} dx + 2\int_{\mbR^3} r^{b_1-1} u_r \om_{\th} \eta \eta'\f{z}{\sqrt{r^2+z^2}} \p_z \om_{\th} dx\\\no
%&\quad&+ \int_{\mbR^3} 2r^{b_1-2} u_{\th}\p_z u_{\th} \eta^2\p_z^2\om_{\th} dx+ 4 \int_{\mbR^3} r^{b_1-1} u_{\th}\p_z u_{\th}\eta\eta' \f{z\p_z\om_{\th}}{\sqrt{r^2+z^2}} dx \co \sum_{k=1}^6 I_k',\\\no
%RHS%&=& \int_{\mbR^3} r^{b_1} \eta^2 \p_z\om_{\th} \b(\p_r^2+\p_z^2+\f{1}{r}\p_r+\p_z^2-\f1{r^2}\b)\om_{\th} dx \\\no
%&=&-\int_{\mbR^3} r^{b_1} \eta^2|\nabla \p_z\om_{\th}|^2 dx - \int_{\mbR^3} 2 r^{b_1} \eta\eta' \p_z\om_{\th} \f{r\p_{rz}^2\om_{\th}+ z\p_z^2\om_{\th}}{\sqrt{r^2+z^2}} dx -b_1\int_{\mbR^3} r^{b_1-1} \eta^2 \p_z \om_{\th}\p_{rz}^2\om_{\th} dx\\\no
%&\quad& -\int_{\mbR^3} r^{b_1-2}\eta^2 (\p_z\om_{\th})^2 dx-\int_{\mbR^3} r^{b_1}\p_z(\eta^2 \p_z\om_{\th})(\p_z f_r-\p_r f_{z}) dx \co \sum_{k=1}^5 J_k'.
\be\no
0&=& \int_{\mbR^3}\eta^2 r^{b_1}|\na \p_z\om_{\th}|^2 dx+ \int\eta^2 r^{b_2} \b|\f{\p_z\om_{\th}}{r}\b|^2 dx -\f12\int_{\mbR^3} \p_r^2(\eta^2 r^{b_1})|\p_z\om_{\th}|^2 dx\\\no
&\q&- \f12 \int_{\mbR^3} r^{-1}\p_r(\eta^2 r^{b_1}) |\p_z\om_{\th}|^2 dx+ \int_{\mbR^3} \eta^2 r^{b_1}\p_z\om_{\th}(\p_z f_r -\p_r f_z) dx-\int_{\mbR^3} \eta^2 r^{b_1} \p_z^2\om_{\th}[(u_r\p_r+u_z\p_z)\om_{\th}] dx \\\no
&\q&+ \int_{\mbR^3} \eta^2 r^{b_1}\p_z^2\om_{\th}\cdot\f{u_r}{r}\om_{\th} dx+ 2\int_{\mbR^3} \eta^2 r^{b_1}\p_z^2\om_{\th} \f{u_{\th}}{r}\p_z u_{\th} dx\\\no
&\co& \int_{\mbR^3}\eta^2 r^{b_1}|\na \p_z\om_{\th}|^2 dx+ \int\eta^2 r^{b_2} \b|\f{\p_z\om_{\th}}{r}\b|^2 dx+ \sum_{k=1}^3 J_k' + \sum_{k=1}^3 I_k'.
\ee
We estimate these terms as follows.
\be\no
\sum_{k=1}^2 |J_k'|&\leq&C\int_{r\geq r_0} r^{b_1-2} |\na\om_{\th}|^2 dx\leq C\int_{\mbR^3} r^{1+\de} |\na\om_{\th}|^2dx<\oo,\q \text{if }b_1-2\leq 1+\de,\\\no
|J_3'|&\leq&\int_{\mbR^3} r^{1+\de}|\na\om_{\th}|^2 dx + \int_{\mbR^3} r^{2b_1-(1+\de)}|\p_z f_r-\p_r f_z|^2 dx,\\\no
|I_1'|&\leq&\f18\int \eta^2 r^{b_1}|\na\p_z \om_{\th}|^2 dx + C \int_{\mbR^3} \eta^2 r^{b_1}|(u_r,u_z)|^2 |\na\om_{\th}|^2 dx\\\no
&\leq&\f18\int_{\mbR^3} \eta^2 r^{b_1}|\na\p_z \om_{\th}|^2 dx + C \int_{\mbR^3} r^{1+\de}|\na\om_{\th}|^2 dx,\q \text{if }b_1\leq 1+3\de,\\\no
|I_2'|&\leq&\f18\int \eta^2 r^{b_1}|\na\p_z \om_{\th}|^2 dx + C \int_{\mbR^3} \eta^2 r^{b_1-2}|u_r|^2|\om_{\th}|^2 dx\\\no
&\leq&\f18\int_{\mbR^3} \eta^2 r^{b_1}|\na\p_z \om_{\th}|^2 dx + C \int_{\mbR^3} \eta^2 |\om_{\th}|^2 dx,\text{if }b_1-2\leq 2\de,\\\no
|I_3'|&\leq&\f18\int \eta^2 r^{b_1}|\na\p_z \om_{\th}|^2 dx + C\int_{\mbR^3} \eta^2 r^{b_1-2}|u_{\th}|^2 |\na u_{\th}|^2dx \\\no
&\leq& \f18\int_{\mbR^3} \eta^2 r^{b_1}|\na\p_z \om_{\th}|^2 dx+ C\int_{\mbR^3} |\na u_{\th}|^2 dx,\q \text{if }b_1-2\leq 2\de.
\ee
Then it is easy to see the essential restriction on $b_2$ is $b_2\leq 1+3\de$. Taking $b_2=1+3\de$ and letting $\rho_1\to \oo$, then we obtain (\ref{omega-theta4}).
%\sum_{k=2}^3|J_k'|%&\leq& \int_{\mbR^3}|r\eta'| |r^{b_1/2-1}\p_z\om_{\th}| (\eta r^{b_1/2}|\na\p_z\om_{\th}|) dx \\\no
%&\leq&\|\eta r^{b_1/2}|\na\p_z\om_{\th}|\|_{L^2}\|r^{b_1/2-1}\p_z\om_{\th}\|_{L^2}\leq\e \|\eta r^{b_1/2}|\na\p_z\om_{\th}|\|_{L^2}^2+ C\|r^{a_1/2}\p_z\om_{\th}\|_{L^2}^2,\textit{if } b_1/2-1\leq a_1/2,\\\no
%%|J_3|&\leq&\int_{\mbR^3} \eta r^{b_1/2-1} \p_z\om_{\th} (\eta r^{b_1/2}|\p_{rz}^2 \om_{\th}|) dx \\\no
%%&\leq&\e \|\eta r^{b_1/2}\na\p_z \om_{\th}\|_{L^2}^2+ C(\e)\|r^{b_1/2-1}|\na_z\om_{\th}|\|_{L^2}^2,\\\no
%|J_4'|&\leq& \|r^{b_1/2-1}|\na\om_{\th}|\|_{L^2}^2\leq C\|r^{a_1/2}|\na\om_{\th}|\|_{L^2}^2,\textit{if } b_1/2-1\leq a_1/2,\\\no
%|J_5'|&\leq& C\|\eta r^{b_1/2}\p_z^2\om_{\th}\|_{L^2}\|r^{b_1/2}(\p_z f_r-\p_r f_z)\|_{L^2}+\|r^{a_1/2}\p_z\om_{\th}\|_{L^2}\|r^{b_1-a_1/2-1}(\p_z f_r-\p_r f_z)\|_{L^2}\\\no
%&\leq&\e \|\eta r^{b_1/2}\p_z^2\om_{\th}\|_{L^2}^2+C(\e)\|r^{2}(\p_z f_r-\p_r f_z)\|_{L^2}^2+\|r^{a_1/2}\p_z\om_{\th}\|_{L^2}\|r^{2}(\p_z f_r-\p_r f_z)\|_{L^2},\textit{if }b\leq 4,\\\no
%|I_1'| &\leq& \int_{\mbR^3}\eta r^{b_1/2-a_1/2}(|u_r|+|u_z|) r^{a_1/2}|\na\om_{\th}|\cdot(\eta r^{b_1/2}|\p_z^2\om_{\th}|) dx\\\no
% &\leq& \e\|r^{b_1/2}\eta \na\p_z\om_{\th}\|_{L^2}^2+C(\e)\|r^{a_1/2}\na\om_{\th}\|_{L^2}^2 ,\textit{if } b_1/2-a_1/2\leq \de,\\\no
%|I_2'| &\leq& 2\int_{\mbR^3} |r\eta'| \eta r^{b_1-a_1-1}(|u_r|+|u_z|) r^{a_1} |\na\om_{\th}|^2 dx
%%&\leq&C\|r^{b_1-a_1-1}(|u_r|+|u_z|)\|_{L^{\oo}} \|r^{a_1/2}|\na\om_{\th}|\|_{L^2}^2, \q b_1\leq a_1+1+\de=2+2\de,\\\no
%\leq C\|r^{a_1/2}|\na\om_{\th}|\|_{L^2}^2,\textit{if } b_1-a_1-1\leq\de,\\\no
%|I_3'|&\leq& \int_{\mbR^3} r^{b_1/2-1} |u_r| |\om_{\th}| (\eta r^{b_1/2}|\p_z^2 \om_{\th}|) dx
%%&\leq& \|r^{b_1/2-1} u_r\|_{L^{\oo}} \|\om_{\th}\|_{L^2} \|\eta r |\na\p_z \om_{\th}|\|_{L^2},\\\no
%\leq \e\|\eta r^{b_1/2} |\na\p_z \om_{\th}|\|_{L^2}^2+C\|\om_{\th}\|_{L^2}^2,\textit{if }b_1/2-1\leq \de,\\\no
%|I_4'|&\leq& \int_{\mbR^3} r^{b_1-a_1/2-2} |u_r| |\om_{\th}|(\eta r^{a_1/2} |\p_z\om_{\th}|) dx
%%&\leq& \|r^{b_1-a_1/2-2} |u_r|\|_{L^{\oo}} \|\om_{\th}\|_{L^2} \|r^{a_1/2}\eta |\na \om_{\th}|\|_{L^2},\\\no
%\leq C\|\om_{\th}\|_{L^2} \|r^{a_1/2}\eta |\na \om_{\th}|\|_{L^2},\textit{if }b_1-a_1/2-2\leq\de,\\\no
%|I_5'| &\leq& 2\int_{\mbR^3} \eta r^{b_1/2-1} |u_{\th}| |\p_z u_{\th}||\eta r^{b_1/2}\p_z^2\om_{\th}| dx
%%&\leq& \e\|r^{b_1/2}\eta \na\p_z\om_{\th}\|_{L^2}^2+ C(\e)\|r^{b_1/2-1}u_{\th}\|_{L^{\oo}}^2\|\na u_{\th}\|_{L^2}^2, \q b_1/2-1\leq \de,\\\no
%\leq \e\|r^{b_1/2}\eta \na\p_z\om_{\th}\|_{L^2}^2+ C(\e)\|\na u_{\th}\|_{L^2}^2,\textit{if } b_1/2-1\leq \de\\\no
%|I_6'| &\leq& 2\int_{\mbR^3}|r\eta'| r^{b_1-a_1/2-2} |u_{\th}| |\p_z u_{\th}|\eta |r^{a_1/2}\p_z\om_{\th}| dx
%%&\leq& C\|r^{b_1-a_1/2-2} u_{\th}\|_{L^{\oo}}\|\p_z u_{\th}\|_{L^2} \|r^{a_1/2}\p_z\om_{\th}\|_{L^2},\q b_1-2-a_1/2\leq\de.
%\leq C\|\p_z u_{\th}\|_{L^2} \|r^{a_1/2}\p_z\om_{\th}\|_{L^2},\textit{if }b_1-a_1/2-2\leq\de.
%\int_{\mbR^3}r^{1+3\de}|\nabla\p_z\om_{\th}|^2 dx \leq C(M).
%By Lemma \ref{rdecay}, one can conclude that
%|\om_{\th}(r,z)|\leq C r^{-\f 78-\f 58\de}.
\epf
\br\lab{omege-theta-remark}
{\it By (\ref{cj1})-(\ref{cj2}), we see that (\ref{omega-theta1}) holds for any $0\leq \de<\f{3}{8}$. Hence (\ref{omega-theta8}) implies that
\be\lab{omega-theta5}
|\om_{\th}(r,z)| \leq C(M) (1+r)^{-(\f{71}{64})^-}.
\ee
\er
\br\lab{exterior1}
{\it The case $\de=0$ was obtained in \cite{cj09}. We can also extend these arguments to the exterior domain case by choosing large enough $r_0$ in the definition of the cut-off function $\eta$.
% as $\phi$ in \cite{cj09}. Suppose $0\in \Om^c$, then choose $\rho_1>\text{diam }\Om$, define $\eta=\eta(\rho)\in C_0^{\oo}(\mbR^3), \rho=\sqrt{r^2+z^2}$, satisfying $0\leq \eta\leq 1$, $\eta(\rho)=1$ on $2\rho_1 \leq \rho\leq \rho_2$, $\eta(\rho)=0$ for $\rho\geq 2\rho_2$ or $0\leq \rho\leq \rho_1$, so that $|\na\eta|\leq \f{c}{\rho}$ for $\rho_1\leq \rho\leq 2\rho_1$ and $\rho_2\leq \rho \leq 2\rho_2$.
\er
\bl\lab{grad-u-rz}
{\it Let $({\bf u}, p)$ be an axially symmetric smooth D-solutions to (\ref{sns}) with the forcing term ${\bf f}$ satisfying (\ref{force}). Suppose that
\be\lab{grad-u-rz1}
|u_r(r,z)|+ |u_{\th}(r,z)|+ |u_z(r,z)|\leq C (1+r)^{-\de},\q \forall z\in \mbR
\ee
holds for some $\de\in [0,1]$. Then the following estimates holds
\be\lab{grad-u-rz2}
\int_{\mbR^3}(|\na u_r|^2+|\na u_z|^2) dx &\leq& c(M), \\\lab{grad-u-rz3}
\int_{\mbR^3}r^{1+\de_1}(|\na^2 u_r|^2+ |\na^2 u_z|^2)dx &\leq& c(M),\\\lab{grad-u-rz4}
\int_{\mbR^3}r^{1+\de_2}(|\p_z\na^2u_r|^2+|\p_z \na^2 u_z|^2)dx &\leq& c(M),
\ee
where $\de_1=\de$ if $\de\in [0,1 )$ and $\de_1=1^-$ if $\de=1$, $\de_2=3\de$ if $\de<\f 13$ and $\de_2=1^-$ if $\de\in [\f 13, 1]$.
In particular, by Lemma \ref{rdecay}, we obtain the following decay rate for $\om_{\th}$:
\be\lab{grad-u-rz5}
|\na u_r (r,z)|+|\na u_z(r,z)| \leq C(M) (1+r)^{-\f 78-\f {2\de_1+\de_2}{8}}.
\ee
\bpf
Since $-\Delta (u_r {\bf e}_r + u_z {\bf e}_z)= \text{curl }(\om_{\theta} {\bf e}_{\th})$, then
\be\no
\nabla (u_r {\bf e}_r + u_z {\bf e}_z)= \nabla (-\Delta)^{-1} \text{curl }(\om_{\theta} {\bf e}_{\th}).
\ee
That is to say, $\na u_r$ and $\na u_z$ can be expressed as singular integral operators of $\om_{\th}$, and we apply Lemma \ref{sio5} to complete the proof.
%By Lemma \ref{sio5}, we get
%\int_{\mbR^3} |\nabla u_r|^2 dx <+\oo, \q \int_{\mbR^3} |\nabla u_z|^2 dx <+\oo,\\\lab{u-rz-first}
%\int_{\mbR^3} r^{1+\de}|\nabla^2 u_r|^2 dx <+\oo,\q \int_{\mbR^3}r^{1+\de} |\nabla^2 u_z|^2 dx <+\oo,\\\lab{u-rz-second}
%\int_{\mbR^3} r^{1+3\de} |\nabla^2\p_z u_r|^2 dx <+\oo,\q \int_{\mbR^3} r^{1+3\de} |\nabla^2\p_z u_r|^2 dx <+\oo.
\epf
\br\lab{grad-u-rz6}
{\it By (\ref{cj1})-(\ref{cj2}), we take $\de=(\f 38)^-$, then we have
\be\lab{grad-u-rz7}
|\na u_r (r,z)|+|\na u_z(r,z)| \leq C(M)(1+r)^{-(\f{35}{32})^-}.
\ee
\er
\br\lab{pressure}
{\it It seems difficult to derive weighted estimates of $(u_r, u_z)$ by using the equations of $u_r$ and $u_z$ directly, since we do not have good estimates on the pressure.
\er
\subsection{Estimates for $\om_r$ and $\om_z$}
\bl\lab{omega-rz}
{\it Let $({\bf u}, p)$ be an axially symmetric smooth D-solutions to (\ref{sns}) with the forcing term ${\bf f}$ satisfying (\ref{force}). Suppose that
\be\lab{omega-rz1}
|u_r(r,z)|+ |u_z(r,z)|&\leq& C (1+r)^{-\de},\\\lab{omega-rz1'}
|\na u_r(r,z)|+ |\na u_z(r,z)|&\leq& C (1+r)^{-1-\ga}
\ee
holds for some $\de,\ga\in [0,1]$. Then the following estimates holds
\be\lab{omega-rz2}
\int_{\mbR^3}(|\om_{r}|^2+|\om_z|^2) dx &\leq& c(M), \\\lab{omega-rz3}
\int_{\mbR^3}r^{1+\de\wedge\ga}(|\na\om_{r}|^2+|\na\om_z|^2) dx &\leq& c(M),\\\lab{omega-rz4}
\int_{\mbR^3}r^{1+\de\wedge\ga+2\de}(|\p_z\na\om_{r}|^2+|\p_z\na \om_z|^2) dx &\leq& c(M),
\ee
where $\de\wedge\ga= \min\{\de,\ga\}$. In particular, by Lemma \ref{rdecay}, we obtain the following decay rate for $\om_{\th}$:
\be\lab{omega-rz5}
|\om_{r}(r,z)|+|\om_z(r,z)| \leq C(M) (1+r)^{-\f 78-\f 18(3(\de\wedge\ga)+2\de)}.
\ee
\el
\br\lab{omega-rz6}
{\it By (\ref{cj1})-(\ref{cj2}) and (\ref{grad-u-rz7}), we take $\de=(\f{3}{8})^-$ and $\ga=(\f{3}{32})^-$ in Lemma \ref{omega-rz}, then
\be\lab{omega-rz7}
|\om_{r}(r,z)|+|\om_z(r,z)| \leq C(M) (1+r)^{-(\f{257}{256})^-}.
\ee
\bpf
We use the same cut-off function $\eta$ as in Lemma Multiplying (\ref{vorticity1}) and (\ref{vorticity3}) by $\eta^2 r^{a_2} \om_{r}$ and $\eta^2 r^{a_2}\om_z$ respectively, integrating over $\mbR^3$ and adding them together, we obtain
%LHS%&=& \f 12\int_{\mbR^3} \eta^2 r^{a_2} (u_r\p_r+u_z\p_z)(|\om_r|^2+|\om_z|^2)dx\\\no
%%&\q&- \int_{\mbR^3}\eta^2 r^{a_2}[\om_r(\om_r\p_r+\om_z\p_z)u_r+\om_z(\om_r\p_r+\om_z\p_z)u_z] dx\\\no
%&=&-\f{1}{2}\int_{\mbR^3} a_2 r^{a_2-1}\eta^2 u_r(|\om_r|^2+|\om_z|^2) dx- \int_{\mbR^3} r^{a_2}\eta \eta'(|\om_r|^2+|\om_z|^2)\f{r u_r+ z u_z}{\sqrt{r^2+z^2}} dx \\\no
%&\quad&-\int_{\mbR^3}\eta^2 r^{a_2}[\om_r(\om_r\p_r+\om_z\p_z)u_r+\om_z(\om_r\p_r+\om_z\p_z)u_z] dx \co A_1 +A_2 +A_3,\\\no
%RHS %&=& \int_{\mbR^3} \eta^2 r^{a_2}\om_r \b(\p_r^2+\f1r\p_r+\p_z^2-\f 1{r^2}\b)\om_r +\eta^2 r^{a_2} \om_z \b(\p_r^2+\f 1r\p_r+\p_z^2\b)\om_z dx \\\no
%%&\q&+ \int_{\mbR^3} \eta^2 r^{a_2}(-\om_r\p_z f_{\th}+ \f{\om_z}{r}\p_r(r f_{\th}) dx\\\no
%&=& -\int_{\mbR^3} \eta^2 r^{a_2} (|\na\om_r|^2+|\na\om_z|^2) dx- 2 \int_{\mbR^3} \eta \eta' r^{a_2} \om_r \f{r\p_r\om_r + z\p_z\om_r}{\sqrt{r^2+z^2}} dx \\\no
%&\quad&- 2\int_{\mbR^3} \eta \eta' r^{a_2} \om_z \f{r\p_r\om_z + z\p_z\om_z}{\sqrt{r^2+z^2}} dx -a_2 \int_{\mbR^3}\eta^2 r^{a_2-1} \om_r \p_r\om_r dx - a_2 \int_{\mbR^3} r^{a_2-1} \eta^2 \om_z \p_r\om_z dx \\\no
%&\quad&- \int_{\mbR^3} \eta^2 r^{a_2-2} \om_r^2 dx+ \int_{\mbR^3} \eta^2 r^{a_2}[-\om_r\p_z f_{\th}+ \f{\om_z}{r}\p_r(r f_{\th})] dx\co \sum_{k=1}^7 B_k.
\be\no
0&=& \int_{\mbR^3} \eta^2 r^{a_2}(|\na\om_r|^2+|\na\om_z|^2) dx + \int_{\mbR^3} \eta^2 r^{a_2}\b|\f{\om_r}{r}\b|^2 dx- \f12\int_{\mbR^3} \p_r^2(\eta^2 r^{a_2}) (|\om_r|^2+|\om_z|^2) dx\\\no
&\q&-\f12\int_{\mbR^3} r^{-1}\p_r(\eta^2 r^{a_2})(|\om_r|^2+|\om_z|^2) dx- \int_{\mbR^3} \eta^2 r^{a_2}\b[-\om_r\p_z f_{\th}+ \f{\om_z}{r}\p_r(r f_{\th})\b] dx\\\no
&\q&-\f12\int_{\mbR^3} \p_r(\eta^2 r^{a_2}) u_r (|\om_r|^2+|\om_z|^2) dx -\int_{\mbR^3} \eta^2 r^{a_2} [\om_r(\om_r\p_r + \om_z\p_z) u_r +\om_z (\om_r\p_r + \om_z\p_z) u_z] dx\\\no
&=& \int_{\mbR^3} \eta^2 r^{a_2}(|\na\om_r|^2+|\na\om_z|^2) dx + \int_{\mbR^3} \eta^2 r^{a_2}\b|\f{\om_r}{r}\b|^2 dx + \sum_{k=1}^3 B_k + \sum_{k=1}^2 A_k.
\ee
We can estimate $B_k, k=1,2, 3$ and $A_k, k=1,2$ as follows.
\be\no
\sum_{k=1}^2 |B_k|&\leq& \int_{r\geq r_0} r^{a_2-2} (|\om_r|^2+|\om_z|^2) dx\leq \|\om_r\|_{L^2}^2+\|\om_z\|_{L^2}^2,\q \text{if }a_2\leq 2,\\\no
|B_3|&\leq& C\|(\om_r,\om_z)\|_{L^2}(\|r^2\p_z f_{\th}\|_{L^2}+\|r\p_r (rf_{\th})\|_{L^2})<\oo,\q \text{if }a_2\leq 2,\\\no
|A_1|&\leq& C\int_{r\geq r_0} r^{a_2-1}|u_r|(|\om_r|+|\om_z|)^2dx\leq C\|(\om_r,\om_z)\|_{L^2}^2<\oo,\q \text{if } a_2\leq 1+\de,\\\no
|A_2|&\leq& C\int \eta^2 r^{a_2} |(\na u_r,\na u_z)|(|\om_r|+|\om_z|)^2 dx\leq C\|(\om_r,\om_z)\|_{L^2}^2<\oo,\q \text{if } a_2\leq 1+\ga.
\ee
Therefore, we may take $a_2=1+\de\wedge \ga$, so that
\be\no
\int_{\mbR^3} \eta^2 r^{a_2}(|\na\om_r|^2+|\na\om_z|^2) dx + \int_{\mbR^3} \eta^2 r^{a_2}\b|\f{\om_r}{r}\b|^2 dx\leq C\|(\om_r,\om_z)\|_{L^2}^2 +C (\|r^2\p_z f_{\th}\|_{L^2}+\|r\p_r (rf_{\th})\|_{L^2})<\oo.
\ee
Letting $r_1\to \oo$, we have derived \eqref{omega-rz3}.
%\sum_{k=1}^2|A_k|&\leq&C(\|r^{a_2-1} u_r\|_{L^{\oo}}+\|r^{a_2-1}u_z\|_{L^{\oo}})(\|\om_r\|_{L^2}^2+\|\om_z\|_{L^2}^2)\leq C(\|\om_r\|_{L^2}^2+\|\om_z\|_{L^2}^2),\\\no
%|A_3|&\leq&C(\|r^{a_2} \na u_r\|_{L^{\oo}}+\|r^{a_2}\na u_z\|_{L^{\oo}})(\|\om_r\|_{L^2}^2+\|\om_z\|_{L^2}^2)\leq C(\|\om_r\|_{L^2}^2+\|\om_z\|_{L^2}^2),\\\no
%\sum_{k=2}^5|B_k|&\leq& C\int_{\mbR^3} r^{a_2-1}(|\om_r|+|\om_z|)\eta(|\na\om_r|+|\na\om_z|) dx \\\no
%&\leq&\e \int_{\mbR^3}\eta^2 r^{2a_2-2}(|\na\om_r|^2+|\na\om_z|^2) dx+ C(\e)(\|\om_r\|_{L^2}^2+\|\om_z\|_{L^2}^2),\\\no
%&\leq&\e \int_{\mbR^3}\eta^2 r^{a_2}(|\na\om_r|^2+|\na\om_z|^2) dx+ C(\e)(\|\om_r\|_{L^2}^2+\|\om_z\|_{L^2}^2),\\\no
%B_7&\leq&\|\om_{r}\|_{L^2}\|r^2\p_z f_{\th}\|_{L^2}+\|\om_z\|_{L^2}\|r^2(\p_r f_{\th}+\f{f_{\th}}{r})\|_{L^2}.
%%|I_1|&\leq& \int_{\mbR^3} |u_r|(|\om_r|^2 +|\om_z|^2) dx \leq C\|u_r\|_{L^{\oo}}(\|\om_r\|_{L^2}^2 + \|\om_z\|_{L^2}^2),\\\no
%%|I_2|&\leq& \int_{\mbR^3} (|u_r|+|u_z|)(|\om_r|^2 +|\om_z|^2) dx \leq C\|(u_r, u_z)\|_{L^{\oo}}(\|\om_r\|_{L^2}^2 + \|\om_z\|_{L^2}^2),\\\no
%%|I_3|&\leq& \int_{\mbR^3} r(|\na u_r|+ |\na u_z|)(|\om_r|^2 +|\om_z|^2) dx \leq C\|r(\na u_r, \na u_z)\|_{L^{\oo}}(\|\om_r\|_{L^2}^2 + \|\om_z\|_{L^2}^2),\\\no
%%|J_2|+|J_4|&\leq& \int_{\mbR^3} |\om_r| |\na\om_r| dx \leq \|\om_r\|_{L^2} \|\na\om_r\|_{L^2},\\\no
%%|J_3|+|J_5|&\leq& \int_{\mbR^3} |\om_z| |\na\om_z| dx \leq \|\om_z\|_{L^2} \|\na\om_z\|_{L^2},\\\no
%%|J_6|&\leq& \int_{\mbR^3}\f{|\om_r|^2}{r} dx \leq \|\om_r\|_{L^2}\|\f{\om_r}{r}\|_{L^2},\\\no
%Letting $\rho_0\to \oo$, we have proved (\ref{omega-rz3}).
%\int_{\mbR^3} r^{1+\de\wedge\ga} (|\na \om_r|^2 +|\na\om_z|^2)dx <\oo.
To show (\ref{omega-rz4}), we need to use the equations for $\p_z\om_r$ and $\p_z\om_z$:
\be\lab{omega-rz-partialz1}
&&\p_z\b((u_r\p_r + u_z\p_z) \om_r - (\om_r\p_r +\om_z\p_z) u_r\b)= \b(\p_r^2+\f{1}{r}\p_r+\p_z^2-\f{1}{r^2}\b)\p_z\om_r-\p_z^2 f_{\th},\\\lab{omega-rz-partialz2}
&&\p_z\b((u_r\p_r + u_z\p_z) \om_z - (\om_r\p_r +\om_z\p_z) u_z\b)= (\p_r^2+\f{1}{r}\p_r+\p_z^2)\p_z\om_z+\p_z\b[\f{1}{r}\p_{r}(r f_{\th})\b].
\ee
Multiplying (\ref{omega-rz-partialz1}) and (\ref{omega-rz-partialz2}) by $\eta^2 r^{b_2} \p_z\om_{r}$ and $\eta^2 r^{b_2}\p_z\om_z$ respectively, integrating over $\mbR^3$ and adding them together, after some calculations we get
\be\no
0&=& \int_{\mbR^3} \eta^2 r^{b_2}(|\na\p_z\om_r|^2+|\na\p_z\om_z|^2) dx + \int_{\mbR^3} \eta^2 r^{b_2}\b|\f{\p_z\om_r}{r}\b|^2 dx- \f12\int_{\mbR^3} \p_r^2(\eta^2 r^{b_2}) (|\p_z\om_r|^2+|\p_z\om_z|^2) dx\\\no
&\q&-\f12\int_{\mbR^3} r^{-1}\p_r(\eta^2 r^{b_2})(|\p_z\om_r|^2+|\p_z\om_z|^2) dx- \int_{\mbR^3} \eta^2 r^{b_2}\b[-\p_z\om_r^2\p_z f_{\th}+ \p_z^2\om_z\b[\f{1}{r}\p_r(r f_{\th})\b] dx\\\no
&\q&-\int_{\mbR^3} \eta^2 r^{b_2}[\p_z^2\om_r (u_r\p_r+u_z\p_z)\om_r+\p_z^2\om_z (u_r\p_r+u_z\p_z)\om_z] dx \\\no
&\q&+\int_{\mbR^3} \eta^2 r^{b_2}[\p_z^2\om_r (\om_r\p_r+\om_z\p_z)u_r+\p_z^2\om_z (\om_r\p_r+\om_z\p_z)u_z] dx\\\no
&=& \int_{\mbR^3} \eta^2 r^{a_2}(|\na\p_z\om_r|^2+|\na\p_z\om_z|^2) dx + \int_{\mbR^3} \eta^2 r^{a_2}\b|\f{\p_z\om_r}{r}\b|^2 dx + \sum_{k=1}^3 B_k' + \sum_{k=1}^2 A_k'.
\ee
%LHS %&=&-\int_{\mbR^3}(u_r\p_r+ u_z\p_z)\om_r \p_z(\eta^2 r^{b_2}\p_z\om_r)+ (u_r\p_r+ u_z\p_z)\om_z \p_z(\eta^2 r^{b_2}\p_z\om_z) dx \\\no
%%&\quad&+\int_{\mbR^3}(\om_r\p_r+ \om_z\p_z)u_r \p_z(\eta^2 r^{b_2}\p_z\om_r)+(\om_r\p_r+ \om_z\p_z)u_z \p_z(\eta^2 r^{b_2}\p_z\om_z) dx \\\no
%&=& -\int_{\mbR^3} (u_r\p_r+ u_z\p_z)\om_r \eta^2 r^{b_2}\p_z^2\om_r dx-\int_{\mbR^3} (u_r\p_r+ u_z\p_z)\om_z \eta^2 r^{b_2}\p_z^2\om_z dx \\\no
%&\q&- 2\int_{\mbR^3} (u_r\p_r+u_z\p_z)\om_r \eta \eta' r^{b_2}\f{z\p_z\om_r}{\sqrt{r^2+z^2}} dx - 2\int_{\mbR^3} (u_r\p_r+u_z\p_z)\om_z \eta \eta' r^{b_2}\f{z\p_z\om_z}{\sqrt{r^2+z^2}} dx \\\no
%&\q&+\int_{\mbR^3}(\om_r\p_r +\om_z\p_z) u_r\eta^2 r^{b_2}\p_z^2 \om_r dx +\int_{\mbR^3}(\om_r\p_r +\om_z\p_z) u_z\eta^2 r^{b_2}\p_z^2 \om_z dx \\\no
%&\q&+2\int_{\mbR^3}(\om_r\p_r +\om_z\p_z) u_r\eta \eta' r^{b_2}\f{z \p_z\om_r}{\sqrt{r^2+z^2}} dx +2\int_{\mbR^3}(\om_r\p_r +\om_z\p_z) u_z\eta \eta' r^{b_2}\f{z \p_z\om_z}{\sqrt{r^2+z^2}} dx\\\no
%&\co& \sum_{i=1}^8 A_i',\\\no
%RHS %&=&\int_{\mbR^3} \eta^2 r^{b_2}\p_z\om_r \b(\p_r^2+\f1r\p_r+\p_z^2-\f 1{r^2}\b)\p_z\om_r +\eta^2 r^{b_2} \p_z\om_z \b(\p_r^2+\f 1r\p_r+\p_z^2\b)\p_z\om_z dx \\\no
%%&\q&+\int_{\mbR^3} \eta^2 r^{b_2}[-\p_z\om_r\p_{z}^2 f_{\th}+ \f{\p_z\om_z}{r}\p_{rz}^2(r f_{\th}) dx]\\\no
%&=& -\int_{\mbR^3} \eta^2 r^{b_2} (|\na\p_z\om_r|^2+|\na\p_z\om_z|^2) dx- 2 \int_{\mbR^3} \eta \eta' r^{b_2} \p_z\om_r \f{r\p_{rz}^2\om_r + z\p_z^2\om_r}{\sqrt{r^2+z^2}} dx \\\no
%&\quad&- 2\int_{\mbR^3} \eta \eta' r^{b_2} \p_z\om_z \f{r\p_{rz}^2\om_z + z\p_z^2\om_z}{\sqrt{r^2+z^2}} dx -b_2 \int_{\mbR^3}\eta^2 r^{b_2-1} \p_z\om_r \p_{rz}^2\om_r dx\\\no
%&\quad&- b_2 \int_{\mbR^3} r^{b_2-1} \eta^2 \p_z\om_z \p_{rz}^2\om_z dx- \int_{\mbR^3} \eta^2 r^{b_2-2} (\p_z\om_r)^2 dx\\\no
%&\q& -\int_{\mbR^3} r^{b_2}[-\p_z(\eta^2\p_z\om_r)\p_{z} f_{\th}+\p_z(\eta^2 \p_z\om_z)\f{1}{r}\p_{r}(r f_{\th})] dx \co \sum_{k=1}^7 B_k'.
We estimates these terms as follows.
\be\no
\sum_{k=1}^2|B_k'|&\leq& \int_{r\geq r_0} r^{b_2-2}(|\na\om_r|+|\na\om_z|)^2 dx\leq \int_{\mbR^3} r^{1+\de\wedge\ga}(|\na \om_r|+|\na\om_z|)^2dx<\oo,\q \text{if }b_2\leq 3+\de\wedge\ga,\\\no
|B_3'|&\leq& \f18\int_{\mbR^3} \eta^2 r^{b_2} (|\na\p_z\om_r|^2+ |\na\p_z\om_z|^2) dx + C\int_{\mbR^3} r^{b_2}\b[|\p_z f_{\th}|^2+\b|\f1r\p_r(r f_{\th})\b|^2\b]^2 dx,\\\no
|A_1'|&\leq& \f18\int_{\mbR^3} \eta^2 r^{b_2} (|\na\p_z\om_r|^2+ |\na\p_z\om_z|^2) dx+ C\int_{\mbR^3} \eta^2 r^{b_2}(|u_r|+|u_z|)^2(|\na \om_r|+|\na\om_z|)^2 dx\\\no
&\leq& \f18\int_{\mbR^3} \eta^2 r^{b_2} (|\na\p_z\om_r|^2+ |\na\p_z\om_z|^2) dx+ C\int_{\mbR^3} r^{1+\de\wedge\ga}(|\na \om_r|+|\na\om_r|)^2 dx,\text{if }b_2\leq 1+\de\wedge\ga+ 2\de,\\\no
|A_2'|&\leq& \f18\int_{\mbR^3} \eta^2 r^{b_2} (|\na\p_z\om_r|^2+ |\na\p_z\om_z|^2) dx+ C\int_{\mbR^3} \eta^2 r^{b_2}(|\na u_r|+|\na u_z|)^2(|\om_r|+|\om_z|)^2 dx\\\no
&\leq& \f18\int_{\mbR^3} \eta^2 r^{b_2} (|\na\p_z\om_r|^2+ |\na\p_z\om_z|^2) dx+ C\int_{\mbR^3}(|\om_r|+|\om_r|)^2 dx,\q\text{if }b_2\leq 2+2 \ga.
\ee
Therefore, the essential restriction on $b_2$ is $b_2\leq 1+\de\wedge\ga+ 2\de$, hence we take $b_2=1+\de\wedge \ga+ 2\de$, then we infer
\be\no
\int_{\mbR^3} \eta^2 r^{b_2}(|\na\p_z\om_r|^2+|\na\p_z\om_z|^2) dx\leq \int_{\mbR^3} r^{1+\de\wedge\ga}(|\na \om_r|+|\na\om_z|)^2dx+ C\int_{\mbR^3} r^{3}\b[|\p_z f_{\th}|^2+\b|\f1r\p_r(r f_{\th})\b|^2\b]^2 dx<\oo.
\ee
Letting $r_1\to \oo$, we derive $\int_{r\geq r_0} r^{1+\de\wedge\ga+2\de} (|\na\p_z\om_r|+|\na\p_z\om_z|)^2dx<\oo$. This directly yields \eqref{omega-rz4}.
%\sum_{k=1}^2|B_k'|&\leq& \int_{r\geq r_0} r^{b_2-2}(|\na\om_r|+|\na\om_z|) dx\leq \int r^{1+\de\wedge\ga}|(\na \om_r,\na\om_z)|^2dx<\oo,\q \text{if }b_2\leq 3+\de\wedge\ga,\\\no
%&\leq&\e\|\eta r^{\f{b_2}{2}}(|\na\p_z\om_r|+|\na\p_z\om_z|)\|_{L^2}^2+C(\e)\|r^{\f{a_2}{2}}(|\na\om_r|+|\na\om_z|)\|_{L^2}^2,\q\textit{if }\f{b_2}{2}-1\leq a_2,\\\no
%|B_6'|&\leq& \|\eta r^{\f{a_2}{2}}\na\om_r\|_{L^2}^2,\q \textit{if }b_2-2\leq a_2,\\\no
%|B_7'|&\leq& \|\eta r^{b_2/2}\p_z^2\om_r\|_{L^2}\|r^{b_2/2}\p_z f_{\th}\|_{L^2}+\|\eta r^{b_2/2}\p_z^2\om_z\|_{L^2}\|r^{b_2/2}(\p_r f_{\th}+\f{1}{r}f_{\th})\|_{L^2}\\\no
%&\q&+\|r^{a_2/2}\p_z\om_r\|_{L^2}\|r^{b_2-a_2/2-1}\p_z f_{\th}\|_{L^2}+ \|r^{a_2/2}\p_z\om_z\|_{L^2}\|r^{b_2-a_2/2-1}(\p_r f_{\th}+\f{1}{r}f_{\th})\|_{L^2}\\\no
%&\leq&\e \|\eta r^{b_2/2}(\p_z\na\om_r,\p_z\na\om_z\|_{L^2}^2+\|r^{a_2/2}(\na \om_r,\na\om_z)\|_{L^2}^2+C(\e)\|r^2(\p_z f_{\th},\p_r f_{\th}+\f{1}{r}f_{\th})\|_{L^2}^2.
%\sum_{k=1}^2 |A_k'|&\leq&\int_{\mbR^3} r^{\f{b_2}{2}-\f{a_2}{2}}(|u_r|+|u_z|)\eta r^{\f{a_2}{2}}(|\na\om_r|+|\na\om_z|)\eta r^{\f{b_2}{2}}(|\na\p_z\om_r|+|\na\p_z\om_z|) dx \\\no
%%&\leq&\e\|\eta r^{\f{b_2}{2}}(|\na\p_z\om_r|+|\na\p_z\om_z|)\|_{L^2}^2+C\|r^{\f{b_2}{2}-\f{a_2}{2}}(|u_r|+|u_z|)\|_{L^{\oo}}^2\|\eta r^{\f{a_2}2}(|\na\om_r|+|\na\om_z|)\|_{L^2}^2\\\no
%&\leq&\e\|\eta r^{\f{b_2}{2}}(|\na\p_z\om_r|+|\na\p_z\om_z|)\|_{L^2}^2+C\|\eta r^{\f{a_2}2}(|\na\om_r|+|\na\om_z|)\|_{L^2}^2,\q\textit{if }\f{b_2}{2}-\f{a_2}{2}\leq \de,\\\no
%\sum_{k=3}^4 |A_k'|&\leq&\int_{\mbR^3} r^{b_2-a_2-1}(|u_r|+|u_z|) r^{a_2}(|\na\om_r|^2+|\na\om_z|^2) dx \\\no
%&\leq&C \|r^{\f{a_2}{2}}(|\na\om_r|+|\na\om_z|)\|_{L^2}^2,\q\textit{if }b_2-a_2-1\leq \de,\\\no
%\sum_{k=5}^6|A_k'|&\leq& \int_{\mbR^3} r^{\f{b_2}{2}}(|\na u_r|+|\na u_z|)(|\om_r|+|\om_z|)\eta r^{\f{b_2}{2}}(|\na\p_z u_r|+|\na\p_z u_z|) dx\\\no
%%&\leq&\e \|\eta r^{\f{b_2}{2}}(|\na\p_z u_r|+|\na\p_z u_z|)\|_{L^2}^2+ C(\e)\|r^{\f{b_2}{2}}(|\na u_r|+|\na u_z|)\|_{L^{\oo}}^2(\|\om_r\|_{L^2}^2+\|\om_z\|_{L^2}^2)\\\no
%&\leq&\e \|\eta r^{\f{b_2}{2}}(|\na\p_z u_r|+|\na\p_z u_z|)\|_{L^2}^2+ C(\e)(\|\om_r\|_{L^2}^2+\|\om_z\|_{L^2}^2),\q\textit{if }\f{b_2}{2}\leq 1+\ga,\\\no
%\sum_{k=7}^8|A_k'|&\leq& \int_{\mbR^3} r^{b_2-\f{a_2}{2}-1}(|\na u_r|+|\na u_z|)(|\om_r|+|\om_z|)\eta r^{\f{a_2}{2}}(|\p_z\om_r|+|\p_z\om_z|) dx\\\no
%%&\leq&C\|r^{b_2-\f{a_2}{2}-1}(|\na u_r|+|\na u_z|)\|_{L^{\oo}}(\|\om_r\|_{L^2}+\|\om_z\|_{L^2})\|\eta r^{\f{a_2}{2}}(|\p_z\om_r|+|\p_z\om_z|)\|_{L^2}\\\no
%&\leq&C(\|\om_r\|_{L^2}+\|\om_z\|_{L^2})\|\eta r^{\f{a_2}{2}}(|\p_z\om_r|+|\p_z\om_z|)\|_{L^2},\q\textit{if }b_2-\f{a_2}{2}-1\leq 1+\ga.
%Take $b_2=1$, then
%|I_1|&\leq&\int_{\mbR^3}|(u_r,u_z)|(r^{1/2}|\na\om_r|) (\eta r^{1/2}|\p_z^2\om_r|) dx \\\no
%&\leq& \e \|\eta r^{1/2}|\na\p_z\om_r|\|_{L^2}^2 + C(\e)\|(u_r,u_z)\|_{L^{\oo}}^2 \|r^{1/2}|\na\om_r|\|_{L^2}^2,\\\no
%|I_2|&\leq& \|(u_r, u_z)\|_{L^{\oo}} \|\na \om_r\|_{L^2}^2,\\\no
%|I_3|&\leq&\e \|\eta r^{1/2}|\na\p_z\om_z|\|_{L^2}^2 + C(\e)\|(u_r, u_z)\|_{L^{\oo}}^2 \|r^{1/2}|\na\om_z|\|_{L^2}^2,\\\no
%|I_4|&\leq&\|(u_r, u_z)\|_{L^{\oo}} \|\na \om_z\|_{L^2}^2,\\\no
%|I_5|&\leq&\int_{\mbR^3} r^{1/2}|\na u_r|(|\om_r|+|\om_z|)(\eta r^{1/2}|\na\p_z\om_r|) dx \\\no
%&\leq& \e \|\eta r^{1/2}|\na \p_z\om_r|\|_{L^2}^2 + C(\e) \|r^{1/2}|\na u_r|\|_{L^{\oo}}^2(\|\om_r\|_{L^2}^2 + \|\om_z\|_{L^2}^2),\\\no
%|I_6|&\leq&\|\na u_r\|_{L^{\oo}}(\|\om_r\|_{L^2}+\|\om_z\|_{L^2})(\|\na\om_r\|_{L^2}+\|\na\om_z\|_{L^2}),\\\no
%|I_7|&\leq& \e \|\eta r^{1/2}|\na \p_z\om_z|\|_{L^2}^2 + C(\e) \|r^{1/2}|\na u_z|\|_{L^{\oo}}^2(\|\om_r\|_{L^2}^2 + \|\om_z\|_{L^2}^2),\\\no
%|I_8|&\leq&\|\na u_z\|_{L^{\oo}}(\|\om_r\|_{L^2}+\|\om_z\|_{L^2})(\|\na\om_r\|_{L^2}+\|\na\om_z\|_{L^2}).
%It is easy to see the essential restriction on $b_2$ is $\f{b_2}{2}-\f{a_2}{2}\leq \de$, i.e. $b_2\leq a_2+2\de$. Hence we choose $b_2=1+\de\wedge\ga+2\de$ and get (\ref{omega-rz4}) by letting $\rho_0\to \oo$.
\epf
\subsection{Improved decay estimates on $u_{\th}$}
Since $\textit{curl }(u_{\th}{\bf e}_{\th})= \om_r {\bf e}_r+ \om_z {\bf e}_z$ and $\textit{div }(u_{\th}{\bf e}_{\th})=0$, then
\be\no
-\Delta( u_{\th} {\bf e}_{\th})= \textit{curl }(\om_r {\bf e}_r+ \om_z {\bf e}_z).
\ee
Fix $(r,z)\in \mbR_+\times \mbR$, we choose a smooth cut-off function $\psi\in C_0^{\oo}(\mbR^3)$, which is axially symmetric and satisfies $0\leq \psi\leq 1$,
\be\no
\psi(\rho,\ka)=\begin{cases}
1,\q r/2<\rho <2r,\q -z_1<\ka<z_1,\\
0,\q \rho<r/4, \rho>4r \ \textit{or }\ |\ka|>2 z_1,
\end{cases}
\ee
where $z_1$ is any constant such that $z_1>\max(2|z|,1)$. Since
\be\no
-\Delta(\psi u_{\th}{\bf e}_{\th})= \psi \textit{curl }(\om_r {\bf e}_r+\om_z {\bf e}_z)- 2\na\psi\cdot\na(u_{\th}{\bf e}_{\th})- (\Delta \psi) u_{\th}{\bf e}_{\th},
\ee
we get the integral representation for $u_{\th}$ in terms of $(\om_{r}, \om_z)$ and the fundamental solution $\Ga({\bf x}, {\bf y})=\Ga({\bf x}-{\bf y})=\f{1}{4\pi |{\bf x}-{\bf y}|}$ of the Laplace operator: for ${\bf x}= (r\cos\th, r\sin\th, z)$
\be\lab{u-theta-formula}\begin{array}{ll}
(\psi u_{\th}{\bf e}_{\th})(x)&= \int_{\mbR^3}\Ga(x-y)\psi(y)\textit{curl }(\om_{\rho}{\bf e}_{\rho}+ \om_{\ka}{\bf e}_{\ka}) dy-2\int_{\mbR^3}\Ga(x-y)\na\psi\cdot\na(u_{\phi}{\bf e}_{\phi})(y) dy\\
&\q-\int_{\mbR^3}\Ga(x-y)(\De\psi)(y)(u_{\phi}{\bf e}_{\phi})(y) dy\\
&=-\int_{\mbR^3}\na_y \Ga(x-y)\times[\psi(y)(\om_{\rho}{\bf e}_{\rho}+\om_{\ka}{\bf e}_{\ka})(y)] dy\\
&\q-\int_{\mbR^3}\Ga(x-y)(\na_y\psi)(y)\times (\om_{\rho}{\bf e}_{\rho}+\om_{\ka}{\bf e}_{\ka})(y)] dy\\
&\q+2\int_{\mbR^3}[\na_y\Ga(x-y)\cdot \na_y\psi(y)](u_{\phi}{\bf e}_{\phi})(y) dy+ \int_{\mbR^3}\Ga(x-y)\Delta_y\psi(y)(u_{\phi}{\bf e}_{\phi})(y)dy.
\end{array}\ee
By taking inner product to (\ref{u-theta-formula}) by ${\bf e}_{\th}$, we get the following integral representation for $u_{\th}$: for ${\bf x}= (r\cos\th, r\sin\th, z)$
\be\lab{c31}\begin{array}{ll}
u_{\th}(r,z)&=-\int_{\mbR^3} \f{\p\hat{\Ga}}{\p \ka}\psi(y)\om_{\rho}(y)\cos(\th-\phi)dy+\int_{\mbR^3} \f{\p\hat{\Ga}}{\p \rho}\psi(y)\om_{\ka}(y)\cos(\th-\phi) dy\\
&\q+ \int_{\mbR^3} \f{1}{\rho}\f{\p\hat{\Ga}}{\p \phi}\phi\om_{\ka}\sin(\th-\phi)d y-\int_{\mbR^3} \hat{\Ga}\f{\p\psi}{\p \ka}\om_{\rho}\cos(\th-\phi) dy\\
&\q+ \int_{\mbR^3}\hat{\Ga} \f{\p\psi}{\p\rho}\om_{\ka} \cos(\th-\phi) dy+\int_{\mbR^3}\hat{\Ga} \f{1}{\rho}\f{\p\psi}{\p\phi}\om_{\ka}\sin(\th-\phi) dy\\
&\q+ 2\int_{\mbR^3}\b(\f{\p\hat{\Ga}}{\p\rho}\f{\p\psi}{\p\rho}+\f{\p\hat{\Ga}}{\p\ka}\f{\p\psi}{\p\ka}\b)u_{\phi}\cos(\th-\phi)dy+\int_{\mbR^3}\hat{\Ga}\De \psi u_{\phi} \cos(\th-\phi) dy.
\end{array}\ee
%Hence we get the integral representation for ${\bf v}$ in terms of $\om_{\th}$ and the fundamental solution $\Ga=\Ga({\bf x},{\bf y})=\f{1}{4\pi |{\bf x}-{\bf y}|}$ of the Laplace operator: for ${\bf x}= (r\cos\th, r\sin\th, z)$
%\psi {\bf v}({\bf x}) &= -\int_{\mbR^3} \na_{{\bf y}}\Ga({\bf x},{\bf y}) \times (\om_{\phi}({\bf y})\psi({\bf y}){\bf e}_{\phi}) d {\bf y} -\int_{\mbR^3} \Ga({\bf x},{\bf y})((\na_{{\bf y}}\psi({\bf y})\times {\bf e}_{\phi}))\om_{\phi}({\bf y}) d {\bf y}\\
%&\quad+ \int_{\mbR^3} \Ga({\bf x},{\bf y})(\Delta_{{\bf y}}\psi)({\bf y}) {\bf v}({\bf y}) d{\bf y} + 2 \int_{\mbR^3} (\na_{{\bf y}}\Ga)({\bf x},{\bf y})\cdot (\na_{{\bf y}}\psi)({\bf y}) {\bf v}({\bf y}) d {\bf y}.
%By taking inner product to (\ref{c26}) by ${\bf e}_r$ and ${\bf e}_3$, we get the following integral representation for $u_r$ and $u_3$, respectively; for ${\bf x}= (r\cos\th, r\sin\th, z)$
%u_r(r, z) &= -\int_{\mbR^3} \na_{{\bf y}}\Ga({\bf x},{\bf y}) \times (\om_{\phi}({\bf y})\psi({\bf y}){\bf e}_{\phi})\cdot {\bf e}_r d {\bf y} -\int_{\mbR^3} \Ga({\bf x},{\bf y})((\na_{{\bf y}}\psi({\bf y})\times {\bf e}_{\phi}))\cdot {\bf e}_r\om_{\phi}({\bf y}) d {\bf y}\\
%&\quad+ \int_{\mbR^3} \Ga({\bf x},{\bf y})(\Delta_{{\bf y}}\psi)({\bf y}) {\bf v}({\bf y})\cdot {\bf e}_r d{\bf y} + 2 \int_{\mbR^3} (\na_{{\bf y}}\Ga)({\bf x},{\bf y})\cdot (\na_{{\bf y}}\psi)({\bf y}) {\bf v}({\bf y})\cdot {\bf e}_r d {\bf y},\\
%u_3(r, z) &= -\int_{\mbR^3} \na_{{\bf y}}\Ga({\bf x},{\bf y}) \times (\om_{\phi}({\bf y})\psi({\bf y}){\bf e}_{\phi})\cdot {\bf e}_3 d {\bf y} -\int_{\mbR^3} \Ga({\bf x},{\bf y})((\na_{{\bf y}}\psi({\bf y})\times {\bf e}_{\phi}))\cdot {\bf e}_3\om_{\phi}({\bf y}) d {\bf y}\\
%&\quad+ \int_{\mbR^3} \Ga({\bf x},{\bf y})(\Delta_{{\bf y}}\psi)({\bf y}) {\bf v}({\bf y})\cdot {\bf e}_3 d{\bf y} + 2 \int_{\mbR^3} (\na_{{\bf y}}\Ga)({\bf x},{\bf y})\cdot (\na_{{\bf y}}\psi)({\bf y}) {\bf v}({\bf y})\cdot {\bf e}_3 d {\bf y}.
The cylindrical coordinate representation of $\Ga$ is denoted by $\hat{\Ga}= \hat{\Ga}(r,\rho, \th-\phi, z-\ka)$:
\be\no
\hat{\Ga}= \f{1}{4\pi\sqrt{r^2+\rho^2- 2r \rho \cos (\th-\phi)+ (z-\ka)^2}}.
\ee
Since ${\bf e}_{\rho}$ and ${\bf e}_r$ are different, they cause extra computations involving with $\cos(\phi-\th)$. Direct computations yield that
\be\no
\f{\p\hat{\Ga}}{\p\rho} &=&-\f{1}{4\pi} \f{\rho -r \cos(\th-\phi)}{(r^2+\rho^2- 2r\rho \cos(\th-\phi)+ (z-\ka)^2)^{\f32}},\\\no
\f{\p\hat{\Ga}}{\p \ka} &=&\f{1}{4\pi} \f{(z-\ka)}{(r^2+\rho^2- 2r\rho \cos(\th-\phi)+ (z-\ka)^2)^{\f32}}.
%(\na_{{\bf y}}\Ga\times (\om_{\phi}({\bf y}) {\bf e}_{\phi}))\cdot {\bf e}_r &=& -\f{\p\hat{\Ga}}{\p \ka} \om_{\phi} \cos (\phi-\th),\\\no
%(\na_{{\bf y}}\Ga\times (\om_{\phi}({\bf y}) {\bf e}_{\phi}))\cdot {\bf e}_3 &=& \f{\p\hat{\Ga}}{\p \rho} \om_{\phi},\\\no
%(\na_{{\bf y}}\Ga\times {\bf e}_{\rho})\cdot {\bf e}_{\th}&=&\b(-\f{1}{\rho}\f{\p\hat{\Ga}}{\p\phi}{\bf e}_{\ka}+\f{\p\hat{\Ga}}{\p\ka}{\bf e}_{\phi}\b)\cdot {\bf e}_{\th}=\f{\p\hat{\Ga}}{\p\ka} \cos(\th-\phi),\\\no
%(\na_{{\bf y}}\Ga\times {\bf e}_{\ka})\cdot {\bf e}_{\th}&=&\b(-\f{\p\hat{\Ga}}{\p\rho}{\bf e}_{\phi}+\f{1}{\rho}\f{\p\hat{\Ga}}{\p\phi}{\bf e}_{\rho}\b)\cdot {\bf e}_{\th}\\\no
%&=&-\f{\p\hat{\Ga}}{\p\rho}\cos(\th-\phi)-\f{1}{\rho}\f{\p\hat{\Ga}}{\p\phi} \sin(\th-\phi),\\\no
%(\na_{{\bf y}}\psi\times {\bf e}_{\phi})\cdot {\bf e}_r &=&-\f{\p\psi}{\p z}\cos(\th-\phi),\q (\na_{{\bf y}}\psi\times {\bf e}_{\phi})\cdot {\bf e}_3= \f{\p\psi}{\p r},\\\no
%(\na_{{\bf y}}\psi\times {\bf e}_{\rho})\cdot {\bf e}_{\th} &=&\f{\p\psi}{\p \ka}\cos(\th-\phi),\\\no
%(\na_{{\bf y}}\psi\times {\bf e}_{\ka})\cdot {\bf e}_{\th} &=&-\f{\p\psi}{\p \rho}\cos(\th-\phi)-\f{1}{\rho}\f{\p\psi}{\p\phi}\sin(\th-\phi),\\\no
%{\bf v}\cdot {\bf e}_r&=& u_{\rho}\cos (\th-\phi),\q {\bf v}\cdot {\bf e}_z = u_z.
\ee
%Hence (\ref{c27}) are rewritten in the following way:
%u_r(r,z)&=\int_{\mbR^3} \f{\p\hat{\Ga}}{\p \ka} \cos(\th-\phi)\psi\om_{\th} \rho d\phi d\rho d\ka+ 2\int_{\mbR^3} \b(\f{\p\psi}{\p\rho} u_{\rho} \f{\p\hat{\Ga}}{\p \rho}\cos (\th-\phi)+ \f{\p\psi}{\p \ka} u_{\rho} \f{\p\hat{\Ga}}{\p \ka}\cos (\th-\phi)\b) \rho d\phi d\rho d\ka \\\no
%&\q + \int_{\mbR^3} (\Delta\psi) u_{\rho} \hat{\Ga} \cos(\th-\phi)\rho d\phi d\rho d\ka + \int_{\mbR^3}\f{\p\psi}{\p \ka}\om_{\phi} \hat{\Ga}\cos(\th-\phi) \rho d\phi d\rho d\ka,\\\lab{c30}
%u_3(r,z)&=-\int_{\mbR^3} \f{\p\hat{\Ga}}{\p \rho} \psi\om_{\th} \rho d\phi d\rho d\ka+ 2\int_{\mbR^3} \b(\f{\p\psi}{\p\rho} u_{3} \f{\p\hat{\Ga}}{\p \rho}\cos (\th-\phi)+ \f{\p\psi}{\p \ka} u_{\rho} \f{\p\hat{\Ga}}{\p \ka}\cos (\th-\phi)\b) \rho d\phi d\rho d\ka \\\no
%&\q + \int_{\mbR^3} (\Delta\psi) u_{3} \hat{\Ga}\rho d\phi d\rho d\ka - \int_{\mbR^3}\f{\p\psi}{\p \rho}\om_{\phi}\hat{\Ga} \rho d\phi d\rho d\ka.
Define $\Ga_i= \hat{\Ga}_i(r,\rho, z-\ka)$, $i=0,\cdots, 5$, by
\be\no\begin{array}{ll}
\Ga_0= \int_{0}^{2\pi} \hat{\Ga}(r,\rho,\phi, z-\ka) d\phi,\q &\Ga_1= \int_{0}^{2\pi} \hat{\Ga}(r,\rho,\phi, z-\ka) \cos\phi d\phi,\\\no
\Ga_2= \int_{0}^{2\pi} \f{\p\hat{\Ga}}{\p \rho}(r,\rho,\phi, z-\ka) d\phi,\q &\Ga_3= \int_{0}^{2\pi} \f{\p\hat{\Ga}}{\p \rho}(r,\rho,\phi, z-\ka) \cos\phi d\phi,\\\no
\Ga_4= \int_{0}^{2\pi} \f{\p\hat{\Ga}}{\p \ka}(r,\rho,\phi, z-\ka) d\phi,\q &\Ga_5= \int_{0}^{2\pi} \f{\p\hat{\Ga}}{\p \ka}(r,\rho,\phi, z-\ka) \cos\phi d\phi.
%\Ga_1'= \int_{0}^{2\pi} \hat{\Ga}(r,\rho,\phi, z-\ka) \sin\phi d\phi,\q &\Ga_3'= \int_{0}^{2\pi} \f{\p\hat{\Ga}}{\p \rho}(r,\rho,\phi, z-\ka) \sin\phi d\phi,\\\no
%\Ga_5'= \int_{0}^{2\pi} \f{\p\hat{\Ga}}{\p \ka}(r,\rho,\phi, z-\ka) \sin\phi d\phi,\q &\Ga_6= \int_{0}^{2\pi} \f{1}{\rho}\f{\p\hat{\Ga}}{\p \ka}(r,\rho,\phi, z-\ka) \cos\phi d\phi.
\end{array}\ee
\bl\lab{cl22}
{\it We have the following integral representation of $u_{\th}$ in terms of $\om_r$ and $\om_z$:
\be\lab{u-theta-formula-final}\begin{array}{ll}
u_{\th}(r,z)&=-\int_{-\oo}^{\oo} \int_0^{\oo}\Ga_5\psi \om_{\rho}\rho d\rho d\ka+ \int_{-\oo}^{\oo} \int_0^{\oo}\Ga_3\psi \om_{\ka}\rho d\rho d\ka-\int_{-\oo}^{\oo} \int_0^{\oo}\Ga_1 \f{\p\psi}{\p\ka}\om_{\rho}\rho d\rho d\ka\\
&\q+ \int_{-\oo}^{\oo} \int_0^{\oo}\Ga_1\f{\p\psi}{\p \rho}\om_{\ka} \rho d\rho d\ka+ 2\int_{-\oo}^{\oo} \int_0^{\oo} (\Ga_3\f{\p\psi}{\p\rho}+\Ga_5 \f{\p\psi}{\p\ka}) u_{\phi} \rho d\rho d\ka\\
&\q+ \int_{-\oo}^{\oo} \int_0^{\oo} \Ga_1\Delta \psi u_{\phi}\rho d\rho d\ka.
\end{array}\ee
\el
%For comparison, we also include the integral formulas for $u_r$ and $u_{\th}$ derived in \cite{cj09}.
%u_r(r,z)&=\int_{-\oo}^{\oo} \int_0^{\oo} \psi(\rho,\ka) \Ga_5 \om_{\phi} \rho d\rho d\ka+ 2\int_{-\oo}^{\oo} \int_0^{\oo}\left(\f{\p\psi}{\p\rho} u_{\rho} \Ga_3 + \f{\p\psi}{\p \ka} u_{\rho}\Ga_5\right) \rho d\rho d\ka \\
%&\quad+ \int_{-\oo}^{\oo} \int_0^{\oo} (\Delta \psi) u_{\rho} \Ga_1 \rho d\rho d\ka+ \int_{-\oo}^{\oo} \int_0^{\oo} \f{\p\psi}{\p \ka} \om_{\phi} \Ga_1 \rho d\rho d\ka,\\
%u_3(r,z)&=-\int_{-\oo}^{\oo} \int_0^{\oo} \psi(\rho,\ka) \Ga_2 \om_{\phi} \rho d\rho d\ka+ 2\int_{-\oo}^{\oo} \int_0^{\oo}\left(\f{\p\psi}{\p\rho} u_{3} \Ga_2 + \f{\p\psi}{\p \ka} u_{3}\Ga_4\right) \rho d\rho d\ka \\
%&\quad+ \int_{-\oo}^{\oo} \int_0^{\oo} (\Delta \psi) u_{3} \Ga_0 \rho d\rho d\ka- \int_{-\oo}^{\oo} \int_0^{\oo} \f{\p\psi}{\p \rho} \om_{\phi} \Ga_0 \rho d\rho d\ka.
To our purpose, we need the following estimates for $\Ga_i, i=1,\cdots, 5$.
\be\lab{c213}
|\Ga_i(r,\rho,z-\ka)|&\leq& \f{1}{\sqrt{(r-\rho)^2 + (z-\ka)^2}}\q \q \textit{for }i=0,1,\\\lab{c214}
|\Ga_i(r,\rho,z-\ka)|&\leq& \f{\rho+r}{[(r-\rho)^2 + (z-\ka)^2]^{\f32}}\q \q \textit{for }i=2,3,\\\lab{c215}
|\Ga_i(r,\rho,z-\ka)|&\leq& \f{|z-w|}{[(r-\rho)^2+(z-\ka)^2]^{\f32}}\q\q \textit{for }i=4,5.
\ee
For $\Ga_2,\Ga_3$ and $\Ga_5$, we have extra decay in $r$.
\bl\lab{cl23}
{\it Suppose $\f 14\leq \f{\rho}{r}\leq 16$, then
\be\lab{cl231}
|\Ga_2(r,\rho,z-\ka)|&\leq& \f{c}{\rho \sqrt{(\rho-r)^2+(z-\ka)^2}},\\\lab{cl233}
|\Ga_3(r,\rho,z-\ka)|&\leq& \f{c}{\rho \sqrt{(\rho-r)^2+(z-\ka)^2}},\\\lab{cl232}
|\Ga_5(r,\rho,z-\ka)|&\leq& \f{c}{r}\f{|z-\ka|}{(\rho-r)^2+(z-\ka)^2}.
\ee
\el
\bpf (\ref{cl231}) and (\ref{cl232}) have been proved in Lemma 2.3 in \cite{cj09}. (\ref{cl233}) follows from the following calculation
\be\no
\Ga_3(r,\rho,z-\ka)&=&-\f{1}{4\pi}\int_0^{2\pi}\f{\rho\cos\phi-r}{(r^2+\rho^2- 2r\rho \cos\phi+ (z-\ka)^2)^{\f32}}d\phi\\\no
&\q&-\f{r}{4\pi}\int_0^{2\pi}\f{\sin^2\phi d\phi}{(r^2+\rho^2- 2r\rho \cos\phi+ (z-\ka)^2)^{\f32}}\\\no
&\co&-\Ga_2(\rho,r, z-\ka)+ rJ(r,\rho,z-\ka),\\\no
|J(r,\rho,z-\ka)|&\leq&2\int_{-1}^1 \f{dt}{[r^2+\rho^2-2r\rho t+(z-\ka)^2]^{\f32}}\\\no
&\leq&\f{2}{r\rho \sqrt{(r-\rho)^2+(z-\ka)^2}}.
\ee
\epf
\bl\lab{cl41}
{\it There is a positive constant $c(M)$ such that
\be\lab{c41}
|u_{\th}(r,z)|\leq c(M) \b(\f{\log r}{r}\b)^{\f12}
\ee
for large $r$, uniformly in $z$.
\bpf
From Lemma \ref{cl22}. we have
\be\no\begin{array}{ll}
u_{\th}(r,z)&=\int_{-\oo}^{\oo} \int_0^{\oo}\Ga_3\psi \om_{\ka}\rho d\rho d\ka-\int_{-\oo}^{\oo} \int_0^{\oo}\Ga_5\psi \om_{\rho}\rho d\rho d\ka+ 2\int_{-\oo}^{\oo} \int_0^{\oo} \Ga_3\f{\p\psi}{\p\rho} u_{\phi} \rho d\rho d\ka\\
&\q+2\int_{-\oo}^{\oo} \int_0^{\oo}\Ga_5 \f{\p\psi}{\p\ka}u_{\phi} \rho d\rho d\ka + \int_{-\oo}^{\oo} \int_0^{\oo}\Ga_1\f{\p\psi}{\p \rho}\om_{\ka} \rho d\rho d\ka-\int_{-\oo}^{\oo} \int_0^{\oo}\Ga_1 \f{\p\psi}{\p\ka}\om_{\rho}\rho d\rho d\ka\\\no
&\q+ \int_{-\oo}^{\oo} \int_0^{\oo} \Ga_1\b(\p_{\rho}^2\psi+\f{1}{\rho}\p_{\rho}\psi\b) u_{\phi}\rho d\rho d\ka+ \int_{-\oo}^{\oo} \int_0^{\oo} \Ga_1\p_{\ka}^2\psi u_{\phi}\rho d\rho d\ka \co \di\sum_{k=1}^8 T_k.
\end{array}
\ee
(1){\it Estimates of $T_1, T_2$.} The estimates of $T_1$ and $T_2$ are similar. We define the integration domain $D$ by
\be\no
D=\{r/4<\rho<4r, -z_1<\ka<z_1\}.
\ee
We decompose the domain of $D$ into two subregions. Let
\be\no
A&=&\{(\rho,\ka)\in [r/4,4r]\times [-z_1,z_1]: |r-\rho|\leq 1\},\\\no
B&=&\{(\rho,\ka)\in [r/4,4r]\times [-z_1,z_1]: |r-\rho|> 1\}.
\ee
\be\no
T_1&\leq& r\sup_{\substack{r/4<\rho<4r\\-z_1<\ka<z_1}} |\om_{\ka}(\rho,\ka)\int_{A} |\Ga_3(r,\rho,z-\ka)|d\rho d\ka\\\no
&\q& + \b(\int_{r/4}^{4r}\int_{-z_1}^{z_1} |\om_{\ka}(\rho,\ka)|^2\rho d\rho d\ka\b)^{\f 12} \b(\int_{B}|\Ga_3(r,\rho,z-\ka)|^2 \rho d\rho d\ka\b)^{\f 12}.
\ee
From Lemma \ref{cl23}, we have the following estimates
\be\no
&\q&\int_A |\Ga_3(r,\rho,z-\ka)| d\rho d\ka \leq \f{c}{r} \int_0^1\int_0^{z_1}\f{ds dt}{\sqrt{s^2+t^2}} \\\no
&\leq&\f{c}{r}\int_0^1 dt\b(\int_0^1 \f{d\th}{\sqrt{1+\th^2}}+ \int_1^{\f{z_1}{t}} \f{d\th}{\th}\b) \\\no
&\leq&\f{c}{r}\b(c+\int_0^1(\log z_1-\log t) dt\b)\leq \f{c+c\log z_1}{r},\\\no
&\q&\int_B |\Ga_3(r,\rho,z-\ka)|^2 d\rho d\ka \leq \f{c}{r} \int_1^{5r} \int_0^{z_1} \f{ds dt}{t^2+s^2} \\\no
&\leq& \f{c}{r} \int_1^{5r} \f{d t}{t} \int_0^{z_1/t} \f{d\th}{1+\th^2}\leq \f{c\log r}{r}.
\ee
Hence by (\ref{omega-rz7}), we have
\be\no
|T_1|%&\leq& \f{c+ c\log z_1}{r}+c\b(\f{\log r}{r}\b)^{\f 12}\b(\int_{r/2}^{2r} \int_{z-r}^{z+r} |\om_{\ka}(\rho, \ka)|^2 \rho d\rho d\ka\b)^{\f 12}\\\no
&\leq& \f{c+ c\log z_1}{r}+c\b(\f{\log r}{r}\b)^{\f 12}.
\ee
(2){\it Estimates of $T_3, T_5$ and $T_7$.} Since $|\f{\p\psi}{\p\rho}|\leq \f{c}{\rho}$,
\be\no
|T_3|%&\leq& \f{c}{r}\int_{-z_1}^{z_1} \int_{r/4}^{r/2} |u_{\phi}||\Ga_3|\rho d\rho d\ka +\int_{-z_1}^{z_1} \int_{2r}^{4r} |u_{\phi}||\Ga_3| d\rho d\ka\\\no
&\leq&\f{c}{r}\b[\b(\int_{r/4}^{r/2}+\int_{2r}^{4r} \b)\rho d\rho\int_{-\infty}^{\infty}\f{d\ka}{r^{6/5}[(\rho-r)^2+(z-\ka)^2]^{\f35}}\b]^{\f65}\\\no
\ee
Note that
\be\no
\int_{-\oo}^{\oo} |\Ga_1(r,\rho,z-\ka)|^2 d\ka \leq \f{c}{|r-\rho|},
\ee
then we have
\be\no
|T_5|%&\leq& \f{c}{r}\int_{-z_1}^{z_1} \b(\int_{r/4}^{r/2}+\int_{2r}^{4r}\b) |\om_{\ka}||\Ga_1|\rho d\rho d\ka \\\no
&\leq&\f{c}{r}\|\om_{\ka}\|_{L^2(\mbR^3)}\b[\b(\int_{r/4}^{r/2}+\int_{2r}^{4r}\b)\rho d\rho\int_{-\infty}^{\infty}|\Ga_1|^{2}d\ka\b]^{\f12}\\\no
&\leq&\f{c}{r}\b[\b(\int_{r/4}^{r/2}+\int_{2r}^{4r} \b)\f{\rho d\rho}{|r-\rho|}\b]^{1/2}\leq \f{c}{r^{1/2}},\\\no
|T_7|%&\leq& \f{c}{r^2}\int_{-z_1}^{z_1} \b(\int_{r/4}^{r/2}+\int_{2r}^{4r}\b) |\om_{\rho}||\Ga_1|\rho d\rho d\ka \\\no
&\leq&\f{c}{r^2}\b(\int_{r/4}^{r/2}+\int_{2r}^{4r}\b)\rho d\rho \b(\int_{-\oo}^{\oo} |\Ga_1|^2 d\ka\b)^{1/2}\b(\int_{-\oo}^{\oo}|\om_{\rho}|^2 d\ka\b)^{1/2}\\\no
&\leq& \f{c}{r^2}\b(\int_{r/4}^{r/2}+\int_{2r}^{4r}\b) \f{\rho d\rho}{\sqrt{|r-\rho|}}\leq \f{c}{r^{1/2}}.
\ee
(3){\it Estimates of $T_4, T_6$ and $T_8$.} Since $|\f{\p \psi}{\p w}|\leq \f{c}{z_1}$ and $|\f{\p^2 \psi}{\p w^2}|\leq \f{c}{z_1^2}$, we observe that
\be\no
|T_4|%&\leq& \f{c}{z_1} \int_{r/4}^{4r}\int_{-\oo}^{\oo} |u_{\phi}| |\Ga_5| \rho d\rho d\ka\\\no
&\leq&\f{c}{z_1}\|u_{\phi}\|_{L^6(\mbR^3)}\b(\int_{r/4}^{4r}\int_{-\oo}^{\oo}\f{c}{r^{\f65}}\f{|z-\ka|^{\f65}}{[(r-\rho)^2+(z-\ka)^2]^{\f 35}} d\ka\b)^{\f 56}\\\no
&\leq&\f{c}{r z_1}\b(\int_{r/4}^{4r}\f{1}{|r-\rho|^{\f15}}\rho d\rho\b)^{\f 56}\leq \f{c r^{1/2}}{z_1},\\\no
|T_6|%&\leq& \f{c}{z_1}\int_{r/4}^{4r}\int_{-\oo}^{\oo} |\Ga_1||\om_{\rho}| \rho d\rho d\ka\\\no
&\leq&\f{c}{z_1}\int_{r/4}^{4r} \b(\int_{-\oo}^{\oo} |\Ga_1|^2 d\ka\b)^{1/2}\b(\int_{-\oo}^{\oo}|\om_{\rho}|^2 d\ka\b)^{1/2} \rho d\rho\\\no
&\leq&\f{c}{z_1}\int_{r/4}^{4r}\f{1}{\sqrt{|r-\rho|}} \f{1}{\sqrt{\rho}}\rho d\rho\leq \f{c r}{z_1},\\\no
|T_8|%&\leq& \f{c}{z_1^2}\int_{r/4}^{4r}\int_{-\oo}^{\oo} |\Ga_1||u_{\phi}| \rho d\rho d\ka\\\no
&\leq&\f{c}{z_1^2}\int_{r/4}^{4r} \b(\int_{-\oo}^{\oo} |\Ga_1|^2 d\ka\b)^{1/2}\b(\int_{-\oo}^{\oo}|u_{\phi}|^2 d\ka\b)^{1/2} \rho d\rho\\\no
&\leq&\f{c}{z_1^2}\int_{r/4}^{4r}\f{1}{\sqrt{|r-\rho|}} \rho d\rho\leq \f{c r^{3/2}}{z_1^2}.
\ee
We choose $z_1=r^2$, then combining all the above estimates, we finally obtain (\ref{c41}).
%{\bf Estimates of $u_z(r,z)$.}
%&\q&\int_A |\Ga_2(r,\rho,z-\ka)| d\rho d\ka \leq \f{c}{r} \int_0^1\int_0^{z_1}\f{ds dt}{\sqrt{s^2+t^2}} \\\no
%&\leq&\f{c}{r}\int_0^1 dt\b(\int_0^1 \f{d\th}{\sqrt{1+\th^2}}+ \int_1^{\f{z_1}{t}} \f{d\th}{\th}\b) \\\no
%&\leq&\f{c}{r}\b(c+\int_0^1(\log z_1-\log t) dt\b)\leq \f{c+c\log z_1}{r},\\\no
%&\q&\int_B |\Ga_2(r,\rho,z-\ka)| d\rho d\ka \leq \f{c}{r} \int_1^{5r} \int_0^{z_1} \f{ds dt}{t^2+s^2} \\\no
%&\leq& \f{c}{r} \int_1^{5r} \f{d t}{t} \int_0^{z_1/t} \f{d\th}{1+\th^2}\leq \f{c\log r}{r}.
%&\q&\b|\int \f{\p\psi}{\p\rho} u_z\Ga_2 \rho d\rho d\ka\b|\\\no
%&\leq& c\int_{-z_1}^{z_1} \int_{r/4}^{r/2} |u_z||\Ga_2| d\rho d\ka + c\int_{-z_1}^{z_1} \int_{2r}^{4r} |u_z||\Ga_2| d\rho d\ka\co a)+b).
%a)&=&\f{r}{4}\int_{-z_1}^{z_1} |u_z(r_1,\ka)||\Ga_2(r,r_1,z-\ka)| d\ka\\\no
%&\leq&\f{r}{4}\b(\int_{-\oo}^{\oo}|u_z(r_1,\ka)|^4 d\ka\b)^{1/4} \b(\int_{-\oo}^{\oo}|\Ga_2(r,r_1,z-\ka)|^{4/3}d\ka\b)^{3/4}\\\no
%&\leq&\f{r}{4}\f{c}{r^{1/4}}\f{c}{r}\b(\int_{-\oo}^{\oo}\f{ds}{(t^2+s^2)^{2/3}}ds\b)^{3/4} \\\no
\epf
\bpf[Proof of Theorem \ref{main1}.]
We have proved (\ref{main11}) in Lemma \ref{cl41}. By (\ref{cj1}) and (\ref{main11}), we see that (\ref{omega-theta1}) holds for $\de=(\f 12)^-$, hence by (\ref{omega-theta8}), we have (\ref{main12}). Applying Lemma \ref{grad-u-rz}, we can take $\de_1=(\f 12)^-$ and $\de_2=1^-$, hence by (\ref{grad-u-rz5}), we obtain (\ref{main13}). Then we use Lemma \ref{omega-rz}, where we can take $\de=(\f 12)^-, \ga=(\f 18)^-$, (\ref{omega-rz5}) implies (\ref{main14}). Since $\na u_{\th}$ can be expressed as singular integral operators of $\om_r$ and $\om_{z}$, we can use $A_p$ weight to derive same weighted energy estimates of $\na u_{\th}$ from (\ref{omega-rz2})-(\ref{omega-rz4}). Then by Lemma \ref{rdecay}, we obtain (\ref{main15}).
\epf
%Take a cut-off function $\eta\in C_0^{\oo}(\mbR^3), \eta=\eta(\rho)$, $\rho=\sqrt{r^2+z^2}$, satisfying $0\leq \eta\leq 1$, $\eta(\rho)=1$ on $2\rho_1\leq \rho\leq \rho_2$, $\eta(\rho)=0$ for $\rho\leq \rho_1$ or $\rho\geq 2\rho_2$, so that
%|\na\eta| \begin{cases}
%\leq \f{c}{\rho_1}\q &\text{on}\q \rho_1<\rho< 2\rho_1,\\
%\leq \f{c}{\rho_2}\q &\text{on}\q \rho_2<\rho< 2\rho_2,\\
%Then $\rho |\eta'(\rho)|\leq c$ for $\forall \rho>0$.
%(u_r\p_r+ u_z\p_z)\Om -\p_z\b(\f{u_{\th}^2} {r^2}\b)=\b(\p_r^2+\f{1}{r}\p_r+\p_z^2+\f{2}{r}\p_r\b)\Om.
%LHS&=&\int_{\mbR^3} r^a \Om \eta^2 \b[(u_r\p_r+u_z\p_z)\Om -\p_z\b(\f{u_{\th}^2} {r^2}\b)\b] dx \\\no
%&=&-\pi\int_{-\oo}^{\oo}\int_0^{\oo} r^a 2\eta \eta' \f{r u_r+ z u_z}{\sqrt{r^2+z^2}} \Om^2 r dr dz - \pi\int_{-\oo}^{\oo}\int_0^{\oo} a r^{a-1} \eta^2 \Om^2 u_r r drdz\\\no
%&\quad& - 2\int_{\mbR^3}\eta^2 r^{a-2} u_{\th} \p_z u_{\th}\Om dx\\\no
%&\co& I_1 +I_2 +I_3.
%RHS &=& \int_{\mbR^3} r^a \Om \eta^2 (\p_r^2+\p_z^2+\f{3}{r}\p_r)\Om dx \\\no
%&=&- 2\pi \int_{-\oo}^{\oo}\int_0^{\oo} r^a |\na \Om|^2 \eta^2 rdr dz- 2\pi\int_{-\oo}^{\oo}\int_0^{\oo} 2r^a\Om \eta\eta' \f{r\p_r\Om+ z\p_z\Om}{\sqrt{r^2+z^2}} rdr dz\\\no
%&\quad&+\pi(a-2) \int_{-\oo}^{\oo}\int_0^{\oo} a r^{a-2} \Om^2 \eta^2 r dr dz+ \pi(a-2)\int_{-\oo}^{\oo}\int_0^{\oo}2 r^{a-1}\eta \eta' \Om^2 \f{r}{\sqrt{r^2+z^2}} r dr dz\\\no
%&\co& J_1 +J_2 + J_3 + J_4.
%Then for $a=3$,
%|I_1|+|I_2|&\leq&\int_{\mbR^3} (|u_r|+|u_z|) r^{a-3} r^2\Om^2 dx \leq \|(u_r,u_z)r^{a-3}\|_{L^{\oo}} \|r\Om\|_{L^2}^2,\\\no
%|I_3|&\leq& \int_{\mbR^3} \eta r^{a-3}|u_{\th}| |\p_z u_{\th}| |r\Om| dx \leq C\|r^{a-3}u_{\th}\|_{L^{\oo}} \|\p_z u_{\th}\|_{L^2} \|r\Om\|_{L^2}.
%|J_2|&\leq& 2\pi\|(u_r,u_z)\|_{L^{\oo}}\int_{-\oo}^{\oo}\int_0^{\oo} |r\eta'| |r^{a/2-1}\Om| (r^{a/2}\eta |\na\Om|) rdr dz \\\no
%&\leq& \e \|\eta r^{a/2}\na \Om\|_{L^2}^2 + C\|r^{a/2-1}\Om\|_{L^2}^2,\q a\leq 4,\\\no
%|J_3|+|J_4|&\leq& \|r^{a/2-1}\Om\|_{L^2}^2,\q \text{if}\q a\leq 4.
%Hence if we have $\|r^{\de}(|u_r|+|u_{\th}|+|u_z|)\|_{L^{\oo}}\leq C$ for some $\de\in (0,1]$, then one can take $a=3+\de$, such that
%\int_{\mbR^3} r^2 \Om^2 dx \leq C(M),\q \int_{\mbR^3} r^{3+\de}|\na \Om|^2 dx \leq C(M).
%The equation for $\p_z\Om$:
%\p_z\b((u_r\p_r+ u_z\p_r)\Om -\p_z\b(\f{u_{\th}^2} {r^2}\b)\b)=(\p_r^2+\p_z^2+\f{3}{r}\p_r)\p_z\Om.
%LHS&=&-\int_{\mbR^3}\b[(u_r\p_r+ u_z\p_z)\Om -\p_z\b(\f{u_{\th}^2} {r^2}\b)\b]\p_z (\eta^2 r^b\p_z \Om) dx \\\no
%&=&-\int_{\mbR^3}(u_r\p_r+ u_z\p_z)\Om \eta^2 r^b \p_z^2\Om dx-2\int_{\mbR^3} (u_r\p_r+ u_z\p_z)\Om r^b\eta \eta' \f{z\p_z\Om}{\sqrt{r^2+z^2}} dx\\\no
%&\quad&+ \int_{\mbR^3} 2r^{b-2} u_{\th}\p_z u_{\th} \eta^2\p_z^2\Om dx+ 4 \int_{\mbR^3} r^{b-2} u_{\th}\p_z u_{\th}\eta\eta' \f{z\p_z\Om}{\sqrt{r^2+z^2}} dx\\\no
%&\co& I_1+ I_2 + I_3 +I_4.
%RHS&=& \int_{\mbR^3} r^b \eta^2 \p_z\Om (\p_r^2+\p_z^2+\f{3}{r}\p_r)\p_z\Om dx \\\no
%&=&-\int_{\mbR^3} r^b \eta^2|\nabla \p_z\Om|^2 dx - \int_{\mbR^3} 2 r^b \eta\eta' \p_z\Om \f{r\p_{rz}^2\Om+ z\p_z^2\Om}{\sqrt{r^2+z^2}} dx \\\no
%&\quad&+(b-2)b\pi \int_{-\oo}^{\oo}\int_0^{\oo} r^{b-2} \eta^2 (\p_z\Om)^2 r dr dz + 2\pi(b-2)\int_{-\oo}^{\oo}\int_0^{\oo} r^{b-1}\eta\eta' (\p_z\Om)^2 r dr dz\\\no
%&\co& J_1 +J_2 +J_3 +J_4
%|J_2|&\leq& \int_{\mbR^3}|r\eta'| |r^{b/2-1}\p_z\Om| (\eta r^{b/2}|\na\p_z\Om|) dx \\\no
%&\leq&\e \|\eta r^{b/2}|\na\p_z\Om|\|_{L^2}^2+ C\|r^{b/2-1}\p_z\Om\|_{L^2}^2,\q b\leq 2+a= 5+\de\\\no
%|J_3|&\leq& \|r^{b/2-1}|\na\Om|\|_{L^2}^2,\q |J_4|\leq \|r^{b/2-1}|\na\Om|\|_{L^2}^2,\q b\leq 2+a= 5+\de.
%|I_1| &\leq& \int_{\mbR^3}\eta r^{b/2-a/2}(|u_r|+|u_z|) r^{a/2}|\na\Om|\cdot(\eta r^{b/2}|\p_z^2\Om|) dx\\\no
% &\leq& C\|r^{b/2-a/2}(u_r, u_z)\|_{L^{\oo}} \|r^{a/2}\na\Om\|_{L^2} \|r^{b/2}\eta \na\p_z\Om\|_{L^2},\q b\leq a+2\de=3+3\de,\\\no
%|I_2| &\leq& 2\int_{\mbR^3} |r\eta'| \eta r^{b-a-1}(|u_r|+|u_z|) r^a |\na\Om|^2 dx \\\no
%&\leq&C\|r^{b-a-1}(|u_r|+|u_z|)\|_{L^{\oo}} \|r^{a/2}|\na\Om|\|_{L^2}^2, \q b\leq a+1+\de=4+2\de,\\\no
%|I_3| &\leq& \int_{\mbR^3} \eta r^{b/2-2} |u_{\th}| |\p_z u_{\th}||\eta r^{b/2}\p_z^2\Om| dx\\\no
%&\leq&C\|r^{b/2-2}u_{\th}\|_{L^{\oo}}\|\na u_{\th}\|_{L^2} \|\eta r^{b/2}\na\p_z\Om\|_{L^2}, \q b\leq 4+2\de,\\\no
%|I_4| &\leq& \int_{\mbR^3}|r\eta'| r^{b-a/2-3} |u_{\th}| |\p_z u_{\th}| |r^{a/2}\p_z\Om| dx \\\no
%&\leq& C\|r^{b-a/2-3} u_{\th}\|_{L^{\oo}}\|\p_z u_{\th}\|_{L^2} \|r^{a/2}\p_z\Om\|_{L^2},\q b\leq\f 92+\f 32 \de.
%We have
%\int_{\mbR^3}r^{3+3\de}|\nabla\p_z\Om|^2 dx \leq C(M).
%By Lemma \ref{rdecay}, one can conclude that
%|\om_{\th}(r,z)|\leq C r^{-\f 78-\f 58\de}.
\section{Proof of Theorem \ref{main2}.}\lab{mainsection2}
\bpf[Proof of Theorem \ref{main2}.]
{\bf Step 1.} We have the following weighted estimates for $\Om\co \f{\om_{\th}}{r}$.
{\bf Claim 1.} {\it Suppose that
\be\lab{urz100}
|u_r(r,z)|+ |u_z(r,z)|\leq C (1+\rho)^{-\tau},\q \rho=\sqrt{r^2+z^2}
\ee
for some $\tau\in [0,1]$, then we have
\be\lab{Omega1}
&&\int_{\mbR^3} \rho^{1+\tau} |\nabla\Om(r,z)|^2 rdr dz <\oo,\\\lab{Omega2}
&&\int_{\mbR^3} \rho^{1+3\tau} |\nabla\p_z\Om(r,z)|^2 rdr dz <\oo.
\ee
To prove {\bf Claim 1.}, we see that for the smooth axially symmetric flows with no swirl, $\Om$ satisfies the following equation
\be\lab{Omega-no swirl}
(u_r\p_r+ u_z\p_z)\Om = (\p_r^2+\f{3}{r}\p_r+ \p_z^2)\Om+\f{1}{r}(\p_z f_r- \p_r f_z).
\ee
Choose a cut-off function $\phi\in C_0^{\oo}(\mbR^3)$, $\phi=\phi(\rho)$, satisfying $0\leq \phi\leq 1$, $\phi(\rho)=1$ on $2\rho_0\leq\rho \leq \rho_1$, $\phi(\rho)=0$ on $\rho\leq \rho_0$ or $\rho\geq 2\rho_1$, such that $|\na \phi|\leq \f{C}{\rho_0}$ on $\rho_0<\rho<2\rho_0$ and $|\na \phi|\leq \f{C}{\rho_1}$ on $\rho_1<\rho< 2\rho_1$, and $\na \phi=0$ elsewhere.
Multiplying (\ref{Omega-no swirl}) by $\phi^2 \rho^{d_1} \Om$ and integrating over $\mbR^3$, then we get
\be\no
0&=& \int \phi^2 \rho^{d_1} |\na\Om|^2 dx -\f12\int \p_r(\phi^2 \rho^{d_1}) |\Om|^2 dx-\f12 \int \p_z^2(\phi^2 \rho^{d_1}) |\Om|^2 dx \\\no
&=& 3\int \phi\phi' \rho^{d_1-1} |\Om|^2 dx -\int \phi^2 \rho^{d_1}\Om \f1r(\p_z f_r-\p_r f_z) dx-\f12 \int_{\mbR^3} \b[u_r\p_r(\phi^2 \rho^{d_1})+ u_z\p_z(\phi^2 \rho^{d_1})\b] |\Om|^2 dx\\\no
&=&\int \phi^2 \rho^{d_1} |\na\Om|^2 dx + \sum_{i=1}^3 K_{1i} + L_{11}.
\ee
We estimate these terms as follows.
\be\no
\sum_{i=1}^3 |K_{1i}|&\leq& C\int_{\rho\geq \rho_0} \rho^{d_1-2} |\Om|^2 dx\leq \|\Om\|_{L^2}^2,\q\text{if }d_1\leq 2,\\\no
|L_{11}|&\leq& \int \phi |(u_r,u_z)|\rho^{d_1-1} |\Om|^2 dx\leq \|\Om\|_{L^2}^2,\q\text{if }d_1\leq 1+\tau.
\ee
This yields \eqref{Omega1} by letting $\rho_1\to \oo$.
%an integral identity with left and right hand sides as
%&\q&\int_{\mbR^3} (z^2+1)^{\f{d_1}{2}}\eta^2 \Om (u_r\p_r + u_z\p_z)\Om dx =\int_{\mbR^3} (z^2+1)^{\f{d_1}{2}}\eta^2 \Om (\p_r^2+\f{3}{r}\p_r+ \p_z^2)\Om dx\\\no
%&+& \int_{\mbR^3} (z^2+1)^{\f{d_1}{2}}\eta^2 \Om \f{1}{r}(\p_z f_r-\p_r f_z) dx.
%LHS &=& -2\pi \int_{-\oo}^{\oo}\int_0^{\oo} (z^2+1)^{\f{d_1}{2}} \eta \eta' \f{r u_r + z u_z}{\sqrt{r^2+z^2}} \Om^2 r dr dz\\\no
%&\quad&-\pi \int_{-\oo}^{\oo}\int_0^{\oo}d_1 (z^2+1)^{\f{d_1}{2}-1} z \eta^2 u_z \Om^2 rdr dz \co E_1 +E_2,\\\no
%RHS&=&-2\pi \int_{-\oo}^{\oo}\int_0^{\oo} (z^2+1)^{\f{d_1}{2}} \eta^2|\nabla \Om|^2 rdr dz-4\pi\int_{-\oo}^{\oo}\int_0^{\oo} (z^2+1)^{\f{d_1}{2}} \eta\eta'\f{r\p_r\Om + z\p_z\Om}{\sqrt{r^2+z^2}} \Om r dr dz\\\no
%&\quad&- 2\pi \int_{-\oo}^{\oo}\int_0^{\oo}d_1 (z^2+1)^{\f{d_1}{2}-1} z \eta^2 \Om \p_z\Om r dr dz-2\pi \int_{-\oo}^{\oo}\int_0^{\oo} (z^2+1)^{\f{d_1}{2}} 2\eta\eta' \f{r}{\sqrt{r^2+z^2}} \Om^2 dr dz \\\no
%&\q&+ \int_{\mbR^3} (z^2+1)^{\f{d_1}{2}}\eta^2 \Om \f{1}{r}(\p_z f_r-\p_r f_z) dx \co \sum_{k=1}^5 F_k.
%Take $d_1=1$, since $|\sqrt{z^2+1}\phi'|\leq C$, then
%|E_1|&\leq& C\|\Om\|_{L^2(\mbR^3)}^2,\q\q |E_2|\leq C\|u_z\|_{L^{\oo}(\mbR^3)} \|\Om\|_{L^2(\mbR^3)}^2,\\\no
%\sum_{k=2}^3|F_k| &\leq& C\|\Om\|_{L^2(\mbR^3)}\|\nabla\Om\|_{L^2(\mbR^3)},\\\no
%|F_4| &\leq& C\int_{-\oo}^{\oo}\int_1^{\oo} \Om^2(r,z) dr dz\leq \|\Om\|_{L^2(\mbR^3)}^2,\\\no
%|F_5| &\leq& C\|\Om\|_{L^2(\mbR^3)}\|(|z|+1)\f{\p_z f_r-\p_r f_z}{r}\|_{L^2(\mbR^3)}.
%Finally, we obtain (\ref{Omega1}) by letting $\rho_0\to \oo$.
To derive the estimate (\ref{Omega2}), we use the equation for $\p_z\Om$.
\be\lab{partialz-Omega}
\p_z\b((u_r\p_r+u_z\p_z) \Om\b) =(\p_r^2+\f{3}{r}\p_r + \p_z^2) \p_z\Om+\f{1}{r}\p_z(\p_z f_r- \p_r f_z).
\ee
Multiplying (\ref{partialz-Omega}) by $\phi^2 \rho^{d_2} \p_z\Om$ and integrating over $\mbR^3$, then we get
\be\no
0&=& \int \phi^2 \rho^{d_1}|\na\p_z\Om|^2 dx -\f12 \int [\p_r^2(\phi^2 \rho^{d_2})+\p_z^2(\phi^2 \rho^{d_2})] |\p_z\Om|^2 dx+\f{3}{2}\int r^{-1}\p_r(\phi^2) \rho^{d_2} |\p_z\Om|^2 dx\\\no
&\q&+ \int \phi^2 \rho^{d_2}\b(\p_r^2\Om+\f{\p_r\Om}{r}\b)\b[\f1r(\p_z f_r-\p_r f_z)\b] dx + \int \p_r(\phi^2 \rho^{d_2})\p_r\Om \b[\f1r(\p_z f_r-\p_r f_z)\b] dx\\\no
&\q&-\int \phi^2 \rho^{d_2}\b(\p_r^2\Om+\f{\p_r\Om}{r}\b)(u_r\p_r+u_z\p_z)\Om dx - \int \p_r(\phi^2 \rho^{d_2})\p_r\Om (u_r\p_r+ u_z\p_z)\Om dx\\\no
&=&\int \phi^2 \rho^{d_1}|\na\p_z\Om|^2 dx + \sum_{i=1}^5 K_{2i} + \sum_{j=1}^2 L_{2j}.
\ee
These terms can be bounded as follows.
\be\no
\sum_{i=1}^3 |K_{2i}|&\leq& \int_{\rho\geq \rho_0} \rho^{d_2-2} |\na \Om|^2 dx<\oo,\q\text{if } d_2\leq 3+\tau,\\\no
|K_{24}|&\leq&\f18 \int \phi^2 \rho^{d_2}|\na \p_z\Om|^2 dx + C\int \phi^2 \rho^{d_2}\b[\f1r(\p_z f_r-\p_r f_z)\b]^2 dx,\\\no
|K_{25}|&\leq& \int_{\rho\geq \rho_0} \rho^{d_2-2} |\na\Om|^2 dx + C\int \phi^2 \rho^{d_2}\b[\f1r(\p_z f_r-\p_r f_z)\b]^2 dx,\q \text{if }d_2\leq 3+\tau,\\\no
|L_{21}|&\leq& \f18 \int \phi^2 \rho^{d_2}|\na \p_z\Om|^2 dx + C\int \phi^2 \rho^{d_2}|(u_r,u_z)|^2 |\na\Om|^2 dx,\q \text{if } d_2\leq 1+3\tau,\\\no
|L_{22}|&\leq& \int_{\rho\geq \rho_0} \rho^{d_2-1} |(u_r, u_z)||\na \Om|^2 dx<\oo,\q \text{if }d_2\leq 2(1+\tau).
\ee
From the above estimates, we derive \eqref{Omega2} by letting $\rho_1\to \oo$.
%an integral identity with left and right hand sides as
%&\q&\int_{\mbR^3} (z^2+1)^{\f{d_2}{2}}\eta^2 \p_z\Om \p_z\b((u_r\p_r+u_z\p_z) \Om\b) dx = \int_{\mbR^3} (z^2+1)^{\f{d_2}{2}}\eta^2 \p_z\Om (\p_r^2+\f{3}{r}\p_r+\p_z^2)\p_z\Om dx\\\no
%&+&\int_{\mbR^3}(z^2+1)^{\f{d_2}{2}}\eta^2 \p_z\Om \f{1}{r}\p_z(\p_z f_r- \p_r f_z)dx.
%LHS%&=& -\int_{\mbR^3}(u_r\p_r+u_z\p_z)\Om \p_z\b((z^2+1)^{\f{d_2}{2}}\eta^2 \p_z\Om\b) dx \\\no
%&=&-\int_{\mbR^3}(u_r\p_r+u_z\p_z)\Om (z^2+1)^{\f{d_2}{2}}\eta^2 \p_z^2\Om dx-\int_{\mbR^3}(u_r\p_r+u_z\p_z)\Om d_2(z^2+1)^{\f{d_2}{2}}\eta\eta'\f{z}{\sqrt{r^2+z^2}}\p_z\Om dx \\\no
%&\quad& -\int_{\mbR^3}(u_r\p_r+u_z\p_z)\Om d_2(z^2+1)^{\f{d_2}{2}-1}z\eta^2 \p_z\Om dx \co E_1' +E_2' +E_3',\\\no
%RHS&=&-2\pi \int_{-\oo}^{\oo}\int_0^{\oo} (z^2+1)^{\f{d_2}{2}} \eta^2|\nabla\p_z \Om|^2 rdr dz-4\pi\int_{-\oo}^{\oo}\int_0^{\oo} (z^2+1)^{\f{d_2}{2}} \eta\eta'\f{r\p_{rz}^2\Om + z\p_z^2\Om}{\sqrt{r^2+z^2}} \p_z\Om r dr %dz\\\no
%&\quad&- 2\pi \int_{-\oo}^{\oo}\int_0^{\oo}d_2 (z^2+1)^{\f{d_2}{2}-1} z \eta^2 \p_z\Om \p_z^2\Om r dr dz -2\pi \int_{-\oo}^{\oo}\int_0^{\oo} (z^2+1)^{\f{d_2}{2}} 2\eta\eta' \f{r}{\sqrt{r^2+z^2}} (\p_z\Om)^2 dr dz \\\no
%&\q& -\int_{\mbR^3}(z^2+1)^{\f{d_2}{2}}\eta^2 \p_z^2\Om \f{1}{r}(\p_z f_r- \p_r f_z)dx- 2\int_{\mbR^3}(z^2+1)^{\f{d_2}{2}}\eta \eta'\f{z}{\sqrt{r^2+z^2}} \p_z\Om \f{1}{r}(\p_z f_r- \p_r f_z)dx\\\no
%&\co& \sum_{k=1}^6 F_k'.
%Then take $d_2=1$, we get
%|E_1'|&\leq&\|(u_r,u_z)\|_{L^{\oo}} \|(z^2+1)^{\f14} \nabla \Om\|_{L^2} \|\eta (z^2+1)^{\f 14} |\nabla\p_z\Om|\|_{L^2},\\\no
%|E_2'|+|E_3'|&\leq&\|(u_r,u_z)\|_{L^{\oo}} \|\nabla \Om\|_{L^2}^2,\\\no
%|F_2'|+|F_3'|&\leq& C\|\nabla\Om\|_{L^2}\|\nabla\p_z\Om\|_{L^2},\\\no
%|F_4'| &\leq& \int_{-\oo}^{\oo}\int_1^{\oo} |\p_z\Om|^2 dr dz\leq \|\nabla \Om\|_{L^2}^2,\\\no
%|F_5'| &\leq& \e \|(|z|^2+1)^{\f 14}\eta\na\p_z\Om\|_{L^2}^2+ C(\e)\|(|z|+1)^{\f 12}\f{\p_z f_r-\p_r f_z}{r}\|_{L^2(\mbR^3)}^2,\\\no
%|F_6'| &\leq& C\|\na\Om\|_{L^2(\mbR^3)}\|\f{\p_z f_r-\p_r f_z}{r}\|_{L^2(\mbR^3)}.
%Hence (\ref{Omega2}) is derived by letting $\rho_0\to \oo$.
%Lemma \ref{zdecay} implies that
%|\Om(r,z)|^2= O(|z|^{-\f34} r^{-1}).
{\it Step 2. Now we can derive the decay rate for $\om_{\th}$.}
{\bf Claim 2.} {\it Suppose that \eqref{urz100} holds, we can infer that
\be\lab{omega1}
|\om(r,z)|\leq C(1+\rho)^{-(\f{5}{16}+\f{1}{2}\tau)^-}.
\ee}
Now we prove {\bf Claim 2.} Combining the results in Lemma \ref{omega-theta} and (\ref{Omega1})-(\ref{Omega2}), then
\be\lab{Omega3}
&&\int_{\mbR^3} r^2 |\Om(r,z)|^2 dx<\oo,\\\lab{Omega4}
&&\int_{\mbR^3} (r^{3+\de}+|z|^{1+\tau}) |\na\Om(r,z)|^2 dx<\oo,\\\lab{Omega5}
&&\int_{\mbR^3} (r^{3+3\de}+|z|^{1+3\tau}) |\na\p_z\Om(r,z)|^2 dx<\oo,
\ee
where $\de$ can be any constant less than $\f 12$. Fix $d>1$, then for each $n\in \mb{N}$,
\be\no
\int_{2^n}^{2^{n+1}} \int_d^{\oo} r^2 |\Om(r,z)|^2 rdr dz<\oo.
\ee
By mean value theorem, there exists $z_n\in [2^n, 2^{n+1}]$ such that
\be\no
\int_d^{\oo} r^2 |\Om(r,z_n)|^2 rdr \leq \f{C}{z_n}.
\ee
Then for any $z$, choose $z_n>z$ and
\be\no
\int_d^{\oo} |\Om(r,z)|^2 r dr &=&\int_d^{\oo} |\Om(r,z_n)|^2 r dr- 2\int_d^{\oo} \int_z^{z_n}\Om(r,t)\p_t \Om(r,t) r dr dt\co I_1 +I_2,\\\no
|I_2|&\leq&\b(\int_d^{\oo}\int_z^{z_n} |\Om(r,t)|^2 r drdt\b)^{1/2}\b(\int_d^{\oo}\int_z^{z_n} |\p_t\Om(r,t)|^2 r drdt\b)^{1/2}\leq \f{C}{d |z|^{\f12(1+\tau)}}.
\ee
Letting $z_n\to \oo$, then $I_1\to 0$ and
\be\lab{Omega6}
\int_d^{\oo}|\Om(r,z)|^2 rdr\leq \f{C}{d|z|^{\f12(1+\tau)}}.
\ee
Similarly, one can find $z_n\in [2^n, 2^{n+1}]$ such that
\be\no
\int_d^{\oo} |\na\Om(r,z_n)|^2 rdr &\leq& \f{C}{z_n^2},\\\no
\int_d^{\oo} |\na\Om(r,z)|^2 r dr&=&\int_d^{\oo} |\na\Om(r,z_n)|^2 r dr- 2\int_d^{\oo}\int_z^{z_n} \na\Om(r,t)\cdot \p_t\na\Om(r,t) r dr dt\co J_1+J_2,\\\no
|J_2|&\leq&\b(\int_d^{\oo}\int_z^{z_n} |\na\Om(r,t)|^2 r drdt\b)^{1/2}\b(\int_d^{\oo}\int_z^{z_n} |\p_t\na\Om(r,t)|^2 r drdt\b)^{1/2}\\\no
\f{C}{d^{3+2\de}},\\
\f{C}{|z|^{1+2\tau}}.
\end{cases}
\ee
Letting $n\to \oo$, $J_1\to 0$. Take $\de=(\f 12)^-$ and $J_2\leq \min\{\f{C}{d^{3+2\de}}, \f{C}{|z|^{1+2\tau}}\}$
\be\no
\int_d^{\oo} |\na\Om(r,z)|^2 r dr\leq \b(\f{C}{d^{4^-}}\b)^{\f14} \b(\f{C}{|z|^{1+2\tau}}\b)^{\f{3}{4}}\leq \f{C}{d |z|^{(\f 34(1+2\tau))^-}}.
\ee
\be\no
|\Om(d,z)|^2 &=&\f{1}{r_1-d}\int_d^{r_1} |\Om(r,z)|^2 dr+ (|\Om(r,z)|^2-\f{1}{r_1-d}\int_d^{r_1}|\Om(r,z)|^2 dr)\co H_1 +H_2,\\\no
|H_2|&=&\b||\Omega(d,z)|^2-|\Om(d_*,z)|^2\b| \leq 2\int_d^{r_1} |\Om(r,z)\p_r\Om(r,z)|dr\\\no
&\leq&\f{C}{d}\b(\int_d^{\oo}|\Om(r,z)|^2 rdr\b)^{1/2}\b(\int_d^{\oo}|\na \Om(r,z)|^2 r dr\b)^{1/2}\\\no
&\leq&\f{C}{d}\b(\f{C}{d|z|^{\f12(1+\tau)}}\b)^{1/2}\b(\f{C}{d |z|^{(\f 34(1+2\tau))^-}}\b)^{\f 12}\leq \f{C}{d^2|z|^{(\f58+\tau)^-}},
\ee
which implies that
\be\lab{omega-z}
|\om(d,z)|\leq \f{C}{|z|^{(\f5{16}+\f12\tau)^-}}.
\ee
Together with Theorem \ref{main1}, we have
\be\lab{omega-decay}
|\om(r,z)|\leq \f{C}{\rho^{(\f5{16}+\f12\tau)^-}},\q \forall (r,z)\in \mbR_+\times \mbR, \rho=\sqrt{r^2+z^2}.
\ee
{\bf Step 3.} Now we derive the new decay rate of ${\bf u}$. Fix any ${\bf x}\in\mbR^3\setminus\{0\}$, define a cut-off function $\psi\in C_0^{\oo}(\mbR^3)$ satisfying $\psi({\bf y})\equiv 1$ for $\forall {\bf y}\in B_{\rho/4}({\bf x})$ and $\psi({\bf y})\equiv 0$ for $\forall {\bf y}\not\in B_{\rho/2}({\bf x})$, where $\rho=|{\bf x}|$. One can require that $|\na\psi({\bf y})|\leq \f{C}{|y|}, |\na^2 \psi({\bf y})|\leq \f{C}{|y|^2}$ for $\forall {\bf y}\in D\co B_{\rho/2}({\bf x})\setminus B_{\rho/4}({\bf x})$. Setting ${\bf u}({\bf x})= u_r {\bf e}_r +u_z {\bf e}_z$, since $\text{curl }{\bf v}= \om_{\th} {\bf e}_{\th}$, then
\be\lab{v-omega}\begin{array}{ll}
{\bf v}({\bf x}) &= -\int_{\mbR^3} \na_{{\bf y}}\Ga({\bf x},{\bf y}) \times (\om_{\phi}({\bf y})\psi({\bf y}){\bf e}_{\phi}) d {\bf y} -\int_{\mbR^3} \Ga({\bf x},{\bf y})((\na_{{\bf y}}\psi({\bf y})\times {\bf e}_{\phi}))\om_{\phi}({\bf y}) d {\bf y}\\
&\quad+ \int_{\mbR^3} \Ga({\bf x},{\bf y})(\Delta_{{\bf y}}\psi)({\bf y}) {\bf v}({\bf y}) d{\bf y} + 2 \int_{\mbR^3} (\na_{{\bf y}}\Ga)({\bf x},{\bf y})\cdot (\na_{{\bf y}}\psi)({\bf y}) {\bf v}({\bf y}) d {\bf y}\\
&\co K_1 +K_2 +K_3 +K_4.
\end{array}\ee
We estimate $K_i, i=2,3,4$ as follows.
\be\no
|K_2|&\leq& \f{C}{\rho}\b(\int_D |\Ga({\bf x}-{\bf y})|^2 d{\bf y}\b)^{1/2}\b(\int_D |\om_{\phi}({\bf y})|^2 d {\bf y}\b)^{1/2}\leq \f{C}{\rho^{1/2}},\\\no
|K_3|&\leq& \f{C}{\rho^2}\b(\int_D |\Ga({\bf x}-{\bf y})|^{\f 65} d{\bf y}\b)^{\f 56}\b(\int_D |{\bf v}({\bf y})|^6 d {\bf y}\b)^{\f 16}\leq \f{C}{\rho^{1/2}},\\\no
|K_4|&\leq& \f{C}{\rho}\b(\int_D |\na \Ga({\bf x}-{\bf y})|^{\f 65} d{\bf y}\b)^{\f 56}\b(\int_D |{\bf v}({\bf y})|^6 d {\bf y}\b)^{\f 16}\leq \f{C}{\rho^{1/2}}.
\ee
For the estimate of $K_1$, fix a $d\in (0,\f{\rho}{2})$, which will be determined later, then
\be\no
|K_1|&\leq& \sup_{{\bf y}\in B_d({\bf x})}|\om_{\phi}({\bf y})|\int_{B_d({\bf x})}|\na\Ga({\bf x}-{\bf y})| d{\bf y}\\\no
&\q&+ \b(\int_{B_{\rho/2}({\bf x})\setminus B_{d}({\bf x})} |\na\Ga({\bf x}-{\bf y})|^2 d{\bf y}\b)^{\f 12} \b(\int_{B_{\rho/2}({\bf x})\setminus B_{d}({\bf x})}|\om_{\phi}({\bf y})|^2 d {\bf y}\b)^{\f 12}\\\no
&\leq& C\rho^{-(\f{5}{16}+\f12\tau)^-} d+ C d^{-\f 12}.
\ee
By choosing $d=\rho^{(\f{5}{24}+\f13\tau)^-}$, we obtain the optimal bound for $|K_1|\leq \f{C}{\rho^{(\f5{48}+\f16\tau)^-}}$. Hence we have
\be\lab{urz200}
|(u_r,u_z)(r,z)|\leq C (\rho+1)^{-(\f5{48}+\f16 \tau)^-}
\ee
{\bf Step 4. Iteration.} At the beginning, we have $\tau=0$ in \eqref{urz100}, then by using the arguments developed in {\bf Step 1} to {\bf Step 3}, we have a new $\tau$ in \eqref{urz100}, which will be denoted by $\tau_1=(\f{5}{48})^-$. Run a second iteration of these three steps, we get a new $\tau_2= \tau_1+ \f16 \tau_1$, and after $n$ iteration, we get
\be\no
\tau_n= \tau_{1}+ \f16 \tau_{n-1}= \tau_1 \sum_{i=0}^{n-1}\f1{6^i}.
\ee
Let $n\to \oo$, $\tau_n\to (\f18)^-$ as $n\to \oo$. In a word, we infer the following decay rates
\be\lab{urz300}
|(u_r,u_z)(r,z)|&\leq& C (\rho+1)^{-(\f18)^-},\\\lab{omgea300}
|\om(r,z)| &\leq& C(\rho+1)^{-(\f38)^-}.
\ee
\epf
\br\lab{gamma}
{\it Since there are no improved decay estimates of $\Ga_2,\Ga_3,\Ga_5$ in $w$ in (\ref{cl231})-(\ref{cl233}), it seems difficult to use the argument in Lemma \ref{cl41} to get better decay estimates of $u_{\th}$. So we use (\ref{v-omega}) directly. From the estimates of $K_2, K_3$ and $K_4$, one may also see the difficulties to improve the decay rates in Theorem \ref{main1}.
\er
%{\it The decay rates in Theorem \ref{main2} are not optimal. However, the estimate of $F_4$ prevents us from improving weighted estimates in the $Oz$ direction.
\br\lab{exterior2}
One can also extend Theorem \ref{main2} to the exterior domain case.
\er
{\bf Acknowledgement.} The author would like to thank Prof. Dongho Chae and Prof. Zhouping Xin for the stimulating discussions and constant encouragement and supports. Special thanks also go to the referee for the important suggestions and comments, which make this paper more readable.
%Weng's research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2014047764).
\begin{thebibliography}{00}
%C. J. Amick. {\it On Leray's problem of steady Navier-Stokes flow past a body in the plane.} Acta Math. 161 (1988), 71-130.
\bibitem{babenko73}
K. I. Babenko. {\it On the stationary solutions of the problem of flow past a body of a viscous incompressible fluid.} Math. Sbornik 91 (133) No. 1 (1973); English Transl.: Math. USSR Sbornik 20 1 (1973), 1-25.
\bibitem{cl02}
D. Chae, J. Lee. {\it On the regularity of the axisymmetric solutions of the Navier-Stokes equations.} Math. Z. 239 (2002), no. 4, 645--671.
\bibitem{chae14}
D. Chae. {\it Liouville-Type Theorem for the Forced Euler Equations and the Navier-Stokes Equations.} Commun. Math. Phys.326: 37-48 (2014).
\bibitem{cy13}
D. Chae, T. Yoneda. {\it On the Liouville theorem for the stationary Navier-Stokes equations in a critical space.} J. Math. Anal. Appl. 405 (2013), no. 2, 706--710.
\bibitem{cw15}
D. Chae, S. Weng. {\it Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations.} Submitted, August 2015.
\bibitem{dg00}
P. Deuring and G. P. Galdi. {\it On the asymptotic behavior of physically reasonable solutions to the stationary Navier-Stokes systme in three-dimensional exterior domains with zeri velocity at infinity.} J. Math. Fluid Mech. 2, no. 4 (2000), 353-364.
\bibitem{cj09}
H. Choe, B. Jin. {\it Asymptotic properties of axi-symmetric D-solutions of the Navier-Stokes equations.} J. Math. Fluid. Mech. 11 (2009), 208-232.
\bibitem{farwig98}
R. Farwig. {\it The stationary Navier-Stokes equations in a 3D exterior domain}, in: {\it Recent topics on mathematical theory of viscous incompressible fluid}, 53-115, Lecture Notes Numer. Appl. Anal. 16, Kinokyniya, Tokyo, 1998.
\bibitem{fs98}
R. Farwig and H. Sohr. {\it Weighted estimates for the Oseen equations and the Navier-Stokes equations in exterior domains}, in: {\it Theory of the Navier-Stokes equations, 11-30}, Ser. Adv. Math. Appl. Sci. 47, World Sci. Publ., RIver Edge, NJ, 1998.
\bibitem{finn59}
R. Finn. {\it On Steady-State Solutions of the Navier-Stokes Partial Differential Equations.} Arch. Rational Mech. Anal., Vol. 3, 1959, 381-396.
\bibitem{finn61}
R. Finn. {\it On the steady-state solutions of the Navier-Stokes equations. III,} Acta Math. 105 (1961), 197-244.
\bibitem{finn65}
R. Finn. {\it On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems.} Arch. Rational Mech. Anal. 19 (1965), 363-406.
\bibitem{fujita}
H. Fujita. {\it On the existence and regularity of the steady-state solutions of the Navier-Stokes theorem.} J. Fac. Sci. Univ. Tokyo Sect. I 9(1961), 59-102.
\bibitem{galdi11}
Giovanni P. Galdi. {\it An Introduction to the Mathematical Theory of the Navier-Stokes Equations. In: Steady State problems,} Springer Monographs in Mathematics, Second edition, 2011.
\bibitem{galdi92}
Giovanni P. Galdi. {\it On the Asymptotic Properties of Leary's Solutions to the Exterior Stationary THree-Dimensional Navier-Stokes Equations with Zero Velocity at Infinity.} Degenerate Diffusions, IMA Volumes in Mathematics and Its Applications Vol 47, Ni, W-M., Peletier, L. A. and Vasquez, J. L., Eds. Springer-Verlag, 95-103.
\bibitem{gw74}
D. Gilbarg, H. F. Weinberger. {\it Asymptotic properties of Leray's solutions of the stationary two-dimensional Navier-Stokes equations.} Uspehi Mat. Nauk 29 (1974), no. 2 (176), 109--122. English transl.: Russian Math. Surveys 29, No. 2 (1974), 109-123.
\bibitem{gw78}
D. Gilbarg, H. F. Weinberger. {\it Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral.} Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4) 5 (1978), no. 2, 381--404.
\bibitem{ks11}
A. Korolev, V. Sverak. {\it On the large-distance asymptotics of steady state solutions of the Navier-Stokes equations in 3D exterior domains.} Ann. I. H. Poincare- AN 28 (2011) 303-313.
\bibitem{kpr15}
M. Korobkov, K. Pileckas and R. Russo. {\it The Liouville Theorem for the Steady-State Navier-Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl.} J. Math. Fluid Mech. 17 (2015), 287-293.
\bibitem{lady69}
O. A. Ladyzhenskaya. {\it The mathematical theory of viscous incompressible fulid.} Gordon and Breach, 1969.
\bibitem{leray33}
J. Leray. {\it $\acute{E}$tude de diverses $\acute{e}quations$ $int\acute{e}grales$ non $lin\acute{e}aire$ et de quelques probl$\grave{e}$mes que pose l'hydrodynamique.} J. Math. Pures Appl. 12 (1933), 1-82.
\bibitem{np00}
S. A. Nazarov and K. I. Pileckas. {\it On steady Stokes and Navier-Stokes problems with zero velocity at infinity in a three-dimensional exterior doamin.} Kyoto Univ. Math. J. 40, no. 3 (2000), 475-492.
\bibitem{np95}
A. Novotny, M. Padula. {\it Note on decay of solutions of steady Navier-Stokes equations in 3-D exterior domains.} Differtial and Integral Equations, Vol. 8, no. 7, 1995, 1833-1842.
\bibitem{stein93}
E. M. Stein. {\it Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals.} Princeton University Press, Princeton, 1993.
\end{thebibliography}
\end{document}
|
1511.00697
|
§ INTRODUCTION
Relativistic scalar fields with strongly infrared (IR) initial
conditions have been studied for their possible role in post-inflation
cosmology <cit.>
and as an analogue theory to QCD
to study the problem of thermalization immediately after a heavy ion
collision. There are physical differences between IR-occupied scalars
and nonabelian gauge fields, most notably the role of much more
efficient particle-number changing processes in the gauge theory
<cit.>, which prevent the formation of infrared condensates
<cit.>. Nevertheless, the theory is of considerable
interest for its own merits.
Particularly interesting is the rich infrared physics which is present
in a scalar theory when the initial state features large occupancy of
relatively IR modes. This problem has been studied
extensively for scalar fields with $N$ components,
using the tools of classical (statistical) field theory, which should
be a good approximation in a regime where the occupancy is large.
They have found several scaling behaviors in different wave number
ranges, some of which have clear explanations in terms of kinetic or
2-particle irreducible descriptions <cit.>. Here we will
study in a little more detail the behavior of the most infrared modes, well
below the effective thermal oscillation frequency induced by mode-mode
interactions. Berges et al demonstrated three scaling
regimes <cit.>. Below a scale
$\kcond \propto t^{-1/2}$ the occupancy is very large and at most
weakly $k$-dependent
[Ref. <cit.> find an $f(k)\propto k^{-1/2}$
dependence in the deep infrared. We find $k^0$. At this
time we do not understand the origin of this discrepancy.];
then it falls
steeply, roughly as $k^{-5}$, until it softens to a $k^{-3/2}$
“energy-cascade” scaling. (Here $k$ is the wave number
Since we use a classical framework, we will talk about wave numbers
and avoid conflating them with momenta.]
and $f$ is a particle occupancy estimate
based on the $k$ Fourier component of the field and its time
derivative.) We will be interested in these two most-infrared
regions. The large occupancy in the deep IR can be
understood as a consequence of an approximate particle number
conservation. Thermodynamically, the excess particle number in the
initial conditions tries to organize into an IR condensate, but by
causality it cannot instantly fall into a single $k=0$-mode across a
large system.
In studying this IR occupancy, we will
concentrate on cases with “large” condensates, where the
particle number in the deep IR is comparable to the
particle number in all other modes, so the IR modes dominate the
effective thermal oscillation frequency.
The energy is $\varepsilon \sim \int d^3 k\; \omega_k f(k)$;
particle number is $n \sim \int d^3 k\; f(k)$, and contribution to
dispersion corrections is $\sim \int d^3 k\; \omega^{-1} f(k)$.
If an $\OO(1)$ fraction of particles are in the IR, they play a
small role in energy but they dominate dispersion corrections.]
We will refer to the large occupancy in very IR wave numbers
as a “local condensate,” since by local probes it appears to be a
condensate and it occurs for the same (particle-number storage)
reasons as a true condensate, but it lacks the system-scale long-range
order of a true IR condensate, which should reside in one mode.
This local condensate then undergoes ordering dynamics, in which it
evolves towards higher long-range order. The $k$-spectrum is a
Fourier representation of this ordering dynamics.
We will show that the dynamics of the local condensate depends
nontrivially on the number of field components $N$ (we only consider
multi-component fields which possess global $O(N)$ symmetry). The
case $N=1$ is qualitatively different than all cases $N \geq 2$. We
will explain why, for $N \geq 2$,
the condensate locally carries large – nearly
maximal – conserved charge density. This phenomenon has previously
been observed in the $N=2$ case <cit.>.
The large local charge density causes the condensate to
decay much more slowly for $N>1$ than for the $N=1$ case. It also
means that, in a finite box, the condensate eventually becomes spatially
homogeneous for $N=1$ but it remains spatially varying for $N\geq 2$,
with overall charge neutrality enforcing that the condensate vary in
such a way that its total charge is zero. For several $N$ values the
ordering dynamics include networks of topological defects, including
domain walls for $N=2$ <cit.> and strings for
$N=1,2,3$ (as we will show). These topological features do not
control the high-$k$ tail of the power spectrum, however; this tail is
the same when comparing theories with such defects to theories with no
defects such as $N=8$.
In the remaining sections we will explain the physics of the results
described above, and we will support the description with results from
3+1 dimensional classical (statistical) lattice field theory
simulations using $N=1$, 2, 4, and 8, with lattices of $512^3$ and
$128^3$ points (the former for spectra, the latter to study how the
condensate moves into the $k=0$ and $\vec k=\frac{2\pi}{L}[1,0,0]$
§ GENERAL PICTURE
Consider $O(N)$ scalar field theory with Lagrangian
- ℒ =
∑_a ( ∂_μϕ_a ∂^μϕ_a )
+ ∑_a m^2/2 ϕ_a^2 + λ/8
( ∑_a ϕ_a^2 )^2 ,
with initial conditions which put a large initial occupancy in IR
modes. For suitably large occupancy the fields can be treated as
classical. At the lowest order, each $k$-mode oscillates
independently at frequency $\omega_{k,a} = \sqrt{k^2 + \mtha^2}$,
where the effective thermal oscillation frequency of species $a$ is
$\mtha^2 \simeq m^2 + \frac{\lambda}{2}
\langle 2\phi_a^2+\sum_b \phi_b^2 \rangle$.
At next order, the modes interact with each other,
exchanging energy between modes and into formerly unoccupied modes.
Entropy considerations tell us that the energy will spread
out into all available modes, which for phase-space reasons are
dominated by larger-$k$ modes. Scalar theory has an
approximately-conserved particle number; for a $k$-mode with
oscillation frequency $\omega_k = \sqrt{k^2+\mth^2}$ and carrying
energy $\varepsilon(k)$, the particle number is
$f(k) = \varepsilon(k)/\omega_k$. This particle number is not
strictly conserved, but it decays on a much longer
time scale than the kinetic equilibration needed to re-arrange the
energy between $k$-modes. Now $\omega_k$ in UV modes is larger
than in the modes occupied in the initial conditions; so as these
modes absorb the system's energy, they take little of its particle
number, leaving an excess in the IR modes.
Naively, we might expect the modes to take an occupancy of form
f(k) ∼T/ω_k - μ
with $T$ and $\mu$ the Lagrange multipliers to conserve the
system's energy and particle number. For sufficiently large initial
$n/\varepsilon$ ratio, $\mu=\mth$ is not enough to contain all the
particle number in the finite-$k$ modes, and an $\OO(1)$ fraction of
the particle number can go into the mode or modes with the smallest
$\omega_k$ value, which is permitted by fakeEQ if
$\mu=\omega_0$. In practice, fakeEQ can occur at late times if a
lattice or other physics provides a UV cutoff, but generally the
occupancies take a more complicated form. But the “need” to store a
large particle number is a fairly general issue, and a large occupancy
in a small $k$-space region is a general solution.
For statistically uniform initial conditions over a large system,
this thermodynamic preference for large occupancy at small $k$ occurs
everywhere. Therefore, locally in the system – on scales large
compared to $\mth^{-1}$ but small compared to the system size – particle
number should move into a nearly-uniform condensate. But since this
occurs due to local physics in a time scale too short for information
to be exchanged throughout the system, this condensate will not
instantly form coherently – in the same field direction and with the
same oscillation phase – everywhere in the system at once. Instead, it
will generically form with an independent field direction and
oscillation phase at widely spatially separated points. This similar
to a quench process which leads to a vacuum expectation value (VEV), a
common phenomenon in condensed matter physics (see for instance
<cit.>) which has also been studied in the context
of cosmological phase transitions
A uniform true condensate would have a common oscillation phase and
field direction throughout the system. This would spontaneously break
the symmetry between different oscillation phases and field
directions. Therefore the formation of a condensate is a type of
symmetry-breaking transition. This is another feature in common with
quench processes. But, just as for the formation of a VEV, the
local condensate initially chooses its phase and direction
independently at widely spaced points. The subsequent dynamics
involve the organization of the initially nonuniform features. It is
well known that the details of this field organization depend on
the space of possible local values (field direction and oscillation
phase) for the condensate, which plays the role of the
vacuum manifold in the usual case. To simplify the language, we will
call the space of possible local values for the condensate the
CCspace (condensate configuration space). The topology of this space
determines what types of topological structures can occur and impede
the ordering dynamics.
Therefore our first order of business is to determine the possible
values a condensate can take locally, and to understand the topology
of the CCspace.
§ DESCRIBING THE LOCAL CONDENSATE
The space of field directions and phases which the condensate can take
depends in an important way on $N$ the number of field components.
For $N=1$ the theory has no continuous global symmetry, which makes
this case essentially different from $N \geq 2$. Therefore we start
by considering $N=1$ and then turn to $N \geq 2$.
§.§ N=1
The condensate is described instantaneously by the field value and its
time derivative. This is equivalent to the field's peak amplitude
$\phi_0$ and the phase of its oscillation,
$\phi = \phi_0 \Re e^{i(\varphi+\omega t)}$, where $\varphi$ is
defined modulo $2\pi$. The amplitude is fixed by the particle
number density which must enter the condensate, so the variable which
can differ through space is the phase $\varphi$. Therefore the
CCspace has the topology of the circle $S^1$.
We also compute the relation between the condensate's peak amplitude
$\phi_0$, particle number density $n$, energy density $\varepsilon$,
and oscillation frequency $\omega$, assuming that the condensate's
self-interactions dominate its interactions with other fluctuations in
establishing the oscillation frequency. Saving the details for
Appendix <ref>, we find that
ω = √(π) Γ(3/4)/Γ(1/4)
√(λ) ϕ_0 ≃0.59907 √(λ) ϕ_0
= 1.00751 λ^1/4 ε^1/4 ,
ε = 3 √(π) Γ(3/4)/2^5/4 Γ(1/4)
λ^1/3 n^4/3 ≃0.68825 λ^1/3 n^4/3 .
§.§ N greater than 1
Next consider the case of a field with $N\geq 2$ components. The
instantaneous value of the condensate is determined by the (locally
space-averaged) instantaneous value of $\phi_a$ and $\dot{\phi}_a$.
One possibility is that $\phi_a$ and $\dot{\phi}_a$ lie in the same
field direction. In this case the field oscillates along one field
direction as in the $N=1$
case. The other extreme is for $\dot\phi_a$ to be orthogonal in field
space to $\phi_a$, and of a magnitude which keeps $|\phi|$ unchanged
with time. That is, the scalar field can follow a circular orbit
through $O(N)$ field space.
The condensate has to carry a certain particle number density.
Statistical mechanics arguments favor whichever form for the
condensate can do so with minimum energy. Therefore it is important
to repeat the calculation of $\omega$, $\varepsilon$, and $n$ as a
function of $\phi_0$ for the circular orbit case. As shown in
Appendix <ref>, for this case we have
ω = λ^1/2/2^1/2 ϕ_0 =
2^1/4/3^1/4 λ^1/4 ε^1/4
≃0.9036 λ^1/4 ε^1/4 ,
ϵ = 3/2^7/3 λ^1/3 n^4/3
≃0.5953 λ^1/3 n^4/3 .
The energy cost of a circular orbit is about $13\%$ lower, at fixed
particle number density, than back-and-forth oscillation. Of course,
there are also possibilities intermediate between these, corresponding
to (precessing) elliptical orbits in field space. These carry energy
per particle number strictly intermediate between the two limiting
cases we have considered, so the circular-orbit case is the most
energetically efficient way to store particle number. Therefore it
will be favored on statistical-mechanical grounds.
[Another way to see this is to consider a generic elliptical
orbit and to consider the energy cost of adding a quantum in the
direction which stretches the ellipse, versus adding a quantum in
the direction which makes it more circular. The energy cost is
lower for the quantum which makes the oscillation more circular.]
The circular orbit carries a large local density of at least one of
the conserved charges
$\rho_{ab} = \dot\phi_a \phi_b - \dot\phi_b \phi_a$ ($a<b$).
One might expect this to forbid circular orbits, since it requires
large charge densities and charge is conserved.
But charge conservation is global, and the condensate can
exchange charge with fluctuations, which propagate freely into other
regions and deposit charge density with the condensate there.
Therefore the condensate can take on a large charge density
locally, provided that the charge density vary through space such that
its space integral vanishes. In the next section we will present
numerical evidence showing that this is what occurs.
How do we describe the condensate locally? It is described by the
field direction $\phi_a$ and the field-derivative direction
$\dot\phi_a$, with the constraints that each is of fixed
magnitude (to carry the correct particle number density and maintain
minimum energy cost) and that $\dot\phi_a$ is orthogonal to $\phi_a$.
A fixed-length $\phi_a$ is an element of the
$(N{-}1)$-sphere $S^{N-1}$, while a fixed-length orthogonal $\dot\phi_a$
is an element of the fixed-length tangent bundle. Therefore the
CCspace is topologically the unit-tangent bundle of $S^{N-1}$,
$UT(S^{N-1})$, which is a fibration of $S^{N-2}$ over $S^{N-1}$.
§.§ Topological considerations
A nontrivial CCspace can have consequences for the ordering dynamics.
If the CCspace is not connected – if it has a nontrivial $\pi_0$
homotopy group – the condensate in different regions may be separated
by domain walls. If it is not simply connected – if the first
homotopy group $\pi_1$ is nontrivial – it can vary in a topologically
nontrivial way around a loop in field space, guaranteeing the
existence of string defects. If $\pi_2$ is nontrivial, there can be
monopole defects. Let us see which occur for some values of $N$:
For $N=1$ the CCspace is topologically the circle $S^1$. This has
$\pi_1(S^1) = \ZZ$ the integers. Therefore there are string
defects, corresponding to lines where the condensate's phase changes
by $2\pi$ as one goes around the line.
The case $N=2$ is equivalent to a complex scalar field. The CCspace
is the unit-tangent bundle of the circle $S^1$. The unit-tangent
space has two points, corresponding to the condensate rotating
clockwise or counterclockwise around the complex plane. So the
unit-tangent bundle is two copies of the circle, $S^1 \times \ZZ_2$.
Again $\pi_1 = \ZZ$ and there are strings; but also $\pi_0 = \ZZ_2$
and there are domain walls, separating regions where the condensate
revolves clockwise from regions where it revolves counterclockwise.
This is the only $N$ value for which there are domain walls.
For $N=3$ the unit-tangent bundle $UT(S^2)$ is equivalent to $SO(3)$
the group of 3-dimensional rotations. To see why, note that
$\phi_a$ is a direction in $\RR^3$, while $\dot\phi_a$ must be
another $\RR^3$ direction orthogonal to the first. Together with
their cross product, they define an orthogonal coordinate frame. The
space of coordinate frames is the same as the space of rotations
(think of the rotation from a standard frame to the desired frame),
which is $SO(3)$.
It is important that $SO(3) \neq SU(2)$ its double cover. In
particular, since it is double-covered, $\pi_1(SO(3)) = \ZZ_2$ and
there are string defects. However $\pi_0(SO(3))$ and $\pi_2(SO(3))$
are trivial so there are no domain walls or monopoles.
For $N=4$, the tangent bundle of $S^3$ is trivial, so
$UT(S^3) = S^2 \times S^3$. Therefore $\pi_0$ and $\pi_1$ are
trivial, but $\pi_2(UT(S^3)) = \ZZ$ and there are monopole defects.
For general $N$ the unit-tangent bundle is
$UT(S^{N-1}) = \mathrm{Spin}(N)/\mathrm{Spin}(N-2)$.
For $N>4$ this space has $\pi_0$, $\pi_1$, and $\pi_2$ trivial, so
there are no walls, strings, or monopoles. In some dimensions
$\pi_3$ is nontrivial and there are textures, but for the special
case $N=8$, $UT(S^7) = S^6 \times S^7$, which has no homotopy below
$\pi_6$ and so is free of defects
in 3+1 dimensional space.
I thank Johannes Walcher for a refresher on unit tangent bundles.]
§ EFFECTS OF CONDENSATE STRUCTURE
Here we will look at the consequences of the form of the condensate,
and check them against numerical investigations.
§.§ Local charge density
The most convincing evidence that the above description of
condensation is true is to look at the process in a small enough
volume that it reaches completion. Then we analyze the late-time
behavior and see that it is qualitatively different for the cases
$N=1$, $N=2$, and $N>2$.
For $N=1$ there is no obstacle to the condensate developing completely
in the $k=0$ mode. But for $N>1$ the condensate locally carries a net
abundance of conserved charge, which should average to zero globally.
Therefore the condensate should not become completely
uniform, but should always vary in such a way that the global charge
vanishes. For $N=2$ this requires a pair of domain walls, separating
the regions with positive and negative charge density. Such domain
walls were observed and characterized in <cit.>. For
$N>2$ there are several charges, and the charge density can revolve
smoothly through the $N(N-1)/2$ dimensional space of possibilities.
0.32kpow1 0.32kpow2 0.32kpow3
Power in the lowest $k$-modes which point along a lattice
direction, for $N=1,2,3$ (left to right). For $N=1$ the power
all concentrates into the lowest mode. For the other cases, the
power is shared equally between the $k=0$ mode and the next-lowest
mode; for $N=2$ there is also power in the $k=(3,0,0)$ mode.
First we look for this behavior in Fourier space, by evolving
$N=1,2,3$ systems in $128^3$ boxes, which allows us to achieve times
where the condensate completes its evolution towards the infrared.
k0123 shows the total power in the $k=\frac{2\pi}{L}(0,0,0)$
mode, and the sum of power in
$k=\frac{2\pi}{L}(\pm l,0,0)$, $\frac{2\pi}{L}(0,\pm l,0)$
and $\frac{2\pi}{L}(0,0,\pm l)$ for
$l=1,2,3$, for a typical simulation. That is, it shows the total
power in each of the four
lowest Fourier modes for which $k$ lies purely along a lattice
direction. For the case $N=1$ we see that the power all moves into
the $k=0$ mode, which then decays with time. For $N=3$ the power is
shared equally between the $k=0$ mode and the modes of form
$k=(\pm 1,0,0)\frac{2\pi}{L}$. It is also shared equally between
$\phi^2$ an $\dot{\phi}^2$, in that the ratio
$\langle \dot\phi^2\rangle / \langle \phi^2 \rangle$ summed over the
lowest modes remains fixed, rather than oscillating as it does for
$N=1$. The case $N=2$ is superficially similar to the $N=3$ case,
except that the power in $l=3$ does not decay but remains much larger
than the power in $l=2$ (or any of the other Fourier modes, such as
$k=\frac{2\pi}{L}(1,1,0)$, which are not along lattice directions).
This is because, for $N=2$, the condensate is discontinuous across a
domain wall, which leaves power in all odd harmonics, while for
$N=3$ the charge density is smooth. We also studied $N=4$, which
shows the same behavior as $N=3$.
Power in the lowest nonzero Fourier mode along the $x$, the $y$,
and the $z$ axis, as well as the $(3,0,0)$-type modes. Left:
$N=2$. Right: $N=3$. In each case, the power along two axes
dies away, while the third axis persists. For $N=2$ it is the
same axis where there is persistent power in the $(0,3,0)$ modes.
What k0123 does not show is that the power in $(1,0,0)$ type
modes is in fact all concentrated in the mode along one axis. To see
this, in kxyz we plot separately the power in
$k=\frac{2\pi}{L}(\pm 1,0,0)$ and that in
$k=\frac{2\pi}{L}(0,\pm 1,0)$ and $(0,0,\pm 1)$, for $N=2$ and $N=3$.
We also plot the $(3,0,0)$ power, to see that it is supported along
the same lattice direction as the $(1,0,0)$ power for the $N=2$ case
where it does not die off. The fact that the power
ended up along one axis indicates that the condensate remains
spatially asymmetric in this, but not the other two, lattice
directions. We performed each simulation several times with different
random number seeds, and found that the lattice axis where the
breaking occurs is randomly different, but the pattern shown above is
the same for each simulation; for $N>1$ the power equipartitions
between the $(0,0,0)$ mode and one set of $(\pm 1,0,0)$ modes and is
small in the others.
0.4lineO2 0.4lineO3 $\vphantom{.}$
plane-averaged charge density plotted along the remaining axis,
in $128^3$ box at $t=10,000$ when the condensate has settled into
its final form. Left: $N=2$ case, where the charge density varies
along the $y$ axis. Right: $N=3$ case, where the three charge
densities vary primarily in the $z$ direction.
To see it another way, let us define the charge density averaged
over two lattice directions as a function of the third:
$\rho_{ab;x} \equiv L^{-2} \int dy\,dz\: \rho_{ab}(x,y,z)$ and similarly for
$\rho_{ab;y}$ and $\rho_{ab;z}$. We plot this, for each axis and at a
fixed late time $t=10,000$ lattice units, in rhoxyz. On the
left, we see that the $N=2$ theory has a pair of sharp discontinuities
in $\rho_{y}$, with plateaux between. For $N=3$ we see that each
charge component varies smoothly such that
$\sum_{ab} \rho^2_{ab;z}$ is nearly constant and much larger than in
the other two directions. Note that these measurements were made
directly on the fields without any smearing and no averaging except
for the averaging over lattice planes. Averaging over 2D planes is
already enough to suppress the contribution from UV modes, without any
field smearing.
Histogram of the condensate's local charge density, for the $N=2$
theory in a $512^3$ box after smearing $\phi$ and $\dot\phi$ to
eliminate the contribution of UV fluctuations.
We can also form a histogram of the value of $\rho$ at each point on
the lattice. The result looks fairly Gaussian, because of the large
site-by-site contribution from UV fluctuations. But if we first
smooth the fields, the result is different. Sticking with the $N=2$
theory, we Fourier transformed $\phi$ and $\dot\phi$, multiplied by
$\phi(k) \to \phi(k) \exp(-k^2/\omega^2)$,
$\dot\phi(k) \to \dot\phi(k) \exp(-k^2/\omega^2)$ (with $\omega$
determined from the data as described in Appendix <ref>),
and Fourier transformed back, in order to eliminate the contribution
of UV fluctuations. Then we determined the charge density due to IR
fields at each lattice point by finding
$\rho(x)=i(\phi^* \dot\phi(x) - \phi\dot\phi^*(x))$ as usual.
rhohist shows a histogram of how common
each possible value of the charge density is, for several times and in
a $512^3$ box. The curves are normalized so the area under each curve
is 1. At late times there are two sharp peaks with a plateau
in between; we interpret the peaks as arising from points within
regions of one or the other charge phase, and the plateau as the
contribution from the domain walls separating these phases.
§.§ Ordering dynamics
Next consider a large lattice volume $512^3$ and somewhat shorter
times, which allows us to see the field organization dynamics of the
local condensate. As we have emphasized, for $N\leq 4$ the condensate
can carry topological defects. A
topological defect is a location where the condensate field varies
abruptly. Therefore it is guaranteed to create power-law tails in the
high-$k$ part of the power spectrum for the condensate. Specifically,
a network of domain walls should create $k^{-4}$ tails in the power
spectrum; strings should create $k^{-5}$ tails, and monopoles should
create $k^{-6}$ tails.
[In general the power is $k^{-d-c}$ with $d$ the space
dimension and $c$ the codimension of the defect. Consider a defect
with codimension $c$ and dimension $d-c$. A function in $c$
dimensions which is discontinuous at a point has a Fourier spectrum
with $k^{-2c}$ tails; but each of the remaining
$d-c$ dimensions “along” the defect dilute this by an additional
To illustrate this behavior, consider Lagrangian with $m^2<0$ and
adding a damping term to the dynamics such that
∂_t^2 ϕ_a = -τ∂_t ϕ_a
+ ∇^2 ϕ_a + V_,ϕ_a .
In our numerical implementation we chose $\tau = a/2$ so the damping
dynamics are strong, and we chose $a^2m^2=-0.5$ and $\lambda=1$ so the VEV
is $|\phi|=1$ in lattice units and the radial fluctuations have
oscillation frequency
$1/a$. In this case the field forms a true condensate, but we choose
random initial conditions, so the field starts out disordered and must
follow ordering dynamics. The ordering dynamics are impeded by
topological defects with codimension $N$: walls for $N=1$, strings for
$N=2$, monopoles for $N=3$, textures for $N=4$, but no topological
structures for $N=5$. By the arguments above, we expect the power
spectrum for $N=1,2,3$ to display $k^{-(3+N)}$ tails, and indeed it does.
decayfig shows the power spectrum at several times for the case
$N=2$, first with fixed axes and then with axes rescaled such that the
curves collapse onto a single scaling behavior. Finally, the scaling
behavior is compared for $N=1,2,3,5$ (at time $t=960a$). The IR
behavior after scaling
to $\kcond$ looks similar for all cases, but the larger-$k$ behavior
is different. For $N=1,2,3$ we see power laws with $k^{-3-N}$ power,
and for $N=5$ the tail does not asymptote to a power, all as
expected. Note that for $N=1,2$ the spectrum is actually steeper than
$k^{-3-N}$ around $k=1.8\kcond$, before taking the power-law form
at $k>2\kcond$.
0.32O2ktdamp 0.32O2ktscaledamp 0.32disscompare
Power spectrum for broken-symmetry theory under dissipative
dynamics, to illustrate the effect of defects on the spectrum.
Left: $N=2$ theory at several times. Middle:
rescaling $k\to k/\kcond$ and power by $2\pi^2\kcond^{-3}$ so the
curves collapse onto a scaling form. Right: scaling form for
$N=1,2,3,5$ superposed, showing different power-law tails.
We also see in the two left plots of decayfig that the power-law
tails break off at a larger-$k$ scale which stays fixed in physical
units. This scale is set by the thickness of the defects; at this
scale the spatial features become smooth so the power spectrum changes
from power to exponential behavior. Again we emphasize that this
discussion is only to show the effect of defects on the power-law
Returning to the theory with a particle-number condensate, we expect
the topological structures to be one contribution to
the IR tail of the condensate's part of the power spectrum.
There can also be ordinary nontopological fluctuations in the
condensate, which will obscure the topological contribution if they
have a softer power law or substantially larger amplitude. We also
expect the defect core size to depend on $N$, now differing between
the $N=2$ case and other cases. For string or monopole defects we
expect the core size to be $\sim \omega^{-1}$. But for domain walls,
we argue that it should be at least 3 times larger. Generally the
thickness of a domain wall is set by the potential energy cost of the
state at the center of the wall, with inverse width
$k_{\mathrm{wall}} \propto \sqrt{V}$. The core of the domain wall
contains condensate which is not locally charged but oscillates
straight back-and-forth. The excess energy cost of this state is only
about $13\%$ of the condensate's energy density, so we expect the
wall's inverse thickness to have an extra power of $\sqrt{0.13}$.
0.45O1kt 0.45O2kt $\phantom{.}$
0.45O2ktscale 0.45compareOs $\phantom{.}$
Power spectrum of the IR condensate. Top: power spectrum at
several times for $N=1$ theory (left) and $N=2$ theory (right).
Bottom left: rescaled version for $N=2$ showing scaling
behavior. Bottom right: scaled spectrum at $t=2400$ for
power shows the analogous figures to decayfig, but for the
theory with a condensate. The time-dependence of the spectra in the
infrared is superficially similar to that in decayfig.
As expected, this behavior flattens out at larger $k$ rather than
getting steeper. In the $N=1$ case, but not for any other $N$, there
is also a feature around $ak=0.9$. This peak is the freshly-produced
excitations of frequency $2\omega_0$, generated from the decay of the
condensate, see <cit.> and the next subsection.
power also shows the spectra rescaled to have the same total
power and characteristic wave-number $\kcond$. It shows excellent
collapse onto a scaling solution, broken at a larger-$k$ scale which
grows, in units of $\kcond$, as time progresses.
[The physical scale decreases, as we see in the upper plots;
but $\kcond$ decreases faster.]
This collapse onto a scaling function has already been
observed by previous authors <cit.>.
In <cit.> it is argued that the steep part
of the spectrum below $\kcond$ should behave as $f\propto k^{-5}$. We
find $k^{-4.5}$, which is the slope of the straight line in the
lower-left frame of the figure, which agrees with the numerical
results in <cit.>. A $k^{-5}$ line visibly fails to
fit (not shown).
Finally, in the lower right we show the scaling solutions for
$N=1,2,3,4,8$ all superposed. Beyond about $k=7\kcond$ the curves
deviate from scaling, so this behavior should be ignored. The curves
agree strikingly well, in strong contrast to the symmetry-broken case
of decayfig. This collapse of different times and $N$ values
onto a single scaling solution has previously been pointed out by
<cit.>. These authors argued that the
concordance indicates a universality in the physical origin of the
spectrum. This is rather surprising, given the differences in
condensate structure and topological obstructions which we have
previously discussed. Therefore it does
not appear that topological structures control the tail's
behavior, at least for the cases other than $N=2$.
It is a little puzzling that a $k^{-4}$ tail does not emerge for the
$N=2$ case, since we have already seen that there are domain walls in
the charge-sign of the condensate. One possibility is that a $k^{-4}$
region would emerge if we could achieve a much wider scaling window.
After all, for the symmetry-breaking theory in decayfig, the
scaling emerges relatively late, and as discussed above we expect it
to break down by $k\sim \omega/3$ because the walls are quite thick.
Unfortunately, achieving a significantly wider scaling window is
prohibitively numerically costly.
§.§ Condensate decay
The presence of a large local charge density has consequences for how
the condensate decays with time. To see this, consider the Feynman
diagrams responsible for condensate decay; $4\to 2$ scattering as
shown in 42diagrams. For the $N=1$ theory, the leading cause of
condensate decay is the case where the 4 incoming particles are drawn
from the condensate and the two outgoing particles are finite-$k$
excitations, each carrying $2\omega$ energy. The two diagrams
partially cancel, since one has a spacelike propagator and one has a
timelike propagator above the mass shell.
Feynman diagrams for $4\to 2$ number-changing scattering.
It is actually not
straightforward to compute the diagram, since the dispersion relation
for the outgoing particles and virtual propagators are not simply
those of a massive particle – the coherence of the condensate is not
the same as $\langle \phi^2 \rangle$ arising from many independent
fluctuations. But we can parametrically estimate the rate fairly
easily. Though the calculation can be done within the classical
theory, we will use the notation of the quantum theory because it is
probably more familiar to our readers. The rate of particle number
change from the condensate is determined by the Boltzmann equation for
the condensate particles:
d/dt ∼
∫d^3p_1,2,3,4 d^3 k_1,2/p_1 p_2 p_3 p_4 k_1 k_2
|ℳ|^2 δ^4( ∑p_i - ∑k_i )
×( f_p_1f_p_2f_p_3f_p_4 [1+f_k_1][1+f_k_2]
- [1+f_p_1][1+f_p_2][1+f_p_3][1+f_p_4] f_k_1
f_k_2 )
∫d^3p_1,2,3,4/p_1 p_2 p_3 p_4 f_p_1f_p_2f_p_3f_p_4
d^3 k_1,2/k_1 k_2 (f_k_1+f_k_2)
|ℳ|^2 δ^4( ∑p_i - ∑k_i )
λ^4 ^4/ω^8 f(2ω) .
Here $\ncond = \int d^3 p f(p)$ is the particle number
density in the condensate, $\lambda^4$ arises from the squared matrix
element, and the powers of $\omega$ enter on dimensional grounds from
the remaining phase space integrals and the matrix element.
Naively this result seems to state that the condensate's decay rate is
a very steep power of the condensate's size $\ncond$. But
the presence of $\omega^{-8}$ in the final expression moderates this,
because the dominant contribution to the oscillation frequency is
actually the condensate itself. Indeed, if we ignore the contribution
from all other fluctuations, we have, from N1omega2 and
ω∼λ^1/4 ε^1/4 ε∼λ^1/3 n^4/3 ⇒ ω^3 ∼λ .
Therefore we expect weaker $\ncond$ dependence:
$d\ncond/dt \propto \ncond^{4/3} f(2\omega)$. If, for instance,
$f(2\omega) \propto \omega^{-3/2}$, this would give
$d\ncond/dt \propto \ncond^{5/6}$. In general $f(2\omega)$ also
varies (decreases) with time. And
any other contribution to the oscillation frequency, such as
an explicit $m^2$ term or a contribution from other excitations, can
also substantially slow the condensate's decay, especially once
$\ncond$ has already gotten smaller.
We actually see the consequences of this decay process in
power. In the upper left panel, the power spectrum has a bump
around $ak=0.9$ which moves to lower $k$ at late time as $\omega$
shrinks (since $\omega \propto \ncond^{1/3}$). These are the
condensate decay products which have not yet rescattered into other
wave numbers – as already realized in <cit.>. The same peak
does not occur for $N=2$, as the figure shows and for reasons we will
now see.
The situation is quite different for $N \geq 2$ because the condensate
is maximally charged. The process in 42diagrams cannot occur if
all 4 initial particles are from the condensate, because all incoming
particles have the same $O(N)$ charge, so the process
violates charge conservation and the relevant diagrams vanish
identically. Instead, 3 condensate particles must
pick up one particle of opposite charge from the thermal bath to
produce two final-state particles with the same charge as the
condensate. These are then free to propagate to a different region to
keep the fluctuations locally charge-neutral. In decayest1, one
of the $\int d^3 p f_p$ must return the number of normal excitations,
$\ncond \to \omega^3 f(\sim 1 \omega)$. The decay rate of the
condensate is estimated as
d/dt ∼λ^4 ^3/ω^5
f^2(∼1 ω)
which is smaller than the previous estimate by a factor of
$\sim \omega^3 f(\sim 1 \omega)/\ncond$, which is the ratio of
infrared but non-condensate excitations to condensate excitations.
The condensate should decay much more slowly for
$N\geq 2$ than for $N=1$, especially in the regime where the
condensate is large. It also does not decay into particles of a
specific $k$ value, since there is an energy and momentum range for
the particle picked up from the medium – hence the absence of a peak
around $k\sim \sqrt{3}\omega$ in the upper-right plot in power.
0.45specNdecay 0.45specdecayN1
Condensate as a function of time for the $N=1,2,3,4,8$ theories,
left; condensate as a function of time as the explicit $m^2$
term is increased for the case $N=1$, right.
We explore condensate decay as a function of $N$
in cond_decay, which shows the particle number
in the condensate (total particle number at small $k$ as
defined in Appendix <ref>)
as a function of time for the $N=1,2,3,4,8$
theories. The figure shows a stark difference between the $N=1$ case,
where there is no charge-conservation constraint, and the $N>1$ cases,
where there is. Note that the initial particle number, and the
particle number which needs to move into the condensate, scales as $N$
for our initial conditions; we have explicitly removed this factor in
the figure. We also study the effect of including an explicit $m^2$
term, which raises the oscillation frequency and therefore
suppresses the condensate decay according to decayest1, for the
$N=1$ case. For the parameters studied, the $m^2=0$ case has
$a^2\omega^2 \sim 0.3$, so the first two $m^2>0$ curves still have the
oscillation frequency dominated by the condensate's self-interactions;
nevertheless, the addition of an explicit $m^2$ term raises $\omega^2$
enough to substantially suppress the condensate's decay.
We have not yet tried to explain these curves
quantitatively, though it would be interesting to do so in future.
They certainly support qualitatively the expectations explained
§ DISCUSSION
We have considered the deep IR behavior of classical scalar field
theory, where the approximate conservation of particle number causes
the development of a particle-number storing condensate. Because
condensate formation occurs locally throughout the system, the
condensate is initially incoherent, and must develop long-range
order. Our most interesting result is that, in the case that the
condensate is large enough that its self-interactions dominate its
oscillation frequency, the condensate's local structure is
qualitatively different in single-component ($N=1$) theory than in
$O(N)$ symmetric $N\geq 2$ component theories. In the latter case,
the condensate locally carries the maximal possible charge density,
with overall charge neutrality maintained by spatial inhomogeneity of
the condensate. This inhomogeneity persists indefinitely in finite
volume, and the charge density impedes the decay of the condensate,
which is much slower for $N>1$ than for $N=1$. Finally, we have
considered the large-$k$ tail of the condensate, which appears to fall
as a power law which is not determined by the topological
structures present due to the long-range disorder of the condensate.
§ ACKNOWLEDGMENTS
I would like to thank Thomas Epelbaum and Jürgen Berges for helpful
conversations, Johannes Walcher for help on homotopy for some specific
spaces, and Paul Mercure for his patience.
§ LATTICE DETAILS
We give a few details of our lattice implementation. We discretize
the theory of Lagrangian on the lattice with a standard nearest-neighbor
implementation of the gradient term and leapfrog update rule, with
temporal step equal to $1/10$ of the spacing. Our initial conditions
are $\phi_a=0$ and each $k$-mode $\dot{\phi}_a(k)$ drawn from a
Gaussian distribution of fixed width for $k < \kinit$ (technically,
$\tilde{k}^2 \equiv \sum_{i=1,2,3} 2-2\cos(k_i a) < \kinit^2$) and zero for
$k>\kinit$. In practice we use $\kinit = 1/a$, $a$ the lattice
spacing. This provides large occupancy per unit energy, so that as the
energy fills into all available modes and in particular into
higher-$k$ modes, there is excess particle number which must fill the
IR condensate. We choose $m^2=0$ except in a few simulations where we
study its effect on the condensate's time evolution.
The field normalization and $\lambda$ value overdefine the
system; one is free to rescale $\phi \to \xi \phi$, with $\xi$ a
constant, simultaneously rescaling $\lambda \to \xi^{-2} \lambda$.
We normalize the field so the total energy is $N$ per lattice site.
We chose this convention because, to the extent that the system is
weakly coupled, the eventual equilibrium (after the condensate decays)
has energy equipartitioned between kinetic and potential+gradient
energy, and this corresponds to an energy of $\half$ per degree of
freedom, and hence to a temperature of 1. With this convention,
our simulations all used $\lambda=0.361$.
This value kept the thermal oscillation frequency somewhat below
$a\mth=1$ so that the
IR behavior should not be too contaminated by lattice effects but the
dynamic range is still relatively good.
When we Fourier transform the $\phi$ and $\dot\phi$ fields to form a
power spectrum, we treat $k^2 < 8/aL$ separately from $k^2 > 8/aL$
(where $a$ is the lattice spacing and $L$ is the box length). For the
former, we bin all Fourier components with the same value of $k^2$,
eg, $k=\frac{2\pi}{L}(2,2,1)$ is combined with its
cubic-invariance equivalents and with $k=\frac{2\pi}{L} (3,0,0)$ but
not with $(3,1,0)$. For $k^2>8/aL$ we use $\tilde k^2$ bins
of width $2/aL$.
We define a range of wave numbers as being the range which contains
the condensate by finding the bin with the largest average Fourier
power in
$\omega^2 \langle \phi^2(k)\rangle + \langle \dot{\phi}^2(k)\rangle$,
and considering all bins where this quantity is at least $1/300$ of
this maximum value. We then find the integral of
$\langle \phi^2(k) \rangle$ and $\langle \dot\phi^2 \rangle$ over this
range, and find $\omega^2$ self-consistently as
ω^2 ≡∑_cond.modes ϕ̇^2(k)/∑_cond.modes ϕ^2(k) ,
re-determining the range of modes in the condensate until we reach
self-consistency. The occupancy at $k$ is estimated as
f(k) = ω⟨ϕ^2(k) ⟩+ ω^-1 ⟨ϕ̇^2(k) ⟩ .
These steps assume that the condensate is carried in a range of $k$
such that $k^2 \ll \omega^2$, which becomes true quite quickly; if it
is not true it does not make sense to speak of a local condensate.
Finally, we define the characteristic wave-vector for the local
condensate as
k_cond = ∑_cond.modes k f(k)/∑_cond.modes f(k) .
§ OSCILLATION FREQUENCIES
Here we complete the details in the calculation of oscillation
frequencies for single-component and “circular” multi-component
Consider a scalar oscillating back-and-forth in a $(\lambda/8) \phi^4$
potential. If the field oscillates with amplitude $\phi_0$ then the
energy is
λ/8 ϕ_0^4 = 1/2 ϕ̇^2 + λ/8
ϕ^4 → ϕ̇= √(λ)/2 √( ϕ_0^4 - ϕ^4 )
and the period of the oscillation is
T/4 = ∫_0^ϕ_0 dt/dϕ dϕ= ∫_0^ϕ_0 1/√(2ε-2V(ϕ)) dϕ= 2/ϕ_0 √(λ)
∫_0^1 dx/√(1-x^4)
= √(π) Γ(1/4)/2 Γ(3/4) ϕ_0 √(λ)
which gives a frequency of
This is not the same as the naive estimate
$\omega^2 = \frac{3\lambda}{2} \langle \phi^2 \rangle$,
\langle \phi^2 \rangle = \frac{\int_0^{\phi_0} d\phi \frac{\phi^2}
{d\phi/dt}}{\int_0^{\phi_0} d\phi \frac{1}{d\phi/dt}} =
\frac{\phi_0^2 \int_0^1 dx\,x^2/\sqrt{1-x^4}}
{\int_0^1 dx/\sqrt{1-x^4}}
= \phi_0^2 \frac{\Gamma^2(3/4)}{\Gamma(5/4)\Gamma(1/4)}
which gives $\omega = .8279 \sqrt{\lambda} \phi_0$. This estimate
may work for larger-$k$ excitations, but it does not work at $k=0$
because the field is time-coherent with itself.
ω= 2π/T = √(π) Γ(3/4)/Γ(1/4)
√(λ) ϕ_0 ≃0.59907 √(λ) ϕ_0 .
We can re-cast this in terms of the energy density using
$\varepsilon = \lambda \phi_0^4/8$:
ω= (8π^2)^1/4 Γ(3/4)/Γ(1/4)
λ^1/4 ε^1/4
= 1.00751 λ^1/4 ε^1/4 .
When we add one particle to the condensate,
we add an energy of $\omega$.
So the particle number stored in the condensate is
If this looks strange, recall the behavior of a quantum system with
potential $V$ in the WKB approximation, valid for high levels.
The level number $f$ is given in terms of the level energy $E$ by
$f = \frac{1}{\pi}\int_{\phi|V(\phi)<E} \sqrt{2(E-V)} d\phi$. The
level spacing $dE/df$ sets the oscillation frequency of a
superposition of levels,
So the period is given by $T = 2\pi/\omega = 2\pi df/dE$
which is $T = 4\int (2(E-V))^{-1} d\phi$, agreeing with N1T.]
n(ε) = ∫_0^ε dε'/ω
= 4/3 ε/ω =
4/3 (1.00751) λ^1/4 ε^3/4 .
ε≃0.68825 λ^1/3 n^4/3 .
Now consider instead a scalar in 2 or more components, with $\dot\phi$
orthogonal to $\phi$ and large enough that the field's amplitude
remains fixed at $\phi_0$ while changing direction in field space.
This is like circular orbital motion in an $r^4$ potential. The energy
density is
ε= 1/2 ϕ̇_0^2 + λ/8 ϕ_0^4
and the Virial relation
$\frac{1}{2} \dot{\phi}^2 = 2 \frac{\lambda}{8} \phi_0^4$ gives
ε= 3/4 ϕ̇_0^2 = 3ω^2/4 ϕ_0^2
= 3λ/8 ϕ_0^4
ω^2 = λ/2 ϕ_0^2
from which we easily find
ω= 2^1/4/3^1/4 λ^1/4 ε^1/4
≃0.9036 λ^1/4 ε^1/4 .
Following the same steps as before, we find
n(ε) = 4 ε/3ω =
2^7/4/3^3/4 λ^1/4 ε^3/4
which equals $n=\phi_0 \dot{\phi}_0 = \omega \phi_0^2$ as expected,
ε≃3/2^7/3 λ^1/3 n^4/3
≃.59528 λ^1/3 n^4/3 .
Therefore the energy associated with a condensate which makes a circular
rotation in field space is lower, at fixed number density, than the
energy for the condensate to oscillate straight back-and-forth.
|
1511.00756
|
Mathematics Department
West Virginia University
Morgantown, WV 26506
A Riemann Solution with Singular Shocks]Singular Shocks in a Chromatography Model
[2000]Primary 35L65, 35L67; Secondary 34E15, 34C37
September 12, 2025
We consider a system of two equations that can be used to describe nonlinear chromatography and produce a coherent explanation and description of the unbounded solutions (singular shocks) that appear in Mazzotti's model <cit.>. We use the methods of Geometric Singular Perturbation Theory, to show existence of a viscous solution to Dafermos-DiPerna regularization.
§ INTRODUCTION
The aim of this paper is to show existence of no classical Riemann solutions to a physical model with important applications in modern industry. It has been already shown in carefully designed experiments by Mazzotti et al. <cit.>, that this model exhibits singular shocks.
Singular shocks, a type of weak solutions of very low regularity have been studied before. There were originally discovered by Keyfitz and Kranzer <cit.>, and later studied in greater depth by Sever <cit.>. Keyfitz and Kranzer <cit.> worked with a strictly hyperbolic, genuinely nonlinear system derived from a $1$-dimensional model for isothermal, isentropic gas dynamics and they observed that there is a large region, where the Riemann problem cannot be solved using shocks and rarefactions. They produced approximate unbounded solutions which do not satisfy the equation in the classical weak-solution sense and showed that only the first component of the Rankine-Hugoniot relation is satisfied, giving a unique speed $s$ for which any given two states $U_L$ and $U_R$ can be joined. Later on, Schecter <cit.> proved existence of a viscous solution following Dafermos's approach <cit.>, under the condition that the singular shock is overcompressive. Schecter used a geometric method and dynamical systems theory (blowing-up approach to geometric singular perturbation problems that lack normal hyperbolicity, see Fenichel <cit.> and Jones <cit.>).
Keyfitz and Tsikkou <cit.>, showed existence of approximations to singular shock solutions by the same method, for a non hyperbolic system (change of type) derived from isentropic gas dynamics for an ideal fluid with $1<\gamma<\frac{5}{3},$ conserving velocity and entropy. Singular shocks also appear in a two-fluid model for incompressible two-phase flow, see Keyfitz et al. <cit.>, in a model describing gravity-driven, particle-laden thin-film flow, see Wang and Bertozzi <cit.>, Mavromoustaki and Bertozzi <cit.>, in the Brio system appearing in the study of plasma and the classical shallow-water system see Kalisch and Mitrovic <cit.> and possibly in a model for chemotaxis, see Levine and Sleeman <cit.>.
Naturally, questions then arise about whether it is possible to predict singular shock solutions to systems, find a physical interpretation of their significance, explain the sense in which they satisfy the equation and find a better definition which will describe some wider collection of examples, check for connection between singular shocks, genuinely nonlinear systems and change of type. A few of these questions will be subject of future work.
Investigation of singular solutions was mostly focused on the case when only one state variable develops the Dirac delta function and the others are functions with a bounded variation. We have though other physically important systems with delta functions in more than one state variables. For example, Mazzotti <cit.> in his recent work, showed numerically and experimentally that the following model, in a single space dimension and time, arising in two-component chromatography (concentration $u_i$ for chemical i)
\begin{equation}
\frac{\partial}{\partial t}(u_i+\frac{\alpha_i u_i}{1-u_1+u_2})+\frac{\partial u_i}{\partial x}=0, \ i=1, 2, \ \alpha_1<\alpha_2.
\label{chromatography}
\end{equation}
exhibits singular solutions. He obtained approximate solutions, using a linear combination of $\delta$-functions, with an error that converged to zero and showed that neither of the Rankine-Hugoniot equations is satisfied. In system (<ref>) that results when some assumptions in the traditional Langmuir equilibrium model are changed, the conserved quantities are the masses of two components flowing at constant speed along a column, cooperating for adsorption sites and is a system which exhibits change of type (hyperbolic and elliptic).
In this paper, we obtain useful information from the Dafermos-DiPerna self-similar regularization and produce an explanation/description of the singular solution in Mazzotti's work.
In the next section, we derive a simpler system of equations which we will study, by rescaling time and changing the dependent variables. These changes are linear in the conserved quantities so that the form of the system is maintained. Derivation of alternative models will be also subject of future work. In Section 3 we give a formal description of the Riemann solutions, including the cases that include vacuum states. As in Keyfitz et al. <cit.>, we are led to the form of the solutions by using a self-similar viscous perturbation of the system. The new system has now similar properties to those in <cit.>, as only the first component of the Rankine-Hugoniot relation is satisfied. In Section 4, we use the theory of dynamical systems in the same spirit as in Schecter <cit.>, Keyfitz and Tsikkou <cit.> and more specifically geometric singular perturbation theory (GSPT), see Fenichel <cit.>, Jones <cit.>, Krupa and Szmolyan <cit.>, Jones and Kopell <cit.>, Schecter and Szmolyan <cit.>, to construct orbits that connect the left and right states given by
\begin{equation}
\begin{array}{ll}
U_L, & \hbox{$x<0$;} \\
U_R, & \hbox{$x\geq 0.$}
\end{array}
\right.
\label{initialdata}
\end{equation}
We also prove existence of self-similar viscous profiles for overcompressive singular shocks for the chromatography model. It should be noted, however, that the symmetry in the orbits is lost and the solutions differ significantly from those of previous models exhibiting singular shocks.
There is a body of literature on all kinds of chromatography systems of the form
\begin{equation}
\label{kinds}
(u_j)_t+(u_jf_j(\mu_1 u_1+\ldots+\mu_n u_n))_x=0, \ \ x\in\mathbb{R}, \ \ t \geqslant 0, \ \ j=1, 2,
\end{equation}
but all the known results are for simplified, everywhere hyperbolic, systems which also belong to Temple class. For recent developments in this direction see Shen <cit.>, Li and Shen <cit.> and Sun <cit.> for a system with $f_j(w)=\frac{1}{1+w}, \ \ \mu_1=1, \ \ \mu_2=0$ (the second characteristic family is linearly degenerate); Guo, Pan and Yin <cit.>, Cheng and Yang <cit.> for a system with $f_j(w)=1+\frac{1}{1+w}, \ \ \mu_1=-1, \ \ \mu_2=1$ (the first characteristic family is linearly degenerate); Wang <cit.> for a system with $f_j(w)=\frac{1}{1+w}, \ \ \mu_1=-1, \ \ \mu_2=1$ (the first characteristic family is linearly degenerate), and the references cited therein. See also Shelkovich <cit.>
for a class of systems with a different definition of solutions whose components contain Dirac delta functions. These include the system of nonlinear chromatography for $f_j(w)=1+\frac{a_j}{1+w}, \ \ \mu_j=1$ and $a_j$ is Henry's constant.
§ PRELIMINARIES
In this section we derive a simpler system of equations from (<ref>) which we study in this paper. Since these changes are linear in the conserved quantities we are not changing the form of the system. We also analyze its basic properties (hyperbolicity, genuine nonlinearity, and the shock and rarefaction curves sketched in Figure <ref>). Finally, we identify the regions where classical Riemann solutions exist.
§.§ Derivation of the model from chromatography
We start out with the equations in the form used by Mazzotti <cit.>,
\begin{equation}
\left\{
\begin{array}{ll}
(u_1+\frac{\alpha_1 u_1}{1-u_1+u_2})_{\tau}+(u_1)_x=0,\\ \\
(u_2+\frac{\alpha_2 u_2}{1-u_1+u_2})_{\tau}+(u_2)_x=0,
\end{array}
\right.
\label{2.1.1}
\end{equation}
with $\alpha_1<\alpha_2.$ To create a system more conventional to conservation laws researchers, we make some changes of variables. First, we change to a moving coordinate system, or rescale time:
$$x'=x, \ \ t=\tau-x,$$
so that the system becomes
\begin{equation}
\left\{
\begin{array}{ll}
(\frac{\alpha_1 u_1}{1-u_1+u_2})_{t}+(u_1)_{x'}=0,\\ \\
(\frac{\alpha_2 u_2}{1-u_1+u_2})_{t}+(u_2)_{x'}=0.
\end{array}
\right.
\label{2.1.2}
\end{equation}
We then drop the prime in $x.$ The aim is to focus on the conserved quantities $v_1=\frac{\alpha_1 u_1}{1-u_1+u_2}$ and $v_2=\frac{\alpha_2 u_2}{1-u_1+u_2}$ so we also change the dependent variables. If we let
$$\omega_1=\frac{u_1}{1-u_1+u_2}, \ \ \omega_2=\frac{u_2}{1-u_1+u_2},$$
then we have
\begin{equation}
\left\{
\begin{array}{ll}
(\omega_{1})_t+(\frac{u_1}{\alpha_1})_x=0, \\ \\
\end{array}
\right.
\label{2.1.3}
\end{equation}
Looking then at (<ref>), we let
$$v=(\alpha_1 \alpha_2)^{1/3}(1+\omega_1-\omega_2), \ \ y=\frac{1}{(\alpha_1 \alpha_2)^{1/3}}[\alpha_2 \omega_1-\alpha_1 \omega_2-(\alpha_1+\alpha_2)v],$$
and we find
\begin{equation}
\left\{
\begin{array}{ll}
v_{t}+(\frac{y}{v})_x=0, \\ \\
\end{array}
\right.
\label{2.1.4}
\end{equation}
In the original variables $u_1$ and $u_2$ the new variables can be expressed as
$$\frac{v}{(\alpha_1 \alpha_2)^{1/3}}=\frac{1}{1-u_1+u_2}, \ \ (\alpha_1 \alpha_2)^{1/3}y=\frac{\alpha_2 u_1-\alpha_1 u_2-(\alpha_1+\alpha_2)}{1-u_1+u_2}.$$
This system, equivalent to (<ref>) for smooth solutions, but possessing different weak solutions, expresses conservation of $v$ and $y.$ We define $U=(v,y)^\intercal$ and $F=F(U)=(\frac{y}{v},\frac{1}{v})^\intercal$ the flux function.
We work with the system (<ref>) and Riemann data
\begin{equation}
\begin{array}{c}
v \\
y \\
\end{array}
\right)
\begin{array}{ll}
U_L, & \hbox{$x<0$;} \\
U_R, & \hbox{$x\geq 0$.}
\end{array}
\right.
\label{2.1.5}
\end{equation}
to show existence of singular shocks. Attention is drawn to the limit $v\rightarrow 0$ where the variables $u_1,$ $u_2$ of (<ref>) become singular.
§.§ Hyperbolicity and Genuine Nonlinearity
The Jacobian of (<ref>) is
\begin{equation}
\left(
\begin{array}{cc}
-\frac{y}{v^2} & \frac{1}{v} \\ \\
-\frac{1}{v^2} & 0 \\
\end{array}
\right).
\label{2.2.1}
\end{equation}
The eigenvalues of (<ref>) are
\begin{align}
\lambda_1(v,y)=\frac{-y-\sqrt{y^2-4v}}{2v^2}, \label{2.2.2} \\
\lambda_2(v,y)=\frac{-y+\sqrt{y^2-4v}}{2v^2}. \label{2.2.3}
\end{align}
The eigenvectors are
\begin{align}
\begin{array}{c}
2v \\ \\
y-\sqrt{y^2-4v} \\
\end{array}
\right), \label{2.2.4} \\ \nonumber \\
\begin{array}{c}
2v \\ \\
y+\sqrt{y^2-4v} \\
\end{array}
\right)
. \label{2.2.5}
\end{align}
The system (<ref>) is strictly hyperbolic when $4v<y^2,$ and non-hyperbolic when $4v>y^2.$ On $y^2=4v,$ $\lambda_1=\lambda_2,$ and $r_1=r_2.$
For the system (<ref>), since $v_i$ as well as $u_i$ must be positive, Mazzotti, considered only states with $1-u_1+u_2>0$ and data in the hyperbolic part of state space in the closure of the open component neighboring the origin. This physically meaningful experimental situation for (<ref>) corresponds to the region bounded by a curvilinear triangle with vertices
$$O=\left(\alpha, -\frac{\alpha_1+\alpha_2}{\alpha}\right), \ \ A=\left(\frac{\alpha_1}{\alpha_2}\alpha, -\frac{2\alpha_1}{\alpha}\right), \ \ B=\left(\frac{\alpha_2}{\alpha_1}\alpha, -\frac{2\alpha_2}{\alpha}\right),$$
where $\alpha=(\alpha_1 \alpha_2)^{1/3}$ and sides
$$\text{OA:}\ \ y=-\frac{\alpha_2 v}{\alpha^2}-\frac{\alpha_1}{\alpha},$$
$$\text{OB:}\ \ y=-\frac{\alpha_1 v}{\alpha^2}-\frac{\alpha_2}{\alpha}.$$
Therefore $v>0,$ $y<0$ and $\lambda_1(v,y), \ \ \lambda_2(v,y)>0.$
Since $D \lambda_i r_i\neq 0$ if $y^2\neq\frac{16}{3}v$ then the states below $y^2=4v$ and above $y^2=\frac{16}{3}v$ are genuinely nonlinear for both $i$-characteristic families. To stay in the strictly hyperbolic, genuinely nonlinear physically feasible region we need $\frac{\alpha_2}{3}<\alpha_1<3\alpha_2.$
§.§ Rarefaction Curves Through the Left State $U_L$ in the Hyperbolic Region
For $i=1$ or $2,$ the $i$-rarefaction curves are solutions of the system
\begin{equation}
\left(\begin{array}{c}
\dot{v} \\
\dot{y} \\
\end{array}
\right)=\left(
\begin{array}{c}
2v \\
y\mp\sqrt{y^2-4v} \\
\end{array}
\right),
\label{2.3.1}
\end{equation}
where overdot denotes derivative with respect to $\xi=\lambda_i(v,y).$
By the change of variables $w=\sqrt{\frac{y^2-4v}{v}},$ we get $\frac{d}{dv}(w)=\pm \frac{\sqrt{w^2+4}}{2v}$ Upon separation of the variables, integration, further calculations, and returning to the $U$ variables we derive
\begin{align}
R_1(v_L,y_L): \sqrt{y^2-4v}-y=& \frac{v}{v_L}(\sqrt{y_L^2-4v_L}-y_L),
\label{2.3.2} \\
R_2(v_L,y_L): \sqrt{y^2-4v}-y=& \sqrt{y_L^2-4v_L}-y_L.
\label{2.3.3}
\end{align}
The curves $R_1$ and $R_2$ lie in the closure of the hyperbolic region and intersect only at $U_L.$ The curves $R_1$ and $y^2=4v$ intersect (tangentially) at
\begin{equation}
\frac{4v_L}{\sqrt{y_L^2-4v_L}-y_L} \right);
\label{2.3.4}
\end{equation}
the curve $R_1$ and the line $\text{OB}$ intersect at
\begin{equation}
U_H=(v_H,y_H)=\left(\frac{-4\alpha^2 v_L^2+2\alpha \alpha_2 v_L(\sqrt{y_L^2-4v_L}-y_L)}{\alpha^2(\sqrt{y_L^2-4v_L}-y_L)^2-2\alpha_1 v_L(\sqrt{y_L^2-4v_L}-y_L)}, -\frac{\alpha_1v_H}{\alpha^2}-\frac{\alpha_2}{\alpha} \right);
\label{2.3.5}
\end{equation}
the curve $R_2$ and the line $\text{OA}$ intersect at
\begin{equation}
U_C=(v_C,y_C)=\left(\frac{(2\alpha \alpha_1-\alpha^2\sqrt{y_L^2-4v_L}+\alpha^2 y_L)(\sqrt{y_L^2-4v_L}-y_L)}{4\alpha^2-2\alpha_2(\sqrt{y_L^2-4v_L}-y_L)}, -\frac{\alpha_2 v_C}{\alpha^2}-\frac{\alpha_1}{\alpha}\right);
\label{2.3.6}
\end{equation}
the curves $R_2$ and $y^2=4v$ intersect at
\begin{equation}
U_D=(v_D,y_D)=\left(\frac{(y_L-\sqrt{y_L^2-4v_L})^2}{4},y_L-\sqrt{y_L^2-4v_L} \right);
\label{2.3.7}
\end{equation}
the curve $R_2$ and the line $\text{OB}$ intersect at
\begin{equation}
U_E=(v_E,y_E)=\left(\frac{(2\alpha \alpha_2-\alpha^2\sqrt{y_L^2-4v_L}+\alpha^2 y_L)(\sqrt{y_L^2-4v_L}-y_L)}{4\alpha^2-2\alpha_1(\sqrt{y_L^2-4v_L}-y_L)}, -\frac{\alpha_1 v_E}{\alpha^2}-\frac{\alpha_2}{\alpha} \right).
\label{2.3.8}
\end{equation}
The portion of $R_i$ with $v<v_{L}$ corresponds to an admissible rarefaction of the $i$th family, $i=1$ or $2.$
§.§ Shock Curves Through the Left State $U_L$ in the Hyperbolic Region
Using the Rankine-Hugoniot jump conditions,
\begin{align}
\label{2.4.1} \\
\label{2.4.2}
\end{align}
we derive
\begin{align}
y&=\frac{v y_L}{2v_L}+\frac{y_L}{2}\pm \frac{1}{2}\frac{(v-v_L)}{v_L}\sqrt{y_L^2-4v_L}; \label{2.4.3}
\end{align}
The choice of sign for $S_1$ and $S_2$ is found by calculating
\begin{align*}
\frac{dy}{dv}|_{u=u_{L}}&=\frac{y_L}{2v_L}\pm \frac{\sqrt{y_L^2-4v_L}}{2v_L}, \\
\frac{dR_1}{dv}|_{u=u_{L}}&=\frac{y_L}{2v_L}-\frac{\sqrt{y_L^2-4v_L}}{2v_L}, \\
\frac{dR_2}{dv}|_{u=u_{L}}&=\frac{y_L}{2v_L}+\frac{\sqrt{y_L^2-4v_L}}{2v_L}.
\end{align*}
Since shock and rarefaction curves have second order contact at $U_L,$ we conclude that the states that can be connected to $U_L$ by a 1-shock or 2-shock lie on the curves
\begin{align}
S_1(v_L,y_L): y&=v(\frac{y_L}{2v_L}-\frac{\sqrt{y_L^2-4v_L}}{2v_L})+\frac{y_L}{2}+\frac{\sqrt{y_L^2-4v_L}}{2} \label{2.4.4}
\end{align}
\begin{align}
S_2(v_L,y_L): y&=v(\frac{y_L}{2v_L}+\frac{\sqrt{y_L^2-4v_L}}{2v_L})+\frac{y_L}{2}-\frac{\sqrt{y_L^2-4v_L}}{2} \label{2.4.5}
\end{align}
respectively. The curves $S_1$ and $S_2$ intersect at $U_L.$
§.§ The Lax Shock Admissibility Criterion and Classical Riemann Solutions
By (<ref>)-(<ref>)
\begin{align}
s_1 &=\frac{-y_L-\sqrt{y_L^2-4v_L}}{2v v_L},\label{2.5.1} \\
s_2 &=\frac{-y_L+\sqrt{y_L^2-4v_L}}{2v v_L}.\label{2.5.2}
\end{align}
From the eigenvalues (<ref>)-(<ref>), we conclude that $\lambda_1(v_L,y_L)>s_1>\lambda_1(v,y)$ and $\lambda_2(v_L,y_L)>s_2>\lambda_2(v,y)$ when $v>v_{L}.$ Therefore the admissible parts of the shock curves consist of points with $v>v_{L}$ and the curves of admissible rarefactions, as stated in the previous section, consist of points with $v<v_{L}$ (if $U_L$ is the state on the left).
The curve $S_1$ and the line $\text{OB}$ intersect at
\begin{equation}
U_H=(v_H,y_H)=\left(\frac{-4\alpha^2 v_L^2+2\alpha \alpha_2 v_L(\sqrt{y_L^2-4v_L}-y_L)}{\alpha^2(\sqrt{y_L^2-4v_L}-y_L)^2-2\alpha_1 v_L(\sqrt{y_L^2-4v_L}-y_L)}, -\frac{\alpha_1v_H}{\alpha^2}-\frac{\alpha_2}{\alpha} \right);
\label{2.5.3}
\end{equation}
the curves $S_2$ and $y^2=4v$ intersect at
\begin{equation}
U_D=(v_D,y_D)=\left(\frac{(y_L-\sqrt{y_L^2-4v_L})^2}{4},y_L-\sqrt{y_L^2-4v_L} \right);
\label{2.5.4}
\end{equation}
the curve $S_2$ and the line $\text{OB}$ intersect at
\begin{equation}
U_E=(v_E,y_E)=\left(\frac{(2\alpha \alpha_2-\alpha^2\sqrt{y_L^2-4v_L}+\alpha^2 y_L)(\sqrt{y_L^2-4v_L}-y_L)}{4\alpha^2-2\alpha_1(\sqrt{y_L^2-4v_L}-y_L)}, -\frac{\alpha_1 v_E}{\alpha^2}-\frac{\alpha_2}{\alpha} \right).
\label{2.5.5}
\end{equation}
Figure <ref> sketches these curves.
Rarefaction and Shock Curves.
Using the results of Sections 2.3 and 2.4 and equations (<ref>)-(<ref>), we see that in a neighborhood of $U_L$ there exist the usual four regions for the solution of the Riemann problem.
Specifically, we have
* Region 1: the unique solution consist of a 1-shock followed by a 2-shock. The region is bounded by $S_1(U_L),$ $S_2(U_L)$ and the line $\text{HE}$ with $v>v_{L}.$
* Region 2: the unique solution consist of a 1-rarefaction followed by a 2-rarefaction. We observe that for any $U_L$ the curve $R_1(U_L)$ becomes tangent to $y^2=4v$ at the point $U_G$ identified in equation (<ref>). The (smooth) continuation of this curve is in fact an $R_2$ curve. This curve and the line $\text{OA}$ intersect at
\begin{equation}
U_F=(v_F,y_F)=\left(\frac{-4\alpha^2 v_L^2+2\alpha \alpha_1 v_L(\sqrt{y_L^2-4v_L}-y_L)}{\alpha^2(\sqrt{y_L^2-4v_L}-y_L)^2-2\alpha_2 v_L(\sqrt{y_L^2-4v_L}-y_L)}, -\frac{\alpha_2v_F}{\alpha^2}-\frac{\alpha_1}{\alpha} \right);
\label{2.5.6}
\end{equation}
The region is bounded by $R_2(U_L),$ the line $\text{FC}$ and the curve which begins as $R_1(U_L)$ and continues as $R_2(U_G).$
* Region 3: the unique solution consist of a 1-rarefaction followed by a 2-shock. The region is bounded since only the finite interval of $R_1(U_L)$ between $U_L$ and $U_G$ is available for the intermediate state $U_M.$ Furthermore, the interval of admissible points $U_R\in S_2(U_M)$ terminates at a point $U_D(U_M)$ at which the shock speed $s_2=\lambda_1(U_M).$ The upper boundary of Region 3 is the curve $y^2=4v.$ This curve is tangent to $S_2(U_L)$ at the point $U_D.$
* Region 4: the unique solution consist of a 1-shock followed by a 2-rarefaction.The region is bounded by the lines $\text{OH},$ $\text{CO},$ $S_1(U_L)$ and the curve $R_2(U_L).$
Rarefaction and Shock Curves, Regions
§.§ Solutions with a Vacuum State
We now observe that $y^2=4v$ is an invariant curve for (<ref>), and if $(v,y)(x,t)$ is a smooth solution on this curve then $v$ satisfies the equation
\begin{equation}
\end{equation}
Therefore, if $U_R$ is in Region 5 of Figure <ref>, the solution consists of a 1-rarefaction from $U_L$ to $U_G,$ a rarefaction solution to (<ref>) from $U_G$ to a point $U_{AB}(U_R),$ and a 2-rarefaction from $U_{AB}(U_R)$ to $U_R,$ where $U_{AB}(U_R)$ is the point where $R_2(U_R)$ is tangent to $y^2=4v.$
In region 6, outside these five regions, no classical Riemann solution exists. In the rest of this paper, we show that a solution containing a singular shock can be constructed.
§ THE FORMAL CONSTRUCTION OF SINGULAR SHOCKS
This section begins the construction of singular solutions by examining a
self-similar approximation to (<ref>), which provides valuable insight in the GSPT analysis. This will become evident in Section 4.
§.§ Dafermos Regularization
We study systems that approximate (<ref>)-(<ref>). Following Dafermos <cit.>, Dafermos and DiPerna <cit.>, and Keyfitz and Kranzer <cit.>, we analyze the regularization of
by a viscous term following Dafermos's approach:
\begin{equation}
\varepsilon t U_{xx}=U_t+F(U)_x.
\label{3.1.1}
\end{equation}
Using $\xi=\frac{x}{t},$ the initial value problem (<ref>)-(<ref>)
becomes a nonautonomous second-order ODE
\begin{equation}
\varepsilon \frac{d^2 U}{d \xi^2}=\left(DF(U)-\xi I\right)\frac{d U}{d \xi},
\label{3.1.2}
\end{equation}
with boundary conditions
\begin{equation}
U(-\infty)=U_L, \ \ \ U(+\infty)=U_R.
\label{3.1.3}
\end{equation}
Since in the region of interest there are no classical solutions, we seek solutions
that are not uniformly bounded in $\varepsilon$ for $\xi$ near some value $s$.
The following technique, motivated by
Keyfitz and Kranzer <cit.>, provides a formal solution.
We develop this and then show in Section 4, following Schecter <cit.>,
that for sufficiently small $\varepsilon>0$, (<ref>) possesses solutions with the
qualitative behavior we predict in this section.
\begin{equation}
U(\xi)= \begin{pmatrix}
v(\xi) \\ \\
\end{pmatrix},
\label{3.1.4}
\end{equation}
$$v(\xi)=\frac{\varepsilon^2 \tilde{u}_2(\frac{\xi-s}{\varepsilon^q})}{(\tilde{u}_1^{2/3}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_3})^{3/2}}-\varepsilon^{\beta_4}, \ \ \ y(\xi)=\frac{\varepsilon \tilde{u}_2^2(\frac{\xi-s}{\varepsilon^q})}{(\tilde{u}_1^{16/15}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_2})^{3/2}}-\varepsilon^{\beta_1},$$
where $\beta_1>1,$ $\beta_4>\frac{41}{15}$ (the values of $\beta_i,$ $i=1,\ldots,4$ are not unique and are chosen so as to ensure the desired behavior) and define $\eta=\frac{\xi-s}{\varepsilon^q}$.
Then (<ref>) becomes
\begin{equation}
\left\{
\begin{array}{ll}
\varepsilon^{3-q}\left(\dfrac{\tilde{u}_2}{(\tilde{u}_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right)_{\eta\eta}\ \ &=-(\varepsilon^q \eta+s)\varepsilon^2\left(\dfrac{\tilde{u}_2}{(\tilde{u}_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right)_{\eta}\\
\left(\dfrac{\left[{ \tilde{u}_2^2(\frac{\xi-s}{\varepsilon^q})}-\varepsilon^{\beta_1-1}(\tilde{u}_1^{16/15}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_2})^{3/2}\right](\tilde{u}_1^{2/3}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_3})^{3/2}}{\left[{ \tilde{u}_2(\frac{\xi-s}{\varepsilon^q})}-\varepsilon^{\beta_4-2}(\tilde{u}_1^{2/3}(\dfrac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_3})^{3/2}\right](\tilde{u}_1^{16/15}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_2})^{3/2}}\right)_{\eta}, \\ \\
\varepsilon^{2-q}\left(\dfrac{\tilde{u}_2^2}{(\tilde{u}_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right)_{\eta\eta}&=-(\varepsilon^q \eta+s)\varepsilon\left(\dfrac{\tilde{u}_2^2}{(\tilde{u}_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right)_\eta\\
&+\varepsilon^{-2}\left(\dfrac{(\tilde{u}_1^{2/3}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_3})^{3/2}}{{ \tilde{u}_2(\frac{\xi-s}{\varepsilon^q})}-\varepsilon^{\beta_4-2}(\tilde{u}_1^{2/3}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_3})^{3/2}}\right)_\eta.
\end{array}
\right.
\label{3.1.5}
\end{equation}
We balance at least two terms in each equation, so that nontrivial solutions can be found. Thus we set $3-q=-1$ in the first equation, and $2-q=-2$ in the second.
This gives $q=4$ and hence
\begin{equation}
\left\{
\begin{array}{ll}
\left(\frac{\tilde{u}_2}{(\tilde{u}_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right)_{\eta\eta}\ \ &=-(\varepsilon^q \eta+s)\varepsilon^3\left(\frac{\tilde{u}_2}{(\tilde{u}_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right)_{\eta}\\
\left(\dfrac{\left[{ \tilde{u}_2^2(\frac{\xi-s}{\varepsilon^q})}-\varepsilon^{\beta_1-1}(\tilde{u}_1^{16/15}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_2})^{3/2}\right](\tilde{u}_1^{2/3}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_3})^{3/2}}{\left[{ \tilde{u}_2(\frac{\xi-s}{\varepsilon^q})}-\varepsilon^{\beta_4-2}(\tilde{u}_1^{2/3}(\dfrac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_3})^{3/2}\right](\tilde{u}_1^{16/15}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_2})^{3/2}}\right)_{\eta}, \\ \\
\left(\frac{\tilde{u}_2^2}{(\tilde{u}_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right)_{\eta\eta}&=-(\varepsilon^q \eta+s)\varepsilon^3\left(\frac{\tilde{u}_2^2}{(\tilde{u}_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right)_\eta\\
&+\left(\dfrac{(\tilde{u}_1^{2/3}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_3})^{3/2}}{{ \tilde{u}_2(\frac{\xi-s}{\varepsilon^q})}-\varepsilon^{\beta_4-2}(\tilde{u}_1^{2/3}(\frac{\xi-s}{\varepsilon^q})+\varepsilon^{\beta_3})^{3/2}}\right)_\eta.
\end{array}
\right.
\label{3.1.6}
\end{equation}
The singular region is narrower than a standard shock profile.
When we expand $\tilde{u}_1$, $\tilde{u}_2$ as series in $\varepsilon$
$$\tilde{u}_1=y_1(\eta)+o(1), \ \ \tilde{u}_2=y_2(\eta)+o(1),$$
we obtain
\begin{equation}
\left\{
\begin{array}{ll}
\left(\frac{y_2}{y_1}\right)_{\eta\eta}\ \ \ =\left(\frac{y_2}{y_1^{3/5}}\right)_{\eta}, \\ \\
\left(\frac{y_2^2}{y_1^{8/5}}\right)_{\eta\eta}=\left(\frac{y_1}{y_2}\right)_\eta.
\end{array}
\right.
\label{3.1.7}
\end{equation}
We note that from (<ref>) we must have $y_1, y_2 \rightarrow 0$ as $|\eta|\rightarrow \infty,$ and $\dfrac{y_2}{y_1^{3/5}}, \dfrac{y_1}{y_2}\rightarrow 0$ as $\eta \rightarrow \infty.$ Assuming that the singular behavior is restricted to a neighborhood of $\xi=s$ we also have $\left(\dfrac{y_2}{y_1}\right)_\eta, \left(\dfrac{y_2^2}{y_1^{8/5}}\right)_{\eta} \rightarrow 0$ as $\eta \rightarrow \infty.$ We integrate (<ref>) once, and now focus attention on solutions of
\begin{equation}
\left\{
\begin{array}{ll}
\frac{d y_1}{d\eta}=\frac{5}{2}\left(\frac{y_1^{18/5}}{y_2^3}-2y_1^{7/5}\right),\\ \\
\frac{d y_2}{d\eta}=\frac{5}{2}\frac{y_1^{13/5}}{y_2^2}-4y_2y_1^{2/5},
\end{array}
\right.
\label{3.1.8}
\end{equation}
which approach $(0,0)$ as $|\eta|\rightarrow \infty$.
The phase portrait of the 2-dimensional system (<ref>) is shown in Figure <ref>.
The origin is the unique equilibrium.
$y_2=2^{1/3}y_1^{11/15}$ is an invariant parabola (in $(v,y)$ coordinates this curve is $y^2=2v$). The line $y_2=0$ corresponds to the point $(0,0)$ in $(v,y)$ coordinates. The homoclinic orbits, which are of greatest interest
to us, are solutions $(y_1(\eta),y_2(\eta))$ which can be determined
uniquely by setting $y_1(0)>0$, $y_2(0)>0.$
Integral Curves and Orbits of (<ref>).
We will need to know the asymptotic behavior of $Y=(y_1,y_2)$ as
$|\eta|\rightarrow \infty$.
\begin{align}
y_1&=\frac{c}{|\eta|^p}+{\mathcal O}\left(\frac{1}{|\eta|^{p+1}}\right),\label{3.1.9}\\
y_2&=\frac{d}{|\eta|^r}+{\mathcal O}\left(\frac{1}{|\eta|^{r+1}}\right),
\label{3.1.10}
\end{align}
we substitute (<ref>)-(<ref>) into (<ref>) and then solve for $c,$ $d,$ $p$ and $r$ to obtain
$$c=\left(\frac{2}{3}\right)^{5/2}, \ \ d=3^{1/3}\left(\frac{2}{3}\right)^{13/6}, \ \ r=\frac{11}{6}, \ \ p=\frac{5}{2}$$
as $\eta\rightarrow \infty.$
We also have
\begin{equation}
\label{be}
y_2\approx 2^{1/3}y_1^{11/15}
\end{equation}
as $Y\to 0$.
This describes the asymptotic behavior of $Y$ as $Y\to 0$.
Therefore the homoclinic orbits are tangent to the invariant parabola $y_2\approx 2^{1/3}y_1^{11/15}$ at one end. In addition we have $c=d=0$ as $\eta\rightarrow -\infty.$
The singular solution (<ref>), has its essential support in a layer of width $|\xi-s|=
O(\varepsilon^q)$ with $q>1,$ and tends to zero away from $\xi=s$.
As in Keyfitz and Kranzer <cit.> we embed the singular shock in
a shock profile of the usual type: a solution $\bar{U}(\tau)=\bar{U}(\frac{\xi-s}{\varepsilon})$
which is bounded in the layer $\varepsilon^{q}<|\xi-s|<\varepsilon$, has an expansion
\begin{equation}
\bar{U}=\bar{U}_0+o(1),
\label{3.1.11}
\end{equation}
and whose derivatives are $O(\varepsilon^{-1})$.
Writing (<ref>) in terms of $\tau=\frac{\xi-s}{\varepsilon}$ we have
\begin{equation}
\frac{d}{d\tau}\left(\frac{d\bar{U}}{d \tau}-F(\bar{U})+s\bar{U}\right)=-\varepsilon \tau\frac{d \bar{U}}{d\tau}.
\label{3.1.12}
\end{equation}
Using the expansion (<ref>) we have
\begin{equation}
\frac{d}{d\tau}\left(\frac{d\bar{U}_0}{d \tau}-F(\bar{U}_0)+s\bar{U}_0\right)=0,
\label{3.1.13}
\end{equation}
in each separate interval $\tau<0$ or $\tau>0$ outside the boundary layer.
Hence, we may write
\begin{equation}
\frac{d\bar{U}_0}{d \tau}-F(\bar{U}_0)+s\bar{U}_0=k_{\mp}.
\label{3.1.14}
\end{equation}
On the other hand, integrating (<ref>) over an interval surrounding $\tau = 0$
(the boundary layer), we obtain
\begin{equation}
\left[\frac{d\bar{U}}{d \tau}-F(\bar{U})+s\bar{U}\right]_{\tau<0}^{\tau>0}=-\varepsilon \int_{\tau<0}^{\tau>0}\tau\frac{d\bar{U}}{d\tau}d\tau.
\label{3.1.15}
\end{equation}
Now, from (<ref>),
and we change the variable to $\eta = \tau/\varepsilon^3$
in (<ref>), which yields
\begin{align}
k_{+}-k_{-}&=\lim_{\varepsilon\rightarrow 0}\left\{-\varepsilon \int\varepsilon^3 \eta\left(
\begin{array}{c}
\frac{dv}{d\eta} \\ \\
\frac{dy}{d\eta} \\
\end{array}
\right)d\eta \ \right\}=\lim_{\varepsilon\rightarrow 0}
\begin{pmatrix}
-\varepsilon^6 \int\eta \frac{d}{d\eta}\left(\frac{\tilde{u}_2}{(\tilde{u}_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right) d\eta\\ \\
-\varepsilon^{5} \int\eta \frac{d}{d\eta}\left(\frac{\tilde{u}_2^2}{(\tilde{u}_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right) d\eta
\end{pmatrix}\nonumber \\
&=\lim_{\varepsilon\rightarrow 0}
\begin{pmatrix}
-\varepsilon^6 \int\eta \frac{d}{d\eta}\left(\frac{y_2}{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right) d\eta\\ \\
-\varepsilon^{5} \int\eta \frac{d}{d\eta}\left(\frac{y_2^2}{(y_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right) d\eta
\end{pmatrix}\nonumber\\
&=\lim_{\varepsilon\rightarrow 0}
\begin{pmatrix}
-\varepsilon^6 \int_{\text{finite} \ \eta}\eta \frac{d}{d\eta}\left(\frac{y_2}{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right) d\eta-\varepsilon^6 \int_{\text{infinite} \ \eta}\eta \frac{d}{d\eta}\left(\frac{y_2}{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right) d\eta\\ \\
-\varepsilon^{5} \int_{\text{finite} \ \eta}\eta \frac{d}{d\eta}\left(\frac{y_2^2}{(y_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right) d\eta-\varepsilon^{5} \int_{\text{infinite} \ \eta}\eta \frac{d}{d\eta}\left(\frac{y_2^2}{(y_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right) d\eta
\end{pmatrix}.
\label{3.1.16}
\end{align}
When $\eta$ is finite we notice that for values of $y_1$ and $y_2$ away from the origin $y$ and $v$ are close to zero, therefore we can focus on the case of $y_1, y_2\rightarrow 0.$ If $v\rightarrow 0$ and $y\rightarrow \infty$ then $\varepsilon^4 v\rightarrow 0.$ If $v, y\rightarrow \infty$ then by (<ref>)
In addition $v\simeq \varepsilon^k\sqrt{y}$ where $-1<k<2.5$ and
$$\varepsilon^6\frac{d}{d\eta}\left(\frac{y_2}{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right)=\varepsilon^7 \frac{y}{v}=\frac{\varepsilon^6}{\varepsilon^2}\frac{dv}{d\eta}\approx\frac{\varepsilon^4\varepsilon^k}{2\sqrt{y}}\frac{dy}{d\eta}=\frac{\varepsilon^5\varepsilon^k}{2\sqrt{y}}\frac{d}{d\eta}\left(\frac{y_2^2}{(y_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right)$$
so either after short calculations or integration by parts all cases give
$$\lim_{\varepsilon\rightarrow 0} \varepsilon^6\int_{\text{finite} \ \eta}\eta \frac{d}{d\eta}\left(\frac{y_2}{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right) d\eta=0.$$
The interesting behavior which will give us the generalized Rankine-Hugoniot condition emerges as $\eta \rightarrow \infty.$ We use (<ref>)-(<ref>), ignoring the constants $c$ and $d,$ without loss of generality, and letting
$$\frac{1}{\eta^{5/3}}=\varepsilon^{\beta_3} \tan^2\theta$$
to get
\begin{align*}
-\varepsilon^6 \int_{\text{infinite} \ \eta}\eta \frac{d}{d\eta}\left(\frac{y_2}{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\right) d\eta&=-\varepsilon^6\frac{\frac{\eta}{\eta^{11/6}}}{\left(\frac{1}{\eta^{5/3}}+\varepsilon^{\beta_3}\right)^{3/2}}\big|_{\text{infinite} \ \eta}+\varepsilon^6\int_{\text{infinite} \ \eta} \frac{\frac{1}{\eta^{11/6}}}{\left(\frac{1}{\eta^{5/3}}+\varepsilon^{\beta_3}\right)^{3/2}} \ d\eta\\
&=\varepsilon^{6-\beta_3}\sin \theta_0 \cos^2 \theta_0+ \frac{6}{5}\cdot\varepsilon^{6-\beta_3}\int_0^{\theta_0} \cos \theta \ d\theta\simeq\varepsilon^{6-\beta_3},
\end{align*}
for some $\theta_0.$
On the other hand, if we let
$$\frac{1}{\eta^{8/3}}=\varepsilon^{\beta_2}\tan^2 \theta$$
we get
\begin{align*}
-\varepsilon^{5} \int_{\text{infinite} \ \eta}\eta \frac{d}{d\eta}\left(\frac{y_2^2}{(y_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right) d\eta&=-\varepsilon^5 \frac{\frac{\eta}{\eta^{11/3}}}{\left(\frac{1}{\eta^{8/3}}+\varepsilon^{\beta_2}\right)^{3/2}} \big|_{\text{infinite} \ \eta}+
\varepsilon^5\int_{\text{infinite} \ \eta} \frac{\frac{1}{\eta^{11/3}}}{\left(\frac{1}{\eta^{8/3}}+\varepsilon^{\beta_2}\right)^{3/2}} \ d\eta\\
&=\varepsilon^{5-\frac{\beta_2}{2}}\sin^2\theta_1\cos \theta_1+\frac{3}{4}\cdot \varepsilon^{5-\frac{\beta_2}{2}}\int_0^{\theta_1}\sin \theta \ d\theta\simeq \varepsilon^{5-\frac{\beta_2}{2}},
\end{align*}
for some $\theta_1.$
§.§ Possible Cases.
* If $\beta_3=6,$ $\beta_2<10$ then
\begin{equation}
\begin{array}{c}
\ast \\
0 \\
\end{array}
\right).
\nonumber
\end{equation}
By (<ref>), we have $\bar{U}_0(-\infty)=U_L$,
$ \bar{U}_0(+\infty)=U_R$ and $\frac{d\bar{U}_0}{d\tau}(\pm\infty)=0$.
Therefore, from (<ref>) we get the generalized Rankine-Hugoniot condition for singular shocks:
\begin{align}
s_{\text{singular}}(U_L,U_R)&=s=\frac{F_2(U_L)-F_2(U_R)}{y_{L}-y_{R}}, \label{3.2.1} \\
\label{3.2.2}
\end{align}
We notice that we have a deficit in the first component. This does not agree with Mazzotti <cit.>. In addition if we check for the overcompressive region
we see that region 6 is overcompressive but the slope of the curve $s=\lambda_1(v_L,y_L)$ is negative. Therefore the region does not look like the required one, which should cover all possible solutions of the Riemann problem in the plane.
* If $\beta_3<6,$ $\beta_2<10$ then we get the Rankine-Hugoniot confition for both components but this does not give a singular shock.
* If $\beta_3=6,$ $\beta_2=10$ then
\begin{equation}
\begin{array}{c}
\ast \\
\ast \\
\end{array}
\right).
\nonumber
\end{equation}
This means we have a deficit for both components. As $\eta \rightarrow \infty$ the solution (<ref>) in this case behaves like
$$v=\frac{\varepsilon^2 \cdot \frac{1}{\eta^{11/6}}}{\left(\frac{1}{\eta^{5/3}}+\varepsilon^6\right)^{3/2}}-\varepsilon^{\beta_4}, \ \ y=\frac{\varepsilon \cdot \frac{1}{\eta^{11/3}}}{\left(\frac{1}{\eta^{8/3}}+\varepsilon^{10}\right)^{3/2}}-\varepsilon^{\beta_1}.$$
$$\frac{1}{\eta^{1/3}}= \tan\theta,$$
then as $\theta\rightarrow 0$
$$v=\frac{\varepsilon^2\tan^{11/2} \theta}{(\tan^5\theta+\varepsilon^6)^{3/2}}-\varepsilon^{\beta_4}, \ \ y=\frac{\varepsilon\tan^{11}\theta}{(\tan^8 \theta+\varepsilon^{10})^{3/2}}-\varepsilon^{\beta_1}.$$
As $\varepsilon\rightarrow 0$ one should expect $y-v$ to have a bigger maximum value than $v-y$ (as we will see in Figure <ref>). However, this is not the case here. In addition this would not agree with Mazzotti <cit.>.
* If $\beta_3\leqslant 5,$ $\beta_2=10$ then to see how this solution behaves for small $\varepsilon$
as $\eta\rightarrow\infty$ we may let
$$\frac{1}{\eta^{1/3}}=\varepsilon \tan\theta,$$
$$v=\frac{\tan^{11/2} \theta}{(\tan^5\theta+1)^{3/2}}-\varepsilon^{\beta_4}.$$
$v$ remains bounded but since $y$ is unbounded one should expect $v$ to be unbounded as well by (<ref>).
* If $5<\beta_3<6,$ $\beta_2=10$ then
\begin{array}{c}
0 \\
k \\
\end{array}
\right),$$
$$k=-\lim_{\varepsilon\rightarrow 0} \varepsilon^{5} \int\eta \frac{d}{d\eta}\left(\frac{y_2^2}{(y_1^{16/15}+\varepsilon^{\beta_2})^{3/2}}\right) d\eta,$$
defined uniquely by each orbit.
Finally, from (<ref>) we get the generalized Rankine-Hugoniot condition for singular shocks:
\begin{align}
s_{\text{singular}}(U_L,U_R)&=s=\frac{F_1(U_L)-F_1(U_R)}{v_{L}-v_{R}}, \label{3.2.3} \\
\label{3.2.4}
\end{align}
The restriction on the sign of $k$ is consistent with having $U_R$ in region 6
with respect to $U_L$.
We now introduce two curves, as shown in Figure <ref>, namely $J_5$
and $J_6$ determined by
We find
\begin{align}
J_5: y=\frac{y_L}{v_L}v+v(v-v_L)\cdot (\frac{-y_L-\sqrt{y_L^2-4v_L}}{2v_L^2})
\label{3.2.5}
\end{align}
The curve
$J_5$ passes through the point $U_L$ and intersects $y^2=4v$ at a point $U_D.$
The second curve is $J_6$, given by
\begin{align}
J_6: y=\frac{vy_L(2v-v_L)+v^2y_L}{2v_L(2v-v_L)}+\frac{(v-v_L)}{2v_L(2v-v_L)}\sqrt{\left( v y_L-4\frac{v_L^2}{y_L}\right)^2+4v_L^3\frac{(y_L^2-4v_L)}{y_L^2}}
\label{3.2.6}
\end{align}
The curve $J_6$ passes through the point $U_L$ and does not intersect $y^2=4v.$ $\beta_3$ is chosen such that $y$ is unbounded as $y_1, y_2\rightarrow 0$ and $v$ passes from a neighborhood of $0$ (where the variables $u_1,$ $u_2$ of (<ref>) become singular) before becoming unbounded. In addition $y-v$ has a bigger maximum value than $v-y.$
Regions of Singular Shocks, Additional Curves.
We conclude that the forth case agrees with Mazzotti <cit.> whereas all other cases fail.
In the remainder of this paper we show existence of self-similar singular shock
solutions to (<ref>).
Our main result is the following theorem.
In the system of conservation laws (<ref>) with Riemann data (<ref>),
assume that $U_R$ is in the interior of region 6 with respect to $U_L$, so that with
\begin{equation}
\frac{F_1(U_L)-F_1(U_R)}{v_{L}-v_{R}}, \label{3.2.7}
\end{equation}
we have
\begin{equation}
\label{3.2.8}
\end{equation}
and the strict inequalities
* $s_{\text{singular}}(U_L,U_R)<\lambda_1(U_L)$
* $\lambda_2(U_R)<s_{\text{singular}}(U_L,U_R)$
Then there exists a singular shock connecting $U_L$ and $U_R$ passing from points very close to the $y$-axis (thus the chromatography model (<ref>) exhibits singular shocks); that is, a solution
$U_\varepsilon$ of (<ref>)-(<ref>) which becomes
unbounded as $\varepsilon\rightarrow 0$.
§.§ Remarks.
Since we are only interested in the curvilinear triangle $OAB,$ proving existence of a self-similar approximate solution in region 6 – in which the Riemann problem is solved by a strictly overcompressive singular shock alone – completes the list of solutions in regions 1-5, given in Sections <ref> and <ref>.
In Section <ref>, we prove Theorem <ref> by showing existence of solutions
to equations (<ref>) and (<ref>) for small $\varepsilon$.
We use the approach of Schecter <cit.>, which proceeds by modifying
GSP theory <cit.> to take into account that normal hyperbolicity fails
in parts of the construction.
A method for handling loss of normal hyperbolicity, known as “blowing up",
was developed by Krupa and Szmolyan,
<cit.>, and applied by Schecter. Strict overcompressibility is needed, as will be seen, to carry out the construction.
§ EXISTENCE OF APPROXIMATIONS TO SINGULAR SOLUTIONS
We use GSPT to prove Theorem <ref> by showing that self-similar regularized solutions exist for sufficiently small $\varepsilon>0.$ The approach was laid out by Schecter <cit.> and was also employed by Keyfitz and Tsikkou <cit.>.
Basically, the situation described in Section <ref> consist of an “outer” part (which includes the two constant states $U_L$ and $U_R$) and an “inner” part (the scaled homoclinic orbit) with no indication how to connect them. The treatment following (<ref>) did not prove that a solution exists, but just simply suggests a mechanism whereby the two parts of the solution could be connected. This is corrected by Geometric singular perturbation theory (GSPT), using the theory of
dynamical systems to prove that smooth systems, under the appropriate nondegeneracy conditions,
do possess connecting orbits, and even that these orbits are unique.
GSPT was developed by Fenichel <cit.> (see also the exposition by Jones
<cit.>), and despite many efforts, points where normal hyperbolicity breaks down, as in our case, remained a major obstacle to the geometric theory. (A flow is normally hyperbolic with respect to an invariant manifold if any manifold transverse to the flow can be factored into stable and unstable directions. More precisely, if the system is linearized at a point on the invariant manifold, then only the eigenvalues with eigenvectors tangent to the invariant manifold have zero real part.) Krupa and Szmolyan <cit.> applied their technique of “blowing up” to some
examples, but it was Schecter who showed how it could also apply to the system
(<ref>). The insight of GSPT is that one can study the systems when $\varepsilon=0$ and then piece the information together to prove the existence of a genuine orbit when $\varepsilon >0$. Within the framework of GSPT, and following Schecter, we find a way to connect the homoclinic orbit produced in the previous section with the skeleton that joins $U_L$ and $U_R.$
The objective of this section is to apply the theory of dynamical systems to prove existence of an orbit when $\varepsilon>0.$ The important tool is the Exchange Lemma of Jones and Kopell <cit.>, and an extension called by
Schecter <cit.> the Corner Lemma. GSPT approach replaces a dynamical problem, here
(<ref>) and (<ref>), in which a singular limit occurs, with a higher-dimensional
dynamical system in which $\{\varepsilon = 0\}$ is merely a subspace, and behavior
near that subspace can be determined by continuity if the hypotheses of the Exchange
and Corner Lemmas are satisfied. We will describe the pieces of the solution in the singular limit and verify the nondegeneracy hypotheses needed to carry out the perturbation. As could be seen already in Section <ref>, some rescaling of the variables is needed to exhibit any of the dynamics on the fast time scale. In addition, the technique of “blowing up” which involves a change of variables to desingularize the invariant manifold will be used to reveal essential information about the flow and gain additional hyperbolicity.
§.§ Creating the Dynamical Problem
We start from (<ref>)-(<ref>) introducing $V=\left(
\begin{array}{c}
v_1 \\
v_2 \\
\end{array}
\right)=\varepsilon \frac{dU}{d\xi}$, and
$\theta=\xi-s_{\text{singular}}.$ It is also convenient to treat $\xi$ as a state variable. This increases the dimension, but yields an
autonomous system.
Therefore the problem in the original self-similar variable (the slow time $\theta$) is
\begin{equation}
\begin{aligned}
\varepsilon\frac{dv}{d\theta}&=v_1, \\ \varepsilon\frac{dy}{d\theta}&=v_2, \\
\varepsilon\frac{dv_1}{d\theta}&=\frac{v_2}{v}-\frac{y v_1}{v^2}-\xi v_1, \\ \varepsilon\frac{dv_2}{d\theta}&=-\frac{v_1}{v^2}-\xi v_2, \\
\frac{d\xi}{d\theta}&=1.
\end{aligned}
\label{4.1.1}
\end{equation}
As written, this is singular as $\varepsilon \to 0$.
Replacing $\theta$ with $\tau$, where
$\theta=\varepsilon \tau$, we will work in the fast time system
\begin{equation}
\begin{aligned}
\frac{dv}{d\tau}&=v_1, \\ \frac{dy}{d\tau}&=v_2, \\
\frac{dv_1}{d\tau}&=\frac{v_2}{v}-\frac{yv_1}{v^2}-\xi v_1, \\ \frac{dv_2}{d\tau}&=-\frac{v_1}{v^2}-\xi v_2, \\
\frac{d\xi}{d\tau}&=\varepsilon,
\end{aligned}
\label{4.1.2}
\end{equation}
We note that in this problem “slow” and “fast” do not correspond to
“outer” and “inner”.
In fact, we will need an inner, faster time variable ($\eta =\tau/\varepsilon^3$) to describe the inner solution, as done formally in the previous section.
The boundary conditions are
\begin{equation}
(U, V, \xi)(-\infty)=(U_L, 0, -\infty), \quad (U, V, \xi)(+\infty)=(U_R, 0, +\infty).
\label{4.1.3}
\end{equation}
We now let $\varepsilon=0$ in (<ref>),
noting that (<ref>) is now a regularly perturbed problem.
With $\xi = \textit{const.}$ for all solutions, the states $V=0$ are all
equilibria, and they are the only equilibria.
Using the eigenvalues (<ref>)-(<ref>) we identify two subsets of $S$:
For $\delta>0$, we define $3$-dimensional manifolds
\begin{equation*}
\begin{aligned}
S_0&=\{(U,V,\xi): \|U\|\leq\frac{1}{\delta}, \ \ V=0, \ \ \text{and} \ \ \xi\leq \lambda_1(U)-\delta\}, \\
S_2&=\{(U,V,\xi): \|U\|\leq\frac{1}{\delta}, \ \ V=0, \ \ \text{and} \ \ \lambda_2(U)+\delta\leq\xi\}.
\end{aligned}
\end{equation*}
which are normally hyperbolic
since the lines $\xi=\lambda_1(U), \ \ \xi=\lambda_2(U)$ are not included in the sets $S_i$.
In fact, if we linearize (<ref>) and set $\varepsilon=0, \ \ V=0$,
there are $3$ eigenvalues of zero, with a full set of eigenvectors in the space of equilibria.
The remaining eigenvalues, $-\xi+\lambda_1(U)$ and $-\xi+\lambda_2(U)$,
are real and nonzero.
In $S_0$, both are positive, so there is an unstable manifold of dimension $2$;
and in $S_2$ a stable manifold of dimension $2$.
The boundary value $(U_L,0,-\infty)$ is an $\alpha$-limit of points in $S_0$, and
$(U_R,0,+\infty)$ an $\omega$-limit in $S_2$.
By Fenichel <cit.>, and as stated in Schecter <cit.>,
a system with normally hyperbolic manifolds of equilibria has perturbed
normally hyperbolic invariant manifolds nearby.
That is the case here: For $\varepsilon>0$ and near $0$, by Fenichel theory <cit.>,
the system (<ref>) has normally hyperbolic invariant manifolds near
each $S_i$. Since the $3$-dimensional space $S\equiv \{(U,V,\xi): V=0\}$ is invariant under (<ref>) for every $\varepsilon,$ the perturbed manifolds may be taken to be the $S_i$ themselves.
For a given $U_L$, we define the $1$-dimensional invariant set
\begin{equation}
S_0(U_L)=\{(U,V,\xi): U=U_L, \ \ V=0, \ \ \xi< \lambda_1(U_L)\}\,.
\nonumber
\end{equation}
The line $S_0(U_L)$ possesses a $3$-dimensional unstable manifold
$W_{\varepsilon}^{u}(S_0(U_L))$, the perturbation of
\begin{equation}
W_{0}^{u}(S_0(U_L))=\{(U,V,\xi): U\in\Omega_{\xi}, \ \ V=V(U), \ \ \xi< \lambda_1(U_L)\},
\nonumber
\end{equation}
where $\Omega_{\xi}$ is an open subset of $U$-space that depends on $\xi$ and $U_L$.
(The linearization of $W_0^u$ at a point in $S_0$ has a basis of eigenvectors, but
we can ignore them for now, noting only that the projection of $W_0^u$ onto $U$-space
contains a full neighborhood of $U_L$.
The function $V(U)$ is determined by solving the system (<ref>).)
\begin{equation}
S_2(U_R)=\{(U,V,\xi): U=U_R, \ \ V=0, \ \ \lambda_2(U_R)<\xi\}
\nonumber
\end{equation}
is a $1$-dimensional set, which has a $3$-dimensional stable manifold,
$W_{\varepsilon}^{s}(S_2(U_R))$, the perturbation of
\begin{equation}
W_{0}^{s}(S_2(U_R))=\{(U,V,\xi): U\in\Omega_{\xi}, \ \ V=V(U), \ \ \lambda_2(U_R)<\xi\}\,.
\nonumber
\end{equation}
Since every trajectory in $W_{\varepsilon}^u(S_0(U_L))\cap W_{\varepsilon}^s(S_2(U_R))$
tends to $U_R$ as $\tau \to \infty$ and to $U_L$ as $\tau \to -\infty$, our objective is to
show that these two $3$-dimensional manifolds intersect in the $5$-dimensional state space.
As an alternative for the same purpose, we focus attention on the shock layer, and specifically on the difficulties surrounding
the Rankine-Hugoniot relation, which normally is derived from equations (<ref>)
and (<ref>), and replace $V$ in (<ref>) by
$$ W= -V+F(U)-\xi U\,.$$
Also, from now on we treat $\varepsilon$ as a dynamical variable. Then we have the system
\begin{equation}
\begin{aligned}
\frac{dv}{d\tau}&=\frac{y}{v}-\xi v-w_1, \\
\frac{dy}{d\tau}&= \frac{1}{v}-\xi y-w_2, \\
\frac{dw_1}{d\tau}&=-\varepsilon v, \\
\frac{dw_2}{d\tau}&=-\varepsilon y, \\
\frac{d\xi}{d\tau}&=\varepsilon, \\
\frac{d\varepsilon}{d\tau}&=0.
\end{aligned}
\label{4.1.4}
\end{equation}
Each subspace $\varepsilon=$constant is invariant. Corresponding to the $3$-dimensional subsets $S_0$ and $S_2$ we have now $4$-dimensional normally hyperbolic subsets which we write as
\begin{equation*}
\begin{aligned}
T_0&=\{(U,W,\xi, \varepsilon): \ \ \|U\|\leq\frac{1}{\delta}, \ \ W=F(U)-\xi U,
\xi\leq \lambda_1(U)-\delta\},\\
T_2&=\{(U,W,\xi, \varepsilon): \ \ \|U\|\leq\frac{1}{\delta}, \ \ W=F(U)-\xi U,
\lambda_2(U)+\delta\leq\xi\}\,.
\end{aligned}
\end{equation*}
The $1$-dimensional sets $S_0(U_L)$ and $S_2(U_R)$ are now
\begin{equation*}
\begin{aligned}
T_0^{\varepsilon}(U_L)&=\{(U,W,\xi,\varepsilon): \ U=U_L, W=F(U_L)-\xi U_L,
\xi\leq \lambda_1(U_L)-\delta, \ \varepsilon \ \text{fixed} \},\\
T_2^{\varepsilon}(U_R)&=\{(U,W,\xi,\varepsilon): \ U=U_R, W=F(U_R)-\xi U_R,
\xi\geq \lambda_2(U_R)+\delta, \ \varepsilon \ \text{fixed} \},
\end{aligned}
\end{equation*}
and we rewrite the $3$-dimensional unstable manifold $W_\varepsilon^u(S_0(U_L))$ as
\begin{equation*}
W^{u}(T_0^{\varepsilon}(U_L))=\{(U,W, \xi, \varepsilon): \ \
U\in \Omega_{\xi}, \ \ W=W(U), \ \xi<\lambda_1(U_L), \ \ \varepsilon \ \ \text{fixed}\},
\end{equation*}
where now $W(U)$ denotes the solution of (<ref>) corresponding to $U$.
Finally, the $3$-dimensional stable manifold $W^s_\varepsilon(S_2(U_R))$ becomes
a $3$-dimensional space
\begin{equation*}
W^{s}(T_2^{\varepsilon}(U_R))=\{(U,W,\xi, \varepsilon):
U\in\Omega_{\xi}, \ \ W=W(U), \ \lambda_2(U_R)<\xi, \ \ \varepsilon \ \ \text{fixed} \}\,.
\end{equation*}
As with the previous coordinates,
we look for a solution for fixed $\varepsilon>0$ that lies in the intersection of
$W^u(T_0^{\varepsilon}(U_L))$ and $W^s(T_2^{\varepsilon}(U_R))$.
Now we write down an expression for the inner solution,
motivated by the formal derivation given in Section <ref>. The scaling (<ref>) introduces a new variable
\begin{array}{c}
y_1 \\
y_2 \\
\end{array}
\right)$
such that
$$v=\frac{\varepsilon^2 y_2}{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}-\varepsilon^{\beta_4}, \ \ \ y=\frac{\varepsilon y_2^2}{(y_1^{16/15}+\varepsilon^{10})^{3/2}}-\varepsilon^{\beta_1}.$$
The system, with a time variable
$\eta = \tau/\varepsilon^3$ is now
\begin{equation}
\begin{aligned}
\frac{dy_1}{d\eta}&=\frac{5\varepsilon A(y_1, y_2, w_1, w_2, \xi, \varepsilon)}{2(4\varepsilon^{\beta_3}y_2 y_1^{6/15}-5\varepsilon^{10}y_2-y_2y_1^{16/15})}, \\
\frac{dy_2}{d\eta}&=\frac{\varepsilon B(y_1, y_2, w_1, w_2, \xi, \epsilon)}{(4\varepsilon^{\beta_3}y_2 y_1^{6/15}-5\varepsilon^{10}y_2-y_2y_1^{16/15})}, \\
\frac{dw_1}{d\eta}&=-\frac{\varepsilon^6 y_2}{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}+\varepsilon^{\beta_4+4},\\
\frac{dw_2}{d\eta}&=\varepsilon^{\beta_1+4}-\frac{\varepsilon^5y_2^2}{(y_1^{16/15}+\varepsilon^{10})^{3/2}}, \\
\frac{d\xi}{d\eta}&=\varepsilon^4, \\
\frac{d\varepsilon}{d\eta}&=0,
\end{aligned}
\label{4.1.5}
\end{equation}
\begin{align*}
A(y_1, y_2, \varepsilon, w_1, w_2, \xi)&=\frac{2(y_1^{2/3}+\varepsilon^{\beta_3})^4 y_1^{1/3}y_2^2}{\varepsilon(y_1^{16/15}+\varepsilon^{10})^{1/2}[y_2-\varepsilon^{\beta_4-2}(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}]}-\xi \varepsilon^2 y_1^{1/3}y_2(y_1^{2/3}+\varepsilon^{\beta_3})(y_1^{16/15}+\varepsilon^{10})\\
&-\frac{y_1^{1/3}(y_1^{2/3}+\varepsilon^{\beta_3})^{5/2}(y_1^{16/15}+\varepsilon^{10})^{5/2}}{\varepsilon y_2[y_2-\varepsilon^{\beta_4-2}(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}]}+2\xi y_1^{1/3}\varepsilon^{\beta_4}(y_1^{2/3}+\varepsilon^{\beta_3})^{5/2}(y_1^{16/15}+\varepsilon^{10})\\
&+\frac{\varepsilon w_2 y_1^{1/3}(y_1^{2/3}+\varepsilon^{\beta_3})(y_1^{16/15}+\varepsilon^{10})^{5/2}}{y_2}-\frac{\varepsilon^{1+\beta_1} \xi y_1^{1/3}(y_1^{2/3}+\varepsilon^{\beta_3})(y_1^{16/15}+\varepsilon^{10})^{5/2}}{y_2},\\
B(y_1, y_2, \varepsilon, w_1, w_2, \xi)&=(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}(4\varepsilon^{\beta_3}y_2y_1^{6/15}+4y_2y_1^{16/15})\\
&\cdot\bigg[\frac{y_2^2(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}{\varepsilon(y_1^{16/15}+\varepsilon^{10})^{3/2}[y_2-\varepsilon^{\beta_4-2}(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}]}-\frac{\varepsilon^2\xi [y_2-\varepsilon^{\beta_4-2}(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}]}{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\\
&-\frac{\varepsilon^{\beta_1-1}(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}{\varepsilon [y_2-\varepsilon^{\beta_4-2}(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}]}-w_1\bigg]\\
&-\frac{5}{2}\varepsilon(y_1^{16/15}+\varepsilon^{10})^{5/2}\bigg[\frac{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}{\varepsilon^2 [y_2-\varepsilon^{\beta_4-2}(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}]}-\frac{\xi \varepsilon y_2^2}{(y_1^{16/15}+\varepsilon^{10})^{3/2}}+\varepsilon^{\beta_1} \xi-w_2\bigg].
\end{align*}
The difficulty lies in matching the two outer solutions, expressed in $v$ and $y,$ satisfying the boundary conditions (<ref>), with an inner solution, expressed in $y_1,$ $y_2.$ System <ref> is of fundamental importance since it is not clear that one could use GSPT without some prior information about the asymptotics of the inner solution.
When $\varepsilon = 0$, the equation for $Y$ decouples from the rest of the
system, and is exactly (<ref>). Thus, (<ref>) when $\varepsilon = 0$ is
\begin{equation}
\begin{aligned}
\frac{dy_1}{d\eta}&=\frac{5}{2}(\frac{y_1^{18/5}}{y_2^3}-2y_1^{7/5}), \\
\frac{dy_2}{d\eta}&=\frac{5}{2}\frac{y_1^{13/5}}{y_2^2}-4y_2y_1^{2/5}, \\
\frac{dw_1}{d\eta}&=-\frac{\varepsilon^6y_2}{(y_1^{2/3}+\varepsilon^{\beta_3})^{3/2}}\bigg|_{\varepsilon=0}=C(y_1,y_2),\\
\frac{dw_2}{d\eta}&=-\frac{\varepsilon^5 y_2^2}{(y_1^{16/15}+\varepsilon^{10})^{3/2}}\bigg|_{\varepsilon=0}=D(y_1, y_2), \\
\frac{d\xi}{d\eta}&=0, \\
\frac{d\varepsilon}{d\eta}&=0.
\end{aligned}
\label{4.1.6}
\end{equation}
The fact that $w_1$ and $w_2$ behave differently from each other is an
indication that the asymmetry in the generalized Rankine-Hugoniot relation will
enter into the analysis.
Desingularization of the system (by rescaling the time variable) on the set $y_1=0, y_2=0, \varepsilon=0$ shows that $E=\{(Y,W,\xi,\varepsilon): Y=0, \ \ \varepsilon=0\}$ is a $3$-dimensional space consisting
entirely of equilibria.
If we linearize at a point in $E$, we find that all $6$
eigenvalues are zero.
This is exactly the situation found by Schecter in <cit.> and a blow-up is necessary to resolve the behavior of the system near $E.$
§.§ The Blow-Up Construction
Under the change of variables
\begin{equation}
\begin{aligned}
y_1&=\bar{r}\bar{y}_1, \\
y_2&=\bar{r}^{11/15}\bar{y}_2, \\
w_1&=w_1, \\
w_2&=w_2, \\
\xi&=\xi, \\
\varepsilon&=\bar{r}\bar{\varepsilon}.
\end{aligned}
\label{4.2.1}
\end{equation}
with $|\bar{Y}|^2+\bar{\varepsilon}^2=1$,
the set $E$ becomes the set $\{\bar{r}=0\}$.
This set is now $5$-dimensional, in the $6$-dimensional $(\bar{Y},\bar{\varepsilon},\bar{r},W,
\xi)$-space
$\mathbf{X} = S^2\times \mathbb{R}_{+} \times \mathbb{R}^3$.
The system is also highly singular at $\{\bar{r}=0\}$,
but becomes non-singular upon division by $\bar{r}^{2/5}$.
Thus, we can study the dynamics of the transformed system on $\mathbf{X}$.
In terms of asymptotic structure, the change of variables (<ref>) couples
the growth of $U$ to the limit $\varepsilon \to 0$ in the fashion predicted by the
formal asymptotics.
The range of $\bar{Y}$ and $\bar{\varepsilon}$ is confined to the unit sphere, but the dynamics of these variables can be explored since we can find invariant sets of low dimension of $\mathbf{X}$ and establish normally hyperbolicity. This will explain the connection between the bounded and unbounded parts of the singular shock. The homoclinic solution of Section <ref> provides the inner dynamics and connecting the inner solution to the limit points $U_L$ and $U_R$
can now be pursued.
We now define two intermediate points $q_L$ and $q_R$ which serve as bridge columns connecting the inner and outer solutions. The connection between the homoclinic orbit, which can be identified as
the unique solution to (<ref>) for which $w_{L2}-w_{R2} = k$ (the Rankine-Hugoniot
deficit, from equation (<ref>)), and the states $U_L$ and $U_R$, which are limit
points of the manifolds $W^u(T_0^\varepsilon(U_L))$ and $W^s(T_2^\varepsilon(U_R))$
(for $\varepsilon \geq 0$), will be described. By making the transition from the unscaled variables $(U,W,\xi,\varepsilon)$
to the coordinate system in $\mathbf{X}$ we shall show that there is a unique orbit connecting $U_L$ with $q_L$. The connection between
$q_L$ and $q_R$ is via the homoclinic orbit and finally, $q_R$
connects to $U_R$ in the same manner as $U_L$ to $q_L$.
Because the beginning and ending connections are similar, in the sequel we will look only
at the first two steps.
Figure <ref> gives a sketch of the key parts of the solution.
We begin with the definition of the intermediate points $q_L$ and $q_R$. In the coordinate system just introduced on $\mathbf{X}$, they are
\begin{align}
q_L&=(\frac{\bar{y}_2^{15/11}}{a_3},\bar{y}_2,0,0,W_L,s) \label{4.2.2}\\
q_R&=(\frac{\bar{y}_2^{15/11}}{a_2},\bar{y}_2,0,0,W_R,s) \label{4.2.3}
\end{align}
where we have written the coordinates in the order $(\bar{Y},\bar{\varepsilon},\bar{r},W,\xi)$;
$s$ is the speed of the singular shock, from (<ref>);
$a_2$ and $a_3$ are the two roots
(in decreasing order) of
\begin{equation}\label{4.2.4}
a(a^{11/5}-2)=0 \;
\end{equation}
and $\bar{y}_2$ is the positive root of $\bar{y}_2^2+\frac{\bar{y}_2^{30/11}}{a_i^2}-1=0$
(so that $|\bar{Y}|^2+\bar{\varepsilon}^2=1$).
\begin{equation}\label{4.2.5}
W_L=F(U_L)-sU_L, \quad W_R=F(U_R)-sU_R\,;
\end{equation}
we recall that $W=F(U_i)-\xi U_i$ ($i=L,R$) is the value of $W$ on the
invariant sets $T_0(U_L)$ and $T_2(U_R)$, so $q_L$ and $q_R$ are
specified by selecting the shock speed for $\xi$.
§.§ The First Stage of the Flow
From the description of the underlying planar system $U'=F(U)$ or $Y'=F(Y)$
and the sketch in Figure <ref>, it is intuitively clear that the flow trajectories
are roughly parabolic.
Specifically, if we consider (<ref>) with $\varepsilon =0$, $\xi =s$ and $W=W_L=
F(U_L)-sU_L$, then the equilibrium $U_L$ is a source.
The planar system $U'=F(U)-sU -W_L$ contains a negatively invariant region to the left of $U_L$, bounded by
\begin{align*}
\phi_1(v)&=y_L-E(v-v_L),\\
\phi_2(v)&=\frac{1}{s}\left(\frac{1}{v}-\frac{1}{v_L}\right)+y_L,
\end{align*}
where $E$ is such that
\begin{align*}
v_L \lambda_1(v_L,y_L)<E<v_L \lambda_2(v_L,y_L).
\end{align*}
A calculation of $U'$ along the curves $\phi_i$, similar to Lemma 3.2 in <cit.>,
gives the result.
If we now consider (<ref>) with $\varepsilon =0$, $\xi =s$ and $W=W_R=
F(U_R)-sU_R$, then the equilibrium $U_R$ is a sink.
The planar system $U'=F(U)-sU -W_R$ contains a positively invariant region to the right of $U_R$, bounded by
\begin{align*}
\phi_1(v)&=y_R-E(v-v_R),\\
\phi_2(v)&=sv(v-v_R)+\frac{y_R}{v_R}v,
\end{align*}
where $E$ is such that
\begin{align*}
v_R \lambda_1(v_R,y_R)<E<v_R \lambda_2(v_R,y_R),
\end{align*}
and a negatively invariant region to the left of $U_R$ bounded by
\begin{align*}
\phi_3(v)&=\frac{1}{s}\left(\frac{1}{v}-\frac{1}{v_R}\right)+y_R,
\end{align*}
and the coordinate axes.
A calculation of $U'$ along the curves $\phi_i$, similar to Lemma 3.2 in <cit.>,
gives the result.
Invariant Regions.
In particular, this means that trajectories within the curvilinear wedge between the
two curves of Proposition <ref> and the $v$-axis all have $U_L$ as their $\alpha$-limits and similarly the trajectories within the open curvilinear wedge between the two curves $\phi_1$ and $\phi_2$ of Proposition <ref> all have $U_R$ as their $\omega$-limits.
The trajectory beginning near $U_L$ becomes unbounded but the ratio $\frac{y_2^{15/11}}{y_1}$
remains bounded.
This motivates introducing a new coordinate chart on $\mathbf{X}$, which we will call
Chart 2, following Schecter's terminology in <cit.>.
2$a_2, \ \xi=s_{\text{singular}}, \ \bar{q}_R$
20$a_2, \ \xi=s_{\text{singular}}, \ {q}_R$
5$W^u(T_0^0(U_L)), \ \xi<s_{\text{singular}}$
6$W^u(N_0^0(U_L)), \ \xi<s_{\text{singular}}$
11$W^s(N_2^0(U_R)), \ \xi>s_{\text{singular}}$
12$W^s(T_2^0(U_R)), \ \xi>s_{\text{singular}}$
Chart 1 and 2.
In terms of the coordinates $(\bar{Y},\bar{\varepsilon},\bar{r})$
(and, for reference, the scaled coordinates $(Y,\varepsilon)$ and the original
coordinates $(U,\varepsilon)$), we define, on the portion of
$\mathbf{X}$ where $\bar{y}_1$ $\bar{y}_2$ are positive,
\begin{equation}
\begin{aligned}
&a=\frac{\bar{y}_2^{15/11}}{\bar{y}_1}=\frac{y_2^{15/11}}{{y_1}}\left(\sim \frac{(y+\varepsilon^{\beta_1})^{10/11}}{(v+\varepsilon^{\beta_4})^{5/11}}\ \text{when} \ v, y \ \text{are large}\right), \\
&r=\bar{r}\bar{y}_2^{15/11}=y_2^{15/11}\left(\sim\frac{\varepsilon^{15/2}(y+\varepsilon^{\beta_1})^{75/22}}{(v+\varepsilon^{\beta_4})^{60/11}}\ \text{when} \ v, y \ \text{are large}\right),\\
&b=\frac{\bar{\varepsilon}}{{\bar{y}_2}^{15/11}}=\frac{\varepsilon}{{y_2}^{15/11}}\left(\sim\frac{(v+\varepsilon^{\beta_4})^{60/11}}{{\varepsilon^{13/2}(y+\varepsilon^{\beta_1})^{75/22}}}\ \text{when} \ v, y \ \text{are large}\right),
\end{aligned}
\label{4.3.1}
\end{equation}
and rescale the time variable to $\dfrac{r^{2/5}}{a^{39/15}}\eta$, which we will call $\zeta.$ This desingularizes the system (necessary to obtain a nontrivial flow) on the set $r=0, a=0$ but leaves it invariant. In these coordinates,
the system (<ref>) becomes
\begin{equation}
\begin{aligned}
\frac{d a}{d\zeta}&=\frac{a}{(4F-5G-1)}\bigg\{-\frac{75}{22}(1+G)^{5/2}\cdot\left(\frac{(1+F)^{3/2}}{(1-\Theta)}-\frac{\xi r^{39/15}a^{13/5}b^3}{(1+G)^{3/2}}-r^{26/15}ab^2w_2+r^{\beta_1}r^{26/15}ab^{2+\beta_1} \xi\right)\\
&+\frac{60}{11}a^{24/15}(1+F)^{5/2}\left(\frac{a^{3/5}(1+F)^{3/2}}{(1+G)^{3/2}(1-\Theta)}-\frac{r^{13/5} ab^3\xi(1-\Theta)}{(1+F)^{3/2}}-r^{13/15}bw_1-\frac{r^{2/15}r^{\beta_1-1}b^{\beta_1-1}}{a(1-\Theta)}(1+F)^{3/2}\right)\\
&-\frac{5a^{33/15}(1+F)^4}{(1+G)^{1/2}(1-\Theta)}+\frac{5}{2}\xi r^{39/15}a^{39/15}b^3(1+F)(1+G)+\frac{5}{2}(1+F)^{5/2}(1+G)^{5/2}\\
&+5bw_1 r^{13/15}a^{24/15}(1+F)^{5/2}(1+G)-\frac{5}{2}w_2r^{26/15}ab^2(1+F)(1+G)^{5/2}\\
&+\frac{5r^{\beta_1-1}r^{2/15}a^{3/5}b^{\beta_1-1}(1+F)^4(1+G)}{(1-\Theta)}+\frac{5}{2}\xi r^{\beta_1+1}r^{11/15}a b^{2+\beta_1}(1+F)(1+G)^{5/2}\bigg\},
\end{aligned}
\label{4.3.2}
\end{equation}
\begin{align*}
\frac{dr}{d\zeta}&=\frac{15r}{11(4F-5G-1)}\bigg\{-\frac{5}{2}(1+G)^{5/2}\\
&\cdot\left(\frac{(1+F)^{3/2}}{(1-\Theta)}-\frac{\xi r^{39/15}a^{13/5}b^3}{(1+G)^{3/2}}-r^{26/15}ab^2w_2+r^{\beta_1}r^{26/15}ab^{2+\beta_1} \xi\right)\\
&+4a^{24/15}(1+F)^{5/2}\left(\frac{a^{3/5}(1+F)^{3/2}}{(1+G)^{3/2}(1-\Theta)}-\frac{r^{13/5} ab^3\xi(1-\Theta)}{(1+F)^{3/2}}-r^{13/15}bw_1-\frac{r^{2/15}r^{\beta_1-1}b^{\beta_1-1}}{a(1-\Theta)}(1+F)^{3/2}\right)\bigg\}, \\
\frac{dw_1}{d\zeta}&=-\frac{r^{16/3}a^{54/15}b^6}{(1+F)^{3/2}}+a^{39/15}b^{4+\beta_4}r^{\beta_4}r^{54/15}, \\
\frac{dw_2}{d\zeta}&=a^{39/15}b^{4+\beta_1}r^{\beta_1}r^{54/15}-\frac{r^{67/15}a^{21/5}b^5}{(1+G)^{3/2}}, \\
\frac{d\xi}{d\zeta}&=r^{18/5}a^{39/15}b^4, \\
\frac{db}{d\zeta}&=\frac{15b}{11(4F-5G-1)}\bigg\{\frac{5}{2}(1+G)^{5/2}\\
&\cdot\left(\frac{(1+F)^{3/2}}{(1-\Theta)}-\frac{\xi r^{39/15}a^{13/5}b^3}{(1+G)^{3/2}}-r^{26/15}ab^2w_2+r^{\beta_1}r^{26/15}ab^{2+\beta_1} \xi\right)\\
&-4a^{24/15}(1+F)^{5/2}\left(\frac{a^{3/5}(1+F)^{3/2}}{(1+G)^{3/2}(1-\Theta)}-\frac{r^{13/5} ab^3\xi(1-\Theta)}{(1+F)^{3/2}}-r^{13/15}bw_1-\frac{r^{2/15}r^{\beta_1-1}b^{\beta_1-1}}{a(1-\Theta)}(1+F)^{3/2}\right)\bigg\},
\end{align*}
$$F(a,r,b)=r^{\beta_3-1}r^{1/3}a^{2/3}b^{\beta_3}, \ \ G(a,r,b)=r^{134/15}a^{16/15}b^{10}, \ \ \Theta(a,r,b)=\frac{b^{\beta_4-2}r^{\beta_4-2}r^{4/15}}{a}(1+F)^{3/2}.$$
System (<ref>) plays a key role, since it contains all the dynamics of the
problem, scaled in a way that emphasizes the region where
the singular shock is formed.
In addition, this system also possesses an invariant manifold, which is
normally hyperbolic, and we are able to prove existence of a solution to the Dafermos
regularization, for small $\varepsilon$, by exhibiting a solution which is close to this invariant
manifold during part of its trajectory.
In the region of interest we require $r=0$ (which corresponds to $\varepsilon =0$) and $b=0$ to
find invariant manifolds, and then we have an equilibrium of (<ref>) when $\frac{da}{d\zeta}=0$;
that is, when
$a$ is a root of the equation (<ref>) introduced in the definition of $q_L$
and $q_R$.
The two roots of (<ref>) are
\begin{equation*}
\begin{aligned}
a_2=2^{5/11}, \ \ a_3=0.
\end{aligned}
\end{equation*}
Using these roots, we define
$$P_j=\{(a,r,W,\xi,b): a=a_j, r=0, b=0\} \quad \textrm{for }j=2, 3.$$
Each of these sets
is a $3$-dimensional manifold of equilibria, corner equilibria in Schecter's
definition <cit.>.
If we linearize (<ref>) at $a=a_j$, $r=b=0$, we find
a zero eigenvalue of multiplicity 3, with 3 linearly independent eigenvectors
lying in $P_j$.
There are three additional eigenvalues,
\begin{align*}
\lambda_2&=-\frac{16}{11}a_j^{11/15}+\frac{10}{11}\,,\\
\lambda_3&=-\frac{60}{11}a_j^{11/5}+\frac{75}{22}\,,\\
\lambda_4&=\frac{60}{11}a_j^{11/5}-\frac{75}{22}\,,
\end{align*}
and since
the corresponding eigenvectors, which are
\begin{align}
&R_2=(1, 0, 0, 0 , 0 , 0)\,,\nonumber\\
&R_3=(0, 1, 0, 0, 0, 0)\,,\label{4.3.3}\\
&R_4=(0, 0, 0, 0, 0, 1)\nonumber
\end{align}
respectively, are transversal to $P_j$, the
$P_j$ are normally hyperbolic manifolds.
We fix a point
$(a_3, 0, W_0, s_{\text{singular}}, 0)$ in $P_3$.
Then $\lambda_4<0<\lambda_2, \lambda_3$
so the point has a 1-dimensional stable manifold tangent to $R_4$.
Indeed, the stable manifold of any point with $r=b=0$ is
contained in the $2$-dimensional plane
$$\{(a,r,w_1,w_2,\xi,b): r=0, \ W=W_0, \ \xi=s_{\text{singular}}\}\,,$$
which is invariant under the flow (<ref>).
Thus the stable manifold of $P_3$ is tangent to
\begin{equation} \label{4.3.4}
\{(a,r,W,\xi,b): r=0, \ a=a_3\}
\end{equation}
at $P_3$.
Since $\lambda_2$ and $\lambda_3$ are positive at points of $P_3$,
each point has a $2$-dimensional unstable manifold tangent to the plane
spanned by $R_2$ and $R_3.$
(The same two eigenvalues, $\lambda_2$ and $\lambda_3$, are
negative on $P_2$.)
Thus $P_3$ has the $5$-dimensional unstable manifold
$$W^u(P_3)=\{(a,r,w_1,w_2,\xi,b): b=0\}.$$
The point $q_L$, identified earlier, is a particular point of $P_3$, with $W=W_L$
and $\xi = s_{\text{singular}}$.
(In Chart 2 coordinates, $q_L= (a_3,0,W_L,s_{\text{singular}},0)$ and $v=0,$ $y=0$ in the original variables.)
Through the $1$-dimensional stable manifold of $q_L\in P_3$, there is a unique
connection backwards in time to $U_L$, and through the $2$-dimensional unstable
manifold, $q_L$ connects forward to the singular orbit.
We state
There is a unique orbit in the $2$-dimensional invariant plane
that connects $q_L$ as $\zeta\to\infty$
with $U_L$ as $\zeta\to -\infty$.
Furthermore, in a neighborhood of $q_L$, we have $b>0$ along the orbit.
The proof is similar to the result of Schecter <cit.>, with details
motivated by
Theorem 3.1 of <cit.>.
One can verify that, in one direction,
the stable manifold of $q_L$ is in the interior of the
negatively invariant region for $U_L$.
The inequality for $b$ follows from examining the eigenvector tangent to
the manifold at $q_L$. The manifolds are described in different coordinate
systems since the coordinate system of Chart 2 is not suitable for describing the entire
trajectory because $y$ (or $y_1,$ $y_2$ or $\bar{y}_1,$ $\bar{y}_2$) need not remain positive throughout
the trajectory.
We now fix a point
$(a_2, 0, W_0, s_{\text{singular}}, 0)$ in $P_2$.
Then $\lambda_2, \lambda_3<0<\lambda_4$
so the point has a 1-dimensional unstable manifold tangent to $R_4$.
Indeed, the unstable manifold of any point with $r=b=0$ is
contained in the $2$-dimensional plane
$$\{(a,r,w_1,w_2,\xi,b): r=0, \ W=W_0, \ \xi=s_{\text{singular}}\}\,,$$
which is invariant under the flow (<ref>).
Thus the unstable manifold of $P_2$ is tangent to
\begin{equation} \label{4.3.4}
\{(a,r,W,\xi,b): r=0, \ a=a_2\}
\end{equation}
at $P_2$.
Since $\lambda_2$ and $\lambda_3$ are negative at points of $P_2$,
each point has a $2$-dimensional stable manifold tangent to the plane
spanned by $R_2$ and $R_3.$ Thus $P_2$ has the $5$-dimensional stable manifold
$$W^s(P_2)=\{(a,r,w_1,w_2,\xi,b): b=0\}.$$
The point $q_R$, identified earlier, is a particular point of $P_2$, with $W=W_R$
and $\xi = s_{\text{singular}}$. The point $q_R$ corresponds to $(v,y),$ $y^2=2v$ in the original variables. Through the $1$-dimensional unstable manifold of $q_R\in P_2$, there is a unique connection forward in time to $U_R.$
On the other hand, we need to show that through the $2$-dimensional stable manifold, $q_R$ connects backwards to the singular orbit. It should be noted that the connections between $U_L,$ $q_L,$ $q_R$ and $U_R$
do not solve the problem, since for example $q_L$ and $U_L$ are the $\omega$- and $\alpha$-limits
of a unique orbit, and thus are not themselves part of a longer connection between $U_L$
and $U_R$. To demonstration that connecting
orbits exist in the neighborhood of these invariant manifolds we use the
Corner Lemma to show that $U_L$ and $U_R$ can be connected when $\varepsilon>0$.
For this, we introduce an $1$-dimensional set that contains $q_L.$ We recall the definitions of $W_L$ and $W_R$, (<ref>), and of $q_L$ and $q_R$
in the Chart $2$ coordinate system
$$q_L=(a_3,0,W_L,s_\text{singular},0), \ \ q_R=(a_2,0,W_R,s_\text{singular},\infty).$$
In addition, we note that
using (<ref>) $w_{L1}=w_{R1},$ and (<ref>), $w_{R2}=w_{L2}-k<w_{L2}$.
If we express
$q_L$ and $q_R$ in $Y,\varepsilon$ coordinates, they are points in $E$
(the invariant set of equilibria of (<ref>)).
$q_L=(0,W_L,s_{\text{singular}},0)$ and $q_R=(0,W_R,s_{\text{singular}},0)$.
Following the discussion of the homoclinic orbits in Section <ref>,
there is a unique solution of (<ref>) that connects the two points such that $k=\lim_{\varepsilon\rightarrow 0} \varepsilon^5 \int \frac{y_2^2}{(y_1^{16/15}+\varepsilon^{10})^{3/2}}\ d\eta.$
Write the solution as
$$w_1(\eta)=w_{L1}=w_{R1},\quad w_2(\eta)=w_{L2}-\lim_{\varepsilon\rightarrow 0} \varepsilon^5 \int_{-\infty}^{\eta} \frac{y_2(t)^2}{(y_1(t)^{16/15}+\varepsilon^{10})^{3/2}}\ dt=w_{L2}-k(\eta).$$
This can be
written in the coordinates of
Chart 2, $(a, r ,W, \xi, b)$ as
\begin{equation}
q(\zeta)=(a(\zeta),r(\zeta),W(\zeta), s_{\text{singular}},b(\zeta))\,.
\label{4.4.3}
\end{equation}
Here $r(\pm\infty)=0$, $a(-\infty)=a_3$, $a(+\infty)=a_2$.
We note that
$q(-\infty)=q_L$, $q(+\infty)=q_R.$
Geometrically, $q(\zeta)$ lies in the $4$-dimensional subspace of $\mathbb{R}^6$
(in Chart 2 coordinates) with $w_1=w_{1L}=w_{2L}$ and $\xi=s.$ In addition there exists $q_M=(a_M, r_M, w_{1L},w_{2M}, s_{\text{singular}}, 0)$ which corresponds to $(v,y)=(0,\infty)$ in the original variables.
We define
\begin{align*}
C_3&=\{(a,r,W, \xi,b): a=a_3, \ r=0, W=F(U_L)-\xi U_L, \xi<\lambda_1(U_L), \ \ b=0\}\subseteq P_3,\\
D_3&=\{(a,r,W, \xi,b): a=a_3, \ r=0, w_1=w_{L1}, \ w_2=w_{L2}-\lim_{\varepsilon\rightarrow 0} \varepsilon^5 \int_{-\infty}^{\zeta} \frac{y_2(t)^2}{(y_1(t)^{16/15}+\varepsilon^{10})^{3/2}}\ dt, \\
& \ \ \ \ \ \xi=s_{\text{singular}}, \ b=b(\zeta), \ \zeta\in\mathbb{R}\}, \\
E_3&=\{(a,r,W, \xi,b): a=a_3, \ r=0, w_1=w_{L1}, \ w_2=w_{L2}-\lim_{\varepsilon\rightarrow 0} \varepsilon^5 \int_{-\infty}^{\zeta} \frac{y_2(t)^2}{(y_1(t)^{16/15}+\varepsilon^{10})^{3/2}}\ dt, \\
&\ \text{with} \ \zeta \ \text{such that} \ b=0, \ \xi=s_{\text{singular}}\}\subseteq D_3, \\
C_2&=\{(a,r,W, \xi,b): a=a_2, \ r=0, W=F(U_R)-\xi U_R, \lambda_2(U_R)<\xi, \ \ b=\infty\},
\end{align*}
where we have not fixed the values of $W$ as we did to define $q_i$.
The stable manifold of $C_3$ is a $2$-dimensional surface in the $5$-dimensional space $r=0$;
it is the union of the stable manifolds of the points of $C_3.$ Since up to now we have not made use of the specific value of $\xi$ (beyond its relation to the eigenvalues of $dF(U_L)$), the results of Proposition <ref> hold
at each point of $C_3$, and we have
(recalling that $T_0^0(U_L)$ is precisely the 1-dimensional set in which $\xi$ is
allowed to vary)
In the coordinate system of Chart 2, the set
$W^u(T_0^0(U_L))$ takes the form
\begin{equation}\label{4.3.5}
W^u(N_0^0(U_L))=\{(a_\xi(\tau),0,W,\xi, b_\xi(\tau))\}\,,
\end{equation}
where $W=F(U_L)-\xi U_L$ for a fixed $\xi<\lambda_1(U_L)$ and $(a_\xi,b_\xi)$,
$$a=\frac{y_2^{15/11}(\tau)}{{y_1(\tau)}} \ \text{and} \ b=\frac{\varepsilon(\tau)}{y_2^{15/11}(\tau)},$$
$$v(\tau)=\frac{\varepsilon^2(\tau) y_2(\tau)}{(y_1^{2/3}(\tau)+\varepsilon^{\beta_3}(\tau))^{3/2}}-\varepsilon^{\beta_4}(\tau), \ \ \ y(\tau)=\frac{\varepsilon(\tau) y_2^2(\tau)}{(y_1^{16/15}(\tau)+\varepsilon^{10}(\tau))^{3/2}}-\varepsilon^{\beta_1}(\tau),$$
is the expression in Chart
2 coordinates of the solution of (<ref>) with $\omega$-limit in $C_3$
for fixed $\xi$.
The intersection of $W^u(N^0_0(U_L))$ and $W^s(P_3)$ is an open
subset $Q_3$ of $W^s(C_3)$, namely the points of $W^s(C_3)$ with $b>0$.
The conclusion of Proposition <ref>, which holds at each point of $C_3$, implies this
The positivity of $b$ follows from the explicit scaling.
The analogous result for $C_2$, and corresponding space
$$W^s(N_2^0(U_R))=\{((a, r, W, \xi, b): (a,b)\in V_{\xi}, \ r=0, \ W=F(U_R)-\xi U_R, \ \lambda_2(U_R)<\xi \} ,$$
are used to construct and analyze the
second half of the orbit.
For this purpose,
we note that $P_2$ has a $5$-dimensional stable manifold
$$W^s(P_2)=\{(a,r,W,\xi,b): b=0\}.$$
§.§ The Inner Solution
We now seek the connection between $q_L$ and $q_R$. The curves $D_3$ of equilibria, in $P_3$ and $C_2$ are useful. The overcompression condition $s<\lambda_1(U_L)$ and $s>\lambda_2(U_R)$ is needed in this part or else the construction fails, because then $q_L$ is an endpoint of $C_3$ and we cannot verify Proposition <ref>, which we will need to apply the Corner Lemma at $q_L$ to match the inner with the outer solution.
The unstable manifold of $D_3$ has dimension three, and we have a description of its tangent space. It is spanned by the eigenvectors $R_2$ and $R_3$ of (<ref>),
and can be written
\begin{equation}
\begin{aligned}
W^u(D_3)=\{(a,r,W,& \xi,b): w_1=w_{L1}, \ w_2=w_{L2}-\lim_{\varepsilon\rightarrow 0} \varepsilon^5 \int_{-\infty}^{\zeta} \frac{y_2(t)^2}{(y_1(t)^{16/15}+\varepsilon^{10})^{3/2}}\ dt, \\
& \ \ \ \ \ \xi=s_{\text{singular}}, \ b=b(\zeta), \ \zeta\in\mathbb{R}\}.
\end{aligned}
\label{4.4.1}
\end{equation}
7$a=a_3, r=b=0$
8chart 1, outer solution
9$a=a_2, r=b=0$
10chart 2, inner solution
11chart 2, inner solution
12$a=a_2, r=0, b\neq 0$
13chart 2, inner solution
14chart 1, outer solution
Solution when $\varepsilon=0$ in the $vy$-plane.
5$a=a_2, r=b=0$
6$a=a_2, r=0, b\neq 0$
7$(\ref{2.1.4})$ has singular shocks
8$(\ref{2.1.1})$ has singular shocks
Solution when $\varepsilon=0$ in $arb$-space.
We observe that as $\zeta\rightarrow \infty,$ $r\rightarrow 0,$ $a\rightarrow a_2$ then $W^u(D_3)$ is tangent to
\begin{equation}
\{(a,r,W,\xi,b): a=a_2, \ r=0, \ W=W_R, \ \xi=s_\text{singular} \}
\end{equation}
therefore $W^u(D_3)\cap W^s(N_2^0(U_R))\neq \emptyset.$
$W^u(D_3) \supseteq W^u(C_3)\cap W^u(E_3)\neq \emptyset .$
§.§ Completion of the Result
The ingredients to be combined so as to synthesize the solution of the problem are now prepared. Three particular orbits have been constructed, each corresponding to the limit
$\varepsilon =0$: $A_1$, joining $U_L$ to $q_L$, $A_2$ joining $q_L$ to $q_R$,
and $A_3$ joining $q_R$ to $U_R$.
To show that a solution exists for $\varepsilon >0$, that will actually connect $U_L$
and $U_R$ via a solution of the equation, we need to
show that there is a solution, with $\varepsilon >0$, that is close to
the union of these three orbits.
The technique is to show that a solution close to $A_1$, in
$W^u(T_0^{\varepsilon}(U_L))$, will enter
$W^u(C_3)$, and similarly to match $W^u(D_3)$ with $W^s(T_2^{\varepsilon}(U_R))$.
We do this by verifying the conditions of the Corner Lemma
(Theorem 5.1 of Schecter <cit.>).
In the coordinate system of Chart 2, the sets
$W^u(T_0^{\varepsilon}(U_L))$ and $W^s(T_2^{\varepsilon}(U_R))$ will be denoted by $W^u(N_0^{\varepsilon}(U_L))$ and $W^s(N_2^{\varepsilon}(U_R)),$ respectively.
The $4$-dimensional set $W^u(N_0(U_L))=\cup_{0\leq \varepsilon \leq \varepsilon_0}W^u(N_0^{\varepsilon}(U_L))$ is transverse to $W^s(P_3)$ along $Q_3.$
When we calculate $W^u(N_0(U_L))$ at $Q_3$ in the coordinate
system of Chart 2, we find
the tangent space to $W^u(N_0(U_L))$ is spanned by
\begin{align*}&(1,0,0,0,0,0), \\ &(0,0,0,0,0,1),\\
&(0,0,-v_{L},-y_{L},1,0), \\ &(0,1,0,0,0,0).
\end{align*}
The tangent space to $W^s(P_3)$ at the same point is spanned by
\begin{align*}&(0,0,1,0,0,0), \\ &(0,0,0,1,0,0).
\end{align*}
These six vectors are linearly independent; therefore transversality follows.
Corner Lemma.
Proposition <ref> establishes the hypotheses of the
Corner Lemma <cit.>.
As Schecter showed in <cit.>, we have the 1-dimensional space
\begin{equation*}
W^s(q_L)=\{(a,r,W,\xi,b): r=0, \ \ W=W_L, \xi=s_{\text{singular}}, \ \ a=a_3\}\,.
\end{equation*}
See Figure <ref>.
We let $p \in W^s(q_L)\backslash\{q_L\}$, let $N$ be a 3-dimensional slice of
transverse to the vector field and to $W^s(P_3)$ at the point $p$;
let $N_{\delta}=N\cap\{r=\delta\}$, a 2-dimensional manifold;
let $q$ be in $W^u(C_3)$ with positive $r$ coordinate, and
let $U$ be a small neighborhood of $q$.
Then under the flow, $N_{\delta}$ becomes a 3-dimensional manifold $\tilde{N}_{\delta}$
(like $W^u(N_0^{\delta}(U_L))$) that passes near $q$. By the Corner Lemma,
\begin{equation}
\text{as} \ \delta \rightarrow 0, \ \tilde{N}_{\delta}\cap U
\rightarrow W^u(C_3)\cap U \ \text{in the} \ C^1 \ \text{topology}.
\nonumber
\end{equation}
With the Lemma and Remarks in Section <ref> we make the final match for the solution since
$W^u(N_0^{\varepsilon}(U_L))$ passes $q_L$ and arrives near
$q(-T)$ for $T>0,$ where
$q(\cdot)$ is given by (<ref>).
We then have a solution connecting $U_L$ and $U_R$.
As $\varepsilon \to 0$, this solution is unbounded.
This completes the proof of Theorem <ref>.
§ ACKNOWLEDGMENTS
Foremost, the author would like to express her deepest thanks to her postdoctoral advisor Prof. Barbara Keyfitz for suggesting the problem and the change of variables leading to the system of equations (<ref>), for many illuminating discussions, and for her support and encouragement during this project. This work was started during the author's post-doctoral studies at the Ohio State University, whose support she also gratefully acknowledges.
B1 Bressan, A., Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem, Oxford: Oxford University Press, 2000. MR1816648 (2002d:35002)
CY Cheng, H. and H. Yang, Delta shock waves in chromatography equations, Journal of Mathematical Analysis and Applications, 380 (2011), no. 2, 475–485. MR2794406 (2012d:35221)
C.M.Dafermos Dafermos, C. M., Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Rational Mech. Anal., 52 (1973), 1–9. MR0340837 (49 #5587)
DafermosDiPerna Dafermos, C. M. and R. J. DiPerna, The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Differential Equations, 20 (1976), no. 1, 90–114. MR0404871 (53 #8671)
Deng Deng, B. Homoclinic bifurcations with nonhyperbolic equilibria,
SIAM J. Math. Anal., 21 (1990), no. 3, 693–720. MR1046796 (91g:58200)
DiP1 DiPerna, R. J. Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys., 91 (1983), no. 1, 1–30. MR0719807 (85i:35118)
DiP2 DiPerna, R. J. Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc., 292 (1985), no. 2, 383–420. MR0808729 (87g:35148)
F Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), no. 1, 53–98. MR0524817 (80m:58032)
GPY Guo, L., Pan, L. and G. Yin, The perturbed Riemann problem and delta contact discontinuity in chromatography equations, Nonlinear Analysis: Theory, Methods & Applications 106 (2014), 110–123. MR3209688
J Jones, C. K. R. T., Geometric singular perturbation theory. Dynamical systems (Montecatini Terme, 1994), Lecture Notes in Mathematics, Vol. 1609, Springer, Berlin, (1995), 44–118. MR1374108 (97e:34105)
JKap Kaper, T. J. and C. K. R. T. Jones, A primer on the exchange lemma for fast-slow
systems, Multiple-time-scale dynamical systems (Minneapolis, MN, 1997), Springer, New York, IMA Vol. Math. Appl., 122 (2001), 65–87. MR1846573 (2002g:37022)
JK Jones, C. K. R. T. and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations, 108 (1994), no. 1, 64–88. MR1268351 (95c:34085)
KaMi Kalisch, H. and D. Mitrovic, Singular solutions of a fully nonlinear $2\times2$ system of conservation laws, Proceedings of the Edinburgh Mathematical Society, volume 55 (2012), no. 3, 711–729. MR2975250
delta Keyfitz, B. L. Conservation laws, delta-shocks and singular shocks, Nonlinear Theory of Generalized Functions (Vienna, 1997), Chapman & Hall/CRC Res. Notes Math., 401 (1999), 99–111. MR1699874
Ke Keyfitz, B. L., Mathematical properties of non hyperbolic models for incompressible two-phase flow, Proc. Fourth Int. Conf. Multiphase Flow, New Orleans (CDROM), ed. E. E. Michaelides, ICMF 2001, Tulane University, 2001.
michelle Keyfitz, B. L., A new look at singular shocks, Confluentes Matematici, to appear, 2012.
B.L.KeyfitzandH.C.Kranzer Keyfitz, B. L. and H. C. Kranzer, A viscosity approximation to a system of conservation laws with no classical Riemann solution, Nonlinear hyperbolic problems, Bordeaux 1988, Lecture Notes in Math., Springer, Berlin, vol. 1402 (1989), 185–197. MR1033283 (90k:35168)
wgt Keyfitz, B. L. and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations, 118 (1995), no. 2, 420–451. MR1330835 (96b:35138)
KeSaSe Keyfitz, B. L, Sanders, R. and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow, Disc. Cont. Dynam. Syst., 3 (2003), no. 4, 541–563. MR2036001 (2004k:76123)
KeSeZh Keyfitz, B. L., Sever, M. and F. Zhang, Viscous singular shock structure for a nonhyperbolic two-fluid model, Nonlinearity, 17 (2004), no. 5, 1731–1747. MR2086148 (2005j:35154)
KT Keyfitz, B. L. and C. Tsikkou, Conserving the Wrong Variables in Gas Dynamics: A Riemann Solution with Singular Shocks, Quart. Appl. Math., 70 (2012), no. 3, 407–436. In the special issue in honor of Dafermos' 70th birthday. MR2986129
sing Kranzer, H. C. and B. L. Keyfitz, A strictly hyperbolic system of conservation laws admitting singular shocks, Nonlinear Evolution Equations that Change Type, Springer, New York, IMA Vol. Math. Appl., 27 (1990), 107–125. MR1074189 (92g:35133)
KS Krupa, M. and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), no. 2, 286–314 (electronic). MR1857972 (2002g:34117)
LeSl Levine, H. A. and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), no. 3, 683–730. MR1450846 (98g:35106)
LS Li, X. and C. Shen, Viscous Regularization of Delta Shock Wave Solution for a Simplified Chromatography System, Abstract and Applied Analysis, vol. 2013, Article ID 893465, 10 pages, (2013). MR3111823
MaBe Mavromoustaki, A. and A. L. Bertozzi, Hyperbolic systems of conservation laws in gravity-driven, particles-laden thin-film flows, J. Eng. Math., 88 (2014), 29–48. MR3254624
Mazzotti1 Mazzotti, M., Local equilibrium theory for the binary chromatography of species subject to a generalized Langmuir isotherm, Indust. Eng. Chem. Res., 45 (2006), 5332–5350.
Mazzotti2 Mazzotti, M., Non-classical composition fronts in nonlinear chromatography - Deltashock, Indust. & Eng. Chem. Res., 48 (2009), 7733–7752.
Mazzotti3 Mazzotti, M., Tarafder, A., Cornel, J., Gritti, F. and G. Guiochon, Experimental evidence of a delta-shock in nonlinear chromatography, J. Chromatography A,
1217 (2010), issue 13, 2002–2012.
SSS Schaeffer, D. G, Schecter, S. and M. Shearer, Non-strictly hyperbolic conservation
laws with a parabolic line, J. Differential Equations, 103 (1993), 94–126. MR1218740 (94d:35102)
Schecter Schecter, S., Existence of Dafermos profiles for singular shocks, J. Differential Equations, 205 (2004), no. 1, 185–210. MR2094383 (2005k:35269)
SS Schecter, S. and P. Szmolyan, Composite waves in the Dafermos regularization, J. Dynamics and Differential Equations, 16 (2004), no. 3, 847–867. MR2109169 (2005h:35234)
Sever Sever, M., Distribution solutions of nonlinear systems of conservation laws, Mem. Amer. Math. Soc., 190 (2007), no. 889, 1–163. MR2355635 (2008k:35313)
M.Sever Sever, M., Large-data solution of a model system for singular shocks, J. Hyperbolic Differential Equations, 7 (2010), 775–840. MR2746206 (2012c:35270)
S2 Shelkovich, V. M., One Class of Systems of Conservation Laws Admitting Delta-shocks, Hyperbolic problems-theory, numerics and applications, Ser. Contemp. Appl. Math. CAM, World Sci. Publishing, Singapore, 18 (2012), no. 2, 667–674. MR3098648
Shen Shen, C., The Asymptotic Behaviors of Solutions to the Perturbed Riemann Problem near the Singular Curve for the Chromatography System, J. Nonlinear Math. Phys., 22 (2015), no. 1, 76–101. MR3286735
Sun1 Sun, M., Delta shock waves for the chromatography equations as self-similar viscosity limits, Quarterly of Applied Mathematics, 69 (2011), no. 3, 425–443. MR2850739
Sun2 Sun, M., Interactions of delta shock waves for the chromatography equations, Applied Mathematics Letters, 26 (2013), no. 6, 631–637. MR3028067
Wang Wang, G., One-dimensional nonlinear chromatography system and delta-shock waves, Z. Angew. Math. Phys., 64 (2013), no. 5, 1451–1469. MR3107574
WB Wang, L. and A. L. Bertozzi, Shock solutions for high concentration particle-laden thin films, SIAM J. Appl. Math., 74 (2014), no. 2, 322–344. MR3179561
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1511.01047
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In a variety of applications, one desires to detect groups of anomalous data samples,
with a group potentially manifesting its atypicality (relative to a reference model) on a low-dimensional subset
of the full measured set of features. Samples may only be weakly atypical individually,
whereas they may be strongly atypical when considered jointly.
What makes this group anomaly detection problem quite challenging is that
it is a priori unknown which subset of features jointly manifests a particular
group of anomalies. Moreover, it is unknown how many anomalous groups are present in a given data batch.
In this work,
we develop a group anomaly detection (GAD) scheme to identify the subset of samples and subset of features that jointly specify an anomalous cluster. We apply our approach to
network intrusion detection to detect BotNet and peer-to-peer flow clusters.
Unlike previous studies, our approach captures and exploits statistical dependencies
that may exist between the measured features.
Experiments on real world network traffic data demonstrate the advantage of our proposed system, and
highlight the importance of exploiting feature dependency structure, compared to the feature (or test) independence assumption made in previous studies.
Bonferroni correction, group anomaly detection, Gaussian Mixture Model, p-value, network intrusion detection, BotNet, dependence tree
§ INTRODUCTION
Group anomaly detection has recently attracted much attention, with applications in astronomy <cit.>,
social media <cit.>, disease/custom control <cit.><cit.> and network intrusion detection <cit.><cit.><cit.>.
In this work, we focus on group anomaly detection applied to network intrusion detection,
where the anomalous groups are either distributed Botnet (Zeus) or peer-to-peer (P2P) nodes generating traffic that deviates from the normal (Web traffic) behavior.
Many existing intrusion detection systems (IDSs) only make sample-wise anomaly detections, e.g., in <cit.>, the samples which deviate most from a normal (reference) model are flagged as anomalies/outliers.
However, such an approach does not identify anomalous groups (e.g., a collection
of BotNet flows), whose samples all exhibit similar behavior. Identifying such groups
could be essential for mounting some form of system response or defense.
individual samples may only be weakly atypical. Thus, a sample-wise IDS may either
fail to detect most of the anomalous samples, or may incur high false positives
when a low detection threshold is used.
By contrast, (weakly) anomalous samples whose anomalies are all “similar to each other”
may be strongly atypical when considered in aggregate, i.e. jointly.
For example, for an $N=100$-dimensional feature space,
suppose there is a sizeable collection of samples in the captured data batch that are all (even only weakly) atypical with respect to the
same feature or the same (small) feature subset. There is a low probability that this occurs by chance, (i.e.,under the null). Thus, such clusters of anomalies, each defined by a sample subset
and a feature subset, may be strongly atypical, and hence more convincing anomalies, than
individual sample anomalies.
It should be noted that there is an enormous number of candidate anomalous clusters, considering the
conjoining of all possible sample subsets and all possible feature subsets.
Thus, a GAD scheme will require some type of heuristic search over this huge space,
aiming to detect the most statistically significant cluster candidates.
In the sequel, we propose such a GAD scheme.
Rather than assuming individual features or outlier events
are statistically independent under the null
as in <cit.>, in our approach, as in <cit.>,
we capture and exploit
statistical dependencies amongst the features defining a candidate cluster.
Compared to previous works, as shown in our experiments,
the proposed scheme is more effective in detecting group anomalies.
The paper is organized as follows. Section II defines the problem and elaborates on related works.
Section III describes the proposed model.
Section IV evaluates the system performance, and compares with some recent works.
We then discuss some extensions of our system and future works in section V, followed by conclusions.
§ PROBLEM DEFINITION AND RELATED WORK
We assume there is a batch of normal web traffic available at the outset as training set,
i.e. $X_l = \{\underline{\tilde{x}}_i, i=1,...,T_l, \underline{\tilde{x}}_i\in R^D\}$, where $\underline{\tilde{x}}_i$
is a $D$-dimensional feature vector representing the $i$-th training traffic flow[A flow is a bidirectional communication sequence
between a pair of nodes in a network.], and where we assume the number of training flows $T_l$ is large enough to learn an accurate reference model (null hypothesis).
These traffic flows can either be generated and captured in a sandbox environment, or sampled from a domain of interest (data warehouse, enterprise network) in real time under normal operating conditions.
Given a model of normal network traffic learned based on ${\cal X}_l$, our goal is to interrogate
a capture batch of unknown traffic flows ${\cal X}_u= \{\underline{x}_i, i=1,...,T_u, \underline{x}_i\in R^D\}$[Unknown in the sense that we do not know
which if any of these flows represent outliers or attacks.],
seeking to identify latent groups of Botnet or P2P traffic, with the flows in each such group exhibiting similar behavior.
This has been previously considered in <cit.>, where the authors used the samples in ${\cal X}_l$ to estimate bivariate Gaussian Mixture Models (GMMs), on all feature pairs, representing the null hypothesis.
These bivariate GMMs were used to evaluate mixture-based p-values[A p-value is the probability that an event is more extreme than the given observation.] for all pairs of features.
Assuming the features (tests) are statistically independent, a joint significance score function was defined for a given candidate cluster, specified by
its sample subset and feature subset, with a Bonferroni correction used to account for multiple testing.
Instead of exhaustively searching over feature subset candidates at order $K$ [We use “order” to denote the maximum feature dimension considered.],
the authors proposed to trial-add individual features only to the top-ranking candidate feature subsets (in terms of the Bonferroni corrected score) at order $K-1$.
Furthermore, the authors showed that the computational complexity of determining the optimal (in terms of the joint score) sample subset given the
feature subset fixed is linear in $T_u$, once the samples in a given feature subset are ranked by their aggregate p-values.
However, the independent test assumption used in <cit.> becomes grossly invalid as more and more features are included in
a cluster, which limits the proposed model's detection accuracy for increasing $K$.
A related framework was also proposed in <cit.>, albeit assuming categorical attributes. Here, the authors built a single, global null hypothesis Bayesian network based on ${\cal X}_l$. They then assigned categorical-based p-values to samples in $X_u$,
with a cross entropy based scoring criterion used to efficiently search for the best feature and sample subset candidates.
A limitation of this approach is that the statistical tests are again assumed to be independent.
We herein describe and experiment with a method of anomaly detection
that extends <cit.> and is closely related to
<cit.>. The method
captures dependencies between the features in a candidate cluster by a
dependence tree structure, and uses this model to help evaluate joint p-values for cluster candidates.
As in <cit.>, the Bonferroni corrected score is used as the objective function for evaluating the best cluster candidates (defined by their sample and feature subsets). The candidate with the best such score is detected as a cluster of anomalies.
Whereas in <cit.> a single global (null model) Bayesian network is used to assess candidate clusters, in <cit.> and in
the current work a local, customized cluster-specific dependence tree model is used to assess each candidate cluster.
§ PROPOSED MODEL
§.§ Mixture-based P-values for Singletons and Feature Pairs
Consider a (sample, feature) index pair $(i,j)$ and let $I_i^{(j)}$ be an indicator variable for the event that the $j^{th}$ feature value of the $i^{th}$ sample, $x_i^{(j)}$,
is an outlier with respect to the null distribution for feature $X^{(j)}$.
Let $O^{(j)}(x_i^{(j)})$ be a subset of the real line such that, $\forall y^{(j)} \in O^{(j)}(x_i^{(j)})$, $y^{(j)}$ is “more extreme” than the given observation $x_i^{(j)}$.
One good definition for this set, consistent with evaluating a 2-sided p-value for a unimodal, symmetric null for $X^{(j)}$, is:
\begin{eqnarray}
\label{2-sided}
O^{(j)}(x_i^{(j)};\mu^{(j)}) = \{y^{(j)}:|y^{(j)}-\mu^{(j)}|\geq |x_i^{(j)}-\mu^{(j)}|\}, \nonumber
\end{eqnarray}
where $\mu^{(j)}$ is a representative (mean) value for feature $X^{(j)}$.
Given the component means $\mu_l^{(j)}, l=1,...,L_j$, of an $L_j$-component Gaussian mixture null, let $M^{(j)}(x)$ be a function that maps $x$ to the mixture component index set $\{1,2,...,L_j\}$,
i.e., it indicates which mixture component generated $x$.
Also, let $Y_j$ be a random variable distributed according to the mixture density $f_{X_j}(x)$.
Then, for a given observation $x_i^{(j)}$, we define the binary random variable $I_i^{(j)}$, where
$I_i^{(j)}=1$ if $Y_j$ is more extreme under the null than $x_i^{(j)}$.
Then, we can write
singleton mixture p-value as:
\begin{eqnarray}
\lefteqn{ P[I_i^{(j)}=1]} & & \nonumber \\
& = & P[Y_j \in \cup_{l=1}^L ((O^{(j)}(x_i^{(j)};\mu_l^{(j)})) \cap (M^{(j)}(x_i^{(j)})=l))] \nonumber
\\
& = & \sum\limits_{l=1}^L P[Y_j \in O^{(j)}(x_i^{(j)};\mu_l^{(j)})] P[M^{(j)}(x_i^{(j)})=l]. \label{mixture}
\end{eqnarray}
Here, an extreme outlier event is conditioned on $x_i^{(j)}$ having been generated by component density $l$.
The probability $P[Y_j \in O^{(j)}(x_i^{(j)};\mu_l^{(j)})]$ is the two-sided Gaussian p-value, integrating over the region
$|y-\mu_l^{(j)}|\geq |x_i^{(j)}-\mu_l^{(j)}|$, while $P[M^{(j)}(x_i^{(j)})=l]$ is the a posteriori probability
that $x_i^{(j)}$ was generated by component $l$.
Similarly, for a pair of observations $(x_i^{(j)},x_i^{(k)})$, we have the second order mixture p-value:
\begin{eqnarray*}
\lefteqn{P[I_i^{(j)}=1,I_i^{(k)}=1]~} & & \\
& = & \sum\limits_{l=1}^L
P[Y_j \in O^{(j)}(x_i^{(j)};\mu_l^{(j)}),Y_k \in O^{(k)}(x_i^{(k)};\mu_l^{(k)})]
\\
& & ~~~~~\cdot P[M^{(j,k)}(x_i^{(j)},x_i^{(k)})=l].
\end{eqnarray*}
Here, $P[Y_j \in O^{(j)}(x_i^{(j)};\mu_l^{(j)}),Y_k \in O^{(k)}(x_i^{(k)};\mu_l^{(k)})]$ integrates the $l$-th component bivariate Gaussian density over the
$$\{(y_j,y_k): |y_j - \mu_l^{(j)}| \geq |x_i^{(j)} - \mu_l^{(j)}|, |y_k - \mu_l^{(k)}| \geq |x_i^{(k)} - \mu_l^{(k)}|\}.$$
This region consists of the union of four unbounded rectangular regions in the plane, as illustrated in Figure <ref>.
Illustrative figure: bivariate Gaussian joint p-value measure coresponds to
the four (unbounded) shaded corners in grey, with mean $\mu$ and a given observation $x$
In this work, a sample's anomalousness on a given feature subset is estimated by a joint p-value, with statistical
dependencies between features accounted for by a dependence tree (DT) structure <cit.>.
Since the dependence tree <cit.> is based on first and second order probabilities, the joint p-value will
be based on the singleton and second order mixture p-values, as given above.
A smaller joint p-value indicates a sample is more anomalous under the given feature subset.
§.§ Scoring Clusters
Let $\{I_c, J_c\}$ denote cluster candidate $c$, $I_c$ its sample subset and $J_c$ its feature subset.
Let $T_c = |I_c|, N_c=|J_c|$. Note that p-values are uniformly distributed on $[0,1]$ under the null.
Thus, given a cluster with feature subset $J_c$, from a test batch of size $T_u$, the probability that at least one cluster with $T_c$ samples has a smaller p-value than $P[\underset{j\in J_c}{\cap}(I_i^{(j)})=1]$ is:
\begin{eqnarray}
\label{bonf}
1-(1-\prod_{i} P[\underset{j\in J_c}{\cap}(I_i^{(j)})=1])^{C(T_u,T_c)}
\end{eqnarray}
Here, $C(T_u,T_c) = \dbinom{T_u}{T_c}$, i.e. it is the number of combinations and implements multiple testing correction, accounting for all possible sample subset configurations in a cluster with $T_c$ samples, from a test batch of size $T_u$.
In principle, (<ref>) provides a sound basis at least for directly comparing all cluster candidates with the same feature subset $J_c$.
However, it does not allow comparing pairs of cluster candidates with any configurations of $(T_c, N_c)$,
because all possible feature subset configurations at a given order, $N_c$,
have not yet been properly multiple-testing corrected.
Also, (<ref>) requires evaluation of the joint p-value $P[\underset{j\in J_c}{\cap}(I_i^{(j)})=1]), \forall i \in I_c$,
which in general depends on the joint density function for $(X_{j_1},X_{j_2},...,X_{j_{N_c}}), j_m\in J_c, m=1,...,N_c$.
When $D$ is large, it is not practically feasible to learn and store these $\dbinom{D}{N_c}$ joint null density functions,
i.e., for all possible combinations of features up to order $N_c$.
Thus, it appears some tractable representation of $P[\underset{j\in J_c}{\cap}(I_i^{(j)})=1])$ is needed.
An obvious temptation is to assume that $I_i^{(j)}$ and $I_i^{(j')}$ are statistically independent $\forall j,j'\in J_c, j'\neq j$.
But this is a very poor assumption, consistent with assuming the features are independent.
To address the above problems, we seek to modify (<ref>) in two respects.
First, we propose to multiple test correct both for the different sample and the different feature subsets, given a cluster candidate with $(T_c,N_c)$.
In this approach, instead of the exponent being the number of combinations, it becomes the product of combinations on samples and combinations on features.
Based on the Bonferroni approximation of (<ref>), we have the joint score function $S(I_c,J_c) = \dbinom{D}{N_c}\dbinom{T}{T_c}\prod_{i \in I_c}P[\underset{j\in J_c}{\cap}(I_i^{(j)})=1])$. For
this joint significance measure, we can efficiently determine the optimal sample subset, given a fixed feature subset, by greedy sequential sample inclusion, in sorted joint p-value order.
This is due to the unimodality of this Bonferroni approximated joint significance measure, as a function of the number of samples included in a cluster's sample subset (see next subsection).
Second, a rich, tractable, joint probability mass function model that does capture statistical dependencies is a restricted form of Bayesian network, based exclusively on first and second order distributions,
i.e., the dependence tree (DT), which factorizes the joint distribution $P[\underset{j\in J_c}{\cap}(I_i^{(j)})=1])$ as a product of first and second order probabilities <cit.>.
In <cit.>, it was shown that, even though there is an enormous number of unique dependence tree structures, one can efficiently find the globally optimal dependence tree, over all such structures,
maximizing the dataset's log-likelihood, by realizing that this can be recast as a maximum weight spanning tree problem,
with the pairwise weights defined as the mutual information between the pairs of random variables.
The maximum weight spanning tree can be efficiently solved via Kruskal's algorithm, with complexity $O(N_c^2log(N_c))$.
Hence, given any candidate feature subset $J_c$, Kruskal's algorithm can be applied to determine the DT that maximizes the likelihood measured on ${\cal X}_l$,
i.e., the null hypothesis is determined, consistent with the given candidate feature subset $J_c$.
Based on a given DT structure, $P[\underset{j\in J_c}{\cap}(I_i^{(j)})=1])$ factorizes as a product of first and second order distributions, i.e., $\forall i \in I_c$:
\begin{eqnarray}
\label{factorize}
P[\underset{j\in J_c}{\cap}(I_i^{(j)})] = P[I_i^{(j_1)}]P[I_i^{(j_2)}|I_i^{(j_1)}]...P[I_i^{(j_{N_c})}|I_i^{(j_{N_c-1})}],
\end{eqnarray}
where we use $j_1$ to denote the root node of the DT representing $J_c$.
It is apparent from (<ref>) that, for any feature subset, one can represent the joint p-value of a given sample by its first and second order mixture p-values.
That is, for any feature pair $(j,k)$, $P[I_i^{(j)}|I_i^{(k)}] = \frac{P[I_i^{(j)},I_i^{(k)}]}{P[I_i^{(k)}]}$.
The numerator and denominator are, respectively, the second and first order mixture-based p-values that we defined
earlier. Also note that, in order to evaluate the first order mixture p-value $P[I_i^{(k)}]$,
we marginalize feature $j$ from the bivariate GMM for the feature pair $(j,k)$. This gives us the GMM for feature $k$.
§.§ Identifying the Optimal Sample Subset $I_c$, Given Fixed $J_c$
Given a fixed $J_c$ and associated DT, we would like to choose the sample subset $I_c$ to minimize (<ref>).
Applying the Bonferroni correction, this is essentially equivalent to choosing $I_c$
to minimize the joint score function:
\begin{eqnarray}
\label{bonf_corr}
S(I_c,J_c)=\dbinom{D}{N_c}\dbinom{T_u}{T_c}\prod_{i \in I_c}P[\underset{j\in J_c}{\cap}(I_i^{(j)})=1]).
\end{eqnarray}
It is in fact easily shown that this objective function is globally minimized by the following procedure:
i) sort the samples in increasing order of their joint p-values $P[\underset{j\in J_c}{\cap}(I_i^{(j)})=1]$;
ii) sequentially include the samples on the sorted list into $I_c$, until the objective function no longer decreases. This procedure globally minimizes over $I_c$ given fixed $J_c$.
§.§ Overall Search Algorithm
First, using the normal samples in ${\cal X}_l$, all the first and second order null GMMs are separately trained
[Separately learning each marginal and pairwise feature GMM using the common training set ${{\cal X}_l}$ will not ensure consistency with respect to feature marginalizations.
Specifically, a marginal-consistent collection of univariate and bivariate density functions should satisfy the following: if we consider any feature pairs $(i,j)$ and ($j,k)$, marginalizing out
feature $i$ from the $(i,j)$ bivariate density and marginalizing out feature $k$ from the $(j,k)$ bivariate density should lead to the same marginal density for feature $j$.
However, when the univariate and bivariate distributions are Gaussian mixtures, with a non-convex log-likelihood function (and with BIC-based model order selection separately applied to choose the number of components for each GMM), separate application of EM-plus-BIC to learn each GMM density function does not ensure a set of marginal-consistent distributions.
This property is not centrally important here, however, since our main concern is only to learn marginal and pairwise density functions that allow accurate assessment of p-values.
Accordingly, in this work we will apply EM-plus-BIC separately, to learn each low-order GMM.
One approach to obtain marginal-consistent low-order distributions is to simply learn the single GMM for the joint distribution on the full feature vector,
$\underline{X}$. This determines (via marginalization) all lower-order distributions (which are also GMMs, and which are
guaranteed to be marginal-consistent). However, this strategy suffers from the curse of dimensionality.
Alternatively, we refer the interested reader to <cit.>, where a procedure for directly, jointly learning a marginal-consistent set of low-order GMMs is elaborated.
Mutual information for all feature pairs is then calculated based on the bivariate GMMs.
This is achieved by generating $M=10^6$ samples from a given bivariate GMM distribution, and then estimating the mutual information
by $\frac{1}{M}\sum\limits_{n=1}^M \log(\frac{f_{X_1 X_2}(x_1^{(n)},x_2^{(n)})}{f_{X_1}(x_1^{(n)}) f_{X_2}(x_2^{(n)})}$.
We then detect clusters in ${\cal X}_u$ sequentially, in a rank-prioritized fashion, according to the joint score $S(I_c,J_c)$.
The algorithm operates on an enormous space of candidate clusters even if the feature space itself is only modestly sized $(D)$.
We start by sweeping over feature subset candidates at low orders and, for tractability, only the “most promising” candidates at higher orders, with candidate feature subsets at order $K$ formed by “accreting”
new features to the best-scoring candidates at order $K-1$.
For each candidate feature subset $J_c$, its DT is first learned and its associated, optimal subset $I_c$ is then determined using
the method described in section III.C.
Evaluating all candidates at all feature subset orders, the one with the best score function value at each order $N_c$ is recorded.
The cluster with smallest Bonferroni-corrected score $S(I_c,J_c)$ is then forwarded as detected.
Its samples are then removed from the test batch.
Subsequent cluster detections can then be made following the same procedure.
Cluster detections are thus made (in general) in order of decreasing joint significance.
§ EXPERIMENTAL SETUP AND RESULTS
Our experiments focus on detecting Zeus
botnet and P2P traffic among normal Web traffic.
The Web packet-flows are
obtained from the LBNL repository <cit.>.
This dataset contains Web traffic on TCP port 80,
with specified time-of-day information.
Specifically, the experiments in this paper are based on three datasets named
“20041215-1343.port008” and “20041215-1443.port010”.
The protocols to obtain normal, P2P and BotNet network traffic are the same as in <cit.>,
i.e., we used the port-mapper in <cit.> to identify P2P traffic in these files by a C4.5 decision tree pre-trained in another domain (the Cambridge dataset <cit.>).
The Zeus Botnet traffic are obtained from another domain <cit.>.
§.§ Feature Space Selection and Representation
Firstly, we did not use layer-4 port number features for purposes of
detection <cit.>.
Also, we did not consider timing
information herein because the Zeus activity
was recorded on another domain <cit.>.
In <cit.>, previous efforts were made to detect BotNet and P2P traffic using the well-known feature representation for network intrusion detection from <cit.>.
The authors found that these features, though able to detect some
attack activity,
could not successfully discriminate BotNet or P2P from normal
Web traffic,
i.e., BotNet and P2P traffic appear as “normal"
Web activity according to the
features of <cit.>.
To capture the intrinsic behavior of BotNet and P2P
packet-traffic, we note that most Zeus BotNet traffic involves masters giving command (control) messages, while slaves execute the given commands.
In the case of P2P, nodes often
communicate in a bidirectional manner, exchanging relatively large packets
in both directions.
Normal/background Web traffic, on the other hand, tends to involve
server-to-client communications.
Hence, we seek to preserve the bidirectional packet size sequence information as feature representation for different traffic flows.
This feature representation was previously considered in <cit.>.
The authors used the first $N$ (we set $N=10$ in our experiments) packets after the three-way hand shake of each TCP flow.
Then a feature vector of dimension $2N$ is defined, specified by the sizes and directionalities of these $N$ packets.
Traffic are assumed to be alternating between client-to-server (CS) and server-to-client (SC).
A zero packet size is thus inserted between two consecutive packets in the same direction to indicate an absence of a packet in the other direction.
For example, if the bidirectional traffic is strictly SC, a zero will be inserted after each SC packet size.
This $2N$-dimensional feature representation preserves bidirectional information of a given TCP flow, which is essential for
discriminating between P2P, Zeus and normal Web traffic.
§.§ Performance Metrics
Our algorithm detects clusters (groups) in a sequential fashion.
For each extracted group, we rank the samples in the group by their associated joint p-values on the given feature subset.
These samples will be sequentially removed from the test batch, with the system then continuing to extract groups until the test set is depleted.
Then we sweep out an ROC curve based on these rank-ordered detected samples.
A larger area under the ROC curve indicates earlier detections of anomalous groups, which implies the effectiveness of the intrusion detection system.
We compare our system's performance with a GMM based anomaly detector, trained by normal samples, on the whole feature space.
For this detector, we rank the test samples based on their data likelihood under the GMM, and sweep out an ROC curve.
We also compare with the approach presented in <cit.>, which assumes significance tests are independent (denoted “Independence tests”),
and with the recent work presented in <cit.> with a slight modification –
instead of discretizing feature values consistent with <cit.>, we use a single dependence tree null distribution learned on
${\cal X}_l$ and our proposed joint p-value for continuous features, $P[\underset{j\in J_c}{\cap}(I_i^{(j)})=1]$.
We denote this variation on the approach in <cit.> by “single Bayesian Net.”
There are two generalization performance measures of interest on the test set:
one is the aforementioned ROC area under curve (ROC AUC) as a function of the maximum feature subset size for a cluster, $K_{\rm max}$.
The other is the top 100 precision rate, defined as the fraction of anomalous samples amongst the first 100 detected samples.
Lastly, instead of exhaustively searching over all feature subsets at order $K$, we trial-add individual features to the top candidate feature subsets from order $K-1$.
At each order $K$, starting from order 2, we only consider the top 500 candidates from order $K-1$.
Two different sets of experiments were performed, one on synthetic data, and the other on the network data mentioned earlier.
In the synthetic dataset experiment, we used one unimodal Gaussian with 10 dimensions to generate normal samples
and two additional unimodal Gaussians to generate two distinct anomalous clusters. The two anomalous clusters use the same
distribution as the normal distribution for nine of the ten features. Thus, they deviate from the normal (null) distribution
only on a single feature dimension (this “informative” feature dimension was different for the two clusters).
Their corresponding sample subsets consist of 2.5% of the whole data batch ${\cal X}_u$ (so the proportion of anomalous samples in ${\cal X}_u$ is 5% of the total).
The variance of the informative features was chosen to be the same as that of the normal features, $\sigma_n^2$.
Moreover the mean of the informative feature under an anomalous cluster was chosen to be two standard deviations
away from the mean under the normal class, i.e. $|\mu_n - \mu_a| = 2 \sigma_n$,
where we use subscripts $n$ and $a$ to denote `normal' and `anomalous', respectively.
Thus, if we consider only the informative feature dimension, the Bayes error rate in discriminating normal from anomalous
is 15.87%.
After generating the synthetic data batch (with a size of ten thousand samples), we randomly chose 20% of normal samples as ground-truth and used them to train the null hypothesis.
The remaining normal samples were used as part of the test batch, along with the samples from the two anomalous clusters.
This was repeated 10 times, with the performance averaged.
For the network data, all the normal web flows from the three files were combined, making nearly ten thousand normal web flows.
We randomly selected 20% of these flows as ground-truth normal samples to train the null,
and treated the remaining normal flows as part of the test batch, combined with either P2P or Zeus anomalous flows.
We separately experimented with P2P and Zeus flows.
There were roughly 5 % of either P2P or Zeus flows in a given test batch.
Experiments for each scenario were averaged over 10 random train-test splits.
§.§ Experimental results
In Figure <ref>, we show the performance on the synthetic data.
Note that both the proposed scheme and <cit.> effectively capture groups of anomalies when the maximum feature subset order is two.
The first captured cluster (sample subset) consists of more than 95% anomalous samples on average.
However, as the maximum feature subset order increases, the
“independence tests" approach drops significantly in performance.
This is because too many (assumed to be independent) pairwise tests create many redundant features that are all used to evaluate cluster anomalousness; use of these redundant features de-emphasizes, within the score function, the important (low-order) feature subset.
Also, we see an early advantage of using cluster-specific DTs, compared to the single Bayesian Net approach.
It appears that if an anomalous process is strictly generated from a low order subspace and normal in other feature dimensions (as is the
case in this experiment) our cluster-specific DT approach
outperforms a single Bayesian Net approach.
Synthetic data experiment: comparison of different schemes with 2 independent Gaussian based anomalous feature subsets in (separate) 1-dim subspace
In Figure <ref> a), we show the performance for normal-P2P discrimination.
Compared to <cit.>, which degrades in performance as more and more tests are included, we see superior performance for the proposed method.
There is a large batch of anomalous samples captured at maximum order 6 by the proposed method, but both <cit.> and <cit.> did not capture this group effectively,
as seen in the top 100 precision figure.
Also, both of these methods are outperformed by the GMM baseline method.
In Figure <ref> b), we show the performance for normal-Zeus discrimination.
Again, at maximum feature subset order 6 the proposed method captures a large portion of the anomalous flows –
more than 50 Zeus flows were captured out of the first 100 flows detected by the proposed method.
<cit.> performs poorly in this experiment, and again we observed that as the number of tests increase,
the independence assumption degrades the detection performance.
The single Bayesian Net approach in <cit.> also performs relatively poorly on this dataset.
(a) P2P
(b) Zeus
Network traffic data experiment: comparison of different schemes with P2P or Zeus anomalies
§ EXTENSIONS AND FUTURE WORK
In this work, we
used the Bonferroni corrected score function to directly evaluate cluster candidates. Alternatively, we could try to
evaluate empirical p-values for this decision statistic, by applying our detection strategy to (many) bootstrap test batches drawn from the null distribution.
It would be interesting to see whether such an approach gives comparable (or even better) detection accuracy than
use of the Bonferroni corrected score by itself. Such an approach could also be used to determine whether any detected clusters
are truly statistically significant.
In this work we showed detection accuracy as a function of the maximum feature subset size for a cluster.
As the maximum feature subset size continues to increase, we observed that false positives also increase in the first detected cluster,
and the objective in (<ref>) tends to favor the maximum feature dimension over use of fewer dimensions.
In future, we should propose and investigate criteria for choosing this maximum feature subset size.
§ CONCLUSION
In this work, we proposed a GAD scheme to identify anomalous sample and feature subsets, accounting for dependencies between the
features in a given subset.
The proposed model outperforms previous works that assume statistical tests are independent under the null.
We demonstrated the effectiveness of our proposed system on both synthetic and real world data, with the latter
drawn from
the network intrusion detection domain, aiming to discriminate between normal and P2P/Zeus traffic.
Our future work includes empirical p-value assessment and automatic determination of the maximum feature subset size of a cluster.
|
1511.00661
|
Given a stream $p_1, \ldots, p_m$ of items from a universe
$\mathcal{U}$, which, without loss of generality we identify with
the set of integers $\{1, 2, \ldots, n\}$,
we consider the problem of returning all
$\ell_2$-heavy hitters, i.e., those items $j$ for which
$f_j \geq \eps \sqrt{F_2}$, where $f_j$ is the number of occurrences
of item $j$ in the stream, and $F_2 = \sum_{i \in [n]} f_i^2$.
Such a guarantee is considerably stronger than the
$\ell_1$-guarantee, which finds those $j$ for which
$f_j \geq \eps m$. In
2002, Charikar, Chen, and Farach-Colton suggested the CountSketch
data structure, which finds all such $j$ using $\Theta(\log^2 n)$ bits of space
(for constant $\eps > 0$). The only
known lower bound is $\Omega(\log n)$ bits of space, which comes from
the need to specify the identities of the items found.
In this paper we show it is possible to achieve $O(\log n \log \log n)$ bits of space for this problem.
Our techniques, based on Gaussian processes, lead to a number of other new results for data streams, including
* The first algorithm
for estimating $F_2$ simultaneously at all points in a stream
using only $O(\log n\log\log n)$ bits of space, improving
a natural union bound and the algorithm of Huang, Tai, and Yi (2014).
* A way to estimate the $\ell_{\infty}$ norm of a stream up to
additive error $\eps \sqrt{F_2}$ with $O(\log n\loglog n)$ bits of space, resolving
Open Question 3 from the IITK 2006 list for insertion only streams.
§ INTRODUCTION
There are numerous applications of data streams, for which the elements $p_i$ may be numbers, points, edges in a graph, and so on. Examples include internet search logs, network traffic, sensor networks, and scientific data streams (such as in astronomy, genomics, physical simulations, etc.).
The sheer size of the dataset often imposes very stringent requirements on an algorithm's resources.
In many cases only a single pass over the data is feasible, such as in network applications, since if the data on a network is not physically stored somewhere, it may be impossible to make a second pass over it.
There are multiple surveys and tutorials in the algorithms, database, and networking communities on the recent activity in this area; we refer the reader to <cit.> for more details and motivations underlying this area.
Finding heavy hitters, also known as the top-$k$, most popular items, elephants, or iceberg queries, is arguably one of the most fundamental problems in data streams.
It has applications in flow identification at IP routers <cit.>, iceberg queries <cit.>, iceberg datacubes <cit.>, association rules, and frequent itemsets <cit.>.
Formally, we are given a stream $p_1, \ldots, p_m$ of items from a universe $\mathcal{U}$, which, without loss of generality we identify with the set $\{1, 2, \ldots, n\}$. We make the common assumption that $\log m = O(\log n)$, though our results generalize naturally to any $m$ and $n$. Let $f_i$ denote the frequency, that is, the number of occurrences, of item $i$. We would like to find those items $i$ for which $f_i$ is large, i.e., the “heavy hitters”.
In this paper we will consider algorithms that are allowed one pass over the stream and must use as little space (memory) in bits as possible, to maintain a succinct summary (“sketch”) so that after processing the stream, we can output the identities of all of the heavy hitters from the summary with large probability.
There are various notions of what it means for $f_i$ to be large. One such notion is that we should return all indices $i \in [n]$ for which $f_i \geq \epsilon m$ for a parameter $\epsilon \in (0,1)$, and no index $i$ for which $f_i \leq (\epsilon - \phi)m$, for a parameter $\phi$.
It is typically assumed that $\phi \geq c \epsilon$ for an absolute constant $c > 0$, and we will do so in this paper.
This notion has been extensively studied, so much so, that the same streaming algorithm for it was re-invented multiple times. The first algorithm was given by Misra and Gries <cit.>, who achieved $O((\log n)/\epsilon)$ bits of space.
The algorithm was rediscovered by Demaine et al. <cit.>, and then later rediscovered by Karp et al. <cit.>.
Cormode and Muthukrishan <cit.> state that “papers on frequent items are a frequent item!”. While these algorithms are deterministic,
there are also several randomized algorithms, including the Count-Min sketch <cit.>, sticky sampling <cit.>, lossy counting <cit.>, sample and hold <cit.>, multi-stage bloom filters <cit.>, sketch-guided sampling <cit.>, and CountSketch <cit.>.
A useful (slightly suboptimal) intuition is that one can sample $O((\log 1/\epsilon)/\epsilon)$ random stream positions, remember the identities of these positions, and then maintain the counts of these items.
By coupon collector arguments, all heavy hitters will be found this way, and one can filter out the spurious ones (those with $f_i \leq (\epsilon-\phi)m$).
One of these techniques, CountSketch <cit.>, refined in <cit.>, gives a way of finding the $\ell_2$-heavy hitters of a stream.
Those are the items for which $f_i^2 \geq \eps^2 F_2$.
Notice that this guarantee is significantly stronger than the aforementioned guarantee that $f_i \geq \epsilon m$, which we will call the $\ell_1$-guarantee.
Indeed, if $f_i \geq \epsilon m$, then $f_i^2 \geq \epsilon^2 m^2 \geq \epsilon^2 F_2$.
So, an algorithm for finding the $\ell_2$-heavy hitters will find all items satisfying the $\ell_1$-guarantee.
On the other hand, given a stream of $n$ distinct items in which $f_i = \sqrt{n}$ for an $i \in [n]$, yet $f_j = 1$ for all $j \neq i$, an algorithm satisfying the $\ell_2$-heavy hitters guarantee will identify item $i$ with constant $\epsilon$, but an algorithm which only has the $\ell_1$-guarantee would need to set $\epsilon = 1/\sqrt{n}$, using $\Omega(\sqrt{n} \log n)$ bits of space.
In fact, $\ell_2$-heavy hitters are in some sense the best one can hope for with a small amount of space in a data stream, as it is known for $p > 2$ that finding those $i$ for which $f_i^p \geq \eps^p F_p$ requires $n^{1-2/p}$ bits of space <cit.>.
The CountSketch has broad applications in compressed sensing <cit.> and numerical linear algebra <cit.>, and are often used as a subroutine in other data stream algorithms, such as $\ell_p$-sampling <cit.>, cascaded aggregates <cit.>, and frequency moments <cit.>.
Given the strong guarantees and many applications of $\ell_2$-heavy hitter algorithms, it is natural to ask what the best space complexity for them is.
Both the original algorithm of <cit.> and the followup of <cit.> achieve $\Theta(\log^2 n)$ bits of space for constant values of $\epsilon$.
On the other hand, the only known lower bound is $\Omega(\log n)$ bits, which is needed just to identify the heavy hitter.
Despite the success we have had in obtaining space-optimal streaming algorithms for estimating moments and $p$-norms, this has remained a glaringly open problem.
It is known that if one allows deletions in the stream, in addition to insertions, then $\Theta(\log^2 n)$ bits of space is optimal <cit.>.
However, in many cases we just have a stream of insertions, such as in the model studied in the seminal paper of Alon, Matias, and Szegedy <cit.>.
§.§ Our Contributions
The main result of this paper is the near resolution of the open question above. We show:
The intuition of the proof is as follows.
Suppose there is a single $\ell_2$-heavy hitter $H$, $\eps > 0$ is a constant, and we are trying to find the identity of $H$.
Suppose further we could identify a substream $S'$ where $H$ is very heavy, specifically we want that the frequencies in the substream satisfy $\frac{f_H^2}{\poly(\log n)}\geq \sum_{j \in S',j \neq H} f_j^2$.
Suppose also that we could find certain $R=O(\log{n})$ “breakpoints” in the stream corresponding to jumps in the value of $f_H$, that is, we knew a sequence $p_{q_1} < p_{q_2} < \cdots < p_{q_{R}}$ which corresponds to positions in the stream for which $f_H$ increases by a multiplicative factor of $\left (1 + \frac{1}{\Theta(R)} \right)$.
Given all of these assumptions, in between breakpoints we could partition the universe randomly into two pieces and run an $F_2$-estimate <cit.> (AMS sketch) on each piece.
Since $f_H^2$ is more than a $\poly(\log n)$ factor times $\sum_{j \in S', j \neq H} f_j^2$, while in between each breakpoint the squared frequency of $H$ is $\Omega \left(\frac{f_H^2}{\log n} \right)$, it follows that $H$ contributes a constant fraction of the $F_2$-value in between consecutive breakpoints, and so, upon choosing the constants appropriately, the larger of the magnitudes of the two AMS sketches will identify a bit of information about $H$, with probability say 90%.
This is our algorithm Sieve.
Since we have $\Theta(\log n)$ breakpoints, in total we will learn all $\log n$ bits of information needed to identify $H$.
One persepective on this algorithm is that it is a sequential implementation of the multiple repetitions of CountSketch, namely, we split the stream at the breakpoints and perform one “repetition” on each piece while discarding all but the single bit of information we learn about $H$ in between breakpoints.
However, it is not at all clear how to (1) identify $S'$ and (2) find the breakpoints.
For this, we resort to the theory of Gaussian and Bernoulli processes.
Throughout the stream we can maintain a sum of the form $X_t = \sum_{i=1}^n f^{(t)}_i Z_i$, where the $Z_i$ are independent Normal$(0,1)$ or Rademacher random variables.
Either distribution is workable.
One might think as one walks through a stream of length $\poly(n)$, there will be times for which this sum is much larger than $\sqrt{F_2}$; indeed, the latter is the standard deviation and a naïve union bound, if tight, would imply positions in the stream for which $|X_t|$ is as large as $\sqrt{F_2 \log n}$.
It turns out that this cannot happen!
Using a generic chaining bound developed by Fernique and Talagrand <cit.>, we can prove that
there exists a universal constant $C'$ such that
\[\E\sup_t|X_t| \leq C'\sqrt{F_2}.\]
We call this the Chaining Inequality.
We now randomly partition the universe into $O(\frac{1}{\epsilon^2})$ “parts”, and run our algorithm independently on each part.
This ensures that, for a large constant $C$, $H$ is $C$-heavy, meaning, $f_H^2 \geq C(F_2-f_H^2)$, where here we abuse notation and use $F_2$ to denote the moment of the part containing $H$.
We run the following two stage algorithm independently on each part.
The first stage, called Amplifier, conists of $L=O(\log \log n)$ independent and concurrent repetitions of the following: randomly split the set of items into two buckets and maintain a two Bernoulli processes, one for the updates in each bucket.
By the Chaining Inequality, a Markov bound, and a union bound, the total $F_2$ contribution, excluding that of $H$, in each piece in each repetition at all times in the stream will be $O(\sqrt{F_2-f_H^2})$.
Since $H$ is sufficiently heavy, this means after some time $t^*$, its piece will be larger in magnitude in most, say 90%, of the $L$ repetitions.
Furthermore, $H$ will be among only $n/2^{\Omega(L)}=n/\poly\log n$ items with this property.
At this point we can restrict our attention to a substream containing only those items.
The substream has the property that its $F_2$ value, not counting $H$, will be a factor $\frac{1}{\log^2 n}$ times the $F_2$ value of the original stream, making $H$ $\Omega(\log^2 n)$-heavy.
Finally, to find the breakpoints, our algorithm Timer maintains a Bernoulli process on the substream, and every time the Bernoulli sum increases by a multiplicative $\left(1+\frac{1}{\theta(R)} \right)$ factor, creates a new breakpoint.
By the Chaining Inequality applied in each consecutive interval of breakpoints, the $F_2$ of all items other than $H$ in the interval is at most $O(\log n)$ larger than its expectation; while the squared frequency of $H$ on the interval is at least $\frac{f_H^2}{\log n}$.
Since $H$ is $\Omega(\log^2 n)$-heavy, this makes $f_H^2$ to be the dominant fraction of $F_2$ on the interval.
One issue with the techniques above is they assume a large number of random bits can be stored.
A standard way of derandomizing this, due to Indyk <cit.> and based on Nisan's pseudorandom generator PRG <cit.>, would increase the space complexity by a $\log n$ factor, which is exactly what we are trying to avoid.
Besides, it is not clear we can even apply Indyk's method since our algorithm decides at certain points in the stream to create new $F_2$-sketches based on the past, whereas Indyk's derandomization relies on maintaining a certain type of linear sum in the stream, so that reordering of the stream does not change the output distribution.
A first observation is that the only places we need more than limited independence are in maintaining a collection of $O(\log{n})$ hash functions and the stochastic process $\sum_{i=1}^n f_i Z_i$ throughout the stream.
The former can, in fact, be derandomized along the lines of Indyk's method <cit.>.
In order to reduce the randomness needed for the stochastic process we use a Johnson-Lindenstrauss transformation to reduce the number of Rademacher (or Gaussian) random variables needed.
The idea is to reduce the frequency vector to $O(\log n)$ dimensions with JL and run the Bernoulli process in this smaller dimensional space.
shortversionThe Bernoulli process becomes $\sum_{i=1}^{O(\log n)}Z_i (T f)_i$, where $T$ is the JL matrix.
The same technique is used by Meka for approximating the supremum of a Gaussian process <cit.>.
It works because the Euclidean length of the frequency vector describes the variance and covariances of the process, hence the transformed process has roughly the same covariance structure as the original process.
shortversionAn alternative perspective on this approach is that we use the JL transformation in reverse, as a pseudorandom generator that expands $O(\log n)$ random bits into $O(n)$ random variables which fool our algorithm using the Bernoulli process.
In the full version
In Section <ref>
we also use our techniques to prove the following.
In Section <ref>, we give preliminaries and define our notation.
In Section <ref> we prove Theorem <ref>.
The proof of the Chaining Inequality for Gaussian and Bernoulli processes, the central tool used in Section <ref>, appears in Section <ref>.
In the full version we give complete proofs, including details about how to implement the algorithm with a reduced number of random bits, and we prove Theorem <ref>.
In Section <ref> we give details about how to implement the algorithm with a reduced number of random bits.
In Section <ref> we prove Theorem <ref>.
§.§ Preliminaries
$L$ amplifier size $O(\log\log{n})$
$\tau$ round expansion $100(R+1)$
$\delta$ small constant $\Omega(1)$
$S^{t_1:t_2}$ interval of the stream $(p_{t_1+1},\ldots,p_{t_2})$
$H$ heavy hitter id $\in[n]$
$e_j$ $j$th unit vector
$T$ JL transformation $\in\R^{k\times n}$
$f_H^{(k)}$ frequency on $S^{0:k}$
$m$ stream length $\poly(n)$
$f^{(k_1:k_2)}$ frequency on $S^{k_1:k_2}$ $f^{(k_2)} - f^{(k_1)}$
$n$ domain size
$R$ $\#$ of Sieve rounds $O(\log{n})$
$k$ JL dimension $O(\log n)$
$C'$ Chaining Ineq. const. $O(1)$
$d$ dim. of Bern. proc. $O(\log \delta^{-1})$
$C$ large const. $\geq d^{\frac{3}{2}}C'/\delta$
Notation and parameters used throughout the paper.
Given a stream $S=(p_1,p_2,\ldots,p_m)$, with $p_i\in[n]$ for all $i$, we define the frequency vector at time $0\leq t\leq m$ to be the vector $f^{(t)}$ with coordinates $f_j^{(t)} := \#\{t'\leq t\mid p_{t'}=j\}$.
When $t=m$ we simply write $f:=f^{(m)}$.
Given two times $t_1\leq t_2$ we use $f^{(t_1:t_2)}$ for the vector $f^{(t_2)}-f^{(t_1)}$.
Notice that all of these vectors are nonnegative because $S$ has no deletions.
An item $H\in[n]$ is said to be an $\alpha$-heavy hitter, for $\alpha>0$, if $f_H^2\geq\alpha\sum_{j\neq H}f_j^2$.
The goal of our main algorithm, CountSieve, is to identify a single $\alpha$-heavy hitter for $\alpha$ a large constant.
We will assume $\log{m}=O(\log{n})$, although our methods apply even when this is not true.
It will be occasionally helpful to assume that $n$ is sufficiently large.
This is without loss of generality since in the case $n=O(1)$ the problem can be solved exactly in $O(\log{m})$ bits.
A streaming algorithm is allowed to read one item at a time from the stream in the order given.
The algorithm is also given access to a stream of random bits, it must pay to store any bits that it accesses more than once, and it is only required to be correct with constant probability strictly greater than $1/2$.
Note that by repeating such an algorithm $k$ times and taking a majority vote, one can improve the success probability to $1-2^{-\Omega(k)}$.
shortversionWe measure the storage used by the algorithm on the worst case stream, i.e. worst case item frequencies and order, with the worst case outcome of its random bits.
The AMS sketch <cit.> is a linear sketch for estimating $F_2$.
The sketch contains $O(\epsilon^{-2}\log{\delta^{-1}})$ independent sums of the form $\sum_{j=1}^n S_jf_j$, where $S_1,S_2,\ldots,S_n$ are four-wise independent Rademacher random variables.
By averaging and taking medians it achieves a $(1\pm\epsilon)$-approximation to $F_2$ with probability at least $(1-\delta)$.
A Gaussian process is a stochastic process $(X_t)_{t\in T}$ such that every finite subcollection $(X_t)_{t\in T'}$, for $T'\subseteq T$, has a multivariate Gaussian distribution.
When $T$ is finite (as in this paper), every Gaussian process can be expressed as a linear transformation of a multivariate Gaussian vector with mean 0 and covariance $I$.
Similarly, a Bernoulli process $(X_t)_{t\in T}$, $T$ finite, is a stochastic process defined as a linear tranformation of a vector of i.i.d. Rademacher (i.e. uniform $\pm1$) random variables.
Underpinning our results is an analysis of the Gaussian process $X_t = \sum_{j\in[n]} Z_j f_j^{(t)}$, for $t=0,\ldots,m$, where $Z_1,\ldots,Z_n\iidsim\calN(0,1)$ are independent standard Normal random variables.
The Bernoulli analogue to our Gaussian process replaces the distribution of the random vector $Z$ as $Z_1,\ldots,Z_n\iidsim\Rademacher$.
shortversionProperties of the Normal distribution make it easier for us to analyze the Gaussian process rather than its Bernoulli cousin.
On the other hand, we find Bernoulli processes more desirable for computational tasks.
Existing tools, which we discuss further in Section <ref> and shortversion
the full version
Section <ref>, allow us to transfer the needed properties of a Gaussian process to its Bernoulli analogue.
A $k\times n$ matrix $T$ is a $(1\pm\gamma)$-embedding of a set of vectors $X\subseteq\R^n$ if
$(1-\gamma)\|x-y\|_2\leq \|Tx-Ty\|_2\leq (1+\gamma)\|x-y\|_2$,
\[(1-\gamma)\|x-y\|_2\leq \|Tx-Ty\|_2\leq (1+\gamma)\|x-y\|_2,\]
for all $x,y\in X\cup\{0\}$.
We also call such a linear transformation a JL Transformation.
It is well-known that taking the entries of the matrix $T$ to be i.i.d. Normal random variables with mean 0 and variance $1/k$ produces a JL transformation with high probability.
Many other randomized and deterministic constructions exist, we will use the recent construction of Kane, Meka, and Nelson <cit.>.
The development and analysis of our algorithm relies on several parameters, some of which have already been introduced.
Table <ref> lists those along with the rest of the parameters and some other notation for reference.
In particular, the values $C$, $d$, $\delta$, and $\gamma$ are constants that we will choose in order to satisfy several inequalities.
We will choose $\delta$ and $\gamma$ to be small, say $1/200$, and $d=O(\log 1/\delta)$.
$C$ and $C'$ are sufficiently large constants, in particular $C\geq dC'/\delta$.
§ $\ELL_2$ HEAVY HITTERS ALGORITHM
This section describes the algorithm CountSieve, which solves the heavy-hitter problem for the case of a single heavy hitter, i.e. top-1, in $O(\log{n}\log\log n)$ bits of space and proves Theorem <ref>.
By definition, the number of $\epsilon$-heavy hitters is at most $1+1/\epsilon$, so, upon hashing the universe into $O(1/\epsilon^2)$ parts, the problem of finding all $\epsilon$-heavy hitters reduces to finding a single heavy hitter in each part.
When $\epsilon=\Omega(1)$, using this reduction incurs only a constant factor increase in space over the single heavy hitter problem.
Suppose the stream has only a single heavy hitter $H\in[n]$.
Sequentially, over the course of reading the stream, CountSieve will hash the stream into two separate substreams for $O(\log{n})$ repetitions, and in each repetition it will try to determine which of the two substreams has the heavy hitter using the AMS Sketch.
With high probability, $H$ has a unique sequence of hashes, so if we correctly identify the stream containing $H$ every time then we can correctly identify $H$. This holds even if we only correctly identify the stream containing $H$ a large constant fraction of the repetitions.
CountSketch accomplishes this by performing the $O(\log{n})$ rounds of hashing in parallel, with $\Omega(\log^2n)$ bits of storage.
One of our innovations is to implement this scheme sequentially by specifying intervals of updates, which we call rounds, during each of which we run the two AMS Sketches.
In total there could be as many as $\Theta(\log^2{n})$ of these rounds, but we will discard all except the last $R=O(\log n)$ of them.
Algorithm <ref> is a simplified version of the Bernoulli process used by CountSieve.
It has all of the properties we need for correctness of the algorithm, but it requires too many random bits.
Chief among these properties is the control on the supremum of the process.
BPStream $S$
Sample $Z_1,\ldots,Z_n\iidsim\Rademacher$
$\langle Z, f^{(t)}\rangle$ at each time $t$
One Bernoulli process.
The Chaining Inequality gives us a uniform bound on the maximum value of the BP process in terms of the standard deviation of the last value.
This property is formalized by the definition of a tame process.
Let $f^{(t)}\in\R^n$, for $t\in[m]$, and let $T:\R^n\to\R^k$ be a matrix.
Let $Z$ be a $d\times k$ matrix of i.i.d. Rademacher random variables.
A $d$-dimensional Bernoulli process $y_t = d^{-\frac{1}{2}}ZTf^{(t)}$, for $t\in[m]$, is tame if, with probability at least $1-\delta$,
\begin{equation}\label{eq: tame definition}
\|y_t\|_2 \leq C\sqrt{\sum_{j=1}^n f_j^2},\quad\text{for all }t\in[m].
\end{equation}
The definition anticipates our need for dimension reduction in order to reduce the number of random bits needed for the algorithm.
Our first use for it is for BP, which is very simple with $d=1$ and $T$ the identity matrix.
BP requires $n$ random bits, which is too many for a practical streaming algorithm.
JLBP, Algorithm <ref>, exists to fix this problem.
Still, if one is willing to disregard the storage needed for the random bits, BP can be substituted everywhere for JLBP without affecting the correctness of our algorithms because our proofs only require that the processes are tame, and BP produces a tame process, as we will now show.
We have a similar lemma for JLBP.
Let $f^{(t)}$, for $t\in[m]$, be the frequency vectors of an insertion-only stream.
The sequence $Zf^{(t)}$ returned by the algorithm BP is a tame Bernoulli process.
By the Chaining Inequality, Theorem <ref> below, there exists a constant $C'$ such that $\E\sup_t|X_t|\leq C'(\sum_{j}f_j^2)^{1/2}$.
Let $F$ be the event that the condition (<ref>) holds.
Then for $C\geq C'/\delta$ we have, by Markov's Inequality,
$\Pr (F) = \Pr \left(\sup_t|X_t|\leq C \sqrt{\sum_j f_j^2}\right)\geq (1-\delta)$.
\[\Pr (F) = \Pr \left(\sup_t|X_t|\leq C \sqrt{\sum_j f_j^2}\right)\geq (1-\delta).\]
In order to reduce the number of random bits needed for the algorithms we first apply JL transformation $T$ to the frequency vector.
The intuition for this comes from the covariance structure of the Bernoulli process, which is what governs the behavior of the process and is fundamental for the Chaining Inequality.
The variance of an increment of the Bernoulli process between times $s$ and $t>s$ is $\|f^{(s:t)}\|_2^2$.
The JL-property of the matrix $T$ guarantees that this value is well approximated by $\|Tf^{(s:t)}\|_2^2$, which is the increment variance of the reduced-dimension process.
Slepian's Lemma shortversion(Lemma <ref>) is a fundamental tool in the theory of Gaussian processes that allows us to draw a comparison between the suprema of the processes by comparing the increment variances instead.
Thus, for $Z_1,\ldots,Z_n\iidsim\Rademacher$, the expected supremum of the process $X_t=\sum_{i=1}^n Z_if_i^{(t)}$ is closely approximated by that of $X_t'= \sum_{i=1}^kZ_i(Tf^{(t)})_i$, and the latter uses only $k=O(\log n)$ random bits.
The following lemma formalizes this discussion, its proof is given in shortversionthe full version.Section <ref>.
JLBPStream $S$
Let $T$ be a JL Transformation The same $T$ will suffice for every instance
Sample $Z\in\{-1,1\}^{d\times k}$ with coordinates $Z_{i,j}\iidsim\Rademacher$
$\frac{1}{\sqrt{d}} ZTf^{(t)}$ at each time $t$
A Bernoulli process with fewer random bits.
Now that we have established the tameness of our Bernoulli processes, let us explain how we can exploit it.
We typically exploit tameness in two ways, one works by splitting the stream according to the items and the second splits the stream temporally.
Given a stream and a tame Bernoulli process on that stream, every substream defines another Bernoulli process, and the substream processes are tame as well.
One way to use this is for heavy hitters.
If there is a heavy-hitter $H$, then the substream consisting of all updates except those to the heavy-hitter produces a tame process whose maximum is bounded by $C(F_2-f_H^2)^{1/2}$, so the value of the process in BP is $Z_Hf_H\pm C(F_2-f_H^2)^{1/2}$.
When $H$ is sufficiently heavy, this means that the absolute value of the output of BP tracks the value of $f_H$, for example if $H$ is a $4C^2$-heavy hitter then the absolute value of BP's output is always a $(1\pm\frac{1}{2})$-approximation to $f_H$.
Another way we exploit tameness is for approximating $F_2$ at all points.
We select a sequnece of times $t_1<t_2<\cdots<t_j\in[m]$ and consider the prefixes of the stream that end at times $t_1,t_2,\ldots,$ etc.
For each $t_i$, the prefix stream ending at time $t_i$ is tame with the upper bound depending on the stream's $F_2$ value at time $t_i$.
If the times $t_i$ are chosen in close enough succession this observation allows us to transform the uniform additive approximation guarantee into a uniform multiplicative approximation.
§.§ Description of CountSieve
CountSieve primarily works in two stages that operate concurrently.
Each stage uses independent pairs of Bernoulli processes to determine bits of the identity of the heavy hitter.
The first stage is the Amplifier, which maintains $L=O(\log\log n)$ independent pairs of Bernoulli processes.
The second stage is the Timer and Sieve.
It consists of a series of rounds where one pair of AMS sketches is maintained during each round.
CountSieve and its subroutines are described formally in Algorithm <ref>.
The random variables they use are
shortversionAmplifier hashes $A_{\ell,1},\ldots,A_{\ell,n}\pwsim\Bernoulli$, for $\ell\in[L]$,
one independent copy of the sequence $Z_{1},\ldots,Z_{k}\iidsim \Rademacher$ for each instance of JLBP, Sieve hashes $B_{r,1},\ldots,B_{r,n}\pwsim\Bernoulli$, and Sieve Rademachers for AMS
The algorithm also needs a random seed of length $O(\log n)$ bits for the Kane, Meka, Nelson JL generator <cit.>.
listed in Table <ref>.
Even though we reduce the number of random bits needed for each Bernoulli process to a managable $O(\log n)$ bits, the storage space for the random values is still an issue because the algorithm maintains $O(\log{n})$ independent hash functions until the end of the stream.
We explain how to overcome this barrier in shortversionthe full versionSection <ref> as well as show that the JL generator of <cit.> suffices.
$Z_{1},\ldots,Z_{k}\iidsim \Rademacher$
Random vectors for CountSieve.
Each vector is independent of the others, and $Z=(Z_i)_{i\in[k]}$ is sampled independently for every instance of JLBP.
shortversionWe can now state an algorithm that maintains a pair of Bernoulli processes and prove that the bits that it outputs favor the process in the pair with the heavy hitter.
PairStream $S$, $A_1,\ldots A_n\in\{0,1\}$
For $b\in\{0,1\}$ let $S_b$ be the restriction of $S$ to $\{j\in[n]\mid A_j=b|\}$
$X^{(t)}_0 = $JLBP$(S^{(t)}_0)$ at each time $t$
$X^{(t)}_1 = $JLBP$(S^{(t)}_1)$ at each time $t$
$b_t = \argmax_{b\in\{0,1\}} \|X_b^{(t)}\|_2$
Split the vector $f$ into two parts depending on $A$ and run a Bernoulli process on each part.
Return the identity of the larger estimate at each time.
Let $t_0\in[m]$ be an index such that $(f_H^{(t_0)})^2> 4C^2\sum_{j\neq H}f_j^2$.
Let $A_1,\ldots,A_n\pwsim\Bernoulli$ and $b_1,b_2,\ldots,b_m$ be the sequence returned by Pair$(f,A_1,\ldots,A_n)$.
$\Pr (b_t=A_H\text{ for all }t\geq t_0)\geq 1-3\delta$ and, for every $j\in[n]\setminus\{H\}$ and $t\geq t_0$, $\Pr (b_t=A_j) \leq \frac{1}{2}+3\delta$.
\[\Pr (b_t=A_H\text{ for all }t\geq t_0)\geq 1-3\delta\]
and, for every $j\in[n]\setminus\{H\}$ and $t\geq t_0$,
\[\Pr (b_t=A_j) \leq \frac{1}{2}+3\delta.\]
Furthermore, if each JLBP is replaced by an AMS sketch with size $O(\log{n}\log\delta^{-1})$ then, for all $t\geq t_0$ and $j\neq H$, $P(b_t=A_H)\geq 1-2\delta$ and $P(b_t=A_j)\leq \frac{1}{2}+3\delta$.
Let $X_0^{(t)}=d^{-\frac{1}{2}}ZTf^{(t)}$ and $X_1^{(t)} = d^{-\frac{1}{2}}WTf^{(t)}$ be the two independent Bernoulli processes output by JLBP.
Without loss of generality, suppose that $A_H=1$, let $v = d^{-\frac{1}{2}}WTe_H$, and let $Y^{(t)} = X_1^{(t)}-f_H^{(t)}v$.
By Lemma <ref>, with probability at least $1-2\delta$ all three of the following hold
* $\|X_0^{(t)}\|^2_2\leq C^2\sum_{j:A_j=0}f_j^2$, for all $t$,
* $\|Y^{(t)}\|^2_2\leq C^2\sum_{\substack{j\neq H\\A_j=1}} f_j^2$, for all $t$, and
* $\|v\|_2\geq 1/2$.
If the three events above hold then, for all $t\geq t_0$,
\[\|X_1^{(t)}\|_2-\|X_0^{(t)}\|_2\geq \|v f_H^{(t)}\|_2 -\|Y^{(t)}\|_2 - \|X_0^{(t)}\|_2 \geq \frac{1}{2}f_H^{(t)}-C\sqrt{\sum_{j\neq H} f_j^2}>0,\]
which establishes the first claim.
The second claim follows from the first using
\[\Pr (b_t=A_j) = \Pr (b_t=A_j=A_H) + \Pr (b_t=A_j\neq A_H) \leq \Pr (A_j=A_H) + \Pr (b_t\neq A_H) = \frac{1}{2}+3\delta.\]
The third and fourth inequalities follow from the correctness of the AMS sketch <cit.>.
CountSieveStream $S=(p_1,p_2,\ldots,p_m)$
Maintain $a_t=(a_{1,t},a_{2,t},\ldots,a_{L,t})\gets$Amplifier$(S)$
Let $t_1<t_2<\cdots = \{t\in[n] \mid A_{\ell,p_t}=a_{\ell,t}\text{ for at least }0.9L\text{ values of }\ell\}$
Let $S_0 = (p_{t_1},p_{t_2},\ldots,)$
$q_0,q_1, \dots, q_R \gets $Timer($S_0$)
$b_1,b_2,\ldots,b_R\gets $Sieve$(S_0, q_0, \dots, q_R)$
Selector(${b_1,b_2,\ldots,b_R}$) based on $S_{0}$
AmplifierStream $S$Find a substream where $H$ is $\polylog(n)$-heavy
$a_{1,t}, \dots, a_{L,t}$ at each time $t$
Algorithm for a single $F_2$ heavy hitters.
TimerStream $S$Break the substream into rounds so $H$ is heavy in each
$Y_t \gets $JLBP$(S)$, for $t=1,2,\ldots,$ over $S$
For each $r\geq 1$, find $q'_{r} = \min\{t\mid \|Y_t\|_2 > (1+\frac{1}{\tau})^r\}$
Let $q_0,q_1,\ldots,q_R$ be the last $R+1$ of $q'_0,q'_1,\ldots$
SieveStream $S$, $q_0,\dots,q_R$ Identify one bit of information from each round
$b_{q_{r}+1},\ldots,b_{q_{r+1}}\gets$Pair$(S^{(q_{r}:q_{r+1})},B_{r,1},\ldots,B_{r,n})$ Replace JLBP here with AMS
Selector$b_1,\ldots,b_R$ Determine $H$ from the round winners
Any $j^*\in\argmax_j\#\{r \in [R]: B_{r,j}=b_{r}\}$.
shortversionAlgorithm for a single $F_2$ heavy hitters.
§.§ Amplifier correctness
The $L=O(\log\log n)$ instances of Pair maintained by Amplifier in the first stage of CountSieve serve to identify a substream containing roughly $n2^{-L}=n/\polylog{n}$ elements in which $H$ appears as a $\polylog (n)$-heavy hitter.
Correctness of Amplifier means that, after some “burn-in” period which we allow to include the first $f_H/2$ updates to $H$, all of the subsequent updates to $H$ appear in the amplified substream while the majority of other items do not.
This is Lemma <ref>.
Let $t_0\in[m]$ be such that $(f_H^{(t_0)})^2\geq 4C^2\sum_{j\neq H}f_j^2$, and let $a_t=(a_{1,t},\ldots,a_{L,t})$ denote the length $L$ bit-vector output by the Amplifier at step $t$.
Let $M_{j,t} = \#\{\ell\in[L]\mid a_{\ell,t}=A_{\ell,j}\}$ and $W=\{j\in[n]\setminus\{H\}\mid\exists t\geq t_0, M_{j,t}\geq 0.9L\}$.
Then, with probability at least $(1-2\delta)$, both of the following hold:
shortversion 1.
for all $t\geq t_0$ simultaneously, $M_{H,t}\geq 0.9L$ and 2.
$\sum_{j\in W}f_j^2 \leq \exp(-\frac{L}{25})\sum_{j\neq H}f_j^2$.
* for all $t\geq t_0$ simultaneously, $M_{H,t}\geq 0.9L$ and
* $\sum_{j\in W}f_j^2 \leq \exp(-\frac{L}{25})\sum_{j\neq H}f_j^2$.
Let $N=\#\{\ell\mid \text{for all }t\geq t_0,a_{\ell,t}=A_{\ell,H}\}$.
If $N\geq 0.9L$ then <ref> holds.
Lemma <ref> implies $\E N \geq (1-3\delta)L\geq 0.97L$, so Chernoff's Bound easily implies $P(N<0.9L) = O(2^{-L})\leq \delta$, where $\delta$ is a constant.
Now, let $j\neq H$ be a member of $W$ and suppose that $M_{H,t}\geq 0.9L$.
Let $t\geq t_0$ be such that $M_{j,t}\geq 0.9L$. Then it must be that
\[M'_j:=\#\{\ell\in [L]\mid A_{\ell,j}=A_{\ell,H}\} \geq 0.8L.\]
However, $\E M'_j=\frac{1}{2}L$ by pairwise independence.
Let $E_j$ be the event $\{j\in W\text{ and }M_{H,t}\geq0.9L\}$.
Since the $L$ instances of Pair are independent, an application of Chernoff's Inequality proves that $\Pr (E_j)\leq \Pr (M'_j \geq 0.8L)\leq \exp\{\frac{-0.6^2 L}{6}\}\leq e^{-L/20}$.
We have
\[\E (\sum_{j\in W} f_j^2) = \E (\sum_{j\neq H} 1_{E_j}f_j^2) \leq e^{-L/20}\sum_{j\neq H}f_j^2.\]
Therefore Markov's Inequality yields
\[\Pr \left(\sum_{j\in W}f_j^2 \geq e^{-L/25}\sum_{j\neq H}f_j^2\right)\leq e^{-L/100} \leq \delta.\]
The lemma follows by a union bound.
§.§ Timer and Sieve correctness
The timing of the rounds in the second stage of CountSieve is determined by Timer.
Timer outputs a set of times $q_0,q_1,\ldots,q_R$ that break the stream into intervals so that each interval has roughly a $1/\log{n}$ fraction of the occurrences of $H$ and not too many other items.
Precisely, we want that $H$ is everywhere heavy for $q$, as stated in the following definition.
When this holds, in every round the Pair is likely to identify one bit of $H$, and Sieve and Selector will be likely to correctly identify $H$ from these bits.
Given an item $H\in[n]$ and a sequence of times $q_0<q_1< \cdots< q_R$ in a stream with frequency vectors $(f^{(t)})_{t\in[m]}$ we say that $H$ is everywhere heavy for $q$ if, for all $1\leq r\leq R$,
$(f_H^{(q_{r-1}:q_{r})})^2 \geq C^2 \sum_{j\neq H} (f_j^{(q_{r-1}:q_{r})})^2$.
\[(f_H^{(q_{r-1}:q_{r})})^2 \geq C^2 \sum_{j\neq H} (f_j^{(q_{r-1}:q_{r})})^2.\]
Correctness for Timer means that enough rounds are completed and $H$ is sufficiently heavy within each round, i.e., $H$ is everywhere heavy for $q$.
Let $S$ be a stream with an item $H\in[n]$ such that the following hold:
1. $f_H\geq \tau^4$,
2. $f_H^2\geq 400C^2\sum_{j\neq H} f_j^2$, and
3. $(f_H^{(t^*:m)})^2=\frac{1}{4}f_H^2\geq 25C^2\tau^2\sum_{j\neq H} (f_j^{(t^*:m)})^2$,
* $f_H\geq \tau^4$,
* $f_H^2\geq 400C^2\sum_{j\neq H} f_j^2$, and
* $(f_H^{(t^*:m)})^2=\frac{1}{4}f_H^2\geq 25C^2\tau^2\sum_{j\neq H} (f_j^{(t^*:m)})^2$,
where $t^* = \min\{t\in[m]\mid f_H^{(t)}\geq 0.5 f_H\}$ and $C$ is the constant from Definition <ref>.
If $q_0,q_1,\ldots,q_R$ is the sequence output by Timer$(S)$ then, with probability at least $1-4\delta$, $H$ is everywhere heavy for $q$.
We begin by proving that at least $R$ rounds occur after $t^*$, which shows that $q_0,\ldots,q_R$ is well defined, and then we show that $H$ is everywhere heavy.
Let $Y_t$ be the sequence output by JLBP and let $X_t = Y_t - d^{-\frac{1}{2}}ZTe_Hf_H^{(t)}$.
$Y_t$ and $X_t$ are tame by Lemma <ref> and $\Pr (0.5\leq \alpha\leq 1.5)\geq 1-\delta$ where $\alpha= \|d^{-\frac{1}{2}}ZTe_H\|_2$.
Hereafter, we suppose that $\alpha\geq 1/2$ and the tameness property holds for $Y_t$ and $X_t$.
With probability at least $1-\delta$, simultaneously for all $t\in[m]$, we have
\begin{equation}\label{eq: timer tameness}
\|X_t\|_2^2\leq C^2\sum_{j\neq H}f_j^2\leq \frac{1}{400}f_H^2.
\end{equation}
Therefore, $\|Y_{t^*}\|_2\leq \|X_{t^*}\|_2 + \alpha f_H^{(t^*)} \leq (\frac{\alpha}{2} + \frac{1}{20})f_H$ and $\|Y_{m}\|_2\geq \alpha f_H^{(m)} - \|X_{m}\|_2 \geq (\alpha - \frac{1}{20})f_H$.
This implies that the number of rounds completed after $t^*$, which is
$$\log_{1+1/\tau}\frac{\|Y_m\|_2}{\|Y_{t^*}\|_2} \ge \log_{1+1/\tau}\frac{\alpha - 1/20}{\alpha/2 + 1/20} \ge \log_{1+1/\tau}(3/2),$$
is at least $R+1$ by our choice of $\tau=100(R+1)$. Similarly $\|Y_{t^*}\|_2\ \geq \alpha f_H^{(t^*)} - \|X_{t^*}\|_2 \geq (\frac{\alpha}{2} - \frac{1}{20})f_H$. Therefore we also get $q_i>q_{i-1}$ because $(1+\tau^{-1})\|Y_{t^*}\|_2\geq 1$ by our assumption that $f_H\geq \tau^4$. Hence $q_0,\ldots,q_R$ are distinct times.
Now we show that $H$ is everywhere heavy for $q$.
Let $W_t = X_t-X_{t^*}$, for $t\geq t^*$.
By design, $W_t - W_s = X_t-X_s$, for $s,t\geq t^*$.
By Lemma <ref>, $W_t$ is also a tame process on the suffix of the original stream that has its first item at time $t^*+1$.
Specifically with probability at least $1-\delta$, for all $t\geq t^*$,
\[
\|W_t\|_2^2 \leq C^2 \sum_{j\neq H} (f_j^{(t^*:m)})^2\leq \frac{1}{400\tau^2}f_H^2.
\]
This inequality, with two applications of the triangle inequality, implies
\begin{equation}\label{eq: f Y W triangle}
\alpha f_H^{(q_{i-1}:q_i)} \geq \|Y_{q_i}-Y_{q_{i-1}}\|_2 - \|W_{q_i}-W_{q_{i-1}}\|_2\geq \|Y_{q_i}-Y_{q_{i-1}}\|_2 - \frac{2}{20\tau}f_H.
\end{equation}
To complete the proof we must bound $\|Y_{q_i}-Y_{q_{i-1}}\|_2$ from below and then apply the heaviness, i.e., assumption <ref>.
Equation (<ref>) and the triangle inequality imply that, for every $t\geq t^*$, it holds that $\|Y_{t}\|_2\geq \alpha f_H^{(t)} - \|X_{t}\|_2 \geq (\frac{\alpha}{2} - \frac{1}{20})f_H$.
Recalling the definition of $q_0', q_1', \cdots$ from Timer Procedure, since $t^*\leq q_0<q_1<\cdots<q_R$ and the rounds expand at a rate $(1+1/\tau)$,
\begin{equation}\label{eq: plenty of Hs}
\|Y_{q_{i+1}}-Y_{q_i}\|_2\geq \frac{1}{\tau}\left( \frac{\alpha}{2} - \frac{1}{20}\right) f_H.
\end{equation}
Using what we have already shown in (<ref>) we have
\[\alpha f_H^{(q_i:q_{i+1})} \geq \frac{1}{\tau}\left( \frac{\alpha}{2} - \frac{1}{20} - \frac{2}{20}\right) f_H\]
so dividing and using $\alpha\geq 1/2$ and $C$ sufficiently large we get
\[(f_H^{(q_i:q_{i+1})})^2 \geq \frac{1}{25\tau^2}f_H^2\geq C^2 \sum_{j\neq H}(f^{(t^*:m)}_j)^2\geq C^2 \sum_{j\neq H}(f^{(q_{i}:q_{i+1})}_j)^2.\]
Since this holds for all $i$, $H$ is everywhere heavy for $q$.
We have used the tameness of the three processes ($X$, $Y$, and $W$) and the bounds on $\alpha$.
Each of these fails with probability at most $\delta$, so the total probability that Timer fails to achieve the condition that $H$ is everywhere heavy for $q$ is at most $4\delta$.
During each round, the algorithm Sieve uses a hash function $A$ to split the stream into two parts and then determines which part contains $H$ via Pair.
For these instances of Pair, we replace the two instances of JLBP with two instances of AMS. This replacement helps us to hold down the storage when we later use Nisan's PRG, because computing the JL transformation $T$ from <cit.> requires $O(\log n\log\log n)$ bits.
Applying Nisan's PRG to an algorithm that computes entries in $T$ would leave us with a bound of $O(\log{n}(\log\log n)^2)$. More details can be found in shortversionthe full version.Section <ref>.
A total of $O(\log n)$ rounds is enough to identify the heavy hitter and the only information that we need to save from each round is the hash function $A$ and the last bit output by Pair.
Selector does the work of finally identifying $H$ from the sequence of bits output by Sieve and the sequence of hash functions used during the rounds.
We prove the correctness of Sieve and Selector together in the following lemma.
Let $q_0,q_1,\ldots,q_R=\textsc{Timer}(S)$ and let $b_1,\ldots,b_R=\textsc{Sieve}(S,q_0,\ldots,q_R)$.
If $H$ is everywhere heavy for $q$ on the stream $S$ then, with probability at least $1-\delta$, Selector$(b_1,\ldots,b_R)$ returns $H$.
Lemma <ref> in the AMS case implies that the outcome of round $r$ satisfies $\Pr (b_r = B_{r,H})\geq 1-3\delta$ and $\Pr (b_r=B_{r,j})\leq \frac{1}{2}+3\delta$.
The random bits used in each iteration of the for loop within Sieve are independent of the other iterations.
Upon choosing the number of rounds $R=O(\log n)$ to be sufficiently large, Chernoff's Inequality implies that, with high probability, $H$ is the unique item in $\argmax_j\#\{r\in[R]\mid B_{r,j}=b_r\}$.
Therefore, Selector returns $H$.
$\ell_2$HeavyHittersStream $S=(p_1,p_2,\ldots,p_m)$
$Q\gets O(\log\epsilon^{-1})$, $B\gets O(\epsilon^{-2})$
Select indep. 2-universal hash functions
$h_1,\ldots,h_Q,h'_1,\ldots,h_Q':[n]\to [B]\text{ and }\sigma_1,\ldots,\sigma_Q:[n]\to\{-1,1\}.$
$\hat{F}_2\gets (1\pm\frac{\epsilon}{10})F_2$ using AMS <cit.>
$(q,b)\in Q\times B$
Let $S_{q,b}$ be the stream of items $i$ with $h_{q}(i)=b$
$c_{q,b}\gets \sum_{j:h_q'(j)=b}\sigma_q(j)f_j$ The CountSketch <cit.>
$H\gets \textsc{CountSieve}(S_{q,b})$
Remove from $\hat{\calH}$ any item such that $i$ such that $\text{median}_q\{|c_{q,h_q(i)}|\}\leq \frac{3\epsilon}{4}\hat{F}_2$.
$\ell_2$ heavy hitters algorithm.
§.§ CountSieve correctness
We now have all of the pieces in place to prove that CountSieve correctly identifies a sufficiently heavy heavy hitter $H$.
As for the storage bound and Theorem <ref>, the entire algorithm fits within $O(\log n\log\log n)$ bits except the $R=O(\log n)$ hash functions required by Sieve.
We defer their replacement to shortversionthe full version.Theorem <ref> in Section <ref>.
If $H$ is a $400C^2$-heavy hitter then, with probability at least $0.95$ CountSieve returns $H$.
The algorithm uses $O(\log{n}\log\log{n})$ bits of storage and can be implemented with $O(\log n\log\log n)$ stored random bits.
We use Theorem <ref> to generate the JL transformation $T$.
Each of our lemmas requires that $T$ embeds a (possible different) polynomially sized set of vectors, so, for $\delta=\Omega(1)$, Theorem <ref> implies that, with probability at least $1-\delta$, $T$ embeds all of the necessary vectors with seed length $O(\log{n})$, and the entries in $T$ can be computed in space $O(\log{n}\log\log{n})$ bits of space.
Because of the heaviness assumption, the conclusion of Lemma <ref> fails to hold for $t_0=t^*$ (defined in Lemma <ref>) with probability at most $2\delta$.
When that failure does not occur, the second and third hypotheses in Lemma <ref> hold.
The first hypothesis is that $f_H\geq\tau^4$, suppose it holds.
Then the probability that $H$ fails to be everywhere heavy for the sequence $q$ that is output by Timer is at most $4\delta$.
In this case, according to Lemma <ref>, Sieve and Selector correctly identify $H$ except with probability at most $\delta$.
Therefore, the algorithm is correct with probability at least $1-8\delta\geq 0.95$, by choosing $\delta \leq 1/200$.
If $f_H<\tau^4$, then because $H$ is a heavy hitter, we get $\sum_{j\neq H}f_j^2 \leq \tau^8 = O(\log^8 n)$.
Then we choose the constant factor in $L$ large enough so that, the second conclusion of Lemma <ref> implies $\sum_{j\in W}f_j^2\leq e^{-L/25}<1$.
This means that $H$ is the only item that passes the amplifier for all $t\geq t^*$, and, no matter what is the sequence output by Timer, $H$ is everywhere heavy because it is the only item in the substream.
Thus, in this case the algorithm also outputs $H$.
Now we analyze the storage and randomness.
Computing entries in the Kane-Meka-Nelson JL matrix requires $O(\log n\log\log n)$ bits of storage, by Theorem <ref>, and there is only one of these matrices.
Amplifier stores $L=O(\log\log n)$ counters. Sieve, Timer, and Selector each require $O(\log n)$ bits at a time (since we discard any value as soon as it is no longer needed).
Thus the total working memory of the algorithm is $O(\log n\log\log n)$ bits.
The random seed for the JL matrix has $O(\log n)$ bits.
Each of the $O(\log\log n)$ Bernoulli processes requires $O(\log n)$ random bits.
By Theorem <ref> below, the remaining random bits can be generated with Nisan's generator using a seed of $O(\log n\log\log n)$ bits.
Using Nisan's generator does not increase the storage of the algorithm.
Accounting for all of these, the total number of random bits used by CountSieve, which also must be stored, is $O(\log n\log\log n)$.
Therefore, the total storage used by the algorithm is $O(\log n\log\log n)$ bits.
*hhtheoremTheorem <ref>
[$\ell_2$-Heavy Hitters]
The algorithm is Algorithm <ref>.
It has the form of a CountSketch <cit.> with $Q=O(\log 1/\epsilon)$ “rows” and $B=8(10 C)^2/\epsilon^2$ “buckets” per row, wherein we run one instance of CountSieve in each bucket to identify potential heavy hitters and also the usual CountSketch counter in each bucket.
Finally, the algorithm discriminates against non-heavy hitters by testing their frequency estimates from the CountSketch.
We will assume that the AMS estimate $\hat{F}_2$ is correct with probability at least $8/9$.
Let $\calH_k = \{i\mid f_i\geq \frac{\epsilon}{k}\sqrt{F_2}\}$ and let $\hat{\calH}$ be set of distinct elements returned by Algorithm <ref>.
To prove the theorem, it is sufficient to prove that, with probability at least $2/3$, $\calH_1\subseteq \hat{\calH}\subseteq \calH_2$.
Let $H\in \calH_1$ and consider the stream $S_{q,h_q(H)}$ at position $(q,h_q(H))$.
We have
\[\E(\sum_{\substack{j\neq H \\ h_q(j)=h_q(H)}}f_j^2)\leq \frac{\epsilon^2}{8(10C)^2}F_2.\]
Let $E_{q,H}$ be the event that
\[\sum_{\substack{j\neq H \\ h_q(j)=h_q(H)}}f_j^2\leq \frac{\epsilon^2}{(10C)^2}F_2,\]
so by Markov's Inequality $\Pr(E_{q,H})\geq 7/8$.
When $E_{q,H}$ occurs $H$ is sufficiently heavy in $S_{q,h_q(H)}$ for CountSieve.
By Theorem <ref>, with probability at least $\frac{7}{8}-\frac{1}{20}\geq 0.8$, CountSieve identifies $H$.
Therefore, with the correct choice of the constant factor for $Q$, a Chernoff bound and a union bound imply that, with probability at least $1-1/9$, every item in $\calH_1$ is returned at least once by a CountSieve.
Let $\hat{\calH}'$ denote the set $\hat{\calH}$ before any elements are removed in the final step.
Since CountSieve identifies at most one item in each bucket, $|\hat{\calH}'|=O(\epsilon^{-2}\log\epsilon^{-1})$.
By the correctness of CountSketch <cit.> and the fact that it is independent of $\hat{H}'$, we get that, with probability at least $1-1/9$, for all $i\in\hat{H}'$
\[\left|f_i - \text{median}_q\{|c_{q,h_q(i)}|\}\right|\leq \frac{\epsilon}{10C}\sqrt{F_2}.\]
When this happens and the AMS estimate is correct, the final step of the algorithm correctly removes any items $i\notin\calH_2$ and all items $i\in\calH_1$ remain.
This completes the proof of correctness.
The storage needed by the CountSketch is $O(BQ\log{n})$, storage needed for the CountSieves is $O(BQ\log{n}\log\log{n})$, and the storage needed for AMS is $O(\epsilon^{-2}\log{n})$.
Therefore the total storage is $O(BQ\log{n}\log\log{n}) = O(\frac{1}{\epsilon^2}\log{\frac{1}{\epsilon}}\log{n}\log\log{n})$ bits.
There exists an insertion-only streaming algorithm that returns an additive $\pm\epsilon\sqrt{F_2}$ approximation to $\ell_\infty$, with probability at least $2/3$.
The algorithm requires $O(\frac{1}{\epsilon^2}\log{\frac{1}{\epsilon}}\log{n}\log\log{n})$ bits of space.
Use Algorithm <ref>.
If no heavy-hitter is returned then the $\ell_\infty$ estimate is 0, otherwise return the largest of the CountSketch medians among the discovered heavy hitters.
The correctness follows from Theorem <ref> and the correctness of CountSketch.
§ CHAINING INEQUALITY
We call these inequalities Chaining Inequalities after the Generic Chaining, which is the technique that we use to prove it.
The book <cit.> by Talagrand contains an excellent exposition of the subject.
The full version contains a more detailed discussion of the method.
Let $(X_t)_{t\in T}$ be a Gaussian process.
The Generic Chaining technique concerns the study of the supremum of $X_t$ in a particular metric space related to the variances and covariances of the process.
The metric space is $(T,d)$ where $d(s,t) = (\E(X_s-X_t)^2)^{\frac{1}{2}}$.
The method takes any finite chain of finite subsets $T_0\subseteq T_1\subseteq\cdots\subseteq T_n\subseteq T$ and uses $(X_t)_{t\in T_i}$ as a sequence of successive approximations to $(X_t)_{t\in T}$ wherein $X_t$, for $t\notin T_i$, is approximated by the value of the process at some minimizer of $d(t,T_i)=\min\{d(t,s)\mid s\in T_i\}$.
To apply the Generic Chaining one must judiciously choose the chain in order to get a good bound, and the best choice necessarily depends on the structure of the process.
We will exploit the following lemma.
Let $\{X_t\}_{t\in T}$ be a Gaussian process and let $T_0\subseteq T_1 \dots \subseteq T_n \subseteq T$ be a chain of sets such that
$|T_0|=1$ and $|T_i|\le 2^{2^i}$ for $i\ge 1$.
$\E \sup_{t\in T} X_t \le O(1) \sup_{t\in T} \sum_{i\ge 0}2^{i/2}d(t, T_i)$.
\begin{equation}\label{chain}
\E \sup_{t\in T} X_t \le O(1) \sup_{t\in T} \sum_{i\ge 0}2^{i/2}d(t, T_i).
\end{equation}
The Generic Chaining also applies to Bernoulli processes, but, for our purposes, it is enough that we can compare related Gaussian and Bernoulli processes.
Let $A\in\R^{m\times n}$ be any matrix and let $G$ and $B$ be $n$-dimensional vectors with independent coordinates distributed as $N(0,1)$ and $\Rademacher$, respectively.
Then the Gaussian process $X = AG$ and Bernoulli process $Y=AB$ satisfy
$\E \sup_{t\in T} Y_t \le \sqrt{\frac{\pi}{2}}\E \sup_{t\in T} X_t.$
\begin{equation*}
\E \sup_{t\in T} Y_t \le \sqrt{\frac{\pi}{2}}\E \sup_{t\in T} X_t.
\end{equation*}
Let $Z_1,\ldots,Z_n\ldots\iidsim\calN(0,1)$ and let $(f^{(t)})_{t\in[m]}$ be the sequence of frequency vectors of an insertion-only stream.
There exists a universal constant $C'>0$ such that if $X_t = \sum_{j=1}^n Z_jf_{j}^{(t)}$, for $t\in[m]$, then
shortversion$\E \sup_i|X_i| \leq C'\sqrt{\Var(X_m)} = C'\|f^{(m)}\|_2$.
\begin{equation}\label{eq: gaussian chaining}
\E \sup_i|X_i| \leq C'\sqrt{\Var(X_m)} = C'\|f^{(m)}\|_2.
\end{equation}
If $\bar Z_1,\ldots,\bar Z_n\ldots\iidsim\Rademacher $ and
$Y_t = \sum_{j=1}^n \bar Z_jf_{j}^{(t)}$, for $t\in[m]$, then
shortversion$\E \sup_i|Y_i| \leq C'\sqrt{\Var(Y_m)}= C'\|f^{(m)}\|_2$.
\begin{equation}\label{eq: bernoulli chaining}
\E \sup_i|Y_i| \leq C'\sqrt{\Var(Y_m)}= C'\|f^{(m)}\|_2.
\end{equation}
Let $T=[m]$.
Define $T_0 = \{t_0\}$, where $t_0$ is the index such that $\Var(X_{t_0}) < 0.5\Var(X_{m}) \le \Var(X_{t_0 + 1})$ and $T_i = \{1, t_{i,1}, t_{i,2},\dots\}$ where for each index $t_{i,j}\in T_i$ $\Var(X_{t_{i,j}}) < \frac{j}{2^{2^i}}\Var(X_{m}) \le \Var(X_{t_{i,j + 1}})$.
This is well-defined because $\Var(X_{t})=\|f^{(t)}\|_2^2$ is the second moment of an insertion-only stream, which must be monotonically increasing.
By construction $|T_i| \le 2^{2^i}$ and, for each $t\in T$, there exist $t_{i,j}\in T_j$ such that $d(t, T_i) = \min(d(t, t_{i,j}), d(t, t_{i,j+1})) \le d(t_{i,j}, t_{i,j+1}) = (\E(X_{t_{i,j+1}}-X_{t_{i,j}})^2)^{\frac{1}{2}}$, where the last inequality holds because $E(X_t^2)$ monotonically increasing with $t$.
Notice that every pair of increments has nonnegative covariance because the stream is insertion-only.
Thus, the following is true:
\begin{align*}
d(t,t_{i,j+1})^2 &\le \E(X_{t_{i,j+1}}-X_{t_{i,j}})^2 \leq \E(X_{t_{i,j+1}}-X_{t_{i,j}})^2 + 2 \E X_{t_{i,j}}(X_{t_{i,j+1}}-X_{t_{i,j}})\\
& = \E X_{t_{i,j+1}}^2 -\E X_{t_{i,j}}^2 \le\frac{j+1}{2^{2^i}}\E X_{m}^2 - \frac{j-1}{2^{2^i}}\E X_m^2 = \frac{2}{2^{2^i}}\E X_m^2.
\end{align*}
\begin{align*}
d(t,t_{i,j+1})^2 &\le \E(X_{t_{i,j+1}}-X_{t_{i,j}})^2\\
& \leq \E(X_{t_{i,j+1}}-X_{t_{i,j}})^2 + 2 \E X_{t_{i,j}}(X_{t_{i,j+1}}-X_{t_{i,j}})\\
& = \E X_{t_{i,j+1}}^2 -\E X_{t_{i,j}}^2 \\
&\le\frac{j+1}{2^{2^i}}\E X_{m}^2 - \frac{j-1}{2^{2^i}}\E X_m^2 = \frac{2}{2^{2^i}}\E X_m^2.
\end{align*}
Then we can conclude that
$\sum_{i\ge 0}2^{i/2}d(t, T_i) \le \sum_{i\ge 0}2^{i/2}\frac{2}{2^{2^{i}}}\sqrt{\E X_m^2} = O(1)\sqrt{\Var(X_m)}$.
\[\sum_{i\ge 0}2^{i/2}d(t, T_i) \le \sum_{i\ge 0}2^{i/2}\frac{2}{2^{2^{i}}}\sqrt{\E X_m^2} = O(1)\sqrt{\Var(X_m)}.\]
Applying shortversionLemma <ref>ineqality (<ref>) we obtain
$\E \sup_{t\in T} X_t \le O(1) \sqrt{\Var(X_m)}$.
In order to bound the absolute value, observe
\begin{equation}\label{eq: sup vs absolute sup}
\sup_t|X_t|\leq |X_1|+\sup |X_t-X_1|\leq|X_1|+\sup_{s,t}(X_t-X_s)=|X_1| + \sup_tX_t + \sup_s(-X_s).
\end{equation}
Therefore, $\E \sup_t|X_t|\leq \E |X_1| + 2\E \sup X_t\leq O(1)\sqrt{\Var(X_m)}$, because $-X_t$ is also Gaussian process with the same distribution as $X_t$ and $\E |X_1|=O(\sqrt{\Var(X_m)})$ because $f^{(1)} = 1$.
This establishes shortversionthe Gaussian inequality and the Bernoulli inequality(<ref>) and (<ref>) follows immediately by an application of Lemma <ref>.
Theorem <ref> would obviously not be true for a stream with deletions, since we may have $\Var(X_m)=0$.
One may wonder if the theorem would be true for streams with deletions upon replacing $\Var(X_m)$ by $\max_t \Var(X_t)$.
This is not true, and a counter example is the stream $(e_1,-e_1,e_2,\ldots,e_n,-e_n)$ which yields $\max_t\Var(X_t) = 1$, but $\E\sup_t|X_t| =\Theta(\sqrt{\log{n}})$.
Theorem <ref> does not apply to the process ouput by JLBP, but the covariance structures of the two processes are very similar because $T$ is an embedding.
In the full version, we prove basically the same inequality for the JLBP process by mimicking the stategy in <cit.>.
We can achieve basically the same inequality for the JLBP process by applying Slepian's Lemma, mimicking the stategy in <cit.>.
Let $X_t$ and $Y_t$, for $t\in T$, be Gaussian processes such that $\E (X_s-X_t)^2\leq \E (Y_s-Y_t)^2$, for all $s,t\in T$.
Then, $\E \sup_{t\in T} X_t\leq \E \sup_{t\in T} Y_t$.
Let $T$ be a $(1\pm\gamma)$-embedding of $(f^{(t)})_{t\in [m]}$ and let $Z_1,\ldots,Z_k\ldots\iidsim\calN(0,1)$.
There exists a universal constant $C'>0$ such that if $X_t = \langle Z, Tf^{(t)}\rangle $, for $t\in[m]$, then $\E \sup_i|X_i| \leq C'\|f^{(m)}\|_2$.
If $\bar Z_1,\ldots,\bar Z_k\iidsim\Rademacher $ and $Y_t = \langle \bar{Z},Tf^{(t)}\rangle $, for $t\in[m]$, then $\E \sup_i|Y_i| \leq C'\|f^{(m)}\|_2$.
Let $W_t$ be the Gaussian process from Theorem <ref>.
Since $T$ is a JL transformation
\[\E (X_t-X_s)^2 = \|Tf^{(s:t)}\|_2^2 \leq (1+\gamma)^2\|f^{(s:t)}\|_2^2 = (1+\gamma)^2 \E(W_t-W_s)^2.\]
The first claim of the corollary follows from Slepian's Lemma, Equation (<ref>), and Theorem <ref>.
The second inequality follows from the first and Lemma <ref>.
§ REDUCED RANDOMNESS
This section describes how CountSieve can be implemented with only $O(\log n\log\log n)$ random bits.
There are two main barriers to reducing the number of random bits.
We have already partially overcome the first barrier, which is to reduce the number of bits needed by a Bernoulli process from $n$, as in the algorithm BP, to $O(\log n)$ by introducing JLBP.
JLBP runs $d=O(1)$ independent Bernoulli processes in dimension $k=O(\log n)$ for a total of $dk=O(\log n)$ random bits.
This section proves the correctness of that algorithm.
The second barrier is to find a surrogate for the $R=O(\log{n})$ independent vectors of pairwise independent Bernoulli random variables that are used during the rounds of Sieve.
We must store their values so that Selector can retroactively identify a heavy hitter, but, naïvely, they require $\Omega(\log^2 n)$ random bits.
We will show that one can use Nisan's pseudorandom generator (PRG) with a seed length of $O(\log n \log\log n)$ bits to generate these vectors.
A priori, it is not obvious that this is possible.
The main sticking point is that the streaming algorithm that we want to derandomize must store the random bits it uses, which means that these count against the seed length for Nisan's PRG.
Specifically, Nisan's PRG reduces the number of random bits needed by a space $S$ algorithm using $R$ random bits to $O(S\log R)$.
Because CountSieve must pay to store the $R$ random bits, the storage used is $S\geq R = \Omega(\log^2 n)$, so Nisan's PRG appears even to increase the storage used by the algorithm!
We can overcome this by introducing an auxiliary (non-streaming) algorithm that has the same output as Sieve and Selector, but manages without storing all of the random bits.
This method is similar in spirit to Indyk's derandomization of linear sketches using Nisan's PRG <cit.>.
It is not a black-box reduction to the auxiliary algorithm and it is only possible because we can exploit the structure of Sieve and Selector.
We remark here that we are not aware of any black-box derandomization of the Bernoulli processes that suits our needs.
This is for two reasons.
First, we cannot reorder the stream for the purpose of the proof because the order of the computation is important.
Reordering the stream is needed for Indyk's argument <cit.> for applying Nisan's PRG.
Second, the seed length of available generators is too large, typically in our setting we would require a seed of length at least $\log^{1+\delta}n$, for some $\delta>0$.
§.§ The Bernoulli process with $O(\log{n})$ random bits
The main observation that leads to reducing the number of random bits needed by the algorithm is that the distribution of the corresponding Gaussian process depends only on the second moments of the increments.
These moments are just the square of the Euclidean norm of the change in the frequency vector, so applying a Johnson-Lindenstrauss transformation to the frequency vector nearly preserves the distribution of the process and allows us to get away with $O(\log n)$ random bits.
One trouble with this approach is that the heavy hitter $H$ could be “lost”, whereby we mean that although $\|Te_H\|\approx 1$ it may be that $\langle Z,Te_H\rangle\approx 0$, for the Rademacher random vector $Z$, whereupon $H$'s contribution to the sum $\langle Z, Tf^{(t)}\rangle$ is lost among the noise.
To avoid this possibility we keep $d=O(1)$ independent Bernoulli processes in parallel.
First, we state the correctness of the Johnson-Lindenstrauss transformation that we use and the storage needed for it.
Let $V=\{v_1,\ldots,v_n\}\subseteq \R^n$.
For any constant $\delta>0$ there exists a $k=O(\gamma^{-2}\log (n/\delta)$ and generator $G:\{0,1\}^{O(\log n)}\times[k]\times[n]\to\R$ such that, with probability at least $1-\delta$, the $k\times n$ matrix $T$ with entries $T_{ij}=G(R,i,j)$ is a $(1\pm\gamma)$-embedding of $V$, where $R\in\{0,1\}^{O(\log{n})}$ is a uniformly random string.
The value of $G(R,i,j)$ can be computed with $O(\log n\log\log n)$ bits of storage.
*lemmaJLBPLemma <ref>
[JLBP Correctness]
Let $X_{i,t} = \sum_{j=1}^kZ_{ij}(Tf^{(t)})_j$ and
\[X_t = \|\frac{1}{\sqrt{d}}ZTf^{(t)}\|_2^2 = \frac{1}{d}\sum_{i=1}^dX^2_{i,t},\]
for $t=1,\ldots,m$.
Each process $X_{i,t}$ is a Bernoulli process with $\Var(X_{i,t}) = \|Tf^{(t)}\|^2_2\leq (1+\gamma)^2\|f^{(t)}\|^2_2$ and, for $s<t$, $\E (X_{i,t}-X_{i,s})^2 = \|Tf^{(s:t)}\|^2_2\leq (1+\gamma)^2\|f^{(s:t)}\|_2^2$.
Notice that for all $i$ Gaussian processes $(X_{i,t})_{t\in[m]}$ are from same distribution. Let $X_{t}'$ be a Gaussian process that is identical to $X_{i,t}$, except that the Rademacher random variables are replaced by standard Gaussians.
$X_t'$ and $X_{i,t}$ have the same means, variances, and covariances.
Therefore $\E \sup_t|X_{i,t}|\leq\sqrt{\frac{\pi}{2}}\E \sup_t|X'_t|$, by Lemma <ref>.
Let $N_1,\ldots,N_n\iidsim N(0,1)$. We will compare $X'_{t}$ against the Gaussian process $X''_t = (1+\gamma)\frac{1}{\sqrt{d}}\langle N,f^{(t)}\rangle$.
By the Chaining Inequality, there exists $C'$ such that $\E \sup |X''_t|\leq C' \sqrt{\Var(X''_m)}=\frac{C'(1 + \gamma)}{\sqrt{d}}\|f^{(m)}\|_2$.
We have $\E (X''_t-X''_s)^2 =\frac{1}{d}(1+\gamma)^2\|f^{(s:t)}\|_2^2$, so by Slepian's Lemma applied to $X'_t$ and $X''_t$ and by (<ref>) we have
\[\E \sup_t|X_{i,t}|\leq \sqrt{\frac{\pi}{2}}\E \sup |X'_t|\leq \sqrt{\frac{\pi}{2}} \sqrt{d}\E\sup_t|X''_t|\leq \sqrt{\frac{\pi}{2}} (1 + \gamma) C'\|f^{(m)}\|_2.\]
Now we apply Markov's Inequality to get $\Pr (\sup_t|X_{i,t}|\geq \frac{C}{\sqrt{d}}\|f^{(m)}\|_2)\leq \frac{\delta}{d}$, by taking $C\geq \sqrt{\frac{\pi}{2}}(1 + \gamma) C'd^{3/2}/\delta$.
From a union bound we find $\Pr (\sup_{i,t}|X_{i,t}|\geq \frac{C}{\sqrt{d}}\|f^{(m)}\|_2)\leq \delta$, and that event implies $\sup_t|X_t|\leq C\|f^{(m)}\|_2$, which is (<ref>) and proves that the process is tame.
For the second claim, we note that the matrix $\frac{1}{\sqrt{d}}Z$ is itself a type of Johnson-Lindenstrauss transformation (see <cit.>), hence $\frac{1}{2}\leq \|d^{-1/2}ZTe_H\|\leq \frac{3}{2}$, with probability at least $1-2^{-d}\geq (1-\delta)$.
The last inequality follows by our choice of $d$.
§.§ Sieve and Selector
In the description of the algorithm, the Sieve and Selector use $O(\log{n})$ many pairwise independent hash functions that are themselves independent.
Nominally, this needs $O(\log^2n)$ bits.
However, as we show in this section, it is sufficient to use Nisan's pseudorandom generator <cit.> to generate the hash functions.
This reduces the random seed length from $O(\log^2n)$ to $O(\log n\log\log n)$.
Recall the definition of a pseudorandom generator.
A function $G : \{0,1\}^m \rightarrow \{0, 1\}^n$ is called a pseudorandom generator (PRG) for space($S$) with parameter $\epsilon$ if for every randomized
space($S$) algorithm $A$ and every input to it we have that
$$\|\calD_{y}(A(y)) - \calD_x(A(G(x))\|_1 < \epsilon,$$
where $y$ is chosen uniformly at random in $\{0, 1\}^n$, $x$ uniformly in $\{0, 1\}^m$, and $\calD(\cdot)$ is the distribution of $\cdot$ as a vector of probabilities.
Nisan's PRG <cit.> is a pseudorandom generator for space $S$ with parameter $2^{-S}$ that takes a seed of length $O(S\log{R})$ bits to $R$ bits.
The total space used by Sieve and Selector is $O(\log n)$ bits for the algorithm workspace and $O(\log^2n)$ bits to store the hash functions.
We will be able to apply Nisan's PRG because Sieve only accesses the randomness in $O(\log n)$ bit chunks, where the $r$th chunk generates the 4-wise independent random variables needed for the $r$th round, namely $B_{r1},\ldots,B_{rn}$ and the bits for two instances of the AMS sketch.
We can discard the AMS sketches at the end of each round, but in order to compute its output after reading the entire stream, Selector needs access to the bit sequence $b_1,b_2,\ldots,b_R$ as well as $B_{ri}$, for $r\in[R]$ and $i\in[n]$.
Storing the $B$ random variables, by their seeds, requires $O(\log^2n)$ bits.
This poses a problem for derandomization with Nisan's PRG because it means that Sieve and Selector are effectively a $O(\log^2n)$ space algorithm, even though most of the space is only used to store random bits.
We will overcome this difficulty by derandomizing an auxiliary algorithm.
The auxiliary algorithm computes a piece of the information necessary for the outcome, specifically for a given item $j\in[n]$ in the stream the auxiliary item will compute $N_j:=\#\{r\mid b_r = B_{rj}\}$ the number of times $j$ is on the “winning side” and compare that value to $3R/4$.
Recall that the Selector outputs as the heavy hitter a $j$ that maximizes $N_j$.
By Lemma <ref> for the AMS case, $\E N_j$ is no larger than $(\frac{1}{2}+3\delta)R$, if $j$ is not the heavy element, and $\E N_H$ is at least $(1-3\delta)R$ if $H$ is the heavy element.
When the Sieve is implemented with fully independent rounds, Chernoff's Inequality implies that $N_H> 3R/4$ or $N_j\leq 3R/4$ with high probability.
When we replace the random bits for the independent rounds with bits generated by Nisan's PRG we find that for each $j$ with high probability $N_j$ remains on the same side of $3R/4$.
Here is a formal description of the auxiliary algorithm.
The auxiliary algorithm takes as input the sequence $q_0,q_1,\ldots,q_R$ (which is independent of the bits we want to replace with Nisan's PRG), the stream $S$, and an item label $j$, and it outputs whether $N_j>3R/4$.
It initializes $N_i=0$, and then for each round $r=1,\ldots,R$ it draws $O(\log n)$ random bits and computes the output $b_r$ of the round.
If $b_r=B_{rj}$ then $N_i$ is incremented, and otherwise it remains unchanged during the round.
The random bits used by each round are discarded at its end.
At the end of the stream the algorithm outputs 1 if $N_j>3R/4$.
Let $X\in\{0,1\}$ be the bit output by the auxiliary algorithm, and let $\tX\in\{0,1\}$ be the bit output by the auxiliary algorithm when the random bits it uses are generated by Nisan's PRG with seed length $O(\log n\log\log n)$. Then
\[ |\Pr (X=1)-\Pr (\tX=1)|\leq \frac{1}{n^2}.\]
The algorithm uses $O(\log{n})$ bits of storage and $O(\log^2n)$ bits of randomness.
The claim follows by applying Nisan's PRG <cit.> with $\epsilon=1/n^2$ and seed length $O(\log n\log\log n)$.
Sieve and Selector can be implemented with $O(\log(n)\log\log{n})$ random bits.
Let $N_j$ be the number of rounds $r$ for which $b_r = B_{rj}$ when the algorithm is implemented with independent rounds, and let $\tN_j$ be that number of rounds when the algorithm is implemented with Nisan's PRG.
Applying Lemma <ref> we have for every item $j$ that $|\Pr (\tN_j>3R/4)-P(N_j>3R/4)|\leq 1/n^2$.
Thus, by a union bound, the probability that the heavy hitter $H$ is correctly identified changes by no more than $n/n^2=1/n$.
The random seed requires $O(\log n\log\log n)$ bits of storage, and aside from the random seeds the algorithms use $O(\log n)$ bits of storage.
Hence the total storage is $O(\log n\log\log n)$ bits.
§ $F_2$ AT ALL POINTS
One approach to tracking $F_2$ at all times is to use the median of $O(\log n)$ independent copies of an $F_2$ estimator like the AMS algorithm <cit.>.
A Chernoff bound drives the error probability to $1/\poly(n)$, which is small enough for a union bound over all times, but it requires $O(\log^2n)$ bits of storage to maintain all of the estimators.
The Chaining Inequality allows us to get a handle on the error during an interval of times.
Our approach to tracking $F_2$ at all times is to take the median of $O(\log\frac{1}{\epsilon}+\log\log{n})$ Bernoulli processes.
In any short enough interval—where $F_2$ changes by only a $(1+\Omega(\epsilon^2))$ factor—each of the processes will maintain an accurate estimate of $F_2$ for the entire interval, with constant probability.
Since there are only $O(\epsilon^{-2}\log^2(n))$ intervals we can apply Chernoff's Inequality to guarantee the tracking on every interval, which gives us the tracking at all times.
This is a direct improvement over the $F_2$ tracking algorithm of <cit.> which for constant $\eps$ requires $O(\log n(\log n + \log\log m))$ bits.
The algorithm has the same structure as the AMS algorithm, except we replace their sketches with instances of JLBP.
F2AlwaysStream $S$
$N\gets O(\frac{1}{\epsilon^2})$, $R\gets O(\log(\frac{1}{\epsilon^2}\log n))$
$X_{i,r}^{(t)}\gets\textsc{JLBP}(S)$ for $i\in[N]$ and $r\in[R]$.Use a $(1\pm\frac{\epsilon}{3})$-embedding $T$ in this step.
$Y^{(t)}_r = \frac{1}{N}\sum_{i=1}^N \|X_{i,r}^{(t)}\|_2^2$
$\hat{F}_2^{(t)}=\text{median}_{r\in R}\{Y_r^{(t)}\}$ at each time $t$
An algorithm for approximating $F_2$ at all points in the stream.
Let $N=O(\frac{1}{\delta\epsilon^2})$ and let $X_{i}^{(t)}$, for $i=1,\ldots,N$, be independent copies of the output of $\textsc{JLBP}(S)$ using a fixed $(1\pm\frac{\epsilon}{8})$-embedding $T$ on an insertion only stream $S$.
Let $Y_t = \frac{1}{N}\sum_{i=1}^N\|X_i^{(t)}\|_2^2$.
Suppose that for two given times $1\leq u<v\leq m$ the stream satisfies $256C^2F_2^{(u:v)}\leq \epsilon^2 F_2^{(u)}$, where $F_2^{(u:v)}=\sum_{i=1}^n(f_i^{(u:v)})^2$ is the second moment of the change in the stream.
\[\Pr\left(|Y_t-F_2^{(t)}| \leq \epsilon F_2^{(t)}\text{, for all }u\leq t\leq v\right)\geq 1-2\delta.\]
We first write $|Y_t-F_{2}^{(t)}| \leq |Y_t - Y_u| + |Y_u-F_{2}^{(u)}| + |F_2^{(t)}-F_2^{(u)}|$.
It follows from the arguments of AMS and the fact that $T$ is a $(1\pm\epsilon/8)$-embedding that, with an appropriate choice for $N=O(\frac{1}{\delta\epsilon^2})$, we arrive at
\begin{equation}\label{eq: AMS for F2always}
\Pr(|Y_u-F_2^{(u)}|\leq \frac{\epsilon}{4} F_2^{(u)})\geq 1-\delta.
\end{equation}
For the third term we have $F_2^{(t)}\geq F_2^{(u)}$ because $t\geq u$ and the stream is insertion only.
We can bound the difference with
\begin{align*}
= \| f^{(u)} + f^{(u:t)}\|_2^2
\leq \|f^{(u)}\|_2^2\left(1 + \frac{\|f^{(u:t)}\|_2}{\|f^{u}\|_2}\right)^2
\leq F_2^{(u)}(1+\frac{\epsilon}{4}),
\end{align*}
where the last inequality follows because $C\geq 2$ and $\epsilon\leq 1/2$.
For the first term, since $X_i^{(t)}$, $i\in[n]$, are independent $d$-dimensional Bernoulli processs, it follows that
\[X^{(t)} = \frac{1}{\sqrt{N}}((X_{1}^{(t)})^T,(X_{2}^{(t)})^T,\ldots,(X_{N}^{(t)})^T)^T\]
is an $Nd$-dimensional Bernoulli process.
By Lemma <ref> and due to the fact that $X^{(t)}$ can be represented as an output of JLBP procedure, the process $X^{(u:t)} = X^{(t)}-X^{(u)}$, is a tame process, so with probability at least $1-\delta$, for all $u\leq t\leq v$ we have
\[\|X^{(u:t)}\|_2^2 \leq C^2 \sum_{j=1}^n (f_{j}^{(u:v)})^2.\]
Therefore, assuming the inequality inside (<ref>),
\begin{align*}
Y_t = \|X^{(u)} + X^{(u:t)}\|_2^2
&\leq Y_u\left(1 + \frac{\|X^{(u:t)}\|_2}{\|X^{(u)}\|_2}\right)^2
\leq Y_u\left(1 + \frac{\sqrt{1+\eps}}{\sqrt{1-\eps}}\frac{\|F^{(u:t)}\|_2}{\|F^{(u)}\|_2}\right)^2\\
&\leq Y_u\left(1+\frac{2\epsilon}{16C}\right)^2
\leq F_2^{(u)}(1+\epsilon/4),
\end{align*}
where the last inequality follows because $C\geq 2$ and $\epsilon\leq 1/2$.
The reverse bound $Y_t\geq F_2^{(u)}(1-\epsilon/4)$ follows similarly upon applying the reverse triangle inequality in place of the triangle inequality.
With probability at least $1-2\delta$,
\[|Y_t-F_{2}^{(t)}| \leq |Y_t - Y_u| + |Y_u-F_{2}^{(u)}| + |F_2^{(t)}-F_2^{(u)}|\leq \epsilon F_2^{(u)}\leq \epsilon F_2^{(t)}.\]
Let $S$ be an insertion only stream and, for $t=1,2,\ldots,m$, let $F_2^{(t)} = \sum_{i=1}^n (f_{i}^{(t)})^2$ and let $\hat{F}_2^{(t)}$ be the value that is output by Algorithm <ref>.
\[P(|\hat{F}_2^{(t)}-F_2^{(t)}|\leq \epsilon F_2^{(t)}\text{, for all }t\in[m])\geq 2/3.\]
The algorithm uses $O\left(\frac{1}{\epsilon^2}\log n \left(\log\log n + \log \frac{1}{\eps}\right)\right)$ bits of space.
By Lemma <ref>, the (single) matrix used by all instances of JLBP is a $(1\pm\epsilon/3)$-embedding with probability at least $0.99$, henceforth assume it is so.
Let $q_0=0$ and
\[q_i = \max_t\left\{t\;|F_2^{(t)} \leq (1+\frac{\eps^2}{256C^2})^i\right\},\]
until $q_K = m$ for some $K$.
Notice that $K=O(\frac{1}{\epsilon^2}\log n)$.
Here, $C$ is the constant from Definition <ref>.
By definition of $q_i$ and using the fact that $(a-b)^2 \leq a^2 - b^2$ for real numbers $0\le b\le a$ we have $F_2^{(q_i:q_{i+1})}\leq (F_2^{(q_{i+1})}-F_2^{(q_i)})\leq \frac{\eps^2}{256C^2}F_2^{(q_i)}$.
Applying Lemma <ref> with $\delta=1/10$, we have, for every $r\in[R]$ and $i\geq 0$ that
\[P(|Y_r^{(t)} - F_2^{(t)}| \leq \epsilon F_2^{(t)},\text{ for all }q_i\leq t\leq q_{i+1})\geq 0.8.\]
Thus, by Chernoff bound, the median satisfies
\[P(|\hat{F}_2^{(t)} - F_2^{(t)}| \leq \epsilon F_2^{(t)},\text{ for all }q_i\leq t\leq q_{i+1})\geq 1- e^{-R/12} \geq 1-\frac{1}{4K},\]
by our choice of $R = 12\log{4K} = O(\log(\epsilon^{-2}\log{n}))$.
Thus, by a union bound over all of the intervals and the embedding $T$ we get
\[P(|\hat{F}_2^{(t)} - F_2^{(t)}| \leq \epsilon F_2^{(t)},\text{ for all }t\in[m])\geq \frac{2}{3},\]
which completes the proof of correctness.
The algorithm requires, for the matrix $T$, the JL transform of Kane, Meka, and Nelson <cit.> with a seed length of $O(\log(n)\log(\frac{1}{\epsilon}\log{n}))$ bits, and it takes only $O(\log(n/\epsilon))$ bits of space to compute any entry of $T$.
The algorithm maintains $NR=O(\epsilon^{-2}\log(\frac{1}{\epsilon}\log{n}))$ instances of JLBP which each requires $O(\log n)$ bits of storage for the sketch and random bits.
Therefore, the total storage used by the algorithm is $O(\epsilon^{-2}\log(n)\log(\frac{1}{\epsilon}\log{n}))$.
This immediately implies Theorem <ref>.
§ ACKNOWLEDGEMENTS
The authors would like to thank Raghu Meka for suggesting the JL strategy to reduce the number of random bits needed for the Bernoulli processes.
We would also like to thank several anonymous reviewers for their careful reading of an earlier version of this paper and the valuable suggestions that they made.
|
1511.00780
|
]Liheng Bian,$^1$ Jinli Suo,$^1$ Guohai Situ,$^2$ Ziwei Li,$^1$ Feng Chen,$^1$ and Qionghai Dai$^1$
$^1$Department of Automation, Tsinghua University, Beijing 100084, China
$^2$Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
Current multispectral imagers suffer from low photon efficiency and limited spectrum range. These limitations are partially due to the technological limitations from array sensors (CCD or CMOS), and also caused by separative measurement of the entries/slices of a spatial-spectral data cube. Besides, they are mostly expensive and bulky.
To address above issues, this paper proposes to image the 3D multispectral data with a single bucket detector in a multiplexing way.
Under the single pixel imaging scheme, we project spatial-spectral modulated illumination onto the target scene to encode the scene's 3D information into a 1D measurement sequence. Conventional spatial modulation is used to resolve the scene's spatial information. To avoid increasing requisite acquisition time for 2D to 3D extension of the latent data, we conduct spectral modulation in a frequency-division multiplexing manner in the speed gap between slow spatial light modulation and fast detector response.
Then the sequential reconstruction falls into a simple Fourier decomposition and standard compressive sensing problem.
A proof-of-concept setup is built to capture the multispectral data (64 pixels $\times$ 64 pixels $\times$ 10 wavelength bands) in the visible wavelength range (450nm–650nm) with acquisition time being 1 minute. The imaging scheme is of high flexibility for different spectrum ranges and resolutions. It holds great potentials for various low light and airborne applications, and can be easily manufactured production-volume portable multispectral imagers.
§ INTRODUCTION
Multispectral imaging is a technique capturing a spatial-spectral data cube of a scene, which contains multiple 2D images under different wavelengths. Possessing both spatial and spectral resolving abilities, multispectral imaging is extremely vital for surveying a scene and extracting detailed information <cit.>.
Current multispectral imagers mostly utilize dispersive optical devices (e.g., prism and optical grating) or narrow band filters to separate different wavelengths, and then use an array detector to separately measure them <cit.>. Using the compressive sensing technique, multispectral images can be multiplexed together to reduce the number of shots <cit.>.
Another kind of multispectral imaging method is Fourier spectroscopy technique<cit.>. This approach uses an interferometer to divide the incoming beam into two halves with variable optical path difference, and generate varying interference intensity at each spatial point. The spectral information can be extracted by applying Fourier transform to the intensities measured by an array detector.
Despite the diverse principles and setups of the above multispectral imaging instruments, the photons are detected separately either in the spatial or spectral dimension using array detectors. Therefore, these multispectral imagers are photon inefficient and spectrum range limited. Besides, they are usually bulky <cit.> and highly expensive (for example, more than $\$$50000 for NIR-SWIR range multispectral imagers <cit.>).
These disadvantages prevent them from wide practical applications.
Illustration of the difference between conventional SPI and the proposed MSPI. Due to the response speed gap between a bucket detector (MHz or GHz) and a spatial light modulator (no higher than KHz), the detector can collect a dense sequence of measurements during elapse of each spatially modulated pattern. In conventional SPI, given a spatial pattern, its light intensity and corresponding measurements are constant. Thus no spectral information could be extracted from the sequence. Differently, in MSPI both the intensity and measurements are time-varying, since the intensity of each spectral component changes sinusoidally with their own frequencies over time. The speed gap enables us to multiplex and demultiplex scene's spectral components from the measurement sequence during elapse of each spatial pattern.
Schematic of the proposed single-pixel multispectral imaging system. The broadband light from the high power bulb is spatially modulated by a spatial light modulator (SLM) to generate a series of 2D random patterns. Next, the spectra of the 2D patterns are distributed into a rainbow stripe, and modulated by a rotating film before transformed back to 2D patterns. After both the spatial and spectral modulations, the incident illumination is tailored structurally in three dimensions—random in the 2D spatial dimensions and sinusoidal along the spectral dimension. Then the patterns illuminate the target scene to encode both its spatial and spectral information. Finally a bucket detector is utilized to measure the correlated signals. In the sequential reconstruction process, the spectral response signals are decoded by Fourier decomposition, while the spatial information are demodulated by a compressive sensing based reconstruction algorithm. Details of the modulations and demodulations are shown in the insets.
Differently, single pixel imaging (SPI)<cit.> provides a promising scheme being able to address the above issues of current multispectral imaging instruments.
Using a bucket detector instead of expensive and bulky CCD or CMOS, SPI systems are of low cost, compact, and own wider spectral detection range <cit.>. Besides, SPI collects all the lights interacted with the scene to a single detection unit. Thus it is more photon efficient <cit.>. What's more, SPI is flexible, meaning that it attaches no requirement on the light path between scene and the detector, providing that all the interacted lights are collected to the detector <cit.>.
In the past years, SPI has achieved great success in 2D imaging and various applications <cit.>.
To produce advantages of the SPI scheme in multispectral imaging, there are two intuitional ways. One is to resolve the spectra of the collected measurements at the detector. Existing such methods include i) directly replacing the bucket detector with a spectrometer <cit.>, and ii) using light filters <cit.> or dispersive optical devices <cit.> to separate signals of different wavelengths, and then measure them separately. Another straightforward way is to directly extend the 2D spatial modulation to 3D spatial-spectral modulation using two spatial light modulators. However, this would largely increase requisite projections <cit.> and corresponding computation complexity for reconstruction.
In a word, since a single bucket detector cannot distinguish different spectra, the above methods needs either high commercial cost or geometrically increasing projections and computational cost.
In this paper, we propose a novel single pixel multispectral imaging technique, termed as multispectral single pixel imaging (MSPI), without increasing requisite projections and capturing time compared to conventional SPI. The main difference between conventional SPI and MSPI is illustrated in Fig. <ref>.
Utilizing the fact that the response speed of a bucket detector (MHz) is magnitudes faster than illumination patterning (KHz)<cit.>, we encode the spectral information into this speed gap.
Specifically, the proposed MSPI technique introduces spectrum-dependent sinusoidal intensity modulation to the lights, during the elapse of each spatially modulated pattern.
Thus, different spectrum bands are multiplexed together into the 1D dense measurements at the bucket detector in a frequency-division multiplexing manner.
Since the response signals of different bands displays distinct dominant frequencies in the Fourier domain, we conduct a simple Fourier decomposition to separate multispectral response signals from each other. Last, the compressive sensing algorithm <cit.> is applied to these signals in different wavelength bands to reconstruct the latent multispectral data.
The spectral multiplexing and demultiplexing based on Fourier decomposition can suppress system noise effectively, and thus produces high robustness to noise and ensures high reconstruction quality.
Multispectral imaging results on a color scene. (a) is the target color scene (a printed film of CIE 1931 color space). (b) is the sinusoidal modulation film used in our setup. While the rainbow spectrum is converged along the radius of the film, different wavelengths are modulated with different sinusoidal periods as the film rotates. (c) shows exemplar recorded correlated measurements corresponding to a specific projecting pattern. (d) is the Fourier decomposition of the measurements, which displays several dominant frequencies. The coefficients of the dominant frequencies correspond to the response signals' strengths of specific wavelengths. (e) shows the decomposed sequences for different spectrum bands, while (f) presents the final reconstructed 2D multispectral images ($64\times64$ pixels) corresponding to 10 narrow bands.
MSPI owns a lot of potential applications in various fields of science. Due to its high photon efficiency and robustness to noise, MSPI could be used in low light conditions, such as fluorescence microscopy <cit.> and Raman imaging <cit.>. Besides, the utilized SPI scheme enables MSPI system to be of compact size and low weight. This is beneficial for a lot of airborne applications, including geologic mapping, mineral exploration, agricultural assessment, environmental monitoring, and so on <cit.>. Moreover, MSPI applies to a large spectral range and is of low cost, thus can be used for production-volume portable devices for daily use.
§ RESULTS
Experimental setup. MSPI builds on the SPI scheme. In SPI, the incident uniform illumination is patterned by a spatial light modulator (SLM), and then projected onto the target scene to multiplex its spatial information. Simultaneously, a bucket detector is used to collect the encoded measurements. Afterwards, the compressive sensing algorithm <cit.> retrieves the spatial information of the target scene computationally.
Under a similar architecture, MSPI adds an extra spectral modulation to the incident light to resolve the scene's spectral resolving information. The principle of the proposed MSPI system is sketched in Fig. <ref>. On a whole, MSPI projects spatial-spectral modulated light beam to modulate corresponding information of the target scene, and collects the correlated lights with a single bucket detector. Integrating both spatial and spectral modulation, MSPI could resolve a spatial-spectral 3D data cube of the target scene computationally, as displayed in the bottom right inset of Fig. <ref>.
We built a proof-of-concept setup to verify the functionality of MSPI, as shown in Fig. <ref>. A broadband light source (Epson white 230W UHE lamp) is converged and collimated via a set of optical elements for succeeding modulation. For spatial modulation, we use a digital micromirror device (DMD, Texas Instrument DLP Discovery 4100, .7XGA), which can switch binary patterns at a given frequency (20kHz maximum) with clean-cut pattern transition. The intensity of the spatial illumination pattern is temporally constant for now, as visualized in the top right inset. The illumination pattern is then diverged by a projector lens (Epson, NA 0.27) for successive spectral modulation.
The spectral modulation module is similar to the agile multispectral optical setup <cit.>, with the light path displayed in the top middle inset.
Specifically, an optical grating (600 grooves, $\phi=50mm$) is placed on the focal plane of the spatial illumination patterns. Then a convex lens collects the first order dispersed spectrum, and focuses it onto the rainbow plane, where a round film printed with sinusoidal annuluses owning different periods spectral modulation is placed for spectral modulation. The rainbow spectrum stretches along the film's radius. Driven by an electric motor rotating at a constant speed (around 6000 r/min), the film realizes a wavelength-dependent intensity modulation to the spectra, i.e., different wavelengths own different temporally sinusoidal intensity variations, as visualized in the top left inset of Fig. <ref>. After both spatial and spectral modulation, the illumination patterns interact with the scene, and we use a bucket detector (Thorlabs PDA100A-EC Silicon photodiode, 340-1100$nm$) together with a 14-bit acquisition board ART PCI8514 to capture the correlated lights.
For reconstruction, we first conduct spectral demultiplexing using simple fast Fourier transform (computation complexity is $\mathcal{O}(n \log n)$), and then reconstruct multispectral scene images using the linearized alternating direction method <cit.> (computation complexity is $\mathcal{O}(n^3)$) to solve the compressive sensing model. Readers are referred to the Methods section for reconstruction details.
In the following experiments, 3000 spatially random modulated patterns (each owning 64$\times$64 pixels) are sequentially projected onto the target scene. The frame rate of the DMD is set to be 50Hz, and the sampling rate of the bucket detector is 100kHz. We utilize the novel self-synchronization technique in <cit.> to synchronize the DMD and the detector. It takes us around 1 minute for data acquisition.
Quantitative analysis on the imaging accuracy of MSPI. (a) is the target scene—X-Rite standard color checker, which consists of 24 swatches owning different known spectra. We use MSPI to image the color checker and obtain 10 multispectral images (450nm-650nm), and calculate the recovered spectrum of each swatch as the average of all corresponding pixels' spectra. (b) presents the reconstruction error of the swatches in terms of root mean square error. (c) shows direct comparison between recovered spectra and their ground truth counterparts on several representative swatches. The standard deviation of each band is also calculated and shown as blue bars. Both the small reconstruction error and deviation validate the accuracy and robustness of MSPI.
Multispectral imaging results of MSPI. We first apply the proposed MSPI technique to capture the multispectral images of a scene with rich color. Here we use a printed 'CIE 1931 color space' image with wide spectrum range (see Fig. <ref>(a)) to demonstrate the effectiveness of the proposed approach. In this experiment, the rainbow spectrum ranges from 450nm to 650nm. The length of the rainbow stripe is around 23mm, and we discretize it into 10 narrow bands, by printing 10 2mm annular rings with sinusoidal periods varying from 2 to 20 (as shown in Fig. <ref>(b)).
The size of the projected pattern on the film is around 45mm$\times$45mm.
Given an exemplar spatial pattern, the recorded correlated measurements from the single pixel detector are plotted in Fig. <ref>(c), and its corresponding Fourier coefficients are displayed in Fig. <ref>(d). One can see that there exist several dominant peaks, which comes from the sinusoidal codes of corresponding frequencies (the 60Hz peak comes from the lamp flicker due to voltage fluctuations). The magnitudes of these peaks are exactly the strengths of the response signals of corresponding spectrum bands. The other small fluctuations of the Fourier coefficients are caused by system noise. From this we can see that although the multispectral response signals are corrupted with system noise in the temporal domain, they are clearly distinguishable in Fourier space.
Therefore, we can easily demultiplex multispectral response signals from each other and suppress system noise by a simple Fourier decomposition (see the Methods section for more details), and the results are shown in Fig. <ref>(e). The frequencies match exactly with the multiplexing codes printed on the film.
After response signal demultiplexing, we can recover the single-band images separatively using the compressive sensing based algorithm. The reconstructed 10 multispectral scene images are shown in Fig. <ref>(f), we integrate which with the Canon EOS 5D MarkII camera's RGB response curves <cit.> for better visualization. The pleasant results verify the effectiveness of the proposed MSPI.
Analysis on the performance and robustness of MSPI.
To quantitatively demonstrate the performance of MSPI, we acquire the multispectral data of a X-Rite standard color checker (see Fig. <ref>(a)) using MSPI, and conduct quantitative analysis on the reconstruction accuracy. In implementation, we introduce a pair of cylinder mirrors to match the shape of the light beam with that of the color checker (125mm $\times$ 90mm). For each swatch on the checker, we average all the pixels' reconstructed spectra as the swatch's reconstruction spectrum. Reconstruction error in terms of root mean square error among the 10 spectral bands is calculated for each swatch, and the results of all the 24 swatches are shown in Fig. <ref>(b). For more direct comparison, we show the spectrum comparison between the reconstruction and the ground truth of several representative swatches in Fig. <ref>(c).
From the small deviation compared to the ground truth, especially the ones with large estimation error (e.g., 'Orange' and 'Yellow'), we can see that the reconstructed spectra of the swatches are compliant with the ground truth.
This experiment largely validates the multispectral reconstruction accuracy of MSPI.
The accuracy benefits from the high precision of spectral demultiplexing (clear-cut discrimination between the Fourier coefficients of signals and noise), as well as the optimization reconstruction algorithm.
§ DISCUSSION
This paper proposes a new multispectral imaging technique using a single bucket detector, termed as MSPI. Making use of the speed gap between the slow spatial illumination patterning and the fast detector response, MSPI extends conventional 2D spatial coding to 3D spatial-spectral coding via temporal sinusoidal spectral modulation within each spatial pattern elapse. This technique successfully resolves multispectral information without introducing additional acquisition time and computational complexity to conventional 2D SPI.
The proposed MSPI holds great potential for developing cheap, compact and high photon efficient multispectral cameras.
The specifications of the spectral modulator are flexible and can be easily customized. First, the width of the printed annuluses on the film determines the spectral resolution and can be adjusted for specific resolutions. Second, we can also use a grating with denser grooves to lengthen the rainbow stripe and raise the spectral resolution alternatively. Third, the multiplexing mode can change easily by designing other film graphs. The sinusoidal spectral modulation utilized in current MSPI system is adopted due to its simplicity. We refer readers to <cit.> for more multiplexing methods.
Recalling that the proposed technique is a general scheme for multispectral imaging, it can be conveniently coupled with a variety of imaging modalities (no matter macroscopy or microscopy), by using corresponding optical elements. The scheme is wavelength independent, and users can apply the scheme to other spectrum ranges readily. This is especially important for the wavelengths under which array sensors are costly or unavailable. In addition, similar to the system in <cit.>, the modulation can be conducted after the light beam interacted with the target scene. This enables us to analyze the scene's spatial-spectral information without active illumination. One can refer to the supplementary material for details of MSPI under passive illumination, which is of wider applicability.
Although MSPI owns many advantages over conventional multispectral imaging techniques, these benefits come at the expense of a large number of projections and algorithmic reconstruction. In other words, MSPI makes a trade-off of temporal resolution for spatial and spectral resolution. Fortunately, the imaging speed of MSPI can be accelerated utilizing advanced techniques. In terms of data acquisition, current efficiency is mainly limited by the spectral modulator, and we can use a faster rotation motor or denser sinusoidal patterns for acceleration. In terms of reconstruction, considering there exists abundant redundancy among different color channels<cit.>, we can utilize this cross channel prior in the reconstruction to reduce the requisite projections and thus accelerate imaging speed. The reconstruction time can also be shortened further, because different spectrum bands are reconstructed separately, and we can utilize graphics processing unit (GPU) to reconstruct different channels in a parallel manner.
Besides, current spatial resolution is apparently insufficient for practical applications. Targeting for proof-of-concept and without loss of generalization ability, here we project randomly spatial modulated patterns in the capturing stage, similar to most SPI systems. However, recent studies <cit.> show that projecting structural and adaptive patterns instead of random ones can largely improve the spatial resolution while decreasing projections and lowering computation cost. Hence, we can easily improve the spatial resolution under exactly the same scheme.
§ METHODS
The reconstruction of the proposed MSPI technique consists of two main steps, namely spectral demultiplexing and multispectral reconstruction.
Spectral demultiplexing.
Due to the sinusoidal spectral modulation, for a spatially modulated pattern, its measurement sequence from the bucket detector consists of several response signals of different spectra. These response signals own different frequencies of sinusoidal intensity variations. Thus in the Fourier domain, the measurement sequence is composed of several corresponding dominant frequencies. Besides, there exists system noise in the measurements, we assume which to be stochastic and zero-mean. In the Fourier domain, the noise mainly locates at high frequencies. Adopting simple Fourier decomposition <cit.>, we could separate the response signals from each other and from the measurement noise.
Mathematically, the Fourier decomposition describes a time series as a weighted summation of sinusoidal functions at different frequencies. A captured measurement sequence $\{y_0, \cdots, y_{T-1}\}$ (captured with a given spatial illumination pattern) can be represented by a series of sinusoidal functions as
\begin{eqnarray}\label{eqs:Fourier_1}
y_t = b_0 + \sum_{i=1}^{T/2}\left\{b_i\sin(\frac{2\pi i}{T}t + \phi_i )\right\}.
\end{eqnarray}
In this equation, $b_0 = \frac{1}{T}\sum_{t=0}^{T-1}y_t$, $b_i = \frac{2}{T}\sqrt{\left[\sum_{t=0}^{T-1}y_t\cos(\frac{2\pi i}{T}t)\right]^2+\left[\sum_{t=0}^{T-1}y_t\sin(\frac{2\pi i}{T}t)\right]^2}(i>0$), and $\phi_i = \arctan \frac{\sum_{t=0}^{T-1}y_t\sin(\frac{2\pi i}{T}t)}{\sum_{t=0}^{T-1}y_t\cos(\frac{2\pi i}{T}t)}$.
Specifically, $b_0$ is the direct current component indicating the average of the measurements, while $b_i (i>0)$ indicates the energy of the $i$th sinusoidal function with modulation frequency $\frac{i}{T}$.
As stated before, each spectrum band corresponds to one specific sinusoidal modulation frequency. Thus, the above coefficients at the specific frequencies are exactly the response signals corresponding to the spectral bands.
Here we adopt fast Fourier transform (FFT) to transfer the measurements into Fourier domain, with computation complexity being $\mathcal{O}(n \log n)$. Then we demultiplex the response signals corresponding to different spectrum bands by finding the local maximum coefficients around corresponding Fourier frequencies.
By doing FFT to each measurement sequence, we obtain a set of response signals for each spectral band.
Mathematically, assuming that the wavelength $\lambda$ is modulated with sinusoidal frequency being $\frac{j}{T}$, we can obtain a response signal $b_j$ from the measurement sequence corresponding to one projecting pattern. Considering that we project $m$ patterns, we can get $m$ response signals of the wavelength $\lambda$. In the following, we indicate the response signal set as a row vector ${\bf b}_\lambda\in \mathbb{R}^{m}$. Each entry in ${\bf b}_\lambda$ corresponds to a response signal of the band $\lambda$ for one pattern.
Multispectral reconstruction. After demultiplexing response signals of different wavelengths, the reconstruction is implemented separately for each wavelength band. For band $\lambda$, we assume the spatial pixel number of each illumination pattern is $n$, and denote the pattern set as ${\bf A}\in \mathbb{R}^{m\times n}$ (each pattern is represented as a row vector).
The multispectral scene images own the same spatial resolution as the illumination patterns, and is denoted as ${\bf x}_\lambda\in \mathbb{R}^{n}$ for the wavelength $\lambda$.
To reduce the number of requisite projections, we choose to conduct reconstruction under the framework of compressive sensing<cit.>. The reconstruction is performed by solving the following optimization problem:
\begin{eqnarray}\label{eqs:Model}
\{{\bf x}_\lambda^*\} = \arg\min && ||\psi({\bf x}_\lambda)||_1 \\ \nonumber
s.t. && {\bf Ax}_\lambda = {\bf b}_\lambda.\nonumber
\end{eqnarray}
The definition of the objective comes from a sparsity prior: natural scene images are statistically sparse when represented with an appropriate basis set (e.g. the discrete cosine transform basis) <cit.>.
We use $\psi({\bf x}_\lambda)$ to denote the coefficient vector, with $\psi$ being the mapping operator to the transformed domain, and minimize its $l_1$ norm to force the sparsity. Eq. <ref> is a standard $l_1$ optimization problem, and there exist many effective algorithms to solve it. Here we use the linearized alternating direction method <cit.> to obtain the optimal ${\bf x}_\lambda^*$, with computation complexity being $\mathcal{O}(n^3)$. This results in the final reconstructed scene image corresponding to the specific wavelength band $\lambda$. After doing the above reconstruction to all the wavelength bands, we get multispectral images of the target scene.
We thank Yuwang Wang, Ziyan Wang and Jing Pu for their valuable discussions and help. This work was supported by the National Natural Science Foundation of China (Nos. 61171119, 61120106003, and 61327902).
|
1511.00811
|
Current-vortex sheets]Approximate current-vortex sheets
near the onset of instability
A. Morando] Alessandro Morando
DICATAM, Sezione di Matematica, Università di Brescia,Via Valotti, 9, 25133 BRESCIA, Italy
[email protected], [email protected], [email protected]
P. Secchi]Paolo Secchi
P. Trebeschi]Paola Trebeschi
The paper is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompressible
magneto-hydrodynamics near the transition point between the linearized stability and instability.
In order to study the dynamics of
the discontinuity near the onset of the instability, Hunter and Thoo <cit.> have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. We study such amplitude equation and prove its nonlinear well-posedness under a stability condition given in terms of a longitudinal strain of the fluid along the discontinuity.
[2010]35Q35, 76E17, 76E25, 35R35, 76B03.
The authors are supported by the national research project PRIN 2012 Nonlinear Hyperbolic Partial Differential Equations, Dispersive and Transport Equations: theoretical and applicative aspects.
§ INTRODUCTION AND MAIN RESULTS
We consider the equations of 2-dimensional incompressible magneto-hydrodynamics (MHD)
\begin{equation}
\label{mhd1}
\begin{cases}
\partial_t \u +\nabla \cdot
(\u \otimes \u-\B\otimes \B) +\nabla q =0 \, ,
\\
\partial_t \B -\nabla \times
(\u\times \B) =0 \, ,
\\
\div \u=0\, ,\;\div \B=0\, \qquad\qquad \text{in }(0,T)\times\R^2,
\end{cases}
\end{equation}
$\u=(u_1,u_2)$ denotes the velocity field
$\B=(B_1,B_2)$ the magnetic field,
$p$ is the
pressure, $q= p+\frac{1}{2}|\B|^2$ the
total pressure
(for simplicity the density $\rho\equiv1$).
Let us consider current-vortex sheets solutions of (<ref>) (also called tangential discontinuities), that is weak solutions that are smooth on either side of a smooth hypersurface
$$\Gamma(t)=\{y=f(t,x)\}, \qquad \mbox{where } t\in[0,T], \, (x,y)\in\R^2,\;
and such that at $\Gamma(t)$ satisfy the boundary conditions
\begin{equation}
\label{bc}
\dt f =\u^\pm \cdot N \, ,\quad \B^\pm \cdot N=0 \, ,\quad [q]=0 \, ,
\end{equation}
with $N:=(-\partial_x f, 1)$. In (<ref>) $(\u^\pm,\B^\pm,q^\pm)$ denote the values of $(\u,\B,q)$ on the two sides of $\Gamma (t)$, and
$[q]=q^+_{|\Gamma}-q^-_{|\Gamma}$ the jump across
$\Gamma (t)$.
From (<ref>) the discontinuity front $\Gamma (t)$
is a tangential discontinuity, namely the plasma does not flow through the discontinuity front and the magnetic field is tangent to $\Gamma (t)$.
The possible jump of the tangential velocity and tangential magnetic field gives a concentration of current and vorticity on the front $\Gamma (t)$.
Current-vortex sheets are fundamental waves in MHD and play an important role in plasma physics and astrophysics. The existence of current-vortex sheets solutions is known for compressible fluids <cit.>, but, as far as we know, is still an open problem for incompressible fluids, see <cit.> for partial results.
The necessary and sufficient linear stability condition for planar (constant coefficients) current-vortex
sheets was found a long time ago, see <cit.>.
To introduce it, let us consider a stationary solution of (<ref>), (<ref>) with interface located at $\{y=0\}$ given by the constant states
\begin{equation}
\begin{array}{ll}\label{constant}
\u^\pm=(U^\pm,0)^T, \qquad \B^\pm=(B^\pm,0)^T
\end{array}
\end{equation}
in the $x$-direction.
The necessary and sufficient stability condition for the stationary solution is
\begin{equation}
\label{syrovatskii}
|U^+-U^-|^2 < 2 \, \Big( |B^+| ^2+ |B^-|^2 \Big) \, ,
\end{equation}
see <cit.>.
Equality in (<ref>) corresponds to the transition to
violent instability, i.e. ill-posedness of the linearized problem.
Let $U=(U^+,U^-), B=(B^+,B^-)$ and define
\[
\Delta(U,B):= \frac12 \, \Big( |B^+| ^2+ |B^-|^2 \Big) -\frac14|U^+-U^-|^2.
\]
According to (<ref>), stability/instability occurs when $\Delta(U,B)\gtrless0$.
Hunter and Thoo investigated in <cit.> the transition to instability when $\Delta(U,B)=0$.
Assume that $U^\pm,B^\pm$ depend on a small positive parameter $\eps$ and
\[
U^\pm=U^\pm_0+\eps U^\pm_1+O(\eps^2), \qquad
B^\pm=B^\pm_0+\eps B^\pm_1+O(\eps^2)
\]
as $\eps\to0^+$, where
\[
\Delta(U_0,B_0)=0\,.
\]
\begin{equation}
\begin{array}{ll}\label{Delta}
\Delta(U,B)=\eps\mu+O(\eps^2)
\end{array}
\end{equation}
as $\eps\to0^+$, where
\[
\mu=B^+_0B^+_1+B^-_0B^-_1 - \frac12\left( U^+_0-U^-_0 \right)\left( U^+_1-U^-_1 \right)
\,.
\]
From (<ref>), $\mu$ plays the role of a scaled bifurcation parameter:
for small $\eps>0$, if $\mu>0$ the stationary solution (<ref>) is linearly stable,
while if $\mu<0$, it is linearly unstable.
It is proved in <cit.> that the perturbed location of the interface has the asymptotic expansion
\begin{equation*}
\begin{array}{ll}\label{}
y=f(t,x;\eps)=\eps\vphi( \tau,\th ) + O(\eps^{3/2}) \qquad \mbox{as } \eps\to0^+,
\end{array}
\end{equation*}
where $\tau=\eps^{1/2} t$ is a slowtime variable and
$\th=x-\lambda_0 t$ is a new spatial variable in a reference frame moving with the surface wave, $\lambda_0=( U^+_0+U^-_0 )/2$.
As shown in <cit.>, after a rescaling, and writing again $(t,x)$ for $(\tau,\th)$, the first order term $\vphi$ satisfies the quadratically nonlinear amplitude equation
\begin{equation}\label{onde_integro_diff}
\varphi_{tt}-\mu\varphi_{xx}=\left(\frac12\mathbb H[\phi^2]_{xx}+\phi\varphi_{xx}\right)_{\!\!x}\,,\qquad\phi=\mathbb H[\varphi]\,,
\end{equation}
where the unknown is the scalar function $\varphi=\varphi(t,x)$, whereas $\mathbb H$ denotes the Hilbert transform with respect to $x$.
(<ref>) is an integro-differential equation of order two: in fact, it may also be written as
\begin{equation*}
\begin{array}{ll}\label{equ1bis}
\vphi_{tt}-\mu\vphi_{xx} = \left( [ \HH;\phi
]\partial_x \phi
_{x} + \HH[\phi
^2_x]\right)_x \,,
\end{array}
\end{equation*}
where $[ \HH;\phi
]\partial_x$ is a pseudo-differential operator of order zero.
In <cit.> the authors discuss the linearized well-posedness of (<ref>). Linearizing the operator
\[
\mathbb L[\varphi]:=\varphi_{tt}-\mu\varphi_{xx}-\left( \left[\mathbb H \,;\phi
\right]\phi
_{xx} + \mathbb H[\phi
^2_x] \right)_{x}\,
\]
about a given basic state $\vphi_0$ gives
\begin{equation}\label{linearizzata}
\mathbb{L}^\prime[\varphi_0]\varphi^\prime=\varphi^\prime_{tt} - {\left(\mu-2\phi
-2\left[\mathbb H\,;\phi
^\prime_{xx}-2\mathbb H[\phi
^\prime;\mathbb H\right]\phi
_{0,xx}+{ \left[\phi
_0;\mathbb H\right]\phi
\end{equation}
'=\HH[\vphi'], \phi
Assume that the last term in (<ref>) may be disregarded, even if of order 2 in $\vphi'$.
\begin{equation}
\begin{array}{ll}\label{extstab2}
\mu-2\phi
\end{array}
\end{equation}
the operator $\mathbb{L}^\prime[\varphi_0]$ is elliptic and (<ref>) is locally linearly ill-posed in any Sobolev space.
On the contrary, when
\begin{equation}
\begin{array}{ll}\label{extstab}
\mu-2\phi
\end{array}
\end{equation}
the operator $\mathbb{L}^\prime[\varphi_0]$ is hyperbolic and (<ref>) is locally linearly well-posed. In a sense we can think of (<ref>) as a nonlinear perturbation of the wave equation.
In <cit.> the reader may also find a physical explanation of condition (<ref>) yielding the linearized ill-posedness, or alternatively (<ref>) for well-posedness, which is given in terms of a longitudinal strain of the fluid along the discontinuity.
For simplicity, in the sequel it is assumed that $\varphi^\prime$ and $\varphi_0$ are periodic functions in $x$ (cf. Theorem <ref> below). In this case, the periodicity of $\varphi_0$ implies that $\varphi_{0, x}$ has spatial mean equal to zero; since $\varphi_0$ and $\varphi_{0,x}$ are also real-valued then $\phi_{0,x}=\mathbb H[\varphi_{0, x}]$ is still real-valued with zero spatial mean (see the results collected in the next sections <ref>, <ref>). Therefore $\phi_{0, x}$ (if not identically zero[Because the spatial mean of $\varphi_{0}$ is zero, $\phi_{0, x}$ identically zero should imply that $\varphi_{0}$ is identically zero too.]) should attain either positive or negative values; consequently inequality (<ref>) yields $\mu>0$ (providing linear stability of (<ref>)), while the opposite inequality implies $\mu<0$ (which gives linear instability). It is therefore somehow natural to regard (<ref>) as a stability condition, under which we investigate the nonlinear well-posedness of the equation (<ref>).
It is interesting to observe that the same quadratic operator of (<ref>) appears in the first order nonlocal amplitude equation
\begin{equation}\label{amplie}
\varphi_{t}+\frac12\mathbb H[\phi^2]_{xx}+\phi\varphi_{xx}=0\,,\qquad\phi=\mathbb H[\varphi]\,,
\end{equation}
for nonlinear Rayleigh waves <cit.> and surface waves on current-vortex sheets and plasma-vacuum interfaces in incompressible MHD <cit.>.
Equation (<ref>) is considered a canonical model equation for nonlinear surface wave solutions of hyperbolic conservation laws, analogous to the inviscid Burgers equation for bulk waves.
In this paper we are mainly interested in the nonlinear well-posedness of (<ref>) under assumption (<ref>). More specifically, we will study the local-in-time existence of solutions to the initial value problem for (<ref>) with sufficiently smooth initial data
\begin{equation}\label{id}
\varphi_{\vert\,t=0}=\varphi^{(0)}\,,\qquad \partial_t\varphi_{\vert\,t=0}=\varphi^{(1)}\,,
\end{equation}
satisfying the following “stability” condition
\begin{equation*}\label{stability_nl}
\mu-2\phi^{(0)}_{x}>0\,,\qquad\phi^{(0)}:=\mathbb H[\varphi^{(0)}]\,,
\end{equation*}
which must be understood as a smallness assumption on the size of the initial data $\varphi^{(0)}$ in (<ref>).
For the sake of convenience, in the paper the unknown $\varphi=\varphi(t,x)$ is a scalar function of the time $t\in\mathbb R^+$ and the space variable $x$, ranging on the one-dimensional torus $\mathbb T$ (that is $\varphi$ is periodic in $x$). For all notation we refer to the following Section <ref>.
The main result of the paper is given by the following theorem.
(1) Assume that $\varphi^{(0)}\in H^{11}(\mathbb T)$, $\varphi^{(1)}\in H^{10}(\mathbb T)$ and
\begin{equation}\label{sign-cond}
\mu- 2\mathbb{H}[\varphi^{(0)}]_x\geq \delta>0 \quad {\rm in}\,\,\mathbb{T},
\end{equation}
with some positive constant $\delta$.
Then there exists $T>0$, depending only on $\Vert\varphi^{(0)}\Vert_{H^{11}(\mathbb T)}$, $\Vert\varphi^{(1)}\Vert_{H^{10}(\mathbb T)}$ and $\delta$, such that the initial value problem (<ref>), (<ref>) with initial data $\varphi^{(0)}$, $\varphi^{(1)}$ admits a unique solution $\varphi$ on $[0,T]$ satisfying
\begin{equation*}%\label{sol}
\varphi\in L^2(0, T; H^{9}(\mathbb T))\cap H^1(0, T; H^{8}(\mathbb T))\cap H^2(0, T; H^{7}(\mathbb T))\,,
\end{equation*}
\begin{equation*}
\mu- 2\mathbb{H}[\varphi]_x\geq \delta/2 \quad {\rm in}\,\,[0,T]\times\mathbb{T}\, .
\end{equation*}
(2) If $\nu>10$ and $\varphi^{(0)}\in H^{\nu+1}(\mathbb T)$, $\varphi^{(1)}\in H^{\nu}(\mathbb T)$ satisfy condition (<ref>) then the solution $\varphi$ of (<ref>), (<ref>) with initial data $\varphi^{(0)}$, $\varphi^{(1)}$, considered in the statement (1), satisfies
\begin{equation*}%\label{reg_sol}
\varphi\in L^2(0, T; H^{\nu-1}(\mathbb T))\cap H^1(0, T; H^{\nu-2}(\mathbb T))\cap H^2(0, T; H^{\nu-3}(\mathbb T))\,.
\end{equation*}
It is worth pointing out that the time interval of the existence of the solution $\varphi=\varphi(t,x)$ to the initial value problem (<ref>), (<ref>), computed from the initial data by Theorem <ref>, is the same in both the statements (1) and (2). Even though in (2) the initial data $\varphi^{(0)}$, $\varphi^{(1)}$ have an additional Sobolev regularity $\nu>10$, with respect to the minimal regularity that is required in (1), the final time $T$ depends on those data only through the lower order norms $\Vert\varphi^{(0)}\Vert_{H^{11}(\mathbb T)}$, $\Vert\varphi^{(1)}\Vert_{H^{10}(\mathbb T)}$, see Subsection <ref>.
The paper is organized as follows. After the following Section <ref> about notations and basic tools, in Section <ref> we study the linearized equation about a given reference state $\vphi_0$. The main result of this section is the basic energy estimate (<ref>) obtained by the standard approach of multiplication and integration by parts. Even if the approach is quite natural, in our opinion the result is not at all obvious. In fact, it follows from a very careful analysis, in particular the study of some critical terms containing derivative of higher order. Here we use in a crucial way some fine properties of the Hilbert transform and new commutator estimates.
In Section <ref> we prove a tame estimate in Sobolev spaces of any order for the solution $\vphi'$ to the linearized equation. The proof follows from the basic energy a priori estimate (<ref>) and suitable commutator estimates involving the Hilbert transform and derivatives of higher order. An important feature is the following one. In the basic energy a priori estimate (<ref>), there is no loss of regularity in the sense that the $L^2$-norm of the source term $g=\mathbb{L}^\prime[\varphi_0]$ controls the $L^2$-norms of both $\vphi_t'$ and $\vphi_x'$ (as for the standard wave equation). Thus, one would hope to prove an estimate in Sobolev spaces of higher order again with no loss of derivative, and consequently to solve the nonlinear problem by a standard method like the implicit function theorem or the contraction principle.
Unfortunately, here we find a serious difficulty. Our tame estimate (<ref>) still has no loss of regularity from the source term $g$ to the solution, but it contains the loss of 2 spatial derivatives in the inversion of the operator $\mathbb{L}^\prime[\varphi_0]$,
from the given basic state $\varphi_0$ to $\varphi^\prime$.
For this reason, we can't apply a standard method for the resolution of the nonlinear problem. Instead, (<ref>) is solved by applying the Nash-Moser's theorem. This is done in Section <ref> where we give an equivalent formulation of the Cauchy problem for (<ref>) as an abstract equation in a suitable functional setting, and verify all the assumptions needed for the application of Nash-Moser's theorem.
Finally, in <ref> we prove our commutator estimates involving the Hilbert transform and give other useful estimates. In <ref>, for reader's convenience, we recall the assumptions and the statement of the Nash-Moser's theorem.
§ PRELIMINARY RESULTS AND BASIC TOOLS
§.§ Notations
Throughout the whole paper, the partial derivative of a function $f(t,x)$ with respect to $t$ or $x$ will be denoted appending to the function the subscript $t$ or $x$ as
\begin{equation*}
f_t:=\frac{\partial f}{\partial t}\,,\qquad f_x:=\frac{\partial f}{\partial x}\,.
\end{equation*}
(The notations $\partial_t f$, $\partial_x f$ will be also used.) Higher order derivatives in $(t,x)$ will be denoted by the repeated indices; for instance $f_{tt}$ and $f_{tx}$ will stand respectively for second order derivatives of $f$ with respect to $t$ twice and $t$, $x$.
Let $\mathbb T$ denote the one-dimensional torus defined as
\begin{equation*}%\label{torus}
\mathbb T:=\mathbb R/(2\pi\mathbb Z)\,,
\end{equation*}
that is the set of equivalence classes of real numbers with respect to the equivalence relation $\sim$ defined as
\begin{equation*}%\label{tilde}
x\sim y\qquad\mbox{if and only if}\qquad x-y\in 2\pi\mathbb Z\,.
\end{equation*}
It is customary to identify functions that are defined on $\mathbb T$ with $2\pi-$periodic functions on $\mathbb R$. According to this convention, it will be usual referring to $f:\mathbb T\rightarrow\mathbb C$ as a “periodic function”.
All periodic functions $f:\mathbb T\rightarrow\mathbb C$ can be expanded in terms of Fourier series as
\begin{equation*}%\label{serie_fourier}
f(x)=\frac{1}{2\pi}\sum\limits_{k\in\mathbb Z}\widehat{f}(k)e^{ikx}\,,
\end{equation*}
where $\left\{\widehat{f}(k)\right\}_{k\in\mathbb Z}$ are the Fourier coefficients defined by
\begin{equation}\label{coeff_fourier}
\widehat{f}(k):=\int_{\mathbb T}f(x)e^{-ikx}\,dx\,,\qquad k\in\mathbb Z\,.
\end{equation}
For $1\le p\le +\infty$, we denote by $L^p(\mathbb T)$ the usual Lebesgue space of exponent $p$ on $\mathbb T$, defined as the set of (equivalence classes of) measurable functions $f:\mathbb T\rightarrow\mathbb C$ such that the norm
\begin{equation*}%\label{normaLp}
\Vert f\Vert_{L^p(\mathbb T)}:=
\begin{cases}\left(\int_{\mathbb T}\vert f(x)\vert^p\,dx\right)^{1/p}\,,\quad\mbox{if}\,\,p<+\infty\,\\ \mbox{ess sup}_{x\in\mathbb T}\vert f(x)\vert\,,\quad\mbox{if}\,\,p=+\infty
\end{cases}
\end{equation*}
is finite. We denote
\begin{equation*}
(f,g)_{L^2(\mathbb T)}:=\int_{\mathbb T}f(x)\overline{g(x)}\,dx
\end{equation*}
the inner product of two functions $f, g\in L^2(\mathbb T)$ ($\overline z$ denotes the conjugate of $z\in\mathbb C$).
For all $s\in\mathbb R$, $H^s(\mathbb T)$ will denote the Sobolev space of order $s$ on $\mathbb T$, defined to be the set of periodic functions[The word “function” is used here, and in the rest of the paper, in a wide sense. To be more precise, one should speak about “periodic distributions” on the torus, instead of “periodic functions”, when dealing with real order Sobolev spaces. However, for the sake of simplicity, here we prefer to avoid the precise framework of distributions. We refer the reader to the monograph <cit.> for a thorough presentation of the periodic setting.] $f:\mathbb T\rightarrow\mathbb C$ such that
\begin{equation}\label{normaHs}
\Vert f\Vert_{H^s(\mathbb T)}^2:=\frac1{2\pi}\sum\limits_{k\in\mathbb Z}(1+|k|)^{2s}\vert\widehat{f}(k)\vert^2<+\infty\,.
\end{equation}
The function $\Vert\cdot\Vert_{H^s(\mathbb T)}$ defines a norm on $H^s(\mathbb T)$, associated to the inner product
\begin{equation*}%\label{prodottoHs}
(f,g)_{H^s(\mathbb T)}:=\frac1{2\pi}\sum\limits_{k\in\mathbb Z}(1+|k|)^{2s}\widehat{f}(k)\overline{\widehat{g}(k)}\,,
\end{equation*}
which turns $H^s(\mathbb T)$ into a Hilbert space.
Because of the relation between differentiation and Fourier coefficients, it is obvious that when $s$ is a positive integer $H^s(\mathbb T)$ reduces to the space of periodic functions $f:\mathbb T\rightarrow\mathbb C$ such that
\begin{equation*}
\partial_x^k f\in L^2(\mathbb T)\,,\quad\mbox{for}\,\,0\le k\le s
\end{equation*}
\begin{equation*}%\label{normaderivate}
\sum\limits_{k=0}^s\left\Vert \partial_x^k f\right\Vert_{L^2(\mathbb T)}
\end{equation*}
defines a norm in $H^s(\mathbb T)$ equivalent to (<ref>)[Even though the functions $f$ involved here depend on $x\in \mathbb T$ alone, the partial derivative notation $\partial_x^k:=\partial_x\dots\partial_x$ ($k$ times) is used just in order to be consistent with the notations adopted in the subsequent sections, where functions will also depend on time.].
In the following, we are mainly concerned with real-valued periodic functions $f:\mathbb T\rightarrow\mathbb R$ with zero spatial mean, that is such that
\begin{equation*}%\label{media}
\int_{\mathbb T}f(x)\,dx=0\,.
\end{equation*}
For such functions the Fourier coefficients (<ref>) obey the additional constraints
\begin{equation}\label{coeff_fourier_condizioni}
\widehat{f}(0)=0\,,\qquad \overline{\widehat{f}(k)}=\widehat{f}(-k)\,,\,\,\forall\,k\in\mathbb Z\,.
\end{equation}
In view of (<ref>), for zero mean periodic functions on $\mathbb T$ a norm in $H^s(\mathbb T)$ equivalent to (<ref>) is provided by
\begin{equation*}%\label{normaHs_equiv}
\Vert f\Vert_{s}^2:=\frac1{2\pi}\sum\limits_{k\in\mathbb Z\setminus\{0\}}|k|^{2s}\vert\widehat{f}(k)\vert^2\,.
\end{equation*}
For $s\in\mathbb N$, from the well-known formula
\begin{equation*}
\widehat{\partial^s_xf}(k)=(ik)^s\widehat{f}(k)
\end{equation*}
and Parseval's identity it follows that
\begin{equation}\label{ident_norme}
\Vert f\Vert_{s}=\Vert\partial^s_x f\Vert_{L^2(\mathbb T)}\,.
\end{equation}
Hereafter, we will deal with spaces of functions that depend even on time $t$. It will be convenient to regard real-valued functions $f=f(t,x)$, depending on time and space, as vector-valued functions of $t$ alone taking values in some Banach space $\mathcal X$ of functions depending on $x\in\mathbb T$. For technical reasons, the time variable $t$ will be allowed to run through the whole real line $\mathbb R_t$ (or even a real interval $(-\infty,T)$ for given $T>0$), and the integrability properties of functions in time will be referred to the weighted measure $e^{-\gamma t}\,dt$ on $\mathbb R_t$ (or $(-\infty,T)$), being $\gamma$ a real positive parameter.
For every Banach space $\mathcal X$ (with norm $\Vert\cdot\Vert_{\mathcal X}$) and $\gamma\ge 1$, we denote by $L^2_\gamma(\mathbb R; \mathcal X)$ the space of measurable functions $f:\mathbb R\rightarrow\mathcal X$, such that the real-valued function
\begin{equation*}
t\mapsto e^{-\gamma t}\Vert f(t)\Vert_{\mathcal X}=\Vert f_\gamma(t)\Vert_{\mathcal X}
\end{equation*}
is square integrable on $\mathbb R$; here it is set
\begin{equation}\label{f_gamma}
f_\gamma:=e^{-\gamma t}f\,.
\end{equation}
The space $L^2_\gamma(\mathbb R; \mathcal X)$ is provided with the norm
\begin{equation}\label{norma_l2gamma}
\Vert f\Vert_{L^2_\gamma(\mathbb R; \mathcal X)}^2:=\int_\mathbb R \Vert f_\gamma(t)\Vert_{\mathcal X}^2\,dt\,.
\end{equation}
In the sequel, the Banach space $\mathcal X$ will be always some Sobolev space $H^m(\mathbb T)$ of integer order $m$. When in particular $\mathcal X=L^2(\mathbb T)$ then the space $L^2_\gamma(\mathbb R; L^2(\mathbb T))$ reduces to be the set of functions $f=f(t,x)$, that are measurable both on $t$ and $x$ and such that $e^{-\gamma t}f$ is square integrable in $\mathbb R\times\mathbb T$; accordingly we set $L^2_\gamma(\mathbb R\times\mathbb T)=L^2_\gamma(\mathbb R; L^2(\mathbb T))$, and denote by $\Vert\cdot\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}$ the related norm in (<ref>). For $\gamma=0$, the weighted spaces above reduce of course to the usual Lebesgue spaces that will be simply denoted as $L^2(\mathbb R;\mathcal X)$.
When $\mathcal X$ is an Hilbert space (that will be always the case in the following), being $\Vert\cdot\Vert_{\mathcal X}$ the norm associated to the natural inner product, then integration by parts yields that the following identity
\begin{equation*}%\label{derivata_gamma}
\Vert \partial_t f\Vert^2_{L^2_\gamma(\mathbb R;\mathcal X)}=\gamma^2\Vert f\Vert^2_{L^2_\gamma(\mathbb R; \mathcal X)}+\Vert\partial_t f_\gamma\Vert^2_{L^2(\mathbb R;\mathcal X)}
\end{equation*}
holds true for every $\gamma\ge 1$ and all sufficiently smooth functions $f$. In particular, it follows that
\begin{equation}\label{dis_gamma}
\gamma\Vert f\Vert_{L^2_\gamma(\mathbb R; \mathcal X)}\le \Vert\partial_t f\Vert_{L^2_\gamma(\mathbb R;\mathcal X)}\,.
\end{equation}
More in general for $k\in\mathbb N$ we set $H^k_\gamma(\mathbb R;\mathcal X)$ the space of measurable functions $f:\mathbb R\rightarrow\mathcal X$ such that
\begin{equation*}
\partial^j_t f\in L^2_\gamma(\mathbb R; \mathcal X)\,,\quad\mbox{for}\,\,j=0,\dots,k\,,
\end{equation*}
endowed with the weighted norm $\Vert\cdot\Vert_{H^k_\gamma(\mathbb R; \mathcal X)}$ defined by
\begin{equation}\label{normaHk}
\Vert f\Vert_{H^k_\gamma(\mathbb R; \mathcal X)}^2=\sum\limits_{j=0}^k\gamma^{2(k-j)}\Vert\partial^j_t f\Vert^2_{L^2_\gamma(\mathbb R; \mathcal X)}\,.
\end{equation}
Applying repeatedly the inequality (<ref>), one can see that for every $k\in\mathbb N$, $\gamma\ge 1$ and $f\in H^k_\gamma(\mathbb R; \mathcal X)$ the following holds
\begin{equation*}
\Vert f\Vert_{H^k_\gamma(\mathbb R; \mathcal X)}\le \sqrt {k+1}\Vert\partial_t^k f\Vert_{L^2_\gamma(\mathbb R; \mathcal X)}\,.
\end{equation*}
The previous inequality shows that $f\mapsto \Vert \partial_t^k f\Vert_{L^2_\gamma(\mathbb R; \mathcal X)}$ defines a norm in $H^k_\gamma(\mathbb R; \mathcal X)$ equivalent (uniformly in $\gamma$) to the norm $\Vert\cdot\Vert_{H^k_\gamma(\mathbb R; \mathcal X)}$ defined in (<ref>).
We denote by $L^\infty(\mathbb R; \mathcal X)$ the space of measurable functions $f:\mathbb R\rightarrow \mathcal X$ such that
t\mapsto\Vert f(t)\Vert_{\mathcal X}
is bounded in $\mathbb R$, provided with the norm
\begin{equation*}%\label{norma_inf}
\Vert f\Vert_{L^\infty(\mathbb R;\mathcal X)}:={\rm ess\,sup}_{t\in\mathbb R}\Vert f(t)\Vert_{\mathcal X}\,.
\end{equation*}
More in general, for $k\in\mathbb N$, $W^{k,\infty}(\mathbb R; \mathcal X)$ will be the space of measurable functions $f:\mathbb R\rightarrow \mathcal X$ such that
\begin{equation*}
\partial^j_t f\in L^\infty(\mathbb R; \mathcal X)\,,\quad\mbox{for}\,\,j=0,\dots,k\,,
\end{equation*}
with the norm
\begin{equation*}
\Vert f\Vert_{W^{k,\infty}(\mathbb R; \mathcal X)}:=\sup\limits_{0\le j\le k}\Vert\partial^j_t f\Vert_{L^\infty(\mathbb R; \mathcal X)}\,.
\end{equation*}
By replacing the real line with the real interval $(-\infty,T)$ for $T>0$ in all the above definitions, the vector-valued spaces $H^k_\gamma(-\infty,T;\mathcal X)$, $W^{k,\infty}(-\infty,T;\mathcal X)$ can be defined exactly in the same manner as before, with similar properties.
We recall the following vector-valued counterpart of the standard Sobolev Imbedding Theorem (see <cit.>).
For any $T>0$ and $\gamma\ge 1$, $H^1_\gamma(-\infty,T; \mathcal X)\hookrightarrow L^\infty(-\infty,T; \mathcal X)$ and there exists a positive constant $C_T$, independent of $\gamma$, such that
\begin{equation*}%\label{sobolev_ineq}
\Vert u\Vert_{L^\infty(-\infty, T;\mathcal X)}\le\frac{C_T}{\sqrt{\gamma}}\Vert u\Vert_{H^1_\gamma(-\infty, T; \mathcal X)}\,,\quad\forall\,u\in H^1_\gamma(-\infty,T;\mathcal X)\,.
\end{equation*}
§.§ Some reminds on periodic Fourier multipliers
For a given sequence of real (or complex) numbers $\{A(k)\}_{k\in\mathbb Z}$, we denote by $A$ the linear operator defined on periodic functions $f:\mathbb T\rightarrow\mathbb C$ by setting
\begin{equation}\label{operatore1}
Af(x):=\frac{1}{2\pi}\sum\limits_{k\in\mathbb Z}A(k)\widehat{f}(k)e^{ikx}\,,\qquad x\in\mathbb T\,,
\end{equation}
or equivalently, on the Fourier side, by its Fourier coefficients
\begin{equation}\label{operatore2}
\widehat{Af}(k)=A(k)\widehat{f}(k)\,,\qquad\forall\,k\in\mathbb Z\,.
\end{equation}
We refer to the sequence $\{A(k)\}_{k\in\mathbb Z}$ as the symbol of the operator $A$.
The following continuity result will be useful in the sequel.
Let the sequence $\{A(k)\}_{k\in\mathbb Z}$ satisfy the following assumption
\begin{equation}\label{stima_simbolo}
(1+\vert k\vert)^{-m}\vert A(k)\vert\le C\,,\qquad\forall\,k\in\mathbb Z\,,
\end{equation}
with suitable constants $m\in\mathbb R$, $C>0$; then the operator $A$ with symbol $\{A(k)\}_{k\in\mathbb Z}$, defined by (<ref>), extends as a linear bounded operator
\begin{equation*}%\label{continuita_sobolev}
A:H^{s}(\mathbb T)\rightarrow H^{s-m}(\mathbb T)\,,
\end{equation*}
for all $s\in\mathbb R$; more precisely
\begin{equation*}%\label{stima_sobolev}
\Vert Af\Vert_{H^{s-m}(\mathbb T)}\le C\Vert f\Vert_{H^{s}(\mathbb T)}\,,\qquad\forall\,f\in H^{s}(\mathbb T)\,,
\end{equation*}
where $C$ is the same constant involved in (<ref>).
The proof is a straightforward application of the estimates (<ref>) and the definition of the Sobolev norm in (<ref>); indeed we compute (see (<ref>)):
\begin{align*}
\Vert Af\Vert^2_{H^{s-m}(\mathbb T)}= &\frac1{2\pi}\sum\limits_{k\in\mathbb Z}(1+\vert k\vert)^{2(s-m)}\vert\widehat{Af}(k)\vert^2=\frac1{2\pi}\sum\limits_{k\in\mathbb Z}(1+\vert k\vert)^{2(s-m)}\vert A(k)\widehat{f}(k)\vert^2\\
\\
&\le \frac{C^2}{2\pi}\sum\limits_{k\in\mathbb Z}(1+\vert k\vert)^{2s}\vert\widehat{f}(k)\vert^2=C^2\Vert f\Vert^2_{H^s(\mathbb T)}\,.
\end{align*}
We will refer to an operator $A$, under the assumptions of Proposition <ref>, as a Fourier multiplier of order $m$. Such an operator transforms periodic functions with mean zero into functions of the same type, as it is easily seen by observing that
$ as long as $\widehat f(0)=0$.
As a straightforward consequence of formulas (<ref>), (<ref>), it even follows that the composition $AB$ of two Fourier multipliers $A$ and $B$, whose symbols are respectively $\{A(k)\}_{k\in\mathbb Z}$ and $\{B(k)\}_{k\in\mathbb Z}$, is again a Fourier multiplier whose symbol is given by $\{A(k)B(k)\}_{k\in\mathbb Z}$ (the order of $AB$ being the sum of the orders of $A$ and $B$ separately, because of (<ref>)). We have in particular that $AB=BA$.
An example of a Fourier multiplier of order one is provided by the $x-$derivative, i.e. $Af=f_x$, since indeed
\begin{equation*}%\label{derivata}
\widehat{Af}(k)=\widehat{f_x}(k)=ik\widehat{f}(k)\,,\qquad\forall\,k\in\mathbb Z\,.
\end{equation*}
Another relevant example of a Fourier multiplier is considered in the next section.
§.§ Discrete Hilbert transform
The discrete Hilbert transform of a periodic function $f:\mathbb T\rightarrow\mathbb C$, denoted by $\mathbb H[f]$, is defined on the Fourier side by setting
\begin{equation}\label{hilbert1}
\widehat{\mathbb H[f]}(k)=-i\,{\rm sgn}\,k\widehat{f}(k)\,,\qquad\forall\,k\in\mathbb Z\,,
\end{equation}
\begin{equation}\label{segno}
{\rm sgn}\,k:=
\begin{cases}
1\,,\quad\mbox{if}\,\,k>0\,,\\ 0\,,\quad\mbox{if}\,\,k=0\,,\\ -1\,,\quad\mbox{if}\,\,k<0\,.
\end{cases}
\end{equation}
It is clear that, in view of Proposition <ref>, the Hilbert transform provides a Fourier multiplier of order zero, the condition (<ref>) being satisfied by $A(k)=-i\,{\rm sgn}\,k$ with $m=0$ and $C=1$; then after Proposition <ref> we conclude that
\begin{equation*}%\label{continuita_hilbert}
\mathbb H:H^s(\mathbb T)\rightarrow H^s(\mathbb T)
\end{equation*}
is a linear bounded operator and
\begin{equation}\label{stima_hilbert}
\Vert\mathbb H[f]\Vert_{H^s(\mathbb T)}\le\Vert f\Vert_{H^s(\mathbb T)}\,,\qquad\forall\,f\in H^s(\mathbb T)
\end{equation}
for all $s\in\mathbb R$.
Since $\vert{\rm sgn}\,k\vert=1$ for $k\neq 0$, it holds in particular that for every periodic function $f\in H^s(\mathbb T)$ with mean zero (i.e. $\widehat{f}(0)=0$), one has
\begin{equation*}
\Vert\mathbb H[f]\Vert^2_{s}=\frac1{2\pi}\sum\limits_{k\in\mathbb Z\setminus\{0\}}\vert k\vert^{2s}\vert\widehat{\mathbb H[f]}(k)\vert^2=\frac1{2\pi}\sum\limits_{k\in\mathbb Z\setminus\{0\}}\vert k\vert^{2s}\vert-i\,{\rm sgn}\,k\widehat{f}(k)\vert^2=\frac1{2\pi}\sum\limits_{k\in\mathbb Z\setminus\{0\}}\vert k\vert^{2s}\vert\widehat{f}(k)\vert^2=\Vert f\Vert^2_{s}\,.
\end{equation*}
Here below we collect a few elementary properties of the Hilbert transform that will be useful in the sequel.
1. The Hilbert transform commutes with the $x-$derivative. It is a particular case of the property recalled in Remark <ref>;
2. For all periodic functions $f,g:\mathbb T\rightarrow\mathbb C$ there holds
\begin{equation}\label{prodotto_hilbert}
\mathbb H\left[fg-\mathbb H[f]\mathbb H[g]\right]=f\mathbb H[g]+\mathbb H[f]g\,.
\end{equation}
3. For every periodic function $f:\mathbb T\rightarrow\mathbb C$, with zero mean, and $k\in\mathbb Z$, the following formulas of calculus hold true [Notice that, according to the convention ${\rm sgn}\,0=0$ (see (<ref>)), the Hilbert transform $\mathbb H[f]$ of any periodic function $f$ on $\mathbb T$ has zero mean. Hence, the first formula in (<ref>) is not true when the mean of $f$ is different from zero. In the latter case, that formula should be replaced by $\mathbb H^2[f]=-f +\widehat{f}(0)$.]
\begin{equation}\label{calcolo}
\mathbb H^2[f]=-f\,,\qquad \mathbb H\left[e^{ik\cdot}\right](x)=-i\,{\rm sgn}\,k\,e^{ikx}\,.
\end{equation}
4. For all periodic functions $f,g\in L^2(\mathbb T)$ there holds
\begin{equation}\label{integraleH}
\left(\mathbb H[f], g\right)_{L^2(\mathbb T)}=\left(f, -\mathbb H[g]\right)_{L^2(\mathbb T)}.
\end{equation}
5. For all periodic functions $f,g\in L^2(\mathbb T)$ and $h\in L^\infty(\mathbb T)$ there holds
\begin{equation}\label{aggiunto}
\left(\left[h;\mathbb H\right]f , g\right)_{L^2(\mathbb T)}=\left(f ,\left[h;\mathbb H\right]g\right)_{L^2(\mathbb T)},
\end{equation}
where $\left[h;\mathbb H\right]$ denotes the commutator between the multiplication by the function $h$ and the Hilbert transform $\mathbb H$.
§ THE LINEARIZED EQUATION
It is well-known that a first step in proving the local-in-time existence for the nonlinear equation (<ref>) is the study of the well-posedness of the linearization of this equation about a sufficiently smooth state $\varphi_0=\varphi_0(t,x)$; according to (<ref>) we assume that the reference state $\varphi_0$ satisfies
\begin{equation*}%\label{stability}
\mu-2\phi_{0,x}>0\,,\qquad\phi_0:=\mathbb H[\varphi_0]\,.
\end{equation*}
The above condition ensures the “leading part” $\varphi_{tt}-(\mu-2\phi_{0,x})\varphi_{xx}$ of the linearized equation to be of hyperbolic type (see (<ref>)).
In order to linearize the equation (<ref>), it is firstly convenient to rewrite it in the following equivalent form
\begin{equation}\label{onde_1}
\varphi_{tt}-\mu\varphi_{xx}=\left(\mathbb H[\phi^2_x]-\left[\phi\,;\mathbb H\right]\phi_{xx}\right)_{x}\,,
\end{equation}
\begin{equation*}%\label{commutatore}
\left[\phi\,;\mathbb H\right]:=\phi\mathbb H-\mathbb H\phi
\end{equation*}
denotes the commutator of the multiplication operator by $\phi$ and the Hilbert transform $\mathbb H$. Here we have used the first formula in (<ref>) (it is assumed that $\varphi$ has zero spatial mean), which implies $\varphi_{xx}=-\mathbb H[\phi_{xx}]$, and that $\mathbb H$ commutes with the $x-$differentiation.
Let $\varphi_0=\varphi_0(t,x)$ be a given basic state, with zero spatial mean, obeying suitable regularity assumptions and set $\phi_0=\mathbb H[\varphi_0]$. If we let $\mathbb L[\cdot]$ denote the nonlinear operator
\begin{equation}\label{operatore_nonlin}
\mathbb L[\varphi]:=\varphi_{tt}-\mu\varphi_{xx}-\left(\mathbb H[\phi^2_x]-\left[\phi\,;\mathbb H\right]\phi_{xx}\right)_{x}\,,
\end{equation}
then the linearization of the equation (<ref>) about $\varphi_0$ is defined by
\begin{equation}\label{equazione_lin}
\mathbb{L}^\prime[\varphi_0]\varphi^\prime:=\frac{d}{d\varepsilon}\left.\mathbb L[\varphi_0+\varepsilon\varphi^\prime]\right\vert_{\varepsilon=0}=g\,.
\end{equation}
In the linear equation above the unknown $\varphi^\prime=\varphi^\prime(t,x)$ represents some small perturbation of the basic state $\varphi_0$, with zero spatial mean, while $g=g(t,x)$ is some given nonzero forcing term taking account of lower order perturbation errors arising from the linearization of (<ref>).
An explicit computation leads to the following form of the linear operator $\mathbb{L}^\prime[\varphi_0]$ in (<ref>).
\begin{equation}\label{operatore_lin1}
\mathbb{L}^\prime[\varphi_0]\varphi^\prime=\varphi^\prime_{tt}-\mu\varphi^\prime_{xx}-\left(2\mathbb H[\phi_{0,x}\phi^\prime_x]-\left[\phi^\prime;\mathbb H\right]\phi_{0,xx}-\left[\phi_0;\mathbb H\right]\phi^\prime_{xx}\right)_x\,,\qquad \phi^\prime:=\mathbb H[\varphi^\prime]\,,
\end{equation}
where again $[A\,;B]:=AB-BA$ is the commutator of the operators $A$ and $B$.
§.§ $L^2-$a priori estimate
Following the approach developed in <cit.>, our first goal is to associate to the linear equation (<ref>) an $L^2$-a priori energy estimate of the first order derivatives of any sufficiently smooth solution of such an equation by the forcing term $g$. In this context, the time $t$ will be allowed to span the whole real line $\mathbb R$, and the energy estimate we are looking for will be of weighted type, in the sense that the integrability of the involved functions will be measured in the weighted space $L^2_\gamma(\mathbb R\times\mathbb T)$.
In the following, for a real $\gamma>0$ and every function $f=f(t,x)$ on $\mathbb R\times\mathbb T$, according to (<ref>), we use the shortcut
\begin{equation}\label{shortcut}
f_\gamma:=e^{-\gamma t}f\,,\qquad f_{t,\gamma}:=e^{-\gamma t}f_t\,,\qquad f_{x,\gamma}:=e^{-\gamma t}f_x\,.
\end{equation}
For the reader convenience, let us recall here below the following useful, though trivial, identities
\begin{equation}\label{identita_1}
f_{x,\gamma}=\partial_x f_\gamma\,,\qquad f_{t,\gamma}=\gamma f_\gamma+\partial_t f_\gamma\,.
\end{equation}
The relation
\begin{equation}\label{identita}
\partial_x f_{t,\gamma}=\gamma f_{x,\gamma}+\partial_t f_{x,\gamma}
\end{equation}
follows at once from a combination of (<ref>) and (<ref>); the latter will be repeatedly used later on.
From the second equality in (<ref>) and the integration by parts in $t$, the following estimate can also be proved, see <cit.> for the proof:
\begin{equation}\label{stima_metivier}
\gamma\Vert f\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}\le\Vert f_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}\,.
\end{equation}
Actually, inequality (<ref>) is a particular case of (<ref>) for $\mathcal{X}=L^2(\mathbb T)$.
This section is devoted to the proof of the following result.
Let the basic state $\varphi_0:\mathbb R\times\mathbb T\rightarrow\mathbb R$, with zero spatial mean, satisfy
\begin{equation}\label{regolarita_phi0}
\varphi_0\in L^\infty(\mathbb R; H^3(\mathbb T))\,,\quad\varphi_{0,t}\in L^\infty(\mathbb R; H^2(\mathbb T))
\end{equation}
\begin{equation}\label{stability_unif}
\mu-2\phi_{0,x}\ge\delta/2\quad\mbox{in}\,\,\,\mathbb R\times\mathbb T\,,
\end{equation}
with $\delta>0$ assigned in Theorem <ref>. Then there exist constants $\gamma_0\ge 1$ depending only on $\delta$ and $\varphi_0$ through the norms $\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}$, $\Vert\varphi_{0,t}\Vert_{L^\infty(\mathbb R; H^2(\mathbb T))}$, and $C_0>0$ depending only on $\delta$, such that for all $\gamma\ge \gamma_0$ and every sufficiently smooth function $\varphi^\prime:\mathbb R\times\mathbb T\rightarrow\mathbb R$, with zero spatial mean, the following a priori estimate
\begin{equation}\label{stima_apriori_1}
\gamma\left\{\Vert\varphi^\prime_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\varphi^\prime_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\le\frac{C_0}{\gamma}\Vert g\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}
\end{equation}
is satisfied, where $g:=\mathbb{L}^\prime[\varphi_0]\varphi^\prime$.
Let $\varphi^\prime(t,x)$ be a sufficiently smooth function, with zero spatial mean, according to the statement of Theorem <ref> and set $g:=\mathbb{L}^\prime[\varphi_0]\varphi^\prime$. To simplify the notation, in the following we drop the superscript ${}^\prime$ in the unknown function $\varphi^\prime$.
Just in order to outline the quantity $\mu-2\phi_{0,x}$, involved in (<ref>), as coefficient of $\varphi_{xx}$, it is convenient to expand the last term in the right-hand side of (<ref>); by the rules of calculus collected in Section <ref> we get
\begin{align*}
\left(2\mathbb H\right.&\left.[\phi_{0,x}\phi_x]-\left[\phi;\mathbb H\right]\phi_{0,xx}-\left[\phi_0;\mathbb H\right]\phi_{xx}\right)_x\\
&=2\mathbb H[\phi_{0,x}\phi_{xx}]+2\mathbb H[\phi_{0,xx}\phi_x]-\left(\left[\phi;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\phi_{xx}\right)_x\\
&=2\phi_{0,x}\mathbb H[\phi_{xx}]+2\left[\mathbb H\,;\phi_{0,x}\right]\phi_{xx}+2\mathbb H[\phi_{0,xx}\phi_x]-\left(\left[\phi;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\phi_{xx}\right)_x\\
&=-2\phi_{0,x}\varphi_{xx}+2\left[\mathbb H\,;\phi_{0,x}\right]\phi_{xx}+2\mathbb H[\phi_{0,xx}\phi_x]-\left(\left[\phi;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\phi_{xx}\right)_x\,.
\end{align*}
Then substituting the last expression into (<ref>) gives
\begin{equation}\label{operatore_lin2}
\mathbb{L}[\varphi_0]\varphi=\varphi_{tt}-\left(\mu-2\phi_{0,x}\right)\varphi_{xx}-2\left[\mathbb H\,;\phi_{0,x}\right]\phi_{xx}-2\mathbb H[\phi_{0,xx}\phi_x]+\left(\left[\phi;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\phi_{xx}\right)_x\,.
\end{equation}
In view of (<ref>), the linearized equation (<ref>) takes the form
\begin{equation}\label{equazione_lin1}
\varphi_{tt}-\left(\mu-2\phi_{0,x}\right)\varphi_{xx}=2\left[\mathbb H\,;\phi_{0,x}\right]\phi_{xx}+2\mathbb H[\phi_{0,xx}\phi_x]-\left(\left[\phi;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\phi_{xx}\right)_x+g\,.
\end{equation}
Since the estimate (<ref>) involves the weighted $L^2-$norms of $\varphi_t$, $\varphi_x$, it is also convenient to restate the equation (<ref>) in terms of $\varphi_{t,\gamma}$, $\varphi_{x,\gamma}$. Then (<ref>) becomes equivalent to
\begin{equation}\label{equazione_lin1_w}
\begin{split}
\gamma\varphi_{t,\gamma}&+\partial_t\varphi_{t,\gamma}-\left(\mu-2\phi_{0,x}\right)\partial_x\varphi_{x,\gamma}=2\left[\mathbb H\,;\phi_{0,x}\right]\partial_x\phi_{x,\gamma}+2\mathbb H[\phi_{0,xx}\phi_{x,\gamma}]\\
&-\left(\left[\phi_\gamma;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\right)_x+g_\gamma\,.
\end{split}
\end{equation}
We multiply by $\varphi_{t,\gamma}$ the equation (<ref>) and integrate over $\mathbb R\times\mathbb T$ to get
\begin{equation}\label{equazione_lin2}
\begin{split}
\gamma\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}&+\int_{\mathbb R\times\mathbb T}\partial_t\varphi_{t,\gamma}\varphi_{t,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)\partial_x\varphi_{x,\gamma}\varphi_{t,\gamma}\,dx\,dt\\
&=2\int_{\mathbb R\times\mathbb T}\left[\mathbb H\,;\phi_{0,x}\right]\partial_x\phi_{x,\gamma}\varphi_{t,\gamma}\,dx\,dt+2\int_{\mathbb R\times\mathbb T}\mathbb H[\phi_{0,xx}\phi_{x,\gamma}]\varphi_{t,\gamma}\,dx\,dt\\
&-\int_{\mathbb R\times\mathbb T}\left(\left[\phi_\gamma;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\right)_x\varphi_{t,\gamma}\,dx\,dt+\int_{\mathbb R\times\mathbb T}g_\gamma\varphi_{t,\gamma}\,dx\,dt\,.
\end{split}
\end{equation}
As for the two integrals in the left-hand side, integration by parts gives
\begin{equation}\label{int_1.1}
\int_{\mathbb R\times\mathbb T}\partial_t\varphi_{t,\gamma}\varphi_{t,\gamma}\,dx\,dt=0;
\end{equation}
\begin{equation}\label{int_1.2}
\begin{split}
-\int_{\mathbb R\times\mathbb T}&\left(\mu-2\phi_{0,x}\right)\partial_x\varphi_{x,\gamma}\varphi_{t,\gamma}\,dx\,dt=\int_{\mathbb R\times\mathbb T}\left(\left(\mu-2\phi_{0,x}\right)\varphi_{t,\gamma}\right)_x\varphi_{x,\gamma}\,dx\,dt\\
%=\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)\partial_x\varphi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt+\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)_x\varphi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt\\
=\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)\partial_x\varphi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}2\phi_{0,xx}\varphi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt\,.
\end{split}
\end{equation}
We use the identity (<ref>), with $f=\varphi$, in (<ref>) to get
\begin{equation}\label{int_1.2.1}
\begin{split}
-\int_{\mathbb R\times\mathbb T}&\left(\mu-2\phi_{0,x}\right)\partial_x\varphi_{x,\gamma}\varphi_{t,\gamma}\,dx\,dt\\
&=\gamma\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)\vert\varphi_{x,\gamma}\vert^2\,dx\,dt+\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)\partial_t\varphi_{x,\gamma}\varphi_{x,\gamma}-2\int_{\mathbb R\times\mathbb T}\phi_{0,xx}\varphi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt\,.
\end{split}
\end{equation}
As for the second integral in the right-hand side of (<ref>), integration by parts in $t$ gives
\begin{equation*}
\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)\partial_t\varphi_{x,\gamma}\varphi_{x,\gamma}=\int_{\mathbb R\times\mathbb T}\phi_{0,xt}\vert\varphi_{x,\gamma}\vert^2\,dx\,dt\,.
\end{equation*}
Replacing the latter into (<ref>) then gives
\begin{equation}\label{int_1.2.2}
\begin{split}
-\int_{\mathbb R\times\mathbb T}&\left(\mu-2\phi_{0,x}\right)\partial_x\varphi_{x,\gamma}\varphi_{t,\gamma}\,dx\,dt\\
&=\gamma\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)\vert\varphi_{x,\gamma}\vert^2\,dx\,dt+\int_{\mathbb R\times\mathbb T}\phi_{0,xt}\vert\varphi_{x,\gamma}\vert^2\,dx\,dt-2\int_{\mathbb R\times\mathbb T}\phi_{0,xx}\varphi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt\,.
\end{split}
\end{equation}
Replacing (<ref>), (<ref>) into (<ref>) we get
\begin{equation}\label{equazione_lin3}
\begin{split}
\gamma\Vert\varphi_{t}&\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\gamma\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)\vert\varphi_{x,\gamma}\vert^2\,dx\,dt\\
&=-\int_{\mathbb R\times\mathbb T}\phi_{0,xt}\vert\varphi_{x,\gamma}\vert^2\,dx\,dt+2\int_{\mathbb R\times\mathbb T}\phi_{0,xx}\varphi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt\\
&+2\int_{\mathbb R\times\mathbb T}\left[\mathbb H\,;\phi_{0,x}\right]\partial_x\phi_{x,\gamma}\varphi_{t,\gamma}\,dx\,dt+2\int_{\mathbb R\times\mathbb T}\mathbb H[\phi_{0,xx}\phi_{x,\gamma}]\varphi_{t,\gamma}\,dx\,dt\\
&-\int_{\mathbb R\times\mathbb T}\left(\left[\phi_\gamma;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\right)_x\varphi_{t,\gamma}\,dx\,dt+\int_{\mathbb R\times\mathbb T}g_\gamma\varphi_{t,\gamma}\,dx\,dt\,.
\end{split}
\end{equation}
Now we are going to provide a suitable estimate for each of the integral terms that appear in the right-hand side of the identity above.
Throughout the following, $C$ will always denote some numerical positive constant that may be possibly different from
line to line.
Combining the Sobolev imbedding $H^1(\mathbb T)\hookrightarrow L^\infty(\mathbb T)$ and Poincaré's inequality for functions with zero spatial mean, the following functional inequalities
\begin{equation}\label{sobolev-poincare}
\Vert\psi\Vert_{L^\infty(\mathbb T)}\le C\Vert\psi\Vert_{H^1(\mathbb T)}\le C\Vert\psi_x\Vert_{L^2(\mathbb T)}\,,
\end{equation}
can be easily established for all periodic functions $\psi\in H^1(\mathbb T)$ with spatial mean equal to zero. They will be repeatedly used in the following calculations.
Let us come back to the estimate of the right-hand side of (<ref>). Hölder's inequality gives
\begin{equation*}%\label{stima_int_2.1}
-\int_{\mathbb R\times\mathbb T}\phi_{0,xt}\vert\varphi_{x,\gamma}\vert^2\,dx\,dt\le \int_{\mathbb R}\Vert \phi_{0,xt}\Vert_{L^\infty(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert^2_{L^2(\mathbb T)}\,dt\,;
\end{equation*}
then we apply (<ref>) to $\phi_{0,xt}$ (recall that $\varphi_0$ and $\phi_0=\mathbb H[\varphi_0]$, as well as all their derivatives in $x$ and $t$, have zero spatial mean) and the $L^2-$continuity of the Hilbert transform (see (<ref>) for $s=0$) to find
\begin{equation}\label{stima_int_2.1'}
-\int_{\mathbb R\times\mathbb T}\phi_{0,xt}\vert\varphi_{x,\gamma}\vert^2\,dx\,dt\le C\int_{\mathbb R}\Vert \varphi_{0,xxt}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert^2_{L^2(\mathbb T)}\,dt\le C\Vert \varphi_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\,.
\end{equation}
Similarly, by Hölder and Young's inequalities and making use of (<ref>) and the continuity of the Hilbert transform, one gets
\begin{equation}\label{stima_int_2.2}
\begin{split}
2\int_{\mathbb R\times\mathbb T}&\phi_{0,xx}\varphi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt\le 2\int_{\mathbb R}\Vert\phi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\int_\mathbb R\Vert\varphi_{0,xxx}\Vert_{L^2(\mathbb T)}\left\{\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}^2+\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}^2\right\}\,dt\\
&\le C\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert\varphi_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2+\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2\right\}\,.
\end{split}
\end{equation}
To provide an estimate of the integral $\displaystyle 2\int_{\mathbb R\times\mathbb T}\mathbb H[\phi_{0,xx}\phi_{x,\gamma}]\varphi_{t,\gamma}\,dx\,dt$ in the right-hand side of (<ref>) we use the properties of the Hilbert transform collected in Section <ref>, together with Hölder and Young's inequalities and (<ref>), to get
\begin{equation}\label{stima_int_2.4}
\begin{split}
2\int_{\mathbb R\times\mathbb T}&\mathbb H[\phi_{0,xx}\phi_{x,\gamma}]\varphi_{t,\gamma}\,dx\,dt=-2\int_{\mathbb R\times\mathbb T}\phi_{0,xx}\phi_{x,\gamma}\mathbb H[\varphi_{t,\gamma}]\,dx\,dt=-2\int_{\mathbb R\times\mathbb T}\phi_{0,xx}\phi_{x,\gamma}\phi_{t,\gamma}\,dx\,dt\\
&\le 2\int_{\mathbb R} \Vert\phi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{t,\gamma}\Vert_{L^2(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\le C\int_{\mathbb R}\Vert\varphi_{0,xxx}\Vert_{L^2(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
%&\le C\int_{\mathbb R}\Vert\varphi_{0,xxx}\Vert_{L^2(\mathbb T)}\left\{\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}^2+\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}^2\right\}\,dt\\
&\le C\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert\varphi_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2+\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2\right\}\,.
\end{split}
\end{equation}
To obtain an estimate of $\displaystyle 2\int_{\mathbb R\times\mathbb T}\left[\mathbb H\,;\phi_{0,x}\right]\partial_x\phi_{x,\gamma}\varphi_{t,\gamma}\,dx\,dt$ we use Hölder's inequality, (<ref>) and the estimate (<ref>) of Lemma <ref> with $s=1$, $v=\phi_{0,x}$ and $f=\phi_{x,\gamma}$ (and again the properties of the Hilbert transform) to get
\begin{equation}\label{stima_int_2.3}
\begin{split}
\displaystyle 2\int_{\mathbb R\times\mathbb T}&\left[\mathbb H\,;\phi_{0,x}\right]\partial_x\phi_{x,\gamma}\varphi_{t,\gamma}\,dx\,dt\le 2\int_{\mathbb R}\Vert \left[\mathbb H\,;\phi_{0,x}\right]\partial_x\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\int_{\mathbb R}\Vert \phi_{0,xx}\Vert_{H^1(\mathbb T)}\Vert \phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\le C \int_{\mathbb R}\Vert \varphi_{0,xx}\Vert_{H^1(\mathbb T)}\Vert \varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\\
&\le C\Vert \varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert \varphi_{x}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2+\Vert \varphi_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2\right\}\,.
\end{split}
\end{equation}
As for the integral term involving the source $g$, Hölder and Young's inequalities yield
\begin{equation}\label{stima_g}
\int_{\mathbb R\times\mathbb T}g_\gamma\,\varphi_{t,\gamma}\,dx\,dt\le\Vert g\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}\Vert\varphi_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}\le\frac{\gamma}{2}\Vert\varphi_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2+\frac{1}{2\gamma}\Vert g\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2\,.
\end{equation}
It remains now to treat the last term in the right-hand side of (<ref>), that is
\begin{equation}\label{int_2.5}
\mathcal I:=-\int_{\mathbb R\times\mathbb T}\left(\left[\phi_\gamma;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\right)_x\varphi_{t,\gamma}\,dx\,dt\,.
\end{equation}
Let us firstly decompose $\mathcal I$ above as the sum
\begin{equation}\label{decomp_int_2.5}
\mathcal I=\mathcal I_1+\mathcal I_2\,,
\end{equation}
\begin{align}
&\mathcal I_1:=-\int_{\mathbb R\times\mathbb T}\left(\left[\phi_\gamma;\mathbb H\right]\phi_{0,xx}\right)_x\varphi_{t,\gamma}\,dx\,dt\,,\label{int_2.5.1}\\
&\mathcal I_2:=-\int_{\mathbb R\times\mathbb T}\left(\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\right)_x\varphi_{t,\gamma}\,dx\,dt\,.\label{int_2.5.2}
\end{align}
The estimate of $\mathcal I_1$: we use Leibniz's formula, the definition of the commutator and formula (<ref>) to rewrite $\mathcal I_1$ as
\begin{equation}\label{rappres_int_2.5.1}
\begin{split}
\mathcal I_1=&-\int_{\mathbb R\times\mathbb T}\left[\phi_{x,\gamma};\mathbb H\right]\phi_{0,xx}\varphi_{t,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}\left[\phi_{\gamma};\mathbb H\right]\phi_{0,xxx}\varphi_{t,\gamma}\,dx\\
&=-\int_{\mathbb R\times\mathbb T}\phi_{x,\gamma}\mathbb H[\phi_{0,xx}]\varphi_{t,\gamma}\,dx\,dt+\int_{\mathbb R\times\mathbb T}\mathbb H\left[\phi_{x,\gamma}\phi_{0,xx}\right]\varphi_{t,\gamma}\,dx\,dt\\
&-\int_{\mathbb R\times\mathbb T}\phi_\gamma\mathbb H[\phi_{0,xxx}]\varphi_{t,\gamma}\,dx\,dt+\int_{\mathbb R\times\mathbb T}\mathbb H\left[\phi_\gamma\phi_{0,xxx}\right]\varphi_{t,\gamma}\,dx\,dt\\
&=-\int_{\mathbb R\times\mathbb T}\phi_{x,\gamma}\mathbb H[\phi_{0,xx}]\varphi_{t,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}\phi_{x,\gamma}\phi_{0,xx}\mathbb H[\varphi_{t,\gamma}]\,dx\,dt\\
&-\int_{\mathbb R\times\mathbb T}\phi_\gamma\mathbb H[\phi_{0,xxx}]\varphi_{t,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}\phi_\gamma\phi_{0,xxx}\mathbb H[\varphi_{t,\gamma}]\,dx\,dt\\
&=\int_{\mathbb R\times\mathbb T}\phi_{x,\gamma}\varphi_{0,xx}\varphi_{t,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}\phi_{x,\gamma}\phi_{0,xx}\phi_{t,\gamma}\,dx\,dt\\
&+\int_{\mathbb R\times\mathbb T}\phi_\gamma\varphi_{0,xxx}\varphi_{t,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}\phi_\gamma\phi_{0,xxx}\phi_{t,\gamma}\,dx\,dt\,.
\end{split}
\end{equation}
By using Höl̈d̈ër̈ and Young's inequalities, (<ref>) and the $L^2-$continuity of the Hilbert transform (cf. Section <ref>), the first and the second integrals involved in the representation (<ref>) can be easily estimated by
\begin{equation}\label{stima_int_2.5.1.1-2}
\begin{array}{ll}
\begin{split}
\int_{\mathbb R\times\mathbb T}&\varphi_{0,xx}\phi_{x,\gamma}\varphi_{t,\gamma}\,dx\,dt\le\int_{\mathbb R}\Vert \varphi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\int_{\mathbb R}\Vert \varphi_{0,xxx}\Vert_{L^2(\mathbb T)}\left\{\Vert\varphi_{x\gamma}\Vert_{L^2(\mathbb T)}^2+\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}^2\right\}\,dt\\
&\le C\Vert \varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2+\Vert\varphi_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2\right\}\,,
\end{split}\\
\\
\begin{split}
-\int_{\mathbb R\times\mathbb T}&\phi_{0,xx}\phi_{x,\gamma}\phi_{t,\gamma}\,dx\,dt\le\int_{\mathbb R}\Vert \phi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\phi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\int_{\mathbb R}\Vert \varphi_{0,xxx}\Vert_{L^2(\mathbb T)}\left\{\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}^2+\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}^2\right\}\,dt\\
&\le C\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2+\Vert\varphi_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2\right\}\,.
\end{split}
\end{array}
\end{equation}
Concerning the third and fourth integrals in (<ref>), firstly we use again Höl̈d̈ër̈ and Young's inequalities to get
\begin{equation}\label{stima_int_2.5.1.3-4}
\begin{array}{ll}
\displaystyle\int_{\mathbb R\times\mathbb T}\varphi_{0,xxx}\phi_\gamma\varphi_{t,\gamma}\,dx\,dt\le\int_{\mathbb R}\Vert \varphi_{0,xxx}\Vert_{L^2(\mathbb T)}\Vert\phi_\gamma\Vert_{L^\infty(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\,,\\
\\
\begin{split}
-\int_{\mathbb R\times\mathbb T}&\phi_{0,xxx}\phi_\gamma\phi_{t,\gamma}\,dx\,dt\le\int_{\mathbb R}\Vert \phi_{0,xxx}\Vert_{L^2(\mathbb T)}\Vert\phi_\gamma\Vert_{L^\infty(\mathbb T)}\Vert\phi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le \int_{\mathbb R}\Vert \varphi_{0,xxx}\Vert_{L^2(\mathbb T)}\Vert\phi_\gamma\Vert_{L^\infty(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\,.
\end{split}
\end{array}
\end{equation}
Since the spatial mean of $\phi$ is zero, we apply (<ref>) to $\phi_\gamma$ and the $L^2-$continuity of the Hilbert transform to estimate
\begin{equation}\label{sobolev-poincare-phi}
\Vert\phi_\gamma\Vert_{L^\infty(\mathbb T)}\le C\Vert\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\le C\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,.
\end{equation}
Combining (<ref>), (<ref>) and using once again Young's inequality and the Hilbert transform properties in Section <ref> we get
\begin{equation}\label{stima_int_2.5.1.3-4'}
\begin{array}{ll}
\begin{split}
\int_{\mathbb R\times\mathbb T}&\varphi_{0,xxx}\phi_\gamma\varphi_{t,\gamma}\,dx\,dt\le C\int_\mathbb R\Vert \varphi_{0,xxx}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\Vert \varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2+\Vert\varphi_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2\right\}\,,
\end{split}\\
\\
\begin{split}
-\int_{\mathbb R\times\mathbb T}&\phi_{0,xxx}\phi_\gamma\phi_{t,\gamma}\,dx\,dt\le C \int_{\mathbb R}\Vert \varphi_{0,xxx}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2+\Vert\varphi_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2\right\}\,.
\end{split}
\end{array}
\end{equation}
Gathering estimates (<ref>), (<ref>), we obtain the following estimate of $\mathcal I_1$
\begin{equation}\label{stima_int_2.5.1}
\mathcal I_1\le C\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2+\Vert\varphi_{t}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}^2\right\}\,.
\end{equation}
The estimate of $\mathcal I_2$: The integral $\mathcal I_2$ is the most difficult to handle; indeed, even after integration by parts, it still contains second order derivatives of $\varphi$ that we can't estimate directly. We will manage to represent $\mathcal I_2$ as the sum of integral terms involving only first order derivatives of $\varphi$ to which operators of the kind considered in Lemma <ref> are applied.
We first integrate by parts in $x$ and use the identity (<ref>) (with $f=\varphi$) to get
\begin{equation}\label{rappres_int_2.5.2_prel}
\mathcal I_2=\int_{\mathbb R\times\mathbb T}\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\partial_x\varphi_{t,\gamma}\,dx\,dt
=\mathcal I_{2,1}+\mathcal I_{2,2}\,,
\end{equation}
\begin{align}
&\mathcal I_{2,1}:=\gamma\int_{\mathbb R\times\mathbb T}\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\,,\label{I_21}\\
&\mathcal I_{2,2}:=\int_{\mathbb R\times\mathbb T }\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\partial_t\varphi_{x,\gamma}\,dx\,dt\,.\label{I_22}
\end{align}
Let us consider $\mathcal I_{2,2}$; integration by parts and Leibniz's rule give
\begin{equation}\label{rappres_int_2.5.2}
\begin{split}
\mathcal I_{2,2}&=-\int_{\mathbb R\times\mathbb T}\partial_t\left(\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\right)\varphi_{x,\gamma}\,dx\,dt\\
&=-\int_{\mathbb R\times\mathbb T}\left[\phi_{0,t};\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}\left[\phi_0;\mathbb H\right]\partial_t\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\\
&=-\int_{\mathbb R\times\mathbb T}\left[\phi_{0,t};\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}\partial_x\left(\left[\phi_0;\mathbb H\right]\partial_t\phi_{x,\gamma}\right)\varphi_{x,\gamma}\,dx\,dt\\
&\quad +\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\partial_t\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\\
&=-\int_{\mathbb R\times\mathbb T}\left[\phi_{0,t};\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt+\int_{\mathbb R\times\mathbb T}\left[\phi_0;\mathbb H\right]\partial_t\phi_{x,\gamma}\partial_x\varphi_{x,\gamma}\,dx\,dt\\
&\quad +\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\partial_t\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\,.
\end{split}
\end{equation}
Let us focus on the second integral $\displaystyle\int_{\mathbb R\times\mathbb T}\left[\phi_0;\mathbb H\right]\partial_t\phi_{x,\gamma}\partial_x\varphi_{x,\gamma}\,dx\,dt$ in the above representation of $\mathcal I_{2,2}$. Using formulas (<ref>) and (<ref>) we get
\begin{align*}
\int_{\mathbb R\times\mathbb T}&\left[\phi_0;\mathbb H\right]\partial_t\phi_{x,\gamma}\partial_x\varphi_{x,\gamma}\,dx\,dt\\
&=\int_{\mathbb R\times\mathbb T}\partial_t\phi_{x,\gamma}\left[\phi_0\,;\,\mathbb H\right]\partial_x\varphi_{x,\gamma}\,dx\,dt=-\int_{\mathbb R\times\mathbb T}\partial_t\varphi_{x,\gamma}\mathbb H\left[\left[\phi_0\,;\,\mathbb H\right]\partial_x\varphi_{x,\gamma}\right]\,dx\,dt\\
&=-\int_{\mathbb R\times\mathbb T}\partial_t\varphi_{x,\gamma}\left[\phi_0\,;\,\mathbb H\right]\mathbb H\left[\partial_x\varphi_{x,\gamma}\right]\,dx\,dt-\int_{\mathbb R\times\mathbb T}\partial_t\varphi_{x,\gamma}\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\,dx\,dt\\
&=-\int_{\mathbb R\times\mathbb T}\partial_t\varphi_{x,\gamma}\left[\phi_0\,;\,\mathbb H\right]\partial_x\phi_{x,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}\partial_t\varphi_{x,\gamma}\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\,dx\,dt\,.
\end{align*}
Then we use once again the identity (<ref>) to exchange the $x$ and $t-$derivatives in $\partial_t\varphi_{x,\gamma}$ involved in the two integrals just above and find, after integration by parts in $x$ (see also (<ref>), (<ref>)),
\begin{equation}\label{rappres_int_2.5.2_bis}
\begin{split}
\int_{\mathbb R\times\mathbb T}&\left[\phi_0;\mathbb H\right]\partial_t\phi_{x,\gamma}\partial_x\varphi_{x,\gamma}\,dx\,dt\\
&=-\int_{\mathbb R\times\mathbb T}(-\gamma\varphi_{x,\gamma}+\partial_x\varphi_{t,\gamma})\left[\phi_0\,;\,\mathbb H\right]\partial_x\phi_{x,\gamma}\,dx\,dt\\
&\quad -\int_{\mathbb R\times\mathbb T}(-\gamma\varphi_{x,\gamma}+\partial_x\varphi_{t,\gamma})\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\,dx\,dt\\
&=\gamma\int_{\mathbb R\times\mathbb T}\varphi_{x,\gamma}\left[\phi_0\,;\,\mathbb H\right]\partial_x\phi_{x,\gamma}\,dx\,dt+\gamma\int_{\mathbb R\times\mathbb T}\varphi_{x,\gamma}\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\,dx\,dt\\
&\quad +\int_{\mathbb R\times\mathbb T}\varphi_{t,\gamma}\left(\left[\phi_0\,;\,\mathbb H\right]\partial_x\phi_{x,\gamma}\right)_x\,dx\,dt
+\int_{\mathbb R\times\mathbb T}\varphi_{t,\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt\\
&=\mathcal I_{2,1}+\gamma\int_{\mathbb R\times\mathbb T}\varphi_{x,\gamma}\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\,dx\,dt\\
&\quad -\mathcal I_2+\int_{\mathbb R\times\mathbb T}\varphi_{t,\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt\,.
\end{split}
\end{equation}
Substituting (<ref>) in (<ref>) we get
\begin{equation}\label{rappres_int_2.5.2_ter}
\begin{split}
\mathcal I_{2,2}&=-\int_{\mathbb R\times\mathbb T}\left[\phi_{0,t};\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt+\mathcal I_{2,1}\\
&\quad +\gamma\int_{\mathbb R\times\mathbb T}\varphi_{x,\gamma}\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\,dx\,dt-\mathcal I_2\\
&\quad +\int_{\mathbb R\times\mathbb T}\varphi_{t,\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt +\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\partial_t\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\,.
\end{split}
\end{equation}
Then substituting (<ref>) into (<ref>) gives
\begin{equation*}
\begin{split}
\mathcal I_2&=2\mathcal I_{2,1}-\int_{\mathbb R\times\mathbb T}\left[\phi_{0,t};\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt+\gamma\int_{\mathbb R\times\mathbb T}\varphi_{x,\gamma}\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\,dx\,dt\\
&\quad -\mathcal I_2+\int_{\mathbb R\times\mathbb T}\varphi_{t,\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt +\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\partial_t\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\,,
\end{split}
\end{equation*}
hence, using also (<ref>) with $f=\phi$ and integrating by parts in $x$,
\begin{equation}\label{rappres_int_2.5.2'}
\begin{split}
\displaystyle\mathcal I_2&=\mathcal I_{2,1}-\frac12\int_{\mathbb R\times\mathbb T}\left[\phi_{0,t};\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt+\frac{\gamma}{2}\int_{\mathbb R\times\mathbb T}\varphi_{x,\gamma}\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\,dx\,dt\\
&\quad+\frac12\int_{\mathbb R\times\mathbb T}\varphi_{t,\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt+\frac12\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\partial_t\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\\
&=\mathcal I_{2,1}-\frac12\int_{\mathbb R\times\mathbb T}\left[\phi_{0,t};\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt-\frac{\gamma}{2}\int_{\mathbb R\times\mathbb T}\varphi_{\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt\\
&\quad+\frac12\int_{\mathbb R\times\mathbb T}\varphi_{t,\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt-\frac{\gamma}2\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\\
&\quad+\frac12\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\partial_x\phi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt\,.
\end{split}
\end{equation}
Let us now rewrite the integral $\mathcal I_{2,1}$, see (<ref>). In fact, the coefficient $\gamma$ in front of it does not allow a direct estimate.
We expand the commutator $\left[\phi_0;\mathbb H\right]$, then use (<ref>), (<ref>) and integration by parts to write
\begin{equation}\label{rappres_int_2.5.2.1'}
\begin{split}
\int_{\mathbb R\times\mathbb T}&\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\\
&=\int_{\mathbb R\times\mathbb T}\phi_0\,\mathbb H[\partial_x\phi_{x,\gamma}]\varphi_{x,\gamma}\,dx\,dt-\int_{\mathbb R\times\mathbb T}\mathbb H\left[\phi_0\,\partial_x\phi_{x,\gamma}\right]\varphi_{x,\gamma}\,dx\,dt\\
&=-\int_{\mathbb R\times\mathbb T}\phi_0\,\partial_x\varphi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt+\int_{\mathbb R\times\mathbb T}\phi_0\,\partial_x\phi_{x,\gamma}\phi_{x,\gamma}\,dx\,dt\\
&=-\frac12\int_{\mathbb R\times\mathbb T}\phi_0\,\partial_x\left(\left\vert\varphi_{x,\gamma}\right\vert^2\right)\,dx\,dt+\frac12\int_{\mathbb R\times\mathbb T}\phi_0\,\partial_x\left(\left\vert\phi_{x,\gamma}\right\vert^2\right)\,dx\,dt\\
&=\frac12\int_{\mathbb R\times\mathbb T}\phi_{0,x}\,\left(
\left\vert\varphi_{x,\gamma}\right\vert^2
%\,dx\,dt-\frac12\int_{\mathbb T}\phi_{0,x}\,
\right\vert^2
\right)\,dx\,dt\,.
\end{split}
\end{equation}
We use (<ref>) with $f=g=\varphi_{x,\gamma}$, together with the first formula in (<ref>) (notice that $fg-\mathbb H[f]\mathbb H[g]$ has zero mean as long as $f$ and $g$ have zero mean), to get
\begin{equation*}
\left\vert\phi_{x,\gamma}\right\vert^2=\left\vert\mathbb H[\varphi_{x,\gamma}]\right\vert^2=\left\vert\varphi_{x,\gamma}\right\vert^2+2\mathbb H\left[\varphi_{x,\gamma}\phi_{x,\gamma}\right]\,.
\end{equation*}
Thus substituting into (<ref>) we find
\begin{equation*}%\label{rappres_int_2.5.2.1''}
\begin{split}
\int_{\mathbb R\times\mathbb T}&\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt
%&=\frac12\int_{\mathbb R\times\mathbb T}\phi_{0,x}\,\left\vert\varphi_{x,\gamma}\right\vert^2\,dx-\frac12\int_{\mathbb R\times\mathbb T}\phi_{0,x}\,\left\vert\phi_{x,\gamma}\right\vert^2\,dx\,dt\\
=-\int_{\mathbb R\times\mathbb T}\phi_{0,x}\mathbb H\left[\varphi_{x,\gamma}\phi_{x,\gamma}\right]\,dx\,dt\\
&=\int_{\mathbb R\times\mathbb T}\mathbb H[\phi_{0,x}]\varphi_{x,\gamma}\phi_{x,\gamma}\,dx\,dt
=-\int_{\mathbb R\times\mathbb T}\varphi_{0,x}\varphi_{x,\gamma}\phi_{x,\gamma}\,dx\,dt\,.
\end{split}
\end{equation*}
We may further rewrite (and it is convenient doing so) the last integral above in terms of a commutator operator; indeed we may compute
\begin{equation*}
\begin{split}
-\int_{\mathbb R\times\mathbb T}&\varphi_{0,x}\varphi_{x,\gamma}\phi_{x,\gamma}\,dx\,dt\\
&=-\int_{\mathbb R\times\mathbb T}\varphi_{0,x}\varphi_{x,\gamma}\mathbb H\left[\varphi_{x,\gamma}\right]\,dx\,dt+\int_{\mathbb R\times\mathbb T}\varphi_{x,\gamma}\mathbb H\left[\varphi_{0,x}\varphi_{x,\gamma}\right]\,dx\,dt-\int_{\mathbb R\times\mathbb T}\varphi_{x,\gamma}\mathbb H\left[\varphi_{0,x}\varphi_{x,\gamma}\right]\,dx\,dt\\
&=\int_{\mathbb R\times\mathbb T}\left[\mathbb H;\varphi_{0,x}\right]\partial_x\varphi_{\gamma}\varphi_{x,\gamma}\,dx\,dt+\int_{\mathbb R\times\mathbb T}\varphi_{0,x}\varphi_{x,\gamma}\phi_{x,\gamma}\,dx\,dt
\end{split}
\end{equation*}
\begin{equation*}
-\int_{\mathbb R\times\mathbb T}\varphi_{0,x}\varphi_{x,\gamma}\phi_{x,\gamma}\,dx\,dt=\frac12\int_{\mathbb R\times\mathbb T}\left[\mathbb H;\varphi_{0,x}\right]\partial_x\varphi_{\gamma}\varphi_{x,\gamma}\,dx\,dt
\end{equation*}
\begin{equation*}
\int_{\mathbb R\times\mathbb T}\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt=-\int_{\mathbb R\times\mathbb T}\varphi_{0,x}\varphi_{x,\gamma}\phi_{x,\gamma}\,dx\,dt=\frac12\int_{\mathbb R\times\mathbb T}\left[\mathbb H;\varphi_{0,x}\right]\partial_x\varphi_{\gamma}\varphi_{x,\gamma}\,dx\,dt\,.
\end{equation*}
In conclusion we obtain
\begin{equation}\label{rappres_I_21}
\mathcal I_{2,1}=\frac{\gamma}2\int_{\mathbb R\times\mathbb T}\left[\mathbb H;\varphi_{0,x}\right]\partial_x\varphi_{\gamma}\varphi_{x,\gamma}\,dx\,dt\,,
\end{equation}
then substituting (<ref>) into (<ref>) we get
\begin{equation}\label{rappres_int_2.5.2''}
\begin{split}
\displaystyle\mathcal I_2&=\frac{\gamma}2\int_{\mathbb R\times\mathbb T}\left[\mathbb H;\varphi_{0,x}\right]\partial_x\varphi_{\gamma}\varphi_{x,\gamma}\,dx\,dt-\frac12\int_{\mathbb R\times\mathbb T}\left[\phi_{0,t};\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\\
&-\frac{\gamma}{2}\int_{\mathbb R\times\mathbb T}\varphi_{\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt+\frac12\int_{\mathbb R\times\mathbb T}\varphi_{t,\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt\\
&-\frac{\gamma}2\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\partial_x\phi_{\gamma}\varphi_{x,\gamma}\,dx\,dt+\frac12\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\partial_x\phi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt=\sum\limits_{j=1}^6 I_j\,.
\end{split}
\end{equation}
We are now in the position to provide a suitable estimate of $\mathcal I_2$.
Concerning the first integral $I_1:=\displaystyle\frac{\gamma}2\int_{\mathbb R\times\mathbb T}\left[\mathbb H;\varphi_{0,x}\right]\partial_x\varphi_{\gamma}\varphi_{x,\gamma}\,dx\,dt$ in the right-hand side of (<ref>), we apply Hölder and Young's inequalities and use the estimates (<ref>) (with $s=1$, $v=\varphi_{0,x}$ and $f=\varphi_\gamma$), (<ref>) to get
\begin{equation*}%\label{stima_I1}
\begin{split}
I_1&\le\frac{\gamma}{2}\int_{\mathbb R}\left\Vert\left[\mathbb H\,;\,\varphi_{0,x}\right]\partial_x\varphi_{\gamma}\right\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\gamma\int_{\mathbb R}\Vert\varphi_{0,xx}\Vert_{H^1(\mathbb T)}\Vert\varphi_\gamma\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\int_{\mathbb R}\Vert\varphi_{0,xx}\Vert_{H^1(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R; H^3(\mathbb T))}\left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\,.
\end{split}
\end{equation*}
To estimate the fifth integral $I_5:=\displaystyle-\frac{\gamma}2\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\partial_x\phi_{\gamma}\varphi_{x,\gamma}\,dx\,dt$ in the right-hand side of (<ref>) we use exactly the same arguments as before to obtain
\begin{equation*}%\label{stima_I5}
I_5\le C\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R; H^3(\mathbb T))}\left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\,.
\end{equation*}
The integrals $I_2:=\displaystyle-\frac12\int_{\mathbb R\times\mathbb T}\left[\phi_{0,t};\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt$ and $I_6:=\displaystyle\frac12\int_{\mathbb R\times\mathbb T}\left[\phi_{0,x};\mathbb H\right]\partial_x\phi_{t,\gamma}\varphi_{x,\gamma}\,dx\,dt$ in the right-hand side of (<ref>) can be treated exactly in the same manner as above, applying Hölder and Young's inequalities, using the estimate (<ref>) (with $s=1$, and $v=\phi_{0,t}$, $f=\phi_{x,\gamma}$ or $v=\phi_{0,x}$, $f=\phi_{t,\gamma}$ respectively in $I_2$ or $I_6$) and the $L^2-$continuity of $\mathbb H$ to get
\begin{equation*}%\label{stima_I2_I6}
\begin{split}
\begin{split}
I_2&\le\frac12\int_{\mathbb R}\Vert\left[\phi_{0,t};\mathbb H\right]\partial_x\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\le C\int_{\mathbb R}\Vert\phi_{0,tx}\Vert_{H^1(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
%&\le C\int_{\mathbb R}\Vert\varphi_{0,tx}\Vert_{H^1(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert^2_{L^2(\mathbb T)}\,dt\\
&\le C\Vert\varphi_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\,;
\end{split}\\
\\
\begin{split}
I_6&\le\frac12\int_{\mathbb R}\Vert\left[\phi_{0,x};\mathbb H\right]\partial_x\phi_{t,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\le C\int_{\mathbb R}\Vert\phi_{0,xx}\Vert_{H^1(\mathbb T)}\Vert\phi_{t,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
%&\le C\int_{\mathbb R}\Vert\varphi_{0,xx}\Vert_{H^1(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\,.
\end{split}
\end{split}
\end{equation*}
To estimate
\begin{equation*}
I_3:=-\frac{\gamma}{2}\int_{\mathbb R\times\mathbb T}\varphi_{\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt\,,\quad I_4:=\displaystyle\frac12\int_{\mathbb R\times\mathbb T}\varphi_{t,\gamma}\left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\,dx\,dt
\end{equation*}
in the right-hand side of (<ref>), we use again Hölder's inequality, the estimates (<ref>) (with $s=1$, $v=\phi_{0}$ and $f=\varphi_{x,\gamma}$), (<ref>), (<ref>) and the $L^2-$continuity of $\mathbb H$ to get
\begin{equation*}%\label{stima_I3_I4}
\begin{split}
\begin{split}
I_3&\le\frac{\gamma}2\int_{\mathbb R}\left\Vert \left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\right\Vert_{L^2(\mathbb T)}\Vert\varphi_{\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\gamma\int_{\mathbb R}\Vert\phi_{0,xx}\Vert_{H^1(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\int_{\mathbb R}\Vert\varphi_{0,xx}\Vert_{H^1(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\,;
\end{split}\\
\\
\begin{split}
I_4&\le\frac12\int_{\mathbb R}\left\Vert \left(\left[\mathbb H\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\partial_x\varphi_{x,\gamma}\right)_x\right\Vert_{L^2(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\int_{\mathbb R}\Vert\varphi_{0,xx}\Vert_{H^1(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\Vert\varphi_{t,\gamma}\Vert_{L^2(\mathbb T)}\,dt\\
&\le C\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\,.
\end{split}
\end{split}
\end{equation*}
Gathering the preceding estimates yields the following estimate of $\mathcal I_2$
\begin{equation*}%\label{stima_int_2.5.2}
\mathcal I_2\le C\left\{\Vert\varphi_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}+\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\right\}\left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\,,
\end{equation*}
from which, in view of (<ref>) and using also (<ref>), we get for the integral $\mathcal I$ in (<ref>) the estimate
\begin{equation}\label{stima_int_2.5}
\mathcal I\le C\left\{\Vert\varphi_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}+\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\right\}\left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\,.
\end{equation}
Gathering the estimates (<ref>)-(<ref>), (<ref>), from (<ref>) we derive
\begin{equation}\label{stima_lineare1}
\begin{split}
\gamma &\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\gamma\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)\left\vert\varphi_{x,\gamma}\right\vert^2\,dx\,dt\\
&\le C\left\{\Vert\varphi_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}+\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\right\}\left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\\
&\quad +\frac{\gamma}{2}\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\frac{1}{2\gamma}\Vert g\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\,.
\end{split}
\end{equation}
The estimate (<ref>) follows at once from (<ref>) by absorbing in the left-hand side the term $\displaystyle\frac{\gamma}{2}\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}$, then using the assumption (<ref>) to bound from below the integral $\displaystyle\int_{\mathbb R\times\mathbb T}\left(\mu-2\phi_{0,x}\right)\left\vert\varphi_{x,\gamma}\right\vert^2\,dx\,dt$ in the left-hand side of (<ref>) by
\begin{equation*}
\frac{\delta}{2}\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\,,
\end{equation*}
and finally taking $\gamma_0=\gamma_0(\delta, \Vert\varphi_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}, \Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))})$ sufficiently large such that
\begin{equation*}
\min\left\{1,\delta\right\}\gamma-2C\left\{\Vert\varphi_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}+\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\right\}\ge\frac12\min\left\{1,\delta\right\}\gamma\,.
\end{equation*}
We can take for instance $\gamma_0=\displaystyle\frac{4C\left\{\Vert\varphi_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}+\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\right\}}{\min\left\{1,\delta\right\}}$ and $C_0=\displaystyle\frac{2}{\min\{1,\delta\}}$.
§.§ Well-posedness of the linearized problem
Theorem <ref> only gives an a priori estimate for any sufficiently smooth function $\varphi^\prime$ in terms of $g:=\mathbb{L}^\prime[\varphi_0]\varphi^\prime$. Using standard methods we can also obtain the existence of the solution.
Let the basic state $\varphi_0:\mathbb R\times\mathbb T\rightarrow\mathbb R$, with zero spatial mean, satisfy
(<ref>), (<ref>), and let $\gamma_0\ge 1$, $C_0>0$ be the constants of Theorem <ref>.
For every $\gamma\ge \gamma_0$ and $g\in{L^2_\gamma(\mathbb R\times\mathbb T)}$ there exists a unique solution $\varphi^\prime$ with zero spatial mean of (<ref>) (expanded form in (<ref>)) such that $\varphi^\prime_{t}\in{L^2_\gamma(\mathbb R\times\mathbb T)}, \varphi^\prime_{x}\in{L^2_\gamma(\mathbb R\times\mathbb T)}$. Moreover, $\varphi^\prime$
the a priori estimate
The proof follows from a standard Galerkin approximation. Given any function $f:\mathbb T\rightarrow\mathbb C$ expanded in terms of Fourier series as
\begin{equation*}\label{}
f(x)=\frac{1}{2\pi}\sum\limits_{k\in\mathbb Z}\widehat{f}(k)e^{ikx}\,,
\end{equation*}
we define the finite dimensional orthogonal projection
\begin{equation*}\label{}
f^N(x)=P_Nf(x)=\frac{1}{2\pi}\sum\limits_{|k|\leq N}\widehat{f}(k)e^{ikx}\,.
\end{equation*}
Let us consider the Galerkin approximation $\vphi^N=P_N\varphi^\prime$, defined as the solution of the approximate ODE
\begin{multline}\label{equazione_lin1approx}
\varphi^N_{tt}-P_N\left( \left(\mu-2\phi_{0,x}\right)\varphi^N_{xx} \right)\\
=P_N\left\{ 2\left[\mathbb H\,;\phi_{0,x}\right]\phi^N_{xx}+2\mathbb H[\phi_{0,xx}\phi^N_x]
-\left(\left[\phi^N;\mathbb H\right]\phi_{0,xx}+\left[\phi_0;\mathbb H\right]\phi^N_{xx}\right)_x \right\}+g^N\,.
\end{multline}
We can repeat for $\vphi^N$ the same calculations in the proof of Theorem <ref>, leading to the analogue of the a priori estimate (<ref>), uniform in $N$. By standard arguments we can extract a subsequence $\{\vphi^N\}$ and pass to the limit in (<ref>) to obtain a solution of (<ref>). The uniqueness of the solution follows from (<ref>) and the linearity of the problem.
§ TAME ESTIMATE FOR THE LINEARIZED EQUATION
This section is devoted to associate to all sufficiently smooth solutions of the equation (<ref>) an appropriate a priori estimate in higher order Sobolev norms. In agreement with the above section, the norms involved in the found estimate are weighted in time by the exponential function $e^{-\gamma t}$.
The goal of this section is to prove the following result.
Let $m\ge1$ be an integer. Assume that the basic state $\varphi_0:\mathbb R\times\mathbb T\rightarrow\mathbb R$, with zero spatial mean, satisfies
\begin{equation*}%\label{regolarita_phi0_m}
\varphi_0\in L^\infty(\mathbb R; H^5(\mathbb T))\cap L^\infty(\mathbb R; H^{m+3}(\mathbb T))\,,\quad\varphi_{0,t}\in L^\infty(\mathbb R; H^2(\mathbb T))
\end{equation*}
condition (<ref>). Then there exist $\gamma_1\ge 1$, and $C>0$, depending boundedly and increasingly on $m$, $\delta$, $\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}$, $\Vert\varphi_{0,t}\Vert_{L^\infty(\mathbb R; H^2(\mathbb T))} $, such that for all $\gamma\ge \gamma_1$ and $g\in{L^2_\gamma(\mathbb R; H^m(\mathbb T))}$ there exists a unique solution $\varphi^\prime$ with zero spatial mean of (<ref>) (expanded form in (<ref>)) such that $\varphi^\prime_{t}\in{{L^2_\gamma(\mathbb R; H^m(\mathbb T))}}, \varphi^\prime_{x}\in{{L^2_\gamma(\mathbb R; H^m(\mathbb T))}}$. Moreover, the following a priori estimate holds true
\begin{equation}\label{stima_tame_1}
\begin{split}
\gamma & \left\{\Vert\varphi^\prime_{t}\Vert^2_{L^2_\gamma(\mathbb R; H^m(\mathbb T))}+\Vert\varphi^\prime_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\right\}\\
&\le \frac{C}{\gamma}\left\{\Vert\varphi_{0,x}\Vert^2_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}\Vert g\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))} +\Vert g\Vert^2_{L^2_\gamma(\mathbb R; H^m(\mathbb T))}\right\}\,.
\end{split}
\end{equation}
The a priori estimate (<ref>) shows the loss of 2 spatial derivatives in the inversion of the operator $\mathbb{L}^\prime[\varphi_0]$,
from the given basic state $\varphi_0$ to $\varphi^\prime$. For this reason the equation (<ref>) cannot be solved by standard methods like the implicit function theorem or the contraction principle. Instead, (<ref>) will be solved by applying the Nash-Moser's theorem.
In view of (<ref>), to obtain an estimate of the $L^2_\gamma(\mathbb R;H^m(\mathbb T))-$norm of the derivatives $\varphi^\prime_t$ and $\varphi^\prime_x$ of a smooth solution $\varphi^\prime$ to the equation (<ref>), with an arbitrary (sufficiently large) order $m\ge 1$, let us first concentrate on the derivative $\partial_x^m\varphi^\prime$. To simplify the notation, in the following we drop the superscript ${}^\prime$ in the unknown function $\varphi^\prime$.
Applying the derivative $\partial_x^m$ to (<ref>) (see also (<ref>)), after some calculations we find that $\partial_x^m\varphi$ must solve the linear equation
\begin{equation*}%\label{equazione_lin_dm}
\begin{split}
\mathbb{L}^\prime[\varphi_0]&\left(\partial_x^m\varphi\right)=2\mathbb H\left[\left[\partial_x^m\,;\,\phi_{0,x}\right]\phi_{xx}\right]+2\mathbb H\left[\left[\partial_x^m\,;\,\phi_{0,xx}\right]\phi_x\right]\\
&-\left(\left[\partial_x^m\,;\,\mathbb H\left[\phi_{0,xx}\right]\right]\phi-\mathbb H\left[\left[\partial_x^m\,;\,\phi_{0,xx}\right]\phi\right]+\left[\left[\partial_x^m\,;\,\phi_0\right]\,;\,\mathbb H\right]\phi_{xx}\right)_x+\partial_x^mg\\
\end{split}
\end{equation*}
which looks like the original linear equation (<ref>) where the forcing term $g$ is replaced by $\sum\limits_{j=1}^5G_j+\partial_x^mg$.
Hence applying the $L^2-$estimate (<ref>) we get for $\gamma\ge\gamma_0$
\begin{equation}\label{stima_apriori_2_0}
\gamma\left\{\Vert\partial_x^m\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\partial_x^m\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\le\frac{C_0}{\gamma}\left\{\sum\limits_{j=1}^5\Vert G_j\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert \partial_x^m g\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\,,
\end{equation}
where the threshold $\gamma_0$ and the constant $C_0$ are provided by Theorem <ref>.
We have now to provide suitable estimates for the $L^2_\gamma-$norms of the $G_j$'s. As in the preceding section, throughout the following $C$ will always denote a positive constant, only depending on $m$, that may be different from line to line.
The estimate of $G_1:=2\mathbb H\left[\left[\partial_x^m\,;\,\phi_{0,x}\right]\phi_{xx}\right]$: for fixed $t\in\mathbb R$ we first compute, by using the commutator estimate (<ref>), the Sobolev continuity of $\mathbb H$ and the Sobolev imbedding $H^1(\mathbb T)\hookrightarrow L^\infty(\mathbb T)$
\begin{equation*}
\begin{split}
\Vert & G_{1,\,\gamma}(t)\Vert_{L^2(\mathbb T)}=2\Vert \mathbb H\left[\left[\partial_x^m\,;\,\phi_{0,x}\right]\phi_{xx,\,\gamma}\right]\Vert_{L^2(\mathbb T)}= 2\Vert \left[\partial_x^m\,;\,\phi_{0,x}\right]\phi_{xx,\,\gamma}\Vert_{L^2(\mathbb T)}\\
&\le C\left\{\Vert\phi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{xx,\gamma}\Vert_{H^{m-1}(\mathbb T)}+\Vert\phi_{xx,\gamma}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{0,x}\Vert_{H^{m}(\mathbb T)}\right\}\\
&\le C\left\{\Vert\phi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}+\Vert\phi_{xx,\gamma}\Vert_{L^\infty(\mathbb T)}\Vert\varphi_{0,x}\Vert_{H^{m}(\mathbb T)}\right\}\\
&\le C\left\{\Vert\varphi_{0}\Vert_{H^3(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}+\Vert\varphi_{x,\gamma}\Vert_{H^2(\mathbb T)}\Vert\varphi_{0,x}\Vert_{H^{m}(\mathbb T)}\right\}\,.
\end{split}
\end{equation*}
Integrating over $\mathbb R_t$ we get
\begin{equation}\label{stima_G1_fin}
\Vert G_{1}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}\le C\left\{\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m}(\mathbb T))}+\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}\Vert\varphi_{0,x}\Vert_{L^\infty(\mathbb R;H^{m}(\mathbb T))}\right\}\,.
\end{equation}
The estimate of $G_2:=2\mathbb H\left[\left[\partial_x^m\,;\,\phi_{0,xx}\right]\phi_{x}\right]$: for fixed $t\in\mathbb R$ we first compute by using the commutator estimate (<ref>) and, for the rest, arguing as before
\begin{equation*}
\begin{split}
\Vert & G_{2,\,\gamma}(t)\Vert_{L^2(\mathbb T)}=2\Vert \mathbb H\left[\left[\partial_x^m\,;\,\phi_{0,xx}\right]\phi_{x,\,\gamma}\right]\Vert_{L^2(\mathbb T)}= 2\Vert \left[\partial_x^m\,;\,\phi_{0,xx}\right]\phi_{x,\,\gamma}\Vert_{L^2(\mathbb T)}\\
&\le C\left\{\Vert\phi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}+\Vert\phi_{x,\gamma}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{0,xx}\Vert_{H^{m}(\mathbb T)}\right\}\\
&\le C\left\{\Vert\varphi_{0}\Vert_{H^3(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}+\Vert\varphi_{x,\gamma}\Vert_{H^1(\mathbb T)}\Vert\varphi_{0,x}\Vert_{H^{m+1}(\mathbb T)}\right\}\,.
\end{split}
\end{equation*}
Integration over $\mathbb R_t$ then gives
\begin{equation}\label{stima_G2_fin}
\Vert G_{2}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}\le C\left\{\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m}(\mathbb T))}+\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^1(\mathbb T))}\Vert\varphi_{0,x}\Vert_{L^\infty(\mathbb R;H^{m+1}(\mathbb T))}\right\}\,.
\end{equation}
The estimate of $G_3:=-\left(\left[\partial_x^m\,;\,\mathbb H\left[\phi_{0,xx}\right]\right]\phi\right)_x$: for fixed $t\in\mathbb R$ we first compute by using again the commutator estimate (<ref>) and the Sobolev continuity of $\mathbb H$:
\begin{equation}\label{stima_G3_0}
\begin{split}
\Vert & G_{3,\,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert -\left(\left[\partial_x^m\,;\,\mathbb H\left[\phi_{0,xx}\right]\right]\phi_\gamma\right)_x\Vert_{L^2(\mathbb T)}\\
&\le\Vert\left[\partial_x^m\,;\,\mathbb H\left[\phi_{0,xxx}\right]\right]\phi_\gamma\Vert_{L^2(\mathbb T)}+\Vert \left[\partial_x^m\,;\,\mathbb H\left[\phi_{0,xx}\right]\right]\phi_{x, \gamma}\Vert_{L^2(\mathbb T)}\\
&=\Vert\left[\partial_x^m\,;\,\varphi_{0,xxx}\right]\phi_\gamma\Vert_{L^2(\mathbb T)}+\Vert \left[\partial_x^m\,;\,\varphi_{0,xx}\right]\phi_{x, \gamma}\Vert_{L^2(\mathbb T)}\\
&\le C\left\{\Vert\varphi_{0,xxx}\Vert_{L^\infty(\mathbb T)}\Vert\phi_\gamma\Vert_{H^m(\mathbb T)}+\Vert\varphi_{0,xxx}\Vert_{H^m(\mathbb T)}\Vert\phi_\gamma\Vert_{L^\infty(\mathbb T)}\right.\\
&\quad\left.+\Vert\varphi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{H^m(\mathbb T)}+\Vert\varphi_{0,xx}\Vert_{H^m(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{L^\infty(\mathbb T)}\right\}\\
&\le C\left\{\Vert\varphi_{0,xxx}\Vert_{L^\infty(\mathbb T)}\Vert\varphi_\gamma\Vert_{H^m(\mathbb T)}+\Vert\varphi_{0,x}\Vert_{H^{m+2}(\mathbb T)}\Vert\phi_\gamma\Vert_{L^\infty(\mathbb T)}\right.\\
&\quad\left.+\Vert\varphi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^m(\mathbb T)}+\Vert\varphi_{0,x}\Vert_{H^{m+1}(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{L^\infty(\mathbb T)}\right\}\,.
\end{split}
\end{equation}
On the other hand from Poincaré's inequality (recall that $\varphi$ has zero spatial mean) and the Sobolev imbedding $H^1(\mathbb T)\hookrightarrow L^\infty(\mathbb T)$ we get
\begin{eqnarray}
\Vert\varphi_\gamma\Vert^2_{H^m(\mathbb T)}=\sum\limits_{k=0}^m\Vert\partial_x^k\varphi_\gamma\Vert^2_{L^2(\mathbb T)}\le C\sum\limits_{k=0}^m\Vert\partial_x^{k+1}\varphi_\gamma\Vert^2_{L^2(\mathbb T)}=C\Vert\varphi_{x,\gamma}\Vert^2_{H^m(\mathbb T)}\,;\label{magg_1}\\
\Vert\phi_\gamma\Vert_{L^\infty(\mathbb T)}\le C\Vert\varphi_\gamma\Vert_{H^1(\mathbb T)}\le C\Vert\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\le C\Vert\varphi_{x,\gamma}\Vert_{H^1(\mathbb T)}\,;\label{magg_2}\\
\Vert\varphi_{0,xxx}\Vert_{L^\infty(\mathbb T)}\le C\Vert\varphi_{0,xxx}\Vert_{H^1(\mathbb T)}\le C\Vert\varphi_{0}\Vert_{H^4(\mathbb T)}\,;\label{magg_3}\\
\Vert\varphi_{0,xx}\Vert_{L^\infty(\mathbb T)}\le C\Vert\varphi_{0,xx}\Vert_{H^1(\mathbb T)}\le C\Vert\varphi_{0}\Vert_{H^3(\mathbb T)}\,.\label{magg_4}
\end{eqnarray}
Hence gathering estimates (<ref>), (<ref>)–(<ref>) we get
\begin{equation*}%\label{stima_G3_1}
\begin{split}
\Vert & G_{3,\,\gamma}(t)\Vert_{L^2(\mathbb T)}\le C\left\{\Vert\varphi_{0}\Vert_{H^4(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^m(\mathbb T)}+\Vert\varphi_{0,x}\Vert_{H^{m+2}(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^1(\mathbb T)}\right\}\,.
\end{split}
\end{equation*}
Finally, integration over $\mathbb R_t$ yields
\begin{equation}\label{stima_G3_fin}
\Vert G_{3}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}\le C\left\{\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^4(\mathbb T))}\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m}(\mathbb T))}+\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^1(\mathbb T))}\Vert\varphi_{0,x}\Vert_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}\right\}\,.
\end{equation}
The estimate of $G_4:=\left(\mathbb H\left[\left[\partial_x^m\,;\,\phi_{0,xx}\right]\phi\right]\right)_x$: for fixed $t\in\mathbb R$ we compute once again
\begin{equation*}%\label{stima_G4_0}
\begin{split}
\Vert & G_{4,\,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert \left(\mathbb H\left[\left[\partial_x^m\,;\,\phi_{0,xx}\right]\phi_\gamma\right]\right)_x\Vert_{L^2(\mathbb T)}=\Vert\left(\left[\partial_x^m\,;\,\phi_{0,xx}\right]\phi_\gamma\right)_x \Vert_{L^2(\mathbb T)}\\
&\le\Vert\left[\partial_x^m\,;\,\phi_{0,xxx}\right]\phi_\gamma\Vert_{L^2(\mathbb T)}+\Vert \left[\partial_x^m\,;\,\phi_{0,xx}\right]\phi_{x, \gamma}\Vert_{L^2(\mathbb T)}\,.
\end{split}
\end{equation*}
Then we argue as for $G_3$ (where the derivatives of $\varphi_0$ are replaced by the same derivatives of $\phi_0$) to get
\begin{equation}\label{stima_G4_fin}
\Vert G_{4}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}\le C\left\{\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^4(\mathbb T))}\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m}(\mathbb T))}+\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^1(\mathbb T))}\Vert\varphi_{0,x}\Vert_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}\right\}\,.
\end{equation}
The estimate of $G_5:=\left(-\left[\left[\partial_x^m\,;\,\phi_0\right]\,;\,\mathbb H\right]\phi_{xx}\right)_x$: for fixed $t\in\mathbb R$ we compute
\begin{equation}\label{stima_G5_0}
\Vert G_{5,\gamma}(t)\Vert_{L^2(\mathbb T)}\le \left\Vert\left[\left[\partial_x^m\,;\,\phi_{0,x}\right]\,;\,\mathbb H\right]\phi_{xx,\gamma}\right\Vert_{L^2(\mathbb T)}+\left\Vert\left[\left[\partial_x^m\,;\,\phi_0\right]\,;\,\mathbb H\right]\phi_{xxx,\gamma}\right\Vert_{L^2(\mathbb T)}\,.
\end{equation}
Now we treat separately the two $L^2-$norms in the right-hand side above. As for the first norm, we expand the commutator with $\mathbb H$ and use the commutator estimate (<ref>) to get
\begin{equation}\label{stima_G5_norma1}
\begin{split}
\left\Vert\left[\left[\partial_x^m\,;\,\phi_{0,x}\right]\,;\,\mathbb H\right]\phi_{xx,\gamma}\right\Vert_{L^2(\mathbb T)} & \le \Vert \left[\partial_x^m\,;\,\phi_{0,x}\right]\mathbb H\left[\phi_{xx,\gamma}\right]\Vert_{L^2(\mathbb T)}+\Vert\mathbb H\left[\left[\partial_x^m\,;\,\phi_{0,x}\right]\phi_{xx,\gamma}\right]\Vert_{L^2(\mathbb T)}\\
& =\Vert \left[\partial_x^m\,;\,\phi_{0,x}\right]\varphi_{xx,\gamma}\Vert_{L^2(\mathbb T)}+\Vert\left[\partial_x^m\,;\,\phi_{0,x}\right]\phi_{xx,\gamma}\Vert_{L^2(\mathbb T)}\\
& \le C\left\{\Vert\phi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\varphi_{xx,\gamma}\Vert_{H^{m-1}(\mathbb T)}+\Vert\phi_{0,x}\Vert_{H^{m}(\mathbb T)}\Vert\varphi_{xx,\gamma}\Vert_{L^\infty(\mathbb T)}\right.\\
&\qquad\left. +\Vert\phi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{xx,\gamma}\Vert_{H^{m-1}(\mathbb T)}+\Vert\phi_{0,x}\Vert_{H^{m}(\mathbb T)}\Vert\phi_{xx,\gamma}\Vert_{L^\infty(\mathbb T)} \right\}\\
&\le C\left\{\Vert\phi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}+\Vert\phi_{0,x}\Vert_{H^{m}(\mathbb T)}\Vert\varphi_{xx,\gamma}\Vert_{L^\infty(\mathbb T)}\right.\\
&\qquad\left. +\Vert\phi_{0,x}\Vert_{H^{m}(\mathbb T)}\Vert\phi_{xx,\gamma}\Vert_{L^\infty(\mathbb T)}\right\}\\
&\le C\left\{\Vert\varphi_{0}\Vert_{H^3(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}+\Vert\varphi_{0,x}\Vert_{H^{m}(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^2(\mathbb T)}\right\}\,.
\end{split}
\end{equation}
Concerning the second $L^2-$norm in the right-hand side of (<ref>), we use Lemma <ref> to get
\begin{equation}\label{stima_G5_norma2}
\left\Vert\left[\left[\partial_x^m\,;\,\phi_0\right]\,;\,\mathbb H\right]\phi_{xxx,\gamma}\right\Vert_{L^2(\mathbb T)}\le C\Vert\phi_{0,x}\Vert_{H^m(\mathbb T)}\Vert\phi_{xx,\gamma}\Vert_{H^1(\mathbb T)}\le C\Vert\varphi_{0,x}\Vert_{H^m(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^2(\mathbb T)}.
\end{equation}
Summing up (<ref>), (<ref>) we then obtain
\begin{equation*}%\label{stima_G5_1}
\Vert G_{5,\gamma}(t)\Vert_{L^2(\mathbb T)}\le C\left\{\Vert\varphi_{0}\Vert_{H^3(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}+\Vert\varphi_{0,x}\Vert_{H^{m}(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^2(\mathbb T)}\right\}
\end{equation*}
and integrating in $\mathbb R_t$
\begin{equation}\label{stima_G5_fin}
\Vert G_{5}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}\le C\left\{\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m}(\mathbb T))}+\Vert\varphi_{0,x}\Vert_{L^\infty(\mathbb R;H^{m}(\mathbb T))}\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R; H^2(\mathbb T))}\right\}\,.
\end{equation}
Using (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) to estimate the right-hand side of (<ref>) we have
\begin{equation}\label{stima_apriori_2_1}
\begin{split}
\gamma &\left\{\Vert\partial_x^m\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}+\Vert\partial_x^m\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\le \frac{C}{\gamma}\left\{\Vert\varphi_{0}\Vert^2_{L^\infty(\mathbb R;H^4(\mathbb T))}\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{m}(\mathbb T))}\right.\\
&\qquad \left.+\Vert\varphi_{0,x}\Vert^2_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{2}(\mathbb T))}+\Vert \partial_x^m g\Vert^2_{L^2_\gamma(\mathbb R\times\mathbb T)}\right\}\,,\qquad\forall\,\gamma\ge\gamma_0\,.
\end{split}
\end{equation}
Because of (<ref>) and $\varphi$ has spatial zero mean, the left-hand side of (<ref>) is equivalent, uniformly in $\gamma$, to $\gamma\left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R; H^m(\mathbb T))}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\right\}$, hence the estimate can be restated as
\begin{equation*}
\begin{split}
\gamma & \left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R; H^m(\mathbb T))}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\right\}\le \frac{C}{\gamma}\left\{\Vert\varphi_{0}\Vert^2_{L^\infty(\mathbb R;H^4(\mathbb T))}\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{m}(\mathbb T))}\right.\\
&\qquad\left.+\Vert\varphi_{0,x}\Vert^2_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{2}(\mathbb T))}+\Vert g\Vert^2_{L^2_\gamma(\mathbb R; H^m_\gamma(\mathbb T))}\right\}\,,\qquad\forall\,\gamma\ge\gamma_0\,.
\end{split}
\end{equation*}
Then the first term in the right-hand side can be absorbed into the left-hand side, provided that $\gamma$ is larger than some positive $\widetilde{\gamma}_0=\widetilde{\gamma}_0(\Vert\varphi_0\Vert_{L^\infty(\mathbb R; H^4(\mathbb T))})$, to get
\begin{equation}\label{stima_apriori_2_2}
\begin{split}
\gamma & \left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R; H^m(\mathbb T))}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\right\}\\
&\le \frac{C}{\gamma}\left\{\Vert\varphi_{0,x}\Vert^2_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{2}(\mathbb T))}+\Vert g\Vert^2_{L^2_\gamma(\mathbb R; H^m(\mathbb T))}\right\}\,,\qquad\forall\,\gamma\ge\max\{\gamma_0,\widetilde{\gamma}_0\}\,.
\end{split}
\end{equation}
Now we write (<ref>) for $m=2$
\begin{equation*}%\label{stima_apriori_2_3}
\begin{split}
\gamma & \left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}\right\}\\
&\le \frac{C}{\gamma}\left\{\Vert\varphi_{0,x}\Vert^2_{L^\infty(\mathbb R;H^{4}(\mathbb T))}\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{2}(\mathbb T))}+\Vert g\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))}\right\}\\
&\le \frac{C}{\gamma}\left\{\Vert\varphi_{0}\Vert^2_{L^\infty(\mathbb R;H^{5}(\mathbb T))}\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{2}(\mathbb T))}+\Vert g\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))}\right\}
\end{split}
\end{equation*}
and again we absorb into the left-hand side the first term in the right-hand side above for $\gamma\ge\widetilde{\gamma_1}=\widetilde{\gamma}_1(\Vert\varphi_{0}\Vert_{L^\infty(\mathbb R;H^{5}(\mathbb T))})$ to find that
\begin{equation}\label{stima_apriori_2_4}
\begin{split}
\gamma & \left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}\right\}\le \frac{C}{\gamma}\Vert g\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))}\,.
\end{split}
\end{equation}
Then we use (<ref>) to estimate the norm $\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}$ in the right-hand side of (<ref>) to find that
\begin{equation*}%\label{stima_apriori_2_4.1}
\begin{split}
\gamma & \left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R; H^m(\mathbb T))}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\right\}\\
&\le \frac{C}{\gamma}\left\{\Vert\varphi_{0,x}\Vert^2_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}\Vert g\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))} +\Vert g\Vert^2_{L^2_\gamma(\mathbb R; H^m(\mathbb T))}\right\}\,,\quad\forall\,\gamma\ge\max\{\gamma_0,\widetilde{\gamma}_0,\widetilde{\gamma}_1\}=:\gamma_1\,.
\end{split}
\end{equation*}
Under the same assumptions of Proposition <ref> we obtain an estimate of the $L^2_\gamma(\mathbb R; H^{m-1}(\mathbb T))$-norm of $\varphi^\prime_{tt}$, namely we prove the following
Under the same assumptions of Proposition <ref> there exists a positive constant $C_1=C_1\left(\mu,\delta, m\right)$ such that for all sufficiently smooth functions $\varphi^\prime:\mathbb R\times\mathbb T\rightarrow\mathbb R$, with zero spatial mean, and for all $\gamma\ge 1$ the following estimate holds true
\begin{equation}\label{stima_dm-1phitt}
\begin{split}
&\Vert\varphi^\prime_{tt}\Vert_{L^2_\gamma(\mathbb R;H^{m-1}(\mathbb T))}\le C_1\left\{\Vert\varphi^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}+\Vert\varphi_0\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\Vert\varphi^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\right.\\
&\qquad\qquad\qquad\quad \left.+\Vert\varphi_{0,x}\Vert_{L^\infty(\mathbb R;H^{m}(\mathbb T))}\Vert\varphi^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}+\Vert g\Vert_{L^2_\gamma(\mathbb R;H^{m-1}(\mathbb T))}\right\}\,,
\end{split}
\end{equation}
where $g:=\mathbb{L}^\prime[\varphi_0]\varphi^\prime$.
As in the proof of Proposition <ref>, to simplify the notation we drop the superscript ${}^\prime$ in the unknown function $\varphi^\prime$. Because of (<ref>), it is enough providing an estimate of the $L^2_\gamma-$norm of $\partial^{m-1}_x\varphi_{tt}$; this can be obtained by applying the differential operator $\partial_x^{m-1}$ to the linearized equation (<ref>) and solving the resulting equation for $\partial^{m-1}_x\varphi_{tt}$.
Applying $\partial^{m-1}_x$ to (<ref>) we get
\begin{equation}\label{eq_dm-1phitt}
\begin{split}
\partial^{m-1}_x&\varphi_{tt}=\mu\partial^m_x\varphi_x-2\phi_{0,x}\partial_x^m\varphi_x-2\left[\partial^{m-1}_x\,;\,\phi_{0,x}\right]\varphi_{xx}+2\left[\mathbb H\,;\,\phi_{0,x}\right]\partial_x^m\phi_x\\
&+2\left[\partial^{m-1}_x\,;\,\left[\mathbb H\,;\,\phi_{0,x}\right]\right]\phi_{xx}+2\mathbb H\left[\phi_{0,xx}\partial_x^{m-1}\phi_x+\left[\partial_x^{m-1}\,;\,\phi_{0,xx}\right]\phi_x\right]\\
&-\left(\left[\phi\,;\,\mathbb H\right]\partial^m_x\phi_{0,x}+\left[\partial^{m-1}_x\,;\,\left[\phi\,;\,\mathbb H\right]\right]\phi_{0,xx}+\left[\phi_0\,;\,\mathbb H\right]\partial_x^m\phi_x+\left[\partial_x^{m-1}\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\phi_{xx}\right)_x\\
&+\partial_x^{m-1}g=\sum\limits_{j=1}^{11}G'_j+\partial^{m-1}_x g\,.
\end{split}
\end{equation}
In order to obtain the desired estimate, we provide a suitable bound of each term in the right-hand side above. Let $t\in\mathbb R$ and $\gamma\ge 1$ be arbitrarily fixed. In all the following estimates the $L^2-$continuity of $\mathbb H$ is used.
Estimate of $G'_1$:
\begin{equation}\label{stima_1}
\Vert G'_{1,\gamma}(t)\Vert_{L^2(\mathbb T)}\le\vert\mu\vert\Vert\partial^m_x\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\le \vert\mu\vert\Vert\varphi_{x,\gamma}\Vert_{H^m(\mathbb T)}\,.
\end{equation}
Estimate of $G'_2$: Hölder's inequality and Sobolev's imbedding $H^1(\mathbb T)\hookrightarrow L^\infty(\mathbb T)$ yield
\begin{equation}\label{stima_2}
\begin{split}
\Vert & G'_{2,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert -2\phi_{0,x}\partial_x^m\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\le C\Vert\phi_{0,x}\Vert_{L^\infty(\mathbb T)}\Vert\partial^m_x\varphi_{x,\gamma}\Vert_{L^2(\mathbb T)}\\
&\le C\Vert\varphi_{0,x}\Vert_{H^1(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^m(\mathbb T)}\le C\Vert\varphi_0\Vert_{H^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^m(\mathbb T)}\,.
\end{split}
\end{equation}
Estimate of $G'_3$: estimate (<ref>) and Sobolev's imbedding $H^1(\mathbb T)\hookrightarrow L^\infty(\mathbb T)$ yield
\begin{equation}\label{stima_3}
\begin{split}
\Vert & G'_{3,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert -2\left[\partial_x^{m-1}\,;\,\phi_{0,x}\right]\varphi_{xx,\gamma}\Vert_{L^2(\mathbb T)}\\
&\le C\left\{\Vert\phi_{0,x}\Vert_{L^\infty(\mathbb T)}\Vert\varphi_{xx,\gamma}\Vert_{H^{m-1}(\mathbb T)}+\Vert\phi_{0,x}\Vert_{H^{m-1}(\mathbb T)}\Vert\varphi_{xx,\gamma}\Vert_{L^\infty(\mathbb T)}\right\}\\
&\le C\left\{\Vert\varphi_{0,x}\Vert_{H^1(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}+\Vert\varphi_{0,x}\Vert_{H^{m-1}(\mathbb T)}\Vert\varphi_{xx,\gamma}\Vert_{H^1(\mathbb T)}\right\}\\
&\le C\left\{\Vert\varphi_{0}\Vert_{H^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}+\Vert\varphi_{0,x}\Vert_{H^{m-1}(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^2(\mathbb T)}\right\}\,.
\end{split}
\end{equation}
Estimate of $G'_4$: applying estimate (<ref>) gives
\begin{equation}\label{stima_4}
\begin{split}
\Vert & G'_{4,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert 2\left[\mathbb H\,;\,\phi_{0,x}\right]\partial^m_x\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\le C\Vert\phi_{0,x}\Vert_{H^1(\mathbb T)}\Vert\partial^m_x\phi_{x,\gamma}\Vert_{L^{2}(\mathbb T)}\\
&\le C\Vert\varphi_{0}\Vert_{H^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}\,.
\end{split}
\end{equation}
Estimate of $G'_5$: in view of Lemma <ref> and using that $\partial^{m-1}_x$ and $\mathbb H$ commute, we get
\begin{equation}\label{stima_5}
\begin{split}
\Vert & G'_{5,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert 2\left[\partial^{m-1}_x\,;\,\left[\mathbb H\,;\,\phi_{0,x}\right]\right]\phi_{xx,\gamma}\Vert_{L^2(\mathbb T)}=\Vert -2\left[\left[\partial^{m-1}_x\,;\,\phi_{0,x}\right]\,;\,\mathbb H\right]\phi_{xx,\gamma}\Vert_{L^2(\mathbb T)}\\
&\le C\Vert\phi_{0,xx}\Vert_{H^{m-1}(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{H^{1}(\mathbb T)}\le C\Vert\varphi_{0,x}\Vert_{H^{m}(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{1}(\mathbb T)}\,.
\end{split}
\end{equation}
Estimate of $G'_6$: Hölder's inequality and Sobolev's imbedding $H^1(\mathbb T)\hookrightarrow L^\infty(\mathbb T)$ yield
\begin{equation}\label{stima_6}
\begin{split}
\Vert & G'_{6,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert 2\mathbb H\left[\phi_{0,xx}\partial^{m-1}_x\phi_{x,\gamma}\right]\Vert_{L^2(\mathbb T)}\le 2\Vert\phi_{0,xx}\partial^{m-1}_x\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\\
&\le 2\Vert\phi_{0,xx}\Vert_{L^{\infty}(\mathbb T)}\Vert\partial^{m-1}_x\phi_{x,\gamma}\Vert_{L^{2}(\mathbb T)}\le C\Vert\varphi_{0,xx}\Vert_{H^{1}(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m-1}(\mathbb T)}\\
&\le C\Vert\varphi_{0}\Vert_{H^{3}(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m-1}(\mathbb T)}\,.
\end{split}
\end{equation}
Estimate of $G'_7$: estimate (<ref>) and Sobolev's imbedding $H^1(\mathbb T)\hookrightarrow L^\infty(\mathbb T)$ yield
\begin{equation}\label{stima_7}
\begin{split}
\Vert & G'_{7,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert 2\mathbb H\left[\left[\partial_x^{m-1}\,;\,\phi_{0,xx}\right]\phi_{x,\gamma}\right]\Vert_{L^2(\mathbb T)}\le 2\Vert \left[\partial_x^{m-1}\,;\,\phi_{0,xx}\right]\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\\
&\le C\left\{\Vert\phi_{0,xx}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{H^{m-1}(\mathbb T)}+\Vert\phi_{0,xx}\Vert_{H^{m-1}(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{L^\infty(\mathbb T)}\right\}\\
&\le C\left\{\Vert\varphi_{0,xx}\Vert_{H^1(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m-1}(\mathbb T)}+\Vert\varphi_{0,xx}\Vert_{H^{m-1}(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^1(\mathbb T)}\right\}\\
&\le C\left\{\Vert\varphi_{0}\Vert_{H^3(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m-1}(\mathbb T)}+\Vert\varphi_{0,x}\Vert_{H^{m}(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^1(\mathbb T)}\right\}\,.
\end{split}
\end{equation}
Estimate of $G'_8$: Leibniz's formula and estimates (<ref>), (<ref>) give
\begin{equation}\label{stima_8}
\begin{split}
\Vert & G'_{8,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert -\left(\left[\phi_\gamma\,;\,\mathbb H\right]\partial_x^m\phi_{0,x}\right)_x\Vert_{L^2(\mathbb T)}\\
&\le \Vert \left[\phi_{x,\gamma}\,;\,\mathbb H\right]\partial_x^m\phi_{0,x}\Vert_{L^2(\mathbb T)}+\Vert \left[\phi_\gamma\,;\,\mathbb H\right]\partial_x\partial_x^{m}\phi_{0,x}\Vert_{L^2(\mathbb T)}\\
&\le C\left\{\Vert\phi_{x,\gamma}\Vert_{H^1(\mathbb T)}\Vert\partial_x^m\phi_{0,x}\Vert_{L^{2}(\mathbb T)}+\Vert\phi_{x,\gamma}\Vert_{H^{1}(\mathbb T)}\Vert\partial_x^m\phi_{0,x}\Vert_{L^2(\mathbb T)}\right\}\\
&\le C\Vert\varphi_{x,\gamma}\Vert_{H^1(\mathbb T)}\Vert\varphi_{0,x}\Vert_{H^{m}(\mathbb T)}\,.
\end{split}
\end{equation}
Estimate of $G'_9$: by Leibniz's formula, Lemma <ref> and using that $\partial^{m-1}_x$ and $\mathbb H$ commute, we get
\begin{equation}\label{stima_9}
\begin{split}
\Vert & G'_{9,\gamma}(t)\Vert_{L^2(\mathbb T)}=\left\Vert -\left(\left[\partial^{m-1}_x\,;\,\left[\phi_\gamma\,;\,\mathbb H\right]\right]\phi_{0,xx}\right)_x\right\Vert_{L^2(\mathbb T)}=\left\Vert -\left(\left[\left[\partial_x^{m-1}\,;\,\phi_{\gamma}\right]\,;\,\mathbb H\right]\phi_{0,xx}\right)_x \right\Vert_{L^2(\mathbb T)}\\
&\le \left\Vert\left[\left[\partial_x^{m-1}\,;\,\phi_{x,\gamma}\right]\,;\,\mathbb H\right]\phi_{0,xx} \right\Vert_{L^2(\mathbb T)}+\left\Vert \left[\left[\partial_x^{m-1}\,;\,\phi_{\gamma}\right]\,;\,\mathbb H\right]\phi_{0,xxx} \right\Vert_{L^2(\mathbb T)}\\
&\le C\Vert\phi_{xx,\gamma}\Vert_{H^{m-1}(\mathbb T)}\Vert\phi_{0,x}\Vert_{H^1(\mathbb T)}+C\Vert\phi_{x,\gamma}\Vert_{H^{m-1}(\mathbb T)}\Vert\phi_{0,xx}\Vert_{H^1(\mathbb T)}\le C\Vert\varphi_{x,\gamma}\Vert_{H^m(\mathbb T)}\Vert\varphi_0\Vert_{H^3(\mathbb T)}\,.
\end{split}
\end{equation}
Estimate of $G'_{10}$: Leibniz's formula and estimates (<ref>), (<ref>) give
\begin{equation}\label{stima_10}
\begin{split}
\Vert & G'_{10,\gamma}(t)\Vert_{L^2(\mathbb T)}=\Vert -\left(\left[\phi_0\,;\,\mathbb H\right]\partial_x^m\phi_{x,\gamma}\right)_x\Vert_{L^2(\mathbb T)}\\
&\le \Vert \left[\phi_{0,x}\,;\,\mathbb H\right]\partial_x^m\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}+\Vert \left[\phi_0\,;\,\mathbb H\right]\partial_x\partial_x^{m}\phi_{x,\gamma}\Vert_{L^2(\mathbb T)}\\
&\le C\Vert\phi_{0,x}\Vert_{H^1(\mathbb T)}\Vert\partial_x^m\phi_{x,\gamma}\Vert_{L^{2}(\mathbb T)}\le C\Vert\varphi_{0}\Vert_{H^2(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^{m}(\mathbb T)}\,.
\end{split}
\end{equation}
Estimate of $G'_{11}$: by Leibniz's formula, Lemma <ref> and using that $\partial^{m-1}_x$ and $\mathbb H$ commute, we get
\begin{equation}\label{stima_11}
\begin{split}
\Vert & G'_{11,\gamma}(t)\Vert_{L^2(\mathbb T)}=\left\Vert -\left(\left[\partial^{m-1}_x\,;\,\left[\phi_0\,;\,\mathbb H\right]\right]\phi_{xx,\gamma}\right)_x\right\Vert_{L^2(\mathbb T)}=\left\Vert -\left(\left[\left[\partial_x^{m-1}\,;\,\phi_{0}\right]\,;\,\mathbb H\right]\phi_{xx,\gamma}\right)_x \right\Vert_{L^2(\mathbb T)}\\
&\le \left\Vert\left[\left[\partial_x^{m-1}\,;\,\phi_{0,x}\right]\,;\,\mathbb H\right]\phi_{xx,\gamma} \right\Vert_{L^2(\mathbb T)}+\left\Vert \left[\left[\partial_x^{m-1}\,;\,\phi_{0}\right]\,;\,\mathbb H\right]\phi_{xxx,\gamma} \right\Vert_{L^2(\mathbb T)}\\
&\le C\Vert\phi_{0,xx}\Vert_{H^{m-1}(\mathbb T)}\Vert\phi_{x,\gamma}\Vert_{H^1(\mathbb T)}+C\Vert\phi_{0,x}\Vert_{H^{m-1}(\mathbb T)}\Vert\phi_{xx,\gamma}\Vert_{H^1(\mathbb T)}\le C\Vert\varphi_{0,x}\Vert_{H^m(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^2(\mathbb T)}\,.
\end{split}
\end{equation}
Using (<ref>)–(<ref>) above to estimate the right-hand side of (<ref>) yields
\begin{equation*}%\label{stima_dm-1phitt_0}
\begin{split}
&\Vert\partial^{m-1}_x\varphi_{tt,\gamma}\Vert_{L^2(\mathbb T)}\le\vert\mu\vert\Vert\varphi_{x,\gamma}\Vert_{H^m(\mathbb T)}+C\left\{\Vert\varphi_0\Vert_{H^3(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^m(\mathbb T)}\right.\\
&\qquad\qquad\qquad\quad \left.+\Vert\varphi_{0,x}\Vert_{H^{m}(\mathbb T)}\Vert\varphi_{x,\gamma}\Vert_{H^2(\mathbb T)}\right\}+\Vert \partial^{m-1}_xg_\gamma\Vert_{L^{2}(\mathbb T)}\,,
\end{split}
\end{equation*}
then integration in time and Hölder's inequality give
\begin{equation*}%\label{stima_dm-1phitt_1}
\begin{split}
&\Vert\partial^{m-1}_x\varphi_{tt}\Vert_{L^2_\gamma(\mathbb R\times\mathbb T)}\le\vert\mu\vert\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}+C\left\{\Vert\varphi_0\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\right.\\
&\qquad\qquad\qquad\quad \left.+\Vert\varphi_{0,x}\Vert_{L^\infty(\mathbb R;H^{m}(\mathbb T))}\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}\right\}+\Vert g\Vert_{L^2_\gamma(\mathbb R;H^{m-1}(\mathbb T))}\,,
\end{split}
\end{equation*}
which provides the desired result, in view of (<ref>).
§ ESTIMATE OF THE SECOND ORDER DERIVATIVE OF $\MATHBB{L}$
In order to apply the Nash-Moser method, we need to have a suitable estimate for the second order derivative of the nonlinear operator (<ref>) at a given state $\varphi_0$. From (<ref>) one computes
\begin{equation*}%\label{operatore_bil}
\begin{split}
\mathbb L^{\prime\prime}&[\varphi_0](\varphi, \psi):=\frac{d}{d\varepsilon}\left\{\mathbb L^\prime[\varphi_0+\varepsilon\psi]\varphi\right\}_{\vert\varepsilon=0}\\
&=\frac{d}{d\varepsilon}\big\{\varphi_{tt}-\mu\varphi_{xx}-\big(2\mathbb H\left[(\phi_{0,x}+\varepsilon\Psi_x)\phi_x\right]-\left[\phi\,;\,\mathbb H\right]\left(\phi_{0,xx}+\varepsilon\Psi_{xx}\right)-\left[\phi_0+\varepsilon\Psi\,;\,\mathbb H\right]\phi_{xx}\big)_x\big\}_{\vert\varepsilon=0}\\
&=\big(-2\mathbb H\left[\Psi_x\phi_x\right]+\left[\phi\,;\,\mathbb H\right]\Psi_{xx}+\left[\Psi\,;\,\mathbb H\right]\phi_{xx}\big)_x\,,
\end{split}
\end{equation*}
where it is set
\begin{equation*}
\phi_0:=\mathbb H[\varphi_0]\,,\quad\phi:=\mathbb H[\varphi]\,,\quad \Psi:=\mathbb H[\psi]\,.
\end{equation*}
We look now for an estimate of the second derivative $\mathbb L^{\prime\prime}[\varphi_0](\varphi, \psi)$ in the space $L^2_\gamma(\mathbb R; H^m(\mathbb T))$, with given integer $m$ and $\gamma$ sufficiently large. To this end we need to provide a suitable estimate of each term involved in the expression of $\mathbb L^{\prime\prime}[\varphi_0](\varphi, \psi)$ above.
Estimate for the 1st term $\left(-2\mathbb H\left[\Psi_x\phi_x\right]\right)_x$: for fixed $t\in\mathbb R$ and $\gamma\ge 1$, we first commute $\mathbb H$ and $\partial_x$, then we use the continuity of $\mathbb H$ in $H^m(\mathbb T)$, Leibniz's formula, Lemma <ref> (estimate (<ref>)) and the Sobolev imbedding $H^1(\mathbb T)\hookrightarrow L^\infty(\mathbb T)$ to get
\begin{equation}\label{stima_punt_1}
\begin{split}
\Vert &\left(-2\mathbb H\left[\Psi_x\phi_x\right]\right)_x\Vert_{H^m(\mathbb T)}\le C\Vert\left(\Psi_x\phi_x\right)_x\Vert_{H^m(\mathbb T)}\le C\left\{\Vert\Psi_{xx}\phi_x\Vert_{H^m(\mathbb T)}+\Vert\Psi_{x}\phi_{xx}\Vert_{H^m(\mathbb T)}\right\}\\
&\le C\left\{\Vert\Psi_{xx}\Vert_{H^m(\mathbb T)}\Vert\phi_{x}\Vert_{L^\infty(\mathbb T)}+\Vert\Psi_{xx}\Vert_{L^\infty(\mathbb T)}\Vert\phi_{x}\Vert_{H^m(\mathbb T)}+\Vert\phi_{xx}\Vert_{H^m(\mathbb T)}\Vert\Psi_{x}\Vert_{L^\infty(\mathbb T)}\right.\\
&\qquad\left. +\Vert\phi_{xx}\Vert_{L^\infty(\mathbb T)}\Vert\Psi_{x}\Vert_{H^m(\mathbb T)}\right\}\\
%&\le C\left\{\Vert\psi_{xx}\Vert_{H^{m}(\mathbb T)}\Vert\varphi_{x}\Vert_{H^1(\mathbb T)}+\Vert\psi_{xx}\Vert_{H^1(\mathbb T)}\Vert\varphi_{x}\Vert_{H^m(\mathbb T)}+\Vert\varphi_{xx}\Vert_{H^m(\mathbb T)}\Vert\psi_{x}\Vert_{H^1(\mathbb T)}\right.\\
%&\qquad\left. +\Vert\varphi_{xx}\Vert_{H^1(\mathbb T)}\Vert\psi_{x}\Vert_{H^m(\mathbb T)}\right\}\\
&\le C\left\{\Vert\psi_{x}\Vert_{H^{m+1}(\mathbb T)}\Vert\varphi_{x}\Vert_{H^1(\mathbb T)}+\Vert\psi_{x}\Vert_{H^2(\mathbb T)}\Vert\varphi_{x}\Vert_{H^m(\mathbb T)}+\Vert\varphi_{x}\Vert_{H^{m+1}(\mathbb T)}\Vert\psi_{x}\Vert_{H^1(\mathbb T)}\right.\\
&\qquad\left. +\Vert\varphi_{x}\Vert_{H^2(\mathbb T)}\Vert\psi_{x}\Vert_{H^m(\mathbb T)}\right\}\\
&\le C\left\{\Vert\psi_{x}\Vert_{H^{m+1}(\mathbb T)}\Vert\varphi_{x}\Vert_{H^2(\mathbb T)}+\Vert\psi_{x}\Vert_{H^2(\mathbb T)}\Vert\varphi_{x}\Vert_{H^{m+1}(\mathbb T)}\right\}\,.
\end{split}
\end{equation}
Then we multiply the square of the $H^m(\mathbb T)-$norm by $e^{-2\gamma t}$, integrate over $\mathbb R_t$ and use Hölder's inequality to find
\begin{equation}\label{stima_2der_1}
\begin{split}
\Vert &\left(-2\mathbb H\left[\Psi_x\phi_x\right]\right)_x\Vert_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\\
&\le C\left\{\Vert\psi_{x}\Vert_{L^2_\gamma(\mathbb R; H^{m+1}(\mathbb T))}\Vert\varphi_{x}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}+\Vert\psi_{x}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}\right\}\,.
\end{split}
\end{equation}
Estimates for the 2nd and the 3rd terms $\left(\left[\phi\,;\,\mathbb H\right]\Psi_{xx}\right)_x$, $\left(\left[\Psi\,;\,\mathbb H\right]\phi_{xx}\right)_x$: since the role of $\varphi$ and $\psi$ in the two terms before is exchanged, it is enough to exhibit the estimate for one of them. We compute the estimate for the first one of the two. In view of Lemma <ref> and the Sobolev continuity of $\mathbb H$, we find for fixed $t\in\mathbb R$:
\begin{equation}\label{stima_punt_2}
\begin{split}
\Vert &\left(\left[\phi\,;\,\mathbb H\right]\Psi_{xx}\right)_x\Vert_{H^m(\mathbb T)}\le C\Vert \left[\phi\,;\,\mathbb H\right]\Psi_{xx}\Vert_{H^{m+1}(\mathbb T)}\\
&\le C\Vert\partial^{m+1}_x\phi\Vert_{L^2(\mathbb T)}\Vert\Psi_{xx}\Vert_{H^1(\mathbb T)}\le C\Vert\varphi_{x}\Vert_{H^m(\mathbb T)}\Vert\psi_x\Vert_{H^2(\mathbb T)}\,.
\end{split}
\end{equation}
Then we multiply the square of the $H^m(\mathbb T)-$norm by $e^{-2\gamma t}$, integrate over $\mathbb R_t$ and use Hölder's inequality to find
\begin{equation}\label{stima_2der_2}
\Vert\left(\left[\phi\,;\,\mathbb H\right]\Psi_{xx}\right)_x\Vert_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\le C\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\Vert\psi_x\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}\,.
\end{equation}
The same arguments can be applied to estimate the third term $\left(\left[\Psi\,;\,\mathbb H\right]\phi_{xx}\right)_x$, where the role of $\varphi$ and $\psi$ is changed; thus
\begin{equation}\label{stima_2der_3}
\Vert\left(\left[\Psi\,;\,\mathbb H\right]\phi_{xx}\right)_x\Vert_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\le C\Vert\psi_{x}\Vert_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\Vert\varphi_x\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}\,.
\end{equation}
Collecting (<ref>), (<ref>), (<ref>) we have proved the following proposition.
For all $m\geq 1$ there exists a constant $C_m>0$ such that, for all sufficiently smooth $\varphi$, $\psi$ and all $\gamma\geq 1$, the following estimate holds true
\begin{equation}\label{stima_der2}
\begin{split}
\Vert\mathbb L^{\prime\prime}[\varphi_0]&(\varphi, \psi)\Vert_{L^2_{\gamma}(\mathbb R;H^m(\mathbb T))}\\
&\leq C_m\left\{\Vert\varphi_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}\Vert\psi_x\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))} + \Vert\psi_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}\Vert\varphi_x\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}\right\}\,.
\end{split}
\end{equation}
Since (<ref>) is obtained from the point-wise in $t$ estimates (<ref>), (<ref>), by repeating the same arguments above where the integration on $\mathbb R_t$ is replaced by integration on $(-\infty, T]$ (for given $T>0$), we get that (<ref>) can be restated in the framework of the spaces $L^2_\gamma(-\infty,T;H^m(\mathbb T))$, $L^\infty(-\infty,T; H^m(\mathbb T))$; namely one has that
\begin{equation}\label{stima_der2_T}
\begin{split}
\Vert\mathbb L^{\prime\prime}[\varphi_0]&(\varphi, \psi)\Vert_{L^2_{\gamma}(-\infty,T;H^m(\mathbb T))}\\
&\leq C_m\left\{\Vert\varphi_{x}\Vert_{L^2_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\Vert\psi_x\Vert_{L^\infty(-\infty,T;H^2(\mathbb T))} + \Vert\psi_{x}\Vert_{L^2_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\Vert\varphi_x\Vert_{L^\infty(-\infty,T;H^2(\mathbb T))}\right\}\,,
\end{split}
\end{equation}
where $C_m>0$ is independent of $T$. Moreover, using suitable extension arguments (see Subsection <ref>), also estimates (<ref>), (<ref>) can be restated on $(-\infty, T]$ (for given $T>0$).
§ THE NONLINEAR PROBLEM
In this section we prove the existence of a solution to the nonlinear Cauchy problem
\begin{equation}\label{cp}
\begin{cases}
\varphi_{tt}-\mu\varphi_{xx}=\left(\mathbb H[\phi^2_x]-\left[\phi\,;\,\mathbb H\right]\phi_{xx}\right)_{x}\,, \quad {\rm in}\,\, [0,T]\times\mathbb{T}, \qquad\phi=\mathbb H[\varphi]\,,\\
\varphi(x,0)=\varphi^{(0)}(x) \quad {\rm in}\,\, \mathbb{T},\\
\varphi_t(x,0)=\varphi^{(1)}(x) \quad {\rm in}\,\, \mathbb{T},
\end{cases}
\end{equation}
as stated in our main Theorem <ref>.
In the following, we prove the result by applying the Nash–Moser's theorem for the resolution of the nonlinear problem above, see <cit.>, <cit.> for a thorough description of the method. In the following for the explanation of the Nash–Moser's method we will refer to the setting and notation of <cit.>.
Assume that $\varphi^{(0)}\in H^{\nu+1}(\mathbb T)$, $\varphi^{(1)}\in H^{\nu}(\mathbb T)$, for a general integer $\nu\ge 0$, and the sign condition (<ref>) is satisfied.
We first consider a suitable lifting $\varphi^{a}:\mathbb R\times \mathbb{T} \rightarrow \mathbb{R}$ of the data $\varphi^{(0)}$, $\varphi^{(1)}$ such that
\begin{equation}\label{reg_phia}
\begin{split}
&\varphi^{a}\in \bigcap \limits_{k=0}^2 W^{k,\infty}(\mathbb R;H^{\nu+1-k}(\mathbb T)),\\
&\varphi^{a}_{|t=0}=\varphi^{(0)}, \quad \partial_t\varphi^{a}_{|t=0}=\varphi^{(1)}\quad {\rm in}\,\,\mathbb{T}\,, \\
&\mu- 2\mathbb{H}[\varphi^{a}]_x\geq 3\delta/4>0 \quad {\rm in}\,\,\,\,\mathbb R\times \mathbb{T}\,,\\
&\sum\limits_{k=0}^2\Vert\partial^k_t\varphi^a\Vert_{L^\infty(\mathbb R; H^{\nu+1-k}(\mathbb T))}\le C\left\{\Vert\varphi^{(0)}\Vert_{H^{\nu+1}(\mathbb T)}+\Vert\varphi^{(1)}\Vert_{H^\nu(\mathbb T)}\right\}\,,
\end{split}
\end{equation}
where $C>0$ is a suitable constant independent of $\varphi^{(0)}$, $\varphi^{(1)}$ and $\nu$.
We define $F^{a}:\mathbb R\times \mathbb{T} \rightarrow \mathbb{R}$ by setting
\begin{equation}\label{Fa}
\begin{cases}
-\mathbb L[\varphi^{a}] \quad {\rm for}\quad t>0\,,\\
0 \quad {\rm for}\quad t<0,
\end{cases}
\end{equation}
where $\mathbb L$ is the nonlinear operator (<ref>). Using the regularity of $\varphi^{a}$ we get the following result.
Let the functions $\varphi^a=\varphi^a(t,x)$, $F^a=F^a(t,x)$ be defined as in (<ref>), (<ref>) and $\nu>1$. Then $F^{a}\in L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))$ and we get the following estimate
\begin{equation}\label{stima_Fa}
\Vert F^a\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T)}\le\mathcal P(\Vert\varphi^{(0)}\Vert_{H^{\nu+1}(\mathbb T)}, \Vert\varphi^{(1)}\Vert_{H^{\nu}(\mathbb T)})\,,
\end{equation}
where $\mathcal P=\mathcal P(x,y)$ is a quadratic polynomial such that $\mathcal P(0,0)=0$.
From the regularity of $\varphi^a$ we obtain the following:
\begin{equation*}
\begin{split}
&\varphi^a\in W^{2,\infty}(\mathbb R; H^{\nu-1}(\mathbb T))\,\,\Rightarrow\,\,\partial_{tt}\varphi^a\in L^{\infty}(\mathbb R; H^{\nu-1}(\mathbb T))\\
&\varphi^a\in L^{\infty}(\mathbb R; H^{\nu+1}(\mathbb T))\,\,\Rightarrow\,\,\partial_{xx}\varphi^a\in L^{\infty}(\mathbb R; H^{\nu-1}(\mathbb T))\,;
\end{split}
\end{equation*}
moreover we compute, for $\phi^a:=\mathbb H\left[\varphi^a\right]$,
\begin{equation}\label{parte_nonlin}
\begin{split}
&\left(\mathbb H\left[\left(\phi^a_x\right)^2\right]-\left[\phi^a\,;\,\mathbb H\right]\phi^a_{xx}\right)_x=2\mathbb H\left[\phi^a_x\,\phi^a_{xx}\right]-\left[\phi^a_x\,;\,\mathbb H\right]\phi^a_{xx}-\left[\phi^a\,;\,\mathbb H\right]\phi^a_{xxx}\\
&\qquad=3\mathbb H\left[\phi^a_x\,\phi^a_{xx}\right]-\phi^a_x\,\mathbb H\left[\phi^a_{xx}\right]-\left[\phi^a\,;\,\mathbb H\right]\phi^a_{xxx}\,.
\end{split}
\end{equation}
Hence $\left(\mathbb H\left[\left(\phi^a_x\right)^2\right]-\left[\phi^a\,;\,\mathbb H\right]\phi^a_{xx}\right)_x\in L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))$ in view of the continuity of $\mathbb H$ in Sobolev spaces, the inclusion $H^\nu(\mathbb T)\cdot H^{\nu-1}(\mathbb T) \subset H^{\nu-1}(\mathbb T)$, that holds since $\nu>1$ (cf. Lemma <ref>), and Lemma <ref> (for $m=\nu-1$, $p=2$, $v=\phi^a$ and $f=\phi^a_x$). The above calculations yield that $\mathbb L(\varphi^a)$, hence $F^a$, belong to $L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))$.
As regards to the estimate (<ref>), from (<ref>), (<ref>) and (<ref>) and using the Sobolev continuity of the Hilbert transform we first get
\begin{equation}\label{stima_Fa_resto}
\begin{split}
\Vert F^a &\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}\le \Vert\mathbb L(\varphi^a)\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}\le\Vert\varphi^a_{tt}\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}+\vert\mu\vert\Vert\varphi^a_{xx}\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}\\
&+3\Vert\mathbb H\left[\phi^a_x\,\phi^a_{xx}\right]\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}+\Vert\phi^a_x\,\mathbb H\left[\phi^a_{xx}\right]\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}+\Vert\left[\phi^a\,;\,\mathbb H\right]\phi^a_{xxx}\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}\\
&\le\Vert\varphi^a_{tt}\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}+\vert\mu\vert\Vert\varphi^a\Vert_{L^\infty(\mathbb R; H^{\nu+1}(\mathbb T))}+3\Vert\phi^a_x\,\phi^a_{xx}\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}\\
&+\Vert\phi^a_x\,\mathbb \varphi^a_{xx}\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}+\Vert\left[\phi^a\,;\,\mathbb H\right]\phi^a_{xxx}\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}\,.
\end{split}
\end{equation}
For fixed $t\in\mathbb R$ from Lemma <ref> (with $s=\nu$ and $m=\nu-1$) and using once again the Sobolev continuity of $\mathbb H$, we get
\begin{equation*}
\Vert\phi^a_x(t)\,\phi^a_{xx}(t)\Vert_{H^{\nu-1}(\mathbb T)}\le C\Vert\phi^a_x(t)\Vert_{H^{\nu}(\mathbb T)}\Vert\phi^a_{xx}(t)\Vert_{H^{\nu-1}(\mathbb T)}\le C\Vert\phi^a(t)\Vert^2_{H^{\nu+1}(\mathbb T)}\le C \Vert\varphi^a(t)\Vert^2_{H^{\nu+1}(\mathbb T)}\,.
\end{equation*}
Then taking the supremum over $\mathbb R_t$ we obtain
\begin{equation}\label{stima_Fa_3}
\Vert\phi^a_x\,\phi^a_{xx}\Vert_{L^\infty(\mathbb R;H^{\nu-1}(\mathbb T))}\le C \Vert\varphi^a\Vert^2_{L^\infty(\mathbb R;H^{\nu+1}(\mathbb T))}\,.
\end{equation}
The same arguments above can be used to estimate the term $\Vert\phi^a_x\,\mathbb \varphi^a_{xx}\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}$ by
\begin{equation}\label{stima_Fa_4}
\Vert\phi^a_x\,\varphi^a_{xx}\Vert_{L^\infty(\mathbb R;H^{\nu-1}(\mathbb T))}\le C \Vert\varphi^a\Vert^2_{L^\infty(\mathbb R;H^{\nu+1}(\mathbb T))}\,.
\end{equation}
As for the subsequent commutator term $\Vert\left[\phi^a\,;\,\mathbb H\right]\phi^a_{xxx}\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}$, for fixed $t\in\mathbb R$ we apply Lemma <ref> (for $m=\nu-1$, $p=2$, $v=\phi^a$ and $f=\phi^a_x$), the imbedding $H^\nu(\mathbb T)\hookrightarrow H^1(\mathbb T)$ (as $\nu>1$) and the Sobolev continuity of $\mathbb H$ to get
\begin{equation*}
\begin{split}
\Vert\left[\phi^a(t)\,;\,\mathbb H\right]&\phi^a_{xxx}(t)\Vert_{H^{\nu-1}(\mathbb T)}\le C\Vert\partial^{\nu+1}_x\phi^a(t)\Vert_{L^2(\mathbb T)}\Vert\phi^a_x(t)\Vert_{H^1(\mathbb T)}\\
&\le C\Vert\phi^a(t)\Vert_{H^{\nu+1}(\mathbb T)}\Vert\phi^a_x(t)\Vert_{H^\nu(\mathbb T)}\le C\Vert\varphi^a(t)\Vert^2_{H^{\nu+1}(\mathbb T)}\,.
\end{split}
\end{equation*}
Again taking the supremum over $\mathbb R_t$ gives
\begin{equation}\label{stima_Fa_5}
\Vert\left[\phi^a\,;\,\mathbb H\right]\phi^a_{xxx}\Vert_{L^\infty(\mathbb R;H^{\nu-1}(\mathbb T))}\le C\Vert\varphi^a\Vert^2_{L^\infty(\mathbb R;H^{\nu+1}(\mathbb T))}\,.
\end{equation}
Gathering estimates (<ref>)-(<ref>) we obtain
\begin{equation}\label{stima_Fa_6}
\begin{split}
\Vert F^a\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}\le\Vert\varphi^a_{tt}\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}+\vert\mu\vert\Vert\varphi^a\Vert_{L^\infty(\mathbb R; H^{\nu+1}(\mathbb T))}+C\Vert\varphi^a\Vert^2_{L^\infty(\mathbb R;H^{\nu+1}(\mathbb T))}\,.
\end{split}
\end{equation}
Combining (<ref>) with $\eqref{reg_phia}_4$ we finally get
\begin{equation*}%\label{stima_Fa_fin}
\begin{split}
\Vert & F^a\Vert_{L^\infty(\mathbb R; H^{\nu-1}(\mathbb T))}\le (1+\vert\mu\vert)\left\{\Vert\varphi^{(0)}\Vert_{H^{\nu+1}(\mathbb T)}+\Vert\varphi^{(1)}\Vert_{H^\nu(\mathbb T)}\right\}\\
&+C\left\{\Vert\varphi^{(0)}\Vert_{H^{\nu+1}(\mathbb T)}+\Vert\varphi^{(1)}\Vert_{H^\nu(\mathbb T)}\right\}^2=:\mathcal P(\Vert\varphi^{(0)}\Vert_{H^{\nu+1}(\mathbb T)}, \Vert\varphi^{(1)}\Vert_{H^{\nu}(\mathbb T)})\,.
\end{split}
\end{equation*}
It is worth to remark that in view of the definition of $F^a$, for $\nu>1$ we have the following:
(i) for $t<0$, $F^a\equiv 0$ then $F^a\in L^2(-\infty, 0; H^{\nu-1}(\mathbb T))$;
(ii) for every $T>0$, $L^\infty(0,T; \mathcal X)\subset L^2(0,T; \mathcal X)$ (where $\mathcal X$ is any Banach space), hence $F^a\in L^\infty(0,T; H^{\nu-1}(\mathbb T))\subset L^2(0,T; H^{\nu-1}(\mathbb T))$.
From (i), (ii) above easily follows that $F^a\in L^2(-\infty,T; H^{\nu-1}(\mathbb T))$. Moreover for $\gamma\ge 1$ fixed we get also that $F^a\in L^2_\gamma(-\infty,T; H^{\nu-1}(\mathbb T))$; indeed from (<ref>) we compute
\begin{equation*}
\begin{split}
\Vert & F^a\Vert^2_{L^2_\gamma(-\infty,T; H^{\nu-1}(\mathbb T))}=\int_{-\infty}^T e^{-2\gamma t}\Vert F^a(t)\Vert^2_{H^{\nu-1}(\mathbb T)}\,dt=\int_{0}^T e^{-2\gamma t}\Vert F^a(t)\Vert^2_{H^{\nu-1}(\mathbb T)}\,dt\\
&\le \Vert F^a\Vert^2_{L^{\infty}(\mathbb R; H^{\nu-1}(\mathbb T))}\int_0^T e^{-2\gamma t}\,dt=\frac{1}{2\gamma}\left(1-e^{-2\gamma T}\right)\Vert F^a\Vert^2_{L^{\infty}(\mathbb R; H^{\nu-1}(\mathbb T))}\\
&\le \frac{1}{2\gamma}\left(1-e^{-2\gamma T}\right)\mathcal P^2(\Vert\varphi^{(0)}\Vert_{H^{\nu+1}(\mathbb T)}, \Vert\varphi^{(1)}\Vert_{H^{\nu}(\mathbb T)})\,.
\end{split}
\end{equation*}
The estimate above shows that for fixed $\gamma\ge 1$ we can make the norm of $F^a$ in $L^2_\gamma(-\infty,T; H^{\nu-1}(\mathbb T))$ as small as we want by choosing a suitable value of $T$ depending on the norms of the initial data $\varphi^{(0)}$, $\varphi^{(1)}$ respectively in $H^{\nu+1}(\mathbb T)$ and $H^{\nu}(\mathbb T)$.
§.§ An equivalent formulation of the problem (<ref>)
In order to solve problem (<ref>) by the Nash-Moser's theorem, it is convenient to recast this problem in an equivalent form. We first look for a solution of the nonlinear problem
\begin{eqnarray}%\label{cp1}
\mathbb L[\varphi]=0\,, \quad {\rm in}\,\, [0,T]\times\mathbb{T}\,,\label{cp1}\\
\varphi_{\vert\,t=0}=\varphi^{(0)}\,\quad {\rm in}\,\, \mathbb{T},\label{cp2}\\
\partial_t\varphi_{\vert\,t=0}=\varphi^{(1)} \quad {\rm in}\,\, \mathbb{T},\label{cp3}
\end{eqnarray}
(see (<ref>) for the definition of $\mathbb L$) in the form of a perturbation of the function $\varphi^a(t,x)$ defined in the previous section, namely
\begin{equation}\label{forma_sol}
\varphi=\varphi^a+\varphi^\prime\,,
\end{equation}
where, according to $\eqref{reg_phia}_2$, we must have
\begin{equation}\label{ic}
\varphi^\prime_{\vert\,t=0}=0\,,\quad \partial_t\varphi^\prime_{\vert\,t=0}=0\,,\quad\mbox{in}\,\,\mathbb T\,.
\end{equation}
Replacing (<ref>) into (<ref>) we find
\begin{equation}\label{equiv}
\mathbb L[\varphi^a+\varphi^\prime]=0\quad\Leftrightarrow\quad\mathcal L[\varphi^\prime]=-\mathbb L [\varphi^a]\,,\quad\mbox{in}\,\,[0,T]\times\mathbb T\,,
\end{equation}
where we have set
\begin{equation}\label{oprt_L}
\mathcal L[\varphi^\prime]:=\mathbb L[\varphi^a+\varphi^\prime]-\mathbb L [\varphi^a]\,.
\end{equation}
In agreement with the functional setting introduced in Section <ref>, it is convenient to “extend” the problem (<ref>) also for negative time; then, in view of (<ref>), we are led to solve the nonlinear problem
\begin{eqnarray}
\mathcal L[\varphi^\prime]=F^a\,,\quad\mbox{in}\,\,(-\infty,T]\times\mathbb T\,,\label{cp1_equiv}\\
\varphi^\prime=0\,,\quad\mbox{for}\,\,t<0\,.\label{cp2_equiv}
\end{eqnarray}
§.§ The functional setting
In this section we fix the functional setting where the Nash-Moser's theorem will be applied.
For $m\ge 1$, we define $X_m$ to be the space of measurable functions $\varphi:(-\infty,T]\times\mathbb T\rightarrow\mathbb R$ such that
i) $\varphi$ has zero spatial mean, i.e. $\widehat{\varphi}(0)=\displaystyle\int_{\mathbb T}\varphi\,dx=0$;
ii) $\varphi_{\vert\{t<0\}}=0$;
iii) the function $\varphi$ enjoys the following regularity assumptions
\begin{equation}\label{Xm}
\varphi\in L^2_\gamma(-\infty, T; H^{m+2}(\mathbb T))\cap H^1_\gamma(-\infty, T; H^{m+1}(\mathbb T))\cap H^2_\gamma(-\infty, T; H^m(\mathbb T))\,,
\end{equation}
where $\gamma\ge 1$ will be fixed in a suitable way later on.
In view of the equality (<ref>) the space $X_m$ can be provided with the norm
\begin{equation}\label{normaXm}
\Vert\varphi\Vert_{X_m}^2:=\Vert\varphi_{x}\Vert^2_{L^2_\gamma(-\infty, T; H^{m+1}(\mathbb T))}+\Vert\varphi_t\Vert^2_{L^2_\gamma(-\infty, T; H^{m+1}(\mathbb T))}+\Vert\varphi_{tt}\Vert^2_{L^2_\gamma(-\infty, T; H^{m}(\mathbb T))}\,.
\end{equation}
We also define $Y_m$ to be
\begin{equation*}%\label{Ym}
Y_m:=\{F: (-\infty,T]\times\mathbb T\rightarrow\mathbb R\,:\,\,F_{\vert\,\{t<0\}}=0\,,\,\,F\in L^2_\gamma(-\infty, T; H^m(\mathbb T))\}\,,
\end{equation*}
with the natural norm
\begin{equation*}%\label{normaYm}
\Vert F\Vert_{Y_m}:=\Vert F\Vert_{L^2_\gamma(-\infty, T; H^m(\mathbb T))}\,.
\end{equation*}
From Remark <ref>, for $\nu>1$ we get that $F^a\in Y_{\nu-1}$ with $Y_{\nu-1}-$norm as small as we want for $T>0$ sufficiently small, depending on the initial data $\varphi^{(0)}$, $\varphi^{(1)}$.
§.§ Nash-Moser's theorem
Now we are going to apply the Nash-Moser theorem to solve the problem (<ref>), (<ref>) in the functional framework introduced in the previous section. Our goal is to find $\varphi^\prime\in X_m$ for a suitable $m\geq 0$ satisfying (<ref>)–(<ref>).
The regularity assumptions defining the spaces $X_m$ (see (<ref>)), imply that the solution to (<ref>), (<ref>) belonging to $X_m$ for some $m\geq 0$ is such that
\begin{equation*}
\begin{split}
\varphi^\prime \in H^1_\gamma(-\infty,T;H^{m+1}(\mathbb T))\hookrightarrow C((-\infty,T]; H^{m+1}(\mathbb T))\\
\partial_t\varphi^\prime \in H^1_\gamma(-\infty,T;H^{m}(\mathbb T))\hookrightarrow C((-\infty,T]; H^{m}(\mathbb T)).
\end{split}
\end{equation*}
Hence, the condition (<ref>) together with the continuity in time of $\varphi^\prime, \partial_t\varphi^\prime$ gives that (<ref>) are satisfied, i.e. $\varphi^\prime_{|\, t=0}=0, \partial_t\varphi^\prime_{|\, t=0}=0$.
Here we closely follow the presentation of the method in <cit.>. According to <cit.> we will use the following notations
\begin{equation*}%\label{spazi_inf}
X_\infty:=\bigcap\limits_{m\ge 0}X_m\,,\quad Y_{\infty}:=\bigcap\limits_{m\geq 0}Y_m\,,\quad H^\infty(\mathbb T):=\bigcap\limits_{m\geq 0}H^m(\mathbb T)\,,
\end{equation*}
where $X_m$, $Y_m$ are the functional spaces introduced in Section <ref>, and $\{X_m\}_{m\ge 0}$, $\{Y_m\}_{m\ge 0}$ are decreasing families of Banach spaces satisfying the smoothing hypothesis, see <cit.>.
We first need to collect a number of properties of the nonlinear operator $\mathcal L$ in (<ref>), as well as its first and second derivatives at a given point $\varphi_0\in X_\infty$. In order to do so, here we assume that the initial data $\varphi^{(0)}$, $\varphi^{(1)}$ in (<ref>), (<ref>) belong to $H^\infty(\mathbb T)$, so that $\varphi^a$ has $H^\infty$-regularity in $x$ and $F^a\in Y_\infty$. Under these assumptions, we have that for every $m\ge 0$
\begin{equation*}
\mathcal L:X_m\rightarrow Y_m\,.
\end{equation*}
The operator $\mathcal L$ must satisfy all the assumptions in <cit.>, see for the reader's convenience <ref>, Theorem <ref>; more precisely we have to check that the first order differential $d\mathcal L[\varphi_0]$ and the second order differential $d^2\mathcal L[\varphi_0]$ obey suitable estimates (see the Assumptions 2.1, 2.2 in <cit.>), as long as $\varphi_0$ belongs to $U\cap X_{\infty}$, being $U$ a suitable bounded open neighborhood of $0$ in $X_{m_0}$ for some $m_0\ge 0$.
Let us observe that, for every $\varphi_0\in X_\infty$, from (<ref>) one computes
\begin{equation}\label{diffL}
\begin{split}
d\mathcal L[\varphi_0](\varphi^\prime)=\mathbb L^\prime[\varphi^a+\varphi_0]\varphi^\prime\,,\\
d^2\mathcal L[\varphi_0](\varphi^\prime, \psi^\prime)=\mathbb L^{\prime\prime}[\varphi^a+\varphi_0](\varphi^\prime, \psi^\prime)\,.
\end{split}
\end{equation}
§.§.§ Estimate for the first order differential of $\mathcal L$
In view of (<ref>), let us consider the equation
\begin{equation}\label{eq_diff}
\mathbb L^\prime[\varphi^a+\varphi_0]\varphi^\prime=g, \quad \mbox{in} \,\,(-\infty,T)\times \mathbb{T}.
\end{equation}
In order to have that the hypothesis of Nash-Moser's theorem is satisfied (see <cit.>) we need equation (<ref>) to admit a unique solution in $X_\infty$ as long as $g\in Y_\infty$ (i.e. $d\mathcal L[\varphi_0](\varphi^\prime): X_{\infty} \rightarrow Y_{\infty}$ admits a right inverse operator). This comes as a consequence of Theorem <ref>.
In order to use the tame estimate (<ref>) in Proposition <ref>, we have to consider the equation (<ref>) with the time extended to the whole real line (i.e. on $\mathbb{R}\times \mathbb{T}$). To this end, let us consider suitable extensions $\tilde g$ and $\tilde\varphi_0$ respectively of the source term $g$ and the basic state $\varphi_0$ in such a way that
\begin{equation}\label{ext_cont}
\begin{split}
&\tilde g\in L^2_\gamma(\mathbb R;H^\infty(\mathbb T)), \quad \tilde\varphi_0\in L^\infty(\mathbb R; H^\infty(\mathbb T))\,,\\
&\Vert\tilde{g}\Vert_{L^2_\gamma(\mathbb R;H^m(\mathbb T))}\le C_{m} \Vert g\Vert_{L^2_\gamma(-\infty, T;H^m(\mathbb T))}\,,\\
&\Vert\tilde{\varphi}_0\Vert_{L^\infty(\mathbb R;H^m(\mathbb T))}\le C_{m} \Vert \varphi_0\Vert_{L^\infty(-\infty, T;H^m(\mathbb T))}\,,\\
&\Vert\tilde{\varphi}_{0,x}\Vert_{L^\infty(\mathbb R;H^m(\mathbb T))}\le C_{m} \Vert \varphi_{0,x}\Vert_{L^\infty(-\infty, T;H^m(\mathbb T))}\,,\quad\forall\,m\ge 0\,,\,\,\forall\,\gamma\ge 1\,,
\end{split}
\end{equation}
where for every $m$, $C_{m}$ is some positive constant depending on $m$ and independent of $\gamma$. The constant $C_{m}$ may be chosen independently of $T$.
Let us assume that the linearized time-extended equation
\begin{equation}\label{ext_eq}
\mathbb L^\prime[\varphi^a+\tilde\varphi_0]\tilde{\varphi}^\prime=\tilde g, \quad \mbox{in} \,\,\mathbb R\times \mathbb{T}
\end{equation}
has a unique solution: we want to derive an energy estimate for it. Let $\tilde{\varphi}^\prime$ be such a solution. From Proposition <ref>, by estimate (<ref>) written for $m+1$ instead of $m$, we get that $\tilde\varphi^\prime$ of (<ref>) satisfies the estimate
\begin{equation}\label{stima_diff}
\begin{split}
\gamma & \left\{\Vert\tilde{\varphi}^\prime_{t}\Vert^2_{L^2_\gamma(\mathbb R; H^{m+1}(\mathbb T))}+\Vert\tilde{\varphi}^\prime_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}\right\}\\
&\le \frac{C}{\gamma}\left\{\Vert\varphi^a_x+\tilde{\varphi}_{0,x}\Vert^2_{L^\infty(\mathbb R;H^{m+3}(\mathbb T))}\Vert \tilde g\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))} +\Vert \tilde g\Vert^2_{L^2_\gamma(\mathbb R; H^{m+1}(\mathbb T))}\right\}\,,
\end{split}
\end{equation}
for all $\gamma\geq \gamma_1$ and $C$ and $\gamma_1$ depend increasingly and boundedly on $m,\delta$ and $\Vert\varphi^a+\tilde{\varphi}_0\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}$ and $\Vert\varphi^a_t+\tilde{\varphi}_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}$, provided that the sign condition
\begin{equation}\label{sign_phia_phi0}
\mu-2\mathbb{H}[\varphi^a+\tilde{\varphi}_0]_x\geq \delta/2\,,\quad\mbox{in}\,\,\mathbb R\times\mathbb T
\end{equation}
holds (see Proposition <ref>). By construction, the function
\begin{equation*}
\varphi^\prime:=\tilde\varphi^{\prime}_{\vert\,\,(-\infty, T]}
\end{equation*}
provides a solution to the equation (<ref>) and, from (<ref>) and (<ref>) we get
\begin{equation}\label{stima_diff_T}
\begin{split}
\gamma & \left\{\Vert\varphi^\prime_{t}\Vert^2_{L^2_\gamma(-\infty, T; H^{m+1}(\mathbb T))}+\Vert\varphi^\prime_{x}\Vert^2_{L^2_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\right\}\\
&\le \gamma\left\{\Vert\tilde{\varphi}^\prime_{t}\Vert^2_{L^2_\gamma(\mathbb R; H^{m+1}(\mathbb T))}+\Vert\tilde{\varphi}^\prime_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}\right\}\\
&\le \frac{C}{\gamma}\left\{\Vert\varphi^a_x+\tilde{\varphi}_{0,x}\Vert^2_{L^\infty(\mathbb R;H^{m+3}(\mathbb T))}\Vert \tilde g\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))} +\Vert \tilde g\Vert^2_{L^2_\gamma(\mathbb R; H^{m+1}(\mathbb T))}\right\}\\
&\le \frac{C}{\gamma}\left\{\left(\Vert\varphi^a_x\Vert^2_{L^\infty(\mathbb R;H^{m+3}(\mathbb T))}+\Vert\varphi_{0,x}\Vert^2_{L^\infty(-\infty, T;H^{m+3}(\mathbb T))}\right)\Vert g\Vert^2_{L^2_\gamma(-\infty,T; H^2(\mathbb T))} +\Vert g\Vert^2_{L^2_\gamma(-\infty,T; H^{m+1}(\mathbb T))}\right\}\,,
\end{split}
\end{equation}
for all $\gamma\ge\gamma_1$, being $\gamma_1$, $C>0$ defined as in (<ref>) and again under the condition (<ref>).
Concerning the condition (<ref>), in view of $\eqref{reg_phia}_3$, (<ref>) and the Sobolev imbeddings $H^1(\mathbb T)\hookrightarrow L^\infty(\mathbb T)$, $H^1_\gamma(-\infty, T; \mathcal X)\hookrightarrow L^\infty(-\infty,T; \mathcal X)$ (cf. Lemma <ref>) we get
\begin{equation}\label{sgn_restr}
\begin{split}
\mu &-2\mathbb{H}[\varphi^a+\tilde{\varphi}_0]_x\geq 3\delta/4-2\mathbb H[\tilde{\varphi}_0]_x\ge 3\delta/4-2\Vert\mathbb H[\tilde{\varphi}_0]_x\Vert_{L^\infty(\mathbb R\times\mathbb T)}\\
&\ge 3\delta/4-C\Vert\mathbb H[\tilde{\varphi}_0]_x\Vert_{L^\infty(\mathbb R;H^1(\mathbb T))}\ge 3\delta/4-C\Vert\tilde{\varphi}_{0,x}\Vert_{L^\infty(\mathbb R;H^1(\mathbb T))}\ge 3\delta/4-C\Vert\varphi_{0,x}\Vert_{L^\infty(-\infty,T;H^1(\mathbb T))}\\
&\ge 3\delta/4-C\Vert\varphi_{0,x}\Vert_{H^1_\gamma(-\infty,T;H^1(\mathbb T))}\ge 3\delta/4-C\Vert\varphi_0\Vert_{X_1}\,.
\end{split}
\end{equation}
From (<ref>) we deduce that condition (<ref>), and consequently the estimate (<ref>), are satisfied provided that
\begin{equation*}%\label{int_0}
\Vert\varphi_0\Vert_{X_1}\le\delta/4C\,.
\end{equation*}
On the other hand, from (<ref>), the Sobolev imbedding and $\eqref{reg_phia}_4$ we find that
\begin{equation}\label{CX4}
\begin{split}
\Vert &\varphi^a+\tilde{\varphi}_0\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}\le\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}+\Vert\tilde{\varphi}_0\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}\\
&\le \Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}+C\Vert\varphi_0\Vert_{L^\infty(-\infty,T;H^5(\mathbb T))}\le \Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}+C\Vert\varphi_0\Vert_{H^1_\gamma(-\infty,T;H^5(\mathbb T))}\\
&\le \Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}+C\Vert\varphi_0\Vert_{X_4}\le C\left\{\Vert\varphi^{(0)}\Vert_{H^5(\mathbb T)}+\Vert\varphi^{(1)}\Vert_{H^4(\mathbb T)}+\Vert\varphi_0\Vert_{X_4}\right\} \,,
\end{split}
\end{equation}
and similarly
\begin{equation}\label{CX41}
\begin{split}
\Vert\varphi^a_t &+\tilde{\varphi}_{0\,,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}\le \Vert\varphi^a_t\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}+ C \Vert\varphi_0\Vert_{X_2}\\
&\le C\left\{\Vert\varphi^{(0)}\Vert_{H^3(\mathbb T)}+\Vert\varphi^{(1)}\Vert_{H^2(\mathbb T)}+\Vert\varphi_0\Vert_{X_2} \right\} \,.
\end{split}
\end{equation}
Finally, we observe that the inequalities
\begin{equation}\label{imbeddingX}
\Vert\varphi\Vert_{X_1}\le\Vert\varphi\Vert_{X_2}\le\Vert\varphi\Vert_{X_4}\,,\quad\forall\varphi\in X_4\,,
\end{equation}
follow at once from the definition (<ref>) of the norm in $X_m$.
Let us define the bounded neighborhood $\mathcal U$ of $0$ in $X_4$ be setting
\begin{equation}\label{neighbrhd}
\mathcal U:=\left\{\varphi\in X_4\,:\,\,\Vert\varphi\Vert_{X_4}\le\delta/4C\right\}\,.
\end{equation}
Because of (<ref>)-(<ref>), one has
\begin{equation}\label{CX.1}
\begin{split}
\Vert &\varphi^a+\tilde{\varphi}_0\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}+\Vert\varphi^a_t +\tilde{\varphi}_{0\,,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}\\
&\le C\left\{\Vert\varphi^{(0)}\Vert_{H^5(\mathbb T)}+\Vert\varphi^{(1)}\Vert_{H^4(\mathbb T)}\right\}+\delta/C\,,
\end{split}
\end{equation}
and the sign condition (<ref>) holds true, as long as $\varphi_0\in\mathcal U\cap X_\infty$. According to the previous observations, let us set
\begin{equation}\label{Cgamma}
\begin{split}
\widehat{\gamma}_1=\widehat{\gamma}_1(m,\delta,\varphi^{(0)},\varphi^{(1)}):=\sup\limits_{\varphi_0\in\mathcal U\cap X_\infty} \gamma_1(m,\delta, \Vert\varphi^a+\tilde{\varphi}_0\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}, \Vert\varphi^a_t+\tilde{\varphi}_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))})\,,\\
\widehat{C}_1=\widehat{C}_1(m,\delta,\varphi^{(0)},\varphi^{(1)}):=\sup\limits_{\varphi_0\in\mathcal U\cap X_\infty} C(m,\delta, \Vert\varphi^a+\tilde{\varphi}_0\Vert_{L^\infty(\mathbb R;H^5(\mathbb T))}, \Vert\varphi^a_t+\tilde{\varphi}_{0,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))})\,,
\end{split}
\end{equation}
where $\gamma_1$ and $C$ in the right-hand sides are the constants involved in (<ref>). From (<ref>) and the above definitions we derive that
\begin{equation}\label{stima_diff_T1}
\begin{split}
\gamma & \left\{\Vert\varphi^\prime_{t}\Vert^2_{L^2_\gamma(-\infty, T; H^{m+1}(\mathbb T))}+\Vert\varphi^\prime_{x}\Vert^2_{L^2_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\right\}\\
&\le \frac{\widehat{C}_1}{\gamma}\left\{\left(\Vert\varphi^a_x\Vert^2_{L^\infty(\mathbb R;H^{m+3}(\mathbb T))}+\Vert\varphi_{0,x}\Vert^2_{L^\infty(-\infty, T;H^{m+3}(\mathbb T))}\right)\Vert g\Vert^2_{L^2_\gamma(-\infty,T; H^2(\mathbb T))} +\Vert g\Vert^2_{L^2_\gamma(-\infty,T; H^{m+1}(\mathbb T))}\right\}\,,
\end{split}
\end{equation}
holds true for all $\varphi^{\prime}\in X_\infty$, $\varphi_0\in\mathcal U\cap X_{\infty}$ and $\gamma\ge\widehat{\gamma}_1$. Note that $\widehat{\gamma}_1$ and $\widehat{C}_1$ in (<ref>) now depend only on $m$, $\delta$ and the initial data $\varphi^{(0)}$, $\varphi^{(1)}$.
From now on, for every $m\ge 1$ let $\gamma$ be fixed such that $\gamma\ge\widehat{\gamma}_1$, being $\widehat{\gamma}_1$ defined as in (<ref>). For such $\gamma$, we consider the estimate (<ref>) (with $m+1$ instead of $m$), written for $\tilde{\varphi}^\prime$; we use (<ref>) and Sobolev imbedding $H^1_\gamma(-\infty,T;\mathcal X)\hookrightarrow L^\infty(-\infty,T; \mathcal X)$ (see Lemma <ref>), to get
\begin{equation}\label{stima_dm-1tildephitt}
\begin{split}
&\Vert\varphi^\prime_{tt}\Vert_{L^2_\gamma(-\infty,T;H^{m}(\mathbb T))}\le\\
&\Vert\tilde{\varphi}^\prime_{tt}\Vert_{L^2_\gamma(\mathbb R;H^{m}(\mathbb T))}\le C_1\left\{\Vert\tilde{\varphi}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}+\Vert\varphi^a+\tilde{\varphi}_0\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}\Vert\tilde{\varphi}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}\right.\\
&\qquad\qquad\qquad\quad \left.+\Vert\varphi^a_x+\tilde{\varphi}_{0,x}\Vert_{L^\infty(\mathbb R;H^{m+1}(\mathbb T))}\Vert\tilde{\varphi}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}+\Vert \tilde g\Vert_{L^2_\gamma(\mathbb R;H^{m}(\mathbb T))}\right\}\\
%&\le C_1\left\{\Vert{\widetilde{\varphi}}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}+\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}+\Vert{\varphi}_0\Vert_{L^\infty(-\infty,T;H^3(\mathbb T))}\right)\Vert{\widetilde{\varphi}}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}\right.\\
%&\qquad\qquad\qquad\quad \left.+\left(\Vert\varphi^a_x\Vert_{L^\infty(\mathbb R;H^{m+1}(\mathbb T))}+\Vert{\varphi}_{0,x}\Vert_{L^\infty(-\infty,T;H^{m+1}(\mathbb T))}\right)\Vert{\widetilde{\varphi}}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}+\Vert g\Vert_{L^2_\gamma(-\infty,T;H^{m}(\mathbb T))}\right\}\\
&\le C_1\left\{\Vert{\widetilde{\varphi}}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}+\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}+\Vert{\varphi}_0\Vert_{H^1_\gamma(-\infty,T;H^3(\mathbb T))}\right)\Vert{\widetilde{\varphi}}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}\right.\\
&\qquad\qquad\qquad\quad \left.+\left(\Vert\varphi^a_x\Vert_{L^\infty(\mathbb R;H^{m+1}(\mathbb T))}+\Vert{\varphi}_{0,x}\Vert_{H^1_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\right)\Vert{\widetilde{\varphi}}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}+\Vert g\Vert_{L^2_\gamma(-\infty,T;H^{m}(\mathbb T))}\right\}\\
&\le C_1\left\{\left(1+\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}+\Vert{\varphi}_0\Vert_{X_2}\right)\Vert{\widetilde{\varphi}}^\prime_{x}
\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}\right.\\
&\qquad\qquad\qquad\quad \left.+\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}+\Vert{\varphi}_{0}\Vert_{H^1_\gamma(-\infty,T;H^{m+2}(\mathbb T))}\right)\Vert{\widetilde{\varphi}}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}+\Vert g\Vert_{L^2_\gamma(-\infty,T;H^{m}(\mathbb T))}\right\}\,,
%&\le C_1\left\{\left(1+\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}+\Vert{\varphi}_0\Vert_{X_2}\right)\left(\Vert g\Vert_{Y_2}\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+4}(\mathbb T))} + \Vert{\varphi}_0\Vert_{X_{m+3}}\right) + \Vert g\Vert_{Y_{m+1}}\right)\right.\\
%&\qquad\qquad\qquad\quad \left.+\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}+\Vert{\varphi}_{0}\Vert_{X_{m+2}}\right)\left(\Vert g\Vert_{Y_2}\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{5}(\mathbb T))} + \Vert{\varphi}_0\Vert_{X_{4}}\right) + \Vert g\Vert_{Y_{2}}\right)+\Vert g\Vert_{Y_2}\right.
\end{split}
\end{equation}
where $C_1$ only depends on $m$, $\delta$, $\mu$, see Proposition <ref>. Now we use (<ref>) together with the Sobolev imbedding and the definition of the spaces $X_m$ and $Y_m$ to bound, for fixed $\gamma$, $\Vert\widetilde{\varphi}^\prime_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}$ and $\Vert\widetilde{\varphi}^\prime_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^{2}(\mathbb T))}$ in the above inequality, finding
\begin{equation}\label{stima_diff_T_NM}
\begin{split}
& \Vert\widetilde{\varphi}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^{m+1}(\mathbb T))}\\
&\le \widehat{C}_1\left\{\left(\Vert\varphi^a_x\Vert_{L^\infty(\mathbb R;H^{m+3}(\mathbb T))}+\Vert\varphi_{0,x}\Vert_{L^\infty(-\infty, T;H^{m+3}(\mathbb T))}\right)\Vert g\Vert_{L^2_\gamma(-\infty,T; H^2(\mathbb T))} +\Vert g\Vert_{L^2_\gamma(-\infty,T; H^{m+1}(\mathbb T))}\right\}\\
%&\le \widehat{C}_1\left\{\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+4}(\mathbb T))}+\Vert\varphi_{0}\Vert_{L^\infty(-\infty, T;H^{m+4}(\mathbb T))}\right)\Vert g\Vert_{Y_2} +\Vert g\Vert_{Y_{m+1}}\right\}\\
&\le \widehat{C}_1\left\{\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+4}(\mathbb T))}+\Vert\varphi_{0}\Vert_{H^1_\gamma(-\infty, T;H^{m+4}(\mathbb T))}\right)\Vert g\Vert_{Y_2} +\Vert g\Vert_{Y_{m+1}}\right\}\\
&\le \widehat{C}_1\left\{\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+4}(\mathbb T))}+\Vert\varphi_{0}\Vert_{X_{m+3}}\right)\Vert g\Vert_{Y_2} +\Vert g\Vert_{Y_{m+1}}\right\},
\end{split}
\end{equation}
which gives, for $m=1$
\begin{equation}\label{stima_diff_T_2_NM}
\Vert\widetilde{\varphi}^\prime_{x}\Vert_{L^2_\gamma(\mathbb R;H^{2}(\mathbb T))}\\
\le \widehat{C}_1\left\{\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{5}(\mathbb T))}+\Vert\varphi_{0}\Vert_{X_{4}}\right)\Vert g\Vert_{Y_2} +\Vert g\Vert_{Y_{2}}\right\}.
\end{equation}
The constant $\widehat{C}_1$ involved in (<ref>), (<ref>) is the value defined in (<ref>).
Inserting (<ref>) and (<ref>) in (<ref>) we get
\begin{equation}\label{stima_dm-phitt_fine}
\begin{split}
&\Vert\varphi^\prime_{tt}\Vert_{L^2_\gamma(-\infty,T;H^{m}(\mathbb T))}\le\\
&\le \widetilde{C}_1\bigg\{\big(1+\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}+\Vert{\varphi}_0\Vert_{X_2}\big)\bigg(\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+4}(\mathbb T))} + \Vert{\varphi}_0\Vert_{X_{m+3}}\right)\Vert g\Vert_{Y_2} + \Vert g\Vert_{Y_{m+1}}\bigg)\bigg.\\
&\quad\ \bigg.+\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}+\Vert{\varphi}_{0}\Vert_{X_{m+1}}\right)\bigg(\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{5}(\mathbb T))} + \Vert{\varphi}_0\Vert_{X_{4}}\right)\Vert g\Vert_{Y_2} + \Vert g\Vert_{Y_{2}}\bigg)+\Vert g\Vert_{Y_{m}}\bigg\}\,,
\end{split}
\end{equation}
with $\widetilde{C}_1=\widetilde{C}_1(m,\delta,\mu,\varphi^{(0)},\varphi^{(1)})$. Adding (<ref>) to (<ref>), and recalling (<ref>), we get
\begin{equation}\label{stima_phiprimeXm}
\begin{split}
&\le C\bigg\{\left(1+\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}+\Vert{\varphi}_0\Vert_{X_2}\right)\bigg(\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+4}(\mathbb T))} + \Vert{\varphi}_0\Vert_{X_{m+3}}\right)\Vert g\Vert_{Y_2} + \Vert g\Vert_{Y_{m+1}}\bigg)\bigg.\\
&\quad \bigg.+\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+2}(\mathbb T))}+\Vert{\varphi}_{0}\Vert_{X_{m+1}}\right)\bigg(\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{5}(\mathbb T))} + \Vert{\varphi}_0\Vert_{X_{4}}\right)\Vert g\Vert_{Y_2} + \Vert g\Vert_{Y_{2}}\bigg)+\Vert g\Vert_{Y_{m}}\bigg\}\,,
\end{split}
\end{equation}
with $C=C(m,\delta,\mu,\varphi^{(0)},\varphi^{(1)})$.
Now we observe that for $\varphi_0\in\mathcal U\cap X_\infty$ (see (<ref>)) in the above inequality the coefficients
\begin{equation*}
\left(1+\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^3(\mathbb T))}+\Vert{\varphi}_0\Vert_{X_2}\right)\,,\quad \left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{5}(\mathbb T))} + \Vert{\varphi}_0\Vert_{X_{4}}\right)
\end{equation*}
can be bounded by some constant $C=C(\delta,\Vert\varphi^{(0)}\Vert_{H^5(\mathbb T)},\Vert\varphi^{(1)}\Vert_{H^4(\mathbb T)})$, see (<ref>). Hence from (<ref>) we get
\begin{equation*}%\label{stima_phiprimeXm_fine1}
\Vert\varphi^\prime\Vert_{X_m}\le C\left\{\left(\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+4}(\mathbb T))} + \Vert{\varphi}_0\Vert_{X_{m+3}}\right)\Vert g\Vert_{Y_2} + \Vert g\Vert_{Y_{m+1}}\right\}\,,
\end{equation*}
where $C=C(m,\mu,\delta,\varphi^{(0)},\varphi^{(1)})$. Using also $\Vert g\Vert_{Y_2}\leq\Vert g\Vert_{Y_{m+1}}$ and that (see (<ref>))
\begin{equation*}
\Vert\varphi^a\Vert_{L^\infty(\mathbb R;H^{m+4}(\mathbb T))}\le C\left\{\Vert\varphi^{(0)}\Vert_{H^{m+4}(\mathbb T)}+\Vert\varphi^{(1)}\Vert_{H^{m+3}(\mathbb T)}\right\}
\end{equation*}
for all $m\geq 1$, we finally get
\begin{equation}\label{stima_phiprimeXm_fine2}
\Vert\varphi^\prime\Vert_{X_m}\le \tilde C\left\{\Vert{\varphi}_0\Vert_{X_{m+3}}\Vert g\Vert_{Y_2} + \Vert g\Vert_{Y_{m+1}}\right\}\,
\end{equation}
where now $\tilde C$ depends only on $m$, $\delta$, $\mu$ and the initial data $\varphi^{(0)}$, $\varphi^{(1)}$; in particular $\tilde C$ is bounded for $m$ bounded and sufficiently regular initial data.
The above estimate implies that Assumption 2.2 in <ref> (see also <cit.>) is satisfied with $m_0=4$, $s=1$ and $s^\prime=3$ (recall that (<ref>) requires $\varphi_0\in\mathcal U\cap X_\infty$, where $\mathcal U$ was defined in (<ref>), and $\Vert g\Vert_{Y_2}\le\Vert g\Vert_{Y_4}$).
§.§.§ Estimate for the second order differential of $\mathcal L$
We need to derive an estimate (see Assumption 2.1 in <ref>; see also <cit.>) for the second order derivative $\mathbb L^{\prime\prime}[\varphi^a+\varphi_0](\varphi^\prime, \psi^\prime)$ (see (<ref>)).
From estimate (<ref>) (see Remark <ref>), for fixed $\gamma\ge\widehat{\gamma}_1$ we find
\begin{equation*}%\label{stima_2der_NM}
\begin{split}
\Vert\mathbb L^{\prime\prime}&[\varphi^a+\varphi_0](\varphi^\prime, \psi^\prime)\Vert_{L^2_{\gamma}(-\infty,T;H^m(\mathbb T))}\\
&\leq C_m\left\{\Vert\varphi^\prime_{x}\Vert_{L^2_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\Vert\psi^\prime_x\Vert_{L^\infty(-\infty,T;H^2(\mathbb T))} + \Vert\psi^\prime_{x}\Vert_{L^2_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\Vert\varphi^\prime_x\Vert_{L^\infty(-\infty,T;H^2(\mathbb T))}\right\}\\
%&\leq C_m\left\{\Vert\varphi^\prime_x\Vert_{L^2_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\Vert\psi^\prime\Vert_{L^\infty(-\infty,T;H^3(\mathbb T))} + \Vert\psi^\prime_x\Vert_{L^2_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\Vert\varphi^\prime\Vert_{L^\infty(-\infty,T;H^3(\mathbb T))}\right\}\\
&\leq C_m\left\{\Vert\varphi^\prime_x\Vert_{L^2_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\Vert\psi^\prime\Vert_{H^1_\gamma(-\infty,T;H^3(\mathbb T))} + \Vert\psi^\prime_x\Vert_{L^2_\gamma(-\infty,T;H^{m+1}(\mathbb T))}\Vert\varphi^\prime\Vert_{H^1_\gamma(-\infty,T;H^3(\mathbb T))}\right\}\\
&\leq C_m\left\{\Vert\varphi^\prime\Vert_{X_m}\Vert\psi^\prime\Vert_{X_2} + \Vert\psi^\prime\Vert_{X_m}\Vert\varphi^\prime\Vert_{X_2}\right\}\,,
\end{split}
\end{equation*}
with the constant $C_m$ bounded for $m$ bounded (again we used the Sobolev imbedding $H^1_\gamma(-\infty,T;\mathcal X)\hookrightarrow L^\infty(-\infty,T;\mathcal X)$).
Estimate above is exactly Assumpion 2.1 in <ref> (see also <cit.>) with $m_0\geq 2$, $r=0$, $r^\prime =0$.
We need that Assumptions 2.1 and 2.2 in <ref> (see also <cit.>) are simultaneously satisfied; hence, from now on, we assume $m_0=4$.
§.§ Proof of Theorem <ref>
We are now in the position to verify that all the assumptions in <cit.> are satisfied (see <ref>, Theorem <ref>) in order to get that the nonlinear equation (<ref>) has a solution.
From the values of $r,r^\prime, s, s^\prime$ obtained in Sections <ref> and <ref>, we compute
\begin{equation*}
m^\prime=m_0+\max{\{r,r^\prime\}} + \max{\{s,s^\prime\}}=4+\max{\{0,0\}} + \max{\{1,3\}}=7.
\end{equation*}
By (i) in <cit.> (see <ref>, Theorem <ref>), there exists $\varepsilon$ such that, if $F^a\in Y_{m^\prime+s+1}=Y_9$ with
\begin{equation}\label{norma-piccola-nostro-caso}
\Vert F^a\Vert_{Y_9}=\Vert F^a\Vert_{L^2_\gamma(-\infty,T; H^9(\mathbb T))}\leq \varepsilon\,,
\end{equation}
there exists a solution $\varphi^\prime \in X_{m^\prime}=X_7$ of (<ref>), provided that Assumptions 2.1.and 2.2 are satisfied. In view of Remark <ref>, the assumptions $\varphi^{(0)}\in H^{11}(\mathbb T)$ and $\varphi^{(1)}\in H^{10}(\mathbb T)$ imply $F^a\in Y_{9}$. On the other hand, the smallness assumption (<ref>) is guaranteed if we take a positive $T$ sufficiently small, depending on the size of the initial data $\varphi^{(0)}\in H^{11}(\mathbb T)$ and $\varphi^{(1)}\in H^{10}(\mathbb T)$, see again Remark <ref>. In view of (<ref>) and (<ref>), the function $\varphi=\varphi^a_{\vert\,[0,T]}+\varphi^\prime_{\vert [0,T]}$ provides a solution of (<ref>) with the regularity required. The uniqueness follows from standard arguments, based on the regularity of the solution. This ends the proof of the statement (1).
Moreover, by (ii) in <cit.> (see also <ref>, Theorem <ref>), if (<ref>) holds and if in addition $F^a\in Y_{m^{\prime\prime}+s+1}=Y_{m^{\prime\prime}+2}$ with $m^{\prime\prime}>m^\prime=7$, then the solution $\varphi^\prime$ of (<ref>) belongs to $X_{m^{\prime\prime}}$, again under the condition that Assumptions 2.1.and 2.2 are satisfied.
Observe now that if $\varphi^{(0)}\in H^{\nu+1}(\mathbb T)$ and $\varphi^{(1)}\in H^\nu(\mathbb T)$, in view of Remark <ref>, we know that $F^a\in Y_{\nu-1}$. Hence, in order to satisfy the above case (ii) of <cit.>, we need to require that
\begin{equation*}
\nu-1=m^{\prime\prime}+2>9
\end{equation*}
that is ensured by the assumption $\nu>10$. For such $\nu>10$, Nash-Moser's theorem implies that the solution of (<ref>) belongs to $X_{m^{\prime\prime}}=X_{\nu-3}$, which yields for the solution $\varphi=\varphi^a_{\vert\,[0,T]}+\varphi^\prime_{\vert [0,T]}$ of (<ref>) the required regularity. Notice in particular that the final time $T$ is just defined from the requirement (<ref>), where the norm of $F^a$ in $L^2_\gamma(-\infty,T; H^{9}(\mathbb T))$ is involved; because of Remark <ref> the time $T$ may be chosen to depend on the norms of the data $\varphi^{(0)}$, $\varphi^{(1)}$ respectively in $H^{11}(\mathbb T)$ and $H^{10}(\mathbb T)$, in spite of the augmented regularity $\varphi^{(0)}\in H^{\nu+1}(\mathbb T)$, $\varphi^{(1)}\in H^{\nu}(\mathbb T)$.
§ SOME COMMUTATOR AND PRODUCT ESTIMATES
For $s>1/2$ there exists a constant $C_s>0$ such that
\begin{eqnarray}
\Vert \left[\mathbb H\,;\,v\right]f\Vert_{L^2(\mathbb T)}\le C_s\Vert v\Vert_{H^s(\mathbb T)}\Vert f\Vert_{L^2(\mathbb T)}\,,\quad\forall\,v\in H^s(\mathbb T)\,,\,\,\forall\,f\in L^2(\mathbb T)\,;\label{stima_comm_1}\\
\Vert \left[\mathbb H\,;\,v\right]f_x\Vert_{L^2(\mathbb T)}\le C_s\Vert v_x\Vert_{H^s(\mathbb T)}\Vert f\Vert_{L^2(\mathbb T)}\,,\quad\forall\,v\in H^{s+1}(\mathbb T)\,,\,\,\forall\,f\in L^2(\mathbb T)\,;\label{stima_comm_2}\\
\left\Vert \left(\left[\mathbb H\,;\,\left[\mathbb H\,;\,v\right]\right]f_x\right)_x\right\Vert_{L^2(\mathbb T)}\le C_s\Vert v_{xx}\Vert_{H^s(\mathbb T)}\Vert f\Vert_{L^2(\mathbb T)}\,,\quad\forall\,v\in H^{s+2}(\mathbb T)\,,\,\,\forall\,f\in L^2(\mathbb T)\,,\label{stima_comm_3}
\end{eqnarray}
where $\left[\mathbb H\,;\,v\right]$ is the commutator between the Hilbert transform $\mathbb H$ and the multiplication by $v$.
We show here only the estimate (<ref>), the proof of estimates (<ref>), (<ref>) following from similar arguments. Let us first recall that, whenever $f$ and $g$ are sufficiently smooth periodic functions on $\mathbb T$, there holds
\begin{equation}\label{convoluzione}
\widehat{f\cdot g}=\frac1{2\pi}\widehat{f}\ast\widehat{g}\,,\quad\forall\,k\in\mathbb Z\,,
\end{equation}
where $\widehat f\ast\widehat g$ is the discrete convolution of the sequences $\widehat f:=\left\{\widehat f(k)\right\}_{k\in\mathbb Z}$ and $\widehat g:=\left\{\widehat g(k)\right\}_{k\in\mathbb Z}$ defined by
\begin{equation}\label{conv}
\widehat f\ast\widehat g(k):=\sum\limits_{\ell}\widehat{f}(k-\ell)\widehat{g}(\ell)\,,\qquad\forall\,k\in\mathbb Z\,.
\end{equation}
Making use of (<ref>), (<ref>) for $k\neq 0$ we compute
\begin{align*}\label{stima_comm_3.1}
\big(\left(\left[\mathbb H\,;\,\left[\mathbb H\,;\,v\right]\right]f_x\right)_x\big)^{\wedge}&(k)=ik\big(\left[\mathbb H\,;\,\left[\mathbb H\,;\,v\right]\right]f_x\big)^{\wedge}(k)=ik\left\{\big(\mathbb H[\left[\mathbb H\,;\,v\right]f_x]\big)^{\wedge}(k)-\big(\left[\mathbb H\,;\,v\right]\mathbb H[f_x]\big)^{\wedge}(k)\right\}\\
&=ik\left\{-i\,{\rm sgn}\,k\,\big(\left[\mathbb H\,;\,v\right]f_x\big)^\wedge(k)+\big(v\,\mathbb H^2[f_x]\big)^\wedge(k)-\big(\mathbb H[v\,\mathbb H[f_x]]\big)^\wedge(k)\right\}\\
&=ik\left\{-i\,{\rm sgn}\,k\,\left(\widehat{\mathbb H[v\,f_x]}(k)-\widehat{v\,\mathbb H[f_x]}(k)\right)-\widehat{v\,f_x}(k)+i\,{\rm sgn}\,k\,\widehat{v\,\mathbb H[f_x]}(k)\right\}\\
&=ik\left\{-i\,{\rm sgn}\,k\,\left(-i\,{\rm sgn}\,k\,\widehat{v\,f_x}(k)-\frac1{2\pi}\sum\limits_{\ell}\widehat{v}(k-\ell)\widehat{\mathbb H[f_x]}(\ell)\right)-\widehat{v\cdot f_x}(k)\right.\\
&\quad\left.+\frac1{2\pi}\,i\,{\rm sgn}\,k\,\sum\limits_{\ell}\widehat{v}(k-\ell)\widehat{\mathbb H[f_x]}(\ell)\right\}\\
&=ik\left\{-({\rm sgn}\,k)^2\,\widehat{v\, f_x}(k)+\frac1{2\pi}i\,{\rm sgn}\,k\,\sum\limits_{\ell}\widehat{v}(k-\ell)\widehat{\mathbb H[f_x]}(\ell)-\widehat{v\, f_x}(k)\right.\\
&\quad\left. +\frac1{2\pi}\,i\,{\rm sgn}\,k\,\sum\limits_{\ell}\widehat{v}(k-\ell)\widehat{\mathbb H[f_x]}(\ell)\right\}\\
&=ik\left\{-2\widehat{v\, f_x}(k)+\frac1{\pi}i\,{\rm sgn}\,k\,\sum\limits_{\ell}\widehat{v}(k-\ell)\widehat{\mathbb H[f_x]}(\ell)\right\}\\
&=ik\left\{\frac1{\pi}i\,{\rm sgn}\,k\,\sum\limits_{\ell}\widehat{v}(k-\ell)\widehat{\mathbb H[f_x]}(\ell)-\frac1{\pi}\sum\limits_{\ell}\widehat{v}(k-\ell)\widehat{f_x}(\ell)\right\}\\
&=ik\left\{\frac1{\pi}i\,{\rm sgn}\,k\,\sum\limits_{\ell}\widehat{v}(k-\ell)\,\ell\,{\rm sgn}\,\ell\,\widehat{f}(\ell)-\frac1{\pi}\sum\limits_{\ell}\widehat{v}(k-\ell)i\,\ell\,\widehat{f}(\ell)\right\}\\
\end{align*}
where we also used $({\rm sgn}\,k)^2=1$, $k\,{\rm sgn}\,k=|k|$ and we have set
\begin{equation*}
\Lambda_1(k,\ell)=2(k\ell-|k|\,|\ell|)\,.
\end{equation*}
Observing that $\Lambda_1(k,\ell)=0$ for $k\ell\ge 0$ and using the numerical inequality $-2k\ell\le (k-\ell)^2$ we get
\begin{equation*}%\label{stima_comm_3.2}
\begin{array}{ll}
\displaystyle\left\vert\big(\left(\left[\mathbb H\,;\,\left[\mathbb H\,;\,v\right]\right]f_x\right)_x\big)^\wedge(k)\right\vert\le \frac1{2\pi}\sum\limits_{\ell\,:\,\,k\ell< 0}(-4k\ell)\,\vert\widehat{v}(k-\ell)\vert\,\vert\widehat{f}(\ell)\vert\le \frac1{\pi}\sum\limits_{\ell}(k-\ell)^2\,\vert\widehat{v}(k-\ell)\vert\,\vert\widehat{f}(\ell)\vert\\
\displaystyle\quad =\frac1{\pi}\sum\limits_{\ell}\,\vert (i(k-\ell))^2\widehat{v}(k-\ell)\vert\,\vert\widehat{f}(\ell)\vert=\frac1{\pi}\sum\limits_{\ell}\,\vert \widehat{v_{xx}}(k-\ell)\vert\,\vert\widehat{f}(\ell)\vert=\frac1{\pi}\left(\vert\widehat{v_{xx}}\vert\ast\vert\widehat{f}\vert\right)(k)\,.
\end{array}
\end{equation*}
Because of Parseval's identity and using Young's inequality we derive
\begin{align*}%\label{stima_comm_1.3}
\Vert &\left(\left[\mathbb H\,;\,\left[\mathbb H\,;\,v\right]\right]f_x\right)_x\Vert^2_{L^2(\mathbb T)}=\frac1{2\pi}\sum\limits_{k\in\mathbb Z}\left\vert \big(\left(\left[\mathbb H\,;\,\left[\mathbb H\,;\,v\right]\right]f_x\right)_x\big)^\wedge(k)\right\vert^2\le\frac{1}{2\pi^2}\sum\limits_{k\in\mathbb Z}\left\vert\left(\vert\widehat v_{xx}\vert\ast\vert\widehat f\vert\right)(k)\right\vert^2\\
&\le \frac{1}{2\pi^2}\left(\sum\limits_{k\in\mathbb Z}\vert\widehat v_{xx}(k)\vert\right)^2\sum\limits_{k\in\mathbb Z}\vert\widehat{f}(k)\vert^2=\frac{1}{\pi}\left(\sum\limits_{k\in\mathbb Z}\vert\widehat v_{xx}(k)\vert\right)^2\Vert f\Vert^2_{L^2(\mathbb T)}\,.
\end{align*}
Estimate (<ref>) follows at once from the above, since for $s>1/2$
\begin{equation}\label{imm_sobolev}
\sum\limits_{k\in\mathbb Z}\vert\widehat v_{xx}(k)\vert\le c_s\Vert v_{xx}\Vert_{H^s(\mathbb T)}
\end{equation}
holds true with some positive constant $c_s$, depending only on $s$.
For every integer $m\ge 1$ there exists a constant $C_m>0$ such that for all functions $v\in H^m(\mathbb T)$ and $f\in H^1(\mathbb T)$
\begin{equation*}%\label{stima_comm_p}
\Vert \left[\mathbb H\,;\,v\right]f\Vert_{H^m(\mathbb T)}\le C_m\Vert \partial^m_x v\Vert_{L^{2}(\mathbb T)}\Vert f\Vert_{H^1(\mathbb T)}\,.
\end{equation*}
In view of (<ref>) we compute
\begin{equation*}%\label{stima_comm_1.1}
\begin{array}{ll}
\displaystyle\widehat{\left[\mathbb H\,;\,v\right]f}(k)=\widehat{\mathbb H[vf]}(k)-\widehat{v\mathbb H[f]}(k)=-i\,{\rm sgn}\,k\,\widehat{vf}(k)-\frac1{2\pi}\sum\limits_{\ell}\widehat{v}(k-\ell)\widehat{\mathbb H[f]}(\ell)\\
\displaystyle\quad =-i\,{\rm sgn}\,k\,\frac1{2\pi}\sum\limits_{\ell}\widehat{v}(k-\ell)\widehat{f}(\ell)+\frac1{2\pi}\sum\limits_{\ell}i\,{\rm sgn}\,\ell\,\widehat{v}(k-\ell)\widehat{f}(\ell)\\
\displaystyle\quad =\frac1{2\pi}\sum\limits_{\ell}\Lambda(k,\ell)\,\widehat{v}(k-\ell)\widehat{f}(\ell)\,,
\end{array}
\end{equation*}
\begin{equation}\label{valori_lambda}
\Lambda(k,\ell):=i\,\left(-{\rm sgn}\,k+{\rm sgn}\,\ell\right)=
\begin{cases}
\mp i\,,\qquad\mbox{if}\,\,\pm k>0\,\,\mbox{and}\,\,\ell=0\,,\\
\end{cases}
\end{equation}
\begin{equation}\label{simmetria_lambda}
\Lambda(\ell,k)=-\Lambda(k,\ell)\,,\quad\forall\,(k,\ell)\in\mathbb Z^2\,.
\end{equation}
In view of (<ref>) and (<ref>) we have more explicitly that
\begin{equation}\label{casi}
\widehat{\left[\mathbb H\,;\,v\right]f}(k)=\begin{cases}\displaystyle\frac{-i}{2\pi}\widehat{v}(k)\widehat{f}(0)-\frac{i}{\pi}\sum\limits_{\ell<0}\widehat{v}(k-\ell)\widehat{f}(\ell)\,,\quad\mbox{if}\,\,k>0\,,\\
\displaystyle-\frac{i}{2\pi}\sum\limits_{\ell}{\rm sgn}\,\ell\,\widehat{v}(-\ell)\widehat{f}(\ell)\,,\quad\mbox{if}\,\,k=0\,,\\
\displaystyle\frac{i}{2\pi}\widehat{v}(k)\widehat{f}(0)+\frac{i}{\pi}\sum\limits_{\ell>0}\widehat{v}(k-\ell)\widehat{f}(\ell)\,,\quad\mbox{if}\,\,k<0\,.
\end{cases}
\end{equation}
On the other hand, in view of (<ref>)
\begin{equation}\label{stima_paolo_1}
\begin{split}
\Vert &\left[\mathbb H\,;\,v\right]f\Vert^2_{H^m(\mathbb T)}=\frac1{2\pi}\sum\limits_{k\in \mathbb Z}(1+|k|)^{2m}\vert\widehat{\left[\mathbb H\,;\,v\right]f}(k)\vert^2\\
&\le C_m\left\{\vert\widehat{\left[\mathbb H\,;\,v\right]f}(0)\vert^2+\sum\limits_{k\in \mathbb Z\setminus\{0\}}|k|^{2m}\vert\widehat{\left[\mathbb H\,;\,v\right]f}(k)\vert^2\right\}\,.
\end{split}
\end{equation}
From (<ref>) we get for $k=0$:
\begin{equation}\label{coeff_0}
\vert \widehat{\left[\mathbb H\,;\,v\right]f}(0)\vert\le\frac{1}{2\pi}\sum\limits_{\ell\neq 0}\vert\widehat{v}(-\ell)\vert\,\vert\widehat{f}(\ell)\vert\le \frac{1}{2\pi}\sum\limits_{\ell\neq 0}\vert -i\ell\vert^m \vert\widehat{v}(-\ell)\vert\vert\widehat{f}(\ell)\vert\le\frac1{2\pi}\left(\vert\widehat{\partial_x^m v}\vert\ast\vert \widehat f\vert\right)(0)\,.
\end{equation}
For $k>0$ we obtain
\begin{equation}\label{coeff_k+}
\begin{split}
\vert k\vert^m\,\vert & \widehat{\left[\mathbb H\,;\,v\right]f}(k)\vert\le \frac{1}{2\pi}\vert k\vert^m \vert\widehat{v}(k)\vert\,\vert\widehat{f}(0)\vert+\frac{1}{\pi}\sum\limits_{\ell<0}\vert ik\vert^m\,\vert\widehat{v}(k-\ell)\vert\,\vert\widehat{f}(\ell)\vert\\
&\le \frac{1}{2\pi}\vert(ik)^m\widehat{v}(k)\vert\,\vert\widehat{f}(0)\vert+\frac{1}{\pi}\sum\limits_{\ell<0}\vert i(k-\ell)\vert^m\,\vert\widehat{v}(k-\ell)\vert\,\vert\widehat{f}(\ell)\vert\\
&\le\frac1{\pi}\sum\limits_{\ell\le 0}\vert\widehat{\partial^m_x v}(k-\ell)\vert\,\vert\widehat{f}(\ell)\vert\le\frac1{\pi}(\vert\widehat{\partial^m_x v}\vert\ast\vert\widehat{f}\vert)(k)\,,
\end{split}
\end{equation}
where we used that $\vert k\vert<\vert k-\ell\vert$ for $k>0$ and $\ell<0$. The same estimate as above can be extended to the coefficients $\vert\widehat{\left[\mathbb H\,;\,v\right]f}(k)\vert$, for $k<0$, by repeating the same arguments and since the inequality $\vert k\vert<\vert k-\ell\vert$ is still true for $k<0$ and $\ell>0$.
Using (<ref>), (<ref>) to estimate the right-hand side of (<ref>), by Young's inequality and (<ref>) (with $f$ instead of $v_{xx}$ and $s=1$) we obtain
\begin{equation*}\label{stima_paolo_2}
\begin{split}
\Vert &\left[\mathbb H\,;\,v\right]f\Vert^2_{H^m(\mathbb T)}\le C_m\sum\limits_{k\in \mathbb Z}\left\vert\left(\vert\widehat{\partial^m_x v}\vert\ast\vert\widehat{f}\vert\right)(k)\right\vert^2\le C_m\left(\sum\limits_{k\in \mathbb Z}\vert\widehat{\partial^m_x v}(k)\vert^2\right)\left(\sum\limits_{k\in \mathbb Z}\vert\widehat{f}(k)\vert\right)^2\\
&\le C_m\Vert\partial^m_x v\Vert^2_{L^2(\mathbb T)}\Vert f\Vert^2_{H^1(\mathbb T)}\,.
\end{split}
\end{equation*}
Following the same arguments as in the case of Lemma <ref>, we may prove the following more general result.
For all integers $m, p\ge 1$ there exists a constant $C_{m,p}>0$ such that for all functions $v\in H^{m+p}(\mathbb T)$ and $f\in H^1(\mathbb T)$
\begin{equation*}%\label{stima_comm_p}
\Vert \left[\mathbb H\,;\,v\right]\partial^p_x f\Vert_{H^m(\mathbb T)}\le C_{m,p}\Vert \partial^{m+p}_x v\Vert_{L^{2}(\mathbb T)}\Vert f\Vert_{H^1(\mathbb T)}\,.
\end{equation*}
The following product and commutator estimates are a consequence of the well known Gagliardo-Nirenberg inequalities, see f.i. <cit.>.
For every integer $m\ge 1$ there exists a constant $C_m>0$ such that the following holds:
i. for all functions $f\,,v\in H^m(\mathbb T)\cap L^\infty(\mathbb T)$
\begin{equation}
\Vert vf\Vert_{H^m(\mathbb T)}\le C_m\left\{\Vert v\Vert_{L^\infty(\mathbb T)}\Vert f\Vert_{H^m(\mathbb T)}+\Vert v\Vert_{H^m(\mathbb T)}\Vert f\Vert_{L^\infty(\mathbb T)}\right\}\,;\label{stima_prod_4}
\end{equation}
ii. for all positive integers $k\le m$ and functions $f\,,v\in H^m(\mathbb T)\cap L^\infty(\mathbb T)$
\begin{equation}
\Vert \left[\partial_x^k\,;\,v\right]f\Vert_{L^2(\mathbb T)}\le C_m\left\{\Vert v\Vert_{L^\infty(\mathbb T)}\Vert f\Vert_{H^k(\mathbb T)}+\Vert v\Vert_{H^k(\mathbb T)}\Vert f\Vert_{L^\infty(\mathbb T)}\right\}\,;\label{stima_comm_4}
\end{equation}
iii. for all positive integers $k\le m$ and functions $f\in H^{m-1}(\mathbb T)\cap L^\infty(\mathbb T)$ and $v\in H^m(\mathbb T)$ such that $v_x\in L^\infty(\mathbb T)$
\begin{equation}
\Vert \left[\partial_x^k\,;\,v\right]f\Vert_{L^2(\mathbb T)}\le C_m\left\{\Vert v_x\Vert_{L^\infty(\mathbb T)}\Vert f\Vert_{H^{k-1}(\mathbb T)}+\Vert v\Vert_{H^k(\mathbb T)}\Vert f\Vert_{L^\infty(\mathbb T)}\right\}\,.\label{stima_comm_5}
\end{equation}
For all real numbers $s>1/2$ and $m\in\mathbb N$, the set inclusion $H^s(\mathbb T)\cdot H^m(\mathbb T)\subset H^m(\mathbb T)$ holds with continuous imbedding. In particular, for $m=s\ge1$ the space $H^s(\mathbb T)$ is an algebra for the point-wise product of functions.
For every integer $m\ge 1$ there exists a positive constant $C_m$ such that for all functions $v\in H^{m+1}(\mathbb T)$, $f\in H^2(\mathbb T)$
\begin{equation}\label{stima_comm_6}
\left\Vert \left[\mathbb H\,;\,\left[\partial_x^m\,;\,v\right]\right]\partial^2_x f\right\Vert_{L^2(\mathbb T)}\le C_m\Vert v_x\Vert_{H^m(\mathbb T)}\Vert f_x\Vert_{H^1(\mathbb T)}\,.
\end{equation}
We proceed again as in the proof of Lemma <ref>, by computing explicitly the Fourier coefficients of the function $\left[\mathbb H\,;\,\left[\partial_x^m\,;\,v\right]\right]\partial^2_x f$. For every integer $k$ we find
\begin{equation}\label{stima_comm_6_1}
\begin{split}
\left(\left[\mathbb H\,;\,\left[\partial_x^m\,;\,v\right]\right]\partial_x^2 f\right)^{\wedge}(k)=\bigg(\mathbb H\left[\left[\partial_x^m\,;\,v\right]\partial^2_x f\right]-\left[\partial_x^m\,;\,v\right]\mathbb H\left[\partial^2_x f\right]\bigg)^{\wedge}(k)\\
=-i{\rm sgn}\,k\,\left(\left[\partial_x^m\,;\,v\right]\partial^2_x f\right)^{\wedge}(k)-\left(\left[\partial_x^m\,;\,v\right]\partial_x^2\mathbb H\left[f\right]\right)^{\wedge}(k)\,.
\end{split}
\end{equation}
Let us first develop the Fourier coefficients of $\left[\partial_x^m\,;\,v\right]\partial^2_x f$; we compute
\begin{equation*}
\begin{split}
&\left(\left[\partial_x^m\,;\,v\right]\partial^2_x f\right)^{\wedge}(k)=\left(\partial_x^m\left(v\partial_x^2 f\right)-v\partial^{m+2}_x f\right)^{\wedge}(k)\\
&\qquad=(ik)^m\left(v\partial_x^2 f\right)^{\wedge}(k)-\frac1{2\pi}\sum\limits_{\ell}\widehat{v}(k-\ell)\left(\partial^{m+2}_x f\right)^{\wedge}(\ell)\\
&\qquad=-\frac1{2\pi}i^m\sum\limits_{\ell}\ell^2(k^m-\ell^m)\widehat{v}(k-\ell)\widehat{f}(\ell)=-\frac1{2\pi}i^m\sum\limits_{\ell\neq 0}\ell^2(k^m-\ell^m)\widehat{v}(k-\ell)\widehat{f}(\ell)\,.
\end{split}
\end{equation*}
Substituting the above expression in (<ref>) we get
\begin{equation*}%\label{stima_comm_6_2}
\begin{split}
&\left(\left[\mathbb H\,;\,\left[\partial_x^m\,;\,v\right]\right]\partial_x^2 f\right)^{\wedge}(k)=\\
&\qquad =\frac{i^{m+1}}{2\pi}{\rm sgn}\,k\,\sum\limits_{\ell\neq 0}\ell^2(k^m-\ell^m)\widehat{v}(k-\ell)\widehat{f}(\ell) +\frac1{2\pi}i^m\sum\limits_{\ell\neq 0}\ell^2(k^m-\ell^m)\widehat{v}(k-\ell)\widehat{\mathbb H\left[f\right]}(\ell)\\
&\qquad =\frac{i^{m+1}}{2\pi}{\rm sgn}\,k\,\sum\limits_{\ell\neq 0}\ell^2(k^m-\ell^m)\widehat{v}(k-\ell)\widehat{f}(\ell) -\frac{i^{m+1}}{2\pi}\sum\limits_{\ell\neq 0}\ell^2(k^m-\ell^m)\widehat{v}(k-\ell)\,{\rm sgn}\,\ell\,\widehat{f}(\ell)\\
&\qquad =\frac{i^{m+1}}{2\pi}\sum\limits_{\ell\neq 0}\left({\rm sgn}\,k-{\rm sgn}\,\ell\right)\ell^2(k^m-\ell^m)\widehat{v}(k-\ell)\,\widehat{f}(\ell)\,.
\end{split}
\end{equation*}
In view of (<ref>) we have for $k>0$
\begin{equation*}\label{casi+}
{\rm sgn}\,k-{\rm sgn}\,\ell=\begin{cases}0\,,\qquad\mbox{if}\,\,\ell>0\,,\\ 2\,,\qquad\mbox{if}\,\,\ell<0\,;\end{cases}
\end{equation*}
for $k=0$
\begin{equation*}\label{casi0}
{\rm sgn}\,k-{\rm sgn}\,\ell=\begin{cases}-1\,,\qquad\mbox{if}\,\,\ell>0\,,\\ 1\,,\qquad\mbox{if}\,\,\ell<0\,;\end{cases}
\end{equation*}
for $k<0$
\begin{equation*}\label{casi-}
{\rm sgn}\,k-{\rm sgn}\,\ell=\begin{cases}-2\,,\qquad\mbox{if}\,\,\ell>0\,,\\ 0\,,\qquad\mbox{if}\,\,\ell<0\,.\end{cases}
\end{equation*}
Consequently we obtain
\begin{equation}\label{stima_comm_6_3}
\begin{split}
\left(\left[\mathbb H\,;\,\left[\partial_x^m\,;\,v\right]\right]\partial_x^2 f\right)^{\wedge}(k)=\begin{cases}\displaystyle\frac{i^{m+1}}{\pi}\sum\limits_{\ell<0}\ell^2(k^m-\ell^m)\widehat{v}(k-\ell)\,\widehat{f}(\ell)\,,\quad\mbox{if}\,\,k>0\,,\\ \displaystyle-\frac{i^{m+1}}{\pi}\sum\limits_{\ell>0}\ell^2(k^m-\ell^m)\widehat{v}(k-\ell)\,\widehat{f}(\ell)\,,\quad\mbox{if}\,\,k<0\,,\\
\displaystyle\frac{i^{m+1}}{2\pi}\sum\limits_{\ell\neq 0}{\rm sgn}\,\ell\,\ell^{m+2}\widehat{v}(-\ell)\,\widehat{f}(\ell)\,,\quad\mbox{if}\,\,k=0\,.\end{cases}
\end{split}
\end{equation}
In view of the above formulas, for $k=0$ we immediately get:
\begin{equation*}\label{stima_comm_6_4}
\begin{split}
&\left\vert\left(\left[\mathbb H\,;\,\left[\partial_x^m\,;\,v\right]\right]\partial_x^2 f\right)^{\wedge}(0)\right\vert\le\frac1{2\pi}\sum\limits_{\ell\neq 0}\vert(-i\ell)^m\widehat{v}(-\ell)\vert\vert(i\ell)^2\widehat{f}(\ell)\vert\\
&\quad =\frac1{2\pi}\sum\limits_{\ell\neq 0}\vert\widehat{\partial_x^m v}(-\ell)\vert\widehat{\partial_x^2 f}(\ell)\vert\le\frac1{2\pi}\left(\vert\widehat{\partial_x^mv}\vert\ast\vert\widehat{\partial^2_x f}\vert\right)(0)\,.
\end{split}
\end{equation*}
If $k>0$ we expand the factor $(k^m-\ell^m)$ in the corresponding expression in (<ref>) by Newton's formula to get
\begin{equation*}\label{stima_comm_6_5}
\begin{split}
&\left(\left[\mathbb H\,;\,\left[\partial_x^m\,;\,v\right]\right]\partial_x^2 f\right)^{\wedge}(k)=\frac{i^{m+1}}{\pi}\sum\limits_{\ell<0}\ell^2(k^m-\ell^m)\widehat{v}(k-\ell)\,\widehat{f}(\ell)\\
\end{split}
\end{equation*}
For $k>0$ and $\ell<0$ one has $0<\vert\ell\vert=-\ell\le k-\ell$, hence there exists $C>0$, depending only on $m$, such that
\begin{equation*}
\left\vert\sum\limits_{h=0}^{m-1}\binom{m}{h}(k-\ell)^{m-h}\ell^h\right\vert\le C(k-\ell)^{m}
\end{equation*}
\begin{equation*}\label{stima_comm_6_6}
\begin{split}
&\left\vert\left(\left[\mathbb H\,;\,\left[\partial_x^m\,;\,v\right]\right]\partial_x^2 f\right)^{\wedge}(k)\right\vert\le\frac{C}{\pi}\sum\limits_{\ell<0}\vert (i(k-\ell))^m\widehat{v}(k-\ell)\vert\,\vert\widehat{(i\ell)^2 f}(\ell)\vert\le\frac{C}{\pi}\left(\vert\widehat{\partial_x^m v}\vert\ast\vert\widehat{\partial_x^2 f}\vert\right)(k)\,.
\end{split}
\end{equation*}
If $k<0$ and $\ell>0$ we have again $\vert\ell\vert=\ell<\ell-k=\vert k-\ell\vert$ and all the arguments used in the case $k>0$ can be repeated to get the same estimates as above for the Fourier coefficients $\left(\left[\mathbb H\,;\,\left[\partial_x^m\,;\,v\right]\right]\partial_x^2 f\right)^{\wedge}(k)$.
Then we end the proof of the estimate (<ref>) by arguing as in the proof of Lemma <ref>, by using Parseval and Young's inequalities.
§ NASH-MOSER'S THEOREM
In this section, for reader's convenience, we report the assumptions and the Nash-Moser's theorem in <cit.> (we adopt the same notation in <cit.>) that we apply in order to get the existence of the solution of the nonlinear problem (<ref>).
Let $\{X_m\}_{m\geq 0}$ and $\{Y_m\}_{m\geq 0}$ be two decreasing families of Banach spaces, each satisfying the smoothing hypothesis (see <cit.>). Let $\mathcal L:X_m\rightarrow Y_m$ be twice differentiable for every $m\geq 0$, with $\mathcal L (0)=0$.
Assumption 2.1: For all $u\in U\cap X_{\infty}$, where $U$ is a bounded open neighborhood of $0$ in $X_{m_0}$ for some $m_0\geq 0$, the function $\mathcal L:X_m\rightarrow Y_m$ satisfies the tame estimate
\begin{equation*}\label{assumption2.1}
\Vert d^2\mathcal L (u)(v_1,v_2)\Vert_{Y_m} \leq C\left( \Vert v_1\Vert_{X_{m+r}} \Vert v_2\Vert_{X_{m_0}} + \Vert v_1\Vert_{X_{m_0}} \Vert v_2\Vert_{X_{m+r}} +\Vert v_1\Vert_{X_{m_0}} \Vert v_2\Vert_{X_{m_0}} \left(1+\Vert u\Vert_{X_{m+r^\prime}} \right)\right)\,,
\end{equation*}
for all $m\geq 0$ and for all $v_1,v_2\in X_{\infty}$, for some fixed integers $r,r^\prime \geq 0$. The constant $C$ is bounded for $m$ bounded.
Assumption 2.2: For all $u\in U\cap X_{\infty}$, there exists a linear mapping $\Psi(u):Y_\infty \rightarrow X_\infty$ such that $d\mathcal L(u)\Psi(u)=Id$ and satisfying the tame estimate
\begin{equation*}\label{assumtion2.2}
\Vert\Psi(u)g\Vert_{X_m}\leq C\left( \Vert g\Vert_{Y_{m+s}} + \Vert g\Vert_{Y_{m_0}}\Vert u\Vert_{X_{m+s^\prime}}\right)
\end{equation*}
for all $m\geq 0$ and some fixed integers $s,s^\prime \geq 0$. The constant $C$ is bounded for $m$ bounded.
The Nash-Moser's theorem requires a family of smoothing operators (see <cit.>). We recall here the statement of <cit.> which is a suitable version of Nash-Moser's theorem.
Let $\{X_m\}_{m\geq 0}$ and $\{Y_m\}_{m\geq 0}$ be two decreasing families of Banach spaces, each satisfying the smoothing hypothesis, and assume that both Assumptions 2.1 and 2.2 hold. Let $m^\prime$ be a positive integer such that $m^\prime\geq m_0+\max{\{r,r^\prime\}}+ \max{\{s,s^\prime\}}$.
(i) There exists a constant $0<\varepsilon \leq 1$ such that if $f\in Y_{m^\prime+s+1}$ with
\begin{equation*}\label{norma-piccola}
\Vert f\Vert_{Y_{m^\prime+s+1}}\le\varepsilon,
\end{equation*}
the equation $\mathcal L(u)=f$ has a solution $u\in X_{m^\prime}$, in the sense that there exists a sequence $\{u_n\}\subset X_\infty$ such that $u_n\rightarrow u$ in $X_{m^\prime}$, $\mathcal L(u_n) \rightarrow f$ in $Y_{m^\prime+s}$, as $n\rightarrow \infty$.
(ii) If (i) holds and if there exists $m^{\prime\prime}>m^\prime$ such that $f\in Y_{m^{\prime\prime}+s+1}$, then the solution constructed $u\in X_{m^{\prime\prime}}$.
ali-hunter G. Alì, J.K. Hunter, Nonlinear surface waves on a tangential discontinuity in magnetohydrodynamics, Quart. Appl. Math. 61 (3) (2003) 451–474.
ali-hunter-parker G. Alì, J.K. Hunter, D.F. Parker, Hamiltonian equations for scale-invariant waves, Stud. Appl. Math. 108 (3) (2002) 305–321.
alinhacgerard07 S. Alinhac, P. Gérard, Pseudo-differential operators and the Nash-Moser theorem, Graduate Studies in Mathematics 82, American Mathematical Society, Providence, RI 2007.
axford W.I. Axford, Note on a problem of magnetohydrodynamic stability, Canad. J. Phys. 40 (1962) 654–655.
benzoni-serre S. Benzoni-Gavage, D. Serre, Multidimensional hyperbolic partial differential equations. First-order systems and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
ChenWang G.-Q. Chen, Y.-G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal. 187 (3) (2008) 369–408.
cmst J.-F. Coulombel, A. Morando, P. Secchi, P. Trebeschi, A priori estimates for 3D incompressible current-vortex sheets, Comm. Math. Phys. 311 (1) (2012) 247–275.
hamilton-et-al M.F. Hamilton, Yu.A. Il'insky, E.A. Zabolotskaya, Evolution equations for nonlinear Rayleigh waves, J. Acoust. Soc. Am. 97 (1995) 891–897
hunter-thoo J.K. Hunter, J.B. Thoo, On the weakly nonlinear Kelvin–Helmoltz instability of tangential discontinuities in MHD, J. Hyperbolic Differ. Equ. 8 (4) (2011) 691–726.
majda A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984.
metivier2 G. Métivier, Stability of multidimensional shocks, Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl. 47, 25–103, Birkhäuser Boston, Boston, MA, 2001.
michael D.H. Michael, The stability of a combined current and vortex sheet in a perfectly conducting fluid, Proc. Cambridge Philos. Soc. 51 (1955) 528–532.
morando-trebeschi2013 A. Morando, P. Trebeschi, Weakly well posed hyperbolic initial-boundary value problems with non characteristic boundary, Methods Appl. Anal. 20 (1) (2013) 1–31.
morandotrakhinintrebeschi A. Morando, Y. Trakhinin, P. Trebeschi, Stability of incompressible current-vortex sheets, J. Math. Anal. Appl. 347 (2) (2008) 502–520.
ruzhansky-turunen M. Ruzhansky, V. Turunen, Pseudo-differential operators and symmetries, Pseudo-Differential Operators. Theory and Applications, 2, Background analysis and advanced topics, Birkhäuser Verlag, Basel 2010.
secchi-quart P. Secchi, Nonlinear surface waves on the plasma-vacuum inteface, Quart. Appl. Math. 2015, to appear.
secchi-nash P. Secchi, On the Nash-Moser iteration technique, Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, Birkhäuser-Verlag, to appear.
syrovatskii S.I. Syrovatskij, The stability of tangential discontinuities in a magnetohydrodynamic medium, Zh. Eksper. Teor. Fiz. 24 (1953) 622–629.
trakhinin09arma Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal.191 (2) (2009) 245–310.
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1511.00859
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Holographic thermalization in a top-down confining model
B. Craps$\,^{a}$, E. J. Lindgren$\,^{a,b}$, A. Taliotis$\,^{a}$
$^a$ Theoretische Natuurkunde, Vrije Universiteit Brussel, and
International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium
$^b$ Physique Théorique et Mathématique, Université Libre de Bruxelles,
Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium
[email protected], [email protected], [email protected]
It is interesting to ask how a confinement scale affects the thermalization of strongly coupled gauge theories with gravity duals. We study this question for the AdS soliton model, which underlies top-down holographic models for Yang-Mills theory and QCD. Injecting energy via a homogeneous massless scalar source that is briefly turned on, our fully backreacted numerical analysis finds two regimes. Either a black brane forms, possibly after one or more bounces, after which the pressure components relax according to the lowest quasinormal mode. Or the scalar shell keeps scattering, in which case the pressure components oscillate and undergo modulation on time scales independent of the (small) shell amplitude. We show analytically that the scattering shell cannot relax to a homogeneous equilibrium state, and explain the modulation as due to a near-resonance between a normal mode frequency of the metric and the frequency with which the scalar shell oscillates.
§ INTRODUCTION
What happens to a strongly coupled field theory when it is brought far from equilibrium? This question is important in many areas of physics, including the formation of a quark gluon plasma in ultrarelativistic heavy ion collisions and quantum quenches in cold atom systems. It is a difficult question, however, because conventional techniques fail in the strongly coupled, far-from-equilibrium regime. In recent years, progress has been made using the gauge/gravity duality, also known as “holography”. The simplest AdS/CFT models describe conformal field theories, and the original studies of holographic thermalization focused on those. Interestingly, when extrapolating to heavy ion collisions, one typically finds thermalization times that are short enough to be compatible with experiment <cit.>. Another noteworthy result is that short-
wavelength modes
thermalize first in the simplest holographic models <cit.>. An obvious question is whether there are interesting new effects for non-conformal models, in particular for confining ones.
The study of holographic thermalization in confining models was initiated in <cit.>, where the hard wall model was considered, first in a weak field approximation <cit.> and then using fully backreacted numerical simulations <cit.>. Following <cit.> (see also <cit.>), starting from the ground state, energy was injected by turning on a homogeneous scalar source of amplitude $\epsilon$ for a brief time interval $\delta t$. In the bulk, this leads to a planar shell falling towards the interior of AdS, which in the non-confining context of <cit.> always led to the formation of a black brane, corresponding to thermalization in field theory. In the hard wall model, however, two regimes were found <cit.>. Certain shells collapsed into large black branes, while others kept scattering between the hard wall and the AdS boundary. In the scattering phase, for certain boundary
conditions at the hard wall, the oscillating scalar expectation values underwent interesting modulation on time scales scaling like the inverse amplitude squared, due to resonant transfer of energy <cit.> similar to that discovered in <cit.> for collapse in global AdS.
The hard wall model is simple, but is sometimes criticized for being crude and ad hoc (an interior region of AdS being artificially cut away by the hard wall), and for being rather different from large-$N$ Yang-Mills theory in certain respects (see, for instance, Section 1.1 of <cit.> for a brief discussion). We therefore decided to re-examine these issues in the context of the top-down AdS soliton model <cit.>, which underlies top-down holographic models for Yang-Mills theory <cit.> and QCD <cit.>. In the bulk spacetime corresponding to the ground state, the radial direction and a circle combine into a cigar-shaped geometry, causing the radial direction to cap off smoothly at a radius that sets the confinement scale. While the starting point of the construction in <cit.> was the AdS$_7$ soliton (which after compactification on two circles left the radial direction and 3+1 large field theory dimensions), we will consider AdS solitons in 4,
5, 6 and 7 dimensions.
Starting from the AdS soliton spacetime, we will again inject energy using a minimally coupled massless scalar field with amplitude of order $\epsilon$. An important difference with the hard wall model is that now the metric itself contains dynamics (in the sense that it is not completely determined by constraints), because there is no isotropy between the circle and the other spatial field theory dimensions. If and when a black brane forms, isotropy is restored in the metric components. Given a bulk solution, holographic renormalizaton can be used to extract field theory quantities such as the expectation values of the energy and of the pressure components.
We perform a fully backreacted numerical analysis, and identify a regime in which the infalling shell collapses into a black brane, possibly after one or more bounces, as well as a regime in which the infalling shell keeps scattering between the tip of the cigar (which we will henceforth refer to as the IR) and the AdS boundary. In the former case, we find that the pressure components relax to their (isotropic) equilibrium values according to their lowest quasinormal mode. In particular, the difference in pressure components along the noncompact and compact spatial dimensions of the AdS boundary relaxes to zero as an oscillating exponential. In the scattering phase, when the injected energy is sufficiently small, we show analytically that the shell cannot relax to a homogeneous equilibrium state, and we find numerically that the pressure anisotropy oscillates and undergoes modulation on a time scale that is $\epsilon$-independent in the limit of small amplitude $\epsilon$ and short injection time. This
modulation time scale is very different from the $1/\epsilon^2$ time scale found for the hard wall model, and indeed the physical mechanism is different as well: the oscillations are due to an almost resonance between the oscillation frequency of the scalar shell and the lowest normal mode frequency of the dynamical metric component. Just above the threshhold for black hole formation, we also find solutions that bounce a few times against the AdS boundary before collapsing into a black brane, similar to solutions found in global AdS <cit.>.
Recently, several other papers have studied holographic thermalization in non-conformal models. Based on a quasinormal mode analysis of top-down non-conformal (but gapless) models, it was conjectured in <cit.> that, as soon as a horizon is formed in the bulk, deviations from conformality do not significantly affect thermalization times. Similarly, the numerical analysis in <cit.> found that the equilibration dynamics of $N=4$ SYM theory does not change much when chemical potentials or magnetic fields are added. In <cit.>, quasinormal modes were computed for bottom-up models mimicking the equation of state of QCD, and a non-trivial dependence on the equation of state was found. A confining bottom-up model for QCD was studied nonlinearly in <cit.>, but only for initial conditions that already contain a small black hole; in this case, good agreement with a quasinormal mode analysis was found. In the present paper, we study top-down confining models at the
nonlinear level, in regimes with and without horizons. If and when a horizon forms, the subsequent dynamics is well-described by a quasinormal mode analysis, and the confinement scale does not play much of a role. In the parameter regime in which no horizon forms, the dynamics is dramatically different.
The remainder of this paper is organized as follows. In Section <ref> we describe our setup, including our metric ansatz, the equations of motion and how to extract field theory quantities from the bulk solutions. In Section <ref> we briefly discuss the numerical methods we have used. Section <ref> starts introducing the reader to our numerical results and to the two different phases. In Section <ref> we discuss the black hole phase and in Section <ref> the scattering phase. We conclude in Section <ref>. A number of technical details are contained in three appendices.
§ HOLOGRAPHIC SETUP
The model we will consider is the Einstein-Hilbert action with a minimally coupled massless scalar field,
\begin{equation}
S=\frac{1}{2\kappa^2}\int \rd^{d+1}x\sqrt{-g}\left(R-2\Lambda-\frac{1}{2}(\partial \phi^2)\right)-\frac{1}{\kappa^2}\int_{\partial}\rd^dx\sqrt{-\gamma}\mathcal{K},\label{action}
\end{equation}
where the cosmological constant $\Lambda$ is related to the AdS radius $L$ by $\Lambda=-d(d-1)/2L^2$.
The boundary term is the Gibbons-Hawking-York term, which is necessary to render the variational principle well defined <cit.>, but it will not play a role in this work. We will start with the AdS soliton as an initial condition, and then inject energy into the system by perturbing the scalar field.
The AdS soliton background corresponds to a Euclidean black brane with an extra time direction. An explicit metric can be written as
\begin{equation}
\rd s^2=\frac{L^2}{z^2}\left(-\rd t^2+\frac{\rd z^2}{1-\frac{z^d}{z_0^d}}+(1-\frac{z^d}{z_0^d})\rd \theta^2+\rd \vec{x}_{d-2}^2\right).\label{solitonmetric}
\end{equation}
We will henceforth work in units with $L=1$. The AdS boundary is located at $z=0$ and the confinement scale is set by $z_0$ (which would correspond to the horizon if we Wick rotated back the above to a black brane). Note that the $\theta$ coordinate is compact in order to avoid a conical singularity, and this metric breaks rotational invariance between the $\vec{x}$ coordinates and the $\theta$ coordinate. This will have the implication that in order to solve for the time-dependence of the metric, we will need the second order dynamical Einstein equations. This should be contrasted with rotationally invariant metrics, for which the metric can be determined using first order constraint equations alone.
The massless scalar field will be dual to a marginal operator in the dual field theory. To quench the system we will use a source $J$ coupled to this operator and turn it on for a short period of time. While we imagine the source to vanish outside a finite time interval, in our numerical computations we choose for simplicity a Gaussian profile of the form
\begin{equation}
J(t)=\epsilon e^{-\frac{t^2}{\delta t^2}},\label{source}
\end{equation}
which is indeed negligibly for $|t|\gg \delta t$.
The total injected energy will scale like $E\sim \epsilon^2/\delta t^d$ for small $\epsilon$ and small $\delta t$ <cit.>. In the dual gravitational description, this source term corresponds to the value of $\phi$ at the AdS boundary. After the source has been turned off, the system's energy will have increased and the gravitational bulk solution will have a nontrivial time dependence, governed by the action (<ref>). The main question is how this time-dependent solution behaves, in particular if it collapses into a black brane solution or not. To avoid an extra scalar field, we will also consider quenching the metric, and compare this to the case of perturbing the scalar. We can inject energy by turning on a short time dependence for $\eta_{\theta\theta}/\eta_{x_jx_j}$, where $\eta$ is the boundary metric, which breaks the isotropy of the boundary metric between the $\theta$ and $\vec{x}$ coordinates (see Section <ref>, in particular (<ref>)). This can also be
interpreted as quenching the size of the compactified dimension. In this case, only the gravitational mode will be turned on, and the dynamics will be qualitatively different from the case where both the scalar and the metric mode are turned on. Although we will just very briefly consider such quenches in the metric, it is important to remember that it is possible in this setup to inject energy via the metric without a scalar field and without breaking additional rotational symmetries.
§.§ Ansatz and equations of motion
To solve the Einstein equations we need to choose specific coordinates. We can constrain the form of the metric by using the symmetries of the problem and by suitable gauge transformations (diffeomorphisms), but otherwise the metric will be completely general. In particular, our metric will only depend on time and on the radial bulk coordinate. Also, note in particular that due to parity invariance in the $\theta$ and $\vec{x}$ coordinates, we can set all off-diagonal terms involving these coordinates to zero. We may then use our gauge transformation to bring the metric to a diagonal form with three free functions. Note that the absence of rotational symmetry between the $\theta$ and $\vec{x}$ coordinates in the AdS soliton background forces us to choose a more general ansatz than in setups with rotational symmetry in all spatial coordinates <cit.>, in which case there are usually only two free functions in the metric, which are
completely determined by the constraint equations (there are no propagating degrees of freedom in the metric). We have found that the following ansatz is useful:
\begin{equation}
\rd s^2=\frac{L^2}{z^2}\left(-h(z,t)^2\rd t^2+ \frac{f(z,t)^2}{1-\frac{z^d}{z_0^d}}\rd z^2+ (1-\frac{z^d}{z_0^d})e^{(d-2)b(z,t)}\rd \theta^2+e^{-b(z,t)}\rd \vec{x}_{d-2}^2\right),\label{zansatz}
\end{equation}
with the initial conditions that $f=1$, $h=1$ and $b=0$ before the injection. We will refer to the boundary at $z=0$ as the UV and the point $z=z_0$ as the IR. The coordinate $\theta$ is periodic with period $4\pi z_0/d$ to avoid a conical singularity. This form of the metric has a remaining gauge symmetry corresponding to rescaling of all coordinates. In the numerics, we will use this to set $z_0=1$, but for now we will keep $z_0$ explicit. The coordinates in (<ref>) are very badly behaved at $z=z_0$, however, so for numerics we will use a different ansatz, see Section <ref> and Appendix <ref>.
To inject energy via the metric, we do this via the function $b(z,t)$, namely we can assume a boundary profile on the form
\begin{equation}
b(0,t)=\epsilon e^{-\frac{t^2}{\delta t^2}},\label{bsource}
\end{equation}
and turn off the scalar. It turns out to be convenient to define the following variables
\begin{equation}
\begin{array}{ccc}\label{PPi}
\Pi=\dot{\phi}\frac{f}{h}, &\hspace{20pt} P=\dot{b}\frac{f}{h},\\
\Phi=\phi', &\hspace{20pt} B=b',\\
\end{array}
\end{equation}
where $'$ means derivative with respect to the $z$ coordinate.[We warn the reader that $'$ will have a different meaning in Section <ref> and Appendix <ref>.]
\begin{equation}
\end{equation}
the equations of motion following from the ansatz (<ref>) are
\begin{align}
\dot{f}=&z^{1-2d}\frac{(d-2)(d-1)Gh}{2(G z^{-2(d-1)})'}(PBz+2P)+\frac{z^{2-2d}\Phi\Pi Gh}{(G z^{-2(d-1)})'}+\frac{d-2}{2}Ph\label{zdotf},\\\nonumber
\frac{h'}{h}=&\frac{1}{z^{2(d-1)}(Gz^{-2(d-1)})'}\left(\frac{(d-1)(d-2)}{2}(P^2+GB^2+\frac{4GB}{z})+G\Phi^2+\Pi^2\right)+\\
\frac{h'}{h}=&\frac{2d(d-1)(f^2-1)}{z^{2d}(Gz^{-2(d-1)})'}+\frac{f'}{f},\\
\dot{P}=&\frac{1}{d-1}\left(\frac{(Ge^{(d-1)b})'he^{-(d-1)b}}{fz^{d-1}}\right)'z^{d-1},\\
\dot{\Pi}=&\left(\frac{hG\Phi}{fz^{d-1}}\right)'z^{d-1},\\
\dot{B}=&\left(\frac{Ph}{f}\right)',\\
\dot{\Phi}=&\left(\frac{\Pi h}{f}\right)'.
\end{align}
Note that in this gauge, only derivatives of $b$ appear in the equations of motion, so we do not need to integrate at every time step to obtain $b$. This is the reason for the particular parametrization in (<ref>), which makes the equations of motion decouple nicely. A similar ansatz is used in <cit.>.
Evaluating equation (<ref>) at the point $z=z_0$, and using (<ref>), we obtain
\begin{equation}
\dot{f}_{z=z_0}=(\frac{d-2}{2}P h)_{z=z_0}=(\frac{d-2}{2}\dot{b}f)_{z=z_0},
\end{equation}
so that
\begin{equation} \label{fB}
f_{z=z_0}=C {e^{\frac{d-2}{2}b}}{\Large |}_{r=0}
\end{equation}
for some constant $C$. Since initially $f=1$ and $b=0$, we have that $C=1$. Thus we can state this result as
\begin{equation}
\left(fe^{-\frac{d-2}{2}b}\right)_{z=z_0}=1,\label{feB2}
\end{equation}
which will be crucial for the analysis in Section <ref>. This condition actually is the statement that the regularity (absence of a conical singularity) at $z=z_0$ is preserved in time. This can be easily seen in the ansatz (<ref>), which is used in the numerical analysis, and we refer the reader to Section <ref> for further discussion.
§.§ Boundary expansion and holographic renormalization
To compute field theory observables, one resorts to a process called “holographic renormalization” <cit.>, which requires adding counterterms to the action to cancel divergences from the near-boundary region. These counterterms, which in odd dimensions give finite contributions to the various one-point functions, must be evaluated explicitly for every dimension and quickly become quite involved for increasing dimension. In addition, these contributions make the one-point functions scheme-dependent. However, when the source is turned off, the first non-trivial term in the boundary expansion is of order $z^d$ and no counterterms are needed. Since we will be interested in the evolution of the one-point functions after the source has been turned off, we will therefore be able to ignore the counterterms. In even dimensions the counterterms do not give finite contributions to the one-point functions even when the source is nonzero and thus in this case the counterterms can always
be ignored <cit.>. For $d=3$ we provide the full asymptotic boundary expansion in Appendix <ref>, which will be used in some of the figures.
The asymptotic behaviour of the various fields after the source has been turned off is given by
\begin{align}
\phi=&\phi_dz^d+\ldots,
\end{align}
where the $z^d$ coefficients of $f$ and of $h$ have been related by the equations of motion. We will see later that $E$ will be the total injected energy, while the coefficient $\phi_d$ is related to the vacuum expectation value of the dual operator. We also have that $\dot{E}=0$, which follows from the equations of motion or from holographic Ward identities.
To identify the stress energy components at the boundary, we want to write the metric in the Fefferman-Graham gauge
\begin{equation}
\rd s^2=\frac{\rd \zeta^2}{\zeta^2}+\frac{1}{\zeta^2}g_{\alpha\beta}\rd x^\alpha \rd x^\beta.
\end{equation}
Doing this asymptotically, we can identify $z=\zeta-\zeta^{d+1}\frac{1}{2d}(\frac{E}{d-1}+\frac{1}{z_0^d})+O(\zeta^{d+2})$, which gives us the metric
\begin{align}
\rd s^2=\frac{\rd \zeta^2}{\zeta^2}-\frac{1}{\zeta^2}{\big\{}& \left(1-(E-\frac{1}{z_0^d})\frac{\zeta^d}{d}+O(\zeta^{d+1})\right)\rd t^2\nonumber\\&+\bigg(1+(\frac{E}{d-1}+\frac{1-d}{z_0^d})\frac{\zeta^d}{d}+(d-2)b_d\zeta^d+O(\zeta^{d+1})\bigg)\rd \theta^2\nonumber\\&+\bigg(1+(\frac{E}{d-1}+\frac{1}{z_0^d})\frac{\zeta^d}{d}-b_d\zeta^d+O(\zeta^{d+1})\bigg)\rd \vec{x}^2 {\big\}}. \label{asymptmetric}
\end{align}
Now it is easy to read off the non-zero stress energy components of the boundary field theory <cit.>:
\begin{align}
\langle T_{tt}\rangle=& \frac{1}{16\pi G_N}(E-\frac{1}{z_0^d}),\\
\langle T_{\theta\theta}\rangle=&\frac{1}{16\pi G_N}(\frac{E}{d-1}+\frac{1-d}{z_0^d}+d(d-2)b_d),\\
\langle T_{xx}\rangle=& \frac{1}{16\pi G_N}(\frac{E}{d-1}+\frac{1}{z_0^d}-db_d),
\end{align}
from which we see that $E$ is indeed the total injected energy (up to a factor of $1/16\pi G_N$), and $-\frac{1}{16\pi G_N}\frac{1}{z_0^d}$ is the initial AdS soliton energy density. Note also that $\langle T_\mu^\mu \rangle=0$.[This is not generally true while the source is turned on; see, for example, Eqs (20)-(23) of <cit.>.] The vacuum expectation value of the scalar is
\begin{equation}\label{vev}
\langle \mathcal{O} \rangle=\frac{1}{16\pi G_N}\phi_d.
\end{equation}
Note that taking the difference $\langle T_{\theta\theta}\rangle-\langle T_{xx}\rangle$ cancels the total injected energy $E$ and isolates the dynamical mode $b$, which is why we will prefer to plot this quantity instead of the individual pressure components.
§.§.§ Temperature of black brane solutions
As seen in (<ref>), the energy density will be positive for energies $E>1/z_0^d$, and we expect that black branes will form. A black brane can be written as the metric
\begin{equation}
\rd s^2=\frac{1}{\xi^2}\left(-\rd t^2(1-\frac{\xi^d}{\xi_h^d})+\frac{\rd \xi^2}{1-\frac{\xi^d}{\xi_h^d}}+\rd\theta^2+\rd \vec{x}_{d-2}^2\right).\label{bhmetric}
\end{equation}
Note in particular that in the case of dynamically evolving from the AdS soliton background into the black brane (<ref>), isotropy between the $\vec{x}$ and $\theta$ coordinates must then be restored. From (<ref>) this means that we must have $b_d=\frac{1}{(d-1)z_0^d}$ and this is indeed verified numerically. The temperature of such a black brane, as obtained by the standard procedure of requiring the absence of a conical singularity for the Euclidean version of (<ref>), is given by $T=d/4\pi\xi_h$. Asymptotically, the radial coordinates $\xi$ and $\zeta$ are related by $\xi=\zeta-\zeta^{d+1}/2d\xi_h^d$, from which, comparing with (<ref>), we can obtain the temperature of the black brane,
\begin{equation}
T=\frac{d}{4\pi \xi_h}=\frac{d}{4\pi}\left[\frac{E-\frac{1}{z_0^d}}{d-1}\right]^{1/d}.
\end{equation}
§ NUMERICAL METHODS
In this section we will list some important tricks that we had to employ to achieve stable numerical evolution. We used a fourth order finite difference method to discretize the radial direction, and then we used the ordinary differential equation solver from the library
<cit.> to evolve the resulting system of ordinary differential equations in time. We have as initial conditions $f=1$, $h=1$, $b=0$ and $\phi=0$, corresponding to the AdS soliton geometry. The boundary conditions we impose in the UV are $f(0,t)=1$ and $h(0,t)=1$ as well as $\phi(0,t)=J(t)$ and $b(0,t)=0$ ($b(0,t)=J(t)$ and $\phi\equiv0$ if we quench the metric instead of the scalar), and the source is always taken as a gaussian $J(t)=\epsilon e^{-t^2/\delta t^2}$. In the IR, we do not have to impose any boundary conditions, since regularity already follows from the equations of motion. However, there are some potential sources for numerical instability and inaccuracy, the coordinate singularity in the IR and the AdS boundary being two examples.
§.§ The coordinate singularity
The $z$ coordinate ansatz (<ref>) is very inconvenient in the IR. The reason is that at this point the geometry looks locally like Minkowski space in cylinder coordinates, with rotational invariance in the $(z,\theta)$ plane. However, the relation to the radial coordinate in this locally flat space is $z=z_0(1-r^2)$, and thus $\rd z=-2rz_0 \rd r$. This means that a small grid spacing in $z$ will be mapped to a very large grid spacing in $r$ (which is the natural coordinate around the point $z=z_0$), so a linearly spaced discretization in the $z$ coordinate will become incredibly bad at this point. We thus found it convenient to instead work with the coordinate $r=\sqrt{1-z/z_0}$, and use the metric ansatz
\begin{equation}\label{ransatz}
\rd s^2=\frac{1}{s(r)^2}(-h(r,t)^2\rd t^2+\frac{4f(r,t)^2}{dg(r)} \rd r^2+ r^2g(r)e^{(d-2)b(r,t)}\rd \tilde{\theta}^2+e^{-b(r,t)}\rd \vec{x}_{d-2}^2),
\end{equation}
where $s(r)=z_0(1-r^2)$ and $g(r)=(1-(1-r^2)^d)/(z_0^2r^2d)$. The advantage of this parametrization of the metric is that now $g(r)$ is a finite slowly varying non-zero function and $g(0)=1/z_0^2$. The new periodic coordinate $\tilde{\theta}$ now has period $4\pi z_0^2/d^{3/2}$. While the coordinate system (<ref>) is convenient to derive analytic results and to extract the boundary field theory observables, the coordinate system (<ref>) will be used for the numerical evolution. It is also clear now in these coordinates, that the regularity condition (absence of conical singularity), means that $fe^{(d-2)b/2}$ remains constant in time, which is exactly the statement (<ref>). The equations of motion for the ansatz (<ref>) can be found in Appendix <ref>. The functions $f$, $P$, $\Pi$, $\Phi$ and $B$ are evolved in time, while the function $h$ is solved for at each time step using equation (<ref>), and equation (<ref>) is checked for consistency during the time
There is also a convenient trick that can be employed to compute derivatives close to an origin of a polar coordinate grid. Usually if one were to employ finite differences close to a boundary, one would have to resort to non-symmetric stencils which can induce instabilities or numerical inaccuracies. However, at $r=0$ we do not have a boundary, and we can imagine continuing $r$ past $r=0$ to negative values and thus it is possible to still use central difference schemes when computing derivatives close to $r=0$. An equivalent way of reaching the same result is to use the fact that all functions must be even functions of $r$ when computing the derivatives.
§.§ Radial discretization
We have found that high order finite difference discretization has worked well. However, to avoid high frequency spurious oscillations, we have found that it is convenient to put different functions on two different grids. To motivate this, consider first a function $v(t,z)$ satisfying the free wave equation $\ddot{v}=v''$. Defining $V=v'$ and $P=\dot{v}$ we obtain
\begin{align}
\dot{V}=P',\hspace{20pt}\dot{P}=V',
\end{align}
which should be compared to the equations of motion for the scalar field and the metric component $b$. Now, if we discretize the $z$ coordinate by $\{z_j\}_{j=0}^n$, and consider the derivative approximation $(z_{j+1}-z_{j})/\Delta z$, this will compute an approximation to the derivative at the point $(z_j+z_{j+1})/2$. We thus see that it might be convenient to put $V$ and $P$ on two different grids, one on $\{z_j\}_{j=0}^n$, and one one $\{x_j\}_{j=0}^n$ where $x_j=(z_j+z_{j+1})/2$, to improve the accuracy of the derivative approximations. If one were to use a central difference scheme, we find that it typically induces high frequency noise. This high frequency noise is still present when using higher order central difference schemes, but disappears when putting $V$ and $P$ on different grids (also when we use a higher order finite difference scheme).
In our more complicated setup, the same reasoning holds for the free wave equation in AdS, and we have found it very useful to employ the same trick even when including backreaction. Thus we have put $\Phi$ and $B$ on one grid, and $\Pi$, $P$ and $f$ on the other. Function values are then interpolated to the other grid when necessary. This proves to result in very stable evolution and the high frequency noise that is present when using central difference schemes with all functions on the same grid disappears.
§.§ Extracting boundary data
To extract the boundary data, we will have to compute quantities like $(f(z)-f(0))/z^d$ when $z\rightarrow0$. This becomes increasingly difficult when the dimension increases, since we are taking the ratio of two very small numbers. In particular for $d=6$, there is high frequency noise which makes it difficult to extract the observables. For the simulations of black hole formation (Fig. <ref>), we therefore found it appropriate to use a Savitzky-Golay <cit.> filter to get rid of this noise and to make the boundary observables more smooth in time.
§ PHASE DIAGRAM
When injecting energy into (the Poincaré patch of) vacuum AdS, we always form a black brane. However, since the energy density of the AdS soliton is negative, and any black brane has positive energy density, there should be a threshold for black hole formation when injecting energy into the AdS soliton. The obvious question is then, what solution do we obtain below the threshold? In the probe limit, the scalar field will just bounce forever between the boundary and the IR, so one could ask if this behaviour will still remain when turning on backreaction, or if the system will equilibrate into some static solution after a long time. In Section <ref>, we prove that the system cannot equilibrate into any static solution. We will thus refer to these solutions as the “scattering phase”, and the solutions that thermalize into black holes as the “black hole phase”. In Fig. <ref>, we show the separation between the two different phases, in terms of the parameters $\epsilon$ and $\delta t$.
For small $\delta t$ we have the relation $\epsilon\sim\delta t^{d/2}$, which is expected since the injected energy (which is the only parameter associated to the shell in the thin shell limit) goes like $E\sim\epsilon^2/\delta t^d$ <cit.>. The shapes of the phase diagrams resemble those found for the hard wall model in <cit.>. In particular, for large $\delta t$ we find numerically the relation $\epsilon\sim\delta t$, which is the same as in the hard wall model with Neumann boundary conditions.
The separation between the black hole phase and the scattering phase. For small $\delta t$, we see that $\epsilon\sim\delta t^{d/2}$, which is expected since the total injected energy goes like $E\sim\epsilon^2/\delta t^d$ <cit.>. For large $\delta t$, we find the relation $\epsilon\sim\delta t$.
Another interesting question is if we can find scattering solutions above the energy threshold. Intuitively, right above the threshhold, a wave packet should bounce before collapsing into a black brane due to the finite width of the wave packet, and this is indeed what we find: Right above the threshhold when black brane formation is possible (the energy density is positive), there is a region where solutions reflect many times against the boundary without collapsing (although we are not able to say whether they eventually collapse, due to numerical difficulties in following the solutions for a long time). We have also found solutions that bounce a few times and then collapse into a black hole, similar to what was found in global AdS <cit.>. In Fig. <ref> we plot the number of scatterings before collapse, as a function of amplitude $\epsilon$ for fixed $\delta t=0.24 z_0$. We see that when decreasing $\epsilon$ the number of reflections against the boundary before collapse varies
between 0 and 3, and then for smaller $\epsilon$ there is a large region where the solutions do not seem to collapse.
In Fig. <ref>, we show the vacuum expectation value of the scalar operator and min$\{f/h\}$ as a function of time, for a solution that bounces twice before collapsing into a black hole.
Example of a quench where the scalar wave packet reflects twice at the boundary before collapsing into a black brane. Time is here in units of $z_0$. The parameters are $d=3$, $\epsilon=0.06305472$ and $\delta t=0.24 z_0$. The left axis is for $h/f$ and the right axis is for $\langle \mathcal{O} \rangle$ in units of $1/(16\pi G_N z_0^d)$. Vanishing of $h/f$ signals the formation of an apparent horizon.
After two reflections (identified by the sharp peaks in the vacuum expectation value) we see that min$\{f/h\}$ approaches zero, which indicates the formation of an apparent horizon. If the wave packet is very close to collapsing to a black hole while it scatters in the IR, the wave packet usually becomes very squeezed and comes out almost like a shock wave, resulting in the very sharp peaks in the expectation value $\langle \mathcal{O} \rangle$.
Number of reflections at the boundary of the scalar field wave packet before black brane formation, as a function of $\epsilon$ for $d=3$ and $\delta t=0.24 z_0$. Note that for $\epsilon$ smaller than $0.06304$, there is a parameter region where the injected energy density is above the black brane threshold, but nevertheless the solutions seem to scatter for as long as we have been able to follow them. This region is relatively large, since the threshhold where the energy density becomes negative is $\epsilon\approx0.0607$.
§ BLACK HOLE PHASE
In the black hole phase, the space-time will collapse into a black brane, and a horizon will form. The resulting solution will be an AdS$_{d+1}$ black brane. This in particular means that isotropy between the $\theta$ coordinate and the $\vec{x}$ coordinates will be restored, which in particular means from equation (<ref>) that $b_d=\frac{1}{(d-1)z_0^d}$, and this is indeed verified numerically. Thus the pressure anisotropy $\langle T_{\theta\theta}-T_{xx}\rangle$ will dynamically evolve from $-d/16z_0^d\pi G_N$ to 0. A relevant question is what this isotropization process looks like. In Fig. <ref>, we show a typical evolution of the pressure anisotropy for $d=3$. We see that the system quite rapidly enters a regime where it is isotropic up to some small fluctuations. In Section <ref>, we will compare these small fluctuations with the quasinormal modes of the resulting black brane. Our numerics will not allow us to follow the evolution for very long times after a black hole has
but long enough to see the quasinormal mode behaviour.
Example of a quench ($\delta t=0.1 z_0$, $\epsilon=0.02$) resulting in black hole formation for $d=3$. Time is here in units of $z_0$, and the vacuum expectation values are in units of $1/(16\pi G_N z_0^d)$. The temperature of the black brane in this example is $T\approx0.15/z_0$.
§.§ Quasinormal modes and late time behaviour
For late times we can view the solution as being composed of a black brane background with small fluctuations. The late-time relaxation is thus expected to be governed by the lowest lying quasinormal modes for this black brane. A standard way to illustrate this behaviour is to plot the logarithm of the absolute value of the deviation of some observable from its final value. In Fig. <ref>, a few examples of the deviation of the pressure difference $\langle T_{\theta\theta} \rangle-\langle T_{xx} \rangle$ from 0 are shown. We see that, as expected, the decay time is set by the lowest quasinormal mode, since the decay constants $10.97T$ (for $d=3$), $8.71T$ (for $d=4$) and $5.66T$ (for $d=6$) are in good agreement with the values for the lowest quasinormal mode frequencies of AdS Schwarzschild black branes obtained in <cit.>, namely $11.16T$, $8.63T$ and $5.47T$, respectively.
Log scaled plots of the absolute deviations from zero of the pressure differences in the late time regime of some black hole collapse processes for various dimensions. Time is expressed in units of $z_0$, $\delta t=0.1z_0$, and the vacuum expectation values are given in units of $1/(16\pi G_N z_0^d)$. The decay constants $10.97T$, $8.71T$ and $5.66T$, where $T$ is the temperature, are in good agreement with the lowest quasinormal modes ($11.16T$, $8.63T$ and $5.47T$, respectively) quoted in <cit.>.
§ SCATTERING PHASE
In the scattering phase, the scalar field wave packet that falls from the boundary, will bounce in the deep IR and return to the boundary. When it reaches the boundary there will typically be some excitation of the boundary observables. The wave packet then reflects from the boundary and the scattering repeats. There will be a similar quasiperiodic behavior in the metric, since due to the broken rotational symmetry between the $\vec{x}$ and $\theta$ coordinates the metric has dynamical degrees of freedom of its own. For all figures we have varied the grid spacing to make sure that the results are not numerical artifacts.
In Fig. <ref>, we show a typical scattering solution. As we can see, every time the scalar field wave packet reaches the boundary, there is a bump in the expectation value, and this oscillation goes on forever as far as we know. We can also see that the dynamical degrees of freedom in the metric are excited, as expected, leading to non-trivial behavior in the boundary pressure components.
The pressure components and vacuum expectation value of the scalar operator (in units of $1/(16\pi G_Nz_0^d)$) for a scattering solution in $d=3$ with parameters $\epsilon=0.01$ and $\delta t=0.1z_0$.
One interesting feature is that the interpretation of the scattering solution as a localized wave packet persists for very long times, even for solutions where the non-linearities play a significant role. This is not obvious; one could have imagined that the wave packet would broaden and that at late times we would have seemingly random fluctuations, but instead we see that the wave packet remains approximately localized for long times. However, the shape of the wave packet can change with time due to the non-trivial dynamics of the full Einstein equations, as is reflected in Fig. <ref> for a scattering solution close to the black hole threshold.
The scalar expectation value compared with the probe limit result (in units of $1/(16\pi G_Nz_0^d)$). There is a clear distortion of the wave packet which is due to the non-trivial dynamics in the full Einstein equations and can not be seen in the probe limit, although the wavepacket remains fairly localized. This example is for $d=3$, with parameters $\epsilon=0.01$ and $\delta t=0.1z_0$, and time is in units of $z_0$. Note that this is already quite far into the non-linear regime since black hole formation occurs around $\epsilon\approx0.016$.
In Fig. <ref> we show a typical long time scattering solution when the metric is quenched according to (<ref>). We notice that the pressure anisotropy develops increasingly sharp features after long times, which suggests transfer of energy to high frequency modes. At first sight, this might seem reminiscent of what happens to small-amplitude spherical scalar perturbations in global AdS$_3$ <cit.>, where turbulent transfer of energy to short wavelengths was interpreted as an instability of AdS$_3$. In Fig. <ref> we repeat our analysis for a smaller-amplitude source. While a Fourier analysis of the early and late time behavior in Fig. <ref> confirmed the transfer of energy to higher frequencies, a similar analysis for Fig. <ref> showed no significant transfer to higher frequencies in the time range studied: in the latter case, the spectrum is dominated by normal mode frequencies, with roughly the same strength at early and late times. Decreasing the amplitude of the source further would simply rescale the vertical axis in Fig. <ref>, showing that for small amplitude the dynamics we see happens on timescales independent of the amplitude. The limited time range of our numerical simulation does not allow us to exclude transfer of energy on longer time scales (e.g., scaling as the inverse amplitude). However, based on the absence of resonances in the normal mode spectrum in our setup, we expect no such energy transfer for small amplitudes. If so, the energy transfer observed in Fig. <ref> is the result of strong nonlinearity and quite different from that of <cit.>.
One important question is whether or not these scattering solutions will go on forever, or whether the system will approach some static solution. In section <ref> we will show that, if the injected energy density is below the black brane threshold, the system must keep scattering forever.
The pressure difference after quenching the metric (in units of $1/(16\pi G_Nz_0^d)$), with $\epsilon=0.008$ and $\delta t=0.1z_0$ for $d=3$. Time is in units of $z_0$. We see signs of transfer to high frequency modes for late times.
The pressure difference after quenching the metric (in units of $1/(16\pi G_Nz_0^d)$), with $\epsilon=10^{-4}$ and $\delta t=0.1z_0$ for $d=3$. Time is in units of $z_0$. This figure is unchanged (except for an overall rescaling of the deviations from $-3$), when $\epsilon$ is decreased further. For these small amplitudes, we find no significant transfer to high frequency modes.
§.§ Static solutions and non-thermalization
From equation (<ref>) we see that we are not able to form a black brane if $E<1/z_0^d$. However, one could imagine that there are other static solutions that the system can end up in. We will in this section show that if $E<1/z_0^d$ there are no static solutions that can be obtained through time evolution. To summarize the argument, the key information we get from the dynamical equations is the relation (<ref>). This condition is essentially the requirement that the spacetime should be regular at $z=z_0$ (such that a conical singularity can not be formed at this point during time evolution). We will then consider static solutions, by looking at the static equations of motion, and show that any possible static solution is incompatible with (<ref>). Actually, most of the solutions have a completely different asymptotic behavior at $z=z_0$ and are trivially excluded. The only solutions for which $fe^{-\frac{d-2}{2}b}$ goes to a constant, turns out to be the AdS soliton solutions. However, as we
will see, all AdS solitons except our initial condition soliton will have $fe^{-\frac{d-2}{2}b}$ approaching a different constant than 1, violating (<ref>), so if the injected energy is non-zero, no static solutions can form. This reasoning is reminiscent of the argument for non-thermalization in the hard wall model, given in <cit.>, where we also had a relation similar to (<ref>).
To investigate this we will start by considering a different coordinate system, such that the metric takes the form
\begin{equation}
\rd s^2=\frac{1}{\hz^d}(-\hat{h}^2\rd t^2+\rd \hat{z}^2 \hat{f}^2+\rd \theta^2 e^{(d-2)\hat{b}}+\rd \vec{x}_{d-2}^2e^{-\hat{b}}). \label{newansatz}
\end{equation}
This can be obtained by the coordinate transformations and field redefinitions given by
\begin{equation}
\begin{array}{ccc}
b=\hb-\frac{\log G}{d-1}, &\hspace{10pt} z=\hz G^{\frac{1}{2(d-1)}},\\
\hf^2=\frac{f^2}{G(1-z \frac{G'}{2(d-1)G})^2}, &\hspace{10pt} \hh^2=h^2G^{-\frac{1}{d-1}},\\
\end{array}\label{transformations}
\end{equation}
where $G(z)=1-z^d/z_0^d$ and $G'(z)=-dz^{d-1}/z_0^d$. The $\hz$ coordinate now ranges from $0$ to $\infty$. The (static) equations of motion for such an ansatz are
\begin{equation}
\frac{\hh'}{\hh}=\frac{\hf'}{\hf}-\frac{d}{\hz}(\hf^2-1),\label{hfeq1}
\end{equation}
\begin{equation}
\frac{\hh'}{\hh}=-\frac{\hf'}{\hf}-\frac{d-2}{4}\hz\hat{B}^2-\frac{1}{2(d-1)}\hz\hat{\Phi}^2,\label{hfeq2}
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\end{equation}
where $\hB=\hat{b}'$, $\hat{\Phi}=\phi'$ and prime now denotes derivative with respect to $\hz$. We can integrate (<ref>) and (<ref>) to obtain
\begin{equation}
\hat{\Phi}=C_\phi \frac{\hat{f}}{\hat{h}} \hz^{d-1},
\end{equation}
\begin{equation}
\hat{B}=C_b \frac{\hat{f}}{\hat{h}} \hz^{d-1}.
\end{equation}
where $C_b$ and $C_\phi$ are integration constants, tuning the UV behavior. From (<ref>) we have $\hat{f}/\hat{h}=e^{\int_0^{\hz} \frac{d}{\hz'}(\hat{f}^2-1)}$, so that we obtain the following formulas for $\hat{B}$ and $\hat{\Phi}$
\begin{equation}
\hat{\Phi}=C_\phi e^{\int_0^{\hz} \frac{d}{\hz'}(\hat{f}^2-1)} \hz^{d-1},\label{Phisol}
\end{equation}
\begin{equation}
\hat{B}=C_b e^{\int_0^{\hz} \frac{d}{\hz'}(\hat{f}^2-1)} \hz^{d-1}.\label{Bsol}
\end{equation}
By eliminating $\hat{h}'/\hat{h}$ from (<ref>) and (<ref>) and substituting the expressions in (<ref>) and (<ref>) for $\hat{B}$ and $\hat{\Phi}$ we obtain that $\hat{f}$ must satisfy
\begin{equation}
2\frac{\hf'}{\hf}-\frac{d}{\hz}(\hf^2-1)=-\frac{d-2}{4}C_b^2\hz^{2d-1}e^{\int_0^{\hz}\frac{2d}{\hz'}(\hf^2-1)\rd \hat{z}'}-\frac{1}{2(d-1)}C_\phi^2\hz^{2d-1}e^{\int_0^{\hz}\frac{2d}{\hz'}(\hf^2-1)\rd \hz'}.\label{staticsolutions}
\end{equation}
With the boundary expansion of $f$ being $f=1+\frac{E}{2(d-1)}z^d+\ldots$ (see (<ref>)), we obtain the boundary expansion of $\hf$ as $\hf=1+\frac{(E-1/z_0^d)}{2(d-1)}\hz^d+\ldots$. We expect that black branes will form when the total energy density is positive, which from (<ref>) corresponds to $E-1/z_0^d>0$. Here we will now consider the case $E<1/z_0^d$ (negative energy density) and show that any possible static solutions with negative energy density cannot be obtained dynamically. We emphasize that some solutions of (<ref>) might have singular behaviours and should be excluded as relevant solutions by other arguments, but we will not care about such issues here, and just directly show that any static solutions, which must satisfy (<ref>), cannot be formed dynamically with the AdS soliton as initial condition. Recall the relation (<ref>), which says that we must have $fe^{-\frac{d-2}{2}b}=1$ in the IR when $\hat{z}\rightarrow\infty$ or equivalently $z\
rightarrow z_
0$. The idea is now to show that all solutions obtained by solving (<ref>) are inconsistent with this requirement. We will use the notation $\approx$ to mean that two quantities are equal asymptotically, while $\sim$ means that they are equal asymptotically up to an overall constant.
To show this we will have to compute the IR asymptotic behaviour of the solutions (<ref>). We first note that the derivative $\hat{f}'$ in (<ref>) is negative if $0<\hf<1$. Since $\hf=1+(E-1/z_0^d)\hz^d/2(d-1)+\ldots<1$ close to the boundary, we obtain that $\hf<1$ for all $\hz$.
It is also easy to see that $\hf$ cannot become negative (because (<ref>) implies that if $\hat{f}=0$ then around that point $\hat{f}\sim \hat{z}^{-\alpha}$ for an $\alpha>0$ so $\hat{f}$ can only go to zero asymptotically). Also, $\hat{f}$ can not asymptote to any other constant than zero. This can be seen by assuming that $\hat{f}\rightarrow c>0$, and then (<ref>) implies that $\hat{f}\sim e^{-\alpha\hz^{\beta}}$ for some $\alpha,\beta>0$ which is inconsistent with $\hat{f}\rightarrow c$ (unless if $C_\phi=C_b=0$, in which case $\hat{f}\sim \hat{z}^{-\alpha}$ for $\alpha>0$, which is also inconsistent, or if $\hat{f}\equiv1$). Since $\hat{f}$ is strictly decreasing, it thus follows that we must have $\hf\rightarrow0$ when $\hz\rightarrow\infty$. When $\hz\rightarrow\infty$ we thus have that
\begin{equation}
e^{\int_0^{\hz}\frac{d}{\hz'}(\hf^2-1)\rd \hat{z}'}\approx C' \hz^{-2d}
\end{equation}
for some constant $C'$. For simplicity of notation we can thus redefine $C_b C'=C_{b, IR}$ and $C_\phi C'=C_{\phi, IR}$. The asymptotic behaviour of $\hat{b}$ is then
\begin{equation}
\hat{b}\approx C_{b,IR}\log \hz. \label{bIR}
\end{equation}
Equation (<ref>) now becomes in the IR
\begin{equation}
\end{equation}
from which we can obtain the asymptotic behaviour
\begin{equation}
\hat{f}\sim \hat{z}^{-\frac{d-2}{8}C_{b,IR}^2-\frac{1}{4(d-2)}C_{\phi,IR}^2-\frac{d}{2}}.\label{hatfIR}
\end{equation}
We must now compute the asymptotic relations between ($f$, $b$) and ($\hat{f}$, $\hat{b}$), by using the expressions in (<ref>). We have that $\hz\approx z_0G^{-1/(2(d-1))}$, which directly implies the asymptotic relations
\begin{equation}
b\approx \hat{b}+2\log\frac{\hz}{z_0}\ \
\end{equation}
\begin{equation}
f\approx \frac{d}{2(d-1)}\hat{f}\left(\frac{\hz}{z_0}\right)^{d-1},\label{IRfehatf}
\end{equation}
which imply
\begin{equation}
fe^{-\frac{d-2}{2}b}\approx \frac{d}{2(d-1)}\frac{\hz}{z_0}\hat{f}e^{-\frac{d-2}{2}\hat{b}} .\label{IRfeb}
\end{equation}
From the above relations and (<ref>) and (<ref>) we now obtain the asymptotic behaviour
\begin{equation}
b\approx(C_{b,IR}+2)\log \hz,\label{BIR}
\end{equation}
\begin{equation}
f^2\sim \hz^{\left(2(d-1)-\frac{d-2}{4}C_{b,IR}^2-\frac{1}{2(d-2)}C_{\phi,IR}^2-d\right)},
\end{equation}
so that we finally obtain
\begin{equation}
\end{equation}
We thus see that $fe^{-\frac{d-2}{2}b}$ will generically vanish in the IR, trivially violating the condition that $fe^{-\frac{d-2}{2}b}\rightarrow1$. The only way to have $fe^{-\frac{d-2}{2}b}$ approach a constant in the IR, is when the power in (<ref>) vanishes. This only happens when $C_{\phi,IR}=0$ and $C_{b,IR}=-2$, which in particular implies that the scalar identically vanishes. Only for these particular IR parameters will $fe^{-\frac{d-2}{2}b}$ go to a constant. We will now show, however, that it will go to a constant different from 1.
To specify a solution in the bulk, it would be customary to specify the UV behavior, meaning that we specify $E$ and $C_{b}$ and then integrate to the IR, which should give a unique solution. Specializing to $C_{b,IR}=-2$ should leave us a one parameter family of static solutions. Below we will construct this one parameter family of solutions, which turns out to be all the AdS solitons.
An AdS soliton solution with a general confinement scale $z_1$, can be given by the metric (<ref>) with $b=0$ and $f=h=1$ with $z_0$ replaced by $z_1$. After transforming to the metric (<ref>), by using the transformations in (<ref>), we can obtain the asymptotic behavior for $\hf$ and $\hat{b}$ as $\hat{b}\approx-2 \log \frac{\hz}{z_1}$ and $\hf\approx\frac{2(d-1)}{d} \left(z_1/\hz\right)^{d-1}$.
We can now easily obtain from (<ref>) that $fe^{-\frac{d-2}{2}b}\rightarrow z_1/z_0$. Thus, the only possible solution that can be obtained is $z_1=z_0$ which corresponds to our initial condition, and which corresponds in the UV to $E=0$.
To conclude, when the total energy density lies between that of the AdS soliton and zero (the threshold for black brane formation), no static solutions exists.
§.§ Long time amplitude modulation
For small-amplitude scattering solutions (small $\epsilon$), we observe an amplitude modulation in the pressure anisotropy on a long time scale, see Fig. <ref>. (The relevant timescale is actually hard to see for $d=6$, for reasons we will explain below.) The time scale can be seen to be independent of $\epsilon$ and $\delta t$ as long as both parameters are sufficiently small. This is different from the $1/\epsilon^2$ modulation time scale observed in <cit.> for the $d=3$ hard wall model with Neumann boundary conditions. As we will now explain, in the present case the modulation is due a near-resonance between a metric mode and the bouncing scalar shell. (In the $d=3$ hard wall model with Neumann boundary conditions, no dynamical metric modes were excited, and the modulation was due to a resonant spectrum of scalar field normal mode frequencies.)
In the small $\epsilon$ regime, the scalar field is $\mathcal{O}(\epsilon)$. Thus the metric is of order $\mathcal{O}(\epsilon^2)$ and the next order corrections to the scalar are $\mathcal{O}(\epsilon^3)$. Working to order $\mathcal{O}(\epsilon^2)$, we can therefore consider $\phi$ as a probe scalar acting as an external source on the metric. Since the scalar $\phi$ bounces back and forth between the IR and the boundary, the source for the metric backreaction can be characterized by a frequency[Not to be confused with the metric component $f$ from previous sections.]
$f_\phi$. In the limit of small $\delta t$ (the thin shell limit), this frequency will be the same as for a lightlike particle (following lightlike geodesics) that bounces between the boundary and the interior. In particular, for small $\delta t$ the bounce frequency becomes independent of $\delta t$.
However, the metric also has some intrinsic frequencies $f_i$ (the normal mode frequencies, see Appendix <ref>). Every time the scalar crosses the space-time it kicks the metric. It is useful to decompose the metric fluctuation into its normal modes. If $f_\phi=f_j$ for some $j$, we would expect a resonance, such that the amplitude of the $j$'th normal mode will increase linearly with time. But if $f_\phi\approx f_j$, such that we are close to resonance, it would be natural to expect another time scale showing up, namely $T=1/|f_\phi-f_j|$. The results are summarized in table <ref>, and can be compared with the numerical results in Fig. <ref> and Fig. <ref>. The latter figure shows the decomposition of $b$ in its normal modes, where the decomposition is of the form $b(z,t)=\sum_{n\geq0} a_n(t)Q_n(z)$. The functions $Q_n$ (corresponding to the $n$th normal mode with frequency $\omega_n$) satisfy the equation (<ref>) with $\omega=\omega_n$,
which constitutes a Sturm-Liouville problem (which makes this decomposition possible), and they are normalized with respect to the inner product $\int_0^{z_0} Q_n(z)Q_m(z)z^{-(d-1)}dz=\delta_{mn}$. Note that replacing the frequency of the scalar wave packet by that of a light-like thin shell still gives decent result, but using the true frequency is required to get accurate results, especially for $d=5$ (where the system is very close to resonance). To summarize, the modulation can be traced back to a near resonance between the lowest normal mode frequency of metric perturbations and the frequency of a bouncing scalar shell. As expected from this picture, this type of modulation does not show up when we quench the metric instead of the scalar field, as can be seen in Fig. <ref>.
One can see from the numerics, however, that the metric perturbation consists not only of a few normal modes: it has a slowly moving normal mode part and a rapidly moving wave packet part. The wave packet part is in general smaller than the normal mode part. However, close to the boundary, the wave packet part can still give large contributions to the boundary observables. The intuitive explanation is as follows. Close to the boundary a wave packet looks typically like $~\psi((z-t)/\delta t)$, where $\psi$ is some localized profile and $\delta t$ is the width. Thus when extracting the $z^d$ coefficient when computing the boundary observables, this will be proportional to $\partial_z^d\psi((z-t)/\delta t)\sim 1/\delta t^d$, while the derivatives of the normal modes are of $\mathcal{O}(1)$. We thus see that for larger dimensions, the wave packet part is expected to become more important. These are exactly the sharp peaks one can see in Fig. <ref>, and indeed they become larger for larger
dimansions. For $d=6$
they completely dominate and this is the reason why we cannot see the modulation due to the first normal mode in the vacuum expectation value in $d=6$. However, it can still be seen in the normal mode decomposition in Fig. <ref>, since here the contribution from the wave packet part is still small.
The pressure anisotropy after a weak scalar perturbation with small $\epsilon$ and $\delta t=0.1z_0$. Time is measured in units of $z_0$, and the vertical axis has been rescaled by $\epsilon^2/\delta t^d$, which is the expected dependence of the total energy of the system for small $\epsilon,\delta t$. For $d=3,4,5$ we see that the amplitude undergoes an amplitude modulation on a much longer time scale which is in excellent agreement with the result in table <ref>. For $d=6$, the modulation due to the first normal mode is hidden by the peaks from the bouncing wave packet part; it is clearly visible, however, in the normal mode decomposition in Fig. <ref>.
The metric function $b$ decomposed in normal modes, after a weak scalar perturbation with small $\epsilon$ and $\delta t=0.1z_0$. Time is measured in units of $z_0$. We see that, as expected, the lowest mode is more excited than higher modes, and undergoes an amplitude modulation which agrees with the result in Table <ref>.
$d$ 3 4 5 6
$z_0f_0$ 0.34195 0.37177 0.40151 0.43004
$z_0f$ 0.35682(19) 0.38190(36) 0.39944(16) 0.41263(43)
$z_0f'$ 0.3564 0.3807 0.3986 0.4117
$T/z_0$ 67.2(8) 98.7(35) 484(37) 57.4(14)
$T'/z_0$ 69.2 111.9 344 54.5
The lowest normal mode frequency $f_0$, the oscillation frequency of the scalar wave packet $f$ for $\delta t=0.1z_0$, the oscillation frequency of a lightlike thin shell $f'$ and the expected modulation times $T=1/|f_0-f|$ and $T'=1/|f_0-f'|$ using $f$ respectively $f'$. Note that the frequency of a thin shell is extremely close to the frequency of the bouncing scalar field. However, note also that in $AdS_6$ ($d=5$) we are extremely close to resonance, and to get an accurate modulation time we must use the true frequency of the scalar wave packet (compare with Fig. <ref> and Fig. <ref>). The estimated error comes from reading off the oscillation frequencies of the wave packet from the numerical simulations, while the errors of $f_0$ and $f'$ are negligible.
§.§.§ Harmonic oscillator toy model
To develop a better understanding of the modulations we have just described, we now study a sourced harmonic oscillator which is conceptually similar to our gravitational setup (in the small-amplitude scattering phase) and which experiences a similar amplitude modulation phenomenon.
Consider a harmonic oscillator with angular frequency $\omega$, sourced by a sequence of local kicks (modelled by delta functions) with period $T$. (We denote the frequency of the kicks by $f=1/T$.) The equation of motion is
\begin{equation}
\ddot{x}+\omega^2x=\sum_{n\geq0} \delta(t-nT),\label{oscillator}
\end{equation}
subject to the initial condition that $x(t)$ should vanish for $t<0$. To compare with our gravitational setup, $x$ is the analogue of $B$ (the metric backreaction), the delta functions are analogous to the stress energy tensor for the scalar $\phi$ which sources the metric, the frequency $f=1/T$ is analogous to the oscillation frequency $f_\phi$ of $\phi$, and $\omega$ is analogous to the normal mode frequencies of the metric perturbations. We can solve (<ref>) by performing a Laplace transform. For the Laplace transformed field $X$ we have
\begin{equation}
s^2X(s)+\omega^2X(s)=\sum_{n\geq0} e^{-nTs}\Rightarrow X(s)=\frac{1}{2i\omega}\sum_{n\geq0}\left(\frac{e^{-nTs}}{s-i\omega}-\frac{e^{-nTs}}{s+i\omega}\right).
\end{equation}
It is now easy to do the inverse Laplace transform, to obtain
\begin{equation}
\end{equation}
where $\theta(t)$ is the Heaviside step function. By letting $N=\lfloor t/T \rfloor$ (the largest integer less than or equal to $t/T$), we can write this as
\begin{equation}
\omega x(t)=\sum_{n=0}^N\sin(\omega(t-nT))=\mathrm{Im }\enspace e^{i\omega t}\sum_{n=0}^Ne^{-i\omega nT}=\mathrm{Im }\enspace e^{i\omega t}\frac{1-e^{-i\omega (N+1)T}}{1-e^{-i\omega T}},\label{geosum}
\end{equation}
under the assumption that $T\omega\not\equiv 0 \pmod{2\pi}$. Extracting the imaginary part and using some trigonometric identities, we obtain
\begin{equation}
x(t)=\frac{2\sin\left[\frac{\omega T}{2}\right]\sin\left[\omega(t-\frac{NT}{2})\right]\sin\left[\omega\frac{N+1}{2}T\right]}{\omega(1-\cos\left[\omega T\right])}.\label{3sine}
\end{equation}
If the system is almost at resonance, $T\omega\approx2\pi$, the middle factor in (<ref>) will give rise to fast oscillations, while the last factor gives rise to slow amplitude modulations. To see this, we write $f-\omega/2\pi=\epsilon\ll f$ (so $\epsilon$ is the difference between the source frequency and the oscillator frequency). The third factor in (<ref>) now becomes
\begin{align}
\sin\left[\omega\frac{N+1}{2}T\right]&=\sin\left[\pi(1-\epsilon T)(N+1)\right]=\pm \sin \left[\pi \epsilon T (N+1)\right]\nonumber\\
&\approx\pm \sin \left[\pi \epsilon (t+T)\right],
\end{align}
where in the last step we approximated $N=\lfloor t/T \rfloor\approx t/T$, and we see that we indeed obtain an overall amplitude modulation with period $1/\epsilon$. An example with $\omega=1$ and $\epsilon=0.05/2\pi$ is shown in Fig. <ref>. Further, we note that it is the small denominator in (<ref>) that causes a near-resonant normal mode to dominate the other normal modes.
If $T\omega =2\pi k$, for some non-zero integer $k$, the summation of the geometric series in (<ref>) yields instead
\begin{equation}
x(t)=\frac{N+1}{\omega}\sin\omega t\approx \frac{t+T}{2\pi k}\sin\omega t,
\end{equation}
which is a sine function with a (step-wise) increasing amplitude, as expected when we are at resonance. This can also be obtained as a limit $T\omega\rightarrow 2\pi k$ in (<ref>).
Evolution of the sourced harmonic oscillator given by equation (<ref>), close to resonance with parameters $\omega=1$ and $T=2\pi/1.05$. We see that the result is a rapidly oscillating solution with a long time modulation with period $2\pi/0.05\approx125.7$.
§ CONCLUSIONS AND OUTLOOK
A common feature of the hard wall model and the AdS soliton is the energy gap between the ground state and the lightest black hole. This feature underlies the dynamical phase structure uncovered in <cit.> (for the hard wall model) and in the present work (for the AdS soliton).
In the black hole phase, we have found that the late-time decay is governed by the lowest quasinormal mode, as could have been expected. It was nice, however, to find numerical solutions in this model that collapse after one or more bounces, which is reminiscent of spherical collapse in global AdS <cit.> – it would be interesting to study the similarities and differences in more detail. A good starting point could be the observation that the AdS$_3$ soliton is identical to global AdS$_3$.
In the scattering phase, we have established that, if the injected energy density is below the black hole threshold, relaxation to a static solution that is translationally invariant in the spatial directions along the boundary is impossible. An interesting question for future work is whether there are instabilities towards the formation of inhomogeneities in the $\vec{x}$ or $\theta$ directions.[We thank Mark Van Raamsdonk for suggesting such a possibility in a discussion on the hard wall model.] In addition to the mere existence of scattering solutions, we have found that the pressure components exhibit clear amplitude modulation, and we have explained this as due to a near-resonance between a scalar shell bouncing frequency and a metric normal mode. It would be interesting to see whether scattering solutions can also be established for more phenomenological holographic models, and if so, whether they exhibit similar features.
§ ACKNOWLEDGEMENTS
We would like to thank Ioannis Papadimitriou, Joris Vanhoof and Hongbao Zhang for useful discussions. This work was supported in part by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P7/37, by FWO-Vlaanderen through project G020714N, and by the Vrije Universiteit Brussel through the Strategic Research Program “High-Energy Physics”. EJL is supported by a PhD fellowship from the Research Foundation Flanders (FWO); his work was also partially supported by the ERC Advanced Grant “SyDuGraM", by IISN-Belgium (convention 4.4514.08) and by the “Communauté Française de Belgique" through the ARC program.
§ EQUATIONS OF MOTION SUITABLE FOR NUMERICS
In this appendix we will present the equations of motion for the ansatz (<ref>) used for numerics. We stress that here $'$ will mean derivative with respect to the $r$ coordinate, which is different from the notation in the main text. Also, the $B$ and $\Phi$ below are not the same as in the main text. Defining
\begin{equation}
\begin{array}{ccc}
P=\dot{b}\frac{f}{h}, &\hspace{20pt} \Pi=\dot{\phi}\frac{f}{h},\\
\Phi=\phi', &\hspace{20pt} B=b',\\
\end{array}
\end{equation}
the equations of motion following from (<ref>) are
\begin{align}
\frac{h'}{h}=&\frac{r^2}{s^{2d-2}(\frac{gr^2}{s^{2d-2}})'}\Big(\frac{2(d-1)(d-2)}{d}P^2+\frac{4}{d}\Pi^2+g\Phi^2+\\\nonumber
&+\frac{(d-1)(d-2)}{2}gB^2+\frac{2(d-2)(d-1)gB s'}{s}\Big)+(d-2)B-\frac{f'}{f},\label{heq}\\
\dot{P}=&\frac{d}{4(d-1)r}\left(\frac{\left(ge^{(d-1)b}r^2\right)'he^{-(d-1)b}}{fs^{d-1}r}\right)'s^{d-1},\\
\dot{f}=&\frac{(d-2)(d-1)gr^2P h B s + 2(d-2)(d-1) gr^2 P h s'+2gr^2\Pi\Phi hs}{2(\frac{gr^2}{s^{2d-2}})'s^{2d-1}}\nonumber\\
&+\frac{d-2}{2}P h,\\
\frac{h'}{h}=&\frac{8(d-1)(f^2-1)r^2}{(r^2gs^{-2(d-1)})'s^{2d}}+\frac{f'}{f},\label{constraint}\\
\dot{B}=&\left(\frac{P h}{f}\right)',\\
\dot{\Pi}=&\frac{d}{4r}\left(\frac{hg\Phi r}{fs^{d-1}}\right)'s^{d-1},\\
\dot{\Phi}=&(\frac{\Pi h}{f} )',
\end{align}
where $s(r)=z_0(1-r^2)$ and $g(r)=(1-(1-r^2)^d)/(z_0^2r^2d)$.
§ NORMAL MODES
To get some analytic understanding of the dynamics of the metric, we will look for normal modes of the metric perturbations. These are solutions of the linearized equations of motion (small perturbations around the soliton background) that can be written as a product of a radial function and a harmonically oscillating function of time. The normal mode spectrum is expected to be discrete in confined geometries, and we saw in Section <ref> that these normal mode frequencies explain the amplitude modulation of the pressure anisotropy.
To find such solutions, we assume that $b=\mathcal{O}(\mu)$, for some small parameter $\mu$, and that the scalar field vanishes. We then solve the equations to linear order in $\mu$. So letting $f=1+\mu f_1+\ldots$, $h=1+\mu h_1+\ldots$, $B=1+\mu B_1+\ldots$ and $P=1+\mu P_1+\ldots$ we obtain
\begin{align}
\dot{f}_1&=z^{1-2d} \frac{(d-2)(d-1)}{(G z^{-2(d-1)})'}P_1G+\frac{d-2}{2}P_1,\\
\dot{P}_1&=\frac{1}{d-1}(G'(h_1-f_1)z^{-(d-1)}+(d-1)B_1Gz^{-(d-1)})'z^{d-1},\\
\dot{B}_1&=P_1',\\
\end{align}
where $G(z)=1-z^d/z_0^d$. To look for normal modes, we make the ansatz $P_1=Q(z)\cos \omega t$. This implies that $B_1=Q'(z) \sin \omega t /\omega$, $f_1=F(z)\sin\omega t$ and $h_1=H(z)\sin\omega t$, and the functions $Q$, $F$ and $H$ satisfy the ordinary differential equations
\begin{align}
\omega F &= z^{1-2d} \frac{(d-2)(d-1)}{(G z^{-2(d-1)})'}QG+\frac{d-2}{2}Q,\label{Feq}\\
-\omega Q&=\frac{1}{d-1}(G'(H-F)z^{-(d-1)}+\frac{(d-1)Q'G}{\omega }z^{-(d-1)})'z^{d-1},\label{Qeq}\\
\end{align}
Actually it is possible to extract from these equations a single ordinary differential equation for $Q$. Since $(G'(H-F)z^{-(d-1)})'=G'z^{-(d-1)}(H'-F')$, we can eliminate $H'-F'$ from equation (<ref>) by using equation (<ref>), and then use (<ref>) to eliminate the remaining $F$ such that we end up with
\begin{equation}
-\omega^2 Q=G'\frac{4d}{z^{2d}(Gz^{-2(d-1)})'}\left(z^{1-2d} \frac{(d-2)(d-1)}{(G z^{-2(d-1)})'}QG+\frac{d-2}{2}Q\right)+\left(\frac{Q'G}{z^{d-1}}\right)'z^{d-1},\label{Qeqbis}
\end{equation}
which is a second order ordinary differential equation for $Q$. Demanding regularity in the IR, the only free parameter is $\omega$ (since we can set $Q(z_0)=1$ by an overall rescaling). Then demanding that $Q$ should be normalizable at the boundary (equivalent to $Q(0)=0$), gives us a discrete set of allowed frequencies $\omega$. These are the frequencies of the normal modes, and can be seen in Table <ref>.
$d$ 3 4 5 6
$z_0\omega_0$ 2.14853 2.33587 2.52274 2.70203
$z_0\omega_1$ 4.790 5.517 6.200 6.854
$z_0\omega_2$ 7.116 8.069 8.925 9.719
The lowest normal mode frequencies for metric perturbations in various dimensions. The normal mode frequencies are inversely proportional to the confinement scale $z_0$.
§.§ Normal modes for the scalar field
Although not relevant for this work, it is interesting to note that the normal modes of the scalar field satisfy (<ref>) but with the first term in the RHS, which is proportional to $Q$, removed:
\begin{equation}
-\omega^2 Q=\left(\frac{Q'G}{z^{d-1}}\right)'z^{d-1},\label{Qeqscalar}
\end{equation}
if we assume that $\phi=Q\cos\omega t$. Given that the omitted term is proportional to $Q$, and therefore combines with the LHS, the normal modes of the scalar and the metric fluctuations can be expected to approach each other for large $\omega$. In addition, we have observed numerically that the spectrum becomes linear for large $\omega$. Global AdS$_3$ is actually identical to the AdS$_3$ soliton, which leads us to expect that (<ref>) must give a linear spectrum for $d=2$. Indeed, in this case if we redefine $Q=z^2q$ and $z^2/z_0^2=(x+1)/2$, we obtain the equation
\begin{equation}
\end{equation}
which is solved by the Jacobi polynomials $q(x)=P_n^{(\alpha,\beta)}(x)$ with $\alpha=0,\beta=1$ and with $\omega=2(n+1)$ for $n=0,1,2,\ldots$.
§ ASYMPTOTIC EXPANSION FOR $D=3$
Here we provide the complete asymptotic expansion for $d=3$, including the time window when the source is turned on. We will thus assume a source $J_b$ for the function $b$ and a source $J_\phi$ for the scalar field $\phi$. The asymptotic expansions for the various functions, following from the equations of motion, are then
\begin{equation}
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\phi=J_\phi-\frac{1}{2}\ddot{J}_\phi z^2+\phi_3z^3+O(z^4).
\end{equation}
We also obtain the Ward identity
\begin{equation}
3(2b_3-\frac{1}{z_0^3})\dot{J}_b+6\dot{J}_\phi \phi_3+2\dot{E}=0.
\end{equation}
When going to Fefferman-Graham gauge as in Section <ref>, the intermediate $z^2$ terms will not affect the $z^3$ terms. Moreover, since in even dimensional AdS spaces the counterterms do not affect the field theory observables, the boundary observables are given by the same formulas (<ref>) and (<ref>), even when the sources are turned on. The Ward identity then takes the form
\begin{equation}
3\left(\langle T_{\theta\theta}\rangle-\langle T_{xx}\rangle\right)\dot{J}_b+6\langle \mathcal{O}\rangle\dot{J}_\phi+2\langle T_{tt} \rangle=0.
\end{equation}
Y. V. Kovchegov and A. Taliotis,
Phys. Rev. C 76 (2007) 014905
[arXiv:0705.1234 [hep-ph]].
J. L. Albacete, Y. V. Kovchegov and A. Taliotis,
JHEP 0807 (2008) 100
[arXiv:0805.2927 [hep-th]].
P. M. Chesler and L. G. Yaffe,
Phys. Rev. Lett. 102 (2009) 211601
[arXiv:0812.2053 [hep-th]].
G. Beuf, M. P. Heller, R. A. Janik and R. Peschanski,
JHEP 0910 (2009) 043
[arXiv:0906.4423 [hep-th]].
J. Abajo-Arrastia, J. Aparicio and E. Lopez,
JHEP 1011 (2010) 149
[arXiv:1006.4090 [hep-th]].
T. Albash and C. V. Johnson,
New J. Phys. 13 (2011) 045017
[arXiv:1008.3027 [hep-th]].
V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps, E. Keski-Vakkuri, B. Muller and A. Schafer et al.,
Phys. Rev. Lett. 106 (2011) 191601
[arXiv:1012.4753 [hep-th]].
M. P. Heller, R. A. Janik and P. Witaszczyk,
Phys. Rev. Lett. 108 (2012) 201602
[arXiv:1103.3452 [hep-th]].
M. P. Heller, D. Mateos, W. van der Schee and D. Trancanelli,
Phys. Rev. Lett. 108 (2012) 191601
[arXiv:1202.0981 [hep-th]].
V. Balasubramanian, A. Bernamonti, J. de Boer, B. Craps, L. Franti, F. Galli, E. Keski-Vakkuri and B. Müller et al.,
Phys. Rev. Lett. 111 (2013) 231602
[arXiv:1307.1487 [hep-th]].
S. Lin and E. Shuryak,
Phys. Rev. D 78 (2008) 125018
[arXiv:0808.0910 [hep-th]].
B. Craps, E. Kiritsis, C. Rosen, A. Taliotis, J. Vanhoof and H. Zhang,
JHEP 1402 (2014) 120
[arXiv:1311.7560 [hep-th]].
B. Craps, E. J. Lindgren, A. Taliotis, J. Vanhoof and H. b. Zhang,
Phys. Rev. D 90, no. 8, 086004 (2014)
[arXiv:1406.1454 [hep-th]].
S. Bhattacharyya and S. Minwalla,
JHEP 0909 (2009) 034
[arXiv:0904.0464 [hep-th]].
P. Bizon and A. Rostworowski,
Phys. Rev. Lett. 107 (2011) 031102
[arXiv:1104.3702 [gr-qc]].
E. Witten,
Adv. Theor. Math. Phys. 2 (1998) 505
T. Sakai and S. Sugimoto,
Prog. Theor. Phys. 113 (2005) 843
A. Buchel, M. P. Heller and R. C. Myers,
Phys. Rev. Lett. 114 (2015) 25, 251601
[arXiv:1503.07114 [hep-th]].
J. F. Fuini and L. G. Yaffe,
JHEP 1507 (2015) 116
[arXiv:1503.07148 [hep-th]].
R. A. Janik, G. Plewa, H. Soltanpanahi and M. Spalinski,
Phys. Rev. D 91 (2015) 12, 126013
[arXiv:1503.07149 [hep-th]].
T. Ishii, E. Kiritsis and C. Rosen,
JHEP 1508 (2015) 008
[arXiv:1503.07766 [hep-th]].
G. W. Gibbons and S. W. Hawking,
Phys. Rev. D 15, 2752 (1977).
B. Wu,
JHEP 1210 (2012) 133
[arXiv:1208.1393 [hep-th]].
S. de Haro, S. N. Solodukhin and K. Skenderis,
Commun. Math. Phys. 217 (2001) 595
K. Skenderis,
Class. Quant. Grav. 19 (2002) 5849
I. Papadimitriou,
JHEP 1108 (2011) 119
[arXiv:1106.4826 [hep-th]].
Eric Jones and Travis Oliphant and Pearu Peterson and others,
SciPy: Open source scientific tools for Python,
Savitzky, A. and Golay, M.J.E. (1964). "Smoothing and Differentiation of Data by Simplified Least Squares Procedures". Analytical Chemistry 36 (8): 1627–39.
G. T. Horowitz and V. E. Hubeny,
Phys. Rev. D 62 (2000) 024027
P. Bizoń and J. Jałmużna,
Phys. Rev. Lett. 111 (2013) 4, 041102
[arXiv:1306.0317 [gr-qc]].
|
1511.01079
|
$^1$ Max-Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1, 85748 Garching, Germany
$^2$ Space Telescope Science Institute, 3700 San Martin Drive, Baltimore MD 21218
There is now strong evidence that Long-duration Gamma-Ray Bursts (LGRBs) are preferentially formed in low-metallicity environments. However, the magnitude of this effect, and its functional dependence on metallicity have not been well characterized. In our previous paper, <cit.>, we compared the metallicity distribution of LGRB host galaxies to the that of star forming galaxies in the local universe. Here we build upon this work by in effect dividing one distribution by the other, and thus directly determine the relative rate of LGRB formation as a function of metallicity in the low-redshift universe. We find a dramatic cutoff in LGRB formation above a metallicity of $ {\rm log(O/H)}+12 \approx 8.3 $ in the KK04 scale, with LGRBs forming between ten and fifty times more frequently per unit star-formation below this cutoff than above. Furthermore, our data suggests that the rate of LGRB formation per unit star formation continues to fall above this break. We estimate the LGRB formation rate per unit star formation may drop by as much as a factor of one hundred between one-third solar and solar metallicity.
§ INTRODUCTION
Shortly after Long-soft Gamma-Ray Bursts (LGRBs) were identified as extragalactic events spanning cosmological distances, it became apparent that they predominantly occur in blue, highly starforming and often irregular galaxies <cit.>. To determine whether this distribution of hosts was different from that expected from a sample of galaxies drawn randomly according to their rate of massive star formation, <cit.> compared the hosts of LGRBs with those of Core-Collapse Supernovae (CCSNe) found in the Great Observatories Origins Deep Survey (GOODS). They found that while half of the GOODs CCSNe occurred in grand design spirals (with the other half in irregulars), only one out of the 18 LGRB host galaxies of a comparable redshift distribution was in a grand design spiral. Using an enlarged sample, <cit.> found a very similar result. Massive stellar progenitors should be just as available per unit star-formation in spirals as they are in irregulars, as it appears the stellar IMFs of blue irregulars and spirals are largely similar <cit.>, However, due to the the mass-metallicity relationship of galaxies <cit.>, blue irregulars are typically far less-metal rich than spirals. This lead <cit.> to conclude that LGRB formation is much more likely in low-metallicity environments.
A similar conclusion was reached by <cit.>, who showed that the very nearest LGRB hosts all have low metallicity when compared to similar magnitude galaxies in the Sloan Digital Sky Survey (SDSS) sample. Furthermore, <cit.> found the LGRB host sample to be comparable to extremely metal-poor galaxies in luminosity-metallicity relation, star-formation rate (SFR), and internal extinction.
<cit.> dramatically strengthened this result by taking advantage of the fact that a broad-lined Type Ic (Ic-bl) supernova has been found underlying the light of nearly every LGRB in which a deep spectroscopic search was performed (). <cit.> showed that Ic-bl SNe with associated LGRBs are observed to occur in host galaxies with much lower metallicities than either the hosts of Type Ic-bl SNe without associated LGRBs or the bulk of the star-forming galaxies in the SDSS. This dramatic metallicity difference between the Ic-bl and LGRB samples suggests a metallicity dependent step in either the formation of the gamma-ray jet or in its ability to escape the progenitor which has either burned or lost its outer hydrogen and helium layers <cit.>.
More recently, however, <cit.> has suggested that the metallicity aversion of LGRBs is not intrinsic to their formation, but rather a consequence of a fundamental relationship between the mass, metallicity, and star-formation rates of galaxies <cit.>. In this relationship the metallicity of a galaxy of a given stellar mass is anti-correlated with its SFR. Thus <cit.> argued that the LGRB hosts are low-metallicity because they are effectively selected based on the basis of their higher then average star-formation. However this argument does not explain why LGRBs should preferentially choose irregular hosts more frequently than the general population of core collapse SNe <cit.> or why the Type Ic-bl SNe without associated LGRBs do not show a preference for low-metallicity hosts comparable to the Type Ic-bl SNe associated with LGRBs (c.f. ).
In our preceding work, <cit.>, we compared the metallicity distribution of the hosts of LGRBs with that of the hosts of several similar indicators of star-formation: LGRBs, Type Ic-bl, and Type II SNe. We found that three quarters of the LGRB hosts have metallicities below 12+log(O/H) $<$ 8.6 in the KK04 metallicity scale <cit.>, while less than a tenth of local star-formation is at similarly low metallicities. However, our supernova samples were statistically consistent with the metallicity distribution of the general galaxy population. Furthermore, we were able to show that as all the LGRBs in our sample are at redshifts lower than one, the general decrease of galaxy metalicities with redshift is far too weak an effect to account for the observed metallicity difference. The fact that LGRBs nearly always are associated with a Type Ic-bl SNe would suggest that LGRB progenitors probably have similar masses to those of regular Type Ic-bl, thus largely eliminating the possibility that the observed LGRB metallicity bias is somehow a byproduct of a difference in the initial stellar mass functions. Rather, metallicity below half-solar must be a fundamental component of the evolutionary process that separates LGRBs from the vast majority of Type Ic-bl SNe and from the bulk of local star-formation.
While this work shows that LGRBs exhibit a strong and apparently intrinsic preference for low metallicity environments some exceptions to this trend do exist — three of the 14 LGRB in the sample possess abundances of about solar and above. These exceptions show that is it still possible to form an LGRB in a high metallicity environment albeit with greater rarity. If we wish, for instance, to use LGRBs to trace the star-formation of the Universe (cf. ) we must understand the conditions required for their production and thus the selection effects that could substantially bias our estimates. The implications of these high metallicity bursts are important not only for understanding the formation of LGRBs but also for our being able to use them as cosmological probes.
Metallicity is certainly critical in LGRB formation, but it might not be the only environmental factor of relevance. An additional observation of <cit.> was that LGRBs are far more likely to occur in the brightest regions of their hosts than if they simply traced the light of their hosts. However, the CCSNe followed their hosts blue light distribution. <cit.> showed that Type Ic-bl trace the blue light of their hosts far more like LGRBs than typical SNe. This suggests that LGRBs (and Type Ic-bl in general) are formed from very massive progenitors which generally do not have time to travel far from their birth sites before exploding, Typical CC-SNe, on the other hand, come from a wider range of masses, and are by number heavily weighted towards lower initial masses. Thus the shape of the IMF, and in particular, the relative rate of formation of the most massive stars is critical. It is thus quite interesting that the work of <cit.> suggests (but does not conclusively show) that the ratio of Ibc SN to Type II SNe is higher in galaxies with disturbed morphology than in undisturbed galaxies. The work of <cit.> shows that Ib SNe come from more massive progenitors than Type II SNe but somewhat less massive than Type Ic SNe. Thus the result of <cit.> could be due to a change in the IMF, though it should be pointed out this effect is roughly a factor of three, or an order of magnitude less than the metallicity effect we report here.
Additionally, <cit.> have shown that in comparison to SDSS galaxies the hosts of Ic-bl SNe and LGRBs of a given stellar mass exhibit high stellar mass and star formation rate densities, and as well as high gas velocity dispersions. CCSNe hosts show no such preference. Furthermore, <cit.> find that this preference cannot be explained as a byproduct of a preference for low metallicity environments. While the surface brightness of LGRB hosts may be biased by the general increase in specific star-formation rate with redshift <cit.>, the hosts of Type Ic-bl sample are all at low redshift and thus would not be affected by this bias. Thus while metallicity may not be the only environmental factor to affect the rate of LGRB formation, it does appear to be the dominant factor.
Recently, <cit.> analyzed a large sample of LGRB host galaxy photometry and came to the conclusion that the LGRB formation rate drops precipitously at metallicities above solar. This is very similar to the result of <cit.> who, using the original <cit.> sample of hosts, argued that if the observed preference for dwarf hosts was produced by a sharp cutoff in the host metallicity distribution, that cutoff would be at about $12 + \rm{log}(\rm{O}/\rm{H}) \approx 8.7$. However, because neither group had spectroscopic metallicity measurements for their host galaxies, both groups used the standard mass-metallicity relationship for galaxies ( used whereas used ) to convert host photometry to metallicities. This is problematic, as LGRB hosts are systematically biased low in metallicity for a given galaxy mass <cit.>. Assuming that LGRB hosts follow the field mass-metallicity relationship causes one to substantially overestimate their metallicity and thus the value of any metallicity cutoff.
Here, we build upon the results derived in <cit.> to estimate the relative LGRB formation rate per unit star formation rate as a function of metallicity. In order to do this we must normalize the LGRB rate to the rate of underlying star formation across different metallicities. However, we show that this can be done as a straightforward extension of the work presented in <cit.>. We combine our estimates of the LGRB and star formation rates as a function of metallicity to answer a fundamental question of this field: how much more likely is an LGRB to form at one metallicity as compared with another?
We address the implications of this metallicity induced rate difference on the absolute LGRB formation rate in <cit.>.
§ THE METALLICITY DEPENDENCE OF THE LGRB FORMATION RATE
In this section we estimate the relative rate of LGRB formation as a function of metallicity. To do this, we use the LGRB sample along with the comparison sample of Sloan Digital Sky Survey (SDSS) <cit.> star-forming galaxies used in <cit.>. The metallicity distributions of these samples are shown in the left hand side of Figure <ref>.
We adopt the LGRB sample of our prior work over the new and larger sample of <cit.> because we have already considered potential biases on the LGRB population (see section 2.4). We will briefly discuss possible selection effects in the following sections.
Our LGRB sample used all LGRBs with published spectra or line lists available (at the time) with the lines sufficient to obtain an R$_{23}$ metallicity with express [N II]/Halpha degeneracy breaking. All metallicities that we present here are determined using the R$_{23}$ diagnostic in the KK04 scale, employing the iterative scheme of <cit.>. The SDSS galaxy sample comprises spectroscopy of photometric catalogue galaxies complete down to an r-band Petrosian apparent magnitudes of 17.77 <cit.> to which we have applied a 0.02 $<$ z $<$ 0.04 redshift cut providing a complete volume limited sample of galaxies brighter than -18 B-band absolute magnitude.
Whereas each LGRB host in Figure <ref> contributes equally to the cumulative LGRB host distribution, each SDSS galaxy contributes by its amount of star formation (estimated from its H$_\alpha$ emission, see ). This sample of SDSS galaxies comprises all the SDSS star-forming galaxies in the redshift range 0.02 $<$ z $\le$z 0.04
with M$_B \le -18$ with spectroscopy suitable for metallicity measurement. The lower bound of this redshift range is necessary for the 3727 Å [O II] line to enter the SDSS spectroscopic wavelength coverage, thus allowing us to apply the R$_{23}$ metallicity diagnostic (see for a description of the diagnostic, code, and the KK04 scale). At the upper bound, the SDSS spectroscopy is complete to M$_B$ $\le$ -18 with partial coverage of dimmer galaxies. Twelve of the fourteen LGRB hosts in our sample are brighter than M$_B$ = -18, so this SDSS sample provides a good comparison. In the next section of this paper, we will discuss estimating an extension of the Sloan sample two magnitudes fainter, to provide an even better match to the LGRB sample. For an in depth discussion of the sample choice and the spectroscopic methods used in this paper, please see , and in particular section 2.1 therein.
To estimate the relative rate of LGRB formation per unit star formation as a function of metallicity, we first introduce a bit of formalism. Define $f_{GRB}(Z)$ to be the LGRB formation rate per co-moving volume at metallicity $Z$ normalized so that $ \int_{-\infty}^\infty f_{LGRB}(Z)\, dZ = 1 $ and similarly let $f_{SFR}(Z)$ be the fractional star-formation rate at metallicity $Z$ per co-moving volume again such that $ \int_{-\infty}^\infty f_{SFR}(Z)\, dZ = 1 $. We then define
\begin{equation}
{\cal R}^-(Z) = \frac{\int_{-\infty}^Z f_{GRB}(Z')\, dZ'}{\int_{-\infty}^Z f_{SFR}(Z')\, dZ'}
\end{equation}
where ${\cal R}^-(Z)$ is the ratio of these fractional rates up to a given metallicity $Z$, and the superscript $-$ on the ${\cal R}$ indicates we are taking a ratio from $-\infty$ to $Z$. Later, we will use a $+$ superscript to indicate the same ratio with the integrals are taken from $Z$ to $\infty$. The cumulative plots in Figure <ref> are in fact plots of the numerator and denominator of this function. We plot the function ${\cal R}^-(Z) $ in the right hand side of Figure <ref>. For any metallicity $Z$, this function is the star-formation normalized LGRB rate below $Z$. This plot shows that the star-formation normalized LGRB rate is very high at low metallicities. This rate plunges at about $12 + \rm{log}(\rm{O}/\rm{H}) \approx 8.3$. By definition ${\cal R}^-(Z)$ converges to 1.0 as $Z \Rightarrow \infty$. At low metallicities, then, using this rough comparison, we appear to be seeing rates of LGRB formation at low metallicities perhaps fifty times greater than the average over the entire range of metallicity.
We call the present comparison rough because while we went to great lengths in <cit.>, to produce a magnitude limited sample of SDSS galaxies for comparison, we used whatever LGRB galaxies for which we could obtain R$_{23}$ metallicities. This means that our results could be subject somewhat to the vagaries of our LGRB sample. In the following two sections, we will attempt to adjust our LGRB sample for potential biases, and will show that while the shape and the magnitude of ${\cal R}^-(Z)$ change in detail, its overall general properties appear to be largely independent of how we weight our LGRB host galaxies, or the exact comparison sample used. The methods we develop here will also be useful for application to the larger samples that are now becoming available.
Left: Cumulative fraction of population or total star-formation vs.galaxy central metallicity (see Figure 5). The LGRB sample from <cit.> is shown in red. The black line shows the local sample of SDSS star-forming galaxies also from <cit.>. Each galaxy contributes according to its star-formation rate, as measured by its H$_\alpha$ emission. Right: Cumulative fraction of LGRBs divided by the cumulative SDSS star-formation fraction, or ${\cal R}^-(Z)$ as defined in Equation 1. For each $Z$, we show the average rate of LGRB formation per unit star formation for all $Z' \le Z$ normalized by the average rate of LGRB formation per unit star formation over the entire metallicity span. Note the sharp cutoff in the LGRB formation rate at about log(O/H)+12 $\sim $8.3 or at about 40% of solar metallicity.
§.§ The Effect of Dark Bursts
Inclusion in this sample requires (high-energy) detection of the burst, localization of the burst to a clearly associated host galaxy, and then suitable spectroscopy of the host galaxy. Fortunately LGRBs themselves are detected by gamma-ray instruments which are not believed to be affected by the properties of their hosts. For example, <cit.> finds no correlation between host metallicity and burst luminosity. Burst localization is however complicated by a subset of bursts which lack the optical counterpart usually used for determining the burst position to sub-arcsecond levels. Fortuitously the nature of these bursts, called “dark bursts”, was of considerable interest when our LGRB sample was compiled and thus they received substantial attention.
Dark bursts are now thought to simply be typical LGRB events obscured by dust in the host galaxy. They comprise twenty to perhaps up to thirty percent of of LGRBs <cit.>. Dust extinguished LGRBs are found to preferentially reside in more massive galaxies, which under the mass metallicity relation are typically more metal rich <cit.>. We have 2 dark bursts (LGRBs 020819B and 051022) in our sample of 14 objects which is roughly consistent with the lower end of this rate. Our two dark bursts are among the three highest metallicity LGRBs in our sample. Thus if the sample were to contain something closer to $\approx$30% dark bursts, we would have four dark bursts, and five rather than three very high metallicity hosts in the sample. Such a sample would produce results that were noticably, but not dramatically, different from those we will derive directly from the <cit.> sample. However, it is generally straightforward to estimate the how the results would change with a higher proportion of dark (metal-rich) bursts, and we do so throughout the paper.
§.§ Luminosity Completeness Adjustment of the Comparison Sample
Left: Histogram of star formation in the SDSS sample. The black histogram shows the star formation per unit absolute magnitude in the volume limited SDSS comparison sample. This sample is complete only to M$_B$ -18. The purple line shows a fit to the raw sample at absolute magnitudes fainter than M$_B$ -18. The green line shows the power-law of the Schechter fit to this star-forming sample (see text of this paper and for more details). We use the ratio of the Schechter power-law to the raw fit to boost the estimate the star-formation per unit magnitude down to M$_B$ -16 (blue line). Right: Cumulative fraction of LGRB population or total star-formation vs.galaxy central metallicity. The red line shows the LGRB metallicity distribution already seen in Figure <ref> left. Similarly, the black line shows the distribution of star-formation versus metallicity for the SDSS sample down to our completeness limit of M$_B$ = -18, as in Figure <ref> left. The blue line shows the star-formation distribution when the sample is extended to M$_B$ = -16. Thus the blue and black lines represent the raw and extended star-forming sample respectively in both sides of the figure.
The LGRB hosts in the present sample are as faint as M$_B \sim -16$, while our SDSS comparison sample is only complete down to M$_B =$ -18. Thus if we want to take full advantage of our LGRB sample, we need to correct for the incompleteness of our SDSS sample in the range $-16 < {\rm M}_B < -18$. As discussed in <cit.>, the star-forming sample can be fit to a Schechter luminosity function with a power-law $\alpha = -1.3$. It is this power-law which determines the luminosity function at the faint end of the distribution. This power law is shown as the green line in Figure <ref> left. Note how the slope of the histogram changes sharply for galaxies fainter than our completeness limit of M$_B =$ -18. To correct this, a fit to the actual measured star formation in the incomplete sample at magnitudes fainter than M$_B = -18$ is shown as the purple line in that figure. To extend the sample below M$_B = -18$, we multiply the star-formation of any galaxy in the sample in that magnitude range by the ratio of the green (Schecter) fit to the purple (actual) fit at the galaxy's magnitude. That is we are boosting the star-formation in each galaxy in the sample between $-16 < {\rm M}_B < -18$ by the ratio of the shortfall of galaxies at that magnitude compared to the power-law extension of the luminosity function. The extended sample of galaxies can be seen as the blue continuation to the SDSS sample in Figure <ref> right. As each galaxy has a measured metallicity, the amount of star formation it contributes to its metallicity bin is also increased by the ratio of the two power laws. In Figure <ref> right we show the effect of this correction on the metallicity distribution of the SDSS star-formation. In particular the amount of star formation below a metallicity of log(O/H)+12 $ = $8.3 is roughly doubled. Nonetheless, the SDSS star formation curve remains dramatically below that of the LGRBs at low metallicities.
K-S tests confirm what is obvious to the eye in Figure <ref> right. Comparing the LGRB and original SDSS star-formation distributions we find a K-S value of 0.70 and a probability of 6.3 $\times$ 10$^{-7}$. Instead comparing the LGRBs to our extrapolated SDSS star-formation distribution gives a K-S value of 0.66 and a probability of 4.3 $\times$ 10$^{-6}$. Thus the LGRBs clearly exhibit a preference for lower metallicity environments. This low metallicity preference grossly exceeds the metallicity gradients available within galaxies. In <cit.> we found that SNe host galaxies have the same metallicity distribution as the SDSS star-formation and estimate the metallicity difference between the SNe site locations and the galaxy centers. The average difference of 0.11 dex we would expect to also be roughly consistent with the correction between the central galaxy metallicities as measured by the SDSS fibers and the average metallicity of the star-formation in these galaxies. Even shifting the SDSS star-formation distribution metallicities down by three times this (which also happened to be the largest single expected gradient in the SNe sample) still gives a K-S value of 0.49 and a probability of only 1.6 $\times$ 10$^{-3}$. While a difference between central galaxy and GRB site metallicity would also be expected, the LGRBs are much more closely distributed on the brightest regions of their typically much smaller host galaxies <cit.> and thus this effect would be reduced.
We also note the existence of a surprisingly flat region in the SDSS star-formation distribution between approximately 12+log(O/H) of 8.37 to 8.5 suggesting a sparsity of star-formation within this metallicity range. We believe this effect is actually caused by measurement error on the R$_{23}$ value. The range where this is observed (i.e. 8.37-8.5) is right at the intersection between the upper and lower branches of the R$_{23}$ diagnostic where a small change in R$_{23}$ reflects a large change in the metallicity (see Figure 7). This scatter is observed in other works using the R$_{23}$ diagnostics on large samples (e.g. Figures 5 & 6). As this effect is equivalently present in both the SDSS and LGRB samples it should not effect the analysis which depends on the ratio of these two samples.
We now use the original and extended SDSS star formation distributions to recompute the cumulative fraction of LGRBs versus star formation as a function of metallicity or $\cal{R}^-(Z)$. In Figure <ref> left we show $\cal{R}^-(Z)$ where we only consider those LGRB hosts brighter than M$_B = -18$. In this way, the LGRB hosts have the same magnitude limit as the SDSS star-formation sample, and the LGRB hosts will not be biased to lower metallicity due to a luminosity metallicity correlation. In Figure <ref> right, instead of cropping the LGRB host sample we use our extension of the SDSS star-formation distribution down to M$_B = -16$ in the computation of $\cal{R}^-(Z)$. Again while the details of the curve vary, the same overall shape is revealed: ${\cal{R}}^-(Z) > 30$ for $Z < 8.3$, while falling sharply above that metallicity. Interestingly the rapid decline slows substantially as the metallicity increases further, and the decline in relative LGRB formation with increasing metallicity appears to be rather minor above a metallicity of about log(O/H)+12 $ = $8.7, a point we shall return to later.
Left: $\cal{R}^-(Z)$ for a magnitude limited sample. Here we show $\cal{R}^-(Z)$ where we restrict the LGRBs to those with hosts brighter than M$_B = -18$. In this case we do not need to use the extended SDSS sample, as the magnitude limits of the two sample are the same, but we must exclude two of our LGRBs. Right: The extended sample. Here we plot $\cal{R}^-(Z)$ for the full LGRB sample compared to the SDSS extended down to M$_B = -16$. Note the cutoff in the LGRB formation rate apparent in Figure <ref> is still present in both versions of this figure, though the exact value of $\cal{R}^-(Z)$ at any particular metallicity does vary somewhat depending upon the sample used.
§.§ LGRB Host Observably Adjustments
The distance to which an LGRB host can be observed and its metallicity measured will depend on its brightness, and in particular on the strength of its emission lines. As our LGRB sample was limited by our ability to get host metallicites, which requires bright emission lines, the sample is to first order a magnitude limited sample and thus more luminous hosts will be overrepresented. However, we can adjust for this effect on ${\cal R}^-(Z)$ by weighting the contribution of each LGRB inversely by the volume over which it could have been discovered.
To calculate the distances over which a host might be detected, we estimate the luminosities of the 3727 Å [O II], 4959 Å H$\beta$, 5007 Å [O III] lines[Other lines used in the metallicity diagnostics are not similarly considered for the following reasons: the 4959 Å [O III] line has $\frac{1}{3}$ the 5007 Å lines flux as quantum mechanically required (see ) thus only the brighter line is needed. The Balmer decrement sets the H$\alpha$ line at 2.87 times the lines H$\beta$ flux as required by Case B recombination (see ) with the value increasing with extinction, and the 6584 Å [N II] line is highly dependent on metallicity such that it is typical excluded from S/N cuts to avoid introducing a metallicity bias (see ).], and for the least luminous line determine the redshift (and corresponding comoving volume) where it would be detected with a flux of 1$\times$10$^{-17}$ erg sec$^{-1}$ cm$^{-2}$ (assuming standard cosmological parameters: $\Omega _m$ = 0.27, $\Omega _\lambda$ = 0.73, & H$_\circ$ = 0.71 km s$^{-1}$ Mpc$^{-1}$). An emission line with this flux would have about a five sigma detection in a four hour integration on an eight-meter class telescope.
LGRB Sample.
2* LGRB host 2*Metallicity 2*M$_B$ 2*z (observed) 2*z (limit) CMV (at limit) CMV (capped)
Mpc$^3$ Mpc$^3$
GRB 991208 8.05 -18.68 0.706 1.31 273 158
GRB 030329 8.12 -16.52 0.1685 0.94 132 132
GRB 070612A 8.17 -20.86 0.671 3.44 1360 158
GRB 011121 8.20 -19.75 0.362 2.91 1080 158
GRB 060218 8.24 -15.92 0.034 0.44 19.9 19.9
GRB 031203 8.27 -18.52 0.1055 1.42 321 158
GRB 010921 8.34 -19.87 0.451 1.29 264 158
GRB 020903 8.38 -19.34 0.251 1.40 312 158
GRB 050824 8.39 -19.02 0.828 1.43 325 158
GRB 980425 8.55 -18.09 0.0085 0.72 70.6 70.6
GRB 060505 8.64 -19.38 0.0889 1.18 219 158
GRB 051022 8.77 -21.23 0.80625 3.24 1250 158
GRB 050826 8.83 -20.28 0.296 1.40 312 158
GRB 020819B 8.97 -21.53 0.411 1.25 247 158
In this table we present the LGRB hosts in our sample, sorted by redshift. We report the host M$_B$, observed redshift and the maximum redshift at which all of the lines necessary for determining the metallicity could be detected with a signal to noise ratio of at least five (See Section <ref>). The CoMoving Volumes (CMV) are given for both the limiting redshift and the redshift capped at z = 1.
weighted by CMV. In this plot, LGRBs are weighted by the inverse of the CoMoving Volume (CMV) when calculating out to the redshift to which their hosts' metallicities could have been measured. The CMV is capped using the lesser of a redshift of $z=1$ or the maximum redshift at which the faintest host line necessary for determining the metallicity would be expected to have a flux above 1$\times$10$^{-17}$ erg sec$^{-1}$ cm$^{-2}$. The hard limit of $z=1$ is a function of
the surveys we used in compiling the sample and the need for deep spectroscopy in the near-IR at redshifts above one. More recent surveys now coming available will allow an extension of the techniques we describe here to higher redshift.
However, the surveys we have used gave either line equivalent widths or line fluxes uncorrected for slit loss. For the two closest objects (LGRBs 980425 and 060505) we use star-formation rates from the literature ( and respectively) to estimate the total H$\alpha$ flux and then rescale the other lines accordingly. The remaining objects are sufficiently distant that slit losses are not large, and we are able to estimate the slit losses as discussed in . For objects where we have a measure of the H$\beta$ line equivalent width and B & V band absolute magnitude values, we can estimate the H$\beta$ flux independently and scale the spectrum according. Although we do not present numbers from this method here, they are comparable to those obtained using our slit loss estimates. In many cases, the redshift out to which we estimate we could accurately determine the host metallicity is quite high. However, none of our objects are much above $z=0.8$. This is because of the surveys which were used to compile our sample, and the technical difficulties of determining galaxy metallicities above $z=1$. While surveys are now becoming available that will be able to reach these higher redshifts, we estimate that our sample was effectively limited at $z=1$ and thus we cap the comoving volume associated with any host to the comoving volume out to $z=1$. In Table <ref>, we give the observed and limiting redshifts for the hosts as well as their estimated and capped co-moving volumes. The LGRB hosts are sorted by their redshift to allow easier identification of specific LGRBs with features in the plots. In Figure <ref> we plot the relative rate of LGRB formation versus metallicity, , with LGRBs weighted inversely to their capped CVM. While the peak of just before the fall is higher in this plot than earlier ones, the general shape of the plot,
and the location of the sharp cutoff are essentially unchanged.
§ RELATIVE RATES OF LGRB PRODUCTION
So far we have entirely relied upon using the function ${\cal R}^-(Z)$ to represent the relative rate of formation of LGRBs as a function of metallicity. However, this is a somewhat blunt, if powerful, tool. By definition ${\cal R}^-(Z)$ converges to 1.0 as $Z \Rightarrow \infty$. Thus the rate of LGRB formation over a range of metallicity ${\infty, Z}$ is measured relative to the entire range, i.e. ${-\infty, \infty}$. To allow us to better compare rates in differing ranges of metallicity, we introduce ${\cal R}^+(Z)$, where
\begin{equation}
\label{R+}
{\cal R}^+(Z) = \frac{\int_Z^{\infty} f_{GRB}(Z')\, dZ'}{\int_Z^{\infty} f_{SFR}(Z')\, dZ'}.
\end{equation}
This is the normalized star-formation rate of LGRB production above metallicity $Z$. We plot ${\cal R}^+(Z)$ in Figure <ref>.
Then if we define
\begin{equation}
{\cal R}^D(Z_1,Z_2) = \frac{{\cal R}^-(Z_1)}{{\cal R}^+(Z_2)} ,
\end{equation}
${\cal R}^D(Z_1,Z_2)$ gives us the star-formation normalized rate of LGRB formation below $Z_1$ divided by the star-formation normalized LGRB formation above a metallicity of $Z_2$. This allows us to directly compare the rate of LGRB formation at low metallicities to that at high metallicities.
The function $\cal{R}^+(Z)$. This function is described Eq. <ref>. It is the integral of the fraction of LGRBs in the sample over the integral of the fraction of total star formation, where both integrals are taken over the range $Z \rightarrow \infty$. Each step in the plot is a separate LGRB. In both cases shown here we use the SDSS sample extended to $\rm M_B =-16$ On the left we show the value of this function where each LGRB is weighted equally. On the right we show the value of this function where the LGRBs are weighted by the their comoving volume, capped by the voulume within the sphere out to $z = 1$, and where LGRBs above $z=0.5$ are weighted by an additional factor of 2.5 to take into account their apparent underrepresentation in our sample (see Figure <ref>). The curvature in the individual steps (which is particularly pronounced at higher values of $Z$) is caused by the integrated star formation between the discrete metallicities of the different LGRBs.
When calculating these functions, we need to choose between the weighting schemes we have discussed in the previous sections. Our most conservative estimate of the LGRB formation rate at low metallicities is produced by comparing the complete sample of unweighted LGRBs to the extended star-formation sample (see Figure <ref> right). A more aggressive approach is that shown in Figure <ref> right, where the LGRBs are weighted by their (capped) co-moving volumes. Therefore for the rest of this section we will calculate values for both these approaches, with the difference between them giving some estimate of the importance of systematic effects in determining the result.
A clear division between low and high-formation rates is created by the rapid drop in ${\cal R}^-(Z)$ centered on
${\rm log(O/H)} + 12 \approx 8.3$. If we then compute ${\cal R}^D(8.3,8.3)$, we obtain the relative rate of LGRB formation per unit star formation above 8.3 compared to that below 8.3. If we do not weight the LGRBs, and compare them to the SDSS star-formation rate extended down to an absolute magnitude $\rm M_B =-16$, i.e. our conservative case, we find ${\cal R}^D(8.3,8.3) \approx 27.1$. As there are seven LGRBs in the sample with metallicity below 8.3 and seven above that metallicity, the statistical errors in this estimate are large, about 40%. If we instead use the LGRBs weighted by their (capped) co-moving volumes, we find ${\cal R}^D(8.3,8.3) \sim 150$. Here, however, one LGRB in particular, GRB 060218, makes a substantial contribution because of the relatively low comoving volume over which the metallicity of its host could be measured. Removing this one object drops the estimate of ${\cal R}^D(8.3,8.3)$ down to $\sim$65 and we use this reduced LGRB sample in our subsequent CMV estimates.
We also examine the effects of our small LGRB sample size through the use of resample and replace. We recreated a 1000 GRB host metallicity samples by randomly choosing metallicities from our present population. In Figure <ref> left we plot the resulting $\cal{R}^-(Z)$ distributions. The general shape of the distribution is largely invariant to the resampling, and in particular the sharp drop at log(O/H)+12 = 8.3 remains. On the right hand side of the same figure we show the cumulative distribution of values of ${\cal R}^D(8.3,8.3)$ under the resampling. We find a population mean of ${\cal R}^D(8.3,8.3)$ of 30.9, with ${\cal R}^D(8.3,8.3)$ lying between 10 and 50 in over 90% of the trials. Use of the CMV weighted sample, on the other hand, causes the mean of and the limits on the ${\cal R}^D(8.3,8.3)$ distribution to go up by more than a factor of two.
Left: $\cal{R}^-(Z)$ shown for 1000 iterations of re-sample with replacement using our LGRB population and the SDSS with cutoff of$\rm M_B =-18$. The dark line is $\cal{R}^-(Z)$ shows the result when the resampling returns our original sample. This line is in fact Figure <ref> right.
Note that a significant dependence on metallicity remains for all iterations, though the strength of the effect varies. Right: A cumulative distribution of the ${\cal R}^D(8.3,8.3)$ values generated in the sample and replace. This effectively give us a cumulative probably distribution of the ${\cal R}^D(8.3,8.3)$ values. Based on the resampllng there is only a 2.5% chance that ${\cal R}^D(8.3,8.3)$ is below a factor of ten, and over a 90% chance that it is between 10 and 50. This is, however, likely a conservative estimate, as use of the CMV weighted sample causes the rate estimates to roughly double.
The effect of adjusting these numbers for our possibly low proportion of dark bursts is small in comparison to the statistical errors. If one assumes that instead of two dark bursts our sample should have four, and if one further assumes that these are high metallicity objects, and that they have weights on average equal to those of the high metallicity objects, then the numbers given above should be multiplied by a factor of $\sim 7/9$. Thus, we would find ${\cal R}^D(8.3,8.3) \sim 20$ in our conservative estimate, and $\sim$50 for the co-moving volume weighted estimate.
If we are willing to work with somewhat smaller statistics we can make a greater separation between the low and high regions of metallicity we use for comparison. The normalized rate of LGRB production is roughly constant (and high) for $Z < 8.2$ in the uniformly weighted sample. If we take the region $Z > 8.6$ for our upper range, we will be comparing to the LGRB production rate at roughly solar metallicity. We find that ${\cal R}^D(8.2,8.6) \sim 100$ for the uniformly weighted LGRBs, and ${\cal R}^D(8.2,8.6) \sim 300$ for the sample weighted by the capped CMVs. Here we only have four LGRBs in each of the two samples, and thus statistical errors will be in the range of 70%. In this case, the effect of correcting for dark bursts would be somewhat larger, as we would add two additional bursts to the four in the high metallicity subset. Here the results would be multiplied by a factor of $\sim 2/3$, meaning ${\cal R}^D(8.2,8.6) \sim 66$ for the uniformly weighted sample, and $\sim 200$ for the sample weighted by capped CMVs.
§.§ Correcting for Metallicity Evolution
While galaxies in our SDSS sample have redshifts $0.02< z < 0.04$, the LGRB hosts in our sample go out to $z \sim 0.8$. However, the metallicities of galaxies of a given mass evolve with redshift, with the least massive galaxies showing the greatest change in metallicity with redshift. Indeed, at a redshift of $z \sim 0.8$ galaxies with a stellar mass of $3 \times 10^9$ M$_\odot$ have metallicities more than $0.15$ dex below that of galaxies of a similar mass at redshift zero <cit.>. This may seem like a small change, but in our extended SDSS sample the fraction of star formation with $ \metal < 8.2 $ is only one quarter that with metallicities $ \metal < 8.3 $. As a result, our estimates of relative rates of LGRB production could be significantly affected by ignoring this effect.
We do not know the distribution of star formation as a function of metallicity at higher redshifts. However, as a first approximation it is reasonable to estimate that the fraction of star formation below a metallicity $Z$ at a redshift of $z$ is equal to the fraction of star formation below a metallicity of $Z + \Delta_z$ at redshift zero, where $\Delta_z$ is the change in metallicity of galaxies between redshift $0$ and $z$. $\Delta_z$ depends both on the redshift under consideration and the mass of the galaxy. The smaller the galaxy mass, the larger the effect. For galaxies of mass of only $\sim 10^9 \, {\rm \Msun}$ at $z = 0.8$, <cit.> find $\Delta_z \sim 0.15$. However, there is a small difference between our sample and that of <cit.>: our sample only goes out to $z=0.04$ while theirs extends to $z=0.08$. Now, essentially all of the galaxies fainter than $\rm M_B = -18$ in the <cit.> sample are also in our sample (due to their limiting magnitude in the SDSS
falling inside $z=0.04$). However, our samples will differ somewhat for brighter galaxies. The more distant galaxies in <cit.> will be smaller on the SDSS spectroscopic fiber than equivalent ones in the nearer sample. The fiber will therefore obtain a central metallicity over a larger area of the galaxy, slightly reducing the estimated metallicity. We have compared galaxies brighter than $\rm M_B = -18$ in the SDSS out to the two different limiting redshifts, and find that this effect only produces a $\Delta_z \approx 0.05$. Again, this effect is only important for the brighter galaxies, where the metallicity evolution seen by <cit.> is quite small. However, we find that when taking these two effects into account for all the galaxies in our sample with metallicities below $8.3$, these galaxies all remain below roughly equal to 8.3. As the functions and do not depend on the distribution of LGRBs above or below a metallicity $Z$, but only the (weighted) fraction of LGRBs below or above that metallicity, the estimates of ${\cal R}^D(8.3,8.3)$ do not change. Thus our estimates of ${\cal R}^D(8.3,8.3)$ are quite robust.
Another quick way to check this calculation is to examine the metallicity of the five bursts with $z < 0.2$. These low redshift bursts would be expected to be subject to minimal metallicity evolution. Three of these five bursts are are below $\metal \sim 8.3$ even though less than three percent of the star formation in our extended sample is in this range. This small sample gives a value of ${\cal R}^D(8.3,8.3)$ about a factor of two larger than that reported for the entire sample. Although the statistical error is large, the fact that this estimate suggests an even greater bias in LGRB formation again supports our claim that our estimate of ${\cal R}^D(8.3,8.3)$ is robust.
However, we cannot make a similar claim for our estimates of ${\cal R}^D(8.2,8.6)$. This result could vary by up to a factor of four, depending how one attempts to take metallicity evolution into account. However a reduction of a factor of four brings ${\cal R}^D(8.2,8.6)$ down only to the level of ${\cal R}^D(8.3,8.3)$. As we would expect ${\cal R}^D(8.2,8.6)$ to be the larger of the two, this suggests that our estimates of the effect of metallicity on LGRB production are indeed quite robust, and that at a very minimum the rate of LGRB production increases by a factor of twenty-five higher between metallicities $ \metal < 8.3 $ and solar metallicity.
§ SUMMARY AND CONCLUSIONS
We have shown a dramatic cutoff in the rate of LGRB formation per unit star-formation at metallicites above log(O/H) + 12 $\approx$ 8.3, where our metallicity is determined using the KK04 scale of the R$_{23}$ metallicity diagnostic. To test the stability of this result we have subjected our samples to a number of corrections intended to remove possible biases.
First we address a known limitation of the SDSS survey, that the SDSS fiber placement is complete only for objects brighter than about 18th magnitude in the B band. This corresponds to a galactic luminosity of M$_B \sim$ -18 at z = 0.04, our highest volume limited redshift. The simplest correction is to just discard the two LGRB hosts and all the SDSS galaxies fainter than this absolute luminosity. We also employ a more detailed correction were we model the SFR as a function of galactic luminosity, extrapolate the missing star-formation, estimate how under surveyed the fainter galaxies in the SDSS are, and overweight them such that their total SFR matches our extrapolations. This allows us to preserve our estimates of star-formation as a function of metallicity down to a luminosity of -16 magnitude consistent with the luminosity range of our LGRB sample. Both of these changes leave the result intact.
Next we weight the hosts by the volume searched. Based on the LGRB hosts in the sample we conclude that the hosts would have remained in the sample so long as the key lines for determining metallicity were all brighter than 1$\times$10$^{-17}$ erg sec$^{-1}$ cm$^{-2}$. We then calculate the maximum redshift where the weakest line necessary for determining the host metallicity would thus be observable. We can then weight each object in our LGRB sample by the inverse of the comoving volume for its maximum observable redshift. However, as our sample relied upon line measurements obtained in the optical, no objects in the sample are at a redshift greater than one, and thus we limit the comoving volume to that of $z=1$.
Additionally, our sample fourteen hosts contains two dark bursts, or a dark burst fraction of fourteen percent, somewhat below the estimated rate of dark bursts of twenty to perhaps thirty percent <cit.>. Where we cite LGRB formation rate, we include a correction that would effectively bring our dark burst fraction up to thirty percent, yet this correction is in all cases is no larger than our statistical errors.
Throughout all of these correction s, we continue to find a sharp cutoff at log(O/H) + 12 $\approx$ 8.3 in ${\cal R}^-(Z)$, the relative rate of LGRB formation below metallicity $Z$.
The dramatic cutoff is followed by a more gradual decline, which itself again appears to be greatly reduced above a metallicity of log(O/H) + 12 $\approx$ 8.7. However, the small number of LGRBs in our sample above this metallicity makes it difficult to accurately determine the slope of this fall-off.
However, we
note that this result agrees well with the claim of <cit.> that the three highest metallicity objects in this sample are completely consistent with the mass metallicity relationships at their redshifts. Nor do these results necessarily conflict with the result of <cit.> that LGRB hosts appear to show little metallicity dependence at 3 $<$ z $<$ 5: presumably, by these redshifts, there would be relatively little star-formation above the metallicity cutoff.
Given the reasonably sharp nature of the observed cutoff observed at a metallicity of log(O/H) + 12 $\approx$ 8.3, we attempt to quantify how much more likely an LGRB is to form at metallicites below as opposed to above this cutoff. To do this, we determine the relative rate of LGRB formation below this cutoff divided by the relative rate of LGRB formation above the cutoff, or in new formalism, ${\cal R}^D(8.3,8.3)$. We find that this ratio is at least a factor of about twenty and quite possibly significantly larger, depending on which weighting scheme is used. We also see evidence that ${\cal R}^D(8.2,8.6)$, the ratio of the relative rate of LGRB formation below a metallicity of log(O/H) + 12 $\approx$ 8.2 to that above 8.6 may be close to a factor of one hundred.
Indeed, the enhancement in the rate of LGRB production at low-metallicity is in fact so great that a substantial fraction of Type Ic-bl SNe at low metallicities may be required to supply LGRB production, a point made in detail in <cit.>.
However, it is also becoming clear that this extraordinary dependence on metallicity may not be restricted to LGRBs. Hydrogen poor superluminous supernovae (Type I SLSNe) also display a remarkable aversion to near solar metallicities. All of the Type I SLSNe of <cit.> and all of the sample of Type I SLSNe in <cit.> (with perhaps one exception) have metallicities ${\rm log(O/H)} + 12 < 8.5$. Now these samples were largely found in surveys that placed a significant emphasis on following up supernovae in faint (or non-visible) hosts. Thus this sample may exaggerate the dependence on metallicity, but it nonetheless seems highly likely that SLSNe share at least as strong an aversion to high metallicities as LGRBs.
While we calculate our metallicities using the R$_{23}$ diagnostic in the KK04 scale, there are a number of alternate means of determining metallicity using strong emission lines (e.g. ). While these diagnostics vary in log(O/H) + 12 values by typically a few tenths of a dex, we do not believe our fundamental results will depend strongly on the choice of diagnostic used. <cit.> examined these differences and determined transformation equations between several of these commonly used diagnostics. While there is some scatter, a galaxy with a low, medium, or high metallicity relative to the galaxy population in one diagnostic would still be expected to be at a low, medium, or high metallicity relative to the galaxy population in other diagnostics. This should apply equally to host galaxies and the galaxies in our Sloan comparison sample. Thus the preference of LGRBs to occur much more often per unit star-formation at low metallicities should remain consistent regardless of how the metallicity is measured. Indeed, we have cross checked our results in the T04 scale of <cit.> and found the basic conclusions of this paper to be unchanged (though the exact metallicity of the curoff, for instance, does change, as expected, with a differing choice of metallicity indicator).
Although the mechanism through which metallicity affects the formation of LGRBs is uncertain <cit.>, it is widely believed that LGRBs require high angular momenta to produce jets, and that low-metallicity is likely required to maintain angular momentum through the evolution of the progenitor. Interestingly, a leading model for the engine in SLSNe is rapidly rotating, highly magnetized neutron stars, or magnetars (c.f. ). Indeed, it has been suggested that magnetars may power both LGRBs and SLSNe <cit.>, and one “ultra-long" GRB, is associated with an unusually bright SN <cit.>. GRB 111209A, whose prompt emission lasted for about fifteen thousand seconds, or more than a factor of a hundred longer than typical LGRBs, had an underlying supernova, SN 2011kl, with an absolute magnitude of about ${\rm M}_V = -20$, <cit.> or roughly a magnitude brighter than the SN associated with a typical LGRB, and a magnitude fainter than a typical SLSN. Whether a magnetar model can account for both LGRBs and SLSNe, or whether another process, such as accretion onto a remnant black hole, powers LGRBs, it seems likely that low metallicity plays an important role in the preservation of the crucial angular momentum in both of these rare, powerful explosions.
Metallicity is not the only significant environmental factor in the production of LGRBs. As <cit.> has reported, star formation density may also affect LGRB formation. This may be related to the enhancement in the rate of Type Ibc formation in galaxy mergers found by <cit.>. However, this effect only produces an enhancement by a factor of $\sim$3, not the factor of $\sim$30 we see in LGRBs due to the effect of metallicity Thus while many processes may contribute to the formation of LGRBs, our result strongly suggests that metallicity is the predominant determinant of the rate of LGRB formation relative to the rate of star formation.
We thank Patricia Schady and Jochen Greiner for helpful comments. We also thank Jarle Brinchmann for his help in determining the T04 metallicities of the hosts in our sample, which we used as a cross-check on the KK04 results presented here.
John Graham acknowledges support through the Sofja Kovalevskaja Award to Patricia Schady from the Alexander von Humboldt Foundation of Germany.
Note added in proof: <cit.> has shown that the metallicity value for GRB 020819B published by <cit.> and used by us was in error. This was due to a problem with the slit placement in the original observation. The host of GRB 020819B has the highest metallicity of all the GRB hosts in our sample. The high rates of LGRB formation at low metallicity which we derive here thus would have been even slightly higher had this been know earlier and had GRB 020819B been excluded.
|
1511.00723
|
Gamma Rays from the Milky Way, Part I
Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, SLAC National Accelerator Laboratory, Menlo Park, CA 94025
The complex interplay of processes at the Galactic Center is at the heart of numerous past, present, and (likely) future mysteries. We aim at a more complete understanding of how spectra extending to $>\,$10 TeV result. We first construct a simplified model to account for the peculiar energy and angular dependence of the intense central parsec photon field. This allows for calculating anisotropic inverse Compton scattering and mapping gamma-ray extinction due to $\gamma \gamma \!\rightarrow\! e^+ e^-$ attenuation. Coupling these with a method for evolving electron spectra, we examine several clear and present excesses, including the diffuse hard X-rays seen by NuSTAR and GeV gamma rays by Fermi. We address further applications to cosmic rays, dark matter, neutrinos, and gamma rays from the Center and beyond.
§ INTRODUCTION
The Galactic Center (GC) is an arena for astrophysical phenomena unlike any other in our galaxy. The inner parsec alone is packed with gas streams, dark matter, a puzzling young massive stellar population, and remnants of a long history of star formation, all encircled by a dusty circumnuclear disk <cit.>. In the middle of this is a supermassive black hole (hereafter Sgr A$^*$) that typically emits well below Eddington, though with flares that occasionally reach at least into hard X-rays <cit.>.
It is easy to imagine substantial concentrations of photon emission and energetic particles within this region. Much of the bolometric luminosity has now been identified as originating from the aforementioned massive stars that in turn power the infrared output from dust in the circumnuclear disk (e.g., ). The many matters of central import left to be resolved span the spectrum of photons, cosmic rays, and neutrinos.
Our purpose is to address aspects related to those mysteries that plausibly involve very energetic particles. These include the origin of the gamut of gamma rays reaching to $>\,$10 TeV <cit.> and bright X-ray emission with non-thermal characteristics. Such photons, if only for their prime location at the center of the Galactic halo, are of great interest, such as the significant excess of gamma rays at $\sim\! 10$ GeV that has produced considerable excitement (e.g., ).
Our focus here is not on providing yet another explanation for such anomalies (not entirely anyway), but rather to better understand the behavior of high-energy particles starting at the Center — electrons and gamma rays in particular — via an improved description of the relevant conditions in this unique environment. For example, recent high spatial resolution infrared data has revealed structures within the central parsec. These imply a photon background much denser than typically encountered in the Galaxy with variations in the amplitude of each component throughout this region.
A population of electrons, even if their velocity distribution is isotropic, will thus encounter anisotropic photon backgrounds. Since head-to-head scatterings result in more energy transfer, the resulting inverse Compton spectrum thus depends on the direction to the observer. Moreover, gamma rays produced via this or other processes can in turn be attenuated by interacting with a background photon to produce an electron-positron pair, the probability of which is dependent on the path taken to the telescope.
We construct a phenomenological energy and angle dependent photon field in the central parsec based on recent infrared data to achieve a basic agreement with the measured broadband spectrum and morphology of the various emissions. This is used to better describe the inverse Compton scattering and $\gamma \gamma \!\rightarrow\! e^+ e^-$ extinction, which have a similar dependence on the geometry of the photon background.
We couple these with a convenient method for calculating time-evolved electron spectra in examining several topics of recent interest. These include the diffuse hard X-ray emission extending to $>\,$40 keV discovered by NuSTAR throughout this region that cannot be simply extrapolated from sources prevalent at lower energies <cit.>. We discuss possible attributions, including synchrotron radiation from $\gtrsim\,$100 TeV electrons, and connections to gamma rays.
We also consider contributions from pulsar electrons to the GeV signal seen from the Galactic Center by Fermi <cit.>. <cit.> extends these techniques in detailing potential TeV gamma-ray signatures of the pulsar wind nebula (PWN) G359.95–0.04 situated at a projected distance of 0.3 pc from Sgr A$^*$ <cit.>.
An illustration of the geometry of photon emission components from the inner parsec of the Milky Way used in constructing the photon field used throughout this paper, labelled by temperature corresponding to Table <ref>. The line-of-sight position of PWN G359 is referenced with the blue cone.
§ A PORTRAIT OF GALACTIC CENTER BACKGROUNDS
The cluster of massive stars at the Galactic Center provides $\gtrsim\! 10^7 \, L_\odot$ of UV photons that drive emission over a broad range of wavelengths. While UV radiation can be effectively upscattered by GeV electrons, for TeV electrons scattering is suppressed due to the energy dependence of the Klein-Nishina cross section so that infrared emission is their most relevant inverse Compton (IC) target. Since this cross section depends on the angle between electron and photon, with head-on scattering resulting in a photon with higher energy <cit.>, it is of interest to understand the directional variation of the photon field beyond the integrated intensity.
Constructing a first principles model of the energy/angle dependent photon field in the GC would itself be a tremendous achievement. We rather content ourselves with a satisfactory phenomenological background based on the most recent data. For easy reference, the component parameters are summarized in Table <ref> and the layout illustrated in Fig. <ref>.
Herschel has now resolved cold dust in the circumnuclear disk (CND) in the FIR from $70\!-\!500\,\mu$m <cit.>. <cit.> also utilized ISO-LWS data from $46\!-\!180\,\mu$m, which has less angular resolution, to fill in flux from warmer dust. SOFIA, with shorter wavelength coverage ($19.7\!-\!37.1\,\mu$m) and sharper resolution, was used to resolve warmer locations of the inner CND in greater detail by <cit.>.
We describe these data using two separate rings: one with $T\!=\!90\,$K, $L_{90} \!=\! 2 \times 10^6~L_\odot$, a major radius of $R_{90} \!=\! 1.4\,$pc, and minor radius of $r_{90} \!=\! 0.2\,$pc; the other with $T \!=\! 40\,$K, $L_{40} \!=\! 2 \!\times\! 10^5~L_\odot$, $R_{40} \!=\! 1.7\,$pc, and $r_{40} \!=\! 0.3\,$pc. The inclination follows the orientation derived in <cit.>. We assume optically thin emission that is uniform throughout the volume with a blackbody spectrum
\begin{equation}
\frac{dN_i}{d\epsilon_\gamma} = \frac{1}{\pi^2 (\hbar c)^3} \frac{\epsilon_\gamma^2}{e^{\epsilon_\gamma/k_B T_i}-1}
\,.
\label{BB}
\end{equation}
This is not formally correct, since optically thin dust has a modified blackbody form with an emissivity $\propto\!\nu^\beta$ and $\beta \!\lesssim\! 2$ that results in a steeper long wavelength tail (e.g., ). However, we compensate for this by choosing values for $T$ and $L$ to match the spectral peak for dust of a given temperature (and typically another component becomes more important in the tails).
SOFIA images display warmer emission nearer the GC <cit.>, mostly coinciding with the ionized gas streamers seen in radio (e.g., ). In principle, one can begin from the <cit.> model of Keplerian gas stream orbits to construct a more elaborate model accounting for a heating from a central cluster. We here assume emission with $T\!=\!120\,$K and $L_{120} \!=\! 1.5\!\times\!10^6~L_\odot$ and approximate the multiple streams with a uniform sphere of radius $R_{120} \!=\! 0.75\,$pc. The extinction corrected ISO-SWS spectrum from <cit.>, extending from $2.6\!-\!26\,\mu$m and covering an extended inner portion of the central parsec, as well as radio line measurements (e.g., ) also suggest a warmer component that we ascribe to the same volume with $T\!=\!250\,$K and $L_{250} \!=\! 2\!\times\!10^6~L_\odot$.
$T [K]$ $L~[L_\odot]$ $R$ [pc] $r$ [pc]
35000 $20\!\times\!10^6$ 0.25 —
3500 $30\!\times\!10^6$ — —
250 $2\!\times\!10^6$ 0.75 —
120 $1.5\!\times\!10^6$ 0.75 —
90 $2\!\times\!10^6$ 1.4 0.2
40 $0.2\!\times\!10^6$ 1.7 0.3
2.73 CMB — —
Properties of the GC radiation components used here. $R$ refers to the radius of a sphere or major radius of a ring, $r$ to a ring minor radius.
The IR data are consistent with reprocessing of a fraction of the incident UV flux from a $T \!\approx\! 35000\,$K, $L_{35000} \!\approx\! 2\!\times\!10^7\,L_\odot$ cluster of massive stars at the GC. <cit.> and <cit.> infer a cutoff in the surface brightness by $\sim\!0.5\,$pc for this population, which we approximate with a sphere of $R_{35000} \!=\! 0.25\,$pc. <cit.> concludes that little of the line of sight extinction towards the GC arises from within the central parsec, which we will assume to hold for sight-lines not passing through the major dust structures. We also include a contribution from the much more extended old GC stellar component, using the radial profile from <cit.>, with a 3500 K spectrum normalized to $3 \!\times\! 10^7\,L_\odot$ within 100 arcsec, along with the uniform 2.73 K cosmic microwave background (CMB).
§ GEOMETRY OF EMISSION
Assuming uniform emissivity, the flux arriving from a given direction can be calculated using ray tracing techniques. For instance, we take an equation for a torus in Euclidean space, $f = (x^2 + y^2 + z^2 - r_1^2 - R_1^2)^2 + 4 R_1^2 (z^2 - r_1^2)$, insert the components for a ray ${\bold x}(t)$ starting from the electron position ${\bold r}_e$ and traversing direction ${\bold p}$, ${\bold x}(t) = {\bold r}_e + {\bold p}\, t$, and solve for the roots to find the length through ring 1, $\ell_1(\theta,\phi)$. This involves solving a quartic equation, which can be done fairly quickly numerically. The procedure either interior or exterior to spherical regions is similar.
To arrive at the energy density at a given position from each component, $u_i$, we do this many times en route to integrating over all angles
\begin{equation}
u_i = \frac{L_{i}}{4\pi c\, V_i} \int d\Omega\, \ell_i(\theta,\phi)
\,,
\label{ui}
\end{equation}
with $V_i$ the component volume. Each spectral energy distribution is shown in Fig. <ref> along with the CMB (at $\sim\!10^{-3}\,$eV).
In Fig. <ref> we also compare to the oft-used modeled interstellar radiation field at the GC from <cit.>. Since this model is constructed from stellar contributions over larger scales, it is indicative of contributions within the central parsec from outside. We see that our FIR energy density is larger by a factor of $\sim\! 10^3$ and so should remain dominant out to $\sim\!30\,$pc, corresponding to $\sim\!0.25^\circ$ (not accounting for any additional absorption). Other more explicit contributions include the Arches and Quintuplet stellar clusters, which have luminosities comparable to the central cluster <cit.>, but are relatively distant. We thus assume these to be small in comparison to the local emission in what follows.
Energy spectrum of background photons from our photon field. Shown are the components of Table <ref> at a distance from Sgr A$^*$ of 0.3 pc and their sum (solid line). The dashed line shows the total background at 1 pc (in front and behind Sgr A$^*$ are similar). The GC background of <cit.>, designed to be valid over larger scales, is also shown (PMS; dotted).
§ GAMMA-RAY ATTENUATION
Our first application is to the attenuation of gamma rays due to $\gamma \gamma \rightarrow e^+ e^-$ interactions on intervening photon backgrounds. The cross section depends on the relative angle with a gamma ray of energy $E_\gamma$ through $s \!=\! 2 E_\gamma \epsilon_\gamma (1-\cos{\theta})$ via $q \!=\! \sqrt{1-(2 m_e c^2)^2/s}$ as
\begin{equation}
\sigma_{\gamma \gamma}(s) \!=\! \frac{3}{4} \sigma_T \frac{(m_e c^2)^2}{s} \left[ (3\!-\! q^4) \ln \!\frac{1\!+\! q}{1\!-\! q} \!-\! 2q (2\!-\! q^2) \right]
\!,
\label{sigmapair}
\end{equation}
with $\sigma_T$ the Thomson cross section.
In Fig. <ref>, we show the result of integrating over two paths: one from the GC and a longer beam through the line of sight to PWN G359 (as denoted in Fig. <ref>) to 1 pc behind the GC. Considering photon number density above the pair threshold, the 90 K, 120 K, and 250 K fields are the most important targets. These are displayed for the latter case. To obtain the total extinction, we add the GC attenuation curve from <cit.>, which is based on an interstellar radiation field model describing the galaxy on larger scales (plus the CMB), so double counting relative to our curves should be minimal.
We also display for comparison attenuation from within the inner accretion flow of Sgr A$^*$. We use spectra from <cit.> spanning from radio to IR (model 915h), with the simplifying assumption that this is spherical within a distance from the black hole of $3\, r_g$, with $r_g \!\simeq\! 6 \times 10^{11}$ cm, comparable to the IR emitting regions. We see that this can be more important for any TeV gamma rays arising from within this limited volume around the black hole.
§ ELECTRON ENERGY LOSS SIMPLY STATED
We turn our attention to describing populations of electrons in the central parsec that can upscatter the above photon field into gamma rays. We focus on energy spectra, not attempting to fully describe source morphology (though we remark on this later), evolving an injection spectrum with synchrotron and inverse Compton losses over a specified duration. As far as X-rays from synchrotron are concerned, we will see that only the past few decades are relevant due to rapid cooling.
$T [K]$ $u_{\rm BB}$ [$10^{-9}\,$GeV cm$^{-3}$] $u_{i}/u_{\rm BB}$ (0.3 pc) $u_{i}/u_{\rm BB}$ (1 pc)
35000 $7.1\!\times\!10^{15}$ $2.5\!\times\!10^{-11}$ $1.9\!\times\!10^{-12}$
3500 $7.1\!\times\!10^{11}$ $2.1\!\times\!10^{-8}$ $1.3\!\times\!10^{-8}$
250 $1.8\!\times\!10^{7}$ $3.6\!\times\!10^{-4}$ $8.3\!\times\!10^{-5}$
120 $9.8\!\times\!10^{5}$ $5.1\!\times\!10^{-3}$ $1.2\!\times\!10^{-3}$
90 $3.1\!\times\!10^{5}$ $2.3\!\times\!10^{-3}$ $4.0\!\times\!10^{-3}$
40 $1.2\!\times\!10^{4}$ $3.9\!\times\!10^{-3}$ $2.0\!\times\!10^{-3}$
2.73 $0.26$ 1 1
Energy density normalizations of the GC radiation components.
Gamma-ray attenuation due to our background components from a location 1 pc behind the GC (dashed), the Galactic model of <cit.> (MPS; dotted), and their combination (thick solid). Also, shown is the combined total from the GC position (thin solid), compared to attenuation within the inner accretion flow of Sgr A$^*$ due to mm–IR emission (dotted).
Use of blackbody spectra allows for standard inverse Compton loss methods (dusty spectra will be examined elsewhere). This can be done more or less exactly, although the resulting solution is rather cumbersome. We rather examine first the form of the energy loss rate in the Thomson limit
\begin{equation}
\left.\frac{dE_e}{dt}\right\vert_{\rm T} = - \frac{4}{3} \, \sigma_T \, c \left(\frac{E_e}{m_e c^2}\right)^{\!2} u_{\rm BB}
\,,
\label{dEt}
\end{equation}
where $E_e$ is the electron energy, $m_e$ the electron mass, and $u_{\rm BB}$ the blackbody energy density for a given $T$, while in the extreme Klein-Nishina regime <cit.>,
\begin{equation}
\left.\frac{dE_e}{dt}\right\vert_{\rm KN} \! \!=\! - \frac{\sigma_T}{16} \frac{(m_e k_B T c)^2}{\hbar^3} \left( \ln 4 \kappa_e \!-\! 1.981 \right) \!,
\label{dEkn}
\end{equation}
where $\kappa_e \!=\! E_e k_B T/(m_e c^2)^2$. To obtain $dE_e/dt\vert_{\rm IC}$ over the entire energy range, we find a convenient interpolation valid to $\sim\!1$% below the KN limit,
\begin{equation}
\! \!\! \left.\frac{dE_e}{dt}\right\vert_{\rm H} \!=\! -b_{\rm H} \kappa_e
\left[\left(\frac{\kappa_e}{\kappa_1} \right)^{\!\!A \xi} \!\!+\! \left(\frac{\kappa_e}{\kappa_1} \right)^{\!\!B \xi}
\!\!+\! \left(\frac{\kappa_2}{\kappa_1} \right)^{\!\!B \xi} \!\! \left(\frac{\kappa_e}{\kappa_2} \right)^{\!\!C \xi} \right]^{1/\xi} \!\!\!,
\label{hasanian}
\end{equation}
with $b_{\rm H} \!=\! 3.87 \!\times\! 10^{19}(k_B T)^2 \,$GeV$^{-1}$s$^{-1}$, $A \!=\! 1$, $B \!=\! -0.063$, $C \!=\! -0.855$, $\kappa_{1} \!=\! 0.065$, $\kappa_{2} \!=\! 4.16$, $\xi \!=\! -0.815$, and in which energy is given in terms of GeV.
Now we combine Eqs. (<ref>) and (<ref>) as
\begin{equation}
\!\! \left.\frac{dE_e}{dt}\right\vert_{\rm IC} \! \!= \!
\left.\frac{dE_e}{dt}\right\vert_{\rm H} \Theta[10^{3.3} \!-\! \kappa_e]
+\! \left.\frac{dE_e}{dt}\right\vert_{\rm KN} \Theta[\kappa_e \!-\! 10^{3.3}]
\label{dEic}
\end{equation}
where $\Theta$ are step functions (cf., ). This is to be evaluated for each distinct background component.
We also need consider the rate of energy loss due to synchrotron radiation,
\begin{equation}
\left.\frac{dE_e}{dt}\right\vert_{\rm sync} = - \frac{4}{3} \, \sigma_T \, c \left(\frac{E_e}{m_e c^2}\right)^2 u_B
\,,
\label{dEtsync}
\end{equation}
for magnetic field energy density $u_B \!=\! B^2/8\pi$. Adding this to the sum of the IC loss terms, we arrive at the total losses
\begin{equation}
b_e(E_e) = - \left.\frac{dE_e}{dt}\right\vert_{\rm sync} - \sum\limits_i \frac{u_i}{u_{\rm BB}} \left.\frac{dE_e}{dt}\right\vert_{{\rm IC,}\,i}
\,,
\label{dEtot}
\end{equation}
where each IC term is scaled by the ratio of the energy density of the photon background $u_i$ to the energy density of a pure blackbody $u_{\rm BB}$ for each $T_i$ (see Table <ref>).
In Fig. <ref>, we show the cooling rate, $b_e(E_e)/E_e$, for each component of the photon field at a distance of 1 pc from Sgr A$^*$ and for two different choices of $B$ within the range discussed in <cit.> related to observations of the GC magnetar SGR J1745-29 <cit.>. These demonstrate the change in relative importance of IC versus synchrotron as $B$ is varied as well as the KN suppression via the downturn in the IC curves, e.g., for $E_e \!\gtrsim\! 10$ TeV even the CMB is more relevant than the UV background.
Rate of electron cooling due to synchrotron radiation for two field strengths (dashed lines), summed with inverse Compton losses on each background component of Fig. <ref> at a distance from Sgr A$^*$ of 1 pc (thin solid lines; as labeled) to give the total loss rates (thick solid lines).
§ EVOLVING THE ELECTRON SPECTRUM
We are interested in the present population of electrons, which requires evolving the spectrum injected over all time. To do so, we first determine the time it takes for an electron with initial energy $E_i$ to reach a final energy $E_f$ as
\begin{equation}
t_l(E_i,E_f) = \int_{E_i}^{E_f} -\frac{dE}{b_e(E)}
\,.
\label{tloss}
\end{equation}
In practice, we take a very high energy, $E_h \!=\! 10^8\,$GeV, and evaluate $t_h(E_f) \!=\! t_l(E_h,E_f)$. We then construct the inverse function $E_t[t_h(E_f)]$ numerically. This is used as a convenient way to relate initial and final energies by taking the relative difference between them. Now we integrate the source injection spectrum $dN_e/dE$ from a time $\tau$ up to today
\begin{equation}
\frac{dN_e}{dE_0} = \int_{0}^{\tau} dt\, \left.\frac{dN_e}{dE\,dt}\right\vert_{E_t[t_h(E)-t]} \frac{b_e(E_t[t_h(E)-t])}{b_e(E)}
\,.
\label{spec}
\end{equation}
This maps the source spectrum at each $t$ to the present time accounting for all relevant energy losses.
§ SYNCHROTRON AND INVERSE-COMPTON PRODUCTION
We will consider a few illustrative problems of current interest, both in limits where synchrotron is dominant and where inverse Compton losses are clearly more important. In evaluating the expected spectra of synchrotron and inverse Compton photons to compare with data, we assume that the electron population has a locally isotropic velocity distribution and that relativistic beaming effects are not relevant, though we do consider scattering off of anisotropic photon backgrounds.
Synchrotron can be elegantly calculated in the textbook manner using Bessel functions (e.g., ). Since we are interested in an isotropic electron distribution, we instead follow the simpler approach in <cit.>, with
\begin{equation}
\frac{dN_\gamma}{dE_\gamma} = \frac{\sqrt{3}}{2 \pi} \frac{e^3 B}{m_e c^2 \hbar E_\gamma} G(x) \,e^{-x} \,,
\label{synch}
\end{equation}
where $x \!=\! 3 e \hbar B E_e^2/(2 m_e^3 c^5)$ and $G(x)$ is an interpolation close to the exact solution and faster to compute. This is convolved with the present electron spectrum $dN_e/dE_0$.
Inverse Compton scattering becomes more involved, since we aim to examine the bulk angular dependence of central parsec photon backgrounds rather than assuming isotropy. <cit.> provides a treatment convenient for this purpose (see also ). For a mono-directional, blackbody photon distribution
\begin{equation}
\frac{dN_{\rm ani}}{dE_\gamma} \!=\! \frac{3 \sigma_T \,m_e^2 c}{4 \pi^2 (\hbar c)^3} \frac{(k_B T)^2}{E_e^2} \! \left[ \frac{z^2}{2(1-z)} F_1(y) \!+\! F_2(y) \right] \!,
\label{ani}
\end{equation}
with $z \!=\! E_\gamma/E_e$, $y \!=\! z (m_e c^2)^2/[2 (1-z) E_e k_B T (1 \!-\! \cos{\theta})]$. Here the photons arrive at an angle $\theta$ to the electron, with the gamma ray departing in the electron direction. Formulas for $F_1$ and $F_2$ are given in <cit.>, along with similar fitting equations $F_3$ and $F_4$ in case one is interested in using this technique to find the emission from an isotropic photon background, $dN_{\rm iso}/dE_\gamma$, e.g., the CMB.
Left: Projected distribution of electrons with 100 TeV initial energies continuously injected for 10 yr propagating in a 0.1 mG random magnetic field.
Right: Synchrotron and inverse Compton spectra from hard electron models in a 0.1 mG field with exponential (solid lines) and power-law (dotted) spectral breaks. We show an approximate NuSTAR band and GC source TeV data from HESS <cit.> and VERITAS <cit.> for scale.
Using each angular-dependent photon field, we integrate from the source vantage point over angles with respect to the direction pointing at Earth to obtain the IC spectrum as
\begin{equation}
\frac{dN_i}{dE_\gamma} = \mathcal{E}_i \int_{E_\gamma}^{E_{\rm max}} dE_e \frac{dN_e}{dE_0} \int d\Omega\, \frac{dN_{\rm ani}}{dE_\gamma} \ell_i(\theta,\phi)
\,,
\label{ICspec}
\end{equation}
where $\mathcal{E}_i \!=\! L_i / (4\pi c\, u_{\rm BB} V_i)$. One notable difference from assuming a central source is that the scattering on FIR emission from the rings is seen to vary much less in space. We obtain fluxes $\varphi_i(E_\gamma)$ using a GC distance $d_{\rm GC} \!=\! 8.5\,$kpc.
§ HARD X-RAY SYNCHROTRON AND NUSTAR
NuSTAR has recently discovered a diffuse hard X-ray flux reaching to $\gtrsim\!40\,$keV pervading the central parsecs <cit.>. While this could very well be due to some new class of sources endemic to the GC, synchrotron radiation is a well understood means of photon production, and while these X-ray energies are somewhat extreme, they are not terribly so and GC magnetic fields are unusually strong.
As Fig. <ref> shows, at sufficiently high energies synchrotron dominates over IC. Examining the characteristic energy of synchrotron emission,
\begin{equation}
E_\gamma \sim 20\, \left(\frac{E_e}{20\,{\rm TeV}}\right)^{\!2} \left(\frac{B}{{\rm mG}}\right) \, {\rm keV}
\,,
\label{Echar}
\end{equation}
we see that hard X-rays can be the main product at these energies and field strengths.
Now, Fig. <ref> also shows that the cooling time (taking the inverse of the cooling rate) becomes quite short in this range, so one might expect X-ray emission to be limited to a small region around any such electron source.
However, as <cit.> and <cit.> note, particles tend to propagate anisotropically at early times after injection from a fixed location, i.e., more quickly along the direction of the local magnetic field. Following the arguments in <cit.>, if the cooling time is shorter than the characteristic timescale to reach isotropic diffusion, we would expect synchrotron emission to illuminate a path dependent on the local field structure since the particles only possess large energies for a limited duration. If such a population is present in the GC, their bulk trajectories might be traceable by a hard X-ray telescope like NuSTAR.
To examine the plausibility of extended hard X-ray emission arising from electrons escaping a discrete source, we first consider the behavior of a population with initial energies of $\sim\!100\,$TeV. Using the methods described in <cit.> and <cit.>, we show in Fig. <ref> (left) an example of a possible realization of this scenario. Here, we have injected 100 TeV electrons over a 0.1 pc radius volume in an isotropic random field configuration scaled to $B_{\rm rms} \!=\! 0.1\,$mG with a coherence length $l_c \!\approx\! 4\,$pc. We inject continuously for $\sim\! 10\,$yr, over which time the energy can decrease to $\sim\! 50$ TeV.
While more elaborate simulations are possible, accounting for a spectrum of injected particles and energy dependence of propagation, this serves to illustrate the basic picture if high-energy electrons are not confined and free to propagate with only the local field guiding them, which may well be predominantly along the Galactic plane. Alternatively, extended emission could arise from jet-like structures as seen reaching from some pulsars (e.g., IGR J11014–6103; ).
Models in a 0.1 mG field to mimic a PWN relativistic Maxwellian, with $E_c \!=\! 25\,$GeV or $E_c \!=\! 250\,$GeV and durations $\tau \!=\! 10^3\,$yr and $10^4\,$yr yielding synchrotron (far left lines) and inverse Compton gamma rays (darker lines: isotropic IC at 1 pc; lighter lines: anisotropic IC 1 pc behind the GC). We include here the Fermi GC source 3FGL J1745.6–2859c <cit.>.
If electrons are capable of retaining high energies over such distances near the GC, we can consider the emission from a single source. We now calculate possible X-ray and gamma-ray fluxes using the methods described above. <cit.> claims that while a bulk anisotropy may be present, the local velocity distribution can still be fairly isotropic. In order to account for the hard spectrum of hard X-rays seen by NuSTAR, a hard electron spectrum may be needed. We use a smoothly-broken power law with an exponential cutoff to describe the source spectrum
\begin{equation}
\frac{dN_e}{dE} = f_e
\left[\left(E/E_1\right)^{\alpha \eta} + \left(E/E_1\right)^{\beta \eta} \right]^{1/\eta} e^{-E/E_c}\,,
\label{fit}
\end{equation}
with $\alpha$ and $\beta$ the slopes, a break at $E_1$, cutoff energy $E_c$, and using $\eta \!=\! -10$ to give a sharp break. We assume a constant injection spectrum and luminosity over a duration $\tau \!=\! 1000\,$yr.
We assume $\alpha \!=\! -1$ here. Spectra as hard as this have been displayed recently in reconnection simulations (e.g., , , ). This is also representative of any harder spectra, since an equilibrium $\sim\! E_e^{-2}$ electron spectrum would generically result from continuous injection and cooling, leading to an X-ray spectrum of $\sim\! E_\gamma^{-1.5}$. The spectral cutoff at low energies is not relevant here. We consider cases where the high-energy break is due to an exponential cutoff alone at $E_c \!=\! 1000\,$TeV, with luminosity $\mathcal{L}_e \!=\! 2 \!\times\! 10^{35}\,$erg s$^{-1}$, or a change in index to $\beta \!=\! -2.5$ at $E_1 \!=\! 200\,$TeV with $E_c \!=\! 2000\,$TeV and $\mathcal{L}_e \!=\! 10^{35}\,$erg s$^{-1}$.
Fig. <ref> (right) shows the X-ray and gamma-ray fluxes for a uniform 0.1 mG field compared to an approximated band for NuSTAR. For these hard spectra, there need not be bright radio emission. We also see that the KN suppression leads to gamma rays principally from electrons with energies lower than that yielding synchrotron in the NuSTAR range. In a weaker field, electrons would retain their energy longer. They would more easily travel large distances and, for the same photon field, emit more gamma rays. However, for a location beyond the central parsec, the photon background would be lower and the emission could remain below the HESS data.
As for where the electrons arise, the most likely culprits could be a pulsar associated with G359 or some heretofore unknown young pulsar with a velocity too low or local conditions otherwise unfavorable to yielding a prominent cometary nebula (see ). For this scenario we have assumed that the highest energy electrons are able to escape and freely propagate. The physical conditions that might permit this would depend on the nature of the source, whether one or both of a linear accelerator setup by magnetic reconnection or Fermi shock acceleration is operating, and the magnetic field structure. Excesses could be present close to the pulsar, where the field should be larger, or where a coherent PWN flow ends. This basic setup can also be applied to a population of hard X-ray sources, such as fainter PWNe due a large number of active pulsars near the GC <cit.>, which we defer to elsewhere.
§ PULSAR EXHAUST AND FERMI
Often one simply imposes a sharp break in the electron injection spectrum at some low energy (as we just assumed above). However, depending upon prevailing conditions, models that place the acceleration of particles at the termination shock in the pulsar wind can imply thermalization into a relativistic Maxwellian spectrum based on the bulk Lorentz factor of particles in the wind, with the shock energizing only some fraction of these into a power law component (e.g., ).
If such an exhaust from the electron acceleration process is produced and goes somewhere, though, it should be emitting. We examine two possible outcomes using unbroken $\alpha \!=\! 2$ spectra cutoff with $E_c \!=\! 25\,$GeV (corresponding to a bulk pre-shock wind Lorentz factor $\Gamma \!\sim\!5 \times 10^4$ and pair multiplicity $\mathcal{M} \!\sim\! 10^5$) and present luminosity $\mathcal{L}_0 \!=\! 10^{36}\,$erg s$^{-1}$ or $E_c \!=\! 250\,$GeV ($\Gamma \!\sim\!5 \times 10^5$, $\mathcal{M} \!\sim\! 10^3$, and $\mathcal{L}_0 \!=\! 10^{35}\,$erg s$^{-1}$).
Assuming a continuous luminosity, the equilibrium electron spectrum from this hard injected population will again tend toward $\sim\! E_e^{-2}$. While a fixed $\mathcal{L}_e$ is reasonable for X-rays and TeV gamma rays due to the short cooling times of the emitting particles, at lower energies the accumulated spectrum may be enhanced by the pulsar spin down history. We consider
\begin{equation}
\mathcal{L}_e(t) = \mathcal{L}_0 \left[\frac{1+(\tau-t)/\tau_p}{1+\tau/\tau_p} \right]^{-\frac{n+1}{n-1}} ,
\label{spindown}
\end{equation}
where $\tau$ is the pulsar age, $\tau_p$ is a characteristic spin down time, and we use the canonical dipole $n \!=\! 3$ <cit.>, although measured values for very young pulsars are often less than this (e.g., ) which would imply a different evolutionary history. Our choices of $\tau_p \!=\! 10^3\,$yr and $\tau$, as well as $E_c$, are motivated to illustrate relations to gamma-ray data.
Fig. <ref> shows the gamma-ray and synchrotron spectra assuming injection has occurred for $\tau \!=\! 10^3\,$yr or $10^4\,$yr. The distance is fixed to 1 pc with a 0.1 mG field, although we note that the lower loss rates at these energies would likely result in most gamma rays being produced beyond a nominal PWN. Increasing the injection duration has the effect of accumulating GeV electrons and pushing the sub-GeV gamma-ray flux upwards. The lighter IC lines show an enhancement due to assuming anisotropic IC from 1 pc behind Sgr A$^*$.
This flux is compared to the Fermi source coincident with the Galactic Center, 3FGL J1745.6–2859c, using data points from the 3FGL source catalog <cit.>, which roughly split the previous 2FGL GC source (; cf., ) into two distinct sources. We also show the TeV data for scale, though one must keep in mind that there is a possible mismatch of spatial scales between the gamma-ray data sets.
Using a larger $\tau \!=\! 10^5\,$yr would decrease the energy at which particles accumulate to $\sim\,$100 MeV. The IC flux could be increased into the NuSTAR range and continued to INTEGRAL energies <cit.>. This is though, a rather long duration to expect a high luminosity from a lone pulsar. To go farther back would also, considering typical densities in this region (e.g., ) necessitate accounting for ionization/Coulomb losses, which should overtake IC at some point and deplete the electron population at lower energies (see, e.g., Fig. 1 of ).
The large number of massive stars in the GC implies an enhanced supernova rate and can lead to a typical interval between pulsar births of $\sim\!10^4$–$10^5\,$yr <cit.>. Comparing the fluxes with varying injection periods shows the general behavior for a pulsar population. Relic electrons from inactive pulsars will no longer contribute gamma rays since high-energy particles have lost energy. For fixed luminosity, a higher spectral cutoff means fewer particles accumulating at lower energies with time (compare the two $E_c$ sets in Fig. <ref>). So while the high-energy range is more sensitive to a combination of cutoff and age, the flux of softer gamma rays depends less on age than the total number of electrons injected.
One might hope to use synchrotron to track these GeV particles and constrain the morphology. Though the details will again depend upon the ambient magnetic field, as well as the spin down history of the pulsar, we can make a few rough estimates. An isotropic diffusion coefficient of $D \!\sim\! 10^{26}\,$cm$^2\,$s$^{-1}$ implies a distance scale of $(2D \tau)^{1/2} \!\sim\! 1\,$pc for $\tau \!=\! 10^3\,$yr. Generally, both $D$ and $\mathcal{L}_{\rm sync}$ depend on $B$, with the morphology of the emission depending upon the magnetic field configuration and the photon field geometry. We defer detailed examination of such variations to elsewhere.
§ DISCUSSION AND CONCLUSIONS
The properties of the Galactic Center can lead to unusual phenomena. Consider if you will our two examples. The former examines extremely high-energy electrons, yet results mostly in photons emitted with much lower energies than those from our later example that considers much lower energy electrons. The nominal setup and parameter values appear physically plausible while leading to fluxes of hard X-rays and GeV gamma rays near the observed levels, illustrating the additional lengths yet required to understand this zone and its surroundings at high energies.
We have used a simplified model for the photon field of this complicated region as a starting point for better understanding the production of gamma rays from energetic electrons. This also helps in determining the fate of the gamma rays, produced by whatever process imagined, that may be attenuated by the same photon backgrounds. This explores a tractable middle ground between assuming isotropic backgrounds and following photons from the level of known stars to the heating and emission of dust. The latter course is perhaps difficult, but not impossible (e.g., considered the expected starlight background near the G2 object), and would aid in addressing the following additional implications.
§.§ Electrons and TeV gamma rays
In the hard X-ray range there is a paucity of backgrounds as compared to lower energies, so that emission might be attributable to synchrotron radiation even in a complex region. Evidence for electron acceleration to extremely high energies by pulsars includes the $\gtrsim\,$100 MeV flares from the Crab nebula ascribed to synchrotron from PeV electrons <cit.> and signatures of multi-TeV electrons from pulsars in the solar neighborhood (see, e.g., ). The pulsar wind nebula G359.95–0.04 <cit.> suggests such processes are active near the GC, which may also be quite relevant to TeV gamma-ray data (see for greater detail).
Our photon field model also allows for examination of another distinct scenario involving TeV gamma rays. While the gamma-ray opacity along the sight lines examined in Fig. <ref> ended up not being overwhelming, this did not have to be the case. A larger young stellar flux and/or a larger fraction of dust reprocessing, as may have been present in the past or in more active extragalactic central parsec regions, could easily lead to a more substantial suppression <cit.>.
Any TeV gamma-ray source in this region produces an extended distribution of electrons and positrons due to $\gamma \gamma \rightarrow e^+ e^-$ on the photon field. Comparing the NuSTAR and TeV energetics in Fig. <ref>, these roughly coincide. Although our result suggests that such a process is not currently efficient in our GC, a sufficiently recent outburst of TeV gamma rays would have left an $e^\pm$ detritus yet emitting synchrotron.
§.§ More on Galactic Center Hard X-rays
As a more general point regarding hard X-rays, while absorption is largely irrelevant <cit.>, the unusual gas streams in the central parsec could possess column depths sufficient to cause appreciable Thomson scattering. If so, models of the gas density can be compared to X-ray maps to examine variations in intensity to determine the relative geometry of the X-ray emission and estimate the gas column. This would help to clear up uncertainties over the nature of the CND, between high <cit.> and low <cit.> inferred masses.
We also note that NuSTAR has detected non-thermal hard X-ray emission from the radio filament Sgr A–E, suggesting that the spectrum could be accounted for via injection of electrons from an unknown PWN <cit.>. Comparing to the better resolved radio images of Sgr A–E <cit.>, we see that the tail of PWN G359 extrapolates back to this general vicinity. If related, this would imply a coherent structure of $\sim\! 10\,$pc, not unprecedented in the Milky Way (e.g., ), just not obviously realizable near the GC. This would require a rather low field strength for electrons to retain their energy until they reach the larger fields in the filament.
§.§ Moving Beyond the Center, Dark Matter, and Neutrinos
We have focused on positions within the central parsec, since at larger distances the benefit of bright, compact infrared emission potentially producing an unusually large amount of IC losses in a small volume is lost. The rather generic pulsar wind parameters used lead to a flux within range of Fermi data and allow further room for accommodation. For instance, there may well be other pulsars in this area yielding GeV gamma rays, either pulsed or from a wind. For a local supernova rate of $\sim\! 10^{-4}\,$yr$^{-1}$ these lead to overlapping contributions in the Fermi range, with burn off of electrons due to the steep rate of losses simplifying matters at higher energies.
Beyond the incentives to understand the novel astrophysics at the Galactic Center, there is also the quests for dark matter and neutrinos. Of recent interest are claims of a significant excess of gamma rays at $\sim\! 1\!-\!10\,$GeV. This may or may not be related to dark matter, but it does seem to originate at the Center, the IC scattering of electrons from annihilation or decay (cf., ) is a direct application. Improved understanding of the mechanism behind GC gamma rays will directly affect the expected flux of neutrinos (e.g., ) and whether the PeV neutrino seen from the vicinity of the GC by IceCube <cit.> has a Galactic origin <cit.>.
On larger scales, there should also be energetic electrons present from these and other processes. While the concentrated UV emission most relevant in the central parsec will drop off rapidly, the old stellar component falls off less steeply so its contribution to IC will become relatively more important and may show up at lower gamma-ray energies (cf., ). One can also consider the aforementioned Arches and Quintuplet stellar clusters, although these are rather young and lack the longer history of star formation present in the central parsec, possibly leading to fewer young pulsars. They notably would also not contain a supermassive black hole. Along with the central parsec, these could provide useful checks to discriminate between dark matter, pulsars, and diffuse cosmic-ray background contributions, details of which we will explore elsewhere.
We thank John Beacom, Jason Dexter, Ryan O'Leary, Troy Porter, and Hasan Yuksel for useful discussions and the hospitality of Brandt-Leland during the completion of this paper.
MDK acknowledges support provided by Department of Energy contract DE-AC02-76SF00515, and the KIPAC Kavli Fellowship made possible by The Kavli Foundation.
[Aartsen et al.(2013)]Aartsen2013
Aartsen, M. G., et al. [IceCube Collaboration]
2013a, , 111, 021103
[Abazajian et al.(2014)]Abazajian2014
Abazajian, K. N., Canac, N., Horiuchi, S., & Kaplinghat, M. 2014, , 90, 023526
[Abazajian et al.(2015)]Abazajian2015
Abazajian, K. N., Canac, N., Horiuchi, S., Kaplinghat, M., & Kwa, A. 2015, JCAP, 7, 013
[Abdo et al.(2011)]Abdo2011
Abdo, A. A., et al. [Fermi-LAT Collaboration]
2011, Science, 331, 739
[Abramowski et al.(2016)]Abramowski2016
Abramowski, A., et al. [HESS Collaboration]
2016, Nature, 531, 476
[Acero et al.(2015)]Acero2015
Acero, F.., et al. [Fermi-LAT Collaboration]
2015, , 218, 23
[Aharonian et al.(2004)]Aharonian2004
Aharonian, F., et al. [HESS Collaboration]
2004, , 425, L13
[Aharonian et al.(2009)]Aharonian2009
Aharonian, F., et al. [HESS Collaboration]
2009, , 503, 817
[Aharonian et al.(2010)]Aharonian2010
Aharonian, F. A., Kelner, S. R., & Prosekin, A. Y. 2010, , 82, 043002
[Ahnen et al.(2016)]Ahnen2016
Ahnen, M. L., et al. [MAGIC Collaboration]
2016, arXiv:1611.07095
[Ajello et al.(2016)]Ajello2016
Ajello, M., et al. [Fermi-LAT Collaboration]
2016, , 819, 44
[Albert et al.(2006)]Albert2006
Albert, J., et al. [MAGIC Collaboration]
2006, , 638, L101
[Amato & Arons(2006)]Amato2006
Amato, E., & Arons, J. 2006, , 653, 325
[Archer et al.(2014)]Archer2014
Archer, A., et al. [VERITAS Collaboration]
2014, , 790, 149
[Archer et al.(2016)]Archer2016
Archer, A., et al. [VERITAS Collaboration]
2016, , 821, 129
Arons, J. 2012, , 173, 341
[Barriere et al.(2014)]Barriere2014
Barrière, N. M., Tomsick, J. A., Baganoff, F. K., et al. 2014, , 786, 46
[Belanger et al.(2006)]Belanger2006
Bélanger, G., Goldwurm, A., Renaud, M., et al. 2006, , 636, 275
[Blumenthal & Gould(1970)]Blumenthal1970
Blumenthal, G. R., & Gould, R. J. 1970, , 42, 237
[Calore et al.(2015)]Calore2014
Calore, F., Cholis, I., McCabe, C., & Weniger, C. 2015, , 91, 063003
[Cerutti et al.(2013)]Cerutti2013
Cerutti, B., Werner, G. R., Uzdensky, D. A., & Begelman, M. C. 2013, , 770, 147
[Chernyakova et al.(2011)]Chernyakova2011
Chernyakova, M., Malyshev, D., Aharonian, F. A., Crocker, R. M., & Jones, D. I. 2011, , 726, 60
[Cholis et al.(2015)]Cholis2014
Cholis, I., Hooper, D., & Linden, T. 2015, , 91, 083507
[Christopher et al.(2005)]Christopher2005
Christopher, M. H., Scoville, N. Z., Stolovy, S. R., & Yun, M. S. 2005, , 622, 346
[Crocker et al.(2005)]Crocker2005
Crocker, R. M., Melia, F., & Volkas, R. R. 2005, , 622, L37
[Davidson et al.(1992)]Davidson1992
Davidson, J. A., Werner, M. W., Wu, X., et al. 1992, , 387, 189
[Daylan et al.(2014)]Daylan2014
Daylan, T., Finkbeiner, D. P., Hooper, D., Linden, T., Portillo, S. K. N., Rodd, N. L., & Slatyer, T. R.
2014, arXiv:1402.6703
[Delahaye et al.(2010)]Delahaye2010
Delahaye, T., Lavalle, J., Lineros, R., Donato, F., & Fornengo, N. 2010, , 524, A51
[Dexter & Fragile(2013)]Dexter2013
Dexter, J., & Fragile, P. C. 2013, , 432, 2252
[Dexter & O'Leary(2014)]Dexter2014
Dexter, J., & O'Leary, R. M. 2014, , 783, L7
Draine, B. T. 2003, , 41, 241
[Eatough et al.(2013)]Eatough2013
Eatough, R. P., Falcke, H., Karuppusamy, R., et al. 2013, , 501, 391
[Eatough et al.(2015)]Eatough2015
Eatough, R. P., Lazio, T. J. W., Casanellas, J., et al. 2015, arXiv:1501.00281
[Etxaluze et al.(2011)]Etxaluze2011
Etxaluze, M., Smith, H. A., Tolls, V., Stark, A. A., & González-Alfonso, E. 2011, , 142, 134
[Feldmeier-Krause et al.(2015)]FeldmeierKrause2015
Feldmeier-Krause, A. Neumayer, N., Schodel, R., et al. 2015, , 584, A2
Ferrière, K. 2012, , 540, A50
Figer, D. F. 2008, arXiv:0803.1619
[Fritz et al.(2011)]Fritz2011
Fritz, T. K., Gillessen, S., Dodds-Eden, K., et al. 2011, , 737, 73
[Fritz et al.(2014)]Fritz2014
Fritz, T. K., Chatzopoulos, S., Gerhard, O., et al. 2014, arXiv:1406.7568
[Gaensler & Slane(2006)]Gaensler2006
Gaensler, B. M., & Slane, P. O. 2006, , 44, 17
[Genzel et al.(2010)]Genzel2010
Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, , 82, 3121
[Giacinti et al.(2012)]Giacinti2012
Giacinti, G., Kachelriess, M., & Semikoz, D. V. 2012, , 108, 261101
[Goicoechea et al.(2013)]Goicoechea2013
Goicoechea, J. R., Etxaluze, M., Cernicharo, J., et al. 2013, , 769, L13
[Guo et al.(2014)]Guo2014
Guo, F., Li, H., Daughton, W., & Liu, Y.-H. 2014, , 113, 155005
[Harada et al.(2015)]Harada2015
Harada, N., Riquelme, D., Viti, S., et al. 2015, , 584, A102
[Hinton & Aharonian(2007)]Hinton2007
Hinton, J. A., & Aharonian, F. A. 2007, , 657, 302
[Ho et al.(1985)]Ho1985
Ho, P. T. P., Jackson, J. M., Barrett, A. H., & Armstrong, J. T.
1985, , 288, 575
Jones, F. C. 1968, Phys. Rev., 167, 1159
[Khangulyan et al.(2014)]Khangulyan2014
Khangulyan, D., Aharonian, F. A., & Kelner, S. R. 2014, , 783, 100
[Kistler & Beacom(2006)]Kistler2006
Kistler, M. D., & Beacom, J. F. 2006, , 74, 063007
[Kistler & Yuksel(2009)]Kistler2009
Kistler, M. D., & Yuksel, H. 2009, arXiv:0912.0264
[Kistler et al.(2012)]Kistler2012
Kistler, M. D., Yuksel, H., & Friedland, A. 2012, arXiv:1210.8180
[Kistler et al.(2014)]Kistler2014
Kistler, M. D., Stanev, T., Yuksel, H. 2014, , 90, 123006
Kistler, M. D. 2015, arXiv:1511.01159
Kistler, M. D. 2015, arXiv:1511.01530
Kistler, M. D. 2015, arXiv:1511.05199
[Kistler & Laha(2006)]Kistler2016
Kistler, M. D., & Laha, R. 2016, arXiv:1605.08781
[Krabbe et al.(1995)]Krabbe1995
Krabbe, A., Genzel, R., Eckart, A., et al. 1995, , 447, L95
[Lau et al.(2013)]Lau2013
Lau, R. M., Herter, T. L., Morris, M. R., Becklin, E. E., & Adams, J. D. 2013, , 775, 37
[Linden & Profumo(2012)]Linden2012
Linden, T., & Profumo, S. 2012, , 760, 23
[Livingstone et al.(2011)]Livingstone2011
Livingstone, M. A., Ng, C.-Y., Kaspi, V. M., Gavriil, F. P., & Gotthelf, E. V. 2011, , 730, 66
[Mills et al.(2013)]Mills2013
Mills, E. A. C., Güsten, R., Requena-Torres, M. A., & Morris, M. R. 2013, , 779, 47
[Montero-Castaño et al.(2009)]Montero2009
Montero-Castaño, M., Herrnstein, R. M., & Ho, P. T. P. 2009, , 695, 1477
[Mori et al.(2015)]Mori2015
Mori, K., Hailey, C. J., Krivonos, R., et al. 2015, arXiv:1510.04631
[Morris et al.(2014)]Morris2014
Morris, M. R., Zhao, J.-H., & Goss, W. M. 2014, IAU Symposium, 303, 369
[Moskalenko & Strong(2000)]Moskalenko2000
Moskalenko, I. V., & Strong, A. W. 2000, , 528, 357
[Moskalenko et al.(2006)]Moskalenko2006
Moskalenko, I. V., Porter, T. A., & Strong, A. W. 2006, , 640, L155
[Muno et al.(2008)]Muno2008
Muno, M. P., Baganoff, F. K., Brandt, W. N., Morris, M. R., & Starck, J.-L. 2008, , 673, 251
[Nolan et al.(2012)]Nolan2012
Nolan, P. L., et al. [Fermi-LAT Collaboration]
2012, , 199, 31
[O'Leary et al.(2015)]OLeary2015
O'Leary, R. M., Kistler, M. D., Kerr, M., & Dexter, J. 2015, arXiv:1504.02477
[O'Leary et al.(2016)]OLeary2016
O'Leary, R. M., Kistler, M. D., Kerr, M., & Dexter, J. 2016, arXiv:1601.05797
[Pavan et al.(2014)]Pavan2014
Pavan, L., Bordas, P., Pühlhofer, G., et al. 2014, , 562, A122
[Perez et al.(2015)]Perez2015
Perez, K., Hailey, C. J., Bauer, F. E., et al. 2015, , 520, 646
[Porter et al.(2006)]Porter2006
Porter, T. A., Moskalenko, I. V., & Strong, A. W. 2006, , 648, L29
[Requena-Torres et al.(2012)]RequenaTorres2012
Requena-Torres, M. A., Güsten, R., Weiß, A., et al. 2012, , 542, L21
[Rybicki & Lightman(1979)]Rybicki1979
Rybicki, G. B., & Lightman, A. P. 1979, Radiation Processes in Astrophysics (New York: Wiley)
Shcherbakov, R. V. 2014, , 783, 31
[Sironi et al.(2013)]Sironi2013
Sironi, L., Spitkovsky, A., & Arons, J. 2013, , 771, 54
[Sironi & Spitkovsky(2014)]Sironi2014
Sironi, L., & Spitkovsky, A. 2014, , 783, L21
[Smith & Wardle(2014)]Smith2014
Smith, I. L., & Wardle, M. 2014, , 437, 3159
[Støstad et al.(2015)]Stostad2015
Støstad, M., Do, T., Murray, N., Lu, J. R., Yelda, S., & Ghez, A. 2015, , 808, 106
[Tavani et al.(2011)]Tavani2011
Tavani, M., et al. [AGILE Collaboration]
2011, Science, 331, 736
[Wang et al.(2006)]Wang2006
Wang, Q. D., Lu, F. J., & Gotthelf, E. V. 2006, , 367, 937
[Werner et al.(2014)]Werner2014
Werner, G. R., Uzdensky, D. A., Cerutti, B., Nalewajko, K., & Begelman, M. C. 2014, arXiv:1409.8262
[Wilms et al.(2000)]Wilms2000
Wilms, J., Allen, A., & McCray, R. 2000, , 542, 914
[Yuksel et al.(2009)]Yuksel2009
Yüksel, H., Kistler, M. D., & Stanev, T. 2009, , 103, 051101
[Yuksel et al.(2012)]Yuksel2012
Yüksel, H., Stanev, T., Kistler, M. D., & Kronberg, P. P. 2012, , 758, 16
[Yusef-Zadeh & Morris(1987)]YusefZadeh1987
Yusef-Zadeh, F., & Morris, M. 1987, , 320, 545
[Yusef-Zadeh et al.(2013)]YusefZadeh2013
Yusef-Zadeh, F., Hewitt, J. W., Wardle, M., et al. 2013, , 762, 33
[Zdziarski & Pjanka(2013)]Zdziarski2013
Zdziarski, A. A., & Pjanka, P. 2013, , 436, 2950
[Zhang et al.(2014)]Zhang2014
Zhang, S., Hailey, C. J., Baganoff, F. K., et al. 2014, , 784, 6
[Zhao et al.(2009)]Zhao2009
Zhao, J.-H., Morris, M. R., Goss, W. M., & An, T. 2009, , 699, 186
[Zhao et al.(2010)]Zhao2010
Zhao, J.-H., Blundell, R., Moran, J. M., et al. 2010, , 723, 1097
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$^{1}$Max-Planck-Institut für Plasmaphysik, 85748 Garching, Germany
$^{2}$Department of Physics, Saint Michael's College, Colchester, VT 05439, USA
A consistent guiding-center Hamiltonian theory is derived by Lie-transform perturbation method, with terms up to second order in magnetic-field nonuniformity. Consistency is demonstrated by showing that the guiding-center transformation presented here satisfies separate Jacobian and Lagrangian constraints that have not been explored before. A new first-order term appearing in the guiding-center phase-space Lagrangian is identified through a calculation of the guiding-center polarization. It is shown that this new polarization term also yields a simpler expression of the guiding-center toroidal canonical momentum, which satisfies an exact conservation law in axisymmetric magnetic geometries. Lastly, an application of the guiding-center Lagrangian constraint on the guiding-center Hamiltonian yields a natural interpretation for its higher-order corrections.
October 18, 2015
§ INTRODUCTION
The consistent derivation of a Hamiltonian guiding-center theory that includes second-order effects in magnetic-field nonuniformity is an important problem in magnetic fusion plasma physics. While the derivation of the second-order corrections in the guiding-center Hamiltonian equations of motion yield higher-order corrections that may be ignored in practical applications, they can nonetheless be useful in gaining insights into higher-order perturbation theory.
§.§ Previous works
Recently, Parra and Calvo <cit.> and Burby, Squire, and Qin <cit.> derived guiding-center theories with second-order corrections in the guiding-center Hamiltonian using different methods. Parra and Calvo <cit.> constructed their guiding-center transformation based on a microscopic view that treats the lowest-order gyroradius $\rho_{\rm g}$ as a zeroth-order (nonperturbative) term that is introduced by a preliminary transformation, which introduces explicit gyroangle dependence in the preliminary phase-space Lagrangian. The subsequent derivation of the guiding-center phase-space Lagrangian proceeds through an asymptotic expansion in powers of a small ordering parameter $\epsilon_{\rm B} \equiv \rho_{\rm g}/L_{\rm B} \ll 1$ defined as the ratio of the gyroradius $\rho_{\rm g}$ (which is considered finite in the microscopic view) to the magnetic nonuniformity length scale $L_{\rm B} \gg \rho_{\rm g}$. Burby, Squire, and Qin <cit.>, on the other hand, derived the second-order guiding-center Hamiltonian through a computer-based algorithm that bypassed the issue of gyrogauge invariance.
These two theories were compared in Ref. <cit.> and were found to agree up to a gyroangle-independent gauge term in the guiding-center phase-space Lagrangian. Both works (which assume a vanishing electric field ${\bf E} = 0$) reproduced the first-order results of the pioneering work of Littlejohn <cit.>, which made certain simplifying assumptions on the symplectic part of the guiding-center phase-space Lagrangian (see Ref. <cit.> for a review of Hamiltonian guiding-center theory).
§.§ Present work
The purpose of the present work is to use the standard Lie-transform perturbation method to derive higher-order guiding-center Hamilton equations of motion with as few assumptions about the guiding-center Hamiltonian and Poisson-bracket structure as possible. The consistency of our guiding-center transformation will be checked through Jacobian, Hamiltonian, and Lagrangian constraints. Only results are presented here and details of the calculations are presented elsewhere <cit.>.
In the present work, we recover standard expressions for the guiding-center polarization <cit.>. We also show that a consistent treatment of a guiding-center polarization and its role in providing a more transparent guiding-center representation of the toroidal canonical angular momentum, which is an exact constant of motion in axisymmetric magnetic geometry, both require that a new first-order term be kept in the symplectic part of the guiding-center phase-space Lagrangian <cit.>.
§.§ Organization
The remainder of the paper is organized as follows. In Sec. <ref>, equivalent representations of guiding-center Hamiltonian theory are presented in terms of the guiding-center Hamiltonian (<ref>) and the guiding-center Poisson bracket (<ref>), in which the guiding-center magnetic moment $\mu \equiv J\,\Omega/B$ (expressed in terms of the gyroaction $J$) is uniquely defined and higher-order corrections due to magnetic-field nonuniformity are included in either the guiding-center potential energy $\Psi \equiv J\,\Omega + \cdots$ or the guiding-center symplectic momentum $\vb{\Pi} \equiv p_{\|}\,\bhat + \cdots$. In the Hamiltonian representation $(\vb{\Pi} \equiv p_{\|}\bhat)$, these higher-order corrections appear only in the guiding-center Hamiltonian, while, in the symplectic representation $(\Psi \equiv J\,\Omega)$, they appear only in the guiding-center Poisson bracket.
In Sec. <ref>, the higher-order guiding-center transformation is given up to second order in magnetic-field nonuniformity, and it is shown to simultaneously satisfy several consistency constraints based on the guiding-center Jacobian, Hamiltonian, and Lagrangian. These constraints leave only the perpendicular components of the first-order symplectic momentum $\vb{\Pi}_{1\bot}$ unspecified. In previous works, from Littlejohn's work <cit.> up until recent work <cit.>, the choice $\vb{\Pi}_{1\bot} \equiv 0$ was implicitly assumed. In Ref. <cit.>, it was shown that a new constraint on the choice for $\vb{\Pi}_{1\bot}$ is imposed if the guiding-center transformation must also yield the standard Pfirsch-Kaufman expression for the guiding-center polarization <cit.>. This new choice is shown in Sec. <ref> to lead to a more transparent guiding-center representation for the toroidal canonical momentum, which is an exact constant of motion in axisymmetric tokamak geometry.
§ HIGHER-ORDER GUIDING-CENTER HAMILTONIAN THEORY
In the following perturbation analysis, we use the macroscopic view (i.e., $L_{\rm B}$ is finite and $\rho_{\rm g} \ll L_{\rm B}$), which introduces a dimensionless ordering parameter $\epsilon$ used in renormalizing the electric charge $e \rightarrow e/\epsilon$ (e.g., $\Omega = eB/mc \rightarrow \epsilon^{-1}\Omega$) <cit.>. According to this view, a preliminary phase-space transformation is not required and physical results are recovered by setting $\epsilon = 1$ (while ordering in $]epsilon_{B}$ is simply determined by inspection).
§.§ Guiding-center Hamiltonian and Poisson-bracket structure
Guiding-center Hamiltonian dynamics is expressed in terms of a guiding-center Hamiltonian function that depends on the guiding-center position ${\bf X}$, the guiding-center parallel momentum $p_{\|}$, and the guiding-center gyroaction $J \equiv \mu\,B/\Omega$; it is, however, independent of the gyroangle $\theta$ at all orders. Since the guiding-center phase-space coordinates are non-canonical coordinates, a noncanonical guiding-center Poisson bracket, whose components are also gyroangle-independent, is also needed.
The guiding-center Hamiltonian $H_{\rm gc}$ and the guiding-center symplectic structure defined by the Poincaré-Cartan one-form $\Gamma_{\rm gc}$ (from which the guiding-center Poisson bracket is constructed) are used to construct the guiding-center phase-space Lagrangian:
\begin{eqnarray}
\Lambda_{\rm gc} & \equiv & \Gamma_{\rm gc} \;-\; H_{\rm gc}\,dt \nonumber \\
& = & \left( {\sf T}_{\rm gc}^{-1}\Gamma_{0} \;+\; \exd S\right) \;-\; \left( {\sf T}_{\rm gc}^{-1}H_{0}\right)\,dt,
\label{eq:Gamma_H_gc}
\end{eqnarray}
where ${\sf T}_{\rm gc}^{-1}$ denotes the guiding-center (push-forward) Lie-transform operator and $S$ denotes an arbitrary gauge-function. In addition, the lowest-order Hamiltonian and symplectic structure
\begin{equation}
\left. \begin{array}{rcl}
H_{0} & \equiv & p_{\|0}^{2}/2m + J_{0}\Omega({\bf x}) \\
& & \\
\Gamma_{0} & \equiv & \left[e\,{\bf A}({\bf x})/c \;+\frac{}{} {\bf p}_{0}({\bf x},p_{\|0},J_{0},\theta_{0})\right]\bdot\exd{\bf x}
\end{array} \right\},
\label{eq:HGamma_0}
\end{equation}
are expressed in terms of the lowest-order guiding-center (local particle) coordinates
\begin{equation}
z_{0}^{\alpha} \equiv ({\bf x},p_{\|0},J_{0},\theta_{0}),
\label{eq:z0_def}
\end{equation}
where ${\bf p}_{0} \equiv p_{\|0}\bhat ({\bf x}) + {\bf p}_{\bot 0}(J_{0},\theta_{0},{\bf x})$ denotes the local particle momentum expressed in terms of parallel and perpendicular components defined with respect to the magnetic unit vector $\bhat({\bf x})$ at the particle position ${\bf x}$.
The guiding-center Euler-Lagrange equations are obtained from the guiding-center variational principle $\delta\int\Lambda_{\rm gc} = 0$:
\begin{equation}
\left(\vb{\omega}_{\rm gc}\right)_{\alpha\beta}\;\frac{d_{\rm gc}Z^{\beta}}{dt} \;=\; \pd{H_{\rm gc}}{Z^{\alpha}},
\label{eq:EL_gc}
\end{equation}
where the guiding-center Lagrange two-form $\vb{\omega}_{\rm gc} = \exd\Gamma_{\rm gc}$ has the components $(\vb{\omega}_{\rm gc})_{\alpha\beta} \equiv
\partial_{\alpha}\Gamma_{{\rm gc}\beta} - \partial_{\beta}\Gamma_{{\rm gc}\alpha}$, which form an anti-symmetric matrix. We note that the exact one-form $\exd S$ in
Eq. (<ref>) does not change the guiding-center Lagrange two-form $\vb{\omega}_{\rm gc} = \exd\Gamma_{\rm gc} = {\sf T}_{\rm gc}^{-1}(\exd\Gamma_{0}) = {\sf T}_{\rm gc}^{-1}\vb{\omega}_{0}$, since the exterior derivative $\exd$ satisfies the identity $\exd^{2}S \equiv 0$ (analogous to the vector identity $\nabla\btimes\nabla S = 0$), and $\exd$ commutes with ${\sf T}_{\rm gc}^{-1}$.
§.§.§ Equivalent Hamiltonian theories
In the present work, the guiding-center Hamiltonian in Eq. (<ref>) is defined as
\begin{equation}
H_{\rm gc} \;\equiv\; \frac{p_{\|}^{2}}{2m} \;+\; \Psi,
\label{eq:Hamiltonian_gc}
\end{equation}
where the effective guiding-center potential energy
\begin{equation}
\Psi \;\equiv\; J\,\Omega \;+\; \epsilon\,\Psi_{1} \;+\; \epsilon^{2}\,\Psi_{2} \;+\; \cdots
\label{eq:Psi_def}
\end{equation}
is defined in terms of the gyroangle-independent scalar fields $\Psi_{n}$ ($n \geq 1$), which contain corrections due to magnetic-field nonuniformity.
The guiding-center symplectic structure in Eq. (<ref>), on the other hand, is defined in terms of the Poincaré-Cartan one-form
\begin{eqnarray}
\Gamma_{\rm gc} & \equiv & \left( \frac{e}{\epsilon c}\,{\bf A} \;+\; \vb{\Pi} \right)\bdot\exd{\bf X} \;+\; \epsilon\,J\left(\exd\theta \;-\; {\bf R}\bdot\exd{\bf X}\right),
\label{eq:Gamma_gc}
\end{eqnarray}
where the symplectic guiding-center momentum
\begin{equation}
\vb{\Pi} \;\equiv\; \sum_{n = 0}^{\infty}\epsilon^{n}\,\vb{\Pi}_{n} \;=\; p_{\|}\,\bhat \;+\; \epsilon\,\vb{\Pi}_{1} \;+\; \epsilon^{2}\,\vb{\Pi}_{2} + \cdots
\label{eq:Pi_def}
\end{equation}
is expressed in terms of the gyroangle-independent vector fields $\vb{\Pi}_{n}$ ($n \geq 1$), which contain corrections due to magnetic-field nonuniformity. The presence of the gyrogauge vector ${\bf R}({\bf X})$ guarantees that the guiding-center one-form (<ref>) is gyrogauge-invariant <cit.>.
Using Eqs. (<ref>) and (<ref>), the guiding-center phase-space Lagrangian (<ref>) is thus expressed as
\begin{eqnarray}
\Lambda_{\rm gc} & = & \left[ \left(\frac{e}{\epsilon c}\,{\bf A} \;+\; p_{\|}\,\bhat \;-\; \epsilon\,J\,{\bf R}\right)\bdot\exd{\bf X} \;+\; \epsilon\,J\;\exd\theta \right]
\label{eq:Lambda_gc} \\
& - &\left( \frac{p_{\|}^{2}}{2m} + J\,\Omega \right) dt + \sum_{n=1}^{\infty}\epsilon^{n} \left( \Psi_{n}\frac{}{}dt - \vb{\Pi}_{n}\bdot\exd{\bf X} \right),
\nonumber
\end{eqnarray}
where higher-order corrections $(n \geq 1)$ are either contained in the guiding-center Hamiltonian $(\Psi_{n} \neq 0)$ or the guiding-center symplectic structure
$(\vb{\Pi}_{n} \neq 0)$.
Guiding-center theories are said to be equivalent <cit.> if they have the same definition of the guiding-center gyroaction $J$ but different definitions of the scalar field $\Psi$ and the vector field $\vb{\Pi}$. This equivalence class will be expressed at each order in terms of a relation involving the combination $\Psi_{n} - \Pi_{n\|}\;p_{\|}/m$, where $\Pi_{n\|} \equiv \bhat\bdot\vb{\Pi}_{n}$ denotes the parallel component of $\vb{\Pi}_{n}$.
In a purely Hamiltonian representation ($\vb{\Pi}_{n} \equiv 0)$, the vector field $\vb{\Pi} \equiv p_{\|}\,\bhat$ is independent of the gyroaction $J$, while the scalar field $\Psi \equiv J\,\Omega + \epsilon\,\Psi_{1} + \epsilon^{2}\,\Psi_{2} + \cdots$ contains all the correction terms associated with the nonuniformity of the magnetic field. In a purely symplectic representation ($\Psi_{n} \equiv 0)$, on the other hand, the scalar field $\Psi \equiv J\,\Omega$ is independent of the parallel momentum $p_{\|}$, while the vector field $\vb{\Pi} = p_{\|}\,\bhat + \epsilon\,\vb{\Pi}_{1} + \cdots$ contains all the correction terms associated with the nonuniformity of the magnetic field. Our analysis will show that, while a purely Hamiltonian representation is possible at all orders, a purely symplectic representation is possible only at first order. We note that previous guiding-center Hamiltonian theories were constructed in a mixed representation (i.e., symplectic at first order and Hamiltonian at second order).
§.§.§ Guiding-center Poisson bracket
The guiding-center Poisson bracket obtained from the guiding-center Euler-Poincaré one-form (<ref>) by following the following inversion procedure. First, we construct the guiding-center Lagrange two-form $\vb{\omega}_{\rm gc} \equiv \exd\Gamma_{\rm gc}$. We note that the Lagrange component-matrix is invertible if the
guiding-center Jacobian does not vanish <cit.>
\begin{equation}
{\cal J}_{\rm gc} \equiv \sqrt{{\rm det}(\vb{\omega}_{\rm gc})} = \epsilon\;\bhat^{*}\bdot\left(\frac{e}{\epsilon\,c}\;{\bf B}^{*}\right) \equiv \frac{e}{c}\;B_{\|}^{**} \neq 0,
\label{eq:Jac_gc}
\end{equation}
where we use the following definitions
\begin{eqnarray}
{\bf B}^{*} & \equiv & \nabla\btimes\left[ {\bf A} \;+\; \frac{c}{e}\,\left( \epsilon\,\vb{\Pi} \;-\frac{}{}
\epsilon^{2}\;J\;{\bf R} \right) \right], \label{eq:Bstar_def} \\
\bhat^{*} & \equiv & \pd{\vb{\Pi}}{p_{\|}} \;=\; \bhat \;+\; \epsilon\;\pd{\vb{\Pi}_{1}}{p_{\|}} \;+\; \cdots, \label{eq:bstar_def} \\
{\bf R}^{*} & \equiv & {\bf R} \;-\; \epsilon^{-1}\;\pd{\vb{\Pi}}{J} \;=\; {\bf R} \;-\; \pd{\vb{\Pi}_{1}}{J} \;+\; \cdots, \label{eq:Rstar_def} \\
B_{\|}^{**} & \equiv & \bhat^{*}\bdot{\bf B}^{*} = \left( \bhat + \epsilon\pd{\vb{\Pi}_{1}}{p_{\|}} + \cdots \right)\bdot{\bf B}^{*}.
\label{eq:B||star_def}
\end{eqnarray}
Here, the fields ${\bf B}^{*}$ and $\bhat^{*}$ satisfy the identities $\nabla\bdot{\bf B}^{*} \equiv 0$, $\partial{\bf B}^{*}/\partial p_{\|} \equiv \epsilon\,(c/e)\,\nabla\btimes\bhat^{*}$, and $\partial{\bf B}^{*}/\partial J \equiv -\,\epsilon^{2}(c/e)\,\nabla\btimes{\bf R}^{*}$, which play an important role in the properties of the guiding-center Poisson bracket.
Next, we invert the guiding-center Lagrange matrix $\vb{\omega}_{\rm gc}$ to construct the guiding-center Poisson matrix with components
$J_{\rm gc}^{\alpha\beta}$, such that $J_{\rm gc}^{\alpha\nu}\,(\omega_{\rm gc})_{\nu\beta} \equiv \delta^{\alpha}_{\;\beta}$. Lastly, we construct the guiding-center Poisson bracket $\{F,\; G\}_{\rm gc} \equiv (\partial F/\partial Z^{\alpha})\,J_{\rm gc}^{\alpha\beta}\,(\partial G/\partial Z^{\beta})$:
\begin{eqnarray}
\left\{ F,\frac{}{} G\right\}_{\rm gc} & = & \epsilon^{-1} \left( \pd{F}{\theta}\,\pd{G}{J} \;-\; \pd{F}{J}\,\pd{G}{\theta} \right) \nonumber \\
& &+\; \frac{{\bf B}^{*}}{B_{\|}^{**}}\bdot\left(\nabla^{*}F\;\pd{G}{p_{\|}} \;-\; \pd{F}{p_{\|}}\;\nabla^{*}G \right) \nonumber \\
& &-\; \frac{\epsilon\,c\bhat^{*}}{e\,B_{\|}^{**}}\bdot\nabla^{*}F\btimes\nabla^{*}G,
\label{eq:PB_gc_star}
\end{eqnarray}
where the modified gradient operator $\nabla^{*} \equiv \nabla + {\bf R}^{*}\partial/\partial\theta$ ensures gyrogauge-invariance <cit.>. The derivation procedure of the guiding-center Poisson bracket (<ref>) guarantees that it satisfies the standard Poisson-bracket properties, while the guiding-center Jacobian (<ref>) can be used to write Eq. (<ref>) in phase-space divergence form
\begin{equation}
\left\{ F,\frac{}{} G\right\}_{\rm gc} \;=\; \frac{1}{{\cal J}_{\rm gc}}\;\pd{}{Z^{\alpha}}\left({\cal J}_{\rm gc}\;F\frac{}{} \left\{ Z^{\alpha},\;
G\right\}_{\rm gc}\right).
\label{eq:PBgc_div}
\end{equation}
§.§ Guiding-center Hamilton equations of motion
The Hamiltonian guiding-center equations of motion
\begin{eqnarray}
\frac{d_{\rm gc}Z^{\alpha}}{dt} & = & J_{\rm gc}^{\alpha\nu}\,(\omega_{\rm gc})_{\nu\beta}\,\frac{d_{\rm gc}Z^{\beta}}{dt} \nonumber \\
& = & J_{\rm gc}^{\alpha\nu}\,\pd{H_{\rm gc}}{Z^{\nu}} \;\equiv\; \left\{ Z^{\alpha},\frac{}{} H_{\rm gc}\right\}_{\rm gc}
\end{eqnarray}
are expressed in terms of the guiding-center Hamiltonian (<ref>) and the guiding-center Poisson bracket (<ref>) as
\begin{eqnarray}
\frac{d_{\rm gc}{\bf X}}{dt} & = & \left(\frac{p_{\|}}{m} + \pd{\Psi}{p_{\|}}\right)\;\frac{{\bf B}^{*}}{B_{\|}^{**}}
\;+\; \frac{\epsilon\,c\bhat^{*}}{e\,B_{\|}^{**}}\btimes\nabla\Psi,
\label{eq:Xdot_gc} \\
\frac{d_{\rm gc}p_{\|}}{dt} & = & -\;\frac{{\bf B}^{*}}{B_{\|}^{**}}\bdot\nabla\Psi,
\label{eq:pdot_gc} \\
\frac{d_{\rm gc}\theta}{dt} & = &\epsilon^{-1}\;\pd{\Psi}{J} \;+\; \frac{d_{\rm gc}{\bf X}}{dt}\bdot{\bf R}^{*},
\label{eq:thetadot_gc}
\end{eqnarray}
\begin{equation}
\frac{d_{\rm gc}J}{dt} \;=\; -\,\epsilon^{-1}\,\pd{\Psi}{\theta} \;\equiv\; 0,
\label{eq:Jdot_gc}
\end{equation}
where the last equation follows from the effective guiding-center potential energy $\Psi$ being gyroangle-independent to all orders in $\epsilon$. We note that the Hamiltonian guiding-center equations of motion (<ref>)-(<ref>) satisfy the guiding-center Liouville theorem
\begin{equation}
\nabla\bdot\left( B_{\|}^{**}\;\frac{d_{\rm gc}{\bf X}}{dt}\right) \;+\; \pd{}{p_{\|}}\left(B_{\|}^{**}\;\frac{d_{\rm gc}p_{\|}}{dt}\right) \;=\; 0,
\label{eq:gc_Liouville}
\end{equation}
which shows that the gyromotion action-angle dynamics, represented by Eqs. (<ref>)-(<ref>), is completely decoupled from the reduced guiding-center dynamics represented by Eqs. (<ref>)-(<ref>).
In the guiding-center Hamilton equations (<ref>)-(<ref>), the scalar field $\Psi$ appears explicitly, while the symplectic momentum
vector field $\vb{\Pi}$ appears implicitly in the guiding-center Poisson bracket through the vector fields ${\bf B}^{*}$, $\bhat^{*}$, and ${\bf R}^{*}$. The advantage of the Hamiltonian representation is that the guiding-center Poisson bracket is simplified by the choice $\vb{\Pi} = p_{\|}\,\bhat$, while the advantage of the symplectic representation is that the guiding-center Hamiltonian is simplified by the choice $\Psi = J\,\Omega$.
§ CONSISTENT GUIDING-CENTER TRANSFORMATION
The derivation of the guiding-center phase-space Lagrangian (<ref>) by Lie-transform phase-space Lagrangian perturbation method is based on a phase-space transformation from the local phase-space coordinates (<ref>) to guiding-center coordinates $Z^{\alpha} = ({\bf X}, p_{\|}; J, \theta)$ generated by the vector fields
$({\sf G}_{1}, {\sf G}_{2}, \cdots)$:
\begin{equation}
Z^{\alpha} \;=\; z_{0}^{\alpha} + \epsilon\,G_{1}^{\alpha} + \epsilon^{2}\,\left( G_{2}^{\alpha} + \frac{1}{2}\,{\sf G}_{1}\cdot\exd
G_{1}^{\alpha}\right) + \cdots,
\label{eq:z_bar_z}
\end{equation}
with its inverse defined as
\begin{equation}
z_{0}^{\alpha} \;=\; Z^{\alpha} - \epsilon\,G_{1}^{\alpha} - \epsilon^{2}\,\left( G_{2}^{\alpha} - \frac{1}{2}\,{\sf G}_{1}\cdot\exd
G_{1}^{\alpha}\right) + \cdots.
\label{eq:zz_bar}
\end{equation}
In Eqs. (<ref>)-(<ref>), the lowest-order guiding-center phase-space coordinates $z_{0}^{\alpha}$ are the local phase-space coordinates (<ref>), where ${\bf x}$ denotes the particle position, $p_{\|0} \equiv {\bf p}_0\bdot\bhat({\bf x})$ denotes the local parallel momentum as calculated from the magnetic unit vector $\bhat({\bf x})$ evaluated at the particle position ${\bf x}$, $J_{0} \equiv |{\bf p}_{\bot 0}|^{2}/2m\Omega({\bf x})$ denotes the lowest-order gyroaction, where ${\bf p}_{\bot 0} \equiv \bhat\btimes({\bf p}_0\btimes\bhat)$, and $\theta_{0}$ denotes the lowest-order gyroangle such that $\partial{\bf p}_{\bot 0}/\partial\theta_{0} =
{\bf p}_{\bot 0}\btimes\bhat$. The Jacobian for the transformation to local phase-space coordinates $({\bf x},{\bf p}_0) \rightarrow ({\bf x}, p_{\|0}, J_{0}, \theta_{0})$ is ${\cal J}_{0} = m\Omega = e\,B/c$.
While the derivation of the guiding-center phase-space coordinates some freedom (e.g., choosing a Hamiltonian or a symplectic representation), we must ensure that these coordinates are chosen consistently. For this purpose, a set of constraints is introduced to verify consistency at each order.
§.§ Guiding-center Jacobian constraints
The guiding-center Jacobian (<ref>) associated with the phase-space transformation (<ref>) is defined as
\begin{eqnarray}
{\cal J}_{\rm gc} & = & {\cal J}_{0} \;-\; \left. \pd{}{Z^{\alpha}}\right[ {\cal J}_{0}\frac{}{} \left(\epsilon\,G_{1}^{\alpha} \;+\frac{}{}
\epsilon^{2}\,G_{2}^{\alpha} + \cdots\right) \nonumber \\
& &\left.-\; \frac{\epsilon^{2}}{2}\;G_{1}^{\alpha}\;\pd{}{Z^{\beta}}\left({\cal J}_{0}\frac{}{} G_{1}^{\beta} + \cdots\right)
\;+\; \cdots \right] \nonumber \\
& \equiv & {\cal J}_{0} \;+\; \epsilon\,{\cal J}_{1} \;+\; \epsilon^{2}\;{\cal J}_{2} \;+\; \cdots.
\label{eq:Jacobian_Lie}
\end{eqnarray}
Hence, at first and second orders, the components of the first and second order generating vector fields
${\sf G}_{1}$ and ${\sf G}_{2}$ must satisfy the Jacobian constraints:
\begin{eqnarray}
\frac{{\cal J}_{1}}{{\cal J}_{0}} & = & \pd{\Pi_{1\|}}{p_{\|}} \;+\; \varrho_{\|}\,\tau \;\equiv\; -\;\frac{1}{{\cal J}_{0}} \pd{}{Z^{\alpha}}\left(
{\cal J}_{0}\frac{}{} G_{1}^{\alpha}\right), \label{eq:Jac_1} \\
\frac{{\cal J}_{2}}{{\cal J}_{0}} & = & \pd{\Pi_{2\|}}{p_{\|}} + \varrho_{\|}\;\pd{\vb{\Pi}_{1}}{p_{\|}}\bdot\nabla\btimes\bhat + \frac{c\bhat}{eB}\bdot\nabla\btimes(\vb{\Pi}_{1} - J\,{\bf R}) \nonumber \\
& \equiv & -\;\frac{1}{{\cal J}_{0}} \pd{}{Z^{\alpha}}\left( {\cal J}_{0}\; G_{2}^{\alpha} \;+\;\frac{1}{2}\;{\cal J}_{1}\,G_{1}^{\alpha} \right),
\label{eq:Jac_2}
\end{eqnarray}
where $\varrho_{\|} \equiv p_{\|}/(m\Omega)$ and $\tau \equiv \bhat\bdot\nabla\btimes\bhat$.
§.§ Guiding-center Hamiltonian constraints
Another requirement for the guiding-center transformation (<ref>) is that the definition of the guiding-center gyroaction $J$ must be unique, which leads to the following guiding-center Hamiltonian constraints <cit.>.
§.§.§ First-order Hamiltonian constraint
The second-order $(\epsilon^{2})$ Lie-transform perturbation analysis <cit.> yields the first-order $(\epsilon_{\rm B})$ guiding-center Hamiltonian constraint
\begin{eqnarray}
\Psi_{1} \;-\; \frac{p_{\|}}{m}\;\Pi_{1\|} & \equiv & -\;\Omega\;\langle G_{1}^{J}\rangle \;-\; \frac{1}{2}\;J\,\Omega\;\varrho_{\|}\,\tau \nonumber \\
& = & \frac{1}{2}\;J\,\Omega\;\varrho_{\|}\,\tau,
\label{eq:Ham_constraint_1}
\end{eqnarray}
where $\langle G_{1}^{J}\rangle \equiv -\,J\;\varrho_{\|}\,\tau$ is calculated at order $\epsilon^{3}$ in the Lie-transform perturbation analysis <cit.>. This first-order Hamiltonian constraint, of course, has an infinite number of solutions for $(\Pi_{1\|},\Psi_{1})$. One possible choice for $(\Pi_{1\|},\Psi_{1})$, for example, is $\Pi_{1\|} = \frac{1}{2}\,J\,\tau$ and $\Psi_{1} = J\,\Omega\;(\varrho_{\|}\tau)$, which allows the Baños parallel drift velocity $\partial\Psi_{1}/\partial p_{\|} = J\,\tau/m$ to be included in Eq. (<ref>).
Here, we note that, since the right side of Eq. (<ref>) is linear in $p_{\|}$, we may choose $\Psi_{1} \equiv 0$ without making $\Pi_{1\|}$ singular. We, therefore, choose the first-order symplectic representation
\begin{equation}
\left. \begin{array}{rcl}
\Psi_{1} & \equiv & 0 \\
& & \\
\Pi_{1\|} & \equiv & -\;\frac{1}{2}\,J\,\tau
\end{array} \right\},
\label{eq:Pi1||_def}
\end{equation}
in accordance with standard guiding-center and gyrocenter Hamiltonian theories <cit.>.
§.§.§ Second-order Hamiltonian constraint
The third-order $(\epsilon^{3})$ Lie-transform perturbation analysis <cit.> yields the second-order $(\epsilon_{\rm B}^{2})$ guiding-center Hamiltonian constraint
\begin{eqnarray}
\Psi_{2} \;-\; \frac{p_{\|}}{m}\;\Pi_{2\|} & \equiv & -\,\Omega\;\langle G_{2}^{J}\rangle \;+\; J\Omega\;\varrho_{\|}^{2} \left( \frac{1}{2}\,\tau^{2} -
\langle\alpha_{1}^{2}\rangle \right) \nonumber \\
& &+\; \vb{\Pi}_{1}\bdot{\bf v}_{\rm gc} \;-\; \frac{m}{2}\,|{\bf v}_{\rm gc}|^{2},
\label{eq:Ham_constraint_2_primitive}
\end{eqnarray}
where $\langle G_{2}^{J}\rangle$ is calculated at order $\epsilon^{4}$ <cit.>, we have defined the gyroangle-dependent scalar function
\begin{equation}
\alpha_{1} \;\equiv\; -\,\frac{1}{2} \left(\wh{\bot}\wh{\rho} + \wh{\rho}\wh{\bot}\right):\nabla\bhat
\label{eq:alpha1_def}
\end{equation}
(where we use the rotating unit-vector basis $\wh{\bot} \equiv \wh{\rho}\btimes\bhat = \partial\wh{\rho}/
\partial\theta$), and ${\bf v}_{\rm gc}$ denotes the lowest-order guiding-center (perpendicular) drift velocity
\begin{equation}
{\bf v}_{\rm gc} \;\equiv\; \frac{\bhat}{m\Omega}\btimes\left( J\;\nabla\Omega \;+\; \frac{p_{\|}^{2}}{m}\;\vb{\kappa}\right),
\label{eq:vgc_def}
\end{equation}
where $\vb{\kappa} \equiv \bhat\bdot\nabla\bhat$ denotes the magnetic curvature. We now see that the perpendicular component $\vb{\Pi}_{1\bot}$ makes its appearance in Eq. (<ref>).
When $\langle G_{2}^{J}\rangle$ is calculated at order $\epsilon^{4}$ in the Lie-transform perturbation analysis <cit.>, we find
\begin{eqnarray}
\langle G_{2}^{J}\rangle & = & \frac{J^{2}}{2m\Omega} \left[ \frac{\tau^{2}}{2} + \bhat\bdot\nabla\btimes{\bf R} - \langle\alpha_{1}^{2}
\rangle - \frac{\bhat}{2}\bdot\nabla\btimes(\bhat\btimes\nabla\ln B) \right] \;-\; \frac{J}{2}\,\varrho_{\|}^{2} \left[ \vb{\kappa}\bdot(3\,
\vb{\kappa} - \nabla\ln B) \;+\frac{}{} \nabla\bdot\vb{\kappa} - \tau^{2} \right].
\label{eq:G2_J_Ham}
\end{eqnarray}
which, when inserted into Eq. (<ref>), yields the second-order $(\epsilon_{\rm B}^{2})$ guiding-center Hamiltonian constraint <cit.>
\begin{eqnarray}
\Psi_{2} \;-\; \frac{p_{\|}}{m}\;\Pi_{2\|} & \equiv & J\,\Omega\left( \frac{J}{2\,m\Omega}\;\beta_{2\bot} \;+\; \frac{1}{2}\,\varrho_{\|}^{2}\;\beta_{2\|} \right) \nonumber \\
& &-\; \frac{p_{\|}^{2}}{2m}\;\left(\varrho_{\|}^{2}\frac{}{}|\vb{\kappa}|^{2}\right) \;+\; \vb{\Pi}_{1}\bdot{\bf v}_{\rm gc},
\label{eq:Hamiltonian_constraint_2}
\end{eqnarray}
where the gyroangle-independent scalar fields $\beta_{2\bot}({\bf X})$ and $\beta_{2\|}({\bf X})$ are defined as
\begin{eqnarray}
\beta_{2\bot} & = & -\,\frac{1}{2}\,\tau^{2} \;-\; \bhat\bdot\nabla\btimes{\bf R} \;+\; \langle\alpha_{1}^{2}\rangle \;-\; \left|\bhat\btimes\nabla\ln B\right|^{2} \nonumber \\
& &+\; \frac{1}{2}\;\bhat\bdot\nabla\btimes\left(\bhat\btimes\nabla\ln B\right),
\label{eq:beta2_perp} \\
\beta_{2\|} & = & -\,2\;\langle\alpha_{1}^{2}\rangle \;-\; 3\;\vb{\kappa}\bdot\left(\nabla\ln B \;-\frac{}{} \vb{\kappa}\right) \;+\;
\nabla\bdot\vb{\kappa},
\label{eq:beta2_par}
\end{eqnarray}
with the definitions <cit.>
\begin{equation}
\bhat\bdot\nabla\btimes{\bf R} \;=\; \frac{1}{2}\;\nabla\bdot\left[ \vb{\kappa} \;-\frac{}{} \bhat\;(\nabla\bdot\bhat)\right],
\end{equation}
\begin{equation}
\langle \alpha_{1}^{2}\rangle \;=\; \frac{1}{2}\,\bhat\bdot\nabla\btimes{\bf R} \;+\; \frac{1}{8} \left[ \tau^{2} \;+\; \left(\nabla\bdot\bhat\right)^{2}
\right].
\label{eq:alpha1_square}
\end{equation}
The last term in Eq. (<ref>), which involves $\vb{\Pi}_{1\bot}$, is ignored in all previous works since it was previously assumed that
$\vb{\Pi}_{1\bot} = 0$.
We now note that, in contrast to first-order guiding-center Hamiltonian constraint (<ref>), the right side of Eq. (<ref>) contains terms that are constant, quadratic, and quartic in $p_{\|}$. Hence, since Eq. (<ref>) shows that $\beta_{2\bot} \neq 0$, we cannot choose $\Psi_{2} = 0$ without making $\Pi_{2\|}$ singular in $p_{\|}$, i.e., a purely symplectic representation is no longer possible at second order.
§.§ Previous second-order Hamiltonian representations
In order to compare our results with the results presented in Refs. <cit.>, going back to Littlejohn's work
<cit.>, we choose $\Pi_{2\|} \equiv 0$ and temporarily set $\vb{\Pi}_{1\bot} \equiv 0$ in Eq. (<ref>). Hence, with these simplifying assumptions, our work agrees with the second-order guiding-center Hamiltonian of Burby, Squire, and Qin (BSQ) <cit.>:
\begin{equation}
\Psi_{2(TB)} \;=\; \Psi_{2(BSQ)} \;=\; \Psi_{2(PC)} \;+\; \frac{d_{0}\langle\sigma_{3(PC)}\rangle}{dt},
\label{eq:Psi2_BT_BSQ_PC}
\end{equation}
while it agrees with the second-order guiding-center Hamiltonian of Parra and Calvo (PC) <cit.> only up to the lowest-order guiding-center time derivative of the gyroangle-independent third-order gauge function
\begin{equation}
\langle\sigma_{3(PC)}\rangle \;=\; \frac{1}{2}\,J\;\varrho_{\|}\,(\nabla\bdot\bhat) \;\equiv\; \frac{d_{0}}{dt}\left(\frac{J}{2\Omega}\right)
\label{eq:PC_gauge}
\end{equation}
in the same manner discussed in Ref. <cit.>, where the lowest-order guiding-center time derivative is defined as $d_{0}/dt \equiv (p_{\|}/m)\bhat\bdot\nabla +
J\Omega\,(\nabla\bdot\bhat)\,\partial/\partial p_{\|}$.
We note that the guiding-center phase-space Lagrangian $\Lambda_{\rm gc(PC)} \equiv L_{\rm gc(PC)}\,dt$ of Parra and Calvo <cit.> differs from the other two guiding-center phase-space Lagrangians $L_{\rm gc(BSQ)} = L_{\rm gc(TB)}$ by an exact time derivative $d_{0}\langle\sigma_{3(PC)}\rangle/dt$. Since two Lagrangians $L({\bf q},\dot{\bf q},t)$ and $L^{\prime}({\bf q},\dot{\bf q},t)$ on configuration space ${\bf q}$ that differ by an exact time derivative $L^{\prime} \equiv L + dF/dt$ yield the same Euler-Lagrange equations <cit.> for any function $F({\bf q},t)$, the Lagrangians of Parra and Calvo <cit.> and Burby, Squire, and Qin <cit.> are said to be equivalent <cit.>.
Lastly, in our previous work <cit.>, where $\vb{\Pi}_{1\bot} \equiv 0$ was assumed, we selected the following mixed representation: the second-order symplectic term $\Pi_{2\|}(p_{\|},J,{\bf X}) = \frac{1}{2}\,p_{\|}\,[ \varrho_{\|}^{2}|\vb{\kappa}|^{2} - (J/m\,\Omega)\,\beta_{2\|}]$, and the second-order Hamiltonian term
$\Psi_{2}(J,{\bf X}) \equiv (J^{2}/2m)\,\beta_{2\bot}$, which follows from Eq. (<ref>), was not included in Ref. <cit.>.
§.§ Guiding-center transformation
The full Lie-transform perturbation analysis leading to the present higher-order guiding-center Hamiltonian theory will be presented elsewhere <cit.>. Here, we summarize the guiding-center phase-space transformation $z_{0}^{\alpha} \equiv ({\bf x},p_{\|0},J_{0},\theta_{0}) \rightarrow Z^{\alpha} \equiv ({\bf X}, p_{\|},J,\theta)$,
defined in Eq. (<ref>) by the first-order generating vector-field components
\begin{eqnarray}
G_{1}^{\bf x} & = & -\;\vb{\rho}_{0} \;\equiv\; {\bf p}_{\bot 0}\btimes\bhat/m\Omega, \label{eq:G1_x} \\
G_{1}^{p_{\|}} & = & -\;p_{\|0}\;\vb{\rho}_{0}\bdot\vb{\kappa} \;+\; J_{0} \left( \tau \;+\; \alpha_{1} \right), \label{eq:G1_p} \\
G_{1}^{J} & = & \vb{\rho}_{0}\bdot\left( J_{0}\;\nabla\ln B + \frac{p_{\|0}^{2}\,\vb{\kappa}}{m\Omega}\right) \nonumber \\
& &-\; J_{0}\,\varrho_{\|0}\,(\tau + \alpha_{1}), \label{eq:G1_J} \\
G_{1}^{\theta} & = & \pd{\vb{\rho}_{0}}{\theta_{0}}\vb{\cdot}\left(\nabla\ln B + \frac{p_{\|0}^{2}\vb{\kappa}}{2\,mJ_{0}\Omega}\right) \nonumber \\
& &-\; \vb{\rho}_{0}\bdot{\bf R} \;+\; \varrho_{\|0}\;\alpha_{2}, \label{eq:G1_theta}
\end{eqnarray}
where $\alpha_{1} \equiv \partial\alpha_{2}/\partial\theta_{0}$, and the second-order generating vector-field components
\begin{eqnarray}
G_{2}^{\bf x} & = & \left( 2\,\varrho_{\|0}\;\pd{\vb{\rho}_{0}}{\theta_{0}}\bdot\vb{\kappa} \;+\; \frac{J_{0}\,\alpha_{2}}{m\Omega}\right) \bhat \;-\; \vb{\Pi}_{1}\btimes\frac{\bhat}{m\Omega} \nonumber \\
& &+\; \frac{1}{2} \left[ \frac{p_{\|0}^{2}}{m\Omega}\;(\vb{\rho}_{0}\bdot\vb{\kappa}) + J_{0}\,\varrho_{\|0}\;(3 \tau - \alpha_{1}) \right] \pd{\vb{\rho}_{0}}{J_{0}} \label{eq:G2_x} \\
& &+\; \frac{1}{2} \left[ \varrho_{\|0}\,\alpha_{2} + \pd{\vb{\rho}_{0}}{\theta_{0}}\bdot \left( \nabla\ln B + \frac{p_{\|0}^{2}\,\vb{\kappa}}{2m\Omega\,J_{0}}\right) \right]
\pd{\vb{\rho}_{0}}{\theta_{0}}, \nonumber \\
G_{2}^{p_{\|}} & = & p_{\|0}\;\vb{\kappa}\frac{}{}\bdot G_{2}^{\bf x} \;+\; \bhat\bdot\left[D_{1}^{2}({\bf P}_{3}) + \nabla\sigma_{3} - \vb{\Pi}_{2}\right],
\label{eq:G2_p} \\
G_{2}^{J} & = & -\; \frac{1}{\Omega} \left( \Psi_{2} \;-\; \frac{p_{\|}}{m}\;\Pi_{2\|} \right) - \varrho_{\|}\bhat\bdot
\left[D_{1}^{2}({\bf P}_{3}) + \nabla\sigma_{3}\right] \nonumber \\
& &- G_{2}^{\bf x}\bdot\left(J_{0}\nabla\ln B + \frac{p_{\|0}^{2}\,\vb{\kappa}}{m\,\Omega} \right), \label{eq:G2_J}
\end{eqnarray}
where ${\bf P}_{3} \equiv \frac{1}{2}\,p_{\|0}\bhat + \frac{1}{3}\,{\bf p}_{\bot 0}$. We note that the spatial component
\begin{eqnarray}
G_{3}^{\bf x} & = & G_{3\|}^{\bf x}\;\bhat \;+\; G_{2\|}^{\bf x}\;\left(\varrho_{\|0}\frac{}{}\nabla\btimes\bhat\right) \;-\; G_{2}^{\bf x}\;
\left(\varrho_{\|0}\frac{}{}\tau \right) \nonumber \\
& &-\; \frac{c\bhat}{eB}\btimes\left[D_{1}^{2}({\bf P}_{3}) + \nabla\sigma_{3} - \vb{\Pi}_{2}\right],
\label{eq:G3_x}
\end{eqnarray}
which is determined at third order <cit.>, is not needed in this Section and the remaining components $G_{3\|}^{\bf x}$ and $G_{2}^{\theta}$, which are determined at fourth order, are not needed in what follows. In the expressions above, we used the definition
\begin{eqnarray*}
D_{1}(\cdots) & \equiv & \left(G_{1}^{p_{\|}}\pd{}{p_{\|0}} + G_{1}^{J}\pd{}{J_{0}} + G_{1}^{\theta}\pd{}{\theta_{0}}\right)(\cdots) \\
& &+\; \vb{\rho}_{0}\btimes \nabla\btimes(\cdots),
\end{eqnarray*}
and the gyroangle-dependent gauge function
\begin{equation}
\sigma_{3} \;\equiv\; -\,\frac{1}{3}\,p_{\|0}\;G_{2\|}^{\bf x}
\label{eq:sigma3_def}
\end{equation}
appearing in the third-order Lie-transform perturbation analysis <cit.>.
§.§ Push-forward Lagrangian Constraints
The second-order guiding-center Hamiltonian constraint (<ref>) leads to a complex expression whose interpretation for
$\Psi_{2}$ and $\Pi_{2\|}$ may be difficult to obtain. For this purpose, we wish to explore a new perturbation approach to guiding-center Hamiltonian theory.
We begin with the following remark for the phase-space Lagrangian formulation of single-particle dynamics in a potential $U({\bf x})$, where the particle position ${\bf x}$ and its velocity ${\bf v}$ are viewed as independent phase-space coordinates. From the phase-space Lagrangian
\[ L({\bf x},{\bf v};\dot{\bf x},\dot{\bf v}) = \left(\frac{e}{c}{\bf A} + m{\bf v}\right)\bdot\dot{\bf x} - \left(\frac{m}{2}\,|{\bf v}|^{2} + e\Phi\right), \]
we first obtain the Euler-Lagrange equation for ${\bf x}$: $m\,d{\bf v}/dt = e\,{\bf E} + {\bf v}\btimes e\,{\bf B}/c$. Since the phase-space Lagrangian is independent of $d{\bf v}/dt$, however, the Euler-Lagrange equation for ${\bf v}$ yields the Lagrangian constraint
\begin{equation}
\pd{L}{\bf v} \;=\; m\;\left(\frac{d{\bf x}}{dt} \;-\; {\bf v}\right) \;\equiv\; 0.
\label{eq:Lag_constraint}
\end{equation}
Hence, the guiding-center transformation of the particle velocity ${\bf v}$ is constrained to be also expressed in terms of the guiding-center transformation of $d{\bf x}/dt$.
We would now like to obtain the guiding-center version of the Lagrangian constraint (<ref>):
\begin{equation}
{\sf T}_{\rm gc}^{-1}{\bf p}_{0} \;=\; m\,{\sf T}_{\rm gc}^{-1}\left(\frac{d{\bf x}}{dt}\right) \;\equiv\; {\bf P}_{\rm gc}.
\label{eq:gc_Lag}
\end{equation}
First, using the functional definition for $d_{\rm gc}/dt$:
\begin{equation}
\frac{d_{\rm gc}}{dt} \;\equiv\; {\sf T}_{\rm gc}^{-1}\left(\frac{d}{dt}\;{\sf T}_{\rm gc}\right),
\label{eq:d_gc_def}
\end{equation}
we introduced in Eq. (<ref>) the guiding-center particle-momentum
\begin{equation}
{\bf P}_{\rm gc} \;=\; m\,\frac{d_{\rm gc}}{dt}\left({\sf T}_{\rm gc}^{-1}{\bf x} \right) = m\,\frac{d_{\rm gc}{\bf X}}{dt} + m\,
\frac{d_{\rm gc}\vb{\rho}_{\rm gc}}{dt},
\label{eq:gcLc_id}
\end{equation}
which is expressed as the sum of the guiding-center velocity
\[ \frac{d_{\rm gc}{\bf X}}{dt} \;=\; \frac{d_{0}{\bf X}}{dt} + \epsilon\,\frac{d_{1}{\bf X}}{dt} + \cdots \;=\; \frac{p_{\|}}{m}\;\bhat +
\epsilon\,{\bf v}_{\rm gc} + \cdots \]
and the guiding-center displacement velocity
\[ \frac{d_{\rm gc}\vb{\rho}_{\rm gc}}{dt} = \epsilon^{-1}\;\pd{\Psi}{J}\;\pd{\vb{\rho}_{\rm gc}}{\theta} + \frac{d_{\rm gc}{\bf X}}{dt}\bdot
\nabla^{*}\vb{\rho}_{\rm gc} + \frac{d_{\rm gc}p_{\|}}{dt}\,\pd{\vb{\rho}_{\rm gc}}{p_{\|}}, \]
\[ \frac{d_{\rm gc}p_{\|}}{dt} \;=\; \frac{d_{0}p_{\|}}{dt} + \epsilon\,\frac{d_{1}p_{\|}}{dt} + \cdots \;=\; J\,\Omega\;\left(\nabla\bdot\bhat\right) + \cdots. \]
Here, the guiding-center displacement is expanded as
\begin{equation}
\vb{\rho}_{\rm gc} \;\equiv\; {\sf T}_{\rm gc}^{-1}{\bf x} \;-\; {\bf X} \;=\; \epsilon\,\vb{\rho}_{0} \;+\; \epsilon^{2}\;
\vb{\rho}_{1} \;+\; \epsilon^{3}\,\vb{\rho}_{2} + \cdots,
\label{eq:rho_gc}
\end{equation}
where the higher-order gyroradius corrections are
\begin{eqnarray}
\vb{\rho}_{1} & = & -\;G_{2}^{\bf x} \;-\; \frac{1}{2}\;{\sf G}_{1}\cdot\exd\vb{\rho}_{0}, \label{eq:rho_1} \\
\vb{\rho}_{2} & = & -\;G_{3}^{\bf x} - {\sf G}_{2}\cdot\exd\vb{\rho}_{0} + \frac{1}{6}\,{\sf G}_{1}\cdot
\exd({\sf G}_{1}\cdot\exd\vb{\rho}_{0}). \label{eq:rho_2}
\end{eqnarray}
We note that, in general, we find $\langle\vb{\rho}_{n}\rangle \neq 0$ and $\vb{\rho}_{n}\bdot\bhat \neq 0$ for $n \geq 1$.
§.§.§ First-order Lagrangian constraint
The first-order Lagrangian constraints on the components $(G_{1}^{p_{\|}}, G_{1}^{J}, G_{1}^{\theta})$ are expressed as
\begin{equation}
G_{1}^{p_{\|}}\;\bhat + G_{1}^{J}\,\pd{{\bf p}_{\bot 0}}{J} + G_{1}^{\theta}\,\pd{{\bf p}_{\bot 0}}{\theta} - \vb{\rho}_{0}\bdot\nabla{\bf p}_{0} +
{\bf P}_{{\rm gc}1} \;\equiv\; 0,
\label{eq:gcLc_1}
\end{equation}
\[ {\bf P}_{{\rm gc}1} \equiv m\,\frac{d_{1}{\bf X}}{dt} \;+\; m\,\left(\frac{d_{\rm gc}\vb{\rho}_{\rm gc}}{dt}\right)_{1}, \]
\[ \left(\frac{d_{\rm gc}\vb{\rho}_{\rm gc}}{dt}\right)_{1} \;\equiv\; \Omega\;\pd{\vb{\rho}_{1}}{\theta} \;+\; \frac{d_{0}\vb{\rho}_{0}}{dt}, \]
\[ \frac{d_{0}\vb{\rho}_{0}}{dt} \;\equiv\; \frac{p_{\|}}{m}\bhat\bdot\left[\nabla\vb{\rho}_{0} \;+\; \left({\bf R} + \pd{\vb{\Pi}_{1}}{J}\right)
\pd{\vb{\rho}_{0}}{\theta}\right]. \]
Using the identity
\[ \Psi_{1} \;=\; \left\langle {\bf p}_{\bot 0}\bdot\left(\frac{d_{\rm gc}\vb{\rho}_{\rm gc}}{dt}\right)_{1}\right\rangle \;\equiv\; 0, \]
which follows from the first-order symplectic representation (<ref>), Eq. (<ref>) yields the same condition used in the first-order Hamiltonian constraint (<ref>):
\begin{equation}
\langle G_{1}^{J}\rangle \;=\; \left\langle \vb{\rho}_{0}\bdot\nabla{\bf p}_{0}\bdot\pd{\vb{\rho}_{0}}{\theta}\right\rangle \;=\; -\;J\,\varrho_{\|}\tau,
\label{eq:G1_J_ave}
\end{equation}
which is calculated at order $\epsilon^{3}$ in the Lie-transform perturbation analysis <cit.>.
§.§.§ Second-order Lagrangian constraint
The second-order components $(G_{2}^{p_{\|}}, G_{2}^{J}, G_{2}^{\theta})$ are also constrained by the second-order Lagrangian constraint
\begin{eqnarray}
& &G_{2}^{p_{\|}}\;\bhat \;+\; G_{2}^{J}\,\pd{{\bf p}_{\bot 0}}{J} \;+\; G_{2}^{\theta}\,\pd{{\bf p}_{\bot 0}}{\theta} \;+\; G_{2}^{\bf x}\bdot\nabla{\bf p}_{0} \nonumber \\
& &\;-\; \frac{1}{2}\;{\sf G}_{1}\cdot\exd\left( {\sf G}_{1}\cdot\exd{\bf p}_{0} \right) \;+\; {\bf P}_{{\rm gc}2} \;\equiv\; 0,
\label{eq:gcLc_2}
\end{eqnarray}
\[ {\bf P}_{{\rm gc}2} \;\equiv\; m\,\frac{d_{2}{\bf X}}{dt} + m\,\left(\frac{d_{\rm gc}\vb{\rho}_{\rm gc}}{dt}\right)_{2} \]
\[ \left(\frac{d_{\rm gc}\vb{\rho}_{\rm gc}}{dt}\right)_{2} \equiv \Omega\;\pd{\vb{\rho}_{2}}{\theta} + \pd{\Psi_{2}}{J}\,\pd{\vb{\rho}_{0}}{\theta} +
\frac{d_{1}{\bf X}}{dt}\bdot\nabla_{0}^{*}\vb{\rho}_{0} + \frac{d_{0}\vb{\rho}_{1}}{dt}, \]
\begin{eqnarray*}
\frac{d_{0}\vb{\rho}_{1}}{dt} & = & \frac{p_{\|}}{m}\bhat\bdot\left[\nabla\vb{\rho}_{1} \;+\; \left({\bf R} + \pd{\vb{\Pi}_{1}}{J}\right)
\pd{\vb{\rho}_{1}}{\theta}\right] \\
& &+\; \left[ J\,\Omega\;\left(\nabla\bdot\bhat\right)\right]\;\pd{\vb{\rho}_{1}}{p_{\|}}.
\end{eqnarray*}
In particular, the Lagrangian constraint on $\langle G_{2}^{J}\rangle$ yields
\begin{eqnarray}
\langle G_{2}^{J}\rangle & = & -\;\left\langle G_{2}^{\bf x}\bdot\nabla{\bf p}_{0}\bdot\pd{\vb{\rho}_{0}}{\theta}\right\rangle - m\,\left\langle\left(
\frac{d_{\rm gc}\vb{\rho}_{\rm gc}}{dt}\right)_{2}\bdot\pd{\vb{\rho}_{0}}{\theta}\right\rangle \nonumber \\
& &+\; \frac{1}{2}\;\left\langle\left[{\sf G}_{1}\cdot\exd\frac{}{}\left( {\sf G}_{1}\cdot\exd{\bf p}_{0} \right)\right]\bdot\pd{\vb{\rho}_{0}}{\theta}\right\rangle,
\label{eq:G2_J_ave}
\end{eqnarray}
which yields the same result as Eq. (<ref>) obtained at order $\epsilon^{4}$ in the Lie-transform perturbation analysis <cit.>.
§.§.§ Lagrangian constraint on the guiding-center Hamiltonian
The generating-field components (<ref>)-(<ref>) were shown to satisfy the guiding-center Lagrangian constraints
(<ref>)-(<ref>). This means that the guiding-center Hamiltonian
\begin{eqnarray}
H_{\rm gc} & \equiv & \frac{m}{2} \left\langle \left|\frac{d_{\rm gc}{\bf X}}{dt} \;+\; \frac{d_{\rm gc}\vb{\rho}_{\rm gc}}{dt}\right|^{2}\right\rangle,
\label{eq:Ham_gc_final}
\end{eqnarray}
can also be expressed in terms of guiding-center velocity $d_{\rm gc}{\bf X}/dt$ and the guiding-center displacement velocity
$d_{\rm gc}\vb{\rho}_{\rm gc}/dt$. In the second-order Hamiltonian representation $(\Pi_{2\|} \equiv 0)$, the Lagrangian constraint of the guiding-center Hamiltonian (<ref>) implies that
\begin{eqnarray}
\Psi_{2} \equiv \epsilon^{-2} \left[ \frac{m}{2} \left\langle \left|\frac{d_{\rm gc}{\bf X}}{dt} + \frac{d_{\rm gc}\vb{\rho}_{\rm gc}}{dt}\right|^{2}\right\rangle \;-\; \left( \frac{p_{\|}^{2}}{2m} + J\,\Omega \right) \right],
\end{eqnarray}
which is identical to Eq. (<ref>) (with $\Pi_{2\|} \equiv 0)$.
§ GUIDING-CENTER POLARIZATION AND TOROIDAL CANONICAL MOMENTUM
There is now well-established connection between polarization and the conservation of toroidal canonical momentum in an axisymmetric magnetic field.
We now show how $\vb{\Pi}_{1\bot}$, which was originally chosen by Littlejohn <cit.> to be zero, can be determined by requiring that the guiding-center transformation (<ref>) yields the guiding-center polarization obtained by Pfirsch <cit.> and Kaufman
<cit.>. We will also show that the polarization term $\vb{\Pi}_{1\bot}$ leads to a more transparent guiding-center representation of the toroidal canonical angular momentum in axisymmetric magnetic geometry.
§.§ Guiding-center polarization
The guiding-center transformation (<ref>) can be used to calculate polarization and magnetization effects associated with the guiding-center displacement $\vb{\rho}_{\rm gc}$, defined by Eq. (<ref>).
Since the dipole contribution to the guiding-center polarization <cit.> involves the gyroangle-averaged displacement $\langle\vb{\rho}_{\rm gc}\rangle = \epsilon^{2}\,\langle\vb{\rho}_{1}\rangle + \cdots$ (since $\langle\vb{\rho}_{0}\rangle \equiv 0$), we begin with the gyroangle-averaged first-order displacement calculated from Eq. (<ref>) <cit.>:
\begin{eqnarray}
\langle\vb{\rho}_{1}\rangle & = & -\;\frac{J}{m\Omega} \left[ \frac{1}{2}\,(\nabla\bdot\bhat)\,\bhat \;+\; \frac{3}{2}\,\nabla_{\bot}\ln B\right] \nonumber \\
& &-\; \varrho_{\|}^{2}\,\vb{\kappa} \;+\; \vb{\Pi}_{1}\btimes\frac{\bhat}{m\Omega} \nonumber \\
& \equiv & -\,\frac{1}{m\Omega}\,\left( J\;\nabla_{\bot}\ln B + \frac{p_{\|}^{2}\,\vb{\kappa}}{m\,\Omega}\right) + \nabla\bdot\left(\left\langle
\frac{\vb{\rho}_{0}\vb{\rho}_{0}}{2}\right\rangle\right) \nonumber \\
& &+\; \left( \frac{J}{2} \;\bhat\btimes\vb{\kappa} \;+\; \vb{\Pi}_{1}\right)\btimes\frac{\bhat}{m\Omega},
\label{eq:rho1_ave}
\end{eqnarray}
where we used Eqs. (<ref>)-(<ref>), with
\begin{eqnarray*}
\nabla\bdot\left(\left\langle\frac{\vb{\rho}_{0}\vb{\rho}_{0}}{2}\right\rangle\right) & = & \nabla\bdot\left[ \frac{J}{2\,m\Omega}\;\left({\bf I} - \bhat\bhat\right) \right] \\
& = & -\,\frac{J}{2\,m\Omega} \left[\vb{\kappa} + \nabla_{\bot}\ln B \;+\; (\nabla\bdot\bhat)\,\bhat \right].
\end{eqnarray*}
Next, the guiding-center polarization density is defined as the first-order expression <cit.>
\begin{eqnarray}
\vb{\pi}_{\rm gc}^{(1)} & \equiv & e\,\langle\vb{\rho}_{1}\rangle \;-\; e\;\nabla\bdot\left(\left\langle\frac{\vb{\rho}_{0}\vb{\rho}_{0}}{2}\right\rangle\right) \nonumber \\
& = & -\,\frac{e}{m\Omega}\,\left( J\;\nabla_{\bot}\ln B \;+\; \frac{p_{\|}^{2}\,\vb{\kappa}}{m\,\Omega}\right) \nonumber \\
& &+\; \left( \frac{J}{2} \;\bhat\btimes\vb{\kappa} \;+\; \vb{\Pi}_{1}\right)\btimes\frac{c\bhat}{B},
\label{eq:gc_pol_def}
\end{eqnarray}
which yields the Pfirsch-Kaufman formula <cit.>
\begin{equation}
\vb{\pi}_{\rm gc}^{(1)} \;\equiv\; e\;\bhat\btimes \frac{1}{\Omega}\frac{d_{1}{\bf X}}{dt} \;=\; e\;\bhat\btimes\frac{{\bf v}_{\rm gc}}{\Omega},
\label{gc_pol_PK}
\end{equation}
only if we use the definition <cit.>
\begin{equation}
\vb{\Pi}_{1\bot} \;\equiv\; -\;\frac{J}{2}\;\bhat\btimes\vb{\kappa}.
\label{eq:Pi1_perp_choice}
\end{equation}
Hence, by combining with the condition (<ref>), $\Pi_{1\|} \equiv \bhat\bdot\vb{\Pi}_{1} = -\,\frac{1}{2}\,J\,\tau$, we find
\begin{equation}
\vb{\Pi}_{1} \;=\; -\;\frac{J}{2} \left( \tau\;\bhat \;+\frac{}{} \bhat\btimes\vb{\kappa} \right) \;=\; -\;\frac{J}{2}\;\nabla\btimes\bhat,
\label{eq:Pi1_final}
\end{equation}
and, hence, the guiding-center vector (<ref>) becomes $\bhat^{*} = \bhat + {\cal O}(\epsilon^{2})$, since $\partial\vb{\Pi}_{1}/\partial p_{\|} = 0$.
We note that the Pfirsch-Kaufman formula (<ref>) yields a guiding-center moving-electric-dipole correction $\vb{\mu}_{\rm gc}^{(E)} \equiv p_{\|}{\bf v}_{\rm gc}/B$ to the intrinsic guiding-center magnetic-dipole moment $\vb{\mu}_{\rm gc}^{(B)} \equiv -\,\mu\,\bhat$. Since the total guiding-center magnetic-dipole moment can be obtained from the Lagrangian variational expression <cit.>
\begin{eqnarray}
\vb{\mu}_{\rm gc} & = & -\;\mu\;\bhat \;+\; \epsilon\;p_{\|}\;{\bf v}_{\rm gc}/B \label{eq:mu_gc_total} \\
& \equiv & \fd{}{\bf B}\left[ \left(\frac{e}{\epsilon c}\,{\bf A} + p_{\|}\,\bhat\right)\bdot\dot{\bf X} \;-\; \left( \mu\,B \;+\; \frac{p_{\|}^{2}}{2m}\right) \right],
\nonumber
\end{eqnarray}
where we used $\delta\bhat/\delta{\bf B} = ({\bf I} - \bhat\bhat)/B$ and $\delta B/\delta{\bf B} = \bhat$, we again conclude that the term $\vb{\Pi}_{1\bot}$, defined by
Eq. (<ref>), cannot be ignored in guiding-center theory if polarization effects are to be accounted for correctly.
Lastly, the guiding-center phase-space Lagrangian is expressed as
\begin{eqnarray}
\Gamma_{\rm gc} & = & \left( \frac{e}{\epsilon\,c}\;{\bf A} \;+\; p_{\|}\,\bhat \;-\; \frac{\epsilon}{2}\;J\,\nabla\btimes\bhat \right)\bdot\exd{\bf X}
\nonumber \\
& &+\; \epsilon\,J\;\left(\exd\theta \;-\frac{}{} {\bf R}\bdot\exd{\bf X}\right),
\label{eq:Gamma_gc_primitive}
\end{eqnarray}
when terms up to first order in magnetic-field nonuniformity are retained. In Eq. (<ref>), we have retained the guiding-center polarization contribution
(<ref>) to $\vb{\Pi}_{1} \equiv -\,\frac{1}{2}\,J\;\nabla\btimes\bhat$. We now show that this polarization correction yields a more transparent expression for the guiding-center toroidal canonical momentum up to second order in $\epsilon$ (i.e., first order in magnetic-field nonuniformity).
§.§ Guiding-center toroidal canonical angular momentum
We now construct the guiding-center representation for the toroidal canonical angular momentum in axisymmetric magnetic geometry, for which it is an exact constant of motion. Here, we represent an axisymmetric magnetic field
\begin{equation}
{\bf B} \;=\; B_{\varphi}(\psi)\;\nabla\varphi \;+\; \nabla\varphi\btimes\nabla\psi,
\label{eq:B_axis}
\end{equation}
where $\varphi$ denotes the toroidal angle and $\psi$ denotes the magnetic flux on which magnetic-field lines lie (i.e., ${\bf B}\bdot\nabla\psi
\equiv 0$). Note that we have added a toroidal magnetic field $B_{\varphi}\,\nabla\varphi$ in Eq. (<ref>), with a covariant component
$B_{\varphi}$ that is constant on a given magnetic-flux surface.
We first calculate the guiding-center toroidal canonical momentum from the guiding-center phase-space Lagrangian (<ref>):
\begin{eqnarray}
P_{{\rm gc}\varphi} & \equiv & \left[ \frac{e}{\epsilon\,c}\;{\bf A} + p_{\|}\,\bhat - \epsilon\;J\,\left({\bf R} + \frac{1}{2}\,\nabla\btimes\bhat
\right) \right]\bdot\pd{\bf X}{\varphi} \nonumber \\
& = & -\frac{e\psi}{\epsilon\,c} + p_{\|}\,b_{\varphi} - \epsilon\,J \left[ b_{\sf z} + \bhat\bdot\nabla\btimes\left( \frac{1}{2}\,{\cal R}^{2}\,\nabla\varphi\right) \right] \nonumber \\
& &-\; \epsilon\,J \nabla\bdot\left(\bhat\btimes \frac{1}{2}\,{\cal R}^{2}\,\nabla\varphi\right)
\label{eq:Pgc_phi_def}
\end{eqnarray}
where we used ${\bf R}\bdot\partial{\bf X}/\partial\varphi \equiv b_{\sf z}$ <cit.> (i.e., the component of $\bhat$ along the symmetry axis $\wh{\sf z}$ for toroidal rotations), we wrote $\partial{\bf X}/\partial\varphi \equiv {\cal R}^{2}\,\nabla\varphi$ in terms of the major radius ${\cal R} \equiv |\nabla\varphi|^{-1}$, and we used the identity ${\bf F}\bdot\nabla\btimes{\bf G} \equiv \nabla\bdot({\bf G}\btimes{\bf F}) + {\bf G}\bdot\nabla\btimes{\bf F}$, for arbitrary vector fields ${\bf F}$ and ${\bf G}$. Next, we use
\[ \bhat\bdot\nabla\btimes\left( \frac{1}{2}\,{\cal R}^{2}\,\nabla\varphi\right) \;=\; \bhat\bdot\left(\wh{\cal R}\btimes\wh{\varphi}\right) \;=\;
b_{\sf z}, \]
\[ \bhat\btimes \frac{1}{2}\,{\cal R}^{2}\,\nabla\varphi \;=\; \frac{1}{2B}\;\nabla\psi, \]
so that Eq. (<ref>) becomes
\begin{eqnarray}
P_{{\rm gc}\varphi} & = & -\;\frac{e}{\epsilon\,c}\;\left[\psi \;+\; \epsilon^{2}\;\nabla\bdot\left(\frac{J}{2\,m\Omega}\,\nabla\psi\right) \right]
\nonumber \\
& &+\; p_{\|}\,b_{\varphi} \;-\; 2\,\epsilon\,J\;b_{\sf z}.
\label{eq:Pgc_phi}
\end{eqnarray}
Here, the second term on the first line in Eq. (<ref>) is the second-order finite-Larmor-radius (FLR) correction to the first term.
We now show that Eq. (<ref>) is the exact guiding-center representation of the toroidal canonical angular momentum:
\begin{equation}
P_{{\rm gc}\varphi} \equiv {\sf T}_{\rm gc}^{-1}P_{\varphi} = -\,\frac{e}{\epsilon c}\,{\sf T}_{\rm gc}^{-1}\psi + {\sf T}_{\rm gc}^{-1}\left(m\,
{\bf v}\bdot\pd{\bf x}{\varphi} \right),
\label{eq:Pgc_phi_push}
\end{equation}
which guarantees the conservation of guiding-center toroidal canonical angular momentum
\begin{equation}
\frac{d_{\rm gc}P_{{\rm gc}\varphi}}{dt} \;=\; \frac{d_{\rm gc}}{dt}\left({\sf T}_{\rm gc}^{-1}\frac{}{}P_{\varphi}\right) \;=\; {\sf T}_{\rm gc}^{-1}
\left(\frac{dP_{\varphi}}{dt} \right) \;\equiv\; 0.
\label{eq:Pgcphi_dot}
\end{equation}
First, we note that, while the term ${\sf T}_{\rm gc}^{-1}P_{\varphi}$ in Eq. (<ref>) contains contributions that are gyroangle-independent and contributions that are explicitly gyroangle-dependent, the term $P_{{\rm gc}\varphi}$ is explicitly gyroangle-independent. Hence, the gyroangle-dependent contributions must vanish at all orders in
$\epsilon$, and thus $P_{{\rm gc}\varphi} \equiv \langle{\sf T}_{\rm gc}^{-1}P_{\varphi}\rangle$; this identity, which is equivalent to a toroidal-canonical-momentum constraint on the guiding-center transformation, will be proved elsewhere <cit.>.
Secondly, we therefore introduce the guiding-center magnetic flux $\psi_{\rm gc} \equiv \langle {\sf T}_{\rm gc}^{-1}\psi\rangle$:
\begin{eqnarray}
\psi_{\rm gc} & = & \psi \;+\; \epsilon^{2} \left( \langle\vb{\rho}_{1}\rangle\bdot\nabla\psi \;+\; \frac{1}{2}\,\langle\vb{\rho}_{0}\vb{\rho}_{0}\rangle:\nabla\nabla \psi \right) + \cdots \nonumber \\
& = & \psi + \epsilon^{2}\;\nabla\bdot\left( \frac{J}{2\,m\Omega}\;\nabla\psi\right) + \epsilon^{2}\;\bhat\btimes\frac{{\bf v}_{\rm gc}}{\Omega}\bdot\nabla\psi,
\label{eq:psi_gc}
\end{eqnarray}
where we used Eqs. (<ref>)-(<ref>). In Eq. (<ref>), the second term is an FLR correction to the first term, while the last term is easily recognized as a correction due to the guiding-center polarization (<ref>).
Thirdly, using the identity $\nabla\psi \equiv {\bf B}\btimes\partial{\bf X}/\partial\varphi$, with $\bhat\bdot{\bf v}_{\rm gc} \equiv 0$, we obtain a term proportional to the toroidal component of the guiding-center velocity:
\[ \bhat\btimes\frac{{\bf v}_{\rm gc}}{\Omega}\bdot\nabla\psi \;=\; \frac{B}{\Omega}\;\left({\bf v}_{\rm gc}\bdot\pd{\bf X}{\varphi}\right) \;\equiv\;
\frac{B}{\Omega}\;v_{{\rm gc}\varphi}. \]
Hence, the final expression for the guiding-center toroidal canonical momentum defined by Eq. (<ref>) is
\begin{equation}
P_{{\rm gc}\varphi} = -\frac{e \psi_{\rm gc}}{\epsilon\,c} + m \left( \frac{d_{0}{\bf X}}{dt} + \epsilon\;\frac{d_{1}{\bf X}}{dt}
\right)\bdot\pd{\bf X}{\varphi} - 2\,\epsilon\;J\,b_{\sf z},
\label{eq:Pgc_phi_final}
\end{equation}
where $d_{0}{\bf X}/dt \equiv (p_{\|}/m)\,\bhat$ and $d_{1}{\bf X}/dt \equiv {\bf v}_{\rm gc}$, while
\[ m\;\left( \frac{d_{0}{\bf X}}{dt} \;+\; \epsilon\;\frac{d_{1}{\bf X}}{dt}\right)\bdot\pd{\bf X}{\varphi} \;\equiv\; m\;{\cal R}^{2}\;
\frac{d_{\rm gc}\varphi}{dt} \]
denotes the guiding-center toroidal momentum with first-order corrections due to the guiding-center magnetic-drift velocity.
The last term in Eq. (<ref>) might be puzzling until we consider the guiding-center transformation of the particle toroidal canonical momentum $P_{{\rm gc}\varphi} \equiv \langle{\sf T}_{\rm gc}^{-1}\;p_{\varphi}\rangle$:
\begin{eqnarray}
P_{{\rm gc}\varphi} & = & -\,\frac{e}{\epsilon\,c}\;\langle{\sf T}_{\rm gc}^{-1}\psi\rangle \;+\; m\;\left\langle \left({\sf T}_{\rm gc}^{-1}
\frac{d{\bf x}}{dt}\right)\bdot\left({\sf T}_{\rm gc}^{-1}\pd{\bf x}{\varphi}\right)\right\rangle \nonumber \\
& = & -\;\frac{e\psi_{\rm gc}}{\epsilon\,c} \;+\; m \left( \frac{d_{0}{\bf X}}{dt} \;+\; \epsilon\;\frac{d_{1}{\bf X}}{dt}
\right)\bdot\pd{\bf X}{\varphi} \nonumber \\
& &+\; \epsilon\;m\Omega\left\langle\pd{\vb{\rho}_{0}}{\theta}\bdot\pd{\vb{\rho}_{0}}{\varphi}\right\rangle + \cdots.
\label{eq:P_particle_phi}
\end{eqnarray}
Since $\partial\vb{\rho}_{0}/\partial\varphi \equiv \wh{\sf z}\btimes\vb{\rho}_{0}$ in axisymmetric magnetic geometry, the last term in
Eq. (<ref>) becomes
\[ \epsilon\;m\Omega\left\langle\pd{\vb{\rho}_{0}}{\theta}\bdot\pd{\vb{\rho}_{0}}{\varphi}\right\rangle \;=\; -\; 2\,\epsilon\;J\,b_{\sf z}, \]
and we recover the guiding-center toroidal canonical momentum (<ref>) from the guiding-center transformation of the particle toroidal canonical momentum (<ref>).
§.§ Comparison with Littlejohn's results
By comparison, the guiding-center toroidal canonical momentum obtained by Littlejohn <cit.> and all subsequent guiding-center theories, is calculated with the choice
$\vb{\Pi}_{1\bot} \equiv 0$:
\begin{equation}
(P_{{\rm gc}\varphi})_{\rm RGL} = -\;\frac{e\psi}{\epsilon\,c} + p_{\|}\,b_{\varphi} + \epsilon\,\left(\Pi_{1\|}\,b_{\varphi} \;-\frac{}{} J\,b_{\sf z}\right),
\label{eq:Pgc_phi_RGL}
\end{equation}
where the FLR correction to $\psi$ and the missing additional $b_{\sf z}$-term are hidden in $\Pi_{1\|}\,b_{\varphi} \equiv -\,\frac{1}{2}\,J\,\tau\,b_{\varphi}$:
\begin{eqnarray*}
-\;\frac{1}{2}\,J\;\tau\,b_{\varphi} & = & -\,\frac{1}{2}\,J \left( \nabla\btimes\bhat\bdot\pd{{\bf X}}{\varphi} \;+\;
\bhat\btimes\pd{\bf X}{\varphi}\bdot\vb{\kappa} \right) \\
& = & -\;\nabla\bdot\left(\frac{J}{2B}\;\nabla\psi\right) \;-\; J\;b_{\sf z} \;-\; \frac{J}{2B}\,\vb{\kappa}\bdot\nabla\psi.
\end{eqnarray*}
Hence, the Littlejohn guiding-center toroidal canonical momentum (<ref>) becomes
\begin{eqnarray}
(P_{{\rm gc}\varphi})_{\rm RGL} & = & -\;\frac{e}{\epsilon\,c}\;\left[\psi \;+\; \epsilon^{2}\;\nabla\bdot\left(\frac{J}{2\,m\Omega}\,\nabla\psi\right) \right] \nonumber \\
& &+\; p_{\|}\,b_{\varphi} - \epsilon\left(2\,J\;b_{\sf z} + \frac{J\vb{\kappa}}{2B}\bdot\nabla\psi\right).
\label{eq:Pgc_phi_RGL_FLR}
\end{eqnarray}
The Littlejohn guiding-center toroidal canonical momentum (<ref>), of course, has the same form as Eq. (<ref>) since its associated guiding-center magnetic flux is
\begin{eqnarray}
(\psi_{\rm gc})_{\rm RGL} & = & \psi \;+\; \epsilon^{2}\;\nabla\bdot\left( \frac{J}{2\,m\Omega}\;\nabla\psi\right) \nonumber \\
& &+\; \epsilon^{2}\;\left(\bhat\btimes\frac{{\bf v}_{\rm gc}}{\Omega} \;+\; \frac{J\vb{\kappa}}{2m\Omega}\right)\bdot\nabla\psi \nonumber \\
& \equiv & \psi_{\rm gc} \;+\; \epsilon^{2}\;\frac{J\,\vb{\kappa}}{2\,m\Omega}\bdot\nabla\psi,
\end{eqnarray}
where the extra term associated with the normal magnetic curvature $\vb{\kappa}\bdot\nabla\psi/|\nabla\psi|$ was eliminated by our choice
(<ref>) for $\vb{\Pi}_{1\bot}$. Once again, we point out that this choice is mandated by a correct derivation of the standard guiding-center polarization (<ref>).
We note that Belova et al. <cit.> have shown that the second-order $(\epsilon^{2})$ corrections to the guiding-center toroidal canonical momentum (<ref>) were shown to be crucial in obtaining excellent conservation properties of toroidal canonical momentum in realistic axisymmetric tokamak plasmas.
It remains to be seen whether the additional guiding-center polarization correction (<ref>) yields improved comservation properties.
§ SUMMARY
In conclusion, a systematic derivation of the Hamiltonian guiding-center dynamics has been derived by Lie-transform perturbation analysis. The guiding-center Poisson bracket derived from the guiding-center phase-space Lagrangian (<ref>) and the guiding-center Hamiltonian (<ref>). These guiding-center Hamilton equations have passed several consistency tests along the way.
First, we verified that our guiding-center transformation satisfies the guiding-center Jacobian constraints at first and second orders. Next, we verified that our guiding-center transformation also satisfies the guiding-center Lagrangian constraints at first and second orders. In fact, the use of the Lagrangian constraints on the guiding-center transformation yields a natural expression (<ref>) for the guiding-center Hamiltonian in terms of the guiding-center velocity $d_{\rm gc}{\bf X}/dt$ and the guiding-center displacement velocity $d_{\rm gc}\vb{\rho}_{\rm gc}/dt$. When the polarization term $\vb{\Pi}_{1\bot}$ is ignored in the guiding-center Hamiltonian, our second-order guiding-center Hamiltonian is identical to the Hamiltonian derived by Burby, Squire, and Qin <cit.>.
We also showed that the perpendicular component of $\vb{\Pi}_{1}$, which could not be determined at the perturbation orders considered in this work, could nevertheless not be chosen to be zero, in contrast to the simplifying choice made by Littlejohn <cit.>. The choice (<ref>) defined in the present work not only yields the standard Pfirsch-Kaufman guiding-center polarization (<ref>), but also yields a simpler and more transparent guiding-center representation of the particle toroidal canonical momentum (<ref>).
Work by AJB was partially supported by a U. S. DoE grant under contract No. DE-SC0006721. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
Parra_Calvo_2011 F. I. Parra and I. Calvo, Plasma Phys. Controlled Fusion 53, 045001 (2011).
Burby_SQ_2013 J. W. Burby, J. Squire, and H. Qin, Phys. Plasmas 20, 072105 (2013).
PCBSQ_2014 F. I. Parra, I. Calvo, J. W. Burby, J. Squire, and H. Qin, Phys. Plasmas 21, 104506 (2014).
RGL_1979 R. G. Littlejohn, J. Math. Phys. 20, 2445 (1979).
RGL_1981 R. G. Littlejohn, Phys. Fluids 24, 1730 (1981).
RGL_1983 R. G. Littlejohn, J. Plasma Phys. 29, 111 (1983).
Cary_Brizard_2009 J. R. Cary and A. J. Brizard, Rev. Mod. Phys. 81, 693 (2009).
Brizard_Tronko_2015 A. J. Brizard and N. Tronko, Equivalent higher-order guiding-center Hamiltonian theories, in preparation (2015).
Pfirsch_1984 D. Pfirsch, Zeitschrift Naturforschung Teil A 39, 1 (1984).
CRPW_1986 D. Correa-Restrepo, D. Pfirsch, and H. K. Wimmel, Physica 136A, 453 (1986).
Kaufman_1986 A. N. Kaufman, Phys. Fluids 29, 1736 (1986).
Brizard_2013 A. J. Brizard, Phys. Plasmas 20, 092309 (2013).
MSY_2013 N. Miyato, B. D. Scott, and M. Yagi, Plasma Phys. Contr. Fusion 55, 074011 (2013).
Brizard_Tronko_2012 A. J. Brizard and N. Tronko, arXiv:1205.5772 (2012).
footnote We note here that, since $B_{\|}^{**} = B\,(1 + \epsilon\,\varrho_{\|}\,\bhat\bdot\nabla\btimes\bhat + \cdots)$ up to first order in $\epsilon$, we might be concerned with the possibility of $B_{\|}^{**}$ vanishing if the parallel guiding-center velocity $v_{\|}$ is large enough, i.e., when $|v_{\|}| = |v_{\|{\rm cr}}| \equiv L_{\tau}\,\Omega$, where
$L_{\tau} \equiv |\bhat\bdot\nabla\btimes\bhat|^{-1}$. If we introduce the ordering $L_{\tau} = \rho_{\rm th} \epsilon_{B}^{-1}$ expressed in terms of the thermal gyroradius
$\rho_{\rm th} \equiv v_{\rm th}/\Omega$, we then obtain the ordering $|v_{\|{\rm cr}}|/v_{\rm th} = \epsilon_{B}^{-1} \gg 1$. Hence, only (extreme) superthermal parallel guiding-center motion would cause the guiding-center Jacobian to become singular. Under those circumstances, however, standard guiding-center orderings would also break down (e.g., the curvature-drift motion would now be ordered at $\epsilon_{B}^{-1}$) and, thus, guiding-center theory would become invalid for these superthermal particles. The removal of this singularity can be accomplished by a process of regularization of guiding-center theory, as shown in Refs. <cit.>.
CRW_1985 D. Correa-Restrepo and H. K. Wimmel, Phys. Scr. 32, 552 (1985).
Brizard_Hahm_2007 A. J. Brizard and T. S. Hahm, Rev. Mod. Phys. 79, 421 (2007).
Brizard_Lag A. J. Brizard, An Introduction to Lagrangian Mechanics, 2nd ed. (World Scientific, 2015), Sec. 2.5.
footnote_2 We note, however, that the second-order guiding-center Hamiltonian $\Psi_{2(PC)}$ does not satisfy the second-order Hamiltonian constraint (<ref>), since standard Lie-transform perturbation theory [e.g., see Eq. (<ref>)] explicitly assumes that the Lagrangian gauge-invariance property $\Gamma_{\rm gc}^{\prime} \equiv \Gamma_{\rm gc} + \exd\sigma$ involves gyroangle-dependent gauge functions only (i.e., $\langle\sigma\rangle \equiv 0$).
Brizard_Tronko_2011 In Eq. (56) of a recent paper by A. J. Brizard and N. Tronko [Phys. Plasmas 18, 082307 (2011)], where $\vb{\Pi}_{1\bot}$ is assumed to be zero, the expression for $\langle\vb{\rho}_{1}\rangle$ is erroneously given without the quadrupole correction $\nabla\bdot(\frac{1}{2}\langle\vb{\rho}_{0}\vb{\rho}_{0}\rangle)$. In the absence of the guiding-center polarization correction (<ref>), Eq. (<ref>) agrees with previous expressions for $\vb{\rho}_{1}$ <cit.>.
Tronci C. Tronci, private communication (2015).
Belova E. V. Belova, N. N. Gorelenkov, and C. Z. Cheng, Phys. Plasmas 10, 3240 (2003).
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1511.00734
|
Modeling of Stationary Periodic Time Series]Modeling of Stationary Periodic Time Series
ARMA Representations
A. Lindquist]Anders Lindquist†
G. Picci]Giorgio Picci
† Shanghai Jiao Tong University, Shanghai, China, and Royal Institute of Technology, Stockholm, Sweden
Univerity of Padova, Padova, Italy
Dedicated to Boris Teodorovich Polyak
on the occasion of his 80th birthday
This is a survey of some recent results on the rational circulant covariance extension problem: Given a partial sequence $(c_0,c_1,\dots,c_n)$ of covariance lags $c_k=\E\{y(t+k)\overline{y(t)}\}$ emanating from a stationary periodic process $\{y(t)\}$ with period $2N>2n$, find all possible rational spectral functions of $\{y(t)\}$ of degree at most $2n$ or, equivalently, all bilateral and unilateral ARMA models of order at most $n$, having this partial covariance sequence. Each representation is obtained as the solution of a pair of dual convex optimization problems. This theory is then reformulated in terms of circulant matrices and the connections to reciprocal processes and the covariance selection problem is explained. Next it is shown how the theory can be extended to the multivariate case. Finally, an application to image processing is presented.
§ INTRODUCTION
The rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags has been studied in great detail <cit.>, and it can be formulated as a (truncated) trigonometric moment problem with a degree constraint. Among other things, it is the basic problem in partial stochastic realization theory <cit.> and certain Toeplitz matrix completion problems. In particular, it provides a parameterization of the family of (unilateral) autoregressive moving-average (ARMA) models of stationary stochastic processes with the same finite sequence of covariance lags. We also refer the reader to the recent monograph <cit.>, in which this problem is discussed in the context of stochastic realization theory.
Covariance extension for periodic stochastic processes, on the other hand, leads to matrix completion of Toeplitz matrices with circulant structure and to partial stochastic realizations in the form of bilateral ARMA models
∑_k=-n^n q_k y(t-k) = ∑_k=-n^n p_k e(t-k)
for a stochastic processes $\{y(t)\}$, where $\{e(t)\}$ is the corresponding conjugate process. This connects up to a rich realization theory for reciprocal processes <cit.>. As we shall see there are also (forward and backward) unilateral ARMA representations for periodic processes.
In <cit.> a maximum-entropy approach to this circulant covariance extension problem was presented,
providing a procedure for determining the unique bilateral AR model matching the covariance sequence. However, more recently it was discovered that the circulant covariance extension problem can be recast in the context of the optimization-based theory of moment problems with rational measures developed in <cit.> allowing for a complete parameterization of all bilateral ARMA realizations. This led to a a complete theory for the scalar case <cit.>, which was then extended to to the multivariable case in <cit.>. Also see <cit.> for modifications of this theory to skew periodic processes and <cit.> for fast numerical procedures.
The AR theory of <cit.> has been successfully applied to image processing of textures <cit.>, and we anticipate an enhancement of such methods by allowing for more general ARMA realizations.
The present survey paper is to a large extent based on <cit.>, <cit.> and <cit.>. In Section <ref> we begin by characterizing stationary periodic processes. In Section <ref> we formulate the rational covariance extension problem for periodic processes as a moment problem with atomic measure and present the solution in the context of the convex optimization approach of <cit.>. These results are then reformulated in terms of circulant matrices in Section <ref> and interpreted in term of bilateral ARMA models in Section <ref> and in terms of unilateral ARMA models in Section <ref>. In Section <ref> we investigate the connections to reciprocal processes of order $n$ <cit.> and the covariance selection problem of Dempster <cit.>. In Section <ref> we consider the situation when both partial covariance data and logarithmic moment (cepstral) data is available. To simplify the exposition the theory has so far been developed in the context of scalar processes, but in Section <ref> we show how it can be extended to the multivariable case. All of these results are illustrated by examples taken from <cit.> and <cit.>. Section <ref> is devoted to applications in image processing.
§ PERIODIC STATIONARY PROCESSES
Consider a zero-mean full-rank stationary process $\{y(t)\}$, in general complex-valued, defined on a finite interval $[-N+1,\,N]$ of the integer line $\Zbb$ and extended to all of $\Zbb$ as a periodic stationary process with period $2N$ so that
\begin{equation}
\label{periodic2N}
y(t +2kN) =y(t)
\end{equation}
almost surely. By stationarity there is a representation
\begin{equation}
\label{ }
y(t)=\int_{-\pi}^\pi e^{it\theta}d\hat{y}(\theta), \quad \text{ where $\E \{|d\hat{y}|^2\}=dF(\theta)$},
\end{equation}
(see, e.g., <cit.>), and therefore
\begin{equation}
\label{F2c}
c_k:= \E\{y(t+k)\overline{y(t)}\} = \int_{-\pi}^\pi e^{ik\theta}dF(\theta).
\end{equation}
Also, in view of (<ref>),
∫_-π^πe^itθ(e^i2Nθ-1)dŷ =0,
and hence
∫_-π^π|e^i2Nθ-1|^2dF =0,
which shows that the support of $dF$ must be contained in $\{ k\pi/N; \, k=-N+1, \dots, N\}$. Consequently the spectral density of $\{y(t)\}$ consists of point masses on the discrete unit circle $\mathbb{T}_{2N}:=\{\zeta_{-N+1},\zeta_{-n+2},\dots,\zeta_N\}$, where
\begin{equation}
\label{zetadefn}
\zeta_k=e^{ik\pi/N}.
\end{equation}
More precisely, define the function
\begin{equation}
\label{Phi}
\Phi(\zeta)=\sum_{k=-N+1}^Nc_k\zeta^{-k}
\end{equation}
on $\mathbb{T}_{2N}$. This is the discrete Fourier transform (DFT) of the sequence $(c_{-N+1},\dots,c_N)$, which can be recovered by the inverse DFT
\begin{equation}
\label{discretemoments}
c_k=\frac{1}{2N}\sum_{j=-N+1}^N \Phi(\zeta_j)\zeta_j^k =\int_{-\pi}^\pi e^{ik\theta}\Phi(e^{i\theta})d\nu,
\end{equation}
where $\nu$ is a step function with steps $\frac{1}{2N}$ at each $\zeta_k$; i.e.,
\begin{equation}
\label{nu}
d\nu(\theta) =\sum_{j=-N+1}^N\delta(e^{i\theta}-\zeta_j)\frac{d\theta}{2N}.
\end{equation}
Consequently, by (<ref>), $dF(\theta)=\Phi(e^{i\theta})d\nu(\theta)$. We note in passing that
\begin{equation}
\label{delta}
\int_{-\pi}^\pi e^{ik\theta}d\nu(\theta)=\delta_{k0},
\end{equation}
where $\delta_{k0}$ equals one for $k=0$ and zero otherwise. To see this, note that, for $k\ne 0$,
(1-ζ_k)∫_-π^πe^ikθdν = 1/2N ∑_j=-N+1 ^N(ζ_k^j -ζ_k^j+1)
= 1/2N(ζ_k^-N+1-ζ_k^N+1)=0.
Since $\{y(t)\}$ is stationary and full rank, the Toeplitz matrix
\begin{equation}
\label{Toeplitz}
\Tb_n=\begin{bmatrix} c_0&\bar{c}_1&\bar{c}_2&\cdots&\bar{c}_n\\
c_1&c_0&\bar{c}_1&\cdots& \bar{c}_{n-1}\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
\end{bmatrix}
\end{equation}
is positive definite for all $n\in\mathbb{Z}$. However, this condition is not sufficient for $c_0,c_1,\dots,c_n$ to be a bona-fide covariance sequence of a periodic process, as can be seen from the following simple example. Consider a real-valued periodic stationary process $y$ of period four. Then
=\begin{bmatrix} c_0&c_1&c_2&c_3\\c_1&c_0&c_1&c_2\\c_2&c_1&c_0&c_1\\c_3&c_2&c_1&c_0\end{bmatrix}.$$
Then looking at the covariance matrix for two periods, we obtain
\begin{bmatrix} c_0&c_1&c_2&{\color{red}c_3}&c_0&c_1&c_2&{\color{red}c_3}\\
\end{bmatrix},$$
which is a Toeplitz matrix only when $c_3=c_1$. Therefore the condition $c_3=c_1$ is necessary. Consequently
$$\Tb_8=\begin{bmatrix} c_0&c_1&c_2&c_1&c_0&c_1&c_2&c_1\\
\end{bmatrix} $$
is a circulant matrix, where the columns are shifted cyclically, the last component moved to the top. Circulant matrices will play a key role in the following.
§ THE COVARIANCE EXTENSION PROBLEM FOR PERIODIC PROCESSES
Suppose that we are given a partial covariance sequence $c_0,c_1,\dots,c_n$ with $n< N$ such that the Toeplitz matrix $\Tb_n$ is positive definite. Consider the problem of finding and extension $c_{n+1},c_{n+2},\dots,c_{2N}$ so that the corresponding sequence $c_0,c_1,\dots,c_N$ is the covariance sequence of a stationary process of period $2N$.
In general this problem will have infinitely many solutions, and, for reasons that will become clear later, we shall restrict our attention to spectral function (<ref>) which are rational in the sense that
\begin{equation}
\label{Phi=P/Q}
\Phi(\zeta)=\frac{P(\zeta)}{Q(\zeta)},
\end{equation}
where $P$ and $Q$ are Hermitian pseudo-polynomials of degree at most $n$, that is of the form
\begin{equation}
\label{P}
P(\zeta)=\sum_{k=-n}^n p_k \zeta^{-k}, \quad p_{-k}=\bar{p}_k.
\end{equation}
Let $\mathfrak{P}_+(N)$ be the cone of all pseudo-polynomials (<ref>) that are positive on the discrete unit circle $\mathbb{T}_{2N}$, and let $\mathfrak{P}_+\subset\mathfrak{P}_+(N)$ be the subset of pseudo-polynomials (<ref>) such that $P(e^{i\theta})>0$ for all $\theta\in [-\pi,\pi]$. Moreover let $\mathfrak{C}_+(N)$ be the dual cone of all
partial covariance sequences $\cb=(c_0,c_1,\dots,c_n)$ such that
⟨,⟩:=∑_k=-n^n c_kp̅_k >0 for all $P\in\overline{\mathfrak{P}_+(N)}\setminus\{0\}$,
and let $\mathfrak{C}_+$ be defined in the same way as the dual cone of $\mathfrak{P}_+$. It can be shown <cit.> that $\cb\in\mathfrak{C}_+$ is equivalent to the Toeplitz condition $\Tb_n>0$. Since $\mathfrak{P}_+\subset\mathfrak{P}_+(N)$, we have $\mathfrak{C}_+(N)\subset \mathfrak{C}_+$, so in general $\cb\in\mathfrak{C}_+(N)$ is a stricter condition than $\Tb_n>0$.
The proof of the following theorem can be found in <cit.>.
Let $\cb\in\mathfrak{C}_+(N)$. Then, for each $P\in\mathfrak{P}_+(N)$, there is a unique $Q\in\mathfrak{P}_+(N)$ such that
satisfies the moment conditions
\begin{equation}
\label{momentconditions}
\int_{-\pi}^\pi e^{ik\theta}\Phi(e^{i\theta})d\nu(\theta) =c_k, \quad k=0,1,\dots,n.
\end{equation}
Consequently the family of solutions (<ref>) of the covariance extension problem stated above are parameterized by $P\in\mathfrak{P}_+(N)$ in a bijective fashion. From the following theorem we see that, for any $P\in\mathfrak{P}_+(N)$, the corresponding unique $Q\in\mathfrak{P}_+(N)$ can be obtained by convex optimization. We refer the reader to <cit.> for the proofs.
Let $\cb\in\mathfrak{C}_+(N)$ and $P\in\mathfrak{P}_+(N)$. Then the problem to maximize
\begin{equation}
\label{primal}
\mathbb{I}_P(\Phi) =\int_{-\pi}^\pi P(e^{i\theta})\log \Phi(e^{i\theta})d\nu
\end{equation}
subject to the moment conditions (<ref>) has a unique solution, namely (<ref>), where $Q$ is the unique optimal solution of the problem to minimize
\begin{equation}
\label{dual}
\mathbb{J}_P(Q)= \langle \cb,\qb\rangle -\int_{-\pi}^\pi P(e^{i\theta})\log Q(e^{i\theta})d\nu
\end{equation}
over all $Q\in\mathfrak{P}_+(N)$, where $\qb:=(q_0,q_1,\dots,q_n)$. The functional $\mathbb{J}_P$ is strictly convex.
Theorems <ref> and <ref> are discrete versions of corresponding results in <cit.>.
The solution corresponding to $P=1$ is called the maximum-entropy solution by virtue of (<ref>).
As $N\to\infty$ the process $y$ looses it periodic character, and its spectral density $\Phi_\infty$ becomes continuous and defined on the whole unit circle so that
\begin{equation}
\label{continuousmoments}
\int_{-\pi}^\pi e^{ik\theta}\Phi_\infty(e^{i\theta})\frac{d\theta}{2\pi} =c_k, \quad k=0,1,\dots,n.
\end{equation}
In fact, denoting by $Q_N$ the solution of Theorem <ref>, it was shown in <cit.> that $\Phi_\infty=P/Q_\infty$, where, for each fixed $P$,
is the unique $Q$ such that $\Phi_\infty=P/Q$ satisfies the moment conditions (<ref>).
§ REFORMULATION IN TERMS OF CIRCULANT MATRICES
Circulant matrices are Toeplitz matrices with a special circulant structure
\begin{equation}
\label{ }
\Circ\{ \gamma_0,\gamma_1,\dots, \gamma_\nu\} =
\begin{bmatrix}\gamma_0&\gamma_\nu&\gamma_{\nu-1}&\cdots&\gamma_1\\
\gamma_1&\gamma_0&\gamma_\nu&\cdots&\gamma_2\\
\gamma_2&\gamma_1&\gamma_0&\cdots&\gamma_3\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
\gamma_\nu&\gamma_{\nu-1}&\gamma_{\nu-2}&\cdots&\gamma_0
\end{bmatrix},
\end{equation}
where the columns (or, equivalently, rows) are shifted cyclically, and where $\gamma_0,\gamma_1,\dots,\gamma_\nu$ here are taken to be complex numbers. In our present covariance extension problem we consider Hermitian circulant matrices
\begin{equation}
\label{M_C}
\Mb:=\Circ\{ m_0,m_1,m_2,\dots, m_N,\bar{m}_{N-1},\dots,\bar{m}_2,\bar{m}_1\},
\end{equation}
which can be represented in form
\begin{equation}
\label{S2C}
\Mb =\sum_{k=-N+1}^N m_k\Sb^{-k}, \quad m_{-k}=\bar{m}_k
\end{equation}
where $\Sb$ is the nonsingular $2N\times 2N$ cyclic shift matrix
\begin{equation}
\label{Sb}
\Sb := \left[\begin{array}{cccccc}0 & 1 & 0 & 0 & \dots & 0 \\0 & 0 & 1 & 0 & \dots & 0 \\0 & 0 & 0 & 1 & \dots & 0 \\\vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\0 & 0 & 0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 & 0 & 0\end{array}\right].
\end{equation}
The pseudo-polynomial
\begin{equation}
\label{symbol}
M(\zeta)=\sum_{k=-N+1}^N m_k \zeta^{-k}, \quad m_{-k}=\bar{m}_k
\end{equation}
is called the symbol of $\Mb$. Clearly $\Sb$ is itself a circulant matrix (although not Hermitian) with symbol $S(\zeta)= \zeta$. A necessary and sufficient condition for a matrix $\Mb$ to be circulant is that
\begin{equation}
\label{ }
\Sb\Mb\Sb\Tr=\Mb.
\end{equation}
Hence, since $\Sb^{-1}=\Sb\Tr$, the inverse of a circulant matrix is also circulant. More generally, if $\Ab$ and $\Bb$ are circulant matrices of the same dimension with symbols $A(\zeta)$ and $B(\zeta)$ respectively, then $\Ab\Bb$ and $\Ab+\Bb$ are circulant matrices with symbols $A(\zeta)B(\zeta)$ and $A(\zeta)+B(\zeta)$, respectively. In fact, the circulant matrices of a fixed dimension form an algebra – more precisely, a commutative *-algebra with the involution * being the conjugate transpose – and the DFT is an algebra homomorphism of the set of circulant matrices onto the pseudo-polynomials of degree at most $N$ in the variable $\zeta \in \mathbb{T}_{2N}$. Consequently, circulant matrices commute, and, if $\Mb$ is a circulant matrix with symbol $M(\zeta)$ then $\Mb^{-1}$ is circulant with symbol $M(\zeta)^{-1}$.
The proof of the following proposition is immediate.
Let $\{y(t); \, t=-N+1,\dots,N\}$ be a stationary process with period $2N$ and covariance lags (<ref>), and let $\yb$ be the $2N$-dimensional stochastic vector $\yb=[y(-N+1),y(-N+2), \cdots, y(N)]\Tr$. Then, with $^*$ denoting conjugate transpose,
\begin{equation}
\label{Sigma}
\Sigmab :=\E\{\yb\yb^*\} =\Circ\{ c_0,c_1,c_2,\dots, c_N,\bar{c}_{N-1},\dots,\bar{c}_2,\bar{c}_1\}
\end{equation}
is a $2N\times 2N$ Hermitian circulant matrix with symbol $\Phi(\zeta)$ given by (<ref>).
The covariance extension problem of Section <ref>, called the circulant rational covariance extension problem, can now be reformulated as a matrix extension problem. The given covariance data $\cb=(c_0,c_1,\dots,c_n)$ can be represented as a circulant matrix
\begin{equation}
\label{bandedC}
\Cb = \Circ\{ c_0,c_1,\dots,c_n,0,\dots,0,\bar{c}_n,\bar{c}_{n-1},\dots,\bar{c}_1\}
\end{equation}
with symbol
\begin{equation}
\label{C(z)}
C(\zeta)=\sum_{k=-n}^n c_k \zeta^{-k},
\end{equation}
where the unknown covariance lags $c_{n+1},c_{n+2},\dots,c_N$ in (<ref>), to be determined, here are replaced by zeros. A circulant matrix of type (<ref>) is called banded of order $n$. We recall that $n<N$. From now one we drop the attribute `Hermitian' since we shall only consider such circulant matrices in the sequel. A banded circulant matrix of order $n$ will thus be determined by $n+1$ (complex) parameters.
The next lemma establishes the connection between circulant matrices and their symbols.
Let $\Mb$ be a circulant matrix with symbol $M(\zeta)$. Then
\begin{equation}
\label{Mdiag}
\Mb=\Fb^*\text{\rm diag}\big(M(\zeta_{-N+1}),M(\zeta_{-N+2}),\dots,M(\zeta_N)\big)\Fb,
\end{equation}
where $\Fb$ is the unitary matrix
\begin{equation}
\label{F}
\Fb= \frac{1}{\sqrt{2N}}\left[\begin{array}{cccc}\zeta_{-N+1}^{N-1} & \zeta_{-N+1}^{N-2} & \cdots & \zeta_{-N+1}^{-N} \\ \vdots & \vdots & \cdots & \vdots \\\zeta_{0}^{N-1} & \zeta_{0}^{N-2} & \cdots & \zeta_{0}^{-N} \\ \vdots & \vdots & \cdots & \vdots \\\zeta_{N}^{N-1} & \zeta_{N}^{N-2} & \cdots & \zeta_{N}^{-N} \end{array}\right] .
\end{equation}
Moreover, if $M(\zeta_k)>0$ for all $k$, then
\begin{equation}
\label{logM}
\log\Mb=\Fb^*\text{\rm diag}\big(\log M(\zeta_{-N+1}),\log M(\zeta_{-N+2}),\dots,\log M(\zeta_N)\big)\Fb.
\end{equation}
The discrete Fourier transform $\script F$ maps a sequence $(g_{-N+1},g_{-N+2},\dots,g_N)$ into the sequence of complex numbers
\begin{equation} \label{DFT}
G(\zeta_{j}) := \sum_{k=-N+1 }^N g_k \zeta_{j}^{-k} \,,\qquad j=-N+1,-N+2, \ldots , N .
\end{equation}
The sequence $\gb$ can be recovered from $G$ by the inverse transform
\begin{equation} \label{InvDFTmeas}
g_k = \int_{-\pi}^\pi e^{ik\theta}G(e^{i\theta}) d\nu(\theta),\quad k = -N+1,-N+2, \dots ,N.
\end{equation}
This correspondence can be written
\begin{equation}
\label{ghat=Fg}
\hat{\gb} =\Fb\gb,
\end{equation}
where $\hat{\gb}:=(2N)^{-\frac12}\big(G(\zeta_{-N+1}),G(\zeta_{-N+2}),\dots,G(\zeta_N)\big)\Tr$, $\gb:=(g_{-N+1},g_{-N+2},\dots,g_N)\Tr$, and $\Fb$ is the nonsingular $2N\times 2N$ Vandermonde matrix (<ref>). Clearly $\Fb$ is unitary. Since
= ∑_k=-N+1 ^N m_k^-k
and $[\Sb^{-k}\gb]_j =g_{j-k}$, where $g_{k+2N}=g_k$, we have
F() =∑_j=-N+1^N ζ^-j∑_k=-N+1^N m_kg_j-k
=∑_k=-N+1^N m_k ζ^-k∑_j=-N+1^N g_j-kζ^-(j-k)
= M(ζ)F,
which yields
√(2N)()_j=M(ζ_j)√(2N)()_j, j=-N+1,-N+2,…,N,
from which (<ref>) follows. Finally, since $\log M(\zeta)$ is analytic in the neighborhood of each $M(\zeta_k)>0$, the eigenvalues of $\log\Mb$ are just the real numbers $\log M(\zeta_k)$, $k=-N+1,\dots,N$, by the spectral mapping theorem <cit.>, and hence (<ref>) follows.
We are now in a position to reformulate Theorems <ref> and <ref> in terms of circulant matrices. To this end first note that, in view of Lemma <ref>, the cone $\mathfrak{P}_+(N)$ corresponds to the class of positive-definite banded $2N\times 2N$ circulant matrices $\Pb$ of order $n$. Moreover, by Plancherel's Theorem for DFT, which is a simple consequence of (<ref>), we have
∑_k=-n^n c_kp̅_k =1/2N∑_j=-N+1^N C(ζ_j)P(ζ_j),
and hence, by Lemma <ref>,
\begin{equation}
\label{ }
\langle \cb,\pb\rangle = \frac{1}{2N} \trace(\Cb\Pb).
\end{equation}
Consequently, $\cb\in\mathfrak{C}_+(N)$ if and only if $\trace(\Cb\Pb)>0$ for all nonzero, positive-semidefinite,
banded $2N\times 2N$ circulant matrices $\Pb$ of order $n$. Moreover, if $\Qb$ and $\Pb$ are circulant matrices with symbols $P(\zeta)$ and $Q(\zeta)$, respectively, then, by Lemma <ref>, $P(\zeta)/Q(\zeta)$ is the symbol of $\Qb^{-1}\Pb$. Therefore Theorem <ref> has the following matrix version.
Let $\cb\in\mathfrak{C}_+(N)$, and let $\Cb$ be the corresponding circulant matrix (<ref>). Then, for each
positive-definite banded $2N\times 2N$ circulant matrices $\Pb$ of order $n$, there is unique positive-definite banded $2N\times 2N$ circulant matrices $\Qb$ of order $n$ such that
\begin{equation}
\label{Sigmab=QbinvPb}
\Sigmab=\Qb^{-1}\Pb
\end{equation}
is a circulant extension (<ref>) of $\Cb$.
In the same way, Theorem <ref> has the following matrix version, as can be seen by applying Lemma <ref>.
Let $\cb\in\mathfrak{C}_+(N)$, and let $\Cb$ be the corresponding circulant matrix (<ref>). Moreover, let $\Pb$ be a positive-definite banded $2N\times 2N$ circulant matrix of order $n$. Then the problem to maximize
\begin{equation}
\label{primal_matrix}
\mathcal{I}_{\Pb}(\Sigmab) = \text{\rm trace}(\Pb\log\Sigmab)
\end{equation}
subject to
\begin{equation}
\label{momentcondmatrixb}
\Eb_n\Tr\Sigmab \Eb_n =\Tb_n, \quad \text{where\;} \Eb_n =\begin{bmatrix}\Ib_n\\{\bold 0}\end{bmatrix}
\end{equation}
has a unique solution, namely (<ref>), where $\Qb$ is the unique optimal solution of the problem to minimize
\begin{equation}
\label{dual_matrix}
\mathcal{J}_{\Pb}(\qb)= \text{\rm trace}(\Cb\Qb) - \text{\rm trace}(\Pb\log\Qb)
\end{equation}
over all positive-definite banded $2N\times 2N$ circulant matrices $\Qb$ of order $n$, where $\qb:=(q_0,q_1,\dots,q_n)$. The functional $\mathcal{J}_{\Pb}$ is strictly convex.
§ BILATERAL ARMA MODELS
Suppose now that we have determined a circulant matrix extension (<ref>). Then there is a stochastic vector $\yb$ formed from the a stationary periodic process with corresponding covariance lags (<ref>) so that
:={^*} ={ c_0,c_1,c_2,…, c_N,c̅_N-1,…,c̅_2,c̅_1}.
Let $\hat{\E}\{y(t)\mid y(s),\, s \neq t\}$ be the wide sense conditional mean of $y(t)$ given all $\{y(s),\, s \neq t\}$. Then the error process
\begin{equation} \label{finconjn}
d(t) := y(t)-\hat{\E} \{ y(t)\mid y(s),\, s \neq t\}
\end{equation}
is orthogonal to all random variables $\{ y(s),\, s \neq t\}$, i.e., $\E\{y(t)\,\overline{d(s)}\}= \sigma^2 \, \delta_{ts}$, $t, s \in\Zbb_{2N}:=\{-N+1,-N+2,\dots,N\}$, where $\sigma^2$ is a positive number. Equivalently, $\E\{\yb\db^*\}=\sigma^2 \Ib$, where $\Ib$ is the $2N\times 2N$ identity matrix. Setting $\eb:=\db/\sigma^2$, we then have
\begin{equation}
\label{eystar}
\E\{\eb\yb^*\}=\Ib,
\end{equation}
i.e., the corresponding process $e$ is the conjugate process of $y$ <cit.>. Interpreting (<ref>) in the $\mod 2N$ arithmetics of $\Zbb_{2N}$, $\yb$ admits a linear representation of the form
\begin{equation}
\label{Ay=e}
\Gb\yb= \eb,
\end{equation}
where $\Gb$ is a $2N\times 2N$ Hermitian circulant matrix with ones on the main diagonal. Since $\Gb\E\{\yb\yb^*\}= \E\{\eb\yb^*\} = \Ib$, $\Gb$ is also positive definite and the covariance matrix $\Sigmab$ is given by
\begin{equation}
\label{cov}
\Sigmab = \Gb ^{-1},
\end{equation}
which is circulant, since the inverse of a circulant matrix is itself circulant. In fact, a stationary process $\yb$ is full-rank periodic in $\Zbb_{2N}$, if and only if $\Sigmab $ is a Hermitian positive definite circulant matrix <cit.>.
Since $\Gb$ is a Hermitian circulant matrix, it has a symbol
G(ζ)=∑_k=-N+1^N g_kζ^-k, g_-k=g̅_k,
and the linear equation can be written in the autoregressive (AR) form
\begin{equation}
\label{AR}
\sum_{k=-N+1}^N g_k y(t-k)=e(t).
\end{equation}
However, in general $\Gb$ is not banded and $n<<N$, and therefore (<ref>) is not a useful representation. Instead using the solution (<ref>), we have $\Gb=\Pb^{-1}\Qb$, where $\Pb$ and $\Qb$ are banded of order $n$ with symbols
P(ζ)=∑_k=-n^n p_kζ^-k and Q(ζ)=∑_k=-n^n q_kζ^-k,
and hence (<ref>) can be written
or equivalently in the ARMA form
\begin{equation}
\label{ARMA}
\sum_{k=-n}^n q_k y(t-k) = \sum_{k=-n}^n p_k e(t-k).
\end{equation}
Consequently, by Theorem <ref>, there is a unique bilateral ARMA model (<ref>) for each banded positive-definite Hermitian circulant matrix $\Pb$ of order $n$, provided $\cb\in\mathfrak{C}_+$. Of course,we could use the maximum-entropy solution with $\Pb=\Ib$ leading to an AR model
\begin{equation}
\label{ME-AR}
\sum_{k=-n}^n q_k y(t-k) = e(t).
\end{equation}
Next, to illustrate the accuracy of bilateral AR modeling by the methods described so far we give some simulations from <cit.>, provided by Chiara Masiero. Given an AR model of order $n=8$ with poles as depicted in Figure <ref>, we compute a covariance sequence $\cb=(c_0,c_1,\dots,c_n)$ with $n=8$, which is then used to solve the optimization problem (<ref>) with $\Pb=\Ib$ to obtain a bilateral AR approximations of degree eight for various choices of $N$. In Figure <ref>, the left picture depicts the spectral density for $N=128$ together with the true spectral density (dashed line), and the right picture illustrates how the estimation error decreases with increasing $N$.
Poles of true AR model.
Bilateral AR approximation: (left) spectrum for N = 128 and true spectrum (dashed); (right) errors for N=32, 64, 128, 256, 512 and 1024.
§ UNILATERAL ARMA MODELS AND SPECTRAL FACTORIZATION
As explained in Section <ref>, a periodic process $y$ has a discrete spectrum, and Theorem <ref> provides values of
only in the discrete points $z\in\mathbb{T}_{2N}:=\{\zeta_{-N+1},\zeta_{-n+2},\dots,\zeta_N\}$. Since $\Phi$ takes positive values on $\mathbb{T}_{2N}$, there is a trivial discrete factorization
\begin{equation}
\label{discretefactor}
\Phi(\zeta_k)=W(\zeta_k)W(\zeta_k)^*\quad k=-N+1,\dots, N.
\end{equation}
W_k= 1/2N∑_j=-N+1^N W(ζ_j)ζ_j^k, k=-N+1,…, N,
we can write (<ref>) in the form
\begin{equation}
\label{Phi(zeta)}
\Phi(\zeta)=W(\zeta)W(\zeta)^*.
\end{equation}
where $W(\zeta)$ is the discrete Fourier transform
W(ζ)=∑_k=-N+1^N W_kζ^-k.
Applying Lagrange interpolation to $W$, we obtain a spectral factorization equation
\begin{equation}
\label{Phi(z)}
\tilde\Phi(z)=W(z)W(z)^*,\quad z\in\mathbb{T},
\end{equation}
defined on the whole unit circle, where $\tilde\Phi(\zeta)=\Phi(\zeta)$ on $\mathbb{T}_{2N}$. This is a spectral density of non-periodic stationary process but should not be confused with $\Phi_\infty$ in Remark <ref>, which is the unique continuous $\Phi$ with numerator polynomial $P$ and the same covariance structure as the periodic process $y$, i.e.,
∫_-π^πe^ikθΦ_∞(e^iθ)dθ/2π =c_k, k=0,1,…,n.
In fact, although
\begin{equation}
\label{Phitilde2c}
\int_{-\pi}^\pi e^{ik\theta}\tilde\Phi(e^{i\theta})d\nu(\theta) =c_k, \quad k=0,1,\dots,n ,
\end{equation}
the non-periodic process with spectral density $\tilde\Phi$ has the covariance lags
c̃_k=∫_-π^πe^ikθΦ̃(e^iθ)dθ/2π, k=0,1,…,n ,
which differ from $c_0,c_1,\dots,c_n$. However, setting $\Delta\theta_j:=\theta_j-\theta_{j-1}$ where $e^{\theta_j}=\zeta_j$, we see from (<ref>) that $\Delta\theta_j=\pi/N$ and that the integral (<ref>) with $\tilde\Phi$ fixed is the Riemann sum
∑_j=-N+1^Ne^ikθ_j Φ̃(ζ_j)Δθ_j/2π
converging to $\tilde{c}_k$ for $k=0,1,\dots,n$ as $N\to\infty$.
By Proposition <ref>, $\Phi(\zeta)$ is the symbol of the circulant covariance matrix $\Sigmab$, and hence (<ref>) can be written in the matrix form
\begin{equation}
\label{matrixWW*}
\Sigmab=\Wb\Wb^*,
\end{equation}
where $\Wb$ is the circulant matrix with symbol $W(\zeta)$. The spectral-factorization (<ref>) has a unique outer spectral factor $W(z)$; see, e.g., <cit.>. As explained in detail in <cit.>, this corresponds in the discrete setting to $W(\zeta)$ taking the form
\begin{equation}
\label{analyticW}
W(\zeta)=\sum_{k=0}^N W_k\zeta^{-k},
\end{equation}
which in turn corresponds to $\Wb$ being lower-triangular circulant, i.e.,
\begin{equation}
\label{ }
\Wb=\Circ\{ W_0,W_1,\dots,W_N,0,\dots,0\}.
\end{equation}
Note that a lower-triangular circulant matrix is not lower triangular as the circulant structure has to be preserved.
Since $\Sigmab$ is invertible, then so is $\Wb$.
Next define the periodic stochastic process $\{w(t),\, t=-N+1\dots,N\}$ for which $\wb = [w(-N +1),w(-N +2),\dots,w(N)]\Tr$ is given by
\begin{equation}
\label{wdefn}
\wb=\Wb^{-1}\yb.
\end{equation}
Then, in view of (<ref>), we obtain $\E\{\wb\wb^*\}=\Ib$, i.e., the process $w$ is a white noise process. Consequently we have the unilateral representation
y(t)=∑_k=0^N W_kw(t-k)
in terms of white noise.
To construct an ARMA model we appeal to the following result, which is easy to verify in terms of symbols but, as demonstrated in <cit.>, also holds for block circulant matrices considered in Section <ref>.
A positive definite, Hermitian, circulant matrix $\Mb$ admits a factorization $\Mb=\Vb\Vb^*$, where $\Vb$ is of a banded lower-diagonal circulant matrix of order $n<N$, if and only if $\Mb$ is bilaterally banded of order $n$.
By Theorem <ref>, $\Sigmab=\Qb^{-1}\Pb$, where $\Qb$ and $\Pb$ are banded, positive definite, Hermitian, circulant matrices of order $n$.
Hence, by Lemma <ref>, there are factorizations
=^* and =^*,
where $\Ab$ and $\Bb$ are banded lower-diagonal circulant matrices of order $n$. Consequently, $\Sigmab=\Ab^{-1}\Bb(\Ab^{-1}\Bb)^*$, i.e.,
\begin{equation}
\label{W=AinvB}
\Wb=\Ab^{-1}\Bb,
\end{equation}
which together with (<ref>) yields $\Ab\yb=\Bb\wb$, i.e., the unilateral ARMA model
\begin{equation}
\label{unilateralARMA}
\sum_{k=0}^n a_k y(t-k) = \sum_{k=0}^n b_k w(t-k).
\end{equation}
Since $\Ab$ is nonsingular, $a_0\ne 0$, and hence we can normalize by setting $a_0=1$.
In particular, if $\Pb=\Ib$, we obtain the AR representation
\begin{equation}
\label{unilateralAR}
\sum_{k=0}^n a_k y(t-k) = b_0 w(t).
\end{equation}
Symmetrically, there is factorization
\begin{equation}
\label{matrixWbarWbar*}
\Sigmab=\bar{\Wb}\bar{\Wb}^*,
\end{equation}
where $\bar{\Wb}$ is upper-diagonal circulant, i.e. the transpose of a lower-diagonal circulant matrix, and a white-noise process
\begin{equation}
\label{wbardefn}
\bar\wb=\bar{\Wb}^{-1}\yb.
\end{equation}
Likewise there are factorizations
=^* and =^*,
where $\bar\Ab$ and $\bar\Bb$ are banded upper-diagonal circulant matrices of order $n$. This yields a backward unilateral ARMA model
\begin{equation}
\label{unilateralARMAbar}
\sum_{k=-n}^0 \bar a_k y(t-k) = \sum_{k=-n}^0 \bar b_k \bar w(t-k).
\end{equation}
These representations should be useful in the smoothing problem for periodic systems <cit.>.
§ RECIPROCAL PROCESSES AND THE COVARIANCE SELECTION PROBLEM
Let $\Ab$, $\Bb$ and $\Xb$ be subspaces in a certain common ambient Hilbert space of zero mean second order random variables.
We say that
$\Ab$ and $\Bb$ are conditionally orthogonal given $\Xb$ if
\begin{equation}
\label{condorthPerp}
\alpha -\hat\E\{\alpha\mid\Xb\}\perp \beta -\hat\E\{\beta\mid\Xb\}, \quad \forall\alpha\in\Ab, \forall\beta\in\Bb
\end{equation}
(see, e.g., <cit.>), which we denote $\Ab\perp\Bb\mid\Xb$, and which clearly is equivalent to
\begin{equation}
\label{condorthPerp2}
\E\left\{\hat\E\{\alpha\mid\Xb\}\overline{\hat\E\{\beta\mid\Xb\}}\right\}=\E\{\alpha\overline{\beta}\}, \quad \forall\alpha\in\Ab, \forall\beta\in\Bb.
\end{equation}
Conditional orthogonality is the same as conditional uncorrelatedness, and hence conditional independence in the Gaussian case.
Let $\yb_{[t-n,t)}$ and $\yb_{(t ,t+n]}$ be the $n$-dimensional random column vectors obtained by stacking $y(t-n),y(t-n+1) \ldots, y(t-1)$ and $y(t+1),y(t+2) \ldots, y(t+n)$, respectively, in that order. In the same way, $\yb_{[t-n,t]}$ is obtained by appending $y(t)$ to $\yb_{[t-n,t)}$ as the last element, etc. Here and in the following the sums $t-k$ and $t+k$ are to be understood modulo $2N$. For any interval $(t_1,t_2)\subset [-N+1,N]$, we denote by $(t_1,t_2)^c$ the complementary set in $[1,2N]$.
A reciprocal process of order $n$ on $ (-N,N]$ is a process $\{y(t); \, t=-N+1,\dots,N\}$ such that
\begin{equation}
\label{Rec}
\hat\E\{y(t)\mid y(s),\, s \neq t\} = \hat\E\{y(t)\mid \yb_{[t-n,t)}\vee\yb_{(t ,t+n]}\}
\end{equation}
for $t\in (-N,N]$.
This is a generalization introduced in <cit.> of the concept of reciprocal process, which can be trivially extended to vector processes. In fact, a reciprocal process in the original sense is here a reciprocal process of order one. This concept does not require stationarity, although in the paper it will always be assumed.
It follows from <cit.> that $\{y(t)\}$ is reciprocal of order $n$ if and only if
\begin{equation}\label{RecMod}
\hat\E\{y(t)\mid y(s),\, s \in [t-n, \,t+n]^c \} = \hat\E\{y(t)\mid \yb_{[t-n,t)}\vee\yb_{(t ,t+n]}\}
\end{equation}
for $t \in[-N+1,N]$. In particular,
the estimation error
\begin{equation}
\label{finconjn2}
\begin{split}
d(t) & := y(t)-\hat\E\{y(t)\mid y(s),\, s \neq t\} \\
& = y(t) -\hat\E\{y(t)\mid \yb_{[t-n,t)}\vee\yb_{(t ,t+n]}\}
\end{split}
\end{equation}
must clearly be orthogonal to all random variables $\{y(s),\, s \neq t\}$; i.e. $\E\{d(t)\overline{y(s)}\}=\sigma^2\delta_{st}$, where $\sigma^2$ is the variance of $d(t)$.
Then $e(t):=d(t)/\sigma^2$ is the (normalized) conjugate process of $y$ satisfying (<ref>), i.e.,
\begin{equation}
\label{conjugate}
\E\{e(t)\overline{y(s)}\}=\delta_{ts}.
\end{equation}
Since $e(t+k)$ is a linear combination of the components of the random vector $\yb_{[t+k-n,t+k+n]}$, it follows from (<ref>) that both $e(t+k)$ and $e(t-k)$ are orthogonal to $e(t)$ for $k >n$. Hence the process $\{e(t)\}$ has correlation bandwidth $n$, i.e.,
\begin{equation}\label{MARecipn}
\E\{e(t+k)\,e(t)^*\} = 0 \quad \text{for $n < |k| < 2N-n, \; k\in [-N+1,N]$},
\end{equation}
and consequently $(\yb,\eb)$ satisfies (<ref>), where $\Gb$ is banded of order $n$, which corresponds to an AR representation (<ref>).
Consequently, the AR solutions of the rational circulant covariance extension problem are precisely the ones corresponding to a reciprocal process $\{y(t)\}$ of order $n$. Next we demonstrate how this representation is connected to the covariance selection problem of Dempster <cit.> by deriving a generalization of this seminal result.
Let $J:=\{j_1,\dots,j_p\}$ and $K:=\{k_1,\dots,k_q\}$ be two subsets of $\{-N+1,-N+2,\dots, N\}$, and define $\yb_J$ and $\yb_K$ as the subvectors of $\yb=(y_{-N+1},y_{-N+2}, \cdots, y_N)\Tr$ with indices in $J$ and $K$, respectively. Moreover, let
_J,K:= span { y(t); t∉J, t∉K}= _J∩_K,
where $\check{\Yb}_{J}:=\text{span} \{ y(t); \; t\notin J\}$. With a slight misuse of notation, we shall write
\begin{equation}\label{CovSel}
\yb_J \perp \yb_K \mid \check{\Yb}_{J,K},
\end{equation}
to mean that the subspaces spanned by the components of $\yb_J$ and $\yb_K$, respectively, are conditionally orthogonal given $\check{\Yb}_{J,K}$. This condition can be characterized in terms of the inverse of the covariance matrix $\Sigmab:=\E\{\yb\yb^*\}=\bmat \sigma_{ij} \emat_{i,j=-N+1}^N$ of $y$.
Let $\Gb:=\Sigmab^{-1}=\bmat g_{ij} \emat_{i,j=1}^N$ be the concentration matrix of the random vector $y$. Then the conditional orthogonality relation (<ref>) holds if and only if $g_{jk}=0$ for all $(j,k)\in J\times K$.
Let $E_J$ be the $2N\times 2N$ diagonal matrix with ones in the positions $(j_1,j_1),\dots,\\(j_m,j_m)$ and zeros elsewhere and let $E_K$ be defined similarly in terms of index set $K$. Then $\check{\Yb}_{J}$ is spanned by the components of $\yb-E_J\yb$ and $\check{\Yb}_{K}$ by the components of $\yb-E_K\yb$. Let
_K := _K- {_K|_K},
and note that its $q\times q$ covariance matrix
\begin{equation*}
\tilde{\Sigmab}_K := \E\{\tilde{\yb}_K \tilde{\yb}_K^*\}
\end{equation*}
must be positive definite, for otherwise some linear combination of the components of $\yb_K$ would belong yo $\check{\Yb}_{K}$. Let $\tilde{\yb}_K=G_K\yb$ for some $q\times 2N$ matrix $G_K$. Since $\tilde{\yb}_K\perp\check{\Yb}_{K}$,
\begin{equation*}
\E\{\tilde{\yb}_K(\yb-E_K\yb)^*\}=0
\end{equation*}
and therefore $\E\{\tilde{\yb}_K\yb^*\}= G_K\Sigmab$ must be equal to $\E\{\tilde{\yb}_K(E_K\yb)^*\}$, which by $\tilde{\yb}_K \in \check{\Yb}_{K}^{\perp}$, in turn equals
However, since the nonzero components of $\hat\E\{E_K\yb\mid \check{\Yb}_{K}^{\perp}\}$ are those of $\tilde\yb_K$, there is an $2N\times q$ matrix $\Pi_K$ with the unit vectors $e^{\prime}_{k_i}$, $i=1,\ldots,q$, as the rows such that
and hence
Consequently, $G_K\Sigmab=\tilde{\Sigmab}_K\Pi_K^*$, i.e.,
In the same way, $\tilde{\yb}_J=G_J\yb$, where $G_J$ is the $q\times 2N$ matrix
and therefore
{_J_K^*} =_JΠ_J^*^-1Π_K_K,
which is zero if and only if $\Pi_J^*\Sigmab^{-1}\Pi_K=0$, i.e., $g_{jk}=0$ for all $(j,k)\in J\times K$.
It remains to show that $\E\{\tilde{\yb}_J\tilde{\yb}_K^*\}=0$ is equivalent to (<ref>), which in view of (<ref>), can be written
\begin{equation*}
\E\left\{\hat\E\{\yb_J\mid \check{\Yb}_{J,K}\}\{\hat\E\{\yb_K\mid \check{\Yb}_{J,K}\}^*\right\}=\{\yb_J\yb_K^*\}.
\end{equation*}
E{_J_K^*}=E{_J_K^*} -{{_J|_J}{{_K|_K}^*},
so the proof will complete if we show that
\begin{equation}
\label{Lemma2.6.9}
\E\left\{\hat\E\{\yb_J\mid \check{\Yb}_J\}\{\hat\E\{\yb_K\mid \check{\Yb}_K\}^*\right\}=
\E\left\{\hat\E\{\yb_J\mid \check{\Yb}_{J,K}\}\{\hat\E\{\yb_K\mid \check{\Yb}_{J,K}\}^*\right\}
\end{equation}
the proof of which follows precisely the lines of Lemma 2.6.9 in <cit.>.
Taking $J$ and $K$ to be singletons we recover as a special case Dempster's original result <cit.>.
To connect back to Definition <ref> of a reciprocal process of order $n$, use the equivalent condition (<ref>) so that, with $J=\{t\}$ and $K= [t-n, \,t+n]^c $, $\yb_J=y(t)$ and $\yb_{K}$ are conditionally orthogonal given $\check{\Yb}_{J,K}= \yb_{[t-n,t)}\vee\yb_{(t ,t+n]}$.
Then $J\times K$ is the set $\big \{ t \times [t-n, \,t+n]^c \,;\, t\in (-N,\,N]\,\big \}$, and hence
Theorem <ref> states precisely that the circulant matrix $\Gb$ is banded of order $n$. We stress that in general $\Gb=\Sigmab^{-1}$ is not banded, as the underlying process $\{y(t)\}$ is not reciprocal of degree $n$,
and we then have an ARMA representation as explained in Section <ref>.
§ DETEMINING $\PB$ WITH THE HELP OF LOGARITHMICAL MOMENTS
We have shown that the solutions of the circulant rational covariance extension problem, as well as the corresponding bilateral ARMA models, are completely parameterized by $P\in\mathfrak{P}_+(N)$, or, equivalently, by their corresponding banded circulant matrices $\Pb$. This leads to the question of how to determine the $\Pb$ from given data.
To this end, suppose that we are also given the logarithmic moments
\begin{equation}
\label{cepstrum}
\gamma_k=\int_{-\pi}^\pi e^{ik\theta}\log\Phi(e^{i\theta})d\nu, \quad k=1,2,\dots,n.
\end{equation}
In the setting of the classical trigonometric moment problem such moments are known as cepstral coefficients, and in speech processing, for example, they are estimated from observed data for purposes of design.
Following <cit.> and, in the context of the trigonometric moment problem, <cit.>, we normalize the elements in $\mathfrak{P}_+(N)$ to define $\tilde{\mathfrak{P}}_+(N):=\{P\in\mathfrak{P}_+(N)\mid p_0=1\}$ and consider the problem to find a nonnegative integrable $\Phi$ maximizing
\begin{equation}
\label{I(Phi)}
\mathbb{I}(\Phi) =\int_{-\pi}^\pi \log\Phi(e^{i\theta})d\nu =\frac{1}{2N}\sum_{j=-N+1}^N \log\Phi(\zeta_j)
\end{equation}
subject to the moment constraints (<ref>) and (<ref>). It is shown in <cit.> that
if there is a maximal $\Phi$ that is positive on the unit circle, it is given by
\begin{equation}
\label{Phi2}
\Phi(\zeta)=\frac{P(\zeta)}{Q(\zeta)},
\end{equation}
where $(P,Q)$ is the unique solution of the dual problem to minimize
\begin{equation}
\label{J(P,Q)}
\mathbb{J}(P,Q)=\langle \cb,\qb\rangle -\langle \gammab,\pb\rangle + \int_{-\pi}^\pi P(e^{i\theta})\log\left(\frac{P(e^{i\theta})}{Q(e^{i\theta})}\right)d\nu
\end{equation}
over all $(P,Q)\in\tilde{\mathfrak{P}}_+(N)\times\mathfrak{P}_+(N)$, where $\gammab=(\gamma_0,\gamma_1,\dots,\gamma_n)$ and $\pb=(p_0, p_1,\dots,p_n)$ with $\gamma_0=0$ and $p_0=1$.
The problem is that the dual problem might have a minimizer on the boundary so that there is no stationery point in the interior, and then the constraints (<ref>) will in general not be satisfied <cit.>. Therefore the problem needs to be regularized in the style of <cit.>. More precisely, we consider the regularized problem to minimize
\begin{equation}
\label{J(P,Q)reg}
\mathbb{J}_\lambda(P,Q) =\mathbb{J}(P,Q) -\lambda\int_{-\pi}^\pi \log P(e^{i\theta})d\nu
\end{equation}
for some suitable $\lambda>0$ over all $(P,Q)\in\tilde{\mathfrak{P}}_+(N)\times\mathfrak{P}_+(N)$. Setting $\mathbf{J}_\lambda(\Pb,\Qb):=2N\mathbb{J}_\lambda(P,Q)$, (<ref>) can be written
\begin{equation}
\label{Jlambda}
\mathbf{J}_\lambda(\Pb,\Qb) =\text{\rm tr}\{\Cb\Qb\} -\text{\rm tr}\{\Gammab\Pb\} + \text{\rm tr}\{\Pb\log\Pb\Qb^{-1}\} - \lambda\,\text{\rm tr}\{\log\Pb\},
\end{equation}
where $\Gammab$ is the Hermitian circulant matrix with symbol
\begin{equation}
\label{M(z)}
\Gamma(\zeta)=\sum_{k=-n}^n \gamma_k \zeta^{-k}, \quad \gamma_{-k}=\bar{\gamma}_k.
\end{equation}
Therefore, in the circulant matrix form, the regularized dual problem amounts to minimizing (<ref>) over all banded Hermitian circulant matrices $\Pb$ and $\Qb$ of order $n$ subject to $p_0=1$. It is shown in <cit.> that
\begin{equation}
\label{Sigmab2}
\Sigmab=\Qb^{-1}\Pb,
\end{equation}
or, equivalently in symbol form (<ref>), maximizes
\begin{equation}
\label{ }
\mathbf{I}(\Sigmab)=\text{\rm tr}\{\log\Sigmab\} =\log\det\Sigmab,
\end{equation}
or, equivalently (<ref>), subject to (<ref>) and (<ref>), the latter constraint modified so that the logarithmic moment $\gamma_k$ is exchanged for $\gamma_k+\varepsilon_k$, $k=1,2,\dots,n$, where
\begin{equation}
\label{epsilon}
\varepsilon_k=\int_{-\pi}^\pi e^{ik\theta}\frac{\lambda}{\hat P(e^{i\theta})}d\nu= \frac{\lambda}{2N}\text{\rm tr}\{\Sb^k\hat\Pb^{-1}\},
\end{equation}
$\hat P$ being the optimal $P$.
The following example from <cit.>, provided by Chiara Masiero, illustrates the advantages of this procedure. We start from an ARMA model with $n=8$ poles and three zeros distributed as in Figure <ref>, from which we compute $\cb=(c_0,c_1,\dots,c_n)$ and $\gammab=(\gamma_1,\dots,\gamma_n)$ for various choices of the order $n$.
Poles and zeros of true ARMA model.
First we determine the maximum entropy solution from $\cb$ with $n=12$ and $N=1024$. The resulting spectral function $\Phi$ is depicted in the left plot of Figure <ref> together with the true spectrum. Next we compute $\Phi$ by the procedure in this section using $\cb$ and $\gammab$ with $n=8$ and $N=128$. The result is depicted in the right plot of Figure <ref> again together with the true spectrum. This illustrates the advantage of bilateral ARMA modeling as compared to bilateral AR modeling, as a much lower value on $N$ provides a better approximation, although $n$ is smaller.
Bilateral approximations with true spectrum (dashed): (left) bilateral AR with $n=12$ and $N=1024$; (right) bilateral ARMA with $n=8$ and $N=128$ using both covariance and logarithmic moment estimates.
§ EXTENSIONS TO MULTIVARIATE CASE
To simplify notation we have so far restricted our attention to scalar stationary periodic processes. We shall now demonstrate that most of the results can be simply extended to the multivariate case, provided we restrict the analysis to scalar pseudo-polynomials $P(\zeta)$. In fact, most of the equations in the previous section will remain intact if we allow ourselves to interpret the scalar quantities as matrix-valued ones.
Let $\{y(t)\}$ be a zero-mean stationary $m$-dimensional process $\{y(t)\}$ defined on $\Zbb_{2N}$; i.e., a stationary process defined on a finite interval $[-N+1,\,N]$ of the integer line $\Zbb$ and extended to all of $\Zbb$ as a periodic stationary process with period $2N$. Moreover, let $C_{-N+1},C_{-N+2},\dots,C_{N}$ be the $m\times m$ covariance lags $C_k:=\E\{ y(t+k)y(t)^*\}$, and define its discrete Fourier transformation
\begin{equation} \label{c2Phi}
\Phi(\zeta_{j}) := \sum_{k=-N+1 }^{N }\, C_k \zeta_{j}^{-k} \,,\qquad j=-N+1,\dots , N,
\end{equation}
which is a positive, Hermitian matrix-valued function of $\zeta$. Then, by the inverse discrete Fourier transformation,
\begin{equation} \label{Phi2c}
C_k = \frac{1}{2N}\sum_{j=-N+1 }^{N }\zeta_{j}^k \Phi(\zeta_j)
=\int_{-\pi}^\pi e^{ik\theta}\Phi(e^{i\theta}) d\nu,\quad k = -N+1, \dots ,N,
\end{equation}
where the Stieljes measure $d\nu$ is given by (<ref>). The $m\times m$ matrix function $\Phi$ is the spectral density of the vector process $y$. In fact, let
\begin{equation}
\label{yDFT}
\hat{y}(\zeta_k):= \sum_{t=-N+1}^N y(t)\zeta_k^{-t}, \quad k=-N+1,\dots, N,
\end{equation}
be the discrete Fourier transformation of the process $y$. Since
1/2N∑_t=-N+1^N (ζ_kζ_ℓ^*)^t =δ_kℓ
by (<ref>), the random variables (<ref>) are uncorrelated, and
\begin{equation}
\label{yhatyhat}
\frac{1}{2N}\E\{ \hat{y}(\zeta_k)\hat{y}(\zeta_\ell)^*\}=\Phi(\zeta_{k})\delta_{k\ell}.
\end{equation}
This yields a spectral representation of $y$ analogous to the usual one, namely
\begin{equation}
\label{ }
y(t)=\frac{1}{2N}\sum_{k=-N+1}^N \zeta_k^t\,\hat{y}(\zeta_k)=\int_{-\pi}^\pi e^{ik\theta}d\hat{y}(\theta),
\end{equation}
where $d\hat{y}:=\hat{y}(e^{i\theta})d\nu$.
Next, we define the class $\mathfrak{P}_+^{(m,n)}(N)$ of $m\times m$ Hermitian pseudo-polynomials
\begin{equation}
\label{Qmatrix}
Q(\zeta)=\sum_{k=-n}^n Q_k\zeta^{-k}, \quad Q_{-k}=Q_k^*
\end{equation}
of degree at most $n$ that are positive definite on the discrete unit circle $\mathbb{T}_{2N}$, and let $\mathfrak{P}_+^{(m,n)}\subset\mathfrak{P}_+^{(m,n)}(N)$ be the subset of all (<ref>) such that $Q(e^{i\theta})$ is positive define for all $\theta\in [-\pi,\pi]$. Moreover let $\mathfrak{C}_+^{(m,n)}(N)$ be the dual cone of all $C=(C_0,C_1,\dots,C_n)$ such that
⟨C,Q⟩:=∑_k=-n^n {C_k Q_k^*} >0 for all $Q\in\overline{\mathfrak{P}_+^{(m,n)}(N)}\setminus\{0\}$,
and let $\mathfrak{C}_+^{(m,n)}\supset\mathfrak{C}_+^{(m,n)}(N)$ be defined as the dual cone of $\mathfrak{P}_+^{(m,n)}$. Analogously to the scalar case it can be shown that $C\in\mathfrak{C}_+^{(m,n)}$ if and only if the block-Toeplitz matrix
\begin{equation}
\label{blockToeplitz}
\Tb_n=\begin{bmatrix} C_0&C_1^*&C_2^*&\cdots&C_n^*\\
C_1&C_0&C_1^*&\cdots& C_{n-1}^*\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
\end{bmatrix}
\end{equation}
is positive definite <cit.>, a condition that is necessary, but in general not sufficient, for $C\in\mathfrak{C}_+^{(m,n)}(N)$ to hold.
The basic problem is now the following. Given the sequence $C=(C_0,C_1,\dots,C_n)\in\mathfrak{C}_+^{(m,n)}(N)$ of $m\times m$ covariance lags, find an extension $C_{n+1},C_{n+2},\dots,C_N$ with $C_{-k}=C_k^*$ such that the spectral function $\Phi$ defined by (<ref>) has the rational form
\begin{equation}
\label{Phimatrix=PQinv}
\Phi(\zeta)=P(\zeta)Q(\zeta)^{-1}, \quad P\in\mathfrak{P}_+^{(1,n)}(N), \, Q\in\mathfrak{P}_+^{(m,n)}(N).
\end{equation}
Let $C\in\mathfrak{C}_+^{(m,n)}(N)$. Then, for each $P\in\mathfrak{P}_+^{(1,n)}(N)$, there is a unique $Q\in\mathfrak{P}_+^{(m,n)}(N)$ such that
\begin{equation}
\label{Phi=P/Qmatrix}
\Phi=PQ^{-1}
\end{equation}
satisfies the moment conditions
\begin{equation}
\label{matrixmoments}
\int_{-\pi}^\pi e^{ik\theta}\Phi(e^{i\theta})d\nu =C_k, \quad k=0,1,\dots,n.
\end{equation}
Theorem <ref> is a direct consequence of the following theorem, which also provides an algorithm for computing the solution.
For each $(C,P)\in\mathfrak{C}_+^{(m,n)}(N)\times\mathfrak{P}_+^{(1,n)}(N)$, the problem to maximize the functional
\begin{equation}
\label{matrixprimal}
\mathbb{I}_P(\Phi) =\int_{-\pi}^\pi P(e^{i\theta})\log\det \Phi(e^{i\theta})d\nu
\end{equation}
subject to the moment conditions (<ref>) has a unique solution $\hat{\Phi}$, and it has the form
\begin{equation}
\label{Phiopt}
\hat{\Phi}(z)=P(z)\hat{Q}(z)^{-1},
\end{equation}
where $\hat{Q}\in\mathfrak{P}_+^{(m,n)}(N)$ is the unique solution to the dual problem to minimize
\begin{equation}\label{dual}
\mathbb{J}_P(Q)= \langle C,Q\rangle -\int_{-\pi}^\pi P(e^{i\theta})\log \det Q(e^{i\theta})d\nu
\end{equation}
over all $Q\in\mathfrak{P}_+^{(m,n)}(N)$.
The proofs of Theorems <ref> and <ref> follow the lines of <cit.>.
It can also be shown that the moment map sending $Q\in\mathfrak{P}_+^{(m,n)}(N)$ to $C\in\mathfrak{C}_+^{(m,n)}(N)$ is a diffeomorphism.
To formulate a matrix version of Theorems <ref> and <ref> we need to introduce (Hermitian) block-circulant matrices
\begin{equation}
\label{S2C}
\Mb =\sum_{k=-N+1}^N S^{-k}\otimes M_k, \quad M_{-k}=M_k^*
\end{equation}
where $\otimes$ is the Kronecker product and $S$ is the nonsingular $2N\times 2N$ cyclic shift matrix (<ref>). The notation $\Sb$ will now be reserved for the $2mN\times 2mN$ block-shift matrix
\begin{equation}
\label{Sblarge}
\Sb = S\otimes I_m =\left[\begin{array}{cccccc}0 & I_m & 0 & \dots & 0 \\0 & 0 & I_m & \dots & 0 \\\vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & 0 & I_m \\I_m & 0 & 0 & 0 & 0\end{array}\right].
\end{equation}
As before $\Sb^{2N}=\Sb^0=\Ib :=I_{2mN}$, $\Sb^{k+2N}=\Sb^k$, and $\Sb^{2N-k}=\Sb^{-k}=(\Sb^k)\Tr$. Moreover
\begin{equation}
\label{circulantcondition}
\Sb\Mb \Sb^*=\Mb
\end{equation}
is both necessary and sufficient for $\Mb$ to be $m\times m$ block-circulant. The symbol of $\Mb$ is the $m\times m$ pseudo-polynomial
\begin{equation}
\label{symbol}
M(\zeta)=\sum_{k=-N+1}^N M_k \zeta^{-k}, \quad M_{-k}=M_k^*.
\end{equation}
We shall continue using the notation
\begin{equation}
\label{M_C}
\Mb:=\Circ\{ M_0,M_1,M_2,\dots, M_N,M_{N-1}^*,\dots,M_1^*\}
\end{equation}
also for (Hermitain) block-circulant matrices.
The problem can now be reformulated in the following way. Given the banded block-circulant matrix
\begin{equation}
\label{Cbmatrix}
\Cb =\sum_{k=-n}^n S^{-k}\otimes C_k, \quad C_{-k}=C_k^*
\end{equation}
of order $n$, find an extension $C_{n+1},C_{n+2},\dots,C_N$ such that the block-circulant matrix
\begin{equation}
\label{Sigmabmatrix}
\Sigmab =\sum_{k=-N+1}^N S^{-k}\otimes C_k, \quad C_{-k}=C_k^*
\end{equation}
has the symbol (<ref>).
To proceed we need a block-circulant version of Lemma <ref>.
Let $\Mb$ be a block-circulant matrix with symbol $M(\zeta)$. Then
\begin{equation}
\label{matrixMdiag}
\Mb=\Fb^*\text{\rm diag}\big(M(\zeta_{-N+1}),M(\zeta_{-N+2}),\dots,M(\zeta_N)\big)\Fb,
\end{equation}
where $\Fb$ is the unitary $2mN\times 2mN$ matrix
\begin{equation}
\label{Fmatrix}
\Fb= \frac{1}{\sqrt{2N}}\left[\begin{array}{cccc}\zeta_{-N+1}^{N-1}I_m & \zeta_{-N+1}^{N-2}I_m & \cdots & \zeta_{-N+1}^{-N} I_m\\ \vdots & \vdots & \cdots & \vdots \\\zeta_{0}^{N-1}I_m & \zeta_{0}^{N-2}I_m & \cdots & \zeta_{0}^{-N} I_m \\ \vdots & \vdots & \cdots & \vdots \\\zeta_{N}^{N-1}I_m & \zeta_{N}^{N-2} I_m& \cdots & \zeta_{N}^{-N}I_m \end{array}\right] .
\end{equation}
Moreover, if $M(\zeta_k)$ is positive definite for all $k$, then
\begin{equation}
\label{matrixlogM}
\log\Mb=\Fb^*\text{\rm diag}\big(\log M(\zeta_{-N+1}),\log M(\zeta_{-N+2}),\dots,\log M(\zeta_N)\big)\Fb,
\end{equation}
where $\text{\rm diag}$ stands for block diagonal.
The proof of Lemma <ref> will be omitted, as it follows the same lines as that of Lemma <ref> with straight-forward modification to the multivariate case.
Clearly the inverse
\begin{equation}
\label{matrixMdiaginv}
\Mb^{-1}=\Fb^*\text{\rm diag}\big(M(\zeta_{-N+1})^{-1},M(\zeta_{-N+2})^{-1},\dots,M(\zeta_N)^{-1}\big)\Fb
\end{equation}
is also block-circulant, and
\begin{equation}
\label{matrixSbrepr}
\Sb =\Fb^*\text{\rm diag}\big(\zeta_{-N+1}I_m,\zeta_{-N+2}I_m,\dots,\zeta_N I_m\big)\Fb.
\end{equation}
However, unlike the scalar case block-circulant matrices do not commute in general.
Given Lemma <ref>, we are now in a position to reformulate Theorems <ref> and <ref> in matrix from.
Let $C\in\mathfrak{C}_+^{(m,n)}(N)$, and let $\Cb$ be the corresponding block-circulant matrix (<ref>) and
(<ref>) the corresponding block-Toeplitz matrix. Then, for each positive-definite banded $2mN\times 2mN$ block-circulant matrices
\begin{equation}
\label{matrixPbmatrix}
\Pb =\sum_{k=-n}^n S^{-k}\otimes p_k I_m, \quad p_{-k}=\bar{p}_k
\end{equation}
of order $n$, where $P(\zeta)=\sum_{k=-n}^n p_k \zeta^{-k}\in\mathfrak{P}_+^{(1,n)}(N)$, there is a unique sequence $Q=(Q_0,Q_1,\dots,Q_n)$ of $m\times m$ matrices defining a positive-definite banded $2mN\times 2mN$ block-circulant matrix
\begin{equation}
\label{matrixQbmatrix}
\Qb =\sum_{k=-n}^n S^{-k}\otimes Q_k, \quad Q_{-k}=Q_k^*
\end{equation}
of order $n$ such that
\begin{equation}
\label{mSigmab=QbinvPb}
\Sigmab=\Qb^{-1}\Pb
\end{equation}
is a block-circulant extension (<ref>) of $\Cb$. The block-circulant matrix (<ref>) is the unique maximizer of the function
\begin{equation}
\label{mprimal_matrix}
\mathcal{I}_{\Pb}(\Sigmab) = \trace(\Pb\log\Sigmab)
\end{equation}
subject to
\begin{equation}
\label{mmomentcondmatrixb}
\Eb_n\Tr\Sigmab \Eb_n =\Tb_n, \quad \text{where\;} \Eb_n =\begin{bmatrix}\Ib_{mn}\\{\bold 0}\end{bmatrix}.
\end{equation}
Moreover, $\Qb$ is the unique optimal solution of the problem to minimize
\begin{equation}
\label{dual_matrix2}
\mathcal{J}_{\Pb}(\Qb)= \trace(\Cb\Qb) - \trace(\Pb\log\Qb)
\end{equation}
over all positive-definite banded $2mN\times 2mN$ block-circulant matrices (<ref>) of order $n$. The functional $\mathcal{J}_{\Pb}$ is strictly convex.
For $\Pb=\Ib$ we obtain the maximum-entropy solution considered in <cit.>, where the primal problem to maximize $\mathcal{I}_{\Ib}$ subject to (<ref>) was presented. In <cit.> there was also an extra constraint (<ref>), which, as we can see, is not needed, since it is automatically fulfilled. For this reason the dual problem presented in <cit.> is more complicated than merely minimizing $\mathcal{J}_{\Ib}$.
Next suppose we are also given the (scalar) logarithmic moments (<ref>) and that $C\in\mathfrak{C}_+^{(m,n)}(N)$. Then, if the problem to maximize $\text{\rm tr}\{\log\Sigmab\}$ subject to (<ref>) and (<ref>) over all positive-definite block-circulant matrices (<ref>) has a solution, then it has the form
\begin{equation}
\label{PbQb12Sigmab}
\Sigmab=\Qb^{-1}\Pb
\end{equation}
where the $(\Pb,\Qb)$ is a solution of the dual problem to minimize
\begin{equation}
\label{Jmatrix}
\mathbf{J}(\Pb,\Qb) =\text{\rm tr}\{\Cb\Qb\} -\text{\rm tr}\{\Gammab\Pb\} + \text{\rm tr}\{\Pb\log\Pb\Qb^{-1}\},
\end{equation}
over all positive-definite block-circulant matrices of the type (<ref>) and (<ref>) with the extra constrain $p_0=1$, where $\Gammab$ is the Hermitian circulant matrix with symbol
\begin{equation}
\label{M(z)}
\Gamma(\zeta)=\sum_{k=-n}^n \gamma_k \zeta^{-k}, \quad \gamma_{-k}=\bar{\gamma}_k.
\end{equation}
However, the minimum of (<ref>) may end up on the boundary, in which case the constraint (<ref>) may fail to be satisfied. Therefore, as in the scalar case, we need to regularize the problem by instead minimizing
\begin{equation}
\label{magtrixJlambda}
\mathbf{J}_\lambda(\Pb,\Qb) =\text{\rm tr}\{\Cb\Qb\} -\text{\rm tr}\{\Gammab\Pb\} + \text{\rm tr}\{\Pb\log\Pb\Qb^{-1}\} - \lambda\,\text{\rm tr}\{\log\Pb\}.
\end{equation}
This problem has a unique optimal solution (<ref>) satisfying (<ref>), but not (<ref>). The appropriate logarithmic moment constraint is obtained as in the scalar case by exchanging $\gamma_k$ for $\gamma_k+\varepsilon_k$ for each $k=1,2,\dots,n$, where $\varepsilon_k$ is given by (<ref>).
Again each solution leads to an ARMA model
\begin{equation}
\label{matrixARMA}
\sum_{k=-n}^n Q_k y(t-k) = \sum_{k=-n}^n p_k e(t-k),
\end{equation}
where $\{e(t)\}$ is the conjugate process of $\{y(t)\}$, $Q_0,Q_1,\dots,Q_n$ are $m\times m$ matrices, whereas $p_0,p_1,\dots,p_n$ are scalar with $p_0=1$.
We illustrate this theory with a simple example from <cit.>, where a covariance sequence $C:=(C_0,C_1,\dots C_n)$ and a cepstral sequence $\gammab:=(\gamma_1,\gamma_2,\dots,\gamma_n)$ have been computed from a two-dimensional ARMA process with a spectral density $\Phi:=PQ^{-1}$, where $P$ is a scalar pseudo-polynomial of degree three and $Q$ is a $2\times 2$ matrix-valued pseudo-polynomial of degree $n=6$.
Its zero and poles are illustrated in Fig. <ref>.
Poles and zeros of an ARMA $2\times 2$ model of order $n=6$.
The norm of the approximation error for a bilateral AR of order $12$ for $N=64$ and a bilateral ARMA of order $6$ for $N=32$
Given $C$ and $\gammab$, we apply the procedure in this section to determine a pair $(\Pb,\Qb)$ of order $n=6$.
For comparison we also compute an bilateral AR approximation with $n=12$ fixing $\Pb=\Ib$. As illustrated in Fig. <ref>, the bilateral ARMA model of order $n=6$ computed with $N=32$ outperforms the bilateral AR model of order $n=12$ with $N=64$.
The results of Section <ref> can also be generalized to the multivariate case along the lines described in <cit.>.
§ APPLICATION TO IMAGE PROCESSING
In <cit.> the circulant maximum-entropy solution has been used to model spatially stationary images (textures) <cit.> in terms of (vector-valued) stationary periodic processes. The image could be thought of as an $m\times M$ matrix of pixels where the columns form a $m$-dimensional reciprocal process $\{y(t)\}$, which can extended to
a periodic process with period $M>N$ outside the interval $[0,N]$; see Figure <ref>.
An image modeled as a reciprocal vector process
This imposes the constraint $C_{M-k}=C_k\Tr$ on the covariance lags $C_k:=E\{ y(t+k)y(t)\Tr\}$,
leading to a circulant Toeplitz matrix. The problem considered in <cit.> is to model the process $\{y(t)\}$ given (estimated) $C_0,C_1,\dots,C_n$, where $n<N$ with an efficient low-dimensional model. This is precisely a problem of the type considered in Section <ref>.
Solving the corresponding circulant maximum-entropy problem (with $\Pb=\Ib$), $n=1$, $m=125$ and $N=88$, Carli, Ferrante, Pavon and Picci <cit.> derived a bilateral model of the images at the bottom row of Figure <ref> to compress the images in the top row, thereby achieving a compression of 5:1.
Three images modeled by reciprocal processes (original at bottom).
While the compression ratio falls short of competing with current jpeg standards (typically 10:1 for such quality), our approach suggests a new stochastic alternative to image encoding. Indeed the results in Figure <ref> apply just the maximum entropy solution of order $n=1$. Simulations such as those in Figure <ref> suggest that much better compression can be made using bilateral ARMA modeling.
An alternative approach to image compression using multidimensional covariance extension can be found in the recent paper <cit.>.
Kalman Kalman, R. E.: Realization of covariance sequences, Proc. Toeplitz Memorial Conference, Tel Aviv, Israel, 1981.
T.T. Georgiou: Partial Realization of Covariance Sequences, Ph.D. thesis, CMST, University of Florida, Gainesville 1983.
Georgiou Georgiou, T.T.: Realization of power spectra from partial covariances, IEEE Trans. on Acoustics, Speech and Signal Processing 35, 438-449 (1987)
BLGM1 Byrnes, C.I., Lindquist, A, Gusev, S.V., Matveev, A.V.: A complete parameterization of all positive rational extensions of a covariance sequence, IEEE Trans. Aut. Contr. 40, 1841–1857 (1995)
Byrnes, C. I., Lindquist, A.: On the partial stochastic realization problem, IEEE Trans. Automatic Control 42, 1049–1069 (1997)
BGuL Byrnes, C. I., Gusev, S. V. , Lindquist, A.: A convex optimization approach to the rational covariance extension problem, SIAM J. Contr. Opt. 37, 211–229 (1999)
SIGEST Byrnes, C. I., Gusev, S. V. , Lindquist, A.:
From finite covariance windows to modeling filters: A convex optimization
approach, SIAM Review 43, 645–675 (2001)
Byrnes, C. I.,, Georgiou, T.T., Lindquist, A.: A generalized entropy criterion for Nevanlinna-Pick interpolation with degree constraint, IEEE Trans. on Automatic Control 45, 822-839 (2001)
Byrnes, C. I., Lindquist, A.: The moment problem for rational measures: convexity in the spirit of Krein, In: Modern Analysis and Application: Mark Krein Centenary Conference, Vol. I: Operator Theory and Related Topics, Book Series: Operator Theory Advances and Applications Volume 190, Birkhäuser, 157 – 169 (2009)
Dunford, N., Schwartz, J.T.: Linear Operators, Part I: General Theory, John Wiley & Sons, New York, 1958
LPbook Lindquist, A. Picci, G.: Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification, Springer, Heidelberg, New York, Dordrecht and London, 2015.
LPcirculant Lindquist, A., Picci, G.: The Circulant Rational Covariance Extension Problem: The Complete Solution, IEEE Trans. Automatic Control 58, 2848–2861 (2013)
LMPcirculantMult Lindquist, A. Masiero, C., Picci, G.:
On the multivariate circulant rational covariance extension problem, Proc. 52st IEEE Conf. Decision and Control (2013)
Ringh, A. and Lindquist, A.:
Spectral estimation of periodic and skew periodic random signals and approximation of spectral densities, in 33rd
Chinese Control Conference (CCC), 5322–5327 (2014)
Ringh, A. and Karlsson, J.:
A fast solver for the circulant rational covariance extension problem, in European Control Conference (ECC), July
2015, 727–733 (2015)
Jamison, B.: Reciprocal Processes, Zeitschrift. Wahrsch. Verw. Gebiete 30, 65–86 (1974)
Krener, A.J.: Reciprocal Processes and the stochastic realization problem for acausal systems, In: Byrnes, C. I., Lindquist, A. (eds.): Modeling Identification and Robust Control, North-Holland, 197–211 (1986)
Krener, A. J., Frezza, R., Levy, B. C.: Gaussian reciprocal processes and self-adjoint differential equations of second order, Stochastics and Stochastics Reports 34, 29-56 (1991)
Levy, B. C., Ferrante, A.: Characterization of stationary discrete-time Gaussian reciprocal processes over a finite interval, SIAM J. Matrix Anal. Appl.24, 334-355 (2002)
Levy, B. C., Frezza, R., Krener, A.J.: Modeling and Estimation of discrete-time Gaussian Reciprocal Processes, IEEE Trans. Autom. Contr. 35, 1013–1023 (1990)
Carli, F. P., Ferrante, A., Pavon, M. Picci, G.: A Maximum Entropy Solution of the Covariance Extension Problem for Reciprocal Processes, IEEE Trans. Automatic Control 56, 1999-2012 (2011)
Chiuso, A., Ferrante, A., Picci, G.: Reciprocal realization and modeling of textured images, Proc. of the 44rd IEEE Conference on Decision and Control (2005)
Davis, P.,: Circulant Matrices, John Wiley & Sons (1979)
Masani, P.: The prediction theory of multivariate stochastic processes, III, Acta Mathematica 104, 141-162 (1960)
Musicus, B. R., Kabel, A. M.:
Maximum entropy pole-zero estimation, Technical Report 510, MIT Research Lab. Electronics, Aug. 1985; now available on the internet at http://dspace.mit.edu/bitstream/handle/1721.1/4233/ RLE-TR-510-17684936.pdf.
Byrnes, C.I., Enqvist, P., Lindquist, A.:
Cepstral coefficients, covariance lags and pole-zero models for finite datastrings,
IEEE Trans. on Signal Processing 50, 677-693 (2001)
Byrnes, C.I., Enqvist, P., Lindquist, A.:
Identifiability and well-posedness of shaping-filter parameterizations: A global analysis approach,
SIAM J. Control and Optimization 41, 23-59 (2002)
Enqvist, P.:
A convex optimization approach to ARMA(n,m) model design from covariance and cepstrum data,
SIAM Journal on Control and Optimization43(3), 1011-1036 (2004)
Enqvist, P.:
Spectral estimation by Geometric, Topological and Optimization Methods,
PhD thesis, Optimization and Systems Theory, KTH, Stockholm,
Sweden, 2001.
Blomqvist, A., Lindquist, A., Nagamune, R.:
Matrix-valued Nevanlinna-Pick interpolation with complexity constraint: An optimization approach, IEEE Trans. Automatic Control 48, 2172-2190 (2003)
Byrnes, C.I., Lindquist, A.:
The generalized moment problem with complexity constraint, Integral Equations and Operator Theory 56, 163-180 (2006)
Georgiou, T.T.: Solution of the general moment problem via a one-parameter imbedding, IEEE Trans. Aut. Control 50, 811-826 (2005)
Georgiou, T. T., Lindquist, A.: Kullback-Leibler approximation of spectral density functions, IEEE Trans. Information Theory 49, 2910-2917 (2003)
Pavon, M., Ferrante, A.: On the Geometry of Maximum Entropy Problems, SIAM review 55(3), 415-439 (2013)
Dempster, A. P.: Covariance selection, Biometrics 28(1), 157-175 (1972)
Krein, M. G. and Nudelman,A. A.:
The Markov Moment Problem and Extremal Problems, American Mathematical
Society, Providence, Rhode Island (1977)
Picci, G., Carli, F.: Modelling and simulation of images by reciprocal processes, Proc. Tenth International Conference on Computer Modeling and Simulation UKSIM 2008, 513-518.
Soatto, S., Doretto, G., Wu, Y.: Dynamic Textures, Proc. Intern. Conf. Computer Vision (July 2001), 439-446.
Carli, F. and Picci, G.:
A finite factorization approach to estimation of periodic processes, to appear.
Ringh A., Karlsson, J. and Lindquist, A:
Multidimensional rational covariance extension with applications to spectral estimation and image compression, submitted for publication.
|
1511.00804
|
The hydrodynamic escape problem (HEP), which is characterized by a free boundary value problem of Euler equation with gravity and heat, is crucial for investigating the evolution of planetary atmospheres. In this paper, the global existence of transonic solutions to the HEP is established using the generalized Glimm method. The new version of Riemann and boundary-Riemann solvers, are provided as building blocks of the generalized Glimm method by inventing the contraction matrices for the homogeneous Riemann (or boundary-Riemann) solutions. The extended Glimm-Goodman wave interaction estimates are investigated for obtaining a stable scheme and positive gas velocity, which matches the physical observation. The limit of approximation solutions serves as an entropy solution of bounded variations. Moreover, the range of the hydrodynamical region is also obtained.
MSC: 35L50, 35L60, 35L65, 35L67, 76N10, 85A20, 85A30
Keywords: hydrodynamic escape problem; nonlinear hyperbolic systems of balance laws; generalized Riemann and boundary-Riemann problems; generalized Glimm scheme; hydrodynamic region.
§ INTRODUCTION
Spacecraft exploration of the planets in our solar system and the discovery of exoplanets has attracted considerable attention in the atmospheric escape from planetary objects <cit.>. The Cassini spacecraft currently improves our understanding of the atmospheric escape from Titan <cit.>. The Maven Mission circuits around Mars for studying its atmospheric composition <cit.>. In July 2015, the New Horizons (NH) spacecraft completed its flyby of Pluto and discovered flowing ice and an extended haze on the planet. Pluto already exhibits a planetary geology that comprises flowing ices, exotic surface chemistry, mountain ranges, and vast haze. Analyzing Pluto's atmosphere reveals that Pluto's surface has a reddish hue, a simple hydrocarbon in its atmosphere, and the temperature for hazes to form at altitudes higher than 30 kilometers above Pluto's surface.
The hydrodynamic escape problem (HEP) is crucial for investigating of the evolution of planetary atmospheres. The HEP for a single-constituent atmosphere is governed by the following Euler equations with gravity and heat:
\begin{equation}
\label{3dHEP}
\left\{
\begin{split}
&\partial_t\r+\nabla\cdot(\r\tu)=0, \\
&\partial_t(\r\tu)+\nabla\cdot((\r\tu)\otimes\tu)+\nabla P=-\frac{GM_p\r}{|\br|^3}\br, \\
\end{split}\right.
\end{equation}
where $\br$ is the position vector from the center of the planet to the particle of the gas; $\r,\ \tu,\ P$, and $E$ represent the density, velocity, pressure, and total energy of the gas respectively; and $G,\ M_p,\ q=q(\br)$ are the gravitational constant, mass of the planet, and heating, respectively.
In this paper, we are concerned with the three-dimensional inviscid hydrodynamic equations without thermal conduction in spherical symmetric space-time models, that is, we considered (<ref>) to be of the following form
\begin{equation}
\label{HEP1}
\left\{
\begin{split}
&\partial_t\big(\rho x^2\big)+\partial_x\big(\rho u x^2\big)=0, \\
&\partial_t\big(\rho u x^2\big)+\partial_x\big(\rho u^2 x^2+P x^2\big)=-GM_p\rho+2Px, \quad 0<x_B<x<x_T, \\
& \partial_t\big(E x^2\big)+\partial_x\big((E+P)u x^2\big)=-GM_p\rho u+qx^2.
\end{split}\right.
\end{equation}
Here, $x$ denotes the distance from the center of the planet, $x_B$ and $x_T$ are the altitudes of the inner and outer boundaries of the planetary atmosphere, respectively. Typically, $x_B$ and $x_T$ are the altitudes of the upper thermosphere and exobase. The total energy $E$ is the sum of the kinetic energy and the internal energy of the gas flow,
\begin{equation}
\label{totalE}
E=\frac{1}{2}\rho u^2+\rho e=\frac{1}{2}\rho u^2+\frac{P}{\gamma-1},
\end{equation}
where $\gamma$ is the adiabatic constant with $1<\g<5/3$.
The steady transonic solutions of (<ref>) are crucial because of an almost hydrodynamic equilibrium state near the bottom boundary. The hybrid fluid/kinetic model <cit.> seems to be realistic approach, which comprises the hydrodynamic escape result <cit.> and a drifting Maxwell-Boltzmann distribution function that includes the bulk velocity $u$ in the velocity distribution <cit.>. Tian and Toon <cit.> implemented a numerical simulation using a time-dependent hyperbolic system. A time-independent model experiences singularity at the sonic points <cit.>. For the relaxation methods in <cit.> for free conduction, the achieved numerical solutions depend on close guess of initial data with positive velocity. The first theoretical analysis for a steady HEP was reported in <cit.>. Using the geometric singular perturbation method, the authors constructed smooth transonic stationary solutions issuing from subsonic states to supersonic states and various types of discontinuous stationary solutions for (<ref>). For the time-evolutionary case, the global existence results are yet to be established. In this paper, the global existence of time-evolutionary transonic solutions to the HEP in the hydrodynamic region $\Sg\equiv\{(x,t):x_B\le x\le x_T,\ t\in[0,\infty)\}$ is established. The gravity and heat affecting intensity can be distinguished during the wave interaction, leading us to the effective development of the numerical simulation.
We define the notations as follows:
\begin{equation}
\label{setting1}
m:=\rho u,\quad U:=(\rho,\; m,\; E)^T.
\end{equation}
Using (<ref>), we can rewrite (<ref>) in a compact form
\begin{equation}
\label{3x3system}
\end{equation}
where $h(x)=-2/x$ and
\begin{equation}
\label{setting}
\begin{split}
f(U)&=\Big(m,\ \frac{3-\g}{2}\frac{m^2}{\r}+(\g-1)E,\ \frac{m}{\r}\Big(\g E-\frac{\g-1}{2}\frac{m^2}{\r}\Big)\Big)^T, \\
g(x,U)&=\Big(m,\ \frac{m^2}{\r}+\frac{GM_p}{2x}\r,\ \frac{m}{\r}\Big(\g E-\frac{\g-1}{2}\frac{m^2}{\r}\Big)+\frac{GM_p}{2x}m-\frac{xq}{2}\Big)^T.
\end{split}
\end{equation}
The complete model of the HEP is given by the following free boundary value problem:
\begin{eqnarray}%---------------------{1.4}----------------------%
\label{FBVP}
\text{(HEP)}&&\left\{
\setlength\arraycolsep{0.1em}
\begin{split}
&U_t+f(U)_x=h(x)g(x,U),\quad (x,t)\in\Sg\equiv[x_B,x_T]\times[0,\infty), \\
&U(x,0)=U_0(x)\in\Omega,\ x\in[x_B,x_T], \\
&\r(x_B,t)=\r_B(t),\ m(x_B,t)=m_B(t),\quad t>0, \\
&\r\Big|_{\Sg},\ \frac{m}{\r}\Big|_{\Sg}>0,\ \mathfrak{Kn}(U)\Big|_{\Sg}\le 1,
\end{split}\right.
\end{eqnarray}
where the exobase of the atmosphere $x=x_T$ (as well as $\Sg$) must be determined and $\mathfrak{Kn}(U)$ denotes the Knudsen number of $U$. Physically, the region $\Sg$ is called the hydrodynamic region of (<ref>). The position of the inner boundary $x_B$ may be probed through astronomical observation. However, determining the outer boundary $x_T$ is usually difficult due to the transition of the kinetic and hydrodynamical regions. Determining the position of the outer boundary $x=x_T$ and solving (<ref>) in $\Sg$ simultaneously is basically a free boundary problem, which makes it difficult to establish the global existence result. To overcome this difficulty, we first propose the following associated initial-boundary value problem (IBVP) without vacuum in $\Pi\equiv[x_B,\infty)\times[0,\infty)$:
\begin{eqnarray}%---------------------{1.4}----------------------%
\label{IBVP}
\text{(IBVP)}&&\left\{
\setlength\arraycolsep{0.1em}
\begin{split}
&U_t+f(U)_x=h(x)g(x,U),\quad (x,t)\in\Pi\equiv[x_B,\infty)\times[0,\infty), \\
&U(x,0)=U_0(x)\in\Omega, \\
&\r(x_B,t)=\r_B(t),\ m(x_B,t)=m_B(t),\quad t>0,
\end{split}\right.
\end{eqnarray}
where $U_0(x)=(\rho_0(x),m_0(x),E_0(x))^T$ and $\Omega$ is an open domain centered at some sonic state
\begin{equation*}
U_s\equiv(\rho_s,m_s,E_s)\in \CMcal{T}:=\Big\{(\rho ,m,E)\Big|\; m=\r\sqrt{\g(\g-1)\Big(\frac{E}{\r}-\frac{u^2}{2}\Big)}\Big\}.
\end{equation*}
We call the set $\CMcal{T}$ the transition surface or the sonic states. The vacuum case is excluded from this formula because the atmospheric gas does not act as fluid when the density tends to zero. Whether the Glimm method can be applied to the vacuum case for the general system has remained unsolved for decades. In this paper, a new version of the Glimm method is used for establishing the existence of global entropy solutions of (<ref>) under the following conditions:
($A_1$) $\rho_0(x)$, $m_0(x),$ $E_0(x)$, $\r_B(t)$ and $m_B(t)$ are bounded positive functions with small total variations, and there exists $\vr>0$ sufficiently small such that $\r_0(x)\ge\vr$ and $\r_B(t)\ge\vr$ for $(x,t)\in\Pi$;
($A_2$) $\min\limits_{t\ge 0}\{m_B(t)\}>(1+\e)\TV\{U_0(x)\}+(1+\e+\e^2)^2\CMcal{C}$ for $0<\e<\frac{1}{2}$ and some positive constant $\mathcal{C}$;
($A_3$) $q(x)\in W^{1,1}[x_B,\infty)$.
Under the condition ${\rm(}A_1{\rm)}\sim{\rm(}A_3{\rm)}$, (<ref>) consists of global entropy solutions with positive velocity in $\Pi$ (Main Theorem I). In addition, under a certain constraint of transonic initial data, in the complement of $\Sg$, denoted as $\Sg^c$, the wave speeds of Glimm's approximate solutions $\{U_{\D x}\}$ to (<ref>) are positive, that is, the entropy flow $U$ to which $\{U_{\D x}\}$ converges is supersonic in $\Sg^c$ so that the waves of $U$ in $\Sg^c$ do not move into $\Sg$ to interact with the waves in $\Sg$. Moreover, we prove that the Knudsen number $\mathfrak{Kn}(U)$ of $U$ in $\Sg$ satisfies
\mathfrak{Kn}(U)\le 1,
which implies that $\Sg$ fulfills the physical meaning of the hydrodynamic region <cit.>. Using this strategy, we can prove that the solution $U\big|_{\Sg}$ for (<ref>) is indeed the entropy solution of (<ref>) (Main Theorem II).
Let us review some previous results related to this topic and clarify the motivation of the study. When $g\equiv 0$, the system (<ref>) is reduced to the strictly hyperbolic system,
\begin{equation}%---------------------{1.6}------------------------%
\label{homo}
\end{equation}
The entropy solution to the Riemann problem was first constructed by Lax <cit.>. In particular, the solution is self-similar and consists of constant states separated by elementary waves: rarefaction waves, shocks, and contact discontinuities. Furthermore, the global existence of weak solutions to the Cauchy problem was established by Glimm <cit.>, who considered Lax's solutions as the build blocks of the scheme. For the inhomogeneous hyperbolic systems,
\begin{eqnarray}%-----------------------(1.5.1)---------------------%
\label{balance}
U_t +f(x,U)_x = g(x,U).
\end{eqnarray}
The Cauchy problem was first studied by Liu <cit.>. For the Cauchy problem of the general quasi-linear, strictly hyperbolic system
\begin{eqnarray}%-----------------------(1.5.3)---------------------%
\label{qlinear}
U_t +f(x,t,U)_x = g(x,t,U).
\end{eqnarray}
The existence of entropy solutions was first established by Dafermos and Hsiao <cit.>. In <cit.>, system (<ref>) was studied under dissipative conditions by using the asymptotic expansion of the classical Riemann solutions. The aforementioned conditions contribute considerably in investigating the systems (<ref>) and (<ref>). In the aforementioned studies, the source term was used for generating an extra stationary characteristic field in the Riemann problem. The time-independent wave curves generated by this filed are tangential to the classical 1-wave curves at sonic states, resulting in the nonuniqueness of solutions of a Riemann problem. The total variations of solutions may blow up in a finite time and the blow up phenomenon can be eliminated using a further dissipative assumption.
When $f$ and $g$ are independent of $x$, Bianchini-Bressan <cit.> studied the existence result by using regularization method. Luskin-Temple <cit.> and the authors in <cit.> establish the existence result by combining Glimm's scheme with the method of fractional steps. Base on these studies, we can consider the effect of the source in (<ref>) as the perturbations of the solutions to the homogeneous conservation laws. The appearance of the source terms in (<ref>) breaks up the self-similarity of Riemann waves. But the effect is only up to $O(1)\D t$, that is, the effect of the source terms on self-similar waves is of the order $\D t$ in each Riemann cell. More precisely,
\begin{equation}
\label{1.12t}
(df(U)-\xi I_3)\dot{U}=(t-t_0)h(x)g(x,U)\approx(\D t)h(x)g(x,U).
\end{equation}
Therefore, we can construct the approximate solution for our generalized Riemann problem as $U=\widetilde{U}+\widebar{U}$ in each Riemann cell, where $\widetilde{U}$ solves (<ref>) and $\widebar{U}$ solves the linearized system of (<ref>) around $\widetilde{U}$. We have
for some contraction matrix $S(x,t,\widetilde{U})$ depending on $\widetilde{U}$ (as well as $f,\ g$). This construction of the generalized Riemann solver is in contrast to the fractional step scheme <cit.> and other operator splitting methods <cit.>. For instance, in <cit.>, the effect of the source on the solutions of the classical Riemann problem is decoupled. In our case, the effect of the source on the solutions of the classical Riemann problem is strongly coupled. The estimates of wave interaction are more complicated than in <cit.>.
For the stability of the generalized Glimm scheme, contrary to the methods used in <cit.>, in which the positivity of the gas velocity is assumed, we can demonstrate, through the structure of the generalized Glimm's approximate solution, that the escape velocity of the gas is globally positive, which matches the astronomical observation. Therefore, the uniform bounds of the total variations of the approximate solutions in (<ref>) can be achieved by showing that
(1) the Glimm functionals of $\widetilde{U}$ are nonincreasing in time and
(2) the perturbations have a uniform bound of the total variations in each time step.
In addition, we prove positivity of the gas velocity through a rigorous mathematical proof. Based on the contraction matrix $S$, we can achieve a more accurate formula of wave interaction estimates that lead to the decay result of the Glimm functionals, and a new relation between the velocity and the Glimm functionals in each time step. Consequently, the stability of the generalized Glimm scheme and the global positivity of the gas velocity is obtained, as shown in Section 3. Based on ${\rm(}A_1{\rm)}\sim{\rm(}A_3{\rm)}$ and the estimation of interaction, the existence of global entropy solutions in transonic gas flow without any dissipative condition is established.
We now introduce the definitions of weak solutions and entropy solutions for (<ref>), and state the main theorems.
Consider the initial-boundary value problem in (<ref>). A measurable function $U(x,t)$ is a weak solution of (<ref>) if
\begin{equation*}
\iint_{x>x_B, t>0}\left\{U\phi_t+f(U)\phi_x+h(x)g(x,U)\phi\right\}dxdt +\int^{\infty}_{x_B} U_0(x)\phi(x,0)
\end{equation*}
for any test function ${\phi} \in C^1_0(\Pi)$.
Let $\Omega$ be a convex subset of $\mathbb{R}^3$. A pair $(\eta(U),\om(U))$ is an entropy pair of (<ref>) if $\eta$ is convex on $\Omega$ and
\begin{eqnarray*}%-------------------------(1.8)-------------------------%
d\om=d\eta df\quad\text{on}\quad\Omega.
\end{eqnarray*}
Furthermore, a measurable function $U$ is an entropy solution of (<ref>) if $U$ is a weak solution of (<ref>) and satisfies
\begin{equation}%-------------------------(1.9)-------------------------%
\label{ibvpentropy}
\iint_{x>x_B,t>0}\left\{\eta\phi_t+\om\phi_x +d\eta\cdot hg\phi\right\}dxdt+\int^\infty_{x_B}\eta(U_0(x))\phi(x,0)dx
+\int^{\infty}_{0}\om(U(x_B,t))\phi(x_B,t)dt\geq 0,
\end{equation}
for every entropy pair $(\eta(U),\om(U))$ and any positive test function $\phi\in C^1_0(\Pi)$.
Main Theorem I.
Consider the initial-boundary value problem (<ref>) with transonic initial data $U_0=(\r_0,m_0,E_0)^T$. Assume that the inner boundary data $(\r_B(t),m_B(t))$ satisfies the condition ${\rm(}A_1{\rm)}\sim{\rm(}A_3{\rm)}$ and the heat $q$ satisfies the condition ($A_3$). Let $\{U_{\theta,\Delta x}\}$ be the sequence of approximate solutions of (<ref>) by using the generalized Glimm scheme. Then, there exist a null set $N \subset \Phi$ and a subsequence $\{\Delta x_{i}\}\rightarrow 0$ such that if $ \theta \in \Phi\setminus N$, then
U(x,t):=\lim_{\scriptstyle \Delta x_i\rightarrow 0} U_{\theta,\Delta x_i}(x,t)
is the positive entropy solution of (<ref>). In particular, the gas velocity is positive in $\Pi$.
Main Theorem II.
Assume that the transonic initial data $U_0=(\r_0,m_0,E_0)^T$ such that $\r_0,E_0$ is decreasing and $m_0$ is increasing and $u_0(x_B)<c_0(x_B)$, where $u_0,c_0$ as defined in (<ref>). There exists $x_T>x_B$ depending on the initial and boundary data such that $\Sg\equiv [x_B,x_T]\times[0,\infty)$ is the hydrodynamic region of (<ref>), which means $U(x,t)|_{\Sg}$ is the global entropy solution of (<ref>) satisfying $\mathfrak{Kn}(U)\big|_{\Sg}\le 1$.
This paper is organized as follows. Section 2 presents the generalized solvers for Riemann and boundary-Riemann problems based on the construction of the approximate solutions to these problems through the asymptotic expansion and operator splitting techniques. The residuals of the solutions in each grid are calculated to preserve the consistency of the proposed scheme. Section 3 presents a generalized version of the Glimm scheme. Moreover, the generalized wave interaction estimate, nonincreasing Glimm functional, and estimate for the total variation of the perturbations in each time strip are obtained. The global existence of the entropy solutions for (<ref>) is proved. The hydrodynamic region is determined in the final section.
§ GENERALIZED SOLUTIONS FOR THE RIEMANN AND BOUNDARY-RIEMANN PROBLEMS
In this section, we introduce a new method of constructing the approximate solutions to the Riemann and boundary-Riemann problems of (<ref>), which are the building blocks of the generalized Glimm scheme of the HEP. The residuals of the approximate solutions will be estimated for maintaining the consistency of the generalized Glimm scheme. Let us select the spatial resolution $\Delta x>0$ and the temporal step $\Delta t>0$ sufficiently small, which satisfies the Courant-Friedrichs-Lewy (CFL) condition
\begin{equation}
\label{CFL}
\l_*:=\frac{\Delta x}{\Delta t} > \sup\limits_{(\rho,m,E)\in\Omega}\left\{\frac{m}{\rho}+\sqrt{\g(\g-1)\Big(\frac{E}{\r}-\frac{m^2}{2\r^2}\Big)}\right\}.
\end{equation}
We define the inner region at the location $x_0$ and time $t_0$
\begin{equation}
\label{ingrid}
D(x_0,t_0):=\{(x,t)\mid|x-x_0|<\Delta x, \;t_0<t< t_0+\Delta t\},
\end{equation}
and the boundary region at the lower boundary $x_B$ and time $t_0$
\begin{equation}
\label{bdgrid}
D(x_B,t_0):=\{(x,t)\mid x_B<x<x_B+\Delta x, \; t_0<t< t_0+\Delta t\}.
\end{equation}
The Riemann problem of (<ref>) in $D(x_0,t_0)$, denoted by $\CMcal{R}_G(x_0,t_0; g)$, is given by
\begin{equation}
\label{RP}
\CMcal{R}_G(x_0,t_0;g) :\qquad
\left\{\begin{array}{l}
U_t+f(U)_x=h(x)g(x,U),\quad (x,t)\in D(x_0,t_0), \\
U_L, &\text{if }x_0-\D x<x<x_0, \\
U_R, &\text{if }x_0<x<x_0+\D x,
\end{array}\right.
\end{array}\right.
\end{equation}
and the boundary-Riemann problem of (<ref>) in $D(x_B,t_0)$, denoted by
$\CMcal{BR}_G(x_B,t_0; g)$ can be expressed as
\begin{equation}
\label{BRP}
\CMcal{BR}_G(x_B,t_0; g):\qquad
\left\{\begin{array}{ll}
U_t+f(U)_x=h(x)g(x,U), &(x,t)\in D_B(x_B,t_0), \\
U(x,t_0)=U_R, &x_B\le x\le x_B+\D x, \\
\r(x_B,t)=\r_B,\ m(x_B,t)=m_B, &t_0<t<t_0+\D t,
\end{array}\right.
\end{equation}
where $m,\ U,\ f$, and $g$ are defined in (<ref>) and (<ref>); $U_L=(\rho_L, m_L,E_L)$ and $U_R=(\rho_R, m_R,E_R)$ are the left and right constant states; and $\r_B>0$ and $m_B>0$ are the density and momentum at the boundary $x_B$, respectively. By setting the source term $g \equiv 0$ in (<ref>) and (<ref>), the corresponding classical Riemann and boundary-Riemann problems are denoted by $\CMcal{R}_C(x_0,t_0)=\CMcal{R}_G(x_0,t_0;0)$ and $\CMcal{BR}_C(x_B,t_0)=\CMcal{BR}_G(x_B,t_0;0)$.
§.§ Construction of approximate solutions to the Riemann and boundary-Riemann problems
The system (<ref>) is a strictly hyperbolic system whose Jacobian matrix $df$ has three distinct real eigenvalues:
\lambda_1(U):= u-c(U),\quad \l_2(U):=u,\quad\text{and}\quad\lambda_3(U):=u+c(U),
\begin{equation}
\label{2.5.5}
\end{equation}
The corresponding right eigenvectors of $df$ are
R_1(U)=(-1,c-u,uc-H)^T,\quad R_2(U)=(1,u,\tfrac{1}{2}u^2)^T,\quad\text{and}\quad R_3(U)=(1,c+u,uc+H)^T
where the total specific enthalpy $H$ is
\begin{equation}
\label{enthalpy}
H=H(U)=\frac{\g E}{\r}-\frac{\g-1}{2}u^2,
\end{equation}
and $c(U)$ is the sound speed of the gas. Here, the gas is assumed to be ideal so that the pressure satisfies
\begin{equation}
\label{ideal}
P=\r RT,
\end{equation}
where $R$ is the molar gas constant and $T$ is the absolute temperature. According to (<ref>) and (<ref>), the sound speed can be expressed as
\begin{equation*}
c=\sqrt{\g RT}.
\end{equation*}
Since the laws of thermodynamics indicate that the absolute zero temperature cannot be reached by only the thermodynamic process, it allows us to assume that
\begin{equation}
\label{minc}
\min_{U\in\Omega}c(U)=c_*>0.
\end{equation}
Furthermore, we have
\begin{equation*}
\nabla\lambda_i(U)\cdot R_i(U)=\frac{(\g+1)c}{2\r}>0,\ i=1,3,\quad\text{and}\quad\nabla\l_2(U)\cdot R_2(U)=0,
\end{equation*}
which implies that the first and third characteristic fields are genuinely nonlinear and the second characteristic field is linearly degenerate. Therefore, the entropy solutions for $\CMcal{R}_C(x_0,t_0)$ and $\CMcal{BR}_C(x_B,t_0)$ consist of either shock waves, rarefaction waves first or third characteristic fields, or contact discontinuities from the second characteristic field. For each $i=1,3$, the $i$-rarefaction wave is a self-similar function
U=U(\xi), \quad \xi=\frac{x-x_0}{t-t_0},
which satisfies
\begin{equation*}
(df(U)-\xi I_3)\cdot \frac{dU}{d\zeta}=0,
\end{equation*}
where $I_3$ is the $3\times3$ identity matrix and the admissible $i$-shock is a discontinuous function satisfying the Rankine-Hugoniot condition
\begin{equation}
\label{jumpcond}
s [U]=[f(U)],
\end{equation}
and Lax's entropy condition
\begin{equation*}
\lambda_i(U_R) <s< \lambda_i(U_L),
\end{equation*}
where $s$ is the speed of the shock-front and $[\cdot]$ denotes the difference of states across the shock. According to Lax's method <cit.>, we can obtain the existence and uniqueness of the entropy solution for $\CMcal{R}_C(x_0,t_0)$. The solution consists of at most four constant states separated by shocks, rarefaction waves, or contact discontinuity. However, for $\CMcal{BR}_C(x_B,t_0)$, even under the Rankine-Hugoniot and Lax's entropy conditions, we may not obtain the uniqueness of the weak solutions when $U_R$ is near the transition surface $\CMcal{T}$, see Figure 1.
Figure 1. Two states $U_B$ and $U_B'$ connect to $U_R\in\mathcal{T}$ by two different waves:
$\mathfrak{S}_2+\mathfrak{R}_3$ and $\mathfrak{C}_2$. Both states satisfy the Rankine-Hugoniot and Lax-entropy conditions.
Moreover, the total variation of these solutions can be large even $|\r_R-\r_B|$, $|m_R-m_B|$ are small. To solve this problem, we impose an additional condition on the solutions:
($\CMcal{E}$) A weak solution $U=(\rho, m,E)$ is the entropy solution of $\CMcal{BR}_C(x_B,t_0)$ if $U$ has the least total variation in $\rho$ within all weak solutions of $\CMcal{BR}_C(x_B,t_0)$.
Under the condition $(\CMcal{E})$, we can select the unique entropy solution for $\CMcal{BR}_C(x_B,t_0)$. In addition, the entropy solution does not consist of the 0-speed shock from the first characteristic field attached on the boundary $x=x_B$. The following theorem states the existence and uniqueness of entropy
solutions for $\CMcal{R}_C(x_0,t_0)$ and $\CMcal{BR}_C(x_B,t_0)$.
<cit.> Suppose $U_L \in\Omega$. Then, there is a neighborhood $\Omega_1\subset\Omega$ of $U_L$ such that if $U_R\in\Omega_1$, $\CMcal{R}_C(x_0,t_0)$ has a unique solution consisting of at most four constant states separated by shocks, rarefaction waves, and contact discontinuity. Moreover, under the additional condition $(\CMcal{E})$, there exist a neighborhood $\Omega_2\subset\Omega$ of $U_R$ and $E_B>0$ exist such that $U_B=(\rho_B, m_B,E_B)\in\Omega_2$ and $\CMcal{BR}_C(x_B,t_0)$ admits a unique self-similar solution $U$ satisfying $U(x_B,t)=U_B$.
We can now construct the approximate solutions for $\CMcal{R}_G(x_0,t_0;g)$ and $\CMcal{BR}_G(x_B,t_0;g)$ by using Theorem 2.1. Let $\widetilde{U}=(\tilde{\r},\tilde{m},\widetilde{E})^T$ be the entropy solution of $\CMcal{R}_C(x_0,t_0)$. Then, for $(x,t)\in D(x_0,t_0)$, the approximate solution $U$ of $\CMcal{R}_G(x_0,t_0;g)$ is given by
\begin{equation}
\label{aproxsol}
\end{equation}
where $\widebar{U}(x,t)$ is constructed using the following steps: (1) Linearizing of the system (<ref>) around the homogeneous solution $\widetilde{U}$. (2) Averaging the coefficient of the linearized system. (3) Applying the operator-splitting method to the modified system. The detailed construction of $\widebar{U}(x,t)$ is provided in appendix A. On the basis of these steps and (<ref>) and (<ref>), the approximate solution $U(x,t)$ can be expressed as
\begin{equation}
\label{aproxsol4}
\end{equation}
\begin{equation}
\label{solver}
e^{h\tilde{u}t}\cosh(hvt) & 0 & 0 \\
\vspace{-0.4cm} \\
ve^{h\tilde{u}t}\sinh(hvt) & e^{h\tilde{u}t}\cosh(hvt) & 0 \\
S_{31} & S_{32} & S_{33}
\end{array}\right],
\end{equation}
and $v=v(x)$ is defined as
\begin{equation}
\label{grav}
\end{equation}
\begin{eqnarray*}
&&S_{31}=\frac{-xq}{2\g\tilde{m}}(e^{\g h\tilde{u}t}-1)-\frac{v^3}{v^2-(\g-1)^2\tilde{u}^2}
(ve^{\g h\tilde{u}t}-ve^{h\tilde{u}t}\cosh(hvt)-(\g-1)\tilde{u}e^{h\tilde{u}t}\sinh(hvt)), \\
(e^{\g h\tilde{u}t}-e^{h\tilde{u}t}\cosh(hvt))
+\frac{v(2v^2-(3\g-2)(\g-1)\tilde{u}^2)}{2(v^2-(\g-1)^2\tilde{u}^2)}e^{h\tilde{u}t}\sinh(hvt), \\
&&S_{33}=-\frac{(\g-1)(v^2+(\g-1)\tilde{u}^2)}{v^2-(\g-1)^2\tilde{u}^2}e^{\g h\tilde{u}t}
+\frac{\g v}{v^2-(\g-1)^2\tilde{u}^2}(ve^{h\tilde{u}t}\cosh(hvt)+(\g-1)\tilde{u}e^{h\tilde{u}t}\sinh(hvt)).
\end{eqnarray*}
Furthermore, through complex computation, we have
\begin{equation}
\label{1stubar}
\widebar{U}(x,t)
=-\frac{\tilde{\r}\D t}{x}\Big(2\tilde{u},2(\tilde{u}^2+v^2),\tilde{u}\Big(\tilde{u}^2+2v^2+\frac{2\tilde{c}^2}{\g-1}\Big)-\frac{q}{x\tilde{\r}}\Big)^T+O(1)(\D t)^2.
\end{equation}
\begin{equation}
\label{2.13.1}
|\widebar{U}|=|(S-I_3)\widetilde{U}|=O(1)(\Delta t),
\end{equation}
and $\widebar{U}\rightarrow 0$ as $\D t\to 0$ or $h\to 0$, which is consistent with the entropy solution for homogeneous conservation laws. Moreover, the approximation in (<ref>) continues to be true when $\widetilde{U}$ is a constant state.
The construction of the approximate solution for $\CMcal{BR}_G(x_B,t_0;g)$ is similar to that for $\CMcal{R}_G(x_0,t_0;g)$. Therefore, the approximate solution for $\CMcal{BR}_G(x_B,t_0;g)$ is also given by (<ref>). The perturbation $\widebar{U}$ in $\CMcal{BR}_G(x_B,t_0;g)$ may not satisfy $\widebar{U}(x_B,t)=0$ because of (<ref>), which means that the approximate solution $U$ may not match the boundary condition.
However by (<ref>), the error between the approximation $U$ and the boundary data $\widehat{U}_B:=(\r_B,m_B,0)^T$ can be estimated by
\begin{equation}
\label{2.14}
|U(x_B,t)-\widehat{U}_B|\le|\widebar{U}(x_B,t)|+|\widetilde{U}(x_B,t)-\widehat{U}_B|=O(1)(\Delta t)+\underset{D(x_B,t_0)}{\osc}\{\widehat{U}_B\},
\end{equation}
where $\underset{D(x_B,t_0)}{\osc}\{\widehat{U}_B\}$ denotes the oscillation of a function $\widehat{U}_B$ in the set $D(x_B,t_0)$. This indicates that such approximation does not affect the stability and consistency of the generalized Glimm method, which will be discussed later.
§.§ Residuals of the approximate solutions for Riemann and boundary-Riemann problems
To demonstrate the consistency of the generalized Glimm scheme, it is necessary to calculate the residuals of the approximate solutions for
Riemann and boundary-Riemann problems. Given a measurable function $U$,
region $\Gamma \subset\Pi$, and test function $\phi\in C^1_0(\Gamma)$, the residual of $U$ for (<ref>) in $\Gamma$ is defined as
\begin{equation}%-----------------------------(2.15)--------------------------%
\label{res1}
R(U, \Gamma,\phi):=\iint_{\Gamma}\left\{U\phi_t+f(U)\phi_x+h(x)g(x,U)\phi\right\}dxdt.
\end{equation}
We have the following estimates.
Let $U$ and $U^B$ be the approximate solutions of the Riemann problem $\CMcal{R}_G(x_0,t_0;g)$ and the boundary-Riemann problem $\CMcal{BR}_G(x_B,t_0;g)$ respectively. Let $\phi\in C^1_0(\G)$ be a test function. Suppose that $U=\widetilde{U}+\widebar{U}$ and $U^B=\widetilde{U}^B+\widebar{U}^B$. Then the residuals of $U$ and $U^B$ can be estimated respectively by
\begin{align}%-----------------------------(2.16)--------------------------%
\label{RPres}
&=\int^{x_0+\Delta{x}}_{x_0-\Delta{x}}(U\phi)(x,t)\Big|^{t=t_0+\Delta t}_{t=t_0}dx
+ \int^{t_0+\Delta{t}}_{t_0}(f(U)\phi)(x, t)\Big|^{x=x_0+\Delta x}_{x=x_0-\Delta x}dt\nonumber\\
&\quad+O(1)\left((\Delta t)^2(\Delta
\end{align}
\begin{align*}
&R(U^B, D(x_B,t_0), \phi)\nonumber\\
&=\int^{x_B+\Delta{x}}_{x_B}(U^B\phi)(x, t)\Big|^{t=t_0+\D t}_{t=t_0}dx
+\int^{t_0+\Delta{t}}_{t_0}(f(U^B)\phi)(x, t)\Big|^{x=x_B+\Delta x}_{x=x_B}dt\nonumber\\
&\quad+O(1)\left((\Delta t)^2(\Delta
\end{align*}
where $\underset{\Lambda}{osc}\{w\}$ denotes the oscillation of a function $w$ in the set $\Lambda$, and $D(x_0,t_0)$ and $D(x_B,t_0)$ are given by (<ref>) and (<ref>), respectively.
We only demonstrate the calculation of $R(U,D(x_0,t_0),\p)$; the calculation of $R(U^B,D(x_B,t_0),\p)$ is similar.
Without loss of generality, let $t_0=0,\ D:=D(x_0,t_0)$ with $x_0>x_B$, and $\widetilde{U}$ consists of the 1-shock with speed $s_1$, 2-contact discontinuity with speed $s_2$,
and 3-rarefaction wave with lower speed $s_3^-$ and upper speed $s^+_3$.
By (<ref>) and (<ref>),
\begin{align}%-----------------------------(2.17)--------------------------%
\label{res2}
R(U, D, \phi)&=\iint_D\left\{\widetilde{U}\phi_t+f(\widetilde{U})\phi_x\right\}dxdt
+\iint_D\left\{\widebar{U}\phi_t+(f(U)-f(\widetilde{U}))\phi_x+h(x)g(x,U)\phi\right\} dxdt\nonumber\\
&\equiv Q_1+Q_2.
\end{align}
According to the structure of $\widetilde{U}$, $Q_2$ can be evaluated as
\begin{align}%-----------------------------(2.18)--------------------------%
\label{perturbres}
+\int^{\Delta{t}}_{0}\!\!\!\int_{x_0+s_2t}^{x_0+s^-_3t}+\int^{\D t}_0\!\!\!\int_{x_0+s^-_3t}^{x_0+s^+_3t}
&\qquad\left\{\widebar{U}\phi_t+(f(U)-f(\widetilde{U}))\phi_x+h(x)g(x,U)\phi\right\} dxdt\nonumber\\
&\equiv Q_{21}+Q_{22}+Q_{23}+Q_{24}+Q_{25}.
\end{align}
Figure 2. Typical solution to the classical Riemann problem $\CMcal{R}_C(x_0,t_0)$.
Here $\widetilde{U}$ consists of different constant states in the region $D_i,\ i=1,2,3,5$, and self-similar in $D_4$, where $\{D_i\}_{i=1}^5$ is as shown in Figure 2. Next, according to (<ref>) and (<ref>) and the complicate calculation,
\begin{equation}
\label{diff1}
\begin{split}
\widebar{U}_t-hg(x,U)
&=\partial_t((S-I_3)\widetilde{U})-hg(x,S\widetilde{U})=S_t\widetilde{U}-hg(x,S\widetilde{U}) \\
&=h\tilde{\r}v\big(0,ve^{h\tilde{u}t}\sinh(hvt)\tanh(hvt),\D_3\big)^T=O(1)(\D t)^2,
\end{split}
\end{equation}
\D_3=\tanh(hvt)\cdot
\left\{
\begin{split}
&\frac{xq}{2\tilde{m}}(e^{\g h\tilde{u}t}-1)
+\frac{(v^2+(\g-1)\tilde{u}^2)\tilde{c}^2+\g v^4}{v^2-(\g-1)^2\tilde{u}^2}(e^{\g h\tilde{u}t}-e^{h\tilde{u}t}\cosh(hvt)) \\
&\quad-\frac{\tilde{u}v((\g-3)(\g-1)^2\tilde{u}^2+2\g\tilde{c}^2+(2\g^2-3\g+3)v^2)}{2(v^2-(\g-1)^2\tilde{u}^2)}e^{h\tilde{u}t}\sinh(hvt) \\
\end{split}\right\}.
Since $\sinh(hvt)=O(1)\Delta t$, $\tanh(hvt)=O(1)\Delta t$, and $e^{\gamma h \tilde{u}t}-1=O(1)\Delta t$ when $0<t<\Delta t$. Applying Green's theorem and (<ref>) to $Q_{2i},\ i=1,2,3,5$, we have
\begin{align}%-----------------------------(2.20)--------------------------%
\label{Q21}
\int^{x_0}_{x_0-\Delta{x}}(\widebar{U}\p)(x,0)dx
&\quad+\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U}))\p\big)(x, t)\Big|^{x=s_1t-}_{x=-\Delta{x}}dt+O(1)(\Delta t)^3(\Delta x)\|\phi\|_\infty , \\
\label{Q22}
+\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U})-s_2\widebar{U})\p\big)(s_2t-, t)dt\nonumber\\
&\quad-\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U})-s_1\widebar{U})\p\big)(s_1t+, t)dt+O(1)(\Delta t)^3(\Delta x)\|\phi\|_\infty, \\
\label{Q23}
+\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U})-s_3^-\widebar{U})\p\big)(s_3^-t, t)dt\nonumber\\
&\quad-\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U})-s_2\widebar{U})\p\big)(s_2t+, t)dt+O(1)(\Delta t)^3(\Delta x)\|\phi\|_\infty,
\end{align}
\begin{align}%-----------------------------(2.22)--------------------------%
\label{Q25}
Q_{25}&=\int^{x_0+\D x}_{x_0+s_3^+\D t}(\widebar{U}\p)(x,\Delta{t})dx-
\int_{x_0}^{x_0+\Delta{x}}(\widebar{U}\p)(x,0)dx
&\quad+\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U}))\p\big)(x, t)\Big|^{x=\D x}_{x=s_3^+t}dt+O(1)(\Delta t)^3(\Delta x)\|\phi\|_\infty.
\end{align}
Next, $Q_{24}$ is estimated as follows. In the region $D_4=\{(x,t)\mid\frac{x-x_0}{t}\in[s_3^-,s_3^+]\}$, $\widetilde{U}$ is the 3-rarefaction wave; therefore, by (<ref>) the approximate solution $U$ satisfies
where $\xi=\frac{x-x_0}{t}$. It leads to
\begin{eqnarray}
\label{diff2}
&&\widebar{U}_t-h(x_0+t\xi)g(x_0+t\xi,U) \nonumber\\
-h(x_0+t\xi)g\big(x_0+t\xi,S(x_0+t\xi,t,\widetilde{U}(\xi))\widetilde{U}(\xi)\big) \nonumber\\
-h(x_0+t\xi)g(x_0+t\xi,S\widetilde{U}) \nonumber\\
\end{eqnarray}
where $\dot{\widetilde{U}}:=\frac{d\widetilde{U}}{d\xi}$, and
\begin{align*}
\end{align*}
According to (<ref>) and further calculation, we have
\begin{eqnarray}
\label{Q241}
=\frac{2\xi t}{x_0^2}\Big(\tilde{m},\tilde{\r}(\tilde{u}^2+2v_0^2),\tilde{m}(\widetilde{H}+2v_0^2)\Big)^T
+O(1)(\D t)^2, \\
\label{Q242}
0 & 2 & 0 \\
2(v_0^2-\tilde{u}^2) & 4\tilde{u} & 0 \\
(\g-1)\tilde{u}^3-2\tilde{u}\widetilde{H} & 2(\widetilde{H}-(\g-1)\tilde{u}^2+v_0^2) & 2\g\tilde{u}
\end{array}\right]\dot{\widetilde{U}} \nonumber \\
&&\hspace{2cm}+O(1)\D t,
\end{eqnarray}
where $v_0:=v(x_0)$.
Applying the integration by parts to $Q_{24}$ along with (<ref>), we obtain
\begin{align}%-----------------------------(2.21)--------------------------%
\label{Q24}
+\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U})-s_3^+\widebar{U})\p\big)(s_3^+t, t)dt\nonumber\\
&\quad-\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U})-s_3^-\widebar{U})\p\big)(s_3^-t, t)dt\nonumber\\
&\quad-\int_0^{\D t}\!\!\!\int_{s_3^-}^{s_3^+}\{Q_{24}^1(\xi,t)+Q_{24}^2(\xi,t)\}\p(t\xi,t)d\xi dt.
\end{align}
According to, (<ref>) and (<ref>)-(<ref>)
\begin{align}%-----------------------------(2.27)--------------------------%
\label{Q2}
Q_2&=\sum_{i=1}^5Q_{2i}=\int^{\Delta{x}}_{-\Delta{x}}(\widebar{U}\p)(x,t)\Big|_{t=0}^{t=\D t}dx
+\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U})-s_1\widebar{U})\p\big)(x, t)\Big|^{x=s_1t-}_{x=s_1t+}dt\nonumber\\
&\quad+\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U})-s_2\widebar{U})\p\big)(x, t)\Big|^{x=s_2t-}_{x=s_2t+}dt
+\int^{\Delta{t}}_{0}\big((f(U)-f(\widetilde{U}))\p\big)(x, t)\Big|_{x=-\Delta x}^{x=\Delta{x}}dt\nonumber\\
&\quad+O(1)\left((\Delta t)^3(\Delta x)+(\Delta{t})^3+(\Delta{t})^2(\underset{D}{\osc}\{\widetilde{U}\})\right)\|\phi\|_\infty.
\end{align}
We estimate the second and third terms on the right-hand side of (<ref>). Suppose that the state $\widetilde{U}_1$ is connected to the
state $\widetilde{U}_L=U_L=(\r_L,m_L,E_L)$ by 1-shock on the right and the $\widetilde{U}_2$ by the 2-contact discontinuity on the left. Then, the Rankine-Hugoniot condition (<ref>) gives
\begin{equation}%-----------------------------(2.28)--------------------------%
\label{RHcond}
\end{equation}
According to (<ref>) and (<ref>), we obtain
\begin{equation}
\label{shockres}
\int_0^{\D t}\big((f(U)-f(\widetilde{U})-s_1\widebar{U})\p\big)(x,t)\Big|_{x=s_1t+}^{x=s_1t-}dt
=\Big(O(1)(\D t)^2(\underset{D}{\osc}\{\widetilde{U}\})+O(1)\D t(\underset{D}{\osc}\{\widetilde{U}\})^2\Big)\|\p\|_{\infty},
\end{equation}
\begin{eqnarray}
\label{solverestmate}
&& \int_0^{\D t}\big((f(U)-f(\widetilde{U})-s_1\widebar{U})\p\big)(x,t)\Big|_{x=s_1t+}^{x=s_1t-}dt \nonumber\\
&=& \int_0^{\D t}\big(\big(f(S\widetilde{U}_L)-f(S\widetilde{U}_1)-S(f(\widetilde{U}_L)-f(\widetilde{U}_1))\big)\p\big)(s_1t,t)dt \nonumber\\
&=& \int_0^{\D t}\big(\big(f(S\widetilde{U}_L)-f(S\widetilde{U}_L+S\widetilde{U}_1-S\widetilde{U}_L)\big)\p\big)(s_1t,t)dt \nonumber\\
&& \quad-\int_0^{\D t}\big(S\big(f(\widetilde{U}_L)-f(\widetilde{U}_L+\widetilde{U}_1-\widetilde{U}_L)\big)\p\big)(s_1t,t)dt
+O(1)\D t(\underset{D}{\osc}\{\widetilde{U}\})^2\|\p\|_{\infty}\nonumber\\
&=& \int_0^{\D t}\big(\big(df(S\widetilde{U}_L)\cdot S\cdot(\widetilde{U}_1-\widetilde{U}_L)-S\cdot
df(\widetilde{U}_L)\cdot(\widetilde{U}_1-\widetilde{U}_L)\big) \p\big)(s_1t,t)dt
+O(1)\D t(\underset{D}{\osc}\{\widetilde{U}\})^2\|\p\|_{\infty}\nonumber\\
&=& \int_0^{\D t}\big(\big(df(S\widetilde{U}_L)\cdot S-S\cdot df(\widetilde{U}_L)\big)\cdot(\widetilde{U}_1-\widetilde{U}_L)\p\big)(s_1t,t)dt
+O(1)\D t(\underset{D}{\osc}\{\widetilde{U}\})^2\|\p\|_{\infty}\nonumber\\
&=& \Big(O(1)(\D t)^2(\underset{D}{\osc}\{\widetilde{U}\})+O(1)\D t(\underset{D}{\osc}\{\widetilde{U}\})^2\Big)\|\p\|_{\infty}.
\end{eqnarray}
The aforementioned final equality holds because of the complex computation; therefore, we obtain:
\begin{eqnarray*}
df(S\widetilde{U}_L)\cdot S-S\cdot df(\widetilde{U}_L)
&=& df(S(x_0,t,\widetilde{U}_L)\widetilde{U}_L)\cdot S(x_0,t,\widetilde{U}_L)-S(x_0,t,\widetilde{U}_L)\cdot df(\widetilde{U}_L) \\
&=& \Big(hv^2t,-\frac{(\g-1)xhqt}{2\,\tilde{\r}_L},-\frac{(\g-1)^2h\tilde{u}_L^4t}{4}\Big)\Big|_{x=x_0}+O(1)(\D t)^2.
\end{eqnarray*}
Similarly, we have
\begin{equation}
\label{discontres}
\int_0^{\D t}\big((f(U)-f(\widetilde{U})-s_2\widebar{U})\p\big)(x,t)\Big|_{x=s_2t+}^{x=s_2t-}dt
=\Big(O(1)(\D t)^2(\underset{D}{\osc}\{\widetilde{U}\})+O(1)(\D t)(\underset{D}{\osc}\{\widetilde{U}\})^2\Big)\|\p\|_{\infty}.
\end{equation}
Therefore, by (<ref>) and (<ref>), (<ref>) can be rewritten as
\begin{align}%-----------------------------(2.27)--------------------------%
Q_2&=\int_{-\D x}^{\D x}(\widebar{U}\p)(x,t)\Big|^{t=\D t}_{t=0}dx
+\int_0^{\D t}\big((f(U)-f(\widetilde{U}))\p\big)(x,t)\Big|_{x=-\D x}^{x=\D x}dt\nonumber\\
&\quad+O(1)\left((\Delta t)^3(\Delta x)+(\Delta{t})^3+(\Delta{t})^2(\underset{D}{\osc}\{\widetilde{U}\})\right)\|\phi\|_\infty.
\end{align}
Since $\widetilde{U}$ is the entropy solution for $\CMcal{R}_C(D)$, then according to the results of <cit.> and (<ref>), we have
\begin{align}%-----------------------------(2.31)--------------------------%
\label{Q1}
\begin{split}
Q_1&=\iint_D\left\{\widetilde{U}\phi_t+f(\widetilde{U})\phi_x\right\}dxdt=\int_{-\D x}^{\D x}(\widetilde{U}\p)(x,t)\Big|_{t=0}^{t=\D t}dx
+\int_0^{\D t}\big(f(\widetilde{U})\p\big)(x,t)\Big|_{x=-\D x}^{x=\D x}dt.
\end{split}
\end{align}
Following (<ref>), (<ref>) and (<ref>)-(<ref>), we obtain (<ref>). The proof is complete.
§ EXISTENCE OF GLOBAL ENTROPY SOLUTIONS FOR IBVP (<REF>)
In this section, we establish the stability of the generalized Glimm scheme and consequently the compactness of the subsequences of approximate solutions $\{U_{\th,\D x}\}$ to (<ref>). The stability, which is always the core of the Glimm method, is obtained through the modified wave interaction estimates, nonincreasing property of the Glimm functional, and uniform boundedness of the total variations of the perturbations in approximate solutions. We prove the global existence of entropy solutions to (<ref>) by demonstrating the consistency of the scheme and entropy inequalities for weak solutions at the end of this section.
§.§ Generalized Glimm scheme for (<ref>)
In this subsection, we introduce a nonstaggered generalized Glimm scheme for the initial boundary value problem (<ref>). Let us discretize the domain $\Pi\equiv[x_B,\infty)\times[0,\infty)$ into
\begin{equation*}
x_k=x_B+k\Delta x,\quad t_n=n\Delta t,\quad k, n=0,1,2,\cdots,
\end{equation*}
where $\Delta x$ and $\Delta t$ are small positive constants satisfying the CFL condition (<ref>). The $n$th time strip $T_n$ is denoted by
T_{n}:=[x_B,\infty)\times[t_n,t_{n+1}),\quad n=0,1,2,\cdots.
Suppose that the approximate solution $U_{\theta,\Delta{x}}(x,t)$ has been constructed in $T_n$; then, we choose a random number $\theta_n \in (-1, 1)$ and define the initial data $U_k^n\equiv(\rho_k^n,m_k^n,E_k^n)$ in $T_n$ by
U_k^n:=U_{\theta,\Delta{x}}(x_{2k}+\theta_n\Delta{x},t_n^-),\ k=1,2,\cdots.
To initiate the scheme at $n=0$, we set $t_0^-=0$. The points $\{(x_{2k}+\theta_n\Delta{x},t_n^-)\}_{n=0,k=1}^{\infty}$ are called the $mesh \; points$ of the scheme and the points $\{(x_B,t_n+\frac{\D t}{2})\}_{n=0}^{\infty}$ are the mesh points on the boundary $x=x_B$. Then, $U_{\theta,\Delta{x}}$ in $T_{n+1}$ is constructed by solving the set of Riemann problems $\{\CMcal{R}_G(x_k,t_n;g)\}_{k\in\NN}$ with initial data
\begin{align*}%------------------------(3.0.1)---------------------%
\begin{array}{ll}
U^n_k,&\; \mbox{if } x_{2k-1}\leq x<x_{2k},\\
U^n_{k+1},&\;\mbox{if } x_{2k}<x\leq x_{2k+1},
\end{array}\right.\ k=1,2,\cdots,
\end{align*}
and the boundary-Riemann problem $\CMcal{BR}_G(x_B,t_n;g)$ with initial-boundary data
U(x,t_n)=U^n_1, x_0<x≤x_1,
(̊x_B,t)=^̊n_B:=_̊B(t_n), t_n≤t< t_n+1,
m(x_B,t)=m^n_B:=m_B(t_n), t_n≤t< t_n+1.
Moreover, near the boundary $x=x_B$, the approximate solution for (<ref>) satisfies the entropy condition $(\CMcal{E})$ in Section 2.
Figure 3. Approximate solution for IBVP with mesh curves.
According to (<ref>), the approximate solution $U_{\theta ,\Delta{x}}$ has an explicit representation
\begin{equation*}%------------------------(3.0.2)---------------------%
U_{\theta ,\Delta{x}}(x,t)=S(x,t,\widetilde{U}_{\th,\D x}(x,t))\widetilde{U}_{\th,\D x}(x,t),\quad (x,t) \in T_{n},
\end{equation*}
where $\widetilde{U}_{\theta,\Delta{x}}$ consists of the weak solutions to the corresponding classical Riemann problems $\{\CMcal{R}_C(x_k,t_n)\}$ or boundary-Riemann problems $\CMcal{BR}_C(x_B,t_n)$ in $T_{n}$. The CFL condition (<ref>) ensures that the elementary waves in each $T_{n}$ do not interact with each other before time $t=t_{n+1}$. Repeating this process, we construct the approximate solution $U_{\theta,\Delta x}$ of (<ref>)
in $\Pi$ by using the generalized Glimm scheme with a random sequence $\theta:= (\theta_0, \theta_1, \theta_2,\ldots)$, $\th_i\in(-1,1)$ for all $i$.
To obtain the desired estimates, it is favorable to consider curves comprising line segments joining mesh points rather then horizontal lines. Therefore, we define the mesh curves for the non-local Glimm functionals introduced in <cit.>.
A mesh curve $J$ for (<ref>) is a piecewise linear curve that connects the mesh point $(x_{2k}+\theta_n\Delta x,t_n)$ on the left with $(x_{2k+2}+\theta_{n+1}\Delta x,t_{n+1})$ or $(x_{2k+2}+\theta_{n-1}\Delta x,t_{n-1})$ on its right, $k,n=0,1,2,\cdots$ together with the line segments joining the points $(x_B,t_n+\frac{\D t}{2})$ and $(x_B+\th_n\D x,t_n)$ and some portion of the boundary (see Figure 3 and <cit.>). Simultaneously, the mesh curve $J$ divides the domain $[x_0,\infty)\times[0,\infty)$ into $J^+$ and $J^-$ regions such that $J^-$ contains the line $[x_0,\infty)\times\{0\}$. We can partially order two mesh curves by saying $J_2>J_1$ (or $J_2$ is a successor of $J_1$) if every mesh point of $J_2$ is either on $J_1$ or contained in $J_1^+$. In particular, $J_2$ is an immediate successor of $J_1$ if
$J_2>J_1$ and all mesh points on $J_2$ except one are on $J_1$.
A diamond region is a closed region enclosed by a mesh curve and its immediate successor.
§.§ Wave interaction estimates
In this subsection, several types of nonlinear wave interactions will be described and the classical wave strengths of $\{{U}_{\th,\D x}\}$ in each time step will be estimated through wave interactions between the classical waves and the perturbations in the previous time step.
Connecting all the mesh points through the mesh curves, the domain $\Pi$ is decomposed as a union of the sets of diamond, triangular, and pentagonal regions. The wave interactions can be divided into the following three types:
(I) In each diamond region, the incoming generalized waves from adjacent Riemann problems interact with each other and emerge as the outgoing generalized waves of the Riemann problem in the next time step;
(II) In each triangular region, the incoming generalized waves from the Riemann problem at the boundary interact with each other and emerge as the outgoing generalized waves of the boundary-Riemann problem in the next time step;
(III) In each pentagonal region, two families of incoming generalized waves, one from the boundary-Riemann problem and the other from adjacent Riemann problem, interact with each other and emerge as the outgoing generalized waves of the Riemann problem in the next time step.
In each diamond (or triangular and pentagonal) region, all the generalized waves comprise classical outgoing waves and perturbations. Therefore, the objective of wave interaction estimates is to estimate how the wave strengths of classical outgoing waves are influenced by the interaction or reflection of generalized incoming waves.
We start with the wave interaction estimates of type (I).
Suppose $(x,t)\in (x_B,\infty)\times[0,\infty)$ and let $\CMcal{R}_G(U_R,U_L;x,t)$ denote the generalized Riemann solution of $\CMcal{R}_G(x,t;g)$ connecting the left constant state $U_L$ with the right constant state $U_R$. Moreover let $\CMcal{R}_C(U_R,U_L;x,t)$ be the solution of the corresponding classical Riemann problem $\CMcal{R}_C(x,t)$. Then, the classical wave strength of $\CMcal{R}_G(U_R,U_L;x,t)$ is defined as the wave strength of $\CMcal{R}_C(U_R,U_L;x,t)$, which is expressed as
\begin{equation*}%-----------------(3.1)-------------------%
\ve=\varepsilon(U_R,U_L;x,t)=(\ve_1,\ve_2,\ve_3).
\end{equation*}
In other words, the jump discontinuity $\{U_L,U_R\}$ is resolved into $U_L=\widetilde{U}_0$, $\widetilde{U}_1,\ \widetilde{U}_2$, and $\widetilde{U}_3=U_R$ such that $\widetilde{U}_{j}$ is connected to $\widetilde{U}_{j-1}$ on the right by a $j$-wave of strength $\varepsilon_j$. Note that $\CMcal{R}_C(U_R,U_L;x,t)$ is independent of the choice of $(x,t)$. We say that an $i$-wave and a $j$-wave approach if either $i>j$, or else $i = j$ and at least one wave is a shock. Given another $\CMcal{R}_G(U'_R,U'_L;x',t')$ with classical wave strength $\ve'=(\ve_1',\ve_2',\ve_3')$, the wave interaction potential associated with $\ve$, $\ve'$ is defined as
\begin{equation*}%-------------------------(3.2)-------------------------%
D(\varepsilon,\varepsilon'):=\sum_{App}\{|\varepsilon_i\varepsilon_j'|:\varepsilon_i \mbox{ and } \varepsilon'_j \mbox{ approach}\}.
\end{equation*}
Assume that $J'$ is an immediate successor of $J$. Let $\Gamma_{k,n}$ denote the diamond region centered at $(x_{2k},t_{n})$ and enclosed by $J$ and $J'$. Four vertices of $\Gamma_{k,n}$, see Figure 4, are denoted by
\begin{align*}
\begin{array}{ll}
\mathpzc{N}=(x_{2k}+\theta_{n+1}\Delta x,t_{n+1}),& \mathpzc{E}=(x_{2k}+\theta_{n}\Delta x,t_{n}),\\
\mathpzc{W}=(x_{2k+2}+\theta_{n}\Delta x,t_{n}), & \mathpzc{S}=(x_{2k}+\theta_{n-1}\Delta x, t_{n-1}),
\end{array}
\end{align*}
\begin{align*}
\begin{array}{ll}
\mathpzc{N}=(x_{2k}+\theta_{n+1}\Delta x,t_{n+1}),& \mathpzc{E}=(x_{2k-2}+\theta_{n}\Delta x,t_{n}),\\
\mathpzc{W}=(x_{2k}+\theta_{n}\Delta x,t_{n}), & \mathpzc{S}=(x_{2k}+\theta_{n-1}\Delta x, t_{n-1}),
\end{array}
\end{align*}
Here $\{\theta_{n-1}, \theta_{n},\theta_{n+1} \}$ are random numbers in $(-1,1)$.
Define the matrix $R(U):=[R_1(U),R_2(U),R_3(U)]$, where $R_j$ is the right eigenvector of $df$ associated with the eigenvalue $\lambda_j$, and $R(U)$ is invertible in $\Omega$. Then, we have the following theorems on the wave interaction estimates.
Let $U_L,\ U_R$, and $U_M$ be constant states in some neighborhood contained in $\Omega$ with $U_L=\widetilde{U}_L+\widebar{U}_L$ and $U_R=\widetilde{U}_R+\widebar{U}_R$, where $\widetilde{U}_L(x_L,t_n):=(\widetilde{\rho}_L,\widetilde{m}_L,\widetilde{E}_L)^T,\ \widetilde{U}_R(x_R,t_n):=(\widetilde{\rho}_R,\widetilde{m}_R,\widetilde{E}_R)^T$, and $U_M(x_M,t_n):=(\rho_M,m_M,E_M)^T$
\begin{eqnarray}%------------------(3.2.1)-------------------%
\label{aproxperturb}
&\widebar{U}_L=\big(S(x_L,\D t,\widetilde{U}_L)-I_3\big)\widetilde{U}_L=:(S_L-I_3)\widetilde{U}_L, \\
&\widebar{U}_R=\big(S(x_R,\D t,\widetilde{U}_R)-I_3\big)\widetilde{U}_R=:(S_R-I_3)\widetilde{U}_R,
\end{split} \\
&&x_L:=x_{2k-2}+\th_n\D x,\quad x_R:=x_{2k+2}+\th_n\D x, \nonumber\\
&&x_M=x_{2k}+\th_n\D x.\nonumber
\end{eqnarray}
Suppose that the classical wave strengths of the incoming generalized waves across the boundaries $\mathpzc{WS}$ and $\mathpzc{SE}$ of $\Gamma_{k,n}$ are
\begin{equation*}%------------------(3.3)-------------------%
\a(U_M,\widetilde{U}_L;x_L,t_{n-1})=(\a_1,\a_2,\a_3) \quad\text{and}\quad
\beta(\widetilde{U}_R,U_M;x_R,t_{n-1})=(\beta_1,\beta_2,\b_3),
\end{equation*}
Figure 4. Classical wave strengths in the diamond region $\Gamma_{k,n}$.
respectively, and the classical wave strength of the outgoing generalized waves across the boundary $\mathpzc{WNE}$ (see Figure 4) is
\begin{equation*}%------------------(3.4)-------------------%
\varepsilon(U_R,U_L;x_M,t_n)=(\varepsilon_1,\varepsilon_2,\ve_3),
\end{equation*}
Then there exist constants $C'$ and $C''_{k,n}$ such that
\begin{equation}%-------------------------(3.6)-------------------------%
\label{3.6}
|\varepsilon|\le(1-\z\D t)(|\a|+|\beta|)+C'D(\a,\beta)+C''_{k,n}\D t\D x+O(1)(\D t)^3,\ \text{as}\ |\a|+|\beta|\rightarrow 0,
\end{equation}
where $\z=\frac{(3-\g)|u_M|}{x_M}$ and
In particular, (<ref>) is reduced to the classical wave interaction estimate in <cit.> when the system (<ref>) is without the source term.
According to the results of <cit.>, we have
\begin{eqnarray}%-----------------(3.7)-------------------%
\label{3.7}
&&U_R-U_M=\widebar{U}_R+\sum_{j=1}^3\beta_jR_j+ \sum_{j\leqslant i}\beta_j\beta_i\big(R_j\cdot\nabla
R_i\big)\left(1-\frac{\delta_{ij}}{2}\right)+O(1)|\beta|^3, \\
\label{3.8}
&&U_L-U_M=\widebar{U}_L+\sum_{j=1}^3(-\a_j)R_j+\sum_{j\geqslant i}\a_j\a_i\big(R_j\cdot\nabla R_i\big)\left(1-\frac{\delta_{ij}}{2}\right)+O(1)|\a|^3,
\end{eqnarray}
where $R_j$ is the right eigenvector of $df$ associated with the eigenvalue $\lambda_j$, and $\delta_{ij}$ is the Kronecker delta. In addition, the coefficients in (<ref>) and (<ref>) are all evaluated at $U_M$. Similarly,
\begin{align}%-----------------(3.9)-------------------%
\label{3.9}
U_R-U_L&=\sum_{i=1}^3\varepsilon_iR_i(U_L)+\sum_{j\leqslant i}\varepsilon_j\varepsilon_i\big(R_j\cdot\nabla
&=\sum_{i=1}^3\varepsilon_iR_i(\widetilde{U}_L)+\sum_{j\leqslant i}\varepsilon_j\varepsilon_i\big(R_j\cdot\nabla
&\quad+\left(\sum_{i=1}^3\varepsilon_i\nabla R_i(\widetilde{U}_L)\right)\widebar{U}_L+\CMcal{C}(\Delta x,|\varepsilon|),
\end{align}
where $\CMcal{C}(\Delta x,|\varepsilon|)$ denotes the cubic terms of $\Delta x$ and $|\varepsilon|$. According to (<ref>)-(<ref>), $\varepsilon=0$ when $\a=\beta=0$ and $\widebar{U}_R-\widebar{U}_L=0$. By (<ref>),
\varepsilon=O(1)(\Delta x+|\a|+|\beta|).
According to the Taylor expansion of $R_i$ around $U_M$,
\begin{equation}%-----------------(3.10)-------------------%
\label{3.10}
R_i(\widetilde{U}_L)=R_i(U_M)-\sum_{j=1}^3\a_j(R_j\cdot\nabla R_i)(U_M)+O(1)|\a|^2.
\end{equation}
According to (<ref>) and (<ref>) together with $\varepsilon=O(1)(\Delta x+|\a|+|\beta|)$, the difference of $U_R$ and $U_L$ is expressed as
\begin{align}%-----------------(3.11)-------------------%
\label{3.11}
&=\sum_{i=1}^3\varepsilon_iR_i(U_M)+\sum_{j\leqslant i}\varepsilon_j\varepsilon_i(R_j\cdot\nabla
&\quad-\sum_{\scriptstyle i,j}\varepsilon_i\a_j(R_j\cdot\nabla
R_i)(U_M)+\left(\sum_{i=1}^3(\a_i+\beta_i)\nabla R_i(U_M)\right)\widebar{U}_L\nonumber\\
&\quad+\CMcal{C}(\Delta x,|\a|+|\beta|).
\end{align}
According to (<ref>) and (<ref>), $\widebar{U}_L=O(1)(\Delta t)=O(1)(\Delta x)$ and
\begin{align}
\label{3.14}
\widebar{U}_R-\widebar{U}_L
&=(S_R-I_3)\widetilde{U}_R-(S_L-I_3)\widetilde{U}_L \nonumber\\
\end{align}
By comparing (<ref>) with (<ref>), and (<ref>), and using (<ref>), we obtain
\begin{align}%-------------------------(3.17)-------------------------%
\label{3.17}
\varepsilon^T
&=(\a+\beta)^T+\sum_{j<i}\a_i\beta_jL_{ij}(U_M)-R^{-1}(U_M)\left(\sum_{i=1}^3(\a_i+\beta_i)\nabla R_i(U_M)\right)(S_L-I_3)\widetilde{U}_L\nonumber\\
&\quad+\CMcal{C}(\Delta x,|\a|+|\beta|),
\end{align}
where $R=[R_1,R_2,R_3]$ and $L_{ij}:=R^{-1}(R_i\cdot \nabla R_j-R_j\cdot \nabla R_i)$. To estimate (<ref>), we need to evaluate all terms at the state $U_M$. After a complex calculation and using the Taylor expansion with respect to $\D t$, the term $(S_L-I_3)\widetilde{U}_L$ is estimated as follows:
\begin{align}
\label{3.17-1}
&=(S_M-I_3)U_M+(S_M-I_3)(\widetilde{U}_L-U_M)+(S_L-S_M)\widetilde{U}_L \nonumber \\
&=(S_M-I_3)U_M+O(1)\osc\{U\}\D t+O(1)\D t\D x,
\end{align}
where $S_M:=S(x_M,\D t,U_M)$.
Next, we estimate $(S_R-S_M)\widetilde{U}_R+(S_M-S_L)\widetilde{U}_L$ in (<ref>). Define $\mathscr{F}(x,\D t,\widetilde{U}):=S(x,\D t,\widetilde{U})\widetilde{U}$; then,
\begin{align}
\label{3.17-2}
&(S_R-S_M)\widetilde{U}_R+(S_M-S_L)\widetilde{U}_L \nonumber\\
&=\mathscr{F}(x_R,\D t,\widetilde{U}_R)-\mathscr{F}(x_L,\D t,\widetilde{U}_L)-S_M(\widetilde{U}_R-\widetilde{U}_L) \nonumber\\
&=\mathscr{F}_x(x_M,\D t,U_M)\D x+(\mathscr{F}_U(x_M,\D t,U_M)-S_M)(\widetilde{U}_R-\widetilde{U}_L) \nonumber\\
&=W(U_M)\D t\D x+\Psi(U_M)(\widetilde{U}_R-\widetilde{U}_L)+\CMcal{C}(\Delta x,|\a|+|\beta|),
\end{align}
\begin{align*}
&W(U_M)=\frac{h_M^2}{2}\Big(\r_Mu_M,\r_M(u_M^2+2v_M^2),\r_Mu_M(H_M+2v_M^2)+\frac{x_M^2q_M'}{2}\Big)^T, \\%\label{3.17-3}\\
&\Psi(U_M)=\mathscr{F}_U(x_M,\D t,U_M)-S_M %\label{3.17-4}
\end{align*}
with $h_M:=-\frac{2}{x_M},H_M:=H(U_M)$, and $q_M':=q'(x_M)$. According to (<ref>) and (<ref>), the difference between $\widetilde{U}_R$ and $\widetilde{U}_L$ is
\begin{equation}%-----------------(3.19)-------------------%
\label{3.19}
\widetilde{U}_R-\widetilde{U}_L=R(U_M)(\a+\beta)^T+O(1)(|\a_i||\a_j|+|\beta_i||\beta_j|).
\end{equation}
Applying the Taylor expansion of (<ref>) at $U_M$ along with (<ref>), and using (<ref>) and (<ref>), we have
\begin{align}%-------------------------(3.18)-------------------------%
\label{3.18}
\varepsilon^T
&= \big(I_3-\widebar{D}(U_M)+R^{-1}(S_M-I_3)R\big)(\a+\beta)^T+\sum_{j<i}\a_i\beta_jL_{ij}(U_M) \nonumber\\
&\quad+R^{-1}(U_M)W(U_M)\D t\D x+R^{-1}(U_M)\Psi(U_M)(\widetilde{U}_R-\widetilde{U}_L)+\CMcal{C}(\Delta x,|\a|+|\beta|),
\end{align}
\begin{align*}
\widebar{D}(U_M)
:=R^{-1}(U_M)\big(\nabla R_1(U_M)(S_M-I_3)\widetilde{U}_M,\nabla R_2(U_M)(S_M-I_3)\widetilde{U}_M,\nabla R_3(U_M)(S_M-I_3)\widetilde{U}_M\big).
\end{align*}
\begin{align*}%-------------------------(3.21)-------------------------%
\Phi(U_M):=I_3-\widebar{D}(U_M)-R^{-1}(U_M)(S_M-I_3)R(U_M).
\end{align*}
By a direct calculation of eigenvalues $\nu_1,\nu_2,\text{and}\ \nu_3$ of $\Psi$ and $\m_1,\m_2,\text{and}\ \m_3$ of $\Phi$ evaluated at $x=x_M$, and considering $h_M=-\frac{2}{x_M}$, we obtain
\begin{equation*}
\nu_1=\nu_2=0,\qquad\nu_3=\frac{4v_M^2}{x_M^2}(\D t)^2+O(1)(\D t)^3,
\end{equation*}
\begin{equation*}%-------------------------(3.26)-------------------------%
\begin{split}
\m_1&=1-\frac{2u_M}{x_M}\D t+O(1)(\D t)^2, \\
\m_2&=1-\Big(\frac{(3-\g)u_M}{x_M}+\frac{\g(\g-1)q_M}{2\r_Mc_M^2}\Big)\D t+O(1)(\D t)^2, \\
\m_3&=1-\Big(\frac{2u_M}{x_M}+\frac{\g(\g-1)q_M}{\r_Mc_M^2}\Big)\D t+O(1)(\D t)^2.
\end{split}
\end{equation*}
We obtain that $0<\m_i < 1,\ i=1,2,3$ when $\Delta t>0$ is sufficiently small. Therefore, there exist non-singular matrices $Q_{\D t}(U_M),\text{and}\ T_{\Delta t}(U_M)$ such that
\begin{align*}%-------------------------(3.27)-------------------------%
\Psi(U_M)=Q_{\Delta t}\cdot\diag[\nu_1,\nu_2,\nu_3]\cdot Q_{\Delta t}^{-1}, \\
\Phi(U_M)=T_{\Delta t}\cdot\diag[\m_1,\m_2,\m_3]\cdot T_{\Delta t}^{-1}.
\end{align*}
Since $1<\g<\frac{5}{3}$, if $\z=\frac{(3-\g)|u_M|}{x_M}$ is selected, then,
\begin{equation}%-------------------------(3.28)-------------------------%
\label{3.28}
\begin{split}
|\Psi(U_M)(\widetilde{U}_R-\widetilde{U}_L)|&\le|Q_{\D t}||\diag[\nu_1,\nu_2,\nu_3]||Q_{\D t}^{-1}||R||\a+\b|=\CMcal{C}(\Delta x,|\a|+|\beta|), \\
\left|\Phi(U_M)\right|&\le|T_{\Delta t}||\diag[\m_1,\m_2,\m_3]||T_{\Delta t}^{-1}|\le\max\limits_{1\le i\le3}\m_i\leq 1-\z\D t.
\end{split}
\end{equation}
Finally, by (<ref>) and (<ref>), we obtain (<ref>). The proof is complete.
For the cases (II) and (III), we construct the approximate solution for boundary-Riemann problem by (<ref>). Due to the construction of the approximate solutions to the boundary-Riemann problems, the approximate solutions do not match the boundary conditions. We need to understand how the
the errors (<ref>) on the boundary affect the interaction of waves. Therefore, the wave interaction near the boundary is more complicated than the wave interaction in case (I). On the other hand, to estimate the wave interaction near the boundary, the exact direction of wave for each characteristic field must be known. Because of the positivity of the initial and boundary velocities, we can prove that the velocity is globally positive, as shown in Section 3.3.
Let us denote the boundary data at $n$th time strip by
\widehat{U}_B^n:=(\r(x_B,t_n),m(x_B,t_n),0)^T=(\r_B^n,m_B^n,0)^T
and let $U_B^n=\widetilde{U}_B^n+\widebar{U}_B^n$ be the solution of the generalized boundary-Riemann problem $\CMcal{BR}_G(x_B,t_n)$. We define the strength of the 0-wave on the boundary as
\a_0:=|R^{-1}(\widetilde{U}_B^n)\cdot(U_B^n-\widehat{U}_B^n)|.
According to <cit.>, the following theorem can be proved using the generalized version of Goodman's wave interaction estimates near the boundary:
(Boundary interaction estimate)
Let $\widehat{U}_B^n$ and $U_B^n$ as defined previously, and $U_B^{n+1}:=\widetilde{U}_B^{n+1}+\widebar{U}_B^{n+1}$, where $U_M^n,\ U_R^n,\text{and}\ U_R^{n+1}$
are defined as in Theorem 3.1., (see Figure 5).
Figure 5. Wave strengths in the region near the boundary $x=x_B$.
Moreover, let
represent the solutions of $\CMcal{BR}_G(\widehat{U}_B^n,U_M^n;x_B,t_n)$ and $\CMcal{BR}_G(\widehat{U}_B^{n+1},U_R^{n+1};x_B,t_{n+1})$, respectively. Assume that $(U_M^n,U_R^n):=[(U_M^n,U_R^n)/(\beta_1)]$ is the 1-wave of $\CMcal{R}_G(x_2,t_n)$ right next to $(\widehat{U}_B^n,U_M^n)$ on the $n$th time strip (see Figure. 5). Then there exists a constant $C$ such that
\begin{align}%-------------------------(4.26)-------------------------%
\label{4.26b}
|\varepsilon| &\leq |\a+\beta_1\mathbf{1}| +
\end{align}
where $\mathbf{1}=(1,1,1)$.
The boundary interaction estimate is calculated using the method reported in <cit.>. The estimation is constructed using the following steps.
(1) Decompose the wave interaction into two parts, the transmission part and the reflection part, and evaluate the interacted wave strength of these parts. (2) Estimate the effect of the Riemann solver (<ref>) on the wave strength. (3) Combine the estimation of (1) and (2) to complete the proof.
Step (1): We divide the interaction of the waves into two parts, the transmission part and the reflection part. The incoming wave $\b_1$ interacts with $\a_3,\ \a_2$, and $\a_0$ in order; it generates one penetrative wave $\b_1'$ through the boundary, see Figure 6, and two reflected waves $\a_2'$ and $\a_3'$ from the boundary, see Figure 7. For the transmission part, the wave strength $\b_1'$ and states $U_1'$ and $U_R'$ can be determined through the interaction estimates of waves $\beta_1,\ \a_0,\ \a_2$, and $\a_3$. Furthermore, according to Lemma 4.2 (a) in <cit.> and the triangle inequality, we obtain
\begin{eqnarray}%-------------------------(4.30)-------------------------%
\label{4.30b}
|U'_R-U_R^n|\leq C(|\a_0\beta_1|+|\a_2\beta_1|+|\a_3\b_1|).
\end{eqnarray}
It is evident that
\begin{eqnarray}%-------------------------(4.31)-------------------------%
\label{4.31b}
|U'_1-\widehat{U}^n_B|\leq C_1|\beta_1'|\le C_2|\b_1|,
\end{eqnarray}
for some constants $C_1,C_2$.
Figure. 6: Interaction of waves for the transmission part $\ve=\a+\b+$ higher order term.
Figure 7: Interaction of waves for the reflection part $\ve=\a+\a'+$ higher order term.
For the reflection part, the wave strengths $\a_2'$ and $\a_3'$ can be determined through the generalized boundary-Riemann problem $\CMcal{BR}_G(\widehat{U}_B^n;U_1')$. Therefore, we estimate the interaction of $\a_2'$ and $\a_3'$ and the waves $\a_2$ and $\a_3$. For any state $U=(\r,m,E)^T$, let us define
According to (<ref>), $\a_2'$ and $\a_3'$ satisfy
\begin{eqnarray}%-------------------------(4.32)-------------------------%
\label{4.32b}
|(\a_2',\a_3')|\leq C|U_1'-\widehat{U}_B^n|_{1,2}\leq C|\beta_1|.
\end{eqnarray}
Therefore, according to Lemma 4.2 (a) and (b) in <cit.> and the triangle inequality, we obtain
\begin{align}%-------------------------(4.33)-------------------------%
\label{4.33b}
|U''_R-U'_R|&\leq C(|\a_0\a_2'|+|\a_2\a_2'|+|\a_3\a_2'|+|\a_0\a_3'|+|\a_2\a_3'|+|\a_3\a_3'|)\leq C|(\a_2',\a_3')|,
\end{align}
where $U_R''$ is connected to $\widehat{U}_B'$ on the right by waves $\a_0,\ \a_2'+\a_2$, and $\a_3'+\a_3$. Wave strengths $\a_0$ and $\a_2$ are bounded; using this and (<ref>) and (<ref>), we have
\begin{align}%-------------------------(4.34)-------------------------%
\label{4.34b}
|U''_R-U'_R|\leq C|\beta_1|.
\end{align}
Let $(\widehat{U}^n_B, U^{n+1}_R):=[(\widehat{U}^n_B;U^{n+1}_R)/(\k_0,\k_2,\k_3)]$ be the approximate solution of $\mathcal{BR}_G(\widehat{U}^n_B;U^{n+1}_R)$, for some $\k_i,\ i=0,2,3$. According to the result in Section 2, there exists a smooth function $\Theta=(\Theta_0,\Theta_2,\Theta_3)$ connecting two constant states such that $(\varepsilon_0,\varepsilon_2,\ve_3)=\Theta(U^{n+1}_R;\widehat{U}^{n+1}_B)$ and $(\k_0,\k_2,\k_3)=\Theta(U^{n+1}_R;\widehat{U}^{n}_B)$. By (<ref>) and $|U^{n+1}_R-\widehat{U}^{n}_B|=O(1)|(\k_0,\k_2,\k_3)|$, we obtain
\begin{align}%-------------------------(4.4)-------------------------%
\label{4.27d}
&= \int^1_0 \frac{d}{d\xi}\Theta_j(\widehat{U}^{n+1}_B+\xi(U^{n+1}_R-\widehat{U}^{n+1}_B);\widehat{U}^{n+1}_B) d \xi
-\int^1_0 \frac{d}{d\xi}\Theta_j(\widehat{U}^{n}_B+\xi(U^{n+1}_R-\widehat{U}^{n}_B);\widehat{U}^{n}_B) d \xi\nonumber\\
&= \int^1_0 \{d\Theta_j(\widehat{U}^{n+1}_B+\xi(U^{n+1}_R-\widehat{U}^{n+1}_B);\widehat{U}^{n+1}_B)
\cdot(\widehat{U}^{n}_B-\widehat{U}^{n+1}_B) d \xi,\ j=0,2,3.
\end{align}
According to (<ref>),
\begin{align}%-------------------------(4.28)-------------------------%
\label{4.28b}
&\le O(1)|(\k_0,\k_2,\k_3)|\big(|\rho^{n+1}_B-\rho^{n}_B|+|m^{n+1}_B-m^{n}_B|\big)+O(1)(|\rho^{n+1}_B-\rho^{n}_B|+|m^{n+1}_B-m^{n}_B|)\nonumber \\
&\leq C(|\rho^{n+1}_B-\rho^{n}_B|+|m^{n+1}_B-m^{n}_B|), \quad j=0,2,3.
\end{align}
Step (2): Let us denote $S_B:=S(x_B,t_n,\widehat{U}_B^n)=S(x_B,t_n,\widetilde{U}_B^n)$ and $S_R:=S(x_2,t_n,\widetilde{U}_R^n)$, where $S$ is in (<ref>). According to (<ref>), (<ref>), and (<ref>), we obtain
\begin{align}%-------------------------(4.35)-------------------------%
\label{4.35b}
&\qquad\leq C\big|R^{-1}(\widetilde{U}_B^n)U_R^{n+1}-S_BR^{-1}(\widetilde{U}_B^n)U_R''\big|\nonumber\\
&\qquad\leq C(|U_R^{n+1}-S_RU_R^n|+|S_B||U_R-U_R'|+|S_B||U_R'-U_R''|)\nonumber\\
&\qquad\leq C(e^{h_B\tilde{u}_B^n\D t}\cosh(h_Bv_B\D t)|U_R-U_R'|+e^{h_B\tilde{u}_B^n\D t}\cosh(h_Bv_B\D t)|U_R'-U_R''|)\nonumber\\
&\qquad\leq Ce^{h_B\tilde{u}_B^n\D t}\cosh(h_Bv_B\D t)(|\a_0\beta_1|+|\a_2\beta_1|+|\a_3\b_1|+|\beta_1|),
\end{align}
where $\a_0':=0$, $h_B:=h(x_B)=-\frac{2}{x_B}$ and $v_B=v(x_B)$ as in (<ref>). Finally, by (<ref>) and (<ref>), we have
\begin{align}%-------------------------(4.36)-------------------------%
\label{4.36b}
&\leq|\k-S(x_B,t_n,\widehat{U}_B^n)(\a+\a')^T|+e^{h_B\tilde{u}_B^n\D t}\cosh(h_Bv_B\D t)(|\a'|+|\beta_1|)\nonumber\\
&\leq Ce^{h_B\tilde{u}_B^n\D t}\cosh(h_Bv_B\D t)(|\a_0\beta_1|+|\a_2\beta_1|+|\beta_1|).
\end{align}
Step (3): Finally, according to (<ref>) and (<ref>),
\begin{align*}%-------------------------(4.26)-------------------------%
&|\ve|\le|\ve-\k|+|\k^T-S(x_B,t_n,\widehat{U}_B^n)(\a+\b_1\mathbf{1})^T|+e^{h_B\tilde{u}_B^n\D t}\cosh(h_Bv_B\D t)|\a+\b_1|\nonumber\\
&\quad\leq |\a+\beta_1\mathbf{1}|+C\Big(\sum_{App}|\a_i||\beta_1|+|\beta_1|+|\rho_B^{n+1}-\rho_B^n|+|m_B^{n+1}-m_B^n|\Big).
\end{align*}
We complete the proof of the theorem.
If $\beta_1$ is not the incoming wave of the boundary triangle region, then (<ref>) is reduced by
\begin{align*}%-------------------------(4.36.1)-------------------------%
|\varepsilon| &\leq |\a|+C\big(|\rho_B^{n+1}-\rho_B^n|+|m_B^{n+1}-m_B^n|\big).
\end{align*}
§.§ The stability of the generalized Glimm scheme
In this subsection, we prove the nonincreasing of the Glimm functional and provide the estimate of the total variations of the perturbations, which lead to the compactness of subsequences of the approximate solutions for (<ref>).
Let $U_{\theta, \Delta x}$ denote the approximate solution for (<ref>) by the generalized Glimm scheme described in Section
3.1; $U_{\theta, \Delta x}$ can be decomposed as
\begin{equation*}
U_{\theta, \Delta x}=\widetilde{U}_{\theta, \Delta x}+\widebar{U}_{\theta, \Delta x},
\end{equation*}
where $\widetilde{U}_{\theta, \Delta x}$ is the approximate
solution obtained by solving homogeneous conservation laws in each time
step and $\widebar{U}_{\theta, \Delta x}$ is the perturbation
term; $\widetilde{U}_{\theta, \Delta x}$ and
its total variation are uniformly bounded. According to the results of
<cit.>, it can be accomplished that the
Glimm functional is nonincreasing in time.
Let $J$ be a mesh curve, $J'$ be the immediate successor of $J$, and $\Gamma_{k,n}$ be the diamond region enclosed by $J$ and $J'$, centered at $(x_{2k},t_n)$. The Glimm functional $F$ for $\widetilde{U}_{\theta ,\Delta x}$
over $J$ is defined as
\begin{equation}%------------------------(3.31)---------------------%
\label{glimfunl}
\end{equation}
where $K$ is a sufficiently large constant, which will be determined later, and
\begin{align*}
L(J)&:=\sum \{ |\a_i| : \a_i \ \mbox{crosses} \ J \}+K_1\Big(|\beta_1|+\sum_{k\in B(J)}l_B^k
\Big),\\
Q(J)&:= \sum \{ |\a_i||\a_{i'}|: \a_i,\a_{i'} \ \mbox{cross} \ J \ \mbox{and approach} \},\\
\end{align*}
Here, both constants $K>1$ and $K_1>1$ will be determined later, $B(J):=\{n:P_{x_B,n}=(x_B,t_n+\frac{\Delta t}{2})\in J\}$, $l_b^n$ is evaluated at the mesh point $P_{x_B,n}$, and the presence of $|\beta_1|$ is because $\beta_1$ crosses $J$ and locates in some boundary triangle region (see Figure 5).
We first consider the case that $J$ and $J'$ differ in the diamond region away from the boundary.
According to Theorem 3.1., let $Q(\Gamma_{k,n}):=D(\a,\beta)$ be the wave interaction potential associated with $\a$ and $\beta$ and let
C_1:=\max\limits_{U\in\Omega}\Big|\sum\limits_{j<i}L_{ij}(U)\Big|\ge C'_{k,n},\ \forall\,k,n.
By the condition $(A_2)$, $u_0(x)>0,\ \forall\,x\in[x_B,\infty)$. According to (<ref>), we have the following inequalities
\begin{align}%------------------------(3.32)---------------------%
\label{3.32}
L(J')-L(J)&\leq C_1Q(\Gamma_{k,n})-\l_*^{-1}\z_{k,n}(|\a|+|\beta|)\D x
+\l_*^{-1}C''_{k,n}(\Delta x)^2+O(1)(\Delta x)^3,\\
\label{3.33}%------------------------(3.33)---------------------%
Q(J')-Q(J)&\leq -Q(\Gamma_{k,n})+L(J)\big(C_1Q(\Gamma_{k,n})-\l_*^{-1}\z_{k,n}(|\a|+|\beta|)\Delta{x}\nonumber\\
&\quad+\l_*^{-1}C''_{k,n}(\Delta x)^2+O(1)(\Delta x)^3\big),
\end{align}
where $C''_{k,n},\z_{k,n}$ are in (<ref>) of Theorem <ref>.
By (<ref>), (<ref>), and (<ref>),
\begin{align}%-----------------------(3.34)---------------------%
\label{3.34}
F(J')-F(J)&\leq -\big(K-C_1-KC_1L(J)\big)Q(\Gamma_{k,n})-\l_*^{-1}\z_{k,n}(|\a|+|\beta|)\D x\nonumber\\
&\quad+\big(1+KL(J)\big)\l_*^{-1}C''_{k,n}(\Delta x)^2+O(1)(\Delta x)^3.
\end{align}
If $K$ satisfies $2C_1<K\le\frac{\e}{L(J)}$ for some $0<\e<\frac{1}{2}$, then
\begin{align}
\label{3.38.c}%-----------------------(3.38)---------------------%
F(J)=L(J)+KQ(J)\leq L(J) + K L^2(J)\leq(1+\e)L(J).
\end{align}
Coupling (<ref>) with (<ref>), we obtain the estimate
\begin{align}%-----------------------(3.36)---------------------%
\label{3.36}
F(J')&<F(J)-\l_*^{-1}\z_{k,n}(|\a|+|\beta|)\D x+\l_*^{-1}(1+\e)C''_{k,n}(\Delta x)^2+O(1)(\Delta x)^3.
\end{align}
Now, let $J_n$, $n=1,2,\ldots$, denote the mesh curves that contain all mesh points
$(x_k+\theta_{n-1}\Delta x,t_{n-1})$ at time $t=t_{n-1}$; therefore, $J_{n}$ is located on the time strip $T_n:=(x_B,\infty)\times[t_{n-1},t_n)$. We select the positive number $K$ such that
\begin{equation}%------------------------(3.35)---------------------%
\label{3.35}
\end{equation}
where $\mathcal{C}$ will be determined later. Then, $2C_1<K\le\frac{\e}{L(J_1)}$.
By (<ref>) and adding up recursive relation (<ref>) over all $k$,
and using $\int_{x_B}^{\infty}\frac{dx}{x^2}=\frac{1}{x_B}$ and $q\in W^{1,1}[x_B,\infty)$, we obtain
\begin{align*}%-----------------------(3.37)---------------------%
F(J_{2})&< F(J_1)-\l_*^{-1}C_2F(J_1)\Delta x+\l_*^{-1}C_3\Delta x+O(1)(\Delta x)^2,
\end{align*}
\begin{equation*}
C_2=\frac{3-\g}{x_B}\cdot\min_{t\ge 0}\{u_B(t)\},\quad\ C_3=\sqrt{3}\cdot\max_{U\in\Omega}\Big\{\frac{m}{x_B}+\frac{2\r v_B^2}{x_Bc_*}+\frac{(\g-1)\|q'\|_{L^1[x_B,\infty)}}{c_*^2}\Big\}.
\end{equation*}
Therefore, if $\D x$ sufficiently small, we have
\begin{align*}
L(J_2)\leq F(J_{2})&<\Big(1-\l_*^{-1}\frac{C_2}{1+\e}\Delta x\Big)F(J_1)+\l_*^{-1}(1+\e)C_3\Delta x+O(1)(\Delta x)^2 \nonumber\\
&< F(J_1)+(1+\e)^2\mathcal{C}+O(1)(\D x)^2 \nonumber\\
&\le (1+\e)L(J_1)+(1+\e)^2\mathcal{C}+O(1)(\D x)^2,
\end{align*}
\begin{equation}
\label{const}
\mathcal{C}:=\frac{C_3}{C_2}
=\frac{\sqrt{3}(1+\e)^2}{(3-\g)\min\limits_{t\ge 0}\{u_B(t)\}}\cdot\max\limits_{U\in\Omega}\Big\{m+\frac{\r GM_p}{x_Bc_*}+\frac{(\g-1)x_B\|q'\|_{L^1[x_B,\infty)}}{c_*^2}\Big\},
\end{equation}
and $c_*$ is given in (<ref>).
Define $m_*:=\min\limits_{t\ge 0}\{m_B(t)\}$. By the condition ($A_2$),
\begin{align*}
m(x,t_2)\geq m_B(t_2)-L(J_2)\geq m_*-(1+\e)L(J_1)-(1+\e)^2\mathcal{C}>0.
\end{align*}
This implies that $u_{\theta,\D x}(x,t_2)>0,\ \forall\,x\in(x_B,\infty)$.
Moreover, $L(J_2)\le(1+\e)L(J_1)+(1+\e)^2\mathcal{C}$ and
KL(J_2)\le K\big(L(J_1)+(1+\e)^2\mathcal{C}\big)+\e KL(J_1)+O(1)(\D x)^2\le\e(1+\e).
Therefore, $-K+C_1+KC_1L(J_2)<-(1-\e-\e^2)C_1<0$. According to (<ref>) and (<ref>)
and the similarly argument in the previous step, we further obtain
\begin{align*}
L(J_3)&\leq F(J_{3})\leq\Big(1-\l_*^{-1}\frac{C_2}{1+\e+\e^2}\D x\Big)F(J_2)+\l_*^{-1}(1+\e+\e^2)C_3\D x+O(1)(\Delta x)^2\nonumber\\
&\le \Big(1-\l_*^{-1}\frac{C_2}{1+\e+\e^2}\D x\Big)^2F(J_1)\nonumber\\
&\quad+\l_*^{-1}(1+\e+\e^2)C_3\D x\Big(1+\Big(1-\l_*^{-1}\frac{C_2}{1+\e+\e^2}\D x\Big)\Big)+O(1)(\Delta x)^2\nonumber\\
&\le F(J_1)+(1+\e+\e^2)^2\mathcal{C}+O(1)(\D x)^2\nonumber \\
&\le (1+\e)L(J_1)+(1+\e+\e^2)^2\mathcal{C}+O(1)(\D x)^2,
\end{align*}
\begin{align*}
m_{\theta,\D x}(x,t_3)\geq m_B(t_3)-L(J_3)\geq m_*-(1+\e)L(J_1)-(1+\e+\e^2)^2\mathcal{C}>0,
\end{align*}
which implies $u_{\theta,\D x}(x,t_3)>0,\ \forall\,x\in[x_B,\infty)$. Based on the selection of the constants $K$ and $\e$ in (<ref>), it can be verified that
\begin{align*}
KL(J_3)&\le K\big((1+\e)L(J_1)+(1+\e)^2\mathcal{C}\big)+\e KL(J_1)+\e^2(2+2\e+\e^2)K\mathcal{C}+O(1)(\D x)^2, \\
&\le\e+\e K\big(L(J_1)+(1+\e)^2\mathcal{C}\big)+O(1)(\D x)^2, \\
\end{align*}
Continue this process and by using induction, if (<ref>) holds true for all $J$ with $J_k\leq J<J_{k+1}$, $k=1,\ldots,n-1$, it yields
\begin{align*}%-----------------------(3.41)---------------------%
F(J_{n})&\le\Big(1-\l_*^{-1}\frac{C_2}{1+\e+\e^2}\D x\Big)F(J_{n-1})+\l_*^{-1}(1+\e+\e^2)C_3\D x+O(1)(\Delta x)^2\nonumber\\
&\leq \Big(1-\l_*^{-1}\frac{C_2}{1+\e+\e^2}\D x\Big)^{n-1}F(J_1)\nonumber\\
&\quad+\l_*^{-1}(1+\e+\e^2)C_3\D x\sum_{k=1}^{n-1}\Big(1-\l_*^{-1}\frac{C_2}{1+\e+\e^2}\D x\Big)^{k-1}+O(1)(\Delta x)^2,
\end{align*}
and therefore,
\begin{align}%-----------------------(3.42)---------------------%
\label{3.42}
F(J_{n})&\le F(J_1)+(1+\e+\e^2)^2\mathcal{C}+O(1)(\Delta x)^2\nonumber\\
&\leq (1+\e)\TV\{U_0\}+(1+\e+\e^2)^2\mathcal{C}+O(1)(\Delta x)^2, \\
m_{\theta,\D x}(x,t_n)&\geq m_B(t_n)-L(J_n)\geq m_*-(1+\e)L(J_1)-(1+\e+\e^2)^2\mathcal{C}>0.\nonumber
\end{align}
In particular, $u_{\theta,\D x}(x,t_n)>0,\ \forall\,x\in[x_B,\infty)$.
Therefore, the inequality (<ref>) leads to
\begin{align}%-----------------------(3.43)---------------------%
\label{3.43}
\TV_{J}\{\widetilde{U}_{\theta,\Delta x}\}&\leq O(1)L(J)\leq O(1)F(J)\nonumber\\
&\leq (1+\e)\TV\{U_0(x)\}+(1+\e+\e^2)^2\mathcal{C}+O(1)(\Delta x)^2
\end{align}
for $J_k\leq J<J_{k+1}$, $k=1,\ldots,n-1$.
Next, we consider the case that $J'$ is an immediate successor of $J$ so that they only
differ on boundary $P_{x_B,{n}}$.
According to the conditions ($A_2$), (<ref>), and (<ref>), we obtain
\begin{align*}%-------------------------(4.38)-------------------------%
&\leq O(1)C(|\a_0\beta_1|+|\a_2\beta_1|+|\a_3\beta_1|+|\beta_1|+l_B^k)-K_1(|\beta_1|+l_B^k)\nonumber\\
&\quad-K(|\a_0\beta_1|+|\a_2\beta_1|+|\a_3\beta_1|)+O(\Delta x)^2\nonumber\\
&\leq (-K_1+O(1)C+O(1)CK\cdot F(J))(|\beta_1|+l_B^k)\nonumber\\
&\quad+(-K+O(1)C+O(1)CK\cdot F(J))(|\a_0||\beta_1|+|\a_2||\beta_1|+|\a_3||\beta_1|)\nonumber\\
&\quad+O(1)(\Delta x)^2\leq O(1)(\Delta x)^2
\end{align*}
provided that constants $K_1$, $K\geq 2C_1$, and $ KL(J)\leq\e$. Now, let $J_n$ be the mesh curve located on the time strip $T_n:=(x_B,\infty)\times[t_{n-1},t_n)$ and include the half-ray $\{x=x_B,\;t\geq t_n+\frac{\Delta t}{2}\}$. Moreover, let
$\TV\{U_0(x)\}:=\TV\{\r_0(x)\}+\TV\{m_0(x)\}+\TV\{E_0(x)\}$. If $\Delta x$ and $\TV\{U_0(x)\}$ are sufficiently small,
then we have
\begin{align}%-------------------------(4.41)-------------------------%
\label{4.41b}
F(J_{k+1})\le F(J_{k})-\l_*^{-1}\frac{C_2}{1+\e+\e^2}(\D x)F(J_k)+O(1)(\Delta x)^2,\quad k=1,...,n.
\end{align}
Based on (<ref>) and the analysis similar to that in the interior wave interaction, we show that the Glimm functional $F$ is nonincreasing in time. Therefore, $\widetilde{U}_{\theta,\Delta x}$ is defined for $t > 0$ and $\Delta x \rightarrow 0$.
Next, we verify that the total variation of the perturbation
is bounded in any time step.
Let us denote $S_k:=S(x_k,t,\widetilde{U}_{\th,\D x}(x_k,t))$, where $S$ is given in (<ref>). Then,
\begin{equation*}
\begin{split}
\TV\{\widebar{U}_{\theta,\Delta x}\}
&=\sum_k|\widebar{U}_{\theta,\D x}(x_{k+1})
-\widebar{U}_{\theta,\D x}(x_{k-1})|\le\sum_k|\big((S-I_3)\widetilde{U}_{\theta,\D x}\big)(x_{k+1})
-\big((S-I_3)\widetilde{U}_{\theta,\Delta x}\big)(x_{k-1})| \\
&\le\sum_k|(S_z-I_3)(\widetilde{U}_{\th,\D x}(x_k+1)-\widetilde{U}_{\th,\D x}(x_{k-1})| \\
&\quad+\sum_k|(S_{k+1}-S_z)\widetilde{U}_{\th,\D x}(x_{k+1})+(S_z-S_{k-1})\widetilde{U}_{\th,\D x}(x_{k-1})|
\end{split}
\end{equation*}
According to (<ref>) and the definition of $h(x)=-\frac{2}{x}$, we obtain
\begin{align}
\label{3.45}
\TV\{\widebar{U}_{\th,\D x}\}
&\le\|S_z-I_3\|\sum_k\osc\{\widetilde{U}_{\th,\D x}\}+2\|W\|\sum_k\frac{1}{x_k^2}\D x\D t+\|\Psi\|\sum_k\osc\{\widetilde{U}_{\th,\D x}\} \nonumber\\
&\le\|S_z-I_3\|\TV\{\widetilde{U}_{\th,\D x}\}+\frac{2\|W\|}{x_B}\D t+\|\Psi\|\TV\{\widetilde{U}_{\th,\D x}\}.
\end{align}
Since $\TV\{\widetilde{U}_{\th,\D x}\}$ is finite, the total variation of $\widebar{U}_{\th,\D x}$ is bounded.
Because of the boundedness of the total variation of approximate solutions,
the constant $\mathcal{C}$ in (<ref>) can be easily determined by the initial and boundary data, heat profile, and gravity. By (<ref>), (<ref>), and the results in <cit.>, the following theorem
is achieved.
Let $K,\ \e$ be as chosen in (<ref>), and let $U_{\theta,\Delta x}$ be an approximate solution of (<ref>) based on the generalized Glimm scheme. Then, under the condition ${\rm(}A_1{\rm)}\sim{\rm(}A_3{\rm)}$,
for any given constant state $\widecheck{U}$, there exists a positive constant $d$, depending on the radius $r$ of $\Omega$, such that if
\begin{align*}
\sup_{x \in [x_B,\infty)}|U_0(x)-\widecheck{U}|\leq\frac{r}{2}, \quad\TV\{U_0(x)\}\leq d,
\end{align*}
and the condition
\begin{align*}
\sup_{t\in{\mathbb{R}^+}}|m_B(t)-\widecheck{m}|\leq\frac{r}{2}+(1+\e+\e^2)^2\mathcal{C}
\end{align*}
hold true for (<ref>) with the constant $\mathcal{C}$ in (<ref>),
then $U_{\theta,\Delta x}(x,t)$ is well-defined for $t \geq 0$ and $\Delta x>0$ is sufficiently small.
Furthermore, $U_{\th,\D x}(x,t)$ has a uniform bound on the total variation and satisfies the following properties:
(i) $\displaystyle \|U_{\theta,\Delta{x}}-\widecheck{U}\|_{L^\infty}\leq r+(1+\e+\e^2)^2\mathcal{C}$.
(ii) $\TV\{U_{\theta,\Delta{x}}(\cdot,t)\}\leq \dfrac{r}{2}+(1+\e+\e^2)^2\mathcal{C}$.
(iii) $\displaystyle\int_{x_B}^{\infty}|U_{\theta,\Delta{x}}(x,t_2)-U_{\theta,\Delta{x}}(x,t_1)|dx\leq
(iv) The velocity $u_{\th,\D x}(x,t)>0,\ \forall\,(x,t)\in\Pi$.
(v) The density $\r_{\th,\D x}(x,t)\ge\varrho,\ \forall\,(x,t)\in\Pi$, where $\varrho$ is the constant in ($A_1$).
We prove (ii) first. Choose a fixed $d$ such that $(1+\e)d\le\frac{r}{2}$. According to (<ref>), if
\begin{equation}
\label{3.44_a}
\sup|U_0(x)-\widecheck{U}|\leq \frac{r}{2},\quad\TV\{U_0(x)\}\leq d,
\end{equation}
then, for a sufficiently small $\Delta x>0$,
\TV_{J_n}\{U_{\th,\D x}\}\le(1+\e)\TV\{U_0(x)\}+(1+\e+\e^2)^2\mathcal{C}\le\frac{r}{2}+(1+\e+\e^2)^2\mathcal{C},
where the constant $\mathcal{C}$ is defined in (<ref>). Next, for (i),
\begin{align}%-----------------------(3.44)---------------------%
\label{3.44}
\sup_{J_n}|U_{\theta,\Delta x}-\widecheck{U}|\leq \sup |U_0(x)-\widecheck{U}|+\TV_{J_n}\{U_{\theta,\Delta x}\}.
\end{align}
By (<ref>) and (<ref>),
\begin{align*}%-----------------------(3.44.1)---------------------%
\sup_{\scriptstyle J_n}|\widetilde{U}_{\th,\Delta x}-\widecheck{U}|\leq r+(1+\e+\e^2)^2\mathcal{C}.
\end{align*}
By using the aforementioned constants $K$ and $d$, we obtain that, for a sufficiently small $\Delta x$, $U_{\th,\Delta x}(x,t)$ is defined on $\Pi$ when the condition ${\rm(}A_1{\rm)}\sim{\rm(}A_3{\rm)}$ hold true. In addition, $U_{\th,\Delta x}(x,t)$ and its total variation are uniformly bounded and independent of $\Delta x$.
For (iii), without loss of generality, let $t_2>t_1$, $t_0=\sup\{t\le t_1\mid t=n\D t\text{ for some }n\}$, and let $\ell=\lfloor\frac{t_2-t_0}{\D t}\rfloor+1$. According to (<ref>),
\begin{align*}
|U_{\th,\D x}(x,t_2)-U_{\th,\D x}(x,t_1)|
&\le |U_{\th,\D x}(y,t_0)-U_{\th,\D x}(x,t_0)|+|(S(y,\D t,\widetilde{U}_{\th,\D x}(y,t_0))^{\ell}-I)\widetilde{U}_{\th,\D x}(y,t_0)| \\
&=|U_{\th,\D x}(y,t_0)-U_{\th,\D x}(x,t_0)|+O(1)(\D t)
\end{align*}
for some $y\in[x-\ell\D x,x+\ell\D x]$. Therefore, according to the Corollary 19.8 in <cit.>, the result (iii)
is obtained immediately.
According to Theorem 3.4. and Oleinik's analysis in <cit.>,
the following theorem for the compactness of the subsequence
of $\{U_{\theta,\Delta x}\}$ holds true.
Assume that the condition ${\rm(}A_1{\rm)}\sim{\rm(}A_3{\rm)}$ hold true. Let $\{U_{\theta ,\Delta x}\}$ be a family of approximate solutions (<ref>) obtained using the generalized Glimm scheme. Then, there exist a subsequence
$\{U_{\theta ,\Delta x_i}\}$ of $\{U_{\theta ,\Delta x}\}$ and measurable function $U$ such that
(i) $U_{\theta ,\Delta x_i}(x,t)\rightarrow U(x,t)$ in $L^1_{loc}$ as
$\Delta x_i\rightarrow 0$;
(ii) for any continuous function $f$, we have $f(x,t,U_{\theta,\Delta x_i})\rightarrow f(x,t,U)$
in $L^1_{loc}$ as $\Delta x_i \rightarrow 0$.
§.§ The consistency and existence of the entropy solution
Finally, the global existence of entropy solutions to (<ref>)
is presented by demonstrating the consistency of the scheme and entropy inequalities for the weak solutions.
To achieve the consistency, the convergence of the residual to the approximate
solutions $\{U_{\theta ,\Delta x}\}$ of (<ref>) is given.
Consider the problem (<ref>) with the condition ${\rm(}A_1{\rm)}\sim{\rm(}A_3{\rm)}$. Assume that $\{U_{\th,\D x}\}$ is a sequence of approximate solutions for (<ref>), which are constructed using the generalized Glimm scheme. There exist a null set $N\subset \Phi$ and subsequence $\{\Delta x_i\}$ such that the limit
$$U(x,t):=\displaystyle\lim_{\Delta x_i \rightarrow 0\atop\theta \in \Phi \backslash N} U_{\theta,\Delta{x}_i}(x,t)$$
is an entropy solution of (<ref>).
We first calculate the residual of $\{U_{\theta ,\Delta x}\}$.
According to (<ref>), $U_{\th,\D x}=\widetilde{U}_{\th,\D x}+\widebar{U}_{\th,\D x}$. For convenience,
we omit the symbol $\th$ in $U,\ \widetilde{U}$, and $\widebar{U}$ in the rest of this subsection.
D_{k,n}:=[x_{k-1},x_{k+1}]\times[t_n,t_{n+1}],\ n=0,1,2,\cdots;k=1,2,\cdots,
\begin{equation*}%-----------------------------(4.0)--------------------------%
\lfloor U\rfloor(x,t_n):=U(x,t_n^+)-U(x,t_n^-).
\end{equation*}
We concentrate on the case that $D_{k,n}$ is away from the boundary, that is, $k\ge 1$; the case $k=0$ can be estimated similarly. For a test function
$\phi(x,t)\in C^1_0(\Pi)$,
by summing (<ref>) over all $D_{k,n}$, we have the following estimate
$\{U_{\Delta x}\}$:
\begin{align}
\label{4.1}
&\iint_{x>x_B,t>0}\{U_{\Delta{x}}\phi_t+f(U_{\Delta{x}})\phi_x+h(x)g(x,U_{\Delta{x}})\phi\}dx dt
+\int^{\infty}_{x_B} U_0(x)\phi(x,0) dx+\int^{\infty}_{0} f(\widehat{U}_B(t))\phi(x_B,t) dt\nonumber\\
&=\sum_{\scriptstyle n=0}^\infty\sum_{k=1}^{\infty}R(U_{\Delta{x}}, D_{k,n}, \phi)+\int_{x_B}^\infty U_0(x)\phi(x,0) dx
+\int_{0}^\infty f(\widehat{U}_B(t))\phi(x_B,t) dt\nonumber\\
&=-\sum_{n=1}^\infty\sum_{k=1}^{\infty}\int^{x_{k+1}}_{x_{k-1}}\lfloor\widetilde{U}_{\Delta{x}}+{\widebar{U}_{\Delta{x}}}\rfloor(x, t_n)
\phi(x, t_n)dx\nonumber\\
&=J(\theta , \Delta x , \phi)+O(1)\Delta x ,
\end{align}
J(\theta , \Delta x ,
\phi):=-\sum_{n=1}^\infty\sum_{k=1}^{\infty}\int^{x_{k+1}}_{x_{k-1}}\lfloor\widetilde{U}_{\Delta{x}}+\widebar{U}_{\Delta{x}}\rfloor(x, t_n)\phi(x, t_n)dx.
According to Glimm's argument in <cit.> and
\begin{align}%-----------------------------(4.2)--------------------------%
\label{4.2} J(\theta , \Delta x , \phi) \rightarrow 0\; \mbox{
as }\Delta{x}\rightarrow 0
\end{align}
for almost random sequence $\theta \in \Theta$, where $\Theta$ is a
probability space of the random sequence. Therefore, by (<ref>), (<ref>) and Theorem 3.3.,
there exist a null set $N\subset \Phi$ and subsequence $\{\Delta x_i\}$ such
that the limit
$$U(x,t):=\displaystyle\lim_{\Delta x_i \rightarrow
0\atop\theta \in \Phi \backslash N} U_{\theta,\Delta{x}_i}(x,t)$$
a weak solution
of (<ref>).
Next, we show that the aforementioned weak solution $U$ is indeed an entropy solution satisfying the entropy inequality (<ref>).
Given an entropy pair $(\eta,\om)$, define
\begin{align*}%-------------------------(4.2.1)-------------------------%
\widehat{R}(U_{\Delta{x}},\Pi, \phi)
:=\displaystyle\iint_{x>x_B,t>0}\left\{\eta({U}_{\Delta{x}})\phi_t+\om({U}_{\Delta{x}})\phi_x+d\eta({U}_{\D x})\cdot h(x)g(x,U_{\D x})\phi\right\}
\end{align*}
By using an argument similar to the proof of Theorem 2.2. (see Appendix B), we can estimate
the residuals of $U_{\D x}$ in $D_{k,n}$ as
\begin{align}%-----------------------------(2.16)--------------------------%
\label{RPres1}
&\widehat{R}(U_{\D x},D_{k,n},\phi)\ge\int^{x_{k+1}}_{x_{k-1}}\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(x,t^-_{n+1})dx
-\int^{x_{k+1}}_{x_{k-1}}\big(\eta(\widetilde{U}_{\D x})\p\big)(x,t)\Big|^{t=t^-_{n+1}}_{t=t^+_n}dx\nonumber\\
&\quad+\int^{t_{n+1}}_{t_n}\big(\om(U_{\D x})\p\big)(x,t)\Big|^{x=x_{k+1}}_{x=x_{k-1}}dt
+O(1)\left((\Delta t)^2(\Delta x)+(\Delta{t})^3+(\Delta{t})^2\underset{D_{k,n}}{\osc}\{\widetilde{U}\}\right)\|\phi\|_\infty.
\end{align}
Summing (<ref>) over all $D_{k,n}$, we have
\begin{align}%-----------------------------(4.15)--------------------------%
\label{4.15}
&\sum_{\scriptstyle n=0}^\infty\sum_{k=1}^{\infty}\widehat{R}(U_{\Delta{x}}, D_{k,n}, \phi)+\int_{-\infty}^\infty \eta(U_0(x))\phi(x,0)dx\nonumber\\
+d\eta(\widetilde{U}_{\Delta x})\widebar{U}_{\Delta x}\rfloor(x,t_n)\phi(x, t_n)dx\nonumber\\
&\quad-\int^{\infty}_{-\infty}\big(\eta(\widetilde{U}_{\Delta{x}}(x, 0^+))-\eta\left(U_0(x)\right)\big)\phi(x, 0)dx+O(\Delta{x})
\end{align}
for a sufficiently small $\Delta x$. Based on Glimm's argument, (<ref>) implies that
\begin{align*}%-------------------------(4.18)-------------------------%
&\iint_{x>x_B,t>0}\left\{\eta(U)\phi_t+\om(U)\phi_x+{d\eta}\cdot h(x)g(x,U)\phi\right\} dxdt
&\hspace{9cm}+\int_0^{\infty}\om(\widehat{U}_B(t))\p(x_B,t)dt\geq 0
\end{align*}
for every entropy pair $(\eta,\om)$ and positive test function $\phi\in C^1_0(\Pi)$. The existence of the entropy solution of (<ref>) is established.
§ HYDRODYNAMIC REGIONS
In Section 3, the global existence of the entropy solution to HEP (<ref>) is established on the basis of the initial density with a positive lower bound. However, (<ref>) does not fulfill the physical meaning because of which the atmosphere density reaches vacuum as $x$ approaches infinity. Therefore, it is necessary to determine the hydrodynamic region, a subset of $\Pi\equiv[x_B,\infty)\times[0,\infty)$, in which our solution of (<ref>) is physically well-defined. This section is devoted to establishing the main Theorem II. For a certain constraint on the transonic initial data $U_0(x)=(\r_0(x),m_0(x),E_0(x))^T$, where $\r_0,E_0$ is a decreasing function and $m_0$ is an increasing function in $[x_B,\infty)$, there exists a region $\Sg_1\equiv[x_B,x^*]\times[0,\infty)$ such that the wave speeds of the solutions to (<ref>) are positive in $\Pi\backslash\Sg_1$. Next, by adopting the Knudsen number of the gas, we prove that there also exists a region $\Sg_2=[x_B,x^{**}]\times[0,\infty)$ such that the gas in $\Pi\backslash \Sg_2$ no longer acts like fluid. In other words, (<ref>) fails to model the HEP outside $\Sg_2$, and the governed equations must be replaced by the kinetic equations. According to the assumed initial data described previously, there exists a nonempty hydrodynamic region of HEP (<ref>) such that $\Sg\equiv\Sg_1\cup\Sg_2$.
Let $U(x,t)=(\r(x,t),m(x,t),E(x,t))^T$ be the solution of (<ref>) constructed in Theorem 3.5., with initial data $U_0(x)=(\r_0(x),m_0(x),E_0(x))^T$ and boundary data $(\r_B(t),m_B(t))^T$. Then, for any $(x,t)\in\Pi$, we have
\begin{equation}
\label{globlestimate}
\begin{split}
&|\r(x,t)-\r_0(x)|\le\TV\{\r_0(x)\}+\TV\{\r_B(t)\}, \\
&|m(x,t)-m_0(x)|\le\TV\{m_0(x)\}+\TV\{m_B(t)\}, \\
\end{split}
\end{equation}
For any $(x,t)\in\Pi$, let $k=\lfloor\frac{x}{\D x}\rfloor,\ n=\lfloor\frac{t}{\D t}\rfloor+1$, and let $D_{k,n}$ denote the Riemann cell containing the point $(x,t)$. By (<ref>) and the random choice process, the approximate solution in the $n$th time step satisfies
\begin{equation*}
\begin{split}
&|U_{\th,\D x}(x,n\D t)-U_{\th,\D x}(x,(n-1)\D t)| \\
&\le\big|\big(S(y,(n-1)\D t,\widetilde{U}_{\th,\D x}(y,(n-1)\D t))-I\big)\widetilde{U}_{\th,\D x}(y,(n-1)\D t)\big| \\
&\quad+|\widetilde{U}_{\th,\D x}(y,(n-1)\D t)-U_{\th,\D x}(x,(n-1)\D t)|,
\end{split}
\end{equation*}
for some $y\in D_{k,n}$ such that $U_{\th,\D x}(x,n\D t)=S(y,(n-1)\D t,\widetilde{U}_{\th,\D x}(y,(n-1)\D t))\widetilde{U}_{\th,\D x}(y,(n-1)\D t)$. More precisely, according to (<ref>),
\begin{equation*}
\begin{split}
\r_{\th,\D x}(x,n\D t)&=\tilde{\rho}_{\th,\D x}(y,(n-1)\D t)-\frac{2\tilde{m}_{\th,\D x}(y,(n-1)\D t)}{y} \Delta t +O(1)(\Delta t)^2,\\
m_{\th,\D x}(x,n\D t) &=\tilde{m}_{\th,\D x}(y,(n-1)\D t) \\
&\quad-\frac{2\tilde{\rho}_{\th,\D x}(y,(n-1)\D t)}{y}\big(\tilde{u}_{\th,\D x}(y,(n-1)\D t)^2+v(y)^2\big)\Delta t +O(1)(\Delta t)^2,\\
E_{\th,\D x}(x,n\D t)&=\widetilde{E}_{\th,\D x}(y,(n-1)\D t)+q(y)\D t \\
&\quad-\frac{2\tilde{m}_{\th,\D x}(y,(n-1)\D t)}{y}\Big(\frac{1}{2}\tilde{u}_{\th,\D x}(y,(n-1)\D t)^2+v(y)^2+\frac{\tilde{c}_{\th,\D x}(y,(n-1)\D t)^2}{\g-1}\Big)\Delta t \\
&\quad+O(1)(\Delta t)^2,
\end{split}
\end{equation*}
where $v(y)=\sqrt{\frac{GM_p}{2y}}$. Since the density and momentum are globally positive according to Theorem 3.3.,
\begin{eqnarray*}
&& |\r_{\th,\D x}(x,n\D t)-\r_{\th,\D x}(x,(n-1)\D t)|\le\underset{D_{k,n}}{\osc}\{\tilde{\r}_{\th,\D x}\}, \\
&& |m_{\th,\D x}(x,n\D t)-m_{\th,\D x}(x,(n-1)\D t)|\le\underset{D_{k,n}}{\osc}\{\tilde{m}_{\th,\D x}\}, \\
&& |E_{\th,\D x}(x,n\D t)-E_{\th,\D x}(x,(n-1)\D t)|\le\underset{D_{k,n}}{\osc}\{\widetilde{E}_{\th,\D x}\}+[q(x)+q(y)]\D t,
\end{eqnarray*}
where $q$ is the heat profile. Note that the grid $D_{k,n}$ is the domain of dependence of $(x,t)$ for one time step. By using backward induction on $n$ and passing to the limit as $\D x\to 0,\ \th\in\Phi \backslash N$ in Theorem 3.5., the result (<ref>) is obtained.
Next, we show that there exist $x^*>x_B$ and $\Sg_1\equiv[x_B,x^*]\times[0,\infty)$ such that the wave speeds of $U(x,t)|_{\Pi\backslash \Sg_1}$ are positive. Recall that the Mach number of $U$ is defined as
\mathfrak{Ma}(U):=\frac{|u|}{c}
Assume that the transonic initial data $U_0=(\r_0,m_0,E_0)^T$ satisfying the condition ${\rm(}A_1{\rm)}\sim{\rm(}A_2{\rm)}$, where $\r_0,E_0$ are decreasing, $m_0$ is increasing, and $u_0(x_B)<c_0(x_B)$. Let $U=(\r,m,E)^T$ be the solution of (<ref>) constructed using Theorem 3.5.; then there exist $x^*\in(x_B,\infty)$ and $\Sg_1\equiv[x_B,x^*]\times[0,\infty)$ such that the characteristic speeds of the solution $U(x,t)$ in $\Pi\backslash\Sg_1$ are positive.
According to (<ref>), the Mach number $\mathfrak{Ma}$ satisfies
\mathfrak{Ma}(x,t)^2=\frac{u(x,t)^2}{c(x,t)^2}=\frac{m(x,t)^2}{\g(\g-1)(\r(x,t)E(x,t)-m(x,t)^2)}\ge\frac{m(x,t)^2}{\g(\g-1)\r(x,t)E(x,t)}.
Let $E^*:=\max\limits_{x\in[x_B,\infty)}E_0(x)+\TV\{E_0\}+\TV\{E(x_B,t)\}+2\l_*^{-1}\|q\|_{L^1[x_B,\infty)}$, where $\l_*$ is defined in (<ref>).
According to Lemma 4.1., $E(x,t)\le E^*$ for all $(x,t)\in\Pi$ and therefore,
\mathfrak{Ma}(x,t)\ge\frac{m(x,t)}{\sqrt{\g(\g-1)E^*\r(x,t)}}\text{ for all }(x,t)\in\Pi.
Next, we define
\mathfrak{M}(x):=\frac{m_0(x)-\TV\{m_0\}-\TV\{m_B\}}{\sqrt{\g(\g-1)E^*(\r_0(x)+\TV\{\r_0\}+\TV\{\r_B\})}}.
According to Lemma 4.1.,
\begin{equation*}
\mathfrak{Ma}(x,t)\ge\mathfrak{M}(x),\ \forall\,(x,t)\in\Pi.
\end{equation*}
The function $\mathfrak{M}(x)$ is increasing based on the assumption of $\r_0$ and $m_0$. Since the initial data is transonic with $u_0(x_B)<c_0(x_B)$, $\mathfrak{M}(x_B)\le\mathfrak{Ma}(x_B,0)<1$. On the other hand, according to ($A_1$), the function $\mathfrak{M}(x)$ is greater than 1 in the far field, when $\r_0$ is near $\vr$ which is sufficiently small. There exists $x^*\in(x_B,\infty)$ such that $\mathfrak{M}(x^*)=1$, and $\mathfrak{M}(x)>1$ for $x\in(x^*,\infty)$. Therefore,
\begin{eqnarray*}
&&\mathfrak{Ma}(x^*,t)\ge\mathfrak{M}(x^*)=1,\quad \text{for }t\in[0,\infty), \\
&&\mathfrak{Ma}(x,t)\ge\mathfrak{M}(x)>1,\quad \text{for }(x,t)\in(x^*,\infty)\times[0,\infty).
\end{eqnarray*}
Denote the region $\Sg_1\equiv[x_B,x^*]\times[0,\infty)$. We have shown that $u(x,t)\ge c(x,t)$, for all $(x,t)\in\Pi\backslash\Sg_1$, that is, the characteristic speeds of the solution $U(x,t)$ in $\Pi\backslash\Sg_1$ are positive.
Next, we want to determine the hydrodynamic region $[x_B,x_T]$ such that the constructed solution $U(x,t)$ has mathematical and physical significance. The Knudsen number is defined as $\mathfrak{Kn}=\frac{l}{\mathcal{H}}$, the ratio of the mean free path of the molecules, $l=\frac{1}{\sqrt{2}\t\mathfrak{n}}$, to the density scale height, $\mathcal{H}=\frac{k_BT}{GM_p\mathfrak{m}/x^2}$, of the atmosphere. The region of validity of the hydrodynamic equation is often classified using the Knudsen number $\mathfrak{Kn}$, which is useful for determining whether statistical mechanics or continuum mechanics formulation of fluid dynamics must be used. Here, $\t\approx 10^{14}\pi\ \text{cm}^2$ is the collision cross section used in <cit.>, $\mathfrak{n}$ is the number density, $\mathfrak{m}$ is the mass of a molecule, $T$ is the temperature of the gas, and $k_B$ is the Boltzmann constant. According to (<ref>),
P=\r RT=\mathfrak{n}k_BT.
The Knudsen number can be computed as follows:
\begin{equation*}
\mathfrak{Kn}(x,U)=\frac{l}{\mathcal{H}}=\frac{GM_p\mathfrak{m}}{\sqrt{2}\t\mathfrak{n}x^2k_BT}=\frac{GM_p\mathfrak{m}}{\sqrt{2}\t x^2P}
=\frac{\g GM_p\mathfrak{m}}{\sqrt{2}\t x^2\r c^2}.
\end{equation*}
The hydrodynamic equations are applied appropriately, where $\mathfrak{Kn}<1$, so that many collisions occur over relevant length scales keeping the gas in thermal equilibrium. If $\mathfrak{Kn}\ge 1$, the continuum assumption of fluid mechanics maybe no longer be a good approximation because there are few collisions in this level to inhibit a molecule from escaping.
For a transonic initial data $U_0=(\r_0,m_0,E_0)^T$ as in Theorem 4.2. and $\mathfrak{Kn}(x_B,0)<1$,
\begin{equation}
\label{knudsencond}
x_B^2\r_0(x_B)c_0(x_B)^2>\frac{\g GM_p\mathfrak{m}}{\sqrt{2}\,\t}.
\end{equation}
Let $U=(\r,m,E)^T$ be the solution of (<ref>) constructed using Theorem 3.5.; then, there exist $x^{**}\in(x_B,\infty)$ and $\Sg_2\equiv[x_B,x^{**}]\times[0,\infty)$ such that $\mathfrak{Kn}(x,t)\le 1,\ \forall\,(x,t)\in\Sg_2$.
Let us denote $\psi(x,t):=\r(x,t)c(x,t)^2$ and $\psi_0(x):=\r_0(x)c_0(x)^2$; then, $\psi$ and $\psi_0$ are of bounded variation functions based on the construction of solutions in Theorem 3.5., and the lower boundedness of the density in ${\rm(}A_1{\rm)}$. Moreover, according to (<ref>) and the assumption of the initial data,
\begin{equation*}
\psi'_0(x)=\frac{\g(\g-1)m_0^2}{2\r_0^2}\r_0'(x)-\frac{\g(\g-1)m_0}{\r_0}m_0'(x)+\g(\g-1)E_0'(x)<0,
\end{equation*}
that is, $\psi_0(x)$ is a decreasing function of $x$. Define the function $\psi_*(x)$ as
\begin{equation}
\label{defpsi}
\psi_*(x):=\psi_0(x)-\TV\{\psi_0(x)\}-\TV\{\psi(x_B,t)\}-2\l_*^{-1}\g(\g-1)\|q\|_{L^1[x_B,\infty)}.
\end{equation}
According to (<ref>), for any $(x,t)\in\Pi$, the approximation of $\psi(x,t)$ in the $n$th time step can be evaluated as
\begin{equation*}
\begin{split}
\psi_{\th,\D x}(x,n\D t)&=\widetilde{\psi}_{\th,\D x}(y,(n-1)\D t)+\g(\g-1)q(y)\D t \\
&\quad-\frac{2\g\tilde{m}_{\th,\D x}(y,(n-1)\D t)\tilde{c}_{\th,\D x}(y,(n-1)\D t)^2}{y}\D t+O(1)(\Delta t)^2,
\end{split}
\end{equation*}
where $\psi_{\th,\D x}(x,n\D t):=\r_{\th,\D x}(x,n\D t)c_{\th,\D x}(x,n\D t)^2$ and $k,n$, and $y$ are as defined in Lemma 4.1. Based on the similar argument in Lemma 4.1 and (<ref>), $\psi_*$ is a decreasing function and $\psi_*(x)\le\psi(x,t),\ \forall\,(x,t)\in\Pi$.
Next, define
\mathfrak{K}(x):=\frac{\g GM_p\mathfrak{m}}{\sqrt{2}\t x_B^2\psi_*(x)}.
Since $\psi_*(x)\le\psi(x,t)$ for all $(x,t)\in\Pi$,
\mathfrak{Kn}(x,t)\le\frac{\g GM_p\mathfrak{m}}{\sqrt{2}\t x_B^2\psi(x,t)}\le\mathfrak{K}(x),\ \forall\,(x,t)\in\Pi.
According to (<ref>) and the assumption (<ref>), $\mathfrak{K}(x_B)<1$. On the other hand, according to the decrease of the function $\psi_*$ and the assumption ($A_1$), $\mathfrak{K}(x)>1$ in the far field whenever $\r_0$ near $\vr$. There exists $x^{**}\in(x_B,\infty)$ such that $\mathfrak{K}(x^{**})=1$, and $\mathfrak{K}(x)\le1$ for $x\in[x_B,x^{**}]$. Therefore,
\begin{eqnarray*}
&&\mathfrak{Kn}(x^{**},t)\le\mathfrak{K}(x^{**})=1,\quad \text{for }t\in\times[0,\infty), \\
&&\mathfrak{Kn}(x,t)\le\mathfrak{K}(x)<1,\quad \text{for }(x,t)\in[x_B,x^{**})\times[0,\infty).
\end{eqnarray*}
Denote the region $\Sg_2\equiv[x_B,x^{**}]\times[0,\infty)$. Therefore, the Knudsen number $\mathfrak{Kn}(x,t)\le 1$ for all $(x,t)\in\Sg_2$.
Finally, according to Theorem 4.2. and Theorem 4.3, we define the hydrodynamic region of HEP (<ref>) by using $\Sg\equiv\Sg_1\cup\Sg_2$. The wave speeds of the solution $U(x,t)$, constructed in Theorem 3.5., are positive in the region $\Pi\backslash\Sg$. Moreover, the Knudsen number $\mathfrak{Kn}(x,U)\le 1$ in the region $\Sg$. Therefore, we obtain an entropy solution $U(x,t)$ of (<ref>) that has both mathematical and physical significance in the hydrodynamic region $\Sg$.
§ CONSTRUCTION OF THE RIEMANN SOLVER
Here, we describe the construction of the Riemann solver in (<ref>). Based on Theorem 2.1., let $\widetilde{U}$ be the entropy solution for $\mathcal{R}_C(x_0,t_0)$,
we only need to construct the perturbation $\widebar{U}$ of $U$ from $\widetilde{U}$. Let us denote
\tilde{u}:=\frac{\tilde{m}}{\tilde{\r}},\quad v(x):=\sqrt{\frac{GM_p}{2x}},
and consider the linearized system of (<ref>) around $\widetilde{U}$ with initial data $\widebar{U}(x,0)= 0$:
\begin{equation}
\label{perturb}
\left\{\begin{split}
&\widebar{U}_t+(A(x,t)\widebar{U})_x=B(x,t)\widebar{U}+C(x,t), \\
\end{split}\right.\quad (x,t)\in D(x_0,t_0).
\end{equation}
\begin{eqnarray*}
&& A(x,t)=df(\widetilde{U})=\left[\begin{array}{ccc}
0 & 1 & 0 \\
\frac{\g-3}{2}\tilde{u}^2 & (3-\g)\tilde{u} & \g-1 \\
\vspace{-0.4cm} \\
\frac{\g-1}{2}\tilde{u}^3-\tilde{u}\widetilde{H} & \widetilde{H}-(\g-1)\tilde{u}^2 & \g\tilde{u}
\end{array}\right], \\
&& B(x,t)=h(x)g_U(x,\widetilde{U})=h(x)\left[\begin{array}{ccc}
0 & 1 & 0 \\
v^2-\tilde{u}^2 & 2\tilde{u} & 0 \\
\frac{\g-1}{2}\tilde{u}^3-\tilde{u}\widetilde{H} & \widetilde{H}-(\g-1)\tilde{u}^2-v^2 & \g\tilde{u}
\end{array}\right], \\
&& C(x,t)=h(x)g(x,\widetilde{U})=h(x)\left[\begin{array}{c}
\tilde{m} \\ \tilde{\r}(\tilde{u}^2+v^2) \\ \tilde{m}(\widetilde{H}+v^2)-\frac{xq}{2}
\end{array}\right],
\end{eqnarray*}
and $\widetilde{H}=H(\widetilde{U})$ is given in (<ref>). Due to the appearance of shocks or discontinuity in $\widetilde{U}$, the coefficients in (<ref>) may be discontinuous. To obtain the better regularity of approximate solutions for (<ref>), the averaging process of the coefficients in (<ref>) with respect to $t$ over $[0,\D t]$ is used. For a bounded variation function $z(x,t,U)$, the average of $z(x,t,U)$ is defined as
\begin{equation*}
z_{\star}(x):=\frac{1}{\Delta t}\int_{t_0}^{t_0+\Delta t}z(x,s,U(x,s))ds, \quad x_0-\Delta x\leq x\leq x_0+\Delta x.
\end{equation*}
In addition, $w_{\star}(x)$ is continuous even across the shock and contact discontinuity.
We construct approximate solution for (<ref>) by solving
\begin{equation}
\label{avgpde}
\left\{\begin{split}
&(\widebar{U}_\star)_t+(A_\star(x)\widebar{U}_\star)_x=B_\star(x)\widebar{U}_\star+C_\star(x), \\
\end{split}\right.\quad (x,t)\in D(x_0,t_0).
\end{equation}
Based on the operator-splitting method, solutions for (<ref>) can be approximated through composing solutions for
\begin{equation}
\label{avghomo}
\left\{\begin{split}
&V_t+(A_\star(x)V)_x=0, \\
\end{split}\right.\quad (x,t)\in D(x_0,t_0),
\end{equation}
\begin{equation}
\label{avgode}
\left\{\begin{split}
&Y_t=B_{\star}(x)Y+C_{\star}(x), \\
&Y(x,0)=V(x,\D t),
\end{split}\right.\quad (x,t)\in D(x_0,0).
\end{equation}
Since the system (<ref>) admits zero solution. The solution for (<ref>) can be approximated as the solution of the follows:
\begin{equation}
\label{splitode}
\left\{\begin{split}
&(\widebar{U}_\star)_t=B_\star(x)\widebar{U}_\star+C_\star(x), \\
\end{split}\right.\quad (x,t)\in D(x_0,t_0),
\end{equation}
where $B_{\star}(x)$ and $C_{\star}(x)$ can be derived from $\widetilde{U}_{\star}$. The averaging process is reasonable because
\begin{equation*}
\iint_{D(x_0,t_0)}|\widebar{U}-\widebar{U}_{\star}|dxdt=O(1)\left((\Delta
t)^3+(\Delta t)^2\cdot\TV_{D(x_0,t_0)}\{\widetilde{U}\}\right).
\end{equation*}
The matrix $B_{\star}(x)$ in (<ref>) has eigenvalues
\begin{equation*}
\s_1(x)=h(\tilde{u}_{\star}+v),\quad\s_2(x)=\g h\tilde{u}_{\star},\quad\s_3(x)=h(\tilde{u}_{\star}-v).
\end{equation*}
and the corresponding eigenvectors
P_1(x)=(1,\tilde{u}_{\star}+v,\D_1)^T,\quad P_2(x)=(0,0,1)^T,\quad P_3(x)=(1,\tilde{u}_{\star}-v,\D_2)^T.
\begin{eqnarray*}
\D_1 &=& \frac{v^3}{v-(\g-1)\tilde{u}_{\star}}+\frac{2v^2-(\g-1)\tilde{u}_{\star}(\tilde{u}_{\star}+3v)}{2(v-(\g-1)\tilde{u}_{\star})}\tilde{u}_{\star}
+\frac{\g v}{\tilde{\r}_{\star}(v-(\g-1)\tilde{u}_{\star})}\widetilde{E}_{\star}, \\
\D_2 &=& \frac{v^3}{v+(\g-1)\tilde{u}_{\star}}-\frac{2v^2-(\g-1)\tilde{u}_{\star}(\tilde{u}_{\star}-3v)}{2(v+(\g-1)\tilde{u}_{\star})}\tilde{u}_{\star}
+\frac{\g v}{\tilde{\r}_{\star}(v+(\g-1)\tilde{u}_{\star})}\widetilde{E}_{\star}.
\end{eqnarray*}
The transformation matrix of $B_*(x)$ is then given by $P(x)=[P_1(x),P_2(x),P_3(x)]$ so that $B_*(x)$ can be diagonalized as
\begin{align*}%---------------------------(2.10)-------------------------%
\L(x)=P^{-1}(x)B_{\star}(x)P(x)=h(x)\diag[\tilde{u}_{\star}+v,\g\tilde{u}_{\star},\tilde{u}_{\star}-v].
\end{align*}
Moreover, the fundamental matrix solved using $\dot{X}(t)=\L(x)X(t)$ is
\begin{align*}%---------------------------(2.11)-------------------------%
e^{\L(x)t}=\diag[e^{h(\tilde{u}_{\star}+v)t},e^{\g h\tilde{u}_{\star}t},e^{h(\tilde{u}_{\star}-v)t}].
\end{align*}
Using the transformation matrix $P$, the state transition matrix of $\dot{X}(t)=B(x)X(t)$ is given by
\begin{equation}
\label{transmatrx}
N(x,t,s)=P(x)e^{\L(x)(t-s)}P(x)^{-1}=N_1(x)e^{h(\tilde{u}_{\star}+v)(t-s)}+N_2(x)e^{\g h\tilde{u}_{\star}(t-s)}+N_3(x)e^{h(\tilde{u}_{\star}-v)(t-s)},
\end{equation}
\begin{equation*}
\begin{split}
-(\tilde{u}_{\star}-v) & 1 & 0 \\
-(\tilde{u}_{\star}^2-v^2) & v+\tilde{u}_{\star} & 0 \\
-(\tilde{u}_{\star}-v)\D_1 & \D_1 & 0
\end{array}\right], \\
0 & 0 & 0 \\
0 & 0 & 0 \\
(\tilde{u}_{\star}-v)\D_1-(\tilde{u}_{\star}+v)\D_2 & \D_2-\D_1 & 2v
\end{array}\right], \\
\tilde{u}_{\star}+v & -1 & 0 \\
\tilde{u}_{\star}^2-v^2 & -(\tilde{u}_{\star}+v) & 0 \\
(\tilde{u}_{\star}+v)\D_2 & -\D_2 & 0
\end{array}\right].
\end{split}
\end{equation*}
According to (<ref>) and the variation of constant formula,
the solution for (<ref>) is given by
\begin{eqnarray}
\label{avgperturb}
&& \hspace{-0.7cm}\widebar{U}_{\star}(x,t)=\int_{t_0}^{t_0+t}N(x,t,s)C_{\star}(x)ds
=\sum_{i=1}^3N_i(x)C_{\star}(x)\int_{t_0}^{t_0+t}e^{\s_i(x)(t-s)}ds \nonumber \\
&& \hspace{0.55cm}=\frac{\tilde{\r}_{\star}}{2}(e^{h(\tilde{u}_{\star}+v)t}-1)\left[\begin{array}{c}
1 \\ \tilde{u}_{\star}+v \\ \D_1
\end{array}\right]+\frac{\tilde{\r}_{\star}}{2}(e^{h(\tilde{u}_{\star}-v)}-1)\left[\begin{array}{c}
1 \\ \tilde{u}_{\star}-v \\ \D_2
\end{array}\right] \nonumber \\
&& \hspace{2cm}+\frac{\tilde{\r}_{\star}}{2\g\tilde{u}_{\star}}(e^{\g h\tilde{u}_*t}-1)\left[\begin{array}{c}
0 \\ 0 \\ -(\tilde{u}_{\star}+v)\D_1-(\tilde{u}_{\star}-v)\D_2+2\tilde{u}_{\star}(\widetilde{H}_{\star}+v^2)-\frac{xq}{\tilde{\r}_{\star}}
\end{array}\right] \nonumber \\
%&& \hspace{0.55cm}=\frac{e^{h(\tilde{u}_{\star}+v)t}-1}{2}\left[\begin{array}{c}
%\tilde{\r}_{\star} \\ \tilde{m}_{\star}+v\tilde{\r}_{\star} \\
%+\frac{\g v}{v-(\g-1)\tilde{u}}_{\star}\widetilde{E}_{\star}
%\end{array}\right] \nonumber \\
%&& \hspace{0.8cm}+\frac{e^{\g h\tilde{u}_{\star}t}-1}{2}\left[\begin{array}{c}
%0 \\ 0 \\ \big(\frac{2v^4}{v^2-(\g-1)^2\tilde{u}_{\star}^2}+\frac{xq}{\g\tilde{m}_{\star}}\big)\tilde{\r}_{\star}
%\end{array}\right] \nonumber \\
%&& \hspace{0.8cm}+\frac{e^{h(\tilde{u}_{\star}-v)t}-1}{2}\left[\begin{array}{c}
%\tilde{\r}_{\star} \\ \tilde{m}_{\star}-v\tilde{\r}_{\star} \\
%+\frac{\g v}{v+(\g-1)\tilde{u}_{\star}}\widetilde{E}_{\star}
\end{eqnarray}
where $\widetilde{H}_\star=H(\widetilde{U}_\star)$ and $t\in[0,\D t]$. Finally, replacing $\widetilde{U}_{\star}$ in (<ref>) by $\widetilde{U}$,
the perturbation $\widebar{U}$ is obtained and is given in the form
\begin{equation}
\label{perturbsol}
\widebar{U}(x,t)=(S(x,t,\widetilde{U})-I_3)\widetilde{U},
\end{equation}
\begin{equation*}
e^{h\tilde{u}t}\cosh(hvt) & 0 & 0 \\
\vspace{-0.4cm} \\
ve^{h\tilde{u}t}\sinh(hvt) & e^{h\tilde{u}t}\cosh(hvt) & 0 \\
S_{31} & S_{32} & S_{33}
\end{array}\right],
\end{equation*}
\begin{eqnarray*}
&&S_{31}=\frac{-xq}{2\g\tilde{m}}(e^{\g h\tilde{u}t}-1)-\frac{v^3}{v^2-(\g-1)^2\tilde{u}^2}
(ve^{\g h\tilde{u}t}-ve^{h\tilde{u}t}\cosh(hvt)-(\g-1)\tilde{u}e^{h\tilde{u}t}\sinh(hvt)), \\
(e^{\g h\tilde{u}t}-e^{h\tilde{u}t}\cosh(hvt))
+\frac{v(2v^2-(3\g-2)(\g-1)\tilde{u}^2)}{2(v^2-(\g-1)^2\tilde{u}^2)}e^{h\tilde{u}t}\sinh(hvt), \\
&&S_{33}=-\frac{(\g-1)(v^2+(\g-1)\tilde{u}^2)}{v^2-(\g-1)^2\tilde{u}^2}e^{\g h\tilde{u}t}
+\frac{\g v}{v^2-(\g-1)^2\tilde{u}^2}(ve^{h\tilde{u}t}\cosh(hvt)+(\g-1)\tilde{u}e^{h\tilde{u}t}\sinh(hvt)).
\end{eqnarray*}
§ ESTIMATION OF THE RESIDUAL $\WIDEHAT{R}(U_{\D X},D_{K,N},\P)$
Let $(\eta,\om)$ be an entropy pair of (<ref>). Here, we describe the estimation of the residual $\widehat{R}(U_{\D x},D_{k,n},\p)$
\begin{align*}%-------------------------(4.2.1)-------------------------%
&\widehat{R}(U_{\Delta{x}},D_{k,n}, \phi)
:=\iint_{D_{k,n}}\left\{\eta({U}_{\D x})\phi_t+\om({U}_{\D x})\phi_x+d\eta({U}_{\D x})\cdot h(x)g(x,U_{\D x})\phi\right\}dxdt,
\end{align*}
where $U_{\D x},\ D_{k,n}$, and $\p$ are defined as in Theorem 3.5.
According to the Taylor expansion of $\eta({U}_{\Delta{x}})$ and (<ref>), for any positive test function $\phi\in C^1_0(\Pi)$,
\begin{align}%-----------------------------(4.3)--------------------------%
\label{b.3}
&\widehat{R}(U_{\Delta{x}}, D_{k,n}, \phi)\nonumber\\
&=\iint_{D_{k,n}}\left\{\eta(\widetilde{U}_{\D x})\phi_t+\om(\widetilde{U}_{\D x})\phi_x\right\}dxdt\nonumber\\
&\quad+\iint_{D_{k,n}}\big\{\big(\eta(U_{\D x})-\eta(\widetilde{U}_{\D x})\big)\phi_t+\big(\om(U_{\D x})-\om(\widetilde{U}_{\D x})\big)\phi_x
+d\eta(U_{\D x})\cdot h(x)g(x,U_{\D x})\phi\big\} dxdt\nonumber\\
&\quad+\iint_{D_{k,n}}\{\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\big)\phi_t+\big(\om(U_{\D x})-\om(\widetilde{U}_{\D x})\big)\phi_x
+d\eta(\widetilde{U}_{\D x})\cdot h(x)g(x,U_{\D x})\phi\} dxdt \nonumber\\
&\quad+O(1)(\Delta t)^3 \nonumber\\
&\equiv\widehat{Q}_{k,n}^1+\widehat{Q}_{k,n}^2+O(1)(\Delta t)^3.
\end{align}
In <cit.>, if $(\eta,\om)$ is an entropy pair, then
\begin{align}%-----------------------------(4.4)--------------------------%
\label{b.4}
\frac{\partial \eta(\widetilde{U}_{\D x}(x,t))}{\partial t}+\frac{\partial \om(\widetilde{U}_{\D x}(x,t))}{\partial x}\leq 0
\end{align}
in the sense of distribution on the rectangle $D_{k,n}$.
Multiplying the left hand side of (<ref>) by a positive test function $\phi$, and integrating by parts
over $D_{k,n}$, we obtain
\begin{align}%-----------------------------(4.5)--------------------------%
\label{b.5}
\widehat{Q}_{k,n}^1
&\geq\int^{x_{k+1}}_{x_{k-1}}\eta(\widetilde{U}_{\D x}(x, t))\p\Big|_{t=t_n^+}^{t=t_{n+1}^-}dx
+\int^{t_{n+1}}_{t_n}\om(\widetilde{U}_{\D x}(x, t))\p\Big|_{x=x_{k-1}}^{x=x_{k+1}}dt.
\end{align}
The estimate of $\widehat{Q}_{k,n}^2$ is similar to that of $Q_2$ in Theorem 2.2. Let $(x_k,t_n)=(x_k,0)$, $D=D_{x_k,0}$, and let $\widetilde{U}_{\Delta x}$ in $D$ consist of the 1-shock with speed $s_1$, 2-contact discontinuity with speed $s_2$, and 3-rarefaction wave with lower speed $s_3^-$ and upper speed $s^+_3$. Then,
\begin{align}%-----------------------------(4.6)--------------------------%
\label{b.6}
\widehat{Q}_{k,n}^2
&=\left(\int^{\D t}_0\!\!\!\int^{x_k+s_1t}_{x_k-\D x}+\int_0^{\D t}\!\!\!\int_{x_k+s_1t}^{x_k+s_2t}+\int^{\D t}_0\!\!\!\int^{x_k+s^-_3t}_{x_k+s_2t}
+\int^{\D t}_0\!\!\!\int_{x_k+s_3^-t}^{x_k+s_3^+t}+\int^{\D t}_0\!\!\!\int_{x_k+s_3^+t}^{x_k+\D x}\right)\nonumber\\
+\big(\om(U_{\Delta{x}})-\om(\widetilde{U}_{\Delta{x}})\big)\phi_x+d\eta(\widetilde{U}_{\D x})\cdot h(x)g(x,U_{\D x})\phi\Big\} dxdt\nonumber\\
&\equiv \widehat{Q}^{21}+\widehat{Q}^{22}+\widehat{Q}^{23}+\widehat{Q}^{24}+\widehat{Q}^{25}.
\end{align}
According to (<ref>) and integration by parts, we have
\begin{align}%-----------------------------(4.7)--------------------------%
\label{b.7}
\widehat{Q}^{21}&=\int^{x_k+s_1\D t}_{x_k-\D x}\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(x,\Delta{t})dx-
\int^{x_k}_{x_k-\D x}\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(x,0)dx\nonumber\\
&\quad-\int^{\D t}_{0}s_1\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(s_1t-,t)dt
+\int^{\D t}_{0}\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x}))\p\big)(x,t)\Big|^{x=s_1t-}_{x=-\D x}dt\nonumber\\
&\quad+O(1)(\Delta t)^3(\Delta x)\|\phi\|_\infty, \\
\label{b.8}
\widehat{Q}^{22}&=\int^{x_k+s_2\Delta{t}}_{x_k+s_1\Delta{t}}\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(x,\Delta{t})dx
+\int^{\D t}_{0}\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x})-s_2d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x})\p\big)(s_2t-, t)dt\nonumber\\
&\quad-\int^{\D t}_{0}\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x})-s_1d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x})\p\big)(s_1t+, t)dt
+O(1)(\Delta t)^3(\Delta x)\|\phi\|_\infty, \\
\label{b.8.5}
\widehat{Q}^{23}&=\int^{x_k+s_3^-\D t}_{x_k+s_2\D t}\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(x,\Delta{t})dx
+\int^{\D t}_{0}\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x})-s_3^-d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x})\p\big)(s_3^-t, t)dt\nonumber\\
&\quad-\int^{\D t}_{0}\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x})-s_2d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x})\p\big)(s_2t+, t)dt
+O(1)(\Delta t)^3(\Delta x)\|\phi\|_\infty,
\end{align}
\begin{align}%-----------------------------(2.22)--------------------------%
\label{b.9}
\widehat{Q}^{25}&=\int^{x_k+\D x}_{x_k+s_3^+\D t}\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(x,\Delta{t})dx-
\int_{x_k}^{x_k+\D x}\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(x,0)dx\nonumber\\
&\quad+\int^{\D t}_{0}s_3^+\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(s_3^+t,t)dt
+\int^{\D t}_{0}\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x}))\p\big)(x, t)\Big|^{x=\D x}_{x=s_3^+t}dt\nonumber\\
&\quad+O(1)(\Delta t)^3(\Delta x)\|\phi\|_\infty.
\end{align}
Next, we estimate $\widehat{Q}^{24}$. According to the same argument as that for (<ref>)-(<ref>) and $d\om=d\eta df$, we obtain
\begin{align}%-----------------------------(2.21)--------------------------%
\label{b.10}
\widehat{Q}^{24}&=\int^{x_k+s_3^+\D t}_{x_k+s_3^-\D t}\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(x,\Delta{t})dx
+\int^{\D t}_0\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x})-s_3^+d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x})\p\big)(s_3^+t, t)dt\nonumber\\
&\quad-\int^{\D t}_0\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x})-s_3^-d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x})\p\big)(s_3^-t, t)dt\nonumber\\
&\quad+O(1)\left((\D t)^3(\D x)+(\D t)^2(\underset{D_{k,n}}{\osc}\{\widetilde{U}\})\right)\|\phi\|_\infty.
\end{align}
By (<ref>)-(<ref>),
\begin{align}%-----------------------------(2.27)--------------------------%
\label{b.11}
\widehat{Q}^2_{k,n}&=\sum_{i=1}^5\widehat{Q}^{2i}
=\int^{\D x}_{-\D x}\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(x,t)\Big|_{t=0}^{t=\D t}dx
+\int^{\D t}_0\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x}))\p\big)(x, t)\Big|_{x=-\D x}^{x=\D x}dt\nonumber\\
&\quad+\int^{\D t}_0\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x})
-s_1d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x})\p\big)(x, t)\Big|^{x=s_1t-}_{x=s_1t+}dt\nonumber\\
&\quad+\int^{\D t}_0\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x})
-s_2d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x})\p\big)(x, t)\Big|^{x=s_2t-}_{x=s_2t+}dt
\nonumber\\
&\quad+O(1)\left((\Delta t)^3(\Delta x)+(\Delta{t})^3+(\Delta{t})^2(\underset{D}{\osc}\{\widetilde{U}\})\right)\|\phi\|_\infty.
\end{align}
We estimate the second and third terms on the right-hand side of (<ref>). Suppose that the state $\widetilde{U}_1$ is connected to the state $\widetilde{U}_L=U_L=(\r_L,m_L,E_L)$ by 1-shock on the right, and to the state $\widetilde{U}_2$ by 2-contact discontinuity on the left. By the result of <cit.>, (<ref>) is replaced by
\begin{equation}%-----------------------------(2.28)--------------------------%
\label{b.12}
\om(\widetilde{U}_1)-\om(\widetilde{U}_L)\ge s_1(\widetilde{U}_1-\widetilde{U}_L),\quad
\om(\widetilde{U}_1)-\om(\widetilde{U}_2)\ge s_2(\widetilde{U}_1-\widetilde{U}_2).
\end{equation}
Therefore, according to (<ref>) and similar argument as (<ref>), we obtain
\begin{align}
\label{b.13}
&\int_0^{\D t}\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x})-s_1d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x})\p\big)(x,t)\Big|_{x=s_1t+}^{x=s_1t-}dt
\nonumber\\
&\hspace{3cm}\ge\Big(O(1)(\D t)^2(\underset{D}{\osc}\{\widetilde{U}\})+O(1)\D t(\underset{D}{\osc}\{\widetilde{U}\})^2\Big)\|\p\|_{\infty}, \\
\label{b.14}
&\int_0^{\D t}\big((\om(U_{\D x})-\om(\widetilde{U}_{\D x})-s_2d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x})\p\big)(x,t)\Big|_{x=s_2t+}^{x=s_2t-}dt
\nonumber\\
&\hspace{3cm}\ge\Big(O(1)(\D t)^2(\underset{D}{\osc}\{\widetilde{U}\})+O(1)(\D t)(\underset{D}{\osc}\{\widetilde{U}\})^2\Big)\|\p\|_{\infty}.
\end{align}
According to (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>), we obtain the result of (<ref>):
\begin{align*}%-----------------------------(2.16)--------------------------%
&\widehat{R}(U_{\D x},D_{k,n},\phi)\ge\int^{x_{k+1}}_{x_{k-1}}\big(d\eta(\widetilde{U}_{\D x})\widebar{U}_{\D x}\p\big)(x,t^-_{n+1})dx
-\int^{x_{k+1}}_{x_{k-1}}\big(\eta(\widetilde{U}_{\D x})\p\big)(x,t)\Big|^{t=t^-_{n+1}}_{t=t^+_n}dx\nonumber\\
&\quad+\int^{t_{n+1}}_{t_n}\big(\om(U_{\D x})\p\big)(x,t)\Big|^{x=x_{k+1}}_{x=x_{k-1}}dt
+O(1)\left((\Delta t)^2(\Delta x)+(\Delta{t})^3+(\Delta{t})^2\underset{D_{k,n}}{\osc}\{\widetilde{U}\}\right)\|\phi\|_\infty.
\end{align*}
S. Bianchini, A. Bressan,
Vanishing viscosity solutions of nonlinear hyperbolic systems,
Ann. of Math., 161 (2005), pp. 223–342.
J.W. Chamberlain and D.M. Hunten,
Theory of Planetary Atmospheres,
Academic, Orlando, Fla (1987).
G.-Q. Chen, M. Slemrod, D. Wang,
Vanishing viscosity method for transonic flow,
Arch. Rational Mech. Anal., 189 (2008), pp. 159–188.
S.-W. Chou, J. M. Hong, Y.-C. Su,
An extension of Glimm's method to the gas dynamical model of transonic flows,
Nonlinearity, 26 (2013), pp. 1581–1597.
S.-W. Chou, J. M. Hong, Y.-C. Su,
Global entropy solutions of the general nonlinear hyperbolic balance laws with time-evolution flux and source,
Mathods Appl. Anal., 19 (2012), pp. 43–76.
S.-W. Chou, J. M. Hong, Y.-C. Su,
The initial-boundary value problem of hyperbolic integro-differential systems of nonlinear balance laws,
Nonlinear Analysis: Theory, Methods and Applications, 75 (2012), pp. 5933–5960.
C. M. Dafermos, L. Hsiao,
Hyperbolic systems of balance laws with inhomogeneity and dissipation,
Indiana Univ. Math. J., 31 (1982), pp. 471–491.
G. Dal Maso, P. LeFloch, F. Murat,
Definition and weak stability of nonconservative products,
J. Math. Pure Appl., 74 (1995), pp. 483–548.
J. Erwin, O.J. Tucker and R.E. Johnson,
Hybrid fluid/kinetic modeling of Pluto's escaping atmosphere,
Icarus, 226 (2013), pp. 375–384.
J. Glimm,
Solutions in the large for nonlinear hyperbolic systems of equations,
Commun. Pure Appl. Math., 18 (1965), pp. 697–715.
J. B. Goodman,
Initial boundary value problems for hyperbolic systems of conservation laws,
Thesis (Ph. D.)–Stanford University., (1983).
P. Goatin, P.G. LeFloch,
The Riemann problem for a class of resonant nonlinear systems of balance laws,
Ann. Inst. H. Poincare-Analyse Non-lineaire, 21 (2004), pp. 881–902.
J. Groah, J. Smoller, B. Temple,
Shock Wave Interactions in General Relativity,
Monographs in Mathematics, Springer, Berlin, New York, 2007.
Y. Guo, R.W. Farquhar,
New horizons Pluto-Kuiper belt mission: Design and simulation of the Pluto-Charon encounter,
Acta Astro., 56 (2005), pp. 421-429.
James R. Holton,
An introduction to dynamical meteorology,
Elsevier Academic Press 200, 4th ed., Wheeler Road, Burlington, MA 01803, USA, 2004.
J. M. Hong,
An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem,
J. Diff. Equ., 222 (2006), pp. 515–549.
J. M. Hong, P.G. LeFloch,
A version of Glimm method based on generalized Riemann problems,
J. Portugal Math., 64, (2007) pp. 199–236.
J. M. Hong, B. Temple,
The generic solution of the Riemann problem in a neighborhood of a point of resonance for systems of nonlinear balance laws,
Methods Appl. Anal., 10 (2003), pp. 279–294.
J. M. Hong, B. Temple,
A bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law,
SIAM J. Appl. Math., 64 (2004), pp. 819–857.
J. M. Hong, C.-C. Yen, B.-C. Huang,
Characterization of the transonic stationary solutions of the hydrodynamic escape problem,
SIAM J. Appl. Math., 74 (2014), pp. 1709–1741.
D.M. Hunten,
Space Sci., 30 (1982), pp. 773-783.
D.M. Hunten, R.O. Pepin, J.C.G. Walker,
Mass fractionation in hydrodynamic escape,
Icarus, 69 (1987), pp. 532-549.
E. Isaacson, B. Temple,
Nonlinear resonance in systems of conservation laws,
SIAM J. Appl. Anal., 52 (1992), pp. 1260–1278.
E. Isaacson, B. Temple,
Convergence of the $2\times2$ Godunov method for a general resonant nonlinear balance law,
SIAM J. Appl. Math., 55, (1995), pp. 625–640.
P. D. Lax,
Hyperbolic system of conservation laws II,
Commun. Pure Appl. Math., 10 (1957), pp. 537–566.
P. G. LeFloch,
Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form,
Commun. Part. Diff. Equ., 13 (1988) pp. 669–727.
P. G. LeFloch,
Shock waves for nonlinear hyperbolic systems in nonconservative form,
Institute for Math. and its Appl., Minneapolis, Preprint 593, 1989.
P. G. LeFloch, T.-P. Liu,
Existence theory for nonlinear hyperbolic systems in nonconservative form,
Forum Math., 5 (1993), pp. 261–280.
P. G. LeFloch, P. A. Raviart,
Asymptotic expansion for the solution of the generalized Riemann problem, Part 1,
Ann. Inst. H. Poincare, Nonlinear Analysis, 5 (1988) pp. 179–209.
Randall J. LeVeque,
Finite Volume Methods for Hyperbolic Problem,
Cambridge Texts in Applied Mathematics (2002), Cambridge.
Mao-Chang Liang, Alan N. Heays, Brenton R. Lewis, Stephen T. Gibson, and Yuk L. Yung,
Source of Nitrogen Isotope Anmaly in HCN in the Atmosphere ot Titan,
The Astrophysical Journal, 664 (2007), pp. 115-118.
R.P. Lin, B. Jakosky,
The 2013 Mars Atmosphere and Volatile Evolution (MAVEN) Mission to Mars. In: 39th COSPAR Scientific Assembly,
COSPAR Meeting, 39 (2012), pp. 1089.
T.-P. Liu,
Quasilinear hyperbolic systems,
Commun. Math. Phys., 68 (1979), pp. 141–172.
T.-P. Liu,
Nonlinear stability and instability of transonic flows through a nozzle,
Commun. Math. Phys., 83 (1982), pp. 243–260.
T.-P. Liu,
Nonlinear resonance for quasilinear hyperbolic equation,
J. Math. Phys., 28 (1987), pp. 2593–2602.
M. Luskin, B. Temple,
The existence of global weak solution to the nonlinear waterhammer problem,
Commun. Pure Appl. Math., 35 (1982), pp. 697–735.
C. S. Morawetz,
On a weak solution for a transonic flow problem,
Commun. Pure Appl. Math., 38 (1985), pp. 797–817.
R.A. Murray-Clay, E.I. Chiang and N. Murray,
Atmospheric Escape From Hot Jupiters,
The Astrophysical Journal, 693 (2009), pp. 23-43.
E.N. Parker,
Dynamical properties of stellar coronas and stellar winds I. Integration of the momentum equation,
Astrophys. J., 139 (1964), pp. 72-92.
E.N. Parker,
Dynamical properties of stellar coronas and stellar winds II. Integration of the heat-flow equation,
Astrophys. J., 139 (1964), pp. 93-122.
J. Smoller,
On the solution of the Riemann problem with general stepdata for an extended class of hyperbolic system,
Mich. Math. J., 16, pp. 201–210.
J. Smoller,
Shock Waves and Reaction-Diffusion Equations,
2nd ed., Springer-Verlag, Berlin, New York, 1994.
Darrell F. Strobel,
Titan's hydrodynamically escaping atmosphere,
Icarus, 193 (2008), pp. 588–594.
Darrell F. Strobel,
$N_2$ escape rates from Pluto's atmosphere,
Icarus, 193 (2008), pp. 612–619.
B. Temple,
Global solution of the Cauchy problem for a class of 2$\times$2 nonstrictly hyperbolic conservation laws,
Adv. Appl. Math., 3 (1982), pp. 335–375.
F. Tian, J.F. Kasting, H.L. Liu, R.G. Roble,
Hydrodynamic planetary thermosphere model 1: Response of the earth's thermosphere to extreme solar euv conditions and the significance of adiabatic cooling,
J. Geophys. Res.: Planets, 113 (2008), E05008.
Feng Tian, Owen B. Toon, Alexander A. Pavlov, and H. De Sterck,
Transonic hydrodynamic escape of hydrogen from extrasolar planetary atmospheres,
The Astrophys. J., 621 (2005), pp. 1049–1060.
Feng Tian, Owen B. Toon, Alexander A. Pavlov, and H. De Sterck,
A hydrogen-rich early Earth atmosphere,
Scicence, 308 (2005), pp. 1014–1017.
O. J. Tucker, J. T. Erwin, J. I. Deghan, A. N. Volkov, and R. E. Johnson,
Thermally driven escape from Pluto's atmosphere: A combined fluid/kinetic model,
Icarus, 217 (2012), pp. 408–415.
A. N. Volkov, R. E. Johnson, O. J. Tucker, and J. T. Erwin,
Thermally driven atmospheric escape: transition from hydrodynamic to Jeans escape,
The Atrophys. J. Lett., 729:L24 (5pp) (2011).
R.V. Yelle,
Aeronomy of extra-solar giant planets at small orbital distances,
Icarus, 170 (2004), pp. 167–179.
Xum Zhu, Darrell F. Strobel, and Justin T. Erwin,
The density and thermal structure of Pluto's atmosphere and associated escape processes and rates,
Icarus, 228 (2014), pp. 301–314.
|
1511.00609
|
add2]Enrico Calloni
add3]S. Caprara
add2]Martina De Laurentis
add2]Giampiero Esposito
add3]M. Grilli
add3]Ettore Majorana
add2]G.P. Pepe
add3]S. Petrarca
add3]Paola Puppo
add3]P. Rapagnani
add3]Fulvio Ricci
add2]Luigi Rosa
add4]Carlo Rovelli
add1]P. Ruggi
add3]N.L. Saini
add2]Cosimo Stornaiolo
add2]Francesco Tafuri
[add1]European Gravitational Observatory (EGO), Cascina (Pisa)
[add2]University of Napoli Federico II and INFN Napoli
[add3]University of Roma Sapienza and INFN Roma
[add4]University of Aix-Marseille
Archimedes is an INFN-funded pathfinder experiment aimed at verifying the feasibility of measuring the interaction
of vacuum fluctuations with gravity. The final experiment will measure the force exerted by the gravitational field
on a Casimir cavity whose vacuum energy is modulated with a superconductive transition, by using a balance as a small
force detector. Archimedes is two-year project devoted to test the most critical experimental aspects, in particular
the balance resonance frequency and quality factor, the thermal modulation efficiency and the superconductive sample realization.
Experimental test of gravitational theories Quantum fluctuations
04.80.Cc 42.50.Lc
§ INTRODUCTION
The cosmological constant problem belongs to the main fundamental unsolved questions
which already puzzle the modern physics.
Does the enormous energy of the vacuum fluctuations foreseen by quantum mechanics contribute to gravity?
If it does, the vacuum energy density suggested by the quantum field theory is enormously larger than the value
constrained from General Relativity, by considering the radius of our universe and its accelerated expansion.
At present, although there is a long list of detailed and important theoretical works <cit.>
on this argument, there is no agreement on the real contribution of vacuum fluctuations to gravity on the theoretical side.
Moreover no experiment has been performed to finally verify or discard this assumption so far.
Taking into account the present status of the small forces measurement techniques and of the technologies of the high
temperature superconductors, an experimental device to make such a measurement is on the designing path <cit.>.
Scheme of the balance suspended to the inverted pendulum.
§ THE WEIGHT OF THE VACUUM
The idea is to weigh the vacuum energy stored in a rigid Casimir cavity formed by parallel conductive plates. The Casimir
effect is a macroscopic manifestation of vacuum fluctuations. When a Casimir cavity is formed only some modes can resonate
inside it. The ones which do not satisfy specific boundary conditions are expelled and the total vacuum energy changes.
If the vacuum energy does interact with gravity, a force directed upwards acts on the cavity and can be interpreted as the
lack of weight of the modes expelled by it, in similarity with the Archimedes buoyancy of fluid. The force is equal to <cit.>:
\begin{equation}
\vec{F}=-\frac{\left| E_{C}\right|}{c^2} \vec{g}
=-{A} \frac {\pi^2 \hbar}{720~a^3}
\frac {\vec{g}}{ c}
\end{equation}
where $E_C$ is the vacuum energy stored in the cavity, $a$ is the gap between the plates and $A$ is their surface, $c$ is
the speed of light and $\vec{g}$ is the earth gravitational acceleration directed
The measurement strategy is to modulate the reflectivity of the plates in such a way as to periodically expel the vacuum energy
from the Casimir cavity and consequently modulate its weight.
A possible way to modulate the reflectivity is by performing a superconducting transition of the plates: when the plates become
superconductive the reflectivity changes and so does the vacuum energy contained between the plates <cit.>.
The variation of Casimir energy could be particularly relevant in case of type II high layered $T_c$ superconductors,
like cuprates which behave as natural multi Casimir-cavities. The advantage is due to the fact that in normal state the
planes (that will become superconducting) are very poorly conductive.
Even though a complete study is still lacking, and it is one of the problems to be faced inside the Archimedes project,
order of magnitude expectations suggest that the amplitude $F$ of the modulated force is of $F\sim10^{-16} N$.
Expected signal and main noises. The detection bandwidth is within the resonance of the balance, limited by the suspension
thermal (dashed line) and seismic noise (dotted line)
§ THE EXPERIMENT
The experiment aim is to measure forces of $F\sim10^{-16} N$. Recently <cit.> it has been shown that both
the gravitational wave detectors and balances could be suitable for detecting the Archimedes force.
However, other aspects must be taken into account. The frequency bandwidth favored for detecting the effect at the periodical
transition from normal to superconducting state obtained by a temperature change is in the region of 1-100 mHz,
typical of the thermal processes. As a consequence the use of balances is the more suitable choice for this application.
§.§ The balance and its seismic isolation
A balance capable of measuring forces of the order of $\sim10^{-16} N$ requires an excellent isolation from the seismic
noise at the very low frequency regime of 1-100 mHz. The use of a seismic-attenuation device based on the use of an inverted
pendulum and a suitably tuned resonator hung to it (see fig. fig. <ref>) can fullfil this requirement.
The read-out system can be an optical lever or an interferometric sensor.
A challenging point is the design of the balance whose center of mass must be positioned on its bending point in order not
to reinject the residual seismic noise in the system.
The major sources of noise in this scheme are represented by the thermal
noise and the seismic noise. As reported in fig. <ref>, for an integration
time of several months, this would permit to reach a good signal-to-noise
§.§ Thermal modulation study
Another challenging point is related to the modulation of
the superconductor's temperature. Only the radiative mechanism must be used to remove or add thermal energy to the sample
as it must be isolated from any external interaction that could add energy other than the vacuum one. The main difficulties
are related to heating samples with mass of hundreds of $g$ within thermal transition time scales of hundreds seconds.
To reduce the transition times and increase the frequency bandwidth the choice of the sample shape and of the heating system
is crucial. With the help of a finite element analysis we have obtained a temperature modulation of about 1 K on a 150 mm
diameter 0.5 mm thick disk. The shape is compatible with the present YBCO technology and is usefull to have
signal within the expected sensitivity.
§ CONCLUSION
We have presented the Archimedes experiment aiming to verify the feasibility to measure the weigh of the vacuum.
Three crucial points are focused with this project, the construction of a balance suitable to measure forces at the level
of $10^{-16} ~ N$, a thermal modulation system using only a radiative heat exchange mechanism and the detailed study of
high $T_c$ superconductive systems as the main source of vacuum energy.
S. Weinberg, Rev. Mod. Phys. 61 (1989) 1 ; M. Ishak, Found. Phys. 37 (2005) 1470.
B.S. DeWitt and G Esposito, Int. J. Geom Methods Mod. Phys. 5 (2008) 101.
T. Padmanabhan, Int. J. Mod. Phys. D 15 (2006) 2029.
E. Calloni et al, arXiv:1409.6974 [gr-qc].
E. Calloni et al, Phys. Rev. D 90 (2014) 022002.
E. Calloni et al, Phys. Lett. A 297 (2002) 328 ; G. Bimonte et al, Phys. Rev. D 74 (2006) 085011; G. Bimonte et al, Phys. Rev. D 76 (2007) 025008.
|
1511.00791
|
Li-Yau gradient bounds under nearly optimal curvature conditions]Li-Yau gradient bounds under nearly optimal curvature conditions
Department of Mathematics, University of California, Riverside, Riverside, CA 92521, USA
Department of Mathematics, East China Normal University, Shanghai 200241, China - and Department of Mathematics, University of California, Riverside, Riverside, CA 92521, USA
We prove Li-Yau type gradient bounds for the heat equation either on manifolds with fixed metric or
under the Ricci flow. In the former case the curvature condition is $|Ric^-| \in L^p$ for some
$p>n/2$, or $\sup_\M \int_\M |Ric^-|^2(y)d^{2-n}(x,y)dy<\infty$, where $n$ is the dimension of the manifold. In the later case, one only needs scalar curvature being bounded. We will explain why the conditions are nearly optimal and give an application. The Li-Yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing Ricci curvature not bounded from below.
§ INTRODUCTION
Let $(\M^n, g_{ij})$ be a complete Riemannian manifold. In <cit.>, P. Li an S.T. Yau discovered the following celebrated Li-Yau bound, for positive solutions of the heat equation
Suppose $Ric\geq-K$, where $K\geq0$ and $Ric$ is the Ricci curvature of $\M$. Then any positive solution of (<ref>) satisfies
\begin{equation}\label{Li-Yau>1}
\frac{|\nabla u|^2}{u^2} - \alpha\frac{u_t}{u} \leq \frac{n\alpha^2K}{2(\alpha-1)}+\frac{n\alpha^2}{2t},\quad \forall \alpha>1.
\end{equation}
In the special case where $Ric\geq 0$, one has the optimal Li-Yau bound
\begin{equation}\label{Li-Yau}
\frac{|\nabla u|^2}{u^2} - \frac{u_t}{u} \leq \frac{n}{2t}.
\end{equation}
In the same paper, many applications of (<ref>) and (<ref>) have also been demonstrated by the authors, including the classical parabolic Harnack inequality, optimal Gaussian estimates of the heat kernel, estimates of eigenvalues of the Laplace operator, and estimates of the Green's function. Moreover, (<ref>) and
(<ref>) can even imply the Laplacian Comparison Theorem (see e.g. <cit.> page 394).
The Li-Yau bound (<ref>) was later improved for small time by Hamilton in <cit.>, where he proved under the same assumptions as above that
\begin{equation}\label{Hamilton}
\frac{|\nabla u|^2}{u^2} - e^{2Kt}\frac{u_t}{u} \leq e^{4Kt}\frac{n}{2t}.
\end{equation}
Hamilton <cit.> further showed a matrix Li-Yau bound for the heat equation. Similar matrix Li-Yau bound was subsequently obtained by Cao-Ni <cit.> on Kähler manifolds.
For the past three decades, many Li-Yau type bounds have been proved not only for
the heat equation, but more generally, for other linear and semi-linear parabolic equations
on manifolds with or without weights. Let us mention the result by Bakry and Ledoux <cit.> who
derived the Li-Yau bound for weighted manifolds by an ordinary differential inequality
involving the entropy and energy of the backward heat equation. For most recent development, see the papers <cit.>, <cit.>, <cit.>,
<cit.>, <cit.>, <cit.>, <cit.> and the latest <cit.> and its references. In all of these results, the essential assumption is that the Ricci curvature or the corresponding Bakry-Emery Ricci curvature is bounded from below by a constant.
In many situations, it is highly desirable to weaken this assumption.
On the other hand, Li-Yau bounds have been extended to situations with moving metrics.
Let $g_{ij}(t)$, $t\in[0, T]$, be a family of Riemannian metrics on $\M$ which solves the Ricci flow:
\begin{equation}\label{RF1}
\frac{\partial }{\partial t}g_{ij}(t)=-2R_{ij}(t),
\end{equation}
where $R_{ij}(t)$ is the Ricci curvature tensor of $g_{ij}(t)$. One may still consider linear and semi-linear parabolic equations under the Ricci flow in the sense that in the heat operator $\frac{\partial}{\partial t}-\Delta$, we have $\Delta=\Delta_t$ which is the Laplace operator with respect to the metric $g_{ij}(t)$ at time $t$. The two most prominent examples are the heat equation
(Δ-∂/∂t)u= 0, ∂_t g_ij= - 2 R_ij
and the conjugate heat equation
(Δ-R + ∂/∂t)u= 0, ∂_t g_ij= - 2 R_ij.
The study of Li-Yau bound for heat type equations under the Ricci flow was initiated by Hamilton. In <cit.>, he obtained a Li-Yau bound for the scalar curvature along the Ricci flow on 2-sphere. This result was later improved by Chow <cit.>. In higher dimensions, both matrix and trace Li-Yau bounds for curvature tensors, also known as Li-Yau-Hamilton inequalities, were obtained by Hamilton <cit.> for the Ricci flow with bounded curvature and nonnegative curvature operator. These estimates played a crucial role in the study of singularity formations of the Ricci flow on three-manifolds and solution to the Poincaré conjecture. We remark that
Brendle <cit.> has generalized Li-Yau-Hamilton inequalities under weaker curvature assumptions. The Li-Yau-Hamilton inequality for the Kähler-Ricci flow with nonnegative holomorphic bisectional curvature was obtained by H.-D. Cao <cit.>. In addition, in <cit.>, Perelman showed a Li-Yau type bound for the fundamental solution of the conjugate heat equation (<ref>) under the Ricci flow (see also <cit.>). Recently, there have been many results on Li-Yau bounds for positive solutions of the heat or conjugate heat equations under the Ricci flow. For example authors of <cit.> and <cit.> proved Li-Yau type bound
for all positive solutions of the conjugate heat equation with out any curvature condition, just like
Perelman's aforementioned result for the fundamental solution.
In <cit.> and <cit.> the authors proved various Li-Yau type bounds for positive solutions of (<ref>)
under either positivity condition of the curvature tensor or boundedness of the Ricci curvature. So there is a marked difference between these results on the conjugate
heat equation and the heat equation in the curvature conditions. In view of the absence of curvature
condition for the conjugate heat equation, one would hope that the curvature conditions for the heat equation can be weakened.
Recently, in <cit.>, the authors proved the following gradient estimate for bounded positive solutions $u$ of the heat equation (<ref>),
\begin{equation}\label{BZ}
|\Delta u|+\frac{|\nabla u|^2}{u}-aR\leq \frac{Ba}{t},
\end{equation}
where $R=R(x,t)$ is the scalar curvature of the manifold at time $t$, and $B$ is a constant and $a$ is an upper bound of $u$ on $M\times[0,T]$. Although this result requires no curvature condition and it has some other applications, it is not a
Li-Yau type bound.
The goal of this paper is to prove Li-Yau bounds for positive solutions for both the fixed metric case (<ref>) and the Ricci flow case
(<ref>) under essentially optimal curvature conditions.
The first theorem is for the fixed metric case, we will have two independent conditions
and two conclusions. The conditions are motivated by different problems such as studying
manifolds with integral Ricci curvature bound and the Kähler-Ricci flow. The conclusions
range from long time bound with necessarily worse constants, to short time
bound with better constants.
Let $(\M, g_{ij})$ be a compact $n$ dimensional Riemannian manifold, and $u$ a positive solution of (<ref>). Suppose either one of the following conditions holds.
(a) $\int_\M |Ric^-|^p dy \equiv \sigma <\infty$ for some $p>\frac{n}{2}$, where $Ric^-$ denotes the nonpositive part of the Ricci curvature; and the manifold is noncollapsed under scale 1, i.e., $|B(x, r)| \ge \rho r^n$ for $0<r\leq 1$ and some
(b) $\sup_\M \int_\M |Ric^-|^2 d^{-(n-2)}(x,y)dy \equiv \sigma <\infty$ and the heat kernel of (<ref>)
satisfies (<ref>): the Gaussian upper bound. Here $d(x,y)$ is the distance from $x$ to $y$.
(1) for any constant $\alpha\in(0,1)$, we have
J(t) |ụ|^2/u^2 - _t u/u ≤n/(2 - δ)J(t) 1/t
for $t\in (0, \infty)$, where
2^-1/(5δ^-1-1)e^-(5δ^-1-1)^n/2p-n[4σĈ(t)^1/p]^2p/2p-nt, under condition (a);
2^-1/(10δ^-1-2)e^-C_0(5δ^-1-1)σĈ(t)t, under condition (b),
with $C_0$ being a constant depending only on $n$, $p$ and $\rho$, and $\hat C(t)$ the increasing function on the right hand side of (<ref>).
(2) in particular, for any $\beta\in(0,1)$, there is a $T_0=T_0(\beta,\sigma, p, n, \rho)$ such that
β |ụ|^2/u^2 - _t u/u ≤n/2β 1/t
for $t\in (0, T_0]$. Here $T_0 = c (1-\beta)^{4p/(2p-n)}$ and $c (1-\beta)^{4}$
under conditions (a) and (b), respectively; and $c$ is a positive constant depending only
on the parameters of conditions (a) and (b), i.e., $c=c(\sigma, p, n, \rho) $.
Condition (a) actually implies that the heat kernel of (<ref>) has a Gaussian upper bound for all time,
G(x, t; y, 0) ≤Ĉ(t)/t^n/2 e^ - c̅ d^2(x, y)/t, t ∈(0, ∞)
for some positive constant $\bar c$ and positive increasing function $\hat C(t)$ which grows to infinity as $t\to \infty$. For short time interval $(0,1]$, the function $\hat C(t)$ can be replaced by a constant $\hat C$. This is proven in <cit.> Section 2 and <cit.>. Longer time bound follows from the reproducing formula of heat kernels.
In addition, a volume upper bound $|B(x, r)| \le C r^n$ follows from Petersen-Wei <cit.>.
We will use these facts during the proof of the theorem.
The condition $|Ric^-|\in L^p$ for some $p>\frac{n}{2}$ is nearly optimal in the sense that it is not clear $|\nabla u|$ will stay bounded when $|Ric^-|\in L^{\frac{n}{2}}$, which is the well known border
line condition where regularity may fail.
Professor Guofang Wei kindly informed us that the noncollapsing condition in (a) may possibly be removed
by a recent result of Dai-Wei-Z. L. Zhang.
The motivation of assuming (b) is that it is preserved under the Kähler-Ricci flow as proved in <cit.> and <cit.>.
The Li-Yau bound in the above theorem seems to be the first one allowing Ricci curvature not bounded from below.
Moreover, as an application, we use (<ref>) to extend some results in <cit.> on the parabolic approximations of the distance functions to the case where $|Ric^-|\in L^p$ for some $p>\frac{n}{2}$. The main extension results were first proved in <cit.>.
Next we turn to the heat equation coupled with the Ricci flow (<ref>) for which we prove
Let $\M$ be a compact $n$ dimensional Riemmannian manifold, and $g_{ij}(t)$, $t\in [0,T)$, a solution of the Ricci flow (<ref>) on $\M$. Denote by $R$ the scalar curvature of $\M$ at $t$, and $R_1$ a positive constant. Suppose that $-1\leq R\leq R_1$ for all time $t$, and $u$ is a positive solution of the heat equation (<ref>).
Then, for any $\delta\in[\frac{1}{2},1)$, we have
\begin{equation}\label{LYRF}
\delta\frac{|\nabla u|^2}{u^2}-\frac{\partial_t u}{u} \le \delta\frac{|\nabla u|^2}{u^2}-\frac{\partial_t u}{u}-\alpha R+\frac{\beta}{R+2}\leq
\frac{1}{t} \left(\frac{n}{2\delta}+\frac{4n\beta T}{\delta(1-\delta)} \right)
\end{equation}
for $t\in(0,T)$, where $\alpha=\frac{n}{2\delta(1-\delta)^2}$ and $\beta=\alpha (R_1+2)^2$.
Note that the curvature assumption is made only on the scalar curvature rather than on the Ricci or curvature
tensor. In this sense, this assumption is essentially optimal. Under suitable assumptions, the
result in the theorem still holds when $\M$ is complete noncompact.
For the Ricci flow on a compact manifold $\M$, one can always rescale a solution so that the scalar curvature to be bounded from below by $-1$.
The Li-Yau bound in the above theorem actually is scaling invariant. Readers can refer to Theorem <ref> in section 3 for the corresponding version before rescaling the metrics.
In case the scalar curvature is 0, the Ricci curvature is also $0$ by the maximum principle. Then, by scaling,
we can let $\delta=1$ in the theorem and the bound becomes the optimal Li-Yau bound for Ricci flat case.
This theorem clearly implies a Harnack inequality for positive solutions of (<ref>)
if the scalar curvature is bounded.
This paper is organized as follows: the main Theorems <ref> and <ref> are proved in sections 2 and 3, respectively. The main technical hurdle is to construct certain auxiliary functions to
cancel various curvature terms arising from commutation formulas. For example, in order to
prove Theorem <ref>, one needs to deal with the bad term $| Ric^-| \frac{|\nabla u|^2}{u}$.
If one only imposes integral conditions on $|Ric^-|$, then this term can not be bounded by good terms coming out of the Bochner's formula. The auxiliary functions for Theorem <ref> are obtained by solving
a nonlinear evolution equation, which is used to cancel the bad term.
When proving Theorem <ref> in section 3, an additional bad term $< \nabla^2 u, Ric >$
We will use the good terms coming from the equation of $\frac{\beta}{R}$ to
control it.
In section 4, we deduce from Theorem <ref> the extended parabolic approximations of the distance functions.
§ FIXED METRIC CASE
In this section, we work on a compact Riemannian manifold $\M$ with a fixed
metric $g$. For the Ricci curvature, we assume either
| Ric^- | ∈L^p(), p>n/2, or sup_x ∈
∫_ |Ric^-(y)|^2/d^n-2(x, y) dy < ∞.
By direct computation, we have
\[
(\Delta - \p_t) \frac{ |\d u|^2}{u} = \frac{2}{u} \left| u_{ij} - \frac{u_i u_j}{u}
\right|^2 + 2 R_{ij} \frac{u_i u_j}{u}.
\]
Let $J=J(x, t)$ be a smooth positive function and $\a \in (0, 1)$ be a parameter. Then
\[
\al
&(\Delta - \p_t) \left[ \a J \frac{ |\d u|^2}{u} - \p_t u \right] \\
&= \a \left[ \frac{2}{u} \left| u_{ij} - \frac{u_i u_j}{u}
\right|^2 J + 2 R_{ij} \frac{u_i u_j}{u} J + \Delta J \frac{ |\d u|^2}{u}
+ 2 \d J \d \frac{ |\d u|^2}{u} -(\p_t J) \frac{ |\d u|^2}{u} \right].
\eal
\]
Denote the heat operator $\Delta-\frac{\partial}{\partial t}$ by $\mathcal{L}$. Recall the quotient formula for the heat operator.
\[
\mathcal{L} \left(\frac{F}{G}\right) + 2 \d \ln G \d \frac{F}{G} =
\frac{\mathcal{L}F}{G} - \frac{F \mathcal{L}G}{G^2}.
\]
\[
F= \a J \frac{ |\d u|^2}{u} - \p_t u, \quad G=u,
\]
Q ≡J |ụ|^2/u^2 - _t u/u.
We find that
(Δ- _t) Q
+ 2 ụ/u Q̣
= [ 2/u^2 | u_ij - u_i u_j/u
|^2 J + 2 R_ij u_i u_j/u^2 J + ΔJ |ụ|^2/u^2
2 J̣ ( |ụ|^2/u ) 1/u
-(_t J) |ụ|^2/u^2 ].
Let $f=\ln u$. Using the identities
\[
\frac{1}{u^2} \left| u_{ij} - \frac{u_i u_j}{u}
\right|^2 = | f_{ij}|^2,
\]
\[
\d \left(\frac{ |\d u|^2}{u} \right) \frac{1}{u}
= \d \left(\frac{ |\d u|^2}{u^2} \right) + \frac{ |\d u|^2}{u} \frac{\d u}{u^2}
= \d \left( |\d f|^2 \right)+ \frac{ |\d u|^2}{u} \frac{\d u}{u^2},
\]
we can turn (<ref>) into
(Δ- _t) Q
+ 2 ụ/u Q̣
= [ 2 | f_ij|^2 J + 2 R_ij u_i u_j/u^2 J
+ ΔJ |ụ|^2/u^2
2 J̣ |̣f̣|^2 + 2 J̣ ụ/u ( |ụ|^2/u^2 )
-(_t J) |ụ|^2/u^2 ].
Observe that, in local coordinates,
\[
2 \d J \d |\d f|^2 =2 J_i (f^2_j)_i = 4 J_i f_{ji} f_j
\ge - \delta | f_{ij} |^2 J - \frac{| \d J|^2}{J} \, |\d f|^2 4 \delta^{-1}.
\]
Therefore, we deduce the following inequality
(Δ- _t) Q
+ 2 ụ/u Q̣
≥[ (2 J - δJ) | f_ij|^2 - 2 |Ric^-| |ụ|^2/u^2 J
+ ΔJ |ụ|^2/u^2
-| J̣|^2/J |f̣|^2 4 δ^-1.
. -2 |J̣| |ụ|^3/u^3
-(_t J) |ụ|^2/u^2 ].
Using the inequality, for any $\delta>0$,
\[
2 |\d J| \frac{ |\d u|^3}{u^3} =
2 |\d J | \, |\d f|^3 \le \delta J |\d f|^4 + \delta^{-1} \frac{| \d J|^2}{J} \, |\d f|^2,
\]
we can turn the above inequality into
\[
\al
&(\Delta - \p_t) Q
+ 2 \frac{\d u}{u} \d Q \\
&\ge \a (2 J - \delta J) \left| f_{ij}\right|^2 +
\a \left[\Delta J - 2 |Ric^-| J
5\delta^{-1} \frac{| \d J|^2}{J}
-\p_t J \right] |\d f|^2 - \delta \a J |\d f|^4.
\eal
\]
From (<ref>), we know
$$\Delta f-\partial_t f=|\nabla f|^2.$$
(Δ- _t) ( t Q )
+ 2 ụ/u (̣t Q)
≥t (2 J - δJ) 1/n ( | f̣ |^2 - _t f )^2 +
[ΔJ - 2 V J
5 δ^-1 | J̣|^2/J
-_t J ] t |f̣|^2
- δt J |f̣|^4 -Q,
where we have written $| Ric^-|=V$.
For any given parameter $\delta>0$ such that $5\delta^{-1}>1$, we make the following
the problem
ΔJ - 2 V J
5 δ^-1| J̣|^2/J
-_t J =0, on ×(0, ∞);
J(·, 0) = 1,
has a unique solution for $t\in[0,\infty)$, which satisfies
2^-1/(5δ^-1-1)e^-(5δ^-1-1)^n/2p-n[4σĈ(t)^1/p]^2p/2p-nt, under condition (a);
2^-1/(10δ^-1-2)e^-C_0(5δ^-1-1)σĈ(t)t, under condition (b),
with $C_0$ being a constant depending only on $n$, $p$ and $\rho$, and $\hat C(t)$ the increasing function in (<ref>).
In the following steps, we will prove the claim.
step 1. Conversion into an integral equation.
Let $a=5\delta^{-1}$, and
w = J^-(a-1).
It is straightforward to check that $w$ satisfies
Δw -_t w +2(a-1) V w =0 , on ×(0, ∞);
w(·, 0) =1.
Since $V$ is a smooth function, (<ref>) has a long time solution.
To show that $J$ exists for all time and derive the bounds for $J$, we derive the bounds for $w$ first. Via the Duhamel's formula, (<ref>) can be transformed to the following
integral equation,
w(x, t) = 1+ 2(a-1)∫^t_0 ∫_G(x, t; y, s) V(y) w (y, s) dyds.
Here $G(x,t;y,s)=G(y,t;x,s)$ is the heat kernel on $\M$.
step 2. long time bounds
Here we prove long time bounds for solutions of (<ref>).
Let $w$ be a solution of (<ref>). For a lower bound of $w$, we can show that
w ≥1
for any $t>0$.
In fact, let $\e>0$ be a small positive number, which will be taken to $0$ eventually. Then the
function $Z_\e = e^{\e t} w$ satisfies the equation
ΔZ_+ 2(a-1) V Z_-_t Z_= - Z_.
First, by continuity, since $w(\cdot, 0)=1$, we know that $w\geq0 $ at least for a short time. Applying the maximum principle on (<ref>), we see that (<ref>) holds
at least for a short time. So $Z_\e > 1$ at least for a short time. We now show that
Z_> 1
for all time $t>0$ as long as the
solution exists. Suppose not. Then there exists a first time $t_0$ and point $x_0 \in \M$ such that
$Z_\e(x_0, t_0) = 1$. At this point $(x_0, t_0)$, the following holds
\[
\Delta Z_\e \ge 0, \quad \partial_t Z_\e \le 0, \quad 2(a-1) V Z_\e \ge 0.
\]
This is a contradiction to equation (<ref>). Letting $\e \to 0$ in (<ref>), we know (<ref>) holds for all time.
Notice that (<ref>) implies that $J\leq 1$ as long as the solution exits, which is not obvious to see from (<ref>).
Next, for any fixed $T>0$, we derive an upper bound for $w(x,t)$ on $[0,T]$. We will treat condition (a) and (b) separately.
First, under condition (b), since
\[
\al
w(x, t) &= 1+2(a-1)\int_0^t\int_\M G(x,t;y,s)V(y)w(y,s)dyds\\
&\le 1 +2(a-1) \left(\int^t_0 \int_\M G(x, t; y, s) V^2(y)dyds\right)^{1/2} \left( \int^t_0 \int_\M G(x, t; y, s) w^2(y,s)dyds\right)^{1/2}\\
& \le 1+ 2(a-1) \left(\int^t_0 \int_\M \frac{\hat C(t)}{(t-s)^{n/2}}e^{-\frac{\bar{c} d^2(x,y)}{t-s}}V^2(y)dyds\right)^{1/2}\left( \int^t_0 \int_\M G(x, t; y, s) m^2(s)dyds\right)^{1/2}\\
&\leq 1+C_0(a-1)\sqrt{\hat C(t)}\left( \int_\M \frac{V^2(y)}{d^{n-2}(x,y)}dy\right)^{1/2}\left(\int^t_0m^2(s)ds\right)^{1/2}\\
&\leq 1+C_0(a-1)\sqrt{\sigma\hat C(T)}\left(\int^t_0m^2(s)ds\right)^{1/2}
\eal
\]
for all $t\in[0,T]$ and $x\in\M$, we have
m(t)\leq 1+C_0(a-1)\sqrt{\sigma\hat C(T)}\left(\int^t_0m^2(s)ds\right)^{1/2},
and hence
m^2(t)\leq 2+2C_0(a-1)^2\sigma\hat C(T)\int^t_0m^2(s)ds,
which is the Grönwall inequality.
Therefore, we get
\begin{align}
m^2(t)\leq 2e^{2C_0(a-1)^2\sigma\hat C(T)t},
\end{align}
\begin{align}
m(t)\leq \sqrt{2}e^{C_0(a-1)^2\sigma\hat C(T)t}.
\end{align}
Especially, we have shown
\begin{align}
w(x,t)\leq \sqrt{2}e^{C_0(a-1)^2\sigma\hat C(t)t},
\end{align}
for any $t\in[0,\infty)$.
From (<ref>), we have
$$ 2^{-1/(2a-2)}e^{-C_0(a-1)\sigma \hat C(t)t}\leq J(x,t)\leq 1.$$
Under condition (a),
\[
\al
w(x, t) &= 1+2(a-1)\int_0^t\int_\M G(x,t;y,s)V(y)w(y,s)dyds\\
&\le 1 +2(a-1)\int^t_0 \int_\M G(x, t; y, s) V(y) m(s)dyds.
\eal
\]
m(t) ≤1 +2(a-1)∫^t_0 ∫_G(x, t; y, s) V(y) m(s)dyds
= 1+ 2(a-1)∫^t-_0 ∫_G(x, t; y, s) V(y) m(s)dyds
+ 2(a-1)∫^t_t- ∫_G(x, t; y, s) V(y) m(s)dyds.
Notice that
\[
\al
\int_\M G(x, t; y, s) V(y)dy&\leq ||V||_{L^p}\left(\int_\M G^{\frac{p}{p-1}}(x,t;y,s)dy\right)^{(p-1)/p}\\
&= ||V||_{L^p}\left(\int_\M G^{\frac{1}{p-1}}\cdot Gdy\right)^{(p-1)/p}\\
&\leq \sigma\hat C(t)^{1/p} \frac{1}{(t-s)^{\frac{n}{2p}}}.
\eal
\]
Therefore, (<ref>) can be further written as
\[
\al
&\leq 1+ 2(a-1)\sigma\hat C(T)^{1/p}\int^{t-\e}_0\frac{1}{(t-s)^{\frac{n}{2p}}} m(s)ds+2(a-1)\sigma\hat C(T)^{1/p}m(t)\e^{1-\frac{n}{2p}}.
\eal
\]
Moving the third term on the right hand side to the left hand side, we get
[1-2(a-1)σĈ(T)^1/p^1-n/2p]m(t) ≤1+ 2(a-1)σĈ(T)^1/p∫^t-_01/(t-s)^n/2p m(s)ds
≤1+ 2(a-1)σĈ(T)^1/p^-n/2p∫^t-_0m(s)ds.
$$\e=\left(4(a-1)\sigma\hat C(T)^{1/p}\right)^{-\frac{2p}{2p-n}},$$
we have
$$1-2(a-1)\sigma\hat C(T)^{1/p}\e^{1-\frac{n}{2p}}=\frac{1}{2}.$$
Therefore, (<ref>) becomes
\begin{align*}
m(t)\leq 2+ \left(4(a-1)\sigma\hat C(T)^{1/p}\right)^{\frac{2p}{2p-n}}\int^{t}_0m(s)ds,
\end{align*}
which again is the Grönwall inequality.
Hence, we have
for all $t\in[0,T]$. Especially, we have
for any $t\in[0,\infty)$, i.e.,
$$2^{-1/(a-1)}e^{-(a-1)^{\frac{n}{2p-n}}\left[4\sigma\hat C(t)^{1/p}\right]^{\frac{2p}{2p-n}}t}\leq J(x,t)\leq 1.$$
Therefore, $J$ exists for $t\in [0, \infty)$. This completes the proof of the claim <ref>.
Now we continue with the proof of part (1).
In (<ref>), choosing $J$ as in Claim <ref>, then we deduce
(Δ- _t) ( t Q )
+ 2 ụ/u (̣t Q)
≥t (2 J - δJ) 1/n ( | f̣ |^2 - _t f )^2 - δt J|f̣|^4 -Q
For any $T>0$, let $(x_0, t_0)$ be a maximum point of
$t Q = t \left( \a J \frac{ |\d u|^2}{u^2} - \frac{\p_t u}{u} \right)$ in ${\M} \times [0, T]$.
Then at this point, the above inequality induces
\[
0 \ge \a t (2 J - \delta J) \frac{1}{n} \left( | \d f |^2 - \p_t f \right)^2
- \delta \a t J|\d f|^4 -Q.
\]
Clearly we can assume $Q \ge 0$ at $(x_0, t_0)$ since the result is already proven
otherwise. Then
\[
\left( | \d f |^2 - \p_t f \right)^2 \ge
\left( \a J \frac{ |\d u|^2}{u^2} - \frac{\p_t u}{u} \right)^2 + (1-\a J)^2 | \d f|^4.
\]
Plugging this into the previous inequality, we find that
\[
0 \ge \a \frac{2 J - \delta J}{n} t Q^2 + \left[ \frac{2-\delta }{n} (1-\a J)^2 - \delta \right]
\a t J| \d f|^4 - Q.
\]
By choosing
one has
2-δ/n (1-)^2 - δ= 0.
Since $J\leq 1$, we derive from above that
$$\frac{2-\delta}{n} (1-\a J)^2 - \delta \ge 0\quad \text{on} \quad {\M} \times [0, \infty).$$
Therefore, at $(x_0, t_0)$,
\[
0 \ge \a \frac{2 J - \delta J}{n} t^2 Q^2 - t Q,
\]
which infers
\[
t Q \le t Q |_{(x_0, t_0)} \le \frac{n}{(2 - \delta ) \a J}\le\frac{n}{(2 - \delta ) \a \underline{J}(T)},
\]
J(t) |ụ|^2/u^2 - _t u/u ≤n/(2 - δ) J(t) 1/t.
This proves part (1) of the theorem.
For part (2), we first prove an improved short time bound for $J$.
Consider the closed ball in $L^\infty({\M} \times [0, T_0])$
X = { w ∈L^∞( ×[0, T_0])
| 1≤w ≤1+η}.
Here $\eta$ is a positive number in $(0, 1)$, and $T_0$ is a constant to be determined. Let $w_0=w(\cdot, 0)=1$, and $P$ the map
P w = w_0 +2(a-1) ∫^t_0 ∫_G(x, t; y, s) V(y) w(y, s) dyds.
For any $w \in X$, since $w\geq w_0=1$, we have
$$Pw\geq w_0.$$
P w -w_0
≤2(a-1) ∫^t_0 ∫_G(x, t; y, s) V(y) w(y, s) dyds
≤2(a-1)(1+η) w_0∫^t_0 ∫_G(x, t; y, s) V(y) dyds .
Notice that, under condition (b), we have $\displaystyle \sup_x \int_\M \frac{| Ric^-(y)|^2} {d(x, y)^{n-2}} dy< \infty$. Then by using the Gaussian upper bound of $G$,
\[
\al
&\int^t_0 \int_M G(x, t; y, s) V(y) dyds \\
\le& \left(\int^t_0 \int_\M G(x, t; y, s) dyds \right)^{1/2} \, \left(\int^t_0 \int_\M G(x, t; y, s) V^2(y) dy \right)^{1/2}\\
\le& C \sqrt{t} \left(\int^t_0 \int_\M \frac{1}{(t-s)^{n/2}} e^{- c d^2(x, y)/(t-s)} V^2(y) dyds \right)^{1/2}\\
\le& C \sqrt{t} \left(\int_\M \frac{1}{d^{n-2}(x, y)} V^2(y) dy\right)^{1/2} \\
=& C \sqrt{t} \sqrt{\sigma} \equiv C_0 \sqrt{t}.
\eal
\]
∫^t_0 ∫_G(x, t; y, s) V(y) dyds ≤C_0 √(t).
If $|Ric^-| \in L^p$ with $p>n/2$, the $1/2$ power on $t$ on the right hand side above should be replaced by $1-\frac{n}{2p}$. Here is a quick proof. By Remark 1.2, the heat kernel $G$ also has an
Gaussian upper bound and $|B(x, r)| \le C r^n$. So
∫^t_0 ∫_G(x, t; y, s) V(y) dyds
≤ C ∫^t_0 ∫_1/(t-s)^n/2 e^- c d^2(x, y)/(t-s) V(y) dyds
≤ C ∫^t_0 ( ∫_1/(t-s)^np/[2(p-1)] e^- c p/(p-1) d^2(x, y)/t-s dy )^(p-1)/p ds ‖V ‖_L^p
≤ C_0 t^1-n/2p .
In the following, we prove the theorem under the condition (b), so that (<ref>) holds. The proof
under condition (a) works verbatim after replacing (<ref>) by (<ref>).
From (<ref>) and (<ref>), we see that
\[
P w -w_0 \le C_0(a-1) \sqrt{t} w_0.
\]
If we choose
T_0 = [ C_0 (a-1) ]^-2 η^2=[ C_0 (5δ^-1-1) ]^-2 η^2,
P w -w_0 ≤ηw_0.
Thus $P$ maps $X$ into $X$.
Next we show that $P$ is a contraction mapping on $X$ when $T_0$ is chosen as in (<ref>).
Let $w_1$ and $w_2$ be two elements in $X$. Then (<ref>) implies
\[
\al
&|P w_2 - P w_1|(x, t) =
2(a-1)\left| \int^t_0 \int_\M G(x, t; y, s) V(y)
\left( w_2 - w_1\right)(y, s) dyds \right|\\
&\le 2(a-1)\int^t_0 \int_\M G(x, t; y, s) V(y) dyds \,
\Vert w_2-w_1 \Vert_\infty\\
& \le C_0 (a-1)\sqrt{t}
\Vert w_2-w_1 \Vert_\infty.
\eal
\]
By (<ref>), we know that under condition (b) of the theorem,
(<ref>) holds and also
‖P w_2 - P w_1 ‖_∞≤η‖w_2-w_1 ‖_∞.
Hence $P$ is a contraction map from $X$ to $X$. The unique fixed point, named
$w$, is a solution to (<ref>) and (<ref>). By the definition of $X$,
we already know that on ${\M} \times [0, T_0]$,
\[
1 \le w \le 1+\eta.
\]
From the relations (<ref>), we know that
\[
J = w^{\frac{1}{a-1}}=w^{-\frac{\delta}{5-\delta}}.
\]
(1+η)^-δ/5-δ≤J ≤1.
β=n/n+(1-)^2 (1+η)^-δ/5-δ.
Then, (<ref>) can be rewritten as
\[\beta\frac{n+(1-\a)^2}{n}\frac{|\nabla u|^2}{u^2}-\frac{\partial_t u}{u}\leq \frac{n}{2\beta}\frac{1}{t},\]
which obviously implies (<ref>).
Moreover, from (<ref>), (<ref>), and (<ref>), we see that
T_0 = c (1-β)^4
under condition (b) of the theorem.
Similarly, under condition (a), one can get
and hence
T_0 = c (1-β)^4p/(2p-n).
§ RICCI FLOW CASE
In this section, we consider the Li-Yau bound in the Ricci flow case and prove Theorem <ref>. The main tool is still the maximum principle applied on a differential inequality involving
Li-Yau type quantity. However, due to the Ricci flow, extra terms involving the Ricci curvature and
Hessian of the solution will come out. In order to proceed we need to create a new term with
the scalar curvature in the denominator.
Before proving the theorem, we carry out some basic computations.
and operator $\mathcal{L}=\Delta -\frac{\partial }{\partial t}$, where $\delta$, $\alpha$, and $\beta$ are arbitrary constants and $C$ is a constant so that $R+C>0$. Then
ℒF = 1/u|u_ij-u_iu_j/u+uR_ij|^2+2δ-1/u|u_ij-u_iu_j/u|^2+1/(2α-1)u|(2α-1)uR_ij+u_iu_j/u|^2
-α(R+C)|∇u|^2/u+2βu|R_ij|^2/(R+C)^2+βu/R+C| ∇R/R+C-∇u/u|^2-β|∇u|^2/u(R+C).
It follows from (<ref>) that
\begin{align}\label{delta u}
\mathcal{L}(\Delta u)=-2R_{ij}u_{ij},
\end{align}
\begin{align}\label{nabla u}
\mathcal{L}(|\nabla u|^2)=2|u_{ij}|^2.
\end{align}
Also, it is well known that under the Ricci flow we have
\begin{equation}\label{R}
\mathcal{L}R=-2|R_{ij}|^2.
\end{equation}
On the other hand, it is straightforward to check that for any smooth functions $f$ and $g$, one has
\begin{align}\label{f/g}
\mathcal{L}(\frac{f}{g})=\frac{1}{g}\mathcal{L}f-\frac{f}{g^2}\mathcal{L}g-\frac{2}{g}\nabla_i\frac{f}{g}\nabla_i g,
\end{align}
\begin{align}\label{fg}
\mathcal{L}(fg)=f\mathcal{L}g+g\mathcal{L}f+2\nabla_i f\nabla_i g.
\end{align}
It then follows from (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) that
\begin{equation}\label{nabla u/u}
\begin{aligned}
\mathcal{L}\left(\frac{|\nabla u|^2}{u}\right)&=\frac{1}{u}\mathcal{L}|\nabla u|^2-\frac{2}{u}\nabla_i\frac{|\nabla u|^2}{u}\nabla_i u\\
&=\frac{2}{u}|u_{ij}|^2-\frac{4}{u^2}u_{ij}u_iu_j+\frac{2|\nabla u|^4}{u^3}\\
\end{aligned}
\end{equation}
\begin{equation}\label{Ru}
\mathcal{L}(Ru)=u\mathcal{L}R+2\nabla_iR\nabla_iu=-2u|R_{ij}|^2+2\nabla_iR\nabla_i u,
\end{equation}
\begin{equation}\label{u/R+C}
\begin{aligned}
\mathcal{L}\left(\frac{u}{R+C}\right)&=-\frac{u}{(R+C)^2}\mathcal{L}R-\frac{2}{R+C}\nabla_i\frac{u}{R+C}\nabla_iR\\
&=\frac{2u|R_{ij}|^2}{(R+C)^2}+\frac{2u|\nabla R|^2}{(R+C)^3}-\frac{2}{(R+C)^2}\nabla_i R\nabla_i u.
\end{aligned}
\end{equation}
Thus, by (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we have, after splitting zeros in four occasions, that
\begin{align*}
\mathcal{L}F&=2R_{ij}u_{ij}+\frac{2\delta}{u}\left|u_{ij}-\frac{u_iu_j}{u}\right|^2+2\alpha u|R_{ij}|^2-2\alpha\nabla_iR\nabla_iu\\
& \quad +\frac{2\beta u|R_{ij}|^2}{(R+C)^2}+\frac{2\beta u|\nabla R|^2}{(R+C)^3}-\frac{2\beta}{(R+C)^2}\nabla_i R\nabla_i u\\
&\quad +\frac{1}{2\alpha-1}\frac{|\nabla u|^4}{u^3}-\frac{1}{2\alpha-1}\frac{|\nabla u|^4}{u^3}-2\alpha\nabla_iR\nabla_iu+\frac{\alpha u|\nabla R|^2}{R+C}+\frac{\alpha (R+C)|\nabla u|^2}{u}\\
&\quad +(\frac{\beta}{(R+C)^3}-\frac{\alpha}{R+C})u|\nabla R|^2-\frac{\alpha (R+C)|\nabla u|^2}{u}+\frac{2\beta u|R_{ij}|^2}{(R+C)^2}-\frac{2\beta}{(R+C)^2}\nabla_i R\nabla_i u\\
&\quad +\frac{\beta u|\nabla R|^2}{(R+C)^3}+\frac{\beta|\nabla u|^2}{u(R+C)}-\frac{\beta|\nabla u|^2}{u(R+C)}.
\end{align*}
Observe that the 3rd, 4th and 5th terms, 7th, 8th and 9th terms and 13th, 14th and 15th
terms form complete squares, respectively. Hence we get (<ref>).
Now we are ready to prove Theorem <ref>.
Assume that $1>\delta \geq \frac{1}{2}$ and $\alpha>1$. By choosing $C=2$ and $\beta= \alpha (R_1+2)^2$ in the above lemma, we have
\begin{equation}\label{LF1}
\begin{aligned}
\mathcal{L}F&\geq \frac{1}{nu}\left|\Delta u-\frac{|\nabla u|^2}{u}+uR\right|^2+\frac{2\delta-1}{nu}\left|\Delta u-\frac{|\nabla u|^2}{u}\right|^2 -\frac{1}{2\alpha-1}\frac{|\nabla u|^4}{u^3}\\
&\quad -\frac{\alpha (R+2)|\nabla u|^2}{u}-\frac{\beta|\nabla u|^2}{u(R+2)}.
\end{aligned}
\end{equation}
Notice that
$$\Delta u-\frac{|\nabla u|^2}{u}=\left(\Delta u-\frac{|\nabla u|^2}{u}+uR\right)-uR.$$
We rewrite (<ref>) as
\begin{align*}
\mathcal{L}F&\geq\frac{2\delta}{nu}\left|\Delta u-\frac{|\nabla u|^2}{u}+uR\right|^2-\frac{2(2\delta-1)R}{n}\left(\Delta u-\frac{|\nabla u|^2}{u}+uR\right) +\frac{(2\delta-1)R^2u}{n}\\
&\quad -\frac{1}{2\alpha-1}\frac{|\nabla u|^4}{u^3}-\frac{\alpha (R+2)|\nabla u|^2}{u}-\frac{\beta|\nabla u|^2}{u(R+2)}.
\end{align*}
According to the definition of $F$ in (<ref>), the above inequality becomes
\begin{align*}
\mathcal{L} F &\geq\frac{2\delta}{nu}\left|F+(1-\delta)\frac{|\nabla u|^2}{u}+(\alpha-1)Ru-\frac{\beta u}{R+2}\right|^2\\
&\quad +\frac{2(2\delta-1)R}{n}\left(F+(1-\delta)\frac{|\nabla u|^2}{u}+(\alpha-1)Ru-\frac{\beta u}{R+2}\right)\\
&\quad +\frac{(2\delta-1)R^2u}{n}-\frac{1}{2\alpha-1}\frac{|\nabla u|^4}{u^3}
-\left[\alpha(R+2)+\frac{\beta}{R+2}\right]\frac{|\nabla u|^2}{u}.
\end{align*}
Let $Q=tF-\theta u$. Then at $t=0$, we have $Q<0$. Suppose that at time $t_0>0$ and point $x_0\in \M$, $Q$ reaches $0$ for the first time. Then at $(x_0, t_0)$, we have $t_0 F=\theta u$ and
\begin{align*}
0&\geq t_0\mathcal{L}Q(x_0,t_0)\\
%&\geq -t_0F+\frac{2\delta}{nu}\left|t_0F+(1-\delta)\frac{|\nabla u|^2}{u}t_0+(\alpha-1)Rut_0-\frac{\beta ut_0}{R+2}\right|^2\\
%&\quad +\frac{2(2\delta-1)Rt_0}{n}\left(t_0F+(1-\delta)\frac{|\nabla u|^2}{u}t_0+(\alpha-1)Rut_0-\frac{\beta u t_0}{R+2}\right)\\
%&\quad +\frac{(2\delta-1)R^2u}{n}t_0^2-\frac{1}{2\alpha-1}\frac{|\nabla u|^4}{u^3}t_0^2 -2\beta\frac{|\nabla u|^2}{u}t_0^2\\
&\geq-\theta u +\frac{2\delta}{nu}\left|\theta u+(1-\delta)\frac{|\nabla u|^2}{u}t_0+(\alpha-1)Rut_0-\frac{\beta ut_0}{R+2}\right|^2\\
&\quad +\frac{2(2\delta-1)Rt_0}{n}\left(\theta u+(1-\delta)\frac{|\nabla u|^2}{u}t_0+(\alpha-1)Rut_0-\frac{\beta u t_0}{R+2}\right)\\
&\quad +\frac{(2\delta-1)R^2u}{n}t_0^2-\frac{1}{2\alpha-1}\frac{|\nabla u|^4}{u^3}t_0^2 -\left[\alpha(R+2)+\frac{\beta}{R+2}\right]\frac{|\nabla u|^2}{u}t_0^2
\end{align*}
After expanding the first square, we deduce
\begin{align*}
&0 \ge -\theta u+\frac{2\delta}{n}\theta^2u+\frac{2\delta(1-\delta)^2}{n}\frac{|\nabla u|^4}{u^3}t_0^2+\frac{2\delta(\alpha-1)^2R^2u}{n}t_0^2\\
&\quad +\frac{2\delta\beta^2u}{n(R+2)^2}t_0^2+\frac{4\delta(1-\delta)\theta}{n}\frac{|\nabla u|^2}{u}t_0+\frac{4\delta(\alpha-1)\theta Ru}{n}t_0-\frac{4\delta\beta \theta u}{n(R+2)}t_0\\
&\quad+\frac{4\delta(1-\delta)(\alpha-1)R}{n}\frac{|\nabla u|^2}{u}t^2_0-\frac{4\delta(1-\delta)\beta}{n(R+2)}\frac{|\nabla u|^2}{u}t_0^2-\frac{4\delta(\alpha-1)\beta R u}{n(R+2)}t_0^2\\
&\quad +\frac{2(2\delta-1)\theta Ru}{n}t_0+\frac{2(2\delta-1)(1-\delta)R}{n}\frac{|\nabla u|^2}{u}t_0^2+\frac{2(2\delta-1)(\alpha-1)R^2u}{n}t_0^2\\
&\quad -\frac{2(2\delta-1)\beta R u}{n(R+2)}t_0^2+\frac{(2\delta-1)R^2u}{n}t_0^2-\frac{1}{2\alpha-1}\frac{|\nabla u|^4}{u^3}t_0^2 -\left[\alpha(R+2)+\frac{\beta}{R+2}\right]\frac{|\nabla u|^2}{u}t_0^2.
\end{align*}
This becomes, after combining similar terms,
\begin{align*}
&0 \geq -\theta u+\frac{2\delta}{n}\theta^2u-\frac{4\delta\beta \theta u}{n(R+2)}t_0-\frac{(4\delta\alpha-2)\theta u}{n}t_0-\frac{(4\delta\alpha-2)\beta R_1u}{n(R+2)}t_0^2 \\
&\quad +\left(\frac{2\delta(1-\delta)^2}{n}-\frac{1}{2\alpha-1}\right)\frac{|\nabla u|^4}{u^3}t_0^2\\
&\quad +\left(\frac{4\delta(1-\delta)\theta}{n}t_0-\frac{4\delta(1-\delta)\beta}{n(R+2)}t_0^2-\frac{2\beta}{(R+2)} t_0^2-\frac{(4\delta\alpha-2)(1-\delta)}{n}t_0^2\right)\frac{|\nabla u|^2}{u}.
\end{align*}
It is straightforward to check that by choosing
$$\alpha=\frac{n}{2\delta(1-\delta)^2}, \qquad \textrm{and}\qquad \theta=\frac{n}{2\delta}+\frac{4n\beta T}{\delta(1-\delta)}$$
one has
\begin{equation}
\frac{2\delta(1-\delta)^2}{n}-\frac{1}{2\alpha-1}>0,
\end{equation}
\begin{equation}
\frac{4\delta(1-\delta)\theta}{nT}\geq \left[\frac{4\delta(1-\delta)}{n}+2\right] \beta+\frac{(4\delta\alpha-2)(1-\delta)}{n},
\end{equation}
\begin{equation}
\frac{2\delta}{nT^2}\theta^2-(\frac{1}{T^2}+\frac{4\delta\beta }{nT}+\frac{4\delta\alpha }{nT})\theta-\frac{4\delta\alpha\beta R_1 }{n}>0.
\end{equation}
Therefore, we have a contradiction. It follows that
$$-\Delta u+\delta\frac{|\nabla u|^2}{u}-\alpha Ru+\frac{\beta u}{R+2}\leq\frac{\theta u}{t}$$
for any $t\in(0, T)$, which is (<ref>).
In general, along the Ricci flow we have
$$-\sup_\M R(x,0)\leq R(x,t)\leq \sup_{\M \times[0,T)} R(x,t).$$
Denote by $R_1=sup_{\M \times[0,T)}R(x,t)$ and
\begin{equation*}
R_0=\left\{\begin{aligned}&\sup_\M R^{-}(x,0),\ if\ \sup_\M R^{-}(x,0)>0\\
&\inf_\M R(x,0), \ if\ \sup_\M R^{-}(x,0)=0.
\end{aligned}
\right.
\end{equation*}
It is not hard to check that by choosing $C=2R_0$ and $\beta=\alpha(R_1+2R_0)^2$ in Lemma <ref> and repeating the proof of Theorem <ref>, we can get the following scaling invariant Li-Yau bounds.
Let $\M$ be a compact n dimensional Riemmanian manifold, and $g_{ij}(t)$, $t\in [0,T)$, a solution of the Ricci flow (<ref>) on $\M$. Suppose that $u$ is a positive solution of the heat equation (<ref>). Then, for any $\delta\in[\frac{1}{2},1)$, when $\sup_\M R^-(x,0)>0$, we have
\begin{equation}
\delta\frac{|\nabla u|^2}{u^2}-\frac{\partial_t u}{u}-\alpha R+\frac{\beta}{R+2R_0}\leq\frac{\theta}{t},
\end{equation}
and when $\sup_\M R^-(x,0)=0$, we have
\begin{equation}
\delta\frac{|\nabla u|^2}{u^2}-\frac{\partial_t u}{u}-\alpha R+\frac{\beta}{R}\leq\frac{\theta}{t},
\end{equation}
for any $t\in(0,T)$, where $\alpha=\frac{n}{2\delta(1-\delta)^2}$, $\beta=\alpha (R_1+2R_0)^2$, and $\theta=\frac{n}{2\delta}+\frac{4n\beta T}{\delta(1-\delta)R_0}$.
From (<ref>), one can see that if $R>0$, then there are both a forward inequality $u_t\geq -\frac{Ba}{t}$, and a backward inequality $u_t\leq\frac{Ba}{t}$.
The Li-Yau bound (<ref>) obtained above gives us a stronger forward Harnack inequality $\frac{u_t}{u}\geq -\frac{k}{t}$ when $R>0$. However, it seems that a backward Harnack inequality of the form
$\frac{u_t}{u}\leq \frac{k}{t}$ cannot be expected. Because if this were the case, then one would have
$u(x,t_2)\leq u(x,t_1)(\frac{t_2}{t_1})^k$. Now suppose that $M$ is an Einstein manifold $R_{ij}=\rho g_{ij}$ with $\rho>0$ and $u(x,t)=G(x,t;x_0,0)$ the heat kernel under the Ricci flow. According to a result in <cit.>, we have the Gaussian lower and upper bounds of $G$, i.e.,
$$Ct^{-\frac{n}{2}}e^{-\frac{cd^2_{t}(x,x_0)}{t}}\leq G(x,t;x_0,0)\leq Ct^{-\frac{n}{2}}e^{-\frac{d_t^2(x,y)}{ct}}.$$
It then follows that
$$Ct_2^{-\frac{n}{2}}e^{-\frac{cd^2_{t_2}(x,x_0)}{t_2}}\leq G(x,t_2;x_0,0)\leq G(x,t_1;x_0,0)(\frac{t_2}{t_1})^k\leq C(\frac{t_2}{t_1})^kt_1^{-\frac{n}{2}}e^{-\frac{d_{t_1}^2(x,y)}{ct_1}},$$
$$e^{-\frac{(1-2\rho t_2)d^2_0(x,x_0)}{t_2}+\frac{(1-2\rho t_1)d^2_0(x,x_0)}{t_1}}\leq C(\frac{t_2}{t_1})^{\frac{n}{2}+k}.$$
Obviously, when $t_2=2t_1$ and $x\neq x_0$, we get a contradiction for $t_1$ small enough.
§ APPLICATIONS ON EXTENDING COLDING-NABER RESULT
In this section, we mainly apply the Li-Yau bound (<ref>) to extend parabolic approximations of distance functions of Colding-Naber <cit.> to the case where $|Ric^-|\in L^{p}$ for some $p>\frac{n}{2}$. Meanwhile, some of the intermediate results can also be proved by replacing the condition that $|Ric^-|\in L^{p}$ by $|Ric^-|\in K^{2,n-2}$, where $K^{p,\lambda}$ denotes the Kato type space with the norm
$$\|w\|_{K^{p,\lambda}}=\left(\sup_\M\int_\M \frac{|w|^p}{d^{\lambda}(x,y)}dy\right)^{\frac{1}{p}}.$$
Let $(\M, g_{ij})$ be a compact $n$ dimensional Riemannian manifold.
Parts of the following three assumptions will be used in the results of this section.
A1: $M$ is $\kappa$-noncollapsed for some constant $\k$, i.e.,
\begin{equation}\label{noncollapsing}
\v(B_r(x))\geq \k r^n,\quad \forall x\in \M,\ \textrm{and}\ r\leq 1.
\end{equation}
\textit{A2}: $||Ric^-||_L^p≤Λ$ for some $p>n/2$.
\textit{A3}: $||Ric^-||_K^2,n-2≤Γ$, the heat kernel of \eqref{heatequation}
has a Gaussian upper bound for $0<t ≤1$ as in \eqref{GUP}, and $|B(x, r)| ≤C r^n$.\\
We mention that conditions $A3$ holds on each time slice of the normalized K\"ahler-Ricci flow, which is the motivation for imposing such a condition (\cite{TZq1} and \cite{TZq2}).
Let $h^±_t(x)$ be the parabolic approximations of the local distance functions as defined in \eqref{defh}. The main result of this section is
\begin{theorem} (Tian-Z. Zhang in \cite{TZz:1})
\label{distance approximation}
Assume that A1 and A2 are satisfied. Let $O^+$ and $O^-$ be two fixed points in $\M$. Denote by $d_0=d(O^+, O^-)$. Then for some fixed $\delta>0$, there exist constants $C=C(n,p,\k,\Lambda,\delta)$ and $\overline{\e}=\overline{\e}(n,p,\delta)$, such that for any $0<\e\leq\overline{\e}$,
\[
x\in M_{\delta, 2} \equiv \{x \in \M \, | \,
\delta d_0 < d(x, \{O^+, O^-\}) \le 2 d_0 \}
\] with
\[
e(x) \equiv d(O^-,x)+d(O^+,x)-d(O^+,O^-) \leq \e^2d_0,
\] and any $\e$-geodesic $\sigma$ connecting $O^+$ and $O^-$, there exists $r\in[\frac{1}{2}, 2]$ satisfying\\
(1) $\displaystyle \left|h^{\pm}_{r{\e}^2d_0^2}-d^{\pm}\right|\leq C d_0(\e^2+\e^{2-\frac{n}{2p}})$.\\
(2) $\displaystyle \oint_{B_{{\e}d_0}(x)}\left||\nabla h^{\pm}_{r{\e}^2d_0^2}|^2-1\right|\leq C(\e+\e^{1-\frac{n}{2p}})$.\\
(3) $\displaystyle \oint_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{{\e}d_0}(\sigma(s))}\left||\nabla h^{\pm}_{r{\e}^2d_0^2}|^2-1\right|\leq C(\e^2+\e^{2-\frac{n}{p}})$.\\
(4) $\displaystyle \oint_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{{\e}d_0}(\sigma(s))}\left|\nabla^2 h^{\pm}_{r{\e}^2d_0^2}\right|^2\leq \frac{C(1+\e^{-\frac{n}{p}})}{d_0^2}$.
\end{theorem}
More explanations of the notations in the above theorem can be found in the following context.
The theorem was first obtained by Tian-Z. Zhang in \cite{TZz:1}. Here by using the Li-Yau bound \eqref{eqLY2}, following the
original route in \cite{CoNa} for the case where the Ricci curvature is bounded from below,
we are able to exemplify some of the results. Alternatively, one may also derive a
Gaussian estimate of $|∂G/∂t|$ which then can be used in place of the Li-Yau bound.
%and the Ricci curvature tensor satisfies either
%\int_\M |Ric^{-}|^p\leq \Lambda
%for some constants $\Lambda$ and $p>\frac{n}{2}$.
A very important tool that will be used repeatedly in this section is the following volume comparison theorem proved by Petersen-Wei.
\begin{theorem}{(Petersen-Wei \cite{PW1})}\label{volumecomparison}
If A2 is satisfied, then there exists a constant $C=C(n,p)$ which is nondecreasing in $R$ such that for all $r\leq R$ and $x\in \M$, we have
\begin{equation}
\left(\frac{\v(B_R(x))}{R^n}\right)^{1/2p}-\left(\frac{\v(B_r(x))}{r^n}\right)^{1/2p}\leq C\Lambda^{1/2p}R^{1-\frac{n}{2p}},
\end{equation}
where $B_r(x)$ denotes the geodesic ball centered at $x$ with radius $r$.
\end{theorem}
A very important corollary of the above theorem is the following volume doubling property (see \cite{PW2} Theorem 2.1).
\begin{theorem}{(Petersen-Wei \cite{PW2})}
Given $\alpha<1$ and $p>n/2$. Assume that A1 and A2 are satisfied. Then there exists an $R=R(\alpha, p, n, \Lambda)>0$ such that for any $0<r_1\leq r_2\leq R$, we have
\begin{equation}\label{Lpvolumedoubling}
\alpha\frac{r_1^n}{r_2^n}\leq \frac{\textrm{vol}\, B_{r_1}(x)}{\textrm{vol}\, B_{r_2}(x)}.
\end{equation}
\end{theorem}
By using the above theorem, Petersen-Wei also obtained the following cut-off function, which was first observed by Cheeger-Colding in \cite{ChCo} for manifolds with Ricci curvature bounded from below.
\begin{lemma}{(Petersen-Wei \cite{PW2})}\label{cutoffball}
Suppose that A1 and A2 are satisfied. There exist $r_0=r_0(n,p,\k,\Lambda)$ and $C=C(n,p,\k,\Lambda)$ such that on any geodesic ball $B_r(x)$, $r\leq r_0$, there exists a function $\phi\in C_0^{\infty}(B_r(x))$ such that
$$\phi\geq0,\quad\ \phi=1\ \textrm{in}\ B_{r/2}(x),$$
$$|\nabla \phi|^2+|\Delta \phi|\leq Cr^{-2}.$$
\end{lemma}
Let $E$ be a closed subset of $M$. Denote the $r$-tubular neighborhood of $E$ by
$$T_r(E)=\{x\in \M|\ d(x,E)\leq r\}.$$
For $0<r_1<r_2$, define the annulus $A_r_1, r_2(E)=T_r_2(E)∖T_r_1(E)$.
Using the lemma above and a similar argument as in the proof of Lemma 2.6 in \cite{CoNa}, one has
\begin{lemma}{(Tian-Z. Zhang \cite{TZz:1})}\label{cutoffannulus}
Suppose that A1 and A2 are satisfied. For any $R>0$, there exists $C=C(n,p,\k,\Lambda, R)$ such that the following holds.
Let $E$ be any closed subset and $0<r_1<10r_2<R$. There exists a function $\phi\in C^{\infty}(B_R(E))$ satisfying
$$\phi\geq0,\quad\ \phi=1\ \textrm{in}\ A_{3r_1,r_2/3}(E),\quad \phi=0\ \textrm{outside}\ A_{2r_1, r_2/2}(E),$$
$$|\nabla \phi|^2+|\Delta \phi|\leq Cr_1^{-2}\ \textrm{in}\ A_{2r_1, 3r_1}(E),$$
$$|\nabla\phi|^2+|\Delta\phi|\leq Cr_2^{-2} \ \textrm{in}\ A_{r_2/3, r_2/2}(E).$$
\end{lemma}
%By Theorem 7.4 in \cite{Yang}, we have uniform control of the Sobolev constant. Then by using Moser iteration and a method of Li-Schoen \cite{LiSc}, one can get an $L^1$ mean value inequality for positive solutions of the heat equation,
Let $G(y,t;x,0)=G(x,t;y,0)$ be the heat kernel on $M$. It can be showed that $G(y,t;x,0)$ has both Gaussian upper and lower bounds as follows%In \cite{TZz:1}, the authors proved the following upper and lower bounds of $G(y,t;x,0)$.
\begin{lemma}{(Tian-Z. Zhang \cite{TZz:1} ) }
\label{HK}
Suppose that A1 and A2 are satisfied. There exist positive constants $C_i=C_i(n,p,\k,\Lambda)$, $i=1,2,3,4$, such that
\begin{equation}\label{HKbound}
C_1t^{-\frac{n}{2}}e^{\frac{-C_2d^2(x,y)}{t}}\leq G(y,t;x,0)\leq C_3t^{-\frac{n}{2}}e^{-\frac{d^2(x,y)}{C_4t}},\ \forall x,y\in \M,\ \textrm{and}\ 0<t\leq 1.
\end{equation}
\end{lemma}
Actually, the Gaussian upper bound can be obtained by an $L^1$ mean value inequality for $G(y,t;x,0)$ and Grigor'yan's method in \cite{Gr1997}. Then the lower bound follows from the upper bound and an on-diagonal gradient bound for $G(y,t;x,0)$.
By using Duhamel's principle, it is not hard to prove the following $L^1$ Harnack inequalities (see e.g. \cite{TZz:1}).
\begin{lemma}\label{parabolicL1harnack}
Let $u(x,t)$ be a nonnegative function satisfying
\begin{equation}
\frac{\partial u}{\partial t}\geq\Delta u -\xi,
\end{equation}
where $\xi=\xi(x)\geq0$ is a smooth function.
(i) If A1 and A2 are satisfied, then for any $q>\frac{n}{2}$, there exists a constant $C=C(n,p,q,\kappa,\Lambda)$ such that
\begin{equation}\label{L1harnack1}
\oint_{B_r(x)}u(y,0)dy\leq C\left(u(x,r^2)+r^{2-\frac{n}{q}}\|\xi\|_{L^q}\right)
\end{equation}
holds for any $x\in \M$ and $0<r\leq 1$.
More generally, we have
\begin{equation}
\oint_{B_r(x)}u(y,0)dy\leq C\left(\inf_{B_r(x)}u(\cdot,r^2)+r^{2-\frac{n}{q}}\|\xi\|_{L^q}\right).
\end{equation}
(ii) If A1 and A3 are satisfied, then for any $q>0$ and $\lambda>n-2q$, there exists a constant $C=C(n,q,\lambda,\kappa, \Gamma)$ such that
\begin{equation}\label{L1harnack2}
\oint_{B_r(x)}u(y,0)dy\leq C\left(u(x,r^2)+r^{2-\frac{n-\lambda}{q}}\|\xi\|_{K^{q,\lambda}}\right)
\end{equation}
holds for any $x\in \M$ and $0<r\leq 1$.
More generally, we have
\begin{equation}
\oint_{B_r(x)}u(y,0)dy\leq C\left(\inf_{B_r(x)}u(\cdot,r^2)+r^{2-\frac{n-\lambda}{q}}\|\xi\|_{K^{q,\lambda}}\right).
\end{equation}
\end{lemma}
\begin{proof}
By Duhamel's principle, we have
\begin{align*}
u(x,t)-\int_\M u(y,0)G(y,t;x,0)dy\geq -\int_0^t\int_\M \xi(y)G(y,t-s;x,0)dyds.
\end{align*}
If {\it A1} and {\it A2} are satisfied, from Lemma \ref{HK}, we have
\begin{align*}
\int_{B_r(x)}u(y,0)G(y,t;x,0)dy&\leq u(x,t)+\int_0^t\int_\M \xi(y)G(y,t-s;x,0)dyds\\
&\leq u(x,t)+C\int_0^t\|\xi\|_{L^q}\left(\int_\M (t-s)^{-\frac{nq}{2(q-1)}}e^{-\frac{d^2(x,y)}{C(t-s)}}dy\right)^{\frac{q-1}{q}} ds\\
&\leq u(x,t)+Ct^{1-\frac{n}{2q}}\|\xi\|_{L^q}.
\end{align*}
By \eqref{noncollapsing} and the lower bound of $G(y,t;x,0)$ in \eqref{HKbound}, we have
$$\oint_{B_r(x)}u(y,0)dy\leq C\int_{B_r(x)}u(y,0)G(y,r^2;x,0)dy,$$
from which \eqref{L1harnack1} follows easily.
If {\it A1} and {\it A3} are satisfied, by a similar argument as above, we get
\begin{align*}
\int_{B_r(x)}u(y,0)G(y,t;x,0)dy&\leq u(x,t)+\int_0^t\int_\M \xi(y)G(y,t-s;x,0)dyds\\
&\leq u(x,t)+\int_0^t\int_\M \frac{\xi(y)}{d^{\lambda/q}(x,y)}\frac{Cd^{\lambda/q}(x,y)}{(t-s)^{\frac{n}{2}}}e^{-\frac{d^2(x,y)}{C(t-s)}}dyds\\
&\leq u(x,t)+ C||\xi||_{K^{q,\lambda}}\int_0^t \left( \int_\M \frac{d^{\lambda/(q-1)}(x,y)}{(t-s)^{\frac{nq}{2(q-1)}}}e^{-\frac{d^2(x,y)}{C(t-s)}}dy \right)^{\frac{q-1}{q}}ds\\
&\leq u(x,t)+C||\xi||_{K^{q,\lambda}}\int_0^t\frac{1}{(t-s)^{\frac{n-\lambda}{2q}}}ds\\
&=u(x,t)+C t^{1-\frac{n-\lambda}{2q}}||\xi||_{K^{q,\lambda}},
\end{align*}
$$\oint_{B_r(x)}u(y,0)dy\leq C\int_{B_r(x)}u(y,0)G(y,r^2;x,0)dy.$$
Hence, \eqref{L1harnack2} follows.
\end{proof}
When the function $u$ does not depend on $t$, the above lemma becomes
\begin{corollary}\label{ellipticL1harnack}
Let $u(x)$ be a nonnegative function satisfying
\begin{equation}
\Delta u\leq\xi(x),
\end{equation}
where $\xi(x)\geq0$ is a smooth function.
(i) If A1 and A2 are satisfied, then for any $q>\frac{n}{2}$, there exists a constant $C=C(n,p,q,\kappa,\Lambda)$ such that
\begin{equation}
\oint_{B_r(x)}u(y)dy\leq C\left(u(x)+r^{2-\frac{n}{q}}\|\xi\|_{L^q}\right)
\end{equation}
holds for any $x\in M$ and $0<r\leq 1$.
More generally, we have
\begin{equation}
\oint_{B_r(x)}u(y)dy\leq C\left(\inf_{B_r(x)}u(\cdot)+r^{2-\frac{n}{q}}\|\xi\|_{L^q}\right).
\end{equation}
(ii) If A1 and A3 are satisfied, then for any $q>0$ and $\lambda>n-2q$, there exists a constant $C=C(n,q,\lambda,\kappa,\Gamma)$ such that
\begin{equation}
\oint_{B_r(x)}u(y)dy\leq C\left(u(x)+r^{2-\frac{n-\lambda}{q}}\|\xi\|_{K^{q,\lambda}}\right)
\end{equation}
holds for any $x\in M$ and $0<r\leq 1$.
More generally, we have
\begin{equation}
\oint_{B_r(x)}u(y)dy\leq C\left(\inf_{B_r(x)}u(\cdot)+r^{2-\frac{n-\lambda}{q}}\|\xi\|_{K^{q,\lambda}}\right).
\end{equation}
\end{corollary}
Now let $O^+$ and $O^-$ be two fixed points in $$. Following \cite{CoNa}, define
\begin{equation}
d^-(x)=d(O^-,x),\ d^+(x)=d(O^+,O^-)-d(O^+,x),
\end{equation}
\begin{equation}
\end{equation}
First of all, we have in barrier sense that
\begin{align}
\Delta d^-(x)&\leq \frac{n-1}{d^-}+\psi^-\\
-\Delta d^+(x)&\leq \frac{n-1}{d^+}+\psi^+,
\end{align}
where $ψ^-=max{Δd^-(x)-n-1/d^-,0}$, and $ψ^+=max{-Δd^+(x)-n-1/d^+,0}$.
Moreover, it follows from Lemma 2.2 in \cite{PW1} that
\begin{equation}
\int_{B_r(x)}|\psi^{\pm}|^{2p}(y)dy\leq C(n,p)\int_{B_r(x)}|Ric^-|^p(y)dy.
\end{equation}
Denote by
$$d_0=d(O^+,O^-)\ \textrm{and}\ M_{r_1,r_2}=A_{r_1d_0,r_2d_0}(\{O^+, O^-\}).$$
With out loss of generality, we may assume that $d_0≤1$.
\begin{lemma}\label{excess}
For some fixed $\delta>0$,
i) if A1 and A2 are satisfied, then there exist a small constant $\overline{\e}=\overline{\e}(n,p,\delta)$, and a constant $C=C(n,p,\k,\Lambda,\delta)$ such that for any $0<\e\leq\overline{\e}$, we have
\begin{equation}\label{general excess estimate}
\oint_{B_{\e d_0}(x)}e(y)dy\leq C\left[e(x)+\e^2d_0+(||\psi^+||_{L^{2p}}+||\psi^-||_{L^{2p}})\e^{2-\frac{n}{2p}}d_0\right]\leq C(e(x)+\e^{2-\frac{n}{2p}}d_0),
\end{equation}
for all $x\in M_{\frac{\delta}{4},16}$.
In particular, this implies the excess estimate of Abresch-Gromoll \cite{AbGr}, i.e.,
\begin{equation}\label{excess estimate}
e(y)\leq C\e^{1+\alpha(n,p)}d_0,\ \forall y\in B_{\frac{1}{2}\e d_0}(x)
\end{equation}
whenever $e(x)\leq \e^{2-\frac{n}{2p}}d_0$, where $\alpha(n,p)=\frac{1}{n+1}(1-\frac{n}{2p}).$
ii) if A1 and A3 are satisfied, then for any $q>0$ and $\lambda> n-2q$, there exist a small constant $\overline{\e}=\overline{\e}(n,\delta)$, and a constant $C=C(n,q,\lambda,\k,\Gamma,\delta)$ such that for any $0<\e\leq\overline{\e}$, we have
\begin{equation}\label{general excess estimate1}
\oint_{B_{\e d_0}(x)}e(y)dy\leq C\left[e(x)+\e^2d_0+(||\psi^+||_{K^{q,\lambda}}+||\psi^-||_{K^{q,\lambda}})\e^{2-\frac{n-\lambda}{q}}d_0^{2-\frac{n-\lambda}{q}}\right],
\end{equation}
for all $x\in M_{\frac{\delta}{4},16}$.
\end{lemma}
\begin{proof}
Inequalities \eqref{general excess estimate} and \eqref{general excess estimate1} follow directly from Corollary \ref{ellipticL1harnack}, since
$$\Delta e(x)=\Delta d^--\Delta d^+\leq \frac{C}{d_0}+\psi^{-}+\psi^{+}.$$ To see that \eqref{general excess estimate} implies \eqref{excess estimate}, notice that for some $q>1$ satisfying $2\e^q\leq \e$, and any $y\in B_{(\e-\e^q)d_0}(x)$, we have
\int_{B_{\e^qd_0}(y)}e(z)dz\leq \int_{B_{\e d_0}(x)}e(z)dz\leq C(e(x)+\e^{2-\frac{n}{2p}}d_0)\v(B_{\e d_0}(x))\leq C\e^{2-\frac{n}{2p}}d_0(\e d_0)^n.
\begin{equation}
\oint_{B_{\e^qd_0}(y)}e(z)dz\leq C\e^{2-\frac{n}{2p}}d_0 \frac{(\e d_0)^n}{(\e^{q}d_0)^n}=C\e^{2-\frac{n}{2p}+n-nq}d_0.
\end{equation}
This means that there exists a point $y^{\prime}\in B_{\e^q}(y)$ such that
$$e(y^{\prime})\leq C\e^{2-\frac{n}{2p}+n-nq}d_0.$$
$$e(y)\leq e(y^{\prime})+2d(y,y^{\prime})\leq C(\e^{2-\frac{n}{2p}+n-nq}+\e^q)d_0.$$
By choosing $q=1+\alpha(n,p)=1+\frac{1}{n+1}(1-\frac{n}{2p})$, one has
$$e(y)\leq C\e^{1+\alpha(n,p)}d_0,$$
for any $y\in B_{\frac{1}{2}\e d_0}(x)\subset B_{(\e-\e^q)d_0}(x)$.
\end{proof}
Under the assumptions \textit{A1} and \textit{A2}, according to Lemma \ref{cutoffannulus}, we can construct a cut-off function $ϕ≥0$ such that
\begin{equation}\label{cutoff}
\phi=1\ \textrm{on}\ M_{\frac{\delta}{4},8},\ supp(\phi)\subset M_{\frac{\delta}{16},16},\ \textrm{and}\ |\Delta \phi|+|\nabla \phi|^2\leq \frac{C}{d_0^2}.
\end{equation}
%where $\delta$ is the constant in Lemma \ref{excess}.
Define $h_0^±(x)=ϕd^±(x)$, and $e_0(x)=ϕe(x)$. Also, denote by $h_t^±(x)$ and $e_t(x)=h_t^–h_t^+$ the solutions of the equations
\begin{equation}\label{defh}
\left\{\begin{aligned}
&(\frac{\partial}{\partial t}-\Delta)h^{\pm}(x,t)=0\\
\end{aligned}\right.
\end{equation}
\begin{equation}\label{defe}
\left\{\begin{aligned}
&(\frac{\partial}{\partial t}-\Delta)e(x,t)=0\\
\end{aligned}\right.
\end{equation}
If only \textit{A1} and \textit{A3} are satisfied, assuming further that there exists a cut-off function as in \eqref{cutoff}, we can still construct $h^±_t(x)$ and $e_t(x)$ as above.
In the following, we derive estimates of $h^±_t(x)$ and $e_t(x)$. We will use the notation
The following lemmas that we obtained are under the assumptions that
either \textit{A1} and \textit{A2} are satisfied, or \textit{A1} and \textit{A3} are satisfied, and there exists a cut-off function as in \eqref{cutoff}. But we will only present the proofs for the former case since the proofs for the latter case follow in a similar way.
\begin{lemma}\label{laplacian upper bound}
i) If A1 and A2 are satisfied, then there exists a constant $C=C(n,p,\k,\Lambda,\delta)$ such that
$$\Delta h_t^-, -\Delta h_t^+, \Delta e_t\leq C\left(\frac{1}{d_0}+(||\psi^+||_{L^{2p}}+||\psi^-||_{L^{2p}})t^{-\frac{n}{4p}}\right)\leq C\left(\frac{1}{d_0}+t^{-\frac{n}{4p}}\right)$$
in $M_{\frac{\delta}{16}, 16}$.\\
ii) If A1 and A3 are satisfied, and there exists a cut-off function as in \eqref{cutoff}, then for any $q>0$ and $\lambda>0$, there exists a constant $C=C(n,q,\lambda, \k,\Gamma,\delta)$ such that
$$\Delta h_t^-, -\Delta h_t^+, \Delta e_t\leq C\left(\frac{1}{d_0}+||\psi^{\pm}||_{K^{q,\lambda}} t^{\frac{n-\lambda}{2q}}\right).$$
in $M_{\frac{\delta}{16}, 16}$.
\end{lemma}
\begin{proof}
Following Tian-Zhang \cite{TZz:1}, here we only prove the estimate for $\Delta e_t$. The proofs of the other two estimates are similar. First, notice that for $x\in M_{\frac{\delta}{16}, 16}$, we have
\begin{align*}
\Delta e_0(x)&=\Delta \phi e(x)+2\nabla\phi\cdot\nabla e(x)+\phi\Delta e(x)\\
&\leq \frac{C}{d_0}+\psi^-+\psi^+.
\end{align*}
\begin{align*}
\Delta e_t(x)&=\int_{M_{\frac{\delta}{16},16}}\Delta_x G(y,t;x,0)e_0(y)dy\\
&=\int_{M_{\frac{\delta}{16},16}}\Delta_y G(y,t;x,0)e_0(y)dy\\
&=\int_{M_{\frac{\delta}{16},16}}G(y,t;x,0)\Delta e_0(y)dy\\
&\leq \int_{M_{\frac{\delta}{16},16}}G(y,t;x,0)(\frac{C}{d_0}+\psi^-(y)+\psi^+(y))dy\\
&\leq \frac{C}{d_0}+(\|\psi^-\|_{L^{2p}}+\|\psi^+\|_{L^{2p}})\|G(y,t;x,0)\|_{L^{\frac{2p}{2p-1}}}.
%&\leq C(\frac{1}{d_0}+t^{-\frac{n}{4p}}).
\end{align*}
In the last step above, we have used the Gaussian upper bound of $G(y,t;x,0)$ in Lemma \ref{HKbound}. This is part (i). Part (ii) can then be proven by the argument at the
end of the proof of Lemma \ref{parabolicL1harnack}.
\end{proof}
\begin{lemma}
i) If A1 and A2 are satisfied, there exists a constant $C=C(n,p,\k,\Lambda,\delta)$, such that for any $x\in M_{\frac{\delta}{2},4}$ and $0<t\leq \overline{\e}^2d_0^2$, the following estimates hold for $y\in B_{\sqrt{t}}(x)$,\\
(1) $\displaystyle |e_t(y)|\leq C\left(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}\right)$. \\
(2)$\displaystyle |\nabla e_t|(y)\leq \frac{C}{\sqrt{t}}\left(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}\right)$.\\
(3) $\displaystyle \left|\frac{\partial}{\partial t}e_t(y)\right|=|\Delta e_t(y)|\leq \frac{C}{t}\left(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}\right)$.\\
(4) $\displaystyle \oint_{B_{\sqrt{t}}(y)}|\nabla^2 e_t|^2\leq \frac{C}{t^2}\left(e(x)+td^{-1}_0+t^{1-\frac{n}{4p}}\right)^2$.\\
ii) If If A1 and A3 are satisfied, and there exists a cut-off function as in \eqref{cutoff}, then for any $q>0$ and $\lambda>n-2q$, there exists a constant $C=C(n,q,\lambda,\Gamma,\delta)$ such that\\
(1') $\displaystyle |e_t(y)|\leq C\left(e(x)+td_0^{-1}+||\psi^{\pm}||_{K^{q,\lambda}} t^{1-\frac{n-\lambda}{2q}}\right)$. \\
(2')$\displaystyle |\nabla e_t|(y)\leq \frac{C}{\sqrt{t}}\left(e(x)+td_0^{-1}+||\psi^{\pm}||_{K^{q,\lambda}} t^{1-\frac{n-\lambda}{2q}}\right)$.\\
(3') $\displaystyle \left|\frac{\partial}{\partial t}e_t(y)\right|=|\Delta e_t(y)|\leq \frac{C}{t}\left(e(x)+td_0^{-1}+||\psi^{\pm}||_{K^{q,\lambda}} t^{1-\frac{n-\lambda}{2q}}\right)$.\\
(4') $\displaystyle \oint_{B_{\sqrt{t}}(y)}|\nabla^2 e_t|^2\leq \frac{C}{t^2}\left(e(x)+td^{-1}_0+||\psi^{\pm}||_{K^{q,\lambda}} t^{1-\frac{n-\lambda}{2q}}\right)^2$.\\
Here $\overline{\e}$ is the constant in Lemma \ref{excess}.
\end{lemma}
\begin{proof} Again we will only prove (i). The main idea of the proof is from \cite{CoNa}. Here we use our Li-Yau bound in section 2 instead of the original Li-Yau bound used in \cite{CoNa}.
For any $x\in M$, Lemma \ref{laplacian upper bound} implies that
\begin{equation}
e_t(x)=e_0(x)+\int_0^t\Delta e_s(x) ds\leq e(x)+Ctd_0^{-1}+Ct^{1-\frac{n}{4p}},
\end{equation}
which is (1).
%Therefore, for any $t\in[d_{\e}^2/4, 4d_{\e}^2]$, we have
%$$e_t(x)\leq e(x)+C(\e^2d_0+\e^{2-\frac{n}{2p}}d_0).$$
It is not hard to check that the Li-Yau bound \eqref{eqLY2} gives the following Harnack inequality
%$$\alpha J\frac{|\nabla u|^2}{u^2}-\frac{u_t}{u}\leq \frac{n}{2J-\e}\cdot\frac{1}{t}.$$
\begin{equation}\label{harnack}
u(x_1, t_1)\leq u(x_2, t_2)\left(\frac{t_2}{t_1}\right)^{\frac{n}{k}}e^{\frac{d^2(x_1, x_2)}{C(t_2-t_1)}}.
\end{equation}
for any positive solution $u(x,t)$ for the heat equation, where $0<t_1\leq t_2$, and $k=k(n)$ is a constant.
By applying \eqref{harnack} to $e_t$, one has
$$e_t(y)\leq Ce_{2t}(x)\leq C\left(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}\right).$$
Also, from \eqref{eqLY2}, we have
\begin{align*}
\frac{\partial}{\partial t}e_t(y)\geq -\frac{n}{kt}e_t(y)\geq -\frac{C}{t}\left(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}\right).
\end{align*}
Therefore, we have
\begin{equation}
\left|\frac{\partial}{\partial t}e_t(y)\right|=|\Delta e_t(y)|\leq \frac{C}{t}\left(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}\right),
\end{equation}
which is (3).
Again, \eqref{eqLY2} implies that
\begin{align*}
|\nabla e_t|^2(y)&\leq C\left(\frac{1}{t}+\frac{\frac{\partial}{\partial t}e_t(y)}{e_t(y)}\right)e_t^2(y)\\
&\leq \frac{C}{t}\left(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}\right)^2+C\left(e(x)+t^{-1}d_0+t^{1-\frac{n}{4p}}\right)\left(\frac{1}{d_0}+t^{-\frac{n}{4p}}\right)\\
&\leq \frac{C}{t}\left(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}\right)^2.
\end{align*}
Thus, (2) follows immediately.
%$$|\nabla e_t|(y)\leq \frac{C}{\sqrt{t}}\left(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}\right).$$
%$$|\nabla e_{d^2_{\e}}|(y)\leq C\left(\frac{e(x)}{\e d_0}+\e+\e^{1-\frac{n}{2p}}\right).$$
For the last inequality, assume that $\eta\geq0$ is a cut-off function so that
$$\eta=1\ \textrm{in}\ B_{\sqrt{t}}(y),\ \eta=0\ \textrm{outside of}\ B_{2\sqrt{t}}(y),$$
$$|\Delta\eta|+|\nabla\eta|^2\leq \frac{C}{t}.$$
\begin{align*}
\oint_{B_{\sqrt{t}}(y)}|\nabla^2 e_t|^2&\leq \frac{1}{\v(B_{\sqrt{t}}(y))}\int_{M}\eta|\nabla^2 e_t|^2\\
&=\frac{1}{\v(B_{\sqrt{t}}(y))}\int_{M}\frac{1}{2}\eta\Delta|\nabla e_t|^2-\eta\nabla(\Delta e_t)\cdot\nabla e_t-\eta R_{ij}\nabla_i e_t\nabla_j e_t\\
&\leq \frac{1}{\v(B_{\sqrt{t}}(y))}\int_{B_{2\sqrt{t}}(y)}\Delta\eta |\nabla e_t|^2+\Delta e_t\nabla e_t\cdot\nabla\eta+\eta(\Delta e_t)^2+ \eta|Ric^-||\nabla e_t|^2\\
&\leq \frac{C}{t^2}\left(e(x)+td^{-1}_0+t^{1-\frac{n}{4p}}\right)^2.
\end{align*}
In the last step above, we have used (2) and (3) to bound $|\nabla e_t|^2$ and $|\Delta e_t|^2$, separately.
\end{proof}
Alternatively, one may also use the Gaussian estimate of $|∂G/∂t|$ to obtain the estimates in the above lemma.
From the lemma above, one gets
\begin{lemma}\label{c0approximation}
i) If A1 and A2 are satisfied, then for any $x\in M_{\frac{\delta}{2}, 4}$, we have
$$|h_t^{\pm}(x)-d^{\pm}(x)|\leq C(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}).$$
ii) If If A1 and A3 are satisfied, and there exists a cut-off function as in \eqref{cutoff}, then for any $q>0$, $\lambda>n-2q$ and $x\in M_{\frac{\delta}{2}, 4}$, we have
$$|h_t^{\pm}(x)-d^{\pm}(x)|\leq C(e(x)+td_0^{-1}+||\psi^{\pm}||_{K^{q,\lambda}} t^{1-\frac{n-\lambda}{2q}}).$$
\end{lemma}
\begin{proof} As before we will only prove (i).
Here we only present the proof for $h_t^{-}(x)$. The proof for $h^+_t(x)$ follows similarly. Firstly, we have
\begin{align*}
h^-_t(x)&=h^-_0(x)+\int_0^t\Delta h_s^-(x)ds\\
&\leq d^-(x)+Ctd_0^{-1}+Ct^{1-\frac{n}{4p}}.
\end{align*}
$$h_t^-(x)-d^-(x)\leq C(td_0^{-1}+t^{1-\frac{n}{4p}}).$$
Similarly, we have
\begin{align*}
h_t^+(x)\geq d^+(x)-Ctd_0^{-1}-Ct^{1-\frac{n}{4p}}.
\end{align*}
$$|e_t(x)|\leq C\left(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}\right),$$
we have
$$h_t^-(x)-d^-(x)\geq -C(e(x)+td_0^{-1}+t^{1-\frac{n}{4p}}).$$
\end{proof}
Recall from \cite{CoNa} that an $$-geodesic connecting $O^+$ and $O^-$ is a unit speed curve $σ$ such that $||σ|-d_0|≤^2d_0$. Moreover, one has
\begin{lemma}{(Colding-Naber \cite{CoNa})} \label{egeodesic}\\
1) Let $\sigma$ be an $\e$-geodesic connecting $O^+$ and $O^-$. Then for any $z\in \sigma$, we have $e(z)\leq \e^2d_0$.\\
2) Let $x\in M$ such that $e(x)\leq \e^2d_0$. Then there exists an $\e$-geodesic $\sigma$ such that $x\in \sigma$.
\end{lemma}
From Lemma \ref{c0approximation} and Lemma \ref{egeodesic}, we immediately have
\begin{corollary}\label{c0egeodesic}
For any $\e$-geodesic $\sigma$ connecting $O^+$ and $O^-$, any $x\in \sigma\bigcap M_{\delta/2, 4}$, and $0<\e\leq\overline{\e}$, we have\\
i) when A1 and A2 are satisfied, $$\left|h^{\pm}_{d^2_{\e}}-d^{\pm}\right|\leq C(\e^2d_0+\e^{2-\frac{n}{2p}}d_0^{2-\frac{n}{2p}}).$$
ii) when If A1 and A3 are satisfied, and there exists a cut-off function as in \eqref{cutoff}, for $q>0$ and $\lambda>n-2q$ we have
$$\left|h^{\pm}_{d^2_{\e}}-d^{\pm}\right|\leq C(\e^2d_0+||\psi^{\pm}||_{K^{q,\lambda}} \e^{2-\frac{n-\lambda}{q}}d_0^{2-\frac{n-\lambda}{q}}).$$
Here $\overline{\e}$ is the constant in Lemma \ref{excess}.
\end{corollary}
To prove that $h^±_t$ are $L^1$ close to $d^±$, we first need the following lemma.
\begin{lemma}
For any $x\in M_{\frac{\delta}{2}, 4}$, we have
$$\int_{M_{\frac{\delta}{16}, 16}\setminus M_{\frac{\delta}{4},8}} G(y,t;x,0)dy\leq \frac{C}{d_0^2}t.$$
\end{lemma}
\begin{proof}
Let $k$ be a positive integer, and $\phi_k\geq0$ a cut-off function such that $\phi_k=1$ in $M_{\frac{\delta}{k},2k}$, $\phi_k=0$ outside of $M_{\frac{\delta}{2k},4k}$, and $|\Delta \phi_k|+|\nabla \phi_k|^2\leq \frac{C}{d_0^2}$.
Then, for any $x$, we have
\begin{align*}
\left|\frac{d}{dt}\int_\M \phi_k(y)G(y,t;x,0)dy\right|&=\left|\int_\M \phi_k(y)\Delta_yG(y,t;x,0)dy\right|\\
&=\left|\int_\M \Delta\phi_k(y) G(y,t;x,0)dy\right|\\
&\leq \frac{C}{d_0^2}.
\end{align*}
Thus, we have
$$-\frac{C}{d_0^2}t\leq -\phi_k(x)+\int_{M} \phi_k(y)G(y,t;x,0)dy\leq \frac{C}{d_0^2}t.$$
It follows that for $x\in M_{\frac{\delta}{2}, 4}$,
$$\int_{M_{\frac{\delta}{16}, 32}}G(y,t;x,0)dy\leq 1+\frac{C}{d_0^2}t,$$
$$\int_{M_{\frac{\delta}{4}, 8}}G(y,t;x,0)dy\geq 1-\frac{C}{d_0^2}t.$$
$$\int_{M_{\frac{\delta}{16}, 16}\setminus M_{\frac{\delta}{4},8}} G(y,t;x,0)dy\leq \frac{C}{d_0^2}t.$$
\end{proof}
By using the above lemma, we can get
\begin{lemma}\label{c1approximation}
For any $x\in M_{\frac{\delta}{2}, 4}$, and $0<t<\overline{\e}^2d_0^2$\\
i) if A1 and A2 are satisfied, then we have
\begin{align*}
|\nabla h^{\pm}_t|^2(x)\leq 1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}}.
\end{align*}
ii) if If A1 and A3 are satisfied, and there exists a cut-off function as in \eqref{cutoff}, then
\begin{align*}
|\nabla h^{\pm}_t|^2(x)\leq 1+\frac{C}{d_0^2}t+C\sqrt{t}.
\end{align*}
Here $\overline{\e}$ is the constant in Lemma \ref{excess}.
\end{lemma}
\begin{proof} Again we will only prove (i).
Notice that
$$\frac{\partial}{\partial t}|\nabla h^{\pm}_t|^2\leq \Delta|\nabla h^{\pm}_t|^2+|Ric^-||\nabla h^{\pm}_t|^2.$$
$$|\nabla h^{\pm}_0|=1,\ \textrm{in}\ M_{\delta/4, 8},$$
$$|\nabla h^{\pm}_0|\leq C,\ \textrm{in}\ M_{\delta/16, 16}\setminus M_{\delta/4, 8},$$
$$|\nabla h^{\pm}_0|=0, \textrm{outside of}\ M_{\delta/16, 16}.$$
By Duhamel's principle, we have
\begin{equation}\label{eq1}
\begin{aligned}
|\nabla h^{\pm}_t|^2(x)&\leq \int_{M_{\delta/16, 16}}|\nabla h^{\pm}_0|^2(y)G(y,t;x,0)dy+\int_0^t\int_\M |Ric^-||\nabla h^{\pm}_s|^2G(y,t-s;x,0)dyds\\
%&\leq \int_{M_{\delta/4, 8}}|\nabla h^{\pm}_0|^2(y)G(y,t;x,0)dy+\int_{M_{\delta/16, 16}\setminus M_{\delta/4, 8}}|\nabla h^{\pm}_0|^2(y)G(y,t;x,0)dy\\
%&\quad +\int_0^t\int_\M |Ric^-||\nabla h^{\pm}_s|^2G(y,t-s;x,0)dyds\\
&\leq C+\sup_{M\times[0,t]}|\nabla h^{\pm}_s|^2\int_0^t\|Ric^-\|_{L^p}\left(\int_\M G^{\frac{p}{p-1}}dy\right)^{\frac{p-1}{p}}ds.
\end{aligned}
\end{equation}
Then the Gaussian upper bound of $G(y,t;x,0)$ in \eqref{HKbound} gives
\begin{equation}
|\nabla h^{\pm}_t|^2(x)\leq C+\sup_{M\times[0,t]}|\nabla h^{\pm}_s|^2\cdot Ct^{1-\frac{n}{2p}}.
\end{equation}
%(1-Ct^{1-\frac{n}{2p}})\sup_{M\times[0,t]}|\nabla h^{\pm}_s|^2\leq C.
Since $Ct^{1-\frac{n}{2p}}\leq C\overline{\e}^{2-\frac{n}{p}}\leq \frac{1}{2}$, it follows that
\begin{equation}\label{UB}
\sup_{M\times[0,t]}|\nabla h^{\pm}_s|^2\leq C.
\end{equation}
By plugging \eqref{UB} in \eqref{eq1} and using the Gaussian upper bound of $G$ again, we get
\begin{align*}
|\nabla h^{\pm}_t|^2(x)&\leq \int_{M_{\delta/16, 16}}|\nabla h^{\pm}_0|^2(y)G(y,t;x,0)dy+\int_0^t\int_\M |Ric^-||\nabla h^{\pm}_s|^2G(y,t-s;x,0)dyds\\
&\leq \int_{M_{\delta/4, 8}}|\nabla h^{\pm}_0|^2(y)G(y,t;x,0)dy+\int_{M_{\delta/16, 16}\setminus M_{\delta/4, 8}}|\nabla h^{\pm}_0|^2(y)G(y,t;x,0)dy\\
&\quad +\int_0^t\int_\M |Ric^-||\nabla h^{\pm}_s|^2G(y,t-s;x,0)dyds\\
&\leq 1+\frac{C}{d_0^2}t+C\int_0^t\|Ric^-\|_{L^p}\left(\int_\M G^{\frac{p}{p-1}}dy\right)^{\frac{p-1}{p}}ds\\
\end{align*}
\end{proof}
The above $C^1$ bound of $h^±_t$ can be applied to show that
\begin{lemma}\label{L1approximation}
i) If A1 and A2 are satisfied, then there exists a constant $C=C(n,p,\k,\Lambda,\delta)$, such that for any $0<\e\leq\overline{\e}$ and $0<\sqrt{t}<\e^2d_0^2$, we have\\
(1) If $x\in M_{\delta, 2}$, and $e(x)\leq \e^2d_0$, then $\displaystyle \oint_{B_{10\sqrt{t}}(x)}\left||\nabla h^{\pm}_{t}|^2-1\right|\leq Ct^{-\frac{1}{2}}(\e^2d_0+td_0^{-1}+t^{1-\frac{n}{4p}})$.\\
(2) If $\sigma$ is an $\e$-geodesic connecting $O^+$ and $O^-$, then
$$\oint_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{10\sqrt{t}}(\sigma(s))}\left||\nabla h^{\pm}_{t}|^2-1\right|\leq C(\e^2d_0+td_0^{-1}+t^{1-\frac{n}{2p}}d_0^{\frac{n}{2p}}).$$
ii) If A1 and A3 are satisfied, and there exists a cut-off function as in \eqref{cutoff}, then for any $q>0$ and $\lambda>n-2q$, there exists a constant $C=C(n,q,\lambda,\k,\Gamma,\delta)$, such that for any $0<\e\leq\overline{\e}$ and $0<\sqrt{t}<\e^2d_0^2$, we have\\
(1') If $x\in M_{\delta, 2}$, and $e(x)\leq \e^2d_0$, then $$\displaystyle \oint_{B_{10\sqrt{t}}(x)}\left||\nabla h^{\pm}_{t}|^2-1\right|\leq Ct^{-\frac{1}{2}}(\e^2d_0+td_0^{-1}+||\psi^{\pm}||_{K^{q,\lambda}}t^{1-\frac{n-\lambda}{2q}}).$$
(2') If $\sigma$ is an $\e$-geodesic connecting $O^+$ and $O^-$, then
$$\oint_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{10\sqrt{t}}(\sigma(s))}\left||\nabla h^{\pm}_{t}|^2-1\right|\leq C(\e^2d_0+td_0^{-1}+\sqrt{t}d_0+||\psi^{\pm}||_{K^{q,\lambda}}t^{1-\frac{n-\lambda}{2q}}).$$
Here $\overline{\e}$ is the constant in Lemma \ref{excess}.
\end{lemma}
\begin{proof} Let us just prove (i).
$$w_t=1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}}-|\nabla h^-_t|^2,$$
and $\phi\geq0$ a cut-off function such that $\phi=1$ in $M_{\delta, 2}$, $\phi=0$ outside of $M_{\delta/2, 4}$, and
$|\Delta\phi|+|\nabla\phi|^2\leq \frac{C}{d_0^2}.$
By direct computation and Lemma \ref{c1approximation}, one has
\begin{align*}
(\frac{\partial}{\partial t}-\Delta)[\phi^2w_t]&=\phi^2\left(\frac{C}{d_0^2}+Ct^{-\frac{n}{2p}}+2|\nabla^2h^-_t|^2+2R_{ij}\nabla_ih^-_t\nabla_jh^-_t\right)\\
&\quad -2\phi\Delta\phi w_t-2|\nabla \phi|^2w_t+4\phi\nabla\phi\cdot\nabla|\nabla h^-_t|^2\\
&\geq 2\phi^2|\nabla^2h^-_t|^2-2\phi^2|Ric^-||\nabla h^-_t|^2-\frac{C}{d_0^2}(1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}})\\
&\quad -\frac{C}{d_0}(1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}})^{\frac{1}{2}}\phi|\nabla^2 h^-_t|\\
&\geq -\frac{C}{d_0^2}(1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}})-|Ric^-|(1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}}).
\end{align*}
It follows from Lemma \ref{parabolicL1harnack} that
\begin{align}\label{eq2}
\oint_{B_{10\sqrt{t}}(x)}w_t(y)dy\leq C\left(\inf_{B_{10\sqrt{t}}(x)}w_{2t}+\frac{t}{d_0^2}(1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}})+(1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}})t^{1-\frac{n}{2p}}\right).
\end{align}
Assume that $\sigma$ is a $\e$-geodesic connecting $O^+$ and $O^-$. For (1), since $e(x)\leq \e^2d_0$, we may further choose $\sigma$ to be a piecewise geodesic passing through $x$. Then, Lemma \ref{egeodesic} implies that $e(z)\leq \e^2d_0$ along $\sigma$. Moreover, we can get the following estimate from Lemma \ref{c0approximation}.
\begin{align*}
\leq&\left|d^-(x)-d^-(\sigma(d^-(x)-10\sqrt{t}))-10\sqrt{t}\right|+C(\e^2d_0+td_0^{-1}+t^{1-\frac{n}{4p}})\\
\end{align*}
\begin{align*}
&\int_{d^-(x)-10\sqrt{t}}^{d^-(x)} w_{2t}(\sigma(s))ds\\
=&\int_{d^-(x)-10\sqrt{t}}^{d^-(x)} \left[1+C\frac{t}{d_0^2}+Ct^{1-\frac{n}{2p}}-|\nabla h^-_{2t}|^2\right] ds\\
\leq& 10\sqrt{t}+C\frac{t^{3/2}}{d_0^2}+Ct^{\frac{3}{2}-\frac{n}{2p}}-\frac{1}{\sqrt{t}}\left(\int_{d^-(x)-10\sqrt{t}}^{d^-(x)} \nabla_{\dot{\sigma}(s)}h^{-}_{2t}ds\right)^2\\
\leq& 10\sqrt{t}+C\frac{t^{3/2}}{d_0^2}+Ct^{\frac{3}{2}-\frac{n}{2p}}-\frac{1}{10\sqrt{t}}\left(h^-_{2t}(x)-h^-_{2t}(\sigma(d^-(x)-10\sqrt{t}))\right)^2\\
\leq&C(\e^2d_0+td_0^{-1}+t^{1-\frac{n}{4p}}).
\end{align*}
This means that there exists a point $y\in \sigma$ such that
$$w_{2t}(y)\leq Ct^{-\frac{1}{2}}(\e^2d_0+td_0^{-1}+t^{1-\frac{n}{4p}}).$$
Substituting the inequality above in \eqref{eq2}, one finishes the proof of (1) for $h_t^-$.
To prove statement (2) for $h^-_t$, by Lemma \ref{c0approximation}, for any $s$ with $\delta d_0<s<(1-\delta)d_0$, we have
$$\left|h_{2t}^-(\sigma(s))-h_{2t}^-(\sigma(\delta d_0))-(s-\delta d_0)\right|\leq C(\e^2 d_0 +td_0^{-1}+t^{1-\frac{n}{4p}}).$$
\begin{align*}
&\int_{\delta d_0}^{(1-\delta)d_0}w_{2t}(\sigma(s))ds\\
\leq & (1-2\delta)d_0+\frac{C}{d_0}t+Ct^{1-\frac{n}{2p}}d_0-\frac{1}{(1-2\delta)d_0}\left(\int_{\delta d_0}^{(1-\delta)d_0}\nabla_{\dot{\sigma}(s)}h^-_{2t}ds\right)^2\\
\leq & C(\e^2d_0+td_0^{-1}+t^{1-\frac{n}{2p}}d_0^{\frac{n}{2p}}).
\end{align*}
\begin{align*}
&\int_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{10\sqrt{t}}(\sigma(s))}\left||\nabla h^-_{t}|^2-1\right|\\
\leq& \int_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{10\sqrt{t}}(\sigma(s))} \left(w_{t}+Ctd_0^{-2}+Ct^{1-\frac{n}{2p}}\right)\\
\leq& C\int_{\delta d_0}^{(1-\delta)d_0}\inf_{B_{10\sqrt{t}}(\sigma(s))}w_{2t}+Ctd_0^{-1}+Ct^{1-\frac{n}{2p}}d_0\\
\leq& C(\e^2d_0+td_0^{-1}+t^{1-\frac{n}{2p}}d_0^{\frac{n}{2p}}).
\end{align*}
The proofs of both (1) and (2) for $h^+_t$ can be carried out similarly.
\end{proof}
Now we are ready to Theorem \ref{distance approximation}.
%There exists a constant $C=C(n,p,\k,\lambda,\delta)$, such that for any $0<\e\leq\overline{\e}$, $x\in M_{\delta, 2}$ with $e(x)\leq \e^2d_0$, and any $\e$-geodesic $\sigma$ connecting $O^+$ and $O^-$, there exists $r\in[\frac{1}{2}, 2]$ satisfying\\
%(1) $\displaystyle \left|h^{\pm}_{r{\e}^2d_0^2}-d^{\pm}\right|\leq C d_0(\e^2+\e^{2-\frac{n}{2p}})$.\\
%(2) $\displaystyle \oint_{B_{{\e}d_0}(x)}\left||\nabla h^{\pm}_{r{\e}^2d_0^2}|^2-1\right|\leq C(\e+\e^{1-\frac{n}{2p}})$.\\
%(3) $\displaystyle \oint_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{{\e}d_0}(\sigma(s))}\left||\nabla h^{\pm}_{r{\e}^2d_0^2}|^2-1\right|\leq C(\e^2+\e^{2-\frac{n}{p}})$.\\
%(4) $\displaystyle \oint_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{{\e}d_0}(\sigma(2))}\left|\nabla^2 h^{\pm}_{r{\e}^2d_0^2}\right|^2\leq \frac{C(1+\e^{-\frac{n}{p}})}{d_0^2}$.
\begin{proof}[Proof of Theorem \ref{distance approximation}:]
Estimates (1), (2), and (3) are contained in Lemmas \ref{c0approximation} and \ref{L1approximation}, respectively. In the following, we prove (4).
For any $\sigma(s)$, let $\eta(x)\geq0$ be the cut-off function satisfying $\eta=1$ in $B_{d_{\e}}(\sigma(s))$,$\eta=0$ outside of $B_{3d_{\e}}(\sigma(s))$, and $|\Delta\eta|+|\nabla\eta|^2\leq \frac{C}{d_{\e}^2}$, where $d_{\e}=\e d_0$.
Let $a(t)$ be a smooth function in time such that $0\leq a(t)\leq 1$, $a(t)=1$ for $t\in[\frac{1}{2}d_{\e}^2, 2d_{\e}^2]$, $a(t)=0$ for $t\not\in[\frac{1}{4}d^2_{\e}, 4d^2_{\e}]$, and $|a^{\prime}(t)|\leq\frac{C}{d^2_{\e}}$.
Recall that $$(\frac{\partial}{\partial t}-\Delta)\left(|\nabla h^{\pm}_t|^2-1\right)=-2|\nabla^2 h^{\pm}_{t}|^2-2R_{ij}\nabla_i h^{\pm}_t\nabla_j h^{\pm}_t.$$
Hence, we have
\begin{align*}
&2\int_\M a(t)\eta|\nabla^2 h^{\pm}_t|^2\\
=&\int_\M a(t)\eta\Delta(|\nabla h^{\pm}_t|^2-1)-2\int_\M a(t)\eta R_{ij}\nabla_i h^{\pm}_t\nabla_j h^{\pm}_t-\int_\M a(t)\eta\frac{\partial}{\partial t}(|\nabla h^{\pm}_t|^2-1)\\
\leq & \frac{C}{d_{\e}^2}\int_{B_{3d_{\e}}(\sigma(s))}a(t)||\nabla h^{\pm}_t|^2-1|+2\int_{B_{3d_{\e}}(\sigma(s))}a(t)|Ric^-||\nabla h^{\pm}_t|^2-\int_\M a(t)\eta\frac{\partial}{\partial t}(|\nabla h^{\pm}_t|^2-1).
\end{align*}
\begin{align*}
&\frac{1}{\v(B_{3d_{\e}}(\sigma(s)))}\int_{\frac{1}{2}d_{\e}^2}^{2d_{\e}^2}\int_{B_{d_{\e}}(\sigma(s))}|\nabla^2 h^{\pm}_t|^2 dydt\\
\leq & \frac{C}{d_{\e}^2}\int_{\frac{1}{4}d_{\e}^2}^{4d_{\e}^2}\oint_{B_{3d_{\e}}(\sigma(s))}||\nabla h^{\pm}_t|^2-1|dydt+C\int_{\frac{1}{4}d_{\e}^2}^{4d_{\e}^2}\oint_{B_{3d_{\e}}(\sigma(s))}|Ric^-|(1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}})dydt\\
%\leq & \frac{C}{d_{\e}^2}\int_{\frac{1}{4}d_{\e}^2}^{4d_{\e}^2}(\frac{t}{d_0^2}+t^{1-\frac{n}{2p}})dt+C\int_{\frac{1}{4}d_{\e}^2}^{4d_{\e}^2}(1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}})\left(\v(B_{3d_{\e}}(\sigma(s)))\right)^{-\frac{1}{p}}dt\\
%\leq& \frac{C}{d_{\e}^2}\int_{\frac{1}{4}d_{\e}^2}^{4d_{\e}^2}(\frac{t}{d_0^2}+t^{1-\frac{n}{2p}})dt+Cd_{\e}^{-\frac{n}{p}}\int_{\frac{1}{4}d_{\e}^2}^{4d_{\e}^2}(1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}})dt.
\end{align*}
It follows immediately from \eqref{noncollapsing} and Theorem \ref{volumecomparison} that
\begin{align*}
&\int_{\frac{1}{2}d_{\e}^2}^{2d_{\e}^2}\int_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{d_{\e}}(\sigma(s))}|\nabla^2 h^{\pm}_t|^2 dydsdt\\
\leq & \frac{C}{d_{\e}^2}\int_{\frac{1}{4}d_{\e}^2}^{4d_{\e}^2}(\e^2d_0+\frac{t}{d_0}+t^{1-\frac{n}{2p}}d_0^{\frac{n}{2p}})dt+Cd_0d_{\e}^{-\frac{n}{p}}\int_{\frac{1}{4}d_{\e}^2}^{4d_{\e}^2}(1+\frac{C}{d_0^2}t+Ct^{1-\frac{n}{2p}})dt\\
%\leq& C(\e^2d_0+\e^{2-\frac{n}{p}}d_0)+C(\e^{2-\frac{n}{p}}d_0+\e^{4-\frac{n}{p}}d_0+\e^{4-\frac{2n}{p}}d_0)\\
\leq& Cd_0(\e^2+\e^{2-\frac{n}{p}}).
\end{align*}
Therefore, there exists an $r\in[\frac{1}{2}, 2]$ such that
\begin{equation}
\int_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{d_{\e}}(\sigma(s))}|\nabla^2 h^{\pm}_{rd_{\e}^2}|^2 dyds\leq \frac{C(1+\e^{-\frac{n}{p}})}{d_0}.
\end{equation}
\end{proof}
Similarly, we can prove
\begin{theorem}
Assume that A1 and A3 are satisfied, and there is a cut-off function as in \eqref{cutoff}. Then for some fixed $\delta>0$, any $q>0$ and $\lambda>n-2q$, there exist constants $C=C(n,q,\lambda,\k,\Gamma,\delta)$ and $\overline{\e}=\overline{\e}(n,\delta)$, such that for any $0<\e\leq\overline{\e}$,
$x\in M_{\delta, 2}$ with
$e(x)\leq \e^2d_0$ and any $\e$-geodesic $\sigma$ connecting $O^+$ and $O^-$, there exists $r\in[\frac{1}{2}, 2]$ satisfying\\
(1) $\displaystyle \left|h^{\pm}_{r{\e}^2d_0^2}-d^{\pm}\right|\leq C d_0(\e^2+||\psi^{\pm}||_{K^{q,\lambda}}\e^{2-\frac{n-\lambda}{q}}d_0^{2-\frac{n-\lambda}{q}})$.\\
(2) $\displaystyle \oint_{B_{{\e}d_0}(x)}\left||\nabla h^{\pm}_{r{\e}^2d_0^2}|^2-1\right|\leq C(\e+||\psi^{\pm}||_{K^{q,\lambda}}\e^{1-\frac{n-\lambda}{q}}d_0^{1-\frac{n-\lambda}{q}})$.\\
(3) $\displaystyle \oint_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{{\e}d_0}(\sigma(s))}\left||\nabla h^{\pm}_{r{\e}^2d_0^2}|^2-1\right|\leq C(\e^2d_0+\e d_0^2+||\psi^{\pm}||_{K^{q,\lambda}}\e^{2-\frac{n-\lambda}{q}}d_0^{2-\frac{n-\lambda}{q}})$.\\
(4) $\displaystyle \oint_{\delta d_0}^{(1-\delta)d_0}\oint_{B_{{\e}d_0}(\sigma(s))}\left|\nabla^2 h^{\pm}_{r{\e}^2d_0^2}\right|^2\leq \frac{C(1+\e^{-1}+||\psi^{\pm}||_{K^{q,\lambda}}\e^{-\frac{n-\lambda}{q}})}{d_0^2}$.
\end{theorem}
Moreover, from Lemma \ref{c0approximation} and Lemma \ref{L1approximation}, we can also get
\begin{lemma}
Assume that A1 and A2 are satisfied. Let $x\in M_{\delta, 2}$ with $\sigma$ a unit speed minimizing geodesic from $O^-$ to $x$. Then for any $\delta\leq \tau_1<\tau_2\leq d^-(x)$, the following estimates hold:\\
1) $\displaystyle \int_{\delta}^{d^-(x)}\left||\nabla h^-_t|^2-1\right|\leq \frac{C}{d_0}(e(x)+t+t^{1-\frac{n}{4p}})$.\\
2) $\displaystyle \int_{\delta}^{d^-(x)}\left|<\nabla h^-_t, \nabla d^->-1\right|\leq \frac{C}{d_0}(e(x)+t+t^{1-\frac{n}{4p}})$.\\
3) $\displaystyle \int_{\tau_1}^{\tau_2}\left|\nabla h^-_t-\nabla d^-\right|\leq \frac{C\sqrt{\tau_2-\tau_1}}{\sqrt{d_0}}(\sqrt{e(x)}+\sqrt{t}+t^{\frac{1}{2}-\frac{n}{8p}})$.
\end{lemma}
%\section{Distance comparison under the Ricci flow}
\section*{Acknowledgements}
The paper was originally intended to be a joint one with Richard Bamler who made an important contribution.
However, Richard has withdrawn from authorship to concentrate on other projects. We sincerely thank him for his generosity.
M. Zhu would like to express his great appreciation to Prof. Huai-Dong Cao for constant encouragement and support. We are grateful to Professors H.-D. Cao, X.Z. Dai, H.Z. Li,
G.F. Wei for their interest and comments on the result.
Q.S.Z. gratefully acknowledges the support of Simons'
Foundation and Siyuan Foundation
through Nanjing University.
\begin{thebibliography}{99}
\bibitem[AbGr]{AbGr} Abresch, Uwe; Gromoll, Detlef, {\it On complete manifolds with nonnegative Ricci curvature}, J. Amer. Math. Soc. 3 (1990), no. 2, 355-374.
%\bibitem[And]{Anderson} Anderson, Michael T., {\it A survey of Einstein metrics on 4-manifolds}, Handbook of geometric analysis, No. 3, 1-39, Adv. Lect. Math. (ALM), 14, Int. Press, Somerville, MA, 2010, http://arxiv.org/abs/0810.4830.
%\bibitem[ArBe]{ArBe} Aronson, Donald G.; B\'enilan, Philippe, {\it R\'egularit\'e des solutions de l'\'equation des milieux poreux dans RN}, C. R. Acad. Sci. Paris S\'er. A-B 288 (1979), no. 2, A103-A105.
\bibitem[BCP]{BCP} Bailesteanu, Mihai; Cao, Xiaodong; Pulemotov, Artem Gradient estimates for the heat equation under the Ricci flow. J. Funct. Anal. 258 (2010), no. 10, 3517-3542.
\bibitem[BBG]{BBG} Bakry, Dominique; Bolley, Francois; Gentil Ivan, {\it The Li-Yau inequality and applications under a curvature-dimension condition}, arXiv:1412.5165, 2014.
\bibitem[BL]{BL} Bakry, Dominique and Ledoux, Michel,
{\it A logarithmic Sobolev form of the Li-Yau parabolic inequality},
Rev. Mat. Iberoamericana, Volume 22, Number 2 (2006), 683-702.
\bibitem[BZ]{BZ} Bamler, Richard; Zhang, Qi S., {\it Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature}, http://arxiv.org/abs/1501.01291 (2015).
%\bibitem [BCG]{BCG:1} Barlow, Martin; Coulhon, Thierry; Grigor'yan, Alexander, {\it Manifolds and graphs with slow heat kernel decay}, Invent. Math. 144 (2001), no. 3, 609-649.
\bibitem[Bre]{Bre} Brendle, Simon, {\it A generalization of Hamilton's differential Harnack inequality for the Ricci flow}, J. Differential Geom. 82 (2009), no. 1, 207-227.
\bibitem[Cao]{Cao}Cao, Huai-Dong, {\it On Harnack's inequalities for the K\"ahler-Ricci flow}, Invent. Math. 109 (1992), no. 2, 247-263.
\bibitem[CaNi]{CaNi} Cao, Huai-Dong; Ni, Lei, {\it Matrix Li-Yau-Hamilton estimates for the heat equation on K\"ahler manifolds}, Math. Ann. 331 (2005), no. 4, 795-807.
\bibitem[CTZ]{CTZ} Cao, Huai-Dong; Tian, Gang; Zhu, Xiaohua, {\it K\"ahler-Ricci solitons on compact complex manifolds with $C_1(M)>0$}. Geom. Funct. Anal. 15 (2005), no. 3, 697-719.
\bibitem[Cx]{Cx} Cao, Xiaodong, {\it Differential Harnack estimates for backward heat equations with potentials under the Ricci flow.} J. Funct. Anal. 255 (2008), no. 4, 1024-1038.
%\bibitem[CaHa]{CaHa} Cao, Xiaodong; Hamilton, Richard S., {\it Differential Harnack estimates for time-dependent heat equations with potentials}, Geom. Funct. Anal. 19 (2009), no. 4, 989-1000.
\bibitem [CH]{CH:1} Cao, Xiaodong; Hamilton, Richard S., {\it Differential Harnack estimates for time-dependent heat equations with potentials}, Geom. Funct. Anal. 19 (2009), no. 4, 989-1000.
\bibitem[CZ]{CZ} Cao, Xiaodong; Zhang, Qi S. {\it The Conjugate Heat Equation and Ancient Solutions of the Ricci Flow}, Adv. Math., Vol. 228 (2011), no. 5, 2891-2919.
%\bibitem[CG]{CG} Cheeger, Jeff; Gromoll, Detlef, {\it The splitting theorem for manifolds of nonnegative Ricci curvature}, J. Differential Geometry 6 (1971/72), 119-128.
\bibitem[ChCo]{ChCo} Cheeger, Jeff; Colding, Tobias H., {\it Lower bounds on Ricci curvature and the almost rigidity of warped products}, Ann. of Math. (2) 144 (1996), no. 1, 189-237.
\bibitem[Chow]{Chow} Chow, Bennett, {\it The Ricci flow on the 2-sphere}, J. Differential Geom. 33 (1991), no. 2, 325-334.
\bibitem [ChH]{ChH:1} Chow, Bennett; Hamilton, Richard S., {\it Constrained and linear Harnack inequalities for parabolic equations}, Invent. Math. 129 (1997), no. 2, 213-238.
%\bibitem[CN]{Cheeger-Naber-codim-4} Cheeger, Jeff; Naber, Aaron, {\it Regularity of Einstein Manifolds and the Codimension 4 Conjecture}, http://arxiv.org/abs/1406.6534 (2014).
%\bibitem[CT]{CH:2} Cao, Xiaodong; Tran, Hung, {\it Mean Value Inequalities and Conditions to Extend Ricci Flow}, http://arxiv.org/abs/1303.4492v1 (2013).
%\bibitem [CW1]{CW:1} Chen, Xiuxiong; Wang, Bing, {\it Space of Ricci flows I}, Comm. Pure Appl. Math. 65 (2012), no. 10, 1399-1457, http://arxiv.org/abs/0902.1545
%\bibitem [CW2]{CW:2} Chen, Xiuxiong; Wang, Bing, {\it On the conditions to extend Ricci flow(III)},
%Int. Math. Res. Not. IMRN 2013, no. 10, 2349-2367, http://arxiv.org/abs/1107.5110
%\bibitem [CW3]{CW:3} Chen, Xiuxiong; Wang, Bing, {\it Space of Ricci flows (II)}, http://arxiv.org/abs/1405.6797
\bibitem[Chowetc]{Chowetc} Chow, Bennett; Lu, Peng; Ni, Lei, {\it Hamilton's Ricci flow}. Graduate Studies in Mathematics, 77. American Mathematical Society, Providence, RI; Science Press, New York, 2006. xxxvi+608 pp.
\bibitem[CoNa]{CoNa} Colding, Tobias Holck; Naber, Aaron, {\it Sharp H\"older continuity of tangent cones for spaces with a lower Ricci curvature bound and applications}, Ann. of Math. (2) 176 (2012), no. 2, 1173-1229.
%\bibitem[GH]{GH:2} Grigor'yan, Alexander; Hu, Jiaxin, {\it Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces}, Invent. Math. 174 (2008), no. 1, 81-126.
\bibitem[Dav]{Dav} E. B. Davies, {\it
Heat kernels and spectral theory}, volume 92 of
Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1989
\bibitem[GM]{GM} N. Garofalo and A. Mondino, {\it Li-Yau and Harnack type inequalities in
$RCD^*(K;N)$ metric measure spaces.} Nonlinear Anal., 95: 721-734, 2014.
\bibitem[Gri]{Gr1997} Grigor'yan, Alexander, {\it Gaussian upper bounds for the heat kernel on arbitrary manifolds}, J. Differential Geom. 45 (1997), no. 1, 33-52.
%\bibitem[Gr]{Gr:2} Grigor'yan, Alexander, {\it Heat kernel and analysis on manifolds}, AMS/IP Studies in Advanced Mathematics, 47. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. xviii+482 pp.
%\bibitem [Ha1]{Ha:0}Hamilton, Richard S., {\it Three-manifolds with positive Ricci curvature}, J. Differential Geom. 17 (1982), no. 2, 255-306.
%\bibitem [Ha2]{Ha:1} Hamilton, Richard S., {\it The formation of singularities in the Ricci flow}, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7-136, Int. Press, Cambridge, MA, 1995.
\bibitem[Ha3]{Ha:2} Hamilton, Richard S., {\it A matrix Harnack estimate for the heat equation}, Comm. Anal. Geom. 1 (1993), no. 1, 113-126.
\bibitem[Ha4]{Ham1988} Hamilton, Richard S., {\it The Ricci flow on surfaces}, Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988.
\bibitem[Ha5]{Ham1993jdg} Hamilton, Richard S., {\it The Harnack estimate for the Ricci flow}, J. Differential Geom. 37 (1993), no. 1, 225-243.
%\bibitem[HN]{HN:1} Hein, Hans-Joachim; Naber, Aaron, {\it New logarithmic Sobolev inequalities and an $\varepsilon$-regularity theorem for the Ricci flow}, Comm. Pure Appl. Math. 67 (2014), no. 9, 1543-1561.
\bibitem[LX]{LX} J. Li and X. Xu. {\it Differential Harnack inequalities on Riemannian manifolds I: linear
heat equation.} Adv. Math., 226(5):4456-4491, 2011.
\bibitem[KuZh]{KuZh} Kuang, Shilong; Zhang, Qi S., {\it A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow}, J. Funct. Anal. 255 (2008), no. 4, 1008-1023.
%\bibitem[L]{L:1} Li, Peter, {\it Geometric analysis}, Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press, Cambridge, 2012.
%\bibitem[LT]{LT:1} Li, Peter; Tian, Gang, {\it On the heat kernel of the Bergmann metric on algebraic varieties}, J. Amer. Math. Soc., 8 (1995), 857-877.
\bibitem [LY]{LY:1} Li, Peter; Yau, Shing-Tung, {\it On the parabolic kernel of the Schr\"odinger operator.} Acta Math. 156 (1986), no. 3-4, 153-201.
\bibitem[Ni]{Ni2006} Ni, Lei, {\it A note on Perelman's LYH-type inequality}, Comm. Anal. Geom. 14 (2006), no. 5, 883-905.
\bibitem[P1]{P:1} Perelman, Grisha, {\it The entropy formula for the Ricci flow and its
geometric applications}, http://arxiv.org/abs/math/0211159 (2002).
%\bibitem[P2]{P:2} Perelman, Grisha, {\it Ricci flow with surgery on three-manifolds}, http://arxiv.org/abs/math/0303109 (2003).
\bibitem[PW1]{PW1} Petersen, Peter; Wei, Guofang, {\it Relative volume comparison with integral curvature bounds}, Geom. Funct. Anal. 7 (1997), no. 6, 1031-1045.
\bibitem[PW2]{PW2} Petersen, Peter; Wei, Guofang, {\it Analysis and geometry on manifolds with integral Ricci curvature bounds. II}, Trans. Amer. Math. Soc. 353 (2001), no. 2, 457-478.
\bibitem[QZZ]{QZZ}
Z. Qian, H.-C. Zhang, and X.-P. Zhu. {\it Sharp spectral gap and Li-Yau's estimate on
Alexandrov spaces}.
Math. Z., 273(3-4):1175-1195, 2013
%\bibitem[Se]{Sesum-Ricci}Sesum, Natasa, {\it Curvature tensor under the Ricci flow}, Amer. J. Math. 127 (2005), no. 6, 1315-1324.
%\bibitem[Sh]{Sh:1} Shi, Wan-Xiong, {\it Deforming the metric on complete Riemannian manifolds}, J. Differential Geom. 30 (1989), no. 1, 223-301.
%\bibitem[Si]{Si:1} Simon, Miles, {\it Ricci flow of almost non-negatively curved three manifolds}, J. Reine Angew. Math. 630 (2009), 177-217.
%\bibitem[ST]{ST:1} Sesum, Natasa; Tian, Gang, {\it Bounding scalar curvature and diameter along the K\"ahler Ricci flow (after Perelman)}, J. Inst. Math. Jussieu 7 (2008), no. 3, 575-587.
%\bibitem[TW]{Tian-Wang} Tian, Gang; Wang, Bing, {\it On the structure of almost Einstein manifolds}, http://arxiv.org/abs/1202.2912 (2012).
\bibitem[TZq1]{TZq1} Tian, Gang; Zhang, Qi S. {\it Isoperimetric inequality under K\"ahler Ricci flow}.
Amer. J. Math. 136 (2014), no. 5, 1155-1173.
\bibitem[TZq2]{TZq2} Tian, Gang; Zhang, Qi S., {\it A compactness result for Fano manifolds and K\"ahler Ricci flows}, Math. Ann. 362 (2015), no. 3-4, 965-999.
\bibitem[TZz]{TZz:1} Tian, Gang; Zhang, Zhenlei,
{\it Regularity of K\"ahler Ricci flows on Fano manifolds}, http://arxiv.org/abs/1310.5897v1 (2013).
%\bibitem[Wa]{Wa:1} Wang, Bing, {\it On the conditions to extend Ricci flow (II)}, Int. Math. Res. Not. IMRN 2012, no. 14, 3192-3223.
%\bibitem[Ye]{Ye:1} Rugang Ye, {\it The logarithmic Sobolev inequality along the Ricci flow}, http://arxiv.org/abs/0707.2424 (2007).
\bibitem[Wan]{Wan} F.-Y. Wang. {\it Gradient and Harnack inequalities on noncompact manifolds with boundary},
Pacific J. Math., 245(1):185-200, 2010.
\bibitem[WanJ]{WanJ}Wang, Jiaping, {\it Global heat kernel estimates.} Pacific J. Math. 178 (1997), no. 2, 377-398.
\bibitem[Z06]{Z06:1} Zhang, Qi S., {\it Some gradient estimates for the heat equation on domains and for an equation by Perelman}, Int. Math. Res. Not., 39 pages Art. ID 92314, 39, 2006.
%\bibitem[Z07]{Z07:1} Zhang, Qi S., {\it A uniform Sobolev inequality under Ricci flow}, IMRN 2007, ibidi Erratum, Addendum.
%\bibitem[Z11]{Z11:1} Zhang, Qi S., {\it Bounds on volume growth of geodesic balls under Ricci flow}, Math. Res. Lett. 19 (2012), no. 1, 245-253, http://arxiv.org/abs/1107.4262
%\bibitem[ZZh]{ZhouZhang-scal} Zhang, Zhou, {\it Scalar curvature behavior for finite-time singularity of K\"ahler-Ricci flow}, Michigan Math. J. 59 (2010), no. 2, 419-433.
\end{thebibliography}
\end{document}
|
1511.00905
|
University of Alabama at Birmingham, USA University of Helsinki, Finland Aalto University, Finland
Contextual proximity detection (or, co-presence detection) is a promising
approach to defend against relay attacks in many mobile authentication
We present a systematic assessment of co-presence detection in
the presence of a context-manipulating attacker. First, we
show that it is feasible to manipulate, consistently control and stabilize the
readings of different acoustic and physical environment sensors (and even
multiple sensors simultaneously) using low-cost, off-the-shelf
Second, based on these capabilities, we
show that an attacker who can manipulate the context gains a significant
advantage in defeating context-based co-presence detection. For systems that
use multiple sensors, we investigate two sensor fusion approaches based on machine learning techniques –
features-fusion and
decisions-fusion, and show that both are vulnerable to
contextual attacks but the latter approach can be more resistant in some cases.
§ INTRODUCTION
Authentication is critical to many mobile and wireless systems where one
communicating device (prover ) needs to validate its identity to the
other (verifier ). Traditional cryptographic authentication typically
involves a challenge-response protocol whereby proves the possession of
the key $K$ that it pre-shares with by constructing a valid response
to a random challenge sent by .
Examples of systems where such
authentication is deployed include payment transactions between NFC/RFID
devices and point-of-sale systems, and zero-interaction authentication
<cit.> scenarios between a token and a terminal (e.g., phone-laptop, or key-car). Unfortunately, the security and usability benefits
provided by these authentication systems can be subverted by means of relay attacks, as demonstrated by prior research (e.g.,
<cit.>), which involve two non
co-present colluding attackers, one near and one near , simply
relaying protocol messages back and forth between and .
A known defense to relay attacks is
distance bounding, where a challenge-response
authentication protocol allows to measure an upper-bound of its
distance from <cit.>. Using this protocol,
can verify whether is within a close proximity thereby
detecting the presence of relay attacks
<cit.>. However, distance bounding
systems may not be currently feasible on commodity devices (such as
smartphones or payment tokens) due to their
sensitivity to measurement errors (of elapsed time).
The presence of ubiquitous and low-cost sensing capabilities on many
modern mobile devices has facilitated a potentially more viable relay attack defense
This defense leverages the notion of “context” derived from on-board device
sensors based on which -proximity, or lack of it, could be
determined. In other words, in a benign setting, where and are co-present, both would record a similar context with a high probability.
In contrast, if the system is subject to a relay attack, and and
are non co-present, devices' context should be different with a high
Extensive recent prior work demonstrated the feasibility of using different
types of sensor modalities for such contextual co-presence detection,
including audio <cit.>, radio (WiFi
<cit.>, Bluetooth <cit.> and
GPS <cit.>, and the physical environment
(temperature, humidity, gas and
altitude/pressure) <cit.>. Many single modalities,
such as audio and WiFi, were shown to be performing quite well for contextual
co-presence detection resulting in low false negatives (i.e.,
rejecting a co-presence instance; a measure of usability) and low false
positives (accepting a non co-presence instance; a measure of security). In
addition, fusion of multiple modalities, including combination of
audio-radio <cit.>, and combination of physical sensors
<cit.>, has been shown to further reduce false negatives and
false positives.
Our Contributions:
The focus of prior work on contextual co-presence detection largely centered on
evaluating the system's security under the assumption that it is very hard to
manipulate the contextual environment (i.e., it considered only a Dolev-Yao
attacker <cit.>).
In this paper, we are extending this model to the realm of a
context-manipulating attacker.
There are two main parts and contributions of the paper, as summarized below:
1. Novel Context Manipulation
Attacks: We show that it is feasible to manipulate the readings of different
sensors (and combinations thereof) using
low-cost, off-the-shelf equipment, representing a
realistic attacker. We demonstrate attacks against a variety of modalities
studied in prior work including audio, radio (Bluetooth/WiFi), and physical
(temperature, humidity, gas and altitude).
work is the first to consider context-manipulation attacks against
audio and physical modalities. In particular,
we demonstrate how an attacker in
close proximity of the sensors can successfully manipulate the
physical environment “seen” by these sensors, without the need to manipulate
the global surrounding environment or compromise the devices/sensors
Our attacks are described in Section <ref>.
2. Co-Presence Detection with Context
Manipulations: Based on the above manipulation capabilities, we
comprehensively examine and quantify the advantage a
attacker, who can manipulate multiple sensor
modalities simultaneously, can have in defeating co-presence detection over a
attacker (one studied in prior work). To accomplish this, we
re-orchestrated the co-presence detection approaches based on machine learning techniques in
audio-only <cit.>, audio-radio
<cit.>, physical <cit.> and a
(newly-proposed) audio-radio-physical systems, in a way that non
co-present data samples were manipulated for different modality combinations.
Our results show that the attacker advantage increases many-folds in several
cases (Table <ref> quantifies the attacker success rates).
For systems that use multiple modalities, we investigate two different sensor
fusion approaches – features-fusion (proposed in
<cit.>) and decisions-fusion based on majority voting,
and show that both approaches are vulnerable to contextual attacks but the
latter can be more resistant in some cases, at the cost of slight degradation
in usability. Our detailed analysis is presented in Section
Broader Impact and Lessons Learned: Our work
represents the first concrete step towards analyzing, extending and
systematizing prior work on contextual co-presence detection under a stronger,
but realistic adversarial model.
It suggests that tampering with context may
not be as difficult as previously believed, and the security offered by
contextual co-presence detection is therefore weaker. Although a sophisticated
attacker would likely fare better at manipulating the context (compared to our
attacks), we also suggest potential strategies (including decisions
fusion) that may still be used to strengthen the security of co-presence
detection against a attacker (Section <ref>).
At a broader level, our work calls the security of contextual co-presence detection
into question, and motivates the need of re-evaluating the security of other
context-centric systems in the face of context manipulation.
For instance, our work may be extended to analyze the security of other
promising context-based systems such as contextual access control
<cit.> with respect to context-manipulating adversaries.
§ BACKGROUND AND MODELS
§.§ Relay Attacks and Contextual Co-Presence Detection
The goal of the adversary against a challenge-response authentication system
is to fool into
concluding that is nearby and thus needs access to even when
is actually far away. The attacker possesses standard Dolev-Yao
capabilities <cit.>: it has complete control of the
communication channel over which the authentication protocol between and is run but does not have physical possession of nor is
able to compromise (e.g., through malware) either or .
The attacker could take the form of a
“ghost-and-leech” <cit.> duo ($A_p$, $A_v$)
such that $A_p$ (respectively $A_v$) is physically close to (), and $A_p$ and $A_v$ communicate over a high-speed connection.
Such an adversary pair can
compromise the security of traditional challenge-response authentication by
simply initiating a protocol session between and , relaying
messages between them, leading to conclude that is in
proximity. This is an attack applicable to zero-interaction authentication
systems. A similar attack applies to proximity-based payment systems
model of proximity based authentication with contextual co-presence. In our work, the unidirectional single-modal attacker model
<cit.> is extended to bidirectional and multiple-modality attackers (highlighted with blue arrows). Radio sensors and Gas are subject to bidirectional attacks in our new model.
Co-presence detection schemes aim to address such relay attacks.
Figure <ref>(a) shows a typical system model of an
authentication/authorization protocol using contextual co-presence, adapted
from <cit.>.
In this defense, (respectively ) pre-shares a key
$K$ ($K'$) with a “comparator” (which may be part of or
a separate entity, depending on the scenario). When sends a trigger to
, it responds with a challenge $ch$. and then
initiate context sensing for a fixed duration $t$. computes a response
$rsp$ (using $K$), appends it to the sensed context information $CP$ and sends
both , protected by $K$. forwards this to .
In the meantime, finishes sensing its own context and sends the
resulting context data $CV$ protected using $K'$ to .
then recovers $CP$, $CV$, $ch$ and $rsp$. It checks the
validity of $rsp$ and compares if $CP$ is sufficiently similar to $CV$. If
both checks succeed, concludes that and are
co-present. When is integrated with , $K'$ is not used.
Figure <ref>(b) shows how contextual co-presence can thwart
a Dolev-Yao relay attacker.
Prior work has proposed the use of different sensor modalities for such
co-presence detection: ambient audio – <cit.>, radio context including WiFi – and
Bluetooth – <cit.>, and physical environmental attributes, temperature – , humidity – ,
concentration of gases – and altitude – <cit.>.
§.§ Threat Model for a Contextual Attacker
Our focus is on a context-manipulating attacker against co-presence detection (going beyond a Dolev-Yao attacker).
Truong et al. <cit.> briefly explored the problem of
characterizing such a contextual attacker. They only consider an attacker who
is capable of manipulating a single sensor modality at a time
(“attacker”, in our parlance). Again, in this model, an attacker cannot
compromise and devices.
Based on the rationale that is often unattended, whereas is in the possession of a human user, they speculated that the context
attacker can manipulate context without detection only in one direction.
More precisely, they modeled a attacker as follows:
* $A_p$, $A_v$ can measure the context information that , would sense, respectively.
* $A_v$ can fool into sensing the context information $A_v$
chooses. Specifically $A_v$ can receive context information from
$A_p$ and reproduce it near .
* $A_v$ ($A_p$) cannot suppress any contextual
information from being sensed by ().
Figure <ref>(c) illustrates this threat model.
Later in
Section <ref>, based on our context-manipulation attacks presented
in Section <ref>, this current model will be extended,
to incorporate attackers, who can perform the above ()
tasks corresponding to multiple modalities simultaneously.
We reiterate that this model assumes that is unattended,
and as such, the attacks we will present in this paper are based on this
assumption. Vehicles parked in underground parking lots/decks represent an apt
example of unattended verifiers. Relay attacks against such vehicles have
already been demonstrated in <cit.>. Other examples
include stolen laptops in a zero-interaction authentication system. Payment
scenarios, such as those involving parking meters or remote gas station pumps,
also involve unattended payment terminals and are thus also vulnerable to our
§ CONTEXT MANIPULATION ATTACKS
§.§ Manipulating Audio Sensor Modality
To manipulate ambient audio, an adversary must find a way to make ambient audio
on one side similar to that on the other side. Recall from
Section <ref> that our threat model allows the attacker to add
to the ambient audio at 's side without being noticed, allowing him to
relay/stream the ambient audio in real-time from 's side to 's
side thereby causing the features used for audio correlation almost match at
both sides. The assumption that manipulating audio at 's side can
go undetected is valid since may be unattended in many scenarios
(as our model in Section <ref> assumed). The attacker duo can
use any reliable audio streaming tool
to stream the audio from 's side to 's side. They can execute
this attack conveniently using mobile phones and wireless data connection.
We evaluated how well such an attacker can succeed in fooling audio-based
co-presence detection by streaming ambient audio
using Skype <cit.>. We use the features and classifier described in prior work
<cit.>. Our results are presented in Section
We consider the one-side context adversary model described in previous section.
In that model, an attacker manipulates ambient audio at 's side by
relaying ambient audio from 's side. We designed different experiments
which take into account various types of ambient noise and different streaming
channels. Two locations in Finland and the USA were set up for audio streaming
experiment. and used a Samsung Galaxy SIII phone and a Samsung
Nexus phone, respectively, for recording ambient audio. Streaming channels were
established by computer to computer via WiFi and by phone to phone via 3G/4G.
Similarity (or difference) of ambient audio was based on features proposed by
Halevi et al. <cit.>. Those features mainly rely
on the time and frequency of audio samples. We used audio samples with various
frequencies ranging from low level (less than 100Hz) to medium (human hearable
range about 500Hz) and high or very high level (equal or more than 5000Hz). For
a scenario of specific level of ambient noise at and , we
recorded five samples, each sample in ten seconds. As result of nine
combinations, we obtained 45 samples for each type of streaming channel (WiFi
and 3G/4G). In total, we collected 90 samples with non co-presence ground
§.§ Manipulating Radio-Frequency Sensor Modalities
Prior work suggests that
manipulating the radio context is possible in general.
The work presented in <cit.> describes attacks on a
public WiFi-based positioning system. They used a Linux laptop as an Access
Point (AP) with the Scapy packet manipulation program <cit.> to spoof WiFi APs.
Similarly, spoofing bluetooth device addresses has already been demonstrated in
prior work <cit.>, both of which
reported bluetooth-based relay attacks. An attacker can control the received
signal strength by controlling the transmission power of his masquerading
devices. Therefore, we conclude that the threat model assumed in
<cit.> (see Section <ref>) is reasonable.
Furthermore, in the case of RF sensor modalities, it is reasonable to assume
that an attacker can also manipulate the RF environment at 's end
without being noticed (since radio waves are imperceptible to human users).
Therefore, limiting the attacker to unidirectional manipulation only is too
We tested the feasibility of WiFi spoofing ourselves, and studied how it can be
used to match the WiFi context at two ends. In our experiment, we used a
Linksys router (WRT54G) to create a spoofed hotspot. We flashed DD-WRT
firmware <cit.> to the router since the default firmware did not allow us to
spoof the Basic Service Set Identifier (BSSID).
The router used in our experiment is portable, easily available in the market, and much
cheaper than other devices which can also be used to spoof the
hotspot such as laptops or smartphones.
The DD-WRT control panel also provides an option to change the transmission
(TX) power with which we can increase/decrease the signal strength.
The normal signal strength for the router detected by our target device (a
MacBook Air laptop) was around -39 dBm. The router and the target device were
located around 30 cm apart.
Merely by adjusting router settings, we were able to vary the signal strength
of the router, as sensed by the target device, between -25 dBm and -48 dBm.
By changing the distance between the target device and the spoofed router, we
were able to further reduce the signal strength down to -87 dBm. This suggests
that the adversary has a high degree of control in manipulating sensed signal
Based on this spoofing and Received Signal Strength Indicator (RSSI) manipulation capability, the WiFi context
matching attack becomes rather straightforward. The attacker can even have
advantage in environments where number of WiFi APs is low. For example, we
observed that there are less than five APs in outdoors such as parking lot. In
such cases, the attacker would only need to spoof 's side.
Let us assume that victim devices (and ) are non co-present,
and detects ($P_1$, $P_2$, $\cdots$, $P_m$) APs with different signal
strengths while detects ($V_1$, $V_2$, $\cdots$, $V_n$) APs with
different signal strengths. The attacker knows where the victim devices are
located, and what WiFi APs and signal strength they are receiving at those
locations at a given time (the attacker can simply scan this information with a WiFi device by
being in proximity of the victim devices). Now, the attacker just needs to
bring $(m+n)$ APs to spoof the WiFi context at both sides. He places $n$ APs at
's side with spoofed addresses as ($V_1$, $V_2$, $\cdots$, $V_n$) and the corresponding signal strengths,
while he places $m$ APs at 's side with spoofed addresses as ($P_1$,
$P_2$, $\cdots$, $P_m$) and the corresponding signal strengths. Now, both and will report ($P_1$,
$P_2$, $\cdots$, $P_m$, $V_1$, $V_2$, $\cdots$, $V_n$). Since both report the
set of APs with almost similar signal strengths, the attacker would succeed in
matching the WiFi context at both ends (although the devices were not co-present).
Let's assume that the system is using WPS system with location database such as WiGLE<cit.> to identify the location of where the device is then the attack described above may not work. Since, P and V will notice abnormal APs such as some APs from Europe and others from America or New York and California, the P and V devices may get suspicious if their system use this as a preventive feature. An attack described
by Tippenhauer <cit.> can successfully invade this prevention
by jamming the signals from valid APs and spoofing APs signal from other side. This means
both P and V will see APs from one side either ($P_1$, $P_2$, $\cdots$, $P_m$) or ($V_1$, $V_2$, $\cdots$, $V_n$) whichever an attacker feels easy and inexpensive.
§.§ Manipulating Physical Environment Sensor Modalities
As discussed in <cit.>, it may seem hard to manipulate physical
modalities, Temperature , Humidity , Gas and Altitude .
For example, it appears that an adversary has to change the temperature or
humidity of the entire environment surrounding the victim device which may be
quite challenging or detected easily. However, in this section, we show
that, by using off-the-shelf devices, manipulating physical context is not only
feasible but also realistic and effective by tampering with the “local”
environment close to one of the devices (e.g., an unattended ). Our
attacks do not require the compromise of the devices (or ),
but rather only manipulation of environment close to their sensors. In order to
monitor the current ambient readings as they are being changed, the attacker
has to use his sensors. These ambient readings serve as a feedback for the
attacker while he attempts to change the current 's ambience. The
feedback sensor needs to be placed very close to the victim sensor so that the
two provide similar readings.
Our experiments demonstrate how different contextual modalities can be
manipulated, controlled and stabilized to enable successful
relay attacks. Arbitrarily changing a sensor's readings, at the verifier's
side, based on a physical activity may be straightforward but consistently
maintaining and controlling these readings to match those at the prover's side,
is non-trivial. For example, it may be obvious that temperature can be
increased using a hair dryer (a simple tool used in our temperature
manipulation experiments), but how to maintain it at a desired level for a
reasonable period of time (during which the attack can be launched) is not
obvious. While we present several direct/explicit ways to manipulate many
modalities, we also demonstrate some indirect/implicit techniques. For example,
we show how altitude can be manipulated by changing pressure (i.e., without
relocating the device to a different altitude).
We demonstrate several indirect ways to manipulate some of the sensor
modalities, such as applying pressure on the sensors to change the
altitude readings (i.e., not changing the altitude directly, which may not be
possible in many cases), or using an aerosol spray (which does not contain CO as
such) to alter CO readings. In general, we demonstrate the capability
of the attacker to consistently control the readings of a given sensor
(not just manipulate them arbitrarily).
The physical sensing devices have an operational weakness in that the
information about the environment they measure corresponds to the environment
close to the sensors. For example, the temperature may be 20 °C
inside a room, but if the sensor is near a bonfire or a stove in the kitchen,
the temperature it measures is higher than the actual ambient temperature.
Hence, to change these sensor readings, the adversary does not need to change
the entire environment around the devices. He can simply change the ambient
properties around the sensors locally to fool them to record the wrong information.
When performing the attacks, we need to consider that the attacker will not have
access to the direct readings from the actual () device
and hence has to use his own sensors to monitor the current ambient readings during the attack.
These ambient readings serve as a
feedback for the attacker while he attempts to change the current 's
ambience. The feedback sensor needs to be
placed close to the victim sensor so that both provide similar readings.
Certain factors may play an important role in changing a target modality to a
desired level, especially the current ambient environment at the attack site.
The attacker also needs to be confident enough that he can manipulate the
sensors to the desired level within a certain threshold for certain duration of
time window. Considering these factors, we performed all the attack
activities, described in the following sections, for more than three times to
obtain consistent results.
§.§.§ Temperature Manipulation
§.§.§ Humidity Manipulation
§.§.§ Gas Manipulation
§.§.§ Altitude Manipulation
§.§ Manipulating Multiple Sensor Modalities Simultaneously
As demonstrated by prior work
<cit.>, a
contextual co-presence detection system can use
combinations of several sensor modalities.
In such cases, the attacker needs to manipulate multiple modalities at the same
time (attacker).
However, performing one
activity may be altering not only the target modality but also
one or more other modalities
that a system might be using for context detection, such as (and ) or (and ) even though they are not directly
For example, hair dryer increases temperature but also dries-up the air (i.e.,
potentially reduces the humidity) around the sensor where it is applied. It
also changes the ambient noise. An attacker needs to manipulate in such a way
that if the multiple modalities are involved in the system he should change the
target modality without altering other modalities by effective amount.
We also found that hair dryer activity results in a huge momentary change in
gas level. However, the reading comes back to normal when hair dryer
is applied for a long period of time. Altitude and pressure did not change with
the hair dryer activity. Hair dryer activity also does not impact on RF
signals. Hence, hair dryer activity can be used to manipulate the system which
uses either temperature or humidity along with gas, altitude and RF signals.
Using aerosol spray to increase the gas content does not have effective change on
any other modalities besides humidity. Similarly, updating RF signals does not
seem to have any effect on physical modalities. Therefore, an attacker can
simultaneously manipulate radio, temperature and gas while he hopes that audio,
altitude and humidity either match the minimum criteria from both sides or is
not used by the system.
Using an ice cube to decrease the temperature does not affect other modalities
However, if the ice melts then
it may affect the humidity of the space near the sensors. In our experiment, we
saw that humidity fluctuates when we tried to decrease the temperature using an
ice cube. Hence, using an ice cube to decrease temperature activity can be used
with all other modalities except altitude and humidity.
Hot coffee cup changes the humidity along with the temperature,
while other modalities remain unchanged. In this case, an attacker can
manipulate humidity along with radio, audio and gas while he cannot control
temperature and humidity together.
To manipulate gas, we used an aerosol spray. While performing this
activity, only humidity seems to have been altered while other physical
modalities did not have any effective change. Hence, if the system does not use
humidity, gas can be changed along with radio, temperature, and audio.
When an attacker has to use an air pump or vacuum cleaner to increase or
decrease the altitude, it affects ambient noise. Also, an air pump was used in
conjunction with a Ziploc bag where the sensors were wrapped to create an
enclosed space. When an attacker performs such activity with an enclosed
space, it will be very difficult for him to change gas, temperature or
humidity. We thus may only claim that the attacker can manipulate altitude
along with radio modalities.
To summarize, our attacks support the following combinations of manipulations:
(1) , , ;
(2) , , , (increase for ), ;
(3) , , , (decrease for ), ;
(4) , , , ;
(5) , , , ;
(6) , , , .
However, a more sophisticated attacker (than the one we considered) may use
different techniques to possibly attack other combinations too.
* , ,
* , , , (increase),
* , , , (decrease),
* , , ,
* , , ,
* , , ,
§ PERFORMANCE OF CO-PRESENCE DETECTION SYSTEMS UNDER CONTEXT MANIPULATION
In light of the attacks presented in
Section <ref>, we first extend the rudimentary
context attacker model from <cit.> as follows:
* We allow attackers who can simultaneously control
multiple sensor modalities, in addition to the attacker of
We assume that a context attacker can
manipulate radio contexts in both directions. The same
assumption applies to gas sensors in light of our aerosol spray attack.
§.§ Analysis Methodology
To fairly evaluate the resilience of co-presence detection systems in the
presence of our contextual attacker, we used the same datasets and the same set
of features originally used to evaluate the systems in question. The
audio-radio system <cit.> used a dataset
to evaluate resistance against attackers. The physical system
<cit.> used a dataset
to model a attacker. We use these datasets to evaluate the
resistance of the respective systems against attackers. In
addition, we conducted a set of new audio relaying experiments to collect a
to evaluate the performance of contextual co-presence detection based on audio.
Furthermore, we collected a new dataset corresponding to the
audio-radio-physical system (which was not considered in prior work).
When evaluating prior systems, we used the same classification techniques used
in the original evaluations (Decision Tree and Random Forest), implemented in
Scikit-learn <cit.>. The results are reported after running
ten-fold cross validation.
We use False Positive Rate (FPR) as a metric to
represent the attacker's success probability. FPR corresponds to “non
co-presence” samples which are mislabeled as “co-presence”, reflecting the
security of the system (higher the FPR, lower the security). We use False
Negative Rate (FNR) as a metric to represent the usability of the system. FNR
represents to “co-presence” samples that are mislabeled as “non co-presence”
(lower the FNR, better the usability). F1 score is reported only for the
overall performance of the classification model under attack.
Whenever multiple sensor modalities are used, we fuse the data from these modalities before feeding it to
the classifier. We considered the following fusion approaches:
* Features-fusion: The features of all sensor modalities are
together fed to the classifier. The decision of co-presence or non co-presence
is made one-time only based on the output of the prediction model. This is
the approach followed in prior work <cit.>.
* Decisions-fusion: Each of the $n$ sensors (with all its
features) is used separately by the classifier. As result there are $n$
decisions made. All decisions are then combined based on majority voting to
produce a final decision. This is a new approach we are exploring in this
paper. Decisions-fusion can aggregate decisions from single sensor modalities
or from subsets of sensor modalities, for example, three subsets can be built
on top of seven sensors: acoustic = {}, radio = {, },
physical = {, , , }. In the latter fusion
approach, classifiers of subsets are built using features-fusion.
§.§ Audio-Only System
Halevi et al. <cit.> proposed the use of (only) audio
for co-presence detection. Their work showed that audio is a good ambient
context resulting in 100% accuracy and 0% False Positive Rate (FPR).
To assess how an attacker can manipulate ambient audio via the
streaming attack (Section <ref>), we conducted a
set of experiments to collect about 100 audio samples for the non
co-presence case. The audio streaming was done over two different channels:
WiFi and cellular data. was a Galaxy Nexus device while
was a Galaxy S3 device. Unidirectional streaming of the audio from
's side to 's side was done between a pair of devices (from
a Galaxy S4 to an iPhone 5 in the case of the cellular data channel,
and from a MacBook Air to a ThinkPad Carbon X1 in the WiFi
channel). The attacker devices used a Skype connection as the audio
relay channel.
The audio features used in <cit.> are based on
audio frequency. Therefore, to evaluate the impact frequency on the attack
feasibility, we tested three different levels of ambient audio frequencies
collected by controlled experiments where we set up the ambient noise
surrounding recording devices falling into different categories. Low
ambient audio (frequency less than 100 Hz); Medium ambient audio
(frequency in the human audible range, at around 500 Hz); High ambient
audio (frequency 5000 Hz or more).
Relay attack success rate (FPR) for audio streaming via WiFi and
Cellular networks
Acoustic relaying environments WiFi Cellular
(freq $\rightarrow$ freq)
High $\rightarrow$ Medium 100% 40%
High $\rightarrow$ Low 100% 20%
Medium $\rightarrow$ Medium 100% 0%
Medium $\rightarrow$ Low 100% 60%
Low $\rightarrow$ Low 20% 0%
Others 0% 0%
We used the dataset for ambient audio of previous work <cit.>
which collected ambient acoustic data to build the classification model
(F1 of 0.86 and FPR of 9.3%). The 100 samples we collected via audio streaming
channels are fed to the classifier for prediction.
Table <ref> presents the FPR of non co-presence detection
under the streaming attacks over WiFi and cellular data channels. The results
indicate that the attacker (1) has a higher chance of success using the WiFi
channel and (2) could be thwarted when either the ambient audio at is low
frequency or if the ambient audio at is high frequency.
This simple streaming attack with commodity devices shows that the
audio-only system is highly vulnerable to relay attacks, especially via the
WiFi channel. The attack has very high success rate regardless of hardware
variations and network delays inherent to streaming. However, an attacker can
succeed only when relaying ambient audio from a higher frequency acoustic
environment to a similar or lower frequency acoustic environment, such that,
the higher frequency dominates the lower frequency, and makes falsely
record 's ambient noise instead of the real “localized” ambient noise.
The audio features we used, i.e., the ones proposed in
<cit.>, are not sensitive to time
synchronization. This is effective in terms of co-presence detection (i.e.,
results in very low FNR). However, as we can see from our experiments, these
features also enable the attacker to succeed in the relay attack with a very
high chance. Other audio features, such as the ones proposed in
<cit.>, require tight synchronization and could be more
resistant to relaying. Unfortunately, because of their high sensitivity to
synchronization, these features did not perform well in the benign
(co-presence) case based on our experiments (i.e., resulted in high FNR).
§.§ Audio-Radio System
Truong et al. <cit.>, evaluated the performance of an
audio-radio system against a unidirectional, attacker. They showed
that while the system achieves good performance (F1 of 0.98) and high security
(FPR of 2.0%), a context attacker could increase the FPR: from 0.18% to 65.8%
(manipulating ), from 1.1% to 1.2% (); from 1.62% to 3.01%
(audio). Now, we will analyze the same system against a bi-directional (for
radio), attacker. To model the attack, in each run, the non
co-presence samples in the test were transformed as below.
Audio: Because raw audio data is additive, and one-side
context manipulation for audio is tested, an adversary can be modelled by
replacing side audio ($X_a$) to be the sum of its own ambient
audio and side audio ($X_a + X_b$).
Radio (and ): In <cit.>, the
set of radio records from two devices $A$ and $B$ are defined as: $S_{a}
= \{(m^{(a)}_i,s^{(a)}_i)\ |\ i \in \mathbb{Z}_{n_a-1} \}$, and $S_{b}
= \{(m^{(b)}_i,s^{(b)}_i)\ |\ i \in \mathbb{Z}_{n_b-1} \}$, where ($m,s$) with
$m$ is an identifier and $s$ is associated signal strength of a beacon; $n_a$
and $n_b$ denote the number of different beacons (i.e., WiFi access points or
Bluetooth devices). The both-sides context adversary can be modeled by
replacing $S_a$ with $S_a \cup \{(m, s) ~\forall (m, s) \in S_b, m \not \in
S^{(m)}_a\}$, and $S_b$ with $S_b \cup \{(m, s) ~\forall (m, s) \in S_a, m \not
\in S^{(m)}_b\}$.
We considered two approaches of fusing sensor data against bi-directional relay
attacks and showed which of them is more suitable for resisting against the
presence of context attackers.
FPRs with/without
different contextual attacks in various audio/radio/physical
systems. Notations: Sets of manipulated sensors are put
inside curly braces {}. {$\widetilde{X}$} denotes an arbitrary
set of sensor modalities. Fuse-F: features-fusion, Fuse-D-S:
decisions-fusion from single modalities, Fuse-D-M: decisions-fusion
from subsets of modalities. Result highlights:
Manipulation of sensor modalities, especially multiple of them, can
significantly reduce security (increase FPR) in most cases.
Decisions-fusion can help improve security when dominant sensors are
manipulated, but it may reduce usability (increase FNR).
2cAudio-Radio aaa
2cPhysical aaa
3-4 6-7 9-11
(1) (2) (3) (4) (5) (6) (7)
3-4 6-7 9-11
-1*90 Zero-modality
(FNR: 1.4%)
(FNR: 12.0%)
(FNR: 3.9%)
(FNR: 14.5%)
(FNR: 0.0%)
(FNR: 0.3%)
(FNR: 0.0%)
(F1: 0.977)
(F1: 0.925)
(F1: 0.928)
(F1: 0.861)
(F1: 0.923)
(F1: 0.980)
3-4 6-7 9-11
{}: 3.0%
{}: 3.0%
{}: 8.3%
{}: 17.0%
{}: 87.7%
{}: 45.3%
{}: 36.9%
{}: 2.7%
{}: 9.0%
{}: 11.9%
{}: 20.0%
{}: 100%
{}: 45.8%
{}: 36.9%
{}: 99.8%
{}: 8.0%
{}: 15.3%
{}: 24.4%
{}: 12.3%
{}: 44.8%
{}: 35.0%
{}: 55.1%
{}: 33.1%
{}: 5.4%
{}: 37.9%
{}: 6.9%
{}: 5.9%
{}: 29.6%
{}: 6.9%
{}: 3.4%
{}: 29.1%
{}: 6.9%
-3*90 Single-modality
{}: 3.4%
{}: 31.5%
{}: 6.9%
3-4 6-7 9-11
{, }: 3.6%
{, }: 96.0%
{, }: 13.9%
{,}: 40.1%
{}$\cup${$\widetilde{X}$}: 100%
{2 sensors}:
{, }$\cup${$\widetilde{X}$}:
{, }: 99.8%
{, }: 96.0%
{, }: 15.7%
{, }: 41.9%
{, }: 100%
{, }: 100%
{, }: 29.6%
{, }: 50.6%
{3 sensors}:
{, }$\cup${$\widetilde{X}$}:
{, , }: 100%
{, , }: 100%
{, , }: 31.1%
{, }: 57.5%
{, }: 61.2%
{$\widetilde{X}$}$\backslash${, }:$<12.3\%$
{4 sensors}:
{, , , }:
{, }: 65.5%
rest: 100%
rest: 10%
{, }: 36.9%
-4*90 Multi-modality
rest: 6.9-87.7%
Table <ref> (columns 1 and 2) presents the analysis results of
training model combining all three audio-radio modalities (, and ) and testing with different attacks. Zero-modality attack shows the
very low FPR with both fusion methods. The FNR for decisions-fusion is higher
compared to that for features-fusion.
For features-fusion, the results are aligned with the ones reported in
In attack, manipulating WiFi, the dominant feature, results in a
very high success rate with features-fusion. The results change when
decisions-fusion was applied. Here, each sensor contributed equally to the
voting process. In such case, manipulating any single sensor, even the most
powerful one, does not significantly degrade the overall security. The FPR in
case was manipulated decreases from 99.8% (features-fusion) down to 8%
(decisions-fusion). We recall that the performance difference of audio and
radio sensors is not large (as reported in <cit.>, F1 ranges
from 0.857 for to 0.989 for ).
This explains why decisions-fusion reduces the overall performance
slightly (F1 reduces from 0.977 to 0.925) in case of attack but
significantly improves the security under a single-modality attack. The
security is very low in attack, and neither of the fusion
approaches could restore the security level when majority of the sensors are
under attacker's control. When manipulating any modality along with WiFi, the
FPR is above 95%. We earlier argued that audio and radio modalities can be
manipulated simultaneously.
§.§ Physical System
In <cit.>, four physical modalities (, , ,
and ) were introduced for co-presence detection. The performance of
the features-fusion based classifier trained with their dataset is good (F1 of
0.957, FPR of 5.81%) against a adversary.
Based on our attacks against physical modalities (Section <ref>),
we consider an adversary model where an attacker can manipulate the physical
context on one side (unattended verifier) to match the sensor readings at the
other side (prover). To model this attack, all non co-presence samples in the
test set were transformed to the “attack” value (distance 0). The distance
is set to 0 as data collection in <cit.> was done by a single
device at a given point of time, hence, no hardware effect or calibration error
was taken into account. The non co-presence class in the dataset is about 18
times larger than co-presence class. To correct this imbalance, we applied the
same under-sampling as in <cit.>: we divided the non co-presence
samples into 19 subsets, and ran several rounds of cross validation taking 10
subsets in each round and aggregating the results in the end. In addition to
the features-fusion employed in <cit.>, we tested the decisions
fusion similar to our audio-radio system analysis in the previous section.
Table <ref> (columns 3 and 4) shows our analysis results. The
system performance in zero-modality attack is well-aligned with the one
reported in <cit.>. As in <cit.>, among four
physical modalities, performs the best. Consequently, manipulating
only degrades the security vastly with features-fusion (FPR increases
to over 50%). Decisions-fusion in general brings lower security and lower
performance/usability in attack and attack. However,
it avoids the dominance of sole sensor in case the attacker can control such
sensor (in this case). Decisions-fusion can also help improve
security against a attacker who manipulates along with
other sensors. Compared to audio-radio system, in physical system, attacking
each single modality results in higher success rate.
§.§ Audio-Radio-Physical System
We extended the data collector used in <cit.> to record physical
sensor data using an attached Sensordrone device (as used in
<cit.>). Different device models were used to record sensor
data. Each device, in a pair of devices, was connected to its own Sensordrone
device. Two users were involved in the data collection. Data was collected at
different locations in two countries for ten days. The resulting dataset has
203 non co-presence samples and 335 co-presence samples.
Unlike the dataset for physical sensors (<cit.>) which was
collected from one device at a time only, we collected data from pairs of
devices, and therefore hardware variance and calibration errors between
co-presence device sensors need to be taken into account. When we try to model
the contextual attack on given sensor(s), distance 0 does not ensure that the attack will succeed. As
the classifier is trained with data which may contains noise, we compute the
mode of the histogram for distance values for the co-presence samples. As the data
aggregated is from two participants, histograms of distance values are not
uninomial but multinomial. Multinomial distribution implies several modes. For
each physical sensor, we choose a mode value and assign it as the distance value.
The mode values for , , and are 13.54, 0.3,
6.61 and 0.153, respectively.
As the manipulation by replacing the radio data at
both sides has to be identical, the distance features for radio sensors are set to 0.
Table <ref> (columns 5, 6, 7) reports our analysis results with
different fusion methods. Under attack, features-fusion performs the
best while decisions-fusion from single modalities performs the worst.
Features-fusion uses all possible features for training so that the classifier
can be built based on the best features or best combination of features
(and with our current dataset). Thus, it returns the
best results (in the absence of context manipulation) compared to any other
ways of fusing sensor data. Decisions-fusion based on single modalities lets
the worst sensors being able to contribute to the voting scheme, thus bringing
down the overall performance. This is the case in our dataset where radio
sensors and audio sensor perform better than physical sensors. Note that if all
sensors perform equally well, features-fusion and decisions-fusion would not
differ much. Decisions-fusion from subsets of sensors has a moderate
performance, worse than features-fusion but better than decisions-fusion from
single modalities. This hybrid approach avoids mis-learning as in the case of
using a single modality only.
Let us now assess the security of this co-presence detection system when any
single modality is controlled by the attacker. Depending on how sensors are
fused, the impact of manipulated sensor varies. In features-fusion, as the
classifier decision relies on the best features of dominant sensors, the FPR
increases drastically when such sensors are manipulated (i.e. or
in our dataset). In contrast, when weaker sensors (physical or
) are manipulated, it has a relatively small impact on the security as
the resulting FPR increases a bit compared to a zero-modal attack (especially
for ). Decisions-fusion reduces attacker success rate when single sensor is
manipulated, for example, FPR of manipulating decreases from 100%
to 36.9%. Recall that manipulating single sensor is not difficult as we
demonstrated in Section <ref>.
An attacker has the highest chance to succeed if he can control the dominant
sensors or a subset of sensors that contain the dominant sensors. In such case,
the success rate could reach 100% with only one single dominant sensor (i.e.
in our dataset) if the system uses features-fusion or with
majority dominant sensors (i.e. and ). In most cases,
attacking the set of weak sensors (e.g. {, , ,
}) does not impact the security much, except when system uses
decision fusion from single modalities.
We have seen that decisions-fusion reduces attack success rates in cases where
the minority of the sensors are manipulated. However, this may come at the
cost of higher FNR which represents the usability of co-presence systems.
Decisions-fusion from single sensors improves security when individual sensors
perform well. However, it increases the attack success rate for weak sensors.
For example, to the audio-radio-physical system, attacking weak sensors such as
or brings relatively high success rate compared to
features-fusion. Decisions-fusion from subsets of sensors reduces the FPR in
general especially when dominant sensors are controlled by the attacker.
The dataset we used for analyzing the attacks is relatively small. It was
collected from limited number of devices and might not represent all possible
scenarios and environments. However, it was sufficient to demonstrate the
impact of attacks and defensive solutions.
Typically, during the authentication/deauthentication process, the prover moves
nearer to/farther away from the verifier. In this case, all the radio signals
will be changing gradually, i.e., if prover and verifier are moving towards
APs, new APs will be shown, or their signal strengths will continuously grow,
while if they are moving further away from APs, their strengths will decrease
or the APs may be not visible at all. If the verifier or prover device detects
a lot more APs (or Bluetooth devices) nearby all of a sudden, it may be
indicative of a radio manipulation attack. The system can be made aware of
such situations.
We noticed that when the verifier is in an environment which has high frequency
noise, an attacker tends to fail with audio streaming. This can be used to
design an active defense mechanism such that whenever audio contextual
information is requested, the verifier can emit a high frequency audio to
reduce the chances of attacker success.
§ DISCUSSION
Reducing Attack Success with
Decisions Fusion:
In the previous section on analysis of an audio-radio-physical system, we showed that decisions-fusion reduces attack success rates in cases where the
minority of the sensors are manipulated. However, this may come at the cost of
higher FNR which represents the usability of co-presence systems.
Decisions-fusion from single sensors improves security when individual
sensors perform well. However, it increases the attack success rate for weak
sensors as they equally contribute to the voting. For example, to the
audio-radio-physical system, attacking weak sensors such as or brings relatively high success rate compared to features fusion.
Decisions-fusion from subsets of sensors reduces the FPR in general especially
when dominant sensors are controlled by the attacker.
Other Potential Countermeasures:
Typically, during the authentication/deauthentication process, the prover moves
nearer to/farther away from the verifier. In this case, the radio signals
changes gradually, i.e., if prover and verifier move towards APs, then new APs
will be shown, or their signal strengths will continuously grow, while if they
move further away from APs, their strengths will decrease or the APs will not
be visible at all. If the verifier or prover device detects much more APs (or
Bluetooth devices) nearby all of a sudden, it probably indicates a radio
manipulation attack. The system can be made aware of such situations.
We noticed that when the verifier is in an environment which has high frequency
noise, an attacker tends to fail with audio streaming. This can be used to
design an active defense mechanism such that whenever audio contextual
information is requested, the verifier can emit a high frequency audio. This
audio signal can be for a short duration, and does not need to be loud (not
high amplitude). As a result, the chances of attacker succeeding in a
relay attack could be reduced.
Limitations and Future Work:
There are
certain limitations of our work.
The dataset we used for analyzing the attacks in audio-radio-physical system is
relatively small. It was collected from limited number of devices. It might not
represent all possible scenarios and environments. However, it
was sufficient to demonstrate the impact of attacks and defensive solutions. It
gave insights for better understanding of the contextual co-presence
detection system and possible defenses to improve security against different
contextual attacks. Further work may be needed to collect and analyze a
larger scale dataset to evaluate this system. The decisions-fusion from
subsets of sensors seems to be the most appropriate solution for improving
security against context manipulation attacks. However, we have analyzed it
only with three subsets: acoustic (), radio (, ) and
physical subsets (, , , ). In design of a
real system in the future, we would like to test different subsets combinations
to find the best candidate for fusion.
§ CONCLUSIONS
Contextual co-presence detection has been shown to be a very promising relay
attack defense in many mobile authentication settings suitable for
off-the-shelf, sensor-equipped devices. We presented a systematic assessment
of co-presence detection in the presence of a context-manipulating attacker.
Our work suggests that tampering with the context can be achieved with simple
yet effective strategies, and the security offered by co-presence detection is
therefore weaker than previously believed.
We also suggested potential countermeasures (e.g., decisions fusion based
machine learning) that may be used to strengthen the security of co-presence
detection against a attacker. Some of these countermeasure may
require a thorough future investigation, which we plan to pursue.
§ APPENDIX
§.§ Increasing the temperature when the attacker does not know $VS$'s location
An attacker who doesn't know the location of $VS$ will try to keep the $FS$ as close as
possible and perform the attack activity. In our
experiment, we placed the $FS$ 10 cm apart from the $VS$ and performed
1) when the hair dryer is closer to $VS$ as shown in Fig. <ref>, and
2) when the hair dryer is closer to $FS$ as shown in Fig. <ref>.
Increasing the temperature; location of $VS$ unknown to the attacker; $VS$ is
10 cm closer to hair dryer than $FS$; the attacker trying to increase temperature to 35
Increasing the temperature; location of $VS$ unknown to the attacker; $FS$ is
10 cm closer to hair dryer than $VS$; the attacker trying to increase temperature to 35
§.§ Increasing the CO gas level
We effectively manipulated the CO gas sensor using cigarette and car exhaust. The increase
in the gas level due to the activity is abrupt when CO is blown onto the sensors, however, it takes
a while for the sensors to fall back to normal readings. This provides an enough time window for the attacker as depicted in Figs <ref> and <ref>.
Effect of cigarette in CO level; increasing the gas content to an arbitrary value and waiting to decrease to desired level.
Effect of car exhaust in CO level; increasing the CO gas level to arbitrary
value and wait to decrease to desired level.
§.§ Increasing the altitude using a car vacuum
As an alternative to air pump, we tried a portable car vacuum cleaner for
inducing an altitude increase. When we hovered the vacuum cleaner pipe around
the sensors, it did not have any effect. However, when we put the pipe just on
top of the sensor, it increased the altitude by 10-11 meters as shown in Fig.
<ref>. An attacker can adjust the altitude to a desired level by
changing the power level of the vacuum cleaner, similar to the air pump
manipulation. The earlier part of the Fig. <ref> shows a little
fluctuation in altitude when we hovered the pipe around the sensors while the
later spikes clearly show that there was an increase of almost 10 meters when
the pipe was touched to the sensors. A video demo of our attack has been
uploaded to YouTube <cit.> to show the effect of portable car vacuum
cleaner on the pressure/altitude sensors.
Using a car vacuum cleaner to reduce pressure around the sensor and increase the altitude.
|
1511.00835
|
DIVISION V
COMMISSION 37 STAR CLUSTERS AND ASSOCIATIONS
(STAR CLUSTERS AND ASSOCIATIONS)
PRESIDENT Giovanni Carraro
VICE-PRESIDENT Richard de Grijs
PAST PRESIDENT Bruce Elmegreen
ORGANIZING COMMITTEE Peter Stetson, Barbara Anthony-Twarog,
Simon Goodwin, Douglas Geisler,
Dante Minniti
HIGHLIGHTS of COMMISSION 37 science results.
§ ABSTRACT
It is widely accepted that stars do not form in isolation but result from the fragmentation of molecular clouds, which in turn leads to star cluster formation. Over time, clusters dissolve or are destroyed by interactions with molecular clouds or tidal stripping, and their members become part of the general field population. Star clusters are thus among the basic building blocks of galaxies.
In turn, star cluster populations, from young associations and open clusters to old globulars, are powerful tracers of the formation, assembly, and evolutionary history of their parent galaxies. Although their importance (e.g., in mapping out the Milky Way) had been recognised for decades, major progress in this area has only become possible in recent years, both for Galactic and extragalactic cluster populations. Star clusters are the observational foundation for stellar astrophysics and evolution, provide essential tracers of galactic structure, and are unique stellar dynamical environments. Star formation, stellar structure, stellar evolution, and stellar nucleosynthesis continue to benefit and improve tremendously from the study of these systems. Additionally, fundamental quantities such as the initial mass function can be successfully derived from modelling either the Hertzsprung$-$Russell diagrams or the integrated velocity structures of, respectively, resolved and unresolved clusters and cluster populations. Star cluster studies thus span the fields of Galactic and extragalactic astrophysics, while heavily affecting our detailed understanding of the process of star formation in dense environments.
This report highlights science results of the last decade in the major fields covered by IAU Commission 37: Star clusters
and associations. Instead of focusing on the business meeting - the out-going president presentation can be found here:
www.sc.eso.org/ gcarraro/splinter2015.pdf - this legacy report contains highlights of the most important scientific achievements
in the Commission science area, compiled by 5 well expert members.
§ EMBEDDED AND MASSIVE STAR CLUSTERS IN THE MILKY WAY: PREPARED BY IGNACIO NEGUERUELA, UNIVERSIDAD DE ALICANTE, SPAIN
The past decade has been an era of discovery in the Milky Way, ushered in by our increasing capability to see through dust. This text is a short summary of the observations that have helped shape our understanding of massive star formation and young clusters in the Galaxy. I must apologise from the beginning for all the very relevant work that I have not been able to review. My reference list is just intended as a collection of examples to illustrate the broad range of excellent research carried out by our community. It is only natural that I concentrate on what I know best.
Deep infrared surveys of the Galactic Plane have revealed hundreds of new open clusters hidden by high obscuration. The publication of the 2MASS catalogue was followed by a number of dedicated searches, some by eye (e.g. ), others using automated methods (e.g. ), that resulted in close to one thousand new cluster candidates. The Spitzer/GLIMPSE survey () presented us with a mid-IR view of the Galactic Plane at high spatial resolution, leading to a new collection of even more highly absorbed clusters (e.g. ). The process has continued as new, deeper surveys, such as VVV, have come on line (e.g. ; ).
Many of these clusters are low-mass stellar groups still embedded in their parental clouds (e.g. ). Others are more massive young clusters with high-mass stars (e.g. ; ). Finally, a significant number are simply old open clusters affected by interstellar extinction. Discriminating between these types is not always easy without more detailed investigation. Even though some semi-automated methods have been proposed to discern their nature (e.g. ), follow-up deep photometry (e.g. ) and, above all, (in most cases, near-IR) spectroscopy is necessary for a good characterisation (e.g. ; ). Because of this, a large number of these candidates still remain unexplored.
Infrared imaging has also resulted in a novel picture of the interstellar medium. Large numbers of bubble-like structures have been found (; ), likely marking sites of massive star formation (). Highly reddened high-mass stars have been identified in the vicinity of many them (e.g. ; ). In less obscured environments, bright-rimmed clouds also seem to mark sites of ongoing star formation in the immediate neightbourhood of high-mass stars (e.g. ). Herschel observations have shown the concentration of proto-stellar cores along filaments. This increasing body of data supports a view of massive star formation as a process extending over large spatial scales, and thus resulting in measurable age spreads in many young regions (e.g. ). Infrared imaging has also allowed the identification of bow-shocks produced by high-mass stars ejected from massive clusters (e.g. ), giving rise to a lively theoretical discussion on the origin of isolated high-mass stars.
Adopting a more global point of view, statistical studies of the distribution of mid-IR sources have provided strong constraints on massive star formation in the Milky Way, providing estimates of the current global star-formation rate (e.g. ), typical duration of different phases (e.g. ; ) or incidence of triggered star formation (e.g. ). More recently, and in contrast, Herschel has provided a very detailed view of gas and dust around a few massive star-formation sites (e.g. ).
But the opportunities for discovery do not only come from infrared imaging. The combination of X-ray imaging of young clusters with IR catalogues has become the preferred method to investigate the formation of low-mass stars in environments dominated by high-mass stars (e.g. ; ), providing valuable input for theories of high-mass star formation (e.g. ; ). Radio surveys have detected large numbers of new H ii regions (e.g. ), whose ionising stars or clusters remain in most cases still unknown. Radio observations have also produced one of the most significant advances in the past few years: the generalised use of geometric parallaxes to masers for determination of distances to star-forming regions, an extremely powerful tool to probe the structure of the Milky Way (see references in ).
The current view of large-scale star formation in the Milky Way, emerging from all these developments, is complex. A typical configuration may be illustrated by the G305 star-forming region. High-mass stars in two moderately massive young clusters, Danks 1 and 2, drive a huge wind-blown bubble, whose rim is teeming with star-formation sites, where embedded lower-mass proto-clusters are forming just now. A diffuse population of proto-stars is scattered over the whole region ( and references). The output of such a process is a classical OB association with massive central clusters, such as Per OB1 or Cas OB8. Many giant molecular clouds seem to harbour similar configurations, among them the Carina Nebula (e.g. ), where Trumpler 14 has a mass $M_{{\rm cl}}\approx 10^{4}\:M_{\odot}$ (<cit.>), or the W3 region () that seems to form a larger structure with W4 and W5, included within Cas OB6. Others, such as W33 () or W51, lack the central massive clusters, and will very likely evolve into dispersed associations, similar to Cyg OB2 (). This may also be the future of one of the most powerful star-forming regions known in the Milky Way, W49A ().
As a result of spatially concentrated star formation, we find massive young clusters. For many years, it was believed that the Milky Way lacked clusters with masses $\geq10^{4}\:M_{\odot}$. Then the obscured open cluster Westerlund 1 (Wd1) was found to host a population of $>70$ supergiants of spectral types ranging from O to M (; ). A direct extrapolation of the number of massive stars detected would suggests a mass $M_{{\rm cl}}\approx 10^{5}\:M_{\odot}$ for a standard initial mass function(IMF). Direct star counts in the infrared give a lower limit of $5\times 10^{4}\:M_{\odot}$ (). The cluster seems to have formed monolithically in a single burst lasting less than 0.4 Ma ().
Many clusters have joined the list of massive clusters over the past decade. Some of them were already known, but their masses have been revised upwards. For example, Trumpler 14, mentioned above, or NGC 3603, for which <cit.> derive a mass in the range $M_{{\rm cl}}= 1$ – $1.6\times 10^{4}\:M_{\odot}$ with indications of a top-heavy IMF. These mass estimates depend strongly on the distance adopted to the cluster and, hence, the extinction law. This problem becomes especially acute for the clusters close to the Galactic centre, affected by very heavy and variable reddening, with an extinction law that deviates strongly from the Galactic average (e.g. ). For example, the mass of the Arches cluster is hotly debated. The extinction law adopted or even possible colour terms due to the very high reddening affect the luminosity function, which has to be later translated into a mass function. Masses ranging from $M_{{\rm cl}}<10^{4}\:M_{\odot}$ to several $10^{4}\:M_{\odot}$ have been found depending on assumptions (see references in ). But even much less reddened clusters are subject to such uncertainties. An extreme case is Westerlund 2, for which distances ranging from $\sim3$ to $\sim8$ kpc have been claimed (), because of very different estimates of the reddening law.
Besides W49A, the strongest candidate for the most massive star-forming region in the Milky Way is W43 (; ), located close to the point where the Galactic bar joins the Scutum-Crux arm. The vigour of star formation in this area is likely related to this location, as there are indications of inflows from the spiral arm towards the densest regions (<cit.>). Not far from W43, and at about the same distance from the Sun, several clusters rich in red supergiants have been found (e.g. ; ). These clusters have ages in the 10 – 20 Ma range, and estimated masses between 2 and $>5\times10^{4}\:M_{\odot}$. At least two large associations with $>10^{5}\:M_{\odot}$ have been found around them (). The presence of so many extreme examples of present or recent star formation within a small span of the Galactic Plane ($\sim6^{\circ}$) seems to hint at a very privileged location. Further support for this interpretation would come from the detection of a similarly hefty complex at the opposite end of the Bar. Even though at least one massive cluster has been found in that general direction (), the exact position where we should expect this starburst region is not certain. The young massive cluster vdBH 222, though originally thought to mark this far end of the bar (), seems to be closer to us (and the Galactic Centre), suggesting that very large associations do not necessarily form at special places, and so hinting at the possibility that star formation in the inner Milky Way may be rather more vigorous than in the outer spiral arms.
Finding massive young clusters is not an end in itself. They not only inform us about star formation in violent environments, but also play a significant role as laboratories for the study of high-mass star evolution. With its huge population of evolved high-mass stars spanning all stages of evolution, Wd1 is the prime example. The different evolutionary paths of isolated and binary high-mass stars can be directly observed and documented within its co-eval population ( and references therein). Similarly, multi-epoch spectroscopic surveys of young open clusters have become fundamental tools to understand the physical properties of high-mass stars (, and related references). As we move into the era of large spectroscopic surveys, this potential will be fullfilled. At the same time, ALMA is starting to look at the sites of massive star formation with unprecedented resolution. In the near future, more advanced instrumentation on the 10-m class telescopes, the launch of JWST and the advent of giant telescopes with adaptive-optics-assisted instrumentation offer the promise of not only an extended era of discovery, but also profound advances in our understanding of how star formation on large scales proceeds, and how high-mass stars are born, evolve and return to the interstellar medium the enriched material that will feed the future stellar generations.
§ OPEN CLUSTERS IN THE MILKY WAY: PREPARED BY ELENA GLUSHKOVA, STERNBERG OBSERVATORY, MOSCOW, RUSSIA
Open star clusters (OCs) have always received
special interest, because by studying these objects, it is
possible to make suggestions on the structure and kinematics of
the Galactic disk and its dynamical and chemical evolution. However,
there are two major obstacles to achieve these goals: 1) the
sampling of OCs is complete only up to distances of 800 – 1000
pc and 2) no unbiased, homogeneous collection of
fundamental parameters are available in the literature.
Concentration of interstellar gas and dust toward the Galactic
plane makes the detection of new OCs difficult especially in the
visible range. To build a homogeneous catalog of OC parameters,
homogeneous observational data and reliable methods of
determination of the distances, ages, and reddening of the
clusters are required. Nonetheless, these obstacles are cleared
step by step in numerous studies, each considering few, sometimes
a dozen, clusters at once.
The situation dramatically changed in the beginning of the 2000s,
when large sky surveys became available. They triggered a new wave
of interest to discover new, and investigate already
known, open clusters. The greatest attention
was drawn by Two-Micron All Sky Survey (2MASS, ),
whose data were collected in three near-$IR$ bands.
Searches for clusters were conducted either visually, on images
recorded in one of the bands, or from the Point Source Catalog using an
automated routine (; ; ; ;
; ; ;
; ). They found about
2000 candidates, visible in
infrared only. One thousand cluster candidates more
were mined in other surveys: DENIS (),
USNO-A2.0 (<cit.>), GLIMPSE (; ),
UKIDSS GPS (), VVV (; ), WISE
(). <cit.> started the program
of searching for MAssive Stars in Galactic Obscured MAssive
clusterS (MASGOMAS) and developed a new automatic tool,
which allows the identification of a large number of massive ( a few
1000 solar masses) cluster candidates from the 2MASS and
VVV catalogs.
As 2MASS is a homogeneous and all-sky catalog, many attempts
were made to derive a homogeneous list of accurate physical
parameter for OCs out of $JHK_S$ data: refer, for example, to the
papers by <cit.>, <cit.>,
<cit.>, <cit.>,
<cit.>. However, the
comparison of parameters obtained in these investigations with
respect to the largest catalog by <cit.>, gives
mean standard deviations for the distance, age and reddening as
1.2 kpc, 0.5 dex, and 0.27 mag, respectively (,
which are definitely insufficient for comprehensive studies
of the structure of the Galactic disc. Involving data from large
surveys, scientists undertook massive determinations of other OC
characteristics: the radial velocities of 110 OCs and the
metallicities of 89 OCs were derived using data from RAdial
Velocity Experiment (RAVE, ), the mean
proper motions and stellar membership probabilities for 1805 open
clusters were found using UCAC4 ().
Since open clusters provide information about the chemical
pattern of Milky Way disk, the study of their chemical composition
attracts special interest. <cit.> compiled data from the
available sources in the literature to make up the catalog of OC
metallicities based on photometric data, which lists 188 clusters
() and set of high-quality cluster metallicities
based on high-resolution
spectroscopic studies for 78 clusters ().
show that none of the current models are able to
satisfactorily describe the OC's metallicity as function of
galactocentric distance. The present high-resolution spectroscopic
Galactic surveys include OCs among their targets: Apache Point Observatory
Galactic Evolution Experiment (APOGEE,
the Gaia-ESO Survey (GES, ), the GALactic
Archaeology with HERMES (GALAH, ) supply accurate
radial velocities and detailed chemical abundances. Using 100 OCs
from the uniformly observed complete SDSS-III/APOGEE-1
DR12 dataset (OCCAM survey), presented
age and multi-element abundance gradients for the disk of Milky Way.
The INfrared Survey of Young Nebulous Clusters (IN-SYNC,
) leverages the stability and multiplex capability
of the APOGEE spectrograph to obtain high resolution spectra
at near-infrared wavelengths and to study the dynamics and star
formation history of young clusters. Four young stellar groups in Perseus
and Orion molecular complexes were investigated ()
based on homogeneous stellar parameters derived from APOGEE
spectra for thousands of pre-main sequence stars ().
OCs are also included in the list for the space missions Gaia and Kepler.
Some projects were delivered especially to investigate OCs
and derive their physical parameters in a precise and
homogeneous way: for example, within the framework of Bologna Open Clusters
Chemical Evolution project (BOCCE, )
about 50 clusters were studied. BOCCE uses both
comparison between observed color-magnitude diagrams (CMDs)
and stellar evolutionary models, and the analysis of
high-resolution spectra to derive age, distance and
chemical composition. The Open Clusters Chemical Abundances from
Spanish Observatories survey (OCCASO, )
plans to derive abundances for more than 20 chemical species in at
least 6 Red Clump stars in 30 northern OCs. Nine clusters have been studied by OCCASO.
Sejong Open Cluster Survey (SOS, ) – a project dedicated
to providing homogeneous photometry of a large number of OCs
in the SAAO Johnson Cousins' $UBVI$ system, has many relatively small
sparse unstudied clusters among its targets. The OPD survey
() is UBVRI photometric survey of southern open OCs,
which is complementary to OAN-SPM UBVRI survey of 406
northern clusters (Michel et al. (2016)).
Lasting 17 years, WIYN Open Cluster Study (WOCS, )
is dedicated to comprehensive photometric, astrometric, and
spectroscopic studies of select OCs. In their recent investigation
() new deep wide-field optical and near-infrared
photometry ($UBVRJHK_S$) of the cluster M35 is presented, against which
several isochrone systems are compared: Padova, PARSEC, Dartmouth, and $Y^2$.
Two different atmosphere models are applied to each isochrone: ATLAS9 and BT-Setti.
For any isochrone set and atmosphere model, observed data are accurately
reproduced for all stars more massive than $0.7\:M_{\odot}$. For
less massive stars, Padova and PARSEC isochrones consistently produce higher
temperature than observed. Dartmouth and $Y^2$ isochrones with BT-Setti
atmospheres reproduce optical data accurately; however, they appear too blue
in IR colors. WIYN's extension – the Southern Open Cluster Study (SOCS,
) includes 24 clusters, wide-field photometry results are
already available for three of them.
determined the radial velocity of stars belonging to a group of
open clusters using spectra with spectral resolution of 4000 and plan
to calculate the mean radial velocities of a number of OCs. To
investigate star formation processes, <cit.> initiated
a photometric survey of young open clusters in the Galaxy and
already studied 13 famous OCs having a wide range of surface
densities (log($\sigma$)=-1 – 3 $stars/pc^2$) and total
masses ($500 \-- 50000\:M_{\odot}$) and also distributed in five different
spiral arms in the Galaxy. <cit.> found that the slope
of the IMFs in the high-mass regime appears to be shallow for
massive compact clusters, and the mass of the most massive star in
a given cluster also has a tendency to be large in massive
<cit.> started a program that determines the properties
of Local (Orion) spiral arm. They plan to carry on a comprehensive study
of 29 young OCs which includes a $UBVRI$ photometric analysis
and determination of their kinematics. The first cluster NGC 2302 has already been investigated.
When large catalogs and data sets became available in the
literature, and massive determination of parameters of clusters
became possible, numerous automated and semi-automated techniques
were developed to retrieve these parameters primarily by using
color-magnitude diagrams. One of the most powerful tools
is the Automated Stellar Cluster Analysis package (ASteCA,
, which makes use of positional and photometric
data to provide accurate estimates of the cluster's metallicity,
age, extinction and distance values, and robust stellar cluster
image and photometry simulation package MASSCLEAN (),
which creates synthetic clusters and generate CMD
templates for a variety of cluster masses and ages, and which
mimic the observational photometric errors when using isochrone
fitting ().
Thanks to All Sky Automated Survey (ASAS, ),
a large number of new Galactic Cepheids have been
discovered during the last dozen years. That is why new attempts were
undertaken to find Cepheids attributed to open star clusters.
Taking into account all possible characteristics of these variable
stars and OCs, <cit.> found five new genetic
relations between Cepheids and clusters, and <cit.>
reported 8 new Chepheid-cluster pairs. Some papers were devoted to
photometric and spectroscopic observations of known clusters
hosting the Cepheids, in order to confirm membership of Cepheids in
OCs and refine cluster's physical parameters (, ).
One of the most interesting results of the studies of individual
clusters was published by <cit.> on GLIMPSE-C01
referred to in the literature as an old globular cluster
traversing the Galactic disk. The authors obtained high-resolution
near-infrared spectroscopy of over 50 stars in the cluster and
found the average radial velocity is consistent with being part of
the disk, and determined the cluster's dynamical mass to be
$8\times 10^{4}\:M_{\odot}$. From analysis of the cluster's $M/L$ ratio
and location of the red clump, <cit.> suggested the
cluster's age to be 400–800 Myr and concluded that GLIMPSE-C01 is
the most massive Galactic intermediate-age cluster discovered to
This way, the number of known Milky Way open clusters increased
from about 1500 to almost 4000 during the last 10 – 15 years. The main
physical parameters were derived for most of them but quality of these
determinations is insufficient to study the Galactic disk comprehensively.
That is why a lot of new different surveys were started to measure a large range
of open cluster's properties.
§ GLOBULAR CLUSTERS IN THE MILKY WAY: PREPARED BY ANGELA BRAGAGLIA AND EUGENIO CARRETTA, OSSERVATORIO DI BOLOGNA, ITALY
The last ten years have seen a renewed interest in Galactic globular clusters
(GC), mainly because they have been demonstrated to be much more complex and
intriguing than believed in the past. The main reason is that spectroscopic and,
later on, photometric observations have driven a dramatic shift from
considering GCs as the best approximation of simple stellar populations (see
e.g. the review by ) to the simplest example of $multiple$
stellar populations (e.g. the review by ). We use here the
words “populations" and “generations" as synonym, implying that in the same GC
stars of (slightly) different age coexist. This is the generally accepted
scenario, although there are still many problems in explaining the mechanism of
GC formation and internal self-enrichment (in light elements for all GCs, in
heavier elements only for a fraction). For recent reviews see for instance
<cit.> for spectroscopic results, <cit.> for photometry, and
<cit.> for theoretical challenges.
We present here selected highlights in Galactic GC work of the last decade, with
a strong bias towards multiple populations and observations.
The abundance of light elements in GC stars shows large star-to-star scatter, at
variance with what happens for the bulk of field halo stars; these variations
are anti-correlated (C and N, O and Na, and Mg and Al are depleted and enhanced,
respectively). These so called “anomalies" in light elements had already been
detected in many GCs, but generally only in giant stars. However, the presence
of products of hot H-burning also in main sequence stars called for more massive
stars as original polluters (see subsection on models) and we now speak of
first-generation (FG) and second-generation (SG) stars in GCs. After the first
pioneering works, light element (anti-)correlations were routinely found also
among main sequence and unevolved stars, using also high-resolution spectra, see
e.g. <cit.>
for C, N, and
<cit.> for Na, O,
Mg, Al, and Li.
This decade saw a wealth of observations on large samples of stars and of
clusters, also thanks to multi-object high-resolution spectrographs, like Hydra
or FLAMES, so that a quantitative analysis of the light elements
anti-correlation became possible. Large scale studies were conducted, see the
long list of GCs and papers in , among which we have for
instance the closest GCs, i.e., M 4, NGC 6752, NGC 6397, and M 22
<cit.>, very
massive clusters like $\omega$ Cen <cit.> and M 54
<cit.>, and low mass GCs like NGC6838 <cit.>. Recently,
also high-resolution surveys produced results in this field <cit.>.
All evolutionary phases were targeted. The red giant branch (RGB) was the
favourite, but also the horizontal branch (HB) was observed and in some GCs also
He was measured <cit.>, an important
diagnostics of the multipopulation scenario. Helium has also been measured in
RGB stars, but only in NGC 2808 <cit.> and $\omega$ Cen
<cit.>. More recently, the Asymptotic Giant Branch (AGB) observations
showed that at least in some GCs the AGB stars do not show the same level of
modification in light element abundances (, but
see also ).
Briefly, in all the examined MW GCs (and interestingly, also in GCs of the LMC
( and Fornax ), a prominent
Na-O anti-correlation has been found. Possible, rare, exceptions are very
low-mass GCs, like the two Sgr clusters Ter 7 and Pal 12 <cit.>, and
Rup106 <cit.>. The Na-O anti-correlation appears almost a defining
properties of (massive) GCs and is not present among field
stars or open clusters (, but
see for a different opinion on NGC6791).
An even more extreme departure from the simple stellar population paradigma
comes from the finding of GCs with dispersion in iron and heavy elements. After
$\omega$ Cen, also M 22 was found to display an intrinsic metallicity spread by
<cit.> (recently challenged by ),
correlated with a spread in neutron-capture elements. Among GCs displaying iron
and n-capture element spreads are for instance M 54, NGC 1851, and M 22.
The impact of different abundances of light element ions the photometric
properties of stars in GCs (especially in the filters containing molecular
features of CNO elements, see ) is exploited to
better understand the origin of GCs. Different abundances between FG and SG
stars translate into spread and even split sequences along the whole CMD of GCs.
These multiple sequences are observed with several photometric systems,
including broad band <cit.>, intermediate
band <cit.>, and narrow band <cit.>. The recent
exploitation of the UV Hubble Space telescope (HST) filters allows to reach also features of OH
hydride (see the UV GC survey described in ). HST observations
reveal split sequences all the way from the main sequence (MS) up to the red giant branch (RGB) and
horizontal brand (HB, see for instance the spectacular main sequence (MS) of NGC 2808, ). The
large photometric samples allow to study the radial distribution of stellar
populations across the cluster area <cit.>.
Usually SG stars are found more centrally concentrated, as predicted by most
scenarios of GC formation. Photometry
permits to detect even discrete populations in a single GC, like NGC 2808
<cit.>, confirmed by abundance analysis of individual stars
The coupling between precise spectroscopy and photometry unravels a variegated
landscape for Galactic GCs, that come in different flavours. Most GCs are
monometallic (concerning iron and heavier elements). A growing number of objects
shows a spread in Fe (see above). They also share a few other common
properties: a correlation between Ca and Fe (suggesting enrichment by type II
SNe), and an enhancement of elements from slow neutron-capture process observed
among stars of the more metal-rich cluster component. This suggests that, like
$\omega$Cen and M 54, the remnant nuclei of former dwarf galaxy
<cit.>, these GCs may be the final products of cluster
formation in a dwarf galaxy environment <cit.>.
Maybe related to the multiple populations in GCs, in the LMC and SMC many
intermediate-age clusters with extended or even split main sequence turn-off's
and red clumps have been found <cit.>.
This may be due to extended star formation <cit.> or to stellar
rotation <cit.> or binarity <cit.>. No light
elements anti-correlation has yet been found in these clusters
Observational evidence of multiple stellar populations calls for some FG stars
to have polluted material from which the SG stars formed. The most commonly
discussed FG polluters are intermediate mass asymptotic giant branch (AGB) stars <cit.>
and fast rotating massive stars <cit.>, but also
interactive massive binaries have been proposed <cit.>, very massive
stars <cit.>, or early disk accretion <cit.>. However, multiple
populations also pose tight constraints that challenge (m)any model(s) of GC
formation. The currently observed ratio of FG to SG stars (about 1/3 and 2/3,
) is not easily reconciled with the amount of yields
provided by any candidate FG polluter (see for instance ).
This “mass budget" problem was often circumvented by scenarios assuming that
GCs were initially from 10 up to 100 times more massive than present-day GCs
<cit.>. However,
evidence is growing that in external dwarf galaxies harbouring old GCs (showing
multiple populations) these objects cannot have been more than 3-4 times more
massive at their formation epoch, since they already account for about 25$\%$ of
the galaxy mass in metal-poor stars <cit.>.
Another severe challenge is posed by the very same existence of the observed
anti-correlations, which require a certain amount of pristine gas to be mixed
with ejecta of polluters to reproduce the observations. In some scenario (e.g.
with AGBs, ) dilution is mandatory since AGBs produce a
correlation between Na and O abundance. From where this gas with primordial
composition came after the GCs were swept by the first type II Supernovae is an
issue still open, widely discussed but with no completely satisfactory answer
up to now. A recent scenario proposed by shows a possible
way out, with blobs of pristine gas nearby the outcome of a major mini-halo
mini-halo merger at high redshift, which could be later accreted refurbishing
the evolving proto-GC with fresh reservoir of diluting matter. This mechanism
could also be an attractive way to explain the discreteness observed both in
CMDs and in the anti-correlations (see e.g. and
, for NGC 6752 and NGC 2808).
Clusters lose mass and stars during their evolution (see, e.g.,
). Stars probably lost by GCs have been searched for in
halo samples using the peculiar SG chemistry. CN band strength was used by
<cit.> on Segue SDSS spectra, while enhanced Na and/or depleted O
abundances were considered by <cit.>. These papers provided
a fraction of SG-like halo stars of about 1.5 to 3$\%$. Recently, using data from
the public spectroscopic survey Gaia-ESO <cit.> found a probable GC
escapee. For a discussion on the contribution of GC stars to the halo, see for
instance <cit.>.
With the advent of large photometric surveys such as SDSS, Pan-STARRS, DES, VST
ATLAS, VVV, etc new GCs have been found, see e.g.,
<cit.>, even if the objects found are
sometimes classified as GCs or ultrafaint dwarfs depending on the study. The
Galactic GC populations has increased in number, but the newly discovered
clusters are all low-mass systems.
Wide field imaging has been used systematically to look for extended and
extra-tidal structures <cit.> In a few cases, tidal
tails or streams still connected to the originating GC have been found (e.g.,
NGC 288, or Pal 14, ), but there are
many more without a clearly associated progenitor cluster.
Furthermore, streams, moving groups, common proper motions groups have been
sometimes associated to now dissolved clusters, in some cases using chemical
tagging. Examples of a positive and negative identification, respectively, are
the Aquarius stream (consistent with being GC debris, ) and the
Kapteyn group (not associated with $\omega$ Cen, ).
“Exotic” objects in GCs comprise for instance blue stragglers stars (BSS),
low-mass X-rays binaries, and millisecond pulsars (MSP), all very good tracers
of the evolution of close binary systems in dense environments and of the
dynamical history of the parent cluster.
The study of BSS proceeded with the collection of large samples covering the
entire extension of the GCs <cit.>, in the
determination of their dynamical status
<cit.> and chemical composition
<cit.>. For more detailed and recent highlights on BSS, see for
instance the contributions in `Ecology of Blue Straggler Stars', held in 2012.
<cit.> found 21 new MSP in tbe massive GC Terzan 5, which shows the
largest number of X-ray sources among GCs. This cluster is particularly
interesting also because <cit.> found it hosts two stellar
populations with different iron contents and ages and proposed it to be the
remnant of one of the primordial building blocks that formed the bulge (and not
a true GC). The different metallicity was later confirmed by
§ EXTRA-GALACTIC STAR CLUSTERS: PREPARED BY TOM RICHTLER, UNIVERSIDAD DE CONCEPCIÓN, CHILE
The following pages try to extract the results from 10 years of world wide intensive research using the extremely short style of a conference summary and applying a strong bias to observational work.
<cit.> review "Globular clusters and Galaxy Formation", <cit.> "massive star clusters in galaxies", <cit.> "young massive clusters" , <cit.> concepts of GC formation.
New M31 GCs have been detected, many within the "Pan-Andromeda Archaeological Survey" <cit.> and the data base for M31 GCs has been significantly increased both in quantity and quality <cit.>.
A updated analysis of the properties of the M31 GCS is given by <cit.>. A flattening of the surface density profile at a radius of about 30 kpc corresponds to a
flattening in the stellar surface brightness profile, which might indicate an accretion of the outer halo. More direct evidence for accreted clusters is found in <cit.>.
Structural parameters are presented by <cit.>. 300 metallicities of old GCs have been derived by <cit.>. The metallicity distribution is unimodal in contrast to
that of the Milky Way and giant ellipticals. Detailed element abundances for GCs from integrated light have been derived by <cit.>.
<cit.> combine kinematic data and structure parameters to derive M/L-ratios for 200 GCs and confirmed previous findings that the M/L$_V$-values decline with increasing metallicity,
contrary to what naively is expected from stellar models. Shallower mass functions of metal-rich clusters can explain this. M/L$_V$-values also increase with cluster mass, possibly as a consequence of mass segregation. This is disputed by <cit.>.
The kinematics of the outer halo GCs in M31 is studied by <cit.>. Groups of GCs are related to the debris of stellar streams.
The Fornax dwarf spheroidal shows an extremely large ratio of stars in GC to field stars of a similar low metallicity. This constrains the loss of first generation stars in GCs <cit.>.
Nearby spirals with high star formation rates have been surveyed for clusters by <cit.> (M83),
<cit.> used HST data to measure colour magnitude diagrams for resolved young massive clusters in nearby spiral galaxies. There is no gap visible between the H-burning main sequence
stars and the He-burning supergiants like in canonical isochrones. Age spreads of a few Myrs are able to fill this gap, may be also interacting binaries.
The Initial Cluster Mass Function (ICMF) of young clusters in spiral galaxies has been investigated by <cit.>. A original Schechter function with a cut-off mass of $\approx 2\times10^5 M_\odot$
describes the GC luminosity functions well, if the luminosity evolution is only secular. <cit.> provides insight into the similarity of cluster and stellar IMFs.
Based on a sample of 37 nearby dwarf galaxies, <cit.> study the relation between SFR and cluster formation.
Galaxy mergers can host a plethora of massive clusters.
The Antennae galaxies are prominent targets with age distribution of GCs studied by , new spectroscopic data <cit.>,
general demographic model and application to the antennae <cit.>. The latter work demonstrates that the enhanced number of massive clusters in mergers is a sample size effect due to the high star
formation rate. The Antennae are also a test ground for the evolution of clusters, in particular the processes of disruption <cit.>. Some young clusters in the Antennae have been observed with ALMA, but large reservoirs of molecular gas in the clusters have not been found <cit.>. Cluster formation and disruption in mergers through simulations have been investigated by
<cit.>. <cit.> give structural parameters for 36 clusters in NGC 7572. Among them is W3, the most massive cluster known, whose profile extends out to 500 pc.
The nearest merger remnant (and giant elliptical) is NGC 5128 (CenA).
Ages, metallicities are known for about 400 objects <cit.>. One third of this sample
show ages less than 8 Gyr. <cit.> find an increase of the M/L-values with the dynamical mass.
The GCS has been searched over an area of 1.5deg$^2$ <cit.>. More than 1000 GCs are now known.
NGC1316 (Fornax A) is after CenA the closest merger remnant. Its GCS host clusters of a wide age-range down to 0.5 Gyr (). Among them are objects as massive as $1.6\times10^7 M_\odot$
(). There is an isolated young star cluster complex which demonstrates GC formation outside periods of high star formation rate <cit.>.
The nearby galaxy clusters Virgo and Fornax have been intensively surveyed with HST/ACS which produced a bulk of papers. From the Virgo survey: <cit.> study formation efficiencies of GCs. The GC
mass fraction is lowest at intermediate host luminosity, the specific frequency is dominated by blue clusters. Dwarf galaxies near to M87 seem to be tidally stripped of their GCs.
<cit.> on the GC luminosity function and distance determination: the dispersion of the GCLF correlated tightly with the host's absolute magnitude.
<cit.> on colour-magnitude relations in GCSs, <cit.> on colour gradients within GCSs.
<cit.> find that the total mass in GCs scales with the halo mass of the host galaxy.
A new catalogue by <cit.> updates our current knowledge of GCS properties and relations with host galaxy properties. The number of GCs obeys a fundamental plane-like relation $N_{GC} \sim (R_e \sigma_e)^{1.3}$
for galaxies of all luminosities. On the other hand the ratio mass in GCs/halo mass is essentially constant
<cit.>, the GC number also scales with halo dark mass and <cit.> even find a strict proportionality between the number of blue clusters and the halo mass.
<cit.> compare the GC luminosity functions of seven brightest cluster galaxies and find identical shapes.
Dwarf galaxies show a tendency of increasing specific frequency with decreasing luminosity <cit.>.
How the GCSs of dwarf galaxies are affected by galaxy harassment in clusters, is studied by <cit.>. The critical parameter is the dark matter fraction that remains after interaction
processes. <cit.> explain both compact and extended clusters by dynamical evolution in star cluster complexes.
Massive young GCs are also seen in star forming dwarf galaxies <cit.> which in the context of hierarchical clustering may have been important for the assembly of GCSs.
The richest GCs of nearby galaxies are found in the central galaxies of the Virgo <cit.> and Fornax galaxy clusters. As population and dynamical
tracers GCs have a high significance for investigating the dark matter content and distribution as well as the population structure and formation
history of M87 and NGC 1399.
Regarding NGC 1399, <cit.> presented about 700 GC radial velocities within 80 kpc of galactocentric radius. Blue and red clusters show distinct kinematical properties
with a sharp transition. The red clusters resemble the galaxy light, while blue clusters behave more erratic and are probably accreted.
The best fitting dark halo agrees reasonably well with that from X-ray studies, but a substructure within the dark halo that
has been suspected in earlier X-ray studies, has not been confirmed.
M87 also received particular attention. <cit.> provide a wealth of kinematical data for over 700 GCs
that <cit.> use for a dynamical analysis. They found a dark matter fraction of 0.95 within a radius of 135 kpc and an inner dark matter profile that is steeper
than predicted by cosmological simulations. Such a big sample also permits to analyse the orbital properties of GCs. Remarkably, the anisotropy seems to be
mainly tangential which supports the idea that many objects on radial orbits have been dissolved.
Near M87 has been found the object with the highest negative radial velocity detected so far, a GC with -1025 km/s (). Its dynamical history
is mysterious.
A significant increase of the database regarding metallicities <cit.> and kinematics of GCs in early-type galaxies <cit.> was achieved by the still ongoing "The SAGES Legacy Unifying Globulars and GalaxieS Survey (SLUGGS)" <cit.>. Until now, almost 1000 metallicities and 2500 velocities of GCs in a dozen early-type galaxies have been published.
Highlighting a few galaxies: The disputed metallicity bimodality of GCs in NGC 3115 has been confirmed by <cit.> through CaII triplet strengths. Breaking the degeneracy in
dynamical models between potential and orbital anisotropy, <cit.> find the dark halo of NGC 5846 to be consistent
with cosmological simulations, that its stellar IMF is Salpeter-like, and that the GC orbits are isotropic in the central parts and slightly radial at larger distances <cit.>.
More than 400 GCs with photometry and velocities build the database for M60 <cit.>.
The correlation between central supermassive black hole masses and properties of GCSs have been discussed in several papers <cit.>. The latter two
contributions could investigate larger galaxy samples and find the correlations weaker than described before.
Massive nuclear GCs host supermassive black holes. <cit.> provides a relation between the masses of a nuclear star cluster, the bulge and the black hole.
<cit.> show how erosion by binary BHs may change the mass of a nuclear star cluster. <cit.> provides a catalogue of 228 nuclear star clusters in nearby spirals.
The discussion regarding the characteristic parameters and the nature of UCDs is continuing. <cit.> find, on the basis of a survey in M87,
that the colour-magnitude relation of UCDs is offset from that of GCs. They conclude that the majority of UCDs are stripped nuclei of former dwarf
galaxies. <cit.> point to the dual nature of UCDs characterised by mass. Above $7\times10^7 M_\odot$, UCDs are predominantly
stripped nuclei, while at lower masses, many "normal" GCs may be mixed in.
An interesting finding were the enhanced M/L-values of some UCDs in the Virgo cluster which could not be explained by stellar population models ,
perhaps indicative of dark matter <cit.>. Later work on UCDs in the Fornax cluster did not confirm this <cit.>. In particular,
the brightest and resolved UCD in the Fornax cluster shows population properties consistent with existing models <cit.>, but the discussion
is going on <cit.>.
The study of a large number of UCDs around M87 suggests that UCDs are distinct from GCs by their sizes rather than by their masses. The UCDs in
M87 show both radial distributions and orbital properties different from those of GCs <cit.>.
<cit.> explain both compact and extended clusters by dynamical evolution in star cluster complexes.
Convincing evidence that some UCDs evolve from bigger parent galaxies, is the detection of a supermassive black hole
in a UCD in M60 that has 15% of the mass of its host system <cit.>.
Cosmological formation of GCs has not yet been identified, but GCs populating a galaxy cluster rather than an individual galaxy, have been
found by <cit.> in the Coma cluster, <cit.> in Abell 1189, and by <cit.> in Abell 1689. For the latter cluster, the authors
quote a number of 160 000 GC within 400 kpc.
The other extreme appears as "the most isolated globular clusters in the Local Universe" in the
vicinities of M81 and M82 <cit.>. The Local Group does not seem to host very isolated GCs <cit.>.
|
1511.00715
|
Distributed Selection and AKS Sorting Network
Department of Computer Science and Engineering
The Pennsylvaina State University
[email protected], [email protected]
We consider the selection problem on a completely connected network of $n$ processors with no shared memory. Each processor initially holds a given numeric item of $b$ bits, allowed to send a message of $\max(b,\lg n)$ bits to another processor at a time. On such a communication network ${\cal G}$, we show that the $k^{th}$ smallest of the $n$ inputs can be detected in $O \lp \log n \rp$ time with $O \lp n \log \log n \rp$ messages. The possibility of such a parallel algorithm for this distributed $k$-selection problem has been unknown despite the intensive investigation on many variations of the selection problem carried out since 1970s.
Satisfying the constraint of total $O \lp n \log \log n \rp$ messages, it improves on ${\cal G}$ the asymptotic running time of Kuhn, Locker and Wattenhofer's algorithm.
Our parallel algorithm simulates the comparisons and swaps performed by the AKS sorting network, the $n$-input sorting network of logarithmic depth discovered by Ajtai, Komlós and Szemerédi in 1983.
Simulation of such a network is our main trick to achieve $O \lp \log n \rp$ time and $O \lp n \log \log n \rp$ messages simultaneously.
Extending its correctness proof, we will be able improve by about 47% the upper bound found by Seiferas on the constant factor of the $O \lp \log n \rp$ depth of an $n$-input sorting network.
Furthermore, we show the universal time lower bound $\lg n$ for many basic data aggregation problems on ${\cal G}$.
The class of problems having this lower bound is huge including the selection problem, the problem of finding the sum of $n$ items, and that of counting items exceeding a threshold. Thus the universal lower bound means the asymptotic time optimality of our parallel algorithm.
§ INTRODUCTION
The classical $k$-selection problem finds the $k^{th}$ smallest element of given $n$ numeric items. We consider the problem on a completely connected network ${\cal G}$ of $n$ processors with no shared memory, each holding exactly one item of $b$ bits initially. A processor node in ${\cal G}$ may send a message of $\max\lp b, \lg n \rp$ bits at any parallel step. As the performance metrics, we minimize the parallel running time, and/or the total number of messages as the measure of amount of information transmitted on ${\cal G}$.
The parallel selection problem for connected processors with no shared memory has been extensively investigated for various network topologies, cases of how $n$ inputs are distributed, and other constraints <cit.>, as well as on the parallel comparison tree (PCT) and parallel random access machine (PRAM) models with shared memories <cit.>. In this paper we focus on the above case, calling it the $k$-distributed selection problem on a communication network ${\cal G}$.
As suggested in <cit.>, this case of ${\cal G}$ has become increasingly significant for the contemporary distributed computing applications such as sensor networks and distributed hash tables: It is common in their performance analysis to count the number of messages delivered at the designated destinations assuming constant time per delivery (called hops), rather than count how many times messages are forwarded by processor nodes. The considered network ${\cal G}$ models it well.
The distributed selection problem in this particular network case is one of the long time research topics in parallel algorithms. The results are included in the work such as <cit.>. The following summarizes only a few most closely related to our interest in the paper: Let $w$ stand for the number of processors initially holding one or more inputs.
The algorithm Frederickson and Johnson developed in <cit.> finds the $k^{th}$ smallest element with $O \lp w \log \frac{k}{w} \rp$ messages on a completely connected or star-shaped processor network. Santoro et al <cit.> discovered in 1992 an algorithm with the expected number of messages bounded by
$O \lp w \log \log \min(k, n-k) + w \log w \rp$.
The more recent result in <cit.> explores a case when the processor network has a general diameter $D$; it
presents a parallel algorithm with the average time bound $O \lp D \log_D n \rp$ that is asymptotically optimal under some probabilistic assumptions, and one with the deterministic time bound $O \lp D \log_D^2 n \rp$.
The first contribution of this paper is a parallel algorithm for the distributed $k$-selection problem that runs in time $O \lp \log n \rp$ with total $O \lp n \log \log n \rp$ messages on ${\cal G}$. We will prove the following theorem after formulating the problem.
The $k^{th}$ smallest of $n$ inputs can be computed distributedly on a communication network ${\cal G}$ in time $O \lp \log n \rp$ with $O \lp n \log \log n \rp$ messages.
Such a parallel algorithm has been unknown despite the long research history on the selection problem. Satisfying the constraint of total $O \lp n \log \log n \rp$ messages, it improves the parallel time bound $O \lp D \log_D^2 n \rp$ in <cit.> when $\log^2 D \ll \log n$: On a processor network with diameter $D$, a message can be sent to anywhere forwarded by
$O \lp D \rp$ processors. So the theorem means the parallel time bound $O \lp D \log n \rp \ll D \log_D^2 n$.
Our algorithm simulates the comparisons and swaps performed by the AKS sorting network, the $n$-input sorting network of $O \lp \log n \rp$ depth discovered by Ajtai, Komlós and Szemerédi in 1983 <cit.>. Known for the difficulty of its performance analysis, the AKS sorting network itself has been a research subject in parallel algorithm since then. Paterson <cit.> simplified its construction. Seiferas <cit.> further clarified it with the estimate that the depth can be at most $7 \cdot 6.07 \lg n=48.79 \lg n$ layers of $1/402.15$-halvers. Here an $\ve$-halver $\lp \ve \in \lp 0,\frac{1}{2} \rp \rp$ is a comparator network defined in <cit.>, such that for every positive integer $z \le \frac{m}{2}$, the left half of the output includes at most $\ve z$ items among the $z$ largest inputs, and the right half at most $\ve z$ among the $z$ smallest inputs.
In the recent work by Goodrich <cit.>, the constant factor of the total $O \lp n \log n \rp$ nodes is significantly reduced although its depth bound is $O \lp n \log n \rp$. The overall research interest on the AKS sorting network has been simpler understanding and reduction of the constant factor of the asymptotic quantities.
In order to design our parallel algorithm for the distributed selection problem, we show that if the depth of the AKS sorting network is reduced from $\Theta \lp \log n \rp$ to $\Theta \lp \log \log n \rp$, the number of items in the wrong half of the output is $O \lp n \log^{-c} n \rp$ for any given constant $c>0$. The parallel algorithm simulates it, being our main trick to achieve $O \lp \log n \rp$ time and $O \lp n \log \log n \rp$ messages simultaneously.
Formally we will show:
For each $c \in \R^+$ and sufficiently large power $n$ of 2,
there exists a comparator network ${\cal H}$ to re-order $n$ inputs satisfying the following three.
* ${\cal H}$ is an AKS sorting network of reduced depth.
* ${\cal H}$ is a weak $\lg^{-c} n$-halver, $i.e.$, the first half of the ordered $n$ outputs includes at most $n \lg^{-c} n$ items larger than the median, and the second half at most $n \lg^{-c} n$ smaller than the median.
* There are no more than $22. 34 c \lg \lg n$ layers of $1/175$-halvers in ${\cal H}$.
Here an AKS sorting network of reduced depth
means that the lower nodes of depth more than $O \lp \log \log n \rp$ are simply removed. We will define it exactly when we construct ${\cal H}$ in SecHalver.
The proof of Halver is the second contribution of the paper.
Thirdly, we will show a smaller constant factor of the $O \lp \log n \rp$ depth of the $n$-input sorting network. Extending the proof of Halver, we will confirm:
For each sufficiently large $n \in \Z^+$, there exists a sorting network on $n$ inputs with at most $25.54 \lg n$ layers of $1/395$-halvers.
This improves the above Seiferas's estimate by about 47%.
Our fourth result is the running time lower bound $\lg n$ for many data aggregation problems on ${\cal G}$.
The class of problems having the lower bound is huge including the selection problem and many others. The following statement will be confirmed as a corollary to the theorem we will show in LB.
Any parallel algorithm takes at least $\lg n$ steps in the worst case to compute each of the following three problems on a communication network distributedly: i) the $k$-selection problem on $n$ inputs each of at least $\lc \lg n \rc+1$ bits; ii) the problem of finding the sum of $n$ inputs; iii) the problem of counting items among $n$ inputs, each exceeding a given threshold.
By our formulation in Def, the statement assumes that each of the $n$ processor nodes holds an input at time 0.
The corollary shows the asymptotic time optimality of our parallel algorithm as well.
The rest of the paper consists as follows. In Section 2, we define general terminology showing related facts. We will prove Halver in Section 3, which is extended to our improved constant estimate in Section 4. MainClaim is verified in Section 4. We prove the universal lower bound $\lg n$ in Section 5, followed by concluding remarks in Section 6.
§ GENERAL TERMINOLOGY AND RELATED FACTS
In this paper, a communication network ${\cal G}$ means a collection of $n$ processor nodes each two of which are connected, capable of exchanging a message of $\max\lp b, \lg n \rp$ bits at a parallel step. We consider a computational problem $P$ whose input is a set of $n$ numeric items of $b$ bits distributed over ${\cal G}$, $i.e.$, each processor holds an item at time 0. A parallel algorithm $A$ is said to compute $P$ on ${\cal G}$ distributedly in time $t$ with $m$ messages, if all the bits of the computed result from $n$ inputs distributed over ${\cal G}$ is stored at a designated processor node after the $t^{th}$ parallel step with total $m$ messages. Messages may be exchanged asynchronously on ${\cal G}$.
A Separator $S$ in the AKS Sorting Network
A comparator network ${\cal H}$ is a directed acyclic graph consisting of comparators as nodes of positive depth, and nodes of depth $0$ each storing an input numeric item.
Such a comparator of depth $d \ge 1$ can receive two items from nodes of depth $d-1$ sending their maximum and/or minimum to other comparators of depth $d+1$. It can thus swap two items, or copy by receiving a same item.
For notational convenience, we say that the width of ${\cal H}$ is the maximum number of nodes with a same depth. In the standard terminology, a numeric item input to a comparator is called wire. We may call an ordered set of wires array, for which the set theoretical notation is used.
The AKS sorting network is a comparator network to sort $n$ inputs, whose depth is $\Theta \lp \log n \rp$ and width $n$, performing $O \lp n \log n \rp$ comparisons.
We observe here that the $k^{th}$ smallest of $n$ items distributed over ${\cal G}$ can be detected in
$O \lp \log n \rp$ time with $O \lp n \log n \rp$ messages the following way. We design such a parallel algorithm $A$ so that a processor node $v$ in ${\cal G}$ simulates from time $ct$ to $c(t+1)$ ($c:$ a constant, $t \in \N$) a comparator of depth $t$ in the AKS sorting network ${\cal H}$. We mean by “simulate" that $v$ receives two items from other nodes to send their minimum and/or maximum to anywhere in ${\cal G}$.
Due to the width of the AKS sorting network, and since ${\cal G}$ is completely connected,
$n$ processors in ${\cal G}$ can simulate all the swaps and copying performed by ${\cal H}$.
This way $A$ sorts the $n$ inputs in $O \lp \log n \rp$ steps, then fetches the $k^{th}$ smallest one. We can therefore achieve $O \lp \log n \rp$ time and $O \lp n \log n \rp$ messages to compute the distributed $k$-selection problem on ${\cal G}$.
Our parallel algorithm improves the above so that the number of messages is reduced to $O \lp n \log \log n \rp$. To modify the AKS sorting network, we look into the details of its unit component called separator given in a textbook such as <cit.>: A separator illustrated as $S$ in fig0 consists of the seven $\ve$-halvers $H_0, H_1, \ldots, H_6$, for a sufficiently small constant $\ve \in (0, 1)$.
Let $I_1$ and $I_2$ be the left and right halves of the input array to $S$, respectively.
The separator $S$ re-orders $I_1 \cup I_2$ into four disjoint output arrays $A_1$, $A_2$, $A_3$ and $A_4$ such that
\[
|A_1|= |A_4|=\frac{1}{16}m, \eqand |A_2|= |A_3|=\frac{7}{16}m.
\]
The way numeric items are compared in each $\ve$-halver is represented by a $\lp d_0, \ve \rp$-expander graph $\lp d_0 \in \Z^+ \rp$. It is a $d_0$-regular bipartite graph $(V_1, V_2, E)$ such that $|V_1|=|V_2|$ and
ε|ΓU | ≥(1- ε) minε|V_i|, |U| ,
for every subset $U$ of $V_1$ or $V_2$, where $\Gamma(U)$ stands for the neighbor set of $U$.
The input to the halver is an array of $|V_1 \cup V_2|$ elements where each comparison is represented by an edge in $E$. When two items are compared, they are swapped to correct the order if necessary.
We can choose $\ve$ arbitrarily small by increasing $d_0$ to a sufficiently large constant.
The seven components $H_0, H_1, \cdots, H_6$ of $S$ all satisfy Expander, meeting the condition to be an $\ve$-halver. Suppose the left output of $H_0$ contains a set $U$ of more than $\ve z$ elements each at least the $z^{th}$ largest element of $I_1 \cup I_2$. By Expander, $\left| \Gamma \lp U \rp \right| \ge (1-\ve ) z> z - |U|$, meaning that $\Gamma \lp U \rp$ includes an element less than the $z^{th}$ largest input. A contradiction against correct swapping. So the left output of $H_0$ includes at most $\ve z$ elements among the $z$ largest inputs.
We also have the facts below. We will use them in our proof in SecHalver.
Let $S$ be a separator on $m$ inputs with the seven $\ve$-halvers as in fig0, where $m \in 16 \Z^{+}$, and $\ve \in \lp 0, \frac{1}{100} \rp$ is a constant.
The following three hold true for every $g \in \R$:
* Suppose there are $l$ items larger than $g$ in $I_1$, and $r$ items not exceeding $g$ in $I_2$. Then $A_1 \cup A_2$ includes at most $\frac{1}{2} \ve m+ \max \lp l-r, 0 \rp$ items larger than $g$, and $A_3 \cup A_4$ includes at most $\frac{1}{2} \ve m + \max \lp r-l, 0 \rp$ items not exceeding $g$.
* Suppose the $A_4$ output includes $p$ items larger than $g$ where $p \le \frac{m}{32}$. Then
$A_1 \cup A_2 \cup A_3$ includes at most $\frac{4 \ve p}{1-4\ve}$ items larger than $g$.
* If the $A_1$ output includes $p' \le \frac{m}{32}$ items not exceeding $g$, then $A_2 \cup A_3 \cup A_4$ includes at most $\frac{4 \ve p'}{1-4\ve}$ items not exceeding $g$.
i): WLOG assume $r \ge l$.
The top halver $H_0$ of $S$ swaps so that there are $r-y$ items larger than $g$ in $A_1 \cup A_2$, and $l-y$ items not exceeding $g$ in $A_3 \cup A_4$, for some $y \ge 0$.
It suffices to verify $l-y \le \frac{1}{2} \ve m$ to have i). Suppose not.
Then there are more than $\frac{1}{2} \ve m$ elements of $A_1 \cup A_2$ larger than $g$, and more than $\frac{1}{2} \ve m$ of $A_3 \cup A_4$ not exceeding $g$.
Let $U$ be the set of those elements of $A_1 \cup A_2$. By Expander,
\left| \Gamma \lp U \rp \right| \ge \\ \frac{1-\ve}{\ve} \min \lp \frac{\ve m}{2}, |U| \rp > \frac{(1-\ve)m}{2}.
So the items in $U$ would be compared more than $\frac{m}{2} - \frac{1}{2} \ve m$ elements of $A_3 \cup A_4$, which contradicts the last sentence of the first paragraph. This proves i).
ii), iii): We only show ii) as the proof of iii) is similar.
Let $z$ be the number of items larger than $g$ in $I_1 \cup I_2$. We claim that $z \le \lp \frac{1}{32} + 2 \ve \rp m$. By the same argument as above, the right output of the halver $H_0$ includes at least $\min \lp \frac{m}{2}, z \rp - \frac{1}{2} \ve m$ items larger than $g$. Also those of $H_2$, $H_4$ and $H_6$ include at least $\min \lp \frac{m}{4}, z \rp - \ve m$, $\min \lp \frac{m}{8}, z \rp - \frac{3}{2} \ve m$ and $\min \lp \frac{m}{16}, z \rp - 2 \ve m$ such elements, respectively. So
$\min \lp \frac{m}{16}, z \rp - 2 \ve m \le p \le \frac{m}{32}$, meaning $\frac{m}{32} \ge z - 2 \ve m$. The claim follows this.
Because of $z \le \lp \frac{1}{32} + 2 \ve \rp m$, and since $H_0$ is an $\ve$-halver, its left output includes at most $\ve z$ items larger than $g$. Considering $H_2$, $H_4$ and $H_6$ similarly, $A_1 \cup A_2 \cup A_3$ contains at most $4 \ve z$ such items. Its ratio to $p$ is no more than $\frac{4\ve}{1-4\ve}$, proving ii).
In AKSImprovement, we will have a situation where the input size $m=|I_1 \cup I_2|$ may not be a multiple of 16. We modify the $\ve$-halvers in $S$ in such a case.
Construction of a separator $S$ for any input size $m \in \Z^+$: If $m$ is odd, add an extra element, which is compared with no other elements in $H_0$, to the left input of $H_0$, and remove it from the output. Similarly, add an extra element to the right input to $H_1$, $H_3$ or $H_5$ if the input size is odd. Add an extra element to the left input to $H_2$, $H_4$ or $H_6$ if the input size is odd. Remove them from their output arrays.
As a result,
m/16 - 1/16 ≤|A_1| ≤m/16 + 7/8,
7m/16 - 21/16 ≤|A_2| ≤7m/16,
m/16 ≤|A_4| ≤m/16 + 15/16,
7m/16 - 7/8 ≤|A_3| ≤7m/16 + 7/16.
The upper bound in the first case is due to
|A_1| \le
\frac{1}{2} \lp
\frac{1}{2} \lp
\frac{1}{2} \lp \frac{m}{2} + 1
\rp
+ 1
\rp
+ 1
\rp
The other bounds are found similarly.
For this $S$, the three statements of L0 can be shown true if we replace $\ve$ by
= + 300/299 ^2,
in i) when $\ve = 1/395$.
We see it as a corollary to the lemma.
Let $m \in \Z^{+}$, $\ve = 1/395$, and $S$ be a separator on $m$ inputs with the seven $\ve$-halvers. The three statements i), ii) and iii) of L0 hold true if $\ve$ is replaced by $\hat{\ve}$ in i).
By the same argument as the lemma, we can show the bounds $\frac{m+1}{2} + \max \lp l-r, 0 \rp$ and $\frac{m+1}{2} + \max \lp r-l, 0 \rp$ for the left and right halves, respectively. If $m \ge \frac{299}{300\ve}$, then
\ve \lp m+1 \rp \le \ve \lp m+ \frac{300}{299} \ve m \rp
=\hat{\ve} m
$, so i) holds in this case.
If $m < \frac{299}{300\ve}$, then $\ve (m+1) \le \frac{299}{300} + \ve < 1$. The number $\ve (m+1)$ in place of $\ve m$ is an upper bound for an integral value, so the bounded value is zero. Thus i) holds in every case with $\hat{\ve}$.
ii), iii): We saw $\min \lp \frac{m}{16}, z \rp - 2 \ve m \le p \le \frac{m}{32}$ in the lemma. Instead we have
\[
\min \lp \frac{m}{16} - \frac{1}{16} , z \rp
- 2 \ve \lp m+ 1 \rp
\le p.
\]
Here $-2 \ve m$ is replaced by $-2 \ve \lp m+ 1 \rp$ since the input size of any $H_i$ does not exceed $m+1$. Also $\frac{m}{16}$ is replaced by $\min \lp |A_1|, |A_4|\rp \ge \frac{m}{16} - \frac{1}{16}$.
Thus $\min \lp \frac{m}{16} - \frac{1}{16} , z \rp
- 2 \ve \lp m+ 1 \rp
\le p \le \frac{m}{32}$, meaning $z \le \lp \frac{1}{32} + 2 \hat{\ve} \rp m$: If $\frac{m}{16} - \frac{1}{16} - 2 \ve \lp m+ 1 \rp \le \frac{m}{32}$, then $m \le 2$ so $p \le \frac{m}{32}$ is zero. Otherwise $z \le \frac{m}{32} + 2 \ve (m+1) \le \lp \frac{1}{32} + 2 \hat{\ve} \rp m$, seen similarly to i).
The remaining argument is the same as the lemma. The array $A_1 \cup A_2 \cup A_3$ includes at most $4 \ve z$ items larger than $g$. Its ratio to $p$ does not exceed $\frac{4 \ve}{1 - 4\ve}$. The statement ii) follows. Similarly for iii).
§ THE AKS SORTING NETWORK AS A WEAK HALVER
We show Halver in this section. Below we construct such an AKS sorting network ${\cal H}$ of reduced depth $O \lp \log \log n \rp$.
Our main task is to bound the number of items in the wrong half of the output array, called strangers, by $n \lg^{-c} n$ where the constant $c>0$ is given by the theorem. L01 below will lead to the depth bound, $i.e.$, at most $22.34 c \lg \lg n$ layers of $1/175$-halvers.
Assume without loss of generality that $n$ inputs have distinct values[For $s$ elements of a same value $g$, regard that they are valued $g+\delta, g+2 \delta, \ldots, g+ s \delta$ for an infinitesimal number $\delta>0$.
All the statements in our proof remain true with this change.
There are $j_{max}+1$ layers in ${\cal H}$, where
\[
j_{max} = \frac{c}{\lg (4/e)} \lg \lg n,
\]
with $e = 2.71828 \ldots$ being the natural logarithm base. In this section, obvious ceiling/floor functions are omitted.
Each layer is a complete binary tree of separators of depth $d_{max} - j$ where
\[
d_{max} = \frac{c}{5} \lg \lg n + j_{max}
= c \lp \frac{1}{\lg (4/e)} + \frac{1}{5} \rp \lg \lg n.
\]
Every separator consists of the seven $\ve$-halvers as in fig0 where
\[
\ve = \frac{1}{175}.
\]
Also we create two arrays $L$ and $R$ that will be the left and right halves of the output array of ${\cal H}$, respectively.
Denote by $S_{d, j}$ ($0 \le j \le j_{max}$, $0 \le d \le d_{max}-j$) a separator node of depth $d$ in the $j^{th}$ layer of ${\cal H}$. Here are the main construction rules.
The Construction Rules of the AKS Sorting Network ${\cal H}$ of Reduced Depth:
* If a separator $S_{d, j}$ ($d\ge 1$, $j \le j_{max}-1$) is the left child of its parent $S_{d-1, j}$, the $A_1$ and $A_4$ outputs of $S_{d, j}$ are sent to the left of the copy of $S_{d-1, j}$ in the $j+1^{st}$ layer. That is, the sent array is concatenated with the first element of the destination array. Also if $S_{d, j}$ is the right child of $S_{d-1, j}$, its $A_1 \cup A_4$ is sent to the right of the copy of $S_{d-1, j}$ in the the $j+1^{st}$ layer.
* The $A_2$ and $A_3$ outputs of $S_{d, j}$ such that $d<d_{max}-j$ are sent to its left and right children in the same layer, resp.
* The $A_2$ and $A_3$ outputs of $S_{d_{max}-j, j}$ for every $j$ are sent to $L$ and $R$, resp.
* The $A_1$ and $A_4$ outputs of $S_{0, j}$ such that $j<j_{max}$ are sent to the left and right of $S_{0, j+1}$, resp.
* The $A_1$ and $A_4$ outputs of $S_{d, j_{max}}$ for all $d$ are sent to $L$ and $R$, resp.
* The union $L \cup R$ of the finally obtained $L$ and $R$ is the output of ${\cal H}$ as a weak $\lg^{-c} n$-halver.
As mentioned in Introduction, we call the comparator network ${\cal H}$ constructed by this set of rules an AKS sorting network of reduced depth. It is identical with the standard AKS sorting network given in <cit.> except for its depth and the two output arrays $L$ and $R$.
We introduce extra terminology. A stranger is a numeric item greater than the median of $n$ inputs to ${\cal H}$ existing in the left half of ${\cal H}$, or one less than or equal to the median existing in the right half.
\(
m_{d, j} &:& \textrm{the number of inputs to each $S_{d, j}$, $i.e.$, $|I_1 \cup I_2|$ of $S_{d, j}$.}
\\
p_{d, j} &:& \textrm{the number of strangers existing in $\bigcup_{i=1}^4 A_i$ of all $S_{d, j}$, and }
\\ &&
\Delta = \frac{1}{32}.
\)
We observe the following four.
The number of layers of $\ve$-halvers in ${\cal H}$ does not exceed
\[
4 d_{max} + 8 j_{max}
= c \lp \frac{12}{\lg (4/e)} + \frac{4}{5} \rp \lg \lg n
< 22.34 c \lg \lg n,
\]
as desired.
It is confirmed by the first lemma below.
* We have
\[
\lp \frac{7}{16} \rp^{d} m_{0, j}
\le
m_{d, j} \le n \lp \frac{7}{16} \rp^d 8^{-j} \frac{(d+2j)^j}{j!},
\]
for each $j \le j_{max}$ and $d \le d_{max}$, regarding $0^0=0!=1$.
The lower bound is due to $m_{d, j} \ge \frac{7}{16} m_{d-1, j}$, seen by construction.
The upper bound is proved in L01 below.
* By Stirling's approximation <cit.>, $j!> \lp \frac{j}{e} \rp^j$, so <ref>) means
\[
m_{d, j} \le \lp \frac{7}{16} \rp^d \lp \frac{e}{8} \lp 2+ \frac{d}{j} \rp \rp^j n,
\]
when $j \ge 1$. Thus $m_{0, j} \le \lp \frac{e}{4} \rp^{j} n$.
Our invariant for $j \le j_{max}$ and $d \le d_{max}-j$ is
\[
p_{d, j} \le \Delta^{d+1} m_{0, j}.
\]
We prove it in the third lemma below. With this and <ref>), the number of strangers in $L \cup R$ is upper-bounded by
\(
\sum_{j=0}^{j_{max}} p_{d_{max}-j, j} +
\sum_{d=0}^{d_{max}-j_{max}} p_{d, j_{max}}
\\ &<&
2 \Delta^{d_{max} - j_{max}+1} n +
2 \Delta \lp \frac{e}{4} \rp^{j_{max}} n
< n \lg^{-c} n,
\)
as $j_{max} = \frac{c}{\lg (4/e)} \lg \lg n$ and $d_{max} -j_{max}=
\frac{c}{5} \lg \lg n = \frac{c}{- \lg \Delta} \lg \lg n
By <ref>) and <ref>), the comparator network ${\cal H}$ satisfies all the desired properties. It remains to prove the three lemmas mentioned above. We will have Halver after it.
Let $h_{d, j}$ be the number of layers of $\ve$-halvers in the sub-network of ${\cal H}$ rooted at $S_{d, j}$. Then $h_{d, j} \le 4 \lp d_{max}-d + 2 j_{max} - 2 j \rp$ for every $j \le j_{max}$ and $d \le d_{max}-j$.
Proof by double induction on $j$ and $d$. We show the general induction step first. Assume true for $j+1$ and all $d$, and for $j$ and $d+1$. Prove true for $j$ and $d$.
By induction hypothesis, $h_{d-1, j+1} \le 4 \lp d_{max}-(d-1) + 2 j_{max} - 2 (j+1) \rp$
and $h_{d+1, j} \le 4 \lp d_{max}-(d+1) + 2 j_{max} - 2 j \rp$.
By construction, the depth $h_{d, j}$ does not exceed their maximum plus 4, as the separator $S_{d, j}$ has four layers of $\ve$-halvers. Thus, $h_{d, j} \le 4 \lp _{max}-d + 2 j_{max} - 2 j \rp$, proving the induction step for $j \ge 1$ and $d \ge 1$.
The claim is clearly true when $j=j_{max}$, and can be shown similarly for the case $j<j_{max}$ and $d=d_{max}$. The lemma follows.
\[
m_{d, j} \le n \lp \frac{7}{16} \rp^d 8^{-j} \frac{(d+2j)^j}{j!},
\]
for every $j \le j_{max}$ and $d \le d_{max}-j$.
Proof by double induction on $j$ and $d$. The basis $j=d=0$ is trivially true as $0^0=0!=1$. Assume true for $j=0$ and $d-1$. The desired bound holds for $j=0$ and $d$ due to $m_{d, 0} = \frac{7}{16} m_{d-1, 0}$ and the induction hypothesis. Thus it is true for $j=0$ and all $d$.
Assume true for $j-1$ and prove true for $j$ and $d=0$. We have the recurrence
\[
m_{0, j} = \frac{1}{8} m_{0, j-1} + \frac{1}{4} m_{1, j-1},
\]
by construction.
Find with the induction hypothesis that
\[
j (2j-2)^{j-1} + \frac{7j}{8}(2j-1)^{j-1} \le (2j)^j
\]
suffices. The inequality holds true since when $j \ge 2$, we have $(2j)^j \ge (2j-1)^j + j(2j-1)^{j-1}$ and $(2j-1)^j \ge (2j-2)^j + j(2j-2)^{j-1}$ by the binomial theorem. This proves the lemma for $j \ge 1$ and $d =0$.
Assume true for $j-1$ and all $d$, and for $j$ and $d-1$. Prove true for $j$ and $d$. We similarly find from
m_{d, j} = \frac{1}{8} m_{d+1, j-1} + \frac{7}{16} m_{d-1, j}
\frac{7}{16} j (d+2j-1)^{j-1} + (d+2j-1)^j \le (d+2j)^j
suffices. It is true by the binomial theorem again.
This completes the induction step.
$p_{d, j} \le \Delta^{d+1} m_{0, j}$ for each $j \le j_{max}$ and $d \le d_{max}-j$.
Proof by double induction on $j$ and $d$. For the basis $j=d=0$, we observe that the number of strangers in $I_1$ of $S_{0, 0}$ equals that in $I_2$. By L0 i), the separator $S_{0, 0}$ reduces strangers to at most $\ve m_{0, 0} < \Delta m_{0, 0}$, proving the basis.
Assume true for $j=0$ and $d-1$ and prove true for $j=0$ and $d$.
Consider each $S_{d, 0}$. There are at most
\Delta^{d} m_{0, 0}
strangers in $A_1 \cup A_4$ of its parent $S_{d-1, 0}$ by induction hypothesis.
This number does not exceed $\Delta m_{d-1, 0} \ge \Delta \lp \frac{7}{16} \rp^{d-1} m_{0, 0}$. By L0 ii) or iii), if $p$ strangers are output from $A_1 \cup A_4$ of $S_{d, j-1}$, its $A_2 \cup A_3$ includes at most $\frac{4 \ve}{1- 4 \ve} p$ strangers.
Consider all $S_{d, 0}$ and $S_{d-1, 0}$. By induction hypothesis and the above, there are no more than $\frac{4 \ve}{1- 4 \ve} p_{d-1, 0} \le \frac{4 \ve}{1- 4 \ve} \Delta^{d} m_{0, 0} \le \Delta^{d+1} m_{0, 0}$ strangers in $A_2 \cup A_3$ of all $S_{d-1, 0}$. (This holds for $\ve = 1/175$ and $\Delta = 1/32$.) So
p_{d, 0} < \Delta^{d+1} m_{0, 0},
proving the induction step. We have the lemma for $j=0$ and all $d$.
Assume true for $j-1$ and prove true for $j$ and $d=0$.
By induction hypothesis,
p_{d', j-1} \le \Delta^{d'+1} m_{0, j-1}
$ for all $d' \le d_{max}- j+1$. So there are at most
\[
\sum_{d' = 2}^{d_{max}-j+1} p_{d', j-1}
2 \Delta^{3} m_{0, j-1}
\]
strangers not input to $S_{0, j}$. This bounds the difference between the numbers of strangers in $I_1$ and $I_2$ of $S_{0, j}$. By L0 i), $S_{0, j}$ reduces strangers to at most
\[
\ve m_{0, j}
+ 2 \Delta^{3} m_{0, j-1}
\le
\ve m_{0, j}
+ 16 \Delta^{3} m_{0, j}
\le \Delta m_{0, j},
\]
where $m_{0, j} \ge \frac{1}{8} m_{0, j-1}$ by construction.
This proves the lemma for $j$ and $d=0$.
Assume true for $j-1$, and $j$ and $d-1$. Prove true for $j$ and $d$. Consider each $S_{d, j}$ and its parent $S_{d-1, j}$. Suppose $S_{d-1, j}$ outputs $p$ strangers from $A_1 \cup A_4$.
By induction hypothesis, $p \le \Delta^{d} m_{0, j}$ that does not exceed $\Delta m_{d-1, j} \ge \Delta \lp \frac{7}{16} \rp^{d-1} m_{0, j}$.
L0 again assures that $A_2 \cup A_3$ of $S_{d-1, j}$ includes at most $\frac{4 \ve}{1- 4 \ve} p$ strangers.
\[
p_{d, j} \le p_{d+1, j-1} + \frac{4 \ve p_{d-1, j}}{1- 4 \ve}
\le
\Delta^{d+2} m_{0, j-1} + \frac{4 \ve}{1- 4 \ve} \Delta^{d} m_{0, j}
\le
\Delta^{d+1} m_{0, j},
\]
by induction hypothesis and $m_{0, j} \ge \frac{1}{8} m_{0, j-1}$.
This verifies the induction step.
The lemmas complete the proof of Halver.
§ IMPROVED CONSTANT FACTOR OF THE $O \LP \LOG N \RP$ DEPTH OF AN $N$-INPUT SORTING NETWORK
Modify the AKS sorting network ${\cal H}$ constructed in Section 3 so that
j_max = n/(4/e) ,
d_max = n + j_max+ 6,
= 1/395,
while $\Delta= 1/32$ remains the same. Create separators $S_{d, j}$ for all $j$ and $d$ such that $0 \le j \le j_{max}$ and $0 \le d \le d_{max}$ rather than $d \le d_{max}-j$.
With this setting for sufficiently large $n$, the depth of ${\cal H}$ as a graph of $\ve$-halvers does not exceed
\[
\lp d_{max} + 2 j_{max} + 1 \rp \le 4 \lp 1 + 3 \lg^{-1} \frac{4}{e} \rp \lg n + 18 <25.54 \lg n,
\]
seen similarly to L2.
In this section, we prove that the AKS sorting network with the setting sorts the $n$ inputs correctly. This will confirm ConstantImprovement improving the aforementioned estimate in <cit.> that the depth can be at most $7 \cdot 6.07 \lg n=48.79 \lg n$ layers of $1/402.15$-halvers.
Due to the new depth of ${\cal H}$, the input array sizes $m_{d, j}$ of some separators $S_{d, j}$ may not be divisible by $16$. Construct $S_{d, j}$ by the rule given below L0.
The changes affect ${\cal H}$ as follows.
* The input sizes of two $S_{d, j}$ may differ due to varying rounding errors generated by construction. For simplicity, we regard $m_{d, j}$ as $|I_1 \cup I_2|$ of the currently considered separator $S_{d, j}$.
* The claim of L01 is modified into $
m_{d, j} \le n \lp \frac{7}{16} \rp^d 8^{-j} \frac{(d+2j)^j}{j!} + 13
$, proved in SizeBound in Appendix.
When $j \ge 1$,
m_d, j
≤7/16 ^d e/8 2+ d/j ^j n + 13,
by Stirling's approximation.
* It means $m_{d, j_{max}} \le 14$ for any $d$: In
we have $
\lp \frac{7}{16} \rp^\frac{d}{j} \frac{e}{8} \lp 2+ \frac{d}{j} \rp
approaching zero as $\frac{d}{j}$ grows larger, and not exceeding $\frac{e}{4}$. As $j_{max} \ge \frac{\lg n}{\lg (4/e)}$, we have$m_{d, j_{max}} \le \lp \frac{e}{4} \rp^{j_{max}} n + 13 \le 14$.
With these we see:
All the $n$ items input to ${\cal H}$ are also input to each layer.
Observe the following.
* By construction, $|A_4| \ge 1$ if $m_{d, j} \ge 1$, and $|A_1| \ge 1$ if $m_{d, j} \ge 2$ for every $S_{d, j}$.
* $m_{d, j} \le 13$ if $d \ge \lg n + j$. It is true for $j=0$ by SizeBound.
When $j \ge 1$, by InputSizeBound, $m_{d, j}-13$ is at most
\[
\lp \frac{7}{16} \rp^d \lp \frac{e}{8} \lp 2+ \frac{d}{j} \rp \rp^j n
\le n 2^{-\lg n} \lp \frac{e}{2 \cdot 8} \lp 2+ \frac{d}{j} \rp \lp \frac{7}{8} \rp^{\frac{d}{j}} \rp^j < 1,
\]
whose second inequality is due to $\frac{e}{16} \lp 2+ x \rp \lp \frac{7}{8} \rp^{x}<1$ for every $x\ge 0$.
* So $|A_1 \cup A_4| \le 2$ if $d \ge \lg n + j$.
* The above two mean $m_{d, j} \le 10$ for every $S_{d, j}$ such that $d \ge \lg n + j+1$: The separator receives at most $\lc \frac{13-2}{2} \rc$ elements from the parent and 4 elements from the two $S_{d+1, j-1}$.
* Likewise, $m_{d, j} \le 8$ if $d \ge \lg n + j+2$, and $m_{d, j} \le 7$ if $d \ge \lg n + j+3$.
We show by induction on $j$ that:
* $m_{d, j} \le 3$ if $d = \lg n + j+4$,
* $m_{d, j} \le 1$ if $d = \lg n + j+5$, and
* $m_{d, j} =0$ if $d \ge \lg n + j+6$.
We only show the induction step as the basis $j=0$ is proved similarly. Consider $S_{d, j}$ such that $d \ge \lg n + j+4$. We have $m_{d-1, j} \le 7$ from the above, and $m_{d+1, j-1} \le 0$ by induction hypothesis. So $m_{d, j} \le \lc \frac{7-2}{2} \rc=3$, proving i). The other two are shown similarly.
The lemma follows iii) since $d_{max} = \lg n + j_{max} + 6$.
Define arrays associated with a considered separator $S_{d, j}$ the following way.
Fix a given input array to ${\cal H}$. Denote by $S_L$ and $S_R$ the left and right children of $S_{d, j}$, respectively.
Let $B(S_{d, j})$ be the array recursively defined by
\[
B(S_{d, j}) =
\lp \textrm{$A_1$ of $S_{d, j}$} \rp
\cup
B \lp S_L \rp \cup B \lp S_R \rp
\cup \lp \textrm{$A_4$ of $S_{d, j}$} \rp,
\]
if $d<d_{max}$. It equals $\bigcup_{i=1}^4 A_i$ of $S_{d, j}$ if $d=d_{max}$.
It can be seen by induction on $j$ that $B(S_{d, j})$ consists of the items input to the subtree rooted at $S_{d, j}$ in the $j^{th}$ layer. With AllItems, we define $B \lp S_{0, j_{max}} \rp$ as the output of ${\cal H}$.
Consider the complete binary tree of separators in the $j^{th}$ layer. Initially, let $I$ be the sorted array of the $n$ inputs. We say that it is correct for $S_{0,j}$. Recursively, a sorted array $I$ correct for a separator $S_{d, j}$ is given so that $|I|=|B \lp S_{d, j} \rp|$.
Split $I$ into the four sub-arrays of sizes $|A_1|$, $\left| B \lp S_L \rp \right|$, $\left| B \lp S_R \rp \right|$, and $|A_4|$ in the order. They are correct for $A_1$, $S_L$, $S_R$ and $A_4$, respectively.
We generalize the definition of strangers.
A numeric item $x$ is said to be an $l$-stranger if there exists $S_{l, j}$ satisfying one of the following two:
* $x$ is in the array correct for $A_1$ ($A_4$) of $S_{l, j}$ or $S_L$ ($S_R$), but is incorrectly output from
$A_4$ ($A_1$) of $S_{l, d}$ or its descendant.
* $x$ is in the array correct for $A_1$ or $A_4$ of $S_{l, j}$ but is incorrectly output from a proper descendant of $S_{l, d}$.
The following statement is our invariant.
For each separator $S_{d, j}$ and index $l \le d$, there are $\Delta^{d-l + 1} m_{d, j}$ or less $l$-strangers output from $S_{d, j}$.
Its proof is found in Appendix.
The lemma means that $B\lp S_{0, j_{max}} \rp$ is sorted, proving that ${\cal H}$ satisfying Parameters correctly re-orders the $n$ inputs. Verify it as follows:
Let $j=j_{max}$. There are no strangers in the $j^{th}$ layer by Invariant2 and $m_{d, j} \le 14$.
This implies that
$B\lp S_{d, j} \rp$ for every $S_{d, j}$ consists of the elements in the array correct for $S_{d, j}$. It also means that $B(S_{d, j})$ is sorted if both $B \lp S_L \rp$ and $B \lp S_R \rp$ are sorted.
For a leaf $S_{d_{max}, j}$, the array $B \lp S_{d_{max}, j} \rp$ is empty thus sorted, by iii) in the proof of AllItems.
Hence $B\lp S_{0, j} \rp$ is sorted, completing our proof of ConstantImprovement.
§ SELECTION IN $O \LP \LOG N \RP$ TIME WITH $O \LP N \LOG \LOG N \RP$ MESSAGES ON A COMMUNICATION NETWORK
We prove MainClaim in this section.
Our parallel algorithm for the distributed $k$-selection problem on a communication network ${\cal G}$ uses the weak halver ${\cal H}$ given by Halver and three other AKS sorting networks as sub-components. Keep assuming that $n$ is a large power of 2 and the input array has elements of distinct values. Let
\[
q=\lg n.
\]
Choose $c=2$ in Halver, so ${\cal H}$ is a weak $n \lg^{-2} n$-halver.
All the expressions of array sizes in this section omit the floor or ceiling function.
We first describe the algorithm focusing on the case $k=n/2$.
Algorithm 1 for Finding the Median of $n$ Inputs:
* Apply ${\cal H}$ to the given $n$ inputs. Let $L$ and $R$ be its left and right halves of the output, respectively, and $C = \emptyset$.
* Arbitrarily partition $L$ into $\frac{n}{2q}$ subarrays each of $q$ items naming them $L_1, L_2, \ldots, L_{\frac{n}{2q}}$.
* Find the maximum value $l_i$ in each $L_i$.
* Sort the $\frac{n}{2q}$ items $l_1, l_2, \ldots, l_{\frac{n}{2q}}$ by an AKS sorting network.
Add the elements of $L_i$ to $C$ for every $i$ such that $l_i$ is among the largest $n \lg^{-2} n$ of $l_1, l_2, \ldots, l_{\frac{n}{2q}}$.
* Perform Steps 2–4 to $R$ symmetrically.
* Sort the elements of $C$ by another AKS sorting network. Find its median, returning it as the median of all $n$.
Each AKS sorting network used above re-orders at most $2 n \lg^{-1} n$ inputs. This allows us to detect the median with a sufficiently small number of comparisons, leading to $O \lp n \log \log n \rp$ messages. We confirm the correctness of the algorithm first.
Algorithm 1 correctly finds the median of all $n$ inputs.
We claim that no element of $L-C$ is a stranger or the median of $n$ after Step 4.
Let $i$ be an index such that $l_i$ is not among the largest $n \lg^{-2} n$ of $l_1, l_2, \ldots, l_{\frac{n}{2q}}$. If an element of $L_i$ were a stranger or the median, the largest $n \lg^{-2} n$ items would be all strangers, contradicting Halver. This proves the claim.
By symmetry, no element of $R-C$ is a stranger. Steps 4 and 5 remove the same number of items from the both halves, each neither a stranger nor the median. The array $C$ includes the same number of strangers in the both halves.
Hence Step 6 correctly finds the median of all.
We now describe how to run Algorithm 1 on the communication network ${\cal G}$.
Find our implementation below noting two remarks.
* Regard that there are $2n$ processor nodes in ${\cal G}$ instead of $n$, since any of them at odd and even time slots can play two different roles.
* Each processor in ${\cal G}$ can simulate any of the four comparator networks the way mentioned in Def: A processor receives two numeric items from other nodes to send their minimum and/or maximum to anywhere in ${\cal G}$.
Implementation of Algorithm 1 on ${\cal G}$:
Simulate two AKS sorting networks in Steps 4 and 5 with extra $n$ processors. Then select the elements of $C$ in $O \lp \log n \rp$ time as follows. Initially the first node of each $L_i$ is notified if $l_i \in C$. Move items in $L \cap C$ by sending messages so that $L \cap C$ is held by the consecutive smallest numbered nodes.
To find the message destinations, construct,
right after $l_i$ are sorted in Step 4, a complete binary tree $T$ of $\frac{n}{q}$ nodes whose leaves are the first nodes of all $L_1, L_2, \ldots, L_{\frac{n}{2q}}$. (For simplicity identify the first node with $L_i$ itself.) Started from the leaves, recursively compute the number of $L_i \subset C$ existing in the subtree $T'$ rooted at each node of $T$. Then, started at the root of $T$, compute the number of $L_i \subset C$ existing outside $T'$ to the left: It is recursively sent from the parent of the current node. Pass it to its left child. Add the total number of $L_i \subset C$ under the left child and send the value to the right child.
This way every leaf of $T$ is informed of the number of $L_i \subset C$ to the left. The leaf disseminates the information to all the nodes in the same $L_i$. Each node is now able to compute the message destination so $L \cap C$ is held by the consecutive smallest numbered nodes.
Merge $L \cap C$ with $R \cap C$ similarly. Perform Step 6 with a simulated AKS sorting network on the obtained $C$. This completes the description of our implementation.
Algorithm 1 correctly runs on ${\cal G}$ in $O \lp \log n \rp$ time with $O \lp \log \log n \rp$ messages. One can check it with L3 noting that:
* The simulated ${\cal H}$ runs in $O \lp \log \log n \rp$ time with $O \lp n \log \log n \rp$ messages.
* Each of the three simulated AKS sorting network runs in $O \lp \log n \rp$ time with $O \lp n \rp$ messages.
* It takes $O \lp n \rp$ messages each of less than $\lg n$ bits to construct $C$.
This verifies MainClaim when $k=\frac{n}{2}$.
Selecting an item but the median is done similarly. Assume without loss of generality that we want the $k$th smallest item such that $k < \frac{n}{2}$. We can detect it by adding extra $n-2k$ elements valued $-\infty$ to the input array. This changes no asymptotic bounds we showed so far.
We have constructed a parallel algorithm on ${\cal G}$ that distributedly computes the $k$th smallest of the $n$ inputs in $O \lp \log n \rp$ time with $O \lp n \log \log n \rp$ messages.
We now have MainClaim.
§ THE UNIVERSAL PARALLEL TIME LOWER BOUND $\LG N$ ON A COMMUNICATION NETWORK
In this section, we show that it takes at least $\lg n$ steps to compute many basic data aggregation problems on a communication network ${\cal G}$ including the distributed selection problem. Consider a parallel algorithm to compute a function $f(x_1, x_2, \ldots, x_n) \in \lb 0, 1 \rb$ where $x_1, x_2, \ldots, x_n$ are input numeric items of $b$ bits distributed over ${\cal G}$.
Such $f$ is said to be critical everywhere if there exist $x_1, x_2, \ldots x_n$ such that
f x_1, x_2, …, x_i-1, x̂_i, x_i+1, …, x_n f x_1, x_2, …, x_i-1, x_i, x_i+1, …, x_n ,
for each index $i=1, 2, \ldots, n$ and some $b$-bit numeric item $\hat{x}_i$ depending on $i$.
Below we prove the time lower bound $\lg n$ to compute such $f$ as the following theorem.
Let $f$ be a function on $n$ inputs critical everywhere.
It takes at least $\lg n$ parallel steps in the worst case to compute $f$ on a communication network distributedly.
By the theorem, we will have the statement mentioned in Introduction.
Corollary <ref>.
Any parallel algorithm takes at least $\lg n$ steps in the worst case to compute each of the following three problems on a communication network distributedly: i) the $k$-selection problem on $n$ inputs each of at least $\lc \lg n \rc + 1$ bits; ii) the problem of finding the sum of $n$ inputs; iii) the problem of counting numeric items among $n$ inputs, each exceeding a given threshold.
i): Given such an algorithm on $n$ inputs $x_1, x_2, \ldots, x_n$, let $f(x_1, x_2, \ldots, x_n)$ be the least significant bit of the $k$th smallest input. Such a function $f$ is critical everywhere: Choose $x_i= 2(i-1)$ for $i=1, 2, \ldots, n$. If $k>1$, let $\hat{x}_i = x_k - 1$, otherwise $\hat{x}_i = 1$.
(All of these numbers are $\lc \lg n \rc + 1$-bit integers.)
These satisfy CriticalEverywhere. By the theorem, the algorithm takes at least $\lg n$ steps to compute such $f$ on a communication network distributedly.
ii), iii): Shown similarly to i) by choosing $f$ as the least significant bit of the sum of $n$ inputs, and that of the number of inputs exceeding a given threshold, respectively.
Consider the class of problems to compute on a communication network distributedly such that any particular bit of the computed result is a function critical everywhere. It is very large containing many aggregation problems as the above three, each with the parallel running time lower bound $\lg n$ on ${\cal G}$.
We prove the theorem. Denote by $v_i$ the $i$th processor node of ${\cal G}$, and by $x_i$ the numeric item held by $v_i$ at time 0. Also let $A$ be a parallel algorithm to compute $f$ on ${\cal G}$ distributedly. Since $f$ is critical everywhere, there exists an input set $X= \lb x_1, x_2, \ldots, x_n \rb$ such that CriticalEverywhere.
Assume $f(x_1, x_2, \ldots, x_n)=1$ without loss of generality, written as $f(X)=1$.
We focus on the computation performed by $A$ on the input set $X$. For time $t \ge 0$ and index $i=1, 2, \ldots, n$, define the set $V_{i, t}$ of nodes in ${\cal G}$ recursively as follows.
* $V_{i, 0} = \lb v_i \rb$ for time $t=0$.
* $V_{i, t}=V_{i, t-1} \cup V_{j, t-1}$ if $v_j$ sends a message to $v_i$ at time $t \ge 1$. Otherwise $V_{i, t}=V_{i, t-1}$.
$V_{i, t}$ is the set of processors $v_j$ such that the values of $x_j$ can possibly affect the computational result at $v_i$ at time $t$.
Let $t_{max}$ be the time when $A$ terminates on $X$, and $v_i$ be the processor that decides $f(X)=1$ at time $t_{max}$. The size of $V_{i, t_{max}}$ must be $n$; otherwise $A$ decides $f(X)=1$ at $v_i$ with no information on some input $x_j$, contradicting that $f$ is critical everywhere. The following lemma formally confirms it.
$|V_{i, t_{max}}|=n$.
By the equivalence between an algorithm and Boolean circuit <cit.>, there exists a Boolean circuit $C$ to compute $f(X)$, which is converted from $A$ by the reduction algorithm. Construct the subgraph of $C$ that decides $f(X)=1$ at $v_i$ as follows: For each $t=0, 1, \ldots, t_{max}$, with the algorithm running on $X$ at each processor $v_j$ and the exchanged messages,
inductively construct a Boolean circuit to decide each bit of the computed result stored at $v_j$ at time $t$. For the time $t=t_{max}$ and processor $v_i$, we have a circuit $C'$ that decides $f(X)=1$ only from $x_j$ such that $v_j \in V_{i, t_{max}}$.
The existence of $C'$ means $f(X)=1$ if
all $x_j$ such that $v_j \in V_{i, t_{max}}$ have the values specfied by $X$. If $|V_{i, t_{max}}|<n$, this contradicts that $f$ is critical everywhere satisfying CriticalEverywhere. Hence $|V_{i, t_{max}}|=n$.
We also have $|V_{j, t}| \le 2^t$ for each $t$ and $j$. It is because $
|V_{j, t}| = \left| V_{j, t-1} \cup V_{l, t-1} \right|
\le 2 \cdot 2^{t-1} = 2^t
$ if a processor $v_l$ sends a message to $v_j$ at time $t \ge 1$.
\lg n = \lg |V_{i, t_{max}}| \le t_{max}
LowerBound follows.
§ CONCLUDING REMARKS AND OPEN PROBLEMS
We have shown $25.54 \lg n$ layers of $1/395$-halvers as an upper bound on the depth of the AKS sorting network, and $22.34 c \lg \lg n$ layers of $1/175$-halvers as that on the depth of a weak $\lg^{-c} n$-halver.
The obtained bounds can be further improved by choosing another parameter set.
It is a question if we can significantly reduce these values. Regarding the number $O \lp n \log n \log n \rp$ of messages to find the $k^{th}$ smallest item in $O \lp \log n \rp$ time, it remains open whether or not it is asymptotically optimal on a communication network. We conjecture it positively.
§ APPENDIX: PROOF OF INPUTSIZEBOUND AND INVARIANT2
We have InputSizeBound when $j \ge 1$ as a result of SizeBound below. As in AKSImprovement, $m_{d, j}$ denotes the input size $|I_1 \cup I_2|$ of the currently considered separator $S_{d, j}$ of depth $d$ in the $j^{th}$ layer of the AKS sorting network ${\cal H}$ satisfying Parameters.
m_{d, j} \le n \lp \frac{7}{16} \rp^d 8^{-j} \frac{(d+2j)^j}{j!} + 13
$ for each $j \le j_{max}$ and $d \le d_{max}$.
Denote by $m^*_{d, j}$ the maximum of $m_{d, j}$ for all $S_{d, j}$. It satisfies the following recursive relation.
\[
m^*_{d, j} \le
\lb \begin{array}{cc}
\frac{m^*_{d+1, j-1}}{4} + \frac{7m^*_{d-1, j}}{16} + \frac{65}{16}, & \textrm{~if $j \ge 1$ and $d \ge 1$.} \\
\frac{m^*_{0, j-1}}{8}+\frac{m^*_{1,j-1}}{4} + \frac{87}{16}, & \textrm{~if $j \ge 1$ and $d=0$.} \\
\frac{7m^*_{d-1, j}}{16}+\frac{7}{16}, & \textrm{~if $j=0$ and $d \ge 1$.} \\
n, & \textrm{~if $j=d=0$.} \\
\end{array} \right.
\]
We have the upper bound in the first case since
\[
m^*_{d, j} \le 2 \lp \frac{m^*_{d+1, j-1}}{8} + \frac{29}{16} \rp + \lp \frac{7m^*_{d-1, j}}{16} + \frac{7}{16} \rp \le \frac{m^*_{d+1, j-1}}{4} + \frac{7m^*_{d-1, j}}{16} + \frac{65}{16}.
\]
It is true by OutputSizes, since it means $|A_1|+ |A_4|$ of $S_{d+1, j-1}$ does not exceed $\frac{m_{d+1, j-1}}{8} + \frac{29}{16}$. Find the other inequalities similarly.
To verify the lemma, we show by double induction on $j$ and $d$ that
m^*_d, j ≤n 7/16 ^d 8^-j (d+2j)^j/j! + 13.
We first prove the general induction step. Assume true for $j-1$ and all $d$, and $j$ and $d-1$. Prove true for $j$ and $d$.
The inequality in the first case above is equivalent to
\[
m^*_{d, j} - 13 \le \frac{1}{4} \lp m^*_{d+1, j-1} - 13 \rp + \frac{7}{16} \lp m^*_{d-1, j} - 13 \rp.
\]
By induction hypothesis,
m^*_{d+1, j-1} - 13 \le \lp \frac{7}{16} \rp^{d+1} 8^{-j+1} \frac{(d+2j-1)^{j-1}}{(j-1)!},
m^*_{d-1, j} - 13 \le \lp \frac{7}{16} \rp^{d-1} 8^{-j} \frac{(d+2j-1)^j}{j!}.
Similarly to the proof of L01, we find that it suffices to verify
\frac{7}{16} j (d+2j-1)^{j-1} + (d+2j-1)^j \le (d+2j)^j
which is true by the binomial theorem. This proves eqSizeBound for $j \ge 1$ and $d \ge 1$.
We show eqSizeBound for $j \ge 1$ and $d=0$ assuming true for $j-1$. With
m^*_{0, j} - 13 \le \frac{1}{8} \lp m^*_{0, j-1} - 13 \rp + \frac{1}{4} \lp m^*_{1, j-1} - 13 \rp
derived from the second case of the recursive relation, we find that
j (2j-2)^{j-1} + \frac{7j}{8}(2j-1)^{j-1} \ge (2j)^j
suffices. It is true since $(2j)^j \ge (2j-1)^j + j(2j-1)^{j-1}$ and $(2j-1)^j \ge (2j-2)^j + j(2j-2)^{j-1}$, proving eqSizeBound for $j \ge 1$ and $d =0$.
eqSizeBound is true for $j=d=0$ by $0^0=0!=1$, and shown similarly for $j=0$ and $d \ge 1$.
This completes the proof.
The following lemma is a preparation to verify Invariant2.
Let $m_{d, j}$ and $m_{d-1, j}$ be the input sizes of a separator $S_{d, j}$ and its parent $S_{d-1, j}$, respectively. Then $m_{d, j} \le m_{d-1, j}+14$.
Proof by induction on $j \le j_{max}$. Assuming true for $j-1$, we first show the induction step by inner induction on $d$.
We show the basis $d=1$ of inner induction. For the given $S_{d, j}$, consider its child $S_{1, j}$ and the copy $S_{1, j-1}$ in the $j-1^{st}$ layer. From OutputSizes,
m_0, j ≥ 1/8 m_0, j-1 - 1/16 +
2 1/87/16 m_0, j-1 - 21/16 - 1/16 ,
m_1, j ≤ 7/16 m_0, j + 7/16 +
2 1/8 7/16 m_1, j-1 + 7/16 + 29/16 .
\[
m_{0, j} \le
\lp \frac{1}{8} m_{0, j-1} + \frac{29}{16} \rp +
2 \lp \frac{1}{8} \lp \frac{7}{16} m_{0, j-1} +\frac{7}{16} \rp + \frac{29}{16} \rp
= \frac{15}{64} m_{0, j-1} + \frac{355}{64}.
\]
Substitute this into the second line of eqInputSize, from which we find that
\[
m_{1, j} - m_{0, j} - 14 \le
\frac{7}{64} \lp m_{1, j-1} - m_{0, j-1} - 14 \rp.
\]
By induction hypothesis, the both hand sides are negative, proving the basis $d=1$.
Assume true for $d-1$ and prove true for $d$. For the given $S_{d, j}$ and $S_{d-1, j}$, let $S_{d-2, j}$, $S_{d-1, j-1}$ and $S_{d, j-1}$ be the parent of $S_{d-1, j}$, the copy of $S_{d-1, j}$ in the $j-1^{st}$ layer, and that of $S_{d, j}$, resp. For these separators,
\(
m_{d-1, j} \ge
2 \lp \frac{1}{8}\lp \frac{7}{16} m_{d-1, j-1} - \frac{21}{16} \rp - \frac{1}{16} \rp
\lp \frac{7 m_{d-2, j}}{16} - \frac{21}{16} \rp, \eqand
\\ &&
m_{d, j} \le 2 \lp \frac{1}{8} \lp \frac{7}{16} m_{d, j-1} + \frac{7}{16} \rp
+ \frac{29}{16} \rp +
\frac{7 m_{d-1, j}}{16}+\frac{7}{16},
\)
from which we find
\[
m_{d, j} - m_{d-1, j} - 14 < \frac{7}{64} \lp m_{d, j-1} - m_{d-1, j-1} - 14 \rp + \frac{7}{16} \lp m_{d-1, j} - m_{d-2, j} - 14 \rp \le 0,
\]
proving the induction step.
It remains to show the claim when $j=0$. It is similarly proved by inner induction on $d$ with
m_{d-1, 0} \le \frac{7}{16} m_{d, 0} + \frac{7}{16}
The lemma follows.
We now prove our invariant with
\[
\Delta = \frac{1}{32},~~~
\ve = \frac{1}{395}, \eqand
\hat{\ve} = \ve + \frac{300}{299} \ve^2 < \frac{1}{393},
\]
from L0Variant.
Lemma <ref>.
For each separator $S_{d, j}$ and index $l \le d$, there are $\Delta^{d-l + 1} m_{d, j}$ or less $l$-strangers output from $S_{d, j}$, $i.e.$, in $\bigcup_{i=1}^4 A_i$ of $S_{d, j}$.
We show the lemma by induction on $j$, $l$ and $d$. We first show the general induction step for $j \ge 1$ and $l \ge 1$. Assume true for $j-1$ and all $l$, and for $j$ and $1, 2, \ldots, l-1$. Prove true for $j$ and $l$ by inner induction on $d$.
The basis of the inner induction occurs when $d=l$. Consider each $S_{d, j}$. Let $\alpha$ be the total number of $l'$-strangers ($0 \le l' \le l$) sent to all the proper descendants $S_{d', j}$ of $S_{d, j}$ from the $j-1^{st}$ layer. By induction hypothesis,
α≤∑_d+1 ≤d' ≤d_max
0 ≤l' ≤l=d 2^(d'+1)-d Δ^(d'+1)-l'+1 m^*_d'+1, j-1,
where $m^*_{d'+1, j-1}$ denotes the maximum value of $m_{d+1, j-1}$ under consideration ($i.e.$, of all $S_{d'+1, j-1}$ sending items to $S_{d', j}$).
Defining $m^*_{d', j}$ similarly, we have $m^*_{d', j} \ge \frac{1}{8} m^*_{d'+1, j-1} - \frac{1}{16}$ from OutputSizes, and $m^*_{d', j} \le m_{d, j} + 14(d' - d)$ by InputSize. So
m^*_d'+1, j-1 ≤8 m_d, j + 14(d' - d) + 1/16 ≤141 m_d, j ·1.35^d'-d:
If $m_{d, j} \ge 1$, the right inequality is implied by $141 m \cdot 1.35^x \ge 8 (m+14x+1/16)$ for every $m \ge 1$ and $x \ge 0$. Otherwise $m_{d, j} =0$ so $S_{d, j}$ outputs no $l$-strangers.
By eq00Invariant2 and eq01Invariant2,
α ≤ ∑_d+1 ≤d' ≤d_max
0 ≤l' ≤l=d 2.7^d'-d
Δ^d'-l' ·282 Δ^2 m_d, j
2.7 Δ/1 - 2.7 Δ
·282 Δ^2/1- Δ m_d, j
< 0.839 Δm_d, j,
holding true for $\Delta=1/32$.
Let $\beta$ be the number of $l''$-strangers ($0 \le l'' \le l-1$) input to $S_{l, j}$. We claim that
β≤1 + 4/1-4 Δ^2/1- Δ m_d, j<0.0326 Δm_d, j.
By induction hypothesis, there are
\sum_{l''=0}^{l-1} \Delta^{d-l''+1} m_{d, j} < \frac{\Delta^2}{1- \Delta} m_{d, j}
or less $l''$-strangers output from $S_{d, j}$.
Denote by $I$ the array correct for $S_{d, j}$, by $p_1$ the number of $l''$-strangers in $A_1 \cup A_4$ exceeding the maximum in $I$, and by $p_2$ the number of those less than the minimum in $I$. By ii) and iii) of L0Modified with $p_i<\frac{\Delta^2}{1- \Delta} m_{l, j} < \Delta m_{l, j}$, the total number of $l''$-strangers input to $S_{d, j}$ is no more than
(p_1 + p_2)\lp 1 + \frac{\ve}{1-4 \ve} \rp <
\lp 1 + \frac{4\ve}{1-4 \ve} \rp \frac{\Delta^2}{1- \Delta} m_{d, j},
proving eq1Invariant2.
Note that this counts all the $l''$-strangers satisfying I) or II) in AKSImprovement.
To complete the basis $d=l$, we will apply L0Modified i) to $I_1 \cup I_2$ of $S_{d, j}$. Set $g$ as the $|I_1|^{th}$ smallest element of $I$. Let $q_1$ and $q_2$ be the number of elements in $I_1$ greater than $g$ and that of elements in $I_2$ not exceeding $g$, resp. Then
| q_1 - q_2 | ≤α+ β:
All that may contribute to $|q_1 - q_2|$ are $l''$-strangers input to the subtree rooted at $S_{d, j}$ ($i.e.$, the elements not belonging to $I$ as $l''<l =d$), and $l$-strangers sent to the proper descendants of $S_{d, j}$ from the $j-1^{st}$ layer. Their total number is bounded by $\alpha + \beta$.
Hence, by eq0Invariant2, eq1Invariant2, eq2Invariant2 and L0Modified i), there are
\[
|q_1 - q_2| + \hat{\ve} m_{l, j} \le \alpha+ \beta + \hat{\ve} m_{l, j} \le \Delta m_{l, d}
\]
or less $l$-strangers output from $S_{d, j}$.
This proves
the basis $d=l$ of the inner induction.
We show the induction step.
Assume true for $d-1$ and prove true for $d$. Consider each $S_{d, j}$ and its parent $S_{d-1, j}$. There are
\Delta^{(d-1)-l+1} m_{d-1, j}
\le \Delta m_{d-1, j}
or less $l$-strangers in $A_1 \cup A_4$ of $S_{d-1, j}$ by induction hypothesis. By L0Modified ii) and iii), and also OutputSizes, the number of $l$-strangers sent from $S_{d-1, j}$ to $S_{d, j}$ is upper-bounded by
\[
\frac{4\ve}{1-4 \ve} \Delta^{d-l} m_{d-1, j}
\le \frac{4 \ve}{1-4\ve}\Delta^{d-l} \cdot \frac{16}{7} \lp m_{d, j} + \frac{21}{16} \rp
\le
\frac{9.152 \ve}{1-4 \ve}\Delta^{d-l} m_{d, j}
\]
If $\frac{4 \ve}{1-4 \ve} \Delta m_{d-1, j}<1$, its left hand size is zero. So $m_{d-1, j} \ge \frac{1-4 \ve}{4 \Delta \ve} \ge 3128$, meaning $m_{d, j} \ge \frac{7}{16}m_{d-1, j} - \frac{21}{16}>1367$. The second inequality above is derived from $\frac{16}{7} \lp m_{d, j} + \frac{21}{16} \rp<\frac{16}{7} \lp 1+ \frac{1}{1367} \cdot \frac{21}{16} \rp m_{d, j}$.
Also there are no more than
\[
\Delta^{(d+1)-l + 1} \lp m^1_{d+1, j-1} + m^2_{d+1, j-1} \rp
\le 8 \Delta^{d-l+2} \lp m_{d, j} + \frac{1}{8} \rp
\le 8.008 \Delta^{d-l+2} m_{d, j}
\]
$l$-strangers sent from the $j-1^{st}$ layer. Here $m^i_{d+1, j-1}$ ($i=1, 2$) denotes the input sizes of the two $S_{d+1, j-1}$ sending items to $S_{d, j}$. Assume
$\Delta^2 \lp m^1_{d+1, j-1} + m^2_{d+1, j-1} \rp \ge 1$ to derive the second inequality similarly[
$m^1_{d+1, j-1} + m^2_{d+1, j-1} \ge \Delta^{-2} \ge 1024$ so $m_{d, j} \ge \sum_{i=1}^2 m^2_{d+1, j-1} > \frac{1024}{8}- \frac{1}{8}>127$ by OutputSizes. Then $8 \lp m_{d, j} + \frac{1}{8} \rp < 8 \lp 1 + \frac{1}{8 \cdot 127} \rp m_{d, j} <8.008$.
As a result, the number of $l$-strangers output from $S_{d, j}$ is upper-bounded by
\[
\frac{9.152 \ve}{1-4 \ve}\Delta^{d-l} m_{d, j}
+8.008 \Delta^{d-l+2} m_{d, j}
\le \Delta^{d-l + 1} m_{d, j},
\]
true with $\ve = 1/395$ and $\Delta= 1/32$.
This completes the induction step for $d$, as well as that for $j$ and $l$.
It remains to show the claim for the base cases. The arguments are all similar to the general induction step. One can prove it with minor changes noting that:
* For $j \ge 1$ and $l=0$, the differences from the above are that $S_{0, j}$ receives items from $S_{0, j-1}$ instead of $S_{l-1, j}$, and that there are no $l'$-strangers such that $l'<l$.
* When $j=l=d=0$, the claim holds by L0Modified i), since there are the same number of $0$-strangers in both $I_1$ and $I_2$ of $S_{0, 0}$. This requires $\hat{\ve} \le \Delta$.
* For the other cases of $j=0$, the difference from the above is that there are no nodes in the $j-1^{st}$ layer sending items to $S_{d, 0}$. This requires $\frac{4 \ve}{1- 4 \ve} \le \Delta$.
The lemma follows.
KLW07 Kuhn, F., Locher, T., Wattenhofer, R.: Tight bounds for distributed selection. In: Proc. of the 19th Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 145-153. ACM Press (2007)
KDG03 Kempe, D., Dobra, A., Gehrke, J.: Gossip-based computation of aggregate information. In: Proc. of the 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 482–491. IEEE Press (2003)
SSS92 Santoro, N., Sidney, J.B., Sidney S.J.: A distributed selection algorithm and its expected communication complexity. Theoretical Computer Science, vol. 100, pp. 185–204. Elsevier (1992)
F83 Frederickson, G.N.: Tradeoffs for Selection in distributed networks. In: Proc. of the 2nd Annual ACM
Symposium on Principles of Distributed Computing
(PODC), pp. 154–160. ACM Press (1983)
Textbook2 Pardalos, P., Rajasekaran, S., Reif, J., Rolim, J.: Handbook on randomized computing. Kluwer Academic Publishers (2001)
CHR93 Chaudhuri, S., Hagerup, T., Raman, R.: Approximate and exact deterministic parallel selection. LNCS, vol. 711, pp. 352–361. Springer (1993)
AKSS89 Ajtai, M., Komlós, J., Steiger, W.L., Szemerédi, E.: Optimal parallel selection has complexity $O \lp \log \log n \rp$. Journal of Computer and System Sciences, vol. 38, pp. 125–133. Academic Press (1989)
FJ82 Frederickson, G.N., Johnson, D.B.: The complexity of selection and ranking in X + Y and matrices with sorted columns, J. Comput. System Sci., vol. 24 (2), pp. 197–208 Elsevier (1982).
SS89 Santoro, N., Suen, E.: Reduction techniques for selection in distributed files. IEEE Transactions on Computers, vol. 38, pp. 891–896. IEEE Press (1989)
AKS Ajtai, M., Komlós, J., Szemerédi, E.: Sorting in $c \log n$ parallel steps. Combinatorica, vol. 3, pp. 1–19. Springer (1983)
Paterson Paterson, M.: Improved sorting networks with O(log N) depth. Algorithimica, vol. 5, pp. 65–92. Springer (1990)
Seiferas Seiferas, J. L.: Sorting networks of logarithmic depth, further simplified. Algorithmica, vol. 53, pp. 374–384. Springer (2007)
Textbook Baddar, S. W. A., Batcher, K. E.: Designing sorting networks. Springer (2011)
Goodrich Goodrich, M.T.: Zig-zag sort: a simple deterministic data-oblivious sorting algorithm running in O(n log n) time. In: Proc. of Symposium on Theory of Computing (STOC), pp. 684–693. ACM Press (2014)
ConcreteMath Graham, R. L., Knuth, D. E., Patashnik, O.: Concrete mathematics. Addison-Wesley (1994)
papa Papadimitriou, C. H.: Computational Complexity. Addison-Wesley (1994)
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1511.01040
|
Efficient Photo-heating Algorithms]
Efficient Photo-heating Algorithms in Time-dependent Photo-ionization Simulations
Lee et al.]Kai-Yan Leee–mail: [email protected],
Garrelt Mellema,
and Peter Lundqvist
Department of Astronomy & Oskar Klein Centre, AlbaNova, Stockholm University, SE-106 91
Stockholm, Sweden
2015 000 0
|
1511.00865
|
§ INTRODUCTION
Proposed in the mid 80s, the Georgi-Machacek (GM) model <cit.> augments the SM Higgs sector by adding a complex triplet of hypercharge $Y = 1$ and a real triplet of $Y = 0$ under the SM $SU(2)_L \times U(1)_Y$ gauge symmetry. It has many intriguing properties. First, the triplet fields can develop vacuum expectation values (VEV's), as automatically induced by SM electroweak symmetry breaking (EWSB) through a trilinear interaction term with the SM Higgs doublet field. With the triplet VEV's, it is possible to give Majorana mass to the left-handed neutrinos through the so-called type-II seesaw mechanism.
Secondly, the model predicts the existence of several Higgs multiplets under the custodial symmetry: two singlets, one triplet, and one quintet <cit.>. In particular, the quintet contains a doubly-charged Higgs boson that can mediate lepton number-violating or even lepton flavor-violating processes. Recently, there have been many phenomenological studies about searching for the exotic Higgs bosons at colliders <cit.> and their effects in enhancing the strength of phase transition in electroweak baryogenesis <cit.>.
Thirdly, with the assumption of vacuum alignment between the complex and real triplet VEV's in the tree potential, the model preserves the electroweak $\rho$ parameter at unity even with a VEV as large as up to a few tens of GeV. [It is noted that divergences for the $\rho$ parameter and certain mixings among the Higgs bosons have been studied at loop levels and found to have a similar naturalness issue as the SM Higgs mass <cit.>.] The possibility of a large triplet VEV leads to enhanced couplings between the exotic Higgs bosons and the weak gauge bosons and thus a plethora of interesting collider phenomena. For example, without a significant mass hierarchy among different Higgs multiplets, the doubly-charged Higgs boson decays dominantly into a pair of like-sign $W$ bosons rather than like-sign leptons.
Yet another feature impossible for models extended with only $SU(2)_L$ singlet and/or doublet fields is that the coupling between the SM-like Higgs boson and the weak gauge bosons can be stronger than in the SM as a result of mixing between the SM doublet and the triplet fields <cit.>. This, for example, can be tested through a precise determination of the SM-like Higgs signal strengths at the LHC.
Finally, the model predicts the existence of a singly charged Higgs boson coupling with the $W$ and $Z$ bosons at tree level through mixing, while such a vertex is induced only at loop levels in singlet- and/or doublet-extended models <cit.>, such as the two-Higgs doublet model <cit.>.
The structure of this paper is as follows. Section <ref> briefly reviews the GM model, paying particular attention to the mass spectrum, some tree-level theoretical constraints, and indirect experimental constraints. In section <ref>, we perform a comprehensive scan of the mass spectrum allowed by the above-mentioned constraints. We here incorporate the most general case in which there can be a mass hierarchy among the different Higgs multiplets. The Higgs masses, signal strengths of the SM-like Higgs boson decays into $\gamma\gamma$ and $\gamma Z$, and branching ratios of (cascade) decays of the exotic Higgs bosons are plotted.
In section <ref>, we concentrate on one of the signals of the GM model, namely, the vector boson fusion production of the doubly-charged Higgs boson that decays into final states with a same-sign lepton pair at the LHC, for which we evaluate the production cross section and the acceptance times efficiency with a certain set of selection criteria. Combining them with the branching ratios of the exotic Higgs boson decays evaluated in the previous section, and comparing them with SM background estimates, we obtain the prospect for the discovery of the GM model through this channel for most general mass spectra.
We also comment on the phenomenology at a 100-TeV hadron collider. Finally, section <ref> summarizes our findings in this work.
§ REVIEW ON THE GEORGI-MACHACEK MODEL
In this section, we review the basics of the Higgs sector in the GM model, theoretical constraints of the vacuum stability and perturbative unitarity, both at tree level, and indirect experimental constraints, such as the oblique corrections, the $Zb\bar b$ vertex, and 125-GeV Higgs signal strengths.
§.§ Higgs Sector and Mass Spectrum
The EWSB sector of the GM model <cit.> comprises one isospin doublet scalar field with hypercharge $Y=1/2$, one isospin triplet scalar field with $Y=1$, and one isospin triplet scalar field with $Y=0$ [Here the normalization for the hypercharge quantum number $Y$ is such that the electric charge $Q = I_3 + Y$, where $I_3$ denotes the third component of the weak isospin number.].
These fields are denoted respectively by [Here we use the convention that $\chi^{--}=(\chi^{++})^*$, $\chi^{-}=(\chi^{+})^*$, $\xi^-= (\xi^+)^*$ and $\phi^-= (\phi^+)^*$.]
\begin{align}
\begin{split}
\phi=
\begin{pmatrix}
\phi^+ \\ \phi^0
\end{pmatrix}
\chi=
\begin{pmatrix}
\chi^{++} \\ \chi^+ \\ \chi^0
\end{pmatrix}
\xi=
\begin{pmatrix}
\xi^+ \\ \xi^0 \\ -(\xi^+)^*
\end{pmatrix}
\\
\mbox{with }
\phi^0 = \frac{1}{\sqrt{2}}(h_{\phi}+ia_{\phi}) ~,~
\chi^0 = \frac{1}{\sqrt{2}}(h_{\chi}+ia_{\chi}) ~,~
\xi^0 = h_{\xi} ~,
\end{split}
\label{eq:component_fields}
\end{align}
where the neutral components have been further decomposed into CP-even ones ($h_{\phi}, \, h_{\chi}, \, h_{\xi}$) and CP-odd ones ($a_{\phi}, \, a_{\chi}$).
The global SU(2)$_L \times$SU(2)$_R$ symmetry is imposed on the Higgs potential at tree level, which is explicitly broken by the Yukawa and the hypercharge gauge interactions.
To make this symmetry manifest, it is convenient to introduce the SU(2)$_L \times$SU(2)$_R$-covariant forms of the fields:
\begin{align}
\begin{split}
\Phi \ &\equiv \ \left( \epsilon_2 \phi^*, \, \phi \right)
\ = \ \left(
\begin{array}{cc}
(\phi^0)^* & \phi^+ \\
-(\phi^+)^* & \phi^0
\end{array}
\right) ~,~~ \mbox{with }
\epsilon_2 = \left( \begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}
\right) ~,
\\
\Delta \ &\equiv \ \left( \epsilon_3 \chi^*, \, \xi, \, \chi \right)
\ = \ \left(
\begin{array}{ccc}
(\chi^0)^* & \xi^+ & \chi^{++} \\
-(\chi^+)^* & \xi^0 & \chi^+ \\
(\chi^{++})^* & -(\xi^+)^* & \chi^0
\end{array}
\right) ~,~~
\mbox{with }
\epsilon_3 = \left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & -1 & 0 \\
1 & 0 & 0
\end{array}
\right) ~.
\end{split}
\end{align}
Under an SU(2)$_L$ $\times$ SU(2)$_R$ transformation, $\Phi \rightarrow U_{2L} \Phi U^{\dagger}_{2R}$ and $\Delta \rightarrow U_{3L} \Delta U^{\dagger}_{3R}$, where $U_{2L}$ $(U_{2R})$ is the two-dimensional representation of the SU(2)$_L$ (SU(2)$_R$) group component and $U_{3L}$ $(U_{3R})$ is the corresponding three-dimensional one.
Using $\Phi$ and $\Delta$, the Lagrangian of the EWSB sector is succinctly given by
\begin{align}
{\cal L} \ =& \
\frac{1}{2} {\rm tr}[ (D^{\mu}\Phi)^{\dagger} D_{\mu}\Phi ]
\ + \
\frac{1}{2} {\rm tr}[ (D^{\mu}\Delta)^{\dagger} D_{\mu}\Delta ]
\ - \ V(\Phi, \, \Delta) ~,
\end{align}
where $D_{\mu}$ denotes the covariant derivative for $\Phi$ or $\Delta$.
The potential term, $V(\Phi, \, \Delta)$, is given by
\begin{align}
\begin{split}
V(\Phi, \, \Delta) \ =& \ \frac{1}{2} m_1^2 \, {\rm tr}[ \Phi^{\dagger} \Phi ] +
\frac{1}{2} m_2^2 \, {\rm tr}[ \Delta^{\dagger} \Delta ]
+ \lambda_1 \left( {\rm tr}[ \Phi^{\dagger} \Phi ] \right)^2
+ \lambda_2 \left( {\rm tr}[ \Delta^{\dagger} \Delta ] \right)^2
\\
+ \lambda_3 {\rm tr}\left[ \left( \Delta^{\dagger} \Delta \right)^2 \right]
+ \lambda_4 {\rm tr}[ \Phi^{\dagger} \Phi ] {\rm tr}[ \Delta^{\dagger} \Delta ]
+ \lambda_5 {\rm tr}\left[ \Phi^{\dagger} \frac{\sigma^a}{2} \Phi \frac{\sigma^b}{2} \right]
{\rm tr}[ \Delta^{\dagger} T^a \Delta T^b]
\\
&+ \mu_1 {\rm tr}\left[ \Phi^{\dagger} \frac{\sigma^a}{2} \Phi \frac{\sigma^b}{2} \right]
(P^{\dagger} \Delta P)_{ab}
+ \mu_2 {\rm tr}[ \Delta^{\dagger} T^a \Delta T^b]
(P^{\dagger} \Delta P)_{ab} ~,
\end{split}
\label{potential}
\end{align}
where summations over $a,b = 1,2,3$ are understood, $\sigma$'s and $T's$ are the $2\times2$ (Pauli matrices) and $3\times3$ matrix representations of the SU(2) generators, respectively, and
\begin{align}
P &= \frac{1}{\sqrt{2}} \left( \begin{array}{ccc}
-1 & i & 0 \\
0 & 0 & \sqrt{2} \\
1 & i & 0
\end{array}
\right) \nonumber
\end{align}
diagonalizes the adjoint representation of the SU(2) generator. It is noted that all parameters in the Higgs potential are real and do not allow CP violation.
The EWSB vacuum is derived from the tadpole conditions:
\begin{align}
\frac{\partial V(\Phi,\Delta)}{\partial h_{\phi}} \ &= \ \frac{\partial V(\Phi,\Delta)}{\partial h_{\chi}}
\ = \ \frac{\partial V(\Phi,\Delta)}{\partial h_{\xi}} \ = \ 0 ~,
\label{vacuum}
\end{align}
where the fields other than $h_{\phi},h_{\chi}$, and $h_{\xi}$ take zero VEV's.
In Eq. (<ref>), we select the solution satisfying the relation $\langle h_{\chi} \rangle=\sqrt{2} \langle h_{\xi} \rangle$, by which the EWSB vacuum maintains the diagonal SU(2)$_{L+R}$ or SU(2)$_V$ symmetry.
We denote the VEV's of $h_{\phi}, h_{\chi}, h_{\xi}$ by
$\langle h_{\phi} \rangle = v_{\Phi}, \ \langle h_{\chi} \rangle = \sqrt{2}v_{\Delta}, \ \langle h_{\xi} \rangle = v_{\Delta}$, respectively, which are related to the SM Higgs boson VEV, $v \simeq 246$ GeV, by
$\vert\langle h_{\phi} \rangle\vert^2 + 2\vert\langle h_{\chi} \rangle\vert^2 + 4\vert\langle h_{\xi} \rangle\vert^2 = v_{\Phi}^2+8v_{\Delta}^2 = v^2$.
In a fashion similar to the two-Higgs doublet model, we define $\tan \beta$ as the VEV ratio, $\tan \beta \equiv v_{\Phi}/ \left( 2\sqrt{2}v_{\Delta} \right)$.
Assuming $v_{\Phi},v_{\Delta} \neq 0$ [As alluded to earlier, the triplet VEV can be automatically induced by the $\mu_1$ term once the doublet gets a VEV to break the electroweak symmetry.], we can rewrite $m_1^2, m_2^2$ in terms of $v_{\Phi},v_{\Delta}$ as
\begin{align}
m_1^2 \ &= \ -4\lambda_1 v_{\Phi}^2 - 6\lambda_4 v_{\Delta}^2 - 3\lambda_5 v_{\Delta}^2
- \frac{3}{2} \mu_1 v_{\Delta} ~, \nonumber \\
m_2^2 \ &= \ -12\lambda_2 v_{\Delta}^2 - 4\lambda_3 v_{\Delta}^2 - 2\lambda_4 v_{\Phi}^2
- \lambda_5 v_{\Phi}^2 - \mu_1 \frac{v_{\Phi}^2}{4v_{\Delta}} - 6\mu_2 v_{\Delta} ~.
\label{m1m2}
\end{align}
For later convenience, we define
\begin{align}
M_1^2 \ &\equiv \ -\frac{v}{\sqrt{2} \cos\beta} \mu_1 ~,
\ \ \ M_2^2 \ \equiv \ -3\sqrt{2} \cos\beta \, v \mu_2 ~.
\end{align}
It is noted that $|\mu_1|$ or $M_1^2 \to \infty$ corresponds to the decoupling limit of the model <cit.>. On the other hand, no decoupling limit exists once one imposes the $Z_2$ symmetry $\Delta \to - \Delta$. Also, this symmetry does not allow the desired interaction between left-handed neutrinos and the triplet Higgs field for neutrino mass generation.
Because of the SU(2)$_V$ symmetry of the (tree-level) EWSB vacuum, the physical mass eigenstates form one 5-plet, one 3-plet and two singlets, where the components in each of the multiplets are degenerate in mass at the tree level. Mass splitting within each multiplet due to custodial symmetry breaking is expected to be at the ${\cal O}(100)$ MeV level.
We denote the 5-plet, 3-plet and two singlets by $H_5 = (H_5^{++},H_5^{+},H_5^{0},H_5^{-},H_5^{--})^T$, $H_3=(H_3^{+},H_3^{0},H_3^{-})^T$, $H_1$ and $h$, respectively, with $h$ identified as the 125-GeV SM-like Higgs boson observed at the LHC. In terms of the fields $\phi$, $\xi$ and $\chi$ introduced in Eq. (<ref>), the physical states are expressed as follows:
\begin{align}
\begin{split}
H_5^{++} = \chi^{++} ~, \ \ \ H_5^+ = \frac{1}{\sqrt{2}} \left( \chi^+ - \xi^+ \right) ~,
\ \ \ H_5^0 = \sqrt{\frac{1}{3}} h_{\chi} - \sqrt{\frac{2}{3}} h_{\xi} ~,
\\
H_3^+ = -\cos \beta \, \phi^+ + \sin \beta \, \frac{1}{\sqrt{2}} \left( \chi^+ + \xi^+ \right) ~,
\ \ \ H_3^0 = -\cos \beta \, a_{\phi} + \sin \beta \, a_{\chi} ~,
\\
h = \cos \alpha \, h_{\phi} - \frac{\sin \alpha}{\sqrt3} \, \left( \sqrt{2} h_{\chi} + h_{\xi} \right) ~,
\ \ \ H_1 = \sin \alpha \, h_{\phi} + \frac{\cos \alpha}{\sqrt3} \, \left( \sqrt{2} h_{\chi} + h_{\xi} \right) ~,
\end{split}
\end{align}
where the mixing angle $\alpha$ between the singlets takes a value in the range $-\pi/2 \le \alpha \le \pi/2$, and is given through
\begin{align}
\tan 2\alpha &= \frac{2 (M^2)_{12}}{(M^2)_{22} - (M^2)_{11}} ~,
\end{align}
\begin{align}
\begin{split}
(M^2)_{11} &= 8 \lambda_1 v^2 \sin^2 \beta ~,
\\
(M^2)_{22} &= (3\lambda_2+\lambda_3) v^2 \cos^2 \beta + M_1^2 \sin^2 \beta - \frac{1}{2} M_2^2 ~,
\\
(M^2)_{12} &= \sqrt{\frac{3}{2}} \sin \beta \cos \beta \, \left[ (2\lambda_4+\lambda_5) v^2 - M_1^2 \right] ~.
\end{split}
\end{align}
The mass eigenvalues are given by
\begin{align}
\begin{split}
m_{H_5}^2 &\equiv m_{H_5^{++}}^2=m_{H_5^+}^2=m_{H_5^0}^2
= (M_1^2-\frac{3}{2}\lambda_5v^2)\sin^2\beta+\lambda_3v^2\cos^2\beta+M_2^2 ~,
\\
m_{H_3}^2 & \equiv m_{H_3^+}^2 = m_{H_3^0}^2=M_1^2-\frac{1}{2}\lambda_5v^2 ~,
\\
m_{H_1^0}^2 &= M_{11}^2\sin^2 \alpha+M_{22}^2\cos^2 \alpha+2M_{12}^2\sin \alpha\cos\alpha ~,
\\
m_h^2 &=M_{11}^2\cos^2\alpha+M_{22}^2\sin^2\alpha-2M_{12}^2\sin \alpha\cos\alpha ~.
\end{split}
\label{massformulas}
\end{align}
It is noted that these masses are generally different, and the mass differences are of ${\cal O}(100)$ GeV if one naïvely takes $\mu_{1,2} \sim {\cal O}(100)$ GeV and the quartic couplings $\lambda$'s $\sim {\cal O}(1)$. In our numerical analysis, we will assume a general mass hierarchy among these mass eigenvalues (but neglecting the smaller mass splitting within each multiplet), subject to the constraints to be discussed below, and analyze the prospects of detecting the exotic Higgs bosons at the 14-TeV LHC and future 100-TeV hadron collider.
§.§ Theoretical Constraints
We will take into account two theoretical constraints on the parameters of the GM Higgs potential. One comes from the stability of the electroweak vacuum, and the other from the unitarity of the perturbation theory. We satisfy ourselves with these constraints at the tree level for the consistency with the masses given above.
When requiring the electroweak vacuum to be stable (i.e., bounded from below), one obtains the following constraints for the quartic couplings <cit.>:
\begin{align}
\begin{split}
& \lambda_1 > 0 ~, \ \lambda_2+\lambda_3 > 0 ~, \ \lambda_2 + \frac{1}{2}\lambda_3 > 0 ~,
\ -\vert \lambda_4 \vert + 2\sqrt{\lambda_1(\lambda_2+\lambda_3)} > 0 ~,
\\
& \lambda_4 - \frac{1}{4}\vert \lambda_5 \vert + \sqrt{2\lambda_1(2\lambda_2+\lambda_3)} > 0 ~.
\end{split}
\label{stability}
\end{align}
From the perturbative unitarity, we have another set of constraints <cit.>:
[See also Ref. <cit.> for constraints on additional Higgs bosons based on Higgs data and unitarity.]
\begin{align}
\begin{split}
\left\vert \, 6 \lambda_1 + 7 \lambda_3 + 11\lambda_2\, \right\vert +\sqrt{(6\lambda_1-7\lambda_3-11\lambda_2)^2+36\lambda_4^2}< 4\pi ~,
\\
\left\vert \, \lambda_4 - \lambda_5 \, \right\vert < 2\pi ~,
~~~ \left\vert \, 2 \lambda_3 + \lambda_2 \, \right\vert < \pi ~,
\\
\left\vert \, 2 \lambda_1 - \lambda_3 + 2\lambda_2\, \right\vert
+ \sqrt{(2\lambda_1+\lambda_3-2\lambda_2)^2+\lambda_5^2} < 4\pi ~.
\end{split}
\label{unitarity}
\end{align}
Since one can trade the quartic couplings and $M_{1,2}^2$ with the four physical masses, $\alpha$, $\beta$ and $v$, the above two sets of constraints can be turned into constraints on the unknown masses and mixing angles.
§.§ Experimental Constraints
We now turn to the discussion of constraints on the GM model derived from measurements of SM quantities in collider experiments. These include the electroweak precision tests, the determination of the $Z b \bar{b}$ coupling, and the measurement of the Higgs boson signal strengths.
The GM model is subject to constraints from the $S$ and $U$ parameters <cit.> of electroweak precision tests.
The $\rho$ parameter can take any value in the GM model if we add a term that explicitly breaks the SU(2)$_V$ symmetry, and hence the $T$ parameter does not impose any restriction on the model.
Since the absolute value of the $U$ parameter is found to be below $0.01$ in all the mass spectra generated in the next section, we will only consider the constraint from the $S$ parameter by fixing the $U = 0$ and taking the $T$ parameter to be free.
The latest experimental data <cit.> report the following $1\sigma$ range for the $S$ parameter:
\begin{align}
S \vert_{U=0, \, T \, {\rm free}} \ &= \ 0.00 \pm 0.08 ~.
\label{Sparameter}
\end{align}
In the GM model, the 3-plet Higgs bosons couple with the SM quarks through mixing with the Higgs doublet, as explicitly given, for example, in Ref. <cit.>. Therefore, $H_3^+$ can give rise to significant radiative corrections to the $Z b \bar{b}$ coupling through a triangular one-loop diagram involving the top quark and $H_3^+$, as the $\bar{t} b H_3^+$ coupling has an overall factor proportional to $v_\Delta^2$ and a term proportional to the top quark Yukawa coupling.
The data on the $Z b \bar{b}$ coupling therefore impose a constraint on the triplet VEV $v_{\Delta}$ and the SU(2)$_V$ triplet mass $m_{H_3}$, which has been evaluated in Ref. <cit.>.
It was found that mass spectra with $v_{\Delta} \lesssim 50$ GeV and $m_{H_3}$ above 100 GeV were consistent at $2\sigma$ level with the current data <cit.>.
The signal strengths of the SM-like Higgs boson production and decay in various channels have been measured in the LHC 7-TeV and 8-TeV runs by the ATLAS <cit.> and CMS <cit.> Collaborations, and provide significant constraints on the couplings of $h$ to SM particles in the GM model.
Here we consider the following six channels of the Higgs boson production and decay: the gluon-gluon fusion (GGF) production of $h$ decaying into $ZZ$, $WW$ and $\tau^+ \tau^-$, the vector boson fusion (VBF) production of $h$ decaying into $WW$ and $\tau^+ \tau^-$, and the vector boson associated (VBA) production of $h$ decaying into $b \bar{b}$.
The modification of the signal strengths in these channels depends only on the triplet VEV, $v_{\Delta}$ (or $\beta$), and the mixing angle of the SU(2)$_V$ singlets, $\alpha$. Hence we can directly constrain $v_{\Delta}$ and $\alpha$ from the data, without specifying other parameters including the mass spectrum. Note that we avoid using the diphoton signal strength because it is a loop-mediated process that has additional dependences on the masses of heavy charged Higgs bosons and their couplings with $h$. Although such uncertainties in the diphoton channel will also enter the signal strengths of the above-mentioned six tree-level decay channels through modifications in the branching ratios, the effects are expected to be negligible because of the relatively small $h \to \gamma\gamma$ decay rate. Throughout this paper, we employ the narrow width approximation when calculating the signal strengths.
$1\sigma$ (solid) and $2\sigma$ (dashed) contours on the $v_\Delta$-$\alpha$ plane through a $\chi^2$ fit to the current data of six Higgs signal strengths detailed in the main text. The red cross marks the point with the $\chi^2$ minimum.
We perform a $\chi^2$ fit on $v_{\Delta}$-$\alpha$ plane by using the signal strength data of the above-mentioned six channels obtained in the LHC 7-TeV and 8-TeV runs <cit.>. The $1\sigma$ and $2\sigma$ contours along with the best-fit point are displayed in Fig. <ref>. From the figure, we select the following twelve sets of parameters that are consistent with the data at the $2\sigma$ level: $(v_\Delta,\alpha) = (10, -30^\circ)$, $(10, -10^\circ)$, $(10, +10^\circ)$, $(20, -30^\circ)$, $(20, -10^\circ)$, $(20, +10^\circ)$, $(30, -30^\circ)$, $(30, -10^\circ)$, $(30, +10^\circ)$, $(40, -10^\circ)$, $(50, -10^\circ)$, and $(1, 0^\circ)$ (close to the decoupling limit), where the values of $v_\Delta$ are given in units of GeV. These parameter choices will be used in the next section for numerical studies.
$1\sigma$ (solid) and $2\sigma$ (dashed) contours on the $v_\Delta$-$\alpha$ plane through a $\chi^2$ fit to the SM signal strengths of the six channels with precisions expected to reach at 14-TeV LHC with 300 fb$^{-1}$ (left) and 3000 fb$^{-1}$ (right) of data.
As a reference, we also present in Fig. <ref> the future prospects for confining the $v_{\Delta}$-$\alpha$ parameter space as derived from the same six signal strength measurements at the 14 TeV LHC with the integrated luminosities of 300 fb$^{-1}$ (left plot) and 3000 fb$^{-1}$ (right plot), assuming that the central values of the signal strength data are all unity.
We here use the uncertainty estimates given in Ref. <cit.>. Although the constraint in the left plot of Fig. <ref> looks comparable to that in Fig. <ref>, such a comparison is meaningless because the former assumes the SM signal strengths. The constraint on $v_{\Delta}$, $\alpha$ does not improve significantly with 3000 fb$^{-1}$ of data compared to the case with 300 fb$^{-1}$ of data. This is because, for the $h\to WW$, $ZZ$ and $\tau\tau$ channels, theoretical systematic uncertainties and experimental systematic uncertainties give major contributions to the overall uncertainty for the 300 fb$^{-1}$ data. Hence, larger statistics does not lead to a significant reduction in uncertainties of Higgs boson signal strengths.
The signal strength of GGF production of the Higgs boson decaying into $\gamma \gamma$ has also been measured at the 7 TeV and 8 TeV LHC. However, as alluded to before, the branching ratio of $h \to \gamma \gamma$ can be altered by the loop diagrams involving the charged Higgs bosons $(H_5^{++}, \, H_5^+, \, H_3^+)$ and hence depends significantly on details of the mass spectrum and triple scalar couplings. Therefore, we will discuss the constraint from this channel after we perform a parameter scan for the Higgs mass spectrum in the next section.
We now comment on constraints from searches for an extra neutral Higgs boson through the $H_1\to\gamma\gamma$ process, as this mode yields the strongest bound. The ATLAS Collaboration has already given a bound on this process for the mass range of 65 GeV to 600 GeV <cit.>. However, we will not use this constraint in our analysis in the next section, because $BR(H_1\to\gamma\gamma)$ varies sensitively with the values of $M_1^2$ and $M_2^2$ while these parameters are taken to be free in our parameter scan.
§ SEARCH OF VIABLE EXOTIC HIGGS BOSON MASS SPECTRA AND DECAY BRANCHING RATIOS OF EXOTIC HIGGS BOSONS
We now conduct a comprehensive parameter scan for viable exotic Higgs boson mass spectra of the GM model by using the most general set of parameters. From randomly generated mass spectra, we select those that pass the theoretical constraints given in Eqs. (<ref>) and (<ref>).
Also, the mass spectra are required to satisfy at the $2\sigma$ level the experimental constraints derived from electroweak precision tests in Eq. (<ref>) and the $Z b \bar{b}$ coupling measurement. As discussed in the previous section, we adopt the twelve sets of $(v_\Delta,\alpha)$ selected based on Fig. <ref> for further numerical analyses.
We calculate the following quantities for each viable mass spectrum and plot them on a two-dimensional plane spanned by the 5-plet mass $m_{H_5}$ and the 3-plet mass $m_{H_3}$. In the case of decays, we only show the results for positively charged Higgs bosons while those for the conjugate particles should be obvious.
* The mass of the heavier SU(2)$_V$ singlet $H_1$, $m_{H_1}$, shown in Fig. <ref>.
* The signal strength of the GGF production of $h$ followed by a decay into $\gamma \gamma$,
\begin{align*}
\mu^{\text{GGF}}_{h\gamma\gamma} &= \frac{\sigma(g_{/p} g_{/p} \rightarrow h)_{{\rm GM}} BR(h \rightarrow \gamma \gamma)_{{\rm GM}}}{\sigma(g_{/p} g_{/p} \rightarrow h)_{{\rm SM}} BR(h \rightarrow \gamma \gamma)_{{\rm SM}}}~,
\end{align*}
shown in Fig. <ref>.
* The signal strength of the GGF production of $h$ followed by a decay into $\gamma Z$,
\begin{align*}
\mu^{\text{GGF}}_{h\gamma Z} &= \frac{\sigma(g_{/p} \, g_{/p} \rightarrow h)_{{\rm GM}} BR(h \rightarrow \gamma Z)_{{\rm GM}}}{\sigma(g_{/p} g_{/p} \rightarrow h)_{{\rm SM}} BR(h \rightarrow \gamma Z)_{{\rm SM}}}~,
\end{align*}
shown in Fig. <ref>.
* The total decay widths of $H_5^{++}$ and $H_3^+$ divided by their corresponding masses,
\begin{align*}
\frac{\Gamma_{H_5^{++}}}{m_{H_5}}~, \ \ \frac{\Gamma_{H_3^+}}{m_{H_3}}~,
\end{align*}
shown respectively in Figs. <ref> and <ref>.
* The branching ratio of the direct decay of $H_5^{++}$ into $W^+ W^+$ followed by the leptonic decay of each $W^+$ (summed over all flavors), including contributions from off-shell $W^+$,
\begin{align*}
BR(H_5^{++} \rightarrow W^+(\rightarrow \ell^+ \nu_{\ell}) \, W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}) )~,
\end{align*}
shown in Fig. <ref>.
* The branching ratio of the cascade decay of $H_5^{++}$ into $H_3^+ W^+$ followed by the $H_3^+$ decay into $h W^+$ where each $W^+$ decays leptonically,
including contributions from off-shell $W^+$,
\begin{align*}
BR(H_5^{++} \rightarrow H_3^+ W^+(\rightarrow \ell^+ \nu_{\ell}) ) BR(H_3^+ \rightarrow h W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}))~,
\end{align*}
shown in Fig. <ref>. The plot is restricted to the region with $m_{H_5} > m_{H_3}$ where this process is possible.
* The branching ratio of the cascade decay of $H_5^{++}$ into $H_3^+ W^+$ followed by the $H_3^+$ decay into $H_1 W^+$ where each $W^+$ decays leptonically, including contributions from off-shell $W^+$,
\begin{align*}
BR(H_5^{++} \rightarrow H_3^+ W^+(\rightarrow \ell^+ \nu_{\ell}) ) BR(H_3^+ \rightarrow H_1 W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}))~,
\end{align*}
shown in Fig. <ref>. The plot is restricted to the parameter points with $m_{H_5} > m_{H_3} > m_{H_1}$ where this process is possible.
* The branching ratio of the cascade decay of $H_3^+$ into $H_5^{++} W^-$ followed by the $H_5^{++}$ decay into $W^+ W^+$ with each $W^+$ further decaying leptonically, including contributions from off-shell $W^{\pm}$,
\begin{align*}
BR(H_3^+ \rightarrow H_5^{++} W^-) BR(H_5^{++} \rightarrow W^+ (\rightarrow \ell^+ \nu_{\ell})W^+(\rightarrow \ell^+ \nu_{\ell})) ~,
\end{align*}
shown in Fig. <ref>. The plot is restricted to the region with $m_{H_3} > m_{H_5}$ where this process is possible.
* The branching ratios of $H_3^+$ decaying into $hW^+$ and $H_1W^+$, including contributions from off-shell $W^+$,
\begin{align*}
BR(H_3^+\to hW^+)~, \ \ BR(H_3^+\to H_1W^+)~,
\end{align*}
shown respectively in Figs. <ref> and <ref>.
* The branching ratios of $H_1$ decaying into $hh$ and $W^+ W^-$, including contributions from off-shell $W^{\pm}$ in the latter case,
\begin{align*}
BR(H_1 \rightarrow hh)~, \ \ BR(H_1 \rightarrow W^+ W^-)~,
\end{align*}
shown respectively in Figs. <ref> and <ref>.
$m_{H_1}$ for various values of ($v_\Delta,\alpha$). The values of $m_{H_1}$ for those magenta points are either equal or smaller than $m_h$.
$\mu^{\text{GGF}}_{h\gamma\gamma}$ for various values of ($v_\Delta,\alpha$).
$\mu^{\text{GGF}}_{h\gamma Z}$ for various values of ($v_\Delta,\alpha$).
The total decay width of $H_5^{++}$ divided by $m_{H_5}$ for various values of ($v_\Delta,\alpha$).
The total decay width of $H_3^+$ divided by $m_{H_3}$ for various values of ($v_\Delta,\alpha$).
$BR(H_5^{++} \rightarrow W^+(\rightarrow \ell^+ \nu_{\ell}) \, W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}) )$ for various values of ($v_\Delta,\alpha$).
$BR(H_5^{++} \rightarrow H_3^+ W^+(\rightarrow \ell^+ \nu_{\ell}) ) BR(H_3^+ \rightarrow h W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}))$ for various values of ($v_\Delta,\alpha$).
$BR(H_5^{++} \rightarrow H_3^+ W^+(\rightarrow \ell^+ \nu_{\ell}) ) BR(H_3^+ \rightarrow H_1 W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}))$ for various values of ($v_\Delta,\alpha$).
$BR(H_3^+ \rightarrow H_5^{++} W^-) BR(H_5^{++} \rightarrow W^+ (\rightarrow \ell^+ \nu_{\ell})W^+(\rightarrow \ell^+ \nu_{\ell}))$ for various values of ($v_\Delta,\alpha$).
$BR(H_3^+\to hW^+)$ for various values of ($v_\Delta,\alpha$).
$BR(H_3^+\to H_1W^+)$ for various values of ($v_\Delta,\alpha$).
$BR(H_1\to hh)$ for various values of ($v_\Delta,\alpha$). We do not show the trivial plot for the case of $(v_\Delta,\alpha)=(1~\text{GeV},0^\circ)$.
$BR(H_1\to W^+W^-)$ for various values of ($v_\Delta,\alpha$).
One can use the plot of $\mu^{\text{GGF}}_{h\gamma\gamma}$ (i.e., Fig. <ref>) to compare the prediction of the GM model with the corresponding 7-TeV and 8-TeV LHC data, thereby studying which mass spectra are consistent with experiments.
The plot of $\mu^{GGF}_{h \gamma Z}$ (i.e., Fig. <ref>) enables one to estimate the possibility of indirect search of the GM model through the $h \rightarrow Z \gamma$ process.
We will make use of the plots of $H_5^{++}$ branching ratios (i.e., Figs. <ref>, <ref>, and <ref>) to examine prospects for discovering the GM model at the 14-TeV LHC in the next section.
The method of our parameter scan is explicitly described as follows.
The electroweak VEV and the mass of the SM-like Higgs boson are fixed at $v=246$ GeV and $m_h=125$ GeV, respectively.
We choose the following seven independent parameters to scan:
the VEV ratio $\tan\beta$,
the mixing angle $\alpha$,
the three mass eigenvalues $m_{H_5}, m_{H_3}, m_{H_1}$, and
the parameters $M_1^2$ and $M_2^2$.
Note that the signs of $M_1^2$ and $M_2^2$ could be either positive or negative.
Using the twelve sets of ($v_{\Delta}$, $\alpha$) selected within the $2\sigma$ bound in Fig. <ref>, we randomly generate the rest five parameters ($m_{H_5}$, $m_{H_3}$, $m_{H_1}$, $M_1^2$ and $M_2^2$), and check whether they satisfy the constraints in Eqs. (<ref>) and (<ref>) and the $2\sigma$ bound of Eq. (<ref>). Note that at this stage, we do not assume any mass hierarchy among the Higgs bosons. To be phenomenologically interesting at the LHC, the ranges of $m_{H_5}$ and $m_{H_3}$ are both fixed as 100 GeV $< m_{H_{3,5}} <$ 1 TeV.
The ranges of $m_{H_1}$, $M_1^2$ and $M_2^2$ are determined according to the generated values of $m_{H_5}, m_{H_3}$, by taking advantage of the following inequalities that are deduced from the mass formulas in Eq. (<ref>) and the theoretical constraints in Eqs. (<ref>) and (<ref>):
\begin{align}
& \frac{2}{v^2} | M_1^2 - m_{H_3}^2 | < \frac{8(\sqrt{3}+1)\pi}{3} ~,
\label{M1} \\
& 0 < \frac{1}{3 \cos^2 \beta \, v^2} \left\{ m_{H_5}^2 - 3\sin^2 \beta \, m_{H_3}^2 + 2(m_{H_1}^2 \cos^2\alpha + m_h^2 \sin^2\alpha) \right\} < \pi ~,
\label{mH1} \\
& 0 < \frac{1}{6 \cos^2 \beta \, v^2} \left\{ 4 m_{H_5}^2 - 12 \sin^2 \beta \, m_{H_3}^2 + 2(m_{H_1}^2 \cos^2\alpha + m_h^2 \sin^2\alpha) \right.
\nonumber \\
& \qquad\qquad\qquad\qquad\qquad \left. + 6 \sin^2 \beta \, M_1^2 -3 M_2^2 \right\} < \frac{4\pi}{3} ~.
\label{M2}
\end{align}
We generate 8000 mass spectra for each set of $(v_\Delta,\alpha)$ and plot the results in Fig. <ref>, with different colors representing $m_{H_1}$ falling in different mass ranges. The magenta colored points are those with $m_{H_1} \le m_h$. It should be noted that for a given point $(m_{H_5},m_{H_3})$ in each scatter plot, the value of $m_{H_1}$ actually varies with $M_1^2$ and $M_2^2$ over a small range. It is seen that the parameter spaces for $\alpha = 10^\circ$ or the close-to-decoupling limit $(v_\Delta,\alpha) = (1~\mbox{GeV},0^\circ)$ (plots on the right hand side of the figure) are relatively limited, with $m_{H_5} \lesssim 600$ GeV, $m_{H_3} \lesssim 350$ GeV and $m_{H_1} \lesssim 300$ GeV. In a certain region of $(v_\Delta,\alpha)$ (upper left plots), some or all the exotic Higgs boson masses can be in the TeV regime. These spectra serve as the basis of Figs. <ref> to <ref>.
There is an upper bound on $m_{H_3}$ when $\alpha > 0$, as one can readily observe in the plots with $\alpha=10^\circ$ in Figs. <ref> to <ref>. The origin of this bound is understood as follows.
The combination of the couplings $4\lambda_4 + \lambda_5$ can be expressed, with the help of Eq. (<ref>), as
\begin{align}
4\lambda_4 + \lambda_5 = \frac{2}{v^2} \left[ m_{H_3}^2 - \sqrt{\frac{2}{3}} \frac{\sin \alpha \cos \alpha}{\sin \beta \cos \beta} (m_h^2 - m_{H_1}^2) \right] ~.
\end{align}
Since $m_{H_1}^2 > 0$, we obtain the following inequality when $\alpha>0$:
\begin{align}
m_{H_3}^2 \leq \frac{1}{2}(4\lambda_4 + \lambda_5)v^2 + \sqrt{\frac{2}{3}} \frac{\sin \alpha \cos \alpha}{\sin \beta \cos \beta} m_h^2 ~.
\end{align}
Larger $\lambda_4$ or $\lambda_5$ would lead to violation of the vacuum stability conditions in Eq. (<ref>) or the perturbative unitarity conditions in Eq. (<ref>). Therefore, $m_{H_3}$ is bounded from above, though the exact value of the upper bound cannot be expressed analytically.
Also explicitly shown in Fig. <ref> is that all the six mass hierarchies for the exotic Higgs bosons are possible according to the parameter scan. Nevertheless, the most probable ones are either $m_{H_5}>m_{H_3}>m_{H_1}$, dubbed the normal hierarchy, or $m_{H_1}>m_{H_3}>m_{H_5}$, dubbed the inverted hierarchy.
The average signal strength of the SM-like Higgs boson production and decay into two photons from the ATLAS Collaboration <cit.> and CMS Collaboration <cit.> is given by $\mu^{\text{GGF}}_{h\gamma\gamma}=1.12\pm0.22$. In Fig. <ref>, we see that the predicted signal strength ranges from $\sim 0.6 - 1.4$ for almost all the obtained mass spectra. On the other hand, the constraint from $\mu^{GGF}_{h\gamma Z}=2.7^{+4.5}_{-4.3}$ <cit.> is not constraining at all. Most of the predicted $\mu^{GGF}_{h\gamma Z}$ values tend to be bigger than or about 1.
The total decay widths of $H_5^{++}$ and $H_3^+$ normalized to their corresponding masses are shown in Figs. <ref> and <ref>. Since $H_5^{++}$ is a quark-phobic scalar, its only decay channels are $H_3^+W^+$, $W^+W^+$ and $H_3^+H_3^+$ at tree level, provided allowed by kinematics. It is seen that its decay width is $\lesssim 1\%$ of its mass in almost all cases. On the other hand, $H_3^+$ is a gauge-phobic scalar boson. The only decay channels of $H_3^+$ are $hW^+$, $H_1W^+$, $H_5^{++}W^-$ and $t\overline{b}$ at tree level as long as it is kinematically allowed. Compared to $H_5^{++}$, the $H_3^+$ boson has a slightly larger value of the total width-to-mass ratio in most allowed $(v_\Delta,\alpha)$ space and can sometimes reach $\sim 10\%$. In general, Figs. <ref> and <ref> verify that the narrow width approximation employed in our numerical analysis is valid in most spectra.
We show plots for the branching ratios of the decays of $H_5^{++}$ into several different final states in Fig. <ref> to <ref>. One distinct feature of the GM model is that thanks to the custodial symmetry, $v_\Delta$ can be larger compared to the model extended with only one complex Higgs triplet field. Hence same-sign dilepton events coming from the process of $H_5^{++}\to W^+(\to\ell^+\nu_\ell)W^+(\to\ell'^+\nu_{\ell'})$ with the inclusive $BR(W^+\to\ell^+\nu)=10.86\%$ <cit.> provide a distinguished way to test the model because the $H_5^{++}W^-W^-$ vertex is proportional to $v_\Delta$. The resulting branching ratio for different sets of $v_\Delta$ and $\alpha$ are shown in Fig. <ref>. It is observed that the $H_5^{++}\to W^+(\to\ell^+\nu_\ell)W^+(\to\ell'^+\nu_{\ell'})$ decay is often a major one in a significant portion of the allowed region in each plot.
This is because most cases either do not have a hierarchy with $m_{H_5} > m_{H_3}$ in the mass spectrum or do not have sufficient mass splitting.
When cascade decays are allowed, $H_5^{++}$ can also decay into $H_3^+W^+$ with $H_3^+$ further decaying into $hW^+$ or $H_1W^+$.
Their corresponding branching ratios are respectively shown in Fig. <ref> and Fig. <ref>, where only those mass spectra with $m_{H_5} > m_{H_3}$ are plotted. A detailed collider phenomenology study of these scenarios is given in the next section.
If instead $m_{H_5}$ is sufficiently lighter than $m_{H_3}$, $H_3^+$ can decay into $H_5^{++}W^-$ with $H_5^{++}$ further decaying into $W^+W^+$. Such a result is shown in Fig. <ref>, where only the mass spectra with $m_{H_5}<m_{H_3}$ are shown. In addition, $H_3^+$ can also decay into $hW^+$ or $H_1W^+$ as shown in Fig. <ref> and Fig. <ref>, respectively. In the latter case, $m_{H_1}$ must be smaller than $m_{H_3}$ while the mass relation between $m_{H_1}$ or $m_{H_3}$ and $m_{H_5}$ remains arbitrary in both cases. As shown in Fig. <ref>, $BR(H_3^+\to H_1W^+)$ increases with $m_{H_5}$ for a fixed $m_{H_3}$.
Finally, we discuss the decay channels of $H_1$. In addition to the same decay channels as the SM-like Higgs boson, it can also decay into $H_3^\pm W^{\mp}$ and/or a pair of other Higgs bosons, provided these are kinematically allowed. As two promising channels in the search of an additional neutral Higgs boson, we discuss the $H_1 \to hh / W^+W^-$ decays. The branching ratio of $H_1 \to hh$ is shown in Fig. <ref>, where we have omitted the plot for $(v_\Delta,\alpha)=(1~\text{GeV},0^\circ)$ because the value is diminishing in such a decoupling limit. In the $\alpha = 10^\circ$ cases, there is only a very tiny portion of the allowed spectra that can have this decay channel, and the branching ratio is generally less than $\sim 40\%$. In cases with negative $\alpha$, however, the branching ratio can sometimes reach above $\sim 90\%$.
[See also Ref. <cit.> for a similar finding under the consideration of a simplified Higgs potential of the GM model.]
The branching ratio of the $H_1$ decaying into $W^+W^-$ is plotted in Fig. <ref>. The $H_1W^+W^-$ vertex has the coupling $(g^2/6)(3\sin\alpha\sin\beta+2\sqrt6\cos\alpha\cos\beta)$. With appropriate $\alpha$ and $v_\Delta$, the $H_1 \to W^+W^-$ mode can be the dominant one.
§ PROSPECTS FOR OBSERVING SIGNATURES OF GEORGI-MACHACEK MODEL AT 14-TEV LHC
We evaluate the prospects for observing a signature of the GM model in the 14-TeV run of the LHC.
One of the promising channels for discovering the GM model is the production of a doubly-charged Higgs boson $H_5^{\pm \pm}$ via the VBF process, followed by its decay into final states containing a pair of same-sign leptons.
This channel has three advantages:
First, the SM background for events with two same-sign leptons is suppressed compared to those with opposite-sign leptons or only one lepton.
Secondly, the VBF process is the dominant production mechanism of $H_5^{\pm \pm}$ if its mass is above $\sim 300$ GeV and $v_{\Delta} \gtrsim 10$ GeV (see the left plot of Fig. <ref>). Hence, it is a good strategy to concentrate on its production via the VBF process.
Thirdly, in the VBF production of $H_5^{\pm \pm}$, leptons possibly arise only from the decay of the singly-produced $H_5^{\pm \pm}$.
This fact allows a less parameter-dependent estimation of the acceptance and efficiency of events with two same-sign leptons, in comparison with the Drell-Yan (DY) production of $H_5^{\pm \pm} H_5^{\mp \mp}$ or $H_5^{\pm \pm} H_3^{\mp}$ and the associated production of $H_5^{\pm \pm} W^{\mp}$.
For these reasons, we hereafter focus on the process of the VBF production of $H_5^{\pm \pm}$ followed by its decay into a pair of same-sign leptons through same-sign $W$ bosons [We neglect the direct decays into like-sign leptons because the couplings of $H_5^{\pm \pm}$ to SM leptons are strongly suppressed when $v_{\Delta} \gtrsim 1$ GeV.]. The search for this process can be most easily done by selecting events containing a pair of same-sign light leptons, $\mu^\pm \mu^\pm$, $e^\pm e^\pm$ and $e^\pm \mu^\pm$.
It is not necessary to impose any selection cut on the jets associated with the VBF process, as our purpose is to observe the production of $H_5^{\pm \pm}$ rather than identifying the production process.
Left: The production cross sections (fb) of $H_5^{\pm\pm}$ for various channels in $pp$ collisions with $\sqrt{s}=14$ TeV as a function of $m_{H_5}$. The red curves correspond to those for the vector boson fusion, the blue curves to those for the associated production of $H_5^{\pm\pm} W^\mp$, and the black curve to that for the Drell-Yan production of $H_5^{++}H_5^{--}$. The solid red and short-dashed blue curves are for $H_5^{++}$ while the dotted red and long-dashed curves are for $H_5^{--}$. Here we take $v_{\Delta}=10$ GeV.
Right: The same as left, but with $\sqrt{s}=100$ TeV.
We can evaluate the significance of a signal of the GM model by comparing the number of events with two same-sign light leptons coming from the production and decay of $H_5^{\pm \pm}$, $N_{H_5^{++}; {\rm SS \, light \, leptons}}$, where $\ell,\ell'$ denote SM leptons, with the number of those coming from SM processes. The former can be expressed as (the expression for the number of events with two negatively-charged leptons is similar):
\begin{align}
N_{H_5^{++}; {\rm SS \, light \, leptons}} \ &= \ L \, \sigma(p p \rightarrow H_5^{++} + X) \, BR(H_5^{++} \rightarrow \ell^{+} \ell^{\prime +} + X^{\prime})
\, (A \times \epsilon)~,
\label{numberofsignals}
\end{align}
where $L$ denotes the integrated luminosity, $\sigma(p p \rightarrow H_5^{++} + X)$ the inclusive production cross section of $H_5^{++}$, $BR(H_5^{++} \rightarrow \ell^{+} \ell^{\prime +} + X^{\prime})$ the branching ratio of $H_5^{++}$ decaying into final states containing two same-sign leptons, and $A \times \epsilon$ the acceptance times efficiency for events with two same-sign leptons arising from processes involving $H_5^{++}$ with certain event selection criteria. When multiple processes contribute to such events, one should take an average over these processes for the calculation of $A \times \epsilon$. We include the decays into tau leptons in the definition of $BR(H_5^{++} \rightarrow \ell^{+} \ell^{\prime +} + X^{\prime})$, where the tau leptons further decay leptonically.
At this stage, we do not specify the production process or the decay process of $H_5^{\pm \pm}$.
Later on, however, we will find numerically that the dominant production process is the VBF mechanism,
and the dominant decay channel whose final state involves two same-sign leptons is either
$H_5^{\pm \pm} \rightarrow W^{\pm}(\rightarrow \ell^{\pm} \nu) W^{\pm}(\rightarrow \ell^{\prime \pm} \nu)$ or
$H_5^{\pm \pm} \rightarrow W^{\pm}(\rightarrow \ell^{\pm} \nu) H_3^{\pm}, \, H_3^{\pm} \rightarrow W^{\pm}(\rightarrow \ell^{\prime \pm} \nu) h/H_1$.
By focusing on these specific production and decay channels, it becomes simple and straightforward to estimate the acceptance times efficiency, $A \times \epsilon$, that eventually yields $N_{H_5^{++}; {\rm SS \, light \, leptons}}$.
In Fig. <ref>, we display the cross sections for the VBF production of $H_5^{\pm \pm}$ in $pp$ collisions with $\sqrt{s}=14$ GeV (left plot) and $100$ TeV (right plot), as well as the cross sections for the DY production of $H_5^{\pm \pm} H_5^{\mp \mp}$ and the associated production of $H_5^{\pm \pm} W^{\mp}$. The DY production of $H_5^{\pm\pm}H_3^\mp$ is not taken into account in our analysis. All these production cross sections are independent of $\alpha$, as $\alpha$ is the mixing angle between the two singlets. Here we do not impose any selection cut on the jets associated with the VBF process. The figure tells us that the VBF mechanism is the dominant production process for $H_5^{++}$ and $H_5^{--}$ when $v_{\Delta}$ is above 10 GeV and $m_{H_5}$ is above $\sim 300$ (400) GeV and $\sim400$ (500) GeV with $\sqrt s=14$ $(100)$ GeV, respectively.
The cross sections for the VBF production and the associated production with different values of $v_\Delta$ can be readily obtained by rescaling, since both of them are proportional to $v_\Delta^2$. On the other hand, the cross sections for the Drell-Yan production of $H_5^{++}H_5^{--}$ are independent of $v_\Delta$.
Regarding the calculation of $BR(H_5^{\pm \pm} \rightarrow \ell^{\pm} \ell^{\pm} + X^{\prime})$, we note that $H_5^{\pm \pm}$ has only two decay channels for sufficiently large $v_\Delta$.
It decays into either $W^{\pm} W^{\pm}$ or $H_3^{\pm} W^{\pm}$, where $W^{\pm}$ can be off-shell, and $H_3^{\pm}$ further decays into SM particles, possibly involving $H_1$ at an intermediate stage.
The $W^{\pm}$ boson and the decay products of $H_3^{\pm}$ can decay into SM leptons, thereby giving rise to two-same-sign-lepton events.
In Fig. <ref>, we present scatter plots of the branching ratio of $H_5^{++}$ decaying into $W^{+} W^{+}$ and the $W^+$'s further decaying leptonically, $BR(H_5^{++} \rightarrow W^+(\rightarrow \ell^+ \nu_{\ell}) \, W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}) )$, with one of the $W^{+}$'s possibly off-shell, on the plane spanned by $m_{H_5}$ and $m_{H_3}$ for various values of $\alpha$ and $v_{\Delta}$.
The $H_3^{\pm}$ boson has a variety of decay channels, but the one with the dominant branching fraction is either $H_3^{+} \rightarrow h W^+$ or $H_3^{+} \rightarrow H_1 W^+$, depending on the mass spectrum and other parameters.
We thus present in Fig. <ref> and <ref> scatter plots of the products of branching ratios, $BR(H_5^{++} \rightarrow H_3^+ W^+(\rightarrow \ell^+ \nu_{\ell}) ) BR(H_3^+ \rightarrow h W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}))$ and $BR(H_5^{++} \rightarrow H_3^+ W^+(\rightarrow \ell^+ \nu_{\ell}) ) BR(H_3^+ \rightarrow H_1 W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}))$,
where the $W^{+}$'s can be off-shell.
We estimate the acceptance times efficiency, $A \times \epsilon$, for the processes of $H_5^{\pm \pm}$ production followed by its decay into final states containing two same-sign leptons.
We define the acceptance times efficiency as
\begin{align}
A \times \epsilon \equiv \frac{N_{\text{pass}}}{N_{\text{all}}} ~,
\end{align}
where $N_{\text{pass}}$ is the number of events that pass the selection criteria (a) through (e) defined below, and $N_{\text{all}}$ is the number of events for the processes of $p p \rightarrow H_5^{\pm \pm} + X, \, H_5^{\pm \pm} \rightarrow \ell^{\pm} \ell^{\prime \pm} + X^{\prime}$.
Note that $A \times \epsilon$ is almost the same for both $H_5^{++}$ and $H_5^{--}$.
In our simulation study, we consider the following criteria for selecting events with two same-sign leptons, based on which we estimate $A \times \epsilon$. These criteria mimic part of the selection criteria used in the analysis of Ref. <cit.>:
(a) An electron is identified when its transverse momentum satisfies $p_{eT} > 10$ GeV and its pseudo-rapidity satisfies $\vert \eta_e \vert < 1.37$ or $1.52 < \vert \eta_e \vert < 2.47$.
A muon is identified when its transverse momentum satisfies $p_{\mu T} > 10$ GeV and its pseudo-rapidity satisfies $\vert \eta_{\mu} \vert < 2.5$.
(b) The event should contain $e^{\pm} e^{\pm}$, $e^{\pm} \mu^{\pm}$, or $\mu^{\pm} \mu^{\pm}$. The harder lepton $\ell_1$ should have a transverse momentum above 25 GeV, $p_{\ell_1 T} > 25$ GeV, and the other lepton $\ell^{\prime}_2$ should have a transverse momentum above 20 GeV, $p_{\ell^{\prime}_2 T} > 20$ GeV.
(c) If the event further contains a lepton with a sign opposite to the same-sign lepton pair found above, then the event is vetoed.
(d) The invariant mass of the two same-sign leptons should be larger than 15 GeV, $m(\ell, \ell^{\prime}) > 15$ GeV.
(e) If the two same-sign leptons are electrons, their invariant mass should be below 70 GeV or above 110 GeV, $m(\ell_1, \ell_2) \, <$ 70 GeV or $m(\ell_1, \ell_2) \, >$ 110 GeV.
Based on the selection criteria (a) through (e), we estimate $A \times \epsilon$ for the following processes:
\begin{align}
p \, p \ &\rightarrow \ H_5^{++} \, j \, j~, \ \ \ H_5^{++} \ \rightarrow \ W^+(\rightarrow \ell^+ \nu) \, W^+(\rightarrow \ell^{\prime +} \nu)~; \label{wwdecay} \\
p \, p \ &\rightarrow \ H_5^{++} \, j \, j~, \ \ \ H_5^{++} \ \rightarrow \ H_3^+ \, W^+(\rightarrow \ell^+ \nu)~, \ \ \ H_3^+ \, \rightarrow \, h \, W^+(\rightarrow \ell^{\prime +} \nu)~; \label{wwhdecay} \\
p \, p \ &\rightarrow \ H_5^{++} \, j \, j~, \ \ \ H_5^{++} \ \rightarrow \ H_3^+ \, W^+(\rightarrow \ell^+ \nu)~, \ \ \ H_3^+ \, \rightarrow \, H_1 \, W^+(\rightarrow \ell^{\prime +} \nu)~. \label{wwh1decay}
\end{align}
Here $j$ denotes a jet originating from a quark involved in the VBF process, and $\ell, \ell^{\prime}$ represent an electron, a muon or a tau lepton, where the tau lepton will further decay leptonically.
Contributions from decays involving an off-shell $W$ boson are also taken into account.
We neglect the decay products of $h$ and $H_1$. Although electrons and muons that come from the decay of $h$ or $H_1$ may affect the chance for an event to pass the selection criteria, we expect that their impact is negligibly small because more than 90% of $h$'s decay into final states without electron or muon.
As for $H_1$, it mainly decays into $hh$, $t \bar{t}$, $W^+ W^-$ and $ZZ$ when $m_{H_1}$ is below $m_{H_3}$ and $m_{H_5}$. Thus, more than 60% of $H_1$'s decay into final states without electron or muon.
To evaluate $A \times \epsilon$, we perform a realistic detector-level simulation by using the Monte Carlo event generator MadGraph5 <cit.>, interfaced with PYTHIA6 <cit.> for simulating parton showering and hadronization and with DELPHES3 <cit.> for simulating detector responses and object reconstruction.
Our simulation of $H_5^{\pm \pm}$ production events is based on the leading-order (LO) QCD calculation of matrix elements.
In the following, we present the acceptance times efficiency for the processes of Eq. (<ref>) and Eq. (<ref>).
The acceptance times efficiency, $A \times \epsilon$, for the process of Eq. (<ref>) in 14-TeV $pp$ collisions with the selection criteria (a) through (e) as a function of $m_{H_5}$.
The solid curve corresponds to the final state with $\mu^\pm \mu^\pm$, the dashed curve to that with $\mu^\pm e^\pm$, and the dotted curve to that with $e^\pm e^\pm$.
In Fig. <ref>, we plot the acceptance times efficiency for the process of Eq. (<ref>) in 14-TeV $pp$ collisions with the selection criteria (a) to (e) as a function of the $H_5^{\pm \pm}$ mass.
In Table <ref>, we present $A \times \epsilon$ for the process of Eq. (<ref>) in 14-TeV $pp$ collisions with the selection criteria (a) to (e), for various benchmark values of $(m_{H_5}, \, m_{H_3})$, all of which can be realistic mass spectra consistent with all theoretical and experimental constraints, as can be read from Fig. <ref> and other figures.
The acceptance times efficiency is calculated for the three channels with $e^{\pm} e^{\pm}$, $e^{\pm} \mu^{\pm}$ and $\mu^{\pm} \mu^{\pm}$.
We do not show $A \times \epsilon$ for the process of Eq. (<ref>) because it depends on $m_{H_5}$, $m_{H_3}$, and $m_{H_1}$ and hence is highly mass spectrum-dependent. This fact renders the process of Eq. (<ref>) an ineffective channel for the discovery of the GM model.
$(m_{H_5}, \, m_{H_3})$ GeV (300, 200) (400, 250) (400, 300) (500, 300) (500, 350)
$A \times \epsilon$ for $e^{\pm} e^{\pm}$ channel 0.021 0.032 0.029 0.046 0.041
$A \times \epsilon$ for $e^{\pm} \mu^{\pm}$ channel 0.067 0.11 0.10 0.13 0.12
$A \times \epsilon$ for $\mu^{\pm} \mu^{\pm}$ channel 0.038 0.063 0.059 0.075 0.074
$(m_{H_5}, \, m_{H_3})$ GeV (500, 400) (600, 400) (600, 450) (600, 500) (700, 500)
$A \times \epsilon$ for $e^{\pm} e^{\pm}$ channel 0.036 0.051 0.050 0.040 0.058
$A \times \epsilon$ for $e^{\pm} \mu^{\pm}$ channel 0.11 0.14 0.13 0.12 0.15
$A \times \epsilon$ for $\mu^{\pm} \mu^{\pm}$ channel 0.069 0.081 0.076 0.069 0.083
$(m_{H_5}, \, m_{H_3})$ GeV (700, 550) (700, 600) (800, 600) (800, 650) (800, 700)
$A \times \epsilon$ for $e^{\pm} e^{\pm}$ channel 0.053 0.045 0.062 0.056 0.046
$A \times \epsilon$ for $e^{\pm} \mu^{\pm}$ channel 0.14 0.13 0.15 0.14 0.13
$A \times \epsilon$ for $\mu^{\pm} \mu^{\pm}$ channel 0.083 0.074 0.087 0.081 0.077
$A \times \epsilon$ for the process of Eq. (<ref>) in 14-TeV $pp$ collisions with the selection criteria (a) to (e) for various benchmark values of $(m_{H_5}, \, m_{H_3})$ GeV.
Finally, we estimate the number of background events that arise from SM processes and pass the selection criteria (a) through (e).
The dominant sources of background events are the production of $W^{\pm} Z$ and $ZZ$ followed by their decays into leptons including tau leptons:
\begin{align}
p \, p \ &\rightarrow \ W^+(\rightarrow \ell^+ \nu) \, Z(\rightarrow \ell^{\prime +} \ell^{\prime -})~, \label{wzbkg} \\
p \, p \ &\rightarrow \ Z(\rightarrow \ell^+ \ell^-) \, Z(\rightarrow \ell^{\prime +} \ell^{\prime -})~, \label{zzbkg}
\end{align}
where $\ell, \ell^{\prime}$ represent an electron, a muon or a tau lepton.
We further take into account the background coming from the $W^{\pm} W^{\pm}$ production process:
\begin{align}
p \, p \ &\rightarrow \ W^+(\rightarrow \ell^+ \nu) \, W^+(\rightarrow \ell^{\prime +} \nu) \, j \, j~, \label{wwbkg}
\end{align}
where $j$ denotes a jet originating from a quark in the subprocesses of $u u \rightarrow W^+ W^+ d d$ or $d d \rightarrow W^- W^- u u$. Although its contribution is subdominant, this process gives an irreducible background to the same-sign lepton signal defined in terms of the selection criteria (a) to (e).
We note that charge misidentification of electrons can be another dominant source of backgrounds for the channels involving an electron.
Nevertheless, we do not estimate its contribution as it is beyond the scope of our theoretical study.
To assess the number of background events, we also perform a realistic detector-level simulation by using MadGraph5 <cit.> interfaced with PYTHIA6 <cit.> and DELPHES3 <cit.>.
The generation of background events is based on the LO QCD calculation of matrix elements, but we take into account the next-to-leading order (NLO) QCD effects by multiplying the number of background events with a $K$-factor derived as the ratio of the $W^{\pm} W^{\pm}$, $W^{\pm} Z$ or $ZZ$ production cross section calculated at the NLO divided by the corresponding one at the LO.
In Table <ref>, we present the LO and NLO production cross sections, $\sigma_{{\rm LO}}$ and $\sigma_{{\rm NLO}}$, the $K$-factor estimated as above, and the number of background events that pass the selection criteria (a) through (e) divided by the integrated luminosity, $N_{bkg}/L$ (the $K$-factor is multiplied already), for each of the $W^{\pm} W^{\pm}$, $W^{\pm} Z$ and $ZZ$ production processes, for each final state containing $e^{\pm}e^{\pm}$, $e^{\pm}\mu^{\pm}$ or $\mu^{\pm}\mu^{\pm}$.
Process $W^{+} Z$ $W^{-} Z$ $ZZ$ $W^{+} W^{+}$ $W^{-} W^{-}$
$\sigma_{{\rm LO}}$ $1.72\times 10^4$ fb $1.05\times 10^4$ fb $1.06\times 10^4$ fb 257 fb 113 fb
$\sigma_{{\rm NLO}}$ $3.01\times 10^4$ fb $1.94\times 10^4$ fb $1.52\times 10^4$ fb 340 fb 161 fb
$K$-factor 1.75 1.84 1.43 1.33 1.42
$N_{bkg}(e^{\pm}e^{\pm})/L$ 245 fb 158 fb 110 fb 6.70 fb 3.17 fb
$N_{bkg}(e^{\pm}\mu^{\pm})/L$ 723 fb 466 fb 379 fb 19.3 fb 9.12 fb
$N_{bkg}(\mu^{\pm}\mu^{\pm})/L$ 370 fb 239 fb 178 fb 11.5 fb 5.45 fb
The production cross sections calculated at the LO, $\sigma_{{\rm LO}}$, and at the NLO, $\sigma_{{\rm NLO}}$, the estimated $K$-factor, and the number of background events that pass the selection criteria (a) to (e) divided by the integrated luminosity, $N_{bkg}/L$ (with the $K$-factor multiplied) for each of the $W^{\pm} W^{\pm}$, $W^{\pm} Z$ and $ZZ$ production processes and for each final state containing $e^{\pm}e^{\pm}$, $e^{\pm}\mu^{\pm}$ or $\mu^{\pm}\mu^{\pm}$ at the 14-TeV LHC.
With Figs. <ref>, <ref>, <ref>, <ref>, <ref>, <ref> and Tables <ref>, <ref>, we can evaluate the significance of the same-sign lepton signal for the most general mass spectra of the GM model.
This is done in the following manner.
First, we take a set of parameters $(v_{\Delta}, \alpha, m_{H_5}, m_{H_3})$ for which we want to study the discovery potential of the GM model at the LHC.
From Figs. <ref> and <ref>, we check if there exists a mass spectrum that satisfies all the theoretical and experimental constraints.
We then look up the corresponding values of $BR(H_5^{++} \rightarrow W^+(\rightarrow \ell^+ \nu_{\ell}) \, W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}) )$, $BR(H_5^{++} \rightarrow H_3^+ W^+(\rightarrow \ell^+ \nu_{\ell}) ) BR(H_3^+ \rightarrow h W^+(\rightarrow \ell^{\prime +} \nu_{\ell^{\prime}}))$ in Figs. <ref> and <ref> and the VBF production cross section of $H_5^{\pm \pm}$ in $pp$ collisions in Fig. <ref>.
The acceptance times efficiency $A\times \epsilon$ for these values of $m_{H_5}, m_{H_3}$ can be estimated using Fig. <ref> and Table <ref> for the two processes of $p p \rightarrow H_5^{++} \, j \, j, \, H_5^{++} \rightarrow W^+(\rightarrow \ell^+ \nu) \, W^+(\rightarrow \ell^{\prime +} \nu)$ and $p p \rightarrow H_5^{++} \, j \, j, \, H_5^{++} \rightarrow H_3^+ \, W^+(\rightarrow \ell^+ \nu), \, H_3^+ \rightarrow h \, W^+(\rightarrow \ell^{\prime +} \nu)$, respectively.
Finally, we evaluate the number of events with a same-sign light lepton pair arising from the production and decay of $H_5^{\pm \pm}$ by Eq. (<ref>), and compare it with the number of SM background events that can be extracted from Table <ref> to derive the significance of the signal for some value of the integrated luminosity.
We note in passing that the true value of the significance of the same-sign light lepton signal can be larger than evaluated above, because the vector boson associated and Drell-Yan productions of $H_5^{\pm \pm}$ as well as the process of Eq. (<ref>), which are neglected in the above evaluation, also contribute to the signal.
Hence the significance evaluated following the above-described procedure actually corresponds to the lower bound.
The results given in the previous section can be extended to a 100 TeV hadron collider. Since the cross section for the VBF production of $H_5^{++}$ is much larger than that for the associated production (see Fig. <ref>), the process of $pp\to H_5^{++}jj$ followed by $H_5^{++}$ decays can be used to estimate the observability of $H_5^{++}$. The transverse momenta of $H_5^{++}$'s in the VBF process tend to zero. Therefore, the acceptance times efficiency for the decay products of $H_5^{++}$ depends only on the mass of $H_5^{++}$. Note that no selection cuts are put on the two forward jets in the VBF process. Otherwise, it will depend on the collision energy. Thus, we can safely assume that the acceptance times efficiency does not vary significantly from 14-TeV to 100-TeV colliders. In other words, the values of the acceptance times efficiency given in Fig. <ref> can simply be applied to the case with $\sqrt{s}=100$ TeV.
For a future $pp$ collider with 100-TeV collision energy, it is sufficient to observe the production of $H_5^{++}$ to test the GM model.
§ CONCLUSIONS
In this work, we have studied the most general mass spectrum of the exotic Higgs bosons of the Georgi-Machacek model that is allowed by theoretical and experimental constraints.
As theoretical constraints, we have taken into account the unitarity of the perturbation theory and the stabiliy of the electroweak symmetry breaking vacuum.
On the other hand, the experimental constraints we have considered are the electroweak precision tests, the $Z b \bar{b}$ vertex measurement and the Higgs boson signal strength data.
Here we used the latest Higgs boson signal strength data to find the allowed region at $1\sigma$ and $2\sigma$ confidence level on the plane of the triplet VEV $v_\Delta$ and the singlet mixing angle $\alpha$.
The diphoton channel was not included in the above analysis, because its strength depends on the charged Higgs boson mass spectrum, in addition to $v_\Delta$ and $\alpha$.
From the $2\sigma$ region, we identified twelve example sets of $(v_\Delta,\alpha)$ for subsequent analyses.
We have performed a comprehensive parameter scan for the exotic Higgs boson mass spectrum allowed by these constraints.
We found that the most probable spectra are either the normal hierarchy ($m_{H_5} > m_{H_3} > m_{H_1}$) or the inverted hierarchy ($m_{H_5} < m_{H_3} < m_{H_1}$), though other scenarios are generally possible as well.
We worked out the signal strengths of diphoton and $Z\gamma$ channels of the SM-like Higgs boson via the gluon-gluon fusion process, the decay widths of $H_5^{\pm\pm}$ and $H_3^\pm$, branching ratios of various cascade decays of $H_5^{++}$ and $H_3^{+}$, and branching ratios of the $H_1 \to hh / W^+W^-$ decay.
As to collider signatures of the model, we focused on the production and decays of the $H_5^{\pm\pm}$ boson.
We computed the cross sections of the vector boson fusion, vector boson associated, and Drell-Yan productions of the $H_5^{\pm\pm}$ boson for the 14-TeV LHC and a future 100-TeV $pp$ collider.
In accord with our selection criteria set for signal events, acceptance times efficiency for signal events was evaluated for the three processes: $p p \rightarrow H_5^{++} j j$ with $H_5^{++} \rightarrow W^+(\rightarrow \ell^+ \nu) W^+(\rightarrow \ell^{\prime +} \nu)$;
$p p \rightarrow H_5^{++} j j$ with $H_5^{++} \rightarrow H_3^+ W^+(\rightarrow \ell^+ \nu)$ and $H_3^+ \rightarrow h W^+(\rightarrow \ell^{\prime +} \nu)$; and
$p p \rightarrow H_5^{++} j j$ with $H_5^{++} \rightarrow H_3^+ W^+(\rightarrow \ell^+ \nu)$ and $H_3^+ \rightarrow H_1 W^+(\rightarrow \ell^{\prime +} \nu)$. This part was done at the leading order in QCD and with a realistic detector-level simulation. For background events, a similar simulation was carried out at the leading order of QCD and then scaled by the $K$-factor to the next-to-leading order.
We have argued that, by combining our estimates on the production cross section of $H_5^{\pm\pm}$, acceptance times efficiency for the signal and the branching ratios of cascade decays of $H_5^{\pm\pm}$, and by comparing them with SM background estimates,
we can evaluate prospects for observing a signal of the GM model for its most general mass spectrum.
Finally, we have also made a remark that the same acceptance times efficiency that we computed for the 14-TeV LHC can be applied to the case of a future 100-TeV $pp$ collider to a good approximation because no forward jet tagging is required in our proposed selection cuts.
Note Added:
Recently, the ATLAS and CMS Collaborations have observed through the diphoton decay mode a hint of a resonance at about 750 GeV and a width of about 45 GeV <cit.>. In the GM model, $H_1$ is a good candidate for the 750-GeV resonance because it can be produced in the s-channel in $pp$ collisions and decay into diphotons. The mass of $H_1$ can be read from Fig. <ref>. For example, there is some parameter space in the plot for $v_\Delta = 10$ GeV and $\alpha = -10^\circ$ that gives $m_{H_1} \simeq 750$ GeV. A quick estimate shows that its decay width is about 10 GeV and the cross section of the diphoton channel at the 13-TeV LHC is about 1 fb. We leave the detailed analysis to a future work.
This work was supported in part by the Ministry of Science and Technology of Taiwan under Grant Nos. MOST-100-2628-M-008-003-MY4, 104-2628-M-008-004-MY4, and 103-2811-M-008-058,
and by the National Research Foundation of Korea (NRF) Research Grant NRF-2015R1A2A1A05001869 (TY).
H. Georgi and M. Machacek,
Nucl. Phys. B 262, 463 (1985).
M. S. Chanowitz and M. Golden,
Phys. Lett. B 165, 105 (1985).
J. F. Gunion, R. Vega and J. Wudka,
Phys. Rev. D 42, 1673 (1990).
H. E. Haber and H. E. Logan,
Phys. Rev. D 62, 015011 (2000)
S. Godfrey and K. Moats,
Phys. Rev. D 81, 075026 (2010)
[arXiv:1003.3033 [hep-ph]].
C. W. Chiang, T. Nomura and K. Tsumura,
Phys. Rev. D 85, 095023 (2012)
[arXiv:1202.2014 [hep-ph]].
C. W. Chiang and K. Yagyu,
JHEP 1301, 026 (2013)
[arXiv:1211.2658 [hep-ph]].
C. Englert, E. Re and M. Spannowsky,
Phys. Rev. D 87, no. 9, 095014 (2013)
[arXiv:1302.6505 [hep-ph]].
C. Englert, E. Re and M. Spannowsky,
Phys. Rev. D 88, 035024 (2013)
[arXiv:1306.6228 [hep-ph]].
C. W. Chiang, A. L. Kuo and K. Yagyu,
JHEP 1310, 072 (2013)
[arXiv:1307.7526 [hep-ph]].
K. Hartling, K. Kumar and H. E. Logan,
Phys. Rev. D 90, 015007 (2014)
[arXiv:1404.2640 [hep-ph]].
C. W. Chiang, S. Kanemura and K. Yagyu,
Phys. Rev. D 90, 115025 (2014)
[arXiv:1407.5053 [hep-ph]].
K. Hartling, K. Kumar and H. E. Logan,
arXiv:1410.5538 [hep-ph].
C. W. Chiang, S. Kanemura and K. Yagyu,
arXiv:1510.06297 [hep-ph].
C. W. Chiang and T. Yamada,
Phys. Lett. B 735, 295 (2014)
[arXiv:1404.5182 [hep-ph]].
J. F. Gunion, R. Vega and J. Wudka,
Phys. Rev. D 43, 2322 (1991).
H. E. Logan and M. A. Roy,
Phys. Rev. D 82, 115011 (2010)
[arXiv:1008.4869 [hep-ph]].
A. Falkowski, S. Rychkov and A. Urbano,
JHEP 1204, 073 (2012)
[arXiv:1202.1532 [hep-ph]].
S. Chang, C. A. Newby, N. Raj and C. Wanotayaroj,
Phys. Rev. D 86, 095015 (2012)
[arXiv:1207.0493 [hep-ph]].
J. A. Grifols and A. Mendez,
Phys. Rev. D 22, 1725 (1980).
M. Capdequi Peyranere, H. E. Haber and P. Irulegui,
Phys. Rev. D 44, 191 (1991).
S. Kanemura,
Phys. Rev. D 61, 095001 (2000);
A. Arhrib, R. Benbrik, M. Chabab, G. Moultaka, M. C. Peyranere, L. Rahili and J. Ramadan,
Phys. Rev. D 84 (2011) 095005
[arXiv:1105.1925 [hep-ph]].
M. Aoki and S. Kanemura,
Phys. Rev. D 77, no. 9, 095009 (2008)
[Phys. Rev. D 89, no. 5, 059902 (2014)]
[arXiv:0712.4053 [hep-ph]];
B. Grinstein, C. W. Murphy, D. Pirtskhalava and P. Uttayarat,
JHEP 1405, 083 (2014)
[arXiv:1401.0070 [hep-ph]].
M. E. Peskin and T. Takeuchi,
Phys. Rev. Lett. 65 (1990) 964;
M. E. Peskin and T. Takeuchi,
Phys. Rev. D 46 (1992) 381.
K. A. Olive et al. [Particle Data Group Collaboration],
Chin. Phys. C 38, 090001 (2014).
The ATLAS collaboration,
ATLAS-CONF-2015-007, ATLAS-COM-CONF-2015-011.
V. Khachatryan et al. [CMS Collaboration],
Eur. Phys. J. C 75, no. 5, 212 (2015)
[arXiv:1412.8662 [hep-ex]].
The ATLAS collaboration,
G. Aad et al. [ATLAS Collaboration],
Phys. Rev. Lett. 113, no. 17, 171801 (2014)
[arXiv:1407.6583 [hep-ex]].
S. I. Godunov, M. I. Vysotsky and E. V. Zhemchugov,
Phys. Lett. B 751, 505 (2015)
[arXiv:1505.05039 [hep-ph]].
The ATLAS collaboration,
JHEP 1503 (2015) 041 [arXiv:1412.0237 [hep-ex]].
J. Alwall et al., JHEP 0709 (2007) 028 [arXiv:0706.2334[hep-ph]];
J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer,
JHEP 1106 (2011) 128 [arXiv:1106.0522[hep-ph]];
J. Alwall et al.,
JHEP 1407 (2014) 079
[arXiv:1405.0301 [hep-ph]].
T. Sjostrand, S. Mrenna and P. Z. Skands,
JHEP 0605 (2006) 026 [arXiv:hep-ph/0603175].
J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaitre, A. Mertens, M. Selvaggi
(The DELPHES 3 collaboration),
JHEP 1402 (2014) 057 [arXiv:1307.6346 [hep-ex]].
The ATLAS collaboration,
The CMS Collaboration,
|
1511.01024
|
$^{1}$Institute of Solid State Physics, Russian
Academy of Sciences, Chernogolovka, 142432 Russia.
$^{2}$Grenoble High Magnetic Field Laboratory,
MPI-FKF and CNRS, 38042 Grenoble, Cedex, France.
$^{3}$Microelectronic Centre, The Technical University of
Denmark, DK 2800 Lyngby, Denmark
Strong increase in the intensity of the peaks of
excited magneto-exciton (ME) states in the photoluminescence excitation
(PLE) spectra recorded for the ground heavy-hole magneto-excitons
(of the 1sHH type) has been found in a GaAs/AlGaAs superlattice in
strong magnetic field B applied normal to the sample layers. While
varying B the intensities of the PLE peaks have been measured as functions
of energy separation $\Delta E$ between excited ME peaks and the ground state
of the system. The resonance profiles have been found to have maxima at
$\Delta E_{\mbox{\scriptsize max}}$ close to the energy of the GaAs LO-phonon.
However, the value of $\Delta E_{\mbox{\scriptsize max}}$ depends on quantum
numbers of the excited ME state. The revealed very low quantum efficiency of
the investigated sample allows us to ascribe the observed resonance to the
enhancement of the non-radiative magneto-exciton relaxation rate arising due to
LO-phonon emission. The presented theoretical model, being in a good agreement
with experimental observations, provides a method to extract 1sHH
magneto-exciton “in-plane" dispersion from the dependence of
$\Delta E_{\mbox{\scriptsize max}}$ on the excited ME state quantum numbers.
PACS numbers: 78.66.Fd, 78.55.Cr, 73.61.Ey
§ INTRODUCTION
Optical transitions in semiconductor superlattices (SLs) have received
great attention recently. In contrast to the quasi
two-dimensional (2D) case of quantum wells (QWs), quasi-particles in SLs
are not fully confined in the growth direction $\hat{z}$. Moreover,
periodicity in the distribution of the materials with
different band gaps and elastic constants leads to
the formation of minibands in the case of free carriers
and excitons <cit.>, and the appearance of additional
optical and acoustic phonon modes <cit.>.
Numerous investigations concerning optical observation of
excitons and magnetoexcitons (MEs) in QWs (e.g. see Refs. ba82,gr84,vi90,ba98) and in Sls <cit.>
usually leave aside the questions
related to the non-radiative excitonic relaxation. At the same time,
non-radiative excitonic transitions such as phonon emission
or absorption allow one to probe indirectly the exciton-energy dispersion
${\cal E}(\mbox{\boldmath $k$})$ ( is the exciton
wave vector). Furthermore, studies of these processes may provide
the only way of the experimental measurement of the excitonic dispersion.
Indeed, such a powerful method as the hot-luminescence technique (reported
for the first time in Ref. mirl81), which was widely used
for measurements of the hole band-structure in bulk
GaAs <cit.> and in GaAs/AlGaAs QWs <cit.>, is
inefficient for the study of the excitonic dispersion. Although excited ME
states were observed in hot magneto-luminescence measurements <cit.>,
such experiments can hardly reveal any information on the function
${\cal E}(\mbox{\boldmath $k$})$. This is because hot-luminescence measurements
can only probe excitons in their radiative states, i.e. when $k$ is very
small, so that ${\cal E}(\mbox{\boldmath $k$})\approx {\cal E}(0)$.
The theoretical investigations of excitonic dispersion have attracted large
efforts ever since Gor'kov and
Dzyaloshinskii's work devoted to three-dimensional ME <cit.>.
Later, 2D ME was studied in the paper by Lerner and Lozovik
<cit.> and also in the works concerning 2D excitons without magnetic
field <cit.>. Exciton dispersion relations in SLs
presenting the dependencies ${\cal E}(\mbox{\boldmath $k$})$ where
$\mbox{\boldmath $k$}\parallel \hat{z}$
(i. e. minibands) were calculated in Ref. di90.
In parallel, the excitonic binding energy in SLs were studied
theoretically and experimentally in Ref. ch87 (see also Refs.
chu89,ch89), and later the binding energies of ground and excited
states in SLs were calculated in magnetic and electric fields <cit.>.
It is worth noting that two-dimensionality and
strong magnetic fields are the features which usually allow the
separation of transverse variables $x$ and $y$ from the longitudinal
one $z$ ($\mbox{\boldmath $B$}\parallel\hat{z}$) and in addition permit
a simplification of the model for valence band due to the removal of degeneracy
The present paper is the result of experimental and theoretical
studies on the observation of a resonant behaviour in relaxation
of MEs in type-I GaAs/Al${}_{0.3}$Ga${}_{0.7}$As SLs in a high magnetic
field perpendicular to the SL layers. In our experiments we detected only
photoluminescence (PL) signal from the ground heavy hole exciton state 1sHH
(we employ the usual notation for 2D exciton states <cit.>)
while the energy of the laser excitation was continuously varied in a
range of 80 meV above the 1sHH PL peak. At particular magnitudes
of magnetic field we have observed very strong resonant increase of
intensity of peaks corresponding to the excited ME states in PLE spectra.
We interpret this effect as a manifestation of the magneto-phonon resonance
when strong relaxation of the excited ME state takes place via
longitudinal-optic-(LO-) phonon emission. This occurs when frequency
$\omega_{\mbox{\scriptsize LO}}$ is equal to an approximate multiple of
the excitonic cyclotron frequency $\omega_c$ (a similar effect for free
electrons in heterojunctions is reported in Refs.ha89,ba91,va96):
\hbar\omega_{\mbox{\scriptsize LO}}=N\hbar\omega_c,\quad N=1,2,3,...
\eqno (1.1)
Here $\omega_c=eB/\mu c$, where $\mu$ is the reduced excitonic mass for
transverse motion in the layer plane. To our best knowledge this is the
first observation of the magneto-phonon resonance for excitons.
The effect has been found only in one SL sample of the investigated
series of SL structures. According to our measurements, the quantum
efficiency of this sample is about two orders of magnitude less
than that for the other investigated structures. This fact
allows us to treat the considered phenomenon as a feature of the enhanced
relaxation of the excited ME states rather than the peculiarity of light
The observed resonance profiles (i.e. integrated intensity of enhanced
PLE peaks as a function of their energy separation $\Delta E$ from the
ground state) were found to be rather broad (with half-widths above $5\,$meV)
and have strong maxima at $\Delta E_{\mbox{\scriptsize max}}$ dependent
on the quantum numbers of excited ME states. These facts indicate that
transitions to the ground ME state occur via intermediate non-radiative
states of the ground excitonic 1sHH band. The theoretical treatment
developed in our report allows us to extract certain information about
the exciton energy dependence on “in-plane" wave-vector component
$\mbox{\boldmath $q$}$ from the experimental data.
After the description of the experimental results in Section II we
present the theoretical study of the phenomenon in Section III. Finally,
in the discussion of Section IV we demonstrate the comparison of the PLE
data with the theoretical results and present dispersion curves for
the “transverse" 1sHH band
$\epsilon(q) \equiv{\cal E}(\mbox{\boldmath $q$},k_z=0)$ extracted from
our experiment.
§ EXPERIMENT
Samples used in our investigations are molecular beam
epitaxy grown type-I GaAs/AlGaAs SLs. The structures
are not intentionally doped and the flat band regime is
realized. All the samples have $L_{w}=8\,$nm wide GaAs QWs, while
the width of AlGaAs barrier is varied from sample to sample
as $L_{b}=2, 3, 5, 10, 20\,$nm. All the SLs consist
of 20 periods $d=L_{b}+L_{w}$.
The PLE experiments were performed in a He cryostat
with a superconducting magnet providing a magnetic field up to
23 T normal to the SL layers. A technique based on employing optical
fibers was used for the sample excitation and collection of the PLE
signal. In the geometry of our experiment the incident laser light
propagates in the direction close to the normal to the SLs layers.
In order to measure the PLE spectra we tuned a double 1 m monochromator
slightly lower or directly to the ground heavy-hole exciton photoluminescence
(PL) peak and scanned the excitation energy of the Ar$^{+}$-pumped
Ti-Sapphire laser. The main features of the PLE spectra remained
unchanged when the detection position was moved in the limits
of the linewidth around the 1sHH PL peak. PLE of two different polarizations,
$\sigma^{+}$ and $\sigma^{-}$, was detected by a cooled GaAs detector
in the photon counting regime.
Fig. 1 displays typical heavy-hole exciton PLE spectra
for the SL with $L_w=8\,$nm and $L_b=3\,$nm (referred to as sample 8/3
below) recorded in magnetic field applied normal to the SL layers. We present
here the data for $\sigma^+$
polarization only, since the observed behaviour is very similar to that
for $\sigma^-$ polarization. At $B=0$ the most pronounced peaks in the
spectrum are the direct heavy hole 1sHH-exciton (its
position is not clearly resolved in the shown series of spectra
because the laser wavelength was scanned from the high energy
side of the 1sHH line), the indirect heavy hole $I$(1sHH)-exciton
at $1.586\,$eV <cit.> and the light hole 1sLH-exciton at
$1.5955\,$eV <cit.>. When $B$ is increased to 6 T, new features
become clearly resolved in the spectra above the 1sLH peak.
Energies of these excited states rapidly increase with $B$. The
origin of the new peaks can be revealed with the help of an elaborate
theoretical analysis of the magneto-exciton band structure.
However, this task lies beyond the scope of our investigation.
In order to understand the nature of the strongest PLE peaks
we carried out a simplified analysis of PLE spectra recorded at
high magnetic fields. As a result it has been found that the peaks
plotted in Fig. 1 by thick lines correspond to MEs formed by electrons
and heavy holes from Landau levels with equal $N=2, 3$
(2sHH, 3sHH) <cit.>. In what follows we restrict our
investigation to the study of the resonant behaviour of these
ME states.
As it is seen in Fig. 1, already starting from $B$
of several Teslas the energies of 2sHH and 3sHH ME
peaks increase quasi-linearly with $B$. At $B<6\,$T the intensity of
the 2sHH ME peak is weak. Then starting from $B=12\,$T the intensity
of the 2sHH peak grows rapidly, reaching its maximum at $B=14\,$T and
then decreases more slowly. A similar resonant behaviour is clearly
observed for the 3sHH ME at $B \approx 9\,$T.
Similar resonances have been also observed in $\sigma^-$ polarization,
however they occur at slightly larger $B$: at $B=17\,$T for 2sHH and at
$B=9.5\,$T for 3sHH.
To summarize the PLE data of Fig. 1 (and of the similar series for
$\sigma^-$) the integrated intensities of the 2sHH and 3sHH MEs are
plotted in Figs. 2a,b (black and open squares, respectively) for
both polarizations versus the energy separation $\Delta E$ between
their positions and the location of the 1sHH peak. The energy of 1sHH
peak is extracted from another series of PLE measurements.
The variation of $\Delta E$ with increasing magnetic field occurs
due to the stronger diamagnetic shifts of the excited ME peaks with
respect to that of the 1sHH line (see insets of Figs. 2a,b, where it is
seen that $\Delta E$ changes almost linearly with the magnetic field
both for 2sHH and 3sHH MEs). Figs. 2a,b show that a very strong
increase (by a factor of $10\,-\,20$) of ME peak intensities arises
when $\Delta E\approx 35\,$ and $\approx40\,$ meV for the 2sHH and 3sHH
peaks, respectively. These values are very close to the energy of optical
phonons in GaAs/AlGaAs <cit.>. Note however that the resonance for the 2sHH peak appears at smaller $\Delta E$ than that for the 3sHH
peak. At the same time the resonant
enhancement for the 2s$\to$1s transition naturally occurs in stronger
magnetic field than the resonance for the 3s$\to$1s one.
The following features of the resonances should be noted as well:
(i) their decay as a function of $\Delta E$ is slower than
their build-up; (ii) a structure is
observed at $\Delta E\approx 37\,$meV for 2sHH ME and at
$\Delta E\approx 45\,$meV for 3sHH state.
As it is mentioned above, the precise comparative experiments
showed that sample 8/3 had quantum efficiency
$\eta$ about two orders of magnitude lower than those of all other
SL samples investigated. Meanwhile, other features characterising the
quality of the samples such as PL linewidths and Stokes shifts in
PLE spectra are very similar for all samples (1.5-2 meV and 1-1.5
meV, respectively). Moreover, the heavy-hole exciton binding energy
was found to decrease continuously from the sample with $L_b=20\,$nm
to the sample with $L_b=2\,$nm without any peculiarity for the structure
with $L_b=3\,$nm <cit.>. These two facts imply that the band
structure of sample 8/3 in the energy range close to the energy
of the superlattice ground state is as yet unperturbed. We can suppose
that strong non-radiative recombination channels are most likely
due to deep trapping centers, which originate from native lattice
defects <cit.>. However, the nature of non-radiative centers which
is caused by growth procedure details of a particular sample plays
no significant role in our study.
Closing this Section we would like to note that the intensity
of PLE peaks would reflect the absorption efficiency only in the
case of $\eta=1$. This condition
together with the condition of carriers radiative lifetime being long
compared with their relaxation time would permit the excited MEs to
relax into the ME ground state without scattering into other states which
relax later non-radiatively.
On the other hand, in our case of $\eta \ll 1$ the intensity of some
peaks can be resonantly enhanced by the mechanism which strongly reduces
the relaxation time of the ME transition from the excited state to the
radiative ground state and hence decreases to some extent the probability of
non-radiative ME escape.
§ THEORY
§.§ Qualitative consideration of the transition
and formulation of the problem
First note that both states, namely initial $|i\rangle$,
which is 2sHH
or 3sHH, and final $|f_0\rangle=$1sHH have very small exciton
wave-vectors, since $|i\rangle$ arises by virtue of direct light
absorption, whereas $f_0$-relaxation directly provides the optical
PLE signal. More accurately if
$\mbox{\boldmath $q$}=(k_x,k_y)$ and $k_z$ are the transverse and
components, then for these states we find for the used experimental
geometry that
q\lesssim 10^4\,\mbox{cm}^{-1},\quad k_z\lesssim 10^5\,\mbox{cm}^{-1}.
\eqno(3.1)
Here the right sides are determined by the
homogeneity breakdown due to impurities or by momenta of absorbed
and emitted photons. Meanwhile the actual extent
of the exciton wave functions is of the order of $10\,$nm, since,
it is determined by three values: the
magnetic length $l_B=(c\hbar/eB)^{1/2}$, the
effective exciton Bohr radius $a_B=\hbar^2\varepsilon_0/\mu e^2$, and
the period $d$. In the scale of inverse lengths this reads
k_0\sim 2\pi/l_B, 2\pi/a_B, 2\pi/d \sim 10^6\,-\,10^7\,\mbox{cm}^{-1}.
\eqno (3.2)
The significant difference between the values (3.1) and (3.2) allows
one to conclude that the considered resonant transition $i\to f_0$ is
indirect one. Indeed, due to momentum conservation, in a direct transition
the emitted optical phonon would have a negligibly
small wave-vector (3.1). As a
result the macroscopic LO polarization field applied to ME may be
considered to be homogeneous, and in the limit $k/k_0\to 0$ the
corresponding transition matrix element would be simply proportional
to $|\langle i|\mbox{\boldmath $r$}|f_0\rangle|$
(with $\mbox{\boldmath $r$}=\mbox{\boldmath $r$}_2-\mbox{\boldmath $r$}_1$
being the difference between electron and hole positions), which turns out
to be equal to zero because of the identical symmetry of the initial
and final states with respect to inversion. Thus we consider
the LO transition not right into $|f_0\rangle$ but initially
into $|f\rangle$ which is a 1sHH exciton state with the wave vector
$k\sim k_0$. Finally, the transition $f\to f_0$ is a non-radiative
process, e.g. provided by acoustic phonon emission.
The assumed transitions are schematically
demonstrated in Fig. 3. The transitions resulting in
non-radiative exciton relaxation are also shown in this diagram.
One can see that in the assumed scheme the PLE signal is
proportional to the rate of the allowed LO phonon emission.
Among the others the diagram reflects one essential
simplification used in our calculations presented below: we
ignore the broadening of the exciton peaks, which naturally
occurs due to crystal and SL imperfections (the curves in
Fig. 3 have zero widths). The diagram implies that
$\Delta E=E_i-E_{f_0}$ should be larger than
$\hbar\omega_{\mbox{\scriptsize LO}}$. However, in the experiment
the beginning of the enhanced relaxation for 2s-exciton occurs
even at $\Delta E \approx 33\,$meV (see Fig. 2), which is lower
than the LO phonon energy in a bulk GaAs crystal $\approx 36\,$meV.
Here we should note that the value of $\Delta E$ (extracted from
PLE spectra) corresponds to the separation of the excited and
ground state exciton absorption maxima. Meanwhile the density
of states in the vicinity of the ground state is presented by a
rather wide band, and at low temperatures the emission comes
mostly from the lower states. This fact provides an effective
Stokes shift; so that the resonant enhancement of the ground-state
luminescence due to LO-phonon mediated relaxation may start at
$\Delta E \approx \hbar \omega_{\mbox{\scriptsize LO}} -
\Delta_{\mbox{\scriptsize PL}} - \Delta_{\mbox{\scriptsize St}}$,
where the PL linewidth $\Delta_{\mbox{\scriptsize PL}} \approx 2\,$meV
and the Stokes shift $\Delta_{\mbox{\scriptsize LO}} \approx 1\,$meV.
Both values $\Delta_{\mbox{\scriptsize PL}}$ and
$\Delta_{\mbox{\scriptsize St}}$ are determined by disorder
In order to describe the data presented in Fig. 2 in the
approximation of zero exciton-level width we have to employ an
effective energy of LO-phonon which is lower than the tabulated
bulk value. As it was described above this disagreement can be
easily eliminated by the consideration of the finite exciton peak
widths. However, for the simplicity of the model this procedure
is omitted in our calculations. The deviation of the LO-phonon
energy from the bulk value can also be related to an inevitable
effective averaging of the SL multi-mode phonon spectrum in the
single-mode approach employed in our model. Later in Sec. IV we
will briefly discuss the role of the multi-mode spectrum in the
context of the phenomenon studied.
Thus our approach should be considered as a theoretical model,
which simplifies the analytical calculation of the relaxation
rate as follows:
i) results are obtained in the strong magnetic field
approximation, which enables to separate the transverse and
longitudinal variables.
ii) we consider only the heavy-hole band ignoring its
non-parabolicity and the difference in the
effective hole masses in GaAs and Al${}_{0.3}$Ga${}_{0.7}$As layers,
though we take into account the anisotropy of effective hole mass.
iii) we also ignore the difference of the effective electron masses
in GaAs and Al${}_{0.3}$Ga${}_{0.7}$As layers.
iv) we consider only one LO-phonon mode with the effective energy
$\hbar\omega_0=33\,$meV independent of phonon
wave-vector direction (to avoid the possible misunderstandings we replace
everywhere below $\omega_{\mbox{\scriptsize LO}}$ by $\omega_0$). Besides,
as in the case of bulk GaAs <cit.> we use only the Fröhlich-type
Hamiltonian for electron-LO-phonon as well as for hole-LO-phonon
interactions (c.f. Refs.
iv95 and na89).
v) we ignore possible momentum conservation breakdown and finite widths
of the excitonic peaks which occur
due to random impurity potential or quantum well and interface roughness.
vi) finally, any spin-orbit terms in the used Hamiltonians
are disregarded, therefore the presented theory does not take into
consideration the effects of “fine structure" in the PLE signal dependent
on light polarization (such effects can be rather peculiar, see e.g.
Ref. ba98).
In spite of these essential assumptions we believe that
such a simplified approach accounts for the most important aspects
of the transition shown in Fig. 3 and
yields reliable information about the ME relaxation rate.
According to Fig. 3 a considerable enhancement takes place when the
intermediate state $|f\rangle$ is a real (not virtual) state of the lowest
excitonic band. The general formula for the total probability of the LO
phonon emission is
W_{\mbox{\scriptsize LO}}=\sum_{f} W_{if}\,, \eqno (3.3)
where $W_{if}$ is the probability of the transition into the
state $|f\rangle$. Meanwhile subsequent processes of relaxation
$f\to f_0$ are not the matter of our interest here.
Evidently the energy conservation leads to the equations
E_i(B)-E_0(B)= \hbar\omega_0+{\cal E}(\mbox{\boldmath $k$})\,,
\qquad
E_f-E_0={\cal E}(\mbox{\boldmath $k$})\,. \eqno (3.4)
Here the intermediate exciton state energy is written as
$E_{f}=E_0+{\cal E}(\mbox{\boldmath $k$})$ where $E_0=E_{f_0}$ is the
ground state energy and ${\cal E}(\mbox{\boldmath $k$})$ is the
excitonic kinetic energy. The left side in the first Eq. (3.4)
depends quasi-linearly on $B$ (c.f. the insets in Figs. 2a,b)
because $E_i-E_0\equiv
N\hbar\omega_c+\delta U(B)$, where $\delta U$, being much less than
$\hbar\omega_c$, is the difference of
binding energies in $|i\rangle$ and $|f_0\rangle$ states.
At the same time we will see that in strong
magnetic field the matrix element for the $i\to f$
transition has a rather sharp maximum in the vicinity of
$q=q_m \sim l_B^{-1}$,
which provides really the resonant
dependence of PLE signal on the magnetic field.
Let us employ the same material
parameters as in Refs. di91,di90, i.e. for the
transverse (in the layer of the well) and for the longitudinal
hole masses we get $m^*_{h\perp}=0.18m_e$ and
$m^*_{h\parallel}=0.34m_e$, respectively. The electron mass is
$m^*_{e\perp}=m^*_{e\parallel}=0.067m_e$, and the dielectric
constant $\varepsilon_0$ is equal to 12.5. Then
the “transverse" excitonic Bohr radius
$a_0=\hbar^2\varepsilon_0/\mu e^2$ is $14\,$nm, and
for the actual magnetic fields we obtain $l_B\le L_w<a_0$. This
fact justifies the employment of the strong magnetic field
l_B \ll a=\min{(a_0,\: L_w)}
($a$ is the characteristic distance between an electron and a hole in
${\hat z}$-direction).
The specific dependence ${\cal E}(\mbox{\boldmath $k$})$ is
unknown. Nevertheless, the calculations for “free"
exciton<cit.> in a strong magnetic field and for the
exciton in SL with $L_w=L_b$ and $B=0$ <cit.> make it
possible to estimate this value and to find that
$\mbox{\footnotesize $\partial $}{\cal E}/{\mbox{\footnotesize
$\partial $}q}
\sim \mu e^4 l_B^2 q/\varepsilon_0^2\hbar^2 \gg
\mbox{\footnotesize $\partial $}{\cal E}/
{\mbox{\footnotesize $\partial $}k_z}$
Indeed according to
Ref. di90 the miniband width for a SL with
$L_w=L_b=3\,$nm is approximately $6\,$meV. Consequently for our SL
with the same $L_b$ and with $L_w=8\,$nm the miniband width should
be smaller than $1\,$meV <cit.>.
In a strong perpendicular magnetic field this value is to be even more
strongly reduced and thus it becomes negligible in comparison
with the expected characteristic energy of dependence $\epsilon(q)$.
Further, the summation in Eq. (3.3) leads to the result which
contains the density of allowed states $|f\rangle$. This value
is inversely proportional to the Jacobian of the change from
variables of integration over phase
space to the integration over $f$-exciton energy and over
$k_z$ wave-vector component:
\frac{\mbox{\footnotesize $\partial $}({\cal E},\: k_z)}{
\mbox{\footnotesize $\partial $}(q,\:k_z)}\approx
d\epsilon/dq\,. \eqno (3.5)
The transition probabilities in Eq. (3.3) are expressed in terms of
the relevant matrix element ${\cal M}_{i\to f}$,
W_{if}=\frac{2\pi}{\hbar}|{\cal M}_{i\to f}|^2\delta
(E_i-E_{f}-\hbar\omega_0)\,, \eqno (3.6)
which in their turn is calculated using the wave functions of
the excitonic states.
§.§ Excitonic wave functions
We can write the excitonic wave functions
in the following manner (c.f. Refs. go68,le80):
\Psi(\mbox{\boldmath $r$}_1,\mbox{\boldmath $r$}_2,z_1,z_2)=
{\cal L}^{-1}\cdot \exp{\left(i\mbox{\boldmath $R$}\mbox{\boldmath $q$}+
\frac{i}{2l_B^2}[\mbox{\boldmath $r$}_1\times \mbox{\boldmath $r$}_2]
\mbox{\boldmath $B$}/B\right)}
\Phi(\mbox{\boldmath $r$}-\mbox{\boldmath $r$}_0)F(z_1,z_2).
\eqno (3.7)
Here $j=1,\:2$ denotes electron and hole;
$\mbox{\boldmath $r$}_j=(x_j,y_j)$ is the 2D vector;
$\mbox{\boldmath $r$}=\mbox{\boldmath $r$}_1-\mbox{\boldmath $r$}_2,\quad$
$\mbox{\boldmath $R$}=(\mbox{\boldmath $r$}_1+\mbox{\boldmath
$r$}_2)/2,\quad$ $\mbox{\boldmath $r$}_0=\mbox{\boldmath
$B$}\times\mbox{\boldmath $q$}l_B^2/B;\quad$ ${\cal L}$ is the sample
size in
the plane $({\hat x},\hat{y})$. $\Phi(\mbox{\boldmath $r$})$ obeys
two dimensional Srödinger equation in the main approximation of which
the Coulomb interaction can be neglected <cit.>. In this
$\Phi(\mbox{\boldmath $r$})\approx |N,m,\mbox{\boldmath $r$}\rangle$
($N$ is the Landau level number,
$m$ is the magnetic quantum number), where
|N,m,\mbox{\boldmath $r$}\rangle=
\left[\frac{N!}{2^{|m|+1}(N+|m|)!\pi}\right]^{1/2}
\eqno (3.8)
($L_N^m$ are Laguerre polynomials). The energies corresponding to these
functions are
\hbar\omega_c[N+1/2(|m|+\gamma m+1)],
where $\gamma=(m_{h\perp}^*-m_e^*)/(m_{h\perp}^*+m_e^*)$.
In the next approximation Coulomb interaction can be taken into account
with the use of the operator
{\cal H}_{\mbox{\scriptsize int}}=
(\mbox{\boldmath $r$}+\mbox{\boldmath $r$}_0)^2+w^2}\right.,\quad
\mbox{where}\qquad
w=z_1-z_2\,, \eqno (3.9)
in perturbation theory<cit.>. Coulomb interaction in
function $F(z_1,\,z_2)$ should be included from the very
first step. The Bloch theorem for this function takes place
if one changes the variables; namely if
{\cal F}_{k_z}(Z,w)= F(Z+\gamma_1 w,\; Z-\gamma_2 w)\,, \quad
\mbox{where}\quad \gamma_1=m_{h\parallel}^*/M,\quad \gamma_2=m_e^*/M
\,, \eqno (3.10)
{\cal F}_{k_z}(Z,w)={\cal L}_z^{-1/2}e^{ik_zZ}v_{k_z}(Z,w)\,,\qquad
\mbox{where}\quad v_{k_z}(Z+d,w)= v_{k_z}(Z,w). \eqno (3.11)
(${\cal L}_z$ is the sample size along ${\hat z}$). We restrict ourselves
to the one-band approximation assuming for all $i$ and $f$ states that
$v_{k_z}(Z,w)$ presents the ground state function of the two-particle
motion in $\hat{z}$ direction. This function can evidently be
normalized so that $\int_{-\infty}^{+\infty}\int_{Z_0}^{Z_0+d}
§.§ Matrix element calculation and inverse transition
Optical phonons in SLs have considerable energy dependence on
their wave-vector direction. Moreover, for arbitrary direction the
classification of the optical phonon branches as longitudinal and transverse
is impossible due to the inhomogeneity of the superlattice medium along
the $z$-axis.
We choose the simplified model and calculate ${\cal M}_{i\to f}$
employing the Hamiltonian of
exciton-LO-phonon interaction in the following form<cit.>:
{\cal H}_{\mbox{\scriptsize opt}}=\frac{1}{{\cal L}}\left(\frac{\hbar}{{\cal L}_z}\right)^{1/2}
e^{-i\omega_{\mbox{\tiny 0}}t}\sum_{\mbox{\footnotesize{{\boldmath $k$}}}}
U_{\mbox{\scriptsize opt}}(\mbox{\boldmath $k$})\left(
e^{i{\mbox{\footnotesize{{\boldmath $qr$}${}_1$}}}+ik_zz_1} -
e^{i{\mbox{\footnotesize{{\boldmath $qr$}${}_2$}}}+ik_zz_2}
\right) + \mbox{H.c.}\,. \eqno (3.12)
Final results include only the squared modulus of the vertex, which is
|U_{\mbox{\scriptsize opt}}|^2=\frac{2\pi e^2\omega_0}{\overline{\varepsilon}k^2}\,.
\eqno (3.13)
Here ${\overline\varepsilon}^{-1}= \varepsilon^{-1}_{\infty}-
\varepsilon^{-1}_0$ (the standard notations<cit.> are
Note that the calculation of $\langle i|{\cal H}_{\mbox{\scriptsize opt}}|f\rangle$ with the functions (3.8) without Coulomb interaction
gives exactly zero in the result.
Indeed the factorization in the form of the product of one-particle
$N$-Landau-level functions is always possible for these functions.
Therefore if we are interested in the transition between the
different levels $N_1$ and $N_2$, then the matrix element of a
one-particle operator always includes the convolution over the
transverse variables $\mbox{\boldmath $r$}_j$ of one of the
particles $\langle N_1,m|m,N_2\rangle$ which is zero because $N_1\not=N_2$.
Thus ${\cal M}_{i\to f}$ is defined by Coulomb corrections
to $\Phi(\mbox{\boldmath $r$})$, which were discussed above.
Taking into account this comment, the final expression for the
matrix element is
{\cal M}_{i\to f}=\sum_v\frac{\langle v,\mbox{\boldmath $k$}|
{}_{\perp}\!\langle f,\mbox{\boldmath $k$}|{\cal H}_{\mbox{\scriptsize int}}|
v,\mbox{\boldmath $k$}\rangle\!{}_{\perp}|{\cal H}_{\mbox{\scriptsize opt}}|i,
0\rangle}{E_i^{(0)}-E_v^{(0)}-\hbar\omega_0} +
\sum_v\frac{\langle f,\mbox{\boldmath $k$}|{\cal H}_{\mbox{\scriptsize opt}}|
{}_{\perp}\!\langle v,0|{\cal H}_{\mbox{\scriptsize int}}
\,.\eqno (3.14)
Here to calculate the expectations one should use the following rules:
Brackets ${}_{\perp}\!\langle...,\mbox{\boldmath $k$}|...|...,
\mbox{\boldmath $k$}\rangle\!{}_{\perp}$
mean the integration over the transverse variables of Coulomb energy
with the functions (3.7); if $\mbox{\boldmath $q$}=0$ then
in Eq. (3.9) $\mbox{\boldmath $r$}{}_0=0$.
Usual brackets $\langle...\rangle$
mean full expectation (the integration over
$\mbox{\boldmath $r$}{}_1, z_1, \mbox{\boldmath $r$}{}_2, z_2$).
Denominators in (3.14) contain the energy values in zero-order
approximation in the interaction and hence one should assume that
$E_i^{(0)}-E_{f}^{(0)} = N\hbar\omega_c \approx
\hbar\omega_0$.
Indexes $i$, $v$ $f$ are two-dimensional (each of them is a pair $N,m$).
In our case $i=(N,0)$, $f=(0,0)$. The allowed $v$ in the first sum are
$(N,\pm m)$ or $[s,\pm (N-s)]$ with $s=0,\,1,...N-1$; and in the
second sum $v=(0,0)$. Note also that for $\mbox{\boldmath $q$}=0$
the value (3.14) becomes zero.
After all the integrals are calculated, the matrix element
to the first order in $l_B/a$ takes the form:
{\cal M}_{i\to f}=\left(\frac{\hbar}{V}\right)^{1/2}
\frac{e^2}{\hbar\omega_c\varepsilon_0}U_{\mbox{\scriptsize opt}}
T(k_z){\cal G}_N(ql_B)\,,\eqno (3.15)
where $V={\cal L}^2{\cal L}_z$,
T(k_z)=\langle{\cal F}_{k_z}(Z,0)|e^{ik_zZ}|{\cal F}_0(Z,0)\rangle=
\frac{1}{a}\,, \eqno (3.16)
{\cal G}_1(p)=-\frac{4\gamma}{1-\gamma^2}e^{-p^2/4}(1-e^{-p^2/2})+
\sum_{m=1}^{\infty}\frac{\gamma e^{-3p^2/4}mp^{2m}(m+1-p^2)}{2^m(m+1)!
[(1+m/2)^2-m^2\gamma^2/4]}\,, \eqno (3.17)
{\cal G}_{2}(p)=\frac{-\gamma(1-p^2)}{1-\gamma^2}e^{-p^2/4}
\left[1-e^{-p^2/2}(1+p^2/2)\right]-
\frac{2\gamma}{9-\gamma^2}p^2(1+p^2/4)e^{-3p^2/4}
+\sum_{m=1}^{\infty}\frac{\gamma me^{-3p^2/4}(p^2/2)^m
\left[(2+m/2)^2-m^2\gamma^2/4\right]}\,. \eqno (3.18)
Now we have to substitute the expression (3.15) into Eq. (3.6).
Then in Eq. (3.3) with the help of Eq. (3.5) we change
from the summation over states $f$ to the integration over phase
space and further to the integration over $\epsilon$ and $k_z$. We find
then that the $\delta$-function in Eq. (3.6) removes the
integration over $\epsilon$. Finally,
the result for the total probability of the
transition from the excited ME state to some state of the ground ME band
W_{\mbox{\scriptsize LO}}(B)=\frac{e^6\omega_0\Lambda(q)G_{N}(ql_B)}
d\epsilon/dq}\,, \eqno (3.19)
\Lambda(q)=\frac{q}{\pi}\int_{-\infty}^{-\infty}
\frac{|T|^2dk_z}{q^2+k_z^2}\,, \eqno (3.20)
and the functions
G_N(p)=\pi{\cal G}_N^2(p) \eqno (3.21)
are plotted in Fig. 4. One should bear in mind that $q$
in Eq. (3.19) is not independent value but $q=q(B)$
is the root of first Eq. (3.4) with
${\cal E}(\mbox{\boldmath $k$})\approx \epsilon(q)$
Therefore $W_{\mbox{\scriptsize LO}}$ is the function of magnetic field which
may be converted to the function of $(N+1)$sHH-exciton peak position
(see insets in Figs. 2a,b).
Note also that the maximum of function
$G_2$ is shifted substantially to higher $q$ than that of
function $G_1$. This fact accounts for the smaller experimentally
observed $\Delta E_{\mbox{\scriptsize max}}$ for the 2s$\to$1s transition
with respect to that of the 3s$\to$1s one.
Finally one can estimate with the help of Eq. (3.19) the inverse time
of relaxation $i\to f$ in the vicinity of the
resonance when $q\simeq q_{Nm}$.
If $B\simeq 10\,$T,
$d\epsilon/dq\sim 10^{-5}\,$meV$\cdot$cm and $|T|\sim 10^{6}\,$cm${}^{-1}$,
then this time is
\tau_{N}^{\pm}\sim W_{\mbox{\scriptsize LO}}^{-1}\sim 0.01\,\mbox{ns}\,,
\eqno(3.22)
where superscript $+$ or $-$ labels exciton spin quantum
numbers $S_z=\pm 1$ which associated with $\sigma^{\pm}$ luminescence
polarizations. Generally, the considerable spin-orbit coupling manifests
itself in the experimental data, and accordingly we should
label by “$+$" or “$-$" all the
quantities $T$, $\Lambda$, $\epsilon$, $q$ (considered in their turn as
the functions
of either $B$ or the peak position $\Delta E=E_i-E_{0})$.
§ DISCUSSION
The intensity of
the PLE signal under the resonant conditions should be proportional to
the inverse time (3.22). Nevertheless an
immediate comparison with experimental data of Figs. 2a,b is
impossible as long as the functions $\epsilon_{\pm}(q)$
are unknown. The alternative approach is to find these
functions being guided by this comparison. Note that the following
results for $\epsilon_{\pm}$ obtained below are rather qualitative
and should be considered as an estimate of the energy dependence on the
component $q$.
Let us specify the
form of energy dispersion phenomenologically as
\epsilon_{\pm}(q)={\cal E}_b\frac{(g^{\pm}_1+
{1+g^{\pm}_3(ql_B)^2}\,, \eqno (4.1)
where ${\cal E}_b$ is a parameter of the order of the exciton binding
energy<cit.> and $B$ is
measured in Teslas. Naturally the parameters
$g^{\pm}_i$ should be the same
for the both resonant peaks (that is they are independent
of excited exciton quantum number $N$), but the set of these
parameters varies with the spin quantum number.
The functions (4.1) with ${\cal E}_b=5\,$meV are presented
in Fig. 5 for the
specific sets of $g^{\pm}_i$ (indicated in the caption)
and for various
magnetic fields. Now we can find the values $q_{\pm}$ from
Eqs. (3.4) as functions of $\Delta E$ (see Figs. 6a,b). Meanwhile
the dependence $B(\Delta E)$ for magnetic field entering
Eq. (4.1) (and also indirectly through $l_B$) is extracted from
the experiment. Therefore Eqs. (3.4), (3.19) and (4.1) lead to
the formula of relevant PLE intensities $I^{\pm}_N \propto
W_{\mbox{\scriptsize LO}}$, namely, in arbitrary units (a.u.) of Figs. 2a,b:
I^{\pm}_N(\Delta E)=\frac{C_{\pm}{\cal E}_b\Lambda_{\pm}(q_{\pm})}
G_N(q_{\pm}l_B)\,. \eqno(4.2)
Here the additional parameters $C_{\pm}$ arise, which are required
for fitting the experimental data presented in arbitrary units.
Besides, one should specify the functions $\Lambda_{\pm}$. We do it
with the help of Eq. (3.20) and two-harmonic expansion for the periodic
functions $T_{\pm}(k_z)$,
T_{\pm}\propto 1+h_1^{\pm}\cos{(k_zd)}+h_2^{\pm}\cos{(2k_zd)}\,.
\eqno(4.3)
The solid lines in Figs. 2a,b correspond to the dependencies
(4.2) with $C_+=6.1\,$a.u.$\times$T${}^{3/2}$ and
$C_-=10.6\,$a.u.$\times$T${}^{3/2}$. In our calculations we
use $h_1^+=1.68$, $h_2^+=0.902$, $h_1^-=1.79$, $h_2^-=1.34$ and
the sets of functions $g_i^{\pm}$ which are presented in
Fig. 5. These parameters are found during the fitting of the
experimental points of Figs. 2a,b.
Meanwhile the picture of such a comparison remains qualitatively
the same even in the case when $h_1^{\pm}=h_2^{\pm}=0$. It is also
important that all the optimum parameters $C_{\pm},g_i^{\pm}$ and
$h_i^{\pm}$ turn out to be of the
order of $0.1\,-\,1$. This confirms the validity of the
estimate (3.22) and indirectly the choice of the
functions (4.1) and (4.3).
Summarizing the results of the paper we see
that our theory is in satisfactory agreement with
the experimental data. Comparison with experiment
leads to reasonable dependencies $\epsilon(q)$ which are
presented in Fig. 5. At the same time, a more detailed theory
taking into account real multi-mode phonon
spectrum in SL <cit.> is yet to be developed.
In the real situation we can expect that with changing
$\Delta E$ ME relaxation mediated by different dominant
optic-phonon modes should occur. We think that this change
to another type of optic phonons explains the appearance of the
shoulders in resonance profiles mentioned in Sec. 2. Also one can expect
a quasi-continuous increase of the frequency of emitted phonons in
comparison with the employed parameter $\omega_0=33\,$meV
with increasing $\Delta E$. This implies that the real bands $\epsilon(q)$
are more narrow than those calculated in the frame of our model.
Actually only the initial portions of the Fig. 5 curves [it seems
for $\epsilon(q)<10\,$meV] should reflect a real exciton dispersion.
Finally, note that there are other hypothetical ways for the
excited ME states to increase their intensity in PLE spectra. First,
the increase in the light absorption (not in the ME relaxation) can
be caused by a resonant increase in oscillator strengths of the direct
radiative transition. This can occur due to the mixing of MEs states
with some other quasi-particle states in SL. Second, the increase in the
relaxation rate may appear because the intermediate
$f$-state overlaps ME states of the region of increased density of
states, namely, at SL miniband edge. However, all these opportunities
can not lead to the observed increase in the intensity by a factor of
more than 20 times. Moreover, in these cases similar resonances also
should be observed in the samples with $L_b=2$ and $5\,$nm (of
course at different magnetic fields), which does not occur.
The authors thank V. D. Kulakovskii for useful discussions and
R. M. Stevenson for the critical reading of the manuscript.
This work is supported by Russian Foundation for Basic Research.
A. I. T. and V. B. T. thank also INTAS and the Sci.-Technical
Program on the Physics of Solid State Nanostructures for the support.
G. Bastard, Wave Mechanics Applied to Semiconductor
Heterojunctions (Les éditions de physique, Les Ulis, 1988).
E. L. Ivchenko, and G. E. Pikus, Superlattices and Other
Heterostructures: Symmetry a. Optical Phenomena (Springer-Verlag,
Berlin, 1995).
B. Jusserand, D. Paquet, A. Regreny, Phys. Rev. B 30, 6245 (1984).
A. K. Sood, J. Menendez, M. Cardona, and K. Ploog, Phys. Rev. Lett. 54,
2111 (1985); ibid 54, 2115 (1985).
T. Tsuchiva, H. Akera, T. Ando, Phys. Rev. B 39, 6025 (1989).
J. Menéndez, J. of Lum. 44, 285 (1989).
V. F. Sapega, M. P. Chamberlain, T. Ruf, M. Cardona,
D. N. Mirlin, K. Totemeyer, A. Fischer, and K. Eberl, Phys. Rev.
B 52, 14144 (1995).
Yu. A. Pusep, S. W. da Silva, J. C. Galzerani, A. G. Milekhin,
V. V. Preobrazhenskii, B. R. Semyagin, and I. I. Marahovka, Phys.
Rev. B 52, 2610 (1995).
G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. 26,
1974 (1982).
R. L. Greene, K. K. Bajaj, and D. E. Phelps, Phys. Rev. B 29, 1807
L. Vi$\tilde{\mbox{n}}$a , G. E. W. Bauer, M. Potemski, J. C. Maan,
E. E. Mendez, and W. I. Wang, Phys. Rev. B 41, 10767 (1990).
M. Bayer, T. L. Reinecke, S. N. Walck, V. B. Timofeev, and A. Forchel,
Phys. Rev. B 58, 9648 (1998).
K. Fujiwara, K. Kawashima, T. Yamamoto, N. Sano, R. Gingolani, H. T. Grahn,
and K. Ploog, Phys. Rev. B 49, 1809 (1994).
V. Mizeikis, D. Birkedal, W. Langebein, V. G. Lysenko, and J. M. Hvam, Phys.
Rev. B 55, 5284 (1997).
A. I. Tartakovskii, V. B. Timofeev, V. G. Lyssenko, D. Birkedal,
and J. V. Hvam, JETP 85, 601 (1997).
mirl81 D. N. Mirlin, I. Ja. Karlik, L. P. Nikitin, I. I. Reshina,
and V. F. Sapega, Solid State Commun. 37, 757 (1981).
ruf T. Ruf, in Phonon Raman Scattering in
Semiconductors, Quantum Wells and Superlattices, Vol. 142 of
Springer Tracts in Modern Physics
(Springer-Verlag, Berlin, 1998) p.163.
kash93 J. A. Kash, Phys. Rev. B 47, 1221 (1993).
kashzach J. A. Kash, M. Zachau, M. A. Tischler, and U. Ekenberg,
Phys. Rev. Lett. 69, 2260 (1992); M. Zachau, J. A. Kash,
and W. T. Masselink, Phys. Rev. B 44, 4048 (1991).
mes90 F. Meseguer, F. Calle, C. López, J. M. Calleja,
L. Vi$\tilde{\mbox{n}}$a, C. Tejedor, and K. Ploog, in Proceedings of
the 20th International Conference on the Physics of Semiconductors,
edited by E. M. Anastassakis and J. D. Joannopoulos (World Scietific,
Singapore, 1990), p. 1461.
L. P. Gor'kov and I. E. Dzyaloshinskii, Sov. Phys. JETP 26,
449 (1968).
I. V. Lerner and Yu. E. Lozovik, Sov. Phys. JETP 51, 588 (1980).
J. A. Brum and G. Bastard, J. Phys. C 18, L789 (1985).
M. M. Dignam and J. E. Sipe, Phys. Rev. B 41, 2865 (1990).
A. Chomette, B. Lambert, D. Deveaud, F. Clerot, A. Regreny, and G. Bastard,
Europhys. Lett. 4, 461 (1987).
H. Chu and Y-C. Chang, Phys. Rev. B 39, 10861 (1989).
A. Chomette, B. Deveaud, F. Clérot, B. Lambert, and A. Regreny, J. of Lum.
44, 265 (1989).
D. M. Whittaker, Phys. Rev. B 41, 3238 (1990).
M. M. Dignam and J. E. Sipe, Phys. Rev. B 43, 4097 (1991).
P. Hawker, A. J. Kent, L. J. Challis, M. Henini, and O. H. Hughes, J. Phys.:
Condensed Matter 1, 1153 (1989).
D. J. Barnes, R. J. Nicholas, F. M. Peeters, X-G. Wu, J. T. Devreese, J.
Singleton, C. J. G. M. Langerak, J. J. Harris, and C. T. Foxon, Phys. Rev.
Lett. 66, 794 (1991).
T. A. Vaughan, R. J. Nicholas, C. J. G. M. Langerak B. N. Murdin, C. R.
Pidgeon, N. J. Mason, and P. J. Walker, Phys. Rev. B 53, 16481
The energy of 1sHH which we will use in the following cannot be extracted
from this set of spectra but its dependence on magnetic field was obtained
from the similar experiment previously (see Ref. [14]). In particular, for
$B=0$ this energy is $1.5805\,$meV.
For example see D. Wong, H. K. Kim, Z. Q. Fang, T. E. Schlesinger, and
A. G. Milnes J. Appl. Phys. 66, 2002 (1989);
C. V. Reddy, S. Fung, and C. D. Beling, Phys. Rev. B 54, 11290 (1996).
V. F. Gantmakher and Y. B. Levinson, Carrier Scattaring in Metals
and Semiconductors (North-Holland, Amsterdam, 1987).
K. J. Nash and D. J. Mowbray, J. of Lum. 44, 315 (1989).
A simple estimate for the miniband width is
L_b)}$, where the factor $\alpha$ characterizes the wave function
attenuation in the barrier which depends on the barrier height. If only
$L_w$ is changed by factor 8/3, we arrive at the indicated estimate.
$\sigma^+$ PLE spectra recorded for ground
magneto-exciton state for various magnetic fields.
The excited magneto-exciton peaks are indicated by arrows.
Integrated intensities of the 2sHH and 3sHH PLE peaks
for the both polarizations (a – $\sigma^+$, b – $\sigma^-$)
as a function of the peak
position $\Delta E$ measured from the 1sHH peak energy.
Black and open squares show the experimental data.
The insets present the experimental
dependences of $\Delta E(B)$ for the 2sHH and 3sHH peaks.
Diagram of possible transitions. LO-phonon-emission $i\to f$ gives
rise to the studied effect. Transitions $i\to 1$, $i\to 2$, and
$f\to 3$, lead to nonradiative exciton annihilation.
Transitions $i\to f\to f_0$ and $i\to f\to v\to f_0$
are the examples
of nonradiative 2(or 3)sHH$\to$1sHH relaxation yielding the measured
luminescence signal.
Functions $G_N$ versus the dimensionless “in plane"
wave-vector component. Landau level numbers $N=1,\,2$
correspond to 2s$\to$1s and 3s$\to$1s transitions respectively.
Transverse energy dispersion functions of Eq. (4.1) when
${\cal E}_b=5\,$meV;
$g_1^+=0.338$, $g_2^+=0.019$, $g_3^+=0.0561$, for $\sigma^+$-polarization
(a), and
$g_1^-=-0.054$, $g_2^-=0.108$, $g_3^-=0.0459$, for $\sigma^-$-polarization
Values $q^{\pm}(\Delta E)$ found from the equation
$\epsilon_{\pm}(q)=\Delta E-\hbar\omega_0$ for all
transitions of $\sigma^+$ (a) and $\sigma^-$ (b) polarizations.
|
1511.00887
|
To characterize the thermodynamical equilibrium of DNA chains interacting with a solution of non-specific binding proteins, a Flory-Huggins free energy model was implemented. We explored the dependence on DNA and protein concentrations of the DNA collapse. For physiologically relevant values of the DNA-protein affinity, this collapse gives rise to a biphasic regime with a dense and a dilute phase; the corresponding phase diagram was computed. Using an approach based on Hamiltonian paths, we show that the dense phase has either a molten globule or a crystalline structure, depending on the DNA bending rigidity, which is influenced by the ionic strength. These results are valid at the thermodynamical equilibrium and should therefore be consistent with many biological processes, whose characteristic timescales range typically from 1 ms to 10 s. Our model may thus be applied to biological phenomena that involve DNA-binding proteins, such as DNA condensation with crystalline order, which occurs in some bacteria to protect their chromosome from detrimental factors; or transcription initiation, which occurs in clusters called transcription factories that are reminiscent of the dense phase characterized in this study.
§ INTRODUCTION
Predicting the three-dimensional structure of chromosomes from the primary DNA sequence has become an important goal, as genomic and transcriptomic data are now generated at an elevated pace. In eukaryotes and prokaryotes, transcription of highly active genes has been shown through morphological evidence to occur within discrete foci containing RNA Polymerases (RNAPs). It was later demonstrated that one given focal point was enriched in one type of dedicated transcription factor (TF) <cit.> and one type of gene promoter <cit.>, as well as nascent transcripts <cit.>, thus justifying to name such foci "transcription factories". A thermodynamic model allowed to show that the stiff DNA polymer and properly located attractive sites mimicking TF bridges were necessary and sufficient ingredients to produce a transcription factory through DNA looping <cit.>. Indeed, there is now convincing evidence that chromosomes are organized into loops (Hi-C, 3C, etc...) <cit.>, and that looping brings distant genes together so that they can bind to elevated local concentrations of RNAPs (FISH, 3C, etc...) <cit.>. DNA-binding proteins such as TFs are generally positively charged, thus providing a non-specific interaction with the negatively charged DNA polymer. DNA sequence-dependent binding offers specific interactions. Together, non-specific and specific associations allow proteins to search their target DNA sequences more efficiently via facilitated diffusion <cit.>, which combines three-dimensional diffusion in the bulk volume and mono-dimensional diffusion along the DNA. These considerations motivated studies to characterize the time-scale of the dynamics or anomalous diffusion. Molecular dynamics simulations are used to model proteins that diffuse to DNA, bind, and dissociate. The time scales reached in numerical simulations are usually several orders of magnitude smaller than the biological ones, and thus the phenomena observed during such simulations might be transient and irrelevant biologically. In this paper, we present a study of the properties and phase diagram of a DNA-protein solution, at thermodynamic equilibrium, which entails DNA condensation into compact structures induced by non-specific DNA-binding proteins. The calculated phase diagram is thus expected to be relevant at biological time scales. To do so, we will consider a simplified model in which the nucleus (or bacterial nucleoid) is represented by a closed volume $V$ (Fig. <ref>). The double-stranded DNA chains are modeled as $M$ semi-flexible polymer chains (polymerization index $N$) which interact with $P$ spheres, which represent either transcription factors or structural proteins. We consider the nucleus (or bacterial nucleoid) to be a good solvent for DNA chains, so that monomers experience a repulsive interaction between themselves. Conversely, we assume that there is an attractive interaction between proteins and DNA which allow the proteins to bind to DNA. As for the protein-protein interaction, we will consider a repulsive (hard-core) interaction, but the case of an attractive (e.g. complexation, dimerization...) interaction could be treated in the same way. Finally, we make the assumption that all interactions are non-specific. In the sequel, subscripts $D$ and $P$ will stand for DNA and protein respectively. We will first describe the phase diagram of such a system in the mean-field approximation and show that there is a phase transition from a dilute phase at high temperature, to a concentrated phase of the DNA and proteins at lower temperature, which can be identified as the transcription factory phase. In a second step, we characterize the structure of the dense phase and show that it can adopt a crystalline order, suggesting an interesting parallel with the existence of some DNA biocrystals in vivo. This method is general and can be applied to many genome architecture problems.
A: model of DNA represented as beads-on-string polymers (blue) interacting with proteins (red). Dotted circles stand for clusters with high concentrations of DNA monomers and proteins. B: monomer-monomer interaction is repulsive. C: protein-protein interaction is repulsive. D: DNA-protein interaction is attractive.
§ FLORY-HUGGINS THEORY
§.§ Free energy and thermodynamic functions
In the following, we will study the phase diagram of the bulk of the bacterial cell (or nucleus) in the mean-field approximation. In the context of polymer theory, this approximation is also called the Flory-Huggins theory <cit.>. A similar kind of approach has been used to study the demixion of a mixture of polymers and colloids, in which the interaction is repulsive <cit.>. By contrast, in the present work, the polymer-colloid interaction will be taken as attractive. We will denote by $c_D$ and $c_P$ the concentrations of DNA monomers and proteins, and by $\sigma_D$ and $\sigma_P$ the molecular volume of a DNA monomer and of a protein. The (Flory-Huggins) free energy per unit volume reads:
\begin{align} \label{eq::mean_field_free_energy_volumic}
\begin{split}
\beta f(c_D, c_P) &= \dfrac{1}{2} \alpha_D c_D^2 + \dfrac{1}{2} \alpha_P c_P^2 + v c_D c_P + \dfrac{1}{6} w (c_D+c_P)^3 + c_P \log \dfrac{c_P \sigma_P}{e} + \dfrac{c_D}{N} \log \dfrac{c_D \sigma_D}{e N} \\
\end{split}
\end{align}
where $\alpha_D$, $\alpha_P$ and $v$ are second order virial coefficients denoting respectively the DNA-DNA, protein-protein and DNA-protein interactions, and $w$ is the third virial coefficient, necessary to avoid the collapse of the system. Note that this last term comes mostly from the entropy of the solvent. Indeed, if solvent molecules were present with concentration $c_S$ and molecular volume $\sigma_S$, the solvent translational entropy would be $c_S \log {c_S \sigma_S/e} $. If we assume incompressibility of the DNA-protein-solvent mixture (i.e. $c_D + c_P + c_S = c_0$), the solvent entropy can be written in mean-field as $(c_0-c_D-c_P) \log {(c_0-c_D-c_P)\sigma_0/e}$ which, when expanded to 3rd order in $(c_D+c_P)$, yields the cubic term in Eq. <ref>.
The Gibb's free energy per unit volume is the Legendre transform of Eq. <ref>:
\begin{align} \label{eq::mean_field_gibbs_volumic}
\beta g(c_D, c_P)= \beta f(c_D, c_P) - \mu_D c_D - \mu_P c_P = -\beta \Pi
\end{align}
where $\Pi$ is the osmotic pressure, and where $\mu_D$ and $\mu_P$ are the chemical potentials of DNA monomers and proteins. The total number of particles of the system is fixed, but as we shall see later, these chemical potentials play a useful role when the system separates into two phases at equilibrium. At thermal equilibrium, the Gibb's energy is an extremum: $\partial \beta g / \partial c_D = 0$ and $\partial \beta g / \partial c_P = 0$, from which we deduce the chemical potentials:
\begin{align}
\begin{split}
\mu_D (c_D,c_P) &= \dfrac{\partial \beta f}{\partial c_D} = \alpha_D c_D + v c_P + \dfrac{1}{2} w (c_D + c_P)^2 + \dfrac{1}{N} \ln{\left( \dfrac{c_D \sigma_D}{N}\right) } \\
\mu_P (c_D,c_P) &= \dfrac{\partial \beta f}{\partial c_P} = \alpha_P c_P + v c_D + \dfrac{1}{2} w (c_D + c_P)^2 + \ln{(c_P \sigma_P)}
\end{split}
\end{align}
By inserting the last expressions in Eq. <ref>, it is straightforward to see that $\alpha_D$, $\alpha_P$ and $v$ are indeed identified to the coefficients of a virial expansion.
The free energy in Eq. <ref> is quite general and holds for arbitrary interactions between the constituents. It is possible to relate $\alpha_D$, $\alpha_P$ and $v$ to any pair potential, say $u(r)$, used to model the corresponding interaction. Indeed, these virial coefficients can be computed to lowest order (in the density) using the well known Mayer relation:
\begin{align} \label{eq::mayer_formula}
\alpha = - \int {\mathrm{d}^3 \mathbf{r} \, \left( e^{- \beta u(\mathbf{r})} - 1 \right)}
\end{align}
and the same for $v$.
In physiological conditions, salt (e.g. $NaCl$, $KCl$) and ions (e.g. $Ca^{2+}$, $Mg^{2+}$) are present in solution, giving rise to screened electrostatic interactions. The interactions are therefore short-ranged with a range given by the Debye-Hückel length. Yet , at the mean field level, the specific shapes of the interaction potentials is irrelevant, and the effect of ions in solution only arises through an adjustment of the Mayer coefficients $\alpha_D$, $\alpha_P$ and $v$. The DNA excluded volume coefficient $\alpha_D$, which accounts also for the electrostatic repulsion between negatively charged monomers is positive. The protein-protein coefficient $\alpha_P$ is in general positive (repulsive), due to electrostatic repulsion between identically charged proteins, but can be attractive (negative) for specific molecules, in particular they may undergo dimerization or hybridization. For hard spheres for instance, $\alpha = 2^3 \sigma$, $\sigma$ being the volume of one sphere. We hereafter restrict ourselves to the case where DNA-DNA and protein-protein interactions are purely repulsive (steric).
Conversely, we assume $v<0$, i.e. the DNA-protein interaction has an attractive tail, which is temperature independent (in first approximation). As we shall see, this Flory-Huggins theory predicts the existence of a critical temperature $T^c$. We will assume that $v(T)$ is analytic in $\mid T - T^c \mid$ and can be written to leading order as:
\begin{align} \label{eq::flory_expression_v}
v(T) = v(T^c) \dfrac{\theta - T}{\theta - T^c}
\end{align}
where $\theta$ is the Flory temperature for which $v(\theta)=0$, i.e. the interaction vanishes.
§.§ The regime of phase separation
As mentioned previously, there will be a phase transition when the homogeneous solutions $c_D=MN/V$ and $c_P=P/V$ become unstable. It is well known that generically, when the temperature decreases, the system separates into two phases. At the phase separation point, the homogeneous high temperature phase may stay metastable down to a point, called the spinodal point, where the homogeneous phase becomes totally unstable. The so-called spinodal condition is given by the equation:
\begin{align} \label{eq::spinodal_condition}
\left| \dfrac{\partial^2(\beta f)}{\partial(c_D,c_P)} \right| =
\left|
\begin{array}{l l}
\partial^2 \beta f / \partial c_D^2 & \partial^2 \beta f / \partial c_D \partial c_P \\
\partial^2 \beta f / \partial c_D \partial c_P & \partial^2 \beta f / \partial c_P^2
\end{array}
\right| \le 0
\end{align}
where the array denotes the determinant of the matrix.
In general for $v(T)$ fixed, Eq. <ref> with the equality determines a line of spinodal points, delimiting the region where the homogeneous mean field solution is stable from the region where it is not.
In the unstable regions, the system undergoes a phase separation.
If $T$ is increased, $v(T)$ becomes less negative. At some point, the spinodal lines merge into a point when $T$ reaches a critical value $T^c$ (Fig. <ref>). This is a tricritical point. For $T > T^c$ the homogeneous solution is stable for any value of $c_D^*$ and $c_P^*$, where we used the $\ast$ superscript to emphasize that these concentrations are the mean field solutions in the absence of phase separation. There are critical lines emerging from the tricritical point when the temperature is decreased, as will be seen later.
In a biphasic regime, the concentrations will be different but uniform in each of the two phases, separated by an interface whose energy is not extensive (the interfacial free energy is proportional to the surface of the interface). We label the dilute phase (resp. dense phase) by $I$ (resp. $II$). The total system free energy then reads:
\begin{align} \label{eq::mean_field_biphasic_free_energy}
\dfrac{\beta F^{tot}}{V} = \phi^{I} \beta f(I) + \phi^{II} \beta f(II)
\end{align}
where $f(I)$ is a short-hand for $f(c_D^{I}, c_P^{I})$ and $\phi^I$ and $\phi^{II}$ denote the volume fraction of the dilute and dense phase.
A straightforward minimization of Eq. <ref>, with the constraints of conservation of volume and particles number of $D$ and $P$, yields the usual equations of coexistence between phase $I$ and phase $II$:
\begin{align} \label{eq::mean_field_coexistence_equations}
\left\lbrace
\begin{array}{l l l}
\mu_D(I) &=& \mu_D(II) \\
\mu_P(I) &=& \mu_P(II) \\
\Pi(I) &=& \Pi(II) \\
\end{array}
\right.
\end{align}
where $\Pi$ denote the osmotic pressures of each phase.
The above equations are simply the equalities of the chemical potentials and the osmotic pressures. It trivially implies $\phi^{II} = 1 - \phi$ with $\phi = \phi^I$. Note that Eq. <ref> is a system of 3 equations with 5 variables $(c_P^I,c_D^I,c_P^{II},c_D^{II},T)$, thus it determines a surface of coexistence.
§.§ Results
We present here the results of the mean field theory and defer the discussion of the actual values of the parameters. We will assume that DNA and protein spheres have the same size (diameter $a$) which we will use as the new unit length. The temperature $T$ and the coefficient $v(T)$ are related through Eq. <ref>, and we therefore introduce the order parameter $t$:
\begin{equation} \label{eq::order_parameter}
v(T) = v(T^c) \left( 1 + t \right)
\end{equation}
We choose to discuss the phase separation in terms of $t$ and of the densities $\eta_D = c_D \sigma_D$ and $\eta_P = c_P \sigma_P$. The coexistence surface is then computed numerically by solving Eq. <ref> and is shown in Fig. <ref>. At the critical point, Eq. <ref> has a unique solution, namely the triplet $(\eta_D^c, \eta_P^c, T^c)$. As previously reported for such systems <cit.>, we find that
\begin{align}
\eta_D^c \sim \dfrac{1}{\sqrt{N}}
\end{align}
This is the same scaling as the overlap density for polymer chains of size $N$, which is the density at which polymer coils begin to interpenetrate (see for instance <cit.>). Given that DNA is present in the nucleoid at concentrations close to the overlap density, this suggests that biological systems may function at the vicinity of this tricritical point.
3D representation of the coexistence surface in a $ \left(\eta_D, \, \eta_P, \, t \right)$ coordinate system. Coexistence lines are shown for $\eta_P^*=0.0015$ and $0.015$, with $\eta_D^*= 0.01$ (red lines). Critical lines (black lines) emerge from the tricritical point (black dot).
For $T < T^c$ we have $ v(T) < v(T^c) < 0$ which corresponds to a DNA-protein interaction more attractive than at the critical point and the solutions of Eq. <ref> are distributed on a closed curve. This closed curve is in fact the collection of all pairs of coexisting dilute $(c_D^I,c_P^I)$ and dense $(c_D^{II},c_P^{II})$ phases. They are represented in Fig. <ref>, where the pairs of coexisting phases are connected by tie lines. Although there is an infinite set of possible states of coexistence, the total number of DNA and protein spheres selects a unique solution pair. The resulting coexistence state is uniquely determined by the following relation, whose graphical interpretation is shown in Fig. <ref>:
\begin{align} \label{eq::mean_field_maxwell_rule}
\phi \left(\begin{array}{l} c_D^{I} \\ c_P^{I} \end{array} \right) + (1 - \phi) \left(\begin{array}{l} c_D^{II} \\ c_P^{II} \end{array} \right) = \left(\begin{array}{l} c_D^{*} = MN / V \\ c_P^{*} = P / V \end{array} \right)
\end{align}
Coexistence lines for $t = 0.05, \, 0.5, \, 1.0$. The coexistence line shrinks toward the tricritical point (red dot) when $t \to 0$. For each curve, the dilute phase is shown in green and the concentrated phase is shown in blue. Coexisting states are connected by tie lines (dotted segments). The volume fraction of each phase is determined (black arrows) according to Eq. <ref>.
An important point to note is that in general, the concentration of DNA in the dilute phase is very small.
Indeed, the translational entropy of DNA is small, due to the factor $1/N$, and thus there is no entropic gain for the DNA to be in the dilute phase, whereas it has an important enthalpic advantage to be in the concentrated phase. To illustrate how one can determine the composition of the system, we now explain how phase separation takes place when cooling the system from high to low temperature. We consider the case of DNA and protein densities:
\begin{align} \label{eq::physiological_densities}
\eta_D^* &=0.01\\
\eta_P^* &=0.0015
\end{align}
Again, we defer the justification for this choice of the parameters. The system splits into two phases at a temperature $T^*$. For $T>T^*$ the system is homogeneous with values of the concentration given by Eq. <ref>, whereas for $T<T^*$, the system splits into two phases whose composition is determined by Eq. <ref> and <ref>. The line of coexistence obtained is shown on Fig. <ref>. Note that at $T=T^*$, the phase transition is first order, except when $T^*$ is on a critical line, in which case it is second order. Let us now assume that the concentration of proteins is increased by a factor of ten to $\eta_P^*=1.5 \, 10^{-2}$. When the system is cooled from high temperatures, it splits into two phases, and as before, the coexisting states are distributed on the surface of coexistence. However the mass conservation requirement (Eq. <ref>) yields a different line of coexistence. The new line of coexistence is shown with dashed lines in Fig. <ref> and Fig. <ref>. As might have been expected, an augmentation of the protein concentration results in an increased protein concentration in both the dilute and concentrated phase (Fig. <ref>A). The DNA concentration, however, shows a two-step pattern. When further proteins are added to the solution, the "free" DNA monomers of the dilute phase are transferred to the concentrated one, and consequently, the DNA concentration in the dense phase first increases. But at some point, the DNA concentration in the dense phase reaches a maximum and starts to decrease (Fig. <ref>B-C). Indeed, a fraction of the newly added proteins will populate the concentrated phase and make it swell, while the amount of DNA remains the same. Therefore varying the total quantity of proteins can induce non monotonous variations of the DNA concentration in the phases of the system.
A and B: coexistence lines for $\eta_P^*=0.0015$ and $0.015$, with $\eta_D^*= 0.01$. These are the projections of the coexistence lines of Fig. <ref> as a function of the density of proteins (A) or DNA (B). C: section of the phase diagram at $t=1.5$ (corresponding to the parameters given in Tab. <ref>). A path corresponding to a fixed DNA density of $\eta_D^*= 0.01$ is drawn (black line). D: density ratio $(\eta_D + \eta_P) / (\eta_D^* + \eta_P^*)$ for the dilute phase and dense phase, for a density of DNA fixed to $\eta_D^*= 0.01$.
§ STRUCTURE OF THE DENSE PHASE
§.§ Example of structures
In the last section, we saw that the Flory-Huggins theory predicts the existence of a phase separation between two homogeneous phases. It is well known that within the Flory-Huggins approximation, the chain structure is not taken into account, except through the suppression of the translational entropy of the chains. In particular, the fact that chains may have a strong bending rigidity (long persistence length) does not play any role at this level. Therefore, the predicted structure of the dense phase is that of a melt of collapsed polymer with spheres. However, several studies have highlighted that the bending rigidity of the polymer has an influence on the microstructure of the dense phase <cit.>. This is well characterized (Fig. <ref>). A standard way to characterize the effect of the chain structure is to use the Random Phase Approximation (RPA) (see reference <cit.>). We have performed such RPA calculations, but we don't report it here, because it did not show any interesting instability in the dense phase. The reason for the failure of RPA is that the phase transition from the homogeneous to the separated phases is first order and thus is not driven by critical fluctuations. This is a typical case when RPA fails to give insights about the dense phase structure.
Two equilibrium configurations of a single polymer chain (blue) displaying the coexistence of a dense and a dilute phase, with persistence length $l_p=1$ (A), and $l_p=20$ (B) interacting with proteins (red). For small bending rigidity, the structure of the dense phase is globular whereas it is cylindrical in the other case. We performed simulations with $P=10$ spheres (see methods).
§.§ Theory of Hamiltonian paths
Since RPA is not appropriate to describe the system in the dense phase, we adopt an other approach. Because of their attractive interactions with the DNA, the spheres in which the polymer is immersed, play the role of colloid particles which bridge various parts of the polymers. Consequently, these spheres induce an effective attraction between the monomers.
We thus turn to a model of a semi-flexible polymer chain on a lattice that has been proposed initially to explain the folding of a protein in compact structures <cit.> (see Fig. <ref>). An attraction energy $\epsilon_v$ between non-bonded nearest neighbors is included, which favors compact configurations. A bending energy of the chain is introduced as a corner penalty. It penalizes corners by an energy $\varepsilon_h$ and thus plays the role of a bending rigidity. As we will see, this term induces an ordering transition between a random (molten) globule where corners are mobile in the bulk, and a crystalline phase, where corners are expelled to the surface of the globule. Using a mean field theory, it was shown that depending on the temperature and chain stiffness, three phases can exist, namely a dilute phase where the polymer is swollen, a condensed phase, which we call a molten globule, where the polymer is collapsed and disordered and finally a second condensed phase where the polymer is collapsed but with a local crystalline ordering. The phase diagram is described simply by the two parameters $\epsilon_v$ and $\epsilon_h$. For fixed small $\epsilon_h$, there is a second-order phase transition at a temperature $T=T_\theta$ between a dilute and a disordered condensed phase, followed by a first-order freezing transition at $T_F$ between the disordered condensed phase and a locally ordered condensed phase of the polymer. Upon increasing the chain stiffness $\epsilon_h$, the molten globule region shrinks until it eventually vanishes. For larger stiffness, the polymer goes abruptly from a swollen to a frozen configuration ($T_F > T_\theta$) through a direct first order transition (Fig. <ref>). These theoretical results were readily confirmed and improved by Monte-Carlo simulations <cit.>.
In its simplest form, this model considers a completely collapsed polymer on a lattice. It is represented as a Hamiltonian path (HP) on a lattice, that is a path which visits each site once and only once. Thus the density of monomer is $\eta=1$. A good approximation to the total number ${\cal N}$ of HP on a lattice was shown to be <cit.>:
\begin{equation} \label{eq::hamiltonian_path}
{\cal N} = \left(\frac{q}{e}\right)^N
\end{equation}
where $N$ is the total number of points of the lattice, and $q$ is the coordination number of the lattice, e.g. $q=2d$ on a $d$-dimensional cubic lattice.
In the case of semi-flexible polymers (HP with corner penalty), the partition function is <cit.>:
\begin{align}
\mathcal{Z} = \sum \limits_{ \{ HP \} } e^{-\beta \epsilon_h {N}_C(HP)}
\label{eq::HP_partition_function}
\end{align}
where ${N}_C(HP)$ counts the number of corners of a HP realization.
Two realization of Hamiltonian paths on a cubic lattice. The globular state contains an extensive number of corners whereas the crystalline state contains a non-extensive number of corners (proportional to the surface).
A saddle-point approximation gives the corresponding free energy per monomer:
\begin{align} \label{eq::free_energy_HP}
\beta f = - \ln \dfrac{q(\beta)}{e}
\end{align}
\begin{align}
q(\beta)=2 + 2(d-1)e^{-\beta \epsilon_h}
\end{align}
is an effective coordination number,
$e=2.718\, 28 ...$ and $d$ is the dimensionality of the lattice. Note that if the corner penalty vanishes, we recover the result of Eq. <ref>.
As the temperature decreases, the effective coordination number $q(\beta)$ decreases, and the free energy increases. There is a temperature $T_F$ for which $q(\beta)=e$ giving a free energy per monomer $f(T_F)=0$. For $T<T_F$, $f(T)$ would become positive in Eq. <ref> if the saddle-point approximation were still valid. However $f(T$) is a negative quantity <cit.> and therefore remains zero below the freezing temperature $T_F$. Consequently for temperatures $T>T_F$, corners are mobile in the bulk, leading to a liquid-like structure for the corners, whereas for $T<T_F$, the polymer is frozen in stretched configurations with $f(T)=0$, in which corners are expelled to the surface and polymer segments tend to be aligned inside (Fig. <ref>). These configurations have been studied previously: they are elongated neck structures or toroïds <cit.>, whose typical size is given by
\begin{align}
\dfrac{\epsilon_h}{U(\beta)} \sim l_p
\end{align}
where $U(\beta) = \partial (\beta f) / \partial \beta$ is the internal energy.
This simple model can be extended to the case where the volume fraction $\eta < 1$. It requires the introduction of the parameter $\epsilon_v = -\chi$, where $\chi$ is the Flory parameter of a polymer chain in a solvent and denotes the effective attraction between monomers induced by the proteins. This leads to the computation of the above mentioned phase diagram in terms of the two parameters $\epsilon_v$ and $\epsilon_h$.
§.§ Results
To make the Hamiltonian paths approach more quantitative, it is interesting to relate its parameters to our simplified picture of DNA interacting with proteins. Namely, we would like to relate $\epsilon_v$ and $\epsilon_h$ to the parameters of the Flory-Huggins free energy in Eq. <ref>. But in the last one, the monomer-monomer attraction is mediated by spheres. In the dense phase, the total concentration of monomers and spheres: $c=c_D + c_P$ is essentially fixed to the close packing concentration $c_0$. By inserting this in Eq. <ref> we obtain
\begin{align}
\beta f = \left(\dfrac{\alpha_D + \alpha_P}{2} + v(T) \right) c_D^2 + \dfrac{1}{6} c^3 + (c-c_D) \ln\dfrac{c-c_D}{e}
\end{align}
where we neglected the translational entropy of the polymer in the dense phase and dropped the linear terms in $c_D$ as this results in an adjustment of the chemical potentials. We then have the correspondence:
\begin{align}
\epsilon_v \equiv - v(T) - \dfrac{\alpha_D + \alpha_P}{2}
\end{align}
For low temperature, $v(T)$ can reach large negative values. This is mapped to a large $\epsilon_v$ in the HP model. Depending on the rigidity of the chain, the dense phase might be globular (low $l_p$) or crystalline (large $l_p$).
The effective monomer density in the dense phase is given by
\begin{align}
\eta \equiv \dfrac{c_D}{c_D + c_P}
\end{align}
One important result obtained using a HP model is the phase diagram of a polymer on a lattice (implicit solvent) with bending rigidity, obtained in mean-field in ref. <cit.> and then supplemented by Monte Carlo studies <cit.>. We show here that the phase diagram of a semi-flexible polymer interacting explicitely with spheres in an off-lattice volume is very similar. We performed Brownian Dynamics simulations with a polymer chain of $N=400$ beads and $P=100$ protein spheres in a cubic volume of size $L=100$ with periodic boundary conditions (see methods). Polymer beads and protein spheres interact through a Lennard-Jones potential with a well depth given by the energy scale $\epsilon$ (in $k_B T$). We used a Kratky-Porod model of polymer, with bending rigidity characterized by the persistence length $l_p$. By varying $l_p$ and $\epsilon$ independently, we were able to explore the phase behaviour of this system. We monitored the coil-globule transition by looking at the quantity:
\begin{align}
q=\dfrac{\log R_g}{\log N}
\end{align}
where $R_g$ is the radius of gyration of the polymer. For a self-avoiding polymer with scaling law $R_g \sim b N^{\nu}$, $q = \nu + cst / \log N$. In a good solvent, the polymer is swollen with $\nu=0.588$ whereas in a bad solvent it collapses with $\nu=1/3$ . It is clear that $q$ varies like $\nu$.
Following the same authors <cit.>, we define the quantity $n_\alpha = \sum \mid \mathbf{u}_i \cdot \mathbf{e}_\alpha \mid$ for $\alpha=x,y,z$, in which $i$ runs over all the bonds of the polymer, $\mathbf{u}_i$ is the unit vector having the same direction as the bond $i$ and $\mathbf{e}_\alpha$ is the unit vector of the corresponding $\alpha$-axis. We then define $n_{min}=\min_{\alpha} (n_\alpha)$, $n_{max}=\max_{\alpha} (n_\alpha)$ and
\begin{align}
p=1 - \dfrac{n_{min}}{n_{max}}
\end{align}
For an isotropic configuration, $n_x=n_y=n_z$ resulting in $p=0$. Conversely, for a configuration streched in one direction, say along the x-axis, $n_x=1$ and $n_y=n_z=0$, resulting in $p=1$. Thus $p$ measures the directional order of the polymer.
We plotted the phase diagram obtained as a function of $k_B T / \epsilon$ and $l_p$ (Fig. <ref>). There is a clear similarity with the case of a polymer on a lattice without explicit proteins. However we observe that the coil-globule and globule-crystal transitions occur at a higher interaction energy $\epsilon$. This might be a consequence of going from a lattice model to a continuous model. It might also be due to the fact that in the high temperature regime, the concentration of spheres in solution is smaller than the close packing, therefore making it hardly comparable to an actual solvent. There is a specific persistence length $l_p^c \simeq 10$ such that:
* for $l_p < l_p^c$, the polymer collapses through a second order coil-globule transition, followed by a first order globule-crystal transition when $\epsilon$ increases;
* for $l_p > l_p^c$, the coil-globule transition no longer exists and the polymer collapses directly from a coil to a crystalline phase through a first order phase transition.
Phase diagram obtained for a polymer chain interacting with spheres. The phase diagram is plotted as a function of $k_B T /\epsilon$ and $l_p$, where $\epsilon$ is the strength of the Lennard-Jones DNA-protein interation and $l_p$ is the persistence length. We performed simulations with $P=100$ spheres (see methods).
§ PARAMETERS AND METHODS
Molecular dynamics simulations
In order to sample configurations of our system we used the LAMMPS (Large-Scale Atomic / Molecular Massively Parallel Simulator) software package. The system is coarse-grained so that DNA is modeled as a beads-on-string polymer and proteins as spheres. The simulations are run in the Brownian Dynamics (BD) mode. The resulting dynamics consists in the integration of the Langevin equation:
\begin{align}
m_i \dfrac{\mathrm{d}^2 \mathbf{r}_i}{\mathrm{d}t^2} = \mathbf{F}_i - \gamma_i \dfrac{\mathrm{d} \mathbf{r}_i}{\mathrm{d}t} + \sqrt{2 k_B T \gamma_i} \mathbf{\eta}_i (t)
\end{align}
where $i$ is the index of one bead, $\mathbf{r}_i$ is the position of the bead, $\gamma_i$ the friction coefficient, $\mathbf{F}_i$ is the resulting force exerted on the bead from the rest of the system, $k_B$ is the Boltzmann constant and $T$ is the temperature. The last term is a stochastic force in which $\mathbf{\eta}_i(t)$ is a Gaussian white noise such as: $\left\langle \mathbf{\eta}_i(t) \mathbf{\eta}_j(t') \right\rangle=\delta_{ij} \delta(t-t')$. Simulations were run with $m_i=1$, $\gamma_i=1$, $k_B T=1$, in a cubic volume of size $L=100$ with periodic boundary conditions and we took a polymer with $N=400$ beads. The diameter $a$ of the beads was taken as the unit length.
As for the polymer chain model, beads $i$ and $i+1$ are connected with a finetely extensible non-linear potential (FENE) such that:
\begin{align}
U_{el} (\mathbf{u}_i) = -\dfrac{K_{el}}{2} r_0^2 \ln{\left( 1 - \dfrac{u_i^2}{r_0^2} \right)}
\end{align}
where $\mathbf{u}_i=\mathbf{r}_i - \mathbf{r}_{i-1}$. Simulations were run with $K_{el}= 30 \, k_B T / a^2$ and $r_0=1.5 \, a$.
The bending rigidity of the polymer chain is introduced through the Kratky-Porod potential:
\begin{align}
U_{b}(\theta_i)= K_{b} \left(1 - \cos \theta_i \right)
\end{align}
where $\theta_i$ is the angle between vectors $\mathbf{u}_i$ and $\mathbf{u}_{i+1}$. The bending coefficient is related to the persistence length by $K_b=l_p \times k_B T$.
We restricted our analysis to the case of DNA monomers and protein spheres of same dimension (diameter $a$). This assumption does not alter the main qualitative features of the physics but makes the discussion simpler. Steric as well as attractive interactions between beads are introduced using a truncated Lennard-Jones potential such as:
\begin{align}
U_{LJ}(\mathbf{r}_{ij}) =
\left\lbrace
\begin{array}{l l}
4 \epsilon \left[ \left(\dfrac{a}{r_{ij}} \right)^{12} -\left(\dfrac{a}{r_{ij}} \right)^6 - \left(\dfrac{a}{d^{tr}} \right)^{12} +\left(\dfrac{a}{d^{tr}} \right)^6 \right] & \text{ if } r_{ij} < d^{tr} \\
0 & \text{ otherwise }
\end{array} \right.
\end{align}
Unless stated otherwise, we took the values indicated in Tab. <ref>. The DNA-DNA interaction being purely repulsive, we retrieve $\alpha_D \simeq 2^3 \sigma_D$ where $\sigma_D$ is the volume of one monomer. The same argument holds for the proteins. Conversely, the monomer-sphere coefficient $v(T)$ is largely negative (Tab. <ref> and Fig. <ref>).
1 2 $\epsilon$ $a$ $d^{tr}$ Mayer coefficient $\alpha$
DNA DNA $1.00$ $1.00$ $1.12$ $4.40$
Protein Protein $1.00$ $1.00$ $1.12$ $4.40$
DNA Protein $3.00$ $1.00$ $2.00$ $-62.6$
Parameters for the truncated Lennard-Jones potential modeling the DNA-DNA, protein-protein and DNA-protein interactions. The mean field pair coefficients are computed using Mayer formula.
Truncated Lennard-Jones potential modeling an attractive interaction between DNA and proteins with parameters indicated in Tab. <ref>. A: potential shape. B: Mayer function.
We systematically performed equilibration runs of $10^5$ time units to reach thermal equilibrium. We then performed sampling runs of $10^6$ time units which we used to measure physical quantities. One simulation time units can be understood as follows. The diffusion coefficient can be expressed as $D=a^2 /\tau_B$ where $\tau_B$ is the Brownian time corresponding to one simulation time unit. Therefore, we can do a mapping to physical units. Assuming a diffusion coefficient of $D \simeq 10 \, \mu m^2 s^{-1}$ for a protein in the bacterial nucleoid <cit.>, we find $\tau_B = 600 \, ns$ for $a=6 \, nm$ and $\tau_B=2.0 \, \mu s$ for $a=20 \, nm$.
Mean-field concentrations
To assess physiological concentrations, we assimilate an E. coli bacteria to a cylinder of radius $0.5 \, \mu m$ and length $1 \, \mu m$. We took a genome length of $4.6 \, 10^6$ base pairs ($bp$). We consider that the diameter of one monomer is $1 \, a = 6 \, nm \equiv 17 \, bp $, so that one such chromosome is modeled as a polymer of $N=2.6 \, 10^5$ monomers of diameter $1 \, a$. This leads to a density of DNA $\eta_D \sim 10^{-2}$. As for the proteins, there are several DNA-binding proteins, called nucleoid-associated proteins, which act on the structure of the DNA. In E. coli for instance HU, H-NS, Fis, RecA, Dps and other proteins have this stucturing function. We choose to use Fis as a reference because it is a well known structural protein with a large number of target sites <cit.> and binds to DNA sequences of $17$ nucleotides <cit.>. In E. coli, there are approximately $75,000$ Fis proteins per genome copy <cit.> in the early exponential growth phase, yielding a ratio $\eta_P / \eta_D \simeq 0.3$. This is also consistent with previous numerical studies for which $0.1 < \eta_P / \eta_D < 0.5$ <cit.>. Eventually we suggest that this ratio might be quite general and can also be mapped to eukaryotes. Indeed, we could either consider a human cell (genome length $\sim 3.3 \, 10^9 \, bp$ and nucleus diameter of $10 \, \mu m$) and take $1 \, a = 20 \, nm$ as a unit length (size of protein complex). Considering that the DNA would be packed as euchromatin with linear packing fraction $\nu = 100 \, bp/nm$ <cit.>, the unit length is $1 \, a \equiv 1.7 10^6 \, bp$ and we get $\eta_D \sim 10^{-2}$. In the former case the proteins are assumed to be roughly $6 \, nm$ in diameter while the latter case better describes the interaction of DNA with large protein complexes.
Computation of the phase diagram
The coexistence surface is computed by solving Eq. <ref>. Due to numerical accuracy limitations, we computed the phase diagrams for chains with polymerization index $N=5000$. Since the critical values scale like $1 / \sqrt{N}$ (Fig. <ref>), this arbitrary choice captures the essential features of the $N\to \infty$ limit. Finally, the model relies on a free parameter $w$, which can be used to fit the model. In Flory theory, $w$ is extracted from the development of the entropy of mixing and would be $w_F = {1}/{c_0^2}$, where $c_0$ is the close packing concentration. However cells are crowded environments, containing other species, metabolites or organelles. Furthermore, biological compartments can also restrict the accessible volume. Eventually, this value is underestimated and we arbitrarily choose $w = 10 \times w_F$. Although it could be much larger, the variations of $c_P^c$ above this value are quite slow and will not alter dramatically the previous phase diagrams (Fig. <ref>). With all these parameters, we find for the tricritical point coordinates: $c_P^c = 2.2 \, 10^{-2} \, a^{-3}$, $c_D^c = 7.7 \, 10^{-4} \, a^{-3}$ and $\beta^c v(T^c)=-25.67 \, a^{3}$.
A: the DNA critical concentration $c_D^c \sim 1 / \sqrt{N}$. B: for $w > 10 w_F$ the variations of $c_P^c$ and of the other critical parameters are very slow.
§ DISCUSSION AND CONCLUSION
DNA condensation in vitro
The condensation of DNA induced by DNA-binding proteins or ions has been thoroughly studied. It is well known that DNA collapses from disperse structures corresponding to swollen coil configurations into ordered, highly condensed states. This has been the focus of several in vitro experimental works <cit.>. One important conclusion from these studies is that during its collapse, DNA undergoes phase transitions through the following three phases: isotropic fluid, cholesteric and crystalline (hexagonal), in agreement with our results. As stated above, within the Flory-Huggins theory, the phase transition induced by ions or DNA-binding proteins appears to be first order, except at the tricritical point and on the critical lines. Therefore the transition from the swollen to the condensed state should be discontinuous and present hysterisis effects, which was indeed observed <cit.>. Interestingly, the Flory-Huggins theory predicts another effect. At fixed temperature, there is a line of possible coexisting states. Given a certain amount of DNA let us discuss the consequences of adding proteins to the system. The protein concentration would increase from zero until it reaches a value for which the system splits into two phases. If we keep adding proteins, the system will at some point exit the biphasic regime (Fig. <ref>C-D). This phenomenon called reentrance has been observed in some experimental work <cit.>.
DNA condensation in vivo
Although it is premature to draw any clear biological conclusion, it is tempting to discuss at least qualitatively the effect of DNA condensation on biological functions. In eukaryotes, nucleosomal organization provides an effective protection against detrimental factors. This organization is absent in prokaryotes, which have a significantly lower ratio of DNA-binding proteins <cit.>. However, in harsh environmental conditions (radiations, temperature, oxidating agents and radicals), several bacteria resort to DNA condensation mechanisms to protect their genome. Maybe the most spectacular case is the appearance of macroscopical DNA aggregates with crystal-like order in starved E. coli cells. In stressful conditions, the alternative $\sigma^S$ factor is expressed, in response to low temperature, cell surface stress or oxidative shock. This in turn induces the expression of the DNA-binding protein Dps <cit.>. In starved cells, Dps is the most abundant DNA-binding protein, with approximately 20,000 Dps protein per cell. Consequently, DNA is condensed into crystal-like aggregates, which make it less accessible to damaging factors. This process is reversible and wild-type E. coli cells starved for three days remain unaffected by a high dose of oxidating agents whereas mutants lacking Dps lose viability <cit.>. Interestingly, Dps binds non-specifically to DNA. In regard to what has been discussed in this paper, we may infer that when Dps concentration increases, a dense phase appears. But at a scale of Dps size ($\sim 10 \, nm$), the apparent rigidity of DNA is large ($\sim 50 \, nm$). Therefore, as seen in Fig. <ref>, we might be in a case where the coil-globule transition is precluded by the freezing transition. Other examples of DNA compaction by non-specific proteins seem to exist <cit.>.
Local concentration effects and transcription
Increasing evidence suggests that transcription proceeds from nucleation points called transcription factories, which are formed from the interaction of DNA with general and dedicated transcription factors. Although the non-specific hypothesis is not guaranteed, it is true that RNAP can bind widely onto DNA thanks to its $\sigma$-unit. The Flory-Huggins results from this paper suggest that a biphasic regime can exist, with a dense phase spanning a volume of size $(1-\phi) V$ and with local concentrations of DNA and RNAP increased by a factor of 4-to-8 with respect to the mean-field ones (Fig. <ref>D). This would result in shifting the equilibrium of complexation reactions such as:
\text{ DNA } + \text{ protein } \rightleftarrows RNAP \text{ bound to } DNA
towards the formation of complexes and may favour transcription initiation. This is consistent with some experimental work showing that RNAP clusters are formed during preinitiation and initiation of transcription <cit.>. The same authors also proposed that crowding of enzymes, i.e. higher local concentrations, may aid in rate-limiting steps of gene regulation. From a dynamical standpoint, the confinment of unbound RNAP in a restricted volume of size $(1-\phi)V$ can reduce the search time for a promoter. To this extent, it is worthwhile to point out a recent study claiming that the promoter search mechanism is indeed dominated by 3D diffusion of RNAP over the 1D diffusion along DNA <cit.>.
Structure of the dense phase
Earlier studies have demonstrated that the frozen phase can present various metastable states <cit.>. In the $N \to \infty$ limit ($N$ is the length of one chain), the transition time scale from one to another could be very large, and the system might well never equilibrate within biological timescales. Finally, the parallel drawn between the Hamiltonian paths theory and the Flory-Huggins theory does not pretend to mathematical rigor. One essential difference is that in our case the attractive interaction between monomers is mediated by spheres. A way to compute more precisely the structure of the dense phase would be to go beyond the homogeneous saddle point approximation, for instance by using the so-called self-consistent field theory method <cit.>, which is a very complex method in the case of semi-flexible polymers.
We presented here two complementary frameworks to describe the phase diagram of polymeric fluids induced by colloids, and applied it to a DNA chain interacting with DNA-binding proteins. Starting from a Flory-Huggins free energy, we first computed the mean-field phase diagram and found that at low temperature (i.e. high DNA-protein affinity) a biphasic regime exists, consisting of the coexistence of a dilute phase and a concentrated phase. The dilute phase may correspond to swollen configurations of the DNA whereas the concentrated phase is a model for condensed states of DNA. The theory may also apply to DNA condensation by multivalent ions or proteins in general. Second, we addressed the characterization of the dense phase structure and showed that the chain bending rigidity can have dramatic effects. Without bending rigidity, the dense phase has no directional order and is a molten globule. However, when the chain bending rigidity is large enough, there is a freezing transition from the globular to crytalline phase. Eventually for very rigid chains, the coil-globule transition is precluded by the freezing transition and the phase transition predicted in the Flory-Huggins framework does not occur.
In the cell, the existence of a dense phase could be a good approximation for the transcription factories observed experimentally. It is conjectured that this may increase the rate of success in transcription initiation by means of protein crowding and by enhancing the promoter search mechanism. Note that at a scale coarse-grained to several thousand base-pairs (gene scale), the chromosome is flexible and the dense phase has the structure of a molten globule. Conversely, at a scale of a few base-pairs, the apparent rigidity of DNA is much higher. Thus, the Dps protein, which binds non-specifically to DNA, can induce the collapse of the E. coli chromosome into crystal-like aggregates; the dense phase is then frozen. This is not an efficient state for a searching mechanism. But on the contrary, it is very adequate to protect DNA.
The two frameworks are quite general and can be used to describe biological phenomena where DNA compaction occurs under the cooperative effect of binding proteins. In the future, we plan to apply it to other biological cases when more quantitative experiments become available.
§ ACKNOWLEDGMENTS
The authors thank the MEGA team members at iSSB for excellent discussions. This work was supported by the IDEX Paris-Saclay grant, CNRS Genopole and the ANR project "Synpathic".
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%\markboth{R. Dridi, L. Loveridge}
%Instructions for Typing Manuscripts (Paper's Title)
\title{The many mathematical faces of Mermin's proof of the Kochen-Specker theorem}
%[Approximating relational observables by absolute quantities]
%More than one Author with different Affiliations \par \bigskip
\author{%footnotes
Leon Loveridge$^{a,}$\thanks{[email protected]} {\ and} %\textsuperscript{1},
Raouf Dridi$^{b,}$\thanks{[email protected]}\\ %\textsuperscript{2} and
{\small\sl a. Department of Computer Science, University of Oxford,}\\
{\small\sl Wolfson Building, Parks Rd, Oxford, UK OX1 3QD}\\ %\par
{\small\sl b. Department of Mathematics, University of British Columbia,}\\
{\small\sl Vancouver, BC V6T 1Z2, Canada}%\par \bigskip
\date{\today}
\maketitle
\begin{abstract}
Mermin's simple ``pentagram" proof of the Kochen-Specker theorem is examined from various perspectives. We emphasise the many mathematical structures intimately related to Kochen-Specker proofs, ranging through functional analysis, sheaf theory and topos theory, Coxeter groups and algebraic geometry.
Some novel results are presented along the way.
\end{abstract}
%\author{LEON LOVERIDGE}
%\address{Quantum Group, Department of Computer Science, University of Oxford\\
%Oxford, Oxfordshire \,OX1 3QD, United Kingdom}
%\email{[email protected] }
%\author{RAOUF DRIDI}
%\address{Department of mathematics, University of British Columbia\\
% Vancouver, BC V6T 1Z2, Canada\\}%\footnote{State completely without
%abbreviations, the affiliation and mailing address, including country.
%Typeset in 8 pt italic.}
%\keywords{Kochen-Specker theorem; C*-algebras; root systems; toposes.}
%\ccode{Mathematics Subject Classification 2000: 22E46, 53C35, 57S20}
\section{Introduction}
The assortment of results collectively referred to as the \emph{Kochen-Specker theorem}, discovered by Bell \cite{bell} and Kochen and Specker \cite{KS1} in the 1960s, still occupies a prominent position in the foundations of quantum mechanics. The physical and philosophical content of these theorems expound essential differences between classical and quantum mechanics,
{\it grosso modo} informing us that the mathematical structure of quantum theory is incompatible with a realist interpretation of the theory in the spirit of classical statistical mechanics.
There now exists a vast literature on many aspects of the Kochen-Specker theorem. On the mathematical side this includes ``small"/efficient proofs (e.g., \cite{peres2, penrose1} for proofs in $\mathbb{R}^3$, the record there for the smallest number of rays being held by Conway and Kochen---see \cite{peres1}), general graph theoretic treatments \cite{csw1, csw2, afls1}, generalisations to unsharp observables (e.g., \cite{spek1, paul1, cab1}), preparation and transformation contextuality \cite{spek1}, operator algebras \cite{dor1}, (co-)sheaf and\linebreak (co-)presheaf \cite{ib1,ib2,ib3,ib4,hls1,abram1} approaches and geometry and cohomology \cite{abram2, abrampara}, ``logical" Bell inequalities \cite{ah1}, topos theory \cite{id1, ldr1, hls1}, and non-contextual inequalities \cite{kly1,csw1} akin in style to the Bell inequality \cite{bell1, chsh1}, whose violation detects contextuality, to name but a few.
Contextuality has also recently been pinpointed as a resource for quantum
computation \cite{rau1,rau2,hwve1}.
In this paper we use Mermin's simple ``pentagram" proof \cite{Mermex} of the Kochen-Specker theorem in Hilbert space dimension $8$, along with a corresponding state-dependent version, to emphasise the many
mathematical structures intimately related to Kochen-Specker proofs.
This functions as an introduction to the subject of Kochen-Specker proofs as well as a
survey of the many mathematical viewpoints that now exist on the Kochen-Specker theorem.
After providing some background and motivating the problem that Bell and Kochen and Specker addressed, we give Mermin's state-independent proof based on the non-existence of a valuation for 10-qubit observables, as well as the corresponding state-dependent version. This gives rise to colouring proofs, based on the impossibility of assigning values to projections/rays arising from the spectral resolution of the given observables. We mention another general form of Kochen-Specker scenario called ``all-versus-nothing" arguments \cite{abrampara} of which the Mermin system is an example. We also present a result due to Clifton \cite{clif1}
showing that Mermin's proof may be adapted to the case of position-momentum contextuality for three degrees of freedom. An algebraic perspective due to D\"{o}ring \cite{dor1} is then presented,
before a discussion of Coxeter groups, $E_8$ in particular, which is seen to naturally arise from the
Mermin system.
Three sheaf-theoretic formulations are then given, based on the work of Isham and collaborators and focussing on the so-called spectral presheaf (e.g., \cite{id1}), followed by a covariant approach of Heunen, Landsman, and Spitters \cite{hls1}, and finally an operational approach initiated by Abramsky and Brandenburger \cite{abram1}. A novel algebraic geometric framework is then provided, followed by a short section providing some conclusions and avenues for further investigation.
\section{Background on the Kochen-Specker Theorem}
A basic question in the foundations of quantum mechanics is whether the probabilistic structure
of the orthodox formalism is a reflection of a fundamental property of the quantum world, or comes about as an effective description arising from the theorist's lack of detail about the inner workings of quantum phenomena.
If it is the latter, it is compelling to believe that quantum mechanical observables should have values prior to, and independently of, measurement. Although these values cannot, in general, be realised by a quantum state, they should, in principle, be realised by a ``microstate". In Newtonian physics,
an observable quantity is represented by a\linebreak (Borel/continuous/smooth) function $f:T^*\mathcal{M} \to \mathbb{R}$ on the phase space $T^*\mathcal{M}$. The value $\nu_s(f)$ of $f$ in a state $s \in T^*\mathcal{M}$ is given by $\nu_s(f)=f(s)$. Therefore, the possible values of an observable $f$ are the spectral values of $f$ in the sense of $C^*$-algebras. Moreover, given a function
$h:\mathbb{R} \to \mathbb{R}$, the observable $h(f) :=h \circ f : T^*\mathcal{M} \to \mathbb{R}$
has value $\nu_s(h \circ f) = h(f(s))$ in the state $s \in T^*\mathcal{M}$.
It is then natural to look for a ``hidden state" underlying the quantum description that behaves analogously to a state
in classical physics, and specifically that values quantum observables in an appropriate way.
With $\mathcal{H}$ a complex separable Hilbert space and $\mathcal{L(H)}$ the collection of bounded linear operators (to be viewed as either a von Neumann or $C^*$-algebra), the hidden state
would then be given as a valuation obeying analogous conditions to the classical case, namely
$\nu : \mathcal{L(H)} \to \mathbb{C}$ such that for any self-adjoint element $a \in \mathcal{L(H)}$ and function
$f: \mathbb{R} \to \mathbb{R}$,
\begin{enumerate}
\item $\nu(a) \in spec(a)$ (SPEC),
\item $\nu (f(a)) = f(\nu(a))$ (FUNC),
\end{enumerate}
where condition (2) (FUNC) is understood in the sense of the appropriate functional calculus. An immediate consequence of FUNC is that $\nu$ is linear and multiplicative on commuting operators.
% i.e., if $[a_1,a_2]=0$,
%$\nu(\alpha_1 a_1 + \alpha_2 a_2) = \alpha_1 \nu(a_1) + \alpha_2 \nu(a_2)$
%(with $\alpha_{1,2} \in \mathbb{R}$) and $\nu(a_1a_2) = \nu(a_1)\nu(a_2)$.
Alternatively, one can say that $\nu$ is a multiplicative linear
functional on $C^*(a)$ (with the natural extension to complex combinations), and is therefore
a character of $C^*(a)$. In a finite dimensional Hilbert space, the collection of all such characters are in one-to-one correspondence with the set of non-zero eigenvalues of $a$ and, therefore, characters are generic as valuations.
We also note that $\nu$ can be extended naturally to the whole of $\mathcal{L(H)}$ and FUNC
to complex-valued functions of the appropriate type.
\section{Mermin's Pentagram: Combinatoric Proofs}
\subsection{Colouring and parity proofs}
Mermin provided a simple construction for ruling out a valuation $\nu$ on a collection of ten
self-adjoint operators in $\mathbb{C}^8$, arranged in five \emph{contexts}, given here as collections of commuting observables. With
$\sigma_{x,z} \in M_2(\mathbb{C})$ denoting the Pauli operators and writing $X_1 := \sigma_x \otimes \id \otimes \id$, etc., applying FUNC (and for the final equality, also SPEC) yields
\begin{eqnarray}\label{eq:mermcon}
\nu(X_1X_2X_3) = \nu(X_1)\nu(X_2)\nu(X_3),\label{eq:1}\\
\nu(X_1Z_2Z_3) = \nu(X_1)\nu(Z_2)\nu(Z_3),\label{eq:2}\\
\nu(Z_1X_2Z_3) = \nu(Z_1) \nu (X_2) \nu (Z_3),\label{eq:3}\\
\nu(Z_1Z_2X_3) = \nu(Z_1)\nu (Z_2) \nu (X_3),\label{eq:4} \\
\nu(X_1X_2X_3)\nu(X_1Z_2Z_3) \nu(Z_1X_2Z_3) \nu(Z_1Z_2X_3)=-1 \label{eq:5};
\end{eqnarray}
for the final equation each observable
appearing as an argument in $\nu$ commutes with all others inside a
$\nu(\cdot)$, and we exploit the anticommutativity of $X_i$ and $Z_i$.
On the other hand, taking products of the left-hand sides and right-hand sides of \eqref{eq:1}--\eqref{eq:4} gives
\begin{equation}\label{eq:5-}
\nu(X_1X_2X_3)\nu(X_1Z_2Z_3) \nu(Z_1X_2Z_3) \nu(Z_1Z_2X_3)=1
\end{equation} by SPEC, and therefore there exists no context-independent valuation $\nu$ compatible
with the above constraints, yielding a combinatoric obstruction to a hidden state acting as a valuation. This contradiction
constitutes a so-called \emph{parity proof} of the Kochen-Specker theorem.
\emph{Colouring proofs} can also be extracted from the Mermin system in more than one way;
Kerhaghan and Peres \cite{kpe1}, Waegell and Aravind \cite{arw1}, and Toh \cite{toh1} have each given an example.
Kernaghan and Peres note that each context generates 8 mutually orthogonal vectors, given as the simultaneous eigenvectors of the (commuting) observables appearing in the context. Following Waegell and Aravind \cite{arw1}, we may form the rank-one
projections $P[v_i]\equiv P_i$, and given that for the first context, for example,
$\sum_{i=1}^8 P_i = \id$, we have $\sum_{i=1}^8 \nu(P_i)=1$. Therefore, precisely
one $P_i$ must have value $\nu(P_i)=1$, with all others having a value of zero. This amounts
to colouring precisely one vector/ray out of a complete orthogonal set green, for instance, and all of the others red. The assumption of non-contextuality manifests as the requirement that the given colour must be independent of the context with which the projection is associated (since the contexts have non-zero intersections). It is found that such a colouring is impossible, yielding another combinatoric proof.
To conclude this subsection, we also mention state-dependent versions of the Kochen-Specker theorem
based on the Mermin system. Equation \eqref{eq:5} is consistent with
$\nu(X_1X_2X_3) = 1$, $\nu(X_1Z_2Z_3) = -1$, $\nu(Z_1X_2Z_3) = -1$, and $\nu(Z_1Z_2X_3) = -1$, yielding
the equations
\begin{eqnarray}\
\nu(X_1)\nu(X_2)\nu(X_3)=1, \label{eq:MS1}\\
\nu(X_1)\nu(Z_2)\nu(Z_3)=-1, \label{eq:MS2}\\
\nu(Z_1) \nu(X_2)\nu(Z_3) = -1, \label{eq:MS3} \\
\nu (Z_1) \nu(Z_2) \nu(X_3) = -1. \label{eq:MS4}
\end{eqnarray}
Again, by taking products of the left-hand side and the right-hand side, a contradiction arises.
This is clearly weaker than the contradiction implied by equations \eqref{eq:1}--\eqref{eq:5}.
The state-dependence comes from observing that the numbers on the right-hand side are eigenvalues of the products $X_1X_2X_3$, $X_1Z_2Z_3$, etc., in the Greenberger-Horne-Zeilinger (GHZ) state
$\Psi = \frac{1}{\sqrt{2}}\left(\state{000}+\state{111} \right)$ \cite{GHZ} . Thus the contradiction may be understood as the impossibility of the given observables having values compatible with the quantum predictions in
a specific state, in this case the GHZ state $\state{\Psi}$.
The Hasse diagram illustrating the state-dependent Mermin poset $\mathcal{P}$ is as follows:
\begin{small}
\begin{center}
\begin{equation}\label{d:ghz}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (g1) at (-5,3) {$\{X_1,X_2,X_3\}$};
\node (g2) at (-2,3) {$\{X_1, Z_2, Z_3 \}$};
\node (g3) at (2,3) {$\{Z_1, X_2, Z_3\}$};
\node (g4) at (5,3) {$\{Z_1, Z_2, X_3 \}$};
\node (x1)at (-5,0) {${X_1}$};
\node (x2) at (-3,0) {${X_2}$};
\node (x3) at (-1,0) {${X_3}$};
\node (y3) at (1,0) {$Z_3$};
\node (y2)at (3,0) {$Z_2$};
\node (y1)at (5,0) {$Z_1$};
% \node (min) at (0,-2) {$\{\}$};
\draw (g1)--(x1);
\draw (g1)--(x2) ;
\draw (g1)--(x3) ;
\draw (g2)--(x1) ;
\draw (g2)--(y2);
\draw (g2)--(y3);
\draw (g3)--(x2);
\draw (g3)--(y3) ;
\draw (g3)--(y1);
\draw (g4)--(y2) ;
\draw (g4)--(x3);
\draw (g4)--(y1);
% \draw[preaction={draw=white, -,line width=6pt}] (g1) -- (x2) -- (g3);
\end{tikzpicture}
\end{equation}
\end{center}
\end{small}
\subsection{``All-versus-nothing" arguments}
Mermin \cite{Mermex} dubbed the contradiction obtained from the GHZ state (i.e., \eqref{eq:MS1}--\eqref{eq:MS4})
an ``all-versus-nothing" (AvN) argument. Similar strong forms of contextuality (to be discussed further in Section \ref{subsec:Abram}) have appeared in the literature, and have been generalised in \cite{abrampara}, where a general form of AvN argument is provided based on $mod-n$ linear
equations and is independent of the quantum formalism.
A rich source of examples of quantum AvN arguments is provided in the setting of stabiliser quantum mechanics. $\mathcal{P}_n$ denotes the $n$-qubit Pauli group,
i.e., products of $\mathcal{P}_1 = \left\langle j X, jY , j Z, j \id\right\rangle$, with
$j \in \{\pm 1, \pm i \}$. Then, an AvN argument, in direct analogy with the GHZ case already considered, is a system of local equations of parity type which have no global solution.
The main theorem on this subject in \cite{abrampara} shows that any
``AvN triple" gives rise to an AvN argument.
\begin{definition}
An AvN triple $\left\langle e,f,g \right\rangle$ in $\mathcal{P}_n$ is a triple $\left\langle e,f,g \right\rangle$ of elements of $\mathcal{P}_n$ with global phases +1 which pairwise commute, and which satisfy the following conditions:
\begin{enumerate}
\item For each $i \in \{1,...,n\}$ at least two of $e_i$, $f_i$, $g_i$ are equal.
\item The number of $i$ such that $e_i = g_i = f_i$, all distinct from $\id$, is odd.
\end{enumerate}
\end{definition}
\begin{theorem}
Let S be the subgroup of $\mathcal{P}_n$ generated by an AvN triple, and $V_S$ the subspace stabilised by $S$. For every state $\Psi$ in $V_S$, the empirical model (a precise definition is given in \ref{subsec:Abram}) realised by $\Psi$ under the Pauli measurements admits an all-versus-nothing argument.
\end{theorem}
Here, ``empirical model" refers to the probability table arising from joint distributions defined by $\Psi$ and the observables appearing in each context.
\subsection{Position-Momentum contextuality for three spacetime degrees of freedom}
Clifton \cite{clif1} has observed that the Mermin proof encapsulated in equations
\eqref{eq:1}--\eqref{eq:5} may be adapted to provide a proof of contextuality based on the position and momentum of a spinless particle in three spacetime dimensions. He considers the Weyl form of the canonical commutation relation (CCR) in $\mathcal{H} \equiv L^2(\mathbb{R}^3)$. With $Q$ and $P$ the usual position and momentum operators acting in (some dense domain of)
$\mathcal{H}$ (i.e., $Q=(Q_1, Q_2, Q_3)$, etc.), $p,q \in \mathbb{R}^3$, and writing $U(p)\equiv e^{ip.Q}$ and
$V(q) \equiv e^{iq.P}$, the CCR takes the form
\begin{equation}\label{eq:Weyl}
\end{equation}
The component operators $U(p_i) \equiv e^{ip_iQ_i}$ and $V_i = e^{i q_i P_i}$ satisfy the same
form of CCR as \eqref{eq:Weyl} for each $i$; furthermore,
\begin{equation}\label{eq:com}
[U(p_i),V(q_i)]=0 \text{~whenever~} p_iq_i = 2ni\pi,
\end{equation}
and, with $[\cdot,\cdot]_+$ denoting the anticommutator,
\begin{equation}\label{eq:acom}
[U(p_i),V(q_i)]_+=0 \text{~whenever~} p_iq_i = (2n+1)i\pi.
\end{equation}
Then, as Clifton observed, one may consider once more a valuation $\nu$, defined on self-adjoint
operators in $\mathcal{L(H)}$ and naturally extended to all bounded operators in the obvious way.
The commutativity and anticommutativity displayed by equations \eqref{eq:com} and \eqref{eq:acom}
suggest that one may find suitable values of $p_i,q_i$ to reproduce the Mermin proof, and this is precisely what is done in \cite{clif1}, where Clifton chooses $p_iq_i=(2n+1)\pi$ for all $i$.
The Mermin contradiction stemming from equations \eqref{eq:1}--\eqref{eq:5} relies on the fact that the operators in question are involutions, and therefore that $\nu(X_i^2) = \nu(Z_i^2)=1$. Such a property is, however, lost for $U(p_i)$ and $V(q_i)$. Nevertheless, with $U(-q_i)$, the inverse of
$U(q_i)$, etc., the equations
\begin{eqnarray}
\nu \left( U(-p_1)U(-p_2)U(-p_3)) = \nu(U(-p_1))\nu(U(-p_2))\nu(U(-p_3) \right),\label{eq:1''}\\
\nu(V(q_1)V(q_2)U(p_3)) = \nu(V(q_1))\nu(V(q_2))\nu(U(p_3)),\label{eq:2''}\\
\nu(V(-q_1)U(p_2)V(-q_3)) = \nu(V(-q_1))\nu(U(p_2))\nu(V(-q_3)),\label{eq:3''}\\
\nu(U(p_1)V(-q_2)V(-q_3)) = \nu(U(p_1))\nu(V(-q_2))\nu(V(-q_3)),\label{eq:4''}
\end{eqnarray}
hold by FUNC.
Precisely akin to the Mermin proof, taking products of the right-hand sides and left-hand sides therefore gives
\begin{multline}
\nu(U(-p_1)U(-p_2)U(-p_3))\nu(V(q_1)V(q_2)U(p_3))\nu(V(-q_1)U(-p_2)V(-q_3)) \\ \nu(U(p_1)V(-q_2)V(-q_3))=1,
\end{multline}
and, in direct analogy to Mermin, noticing that each operator appearing as an argument of $\nu$ commutes with all others appearing in $\nu$, the anticommutativity yields
\begin{multline}
\nu(U(-p_1)U(-p_2)U(-p_3))\nu(V(q_1)V(q_2)U(p_3))\nu(V(-q_1)U(-p_2)V(-q_3)) \\ \nu(U(p_1)V(-q_2)V(-q_3))=-1,
\end{multline}
giving the required contradiction.
\section{An Operator Algebraic Perspective: D\"{o}ring's Theorem}
The most general form of the Kochen-Specker theorem is due to D\"{o}ring \cite{dor1} and
concerns valuations on a general von Neumann algebra $\mathcal{R}$ (viewed as a von Neumann subalgebra of $\mathcal{L(H)}$ for some separable Hilbert space $\mathcal{H}$).
\begin{theorem} (D\"{o}ring)\label{th:dor}\\
Let $\mathcal{R}$ be a von Neumann algebra, with self-adjoint part $\mathcal{R}_{\rm sa}$, without type $I_1$ or $I_2$ summand.
Then there is no $\nu : \mathcal{R}_{\rm sa} \to \mathbb{C}$ satisfying SPEC and FUNC.
\end{theorem}
D\"{o}ring actually provides a number of different Kochen-Specker-type theorems in \cite{dor1} which arise from two Gleason-type theorems.
The simplest, corresponding closely in its conclusion to the original Kochen-Specker result
(in which $\mathcal{R} = M_3(\mathbb{R})$), and of which the situation arising from the Mermin system is an example,
is a direct consequence of Gleason's theorem \cite{glea1}, which we recall
states that completely additive measures on the projection lattice $\mathcal{P(H)}$ are given by normal states and thus by density operators via the usual trace formula. The lack
of a dispersion-free (i.e., $\{0,1\}$-valued) state on the projection lattice is then enough to
rule out a valuation $\nu$. The form of Kochen-Specker theorem thus arising
is valid on type $I$ factors.
Remarkably, as D\"{o}ring demonstrates,
any non-abelian von Neumann algebra without summands of type $I_1$ and $I_2$ has enough structure to display
a Kochen-Specker contradiction. We refer to \cite{dor1} for a proof which relies on
a generalised version of Gleason's theorem (see \cite{chr1,yea1,yea2,mae1}; given also in \cite{ham1}) applicable to general von Neumann algebras (i.e., not necessarily factors, and certainly not of type $I$). Roughly speaking, this version
of Gleason's theorem shows that additive measures extend to states provided that the given
algebras have no type $I_2$ summand.
It is immediately obvious that the von Neumann algebra generated by
the state-independent Mermin proof falls within the remit of D\"{o}ring' theorem.
Indeed, it is of the simplest form: consider the ten observables $A_i$ appearing in Mermin's proof, and form $\A := \{A_i\}^{\prime \prime}$ (prime denoting commutant). Of course, $\A=\{ X_1, X_2, X_3, Z_1, Z_2, Z_3\}^{\prime \prime}$, and it is readily verified that $\A=M_2(\mathbb{C})\otimes M_2(\mathbb{C}) \otimes M_2(\mathbb{C}) \cong M_8(\mathbb{C})$. Clearly, $\A$ is a type $I_8$ factor and and hence there is no valuation on $\A_{\rm sa}$.
\section{Geometry: Root Systems, Weyl Chambers and the Emergence of $E_8$}
Mermin's proof connects to the Coxeter group $E_8$ in two simple steps: one collects together the eigenstates defined by the contexts, and then a process of ``completion" is effected as described below. The final set turns out to be the root system of $E_8$, which can be verified
using Coxeter diagrams. Additionally, one can exploit the simple transitive action of $E_8$ on its set of chambers and get a description
of Mermin's proof in terms of galleries.
We briefly review the mathematics involved and refer the reader to \cite{humphreys, nicolas} for details. A
{Coxeter matrix} indexed by $S$, with $S$ a finite set, is a function
$m_{i j}:S \times S \into \{1, 2, \cdots\} \cup \{+\infty\}$ satisfying
$m_{ii}=1$ and $m_{i j}=m_{ji} >1$ for $i\neq j$. Associated to a Coxeter matrix is a
{Coxeter diagram}. Its set of nodes is $S$ and the two nodes $i$ and $j$ are connected if $m_{i j}\geq 3$: we assign 0, 1, 2, or 3 edges between $i$ and $j$ when $m_{ij}$ is 2, 3, 4, or 6, respectively.
The {Coxeter group} $W=W_S$ associated to a Coxeter matrix $m$ is the group with generators $s_i, \, i\in S$
and relations: (1) $s_i^2=1$ and (2) $(s_is_j)^{m_{ij}}=1$ (the braid relations). The pair $(W, S)$ is called a {Coxeter system}. The canonical geometric realisation of Coxeter groups uses root systems.
There, the elements of $S$ are represented by real reflections.
A root system $\Phi \subset \R^n$ (with $n=card(S))$ is a finite set of vectors (roots) satisfying the two simple properties:
\begin{enumerate}
\item For all $\alpha \in \Phi$: $\lambda \alpha\in \Phi$ iff $\lambda=\pm 1$,
\item For all $\alpha, \, \beta \in \Phi$:
$$s_\alpha(\beta): = \beta - 2\frac{(\alpha, \beta)}{(\alpha, \alpha)}\, \alpha \in \Phi.$$
\end{enumerate}
The pairing is the standard inner product on $\R^n$. The transformation $s_\alpha$
is the { reflection} with respect to the {hyperplane (wall)} $ \{x: (x, \alpha)=0\}$; it is an involution. Reflections $s_\alpha$ for $\alpha\in \Phi$ generate the group $W$ (more precisely, generate the Tits realisation of $W$ as a reflection group). By property (2), it is a symmetry group of $\Phi$ and $W$ is a subgroup of the group $\subset O(\R^n)$ of all
orthogonal transformations on $\R^n$. Hence, $W(\Phi)$ preserves angles and lengths.
\medskip
We return now to Mermin's proof and explain the emergence of the group $E_8$. Each context in the proof yields a basis for the Euclidean space $\R^8$. We get 40 vectors in total, corresponding to the five bases. By reflections, this set of vectors is completed to
240 vectors $\alpha_i$ with $\alpha_{i+120} = - \alpha_i$ for all $1\leq i \leq 120$. Coxeter's diagram computation yields the following diagram,
\begin{figure}[!h]
\centering
\POS (-10,0) ="z",
\POS (0,0) *\cir<3pt>{} ="a",
\POS (10,0) *\cir<3pt>{} ="b",
\POS (20,0) *\cir<3pt>{} ="c",
\POS (30,0) *\cir<3pt>{} ="d",
\POS (20,10) *\cir<3pt>{} ="e",
\POS (40, 0) *\cir<3pt>{} ="f",
\POS (50, 0) *\cir<3pt>{} ="g",
\POS (60, 0) *\cir<3pt>{} ="h",
%\POS (40,0) *\cir<3pt>{} ="e",
\POS (70,0) ="y",
\POS "a" \ar@{-}^<<{12} "b",
\POS "b" \ar@{-}^<<{164} "c",
\POS "c" \ar@{-}^<<{48} "d",
\POS "e" \ar@{-}^<<{225} "c",
\POS "d" \ar@{-}^<<{222} "f",
\POS "f" \ar@{-}^<<{60} "g",
\POS "g" \ar@{-}^<<{188} "h",
\POS "h" \ar@{}^<<{73} "y",
%\POS "e" \ar@{}^<<<<{\alpha_4} "f",
\POS "z" \ar@{}^<<<<{} "a",
\endxy$
\caption{Coxeter diagram of $E_8.$}
\end{figure}
which uniquely identifies the Coxeter group $E_8$ (a fact about finite irreducible Coxeter groups).
A Weyl chamber of a root system $\Phi$ is the geometrical realisation of the generating set $S$; it is the connected region of $\R^n$ bounded by the walls of the different generating reflections in $S$.
%For all $\alpha_i, \alpha_j\in \Delta$, we have
%(\alpha_i, \alpha_j)= ||\alpha_i|| ||\alpha_j|| \cos (\pi+ \frac{\pi}{m_{ij}} ).
%The value $\pi+ \frac{\pi}{m_{ij}} $ is the angle between the roots $\alpha_i$ and $\alpha_j$. We recover the integers $m_{ij}$ and the set $S := \{ s_{\alpha_i}, \, {\alpha_i}\in \Delta\}$.
A {gallery} is a sequence of chambers $S_i$ with each successive pair sharing
a face of codimension one. For instance, if $w=s_1\cdots s_n$, then the sequence $S, s_1S, s_1s_2S, \cdots,s_1s_2\cdots s_n S$ is a gallery.
%The chambers $S$ and $sS$ share a face labeled by $s$.
The Coxeter group $W$ acts transitively on its set of chambers; this will help us to translate Mermin's proof from the level of roots to that of chambers.
%(likewise translating KS into topos, no practical gain for now!).
First, we fix a reference chamber $S_0$.
%Each of the 40 reflections is represented by precisely one chamber. This is because of the simple transitive action
%of $E_8$ on the chambers.
Suppose $S_0$ is the chamber given by the diagram of Figure 1.
%$$S_0 := (12, 164, 48, 225, 222, 60, 188, 73)$$
The reflection $s_1$ (with respect to the first root)
is then represented by the chamber
\begin{center}
\POS (-10,0) ="z",
\POS (0,0) *\cir<3pt>{} ="a",
\POS (10,0) *\cir<3pt>{} ="b",
\POS (20,0) *\cir<3pt>{} ="c",
\POS (30,0) *\cir<3pt>{} ="d",
\POS (20,10) *\cir<3pt>{} ="e",
\POS (40, 0) *\cir<3pt>{} ="f",
\POS (50, 0) *\cir<3pt>{} ="g",
\POS (60, 0) *\cir<3pt>{} ="h",
%\POS (40,0) *\cir<3pt>{} ="e",
\POS (70,0) ="y",
\POS "a" \ar@{-}^<<{44} "b",
\POS "b" \ar@{-}^<<{132} "c",
\POS "c" \ar@{-}^<<{20} "d",
\POS "e" \ar@{-}^<<{225} "c",
\POS "d" \ar@{-}^<<{222} "f",
\POS "f" \ar@{-}^<<{60} "g",
\POS "g" \ar@{-}^<<{188} "h",
\POS "h" \ar@{}^<<{73} "y",
%\POS "e" \ar@{}^<<<<{\alpha_4} "f",
\POS "z" \ar@{}^<<<<{} "a",
\endxy.$
\end{center}
%$S_1 := s_1 S_0 = (44, 132, 20, 225, 222, 60, 188, 73).$
The second reflection $s_2$ is represented by
%$S_2 := s_2 S_0 = (163, 11, 139, 82, 85, 32, 188, 73),$
\begin{center}
\POS (-10,0) ="z",
\POS (0,0) *\cir<3pt>{} ="a",
\POS (10,0) *\cir<3pt>{} ="b",
\POS (20,0) *\cir<3pt>{} ="c",
\POS (30,0) *\cir<3pt>{} ="d",
\POS (20,10) *\cir<3pt>{} ="e",
\POS (40, 0) *\cir<3pt>{} ="f",
\POS (50, 0) *\cir<3pt>{} ="g",
\POS (60, 0) *\cir<3pt>{} ="h",
%\POS (40,0) *\cir<3pt>{} ="e",
\POS (70,0) ="y",
\POS "a" \ar@{-}^<<{163} "b",
\POS "b" \ar@{-}^<<{11} "c",
\POS "c" \ar@{-}^<<{139} "d",
\POS "e" \ar@{-}^<<{82} "c",
\POS "d" \ar@{-}^<<{85} "f",
\POS "f" \ar@{-}^<<{32} "g",
\POS "g" \ar@{-}^<<{188} "h",
\POS "h" \ar@{}^<<{73} "y",
%\POS "e" \ar@{}^<<<<{\alpha_4} "f",
\POS "z" \ar@{}^<<<<{} "a",
\endxy,$
\end{center}
etc. The fact that $m_{1, 2}=2$ means that two galleries $S_0, S_1, s_2 S_1$ and $S_0, S_2, s_1 S_2$ intersect
and define a cycle.
%The orthogonality of
%$s_1$ to $s_2$ is translated into intersection of the corresponding galleries $S_0, S_1, s_2 S_1$ and $S_0, S_2, s_1 S_2$. [nb: orthogonality = commutativity since $m_{1, 2}=2$].
Now, suppose the first root is picked (coloured black); then we have a valuation which assigns 1 to $s_1$ and 0 to $s_2$ (root 2 coloured white) and so on. This valuation also assigns 1 to $s_1s_2$. The gallery $S_0, S_1, s_2 S_1$ is then coloured black. This is also valid for any gallery $S_0, S_1, s_i S_1$ with $(s_1s_i)^2=1$. The gallery $S_0, S_2$ is white, which prevents the whole cycle from being coloured black.
For each context, we construct all such galleries. We then group them all together to get a centred network (all galleries start from $S_0$).
The translation of the colouring proof is now straightforward.
%\blue{Nice to show it explicitly! Also, is there a good way to see the contradiction on the level of the Dynkin diagram? I think we tried it once, but a 2-colouring of course does exist there so we need some more structure.}
\section{(Co-)presheaves and (Co-)sheaves}
The Kochen-Specker theorem can be elegantly phrased in terms of functors from a category
to the category of sets. In this section we review three such formulations: the spectral presheaf
$\Sigma$ of Isham {\it et al.}, where the (state-independent) Kochen-Specker theorem concerns the\linebreak (non-)existence of points of $\Sigma$; the covariant approach of Heunen {\it et al.}, and the operational approach of Abramsky {\it et al.}
\subsection{Isham et al.}\label{subsec:isham}
The base category under consideration in the work of Isham, Butterfield, and Hamilton (e.g., \cite{ib3}) is
$\mathcal{V(A)}$, with $\mathcal{A}$ a $C^*$-algebra, $\mathcal{V(A)}$ denoting the partially ordered set (poset) of unital abelian $C^*$-subalgebras, and elements of $\mathcal{V(A)}$
now referred to as contexts. The morphisms in $\mathcal{V(A)}$ are inclusion relations; $U \to V$
if and only if $U \subseteq V$ as abelian algebras (we will also use $\hookrightarrow$ for inclusion arrows).
A presheaf of particular relevance to the Kochen-Specker theorem is the spectral presheaf $\Sigma : \mathcal{V(A)} \to \mathsf{Set}$. For each $V \in \mathcal{V(A)}$,
\begin{itemize}
\item $\Sigma (V):= \{\lambda_V : \lambda_V \text{is a character of } V\}$ equipped with the topology of pointwise convergence, i.e., $\Sigma (V)$ is the Gelfand spectrum of $V$ (in the case of the Mermin proof, the topology is discrete);
\end{itemize}
on inclusions $U \hookrightarrow V$,
\begin{itemize}
\item $\Sigma (U \subseteq V) : \Sigma (V) \to \Sigma (U)$; $\lambda_V \mapsto \lambda_V |_U \equiv \lambda _U$, where the restriction is understood in the sense of function restriction.
\end{itemize}
The state-independent Kochen-Specker theorem concerns the existence of a global section (or \emph{point}) $\tau:1 \to \Sigma$ ($1$ is the terminal element). If such a section were to exist, for each $V$, $\tau_V : \{*\} \to \Sigma (V)$, and
for each $U \hookrightarrow V$, $\tau_V(*)(\lambda_V)=\lambda_V|_U \equiv \lambda_U$. In other words, we would have a valuation for each context giving rise to a global valuation.
The Mermin proof can be phrased in these terms when restricted to the poset $\mathcal{P}$. Let $\tau$ be such a putative global section.
Then, since $\tau_{V_i}(*)$ is a character of each abelian algebra $V_i$,
\begin{eqnarray}
\tau_{V_1}(*)(X_1X_2X_3)\tau_{V_1}(*)(X_1)\tau_{V_1}(*)(X_2)\tau_{V_1}(*)(X_3) = 1,\label{eq:1'}\\
\tau_{V_2}(*)(X_1Z_2Z_3)\tau_{V_2}(*)(X_1)\tau_{V_2}(*)(Z_2)\tau_{V_2}(*)(Z_3) = 1,\label{eq:2'}\\
\tau_{V_3}(*)(Z_1X_2Z_3)\tau_{V_3}(*)(Z_1) \tau_{V_3}(*) (X_2) \tau_{V_3}(*)(Z_3) = 1,\label{eq:3'}\\
\tau_{V_4}(*)(Z_1Z_2X_3)\tau_{V_4}(*)(Z_1)\tau_{V_4}(*)(Z_2) \tau_{V_4}(*)(X_3) = 1,\label{eq:4'}\\
\tau_{V_5}(*)(X_1X_2X_3)\tau_{V_5}(*)(X_1Z_2Z_3) \tau_{V_5}(*) (Z_1X_2Z_3) \tau_{V_5}(*)(Z_1Z_2X_3)=-1\label{eq:5'}.
\end{eqnarray}
However, the choice of characters on each algebra is constrained by the requirement
that $\tau_{V_1}(*)(X_1)= \tau_{V_2}(*)(X_1)$,
$\tau_{V_5}(*)(X_1X_2X_3) = \tau_{V_1}(*)(X_1X_2X_3)$, etc. After making these replacements in equations \eqref{eq:1'}--\eqref{eq:5'}, one again meets with a contradiction.
Given a unit vector $\varphi \in \mathbb{C}^8$ one can define a subpresheaf $\mathfrak{w}^{\varphi}$ of $\Sigma$ in which state-dependent proofs may be considered. Indeed, as demonstrated in
\cite{ldr1}, the GHZ state $\state{\Psi}$ gives rise to the presheaf $\mathfrak{w}^{\Psi}$
which lacks a global section.
A primary motivation for Isham {\it et al.} for constructing $\Sigma$ and $\mathfrak{w}^{\varphi}$ as they do is that they may be viewed as objects in the topos
$\mathsf{Set}^{\mathcal{V(A)}}$, i.e., the topos of all set-valued presheaves on $\mathcal{V(A)}$. A topos
has a multivalued ``truth object" provided by the subobject classifier; Isham and D\"{o}ring \cite{ido1}
then argue for a connection between the quantum probabilities given by the Born rule and
the truth values in $\mathsf{Set}^{\mathcal{V(A)}}$. Furthermore, they argue, the relaxation
of a Boolean truth system to a more general, many-valued intuitionistic one paves the way for
a ``neo-realist" interpretation of quantum theory.
\subsection{Heunen-Landsman-Spitters}
Heunen, Landsman, and Spitters \cite{hls1} have provided another topos-theoretic view on the
Kochen-Specker theorem. With $\mathcal{V(A)}$ denoting the poset of unital, abelian $C^*$-subalgebras of
some fixed $C^*$-algebra $\mathcal{A}$, they construct the topos $\mathcal{T(A)}$---the topos of all \emph{covariant} set-valued functors on $\mathcal{V(A)}$.
The point is to \emph{internalise} the (in general) non-abelian algebra $\mathcal{A}$, yielding
$\underline{\mathcal{A}}$ internal to $\mathcal{T(A)}$ (in this section we follow the convention of the authors of \cite{hls1} in underlining internal entities) which \emph{is} abelian, and is referred to
as the \emph{Bohrification} of $\mathcal{A}$.
$\underline{\mathcal{A}}$ is the ``tautological functor", defined on objects by
\begin{equation}
\underline{\mathcal{A}}(C)=C,
\end{equation}
and on any arrow $C \subseteq D$, $\underline{\mathcal{A}}(C) \hookrightarrow \underline{\mathcal{A}}(D)$. The $C^*$-algebra $\mathcal{A}$ is a (not necessarily commutative) $C^*$-algebra in the topos $\mathsf{Set}$, whereas, as proved in \cite{hls1}, $\underline{\mathcal{A}}$ is a commutative $C^*$-algebra in the ``universe of discourse" provided by
The covariant ``version" of the spectral presheaf $\Sigma$ from Section \ref{subsec:isham}, which will be denoted $\underline{\Sigma}$, gives the internal Gelfand spectrum
$\underline{\Sigma}(\underline{\mathcal{A}})$ of the internal abelian algebra $\underline{\mathcal{A}}$. $\underline{\Sigma}(\underline{\mathcal{A}})$ is an internal locale;
the points of a general locale $X$ in a topos are the frame maps $\mathcal{O}(X) \to \Omega$, with $\Omega$ being the subobject classifier of the topos in question. As shown in \cite{hls1}, the Kochen-Specker theorem now reads:
\begin{theorem}\label{th:hls1} (Heunen-Landsman-Spitters)\\
Let $\mathcal{H}$ be a Hilbert space for which $dim \mathcal{H} \geq 3$ and let $\mathcal{A} = \mathcal{L(H)}$. Then $\underline{\Sigma}(\underline{\mathcal{A}})$ has no points.
\end{theorem}
The proof is roughly as follows (see \cite{hls1}, Theorem 6, and the accompanying proof):
internally, a point $\underline{\rho}: \underline{*}\rightarrow \underline{\Sigma}$ of the locale
$\underline{\Sigma}$ can be combined with a self-adjoint operator $a\in \underline{A}_{\mathrm{sa}}$ through its Gelfand transform
$\hat a:\underline{\Sigma}\rightarrow \underline{\mathbb R}$ to give a point
$\hat a\circ \underline{\rho}: \underline{*}\rightarrow \underline{\mathbb R}$, the latter denoting the locale of Dedekind reals, again viewed internally. This defines an internal multiplicative linear map (natural transformation) $v_\rho: \underline{A}_{\mathrm{sa}} \rightarrow {\mathrm{Pt}( \underline{\mathbb R})}$ with components $v_\rho(V): \underline{A}_{\mathrm{sa}}(V) \rightarrow {\mathrm{Pt}( \underline{\mathbb R})}(V)$, which coincides externally with $C_{\mathrm{sa}} \to \mathbb{R}$, and
is precisely the valuation that is ruled
out by the Kochen-Specker theorem \cite{KS1}.
With regard to the Mermin system, replacing $\mathcal{V(A)}$ by $\mathcal P$ generated by the five contexts involved, we may view the (non-commutative) $C^*$-algebra $\mathcal{A_P}$ again
as an internal commutative $C^*$-algebra $\underline{\mathcal{A_P}}$ in the topos
$Sets^P \equiv \mathcal{T}(\mathcal{A_P})$ with associated internal locale
$\underline{\Sigma}(\underline{\mathcal{A_P}})$. That this locale has no points follows from
considering valuations $v_\rho(V)$ for the abelian algebras of each context, yielding
again the insoluble set of equations \eqref{eq:1'}--\eqref{eq:5'}, with $v_\rho(V)$ coinciding with $\tau(*)_V$.
\subsection{Abramsky et al.}\label{subsec:Abram}
Abramsky and Brandenburger \cite{abram1} have presented an operational, sheaf-theoretic description of contextuality, and Abramsky, with Mansfield and Barbosa, have developed cohomological techniques \cite{abrampara, abram2} for identifying its presence.
Their approach is independent of the Hilbert space formalism, thereby doing away with
SPEC and FUNC and instead focussing directly on the (im-)possibility of a global section
for a compatible family of no-signalling distributions.
Following Abramsky and Brandenburger, we refer to a finite collection of observables $X$, understood as a topological space with the discrete topology
and distinguished open sets (the \emph{contexts} here) $\{C_i \} \equiv \mathcal{M}; \bigcup C_i = X$ as a \emph{measurement cover}. Each $C_i$ is a maximal set of observables which may be measured jointly.
The outcomes of a measurement of any observable $A \in X$ are given by the set $O$ and, therefore, under the assumption that all observables have the same outcome sets,
for a measurement of $n$ observables in $X$, outcomes are in $O \times O \times ... \times O \equiv O^n$. The \emph{event sheaf} $\mathcal{E}$
is defined
by $\mathcal{E}(U):= O^U \equiv \{ f:U \to O\}$ for $U \subset X$, i.e., the collection of assignments for observables in $U$. Elements of $\mathcal{E}^U$,
or \emph{sections} above $U$, are therefore viewed as possible assignments of observables in
$U$. An \emph{empirical model} $e$ specifies joint probability distributions $e(C)$ over assignments
in $\mathcal{E}(C)$, where $C \in \mathcal{M}$, to be compatible in the sense of the no-signalling principle (or, equivalently, the sheaf condition: see \cite{abram1}).
The \emph{support} presheaf $S_e$
of $e$ specifies a subpresheaf of $\mathcal{E}$ defined by $S_e(U)=\{s \in \mathcal{E}(U)~:~s \in \text{supp}e_U\}$.
Abramsky and Brandeburger \cite{abram1} identify three levels of contextuality/non-locality arranged in a proper hierarchy---probabilistic non-locality, possibilistic non-locality, and strong contextuality---in which the state-dependent Mermin system, and indeed all AvN arguments, occupies the strongest level of strong contextuality. The next level
down corresponds to the situation where outcome probabilities can be neglected and all the matters is what is \emph{possible}, i.e., on the level of supports; the Hardy model occupies this level and is \emph{not} strongly contextual. In turn, the Bell model is probabilistically non-local but \emph{not} possibilistically non-local, thus showing that each level in the hierarchy can be realised by quantum mechanics. It should be noted, however, that quantum mechanics is not a necessary requirement---for example, Podolosky-Rosen boxes are also strongly contextual \cite{abram1}.
\begin{definition}
The model $e$ is called \emph{strongly contextual} if $S_e(X) = \emptyset$.
\end{definition}
As shown in \cite{abram1}, the \emph{GHZ models} (for at least three parties) are strongly contextual; We also note that all AvN arguments are strongly contextual. To exhibit strong contextuality in our familiar setting
we turn our attention to the Mermin state-dependent proof, with the following table:
\begin{center}
\begin{tabular}{| l | l | l | l | l | l | l | l | l |}
\hline
& $+++$ & $++-$ & $+-+$ & $+--$ & $-++$ & $-+-$ & $--+$ & $---$ \\ \hline
$C_1 = X_1, X_2, X_3$ & $1$ & $0$ & $0$ & $1$ & $0$ & $1$ & $1$ & $0$ \\ \hline
$C_2=X_1, Y_2, Y_3$ & $0$ & $1$ & $1$ & $0$ & $1$ & $0$ & $0$ & $1$ \\ \hline
$C_3=Y_1, X_2, Y_3$ & $0$ & $1$ & $1$ & $0$ & $1$ & $0$ & $0$ & $1$ \\
\hline
$C_4 = Y_1, Y_2, X_3$ & $0$ & $1$ & $1$ & $0$ & $1$ & $0$ & $0$ & $1$ \\
\hline
\end{tabular}
\end{center}
There are four ``potential" global sections compatible with the assignments for $C_2$, $C_3$, and $C_4$,
\begin{eqnarray}
s_1 : (X_1, X_2, X_3, Y_1, Y_2, Y_3) \mapsto (++-++-),\\
s_2 : (X_1, X_2, X_3, Y_1, Y_2, Y_3) \mapsto (+-++-+),\\
s_3 : (X_1, X_2, X_3, Y_1, Y_2, Y_3) \mapsto (-++-++),\\
s_4 : (X_1, X_2, X_3, Y_1, Y_2, Y_3) \mapsto (------).
\end{eqnarray}
As can be seen from the table, none of these are compatible with the constraints imposed
by $C_1$, thus yielding the required contradiction.
To conclude this section, we briefly remark that Abramsky {\it et al.} \cite{abram2, abrampara}
have observed cohomological witnesses for contextuality, including strong contextuality,
by imposing a ring structure on the outcome spaces. These tools are strong enough
to identify the presence of an AvN argument, for instance. In general, their cohomological
construction yields sufficient conditions under which a Kochen-Specker contradiction occurs;
unfortunately, these are sometimes not necessary \cite{man1}.
%\remark{Mention cohomology - sometimes works, sometimes not, but holds for all AvN (\cite{abrampara})}
%\section{Measurement-based quantum computation}
%{\color{red}(RD) Not very fan for having MBQC here. We should keep it KS}
%We can use this to tie many things together: give the "standard" l2-MBQC, discussions of root systems, sausage, etc.
\section{Algebraic Geometry}
It was observed in subsection \ref{subsec:isham} that the topology on $\Sigma(V)$ is discrete
in the Mermin example. If one views the discrete topology as the Zariski topology, then
one can phrase the contradiction in the framework of algebraic geometry. For that, let $A$ be a commutative ring with identity. It will be, for us, of the form
$$\Z_2[i, j, \cdots, k]/ \p := \{\mbox {polynomials in } i, j, \cdots, k \mod \p\},$$
i.e., a coordinate ring of an affine variety $V(\p)\subset \Z_2^n$. The ideal $\p$,
in our context, represents the algebraic constraints involved in Mermin's type of proofs. The discreteness of
the situation allows the replacement of the Gelfand spectrum by the prime spectrum of an algebraic variety to be defined next.
Specifically, the spectrum $\Sigma(\mathcal A)$ of the commutative algebra $\mathcal A$ is replaced by
the prime spectrum of a ring $A$ associated to $\mathcal A$.
%The local sections we have seen before
%will play the role of local coordinate system for the variety $V(\p)$.
\begin{definition}[Prime spectrum of a ring]
The prime spectrum of a ring $A$ is the set $\mathrm{PSpec}\, A$ of all prime ideals in $A$. %, i.e.,
%\mathrm{PSpec}\, A := \left\{\p: \, \p \mbox{prime ideal of } A \right\}.
%The notation $[\p]$ is used to distinguish between the prime ideal $\p$
%and the point it represents inside $\Spec A$.
\end{definition}
In our case of coordinate rings, $\mathrm{PSpec}\, A$ has two types of elements: maximal ideals and non-maximal (but prime) ideals.
The first type corresponds to points in the affine space $\Z^n$, whereas prime ideals which are not maximal are irreducible subvarieties
of our variety. Each element $f\in A$ defines a function on the prime spectrum of $A$. Let $\p$ be an element of $\mathrm{PSpec}\, A$.
We denote by $\kappa(\p)$ the quotient field $A/\p$, called the {residue} field
of $\mathrm{PSpec}\, A$ at $\p$. We can define the value of $f$ at $\p$ to be the image of $f$ via the ring homomorphism
v_\p:A\into \kappa(\p).
If $A$ is a coordinate ring of an affine variety over an algebraically closed field $K$ and $\p$ is the maximal
ideal corresponding to a point of the variety, then $\kappa(\p) =K$ and $f(\p)$ is just the valuation of $f$
at this point.
\medskip
The goal of the remainder of this section is to explain contextuality of Mermin's system in the language of algebraic geometry we have just reviewed.
Notation is as defined in the previous section. We ``identify" each set $\mathcal S_e(C)$, where $C\in \mathcal P$ and $\mathcal P$ is the poset generated by $\mathcal M$,
as an affine variety, and local sections in $\mathcal S_e(C)$ as maximal (ring) ideals
in $\mathrm{PSpec}\, A_C$. % ({\color{blue} Zariski topology on the $\Spec A_C$ we have here is just the discrete topology}).
This yields for each $C$ a coordinate ring $A_C$. The set of rings $A_C$, which are partially ordered by inclusion, define the poset ${\bf CoordRings}(\mathcal P)$.
We can define the functor
\begin{eqnarray}\nonumber
\mathcal F : & {\bf CoordRings}(\mathcal P) & \longrightarrow \Sets\\
& A & \longmapsto \left\{ v_\p, \quad \p\in \mathrm{PSpec}\, A\right\}.
\end{eqnarray}
For Mermin-type Kochen-Specker proofs, the functor $\mathcal F$ has no point. We explain this in the Mermin example, which should be enough to convince
the reader of the validity of the latter statement.
For the first context, we have the coordinate ring
A_1 := \Z_2 \left[ s_1, s_2, s_3\right]/\left(s_1s_2s_3-1\right)
where, for instance, the local section $(s_1, s_2, s_3) \mapsto (-1, -1, 1)$ {of $\mathcal S_e$ at the first context} defines a maximal
ideal $\p =\left(s_1+1, s_2+1, s_3-1 \right) \in \mathrm{PSpec}\, A_1$. For the second context, we have the coordinate ring
A_2 := \Z_2 \left[ s_1, \tilde s_2, \tilde s_3\right]/\left( s_1\tilde s_2\tilde s_3+1\right),
and similarly for the two other maximal contexts.
For the context $\{x_1\}$, the coordinate ring
A_{1, 2} := \Z_2 \left[ s_1\right]
with two proper maximal ideals $(s_1-1)$ and $(s_1+1)$; i.e., the corresponding affine variety is the whole line $\Z_2$.
For each ring $A_i$ and each maximal ideal $\p \in \mathrm{PSpec}\, A_i$, the local valuation
$v_\p:A_i \into \kappa(\p)= A_i/\p$ takes $x\in A_i$ and returns $ v_\p(x) = x \mod \p$.
In the case of $A_1$ and $\p = \left(s_1+1, s_2+1, s_3-1 \right)$, we have
\begin{eqnarray}
v_\p\left( x_1(s_1, s_2, s_3)\right) &=& v_\p(s_1) = s_1 \mod \p =-1, \\\nonumber
v_\p\left( x_2(s_1, s_2, s_3)\right) &=& v_\p(s_2) = s_2 \mod \p =-1, \\\nonumber
v_\p\left( x_3(s_1, s_2, s_3)\right) &=& v_\p(s_3) = s_3 \mod \p =~~1. \\\nonumber
\end{eqnarray}
% If $\p = \left(s_1-1, s_2-1, s_3-1 \right)$ we get
% \begin{eqnarray}
% v_\p\left( x_1(s_1, s_2, s_3)\right) &=& v_\p(s_1) = s_1 \mod \p =1, \\\nonumber
% v_\p\left( x_2(s_1, s_2, s_3)\right) &=& v_\p(s_2) = s_2 \mod \p =1, \\\nonumber
% v_\p\left( x_3(s_1, s_2, s_3)\right) &=& v_\p(s_3) = s_3 \mod \p =1. \\\nonumber
% \end{eqnarray}
% If the affine scheme was $A_{1, 2} = \Z_2 \left[ s_1\right] $ and $\p =(s_1-1)$ we have
% v_\p\left( x_1(s_1, s_2, s_3)\right) = s_1 \mod \p =1.
% \subsection{$\mathcal T$}
%We replace the poset $\mathcal P$ by the poset ${\bf CoordRings}(\mathcal P) $ (each context
%$C$ corresponds to the coordinate ring $A_C$) and define the new functor
% \begin{eqnarray}\nonumber
% \mathcal T : & {\bf CoordRings}(\mathcal P) & \longrightarrow \Sets\\
% & A & \longmapsto \left\{ v_\p, \quad \p\in \mathrm{PSpec}\, A\right\}
% \end{eqnarray}
The functor $\mathcal F$ has no points for GHZ3: for any maximal ideal $\p_1\in A_1$, one has
\begin{equation}
v_{\p_1} \left(s_1 s_2 s_3 \right) = v_{\p_1}(s_1) \, v_{\p_1}(s_2) \,v_{\p_1}(s_3) =1,
\end{equation}
which is nothing but the equation (\ref{eq:mermcon}). For the three other maximal rings, we have similar equations
valid for all maximal ideals $\p_i\in \Spec A_i$. The system is inconsistent and the functor $ \mathcal F$ is pointless. % I might talk about structure sheaf and pb localization
\section{Discussion and Outlook}
As has been presented, there is a remarkable variety %(OF WHAT)
and mathematical richness surrounding
Kochen-Specker proofs, much of which is exhibited by Mermin's simple system. We saw that the collection of observables given in that example, along with a given state where appropriate,
yield naturally combinatoric contradictions in the form of insoluble equations, impossible colourings, and so on. Mermin's system, as Clifton showed, could also be adapted to the case of position and momentum observables.
D\"{o}ring provided a non-combinatorial proof of the Kochen-Specker theorem in the language
of von Neumann algebras, thereby providing a genuine generalisation. We then provided
some geometric insight on the Mermin proof through Coxeter groups and their root systems,
and observed that $E_8$ arises naturally in this context. The presheaf perspectives presented
then provide another way of phrasing Kochen-Specker-type contradictions---three such approaches were given. Finally, an
algebraic geometric framework was outlined. It was observed that in the finite scenario,
the Gelfand spectrum of an abelian algebra can be replaced by the prime spectrum of an algebraic
variety, and the statement that the spectral presheaf of Isham and Butterfield has no
global section ``translates" into the statement that the prime spectrum functor has no global section. This algebraic geometric approach seems to warrant further exploration.
We only briefly touched upon the role played by cohomology in the Kochen-Specker theorem.
We believe this to be an area for much further work, particularly when phrased in the language
of toposes \cite{mord1}. Let % SUDDENLY JUMP INTO A LET STATEMENT
$\mathcal E$ be the topos ${\sf Sets}^{\mathcal P^{op}}$ and
$\mathcal{AB}\left (\mathcal E\right)$ the category of of abelian group objects, i.e., presheaves from the poset $\mathcal P$ into the category $\mathcal{AB}$ of abelian groups. The Giraud axiom for generators implies that the abelian category $\mathcal{AB}(\mathcal E)$ has enough injectives. The global sections functor $\Gamma: \mathcal E\rightarrow {\sf Sets}$ induces a functor (again denoted) $\Gamma: \mathcal{AB}(\mathcal E)\rightarrow \mathcal{AB},$ which is left exact and preserves injectives. For any abelian group
object $A\in \mathcal E,$ the cohomology groups $H^n(\mathcal E, A)$ are defined as the right derived functors of $\Gamma$.
Now, the functors
$\Sigma$ and $S_e$ are in $\mathcal{AB}\left (\mathcal E\right)$ in the case of GHZ3; i.e., they are functors of vector spaces. Mermin's proof translates into the functors $H^n$, for both $\Sigma$ and $S_e$, being all equal to the zero functor.
%--------------------------PREVIOUS CONCLUSION (FROM FACES 5.tex)-----------------
%We briefly mention here an ensuing cohomological obstruction. This takes a natural form when phrased %in the language of toposes \cite{mord1}. Let $\mathcal E$ be the topos ${\sf Sets}^{\mathcal P^{op}}$ %and
%$\mathcal{AB}\left (\mathcal E\right)$ the category of of abelian group objects, i.e., presheaves from the poset $\mathcal P$ into the category $\mathcal{AB}$ of abelian groups. The Giraud axiom for generators implies that the abelian category $\mathcal{AB}(\mathcal E)$ has enough injectives. The global sections functor $\Gamma: \mathcal E\rightarrow {\sf Sets}$ induces a functor (again denoted) $\Gamma: \mathcal{AB}(\mathcal E)\rightarrow \mathcal{AB},$ which is left exact and preserves injectives. For any abelian group
%object $A\in \mathcal E,$ the cohomology groups $H^n(\mathcal E, A)$ are defined as the right derived functors of $\Gamma$.
%Now, the functors
%$\Sigma$ and $S_e$ are in $\mathcal{AB}\left (\mathcal E\right)$ in the case of GHZ3; i.e., they are functors of vector spaces. Mermin's proof translates into the functors $H^n$, for both $\Sigma$ and $S_e$, being all equal to the zero functor.
%--------------------------==END== PREVIOUS CONCLUSION (FROM FACES 5.tex)-----------------
\section*{Acknowledgements}
Thanks are due to Shane Mansfield for useful discussions and a careful reading of this manuscript.
We also thank Marko Bucyk for his help. Leon Loveridge acknowledges funding from the grant \emph{Quantum Mathematics and Computation}.
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\begin{thebibliography}{99}
%\bibitem{bell} Bell J S, On the Problem of Hidden Variables in Quantum Mechanics. 1966 Rev. Mod. Phys. 38 447
\bibitem{bell} J. S. Bell, On the problem of hidden variables in quantum mechanics, {\it Rev. Mod. Phys.} {\bf 38} (1966) 447--452.
%\bibitem{KS1} Kochen S and Specker E P The problem of hidden variables in quantum mechanics. 1967 J. Math. Mech. 17 59
\bibitem{KS1} S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, {\it J. Math. Mech.} {\bf 17} (1967) 59-87.
%\bibitem{peres2} A.Peres, "Two simple proofs of the Kochen-Specker theorem", J.Phys. A24, 174-8 (1991)
\bibitem{peres2} A. Peres, Two simple proofs of the Kochen-Specker theorem, {\it J. Phys. A} {\bf 24} (1991) 174-178.
%\bibitem{penrose1} R. Penrose, ``On Bell non-locality without probabilities: some curious geometry", in Quantum Reflections J. Ellis and D. Amati, Eds, CUP, New York, 2000
\bibitem{penrose1} J. Zimba and R. Penrose, On Bell non-locality without probabilities: some curious geometry, in {\it Quantum Reflections}, eds. J. Ellis and D. Amati (Cambridge University Press, 2000), pp. 697-720.
\bibitem{peres1} A. Peres, {\it Quantum Theory: Concepts and Methods} (Kluwer, 1993).
%\bibitem{csw1} Ad\'{a}n Cabello, Simone Severini, and Andreas Winter arXiv:1010.216
\bibitem{csw1} A. Cabello, S. Severini and A. Winter, (Non-)Contextuality of physical theories as an axiom; arXiv:1010.216.
%\bibitem{csw2} Ad\'{a}n Cabello, Simone Severini, Andreas Winter Phys. Rev. Lett. 112, 040401 2014(DOI http://dx.doi.org/10.1103/PhysRevLett.112.040401)
\bibitem{csw2} A. Cabello, S. Severini and A. Winter, Graph-theoretic approach to quantum correlations, {\it Phys. Rev. Lett. } {\bf 112} (2014) 040401.
%\bibitem{afls1} Antonio Ac\'{i}n, Tobias Fritz, Anthony Leverrier, Ana Bel\'{e}n Sainz A Combinatorial Approach to Nonlocality and Contextuality Communications in Mathematical Physics
%March 2015, Volume 334, Issue 2, pp 533-628
\bibitem{afls1} A. Ac\'{i}n, T. Fritz, A. Leverrier and A. B. Sainz, A combinatorial approach to nonlocality and contextuality,
{\it Comm. Math. Phys.} {\bf 334}(2) (2015) 533-628.
%\bibitem{spek1} R. W. Spekkens Contextuality for preparations, transformations, and unsharp measurements Phys. Rev. A 71, 052108 2005 (DOI 10.1103/PhysRevA.71.052108)
\bibitem{spek1} R. W. Spekkens, Contextuality for preparations, transformations, and unsharp measurements, {\it Phys. Rev. A} {\bf 71} (2005) 052108.
%(DOI 10.1103/PhysRevA.71.052108)
%\bibitem{paul1} P. Busch Phys. Rev. Lett. 91, 120403, 2003 Quantum States and Generalized Observables: A Simple Proof of Gleason's Theorem
\bibitem{paul1} P. Busch, Quantum states and generalized observables: A simple proof of Gleason's theorem, {\it Phys. Rev. Lett.} {\bf 91} (2003) 120403.
%\bibitem{cab1} Ad\'{a}n Cabello Phys. Rev. Lett. 90, 190401 2003 http://dx.doi.org/10.1103/PhysRevLett.90.190401
\bibitem{cab1} A. Cabello, Kochen-Specker theorem for a single qubit using positive operator-valued measures, {\it Phys. Rev. Lett.} {\bf 90} (2003) 190401.
\bibitem{dor1} A. D\"{o}ring, Kochen-Specker theorem for von Neumann algebras, \emph{Int. J. Th. Phys.} {\bf 44}(2) (2005) 139-160.
%\bibitem{ib1} Isham, C. J., Butterfield, J. 1998 Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations. \emph{Int. J. Th. Phys.} {\bf 37}, issue 11, 2669-2733. (DOI 110.1023/A:1026680806775.)
\bibitem{ib1} C. J. Isham and J. Butterfield, Topos perspective on the Kochen-Specker theorem: I. Quantum states as generalized valuations, \emph{Int. J. Th. Phys.} {\bf 37}(11) (1998) 2669-2733.
\bibitem{ib2} C. J. Isham and J. Butterfield, Topos perspective on the Kochen-Specker theorem: II. Conceptual aspects and classical analogues, \emph{Int. J. Th. Phys.} {\bf 38}(3) (1999) 827-859. %(DOI 10.1023/A:1026652817988.)
\bibitem{ib3} C. J. Isham, J. Butterfield and A. Hamilton, Topos perspective on the Kochen-Specker theorem: III. Von Neumann algebras as the base category, \emph{Int. J. Th. Phys.} {\bf 39} (2000) 1413-1436. %(DOI 10.1023/A:1003667607842.)
\bibitem{ib4} C. J. Isham and J. Butterfield, Topos perspective on the Kochen-Specker theorem: IV. Internal valuations, \emph{Int. J. Th. Phys.} {\bf 41}(4) (2000) 613-639.
% (DOI 10.1023/A:1015276209768.)
\bibitem{hls1} C. Heunen, N. P. Landsman and S. Spitters, A topos for algebraic quantum theory, \emph{Comm. Math. Phys.} {\bf 291} (2009) 63-110. %(DOI 10.1007/s00220-009-0865-6.)
%\bibitem{abram1} Abramsky, S. \& Brandenburger, A. 2011 The sheaf-theoretic structure of non-locality and contextuality. \emph{N. J. Phys.} {\bf 13}. (DOI10.1088/1367-2630/13/11/113036.)
\bibitem{abram1} S. Abramsky and A. Brandenburger, The sheaf-theoretic structure of non-locality and contextuality, \emph{N. J. Phys.} {\bf 13} (2011) 113036.
%\bibitem{abram2} Abramsky, S., Mansfield, S. \& Barbosa, R. S. 2012 The Cohomology of Non-Locality and Contextuality.
%\emph{Proceedings 8th International Workshop on Quantum Physics and Logic Nijmegen, Netherlands} {\bf 95}, 1-14. Op. Pub. Ass. (DOI 10.4204/EPTCS.95.1.)
\bibitem{abram2} S. Abramsky, S. Mansfield and R. S. Barbosa, The Cohomology of non-locality and contextuality,
in \emph{Proceedings 8th International Workshop on Quantum Physics and Logic, Nijmegen, Netherlands} (2012), pp. 1-14. %. Op. Pub. Ass. (DOI 10.4204/EPTCS.95.1.)
%\bibitem{abrampara} S.~Abramsky, R.~S.~Barbosa, K.~Kishida, R.~Lal, S.~Mansfield Contextuality, Cohomology and Paradox arXiv:1502.03097
\bibitem{abrampara} S.~Abramsky, R.~S.~Barbosa, K.~Kishida, R.~Lal and S.~Mansfield, Contextuality, cohomology and paradox; arXiv:1502.03097.
%\bibitem{ah1} Samson Abramsky and Lucien Hardy Logical Bell Inequalities Phys. Rev. A 85, 062114
\bibitem{ah1} S. Abramsky and L. Hardy, Logical Bell inequalities, {\it Phys. Rev. A } {\bf 85} (2012) 062114-062121.
%\bibitem{id1} Isham, C. J. \& D\"{o}ring, A. 2011 `What is a thing'?: Topos Theory in the Foundations of Physics. In \emph{New Structures for Physics, Springer Lecture Notes in Physics} (ed. Bob Coecke). {\bf 813}, pp. 753-940. Springer, Heidelberg.
\bibitem{id1} C. J. Isham and A. D\"{o}ring, `What is a thing'?: Topos theory in the foundations of physics, In \emph{New Structures for Physics}, ed. Bob Coecke (Springer, 2011), pp. 753-940.
\bibitem{ldr1} L. Loveridge, R. Dridi and R. Raussendorf, Topos logic in measurement-based quantum computation, \emph{Proc. R. Soc. A.} {\bf 471} (2015) 2176.
%(DOI: 10.1098/rspa.2014.0716.)
%\bibitem{kly1} A. A. Klyachko, M. A. Can, S. Binicioglu, and A. S. Shumovsky, Phys. Rev. Lett. 101, 020403 (2008).
\bibitem{kly1} A. A. Klyachko, M. A. Can, S. Binicioglu and A. S. Shumovsky, Simple test for hidden variables in spin-1 systems, {\it Phys. Rev. Lett. } {\bf 101} (2008) 020403.
\bibitem{bell1} J. Bell, On the Einstein-Podolsky-Rosen paradox, {\it Physics} {\bf 1} (1964) 195-200.
%\bibitem{chsh1} John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt Proposed Experiment to Test Local Hidden-Variable Theories
%Phys. Rev. Lett. 23, 880; Erratum Phys. Rev. Lett. 24, 549 (1970)
\bibitem{chsh1} J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-variable theories,
{\it Phys. Rev. Lett.} {\bf 23} 880 (1970); Erratum {\it Phys. Rev. Lett.} {\bf 24} (1970) 549.
%\bibitem{rau1} Raussendorf, R. 2013 Contextuality in measurement based quantum computation. \emph{Phys. Rev. A} {\bf 88} (DOI 10.1103/PhysRevA.88.022322.)
\bibitem{rau1} R. Raussendorf, Contextuality in measurement-based quantum computation, \emph{Phys. Rev. A} {\bf 88} (2013) 022322. % (DOI 10.1103/PhysRevA.88.022322.)
%\bibitem{rau2} Nicolas Delfosse, Philippe Allard Guerin, Jacob Bian, and Robert Raussendorf
%Phys. Rev. X 5, 021003 2015 Wigner Function Negativity and Contextuality in Quantum Computation on Rebits (DOI http://dx.doi.org/10.1103/PhysRevX.5.021003)
\bibitem{rau2} N. Delfosse, P. A. Guerin, J. Bian and R. Raussendorf, Wigner function negativity and contextuality in quantum computation on rebits,
{\it Phys. Rev. X} {\bf 5} (2015) 021003.
%\bibitem{hwve1} Howard, M., Wallman, J. J., Veitch, V. \& Emerson, J. 2014 Contextuality supplies the magic for quantum computation. \emph{Nature} {\bf 510}, 351-355. (DOI 10.1038/nature13460.)
\bibitem{hwve1} M. Howard, J. J. Wallman, V. Veitch and J. Emerson, Contextuality supplies the magic for quantum computation, \emph{Nature} {\bf 510} (2014) 351-355.
%\bibitem{Mermex} N. David Mermin, Extreme quantum entanglement in a superposition of macroscopically distinct states, Phys. Rev. Lett. 65 (1990), no. 15, 1838-1840
\bibitem{Mermex} N. D. Mermin, Extreme quantum entanglement in a superposition of macroscopically distinct states, {\it Phys. Rev. Lett.} {\bf 65}(15) (1990) 1838-1840.
%\bibitem{clif1} R. Clifton Complementarity between position and momentum as a consequence of Kochen-Specker arguments, Physics Letters A Volume 271, Issues 1-2, June 2000, pages 1-7
\bibitem{clif1} R. Clifton, Complementarity between position and momentum as a consequence of Kochen-Specker arguments, {\it Phys. Lett. A} {\bf 271}(1-2) (2000) 1-7.
%\bibitem{kpe1} M. Kernaghan and A. Peres Kochen-Specker theorem for eight-dimensional space
%Physics Letters A 198, 1, pages 1-5, 1995
%DOI 10.1016/0375-9601(95)00012-R
\bibitem{kpe1} M. Kernaghan and A. Peres, Kochen-Specker theorem for eight-dimensional space,
{\it Phys. Lett. A} {\bf 198}(1) (1995) 1-5.
%\bibitem{arw1} M. Waegell, P.K. Aravind J. Phys. A: Mathematical and Theoretical
%2012 J. Phys. A: Math. Theor. 45 405301 doi:10.1088/1751-8113/45/40/405301Proofs of the Kochen-Specker theorem based on a system of three qubits
\bibitem{arw1} M. Waegell, P. K. Aravind, Proofs of the Kochen-Specker theorem based on a system of three qubits, {\it J. Phys. A: Math. Theor.}
{\bf 45} (2012) 405301. % doi:10.1088/1751-8113/45/40/405301
%\bibitem{toh1} (CHECK) S. P. Toh Kochen-Specker Sets with a Mixture of 16 Rank-1 and 14 Rank-2 Projectors for a Three-Qubit System
%Chin. Phys. Lett. Vol.30., No.10(2013)100302
\bibitem{toh1}S. P. Toh, Kochen-Specker sets with a mixture of 16 rank-1 and 14 rank-2 projectors for a three-qubit system,
{\it Chin. Phys. Lett. } {\bf 30}(10) (2013) 100302.
%\bibitem{GHZ} Greenberger, D. M., Horne, M. A. \& Zeilinger, A. 1989 Going Beyond Bell's Theorem. In \emph{Bell's Theorem, Quantum Theory, and Conceptions of the Universe} (ed. M. Kafatos), pp. 69 - 72. Kluwer, Dordrecht.
\bibitem{GHZ} D. M. Greenberger, M. A. Horne and A. Zeilinger, Going beyond Bell's theorem, In \emph{Bell's Theorem, Quantum Theory, and Conceptions of the Universe}, ed. M. Kafatos (Kluwer 1989), pp. 69-72.
\bibitem{glea1} A. M. Gleason, Measures on the closed subspaces of a Hilbert space,
{\it J. Math. Mech.} {\bf 6} (1957) 885-893.
\bibitem{chr1} E. Christensen, Measures on projections and physical states, {\it Comm. Math. Phys.} {\bf 86} (1982) 529-538.
\bibitem{yea1}F. J. Yeadon, Measures on projections in $W^*$-algebras of type II1, {\it Bull. London Math. Soc.} {\bf 15} (1983) 139-145.
\bibitem{yea2} F. J. Yeadon, Finitely additive measures on projections in finite $W^*$-algebras, {\it Bull. London Math. Soc.} {\bf 16} (1984) 145-150.
\bibitem{mae1} S. Maeda, Probability measures on projections in von Neumann algebras, {\it Rev. Math. Phys.} {\bf 01}(235) (1989). %: 10.1142/S0129055X89000122
\bibitem{ham1} J. Hamhalter, {\it Quantum Measure Theory} (Springer, 2003).
%ISBN: 978-90-481-6465-3 (Print) !!!!
%\bibitem{humphreys}Humphreys, James E. 1990 Reflection groups and {C}oxeter groups. Cambridge Studies in Advanced Mathematics. Volume 29. Cambridge University Press, Cambridge.
\bibitem{humphreys} J. E. Humphreys, {\it Reflection Groups and {C}oxeter groups} (Cambridge University Press, 1990).
%\bibitem{nicolas} Nicolas Bourbaki. 2002 {{Elements of mathematics. Lie groups and Lie algebras. Chapters 4--6.}} Berlin: Springer
\bibitem{nicolas} N. Bourbaki, {{\it Elements of Mathematics. Lie Groups and Lie Algebras. Chapters 4--6}} (Springer, 2002).
%\bibitem{ido1} Isham, C. J. \& D\"{o}ring, A. 2012 Classical and Quantum Probabilities as Truth Values. \emph{J. Math. Phys} {\bf 53}, 032101. (DOI 10.1063/1.3688627.)
\bibitem{ido1} C. J. Isham and A. D\"{o}ring, Classical and quantum probabilities as truth values, \emph{J. Math. Phys.} {\bf 53} (2012) 032101.
%(DOI 10.1063/1.3688627.)
\bibitem{man1} S. Mansfield, The mathematical structure of non-locality and contextuality, Ph.D. thesis, University of Oxford (2013).
\bibitem{mord1} I. Moerdijk, {\it Classifying Spaces and Classifying Topoi} (Springer, 1995).
\end{thebibliography}
\end{document}
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1511.00668
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]Kunle AdegokeCorresponding author: [email protected]
2]Olawanle Layeni
]Rauf Giwa
]Gbenga Olunloyo
Department of Physics and Engineering Physics, Obafemi Awolowo University, Ile-Ife, Nigeria
[2]Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria
In this paper we give a Casimir Invariant for the Symmetric group $S_n$. Furthermore we obtain and present, for the first time in the literature, explicit formulas for the matrices of the standard representation in terms of the matrices of the permutation representation.
§ INTRODUCTION
The symmetric group $S_n$, whose elements consist of the set of all permutations on $n$ symbols is of central importance to mathematics and physics <cit.>. Cayley's theorem states that every group is isomorphic to a subgroup of the symmetric group on that group. In physics the classification of atomic and nuclear states depends essentially on the properties of $S_n$ <cit.>. The representation theory of the symmetric group is a well studied subject <cit.>. The partitions of $n$ or equivalently Young diagrams of size $n$ are the natural ways in which to parametrize the irreducible representations of $S_n$ <cit.>. This paper is concerned, not with the general irreducible representations of $S_n$, but, more specifically, with the so-called standard representation of the symmetric group, formally obtained from the dimensional subspace of vectors whose sum of coordinates is zero in the basis set of a permutation representation. The path taken in this work shall however be non-group theoretic. For example, we will not be concerned with Young diagrams.
The standard representation of $S_n$ is important for the following reason: For $n \ge 7$, the permutation representation, the trivial (identical) representation, the sign representation, the standard representation and another $n-1$ dimensional irreducible representation found by tensoring with the sign representation are the only lowest-dimensional irreducible representations of $S_n$ <cit.>. All other irreducible representations have dimension at least $n$. While it is a fact that all irreducible representations of $S_n$ can be found, using Frobenius formula (<cit.>, pp 189), for example, there are no known explicit formulas for the standard representation. The main result of this paper is the derivation of such formulas, which now make it possible to write down the standard representation matrices directly from those of the permutation representation.
§ THE PERMUTATION REPRESENTATION
Denote the $n!$ elements of $S_n$ by $A^k$, $k = 1,2, \ldots, n!$, such that, in usual notation,
\begin{equation}
A^k = \left( {\begin{array}{*{20}c}
1 & 2 & 3 & \cdots & \cdots & \cdots & n \\
{a_1^k } & {a_2^k } & {a_3^k } & \cdots & \cdots & \cdots & {a_n^k } \\
\end{array}} \right)\,,
\end{equation}
and $1\le a_i^k\le n$, all $a_i^k$ being distinct. For simplicity, and since no ambiguity can result, we will use the same symbol $A^k$ for the representation matrices. Then, in the permutation representation, the $n\times n$ matrices $A^k$ are given, through their elements, by:
\begin{equation}
(A^k )_{ij} = \delta _{a_i^k ,j} ,\quad i = 1 \ldots n,\;j = 1 \ldots n\,,
\end{equation}
which is clearly a unitary representation, since
\[
\left( {(A^k )^ \dagger A^k } \right)_{ij} = \sum\limits_{r = 1}^n {\left( {(A^k )^ \dagger } \right)_{ir} A_{rj}^k } = \sum\limits_{r = 1}^n {A_{ri}^k A_{rj}^k } = \sum\limits_{r = 1}^n {\delta _{a_r^k ,i} \delta _{a_r^k ,j} } = \delta _{ij}\,.
\]
§.§ A Casimir Invariant for $S_n$ in the permutation representation
The $n\times n$ matrix $C$, with elements $C_{ij}=1-\delta_{ij}$ is a Casimir Invariant of $S_n$ in the permutation representation.
Proof We require to prove that $C$ commutes with every $A^k$.
\begin{equation}
\begin{split}
(CA^k )_{ij} = \sum\limits_{r = 1}^n {C_{ir} (A^k )_{rj} } &= \sum\limits_{r = 1}^n {(1 - \delta _{ir} )(A^k )_{rj} }\\
&= \sum\limits_{r = 1}^n {(A^k )_{rj} } - \sum\limits_{r = 1}^n {\delta _{ir} (A^k )_{rj} }\\
&= \sum\limits_{r = 1}^n {(A^k )_{rj} } - (A^k )_{ij}\\
&= 1 - (A^k )_{ij}\,.
\end{split}
\end{equation}
A similar calculation gives,
\begin{equation}
(A^k C)_{ij} = \sum\limits_{r = 1}^n {(A^k )_{ir} C_{rj} } = \sum\limits_{r = 1}^n {(A^k )_{ir} } - (A^k )_{ij}= 1 - (A^k )_{ij}\,.
\end{equation}
We see, therefore, that $A^kC=CA^k$, so that $C$ is a Casimir Invariant of the symmetric group.
§ THE STANDARD REPRESENTATION
Since the Casimir invariant $C$, obtained in the previous section, is not proportional to the identity, Schur's lemma tells us that the permutation representation is not irreducible, a well-known fact. It therefore remains to find the matrix $P$ which diagonalizes $C$. First we prove a lemma.
Lemma 1 The nonsingular $n\times n$ matrix $P$ with elements has the inverse $P^{-1}$ where
\begin{equation}
\begin{split}
n(PP^{ - 1} )_{ij}&= \sum\limits_{r = 1}^n {P_{ir} n(P^{ - 1} )_{rj} }\\
&= \sum\limits_{r = 1}^n {(\delta _{r1} - \delta _{i1} + \delta _{r1} \delta _{i1} + \delta _{i,n + 2 - r} )}\\
&\qquad\qquad\qquad\times (2\delta _{r1} - 1 + n\delta _{r,n + 2 - j} )
\end{split}
\end{equation}
It is straightforward to write out the terms and evaluate the summation termwise. One merely needs to note that
\[\sum\nolimits_{r = 1}^n {\delta _{r,n + 2 - j} } = 1 - \delta _{j1}\]
\[\sum\nolimits_{r = 1}^n {\delta _{i,n + 2 - r} } \delta _{r,n + 2 - j} = \delta _{ij} - \delta _{i1} \delta _{j1}\,.\]
One then finds , which establishes the claim.
§.§ Diagonal form of the Casimir Invariant
The matrix $P$ given in Lemma 1, diagonalizes the Casimir Invariant, $C$.
Proof. We wish to compute
\[
\]
\[
D_{ij} = \sum\limits_{r = 1}^n {\sum\limits_{s = 1}^n {(P^{ - 1} )_{ir} C_{rs} P_{sj} } }\,.
\]
Substituting the matrix elements, expanding and evaluating the sums, we find after some algebra, that
\begin{equation}
D_{ij}=n\delta _{i1} \delta _{j1}-\delta _{ij}\,.
\end{equation}
Thus we see that $D$ is a diagonal matrix, as claimed, with the entry `$n-1$' in row $1$, column $1$ and the remaining diagonal elements being $-1$.
The matrix $P$, above, which diagonalizes $C$ will block-diagonalize the matrices $A^k$.
§.§ Similarity Transformation of $A^k$: the standard representation
Using the matrix elements of $P$, $P^{-1}$ and $A^k$, it is not difficult to obtain the interesting result:
\begin{equation}\label{equ.m5z7nvb}
\begin{split}
(P^{ - 1} A^k P)_{ij} & = \sum\limits_{r = 1}^n {\sum\limits_{s = 1}^n {(P^{ - 1} )_{ir} A_{rs}^k P_{sj} } }\\
&= \delta _{j1} \delta _{i1} + (1 - \delta _{i1} )(1 - \delta _{j1} )[A_{n + 2 - i,n + 2 - j}^k - A^k_{n + 2 - i,1}]
\end{split}
\end{equation}
We see from (<ref>) that each matrix $P^{ - 1} A^k P$ is block diagonal, being the direct sum of a $1\times 1$ matrix with entry $1$ and an $(n-1)\times (n-1)$ matrix $B^k$, with elements
\begin{equation}
B_{ij}^k = A_{n + 1 - i,n + 1 - j}^k - A_{n + 1 - i,1}^k \;,\quad i= 1, \ldots ,n - 1,\;j = 1, \ldots ,n - 1\,.
\end{equation}
The $1\times 1$ matrices correspond to the identical (trivial) representation in which every element of $S_n$ is sent to the one-dimensional identity matrix, while the $B^k$ matrices correspond to the irreducible $n-1$ dimensional standard representation.
§ CONCLUSION
In this paper we have shown that the operator $C$ with matrix elements is a Casimir Invariant for the symmetric group $S_n$. We also showed that if $A^k,\;k=1,2,\ldots, n!$ are the representation matrices for the elements of $S_n$ in the permutation representation, then the matrices $B^k$ for the standard representation of $S_n$ are given by
\[
B_{ij}^k = A_{n + 1 - i,n + 1 - j}^k - A_{n + 1 - i,1}^k \;,\quad i= 1, \ldots ,n - 1,\;j = 1, \ldots ,n - 1\,.
\]
M. Hamermesh.
Group Theory and Its Application to Physical Problems.
Dover Publications, 1989.
W. Burnside.
Theory of groups of finite order.
Dover Publications, 1955.
W. Fulton and J. Harris.
Representation theory. A first course,.
Springer-Verlag, 1991.
|
1511.01003
|
Departamento de Engenharias e Ciências Exatas,
Universidade Federal do Paraná (UFPR),
Pioneiro, 2153, 85950-000 Palotina, PR, Brazil.
13.85.Tp, 98.70.Sa, 95.30.Qd, 98.62.Js
We investigate the production of magnetic flux from rotating black
holes in active galactic nuclei (AGNs) and compare it with the upper limit of
ultra high energy cosmic ray (UHECR) luminosities, calculated from observed integral flux of GeV-TeV gamma rays
for nine UHECR AGN sources.
We find that, for the expected range of black hole rotations ($0.44<a<0.80$),
the corresponding bounds of theoretical magnetic luminosities from AGNs coincides
with the calculated UHECR luminosity. We argue that such result possibly can
contribute to constrain AGN magnetic and dynamic properties as phenomenological tools
to explain the requisite conditions to proper accelerate the highest energy cosmic rays.
The origin of the highest energy cosmic rays, or ultra high energy cosmic rays (UHECRs), with energies $E > 10^{18}$ eV,
in spite of huge observational/experimental endeavors,
represents one of the greatest puzzles of modern astrophysics <cit.>.
The possibility of accelerating particles up to such extreme energies is addressed by the well-known Hillas source plot <cit.>.
There, a table of possible UHECR sources is presented by bringing the simple yet efficient remark that particles, during acceleration,
are confined in the source on a Larmor timescale (for more details see, e.g., <cit.>).
The prominent extragalactic candidates for accelerating particles (mainly protons) to the highest energies are active galactic nuclei (AGNs) <cit.>,
the most powerful radiogalaxies <cit.>, and also gamma ray bursts, fast spinning newborn
pulsars, interacting galaxies, large-scale structure formation shocks and some other objects <cit.>.
On the other hand, the reconstruction of cosmic ray luminosities, from Earth laboratory observations, possibly can shed some light on radiative bounds of UHECR potential sources.
For example, in <cit.> it is shown that using the methods of UHECR propagation from the source
to Earth and the measured upper limit on the integral flux of GeV-TeV gamma-rays it is possible to infer the upper limits of the proton and total UHECR
(iron) luminosity. This comes from the fact that gamma-rays can be produced as a result of the cosmic ray propagation and contribute to the total flux measured
from the source.
In the present work, some of the potential AGN sources will be investigated as UHECR sources (see Table <ref>).
It will be calculated upper limits of UHECRs luminosities to be compared to the theoretical magnetic/jet luminosity of those AGNs.
To reconstruct, from experiments, the possible UHECR luminosities here it will be used the method described in <cit.>
as a prolific way to calculate upper limits on UHECR luminosities.
In first place, space and ground instruments, as FERMI-LAT <cit.>, VERITAS <cit.>, H.E.S.S. <cit.> and MAGIC <cit.>
provide upper limits on the GeV-TeV gamma-ray integral flux and the
method <cit.> connects those measured upper limits with the source UHECR cosmic ray luminosity ($L_{CR}^{UL}$) by
\begin{equation}
L_{CR}^{UL} = \frac{4\pi D^{2}_s(1+z_s)\langle E \rangle_{0}}{ \ K_{\gamma} {\mathop{\displaystyle
\int_{E_{th}}^{\infty} dE_\gamma\ P_{\gamma}(E_{\gamma})}}}\ I_{\gamma}^{UL}(> E^{th}_{\gamma}),
\label{eq:CRUL}
\end{equation}
where $I_{\gamma}^{UL}(> E^{th}_{\gamma})$ is the upper limit on the integral gamma-ray flux for a given confidence level and energy
threshold, $K_{\gamma}$ is the number of gamma rays generated from the
cosmic ray particles, $P_{\gamma}(E_{\gamma})$ is the energy
distribution of the gamma-rays arriving on Earth, $E_{\gamma}$ is the
energy of gamma-rays, $\langle E \rangle_{0}$ is the mean energy, $D_s$
is the comoving distance and $z_s$ is the redshift of the source. This
method allows one to calculate upper limits on the proton and total
luminosities for energies above $10^{18}$ eV. Also, it illustrates techniques to study the origin of UHECRs
from multi-messenger GeV-TeV gamma-rays and it has been used to calculate
at least upper limits for thirty sources (AGNs), with redshift smaller
than 0.048 and UHECR spectra measured by the Pierre Auger <cit.> and Telescope
Array <cit.> (TA) observatories. In fact, as described by <cit.>, the UHECR upper limit
luminosity is obtained from the integral of the gamma-ray flux from the observed spectrum of UHECRs.
Propagation models and the measured upper limit of gamma-ray flux of a source amongst propagation
models are the fundamental aspects to be considered to perform such calculation. Indeed, this same method will allow,
in the future, from CTA Observatory <cit.>, a range of new UHECR luminosity upper limits.
In second place, it is well-known that radiative mechanisms within AGNs are chiefly powered by their central supermassive black holes (SMBHs) and the companion accretion disk.
The rotation of the system creates twisted magnetic fields that drive jets of relativistic particles. Here it will be considered a standard accretion
mechanism (Bondi accretion model $\dot{M} = \pi \lambda c_{S} \rho_B r_B$ <cit.>,
where $\dot{M}$ is the accretion rate, $\lambda$ takes the value of 0.25 for an adiabatic index 5/3, $c_S$ is the sound speed in the medium,
$r_B$ is the Bondi accretion radius and $\rho_B$ is the gas density at that radius) which produces, by friction and other radiative processes, enormous bolometric luminosities. The injection
of only a moderate fraction of this bolometric luminosity would suffice to reproduce the observed cosmic ray flux above $10^{19}$ eV.
Nevertheless, it appears that cosmic ray flux from AGNs probably comes from jet luminosities. Therefore, the magnetic field of the structure is
the main responsible to produce such perpendicular jet outflows. In this case, they do not all offer as appetizing physical conditions with
respect to particle acceleration as, e.g., gamma ray bursts, since
considerable outflow luminosities, i.e., magnetic luminosities $L_B$ are actually required to accelerate protons to the highest energies observed.
This limits the potential number of AGNs as cosmic ray sources in the nearby universe (unless the highest energy cosmic rays are heavy nuclei) <cit.>.
In the present contribution, we describe the mechanism via powering jets from Blandford-Znajek mechanism <cit.>.
This approach is based on flux accumulation that leads to the formation of a
magnetically chocked accretion flow and strong flux of high energy particle jets and it assumes that any geometrically thick or hot inner region of an accretion flow
can drive magnetic field fluctuations to produce jets. This technique has been proposed to explain the most luminous and radio-loud AGNs,
and consequently could also explain mechanisms behind the production of UHECRs.
In what follows it is derived a description for the production of UHECR luminosities based on
the possible relation between magnetic flux accumulation and jet production
efficiency. Similar schemes have been proposed to describe UHECR luminosity
contribution and sources of cosmic rays from black hole accretion mechanisms
<cit.>. The significant feature of this model is that the dominant factor in the magnetic luminosity
is the powerful jets to therefore determinate the radio loudness of the AGN.
In first place, considering a system with a rotating central BH, the necessary condition to use Blandford-Znajek mechanism is that
$\Phi_d > \Phi_{BH,\hbox{max}}(\dot{M})$, i. e., the net poloidal magnetic flux $\Phi_d$ trapped in the disk
is larger than the maximum that can be confined on the BH caused by pressure of the accreting plasma. Satisfied this condition,
the rate of energy extraction from the rotating BH via the Blandford-Znajek mechanism yields the magnetic luminosity
\begin{eqnarray}\label{lb}
L_{B} &\simeq& 4\times 10^{-3}\Phi_{BH,\hbox{max}}^{2}(\dot{M})\frac{\Omega_{BH}^{2}}{c}f_a(\Omega_{BH})\nonumber\\
&=& 10(\phi/50)x_{a}^{2}f_{a}(x_{a})\dot{M}c^{2},
\end{eqnarray}
where $\Omega_{BH}$ is the angular velocity of the black hole and
\begin{equation}
\end{equation}
\begin{equation}
f_{a}(x_a)\simeq 1 + 1.4x_{a}^{2}-9.2x_{a}^{4},
\end{equation}
where $a$ is the dimensionless angular momentum parameter ($a=J/Mc$, with $J$ the BH angular momentum), $\phi$
is a dimensionless factor which, according to numerical simulations (see, e.g., <cit.>), is typically of order 50, and
$r_g$ is the gravitational radius $r_g=GM/c^2$.
$L_{B}$ as a function of the mass and parameter
$a$. For all plots it was considered $\dot{M}$ as the Bondi accretion rate <cit.>. For comparison, the Eddington luminosities $L_{\mathrm{Edd}}$
for each considered AGN mass are: $L_{\mathrm{Edd}}(M=10^{9.5}M_\odot)=1.26\times10^{47.5}$ erg s$^{-1}$, $L_{\mathrm{Edd}}(M=10^{10.0}M_\odot)=1.26\times10^{48}$ erg s$^{-1}$,
$L_{\mathrm{Edd}}(M=10^{10.5}M_\odot)=1.26\times10^{48.5}$ erg s$^{-1}$, $L_{\mathrm{Edd}}(M=10^{11.0}M_\odot)=1.26\times10^{49}$ erg s$^{-1}$.
In Fig. <ref> it is displayed the effect of black hole spinning on magnetic luminosity with
the variation of mass. The greater the AGN BH spinning, the greater the
AGN magnetic loudness. A similar result is observed with the increasing of the black hole mass. In this approach, $M$ and $a$ are therefore the main
parameters to affect energetic jets and the consequent energetic cosmic ray production at AGN expected sources. For comparison one can calculate
the upper limit of bolometric luminosity for each source, i.e., the Eddington luminosity $L_{\mathrm{Edd}}=\frac{4\pi G M m_p c}{\sigma_T}= 1.26\times10^{45}\frac{M}{10^7M_\odot}$
(where $M$ is the mass of the AGN, $m_p$ is the hydrogen atom mass, $c$ is the light speed and $\sigma_T$ is the Stephen-Boltzmann constant) to see that, as awaited, for each
source $L_B < L_{\mathrm{Edd}}$.
As AGN jets can be the main extragalactic sources of UHECRs
<cit.>, one can write the UHECR luminosity $L_{CR}^{Theory}$ as a
fraction $\eta$ of the magnetic luminosity:
\begin{equation}\label{eq:lumicosmic}
L_{CR}^{Theory} = \eta L_{B}.
\end{equation}
For example, bounds on the fraction $\eta_{pr}$ of $L_B$ to be converted in relativistic protons, as a function of BH spin $a$, come
from eqs. (<ref>) and (<ref>) as
\begin{equation}\label{eq:etapr}
\eta(a)_{pr} = \frac{L_{pr}^{UL}}{10(\phi/50)x_{a}^{2}f_{a}(x_{a})\dot{M}c^{2}}.
\end{equation}
For the spin range $0.44 < a < 0.8$ where most black holes
are expected to lie, the fraction $\eta_{pr}$ varies from as $\sim 5\%$ to $40\%$ for $a=0.45$ and from $2\%$ to $10\%$ for $a=0.8$, see Fig. <ref>.
An important remark is that Fig. <ref> reflects the outter bounds of all nine curves $\eta(a)_{pr}$ produced from the nine sources of Table <ref>.
The sources 2MASX J1145 and NGC 5995 are taken as the critical limits for AGNs in the universe up to redshifts $z_s < 0.048$ and they
plot the threshold curves for eq. (<ref>). See, for instance, that the upper limit $L^{UL}_{pr}$ for these two sources is greater
than the theoretical calculated range (for a spin $a \sim 0.7$). In this case, it is expected that their masses produce low levels of magnetic luminosity.
To explain such unexpected great upper limits, one has to admit, e.g., that possibly those AGNs have a critical spinning black hole, with
$a\rightarrow 1$, since the greater the spin, the greater the luminosity $L_B$.
$\eta_{pr}$ as a function of the parameter
$a$. For all plots it was considered $\dot{M}$ as the Bondi accretion rate <cit.>.
Table <ref> shows the range $L_{CR\hbox{,min}}^{Theory} - L_{CR\hbox{,max}}^{Theory} = \eta_{pr,\hbox{min}}L_B - \eta_{pr,\hbox{max}}L_B$ for nine
AGN sources, assuming a fixed spin of $a=0.45$ and $a=0.7$. The column $L_{pr}^{UL}$ is the calculated upper limit of the proton luminosity for each source from
eq. (<ref>), and it has dependence only with observational [GeV-TeV gamma rays + cosmic rays] constraints <cit.>,
i.e., it has no dependence with BH mass or BH spin of the AGN.
Comparing both results, it is possible to see that NGC 1142 fails to produce a proton upper luminosity that remains in the $L_{CR}^{Theory}$
calculated range. This is explained by the fact that the observed
NGC 1142 mass must drive a great magnetic luminosity $L_B$, since the greater the
mass, the greater the luminosity $L_B$. To explain the low cosmic ray
luminosity from NGC 1142 one can assume that the AGN black hole possibly has a small
spin, since the smaller the spin, the smaller the luminosity $L_B$ and consequently the smaller the cosmic ray luminosity.
The present work compared ultra high energetic cosmic ray luminosities with a theoretical luminosity derived from intrisic AGN properties,
as its mass and central black hole spin. An important remark is that luminosity calculation from method (<ref>) does not require any AGN property,
since they come unically from observed integral flux of GeV-TeV gamma-rays of UHECR AGN sources.
In this aspect, both for proton and iron luminosities it is possible to find phenomenological bounds on the conversion fraction of magnetic luminosities into
energetic particles. Such results logically lead to the question whether AGNs are or not the sources of the high energy cosmic rays.
In general, the Fermi acceleration processes are evoqued to explain if AGNs could indeed reach $\sim 10^{19.5}$ eV. In parallel,
for protons, only the most powerful flat spectrum radio quasars, which are considered to be the jet-on analogs of FR II radio galaxies with relativistic jets,
show a magnetic luminosity in excess of $10^{45}$ ergs s$^{-1}$ that can power sufficient energetic jets. Those are the objects studied here.
In complement, BL Lac objects or TeV blazars, thought to be the analogs of FR I radio galaxies (such as Cen A), typically exhibit magnetic luminosities $L_B$ of the order of $10^{44}$ ergs s$^{-1}$ or less.
In such cases, only Fermi processes and shocks in the jets and the hot spots of the most powerful FRII
radio-galaxies may nevertheless offer the requisite conditions to proper accelerations to reach, e. g., $\sim 10^{20}$ eV cosmic rays <cit.>.
In this manner, the presented results could possibly contribute to constrain magnetic and dynamic properties of AGNs to better understand the requisite conditions to proper accelerate
the highest energy cosmic rays.
The authors are very grateful to the researchers of DEE-UFPR. This work was funded by CNPq under grant 458896/2013-6.
olinto10 K. Kotera and A. V. Olinto, The Astrophysics of Ultrahigh-Energy Cosmic Rays, Ann. Rev. of Astron. and Astrophys. 49, 119 (2011).
lemoine12 M. Lemoine, Acceleration and propagation of ultrahigh energy cosmic rays, JPhCS 409, 012007 (2012).
hillas84 A. M. Hillas, The origin of ultra-high-energy cosmic rays, Ann. Rev. Astron. Astrophys. 22, 425 (1984).
nagano1 M. Nagano and A. A. Watson, Observations and implications of the ultrahigh-energy cosmic rays, Rev. Mod. Phys. 72, 689 (2000).
nagano2 M. Nagano, Search for the end of the energy spectrum of primary cosmic rays, N. J. of Phys. 11, 065012 (2009).
bluemer J. Bluemer, R. Engel, J. R. Hörandel, Cosmic rays from the knee to the highest energies, Prog. in Part. and Nuc. Phys. 63, 293 (2009).
beatty J. Beatty and S. Westerhoff, The Highest-Energy Cosmic Rays, Annu. Rev. Nucl.Sci and Particle Physics 59, 319 (2009).
letessier Letessier-Selvon, A and T Stanev, Ultrahigh energy cosmic rays, Rev. Mod. Phys. 83, 907 (2011).
watson13 A. A. Watson, High-energy cosmic rays and the Greisen–Zatsepin–Kuz'min effect, Rep. Prog. Phys. 77, 036901 (2014).
henri G. Henri, G. Pelletier, P.-O. Petrucci, N. Renaud, Active galactic nuclei as high energy engines, Astropart. Phys. 11, 347 (1999).
takahara F. Takahara, On the origin of highest energy cosmic rays, Prog. Theor Phys. 83, 1071 (1990).
rachen J. P. Rachen, P. L. Biermann, Extragalactic ultra-high energy cosmic-rays-part one-contribution from hot spots in Fr-II radio galaxies,
Astron. Astrophys. 272 161 (1993).
supa A.D. Supanitsky and V. de Souza, An upper limit on the cosmic-ray luminosity of individual sources from gamma-ray observations, JCAP, 12, 023 (2013).
vitor R. C. Anjos, A.D. Supanitsky and V. de Souza,
Upper limits on the total cosmic-ray luminosity of individual sources , JCAP, 07, 049 (2014).
The Fermi-LAT Collaboration, The first catalog of active galactic nuclei detected by the Fermi Large Area Telescope, The Astrophysical Journal, 715, 429 (2010).
N. Galante, for the VERITAS Collaboration, VERITAS Observations of Radio Galaxies, Fermi Symposium (2009) 2 arXiv:0912.3850.
hess F. Aharonian et al., Upper limits from HESS active galactic nuclei observations in 2005-2007, Astronomy & Astrophysics, 478, 387 (2008).
magic The MAGIC Collaboration, Detection of very high energy gamma-ray emission from the Perseus cluster head-tail galaxy IC 310 by the MAGIC telescopes , (2010), arXiv:1009.2155.
A. Schulz for The Pierre Auger Collaboration, The measurement of the cosmic ray spectrum above $3 \times 10^{17}$ eV with the Pierre Auger Observatory, Proceedings of the 33rd International Cosmic Ray Conference, Rio de Janeiro, Brazil, 2013, arXiv:1307.5059.
Douglas Bergman for the Telescope Array Collaboration, TA Spectrum Summary, Proceedings of the 33rd International Cosmic Ray Conference, Rio de Janeiro, Brazil, 2013.
CTA Consortium,Design Concepts for the Cherenkov Telescope
Array, Experimental Astronomy, 32 (2011) 193-316.
bondi H. Bondi, On Spherically Symmetrical Accretion, MNRAS 112, 195 (1952).
marek M. Sikora and M. C. Begelman, Magnetic
flux paradigm for radio-loudness of AGN, APJ 64, L24 (2013).
ioana I. Dutan, Laurentiu I. and I. Caramete,
Ultra-high-energy cosmic rays from low-luminosity active galactic nuclei, Astro. Phys., 62 (2015) 206.
biermann J.K. Becker and P.L. Biermann, Neutrinos from
active black holes, sources of ultra high energy cosmic rays,
Astro. Phys., 31 (2009) 138.
mckinney J. C. McKinney, A. Tchekhovskoy, R. D. Blandford,
General relativistic magnetohydrodynamic simulations of magnetically choked accretion flows around black hole,
MNRAS 423, 3083 (2012).
biermann1 P. L. Biermann and P. A. Strittmatter,
Synchrotron emission from shock waves in active galactic nuclei,
Astrophys. J. 322, 643, 1987.
celotti A. Celotti and G. Ghisellini, The power of blazar jets, Month. Not. Roy. Astron. Soc. 385, 283 (2008).
waxman1 E. Waxman and J. Bahcall, Phys. Rev. Lett. 78, 2292 (1997).
waxman2 E. Waxman, Lect. Notes in Phys. 576, 122 (2001).
icecube R. Abbasi et al. (The Ice Cube Collaboration), Nature 484, 351 (2012).
lisa Lisa M.W., Extragalactic X-ray Surveys of ULXs and
AGNs, dissertation, University of Maryland, 2008.
michael Michael Koss et al, Host Galaxy Properties of
the Swift Bat Ultra Hard X-Ray Selected Active Galactic Nucleus,
ApJ 739 57 (2011).
oh K. Oh, H. Choi, H.-G. Kim, J.-S. Moon, S. K. Yi, Astron. J. 146, 151O (2013).
Comparison between cosmic ray luminosity $L_{pr}^{UL}$ (protons) from the method derived from eq. (<ref>) and the
theoretically calculated range $L_{CRmin}^{Theory}-L_{CRmax}^{Theory}$
of cosmic ray luminosities to $a=0.45$ and $a=0.7$, from many sources. For comparison, it is also calculated the upper limit of
the bolometric luminosity (the Eddington luminosity $L_{\mathrm{Edd}}$) for each case below, in erg s$^{-1}$: $L_{\mathrm{Edd}}[\mathrm{NGC}985]=5.02\times10^{48}$,
$L_{\mathrm{Edd}}[\mathrm{NGC}1142]=6.31\times10^{48}$, $L_{\mathrm{Edd}}[2\mathrm{MASXJ}07]=4.68\times10^{48}$, $L_{\mathrm{Edd}}[\mathrm{CGCG}420]=6.83\times10^{48}$,
$L_{\mathrm{Edd}}[\mathrm{MCG}-01]=1.82\times10^{48}$, $L_{\mathrm{Edd}}[2\mathrm{MASXJ}11]=1.26\times10^{48}$, $L_{\mathrm{Edd}}[\mathrm{LEDA}]=4.92\times10^{48}$,
$L_{\mathrm{Edd}}[\mathrm{NGC5995}]=9.78\times10^{47}$, $L_{\mathrm{Edd}}[\mathrm{Mrk520}]=3.17\times10^{48}$.
The mass source comes from
<cit.>. Redshift and other properties are taken from <cit.>.
$a=0.45$ $a=0.7$
Source name $z_s$ log$M_{\odot}$ $\mathbf{L_{pr}^{UL}}$ (Proton) 2|c|$\mathbf{L_{B}}$[erg s$^{-1} \times 10^{45}$] $\mathbf{L_{CRmin}^{Theory}-L_{CRmax}^{Theory}}$ $\mathbf{L_{CRmin}^{Theory}-L_{CRmax}^{Theory}}$
[erg s$^{-1} \times 10^{45}$] $a = 0.45$ $a = 0.7$ [erg s$^{-1} \times 10^{45}$] [erg s$^{-1} \times 10^{45}$]
NGC 985 0.04353 10.6 1.03 5.14 15.54 0.21 - 2.57 0.77 - 2.79
NGC 1142 0.02916 10.7 0.49 6.47 19.57 0.26 - 3.23 0.97 - 3.52
2MASX J07595347+2323241 0.03064 10.57 1.01 4.80 14.51 0.19 - 2.40 0.72 - 2.61
CGCG 420-015 0.02995 10.63 0.95 5.51 16.66 0.22 - 2.76 0.83 - 2.99
MCG-01-24-012 0.02136 10.16 0.65 1.86 5.64 0.07 - 0.93 0.28 - 1.02
2MASX J11454045-1827149 0.03616 10.0 1.30 1.29 3.90 0.07 - 0.64 0.19 - 0.70
LEDA 170194 0.04024 10.59 1.48 5.02 15.19 0.21 - 2.51 0.75 - 2.73
NGC 5995 0.02834 9.89 0.90 1.00 3.03 0.04 - 0.51 0.15 - 0.54
Mrk 520 0.02772 10.4 0.98 3.24 9.81 0.13 - 1.62 0.49 - 1.77
|
1511.00774
|
This paper presents a new approach in application of the Fourier transform to the complex error function resulting in an efficient rational approximation. Specifically, the computational test shows that with only $17$ summation terms the obtained rational approximation of the complex error function provides the average accuracy ${10^{ - 15}}$ over the most domain of practical importance $0 \le x \le 40,000$ and ${10^{ - 4}} \le y \le {10^2}$ required for the HITRAN-based spectroscopic applications. Since the rational approximation does not contain trigonometric or exponential functions dependent upon the input parameters $x$ and $y$, it is rapid in computation. Such an example demonstrates that the considered methodology of the Fourier transform may be advantageous in practical applications.
Keywords: Fourier transform, complex error function, Faddeeva function, rational approximation
§ INTRODUCTION
The forward and inverse Fourier transforms can be defined as <cit.>
\[
\label{eq_1a}\tag{1a}
F\left( \nu \right) = {\cal{F}}\left\{ {f\left( t \right)} \right\}\left( \nu \right) = \int\limits_{ - \infty }^\infty {f\left( t \right){e^{ - 2\pi i \nu t}}dt}
\]
\[
\label{eq_1b}\tag{1b}
f\left( t \right) = {\cal{F}}^{-1}\left\{ {F\left( \nu \right)} \right\}\left( t \right) = \int\limits_{ - \infty }^\infty {F\left( \nu \right){e^{2\pi i \nu t}}d\nu },
\]
respectfully. Approximation theory based on the Fourier trigonometric series for functions or signals remains a topical subject in mathematical analysis and many new efficient methodologies have been reported in the recent scientific literature (see for example <cit.>).
In our recent publication <cit.> we have shown that a sampling with the Gaussian function of the kind $h{e^{ - {{\left( {t/c} \right)}^2}}}/ \left( {c\sqrt \pi } \right)$ leads to the trigonometric approximations for the forward
\[
\label{eq_2a}\tag{2a}
F\left( \nu \right) = {\cal{F}}\left\{ {f\left( t \right)} \right\}\left( \nu \right) \approx h{e^{ - {{\left( {\pi c\nu } \right)}^2}}}\sum\limits_{n = - N}^N {f\left( {nh} \right){e^{ - 2\pi i\nu nh}}}
\]
and inverse Fourier transforms
\[
\label{eq_2b}\tag{2b}
f\left( t \right) = {\cal{F}}^{-1}\left\{ {F\left( \nu \right)} \right\}\left( t \right) \approx h{e^{ - {{\left( {\pi ct} \right)}^2}}}\sum\limits_{n = - N}^N {F\left( {nh} \right){e^{2\pi itnh}}},
\]
where $h$ is the step between two adjacent sampling points and $c$ is the fitting parameter, $e^{ - 2\pi i\nu nh} = \cos \left( 2\pi i\nu nh \right) - i\sin \left( 2\pi i\nu nh \right)$ and $e^{ 2\pi itnh} = \cos \left( 2\pi itnh \right) + i\sin \left( 2\pi itnh \right)$. The parameters $h$, $c$ and $N$ in the equations (<ref>) and (<ref>) may be the same in the forward and inverse Fourier transforms only when we imply the most favorable conditions $h < < 1$, $c < < 1$ and $N > > 1$. In practical tasks, however, these conditions may be compromised in order to reduce the number of the summation terms. As a result, these parameters may not be necessarily equal to each other in the forward and inverse Fourier transforms. Consequently, it is convenient to rewrite two equations above in form
F\left( \nu \right) = {\cal{F}}\left\{ {f\left( t \right)} \right\}\left( \nu \right) \approx {h_f}{e^{ - {{\left( {\pi {c_f}\nu } \right)}^2}}}\sum\limits_{m = - M}^M {f\left( {m{h_f}} \right){e^{ - 2\pi i\nu m{h_f}}}}
f\left( t \right) = {\cal{F}}^{-1}\left\{ {F\left( \nu \right)} \right\}\left( t \right) \approx {h_i}{e^{ - {{\left( {\pi {c_i}t} \right)}^2}}}\sum\limits_{n = - N}^N {F\left( {n{h_i}} \right){e^{2\pi itnh{_i}}}},
where ${h_f}$, ${c_f}$, $M$ and ${h_i}$, ${c_i}$, $N$ are the steps, the fitting parameters and the integers corresponding to the forward and inverse Fourier transforms, respectively.
The presence of the damping functions ${e^{ - {{\left( {\pi {c_f}\nu } \right)}^2}}}$ and ${e^{ - {{\left( {\pi {c_i}t} \right)}^2}}}$ in the equations above excludes periodicity of the approximated functions $f\left( t \right)$ and $F\left( \nu \right)$. Consequently, a solitary wavelet (or non-periodic pulse) can be effectively approximated in the Fourier transform. However, when we take ${c_f} = {c_i} = 0$, the right side of these equations become periodic with corresponding periods $1/{h_f}$, $1/{h_i}$ and represent the Fourier-type expansion series as follows
\[
\label{eq_3a}\tag{3a}
F\left( \nu \right) \approx {h_f}\sum\limits_{m = - M}^M {f\left( {m{h_f}} \right){e^{ - 2\pi i\nu m{h_f}}}}, \qquad - \frac{1}{{2{h_f}}} \le \nu \le \frac{1}{{2{h_f}}},
\]
\[
\label{eq_3b}\tag{3b}
f\left( t \right) \approx {h_i}\sum\limits_{n = - N}^N {F\left( {n{h_i}} \right){e^{2\pi itn{h_i}}}}, \qquad - \frac{1}{{2{h_i}}} \le t \le \frac{1}{{2{h_i}}}.
\]
It should be noted that if the integral (<ref>) is not analytically integrable, then the function $f\left( t \right)$ can be approximated by substituting equation (<ref>) into (<ref>). This substitution yields
\[
\begin{aligned}
f\left( t \right) \approx {h_i}\sum\limits_{n = - N}^N {\underbrace {\left[ {{h_f}\sum\limits_{m = - M}^M {f\left( {m{h_f}} \right){e^{ - 2\pi in{h_i}m{h_f}}}} } \right]}_{ \approx F\left( {n{h_i}} \right)}{e^{2\pi itn{h_i}}}} \\
= {h_i}{h_f}\sum\limits_{n = - N}^N {\sum\limits_{m = - M}^M {f\left( {m{h_f}} \right){e^{2\pi in{h_i}\left( {t - m{h_f}} \right)}}} } , & & - \frac{1}{{2{h_i}}} \le t \le \frac{1}{{2{h_i}}}.
\end{aligned}
\]
In this work we show a new application methodology of the Fourier transform to the complex error function. Due to representation of the complex error function as a rational approximation, it is rapid in computation. Furthermore, with only 17 summation terms the obtained rational approximation of the complex error function provides accuracy ${10^{ - 15}}$ over the most domain of practical importance $0 \le x \le 40,000 \cap {10^{ - 4}} \le y \le {10^2}$ required for applications utilizing the HITRAN molecular spectroscopic database <cit.>.
§ DERIVATION
§.§ Function overview
The complex error function, also known as the Faddeeva function or the Kramp function, can be defined as <cit.>
w\left( z \right) = {e^{ - {z^2}}}\left( {1 + \frac{{2i}}{{\sqrt \pi }}\int\limits_0^z {{e^{{t^2}}}dt} } \right).
where $z = x + iy$ is the complex argument. The complex error function is a solution of the differential equation <cit.>
w'\left( z \right) + 2zw\left( z \right) = \frac{{2i}}{{\sqrt \pi }},
with initial condition $w\left( 0 \right) = 1.$
The complex error function is closely related to a family of the special functions. Among them the most important one is the complex probability function <cit.>
W\left( z \right) = PV\frac{i}{\pi }\int\limits_{ - \infty }^\infty {\frac{{{e^{ - {t^2}}}}}{{z - t}}dt}
W\left( {x,y} \right) = PV\frac{i}{\pi }\int\limits_{ - \infty }^\infty {\frac{{{e^{ - {t^2}}}}}{{\left( {x + iy} \right) - t}}dt}.
This principal value integral implies that the complex probability function has no discontinuity at $y = 0$ and $x = t$. In particular,
\begin{equation}\label{eq_4}
\lim W\left( {x,y \to 0} \right) = {e^{ - {x^2}}} + \frac{{2i}}{{\sqrt \pi }}{\rm{daw}}\left( x \right),
\end{equation}
where ${\rm{daw}}\left( x \right)$ is the Dawson's integral that will be briefly introduced later. There is a direct relationship between complex error function and complex probability function <cit.>
\begin{equation}\label{eq_5}
W\left( z \right) = w\left( z \right), \qquad {\mathop{\rm Im}\nolimits} \left[ z \right] \ge 0.
\end{equation}
The real part of the complex probability function, denoted as $K\left( {x,y} \right)$, is known as the Voigt function. Mathematically, the Voigt function represents a convolution integral of the Gaussian and Lorentzian distributions <cit.>
K\left( {x,y} \right) = PV\frac{y}{\pi }\int\limits_{ - \infty }^\infty {\frac{{{e^{ - {t^2}}}}}{{{y^2} + {{\left( {x - t} \right)}^2}}}dt,}
where the principal value integral also implies that it has no discontinuity at $y = 0$ and $x = t$. Specifically, from equation (<ref>) it follows that
\lim K\left( {x,y \to 0} \right) = {e^{ - {x^2}}}.
At non-negative argument $y$ the real part of the complex error function is also the Voigt function in accordance with identity (<ref>). The Voigt function is widely used in many spectroscopic applications as it describes the line broadening effects <cit.>. Therefore, the application of the complex error function is very significant in quantitative spectroscopy.
Other closely related functions are the error function of complex argument <cit.>
w\left( z \right) = {e^{ - {z^2}}}{\rm{erfc}}\left( { - iz} \right) = {e^{ - {z^2}}}\left[ {1 - {\rm{erf}}\left( { - iz} \right)} \right] \quad \Leftrightarrow \quad {\rm{erf}}\left( z \right) = 1 - {e^{ - {z^2}}} w\left( {iz} \right),
the plasma dispersion function <cit.>
Z\left( z \right) = PV\frac{1}{{\sqrt \pi }}\int\limits_{ - \infty }^\infty {\frac{{{e^{ - {t^2}}}}}{{t - z}}dt = i\sqrt \pi w\left( z \right)}
the Dawson's integral <cit.>
{\rm{daw}}\left( z \right) = {e^{ - {z^2}}}\int\limits_0^z {{e^{{t^2}}}dt = } \sqrt \pi \frac{{ - {e^{ - {z^2}}} + w\left( z \right)}}{{2i}},
the Fresnel integral <cit.>
\[
\begin{aligned}
F_r\left( z \right) &= \int\limits_{0}^z {{e^{i\left( {\pi /2} \right){t^2}}}dt} \\
&= \left( {1 + i} \right)\left[ {1 - {e^{i\left( {\pi /2} \right){z^2}}}w\left( {\sqrt \pi \left( {1 + i} \right)z/2} \right)} \right]/2
\end{aligned}
\]
and the normal distribution function <cit.>
\[
\begin{aligned}
\Phi \left( z \right) = \frac{1}{{\sqrt {2\pi } }}\int\limits_0^z {{e^{ - {t^2}/2}}dt = \frac{1}{2}{\rm{erf}}\left( {\frac{z}{{\sqrt 2 }}} \right)} \\
= \frac{1}{2}\left[ {1 - {e^{ - {z^2}/2}}w\left( {\frac{{iz}}{{\sqrt 2 }}} \right)} \right].
\end{aligned}
\]
It is not difficult to show that the complex error function can be represented in an alternative form (see equation (3) in <cit.> and <cit.>, see also Appendix A in <cit.> for derivation)
\begin{equation}\label{eq_6}
w\left( {x,y} \right) = \frac{1}{{\sqrt \pi }}\int\limits_0^\infty {\exp \left( { - {t^2}/4} \right)\exp \left( { - yt} \right)\exp \left( {ixt} \right)dt}.
\end{equation}
This representation of the complex error function will be used for derivation of a rational approximation.
§.§ Rational approximation
In our recent publications we have shown a new technique to obtain a rational approximation for the integrals of kind <cit.>
\int \limits_0^\infty {{e^{ - {t^2}}}f\left( t \right)dt}.
We apply this approach together with the Fourier transform methodology discussed above in the Introduction.
We can use either of equation (<ref>) or (<ref>). For example, we may choose the equation (<ref>) corresponding to the inverse Fourier transform. Consider the function $f\left( t \right) = {e^{ - {t^2}/4}}$. Let us find first its forward Fourier transform by substituting $f\left( t \right) = {e^{ - {t^2}/4}}$ into equation (<ref>). These leads to
$$F\left( \nu \right) = \int\limits_{ - \infty }^\infty {{e^{ - {t^2}/4}}{e^{ - 2\pi ivt}}dt} = 2\sqrt \pi {e^{ - {{\left( {2\pi \nu } \right)}^2}}}.
Now substituting $2\sqrt \pi {e^{ - {{\left( {2\pi \nu } \right)}^2}}}$ into equation (<ref>) yields the following approximation for the exponential function
{e^{ - {t^2}/4}} \approx 2\sqrt \pi {h_i}\sum\limits_{n = - N}^N {{e^{ - {{\left( {2\pi n{h_i}} \right)}^2}}}{e^{2\pi itn{h_i}}}}, \qquad - \frac{1}{{2{h_i}}} \le t \le \frac{1}{{2{h_i}}},
\begin{equation}\label{eq_7}
{e^{ - {t^2}/4}} \approx 2\sqrt \pi {h_i}\sum\limits_{n = - N}^N {{e^{ - {{\left( {2\pi n{h_i}} \right)}^2}}}\cos \left( {2\pi tn{h_i}} \right)}, \qquad - \frac{1}{{2{h_i}}} \le t \le \frac{1}{{2{h_i}}}.
\end{equation}
Taking into account that
{e^{ - {{\left( {2\pi 0{h_i}} \right)}^2}}}\cos \left( {2\pi t0{h_i}} \right) = 1
\sum\limits_{n = - N}^{ - 1} {{e^{ - {{\left( {2\pi n{h_i}} \right)}^2}}}\cos \left( {2\pi tn{h_i}} \right)} = \sum\limits_{n = 1}^N {{e^{ - {{\left( {2\pi n{h_i}} \right)}^2}}}\cos \left( {2\pi tn{h_i}} \right)}
the approximation (<ref>) can be simplified as given by
\begin{equation}\label{eq_8}
{e^{ - {t^2}/4}} \approx 2\sqrt \pi {h_i}\left[ {1 + 2\sum\limits_{n = 1}^N {{e^{ - {{\left( {2\pi n{h_i}} \right)}^2}}}\cos \left( {2\pi tn{h_i}} \right)} } \right], \qquad - \frac{1}{{2{h_i}}} \le t \le \frac{1}{{2{h_i}}}.
\end{equation}
The right side limitation $t \le 1/\left( {2{h_i}} \right)$ along the positive $t$-axis in equation (<ref>) can be readily excluded by multiplying both its sides to $\exp \left( { - \sigma t} \right)$ if a constant $\sigma $ is positive and sufficiently large. This can be explained by considering Fig. 1 that shows two functions computed according to right side of equation (<ref>) at $\sigma = 0.1$ (blue curve) and $\sigma = 0.2$ (red curve). For example, at $\sigma = 0.1$ we can observe two additional peaks at $1/{h_i}$ and $2/{h_i}$ (blue curve). However, as $\sigma $ increases the additional peaks are suppressed stronger to zero due to multiplication to the damping exponential function $\exp \left( { - \sigma t} \right)$. As a result, at $\sigma = 0.2$ only a single additional peak at $1/{h_i}$ remains visible (red curve). By $\sigma \gtrsim 1$ all additional peaks completely vanish and, therefore, do not contribute to error in integration. Consequently, if the constant $\sigma $ is large enough, say approximately equal or greater than $1$, we can write the approximation
{e^{ - {t^2}/4}}{e^{ - \sigma t}} \approx 2\sqrt \pi {h_i}\left[ {1 + 2\sum\limits_{n = 1}^N {{e^{ - {{\left( {2\pi n{h_i}} \right)}^2}}}\cos \left( {2\pi tn{h_i}} \right)} } \right]{e^{ - \sigma t}}, \qquad \sigma \mathbin{\lower.3ex\hbox{$\buildrel>\over
{\smash{\scriptstyle\sim}\vphantom{_x}}$}} 1,
that remains always valid without any limitation along the positive $t$-axis. Assuming $y \ge 0$ we, therefore, can write now
\begin{equation}\label{eq_9}
{e^{ - {t^2}/4}}{e^{ - \left( {y + \sigma } \right)t}} \approx 2\sqrt \pi {h_i}\left[ {1 + 2\sum\limits_{n = 1}^N {{e^{ - {{\left( {2\pi n{h_i}} \right)}^2}}}\cos \left( {2\pi tn{h_i}} \right)} } \right]{e^{ - \left( {y + \sigma } \right)t}}, \qquad \sigma \mathbin{\lower.3ex\hbox{$\buildrel>\over
{\smash{\scriptstyle\sim}\vphantom{_x}}$}} 1.
\end{equation}
Fig. 1. Function approximation for ${e^{ - {t^2}/4}}{e^{ - \sigma t}}$ at $\sigma = 0.1$ (blue curve) and $\sigma = 0.2$ (red curve). The dashed and dot-dashed curves are the exponential functions ${e^{ - 0.1t}}$ and ${e^{ - 0.2t}}$, respectively.
Since ${e^{ - {t^2}/4}}{e^{ - yt}} = {e^{{\sigma ^2}}}{e^{ - {{\left( {t - 2\sigma } \right)}^2}/4}}{e^{ - \left( {y + \sigma } \right)t}}$ from approximation (<ref>) we obtain
\begin{equation}\label{eq_10}
{e^{ - {t^2}/4}}{e^{ - yt}} \approx 2\sqrt \pi {h_i}{e^{{\sigma ^2}}}\left[ {1 + 2\sum\limits_{n = 1}^N {{e^{ - {{\left( {2\pi n{h_i}} \right)}^2}}}\cos \left( {2\pi n{h_i}\left( {t - 2\sigma } \right)} \right)} } \right]{e^{ - \left( {y + \sigma } \right)t}}, \quad \sigma \mathbin{\lower.3ex\hbox{$\buildrel>\over
{\smash{\scriptstyle\sim}\vphantom{_x}}$}} 1.
\end{equation}
Once again, due to presence of the rapidly damping exponential multiplier ${e^{ - \left( {y + \sigma } \right)t}}$ this approximation is valid without any limitation along the positive $t$-axis. As the peak of the function ${e^{ - {{\left( {t - 2\sigma } \right)}^2}/4}}$ is shifted towards right with respect to the origin, we may regard to the value $\sigma $ as the shift constant.
Finally, substituting approximation (<ref>) into integral (<ref>) yields
\begin{equation}\label{eq_11}
w\left( z \right) = i\frac{{2{h_i}{e^{{\sigma ^2}}}}}{z + i\sigma } + \sum\limits_{n = 1}^N {\frac{{{A_n} - i\left( {z + i\sigma } \right){B_n}}}{{C_n^2 - {{\left( {z + i\sigma } \right)}^2}}}},
\end{equation}
{A_n} = 8\pi h_i^2n{e^{{\sigma ^2} - {{\left( {2\pi {h_i}n} \right)}^2}}}\sin \left( {4\pi {h_i}n\sigma } \right),
{B_n} = 4{h_i}{e^{{\sigma ^2} - {{\left( {2\pi {h_i}n} \right)}^2}}}\cos \left( {4\pi {h_i}n\sigma } \right)
{C_n} = 2\pi {h_i}n.
As the expansion coefficients ${A_n}$, ${B_n}$ and ${C_n}$ are independent of the argument $z$, the obtained equation (<ref>) is a rational approximation.
In algorithmic implementation it is more convenient to use $\psi $-function defined as
\begin{equation}
\begin{aligned}\label{eq_12}
\psi \left( z \right) = i\frac{{2{h_i}{e^{{\sigma ^2}}}}}{z} + \sum\limits_{n = 1}^N {\frac{{{A_n} - iz{B_n}}}{{C_n^2 - {z^2}}}} \\
\Rightarrow w\left( z \right) \approx \psi \left( {z + i\sigma } \right).
\end{aligned}
\end{equation}
§.§ Computational procedure and error analysis
Due to a remarkable identity of the complex error function <cit.>
\begin{equation}\label{eq_13}
w\left( { - z} \right) = 2{e^{ - {z^2}}} - w\left( z \right),
\end{equation}
it is sufficient to consider only I and II quadrants in order to cover the entire complex plane. This can be seen explicitly by representation of the identity (<ref>) in form
w\left( { \pm x, - \left| y \right|} \right) = 2{e^{ - {{\left( { \mp x + i\left| y \right|} \right)}^2}}} - w\left( { \mp x, + \left| y \right|} \right).
Thus, if the parameter $y$ is negative we can simply take it by absolute value and then compute the complex error function according to right side of this equation. Therefore, further we will always assume that $y \ge 0$.
When the argument $z$ is large enough by absolute value, say $\left| {x + iy} \right| \mathbin{\lower.3ex\hbox{$\buildrel>\over
{\smash{\scriptstyle\sim}\vphantom{_x}}$}} 15$, we can truncate the Laplace continued fraction <cit.>
w\left( z \right) = \frac{{{\mu _0}}}{{z - }}\frac{{1/2}}{{z - }}\frac{1}{{z - }}\frac{{3/2}}{{z - }}\frac{2}{{z - }}\frac{{5/2}}{{z - }}\frac{3}{{z - }}\frac{{7/2}}{{z - }}...\,, \quad {\mu _0} = i/\pi .
Approximation based on the Laplace continued fraction is rapid in computation. However, its accuracy deteriorates as the argument $z$ decreases by absolute value.
There are different approximations for computation of the narrow-band domain $0 \le x \le 15$ and $0 \le y < {10^{ - 6}}$ <cit.>. We can apply, for example, an approximation proposed in our recent work <cit.>
w\left( {x,y < < 1} \right) \approx \left( {1 - \frac{y}{{{y_{\min }}}}} \right){e^{ - {x^2}}} + \frac{y}{{{y_{\min }}}}K\left( {x,{y_{\min }}} \right) + iL\left( {x,{y_{\min }}} \right), \qquad y_{\min} < < 1,
where $L \left( x, y_{\min} \right) = {\mathop{\rm Im}\nolimits} \left[ w \left( x, y_{\min} \right) \right]$ and ${y_{\min }}$ can be taken equal to ${10^{ - 5}}$. It has been shown that this approximation can provide accuracy better than ${10^{ - 9}}$ over the narrow-band domain $0 \le x \le 15$ and $0 \le y < {10^{ - 6}}$.
The domain $\left| {x + iy} \right| \le 15 \cap y \ge {10^{ - 6}}$ is the most difficult for computation. Nevertheless, with only $17$ summation terms (at $N = 16$) the proposed rational approximation (<ref>) covers this domain providing high-accuracy and rapid computation. In computational procedure we have to choose properly the margin value ${\nu _m}$ for the exponential function ${e^{ - {{\left( {2\pi \nu } \right)}^2}}}$ that appears from the forward Fourier transform $F\left( \nu \right) = 2\sqrt \pi {e^{ - {{\left( {2\pi \nu } \right)}^2}}}$. As it has been justified by Melone et al. <cit.>, the margin value for integration involving the exponential function ${e^{ - {t^2}}}$ can be taken as $t = {t_m} = 6$. We can use this result in order to determine the required value by solving the following equation with respect to the variable $\nu $ as follows
{e^{ - {{\left( {2\pi \nu } \right)}^2}}} = {\left. {{e^{ - {t^2}}}} \right|_{t = 6}} \Rightarrow {\left( {2\pi \nu } \right)^2} = 36.
There are two solutions for this equation ${\nu _{1,2}} = \pm 6/\left( {2\pi } \right)$. Consequently, the margin value for the exponential function ${e^{ - {{\left( {2\pi \nu } \right)}^2}}}$ can be taken as ${\nu _m} = 6/\left( {2\pi } \right)$. As a parameter ${h_i}$ is the step between two adjacent sampling points along positive $\nu $-axis (see <cit.> for details), its value can be calculated as ${h_i} = {\nu _m}/N$. Taking $N = 16$ we can find that ${h_i} = {\nu _m}/16 \approx {\rm{5}}{\rm{.968310365946075}} \times {\rm{1}}{{\rm{0}}^{ - 2}}$.
In order to quantify the accuracy of the rational approximation (<ref>) we may define the relative errors
{\Delta _{{\mathop{\rm Re}\nolimits} }} = \left| {\frac{{{\mathop{\rm Re}\nolimits} \left[ {{w_{ref}}\left( {x,y} \right)} \right] - {\mathop{\rm Re}\nolimits} \left[ {w\left( {x,y} \right)} \right]}}{{{\mathop{\rm Re}\nolimits} \left[ {{w_{ref}}\left( {x,y} \right)} \right]}}} \right|
{\Delta _{{\mathop{\rm Im}\nolimits} }} = \left| {\frac{{{\mathop{\rm Im}\nolimits} \left[ {{w_{ref}}\left( {x,y} \right)} \right] - {\mathop{\rm Im}\nolimits} \left[ {w\left( {x,y} \right)} \right]}}{{{\mathop{\rm Im}\nolimits} \left[ {{w_{ref}}\left( {x,y} \right)} \right]}}} \right|,
where ${w_{ref}}\left( {x,y} \right)$ is the reference, for the real and imaginary parts, respectively. The highly accurate reference values can be generated by using, for example, Algorithm 680 <cit.>, recently published Algorithm 916 <cit.> or C++ code from the RooFit package, CERN's library <cit.>.
Figure 2 shows ${\log _{10}}{\Delta _{{\mathop{\rm Re}\nolimits} }}$ for the real part of the complex error function computed over the domain $0 \le x \le 15$ and ${10^{ - 6}} \le y \le 15$ at $N = 16$, $\sigma = 1.5$ and ${h_i} = {\rm{5}}{\rm{.968310365946075}} \times {\rm{1}}{{\rm{0}}^{ - 2}}$. As we can see from this figure, the rational approximation (<ref>) provides accuracy ${10^{ - 15}}$ (blue color) over the most of this domain. Although accuracy deteriorates with decreasing $y$, it remains better than ${10^{ - 9}}$ (red color) in the range ${10^{ - 4}} \le y \le {10^{ - 6}}$. This indicates that at the same $N = 16$ the accuracy of the rational approximation (<ref>) is by several orders of the magnitude higher than the accuracy of the Weideman's rational approximation (see equation 38(I) in <cit.>).
Fig. 2. The logarithm of the relative error ${\log _{10}}{\Delta _{{\mathop{\rm Re}\nolimits} }}$ for the real part of the rational approximation (<ref>) over the domain $0 \le x \le 15 \cap {10^{ - 6}} \le y \le {10^4}$.
Figure 3 illustrates ${\log _{10}}{\Delta _{{\mathop{\rm Im}\nolimits} }}$ for the imaginary part of the complex error function also computed over the domain $0 \le x \le 15$ and ${10^{ - 6}} \le y \le 15$ at $N = 16$, $\sigma = 1.5$ and ${h_i} = {\rm{5}}{\rm{.968310365946075}} \times {\rm{1}}{{\rm{0}}^{ - 2}}$. One can see that in the imaginary part the accuracy is also highly accurate ${10^{ - 15}}$ (blue color) over the most domain. There is only a small area $0 \le x < 1$ and ${10^{ - 6}} \le y \le {10^{ - 4}}$ near the origin where the accuracy deteriorates as the parameters $x$ and $y$ both tend to zero. Nevertheless, the accuracy in this area still remains high and better than ${10^{ - 9}}$ (red color).
Fig. 3. The logarithm of the relative error ${\log _{10}}{\Delta _{{\mathop{\rm Im}\nolimits} }}$ for the imaginary part of the rational approximation (<ref>) over the domain $0 \le x \le 15 \cap {10^{ - 6}} \le y \le {10^4}$.
The computational test reveals that with only $17$ summation terms (at $N = 16$) the rational approximation (<ref>) alone can cover the entire HITRAN domain $0 \le x \le 40,000 \cap {10^{ - 4}} \le y \le {10^2}$ providing average accuracy ${10^{ - 15}}$ for an input array consisting of $3 \times 10^7$ elements. Algorithmic implementation of the rational approximation (<ref>) results to the same computational speed as that of described in our recent work where we proposed a sampling by incomplete cosine expansion of the sinc function to approximate the complex error function <cit.>.
A Matlab subroutine code that covers the HITRAN domain with high-accuracy is presented in Appendix A.
§ CONCLUSION
We present a new efficient rational approximation to the complex error function by application of the Fourier transform that provides computationally rapid and highly accurate results. The computational test we performed with only $17$ summation terms shows that the accuracy of the rational approximation of the complex error function is ${10^{ - 15}}$ over the most domain of practical importance. In particular, the proposed rational approximation of the complex error function alone can cover with high accuracy the entire domain $0 \le x \le 40,000 \cap {10^{ - 4}} \le y \le {10^2}$ required for the HITRAN-based spectroscopic applications.
§ ACKNOWLEDGMENTS
This work is supported by National Research Council Canada, Thoth Technology Inc. and York University.
§ APPENDIX A
R.N. Bracewell, The Fourier transform and its application, $3^\text{rd}$ ed., McGraw-Hill, 2000.
E.W. Hansen, Fourier transforms. Principles and applications, John Wiley & Sons, 2014.
V.N. Mishra and L.N. Mishra, Trigonometric approximation of signals (functions) in $L_{p}$-norm. Int. J. Contemp. Math. Sci., 7 (19) (2012) 909-918. <http://www.m-hikari.com/ijcms/ijcms-2012/17-20-2012/narayanmishraIJCMS17-20-2012.pdf>
V.N. Mishra, H.H. Khan, I.A. Khan, K. Khatri and L.N. Mishra, Trigonometric approximation of signals (functions) belonging to the $Lip \left( \xi \left( t \right),r \right)$, $\left( r > 1 \right)$-class by $\left( E,q \right)$ $\left( q > 0 \right)$-means of the conjugate series of its Fourier series. Advan. Pure Math., 3 (2013) 353-358. <http://dx.doi.org/10.4236/apm.2013.33050>
J.P. Boyd, A comparison of companion matrix methods to find roots of a trigonometric polynomial, J. Comp. Phys., 246 (2013) 96–112. <http://dx.doi.org/10.1016/j.jcp.2013.03.022>
S.M. Abrarov and B. M. Quine, Representation of the Fourier transform as a weighted sum of the complex error functions, arXiv:1507.01241v3. <http://arxiv.org/pdf/1507.01241v3.pdf>
L.S. Rothman, I.E. Gordon, Y. Babikov, A. Barbe, D.C. Benner, P.F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L.R. Brown, A. Campargue, K. Chance, E.A. Cohen, L.H. Coudert, V.M. Devi, B.J. Drouin, A. Fayt, J.-M. Flaud, R.R. Gamache, J.J. Harrison, J.-M. Hartmann, C. Hill, J.T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R.J. Le Roy, G. Li, D.A. Long, O.M. Lyulin, C.J. Mackie, S.T. Massie, S. Mikhailenko, H.S.P. Müler, O.V. Naumenko, A.V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E.R. Polovtseva and C. Richard, The HITRAN2012 molecular spectroscopic database, J. Quant. Spectrosc. Radiat. Transfer, 130 (2013) 4-50. <http://dx.doi.org/10.1016/j.jqsrt.2013.07.002>
V.N. Faddeyeva, and N.M. Terent'ev, Tables of the probability integral $w\left( z \right) = {e^{ - {z^2}}}\left( {1 + \frac{{2i}}{{\sqrt \pi }}\int_0^z {{e^{{t^2}}}dt} } \right)$ for complex argument. Pergamon Press, Oxford, 1961.
W. Gautschi, Efficient computation of the complex error function. SIAM J. Numer. Anal., 7 (1970) 187-198. <http://dx.doi.org/10.1137/0707012>
M. Abramowitz and I.A. Stegun. Error Function and Fresnel Integrals. Handbook of mathematical functions with formulas, graphs, and mathematical tables. $9^{\text{th}}$ ed. New York 1972, 297-309.
B.H. Armstrong and B.W. Nicholls, Emission, absorption and transfer of radiation in heated atmospheres. Pergamon Press, New York, 1972.
F. Schreier, The Voigt and complex error function: A comparison of computational methods. J. Quant. Spectrosc. Radiat. Transfer, 48 (1992) 743-762. <http://dx.doi.org/10.1016/0022-4073(92)90139-U>
J.A.C. Weideman, Computation of the complex error function. SIAM J. Numer. Anal., 31 (1994) 1497-1518. <http://dx.doi.org/10.1137/0731077>
K.L. Letchworth and D.C. Benner, Rapid and accurate calculation of the Voigt function, J. Quant. Spectrosc. Radiat. Transfer, 107 (2007) 173-192. <http://dx.doi.org/10.1016/j.jqsrt.2007.01.052>
G. Pagnini and F. Mainardi, Evolution equations for the probabilistic generalization of the Voigt profile function, J. Comput. Appl. Math., 233 (2010) 1590-1595. <http://dx.doi.org/10.1016/j.cam.2008.04.040>
S.M. Abrarov, B.M. Quine and R.K. Jagpal, High-accuracy approximation of the complex probability function by Fourier expansion of exponential multiplier, Comp. Phys. Commun., 181 (5) (2010) 876-882. <http://dx.doi.org/10.1016/j.cpc.2009.12.024>
D.P. Edwards, GENLN2: A general line-by-line atmospheric transmittance and radiance model, NCAR technical note, 1992. <http://dx.doi.org/10.5065/D6W37T86>
B.M. Quine and J.R. Drummond, GENSPECT: a line-by-line code with selectable interpolation error tolerance J. Quant. Spectrosc. Radiat. Transfer 74 (2002) 147-165. <http://dx.doi.org/10.1016/S0022-4073(01)00193-5>
L.E. Christensen, G.D. Spiers, R.T. Menzies and J.C Jacob, Tunable laser spectroscopy of $\rm{CO}_{2}$ near $2.05 \, {\mu}m$: Atmospheric retrieval biases due to neglecting line-mixing, J. Quant. Spectrosc. Radiat. Transfer, 113 (2012) 739-748. <http://dx.doi.org/10.1016/j.jqsrt.2012.02.031>
A. Berk, Voigt equivalent widths and spectral-bin single-line transmittances: Exact expansions and the MODTRAN5 implementation, J. Quant. Spectrosc. Radiat. Transfer, 118 (2013) 102-120. <http://dx.doi.org/10.1016/j.jqsrt.2012.11.026>
B.M. Quine and S.M. Abrarov, Application of the spectrally integrated Voigt function to line-by-line radiative transfer modelling. J. Quant. Spectrosc. Radiat. Transfer, 127 (2013) 37-48. <http://dx.doi.org/10.1016/j.jqsrt.2013.04.020>
B.D. Fried and S.D. Conte. The plasma dispersion function. New York: Academic Press, 1961.
W.J. Cody, K.A. Paciorek and H.C. Thacher, Chebyshev approximations for Dawson's integral. Math. Comp. 24 (1970) 171-178. <http://dx.doi.org/10.1090/S0025-5718-1970-0258236-8>
J.H. McCabe, A continued fraction expansion with a truncation error estimate for Dawson's integral, Math. Comp. 28 (1974) 811-816. <http://dx.doi.org/10.1090/S0025-5718-1974-0371020-3>
G.B. Rybicki, Dawson's integral and the sampling theorem, Comp. Phys., 3 (1989) 85-87. <http://dx.doi.org/10.1063/1.4822832>
J.P. Boyd, Evaluating of Dawson's integral by solving its differential equation using orthogonal rational Chebyshev functions, Appl. Math. Comput., 204 (2) (2008) 914-919. <http://dx.doi.org/10.1016/j.amc.2008.07.039>
S.M. Abrarov and B.M. Quine, A rational approximation for the Dawson's integral of real argument, arXiv:1505.04683. <http://arxiv.org/pdf/1505.04683.pdf>
S.J. McKenna, A method of computing the complex probability function and other related functions over the whole complex plane. Astrophysics and Space Science, 107 (1) (1984) 71-83. <http://dx.doi.org/10.1007/BF00649615>
E.W. Weisstein, CRC concise encyclopedia of mathematics. Chapman & Hall/CRC, $2^\text{{nd}}$ ed. 2003.
H.M. Srivastava and E.A. Miller, A unified presentations of the Voigt functions, Astrophys. Space Sci., 135 (1987) 111-118. <http://dx.doi.org/10.1007/BF00644466>
H.M. Srivastava and M.P. Chen, Some unified presentations of the Voigt functions. Astrophys. Space Sci., 192 (1) (1992) 63-74. <http://dx.doi.org/10.1007/BF00653260>
S.M. Abrarov and B.M. Quine, Master-slave algorithm for highly accurate and rapid computation of the Voigt/complex error function, J. Math. Research, 6 (2) (2014) 104-119. <http://dx.doi.org/10.5539/jmr.v6n2p104>
S.M. Abrarov and B.M. Quine, Sampling by incomplete cosine expansion of the sinc function: Application to the Voigt/complex error function, Appl. Math. Comput., 258 (2015) 425-435. <http://dx.doi.org/ 10.1016/j.amc.2015.01.072>
M.R. Zaghloul and A.N. Ali, Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Transactions on Mathematical Software, 38 (2011) 15:1-15:22. <http://dx.doi.org/10.1145/2049673.2049679>
W.B. Jones and W.J. Thron, Continued fractions in numerical analysis. Appl. Num. Math., 4(2-4) (1988) 143-230. <http://dx.doi.org/10.1016/0168-9274(83)90002-8>
H. Amamou, B. Ferhat and A. Bois, Calculation of the Voigt Function in the region of very small values of the parameter $a$ where the calculation is notoriously difficult, Amer. J. Anal. Chem., 4 (2013) 725-731. <http://dx.doi.org/10.4236/ajac.2013.412087>
S.M. Abrarov and S.M. Quine, Accurate approximations for the complex error function with small imaginary argument, J. Math. Research 7 (1) (2015) 44-53. <http://dx.doi.org/10.5539/jmr.v1n1p44>
A.A.E. Milone, L.A. Milone and G.E. Bobato, Numerical evaluation of the line broadening function . Astrophysics. and Space Science, 147 (2) (1988) 229-234. <http://dx.doi.org/10.1007/BF00645667>
G.P.M. Poppe and C.M.J. Wijers, More efficient computation of the complex error function. ACM Transact. Math. Software, 16 (1990) 38-46. <http://dx.doi.org/10.1145/77626.77629>
G.P.M. Poppe and C.M.J. Wijers, Algorithm 680: evaluation of the complex error function. ACM Transact. Math. Software, 16 (1990) 47. <http://dx.doi.org/10.1145/77626.77630>
T.M. Karbach, G. Raven and M. Schiller, Decay time integrals in neutral meson mixing and their efficient evaluation, arXiv:1407.0748. <http://arxiv.org/pdf/1407.0748v1.pdf>
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1511.00834
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A system in a Birkhoff normal form with an irregular singularity of Poincaré rank 1 at the origin and a regular singularity at infinity is
through the Borel-Laplace transform dual to a system in an Okubo form.
Schäfke has showed that the Birkhoff system can also be obtained from the Okubo
system by a simple limiting procedure.
The Okubo system comes naturally with two kinds of mixed solution bases,
both of which converge under the limit procedure to the canonical solutions of the limit Birkhoff system on sectors near the irregular singularity at the origin.
One can then define Stokes matrices of the Okubo system as connection matrices between different branches of the mixed solution bases
and use them to relate the monodromy matrices of the Okubo system to the usual Stokes matrices of the limit system at the irregular singularity.
This is illustrated on the example of confluence in the generalized hypergeometric equation.
Key words: Linear differential equations, generalized hypergeometric equation, confluence, Stokes matrices, monodromy.
§ INTRODUCTION
A linear differential system
\begin{equation}\label{eq:hg-1}
(s-B)\frac{dv}{ds}=(A+\rho)v, \qquad (s,v)\in\CP^1\times\C^n,
\end{equation}
where $A,B$ are constant $n\times n$-matrices, $B$ is diagonal, $\rho\in\C$ a parameter,
is called an Okubo system, or also a hypergeometric system.
Such systems appear as a natural generalization of the hypergeometric equation.
It is known <cit.>, that every single Fuchsian differential equation can be reduced to such a system.
The assumption that $B$ is diagonal (or semisimple) assures that the 1-form $(s-B)^{-1}\,ds$ has only simple poles (placed at the eigenvalues of $B$ and at $\infty$),
i.e. that all the singularities of the Okubo system (<ref>) are Fuchsian.
The Okubo system (<ref>) appears also as a dual to a system in Birkhoff normal form
\begin{equation}\label{eq:hg-2}
z^2\frac{d\psi}{dz}=(B+zA)\psi, \qquad (z,\psi)\in\CP^1\times\C^n,
\end{equation}
which has an irregular singular point at 0 and a Fuchsian singular point at $\infty$,
through the Laplace transform
\begin{equation*}
\psi(z)=z^{-1-\rho}\int_0^\infty v(s,\rho)\,e^{-\frac{s}{z}}\,ds, \qquad |\arg s-\arg z|<\tfrac{\pi}{2}.
\end{equation*}
This fact can be used to express the Stokes and connection matrices of the Birkhoff system in terms of connection matrices and monodromies of the dual Okubo system <cit.>.
Schäfke <cit.> has remarked that the the system (<ref>) can be also obtained from (<ref>) by the following
confluence procedure:
$$\text{let}\quad s=\rho z, \quad\text{and}\quad y(z,\rho)=s^{-\rho}v(s,\rho),$$
then $y$ satisfies
\begin{equation}\label{eq:hg-3}
\end{equation}
which becomes (<ref>) at the limit when $\rho\to\infty$.
In case of rank $n=2$ and $B$ with two distinct eigenvalues,
this confluence procedure corresponds exactly to the confluence of the (Gauss') hypergeometric equation to the (Kummer's) confluent hypergeometric equation.
Aside from the usual local Levelt bases at each of the singularities, the Okubo system (<ref>) has two other kinds of natural solution bases,
so called mixed bases <cit.>:
The first one, called Floquet basis, consists of the Floquet solutions (singular Levelt solutions) at different finite singularities $\lambda_j$ (eigenvalues of $B$). The other, called co-Floquet basis, is in a sense dual; a co-Floquet solution at a singularity $\lambda_j$ is one that is
analytic at al other singularities $\lambda_k$, $k\neq j$.
Schäfke <cit.> has studied the limits of these mixed bases in the confluent family (<ref>)
in the case where all the eigenvalues of $B$ are distinct
and has shown that they both tend to the canonical solution basis of the limit system (Borel sum of a formal fundamental solution)
on sectors at the irregular singularity $z=0$:
the Floquet basis when $\rho\to+\infty$, and the co-Floquet basis when $\rho\to-\infty$.
This article exposes these results while extending them to a more general situation, where $B$ is allowed to have multiple eigenvalues, and $\rho$ can go to infinity along any fixed direction in one of two sectors of opening $>\pi$ covering a neighborhood of
$\infty$ on the Riemann sphere $\CP^1$.
In an analogy with <cit.> it is natural to introduce parametric Stokes matrices of the confluent family (<ref>),
as connection matrices between different branches of the Floquet (resp. co-Floquet) basis far from the origin.
These parametric Stokes matrices are closely related to the monodromy
of the family (<ref>):
in general, the monodromy matrices of the Floquet and co-Floquet bases
can be expressed as products of these Stokes matrices and formal monodromy matrices.
While the monodromy matrices diverge when $\rho\to\infty$
(because of their formal monodromy parts which are exponential functions of $\rho$)
these parametric Stokes matrices tend to the usual Stokes matrices of the limit system, and can be easily obtained from them
(Proposition <ref> ).
These results are illustrated in Section 2 on explicit calculations in the case of the generalized hypergeometric equation,
previously studied by Duval <cit.>.
Duval considered the problem of convergence of the monodromy matrices to the Stokes matrices without separating formal monodromy part and the Stokes part. Therefore she could only consider limits when $\rho\to\pm\infty$ following a discrete set of values on which the formal monodromy part is constant.
A different confluence procedure of the type
\begin{equation}\label{eq:hg-4}
(z^2-\epsilon)\frac{dy}{dz}=\Omega(z)y, \qquad\C\ni\epsilon\to 0,
\end{equation}
was investigated in e.g. <cit.>.
In case of $B$ having only two eigenvalues (one of which can be always shifted to 0), the confluence procedure (<ref>)
can be considered as a special case of (<ref>).
In this case our perspective essentially coincides with that of <cit.>.
In particular,
this includes the confluence in the Gauss' hypergeometric equations <cit.> and in the generalized hypergeometric equation <cit.>.
§ GENERAL THEORY
Let the matrix $B$ be diagonal with eigenvalues $\lambda_1,\ldots,\lambda_p$ of respective multiplicities $n_1,\dots,n_p$, and let the matrix $A$ be partitioned into blocks accordingly
\begin{equation*}
\lambda_1I_{n_1} && \\ & \ddots & \\ && \lambda_pI_{n_p}
\end{pmatrix},\qquad
A_{11} &\ldots& A_{1p} \\ \vdots & &\vdots \\ A_{p1} &\ldots & A_{pp}
\end{pmatrix}.
\end{equation*}
The following assumption is made throughout the text:
\begin{equation}\label{eq:hg-asum}
\textit{No two eigenvalues of $A_{jj}$ differ by a non-zero integer, $1\leq j\leq p$.}
\end{equation}
For any $n\!\times\! n$-matrix $X$, let $(X_{ij})_{1\leq i,j\leq p}$ be its bloc-partition according to $B$, and
denote $X_{\cdot j}=$ the $j$-th bloc column of $X$.
§.§ Fundamental solution of the limit system (<ref>)
It is well-known (see for example <cit.>)
that the system (<ref>) can be bloc-diagonalized by means of a formal power series transformation
$\psi=\hat T(z)\phi$, with
$$\hat T(z)=\sum_{k=0}^{+\infty}T^{(k)}z^k, \qquad T^{(0)}=0.$$
Under the assumption (<ref>), the formally transformed system can be given the following Birkhoff form
\begin{equation}\label{eq:hg-2formal}
z^2\frac{d\phi}{dz}=(B+zA_D)\phi, \qquad \text{with}\quad
A_{11} && \\ & \ddots & \\ && A_{pp}
\end{pmatrix}.
\end{equation}
Therefore the system (<ref>) has a formal fundamental solution $\hat\Psi(z)$ whose $j$-th bloc-column is given by
\begin{equation*}
\hat\Psi_{\cdot j}(z)=\hat T_{\cdot j}(z) z^{A_{jj}} e^{-\frac{\lambda_j}{z}}.
\end{equation*}
While $\hat T(z)$ is in general divergent, it is Borel summable.
More precisely each its column $\hat T_{\cdot j}(z)$ is Borel summable
in all directions $\alpha$ with $e^{i\alpha}\R^+$ disjoint from all $\lambda_i-\lambda_j$, $i\neq j$ (such direction $\alpha$ will be called non-singular).
\begin{equation}\label{eq:hg-U}
U_{\cdot j}(s)=\sum_{k=0}^{+\infty} \frac{T^{(k)}_{\cdot j}}{k!}s^k
\end{equation}
be the formal Borel transform of $z\,\hat T_{\cdot j}(z)$, convergent near $s=0$ and extended analytically on the universal covering of
$\C\sminus\{\lambda_i-\lambda_j \mid 1\leq i\leq p,\ i\neq j\}$.
The matrix function $U(s)$ is a solution to linear system
\begin{equation*}
\end{equation*}
with Fuchsian singularities at the points $\lambda_i-\lambda_j$ and $\infty$.
In particular, $U$ has only a moderate growth at each of the singularities.
Therefore the Borel sum of $\hat T_{\cdot j}(z)$ in a non-singular direction $\alpha$ is well-defined by the Laplace integral
\begin{equation*}
T_{[\alpha],\cdot j}(z)=\frac{1}{z}\int_0^{+\infty e^{i\alpha}}\!\!\! U_{\cdot j}(s)\,e^{-\frac{s}{z}} ds,
%=:\frac{1}{z}\cal L_\alpha[U_{\cdot j}](z),
\end{equation*}
which converges and is bounded for $z$ in the open half-plane bisected by $e^{i\alpha}\R^+$,
and whose value is independent of when the direction $\alpha$ varies a bit.
In another words, the sectoral transformation $T_{[\alpha]}$ depends only on the homotopy class $[\alpha]$ of the direction $\alpha\in
\R\sminus\{\text{singular directions}\}$,
and one can consider it as defined on a sector in the $z$-plane
\begin{equation}\label{eq:hg-sector}
\mscr S_{[\alpha]}(\infty):=
\bigcup_{\alpha'\in[\alpha]}\{\Re(e^{-i\alpha'}z)>0\},\end{equation}
of opening $>\pi$.
Once a branch of $\log z$ is fixed, the system (<ref>) has on each of these sectors a canonical solution basis $\Psi_{[\alpha]}(z)$
\begin{equation}\label{eq:hg-Psi_alpha}
\Psi_{[\alpha],\cdot j}(z)=T_{[\alpha],\cdot j}(z)\, z^{A_{jj}} e^{-\frac{\lambda_j}{z}}=
\frac{1}{z}\int_0^{+\infty e^{i\alpha}}\!\!\! U_{\cdot j}(s)\,e^{-\frac{s}{z}} ds\cdot z^{A_{jj}}e^{-\frac{\lambda_j}{z}}.
\end{equation}
For every pair of non-singular directions $\alpha_1,\, \alpha_2$ there is Stokes matrix $S_{[\alpha_1][\alpha_2]}(\infty)$
\begin{equation}\label{eq:hg-stokesmatrices}
\Psi_{[\alpha_2]}=\Psi_{[\alpha_1]}\cdot S_{[\alpha_1][\alpha_2]}(\infty)
\end{equation}
(defined by analytic continuation).
It is an easy fact that for two neighboring direction classes $[\alpha_1],\, [\alpha_2]$ the Stokes matrix $S_{[\alpha_1][\alpha_2]}(\infty)$ is unipotent with only non-zero off-diagonal entries at the positions $(j,i)$ corresponding to the singularity $\lambda_i-\lambda_j$ separating the direction classes $[\alpha_1],\,[\alpha_2]$.
Note that by the Liouville-Ostrogradski formula $\det T_{[\alpha]}$ is constant in $z$ and therefore equal to 1.
§.§ Fundamental solutions of the Okubo system
The Okubo system (<ref>) has $(p+1)$ Fuchsian singularities on $\CP^1$ at the points $\lambda_j$, $1\leq j\leq p$, and $\infty$.
Near each $s=\lambda_j$, the system is written as
\begin{equation*}
(s-\lambda_j)\frac{dv}{ds}=E_j(A+\rho)+{\cal O}(s-\lambda_j),
\end{equation*}
$E_j$ denoting the $j$-th column bloc of the identity matrix,
and ${\cal O}(s\!-\!\lambda_j)$ standing for holomorphic terms that vanish at $\lambda_j$.
Its local “multipliers" are therefore $A_{jj}+\rho$ in the $j$-th bloc and $0$ in the other $(p-1)$ blocs.
The system comes with two kinds of canonical mixed bases that will be of interest in this article.
The first one, which will be denoted $V^+$,
consists of the so called Floquet solutions $V_{\cdot j}^+(s,\rho)$, which behave asymptotically like $E_j(s-\lambda_j)^{A_{jj}+\rho}$ at the respective singularities $\lambda_j$.
The second one, denoted by $V^-$, is in a sense dual to the Floquet basis; it consists of solutions
$V_{\cdot j}^-(s,\rho)$ that are analytic at each other singularity $\lambda_i$, $i\neq j$.
This section describes these two bases in more detail.
Let $P^+$ be a sector at $\infty$ in the parameter $\rho$-space, on which $\arg\rho\in\, ]\!-\!\pi+\eta,\pi-\eta[$, with $0<\eta<\frac{\pi}{2}$ fixed arbitrary,
and $|\rho|$ is sufficiently big so that $\rho\notin -\N_{>0}-(\Spec A\cup\Spec A_D)$.
Symmetrically, let $P^-$ be a sector at $\infty$ on which $\arg\rho\in\, ]\eta,2\pi\!-\!\eta[$, and
$|\rho|$ is sufficiently big so that $\rho\notin \N_{>0}-(\Spec A\cup\Spec A_D)$.
The Floquet bases.
If $\Re(\rho)>0$ and $\rho$ is large enough so that all eigenvalues of $A_{jj}+\rho$ have positive real part,
then the matrix function $(s-\lambda_j)^{A_{jj}+\rho}$ vanishes when $s$ approaches $\lambda_j$ radially.
Correspondingly, consider the space of solutions of (<ref>) that vanish when $s\to \lambda_j$ radially.
[Note that no nontrivial combination of the other solutions corresponding to the multiplier $0$ can vanish at the singularity, they are asymptotically bigger and cannot hide behind the vanishing solutions. This is what makes this subspace of the space of solutions well-defined.]
It is invariant by the local monodromy, and
it follows from the local theory of Fuchsian singularities (cf. <cit.>)
and the assumption (<ref>) on $A_{jj}$,
that this space has a unique basis written as
\begin{equation}\label{eq:hg-V+asympt}
V_{\cdot j}^+(s,\rho)=(E_j+{\cal O}(s\!-\!\lambda_j))\cdot (s-\lambda_j)^{A_{jj}+\rho}.
\end{equation}
This construction can be extended to all parameters $\rho\in P^+$, if instead of letting $s$ approach $\lambda_j$ radially, one lets it approach $\lambda_j$ following a suitable logarithmic spiral along which $(s-\lambda_j)^{A_{jj}+\rho}\to 0$.
More precisely, $s$ should follow a real positive trajectory of the vector field $-e^{i\theta}(s\!-\!\lambda_j)\partial_s$, for some
$\theta\in\, ]\!-\!\frac{\pi}{2},\frac{\pi}{2}[$ with $\Re(e^{i\theta} \rho)>0$.
The Floquet solution $V_{\cdot j}^+$ is closely related to the $j$-th formal canonical solution
(<ref>) of the dual Birkhoff system (<ref>).
In fact, the formal Borel transform (=term-wise inverse Laplace transform) of $z^{\rho+1}\hat\Psi_{\cdot j}(z)$ equals
to the convolution integral <cit.>
\begin{equation}\label{eq:hg-RLI+}
I_{\cdot j}^+(s,\rho):= %{\cal B}[z^{\rho+1}\hat\Psi_{\cdot j}(z)](s)=
\int_{\lambda_j}^{s} U_{\cdot j}(\sigma\!-\!\lambda_j) (s\!-\!\sigma)^{A_{jj}+\rho-1}d\sigma
\cdot\Gamma(A_{jj}\!+\!\rho)^{-1},
\end{equation}
where the matricial Gamma function is defined as usual by the integral
$\Gamma(A_{jj}+\rho)=\int_0^{+\infty}t^{-(A_{jj}+\rho-1)}e^{-t}dt, \,$
and $U(s)$ is given in (<ref>).
The integral (<ref>), also known as Riemann-Liouville integral with base-point at $\lambda_j$,
solves (<ref>), and moreover it satisfies
$\,\frac{d}{ds}I_{\cdot j}^+(s,\rho)=I_{\cdot j}^+(s,\rho-1)$, and therefore solves the difference equation
\begin{equation*}
(s-B)I_{\cdot j}^+(s,\rho\!-\!1)=(A+\rho)I_{\cdot j}^+(s,\rho).
\end{equation*}
The canonical solution $\Psi_{[\alpha]}$ (<ref>) of the Birkhoff system equals
\begin{equation} \label{eq:hg-I+Psi}
\Psi_{[\alpha],\cdot j}(z)=z^{-\rho-1}\int_{\lambda_j}^{+\infty e^{i\alpha}}I_{\cdot j}^+(s,\rho)\,e^{-\frac{s}{z}}ds.
\end{equation}
The Floquet solution
$V_{\cdot j}^+$ is obtained from $I_{\cdot j}^+$ after a normalization:
[The fact that (<ref>) has the asymptotic behavior (<ref>) is easily verified by integrating per partes.]
\begin{equation}\label{eq:hg-V+}
V_{\cdot j}^+(s,\rho)=\int_{\lambda_j}^{s} U_{\cdot j}(\sigma\!-\!\lambda_j) (s\!-\!\sigma)^{A_{jj}+\rho-1}d\sigma
%=\int_0^{s-\lambda_j} U_{\cdot j}(\sigma) (s\!-\!\lambda_j\!-\!\sigma)^{A_{jj}+\rho-1}d\sigma
\cdot(A_{jj}\!+\!\rho).
\end{equation}
The integrating path in (<ref>) is such that $\sigma$ follows a positive real trajectory of the vector field
$e^{i\theta}(s\!-\!\sigma)\partial_\sigma$ from the point $\lambda_j$ to $s$, with suitable $\theta$ as above, avoiding other singularities $\lambda_i$, $i\neq j$, of $U_{\cdot j}(\sigma-\lambda_j)$.
The set of points $s$ which can be reached by such paths with varying $\theta$ then
defines a ramified domain on which (<ref>) is defined.
Note that if $\arg(s-\lambda_j)=\alpha-\theta$, then the integrating trajectory approaches $\lambda_j$ in the asymptotic direction $\alpha$.
Let $\Omega^+_{[\alpha],j}(\rho)$ be a (ramified) domain consisting of those $s$ that can be reached by such trajectory for some direction $\alpha$ in the given homotopy class,
$$\Omega^+_{[\alpha],j}(\rho)\subseteq\{s\in\C\mid \arg(s-\lambda_j)=\alpha'-\theta,\
|\theta|<\tfrac{\pi}{2},\ |\theta+\arg\rho|<\tfrac{\pi}{2},\ \alpha'\in[\alpha]\},$$
and let $\Omega^+_{[\alpha]}(\rho):=\bigcap_j\Omega^+_{[\alpha],j}(\rho)$.
The restriction of $V^+$ to $\Omega^+_{[\alpha]}$ will be denoted $V_{[\alpha]}^+$.
Different homotopy classes of non-singular directions $[\alpha]$ give rise to to different branches $V_{[\alpha]}^+$ of $V^+$ near infinity.
The co-Floquet bases.
For given index $j$, and a direction $\alpha$ such that $\lambda_i-\lambda_j\notin e^{i\alpha}\R^+$, $i\neq j$, define
the co-Floquet solution $V_{[\alpha],\cdot j}^-$ at a singularity $\lambda_j$ as the unique solution analytic
on $\C\sminus \{\lambda_j+e^{i\alpha}\R^+\}$ and having the following asymptotic behavior near $\lambda_j$:
\begin{equation}\label{eq:hg-V-asympt}
V_{[\alpha],\cdot j}^-(s,\rho)=(E_j+{O}(s\!-\!\lambda_j))\cdot (s\!-\!\lambda_j)^{A_{jj}+\rho},
\end{equation}
with $O$ denoting the usual Landau symbol (the corresponding terms may be ramified).
Let's be more precise about where does it comes from.
For each singularity $\lambda_i$ and $\rho\in P^-$ large enough so that $A_{ii}+\rho$ has no positive eigenvalue,
define $\Check W^-_{i}$ as the space of solutions analytic at $\lambda_i$.
It follows from the local theory of Fuchsian singularities (cf. <cit.>) that this space is tangent exactly to the the
$(p-1)$ vector-blocs $E_k$, $k\neq i$, corresponding to the multiplier 0.
For a point $s\in \lambda_j-e^{i\alpha}\R^+ $, continue each solution subspace $\Check W^-_{i}$, $i\neq j$, toward $s$ in the cut plane $\C\sminus \{\lambda_j+e^{i\alpha}\R^+\}$, and
define the subspace $W_{\alpha,j}^-$ as their intersection.
Since it consists of solutions analytic at each $\lambda_k$, $k\neq j$,
it does not depend on the way the $\Check W_{i}^-$ are continued around the singularities $\lambda_k$, only on the direction $\alpha$ of the cut.
Following <cit.>, there is a canonical bloc-solution of (<ref>) generating the space $W_{\alpha,j}^-$
given by the integral:
\begin{equation}\label{eq:hg-RLI-}
I_{[\alpha]}^-(s,\rho):=\int_{\lambda_j}^{+\infty e^{i\alpha}} U_{\cdot j}(\sigma\!-\!\lambda_j) (s\!-\!\sigma)^{A_{jj}+\rho-1}d\sigma \cdot\Gamma(1\!-\!A_{jj}-\rho)e^{-\pi i (A_{jj}+\rho)},
\end{equation}
which satisfies again
$\,\frac{d}{ds}I_{\cdot j}^-(s,\rho)=I_{\cdot j}^-(s,\rho-1)$, and therefore solves the difference equation
\begin{equation}\label{eq:hg-I-}
(s-B)I_{\cdot j}^-(s,\rho\!-\!1)=(A+\rho)I_{\cdot j}^-(s,\rho).
\end{equation}
The integral $I_{[\alpha]}^-$ is a Laplace transform of the canonical solution $\Psi_{[\alpha]}$
\begin{equation*}
I_{[\alpha]}^-(s,\rho):=\int_{0}^{+\infty e^{i\alpha}} z^{\rho-1}\Psi_{[\alpha],\cdot j}(z) e^{\frac{s}{z}} dz,
\end{equation*}
which in turn equals to
\begin{equation}\label{eq:hg-I-Psi}
\Psi_{[\alpha],\cdot j}(z)=z^{-\rho-1} \frac{1}{2\pi i} \int_{\gamma_{j,\alpha}} I_{\cdot j}^-(s,\rho)\,e^{-\frac{s}{z}}ds,
\end{equation}
where the path $\gamma_{j,\alpha}$ encircles the ray $\lambda_j+e^{i\alpha}\R^+$ in positive direction.
While $\Psi_{[\alpha]}$ is defined on a sector at 0 of an opening $>\pi$ bisected by $e^{i\alpha}\R^+$,
the integral $I_{[\alpha]}^-$ is defined on a sector at $\lambda_j$ bisected by $\lambda_j+e^{i(\alpha+\pi)}\R^+$ of an opening $>2\pi$.
The co-Floquet solution is obtained after a normalization
\begin{equation}\label{eq:hg-V-}
V_{[\alpha],\cdot j}^-(s,\rho)=\int_{\lambda_j}^{\infty} U_{\cdot j}(\sigma\!-\!\lambda_j) (s\!-\!\sigma)^{A_{jj}+\rho-1}d\sigma
\cdot(A_{jj}\!+\!\rho).
\end{equation}
In the default situation when $\Re(\rho)<0$ the integration path is the straight ray
$\sigma\in\lambda_j+e^{i\alpha}\R^+$ and the integral is defined for $s\in\lambda_j-e^{i\alpha}\R^+$ and extended analytically from there.
In a general situation, the integration path follows a negative real trajectory of the vector field
$e^{i\theta}(s\!-\!\sigma)\partial_\sigma$ from the point $\lambda_j$ to $\infty$, with a suitable
$\theta\in\, ]\!-\!\frac{\pi}{2},\frac{\pi}{2}[$ such that $\Re(e^{i\theta} \rho)<0$,
that is end-point homotopic to the ray $\lambda_j+e^{i\alpha}\R^+$ in
$\CP^1\sminus\{\lambda_i,\ i\neq j\}$.
The set of points $s$ that can be reached by such paths
defines again a (ramified) sectoral domain
$$\Omega^-_{[\alpha],j}(\rho)\subseteq\{s\in\C\mid \arg(s-\lambda_j)=\alpha'-\theta+\pi,\
|\theta|<\tfrac{\pi}{2},\ |\theta-\pi+\arg\rho|<\tfrac{\pi}{2}, \alpha'\in[\alpha]\},$$
on which the integral (<ref>) is naturally defined. Let $\Omega^-_{[\alpha]}(\rho):=\bigcap_j\Omega^-_{[\alpha],j}(\rho)$.
For $\rho\in P^+\cap P^-$ and $s\in \lambda_j+e^{i \alpha}\R^+$, let $\tilde s=\lambda_j+e^{2\pi i}(s-\lambda_j)$, then
\begin{equation*}
V_{[\alpha],\cdot j}^+(s,\rho)=\big[V_{[\alpha],\cdot j}^-(\tilde s,\rho)-V_{[\alpha],\cdot j}^-(s,\rho)\big]\cdot \big[e^{2\pi i(A_{jj}+\rho)}-1 \big]^{-1},
\end{equation*}
or equivalently
\begin{equation*}
I_{[\alpha],\cdot j}^+(s,\rho)=\frac{1}{2\pi i}\big[I_{[\alpha],\cdot j}^-(\tilde s,\rho)-I_{[\alpha],\cdot j}^-(s,\rho)\big],
\end{equation*}
i.e. $I_{[\alpha],\cdot j}^+$ is a hyperfunction defined by the boundary value of $\frac{1}{2\pi i}I_{[\alpha],\cdot j}^-$ on $\lambda_j+e^{i \alpha}\R^+$.
Follows from the construction.
The following Proposition is due to Okubo and Kohno.
\begin{align}
\det V_{[\alpha]}^+(s,\rho)&=\frac{\det \Gamma(A_D+\rho+1)}{\det\Gamma(A+\rho+1)}\cdot\det\,(s-B)^{A_D+\rho},\\[8pt]
\det V_{[\alpha]}^-(s,\rho)&=\frac{\det \Gamma(-A-\rho)}{\det\Gamma(-A_D-\rho)}\cdot\det\,(s-B)^{A_D+\rho}.
\end{align}
For the sake of completeness we will sketch here the proof in the co-Floquet case; the Floquet case is almost identical
and can be found in <cit.>.
The co-Floquet solution has the following asymptotic behavior w.r.t. $\rho$ (see <cit.>, theorem (4.6)):
\begin{equation}\label{eq:hg-V-as}
V_{[\alpha],\cdot j}^-(s,\rho)\cdot (s\!-\!\lambda_j)^{-A_{jj}-\rho}=E_j+{O}(\frac{1}{|\rho|}),\quad\text{when }\ \Re(\rho)\to -\infty,
\end{equation}
locally uniformly in the cut plane $\C\sminus(\lambda_j+e^{i\alpha}\R^+)$.
Now, for $m\in\N$ it follows from (<ref>) by induction that
$$I_{[\alpha],\cdot j}^-(s,\rho\!-\!m)\cdot (s\!-\!\lambda_j)^{m}=
(A\!+\!\rho\!-\!m\!+\!1)\cdots(A\!+\!\rho)\cdot I_{[\alpha],\cdot j}^-(s,\rho),$$
and hence
\begin{align*}
\Gamma(-A\!-\!\rho)&\cdot\Gamma(-A\!-\!\rho\!+\!m)^{-1}\cdot
V_{[\alpha],\cdot j}^-(s,\rho-m)\cdot (s\!-\!\lambda_j)^{-A_{jj}-\rho+m}=\\
&\qquad = V_{[\alpha],\cdot j}^-(s,\rho) \cdot (s\!-\!\lambda_j)^{-A_{jj}-\rho}\cdot
\Gamma(-A_{jj}\!-\!\rho)\cdot\Gamma(-A_{jj}\!-\!\rho+\!m)^{-1}.
\end{align*}
\begin{align*}
\frac{\det\Gamma(-A_D\!-\!\rho\!+\!m)}{\det\Gamma(-A\!-\!\rho\!+\!m)}
&\cdot \det\left[V_{[\alpha]}^-(s,\rho-m)\cdot (s\!-\!B)^{-A_{D}-\rho+m}\right]=\\
&=\det\left[V_{[\alpha]}^-(s,\rho)\cdot (s\!-\!B)^{-A_{D}-\rho}\right]\cdot
\frac{\det\Gamma(-A_D\!-\!\rho)}{\det\Gamma(-A\!-\!\rho)}.
\end{align*}
Letting $m\to+\infty$ and using (<ref>) and usual formulas for the $\Gamma$-function, one can see that
both expressions on the left side tend to 1.
For $\rho\in P^+$ (resp. $\rho\in P^-$) the Floquet (resp. the co-Floquet) solutions form a basis of the solution space.
§.§ Fundamental matrix solutions of the confluent family
The system (<ref>) has two kinds of canonical fundamental matrix solutions $Y_{[\alpha]}^\pm(z,\rho)$ corresponding to the Floquet
and co-Floquet bases of (<ref>).
In order to obtain a convergence when $\rho\to\infty$, one has to be a bit careful with the choice of their branch.
It is convenient to write them as
\begin{equation}\label{eq:hg-Y}
Y_{[\alpha]}^\pm(z,\rho)=T_{[\alpha]}^\pm(z,\rho)\cdot \Phi(z,\rho),
\end{equation}
is a canonical solution to the bloc-diagonal system
\begin{equation}\label{eq:hg-3formal}
\end{equation}
whose branch needs to be selected so that it converges to the adequate branch of
$$\Phi(z,\infty):=z^{A_{D}} e^{-\frac{B}{z}},$$
when $\rho\to\infty$.
The bloc-diagonalizing transformation $T_{[\alpha]}^\pm$ is defined by
$$T_{[\alpha]}^\pm(z,\rho)=V_{[\alpha]}^\pm(\rho z,\rho)\cdot(\rho z-B)^{-A_{D}-\rho},$$
where the branch of $(\rho z-B)^{-A_{D}-\rho}$ is chosen in accord with the one inside the integral (<ref>), (<ref>).
\begin{equation}\label{eq:hg-T+}
T_{[\alpha],\cdot j}^+(z,\rho)=\int_0^{\rho z-\lambda_j} U_{\cdot j}(\sigma) \Big(\frac{\rho z-\lambda_j-\sigma}{\rho z-\lambda_j}\Big)^{A_{jj}+\rho}\frac{d\sigma}{\rho z-\lambda_j-\sigma} \cdot(A_{jj}+\rho),
\end{equation}
where the integration path follows a positive trajectory of the vector field $\,e^{i\theta}(\rho z\!-\!\lambda_j\!-\!\sigma)\partial_\sigma$ from the point 0 to $\rho z\!-\!\lambda_j$,
and the branch of $\Big(\frac{\rho z-\lambda_j-\sigma}{\rho z-\lambda_j}\Big)^{A_{jj}+\rho}$ is chosen so that it is equal 1 at the endpoint.
Remark, that at the limit, when $\rho\to\infty$ radially with fixed $\arg\rho$,
the trajectories of the given vector field become trajectories of the vector field $e^{i\alpha}\partial_z$ with
$\alpha=\theta+\arg\rho+\arg z$.
Therefore the integral (<ref>) has a well-defined limit
\begin{equation*}
T_{[\alpha],\cdot j}^+(z,\infty)=\frac{1}{z}\int_0^{+\infty e^{i\alpha}} U_{\cdot j}(\sigma)\, e^{-\frac{\sigma}{z}}
\end{equation*}
\begin{equation}\label{eq:hg-T-}
T_{[\alpha],\cdot j}^-(z,\rho)=\int_0^{\infty} U_{\cdot j}(\sigma) \Big(\frac{\rho z-\lambda_j-\sigma}{\rho z-\lambda_j}\Big)^{A_{jj}+\rho}\frac{d\sigma}{\rho z-\lambda_j-\sigma} \cdot(A_{jj}+\rho),
\end{equation}
where the integration path follows a positive trajectory of $\,e^{i\theta-\pi}(\rho z\!-\!\lambda_j\!-\!\sigma)\partial_\sigma$ from the point 0 to $\infty$,
which at the limit, when $\rho\to\infty$ radially, becomes
a trajectory of $\,e^{i\alpha}\partial_z$ with
$\alpha=\theta+\arg\rho-\pi+\arg z$,
and the integral (<ref>) becomes
\begin{equation*}
T_{[\alpha],\cdot j}^-(z,\infty)=\frac{1}{z}\int_0^{+\infty e^{i\alpha}} U_{\cdot j}(\sigma)\, e^{-\frac{\sigma}{z}}
\end{equation*}
The transformations $T_{[\alpha]}^\pm(\cdot,\rho)$ are defined on sectors
$$\mscr S_{[\alpha]}^\pm(\rho):=\tfrac{1}{\rho}\Omega^\pm_{[\alpha]}(\rho),$$
which tend to a subsector of $\mscr S_{[\alpha]}(\infty)$ (<ref>) depending on the radial direction in which $\rho\to\infty$.
Stokes matrices of the confluent family
Fixing a branch of $\Phi(z,\rho)$ near $z=\infty$ and its restrictions to the sectors
$\mscr S_{[\alpha]}^\pm(\rho)$ one obtains a canonical set of fundamental matrix
solutions $Y_{[\alpha]}^\pm(z,\rho)$ (<ref>).
The connection matrices between these solutions near $z=\infty$ corresponding to different non-singular directions $\alpha_1,\, \alpha_2$
$$Y_{[\alpha_2]}^\pm=Y_{[\alpha_1]}^\pm\cdot S_{[\alpha_1][\alpha_2]}^\pm(\rho)$$
will be called Stokes matrices of the family.
\begin{align*}
\\
S_{[\alpha_1][\alpha_2]}^-(\rho)&=\big(e^{-\pi i}\rho\big)^{-A_D}\Gamma(-A_D-\rho)\,
S_{[\alpha_1][\alpha_2]}(\infty)\,\Gamma(-A_D-\rho)^{-1}\big(e^{-\pi i}\rho\big)^{-A_D},
\end{align*}
which tends to the Stokes matrix $S_{[\alpha_1][\alpha_2]}(\infty)$ (<ref>) when $\rho\to\infty$ in
$P^\pm$ respectively.
For two neighboring direction classes $[\alpha_1],\, [\alpha_2]$ the Stokes matrix $S_{[\alpha_1][\alpha_2]}^\pm(\rho)$ is unipotent with only non-zero
off-diagonal blocs at the positions $(i,j)$ corresponding to the direction of $(\lambda_i-\lambda_j)\R+$ separating $\alpha,\,\alpha'$.
This follows from the relation
$$I_{[\alpha_2]}^\pm(s,\rho)=I_{[\alpha_1]}^\pm(s,\rho)\cdot S_{[\alpha_1][\alpha_2]}^\pm(\rho),$$
which is a consequence of the formulas (<ref>), resp. (<ref>).
The connection matrices between different branches of $Y^\pm$ in Example <ref>.
[Figure <ref>]
Suppose $B$ has just three eigenvalues $\lambda_i$, $i=1,2,3$ and assume they are not colinear.
For simplicity we will consider only the default situation when $\rho\in\R\pm$ and restrict the domains of $Y_{[\alpha]}^\pm$ to a smaller
sector consisting of the points $z\in\C$ for which the integration path in (<ref>), resp. (<ref>), can be taken a straight segment.
Near $\infty$, these sectors are are separated by the outer parts of lines through $\lambda_i,\lambda_k$, whose crossing is governed by the Stokes matrices.
For each singularity $\frac{\lambda_j}{\rho}$ or $\infty$ make a cut (wavy line in Figure <ref>) from the origin on which the formal solution $\Phi$ is branched, and therefore changed by its formal monodromy
$$N_j=\left(\begin{smallmatrix}I_{n_1}\hskip-3pt &&&& \\[-6pt] & \ddots &&& \\[1pt] &&\hskip-12pt e^{2\pi i(A_{jj}+\rho)}\hskip-18pt && \\[-4pt] &&& \ddots & \\ &&&& I_{n_3}\end{smallmatrix}\right),\quad j=1,2,3,
\quad\text{and}\quad N_\infty=e^{-2\pi i A_{D}}.$$
§ CONFLUENCE IN GENERALIZED HYPERGEOMETRIC EQUATION
This section illustrates the confluence procedure on the example of the generalized hypergeometric equation, where things can be expressed very explicitly.
Most of the formulas come from <cit.>. To simplify the writing we adopt the following notation.
Let $\bm\alpha=(\alpha_1,\ldots,\alpha_n)$, $\bm\beta=(\beta_1,\ldots,\beta_n)\in\C^n$ and
* $\bmbeta{\,j}\in\C^{n-1}$ obtained from $\bm\beta$ by omitting the $j$-th component, similarly with
* for $c\in\C$, let $\,\bm\alpha-c:=(\alpha_1-c,\ldots,\alpha_n-c),$
* for a function $f:\C\to\C$, write shortly
In the above notation, the generalized hypergeometric equation of order $n$, is written as $\displaystyle D\hg{\bm\alpha}{\bm\beta}{s}w=0$, where
\begin{align}\label{eq:hg-hg}
(\delta_s+\beta_{n}\!-\!1)\ldots(\delta_s+\beta_1\!-\!1) \nonumber \\
\end{align}
and $\delta_s=s\frac{d}{ds}$ is the Euler operator.
It has three regular singular points at $0$, $1$ and $\infty$.
\begin{equation}\label{eq:hg-D1}
D\hg{\bm\alpha}{\bm\beta}{s}(s^c w)=s^cD\hg{\bm\alpha\!+\!c}{\bm\beta\!+\!c}{s}w,
\end{equation}
one can always use the transformation $w\mapsto s^{1-\beta_j} w$, to bring the equation to a more
usual form in which one of the $\beta_j$'s equals 1:
$\displaystyle D\hg{\bm\alpha\!+\!1\!-\!\beta_j}{\bm\beta\!+\!1\!-\!\beta_j}{s}$.
The equation (<ref>) has thus $n$ local solution at $s=0$ given by
the hypergeometric series
$$s^{1-\beta_j} {}_nF_{n-1}\hg{\bm\alpha\!+\!1\!-\!\beta_j}{\bmbeta{j}\!+\!1\!-\!\beta_j}{s}:=
s^{1-\beta_j} \sum_{k=0}^{+\infty}\frac{(\bm\alpha\!+\!1\!-\!\beta_j)_k}{(\bmbeta{j}\!+\!1\!-\!\beta_j)_k(1)_k}s^k, \quad 1\leq j\leq n,$$
convergent for $|s|<1$,
where $(a)_k$ denotes the Pochhammer symbol
$$(a)_k=a\,(a\!+\!1)\ldots(a\!+\!k\!-\!1),\quad (a)_0=1.$$
These solutions are linearly independent if no two $\beta_j$'s differ by an integer.
There is a symmetry between the singular points $0$ and $\infty$ given by the relation
\begin{equation}\label{eq:hg-D2}
D\hg{\bm\alpha}{\bm\beta}{s}= (-1)^ns\, D\hg{1\!-\!\bm\beta}{1\!-\!\bm\alpha}{\frac{1}{s}}.
\end{equation}
In the case of the Gauss' hypergeometric equation ($n=2$)
there is also a symmetry between the two singular points $0,1$
due to the relation
$$D\hg{\alpha_1,\alpha_2}{1,\beta}{s}=\frac{s}{1\!-\!s}D\hg{\alpha_1,\alpha_2}{1,-\gamma}{1\!-\!s},\qquad \gamma=\beta\!-\!1\!-\!\alpha_1\!-\!\alpha_2.$$
This symmetry is broken for $n>2$.
The confluence.
We are interested in the situation when $\beta_n\to\infty$. The situation when $\alpha_n\to\infty$ would be similar due to the symmetry (<ref>).
$$\rho=\beta_n-1,\qquad s=\rho z$$
\begin{equation}\label{eq:hg-hgco}
\tfrac{1}{\rho} D\hg{\bm\alpha}{\bm\beta}{\rho z}=z(\delta_z+\bm\alpha)-(\tfrac{1}{\rho}\delta_z+1)(\delta_z+\bmbeta{n}-1),
\end{equation}
where the regular singularities at $z=0$ and $z=\frac{1}{\rho}$ merge for $\rho\to\infty$ to form an irregular
$y=(y_1,\ldots,y_n)^\msf{T}$, with
$$y_{i+1}(z,\rho)=(\delta_z+\beta_i\!-\!1)y_i(z,\rho),\quad\text{for}\quad 1\leq i\leq n\!-\!1,$$
the equation (<ref>): $\,\displaystyle\tfrac{1}{\rho} D\hg{\bm\alpha}{\bm\beta}{\rho z}y_1=0\,$
is written in the form of a family of systems (<ref>) with
\begin{equation}\label{eq:hg-AB}
0 & & & \\[-3pt]
& \ddots & & \\
& & 0 & \\
& & & 1
\end{pmatrix},\qquad
1\!-\!\beta_1\hskip-9pt & \hskip6pt 1\hskip-6pt & & \\[-3pt]
& \ddots & \ddots & \\
& & \hskip-12pt 1\!-\!\beta_{n-1} \hskip-6pt & 1 \\
* & \ldots & * & \gamma
\end{pmatrix},
\end{equation}
where $A$ has $-\alpha_1,\ldots,-\alpha_n$ as eigenvalues.
The Okubo system.
The corresponding Okubo system
\begin{equation}\label{eq:hg-1x}
\end{equation}
for $v(s,\rho)=s^{-\rho}y(z,\rho)$,
is associated to the generalized hypergeometric equation
\displaystyle D\hg{\bm\alpha\!-\!\rho}{\bm\beta\!-\!\rho}{s}v_1(s,\rho)=0,
\qquad
Suppose now, that
\begin{equation}\label{eq:hg-asump2}
\beta_i-\beta_j\notin\Z\ \text{ for all } i\neq j.
%, \quad\text{and}\quad \alpha_i-\beta_j\notin\Z_{<0}\ \text{ for all }i,j.
\end{equation}
For $\rho\neq\infty$, we have $n-1$ singular solutions of the Okubo system near $s=0$ whose first component is given by
\begin{equation*}
\tilde v_{1j}^+(s,\rho)
=s^{1-\beta_j+\rho}\, {}_nF_{n-1}\hg{\bm\alpha\!+\!1\!-\!\beta_j}{\bmbeta{j}\!+\!1\!-\!\beta_j}{s}, \qquad |s|<1,\quad 1\leq j\leq n-1,
\end{equation*}
and one singular solution at $s=1$ given by Meijer G-function
\begin{align*}
\tilde v_{1n}^+(s,\rho)
&= G_{n,n}^{n,0}\hg{\bm\beta}{\bm\alpha}{\frac{1}{z}} \cdot\Gamma(1\!+\!\gamma\!+\!\rho),\qquad |s|>1,\notag \\
&=(s\!-\!1)^{\gamma+\rho}\sum_{k=0}^{+\infty}\frac{(-1)^k c_k}{(\gamma+\rho+1)_{k+n-1}}(s\!-\!1)^{k+n-1},\qquad |s-1|<1,
\end{align*}
with $c_0=1$ and the coefficients $c_k$ independent of $\rho$ (see <cit.>, p. 601)
\begin{equation}\label{eq:hg-ck}
\,c_k=\text{\footnotesize{$\displaystyle\sum_{i_1+\ldots+i_{n-1}=k}
\prod_{j=1}^{n-1}\frac{
(\beta_1\!-\alpha_1\!+i_1+\ldots+\beta_j\!-\alpha_j\!+i_j)_{i_j}\cdot (\beta_j\!-\!\alpha_{j+1})_{i_j}}{i_j!}$}}.
\end{equation}
It is easy to see that the terms of the fundamental solution matrix
$\tilde V^+=(\tilde v_{ij}^+)$ have the asymptotic behavior
$$\tilde V^+(s,\rho)\sim \left(R+O(1)(s-B)\right)(s-B)^{\tilde A_D+\rho},$$
where the upper-triangular matrix $R=(r_{ij})$
$$r_{1j}=1,\quad r_{ij}=(\beta_1\!-\!\beta_j)\ldots(\beta_{i-1}\!-\!\beta_j),\ i>1,\quad\text{and}\quad
r_{nn}=1,\quad r_{in}=0,\ i< n ,$$
commutes with $B$ and diagonalizes
\begin{equation*}%\label{eq:hg-1}
1\!-\!\beta_1\hskip-9pt & \hskip3pt 1\hskip-6pt & & &\\[-0.7\normalbaselineskip]
& \raisebox{0.15\normalbaselineskip}{$\ddots$} & \hskip-3pt\raisebox{0.7\normalbaselineskip}{$\ddots$} &\hskip-12pt 1 &\\
& & & \hskip-24pt 1\!-\!\beta_{n-1} \hskip-6pt & \\
& & & & \,\gamma
\end{pmatrix},
\qquad
R^{-1}\!A_DR=\tilde A_D:=\begin{pmatrix}
1\!-\!\beta_1\hskip-9pt & \hskip3pt & & \\[-3pt] & \ddots & & \\ & & \hskip-12pt 1\!-\!\beta_{n-1} \hskip-6pt & \\ & & & \,\gamma
\end{pmatrix}.
\end{equation*}
\begin{equation}\label{eq:hg-VV}
V^{+\!}(s,\rho)=\tilde V^{+\!}(s,\rho)R^{-1}
\end{equation}
is the Floquet bases of the Okubo system (<ref>) with the asymptotic behavior (<ref>),
while $R^{-1}\tilde V^+$ is the Floquet basis of the Okubo system with
$\tilde B=B$, $\tilde A=R^{-1}AR$.
The fundamental matrix solutions $\tilde V^\pm$ and their monodromy and transition matrices,
according to the values of $\frac{1}{\rho}$.
Monodromy matrices $\tilde m_0^+(\rho)$ and $\tilde m_1^+(\rho)$ of the solution $\tilde V^+$ around the singularities $0$ and $1$ in the positive direction from a base-point at $s=\frac{1}{2}$ are calculated in <cit.>:
\begin{equation*}
\tilde m_0^+=
\begin{pmatrix}
e_1 & & & \xi_1(e_1\!-\!1)\\[-3pt]
& \hskip-6pt\ddots & & \vdots \\
& & \hskip-6pt e_{n-1} \hskip-3pt & \xi_{n-1}(e_{n-1}\!-\!1)\hskip-3pt\\[3pt]
& & & 1
\end{pmatrix},
\qquad
\tilde m_1^+=
\begin{pmatrix}
1 & & & \\[-3pt]
& \ddots & & \\
& & \hskip-6pt 1 & \\[3pt]
\eta_1(e_n\!-\!1) \hskip-4pt & \ldots & \hskip-4pt \eta_{n-1}(e_n\!-\!1)\hskip-4pt & e_n
\end{pmatrix},
\end{equation*}
$\, e_j=e^{2\pi i(1-\beta_j+\rho)},\,\text{ for }\, j\leq n-1,\, \ e_n=e^{2\pi i(\gamma+\rho)},\,$ and
$$\xi_j=e^{\pi i(\gamma+\rho)} \frac{\Gamma(1\!+\!\gamma\!+\!\rho) \Gamma(\beta_j\!-\!\bmbeta{j})}
\eta_j=e^{-\pi i(\gamma+\rho)} \frac{\Gamma(\!-\gamma\!-\!\rho) \Gamma(1\!-\!\beta_j\!+\!\bmbeta{j})}
The connection matrix between the Floquet and the co-Floquet bases
$\tilde V^-(s,\rho)=\tilde V^+(s,\rho)\tilde C(\rho)$ is also calculated in <cit.>:
\begin{equation*}
\tilde C=\begin{pmatrix}
1 & & & \hskip-6pt -\xi_1 \\[-3pt]
& \ddots & & \vdots\\
& & \hskip-6pt 1 & \hskip-6pt -\xi_{n-1}\hskip-3pt\\[3pt]
-\eta_1 \hskip-3pt & \ldots & \hskip-3pt -\eta_{n-1} \hskip-6pt & 1
\end{pmatrix}.
\end{equation*}
Therefore the corresponding monodromy matrices of $\tilde V^-$ are equal to
$$\tilde m_\iota^-(\rho)=\tilde C(\rho)^{-1}\tilde m_\iota^+(\rho)\tilde C(\rho),\quad \iota=0,1,$$
\begin{equation*}
\tilde m_0^-=
\begin{pmatrix}
e_1 \hskip-12pt & & & \\[-3pt]
& \hskip-6pt\ddots & & \\
& & \hskip-20pt e_{n-1} & \\[3pt]
\eta_1(e_1\!-\!1) \hskip-4pt & \ldots & \hskip-4pt \eta_{n-1}(e_{n-1}\!-\!1)\hskip-4pt & 1
\end{pmatrix},
\qquad
\tilde m_1^-=
\begin{pmatrix}
1 & & & \xi_1(e_n\!-\!1) \\[-3pt]
& \ddots & & \vdots\\
& & 1 & \xi_{n-1}(e_n\!-\!1)\hskip-3pt\\[3pt]
& & & e_n
\end{pmatrix}.
\end{equation*}
The Floquet and co-Floquet bases $\tilde V^\pm$ are both defined and analytic not only on $\rho\in P^\pm$ but for all $\rho\in\C$ except of a discrete set of resonant values accumulating at $\infty$.
The confluent family.
Under the assumption (<ref>) there are $n\!-\!1$ parameter-dependent singular solutions of the confluent equation (<ref>) near $z=0$
are given for $\rho\neq\infty$ by
\begin{align*}
\tilde y_{1j}^+(z,\rho)
&=z^{1-\beta_j}\,{}_nF_{n-1}\hg{\bm\alpha\!+\!1\!-\!\beta_j}{\bmbeta{j}\!+\!1\!-\!\beta_j}{\rho z}\\
&=z^{1-\beta_j}\, G_{n,n}^{n,1}\hg{1,\,\bmbeta{j}\!+\!1\!-\!\beta_j}{\bm\alpha\!+\!1\!-\!\beta_j}{\!-\frac{1}{\rho z}}\cdot \frac{\Gamma(\bmbeta{j}\!+\!1\!-\!\beta_j)}{\Gamma(\bm\alpha\!+\!1\!-\!\beta_j)},\qquad |\rho z|<1,\\
%&= e^{\pm\pi i(1-\beta_j)}\, G_{n,n}^{n,1}\hg{\beta_j,\,\bmbeta{j}}{\bm\alpha}{-\frac{1}{\rho z}}\cdot \frac{\Gamma(\bmbeta{j}\!+\!1\!-\!\beta_j)}{\Gamma(\bm\alpha\!+\!1\!-\!\beta_j)},
&\downarrow \\
\tilde y_{1j}^+(z,\infty)&=z^{1-\beta_j}\,
G_{n-1,n}^{n,1}\hg{1,\,\bmbeta{n,j\!}\!+\!1\!-\!\beta_j}{\bm\alpha\!+\!1\!-\!\beta_j}{\!-\frac{1}{z}} \cdot \frac{\Gamma(\bmbeta{n,j\!}\!+\!1\!-\!\beta_j)}{\Gamma(\bm\alpha\!+\!1\!-\!\beta_j)},\quad |\arg z-\pi|<\frac{3\pi}{2},
%&=e^{\pm\pi i(1-\beta_j)}\, G_{n-1,n}^{n,1}\hg{\beta_j,\,\bmbeta{n,j}}{\bm\alpha}{-\frac{1}{z}} \cdot \frac{\Gamma(\bmbeta{n,j\!}\!+\!1\!-\!\beta_j)}{\Gamma(\bm\alpha\!+\!1\!-\!\beta_j)},
\end{align*}
with the limit asymptotic on compact sub-sectors to the divergent formal series
And the singular solution at $z=\frac{1}{\rho}$ is
is given by
\begin{align*}
\tilde y_{1n}^+(z,\rho)
%&=z^{-\rho}(z\!-\!\tfrac{1}{\rho})^{\gamma+\rho}\,\sum_{k=0}^{+\infty}\frac{(-1)^k(\rho)^{k+n-1} c_k}{(\gamma+\rho+1)_{k+n-1}}(z-\tfrac{1}{\rho})^{k+n-1}\\
&= G_{n,n}^{n,0}\hg{\bm\beta}{\bm\alpha}{\frac{1}{\rho z}} \cdot\rho^{-\gamma} \,\Gamma(1\!+\!\gamma\!+\!\rho),\qquad |\rho z|>1\\
\tilde y_{1n}^+(z,\infty)&= G_{n-1,n}^{n,0}\hg{\bmbeta{n}}{\bm\alpha}{\frac{1}{z}}, \qquad |\arg z|<\frac{3\pi}{2},
\end{align*}
which is asymptotic on compact sub-sectors to the divergent formal series
\begin{equation*}
e^{-\frac{1}{z}}z^{\gamma}\sum_{k=0}^{+\infty}(-1)^k c_k z^{k+n-1},
\qquad\text{with $c_k$ as in \eqref{eq:hg-ck}.}
\end{equation*}
The constructed fundamental matrix solution
$\tilde Y^+=(\tilde y_{ij}^+)$ have the asymptotic behavior
$$\tilde Y^+(z,\rho)\sim \left(R+O(1)(z-\tfrac{B}{\rho})\right)(z-\tfrac{B}{\rho})^{\tilde A_D+\rho}z^{-\rho}.$$
$$Y^{+\!}(z,\rho):=\tilde Y^{+\!}(z,\rho)R^{-1}\sim \left(I+O(1)(z-\tfrac{B}{\rho})\right)(z-\tfrac{B}{\rho})^{A_D+\rho}z^{-\rho}$$
is the corresponding Floquet bases of the confluent family (<ref>) with the right asymptotic behavior.
In the definition of $\tilde Y^+$ above the right choice of branch of $\log z$
and $\log(z-\frac{1}{\rho})$ in $\tilde\Phi(z,\rho)=(z-\tfrac{B}{\rho})^{\tilde A_D+\rho}z^{-\rho}$ is of essential importance.
Let $\alpha$ be a direction, $-\pi<\alpha<0$, and chose the bases $\tilde Y^\pm_{[\alpha]}$ and $\tilde Y^\pm_{[\alpha+\pi]}$ so that they are related to each other as in Figure <ref>,
$$\tilde Y_{[\alpha+\pi]}^+(z,\rho)=(\rho z)^{-\rho}\tilde V^+(\rho z, \rho)\rho^{-\tilde A_D},\quad
\tilde Y_{[\alpha]}^-(z,\rho)=(\rho z)^{-\rho}\tilde V^-(\rho z, \rho)\rho^{-\tilde A_D}.$$
The fundamental matrix solutions $\tilde Y^\pm_{[\alpha]}$,
$\tilde Y^\pm_{[\alpha+\pi]}$ on their natural domains (restricted to a fixed neighborhood of 0) and their transition matrices,
according to the values of $\frac{1}{\rho}$.
The limit system is in the center. (See <cit.> for more details on the construction of these ramified domains.)
Then the monodromy matrices of the fundamental matrix solution
$\tilde Y_{[\alpha+\pi]}^+(z,\rho)$,
resp. $\tilde Y_{[\alpha]}^-(z,\rho)$, around
$0$ and $\frac{1}{\rho}$ ($\rho\neq\infty$) in the positive direction from a base-point at $z=\frac{1}{2\rho}$
are equal
\begin{equation}\label{eq:hg-M+}
\tilde M_0^+=\tilde N_0\tilde S_U^+
=e^{-2\pi i\rho}\rho^{\tilde A_D}\tilde m^+_0\rho^{-\tilde A_D},\qquad
\tilde M_{\frac{1}{\rho}}^+=\tilde S_L^+\tilde N_{\frac{1}{\rho}}
=\rho^{\tilde A_D}\tilde m^+_1\rho^{-\tilde A_D},
\end{equation}
\begin{equation}\label{eq:hg-M-}
\tilde M_0^-=\tilde N_0\tilde S_L^-
=e^{-2\pi i\rho}\rho^{\tilde A_D}\tilde m^-_0\rho^{-\tilde A_D},\qquad
\tilde M_{\frac{1}{\rho}}^-=\tilde N\tilde S_U^-\tilde N_0^{-1}
=\rho^{\tilde A_D}\tilde m^-_1\rho^{-\tilde A_D},
\end{equation}
\begin{equation*}
\tilde N_0=
\begin{pmatrix}
e^{2\pi i(1-\beta_1)} \hskip-18pt & & & \\[-3pt]
& \hskip-6pt\ddots & & \\
& & e^{2\pi i(1-\beta_{n-1})} \hskip-24pt & \\
& & & e^{-2\pi i\rho}
\end{pmatrix},
\qquad
\tilde N_{\frac{1}{\rho}}=
\begin{pmatrix}
1 & & & \\[-3pt]
& \ddots & & \\
& & 1 & \\
& & & e^{2\pi i(\gamma+\rho)}
\end{pmatrix},
\end{equation*}
and $\,\tilde N=\tilde N_0\tilde N_{\frac{1}{\rho}},\,$
are monodromies of the fundamental matrix solution $\tilde\Phi(z,\rho)=z^{-\rho}(z-\frac{B}{\rho})^{\tilde A_D+\rho}$ of the diagonal model system, and
\begin{equation*}
\tilde S_U^\pm:=\tilde S_{\alpha+2\pi,\alpha+\pi}^\pm=
\begin{pmatrix}
1 & & & \tilde s^\pm_{1n}\ \\[-3pt]
& \hskip-3pt\ddots\hskip-3pt & & \hskip3pt\vdots\ \\
& & 1 \hskip-6pt & \tilde s^\pm_{n-1,n}\hskip-3pt \\[3pt]
& & & 1
\end{pmatrix},
\qquad
\tilde S_L^\pm:=\tilde S_{\alpha+\pi,\alpha}^\pm=
\begin{pmatrix}
1 & & & \\[-3pt]
& \ddots & & \\
& & \hskip-6pt 1 & \\[3pt]
\tilde s^\pm_{n1} \hskip-6pt & \ldots & \hskip-6pt\tilde s^\pm_{n,n-1}\hskip-6pt & 1
\end{pmatrix}
\end{equation*}
are the Stokes matrices.
It follows from (<ref>) that the Stokes multipliers $\tilde s^+_{ij}(\rho)$ are equal to:
\begin{align*}
\tilde s^+_{jn}(\rho)
&= \xi_j(\rho)\,(1\!-\!e^{-2\pi i(\rho+1-\beta_j)})\,\rho^{1-\beta_j-\gamma}\\
&=-2\pi i\, e^{\pi i(\gamma+\beta_j+n)} \frac{\Gamma(\beta_j\!-\!\bmbeta{n,j})}{\Gamma(\beta_j\!-\!\bm\alpha)}
\cdot \rho^{1-\beta_j-\gamma} \frac{\Gamma(1\!+\!\gamma\!+\!\rho)}{\Gamma(2\!-\!\beta_j\!+\!\rho)},\\
\tilde s^+_{jn}(\infty)&=-2\pi i\, e^{\pi i(\gamma+\beta_j+n)} \frac{\Gamma(\beta_j\!-\!\bmbeta{n,j})}{\Gamma(\beta_j\!-\!\bm\alpha)},
\end{align*}
$\big(\,\text{since} \ \lim_{P^+\ni\rho\to\infty} \rho^{\gamma+\beta_j-1}\frac{\Gamma(2-\beta_j+\rho)}{\Gamma(1+\gamma+\rho)}=1 \,\big),$
\begin{align*}
\tilde s^+_{nj}(\rho)
&=\eta_j(\rho)\,(e^{2\pi i(\gamma+\rho)}\!-\!1)\,\rho^{\gamma+\beta_j-1}\\
&= -2\pi i \frac{\Gamma(1\!-\!\beta_j\!+\!\bmbeta{n,j})}{\Gamma(1\!-\!\beta_j\!+\!\bm\alpha)}
\cdot \rho^{\gamma+\beta_j-1}\frac{\Gamma(2\!-\!\beta_j\!+\!\rho)}{\Gamma(1\!+\!\gamma\!+\!\rho)},\\
\tilde s^+_{nj}(\infty)&=-2\pi i \frac{\Gamma(1\!-\!\beta_j\!+\!\bmbeta{n,j})}{\Gamma(1\!-\!\beta_j\!+\!\bm\alpha)}.
\end{align*}
From (<ref>) one then obtains the Stokes multipliers $\tilde s^-_{ij}(\rho)$:
\begin{align*}
\tilde s^-_{jn}(\rho)
&=\xi_j(\rho)\, e^{-2\pi i(1-\beta_j+\rho)}(e^{2\pi i(\gamma+\rho)}\!-\!1)\,\rho^{1-\beta_j-\gamma}\\
&=-2\pi i\, e^{\pi i(\gamma+\beta_j+n)} \frac{\Gamma(\beta_j\!-\!\bmbeta{n,j})}{\Gamma(\beta_j\!-\!\bm\alpha)}
\cdot \left(e^{- \pi i}\rho\right)^{1-\beta_j-\gamma} \frac{\Gamma(\beta_j\!-\!1\!-\!\rho)}{\Gamma(\!-\gamma\!-\!\rho)},\\
\tilde s^-_{jn}(\infty)&=-2\pi i\, e^{\pi i(\gamma+\beta_j+n)} \frac{\Gamma(\beta_j\!-\!\bmbeta{n,j})}{\Gamma(\beta_j\!-\!\bm\alpha)},
\end{align*}
$\big(\,\text{since}\ \lim_{P^-\ni\rho\to\infty} (e^{- \pi i}\rho)^{\gamma+\beta_j-1} \frac{\Gamma(-\gamma-\rho)}{\Gamma(\beta_j-1-\rho)}=1\,\big)$,
\begin{align*}
\tilde s^-_{nj}(\rho)
&=\eta_j(\rho)\,(e^{2\pi i(1-\beta_j+\rho)}\!-\!1)\,\rho^{\gamma+\beta_j-1}\\
&= -2\pi i \frac{\Gamma(1\!-\!\beta_j\!+\!\bmbeta{n,j})}{\Gamma(1\!-\!\beta_j\!+\!\bm\alpha)}
\cdot \left(e^{- \pi i}\rho\right)^{\gamma+\beta_j-1} \frac{\Gamma(\!-\gamma\!-\!\rho)}{\Gamma(\beta_j\!-\!1\!-\!\rho)},\\
\tilde s^-_{nj}(\infty)&=-2\pi i \frac{\Gamma(1\!-\!\beta_j\!+\!\bmbeta{n,j})}{\Gamma(1\!-\!\beta_j\!+\!\bm\alpha)}.
\end{align*}
One could also proceed the opposite way:
The Stokes matrices $\tilde S_\bullet(\infty)$, $\bullet=U,L$, of the limit generalized confluent hypergeometric equation have been calculated in <cit.>,
and the Stokes matrices $\tilde S_\bullet^\pm(\rho)$ of the family are related to them via Proposition <ref>.
Note that the confluent Floquet and co-Floquet bases $Y^\pm(z,\rho)=\tilde Y^\pm(z,\rho)R^{-1}$ and their monodromies, resp. Stokes matrices
$M_i^\pm(\rho)=R\tilde M_i^\pm(\rho)R^{-1}$, $\iota=0,\frac{1}{\rho}$,
resp. $S_\bullet^\pm(\rho)=R\tilde S_\bullet^\pm(\rho)R^{-1}$,
are well-defined under a weaker assumption than (<ref>), that no two $\beta_j$'s differ by a non-zero integer.
[Ba]Ba W. Balser, Formal power series and linear systems of meromorphic ordinary differential equations, Springer, 2000.
[BJL]BJL W. Balser, W.B. Jurkat, D.A. Lutz, On the reduction of connection problems with an irregular singularity to ones with only regular singularities, I, II, SIAM J. Math. Anal. 12 (1981), 691–721,
SIAM J. Math. Anal. 19 (1988), 398–443.
[Du]Du A. Duval, Confluence procedures in the generalized hypergeometric family, J. Math. Sci. Univ. Tokyo 5 (1998), 597–625.
[DM]DM A. Duval, C. Mitschi, Matrices de Stokes et groupe de Galois des équations hypergéométriques généralisées, Pacific J. of Math. 138 (1989), 25–56.
[Gar]Gar R. Garnier, Sur les singularités irrégulières des équations différentielles linéaires, J. de math. pures et appl. $8^e$ série (1919), 99–200.
[Gl1]Gl1 A. Glutsyuk, Stokes Operators via Limit Monodromy of Generic Perturbation, J. Dyn. Control Syst. 5 (1999), 101–135.
[Gl2]Gl2 A. Glutsyuk, Resonant Confluence of Singular Points and Stokes Phenomena, J. Dyn. Control Syst. 10 (2004), 253–302.
[Hu]Hu M. Hukuhara, Développements Asymptotiques des Solutions Principales d'un Systeme Différentiel Linéaire du Type Hypergéométrique, Tokyo J. of Math. 05 (1982), 491–499.
[HLR]HLR J. Hurtubise, C. Lambert, C. Rousseau, Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k, Moscow Math. J. 14 (2013), 309–338.
[IY]IY Y. Ilyashenko, S. Yakovenko, Lectures on Analytic Differential Equations, Grad. Studies Math. 86, Amer. Math. Soc., Providence, 2008.
[Kl1]Kl1 M. Klimeš, Analytic classification of families of linear differential systems unfolding a resonant irregular singularity, preprint arXiv:1301.5228.
[Kl2]Kl2 M. Klimeš, Confluence of singularities of non-linear differential equations via Borel-Laplace transformations, to be published in J. Dynam. Contr. Syst. [arXiv:1307.8383]
[Ko1]Ko1 M. Kohno, Frobenius' theorem and Gauss-Kummer's formula, Funkcialaj Ekvacioj 28 (1985), 249–266.
[Ko2]Ko2 M. Kohno, Global analysis in linear differential equations, Math. and its appl. 471, Kluwer Acad. Publ., Dodrecht, 1999.
[KO]KO M. Kohno, S. Ohkohchi, Generalized hypergeometric equations of non-Fuchsian type, Hiroshima Math. J. 13 (1983), 83–100.
[Le]Le A.H.M. Levelt, Hypergeometric functions, dissertation, Nederl. Akad. Wetensch. Proc. Ser. A 64, 1961.
[LR1]LR1 C. Lambert, C. Rousseau, The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation, J. Differential Equations 244 (2008) 2641-–2664.
[LR2]LR2 C. Lambert, C. Rousseau, Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1, Moscow Math. Journal 12 (2012), 77–138.
[Lu]Lu Y.L. Luke, Special functions and their approximations I, II, Math. Sci. Eng. 53, Acad. Press, New York, 1969.
[MR]MR J. Martinet, J.-P. Ramis, Théorie de Galois differentielle et resommation, in: Computer Algebra and Differential Equations (E.Tournier ed.), Acad. Press (1988).
[Ok1]Ok1 K. Okubo, An Extension of Gauss' Formula for Hypergeometric Series, UTF8min 数理解析研究所講究録 105 (1971), 53–57.
[Ok2]Ok2 K. Okubo, Connection problems for systems of linear differential equations, Lect. Notes in Math. 243, Springer, 1971.
[OTY]OTY K. Okubo, K. Takano, S. Yoshida, A connection problem for the generalized hypergeometric equation, Funkcialaj Ekvacioj 31 (1988), 483–495.
[Pa]Pa L. Parise, Confluence de singularités régulières d'équations différentielles en une singularité irrégulière. Modèle de Garnier, thèse de doctorat, IRMA Strasbourg (2001). [http://www-irma.u-strasbg.fr/annexes/publications/pdf/01020.pdf]
[Ra]Ra J.-P. Ramis, Confluence et résurgence, J. Fac. Sci. Univ. Tokyo, Sec. IA Math. 36 (1989), 703–716.
[Sch1]Sch1 R. Schäfke, Über das globale analytische Verhalten der Normallösungen von $(s-B)v'(s)=(B+t^{-1}A)v(s)$ und zweier Arten von assoziierten Funktionen, Math. Nachr. 121 (1985), 123–145.
[Sch2]Sch2 R. Schäfke, Confluence of several regular singular points into an irregular singular one, J. Dyn. Control Syst. 4 (1998), 401–424.
[Zh]Zh C. Zhang, Confluence et phenomène de Stokes, J. Math. Sci. Univ. Tokyo 3 (1996), 91–107.
|
1511.00617
|
Tsao-Hsien Chen was supported in part by the AMS-Simons travel grant.
Department of Mathematics, Northwestern University, USA
Kari Vilonen was supported in part by NSF grants DMS-1402928 & DMS-1069316, the Academy of Finland, the ARC grant DP150103525, the Humboldt Foundation, and the Simons Foundation.
Department of Mathematics, Northwestern University, USA and Department of Mathematics and Statistics, University of Helsinki, Finland
Department of Mathematics and Statistics, University of Helsinki, Finland and School of Mathematics and Statistics, University of Melbourne, Australia
Ting Xue was supported in part by the Academy of Finland and the ARC grant DP150103525.
We continue our study of the Springer correspondence in the case of symmetric spaces initiated in <cit.>. In this paper we introduce a certain class of families of Hessenberg varieties and study their monodromy representations in detail in a special case when the Hessenberg varieties can be expressed in terms of complete intersections of quadrics.
We obtain decompositions of these monodromy representations into irreducibles
compute the Fourier transforms of the IC complexes
associated to these irreducible representations.
§ INTRODUCTION
We continue our study of the Springer correspondence in the case of symmetric spaces initiated in <cit.>. In this paper we introduce a certain class of families of Hessenberg varieties of <cit.> and study their monodromy representations in detail in a special case when the Hessenberg varieties can be expressed in terms of intersections of quadrics.
Let us recall the main question from <cit.>. For further background we refer to the introduction of <cit.>. Let $G$ be a reductive group and $\theta$ an involution of $G$. We write $K= (G^\theta)^0$ for the connected component of the fixed point set. This gives rise to a symmetric pair $(G,K)$. We also have the corresponding decomposition of the Lie algebra $\fg=\fg_0\oplus\fg_1$ where $\fg_0$ is the fixed point set and $\fg_1$ is the $(-1)$-eigenspace of $\theta$, respectively. We write $\cN_1=\cN\cap\fg_1$ where $\cN$ is the nilpotent cone in $\fg$. We address the following question which can be regarded as an analogue of the classical springer correspondence: what are the Fourier transforms of $K$-equivariant IC-sheaves[IC=intersection cohomology] on $\cN_1$? In particular, we would like to understand the case when the Fourier transforms are supported on all of $\fg_1$. We concentrate on the latter questions because we believe, and in fact, conjectured in <cit.>, that the other cases can be reduced to this case via induction from smaller groups.
We work in the context of the split symmetric pair $(G,K)=(SL(N),SO(N))$ where $K=G^\theta$ is given by an involution $\theta:G\to G$ and $N$ is odd. Recall that the $K$-orbits in $\cN_1$ are parametrized by partitions of $N$. In <cit.> we considered the case when the IC-sheaves are supported on nilpotent orbits of order 2, i.e., orbits which correspond to partitions that only involve 2's and 1's. In this paper we treat the case of orbits of order 3.
To this end we proposed in <cit.> a general method of analyzing Fourier transforms of IC-sheaves. We replace the Springer resolution and the Grothendieck simultaneous resolution of the classical Springer correspondence by (several) pairs of families of Hessenberg varieties $\mathcal{X}$ and $\check{\mathcal{X}}$ and obtain the following picture:
𝒳 @>>> 𝒳̌
@VπVV @VVπ̌V
_1 @>>> _1
The image of $\pi$ is a nilpotent orbit closure $\bar\cO$ but neither $\pi$ nor $\check \pi$ are semi-small in general. In fact, their generic fibers are not just points in general but they form smooth families of varieties. The key to analyzing the Fourier transforms of $K$-equivariant IC-sheaves on $\cN_1$ in this manner is that the constant sheaf on $\check{\mathcal{X}}$ is the Fourier transform of the constant sheaf on $\mathcal{X}$. Thus, at least as a first approximation, we are reduced to decomposing the push-forwards $\pi_*\bC_ \mathcal{X} $ and $\check\pi_*\bC_{\check{\mathcal{X}}}$ into direct sums of IC-sheaves; this is possible by the decomposition theorem.
Let us recall the definition of
Hessenberg varieties in our setting following <cit.>. Let $x\in\fg_1$, $P$ a parabolic subgroup of $K$, and $\Sigma\subset \fg_1$ a $P$-invariant subspace. The Hessenberg variety associated to the triple $(x,P,\Sigma)$, denoted by
$\on{Hess}_x(K/P,\fg_1,\Sigma)$, is by definition the
following variety
\[\on{Hess}_x(K/P,\fg_1,\Sigma):=\{g\in K/P\,|\,g^{-1}x\in \Sigma\}.\]
As $x$ varies over $\fg_1$, we get a family of Hessenberg varieties
Our results in this paper and the ones in <cit.> provide evidences for the
following conjecture:
The cohomology of smooth Hessenberg varieties can be expressed in terms of
Hodge classes coming from the cohomology of partial flag varieties and the
cohomology of hyperelliptic curves.
The particular pairs of families of Hessenberg varieties we study here have the following properties.
One of the families in the pair, when restricted to the regular semi-simple locus, is isomorphic to a family
of complete intersections of quadrics (see Theorem <ref>); this is the family $\check\pi: \check{\mathcal{X}}\to \fg_1$ in (<ref>). The other family, corresponding to $\pi: {\mathcal{X}}\to \cN_1$ in (<ref>), is supported on the locus of nilpotent elements of order at most three and the fibers of this family admit affine pavings (see <ref>).
The main results in this paper are in sections <ref> and <ref>. When restricted to the locus of regular semi-simple elements $\fg_1^{rs}$ our particular families of Hessenberg varieties can be interpreted as families of intersections of quadrics in projective spaces. In <ref>,
we study the monodromy representations arising from the primitive cohomology of these families (and their natural double covers). This is accomplished by establishing a relative version of results of T. Terasoma <cit.> in <ref>.
The resulting monodromy representations of $\pi_1^{K}(\fg_1^{rs})$ can be expressed in terms of
monodromy representations of certain families of hyperelliptic curves over $\fg_1^{rs}$.
In particular, we see that the cohomology of those Hessenberg varieties can be
expressed in terms of cohomology of hyperelliptic curves. This can be viewed as an evidence of the conjecture above.
In Theorems <ref> and <ref> we describe these monodromy representations completely by decomposing them into irreducible pieces which we call $E_{ij}^N$ and $\tE_{ij}^N$, respectively.
In <ref>, we study
Fourier transforms of the IC complexes arising from the local systems $E_{ij}^N$ and $\tE_{ij}^N$. Recall that these
were obtained from the primitive cohomology of the particular families of
Hessenberg varieties and their double covers.
We show that their Fourier transforms are
supported on the closed sub-variety $\mathcal N_1^3\subset\mathcal N_1$ consisting of
nilpotent elements of order less than or equal to three. Let $\{(\mO,\mE)\}_{\leq 3}$ be the set of all pairs $(\mO,\mE)$ where $\mO$ is a
$K$-orbit in $\mathcal N_1^3$ and $\mE$ is an irreducible $K$-equivariant local system on
$\mO$ (up to isomorphism). In this manner we obtain
an injective map
{E_ij^N}∪{_ij^N}↪{(,)}_≤3 .
This injection can be interpreted as a generalization of the classical Springer correspondence.
As an interesting corollary
(see Example <ref>), we show that the
Fourier transform of the $\IC$ complex for the unique non-trivial irreducible $K$-equivariant local system on
the minimal nilpotent orbit
has full support and the corresponding local system is given by
the monodromy representation of the universal family of hyperelliptic curves
of genus $n$, where $2n+1=N$.
The paper is organized as follows. In <ref>, we
introduce certain pairs of families of Hessenberg varieties and prove basic fact about them.
In <ref> and <ref>, we establish a relative version of the results of Terasoma <cit.>. We utilize these results to obtain a
decomposition of the monodromy representations into irreducibles.
In <ref>, using the results in previous sections,
we show that
Fourier transforms of the IC complexes
for the local systems arising from
our families of
Hessenberg varieties and their double covers are
supported on the closed sub-variety $\mathcal N_1^3\subset\mathcal N_1$ consisting of
nilpotent elements of order at most three.
In <ref> we give a conjectural (explicit) description of the map in
(<ref>) (see Conjecture <ref> and Conjecture <ref>) for $E_{ij}^N$ and we verify
the conjectures in various examples.
Acknowledgement. We thank Cheng-Chiang Tsai and
Zhiwei Yun for helpful discussions.
KV and TX also thank Manfred Lehn, Anatoly S. Libgober, and Yoshinori Namikawa for helpful discussions.
We thank the Max Planck Institute for Mathematics in Bonn and the Mathematical Sciences Research Institute in Berkeley for support, hospitality, and a nice research environment. Furthermore KV and TX thank the Research Institute for Mathematical Sciences in Kyoto for support, hospitality, and a nice research environment.
§ HESSENBERG VARIETIES
In this section we introduce certain families of Hessenberg varieties
which naturally arise when computing the Fourier transforms of IC complexes supported on
nilpotent orbits of order less than or equal to three, i.e., orbits of the form $\cO_{3^i2^j1^k}$.
Our main theorem
(see Theorem <ref>) says that, generically, these families of Hessenberg varieties
are isomorphic to families of complete intersections of quadrics.
§.§ Definition of Hessenberg varieties
Let $G$ be a reductive group and $V$ a representation of $G$.
Let $P\subset G$ be a parabolic subgroup and $\Sigma\subset V$ a $P$-invariant subspace. Consider the vector bundle $G\times^P\Sigma$ and write
G×^PΣ= Hess(G/P,V,Σ) G/P.
The natural projection to $V$ gives us a projective morphism
Hess(G/P,V,Σ) V .
The fibers of this morphism are called Hessenberg varieties; the fiber over $v$ is given by
Hess_v(G/P,V,Σ)={gP | g^-1v∈Σ}.
§.§ Hessenberg varieties in our set up
As in <cit.>, we consider the following situation.
Let $G=SL(N,\bC)$ and $\theta:G\to G$ the involution such that $K:=G^\theta=SO(N,\bC)$. The pair $(G,K)$ is called a split symmetric pair. As in <cit.>, we will assume, starting with <ref>, that $N=2n+1$ is odd, mainly for simplicity. The involution $\theta$ induces a grading $\Lg=\Lg_0\oplus\Lg_1$ on the Lie algebra $\Lg$ of $G$, where $\Lg_i=\{x\in\Lg\,|\,d\theta(x)=(-1)^ix\}$. The group $K$ acts on $\Lg_1$ by adjoint action.
The Hessenberg varieties we consider are of the following form. The ambient group is $K$ acting on the vector space $\Lg_1$. Let $T_K$ be a maximal torus of $K$ and consider a co-character $\lambda:\bG_m\ra T_K$. We write $P=P(\lambda)$ for the parabolic subgroup of $K$ associated to $\lambda$, $\fp$ for the Lie algebra of $P$,
and $\fg_1=\bigoplus\fg_{1,j}$ for the grading induced by $\lambda$.
For any $i\in\bZ$ we define $\fg_{1,\geq i}=\bigoplus_{j\geq i}\fg_{1,j}$.
Let $\Sigma\subset\fg_1$ be a $P$-invariant subspace. The Hessenberg varieties that we are concerned with are of the form $\Hess_v(K/P,\fg_1,\Sigma)$. We have:
Suppose $\Sigma\supset\fg_{1,\geq i}$ for some $i\leq 0$.
Then the projective morphism $\Hess(K/P,\fg_1,\Sigma)\ra\fg_1$ is smooth over
This is proved in <cit.>. For the reader's convenience, we recall the argument here.
Observe that the Zariski tangent space to
$\Hess_v(K/P,\fg_1,\Sigma)\subset K/P$ at a point
can be identified with the kernel of
\[[v,-]:T(K/P)|_{x}\cong K\times^P(\fg_0/\fp)|_{x}\ra K\times^P(\fg_1/\Sigma)|_{x},\,(k,w)\mapsto (k,[k^{-1}v,w]). \]
So it suffices to show that the map above is surjective on the fibers at each point
$kP\in\Hess_v(K/P,\fg_1,\Sigma)$ if $v\in\fg_1^{rs}$. For this we show that any
$v^*\in\fg_1^*$ that annihilates both $[k^{-1}v,\fg_0]$ and $\Sigma $ is zero.
Since $v^*$ annihilates $\Sigma$ and $\Sigma\supset\fg_{1,\geq i}$ for some $i\leq 0$, there
exists $\delta>0$ such that $v^*\in\fg_{1,\geq\delta}^*$, that is, $v^*$ is $K$-unstable.
Since $k^{-1}v\in\fg_1^{rs}$ is a “good vector
”[ To see that $v\in\fg_1^{rs}$ is a good vector,
we observe that if
$v^*\in\fg_1^*$ is a $K$-unstable vector and annihilates $[v,\fg_0]$,
then the vector $(v^*,0)\in\fg^*\cong\fg_1^*\oplus\fg_0^*$
anihilates $[v,\fg]=[v,\fg_0]\oplus[v,\fg_1]$. However, since $v\in\fg_1^{rs}\subset\fg^{rs}$
it implies $v^*\in\fg_1^*$ is semisimple. This forces $v^*=0$ since any $K$-unstable vector
is nilpotent.
] in $\fg_1$
under the $K$-action, i.e., there is no non-zero $K$-unstable vector $v^*\in\fg_1^*$ that annihilates the subspace $[k^{-1}v,\fg_0]\subset \fg_1$, we have $v^*=0$. The lemma is proved.
§.§ Families of Hessenberg varieties
Let us write $(G,K)=(SL(V),SO(V,Q))$, where $Q$ is a non-degenerate quadratic form on $V$. Denote by $\langle,\rangle_Q$ the non-degenerate bilinear form associated to $Q$. For a subspace $U\subset V$, we write $U^\p=\{v\in V\,|\,\langle v,U\rangle_Q=0\}$.
Let $\cN$ be the nilpotent cone of $\Lg$ and let $\cN_1=\Lg_1\cap\cN$. It is known that the number of $K$-orbits in $\cN_1$ is finite (see <cit.>). Moreover, the $K$-orbits in $\cN_1$ are parametrized as follows (see <cit.>). For $N$ odd (resp. even), each partition of $N$ correspond to one $K$-orbit in $\cN_1$ (resp. except that each partition with only even parts corresponds to two $K$-orbits). In this paper we do not distinguish the two orbits corresponding to the same partition when $N$ is even; thus we write $\cO_\lambda$ for an orbit corresponding to $\lambda$.
Let $\{e_i,\,i=1,\ldots,N\}$ be a basis of $V$ such that $\langle e_i,e_j\rangle_Q=\delta_{i+j,N+1}$. For any $l\leq \frac{N}{2}$,
let $P_{l}$ be the parabolic subgroup of $K$ that stabilizes the
partial flag
\[0\subset V_{l-1}^0\subset V_{l}^0\subset V_l^{0,\p}\subset V_{l-1}^{0,\p}\subset V=\bC^{N},\]
where $V_i^0=\on{span}\{ e_1,...,e_i\}$. Consider the following two subspaces of
$$E_{l}=\{x\in\fg_1\,|\,xV_{l}^0=0,\, xV_{l}^{0\p}\subset
V_{l-1}^0\}\text{ and }O_l=\{x\in\fg_1\,|\,xV_{l}^0=0,\, xV_{l-1}^{0\p}\subset
Note that both $E_l$ and $O_l$ are $P_l$-invariant. We form the
corresponding families of Hessenberg varieties
σ_l^N:_l^O:=(K/P_l,_1,O_l)_1 .
A direct calculation shows that
Im τ_l^N=𝒪̅_3^l-12^11^N+1-3l if 3l≤N+1,
Im τ_l^N=_3^N-2l2^3l-N if 3l>N+1
Im σ_l^N=𝒪̅_3^l-11^N+3-3l if 3l≤N+1,
Im σ_l^N=_3^N-2l2^3l-N if 3l>N+1.
When $3l\leq N+1$, $\tau_l^N$ coincides with
Reeder's resolution of $\bar{\mathcal O}_{3^{l-1}2^11^{N+1-3l}}$ <cit.>.
Let $E_l^\p$ and $O_l^\p$ be the orthogonal complements of $E_l$ and $O_l$
in $\fg_1$ with respect to the non-degenerate trace form, respectively. Let us now consider the following families of Hessenberg varieties
σ̌_l^N:_l^O,:=(K/P_l,_1,O_l^)_1 .
Concretely, we have
\[E_l^\p=\{x\in\fg_1\,|\,xV_{l-1}^0\subset V_l^0,\, xV_l^0\subset V_{l}^{0\p}\},\ \
O_l^\p=\{x\in\fg_1\,|\,xV_{l-1}^0\subset V_l^0\};\]
and then
\[
{\Hess_l^{O,\p}}\is\{(x,0\subset V_{l-1}\subset V_l\subset V_l^\p\subset V_{l-1}^\p\subset \bC^{N})\,|\,x\in\fg_1,\,
xV_{l-1}\subset V_{l}\},\]
\[
{\Hess_l^{E,\p}}\is\{(x,0\subset V_{l-1}\subset V_l\subset V_l^\p\subset V_{l-1}^\p\subset \bC^{N})\,|\,x\in\fg_1,
\,xV_{l-1}\subset V_{l},\, xV_{l}\subset V_l^\p\}.
\]
Finally, note that, as the notation indicates, the bundle $\Hess_l^{E,\p}\to K/P_l$ is the orthogonal complement of the bundle $\Hess_l^{E}\to K/P_n$ in the trivial bundle $\fg_1 \times K/P_l$ and similarly for ${\Hess_l^{O,\p}}$ and ${\Hess_l^{O}}$.
Hence, by functoriality of the Fourier transform, we have:
((σ^N_l)_*[-])≅(σ̌^N_l)_*[-] and ((τ^N_l)_*[-])≅(τ̌^N_l)_*[-].
§.§ Affine pavings
In this subsection we show that
the fibers of $\tau_m^N:\on{Hess}_m^{E}\to\cN_1$ and $\sigma_m^N:\on{Hess}_m^{O}\to\cN_1$ have
a paving by affine spaces.
Let $x\in\cO_{3^i2^j1^{N-3i-2j}}\subset\on{Im}\tau_m^N$ (resp. $\on{Im}\sigma_m^N$) and $x_0\in\cO_{2^j1^{N-3i-2j}}$.
We have
(τ_m^N)^-1(x)≅(τ_m-i^N-3i)^-1(x_0) (resp. (σ_m^N)^-1(x)≅(σ_m-i^N-3i)^-1(x_0)).
We prove the lemma for $\tau_m^N$. The argument for $\sigma_m^N$ is entirely similar and omitted. We have
$$(\tau_m^{N})^{-1}(x)\cong\{0\subset V_{m-1}\subset V_m\subset V_m^\p\subset V_{m-1}^\p\subset\bC^{N}\,|\,xV_m=0,\, xV_{m}^\p\subset V_{m-1}\}.$$
Let $(V_{m-1}\subset V_m)\in(\tau_m^{N})^{-1}(x)$. We have that $\on{Im}\,x\subset(\ker x)^\p\subset V_m^\p$. Thus $\on{Im}\,x^2\subset V_{m-1}$.
Choose a basis $\{x^ku_l,\, k\in[0,2],\, l\in[1,i],\ v_k,\,xv_k, \,k\in[1,j],\ w_l,\, l\in[1,N-3i-2j]\}$ of $V$ as in <cit.>.
$$U^0=\on{span}\{x^lu_k,l\in[0,2],k\in[1,i]\},\,V^0=\on{span}\{v_k,xv_k,k\in[1,j],w_l,\ l\in[1,N-3i-2j]\}.$$
Then $Q|_{U^0}$, $Q|_{V^0}$ are non-degenerate and $V^0=(U^0)^\p$. We have
$$V_m=\on{Im}\,x^2\oplus W_{m-i}\text{ and }V_{m-1}=\on{Im}\,x^2\oplus W_{m-i-1},$$
where $W_{m-i}=V_m\cap V^0\supset W_{m-i-1}=V_{m-1}\cap V^0$. We have $\dim W_{m-i}=m-i$ and $\dim W_{m-i-1}=m-i-1$. Let $x_0=x|_{V^0}$. Then $x_0\in\cO_{2^j1^{N-3i-2j}}$. Note that $xV_m=0$ if and only if $x_0W_{m-i}=0$. Now
$$V_m^\p=\on{span}\{x^lu_k,l=1,2,k\in[1,i]\}\oplus W_{m-i}^{\p_0},$$
where $W_{m-i}^{\p_0}$ denotes the orthogonal complement of $W_{m-i}$ in $V^0$ with respect to $Q|_{V^0}$. Thus $xV_m^\p\subset V_{m-1}$ if and only if $x_0W_{m-i}^{\p_0}\subset W_{m-i-1}$. This gives us the desired isomorphism
(τ_m^N)^-1(x)≅(τ_m-i^N-3i)^-1(x_0), (V_m-1,V_m)↦(pr_V^0(V_m-1),pr_V^0(V_m))
where $\on{pr_{V^0}}$ is the projection from $V$ to $V^0$ with respect to $V=U^0\oplus V^0.$
Let $\on{OGr}(k,N)$ denote the orthogonal Grassmannian variety of $k$-dimensional isotropic subspaces in $\bC^N$ with respect to a non-degenerate bilinear form on $\bC^N$ and $\on{Gr}(k,N)$ denote the Grassmannian variety of $k$-dimensional subspaces in $\bC^N$.
By Lemma <ref>, to describe the fibers $(\tau_m^N)^{-1}(x)$ and $(\sigma_m^N)^{-1}(x)$, it suffices to consider the case when $x\in\cO_{2^j1^{N-2j}}$.
We first introduce some notation. Let $x\in\cO_{2^j1^{N-2j}}$. We write
$$\Sigma:=\ker x/\on{Im}x\text{ and }\bar{U}=U/(U\cap\on{Im}x)\text{ for }U\subset\ker x.$$
Define a bilinear from $(,)$ on $\on{Im} x$ by
(xv,xw):=⟨v,xw⟩_Q.Using $(\on{Im}x)^\p=\ker x$, we see that $(,)$ is non-degenerate. For $U\subset\on{Im}x$, we define
Let us denote
$$\Upsilon_{m,j}^N:=(\tau_m^N)^{-1}(x),\ \ \Gamma_{m,j}^N:=(\sigma_m^N)^{-1}(x),\ \ x\in\cO_{2^j1^{N-2j}}.$$
We partition $\Upsilon_{m,j}^N$ into pieces indexed by the dimension of $V_m\cap \operatorname{Im}x$ by letting
Υ_m,j^N,k={(0⊂V_m-1⊂V_m⊂V_m^⊂V_m-1^⊂^N)∈Υ_m,j^N | (V_m∩Imx)=k}.
We now describe the pieces $\Upsilon_{m,j}^{N,k}$. To this end, let
Θ_m,j^N,k={0⊂V_m⊂V_m^⊂^N | (V_m∩Imx)=k,
xV_m=0, xV_m^⊂V_m, (xV_m^)≤m-1}.
Consider the following map
η:Θ_m,j^N,k→Gr(j-k,Im x)×Gr(m-k,Σ), (V_m)↦((V_m∩Im x)^_(,),V̅_m).
We claim that
Imη≅OGr(j-k,Im x)×OGr(m-k,Σ),
where $\on{Im}\,x$ is equipped with the non-degenerate bilinear form $(,)$ (see (<ref>)), and $\Sigma$ is equipped with the non-degenerate bilinear form induced by $\langle,\rangle_Q$. It is clear that $\bar V_m\subset\on{OGr}(m-k,\Sigma)$ as $\langle,\rangle_Q|_{V_m}=0$. It is easy to check that $x(V_m^\p)\subset(V_m\cap\on{Im}\,x)^{\p_{(,)}}$ and $\dim x(V_m^\p)=\dim\, (V_m\cap\on{Im}\,x)^{\p_{(,)}}=j-k$. Thus $x(V_m^\p)=(V_m\cap\on{Im}\,x)^{\p_{(,)}}$. Therefore the condition $xV_m^\p\subset V_m$ is equivalent to $(V_m\cap\on{Im}\,x)^{\p_{(,)}}\subset V_m\cap\on{Im}\,x$, i.e. $(V_m\cap\on{Im}\,x)^{\p_{(,)}}\in\on{OGr}(j-k,\on{Im}\,x)$.
This proves the claim.
Thus we obtain a surjective map
η:Θ_m,j^N,k→OGr(j-k,Im x)×OGr(m-k,Σ)
and it is easy to see that the fibers of $\eta$ are affine spaces $\mathbb{A}^{k(m-k)}$. Note that the fiber of the natural projection map
Υ_m,j^N,k→Θ_m,j^N,k: (V_m-1,V_m)↦V_m
at $V_m$ is the projective space $\bP(V_m/(xV_m^\p))\cong\bP^{m-j+k-1}$. It is easy to check using the above maps that each piece $\Upsilon_{m,j}^{N,k}$ has an affine paving. Therefore $\Upsilon_{m,j}^{N}$ also has an affine paving.
We can similarly partition $\Gamma_{m,j}^N$ into pieces indexed by the dimension of $V_{m-1}\cap \operatorname{Im}\,x$. Let
\begin{eqnarray*}
&&\Gamma_{m,j}^{N,k}=\{(0\subset V_{m-1}\subset V_m\subset V_m^\p\subset V_{m-1}^\p\subset\bC^{N})\in\Gamma_{m,j}^N\,|\,\dim(V_{m-1}\cap \operatorname{Im}\,x)=k\}\\
&&\Lambda_{m,j}^{N,k}=\{(0\subset V_{m-1}\subset V_{m-1}^\p\subset\bC^N)\,|\,\dim(V_{m-1}\cap \operatorname{Im}\,x)=k,\ xV_{m-1}=0,\ xV_{m-1}^\p\subset V_{m-1}\}.
\end{eqnarray*}
We have a surjective map
η':Λ_m,j^N,k→OGr(j-k,Im x)×OGr(m-k-1,Σ), (V_m-1)↦((V_m-1∩Im x)^_(,),V̅_m-1).
The fibers of $\eta'$ are affine spaces $\mathbb{A}^{k(m-k-1)}$. The fiber of the natural projection map
$$\Gamma_{m,j}^{N,k}\to\Lambda_{m,j}^{N,k}:\ (V_{m-1},V_m)\mapsto V_{m-1}$$
at a given $V_{m-1}$ is the variety of isotropic lines in $(V_{m-1}^\p\cap\ker\,x)/V_{m-1}$ with respect to the quadratic form induced by $Q$. The same argument as before shows that $\Gamma_{m,j}^{N}$ is paved by affines.
In particular, we see from the above discussion that
Finally, in <cit.> we have used the following fact
For $x_i\in\cO_{3^i2^{2m-1-2i}1^{2n-4m+3+i}}$ we have
(τ_m^2n+1)^-1(x_i)≅OGr(m-1-i,2m-1-2i) .
This can be deduced from the results in this subsection as follows. Using Lemma <ref> and (<ref>) we see that $\tau_m^{2n+1}(x_i)\cong\Upsilon_{m-i,\,2m-1-2i}^{2n+1-3i,\,m-i}$. The conclusion follows by considering the maps in (<ref>) and (<ref>).
§.§ Families of complete intersections of quadrics and their identification with Hessenberg varieties.
Let $m\in[1, N-1]$ be a integer.
For any $s\in \fg_1^{rs}$, let
$$X_{m,s}\subset\mathbb P(V)\is\mathbb P^{N-1}$$
be the complete intersection of
$m$ quadrics
$$\langle s^i-,-\rangle_Q=0,\ i=0,\ldots,m-1$$
in $\mathbb P(V)$.
As $s$ varies over $\grs$, we get a family
of complete intersections of
$m$ quadrics in $\mathbb P(V)$.
The families of Hessenberg varieties $\Hess_l^{O,\p}$ and
$\Hess_l^{E,\p}$ over $\Lg_1^{rs}$ are identified with $X_m$'s as follows.
Assume that $k\leq \frac{N-1}{2}$. Then we have
* There is a $K$-equivariant isomorphism $\Hess_k^{O,\p}|_{\grs}\is X_{2k-1}$ of varieties over
* There is a $K$-equivariant isomorphism $\Hess_k^{E,\p}|_{\grs}\is X_{2k}$ of varieties over
We begin with the following simple observation.
Let $s\in\fg_1^{rs}$.
For any isotropic subspace $0\neq U\subset V$,
$\dim (sU\cap U)< \dim(sU)$.
This follows from the fact that $s$ has no isotropic eigenspaces.
We first define a map from $X_{2k-1}$ to $\Hess_n^{O,\p}$.
Let $(s,l)\in X_{2k-1}$, where $s\in \Lg_1^{rs}$ and $l$ is in the
complete intersection of
$2k-1$ quadrics $\langle s^i-,-\rangle_Q=0$, $i=0,\ldots,2k-2$, in $\mathbb P(V)$.
Let $0\neq v\in l$. For $1\leq i\leq k$, consider the subspaces
$$V_{i}=\on{span}\{ v,sv,...,s^{i-1}v\}.$$
Note that $V_{i}$ is isotropic. We show that $\dim V_{i}=i.$ We have $V_{i}=V_{i-1}\cup sV_{i-1}$. Thus $\dim V_{i}=\dim V_{i-1}+\dim sV_{i-1}-\dim(sV_{i-1}\cap V_{i-1})>\dim V_{i-1}$, where in the last inequality we use (<ref>). By induction we see that $\dim V_{i}=i$. Hence
the assignment $(s,l)\mapsto (s,V_{k-1}\subset V_k)$ defines a map
\[\iota:X_{2k-1}\ra \Hess_k^{O,\p}|_{\Lg_1^{rs}}.\]
One checks readily that $\iota$ is $K$-equivariant.
We prove that $\iota$ is an isomorphism by constructing an
explicit inverse map.
Let $(s, V_{k-1}'\subset V_k')\in\Hess_k^{O,\p}$ with $s\in \Lg_1^{rs}$.
We define a sequence of subspaces
$0\subset V_1'\subset V_2'\subset\cdots\subset V_{k-2}'$ recursively. Let us first define $V_{k-2}'$. Consider the map $\bar{s}:V_{k-1}'\xrightarrow{s} V_k'\to V_k'/V_{k-1}'$. Note that by (<ref>), the map $\bar s$ is nonzero, hence surjective as $\dim V'_{k}/V'_{k-1}=1$. Let $V_{k-2}'=\ker\bar s$. We have $\dim V_{k-2}'=k-2$ and $V_{k-1}'=V_{k-2}'\cup sV_{k-2}'$. By induction we can assume that we have defined $V_i'$ such that $\dim V_i'=i$ and
$V_{i+1}'=V_{i}'\cup sV_{i}'$. Let
$$V_{i-1}'=\ker(\bar s:V_{i}'\xrightarrow{s}{V_{i+1}'}\ra V'_{i+1}/V'_{i}).$$
The same argument as before shows that $\dim V_{i-1}'=i-1$ and
$V_{i}'=V_{i-1}'\cup sV_{i-1}'$.
Thus in particular we obtain that $\dim V_1'=1$,
and it is easy to see that the map
\[\Hess_k^{O,\p}|_{\Lg_1^{rs}}\ra X_{2k-1}, (s, V_{k-1}'\subset V_k')\mapsto (s,V_1').\]
defines an inverse of $\iota$.
This finishes the proof of (1).
For (2), we observe that, under the isomorphism $\iota: X_{2k-1}\is\Hess_k^{O,\p}$, the equation $\langle s^{2k-1}v,v\rangle_Q=0$ for the divisor
$X_{2k}\subset X_{2k-1}$ becomes $\langle sV_{k}',V_k'\rangle_Q=0$,
which is the equation for the divisor $\Hess_k^{E,\p}\subset\Hess_k^{O,\p}$. Thus (2) follows.
§ COMPLETE INTERSECTIONS OF QUADRICS
AND THEIR DOUBLE COVERS
In <ref> we have introduced the families $X_m\ra\grs$ of complete intersections of quadrics, which we have identified with families of Hessenberg varieties $\on{Hess}_n^{E,\p}|_{\Lg_1^{rs}}$, $\on{Hess}_n^{O,\p}|_{\Lg_1^{rs}}$. In order to study the monodromy representations of the equivariant fundamental group $\pi_1^K(\Lg_1^{rs})$ associated with the above families of Hessenberg varieties, we introduce families $\mY_m$ of branched covers of $\mathbb P^{N-m-1}$ and
relate them with $X_m$. We also introduce a family of branched double covers of $X_m$, denoted by $\tilde{X}_m$, and relate them with families
of branched covers
of $\mathbb P^{N-m-1}$ which we introduce in <ref>. Our construction can be regarded as a relative version of the construction in <cit.>.
§.§ Some notation
In this section we choose a Cartan subspace $\fa\subset\Lg_1$ that consists of diagonal matrices. Let $\fa^{rs}=\fa\cap\Lg^{rs}$. We write an element $a\in \fa$ with diagonal entries $a_1,\ldots, a_N$ as $a=(a_1,\ldots,a_N)$. Thus $a=(a_1,\ldots,a_N)\in\ars$ if and only if $a_i\neq a_j$ for $i\neq j$.
where we regard $\bZ/2\bZ$ as a subgroup of $(\bZ/2\bZ)^{N}$ via the diagonal embedding.
For any $\chi\in I_N^\vee=\Hom(I_N,\bG_m)$, we define
\[\on{supp}(\chi)=\{i\in [1,N]\,|\,\chi(\xi_i)=-1 \}\text{ and }|\chi|=\#\on{supp}(\chi)\]
where $\xi_i$ is the image of $(0,..,1,...,0)\in(\bZ/2\bZ)^{N}$ in $I_N$.
Note that $|\chi|$ is even.
If we identify the centralizer $Z_K(a)$ of $a\in\ars$ with the
kernel of the map $(\bZ/2\bZ)^{N}\ra \bZ/2\bZ,\ (b_1,...,b_N)\mapsto\sum b_i$,
we obtain a
natural map
Note when $N$ is odd which we will assume from this point on, the map (<ref>)
is an
isomorphism. Therefore, in what follows, we often make the canonical identification $Z_K(a)\cong I_N$.
To emphasize:
From now on we assume that $N$ is odd.
§.§ Family of curves
In this subsection we introduce certain families of curves which will be used to construct the families $\mY_m$ and $\tilde\mY_m$ of branched covers of projective spaces.
For any $a=(a_1,...,a_{N})\in \fa^{rs}$, there are
natural isomorphisms
\[
\pi_1^{ab}(\mathbb P^1-\{a_1,...,a_{N}\})\otimes\bZ/2\bZ\is I_N
\ \ \quad \pi_1^{ab}(\mathbb P^1-\{a_1,...,a_{N},a_{N+1}=\infty\})\otimes\bZ/2\bZ\is I_{N+1}.
\]
The isomorphisms are given by assigning to a small loop around each $a_i$ the element in $I_N$ (resp. $I_{N+1}$) with only non-trivial coordinate in position $i$.
$$\text{$C_a\ra\mathbb P^1$ (resp. $\tilde C_a\ra\mathbb P^1$)}$$
be the abelian covering of $\mathbb P^1$ ramified at $\{a_1,...,a_{N}\}$ (resp. $\{a_1,...,a_{N+1}=\infty\}$)
with Galois groups given by $I_N$ (resp. $I_{N+1}$).
Concretely, $C_a$ (resp. $\tC_a$) is the smooth projective curve corresponding to the
function field
$$\text{$\bC(t)((\frac{t-a_i}{t-a_1})_{i=2,...,N}^{1/2})$ (resp. $\bC(t)((t-a_i)_{i=1,...,N}^{1/2})$).}$$
The group $I_N$ (resp. $I_{N+1}$) acts on $C_a$ (resp. $\tilde C_a$). For any $\chi\in I_N^\vee$ (resp. $\chi\in I_{N+1}^\vee$) we define
\text{$C_{a,\chi}=C_a/\ker\chi$ (resp. $\tilde C_{a,\chi}=\tilde C_a/\ker\chi$),}
which is a branched double cover of $\mathbb P^1$ with branch locus $\{a_i\,|\,i\in\on{supp}(\chi)\}$. Concretely, $C_{a,\chi}$ (resp. $\tC_{a,\chi}$) is isomorphic to
the smooth projective hyperelliptic curve with affine equation
y^2=\prod_{i\in\on{supp}\chi }(x-a_i)\text{ (resp. }
y^2=\prod_{i\in\on{supp}\chi,\ i\neq N+1 }(x-a_i)).
We have $\dim H^1(C_\chi,\bC)=|\chi|-2$ (resp. $\dim H^1(\tilde C_\chi,\bC)=|\chi|-2$).
As $a$ varies over $\ars$, we obtain a family of
$$\text{$C\ra\ars$ (resp. $\tilde C\ra\ars$)}$$
and we similarly obtain families of hyperelliptic curves
$$\text{$C_\chi\ra\ars$ (resp. $\tilde C_\chi\ra\ars$) for any $\chi$.}$$
We also note that the Weyl group $W=S_N$ acts naturally on $C$ (resp. $\tC$) making the
projection $C\ra\ars$ (resp. $\tC\ra\ars$) a $W$-equivariant map.
We will now associate monodromy representations to these families. Let us fix $a\in\ars$ and a character $\chi\in I_N^\vee$ (resp. $\chi\in I_{N+1}^\vee$) and we recall that $\pi_1(\fa^{rs},a)$ is the pure braid group $P_N$. The monodromy representation of the family $C_\chi\ra \ars$ factors through the symplectic group and we denote it by
$\rho_{C_\chi}:P_{N}\ra Sp(H^1(C_{a,\chi},\bC))\is Sp(2m-2)$ where the $m=\frac {|\chi|} 2$. Similarly we obtain a monodromy representation $\rho_{\tilde C_\chi}:P_{N}\ra Sp(H^1(\tilde C_{a,\chi},\bC))\is Sp(2m-2).$
We claim:
The images of the representations $\rho_{C_\chi}$ and $\rho_{\tilde C_\chi}$ are Zariski dense
in $Sp(H^1(C_{a,\chi},\bC))$ and $Sp(H^1(\tilde C_{a,\chi},\bC))$, respectively .
In particular, the representations $\rho_{C_\chi}$ and $\rho_{\tilde C_\chi}$ are irreducible .
We see this as follows. Consider the following subvariety $\fa^{rs}(a,\chi)\subset\fa^{rs}$
^rs(a,χ)={a'∈^rs | a'_i=a_i if i∉supp(χ) }.
It suffices to show that the monodromy representation of the restriction of
$C_\chi\ra\ars$ to $\fa^{rs}(a,\chi)$ has Zariski dense image. Now, $\fa^{rs}(a,\chi)$ is an open subset of the space $\cM_{2m}$ of $2m$ distinct marked points in $\bC$ and the family $C_\chi\times_\ars\fa^{rs}(a,\chi)$ is the restriction of the universal family of hyperelliptic curves parametrized by $\cM_{2m}$. Note further, that $\cM_{2m}$ itself is an open subset of the space $\tilde\cM_{2m}$ of $2m$ distinct marked points in $\bP^1$ carrying its own family of hyperelliptic curves. Now, by <cit.> (see also <cit.>), the monodromy representation on $\tilde\cM_{2m}$ is irreducible and has Zariski dense image. Therefore
$\rho_{C_\chi}$, as a restriction to an open subset, has the same property. The argument in the case $\tilde C_\chi\ra\ars$ is completely analogous except one has to take into account that $a_{N+1}=\infty$.
Finally, there is a unique character $\chi_0\in I_{N+1}^\vee$ with
$|\chi_0|=N+1$ (here we use the assumption that $N$ is odd). The character $\chi_0$ is invariant under the Weyl group action. Thus we can pass to a quotient of $\tC_{\chi_0}\ra\ars$ under the $W$ action and in this way obtain a family $\overline C_{\chi_0}\ra\fc^{rs}=\ars/W$. The family $\overline C_{\chi_0}\ra\fc^{rs}=\ars/W$ is the universal family
of hyperelliptic curves $y^2=\prod_{i=1}^N(x-a_i)$ and $\tC_{\chi_0}\ra\ars$ is a similar universal family with marked ramification points.
§.§ Branched cover $\mY_m$ of projective spaces and $X_m$
where $sum$ is the summation map. Fix $a=(a_1,\ldots,a_N)\in\ars$. Let $C_a\ra\bbP^1$ be the
curve introduced
in <ref>.
The semi-direct product $\bar I_N^{N-m-1}\rtimes S_{N-m-1}$ acts naturally
on $C_a^{N-m-1}$ and we define
$$\mY_{m,a}=C_a^{N-m-1}/\bar I_N^{N-m-1}\rtimes S_{N-m-1}.$$
We have a natural map
\[\iota_a:\mY_{m,a}\ra C_a^{N-m-1}/I^{N-m-1}_N\rtimes S_{N-m-1}\is\mathbb P^{N-m-1}.\]
According to <cit.>,
for a suitable choice of homogeneous coordinates$[x_1,...,x_{N-m}]$
of $\bbP^{N-m-1}$,
each ramification point $a_i$
defines a
in $\bbP^{N-m-1}$ and the map $\iota_a$
is an $I_N$-branched cover of $\mathbb P^{N-m-1}$ with branch locus
$\{H_{a,i}=0\}_{i=1,...,N}$. As $a$ varies over $\ars$, we get a $\ars$-family
of $I_N$-branched
covers of $\mathbb P^{N-m-1}$
[rd][rr]^ ι ℙ^N-m-1_[ld]
C^{N-m-1}/\bar I_N^{N-m-1}\rtimes S_{N-m-1}$$
and the
base change of $\iota$ to $a$ is equal to $\iota_a$.
Observe that the $ W$-action on $C$ induces a $ W$-action on $\mY_m$
making the projection $\mY_m\ra\ars$ a $W$-equivariant map.
Let $X_{m}\ra\grs$ be the family of complete intersections of quadrics introduced in <ref>.
For $a=(a_1,...,a_{N})\in\ars$ the
equation of $X_{m,a}$ is given by
\[a_1^iv_1^2+\cdot\cdot+a_{N}^iv_{N}^2=0,\ \ \ i=0,...,m-1.\]
Consider the map
$$s:\mathbb P(V)\ra\mathbb P(V),\
[v_1,...,v_{N}]\mapsto [v_1^2,...,v_{N}^2].$$
The image
$s(X_{m,a})$ is equal to $\mathbb P(V_{m,a})$, where
V_m,a={v∈V | a_1^iv_1+··+a_N^iv_N=0, i=0,...,m-1}⊂V.
The resulting
\[s_{a}:X_{m,a}\ra\mathbb P(V_{m,a})\]
is an $I_N$-branched cover with branch locus $\{v_i=0\}_{i=1,...,N}$.
As $a$ varies over $\ars$,
we obtain
X_m|_[rd][rr]^s ℙ(V_m)[ld]
Here $V_m\ra\ars$ is the vector bundle over $\ars$ whose fiber over
$a$ is $V_{m,a}$ and $\mathbb P(V_m)$ is the associated projective bundle.
The two families
$X_m|_\ars$ and $\mY_m$ are related as follows. Let
\[
\ c_i^2=d_i:=\prod_{j\neq i}(a_j-a_i)\}.
\]
The projection $(\tilde\fa^{rs})'\ra \ars,\ (a_1,...,a_N,c_1,...,c_N)\mapsto (a_1,...,a_N)$
realizes $(\tilde\fa^{rs})'$ as a
$(\bZ/2\bZ)^N$-torsor over $\ars$.
Consider the following $I_N$-torsor over $\ars$
here we view $\bZ/2\bZ$ as a subgroup of $(\bZ/2\bZ)^N$ via the diagonal embedding. The Weyl group $W$ acts naturally on
$\tilde\fa^{rs}$ making the projection to $\ars$ a $W$-equivariant map. We observe that the $I_N$-torsor $\tilde{\fa}^{rs}$ of (<ref>) gives rise to the following canonical map
We have an $I_N\rtimes{W}$-equivariant isomorphism
where ${W}$ (resp. $I_{N}$) acts on $\mY_m^t$ by the diagonal action (resp.
on the first factor).
Following <cit.>, we
consider the family
\[X_m'|_\ars\ra\ars\]
whose fiber over $a=(a_1,\ldots,a_N)\in\ars$ is the complete intersection of $m$ quadrics in $\mathbb P(V)$
given by
\[\frac{a_1^i}{d_1}v_1^2+\cdot\cdot+\frac{a_{N}^i}{d_N}v_{N}^2=0,\ \ \ i=0,...,m-1,\]
where $d_i:=\prod_{j\neq i}(a_j-a_i)$. One can think of $X_{m}|_\ars$ as
a twist of $X_{m}'|_\ars$. More precisely,
we have a natural map
$$\tilde\fa^{rs}\times_{\ars} X'_{m}|_\ars\ra X_m|_\ars,\ (a,c,[v_1,...,v_N])\mapsto(a,[v_1/c_1,\ldots,v_N/c_N])$$
and it is not hard to see that
it descends to
a canonical $I_N$-equivariant isomorphism
X_m|_(^rs×_ X_m'|_)/I_N.
Here $I_N$ acts on the product via the diagonal action.
The Weyl group ${W}=S_N$ acts naturally on $X_m|_{\ars}$, $X_m'|_\ars$,
$\tilde\fa^{rs}$, and $(\tilde\fa^{rs}\times_{\ars} X_{m}'|_\ars)/I_N$, making the projections to $\ars$ equivariant maps under the $W$-actions. Moreover, the isomorphism in (<ref>) is also $W$-equivariant. In <ref> (see Proposition <ref>),
we show that
there is an $I_N\rtimes W$-equivariant isomorphism
Combining (<ref>) with (<ref>)
we obtain (<ref>).
§.§ Branched double covers $\tilde{X}_m$ of complete intersections of quadrics
We introduce a branched double cover of $X_m$ as follows.
Let $\widetilde{V}=V\oplus\bC$.
For any $s\in\grs$, consider the following quadrics in $\mathbb P(\tV)$,
_i(v, ϵ)=⟨s^iv,v⟩_Q=0, i=0,…,m-1
_m(v, ϵ)=
We define
$\tX_{m,s}$ to be the complete intersection of
$m+1$ quadrics $\tQ_{i}=0$, $i=0,\ldots,m$.
As $s$ varies over $\grs$, we get a family
of complete intersections of $m+1$ quadrics in $\bbP(\tV)$.
The projection $\tV=V\oplus\bC\ra V$,
$(v, \epsilon)\mapsto v$, induces a map
$p_{m}:\tX_{m}\ra X_{m}$
which is
a branched double cover with branch locus $X_{m+1}\subset X_{m}$.
The map $\tV=V\oplus\bC\to\tV$ given by $(v, \epsilon)\mapsto (v, -\epsilon)$ defines an involution on $\tX_m$.
We denote this involution by $\sigma$.
The group $K=SO(V,Q)$ acts naturally on both $X_m$ and $\tX_m$. The maps
$\pi_m:X_m\ra\grs\text{ and }\tilde\pi_m:\tX_m\ra\grs$ are $K$-equivariant.
In particular, the centralizer $Z_{K}(s)$ acts on the fibers $X_{m,s}$ and $\tX_{m,s}$.
§.§ Branched cover $\tilde{\mY}_m$ of projective spaces and $\tX_m$
In this subsection
we generalize Proposition <ref> to the branched double cover
$\tX_m$ introduced in <ref>.
For $a=(a_1,...,a_{N})\in\ars$ the
equations of $\tX_{m,a}\subset\bbP^{N-m-1}(\tV)$ (recall that $\tV=V\oplus\bC$)
are given by
\[a_1^iv_1^2+\cdot\cdot+a_{N}^iv_{N}^2=0,\ \ i=0,...,m-1,\ \
Consider the map
We have
$\tilde s(\tX_{m,a})\is\mathbb P(\tV_{m,a})$,
where $\tV_{m,a}\subset \tV$ is the subspace defined by
the equations
a_1^iv_1+··+a_N^iv_N=0, i=0,...,m-1,
is a $I_{N+1}$-branched cover with branch locus $\{v_i=0\}_{i=1,...,N+1}$.
As $a$ varies over $\ars$,
we obtain
_m|_[dr][rr]^s̃ ℙ(_m)[dl]
here $\tV_m$ is the vector subbundle
of the trivial bundle $\tV\times\ars$
whose fiber
over $a$ is the subspace $\tV_{m,a}$, and $\bbP(\tV_m)$ is the associated projective bundle.
We now introduce another family $\tilde{\mY}_m$ of branched covers of $\bP^{N-m-1}$. Let $sum:I_{N+1}^{N-m-1}\ra I_{N+1}$ be the summation map
and define $\bar I^{N-m-1}_{N+1}=\ker(sum)$. For any $a\in \ars$ let
$\tilde C_a\ra\mathbb P^1$ be the $I_{N+1}$-branched cover of $\mathbb P^1$
introduced in <ref>.
The semi-direct product $\bar I_{N+1}^{N-m-1}\rtimes S_{N-m-1}$ acts naturally
on $(\tilde C_a)^{N-m-1}$. We define
$$\tilde\mY_{m,a}=(\tilde C_a)^{N-m-1}/\bar I_{N+1}^{N-m-1}\rtimes S_{N-m-1}.$$
Similar to the case of $\mY_{m,a}$, the natural map
\[\tilde\iota_a:\tilde\mY_{m,a}\ra (\tilde C_a)^{N-m-1}/I^{N-m-1}_{N+1}\rtimes S_{N-m-1}\is\mathbb P^{N-m-1}\]
is an $I_{N+1}$-branched cover of $\mathbb P^{N-m-1}$ with branch locus
$\{H_{a,i}=0\}_{i=1,...,N+1}$, here
$H_{a,i}=0$ for $i=1,...,N$ are the hyperplanes as before (see (<ref>)) and
$H_{a,N+1}:=x_{N-m}=0$ is the hyperplane corresponding to
the ramification point $a_{N+1}=\infty$.
As $a$ varies over $\ars$, we get an $\ars$-family of $I_{N+1}$-branched
cover of $\mathbb P^{N-m-1}$
\[
\xymatrix{\tilde\mY_m\ar[dr]\ar[rr]^{\tilde\iota\ }&&\mathbb P^{N-m-1}_\ars\ar[dl]
\\&\ars&}
\]
We will again make us of $\tilde\fa^{rs}$ of
(<ref>) to relate the two families
$\widetilde{X}_m|_\ars$ and $\tilde{\mY}_m$. The Weyl group $ W$ acts naturally on $\tX_m|_\ars$ and $\tilde\mY_m$,
making the projections to $\ars$ equivariant with respect to these $W$-actions. We let $I_N$ act on $\tilde\mY_m$ via the map
κ:I_N≅Z_K(a)↪(/2)^N+1 I_N+1,
where the first arrow is given by $(\zeta_1,\ldots,\zeta_N)\mapsto(\zeta_1,\ldots,\zeta_N,0)$.
We also let $W$ act on $I_{N+1}$ by permuting the first $N$
coordinates and we use this convention to form the semi-direct product $I_{N+1}\rtimes W$.
There is an
$I_{N+1}\rtimes W$-equivaraint
where ${W}$ (resp. $I_{N+1}$) acts on $\tilde\mY_m^t$ by the diagonal action (resp.
on the first factor).
Let us consider the following
twist of $\tX_m|_\ars$:
\[\tX_m'|_\ars\ra\ars\]
whose fiber over $a=(a_1,\ldots,a_N)$ is the complete intersection of quadrics
given by
\[\frac{a_1^i}{d_1}v_1^2+\cdot\cdot+\frac{a_{N}^i}{d_N}v_{N}^2=0,\ \ \ i=0,...,m-1,
\ \
\frac{a_1^m}{d_1}v_1^2+\cdot\cdot+\frac{a_{N}^m}{d_N}v_{N}^2-\epsilon^2=0
\]
where $d_i$ is defined as before, i.e., $d_i=\prod_{j\neq i}(a_j-a_i)$. Similar to the case of $X_m$, we have a canonical isomorphism
Here the $I_N$-action on $\tX_m'|_\ars$ is defined as the composition of
$I_N\ra I_{N+1}$ in (<ref>) with
the natural action of $I_{N+1}$ on $\tX_m$. The Weyl group $ W$ acts naturally on $\tX_m$, $\tX_m'$ and $\tilde\mY_m$,
making the projections to $\ars$ equivariant maps under the $W$-actions. Thus we obtain
$I_{N+1}\rtimes W$-actions
on $\tX_m$, $\tX_m'$ and $\tilde\mY_m$ and the projections to
$\ars$ are $I_{N+1}\rtimes W$-equivariant. Moreover, the isomorphism in (<ref>) is $I_{N+1}\rtimes W$-equivariant.
In <ref> (see Proposition <ref>), we show that there is an $I_{N+1}\rtimes W$-equivariant isomorphism $\tX_{m}'|_{\ars}\is\tilde\mY_{m}$. Combining this with (<ref>) we obtain (<ref>).
§.§ The families $\tX_m'$ and $\tilde\mY_m$
In this subsection we state and prove the following proposition which was used in the previous subsections.
We have an $I_{N+1}\rtimes W$-equivariant isomorphism $\tX_{m}'|_{\ars}\is\tilde\mY_{m}$.
In particular, it induces an $I_N\rtimes W$-equivariant isomorphism on the
\[X_m'|_\ars\is\tX_m'|_\ars/(\bZ/2\bZ)\is\tilde\mY_m/(\bZ/2\bZ)\is\mY_m.\]
Here $\bZ/2\bZ$ acts on $\tX_m'$ and $\tilde\mY_m$ via the map
$\bZ/2\bZ\ra I_{N+1}$ given by $1\mapsto {(0,...,0,1)}$.
We follow closely the argument in <cit.>.
We begin by introducing some auxiliary spaces and maps.
Let $\tV_m'\subset \tV\times\ars$ be the vector sub-bundle whose fiber
over $a\in\ars$ is the subspace
$\tV_{m,a}'\subset \tV$
defined by the equations
a_1^i/d_1v_1+··+a_N^i/d_Nv_N=0, i=0,...,m-1,
The map $\tilde{s}:\mathbb P(\tV)\times\ars\ra\mathbb P(\tV)\times \ars$ (see (<ref>))
maps $\tX_m'|_\ars$ to $\mathbb P(\tV_m')$ and the resulting map
\[\tilde s':\tX_m'|_\ars\ra\mathbb P(\tV_m')\]
is an $I_{N+1}$-branched cover with branch locus $\{v_i=0,\ i=1,...,N+1\}$.
Let $\bC(\mathbb P(\tV_m'))$ be the function field of $\mathbb P(\tV_m')$.
Then the function field of $\tX_m'|_\ars$ is given by the following field extension
$$F:=\bC(\mathbb P(\tV_m'))((\frac{v_i}{v_{N+1}})^{1/2}_{i=1,...,N})\supset
\bC(\mathbb P(\tV_m')).$$
Since $\tX_m'|_\ars$ is smooth and $\tilde s'$ is
finite, it follows that
$\tX_m'|_\ars$ is the normalization of $\mathbb P(\tV_m')$
in $F$.[Recall for any irreducible variety $X$ and $K$ a finite extension of the
function field $\bC(X)$, there exists a unique normal variety $Y$ and a finite morphism
$f:Y\ra X$ such that the induced map $\bC(X)\ra\bC(Y)=K$ is the given field extension.
We call $Y$ the normalization of $X$ in $K$.]
The group $I_{N+1}$ acts on $F$ by
$\displaystyle{\zeta:\,v_i^{1/2}\mapsto(-1)^{\zeta_i}\,v_i^{1/2},\ \zeta=(\zeta_1,...,\zeta_{N+1})\in I_{N+1}}$ and
$W$ acts on $F$ by $\displaystyle{w:\ (\frac{v_i}{v_{N+1}})^{1/2}\mapsto(\frac{v_{w(i)}}{v_{N+1}})^{1/2}}$.
Similarly, let
$k_\eta$ be the function field of $\ars$ and let
$\eta=(a_1,...,a_N)\in\ars(k_\eta)$ be the corresponding generic point.
Then the function field of $\tilde\mY_m$ is given by the following field extension
$$F'=\bC(\mathbb P^{N-m-1}_\ars)((\frac{H_{\eta,i}}{H_{\eta,N+1}})^{1/2}_{i=1,...,N})\supset\bC(\mathbb P^{N-m-1}_\ars).$$
$H_{\eta,i},\, i=1,...,N$ are the hyperplanes associated to $\eta\in\ars$ in (<ref>), $H_{\eta,N+1}=x_{N-m}$, and
$\frac{H_{\eta,i}}{H_{\eta,N+1}}$ are rational functions on
$\mathbb P^{N-m-1}_\eta$, regraded as elements in
$\bC(\mathbb P^{N-m-1}_\eta)=\bC(\mathbb P^{N-m-1}_\ars)$.
Since $\tilde \mY_m$ is smooth and $\tilde\iota:\tilde\mY_m\ra\mathbb P_\ars^{N-m-1}$ is finite, it follows that
$\tilde \mY_m$ is the normalization of $
\mathbb P^{N-m-1}_\ars$ in $F'$.
The group $I_{N+1}$ acts on $F'$ by
$\displaystyle{\zeta:\,H_{\eta,i}^{1/2}\mapsto(-1)^{\zeta_i}\,H_{\eta,i}^{1/2},\ \zeta=(\zeta_1,...,\zeta_{N+1})\in I_{N+1}}$ and
$ W$ acts on $F$ by $\displaystyle{w:\,(\frac{H_{\eta,i}}{H_{\eta,N+1}})^{1/2}\mapsto(\frac{H_{\eta,w(i)}}{H_{\eta,N+1}})^{1/2}}$.
By the discussion above, to prove Proposition <ref>,
it is enough to prove the following statement.
The two $\ars$-families of configurations
(ℙ('_m,a),{v_i}_i=1,...N+1) and
,N+1) are
That is,
there is an isomorphism (or trivialization) of
vector bundles $\phi:\bC^{N-m}\times\ars\is \tV_m'$ over $\ars$
such that for any $k$-point $a\in\ars(k)$, $k$ a field,
the induced map on the dual fibers $\phi^*_a:(\tV_{m,a}')^*\is k^{N-m}$
satisfies $\phi^*_a(v_i)=H_{a,i}$ for $i=1,...,N+1$.
To prove (<ref>), we need to construct,
for each $S$-point $a\in\ars(S)$,
a functorial isomorphism
$\phi_a:\bC^{N-m}\times S\is \tV_{m,a}'$ satisfying the desired property.
For notational simplicity we construct such isomorphisms on the level of $k$-points. The argument for general $S$-points is the same.
Consider the following map
\[\psi_a:\tV\otimes k\stackrel{pr}\ra V\otimes k\stackrel{\times d}\is V\otimes k\]
where $pr:\tV\otimes k=(V\otimes k)\oplus k\ra V\otimes k$
is the projection map and the second isomorphism is given by multiplying
the diagonal matrix $d=\on{diag}(d_1^{-1},...,d_N^{-1})\in GL(V\otimes k)$
(recall for $a=(a_1,...,a_N)\in\ars$, $d_i=\prod_{j\neq i} (a_j-a_i)$). One can check that $\psi_a$ maps $\tV_{m,a}'$ isomorphically onto $V_{m,a}$
(see (<ref>) and (<ref>) for the definitions of $\tV_{m,a}'$ and $V_{m,a}$, respectively) and the resulting isomorphism
$\psi_a:\tV_{m,a}'\is V_{m,a}$
$$\text{$\psi_a^*(d_i\cdot v_i)=v_i$ for $i=1,...,N$ and
where $H_\infty:=a_1^mv_1+\cdot\cdot\cdot+a_N^mv_N$. Thus we are reduced to show that
$$\text{$(\bbP(V_{m,a}), \{d_i\cdot v_i\}_{i=1,...,N}\cup H_\infty)$ and
$(\mathbb P(k^{N-m}), \{H_{a,i}\}_{i=1,...N+1})$ are equivalent,}$$
that is,
there is an isomorphism
\[\gamma_a:k^{N-m}\is V_{m,a}\]
such that $\gamma_a^*(d_i\cdot v_i)=H_{a,i}$, $i=1,\ldots,N$ and $\gamma_a^*(H_\infty)=H_{a,N+1}$.
Consider the basis $u_i=(a_1^{i},...,a_N^{i})$, $i=0,...,N-1$
of $V\otimes k$. Then the isomorphism $V\otimes k\is (V\otimes k)^*$, given by the paring
$\langle(v_i),(w_i)\rangle=\sum v_iw_i$, induces the following isomorphism
\[f_1:V_{m,a}^*\is V\otimes k/k\langle u_0,...,u_{m-1}\rangle\is k\langle u_{m},...,u_{N-1}\rangle.\]
Let $s_{i}$ be the elementary symmetric polynomial in $a_1,...,a_N$
of degree $i$ and let
$A=(a_{ij})\in GL_{N-m}(k)$ be the matrix with
entries $a_{ij}=(-1)^{i-1}s_{j-i}$ if $j\geq i$ and $a_{ij}=0$ otherwise.
Consider the following isomorphism
\[f:V_{m,a}^*\stackrel{f_1}\is k\langle u_{m},...,u_{N-1}\rangle\stackrel{f_2}\is k^{N-m}\stackrel{f_3}\is k^{N-m},\]
here $f_2:k\langle u_{m},...,u_{N-1}\rangle\is k^{N-m}$ is
the isomorphism given by $u_{N-i}\mapsto(-1)^{i-1}x_{i}$[Here
we regard $x_i$ as the $i$-th coordinate vector of $k^{N-m}$.] and
$f_3$ is the isomorphism given by right multiplication by $A^{-1}$.
We claim that the dual
$$\gamma_a:=f^*:k^{N-m}\is V_{m,a}$$
is the desired isomorphism, i.e., we have
$f(d_i\cdot v_i)=H_{a,i}$ for $i=1,...,N$, and $f(H_\infty)=H_{a,N+1}$.
Note that the configuration $(\bbP(V_{m,a}), \{d_i\cdot v_i\}_{i=1,...,N})$
(resp. $(\mathbb P(k^{N-m}), \{H_{a,i}\}_{i=1,...N})$)
is equal to the configuration $(P, H_1,...,H_N)$ (resp. $(P',H_1',...,H_N')$)
in <cit.>. Moreover, the map $f$ is the one used in <cit.> to show that
the above configurations are equivalent. Thus
according to <cit.> we have
\[f(d_i\cdot v_i)=H_{a,i}\ \ \text{for}\ \ i=1,...,N.\]
So it remains to show that $f(H_\infty)=H_{a,N+1}$.
For this we
observe that $f_1(H_\infty)=u_m$. Hence
This proves (<ref>).
The proof of Proposition <ref> is complete.
§ MONODROMY OF FAMILIES OF HESSENBERG VARIETIES
In this section we study the monodromy
representation of $\pi_1^K(\grs,a)=Z_K(a)\rtimes B_N$ on the primitive cohomology of
complete intersections of quadrics $X_m$ (and on the primitive cohomology of their branched double covers $\tX_m$). By Theorem <ref> this gives us a
complete description of the monodormy representations of $\pi_1^K(\Lg_1^{rs})$ associated with the families of Hessenberg varieties $\Hess_n^{O,\p}$ and
To state the result, let us recall, from <ref>, the monodromy representations $\rho_{C_\chi}:P_{N}\ra Sp(H^1(C_{a,\chi},\bC))\is Sp(2i-2)$ and $\rho_{\tilde C_\chi}:P_{N}\ra Sp(H^1(\tilde C_{a,\chi},\bC))\is Sp(2i-2)$ where $i=\frac {|\chi|} 2$. Recall further that, by (<ref>), these representations are irreducible with Zariski dense image. Let us consider the irreducible representation of $Sp(2i-2)$ associated to the fundamental weight $\omega_j$. Composing $\rho_{C_\chi}$ and $\rho_{\tilde C_\chi}$ with this fundamental representation we obtain irreducible representations $\on{P}^j_\chi$ and $\tilde{\on{P}}^j_\chi$ of the pure braid group $P_N$.
For a character $\chi$ of an abelian group we write $V_\chi$ for the corresponding one dimensional representation. Recall that the group $Z_K(a)$ can be naturally identified with $I_N$ as explained in (<ref>). We also relate the characters of $I_{N}$ and $I_{N+1}$ using the map $\kappa$ defined in (<ref>).
From these considerations we conclude that
Z_K(a)^∨=I_N^∨ and we have a map κ̌: I_N+1^∨I_N^∨ .
In particular, characters of $I_N$ and $I_{N+1}$ can be regarded as characters of $Z_K(a)$.
To state the main theorems of this section we define two $Z_K(a)\rtimes P_N$-representations as follows:
\[E_{ij}^{N}\is
\bigoplus_{\chi\in I_{N}^\vee,\,|\chi|=2i}\on{P}^j_\chi\otimes V_\chi\qquad\text{and}\qquad \tE_{ij}^N=
\bigoplus_{\substack{\chi\in I_{N+1}^\vee,\,|\chi|=2i,\\N+1\in\on{supp}\chi}}\tilde{\on{P}}^j_\chi\otimes V_\chi,\]
where the $I_N$ acts on $\tE_{ij}^N$ via the map $\check{\kappa}:I_{N+1}^\vee \ra I_N^\vee$ of (<ref>), and $P_N$ acts on $V_\chi$ via the map $\rho:P_N\to I_N$ of (<ref>).
Lemmas <ref> and <ref> show that the $Z_K(a)\rtimes P_N$ actions on $E_{ij}^{N}$ and $\tE_{ij}^N$ extend naturally to $Z_K(a)\rtimes B_N$-actions.
The main results of this section are the following.
For $1\leq m\leq N-1$, the monodromy representation of $\pi_1^K(\grs,a)$ on $P(X_m):=H^{N-m-1}_{prim}(X_{m,a},\bC)$
decomposes into irreducible representations in the following manner:
\[P(X_m)\is
\bigoplus_{i}^{}\bigoplus_{\substack{j\equiv N-m-1\on{mod}2\\ j\in [0,l]}}E_{ij}^N,\]
with $N-m+1\leq 2i\leq N$,
To state the second main result, we set
Recall that there is an involution action $\sigma$ on $\tilde X_{m}$
and the projection map $p_m:\tilde X_m\ra X_m$ is a branched double cover
with Galois group $\langle\sigma\rangle\is\bZ/2\bZ$
(see <ref>). Then $P(\tX_m)= P(\tX_m)^{\sigma=id}\oplus P(\tX_m)^{\sigma=-id}$ and we have
$P(\tX_m)^{\sigma=id}=P(X_m)$. The next theorem describes $P(\tX_m)^{\sigma=-id}$.
For $1\leq m\leq N-1$,
the monodromy representation
of $\pi_1^K(\grs,a)$ on
$P(\tX_m)^{\sigma=-id}$ decomposes into irreducible representations in the following manner:
\[P(\tX_m)^{\sigma=-id}\is
\bigoplus_{i}^{}\bigoplus_{\substack{j\equiv N-m-1\on{mod}2\\ j\in[0,l]}}^{}\tE_{ij}^N,\]
with $N-m+1\leq 2i\leq N+1$,
§.§ Proof of Theorem <ref>
Let us start with the following proposition which is a consequence of Proposition <ref>.
There is an isomorphism
of representations of $\pi_1^{K}(\grs,a)\is
I_N\rtimes B_N$
The group $I_N$ acts on the summand $H^i(\mY_{m,a},\bC)_\chi\otimes V_\chi$
via the character $\chi\in I_N^\vee$.
Observe that the families
$\mY_m\ra\ars$, and $\tilde\fa^{rs}\ra\ars$ are all
$W$-equivariant. Hence
their cohomology groups $H^i(X_{m,a},\bC)$, $H^i(\mY_{m,a},\bC)$, $H^0((\tilde\fa^{rs})_a,\bC)$
carry an action of the braid group
$B_N\is\pi_1^{W}(\ars,a)$. Let
\[
H^i(X_{m,a},\bC)=\bigoplus_{\chi\in I_N^\vee}H^i(X_{m,a},\bC)_\chi\ \ \quad\quad
H^i(\mY_{m,a},\bC)=\bigoplus_{\chi\in I_N^\vee} H^i(\mY_{m,a},\bC)_\chi\]
\[
H^0((\tilde\fa^{rs})_a,\bC)=\bigoplus_{\chi\in I_N^\vee} V_\chi
\]
be the
decompositions with respect to the action of $I_N$; for the last identity we recall that $\tilde\fa^{rs}\to \fa^{rs}$ is an $I_N$-torsor.
For $\chi\in I_N^\vee$ and $b\in B_N$, we write $b\cdot\chi$ for the action of $b$ on $\chi$.
Then the braid group action on $H^i(\mY_{m,a},\bC)$ is described as follows
$$b\in B_N:\,H^i(\mY_{m,a},\bC)_\chi\mapsto H^i(\mY_{m,a},\bC)_{b\cdot\chi}.$$
The $B_N$-actions on $H^i(X_{m,a},\bC)$ and $H^0((\tilde\fa^{rs})_a,\bC)$ are described in the same manner.
By the Künneth formula,
the cohomology of
the fiber of $\mY_m^t=(\mY_m\times_\ars\tilde\fa^{rs})/I_N$ over $a\in\ars$ is canonically isomorphic to
\[H^i(\mY^t_{m,a},\bC)\is\bigoplus_{\chi\in I_N^\vee}H^i(\mY_{m,a},\bC)_\chi\otimes V_\chi.\]
Thus by (<ref>) we obtain the desired $\pi_1^{K}(\grs,a)\is I_N\rtimes B_N$-equivariant isomorphism.
The isomorphism in Proposition <ref> implies
the following isomorphism of monodromy representations
P(X_m)⊕_χ∈I_N^∨ P(_m)_χ⊗V_χ,
$$P(\mY_m):=H^{N-m-1}_{prim}(\mY_{m,a},\bC),\ \ P(\mY_m)_\chi:=H^{N-m-1}(\mY_m,\bC)_\chi\cap P(\mY_m).$$
Our goal is to decompose the representation above
into irreducible representations.
Observe that each summand $P(\mY_m)_\chi$ is invariant under the
action of the pure braid group $P_N$. According to <cit.>,
there is an isomorphism of representations of $P_N$
where $P_N$ acts on $H^1(C_{a,\chi},\bC)$ via the map $\rho_{C_\chi}:P_N\ra Sp(2i-2)$, $i=|\chi|/2$.
As an $Sp(2i-2)$-representation $\wedge^{N-m-1}H^1(C_{a,\chi},\bC)$ decomposes into a direct sum of fundamental representations in a well-known manner. This implies the following decomposition of $P(\mY_m)_\chi$ into irreducible representations of $P_N$:
P(_m)_χ∧^N-m-1H^1(C_a,χ,)=⊕_j≡N-m-1mod2, j∈[0,l]P^j_χwhere $l=\min\{N-m-1,-N+m+|\chi|-1\}$.
Combining (<ref>) with (<ref>), we obtain the following decomposition
P(X_m)⊕_χ∈I_N^∨ P(_m)_χ⊗V_χ⊕_χ∈I_N^∨⊕_jP^j_χ⊗V_χ.
Using the notation from the beginning of this section the decomposition (<ref>) can be rewritten as
P(X_m)⊕_i^⊕_j≡N-m-1mod2, j∈[0,l] E_ij^N ,
where $N-m+1\leq 2i\leq N$ and
We have the following
* Each $E_{ij}^{N}$ is an irreducible
representation of $\pi^{K}_1(\fg_1^{rs},a)$. We denote by $\rho^N_{ij}:
\pi^{K}_1(\fg_1^{rs},a)\ra GL(E_{ij}^N)$ the corresponding map.
* Suppose $j>0$.
Let $H:=\overline{\rho^N_{ij}(P_N)}\subset GL(E_{ij}^N)$ be the
Zariski closure of $\rho^N_{ij}(P_N)$ in $GL(E_{ij}^N)$ (recall $P_N\subset\pi^{K}_1(\fg_1^{rs},a)$ is the pure braid group).
Then we have $\Lie\, H\is\mathfrak{sp}(2i-2)$.
In particular, the image $\rho^N_{ij}(\pi^{K}_1(\fg_1^{rs},a))$ is infinite.
We begin with the proof of (1).
We first show that $E_{ij}^N$ is a $\pi^{K}_1(\fg_1^{rs},a)$-invariant subspace of $P(X_m)$.
For this, we observe that
the decomposition in (<ref>) is compatible with the action of
$B_N$, that is, $\text{ for }b\in B_N$,
\otimes V_\chi\mapsto\on{P}^j_{b\cdot \chi}
\otimes V_{b\cdot\chi}.$$
the braid
group $B_N$ acts transitively on the set
$\{\chi\in I_{N}^\vee\,|\,|\chi|=2i\}$, it follows that the subspace
$$E_{ij}^N=\bigoplus_{\chi\in I_{N}^\vee,|\chi|=2i}\on{P}^j_\chi
\otimes V_\chi$$
is stable under the action of $\pi^{K}_1(\fg_1^{rs},a)$. Now since
each summand
\otimes V_\chi$ is irreducible as a representation of
$P_N$, it follows that each $E_{ij}^N$ is an
representation of $\pi^{K}_1(\fg_1^{rs},a)$.
We prove (2).
For each $\chi\in I_N^\vee$, we define $\rho_\chi:P_N\xrightarrow{\rho} I_N\xrightarrow{\chi}\mu_2$. Here $\rho$ is the map in (<ref>).
$$\psi_1:=(\rho_{C_\chi},\bigoplus_{\chi, |\chi|=2i}\rho_\chi)
:P_N\ra Sp(2i-2)\times\mu_2^{\binom {N}{2i}}.$$
Let $V_{ij}$ denote the irreducible representation of $Sp(2i-2)$ associated to the fundamental weight $\omega_j$.
Then the restriction of $\rho_{ij}^N:\pi^{K}_1(\fg_1^{rs},a)\ra GL(E_{ij}^N)$ to $P_N$ can be identified with
\[\psi:P_N\xrightarrow{\psi_1} Sp(2i-2)\times\mu_2^{\binom {N}{2i}}\xrightarrow{\psi_2}GL(V_{ij})^{\times{N\choose 2i}}\]
where $\psi_2$ maps $Sp(2i-2)$ diagonally into
$GL(V_{ij})^{\times{N\choose 2i}}$ and $\psi_2$ maps $\mu_2=\{\pm1\}$ to $\pm id\in GL(V_{ij})$.
Since $\overline{\rho_{C_\chi}(P_N)}=Sp(2i-2)$, it implies that the
connected component
So to prove (2), it suffices to show that $\Lie(\on{Im}(\psi_2))\is\mathfrak sp(2i-2)$ for $j>0$. This follows
from the fact that the induced map $d\psi_2:\mathfrak sp(2i-2)\ra\bigoplus\mathfrak{gl}(V_{ij})$ on the Lie algebras
is injective.
It follows from the lemma above that
(<ref>) is the decomposition of the monodromy representation
$P(X_m)$ into irreducible subrepresentations. This completes the proof of Theorem
§.§ Proof of Theorem <ref>
The proof is similar to the case of $X_m$. First using the
isomorphism (<ref>) and the same argument as in the case of $X_m$,
we obtain the following proppsition.
There is an isomorphism of $I_{N+1}\rtimes B_N$-representations
where for $V_\chi$, we regard $\chi$ as an element in $I_N^\vee$
via the map
$\check{\kappa}:I_{N+1}^\vee\ra I_N^\vee$ in (<ref>), and the group
$I_{N+1}$ acts on the summand $H^i(\tilde\mY_{m,a},\bC)_\chi\otimes V_\chi$
via the character $\chi\in I_{N+1}^\vee$.
By Proposition <ref>, there is an isomorphism
of $I_{N+1}\rtimes B_N$-representations
P(_m)⊕_χ∈I_N+1^∨ P(_m)_χ⊗V_χ.
For any $\chi\in I_{N+1}^\vee$
let $\tC_{a,\chi}$ be the
hyperelliptic curve defined
in <ref> and let
$\rho_{\tC_\chi}:P_{N}\ra Sp(H^1(\tC_{a,\chi},\bC))\is Sp(2i-2)$ denote
the monodromy representation for the family
Again by <cit.>, we have
an isomorphism of $P_N$-representations
Combining (<ref>) with (<ref>) we obtain the following
\begin{equation}\label{isom tX_m}
P(\tX_m)\is\bigoplus_{\chi\in I_{N+1}^\vee}\wedge^{N-m-1}H^1(\tC_{a,\chi},\bC))\otimes V_\chi.
\end{equation}
We describe the monodromy representation
$ P(\tX_m)^{\sigma=-id}$ (recall that $\sigma$ is the involution on $\tilde X_{m}$).
For this, we first
observe that the involution action of $\langle\sigma\rangle\is\bZ/2\bZ$ on $\tX_m$ is
equal to the composition of
\[i_\infty:\bZ/2\bZ\ra I_{N+1},\ 1\mapsto {(0,...,0,1)},\]
with the action of $I_{N+1}$ on $\tX_m$. Hence
by (<ref>)
we have
N+1∈suppχ P(_m)_χ⊕_χ∈I_N+1^∨
Again, since $\rho_{\tC_\chi}(P_N)$ is Zariski dense in $Sp(H^1(\tC_\chi,\bC))$, we have the following decomposition
$$\wedge^{N-m-1}H^1(\tC_{a,\chi},\bC)=\bigoplus_{ j\equiv N-m-1\on{mod}2,\ j\in[0,l]}\tilde{\on{P}}^j_\chi,$$
where $l=\min\{N-m-1,-N+m+|\chi|-1\}$.
Using the notation from the beginning of this section
the decomposition (<ref>) can be rewritten as
\[P(\tX_m)^{\sigma=-id}\is
\bigoplus_{i}^{}\bigoplus_{\substack{j\equiv N-m-1\on{mod}2\\ j\in[0,l]}}^{}\tE_{ij}^N,\]
where $N-m+1\leq 2i\leq N+1$,
The same argument as in the proof of Lemma <ref> shows the following
* $\tE_{ij}^N$ is an irreducible representation of $\pi_1^K(\grs,a)$.
We denote by $\tilde\rho^N_{ij}:\pi_1^K(\grs,a)\ra GL(E_{ij}^N)$
the corresponding map.
* Suppose $j>0$.
Let $H:=\overline{\tilde\rho^N_{ij}(P_N)}\subset GL(\tE_{ij}^N)$ be the
Zariski closure of $\tilde\rho^N_{ij}(P_N)$ in $GL(\tE_{ij}^N)$.
Then we have
$\Lie\, H\is\mathfrak{sp}(2i-2)$.
In particular, the image $\tilde\rho^N_{ij}(\pi^{K}_1(\fg_1^{rs},a))$ is infinite.
This completes the proof of Theorem <ref>.
§.§ The local systems $E_{ij}^N$ and $\tE_{ij}^N$
In this subsection, we show that from the constructions in previous sections, we have obtained the following set consisting of pairwise non-isomorphic irreducible $K$-equivariant local systems on $\Lg^{rs}_1$
{E_ij^2n+1, i∈[1,n], j∈[0,i-1]; ^2n+1_ij, i∈[1,n+1], j∈[1,i-1], ^2n+1_n+1,0≅}.For this, we first observe that
$$\text{$E_{ij}^{N}\is E_{i'j'}^{N}$ and $\tE_{ij}^N\is\tE_{i'j'}^N$ if and only if $i=i',j=j'$.}$$
In fact, assume that $E_{ij}^N\is E_{i'j'}^N$. Then we must have $i=i'$, otherwise,
the centralizer $Z_{K}(a)\cong I_N$ would act differently on $E_{ij}^N$ and $E^N_{i'j'}$. Now regarding $E_{ij}^N$ and $E^N_{ij'}$ as $P_N$-representations we see that $j=j'$. Similar argument applies to $\tE_{ij}^N$.
It remains to prove the following.
We have $E_{i,j}^{N}\cong\tE_{i',j'}^{N}$ if and only if $i+i'=(N+1)/2$ and $j=j'=0$.
$$E_{ij}^{N}=\bigoplus_{\chi\in I_{N}^\vee,|\chi|=2i}\on{P}^j_\chi
\otimes V_\chi,\,\, \tE_{i'j'}^{N}=\bigoplus_{\substack{\chi'\in I_{N+1}^\vee,\, |\chi|=2i',\\ N+1\in\on{supp}\chi}}\tilde{\on{P}}^{j'}_{\chi'}\otimes V_{\chi'}.$$
For $V_{\chi'}$ we regard $\chi'$ as an element in $I_N^\vee$ via the map
$\check{\kappa}:I_{N+1}^\vee\ra I_{N}^\vee$ in (<ref>).
Observe that for $\chi'\in I_{N+1}^\vee$ and $N+1\in\on{supp}\chi'$, we have
κ̌(χ')=χ if and only if suppχ={1,…,N+1}\suppχ'.
Thus the map $\check{\kappa}$ maps the subset $\{\chi'\in I_{N+1}^\vee,\, |\chi'|=2i',\ N+1\in\on{supp}\chi'\}$ bijectively to the subset $\{\chi\in I_{N}^\vee,|\chi|=2i\}$ where $2i+2i'=N+1$.
Hence we have
⊕_χ∈I_N^∨,|χ|=2iV_χ≅⊕_χ∈I_N+1^∨, |χ'|=2i',
N+1∈suppχV_χ' if and only if $i+i'=(N+1)/2$.
This implies $E_{i,0}^{N}\cong\tE_{i',0}^{N}$ for $i+i'=(N+1)/2$.
Conversely, we observe that
$E_{i,j}^{N}\cong\tE_{i',j'}^{N}$ implies
$\on{P}_\chi^j\otimes V_\chi\cong\tilde{\on{P}}_{\chi'}^{j'}\otimes V_{\chi'}$ (as representations of $I_N\rtimes P_N$)
for some $\chi\in I_{N}^\vee$ and $\chi'\in I_{N+1}^\vee$ with $N+1\in\on{supp}\chi'$.
This implies that $\check\kappa(\chi')=\chi$ and
it follows from (<ref>) that
Therefore the monodromy representation of the restriction of
$C_\chi\ra\ars$ (resp. $\tC_{\chi'}\ra\ars$) to the subvariety $\ars(a,\chi')$ (resp. $\ars(a,\chi)$) in (<ref>)
is trivial. On the other hand,
the monodromy representation of the restriction of $\tC_{\chi'}\ra\ars$ (resp. $C_\chi\ra\ars$) to
$\ars(a,\chi')$ (resp. $\ars(a,\chi)$) has Zariski dense image (see (<ref>)).
This forces $j=j'=0$ and the desired claim follows again from (<ref>).
§.§ The local systems $E^{2n+1}_{i0}$, $\tE_{n+1,j}^{2n+1}$ and the $\calL_i$'s, $\cF_i$'s in <cit.>
Recall that in <cit.>, we have defined the local systems $\calL_i$ and $\cF_i$ on $\Lg_1^{rs}$. We have the following.
We have
E_i,0^2n+1≅_2i if 1≤2i≤n,
E^2n+1_i,0≅_2n-2i+1 if n+1≤2i≤2n,
_n+1,j^2n+1≅_j for 1≤j≤n.
We begin with the proof of (<ref>). Recall from loc. cit. that we have
(π̌_2^n1)_*|__1^rs≅⊕_i=0^n_i and _i=2n+1i
π̌_2^n1:K×^P_K[_P,_P]_1^:={(x,0⊂V_n⊂V_n^⊂^2n+1) | x∈_1, xV_n⊂V_n^}→_1.
On the other hand, recall the $I_N$-torsor over $\fa^{rs}$ in <ref>:
\[\tilde{\pi}:\tilde\fa^{rs}=\{(a,c)\,|\,a=(a_1,...,a_{N})\in\fa^{rs},\,c=(c_1,...,c_N),\, c^2_i=
\prod_{j\neq i}(a_j-a_i)\}/(\bZ/2\bZ)\to\fa^{rs}. \]
We have
π̃_*|__1^rs≅⊕⊕_i=1^nE^2n+1_i,0 and E^2n+1_i,0=2n+12i.
We show that there is a $I_N\rtimes W$-equivariant isomorphism
Then (<ref>) follows from (<ref>), (<ref>), and dimension considerations of
the representations.
Using the identities
$\sum_{i=1}^{2n+1} a_i^k\,c_i^{-2}=0,\ k=0,...,2n-1,$
it is easy to check that the map
\[\tilde\fa^{rs}\ra X_{2n}|_{\fa^{rs}},\ (a,c)\in\tilde\fa^{rs}\mapsto [c_1^{-1},...,c_{2n+1}^{-1}]\]
defines a $I_N\rtimes W$-equivaraint isomorphism
\[\tilde\fa^{rs}\cong X_{2n}|_{\fa^{rs}}.\]
On the other hand, by the description of the
Hessenberg varieties $\on{Hess}_{n}^{E,\bot}$ in <ref>,
we have a natural map
\[\on{Hess}_{n}^{E,\bot}\to K\times^{P_K}[\fn_P,\fn_P]_1^\bot,\ (x, V_{n-1}\subset V_n)\mapsto(x,V_n). \]
It is easy to check that the map above is a $K$-equivariant isomorphism over $\Lg_1^{rs}$.
The desired isomorphism (<ref>) follows from the following compositions of isomorphisms
\[\tilde\fa^{rs}\cong X_{2n}|_{\fa^{rs}}\stackrel{\on{Thm} \ref{H=X}}\cong\on{Hess}_{n}^{E,\bot}|_{\fa^{rs}}\cong K\times^{P_K}[\fn_P,\fn_P]^\bot|_{\fa^{rs}}.\]
This completes the proof of (<ref>).
To prove (<ref>), we observe that
^2n+1_n+1,j≅P̃^j_χ_0⊗V_χ_0≅(∧^j H^1(_a,χ_0,))_prim⊗V_χ_0,
where $\chi_0\in I_{N+1}^\vee$ is the unique character such that
$|\chi_0|=2n+2$, and $\tC_{a,\chi_0}$ is
hyperelliptic curve of genus $n$ with affine equation $y^2=\prod_{i=1}^{2n+1}(x-a_i)$.
By (<ref>),
$\chi_0$, when regarded as an element in $I_N^\vee$ (see (<ref>)), is trivial. Hence
$I_N$ acts trivially on $\tE_{n+1,j}^{2n+1}$ and $V_{\chi_0}$, i.e.
^2n+1_n+1,j≅(∧^j H^1(_a,χ_0,))_prim≅_j
where the last isomorphism follows from the discussion above and the definition of $\cF_j$'s in loc.cit.
In <cit.> we used the fact that among the $\IC(\Lg_1,\calL_i)$'s $(i\geq 1)$, only $\IC(\Lg_1,\calL_{2j-1})$, $1\leq j\leq m$, appear in the decomposition of $(\check\tau_m)_*\bC[-]$, where $\check\tau_m=\check\tau_m^{2n+1}$ and $2m\leq n+1$. To prove this fact, it suffices to show that in the decomposition of the monodromy representation $P(X_{2m})$, only the above mentioned local systems appear. Applying Theorem <ref> to $P(X_{2m}) $ with $N=2n+1$ we see that among the $E_{i0}$'s only those with $n-m+1\leq i\leq n$ appear. The desired conclusion follows from (<ref>) and the fact that $2m\leq n+1$.
§ COMPUTATION OF THE FOURIER TRANSFORMS
Let $\fF:D_K(\Lg_1)\to D_K(\Lg_1)$ denote the Fourier transform, where we identify $\Lg_1$ and $\Lg_1^*$ via a $K$-invariant non-degenerate bilinear form on $\Lg_1$. The Fourier transform $\fF$ induces an equivalence of categories $\fF:\on{Perv}_K(\Lg_1)\to\on{Perv}_K(\Lg_1).$
In this section we
study the
Fourier transforms of
$\IC(\fg_1,E_{ij}^N)$ and $\IC(\fg_1,\tE_{ij}^N)$.
We show that they are supported on
$\mathcal N_1$, more precisely, on $\mN_1^3\subset\mN_1$,
the closed subvariety consisting of nilpotent elements of
order less than or equal to $3$. Thus we obtain many more examples of IC complexes supported on nilpotent orbits whose Fourier transforms have both full support and
infinite monodromy (see also <cit.>). As an interesting corollary (see Example <ref>), we show that
Fourier transform of the $\IC$ extension of the unique non-trivial irreducible $K$-equivariant local system on
the minimal nilpotent orbit has full support and its monodromy is given by
universal family of hyperelliptic curves.
The main result of this section is the following theorem.
Let $\mN_1^3\subset\mN_1$ be
the closed subvariety consisting of nilpotent elements of
order less than or equal to $3$. Then
$\mathfrak F(\IC(\fg_1,E_{ij}^{N}))$ and $\mathfrak F(\IC(\fg_1,\tE_{ij}^N))$
are supported on $\mN^3_1$.
We first argue the case $\mathfrak F(\IC(\fg_1,E_{ij}^{N}))$.
For $m\leq \frac{N-1}{2}$,
consider the families of Hessenberg varieties
$$\sigma_m^N:\on{Hess}_m^O\ra\fg_1,\ \tau_m^N:\on{Hess}_m^E\ra\fg_1$$
$$\check\sigma_m^N:\on{Hess}_m^{O,\p}\ra\fg_1,\ \check\tau_m^{N}:\on{Hess}_m^{E,\p}\ra\fg_1$$
defined in <ref>. We have (see (<ref>))
((σ̌_m^N)_*[-])=(σ_m^N)_*[-], ((τ̌_m^N)_*[-])=(τ_m^N)_*[-].
By Theorem <ref>, over $\fg_1^{rs}$,
we have $\on{Hess}_m^{O,\p}\is X_{2m-1}$, $\on{Hess}_m^E\is X_{2m}$. Hence
the decomposition theorem implies that
is a direct summand of (σ̌_m^N)_*[-](_1,P(X_2m)))
is a direct summand of (τ̌_m^N)_*[-] .
Therefore the Fourier transforms $\mathfrak F(\IC(\fg_1,P(X_{2m-1})))$
and $\mathfrak F(\IC(\fg_1,P(X_{2m})))$
appear as direct summands of $(\sigma_m^N)_*\bC[-]$ and $(\tau_m^N)_*\bC[-]$.
Now in view of (<ref>) and (<ref>), we see that
$\mathfrak F(\IC(\fg_1,P(X_{2m-1})))$
and $\mathfrak F(\IC(\fg_1,P(X_{2m})))$ are supported on
$\mN_1^3$. Since each local system $E^N_{ij}$ appears in $
P(X_m)$ for some $m$ (see Theorem <ref>), we conclude that $\mathfrak F(\IC(\fg_1,E_{ij}^{N}))$ is supported on $\mN_1^3$.
It remains to consider the case $\mathfrak F(\IC(\fg_1,\tE_{ij}^N))$.
Since each local system $\tE_{ij}^N$ appears in $P(\tX_m)^{\sigma=-id}$
for some $m$, we are reduced to proving the following proposition:
$\mathfrak F(\IC(\fg_1,P(\tX_{m})^{\sigma=-id}))$ is supported on
The proof of this proposition occupies the remainder of this section.
§.§ Proof of Proposition <ref> when $m$ is odd
Recall that in <ref> we have introduced the families of Hessenberg varieties
_k^O,:={(x,0⊂V_k-1⊂V_k⊂V_k^⊂V_k-1^⊂V=^N) | x∈_1, xV_k-1⊂V_k}
_k^E,={(x,0⊂V_k-1⊂V_k⊂V_k^⊂V_k-1^⊂V=^N) | x∈_1, xV_k-1⊂V_k, xV_k⊂V_k^}
and the natural projection maps
Our first goal is to show that $\IC(\fg_1,P(\tX_{2k-1}))$ appears as a direct summand in the
push forward of certain intersection cohomology complex on $\Hess_k^{O,\p}$ along $\check{\sigma}_k^N$.
Let $[\bG_a/\bG_m^{[2]}]$
be the stack quotient, where $\bG_m^{[2]}\cong\bG_m$ acts on $\bG_a$ via the
square map, i.e., for $t\in\bG_m$ and $x\in\bG_a$, $t:\,x\mapsto t^2x$.
We first introduce a map
$$\alpha:\Hess_k^{O,\p}\ra [\bG_a/\bG_m^{[2]}].$$
Recall that
such a map is equivalent to a pair $(\widetilde\Hess_k^{O,\p},\phi)$, where
$\widetilde\Hess_k^{O,\p}$ is a $\bG_m$-torsor over $\Hess_n^{O,\p}$ and $\phi:
\widetilde\Hess_k^{O,\p}\ra\bG_a$ is a
map such that
$\phi(t\cdot v)=t^2\phi(v)$ for $v\in\widetilde\Hess_k^{O,\p}$ and
To construct such a pair, we set
\[\widetilde\Hess_k^{O,\p}:=\{(x,V_{k-1}\subset V_k,l)\,|\,(x,V_{k-1}\subset V_k)\in \Hess_k^{O,\p},\ 0\neq l\in V_k/V_{k-1}\is\bC\},\]
where the action of $\bG_m$ on $\widetilde\Hess_k^{O,\p}$ is given by
$t\cdot(x,V_{k-1}\subset V_k,l)=(x,V_{k-1}\subset V_k,tl)$ for $t\in \bG_m$. Define
\[\phi:\widetilde\Hess_k^{O,\p}\ra\bG_a,\ (x,V_{k-1}\subset V_k,l)\mapsto\langle xl,l\rangle_Q.\]
Note that the above pairing is well-defined since $xV_{k-1}\subset V_k$ and $xV_{k}\subset V_{k-1}^\bot$. One checks easily that $\phi$ satisfies (<ref>). This finishes the construction of
$(\widetilde\Hess_k^{O,\p},\phi)$, hence that of the map $\alpha:\Hess_k^{O,\p}\ra[\bG_a/\bG_m^{[2]}].$
By construction, the map $\alpha$
is $K$-equivaraint (where $K$ acts trivially on $[\bG_a/\bG_m^{[2]}]$), moreover it
factors through $\Hess_k^{E,\p}$, i.e.,
[_a/_m^[2]].There is a unique non-trivial irreducible local system $\mL$ on
$[\bG_m/\bG_m^{[2]}]\subset[\bG_a/\bG_m^{[2]}]$. We denote by $\IC([\bG_a/\bG_m^{[2]}],\mL)$ the corresponding
intersection cohomology complex on $[\bG_a/\bG_m^{[2]}]$.
\[\mathcal K:=(\check\sigma_k^N)_*\alpha^*\IC([\bG_m/\bG_m^{[2]}],\mL)\in D_K(\fg_1).\]
The factorization in (<ref>) and
the functorial properties of Fourier transform (see <cit.>)
imply the following
$\mathfrak F(\mathcal K)$ is supported on
Thus to show that $\fF(\IC(\fg_1,P(\tX_{2k-1})))$
is supported on $\mathcal N_1^3$,
it suffices to show that
the complex $\mathcal K$ contains $\IC(\fg_1,P(\tX_{2k-1})^{\sigma=-id})$ as a direct summand.
be the branched double cover of $X_{2k-1}$ and $\sigma$ the involution on $\tX_{2k-1}$ defined in <ref>.
We have that
$((\tilde\pi_{2k-1})_*\bC)^{\sigma=-id}$ contains $P(\tX_{2k-1})^{\sigma=-id}$
as a direct summand.
The statement (<ref>) follows from the following claim
\[\mathcal K|_{\grs}\is((\tilde\pi_{2k-1})_*\bC)^{\sigma=-id}.
\]
To prove the claim,
let $s:[\bG_a/\bG_m]\ra [\bG_a/\bG_m^{[2]}]$
be the descent of the map $\bG_a\ra\bG_a,\,t\mapsto t^2$. Then
from the definitions of $\tX_{2k-1}$ and the map $\alpha$, one can check that,
under the isomorphism $X_{2k-1}\is \on{Hess}_k^{O,\perp}|_{\Lg_1^{rs}}$ in Theorem <ref>,
the branched double cover $\tX_{2k-1}$ can be identified with the following
fiber product
_2k-1[d]^p_2k-1[r] [_a/_m][d]^s
X_2k-1Hess_k^O,⊥|_[r]^-α|_ [_a/_m^[2]].
\bC\oplus\IC([\bG_a/\bG_m^{[2]}],\mL),$$
by proper base change we have
\[(\alpha|_\grs)^*\IC([\bG_a/\bG_m^{[2]}],\mL)\is ((p_{2k-1})_*\bC)^{\sigma=-id}.\]
This implies that
\[\mathcal K|_{\grs}\is(\pi_{2k-1})_*(\alpha|_\grs)^*\IC([\bG_a/\bG_m^{[2]}],\mL)\is
This proves (<ref>).
The construction of the map $\alpha$ was inspired by discussions with Zhiwei Yun. In particular, the idea of making use of the local system $\calL$ on $[\bG_a/\bG_m^{[2]}]$ was explained to one of us by him.
§.§ Proof of Proposition <ref> when $m$ is even
Let us consider the following family of
Hessenberg varieties
\begin{eqnarray*}
&&H=\{(x,0\subset V_{k-1}\subset V_k\subset V_{k+1}\subset V_{k+1}^\bot
\subset V_{k}^\bot
\subset V_{k-1}^\bot\subset V=\bC^N)\\
&&\qquad\qquad|\,x\in\fg_1,\ xV_{k-1}\subset V_k,
xV_{k}\subset V_k^\bot\}.
\end{eqnarray*}
Note that
the natural map
\[p:H\ra \Hess_k^{E,\p},\ (x, V_{k-1}\subset V_k\subset V_{k+1})\mapsto
(x, V_{k-1}\subset V_k)\]
realizes $H$ as a quadric bundle over $\Hess_k^{E,\p}$.
We first construct a map
$\beta:H\ra[\bG_a/\bG_m^{[2]}]$. The construction is very similar to
that of the map $\alpha$ in <ref> and we use the notations there.
\[\widetilde H:=\{(x,V_{k-1}\subset V_k\subset V_{k+1},l)\,|\,(x,V_{k-1}\subset V_k\subset V_{k+1})\in H,\ 0\neq l\in V_k/V_{k-1}\is\bC\},\]
where the action of $\bG_m$ on $\widetilde H$ is given by
$t\cdot(x,V_{k-1}\subset V_k\subset V_{k+1},l)=(x,V_{k-1}\subset V_k
\subset V_{k+1},tl)$. Define
\[\phi:\widetilde H\ra\bG_a,\ (x,V_{k-1}\subset V_k\subset V_{k+1},l)\mapsto\langle xl,xl\rangle_Q.\]
Note that the above pairing is well-defined since $xV_{k-1}\subset V_k$ and $xV_{k}\subset V_{k}^\bot$. One checks that $\phi$ satisfies (<ref>). This finishes the construction of
$(\widetilde H,\phi)$. Hence we obtain a map $\beta:H\ra[\bG_a/\bG_m^{[2]}]$.
Let $f:H\ra\fg_1$ be the natural projection map.
\[\mathcal F:=f_*\beta^*\IC([\bG_a/\bG_m^{[2]}],\mL)\in D_K(\fg_1).\]
We show that
$\mathfrak F(\mathcal F)$ is supported on $\mathcal N_1^3$, and
the complex $\mathcal F$ contains $\IC(\fg_1,P(\tX_{2k})^{\sigma=-id})$ as
a direct summand.
The proposition then follows from (<ref>) and (<ref>).
To prove (<ref>), let
\begin{eqnarray*}
&&H'=\{(x,0\subset V_{k-1}\subset V_k\subset V_{k+1}\subset V_{k+1}^\bot
\subset V_{k}^\bot
\subset V_{k-1}^\bot\subset V=\bC^N)\\
&&\qquad\qquad|\,x\in\fg_1,\ xV_{k-1}\subset V_k,
xV_{k}\subset V_{k+1}\}.
\end{eqnarray*}
Note that $H'\subset H$ is a sub-bundle.
By construction, the map $\beta$ factors through $H'$, i.e.,
\[\beta:H\stackrel{}\ra H/H'\stackrel{\beta'}\ra
Let $\check f'$ be the natural projection map
$$\check f':(H')^\bot:=\{(x,0\subset V_{k-1}\subset V_k\subset V_{k+1}\subset V_{k+1}^\bot
\subset V_{k}^\bot
\subset V_{k-1}^\bot\subset\bC^N)$$
$$\quad\quad\quad\,|\,x\in\fg_1,\ xV_{k}=0,\,xV_{k+1} \subset V_{k-1},\,
xV_{k}^\p\subset V_k\}\ra\mathcal N_1.$$
A direct calculation shows that
Imf̌'=_3^k1^N-3k if 3k≤N and Imf̌'=_3^N-2k2^3k-N if 3k≥N+1.
The standard properties of Fourier transform imply that
$\mathfrak F(\mathcal F)$ is supported on $\on{Im}(\check f')\subset\mN_1^3$.
This proves (<ref>).
It remains to prove (<ref>).
Notice that the map $\beta$ factors as
$\beta:H\stackrel{p}\ra \Hess_k^{E,\p}\xrightarrow{\bar \beta} [\bG_a/\bG_m^{[2]}]$. Consider the following diagram
\[\xymatrix{\beta:H\ar[rd]_f\ar[r]^p&\Hess_k^{E,\p}\ar[d]^{\check\tau_k^N}\ar[r]^-{\bar \beta}&[\bG_a/\bG_m^{[2]}]\\
We have
$$\mathcal F:=f_*\beta^*\IC([\bG_a/\bG_m^{[2]}])\is(\check\tau_k^N)_*p_*p^*\bar \beta^*(\IC([\bG_a/\bG_m^{[2]}],\mL))$$
which is isomorphic to
$(\check\tau_k^N)_*(\bar \beta^*(\IC([\bG_a/\bG_m^{[2]}],\mL))\otimes p_*\bC).$
Since $\bC$ is a direct summand of $p_*\bC$, it implies that
$(\check\tau_k^N)_*(\bar \beta^*(\IC([\bG_a/\bG_m^{[2]}],\mL))$
is a direct summand of $\mathcal F$.
So it is enough to show that
is a direct summand of
$(\check\tau_k^N)_*(\bar \beta^*(\IC([\bG_a/\bG_m^{[2]}],\mL))$.}$$
This follows from the same argument as in the proof of (<ref>), replacing
$X_{2k-1}$ (resp. $\tX_{2k-1}$) there by $X_{2k}$ (resp. $\tX_{2k}$).
Thus the proof of the proposition is complete.
§.§ Matching for $\IC(\bar\cO_{2^i1^{2n+1-2i}},\cE_i)$, $i$ odd
Here we complete the proof of <cit.> by treating the case of odd $i$. In <cit.> we treated the even case of the proposition below and showed that there exists a permutation $s$ of the set $\{2j+1\,|\,1\leq 2j+1\leq n\}$, such that $\fF(\IC(\bar\cO_{2^i1^{2n+1-2i}},\cE_i))=\IC(\Lg_1,\cF_{s(i)})$ (see Theorem 5.2 and Theorem 6.2 in loc.cit.).
We have that
where $\cE_i$ denotes the unique non-trivial irreducible $K$-equivariant local system on $\cO_{2^i1^{2n+1-2i}}$.
It remains to prove the proposition for odd $i$.
Assume that $2m\leq n+1$. By (<ref>) and (<ref>), we see that the Fourier transform of $\IC(\Lg_1,P(\tilde{X}_{2m-1})^{\sigma=-id})$ is supported on $\on{Im}\tau_m^N=\bar\cO_{3^{m-1}2^11^{2n+2-3m}}$ (see (<ref>)). Using Theorem <ref> and (<ref>) we obtain that
(_1,_i) is a direct summand of (_1,P(X̃_2m-1)^σ=-id) if and only if
i is odd and 1≤i≤2m-1.
This implies that the Fourier transform of $\IC(\Lg_1,\cF_{2j-1})$, $1\leq j\leq m$, is supported on $\bar\cO_{3^{m-1}2^11^{2n+2-3m}}$. Now it is easy to check that $\cO_{2^i1^{2n+1-2i}}\subset\bar\cO_{3^{m-1}2^11^{2n+2-3m}}$ if and only if $i\leq 2m-1$. In view of <cit.>, the proposition follows by induction on $m$.
Let $\mO_{min}=\mO_{2^11^{2n-1}}$. By the above proposition, we have
$$\mF_1\is\tE_{n+1,1}^{2n+1}\is H^1(\tC_{a,\chi_0},\bC)\text{ (see \eqref{E_n+1,j's})}$$
is isomorphic to the monodromy representation associated with $\bar C_{\chi_0}\ra\fc^{rs}$, the
universal family of hyperelliptic curves in <ref>.
§ CONJECTURES AND EXAMPLES
Let $N=2n+1$ and let $E^{2n+1}_{ij}$ (resp. $\tE^{2n+1}_{ij}$) be the monodromy representations of $\pi_1^K(\Lg_1^{rs})$ constructed from the
families of complete intersections of quadrics in $\mathbb P^{2n}$ (resp. their double covers), see <ref>. Let $\{(\mO,\mE)\}_{\leq 3}$ denote the set of pairs $(\mO,\cE)$ where $\mO$ is a
$K$-orbit in $\mathcal N_1^3$ and $\cE$ is an irreducible $K$-equivariant local system on
$\mO$ (up to isomorphism). Using Theorem <ref>, we establish
an injective map
𝒮:{E_ij^2n+1, i∈[1,n], j∈[0,i-1];
^2n+1_ij, i∈[1,n+1], j∈[1,i-1], ^2n+1_n+1,0≅}↪{(,)}_≤3,
where $\calS(E_{ij}^{2n+1})=(\cO,\cE)$ if and only if $\fF(\Lg_1,E_{ij}^{2n+1})=\IC(\bar\cO,\cE)$, similarly for $\tE_{ij}^{2n+1}$. Here the $K$-equivariant local systems on $\Lg_1^{rs}$ in the left hand side of (<ref>) are pairwise non-isomorphic, see (<ref>).
In this section we state two conjectures (Conjecture <ref> and Conjecture <ref>)
that describe the map $\calS$ in (<ref>) in the case of $\{E_{ij}^{2n+1}\}$ explicitly.
We verify our conjectures in several examples by studying
various families of Hessenberg varieties.
In what follows we make use of the following observation:
an orbit $\cO_{3^k2^l1^{2n+1-3k-2l}}\subset\cN_1^3$ is odd dimensional $\Leftrightarrow k$ is odd and $l$ is even.
§.§ Complete intersections of even number of quadrics and conjectural matching
Recall that the local systems $E_{i,2j}^{2n+1}$, where $ i\in[1,n]$ and $2j\in[0,i-1]$, are constructed from families of complete intersections $X_{2m}$ of even number of quadrics in $\bP^{2n}$ for $m\in[1,n]$.
We first show that
$\mathfrak{F}(\IC(\Lg_1,E_{i,2j}^{2n+1}))$ is supported on an even dimensional $K$-orbit in $\cN_1^3$ .
To this end we first note that each $\fF(\IC(\Lg_1,E_{i,2j}^{2n+1}))$ is a direct summand of $\fF(\IC(\Lg_1,P(X_{2m})))$ for some $m$, which in turn is a direct summand of $(\tau_m^N)_*\bC[-]$. One readily checks that
Hess_m^E =m(4n-3m+5)-2n-2, which is even.
Note also that $\dim X_{2m,a}$ is even. Now (<ref>) follows from the decomposition theorem and the fact that the fibers of $\tau_m^N$ have non-vanishing cohomology only in even degrees (see <ref>).
Thus (<ref>) puts a restriction on nilpotent orbits which can support $\fF(\IC(\Lg_1,E_{i,2j}^{2n+1}))$.
Our first conjecture is:
We have that
\begin{eqnarray*}
&\mathfrak{F}(\IC(\Lg_1,E_{i,2j}^{2n+1}))\cong\IC({\bar\cO_{3^{2(n-i)+1}2^{2(i+j-n)-1}1^{2i-4j}}},\bC)\text{ if } i+j\geq n+1\\
&\mathfrak{F}(\IC(\Lg_1,E_{i,2j}^{2n+1}))\cong\IC({\bar\cO_{3^{2j}2^{2(n-i-j)+1}1^{4i-2n-2j-1}}},\bC)\text{ if } i+j\leq n\text{ and }2i-j\geq n+1
\\
&\mathfrak{F}(\IC(\Lg_1,E_{i,2j}^{2n+1}))\cong\IC({\bar\cO_{3^{2j}2^{2i-4j}1^{2n-4i+2j+1}}},\bC)\text{ if}\ i+j\leq n\text{ and }2i-j\leq n.
\end{eqnarray*}
The nilpotent orbits appearing in the conjecture above exhaust all the non-zero even dimensional orbits of the form $\cO_{3^i2^j1^k}$, where the partition $3^i2^j1^k$ has no gaps.
Note that the conjecture above holds for $E_{i,0}^{2n+1}$. This follows from (<ref>) and <cit.>, i.e., we have
((_1,E_i,0^2n+1))=(_2^2i1^2n-4i+1,) if 2i≤n
((_1,E_i,0^2n+1))=(_2^2n-2i+11^4i-2n-1,) if 2i≥n+1.
Below we verify the conjecture in a simple case that involves nilpotent orbits of order 3.
§.§ Complete intersection of $4$ quadrics, $n\geq 3$.
In this subsection we show that
Let us write
$$\tau=\tau_2^{2n+1}:\on{Hess}_2^E\to\bar{\cO}_{3\,2\,1^{2n-4}},\ \ \check{\tau}=\check{\tau}_2^{2n+1}:\on{Hess}_2^{E,\p}\to\Lg_1.$$
We have $\fF(\tau_*\bC[-])\cong\check{\tau}_*\bC[-]$ and
τ̌_*[-]=(_1,E_n,2^2n+1⊕E_n,0^2n+1⊕E_n-1,0^2n+1)⊕⊕_a=0^2n-4(_1,)[2n-4-2a]⊕⋯where $\cdots$ is a direct sum of IC complexes with smaller support. We have
_3 2 1^2n-4=_3 2 1^2n-4∪_3^11^2n-2⋃_0≤i≤3_2^i1^2n-2i+1.
In view of Proposition <ref>, Lemma <ref>, (<ref>) and (<ref>), we conclude that $\fF(\IC(\Lg_1,E_{n,2}^{2n+1}))$ is not supported on $\bar\cO_{2^i1^{2n+1-2i}}$'s. Now it follows from (<ref>) and (<ref>) that
$$\text{$\fF(\IC(\Lg_1,E_{n,2}^{2n+1}))$ is supported on $\bar\cO_{3\,2\,1^{2n-4}}$.}$$
Thus (<ref>) follows the fact that the only IC complex supported on $\cO_{3\,2\,1^{2n-4}}$ appearing in $\tau_*\bC[-]$ is $\IC(\bar\cO_{3\,2\,1^{2n-4}},\bC)$ as $\tau$ is a resolution of $\bar\cO_{3\,2\,1^{2n-4}}$.
§.§ Complete intersections of odd number of quadrics and a conjectural matching
Recall that the local systems $E_{i,2j-1}^{2n+1}$, where $ i\in[1,n]$ and $2j\in[2,i]$, are constructed from complete intersections $X_{2m-1}$ of odd number of quadrics in $\bP^{2n}$, $m\in[1,n]$.
Using that $\dim\on{Hess}_m^{O}=m(2n-3m+5)-2n-3$, which is odd, and arguing as in (<ref>), we obtain that
$\mathfrak{F}(\IC(\Lg_1,E_{i,2j-1}^{2n+1}))$ is supported on an odd dimensional $K$-orbit in $\cN_1^3$.
Let $\cO\subset\cN_1^3$ be an odd dimensional $K$-orbit. To describe our second conjecture, let us first label the non-trivial irreducible $K$-equivariant local systems on $\cO$ as follows. By (<ref>), we can assume that $\cO=\cO_{3^{2k-1}2^{2l}1^{2n+4-6k-4l}}.$
Let $x\in\cO_{3^{2k-1}2^{2l}1^{2n+4-6k-4l}}$, $k\geq 1$. We first define representatives for the component group $A_K(x)=Z_K(x)/Z_K(x)^0$.
Take a basis
$$\text{$x^iu_j,i\in[0,2],j\in[1,2k-1]$, $x^iv_j,\ i\in[0,1],\ j\in[1,2l]$ and $w_i,\ i\in[1,2n+4-6k-4l]$}$$
of $V$ as in <cit.>. Define $\gamma_i\in Z_K(x)$, $i=1,2$ as follows
\begin{eqnarray*}
&\gamma_1(w_1)=w_2,\ \gamma_1(w_2)=w_1,\ \gamma_1(x^iu_j)=-x^iu_j,\ i\in[0,2],\ j\in[1,2k-1],\\ &\gamma_2(x^jv_1)=x^jv_2,\ \gamma_2(x^jv_2)=x^jv_1,\ j\in[0,1],\\
&\quad\text{ and }\gamma_1 (\text{resp. }\gamma_2)\text{ acts as identity on all other basis vectors}.
\end{eqnarray*}
Assume that $l\geq 1$ and $2n+4-6k-4l\neq 0$. Then
$$\text{$\cE_{k,l}^1$ (resp. $\cE_{k,l}^2$, $\cE_{k,l}^3$)}$$
denote the irreducible $K$-equivariant local system on $\cO_{3^{2k-1}2^{2l}1^{2n+4-6k-4l}}$ corresponding to the irreducible character of $A_K(x)$
$$\text{$\chi_1$ (resp. $\chi_2$, $\chi_3$) with $\chi(\gamma_1)=-1$ (resp. $-1,1$) and $\chi(\gamma_2)=1$ (resp. $-1,-1$).}$$
Assume that $l=0$ and $2n+4-6k\neq 0$. Then $A_K(x)\cong\{1,\gamma_1\}\cong\bZ/2\bZ$. We denote by $\cE^1_{k,0}$ the irreducible $K$-equivariant local system on $\cO_{3^{2k-1}1^{2n+4-6k}}$ corresponding to the irreducible character $\chi$ of $A_K(x)$ with $\chi(\gamma_1)=-1$.
Assume that $l\geq 1$ and $2n+4-6k-4l\neq 0$. Then $A_K(x)\cong\{1,\gamma_2\}\cong\bZ/2\bZ$. We denote by $\cE^3_{k,l}$ the irreducible $K$-equivariant local system on $\cO_{3^{2k-1}2^{n+2-3k}}$ corresponding to the irreducible character $\chi$ of $A_K(x)$ with $\chi(\gamma_2)=-1$.
We will simply write $\cE^i$, $i=1,2,3$, when the supports of these local systems are clear.
Our second conjecture is the following.
We have that
\begin{eqnarray*}
&\mathfrak{F}(\IC(\Lg_1,E_{i,2j-1}^{2n+1}))\cong\IC({\bar\cO_{3^{2(n-i)+1}2^{2(i+j-n-1)}1^{2i-4j+2}}},\cE^1)\text{ if } i+j\geq n+1\\
&\mathfrak{F}(\IC(\Lg_1,E_{i,2j-1}^{2n+1}))\cong\IC({\bar\cO_{3^{2j-1}2^{2(n-i-j+1)}1^{4i-2j-2n}}},\cE^2)\text{ if } i+j\leq n\text{ and }2i-j\geq n+1\\
&\mathfrak{F}(\IC(\Lg_1,E_{i,2j-1}^{2n+1}))\cong\IC({\bar\cO_{3^{2j-1}2^{2(i-2j+1)}1^{2n-4i+2j}}},\cE^3)\text{ if } i+j\leq n\text{ and }2i-j\leq n.
\end{eqnarray*}
In particular, the conjecture above implies that the set of all Fourier transforms $\mathfrak{F}(\IC(\Lg_1,E_{i,2j-1}^{2n+1}))$ coincides with the set of all IC complexes supported on odd dimensional orbits in $\cN_1^3$, with non-trivial local systems.
In the following subsections we verify the conjecture above in two simple examples, see (<ref>) and (<ref>). We also prove a lemma (Lemma <ref>) that is compatible with our conjecture.
§.§ Complete intersection of 3 quadrics, $n\geq2$
In this subsection we show that
Let us write
$$\sigma=\sigma_2^{2n+1}:\on{Hess}_2^{O}\to\bar{\cO}_{3^11^{2n-2}},\ \ \check{\sigma}=\check\sigma_2^{2n+1}:\on{Hess}_2^{O,\p}\to\Lg_1.$$
The fiber $\sigma^{-1}(x)$ at $x\in\cO_{3^11^{2n-2}}$
is a non-singular quadric in $\mathbb P^{2n-3}$. Thus in the decomposition of $\sigma_*\bC[-]$, we have the following direct summands
We have $\fF(\sigma_*\bC[-])\cong\check{\sigma}_*\bC[-]$ and
σ̌_*[-]≅(_1,E_n,1^2n+1)⊕⋯Note that $\cO_{3^11^{2n-2}}$ is the only odd-dimensional orbit contained in
$\bar{\cO}_{3^11^{2n-2}}$ and there is a unique non-trivial irreducible $K$-equvariant local system on $\cO_{3^11^{2n-2}}$, denoted by $\cE^1$. In view of (<ref>), the equation (<ref>) follows from the fact that the support of
$\fF(\IC(\bar\cO_{3^11^{2n-2}},\bC))$ is a proper subset of $\fg_1$ (see <cit.>).
Here we see that Fourier transform of $\IC$ complexes supported on
nilpotent orbits $\cO_\lambda$, where $\lambda$ has gaps, with
nontrivial local systems can have full support (compare with <cit.>).
§.§ Complete intersection of 5 quadrics, $n\geq 4$
In this subsection, we show that
𝔉((_3^12^21^2n-6,^1))=(_1,E_n,3^2n+1), 𝔉((_3^12^21^2n-6,^2))=(_1,E_n-1,1^2n+1),
Let us write
$$\sigma=\sigma_3^{2n+1}:\on{Hess}_3^{O}\to\bar{\cO}_{3^21^{2n-5}},\ \check\sigma=\check\sigma_3^{2n+1}:\on{Hess}_3^{O,\p}\to\Lg_1.$$
We have $\fF(\sigma_*\bC[-])\cong\check{\sigma}_*\bC[-]$ and
σ̌_*[-]≅(_1,E_n,1^2n+1⊕E_n,3^2n+1⊕E_n-1,1^2n+1)⊕⋯The odd dimensional orbits contained in $\Img\sigma=\bar{\cO}_{3^21^{2n-5}}$ are $\cO_{3^12^21^{2n-6}}$ and $\cO_{3^11^{2n-2}}$. In view of (<ref>), the equation (<ref>) follows from Lemma <ref> (see <ref>) and the following statement.
The IC complexes supported on $\cO_{3^12^21^{2n-6}}$, that appear in the decomposition
of $\sigma_*\bC[-]$, are
\IC({\bar\cO_{3^12^21^{2n-6}}},\cE^1\oplus\cE^2).
It remains to prove (<ref>).
Note that there is no orbit $\cO$ such that $\cO_{3^12^21^{2n-6}}<\cO<\cO_{3^21^{2n-5}}$. The fiber $\sigma^{-1}(x_2)$ at $x_2\in\cO_{3^21^{2n-5}}$ is a nonsingular quadric in $\bP^{2n-6}$. Thus in the decomposition of $\sigma_*\bC[-]$,
the IC complexes supported on $\cO_{3^21^{2n-5}}$ are $\bigoplus_{a=0}^{2n-7}\IC({\bar\cO_{3^21^{2n-5}}},\bC)[2n-7-2a]$.
The fiber at $x_1\in\cO_{3^12^21^{2n-6}}$ is a quadric bundle over $\pi_0^{-1}(x_1)$ with fibers being a quadric $Q$ of rank $2n-8$ in $\bP^{2n-6}$. Here $\pi_0$ is Reeder's resolution of $\bar{\cO}_{3^21^{2n-5}}$, i.e.
$$\pi_0:\{(x,0\subset V_2\subset V_2^\p\subset V)\,|\,x\in\Lg_1,\,xV_2=0,xV_2^\p\subset V_2\}\to\bar{\cO}_{3^21^{2n-5}}.$$
It is easy to check that the map $\pi_0$ is small. Thus we have
Note that $H^{\text{odd}}(\pi_0^{-1}(x_1),\bC)=0$, $H^{\text{odd}}(\sigma^{-1}(x_1),\bC)=0$, and
\begin{eqnarray*}
&H^{2k}(\sigma^{-1}(x_1),\bC)=\bigoplus_{a=0}^{2n-7}H^{2a}(Q,\bC)\otimes H^{2k-2a}(\pi_0^{-1}(x_1),\bC)\\
&\cong\bigoplus_{a=0}^{2n-7}H^{2k-2a}(\pi_0^{-1}(x_1),\bC)\oplus (H_{\on{prim}}^{2n-6}(Q,\bC)\otimes H^{2k-2n+6}(\pi_0^{-1}(x_1),\bC)).
\end{eqnarray*}
We have $\on{codim}_{\on{Hess}_3^O}\cO_{3^12^21^{2n-6}}=2n-6$ and $\pi_0^{-1}(x_1)$ consists of two points. Moreover, $A_K(x_1)$ acts on $H_{\on{prim}}^{2n-6}(Q,\bC)\otimes H^{2k-2n+6}(\pi_0^{-1}(x_1),\bC)$ as $\chi_1(1\oplus\chi_3)=\chi_1\oplus\chi_2$. The equation (<ref>) follows. This finishes the proof of (<ref>).
Note that (<ref>) shows that all three IC complexes supported on $\cO_{3^12^21^{2n-6}}$ with nontrivial local systems correspond to the monodromy representations constructed from complete intersections of odd number of quadrics.
§.§ The case of a curve
In this subsection we prove the following lemma by considering the family $X_{2n-1}$ of complete intersections of quadrics in $\bP^{2n}$.
For each $i\in[1,n-1]$, there exists some $1\leq j\leq [\frac{n-1}{2}]$ and a nontrival local system $\cE_{j}^s$ ($s=2\text{ or }3$) on $\cO_{3^12^{2j}1^{2n-2j-2}}$ such that
Let us write
$$\sigma=\sigma_n^{2n+1}:\on{Hess}_n^{O}\to\bar{\cO}_{3^12^{n-1}}\ \text{ and } \check\sigma=\check\sigma_n^{2n+1}:\on{Hess}_n^{O,\p}\to\Lg_1.$$
Assume that $n\geq 3$.
We have
\dim \on{Hess}_n^{O}=n^2+3n-3.
We show that
\begin{eqnarray}\label{decompositiontau}
&&\sigma_{*}\bC[-]\cong\bigoplus_{a=0}^{n-3}\IC({\bar\cO_{3^12^{n-1}}},\bC)[n-3-2a]\bigoplus _{2j=n-1}\IC({\bar\cO_{3^12^{2j}}},\cE^3)\nonumber\\
&&\qquad\oplus\bigoplus_{2\leq 2j\leq n-2}\IC({\bar\cO_{3^12^{2j}1^{2n-4j-2}}},\cE^2\oplus\cE^3)\oplus \IC({\bar\cO_{3^11^{2n-2}}},\bC\oplus\cE^1)\\
&&\qquad\oplus \IC(\bar\cO_{1^{2n+1}},\bC)[1]\oplus \IC(\bar\cO_{1^{2n+1}},\bC)[-1].\nonumber
\end{eqnarray}
The lemma follows from the decomposition above, the equations $\fF(\check\sigma_*\bC[-])\cong\sigma_*\bC[-]$,
$\check\sigma_*\bC[-]\cong\bigoplus_i\IC(\Lg_1,E_{i,1}^{2n+1})\oplus\cdots$, and (<ref>).
In the remainder of this subsection we prove (<ref>).
Consider first Reeder's resolution of $\bar{\cO}_{3^12^{n-1}}$ given by
ρ:{(x,0⊂V_1⊂V_n⊂V_n^⊂V_1^⊂^2n+1) | x∈_1, xV_n=0,xV_n^⊂V_1}→_3^12^n-1.
It is easy to check that $\rho$ is a small map.
Thus for $x_j\in\cO_{3^12^j1^{2n-2j-2}}$, we have
Now we study the map $\sigma$ and the decomposition of $\sigma_*\bC[-]$. The fiber $\sigma^{-1}(x_{n-1})$ at $x_{n-1}\in\cO_{3^12^{n-1}}$ is a nonsingular quadric in $\bP^{n-2}$ and $\on{codim}_{\on{Hess}_n^{O}}\cO_{3^12^{n-1}}=n-3$. Thus in the decomposition of $\sigma_*\bC[-]$, the following IC complexes supported on $\cO_{3^12^{n-1}}$ appear,
⊕_a=0^n-3(_3^12^n-1,)[n-3-2a] for all n and
(_3^12^n-1,)[-] if n is odd,
where $\cE$ is the unique nontrivial irreducible $K$-equivariant local system on $\cO_{3^12^{n-1}}$.
We have that $\sigma^{-1}(x_j)$ ($x_j\in\cO_{3^12^j1^{2n-2j-2}}$) is a $Q_j$-bundle over $\rho^{-1}(x_j)$ for $j\geq 1$, where $Q_j$ is a quadric $\sum_{k=1}^ja_k^2=0$ in $\bP^{n-2}:=\{[a_1,\cdots,a_{n-1}]\}$.
For $j$ odd, or $j\geq 2$ even and $2k>\on{codim}_{\on{Hess}_n^{O}}\cO_{x_j}$, we have
where in the second isomorphism we use (<ref>). Thus in view of (<ref>) IC complexes supported on $\cO_{3^12^{j}1^{2n-2j-2}}$, for odd $j<n-1$, do not appear in the decomposition of $\sigma_*\bC$.
For $j\geq 2$ even, and $2k=\on{codim}_{\on{Hess}_n^{O}}\cO_{x_j}$, we have
Note that $\rho^{-1}(x_j)$ has two irreducible components. Moreover $A_K(x_j)$ acts on $H^{2n+j-4}_{\text{prim}}(Q_j,\bC)$ via the character $\chi_3$, and acts on $H^{2\dim\rho^{-1}(x_j)}(\rho^{-1}(x_j),\bC)$ via $1\oplus\chi_1$.
In view of (<ref>), we conclude that IC complexes $\IC(\bar\cO_{3^12^j1^{2n-2j-2}},\cE^2)$ and $\IC(\bar\cO_{3^12^j1^{2n-2j-2}},\cE^3)$, for $j$ even, appear in the decomposition of $\sigma_*\bC[-]$.
For $j=0$ and $2k=\text{codim}_{\on{Hess}_n^{O}}\cO_{x_0}=n^2-n-2$, since $2k-2a>2\dim\rho^{-1}(x_0)$ for all $0\leq a\leq n-3$, we have
We have $2\dim\sigma^{-1}(x_0)={\on{codim}_{\on{Hess}_n^{O}}{\cO_{x_0}}}$ and $\sigma^{-1}(x_0)\cong\{0\subset W_{n-2}\subset W_{n-1}\subset W_{n-1}^\p\subset W_{n-2}^\p\subset\bC^{2n-2}\}$. Note that $\sigma^{-1}(x_0)$ has two connected components and $A_k(x_0)$ permutes them. We conclude that the IC complexes supported on $\cO_{x_0}$ appearing in $\sigma_*\bC[-]$ are $\IC(\bar\cO_{3^11^{2n-2}},\bC)\oplus\IC(\bar\cO_{3^11^{2n-2}},\cE^1)$.
The decomposition (<ref>) follows from the above discussion and the fact that none of the IC complexes supported on $\cO_{2^i1^{2n+1-2i}}$, $i\geq 1$ can appear in the decomposition of $\sigma_*\bC[-]$.
The proof of Lemma <ref> is complete.
[A]AA'Campo, Norbert. Tresses, monodromie et le groupe symplectique, Comment. Math. Helv. 54 (1979), no. 2, 318–327.
[CVX]CVXChen, T.H., Vilonen, K., Xue, T. Springer correspondence for symmetric spaces. Arxiv.1510.05986.
[GKM]GKM M. Goresky, R. Kottwitz, R. Macpherson:
Purity of equivalued affine Springer fibers, Representation Theory, 130-146 (2006).
[KaS]KaSKashiwara, Masaki; Schapira, Pierre. Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften, 292. Springer-Verlag, Berlin, 1990.
[KS]KSKatz, Nicholas M.; Sarnak, Peter. Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Publications, 45. American Mathematical Society, Providence, RI, 1999.
[KR]KRKostant, B.; Rallis, S. Orbits and representations associated with symmetric spaces. Amer. J. Math. 93 (1971), 753–809.
[R]RReeder, Mark. Desingularizations of some unstable orbit closures. Pacific J. Math. 167 (1995), no. 2, 327–343.
[S]SSekiguchi, Jirō. The nilpotent subvariety of the vector space associated to a symmetric pair. Publ. Res. Inst. Math. Sci. 20 (1984), no. 1, 155–212.
[T]TTerasoma, Tomohide. Complete intersections of hypersurfaces – the Fermat case and the quadric case. Japan. J. Math. (N.S.) 14 (1988), no. 2, 309–384.
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1511.00902
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KU Leuven Campus Kortrijk - KULAK, Department of Physics, Etienne Sabbelaan 53, 8500 Kortrijk, Belgium
Ghent University, Department of Physics and Astronomy, Krijgslaan 281-S9, 9000 Gent, Belgium
Instituto de Física Teórica, Rua São Francisco Xavier 524, 20550-013, Maracanã, Rio de Janeiro, Brasil
12.38.Aw, 12.38.Lg
In this short note, we come back to the recent proposal put forward by Kharzeev and Levin <cit.>, in which they phenomenologically couple the non-perturbative Veneziano ghost to the perturbative gluon, leading to a modified gluon propagator (the “glost”) of the Gribov type, with complex poles. As such, a possible link was made between the QCD topological $\theta$-vacuum (Veneziano ghost) and color confinement (no physically observable gluons). We discuss some subtleties concerning gauge (BRST) invariance of this proposal, related to the choice of Feynman gauge. We furthermore provide an example in the Landau gauge of a similar phenomenological vertex that also describes the necessary Veneziano ghost but does not affect the Landau gauge gluon propagator.
In the recent Letter <cit.>, see also <cit.>, the issue of the Veneziano ghost <cit.> was revisited. In the absence of massless quarks, the QCD action can explicitly depend on an extra parameter, the $\theta$-angle, closely related to the non-trivial topological nature of the QCD vacuum (cf. instanton dynamics describing the tunneling between different states with different winding number, <cit.>).
For the sake of presentation, we will mostly follow the notations of <cit.>. A fundamental ingredient in the whole discussion is the topological susceptibility $\chi^4$, defined in the pure Yang-Mills case as the zero momentum correlator <cit.>
\begin{eqnarray}\label{eq0}
% \nonumber to remove numbering (before each equation)
i\int \d^4x \braket{\Q(x) \Q(0)} &=& -\chi^4\,,
\end{eqnarray}
where $\Q(x)$ is a pseudoscalar quantity, given by
\begin{equation}\label{eq2}
\mathcal{Q}(x)=\frac{g^2}{32\pi^2} F_{\mu\nu}(x)\tilde F^{\mu\nu}(x)\,,
\end{equation}
with $\tilde F^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ the dual field strength. We refer to <cit.> for more details. The existence of $\chi^4$ received numerous lattice confirmations, see e.g. <cit.>. The interesting part is that the gauge invariant quantity $\mathcal{Q}(x)$ can be written as a total derivative of a gauge variant pseudovector $\mathcal{K}_\mu(x)$,
\begin{equation}\label{eq3}
\mathcal{Q}(x)=\p_\mu \mathcal{K}^\mu\,,\mathcal{K}^\mu = \frac{g^2}{16\pi^2}\epsilon^{\mu\nu\rho\sigma}A_\nu^a\left(\p_\rho A_\sigma^a+\frac{g}{3}f^{abc}A_\rho^b A_\sigma^c\right)\,.
\end{equation}
A non-vanishing $\braket{\Q \Q}$ correlator at zero momentum, that is, a non-vanishing topological susceptibility, is thus only possible if there is a massless pole in the $⟨𝒦𝒦|$⟩ correlator given that $\Q$ is a total derivative. Such a pseudovector pole was introduced first by Veneziano <cit.>, inspired by Witten <cit.>. They were then able to connect the topological susceptibility to the mass of the $\eta'$ particle. More precisely, Veneziano proposed
\begin{equation}\label{eq4}
\mathcal{K}_{\mu\nu}(q)=i \int \d^4x e^{iqx}\braket{\mathcal{K}_\mu(x) \mathcal{K}_\nu(0)}\stackrel{q^2\sim0}{\sim} -\frac{\chi^4}{q^2}g_{\mu\nu}\,.
\end{equation}
The negative sign of residue of the massless pole indicates the new “particle” is indeed a ghost, see also <cit.>.
Kharzeev and Levin then recognized that the current correlator (<ref>) can be interpreted as being sourced by an effective interaction between the glue and the Veneziano ghost. They postulated such a vertex, and then solved the Dyson-Schwinger equation using solely this coupling, leading to a dynamically corrected gluon propagator (the “glost”) given by
\begin{equation}\label{eq5}
\end{equation}
In the following we will first show that this result necessarily breaks the perturbative BRST symmetry of the underlying theory.
They relied on the Feynman gauge to facilitate computations. Let us first look again at the Faddeev-Popov action in a general linear covariant gauge, written as[We switch to Euclidean space time here.]
\begin{equation}\label{4}
S=\int \d^4x \left(\frac{1}{4}F_{\mu\nu}^2+b^a\p_\mu A_\mu^a +\overline c^a \p_\mu D_\mu^{ab} c^b - \frac{\alpha}{2} b^a b^a\right)
\end{equation}
with covariant derivative
\begin{equation}\label{4d}
D_\mu^{ab}=\delta^{ab}\p_\mu -gf^{abc}A_\mu^c\,.
\end{equation}
The auxiliary field $b^a$ enforces the linear covariant gauge, with $\alpha=1$ corresponding to the Feynman gauge.
We recall here that the local gauge invariance gets replaced, after gauge fixing, by the BRST invariance <cit.>, a crucial concept to treat gauge theories at the quantum level, used in proofs of perturbative unitarity, renormalizability or quantum gauge invariance. The BRST variation reads
\begin{equation}\label{4c}
s A_\mu^a=-D_\mu^{ab} c^b\,,\quad sc^a = \frac{g}{2}f^{abc} c^b c^c\,,\quad s\overline c^a = b^a\,,\quad sb^a=0\,.
\end{equation}
\begin{equation}\label{4b}
S=\int \d^4x \left(\frac{1}{4}F_{\mu\nu}^2\right)+ s \int \d^4x \left (\overline c^a\p_\mu A_\mu^a -\frac{\alpha}{2}b^a\overline c^a\right)\,.
\end{equation}
The tree level gluon propagator in momentum (Fourier) space is easily found, viz.
\begin{equation}\label{2}
D_{\mu\nu}(p)=\frac{g_{\mu\nu}}{p^2}-(1-\alpha)\frac{p_\mu p_\nu}{p^4}\,.
\end{equation}
The longitudinal part is thus given by
\begin{equation}\label{3}
\end{equation}
and this is in fact an exact result, as a result of BRST symmetry. One way to appreciate this would be to recall that the gluon self-energy corrections are transverse due to BRST invariance, so the longitudinal sector of its inverse (viz. the gluon propagator) cannot receive any corrections and thus it stays bare. However, as we will discuss later, sometimes one has to be careful when using the gluon self-energy.
Another explicit way to understand (<ref>), not making using of the gluon self-energy but still making use of BRST invariance, goes as follows: we can always exactly compute the $b$-propagator from the action (<ref>); with $\varphi$ representing any other field present in the action and adding to the action the term $\int \d^4 x J^a b^a $, with $J^a$ an external current;
\begin{equation}\label{6}
\braket{b(x) b(y)}=\left.\frac{\delta^2}{\delta J(y) \delta J(x)}\int [\d\varphi \d b]e^{-S}\right\vert_{J=0}\,.
\end{equation}
We can integrate exactly over the $b$-field, giving
\begin{equation}\label{5}
\int [\d\varphi db]e^{-S}= \int [\d\varphi]e^{-\int \d^4x\left(\frac{1}{2\alpha} (\p A)^2+\frac{1}{\alpha}J\p A+\frac{J^2}{2\alpha}+\text{rest}\right)}\,.
\end{equation}
Using (<ref>), this leads to the exact identification
\begin{equation}\label{7}
\braket{b(x) b(y)}=\frac{1}{\alpha^2}\braket{\p A(x) \p A(y)}-\frac{\delta(x-y)}{\alpha}\,.
\end{equation}
Since the l.h.s. must be zero assuming BRST invariance ($\braket{s(\overline c b)}=\braket{bb}$), so must be the r.h.s. As we only relied on the definition of the linear covariant gauge, next to BRST invariance, we can safely state that, according to \eqref{7}, the gluon propagator in that gauge must be of the form
\begin{equation}\label{8}
D_{\mu\nu}(p)=\Delta(p^2)P_{\mu\nu}(p)+\alpha\frac{p_\mu p_\nu}{p^4}
\end{equation}
in Fourier space, where all non-trivial (non-)perturbative information is collected in the form factor $Δ(q^2)$ coupled to the transverse projector $P_μν(p)=g_μν-p_μp_ν/p^2$.
Returning to the glost, it is immediately clear, given the incompatibility of the general form \eqref{8} and the glost-result \eqref{eq5}, that the BRST symmetry must be violated, given that the Feynman gauge definition/choice was not explicitly violated.
Next, let us have a closer look on the alluded connection with the Gribov gauge fixing ambiguity. This was first considered by Gribov in the seminal work \cite{Gribov:1977wm} at leading order, later on generalized to all orders by Zwanziger, see \cite{Zwanziger:1989mf} and the recent review \cite{Vandersickel:2012tz}. These works were focused on either the Landau gauge ($α→0$ limit of the linear covariant gauge, corresponding to $A=0$) or the Coulomb gauge, although it was shown later on by Singer that the Gribov problem is of a quite generic nature \cite{Singer:1978dk}. Only more recently, the Gribov issue was considered for the linear covariant gauge and studied explicitly \cite{Sobreiro:2005vn,Capri:2015pja,Capri:2015ixa}. Let us provide a sketchy overview of the Gribov problem. Assuming a linear gauge fixing of the type
\begin{equation}\label{g1}
\p_\mu A_\mu^a=\alpha b^a\,,
\end{equation}
then the Faddeev-Popov construction implicitly relies on the fact that each gauge orbit only intersects once with the hypersurface defined by \eqref{g1}. Though, Gribov showed this to be wrong at least for $α=0$ (Landau gauge). Assuming an infinitesimally gauge equivalent configuration
\begin{equation}\label{g2}
\tilde A_\mu^a = A_\mu^a + D_\mu^{ab}\omega^b\,,
\end{equation}
then $Ã_μ^a$ can fulfill the same gauge condition \eqref{g1} iff
\begin{equation}\label{g3}
-\p_\mu D_\mu^{ab}\omega^b=0\,,
\end{equation}
that is, there are gauge copies if the Faddeev-Popov operator, $-D$, has non-trivial zero modes. In the case of the Landau gauge, this operator is Hermitian, so its eigenvalues are real. Gribov and Zwanziger therefore suggested to restrict the path integration to a subset of gauge configurations, namely those for which the Faddeev-Popov operator is positive. They managed to implement this restriction explicitly into the partition function. This amounts to the introduction of a dynamical mass scale $λ^4$ into the theory obeying a self-consistent gap equation, and which affects the gluon propagator. At leading order, one finds \cite{Gribov:1977wm,Zwanziger:1989mf}
\begin{equation}\label{eq10}
\end{equation}
The presence of the complex conjugate poles in \eqref{eq10} is why Gribov propagators are frequently used in describing confined degrees of freedom, since these cannot have a physical particle (K\"all\'{e}n-Lehmann) interpretation.
The glost propagator \eqref{eq5} thus appears of the Gribov type, \eqref{eq10}, if we identify the Gribov mass scale $λ^4$ with the topological susceptibility $χ^4$. Note that the derivation of the Gribov propagator \eqref{eq10} is \emph{specific} for the Landau gauge, so it does not apply to the Feynman gauge. The generalization of the Gribov-Zwanziger construction to the linear covariant gauges was discussed in \cite{Sobreiro:2005vn,Capri:2015pja,Capri:2015ixa}, leading to a propagator\footnote{The propagators appearing in \cite{Capri:2015pja,Capri:2015ixa} are more general, taking into account further vacuum corrections, which are however unessential for the discussion here.}
\begin{equation}\label{eq11}
G_{\mu\nu}(p)=\frac{p^2}{p^4+\lambda^4}P_{\mu\nu}(p) + \alpha\frac{p_\mu p_\nu}{p^4}\,.
\end{equation}
As such, the identification of the Feynman gauge glost propagator \eqref{eq5} as the one capable of dynamically curing the Gribov ambiguity based on the topological properties of the QCD vacuum (non-zero $χ^4$) does not necessarily appear to be well founded. Interestingly, the (partial) resolution of the gauge fixing ambiguity by restricting the integration region of the gauge fields \`{a} la Gribov-Zwanziger\footnote{For an alternative approach, also displaying a soft BRST breaking, see \cite{Serreau:2012cg}.} does lead to a (soft) breaking of the ``standard'' BRST symmetry generated by \eqref{4c}, see for instance \cite{Zwanziger:1989mf,Capri:2015ixa,Dudal:2008sp,Schaden:2014bea}. Though, a non-perturbative BRST symmetry can be introduced, depending on the Gribov mass $λ^4$, which allows, for example, to still prove the non-renormalization of the longitudinal gluon propagator. Next to analytical arguments \cite{Capri:2015ixa,Huber:2015ria,Aguilar:2015nqa}, this has also been explicitly observed in lattice simulations \cite{Cucchieri:2009kk,Bicudo:2015rma}, which in principle encompass the topological raison d'\^{e}tre of $⟨|$⟩. It is important to mention that, when using the Gribov-Zwanziger formalism, additional fields are required to maintain locality <cit.>. These extra fields enter the BRST transformation and mix up with e.g. the gauge field. As such, it makes no longer sense to work with only the gluon self energy, since it will be part of a 1PI matrix that defines, after inversion, the (connected) propagator matrix. For an explicit example, see <cit.>.
As the Gribov issue has been studied to more extent in the Landau gauge, we should not refrain from carrying out a similar exercise as <cit.> in this particular gauge. We first notice that (<ref>) is not the unique sensible way to introduce the Veneziano ghost. The most important ingredient is to have the zero mass pole, so
\begin{equation}\label{eq50}
\mathcal{K}_{\mu\nu}(q)=i \int \d^4x e^{iqx}\braket{\mathcal{K}_\mu(x) \mathcal{K}_\nu(0)}\stackrel{q^2\sim0}{\sim} -\frac{\chi^4}{q^2}\frac{q_\mu q_\nu}{q^2}
\end{equation}
will equally well do the job (see also <cit.>), i.e. without changing the gauge invariant $\braket{\Q\Q}$-correlator \eqref{eq0}. Any suitably chosen linear combination of \eqref{eq4} and \eqref{eq50} is also possible. Adding a term transversal in $q$ will not affect $⟨|$⟩ either. We will now benefit from these observations. Following <cit.>, from (<ref>) we may define an effective infrared Veneziano ghost-gluon-gluon vertex $\Gamma_\mu^{\alpha\beta}(q,p)$,
\begin{eqnarray}\label{eq51}
\frac{1}{(2\pi)^4i} \int \d^4p \Gamma_\mu^{\alpha\beta} \Gamma_\nu^{\rho\sigma} \frac{P_{\alpha\rho}(p)}{p^2}\frac{P_{\beta \sigma}(p-q)}{(p-q)^2}=-\frac{\chi^2}{q^2}\frac{q_\mu q_\nu}{q^2}\,,
\end{eqnarray}
where we assumed the Landau gauge condition. From the Landau gauge defining Ward identity $\frac{\delta \Gamma}{\delta b}= \partial_\mu A_\mu$, with $\Gamma$ the 1PI generating functional, it can be easily shown that the Landau gauge gluon propagator is necessarily transverse, thus proportional to the already defined projector $P_{\mu\nu}$.
A (possible) solution to (<ref>) is provided by
\begin{eqnarray}\label{eq52}
&&\Gamma_\mu^{\alpha\beta}(q,p) \propto X q_\mu (p-q)^{\alpha}q^\beta\;;\quad q\leq p \nonumber\\X^2&=&\frac{-\chi^2}{p^2 q^2}\frac{1}{(p-q)_\alpha (p-q)_\rho q_\beta q_\sigma P^{\alpha\rho}(q)P^{\beta\sigma}(p-q)}\,,
\end{eqnarray}
where we omitted some numerical prefactors irrelevant for further purposes. The indices $\mu$, resp. $\alpha,\beta$ refer to the Veneziano ghost, resp. gluons.
Assuming the infrared physics is dominated by the glost[That means, we will omit all information from the already present quark, gluon and ghost vertices, see also <cit.>.], we can compute the one loop gluon self energy $\Sigma$ using the vertex $\Gamma_\mu^{\alpha\beta}$. The diagram shown in FIG. 1 leads to
\begin{eqnarray}\label{int}
% \nonumber to remove numbering (before each equation)
\Sigma_{\mu\nu}(p) &=& \frac{1}{(2\pi)^4i}\int \d^4q \Gamma_\alpha^{\mu\beta}(q,p) \frac{q_{\alpha}q_{\rho}}{q^4}\frac{P_{\beta\sigma}(p-q)}{(p-q)^2}\Gamma_{\rho}^{\nu\sigma}(q,p)\,,\nonumber\\
\end{eqnarray}
where we used for the internal Veneziano ghost a propagator of the form $\frac{q_\alpha q_{\rho}}{q^4}$, consistent with our earlier choice (<ref>).
One loop gluon self energy: dashed line stands for the Veneziano ghost propagator, curly line represents the perturbative gluon, the black dots are vertices.
As in <cit.>, we can approximate the integral (<ref>) by assuming $q\ll p$ as we are considering non-perturbative effects in the very deep infrared. Because of our choice of vertex (<ref>), we would find, for $q\ll p$,
\begin{equation}
\Gamma_\alpha^{\mu\beta}(q,p)\Gamma_{\rho}^{\nu\sigma}(q,p)\stackrel{q\ll p}\propto p_{\mu} p_\nu
\end{equation}
and a fortiori, we would thus have
\begin{eqnarray}\label{intbis}
% \nonumber to remove numbering (before each equation)
\Sigma_{\mu\nu}(p) &\propto& p_{\mu} p_\nu\,,
\end{eqnarray}
i.e. the gluon self energy becomes longitudinal in this case. It is then immediate that the Landau gauge glost propagator will coincide with the perturbative gluon one, since the transverse projector, that is by definition present in the propagator, will always project to zero the self-energy correction (<ref>).
Notice that we are not proclaiming that the vertex defined via (<ref>)-(<ref>) is the correct one in the Landau gauge[Rather the contrary, since the eventual gluon self energy is not transverse either.], as other possibilities exist, as long as the gauge invariant correlation function $\braket{\Q\Q}$ is correctly reproduced. We merely wanted to illustrate that depending on the vertex choice, quite some different glost propagator dynamics could emerge. However, to find out which vertex is effectively realized per gauge is open for future debate. Another strong constraint, next to consistency with $⟨|$⟩, is that the longitudinal part of the gluon propagator in the class of linear covariant gauges should receive no quantum corrections. In brief, a possible strategy to proceed would be to set
\begin{equation}\label{eq50tris}
\mathcal{K}_{\mu\nu}(q)\stackrel{q^2\sim0}{\sim} -\frac{\chi_1^4}{q^2}\delta_{\mu\nu}-\frac{\chi_2^4}{q^2}\frac{q_\mu q_\nu}{q^2}+f(p,q)P_{\mu\nu}(q)\,,
\end{equation}
with $\chi^4=\chi_1^4+\chi_2^4$; and then to propose Ansätze for the vertex $\Gamma_\mu^{\alpha\beta}(q,p)$ such that (<ref>) can be realized in conjunction with a longitudinal gluon propagator projection that remains $\frac{\alpha}{p^2}$.
At last, it is worth mentioning that the Gribov problem has been extensively studied in the Coulomb gauge as well, in which case <cit.> a gluon propagator
\begin{equation}\label{coul}
\end{equation}
was predicted (up to a projector), a form that is indeed pretty consistent with corresponding lattice data <cit.>. It would appear that (<ref>) does not entirely match the Coulomb gauge glost propagator used in <cit.> to analyze a non-perturbative infrared coupling constant. In <cit.>, a link was also made with BRST breaking.
In summary, we have provided some arguments why it appears to be premature to directly link the topological nature of the QCD vacuum to confinement, or more precisely, to link the Veneziano ghost to the issue of Gribov copies and confinement. We discussed the apparent lack of BRST invariance, related to a strong constraint on the longitudinal sector of the gluon/glost propagator that is not fulfilled. As respecting BRST invariance is crucial to extract gauge invariant (or better said, gauge parameter independent) physical information, one does not have the liberty to tamper too much with effective vertices and/or propagators. Therefore, we paid a somewhat closer view at the connection with the Gribov gauge fixing ambiguity, reporting some different behaviour here than from the introduced gluon/glost propagator. Furthermore, we provided an example of a “Veneziano vertex” that does not affect the gluon propagator at all, at least in the Landau gauge.
Though, further research is definitely needed to provide a firmer answer to the premise of <cit.>, and either to confirm or to falsify it. In our opinion, a quite delicate point is the gauge dependent nature of the $\mathcal{K}_\mu$-current and its correlation function. Since this is a genuinely non-perturbative correlation function, it would already be most interesting to have specific information about it using e.g. gauge fixed lattice simulations or perhaps even via functional methods. That such might become possible in the future is not unrealistic, given the recent progress in accessing the linear covariant gauges non-perturbatively <cit.>. This might also put further restrictions on how the effective Veneziano ghost-gluon interaction must be modeled. In any case, lattice Landau gauge is well-matured by now, so that might be the first option to explore the Veneziano ghost in a fully non-perturbative gauge fixed setting.
§ ACKNOWLEDGMENTS
We thank A. Duarte, D. Kharzeev, E. Levin, O. Oliveira, L. F. Palhares, P. J. Silva and S. P. Sorella for discussions. M. S. Guimaraes thanks “The Conselho Nacional de Desenvolvimento Científico e Tecnológico” (CNPq-Brazil) for the financial support and gratefully acknowledges the hospitality at the KU Leuven Campus Kortrijk - KULAK where part of this work was executed.
D. E. Kharzeev and E. M. Levin, Phys. Rev. Lett. 114 (2015) 24, 242001.
D. E. Kharzeev, arXiv:1509.00465 [hep-ph].
G. Veneziano, Nucl. Phys. B 159 (1979) 213.015
G. 't Hooft, Phys. Rept. 142 (1986) 357.
E. V. Shuryak, World Sci. Lect. Notes Phys. 71 (2004) 1 [World Sci. Lect. Notes Phys. 8 (1988) 1].
E. Witten, Nucl. Phys. B 156 (1979) 269.
D. Diakonov and M. I. Eides, Sov. Phys. JETP 54 (1981) 232
[Zh. Eksp. Teor. Fiz. 81 (1981) 434].
K. Kawarabayashi and N. Ohta, Nucl. Phys. B 175 (1980) 477; H. Hata, T. Kugo and N. Ohta, Nucl. Phys. B 178 (1981) 527.
M. Campostrini, A. Di Giacomo and H. Panagopoulos, Phys. Lett. B 212 (1988) 206.
L. Del Debbio, L. Giusti and C. Pica, Phys. Rev. Lett. 94 (2005) 032003.
K. Cichy, A. Dromard, E. Garcia-Ramos, K. Ottnad, C. Urbach, M. Wagner, U. Wenger and F. Zimmermann, PoS LATTICE 2014 (2014) 075.
E. Seiler and I. O. Stamatescu, Report No. MPI-PAE/PTh 10/87 (1987) (unpublished).
C. Becchi, A. Rouet and R. Stora, Phys. Lett. B 52 (1974) 344; Annals Phys. 98 (1976) 287.
I. V. Tyutin, Lebedev Institute preprint (1975).
V. N. Gribov, Nucl. Phys. B 139 (1978) 1.
D. Zwanziger, Nucl. Phys. B 323 (1989) 513; Nucl. Phys. B 399 (1993) 477.
N. Vandersickel and D. Zwanziger, Phys. Rept. 520 (2012) 175.
I. M. Singer, Commun. Math. Phys. 60 (1978) 7.
R. F. Sobreiro and S. P. Sorella, JHEP 0506 (2005) 054.
M. A. L. Capri, A. D. Pereira, R. F. Sobreiro and S. P. Sorella, Eur. Phys. J. C 75 (2015) 479.
J. Serreau and M. Tissier, Phys. Lett. B 712 (2012) 97.
M. A. L. Capri et al., Phys. Rev. D 92 (2015) 4, 045039; work in progress.
D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel and H. Verschelde, Phys. Rev. D 78 (2008) 065047.
M. Schaden and D. Zwanziger, Phys. Rev. D 92 (2015) 2, 025001.
A. R. Zhitnitsky, Annals Phys. 336 (2013) 462.
M. Q. Huber, Phys. Rev. D 91 (2015) 8, 085018.
A. C. Aguilar, D. Binosi and J. Papavassiliou, Phys. Rev. D 91 (2015) 8, 085014.
A. Cucchieri, T. Mendes and E. M. S. Santos, Phys. Rev. Lett. 103 (2009) 141602.
P. Bicudo, D. Binosi, N. Cardoso, O. Oliveira and P. J. Silva, arXiv:1505.05897 [hep-lat].
J. A. Gracey, JHEP 1002 (2010) 009.
D. Zwanziger, Phys. Rev. Lett. 94 (2005) 182301; Phys. Rev. D 76 (2007) 125014.
G. Burgio, M. Quandt and H. Reinhardt, Phys. Rev. Lett. 102 (2009) 032002.
G. Burgio, M. Quandt and H. Reinhardt, Phys. Rev. D 81 (2010) 074502.
F. Siringo, Phys. Rev. D 90 (2014) 9, 094021; Phys. Rev. D 92 (2015) 7, 074034.
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1511.00764
|
[LO]J. E. Johndrow and A. Bhattacharya
[RE]Optimal credible regions for Bayesian log-linear models
In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis–Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. Here we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis–Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models.
credible region; conjugate prior; contingency table; Dirichet–Multinomial; Kullback–Leibler divergence; Laplace approximaton.
§ INTRODUCTION
Contingency table analysis routinely relies on log-linear models, which represent the logarithm of cell probabilities as an additive model <cit.>. With the standard choice of Multinomial or Poisson likelihood, these are exponential family models, and are routinely fit through maximum likelihood estimation <cit.>. However, sparsity in the observed cell counts often makes maximum likelihood estimation infeasible (see <cit.> and <cit.>) in practical applications. In such cases, regularization is often used to obtain unique parameter estimates <cit.>.
A common Bayesian approach to inference in high-dimensional contingency tables is to place a conjugate prior on the parameters of a graphical or hierarchical log-linear model, and an independent prior over the space of all such models (see e.g. <cit.>). This leads to a standard model-averaged posterior <cit.>, where all possible sparse log-linear models in the chosen class are weighted by their posterior evidence. Use of non-conjugate (e.g. Gaussian) priors with computation by Markov chain Monte Carlo <cit.> has also been proposed <cit.>. Although model averaging is generally considered ideal in high dimensional settings, computational algorithms for posterior inference scale exceedingly poorly in $p$. Since the smallest contingency table corresponding to cross-classification of $p$ categorical variables has $2^p$ cells, the corresponding log-linear model has $2^p-1$ free parameters, so the model space grows super-exponentially in $p$. Accordingly, posterior computation is essentially infeasible for $p > 15$, the largest case demonstrated to date in the literature <cit.> to the best of our knowledge.
Alternatively, one can place a Gaussian prior on the parameters of a saturated log-linear model to induce Tikhonov type regularization, and then perform computation by Markov chain Monte Carlo. This approach is well-suited to situations in which the sample size is not tiny relative to the table dimension, but where zero counts nonetheless exist in some cells. In this case, data augmentation Gibbs samplers such as that proposed by <cit.> provide for conditionally conjugate updates. However, this by itself is computationally intensive relative to alternatives such as elastic net <cit.>, and can suffer from poor mixing. In principle, a more scalable Bayesian approach for producing Tikhonov regularized point estimates would be to utilize the Diaconis–Ylvisaker conjugate prior <cit.> on the parameters of the log-linear model, which is essentially computation free. The main drawback is that the resulting posterior distribution is difficult to work with, lacking closed form expressions for even marginal credible intervals or fast algorithms for sampling from the posterior. An accurate and more tractable approximation to this posterior is therefore of practical interest.
Approximations to the posterior distribution have a long history in Bayesian statistics, with the Laplace approximation perhaps the most common and simple alternative <cit.>. More sophisticated approximations, such as those obtained using variational methods <cit.> may in some cases be more accurate but require computation similar to that for generic EM algorithms. Moreover, there exist no theoretical guarantees of the approximation error in finite samples, and these approximations are known to be inadequate in relatively simple models <cit.>.
In this article, we propose a Gaussian approximation to the posterior for log-linear models with Diaconis–Ylvisaker priors. The approximation is shown to be the optimal Gaussian approximation to the posterior in the Kullback–Leibler divergence, and convergence rates to the exact posterior and a finite-sample Kullback–Leibler error bound are provided. The approximation is shown empirically to be accurate even for modest sample sizes; effectively, the empirical results suggest that the approximation is accurate enough to be used in place of the exact posterior within the range of sample sizes for which the posterior is sufficiently concentrated to be statistically useful. We also show how the approximation can be used to perform model selection using the penalized credible region method <cit.>. In a real data application, the method performs favorably in model selection for graphical log-linear models compared to methods requiring vastly greater computational resources.
§ BACKGROUND
We first provide a brief review of exponential families. We then describe the family of conjugate priors for the natural parameter of an exponential family, referred to as Diaconis–Ylvisaker priors. We then provide more detailed background on log-linear models for Multinomial likelihoods and the associated Diaconis–Ylvisaker prior.
§.§ Exponential families
Following <cit.>, let $\mu$ be a $\sigma$-finite measure defined on $(\bb R^p,\mc B)$, where $\mc B$ denotes all Borel sets on $\bb R^p$. Let $\mbox{supp}(\mu) = \{ y \in \bb R^p : d \mu(y) > 0\}$ be the support of $\mu$, and define $\ms Y$ as the interior of the convex hull of $\mbox{supp}(\mu)$. For $\theta \in \bb R^p$, define $M(\theta) = \log \int_{\ms Y} e^{\theta^T y} d\mu(y)$, and let $\Theta = \{\theta \in \bb R^p : M(\theta) < \infty\}$, which we assume is an open set. We refer to $\Theta$ as the natural parameter space. The exponential family of probability measures $\{P(\cdot;\theta)\}$ indexed by a parameter $\theta \in \Theta$ is defined by
\begin{align}
dP(y;\theta) = e^{\theta^T y - M(\theta)} d\mu(y), \qquad \theta \in \Theta.
\end{align}
This family includes many of the probability distributions commonly used as sampling models in likelihood-based statistics. <cit.> develop the family of conjugate priors for the parameter $\theta$ of regular exponential family likelihoods. These Diaconis–Ylvisaker priors are given by
\begin{align}
d \pi(\theta;n_0,y_0) = e^{n_0 y_0^T\theta - n_0 M(\theta)}, \qquad n_0 \in \bb R, y_0 \in \bb R^d.
\end{align}
On observing data $y$ consisting of $n$ observations with sufficient statistics $\bar{y}$, the posterior is then also Diaconis–Ylvisaker, with parameters $n_0 + n, y_0 + \bar{y}$, i.e. $d \pi(\theta \mid y) = d \pi(\theta;n_0 + n,y_0 + \bar{y})$. In the sequel we focus on one member of the exponential family, the multinomial. In the natural parametrization, the ultinomial likelihood gives rise to the log-linear model and the closely related multinomial logit model, which we now describe.
§.§ Log-linear models
Let $\mc S^d = \{(x_1, \ldots, x_d) \in [0, 1]^d : \sum_{j=1}^d x_j \le 1\}$ denote the $d$-dimensional unit simplex. Consider $N$ independent samples from a categorical variable with $(d+1)$ levels. We denote the levels of the variable by $0, 1, \ldots d$, without loss of generality. Let $y_j$ denote the number of times the $j$th level is observed in the $N$ samples and set $y = (y_0, y_1, \ldots, y_d)^{\T}$; clearly $\sum_{j=0}^d y_j = N$. The joint distribution of $y$ is given by a multinomial distribution, denoted $y \sim \Mult{N}{\pi}$, which is parametrized by $\pi = (\pi_1, \ldots, \pi_d)^{\T} \in \mc S^d$, where $\pi_j$ is the probability of observing the $j$th level for $j = 1, \ldots, d$.
The log-linear model is a generalized linear model for multinomial likelihoods obtained by choosing the logistic link function, which also results in the natural exponential family parametrization. Define the logistic transformation $\ell : \bb R^d \to \mc S^d$ and its inverse log ratio transformation $\ell^{-1}: \m S^d \to \mb R^d$ as
\begin{align}\label{eq:logistic_trans}
\pi_j = \frac{e^{\theta_j}}{1 + \sum_{l=1}^d e^{\theta_l}}, \quad \theta_j = \log(\pi_j/\pi_0), \quad (j = 1, \ldots, d),
\end{align}
where $\pi_0 = 1 - \sum_{j=1}^d \pi_j$, and $\theta_0 = 0$. We shall write $\pi = \ell(\theta)$ and $\theta = \ell^{-1}(\pi) = \log(\pi/\pi_0)$, respectively, to denote the transformations in (<ref>). Using (<ref>), the multinomial likelihood in the log-linear parameterization can be expressed as
\begin{align}\label{eq:mult_lik}
f(y \mid \theta) ~\propto~ \frac{\exp\big(\sum_{j=1}^d y_j \theta_j \big)}{\big(1 + \sum_{l=1}^d e^{\theta_l}\big)^N}.
\end{align}
An important motivating case is when $y = \vect{\mathbf{n}}$, with $\mathbf{n}$ a contingency table arising from cross-classification of $N$ independent observations on $p$ categorical variables $y_1, \ldots, y_p$. Suppose that the $v$th variable $y_v$ has $d_v$ many levels, so that the contingency table has $\prod_{v=1}^p d_v$ many cells, and $y$ is a $(d+1)$-dimensional vector of counts with $d = \prod_{v=1}^p d_v - 1$. We refer to the parametrization $\theta = \log (\pi/\pi_0)$ in the contingency table setting as the identity parametrization. Also of particular interest in this setting are reparametrizations of (<ref>) that represent $\log \pi/\pi_0$ as an additive model involving parameters that correspond to interactions among $y_1,\ldots,y_p$. Every identified parametrization of the log-linear model for the multinomial likelihood can be represented by
\begin{align}
\log(\pi/\pi_0) = X \theta^*, \label{eq:llmX}
\end{align}
where $X$ is a $d$ by $d$ non-singular binary matrix and $\theta^* \in \bb R^d$. In the simulations and application, we make a specific choice for $X$ that corresponds to the corner parametrization of the log-linear model <cit.>. We illustrate the identity and corner parameterizations through a $2^3$ contingency table in Example <ref> below.
Details for the general case can be found in the Appendix.
Consider three binary variables $y_1, y_2, y_3$, with $y_v \in \{0, 1\}$ for $v = 1, 2, 3$, and let
\psi_{i_1 i_2 i_3} = \mbox{pr}(y_1 = i_1, y_2 = i_2, y_3 = i_3), \quad (i_1, i_2, i_3) \in \{0, 1\}^3.
A $2^3$ contingency table $\mathbf{n} = (n_{i_1 i_2 i_3})$ is obtained from the cross-classification of $N$ independent observations on $y_1, y_2, y_3$, with $n_{i_1 i_2 i_3}$ denoting the cell count for the cell $(i_1, i_2, i_3)$. Let $y = \vect{\mathbf{n}} = (n_{000}, \ldots, n_{111})^{\T}$ be the vectorized cell counts with $d = 7$.
In the identity parametrization, the vector of log-linear parameters $\theta \in \bb R^7$ is given by
\begin{align*}
\left( \begin{array}{c} \theta_1 \\ \theta_2 \\ \theta_3 \\ \theta_4 \\ \theta_5 \\ \theta_6 \\ \theta_7 \end{array} \right) =
\log \left( \begin{array}{c} \pi_1/\pi_0 \\ \pi_2/\pi_0 \\ \pi_3/\pi_0 \\ \pi_4/\pi_0 \\ \pi_5/\pi_0\\ \pi_6/\pi_0 \\ \pi_7/\pi_0 \end{array} \right) =
\log \left( \begin{array}{c} \psi_{001}/\psi_{000} \\ \psi_{010}/\psi_{000} \\ \psi_{011}/\psi_{000} \\ \psi_{100}/\psi_{000} \\ \psi_{101}/\psi_{000} \\ \psi_{110}/\psi_{000} \\ \psi_{111}/\psi_{000} \end{array} \right).
\end{align*}
On the other hand, in the corner parametrization, we express
\begin{align*}
\theta = \log \left( \begin{array}{c} \psi_{001}/\psi_{000} \\ \psi_{010}/\psi_{000} \\ \psi_{011}/\psi_{000} \\ \psi_{100}/\psi_{000} \\ \psi_{101}/\psi_{000} \\ \psi_{110}/\psi_{000} \\ \psi_{111}/\psi_{000} \end{array} \right) &= \left( \begin{array}{c} \theta^*_{001} \\ \theta^*_{010} \\ \theta^*_{001} + \theta^*_{010} + \theta^*_{011} \\ \theta^*_{100} \\ \theta^*_{001} + \theta^*_{100} + \theta^*_{101} \\ \theta^*_{010} + \theta^*_{100} + \theta^*_{110} \\ \theta^*_{001} + \theta^*_{010} + \theta^*_{100} + \theta^*_{011} + \theta^*_{101} + \theta^*_{110} + \theta^*_{111} \end{array} \right) \\
&= \left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array} \right) \times \left( \begin{array}{c} \theta^*_{001} \\ \theta^*_{010} \\ \theta^*_{011} \\ \theta^*_{100} \\ \theta^*_{101} \\ \theta^*_{110} \\ \theta^*_{111} \end{array} \right) \\
&= X \theta^*.
\end{align*}
The indexing of the elements of $\theta^*$ by binary indices is for ease of interpretation. Indeed, entries of $\theta^*$ with a single $1$ in the binary index are main effects, those with two $1$'s are two-way interactions and $\theta^*_{111} $ is a three-way interaction term. The matrix $X$ can be easily verified to be non-singular, so that the $\theta$ and $\theta^*$ parametrizations are equivalent, with $d = 7$ free parameters in either case.
§.§ Conjugate priors for log-linear models
We now present the Diaconis–Ylvisaker prior for the multinomial likelihood (<ref>) and derive an optimal Gaussian approximation to the corresponding posterior in Kullback–Leibler divergence. Extensions to log-linear models with a non-identity parametrization (i.e., $X \ne \mathrm{I}_d$ in (<ref>)) is straightforward by invariance properties of the Kullback–Leibler divergence and are discussed subsequently. All proofs are deferred to the Appendix.
For the multinomial likelihood (<ref>), the Diaconis–Ylvisaker prior is obtained by applying the inverse logistic transformation $\ell^{-1}$ to a Dirichlet distribution, which not surprisingly is the conjugate prior for $\pi$. Recall that $\pi_0 = 1 - \sum_{j=1}^d \pi_j$. The Dirichlet distribution $\Diri{\alpha}$ on $\m S^d$ with parameter vector $\alpha = (\alpha_0, \alpha_1, \ldots, \alpha_d)^{\T}$ has density
\begin{align}\label{eq:dir_dens}
q(\pi; \alpha) = \frac{\Gamma(\sum_{j=0}^d \alpha_j)}{\prod_{j=0}^d \Gamma(\alpha_j)} \, \prod_{j=0}^d \pi_j^{\alpha_j - 1}, \quad \pi \in \m S^d,
\end{align}
and corresponding probability measure $\mc{Q}(\cdot, \alpha)$ with $\mc Q(A, \alpha) = \int_A q(\pi; \alpha) d \pi$ for Borel subsets $A$ of $\m S^d$.
Suppose $\pi \sim \Diri{\alpha}$ and let $\theta = \log(\pi/\pi_0) \in \bb R^d$. Define $A = \sum_{j=0}^d \alpha_j$. Then $\theta$ has a density on $\bb R^d$ given by
\begin{align}\label{eq:dyprior}
p(\theta; \alpha) = \frac{\Gamma(\sum_{j=0}^d \alpha_j)}{\prod_{j=0}^d \Gamma(\alpha_j)} \, \frac{\exp( \sum_{j=1}^d \alpha_j \theta_j )}{ (1 + \sum_{l=1}^d e^{\theta_l} )^{A} }.
\end{align}
We write $\theta \sim \mc{LD}(\alpha)$ and use $\mc P(\cdot;\alpha)$ to denote the probability measure associated with the density (<ref>), with $\mc P(B;\alpha) = \int_B p(\theta;\alpha) d\theta$ for Borel subsets $B$ of $\mb R^d$. If a non-identity parametrization $\theta = X \theta^*$ as in (<ref>) is employed, then we denote the induced distribution on $\theta^* = X^{-1} \theta$ by $\mc P_X(\cdot; \alpha)$ and the density by $p_X(\theta; \alpha)$.
It is immediate that $\mc{LD}(\alpha)$ is a conjugate family of prior distributions for the likelihood (<ref>), with the posterior $\theta \mid y \sim \mc{LD}(\alpha + y)$. To obtain some preliminary insight into the distribution family $\mc{LD}(\alpha)$, we derive the mean and covariance in Proposition 2 below.
Let $\theta \sim \mc{LD}(\beta)$, with $\beta = (\beta_0, \beta_1, \ldots, \beta_d)^{\T}$ and $\beta_j > 0$ for all $j$. Then,
\begin{align*}
E(\theta_j) &= \psi(\beta_j) - \psi(\beta_0), \quad (j = 1, \ldots, d) \\
\mathrm{cov}(\theta_j, \theta_{j'}) &= \psi'(\beta_j) \delta_{jj'} + \psi'(\beta_0), \quad (j, j' = 1, \ldots, d)
\end{align*}
where $\psi$ and $\psi'$ are the digamma and trigamma functions, respectively, and $\delta_{jj'}=0$ if $j \ne j'$ and $\delta_{jj'} = 1$ otherwise.
The proof of Proposition <ref> is established within the proof of Theorem <ref> in the Appendix. Assume the data $y$ is generated from a $\Mult{N}{\pi^0}$ distribution and let $\theta^0 = \log (\pi^0/\pi^0_0 )$ be the true log-linear parameter, where $\pi_0^0 = 1 - \sum_{j=1}^d \pi_j^0$. If a $\mc{LD}(\alpha)$ prior is placed on $\theta$, one can use Proposition <ref> to show that the posterior mean $E(\theta \mid y)$ converges almost surely to $\theta^0$ with increasing sample size, and the posterior covariance $\mbox{cov}(\theta \mid y)$ converges to the inverse Fisher information matrix as long as the entries of the prior hyperparameter $\alpha$ are suitably bounded. In fact, a Bernstein–von Mises type result can be established, showing that the posterior distribution approaches a Gaussian distribution, centered at the true parameter value and having covariance the inverse Fisher information matrix, in the total variation metric. We do not pursue such frequentist asymptotic validations further in this paper. Our goal rather is to provide a Gaussian approximation to the posterior distribution that can be used in practice, and provide finite sample bounds to the approximation error.
§ MAIN RESULTS
In this section, we provide an optimal Gaussian approximation to a $\mc{LD}(\beta)$ distribution (<ref>) in the Kullback–Leibler divergence, i.e., we exhibit a vector $\mu^* \in \bb R^d$ and a positive definite matrix $\Sigma^*$ such that the Kullback–Leibler divergence between $\mc{LD}(\beta)$ and $\mc N(\mu^*, \Sigma^*)$ is the minimum among all Gaussian distributions. This result provides a readily available Gaussian approximation to the posterior distribution $\mc{LD}(\beta = \alpha + y)$ of the log-linear parameter $\theta$ in (<ref>) with a Diaconis–Ylvisaker prior $\mc{LD}(\alpha)$. We also provide a non-asymptotic error bound for the Kullback–Leibler approximation. Using Pinsker's inequality, the approximation error in the total variation distance can be bounded in finite samples.
For two probability measures $\nu \ll \nu^*$, we write
\begin{align*}
D(\nu \dmid \nu^*) = E_{\nu^*} \log d\nu/d\nu^*
\end{align*}
to denote the Kullback–Leibler divergence between $\nu$ and $\nu^*$.
Given $\beta_j > 0, j = 0, 1, \ldots, d$, let $\beta = (\beta_0, \ldots, \beta_d)^{\T}$, and define
\begin{align}\label{eq:min_mus}
\mu_j^* = \psi(\beta_j) - \psi(\beta_0), \quad \sigma_{jj'}^* = \psi'(\beta_j) \delta_{jj'} + \psi'(\beta_0),
\end{align}
where $\psi$ and $\psi'$ denote the digamma and trigamma functions respectively. Define $\mu^* = (\mu_j^*) \in \mb R^d$ and $\Sigma^* = (\sigma_{jj'}^*) \in \mb R^{d \times d}$. Then,
\begin{align}
D \bigg\{ \mc{LD}(\beta) \dmid \mc{N}(\mu^*, \Sigma^*) \bigg\} = \inf_{\mu, \Sigma} D \bigg\{ \mc{LD}(\beta) \dmid \mc{N}(\mu, \Sigma) \bigg\},
\end{align}
where the infimum is over all $\mu \in \mb R^d$ and all $\Sigma \succ 0 \in \mb R^{d \times d}$. Further, if $\beta_j > 1/2$ for all $j = 0, 1, \ldots, d$, then
\begin{align}
D \bigg\{ \mc{LD}(\beta) \dmid \mc{N}(\mu^*, \Sigma^*) \bigg\} < \frac{1}{2} \sum_{j=0}^d \frac{1}{\beta_j} + \frac{1}{6 B}, \label{eq:klrate}
\end{align}
where $B = \sum_{j=0}^d \beta_j$.
The matrix $\Sigma^*$ has a compound-symmetry structure and is therefore positive-definite. From Proposition <ref>, the parameters of the optimal Gaussian approximation $\mu^*$ and $\Sigma^*$ are indeed the mean and covariance matrix of the $\mc{LD}(\beta)$ distribution. Equation (<ref>) provides an upper-bound to the approximation error. In the posterior, $\beta_j = \alpha_j + y_j$ and $B = \sum_{j=0}^d \alpha_j + N$. The condition $\beta_j \ge 1/2$ is therefore satisfied whenever every category has at least one observation. Since
\begin{align*}
\bb E_{y}[\alpha_j+y_j] = \alpha_j + N \pi_j^0,
\end{align*}
the approximation error is approximately in the order of $\sum_{j=0}^d (\pi_j^0 N)^{-1}$, where as before $\pi_j^0$ denotes the true probability of category $j$. In the best case where all the categories receive approximately equal probability, i.e., $\pi_j^0 \asymp (d+1)^{-1}$, the approximation error is $\mc O(d^2/N)$. However, the convergence rate in $N$ can be slower if some of the $\pi_j^0$s are very small. In other words, the higher the entropy of the data generating distribution, the worse the approximation is, although our simulations suggest that the approximation is practicable even for moderate sample sizes and unbalanced category probabilities. When one considers that the eigenvalues of the covariance matrix enter into the constant in Berry-Esséen convergence rates, and that here the covariance of the data is given by $\mbox{diag}(\pi^0) - \pi^0 (\pi^0)^{\T}$, it appears that a similar phenomenon is at work here.
The main idea behind our proof is to exploit the invariance of the Kullback–Leibler divergence under bijective transformations and transfer the domain of the problem from $\bb R^d$ to $\mc S^d$. Since an $\mc{LD}(\beta)$ distribution is obtained from a Dirichlet $\mc{D}(\beta)$ distribution via the inverse log-ratio transform $\ell^{-1}$, the problem of finding the best Gaussian approximation to $\mc{LD}(\beta)$ is equivalent to finding the best approximation to $\mc{D}(\beta)$ among a class of distributions obtained by applying the logistic transform to Gaussian random variables. If $\theta \sim N(\mu, \Sigma)$, the induced distribution on $\pi = \ell(\theta)$ is called a logistic normal distribution – denoted $\mc L(\mu, \Sigma)$ – and has density on $\m S^d$ given by
\begin{align}\label{eq:ln_dens}
\wt{q}(\pi; \mu, \Sigma) = (2 \pi)^{-d/2} |\Sigma|^{-1/2} \bigg( \prod_{j=0}^d \pi_j \bigg)^{-1} \exp \left[ -\frac{1}{2} \{\log(\pi/\pi_0) - \mu \}^{\T} \Sigma^{-1} \{\log(\pi/\pi_0) - \mu \} \right].
\end{align}
The problem therefore boils down to calculating the Kullback–Leibler divergence between a Dirichlet density $q(\cdot; \beta)$ and a logistic normal density $\wt{q}(\cdot; \mu, \Sigma)$ and optimizing the expression with respect to $\mu$ and $\Sigma$. The details are deferred to the Appendix.
Once the approximation is derived in the identity parametrization, we appeal to the invariance of the Kullback–Leibler divergence under one-to-one transformations to obtain the corresponding approximation in a non-identity parameterization $\theta = X \theta^*$ as in (<ref>) for any non-singular $X$. The result is stated below.
If $\theta \sim \mc{LD}(\beta)$ then
\begin{align}
D\left( \mc P_X(\cdot;\beta) \dmid \mc N(\cdot;X \mu^*,X^T \Sigma^* X) \right) = \inf_{\mu, \Sigma} D\left( \mc P_X(\cdot;\beta) \dmid \mc N(\cdot;\mu,\Sigma) \right)
\end{align}
for any full-rank $d$ by $d$ matrix $X$. Moreover, the bound on the KL divergence as a function of $\beta$ in (<ref>) is attained for $ D\left(\mc P_X(\cdot;\beta) \dmid \mc N(\cdot;\mu^*,\Sigma^*)\right)$
Thus, the best Gaussian approximation to the posterior (in the Kullback–Leibler sense) under the Diaconis–Ylviaker prior is given by $N(X \mu^*,X' \Sigma^* X)$ for any one-to-one linear transformation $X$. We refer to this as the optimal Normal (oN) approximation.
§ SIMULATIONS
We conducted several simulation studies to assess the performance of the approximation in Theorem <ref> and Corollary <ref>. In each study, we simulated 100 realizations from
\begin{align}
\pi \sim \Diri{a,\ldots,a}, \quad y \sim \Mult{N}{\pi}, \label{eq:simmodel}
\end{align}
with the posterior of $\pi$ under a Dirichlet $\Diri{a,\ldots,a}$ prior being $\Diri{y_1+a,\ldots,y_d+a}$.
We chose the dimension $d$ to be $2^8$, corresponding to a $p=8$-way contingency table for binary variables. To obtain a simulation-based approximation to the posterior for $\theta = \log(\pi/\pi_0)$ under the Diaconis–Ylvisaker prior, we sampled $mc$ many $\pi$ values from the $\Diri{y_1+a,\ldots,y_d+a}$ posterior and then transformed to $\theta = \ell^{-1}(\pi)$ to obtain posterior samples of $\theta$; we refer to this procedure as the Monte Carlo approximation. We also computed a Laplace approximation to the posterior under the Diaconis–Ylvisaker prior, which is given by $\No{\hat{\theta}_{MAP}}{\mc I(\hat{\theta}_{MAP})^{-1}}$, where $\hat{\theta}_{MAP}$ is the maximum a-posteriori estimate of $\theta$ and $\mc I(\theta)$ is the Fisher information matrix evaluated at $\theta$. The maximum a-posteriori estimate $\hat{\theta}_{MAP}$ was computed by the Newton–Raphson method.
We compare the accuracy of the proposed Gaussian approximation to the Monte Carlo procedure and the Laplace approximation. In addition to the identity parameterization, i.e., $X = \mr I_d$ in (<ref>), we also consider the corner parameterization given by $\log(\pi/\pi_0) = X \theta^*$ for an appropriate $X$ matrix; see Appendix for more details. For the Monte Carlo samples, each sample of $\theta$ is transformed to $\theta^*$ via $X^{-1} \theta = \theta^*$. For the normal approximations $\theta \sim \No{\mu}{\Sigma}$, the corresponding approximate posterior is given by $\theta^* \sim \No{X^{-1} \mu}{X^{-1} \Sigma X^{-1}}$.
We conduct simulations for different values of $N$ (250, 1000, and 10,000) and $a$ ($1$ and $1/d$). We then assess performance in several ways.
* Proportion of variation unexplained, measured by $\sqrt{\sum_{j=1}^d (\theta - \theta_0)^2}/\mbox{sd}(\theta_0)$, where $\theta_0$ is the true value of $\theta$ (or $\theta^*$, as appropriate).
* Coverage of 95 percent posterior credible intervals for $\theta$ or $\theta^*$.
* The standardized loss in the Frobenius norm for estimates of $\Sigma$, the posterior covariance, given by $||\widehat{\Sigma}-\Sigma||_F/||\Sigma||_F$, where $||S||_F$ is the Frobenius norm of $S$. Note that the covariance in Theorem <ref> is exactly the posterior covariance, so this measure is computed only for the simulation and Laplace approximations.
* The value of the Kolmogorov-Smirnov statistic for comparing the Monte Carlo empirical measure $\frac{1}{mc} \sum_{t=1}^{mc} \delta_{\theta_t}$ to the normal approximation from Theorem <ref>, $\No{\mu}{\Sigma}$.
* The computation time required to compute each posterior approximation.
Table <ref> shows unexplained variation for the Laplace approximation, the Monte Carlo approximation for $mc=10^3, 10^4, 10^5,$ and $10^6$, and the optimal normal approximation. As expected, the optimal normal approximation outperforms the Laplace approximation. Moreover, it is comparable to the Monte Carlo approximation at every sample size and for all of the values of $mc$ considered. Performance for all approximations is noticeably better in the corner parametrization than the identity parametrization.
$\sqrt{\sum_{j=1}^d (\theta - \theta_0)^2}/\mbox{sd}(\theta_0)$ for different values of $mc$, different sample sizes, and two parametrizations. Results are averaged over 100 replicate simulations for each sample size.
Laplace $mc=10^3$ $mc=10^4$ $mc=10^5$ $mc=10^6$ oN
identity, N=250 1.08 0.98 0.98 0.98 0.98 0.98
corner, N=250 0.85 0.81 0.81 0.81 0.81 0.81
identity, N=1000 0.84 0.77 0.77 0.77 0.77 0.77
corner, N=1000 0.67 0.61 0.61 0.61 0.61 0.61
identity, N=10,000 0.40 0.35 0.35 0.35 0.35 0.35
corner, N=10,000 0.31 0.27 0.27 0.27 0.27 0.27
Table <ref> shows coverage of approximate 95 percent credible intervals for the Laplace approximation, optimal Normal approximation, and the Monte Carlo approximation. The intervals derived using the Laplace approximation are universally too wide. Nominal coverage for the Monte Carlo approximation is insensitive to the value of $mc$ in the range tested, and is slightly high at the two smaller sample sizes. The optimal normal approximation has the best coverage; in all cases it is between 0.94 and 0.96 and for $N=10,000$ the coverage is 0.95 in both parametrizations.
coverage of 95% posterior credible intervals
Laplace $mc=10^3$ $mc=10^4$ $mc=10^5$ $mc=10^6$ oN
identity, N=250 0.95 0.97 0.97 0.97 0.97 0.96
corner, N=250 1.00 0.96 0.96 0.96 0.96 0.96
identity, N=1000 0.98 0.96 0.96 0.96 0.96 0.96
corner, N=1000 1.00 0.94 0.94 0.94 0.94 0.94
identity, N=10,000 1.00 0.95 0.95 0.95 0.95 0.95
corner, N=10,000 1.00 0.95 0.95 0.95 0.95 0.95
Table <ref> shows dependence of $||\widehat{\Sigma}-\Sigma||_F/||\Sigma||_F$ on $mc$ for the two different parametrizations and three sample sizes considered. Note that $\Sigma$ is known exactly since $\Sigma=\Sigma^*$, the posterior covariance under the DY prior. The main point of this table is to demonstrate the relatively large number of Monte Carlo samples required to obtain reasonably small error in estimation of the posterior covariance. Even with $10^5$ samples the relative error is on the 1 percent range. Thus, compound linear hypothesis testing and computation of credible regions is very inefficient using the Monte Carlo method.
$||\widehat{\Sigma}-\Sigma||_F/||\Sigma||_F$ for different sample sizes and values of $mc$.
$mc=10^3$ $mc=10^4$ $mc=10^5$ $mc=10^6$
identity, N=250 0.0982 0.0328 0.0093 0.0032
corner, N=250 0.0923 0.0290 0.0086 0.0029
identity, N=1000 0.1045 0.0330 0.0103 0.0035
corner, N=1000 0.0882 0.0277 0.0087 0.0029
identity, N=10,000 0.1231 0.0397 0.0118 0.0040
corner, N=10,000 0.0861 0.0280 0.0084 0.0027
Table <ref> shows the computation time in seconds for each of the three approximations. The Laplace approximation is fast, requiring about 0.03-0.04 seconds to compute at all sample sizes. The optimal normal approximation is about an order of magnitude faster, with the computation time arising mainly in computing the polygamma functions and matrix multiplications. The Monte Carlo approximation is about four orders of magnitude slower than the optimal Normal approximation. Here, only $mc=10^6$ is considered because of the non-negligible error in the posterior covariance for smaller samples; the algorithm scales linearly in $mc$ so for $mc=10^5$ the required time would be approximately 3 seconds. Only about 100 samples could be obtained in the 0.003 seconds required to compute the optimal normal approximation.
Average time (seconds) to compute each approximation, averaged over 100 replicate simulations for each sample size.
Laplace $mc=10^6$ oN
N=250 0.037 32.652 0.003
N=1000 0.031 31.980 0.003
N=10,000 0.035 32.338 0.003
Results in the previous tables make clear that the optimal normal approximation is superior to the other approximations considered in terms of point estimation, estimation of 95 percent credible intervals, covariance estimation, and computation time. However, it is possible that differences between the optimal normal approximation and the exact posterior exist in the tails of the distribution. To assess this, we compare the empirical measure of the Monte Carlo approximation using $mc=10^6$ samples to the optimal normal approximation by computing the Kolmogorov-Smirnov (KS) statistic for the marginal distributions of 20 randomly selected entries of $\theta$. The entries considered were re-selected for each of the 100 replicate simulations and for each of the three sample sizes. Shown in Figure <ref> are histograms of these KS statistics in the corner and identity parametrizations. Most are less than 0.02, and none are greater than 0.07. Considering that the KS statistic is a point estimate of the total variation distance between distributions, this indicates that the optimal normal approximation is an excellent approximation to the posterior marginals. Moreover, we cannot rule out the possibility of residual Monte Carlo error in the marginals from the Monte Carlo approximation, which may account for part of the observed discrepancy.
Kolmogorov-Smirnov – identity Kolmogorov-Smirnov – corner
Distribution of Kolmogorov-Smirnov statistics comparing $\frac{1}{mc} \sum_{t=1}^{mc} \delta_{\theta_t}$ to the oN approximation for 20 randomly selected entries of $\theta$ and over 100 replicate simulations (entries of $\theta$ were re-selected for each replicate).
§ REAL DATA EXAMPLE
We consider the Rochdale data, a $2^8$ contingency table with $N=665$ that is over 50 percent sparse, and for which the top ten cell counts all exceed 20. This dataset is described at length in <cit.>. We first assess the accuracy of the approximation to the full posterior under the Diaconis–Ylvisaker prior in the same manner as in <ref>, by comparing marginal posteriors computed using the approximation to those obtained from large Monte Carlo samples from the exact Dirichlet posterior transformed to the log-linear parametrization. For the log-linear model in the corner parametrization, the distribution of Kolmogorov-Smirnov statistics computed for the 255 entries of $\theta^*$ obtained by comparing $10^6$ Monte Carlo samples from the exact posterior to the optimal Gaussian approximation is shown in Fig. <ref>. The distribution is very similar to that observed for the simulations in <ref>.
Histogram of Kolmogorov-Smirnov statistics for the comparison of $10^6$ Monte Carlo samples from the exact Dirichlet posterior, transformed to $\theta^*$, to the optimal Gaussian approximation to the posterior for $\theta^*$ under the Diaconis–Ylvisaker prior.
Undoubtedly, the Diaconis–Ylvisaker prior is less well-suited to inference on important variable interactions in this dataset than the more sophisticated methods of <cit.> and <cit.>. However, our approximation has the advantage of being essentially computation-free, whereas the methods of <cit.> and <cit.> are computationally intensive even at this small scale. In many settings, particularly with modern large-scale problems, some loss of performance may be acceptable in order to obtain useful inferences instantaneously. Thus, we are interested in the extent to which our method can replicate the results of <cit.>, which were similar to those of <cit.> in many respects. We analyze performance in testing conditional independence hypotheses (i.e. learning an interaction graph).
Sparse $\theta^*$ is a set of measure zero with respect to the posterior under the Diaconis–Ylvisaker prior. To obtain a sparse point estimate of the interaction graph, we employ the penalized credible region approach of <cit.>. This method produces a point estimate by finding the sparsest $\theta^*$ within a $1-\alpha$ credible region for $\theta^*$. Although the exact solution to this problem is intractable, <cit.> show that it can be approximated using a lasso path, and provide software in the package <cit.>. Using the resulting lasso path from , the selected model corresponding to any value of $\alpha \in (0,1)$ can be obtained as follows.
* For the selected value of $\alpha$, find the $1-\alpha$ quantile of a $\chi^2$ distribution with $d-1$ degrees of freedom. Label this $\delta_{\max}$.
* For each model $\theta_0$ in the Lasso path, compute the Mahalanobis distance $\delta(\theta_0) = (\theta^*-\theta_0)^T (\Sigma^*)^{-1} (\theta^*-\theta_0)$.
* Find the sparsest model in the lasso path having $\delta(\theta_0)\le \delta_{\max}$. This is the sparse point estimate.
With 256 cells and 665 observations, the posterior under the saturated model with Diaconis–Ylvisaker prior is very diffuse. To make a reasonable comparison, we obtain the posterior under the Diaconis–Ylvisaker prior for the marginal tables corresponding to all ${8 \choose 4} = 70$ unique subsets of four variables. For each of these marginal tables, we then utilize the penalized credible region procedure of <cit.> to obtain a sparse model. For comparison, we utilize the median probability graphical model from <cit.>, which is shown in Table <ref>. Specifially, for every subset of four variables, we obtain the marginal graph corresponding to the median probability model of <cit.> by removing the complement of the subset of nodes under consideration and moralizing, i.e. placing an edge between nodes that (1) have an edge between them in the full graph or (2) are connected solely by a path through nodes that were removed. We treat the graph obtained in this way as the standard for assessing performance of the penalized credible region applied to our Gaussian posterior approximation.
We compute the true (false) negative and positive counts for the penalized credible region procedure applied to our posterior Gausian approximation to all 70 marginal graphs, treating the corresponding marginal median probability graph from <cit.> as the truth. This produces a total of $70 {4 \choose 2} = 420$ dependent pseudo hypothesis tests. The results for $\alpha = 0.1$ in the penalized credible region procedure are shown in Table <ref>. We obtain a false discovery rate of 0.02, and an $F_1$ score of 0.89, indicating that for marginal tables of size $2^4$, the posterior approximation is useful for model selection on the Rochdale data.
Left, titled CGGM Results: Marginal posterior inclusion probabilities of edges (above the main diagonal) and indicator of edge inclusion in the median probability model (below the main diagonal) from copula Gaussian graphical model estimated on Rochdale data in <cit.>. Rows and columns correspond to the eight binary variables, which are labeled a-h. Right, titled Comparison to oN: table of edge classifications for all marginal tables of size $2^4$ from copula Gaussian graphical model median probability model (columns, labeled CGGM) and penalized credible region for Gaussian approximation to posterior under the DY prior (rows, labeled oN-PCR).
CGGM Results Comparison to oN
a b c d e f g h
a – 0.93 0.67 0.92 0.32 0.42 1 0.26
b 1 – 0.27 1 0.88 0.29 0.70 0.96
c 1 0 – 0.29 0.91 0.35 0.85 0.25
d 1 1 0 – 0.37 0.59 0.66 0.50
e 0 1 1 0 – 0.98 0.58 0.17
f 0 0 0 1 1 – 0.82 0.22
g 1 1 1 1 1 1 – 0.32
h 0 1 0 1 0 0 0 –
0 1
2*oN-PCR 0 4 74
1 7 335
§ DISCUSSION
Outside of linear models, conjugate priors are often non-standard or their multivariate generalizations are difficult to work with. This hampers uncertainty quantification because it is difficult to obtain posterior credible regions for parameters under such priors. Given that automatic and coherent quantification of uncertainty through the posterior is one of the chief advantages of a fully Bayesian approach, this limitation is a significant problem. The optimal Gaussian approximation to the posterior for log-linear models with Dianconis-Ylvisaker conjugate priors derived here offers a highly accurate and essentially computation-free approximation to posterior credible regions for this important class of models. Interestingly, this Gaussian approximation is not the Laplace approximation, and it is faster to compute and offers a better approximation to the posterior than the Laplace approximation. If similar results could be obtained for the posterior in other models, it suggests that the Laplace approximation may not be an appropriate default Gaussian approximation to the posterior. The theoretical result provided here can be easily extended to cases where some categories cannot co-occur, i.e. cases of structural zeros in contingency tables. Extensions to model selection using our approximation are also available by the penalized credible region approach. It seems reasonable that the strategy used here to obtain optimality and convergence rate guarantees could be extended to a larger class of generalized linear models by studying the properties of multivariate Gaussian distributions under inverse link transformations. This may also present a strategy for obtaining approximate credible intervals for parameters in the Bayesian model averaging context for generalized linear models with conjugate priors.
§ ACKNOWLEDGEMENT
The authors thank David Dunson for useful conversations and comments during the preparation of this manuscript.
§ LOG-LINEAR MODEL DETAILS
The discussion here largely follows <cit.> and <cit.> in its presentation. Let $V$ be the set of variables that will be collected into a contingency table. Let $\mc I_{\gamma}, \gamma \in V$ denote the set of possible levels of values of $\gamma$. Without loss of generality, we can take this set to be a finite collection of sequential nonnegative integers. Let $\mc I = \bigtimes_{\gamma \in V} \mc I_{\gamma}$ be the set of all possible combinations of levels of the variables in $V$. Every cell $i$ of the contingency table corresponds to an element of $V$; thus $|\mc I| = d+1$, where $d$ is defined as in the main text.
Following <cit.>, define a cell of the contingency table as $i = (i_{\gamma}, \gamma \in V)$, and let $\pi(i) = \pr[y_1 = i_1,\ldots,y_p = i_p]$. For any $E \subset V$, let $i_E = (i_{\gamma}, \gamma \in E)$ be the cell of the $E$-marginal table corresponding to the values in $i$ of the variables in $E$. Finally, designate the “base” cell $i^* = (0,0,\ldots,0)$. Thus, every $i$ can be written as $i = (i_E, i^*_{E^c})$, where $E$ is the subset of $V$ on which $i \ne 0$. Then, the log-linear model in the corner parametrization is given by
\begin{align*}
\log \frac{\pi(i_E,i^*_{E^c})}{\pi(i^*)} = \sum_{F \subseteq_{\emptyset} E} \theta_F(i_F),
\end{align*}
where for any $F \subset V$, $\theta_F(i_F)$ is a parameter corresponding the the variables in $F$ taking the values in $i_F$, and the notation $\subseteq_{\emptyset}$ means all subsets excluding the empty set. Refer to Proposition 2.1 in <cit.> for a result showing how the model can be expressed in the form in (<ref>).
§ PROOF OF PROPOSITION <REF>
This is readily seen by the change of variable theorem; one only needs some work to calculate the Jacobian term for the change of variable. The matrix of partial derivatives $J = (\partial \theta_j/\partial \pi_r)_{jr}$ is given by
\frac{\partial \theta_j}{\partial \pi_j} = \frac{1 - \sum_{l \ne j} \pi_l }{\pi_j (1 - \sum_{l=1}^d \pi_l) }, \quad \frac{\partial \theta_j}{\partial \pi_r} = - \frac{1}{1 - \sum_{l=1}^d \pi_l}, \quad (1 \le j \ne r \le d).
Write $J = U + u u^{\T}$, where $u = (1 - \sum_{l=1}^d \pi_l)^{-1/2} (1, -1, \ldots, -1)^{\T}$ and $U = \mbox{Diag}(1/\pi_1, \ldots, 1/\pi_d)$. We then have $|J| = |U| (1 + u^{\T} U^{-1} u)$ and therefore,
|J|^{-1} = \pi_1 \ldots \pi_d \bigg(1 - \sum_{l=1}^d \pi_l \bigg) = \frac{e^{\sum_{l=1}^d \theta_l} }{ (1 + \sum_{l=1}^d e^{\theta_l} )^{d+1} }.
The proof is concluded by noting that
$p(\theta; \alpha) = q(\ell(\theta); \alpha) \, |J|^{-1}$.
§ PROOF OF MAIN RESULTS
We first state some preparatory results that are used to prove the main results.
§.§ Preliminaries
The following identity for the Gamma function is well known (see, e.g., <cit.>). For $z > 0$,
\begin{align}\label{eq:gamma_fn_bd}
\Gamma(z) = \frac{\log(2\pi)}{2} + \bigg(z - \frac{1}{2}\bigg) \log z - z + R(z),
\end{align}
where $0 < R(z) < 1/(12 z)$.
The digamma function $\psi(z) = \frac{d}{dz} \log \Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}$ satisfies $\psi(z+1) = \psi(z) + 1/z$ for any $z > 0$.
We use the following bound for the digamma function from Lemma 1 of <cit.>. For any $z > 0$,
\begin{align}\label{eq:digamma_bd1}
\frac{1}{2z} - \frac{1}{12z^2} < \psi(z+1) - \log z < \frac{1}{2z}.
\end{align}
The trigamma function $\psi'(z) = \frac{d}{dz} \psi(z)$ is the derivative of the digamma function. We derive a simple bound for the trigamma function that is used in the sequel.
For any $z > 1/3$,
\begin{align}\label{eq:trigamma_bd_f}
\frac{1}{z} < \psi'(z) < \frac{1}{z} + \frac{1}{z^2}.
\end{align}
The condition $z > 1/3$ is only required for the upper bound.
From <cit.>, the trigamma function admits a series expansion
\psi'(z) = \sum_{j=0}^{\infty} \frac{1}{(z+j)^2}
valid for any $z > 0$. The function $t \mapsto t^{-2}$ is monotonically decreasing on $(0, \infty)$ and hence $x^{-2} > \int_{x}^{x+1} t^{-2} dt$ for any $x > 0$. Therefore, for any $z > 0$, $\psi'(z) > \sum_{j=0}^{\infty} \int_{z+j}^{z+j+1} t^{-2} dt = \int_{z}^{\infty} t^{-2} dt = z^{-1}$. For the upper bound, we use Lemma 1 of <cit.> which states that $1/z - \psi'(z+1) > 1/(2z^2) - 1/(6z^3)$ for any $z > 0$. Since $\psi(z+1) = \psi(z) + 1/z$, $\psi'(z+1) = \psi'(z) - 1/z^2$, which yields $\psi'(z) - 1/z < 1/z^2 - 1/(2z^2) + 1/(6z^3) = 1/(2z^2) + 1/(6z^3)$ for any $z > 0$. The conclusion follows since $1/(6z^3) < 1/(2 z^2)$ for any $z > 1/3$.
Finally, we state a useful result in Lemma <ref>.
Let $X \in \mb R^d$ be a random vector with $E X = \mu_X$ and $\mbox{var}(X) = \Sigma_X$. For $\mu \in \mb R^d$ and $d \times d$ positive definite matrix $\Sigma$, the mapping
\begin{align}
(\mu, \Sigma) \mapsto g(\mu, \Sigma) = \log |\Sigma| + E (X - \mu)^{\T} \Sigma^{-1} (X - \mu)
\end{align}
attains its minima when $\mu = \mu_X$ and $\Sigma = \Sigma_X$. The minimum value of the objective function $g(\mu_X, \Sigma_X) = \log |\Sigma_X| + d$.
To start with, $E\{ (X - \mu_X)^{\T} \Sigma_X^{-1} (X - \mu_X) \} = \tr [ E \{ (X - \mu_X)(X - \mu_X)^{\T} \Sigma_X^{-1} \}] = \tr(\mr I_d) = d$ and hence $g(\mu_X, \Sigma_X) = \log |\Sigma_X| + d$. Fix $\mu \in \mb R^d$ and $\Sigma$ positive definite. We can write
\begin{align*}
E\{ (X - \mu)\Sigma^{-1} (X - \mu) \}
& = \tr[ E \{ (X - \mu) (X - \mu)^{\T} \Sigma^{-1} \} ] \\
& = \tr[ E\{ (X - \mu_X) (X - \mu_X)^{\T} \Sigma^{-1} \} + (\mu_X - \mu)\Sigma^{-1}(\mu_X - \mu) ] \\
& = \tr(\Sigma_X \Sigma^{-1}) + (\mu_X - \mu)^{\T} \Sigma^{-1} (\mu_X - \mu).
\end{align*}
\begin{align*}
g(\mu, \Sigma) - g(\mu_X, \Sigma_X) = \tr(\Sigma_X \Sigma^{-1}) + (\mu_X - \mu)^{\T} \Sigma^{-1} (\mu_X - \mu) - d - \log |\Sigma_X \Sigma^{-1}|.
\end{align*}
The above quantity is non-negative since it equals $2 D\big\{N(\mu_X, \Sigma_X) \dmid N(\mu, \Sigma)\big\}$, i.e., twice the Kullback–Leibler divergence between $N(\mu_X, \Sigma_X)$ and $N(\mu, \Sigma)$. Since $\mu$ and $\Sigma$ were arbitrary, the first part is proved. The second part has been already proved at the beginning.
§.§ Proof of Theorem <ref> and Corollary <ref>
We can now give a proof of Theorem <ref>. Recall the Dirichlet density $q$ from (<ref>) and the logistic normal density $\wt{q}$ from (<ref>). We shall write $q(\pi)$ and $\wt{q}(\pi)$ in place of $q(\pi \mid \beta)$ and $\wt{q}(\pi \mid \mu, \Sigma)$ henceforth for brevity. From (<ref>) and (<ref>),
\begin{align*}
\log \frac{q(\pi)}{\wt{q}(\pi)} = \log B_{\beta} + \frac{d \log (2 \pi)}{2} + \sum_{j=0}^d \beta_j \log \pi_j + + \frac{\log |\Sigma|}{2} + \frac{1}{2} \big\{ \log(\pi/\pi_0) - \mu \big\}^{\T} \Sigma^{-1} \big\{ \log(\pi/\pi_0) - \mu \big\}.
\end{align*}
Observe that $\mu$ and $\Sigma$ appear only in the last two terms in the right hand side of the above display. Invoking Lemma <ref>, it is therefore evident that $D(q \dmid \wt{q}) = E_q \log(q/\wt{q})$ is minimized when $\mu^* = E_q \log(\pi/\pi_0)$ and $\Sigma^* = \mbox{var}_q \{\log(\pi/\pi_0)\}$, and the minimum vaue of the Kullback–Leibler divergence is
\begin{align}\label{eq:min_kl_1}
\log B_{\beta} + \sum_{j=0}^d \beta_j E_q \log \pi_j + \frac{d}{2}\{1 + \log (2 \pi)\} + \frac{\log |\Sigma^*|}{2}.
\end{align}
Using standard properties of the Dirichlet distribution or Exponential family differential identities, with $\beta = \sum_{j=0}^d \beta_j$,
\begin{align}
& E_q \log \pi_j = \psi(\beta_j) - \psi(\beta), \quad j = 0, 1, \ldots d, \label{eq:dir_prop1}\\
& \mbox{cov}_q(\log \pi_j, \log \pi_l) = \psi'(\beta_j) \delta_{jl} - \psi'(\beta), \quad j, l = 0, 1, \ldots, d. \label{eq:dir_prop2}
\end{align}
Therefore, $\mu_j^* = E_q \log \pi_j - E_q \log \pi_0 = \psi(\beta_j) - \psi(\beta_0)$ for $j=1, \ldots d$. Next, $\sigma_{jj'}^* = \mbox{cov}_q(\log \pi_j - \log \pi_0, \log \pi_{j'} - \log \pi_0) = \delta_{jj'} \psi'(\beta_j) + \psi'(\beta_0) $ for $j, j' = 1, \ldots, d$. The expressions for $\mu^*$ and $\Sigma^*$ are identical to (<ref>), proving the first part of the theorem. Note this also establishes Proposition <ref>.
We now proceed to bound each term in the expression for the minimum Kullback–Leibler divergence in (<ref>); refer to them by $T_1, T_2, T_3$ and $T_4$ respectively. First, we have,
\begin{align}
T_1 & := \log B_{\beta} = \log \Gamma(\beta) - \sum_{j=0}^d \Gamma(\beta_j) \notag \\
& < - \frac{d \log (2\pi)}{2} + \bigg( \beta \log \beta - \sum_{j=0}^d \beta_j \log \beta_j \bigg) - \frac{1}{2} \bigg(\log \beta - \sum_{j=0}^d \log \beta_j \bigg) + \frac{1}{12 \beta}. \label{eq:bd_T1}
\end{align}
In the above display, we used (<ref>) to bound $\log \Gamma(\beta)$ from above and $\log \Gamma(\beta_j)$s from below.
The $(-\beta)$ term in upper bound to $\log \Gamma(\beta)$ cancels out the $(-\sum_{j=0}^d \beta_j)$ contribution from the lower bounds to the $\log \Gamma(\beta_j)$s. Next,
\begin{align}
&:= \sum_{j=0}^d \beta_j E_q \pi_j = \sum_{j=0}^d \beta_j \{\psi(\beta_j) - \psi(\beta)\} \notag \\
& = \sum_{j=0}^d \beta_j \{ \psi(\beta_{j+1}) - \psi(\beta+1) \} - \sum_{j=0}^d \beta_j \bigg(\frac{1}{\beta_j} - \frac{1}{\beta} \bigg) \notag \\
& = \bigg\{\sum_{j=0}^d \beta_j \psi(\beta_{j+1}) - \beta \psi(\beta) \bigg\} - d \notag \\
& < \bigg(\sum_{j=0}^d \beta_j \log \beta_j - \beta \log \beta\bigg) - \frac{d}{2} + \frac{1}{12 \beta}. \label{eq:bd_T2}
\end{align}
In the first line of the above display, we used (<ref>). From the first to the second line, we used the identity $\psi(z+1) = \psi(z) + 1/z$. From the second to the third line, we only use $\sum_{j=0}^d \beta_j = \beta$. From the third to the fourth line, we made use of the bound (<ref>) for the digamma function $\psi$. From the upper bound in (<ref>), $\beta_j \psi(\beta_{j+1}) < \beta_j \log \beta_j + 1/2$ and hence $\sum_{j=0}^d \beta_j \psi(\beta_{j+1}) < \sum_{j=0}^d \beta_j \log \beta_j + (d+1)/2$. From the lower bound in (<ref>), $\beta \psi(\beta) > \beta \log \beta + 1/2 - 1/(12 \beta)$.
Finally, from (<ref>), we can write $\Sigma^* = D + \psi'(\beta_0) \bs 1 \bs{1}^{\T}$, with $D = \mbox{diag}(\psi'(\beta_1), \ldots, \psi'(\beta_d))$. Using the fact $|X + u v^{\T}| = |X| (1 + v^{\T} X^{-1} u)$, we obtain
\begin{align*}
|\Sigma^*| = \bigg\{ 1 + \sum_{j=1}^d \psi'(\beta_0)/\psi'(\beta_j) \bigg\} \bigg\{ \prod_{j=1}^d \psi'(\beta_j) \bigg\}
= \bigg\{ \sum_{j=0}^d \frac{\psi'(\beta_0)}{\psi'(\beta_j)} \bigg\} \bigg\{ \prod_{j=1}^d \psi'(\beta_j) \bigg\}.
\end{align*}
From Lemma <ref>, $\psi'(\beta_j) > 1/\beta_j$, implying
\begin{align}
&:= \frac{\log |\Sigma^*|}{2} = \frac{1}{2} \bigg[ \log \bigg\{ \sum_{j=0}^d \frac{\psi'(\beta_0)}{\psi'(\beta_j)} \bigg\} + \sum_{j=1}^d \log \psi'(\beta_j) \bigg] \notag \\
& < \frac{1}{2} \bigg\{ \log \beta + \sum_{j=0}^d \log \psi'(\beta_j) \bigg\}. \label{eq:bd_T4}
\end{align}
Recalling $T_3 = d \{1 + \log (2 \pi)\}/2$ and substituting the bounds for $T_1, T_2$ and $T_4$ from (<ref>), (<ref>) and (<ref>) in (<ref>), we obtain, after plenty of cancellations,
\begin{align*}
\sum_{j=1}^4 T_j
&< \frac{1}{2} \sum_{j=0}^d \log \{ \beta_j \psi'(\beta_j) \} + \frac{1}{6 \beta} \\
& < \frac{1}{2} \sum_{j=0}^d \frac{1}{\beta_j} + \frac{1}{6 \beta}.
\end{align*}
From the first to the second line, we invokeed Lemma <ref> to bound $\beta_j \psi'(\beta_j) < 1 + 1/\beta_j$ and used $\log(1+x) < x$ for $x > 0$. We have obtained the desired bound, concluding the proof.
Now, to show Corollary <ref>, just note that
by the invariance of $D$ under one-to-one transformations, we have that for any full rank matrix $X$,
\begin{align}
D \bigg\{ \mc{LD}(\beta) \dmid \mc{N}(\mu, \Sigma) \bigg\} = D \bigg\{ \mc{P}_X(\cdot;\beta) \dmid \mc{N}(X \mu, X^{\T} \Sigma X) \bigg\}. \label{eq:kltrans}
\end{align}
\begin{align}
\inf_{\mu, \Sigma} \bigg\{ \mc{LD}(\beta) \dmid \mc{N}(\mu, \Sigma) \bigg\} = \inf_{\widetilde{\mu}, \widetilde{\Sigma}} D \bigg\{ \mc{P}_X(\cdot;\beta) \dmid \mc{N}(\widetilde{\mu}, \widetilde{\Sigma}) \bigg\}. \label{eq:klinf}
\end{align}
Since the infimum on the left side in (<ref>) is attained by $\mu^*, \Sigma^*$, we have by (<ref>) that
$$ D\left( \mc P_X(\cdot;\beta) \dmid \mc N(\cdot;X \mu^*,X^T \Sigma^* X) \right) = \inf_{\mu, \Sigma} D\left( \mc P_X(\cdot;\beta) \dmid \mc N(\cdot;\mu,\Sigma) \right), $$
which gives Corollary <ref>.
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1511.00921
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Perrin's century old picture <cit.> where the Brownian motion of a colloid results from the many collisions exerted by the solvent's molecules is a cornerstone of soft-matter physics. Langevin <cit.> modeled the ensuing energy exchanges between the solvent and the colloidal particle in terms of a dissipation channel and energy injection kicks. The key ingredient in the success of that theory was to completely integrate out the "uninteresting" degrees of freedom of the solvent whose properties are gathered in a friction constant and a temperature. In this work we take exactly the reverse stance and ask how, by observing the motion of a tracer embedded in a living medium, one can infer the amount of energy exchange and dissipation with the surrounding medium. The main goal is to quantify the energetic properties of the medium, both injection and dissipation-wise.
This is a stimulating question because there are of course striking differences between a living cell and its equilibrium polymer gel counterpart, to which newly developed <cit.> methods of nonequilibrium statistical mechanics apply. Beyond thermal exchanges that fall within the scope of a Langevin approach, ATP consumption fuels molecular motor activity and drives relentless rearrangement of the cytoskeleton. This chemically driven continuous injection and dissipation of energy adds a nonequilibrium channel that eludes straightforward quantitative analysis. In short, a living cell is not only a fertile playground for testing new ideas from nonequilibrium physics, but also one in which these ideas can lead to a quantitative evaluation of an otherwise ill-understood activity which is of intrinsic biophysical interest. Our work addresses both aspects by a combination of active microrheology, tracking experiments, and theoretical modeling.
One experimental way to access nonequilibrium physics in the intracellular medium is to focus on the deviation from thermal equilibrium behavior of the tracer's position statistics: forming the ratio of the response of the tracer's position to an infinitesimal external perturbation to its unperturbed mean-square displacement leads to a quantity that only reduces to the inverse temperature when in equilibrium, by virtue of the fluctuation-dissipation theorem (FDT). Earlier tracking experiments supplemented by microrheology techniques have allowed the departure from equilibrium to be analyzed in terms of this ratio in a variety of contexts <cit.> ranging from reconstituted actin gels to single cells. However the limitations inherent to this effective temperature are well-known: it bears no universal meaning as it depends on the observable under scrutiny, thus it cannot be equated to a bona fide temperature, and hence it does not connect to the underlying microscopic dynamics. In spite of these caveats, the effective temperature has been widely measured in nonequilibrium systems since it is perhaps the simplest way to assess deviation from equilibrium.
Here we exploit a body of theoretical methods that have been developed over the last ten years to infer quantitative information about the nonequilibrium processes driving intracellular dynamics. Within the realm of stochastic thermodynamics <cit.> –as it strives to extend concepts of macroscopic thermodynamics to small and highly fluctuating systems <cit.>, the Harada-Sasa equality stands out as being particularly suited to our goal <cit.>. Nonequilibrium systems are characterized by the dissipation of energy, which is absorbed by the surrounding thermostat via a transfer from the system to the bath. The Harada-Sasa equality connects the rate of dissipated energy to the spatial fluctuations in a nonequilibrium steady-state system. The feasibility of measuring the various ingredients in the Harada-Sasa framework was demonstrated in model systems such as a micron-sized colloidal particle in a viscous fluid <cit.>, and then later generalized to a viscoelastic medium <cit.>. It has also been used to quantify the efficiency of an isolated molecular motor <cit.>.
The systems to which we apply this equality are micron-sized vesicles that are present in the cytoplasm of mouse unfertilized eggs, known as oocytes. Their motion in the cell is mainly regulated by myosin-V motors on the actin network <cit.>. The use of such vesicles allows us to capture the intrinsic intracellular dynamics without using artificial external particles that may alter the environment. From a physics perspective, oocytes are also major assets since they constitute a rare example of a living cell that remains steady on the timescales of hours. They are spherical in shape, with typical radius of about $40$ $\mu$m, and their nucleus is centrally located at the end of Prophase I <cit.>.
In this paper, we directly access nonequilibrium dissipation within the cell. We first characterize the intrinsic rheology of the medium experienced by the vesicles. Then, we present a minimal microscopic model for the dynamics of the vesicles which is driven by the nonequilibrium reorganization of the cytoskeleton by molecular motor generated forces. Our first main result is to demonstrate that this model is sufficient to capture the available experimental data. As a second step, we use the Harada-Sasa equality to predict and quantify the rate of nonequilibrium dissipated energy in our experimental system. Finally, we employ this new prediction to evaluate how nonequilibrium activity varies across the cell, and we offer interpretations about the role of molecular motors in vesicle motion.
Experimental setup.—Mouse oocytes are collected from 13 week old mice and embedded in a collagen gel between two glass coverslips <cit.>. We measure the local mechanical environment surrounding vesicles in living mouse oocytes using active microrheology <cit.>. We use an optical tweezer to trap vesicles and apply a sinusoidal oscillating force [ <ref>]. The resulting displacement of the vesicle due to the applied force reflects the mechanical response of the system. We deduce the complex modulus of the intracellular environment surrounding the vesicle from the generalized Stokes-Einstein relation $G^*=1/(6\pi a\tilde{\chi})$, where $\tilde{\chi}$ is the Fourier response function, and $a$ is the vesicle's average radius.
(Color online) Experimental setup for measuring oocyte microrheology.
(a) We embed oocytes in a collagen matrix between two glass coverslips to prevent rolling during measurement.
(b) We use optical tweezers to trap intracellular vesicles and perform active microrheology to measure local mechanical properties <cit.>.
(c) Real ($G'$) and imaginary ($G''$) part of the complex modulus, measured from active microrheology. Data at $10$ Hz is used in <cit.>. Best fit curves are in solid and dashed lines for $G'$ and $G''$, respectively.
(d) We use laser tracking interferometry to track spontaneous vesicle motion with high spatiotemporal resolution ($10$ nm, $1$ kHz) <cit.>.
We find that the intracellular mechanics exhibits a power law rheology at high frequencies, and levels off at lower frequencies, as seen in the real and imaginary parts of $G^*$, respectively denoted by $G'$ and $G''$ [ <ref>(c)]. We fit the experimental data with the function
\begin{equation}\label{eq:G}
G^*(\omega) = G_0 (1 + (i\omega\ta)^\alp ) ,
\end{equation}
where $\ta$ is a thermal relaxation time scale <cit.>. To experimentally quantify nonequilibrium dissipation, we must also measure the spontaneous motion of the vesicles by laser interferometry, and extract the power spectral density of the vesicles' position <cit.>, as is done for passive microrheology <cit.> [ <ref>(d)]. These spontaneous fluctuations entangle information about the thermal and nonequilibrium forces applied on vesicles in the oocyte cytoskeleton <cit.>.
Caging model.—We propose a model for the vesicle dynamics and the effect of the surrounding fluctuating actin mesh that takes the observed power law behavior of $G^*$ into account. The model has itself been previously introduced in <cit.>, but it is generalized here to encompass strong memory effects <cit.>. The underlying physical picture is that the vesicle is caged in the cytoskeleton [ <ref>(b)], modeled as a harmonic trap of constant $k$, and we assume that nonequilibrium activity induces rearrangements of the cytoskeletal network resulting in a displacement of the cage. In a medium characterized by a memory kernel $\zeta(t)$, we then describe the one dimensional position $x$ of a vesicle with two coupled generalized Langevin-like equations involving the center of the cage $x_0$:
\begin{equation}\label{eq:model}
\zeta * \f{\dd x}{\dd t} = - k ( x - x_0 ) + \xi , \quad \zeta * \f{\dd x_0}{\dd t} = k \ta \va ,
\end{equation}
where $*$ denotes the convolution product, $\xi$ is a zero mean Gaussian colored noise with correlations $\langle \xi(t)\xi(t')\rangle=\kb T\zeta(\abs{t-t'})$ as provided by the FDT <cit.>, and $T$ is the bath temperature. All the degrees of freedom of the surrounding network are embodied by the cage center $x_0$. We have assumed that the dynamics of the network is not affected by the tracer in the regime of our experiment.
The cage motion is given by the active burst $\va$: a zero mean stochastic process representing the random vesicle motion driven by cellular activity <cit.>. In our further analysis, we consider that this process has a single timescale $\tau$ that governs its decorrelation:
\begin{equation}
\avg{\va (t)\va(0)} = \f{\kb\Ta}{k\ta} \f{\ee^{-|t|/\tau} }{\tau} ,
\end{equation}
where, by analogy with standard Langevin equation, we have defined an active temperature $\Ta$ associated to the amplitude of this process. Notice that $\Ta$ is a scalar quantity which quantifies the amplitude of the active fluctuations. We choose the memory kernel $\zeta$ to recover the observed behavior of the measured $G^*$ by adopting a power law decay:
\begin{equation}\label{eq:zeta}
\zeta(t) = k \pnt{\f{\ta}{t}}^\alp \f{\Theta(t)}{\Gamma(1-\alp)} ,
\end{equation}
where $\Gamma$ is the Gamma function, $\Theta$ is the Heaviside function, and $\alpha<1$. From the generalized Stokes-Einstein relation, we derive that $G^*(\omega)=\brt{k+i\omega\tz(\omega)}/(6 \pi a)$ <cit.>, where the superscript tilde denotes a Fourier transform. The normalization factor in (<ref>) is chosen so that the corresponding $G^*$ has exactly the same expression as the phenomenological function (<ref>) that we use to fit the experimental data, where $k=6 \pi a G_0$. In that respect, the specific form of the memory kernel in (<ref>) reflects the explicit choice of the best fit curve in (<ref>), yet our approach bears a higher level of generality since it can be extended in a straightforward manner to other kinds of rheology. However, it is not possible to capture the complex frequency dependence of $G^*$ shown in Fig. <ref> by discarding memory effects in the dynamics.
Effective temperature.—A standard quantification of the departure of the dynamics from equilibrium relies on a frequency dependent “effective temperature”. Following <cit.>, it has been introduced by analogy with the FDT as
\begin{equation}
\Tf(\omega) = \f{\omega\tC(\omega)}{ 2\kb\tc''(\omega)},
\end{equation}
where $\tC$ and $\tc''$ are the Fourier position autocorrelation function and the imaginary part of the response function, respectively. As a first step, we compute it in terms of the microscopic parameters as
\begin{equation}\label{eq:Tf}
\Tf(\omega) = T + \f{1}{(\omega\ta)^{3\alp-1}\sin\pnt{\f{\pi\alp}{2}
}}\f{\Ta}{1+(\omega\tau)^2} .
\end{equation}
The high frequency value collapses to the bath temperature as for an equilibrium behaviour provided that $3\alpha-1>0$. It constitutes a useful benchmark to delineate a thermal regime where active fluctuations are negligible <cit.>. It also diverges at low frequency as a result of nonequilibrium activity, with a coefficient depending on both the material properties $\{\alp,\ta\}$ and the active temperature $\Ta$. This interplay between mechanics and activity reflects the fact that, in our model, the nonequilibrium processes operating in the system drive motion of the cytoskeletal cage, which in turn affects the vesicle dynamics.
Dissipation spectrum.—A quantification of direct physical relevance is the work done by the vesicle on the thermostat <cit.>, which is the dissipated mechanical energy. The mean rate of energy dissipation $J_\text{diss}$ is the power of the forces exerted by the vesicle on the heat bath, namely the forces opposed to the thermal forces acting on the vesicle by virtue of the action-reaction principle. The thermal forces comprise both the drag force $-\zeta*\dot x$ and the Gaussian noise $\xi$. Therefore, the dissipation rate reads $J_\text{diss}=\avg{\dot x(\zeta*\dot x-\xi)}$, where $\dot x=\dd x/\dd t$ is the vesicle's velocity <cit.>. It is proportional to the rate at which the vesicle exchanges energy with the surrounding environment <cit.>. In equilibrium $J_\text{diss}$ would vanish, thus expressing the fact that the vesicle releases and absorbs on average the same amount of energy from the thermostat. The dissipation rate is equal to the mean rate of entropy production times the bath temperature $T$. Thereby, it directly characterizes the irreversible properties of the dynamics stemming from the active fluctuations.
The Harada-Sasa equality connects the spectral density $I$ of mechanical energy dissipation to $\tC$ and $\tilde{\chi}''$ in a viscous fluid <cit.>. It has been generalized to the case of a complex rheology <cit.>, and we express it in terms of the effective temperature as
\begin{equation}\label{eq:I_teff}
I = \f{ 2 \kb ( T_\text{eff} -T ) }{ 1 + (G'/G'')^2 } .
\end{equation}
This relation allows one to precisely identify the dissipation rate with the nonequilibrium properties of the vesicles' dynamics, since $I$ vanishes at equilibrium. It also enables one to quantify dissipation in the system without any a priori knowledge on the internal source of nonequilibrium fluctuations. The relation (<ref>) between effective temperature and dissipation holds independently of our modeling of intracellular activity; it can be used for a large variety of nonequilibrium dynamics. Within our model, the dissipation spectrum is
\begin{equation}\label{eq:I}
I(\omega)=\f{\pnt{\omega\ta}^{1-\alp}\sin\pnt{\f{\pi\alp}{2}}}{1+2\pnt{\omega\ta}^{\alp}\cos\pnt{\f{\pi\alp}{2}}+\pnt{\omega\ta}^{2\alp}}\f{2\kb\Ta}{1+\pnt{\omega \tau}^2} \,.
\end{equation}
There is no nonequilibrium dissipation when $\Ta=0$ as expected, while in general it depends on both mechanics and activity as for $\Tf$. By integrating the dissipation spectrum over the whole frequency range, we can deduce the total dissipation rate $J_\text{diss} = \int \dd \omega I (\omega)/(2\pi)$. By contrast to $\Tf$, the dissipation spectrum not only quantifies the deviation from equilibrium properties, it is also related to the energy injected by the nonequilibrium processes.
Using the model (<ref>), the nonequilibrium drive is embodied by the $k x_0$ force applied on the vesicle. The dissipation rate precisely equals the mean power of this force: $J_\text{diss} = \avg{ \dot x k x_0}$, reflecting the fact that the mechanical energy dissipated by the vesicle is also the energy provided by the nonequilibrium processes driving the vesicle's motion. In addition, the dissipation spectrum $I$ equals the Fourier transform of the time symmetric correlation between the vesicle velocity $\dot x$ and the driving force $k x_0$.
Energy conversion.—The picture that emerges from our model is that the vesicle motion results from the displacement of the confining cytoskeletal cage, which is due to the active reorganization of the local environment. We denote by $J_\text{env}$ the power of the random force driving the cage's motion. This is the rate of energy injected by the nonequilibrium processes into the environment leading to the cytoskeleton rearrangement in our model. In our model (<ref>), it is given by the mean power injected by the force $k\ta\va$ to the cage: $J_\text{env} = \avg{ \dot x_0 k\ta\va}$, where $\dot x_0=\dd x_0/\dd t$. This can be computed in terms of the microscopic parameters $J_\text{env}=\kb\Ta/\tau(\ta/\tau)^{1-\alpha}$. Note that $J_\text{env}$ can also be regarded as the work per unit time done by the cage on the thermostat, namely on the surrounding medium where the cage is immersed. This interpretation stems from the fact that our model is the limit version of one that features a reaction force of the vesicle upon the cage (for which mechanical interpretations are ambiguity-free), along with thermal fluctuations acting directly on the cage <cit.>.
In our phenomenological picture, the energy $J_\text{env}$ injected by the intracellular active processes serves to relentlessly remodel the cytoskeleton network (represented by $x_0$). This energy is then transduced into the vesicles confined in such network (represented by $x$), which is embodied by $J_\text{diss}$, thus driving their active motion. To quantify the efficiency of this energy transduction we introduce the dimensionless ratio $\rho=J_\text{diss}/J_\text{env}$ of the energy effectively dissipated through active motion of the vesicles over that injected by the nonequilibrium processes into the cage. We find the energetic efficiency $\rho$ to be independent of $\Ta$, and is thus controlled by the time scales $\tau$ and $\tau_\alpha$. We understand such energy transduction as the conversion of the active stirring of the cytoskeleton network into the active dynamics of the intracellular components.
(Color online)
(a) Effective temperature $\Tf$ and (b) dissipation spectrum $I$ as functions of frequency ($\circ$), best fits in solid lines using Eqs. (<ref>) and (<ref>). The horizontal dashed line in (a) is the prediction for a thermal equilibrium system in the absence of activity, for which the dissipation spectrum equals zero. The experimental data is averaged over the whole oocyte.
(c) We distinguish three concentric shells around the central nucleus (black) with a radial extension of about $10$ $\mu$m each: near nucleus (blue), middle (yellow), and cortex (red).
(d) Active temperature, (e) dissipation rate, and (f) power conversion rate estimated from the best fits of $\Tf$ and $I$ at three locations within the cytoplasm: near the nucleus (Nuc), near the cortex (Cor), and in the region in between (Mid).
Quantification of the activity.—We exploit our theoretical predictions to quantify the experimental measurements of nonequilibrium dissipation inside living oocytes. We extract the data for the effective temperature and the dissipation spectrum from a combination of active and passive microrheology. We observe that the experimental effective temperature diverges at low frequency, as a clear evidence that nonequilibrium processes drive the intracellular dynamics in this regime [Fig. <ref>(a)]. It reaches the equilibrium plateau at high frequency as expected. Deviation from thermal equilibrium was already reported in other living systems <cit.>. We use our analytic prediction in Eq. (<ref>) to fit the dissipation spectrum data. As we have already quantified the viscoelastic properties, the remaining two free parameters are the ones characterizing the properties of the nonequilibrium processes, namely the active temperature $\Ta$, and the mean persistence time $\tau$. Our best fit is in very good agreement with the measured dissipation spectrum [Fig. <ref>(b)]. This is the first main result of this paper: our minimal model reproduces very well the available experimental data. It supports the underlying phenomenological picture that the main contribution of the active force driving the vesicle dynamics is mediated by the surrounding network. Moreover, the excellent agreement points to a single dominant active process in the system, characterized by a single time-scale and force magnitude, as captured by the parameters of our model.
The extracted value of the active temperature $\Ta = ( 6.2 \pm 0.5) T $ is larger than the bath temperature $T$. By contrast to $\Tf$, the active temperature is frequency independent, and it quantifies the amplitude of the active fluctuations. Hence, our estimation reveals that the fluctuations due to the nonequilibrium rearrangement of the cytoskeleton have a larger amplitude than the equilibrium thermal fluctuations dominating the short time dynamics. The time scale $\tau = (0.34 \pm 0.04) $ ms that we obtain is of the order of the power stroke time of a single myosin-V motor <cit.>. This supports that the nonequilibrium processes driving the vesicle dynamics are related to the microscopic kinetics of the molecular motors. It is consistent with the fact that nonequilibrium processes are dominant at a higher frequency in our system than in others which were mainly driven by myosin-II <cit.>, for which the power stroke time is about $0.1$ s <cit.>. It is not an obvious result a priori that one can extract the kinetics of individual motors from global measurements of response and fluctuations. Our analytic prediction for $\Tf$ in Eq. (<ref>), for which we use the parameter values $\{\alp,\ta,\Ta,\tau\}$ extracted from the previous fits of $G^*$ [ <ref>(c)] and $I(\omega)$ [ <ref>(b)], is in consistent agreement with the experimental data [Fig. <ref>(a)].
To go beyond the calibration of the model presented above, we now use our predictions to investigate energy transfers within the oocytes. From the best fit parameters, we directly estimate the dissipation rate $J_\text{diss}=(360\pm110)$ $\kb T/\text{s}$, as well as the power conversion rate $\rho=(1.7\pm0.8)10^{-3}$. The definition of $J_\text{diss}$ is independent of any modeling of the underlying activity, whereas $\rho$ depends on our specific model. We find that the conversion of energy from the cytoskeletal network to the vesicle is very low. This is the second main result of this paper. It suggests that a major proportion of the nonequilibrium injected power is dedicated to the network rearrangement, and not necessarily to vesicle dynamics per se. In other words, the injected energy tends to go mostly into elastic stresses, and only a small fraction ends up in kinetic energy <cit.>. Note that the power of the active force driving the vesicle being small compared with the one moving the cage is not in contradiction with our assumption that the vesicle driving is mediated by the cage. Indeed, such a driving already leads to a significant deviation from equilibrium, as shown in Fig. <ref>(a).
It has been reported that a single myosin-V motor does about $3$ $\kb T$ of work during one power stroke <cit.>, from which we deduce that it dissipates approximately $10^4$ $\kb T/\text{s}$ into the intracellular environment during the power stroke. This result is to be compared with our estimation of $J_\text{env}=(2.0\pm0.5)10^5$ $\kb T/\text{s}$. We infer that the power injected by the nonequilibrium processes into the environment represents approximately the activity provided by $20$ myosin-V motors. Assuming that the nonequilibrium processes in oocytes are indeed mainly regulated by myosin-V activity, we infer that $20$ is the typical number of motors involved in the nonequilibrium reorganization of the cytoskeletal cage in the vicinity of a vesicle.
Variability across the oocyte.—One of the main advantages of our energetic approach lies in the ability to compare the same physical quantities across a large variety of living systems, or in different locations of the same system. We consider three concentric shells within the oocyte cytoplasm located near the nucleus, near the cortex, and in between these two regions. Each shell has a radial extension of about $10$ $\mu$m [Fig. <ref>(c)]. We fit the real and imaginary parts of $G^*$ for the three regions, and we use our analysis to quantify the corresponding $\Ta$, $J_\text{diss}$, and $\rho$. Our results hint that nonequilibrium activity is increased near the middle of the cell, and slightly decreased near the nucleus, as quantified by $\Ta$ and $J_\text{diss}$ [Figs. <ref>(d-e)]. This suggests that living oocytes locally regulate the nonequilibrium activity throughout their cytoplasm by injecting different amounts of energy. Note that the relative variation of $J_\text{diss}$ and $\Ta$ are similar, showing the close relation between these quantities as highlighted in Eq. (<ref>).
In comparison, the variation of $\rho$ does not exhibit a clear trend across the oocyte [Fig. <ref>(f)]. Since the $\rho$ definition is a balance of a purely active parameter $\tau$ and the material properties $\{\alpha, \tau_\alpha\}$, this result suggests that the nonequilibrium fluctuations are adapted in each region to the local mechanical properties. It is known that the molecular motors do not only produce active forces in the cell, they also affect its mechanical properties <cit.>. Therefore, we speculate that there might be a feedback between the overall fluctuations induced by the motor activity and the mechanics of the surrounding cytoplasmic network within which they move to find an optimum rate of power conversion, the optimal value being roughly the same for the three locations.
Conclusion.—We quantified the amount of mechanical energy dissipated by the intracellular dynamics. Our analysis utilizes a minimal model describing the effect of the nonequilibrium stochastic forces in living systems with complex rheology. We find the predictions of our model to be in excellent agreement with the experimental results for vesicles in living mouse oocytes, thus allowing us to quantify the main properties of the nonequilibrium dynamics: the amplitude and typical time scale of active fluctuations, the amount of dissipated energy, and the rate of energy transmitted from the cytoskeletal network to the intracellular components. The extracted parameters provide a quantitative support to the experimental picture that the nonequilibrium processes are mainly driven by myosin-V activity <cit.>. The use of general principles in stochastic energetics, together with a minimal microscopic model, makes the results of our study highly relevant to a large variety of nonequilibrium processes in biology and active matter <cit.>.
We warmly thank Gavin Crooks for a critical reading of the manuscript and FvW acknowledges the support of the UC Berkeley Pitzer Center for Theoretical Chemistry. WWA thanks the PGG Fondation and Marie Curie Actions.
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1511.00942
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The authors would like acknowledge the National Science Foundation which has generously supported
the work through grants DMS-1105635 and DMS-1511488. A debt of gratitude also goes to Nsoki Mavinga and the Department of Mathematics and Statistics at Swarthmore College who hosted JDW during much of the research which went into this document.
Additionally, they would like to thank Aaron Hoffman for a huge number of helpful comments and insights.
Finally, they dedicate this article to the memory of Malinda Gilchrist, graduate coordinator of Drexel's math department, colleague, navigator and friend.
We study the existence of solitary waves in a diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) lattice.
For monatomic FPUT the
traveling wave equations are a regular perturbation of the Korteweg-de Vries (KdV) equation's
but, surprisingly, we find that for the diatomic lattice the traveling wave equations are a singular perturbation of KdV's.
Using a method first developed by Beale to study traveling solutions for capillary-gravity waves we
demonstrate that for wave speeds in slight excess of the lattice's speed of sound
there exists nontrivial traveling wave solutions which are the superposition an exponentially localized solitary wave and a periodic wave whose amplitude is extremely small. That is to say, we construct nanopteron solutions. The presence of the periodic wave is an essential part of the analysis and is connected to the fact that
linear diatomic lattices have optical band waves with any possible phase speed.
We consider the problem of traveling waves in a diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. The physical situation is this:
suppose that infinitely many particles are arranged on a line. The mass of the $j$th (where $j \in \Z$) particle
m_j=\begin{cases} m_1 &\text{ when $j$ is odd} \\ m_2 &\text{when $j$ is even.} \end{cases}
Without loss of generality we assume that $m_1 > m_2>0$. The position of the $j$th particle at time $\bar{t}$ is $\bar{y}_j(\bar{t})$.
Suppose that each mass is connected to its two nearest neighbors by a spring and furthermore assume that each spring is identical to every other spring in the sense that the force exerted by said spring when stretched by an amount $r$ from its equilibrium length $l_s$ is given by
F_s( r ) := - k_s r - b_s r^2
where $k_s>0$ and $b_s \ne 0$ are specified constants.
Such a system is called a “diatomic lattice" or “dimer."
Newton's law gives the equations of motion
for the system:
m_j d^2 y̅_j d t̅^2 = - k_s s̅_j-1 - b_ss̅_j-1 + k_ss̅_j + b_s s̅^2_j
\bar{s}_j:=\bar{y}_{j+1}-\bar{y}_j - l_s.
In the setting where $m_1=m_2$ it is well-known that there exist[This result is true for much more general (but still spatially homegeneous) forms of the spring force than the one we have here.] localized traveling wave solutions of (<ref>), see the seminal articles of Friesecke & Wattis <cit.> and
Friesecke & Pego <cit.>.
Here we are interested in extending this result to the diatomic case where $m_1>m_2$. There has been quite a bit of interest in the propagation of
waves through polyatomic FPUT lattices. Such systems represent a paradigm for the evolution of waves through heterogeneous and nonlinear granular media (see <cit.> and <cit.> for an overview).
There are several existence proofs for traveling spatially periodic waves for polyatomic problems <cit.> <cit.>, a number of semi-rigourous asymptotics for solitary wave solutions in various contexts <cit.>, as well as both formal and rigorous results which state that polyatomic FPUT with periodic material coefficients is well-approximated by the soliton bearing Korteweg-de Vries (KdV) equation over very long time scales <cit.> <cit.> <cit.>. While all of this previous work strongly suggests that localized traveling waves for polyatomic FPUT will exist, the question of whether a truly localized traveling wave akin to those developed in <cit.> remains open.
In this article we demonstrate that the answer to this question—at least for waves which travel at a speed just a bit larger than the speed of sound—is
“sort of."
As it happens, the existence problem is inescapably singular. This is particularly surprising because
the existence proof of small amplitude solitary waves for monatomic FPUT in <cit.>
goes through using regular perturbation methods.
In that article,
the equation for the solitary wave's profile $\phi$ is shown to be equivalent to
$\phi = p_c (\phi^2)$ where $p_c$ is a Fourier multiplier operator and $c$ is the speed of propagation. Making the “long wave scaling" $\phi(x) = \ep^2 \Phi(\ep x)$ and $c=c_{sound} + \ep^2$ yields $\Phi = P_\ep (\Phi^2)$. The hinge on which their result turns is the fact that the operator $P_\ep$ converges in the operator norm to $(1-\partial_X^2)^{-1}$ as $\ep \to 0^+$.
Which means that the traveling wave equation at $\ep =0$ can be rewritten as $\Phi'' - \Phi + \Phi^2 =0$. This is the traveling wave equation for KdV and has $\sech^2$ type solutions.
Moreover, using classical results from quantum mechanics, they show that the linearization of the equation $\Phi = P_0 (\Phi^2)$ at the KdV solitary
wave results in an operator which is invertible on even functions. This, with the uniform convergence of $P_\ep$, allows them to extend the wave's existence to $\ep > 0$ using a quantitative inverse function theorem.
This process goes awry in the diatomic setting. In this case the dispersion relation for the linearization of (<ref>) has two parts. The “acoustic" band, which is more or less just like the dispersion relation for the monatomic problem, and the “optical" band[The language “acoustic" and “optical" bands is taken from Brillouin's
foundational text on wave propagation in periodic media <cit.>. This book contains a detailed discussion
of waves in linear diatomic lattices and in particular contains an excellent treatment of the dispersion relation.] which does not exist in the monatomic problem at all.
Roughly speaking, we are able to decompose our problem into a pair of equations,
one for the acoustic part and another for the optical part. The analysis for the acoustic part goes forward
along lines much like in the monatomic case of <cit.>; it limits to a KdV traveling wave equation.
On the other hand, the equation for the optical part is classically singularly perturbed in the sense that the highest derivative
of the unknown is multiplied by the small parameter $\ep$.
The possible outcomes for singularly perturbed problems like this is pretty vast, of course. Many of the approaches for sussing out the consequences are either geometric or dynamical in nature. Our problem is nonlocal and as such it is not obvious to us how to use, say, geometric singular perturbation theory (as in <cit.> for instance) or fast-slow averaging (e.g. <cit.>) to our setting. And so we turn to the functional analytic approach developed by Beale to prove the existence of solitary capillary-gravity waves in his staggering article <cit.>.
Here is what we discover:
Suppose that $m_1>m_2>0$. For wavespeeds $c$ sufficiently close to, but larger than, the speed of sound of the lattice, $c_{sound} := \sqrt{2k_s/(m_1+m_2)}$, there exist traveling wave solutions of the diatomic FPUT problem which are the superposition of two pieces.
One piece is a nonzero exponentially localized function that is a small perturbation of a $\sech^2$ profile which, in turn, solves a KdV traveling wave equation. This whole localized piece has amplitude roughly proportional to $\left(c-c_{\textrm{sound}}\right)$
and has wavelength roughly proportional to $\left(c-c_{\textrm{sound}}\right)^{-1/2}$.
The other piece is a periodic function, called a “ripple." The frequency of the ripple is $\O(1)$ when compared to $\left(c-c_{\textrm{sound}}\right)$. Its amplitude is small beyond all orders of $\left(c-c_{\textrm{sound}}\right)$.
As we shall see, the periodic part
is fundamentally tied to the optical branch of the dispersion relation.
Moreover we expect that the periodic part is exponentially small in $\left(c-c_{\textrm{sound}}\right)$.
Solutions of this type—a localized piece plus an extremely small oscillatory part[Or, equivalently, a heteroclinic connection between small amplitude periodic orbits.]—are sometimes called nanopterons <cit.>.
It is because the solutions we discover do not converge to zero at spatial infinity that we
were cagey about our answer to the existence question earlier; our result raises as many questions as it answers. Chief of these is whether or not the ripple at infinity is genuine or merely a technical byproduct of our proof. After all, we do not provide lower bounds on its size, only upper bounds; perhaps the amplitude is zero! While we do not have the right sort of estimates at this time to answer this question either way we point out that Sun, in <cit.>, showed that the ripple for the capillary-gravity waves studied in <cit.> was in fact non-zero.
And so we conjecture that the same happens here, at least for almost all wave speeds.
This article is structured in the following manner.
* In the next section we nondimensionalize (<ref>) and rewrite the resulting system in terms of the relative displacements.
* In Section <ref> we make the traveling wave ansatz and get the traveling wave equations. We then diagonalize the resulting system using Fourier methods.
It is during the diagonalization that the structure of the branches of the dispersion relation becomes apparent.
Then we make a useful “long wave" rescaling.
It is during this part that the singular nature of the problem comes into sight.
* In Section <ref> we analyze the rescaled system in the limit where $c = c_{sound}$; we find that the problem in this case reduces to a single KdV traveling wave equation.
* In Section <ref> we construct exact traveling wave solutions which are spatially periodic. These will ultimately be the ripples. A major difference between our results and the extant existence results for periodic traveling waves in polyatomic FPUT (<cit.>,<cit.>) is that we prove
estimates on their size and frequency which are uniform in the speed $c$. This is done using a Crandall-Rabinowitz-Zeidler bifurcation analysis <cit.> <cit.>.
* In Section <ref> we make what we call “Beale's ansatz." That is, we assume the solution is the superposition of (a) the KdV solitary wave profile from Section <ref>, (b) a periodic solution from Section <ref> with unknown amplitude and (c) a small, localized remainder. We then derive equations for the remainder and the amplitude of the periodic part. This derivation can be viewed as Liapunov-Schmidt decomposition, albeit a somewhat atypical one.
* In Section <ref> we state the main estimates we need and then, given those estimates, prove our main results using a modified contraction mapping argument.
Specifically we prove Theorem <ref> and Corollary <ref>, which are the technical versions of Theorem <ref>.
* Sections <ref>, <ref> and <ref> contain the proof of the main estimates; these are the technical heart of the paper.
* Finally Section <ref> presents some comments on our results, avenues for further investigation and concluding remarks.
§ NONDIMENSIONALIZATION AND THE EQUATIONS FOR RELATIVE DISPLACEMENTS.
We can simplify (<ref>) somewhat by putting $\bar{y}_j = \bar{x}_j - j l_s$. Then (<ref>) is equivalent to
m_j d^2 x̅_j d t̅^2 = - k_s r̅_j-1 - b_sr̅_j-1 + k_sr̅_j + b_s r̅^2_j
\bar{r}_j:=\bar{x}_{j+1}-\bar{x}_j .
Note that the system is in equilibrium when $\bar{x}_j = 0$ for all $j$.
Next we nondimensionalize by taking
\bar{x}_{j}(\bar{t}) = a_1 x_j(a_2\bar{t})
where $a_1,a_2$ are nonzero constants.
This converts (<ref>) to
m_j a_2^2 d^2 x_j d t^2 = - k_s r_j-1 - a_1 b_sr_j-1 + k_sr_j + a_1 b_s r^2_j
where r_j = x_j+1 - x_j.
Note that here $t = a_2 \bar{t}$.
Selecting $a_1$ and $a_2$ such that
m_1 a_2^2 = k_s$ and $a_1 b_s = k
ẍ_j = -r_j-1 - r_j-1^2 + r_j + r_j^2 when $j$ is odd
1 w ẍ_j = -r_j-1 - r_j-1^2 + r_j + r_j^2 when $j$ is even.
In the above,
w:=m_1 m_2>1because $m_1>m_2$.
It is both traditional and technically advantageous to express the equations of motion for lattices in terms of the relative displacements, $r_j$, instead of in the displacements from equilibrium, $x_j$. We find that
r̈_j =-(1+w)(r_j+r_j^2) + w(r_j+1+r_j+1^2) + (r_j-1+r_j-1^2) when $j$ is odd
r̈_j =-(1+w)(r_j+r_j^2) + (r_j+1+r_j+1^2) + w(r_j-1+r_j-1^2) when $j$ is even.
§ DERIVATION OF THE TRAVELING WAVE EQUATIONS
We are interested in traveling wave solutions
and so we make the ansatz
r_j(t) = p_1(j-ct) when $j$ is odd
p_2(j-ct) when $j$ is even.
Here $c \in \R$ is the wave speed and $p_1, p_2:\R \to \R$.
Putting this into (<ref>) gives us the following advance-delay-differential system of equations for $p_1$ and $p_2$:
c^2p_1” =-(1+w)(p_1+p_1^2) + wS^1(p_2+p_2^2) + S^-1(p_2+p_2^2)
c^2p”_2 =-(1+w)(p_2+p_2^2) + S^1(p_1+p_1^2) + wS^-1(p_1 + p_1^2).
Above, $S^d$ is the “shift by $d$" operator. Specifically:
S^{d} f(\cdot):= f(\cdot+d).
If we let
1+w -(w S^1 + S^-1)
-(w S^-1 + S^1) 1+w
then we can compress[Note the for $2$-vectors $\xb$ and $\yb$ we use the notational convention that $\xb.\yb$ is component-wise multiplication. Likewise $\xb^{.2}$ is component-wise squaring. ] (<ref>) to
c^2 ” + L(+ ^.2) = 0.
§.§ Diagonalization of the linear part
We can diagonalize (<ref>) using Fourier analysis and the first step is to compute the action of $L$ on complex exponentials.
We find that for any vector $\v \in \R^2$ and $k \in \R$ that
L [ e^{ikx} \v] = [\tilde{L}(k) \v] e^{ikx}
\tilde{L}(k):=\left[ \begin{array}{cc} 1+w & -\tilde{\beta}(k) \\ -\tilde{\beta}(-k) & 1+w \end{array} \right]
\mand
\tilde{\beta}(k) := w e^{ik} + e^{-ik}.
A routine calculation shows that the eigenvalues of $\tilde{L}(k)$ are given by
λ̃_±(k):=1+w±ϱ̃(k) where ϱ̃(k):= √(
(1 - w)^2 + 4 w cos^2(k)
The following lemma contains many of the properties of $\tilde{\lambda}_\pm(k)$ we will need (the proof is in Section <ref>).
The following hold for all $w > 1$.
* $\tilde{\lambda}_-(0) = 0$ and $\tilde{\lambda}_+(0) = 2 + 2w.$
* There exists $\tau_0 > 0$ such that $\tilde{\lambda}_\pm(z)$ are uniformly bounded complex analytic functions[We extend $\tlambda_\pm(k)$ and $\tilde{\varrho}(k)$ to functions of $z = k + i \tau \in \C$ in a simple way by using the extension of cosine to complex inputs.] in the closed strip $\overline{\Sigma}_{\tau_0}:=\left\{ z \in \C : |\Im z| \le \tau_0 \right\}$.
* $\tilde{\lambda}_\pm(z)$ are even and $\tilde{\lambda}_\pm(z+\pi) = \tilde{\lambda}_\pm(z)$ for all $z \in \overline{\Sigma}_{\tau_0}$.
* For all $k\in \R$ we have
0≤λ̃_-(k) ≤2<2w ≤λ̃_+(k) ≤2 + 2 w.
* For all $k \in \R$ we have
|λ̃'_±(k)|≤2 |'_±(k)| ≤2c_w^2|k|
c_w:=\sqrt{{1 \over 2} \lambda_-''(0)}=\sqrt{{2 w \over 1+w}}=\text{``the (nondimensionalized) speed of sound."}
Additionally $c_w>1$.
* There exists $c_- \in (0,1)$ and $l_0>0$ such that for all $c \ge c_-$
there exists a unique nonnegative $k_c$ for which
c^2 k_c^2 - _+(k_c) =0.Moreover k_c ∈[√(2w)/c,√(2+2w)/c]and
| 2 c^2 k_c - λ̃_+(k_c)|≥l_0.
Lastly, the map $c \mapsto k_c$ is $C^\infty$.
The eigenvalues $\tlambda_\pm(k)$
are tied to the dispersion relation for (<ref>).
To be precise, we have plane wave solutions for the linearization of (<ref>) at $r = 0$
of the form
r_j(t) = \begin{cases}v_1 e^{i(k j - \omega t)} & \text{when $j$ is odd}\\
v_2 e^{i(k j - \omega t)} & \text{when $j$ is even}
\end{cases}
if and only if $\omega$ and $k$ satisfy the dispersion relation
(ω^2 - (1+w))^2 - (k)(-k)=0
and $(v_1,v_2)^t$ is a an appropriately chosen eigenvector of $\tilde{L}(k)$. (See, for instance, <cit.>.)
The set of such $\omega$ and $k$ which meet (<ref>) has two connected components with $\omega\ge0$. (It is obviously even in $\omega$.) These are
ω = √(_-(k)), in which case we say $\omega$ is on the “acoustic" branch of (<ref>),
and ω = √(_+(k)), in which case we say $\omega$ is on the “optical" branch.
In the monatomic problem, the dispersion relation has but one branch and is akin to $\omega^2 -\sin^2(k)=0$, see <cit.>. Plotting $\tlambda_-(k)$ and $\sin^2(k)$ will show that the two functions are qualitatively much alike. The phase speed of the monatomic problem is maximum at $k = 0$; this maximum speed is called the speed of sound. The same is true for the phase speed
of waves associated to the acoustic branch. Specifically, corollary to the second estimate in (<ref>) is the fact that that the phase speed, $\omega/k$, of plane waves associated to the acoustic branch is no bigger than $c_w$.
The solitary waves for the monatomic problem have speeds which are strictly supersonic; the reason for this is discussed further below. In any case, the nonlinear waves move faster than the linear waves.[This fact is crucial to the proof of the stability of the solitary waves in <cit.> <cit.> <cit.> and <cit.>.]
But in the dimer problem we consider there are optical branch linear waves with all possible phase speeds, as can be seen by plotting $\sqrt{\tlambda_+(k)}/k.$
The point is this: if we search for a localized acoustic wave that travels with a supersonic speed $c>c_w$ then there will necessarily be an optical branch linear wave whose phase is exactly $c$.
See Figure <ref>.
Equating the
solitary wave's speed to the optical phase speed yields the relation
c^2 k^2 - \tlambda_+(k) = 0.
As stated in Lemma <ref>, there is a unique nonnegative solution of this: $k = k_c$.
Sketch of the phase speeds for the different branches of the dispersion relation.
In a very rough sense, then, we expect the localized acoustic wave with speed $c$
to excite a mode in the optical branch with wavenumber $k_c$.
Making this intuition rigorous is, of course, a substantial part of our analysis.
The inequalities in (<ref>) imply $\tilde{\lambda}_-(k) < \tilde{\lambda}_+(k)$ for all $k\in \R$. Thus $\tilde{L}(k)$ is diagonalizable for all $k$.
Towards this end, we compute that the eigenvectors of $\tilde{L}(k)$ are scalar multiples of
\left[ \begin{array}{c} \tilde{\beta}(k) \\ \tilde{\varrho}(k) \end{array} \right]$
(for $\tilde{\lambda}_-(k)$) and
\ds\left[ \begin{array}{c} \tilde{\beta}(k) \\ -\tilde{\varrho}(k) \end{array} \right]
$ (for $\tilde{\lambda}_+(k)$).
We can diagonalize $\tilde{L}$ by dropping these into a matrix. It will be advantageous to renormalize them first, though.
γ̃(k) : =e^-ik+e^ik ϱ̃(k) β̃(k)
and put
γ̃(k) β̃(k) γ̃(k) β̃(k)
γ̃(k) ϱ̃(k) -γ̃(k) ϱ̃(k)
Its inverse is
(2γ̃(k) β̃(k))^-1 (2γ̃(k) ϱ̃(k))^-1
(2γ̃(k) β̃(k))^-1 -(2γ̃(k) ϱ̃(k))^-1
Then we have
\tilde{J}_1(k) \tilde{L}(k) \tilde{J_2}(k) = \tilde{\Lambda}(k) := \left[\begin{array}{cc} \tilde{\lambda}_-(k) & 0 \\ 0 & \tilde{\lambda}_+(k) \end{array}\right].
The reason we choose to normalize the eigenvectors with $\tilde{\gamma}$ is obviously non-obvious. Here is what is special about $\tilde{\gamma}$:
γ̃(-k) β̃(-k) = γ̃(k) ϱ̃(k).
This property—easily checked—will imply a certain symmetry below, specifically in the proof of Lemma <ref>.
A short computation indicates that neither $\tilde{\gamma}$ nor $\tilde{\beta}$ vanish when $k \in \R$.
In fact, we have:
If $w > 1$ then there exists $\tau_1\in(0,\tau_0]$ such that $\tilde{\gamma}(z)$, $\tilde{\gamma}^{-1}(z)$, $\tbeta(z)$ and $\tilde{\beta}^{-1}(z)$ are uniformly bounded analytic functions for $z \in \overline{\Sigma}_{\tau_1}$.
$\tilde{J}_1(z)$ and $\tilde{J}_2(z)$ are
uniformly bounded matrix valued analytic functions for $z \in \overline{\Sigma}_{\tau_1}$.
We do not provide a proof, as it is more or less immediate from the definitions and Lemma <ref>.
Note also that
J̃_1(0) = 1 4(1+w) [ 1 1
1 - 1
] J̃_2(0) = 2(1+w) [ 1 1
1 - 1
] .
Since $\tilde{J}_1$ and $\tilde{J}_2$ diagonalize $\tilde{L}$, we can use Fourier multiplier operators to diagonalize $L$.
We use following normalizations and notations for the Fourier transform and its inverse:
\hat{f}(k):=\Fo[f](k):={1 \over 2 \pi} \int_\R f(x)e^{-ikx} dx \mand
\check{g}(x):=\Fo^{-1}[g](x) := \int_\R g(k)e^{ikx} dk.
Likewise, we use the following normalizations and notations for the Fourier series of
a $2 P$-periodic function:
\hat{f}(k):={1 \over 2P} \int_{-P}^P f(x) e^{-ik\pi x/P} dx \mand {f}(x) = \sum_{k \in \Z} \hat{f}(k) e^{ik \pi x/P}.
We have used the same “hat" notation for the Fourier transform and the coefficients of the Fourier series; context
will always make it clear which we mean.
Suppose that we have $\tmu:\R \to \C$. The “Fourier multiplier with symbol $\tmu$" is defined as follows.
* If $f:\R \to \C$ has a well-defined Fourier transform then
μf (x):= ∫_e^ikx (k) f̂(k) dk.
* If $f:\R \to \C$ is $2P$-periodic then
μf(x) := ∑_k ∈ e^ikπx/P (k π/P) f̂(k).
* If $f = f_1 + f_2$ where $f_1:\R \to \C$ has a well-defined Fourier transform and $f_2:\R \to \C$ is $2P$-periodic then
we have $\mu f = \mu f_1 + \mu f_2$ where $\mu f_1$ is computed with (<ref>) and $\mu f_2$ is computed with (<ref>).
An alternate way to express (<ref>) goes as follows.
Since $f(x)$ is $2P$-periodic we know $f(x) = \phi( \omega x)$ for some
$2 \pi$-periodic function $\phi(y)$ and $\omega = \pi/P$.
In this case $\mu f(x) = (\mu^\omega \phi)(\omega x)$ where
$\mu^\omega$ is a Fourier multiplier with symbol $\tmu^\omega(k) = \tmu(\omega k)$.
The nice thing here is that
\mu^\omega \phi(y) = \sum_{k \in \Z} e^{iky} \tmu^\omega(k ) \hat{\phi}(k),
a slightly less complicated formula than (<ref>).
Note also that if we put a $2P-$periodic function into the Fourier transform integral $\Fo$ then we can interpret the
as a distribution. Specifically, it will be a superposition of delta-functions situated on $(\pi/P) \Z \subset \R$.
In this case we can apply the formula (<ref>). The outcome of this
coincides exactly with (<ref>). In this way we can see that the formula in Part (iii) is the “correct" way to apply
Fourier multipliers to sums of decaying and periodic functions.
So let $\lambda_\pm$, $\varrho$, $\beta$, $\gamma$, $\Lambda$ and $J_n$ be the Fourier multiplier operators with symbols $\tlambda_\pm$, $\tilde{\varrho}$, $\tbeta$, $\tilde{\gamma}$, $\tilde{\Lambda}$ and $\tilde{J}_n$, respectively.
Put $\pb : = J_2 \h$. If $\pb$ solves (<ref>) then we find that $\h$ solves
c^2” + Λ+ ΛB(,) = 0
B(\h,\grave{\h}):=\left( \begin{array}{c} b_1(\h,\grave{\h}) \\ b_2(\h,\grave{\h}) \end{array} \right):= J_1 (J_2 \h.J_2 \grave{\h}).
Written out component-wise this is
c^2 h_1” + λ_- h_1 + λ_- b_1(,)=0 c^2 h_2” + λ_+ h_2 + λ_+ b_2(,)=0.
Since $\lambda_-$ is associated to the acoustic branch of the dispersion relation and $\lambda_+$ to the optical branch, we informally think of $h_1$
as the “acoustic part" of the solution and $h_2$ as the “optical part." What we shall see down the line is that the dominant part of $h_1$ is a $\sech^2$ traveling wave, whereas the periodic
solutions will be largest in $h_2$.
§.§ The Friesecke-Pego cancelation
Applying $\Fo$ to the first equation in (<ref>) gives
(-c^2 k^2 + λ̃_-(k) ) ĥ_1 +λ̃_-(k) b̂_̂1̂(̂,̂)̂ = 0.
Since $\tilde{\lambda}_-(0) = 0$ and is even, it is roughly quadratic in $k$ near the origin. Obviously so is $k^2$.
Which indicates that we can cancel a $k^2$ out of the above if we chose. We do not do precisely this, but instead employ a similar
approach inspired by the proof of the existence of low energy solitary waves for monotomic FPU in <cit.>.
We know that $\tlambda_-(0)=0$ . We also know from (<ref>) that $|\tlambda'_-(k)| \le 2c_w^2 |k|$. And so the FTOC[The Fundamental Theorem of Calculus, of course] implies:
c^2 k^2 - _-(k) > 0 for all $k \ne 0$ provided
$c^2 \ge c_w^2$.
allows us to divide through in (<ref>), not by $k^2$, but rather by $-c^2 k^2 + \tilde{\lambda}_-(k)$.
So put
\tilde{\varpi}_c(k) := -{\tilde{\lambda}_-(k) \over c^2 k^2 - \tilde{\lambda}_-(k)}.
This function has a removable singularity at $k =0$ when $c^2>c_w^2$ and no other singularities for $k \in \R$.
Then (<ref>) is equivalent to
\hat{h}_1 +\tilde{\varpi}_c(k) \hat{b_1(\h,\h)} = 0.
Let $\varpi_c f$ be the Fourier multiplier operator with symbol $\tilde{\varpi}_c $. The above reasoning shows we can rewrite (<ref>) as
h_1 +ϖ_c b_1(,) = 0 c^2 h_2” + λ_+ h_2 + λ_+ b_2(,) = 0or alternately as:
\H_c(\h):= \left[ \begin{array}{cc} 1 & 0 \\
0 & c^2 \partial_x^2 + \lambda_+
\end{array}
\right] \h + \left[ \begin{array}{cc} \varpi_c & 0 \\
0 & \lambda_+
\end{array}
\right] B(\h,\h) = 0.
We will henceforth require $c^2>c_w^2$ so that our map $\varpi_c$ is well-defined; this is the technical
reason why we look for (and why the authors of <cit.> looked for) nonlinear waves which are supersonic.
Unraveling the scalings that lead to (<ref>) from (<ref>) shows that the traveling wave solutions
under investigation here will,
in the physical coordinates, travel with a speed faster than $c_{sound}:=\sqrt{k_s / \bar{m}}$,
where $\bar{m}$ is the average of the two masses.
We have the following nice symmetry result for $\H_c$.
If $h_1$ is even and $h_2$ is odd,
the first and second components of ${\H}_c(\h)$ are, respectively, even and odd.
For a function $f(y)$ let $Rf(y) = f(-y)$. If $f$ is even then $f = Rf$. If $f$ is odd then $f = - Rf$.
If $\mu$ is a Fourier multiplier with symbol $\tilde{\mu}$ then
R (μf)(x) = (Rμ) Rf (x).
By $R \mu$ we mean the Fourier multiplier with symbol $R\tilde{\mu}(k)=\tilde{\mu}(-k)$.
If the symbol $\mu$ is even then this implies $R (\mu f) = \mu (Rf).$ This in turn implies that such a $\mu$ will map even functions to even functions and odd functions to odd functions. Thus $\lambda_\pm$, $\partial_x^2$ and $\varpi_c$ all “preserve parity."
Informally, we say $\h \in E \times O$ if $h_1$ is even and $h_2$ is odd. The preceding comments imply that we will have our result if we can show that $B(\h,\h)$ maps $E \times O$ to itself since the remaining parts of $\H_c$ will not flip an $E$
to an $O$ or vice versa.
Note that
\h \in E \times O$ if and only if $R\h = I_1 \h
where $I_1 = \diag(1,-1)$. Thus our goal is to show that if $R \h = I_1 \h$ then $R B(\h,\h) = I_1 B(\h,\h)$.
So suppose that $R \h = I_1 \h$. Using (<ref>) we have
RB(\h,\h) = (RJ_1) R[(J_2 \h)^{.2}].
It is easy to see that $R(fg) = Rf Rg$ and so the above gives
RB(\h,\h) = (RJ_1) [(R(J_2 \h))^{.2}].
Then we use (<ref>) again to get
RB(\h,\h) = (RJ_1) [((RJ_2) (R\h))^{.2}].
Since $R \h = I_1 \h$ we have
RB(\h,\h) = (RJ_1) [((RJ_2) (I_1 \h))^{.2}].
Then associativity gives:
RB(,) = (RJ_1) [((RJ_2 I_1) )^.2].
The multiplier for $R J_2$ is, using (<ref>),
RJ̃_2(k) = J̃_2(-k)=[
γ̃(-k) β̃(-k) γ̃(-k) β̃(-k)
γ̃(-k) ϱ̃(-k) -γ̃(-k) ϱ̃(-k)
Now we use the special property of $\tgamma$ in (<ref>) to convert this to:
RJ̃_2(k) =[
γ̃(k) ϱ̃(k) γ̃(k) ϱ̃(k)
γ̃(k) β̃(k) -γ̃(k) β̃(k)
If we let $I_2:=\left[\begin{array}{cc}0 & 1 \\ 1 & 0 \end{array} \right]$ then the above relation
implies, after a short calculation, that
R\tilde{J}_2 = I_2 \tilde{J}_2 I_1.
Since $\tilde{J}_1 = \tilde{J}_2^{-1}$, $I_1^{-1} = I_1$ and $I_2^{-1} = I_2$, this gives.
R\tilde{J}_1 = I_1 \tilde{J}_1 I_2.
Using these relations in (<ref>) gives us
RB(\h,\h) = I_1 J_1 I_2 [ (I_2 J_2 I_1^2) \h)^{.2} ].
Since $I_1^2$ is the identity this is
RB(\h,\h) = I_1 J_1 I_2 [ (I_2 J_2) \h)^{.2} ].
Also, it is easy to check that $(I_2 \f)^{.2}= I_2 (\f^{.2})$. Thus we have
RB(\h,\h) = I_1 J_1 I^2_2 [ (J_2 \h)^{.2} ].
Since $I_2^2$ is the identity this is
RB(\h,\h) = I_1 J_1 [ (J_2 \h)^{.2} ] = I_1 B(\h,\h).
This was our goal and so we are done.
In light of this result, we restrict our attention henceforth to looking for solutions which are even in the first component and odd in the second.
§.§ Long wave scaling
Now make the long wave scaling (inspired by the classical multiscale derivation of the Korteweg-de Vries
equation from monotomic FPU in <cit.>)
h_1(x):=\ep^2 \theta_1( \ep x),\quad h_2:= \ep^2 \theta_2(\ep x) \mand c^2 = c_w^2 + \ep^2
where $0 < \ep \ll 1$.
Note that if $\mu$ is a Fourier multiplier operator with symbol $\tilde{\mu}(k)$ and if $f(x) = \phi(\omega x)$ then
\mu f(x) = (\mu^\omega \phi) (\ep x)
where $\mu^\omega $
is a Fourier multiplier with symbol
{\tmu^\omega}(k) := \tilde{\mu}(\omega k).
After the scaling, (<ref>) becomes
θ_1 +ϖ^b_1^(,) = 0 ^2 (c_w^2 +^2) θ_2” + λ_+^θ_2 + ^2 λ^_+b^_2(,) = 0
with $\ds\thetab:=\left(\begin{array}{c} \theta_1\\ \theta_2 \end{array}\right),$
ϖ̃^̃(K):=^2 ϖ̃_√(c^2_w+^2) (K) = -^2 λ̃_-(K) (c_w^2 + ^2)^2K^2 - λ̃_-(K)
B^(,):=( b_1^(,)
b_2^(,) ) := J^_1 [ J_2^. J_2^ ].
Of course $\varpi^\ep$, $\lambda^\ep_\pm$ and $J_n^\ep$ are Fourier multiplier operators with the symbols taken in the obvious way. Note
that since we assume the $\ep \in (0,1)$, the scaling implies, via Lemma <ref> and Corollary <ref>, that $\tilde{\lambda}^\ep_\pm(Z)$
and $\tilde{J}_n^\ep(Z)$ are analytic for $|\Im(Z)|\le \tau_1 \le \tau_1/\ep$.
The equation for $\theta_2$ (which is the part associated to the optical branch) is classically singular when $\ep \sim 0$ because of
the term
$$\ep^2 (c_w^2 +\ep^2) \theta_2''.$$
Had we not performed the Friesecke-Pego cancelation earlier, the $\theta_1$ equation would have a similarly singular term.
But the cancelation desingularizes that equation as we shall demonstrate below. Of course this why they made that cancelation in their work <cit.> and why we do so here. But because $\tlambda_+(0)\ne 0$
there is no chance to make a similar cancelation in the second component.
We can write (<ref>) as:
Θ_():=[ 1 0
0 ^2(c_w^2 + ^2) ∂_X^2 + λ_+^ ]+[ ϖ^ 0
0 ^2λ^_+ ] B^(,) =0.
The long wave scaling does not effect the symmetry mapping properties that $\H_c$ had. To wit:
If $\theta_1$ is even and $\theta_2$ is odd,
the first and second components of ${\Theta}_\ep(\h)$ are, respectively, even and odd.
§ THE FORMAL LONG WAVE LIMIT
In this section we naively set $\ep = 0$ in (<ref>).
This is mostly routine. For instance, given the definitions in (<ref>) and (<ref>), we set
\tilde{\lambda}_+^0(K):=\tilde{\lambda}_+(0)$ and
$\tilde{J}^0_j(K):= \tilde{J}_j(0)$. So we define, (using (<ref>)):
\lambda^0_+ := 2+2w, \quad
J^0_1:={1 \over 4(1 + w)}
\left[\begin{array}{cr} 1 & 1 \\ 1 & -1
\end{array}
\right] \mand J^0_2:=2(1+w) \left[ \begin{array}{rr} 1& 1 \\ 1 & -1 \end{array}\right].
This in turn leads to the definition
B^0(\thetab,\grave{\thetab}):=\left(\begin{array}{c} b_1^0(\thetab,\grave{\thetab}) \\b_2^0(\thetab,\grave{\thetab})\end{array} \right) := J^0_1 \left[ J_2^0 \thetab . J_2^0 \grave{\thetab} \right]
= 2(1+w)\left(\begin{array}{c}
\theta_1 \grave{\theta}_1 + \theta_2 \grave{\theta}_2 \\
\theta_1 \grave{\theta}_2 + \theta_2 \grave{\theta}_1 \end{array}\right).
Blindly setting $\ep=0$ in $\tilde{\varpi}^\ep(K)$ will not work; a “$0/0$" situation occurs.
Computing the Maclaurin expansion of $\tilde{\lambda}_+(k)$ gives
\tilde{\lambda}_-(k) = c_w^2 k^2 - \alpha_w k^4 + \cdots
\alpha_w:={c_w^2 \over 3 } {1 - w + w^2 \over (1 + w)^2}>0.
Thus we have
\tilde{\varpi}^\ep(K) = -{\ep^2 \tilde{\lambda}_-(\ep K) \over (c_w^2 + \ep^2)\ep^2K^2 - \tilde{\lambda}_-(\ep K) }
= - { \ep^4 c_w^2 K^2 + \O_f(\ep^6) \over (c_w^2 + \ep^2)\ep^2 K^2 - \ep^2 c_w^2 K^2 + \alpha_w \ep^4 K^4+ \O_f(\ep^6) }
By “$\O_f(\ep^n)$" we mean terms which are formally of order $\ep^n$.
After some cancelations this becomes
\tilde{\varpi}^\ep(K)
= - { \ep^4 c_w^2 K^2 + \O_f(\ep^6) \over \ep^4 K^2 + \alpha_w \ep^4 K^4+ \O_f(\ep^6) }= - { c_w^2 K^2 + \O_f(\ep^2) \over K^2 + \alpha_w K^4+ \O_f(\ep^2) }.
Now we set $\ep =0$ to get
ϖ̃^0(K) := -c_w^2 1 + α_w K^2 ϖ^0 := -c_w^2(1 - α_w ∂_X^2)^-1.
We will give precise estimates on the operator norms of $\varpi^\ep - \varpi^0$, $\lambda_+^\ep-\lambda_+^0$ and so on in Section <ref>.
With all of this, if we put $\ep = 0$ in (<ref>) we arrive at:
\theta_1 - {4w\over 1 - \alpha_w \partial_X^2} \left[ \theta_1^2 + \theta_2^2 \right]=0 \mand (2+2w)\theta_2 = 0.
To solve this we take $\theta_2 = 0$ and $\theta_1$ a solution of
\theta_1 - {4w \over 1 - \alpha_w \partial_X^2} \left[ \theta_1^2 \right]=0.
Applying $1 - \alpha_w \partial_X^2$ to the above results in
α_w θ_1”-θ_1 +4w θ_1^2 =0.
This is a rescaling of the nonlinear differential equation whose solutions give the profile for the KdV solitary waves.
It has an explicit solution given by
θ_1(X)=σ(X):=σ_0 ^2 (2 q_0 X) where σ_0:=3 8wq_0:=1 2 √(α_w).
Note that if we put $\sigmab:=(\sigma,0)^t$ then we have shown:
σ+ϖ^0 b_1^0(,) = 0.
Observe that in the limit $\ep \to 0^+$ the part of the solution associated to optical branch, $\theta_2$, plays no role. All the action is happening for the acoustic part $\theta_1$.
In <cit.> the authors use homogenization theory to show that solutions of (<ref>) with initial data of the form $r(j,0) = \ep^2 R(\ep j)$ and $\dot{x}(j,0) = \ep^2 V(\ep j)$ are well-approximated over long times by a pair of solutions of KdV. Specifically they show that
r(j,t) = \ep^2 U_-(\ep(j - c_w t),\ep^3t) +\ep^2 U_+(\ep(j + c_w t),\ep^3t) + \O(\ep^{3/2})\quad \text{for all $|t| \le C\ep^{-3}$}
where $U_\pm$ solve the KdV equations
\pm \partial_T U_\pm + \alpha_w \partial_X^3 U_\pm + 4 w \partial_X (U^2_\pm) = 0.
Making a traveling wave ansatz for these of the form $U_\pm(X,T) = \theta(X \mp T)$
results in
(<ref>); the coefficients match exactly. Which is to say the results here are consistent with those in <cit.>.
§ PERIODIC SOLUTIONS
In this section we prove the existence of spatially periodic solutions of (<ref>).
To this end, we first compute the linearization of that equation about $\thetab(X) = 0$ to get
\theta_1 = 0 \mand \ep^2(c_w^2 + \ep^2) \theta_2'' + \lambda_+^\ep \theta_2 = 0.
We are looking for solutions where $\theta_2$ is odd and periodic. Thus we can take $\theta_2(X) = \sin(K_\ep X)$ for some $K_\ep \in \R$.
Inserting this into the second equation above, and recalling that $\lambda_+^\ep$ is a Fourier multiplier operator with symbol $\tlambda_+(\ep K)$, gives
\ep^2 (c_w^2 + \ep^2) K_\ep^2 - \tlambda_+(\ep K_\ep) = 0.
If we put $c = c_\ep:= \sqrt{c_w^2 + \ep^2}$ and
this last equation is exactly (<ref>). Which is to say, by virtue of Part (vi) of Lemma <ref>,
that $K_\ep$ is its unique nonnegative solution.
Thus we have odd periodic solutions of the the linearization of (<ref>) of the form
\thetab(X) = \nub_\ep(X):=\sin(K_\ep X) \jb.
We can extend the existence of periodic solutions for the linear problem to the full nonlinear problem (<ref>) by means of
the technique of “bifurcation from a simple eigenvalue," developed by Crandall & Rabinowitz in <cit.> and Zeidler in <cit.>.
Here is what we find:
For all $w>1$ there exist $\ep_0 > 0$, $a_0>0$ and $0 < C_1 < C_2$ such the following holds for all for $\ep \in (0,\ep_0)$.
There exist maps
K^a_ : [-a_0,a_0] ⟶
ψ_,1^a : [-a_0,a_0] ⟶C^∞_∩{even functions}
ψ_,2^a : [-a_0,a_0] ⟶C^∞_∩{odd functions}
with the following properties.
* Putting
a\left( \begin{array}{c}
\sin(K_\ep^a X)
\end{array}
\right)+a\left( \begin{array}{c}
\psi^a_{\ep,1}(K_\ep^a X)\\
\psi^a_{\ep,1}(K_\ep^a X)
\end{array}
\right)=:a {\nub}(K_\ep^a X) + \psib^a_\ep(K_\ep^a X)
solves (<ref>) for all $|a|\le a_0$.
* $K_\ep^0 = K_\ep$ where $K_\ep$ is the unique positive solution of
_√(c_w^2+^2)(K):= -^2 (c_w^2 + ^2)K^2 + _+(K) =0.
Moreover $K_\ep = \O(1/\ep)$ in the sense that
C_1/< K_< C_2 / for all $\ep \in (0,\ep_0)$.
* $\psi^0_{\ep,1} = \psi^0_{\ep,2} = 0$.
* $\ds \int_{-\pi}^{\pi} \psi^a_{\ep,2}(y) \sin(y) \ dy = 0$ for all $|a|\le a_0$.
* For all $r \ge 0$, there exists $C_r>0$ such that for all $|a|,|\ga|\le a_0$ we have
|K_^a |+ ^a_ _C^r_×C^r_≤C_r
|K_^a -K_^|+ ^a_ - ^__C^r_×C^r_ ≤C_r|a-|.
The remainder of Section <ref> is dedicated to the proof of this theorem.
§.§ Frequency freezing
We begin by making the additional scaling
(X) := (ωX) with :=( ϕ_1
ϕ_2 ), ω∈,
where $\phib(Y)$ is $2\pi$-periodic. By Remark <ref>, our system (<ref>) becomes
_(,ω) :=
[ 1 0
0 ^2ω^2(c_w^2 + ^2) ∂_Y^2 + λ_+^ω ]+[ ϖ^,ω 0
0 ^2λ^ω_+ ] B^ω(,) =0.
where $\varpi^{\ep,\omega}$ is the multiplier with symbol
\tilde{\varpi}^{\ep,\omega}(K) := \tilde{\varpi}^{\ep}(\omega{K}),
and the multipliers $\lambda_+^{\ep\omega}$ and $B^{\ep\omega}$ conform to their prior definitions.
Since $B^{\ep\omega}$ is quadratic in $\phib$, it is easy to see that
D_{\phib}\Phib_{\ep}(0,\omega) = \begin{bmatrix*}
1 &0 \\
0 &\ep^2\omega^2(c_w^2+\ep^2)\partial_Y^2 + \lambda_+^{\ep\omega}
\end{bmatrix*}.
When $\omega = K_{\ep}$, one may show that zero is a simple eigenvalue of $D_{\phib}\Phib_{\ep}(0,K_{\ep})$ when the operator is restricted to a suitable function space;
this is essentially just the calculation carried out at the start of this section. Consequently,
the classical bifurcation results in <cit.> and <cit.> can be used to show that there exists a nontrivial family of solutions to $\Phib_{\ep}(\phib,\omega) = 0$
branching out of $(0,K_\ep)$.
Unfortunately, those classical results do not provide, in an easy way, estimates on the solution which are uniform in $\ep$.
And so, while our strategy is modeled on the proofs of the results in <cit.> and <cit.>, we carry out the proof from scratch and always with our eyes
on how quantities depend on $\ep$.
Our first step is to convert (<ref>) to a fixed point equation.
§.§ Conversion to a fixed-point problem
\Y = (H_{\per}^2 \cap \{\text{even functions}\}) \times (H_{\per}^2 \cap \{\text{odd functions}\}),
where $H_{\per}^r$ is the Sobolev space of $2\pi$-periodic functions $\phi$ such that
\norm{\phi}_{H_{\per}^r} := \left(\sum_{k \in \Z} (1+k^2)^r|\hat{\phi}(k)|^2\right)^{1/2} < \infty.
With $\nub(Y) := \sin(Y)\bf{j}$, we have the direct sum decomposition $\Y = \mathcal{N} \oplus \ortho$, where $\ortho \subseteq \Y$ is the orthogonal complement of $\mathcal{N} := \spn(\{\nub\})$ in the standard $H_{\per}^2 \times H_{\per}^2$ inner product, i.e.,
\ip{\phib}{\psib}_{H_{\per}^2\times H_{\per}^2} := \ip{\phi_1}{\psi_1}_{H_{\per}^2} + \ip{\phi_2}{\psi_2}_{H_{\per}^2} \quad\text{ with }\quad \ip{\phi}{\psi}_{H_{\per}^2} := \sum_{k \in \Z} (1+k^2)^2\hat{\phi}(k)\overline{\hat{\psi}(k)}.
We may then write any $\phib \in \Y$ as
= a+ a for some a ∈, ∈.
Observe that if $\psib \in \ortho$, then $\hat{\psib}(\pm1)\cdot\jb = 0$. Set
= ×.
Since the trivial solution $\phib = 0$ already solves (<ref>) for any choice of $\omega$, we will assume $a \ne 0$. After factoring and dividing by $a$, the problem
\Phib_{\ep}(\phib,K_{\ep}+t) = \Phib_{\ep}(a\nub+a\psib,K_{\ep}+t)=0
ψ_1 + aϖ^,K_+tb_1^(K_+t)(+,+) = 0
(^2(K_+t)^2(c_w^2+^2)∂_Y^2 + λ_+^(K_+t))(sin(Y)+ψ_2) + a^2λ_+^(K_+t)b_2^(K_+t)(+,+)=0.
Let $\Pi_1$ be the Fourier multiplier with symbol
\tilde{\Pi}_1 := \begin{cases}
1, &|k| = 1 \\
0, &|k| \ne 1
\end{cases}
and set $\Pi_2 = 1-\Pi_1$. Then (<ref>) and (<ref>) are equivalent to
ψ_1 + aϖ^,K_+tb_1^(K_+t)(+,+) = 0,
(^2(K_+t)^2(c_2^2+^2)∂_X^2 + λ_+^(K_+t))sin(Y)+ a^2Π_1λ_+^(K_+t)b_2^(K_+t)(+,+)=0
(^2(K_+t)^2(c_2^2+^2)∂_X^2 + λ_+^(K_+t))ψ_2 + a^2Π_2λ_+^(K_+t)b_2^(K_+t)(+,+)=0.
Condition (<ref>) immediately gives a fixed-point equation for $\psi_1$, and we see that (<ref>) holds if and only if the Fourier transform of its left side evaluated at $k=\pm1$ is zero. Because $b_2^{\ep({K_{\ep}}+t)}(\nub+\psib,\nub+\psib)$ is odd by (the proof of) Lemma <ref>, we need only consider this Fourier transform at $k=1$. With
\tilde{\xi}_{\sqrt{c_w^2+\ep^2}}(k) = -(c_w^2+\ep^2)(\ep{k})^2 + \tlambda_+(\ep{k})
as in (<ref>) and $\cep := \sqrt{c_w^2+\ep^2}$, set
^,t(k) := ξ̃_((K_+t)k)
so that (<ref>) is equivalent to
1/2iξ̃^,t(1)+ a^2[Π_1λ_+^(K_+t)b_2^(K_+t)(+,+)](1) = 0.
Taylor's theorem tells us that
\txi^{\ep,1}(1) = \txi_{\cep}(\ep K_{\ep}+ \ep{t}) = \txi_{\cep}'(\ep K_{\ep})(\ep{t}) + R_{\ep}(\ep{t})(\ep{t})^2,
and Part (vi) of Lemma <ref> provides a number $l_0 > 0$ such that
|\txi_{\cep}'(\ep{K_{\ep}})| \ge l_0
for all $\ep$ sufficiently close to zero. So, we may rewrite (<ref>) as
t = -/_'(K_)R_(t)t^2 - 2ia/_'(K_)[Π_1λ_+^(K_+t)b_2^(K_+t)(+,+) ](1)
Finally, we will show that $\txi^{\ep,t}(k) \ne 0$ for $k \ne \pm1$, which means that the multiplier $(\xi^{\ep,t})^{-1}$ with symbol $(\txi^{\ep,t})^{-1}$ is well-defined on the range of $\Pi_2$ for suitably small $\ep$ and $t$. Then (<ref>) becomes
ψ_2 = -a^2(ξ^,t)^-1Π_2λ_+^(K_+t)b_2^(K_+t)(+,+).
We have arrived at our ultimate fixed-point problem. Set $\Psib_{\ep} := (\Psi_{\ep,1}, \Psi_{\ep,2},\Psi_{\ep,3})$ with
\begin{align*}
\Psi_{\ep,1}(\psib,t,a) &:=-a\varpi^{\ep,{K_{\ep}}+t}b_1^{\ep({K_{\ep}}+t)}(\nub+\psib,\nub+\psib) \\
\\
\Psi_{\ep,2}(\psib,t,a) &:= -a\ep^2(\xi^{\ep,t})^{-1}\Pi_2\lambda_+^{\ep({K_{\ep}}+t)}b_2^{\ep({K_{\ep}}+t)}(\nub+\psib,\nub+\psib) \\
\\
\Psi_{\ep,3}(\psib,t,a) &:= -\frac{\ep}{\txi_{\cep}'(\ep{K_{\ep}})}R_{\ep}(\ep{t})t^2 - \frac{2i\ep{a}}{\txi_{\cep}'(\ep{K_{\ep}})}\Fo\big[\Pi_1\lambda_+^{\ep({K_{\ep}}+t)}b_2^{\ep({K_{\ep}}+t)}(\nub+\psib,\nub+\psib)\big]\!(1)
\end{align*}
We will solve this problem by applying the following lemma, whose proof is given in Section <ref>, to the map $\Psib_{\ep}$ for $\ep$ sufficiently small.
Let $\X$ be a Banach space and let $\ball(r) = \{x \in \X : \norm{x} \le r\}$. For $0 < \ep < \ep_0$ let $F_{\ep} \colon \X \times \R \to \X$ be maps with the property that for some $C_1,a_1,r_1 > 0$, if $x,y \in \ball(r_1)$ and $|a| \le a_1$, then
\begin{align}
\sup_{0 < \ep < \ep_0} \norm{F_{\ep}(x,a)} &\le C_1\left(|a|+|a|\norm{x}+\norm{x}^2\right) \label{bound1} \\
\nonumber \\
\sup_{0 < \ep < \ep_0} \norm{F_{\ep}(x,a)-F_{\ep}(y,a)} &\le C_1\left(|a| + \norm{x}+\norm{y}\right)\norm{x-y} \label{bound2}
\end{align}
Then there exist $a_0 \in (0,a_1], r_0 \in (0,r_1]$ such that for each $0 < \ep < \ep_0$ and $|a| \le a_0$, there is a unique $x_{\ep}^a \in \ball(r_0)$ such that $F_{\ep}(x_{\ep}^a,a) = x_{\ep}^a$.
Suppose as well that the maps $F_{\ep}(\cdot,a)$ are Lipschitz on $\ball(r_0)$ uniformly in $a$ and $\ep$, i.e., there is $L_1 > 0$ such that
sup_0 < < _0
x ≤r_0 F_(x,a)-F_(x,à) ≤L_1|a-à|
for all $|a|,|\grave{a}| \le a_0$. Then the mappings $[-a_0,a_0] \to \X \colon a \mapsto x_{\ep}^a$ are also uniformly Lipschitz; that is, there is $L_0 > 0$ such that
sup_0 < < _0 x_^a-x_^à ≤L_0|a-à|
for all $|a|,|\grave{a}| \le a_1$.
§.§ Application of Lemma <ref>
We begin with a general observation about Fourier multipliers. The proof of this lemma follows from direct calculations with the norm
\norm{\psi}_{H_{\per}^r}^2 = \sum_{k \in \Z} (1+k^2)^r|\hat{\psi}(k)|^2,
and so we omit it. Throughout this section, we denote by $\bdop(\mathcal{U},\mathcal{V})$ the space of bounded linear operators between normed spaces $\mathcal{U}$ and $\mathcal{V}$ and set $\bdop(\mathcal{U}) := \bdop(\mathcal{U},\mathcal{U})$.
Let $\mu$ be a Fourier multiplier with symbol $\tmu \in L^{\infty}(\R)$ and let $\omega \in \R$. As in Remark <ref>, let $\mu^{\omega}$ be the Fourier multiplier with symbol $\tmu^{\omega}(k) = \tmu(\omega{k})$. Then
* $\sup_{r,\omega \in \R} \norm{\mu^{\omega}}_{\bdop(H_{\per}^r)} \le \norm{\tmu}_{L^{\infty}(\R)}.$
* If $\tmu$ is Lipschitz, i.e., there is $\Lip(\tmu)> 0$ such that $|\tmu(k) - \tmu(\grave{k})| \le \Lip(\tmu)|k-\grave{k}|$, then
\norm{(\mu^{\omega}-\mu^{\grave{\omega}})\psi}_{H_{\per}^r} \le \Lip(\tmu)|\omega-\grave{\omega}|\norm{\psi}_{H_{\per}^{r+1}}
for all $\omega,\grave{\omega},r \in \R$ and $\psi \in H_{\per}^{r+1}$.
* If there exist $C,p > 0$ such that
|\tmu(k)| \le \frac{C}{(1+k^2)^p}
for all $k \in \R$, then $\norm{\mu}_{\bdop(H_{\per}^r,H_{\per}^{r+2p})} \le C$.
Informally, Part (ii) of Lemma <ref> means that taking a Lipschitz estimate for the map $\omega \mapsto \mu^{\omega}$ costs us a derivative.
The following two lemmas on the Fourier multipliers $\varpi^{\ep,K_{\ep}+t}$ and $(\xi^{\ep,t})^{-1}\Pi_2$ are the keys to our application of Lemma <ref> to the maps $\Psib_{\ep}$. They follow directly from the corresponding results for the symbols $\tvarpi^{\ep,K_{\ep}+t}$ and $(\txi^{\ep,t})^{-1}$, which are stated below as Lemmas <ref> and <ref> and proved in Section <ref>.
* There exist $\ep_{11}, C_{\varpi \max} > 0$ such that
\sup_{\substack{0 < \ep < \ep_{11} \\ |t| \le 1 \\ r \in \R}} \norm{\varpi^{\ep,K_{\ep}+t}}_{\bdop(H_{\per}^r, H_{\per}^{r+2})} \le C_{\varpi \max}.
* There exists $C_{\varpi \Lip} > 0$ such that if $|t|,|\grave{t}| \le 1$, then
\sup_{\substack{0 < \ep < \ep_{11} \\ r \in \R}} \norm{\varpi^{\ep,K_{\ep}+t}-\varpi^{\ep,K_{\ep}+\grave{t}}}_{\bdop(H_{\per}^r)} \le C_{\varpi \Lip}|t-\grave{t}|.
* There exist $\ep_{12}, C_{\xi \max} > 0$ such that
\sup_{\substack{0 < \ep < \ep_{12} \\ |t| \le 1 \\ r \in \R}} \norm{\ep^2(\xi^{\ep,t})^{-1}\Pi_2}_{\bdop(H_{\per}^r,H_{\per}^{r+2})} \le C_{\xi \max}.
* There exists $C_{\xi \Lip} > 0$ such that
\sup_{\substack{0 < \ep < \ep_{12} \\ r \in \R}} \norm{\ep^2(\xi^{\ep,t})^{-1}\Pi_2 - \ep^2(\xi^{\ep,t})^{-1}\Pi_2}_{\bdop(H_{\per}^r)} \le C_{\xi \Lip}|t-\grave{t}|
for all $|t|, |\grave{t}| \le 1$.
There exists $\ep_{11} > 0$ such that the following hold.
* There is $C_{\tvarpi \max} > 0$ such that
sup_0 < < _11
k ∈
|t| ≤1 |^,K_+t(k)| ≤C_max/1+k^2.
* There is $C_{\tvarpi \Lip} > 0$ such that
sup_0 < < _11
k ∈ |^,K_+t(k)-^,K_+t̀(k)| ≤C_|t-t̀|
for all $|t|,|\grave{t}| \le 1$.
There exists $\ep_{12} > 0$ such that the following hold.
* There is $C_{\txi \max} > 0$ such that
sup_0 < < _12
k ∈∖{-1,1}
|t| ≤1 |1/^,t(k)| ≤C_max/1+k^2.
* There is $C_{\txi \Lip} > 0$ such that
sup_0 < < _0
k ∈∖{-1,1} |1/^,t(k) - 1/^,t̀(k)| ≤C_|t-t̀|.
Last, Taylor's theorem provides the following useful decomposition of $\txi^{\ep,t}$, which we prove in Section <ref>.
For $0 < \ep < \ep_{12}$ and $\tau \in \R$, we have
_(K_+τ) = _'(K_)τ+ R_(τ)τ^2,
where the functions $R_{\ep}$ have the following property: there exist $C_{R \max},C_{R \Lip} > 0$ such that when $0 < \ep < \ep_{12}$,
sup_0 < < _12 |R_(τ)| ≤C_R max sup_0 < < _12 |R_(τ)-R_(τ̀)| ≤C_R |τ-τ̀|
for all $\tau,\grave{\tau} \in \R$.
We are now ready to apply Lemma <ref> to our map $\Psib_{\ep}$.
Let $\ep_0 = \min\{\ep_{11},\ep_{12}\}$. The maps $\Psib_{\ep}, 0 < \ep < \ep_0,$ satisfy the conditions (<ref>), (<ref>), and (<ref>) of Lemma <ref> on the space $\X$ defined in (<ref>) when $a_0 = r_0 = 1$.
We begin with some additional notation. Set $\H^r := H_{\per}^r \times H_{\per}^r, \norm{\psib}_r := \norm{\psib}_{\H^r},$ and
T_0^(t) :=
ϖ^,K_+t 0
0 ^2(ξ^,t)^-1Π_2
T_1^(t) :=
1 0
0 λ_+^(K_+t)
T_2^(t) := J_2^(K_+t).
Lemmas <ref> and <ref> combine to produce constants $C_0,C_1,C_2 > 0$ such that the following estimates hold:
sup_0 < < _0
|t| ≤1
r ∈ T_0^(t)_(^̋r,^̋r+2) ≤C_0
sup_0 < < _0
|t| ≤1
r ∈ T_i^(t)_(^̋r) ≤C_i, i = 1,2
sup_0 < < _0
r ∈ T_0^(t)-T_0^(t̀)_(^̋r) ≤C_0|t-t̀|, |t|,|t̀| ≤1
sup_0 < < _0
r ∈ T_i^(t)-T_i^(t̀)_(^̋r,^̋r-1) ≤C_i|t-t̀|, |t|,|t̀| ≤1, i = 1,2.
\begin{align}
\bG_{\ep}(\psib,t) &= T_1^{\ep}(t)(T_2^{\ep}(t)(\nub+\psib))^{.2} \label{big G} \\
\bF_{\ep}(\psib,t) &= T_0^{\ep}(t)\bG_{\ep}(\psib,t) \label{fep}.
\end{align}
The estimates (<ref>) along with the Sobolev embedding estimate
._r ≤C_sob,r_r_r, ,∈^̋r, r ≥1,
give $M_{12,r} > 0$ such that
sup_0 < < _0
|t| ≤1 _(,t)_r ≤M_12,r(_r^2 + _r + 1).
We then use (<ref>) to find
\sup_{\substack{0 < \ep < \ep_0 \\ |t| \le 1}} \norm{\bF_{\ep}(\psib,t)}_{r} \le \sup_{\substack{0 < \ep < \ep_0 \\ |t| \le 1}} \norm{T_0^{\ep}(t)}_{\bdop(\H^{r-2},\H^r)}\norm{\bG_{\ep}(\psib,t)}_{r-2} \le C_0M_{12}(\norm{\psib}_{r-2}^2 + \norm{\psib}_{r-2}+1).
Set $M_{012} = C_0M_{12}$. Since
we find
sup_0 < < _0
|t| ≤1
_r ≤M_012|a|(_r-2^2 + _r-2 + 1).
We will return to the estimate (<ref>) when we prove the bounds (<ref>) for our fixed points. For now, we take $r=2$ to obtain a constant $M_2 > 0$ such that
\sup_{\substack{0 < \ep < \ep_0 \\ \norm{\psib}_2, |t|, |a| \le 1}}\normlarge{\begin{pmatrix*}
\Psi_{\ep,1}(\psib,t,a) \\
\Psi_{\ep,2}(\psib,t,a)
\end{pmatrix*}}_2 \le M_2|a|.
This implies the first estimate (<ref>) of Lemma <ref> for the components $\Psi_{\ep,1}$ and $\Psi_{\ep,2}$.
To prove the second estimate (<ref>) of Lemma <ref>, we first rewrite
_(,t) - _(,t̀) = T_0^(t)(_(,t)-_(,t̀)) + (T_0^(t)-T_0^(t̀))_(,t̀)
and then find
\begin{align*}
\bG_{\ep}(\psib,t)-\bG_{\ep}(\grave{\psib},\grave{t}) &= T_1^{\ep}(t)( [T_2^{\ep}(t)(\nub+\psib)+T_2^{\ep}(\grave{t})(\nub+\grave{\psib})].[(T_2^{\ep}(t)-T_2^{\ep}(\grave{t}))(\nub+\grave{\psib})]) \\
&+T_1^{\ep}(t)[T_2^{\ep}(t)(\nub+\psib)+T_2^{\ep}(\grave{t})(\nub+\grave{\psib})].[T_2^{\ep}(t)(\psib-\grave{\psib})] \\
&+(T_1^{\ep}(t)- T_1^{\ep}(\grave{t}))(T_2^{\ep}(\grave{t})(\nub+\grave{\psib}))^{.2}.
\end{align*}
We estimate the third term above; estimates for the first two terms follow by similar techniques. We have
\begin{align*}
\norm{(T_1^{\ep}(t)- T_1^{\ep}(\grave{t}))(T_2^{\ep}(\grave{t})(\nub+\grave{\psib}))^{.2}}_{r-2} &\le C_1|t-\grave{t}|\norm{(T_2^{\ep}(\grave{t})(\nub+\grave{\psib}))^{.2}}_{r-1} \text{ by \eqref{T12 lip}} \\
\\
&\le C_1C_{\text{sob},r-1}\norm{T_2^{\ep}(\grave{t})(\nub+\grave{\psib})}_{r-1}^2 \text{ by \eqref{sob}} \\
\\
&\le C_1C_{\text{sob},r-1}C_2^2\norm{\nub+\grave{\psib}}_{r-1}^2 \text{ by \eqref{T12 bound}} \\
\\
&\le C_1C_{\text{sob},r-1}C_2^2(\norm{\grave{\psib}}_{r-1}^2+2\norm{\nub}_{r-1}\norm{\grave{\psib}}_{r-1} + \norm{\nub}_{r-1}^2) \\
\\
&\le C_1C_{\text{sob},r-1}C_2^2\max\{2\norm{\nub}_{r-1},1\}(\norm{\grave{\psib}}_{r-1}^2+\norm{\grave{\psib}}_{r-1}+1).
\end{align*}
After comparable work on the other two terms, we ultimately arrive at a constant $L_{12,r} > 0$ such that
sup_0 < < _0 _(,t)-_(,t̀)_r-2 ≤L_12,r(_r^2 + _r + _r+1)(-_r-1 + |t-t̀|).
We will need this estimate below when we work on $\Psi_{\ep,3}$. For now, we return to (<ref>) and find
\norm{\bF_{\ep}(\psib,t) - \bF_{\ep}(\grave{\psib},\grave{t})}_r \le \norm{T_0^{\ep}(t)}_{\bdop(\H^{r-2},\H^r)}\norm{\bG_{\ep}(\psib,t)-\bG_{\ep}(\grave{\psib},\grave{t})}_{r-2} +\norm{T_0^{\ep}(t)-T_0^{\ep}(\grave{t})}_{\bdop(\H^r)}\norm{\bG_{\ep}(\grave{\psib},\grave{t})}_r.
Combining (<ref>), (<ref>), and (<ref>) produces $L_{012,r} > 0$ such that
sup_0 < < _0 _(,t) - _(,t̀)_r ≤L_012,r( _r^2+ _r^2+_r + _r)(-_r-1 + |t-t̀|).
Taking $r = 2$ and assuming $\norm{\psib}_2,\norm{\grave{\psib}}_2 \le 1$, we find $L_2 > 0$ such that
\sup_{0 < \ep < \ep_0} \norm{\bF_{\ep}(\psib,t) - \bF_{\ep}(\grave{\psib},\grave{t})}_2 \le L_2(\norm{\psib-\grave{\psib}}_2 + |t-\grave{t}|).
This together with (<ref>) proves the second estimate (<ref>) of Lemma <ref> for $\Psi_{\ep,1}$ and $\Psi_{\ep,2}$.
Now we proceed to study $\Psi_{\ep,3}$. Set
T_4(t) = \begin{bmatrix*}
0 &0 \\
0 &\Pi_1
\end{bmatrix*}
and keep $T_1$ and $T_2$ as in (<ref>). Here, however, we will only care about the case $r=2$. Using the general bound
|[f](k)| ≤f_L_^∞ ≤C_sob,2f_H_^2,
we find
\begin{align*}
|\Psi_{\ep,3}(\psib,t,a)| &\le \frac{\ep_0C_{R \max}}{l_0}t^2 + \frac{2C_{\text{sob},2}\ep_0|a|}{l_0}\norm{\Pi_1\lambda_+^{\ep(K_{\ep}+t)}b_2^{\ep(K_{\ep}+t)}(\nub+\psib,\nub+\psib)}_{H_{\per}^2} \\
\\
&\le \frac{\ep_0C_{R \max}}{l_0}t^2 + \frac{2C_{\text{sob},2}\ep_0|a|}{l_0}\norm{T_4\bG_{\ep}(\psib,t)}_2 \\
\\
&\le \frac{\ep_0C_{R \max}}{l_0}t^2 + \frac{2C_{\text{sob},r}\norm{T_4}_{\bdop(\H^r)}M_{12}\ep_0|a|}{l_0}(\norm{\psib}_2^2 + \norm{\psib}_2 + 1) \text{ by \eqref{G bound}}.
\end{align*}
Thorough rearrangement of this last line, as well as the assumption $\norm{\psib}_2, |a| \le 1$, produces a constant $M_3 > 0$ such that
\sup_{0 < \ep < \ep_0} |\Psib_{\ep,3}(\psib,t,a)| \le L_3(|a| + \norm{\psib}_2 + t^2), \ \norm{\psib}_2,|t|,|a| \le 1
and this is sufficient to obtain the estimate (<ref>) of Lemma <ref> for $\Psi_{\ep,3}$.
The proof of estimate (<ref>) for $\Psi_{\ep,3}$ is similar to the work above; we omit the details but mention that it uses the Fourier transform estimate (<ref>), the uniform bounds on the functions $R_{\ep}$ from Lemma <ref>, and the Lipschitz estimate (<ref>) for the functions $\bG_{\ep}$.
Last, the final bound (<ref>) of Lemma <ref> is easily established for the components $\Psi_{\ep,i}$ using the uniform bounds on the operators $T_i^{\ep}$ developed above; again, we omit the details.
Lemma <ref> thus provides a number $a_0 > 0$ and, for all $0 < \ep < \ep_0$ and $|a|\le a_0$, a unique pair $(\psib_{\ep}^a,t_{\ep}^a) \in \{(\psib,t) \in \X : \norm{(\psib,t)}_{\X} \le 1\}$ such that $\Psib_{\ep}(\psib_{\ep}^a,t_{\ep}^a,a) = (\psib_{\ep}^a,t_{\ep}^a)$. We may reverse each step of the conversion in Section <ref> and we recall the scaling (<ref>) and the decomposition (<ref>) to find that the function
\thetab(X) := a\varphib_{\ep}^a(X) := a\nub(\ep({K_{\ep}}+t_{\ep}^a)X) + a\psib(\ep({K_{\ep}}+t_{\ep}^a)X)
solves (<ref>). Defining $K_{\ep}^a := {K_{\ep}} + t_{\ep}^a$, we have the maps (<ref>) and property (i) of Theorem <ref>. We prove the rest of the theorem below.
(of Theorem <ref>, Parts (ii), (iii), (iv) and (v)) When $a = 0$, the fixed-point property of $(\psi_{\ep,1}^0,\psi_{\ep,2}^0,t_{\ep}^0)$ and the definition of $\Psib_{\ep}$ give
(ψ_1,^0,ψ_2,^0,t_^0) = _(_^0,t_^0,0) = (0,0,-/_'(K_)R_(t_^0)(t_^0)^2).
We see immediately that $\psi_{\ep,1}^0 = \psi_{\ep,2}^0 = 0$, which is Part (iii), and also
t_{\ep}^0 = -\frac{\ep}{\txi_{\cep}'(\ep{K_{\ep}})}R_{\ep}(\ep{t}_{\ep}^0)(t_{\ep}^0)^2.
Scaling both sides by $\ep$ and rearranging, we find
0 = \tilde{\xi}_{\cep}'(\ep{K_{\ep}})(\ep{t}_{\ep}^0) + R_{\ep}(\ep t_{\ep}^0)(\ep t_{\ep}^0)^2 = \tilde{\xi}_{\cep}(\ep{K_{\ep}}+\ep t_{\ep}^0)
by (<ref>). We may assume that we have taken $\ep_0$ to be so small that $\ep{K_{\ep}} + \ep t > 0$ for any $0 < \ep < \ep_0$ and $|t| \le 1$, thus $\ep{K_{\ep}} + \ep t_{\ep}^0 > 0$. By the uniqueness of positive roots of $\txi_{\cep}$ given in Part (vi) of Lemma <ref>, we have $\ep{K_{\ep}} + \ep t_{\ep}^0 = \ep{K_{\ep}}$, hence $t_{\ep}^0 = 0$ and $K_{\ep}^0 = K_{\ep}$. So, Part (ii) holds.
For Part (iv), since $\psib_{\ep}^a \in \ortho$, we know $\hat{\psib_{\ep}^a}(\pm1)\cdot\jb = 0$, thus
\int_{-\pi}^{\pi} \psi_{\ep,2}^a(y)\sin(y) \ dy = \sum_{k \in \Z} \hat{\psi_{\ep,2}^a}(k)\hat{\sin}(k) = 0.
Last, for Part (v), by (<ref>) in Lemma <ref> we have positive constants $m_*(w)$ and $m^*(w)$, depending only on $w$, such that
m_*(w) \le \ep K_{\ep} \le m^*(w), \ 0 < \ep < 1.
This shows $K_{\ep} = \O(1/\ep)$ and also allows us to estimate
|K_^a| = |K_ + t_^a| ≤m^*(w) + 1.
Next, relying on the notation of the proof of Proposition <ref>, when $r=2$ we have
\sup_{\substack{0 < \ep < \ep_0 \\ |a| \le a_0}} \norm{\psib_{\ep}^a}_2 \le r_0 \le 1
by Lemma <ref>, and when $r > 2$, (<ref>) implies the bootstrap estimate
\norm{\psib_{\ep}^a}_r = \normlarge{\begin{pmatrix*}\Psi_{\ep,1}(\psib_{\ep}^a,t_{\ep}^a,a) \\
\Psi_{\ep,2}(\psib_{\ep}^a,t_{\ep}^a,a)
\end{pmatrix*}}_r
\le M_{012}(\norm{\psib_{\ep}^a}_{r-1}^2 + \norm{\psib_{\ep}^a}_{r-1}+1).
We induct on $r$, bound $|\ep K_{\ep}^a|$ by (<ref>), and use the Sobolev embedding theorem to produce (<ref>).
For the uniform Lipschitz bound (<ref>), we first apply the uniform Lipschitz condition (<ref>) guaranteed by Lemma <ref> to the fixed points $(\psib_{\ep}^a,t_{\ep}^a)$ and compute
|K_^a - K_^à| + _^a - _^à_2 ≤|t_^a - t_^à| + _^a - _^à_2 ≤L_1|a-à|
for some $L_1 > 0$. For $r > 2$, the estimate (<ref>) gives
\begin{align*}
\norm{\psib_{\ep}^a-\psib_{\ep}^{\grave{a}}}_{r} &\le \norm{\bF_{\ep}(\psib_{\ep}^a,t_{\ep}^a)-\bF_{\ep}(\psib_{\ep}^{\grave{a}},t_{\ep}^{\grave{a}})}_{r} \\
&\le L_{012,r}(\norm{\psib_{\ep}^a}_r^2+\norm{\grave{\psib}_{\ep}^a}_r^2+\norm{\psib_{\ep}^a}_r + \norm{\grave{\psib}_{\ep}^a}_r)(\norm{\psib_{\ep}^a-\grave{\psib}_{\ep}^a}_{r-1} + |t_{\ep}^a-\grave{t}_{\ep}^a|)
\end{align*}
for each $0 < \ep < \ep_0$. We bound the factor
\norm{\psib_{\ep}^a}_r^2+\norm{\grave{\psib}_{\ep}^a}_r^2+\norm{\psib_{\ep}^a}_r + \norm{\grave{\psib}_{\ep}^a}_r
by (<ref>) and estimate $|t_{\ep}^a-t_{\ep}^{\grave{a}}| \le L_1|a-\grave{a}|$ as before to find
\norm{\psib_{\ep}^a-\psib_{\ep}^{\grave{a}}}_r \le L_r(\norm{\psib_{\ep}^a-\psib_{\ep}^{\grave{a}}}_{r-1} + |a-\grave{a}|)
for some $L_r > 0$ and all $0 < \ep < \ep_0$. Taking the existing Lipschitz estimate on $|K_{\ep}^a-K_{\ep}^{\grave{a}}|$ from (<ref>), using the Sobolev embedding theorem, and inducting on $r$ produces the final Lipschitz estimate (<ref>) of Part (v).
§.§ Proof of Lemma <ref>
We set
r_0 = \min\left\{\frac{1}{6C_1},r_1\right\} \mand a_0 = \min\left\{\frac{r_0}{6C_1}, \frac{1}{6C_1}, a_1\right\}.
Then whenever $0 < \ep < \ep_0, \norm{x} \le r_0, |a| \le a_0$, we have
\norm{F_{\ep}(x,a)} \le C_1(a_0 + a_0r_0 + r_0^2) \le C_1\left(\frac{r_0}{6C_1} + \frac{r_0}{6C_1} + \frac{r_0}{6C_1}\right) = \frac{r_0}{2} < r_0.
C_1(|a| + \norm{x}+\norm{y}) \le C_1\left(\frac{1}{6C_1} + \frac{1}{6C_1}+\frac{1}{6C_1}\right) = \frac{1}{2}
whenever $|a| \le a_0, \norm{x},\norm{y} \le r_0$. So, (<ref>) gives
F_(x,a)-F_(y,a) ≤1/2x-y
for all such $a,x,y$. Thus have the uniform contraction condition.
We conclude that for each $0 < \ep < \ep_0$ and $|a| \le a_0$, $F_{\ep}(\cdot,a)$ maps $\ball(r_0)$ into itself and is a contraction (with uniform constant 1/2). By Banach's fixed point theorem, for each $0 < \ep < \ep_0$ and $|a| \le a_0$, we then have a unique $x_{\ep}^a \in \ball(r_0)$ such that $F_{\ep}(x_{\ep}^a,a) = x_{\ep}^a$.
For the Lipschitz estimate on the mappings $a \mapsto x_{\ep}^a$, compute, for $|a| \le a_0$,
\begin{align*}
\norm{x_{\ep}^a - x_{\ep}^{\grave{a}}} &= \norm{F_{\ep}(x_{\ep}^a,a) - F_{\ep}(x_{\ep}^{\grave{a}},\grave{a})} \\
\\
&\le \norm{F_{\ep}(x_{\ep}^a,a)-F_{\ep}(x_{\ep}^a,\grave{a})} + \norm{F_{\ep}(x_{\ep}^a,\grave{a})-F_{\ep}(x_{\ep}^{\grave{a}},\grave{a})} \\
\\
&\le L_1|a-\grave{a}| + \frac{1}{2}\norm{x_{\ep}^a-x_{\ep}^{\grave{a}}} \text{ by \eqref{lip bound} and \eqref{fp contr}}.
\end{align*}
\norm{x_{\ep}^a-x_{\ep}^{\grave{a}}} \le 2L_1|a-\grave{a}|
for all $|a| \le a_0$ and $0 < \ep < \ep_0$.
§ THE NANOPTERON EQUATIONS
§.§ Beale's ansatz
Following <cit.>, we let
$$\etab(X):=\left(\begin{array}{c} \eta_1(X) \\ \eta_2(X) \end{array} \right)$$
and look for a solution of (<ref>) of the form
=+a ^a_+ .
In the above there are three unknowns:
* the function $\eta_1$ (which will be an even
exponentially decaying function),
* the function $\eta_2$ (which will be an odd exponentially decaying function) and
* the amplitude of the periodic part, $a \in \R$.
Since $\sigmab=\sigma \ib$ and $\varphib_\ep^0 = \sin(K_\ep X) \jb$, we see that the principal contribution in the first slot is
connected to the acoustic branch and to the optical branch in the second slot, as described above.
One finds that $\etab$ solves the system:
η_1 + 4(1+w) ϖ^0 ( ση_1) = j_1+j_2 + j_3 + j_4+j_5
^2 (c_w^2+^2) η_2” + λ^_+η_2 = ^2 (l_1 + l_2 + l_3 + l_4 + l_5)
\begin{align*}
j_1&:=-\left( \sigma+\varpi^\ep b_1^\ep(\sigmab,\sigmab)\right) &l_1:=- \lambda_+^\ep b_2^\ep(\sigmab,\sigmab)\\
j_2&:=-\left( 2\varpi^\ep b_1^\ep(\sigmab,\etab)- 2 \varpi^0 b_1^0(\sigmab,\etab)\right) &l_2:=-2 \lambda_+^\ep b_2^\ep(\sigmab,\etab)\\
j_3&:=-2a\varpi^\ep b_1^\ep(\sigmab,\varphib^a_\ep) &l_3:=-2a \lambda_+^\ep b_2^\ep(\sigmab,\varphib^a_\ep)\\
j_4&:=-2a \varpi^\ep b_1^\ep(\etab,\varphib^a_\ep) &l_4:=-2a \lambda_+^\ep b_2^\ep(\etab,\varphib^a_\ep)\\
j_5&:=-2\varpi^\ep b_1^\ep(\etab,\etab) &l_5:=-\lambda_+^\ep b_2^\ep(\etab,\etab).
\end{align*}
We used the fact that
2 \varpi^0 b_1^0 (\sigmab,\etab) = {4(1+w)} \varpi^0 \left( \sigma \eta_1\right).
The operator
:= 1 +4(1+w) ϖ^0 ( σ·)
was studied in <cit.> and is invertible the class of even functions. This is made precise below in Theorem <ref>.
Thus we can rewrite the first equation in (<ref>) as
η_1 = ^-1( j_1 + j_2+j_3+j_4+j_5)=:N_1^(,a).
§.§ The solvability condition of $\B_\ep$
On the other hand
_:=^2 (c_w^2+^2) ∂_X^2 + λ^_+
is not so nice.
If we take the Fourier transform of the equation
_f = gwe find
_(K) f̂(K) = ĝ(K)
where $\tilde{\B}_\ep(K)=-\ep^2 (c_w^2 + \ep^2) K^2 + \tilde{\lambda}_+(\ep K).$
In (<ref>) in
Theorem <ref>, we saw that there exists a unique $K_\ep>0$ such that
\tilde{\B}_\ep(\pm K_\ep)=0.
$K_\ep = \O(1/\ep)$.
Since we have $\tilde{\B}_\ep(\pm K_\ep)=0$ we see, by virtue of (<ref>), that
_f = g ĝ(±K_) =0.
Which is to say that $\B_\ep$ in not surjective. (It is injective.)
The appropriate way to view (<ref>) is as a pair of solvability conditions for (<ref>);
it turns out that if the integral conditions are met then there is a solution $f$ of $\B_\ep f = g$. In this case we write $f = \B_\ep^{-1} g$. This is made precise below in Lemma <ref>.
Note that if $f$ is odd, so is $g = \B_\ep f$. And therefore so is $\hat{g}(K)$.
Which means that we can eliminate one of the solvability conditions in (<ref>).
In particular, if $f$ is odd then
_f = g ι_[g]:=∫_g(X) sin(K_X) dX = 0.
§.§ The modified equation for $\eta_2$ and an equation for $a$
Thus (<ref>) implies any solution of (<ref>) has
ι_[l_1 + l_2 + l_3 + l_4 + l_5] = 0.
Following <cit.> and <cit.>, we will use this condition to “select the amplitude $a$."
Toward this end, we let
\chi_\ep(X):=\lambda_+^\ep J_1^\ep \left( J_2^0 \sigmab. J_2^\ep \nub_\ep \right)\cdot \jb\quad \text{where}\quad
\nub_\ep(X):= \varphib^0_\ep(X)= \sin(K_\ep X) \jb.
We claim that
l_{31}:=l_3 + 2a \chi_\ep
is “small", though we hold off on a precise estimate for the time being. Roughly what we mean is that $l_{31}$ contains terms which are either of size comparable to $\ep$, or are quadratic in $a$.
We also claim that
\kappa_\ep := \iota_\ep[\chi_\ep]
is large in the sense that it is strictly bounded away from zero by an amount that does not depend on $\ep$. Both these claims are verified below (in (<ref>) and (<ref>)).
With this definition we can rewrite (<ref>) as
a =1 2 κ_ ι_[l_1 + l_2 + l_31 + l_4 + l_5]=: N^_3(,a).
Next we modify the second equation in (<ref>) to
_η_2 = -2^2 a χ_+ ^2 (l_1 + l_2 + l_31 + l_4 + l_5)
-1 κ_
ι_[-2^2 a χ_+ ^2 (l_1 + l_2 + l_31 + l_4 + l_5) )] χ_.
By design,
$$\iota_\ep \left[-2\ep^2 a \chi_\ep + \ep^2 \left(l_1 + l_2 + l_{31} + l_4 + l_5\right)-{1 \over \kappa_\ep}
\iota_\ep \left[-2\ep^2 a \chi_\ep + \ep^2 \left(l_1 + l_2 + l_{31} + l_4 + l_5) \right)\right] \chi_\ep
\right]=0.
Which is to say that the right hand side of (<ref>) meets the solvability condition (<ref>) and we can apply $\B_\ep^{-1}$ to it.
Also, if (<ref>) is met then the term in the second row of (<ref>) vanishes and so
the right hand side of (<ref>) agrees exactly with the right hand side of the second equation in (<ref>).
Also note that
2\ep^2 a \chi_\ep-{1 \over \kappa_\ep} \iota_\ep[2\ep^2 a \chi_\ep ]\chi_\ep=0.
So if we put
\P_\ep f := \B_\ep^{-1}\left( f - {1 \over \kappa_\ep} \iota[f] \chi_\ep \right)
then (<ref>) is equivalent to
η_2 =^2 _(l_1 + l_2 + l_31 + l_4 + l_5) =:N^_2(,a).
Stating things more abstractly, what we know is that the cokernel of $\diag(\A,\B_\ep)$ is nontrivial, due to the
solvability conditions (<ref>). The classical method for the analysis of nonlinear problems where
the cokernel (or, more typically, the kernel) of the linear part is nontrivial is
the Liapunov-Schmidt decomposition, like we used in the construction of the periodic solutions. But in our case we have the additional complication that $\diag(\A,\B_\ep)$ is injective. Which means that its Fredholm index is negative. It is this feature that results in having different pieces of our problem living in different sorts of function spaces (namely localized and periodic) as opposed to the whole argument taking place in $E^1_q \times O^1_q$.
Another less precise, but perhaps more evocative way, of saying this is to say that we want to do a regular old Liapunov-Schmidt analysis but the function we want to be the basis for the kernel of $\B_\ep$—specifically $\sin(K_\ep X)$—is not in our function space. And so we need to come up with a way to include periodic functions in the solution at the same time as the localized functions. Which leads us to Beale's ansatz (<ref>). At the end of the day, the equation for
the periodic amplitude $a$, (<ref>), can viewed as being the replacement for
“the projection of (<ref>) onto the kernel" which would appear in a more standard Liapunov-Schmidt analysis.
$\eta_1$, (<ref>) and, more relevantly, $\eta_2$, (<ref>) are the replacements
for “the projection onto the orthogonal complement of the kernel."
§.§ The final system
In short, if we can solve the system
η_1 = N_1^(η_1,η_2,a)
η_2 = N_2^(η_1,η_2,a)
a = N_3^(η_1,η_2,a)
then we will have found a solution of our problem.
Observe that (<ref>) is written such that solutions are fixed points of the map $N^\ep:=(N^\ep_1,N^\ep_2,N^\ep_3)$.
We would achieve our goal if we could show that $N^\ep$ is a contraction on a suitable function space. It turns out that the right hand side has some problems
in that regard, due principally to the terms $j_4$ and $l_4$. These have a Lipschitz constant with respect to $a$ that depends
in a bad way on $\etab$. Nevertheless, a modified contraction mapping
argument will get the job done.
But first we need many estimates.
§ EXISTENCE/UNIQUENESS/REGULARITY/MAGNITUDE
§.§ Function spaces
For $r \ge 0$ and $p \in [1,\infty]$, let
$W^{r,p}(\R)$ be the usual Sobolev space of $r$-times (weakly) differentiable functions
in $L^p(\R)$. The norms on these spaces will be denoted by $\|\cdot \|_{W^{r,p}}$.
Put $H^r(\R) := W^{r,2}(\R)$, per the usual convention.
For $r \ge 0$ and $q\ge0$, let
H^r_q:=\left \{ f \in H^r(\R) : \cosh( q \cdot) f(\cdot) \in H^r(\R) \right\}.
$H^r_q$ consists of those functions in $H^r(\R)$ which, roughly speaking, behave like $e^{-q|\cdot|}$ as $|\cdot| \to \infty$.
$$E^r_q:=H^r_q \cap \left\{ \text{even functions}\right\}\mand O^r_q:=H^r_q\cap \left\{ \text{odd functions}\right\}.$$
Each of these is a Hilbert space with inner product given by
(f,g)_{r,q}:= ( \cosh(q \cdot) f, \cosh(q \cdot) g)_{H^r(\R)}
where $(\cdot,\cdot)_{H^r(\R)}$ is the usual $H^r(\R)$ inner product.
Of course we denote $\| f\|_{r,q}:=\sqrt{ (f,f)}_{r,q}$. We abuse notation slightly and, for elements $\ub$ of $H^r_q \times H^r_q$, write $\| \ub \|_{H^r_q \times H^r_q} = \| \ub \|_{r,q}$.
We will show that (<ref>) has a solution in $E^1_q \times O^1_q\times \R$ for some $q > 0$.
§.§ Key estimates
As mentioned above, the existence proof is an iterative argument modeled on the proof of Banach's contraction mapping theorem. The following proposition
contains all the necessary estimates for proving existence and uniqueness. It also contains estimates
which will be used in a bootstrap argument which will show that the solution is smooth and, more interestingly, that the amplitude of the periodic
piece “$a$" is small beyond all orders of $\ep$.
For all $w >1$ there exists $\ep_\star\in(0,1)$, $q_\star>0$ and $C_\star>1$ such that we have the following
* (Mapping estimates)
For all
\etab \in E^1_q \times O^1_q,\quad 0 < \ep \le\ep_\star,\quad {1 \over 2}q_\star \le q \le q_\star \mand -a_0 \le a \le a_0
we have $N_1^\ep(\etab,a) \in E^1_q$ and $N_2^\ep(\etab,a) \in O^1_q$
together with the estimate:
N_1^(,a)_1,q+N_2^(,a)_1,q+|N_3^(,a)|≤C_⋆(+ _1,q+|a| +_1,q^2 + a^2).
* (Lipschitz-type estimates)
For all
\etab,\grave{\etab} \in E^1_{q'} \times O^1_{q'}, \quad
0 < \ep \le\ep_\star,\quad {1 \over 2}q_\star \le q < q' \le q_\star \mand -a_0 \le a \le \ga \le a_0
we have
≤ C_⋆|q-q'|
(+ _1,q'+_1,q'
* (Bootstrap estimates)
For all $r \ge 1$ there exists $C_{\star,r}>1$ such that
for all
\etab \in E^r_q \times O^r_q,\quad 0 < \ep \le\ep_\star,\quad {1 \over 2}q_\star \le q \le q_\star \mand -a_0 \le a \le a_0
we have $N_1^\ep(\etab,a) \in E^{r+1}_q$ and $N_2^\ep(\etab,a) \in O^{r+1}_q$
together with the estimates:
\begin{multline}\label{boot1}
\|N_1^\ep(\etab,a)\|_{r+1,q}+
\|N_2^\ep(\etab,a))\|_{r+1,q}\\
\le C_{\star,r} \left( \ep + \|\etab\|_{r,q} + \ep^{1-r} |a| +\ep^{-r}a^2+\ep^{-r}|a|\|\etab\|_{r,q}+\|\etab\|_{r,q}^2 \right)
\end{multline}
|N_3^(,a) |≤C_⋆,r( ^r+1 +^r _r,q + |a| + a^2 +|a|_r,q + ^r_r,q^2).
The proof of this proposition is lengthy, byzantine and postponed to until Sections <ref> and <ref> below.
Onward to existence.
§.§ Existence
$$\mathcal{X}_q:=E^1_q \times O^1_q \times \R.$$
This is a Banach space with
norm $\| \cdot \|_{\X_q}$ defined in the obvious way.
Fix $w > 1$ and take $\ep_\star$ and $q_\star$ as in Proposition <ref>.
If we put $\nb=(\etab,a)$ and $\grave{\nb}=(\grave{\etab},\ga)$ then the estimate (<ref>) is compressed to
N^()__q ≤C_⋆(+ __q + ^2__q).
Similarly, (<ref>) implies
N^()-N^()__q ≤C_⋆|q-q'|(+ __q' + __q') - __q.
Here we have the same restrictions on $q,q',\ep$ as in the proposition, of course.
:=min(_⋆,1 2(C_⋆+2C_⋆^2) , q_⋆8(C_⋆+4 C^2_*)).
Henceforth we assume that $\ep \in (0,\bar{\ep}]$.
Suppose that
_q_⋆ ≤2C_⋆.Then (<ref>), (<ref>) and the definition of $\bar{\ep}$ imply
≤ C_⋆(+ 2C_⋆^2+ 4C_⋆^2^2)
≤C_⋆(1 + (2C_⋆+ 4C_⋆^2))
Now select $\nb^1 \in \X_{q_\star}$ with $\|\nb^1\|_{q_\star} \le 2C_\star \ep$. For $j \ge 1$, put
^j+1 = N^(^j).
A simple induction argument using (<ref>) shows that, for all $j \in \N$, we have
^j+1__q_⋆ ≤2C_⋆.
Thus we see that $\left\{\nb^{j} \right\}_{j \in \N}$ is a uniformly bounded sequence in $\X_{q_\star}$ (and therefore uniformly bounded
in all spaces $\X_q$ with $q \le q_\star$ too).
We now demonstrate that this sequence is Cauchy in $\X_{3q_\star/4}$. Fix $j \ge 2$. Then (<ref>) and (<ref>) (with $q = 3q_\star/4$ and $q' = q_\star$) imply
^j+1-^j__3q_⋆/4 = N^(^j) - N^(^j-1) __3q_⋆/4
^j__q_⋆+ ^j-1 __q_⋆) ^j-^j-1 __3q_⋆/4.
We use use (<ref>) in the first term to get
4 C_⋆) ^j-^j-1 __3q_⋆/4
Using the fact that $\ep \in (0,\bar{\ep}]$ and the definition of $\bar{\ep}$ we see that
4 C_\star q_\star^{-1}\left(\ep+
4 C_\star \ep\right) \le {1 /2}.
≤1 2 ^j-^j-1 __3q_⋆/4 for $j \ge 2$.
Also, (<ref>) and the triangle inequality give:
\| \nb^{2}-\nb^{1}\|_{\X_{3q_\star/4}} \le 4 C_\star \ep.
A classic induction argument then shows that
^j+1-^j__3q_⋆/4 ≤8 C_⋆2^-j.
for all $j \ge 1$.
Now fix $m >n\ge1$. The triangle inequality, followed by (<ref>) and the geometric series summation formula give:
\| \nb^{m}-\nb^{n}\|_{\X_{3q_\star/4}}
\le \sum_{j=n}^{m-1}\| \nb^{j+1}-\nb^{j}\|_{\X_{3q_\star/4}} \le 8 C_\star \ep\sum_{j=n}^{m-1} 2^{-j}\le 8 C_\star \ep\sum_{j=n}^{\infty} 2^{-j}= {16 C_\star \ep \over 2^n}.
Thus we can make $\| \nb^{m}-\nb^{n}\|_{\X_{3q_\star/4}}$ as small as we like by taken $m,n$ sufficiently large, which means
that the sequence is Cauchy. Which means it converges. Call the limit
\nb_\ep =(\etab_\ep,a_\ep) \in \X_{3q_\star/4}.
Because of (<ref>), we have
___3q_⋆/4 ≤2C_⋆.
Now we claim that
_= N^(_)
which would imply that $\nb_\ep$ is the solution we are looking for.
Since the convergence of $\nb^j$
is in $\X_{3q_\star/4}$, if we knew that $N^\ep$ was continuous on that space we would have our claim by passing the limit
through $N^\ep$ in (<ref>). But $N^\ep$ is not
obviously continuous. One can see this in the fact that the Lipschitz constant in (<ref>) depends on $\| \nb \|_{\X_{q'}}$ with $q'>q$.
We do know that $N^\ep(\nb_\ep) \in \X_{3q_\star/4}$ by virtue of (<ref>).
But nonetheless we have (<ref>). Since $\nb^j$ converges in $\X_{3q_\star/4}$, the scheme (<ref>) implies
N^(^j) →_too. This convergence takes place in $\X_q$ for all $q \in [0,3q_\star/4]$.
So look at
\| N^\ep(\nb_\ep) - \nb_\ep\|_{\X_{q_\star/2}}.
Note that we are estimating this in the bigger space $\X_{q_\star/2}$, not $\X_{3q_\star/4}$. The triangle inequality shows that
\| N^\ep(\nb_\ep) - \nb_\ep\|_{\X_{q_\star/2}} \le \| N^\ep(\nb_\ep) - N^\ep(\nb^j) \|_{\X_{q_\star/2}} + \| N^\ep(\nb^j) - \nb_\ep\|_{ \X_{q_\star/2}}.
The second term can be made as small as we like by taking $j$ big enough because of (<ref>).
For the first term we use (<ref>):
\| N^\ep(\nb_\ep) - N^\ep(\nb^j) \|_{\X_{q_\star/2}} \le 4 C_* q_\star^{-1} \left(\ep + \|\nb_\ep\|_{\X_{3q_\star/4}} + \|\nb^j\|_{\X_{3q_\star/4}} \right)\|\nb_\ep - \nb^j\|_{\X_{q_\star/2}}.
Using (<ref>) and (<ref>) this becomes:
\| N^\ep(\nb_\ep) - N^\ep(\nb^j) \|_{\X_{q_\star/2}} \le 4 C_* q_\star^{-1}\ep \left(1 + 4C_\star \ep \right)\|\nb_\ep - \nb^j\|_{\X_{q_\star/2}}.
Since $\nb^j \to \nb_\ep$ in $\X_{3q_\star/4}$ it also converges in $\X_{q_\star/2}$.
And thus we can make the above term as small as we want by taking $j$ sufficiently large.
Which is to say that
\| N^\ep(\nb_\ep) - \nb_\ep\|_{\X_{q_\star/2}} =0.
Thus we have (<ref>). Which is to say, there exists a solution of (<ref>).
§.§ Uniqueness
Suppose that $\grave{\nb}_\ep \in \X_{3q_\star/4}$ has the property that $\grave{\nb} = N^\ep(\grave{\nb}_\ep)$
and $\| \grave{\nb}_\ep \|_{ \X_{3q_\star/4}} \le 2 C_\star \ep$ and $\ep \in (0,\bar{\ep}]$. Clearly
\gnb_\ep -\nb_\ep = N^\ep(\gnb_\ep) - N^\ep(\nb_\ep).
Applying (<ref>) with $q = q_\star/2$ and $q' = 3q_\star/4$ gives:
\| \gnb_\ep -\nb_\ep \|_{\X_{q_\star/2}} \le 4 C_\star q_\star^{-1}\left(\ep +
\| \gnb_\ep \|_{\X_{3q_\star/4}}+\| \nb_\ep\|_{\X_{3q_\star/4}}\right) \left\| \gnb_\ep - \nb_\ep\right\|_{\X_{q_\star/2}}.
Since $\| \grave{\nb}_\ep \|_{ \X_{3q_\star/4}} \le 2 C_\star \ep$ and $\| {\nb}_\ep \|_{ \X_{3q_\star/4}} \le 2 C_\star \ep$
we have
\| \gnb_\ep -\nb_\ep \|_{\X_{q_\star/2}} \le 4 C_\star q_\star^{-1}\left(\ep +
4 C_\star \ep\right) \left\| \gnb_\ep - \nb_\ep\right\|_{\X_{q_\star/2}}.$$
As above, we saw that $\ep \in (0,\bar{\ep}]$ implies $4 C_\star q_\star^{-1}\left(\ep +
4 C_\star \ep\right) \le 1/2$. Thus we have
\| \gnb_\ep -\nb_\ep \|_{\X_{q_\star/2}} \le {1 \over 2}\| \gnb_\ep -\nb_\ep \|_{\X_{q_\star/2}} $$
which implies $\grave{\nb}_\ep=\nb_\ep.$
And so $\nb_\ep=(\etab_\ep,a_\ep)$ is the unique fixed point of $N^\ep$ in
the ball of radius $2C_\star \ep$ in $\X_{3q_\star/4}$.
§.§ Regularity of $\eta_\ep$ and the size of $a_\ep$
We claim that for
all $r \ge 1$, there exists $C_r > 0$ such that for all $\ep \in (0,\bar{\ep}]$ the fixed points $(\etab_\ep,a_\ep)$ constructed above satisfy
__r,3q_⋆/4 ≤C_r |a_| ≤C_r ^r.
We prove this by induction. The original construction of $(\eta_\ep,a_\ep)$ was done in the ball of radius $2C_\star \ep$ in the space $\X_{{3q_\star/4}}$
and so we have the $r=1$ base case:
\| \etab_\ep\|_{1,{3q_\star/4}} \le 2C_\star \ep \mand |a_\ep| \le 2 C_\star \ep.
Now suppose that (<ref>) holds for some $r\ge1$. We know that
$(\etab_\ep,a_\ep) = N^\ep(\etab_\ep,a_\ep).
Therefore, using (<ref>) we see:
= N_1^(_,a_)_r+1,3q_⋆/4 +N_2^(_,a_)_r+1,3q_⋆/4
C_⋆,r( + __r,3q_⋆/4 + ^1-r |a_|+ ^-r a_^2 + ^-r |a_|__r,3q_⋆/4 + _^2_r,3q_⋆/4).
Using the inductive hypothesis (<ref>) gives:
__r+1,3q_⋆/4 ≤C_⋆,r( + C_r + C_r + C_r^2 ^r + C_r^2 + C_r^2 ^2) ≤C_r+1 .
We are half way done.
Using (<ref>)
we have
|a_\ep|=\left \vert N_3^\ep (\etab_\ep,a_\ep) \right \vert \le C_{\star,r} \left( \ep^{r+1} + \ep^r \|\etab_\ep\|_{r,{3q_\star/4}}
+ \ep|a_\ep| + |a_\ep|^2 + |a_\ep|\|\etab_\ep\|_{r,{3q_\star/4}} + \ep^r \| \etab_\ep\|^2_{r,{3q_\star/4}} \right)
Using the inductive hypothesis (<ref>) gives:
|a_|≤C_⋆,r(^r+1 + C_r ^r+1 + C_r ^r+1 + C_r^2 ^2r + C_r^2 ^r+1 + C_r^2 ^r+2 ) ≤C_r+1 ^r+1.
Thus we have established (<ref>) with for $r+1$ and we are done.
§.§ The main result
Summing up, we have proven our main result, stated here in full technicality.
For all $w>1$ there exists $\bar{\ep}>0$ and $\bar{q}>0$ such that the following holds for all $\ep \in (0,\bar{\ep})$.
* There exists $\ds \etab_\ep \in \cap_{r \ge 0} \left( E^r_{\bar{q}} \times O^r_{\bar{q}} \right)$ and $a_\ep \in [a_0,a_0]$ such that
\thetab(X) = \thetab_\ep(X):= \sigmab(X)+\etab_\ep(X) + a_\ep \varphib^{a_\ep}_\ep(X)
solves (<ref>).
* For all $r \ge 0$ there exists $C_r>0$ such that, for all $\ep \in (0,\bar{\ep})$:
\| \etab_\ep\|_{r,\bar{q}} \le C_r \ep \mand |a_\ep| \le C_r \ep^{r}.
* $\thetab_\ep$ is unique in the sense that $\etab_\ep$ and $a_\ep$ are the only choices for
which $\thetab_\ep$ is a solution of (<ref>) and the estimates in (ii) hold.
The uniqueness result above does not rule out two interesting possibilities.
The first is that there could be a different choice for $\etab$ and $a$ where
$\etab \in E^1_{q} \times O^1_q$ with $q \in[0,\bar{q})$.
That is, $\etab \to 0$ at infinity at a rate slower than $e^{-\bar{q}|X|}$.
We consider this to be unlikely; our conjecture is that the solution is in fact unique in the class of $L^2 \times L^2$ functions.
The other possibility is there are solutions of (<ref>) which converge to $a \varphib_\ep^a (X\pm X_0)$
as $X \to \pm \infty$, for some $X_0 \in \R$. That is to say, the solution $\thetab$ converges to a phase-shifted member of the family of periodic solutions. This almost certainly will happen; the analogous result is shown to be true
for gravity-capillary waves in <cit.> and <cit.> and the singularly perturbed KdV-type equation studied in <cit.>. To prove such a result can be achieved (following <cit.>) by making an adjustment to Beale's ansatz (<ref>). Specifically, replacing $a\varphib_\ep^a(X)$ with $a \varphib_\ep^a(X + \sgn(X) X_0) \Xi(X)$ where $\Xi(X)$ is a smooth positive $C^\infty$ function which is zero at $X =0$ and exactly equal to one for $|X|$ large. Obviously this generates more than a few extra terms in (<ref>) and complications in proving estimates down the line!
Theorem <ref> implies, after undoing all the changes of variables that led from (<ref>) to (<ref>):
For all $w > 1$ there exists $\bar{\ep}>0$ and $\bar{q}$ such that
the following holds for all $\ep \in (0,\bar{\epsilon})$. Let $c_\ep := \sqrt{c_w^2 + \ep^2}$. There is a solution of (<ref>)
of the form
r(j,t)= {3 \over 4} \ep^2(1+w) \sech^2\left({\ep \over \sqrt{\alpha_w}}\left(j\pm c_\ep t\right)\right) + v^\ep_j(\ep(j\pm c_\ep t))
+ p_j^\ep(j \pm c_\ep t)
* $v_{j+2}^\ep (X) = v^\ep_{j}(X)
p^\ep_{j+2}(X) =p^\ep_j (X)
for all $j \in \Z$ and $X \in \R$.
* For all $r \ge 0$ we have $\|v_1^\ep\|_{H^r_{\bar{q}}}+ \|v_2^\ep\|_{H^r_{\bar{q}}} \le C_r \ep^3$.
$C_r>0$ depends only on $r$ and $w$ and not on $\ep$.
* $p_1^\ep$ and $p_2^\ep$ are periodic with period $P_\ep \in I_w$ where $I_w$
is a closed bounded subset of $\R^+$. $I_w$ depends only on $w$ and not on $\ep$.
* For all $r \ge 0$ we have $\|p_1^\ep\|_{W^{r,\infty}}+ \|p_2^\ep\|_{W^{r,\infty}} \le C_r \ep^r$.
$C_r>0$ depends only on $r$ and $w$ and not on $\ep$.
It is this corollary which is paraphrased nontechnically in Theorem <ref>.
§ BASIC ESTIMATES
§.§ Estimates on $\sigmab$.
Since $\sigma(X) = \sigma_0 \sech^2(2 q_0 X)$, for all $r \ge 0$ there exists $C_r>0$ such that
σ_r,q = _r,q ≤C_r
holds for all $q \in [0,q_0]$. In fact $\sigma$ is in $H^r_q$ for all $q \in [0,2 q_0)$, but
by restricting the interval for $q$ we can ensure that the constant $C_r$ does not depend on $q$.
The constant does depend on $r$, of course.
Obviously it does not depend on $\ep$ since $\sigma$ does not.
§.§ Estimates for $a \varphib_\ep^a$.
The estimates for $\varphib_\ep^a$ in Theorem <ref> are valid for rescaled versions which are $2\pi$-periodic.
They are not scaled in this way when they appear in the expressions $j_n$ and $l_n$ and so we need to “translate" the estimates from Theorem <ref>.
The chief difficulty here—which is in fact one of the chief difficulties in the whole argument—is that the frequency of $\varphib^a$ depends on $a$. This frequency mismatch will
ultimate lead to the loss of spatial decay in the Lipschitz estimates (<ref>). Here is the result.
For all $r \ge 0$ there exists $C_r>0$ such that for all $\ep \in (0,1)$ and $a,\ga \in [-a_0,a_0]$ we have
_^a _W^r,∞ + J_2^_^a _W^r,∞ ≤C_r^-r
and, for all $X \in \R$,
|∂_X^r J_2^(_^a - _^) |≤C_r ^-r|a-|(1+|X|).
The estimate (<ref>) follows directly from the estimates in Theorem <ref>, the fact that
$\tilde{J}_2^\ep(k)$ is uniformly bounded and the fact that $K_\ep = \O(1/\ep)$. We skip the details and instead focus on (<ref>).
We make the decomposition
\varphib_\ep^a(X) - \varphib_\ep^{\grave{a}}(X) = \Delta_1 + \Delta_2 + \Delta_3
\begin{multline}\label{Deltas}
\Delta_1 := \nub(K_\ep^a X) - \nub(K_\ep^{\ga} X), \quad \Delta_2:=\psib_\ep^a(K_\ep^a X) - \psib_\ep^\ga(K_\ep^a X)\\ \mand \Delta_3:=\psib_\ep^\ga(K_\ep^a X) - \psib_\ep^\ga(K_\ep^\ga X).
\end{multline}
We start with $J_2^\ep \Delta_1$.
Since $J_2^\ep [\ub e^{i\omega X}] = [\tilde{J}_2(\ep \omega) \ub] e^{i \omega X}$ and since $\nub(X) = \sin(X) \jb $, we see
J_2^[( ωX)] = 1 2i [ J̃_2(ω) ] e^i ωX -1 2i [ J̃_2(-ω)] e^-i ωX.
\begin{multline*}
J_2^\ep \Delta_1 = {1 \over 2i} \left[ \tilde{J}_2(\ep K_\ep^a ) \jb \right] e^{i K_\ep^a X}- {1 \over 2i} \left[ \tilde{J}_2(\ep K_\ep^\ga ) \jb \right] e^{i K_\ep^\ga X}\\ -{1 \over 2i} \left[ \tilde{J}_2(-\ep K_\ep^a )\jb \right] e^{-i K_\ep^a X}
+{1 \over 2i} \left[ \tilde{J}_2(-\ep K_\ep^\ga )\jb \right] e^{-i K_\ep^\ga X}.
\end{multline*}
We add a lot of zeros and do a lot of rearranging to get:
J_2^Δ_1 =
1 2i [( J̃_2(K_^a ) - J̃_2(K_^) ) ]e^i K_^a X
+1 2i [ J̃_2(K_^) ] ( e^i K_^a X -e^i K_^X)
+1 2i [( J̃_2(-K_^a ) - J̃_2(-K_^) ) ]e^-i K_^a X
+1 2i [ J̃_2(-K_^) ] ( e^-i K_^a X -e^-i K_^X).
We know from Corollary <ref> that $\tilde{J}_2(k)$ is analytic and, since it is periodic for $k \in \R$, globally Lipschitz on $\R$.
Thus we can estimate the term in the first line as
\left \vert {1 \over 2i} \left[\left( \tilde{J}_2(\ep K_\ep^a ) - \tilde{J}_2(\ep K_\ep^\ga ) \right) \jb \right]e^{i K_\ep^a X} \right \vert
\le C \ep|K_\ep^a - K_\ep^\ga|.
The uniform Lipschitz estimate (<ref>) for $K_\ep^a$ in Theorem <ref> then gives
\left \vert {1 \over 2i} \left[\left( \tilde{J}_2(\ep K_\ep^a ) - \tilde{J}_2(\ep K_\ep^\ga ) \right) \jb \right]e^{i K_\ep^a X} \right \vert
\le C \ep|a-\ga|.
Exactly the same reasoning leads to the following estimate on the third line:
\left \vert {1 \over 2i} \left[\left( \tilde{J}_2(-\ep K_\ep^a ) - \tilde{J}_2(-\ep K_\ep^\ga ) \right) \jb \right]e^{-i K_\ep^a X}\right \vert \le C \ep |a-\ga|.
To estimate the second line of (<ref>), first we use the fact that $\tilde{J}_2(k)$ is uniformly bounded for $k \in \R$:
\left \vert {1 \over 2i} \left[ \tilde{J}_2(\ep K_\ep^\ga ) \jb \right] \left( e^{i K_\ep^a X} -e^{i K_\ep^\ga X}\right) \right \vert \le C \left \vert e^{i K_\ep^a X} -e^{i K_\ep^\ga X}\right \vert.
Then we use the global Lipschitz estimate for the complex exponential:
$|e^{i y} - e^{i y'}| \le 2|y-y'|$ for $y,y' \in \R$.
This gives
\left \vert {1 \over 2i} \left[ \tilde{J}_2(\ep K_\ep^\ga ) \jb \right] \left( e^{i K_\ep^a X} -e^{i K_\ep^\ga X}\right) \right \vert \le C \left \vert K_\ep^a X -K_\ep^\ga X\right \vert.
Then, as above, the Lipschitz estimate (<ref>) for $K_\ep^a$ gives:
\left \vert {1 \over 2i} \left[ \tilde{J}_2(\ep K_\ep^\ga ) \jb \right] \left( e^{i K_\ep^a X} -e^{i K_\ep^\ga X}\right) \right \vert \le C \left \vert a- \ga \right \vert|X|.
In exactly the same fashion we can estimate the term in the fourth line to get:
\left \vert {1 \over 2i} \left[ \tilde{J}_2(-\ep K_\ep^\ga ) \jb \right] \left( e^{-i K_\ep^a X} -e^{-i K_\ep^\ga X}\right)\right \vert \le C \left \vert a- \ga \right \vert|X|.
Thus all together we have:
|J_2^Δ_1(X)|≤C |a - | + C |a-||X| ≤C|a-| (1 + |X|).
We also want to estimate $\partial_X^r J_2^\ep \Delta_1$.
Each term in $J_2^\ep \Delta_1$ contains $e^{i K_\ep^a X}$ or $e^{i K_\ep^\ga X}$ and thus taking $r$ derivatives with respect to $X$ will produce
additional terms like $(K_\ep^a)^r$.
We know that $K_\ep^0 = \O(1/\ep)$ and the Lipschitz estimate (<ref>) for $K_\ep^a$ implies that $K_\ep^a = \O(1/\ep)$ as well.
Thus $(K_\ep^a)^r = \O(\ep^{-r})$.
This results in the following estimate:
|∂_X^r J_2^Δ_1(X)|≤C_r^-r|a-| (1 + |X|).
Now look at $J_2^\ep \Delta_2$.
We know that $\psib_\ep^a(y)$ is $2\pi-$periodic and, moreover, smooth in $X$. Thus we can expand it in its Fourier series:
\psib_\ep^a(y) = \sum_{ j \in \Z} \hat{\psib_\ep^a}(j) e^{ijy}.
Noting that
both terms in $\Delta_2$ are periodic with the same frequency, we see that:
\Delta_2(X) = \sum_{j\in \Z} (\hat{\psib_\ep^a}(j)-\hat{\psib_\ep^\ga}(j)) e^{i jK_\ep X}.
Applying ${J}^\ep_2$ gives
J_2^\ep \Delta_2(X) = \sum_{j\in \Z} \tilde{J}_2(\ep j K_\ep^a) (\hat{\psib_\ep^a}(j)-\hat{\psib_\ep^\ga}(j)) e^{i j K^a_\ep X}.
Since $\psib_\ep^a$ is smooth, classical Fourier series estimates
\left \vert \hat{\psib_\ep^a}(j)-\hat{\psib_\ep^\ga}(j)\right \vert\le C_r(1+|j|^r)^{-1} \| \psib^a_\ep - \psib^\ga_\ep\|_{C^r_\per \times C^r_\per}
where we make take $r$ as large as we wish.
The uniform Lipschitz estimate (<ref>) for $\psib^a_\ep$ in Theorem <ref> then implies:
\left \vert \hat{\psib_\ep^a}(j)-\hat{\psib_\ep^\ga}(j)\right \vert\le C(1+j^2)^{-1} |a-\ga|.
Thus, since $\left\{(1+j^2)\right\}_{j \in \Z}$ is summable,
|J_2^Δ_2(X) |≤C|a-|.
As above if we differentiate $J_2^\ep \Delta_2$ $r$ times with respect to $X$ (each of which produces one power of $K_\ep^a$) and repeat the same steps we find:
|∂_X^rJ_2^Δ_2(X) |≤C_r^-r|a-|.
To handle $\Delta_3$ is basically a combination of how we dealt with $\Delta_1$ and $\Delta_2$.
Using the Fourier expansion for $\psib_\ep^a$ from above we see that:
J_2^\ep\Delta_3=\sum_{ j \in \Z}\left(
[\tilde{J}_2(\ep jK_\ep^a)\hat{\psib_\ep^\ga}(j)] e^{ijK_\ep^aX}
- [\tilde{J}_2(\ep jK_\ep^\ga)\hat{\psib_\ep^\ga}(j)] e^{ijK_\ep^\ga X}
\right).
Adding zero and rearranging terms gives:
J_2^Δ_3= ∑_ j ∈
[J̃_2(jK_^a)_̂^̂(j)]( e^ijK_^aX- e^ijK_^X
+ ∑_ j ∈
[(J̃_2(jK_^a)-J̃_2(jK_^))_̂^̂(j)] e^ijK_^X.
Using (as we did when estimating $\Delta_1$ above) the fact that $\tilde{J}_2$ and $e^{iy}$ are globally Lipschitz together with the estimate $|K_\ep^a -K_\ep^\ga| \le C|a-\ga|$ implied by Theorem <ref>, we have
\left\vert J_2^\ep\Delta_3(X)\right\vert \le C|a-\ga|(1+|X|)\sum_{j \in \Z}\left \vert \hat{\psib_\ep^\ga}(j)\right \vert |j|.
Next (as we did when estimating $\Delta_2$) we use the rapid decay of the Fourier coefficients of $\psib_\ep^\ga$ to conclude
that $\ds \sum_{j \in \Z}\left \vert \hat{\psib_\ep^\ga}(j)\right \vert |j| \le C$. This gives
In exactly the same fashion, we can establish
|∂_X^r J_2^Δ_3|≤C_r ^-r|a-|(1+|X|).
Thus all together we have shown (<ref>).
§.§ Product estimates
Since our nonlinearity is quadratic we need good estimates for products of functions. In particular we need estimates
that keep track of decay rates.
First we note the famous Sobolev inequality $\|f\|_{L^\infty(\R)} \le \| f\|_{H^1(\R)}$ implies
cosh(q ·) f _W^r,∞ ≤C_rf _r+1,q
for all $r \ge 0$ and $q \ge 0$. Then we have:
For all $r \ge 0$ there exists $C_r>0$ such that following estimates hold for all $q,q' \ge 0$.
If $q \ge q'$ then
f g_r,q ≤C_rf_r,q' cosh(|q-q'| ·) g_W^r,∞ .
If $q \le q'$ then
f g_r,q ≤C_r f_r,q' (|q'-q| ·) g_W^r,∞.
Lastly, if $r \ge 1$ and $0 \le q' \le q$:
f g_r,q ≤C_r f _r,q'g_r,q-q'.
\| f g\|_{r,q} = \| \cosh(q \cdot) f g \|_{H^r}.
We multiply by one inside as follows:
\| f g\|_{r,q} = \| \left( \cosh(q \cdot) \sech(q' \cdot) \cosh((q'-q)\cdot) \right)(\cosh(q'\cdot) f) (\sech((q'-q)\cdot) g) \|_{H^r}.
The estimate $\| uv\|_{H^r} \le C \| u\|_{H^r} \| v\|_{W^{r,\infty}}$ is well-known and using it here gives:
\|f g\|_{r,q} \le \| \cosh(q \cdot) \sech(q' \cdot) \cosh((q'-q)\cdot) \|_{W^{r,\infty}}\| \cosh(q'\cdot) f\|_{H^r} \| \sech((q'-q)\cdot) g\|_{W^{r,\infty}}.
Routine calculus methods shows that the condition $q \le q'$ implies
\| \cosh(q \cdot) \sech(q' \cdot) \cosh((q'-q)\cdot) \|_{W^{r,\infty}} \le C_r
for a constant $C_r$ which depends only on $r$. This gives (<ref>).
If instead we multiply by one inside like:
\| f g\|_{r,q} = \| \left( \cosh(q \cdot) \sech(q' \cdot) \sech((q'-q)\cdot) \right)(\cosh(q'\cdot) f) (\cosh((q'-q)\cdot) g) \|_{H^r}.
then the estimate
\| \cosh(q \cdot) \sech(q' \cdot) \sech((q'-q)\cdot) \|_{W^{r,\infty}} \le C_r,
which holds when $q\ge q,'$
gives (<ref>).
The remaining estimate (<ref>) follows from (<ref>) and (<ref>).
Note that we we sometimes refer to (<ref>) as a “decay borrowing" estimate, since it allows growth in $g$ at the expense of extra decay in $f$.
On the other hand, the estimates (<ref>) and (<ref>) require both $f$ and $g$ to decay.
§.§ Estimates for $\A$
The next result confirms the earlier claim that $\A$ is invertible on even functions.
There exists $q_1>0$ such that $\A$ is a bijection from $E^r_q$ to itself
for all $q \in [0,q_1]$ and $r \ge 1$. Additionally, for each $r\ge1$, there exists $C>0$ such that
^-1 f _r,q ≤C f_r,q
for all $q \in [0,q_1]$ and $f \in E^r_q$.
This is shown to be true in <cit.> for the special case when $q = 0$. The extension to $q>0$ can be achieved by using the now classical technique of operator conjugation <cit.>. We omit the details.
§.§ Estimates for $\iota_\ep$
The following estimate is a version of the famous Riemann-Lebesgue Lemma:
There exists $C>0$ such that for any $f\in H^r_q$, with $r \ge 0$, $q > 0$ and $|\omega|\ge1$ we have:
|∫_f(x) e^i ωx dx |≤C ω^r √(q) f_r,q.
Assume that $f$ is a Schwartz class function and $|\omega|\ge1$. Then integration by parts gives:
\begin{equation*}
\begin{split}
I:=\left \vert \int_\R f(x) e^{i \omega x} dx \right \vert = & \left \vert \int_\R f(x) {1 \over \omega^r} {d^r \over dx^r}[e^{i\omega x}]dx \right \vert = |\omega|^{-r} \left \vert \int_\R f^{(r)}(x) e^{i\omega x}dx \right \vert.
\end{split}
\end{equation*}
Next we use the triangle inequality to get:
I \le |\omega|^{-r} \int_\R |f^{(r)}(x)| dx.
Multliplication by one and Cauchy-Schwartz yields:
I \le |\omega|^{-r} \int_\R |f^{(r)}(x)| \cosh(qx) \sech(qx) dx \le |\omega|^{-r} \| f^{(r)}\|_{0,q} \| \sech(q \cdot)\|_{L^2}.
Of course $ \| f^{(r)}\|_{0,q} \le \| f\|_{r,q}$ and $\| \sech(q \cdot)\|_{L^2}=q^{-1/2} \| \sech( \cdot) \|_{L^2}$.
This establishes the conclusion for Schwartz class functions. A classical density argument completes the proof.
Since $K_\ep = \O(1/\ep)$, Lemma <ref> implies
|ι_[f]|≤C ^r√(q) f_r,q.
§.§ Estimates for Fourier multipliers
The following result of Beale (specifically, Lemma 3 of <cit.>) will be used repeatedly.
Suppose that $\tilde{\mu}(z)$ is a complex valued function which has the following properties:
* $\tilde{\mu}(z)$ is meromorphic on
the closed strip $\overline{\Sigma}_q = \left\{ |\Im z | \le q\right\} \subset \C$ where $q > 0$;
* there exists $m \ge 0$ and $c_*,\zeta_*>0$ such that $|z|>\zeta_*$ and $z \in \overline{\Sigma}_q$ imply
$|\tilde{\mu}(z)|\le c_*/|\Re z|^{m}$;
* the set of singularities of $\tilde{\mu}(z)$ in $\overline{\Sigma}_q$ (which we denote $P_\mu$) is finite
and, moreover, is contained in the interior $\Sigma_q$;
* all singularities of $\tilde{\mu}(z)$ in $\overline{\Sigma}_q$ are simple poles.
U^r_{\mu,q}:= \left\{ f \in H^r_q : z \in P_\mu \implies \hat{f}(z) = 0\right\}.
Then the Fourier multiplier operator $\mu$ with symbol $\tilde{\mu}$
is a bounded injective map from $U^r_{\mu,q}$ into $H^{r+m}_q$. Additionally, for all $m'\in[0,m]$,
we have the estimates:
μf _r+m',q ≤C_μ,m' f_r,q
C_μ,m':= sup_k ∈ |(1+|k|^2)^m'/2 μ̃(k±iq)
The first consequence of this is:
For all $w > 1$ we have the following.
* There exists $C>0$ and $\tau_2 >0$ such that
the following holds for all $\tau \in (0,\tau_2]$ and $r \in \R$.
The operators $\lambda_\pm$ are bounded linear maps
from $H^{r}_{\tau}\to H^r_\tau$.
$J_1$ and $J_2$ are bounded maps from
$H^{r}_{\tau}\times H^r_\tau \to H^r_\tau \times H^r_\tau$.
We have the estimates:
λ_+ f _r,τ + λ_- f _r,τ ≤C f_r,τ J_1 _r,τ + J_2 _r,τ ≤C_r,q.
* There exists $C>0$, $\ep_1\in(0,1)$ and $q_2 >0$ such that
the following holds for all $q \in [0,q_2]$, $r \in \R$ and $\ep \in [0,\ep_1]$.
The operators $\lambda^\ep_\pm$ are bounded linear maps
from $H^{r}_{q}\to H^r_q$.
Likewise $J^\ep_1$ and $J^\ep_2$ are bounded maps from
$H^{r}_{q}\times H^r_q \to H^r_q \times H^r_q$. We have the estimates:
λ^_+f _r,q + λ^_- f _r,q ≤C f_r,q J^_1 _r,q + J^_2 _r,q ≤C_r,q.
We do not provide the details of the proof. All the operators have symbols which are bounded analytic functions on strips (Lemma <ref>, Corollary <ref>)
and thus everything follows directly from Theorem <ref>.
§.§ Estimates for $\chi_\ep$
Restrict $q \in [0,q_2]$.
Using the definition of $\chi_\ep$ and the triangle inequality gives
\| \chi_\ep\|_{r,q}\le\left \|\lambda_+^\ep J_1^\ep \left( J_2^0 \sigmab. J_2^\ep \nub_\ep\right)\right\|_{r,q}.
Corollary <ref> tells us that for $\lambda_+^\ep$ and $J_1^\ep$ are operators from $H^r_q$ to itself which
are bounded independently of $\ep$. Thus
\| \chi_\ep\|_{r,q}\le C\left \|J_2^0 \sigmab. J_2^\ep \nub_\ep\right\|_{r,q}.
Using the product inequality (<ref>) gives
\| \chi_\ep\|_{r,q}\le C\left \|J_2^0 \sigmab \right\|_{r,q} \left\|J_2^\ep \nub_\ep\right\|_{W^{r,\infty}}.
Using (<ref>) and (<ref>) gives:
χ__r,q ≤C_r ^-r
for any $r \ge 0$, $q \in [0,q_2]$ and $\ep \in (0,\ep_1]$.
§.§ Estimates for $\kappa_\ep$
A rather tedious computation shows the unsurprising result that $\chi_\ep(X)$ is an odd function of $X$; we omit it.
Given this, we have
\kappa_\ep = \iota_\ep[\chi_\ep] ={2\pi i} \hat{\chi}_\ep(K_\ep).
Since $\chi_\ep(X) = \lambda_+^\ep J_1^\ep(J_2^0 \sigmab J_2^\ep. \nub_\ep)\cdot \jb$
and $\lambda_\ep^\ep$ and $J^\ep_1$ are Fourier multipliers we have
\hat{\chi}_\ep(K_\ep) = \tilde{\lambda}_+(\ep K_\ep) \left(\tilde{J}_1(\ep K_\ep) \Fo[ J_2^0 \sigmab. J_2^\ep \nub_\ep](K_\ep)\right)\cdot \jb.
By definition
\Fo[J_2^0 \sigmab. J_2^\ep \nub_\ep](K_\ep) = {1 \over 2\pi} \int_\R J_2^0 \sigmab(X). J_2^\ep\nub_\ep(X) e^{- i K_\ep X} dX.
J_2^0 \sigmab(X) = 2(1+w)\sigma(X) \left( \begin{array}{c} 1 \\1 \end{array}\right).
\Fo[J_2^0 \sigmab. J_2^\ep \nub_\ep](K_\ep) = {1+w\over \pi} \int_\R \sigma(X)J_2^\ep\nub_\ep(X) e^{- i K_\ep X} dX.
Since $\nub_\ep(X) =(2i)^{-1} [ e^{iK_\ep X}-e^{-iK_\ep X}] \jb$
and $J_2^\ep$ is a Fourier multiplier, the last becomes
\Fo[J_2^0 \sigmab. J_2^\ep \nub_\ep](K_\ep) = {1+w \over 2\pi i} \int_\R \sigma(X) \left(\tilde{J}_2(\ep K_\ep) \jb e^{i K_\ep X} - \tilde{J}_2(-\ep K_\ep) \jb e^{-i K_\ep X}\right) e^{- i K_\ep X} dX.
After rearranging terms in this we have
\begin{multline*}
\Fo[J_2^0 \sigmab. J_2^\ep \nub_\ep](K_\ep) =
{1+w \over 2\pi i} \left( \int_\R \sigma(X) dX\right) \left( \tilde{J}_2(\ep K_\ep) \jb \right) \\
- {1+w \over 2\pi i} \left( \int_\R \sigma(X) e^{-2iK_\ep X} dX\right) \left( \tilde{J}_2(-\ep K_\ep) \jb \right).
\end{multline*}
Recalling the definition of the Fourier transform, the above can be written as:
\Fo[J_2^0 \sigmab. J_2^\ep \nub_\ep](K_\ep)=
-{i(1+w)} \hat{\sigma}(0)\left( \tilde{J}_2(\ep K_\ep) \jb \right)
+i(1+w)\hat{\sigma}(2K_\ep) \left( \tilde{J}_2(-\ep K_\ep) \jb \right).
Since $\sigma(X)>0$ for all $X$, we have $\hat{\sigma}(0)>0$. Since $\sigma(X)$ is analytic and square integrable, classical Fourier analysis can be used to show that there is a constant $C>0$
for which $|
\hat{\sigma}(k) | \le C e^{-C|k|}.
Since $K_\ep = \O(1/\ep)$ this means
|\hat{\sigma}(2 K_\ep) | \le C e^{-C/\ep}.
That is to say, it is exponentially small in $\ep$. Thus we have shown that
|κ_- κ^*_|≤C e^-C/
\kappa^*_\ep:= 2 \pi(1+w) \hat{\sigma}(0) \tilde{\lambda}_+(\ep K_\ep) [\tilde{J}_1(\ep K_\ep) \tilde{J}_2(\ep K_\ep) \jb] \cdot \jb.
It has been some time, but $\tilde{J}_1 = \tilde{J}_2^{-1}$. Thus
\kappa^*_\ep= 2\pi (1+w)\hat{\sigma}(0) \tilde{\lambda}_+(\ep K_\ep).
We also saw in Lemma <ref> that
$2w < \tilde{\lambda}_+(k)$ for all $k \in \R$. Thus $\kappa^*_\ep$ is strictly bounded away from zero.
This, with $\eqref{most important part}$, demonstrates that there is constant $C>0$ and $\ep_2\in(0,1)$ for which
|κ_| ≥C for all $\ep \in (0,\ep_2]$.
§.§ Estimates and solvability conditions for $\B_\ep$
Theorem <ref>
allows us establish the features of $\B_\ep$
described in the previous section. In particular we have:
There exists $\ep_3\in(0,1)$ and $q_3 >0$ such that for all $q \in (0,q_3]$
there exists $C_q>0$ such that for all $\ep \in (0,\ep_2]$
the following hold.
* There exists $f \in H^{r+2}_q$ such that $\B_\ep f = g \in H^r_q$ if and only if $\hat{g}(\pm K_\ep) =0$.
* If $\hat{g}(\pm K_\ep) =0$ then the solution $f$ is unique. We denote the solution by $f=\B_\ep^{-1} g$.
* If $\hat{g}(\pm K_\ep) =0$ then the solution $f$ satisfies the estimates
f _r+j,q = _^-1 g _r,q ≤C_q ^j+1 g_r,q where $j=0,1$ or $2$.
* For all $g \in O^r_q$ there exists a unique $f\in O^{r+2}_q$ such that
\B_\ep f= g - {1\over \kappa_\ep} \iota_\ep[g] \chi_\ep.
We denote the solution by $f = \P_\ep g$.
* We have the estimates
_g _r+j,q ≤C_q ^j+1 g_r,q where $j=0,1$ or $2$.
* Lastly,
C_q \to \infty\quad \text{as $q \to 0^+$}.
To prove this, we need the following result, which is proved in Section <ref>.
$$\tilde{\xi}_c(z) := -c^2 z^2 + \tilde{\lambda}_+(z).$$
There exists $\delta>0$, $\ell_0>0$, $R>0$, $\tau_3>0$ and $C>0$ such that the following hold when $|c-c_w| \le \delta $ and $|\tau|\le \tau_3$.
* $|\txi_c(z)|$ is analytic on the closed strip $\overline{\Sigma}_{\tau_3}:=\left\{|\Im z| \le \tau_3\right\}$ and is even.
* $|\txi_c(z)| \ge C|z|^2$ for $|z|\ge R$.
* If $j = 0,1$ or $2$ then
inf_k ∈ (1+k^2)^-j/2|_c(k+iτ)| ≥C|τ|.
(of Lemma <ref>)
Suppose that $\ep \in (0,\ep_3)$ where $\ep_3 :=\min(1,\ep_1,\ep_2,{\delta})$ and $q \in (0,q_3]:=\min(\tau_1,\tau_2,\tau_3,q_0,q_1,q_2)$.
The map $\B_\ep$ can be viewed as a Fourier multiplier with symbol
\tilde{\B}_\ep(Z)=\txi_{\sqrt{c_w^2 + \ep^2}}(\ep Z).
From Part (vi) of Lemma <ref> and the estimate (<ref>) we know that
$\tilde{\B}_\ep(Z)$ has exactly two zeros, both real and simple, at $Z = \pm K_\ep= \pm k_{\sqrt{c_w^2 + \ep^2}}/\ep$, in $\overline{\Sigma}_{q} \subset \overline{\Sigma}_{\tau_1/\ep}$.
Thus we see that if $f \in H^{r+2}_q$, with $0< q\le q_3$, then $\hat{\B}_\ep(\pm K_\ep) \hat{f}(\pm K_\ep)=0$; this is “only if" of Part (i). (This is also equivalent to the condition that $\iota_\ep[g] =0$ discussed above, if $g$ is odd.)
And so we see that
has two simple poles at $P_\ep =\left\{ \pm K_\ep \right\}$ and no other poles in $\overline{\Sigma}_q$ when $q \in (0,q_3]$.
Similarly, Part (ii) of Lemma <ref> indicates that $1/|\tilde{\B}_\ep(Z)| \le C |\Re Z|^{-2}$ for $|Z|$ large enough.
Thus $1/\tilde{\B}_\ep(Z)$
satisfies all the conditions of the multiplier in Theorem <ref> for any decay rate $q \in (0,q_3]$, $m = 2$ and pole set $\left\{ \pm K_\ep \right\}$.
And so we have a well-defined map $\B_\ep^{-1}$
$U^r_{\ep,q}:=\left\{g \in H^r_q : \hat{g}(\pm K_\ep) = 0 \right\}$
into $H^{r+2}_q$ which inverts $\B_\ep$. Specifically $\B_\ep \B_\ep^{-1}$ is the identity on $U^r_{\ep,q}$.
Putting $f = \B_\ep^{-1}$ gives the other implication in Part (i). The uniqueness of Part (ii) follows from the injectivity of $\B_\ep$.
The estimates (<ref>) in Part (iii) follow from (<ref>), (<ref>) and the estimate in (<ref>).
Specifically, fix $q \in (0,q_3]$. The formula $\eqref{constant}$ tells that
\|\B_\ep^{-1} g\|_{r+j,q} \le C_{\ep,j} \|g\|_{r,q}
C_{\ep,j}:=\sup_{K \in \R} \left \vert (1+|K|^2)^{j/2} \tB^{-1}_\ep(K+iq)\right \vert
when $j = 0,1$ or $2$. Thus to get (<ref>) we need to show that $C_{\ep,j} \le C_q/\ep^{j+1}$.
Letting $k = \ep K$ we see:
=\sup_{K \in \R} \left \vert (1+|K|^2)^{j/2} \txi^{-1}_{\sqrt{c_w^2 + \ep^2}} (\ep K+i\ep q)\right \vert
= \sup_{k \in \R} \left \vert (1+|k/\ep|^2)^{j/2} \txi^{-1}_{\sqrt{c_w^2 + \ep^2}} (k+i\ep q)\right \vert.
Then we multiply by one on the inside and use elementary estimates to get:
≤sup_k ∈ |(1+|k/|^2)^j/2 (1+k^2)^j/2 |sup_k ∈ | (1+k^2)^j/2 ^-1_√(c_w^2 + ^2) (k+iq)|
≤^-j sup_k ∈ | (1+k^2)^j/2 ^-1_√(c_w^2 + ^2) (k+iq)|Then we use (<ref>) with $\tau = \ep q$ to get
C_{\ep,j} \le C\ep^{-j} |\ep q|^{-1} = C|q|^{-1} \ep^{-j-1}.
Thus we have, using (<ref>) and (<ref>),
\| \B_\ep^{-1} g \|_{r+j,q} \le {C_q \over \ep^{j+1}} \| g \|_{r,q}
which was our goal. Note that $C_q = C/|q|$ and so we have $C_q \to \infty$ as $q \to 0^+$, as stated in Part (vi).
To prove parts (iv) and (v) we first observe that $\B_\ep$ (and therefore $\B_\ep^{-1}$) maps
odd functions to functions. For odd functions, a short computation shows that $\iota_\ep[g] = 2 \pi i \hat{g}(K_\ep)$.
\Fo\left[g - {1 \over \kappa_\ep} \iota_\ep[g] \chi_\ep\right](\pm K_\ep) = 0.
So we can apply parts (i)-(ii) to
get Part (iv). The estimate in Part (v) is shown as follows.
The Riemann-Lebesgue estimate (<ref>) implies that $|\iota_\ep [g]| \le C_q \ep^r \| g \|_{r,q}$.
And so if we use this, the estimate in Part (iii), (<ref>) and (<ref>), we have:
\| \P_\ep g \|_{r+j,q} \le {C_q \over \ep^{j+1}}\left \| g -{1 \over \kappa_\ep}\iota_\ep[g]\chi_\ep\right\|_{r,q}
\le {C_q \over \ep^{j+1}}\left( \| g\|_{r,q} +|\iota_\ep[g]|\ep^{-r}\right) \le {C_q \over \ep^{j+1}}\| g\|_{r,q}.
§.§ Symbol truncation estimates
In this subsection we prove a series of results which give estimates
the operator norm of things like $J^\ep_1 -J^0_1$ when $\ep$ is small.
Suppose that $\tilde{\mu}$ meets the hypotheses of Theorem <ref>
$P_\mu = \left\{\cdot \right\}$ and $m = 0$. (That is to say, $\tmu$ is bounded.)
Let $\tilde{\zeta}_n(z):=\tilde{\mu}(z) - \sum_{j=0}^{n} \mu_j z^j$
where the constants $\mu_j\in\C$ are the coefficients in the Maclaurin series of $\tilde{\mu}$.
Then the Fourier multiplier operators $\zeta_n$ with symbols $\tilde{\zeta}_n$ are bounded
from $H^{r+n+1}_q$ to $H^{r}_q$ and satisfy the estimate
\| \zeta_n f \|_{r,q} \le C_n \| f^{(n+1)} \|_{r,q}.
The constant $C_n>0$ is independent of $r$.
Within the radius of convergence of the Maclaurin series we have:
\tilde{\zeta}_n(z) = \sum_{j=n+1}^\infty \mu_j z^j = z^{n+1} \sum_{j=0}^\infty \mu_{j+n+1} z^j.
So if we put
\tilde{\upsilon}_n(z) = \tilde{\zeta}_n(z)/z^{n+1}
then clearly the singularity at $z = 0$ is removable. This implies that on any closed disk containing the origin and with radius smaller than the radius of convergence, $\tilde{\upsilon}(z)$ is analytic and bounded. Outside this disk, but within $\overline{\Sigma}_q$, we have $\tilde{\upsilon}(z) = (\tilde{\mu}(z) - \sum_{j=0}^{n} \mu_j z^j)/z^{n+1}$. In this case, all of the functions on the right hand side are analytic.
Moreover they are all bounded since $\tilde{\mu}(z)$ is bounded and $|z|$ is smallest on the boundary of the disk. In short, $\tilde{\upsilon}(z)$ meets the hypothesis of Theorem <ref> on $\overline{\Sigma}_q$ with an empty pole set, $m = 0$. Let $\upsilon$ be the operator associated to $\tilde{\upsilon}$.
Observing that $\zeta_n f = (-i)^n \upsilon_n f^{(n+1)}$ and applying the results of Theorem <ref> finishes the proof.
This result implies the following:
There exists $C>0$
such that for $q\in [0,\tau_2]$ and $\ep \in (0,1)$ we have the following
for all $r \ge 0$,
(J_1^- J^0_1)_r,q ≤C _r+1,q
(J_2^- J_2^0)_r,q ≤C _r+1,q.
We prove the estimates for $r = 0$. That multipliers commute with derivatives will extend this case to the general. So
\|(J_n^\ep - J_n^0) \ub \|_{0,q}^2 = \int_\R \left \vert (J_n^\ep-J_n^0)\ub(X) \right \vert^2 \cosh^2(qX) dX.
If we let $\ub^{\ep}(x) = \ub(\ep x)$ then the discussion in Remark <ref> implies that
\|(J_n^\ep - J_n^0) \ub \|_{0,q}^2 = \int_\R \left \vert [(J_n-J_n^0)\ub^\ep] (X/\ep) \right \vert^2 \cosh^2(qX) dX.
We make the change of variables $X = \ep x$ in the integral to get:
\|(J_n^\ep - J_n^0) \ub \|_{0,q}^2 = \ep \int_\R \left \vert [(J_n-J_n^0)\ub^\ep] (x) \right \vert^2 \cosh^2(q\ep x) dX = \ep \| (J_n - J_n^0) \ub^\ep\|_{0,q\ep}^2.
Since $q \in [0,\tau_2$ and $\ep \in (0,1)$ we have $q\ep \le \tau_2$. We know that $\tilde{J}_2$ is analytic on $\overline{\Sigma}_{\tau_2}$ from Corollary <ref>.
Thus we can use Lemma <ref> to get
\|(J_n^\ep - J_n^0) \ub \|_{0,q}^2 \le C \ep \| \partial_x \ub^\ep\|_{0,q\ep}^2.
A routine calculation show that $\| \partial_x \ub^\ep\|_{0,q\ep}^2 = \ep \| \partial_X \ub\|_{0,q}^2$. Thus we have the $r=0$ estimates
\|(J_n^\ep - J_n^0) \ub \|_{0,q} \le C \ep \| \ub\|_{1,q}.
§.§ Estimates for $\varpi^\ep$
In this subsection we shall prove some useful estimates for $\varpi^\ep$ and, in particular, show that
it converges in the operator norm topology to $\varpi^0$ as $\ep \to 0^+$. This result is similar to one employed in <cit.> and stands
in contrast to the results from the previous subsection where the the approximation of $J^\ep_j$ by $J^0_j$
comes at the cost of a derivative. We have:
There exists $C>0$, $\ep_4 \in (0,1)$ and $q_4 >0$ such that following hold for all $\ep \in [0,\ep_4]$
and $q\in [0,q_4]$.
* $\varpi^\ep$ is a bounded map from $H^r_q$ to $H^{r+2}_q$, for any $r \ge 0$.
* For all $r \ge 0$ we have
ϖ^f_r+2,q ≤C f_r,q.
* For all $r \ge 0$ we have ϖ^- ϖ^0_r,q ≤C ^2 f_r,q.
* For all $\omega>1$ we have
ϖ^(f e^i ω·)_1,q ≤Cω^-1 f_2,q.
To prove this we need the following result, which is proved in Section <ref>.
There exists $C>0$, $\ep_4 \in (0,1)$ and $q_4 >0$ such that following hold for all $\ep \in [0,\ep_4]$
and $q\in [0,q_4]$.
* $\tilde{\varpi}^\ep(Z)$ is analytic and bounded in the strip $\overline{\Sigma}_q$.
* $|\tilde{\varpi}^\ep(Z)| \le C/(1+|Z|^2)$ for all $Z \in \overline{\Sigma}_q$.
* $|\tilde{\varpi}^\ep(Z)-\tilde{\varpi}^0(Z)| \le C\ep^2$ for all $Z \in \overline{\Sigma}_q$.
(Of Lemma <ref>)
Parts (i)-(iii) of Lemma <ref> follow from Parts (i)-(iii) of Lemma <ref>
and an application of Theorem <ref>. We spare the details.
Part (iv) is proven using an idea employed from <cit.>.
$(1-\alpha_w \partial_X^2)^{-1} \left(f e^{i \omega X}\right) = g$.
\alpha_w g'' -g = - f e^{i \omega X}.
Put $g_1:=g\ds +{1 \over \alpha_w\omega^2} f e^{i \omega X} $. Then
α_w g_1” -g_1 = α_w g” - g -1 α_w ω^2 f e^i ωX+ f e^iωX +2i ω f' e^iωX + 1 ω^2 f” e^i ωX
= -1 α_w ω^2 f e^i ωX +2i ω f' e^iωX + 1 ω^2 f” e^i ωX
or rather
g_1 = -(1-α_w ∂_X^2)^-1(-1 α_w ω^2 f e^i ωX +2i ω f' e^iωX + 1 ω^2 f” e^i ωX )
= c_w^-2ϖ^0(-1 α_w ω^2 f e^i ωX +2i ω f' e^iωX + 1 ω^2 f” e^i ωX ).
Then we use Part (i) to conclude that
\| g_1\|_{2,q} \le C\omega^{-1} \| f\|_{2,q}.
Next, naive estimates show that $\| f e^{i \omega \cdot} \|_{1,q} \le C \omega \| f\|_{1,q}$. And since
$(1-\alpha_w \partial_X^2)^{-1}(f e^{i \omega X})=g = g_1- \ds {1 \over \alpha_w \omega^2} f e^{i \omega X}$ we have
\|(1-\alpha_w \partial_X^2)^{-1}(f e^{i \omega \cdot})\|_{1,q} \le \| g_1\|_{1,q} + C \omega^{-2}\| f e^{i \omega \cdot} \|_{1,q}\le C \omega^{-1} \| f\|_{2,q}.
This establishes the estimate for $\varpi^0=-c_w^2(1-\alpha_w \partial_X^2)^{-1}$. To establish it for $\varpi^\ep$ observe that the estimates in Part (ii) give $\|(1-\alpha_w \partial_x^2)\varpi^\ep f\|_{r,q} \le C \|f\|_{r,q}$. Thus we can reduce the $\varpi^\ep$ estimate to the $\varpi^0$ case in a simple way.
§.§ Estimates for $B^\ep$
Finally we have several basic estimates for $B^\ep$. First we have some straightforward upper bounds.
For all $r \ge 0$ there exists $C_r>0$ such that for all $q,q' \in [0,q_2]$ and $\ep \in [0,\ep_1]$
we have the following estimates.
If $q \ge q'$ then
B^(,)_r,q ≤C_r_r,q' cosh(|q-q'| ·) J_2^_W^r,∞
for all $r\ge0$.
If $q \le q'$ then
B^(,)_r,q ≤C_r_r,q' (|q-q'| ·) J_2^_W^r,∞
for all $r\ge0$.
Lastly, if $0 \le q' \le q$ and $r\ge1$ then
B^(,)_r,q ≤C _r,q'_r,q-q'.
First use the bound on $J_1^\ep$ in (<ref>) from Corollary <ref> to get
\| B^\ep(\thetab,\grave{\thetab})\|_{r,q} \le \| J_2^\ep \thetab . J_2^\ep \grave{\thetab}\|_{r,q}.
Using the various product estimates in Lemma <ref> followed by the bound
on $J_2^\ep$ in (<ref>) from Corollary <ref> gives the estimates.
The next result deals with approximation of $B^\ep$ by $B^0$.
There exists $\ep_5,q_5>0$ such that
for all $r\ge0$ there exists $C_r>0$ such that for $q\in [0,q_5]$ and $\ep \in (0,\ep_5]$ we have the following inequality
B^(,) - B^0(,)_r,q ≤C_r _r+1,q' _r+1,q”.
Here $q' + q'' = q \in [0,q_4]$ and both are positive.
Let $\ep_5:=\min(1,\ep_1)$ and $q_5:=\min(q_2,\tau_2)$.
The triangle inequality gives
B^(,) - B^0(,)_r,q
= J_1^(J_2^. J_2^)-J_1^0(J_2^0 . J_2^0 ) _r,q
≤ J_1^(J_2^. J_2^)-J_1^(J_2^. J_2^0 ) _r,q
+ J_1^(J_2^. J_2^0 )-J_1^(J_2^0 . J_2^0 ) _r,q
+ J_1^(J_2^0 . J_2^0 )-J_1^0(J_2^0 . J_2^0 ) _r,q.
For the first term, we use the bound on $J_1^\ep$ in (<ref>) from Corollary <ref> to get
\left\| J_1^\ep\left(J_2^\ep \thetab . J_2^\ep \gthetab \right)-J_1^\ep\left(J_2^\ep\thetab . J_2^0 \gthetab \right) \right\|_{r,q}
\le C\left\| J_2^\ep \thetab .\left( J_2^\ep \gthetab-J_2^0 \gthetab\right) \right\|_{r,q}.
Then we use the product inequality (<ref>)
\left\| J_1^\ep\left(J_2^\ep \thetab . J_2^\ep \gthetab \right)-J_1^\ep\left(J_2^\ep\thetab . J_2^0 \gthetab \right) \right\|_{r,q}
\le C\left\| \cosh(q' \cdot) J_2^\ep \thetab\right \|_{W^{r,\infty}}\left \| J_2^\ep \gthetab-J_2^0 \gthetab \right\|_{r,q''}
where we have $q' + q'' = q$ and both are positive.
The Sobolev embedding theorem applied to the first term on the right hand side gives:
\left\| J_1^\ep\left(J_2^\ep \thetab . J_2^\ep \gthetab \right)-J_1^\ep\left(J_2^\ep\thetab . J_2^0 \gthetab \right) \right\|_{r,q}
\le C\left\| J_2^\ep \thetab\right \|_{r+1,q'}\left \| J_2^\ep \gthetab-J_2^0 \gthetab \right\|_{r,q''}.$$
Using the boundedness of $J_2^\ep$ we get
\left\| J_1^\ep\left(J_2^\ep \thetab . J_2^\ep \gthetab \right)-J_1^\ep\left(J_2^\ep\thetab . J_2^0 \gthetab \right) \right\|_{r,q}
\le C\left\| \thetab\right \|_{r+1,q'}\left \| J_2^\ep \gthetab-J_2^0 \gthetab \right\|_{r,q''}.
On the second term we use the estimate (<ref>) from Lemma <ref> to arrive at:
\left\| J_1^\ep\left(J_2^\ep \thetab . J_2^\ep \gthetab \right)-J_1^\ep\left(J_2^\ep\thetab . J_2^0 \gthetab \right) \right\|_{r,q}
\le C \ep \left\| \thetab\right \|_{r+1,q'} \| \gthetab \|_{r+1,q''}.
So the first term in (<ref>) is handled.
The rest of the terms in (<ref>) are estimated in the same way, and we leave out the details.
§ THE PROOF OF PROPOSITION <REF>.
We are now in position to estimate $N^\ep$ and prove Proposition <ref>. Put
$$q_\star := \min(q_0,q_1,q_2,q_3,q_4,q_5)\mand \ep_\star:=\min(\ep_0,\ep_1,\ep_2,\ep_3,\ep_4,\ep_5).$$
In this section we restrict
\ep \in (0,\ep_\star] \mand q \in \left[{q_\star / 2},q_\star\right].
We have the lower bound on $q$ in place so that the constant $C_q$ in the estimates for $\B^{-1}_\ep$ and $\P_\ep$ in Lemma <ref> is bounded above.
In this way, any constant $C>0$ which appears below is independent of $\ep$, $q$, $\etab$ (which is $E^1_q \times O^1_q$) and $a$ (which is in $[-a_0,a_0]$).
Note that it is a consequence of Lemma <ref> that if $\eta_1$ is even and $\eta_2$ is odd that $N_1^\ep$ and $N_2^\ep$ are even and odd, respectively.
§.§ The mapping estimates
In this subsection we prove the estimate (<ref>).
The definitions of $N^\ep_1,N^\ep_2$ and $N^\ep_3$ give us:
\|N^\ep_1(\etab,a)\|_{1,q} \le \| \A^{-1} j_1\|_{1,q} +\| \A^{-1} j_2\|_{1,q}+ \| \A^{-1} j_3\|_{1,q} +\| \A^{-1} j_4\|_{1,q}+ \| \A^{-1} j_5\|_{1,q},
\|N^\ep_2(\etab,a)\|_{1,q} \le \| \ep^2 \P_\ep l_1\|_{1,q} + \| \ep^2 \P_\ep l_2\|_{1,q} + \| \ep^2 \P_\ep l_{31}\|_{1,q} +\| \ep^2 \P_\ep l_4\|_{1,q} + \| \ep^2 \P_\ep l_5\|_{1,q}
|N^\ep_3(\etab,a)| \le | \iota_\ep l_1|+ | \iota_\ep l_2|+| \iota_\ep l_{31}|+| \iota_\ep l_4|+| \iota_\ep l_5|.
Using the uniform bound (<ref>) on $\A^{-1}$ from Lemma <ref> we have
\|N^\ep_1(\etab,a)\|_{1,q} \le C\left( \| j_1\|_{1,q} +\| j_2\|_{1,q}+ \| j_3\|_{1,q} +\| j_4\|_{1,q}+ \| j_5\|_{1,q}\right).
In Lemma <ref>, the estimate (<ref>) gives $\|\P_\ep f\|_{1,q} \le C \ep^{-1} \| f\|_{1,q}$. Thus
\|N^\ep_2(\etab,a)\|_{1,q} \le C\ep\left( \| l_1\|_{1,q} + \| l_2\|_{1,q} + \| l_{31}\|_{1,q} +\| l_4\|_{1,q} + \| l_5\|_{1,q}\right).
And the Riemann-Lebesgue estimate for $\iota_\ep$, (<ref>), following Lemma <ref> gives (for $r=1$)
|N^\ep_3(\etab,a)| \le C\ep\left( \| l_1\|_{1,q} + \| l_2\|_{1,q} + \| l_{31}\|_{1,q} +\| l_4\|_{1,q} + \| l_5\|_{1,q}\right).
Thus we will have (<ref>) if we can show that each of the ten terms
\| j_1\|_{1,q},\ \| j_2\|_{1,q},\ \| j_3\|_{1,q},\ \| j_4\|_{1,q},\ \| j_5\|_{1,q},\
\ep\|l_1\|_{1,q},\ \ep\|l_2\|_{1,q},\ \ep\|l_{31}\|_{1,q},\ \ep\|l_4\|_{1,q}\ \text{and}\ \ep\|l_5\|_{1,q}
is bounded by $C |RHS_{map}|$, where
|RHS_{map}|:= \ep + \ep \|\etab\|_{1,q} + \ep |a|+ \|\etab\|^2_{1,q}+a^2.
§.§.§ Mapping estimates for $j_1$ and $ l_1$
The choice of $\sigma$ was made so that
\sigma+\varpi^0 b_1^0(\sigmab,\sigmab) = 0.
See (<ref>).
Which means that we have
j_1 = \varpi^0 b_1^0(\sigmab,\sigmab) - \varpi^\ep b_1^\ep(\sigmab,\sigmab)
=(\varpi^0-\varpi^\ep)b_1^0(\sigmab,\sigmab) + \varpi^\ep\left( b_1^0(\sigmab,\sigmab) - b_1^\ep(\sigmab,\sigmab)\right).
Call the terms on the right hand side $I$ and $II$ respectively.
To estimate $I$, we first use the
estimate (<ref>) for $\varpi^\ep-\varpi^0$ to get
\|I\|_{1,q} \le C \ep^2 \|b_1^0(\sigmab,\sigmab)\|_{1,q}.
Then we use the estimate for $B^0$ from (<ref>) and get
\|I\|_{1,q} \le C \ep^2 \| \sigmab \|^2_{1,q/2}.
Then the uniform bounds (<ref>) for $\sigmab$ give
\|I\|_{1,q} \le C\ep^2.
For $II$ we use the smoothing property of $\varpi^\ep$ and the associated estimate
(<ref>) to get
$\|II\|_{1,q} \le C\|b_1^0(\sigmab,\sigmab) - b_1^\ep(\sigmab,\sigmab)\|_{0,q}$.
Then we use the approximation estimates for $B^\ep$ by $B^0$ in (<ref>)
to get
$\|II\|_{1,q} \le C\ep\| \sigmab \|^2_{1,q/2}$.
Then the uniform bounds (<ref>) for $\sigmab$ give
\|II\|_{1,q} \le C\ep.
Thus we have
\|j_1\|_{1,q} \le C \ep \le C|RHS_{map}|.
To estimate $l_1=\lambda_+^\ep b_2^\ep(\sigmab,\sigmab)$
is very easy using the bounds for $\lambda_+^\ep$ in (<ref>), the bounds on $B^\ep$
in (<ref>) and the bounds on $\sigmab$ in (<ref>). We get
$\| l_1\|_{1,q} \le C$. And so
\ep \|l_1\|_{1,q} \le C \ep \le C|RHS_{map}|.
§.§.§ Mapping estimates for $j_2$ and $l_2$
By adding zero we see
j_2 = -2 (\varpi^\ep-\varpi^0) b_1^\ep(\sigmab,\etab) - 2 \varpi^0 \left(b_1^\ep(\sigmab,\etab)-b_1^0(\sigmab,\etab) \right).
Call these two terms $I$ and $II$ respectively.
To estimate $I$, we first use the
estimate (<ref>) for $\varpi^\ep-\varpi^0$ to get
\|I\|_{1,q} \le C \ep^2 \|b_1^0(\sigmab,\etab)\|_{1,q}.
Then we use the estimate for $B^0$ from (<ref>) and get
\|I\|_{1,q} \le C \ep^2 \| \sigmab \|_{1,q}\|\etab\|_{1,0}.
Then the uniform bounds (<ref>) for $\sigmab$ give
\|I\|_{1,q} \le C\ep^2\|\etab\|_{1,0}.
For $II$ we use the smoothing property of $\varpi^0$ and the associated estimate
(<ref>) to get
$\|II\|_{1,q} \le C\|b_1^\ep(\sigmab,\etab) - b_1^0(\sigmab,\etab)\|_{0,q}$.
Then we use the approximation estimates for $B^\ep$ by $B^0$ in (<ref>)
to get
$\|II\|_{1,q} \le C\ep\| \sigmab \|_{1,q}\|\etab\|_{1,0}$.
Then the uniform bounds (<ref>) for $\sigmab$ give
\|II\|_{1,q} \le C\ep\|\etab\|_{1,0}.
Thus we have
j_2_1,q ≤C _1,0 ≤C|RHS_map|.
To estimate $l_2=\lambda_+^\ep b_2^\ep(\sigmab,\etab)$, as with $l_1$,
is very straightforward using the bounds for $\lambda_+^\ep$ in (<ref>), the bounds on $B^\ep$
in (<ref>) and the bounds on $\sigmab$ in (<ref>). We get
$\| l_2\|_{1,q} \le C\|\etab\|_{1,0}$. And so
l_2_1,q ≤C _1,0 ≤C|RHS_map|.
§.§.§ Mapping estimates for $j_3$ and $l_{31}$
Recalling the definition of $j_3$, we have
j_3 = -2aϖ^J_1^( J_2^0. J_2^_)·-2 a ϖ^J_1^((J_2^-J_2^0). J^_2_)·-2a ϖ^B^(,_^a-^0_)·
=: j_30 + j_31
j_{30}:=-2a\varpi^\ep J_1^\ep \left( J_2^0\sigmab .J_2^\ep \nub_\ep\right)\cdot \ib
and $j_{31}$ is the rest.
First we will estimate
B^\ep(\sigmab,\varphib_\ep^a - \varphib_\ep^{0}).% = J_1^\ep \left[ J_2^\ep \sigmab . J_2^\ep(\varphib_\ep^a - \varphib_\ep^{0}) \right].
Using the “decay borrowing estimate" (<ref>) for $B^\ep$ we have
\|B^\ep(\sigmab,\varphib_\ep^a - \varphib_\ep^{0}) \|_{r,q} \le C_r\| \sigmab\|_{q_0}\| \sech(|q_0-q| \cdot) J_2^\ep (\varphib_\ep^a - \varphib_\ep^{0})\|_{W^{r,\infty}}.
Using the definition of the $W^{r,\infty}$ norm we get:
\| \sech(|q_0-q| \cdot) J_2^\ep (\varphib_\ep^a - \varphib_\ep^{0})\|_{W^{r,\infty}} \le C_r\sup_{X \in \R} \left \vert
\sech(|q_0-q| X) \sum_{n = 0}^r (\partial_X^n J_2^\ep (\varphib_\ep^a - \varphib_\ep^{0}))\right \vert.
We estimated $\partial_X^n J_2^\ep (\varphib_\ep^a - \varphib_\ep^{0})$ above in
(<ref>) in Lemma <ref>. We use that estimate here to get
\| \sech(|q_0-q| \cdot) J_2^\ep (\varphib_\ep^a - \varphib_\ep^{0})\|_{W^{r,\infty}} \le C_r\ep^{-r}\sup_{X \in \R} \left \vert
\sech(|q_0-q| X) (1+|X|)\right \vert|a|.
Our restrictions on $q$ imply $|q_0 - q|$ is strictly bounded away from zero in a way independent of $q$ or $\ep$.
$\sup_{X \in \R} \left \vert
\sech(|q_0-q| X) (1+|X|)\right \vert \le C.
And so we see that
(|q_0-q| ·) J_2^(_^a - _^0)_W^r,∞ ≤C_r^-r|a|
which in turn gives
B^(,_^a - _^0)_r,q ≤C_r^-r |a|.
This estimate is one of the keys for estimating $j_{31}$ and, as it happens, $l_{31}$.
Estimation of $j_{31}$ goes as follows. First we use the smoothing estimate (<ref>) for $\varpi^\ep$ from
Lemma <ref>:
\begin{align*}
\| j_{31}\|_{1,q} &\le C|a|\| \varpi^\ep J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J^\ep_2\nub_\ep\right)\|_{1,q}
+C |a|\| \varpi^\ep B^\ep(\sigmab,\varphib_\ep^a-\varphib^0_\ep)\|_{1,q} \\
&\le C|a| \| J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J^\ep_2\nub_\ep\right)\|_{0,q}
+C |a|\| B^\ep(\sigmab,\varphib_\ep^a-\varphib^0_\ep)\|_{0,q}.
\end{align*}
Then we use (<ref>), with $r = 0$, on the second term to get
\| j_{31}\|_{1,q} \le C|a|\| J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J^\ep_2\nub_\ep\right)\|_{0,q} +C a^2 .
As for the first term, first we use the boundedness of $J_1^\ep$ from (<ref>) followed by the product estimate (<ref>):
\begin{align*}
\| j_{31}\|_{1,q} \le C|a|\|(J_2^\ep-J_2^0)\sigmab\|_{0,q} \|J^\ep_2\nub_\ep\|_{W^{0,\infty}} +C a^2 .
\end{align*}
Using the estimate for $\nub_\ep=\varphib_\ep^0$ in (<ref>) gives
\begin{align*}
\| j_{31}\|_{1,q} \le C|a|\|(J_2^\ep-J_2^0)\sigmab\|_{0,q} +C a^2 .
\end{align*}
Then we use the estimate (<ref>) for $J^\ep_2-J^0_2$ from Lemma <ref> and the bounds on $\sigmab$ in (<ref>):
\begin{align}\label{j31 est map}
\| j_{31}\|_{1,q} \le C\ep|a| + C a^2 \le C|RHS_{map}|.
\end{align}
Next we estimate $j_{30}$. Estimates like the ones we just used give
j_30_1,q ≤C|a|ϖ^J_1^( J_2^0. J_2^_)_1,q
≤C|a| J_1^( J_2^0. J_2^_)_0,q ≤C |a|.
This is not less than $C |RHS_{map}|$.
It turns out this estimate is not good enough for our purposes; basically, with this estimate $j_{30}$ looks like an $\O(1)$ linear perturbation in our equation and will ruin our contraction mapping argument.
But we can improve this using the estimate (<ref>) from Lemma <ref>.
To wit
\| j_{30}\|_{1,q} = 2|a|\| \varpi^\ep J_1^\ep \left( J_2^0\sigmab .J_2^\ep \nub_\ep\right)\cdot \ib\|_{1,q}.
Using the boundedness (<ref>) of $J^1_\ep$ from Corollary <ref> we have
\| j_{30}\|_{1,q} \le C|a|\| \varpi^\ep ( J_2^0\sigmab .J_2^\ep \nub_\ep) \cdot \ib\|_{1,q}.
Then we apply $J_2^\ep$ to $\nub_\ep$ (as in (<ref>)) and $J_2^0$ to $\sigmab$
to get
\| j_{30}\|_{1,q} \le C|a|\| \varpi^\ep ( \sigma e^{i K_\ep\cdot}) \|_{1,q}.
Then we use the estimate (<ref>) of Lemma <ref> with $\omega = K_\ep$ to see
\| j_{30}\|_{1,q} \le C|a| |K_\ep|^{-1} \|\sigma\|_{2,q}.
And since $K_\ep = \O(1/\ep)$ we have
\| j_{30}\|_{1,q} \le C\ep |a|. $
This is one whole power of $\ep$ better than the naive estimate (<ref>). With (<ref>)
we have
j_3_1,q ≤C |a| + C a^2 ≤C|RHS_map|.
To estimate $l_{3}$ is much the same, though recall that we do not need to estimate all of $l_3$, rather just the term $l_{31}$. As with $j_3$ above, we have:
l_3 = -2a \lambda_+^\ep J_1^\ep \left( J_2^0\sigmab .J_2^\ep \nub_\ep\right)\cdot \jb -2 a \lambda_+^\ep J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J^\ep_2\nub_\ep\right)\cdot\jb -2a \lambda_+^\ep B^\ep(\sigmab,\varphib_\ep^a-\varphib^0_\ep)\cdot\jb .
Since $\chi_\ep:= -2a \lambda_+^\ep J_1^\ep \left( J_2^0\sigmab .J_2^\ep \nub_\ep\right)\cdot \jb$ we see that
l_31 = -2 a λ_+^J_1^((J_2^-J_2^0). J_2^_)·-2a λ_+^B^(,_^a-^0_)·.
Now that we have an explicit formula for $l_{31}$, we use the
boundedness (<ref>) of $\lambda_+^\ep$:
\begin{align*}
\| l_{31}\|_{1,q} \le & C|a|\|J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J^\ep_2\nub_\ep\right)\|_{1,q}
+C |a|\| B^\ep(\sigmab,\varphib_\ep^a-\varphib^0_\ep)\|_{1,q}. &
\end{align*}
Then we use (<ref>), with $r = 1$, on the second term:
\begin{align*}
\| j_{l1}\|_{1,q} \le & C|a|\| J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J^\ep_2\nub_\ep\right)\|_{1,q} +C \ep^{-1}a^2 .
\end{align*}
For the first term, first we use the boundedness of $J_1^\ep$ from (<ref>) followed by the product estimate (<ref>):
\begin{align*}
\| l_{31}\|_{1,q} \le & C|a|\|(J_2^\ep-J_2^0)\sigmab\|_{1,q} \|J^\ep_2\nub_\ep\|_{W^{1,\infty}} +C \ep^{-1}a^2 .
\end{align*}
Using the estimate for $\nub_\ep=\varphib_\ep^0$ in (<ref>), with $r = 1$, gives
\begin{align*}
\| l_{31}\|_{1,q} \le & C\ep^{-1}|a|\|(J_2^\ep-J_2^0)\sigmab\|_{0,q} +C\ep^{-1} a^2 .
\end{align*}
Then we use the estimate (<ref>) for $J^\ep_2-J^0_2$ from Lemma <ref> and the bounds on $\sigmab$ in (<ref>)
\| l_{31}\|_{1,q} \le C|a| + C \ep^{-1} a^2.
l_31_1,q ≤C|a| + C a^2 ≤C|RHS_map|.
and we can move on.
§.§.§ Mapping estimates for $j_4$ and $l_4$
Applying the estimate (<ref>) for $B^\ep$ gives
$\|B^\ep(\etab,\varphib_\ep^a)\|_{r,q} \le C_r\| \etab\|_{r,q}\|J_2^\ep \varphib_\ep^a\|_{W^{r,\infty}}$. Then (<ref>) gives
B^(,_^a)_r,q ≤C_r^-r_r,q.
And so, using the same steps as in the previous subsubsection, we get:
j_4_1,q ≤C|a| ϖ^b_1^(,_^a)_1,q ≤C|a| b_1^(,_^a)_0,q ≤C |a|_0,q≤C|RHS_map|
l_4_1,q ≤C|a| b_2^(,_^a)_1,q ≤C |a|_1,q≤C|RHS_map|.
§.§.§ Mapping estimates for $j_5$ and $l_5$
Using (<ref>) we have
\| B^\ep(\etab,{\etab})\|_{r,q} \le C_r \|\etab\|_{r,q/2}^2.
j_5_1,q ≤C _1,q/2^2≤C|RHS_map|
l_5_1,q ≤C _1,q/2^2≤C|RHS_map|.
With these estimates, the validation of the mapping estimate (<ref>) is complete.
We move on to the Lipschitz estimates.
§.§ The Lipschitz estimates
Now we prove the estimate (<ref>). In this subsection $\grave{j}_n$ is the same as ${j}_n$ but evaluated at $\grave{\etab}$ and $\ga$ instead of at $\etab$ and $a$.
Likewise $\grave{l}_n$ is the same as ${l}_n$ but evaluated at $\grave{\etab}$ and $\ga$ instead of at $\etab$ and $a$. Also we have $q_\star/2 \le q < q' \le q_\star$.
We have by definition and the triangle inequality:
\begin{multline*}
\|N_1^\ep(\etab,a)-N_1^\ep(\grave{\etab},\ga)\|_{1,q}\\ \le \| \A^{-1} (j_2 -\grave{j}_2)\|_{1,q} + \| \A^{-1} (j_3 -\grave{j}_3)\|_{1,q} + \| \A^{-1} (j_4 -\grave{j}_4)\|_{1,q} + \| \A^{-1} (j_5 -\grave{j}_5)\|_{1,q}
\end{multline*}
\begin{multline*}
\|N_2^\ep(\etab,a)-N_2^\ep(\grave{\etab},\ga)\|_{1,q}\\ \le \| \ep^2\P_\ep (l_2 -\grave{l}_2)\|_{1,q} + \| \ep^2\P_\ep (l_{31} -\grave{l}_{31})\|_{1,q} + \| \ep^2\P_\ep (l_4 -\grave{l}_4)\|_{1,q} + \| \ep^2\P_\ep (l_5 -\grave{l}_5)\|_{1,q}
\end{multline*}
|N_3^\ep(\etab,a)-N_2^\ep(\grave{\etab},\ga)| \le \left\vert \iota_\ep [l_2 -\grave{l}_2]\right\vert + \left\vert \iota_\ep [l_{31} -\grave{l}_{31}]\right\vert + \left\vert \iota_\ep [l_4 -\grave{l}_4]\right\vert + \left\vert \iota_\ep [l_5 -\grave{l}_5]\right\vert.
Using the uniform bound (<ref>) on $\A^{-1}$ from Lemma <ref> we have
≤C( j_2 -j̀_2_1,q + j_3 -j̀_3_1,q + j_4 -j̀_4_1,q + j_5 -j̀_5_1,q).
In Lemma <ref>, the estimate (<ref>) gives $\|\P_\ep f\|_{1,q} \le C \ep^{-1} \| f\|_{1,q}$. Thus
≤C ( l_2 -l̀_2_1,q + l_31 -l̀_31_1,q + l_4 -l̀_4_1,q + l_5 -l̀_5_1,q ).
And the Riemann-Lebesgue estimate (<ref>) in Lemma <ref> gives (with $r=1$)
≤C ( l_2 -l̀_2_1,q + l_31 -l̀_31_1,q + l_4 -l̀_4_1,q + l_5 -l̀_5_1,q ).
Thus we will have (<ref>) if we can show that each of the eight terms
\begin{multline*}
\| j_2 -\grave{j}_2\|_{1,q},\
\| j_3 -\grave{j}_3\|_{1,q},\
\| j_4 -\grave{j}_4\|_{1,q},\
\| j_5 -\grave{j}_5\|_{1,q},\\
\ep\| l_2 -\grave{l}_2\|_{1,q},\
\ep\| l_3 -\grave{l}_3\|_{1,q},\
\ep\|l _4 -\grave{l}_4\|_{1,q},\ \text{and}\
\ep\| l_5 -\grave{l}_5\|_{1,q}
\end{multline*}
is bounded by $C |RHS_{lip}|$, where
|RHS_{lip}|:= {1 \over |q-q'|}
\left(\ep + \|{\etab}\|_{1,q'}+\|\grave{\etab}\|_{1,q'}
§.§.§ Lipschitz estimates for $j_2$ and $l_2$
Note that $j_2$ and $l_2$ are linear in $\etab$ and do not depend at all on $a$. Thus we can
use the estimates (<ref>) and (<ref>) to get:
j_2 -j̀_2_1,q ≤C - _1,0 ≤C|RHS_lip|
l_2 -l̀_2_1,q ≤C - _1,0 ≤C|RHS_lip|.
§.§.§ Lipschitz estimates for $j_3$ and $l_3$
Explicit computations give
\begin{multline*}
j_3 -\grave{j}_3= -2(a-\ga)\varpi^\ep J_1^\ep \left( J_2^0\sigmab. J_2^\ep \nub_\ep\right)\cdot \ib -2 (a-\ga) \varpi^\ep J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J^\ep_2\nub_\ep\right)\cdot\ib\\
-2(a-\ga) \varpi^\ep B^\ep(\sigmab,\varphib_\ep^a - \varphib_\ep^0)\cdot \ib + 2\ga \varpi^\ep B^\ep(\sigmab,\varphib_\ep^\ga - \varphib_\ep^a)\cdot \ib
\end{multline*}
\begin{multline*}
l_{31} -\grave{l}_{31}= -2 (a-\ga) \lambda_+^\ep J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J^\ep_2\nub_\ep\right)\cdot\jb\\
-2(a-\ga) \lambda_+^\ep B^\ep(\sigmab,\varphib_\ep^a - \varphib_\ep^0)\cdot \jb + 2\ga \lambda_+^\ep B^\ep(\sigmab,\varphib_\ep^\ga - \varphib_\ep^a)\cdot \jb.
\end{multline*}
We begin with an estimate of
B^\ep(\sigmab,\varphib_\ep^a - \varphib_\ep^{\ga}).% = J_1^\ep \left[ J_2^\ep \sigmab . J_2^\ep(\varphib_\ep^a - \varphib_\ep^{0}) \right].
This estimate parallels the one for $
B^\ep(\sigmab,\varphib_\ep^a - \varphib_\ep^{0})
in (<ref>).
Using the “decay borrowing estimate" (<ref>) for $B^\ep$ we have
\|B^\ep(\sigmab,\varphib_\ep^a - \varphib_\ep^\ga) \|_{r,q} \le C_r\| \sigmab\|_{q_0}\| \sech(|q_0-q| \cdot) J_2^\ep (\varphib_\ep^a - \varphib_\ep^\ga)\|_{W^{r,\infty}}.
Using the definition of the $W^{r,\infty}$ norm we get:
\| \sech(|q_0-q| \cdot) J_2^\ep (\varphib_\ep^a - \varphib_\ep^\ga)\|_{W^{r,\infty}} \le C_r\sup_{X \in \R} \left \vert
\sech(|q_0-q| X) \sum_{n = 0}^r (\partial_X^n J_2^\ep (\varphib_\ep^a - \varphib_\ep^\ga))\right \vert.
We estimated $\partial_X^n J_2^\ep (\varphib_\ep^a - \varphib_\ep^\ga)$ above in
(<ref>) in Lemma <ref>. We use that estimate here to get
(|q_0-q| ·) J_2^(_^a - _^)_W^r,∞ ≤C_r^-rsup_X ∈ |(|q_0-q| X) (1+|X|)||a-|.
We know that
$\sup_{X \in \R} \left \vert
\sech(|q_0-q| X) (1+|X|)\right \vert \le C
and so
(|q_0-q| ·) J_2^(_^a - _^)_W^r,∞ ≤C_r^-r|a-|
which in turn gives
B^(,_^a - _^)_r,q ≤C_r^-r |a-|.
The formulas for $j_3-\grave{j}_3$ and $l_{31}-\grave{l}_{31}$ differ only slightly from those
for $j_3$ and $l_{31}$. Any differences there are can be handled with (<ref>); we leave out the steps.
The results are
j_3-j̀_3_1,q ≤C |a-| + C(|a|+||)|a-| ≤C|RHS_lip|
l_31-l̀_31_1,q ≤C |a-| + C(|a|+||)|a-|≤C|RHS_lip|.
§.§.§ Lipschitz estimates for $j_4$ and $l_4$
Much of this parallels the earlier treatment of
$j_3$ and $l_3$, just swapping out $\sigmab$ for $\etab$.
There is one major wrinkle however: varying $a$ results in a loss of decay rate $q$ for $\etab$.
This is not terribly suprising given the estimate (<ref>).
Thus we must keep careful track of the decay rates. So fix $q \in [q_*/2,q_*)$.
We look at
B^\ep(\etab,\varphib_\ep^a - \varphib_\ep^{\grave{a}}). %= J_1^\ep \left[ J_2^\ep \etab . J_2^\ep(\varphib_\ep^a - \varphib_\ep^{\grave{a}}) \right].
We use the decay borrowing estimate (<ref>) to get, if $q'> q$,
\| B^\ep(\etab,\varphib^a_\ep-\varphib^\ga_\ep)\|_{r,q} \le C_r\|\etab\|_{r,q'} \|\sech(|q-q'|\cdot) J_2^\ep(\varphib^a_\ep-\varphib^\ga_\ep)\|_{W^{r,\infty}}.
As with the estimates that led to (<ref>), we can use (<ref>) to get
(|q-q'| ·) J_2^(_^a - _^)_W^r,∞ ≤C^-rsup_X ∈ |(|q-q'| X) (1+|X|)||a-|.
Now, however, we do not know that $|q-q'|$ is bounded strictly from zero.
Elementary calculus can be used to show that
\sup_{X\in\R} \left \vert (1+ |X|) \sech(|q-q'|X) \right \vert \le C|q-q'|^{-1}.
Thus we have
B^(,^a_-^_)_r,q ≤C^-r|q-q'|^-1_r,q' |a-| when $q'>q$.
Now the triangle inequallity and binliearity of $B^\ep$ give:
\begin{multline*}
\| B^\ep(\etab,a\varphib^a_\ep) - B^\ep(\grave{\etab},\ga\varphib^\ga_\ep)\|_{r,q}
\le |a| \| B^\ep(\etab,\varphib^a_\ep-\varphib^\ga_\ep)\|_{r,q} \\
+ |a-\ga|\| B^\ep(\etab,\varphib^\ga_\ep) \|_{r,q}
+ \| B^\ep(\etab-\grave{\etab},\ga\varphib^\ga_\ep)\|_{r,q}
\end{multline*}
So if we use (<ref>) and (<ref>) we get:
\begin{multline*}%\label{j4l4 est lip}
\| B^\ep(\etab,a\varphib^a_\ep) - B^\ep(\grave{\etab},\ga\varphib^\ga_\ep)\|_{r,q}
\le C\ep^{-r}|q-q'|^{-1} \|\etab\|_{r,q'}|a||a-\ga| \\+ C\ep^{-r} \left(\|\etab\|_{r,q}|a-\ga|+|\ga|\|\etab - \grave{\etab}\|_{r,q}
\right) \quad \text{when $q'>q$}.
\end{multline*}
This estimate, together with the sorts of steps we have used above, lead to:
j_4 - j̀_4_1,q
≤C|q-q'|^-1( _1,q'|a||a-|+_1,q|a-|+||- _1,q) ≤C|RHS_lip|
l_4 - l̀_4_1,q_*/2
≤C|q-q'|^-1( _1,q'|a||a-|+_1,q|a-|+||- _1,q) ≤C|RHS_lip|
so long as $q_*/2 \le q < q' \le q_*$.
§.§.§ Lipschitz estimates for $j_5$ and $l_5$
Using (<ref>) from Lemma <ref> gives
\| B^\ep(\etab,{\etab})-B^\ep(\grave{\etab},\grave{\etab})\|_{r,q} \le C
\|\grave{\etab}\|_{r,q/2})\|\etab-\grave{\etab}\|_{r,q/2}.
j_5-j̀_̀5̀ _1,q ≤C
l_5-l̀_5 _1,q ≤C (_1,q/2+
This completes the estimate that give rise to (<ref>) and we move on to the bootstrap estimates.
§.§ The bootstrap estimates
In this section we prove the estimates (<ref>) and (<ref>).
The triangle inequality gives:
\|N_1(\etab,a)\|_{r+1,q} \le \| \A^{-1} j_1\|_{r+1,q} +\| \A^{-1} j_2\|_{r+1,q}+ \| \A^{-1} j_3\|_{r+1,q} +\| \A^{-1} j_4\|_{r+1,q}+ \| \A^{-1} j_5\|_{r+1,q}
\|N_2(\etab,a)\|_{r+1,q} \le \| \ep^2 \P_\ep l_1\|_{r+1,q} + \| \ep^2 \P_\ep l_2\|_{r+1,q} + \| \ep^2 \P_\ep l_{31}\|_{r+1,q} +\| \ep^2 \P_\ep l_4\|_{r+1,q} + \| \ep^2 \P_\ep l_5\|_{r+1,q}
|N_3(\etab,a)| \le | \iota_\ep l_1|+ | \iota_\ep l_2|+| \iota_\ep l_{31}|+| \iota_\ep l_4|+| \iota_\ep l_5|
Using the bound (<ref>) for $\A^{-1}$ we have
\|N_1(\etab,a)\|_{r+1,q} \le C_r \left (\| j_1\|_{r+1,q} +\| j_2\|_{r+1,q}+ \| j_3\|_{r+1,q} +\| j_4\|_{r+1,q}+ \| j_5\|_{r+1,q}\right).
As seen in Lemma <ref>, the operator $\P_\ep$ is smoothing by up to two derivatives.
However each derivative of smoothing comes at a cost of an additional negative power of $\ep$. Choosing to smooth by just one derivative
we have:
\|N_2(\etab,a)\|_{r+1,q} \le C (\| l_1\|_{r,q}+\| l_2\|_{r,q} + \| l_{31}\|_{r,q} +\| l_4\|_{r,q} + \| l_5\|_{r,q}).
And the Riemann-Lebesgue estimate (<ref>) in Lemma <ref>
≤C ^r( l_1_r,q+ l_2_r,q + l_31_r,q + l_4_r,q + l_5_r,q).
Thus we will have (<ref>) and (<ref>) if we can show that each of the ten terms
\begin{multline*}
\| j_1\|_{r+1,q},\ \| j_2\|_{r+1,q},\ \| j_3\|_{r+1,q},\ \| j_4\|_{r+1,q},\ \| j_5\|_{r+1,q},\\
\| l_1\|_{r,q},\ \| l_2\|_{r,q},\ \| l_{31}\|_{r,q},\ \| l_4\|_{r,q},\ \text{and}\ \| l_5\|_{r,q}
\end{multline*}
is bounded by $C_r |RHS_{boot}|$, where
|RHS_{boot}|:= \ep + \|\etab\|_{r,q} + \ep^{1-r} |a| +\ep^{-r}a^2+\ep^{-r}|a|\|\etab\|_{r,q}+\|\etab\|_{r,q}^2.
§.§.§ Bootstrap estimates $j_1$ and $l_1$
Since $\sigmab$ is a smooth function, the estimates on $j_1$ and $l_1$ can be improved from above more or less for free. Specifically we have, for any $r \ge 0$,
we have
\| j_1 \|_{r+1,q_0} + \ \|l_1\|_{r,q_0} \le C_r\ep\le C_r |RHS_{boot}|.
§.§.§ Bootstrap estimates for $j_2$ and $l_2$
Recall that $j_2 = -2 \varpi^\ep b_1^\ep(\sigmab,\etab) + 2 \varpi^0 b_1^0(\sigmab,\etab)$. From the estimate (<ref>) in
Lemma <ref> we see that the operators $\varpi^\ep$ and $\varpi^0$ smooth by up to two derivatives at no cost in $\ep$. Thus we conclude, with the help of (<ref>),
\| j_1 \|_{r+1,q} \le C_r \|b_1^\ep(\sigmab,\etab)\|_{r,q} + C_r \|b_1^0(\sigmab,\etab)\|_{r,q}\le
C_r \| \etab \|_{r,q}\le C_r |RHS_{boot}|.
Since $l_2 = 2b_2^\ep(\sigmab,\etab)$, we use (<ref>) and see that
\|l_2\|_{r,q} \le C_r\| \etab \|_{r,q}\le C_r |RHS_{boot}|.
§.§.§ Bootstrap estimates for $j_3$ and $l_{31}$
Since $\varpi^\ep$ smooths by up to two derivatives we have, using (<ref>),
\| j_3 \|_{r+1,q} \le C|a|\| b_1^\ep(\sigmab,\varphib^a_\ep)\|_{r-1,q}.
Then using the product inequality for $B^\ep$ in (<ref>) followed by the estimate for $J_2^\ep \varphib_\ep^a$ in (<ref>) gives
\| j_3 \|_{r+1,q} \le C|a|\|\sigmab\|_{r-1,q} \|J_2^\ep \varphib^a_\ep\|_{W^{r-1,\infty}} \le C_r \ep^{1-r}|a|.
We saw above that
l_{31} =-2 a J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J_2^\ep\nub_\ep\right)\cdot\jb -2a B^\ep(\sigmab,\varphib_\ep^a-\varphib^0_\ep)\cdot\jb .
The estimate of this in $H^r_{q}$ is not much different than our earlier estimate in $H^1_q$. Specifically, using (<ref>) and (<ref>):
\|J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J_2^\ep\nub_\ep\right)\cdot\jb\|_{r,q} \le C \| (J_2^\ep - J_2^0)\sigmab\|_{r}\|J_2 \nub_\ep\|_{W^{r,\infty}}.
Using the approximation inequality (<ref>) on the first term and then the bounds on $\varphib_\ep^a$ in (<ref>) for the second gives
\|J_1^\ep \left((J_2^\ep-J_2^0)\sigmab. J_2^\ep\nub_\ep\right)\cdot\jb\|_{r,q} \le C\ep^{1-r}.
Then we recall (<ref>) gives
\|B^\ep(\sigmab,\varphib_\ep^a-\varphib^0_\ep)\|_{r,b} \le C_r \ep^{-r}|a|
and so all together
we find that
\| l_{31} \|_{r,q} \le C_r \ep^{1-r} |a| + C_r\ep^{-r} a^2 \le C|RHS_{boot}|.
§.§.§ Bootstrap estimates $j_4$ and $l_{4}$
Since $\varpi^\ep$ smooths by two derivatives we have, after using (<ref>),
\| j_4 \|_{r+1,q} \le C|a|\| b_1^\ep(\etab,\varphib^a_\ep)\|_{r,q}.
Then using the product inequality (<ref>) and the $\varphib$ bound (<ref>) gives.
\| j_4 \|_{r+1,q} \le C|a|\|\etab\|_{r,q} \|J_2^\ep\varphib^a_\ep\|_{W^{r,\infty}} \le C_r \ep^{-r}|a|\|\etab\|_{r,q} \le C|RHS_{boot}|.
The term $l_4$ has no smoothing operator attached, but is otherwise estimated in the same way. We have
\| l_4 \|_{r,q} \le C|a|\|\etab\|_{r,q} \|\varphib^a_\ep\|_{W^{r,\infty}} \le C_r \ep^{-r}|a|\|\etab\|_{r,q} \le C|RHS_{boot,1}|.
§.§.§ Bootstrap estimates for $j_5$ and $l_{5}$
Since $\varpi^\ep$ smooths by up to two derivatives we have, after using (<ref>) and (<ref>),
\| j_5 \|_{r+1,q} \le C\|\etab\|_{r,q/2}^2 \le C|RHS_{boot}|.
The term $l_4$ has no smoothing operator attached, but is otherwise estimated in the same way. We have
\| l_5 \|_{r,q} \le C\|\etab\|_{r,q/2}^2 \le C|RHS_{boot}|.
That completes our proof of (<ref>), (<ref>) and Proposition <ref>.
§ FUNCTION ANALYSIS
In this section we prove Lemmas <ref>, <ref>, <ref>, <ref>, <ref> and <ref>. Each of these lemmas gives quantitative estimates for some specific meromorophic function.
We use little more than foundational methods from real and complex analysis here though their implementation is sometimes complicated.
§.§ Multiplier properties in $\C$
This subsection contains the proofs of Lemmas <ref>, <ref> and <ref>.
(of Lemma <ref>)
Parts (i), (iii) and (iv) are easily inferred from properties of cosine.
For Part (ii) note that so long as $\Re((1-w)^2 + 4w \cos^2(z))>0$ we can use
the prinicipal square root to extend $\tilde{\varrho}(k)$ analytically into the complex plane. Complex trigonometry identities show
\begin{multline*}
\Re((1-w)^2 + 4w \cos^2(k+i \tau)) = 1+w^2 + 2 w \cos(2k) \cosh(2 \tau) \\
\ge 1+w^2 - 2 w \cosh(2 \tau) =: f(\tau).
\end{multline*}
Note that $f(0) = (1-w)^2$. Since $w>1$, this is strictly positive. Thus we can find $\tau_0 > 0$ such that $f(\tau) > (1-w)^2/2$ when $|\tau| \le \tau_0$.
In turn this implies that $\tilde{\varrho}(z)$ (and thus $\tlambda_\pm(z)$) is analytic when $|\Im(z)| \le \tau_0.$ Since the functions are periodic in the real
direction and the strip $\overline{\Sigma}_{\tau_0}$ is bounded in the imaginary direction, the extreme value theorem implies that the functions (and all their derivatives) are uniformly bounded
on it. Thus we have (ii).
For Part (v), we compute
$$\left \vert \tlambda_\pm'(k)\right \vert = \left \vert {4w \sin(k)\cos(k) \over \sqrt{ (1+w)^2 - 4w \sin^2(k)}}\right \vert =2c_w^2 |\sin(k)|\left \vert {\cos(k) \over \sqrt{1-{4w \over (1+w)^2}\sin^2(k) }}\right \vert.$$
It is clear that
\sup_{s \in [0,1]} \left \vert {{1-s}\over {1-{r}s }} \right \vert \le 1
when $0 \le r \le 1$. The fact that $w>1$ implies that $ 0 < {4w / (1+w)^2}< 1$. Thus
\sup_{k \in \R} \left \vert {\cos(k) \over \sqrt{1-{4w \over (1+w)^2}\sin^2(k) }} \right \vert=
\sup_{k \in \R} \left \vert {\sqrt{1 - \sin^2(k) \over 1-{4w \over (1+w)^2}\sin^2(k) }} \right \vert \le 1.
Since $|\sin(k)| \le |k|$ for all $k$, this gives
$\left \vert \tlambda_\pm'(k)\right \vert \le 2c_w^2 |k|$, the second inequality in Part (v). The first inequality is simpler and omitted.
Since $c_w^2 = 2w/(1+w)$ and $w>1$ we have $c_w^2 > 1$.
For Part (vi), by (iv) we have
$\tilde{\lambda}_+(k) \in [2w,2+2w]$ for all $k$.
We have $c^2 k^2 < 2w$ when $|k|<\sqrt{2w}/c=:k_1$ and $c^2 k^2 > 2+2w$
when $|k|> \sqrt{2+2w}/c=:k_2$. Since our functions are continuous, the intermediate value theorem
implies that there is at least one value of $k$ such that $c^2 k^2 = \tilde{\lambda}_+(k)$ in $[k_1,k_2]$. Likewise, there can be no solutions of
outside $[k_1,k_2]$.
Now put $c_-= 5/4\sqrt{2}\in(0,1)$ and assume $c > c_-$.
If $k\ge k_1$, and
because $w > 1$, we have
2c^2 k \ge 2 c^2 k_1 = 2 c \sqrt{2 w} \ge2 \sqrt{2}c_- = {5 \over 2}.
This implies that
|d dk(c^2 k^2 - λ̃_+(k) ) |= |2c^2 k - λ̃_+'(k)|≥5 2- 2=l_0>0
when $k\ge k_1$. This implies that there can be at most one solution
of $c^2 k^2 - \tilde{\lambda}(k) = 0$ for $k\ge k_1$ and also gives the estimate for $|2c^2 k_c - \tilde{\lambda}_+'(k_c)|$.
The smoothness of the map $c \mapsto k_c$
follows in a routine way from this derivative estimate and the implicit function theorem. Thus we have all of Part (vi).
(of Lemma <ref>)
Take $c_-$ as in the proof of Lemma <ref> and let $c_+=\sqrt{c_w^2 + 1}$ when $c > c_-$.
Note that Part (vi) of that lemma tells us that $\txi_c(k_c)=0$ and $|\txi'(k_c)| \ge l_0$.
Henceforth assume $c \in (c_-,c_+)$. Clearly $c_w \in (c_-,c_+)$.
Parts (i) and (ii) follow immediately from Lemma <ref>.
And so all that remains is to prove the estimate (<ref>) in Part (iii).
Estimates when $|z|$ is large:
First, note that since $\tlambda_+(z)$ is bounded in $\overline{\Sigma}_{\tau_0}$
there exists $k_{big}>0$ such that $|\Re z|\ge k_{big}$ and
$\Im(z) \le \tau_0$ implies
|\txi_c(z)|\ge {1 \over 2} c_w^2 |z|^2
for any $c\in(c_-,c_+)$.
So clearly, for $j =0,1,2$, we have
inf_k≥k_big (1+k^2)^-j/2 |_c(k+iτ)| ≥C_1>0
where $C_1$ does not depend on $c$.
Thus we only need to concern ourselves with $|\Re(z)| \le k_{big}$. Our next stop is near $k_c$.
Estimates when $z\sim k_c$:
Since $\txi_c''(z) = -2c^2 + \tlambda_+''(z)$ there exists $C_2<0$ such that
|\txi_c''(z)| \le C_2
for all $z \in \overline{\Sigma}_{\tau_0}$ and $c \in (c_-,c_+)$. This implies
| \txi_c'(z) - \txi'_c(z')| \le C_2|z-z'|
for all $z,z' \in \overline{\Sigma}_{\tau_0}$.
Thus if $|z-k_c| \le \delta_1:=l_0/2C_2$ this implies $\left \vert\txi_c'(z)-\txi'_c(k_c) \right \vert \le l_0/2$.
Note that $\delta_1$ does not depend on $c$.
The reverse triangle inequality gives
|\txi_c'(z)| = \left\vert |\txi'(k_c)|- \left \vert\txi_c'(z)-\txi'_c(k_c) \right \vert\right \vert \ge {l_0 \over 2}
provided $|z-k_c| \le \delta_1$. The FTOC then implies
|\txi_c(z)| \ge {l_0 \over 2}|z-k_c|
for all $|z-k_c| \le \delta_1$ and $c \in (c_-,c_+)$.
Thus if $z = k + i \tau$ and $|z-k_c| \le \delta_1$ then
|_c(z)| ≥l_0 2|z-k_c|≥l_0 2 |τ|
for any $c \in (c_-,c_+)$.
Estimates for $z$ everywhere else:
Now we know that $k_c$ depends smoothly on $c$. So select $\delta>0$ such that $|c-c_w| \le \delta$ implies $|k_c- k_{c_w}| \le {1 \over 100} \delta_1$.
Let $K:=[0,k_{c_w}-\delta_1/2] \cup [k_{c_w}+\delta_1/2,k_{big}]$. Let
m_*:=\inf_{|k| \in K,k\in\R} \inf_{|c-c_w| \le \delta} |\txi_c(k)|
This number is strictly positive, since the only real zeros for $\txi_c(k)$, by the definition of $\delta$, lie outside of $K$, which is compact.
In the set $\left\{ |\Re(z)| \le k_{big}, |\Im(z)|\le \tau_0\right\}$, which is compact, $\xi_c(z)$ is Lipschitz with a constant (say $C_3>0$) that, so long as $c$ lies in a compact set,
is bounded independent of $c$. Thus, if $|k| \in K$ and $|\tau|\le m_*/2C_3$ we have
|\txi_c(k+i\tau) - \txi_c(k)| \le C_3|\tau| \le m_*/2.
The reverse triangle inequality then gives
|_c(k+i τ)| ≥|_c(k)| - C_3|τ| ≥m_*/2.
Overall estimates:
So put $\tau_3:=\min(\delta_1/2,\tau_0,m_*/2C_3)$. If $z = k + i \tau$ with $|\tau| \le \tau_3$ then notice that we have either
(a) $|k| \in K$, (b) $|k-k_c|\le \delta_1$ or (c) $|k|\ge k_{big}$. Thus we can use either (<ref>), (<ref>) or (<ref>) to see that
(1+k^2)^{-j/2}|\txi_c(k+i \tau)| \ge C|\tau|
where $C>0$. This completes the proof.
(of Lemma <ref>)
We have the series expansion
\tlambda_-(z) = c_w^2z^2 - \alpha_w z^4 + \cdots.
So if we put
$$\tzeta(z):={\tilde{\lambda}_-(z) \over z^2}$$
we see the singularity at $z = 0$ is removable.
Since $\tlambda_-(z)$ is analytic and uniformly bounded in the strip
$\overline{\Sigma}_{\tau_0}$, we have the same for $\tzeta(z)$. With this, we rewrite $\tvarpi^\ep$ as:
\tvarpi^\ep(Z) = -{\ep^2 \tlambda_-(\ep Z) \over (c_w^2+\ep^2) \ep^2 Z^2 - \tlambda(\ep Z)}= -{\ep^2 \tzeta(\ep Z) \over c_w^2 + \ep^2 - \tzeta(\ep Z)}.
\tzeta(z) = c_w^2 - \alpha_w z^2 + \cdots.
And so Taylor's theorem (with the uniform bound) then implies there exists $C_1>0$ such that for all $z$ in the strip $\overline{\Sigma}_{\tau_0}$
we have:
|(z)| ≤C_1, |(z) - c_w^2| ≤C_1|z|^2, |(z) - c_w^2 + α_w z^2| ≤C_1 |z|^4.
Likewise, the uniform bound on $\tzeta'(z)$ implies
|(z) - (z')| ≤C_1|z-z'|
for all $z,z'$ in the strip $\overline{\Sigma}_{\tau_0}$.
Estimates near $Z=0$:
The reverse triangle inequality gives:
\begin{multline}\label{reverse}
|c_w^2 + \ep^2 - \tzeta(\ep Z)| = |\ep^2 + \ep^2 \alpha_w Z^2 + (c_w^2 - \ep^2 \alpha_w Z^2 - \tzeta(\ep Z)|\\
\ge \left \vert \ep^2|1+\alpha_w Z^2| - | c_w^2 - \ep^2 \alpha_w Z^2 - \tzeta(\ep Z)|\right \vert
\end{multline}
If $Z= K + i q$, then we have
|1 + \alpha_w Z^2| \ge |1 + \alpha_w \Re Z^2| = |1 + \alpha_w (K^2-q^2)|.
If we restrict $|q|$ so that
$1-\alpha_w q^2 >1/2$
then we see that
|1+α_w Z^2| ≥1 2 (1 + 2α_w K^2) ≥1 2
for all $Z$.
From (<ref>) we have
| c_w^2 - \ep^2 \alpha_w Z^2 - \tzeta(\ep Z)| \le C_1 \ep^4 |Z|^4.
So let us demand that
|Z| \le {\delta_1 \over \ep}.
We will specify $\delta_1>0$ in a moment.
C_1\ep^4|Z|^4 \le C_1 \delta_1^2 \ep^2|Z|^2 = C_1 \delta_1^2 \ep^2(K^2 + q^2).
Of course there exists $C_2$ such that
K^2 + q^2 \le C_2(1+2 \alpha_w K^2)
C_1\ep^4|Z|^4 \le C_2C_1 \delta^2_1 \ep^2(1+2\alpha_w K^2 ).
Then take $\delta_1 = \ds{1 \over 2 \sqrt{C_1 C_2}}$ so that
C_1 ^4|Z|^4 ≤1 4 ^2(1+2α_w K^2 ).
Putting (<ref>), (<ref>) and (<ref>) together gives, for all $ \ep \in (0,1)$,
|c_w^2 + ^2 - (Z)| ≥1 4 ^2(1+2α_w K^2 ) ≥1 4 ^2
so long as
|(Z) |≤q_51:= min(1 √(2 α_w),τ_0) |Z|≤δ_1 .
And so, if $Z$ meets (<ref>) then (<ref>) and the first estimate in (<ref>) give
|^(Z)| = |^2 (Z) ||c_w^2 + ^2 - (Z)| ≤4C_1 1+2 α_w K^2 ≤C 1+K^2 .
Next look at
\tilde{\rho}^\ep(Z):= \tvarpi^\ep(Z) - \tvarpi^0(Z) = -{\ep^2\tzeta(\ep Z) \over c_w^2 + \ep^2 - \tzeta(\ep Z)}+ {c_w^2 \over 1+\alpha_w Z^2}
Adding zero and the triangle inequality gives:
\left \vert\tilde{\rho}^\ep(Z) \right \vert \le \left \vert {\tzeta(\ep Z) - c_w^2 \over 1+ \alpha_w^2Z^2} \right\vert+ \left \vert \tzeta(\ep Z) \right\vert \left \vert {\ep^2 \over c_w^2 + \ep^2 - \tzeta(\ep Z)} - {1 \over 1+\alpha_w Z^2} \right \vert=:I+II.
The second estimate in (<ref>) gives
I \le {C_1 \ep^2 |Z|^2 \over |1+ \alpha_w^2 Z^2|}.
Then if we assume (<ref>) and apply (<ref>) we have:
I \le C\ep^2 {K^2+q^2 \over 1+ 2 \alpha_w K^2} \le C \ep^2.
Next, combining fractions gives:
II= { \left \vert \tzeta(\ep Z) \right\vert \left \vert \tzeta(\ep Z) - c_w^2 + \ep^2 \alpha_w Z^2 \right \vert\over \left \vert c_w^2 + \ep^2 - \tzeta(\ep Z) \right \vert \left \vert 1+\alpha_w Z^2 \right \vert}
Using the first and third inequalities in (<ref>) gives:
II \le { C\ep^4 |Z|^4\over \left \vert c_w^2 + \ep^2 - \tzeta(\ep Z) \right \vert \left \vert 1+\alpha_w Z^2 \right \vert}
Then if we assume (<ref>) and apply (<ref>) and (<ref>) we have:
II \le { C\ep^2 |Z|^4\over \left( 1+2 \alpha_w K^2 \right )^2 }
Then we see
II \le { C\ep^2 (K^2+q^2)^2\over \left( 1+2 \alpha_w K^2 \right )^2 } \le C \ep^2.
Therefore, if (<ref>) is met, we have
|^(Z) - ^0(Z)|≤C^2.
Estimates far from $Z=0$: We saw in (<ref>) that $c_w^2 k^2 - \tlambda_-(k) \ge 0$ with equality only at $k =0$.
This implies that
(k) < c_w^2for all $k \ne 0$ and $k \in \R$.
Take $\delta_1$ as above and put $\tzeta_1:=\sup_{|k|\ge \delta_1/2} \tzeta(k)$. Because we have (<ref>), we know that
c_w^2 - \tzeta_1 =: \delta_3 >0.
It should be obvious that $\delta_3$ does not depend at all on $\ep$.
Suppose that $Z = K+iq$ with $|K| \ge \delta_1/2\ep$. Then the reverse triangle inequality gives:
|c_w^2 + ^2 - (Z)|≥||c_w^2 + ^2 - (K)|- |(K) - (K + i q)||Since $\ep|K|\ge\delta_1/2$ we have
|c_w^2 + ^2 - (K)|≥δ_3.
Then we use (<ref>) to see that
\left \vert \tzeta(\ep K) - \tzeta(\ep K + i \ep q)\right \vert \le C_1 \ep |q|.
Thus if we restrict $|q| \le q_{52}:=\min(\delta_3/2C_1,\tau_0)$ we have
|(K) - (K + i q)|≤δ_3/2
for all $\ep \in (0,1)$.
Thus if we have
(Z) ≤q_52 (Z) ≥δ_1/2then (<ref>), (<ref>) and (<ref>) give:
|c_w^2 + ^2 - (Z)|≥1 2 δ_3.
This gives
\left \vert \tvarpi^\ep(Z) \right \vert \le C\ep^2 \left \vert \tzeta(\ep Z) \right \vert \le C{\left\vert \tlambda_-(\ep Z)\right \vert\over |Z|^2}.
when we have (<ref>). The uniform bound on $\tlambda_-$ converts this to
\left \vert \tvarpi^\ep(Z) \right \vert \le C\ep^2 \left \vert \tzeta(\ep Z) \right \vert \le {C\over |Z|^2}.
But since we have (<ref>), clearly
|Z|^2 \ge K^2 + q^2 \ge {1 \over 2} K^2 + {\delta_1^2 \over 4\ep^2}
This implies
|^(Z) |≤C ^2 1+ ^2 K^2 ≤C 1+ K^2
Along the same lines we can prove
|^0(Z) |= c_w^2 |1 + α_w Z^2| ≤C ^2 1+ ^2 K^2
provided we have (<ref>).
Overall estimates:
Let $q_{5}:=\min(q_{51},q_{52}).$ If $\Im(Z) \le q_5$ the observe that $Z$ satisfies either (<ref>) or (<ref>).
Thus putting (<ref>) together with (<ref>) yields
\left \vert \tvarpi^\ep(Z) \right \vert \le {C \over 1+ |Z|^2}.
This estimate holds for all $\ep \in (0,1)$.
This is Part (ii) of the lemma and it implies Part (i).
Putting (<ref>) with (<ref>), (<ref>) and the triangle inequality gives
\left \vert \tvarpi^\ep(Z)-\tvarpi^0(Z) \right \vert \le {C \ep^2 \over 1+ \ep^2 |Z|^2}.
This estimate holds for all $\ep \in (0,1)$.
This implies Part (iii) and we are done.
§.§ Multiplier properties in $\R$
This subsection contains the proofs of Lemmas <ref>, <ref> and <ref>. We begin with two more lemmas to prove Lemma <ref>.
$\lambda_-'''(k) < 0$ for $k \in (0,\pi/2)$.
We find
\tlambda_-'''(k) = -\frac{16w\sin(k)\cos(k)}{\tilde{\varrho}(k)^5}q_w(\cos^2(k)),
where $q_w$ is the quadratic
q_w(X) = 4w^2X^2 + (2w^3-4w^2+2w)X + (w^4-w^3-w+1).
The discriminant of $q_w$ is
\Delta(w) := (2w^3-4w^2+2w)-4(4w^2)(w^4-w^3-w+1) = -12w^6+24w^4-12w^2 = -12w^2(w^4-w^2+1),
and when $w > 1$,
w^4-w^2+1 > w^4-2w^2+1 = (w^2-1)^2 > 0,
hence $\Delta(w) < 0$, and so $q_w$ is either strictly positive or strictly negative. Since $q_w$ has positive leading coefficient $4w^2$, $q_w$ is strictly positive, and because $\sin(k)\cos(k) > 0$ on $(0,\pi/2)$, we conclude $\tlambda_-'''(k) < 0$ on $(0,\pi/2)$.
For all $\delta > 0$ there exists $C_{\text{quad},\delta} > 0$ such that $C_{\text{quad},\delta}k^2 \le c_w^2k^2-\tlambda_-(k)$ for all $k \in [\delta,\infty)$.
Without loss of generality, suppose $0 < \delta < \pi/2$. From the proof of Lemma <ref> the function
\tzeta(k) := \begin{cases}
\dfrac{\tlambda_-(k)}{k^2}, &k\ne 0 \\
\\
c_w^2, &k= 0 \\
\end{cases}
is bounded, analytic, and nonnegative on $\R$, and it is an easy computation to see that $\tzeta$ is Lipschitz as well. Next,
\tzeta'(k) = \frac{k^2\tlambda_-'(k)-2k\tlambda_-(k)}{k^2}, k \ne 0.
We will show $k\tlambda_-'(k)-2\tlambda_-(k) < 0$ for all $k \in (0,\pi/2)$, which implies $\tzeta'(k) < 0$ on $(0,\pi/2)$ and therefore that $\tzeta$ is decreasing there. First, Taylor's theorem gives
\tlambda_-(k) = \tlambda_-(0) + \tlambda_-'(0)k + k^2\int_0^1 (1-t)\tlambda_-''(sk) \ ds = k^2\int_0^1(1-t)\tlambda_-''(sk) \ ds.
Then differentiating under the integral, we find
\tlambda_-'(k) = 2k\int_0^1(1-s)\tlambda_-''(sk) \ ds + k^2\int_0^1s(1-s)\tlambda_-'''(sk) \ ds,
k\tlambda_-'(k)-2\tlambda_-(k) = k^3\int_0^1s(1-s)\tlambda_-'''(sk) \ ds < 0
by Lemma <ref>.
So, $\tzeta$ is decreasing on $(0,\pi/2)$, and therefore
c_w^2 = \tzeta(0) > \tzeta(\delta) > \tzeta(k)
for $k \in (\delta,\pi/2)$, from which
0 < c_w^2-(c_w^2-(δ)/10)-(k), k ∈(δ,π/2).
The inequality (<ref>) remains true at $k=\pi/2$ since $\tzeta(\pi/2) = 0$. When $k > \pi/2$, observe that
\tzeta(k) \le \frac{\tlambda_-(k)}{\left(\dfrac{\pi}{2}\right)^2} = \frac{4\tlambda_-(k)}{\pi^2} \le \frac{8}{\pi^2} = \tzeta\left(\frac{\pi}{2}\right)
by (<ref>), thus
\begin{multline*}
c_w^2-\left(\frac{c_w^2-\tzeta(\delta)}{10}\right) - \tzeta(k) \ge c_w^2-\left(\frac{c_w^2-\tzeta(\delta)}{10}\right)-\tzeta\left(\frac{\pi}{2}\right) > \frac{9c_w^2}{10}-\frac{\tzeta(\delta)}{10}-\tzeta\left(\frac{\pi}{2}\right) \\
\\
> \frac{9c_w^2}{10}- \left(1+\frac{1}{10}\right)\frac{8}{\pi^2} = \frac{9}{10}\left(\frac{2w}{w+1}\right) - \frac{11}{10}\left(\frac{8}{\pi^2}\right) > 0
\end{multline*}
since $w>1$. So, we take $C_{\text{quad},\delta} = (c_w^2-\tzeta(\delta))/10$.
(of Lemma <ref>)
We begin with some comments on our choice of $\ep_{12}$. From (<ref>) in Part (vi) of Lemma <ref>, we have
m_*(w) := √(2w/c_w^2+1) ≤K_ ≤√(2+2w)/c_w =: m^*(w), 0 < < 1.
By taking $\ep_* = \ep_*(w)$ close to 0, we will have
K_ ≥2 m_*(w) - _* > 0.
K_ + t ≥1 K_ + t ≥m_*(w) - _* for |t| ≤1, 0 < < _*.
With $\ep_4$ as in Lemma <ref>, set $\ep_{11} = \min\{\ep_*,\ep_4\}$. Then Lemma <ref> gives $C > 0$ such that
\sup_{0 < \ep < \ep_{11}} |\tvarpi^{\ep}(K)| \le \frac{C}{1+K^2}, \ K \in \R.
Then for each $k \in \Z$,
\sup_{0 < \ep < \ep_{11}} |\tvarpi^{\ep,K_{\ep}+t}(k)| = \sup_{0 < \ep < \ep_0} |\tvarpi^{\ep}((K_{\ep}+t)k)| \le \frac{C}{1+((K_{\ep}+t)k)^2}.
Using (<ref>), we have
\frac{C}{1+((K_{\ep}+t)k)^2} \le \frac{C}{1+k^2}
for all $k \in \Z$. This proves the first estimate (<ref>) for the multiplier $\tvarpi^{\ep,K_{\ep}+t}$.
We prove the Lipschitz estimate (<ref>) only when $k \ge 1$ as when $k=0$ the left side of this inequality is zero, and evenness takes care of $k\le -1$. Fix $0 < \ep < \ep_{12}, k \ge 1$, and $|t| \le 1$ and abbreviate $K := \ep({K_{\ep}}+t)k$ and $\grave{K} := \ep({K_{\ep}}+\grave{t})k$ to find
\begin{align*}
\varpi^{\ep,{K_{\ep}}+t}(k) - \varpi^{\ep,{K_{\ep}}+\grave{t}}(k) &= \frac{\ep^2\tlambda_-(K)}{(c_w^2+\ep^2)K^2-\tlambda_-(K)} - \frac{\ep^2\tlambda_-(\grave{K})}{(c_w^2+\ep^2)K^2-\tlambda_-(K)} \\
\\
&+ \frac{\ep^2\tlambda_-(\grave{K})}{(c_w^2+\ep^2)K^2-\tlambda_-(K)} - \frac{\ep^2\tlambda_-(\grave{K})}{(c_w^2+\ep^2)\grave{K}^2-\tlambda_-(\grave{K})} \\
\\
&= \frac{\ep^2(\tlambda_-(K)-\tlambda_-(\grave{K}))}{(c_w^2+\ep^2)K^2-\tlambda_-(K)} \\
\\
&+ \frac{\ep^2\tlambda_-(\grave{K})\big(c_w^2+\ep^2)\grave{K}^2-\tlambda_-(\grave{K})\big)-\ep^2\tlambda_-(\grave{K})\big((c_w^2+\ep^2)K^2-\tlambda_-(K)\big)}{\big((c_w^2+\ep^2)K^2-\tlambda_-(K)\big)\big((c_w^2+\ep^2)\grave{K}^2-\tlambda_-(\grave{K})\big)}
\end{align*}
Call the last two terms above $I$ and $II$. Set $\delta = m_*(w) - \ep_*$ and invoke Lemma <ref> to find $C_{\text{quad},\delta} > 0$ such that
C_quad,δK^2 ≤c_w^2K^2-_-(K) ≤(c_w^2+^2)K^2-_-(K)
for all $K \in [\delta,\infty)$. By (<ref>) we have $\ep(K_{\ep}+t)k \ge \delta$ for all $0 < \ep < 1, |t| \le 1, k \ge 1$. Then using the additional estimates $\Lip(\tlambda_-) \le 2$ from (<ref>) and ${K_{\ep}}+t \ge 1$, which is (<ref>), we estimate I by
|I| \le \left|\frac{\ep^2(\tlambda_-(K)-\tlambda_-(\grave{K}))}{(c_w^2+\ep^2)K^2-\tlambda_-(K)}\right| \le \frac{2\ep^2K-\grave{K}|}{C_{\text{quad},\delta}|K|^2} =\frac{2\ep^3|k||t-\grave{t}|}{C_{\text{quad},\delta}\ep^2({K_{\ep}}+t)^2k^2} \le 2\ep_{11}|t-\grave{t}|.
Next, we rewrite $II$ as
II = \frac{\ep^2(c_w^2+\ep^2)\tlambda_-(\grave{K})(\grave{K}^2-K^2) + \ep^2\tlambda_-(\grave{K})(\tlambda_-(K)-\tlambda_-(\grave{K}))}{\big((c_w^2+\ep^2)K^2-\tlambda_-(K)\big)\big((c_w^2+\ep^2)\grave{K}^2-\tlambda_-(\grave{K})\big)},
|II| \le \frac{\ep^2(c_w^2+\ep^2)|\tlambda_-(\grave{K})(\grave{K}^2-K^2)|}{C_{\text{quad},\delta}^2|K|^2|\grave{K}|^2} + \frac{\ep^2|\tlambda_-(\grave{K})(\tlambda_-(K)-\tlambda_-(\grave{K}))|}{C_{\text{quad},\delta}^2|K|^2|\grave{K}|^2}.
Labeling these two terms as $III$ and $IV$, we find
\begin{align*}
|III| &\le \frac{2\ep^4(c_w^2+\ep^2)k^2|({K_{\ep}}+t)+({K_{\ep}}+\grave{t})||t-\grave{t}|}{C_{\text{quad},\delta}^2\ep^4k^4({K_{\ep}}+t)^2({K_{\ep}}+\grave{t})^2} \\
\\
&\le \frac{2(c_w^2+\ep^2)}{C_{\text{quad},\delta}^2}\left(\frac{|{K_{\ep}}+t|}{({K_{\ep}}+t)^2({K_{\ep}}+\grave{t})^2} + \frac{|{K_{\ep}}+\grave{t}|}{({K_{\ep}}+t)^2({K_{\ep}}+t)^2}\right)|t-\grave{t}| \\
\\
&\le \frac{4(c_w^2+\ep_{11}^2)}{C_{\text{quad},\delta}^2}|t-\grave{t}|
\end{align*}
and, since $\grave{K}^2\tzeta(\grave{K}) = \tlambda_-(\grave{K})$,
|IV| \le \frac{2\ep^2|\grave{K}^2||\tzeta(\grave{K})||K-\grave{K}|}{C_{\text{quad},\delta}^2|K|^2|\grave{K}|^2} \le \frac{2\ep^5c_w^2({K_{\ep}}+\grave{t})^2k^3|t-\grave{t}|}{C_{\text{quad},\delta}^2\ep^4k^4({K_{\ep}}+t)^2({K_{\ep}}+\grave{t})^2} \le \frac{2c_w^2\ep_{11}}{C_{\text{quad},\delta}^2}|t-\grave{t}|.
Together, the estimates on $I$, $III$, and $IV$ give (<ref>) for $k \ge 1$, which satisfies our purposes.
(of Lemma <ref>)
* Since $\tilde{\Pi}_2(\pm1) = 0$ and $\txi_{\cep}(0) = 2+2w$, we show
0 < inf_0 < < _12
|k| ≥2
|t| ≤1 |ξ̃_((K_+t)k)| = :m_
for an appropriate $\ep_{12} >0$ and set
C_min = min{m_,2+2w} C_max = C_min^-1.
We begin with some seemingly unrelated calculations which result in our choice of $\ep_{12}$. Set
f(\gamma,w) := -(1+\gamma)^2\left(\frac{4w^2}{3w+1}\right)+2+2w \mand g(w) := \lim_{\gamma\to1^-} f(\gamma,w) = -\frac{16w^2}{3w+1}+2+2w.
Elementary algebra shows that for $w > 1$, $g(w) < 0$ if and only if $5w^2-4w-1 > 0$, and it is the case that this quadratic is positive on $(1,\infty)$. So, we may find some $\gamma_* = \gamma_*(w) \in (0,1)$ such that $f(\gamma,w) < 0$.
Let $\ep_{12} = \ep_{12}(\gamma,w) > 0$ be so small that the inequalities (<ref>) from the proof of Part (i) of Lemma <ref> above hold with $\ep_* = \ep_{12}$. Furthermore, require $\ep_{12}$ to satisfy
2m_*(w) - 2_* ≥(1+γ_*)m_*(w).
and suppose that $\ep_{12}$ is small enough that $|c_{\ep}-c_w| < \delta$ for $0 < \ep < \ep_{12}$, where $\delta$ is from Lemma <ref>.
Then for $k \ge 2, |t| \le 1, 0 < \ep < \ep_*$, we have
\begin{align*}
\ep(K_{\ep}+t)k &= \ep K_{\ep}(k-1) + \ep{t}k + \ep K_{\ep} \\
&\ge m_*(w)(k-1)-\ep_*k + m_*(w) \\
&= (m_*(w)-\ep_*)k \\
&\ge 2(m_*(w)-\ep_*)k \\
&\ge (1+\gamma_*)m_*(w) \text{ by \eqref{ep gamma}}.
\end{align*}
Now, observe that $\txi_c$ is increasing on $(0,\infty)$ whenever $c \ge c_w$ as from (<ref>), when $k > 0$ we have
_c'(k) = 2c^2k- _+'(k) ≥2c_w^2k- _+'(k) ≥2c_w^2|k|-|_+'(k)| ≥0.
Then the work above shows
\begin{align*}
\txi_{\cep}(\ep({K_{\ep}}+t)k) &\le \txi_{c_w}(\ep({K_{\ep}}+t)k) \\
&\le \txi_{c_w}((1+\gamma_*)m_*(w)) \\
&= -c_w^2[(1+\gamma_*)m_*(w)]^2 + \tilde{\lambda}_+((1+\gamma_*)m_*(w)) \\
&\le -c_w^2[(1+\gamma_*)m_*(w)]^2 +2+2w \\
&\le f(\gamma_*,w) < 0.
\end{align*}
Thus (<ref>) follows with $0 < |f(\gamma_*,w)| \le m_{\txi}$.
* To prove the Lipschitz estimate (<ref>), note that it already holds when $k=0$, so for $k \ge 2$ set $K = \ep({K_{\ep}}+t)k, \grave{K} = \ep({K_{\ep}}+\grave{t})k$ and compute
\begin{align*}
\left|\frac{1}{\txi_{\cep}(K)} - \frac{1}{\txi_{\cep}(\grave{K})}\right| &= \left|\frac{(c_w^2+\ep^2)\grave{K}^2-\tlambda_+(\grave{K})-\big((c_w^2+\ep^2)K^2-\tlambda_+(K)\big)}{\txi_{\cep}(K)\txi_{\cep}(\grave{K})}\right| \\
\\
&\le \frac{(c_w^2+\ep^2)|\grave{K}^2-K^2|}{\left|\txi_{\cep}(K)\txi_{\cep}(\grave{K})\right|} + \frac{|\tlambda_+(K)-\tlambda_+(\grave{K})|}{\left|\txi_{\cep}(K)\txi_{\cep}(\grave{K})\right|}
\end{align*}
Call the two terms above $I$ and $II$. By choice of $\ep_{12}$ above, Lemma <ref> furnishes $C,R > 0$ such that when $k \ge R$ and $0 < \ep < 2$, then
1/|_(k)| ≤1/Ck^2
Since $m_*(w) - \ep_{12} > 0$ by (<ref>), we may set $R_* = R/(m_*(w)-\ep_{12})$ to see that when $k > R_*$, then
R < (m_*(w)-\ep_{12})k \le \ep({K_{\ep}}+t)k.
Estimates when $2 \le k \le R_*$. With $C_{\txi \min}$ as in (<ref>), we bound
\begin{align*}
|I| \le \frac{\ep_{12}^2(c_w^2+\ep^2)|\grave{K}+K|\ep|k||t-\grave{t}|}{C_{\txi \min}^2} &= \frac{\ep_{12}^2(c_w^2+\ep^2)\ep{k}^2|2\ep{K_{\ep}}+\ep{t}+\ep\grave{t}||t-\grave{t}|}{C_{\txi \min}^2} \\
\\
&\le \left(\frac{2\ep_{12}^3R_*^2(c_w^2+\ep_{12}^2)(\beta(w)+\ep_{12})}{C_{\txi \min}^2}\right)|t-\grave{t}|
\end{align*}
and using $\Lip(\tlambda_+) \le 2$,
|II| \le \frac{2\ep_{12}^2|K-\grave{K}|}{C_{\txi \min}^2} = \frac{2\ep|k|||t-\grave{t}|}{C_{\txi \min}^2} \le \left(\frac{2\ep_{12}^3R_*}{C_{\txi \min}^2}\right)|t-\grave{t}|.
Estimates when $k > R_*$. Using (<ref>) we have
|I| \le \frac{\ep^2(c_w^2+\ep^2)|K+\grave{K}||K-\grave{K}|}{C^2|K|^2|\grave{K}|^2} = \frac{\ep^4(c_w^2+\ep^2)k^2|({K_{\ep}}+t) + ({K_{\ep}}+\grave{t})||t-\grave{t}|}{C^2\ep^4k^4({K_{\ep}}+t)^2(K_{\ep}+\grave{t})^2} \le \left(\frac{2(c_w^2+\ep_{12}^2)}{C^2R_*^2}\right)|t-\grave{t}|,
since by (<ref>)
\frac{|({K_{\ep}}+t)+({K_{\ep}}+\grave{t})|}{({K_{\ep}}+t)^2(K_{\ep}+\grave{t})^2} \le 2.
Next, adding and subtracting $K\tzeta(\grave{K})$ and using the triangle inequality in the numerator of II and (<ref>) in the resulting denominators gives
\begin{align*}
|II| &\le \frac{\ep^2|K^2\tzeta(K)-\grave{K}^2\tzeta(\grave{K})|}{C^2|K|^2|\grave{K}|^2} \\
\\
&\le \frac{\ep^2K^2|\tzeta(K)-\tzeta(\grave{K})|}{C^2|{K}|^2|\grave{K}|^2} + \frac{\ep^2|\tzeta(\grave{K})||K^2-\grave{K}^2|}{C^2|K|^2|\grave{K}|^2} \\
\\
&\le \frac{\ep^2\Lip(\tzeta)|K-\grave{K}|}{C^2|\grave{K}|^2} + \frac{\ep^2c_w^2|K+\grave{K}||K-\grave{K}|}{C^2|K|^2|\grave{K}|^2} \\
\\
&\le \frac{\ep^3|k|\Lip(\tzeta)|t-\grave{t}|}{C^2\ep^2|{K_{\ep}}+t|^2|k|^2} + \frac{c_w^2\ep^4k^2|({K_{\ep}}+t)+({K_{\ep}}+\grave{t})||t-\grave{t}|}{C^2\ep^4k^4({K_{\ep}}+t)^2(K_{\ep}+\grave{t})^2} \\
\\
&\le \left(\frac{\Lip(\tzeta)\ep_{12}+2c_w^2}{C^2}\right)|t-\grave{t}|
\end{align*}
by reasoning similar to that above.
(of Lemma <ref>)
Taylor's theorem and some straightforward algebra imply
\txi_{\cep}(\ep{K_{\ep}}+\tau)-\txi_{\cep}'(\ep{K_{\ep}})\tau = \tau^2\left(\int_0^1(1-s)\tlambda_+''(\ep{K_{\ep}}+s\tau) \ ds - (c_w^2+\ep^2)\right)
R_{\ep}(\tau) := \int_0^1(1-s)\tlambda_+''(\ep{K_{\ep}}+s\tau) \ ds - (c_w^2+\ep^2).
The estimates (<ref>) for $R_{\ep}$ follow directly from properties of $\tlambda_-'''$ so long as $\ep$ is bounded.
§ CONCLUSIONS/QUESTIONS/FUTURE DIRECTIONS
We have considered a diatomic lattice where the only spatially variable material property was the particles' masses. We also
took a very basic form for the spring force $F_s(r) = -k_s r - b_s r^2$.
We are confident that the results here can be extended to much more general situations, be it including more complicated and spatially heterogeneous springs or assuming that the number of “species" of masses and springs is greater than two, i.e. a polymer lattice.
As mentioned above, we do not yet have lower bounds for the size of the periodic part's amplitude, though the methods used in <cit.> provide a roadmap
for establishing them.
Again, we expect that those methods will show that that the periodic part is genuinely non-zero, at least for almost all $\ep$.
But if the periodic part is non-zero then necessarily the total mechanical energy of the nanopteron solution will be infinite.
This stands in stark contrast to the solitary waves for monatomic FPUT, which are not only finite energy but also constrained minimizers of an appropriate related energy functional <cit.>.
Nonetheless we contend that the nanopteron solutions we construct are essential for understanding the long time behavior of small amplitude long wave solutions for diatomic FPUT.
It is known that the monatomic solitary waves are asymptotically stable <cit.>-<cit.> and, moreover, there are stable multisoliton-like solutions <cit.> <cit.>. It is widely held that all small, long wave initial conditions for monatomic FPUT will satisfy that the soliton resolution conjecture, which is to say that they will converge to a linear superposition of well-separated solitary waves plus a dispersive tail, also called “radiation."
But if, as we expect, there are no localized traveling long wave solutions in diatomic FPUT then clearly some other asymptotic behavior takes place.
We know, by virtue of the approximation results in <cit.> and <cit.>, that long wave initial data for (<ref>) remains close to suitably scaled solutions of the KdV equation
for very long times. Specifically for times $t$ up to $\O(1/\ep^3)$ where $\ep$ is consistent with its meaning here.
And since the soliton resolution conjecture is known to be true (via integrability) for KdV that means we can expect the solution of diatomic FPUT to resolve, at least temporarily, into a sum of $\sech^2$ like solitary waves. But on time scales beyond this those approximation theorems tell us nothing.
There are any number of possibilities for what happens afterwards. One possibility, which we favor, is that there is a very slow “leak" of energy from the acoustic branch into the optical branch that will eventually erode the solution into nothing but radiation. That is to say, we conjecture the existence of metastable solutions which look for very long times like localized solitary waves but eventually converge to zero.
Another possibility is that there is a heretofore unknown finite energy coherent structure with a more complicated temporal behavior to which the solution converges—for instance something akin to the traveling breathers that exist in modified KdV
or a localized quasi-periodic solution.
Yet another scenario is that there are a discrete set of choices for $\ep$ where the ripple vanishes.
These waves would then, in a rough sense, quantize the possible behavior as $t$ goes to infinity.
These questions are likely very difficult to settle. Note that the time scales are so long that only very careful numerics performed on very large domains will shed any light.
And so, as we stated in the introduction, we feel our work here raises many interesting questions.
|
1511.00969
|
§ INTRODUCTION
Indirect CP violation in neutral kaons is parametrized by $\epsK$
\begin{equation}
\label{eq:epsK_def}
\epsK
\equiv \frac{\mathcal{A}(K_L \to \pi\pi(I=0))}
{\mathcal{A}(K_S \to \pi\pi(I=0))} \,.
\end{equation}
Here, $K_L$ and $K_S$ are the neutral kaon states in nature.
We can also calculate $\epsK$ directly from the standard model (SM)
using tools in lattice QCD.
Hence, we can test the SM through CP violation by comparing the
experimental and theoretical value of $\epsK$.
In order to calculate $\epsK$ directly from the SM, we use input
parameters obtained from lattice QCD and experiments.
In particular, we use lattice QCD inputs for $\BK$, $\abs{V_{cb}}$,
$\abs{V_{us}}$, and $\xi_0$ in this paper.
In addition, in order to avoid unwanted correlation through $\epsK$
between the Wolfenstein parameters of the CKM matrix and the inputs,
we adopt the angle-only fit (AOF) from the UTfit collaboration <cit.> to determine the apex
$(\bar{\rho},\bar{\eta})$ of the unitarity triangle.
§ MASTER FORMULA FOR $\EPSK$
In the SM, the master formula for $\epsK$ is
\begin{align}
\label{eq:epsK_SM_0}
\epsK
=& e^{i\theta} \sqrt{2}\sin{\theta}
\Big( C_{\eps} \; X_\text{SD} \; \hat{B}_{K}
+ \frac{ \xi_{0} }{ \sqrt{2} } + \xi_\text{LD} \Big)
+ \mathcal{O}(\omega \eps^\prime)
+ \mathcal{O}(\xi_0 \Gamma_2/\Gamma_1) \,,
\end{align}
\begin{align}
%= \frac{C}{\sqrt{2} \Delta M_{K}}
&= \frac{ G_{F}^{2} F_K^{2} m_{K^{0}} M_{W}^{2} }
{ 6\sqrt{2} \; \pi^{2} \; \Delta M_{K} } \,,
\quad
\xi_\text{LD} = \frac{m^\prime_\text{LD}}{\sqrt{2} \; \Delta M_K} \,,
% \label{eq:xiLD}
\quad
= -\Im \left[ \mathcal{P} \; \sum_{C}
\frac{\mate{\wbar{K}^0}{H_\text{w}}{C} \mate{C}{H_\text{w}}{K^0}}
\right]
\label{eq:mLD}
\end{align}
Here, $X_\text{SD}$ is the short distance contribution from the box
\begin{align}
X_\text{SD} &= \Im\lambda_t \Big[ \Re\lambda_c \eta_{cc} S_0(x_c)
-\Re\lambda_t \eta_{tt} S_0(x_t) - (\Re\lambda_c -
\Re\lambda_t) \eta_{ct} S_0(x_c,x_t) \Big] \,,
\end{align}
where $S_0$ are the Inami-Lim functions given in Ref. <cit.>, $\lambda_{i} \equiv V_{is}^{\ast}
V_{id} $, and $x_i = m_i^2/M_W^2$ with $m_i = m_i(m_i)$ defined as the
scale invariant $\overline{\text{MS}}$ quark mass <cit.>.
The $\xi_0$ term represents the long distance effect from the
absorptive part of the effective Hamiltonian: $\xi_0 = \Im A_0/\Re
The $\xi_\text{LD}$ term represents the long distance effect from
the dispersive part of the effective Hamiltonian.
Details of how to derive the master formula in
Eq. (<ref>) directly from the standard model using
Wigner-Weisskopf perturbation theory are given in Ref. <cit.>.
§ INPUT PARAMETERS
[Wolfenstein parameters]
CKMfitter UTfit AOF <cit.>
[QCD corrections]
Input Value Ref.
$\eta_{cc}$ $1.72(27)$
$\eta_{tt}$ $0.5765(65)$
$\eta_{ct}$ $0.496(47)$
[$\abs{V_{cb}}$ in units of $10^{-3}$]
Inclusive (Kin.) Inclusive (1S) Exclusive
FLAG SWME
/ <cit.>
/ <cit.>
[Long distance effects]
Input Value Ref.
$\xi_0$ $-1.63(19)(20) \times 10^{-4}$
$\xi_\text{LD}$ $(0 \pm 1.6)\,\%$
[Other input parameters]
Input Value Ref.
$1.1663787(6) \times 10^{-5}$ GeV$^{-2}$
$80.385(15)$ GeV
$1.275(25)$ GeV
$163.3(2.7)$ GeV
$497.614(24)$ MeV
$\Delta M_{K}$
$3.484(6) \times 10^{-12}$ MeV
$156.2(7)$ MeV
Input parameters
The CKMfitter and UTfit groups provide the Wolfenstein parameters
$\lambda$, $\bar{\rho}$, $\bar{\eta}$ and $A$ from the global
unitarity triangle (UT) fit, which are summarized in Table
<ref> tab:in-wolf.
Here, the parameters $\epsK$, $\BK$, and $\abs{V_{cb}}$ are inputs to the
global UT fit.
Hence, the Wolfenstein parameters extracted from the global UT fit
contain unwanted dependence on $\epsK$.
In order to avoid this unwanted correlation and to determine $\epsK$
self-consistently, we take another input set from the angle-only fit
(AOF) in Ref. <cit.>.
The AOF does not use $\epsK$, $\BK$, or $\abs{V_{cb}}$ as input to determine
the UT apex $(\bar{\rho},\bar{\eta})$.
We take $\lambda$ independently from $\abs{V_{us}}$, which has been extracted
from the $K_{\ell 3}$ and $K_{\mu2}$ decays with lattice QCD inputs
The input values for $\abs{V_{cb}}$ are summarized in Table
<ref> tab:in-Vcb.
The inclusive determination takes into account the inclusive decay
modes: $B \to X_c\ell\nu$ (essential) and $B \to X_s \gamma$
Moments of lepton energy, hadron masses, and photon energy (optional)
are measured from the relevant decays.
These moments are fit to the theoretical formula which is a heavy
quark expansion obtained with the aid of the operator product
expansion (OPE) <cit.>.
Here, we use the most updated value, given in Ref. <cit.>.
For the exclusive $\abs{V_{cb}}$, we use the most precise value from
the FNAL/MILC lattice calculation of the form factor $\mathcal{F}(w)$
of the semileptonic decay $\bar{B}\to D^*\ell\bar{\nu}$ at zero recoil
($w=1$) <cit.>.
They combined their lattice result with the HFAG average <cit.> of $\mathcal{F}(1) \abs{\bar{\eta}_\text{EM}}
\abs{V_{cb}}$ to extract $\abs{V_{cb}}$.
There have been a number of lattice QCD calculations of $\BK$ with
$N_f=2+1$ <cit.>.
Here, we use the FLAG average in Ref. <cit.> and the SWME result in Ref. <cit.>, which deviates most from the FLAG average.
They are summarized in Table <ref> tab:in-BK.
The RBC/UKQCD collaboration provides lattice results for $\Im A_2$ and
$\xi_0$ in Ref. <cit.>.
The long distance effect $\xi_0$ is given in Table
<ref> tab:in-LD.
In the master formula in Eq. (<ref>), $\xi_\text{LD}$
represents the long distance effect of $\approx 2\%$ which comes from
the dispersive part of the effective Hamiltonian.
The precise evaluation of $\xi_\text{LD}$ from lattice QCD is not
available yet.
Hence, we do not include this effect in the central value of $\epsK$,
but we take it as a systematic error with the value given in Table
The correction terms $\mathcal{O}(\omega \eps^\prime)$ and
$\mathcal{O}(\xi_0 \Gamma_2/\Gamma_1)$ are of order $10^{-7}$, and we
neglect them in this analysis.
A rough estimate of $\xi_\text{LD}$ is available from Ref. <cit.>.
The $\eta_{ij}$ parameters in Table
<ref> tab:in-eta represent the QCD corrections to
the coefficients of Inami-Lim functions.
The factor $\eta_{tt}$ is given at NLO, whereas $\eta_{cc}$ and
$\eta_{ct}$ are known up to NNLO.
Refer to Ref. <cit.> for more details.
The rest of the input parameters are given in Table
<ref> tab:in-extra.
§ RESULTS
Let us define $\epsK^\text{SM}$ as the theoretical evaluation of
$\abs{\epsK}$ using the master formula of Eq. (<ref>).
We define $\epsK^\text{Exp}$ as the experimental value of
$\abs{\epsK}$: $\epsK^\text{Exp} = (2.228 \pm 0.011) \times 10^{-3}$
Let us define $\Delta \epsK$ as the difference between
$\epsK^\text{Exp}$ and $\epsK^\text{SM}$: $\Delta\epsK \equiv
\epsK^\text{Exp} - \epsK^\text{SM}$.
Here, we assume that the theoretical phase $\theta$ is equal to the
experimental phase $\phi_\varepsilon$, although it is not fully
confirmed in lattice QCD yet.
In Table <ref> tab:epsK, we present results
for $\epsK^\text{SM}$.
They are obtained using the FLAG average for $\BK$ <cit.>, inclusive $\abs{V_{cb}}$ from
Ref. <cit.>, and exclusive
$\abs{V_{cb}}$ from Ref. <cit.>.
The corresponding probability distributions for $\epsK^\text{SM}$ are
presented in Fig. <ref>.
Input Method Inclusive $\Vcb$ Exclusive $\Vcb$
Input Method
Inclusive $\Vcb$
Exclusive $\Vcb$
tab:epsK $\epsK^\text{SM}$ in units of $10^{-3}$,
and tab:DepsK $\Delta\epsK$ in units of $\sigma$. The
$\sigma$ is obtained by combining errors of $\epsK^\text{SM}$ and
$\epsK^\text{Exp}$ in quadrature.
In Table <ref> tab:DepsK, we present results
for $\Delta\epsK$ for both inclusive and exclusive $\abs{V_{cb}}$.
From Table <ref>, we observe no tension in
$\Delta\epsK$ in the inclusive decay channels for $\abs{V_{cb}}$,
which are obtained using QCD sum rules and the heavy quark expansion.
However, from Table <ref>, we find that there exists a
$3.4\sigma$ tension between $\epsK^\text{Exp}$ and $\epsK^\text{SM}$
obtained using the exclusive $\abs{V_{cb}}$, which is determined using
lattice QCD tools.
In other words, $\epsK^\text{SM}$ with exclusive $\abs{V_{cb}}$ and
the most reliable input method (AOF) is only 72% of
The largest contribution that we neglect in our estimate of
$\epsK^\text{SM}$ is much less than 2%.
Hence, the neglected contributions cannot explain the gap
$\Delta\epsK$ of 28% with exclusive $\abs{V_{cb}}$.
[Exclusive $\abs{V_{cb}}$]
[Inclusive $\abs{V_{cb}}$]
Gaussian probability distributions for $\epsK^\text{SM}$
(blue dotted line) and $\epsK^\text{Exp}$ (red solid line). Here,
the results are obtained using the FLAG $\BK$ and the AOF
parameters for the CKM matrix elements.
In Fig. <ref>, we present the chronological evolution of
$\Delta\epsK/\sigma$ as the progress in lattice and perturbative QCD
goes on.
In 2012, RBC/UKQCD reported $\xi_0$ in Ref. <cit.>, and the lattice average for $\BK$
by LLV became available in Ref. <cit.>.
Based on these works, SWME reported $\Delta \epsK = 2.5\sigma$ in
Ref. <cit.> in 2012.
The FLAG average for $\BK$ became available in Ref. <cit.> in 2013.
In 2014, FNAL/MILC reported an updated $\abs{V_{cb}}$ in the exclusive
decay channel, and the NNLO value of $\eta_{ct}$ in Ref. <cit.> became known to us.
In 2014, SWME reported the updated $\Delta \epsK = 3.0\sigma$ in Ref. <cit.>.
In 2015, a remaining issue on the NNLO calculation of $\eta_{cc}$ was
addressed in Refs. <cit.>.
In 2015, SWME reported the updated $\Delta \epsK = 3.4\sigma$ in
Ref. <cit.>.
Recent history of $\Delta \epsK$ along with the theoretical
§ CONCLUSION
Here, we find that there is a substantial $3.4\sigma$ tension in
$\epsK$ between experiment and the SM with lattice QCD inputs.
For the SM estimate of $\epsK$, we use the AOF parameters and lattice
QCD inputs for exclusive $\abs{V_{cb}}$, $\BK$, $\abs{V_{us}}$ and
Since the AOF Wolfenstein parameters do not have unwanted correlation
with the lattice inputs via $\epsK$, the AOF method is relevant to the
data analysis in this paper.
We also find that the tension disappears for the inclusive
$\abs{V_{cb}}$, which is determined using QCD sum rules and the heavy
quark expansion.
In Table <ref>, we present the error budget for
In the second column of the tables, we show the fractional
contribution of each input parameter to the total error of
From this error budget, we find that $\abs{V_{cb}}$ dominates the
error in $\epsK^\text{SM}$.
Therefore, it is essential to reduce the error of $\abs{V_{cb}}$ down
to the sub-percent level.
For this purpose, we plan to extract $\abs{V_{cb}}$ from the exclusive
channel using the Oktay-Kronfeld (OK) action <cit.> for heavy quarks to calculate the form
factors for $\bar{B} \to D^{(*)} \ell \bar{\nu}$ decays.
The first stage ground work for this goal is underway and preliminary
results are reported in Ref. <cit.>.
Several lattice QCD inputs are obtained in the isospin limit, $m_u =
In particular, the isospin breaking effect from $\eps'/\eps$ in
$\xi_0$ could be substantial <cit.>.
The isospin breaking effects on $\xi_0$ and other input parameters are
of order 1% in $\epsK$.
Here we neglect them, but will incorporate them into the evaluation of
$\eps_K$ in the future.
source error (%) memo
$V_{cb}$ 39.3 FNAL/MILC
$\bar{\eta}$ 20.4 AOF
$\eta_{ct}$ 16.9 $c-t$ Box
$\eta_{cc}$ 7.1 $c-c$ Box
$\bar{\rho}$ 5.4 AOF
$m_t$ 2.4
source error (%) memo
$\xi_0$ 2.2 RBC/UKQCD
$\xi_\text{LD}$ 2.0 RBC/UKQCD
$\hat{B}_K$ 1.5 FLAG
$m_c$ 1.0
$\vdots$ $\vdots$
$\vdots$ $\vdots$
Error budget for $\epsK^\text{SM}$ obtained using the AOF
method, the exclusive $\Vcb$, and the FLAG $\BK$. Here, the values
are fractional contributions to the total error obtained using the
formula given in Ref. <cit.>.
Y.C.J. thanks A. Soni for helpful discussion on the unitarity triangle
We thank J. Brod and A. J. Buras for a helpful discussion on
The research of W. Lee is supported by the Creative Research
Initiatives Program (No. 2015001776) of the NRF grant funded by the
Korean government (MEST).
J.A.B. is supported by the Basic Science Research Program of the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education (No. 2015024974).
W. Lee would like to acknowledge the support from the KISTI
supercomputing center through the strategic support program for the
supercomputing application research (No. KSC-2014-G3-002).
Computations were carried out on the DAVID GPU clusters at Seoul
National University.
|
1511.00649
|
We study a weighted low rank approximation that is inspired by a problem of constrained low rank approximation of matrices as initiated by the work of Golub, Hoffman, and Stewart (Linear Algebra and Its Applications, 88-89(1987), 317-327). Our results reduce to that of Golub, Hoffman, and Stewart in the limiting cases. We also propose an algorithm based on the alternating direction method to solve our weighted low rank approximation problem and compare it with the state-of-art general algorithms such as the weighted total alternating least squares and the EM algorithm.
Weighted low rank approximation, singular value decomposition, alternating direction method
65F30, 65K05, 49M15, 49M30
ARITRA DUTTA AND XIN LIWeighted low rank approximation of matrices
§ INTRODUCTION
Let $m$ and $n$ be two natural numbers. For an integer $r\le\min\{m,n\}$ and a matrix $A\in\mathbb{R}^{m\times n}$, the standard low rank approximation problem can be formulated as
\begin{eqnarray}
\label{pca}
\min_{\substack{{X}\in\mathbb{R}^{m\times n}\\{\rm r}({X})\le r}}\|A-{X}\|_F^2,
\end{eqnarray}
where ${\rm r}({X})$ denotes the rank of the matrix ${X}$ and $\|\cdot\|_F$ denotes the Frobenius norm of matrices.
It is well known that the solutions to this problem can be given using the singular value decompositions (SVDs) of $A$ through the hard thresholding operations on the singular values: The solutions to (<ref>) are given by
\begin{align}\label{hardthresholding}
\end{align}
is a SVD of $A$ and $\Sigma_r(A)$ is the diagonal matrix obtained from $\Sigma(A)$ by thresholding: keeping only $r$ largest singular values and replacing other singular values by 0 along the diagonal. This is also referred to as Eckart-Young-Mirsky's theorem (<cit.>) and is closely related to the PCA method in statistics <cit.>. The solutions to (<ref>) as given in (<ref>) suffer from the fact that none of the entries of $X$ is guaranteed to be preserved in $X^*$. In many real world problems, one has good reasons to keep certain entries of $A$ unchanged while looking for a low rank approximation.
In 1987, Golub, Hoffman, and Stewart were the first to consider the following constrained low rank approximation problem <cit.>:
Given $A=(A_1\;\;A_2)\in\mathbb{R}^{m\times n}$ with $A_1\in\mathbb{R}^{m\times k}$ and $A_2\in\mathbb{R}^{m\times (n-k)}$, find $\tilde{A}_2$ such that (with $\tilde{A}_1=A_1$)
\begin{eqnarray}
(\tilde{A}_1\;\tilde{A}_2)=\arg\min_{\substack{X_1,X_2\\{\rm r}(X_1\;\;X_2)\le r\\X_1=A_1}}\|(A_1\;\;A_2)-(X_1\;\;{X}_2)\|_F^2.\label{golub's problem}
\end{eqnarray}
That is, Golub, Hoffman, and Stewart required that certain columns, $A_1,$ of $A$ must be preserved when one looks for a low rank approximation of $(A_1\;\;A_2).$ As in the standard low rank approximation, the constrained low rank approximation problem of Golub, Hoffman, and Stewart also has a closed form solution.
With $k={\rm r}(A_1)$ and $r\ge k$, the solutions $\tilde{A}_2$ in (<ref>) are given by
\begin{align}\label{ghs}
\tilde{A_2}= P_{A_1}(A_2)+H_{r-k}\left(P^{\perp}_{A_1}(A_2)\right),
\end{align}
where $P_{A_1}$ and $P^\perp_{A_1}$ are the projection operators to the column space of $A_1$ and its orthogonal complement, respectively.
According to Section 3 of <cit.>, the matrix $\tilde{A}_2$ is unique if and only if $H_{r-k}\left(P^{\perp}_{A_1}(A_2)\right)$ is unique, which means the $(r-k)$th singular value of $P^{\perp}_{A_1}(A_2)$ is strictly greater than $(r-k+1)$th singular value. When $\tilde{A}_2$ is not unique, the formula for $\tilde{A}_2$ given in Theorem <ref> should be understood as the membership of the set specified by the right-hand side of (<ref>). We will use this convention in this paper.
Inspired by Theorem <ref> above and motivated by applications in which $A_1$ may contain noise, it makes more sense if we require $\|A_1-X_1\|_F$ small instead of asking for $X_1=A_1$. This leads us to consider the following problem: Let $\eta>0$, find $(\hat{X}_1\;\;\hat{X}_2)$ such that
\begin{eqnarray}\label{closeness problem}
(\hat{X}_1\;\;\hat{X}_2)=\arg\min_{\substack{X_1,X_2: \|A_1-X_1\|_F\leq \eta\\{\rm r}(X_1\;\;X_2)\le r}}\| (A_1\;\;A_2)-({X}_1\;\;{X}_2) \|_F^2.
\end{eqnarray}
Or, for a large parameter $\lambda $, consider
\begin{eqnarray}\label{unconstraint closeness}
(\hat{X}_1\;\;\hat{X}_2)=\arg\min_{\substack{X_1,X_2\\{\rm r}(X_1\;\;X_2)\le r}}\left\{ \lambda^2\|A_1-X_1\|_F^2+\| A_2-{X}_2 \|_F^2\right\}.
\end{eqnarray}
This block structure in weight matrix, where very few entries are heavily weighted and most entries stays at 1 (unweighted), is realistic in applications. For example, in the problem of background estimation in a video sequence, each frame is a column in the data matrix and the background is a low rank (ideally of rank 1) component of the data matrix. So, the weight is used to single out which columns are more likely to be the basis of background frames and the low rank constraint enforces the search for other frames that are in the background subspace.
Recent investigations in <cit.> have shown that the above “approximately” preserving (controlled by a parameter $\lambda$) weighted low rank approximation can be more effective in solving the background modeling, shadows and specularities removal, and domain adaptation problems in computer vision and machine learning.
As it turns out, (<ref>) can be viewed as a generalized total least squares problem (GTLS) and can be solved in closed form as a special case of weighted low rank approximation with a rank-one weight matrix by using a SVD of the given matrix $(\lambda A_1\;\;A_2)$ <cit.>. As a consequence of the closed form solutions, one can verify that the solution to (<ref>) is the limit case of the solutions to (<ref>) as $\lambda \to\infty$. Thus, (<ref>) can be viewed as a special case when “$\lambda = \infty$”. A careful reader may also note that, problem (<ref>) can be cast as a special case of structured low rank problems with element-wise weights <cit.>.
More specifically, we observe that (<ref>) is contained in the following more general point-wise weighted low rank approximation problem:
\begin{eqnarray}\label{hadamard problem}
\min_{\substack{X_1,X_2\\{\rm r}(X_1\;\;X_2)\le r}}\|\left((A_1\;\;A_2)-({X}_1\;\;{X}_2)\right)\odot(W_1\;~W_2)\|_F^2,
\end{eqnarray}
for $W_1=\lambda \mathbbm{1}$ and $W_2=\mathbbm{1}$ (a matrix whose entries are equal to 1), where $W\in\mathbb{R}^{m\times n}$ is a weight matrix and $\odot$ denotes the Hadamard product.
This is the weighted low rank approximation problem studied first when $W$ is an indicator weight for dealing with the missing data case (<cit.>) and then for more general weight in machine learning, collaborative filtering, 2-D filter design, and computer vision <cit.>.
For example, if SVD is used in quadrantally-symmetric two-dimensional (2-D) filter design, as explained in <cit.> (see also <cit.>), it might lead to a degraded construction in some cases as it is not able to discriminate between the important and unimportant components of $X$. To address this problem, a weighted least squares matrix decomposition method was first proposed by Shpak <cit.>. Following his idea of assigning different weights to discriminate between important and unimportant components of the test matrix, Lu, Pei, and Wang (<cit.>) designed a numerical procedure to solve (<ref>) for general weight $(W_1\;~W_2)$.
There is another formulation of weighted low rank approximation problem defined as in <cit.>:
\begin{align}
\label{manton}
\min_{\substack{{X}\in\mathbb{R}^{m\times n}}}\|A-X\|_{Q}^2,~~{\rm subject~to~}{\rm r}({X})\le r,
\end{align}
where $Q \in \mathbb{R}^{mn\times mn}$ is a symmetric positive definite weight matrix. Denote $\|A-X\|_Q^2:={\rm vec}(A-X)^TQ{\rm vec}(A-X)$, where ${\rm vec}(\cdot)$ is an operator which maps the entries of $\mathbb{R}^{m\times n}$ to $\mathbb{R}^{mn\times 1}$. It is easy to see that (<ref>) is a special case of (<ref>) with a diagonal $Q$. In this paper, we will not use this more general formulation for simplicity.
Motivated by the limit behavior of (<ref>) as $\lambda\to\infty$, we are interested in finding out the limit behavior of the solutions to problem (<ref>) for general weight $(W_1\;~W_2)$ when $(W_1)_{ij}\to\infty$ and $W_2\to \mathbbm{1}$. One can expect that, with appropriate conditions, the limit should be the solutions to (<ref>). We will verify this with an stronger result with estimation on the rate of convergence (when $W_2=\mathbbm{1}$). The main challenge here is the lack of closed form solutions to problem (<ref>) in general <cit.>. We will also extend the convergence result to the unconstrained version of the problem (<ref>).
In order to make use of the proposed weighted low rank approximation in applications, we will propose a numerical algorithm to solve (<ref>) for the special case when $W_2=\mathbbm{1}$. In view of the existing algorithms for solving the general weighted low rank approximation problem, we want to emphasize that our special algorithm takes advantage of the block structure of our weights that results in better numerical performance as well as the fact that we have detailed convergence analysis for the algorithm.
The rest of the paper is organized as follows. In Section 2, we state our main results on the behavior of the solutions to (<ref>) as $(W_1)_{i,j}\to\infty$ and $(W_2)_{i,j}\to 1$. Their proofs are given in Section 3. In Section 4, we present a numerical solution to problem (<ref>) when $W_2=\mathbbm{1}$ and discuss the convergence of our algorithm. Numerical results verifying our main results are given in Section 5.
The extension of our rate of convergence and numerical algorithm to the more general case when $W_2=\mathbbm{1}$ is replaced by $W_2\to \mathbbm{1}$ is non-trivial and remains open.
§ LIMITING BEHAVIOR OF SOLUTIONS AS $(W_1)_{I,J}\TO\INFTY$ AND $(W_2)_{I,J}\TO 1$
Denote $\mathcal{A}=P_{A_1}^\perp(A_2)$ and $\tilde{\mathcal{A}}=P_{\tilde{X}_1(W)}^\perp(A_2).$ Also denote $s={\rm r}(\mathcal{A})$ and let the ordered non-zero singular values of $\mathcal{A}$ be $\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{s}>0.$
Let $(\tilde{X}_1(W)\;~\tilde{X}_2(W))$ be a solution to (<ref>).
Let $\lambda_j = \displaystyle{\min_{1\le i\le m}(W_1)_{ij}}$ and $\lambda = \displaystyle{\min_{1\le j\le k}\lambda_j}$.
Suppose that $\sigma_{r-k}>\sigma_{r-k+1}$. Then
we have, as $\lambda\to\infty$ and $W_2=\mathbbm{1}$,
where $A_G=(A_1\;\tilde{A}_2)$ is defined to be the unique solution to (<ref>).
(i) The assertion of the uniqueness of $A_G$ is due to the assumption $\sigma_{r-k}>\sigma_{r-k+1}$ (see the Remark <ref>). (ii) The convergence $(\tilde{X}_1(W)\;\;\tilde{X}_2(W))\to A_G$ alone is indeed implied by a general result in <cit.>.
Next, if we do not know $r$ but still want to reduce the rank in our approximation, then we can consider the unconstrained version of (<ref>): for $\tau>0$,
\begin{eqnarray}\label{unconstrained hadamard problem}
\min_{\substack{X_1,X_2}}\left\{\|\left((A_1\;\;A_2)-({X}_1\;\;{X}_2)\right)\odot(W_1\;\;W_2)\|_F^2+\tau{\rm r}({X}_1\;\;{X}_2)\right\}.
\end{eqnarray}
Again, one can expect that the solutions to (<ref>) will converge to the solution of (<ref>) as $(W_1)_{ij}\to\infty$ and $(W_2)_{ij}\to 1$. We will first establish a convergence result for (<ref>) without assuming the uniqueness of the solutions to (<ref>).
Let ${\cal A}_G^r$ be the set of all solutions to (<ref>) and let $(\hat{X}_1(W)\;~\hat{X}_2(W))$ be a solution to (<ref>).
Every accumulation point of $(\hat{X}_1(W)\;~\hat{X}_2(W))$ as $(W_1)_{ij}\to\infty,(W_2)_{ij}\to 1$ belongs to $\displaystyle{\mathop{\cup}_{0\le r\le \min\{m,n\}}{\cal A}_G^r}.$
Next, we have more precise information of the convergence if we assume the uniqueness of (<ref>).
Assume that $\sigma_1>\sigma_2>\cdots>\sigma_s>0$. Denote $\sigma_0:=\infty$ and $\sigma_{s+1}:=0.$
Then the accumulation point of the sequence $(\hat{X}_1(W)\;\hat{X}_2(W)),$ as $(W_1)_{ij}\to\infty$ and $(W_2)_{ij}\to1$ is unique; and this unique accumulation point is given by
\left(A_1\;\;\;\;P_{A_1}(A_2)+H_{r^*}\left(P^{\perp}_{A_1}(A_2)\right)\right)
with $r^*$ satisfying
\sigma_{r^*+1}^2\le\tau<\sigma_{r^*}^2.
For the case when $P^{\perp}_{A_1}(A_2)$ has repeated singular values, we leave it to the reader to verify the following more general statement by using a similar argument: Let $\hat{\sigma}_1>\hat{\sigma}_2>...>\hat{\sigma}_t>0$ be the singular values of $P^{\perp}_{A_1}(A_2)$ with multiplicity $k_1,k_2,\cdots k_t$ respectively. Note that $\sum_{i=1}^tk_i=s.$
Let $\sigma_{p^*+1}^2\le \tau <\sigma_{p^*}^2,$ where $\sigma_{p^*}$ has multiplicity $k_{p^*}.$
Then the accumulation points of the set $(\hat{X}_1(W)\;~\hat{X}_2(W)),$ as $(W_1)_{ij}\to\infty, (W_2)_{ij}\to1$, belongs to the set $\displaystyle{\mathop{\cup}_{r^*}{\cal A}_G^{r^*}},$ where $1+\sum_{i=1}^{p^*-1}k_i\le r^{*}<\sum_{i=1}^{p^*}k_i.$
§ PROOFS OF RESULTS IN SECTION 2
To prove Theorem <ref>, we start with a few well known results from the perturbation theory of singular values. First, we quote the following result of Stewart.
Let $\tilde{{A}}={{A}}+E$ and $\sigma\neq 0$ be a non-repeating singular value of the matrix ${A}$ with $u$ and $v$ being left and right singular vectors respectively. Then as $\|E\| \to 0,$ there is a unique singular value $\tilde{\sigma}$ of $\tilde{{A}}$ such that
\begin{eqnarray}\label{per-singularvalue}
\tilde{\sigma}=\sigma+u^TEv+O(\|E\|^2).
\end{eqnarray}
Let the SVDs of ${B},~\tilde{{B}}\in \mathbb{R}^{m\times n}$ be given by
\begin{eqnarray}\label{SVDA}
{B}=U\Sigma V^T=(U_1\;U_2)\begin{pmatrix}
\Sigma_1 & 0\\
0 &\Sigma_2\\
\end{pmatrix}\begin{pmatrix}
\end{pmatrix}=:{B}_1+{B}_2,
\end{eqnarray}
\begin{eqnarray}\label{SVDA1}
\tilde{{B}}=\tilde{U}\tilde{\Sigma}\tilde{V}^T=(\tilde{U}_1\;\tilde{U}_2)\begin{pmatrix}
\tilde{\Sigma}_1 & 0\\
0 &\tilde{\Sigma}_2\\
\end{pmatrix}\begin{pmatrix}
\tilde{V}_1^T\\\tilde{V}_2^T
\end{pmatrix}=:\tilde{{B}}_1+\tilde{{B}}_2,
\end{eqnarray}
such that $U,~\tilde{U}\in\mathbb{R}^{m\times m},~V,~\tilde{V}\in\mathbb{R}^{n\times n},\;\text{and}\;~ \Sigma,~\tilde{\Sigma}\in\mathbb{R}^{m\times n}$ with $\Sigma$ and $\tilde{\Sigma}$ being diagonal matrices containing singular values of ${B}$ and $\tilde{{B}}$, respectively, arranged in a non-increasing order; $U_1,~\tilde{U}_1\in\mathbb{R}^{m\times s},U_2,~\tilde{U}_2\in\mathbb{R}^{m\times (m-s)},
V_1,~\tilde{V}_1\in\mathbb{R}^{n\times s},\;\text{and}\;~
V_2,~\tilde{V}_2\in\mathbb{R}^{n\times (n-s)}.$
Using (<ref>) and (<ref>) we have, with $E=\tilde{B}-B$,
\begin{eqnarray}\label{per_SVD2}
\tilde{{B}}_1+\tilde{{B}}_2=\tilde{{B}}={B}+E={B}_1+{B}_2+E.
\end{eqnarray}
Next, we state a special form of the sin$\theta$ Theorem of Wedin (see <cit.> and <cit.>) as follows.
With the notations above, let $\eta=\sigma_{min}(\tilde{B}_1)-\sigma_{max}({B}_2)>0$.
\sqrt{\|\sin \Theta(U_1,\tilde{U}_1)\|^2_F+\|\sin \Theta(V_1,\tilde{V}_1)\|^2_F}\leq\frac{\sqrt{2}\|E\|_F}{\eta}.
Finally, using the argument of Wedin (<cit.>), the following result can be achieved.
Assume there exists an $\alpha\ge 0$ and a $\delta>0$ such that
\sigma_{min}(\tilde{{B}}_1)\ge \alpha+\delta\;\;\text{and}\;\;\sigma_{max}({B}_2)\le\alpha,
\begin{align}\label{thresholding inequality}
\|{B}_1-\tilde{{B}}_1\|\le \|E\|(3+\frac{\|{B}_2\|}{\delta}+\frac{\|\tilde{{B}}_2\|}{\delta}).\end{align}
Now, we establish some auxiliary results. Let $P_B$ and $P_{\tilde{B}}$ be the orthogonal projections onto the column spaces of matrices $B$ and $\tilde{B}$, respectively.
Assume that $B$ and $\tilde{B}$ are full rank matrices. Let $\eta$ denote the smallest singular value of $\tilde{B}$. Then
\begin{equation}\label{project estimation}
\|P_B-P_{\tilde{B}}\|_F\leq \frac{2\|B-\tilde{B}\|_F}{\eta}.
\end{equation}
According to <cit.>, we have
\begin{equation}\label{x1}
\|P_B-P_{\tilde{B}}\|_F=\sqrt{2}\|\sin \Theta (B,\tilde{B})\|_F,
\end{equation}
where $\Theta (B,\tilde{B})$ is the canonical angle (diagonal) matrix between the column spaces of $B$ and $\tilde{B}$. Now, by applying Lemma <ref> to the case when $U_2=\tilde{U}_2=\emptyset$ (so that $U_1$ and $\tilde{U}_1$ span the same column spaces as those of $B$ and $\tilde{B}$, respectively, and so $\Theta(B,\tilde{B})=\Theta(U_1,\tilde{U}_1)$ with $\eta=\sigma_{min}(\tilde{B}_1)>0$ (since $\sigma({B}_2)=0$), we have
\begin{equation}\label{wedin1}
\|\sin \Theta (B, \tilde{B})\|_F=\|\sin \Theta (U_1, \tilde{U}_1)\|_F\leq \frac{\sqrt{2}\|B-\tilde{B}\|_F}{\eta}.
\end{equation}
Now, the inequality (<ref>) follows from (<ref>) and (<ref>).
As $(W_1)_{ij}\to\infty$ and $W_2$ stays bounded, we have the following estimates.
* $\tilde{X}_1(W)= A_1+\displaystyle{O(\frac{1}{\lambda})}$.
* $P_{\tilde{X}_1(W)}(A_2)=P_{A_1}(A_2)+\displaystyle{O(\frac{1}{\lambda})}$.
* $P_{\tilde{X}_1(W)}^\perp(A_2)=P_{A_1}^\perp(A_2)+\displaystyle{O(\frac{1}{\lambda})}$.
Note that,
\begin{align*}
&\|(A_1-\tilde{X}_1(W))\odot W_1\|_F^2+\|(A_2-\tilde{X}_2(W))\odot W_2\|_F^2\\
&=\min_{\substack{{X}_1,{X}_2\\{\rm r}({X}_1\;\;{X}_2)\le r}}\left(\|(A_1-{X}_1)\odot W_1\|_F^2+\|(A_2-{X}_2)\odot W_2\|_F^2\right)\\
&\le\|A_2\odot W_2\|_F^2\;(\text{by taking}\;(X_1\;\;X_2)=(A_1\;\;0)).
\end{align*}
Then $\displaystyle{\sum_{\substack{1\le i\le m\\1\le j\le k}}((A_1)_{ij}-(\tilde{X}_1(W))_{ij})^2(W_1)_{ij}^2\le \|A_2\odot W_2\|_F^2}$ and so
$$|(A_1)_{ij}-(\tilde{X}_1(W))_{ij}|\le\frac{\|A_2\odot W_2\|_F}{(W_1)_{ij}};~~1\le i\le m, 1\le j\le k.$$
$$ \tilde{X}_1(W)= A_1+\displaystyle{O(\frac{1}{\lambda})}\;\text{as}\;\lambda\to\infty.$$
(ii). Note that as $\lambda\to \infty$, $\tilde{X}_1(W)\to A_1$. So, $\tilde{X}_1(W)$ will be a full rank matrix (since $A_1$ is assumed to be of full rank).
Applying (i) and Lemma <ref> to $B=A_1$ and $\tilde{B}=\tilde{X}_1(W)$, we have, as $\lambda\to \infty$,
\|P_{A_1}-P_{\tilde{X}_1(W)}\|_F \leq \frac{2}{\eta (\tilde{X}_1(W))} O(\frac{1}{\lambda}).
Since $\eta (\tilde{X}_1(W))\to \eta(A_1)>0$, we have $\eta (\tilde{X}_1(W))\geq \eta(A_1)/2$ as
$\lambda\to \infty$. Thus, (ii) holds.
(iii) We know that
Using (ii)
\begin{align*}
& P_{A_1}(A_2)+\displaystyle{O(\frac{1}{\lambda})}+P_{\tilde{X}_1(W)}^\perp(A_2)=P_{A_1}(A_2)+P_{A_1}^{\perp}(A_2),~\lambda\to\infty.
\end{align*}
\begin{eqnarray}\label{perp}
& P_{\tilde{X}_1(W)}^\perp(A_2)=P_{A_1}^\perp(A_2)+O(\frac{1}{\lambda}),~\lambda\to\infty.
\end{eqnarray}
This completes the proof of Lemma <ref>.
(i) For the case when there is an uniform weight in $(W_1)_{ij}=\lambda>0$, one might refer to <cit.> for an alternative proof of Lemma <ref>. But the proof in <cit.> can not be applied in the more general weight. (ii) There is a more elementary proof of (ii) of Lemma <ref> above by using the Gram-Schmidt process (see <cit.>), instead of using the advanced tools from the perturbation theory of singular values of Stewart and Wedin. We choose to use the shorter proof here because we need to invoke the advanced theory for the proof of the next lemma anyway.
Next, we establish a key lemma on the hard thresholding operator under perturbation. We use the notation introduced at the beginning of Section 2.
Let $W_2$ be bounded. If $\sigma_{r-k}>\sigma_{r-k+1},$ then
\begin{eqnarray}\label{per_SVD1}
\end{eqnarray}
Apply (<ref>) and (<ref>) with $B=\mathcal{A}$ and $\tilde{B}=\tilde{\mathcal{A}}$. Then by (iii) of Lemma <ref>, we know that $E=\tilde{\mathcal{A}}-{\mathcal{A}}=O(\frac{1}{\lambda}).$ Indeed, with the non-increasing arrangement of the singular values in $\Sigma$ and $\tilde{\Sigma}$, and the fact that $E=\displaystyle{O(\frac{1}{\lambda})}$ as $\lambda\to\infty$, Lemma <ref> immediately implies that
\begin{align}\label{svd bound}
\Sigma_1-\tilde{\Sigma}_1=\displaystyle{O(\frac{1}{\lambda})}~\;\;\text{and}\;\;\Sigma_2-\tilde{\Sigma}_2=O(\frac{1}{\lambda})\;\;\text{as}\;\lambda\to\infty.
\end{align}
Note that, ${\rm r}(\mathcal{A}_1)={\rm r}(\tilde{\mathcal{A}}_1)=r-k,$ and, since $\sigma_{r-k}>\sigma_{r-k+1},$ we can choose $\delta$ such that
\delta=\frac{1}{2}(\sigma_{r-k}-\sigma_{r-k+1})>0.
In this way, for all large $\lambda$ the assumption of Lemma <ref> will be satisfied. Since $\mathcal{A}_1=H_{r-k}(\mathcal{A})$ and $\tilde{\mathcal{A}}_1=H_{r-k}(\tilde{\mathcal{A}}),$ (<ref>) can be written as
\begin{eqnarray}\label{thresholding inequality1}
\|H_{r-k}(\mathcal{A})-H_{r-k}(\tilde{\mathcal{A}})\|\le \|E\|(3+\frac{\|\mathcal{A}_2\|}{\delta}+\frac{\|\tilde{\mathcal{A}}_2\|}{\delta}).
\end{eqnarray}
Since ${\mathcal{A}_2}$ is fixed, $\|{\mathcal{A}_2}\|=O(1)$ as $\lambda\to\infty$. On the other hand, by (<ref>), as $\lambda\to\infty$,
Now the unitary invariance of the matrix norm implies,
\begin{align*} \|\tilde{\mathcal{A}_2}\|\le\|\tilde{U}_2\Sigma_2\tilde{V}_2^T\|+O(\frac{1}{\lambda}\|\tilde{U}_2\tilde{V}_2^T\|)=\|\Sigma_2\|+O(\frac{1}{\lambda}),
\end{align*}
which is bounded as $\lambda\to\infty$.
Therefore (<ref>) becomes
\begin{eqnarray}
\|H_{r-k}(\mathcal{A})-H_{r-k}(\tilde{\mathcal{A}})\|\le C\|E\|,
\end{eqnarray}
for some constant $C>0$ and for all large $\lambda\to\infty$.
since $E=\displaystyle{O(\frac{1}{\lambda})}$ as $\lambda\to\infty$. This completes the proof of Lemma <ref>.
Proof of Theorem <ref>. We first note that, for all $X_2$ with $r(\tilde{X}_1(W)\;~X_2)\leq r$,
\|((A_1\;~A_2)-(\tilde{X}_1(W)\;~\tilde{X}_2(W)))\odot (W_1\;~\mathbbm{1})\|_F
\leq \|((A_1\;~A_2)-(\tilde{X}_1(W)\;~X_2))\odot (W_1\;~\mathbbm{1})\|_F.
So, $\tilde{X}_2(W)$ solves
\|A_2-\tilde{X}_2(W)\|_F
=\inf_{X_2:r(\tilde{X}_1(W)\;~X_2)\leq r} \|A_2-X_2\|_F.
Thus, by Theorem <ref>,
\tilde{X}_2(W)=P_{\tilde{X}_1(W)}(A_2)+H_{r-k}(P_{\tilde{X}_1(W)}^\perp(A_2)).$$
Therefore, using (ii) and (iii) of Lemma <ref> and Lemma <ref>, we get, as $\lambda\to\infty$,
\tilde{X}_2(W)=P_{A_1}(A_2)+H_{r-k}(P_{A_1}^\perp(A_2))+O(\frac{1}{\lambda})=\tilde{A}_2+O(\frac{1}{\lambda}).$$
This, together with (i) of Lemma <ref>, yields the desired result. $\Box$
Proof of Theorem <ref>. Let $\hat{X}(W)=(\hat{X}_1(W)\;\;\hat{X}_2(W))$. We need to verify that $\{\hat{X}(W)\}_W$ is a bounded set and every accumulation point (as $(W_1)_{ij}\to\infty$ and $(W_2)_{ij}\to 1$) is a solution to (<ref>) for some $r$. Since $(\hat{X}_1(W)\;\;\hat{X}_2(W))$ is a solution to (<ref>), we have
\begin{align}\label{ineq2}
&\|(A_1-\hat{X}_1(W))\odot W_1\|_F^2+ \|(A_2-\hat{X}_2(W))\odot W_2\|_F^2+\tau{\rm r}(\hat{X}_1(W)\;\;\hat{X}_2(W))\nonumber\\
&\le\|(A_1-{X}_1)\odot W_1\|_F^2+\|(A_2-{X}_2)\odot W_2\|_F^2+\tau{\rm r}({X}_1\;\;{X}_2).
\end{align}
for all $({X}_1\;\;{X}_2).$
By choosing $X_1=A_1,X_2=0$, we can obtain a constant $m_3:=\|A_2\odot W_2\|_F^2+\tau{\rm r}(A_1\;\;0)$ such that $\|(A_1-\hat{X}_1(W))\odot W_1\|_F^2+\|(A_2-\hat{X}_2(W))\odot W_2\|_F^2\le m_3.$ Therefore, $\{\hat{X}_1(W)\;\;\hat{X}_2(W)\}$ is bounded. Let $(X_1^{**}\;\;X_2^{**})$ be an accumulation point of the sequence for some subsequences as $(W_1)_{ij}\to\infty$ and $(W_2)_{ij}\to 1$. We only need to show that $(X_1^{**}\;\;X_2^{**})\in\displaystyle{\mathop{\cup}_{r}{\cal A}_G^r}.$ As in the proof of Lemma <ref> (i), we can show that
\begin{align}\label{2}
\lim_{\substack{(W_1)_{ij}\to\infty\\(W_2)_{ij}\to 1}}\hat{X}_1(W)=A_1.
\end{align}
Now, taking limit and setting $X_1=A_1$ in (<ref>), we obtain,
\begin{align}\label{limitcase}
&\|A_2-X_2^{**}\|_F^2+\tau{\rm r}(A_1\;\;{X}_2^{**})\le\|A_2-{X}_2\|_F^2+\tau{\rm r}(A_1\;\;{X}_2),
\end{align}
for all ${X}_2.$ If we denote $r^{**}={\rm r}(A_1\;\;{X}_2^{**}),$ then for ${X}_2$ with $ {\rm r}(A_1\;\;{X}_2)\le r^{**},$ (<ref>) yields
\begin{align}
\|A_2-X_2^{**}\|_F^2\le\|A_2-{X}_2\|_F^2.
\end{align}
Therefore, $X_2^{**}$ is a solution to the problem of Golub, Hoffman, and Stewart and by Theorem <ref>,
This, together with (<ref>) completes the proof. $\Box$
Proof of Theorem <ref>. For convenience, we will drop the dependence on $W$ in our notations.
Let $\hat{X}=(\hat{X}_1\;\;\hat{X}_2)$ solve the minimization problem (<ref>). As in the proof of Lemma <ref> (i) again we have
\begin{eqnarray}\label{R_1}
\hat{X}_1\to A_1,~{\rm as}~(W_1)_{ij}\to\infty.
\end{eqnarray}
Next, we show the convergence of $\hat{X}_2$: for $r^*$ satisfying $\sigma_{r^*}^2>\tau\geq \sigma_{r^*+1}^2$,
\begin{equation}\label{x2}
\hat{X}_2\to P_{A_1}(A_2)+H_{r^*}(P_{A_1}^\perp(A_2)), ~{\rm as}~(W_1)_{i,j}\to\infty,~(W_2)_{i,j}\to 1.
\end{equation}
By Theorem <ref>, $\hat{X}$ is bounded. So, to establish (<ref>), we will need only to show (i) there is only one accumulation point as $(W_1)_{i,j}\to\infty,~(W_2)_{i,j}\to 1$, and (ii) this unique accumulation point is $P_{A_1}(A_2)+H_{r^*}(P_{A_1}^\perp(A_2))$ for $r^*$ satisfying $\sigma_{r^*}^2>\tau\geq \sigma_{r^*+1}^2$.
Let $X_2^*$ be any accumulation point of $\hat{X}_2$ as $(W_1)_{i,j}\to\infty,~(W_2)_{i,j}\to 1$.
Choosing $X_1=\hat{X}_1$ in (<ref>) we find, for all $X_2$,
\begin{eqnarray}\label{weighted unconstrained problem 5}
\;\;\;\;\;\;\|(A_2-\hat{X}_2)\odot W_2\|_F^2+\tau{\rm r}(
\hat{X}_{1}\;\;\hat{X}_{2})
\le \|(A_2-{X}_2)\odot W_2\|_F^2+\tau{\rm r}(
\hat{X}_{1}\;\;{X}_{2}).
\end{eqnarray}
We will apply some ideas from Golub, Hoffman, and Stewart <cit.>. Since the weight $W_2$ gets in
the way (by destroying the unitary invariance of the norm), we have to take the limit $(W_2)_{i,j}\to 1$ at the right time.
As in <cit.>, assume ${\rm r}(A_1)=k$ and consider a $QR$ decomposition of $A$ and corresponding decomposition of $\hat{X}$ and $X$:
A=QR=(Q_1~~Q_2)\begin{pmatrix}{R}_{11}&{R}_{12}\\0& R_{22}\\0& 0\end{pmatrix}=(Q_1R_{11}~~Q_1R_{12}+Q_2\begin{pmatrix}R_{22}\\ 0\end{pmatrix})=(A_1~~A_2),
and let
Since the rank of a matrix is invariant under an unitary transformation, (<ref>) can be rewritten as
\begin{eqnarray}\label{weighted unconstrained problem 6}
&\|(A_2-\hat{X}_2)\odot W_2\|_F^2+\tau{\rm r}(
&\le \|(A_2-{X}_2)\odot W_2\|_F^2+\tau{\rm r}(
\end{eqnarray}
When $\lambda=\min_{i,j}(W_1)_{i,j}$ is large enough, $\hat{R}_{11}$ is nonsingular by (<ref>). Using ${\rm r}(A_1)=k$
and performing the row and column operations on the second terms on both sides of (<ref>), we get
\begin{align}\label{weighted unconstrained problem 7}
&\|(A_2-\hat{X}_2)\odot W_2\|_F^2+\tau {\rm r}\begin{pmatrix}
\hat{R}_{22}-\hat{R}_{21}\hat{R}_{11}^{-1}\hat{R}_{12} \\
\hat{R}_{32}-\hat{R}_{31}\hat{R}_{11}^{-1}\hat{R}_{12}\\
\end{pmatrix}\nonumber\\&\le\|(A_2-{X}_2)\odot W_2\|_F^2+\tau {\rm r}\begin{pmatrix}
{R}^{\dagger}_{22}-\hat{R}_{21}\hat{R}_{11}^{-1}{R}^{\dagger}_{12} \\
\end{pmatrix},
\end{align}
for all $R^{\dagger}_{12},{R}^{\dagger}_{22},~{\rm and}~{R}^{\dagger}_{32}.$
Note that $\displaystyle{\begin{pmatrix}\hat{R}_{12}^{*}\\\hat{R}_{22}^{*}\\\hat{R}_{32}^{*}\end{pmatrix}}:=Q^T\hat{X}^*_2$ is an accumulation point for a subsequence, $\Lambda$ say, of $\begin{pmatrix}\hat{R}_{12}\\\hat{R}_{22}\\\hat{R}_{32}\end{pmatrix}=Q^T\hat{X}_2.$ From (<ref>), using the fact that $\hat{R}_{11}\to R_{11},\hat{R}_{21}\to 0,~{\rm and}~\hat{R}_{31}\to 0,$ as $(W_1)_{ij}\to\infty, (W_2)_{ij}\to1$ in (<ref>) we get, by taking limit along the subsequence $\Lambda$,
\begin{align}\label{weighted unconstrained problem 8}
&\|A_2-\hat{X}_2^*\|_F^2+\tau {\rm r}\begin{pmatrix}
\hat{R}_{22}^*\\
\hat{R}_{32}^*\\
\end{pmatrix}
\le\|A_2-{X}_2\|_F^2+\tau {\rm r}\begin{pmatrix}
\end{pmatrix},
\end{align}
for all ${R}^{\dagger}_{12},{R}^{\dagger}_{22},~{\rm and}~{R}^{\dagger}_{32}.$ Since Frobenius norm is unitarily invariant, (<ref>) reduces to
\begin{align}\label{weighted unconstrained problem 9}
&\|\begin{pmatrix} R_{12}\\
\end{pmatrix}-\begin{pmatrix} \hat{R}_{12}^*\\
\hat{R}_{22}^*\\
\hat{R}_{32}^*\\
\end{pmatrix}\|_F^2+\tau {\rm r}\begin{pmatrix}
\hat{R}_{22}^*\\
\hat{R}_{32}^*\\
\end{pmatrix}
\le\|\begin{pmatrix} R_{12}\\
\end{pmatrix}-\begin{pmatrix} {R}_{12}^{\dagger}\\
\end{pmatrix}\|_F^2+\tau {\rm r}\begin{pmatrix}
\end{pmatrix},
\end{align}
for all ${R}^{\dagger}_{12},{R}^{\dagger}_{22},~{\rm and}~{R}^{\dagger}_{32}.$
Substituting ${R}^{\dagger}_{22}=\hat{R}_{22}^*, {R}^{\dagger}_{32}=\hat{R}_{32}^*$, and $R^{\dagger}_{12}=R_{12}$, in (<ref>) yields
\begin{align*}
\le 0,
\end{align*}
and so $R^*_{12}=R_{12}$. Now, substituting $R^{\dagger}_{12}=\hat{R}^*_{12}$ in (<ref>) we find
\begin{align}\label{weighted unconstrained problem 11}
\hat{R}_{22}^*\\
\hat{R}_{32}^*\\
\end{pmatrix}\|_F^2+\tau {\rm r}\begin{pmatrix}
\hat{R}_{22}^*\\
\hat{R}_{32}^*\\
\end{pmatrix}
\le\|\begin{pmatrix}{R}_{22}\\0\end{pmatrix}-\begin{pmatrix}
\end{pmatrix}\|_F^2+\tau {\rm r}\begin{pmatrix}
\end{pmatrix},
\end{align}
for all ${R}^{\dagger}_{22},{R}^{\dagger}_{32}$. Let $\bar{R}^*=\begin{pmatrix}
\hat{R}_{22}^{*}\\\hat{R}_{32}^{*}\end{pmatrix}$ and $r^*={\rm r}(\bar{R}^*),$ then (<ref>) implies
\begin{align}\label{R22lowrank}
\|\begin{pmatrix}{R}_{22}\\0\end{pmatrix}-\bar{R}^*\|_F^2\le \|\begin{pmatrix}{R}_{22}\\0\end{pmatrix}-{R}^*\|_F^2,
\end{align}
for all ${R}^*\in\mathbb{R}^{(m-k)\times(n-k)}$ with ${\rm r}({R}^*)\le r^*.$ So $\bar{R}^*$ solves a problem of classical low rank approximation of $\begin{pmatrix}{R}_{22}\\0\end{pmatrix}$. Note that, $Q_2\begin{pmatrix}{R}_{22}\\0\end{pmatrix}=P^{\perp}_{A_1}(A_2)$ (see Theorem 1 in <cit.>). Since $P^{\perp}_{A_1}(A_2)$ has distinct singular values, there exists a unique $\bar{R}^*$ which is given by $\bar{R}^*=H_{r^*}\begin{pmatrix}R_{22}\\0\end{pmatrix}$ as in (<ref>). Therefore,
\lim_{\Lambda} Q^T\hat{X}_2 = Q^T\hat{X}_2^*=\begin{pmatrix}R_{12}\\ H_{r^*}\begin{pmatrix}R_{22}\\0\end{pmatrix}\end{pmatrix},$$
which, together with (<ref>),
\begin{align*}
\displaystyle{\lim_{\Lambda}}(\hat{X}_1\;\;\hat{X}_2)
&=Q\begin{pmatrix} & R_{12}\\
R_1 & H_{r^*}\begin{pmatrix}
\end{pmatrix}
\end{pmatrix}\\
\end{pmatrix}\right)),
\end{align*}
which is the same as
\left(A_1\;\;\;\;P_{A_1}(A_2)+H_{r^*}\left(P^{\perp}_{A_1}(A_2)\right)\right).
Finally, if we can show that $r^*$ does not depend on $\Lambda$ then we know that all accumulation points equal and therefore, we can complete the proof. In fact, $r^*$ depends only on $\tau$ as we see from the argument below.
Assume that
\begin{pmatrix}{R}_{22}\\0\end{pmatrix}=U\Sigma V^T
is a SVD of $\begin{pmatrix}{R}_{22}\\0\end{pmatrix}.$ Then, for any ${R}^*\in\mathbb{R}^{(m-k)\times (n-k)},$ (<ref>) gives
\begin{eqnarray}\label{QR inequality}
&\|\Sigma-U^T\bar{R}^*V\|_F^2+\tau {\rm r}(U^T\bar{R}^*V)\nonumber\\
&\le\|\Sigma-U^T{R}^*V\|_F^2+\tau{\rm r}(U^T{R}^*V),
\end{eqnarray}
Since $r^*={\rm r}(\bar{R}^*)$ and $
U^T\bar{R}^{*} V={\rm diag}(\sigma_1\;\sigma_2\;\cdots\sigma_{r^*}\;0\cdots 0)$,
choosing ${R}^*$ such that
U^T{R}^*V={\rm diag}(\sigma_1\;\sigma_2\;\cdots\sigma_{r^*+1}\;0\cdots 0),
and using (<ref>) we find
\begin{align*}
\sigma_{r^*+2}^2+\cdots+\sigma_n^2+\tau\ge\sigma_{r^*+1}^2+\sigma_{r^*+2}^2+\cdots+\sigma_n^2.
\end{align*}
Next we choose ${R}^*$ such that
U^T{R}^*V={\rm diag}(\sigma_1\;\sigma_2\;\cdots\sigma_{r^*-1}\;0\cdots 0),
and so ${\rm r}(R^*)=r^*-1<r^*$. Now (<ref>) and Eckart-Young-Mirsky's theorem imply the equality in (<ref>) can not hold. So,
\begin{align*}
\sigma_{r^*}^2+\cdots+\sigma_n^2-\tau>\sigma_{r^*+1}^2+\sigma_{r^*+2}^2+\cdots+\sigma_n^2.
\end{align*}
Therefore, we obtain
\begin{eqnarray}\label{sigma inequality}
\sigma_{r^*}^2>\tau\ge\sigma_{r^*+1}^2.
\end{eqnarray}
It is easy to see that if (<ref>) holds then ${\rm r}(\bar{R}^*)=r^*.$
{\rm r}(\bar{R}^*)=r^*\;\text{if and only if}\;\sigma_{r^*}^2>\tau\ge\sigma_{r^*+1}^2.
This completes the proof. $\Box$
§ NUMERICAL ALGORITHM
In this section we propose a numerical algorithm to solve (<ref>) when $W_2=\mathbbm{1}$, which, in general, does not have a closed form solution <cit.>.
Note (<ref>) can be written as (with $W_2=\mathbbm{1}$)
\begin{align*}
\min_{\substack{X_1,X_2\\{\rm r}(X_1\;\;X_2)\le r}}\left(\|(A_1-X_1)\odot W_1\|_F^2+\|A_2-X_2\|_F^2\right).
\end{align*}
We assume that ${\rm r}(X_1)=k$. It can be verified that any $X_2$ such that ${\rm r}(X_1\;\;X_2)\le r$ can be given in the form
for some arbitrary matrices $B\in\mathbb{R}^{m\times (r-k)},$ $D\in\mathbb{R}^{(r-k)\times (n-k)},$ and $C\in\mathbb{R}^{k\times (n-k)}.$ Hence we need to solve
\begin{align}\label{main problem 2}
\min_{X_1,C,B,D}\left(\|(A_1-X_1)\odot W_1\|_F^2+\|A_2-X_1C-BD\|_F^2\right).
\end{align}
Note that, using a block structure, we can write (<ref>) as a weighted low rank approximation (with a special low rank structure):
\begin{align*}
\min_{X_1,C,B,D}\left\{\|\left((A_1\;\;A_2)-(X_1\;\;B)\begin{pmatrix} I_k & C\\0 & D\end{pmatrix}\right)\odot (W_1\;\;\mathbbm{1})\|_F^2\right\},
\end{align*}
which is a form of the alternating weighted least squares problem in the literature <cit.>. But we will not follow the general algorithm proposed in <cit.> for the following reasons: (i) due to the special structure of the weight, our algorithm is more efficient than <cit.> (see Algorithm 3.1, in p. 42 <cit.>), (ii) it allows a detailed convergence analysis which is usually not available in other algorithms proposed in the literature <cit.>, and (iii) it can handle bigger size matrices as we will demonstrate in the numerical result section.
If $k=0,$ then (<ref>) reduces to an unweighted rank $r$ factorization of $A_2$ and can be solved as an alternating least squares problem <cit.>.
§.§ Optimization procedure
Denote $F(X_1,C,B, D)=\|(A_1-X_1)\odot W_1\|_F^2+\|A_2-X_1C-BD\|_F^2$ as the objective function. The above problem can be numerically solved by using an alternating strategy <cit.> of minimizing the function with respect to each component iteratively:
\begin{align}\label{update rule}
\left\{\begin{array}{ll}
\displaystyle{(X_1)_{p+1}=\arg\min_{X_1}F(X_1,C_p,B_p,D_p)},\\
\displaystyle{C_{p+1}=\arg\min_{C}F((X_1)_{p+1},C,B_p,D_p)},\\
\displaystyle{B_{p+1}=\arg\min_{B}F((X_1)_{p+1},C_{p+1},B,D_p)},\\
\text{and,}\; \displaystyle{D_{p+1}=\arg\min_{D}F((X_1)_{p+1},C_{p+1},B_{p+1},D)}.
\end{array}\right.
\end{align}
Note that each of the minimizing problem for $X_1, C, B,$ and $D$ can be solved explicitly by looking at the gradients of $F(X_1,C,B,D)$. But finding an update rule for $X_1$ turns out to be more involved than the other three variables due to the interference of the weight $W_1$. We update $X_1$ element wise along each row. Therefore we will use the notation $X_1(i,:)$ to denote the $i$-th row of the matrix $X_1$. We set $\frac{\partial}{\partial X_1}F(X_1,C_p,B_p,D_p)|_{X_1=(X_1)_{p+1}}=0$ and obtain
\begin{align}\label{x_p}
-(A_1-(X_1)_{p+1})\odot W_1\odot W_1-(A_2-(X_1)_{p+1}C_p-B_pD_p)C_p^T=0.
\end{align}
Solving the above expression for $X_1$ sequentially along each row gives
\begin{align*}
(X_1(i,:))_{p+1}=(E(i,:))_p({\rm diag}(W_1^2(i,1)\;W_1^2(i,2)\cdots W_1^2(i,k))+C_pC_p^T)^{-1},
\end{align*}
where $E_p=A_1\odot W_1\odot W_1+(A_2-B_pD_p)C_p^T$. Note that, for each row $X_1(i,:)$, if $\mathcal{L}_i={\rm diag}(W_1^2(i,1)\;W_1^2(i,2)\cdots W_1^2(i,k))+C_pC_p^T$ then the above system of equations are equivalent to solving a least squares solution of $\mathcal{L}_i(X_1(i,:))_{p+1}^T=(E(i,:))_p^T$ for each $i$.
Next we find $C_{p+1}$ by setting
\frac{\partial}{\partial C}F(X_1,C,B_p,D_p)|_{C = C_{p+1}}=0,
which implies
\begin{align}\label{C_p}
\end{align}
and consequently solving for $C_{p+1}$ gives (assuming $(X_1)_{p+1}$ is of full rank)
Similarly, $B_{p+1}$ satisfies
\begin{align}\label{B_p}
\end{align}
Solving (<ref>) for $B_{p+1}$ obtains (assuming $D_p$ is of full rank)
Finally, $D_{p+1}$ satisfies
\begin{align}\label{D_p}
-B_{p+1}^T(A_2-(X_1)_{p+1}C_{p+1} -B_{p+1}D_{p+1})=0,
\end{align}
and we can write (assuming $B_{p+1}$ is of full rank)
We arrive at the following algorithm.
$A=(A_1\;\;A_2) \in\mathbb{R}^{m\times n}$ (the given matrix); $W= (W_1\;\;\mathbbm{1})\in\mathbb{R}^{m\times n}$ (the weight), threshold $\epsilon>0$
not converged
$E_p=A_1\odot W_1\odot W_1+(A_2-B_pD_p)C_p^T$
$\displaystyle{(X_1(i,:))_{p+1}=(E(i,:))_p({\rm diag}(W_1^2(i,1)\;W_1^2(i,2)\cdots W_1^2(i,k))+C_pC_p^T)^{-1}}$
$(X_1)_{p+1}, (X_1)_{p+1}C_{p+1}+B_{p+1}D_{p+1}.$
WLR Algorithm
§.§ Complexity of the algorithm
In this section we discuss the runtime complexity of Algorithm 1 by making some simplifying assumptions. The update in Step 4 takes $O(mk(n-k))+O(m(r-k)(n-k))$ time. The matrix inversion in Step 6 for updating each row of $X_1$ takes $O(k^3)$ time and the total computational time for $m$ rows is $O(mk^3)+O(mk(2k-1))$. Indeed, the costly procedures in Step 7, 8, and 9 are the matrix inversions. The matrix product and inversion in Step 7 takes $O(k^2(m+k))$ time, where the inner matrix product takes $O(m(r-k)(n-k)+mk(n-k))$ and the final product takes $O(k^2(n-k))$ time. In Step 8, the inner matrix product takes $O(k(n-k)(r-k)+m(r-k)(n-k))$ time. The matrix product and inversion takes $O((r-k)^2(r+n-2k))$ time and the final product takes $O(m(r-k)^2)$ time.
Finally in Step 9, as before the main cost is due to the matrix product and inversion, which takes $O((r-k)^2(m+r-k))$ time. The matrix product in Step 9 takes $O((r-k)m(n-k))+O((r-k)^2(n-k))$ time. In summary, the total complexity of the algorithm is $O(mk^3+mnr)$.
§.§ Convergence analysis
Next we will discuss the convergence of our numerical algorithm. Since the objective function $F$ is convex only in each of the component $X_1,B,C,$ and $D$, it is hard to argue about the global convergence of the algorithm. In Theorem <ref> and <ref>, under some special assumptions when the limit of the individual sequence exists, we show that the limit points are going to be a stationary point of $F$. To establish our main convergence results in Theorem <ref> and <ref>, the following equality will be very helpful.
For a fixed $(W_1)_{ij}>0$ let $m_p=F((X_1)_p,C_p,B_p,D_p)$ for $p=1,2,\cdots$
\begin{align}\label{mj equality}
m_p-m_{p+1}=&\|((X_1)_p-(X_1)_{p+1})\odot W_1\|^2_F+\|((X_1)_p-(X_1)_{p+1})C_p\|^2_F\nonumber\\&+\|(X_1)_{p+1}(C_p-C_{p+1})\|^2_F
\end{align}
\begin{align}\label{update rule 1}
\left\{\begin{array}{ll}
\text{and,}\; F((X_1)_{p+1},C_{p+1},B_{p+1},D_{p})-m_{p+1}=d_4.
\end{array}\right.
\end{align}
\begin{align}\label{update rule calc}
d_1&=\|(A_1-(X_1)_{p})\odot W_1\|_F^2+\|A_2-(X_1)_pC_p-B_pD_p\|_F^2-\|(A_1-(X_1)_{p+1})\odot W_1\|_F^2\nonumber\\
&=\|(X_1)_{p}\odot W_1\|_F^2-\|(X_1)_{p+1}\odot W_1\|_F^2+\|(X_1)_{p}C_p\|_F^2-\|(X_1)_{p}C_{p+1}\|_F^2
+2\langle A_1\odot W_1\odot W_1,\nonumber\\&(X_1)_{p+1}-(X_1)_{p}\rangle-2\langle ((X_1)_{p}-(X_1)_{p+1})C_p,A_2-B_pD_p\rangle.
\end{align}
Note that,
\begin{align*}
(((X_1)_{p+1}-A_1)\odot W_1\odot W_1)((X_1)_{p}-(X_1)_{p+1})^T\\=(A_2-(X_1)_{p+1}C_p-B_pD_p)C_p^T((X_1)_{p}-(X_1)_{p+1})^T,
\end{align*}
as $(X_1)_{p+1}$ satisfies (<ref>).
This, together with (<ref>), will lead us to
\begin{align}\label{update rule calc 1}
d_1=\|((X_1)_p-(X_1)_{p+1})\odot W_1\|^2_F+\|((X_1)_p-(X_1)_{p+1})C_p\|^2_F.
\end{align}
Similarly we find
\begin{align}\label{D}
\left\{\begin{array}{ll}
\end{array}\right.
\end{align}
Combining them together we have the desired result.
From Theorem <ref> we know that the non-negative sequence $\{m_p\}$ is non-increasing. Therefore, $\{m_p\}$ has a limit.
Theorem <ref> implies a lot of interesting convergence properties of the algorithm. For example, we have the following estimates.
We have
* $m_p- m_{p+1}\geq \frac{1}{2}\|B_{p+1}D_{p+1}-B_pD_p\|_F^2$ for all $p.$
* $m_p- m_{p+1}\geq\|((X_1)_p-(X_1)_{p+1})\odot W_1\|^2_F$ for all $p$.
From (<ref>) we can write, for all $p$,
\begin{align*}
& m_p-m_{p+1}\\
&\geq \|(B_p-B_{p+1})D_p\|^2_F+\|B_{p+1}(D_p-D_{p+1})\|^2_F\\
\end{align*}
by parallelogram identity. Therefore,
\begin{align*}
& m_p-m_{p+1}\geq\frac{1}{2}\|B_{p+1}D_{p+1}-B_pD_p\|_F^2.
\end{align*}
This completes the proof of (i).
(ii). This follows immediately from (<ref>).
We now can state our main convergence results as a consequence of Theorem <ref> and Corollary <ref>.
* We have the following: $\sum_{p=1}^{\infty}\|B_{p+1}D_{p+1}-B_pD_p\|_F^2<\infty$, and
$\displaystyle{\sum_{p=1}^{\infty}\left(\|((X_1)_p-(X_1)_{p+1})\odot W_1\|_F^2\right)<\infty}$.
* If $\sum_{p=1}^{\infty}\sqrt{m_p- m_{p+1}}< +\infty,$ then $\displaystyle{\lim_{p\to\infty}B_pD_p}$ and $\displaystyle{\lim_{p\to\infty}(X_1)_p}$ exist. Furthermore if we write $L^*:=\displaystyle{\lim_{p\to\infty}B_pD_p}$ then $\displaystyle{\lim_{p\to\infty}B_{p+1}D_{p}=L^*}$ for all $p.$
(i). From Corollary <ref> (i)
we can write, for $N>0$,
and from Corollary <ref> (ii) and using $\lambda=\displaystyle{\min_{\substack{1\le i\le m\\1\le j\le k}}(W_1)_{ij}}$ we have
$$m_1-m_{N+1}\ge\displaystyle{\sum_{p=1}^{N}\left(\|((X_1)_p-(X_1)_{p+1})\odot W_1\|_F^2\right)}
\ge\displaystyle{\lambda^2\sum_{p=1}^{N}\|(X_1)_p-(X_1)_{p+1}\|_F^2}.$$
Now note that, by Remark <ref>, $\{m_p\}_{p=1}^{\infty}$ is a convergent sequence. Hence the result follows.
(ii). Again using Corollary <ref> (i) we can write, for $N>0$,
\begin{align*}
&\sum_{p=1}^{N}\sqrt{m_p- m_{p+1}}
\ge \frac{1}{\sqrt{2}}\sum_{p=1}^{N}\left(\|B_{p+1}D_{p+1}-B_pD_p\|_F\right),
\end{align*}
which implies $\sum_{p=1}^{\infty}(B_{p+1}D_{p+1}-B_pD_p)$ is convergent if $\sum_{p=1}^{\infty}\sqrt{m_p- m_{p+1}}< +\infty.$ Therefore, $\displaystyle{\lim_{N\to\infty}B_ND_N}$ exists. Similarly, we can conclude that $\displaystyle{\lim_{p\to\infty}(X_1)_p}$ exists.
Further, $\displaystyle{\lim_{p\to\infty}\|B_{p+1}D_{p+1}}-B_{p+1}D_p\|_F^2=0,$ as implied by (i) above.
Therefore $\displaystyle{\lim_{p\to\infty}B_{p+1}D_p}$ exists and is equal to $\displaystyle{\lim_{p\to\infty}B_pD_p}=L^*.$
This completes the proof.
From Corollary <ref>, we can only prove the convergence of the sequence $\{B_pD_p\}$ but not of $\{B_p\}$ and $\{D_p\}$ separately. We next establish the convergence of $\{B_p\}$ and $\{D_p\}$ with stronger assumption. Consider the situation when
\begin{align}\label{mj inequality}
\sum_{p=1}^{\infty}\sqrt{m_p- m_{p+1}}< +\infty.
\end{align}
Assume (<ref>) holds.
* If $B_p$ is of full rank and $B_p^TB_p\ge\gamma I_{r-k}$ for large $p$ and some $\gamma>0$ then $\displaystyle{\lim_{p\to\infty}D_p}$ exists.
* If $D_p$ is of full rank and $D_pD_p^T\ge\delta I_{r-k}$ for large $p$ and some $\delta>0$ then $\displaystyle{\lim_{p\to\infty}B_p}$ exists.
* If $X_1^*:=\displaystyle{\lim_{p\to\infty}(X_1)_{p}}$ is of full rank, then $C^*:=\displaystyle{\lim_{p\to\infty}C_p}$ exists. Furthermore, if we write $L^*=B^*D^*,$ for $B^*\in\mathbb{R}^{m\times(r-k)}, D^*\in\mathbb{R}^{(r-k)\times(n-k)}$, then $({X}_1^*,C^*,B^*,D^*)$ will be a stationary point of $F$.
(i). Using (<ref>) we have, for $N>0$,
\begin{align*}
\sum_{p=1}^{N}\sqrt{m_p-m_{p+1}}&\geq \sum_{p=1}^{N}\|B_{p+1}(D_p-D_{p+1})\|_F\\
&=\sum_{p=1}^{N }\sqrt{{\rm tr}[(D_p-D_{p+1})^TB_{p+1}^TB_{p+1}(D_p-D_{p+1})]},
\end{align*}
where ${\rm tr}(X)$ denotes the trace of the matrix $X$.
Note that $B_p^TB_p\ge\gamma I_{r-k},$ and so we obtain
\begin{align*}
\sum_{p=1}^{N}\sqrt{m_p-m_{p+1}}\geq\sqrt{\gamma}\sum_{p=1}^{N}\|D_p-D_{p+1}\|_F,
\end{align*}
which implies $\sum_{p=1}^{\infty}(D_p-D_{p+1})$ is convergent if (<ref>) holds. Therefore $\displaystyle{\lim_{N\to\infty}D_N}$ exists.
Similarly we can prove $(ii)$.
(iii). Note that, from (<ref>) we have, for $N>0$,
\begin{align*}
\sum_{p=1}^{N}\sqrt{m_p-m_{p+1}}\geq& \sum_{p=1}^{N}\|(X_1)_{p+1}(C_p-C_{p+1})\|_F\\
=&\sum_{p=1}^{N}\sqrt{{\rm tr}[(C_p-C_{p+1})^T(X_1)_{p+1}^T(X_1)_{p+1}(C_p-C_{p+1})]}.
\end{align*}
If $X_1^*:=\displaystyle{\lim_{p\to\infty}(X_1)_{p}}$ is of full rank, it follows that, for large $p$, $(X_1)_{p+1}^T(X_1)_{p+1}\geq\eta I_k,$ for some $\eta> 0.$ Therefore, we have
\begin{align*}
\end{align*}
Following the same argument as in the previous proof we can conclude $\displaystyle{\lim_{p\to\infty}C_p}=C^*$ exists if (<ref>) holds. Recall from (<ref>-<ref>), we have,
\begin{align*}\label{stationary point}
\left\{\begin{array}{ll}
&((X_1)_{p+1}-A_1)\odot W_1\odot W_1-(A_2-(X_1)_{p+1}C_p-B_pD_p)C_p^T=0,\\
& B_{p+1}^T(A_2-(X_1)_{p+1}C_{p+1}-B_{p+1}D_{p+1})=0.
\end{array}\right.
\end{align*}
Taking limit as $p\to\infty$ in above we have
\begin{align*}
\left\{\begin{array}{ll}
\frac{\partial}{\partial X_1}F({X}_1^*,C^*,B^*,D^*)&=(X_1^*-A_1)\odot W_1\odot W_1+(B^*D^*+X_1^*C^*-A_2){C^*}^{T}=0,\\
\frac{\partial}{\partial C}F({X}_1^*,C^*,B^*,D^*)&={{X}_1^*}^T(A_2-{X}_1^*{C}^*-B^*D^*)=0,\\
\frac{\partial}{\partial B}F({X}_1^*,C^*,B^*,D^*)&=(A_2-{X}_1^*C^*-B^*D^*){D^*}^T=0,\\
\frac{\partial}{\partial D}F({X}_1^*,C^*,B^*,D^*)&={B^*}^T(A_2-{X}_1^*C^*-B^*D^*)=0.
\end{array}\right.
\end{align*}
Therefore, $({X}_1^*,C^*,B^*,D^*)$ is a stationary point of $F$. This completes the proof.
§ NUMERICAL RESULTS
In this section we will demonstrate numerical results of our weighted rank constrained algorithm and show the convergence to the solution given by Golub, Hoffman, and Stewart when $W_1\to\infty$ as predicted by our theorems in Section 2. All experiments were performed on a computer with 3.1 GHz Intel Core i7-4770S processor and 8GB memory.
§.§ Experimental setup
To perform our numerical simulations we construct two different variety of test matrix $A$. The first series of experiments were performed to demonstrate the convergence of the algorithm proposed in Section 4 and to validate the analytical result in Theorem <ref>. To this end, we performed our experiments on three full rank synthetic matrices $A$ of size $300\times300$, $500\times500$, and $700\times700$ respectively. We constructed $A$ as low rank matrix plus Gaussian noise such that $A=A_0+\alpha*E_0$, where $A_0$ is the low-rank matrix, $E_0$ is the noise matrix, and $\alpha$ controls the noise level. We generate $A_0$ as a product of two independent full-rank matrices of size $m\times r$ whose elements are independent and identically distributed (i.i.d.) $\mathcal{N}(0,1)$ random variables such that ${\rm r}(A_0)=r$. We generate $E_0$ as a noise matrix whose elements are i.i.d. $\mathcal{N}(0,1)$ random variables as well. In our experiments we choose $\displaystyle{\alpha=0.2\max_{i,j}(A_{ij})}$. The true rank of the test matrices are 10% of their original size but after adding noise they become full rank.
To compare the performance of our algorithm with the existing weighted low-rank approximation algorithms, we are interested in which $A$ has a known singular value distribution. To address this, we construct $A$ of size $50\times 50$ such that ${\rm r}(A)=30$. Note that, $A$ has first 20 singular values distinct, and last 10 singular values repeated. It is natural to consider the cases where $A$ has large and small condition number. That is, we demonstrate the performance of WLR in two different cases: condition number $\kappa(A)$ of $A$: (i) small and (ii) large, where $\kappa(A)=\frac{\sigma_{max}}{\sigma_{min}}.$
§.§ Implementation details
Let $A_{WLR}=(X_1^*\;\; X_1^*C^*+B^*D^*)$ where $(X_1^*, C^*$, $B^*, D^*)$ be a solution to (<ref>). We denote $(A_{WLR})_p$ as our approximation to $A_{WLR}$ at $p$th iteration. Recall that $(A_{WLR})_p=((X_1)_{p}\;\; (X_1)_{p}C_p+B_pD_p).$ We denote $\|(A_{WLR})_{p+1}-(A_{WLR})_{p}\|_F=Error_p$ and use $\frac{Error_p}{\|(A_{WLR})_{p}\|_F}$ as a measure of the relative error. For a threshold $\epsilon>0$ the stopping criteria of our algorithm at the $p$th iteration is $Error_p<\epsilon$ or $\frac{Error_p}{\|(A_{WLR})_{p}\|_F}<\epsilon$ or if it reaches the maximum iteration. The algorithm performs the best when we initialize $X_1$ and $D$ as random normal matrices and $B$ and $C$ as zero matrices. Throughout this section we set $r$ as the target low rank and $k$ as the total number of columns we want to constrain in the observation matrix. The algorithm takes approximately $35.9973$ seconds on an average to perform 2000 iterations on a $300\times 300$ matrix for fixed $r, k$, and $\lambda$.
§.§ Experimental results on algorithm in section 4.1
We first verify our implementation of the algorithm for computing $A_{WLR}$ for fixed weights. Throughout this subsection we set the target low-rank $r$ as the true rank of the test matrix and $k=0.5r$. To obtain the accurate result we run every experiment 25 times with random initialization and plot the average outcome in each case.
A threshold equal to $2.2204\times10^{-16}$ (“machine $\epsilon$”) is set for the experiments in this subsection.
For Figure 5.1, we consider a nonuniform weight with entries in $W_1$ randomly chosen from the interval $[\lambda,\zeta]$, where $\min_{\substack{1\le i\le m\\1\le j\le k}}(W_1)_{ij}=\lambda$ and $\max_{\substack{1\le i\le m\\1\le j\le k}}(W_1)_{ij}=\zeta$ in the first block $W_1$ and $W_2=\mathbbm{1}$ and plot iterations versus relative error. Relative error is plotted in logarithmic scale along $Y$-axis.
Iterations vs Relative error: (a) $\lambda=25,\zeta=75$; (b) $\lambda=100,\zeta=150$.
Iterations vs $\frac{\|A_{WLR}(j)-X_{SVD}\|_F}{\|X_{SVD}\|_F}$: (a) $\lambda = 50$; (b) $\lambda = 200.$
Next, we consider a uniform weight in the first block $W_1$ and $W_2=\mathbbm{1}$. Recall that, in this case the solution to problem (<ref>) can be given in closed form. That is, when $W_1=\lambda\mathbbm{1}$, the rank $r$ solutions to (<ref>) are $X_{SVD}=[\frac{1}{\lambda}\tilde{X}_1\;\;\tilde{X}_2]$, where $[\tilde{X}_1\;\;\tilde{X}_2]$ is obtained in closed form using a SVD of $[\lambda A_1\;\;A_2]$. In Figure 5.2, we plot iterations versus $\frac{\|A_{WLR}(j)-X_{SVD}\|_F}{\|X_{SVD}\|_F}$ in logarithmic scale.
From Figures 5.1 and 5.2 it is clear that the algorithm in Section 4.1 converges. Even for the bigger size matrices the iteration count is not very high to achieve the convergence.
§.§ Numerical results supporting Theorem <ref>
We now demonstrate numerically the rate of convergence as stated in Theorem 2.1 when the block of weights in $W_1$ goes to $\infty$ and $W_2=\mathbbm{1}$. First we use an uniform weight $W_1=\lambda\mathbbm{1}$ and $W_2=\mathbbm{1}$. The algorithm in Section 4 is used to compute $A_{WLR}$ and SVD is used for calculating $A_G$, the solution to (<ref>) when $A=(A_1\;\;A_2)$. We plot $\lambda$ vs. $\lambda\|A_G-A_{WLR}\|_F$ where $\lambda\|A_G-A_{WLR}\|_F$ is plotted in logarithmic scale along $Y$-axis. We run our algorithm 20 times with the same initialization and plot the average outcome. A threshold equal to $10^{-7}$ is set for the experiments in this subsection. For Figure 5.3 we set $\lambda=[1:50:1000]$.
$\lambda$ vs. $\lambda\|A_G-A_{WLR}\|_F$: (a) $(r,k)=(70,50)$, (b) $(r,k)=(60,40)$.
$\lambda$ vs. $\lambda\|A_G-A_{WLR}\|_F$: (a) $(r,k)=(70,50)$, (b) $(r,k)=(60,40)$.
The plots indicate for an uniform $\lambda$ in $W_1$ the convergence rate is at least $O(\frac{1}{\lambda}), \lambda\to\infty.$ Next we consider a nonuniform weight in the first block $W_1$ and $W_2=\mathbbm{1}$. We consider $(W_1)_{ij}\in[2000,2020], [2050,2070]$ and so on. For Figure 5.4, $\lambda\|A_G-A_{WLR}\|_F$ (recall $\min_{\substack{1\le i\le m\\1\le j\le k}}(W_1)_{ij}=\lambda$) is plotted in regular scale along $Y$-axis. The curves in figure 5.4 are not always strictly decreasing but it is encouraging to see that they stay bounded. Figures 5.3 and 5.4 provide numerical evidence in supporting Theorem <ref>. As established in Theorem <ref> the above plots demonstrate the convergence rate is at least $O(\frac{1}{\lambda}), \lambda\to\infty.$
§.§ Comparison with state-of-the-art general weighted algorithms
In this section, we compare the performance of our special weighted algorithm on synthetic data with the standard weighted total alternating least squares (WTALS) method proposed in <cit.> and the expectation maximization (EM) method proposed by Srebro and Jaakkola <cit.>. The existing algorithms are for general weighted case but for our purpose we consider partial weighting in them. Additionally, we compare the performance of our algorithm with the standard alternating least squares, WTALS, and the EM method <cit.> for $k=0$ case.
For the numerical experiments in this section, we are interested to see how the distribution of the singular values affects the performance of our algorithm compare to other state-of-the-art algorithms.
§.§.§ Performance compare to other weighted low-rank approximation algorithms
Comparison of WLR with other methods when $\kappa(A)$ small: (a) $r$ versus RMSE, (b) $r$ versus time. We have $\kappa(A)=1.3736$, $r=[20:1:30]$, $k=10$, and $(W_1)_{ij}\in[50,1000]$.
The weights in the first block are chosen randomly from a large interval. We set $(W_1)_{ij}\in[50,1000]$ and $W_2=\mathbbm{1}$. For WTALS, as specified in the software package, we consider <cit.>. For EM, we choose , and for WLR, we set . For the performance measure, we use the standard root mean square error (RMSE) which is $\|A-\hat{A}\|_F/\sqrt{mn}$, where $\hat{A}\in\mathbb{R}^{m\times n}$ is the low-rank approximation of $A$ obtained by using different weighted low-rank approximation algorithm. The code for the EM method is written by the authors following the algorithm proposed in <cit.>. For computational time of WLR and EM, the authors do not claim the optimized performance of their codes. However, the initialization of $X$ plays a crucial role in promoting convergence of the EM method to a global, or a local minimum, as well as the speed with which convergence is attained.
Comparison of WLR with other methods when $\kappa(A)$ large: (a) $r$ versus RMSE, (b) $r$ versus time. We have $\kappa(A)= 5.004\times10^3$, $r=[20:1:30]$, $k=10$, and $(W_1)_{ij}\in[50,1000]$.
Comparison of WLR with other methods for $k=0$ and $\kappa(A)$ small: (a) $r$ versus RMSE, (b) $r$ versus time. We have $\kappa(A)=1.3736$, $r=[20:1:30]$.
To implement the EM method, as mentioned in <cit.>, first we rescale the weight matrix to $W_{EM}=\frac{1}{\max_{ij}(W_1)_{ij}}(W_1\;\;\mathbbm{1})$. For a given threshold of weight bound $\epsilon_{EM}$, we initialize $X$ to a zero matrix if $\min_{ij}(W_{EM})_{ij}\le \epsilon_{W_{EM}}$, otherwise we initialize $X$ to $A$. Initialization for WLR is same as specified in Section 5.2. To obtain the accurate result we run each experiment 10 times and plot the average outcome. Both RMSE and computational time are plotted in logarithmic scale along $Y$-axis. Figure 5.5 and 5.6 indicate that WLR is more efficient in handling bigger size matrices than WTALS <cit.> with the comparable performance measure. This can be attributed by the fact that WTALS uses a weight matrix of size $mn\times mn$ for the given input size $m\times n$, which is both memory and time inefficient. On the other hand, Figure 5.5 and 5.6 support the fact that as mentioned in <cit.>, the EM method is computationally effective, however in some cases might converge to a local minimum instead of global.
§.§.§ Performance comparison for $k=0$ (Alternating Least Squares)
For $k=0$ we set the weight matrix as $W=\mathbbm{1}$ for all weighted low-rank approximation algorithm. Moreover, we include the classic alternating least squares algorithm to compare between the accuracy of the methods. As specified in Section 5.5.1, the stopping criterion for all weighted low-rank algorithms are kept the same and RMSE is used for performance measure. We run each experiment 10 times and plot the average outcome. Figure 5.7 and 5.8 indicate that WLR has comparable performance. However, the standard ALS, WTALS, and EM method is more efficient than WLR, as for $W=\mathbbm{1}$ case, each method uses SVD to compute the solution.
Comparison of WLR with other methods for $k=0$ and $\kappa(A)$ large: (a) $r$ versus RMSE, (b) $r$ versus time. We have $\kappa(A)=5.004\times10^3$, $r=[20:1:30]$.
§.§.§ Performance compare to constrained low-rank approximation of Golub-Hoffman-Stewart
As mentioned in our analytical results, one can expect, with appropriate conditions, the solutions to (<ref>) will converge and the limit is $A_G$, the solution to the constrained low-rank approximation problem by Golub-Hoffman-Stewart. We now show the effectiveness of our special weighted algorithm compare to other state-of-the-art weighted low rank algorithms when $(W_1)_{ij}\to \infty,$ and $W_2=\mathbbm{1}$. In this section, the weights in $W_1$ are chosen to be large to show effectiveness of our algorithm for large weighted case in the first block. SVD is used for calculating $A_G$, the solution to (<ref>), when $A=(A_1\;\;A_2)$, for varying $r$ and fixed $k$. Considering $A_G$ as the true solution we use the RMSE measure $\|A_G-\hat{A}\|_F/\sqrt{mn}$ as the performance metric, where $\hat{A}\in\mathbb{R}^{m\times n}$ is the low-rank approximation of $A$ obtained by different weighted low-rank algorithm. From Figure 5.9 it is evident that WLR has the superior performance compare to the other state-of-the-art general weighted low-rank approximation algorithms when $W_1\to\infty$.
$r$ vs $\|A_G-\hat{A}\|_F/\sqrt{mn}$ for different methods, $(W_1)_{ij}\in[500,1000], W_2=\mathbbm{1}$, $r=10:1:20$, and $k=10$: (a) $\kappa(A)$ small, (b) $\kappa(A)$ large.
To conclude, WLR has comparable or superior performance compare to the existing general weighted low-rank approximation algorithms for the special case of weight with fairly less computational time. Even when the columns of the given matrix are not constrained, that is $k=0$, its performance is comparable to the standard ALS. Additionally, WLR and EM method can easily handle bigger size matrices and easier to implement for real world problems. On the other hand, WTALS requires more computational time and is not memory efficient to handle large scale data (see table 5.1). Another important feature of our algorithm is that it does not assume any particular condition about the matrix $A$ and performs equally well in every occasion.
Average computation time (in seconds) for each algorithm to converge to $A_G$
$\kappa(A)$ WLR EM WTALS
1.3736 6.5351 6.1454 205.1575
$5.004\times10^3$ 8.8271 8.1073 107.0353
§ ACKNOWLEDGMENTS
We would like to thank the anonymous referees and the associate editor Dr. Ivan Markovsky for providing many useful references, and for their valuable comments and suggestions which improved the presentation and results of this paper. We would also like to thank Dr. Afshin Dehghan, at the Center for Research in Computer Vision, University of Central Florida for his invaluable comments in the numerical results section.
golub G. H. Golub, A. Hoffman, and G. W. Stewart,
A generalization of the Eckart-Young-Mirsky matrix approximation theorem, Linear Algebra and its Applications,
88-89 (1987), pp. 317–327.
A. Dutta, X. Li, B. Gong, and M. Shah, Weighted singular value thresholding and its applications to background estimation, submitted.
A. Dutta and X. Li, Background estimation from video sequences using weighted low-rank approximation of matrices, submitted.
I. T. Jolliffee, Principal Component Analysis,
Second edition, Springer-Verlag, 2002.
Z. Lin, M. Chen, and Y. Ma,
The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices,
arXiv preprint arXiv1009.5055, 2010.
Per-${\rm \AA}$ke Wedin, Perturbation bounds in connection with singular value decomposition,
BIT Numerical Mathematics, 12-1(1972), pp. 99–111.
A. Dutta and X. Li, A Fast Algorithm for a Special Weighted Low Rank
Approximation, submitted.
C. Eckart and G. Young, The approximation of one matrix by another of lower rank,
Psychometrika, 1-3 (1936), pp. 211–218.
N. Srebro and T. Jaakkola, Weighted low-rank approximations,
20th International Conference on Machine Learning (2003), pp. 720–727.
G.W. Stewart, A second order perturbation expansion for small singular values,
Linear Algebra and its Applications, 56 (1984), pp. 231–235.
A. Dutta, Weighted low-rank approximation of matrices: some analytical and numerical aspects, Ph.D. dissertation, Department of Mathematics, University of Central Florida, 2016.
C. Davis and W. Kahan, The rotation of eigenvectors by a perturbation III.,
SIAM Journal on Numerical Analysis, 7 (1970), pp. 1–46.
T. Okatani and K. Deguchi, On the Wiberg algorithm for matrix factorization in the presence of missing components,
International Journal of Computer Vision, 72-3 (2007), pp. 329–337.
T. Wiberg, Computation of principal components when data are missing,
In Proceedings of the Second Symposium of Computational Statistics (1976), pp. 229–336.
N. Srebro, J. D. M. Rennie, and T. S. Jaakola, Maximum-margin matrix factorization, In Proceedings of
Advances in Neural Information Processing Systems, 18 (2005), pp. 1329–1336.
T. Hastie, R. Mazumder, J. Lee, and R. Zadeh, Matrix completion and low-rank SVD via fast alternating least squares, arXiv preprint arXiv1410.2596, 2014.
M. Udell, C. Horn, R. Zadeh, and S. Boyd, Generalized low-rank models,
arXiv preprint arXiv:1410.0342, 2014.
J. Hansohm, Some properties of the normed alternating least squares (ALS) algorithm,
Optimization, 19-5 (1988), pp. 683–691.
A. M. Buchanan and A. W. Fitzgibbon, Damped Newton algorithms for matrix factorization with missing data, In Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2 (2005), pp. 316–322.
H. Liu, X. Li, and X. Zheng, Solving non-negative matrix factorization by alternating least squares with a modified strategy, Data Mining and Knowledge Discovery, 26-3 (2012), pp. 435–451.
I. Markovsky, J. C. Willems, B. De Moor, and S. Van Huffel, Exact and approximate
modeling of linear systems: a behavioral approach, Number 11 in Monographs on Mathematical Modeling and Computation, SIAM, 2006.
I. Markovsky, Low-rank approximation: algorithms, implementation, applications, Communications and Control Engineering. Springer, 2012.
S. Van Huffel and J. Vandewalle, The total least squares problem: computational aspects and analysis, Frontiers in Applied Mathematics 9 , SIAM, Philadelphia, 1991.
K. Usevich and I. Markovsky, Variable projection methods for affinely structured
low-rank approximation in weighted 2-norms, Journal of Computational and Applied Mathematics 272 (2014), pp. 430–448.
G.W. Stewart, On the asymptotic behavior of scaled singular value and QR decompositions,
Mathematics of Computation, 43-168 (1984), pp. 483–489.
J. H. Manton, R. Mehony, and Y. Hua, The geometry of weighted low-rank approximations,
IEEE Transactions on Signal Processing, 51-2 (2003), pp. 500–514.
W. S. Lu, S. C. Pei, and P. H. Wang, Weighted low-rank approximation of general complex matrices and its application in the design of 2-D digital filters, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 44-7 (1997), pp.650–655.
D. Shpak, A weighted-least-squares matrix decomposition with application to the design of 2-D digital filters, In Proceedings of IEEE 33rd Midwest Symposium on Circuits and Systems, (1990), pp. 1070–1073.
K. Usevich and I. Markovsky, Optimization on a Grassmann manifold with application to system identification, Automatica, 50-6 (2014), pp. 1656–1662.
S. Brutzer, B. Höferlin, and G. Heidemann, Evaluation of background subtraction techniques for video surveillance, IEEE Computer Vision and Pattern Recognition (2011), pp. 1937–1944.
J. Wright, Y. Peng, Y. Ma, A. Ganseh, and S. Rao, Robust principal component analysis: exact recovery of corrupted low-rank matrices by convex optimization, Advances in Neural Information Processing systems 22, (2009), pp. 2080–2088.
G.W. Stewart and J. Sun, Matrix Perturbation Theory, Academic Press, Boston, 1990.
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1511.01092
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$\dagger$Based in part on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.
$^{1}$ Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
$^{2}$ California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA
$^{3}$Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, the University of Tokyo, Kashiwa, Japan 277-8583 (Kavli IPMU, WPI)
$^{4}$National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
We investigate the Eddington ratio distribution of X-ray selected broad-line active galactic nuclei (AGN) in the redshift range $1.0<z<2.2$, where the number density of AGNs peaks. Combining the optical and Subaru/FMOS near-infrared spectroscopy, we estimate black hole masses for broad-line AGNs in the Chandra Deep Field-South (CDF-S), Extended Chandra Deep Field-South (E-CDF-S), and the XMM-Newton Lockman Hole (XMM-LH) surveys. AGNs with similar black hole masses show a broad range of AGN bolometric luminosities, which are calculated from X-ray luminosities, indicating that the accretion rate of black holes is widely distributed. We find that a substantial fraction of massive black holes accreting significantly below the Eddington limit at $z\lesssim2$, in contrast to what is generally found for luminous AGNs at high redshift. Our analysis of observational selection biases indicates that the “AGN cosmic downsizing" phenomenon can be simply explained by the strong evolution of the co-moving number density at the bright end of the AGN luminosity function, together with the corresponding selection effects. However, it might need to consider a correlation between the AGN luminosity and the accretion rate of black holes that luminous AGNs have higher Eddington ratios than low-luminosity AGNs in order to understand the relatively small fraction of low-luminosity AGNs with high accretion rates in this epoch. Therefore, the observed downsizing trend could be interpreted as massive black holes with low accretion rates, which are relatively fainter than less massive black holes with efficient accretion.
§ INTRODUCTION
Disentangling the origin and the mass accretion history of black holes is one of the most outstanding issues for understanding the fundamental processes of galaxy formation and evolution. Observations have shown that supermassive black holes are tightly linked with their host galaxies, as revealed by correlations between the black hole mass and the bulge stellar mass, i.e., ${\rm M_{BH}-M_{stellar}}$ relation <cit.> and the velocity dispersion, i.e., ${\rm M_{BH}-\sigma}$ relation <cit.>. Furthermore, it has been widely accepted that the growth of active galactic nuclei (AGN) and the star formation history undergo very similar evolutionary behavior through cosmic time, where the peaks of most luminous AGNs and powerful star-forming galaxies occur at a similar cosmic epoch ($z=2-3$) with a dramatic decline towards low redshift, while the moderate-luminosity AGNs and the bulk of star-forming galaxies peak at lower redshift ($z\lesssim1$) (see e.g. ). This seems to imply that the interaction between the nuclear activity and the star formation in galaxies plays a crucial role in the evolution of black holes and galaxies over cosmic time.
The AGN luminosity function and its evolution are key observational properties for understanding the accretion history onto the black holes. Observational studies have revealed a “cosmic downsizing” or “anti-hierarchical” phenomenon in the black hole growth, which means that the characteristic luminosity of AGNs decreases with time. The co-moving number density of luminous AGNs peaks at higher redshift ($z\sim2$) than moderate-luminosity AGNs, which peaks at $z<1$ <cit.>. This AGN cosmic downsizing trend is seen across a wide range of the electromagnetic spectrum in X-ray, optical, infrared, and radio wavebands <cit.>. If AGN luminosity would strictly correlate with black hole mass, this finding would imply that more massive black holes formed before lower-mass black holes, which is in apparent contradiction to the currently favored hierarchical structure formation paradigm based on the standard cold dark matter model. In the hierarchical framework, more massive halos grow over time hierarchically via subsequent merging and smooth accretion among low mass halos.
The AGN cosmic downsizing, however, is observed in luminosity, and thus the downsizing phenomenon can also be interpreted assuming a relationship between the AGN luminosity and the black hole mass as a function of redshift. Black holes are assumed to undergo several episodes of a significant gas accretion with complex hydrodynamic and magnetic processes, along with relativistic effects during which this accretion powers AGNs (e.g. ). The most luminous AGNs are interpreted as results of major mergers. A substantial starburst occurs as a result of major mergers, and some of the gas eventually reaches the black hole at the center of a galaxy, triggering the AGN activity (see e.g. ). On the other hand, moderate-luminosity AGNs are suggested to be products of modest accretion, in which case the gas accretion via internal, secular processes trigger the AGN activity (e.g. ). An AGN with black hole mass of M$_{\rm BH}$ can produce the maximum luminosity via the Eddington limit (L$_{\rm Edd}$) at which the radiation pressure by the accretion of the infalling matter balances the gravitational attraction of the black hole for spherically symmetric time-invariant accretion. By estimating the Eddington ratios, the ratio of the AGN bolometric luminosity and the Eddington luminosity (L$_{\rm bol}$/L$_{\rm Edd}$), it can be determined whether the accretion rate of black holes can change over cosmic time. One might have expected a correlation between black hole masses and AGN bolometric luminosities, but if there is a range in accretion rates and/or efficiencies, the relation will be weaker. Thus, in order to investigate the observed downsizing trend in black hole growth, it is important to explore the efficiency of gas accretion during the active phases of black holes. Therefore, the black hole mass and the bolometric luminosity are the key parameters to understanding the evolutionary picture for AGNs.
Large, modern photometric and spectroscopic surveys open up a new regime for studying a large sample of AGNs (e.g. Sloan Digital Sky Survey (SDSS), ). Many efforts have been made to describe the properties of thousands of AGNs (e.g. ). In previous studies, the Eddington ratio has been assumed to be close to the Eddington limit regardless of redshift and luminosities. <cit.> suggest that the Eddington ratios of local black holes are in the range between 0.1 and 1.7, suggesting that black hole growth takes place during luminous accretion phases close to the Eddington limit at high redshift. <cit.> present that the AGN population is dominated by narrowly distributed near-Eddington accretion rate objects, with a median of 0.1 and a dispersion of 0.3 dex, also suggesting that supermassive black holes gain most of their mass while radiating close to the Eddington limit. However, it is difficult to draw any conclusions about the underlying distribution of the Eddington ratio because the shallowness of the large wide area surveys imposes severe restrictions on the combinations of AGN luminosities and black hole masses that are observable, especially at $z>1$. Recent studies have shown that there is a wide spread in the range of the Eddington ratios (e.g. ). <cit.> find that the Eddington ratio increases with redshift for AGNs at any given black hole masses. They also show that the Eddington ratio increases with AGN bolometric luminosity, while no clear evolution with redshift is seen. A wide range of Eddington ratios indicates that their luminosity is not directly related to the black hole mass. Therefore, it is necessary to consider a wide range of Eddington ratios with respect to the AGN luminosity and the black hole mass in order to understand the accretion growth history of the black holes.
Unfortunately, the detailed follow-up study in the redshift interval $z=1-2$, where the AGN downsizing appears, has been difficult because of the lack of emission line diagnostics in the optical wavelength range, which is often referred to as the redshift desert. The strong Balmer emission lines, and , are redshifted to 13126Å and 9722Å at z=1, respectively. The advent of the sensitive near-infrared (NIR) spectrograph Fiber Multi Object Spectrograph (FMOS) on the Subaru telescope finally enables us to determine the black hole mass in the key redshift interval $z=1-2$ using the Balmer lines that are the same lines for which the black hole masses are calibrated at low redshift. This redshift range is of particular interest, as it is the epoch that a significant part of the accretion growth of black holes takes place, where the AGN density peaks and where optical spectroscopy cannot easily determine the redshifts and properties of many of the AGNs.
In this paper, we investigate the Eddington ratios for X-ray selected broad-line AGNs in the Chandra Deep Field-South (CDF-S), Extended Chandra Deep Field-South (E-CDF-S), and the XMM-Newton Lockman Hole (XMM-LH) fields. Absorption-corrected X-ray luminosities together with bolometric corrections will allow an estimate of bolometric luminosities of AGNs. The advantage of using X-ray luminosities to derive AGN bolometric luminosities is that they are relatively less affected by the presence of obscuration and contamination effects from the host galaxy. We determine black hole masses using Subaru/FMOS NIR spectroscopic observations and available optical spectroscopy from the literature. We also investigate the possible biases due to systematics and selection effects on the observed data.
Throughout this paper we assume a $\Lambda$CDM cosmology with $\Omega_{m} = 0.3$, $\Omega_{\Lambda}=0.7$ and H$_{0} = 70$ km s$^{-1}$ Mpc$^{-1}$.
Left: The absorption-corrected 0.5-8 keV X-ray luminosity versus redshift (spectroscopic or photometric) from the X-ray surveys as labeled. Right: Survey area coverage as a function of X-ray flux. The black line represent the total combined area of all surveys.
§ SAMPLE SELECTION
An X-ray survey is practically the most efficient way of finding AGNs over a wide range of luminosities and redshifts. The deep surveys with Chandra (i.e., Chandra Deep Field-North and Chandra Deep Field-South; see ) and XMM-Newton (i.e., Lockman Hole; see ) allow us to detect a fair sample of low-luminosity (${\rm 42<log~L_{X}<44}$) AGNs out to $z\sim5$, providing a unique opportunity of studying AGN evolution. Besides, these fields are the best window for the deepest and cleanest images at a variety of wavelengths due to the remarkably low Galactic line-of-sight Hi column density (i.e., ${\rm N_{H}=8.8\times10^{19}~cm^{-2}}$ for CDF-S; , ${\rm N_{H}=5.7\times10^{19}~cm^{-2}}$ for XMM-LH; ).
We start by selecting a sample of AGNs based on comprehensive catalogs of X-ray sources observed in the CDF-S, E-CDF-S, and XMM-LH fields, described below.
§.§ Chandra Deep Field South
The catalog for the 4 Ms CDF-S, which is the deepest Chandra survey covering an area of 464.5 arcmin$^{2}$, contains 740 X-ray sources providing the most sensitive $0.5-8$ keV view of the distant universe <cit.>. The survey reaches flux limits of ${\rm 3.2\times10^{-17},~9.1\times10^{-18},~and~5.5\times10^{-17}~erg~cm^{-2}~s^{-1}}$ for the full ($0.5-8$ keV), soft ($0.5-2$ keV), and hard ($2-8$ keV) bands, respectively. 674 out of the 740 main-catalog sources have either spectroscopic or photometric redshifts, yielding an overall redshift completeness of $\sim$91%.
In addition to the 4Ms CDF-S point source catalog, we make use of the E-CDF-S observations that have been analyzed and cataloged by <cit.> and <cit.>, providing a sample of 762 distinct X-ray point sources with either spectroscopic or photometric redshifts. Of the 762 E-CDF-S main catalog sources, 523 sources were used since 239 sources were also present in the 4 Ms CDF-S catalog. We have an excellent redshift completeness of $\sim$95% (498/523). This survey reaches sensitivity limits of ${\rm 1.1\times10^{-16}~and~6.7\times10^{-16}~ergs~cm^{-2}~s^{-1}}$ for the soft ($0.5-2.0$ keV) and hard ($2-8$ keV) bands, respectively.
§.§ XMM-Newton Lockman Hole
The catalog of the 409 XMM-LH X-ray sources is presented in <cit.>, with sensitivity limits of ${\rm 1.9\times10^{-16},~9\times10^{-16},~and~1.8\times10^{-15}~erg~cm^{-2}~s^{-1}}$ in the soft ($0.5-2$ keV), hard ($2-10$ keV), and very hard ($5-10$ keV) bands, respectively. <cit.> provide spectroscopic or photometric redshifts for the XMM-LH X-ray sources. There is a reasonably high redshift completeness with 92% (376 out of 409). Although the sensitivity limit of the XMM-LH survey is much higher than that of the Chandra survey, the larger field of view of XMM-LH (${\rm 25\times25~arcmin^{2}}$) offers a significant sample of bright AGNs while the CDF-S observation provides the fainter tail of AGNs.
§.§ X-ray selected AGN Sample
We generate a total of 1548 X-ray selected AGNs which have reliable spectroscopic or photometric redshift identifications from the X-ray catalogs containing a total of 1672 X-ray sources. We show the X-ray sources with either spectroscopic or photometric redshift from the X-ray surveys as labeled in Figure 1. The absorption-corrected $0.5-8$ keV X-ray luminosity of AGNs as a function of redshift (spectroscopic or photometric) is shown in the left panel of Figure 1. The luminosity of AGNs in the redshift range $1.0<z<2.2$ is distributed between ${\rm L_{0.5-8~keV} = 10^{42.5}~and~10^{45.5} erg/s}$. In the right panel, we show the sky coverage for the individual surveys and the total sample used. As shown, the total sky area is ${\rm \sim0.7~deg^{2}}$ with the narrow deep CDF-S field and the shallower wide-area E-CDF-S and XMM-LH surveys. The deep CDF-S survey improves the AGN sample at low luminosities, while the E-CDF-S and XMM-LH surveys, of shallower depth but of wider area, effectively supply the more luminous AGNs.
§ SPECTROSCOPIC DATA
§.§ Subaru/FMOS Near-Infrared Observations
We performed NIR spectroscopic observations for the AGN sources with the FMOS <cit.> high-resolution spectrographs on the Subaru telescope. FMOS provides up to 400 1.2$''$ diameter fibers in the circular 30$'$ diameter field of view. In the high resolution mode, the FMOS spectral coverage is divided into four bands, which are J-short ($0.92-1.12~\mu$m), J-long ($1.11-1.35~\mu$m), H-short ($1.40-1.60~\mu$m), and H-long ($1.60-1.80~\mu$m) with a spectral resolution of R$=\lambda/\Delta\lambda\sim2200$. The Cross-Beam Switching (CBS) mode, in which two fibers are allocated to each target, was used for optimal sky subtraction of faint sources. The fibers in each pair are separated by 60 arcseconds, alternating between one for the target and the other one simultaneously placed on the sky, so that sky subtraction is not affected by time variation of sky brightness.
The primary targets are X-ray selected AGNs in the CDF-S, E-CDF-S, and XMM-LH surveys with either spectroscopic or photometric redshifts in the range $1.0<z<2.2$, and J magnitudes brighter than 22.5 mag. The FMOS J-band and H-band observations cover the and/or lines in the redshift range $z=0.7-2.7$. The data was obtained during 2012-2013, shown in Table 1. We observed for a total integration time of 3.5$-$4 hours while accumulating 28-30 frames with an exposure time of 900 seconds per frame. The weather conditions were acceptable, with seeing typically in the range of $0.''6$ to $1.''2$.
SUBARU FMOS Spectroscopic Observations
2012 Mar 25 XMM-LH J-long
2012 Mar 26 XMM-LH H-long
2012 Dec 28 CDF-S J-long
2012 Dec 29 CDF-S, XMM-LH H-short
2012 Dec 30 CDF-S, XMM-LH H-long
2013 Jan 19 CDF-S H-long
2013 Jan 20 CDF-S J-long
2013 Jan 21 CDF-S J-long
2013 Feb 24 XMM-LH H-short
2013 Oct 23 CDF-S J-long
2013 Oct 24 CDF-S H-long
We reduced the data using the publicly available software FIBRE-pac (FMOS Image-Based REduction package; ), which is an IRAF-based reduction tool for FMOS. This procedure includes background subtraction, corrections of detector cross talk, bias difference, bad pixels, the spectral distortion, and the removal of residual airglow lines. Individual frames were combined into an ensemble image, and wavelength and flux calibration were carried out. For the absolute flux calibration, the bright (J$_{\rm AB}$=15-18 mag) stars in each frame were used as a spectral reference. The flux of the reference star was estimated and compared with the photometric data in the catalog. All the spectra were divided by the reference spectrum, and then multiplied by the expected spectrum of the reference star. Apart from the calibration of slit losses through the spectroscopic reference star we do not apply further calibration corrections for our sample of AGNs, since we assume that the reference star corrects most of the slit losses for the point-like sources. While systematic effects like weather conditions, position accuracy still may cause differential flux losses across the field of view, the effect of these systematic errors on black hole masses should be small, since the black hole mass is a function of the square root of the luminosity (see Section 5.2). Finally, the one-dimensional spectrum of each object was extracted from the calibrated image, together with the associated noise spectra.
With the fully reduced 1- and 2-dimensional spectra, we determined the redshift through the identification of prominent emission line features. Each spectrum was visually inspected by Suh and Hasinger individually using the SpecPro <cit.> environment, which is an IDL-based interactive program for viewing and analyzing spectra. We assigned a quality flag to each redshift to indicate the reliability of the redshift determination. Altogether 825 X-ray sources were observed in the combined CDF-S, E-CDF-S, and XMM-LH fields, of which 262 sources are spectroscopically identified. It is noteworthy that we identified new spectroscopic redshifts for 135 X-ray selected AGNs. The redshift identifications are summarized in Appendix Table 2.
§.§ Optical spectroscopy
In addition to NIR spectra, we use existing optical spectroscopy that includes a detection of a broad emission line, shown to be a reliable probe of black hole mass at $z>1$ (e.g. ). Optical spectroscopy has been obtained in the CDF-S, E-CDF-S, and XMM-LH fields <cit.>, providing spectroscopic redshifts for X-ray sources. <cit.> present the results of spectroscopic follow-up for the CDF-S, which were observed at the VLT with the FORS1/FORS2 spectrographs for Chandra sources. Furthermore, <cit.> provide high-quality optical spectra in the E-CDF-S. 283 Chandra sources are observed with deep exposures (2-9 hr per pointing) using multi-slit facilities on both VLT/VIMOS and Keck/DEIMOS. <cit.> offer spectroscopy of the ROSAT Deep Surveys in the Lockman Hole using low-resolution Keck spectra. We compile the existing optical observations of X-ray AGNs from these deep spectroscopic surveys.
§ AGN BOLOMETRIC LUMINOSITY
The bolometric luminosity of AGNs can be estimated from the X-ray luminosity by applying a suitable bolometric correction. In order to estimate an accurate total intrinsic luminosity radiated by the AGN accretion disc, it is necessary to constrain the absorption-corrected intrinsic X-ray luminosity because it is often obscured and also includes reprocessed radiation. We thus derive the absorption corrected rest-frame X-ray luminosity and determine the bolometric luminosity with the bolometric correction. To account for the dependence of the optical to X-ray flux ratio $\alpha_{\rm ox}$ on luminosity, we use the luminosity-dependent bolometric correction factor (see e.g. ). Despite some difference between the luminosity-dependent bolometric correction factor among different studies (e.g. predicted lower bolometric correction at high bolometric luminosity with respect to that predicted by and ), the same trend of increasing bolometric correction at increasing bolometric luminosity is observed within the scatter.
We compute the intrinsic X-ray luminosity of broad-line AGNs following <cit.>. As a first step, we assume the intrinsic X-ray spectrum of AGNs modeled by a power-law component with both intrinsic and Galactic absorption (i.e., $zpow \times wabs \times zwabs$ in XSPEC) to estimate the intrinsic column density. A power-law photon index of $\Gamma=1.8$, which is typical for intrinsic AGN spectra, is assumed and the redshifts of the $zpow$ and $zwabs$ components are fixed to that of the source. We additionally fixed the Galactic column density to N$_{\rm H} = 6.0 \times 10^{19}$ cm$^{-2}$. We then derive the intrinsic column density that reproduces observed hard ($2-8$ keV) and soft ($0.5-2$ keV) band hardness ratios using XSPEC. The intrinsic X-ray luminosity is derived from the equation L$_{\rm X}$ = 4$\pi$d$^{2}_{\rm L}~f_{\rm X, int}$ (1+z)$^{\Gamma -2}$ by correcting both intrinsic and Galactic absorption. $f_{\rm X, int}$ is the absorption-corrected X-ray flux and the d$_{\rm L}$ is luminosity distance. Finally, we derive the bolometric luminosity of AGNs from the absorption-corrected rest-frame intrinsic X-ray luminosity with the luminosity-dependent bolometric correction factor described in <cit.>. They derived the bolometric corrections from an AGN template spectrum of optical-ultraviolet and X-ray luminosities radiated by the accretion disc and hot corona. They considered only the AGN accretion powered luminosity, neglecting the luminosity reprocessed by the dust, which is therefore representative of the AGN accretion power. The scatter is given by $\sim0.1$ for X-ray luminosities.
§ BLACK HOLE MASS ESTIMATION
The black hole mass can be estimated using the broad-line width and the continuum (or line) luminosity from their single-epoch spectra as proxies for the characteristic velocity and the size of the broad-line region (e.g., ). Depending on the redshift, single-epoch virial black hole masses have been estimated from different broad emission lines, such as <cit.>, <cit.>, and <cit.> lines. The virial black hole masses are calibrated against the black hole mass estimated by the reverberation mapping or that from the single-epoch broad-line width of emission line in the local universe. Although there are several systematic uncertainties in these single-epoch virial black hole mass estimators, a number of studies have shown that there is consistency in black hole masses from various estimators. <cit.> point out that there is essentially no difference in black hole mass estimates using and the Balmer lines for high redshift luminous AGNs. <cit.> also show that virial black hole masses based on and emission lines are very similar over a wide range in black hole mass. They suggest that local scaling relations, using or emission lines, are applicable for moderate-luminosity AGNs up to $z\sim2$.
We measure the properties of broad emission lines (e.g., , , and ) present in optical and NIR spectra to derive single-epoch virial black hole mass of broad-line AGNs. The $\lambda$6563Å and the $\lambda$4861Å lines are redshifted to the NIR range, and the $\lambda$2798Å line is present in optical spectra in the redshift range $0.5<z<2.5$.
§.§ Spectral line fitting
We perform a fit to the emission lines using the mpfit routine, which adopts a Levenberg-Marquardt least-squares minimization algorithm to derive the best-fit parameters as well as a measure of the goodness of the overall fit. We specifically measure the width and the luminosity of emission lines in the case of and lines, and the width and the monochromatic continuum luminosity at 3000Å in the case of the line. There might be a non-negligible host galaxy contribution at 3000Å continuum luminosity, but we do not correct for any contamination by the host galaxy and extinction due to dust. While we should be aware of this issue, the impact of these on black hole masses should be small, since the black hole mass scales with the square root of the luminosity (see Section 5.2).
Broad-line AGN spectra in the wavelength region of interest are usually characterized by a power-law continuum, $f_{\lambda} \propto \lambda^{-\alpha}$, and broad (and/or narrow) emission line components. We begin by fitting a power-law continuum with a slope of the continuum as a free parameter. In the case of the line, it is crucial to consider a complex of emission lines because in this wavelength range the lines are strongly blended with the broad emission features (e.g. ). We simultaneously fit the combination of a power-law continuum and emission components. An empirical emission template is adopted from <cit.> and convolved with Gaussian profiles of various widths. We left the width, normalization, and offset from the line center as free parameters during the fit. From the best-fit power-law continuum, we derive an estimate of monochromatic luminosity at 3000Å. Finally, we subtract the best-fit power-law continuum (and/or the emission components) from the spectra.
Comparison of the broad-line fit for the line (top panel) with that of the line (bottom panel) with/without a broad emission component for the same AGN source `XMM-LH 270' at $z=1.576$. The observed spectrum (grey) is shown with the best fit (black). In the top panel, the different components are shown as dotted lines (continuum), red curves (broad-line components), blue curves (narrow-line components of and a pair of lines). In the bottom panel, the fit of the line with emission (upper) and without emission (lower) are shown. The different components are indicated as red curves (individual broad-line components) and a green curve ( emission component).
Examples of the broad-line fits for (top), (middle), and (bottom) emission lines at $z=1.62,~2.13,~{\rm and}~1.88$, respectively. The upper plot of each panel shows the observed spectrum (grey) with the best-fit model (black). The power-law continuum (dotted), narrow-line components (blue), and emission component (green) are also indicated, respectively. The middle plot of each panel shows the only broad-line components after subtraction of the best-fit model of continuum, narrow components and emission. The best-fit broad-line model is shown with the black curve. Each Gaussian broad-line component is also shown with red curves. The residual is shown in the lower plot of each panel.
We further consider individual components to determine the pure broad-line components that enable an accurate determination of the virial black hole masses. The line profile is described by a combination of multiple Gaussian components to best characterize the line shape in the sense that broad emission lines in AGNs can have a complex shape (e.g. Collin et al. 2006). The multiple Gaussian components provide non-Gaussian, asymmetric profiles reproducing the observed broad-line profile smoothly, but we are not concerned with the physical significance of the individual components. We fit the $\lambda$6563Å ( $\lambda$4861Å) line with one or two broad and a narrow Gaussian components, and the $\lambda$6548,6583Å ( $\lambda$4959,5007Å) lines with a pair of Gaussians. The line ratio of the $\lambda$6548,6583Å and the $\lambda$4959,5007Å lines are fixed to the laboratory value of 2.96 and 2.98, respectively. Both the narrow width of the and the lines are fixed to match the narrow component of and , respectively. We left the FWHM of the narrow line components as free parameters but limited to 900 km/s. For the line, we fit with one or two broad Gaussians components. We do not consider the doublet component of the Mg II line because the line separation is small and does not affect the broad-line width.
As a consistency check we compare the fit of the line with emission components to that of the line, since the line is not affected by emission. In Figure 2, we show an example fit of the line and that of the line with/without broad emission component for the same AGN source `XMM-LH 270' at $z=1.576$. We show the observed spectrum (grey) with the best fit (black) of the line (top panel) and the line (bottom panel). The different components are also indicated as red gaussian curves (broad-line components), blue curves (narrow-line components of and a pair of lines), and green curve ( emission). While it is uncertain whether the line is blended with emission or it really has a very broad-component in the bottom panel, we confirm that the line fit with emission is likely to show a similar result with line fit in the upper panel.
In order to guarantee a reliable fit, we compare the fit with only narrow-line components, that with narrow-line and one broad Gaussian components, and that with narrow-line and two broad Gaussian components. We perform an F-test to decide whether an additional broad component is needed. We then subtract the narrow line components from the spectra obtaining a spectrum that contains only broad-line components. Finally, we inspect all fits by eye to check the cases where a broad component is unclear due to the low signal-to-noise ratio (S/N). We only consider spectra having S/N greater than 10 per pixel.
We determine the broad-line width and the line luminosity from the sum of the broad-line components. From the best-fit, the FWHM of the broad , , and lines are computed and corrected for the effect of instrumental resolution to obtain an intrinsic velocity width. We select the broad-line AGNs with broad emission line widths larger than 2000 km/s of FWHM, a secure threshold for truly broadened lines, as compared to the spectral resolution. Additionally, we take into account the uncertainty in the derived FWHM and luminosity. We perform a Monte Carlo simulation comprising 100 realizations adding noise to each spectrum and iterate the whole procedure to find the best-fit model and the errors compatible with the observations, in order to assess the accuracy of the black hole mass measured. Since the best-fit model could have either one or two broad-line components during different Monte Carlo realizations for each spectrum, this could introduce a larger scatter.
In Figure 3, we show examples of broad-line fits for (top), (middle), and (bottom) emission lines at $z=1.62,~2.13,~{\rm and}~1.88$, respectively. The upper plot of each panel shows the observed spectrum (grey) with the best-fit model (black). The power-law continuum (black dotted), narrow-line components (blue), and emission component (green) are also indicated, respectively. The middle plot of each panel shows the broad-line only components after subtraction of the best-fit model of continuum, narrow-line components and emission. The best-fit broad-line model is shown with the black curve. Each Gaussian broad-line component is also shown with red curves. The residual is shown in the lower plot of each panel.
§.§ Black hole masses
We calculate black hole masses from the FWHM and the luminosity of the sum of the broad line components. In the case of and we use the recipes provided by <cit.>. In addition, we specifically estimate the black hole mass based on the FWHM of the broad line and the monochromatic continuum luminosity at 3000Å using the calibration derived by the <cit.>. The black hole mass can be expressed in the forms:
\begin{eqnarray}
\rm M_{BH} & = & 10^{6.301}(\frac{\rm L_{H\alpha}}{\rm 10^{42}~ergs~s^{-1}})^{0.55}(\frac{\rm FWHM_{H\alpha}}{\rm 10^{3}~km~s^{-1}})^{2.06}~\rm M_{\odot}
\end{eqnarray}
\begin{eqnarray}
\rm M_{BH} & = & 10^{6.556}(\frac{\rm L_{H\beta}}{\rm 10^{42}~ergs~s^{-1}})^{0.56}(\frac{\rm FWHM_{H\beta}}{\rm 10^{3}~\rm km~s^{-1}})^{2.0}~\rm M_{\odot}
\end{eqnarray}
\begin{eqnarray}
\rm M_{BH} & = & 10^{0.505}(\frac{\lambda \rm L_{\lambda3000}}{\rm 10^{44}~ergs~s^{-1}})^{0.62}(\frac{\rm FWHM_{MgII}}{\rm km~s^{-1}})^{2.0}~\rm M_{\odot}
\end{eqnarray}
where FWHM is the FWHM of the line in units of 1000 km/s, and L$_{\lambda3000}$ is the continuum
luminosity at 3000Å.
Comparison of black hole masses estimated using the line with that using the line (red circles) or the line (red squares). Our sample of AGNs are shown in red, and the observations from <cit.> and <cit.> are also shown as grey and black symbols, respectively. The black dashed line denotes a one-to-one relation.
We present the comparison of black hole masses estimated using the line with that using the (red square), or (red circles) lines in Figure 4, respectively. We also show the observations from <cit.> and <cit.> as grey and black open circles for comparison, respectively. The black dashed line denotes a one-to-one relation. Our sample of broad-line AGNs spans a range of $7.0<{\rm log~M_{BH}/M_{\odot}}<9.5$ which is consistent with the previous studies of moderate-luminosity AGNs at $z\sim1-2$ <cit.>. The ratios of the mean black hole mass are log(M$_{\rm Mg\,\textsc{ii}}$)/ log(M$_{\rm H\alpha}$)=0.15, and log(M$_{\rm H\beta}$)/ log(M$_{\rm H\alpha})=-0.27$, respectively. The median uncertainty of the black hole mass is $\sim0.1$ dex. While there are offsets between the different black hole mass estimations, it is worth noting that the black hole mass estimated with different calibrations carries a scatter of $\sim0.3$ dex <cit.>. We also note that determination of black hole mass from the emission line are known to be affected by significant systematic uncertainties due to the Balmer decrement. If there are multiple lines measured, we use the lines in order of , Mg II, and for the determination of the black hole mass. There are six objects in our sample that black hole masses are determined with the line.
§.§ Broad-line AGNs
We select the sample of broad-line AGNs for which one or more broad emission lines have been identified in the spectrum. From the NIR/optical spectra, the , , and wavelength regions are covered for 152, 56, and 62 spectra, respectively. Broad , , and lines are detected for 52, 7, and 53 in the NIR and/or optical spectra, respectively, by broad-line widths larger than 2000 km/s of FWHM with the high S/N. For 19 AGNs, broad lines are detected in both and lines (Figure 4). While all AGNs with detection of broad lines are also detected in the broad line, 5 AGNs with broad line show no broad line, mainly due to the low S/N NIR spectra. It is noted that there are quite a number clear broad lines with practically absent lines, indicating a large Balmer decrement. The final sample of broad-line AGNs in the CDF-S, E-CDF-S, and XMM-LH fields consists of 86 objects.
AGN bolometric luminosity versus black hole mass for our sample of broad-line AGNs in the different redshift bins (left). In the right panel, contours at the 1$\sigma$ level are shown in the different redshift bins, together with the published observations from the literature as labeled. As a reference, lines of constant Eddington ratio (${\rm L_{bol}/L_{Edd}}$) equals to 1, 0.1, 0.01, and 0.001 are plotted as dotted lines.
§ EDDINGTON RATIO DISTRIBUTION
Eddington ratio distribution of broad-line AGNs at $1.0<z<2.2$. The different X-ray surveys are shown in different color histograms, and the black histogram represents the combined distribution of all surveys. The grey shade indicates the Eddington limit. The red solid line indicates a log-normal fit with a peak of ${\rm log~L_{bol}/L_{Edd}=-0.6}$ and a dispersion of 0.8 dex.
The mass accretion onto the black hole is important for a better understanding of the AGN evolution. The Eddington ratio, the ratio between the AGN bolometric luminosity and the Eddington luminosity (L$_{\rm bol}$/L$_{\rm Edd}$), provides insight into the black hole growth since the bolometric luminosity reflects the mass accretion rate. We show AGN bolometric luminosity versus black hole mass for our sample of broad-line AGNs in the different redshift bins in the left panel of Figure 5. The different X-ray surveys are shown with different symbols as labeled. The dotted reference lines indicate constant Eddington ratios of 1, 0.1, 0.01, and 0.001, respectively. Our sample of broad-line AGNs covers the black hole mass range ${\rm 7.0<log~M_{BH}/M_{\odot}<9.5}$ and the bolometric luminosity range ${\rm 43<log~L_{bol}<47}$ with a wide dispersion in the Eddington ratio distribution. For comparison, we show published observations in the same redshift range from the literature in the right panel of Figure 5 <cit.>. The SDSS quasar sample (grey points; ) is limited to the high-mass and high-luminosity regime because the SDSS detection limit corresponds to a luminosity of ${\rm log~L_{bol}\sim46}$ at $z\sim1$. Compared to the SDSS quasar sample, our sample of broad-line AGNs show a wider dispersion in the black hole mass, AGN bolometric luminosity and Eddington ratio distribution, consistent with previous studies on deep AGN sample <cit.>, which fill in the low-mass and low-luminosity region. The figure shows contours at the 1$\sigma$ level, together with the literature data, except the SDSS quasar sample. The figure also reveals that only a small number of AGNs exceeds the Eddington limit by a small amount. AGNs with similar black hole masses show a broad range of bolometric luminosities spanning about two orders of magnitude, indicating that the accretion rate of black holes is widely distributed. This suggests that the AGN cosmic downsizing phenomenon could be explained by some more massive black holes with low accretion rates, which are relatively fainter than less massive black holes with efficient accretion. <cit.> suggest that AGNs show higher Eddington ratios at higher redshift at any given ${\rm M_{BH}}$, and the Eddington ratio increases with bolometric luminosity. We confirm that there is a tendency for low-luminosity AGNs (${\rm log~L_{bol}\lesssim45.5}$) with less massive black holes (${\rm log~M_{BH}/M_{\odot}\lesssim8}$) to have lower Eddington ratios than high-luminosity AGNs (${\rm log~L_{bol}\gtrsim45.5}$) with massive black holes (${\rm log~M_{BH}/M_{\odot}\gtrsim8}$), consistent with <cit.>. It is important to note that, when comparing with results in the literature, one should take into account the different methods of spectral line fitting and correction for bolometric luminosities. Nevertheless, they show similar distributions of the accretion rate of black holes over a wide range, consistent with previous studies.
Several studies have found a correlation between the X-ray bolometric correction and the Eddington ratio (e.g. ), which may introduce biases into this diagram. <cit.> found that there is a trend for higher bolometric corrections at higher bolometric luminosities. <cit.> suggest that there appears to be a distinct step change in bolometric correction at an Eddington ratio of $\sim0.1$, below which apply lower bolometric corrections, and above which apply higher bolometric corrections. If one includes this correlation to the trend between bolometric luminosities and black hole masses in Figure 5, which low-luminosity AGNs have lower accretion rates while high-luminosity AGNs show higher accretion rates, would even be more pronounced. However, we note the possibility that there could be the spurious correlations since ${\rm L_{bol}}$ is present on both axes when plotting the bolometric correction against the Eddington ratio.
We show the Eddington ratio distribution of our sample of AGNs in the redshift range $1.0<z<2.2$ in Figure 6. The different X-ray surveys are shown in different color histograms, and the black histogram represents the combined distribution of all surveys. The distribution of Eddington ratios peaks at ${\rm log~L_{bol}/L_{Edd}\sim-1}$ with an extended tail towards low Eddington ratios, down to ${\rm log~L_{bol}/L_{Edd}\sim-3}$. A log-normal fit with a peak of ${\rm log~L_{bol}/L_{Edd}=-0.6}$ and a dispersion of 0.8 dex is shown as red solid line. In previous studies, <cit.> suggest that the Eddington ratios are quite narrowly distributed independent of luminosity (${\rm L_{bol}=10^{45}-10^{47}~erg/s}$) and redshift ($0.3<z<4.0$), with a dispersion of 0.3 dex (see also ). <cit.> also suggest that the distribution of Eddington ratios are nearly Gaussian especially at high-redshift and at high ${\rm L_{bol}/M_{BH}}$, with a dispersion of $\sim$0.35 dex, while the low-redshift and low ${\rm L_{bol}/M_{BH}}$ are more affected by incompleteness. We list our sample of broad-line AGNs in Appendix Table 3, which includes AGN bolometric luminosities, black hole masses, and measurements of emission line properties.
We should emphasize here that the systematic selection effects could certainly be playing a role in determining the distribution of AGN bolometric luminosities and black hole masses. The Eddington ratio distribution, thus, could be a result of the selection bias, mainly the limited X-ray luminosity but also to the broad line width, i.e. the black hole mass. The X-ray luminosity is limited by the X-ray flux limit, depending on redshift and on the limited volume. The detectability of the broad emission line gives rise to a bias against the black hole mass. Also, the black hole mass could be biased by observational limitations to detect the corresponding very broad lines and low signal-to-noise spectra. This is bound to introduce selection biases, which could mimic artificial correlations in the data. Hence, we will further discuss the possible selection effects in the next section.
Monte Carlo simulated data sets from the AGN bolometric luminosity function <cit.> in the different redshift bins with an assumption for the Eddington ratio distribution, which has a peak of ${\rm log~L_{bol}/L_{Edd}=-0.6}$ and a dispersion of 0.8 dex (red curve in Figure 6) regardless of AGN luminosity or redshift. The simulated data sets (left), and those which are affected by the same observed selection effects (right) are shown in grey. As a reference, lines of constant Eddington ratio (${\rm L_{bol}/L_{Edd}}$) equals to 1, 0.1, 0.01, 0.001 are plotted as dotted lines. Contours at the $1\sigma$ level are shown in the different redshift bins. The black solid line indicates the assumed peak of Eddington ratio.
Comparison of the Monte Carlo simulated data sets with the observed AGNs in the different redshift bins. In the top panels, the simulated data sets are shown in grey with the $1\sigma$ contour, and the observed AGNs are shown as purple ($0.5<z<0.8$), green ($0.8<z<1.2$), yellow ($1.4<z<1.6$), and red ($1.8<z<2.2$) symbols, respectively. In the bottom panels, the Poisson likelihood is shown in each bin, which is perpendicular to the Eddington ratio plane. The numbers in parentheses refer to detected (observed) sources over expected (simulated) sources. Reference lines of constant Eddington ratios are plotted as dotted lines.
§ ANALYSIS OF SELECTION BIASES
We investigate the possible bias due to systematics and selection effects on the observed AGN bolometric luminosity and the black hole mass. To explore the effect of these selection biases, we construct Monte Carlo simulations to make artificial datasets, which are affected by the same selection effects. We start from the bolometric luminosity function of AGNs <cit.> in the different redshift bins with an assumption for the Eddington ratio distribution, which has a peak of ${\rm log~L_{bol}/L_{Edd}=-0.6}$ and a dispersion of 0.8 dex, bounded by ${\rm L_{Edd}}$, taken from the observed distribution (see red curve in the top panel of Figure 6). To account for the observed selection biases, we apply the same selection effects based on our combined X-ray surveys. The X-ray flux limit corresponds to a bolometric luminosity of ${\rm log~L_{bol}\sim43}$ at $z\sim1$. We use the survey area of the total combined X-ray surveys (black curve in the right panel of Figure 1). Since it has been known that there is much larger fraction of obscured AGN at lower luminosities <cit.>, we apply the fraction of broad-line AGNs as a function of AGN luminosity from <cit.>, to which the same bolometric correction () was applied to the X-ray luminosity. The black hole mass is biased by the detectability of the broad emission line and the signal-to-noise of spectra. We, thus, apply a `fudge' factor, which is the exponentially decaying function at low-mass (${\rm 6.5<log~M_{BH}<7.5}$) and high-mass (${\rm 8.5<log~M_{BH}<9.5}$). The `fudge' factor brings down the numbers of low-mass (${\rm 6.5<log~M_{BH}<7.5}$) and high-mass (${\rm 8.5<log~M_{BH}<9.5}$) AGNs, which takes into account the spectroscopic incompleteness. For each data set of the different redshift bins, we calculate black hole masses from the AGN bolometric luminosity and the Eddington ratio. The Eddington ratio distribution is assumed to be same regardless of AGN luminosity or redshift, which is a valid assumption for the high-luminosity AGNs <cit.>. <cit.> report that the Eddington ratio distributions are all similar for SDSS quasar populations over a wide range of mass and redshift.
In Figure 7, the Monte Carlo simulated data sets (left panel), and those which are affected by the same observed selection effects (right panel) are shown with grey symbols and contours at the $1\sigma$ level in the different redshift bins. The black solid line in the right panel of Figure 7 indicates the assumed peak of Eddington ratio, ${\rm log~L_{bol}/L_{Edd}=-0.6}$. The AGN downsizing trend is seen in the sense that the characteristic AGN luminosity and black hole mass decrease with redshift. This is primarily due to the strong evolution of the co-moving number density at the bright end of the AGN luminosity function at $0.5<z<2.0$, together with the corresponding selection biases. We compare the simulated data with the observed AGNs in the different redshift bins in Figure 8. The simulated data sets are shown in grey with the contour at the $1\sigma$ level, and the observed AGNs are shown in colored symbols for each redshift bin in the top panels. For each redshift bin we group the data into four sets, using a constant Eddington ratio of ${\rm L_{bol}/L_{Edd}=0.1}$ and a line perpendicular to the Eddington ratio as separation. In the bottom panels, the number of detected (observed) sources over the number of expected (simulated) sources is given in parentheses, as well as the Poisson likelihood calculated from this combination. In bins of high-luminosity AGNs with high Eddington ratio as well as low-luminosity AGNs with low Eddington ratio the detected number of objects agrees with the prediction from the Monte Carlo simulation within the statistical errors. However, for low-luminosity AGNs with high Eddington ratios, especially at high redshift ($1.8<z<2.2$) and low redshift ($0.5<z<0.8$) bins, the simulations systematically predict a larger number of objects, than those observed. Taking all Poisson likelihoods together, there is a difference between the observed and the predicted distributions. We therefore suggest that there is a dependence of AGN luminosities on the Eddington ratios in the sense that luminous AGNs appear to have systematically higher Eddington ratios than low-luminosity AGNs. However, this result is of marginal significance due to the relatively small number of objects in each bin. We note that our sample of high-luminosity X-ray selected AGNs overlaps with the less luminous quasars from the SDSS sample at the highest Eddington ratios (see grey points in the right panel of Figure 5), while the most luminous, most massive SDSS quasars lie further away from their Eddington luminosity <cit.>.
§ DISCUSSION
We now discuss the observed AGN downsizing phenomenon, and possible explanations for the black hole growth over cosmic time. The decrease of the characteristic luminosity of AGNs with redshift has been described as AGN downsizing, implying that the AGN activity at earlier epochs was much more intense. We show that AGNs with similar black hole masses show a broad range of bolometric luminosities, which means the accretion rate of black holes is widely distributed. The average accretion rate of two different AGN fueling mechanisms can play a crucial role for the downsizing interpretation. AGN activity triggered by major mergers is thought to have a higher accretion rate than activity triggered by secular evolution effects. Therefore the luminosity of an AGN with a certain black hole mass may differ widely, depending on the accretion mechanisms.
The colors and morphologies of galaxies may contain a record of their growth history. Several studies have addressed that the majority of AGN host galaxies in the local universe are preferentially in the “green valley" on the color-magnitude diagram, between actively star-forming galaxies in the blue cloud and passively evolving galaxies on the red sequence (e.g. ). Moreover, a large fraction of moderate-luminosity AGNs are likely to live in disk-dominated galaxies <cit.>. <cit.> find that the majority of AGN host galaxies show no significant merger features up to $z\sim2$. It is likely that merger features are visible only for a few Gyrs after major mergers (e.g. ), suggesting that most AGN activity does not seem to be triggered by major mergers since $z\sim2$ (see also ). <cit.> further point out that moderate-luminosity AGNs at $z=0-2$ live in relatively massive dark matter halos (${\rm 10^{13.5}M_{\odot}}$), which corresponds to rich groups of galaxies, independent of redshift. The rich group environment may provide a kind of “goldilocks" zone for AGNs in the sense that on one hand the density is high enough to cause frequent gravitational disturbances bringing cold gas to the center, and on the other hand the gas density in the group is not high enough to remove the cold gas from the galaxies due to ram pressure stripping. This also indicates that major mergers cannot be the main driver of the late evolution of AGNs. This raises interesting questions regarding different fueling mechanisms for the growth of black holes and galaxies at different epochs during cosmic time.
Given these intriguing findings, a possible interpretation for explaining the cosmic downsizing as well as morphologies and colors of AGN host galaxies is that there are two different modes of AGN feedback at different epochs (see ). In an active AGN phase at high redshift, black holes have experienced vigorous growth by major mergers while radiating close to the Eddington limit (see e.g. ). When they reach a critical mass, at which the AGN is sufficient to blow out the surrounding gas, the feedback of the black hole suppresses further star formation and creates a red bulge-dominated remnant (e.g. ). It is likely that only a small fraction of the transient population can be found in between the blue cloud and the red sequence due to rather short merger timescale ($\sim10^{8}$ yrs). The relatively massive galaxies, which have already experienced substantial growth by previous mergers, grow slowly through episodic star formation via secular evolution, leading to a disc surrounding the bulge. The modest AGN activity can be triggered by the gas accretion over cosmic time via internal, secular processes, such as gravitational instabilities in the disc. This secular growth is slow enough, and thus, the presence of AGN host galaxies in the green valley on the color-magnitude diagram could be interpreted as evidence for the on-going star formation in the inner region of low-luminosity AGN host galaxies at lower redshift, coming down from the red sequence. This is also compatible to the weak link between merger features and the AGN activity, as well as the moderate-luminosity AGNs in the relatively massive dark matter halos at $z\lesssim2$, where the number density of most luminous AGNs starts to decline. Finally, the late feedback from AGNs suppresses the late cooling flows of hot gas, keeping the galaxy quiescent. This seems to consistent with dormant supermassive black holes in dynamically hot systems (e.g. massive early-type galaxies) that contain little cold gas and correspondingly little star-formation. All of these seem to be consistent with the hierarchical growth scenario.
§ SUMMARY
We present the Eddington ratio distribution of X-ray selected broad-line AGNs in the CDF-S, E-CDF-S, and the XMM-LH surveys. We calculate AGN bolometric luminosities from absorption-corrected X-ray luminosities, and estimate black hole masses of broad-line AGNs using the optical and Subaru/FMOS near-infrared spectroscopy. Our sample of broad-line AGNs spans the bolometric luminosity range ${\rm L_{bol}\sim10^{43.5-47}~erg/s}$, and the black hole mass range ${\rm M_{BH}\sim10^{6.5-9.5}~M_{\odot}}$ with a broad range of Eddington ratios ${\rm L_{bol}/L_{Edd}\sim0.001-1}$.
We explore the systematics and selection biases, because in general observed distributions are dependent on the X-ray flux limit and the detectability of the broad emission lines. Based on the analysis on these effects, we find that the observed downsizing trend could be simply explained by the strong evolution of the co-moving number density at the bright end of the AGN luminosity function at $0.5<z<2.0$, together with the corresponding selection effects. However, in order to explain the relatively small fraction of low-luminosity AGNs with high accretion rates, we might need to consider a correlation between the AGN luminosity and the accretion rate of black holes that luminous AGNs have higher Eddington ratios than low-luminosity AGNs. We suggest that the AGN downsizing trend can be interpreted as the fraction of AGNs radiating close to the Eddington limit decrease after their peak activity phases, suggesting that the fueling mechanism of growth of black holes might change through the cosmic time.
We thank the anonymous referee for several comments, which helped to improve the quality of the manuscript significantly.
We thank Amy Barger for useful comments that helped improve this paper.
Facilities: Subaru (FMOS)
§ SUBARU/FMOS SPECTROSCOPIC OBSERVATIONS
In Table 2, we present the identified spectroscopic redshifts from the Subaru/FMOS observations. The ID is from the published catalog of CDF-S <cit.>, E-CDF-S <cit.>, and XMM-LH <cit.>. We assign a quality flag that gives the confidence in the redshift measurement. Flag 2 indicates a reliable redshift due to high signal-to-noise ratio (S/N) spectra and multiple spectral features. Flag 1 indicates that a redshift is not securely identified due to either low S/N or the presence of only a single emission line with no additional features. The newly discovered spectroscopic redshifts are marked with a cross sign. “++" if no previous spectroscopic redshift and “+" if the previous spectroscopic redshift is insecure. The redshift column gives our best redshift estimate. We also present the previous spectroscopic and photometric redshifts from the catalog.
In Table 3, we list our sample of broad-line AGNs, which includes AGN bolometric luminosities, black hole masses, and measurements of emission line properties.
l r r r c l c c l l
Subaru/FMOS spectroscopic observations
1cz$_{\rm \bf spec}$
1cz$_{\rm \bf phot}$
1cEmission features
10c – continued.
1cz$_{\rm \bf spec}$
1cz$_{\rm \bf phot}$
1cEmission features
CDF-S 1 52.899 -27.860 1.630 2+ 1.624 1.626 J-long, H-long , , ,
CDF-S 4 52.930 -27.901 1.270 2+ 1.271 1.027 J-long, H-short, H-long ,
CDF-S 7 52.936 -27.865 0.880 1++ -99.0 0.881 H-long
CDF-S 25 52.960 -27.870 1.336 2 1.374 1.386 H-short ,
CDF-S 26 52.960 -27.864 1.447 1++ -99.0 1.438 J-long, H-short, H-long
CDF-S 31 52.963 -27.744 1.608 2++ -99.0 1.829 H-short, H-long , ,
CDF-S 36 52.967 -27.804 1.236 1+ 1.236 1.223 H-short ,
CDF-S 37 52.968 -27.696 1.020 2+ 0.857 1.021 J-long ,
CDF-S 49 52.980 -27.841 1.212 1+ 1.212 1.208 H-short ,
CDF-S 51 52.981 -27.913 0.726 2 0.737 0.724 J-long ,
CDF-S 54 52.983 -27.823 1.219 2++ -99.0 1.208 H-short ,
CDF-S 64 52.991 -27.793 0.856 1++ -99.0 -99.0 J-long, H-long
CDF-S 67 52.993 -27.845 1.544 2++ -99.0 3.309 J-long, H-long
CDF-S 72 52.998 -27.839 1.638 1+ 1.574 1.900 J-long, H-long
CDF-S 76 53.002 -27.722 1.042 2 1.037 1.05 J-long, H-long ,
CDF-S 80 53.003 -27.893 1.549 2++ -99.0 1.593 J-long, H-short, H-long ,
CDF-S 81 53.004 -27.799 0.975 2 0.975 0.951 J-long , ,
CDF-S 83 53.006 -27.780 1.307 2++ -99.0 1.499 H-short, H-long
CDF-S 84 53.006 -27.694 1.414 1 1.406 1.379 H-short ,
CDF-S 88 53.010 -27.767 1.613 1 1.616 1.626 J-long, H-long ,
CDF-S 99 53.016 -27.891 3.039 1++ -99.0 3.877 H-short
CDF-S 101 53.017 -27.624 0.966 2 0.977 1.000 J-long
CDF-S 109 53.020 -27.691 1.524 2+ 0.720 1.562 J-long, H-long ,
CDF-S 112 53.023 -27.757 1.189 1+ 1.189 1.129 H-short ,
CDF-S 113 53.024 -27.746 1.609 1+ 1.608 1.692 J-long, H-long ,
CDF-S 117 53.025 -27.824 1.295 1++ -99.0 1.208 J-long, H-short, H-long
CDF-S 118 53.026 -27.673 1.624 2++ -99.0 1.438 H-long ,
CDF-S 120 53.027 -27.791 1.021 2 1.021 1.023 J-long ,
CDF-S 121 53.027 -27.765 1.329 1 1.329 1.321 H-short ,
CDF-S 127 53.029 -27.936 0.774 2 0.777 0.767 J-long, H-long , , ,
CDF-S 131 53.031 -27.856 1.546 1+ 0.034 1.546 J-long, H-long ,
CDF-S 135 53.033 -27.626 0.977 1 0.976 1.000 J-long, H-short, H-long ,
CDF-S 143 53.036 -27.850 0.736 2 0.736 0.718 J-long ,
CDF-S 152 53.041 -27.887 0.740 1 0.743 0.750 J-long ,
CDF-S 165 53.046 -27.729 0.998 2 0.998 1.000 J-long ,
CDF-S 166 53.046 -27.738 1.611 1 1.605 1.626 J-long, H-long ,
CDF-S 182 53.052 -27.820 1.048 2+ 1.048 0.904 J-long
CDF-S 205 53.060 -27.853 1.540 2 1.544 1.562 J-long, H-long , ,
CDF-S 222 53.066 -27.702 1.691 1++ -99.0 2.779 H-long
CDF-S 226 53.067 -27.817 1.412 1 1.413 1.438 H-short
CDF-S 229 53.068 -27.658 1.326 2 1.324 1.321 J-long, H-short , , , ,
CDF-S 236 53.071 -27.834 1.613 2 1.611 1.579 J-long, H-long ,
CDF-S 257 53.076 -27.849 1.535 2 1.536 1.562 H-long
CDF-S 271 53.081 -27.681 0.681 1 0.761 0.709 H-short
CDF-S 290 53.087 -27.930 2.301 2++ -99.0 2.826 H-short
CDF-S 308 53.094 -27.768 1.729 1 1.735 1.264 J-long, H-long ,
CDF-S 329 53.102 -27.670 0.954 2++ -99.0 1.050 J-long ,
CDF-S 344 53.105 -27.705 1.615 2 1.617 1.626 J-long, H-long , , ,
CDF-S 349 53.106 -27.772 0.896 2 0.896 0.904 J-long ,
CDF-S 368 53.111 -27.824 1.469 2 1.468 1.560 J-long, H-long , ,
CDF-S 369 53.111 -27.670 1.612 2 2.208 2.202 J-long, H-long , , ,
CDF-S 405 53.123 -27.723 1.614 2+ 1.609 1.598 J-long, H-short, H-long ,
CDF-S 409 53.124 -27.863 1.316 2+ 1.316 5.247 H-short
CDF-S 414 53.124 -27.756 0.953 2 0.953 0.951 J-long
CDF-S 415 53.125 -27.717 1.357 2 1.356 1.328 H-short , ,
CDF-S 417 53.125 -27.758 1.222 2 1.209 1.208 H-short , ,
CDF-S 420 53.125 -27.756 0.960 2 0.960 0.951 J-long , ,
CDF-S 424 53.126 -28.005 2.310 2 2.306 2.282 J-long, H-long
CDF-S 429 53.130 -27.655 1.040 1+ 1.038 1.438 J-long, H-long ,
CDF-S 436 53.131 -27.841 1.549 1+ 1.613 1.485 J-long, H-long ,
CDF-S 439 53.132 -27.833 0.980 1 0.98 0.983 J-long
CDF-S 467 53.142 -27.841 1.384 2 1.384 1.395 H-short
CDF-S 473 53.144 -27.654 1.557 2 1.510 1.499 H-long , ,
CDF-S 487 53.147 -27.667 1.473 1++ -99.0 1.490 J-long, H-short, H-long
CDF-S 491 53.149 -27.792 1.223 1 1.223 1.208 H-short
CDF-S 492 53.149 -27.683 0.735 2 0.735 0.724 J-long , ,
CDF-S 499 53.151 -27.857 1.614 2 1.613 1.626 H-long
CDF-S 503 53.151 -27.713 1.542 2+ 1.609 1.582 J-long, H-long , ,
CDF-S 518 53.157 -27.870 1.611 2 1.603 1.692 J-long, H-long , , ,
CDF-S 519 53.158 -27.704 1.791 1+ 1.814 -99.0 J-long, H-long
CDF-S 524 53.160 -27.931 2.043 1++ -99.0 1.829 J-long, H-short, H-long
CDF-S 557 53.171 -27.741 1.299 1 1.298 1.142 H-short ,
CDF-S 580 53.181 -27.783 1.571 1+ 1.570 1.578 J-long, H-long , ,
CDF-S 586 53.184 -27.793 0.738 2 0.738 0.724 J-long
CDF-S 599 53.188 -27.904 1.380 1 1.378 1.328 H-short ,
CDF-S 612 53.192 -27.891 1.381 1+ 1.382 1.428 H-short , ,
CDF-S 619 53.196 -27.730 1.212 1 1.178 1.154 H-short ,
CDF-S 623 53.197 -27.713 0.732 1 0.729 -99.0 J-long, H-long ,
CDF-S 626 53.200 -27.709 0.979 2 0.979 1.000 J-long, H-long , ,
CDF-S 629 53.201 -27.882 0.660 1 0.667 0.601 H-short
CDF-S 634 53.205 -27.681 1.226 2 1.222 1.208 J-long, H-short ,
CDF-S 648 53.214 -27.929 1.448 2++ 0.853 0.857 J-long, H-short, H-long
CDF-S 652 53.216 -27.708 1.023 2 1.020 1.000 J-long ,
CDF-S 656 53.218 -27.762 1.367 2 1.367 1.379 H-short , ,
CDF-S 661 53.226 -27.818 0.987 1+ 0.988 0.999 J-long
CDF-S 680 53.246 -27.861 1.502 2++ -99.0 1.379 J-long, H-short, H-long , ,
CDF-S 683 53.247 -27.816 1.613 2++ -99.0 2.202 J-long, H-long ,
CDF-S 698 53.261 -27.806 2.549 1++ -99.0 3.101 H-long
CDF-S 699 53.261 -27.760 1.257 2+ 1.130 1.301 H-short , ,
CDF-S 706 53.266 -27.841 0.891 2 0.891 0.904 J-long , ,
CDF-S 720 53.282 -27.858 1.609 1 1.609 2.535 J-long, H-long ,
CDF-S 724 53.287 -27.694 1.337 2 1.335 1.321 H-short ,
CDF-S 728 53.292 -27.812 1.013 2 1.034 1.050 J-long, H-long ,
CDF-S 731 53.302 -27.776 0.610 1++ -99.0 0.562 J-long, H-short, H-long
E-CDF-S 61 52.870 -27.983 0.752 2 0.752 0.763 J-long
E-CDF-S 68 52.878 -27.976 1.362 1 1.366 1.382 H-short ,
E-CDF-S 85 52.889 -27.805 0.822 1 0.678 0.669 J-long, H-short
E-CDF-S 89 52.891 -27.767 1.613 2++ -99.0 1.889 J-long, H-long ,
E-CDF-S 105 52.904 -27.969 1.606 1++ -99.0 1.368 J-long, H-short, H-long , ,
E-CDF-S 115 52.911 -27.996 1.190 2++ -99.0 1.255 J-long, H-short, H-long ,
E-CDF-S 118 52.916 -27.699 0.469 2 0.467 -99.0 J-long
E-CDF-S 121 52.917 -27.940 5.137 2++ -99.0 5.315 J-long, H-long
E-CDF-S 137 52.930 -28.020 1.197 2++ -99.0 1.425 J-long, H-short, H-long , ,
E-CDF-S 150 52.935 -27.943 0.727 2++ -99.0 0.879 J-long, H-long ,
E-CDF-S 157 52.942 -27.695 1.322 2+ 1.315 1.184 H-short ,
E-CDF-S 166 52.947 -27.920 1.408 2 1.404 1.186 H-short ,
E-CDF-S 193 52.963 -27.954 1.161 1 1.167 1.153 H-short ,
E-CDF-S 195 52.963 -27.982 1.361 1 1.368 0.831 H-short ,
E-CDF-S 201 52.966 -27.947 1.618 2++ -99.0 1.411 J-long, H-short, H-long ,
E-CDF-S 217 52.975 -28.046 1.602 2++ -99.0 1.578 J-long, H-long , , ,
E-CDF-S 218 52.976 -27.997 0.738 1 0.740 0.731 J-long ,
E-CDF-S 222 52.978 -28.017 0.662 2+ 0.623 0.677 H-short
E-CDF-S 234 52.987 -28.080 1.675 1++ -99.0 -99.0 J-long, H-short, H-long
E-CDF-S 235 52.987 -28.030 1.386 2 1.380 1.889 H-short ,
E-CDF-S 276 53.018 -28.075 1.295 2++ -99.0 -99.0 J-long, H-short, H-long , ,
E-CDF-S 282 53.022 -28.071 1.215 2++ -99.0 -99.0 H-short ,
E-CDF-S 298 53.029 -27.971 0.847 1++ -99.0 1.067 J-long, H-long
E-CDF-S 342 53.066 -28.025 1.641 2++ -99.0 1.651 H-long
E-CDF-S 358 53.085 -28.037 1.626 2 1.624 1.635 J-long, H-long , , , ,
E-CDF-S 372 53.108 -28.013 2.216 2++ -99.0 2.052 J-long, H-short, H-long
E-CDF-S 384 53.116 -28.079 1.531 1++ -99.0 -99.0 J-long, H-short, H-long
E-CDF-S 388 53.122 -28.029 1.555 2+ 0.638 1.591 J-long, H-short, H-long , ,
E-CDF-S 400 53.135 -28.058 1.222 2 1.220 1.102 J-long, H-short, H-long ,
E-CDF-S 411 53.151 -27.589 1.226 2 1.220 1.199 J-long, H-short ,
E-CDF-S 424 53.165 -28.014 1.517 2++ -99.0 1.507 J-long, H-short, H-long ,
E-CDF-S 459 53.194 -27.995 1.667 1++ -99.0 -99.0 J-long, H-short, H-long
E-CDF-S 470 53.205 -28.063 0.680 1 0.680 0.644 J-long, H-short, H-long
E-CDF-S 481 53.213 -28.034 2.529 1++ -99.0 2.695 J-long, H-short, H-long
E-CDF-S 517 53.247 -27.603 1.345 2 1.350 1.337 H-short
E-CDF-S 521 53.248 -27.624 1.608 1++ -99.0 1.484 J-long, H-long
E-CDF-S 538 53.256 -28.048 1.373 2+ 0.200 1.812 J-long, H-short, H-long , ,
E-CDF-S 546 53.264 -27.885 0.892 1+ 1.346 0.954 J-long, H-short ,
E-CDF-S 557 53.271 -27.674 0.311 2 0.311 0.285 J-long, H-short, H-long
E-CDF-S 571 53.278 -27.774 1.705 2 1.705 1.497 J-long, H-long ,
E-CDF-S 588 53.287 -27.974 2.579 2 2.583 1.104 H-long
E-CDF-S 595 53.290 -27.736 1.037 1++ -99.0 1.016 J-long, H-long ,
E-CDF-S 601 53.294 -27.963 1.598 2 1.598 1.726 J-long, H-long , ,
E-CDF-S 621 53.310 -27.975 1.320 2++ -99.0 1.555 J-long, H-short, H-long , ,
E-CDF-S 624 53.311 -27.706 1.320 1++ -99.0 1.440 J-long, H-short, H-long
E-CDF-S 635 53.319 -27.952 1.045 2+ 1.044 1.020 J-long
E-CDF-S 678 53.345 -27.923 1.629 2 1.628 1.633 J-long, H-long , , ,
E-CDF-S 681 53.347 -27.771 0.834 2 0.835 0.809 J-long
E-CDF-S 688 53.349 -27.725 0.787 2 0.787 0.786 J-long ,
E-CDF-S 698 53.357 -27.816 1.413 1++ -99.0 1.464 J-long, H-short, H-long ,
E-CDF-S 701 53.360 -27.722 0.964 2 0.964 0.950 J-long ,
E-CDF-S 705 53.362 -27.755 1.658 1++ -99.0 1.835 H-long , ,
E-CDF-S 709 53.368 -27.972 1.558 1++ -99.0 -99.0 J-long, H-short
E-CDF-S 715 53.371 -27.732 1.280 1++ -99.0 -99.0 J-long, H-short
E-CDF-S 716 53.372 -27.991 0.763 2 0.763 0.768 J-long , ,
E-CDF-S 718 53.374 -27.852 0.946 1 0.951 0.956 H-short, H-long
E-CDF-S 725 53.383 -27.903 1.314 2 1.315 -99.0 J-long, H-short, H-long ,
E-CDF-S 728 53.387 -27.819 1.583 2 1.581 1.568 J-long, H-short, H-long , ,
XMM-LH 5 163.180 57.263 2.138 2 2.144 2.050 H-short, H-long ,
XMM-LH 15 163.184 57.286 2.010 1++ -99.0 2.022 H-short
XMM-LH 19 163.240 57.292 0.746 2++ -99.0 0.730 H-short, H-long
XMM-LH 25 163.348 57.293 1.599 2++ -99.0 1.450 H-short, H-long ,
XMM-LH 34 163.253 57.303 0.517 2++ -99.0 0.520 H-long He i
XMM-LH 35 163.470 57.304 1.727 2++ -99.0 1.441 H-short, H-long ,
XMM-LH 48 163.250 57.317 1.458 1++ -99.0 1.460 H-short, H-long
XMM-LH 50 163.107 57.318 4.470 2 4.449 4.340 J-long, H-short, H-long
XMM-LH 51 163.032 57.318 2.677 1++ -99.0 2.760 H-long ,
XMM-LH 61 162.990 57.328 1.378 2+ 1.379 1.188 H-short , ,
XMM-LH 63 163.151 57.267 1.729 2 1.734 1.649 J-long, H-long , ,
XMM-LH 65 163.231 57.331 1.452 2 1.450 1.296 J-long, H-short, H-long , ,
XMM-LH 72 163.070 57.338 2.741 1 2.710 1.060 H-long ,
XMM-LH 85 163.373 57.351 1.144 2 1.145 0.469 J-long, H-short, H-long , ,
XMM-LH 96 163.187 57.356 2.826 1 2.832 -99.0 J-long, H-short, H-long
XMM-LH 103 163.324 57.364 1.484 2++ -99.0 1.494 H-short, H-long ,
XMM-LH 119 163.510 57.376 1.406 2++ -99.0 2.030 H-short, H-long , ,
XMM-LH 120 163.105 57.385 1.523 2 1.524 1.488 J-long, H-long , , ,
XMM-LH 121 163.243 57.381 0.763 2 0.762 0.520 J-long, H-short, H-long , ,
XMM-LH 124 163.299 57.385 1.534 2 1.552 -99.0 H-long
XMM-LH 131 163.173 57.389 1.005 1 1.013 1.107 J-long, H-long
XMM-LH 136 163.050 57.389 0.743 1++ -99.0 1.170 H-short, H-long
XMM-LH 151 163.483 57.399 1.398 2++ -99.0 1.445 H-short, H-long ,
XMM-LH 154 163.609 57.401 0.963 2+ 0.963 1.049 J-long
XMM-LH 156 162.977 57.402 2.391 2 2.367 0.544 J-long, H-short, H-long
XMM-LH 167 163.574 57.406 1.220 1++ -99.0 1.406 H-short
XMM-LH 168 163.383 57.415 1.958 2 1.956 1.921 J-long, H-short, H-long ,
XMM-LH 171 163.588 57.429 0.205 2 0.205 0.214 J-long , He i
XMM-LH 174 162.837 57.416 1.186 1++ -99.0 1.168 H-short
XMM-LH 176 163.239 57.419 1.533 2 1.527 0.501 J-long, H-long , ,
XMM-LH 179 163.132 57.417 1.475 1++ -99.0 1.480 H-short, H-long ,
XMM-LH 183 163.301 57.418 1.877 2 1.876 1.603 J-long, H-short, H-long
XMM-LH 191 163.396 57.428 0.787 2++ -99.0 0.967 J-long, H-short, H-long , ,
XMM-LH 203 163.500 57.434 1.742 2++ -99.0 1.772 H-long ,
XMM-LH 217 162.799 57.443 0.760 1 0.758 0.734 J-long, H-long
XMM-LH 229 162.935 57.447 1.285 2++ -99.0 1.289 H-short, H-long ,
XMM-LH 247 163.358 57.461 1.367 1++ -99.0 1.379 H-short
XMM-LH 254 163.181 57.466 1.183 1 1.210 0.968 J-long, H-short, H-long
XMM-LH 261 162.937 57.469 3.406 2 3.408 3.022 J-long, H-long ,
XMM-LH 267 163.352 57.472 1.549 2 1.563 1.420 J-long, H-long , ,
XMM-LH 268 163.450 57.471 1.295 1++ -99.0 1.298 H-short
XMM-LH 270 163.289 57.472 1.576 2 1.575 1.049 J-long, H-long , ,
XMM-LH 277 163.437 57.478 1.807 1 1.816 1.945 J-long, H-long
XMM-LH 279 163.192 57.472 1.514 1++ -99.0 1.460 H-short, H-long ,
XMM-LH 300 163.266 57.490 0.789 2 0.788 0.504 J-long, H-short, H-long , , ,
XMM-LH 306 163.028 57.490 0.709 1 0.709 0.677 J-long, H-long
XMM-LH 321 163.102 57.502 1.008 2 1.009 1.543 J-long, H0long ,
XMM-LH 340 163.130 57.504 1.212 2 1.212 1.034 H-short , ,
XMM-LH 342 163.416 57.518 0.587 2 0.586 0.533 J-long, H-short, H-long , He i
XMM-LH 354 162.856 57.514 3.409 2 3.409 2.945 J-long, H-short, H-long ,
XMM-LH 355 163.155 57.518 0.740 1 0.710 0.583 J-long, H-long ,
XMM-LH 359 162.980 57.512 1.102 1++ -99.0 1.087 H-short
XMM-LH 370 163.442 57.518 1.650 1++ -99.0 1.640 H-short, H-long ,
XMM-LH 385 163.176 57.533 1.379 2 1.379 1.288 H-short ,
XMM-LH 386 163.058 57.528 0.894 2 0.896 0.849 J-long ,
XMM-LH 387 163.249 57.532 1.449 2 1.447 1.398 H-short, H-long
XMM-LH 406 163.488 57.545 1.283 2 1.296 1.721 J-long, H-short
XMM-LH 409 163.403 57.550 1.600 2 1.601 0.504 J-long, H-long ,
XMM-LH 419 163.501 57.556 1.491 1++ -99.0 1.491 H-long
XMM-LH 429 163.559 57.558 2.913 1++ -99.0 2.789 H-short, H-long
XMM-LH 430 162.785 57.562 1.553 2 1.539 1.665 H-short, H-long
XMM-LH 443 163.155 57.567 1.877 2 1.877 1.823 J-long, H-short
XMM-LH 450 163.275 57.573 2.952 2 2.945 2.529 J-long, H-short, H-long , ,
XMM-LH 453 163.302 57.573 1.214 2 1.213 1.748 H-short ,
XMM-LH 456 162.977 57.577 0.877 2 0.877 0.648 J-long, H-short, H-long , , ,
XMM-LH 468 163.171 57.580 1.370 2++ -99.0 1.374 H-short ,
XMM-LH 472 163.417 57.588 1.136 1 1.135 0.899 H-short
XMM-LH 475 163.320 57.597 1.205 2++ -99.0 1.219 J-long, H-short, H-long ,
XMM-LH 477 163.283 57.589 1.468 2++ -99.0 0.777 H-long
XMM-LH 488 163.369 57.594 0.690 1 0.700 0.584 H-long
XMM-LH 518 163.401 57.624 2.269 2++ -99.0 2.563 H-short, H-long
XMM-LH 521 163.176 57.621 1.591 1++ -99.0 1.526 H-long
XMM-LH 523 162.875 57.627 1.217 2++ -99.0 1.192 H-short, H-long ,
XMM-LH 527 163.260 57.632 1.885 1 1.881 1.748 J-long, H-short ,
XMM-LH 528 163.151 57.628 2.341 1 2.352 2.076 H-short ,
XMM-LH 529 162.905 57.632 1.940 2+ 1.910 1.997 J-long, H-short, H-long
XMM-LH 530 162.844 57.626 1.474 2++ -99.0 1.426 H-short, H-long ,
XMM-LH 532 163.190 57.629 1.675 2 1.677 0.534 J-long, H-long , , , ,
XMM-LH 553 163.125 57.654 1.440 2 1.437 0.454 J-long, H-short, H-long , , , ,
XMM-LH 555 162.967 57.652 1.674 2++ -99.0 2.080 H-short, H-long ,
XMM-LH 560 163.216 57.652 1.724 1++ -99.0 1.650 H-long ,
XMM-LH 563 163.379 57.656 1.379 2++ -99.0 1.559 H-short, H-long , ,
XMM-LH 569 163.240 57.663 0.947 1++ -99.0 1.044 H-short, H-long
XMM-LH 572 163.343 57.669 0.750 1++ -99.0 0.730 H-short, H-long
XMM-LH 588 163.068 57.686 1.409 2++ -99.0 1.272 H-short, H-long
XMM-LH 591 163.097 57.689 1.535 2 1.534 1.527 J-long, H-short, H-long , , ,
XMM-LH 594 163.202 57.691 0.814 2+ 0.814 1.087 J-long, H-long ,
XMM-LH 595 162.932 57.689 1.602 2++ -99.0 1.583 H-long
XMM-LH 599 163.280 57.251 2.416 2 2.416 2.159 J-long, H-short, H-long , ,
XMM-LH 604 163.299 57.704 2.104 2 2.113 1.932 J-long, H-short, H-long ,
XMM-LH 610 163.166 57.719 0.679 1+ 0.679 0.675 J-long, H-short, H-long
XMM-LH 999 163.105 57.380 0.809 2 0.807 0.659 J-long, H-short, H-long , ,
XMM-LH 1379 163.559 57.462 1.689 1++ -99.0 1.440 H-long
XMM-LH 1395 162.838 57.327 1.649 2++ -99.0 1.672 H-short, H-long
XMM-LH 1443 163.141 57.279 1.391 2++ -99.0 1.415 H-short
XMM-LH 1458 162.855 57.484 0.484 1 0.484 0.502 H-short He i
XMM-LH 1476 163.360 57.651 1.235 1++ -99.0 1.300 H-short ,
XMM-LH 1518 163.525 57.404 0.775 2++ -99.0 0.824 J-long, H-long
XMM-LH 1551 163.461 57.459 1.673 2++ -99.0 1.704 H-short, H-long ,
XMM-LH 1595 163.071 57.610 1.545 1++ -99.0 1.580 H-long
XMM-LH 2015 163.258 57.508 1.357 1++ -99.0 1.370 H-short , ,
XMM-LH 2020 163.459 57.452 1.728 2+ 1.720 1.442 J-long, H-long
XMM-LH 2081 163.035 57.346 1.356 1++ -99.0 1.296 H-short , ,
XMM-LH 2084 163.236 57.400 1.524 2++ -99.0 1.348 H-short, H-long , ,
XMM-LH 2254 163.123 57.726 2.945 2++ -99.0 2.803 H-short, H-long
XMM-LH 2278 163.016 57.452 2.438 2 2.454 2.083 J-long, H-short, H-long ,
XMM-LH 2374 163.460 57.500 2.508 2++ -99.0 -99.0 H-long
XMM-LH 2379 163.181 57.418 0.752 1++ -99.0 0.750 H-long
XMM-LH 2380 163.423 57.507 2.063 1++ -99.0 2.095 H-short, H-long ,
XMM-LH 2506 162.890 57.590 2.010 2++ -99.0 2.016 J-long, H-short
l r r r r r r r r r r
Emission line properties of Broad-line AGNs
1rlog L$_{\rm bol}$
1rlog M$_{\rm BH}$
3clog FWHM (km s$^{-1}$)
3clog L (erg s$^{-1}$)
1c(erg s$^{-1}$)
6c – continued.
1rlog L$_{\rm bol}$
1rlog M$_{\rm BH}$
3clog FWHM (km s$^{-1}$)
3clog L (erg s$^{-1}$)
1c(erg s$^{-1}$)
CDF-S 1 1.630 45.89$\pm$0.07 8.41$\pm$0.08 3.74$\pm$0.02 43.08$\pm$0.06
CDF-S 4 1.270 45.39$\pm$0.14 7.22$\pm$0.24 3.40$\pm$0.05 42.16$\pm$0.12
CDF-S 11 1.888 46.02$\pm$0.06 8.37$\pm$0.16 3.61$\pm$0.02 45.52$\pm$0.00
CDF-S 14 1.370 44.32$\pm$0.05 8.29$\pm$0.05 3.68$\pm$0.13 45.19$\pm$0.01
CDF-S 15 1.065 44.25$\pm$0.04 6.52$\pm$0.07 3.36$\pm$0.33 43.34$\pm$0.03
CDF-S 25 1.336 44.34$\pm$0.13 6.69$\pm$0.09 3.32$\pm$0.04 43.51$\pm$0.02
CDF-S 66 0.575 42.80$\pm$0.06 7.67$\pm$0.58 3.39$\pm$0.60 41.65$\pm$0.00
CDF-S 76 1.042 45.06$\pm$0.02 7.45$\pm$0.07 3.63$\pm$0.08 3.68$\pm$0.02 42.04$\pm$0.17 43.81$\pm$0.00
CDF-S 87 1.437 44.15$\pm$0.06 8.67$\pm$0.06 4.41$\pm$0.17 43.43$\pm$0.02
CDF-S 88 1.613 45.14$\pm$0.02 7.05$\pm$0.02 3.67$\pm$0.01 43.22$\pm$0.01
CDF-S 101 0.966 45.36$\pm$0.05 7.40$\pm$0.16 3.70$\pm$0.05 3.73$\pm$0.02 42.23$\pm$0.11 43.57$\pm$0.00
CDF-S 166 1.608 45.67$\pm$0.01 8.30$\pm$0.19 3.64$\pm$0.07 3.84$\pm$0.16 42.64$\pm$0.19 44.68$\pm$0.00
CDF-S 229 1.326 45.68$\pm$0.01 7.75$\pm$0.02 3.35$\pm$0.33 3.37$\pm$0.05 3.71$\pm$0.14 43.30$\pm$0.00 41.93$\pm$0.14 45.26$\pm$0.01
CDF-S 241 0.566 44.07$\pm$0.02 7.26$\pm$0.06 3.67$\pm$0.14 43.55$\pm$0.01
CDF-S 329 0.954 44.26$\pm$0.13 7.03$\pm$0.86 3.39$\pm$0.09 41.86$\pm$0.27
CDF-S 344 1.615 45.03$\pm$0.02 8.06$\pm$0.09 3.35$\pm$0.23 3.89$\pm$0.01 43.02$\pm$0.16 44.13$\pm$0.02
CDF-S 367 1.041 45.45$\pm$0.00 7.27$\pm$0.38 3.39$\pm$0.02 44.46$\pm$0.00
CDF-S 369 1.612 45.53$\pm$0.02 7.46$\pm$0.82 3.47$\pm$0.16 42.35$\pm$0.15
CDF-S 375 0.742 45.76$\pm$0.00 8.14$\pm$0.02 3.78$\pm$0.24 3.81$\pm$0.01 43.12$\pm$0.03 44.51$\pm$0.01
CDF-S 417 1.222 45.33$\pm$0.01 8.91$\pm$0.14 4.03$\pm$0.15 42.90$\pm$0.12
CDF-S 420 0.960 44.09$\pm$0.03 7.35$\pm$0.30 3.38$\pm$0.65 42.47$\pm$0.00
CDF-S 473 1.557 45.82$\pm$0.02 7.84$\pm$0.52 3.50$\pm$0.59 42.92$\pm$0.00
CDF-S 514 0.664 44.03$\pm$0.03 7.46$\pm$0.61 3.85$\pm$0.27 43.27$\pm$0.00
CDF-S 523 0.838 45.43$\pm$0.01 8.35$\pm$0.03 3.69$\pm$0.40 4.11$\pm$0.29 43.14$\pm$0.00 44.56$\pm$0.03
CDF-S 537 1.216 44.87$\pm$0.02 8.60$\pm$0.16 3.96$\pm$0.17 44.76$\pm$0.02
CDF-S 614 0.664 43.18$\pm$0.06 8.23$\pm$0.03 4.14$\pm$0.02 43.58$\pm$0.02
CDF-S 627 0.736 44.80$\pm$0.01 7.29$\pm$0.12 3.72$\pm$0.02 43.43$\pm$0.02
CDF-S 656 1.367 43.66$\pm$0.07 7.45$\pm$0.49 3.36$\pm$0.49 42.72$\pm$0.00
CDF-S 681 0.733 44.34$\pm$0.02 7.28$\pm$0.11 3.65$\pm$0.11 43.65$\pm$0.00
CDF-S 691 2.005 45.93$\pm$0.05 8.95$\pm$0.39 3.81$\pm$0.15 45.82$\pm$0.01
CDF-S 695 0.622 43.79$\pm$0.03 7.65$\pm$0.05 3.87$\pm$0.04 43.51$\pm$0.02
CDF-S 720 1.609 45.94$\pm$0.02 8.50$\pm$0.12 3.85$\pm$0.04 42.81$\pm$0.08
CDF-S 723 2.072 45.66$\pm$0.05 8.82$\pm$0.02 3.82$\pm$0.01 45.55$\pm$0.02
CDF-S 724 1.337 44.95$\pm$0.07 7.69$\pm$0.16 3.52$\pm$0.30 42.81$\pm$0.00
E-CDF-S 7 1.368 46.16$\pm$0.00 9.30$\pm$0.04 4.12$\pm$0.02 3.90$\pm$0.00 43.27$\pm$0.04 45.29$\pm$0.02
E-CDF-S 53 1.524 45.58$\pm$0.00 9.18$\pm$0.19 4.03$\pm$0.18 45.48$\pm$0.02
E-CDF-S 68 1.362 44.58$\pm$0.03 8.55$\pm$0.06 3.77$\pm$0.03 45.30$\pm$0.02
E-CDF-S 89 1.613 44.44$\pm$0.16 7.32$\pm$0.14 3.44$\pm$0.08 42.22$\pm$0.17
E-CDF-S 100 1.957 45.64$\pm$0.01 8.69$\pm$0.12 3.73$\pm$0.13 45.65$\pm$0.01
E-CDF-S 158 0.717 44.44$\pm$0.01 8.81$\pm$0.08 3.89$\pm$0.02 41.88$\pm$0.03
E-CDF-S 166 1.408 44.81$\pm$0.02 8.31$\pm$0.61 3.70$\pm$0.42 43.02$\pm$0.00
E-CDF-S 193 1.167 44.54$\pm$0.01 8.45$\pm$0.04 3.83$\pm$0.02 44.96$\pm$0.05
E-CDF-S 358 1.626 45.64$\pm$0.00 8.29$\pm$0.02 3.42$\pm$0.43 3.73$\pm$0.09 3.61$\pm$0.00 43.59$\pm$0.00 41.95$\pm$0.21 45.38$\pm$0.01
E-CDF-S 381 0.526 44.28$\pm$0.00 7.72$\pm$0.34 3.63$\pm$0.17 40.87$\pm$0.19
E-CDF-S 517 1.345 44.94$\pm$0.00 7.82$\pm$0.30 3.64$\pm$0.08 42.37$\pm$0.23
E-CDF-S 601 1.598 45.47$\pm$0.02 7.55$\pm$0.35 3.41$\pm$0.59 42.74$\pm$0.00
E-CDF-S 631 2.072 45.49$\pm$0.01 8.67$\pm$0.24 3.79$\pm$0.06 45.42$\pm$0.05
E-CDF-S 678 1.629 45.30$\pm$0.00 8.58$\pm$0.06 3.77$\pm$0.18 3.73$\pm$0.21 43.25$\pm$0.07 45.43$\pm$0.01
E-CDF-S 681 0.834 44.66$\pm$0.00 8.41$\pm$0.06 3.88$\pm$0.02 42.53$\pm$0.04
E-CDF-S 700 2.171 45.78$\pm$0.00 8.31$\pm$0.24 3.91$\pm$0.19 44.45$\pm$0.02
E-CDF-S 712 0.841 45.74$\pm$0.00 8.00$\pm$0.02 3.76$\pm$0.34 3.73$\pm$0.01 42.66$\pm$0.00 44.55$\pm$0.07
E-CDF-S 716 0.763 44.93$\pm$0.00 8.11$\pm$0.43 3.70$\pm$0.33 42.65$\pm$0.00
E-CDF-S 725 1.314 45.74$\pm$0.00 8.31$\pm$0.10 3.59$\pm$0.23 3.71$\pm$0.01 43.11$\pm$0.13 45.12$\pm$0.02
E-CDF-S 728 1.583 45.44$\pm$0.01 8.54$\pm$0.19 3.81$\pm$0.03 43.02$\pm$0.05
E-CDF-S 742 1.762 44.85$\pm$0.03 8.33$\pm$0.18 3.60$\pm$0.03 45.48$\pm$0.01
XMM-LH 5 2.138 46.55$\pm$0.01 8.24$\pm$0.08 3.35$\pm$0.54 42.77$\pm$0.00
XMM-LH 25 1.599 44.52$\pm$0.32 7.52$\pm$0.65 3.50$\pm$0.07 42.36$\pm$0.17
XMM-LH 41 1.653 45.89$\pm$0.08 8.09$\pm$0.20 3.71$\pm$0.15 44.75$\pm$0.02
XMM-LH 85 1.144 45.26$\pm$0.01 8.03$\pm$0.02 3.75$\pm$0.18 44.52$\pm$0.02
XMM-LH 119 1.406 45.25$\pm$0.07 8.55$\pm$0.08 3.83$\pm$0.02 42.97$\pm$0.03
XMM-LH 120 1.523 45.65$\pm$0.02 7.62$\pm$0.16 3.54$\pm$0.09 42.39$\pm$0.25
XMM-LH 148 1.116 46.58$\pm$0.00 8.93$\pm$0.06 3.74$\pm$0.13 46.02$\pm$0.01
XMM-LH 168 1.958 46.95$\pm$0.01 8.85$\pm$0.23 3.74$\pm$0.06 42.49$\pm$0.14
XMM-LH 176 1.533 45.77$\pm$0.01 8.12$\pm$0.04 3.43$\pm$0.01 3.55$\pm$0.01 43.19$\pm$0.02 45.30$\pm$0.01
XMM-LH 191 0.787 45.94$\pm$0.01 7.94$\pm$0.02 3.86$\pm$0.01 3.82$\pm$0.18 42.53$\pm$0.02 44.16$\pm$0.02
XMM-LH 261 3.406 46.77$\pm$0.05 7.58$\pm$0.05 3.37$\pm$0.00 45.02$\pm$0.02
XMM-LH 270 1.576 45.66$\pm$0.01 8.39$\pm$0.06 3.75$\pm$0.01 3.55$\pm$0.05 3.73$\pm$0.14 43.50$\pm$0.02 41.93$\pm$0.11 45.17$\pm$0.01
XMM-LH 321 1.008 45.04$\pm$0.03 7.44$\pm$0.03 3.55$\pm$0.01 44.24$\pm$0.00
XMM-LH 332 1.676 46.31$\pm$0.01 8.08$\pm$0.02 3.74$\pm$0.02 44.64$\pm$0.01
XMM-LH 354 3.409 46.95$\pm$0.02 8.93$\pm$0.17 3.76$\pm$0.16 45.93$\pm$0.02
XMM-LH 364 0.932 44.53$\pm$0.03 7.55$\pm$0.09 3.65$\pm$0.04 44.09$\pm$0.02
XMM-LH 387 1.449 45.04$\pm$0.02 7.58$\pm$0.06 3.38$\pm$0.02 3.47$\pm$0.01 42.90$\pm$0.03 44.61$\pm$0.02
XMM-LH 406 1.283 45.39$\pm$0.06 8.77$\pm$0.02 3.92$\pm$0.01 3.80$\pm$0.03 43.03$\pm$0.01 44.86$\pm$0.01
XMM-LH 430 1.553 46.48$\pm$0.02 8.33$\pm$0.17 3.90$\pm$0.17 44.54$\pm$0.02
XMM-LH 453 1.214 45.02$\pm$0.06 7.41$\pm$0.04 3.52$\pm$0.02 44.26$\pm$0.02
XMM-LH 456 0.877 45.28$\pm$0.02 7.44$\pm$0.30 3.62$\pm$0.06 41.76$\pm$0.15
XMM-LH 475 1.205 47.07$\pm$0.01 8.90$\pm$0.00 3.69$\pm$0.00 3.87$\pm$0.14 44.13$\pm$0.00 45.66$\pm$0.01
XMM-LH 523 1.217 45.65$\pm$0.06 7.84$\pm$0.08 3.60$\pm$0.03 42.57$\pm$0.08
XMM-LH 529 1.940 46.74$\pm$0.04 8.55$\pm$0.11 3.68$\pm$0.09 42.16$\pm$0.20
XMM-LH 532 1.675 45.01$\pm$0.03 8.29$\pm$0.31 3.60$\pm$0.03 3.73$\pm$0.02 42.91$\pm$0.06 45.01$\pm$0.01
XMM-LH 553 1.440 46.36$\pm$0.01 8.68$\pm$0.03 3.76$\pm$0.01 43.49$\pm$0.02
XMM-LH 555 1.674 45.84$\pm$0.03 8.99$\pm$0.13 4.01$\pm$0.04 43.11$\pm$0.07
XMM-LH 591 1.535 45.65$\pm$0.05 8.22$\pm$0.08 3.62$\pm$0.32 43.18$\pm$0.00
XMM-LH 595 1.602 45.08$\pm$0.09 8.50$\pm$0.29 3.78$\pm$0.27 43.09$\pm$0.07
XMM-LH 604 2.104 46.47$\pm$0.02 8.66$\pm$0.83 3.54$\pm$0.62 42.86$\pm$0.00
XMM-LH 2020 1.728 45.96$\pm$0.01 7.98$\pm$0.10 3.55$\pm$0.01 42.99$\pm$0.02
[Allevato et al.(2011)]Allevato11 Allevato, V., Finoguenov, A., Cappelluti, N., et al. 2011, ApJ, 736, 99
[Babić et al.(2007)]Babic07 Babić, A., Miller, L., Jarvis, M. J., et al. 2007, A&A, 474, 755
[Barger et al.(2005)]Barger05 Barger, A. J., Cowie, L. L., Mushotzky, R. F., et al. 2005, AJ, 129, 578
[Barger et al.(2014, priv. comm.)]Barger14 Barger, A. J., et al. 2014, priv. comm.
[Bongiorno et al.(2007)]Bongiorno07 Bongiorno, A., Zamorani, G., Gavignaud, I., et al. 2007, A&A, 472, 443
[Brandt & Hasinger(2005)]Brandt05 Brandt, W. N. & Hasinger, G. 2005, ARA&A, 43, 827
[Brandt & Alexander(2010)]Brandt10 Brandt, W. N. & Alexander, D. M. 2010, PNAS, 107, 7184
[Brunner et al.(2008)]Brunner08 Brunner, H., Cappelluti, N., Hasinger, G., et al. 2008, A&A, 479, 283
[Choi et al.(2012)]Choi12 Choi, E., Ostriker, J. P., Nabb, T., et al. 2012, ApJ, 754, 125
[Choi et al.(2014)]Choi14 Choi, Y., Gibson, R. R., Becker, A., et al. 2014, ApJ, 782, 37
[Cirasuolo et al.(2007)]Cirasuolo07 Cirasuolo, M., McLure, R. J., Dunlop, J. S., et al. 2007, MNRAS, 380, 585
[Cisternas et al.(2011)]Cisternas11 Cisternas, M., Jahnke, K., Inskip, K. J., et al. 2011, ApJ, 726, 57
[Cowie et al.(2003)]Cowie03 Cowie, L. L., Barger, A. J., Bautz, M. W., et al. 2003, ApJ, 584L, 57
[Di Matteo et al.(2005)]DiMatteo05 Di Matteo, T., Springel, V., Hernquist, L. 2005, Nature, 433, 604
[Fabian(1999)]Fabian99 Fabian, A. C. 1999, MNRAS, 308, 39
[Fabian et al.(2008)]Fabian08 Fabian, A. C., Vasudevan, R. V., Gandhi, P. 2008, MNRAS, 385, 43
[Fan et al.(2014)]Fan14 Fan, L., Fang, G., Chen, Y., et al. 2014, ApJ Letters, 784, L9
[Fanidakis et al.(2012)]Fanidakis12 Fanidakis, N., Baugh, C. M., Benson, A. J., et al. 2012, MNRAS, 419, 2797
[Ferrarese & Merritt(2000)]Ferrarese00 Ferrarese, L. & Merritt, D. 2000, ApJ, 539, 9
[Fotopoulou et al.(2012)]Fotopoulou12 Fotopoulou, S., Salvato, M., Hasinger, G., et al. 2012, ApJS, 198, 1
[Gabor et al.(2009)]Gabor09 Gabor, J. M., Impey, C. D., Jahnke, K., et al. 2009, ApJ, 691, 705
[Gavignaud et al.(2008)]Gavignaud08 Gavignaud, I., Wisotzki, L., Bongiorno, A., et al. 2008, A&A, 492, 637
[Gebhardt et al.(2000)]Gebhardt00 Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, 13
[Giacconi et al.(2002)]Giacconi02 Giacconi, R., Zirm, A., Wang, J., et al. 2002, ApJS, 139, 369
[Graham et al.(2011)]Graham11 Graham, A. W., Onken, C. A., Athanassoula, E., et al. 2011, MNRAS, 412, 2211
[Greene & Ho(2005)]Greene05 Greene, J. E., & Ho, L. C. 2005, ApJ, 630, 122
[Gültekin et al.(2009)]Gultekin09 Gültekin, K., Richstone, D., Gebhardt, K., et al. 2009, ApJ, 698, 198
[Harris et al.(2013)]Harris13 Harris, K. A., Williger, G. M., Haberzettl, L., et al. 2013, MNRAS, 435, 3125
[Hasinger et al.(2001)]Hasinger01 Hasinger, G., ltieri, B., Arnaud, M., et al. 2001, A&A, 365L, 45
[Hasinger et al.(2005)]Hasinger05 Hasinger, G., Miyaji, T., Schmidt, M. 2005, A&A, 441, 417
[Hasinger(2008)]Hasinger08 Hasinger, G. 2008, A&A, 490, 905
[Hopkins & Hernquist(2006)]Hopkins06 Hopkins, P. F., & Hernquist, L. 2006, ApJS, 166, 1
[Hopkins et al.(2007)]Hopkins07 Hopkins, P. F., Richards, G. T., Hernquist, L. 2007, ApJ, 654, 731
[Iwamuro et al.(2012)]Iwamuro12 Iwamuro, F., Moritani, Y., Yabe, K., et al. 2012, PASJ, 64, 59
[Ji et al.(2014)]Ji14 Ji, I., Peirani, S., Yi, S. K. 2014, arXiv:1405.1807v1
[Kaspi et al.(2000)]Kaspi00 Kaspi, S., Smith, P. S., Netzer, H., et al. 2000, ApJ, 533, 631
[Kelly et al.(2010)]Kelly10 Kelly, B. C., Vestergaard, M., Fan, X., et al. 2010, ApJ, 719, 1315
[Kimura et al.(2010)]Kimura10 Kimura, M., et al. 2010, PASJ, 62, 1135
[Kocevski et al.(2012)]Kocevski12 Kocevski, D. D., Faber, S. M., Mozena, M. et al. 2012, ApJ, 744, 148
[Kollmeier et al.(2006)]Kollmeier06 Kollmeier, J. A., Onken, C. A., Kochanek, C. S., et al. 2006, ApJ, 648, 128
[Kormendy & Richstone(1995)]Kormendy95 Kormendy, J., & Richstone, D. 1995, ARA&A, 33, 581
[La Franca et al.(2005)]LaFranca05 La Franca, F., Fiore, F., Comastri, A., et al. 2005, ApJ, 635, 864
[Lehmann et al.(2000)]Lehmann00 Lehmann, I., Hasinger, G., Schmidt, M., et al. 2000, A&A, 354, 34
[Lehmann et al.(2001)]Lehmann01 Lehmann, I., Hasinger, G., Schmidt, M., et al. 2001, A&A, 371, 833
[Lehmer et al.(2005)]Lehmer05 Lehmer, B. D., Brandt, W. N., Alexander, D. M., et al. 2005, ApJS, 161, 21
[Lockman et al.(1986)]Lockman86 Lockman, F. J., Jahoda, K., & McCammon, D. 1986, ApJ, 302, 432
[Lotz et al.(2008)]Lotz08 Lotz, J. M., Jonsson, P., Cox, T. J., & Primack, J. R. 2008, MNRAS, 391, 1137
[Lusso et al.(2010)]Lusso10 Lusso, E., Comastri, A., Vignali, C., et al., 2010, A&A, 512, 34
[Lusso et al.(2012)]Lusso12 Lusso, E., Comastri, B. D., Simmons, M. M., et al. 2012, MNRAS, 425, 623
[Madau & Dickinson(2014)]Madau14 Madau, P. & Dickinson, M. 2014, arXiv:1403.0007v2
[Magorrian et al.(1998)]Magorrian98 Magorrian, J., Tremaine, S., Richstone, D., et al. 1998, AJ, 115, 2285
[Mainieri et al.(2011)]Mainieri11 Mainieri, V., Bongiorno, A., Merloni, A. 2011, A&A, 535, A80
[Marconi et al.(2004)]Marconi04 Marconi, A., Risaliti, G., Gilli, R., et al. 2004, MNRAS, 351, 169
[Masters & Capak(2011)]Masters11 Masters, D., & Capak, P. 2011, PASP, 123, 638
[Matsuoka et al.(2007)]Matsuoka07 Matsuoka, Y., Oyabu, S., Tsuzuki, Y., Kawara, K. 2007, ApJ, 663, 781
[Matsuoka et al.(2013)]Matsuoka13 Matsuoka, K., Silverman, J. D., Schramm, M., et al. 2013, ApJ, 771, 64
[McConnell & Ma(2013)]McConnell13 McConnell, N. J., & Ma, C.-P. 2013, ApJ, 764, 184
[McGill et al.(2008)]McGill08 McGill, K. L., Woo, J., Treu, T., Malkan, M. A. 2008, ApJ, 673, 703
[McLure & Dunlop(2004)]McLure04 McLure, R. J., & Dunlop, J. S. 2004, MNRAS, 352, 1390
[McLure & Jarvis(2002)]McLure02 McLure, R. J, & Jarvis, M. J. 2002, MNRAS, 337, 109
[Merloni & Heinz(2008)]Merloni08 Merloni, A., & Heinz, S. 2008, MNRAS, 388, 1011
[Merloni et al.(2010)]Merloni10 Merloni, A., Bongiorno, A., Bolzonella, M., et al. 2010, ApJ, 708, 137
[Merritt & Ferrarese(2001)]Merritt01 Merritt, D., & Ferrarese, L. 2001, ApJ, 547, 140
[Mullaney et al.(2012)]Mullaney12 Mullaney, J. R., Pannella, M., Daddi, E., et al. 2012, MNRAS, 419, 95
[Nobuta et al.(2012)]Nobuta12 Nobuta, K., Akiyama, M., Ueda, Y. et al. 2012, ApJ, 761, 143
[Rafiee & Hall(2011)]Rafiee11 Rafiee, A., & Hall, P. B. 2011, ApJS, 194, 42
[Rosario et al.(2015)]Rosario15 Rosario, D. J., McIntosh, D. H., van der Wel, A., et al. 2015, A&A, 573, 85
[Rovilos et al.(2011)]Rovilos11 Rovilos, E., Fotopoulou, S., Salvato, M., et al. 2011, A&A, 529, 135
[Schawinski et al.(2010)]Schawinski10 Schawinski, K., Urry, C. M., Virani, S., et al. 2010, ApJ, 711, 284
[Schawinski et al.(2011)]Schawinski11 Schawinski, K., Treister, E., Urry, C. M. et al. 2011, ApJ Letters, 727, L31
[Schawinski et al.(2012)]Schawinski12 Schawinski, K., Simmons, B. D., Urry, C. M., et al. 2012, MNRAS, 425, L61
[Schneider et al.(2010)]Schneider10 Schneider, D. P., Richards, G. T., Hall, P. B., et al. 2010, AJ, 139, 2360
[Schramm & Silverman(2013)]Schramm13 Scharamm, M., and Silverman, J. D. 2013, ApJ, 767, 13
[Schulze & Gebhardt(2011)]Schulze11 Schulze, A., & Gebhardt, K. 2011, ApJ, 729, 21
[Schulze & Wisotzki(2010)]Schulze10 Schulze, A., & Wisotzki, L. 2010, A&A, 516, 87
[Shankar et al.(2009)]Shankar09 Shankar, F., Weinberg D, Miralda-Escude, J. 2009, ApJ, 690, 20
[Shen & Liu(2012)]Shen12 Shen, Y., & Liu, X. 2012, ApJ, 753, 125
[Shen et al.(2008)]Shen08 Shen, Y., Greene, J. E., Strauss, M. A., et al. 2008, ApJ, 680, 169
[Shen et al.(2011)]Shen11 Shen, Y., Richards, G. T., Strauss, M. A., et al. 2011, ApJS, 194, 45
[Silverman et al.(2008a)]Silverman08a Silverman, J. D., Mainieri, V., Lehmer B. D., et al. 2008, ApJ, 675, 1025
[Silverman et al.(2008b)]Silverman08b Silverman, J. D., Green, P. J., Barkhouse, W. A., et al. 2008, ApJ, 679, 118
[Silverman et al.(2010)]Silverman10 Silverman, J. D., Mainieri, V., Salvato, M., et al. 2010, ApJS, 191, 124
[Silverman et al.(2011)]Silverman11 Silverman, J. D., Kampczyk, P., Jahnke, K., et al. 2011, ApJ, 743, 2
[Simpson(2005)]Simpson05 Simpson, C. 2005, MNRAS, 360, 565
[Springel et al.(2005)]Springel05 Springel, V., Di Matteo, T., & Hernquist, L. 2005, MNRAS, 361, 776
[Stark et al.(1992)]Stark92 Stark, A. A., Gammie, C. F., Wilson, R. W., et al. 1992, ApJS, 79, 77
[Steffen et al.(2003)]Steffen03 Steffen, A. T., Barger, A. J., Cowie, L. L., et al. 2003, ApJ, 596L, 23
[Steinhardt & Elvis(2010)]Steinhardt10 Steinhardt, C. L., & Elvis, M. 2010, MNRAS, 402, 2637
[Szokoly et al.(2004)]Szokoly04 Szokoly, G. P., Bergeron, J., Hasinger, G., et al. 2004, ApJS, 155, 271
[Tremaine et al.(2002)]Tremaine02 Tremaine, S., Gebhardt, K., Bender, R., et al. 2002, ApJ, 574, 740
[Trump et al.(2011)]Trump11 Trump, J. R., Impey, C. D., Kelly, B. C., et al. 2011, ApJ, 733, 60
[Ueda et al.(2003)]Ueda03 Ueda, Y., Akiyama, M., Ohta, K., et al. 2003, ApJ, 598, 886
[Vasudevan & Fabian(2007)]Vasudevan07 Vasudevan, R. V., and Fabian, A. C. 2007, MNRAS, 381, 1235
[Vestergaard(2002)]Vestergaard02 Vestergaard, M. 2002, ApJ, 571, 733
[Vestergaard & Osmer(2009)]Vestergaard09 Vestergaard, M., & Osmer, P. S. 2009, ApJ, 699, 800
[Vestergaard & Peterson(2006)]Vestergaard06 Vestergaard, M., & Peterson, B. M. 2006, ApJ, 641, 689
[Vestergaard & Wilkes(2001)]Vestergaard01 Vestergaard, M., & Wilkes, B. J. 2001, ApJS, 134, 1
[Vignali et al.(2003)]Vignali03 Vignali C., Brandt W. N., Schneider D. P., 2003, AJ, 125, 433
[Villforth et al.(2014)]Villforth14 Villforth, C., Hamann, F., Rosario, D. J., et al. 2014, MNRAS, 439, 3342
[Wang et al.(2009)]Wang09 Wang, J., Dong, X., Wang, T. et al. 2009, ApJ, 707, 1334
[Woo & Urry(2002)]Woo02 Woo, J., & Urry, C. M. 2002, ApJ, 579, 530
[Woo et al.(2013)]Woo13 Woo, J., Schulze, A., Park, D. et al. 2013, ApJ, 772, 49
[Xue et al.(2011)]Xue11 Xue, Y. Q., Luo, B., Brandt, W. N., et al. 2011, ApJS, 195, 10
|
1511.00776
|
In this work, we discuss techniques for coherently detecting turbo coded
orthogonal frequency division multiplexed (OFDM) signals, transmitted through
frequency selective Rayleigh (the magnitude of each channel tap is Rayleigh
distributed) fading channels having a uniform power delay
profile. The channel output is further distorted
by a carrier frequency and phase offset, besides additive white Gaussian noise
(AWGN). A new frame structure for OFDM, consisting of a known preamble, cyclic
prefix, data and known postamble is proposed, which has a higher throughput
compared to the earlier work. A robust turbo decoder is proposed, which
functions effectively over a wide range of signal-to-noise ratio (SNR).
Simulation results show that it is possible to achieve a bit-error-rate (BER)
of $10^{-5}$ at an SNR per bit as low as 8 dB and throughput of 82.84%, using
a single transmit and two receive antennas. We also demonstrate that the
practical coherent receiver requires just about 1 dB more power compared to
that of an ideal coherent receiver, to attain a BER of $10^{-5}$. The key
contribution to the good performance of the practical coherent receiver is
due to the use of a long preamble (512 QPSK symbols), which is
perhaps not specified in any of the current wireless communication standards.
We have also shown from computer simulations that, it is possible to
obtain even better BER performance, using a better code.
A simple and approximate Cramér-Rao bound on the
variance of the frequency offset estimation error for coherent detection, is
The proposed algorithms are well suited for implementation on a DSP-platform.
OFDM, coherent detection, matched filtering, turbo codes, frequency selective
Rayleigh fading, channel capacity.
§ INTRODUCTION
Future wireless communication standards aim to push the existing data-rates
higher. This can only be achieved with the help of coherent communications,
since they give the lowest bit-error-rate (BER) performance for a given
signal-to-noise ratio (SNR). Conversely, they require the lowest SNR to
attain a given BER, resulting in enhanced battery life. If we look at a
mobile, it indicates a typical received signal strength equal to $-100$ dBm
($10^{-10}$ mW). However this
is not the signal-to-noise ratio! Therefore, the question
is: What is the operating SNR of the mobiles? Would it be possible to achieve
the same performance by transmitting at a lower power? The recent advances in
cooperative communications has resulted in low complexity solutions, that are
not necessarily power efficient <cit.>. In fact, it is
worth quoting the following from <cit.>:
* The Myth: Sixty years of research following Shannon's
pioneering paper has led to telecommunications
solutions operating arbitrarily close to the channel
capacity–“flawless telepresence” with zero error is
available to anyone, anywhere, anytime across the globe.
* The Reality: Once we leave home or the office, even
top of the range iPhones and tablet computers fail to
maintain “flawless telepresence” quality. They also fail
to approach the theoretical performance predictions.
The 1000-fold throughput increase of the best third-
generation (3G) phones over second-generation (2G)
GSM phones and the 1000-fold increased teletraffic
predictions of the next decade require substantial
further bandwidth expansion toward ever increasing
carrier frequencies, expanding beyond the radio-
frequency (RF) band to optical frequencies, where
substantial bandwidths are available.
The transmitter and receiver algorithms proposed in this paper and
in <cit.> are well suited for implementation on a DSP processor
or hardwired and may perhaps not require quantum computers, as mentioned in
<cit.>. The reader is also referred to the brief commentary
on channel estimation and synchronization in page 1351 and also to the
noncoherent schemes in page 1353 of <cit.>, which clearly state
that cooperative communications avoid coherent receivers due to
Broadly speaking, the wireless communication device needs to have the
following features:
* maximize the bit-rate
* minimize the bit-error-rate
* minimize transmit power
* minimize transmission bandwidth
A rather disturbing trend in the present day wireless communication systems is
to make the physical layer very simple and implement it in hardware, and
allot most of the computing resources to the application layer, e.g., for
internet surfing, video conferencing etc. While hardware implementation of
the physical layer is not an issue, in fact, it may even be preferred over
software implementation in some situations, the real cause for concern is the
tendency to make it “simple”, at the cost of BER performance. Therefore,
the questions are:
* was signal processing for coherent communications given a
chance to prove itself, or was it ignored straightaway, due to
“complexity” reasons?
* are the present day single antenna wireless transceivers,
let alone multi-antenna systems, performing anywhere near channel
This paper demonstrates that coherent receivers need not be
restricted to textbooks alone, in fact they can be implemented with
linear (not exponential) complexity. The need of the hour is a paradigm
shift in the way the wireless communication systems are implemented.
In this article, we dwell on coherent receivers based on orthogonal frequency
division multiplexing (OFDM),
since it has the ability to mitigate intersymbol interference (ISI) introduced
by the frequency selective fading channel
<cit.>. The “complexity” of
coherent detection can be overcome by means of parallel processing, for
which there is a large scope. We wish to emphasize
that this article presents a proof-of-concept, and is hence not constrained
by the existing standards in wireless communication.
We begin by first outlining the tasks of a coherent receiver. Next, we scan
the literature on each of these tasks to find out the state-of-the-art, and
finally end this section with our contributions.
The basic tasks of the coherent receiver would be:
* To correctly identify the start of the (OFDM) frame (SoF), such that
the probability of false alarm (detecting an OFDM
frame when it is not present) or equivalently the probability of
erasure/miss (not detecting the OFDM frame when it is
present) is minimized. We refer to this step as timing synchronization.
* To estimate and compensate the carrier frequency offset (CFO),
since OFDM is known to be sensitive to CFO. This task is referred to as
carrier synchronization.
* To estimate the channel impulse/frequency response.
* To perform (coherent) turbo decoding and recover the data.
To summarize, a coherent receiver at the physical layer ensures that the
medium access control (MAC) is not burdened by frequent requests
for retransmissions.
A robust timing and frequency synchronization for OFDM signals transmitted
through frequency selective AWGN channels is presented in <cit.>.
Timing synchronization in OFDM is addressed in <cit.>.
Various methods of carrier frequency synchronization for OFDM are
given in <cit.>.
Joint timing and CFO estimation is discussed in
Decision directed coherent detection of
OFDM in the presence of Rayleigh fading is treated in <cit.>.
A factor graph approach to the iterative (coherent) detection of OFDM
in the presence of carrier frequency offset and phase noise
is presented in <cit.>. OFDM detection in the presence of
intercarrier interference (ICI) using block whitening is discussed in
<cit.>. In <cit.>, a turbo receiver is proposed for detecting
OFDM signals in the presence of ICI and inter antenna interference.
Most flavors of
the channel estimation techniques discussed in the literature are done in the
frequency domain, using pilot symbols at regular intervals in the
time/frequency grid
<cit.>. Iterative joint channel
estimation and multi-user detection for multi-antenna OFDM is discussed in
<cit.>. Noncoherent detection of coded OFDM in the absence of
frequency offset and assuming that the channel frequency response to be
constant over a block of symbols, is considered in <cit.>.
Expectation maximization (EM)-based joint channel estimation and exploitation
of the diversity gain from IQ imbalances is addressed in <cit.>.
Detection of OFDM signals, in the context of spectrum sensing for cognitive
is considered in <cit.>. However, in both these papers,
the probability of false alarm is quite high (5%).
In <cit.>, discrete cosine transform (DCT)
based OFDM is studied in the presence of frequency offset and noise, and
its performance is compared with the discrete Fourier transform (DFT) based
OFDM. It is further shown in <cit.> that the performance of DFT-OFDM
is as good as DCT-OFDM, for small frequency offsets.
A low-power OFDM
implementation for wireless local area networks (WLAN) is addressed in
<cit.>. OFDM is a suggested modulation technique for digital video
broadcasting <cit.>. It has also been proposed for
optical communications <cit.>.
The novelty of this work lies in the use of a filter that is matched to the
preamble, to acquire timing
synchronization <cit.> (start-of-frame (SoF) detection).
Maximum likelihood (ML) channel estimation using the preamble is performed.
This approach does not require any knowledge of the channel and noise
The main contributions of this paper are the following:
* It is shown that, for a sufficiently long preamble, the variance of the
estimator proposed in eq. (28) of <cit.> approaches zero.
* A known postamble is used to accurately estimate the residual frequency
offset for large data lengths, thereby increasing the throughput
compared to <cit.>.
* Turbo codes are used to attain BER performance closer to
channel capacity compared to any other earlier work in the open
literature, for channels having a uniform power delay profile (to the
best of the authors knowledge, there is no similar
work on the topic of this paper, other than <cit.>).
* A robust turbo decoder is proposed, which performs effectively over
a wide range of SNR (0 – 30 dB).
* While most papers in the literature try to attain the channel capacity
for a given SNR, this work tries to attain the minimum SNR for
error-free transmission, for a given channel capacity.
In a multiuser scenario, the suggested technique is OFDM-TDMA. The uplink
and downlink may be implemented using time division duplex (TDD) or
frequency division duplex (FDD) modes.
This paper is organized as follows. Section <ref> describes the
system model. The receiver algorithms are presented in
section <ref>. The bit-error-rate (BER) results from
computer simulations are given in section <ref>. Finally, in
section <ref>, we discuss the conclusions and future work.
§ SYSTEM MODEL
(a) The frame structure. (b) System model. $k$ denotes the frame
index and $n$ denotes the time index in a given frame.
We assume that the data to be transmitted is organized into frames, as
depicted in Figure <ref>.
The frame consists of a known preamble of length $L_p$ symbols, a cyclic
prefix of length $L_{cp}$, followed by data of length $L_d$ symbols.
Thus, the total length of the frame is
\begin{eqnarray}
\label{Eq:Pap8_Eq1}
L = L_p + L_{cp} + L_d.
\end{eqnarray}
Let us assume a channel span equal to $L_h$. The channel span assumed by the
receiver is $L_{hr}$ ($> L_h$). The length of the cyclic prefix is
\begin{eqnarray}
\label{Eq:Pap8_Eq2}
L_{cp} = L_{hr} - 1.
\end{eqnarray}
Throughout the manuscript, we use tilde to denote complex quantities.
However, complex (QPSK) symbols will be denoted without a tilde e.g.
$S_{1,\, n}$. Boldface letters denote vectors or matrices.
The channel coefficients $\tilde h_{k,\, n}$ for the $k^{th}$ frame are
$\mathscr{CN}(0,\, 2\sigma^2_f)$ and independent over time ($n$), that is:
\begin{eqnarray}
\label{Eq:Pap8_Eq3}
\frac{1}{2}
\left[
\tilde h_{k,\, n}
\tilde h_{k,\, n-m}^*
\right] = \sigma^2_f \delta_K(m)
\end{eqnarray}
where “*” denotes complex conjugate and $\delta_K(\cdot)$ is the
Kronecker delta function. This implies a uniform channel power delay profile.
The channel is assumed to be quasi-static, that is $\tilde h_{k,\, n}$
is time-invariant over one frame and varies independently from frame-to-frame,
that is
\begin{eqnarray}
\label{Eq:Pap8_Eq4}
\frac{1}{2}
\left[
\tilde h_{k,\, n}
\tilde h_{j,\, n}^*
\right] = \sigma^2_f \delta_K(k-j)
\end{eqnarray}
where $k$ and $j$ denote the frame indexes. The AWGN noise samples
$\tilde w_{k,\, n}$ for the $k^{th}$ frame at time $n$ are
$\mathscr{CN}(0,\, 2\sigma^2_w)$. The frequency offset
$\omega_k$ for the $k^{th}$ frame is uniformly distributed over
$[-0.04,\, 0.04]$ radian <cit.>.
The phase offset $\theta_k$ for the $k^{th}$ frame is uniformly distributed
over $[0,\, 2\pi)$.
Both $\omega_k$ and $\theta_k$ are fixed for a frame and vary randomly
from frame-to-frame.
Note that:
\begin{eqnarray}
\label{Eq:Pap8_Eq4_1}
\tilde s_{1,\, n} & = & \frac{1}{L_p}
\sum_{i=0}^{L_p-1}
S_{1,\, i}
\mathrm{e}^{\,\mathrm{j}\, 2\pi ni/L_p}
\quad \mbox{for $0\le n \le L_p-1$} \nonumber \\
\tilde s_{k,\, 3,\, n}
& = & \frac{1}{L_d}
\sum_{i=0}^{L_d-1}
S_{k,\, 3,\, i}
\mathrm{e}^{\,\mathrm{j}\, 2\pi ni/L_d}
\quad \mbox{for $0\le n \le L_d-1$} \nonumber \\
\tilde s_{k,\, 2,\, n}
& = & \tilde s_{k,\, 3,\, L_d-L_{cp}+n}
\quad \mbox{for $0 \le n \le L_{cp}-1$}.
\end{eqnarray}
We assume $S_{k,\, 3,\, i}\in \pm 1 \pm \mathrm{j}$. Since we require:
\begin{eqnarray}
\label{Eq:Pap8_Eq4_2}
\left[
\left|
\tilde s_{1,\, n}
\right|^2
\right] & = &
\left[
\left|
\tilde s_{k,\, 3,\, n}
\right|^2
\right] = 2/L_d \stackrel{\Delta}{=} \sigma^2_s
\end{eqnarray}
we must have $S_{1,\, i}\in \sqrt{L_p/L_d}\,(\pm 1 \pm \mathrm{j})$. In other
words, the average power of the preamble part must be equal to the average
power of the data part.
The received signal for the $k^{th}$ frame can be written
as (for $0 \le n \le L+L_h-2$):
\begin{eqnarray}
\label{Eq:Pap8_Eq5}
\tilde r_{k,\, n} & = & \left(
\tilde s_{k,\, n} \star \tilde h_{k,\, n}
\right)\,
\mathrm{e}^{\,\mathrm{j}(\omega_k n+\theta_k)} +
\tilde w_{k,\, n} \nonumber \\
& = & \tilde y_{k,\, n}
\mathrm{e}^{\,\mathrm{j}(\omega_k n+\theta_k)} +
\tilde w_{k,\, n}
\end{eqnarray}
where “$\star$” denotes convolution and
\begin{eqnarray}
\label{Eq:Pap8_Eq5_1}
\tilde y_{k,\, n} = \tilde s_{k,\, n} \star \tilde h_{k,\, n}.
\end{eqnarray}
The set of received samples can be denoted by the vector:
\begin{eqnarray}
\label{Eq:Pap8_Eq5_2}
\tilde\mathbf{r}_k =
\left[
\begin{array}{ccc}
\tilde r_{k,\, 0} & \ldots & \tilde r_{k,\, L+L_h-2}
\end{array}
\right].
\end{eqnarray}
§ RECEIVER
In this section we discuss the key receiver algorithms, namely, start of frame
(SoF), coarse/fine frequency offset, channel and noise variance estimation and
finally data detection.
§.§ Start of Frame and Coarse Frequency Offset Estimation
Let us assume that for the $k^{th}$ frame, the channel impulse response
is known at the receiver. The channel length assumed by the receiver is
$L_{hr}(> L_h)$ such that the first $L_h$ coefficients are identical to the
channel coefficients and the remaining $L_{hr}-L_h$ coefficients are zeros.
Define the $m^{th}$
($0 \le m \le L_{cp}+L_d+L_h+L_{hr}-2$) received vector as:
\begin{eqnarray}
\label{Eq:Pap8_Eq5_3}
\tilde\mathbf{r}_{k,\, m} =
\left[
\begin{array}{ccc}
\tilde r_{k,\, m} & \ldots & \tilde r_{k,\, m+L_p-L_{hr}}
\end{array}
\right].
\end{eqnarray}
The steady-state[By steady-state we mean that all the channel
coefficients are involved in the convolution to generate $\tilde y_{k,\, n}$
in (<ref>)] preamble part of the transmitted signal appearing at
the channel output can be represented by a vector:
\begin{eqnarray}
\label{Eq:Pap8_Eq5_4}
\tilde\mathbf{y}_{k,\, 1} =
\left[
\begin{array}{ccc}
\tilde y_{k,\, L_{hr}-1} & \ldots & \tilde y_{k,\, L_p-1}
\end{array}
\right].
\end{eqnarray}
The non-coherent maximum likelihood (ML) rule for frame detection
can be stated as <cit.>: Choose that time as the start of frame
and that frequency $\hat\omega_k$, which jointly maximize the conditional pdf:
\begin{eqnarray}
\label{Eq:Pap8_Eq6}
\max_{m,\,\hat\omega_k}
\int_{\theta_k=0}^{2\pi}
\left(
\tilde\mathbf{r}_{k,\, m}|\tilde\mathbf{y}_{k,\, 1},\hat\omega_k,\theta_k
\right) p(\theta_k)\, d\theta_k.
\end{eqnarray}
substituting for the joint pdf and $p(\theta_k)$ and defining
\begin{eqnarray}
\label{Eq:Pap8_Eq6_1}
\end{eqnarray}
we get:
\begin{eqnarray}
\label{Eq:Pap8_Eq7}
\lefteqn{
\max_{m,\,\hat\omega_k}
\frac{1}{2\pi}
\frac{1}{(2\pi\sigma^2_w)^{L_1}}} \nonumber \\
& &
\int_{\theta=0}^{2\pi}
\exp
\left(
\frac{
\sum_{i=0}^{L_1-1}
\left|
\tilde r_{m+i}-\tilde y_{k,\, L_{hr}-1+i}
\,
\mathrm{e}^{\,\mathrm{j}(\hat\omega_k i+\theta)}
\right|^2
\right) \nonumber \\
& & \mbox{ } \times \, d\theta.
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Pap8_Eq8}
\theta = \hat\omega_k(L_{hr}-1) + \theta_k
\end{eqnarray}
incorporates the phase accumulated by the frequency offset over the first
$L_{hr}-1$ samples, besides the initial phase $\theta_k$. Observe that $\theta$
is also uniformly distributed in $[0,\, 2\pi)$.
One of the terms in the exponent is:
\begin{eqnarray}
\label{Eq:Pap8_Eq9}
\frac{\sum_{i=0}^{L_1-1}|\tilde r_{m+i}|^2}{2\sigma^2_w}
\end{eqnarray}
is approximately proportional to the average received signal power, for large
values of $L_p$ and $L_p \gg L_{hr}$, and is hence (approximately) independent
of $m$ and $\theta$. The other exponential term
\begin{eqnarray}
\label{Eq:Pap8_Eq10}
\frac{\sum_{i=0}^{L_1-1}|\tilde y_{k,\, L_{hr}-1+i}|^2}{2\sigma^2_w}
\end{eqnarray}
is clearly independent of $m$ and $\theta$. Therefore we are only left with
(ignoring constants):
\begin{eqnarray}
\label{Eq:Pap8_Eq11}
\hspace*{0.1in}
\lefteqn{
\max_{m,\,\hat\omega_k}
\frac{1}{2\pi}} \nonumber \\
& &
\hspace*{-0.15in}
\int_{\theta=0}^{2\pi}
\exp
\left(
\frac{
\Re
\left
\{
\sum_{i=0}^{L_1-1}
\tilde r_{m+i}\,\tilde y_{k,\, L_{hr}-1+i}^*
\,
\mathrm{e}^{-\mathrm{j}(\hat\omega_k i+\theta)}
\right
\}
\right) \nonumber \\
& & \mbox{ } \times \, d\theta
\end{eqnarray}
which simplifies to <cit.>:
\begin{eqnarray}
\label{Eq:Pap8_Eq12}
\max_{m,\,\hat\omega_k}
\left(
\frac{A_{m,\,\hat\omega_k}}
\right)
\end{eqnarray}
where $I_0(\cdot)$ is the modified Bessel function of the zeroth-order and
\begin{eqnarray}
\label{Eq:Pap8_Eq13}
A_{m,\,\hat\omega_k} = \left|
\sum_{i=0}^{L_1-1}
\tilde r_{m+i}\,\tilde y_{k,\, L_{hr}-1+i}^*
\,
\mathrm{e}^{-\mathrm{j}\,\hat\omega_k i}
\right|.
\end{eqnarray}
Noting that $I_0(x)$ is a monotonic function of $x$ and ignoring constants,
the maximization in (<ref>) simplifies to:
\begin{eqnarray}
\label{Eq:Pap8_Eq14}
\max_{m,\,\hat\omega_k}
\left|
\sum_{i=0}^{L_1-1}
\tilde r_{m+i}\,\tilde y_{k,\, L_{hr}-1+i}^*
\,
\mathrm{e}^{-\mathrm{j}\,\hat\omega_k i}
\right|.
\end{eqnarray}
Observe that (<ref>) resembles the operation of demodulation and
matched filtering. The ideal outcome of (<ref>) to estimate
the SoF and frequency offset is:
\begin{eqnarray}
\label{Eq:Pap8_Eq14_1}
m & = & L_{hr} -1 \nonumber \\
\hat\omega_k & = & \omega_k.
\end{eqnarray}
In practice, the receiver has only the estimate of the
channel ($\hat h_{k,\, n}$), hence $\tilde y_{k,\, n}$ must be replaced by
$\hat y_{k,\, n}$, where
\begin{eqnarray}
\label{Eq:Pap8_Eq15}
\hat y_{k,\, n} = \tilde s_{1,\, n} \star \hat h_{k,\, n}
\end{eqnarray}
is the preamble convolved with the channel estimate.
When $\hat h_{k,\, n}$ is not available, we propose a heuristic method
of frame detection as follows:
\begin{eqnarray}
\label{Eq:Pap8_Eq16}
\max_{m,\,\hat\omega_k}
\left|
\sum_{i=0}^{L_p-1}
\tilde r_{m+i}\,\tilde s_{1,\, i}^*
\,
\mathrm{e}^{-\mathrm{j}\,\hat\omega_k i}
\right|
\end{eqnarray}
where again $\tilde s_{1,\, i}$ denotes the preamble as shown in
Figure <ref>. The ideal outcome of (<ref>) is:
\begin{eqnarray}
\label{Eq:Pap8_Eq17}
0 & \le
& m \le L_h - 1 \nonumber \\
\hat\omega_k & = & \omega_k
\end{eqnarray}
depending on which channel coefficient has the maximum magnitude. In practical
situations, one also needs to look at the ratio of the peak-to-average
power of (<ref>) to estimate the SoF <cit.>. When $m$
lies outside the range in (<ref>), the frame is declared as
erased (lost). The probability of frame erasure as a function of the preamble
length is shown in Figure <ref>. Observe that for $L_p=512$, the
probability of erasure is less than $10^{-6}$ and is hence not plotted.
Probability of frame erasure as a function of the preamble length
$L_p$. 2013 IEEE. Reprinted, with permission, from
The coarse frequency offset estimate $\hat\omega_k$ is obtained by dividing the
interval $[-0.04,\, 0.04]$ radian into $B_1$ frequency bins and selecting
that bin which maximizes (<ref>).
§.§ Channel Estimation
Here, we focus on maximum likelihood (ML) channel estimation. We assume that
the SoF has been estimated using (<ref>) with outcome $m_0$
($0 \le m_0 \le L_h-1$) and the frequency offset has been perfectly canceled.
\begin{eqnarray}
\label{Eq:Pap8_Eq18_0}
m_1 = m_0 + L_h -1.
\end{eqnarray}
The steady-state,
preamble part of the received signal for the $k^{th}$ frame can be written as:
\begin{eqnarray}
\label{Eq:Pap8_Eq18}
\tilde\mathbf{r}_{k,\, m_1} = \tilde\mathbf{s}_1 \tilde\mathbf{h}_k +
\tilde\mathbf{w}_{k,\, m_1}
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Pap8_Eq19}
\tilde\mathbf{r}_{k,\, m_1}
& = &
\left[
\begin{array}{ccc}
\tilde r_{k,\, m_1} & \ldots & \tilde r_{k,\, m_1+L_p-L_{hr}}
\end{array}
\right]^T \nonumber \\
& & \mbox{ }
[(L_p-L_{hr}+1)\times 1]
\quad \mbox{vector} \nonumber \\
\tilde\mathbf{w}_{k,\, m_1}
& = &
\left[
\begin{array}{ccc}
\tilde w_{k,\, m_1} & \ldots & \tilde w_{k,\, \, m_1+L_p-{L_{hr}}}
\end{array}
\right]^T \nonumber \\
& & \mbox{ }
[(L_p-L_{hr}+1)\times 1]
\quad \mbox{vector} \nonumber \\
\tilde\mathbf{h}_k
& = &
\left[
\begin{array}{ccc}
\tilde h_{k,\, 0} & \ldots & \tilde h_{k,\, L_{hr}-1}
\end{array}
\right]^T \nonumber \\
& & \mbox{ }
[L_{hr}\times 1]
\quad \mbox{vector} \nonumber \\
\tilde\mathbf{s}_1
& = &
\left[
\begin{array}{ccc}
\tilde s_{1,\, L_{hr}-1} & \ldots & \tilde s_{1,\, 0}\\
\vdots & \ldots & \vdots\\
\tilde s_{1,\, L_p-1} & \ldots & \tilde s_{1,\, L_p-L_{hr}-2}
\end{array}
\right] \nonumber \\
& & \mbox{ }
[(L_p-L_{hr}+1)\times L_{hr}]
\quad \mbox{matrix}
\end{eqnarray}
where again $L_{hr}(> L_h)$ is the channel length assumed by the receiver.
The statement of the ML channel estimation is as follows: find
$\hat\mathbf{h}_k$ (the estimate of $\tilde\mathbf{h}_k$) such that:
\begin{eqnarray}
\label{Eq:Pap8_Eq20}
\left(
\tilde\mathbf{r}_{k,\, m_1} -
\tilde\mathbf{s}_1
\hat\mathbf{h}_k
\right)^H
\left(
\tilde\mathbf{r}_{k,\, m_1} -
\tilde\mathbf{s}_1
\hat\mathbf{h}_k
\right)
\end{eqnarray}
is minimized. Differentiating with respect to $\hat\mathbf{h}_k^*$ and
setting the result to zero yields <cit.>:
\begin{eqnarray}
\label{Eq:Pap8_Eq21}
\hat\mathbf{h}_k =
\left(
\tilde\mathbf{s}_1^H
\tilde\mathbf{s}_1
\right)^{-1}
\tilde\mathbf{s}_1^H
\tilde\mathbf{r}_{k,\, m_1}.
\end{eqnarray}
To see the effect of noise on the channel estimate in (<ref>),
\begin{eqnarray}
\label{Eq:Pap8_Eq22}
\tilde\mathbf{u} =
\left(
\tilde\mathbf{s}_1^H
\tilde\mathbf{s}_1
\right)^{-1}
\tilde\mathbf{s}_1^H
\tilde\mathbf{w}_{k,\, m_1}.
\end{eqnarray}
When $m_0=L_h-1$, observe that
\begin{eqnarray}
\label{Eq:Pap8_Eq22_0}
\hat\mathbf{h}_k = \tilde\mathbf{h}_k + \tilde\mathbf{u}.
\end{eqnarray}
Since $\tilde s_{1,\, n}$ is a zero-mean random sequence with good
autocorrelation properties, it is reasonable to expect
\begin{eqnarray}
\label{Eq:Pap8_Eq23}
\tilde\mathbf{s}_1^H
\tilde\mathbf{s}_1 & = & L_1
\sigma^2_s
\mathbf{I}_{L_{hr}}
\quad \mbox{for $L_p \gg L_{hr}$} \nonumber \\
\Rightarrow
\left(
\tilde\mathbf{s}_1^H
\tilde\mathbf{s}_1
\right)^{-1} & = & 1/(L_1
\sigma^2_s)
\mathbf{I}_{L_{hr}} \nonumber \\
\Rightarrow
\tilde\mathbf{u} & = & 1/(L_1
\sigma^2_s)
\tilde\mathbf{s}_1^H
\tilde\mathbf{w}_{k,\, m_1}
\end{eqnarray}
where $\sigma^2_s$ is defined in (<ref>), $L_1$ is defined in
(<ref>), and $\mathbf{I}_{L_{hr}}$ is an $L_{hr}\times L_{hr}$
identity matrix. It can be shown that
\begin{eqnarray}
\label{Eq:Pap8_Eq24}
\left[
\tilde\mathbf{u}
\tilde\mathbf{u}^H
\right] = \frac{2\sigma^2_w}{L_1\sigma^2_s}
\mathbf{I}_{L_{hr}}
= \frac{\sigma^2_w L_d}{L_1}
\mathbf{I}_{L_{hr}}
\stackrel{\Delta}{=} 2
\sigma^2_u \mathbf{I}_{L_{hr}}.
\end{eqnarray}
Therefore, the variance of the ML channel estimate ($\sigma^2_u$) tends to
zero as $L_1\rightarrow \infty$ and $L_d$ is kept fixed. Conversely, when
$L_d$ is increased keeping $L_1$ fixed, there is noise enhancement.
At this point, it must be mentioned that in the absence of noise, the channel
estimate obtained from (<ref>) depends on the SoF estimate $m_0$
obtained from (<ref>). When $m_0=L_h-1$, the
channel estimate in the absence of noise would be:
\begin{eqnarray}
\label{Eq:Pap8_Eq27}
\hat\mathbf{h}_k =
\left[
\begin{array}{cccccc}
\tilde h_{k,\, 0} & \ldots & \tilde h_{k,\, L_h-1} & 0 & \ldots & 0
\end{array}
\right]^T
\end{eqnarray}
When $m_0=0$, the channel estimate (in the absence of noise) is :
\begin{eqnarray}
\label{Eq:Pap8_Eq29}
\hat\mathbf{h}_k =
\left[
\begin{array}{cccccc}
0 & \ldots & 0 & \tilde h_{k,\, 0} & \ldots & \tilde h_{k,\, L_h-1}
\end{array}
\right]^T.
\end{eqnarray}
Thus we get:
\begin{eqnarray}
\label{Eq:Pap8_Eq30}
L_{hr} = 2 L_h - 1.
\end{eqnarray}
Observe that the channel estimation matrix $\tilde\mathbf{s}_1$ in
(<ref>) remains the same, independent of $m_0$. Therefore, the
pseudoinverse of $\tilde\mathbf{s}_1$ given in (<ref>) can be
precomputed and stored in the receiver.
Magnitude response of the channel at 0 dB SNR per bit after fine
frequency offset compensation, $L_p=512$, $L_d=4096$.
Magnitude response of the channel at 10 dB SNR per bit after fine
frequency offset compensation, $L_p=512$, $L_d=4096$.
The magnitude response of the channel at various SNRs are shown in
Figures <ref> and
§.§ Fine Frequency Offset Estimation
For the purpose of fine frequency offset estimation, we propose to use
(<ref>) with $\tilde y_{k,\, n}$ replaced by $\hat y_{k,\, n}$ as
given in (<ref>). Moreover, since the initial estimate of the
frequency offset $(\hat\omega_k)$ is already available, (<ref>)
must be modified as follows:
\begin{eqnarray}
\label{Eq:Pap8_Eq31}
\max_{m,\,\hat\omega_{k,\, f}}
\left|
\sum_{i=0}^{L_2-1}
\tilde r_{m+i}\,\hat y_{k,\, i}^*
\,
\mathrm{e}^{-\mathrm{j}\,
\left(
\hat\omega_k+\hat\omega_{k,\, f}
\right) i}
\right|
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Pap8_Eq32}
L_2 & = & L_{hr} + L_p - 1 \nonumber \\
0 & \le
& m
\le L_{hr} - 1.
\end{eqnarray}
Observe that the span of $\hat y_{k,\, i}$ is $L_2$. The fine frequency offset
estimate ($\hat\omega_{k,\, f}$) is obtained by dividing the interval
$[\hat\omega_k-0.005,\, \hat\omega_k+0.005]$ radian into $B_2$ frequency bins
<cit.>. The reason for choosing 0.005 radian can be traced to
Figure <ref>. We find that the maximum error in the coarse
estimate of the frequency offset is approximately 0.004 radian over
$10^4$ frames. Thus the probability that the maximum error exceeds 0.005
radian is less than $10^{-4}$.
In Figure <ref>, the coarse frequency offset estimate is
obtained from (<ref>), fine frequency offset estimate from
(<ref>), coherent frequency offset estimate (“RMS coho”) from
(<ref>) and the approximate Cramér-Rao bound from
RMS and maximum frequency offset estimation error for $L_p=512$.
Figure <ref> gives the results for SoF detection, coarse and
fine frequency offset estimation,
for one particular frame at 0 dB SNR per bit, with $B_1=B_2=64$.
SoF detection, coarse and fine frequency offset estimation, for
$L_p=512$, SNR per bit equal to 0 dB, $B_1=B_2=64$.
2013 IEEE. Reprinted, with permission, from
The advantage of the two-stage approach (coarse and fine) for frequency offset
estimation <cit.> is illustrated in
Table <ref>. The complexity
of the two-stage approach is $B_1+B_2=128$ frequency bins. The resolution of
the two-stage approach is $2\times 0.005/B_2=0.00015625$ radian. For obtaining
the same resolution, the single stage approach will require
$2\times 0.04/0.00015625=512$ frequency
bins. Therefore, the two-stage approach is four times more efficient than the
single stage approach.
Complexity comparison between the two-stage and single stage approach
of frequency offset estimation.
At this point, a note on the implementation of the SoF and frequency offset
estimation algorithm is in order. Observe that a 2-D search over both
frequency and time is required and there is a large scope for parallel
processing. Hence, this algorithm is well suited for hardware implementation.
§.§ Noise Variance Estimation
It is necessary to estimate the noise variance for the purpose of turbo
decoding <cit.>. After the channel has been estimated using
(<ref>), the noise variance is estimated as follows:
\begin{eqnarray}
\label{Eq:Pap8_Eq33}
\hat
\sigma^2_w = \frac{1}{2 L_1}
\left(
\tilde\mathbf{r}_{k,\, m_1} -
\tilde\mathbf{s}_1
\hat\mathbf{h}_k
\right)^H
\left(
\tilde\mathbf{r}_{k,\, m_1} -
\tilde\mathbf{s}_1
\hat\mathbf{h}_k
\right)
\end{eqnarray}
where $\tilde\mathbf{s}_1$ is defined in (<ref>) and $L_1$ is
defined in (<ref>).
§.§ Turbo Decoding
The encoder block diagram is shown in Figure <ref>. The overall
rate of the encoder is $1/2$, since $L_{d1}$ data bits generate $2L_{d1}$
coded QPSK symbols.
Encoder block diagram.
The generating matrix for each of the constituent encoders is given by:
\begin{eqnarray}
\label{Eq:Pap8_Eq34}
\mathbf{G}(D) =
\left[
\begin{array}{cc}
1 & \frac{\displaystyle 1+D^2}{\displaystyle 1+D+D^2}
\end{array}
\right].
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Pap8_Eq35}
m_2 = m_1 + L_p
\end{eqnarray}
where $m_1$ is defined in (<ref>). Define
\begin{eqnarray}
\label{Eq:Pap8_Eq36}
\tilde\mathbf{r}_{k,\, m_2} =
\left[
\begin{array}{ccc}
\tilde r_{k,\, m_2} & \ldots & \tilde r_{k,\, m_2+L_d-1}
\end{array}
\right]
\end{eqnarray}
as the data part of the received signal for the $k^{th}$ frame.
After SoF detection, frequency offset compensation and channel estimation,
the receiver block diagram is depicted in Figure <ref>.
OFDM receiver after synchronization.
The output of the FFT can be written as (for $0 \le i \le L_d-1$):
\begin{eqnarray}
\label{Eq:Pap8_Eq36_1}
\tilde R_{k,\, i} = \hat H_{k,\, i} S_{k,\, 3,\, i} +
\tilde W_{k,\, i}.
\end{eqnarray}
Note that $\hat H_{k,\, i}$ and $\tilde W_{k,\, i}$ in
Figure <ref> are the $L_d$-point
DFT of the estimated channel $\hat\mathbf{h}_k$ in (<ref>) and
$\tilde w_{k,\, n}$
in (<ref>) respectively, taken over the time interval
specified in (<ref>), and $S_{k,\, 3,\, i}$ denotes the
data symbols for the $k^{th}$ frame, for $0 \le i \le L_d-1$.
The variance of $\tilde W_{k,\, i}$ is
\begin{eqnarray}
\label{Eq:Pap8_Eq37}
\frac{1}{2}
\left[
\left|
\tilde W_{k,\, i}
\right|^2
\right] = L_d \sigma^2_w
\end{eqnarray}
and the variance of $\hat H_{k,\, i}$ is (assuming perfect channel estimates,
that is $\hat H_{k,\, i}=\tilde H_{k,\, i}$):
\begin{eqnarray}
\label{Eq:Pap8_Eq38}
\frac{1}{2}
\left[
\left|
\tilde H_{k,\, i}
\right|^2
\right] = L_h \sigma^2_f.
\end{eqnarray}
Note that due to multiplication by the channel DFT ($\hat H_{k,\, i}$) in
(<ref>), the data and parity bits of the QPSK symbol cannot be
separated, and the BCJR
algorithm is slightly different from the one given in <cit.>.
This is explained below. Observe also that dividing (<ref>)
by $\tilde H_{k,\, i}$ results in interference
($\tilde W_{k,\, i}/\hat H_{k,\, i}$) having a complex ratio
distribution <cit.>, which is undesirable.
Corresponding to the transition from state $m$ to state $n$,
at decoder 1, for the $k^{th}$ frame, at time $i$ define
(for $0 \le i \le L_{d1}-1$, $L_{d1}$ is defined in
Figure <ref>):
\begin{eqnarray}
\label{Eq:Turbo_Eq10}
\gamma_{1,\, k,\, i,\, m,\, n} =
\exp
\left[-
\frac{
\left(
\tilde R_{k,\, i}-
\hat H_{k,\, i}
S_{m,\, n}
\right)^2}
\right]
\end{eqnarray}
where $S_{m,\, n}$ denotes the QPSK symbol corresponding to the transition
from state $m$ to state $n$ in the trellis. We assume that the data bit maps
to the real part and the parity bit maps to the imaginary part of the QPSK
symbol. We also assume that bit 0 maps to $+1$ and bit 1 maps to $-1$. Observe
that $\hat\sigma^2_w$ is the estimate of $\sigma^2_w$ obtained from
(<ref>). Similarly, for the
transition from state $m$ to state $n$,
at decoder 2, for the $k^{th}$ frame, at time $i$ define
(for $0 \le i \le L_{d1}-1$):
\begin{eqnarray}
\label{Eq:Turbo_Eq10_1}
\gamma_{2,\, k,\, i,\, m,\, n} =
\exp
\left[-
\frac{
\left(
\tilde R_{k,\, L_{d1}+i}-
\hat H_{k,\, L_{d1}+i}
S_{m,\, n}
\right)^2}
\right]
\end{eqnarray}
Let $\mathscr{S}$ denote the number of states in
the encoder trellis. Let $\mathscr{D}_n$ denote the set of states that diverge
from state $n$. For example
\begin{eqnarray}
\label{Eq:Turbo_Eq8_2}
\mathscr{D}_0 = \{0,\, 3\}
\end{eqnarray}
implies that states 0 and 3 can be reached from state 0.
Similarly, let $\mathscr{C}_n$ denote the set of states that converge
to state $n$. Let $\alpha_{i,\, n}$ denote the alpha value at time $i$
($0 \le i \le L_{d1}-2$) at state $n$ ($0 \le n \le \mathscr{S}-1$).
Then the alpha values for decoder 1 can be recursively computed as follows
(forward recursion):
\begin{eqnarray}
\label{Eq:Turbo_Eq9}
\alpha_{i+1,\, n}' & = & \sum_{m \in \mathscr{C}_n}
\alpha_{i,\, m}
\gamma_{1,\, k,\, i,\, m,\, n}
\left(
S_{b,\, i,\, m,\, n}
\right) \nonumber \\
\alpha_{0,\, n} & = & 1
\qquad
\mbox{for $0 \le n \le \mathscr{S}-1$}
\nonumber \\
\alpha_{i+1,\, n} & = & \alpha_{i+1,\, n}'\Big/
\left(
\sum_{n=0}^{\mathscr{S}-1}
\alpha_{i+1,\, n}'
\right)
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Turbo_Eq9_1}
P(S_{b,\, i,\, m,\, n}) =
\left
\{
\begin{array}{ll}
F_{2,\, i+} & \mbox{if $S_{b,\, i,\, m,\, n}=+1$}\\
F_{2,\, i-} & \mbox{if $S_{b,\, i,\, m,\, n}=-1$}\\
\end{array}
\right.
\end{eqnarray}
denotes the a priori probability of the
systematic bit corresponding to the transition from state $m$ to state $n$,
at decoder 1, at time $i$, obtained from the $2^{nd}$ decoder at time $l$,
after deinterleaving (that is, $i=\pi^{-1}(l)$ for some
$0 \le l \le L_{d1}-1$).
The terms $F_{2,\, i+}$ and $F_{2,\, i-}$ are defined similar to
(<ref>) given below.
The normalization step in the last equation of
(<ref>) is done to prevent numerical instabilities
Similarly, let $\beta_{i,\, n}$ denote the
beta values at time $i$ ($1 \le i \le L_{d1}-1$) at state $n$
($0 \le n \le \mathscr{S}-1$).
Then the recursion for beta (backward recursion) at decoder 1
can be written as:
\begin{eqnarray}
\label{Eq:Turbo_Eq11}
\beta_{i,\, n}' & = & \sum_{m \in \mathscr{D}_n}
\beta_{i+1,\, m}
\gamma_{1,\, k,\, i,\, n,\, m}
\left(
S_{b,\, i,\, n,\, m}
\right) \nonumber \\
\beta_{L_{d1},\, n}
& = & 1
\qquad
\mbox{for $0 \le n \le \mathscr{S}-1$} \nonumber \\
\beta_{i,\, n} & = & \beta_{i,\, n}'\Big/
\left(
\sum_{n=0}^{\mathscr{S}-1}
\beta_{i,\, n}'
\right).
\end{eqnarray}
Once again, the normalization step in the last equation of
(<ref>) is done to prevent numerical instabilities.
Let $\rho^+(n)$ denote the state that is reached from
state $n$ when the input symbol is $+1$. Similarly let $\rho^-(n)$ denote
the state that can be reached from state $n$ when the input symbol is $-1$.
Then (for $0 \le i \le L_{d1}-1$)
\begin{eqnarray}
\label{Eq:Turbo_Eq12}
G_{1,\, i+}
& = & \sum_{n=0}^{\mathscr{S}-1}
\alpha_{i,\, n}
\gamma_{1,\, k,\, i,\, n,\, \rho^+(n)}
\beta_{i+1,\, \rho^+(n)} \nonumber \\
G_{1,\, i-}
& = & \sum_{n=0}^{\mathscr{S}-1}
\alpha_{i,\, n}
\gamma_{1,\, k,\, i,\, n,\, \rho^-(n)}
\beta_{i+1,\, \rho^-(n)}.
\end{eqnarray}
Finally, the extrinsic information that is to be fed as a priori
probabilities to the second decoder after interleaving, is computed as:
\begin{eqnarray}
\label{Eq:Turbo_Eq12_2}
F_{1,\, i+}
& = & G_{1,\, i+}/
(G_{1,\, i+} + G_{1,\, i-}) \nonumber \\
F_{1,\, i-}
& = & G_{1,\, i-}/
(G_{1,\, i+} + G_{1,\, i-})
\end{eqnarray}
Equations (<ref>), (<ref>),
(<ref>) and (<ref>) constitute the MAP
recursions for the first decoder. The MAP recursions for the second decoder
are similar.
After a few iterations, (one iteration involves both decoder 1 and 2) the
final a posteriori probabilities of $i^{th}$ bit of the $k^{th}$
frame at the output of decoder 1 is given by:
\begin{eqnarray}
\label{Eq:Turbo_Eq13}
H_{1,\, i+}
& = & \sum_{n=0}^{\mathscr{S}-1}
\alpha_{i,\, n}
\gamma_{1,\, k,\, i,\, n,\, \rho^+(n)}
F_{2,\, i+}
\,
\beta_{i+1,\, \rho^+(n)} \nonumber \\
H_{1,\, i-}
& = & \sum_{n=0}^{\mathscr{S}-1}
\alpha_{i,\, n}
\gamma_{1,\, k,\, i,\, n,\, \rho^-(n)}
F_{2,\, i-}
\,
\beta_{i+1,\, \rho^-(n)}. \nonumber \\
\end{eqnarray}
followed by
\begin{eqnarray}
\label{Eq:Turbo_Eq14}
\left(
S_{b,\, k,\, i}= +1|\mathbf{r}_{k,\, m2}
\right) & = & H_{1,\, i+}/
(H_{1,\, i+} + H_{1,\, i-}) \nonumber \\
\left(
S_{b,\, k,\, i}= -1|\mathbf{r}_{k,\, m2}
\right) & = & H_{1,\, i-}/
(H_{1,\, i+} + H_{1,\, i-}). \nonumber \\
\end{eqnarray}
When puncturing is used to increase the overall rate, e.g. if the QPSK symbol
occurring at odd instants of time in both encoders are not transmitted, then
the corresponding gamma values in (<ref>) and
(<ref>) are set to unity. For the even time instants, the
corresponding gamma values are computed according to (<ref>) and
§.§ Robust Turbo Decoding
At high SNR, the term in the exponent ($b$ is the exponent of $\mathrm{e}^b$)
of (<ref>) and (<ref>) becomes very large
(typically $b>100$) and it becomes unfeasible for the DSP processor or even a
computer to calculate the gammas. We propose to solve this problem by
normalizing the exponents. Observe that the exponents are real-valued and
negative. Let $b_{1,\, j,\, i}$ denote an exponent at decoder 1 due to the
$j^{th}$ symbol in the constellation ($1\le j\le 4$ for QPSK) at time $i$.
\begin{eqnarray}
\label{Eq:Turbo_Eq15}
\mathbf{b}_1 =
\left[
\begin{array}{ccc}
b_{1,\, 1,\, 0} & \ldots & b_{1,\, 1,\, L_{d1}-1}\\
\vdots & \vdots & \vdots \\
b_{1,\, 4,\, 0} & \ldots & b_{1,\, 4,\, L_{d1}-1}
\end{array}
\right]
\end{eqnarray}
denote the matrix of exponents for decoder 1. Let $b_{1,\, \mathrm{max},\, i}$
denote the maximum exponent at time $i$, that is
\begin{eqnarray}
\label{Eq:Turbo_Eq15_1}
b_{1,\,\mathrm{max},\, i} =
\max
\left[
\begin{array}{c}
b_{1,\, 1,\, i}\\
\vdots \\
b_{1,\, 4,\, i}
\end{array}
\right].
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Turbo_Eq15_2}
\mathbf{b}_{1,\,\mathrm{max}} =
\left[
\begin{array}{ccc}
b_{1,\, \mathrm{max},\, 0} & \ldots & b_{1,\,\mathrm{max},\, L_{d1}-1}
\end{array}
\right]
\end{eqnarray}
denote the vector containing the maximum exponents. Compute:
\begin{eqnarray}
\label{Eq:Turbo_Eq16}
\mathbf{b}_1' = \mathbf{b}_1 -
\left[
\begin{array}{c}
\mathbf{b}_{1,\,\mathrm{max}}\\
\vdots\\
\mathbf{b}_{1,\,\mathrm{max}}
\end{array}
\right].
\end{eqnarray}
Note that in (<ref>), the vector $\mathbf{b}_{1,\,\mathrm{max}}$
has to be repeated as many times as the number of symbols in the constellation.
If any element of $\mathbf{b}_1'$ is less than say, $-30$, then set it to
$-30$. Thus we get a normalized exponent vector
whose elements lie in the range $[0,\, -30]$. It has been found from
simulations that normalizing the exponents does not lead to any degradation
in BER performance, on the contrary, it increases the operating SNR range of
the turbo receiver. In practice, we could divide the range $[0,\, -30]$ into
a large number (e.g. 3000) of levels and the exponentials ($\mathrm{e}^b$)
could be precomputed and stored in the DSP processor, and need not be
computed in real-time. The choice of the minimum exponent (e.g. $-30$),
would depend on the precision of the DSP processor or the computer.
§.§ Data Interleaving
Assuming ideal channel estimates, the autocorrelation of the channel DFT at
the receiver is:
\begin{eqnarray}
\label{Eq:Pap8_Eq38_1}
\frac{1}{2}
\left[
\tilde H_{k,\, i}
\tilde H_{k,\, j}^*
\right] = \sigma^2_f
\sum_{n=0}^{L_h-1}
\mathrm{e}^{-\mathrm{j}\, 2\pi n (i-j)/L_d}.
\end{eqnarray}
It has been found from simulations that the performance of the turbo decoder
gets adversely affected due to the correlation in $\tilde H_{k,\, i}$. To
overcome this problem, we interleave the data before the IFFT operation
at the transmitter and deinterleave the data after the FFT operation at the
receiver, before turbo decoding. This process essentially removes any
correlation in $\tilde H_{k,\, i}$ <cit.>.
§.§ Enhanced Frame Structure
The accuracy of the frequency offset estimate depends on the length of the
preamble $L_p$. Increasing the number of frequency bins $B_1$ and $B_2$ in
Figure <ref>, for a given $L_p$, does not improve the
accuracy. From Figure <ref> it can be seen that the RMS value
of the fine frequency offset estimation error is about $2\times 10^{-4}$,
at an SNR per bit equal to 8 dB. The subcarrier spacing with data length
$L_d=4096$ is
equal to $2\pi/4096=1.534\times 10^{-3}$ radians. Therefore, the residual
frequency offset is $0.0002\times 100/0.001534=13\%$ of the subcarrier spacing,
which is quite high and causes severe intercarrier interference (ICI). Note
that the RMS frequency offset estimation error can be reduced by increasing
the preamble length ($L_p$), keeping the data length ($L_d$) fixed, which in
turn reduces the throughput given by:
\begin{eqnarray}
\label{Eq:Pap8_Eq38_2}
\mathscr{T} = \frac{L_{d1}}{L_p+L_{cp}+L_d}.
\end{eqnarray}
Note that for a rate-$1/2$ turbo code $L_d=2L_{d1}$, whereas for a rate-1
turbo code, $L_d=L_{d1}$.
This motivates us to
look for an alternate frame structure which not only solves the frequency
offset estimation problem, but also maintains the throughput at a reasonable
(a) Enhanced frame structure. (b) Processing of the data part at the
Consider the frame in Figure <ref>(a). In addition to the
preamble, prefix and data, it contains “buffer” (dummy) symbols of length $B$
and postamble of length $L_o$, all drawn from the QPSK constellation. In
Figure <ref>(b) we illustrate the processing of $L_d$ symbols
at the transmitter.
Observe that only the data and postamble symbols are interleaved before the
IFFT operation. After interleaving, the postamble gets randomly dispersed
between the data symbols. The buffer symbols are sent directly to the IFFT,
without interleaving. The preamble and the cyclic prefix continue to be
processed according to Figure <ref> and (<ref>). We
now explain the reason behind using this frame structure. In what follows,
we assume that the SoF has been detected, fine frequency offset
correction has been performed and the channel has been estimated.
We proceed by making the following observations:
* Modulation in the time domain results in a shift in the frequency
domain. Therefore, any residual frequency offset after fine
frequency offset correction, results in a frequency shift at the
output of the FFT operation at the receiver. Moreover, due to the
presence of a cyclic prefix, the frequency shift is circular.
Therefore, without the buffer symbols, there is a possibility that
the first data symbol would be circularly shifted to the last
data symbol or vice versa. This
explains the use of buffer symbols at both ends in
Figure <ref>. In order to compute the number of buffer
symbols ($B$), we have to know the maximum residual
frequency offset, after fine frequency offset correction. Referring
to Figure <ref>, we find that the maximum error in
fine frequency offset estimation at 0 dB SNR per bit is about
$\pm 2\times 10^{-3}$ radians. With $L_d=4096$, the subcarrier spacing
is $2\pi/4096=1.534\times 10^{-3}$ radians. Hence, the residual
frequency error would result in a shift of $\pm 2/1.534=\pm 1.3$
subcarrier spacings. Therefore, while $B=2$ would suffice, we have
taken $B=4$, to be on the safe side.
* Since the frequency shift is not an integer multiple of the subcarrier
spacing, we need to interpolate in between the subcarriers, to
accurately estimate the shift. Interpolation can be achieved by
zero-padding the data before the FFT operation. Thus we get a
$2L_d-$point FFT corresponding to an interpolation factor of 2 and
so on. Other methods of interpolation between subcarriers is
discussed in <cit.>.
* After the FFT operation, postamble matched filtering has to be
done, since the postamble and $\hat H_k\approx \tilde H_k$ (in
(<ref>)) are available. The procedure for constructing
the postamble matched filter is illustrated in
Figure <ref>. From simulations, it has been
found that a postamble length $L_o=128$ results in false peaks at
the postamble matched filter output at 0 dB SNR per bit. Therefore we
have taken $L_o=256$. With these calculations, the length of the
data works out as $L_{d2}=L_d-2B-L_o=4096-8-256=3832$ QPSK symbols.
The throughput of the proposed system (with rate-1 turbo code) is
\begin{eqnarray}
\label{Eq:Pap8_Eq38_2_1}
\mathscr{T} & = & \frac{L_{d2}}{L_p+L_{cp}+L_d} \nonumber \\
& = & \frac{3832}{512+18+4096} \nonumber \\
& = & 82.84\%.
\end{eqnarray}
The throughput comparison of various
frame structures is summarized in Table <ref>.
Throughput comparison of various frame structures with
$L_p=L_{d1}=512$, $L_{d2}=3832$, $L_{cp}=18$.
Obtaining the postamble matched filter for $L_d=8$. Buffer symbols
are not shown. The frequency offset ($\pi/L_d$) is half the
subcarrier spacing
($2\pi/L_d$). $H_k$ and $S_k$ are assumed to be real-valued. Noise is
absent. (a) Output of the $L_d$-point FFT in the absence of frequency
offset. The red lines represent postamble and the blue lines
represent data symbols. (b) Output of the $2L_d$-point FFT
in the presence of frequency offset. Observe that the red and blue
lines have shifted to the right by $\pi/L_d$. Green lines denote the
output of the $L_d$-point FFT in the presence of frequency offset.
(c) The postamble matched filter.
§.§ Receiver Diversity
In the presence of receiver diversity, the signal in each diversity
arm ($l$) can be expressed as (see (<ref>)):
\begin{eqnarray}
\label{Eq:Pap8_Eq38_3}
\tilde r_{k,\, n,\, l} & = & \left(
\tilde s_{k,\, n} \star \tilde h_{k,\, n,\, l}
\right)\,
\mathrm{e}^{\,\mathrm{j}
(\omega_k n+\theta_{k,\, l})} +
\tilde w_{k,\, n,\, l} \nonumber \\
& = & \tilde y_{k,\, n,\, l}
\mathrm{e}^{\,\mathrm{j}
(\omega_k n+\theta_{k,\, l})} +
\tilde w_{k,\, n,\, l}
\end{eqnarray}
for $1 \le l \le N$.
The frequency offset is assumed to be identical for all the diversity arms,
whereas the carrier phase and noise are assumed to be independent. The noise
variance is same for all the diversity arms.
Two extreme scenarios are considered in the simulations (a) identical channel
and (b) independent channel in each diversity arm.
The output of the FFT can be written as (for $0 \le i \le L_d-1$):
\begin{eqnarray}
\label{Eq:Pap8_Eq38_3_1}
\tilde R_{k,\, i,\, l} = \hat H_{k,\, i,\, l} S_{k,\, 3,\, i} +
\tilde W_{k,\, i,\, l}
\end{eqnarray}
for $1 \le l \le N$ diversity arms.
The notation in (<ref>) is self explanatory and is based on
In the turbo decoding operation, (for decoder 1, $N$ diversity arms,
rate-$1/2$ turbo code, the enhanced frame structure in
Figure <ref> and $0 \le i \le L_{d2}/2 -1$), we have from
\begin{eqnarray}
\label{Eq:Turbo_Eq38_4}
\gamma_{1,\, k,\, i,\, m,\, n} =
\prod_{l=1}^{N}
\gamma_{1,\, k,\, i,\, m,\, n,\, l}
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Turbo_Eq38_5}
\gamma_{1,\, k,\, i,\, m,\, n,\, l} =
\exp
\left[-
\frac{
\left(
\tilde R_{k,\, i,\, l}-
\hat H_{k,\, i,\, l}
S_{m,\, n}
\right)^2}{2L_d\hat\sigma_w^2}
\right]
\end{eqnarray}
where $\hat\sigma_w^2$ is the average estimate of the noise variance over
all the diversity arms.
Similarly at decoder 2, for $0 \le i \le L_{d2}/2-1$,
we have from (<ref>):
\begin{eqnarray}
\label{Eq:Turbo_Eq38_6}
\gamma_{2,\, k,\, i,\, m,\, n} =
\prod_{l=1}^{N}
\gamma_{2,\, k,\, i,\, m,\, n,\, l}
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Turbo_Eq38_7}
\gamma_{2,\, k,\, i,\, m,\, n,\, l} =
\exp
\left[-
\frac{
\left(
\tilde R_{k,\, j,\, l}-
\hat H_{k,\, j,\, l}
S_{m,\, n}
\right)^2}{2L_d\hat\sigma_w^2}
\right]
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Turbo_Eq38_8}
j = L_{d2}/2 + i.
\end{eqnarray}
For a rate-1 turbo code, alternate gammas have to be set to unity, as
explained in the last paragraph of Section <ref>.
§.§ The Channel Capacity
The communication system model under consideration is given by
(<ref>). The channel capacity is given by <cit.>:
\begin{eqnarray}
\label{Eq:Channel_Cap_Eq1}
C = \frac{1}{2}
\log_2(1+\mbox{SNR})
\qquad \mbox{bits/transmission}
\end{eqnarray}
per dimension (real-valued signals occupy a single dimension, complex-valued
signals occupy two dimensions). The “SNR” in (<ref>)
denotes the minimum average signal-to-noise ratio per dimension, for
error-free transmission. Observe that:
* The sphere packing derivation of the channel capacity formula
<cit.>, does not require noise to be Gaussian. The only
requirements are that the noise samples have to be independent,
the signal and noise have to be independent, and both the signal and
noise must have zero mean.
* The channel capacity depends only on the SNR.
* The average SNR per dimension in (<ref>) is
different from the average
SNR per bit (or $E_b/N_0$), which is widely used in the literature. In
fact, it can be shown that <cit.>:
\begin{eqnarray}
\label{Eq:Channel_Cap_Eq1_1}
\mbox{SNR} = 2C \times
\mbox{SNR per bit.}
\end{eqnarray}
* It is customary to define the average SNR per bit ($E_b/N_0$) over two
dimensions (complex signals). When the signal and noise statistics over
both dimensions are identical, the average SNR per bit over two
dimensions is identical to the average SNR per bit over one dimension.
Therefore (<ref>) is valid, even though the
SNR is defined over one dimension and the SNR per bit is defined
over two dimensions.
* The notation $E_b/N_0$ is usually used for continuous-time, passband
analog signals <cit.>, whereas SNR per
bit is used for discrete-time signals <cit.>. However, both
definitions are equivalent. Note that passband signals are capable of
carrying information over two dimensions, using sine and cosine
carriers, inspite of the fact that passband signals are real-valued.
* Each dimension corresponds to a separate and independent path between
the transmitter and receiver.
* The channel capacity is additive with respect to the number of
dimensions. Thus, the total capacity over $2N$ real dimensions is equal
to the sum of the capacity over each real dimension.
* Each $S_{k,\, 3,\, i}$ in (<ref>) corresponds to
one transmission (over two dimensions, since $S_{k,\, 3,\, i}$ is
* Transmission of $L_{d2}$ data bits in Figure <ref> (for a
rate-1 turbo code),
results in $NL_{d2}$ complex samples ($2NL_{d2}$
real-valued samples) of
$\tilde R_{k,\, i,\, l}$ in (<ref>), for $N^{th}$-order
receive diversity. Therefore, the channel capacity is
\begin{eqnarray}
\label{Eq:Channel_Cap_Eq2}
C & = & \frac{L_{d2}}{2NL_{d2}} \nonumber \\
& = & \frac{1}{2N}
\qquad \mbox{bits/transmission}
\end{eqnarray}
per dimension. In other words, (<ref>) implies that
in each transmission, one data bit is transmitted over $2N$ dimensions.
Similarly, for a rate-$1/2$ turbo code with
$N^{th}$-order receive diversity, transmission of
$L_{d2}/2$ data bits results in $NL_{d2}$ complex samples of
$\tilde R_{k,\, i,\, l}$ in (<ref>), and the channel
capacity becomes:
\begin{eqnarray}
\label{Eq:Channel_Cap_Eq2_1}
C & = & \frac{L_{d2}}{4NL_{d2}} \nonumber \\
& = & \frac{1}{4N}
\qquad \mbox{bits/transmission}
\end{eqnarray}
per dimension. Substituting
(<ref>) and (<ref>) in
(<ref>), and using (<ref>)
we get the
minimum (threshold) average SNR per bit required for error-free
transmission, for a given channel capacity.
The minimum SNR per bit for different code rates and
receiver diversity.
The minimum SNR per bit for various code rates and receiver
diversity is presented in Table <ref>. Note that
<cit.> the minimum $E_b/N_0$ for
error-free transmission is $-1.6$ dB only when $C\rightarrow 0$.
* In the case of fading channels, it may not be possible to achieve
the minimum possible SNR per bit. This is because, the SNR per bit of a
given frame may be less than the threshold average SNR per bit. Such
frames are said to be in outage. The frame SNR per bit can be defined
as (for the $k^{th}$ frame and the $l^{th}$ diversity arm):
\begin{eqnarray}
\label{Eq:Channel_Cap_Eq3}
\mbox{SNR}_{k,\, l,\,\mathrm{bit}} =
\frac{1}{2C}
\frac{<|\tilde H_{k,\, i,\, l} S_{k,\, 3,\, i}|^2>}
{<|\tilde W_{k,\, i,\, l}|^2>}
\end{eqnarray}
where $<\cdot>$ denotes time average over the $L_{d2}$ data symbols.
Note that the frame SNR is different from the average SNR per bit,
which is defined as:
\begin{eqnarray}
\label{Eq:Channel_Cap_Eq4}
\mbox{SNR per bit} =
\frac{1}{2C}
\frac{E\left[\left|\tilde H_{k,\, i,\, l}
S_{k,\, 3,\, i}
\right|^2\right]}
{E\left[\left|\tilde W_{k,\, i,\, l}\right|^2\right]}.
\end{eqnarray}
The $k^{th}$ OFDM frame is said to be in outage when:
\begin{eqnarray}
\label{Eq:Channel_Cap_Eq5}
\mbox{SNR}_{k,\, l,\,\mathrm{bit}} <
\mbox{minimum average SNR per bit}
\end{eqnarray}
for all $l$. The outage probability is given by:
\begin{eqnarray}
\label{Eq:Channel_Cap_Eq6}
P_{\mathrm{out}} = \frac{\mbox{number of frames in outage}}
{\mbox{total number of frames transmitted}}.
\end{eqnarray}
§ SIMULATION RESULTS
In this section, we present the simulation results for turbo-coded OFDM.
In the simulations, the channel length $L_h$ is equal to 10, hence $L_{hr}=19$.
The fade variance $\sigma^2_f=0.5$. The simulation results are presented in
Figure <ref>, for the frame structure in
Figure <ref>(a) with $L_p=512$ and different values of $L_d$.
The term “UC” denotes uncoded, “TC” denotes turbo coded, “data” denotes
$L_{d1}$, “Pr” denotes practical receiver (with acquired synchronization and
channel estimates) and “Id” denotes ideal receiver (ideal synchronization
and channel estimates).
Simulation results without data interleaving, frame structure in
Figure <ref>(a), rate-$1/2$ turbo code. 2013
IEEE. Reprinted, with permission, from <cit.>.
We find that for $L_{d1}=512$, the practical receiver has a performance that is
less than 1 dB inferior to the ideal receiver. However, the throughput of this
system is just 32.95%, since the data length is equal to the preamble length.
Next, for $L_{d1}=1024$, the practical receiver is about 1 dB inferior to the
ideal receiver and the throughput
has improved to 39.72%. When $L_{d1}=4096$, the performance of the practical
receiver is no better than
uncoded transmission. This is due to the fact that the residual RMS frequency
offset estimation error (fine) in Figure <ref> is about
$2\times 10^{-4}$ radian, which is a significant fraction of the
subcarrier spacing ($2\pi/L_d=0.000767$ radian). Note that the frequency
offset estimation error depends only on $L_p$ and the performance of the
ideal receiver is independent of the data length $L_{d1}$.
Simulation results with data interleaving, frame structure in
Figure <ref>(a), rate-$1/2$ turbo code. 2013
IEEE. Reprinted, with permission, from <cit.>
In Figure <ref>, we present the simulation results with
data interleaving, as discussed in Section <ref>. Again, the
performance of the ideal receiver is independent of $L_{d1}$.
We see that the practical receiver exhibits
more than two orders of magnitude improvement in the BER (compared
to the case where there is no data interleaving), at an SNR of 8 dB and
$L_{d1}=512$. When $L_{d1}$ is increased, the performance of the practical
receiver deteriorates.
Simulation results with data interleaving, enhanced frame structure
in Figure <ref>(a) and rate-1 turbo code.
In Figure <ref>, we present simulation results for the
rate-1 turbo code, with enhanced frame structure, $1^{st}$-order
receiver diversity and interpolation factors (ip) equal to 2, 4, 8, 16 and 32.
We find that the performance of the practical receiver is as good as the
ideal receiver. However, there is a 4 dB degradation in performance of the
ideal receiver for the rate-1 turbo code,
with respect to the ideal receiver for the rate-$1/2$
turbo code in Figure <ref>, at a BER of $10^{-5}$. This
degradation in performance
can be compensated by using receiver diversity, which is presented next.
Simulation results with data interleaving, enhanced frame structure
in Figure <ref>(a) and rate-1 turbo code with 2nd order
receive diversity. Identical channel on both diversity arms.
In Figure <ref>, we present simulation results for the
rate-1 turbo code, with enhanced frame structure and $2^{nd}$-order
receiver diversity. The channel in both diversity arms is assumed to be
identical. However, noise in both the diversity arms is assumed to be
independent. Comparing Figure <ref> and
Figure <ref>, we find that the ideal receiver with 2nd-order
diversity is just 2 dB better than the one with 1st-order diversity, at a
BER of $10^{-5}$. Moreover,
the practical receivers, with ip=32 have nearly identical performance. This
is to be expected, since it is well known that diversity advantage is obtained
only when the channels are independent.
Simulation results with data interleaving, enhanced frame structure
in Figure <ref>(a) and rate-1 turbo code with 2nd order
receive diversity. Independent channel on both diversity arms.
In Figure <ref>, we present simulation results for the
rate-1 turbo code, with enhanced frame structure and $2^{nd}$-order
receiver diversity. The channel and noise in both diversity arms are assumed
to be independent. Comparing Figure <ref> and
Figure <ref>, we find that the ideal receiver with 2nd order
diversity exhibits about 5 dB improvement over the one with 1st order
diversity, at a BER of $10^{-5}$. Moreover, the practical receiver with
ip=16, 32 is just 1 dB inferior to the ideal receiver, at a BER of $10^{-5}$.
Simulation results for outage probability with data interleaving,
enhanced frame structure in Figure <ref>(a) and rate-1
turbo code with 1st and 2nd order receive diversity. Independent
channel on both diversity arms.
Finally, in Figure <ref> we present the outage probability
for the rate-1 turbo code with 1st and 2nd order receive diversity. The outage
probability for 1st order receive diversity, at 6 dB SNR per bit is
$3\times 10^{-4}$.
In other words, 3 out of $10^4$ frames are in outage (no error correcting code
can correct errors in such frames). Therefore, in the worst case, the
number of bit errors for the frames in outage would be
$0.5\times 3\times 3832$ (assuming probability of error is 0.5). Let us also
assume that for the remaining ($10000-3=9997$) frames, all errors are
corrected, using a sufficiently powerful error correcting code. Therefore,
in the best case situation, the overall BER at 6 dB SNR per bit, with 1st order
diversity would be $0.5\times 3\times 3832/(10000*3832)=1.5\times 10^{-4}$.
However, from Figure <ref>, even the ideal coherent receiver
exhibits a BER as high as $10^{-2}$ at 6 dB SNR per bit. Therefore, there is
scope for improvement, using perhaps a more powerful error correcting code.
Similarly we observe from Figure <ref> that, with 2nd order
receive diversity, the outage probability is $10^{-4}$ at 3 dB SNR per bit.
This implies that 1 out of $10^{4}$ frames is in outage. Using similar
arguments, the best case overall BER at 3 dB SNR per bit would be
$0.5\times 3832/(10000*3832)=0.5\times 10^{-4}$. From
Figure <ref>, the ideal coherent receiver gives a BER of
$2\times 10^{-2}$, at 3 dB SNR per bit, once again suggesting that there
is large scope for improvement, using a better code.
§ CONCLUSIONS AND FUTURE WORK
This paper deals with linear complexity coherent detectors for turbo-coded OFDM
signals transmitted over frequency selective Rayleigh fading channels.
Simulation results show that it is possible to achieve a BER of $10^{-5}$
at an SNR per bit of 8 dB and throughput equal to 82.84%, using a single
transmit and two receive antennas.
With the rapid advances in VLSI technology,
it is expected that coherent transceivers would drive the future wireless
telecommunication systems.
It may be possible to further improve the performance, using a better code.
§.§ An Approximate and Simple Cramér-Rao Bound on the Variance of
the Frequency Offset Estimation Error
Consider the signal model in (<ref>), which is repeated here for
convenience (for notational simplicity, we drop the subscript $k$, assume
$\theta_k=0$ and $N-1=L_p-L_h+1$):
\begin{eqnarray}
\label{Eq:Ap_Eq1}
\tilde r_n = \tilde y_n
\mathrm{e}^{\,\mathrm{j}\omega n} +
\tilde w_n \quad \mbox{for $0\le n \le N-1$}.
\end{eqnarray}
We assume that the channel is known, and hence $\tilde y_n$ is known at the
receiver. Moreover, we consider only the steady-state preamble part of the
received signal (note that time is suitably re-indexed, such that the first
steady-state sample is considered as time zero, whereas, actually the first
steady-state sample occurs at time $L_h-1$). Define
\begin{eqnarray}
\label{Eq:Ap_Eq2}
\tilde\mathbf{y} & = &
\left[
\begin{array}{ccc}
\tilde y_0 & \ldots & \tilde y_{N-1}
\end{array}
\right] \nonumber \\
\tilde\mathbf{r} & = &
\left[
\begin{array}{ccc}
\tilde r_0 & \ldots & \tilde r_{N-1}
\end{array}
\right].
\end{eqnarray}
The coherent maximum likelihood (ML)
estimate of the frequency offset is obtained as follows: choose that value of
$\hat\omega$ which maximizes the joint conditional pdf
\begin{eqnarray}
\label{Eq:Ap_Eq3}
\max_{\hat\omega \in [-\omega_{\mathrm{max}},\, \omega_{\mathrm{max}}]}
\left(
\tilde\mathbf{r}|\tilde\mathbf{y},\,\hat\omega
\right)
\end{eqnarray}
where $\omega_{\mathrm{max}}$ denotes the maximum possible frequency offset
in radians. Substituting for the joint conditional pdf in (<ref>),
we obtain
\begin{eqnarray}
\label{Eq:Ap_Eq4}
\max_{\hat\omega}
\frac{1}{(2\pi\sigma^2_w)^{N}}
\exp
\left(
\frac{
\sum_{n=0}^{N-1}
\left|
\tilde r_n-\tilde y_n
\,
\mathrm{e}^{\,\mathrm{j}\,\hat\omega n}
\right|^2
\right)
\end{eqnarray}
which simplifies to
\begin{eqnarray}
\label{Eq:Ap_Eq4_1}
\max_{\hat\omega}
\Re
\left
\{
\sum_{n=0}^{N-1}
\tilde r_n\tilde y_n^*
\,
\mathrm{e}^{-\mathrm{j}\,\hat\omega n}
\right
\}.
\end{eqnarray}
Observe that (<ref>) is the non-coherent ML frequency offset
(and timing) estimator, whereas (<ref>) is the coherent ML
frequency offset estimator assuming timing is known.
Since ML estimators are unbiased, the variance of the frequency offset
estimate is lower bounded by the Cramér-Rao bound (CRB):
\begin{eqnarray}
\label{Eq:Ap_Eq5}
\left[
\left(
\hat\omega - \omega
\right)^2
\right] \ge 1\bigg/
\left[
\left(
\frac{\partial}{\partial \omega}
\ln p
\left(
\tilde\mathbf{r}|\tilde\mathbf{y},\,\omega
\right)
\right)^2
\right]
\end{eqnarray}
since $\tilde\mathbf{y}$ is assumed to be known. It can be shown that
\begin{eqnarray}
\label{Eq:Ap_Eq6}
\frac{\partial}{\partial \omega}
\ln p
\left(
\tilde\mathbf{r}|\tilde\mathbf{y},\,\omega
\right) & = & \frac{\mathrm{j}}{2\sigma^2_w}
\sum_{n=0}^{N-1}
\left[
\tilde y_n
\mathrm{e}^{\,\mathrm{j}\,\omega n} \tilde w_n^*
\right. \nonumber \\
& & \left.
\mbox{ } - n
\tilde y_n^*
\mathrm{e}^{-\mathrm{j}\,\omega n} \tilde w_n
\right].
\end{eqnarray}
Substituting (<ref>) in (<ref>) and assuming independent
noise (the real and imaginary parts of noise are also assumed independent),
we obtain:
\begin{eqnarray}
\label{Eq:Ap_Eq7}
\left[
\left(
\frac{\partial}{\partial \omega}
\ln p
\left(
\tilde\mathbf{r}|\tilde\mathbf{y},\,\omega
\right)
\right)^2
\right] = \frac{1}{\sigma^2_w}
\sum_{n=0}^{N-1} n^2
\left|
\tilde y_n
\right|^2
\end{eqnarray}
and hence
\begin{eqnarray}
\label{Eq:Ap_Eq8}
\left[
\left(
\hat\omega - \omega
\right)^2
\right] \ge \left[
\frac{1}{\sigma^2_w}
\sum_{n=0}^{N-1} n^2
\left|
\tilde y_n
\right|^2
\right]^{-1}
\end{eqnarray}
when $\tilde y_n$ is known. When $\tilde y_n$ is a random variable, which is
true in our case, then the right hand side of (<ref>) needs to
be further averaged over $\tilde\mathbf{y}$ <cit.>.
In other words, we need to compute
\begin{eqnarray}
\label{Eq:Ap_Eq9}
\lefteqn{
\left[
\left(
\frac{1}{\sigma^2_w}
\sum_{n=0}^{N-1} n^2
\left|
\tilde y_n
\right|^2
\right)^{-1}
\right]} \nonumber \\
& = &
\int_{\tilde\mathbf{y}}
\left[
\frac{1}{\sigma^2_w}
\sum_{n=0}^{N-1} n^2
\left|
\tilde y_n
\right|^2
\right]^{-1}
p(\tilde\mathbf{y})\, d\tilde\mathbf{y}
\end{eqnarray}
which is complicated. The purpose of this Appendix is to provide an alternate
and a much simpler solution to (<ref>), by assuming that
$\tilde y_n$ is ergodic.
We claim that, for large values of $N$ (in our case $N=504$)
\begin{eqnarray}
\label{Eq:Ap_Eq10}
\sum_{n=0}^{N-1} n^2
\left|
\tilde y_n
\right|^2
& \approx &
\sum_{n=0}^{N-1} n^2
\left[
\left|
\tilde y_n
\right|^2
\right] \nonumber \\
& = & \mbox{a constant}.
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Ap_Eq11}
\tilde y_n = \sum_{i=0}^{L_h-1}
\tilde h_i \tilde s_{n-i}.
\end{eqnarray}
\begin{eqnarray}
\label{Eq:Ap_Eq12}
\left[
\left|
\tilde y_n
\right|^2
\right] & = & E
\left[
\sum_{i=0}^{L_h-1}
\tilde h_i \tilde s_{n-i}
\sum_{j=0}^{L_h-1}
\tilde h_j^* \tilde s_{n-j}^*
\right] \nonumber \\
& = & \sum_{i=0}^{L_h-1}
\sum_{j=0}^{L_h-1}
\left[
\tilde h_i
\tilde h_j^*
\right]
\left[
\tilde s_{n-i}
\tilde s_{n-j}^*
\right]
\end{eqnarray}
where we have assumed
* $\tilde h_n$ and $\tilde s_n$ to be independent
* $\tilde s_n$ (the preamble) varies randomly from frame to frame and is
not a constant.
Hence (<ref>) can be rewritten as:
\begin{eqnarray}
\label{Eq:Ap_Eq13}
\left[
\left|
\tilde y_n
\right|^2
\right] & = & \sum_{i=0}^{L_h-1}
\sum_{j=0}^{L_h-1}
\sigma^2_f \delta_K(i-j)
\sigma^2_s \delta_K(j-i) \nonumber \\
& = & 2
\sigma^2_f
\sigma^2_s L_h.
\end{eqnarray}
where $\sigma^2_f$ is defined in (<ref>), $\sigma^2_s$ is defined
in (<ref>) and $\delta_K(\cdot)$ is the Kronecker delta
function. With these developments (<ref>) becomes
\begin{eqnarray}
\label{Eq:Ap_Eq14}
\left[
\left(
\frac{1}{\sigma^2_w}
\sum_{n=0}^{N-1} n^2
\left|
\tilde y_n
\right|^2
\right)^{-1}
\right]
\approx
\left[
\frac{2\sigma^2_f\sigma^2_s L_h}{\sigma^2_w}
\sum_{n=0}^{N-1} n^2
\right]^{-1}.
\end{eqnarray}
Therefore, the CRB on the variance of the frequency offset estimate is
(assuming $N-1=M$)
\begin{eqnarray}
\label{Eq:Ap_Eq15}
\left[
\left(
\hat\omega - \omega
\right)^2
\right] \ge \left[
\frac{2\sigma^2_f\sigma^2_s L_h}{\sigma^2_w}
\left(
\frac{M^3}{3} +
\frac{M^2}{2} +
\frac{M}{6}
\right)
\right]^{-1}
\end{eqnarray}
|
1511.00657
|
Institute for Quantum Information and Matter
Walter Burke Institute for Theoretical Physics, California Institute of Technology 452-48, Pasadena, CA 91125, USA
Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
National Institute of Standards and Technology,
Gaithersburg, MD, 20899
Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742 USA
Two of the key
properties of quantum physics are the no-signaling principle and the
Grover search lower bound. That is, despite admitting
stronger-than-classical correlations, quantum mechanics does not imply
superluminal signaling, and despite a form of exponential parallelism,
quantum mechanics does not imply polynomial-time brute force solution
of -complete problems. Here, we investigate the degree to which
these two properties are connected. We examine four classes of
deviations from quantum mechanics, for which we draw inspiration from
the literature on the black hole information paradox. We show that in these models, the
physical resources required to send a superluminal signal scale
polynomially with the resources needed to speed up Grover's
Hence the
no-signaling principle is equivalent to the inability to solve -hard problems efficiently by brute force
within the classes of theories analyzed.
03.67.-a, 03.65.-w, 04.70.Dy
§ INTRODUCTION
Recently the firewalls paradox <cit.> has shown that
our understanding of quantum mechanics and general relativity appear to be inconsistent at the event horizon of a black hole. Many of the leading
proposals to resolve the paradox involve modifying quantum
mechanics. For example, the final-state projection model of Horowitz
and Maldecena <cit.> and the state dependence model
of Papadodimas and Raju <cit.> are modifications to quantum
theory which might resolve the inconsistency.
One reason to be skeptical of such modifications of quantum mechanics
is that they can often give rise to superluminal signals, and hence
introduce acausality into the model. For example, Weinberg nonlinearities allow for superluminal signaling <cit.>. This is generally seen as
In contrast, in standard quantum theory, entanglement does not give rise to superluminal signaling.
Another startling feature of such models is that they might allow one to construct computers far more powerful even than conventional quantum computers.
In particular, they may allow one to solve -hard problems in polynomial time.
-hard problems refer to those problems for which the solution can be verified in polynomial time, but for which there are exponentially many possible solutions.
It is impossible for standard quantum computers to solve -hard problems efficiently by searching over all possible solutions. This is a consequence of the query complexity lower bound of Bennett, Bernstein, Brassard and Vazirani <cit.>, which shows one cannot search an unstructured list of $2^n$ items in fewer than $2^{n/2}$ queries with a quantum computer. (Here a query is an application of a function $f$ whose output indicates if you have found a solution. The query complexity of search is the minimum number of queries to $f$, possibly in superposition, required to find a solution.)
This bound is achieved by Grover's search algorithm <cit.>.
In contrast, many modifications of quantum theory allow quantum computers to search an exponentially large solution space in polynomial time.
For example, quantum computers equipped with postselection <cit.>, Deutschian closed timelike curves <cit.>, or nonlinearities <cit.>
all admit poly-time solution of -hard problems by brute force search.
In this paper we explore the degree to which superluminal
signaling and speedups over Grover's algorithm are connected. We consider several
modifications of quantum mechanics which are inspired by resolutions
of the firewalls paradox. For each modification, we show that the
theory admits superluminal signaling if and only if it admits a query
complexity speedup over Grover search.
Furthermore, we establish a quantitative relationship between superluminal signaling and speedups over Grover's algorithm.
More precisely, we show that if one can transmit one classical
bit of information superluminally using $n$ qubits and $m$ operations,
then one can speed up Grover search on a system of poly$(n,m)$ qubits
with poly$(n,m)$ operations, and vice versa. In other words, the
ability to send a superluminal signal with a reasonable amount of
physical resources is equivalent to the ability to violate the Grover
lower bound with a reasonable amount of physical resources.
Therefore the
no-signaling principle is equivalent to the inability to solve -hard problems efficiently by brute force
within the classes of theories analyzed.
Note that in the presence of nonlinear dynamics, density matrices are no longer equivalent to ensembles of pure states. Here, we consider measurements to produce probabilistic ensembles of post-measurement pure states and compute the dynamics of each of these pure states separately. Alternative formulations, in particular Everettian treatment of measurements as entangling unitaries, lead in some cases to different conclusions about superluminal signaling. See e.g. <cit.>.
§ RESULTS
We consider four modifications of quantum mechanics,
which are inspired by resolutions of the firewalls paradox. The first
two are “continuous” modifications in the sense that they have a
tunable parameter $\delta$ which quantifies the deviation from quantum
mechanics. The second two are “discrete" modifications in which standard quantum mechanics is supplemented by one additional operation.
§.§ Final state projection
The first “continuous" modification of quantum theory we consider is the final state projection model of
Horowitz and Maldecena <cit.>, in which the black
hole singularity projects the wavefunction onto a specific quantum
state. This can be thought of as a projective measurement with
postselection, which induces a linear (but not necessarily unitary)
map on the projective Hilbert space. (In some cases it is possible for the
Horowitz-Maldecena final state projection model to induce a perfectly
unitary process $S$ for the black hole, but in general interactions between the collapsing body
and infalling Hawking radiation inside the event horizon
induce deviations from unitarity <cit.>.) Such linear but non-unitary maps allow both superluminal signaling and speedups over Grover
search. Any non-unitary map $M$ of condition number
$1+\delta$ allows for superluminal signaling with channel capacity
$O(\delta^2)$ with a single application of $M$. The protocol for signaling is simple - suppose Alice has the ability to apply $M$, and suppose Alice and Bob share the entangled state
\begin{equation}\frac{1}{\sqrt{2}} \left( \ket{\phi_0} \ket{0} + \ket{\phi_1} \ket{1} \right).\end{equation}
where $\ket{\phi_0}$ and $\ket{\phi_1}$ are the minimum/maximum singular vectors of $M$, respectively. If Alice chooses to apply $M$ or not, then Bob will see a change in his half of the state, which allows signaling with channel capacity $\sim \delta^2$.
Furthermore, it is also possible for Bob to signal superluminally to Alice with the same state - if Bob chooses to measure or not to measure his half of the state, it will also affect the state of Alice's system after Alice applies $M$.
So this signaling is bidirectional, even if only one party has access to the
non-unitary map.
In the context of the black hole information paradox, this implies the acausality in the final state projection model could be present even far away from the black hole.
Also, assuming one can apply the same $M$ multiple
times, one can perform single-query Grover search using $\sim
1/\delta$ applications of $M$ using the methods of <cit.>. More detailed proofs of these results are provided in Appendix A.
We next examine the way in which these results are connected. First,
assuming one can speed up Grover search, by a generalization of the hybrid argument of <cit.>, there is a lower bound on the
deviation from unitarity required to achieve the speedup.
By our
previous results this implies a lower bound on the superluminal
signaling capacity of the map $M$.
More specifically, suppose that one can search an unstructured list of $N$ items using $q$ queries, with possibly non-unitary operations applied between queries.
Then, the same non-unitary dynamics must be capable of transmitting superluminal signals with channel capacity $C$ using shared entangled states, where
\begin{equation}C = \Omega \left( \left( \frac{\eta}{2 q^2} - \frac{2}{N} \right)^2
\right) \end{equation}
Here $\eta$ is a constant which is roughly $\sim 0.42$.
In particular, solving
-hard problems in polynomial time by unstructured search
would imply superluminal signaling with inverse polynomial channel
capacity. This can be regarded as evidence against the possibility of
using black hole dynamics to efficiently solve -hard
problems of reasonable size. A proof of this fact is provided in Appendix A.
In the other direction, assuming one can
send a superluminal signal with channel capacity $C$, there is a
lower bound on the deviation from unitarity which was applied. The proof is provided in Appendix A.
Again by our previous result, this implies one could solve the Grover search problem on a database of size $N$ using a single query and
\begin{equation}O \left( \frac{\log(N)}{\log(1+C^2)} \right)\end{equation}
applications of the nonlinear map.
Combining these results, this implies that if one can send a superluminal signal
with $n$ applications of $M$, then one can beat Grover's algorithm
with $O(n)$ applications of $M$ as well, and vice versa.
This shows
that in these models, the resources required to observe an exponential
speedup over Grover search is polynomially related to the resources
needed to send a superluminal signal. Hence an operational version of
the no-signaling principle (such as “one cannot observe superluminal
signaling in reasonable-sized experiments”) is equivalent to an
operational version of the Grover lower bound (“one cannot observe
violations of the Grover lower bound in reasonable-sized
§.§ Modification of the Born Rule
The next continuous modification of quantum mechanics we consider is
modification of the Born rule. Suppose that quantum states evolve
by unitary transformations, but upon measurement one sees outcome $x$
with probability proportional to some function
$f(\alpha_x)$ of the amplitude $\alpha_x$ on $x$. That is, one sees $x$ with probability
\begin{equation}\frac{f(\alpha_x)}{\sum_y f(\alpha_y)}\end{equation}
Note we have added a normalization factor to ensure this induces a valid probability distribution on outcomes.
This is loosely
inspired by Marolf and Polchinski's work <cit.> which suggests
that the “state-dependence” resolution of the firewalls paradox
<cit.> gives rise to violations of the Born rule. First, assuming some reasonable conditions on $f$ (namely, that $f$ is differentiable, $f'$ changes signs a finite number of times in $[0,1]$, and the measurement statistics of $f$ do not depend on the normalization of the state), we must have
$f(\alpha_x)=|\alpha_x|^p$ for some $p$.
The proof is provided in Appendix B.
Next we study the impact of
such modified Born rules with $p=2+\delta$ for small
$\delta$. Aaronson <cit.> previously showed that such
models allow for single-query Grover search in polynomial time while incurring a
multiplicative overhead $1/|\delta|$, and also allow for superluminal signaling using shared entangled
states of $\sim1/|\delta|$ qubits. (His result further generalizes to the harder problem of counting the number of solutions to an -hard problem, which is a #-hard problem). We find that these relationships hold in the opposite directions as well. Specifically, we show if one can send a superluminal signal with an entangled state on $m$
qubits with probability $\epsilon$, then we must have $\delta =\Omega(\epsilon/m)$.
By the results of Aaronson <cit.> this implies one can search a list of $N$ items using $O(\frac{m}{\epsilon} \log N)$ time.
Hence having the ability to send a superluminal signal using $m$ qubits implies the ability to perform an exponential speedup of Grover's algorithm with multiplicative overhead $m$.
In the other direction, if one
can achieve even a constant-factor speedup over Grover's algorithm using a
system of $m$ qubits, we show $|\delta|$ is at least $1/m$ as
More precisely, by a generalization of the hybrid argument of <cit.>, if there is an algorithm to search
an unordered list of $N$ items with $Q$ queries using $m$ qubits, then
\begin{equation}
\frac{1}{6} \leq \frac{2 Q}{\sqrt{N}} + |\delta| \log (M) +O(\delta^2).
\end{equation}
So if $Q<\sqrt{N}/24$, then we must have $|\delta| \geq \frac{1}{12m}$.
The proofs of these facts are provided in Appendix B.
Combining these results shows that the number of qubits required
to observe superluminal signaling or even a modest speedup over
Grover's algorithm are polynomially related. Hence one can derive an
operational version of the no-signaling principle from the Grover
lower bound and vice versa. This quantitative result is in some sense
stronger than the result we achieve for the final-state projection model,
because here we require only a mild speedup over Grover search to
derive superluminal signaling.
§.§ Cloning, Postselection, and Generic Nonlinearities
We next consider two “discrete” modifications of quantum mechanics in
which standard quantum mechanics is supplemented by one additional
operation. We show that both modifications admit both superluminal signaling with O(1) qubits and exponential speedups over Grover search.
First, we consider a model in which one can clone single
This model can be easily seen to admit superluminal signaling using entangled states, as pointed out by Aaronson, Bouland, Fitzsimons and Lee <cit.>. Indeed, suppose two parties Alice and Bob share the state $\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$. If Alice measures her half of the state, and Bob clones his state $k$ times and measures each copy in the computational basis, then Bob will either see either $0^k$ or $1^k$ as his output. On the other hand, if Alice does not measure her half of the state, and Bob does the same experiment, his outcomes will be a random string in $\{0,1\}^k$. Bob can distinguish these two cases with an error probability which scales inverse exponentially with $k$, and thus receive a signal faster than light.
In addition to admitting superluminal signaling with entangled states, this model also allows the solution of
-hard problems (and even #-hard problems) using a
single query to the oracle.
This follows by considering the following gadget: given a state $\rho$ on a single qubit, suppose one makes two copies of $\rho$, performs a Controlled-NOT gate between the copies, and discards one of the copies. This is summarized in a circuit diagram in Fig. <ref>.
Gadget used to show that cloning allows the poly-time solution of -hard problems.
This performs a non-linear operation $\mathcal{M}$ on the space of density matrices, and following the techniques of Abrams and Lloyd <cit.>, one can use this operation to “pry apart" quantum states which are exponentially close using polynomially many applications of the gadget. The proof is provided in Appendix C. This answers an open problem of
<cit.> about the power of
quantum computers that can clone. Therefore, adding cloning to quantum mechanics allows for both the poly-time solution of -hard problems by brute force search, and the ability to efficiently send superluminal signals.
Second, inspired by the final state projection model
<cit.>, we consider a model in which one can
postselect on a generic state $\ket{\psi}$ of $n$ qubits. Although
Aaronson <cit.> previously showed that allowing for
postselection on a single qubit suffices to solve -hard and #-hard
problems using a single oracle query, this does not immediately imply
that postselecting on a larger state has the same property, because
performing the unitary which rotates $\ket{0}^n$ to $\ket{\psi}$ will
in general require exponentially many gates. Despite this limitation, this model indeed allows the polynomial-time solution of -hard problems (as well as -hard problems) and superluminal signaling.
To see this, first note that given a gadget to postselect on $\ket{\psi}$, one can obtain multiple copies of $\ket{\psi}$ by inputting the maximally entangled state $\sum_i \ket{i}\ket{i}$ into the circuit and postselecting one register on the state $\ket{\psi}$. So consider creating two copies of $\ket{\psi}$, and applying the gadget shown in Figure <ref>, where the bottom
register is postselected onto $\ket{\psi}$, an operation we denote by
$\Qcircuit @C=1em @R=1em { & \gate{\ket{\psi}} & \qw }$.
\[
\Qcircuit @C=1em @R=1em {
& \qw &\qw & \ctrl{1} & \qw \\
\lstick{\ket{\psi}} & {/} \qw & \gate{Z\otimes I} &\multigate{1}{\mathrm{SWAP}} & \qw \\
\lstick{\ket{\psi}} & {/} \qw & \qw & \ghost{\mathrm{SWAP}} & \gate{\ket{\psi}}
\]
Gadget showing postselection onto generic $\ket{\psi}$ is equivalent to postselection onto $\ket{0}$.
For Haar-random $\ket{\psi}$, one can show the quantity $\bra{\psi}Z\otimes I\ket{\psi}$ is exponentially small, so this gadget simulates postselection on $\ket{0}$ on the first qubit.
The complete proof is provided in Appendix D.
Therefore, allowing postselection onto generic states is at least as powerful as allowing postselection onto the state $\ket{0}$, so by Aaronson's results <cit.> this model admits both superluminal signaling and exponential speedups over Grover search.
In addition, we address an open question from <cit.>
regarding the computational implications of general nonlinear maps on
pure states. In <cit.>, Abrams and Lloyd argued that
generic nonlinear maps allow for the solution of -hard problems and -hard
problems in polynomial time, except possibly for pathological
examples. In Appendix E, we prove this result rigorously in the case the map is
differentiable. Thus any pathological examples, if they exist, must
fail to be differentiable. (Here we assume the nonlinearity maps pure states to pure states; as a result it does not subsume our results on quantum computers which can clone, as the cloning operation may map pure states to mixed states. A detailed discussion is provided in Appendix C.) Unfortunately, the action of
general nonlinear maps on subsystems of entangled states are not
well-defined, essentially because they interact poorly with the
linearity of the tensor product. We discuss this in detail in Appendix F. Hence we are unable to connect this
result to signaling in the general case.
§ DISCUSSION
The central question in complexity theory is which computational problems can be solved efficiently and which cannot. Through experience, computer scientists have found that the most fruitful way to formalize the notion of efficiency is by demanding that the resources, such as time and memory, used to solve a problem must scale at most polynomially with the size of the problem instance (i.e. the size of the input in bits). A widely held conjecture, called the quantum Church-Turing thesis, states that the set of computational problems solvable in-principle with polynomial resources in our universe is equal to BQP, defined mathematically as the set of decision problems answerable using quantum circuits of polynomially many gates <cit.>. So far, this conjecture has held up remarkably well. Physical processes which conceivably might be more computationally powerful than quantum Turing machines, such as various quantum many-body dynamics of fermions, bosons, and anyons, as well as scattering processes in relativistic quantum field theories, can all be simulated with polynomial overhead by quantum circuits <cit.>.
The strongest challenge to the quantum Church-Turing thesis comes from quantum gravity. Indeed, many of the recent quantum gravity models proposed in relation to the black hole firewalls paradox involve nonlinear behavior of wavefunctions <cit.> and thus appear to suggest computational power beyond that of polynomial-size quantum circuits. In particular, the prior work of Abrams and Lloyd suggest that such nonlinearities generically enable polynomial-time solution to NP-hard problems, a dramatic possibility, that standard quantum circuits are not generally expected to admit <cit.>. Here, we have investigated several models and found a remarkably consistent pattern; in each case, if the modification to quantum mechanics is in a parameter regime allowing polynomial-time solution to NP-hard problems through brute-force search, then it also allows the transmission of superluminal signals through entangled states. Such signaling allows causality to be broken at locations arbitrarily far removed from the vicinity of the black hole, thereby raising serious questions as to the consistency of the models. Thus, the quantum Church-Turing thesis appears to be remarkably robust, depending not in a sensitive way on the complete Hilbert-space formalism of quantum mechanics, but rather derivable from more foundational operational principles such as the impossibility of superluminal signaling. Some more concrete conjectures on these lines are discussed in Appendix G.
§ ACKNOWLEDGMENTS
We thank Patrick Hayden, Daniel Harlow,
David Meyer, Andrew Childs and Debbie Leung for useful discussions. Portions of
this paper are a contribution of NIST, an agency of the US government,
and are not subject to US copyright. This material is based upon work supported by the DuBridge Postdoctoral Fellowship, by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NFS Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028), by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632, by the NSF Graduate Research Fellowship under grant
no. 1122374, and by the NSF Alan T. Waterman award under grant no. 1249349.
N. B. and A. B. would also
like to thank QuICS for their hospitality during the completion of
this project.
§ APPENDIX A: FINAL-STATE PROJECTION
Recent developments, particularly the AMPS firewall argument
<cit.>, have generated renewed interest in models of black hole
physics in which quantum mechanics is modified. Here, we explore some
difficulties associated one such scheme, namely the
Horowitz-Maldecena final state projection model
<cit.>. In this model, black hole singularities are
thought of as boundaries to spacetime with associated boundary
conditions on the quantum wavefunction <cit.>. That
is, at the singularity, the wavefunction becomes projected onto a
specific quantum state. (This can be thought of as a projective
measurement with postselection.)
If one prepares infalling matter in a chosen initial quantum state
$\ket{\psi} \in V$, allows it to collapse into a black hole, and then
collects all of the the Hawking radiation during the black hole
evaporation, one is left with a new quantum state related to the
original by some map $S:V \to V$. (We assume that black holes do not
alter the dimension of the Hilbert space. Standard quantum mechanics
and the Horowitz-Maldecena proposal share this feature.) Within
standard quantum mechanics, all such $S$ correspond to multiplication
by a unitary matrix, and hence the term $S$-matrix is used. If one
instead drops matter into an existing black hole and collects part of
the outgoing Hawking radiation, one is considering an open quantum
system. We leave the analysis of this more general scenario to future
It is possible for the Horowitz-Maldecena final state projection model
to induce a perfectly unitary process $S$ for the black hole. However,
as pointed out in <cit.>, interactions between the
collapsing body and infalling Hawking radiation inside the event
horizon generically induce deviations from unitarity. In this case,
the action $S$ of the black hole is obtained by applying some linear
but not unitary map $M$, and then readjusting the norm of the quantum
state back to one[Some interpret the final state projection
model as inducing a unitary map for observers who stay outside the
event horizon, while inducing a non-unitary map for infalling
observers <cit.>. Under this interpretation, our arguments
still apply to an infalling observer in the context of a large black
hole.]. Correspondingly, if a subsystem of an entangled state is
collapsed into a black hole and the Hawking radiation is collected
then the corresponding transformation is $M \otimes \id$ followed by
an adjustment of the normalization back to 1. Thus, aside from its
interest as a potential model for black holes, the Horowitz-Maldecena
model provides an interesting example of nonlinear quantum mechanics
in which subsystem structure remains well-defined (i.e. the issues
described in Appendix F do not arise).
In sections <ref> and <ref> we show that if Alice has access to
such a black hole and has foresightfully shared entangled states with
Bob, then Alice can send instantaneous noisy signals to Bob and
vice-versa independent of their spatial separation. We quantify the
classical information-carrying capacity of the communication channels
between Alice and Bob and find that they vanish only quadratically
with the deviation from unitarity of the black hole dynamics, as
measured by the deviation of the condition number of $M$ from
one. Hence, unless the deviation from unitarity is negligibly small,
detectable causality violations can infect the entirety of
spacetime. Furthermore, the bidirectional nature of the communication
makes it possible in principle for Alice to send signals into her own
past lightcone, thereby generating grandfather paradoxes.
In section <ref> we consider the use of the black hole
dynamical map $S$ to speed up Grover's search algorithm
<cit.>. We find a lower bound on the condition number of $M$ as
a function of the beyond-Grover speedup. By our results of sections
<ref> and <ref> this in turn implies a lower bound on the
superluminal signaling capacity induced by the black hole. In section
<ref> we prove the other direction: assuming one can
signal superluminally we derive a lower bound on the condition number
of $M$, which in turn implies a super-Grover speedup[By a
“super-Grover speedup”, we mean an algorithm which searches an
unstructured $N$-element list using fewer queries than Grover's
algorithm.]. We find that the
black-box solution of -hard problems in polynomial time
implies superluminal signaling with inverse polynomial capacity and
vice versa.
§.§ Communication from Alice to Bob
Suppose Alice has access to a black hole described by
the Horowitz-Maldecena final state projection model. Let $M$ be the
linear but not necessarily unitary map describing the dynamics of
the black hole. The non-unitarity of $M$ is quantified by $\delta =
1-\kappa$, the deviation of its condition number from one. Alice can
transmit instantaneous signals to Bob by choosing to drop her half
of a shared entangled state into the black hole or not. The capacity
of the resulting classical communication channel from Alice to Bob
is at least
\[
C \geq \frac{3}{8 \ln 2} \delta^2.
\]
We prove the lower bound on the channel capacity $C$ by exhibiting an
explicit protocol realizing it. Suppose the black hole acts on a
$d$-dimensional Hilbert space and correspondingly $M$ is a $d \times
d$ matrix. Then, $M$ has a singular-value decomposition given by
\begin{equation}
M = \sum_{i=0}^{d-1} \ket{\psi_i} \lambda_i \bra{\phi_i}
\end{equation}
\begin{equation}
\label{ortho}
\braket{\psi_i}{\psi_j} = \braket{\phi_i}{\phi_j} = \delta_{ij}.
\end{equation}
and $\lambda_0,\ldots,\lambda_{d-1}$ all real and nonnegative. We can
choose our indexing so that $\lambda_0$ is the smallest singular value
and $\lambda_1$ is largest singular value. Now, suppose Alice and Bob
share the state
\begin{equation}
\frac{1}{\sqrt{2}} \left( \ket{\phi_0} \ket{0} + \ket{\phi_1} \ket{1} \right).
\end{equation}
Here $\ket{0}$ and $\ket{1}$ refer to Bob's half of the entangled
state, which can be taken to be a qubit. If Alice wishes to transmit
the message “0” to Bob she does nothing. If she wishes to transmit
the message “1” to Bob she applies the black hole dynamical map $S$
to her half of the state. In other words, Alice drops her half of the state into the black hole, and waits for the black hole to evaporate. Correspondingly, one applies $M \otimes \id$
to the above state, yielding the unnormalized state
\begin{equation}
\frac{\lambda_0}{\sqrt{2}} \ket{\psi_0} \ket{0} +
\frac{\lambda_1}{\sqrt{2}} \ket{\psi_1} \ket{1}.
\end{equation}
After normalization, this becomes:
\begin{equation}
\frac{\lambda_0}{\sqrt{\lambda_0^2 + \lambda_1^2}} \ket{\psi_0} \ket{0}
+ \frac{\lambda_1}{\sqrt{\lambda_0^2 + \lambda_1^2}} \ket{\psi_1} \ket{1}.
\end{equation}
Thus, recalling ortho, Bob's reduced density matrix in this case
\begin{equation}
\rho_1 = \frac{\lambda_0^2}{\lambda_0^2 + \lambda_1^2} \ket{0} \bra{0} +
\frac{\lambda_1^2}{\lambda_0^2+\lambda_1^2} \ket{1} \bra{1},
\end{equation}
whereas in the case that Alice's message was “0” his reduced density
matrix is
\begin{equation}
\rho_0 = \frac{1}{2} \ket{0} \bra{0} + \frac{1}{2} \ket{1} \bra{1}.
\end{equation}
If $M$ is non-unitary then $\lambda_1 \neq \lambda_0$ and thus the
trace distance between these density matrices is
nonzero. Consequently, $\rho_1$ is distinguishable from $\rho_0$
and some fraction of a bit of classical information has been
transmitted to Bob.
More quantitatively, one sees that Bob's optimal measurement is in the
computational basis, in which case Alice and Bob are communicating
over a classical binary asymmetric channel. Specifically, if Alice
transmits a 0, the probability of bit-flip error is $\epsilon_0 = 1/2$
whereas if Alice transmits a 1, the probability of bit-flip error is
\begin{equation}
\label{ep1lam}
\epsilon_1 = \frac{\lambda_0^2}{\lambda_0^2 + \lambda_1^2}.
\end{equation}
A standard calculation
(see e.g <cit.>) shows that the classical capacity of
this channel is
\begin{equation}
C = h \left( \frac{1}{1+z} \right) - \frac{\log_2(z)}{1+z} + \epsilon_0
\log_2(z) - h(\epsilon_0),
\end{equation}
\begin{equation}
z = 2^{\frac{h(\epsilon_1)-h(\epsilon_0)}{1-\epsilon_1-\epsilon_0}}
\end{equation}
and $h$ is the binary entropy
\begin{equation}
h(p) = - p \log_2(p) -(1-p) \log_2(1-p).
\end{equation}
Specializing to $\epsilon_0 = \frac{1}{2}$ simplifies the expression to
\begin{equation}
C = h \left( \frac{1}{1+y} \right) - \frac{\log_2(y)}{1+y} +
\frac{1}{2} \log_2 (y) - 1
\end{equation}
\begin{equation}
y = 2^{\frac{h(\epsilon_1) - 1}{1/2-\epsilon_1}}.
\end{equation}
Lastly, we consider the limiting case $\epsilon_1 = \frac{1}{2} -
\Delta$ for $\Delta \ll 1$. In this limit, we get by Taylor
expansion that
\begin{equation}
\label{epscap}
C = \frac{3}{2 \ln 2} \Delta^2 + O(\delta^3).
\end{equation}
By ep1lam, $\Delta = \frac{1}{2} (1-\kappa) + O((1-\kappa)^2)$,
which completes the proof.
§.§ Communication from Bob to Alice
Suppose Alice has access to a black hole described by
the Horowitz-Maldecena final state projection model. Let $M$ be the
linear but not necessarily unitary map describing the dynamics of
the black hole. The non-unitarity of $M$ is quantified by $\delta =
1-\kappa$, the deviation of its condition number from one. Bob can
transmit instantaneous signals to Alice by choosing to measure his
half of a shared entangled state or not. The capacity
of the resulting classical communication channel from Bob to
Alice is at least
\[
C \geq \frac{3}{8 \ln 2} \delta^2.
\]
Suppose again that Alice and Bob share the state $\frac{1}{\sqrt{2}}
\left( \ket{\phi_0} \ket{0} + \ket{\phi_1} \ket{1} \right)$. If Bob
wishes to transmit the message “0” he does nothing, whereas if he
wishes to transmit the message “1” he measures his half of the
entangled state in the $\{\ket{0},\ket{1}\}$ basis. Then, Alice
applies the black hole dynamical map $S$ to her half of the state[That is, Alice drops her half of the shared state into the black hole, and waits for the black hole to evaporate.], and
then performs a projective measurement in the basis
$\{\ket{\psi_1},\ldots,\ket{\psi_d}\}$. We now show that this
procedure transmits a nonzero amount of classical information from Bob
to Alice unless $\lambda_0 = \lambda_1$, in which case $M$ is unitary.
In the case that Bob does nothing, the post-black hole state is again
\begin{equation}
\frac{\lambda_0}{\sqrt{\lambda_0^2 + \lambda_1^2}} \ket{\psi_0} \ket{0}
+ \frac{\lambda_1}{\sqrt{\lambda_0^2 + \lambda_1^2}} \ket{\psi_1} \ket{1}.
\end{equation}
Thus, Alice's post-black-hole reduced density matrix is
\begin{equation}
\frac{\lambda_0^2}{\lambda_0^2 + \lambda_1^2} \ket{\psi_0}
\bra{\psi_0} + \frac{\lambda_1^2}{\lambda_0^2 + \lambda_1^2} \ket{\psi_1}
\bra{\psi_1}.
\end{equation}
Alice's measurement will consequently yield the following probability
distribution, given that Bob's message was “0”:
\begin{eqnarray}
p(0|0) & = & \frac{\lambda_0^2}{\lambda_0^2 + \lambda_1^2} \\
p(1|0) & = & \frac{\lambda_1^2}{\lambda_0^2 + \lambda_1^2}.
\end{eqnarray}
Now, suppose Bob's message is “1”. Then, his measurement outcome
will be either $\ket{0}$ or $\ket{1}$ with equal probability. We must
analyze these cases separately, since the connection between ensembles
of quantum states and density matrices is not preserved under
nonlinear transformations[Elsewhere we have used density
matrices, but only after the application of the nonlinear
operation.]. If he gets outcome zero, then Alice holds the pure
state $\ket{\phi_0}$, which gets transformed to $\ket{\psi_0}$ by the
action of the black hole. If Bob gets outcome one, then Alice holds
$\ket{\phi_1}$, which gets transformed to $\ket{\psi_1}$ by the action
of the black hole. Hence, Alice's measurement samples from the
following distribution given that Bob's message was “1”:
\begin{eqnarray}
p(0|1) & = & 1/2 \\
p(1|1) & = & 1/2.
\end{eqnarray}
Hence, the information transmission capacity from Bob to Alice using
this protocol is the same as the Alice-to-Bob capacity calculated in
section <ref>.
§.§ Super-Grover Speedup implies Superluminal Signaling
Suppose one has access to one or more black holes
described by the Horowitz-Maldecena final state projection model. If
the non-unitary dynamics induced by the black hole(s) allow the
solution of a Grover search problem on a database of size $N$ using $q$
queries then the same non-unitary dynamics could be used transmit
instantaneous signals by applying them to half of an entangled
state. The capacity of the resulting classical communication channel
(bits communicated per use of the nonlinear dynamics) is at least
\[
C = \Omega \left( \left( \frac{\eta}{2 q^2} - \frac{2}{N} \right)^2
\right)
\]
in the regime $0 < \frac{\eta}{2 q^2} - \frac{2}{N} \ll 1$, where
$\eta = (\sqrt{2} - \sqrt{2 - \sqrt{2}})^2 \simeq 0.42$.
Let $V$ be the set of normalized vectors in the Hilbert space
$\mathbb{C}^{N}$. We will let $S:V \to V$ denote the nonlinear map
that a black hole produces by applying the matrix $M$ and then
readjusting the norm of the state to one. We will not assume that all
black holes are identical, and therefore, each time we interact with a
black hole we may have a different map. We denote the transformation
induced on the $k\th$ interaction by $S_k:V \to V$. We treat $S_k$ as
acting on the same state space for all $k$, but this is not actually a
restriction because we can simply take this to be the span of all the
Hilbert spaces upon which the different maps act.
Now suppose we wish to use the operations $S_1,S_2,\ldots$ to speed up
Grover search. Let $x \in \{0,\ldots,N-1\}$ denote the solution to the
search problem on $\{0,\ldots,N-1\}$. The corresponding unitary oracle
on $\mathbb{C}^N$ is[An alternate definition is to use a
bit-flip oracle $U_x \ket{y}\ket{z} = \ket{y} \ket{z \oplus f(y)}$
where $f(y) = 1$ if $y=x$ and $f(y)=0$ otherwise. This choice is
irrelevant since the phase flip oracle can be simulated using a
bit-flip oracle if the output register is initialized to
$(\ket{0}-\ket{1})/\sqrt{2}$, and a bit flip oracle can be
simulated using a controlled-phase-flip oracle.]
\begin{equation}
O_x = \id - 2 \ket{x}\bra{x}.
\end{equation}
The most general algorithm to search for $x$ is of the form
\begin{equation}
\label{finalstate}
S_q O_x \ldots S_2 O_x S_1 O_x \ket{\psi_0}
\end{equation}
followed by a measurement. Here $\ket{\psi_0}$ is any $x$-independent
quantum state on $\mathbb{C}^N$, and $S_k$ is any transformation that
can be achieved on $\mathbb{C}^N$ by any sequence of unitary
operations and interactions with black holes. Note that our
formulation is quite general and includes the case that multiple
non-unitary interactions are used after a given oracle
query, as is done in <cit.>. Also, for some $k$, $S_k$
may be purely unitary. For example,
one may have access to only a single black hole, and the rest of the
iterations of Grover's algorithm must be done in the ordinary unitary
manner. If the final measurement on the state described in
finalstate succeeds in identifying $x$ with high probability for
all $x \in \{0,\ldots,N-1\}$ then we say the query complexity of
Grover search using the black hole is at most $q$.
We now adapt the proof of the $\Omega(\sqrt{N})$ quantum query lower
bound for Grover search that was given in[Our notation is
based on the exposition of this proof given in
<cit.>.] <cit.> to show that any improvement in
the query complexity for Grover search implies a corresponding lower
bound on the ability of $S_k$ for some $k \in \{1,\ldots,q\}$ to “pry
apart” quantum states. This then implies a corresponding lower bound
on the rate of a superluminal classical information transmission
channel implemented using $S_k$.
The sequence of quantum states obtained in the algorithm
finalstate is
\begin{eqnarray}
\ket{\psi_0^x} & = & \ket{\psi_0} \nonumber \\
\ket{\phi_1^x} & = & O_x \ket{\psi_0} \nonumber \\
\ket{\psi_1^x} & = & S_1 O_x \ket{\psi_0} \label{stateseq} \\
\ket{\phi_2^x} & = & O_x S_1 O_x \ket{\psi_0} \nonumber \\
\ket{\psi_2^x} & = & S_2 O_x S_1 O_x \ket{\psi_0} \nonumber \\
& \vdots & \nonumber \\
\ket{\psi_q^x} & = & S_q O_x \ldots S_1 O_x \ket{\psi_0}. \nonumber
\end{eqnarray}
\begin{eqnarray}
\ket{\psi_k} & = & S_k S_{k-1} \ldots S_1 \ket{\psi_0} \\
C_k & = & \sum_{x=0}^{N-1} \| \ket{\phi_k^x} - \ket{\psi_{k-1}} \|^2
\label{ck} \\
D_k & = & \sum_{x=0}^{N-1} \| \ket{\psi_k^x} - \ket{\psi_k} \|^2. \label{dk}
\end{eqnarray}
$\ket{\psi_k}$ can be interpreted as the state which would have been
obtained after the $k\th$ step of the algorithm with no oracle queries
(or of the Grover search problem lacked a solution).
Now, assume that for all $x \in \{0,\ldots,N-1\}$ the search algorithm
succeeds after $q$ queries in finding $x$ with probability at least
$\frac{1}{2}$. Then,
\begin{equation}
| \langle x | \psi_q^x \rangle |^2 \geq \frac{1}{2} \quad \forall x
\in \{0,\ldots,N-1\}
\end{equation}
which implies
\begin{equation}
D_q \geq \eta N, \label{bigomega}
\end{equation}
with $\eta = (\sqrt{2}-\sqrt{2-\sqrt{2}})^2 \simeq 0.42$, as shown in
<cit.> and discussed in <cit.>. By stateseq,
ck, and dk,
\begin{eqnarray}
C_k & = & \sum_{x=0}^{N-1} \| O_x \ket{\psi_{k-1}^x} -
\ket{\psi_{k-1}} \|^2 \\
& \leq & D_{k-1} + 4 \sqrt{D_{k-1}} + 4, \label{ckbound1}
\end{eqnarray}
where the above inequality is obtained straightforwardly using the
triangle and Cauchy-Schwarz inequalities.
Next, let
\begin{equation}
R_k = D_k - C_k. \label{rk}
\end{equation}
Thus, by stateseq, ck, and dk,
\begin{equation}
R_k = \sum_{x=0}^{N-1} \| S_k \ket{\phi_k^x} - S_k \ket{\psi_{k-1}} \|^2
- \sum_{x=0}^{N-1} \| \ket{\phi_k^x} - \ket{\psi_{k-1}} \|^2. \label{rkunpack}
\end{equation}
Hence, one sees that $R_k$ is some measure of the ability of $S_k$ to
“pry apart” quantum states. (In ordinary quantum mechanics $S_k$ would
be unitary and hence $R_k$ would equal zero.)
Combining rk and ckbound1 yields
\begin{equation}
D_k \leq R_k + D_{k-1} + 4 \sqrt{D_{k-1}} + 4. \label{recurrence1}
\end{equation}
\begin{equation}
B = \max_{1 \leq k \leq q} R_k. \label{Bdef}
\end{equation}
Then recurrence1 yields the simpler inequality
\begin{equation}
D_k \leq B + D_{k-1} + 4 \sqrt{D_{k-1}} + 4. \label{recurrence2}
\end{equation}
By stateseq and dk,
\begin{equation}
D_0 = 0. \label{init}
\end{equation}
By an inductive argument, one finds that recurrence2 and init imply
\begin{equation}
D_k \leq (4+B) k^2. \label{payoff}
\end{equation}
Combining payoff and bigomega yields
\begin{equation}
(4+B)q^2 \geq \eta N,
\end{equation}
or in other words
\begin{equation}
B \geq \frac{\eta N}{q^2} -4.
\end{equation}
Thus, by Bdef and rkunpack, there exists some $k \in
\{1,\ldots,q\}$ such that
\begin{equation}
\sum_{x=0}^{N-1} \left( \| S_k \ket{\phi_k^x} - S_k \ket{\psi_{k-1}}
\|^2 - \| \ket{\phi_k^x} - \ket{\psi_{k-1}} \|^2 \right) \geq
\frac{\eta N}{q^2} -4
\end{equation}
Hence, there exists some $x \in \{0,\ldots,N-1\}$ such that
\begin{equation}
\| S_k \ket{\phi_k^x} - S_k \ket{\psi_{k-1}} \|^2 -
\| \ket{\phi_k^x} - \ket{\psi_{k-1}} \|^2 \geq
\frac{\eta}{q^2} - \frac{4}{N}. \label{normpry}
\end{equation}
To simplify notation, define
\begin{eqnarray}
\ket{A} & = & \ket{\phi_k^x} \\
\ket{B} & = & \ket{\psi_{k-1}} \\
\ket{A'} & = & S_k \ket{\phi_k^x} \\
\ket{B'} & = & S_k \ket{\psi_{k-1}}.
\end{eqnarray}
Then normpry becomes
\begin{equation}
\| \ket{A'} - \ket{B'} \|^2 - \| \ket{A} - \ket{B} \|^2 \geq
\frac{\eta}{q^2} - \frac{4}{N}. \label{normpryAB}
\end{equation}
Recalling that $\| \ket{\psi} \| = \sqrt{\langle \psi | \psi
\rangle}$, normpryAB is equivalent to
\begin{equation}
\label{dotpry}
\re \langle A | B \rangle - \re \langle A' | B' \rangle \geq \epsilon
\end{equation}
\begin{equation}
\label{epsdef}
\epsilon = \frac{\eta}{2 q^2} - \frac{2}{N}.
\end{equation}
Next we will show that, within the framework of final-state projection
models, dotpry implies that Alice can send a polynomial fraction
of a bit to Bob or vice versa using preshared entanglement and a
single application of black hole dynamics. Recall that, within the
final state projection model,
\begin{eqnarray}
\ket{A'} & = & \frac{M \ket{A}}{\sqrt{\bra{A} M^\dag M \ket{A}}} \\
\ket{B'} & = & \frac{M \ket{B}}{\sqrt{\bra{B} M^\dag M \ket{B}}}
\end{eqnarray}
Thus, dotpry is equivalent to
\begin{equation}
\re \left[ \bra{A} \left( \id - \frac{M^\dag M}
{\sqrt{\bra{A} M^\dag M \ket{A} \bra{B} M^\dag M \ket{B}}} \right)
\ket{B} \right] \geq \epsilon
\end{equation}
\begin{equation}
\label{Mnorm}
\left\| \id - \frac{M^\dag M}
{\sqrt{\bra{A} M^\dag M \ket{A} \bra{B} M^\dag M \ket{B}}} \right\| \geq \epsilon.
\end{equation}
Again using $\lambda_0$ to denote the smallest singular value of $M$
and $\lambda_1$ to denote the largest, we see that, assuming
$\epsilon$ is nonnegative, Mnorm implies either
Case 1:
\begin{equation}
\frac{\lambda_1^2}{\sqrt{\bra{A} M^\dag M \ket{A} \bra{B} M^\dag M
\ket{B}}} \geq 1 + \epsilon,
\end{equation}
which implies
\begin{equation}
\label{direct}
\frac{\lambda_1^2}{\lambda_0^2} \geq 1 + \epsilon,
\end{equation}
Case 2:
\begin{equation}
\frac{\lambda_0^2}{\sqrt{\bra{A} M^\dag M \ket{A} \bra{B} M^\dag M
\ket{B}}} \leq 1 - \epsilon,
\end{equation}
which implies
\begin{equation}
\label{recip}
\frac{\lambda_0^2}{\lambda_1^2} \leq 1 - \epsilon.
\end{equation}
Examining dotpry, one sees that $\epsilon$ can be at most 2. If
$0 \leq \epsilon \leq 1$ then recip implies direct. If
$1 < \epsilon \leq 2$ then case 2 is impossible. Hence, for any
nonnegative $\epsilon$ one obtains direct. Hence, by the results
of sections <ref> and <ref>, Alice and Bob can communicate in
either direction through a binary asymmetric channel whose bitflip
probabilities $\epsilon_0$ for transmission of zero and $\epsilon_1$
for transmission of one are given by
\begin{eqnarray}
\epsilon_0 & = & \frac{1}{2} \label{epnaught} \\
\epsilon_1 & = & \frac{\lambda_0^2}{\lambda_0^2+\lambda_1^2} \leq
\frac{1}{2+\epsilon}. \label{epone_raw}
\end{eqnarray}
For $0 \leq \epsilon \leq 2$, $\frac{1}{2+\epsilon} \leq \frac{1}{2} -
\frac{\epsilon}{8}$. Thus, epone_raw implies the following more
convenient inequality
\begin{equation}
\label{epone}
\epsilon_1 \leq \frac{1}{2} - \frac{\epsilon}{8}.
\end{equation}
In section <ref> we calculated that the channel capacity in the
case that $\epsilon_0 = \frac{1}{2}$ and $\epsilon_1 = \frac{1}{2} -
\delta$ is $\Omega(\delta^2)$ for $\delta \ll 1$. Thus,
epnaught and epone imply a channel capacity in either
direction of
\begin{equation}
\label{final_capacity}
C = \Omega \left( \left( \frac{\eta}{2 q^2} - \frac{2}{N} \right)^2
\right)
\end{equation}
in the regime $0 < \frac{\eta}{2 q^2} - \frac{2}{N} \ll 1$.
The above scaling of the superluminal channel capacity with Grover
speedup shows that polynomial speedup for small instances or
exponential speedup for large instances imply $1/\mathrm{poly}$
superluminal channel capacity. In particular, to solve in
polynomial time without exploiting problem structure we would need $q
\propto \log^c N$ for some constant $c$. In this setting $N = 2^n$
where $n$ is the size of the witness for the problem in NP. In this
limit, final_capacity implies instantaneous signaling channels in
each direction with capacity at least
\begin{equation}
\label{Cval}
C = \Omega \left( \frac{1}{\log^{4c} N} \right) = \Omega \left(
\frac{1}{n^{4c}} \right).
\end{equation}
If we assume that superluminal signaling capacity is limited to some
negligibly small capacity $C \leq \epsilon$ then, by Cval, NP-hard
problems cannot be solved by unstructured search in time scaling
polynomially with witness size (specifically $n^c$ for some constant
$c$) except possibly for unphysically large instances with
$n = \Omega \left( \left( \frac{1}{\epsilon} \right)^{\frac{1}{4c}} \right)$.
§.§ Signaling implies Super-Grover Speedup
In sections <ref> and <ref> we showed that if final-state
projection can be used to speed up Grover search it can also be used
for superluminal signaling. In this section we show the
converse. Unlike in section <ref>, we here make the
assumption that we can make multiple uses of the same non-unitary map
$S$ (just as other quantum gates can be used multiple times without
variation). Since signaling cannot be achieved by performing unitary
operations on entangled quantum degrees of freedom, superluminal
signaling implies non-unitarity. Furthermore, as shown in Appendix F, iterated application of
any nonlinear but differentiable map allows the Grover search problem
to be solved with only a single oracle query. The nonlinear maps that
arise in final-state projection models are differentiable (provided
$M$ is invertible), and thus within the final-state projection
framework signaling implies single-query Grover search. In the
remainder of this section we quantitatively investigate how many
iterations of the nonlinear map are needed to achieve single-query
Grover search, as a function of the superluminal signaling capacity. We
find that unless the signaling capacity is exponentially small,
logarithmically many iterations suffice. Specifically, our main result
of this section is the following theorem.
Suppose Alice has access to a linear but not necessarily unitary maps
on quantum states, as can arise in the Horowitz-Maldecena final
state projection model. Suppose she achieves instantaneous
classical communication capacity of $C$ bits transmitted per use of
nonunitary dynamics. Then she could solve the Grover search problem on
a database of size $N$ using a single query and
$O \left( \frac{\log(N)}{\log(1+C^2)} \right)$ applications of the
available nonunitary maps.
Suppose Alice has access to black hole(s) and Bob does not. Alice will
use this to send signals to Bob using some shared entangled state
$\ket{\psi}_{AB}$. Her most general protocol is to apply some
map $M_0$ to her half of the state if she wishes to transmit a zero
and some other map $M_1$ if she wishes to transmit a one. (As a special
case, $M_0$ could be the identity.) Here, per the final state
projection model, $M_0$ and $M_1$ are linear but not necessarily
unitary maps, and normalization of quantum states is to be adjusted
back to one after application of these maps. The possible states
shared by Alice and Bob given Alice's two possible messages are
\begin{eqnarray}
\ket{\psi_0}_{AB} & \propto & M_0 \ket{\psi}_{AB} \\
\ket{\psi_1}_{AB} & \propto & M_1 \ket{\psi}_{AB}.
\end{eqnarray}
The signaling capacity is determined by the distinguishability of the
two corresponding reduced density matrices held by Bob
\begin{eqnarray}
\rho_0 & = & \mathrm{Tr}_A \left[ \ket{\psi_0}_{AB} \right] \\
\rho_1 & = & \mathrm{Tr}_A \left[ \ket{\psi_1}_{AB} \right].
\end{eqnarray}
We can define
\begin{equation}
\ket{\psi'} \propto M_0 \ket{\psi}_{AB}
\end{equation}
in which case
\begin{equation}
\ket{\psi_1}_{AB} \propto M_1 M_0^{-1} \ket{\psi'}.
\end{equation}
(We normalize $\ket{\psi'}$ so that $\braket{\psi'}{\psi'} = 1$.)
Thus, the signaling capacity is determined by the distinguishability
\begin{eqnarray}
\rho_0 & = & \mathrm{Tr}_A \left[ \ket{\psi'} \right] \\
\rho_1 & = & \mathrm{Tr}_A \left[ \frac{1}{\eta} M \ket{\psi'} \right]
\end{eqnarray}
\begin{eqnarray}
M & = & M_1 M_0^{-1} \\
\eta & = & \sqrt{\bra{\psi'} M^\dag M \ket{\psi'}}.
\end{eqnarray}
We have thus reduced our analysis to the case that Alice applies some
non-unitary map $M$ to her state if she wants to transmit a one and
does nothing if she wants to transmit a zero. We will next obtain a
lower bound $\kappa_{\min}$ on the condition number of $M$ as a function of
the signaling capacity from Alice to Bob. This then implies that one
of $M_0, M_1$ has a condition number at least $\sqrt{\kappa_{\min}}$
for the general case.
Suppose that $M$ has the following singular value decomposition
\begin{equation}
\label{svd}
M = \sum_i \lambda_i \ket{\psi_i} \bra{\phi_i}.
\end{equation}
We can express $\ket{\psi'}$ as
\begin{equation}
\ket{\psi'} = \sum_{i,j} \alpha_{ij} \ket{\phi_i} \ket{B_j}
\end{equation}
where $\ket{\phi_1}, \ket{\phi_2}, \ldots$ is the basis determined by
the singular value decomposition svd and $\ket{B_1}, \ket{B_2},
\ldots$ is the basis Bob will perform his measurement in when he tries
to extract Alice's message. If Alice wishes to transmit one then she
applies $M$ yielding
\begin{equation}
\ket{\psi_1} \propto \sum_{i,j} \lambda_i \alpha_{ij} \ket{\psi_i} \ket{B_j}.
\end{equation}
\begin{eqnarray}
\rho_0 & = & \sum_{i,j,k} \alpha_{ij} \alpha^*_{ik} \ket{B_j}\bra{B_k}\\
\rho_1 & = & \sum_{i,j,k} \frac{\lambda_i^2}{\eta} \alpha_{ij}
\alpha^*_{ik} \ket{B_j} \bra{B_k}.
\end{eqnarray}
Consequently, Bob's measurement will yield a sample from the following
probability distributions conditioned on Alice's message.
\begin{eqnarray}
p(j|0) & = & \sum_i |\alpha_{ij}|^2 \\
p(j|1) & = & \sum_i \frac{\lambda_i^2}{\eta} |\alpha_{ij}|^2.
\end{eqnarray}
The total variation distance between these distributions, which
determines the capacity of the superluminal channel is
\begin{equation}
\label{deltadef}
\Delta = \frac{1}{2} \sum_j |p(j|0)-p(j|1)| = \frac{1}{2} \sum_j \left| \sum_i
|\alpha_{ij}|^2 \left( 1 - \frac{\lambda_i^2}{\eta} \right) \right|.
\end{equation}
From a given value of this total variation distance we wish to derive
a lower bound on the condition number of $M$, that is, the ratio of
the largest singular value to the smallest. Applying the triangle
inequality to deltadef yields
\begin{equation}
\label{ineq1}
\Delta \leq \frac{1}{2} \sum_{ij} |\alpha_{ij}|^2 \left| 1 -
\frac{\lambda_i^2}{\eta} \right|.
\end{equation}
Because $\alpha_{ij}$ are amplitudes in a normalized quantum state,
\begin{equation}
p(i) = \sum_j |\alpha_{ij}|^2
\end{equation}
is a probability distribution. We can thus rewrite ineq1
\begin{eqnarray}
\label{ineq2a}
\Delta & \leq & \frac{1}{2} \sum_i p(i) \left| 1 -
\frac{\lambda_i^2}{\eta} \right| \\
& \leq & \frac{1}{2} \max_i \left| 1 - \frac{\lambda_i^2}{\eta}
\right|. \label{ineq2b}
\end{eqnarray}
In keeping with the notation from previous sections, we let
$\lambda_0$ denote the smallest singular value of $M$ and $\lambda_1$
the largest. Thus, ineq2b yields
\begin{equation}
\label{ineq3}
\Delta \leq \frac{1}{2} \max \left\{ 1-\frac{\lambda_0^2}{\eta},
\frac{\lambda_1^2}{\eta} - 1 \right\}.
\end{equation}
\begin{eqnarray}
\eta & = & \sum_{jk} |\alpha_{jk}|^2 \lambda_j^2 \\
& = & p(j) \lambda_j^2 \\
& \in & [\lambda_0^2, \lambda_1^2]. \label{etarange}
\end{eqnarray}
Applying etarange to ineq3 yields
\begin{equation}
\label{deltakappa}
\Delta \leq \frac{1}{2} \max \left\{ 1-\frac{\lambda_0^2}{\lambda_1^2},
\frac{\lambda_1^2}{\lambda_0^2} - 1 \right \}.
\end{equation}
As shown in section <ref>, the channel capacity $C$ is related
to the total variation distance $\Delta$ according to
\begin{equation}
C \leq \Delta - \Delta \log_2 \Delta
\end{equation}
for $\Delta \leq 1/e$. For small $\Delta$, the $-\Delta \log_2 \Delta$
term dominates the $\Delta$ term. We can simplify further by noting
that for all positive $\Delta$, $\sqrt{\Delta} >
-\Delta \log_2(\Delta)$. Hence, $C = O(\sqrt{\Delta})$. Thus to
achieve a given channel capacity $C$ we need
\begin{equation}
\Delta = \Omega(C^2).
\end{equation}
By deltakappa, this implies that achieving a channel capacity $C$
\begin{equation}
|1-\kappa_{\min}^2| = \Omega(C^2),
\end{equation}
where $\kappa_{\min}$ is the condition number of the nonlinear map
$M = M_1 M_0^{-1}$. This implies that one of $M_0$ or $M_1$ must have
condition number at least $\kappa = \sqrt{\kappa_{min}} = \Omega \left(
(1-C^2)^{1/4} \right)$. This in turn implies Grover search with one
query and $O(\log_\kappa(N))$ applications of the nonlinear map via
the methods of <cit.>.
§.§ Channel Capacity and Total Variation Distance
Alice wishes to transmit a message to Bob. If she sends zero Bob
receives a sample from $p(B|0)$ and if she sends one Bob receives a
sample from $p(B|1)$. Here, $B$ is a random variable on a finite state
space $\Gamma = \{0,1\ldots,d-1\}$. The only thing we know about this
channel is that
\begin{equation}
\label{tvd}
\left| p(B|0) - p(B|1) \right| = \delta,
\end{equation}
where $| \cdot |$ denotes the total variation distance (i.e.
half the $l_1$ distance). In this section we derive an upper bound on
the channel capacity as a function of $\delta$. Specifically, we show
that the (asymptotic) capacity $C$ obeys
\begin{equation}
C \leq \delta - \delta \log_2 \delta.
\end{equation}
Any strategy that Bob could use for decoding Alice's message
corresponds to a decomposition of $\Gamma$ as
\begin{equation}
\label{decomp}
\Gamma = \Gamma_0 \sqcup \Gamma_1
\end{equation}
where $\Gamma_0$ are the outcomes that Bob interprets as zero and
$\Gamma_1$ are the outcomes that Bob interprets as one.
From tvd it follows that
\begin{equation}
\label{ineq}
\left| p(b \in \Gamma_0|A=0) - p(b \in \Gamma_0|A=1) \right| \leq \delta.
\end{equation}
(The defining property of total variation distance is that this holds
for any set $\Gamma_0$.)
Let $F=0$ whenever $B \in \Gamma_0$ and $F=1$ whenever $B \in
\Gamma_1$. That is, the random variable $F$ is Bob's guess as to
Alice's message. By standard Shannon theory <cit.>, the
channel capacity is the mutual information $I(F;A)$ maximized over
Alice's choice of $p(A)$.
From ineq it follows that
\begin{equation}
\left| p(F|A=0) - p(F|A=1) \right| \leq \delta.
\end{equation}
Let $p_\alpha$ be the probability distribution
\begin{equation}
p_\alpha(F) = \alpha p(F|A=0) + (1-\alpha) p(F|A=1)
\end{equation}
for some $\alpha \in [0,1]$. From the elementary properties of total
variation distance it follows that
\begin{equation}
\left| p_\alpha(F) - p(F|A=0) \right| \leq \delta
\end{equation}
\begin{equation}
\left| p_\alpha(F) - p(F|A=1) \right| \leq \delta
\end{equation}
for any choice of $\alpha$. In particular, we may set $\alpha =
p(A=0)$, in which case we have
\begin{eqnarray}
\left| p(F) - p(F|A=0) \right| & \leq & \delta \label{pf1} \\
\left| p(F) - p(F|A=1) \right| & \leq & \delta. \label{pf2}
\end{eqnarray}
Next, we recall the Fannes inequality. This says that for any two
density matrices $\rho,\sigma$ on a $d$-dimensional Hilbert space
whose trace distance satisfies $T \leq \frac{1}{e}$
\begin{equation}
\left| S(\rho) - S(\sigma) \right| \leq T \log_2 d - T \log_2 T.
\end{equation}
Specializing to the special case that $\sigma$ and $\rho$ are
simultaneously diagonalizable, one obtains the following statement
about classical entropies.
Let $p$ and $q$ be two probability distributions on a state space of
size $d$. Let $T$ be the total variation distance between $p$ and
$q$. Suppose $T \leq \frac{1}{e}$. Then
\begin{equation}
| H(p) - H(q) | \leq T \log_2 d - T \log_2 T.
\end{equation}
Applying corollary <ref> to pf1 and pf2
yields[We have used $H[p]$ to denote the entropy of a
probability distribution $p$ and $H(R)$ to denote the entropy of a
random variable $R$.]
\begin{eqnarray}
\left| H \left[ p(F) \right] - H \left[ p(F|A=0) \right] \right| &
\leq & \delta - \delta \log_2 \delta \\
\left| H \left[ p(F) \right] - H \left[ p(F|A=1) \right] \right| &
\leq & \delta - \delta \log_2 \delta.
\end{eqnarray}
\begin{eqnarray}
I(F;A) & = & H(F) - H(F|A) \\
& = & H \left[ p(F) \right] - p(A=0) H \left[ p(F|A=0) \right]
- p(A=1) H \left[ p(F|A=1) \right]\\
& \leq & \delta - \delta \log_2 \delta,
\end{eqnarray}
which completes the derivation.
§ APPENDIX B: VIOLATIONS OF THE BORN RULE
In this appendix we consider modification of quantum mechanics in which
states evolve unitarily, but measurement statistics are not given by
the Born rule. This is loosely inspired by the “state dependence”
resolution of the firewalls paradox, put forth by Papadodimas and Raju
<cit.>. In this theory, the measurement operators $O$ which
correspond to observables are not fixed linear operators, but rather
vary depending on the state they are operating on,
i.e. $O=O(\ket{\psi})$. (In general such dependencies lead to
nonlinearities in quantum mechanics, but Papadodimas and Raju argue
these are unobservable in physically reasonable experiments.) Recently
Marolf and Polchinski <cit.> have claimed that such modifications
of quantum mechanics lead to violations of the Born rule. We do not
take a position either way on Marolf and Polchinski's claim, but use
it as a starting point to investigate how violations of the Born rule
are related to superluminal signaling and computational complexity.
Here we consider violations of the Born rule of the following form:
given a state $\ket{\psi}=\sum_x \alpha_x \ket{x}$, the probability
$p_x$ of seeing outcome $x$ is given by
\begin{equation} p_x = \frac{f(\alpha_x)}{\sum_{x'} f(\alpha_{x'})}
\end{equation}
for some function $f(\alpha):\mathbb{C} \rightarrow
\mathbb{R}^+$. We assume that states in the theory evolve unitarily as
in standard quantum mechanics. One could consider more general
violations of the Born rule, in which the function $f$ depends not
only on the amplitude $\alpha_x$ on $x$, but on the amplitudes on
other basis states as well. However such a generalized theory seems
impractical to work with, so we do not consider such a theory here.
We first show that, assuming a few reasonable conditions on $f$
(namely that $f$ has a reasonably behaved derivative and that
measurement statistics do not depend on the normalization of the
state), the only way to modify the Born rule is to set
$f(\alpha)=|\alpha|^{2+\delta}$ for some $\delta\neq 0$. We then show
that in theories where the Born rule is modified, superluminal
signaling is equivalent to a speedup to Grover search. More precisely,
we show that if one can send superluminal signals using states on $n$
qubits, then one can speed up Grover search on a system with $O(n)$
qubits, and vice versa. Hence one can observe superluminal signals on
reasonably sized systems if and only if one can speed up Grover search
using a reasonable number of qubits.
We are not the first authors to examine the complexity theoretic
consequences of modifications to the Born rule. Aaronson
<cit.> considered such modifications, and showed that
if $\delta$ is any constant, then such modifications allow for the
solution of -hard problems in polynomial time. Our
contribution is to show the opposite direction, namely that a
significant speedup over Grover search implies the deviation from the
Born rule $\delta$ is large, and to connect this to superluminal
We prove our results in several steps. First, in Theorems
<ref> and <ref>, we show that deviations
in the Born rule by $\delta$ allow the solution of -hard
problems and superluminal signaling using $O(1/\delta)$ qubits. As
noted previously, Theorem <ref> follows from the work of
Aaronson <cit.>, but we include a proof for
In Theorem <ref> we show that, assuming one has a
superluminal signaling protocol using a shared state on $m$ qubits,
the deviation from the Born rule $\delta$ must be $ \geq
\Omega(1/m)$. Likewise in Theorem <ref> we show
that if one can achieve a constant factor super-Grover speedup using
$m$ qubits, that we must have $\delta \geq \Omega(1/m)$ as
well. Combining these with Theorems <ref> and
<ref> shows that a super-Grover speedup on $m$ qubits
implies superluminal signaling protocols with $O(m)$ qubits and vice
versa. Supplementary Figure 1 explains the relationship between
these theorems below.
{\text{Deviation } \delta \text{ from the Born rule}} \ar@<1ex>[d]^{\text{Theorem } \ref{bornimpliesFTL}} \ar@<1ex>[r]^{\text{Theorem \ref{bornimpliesNP}} \hspace{3em} } & {\text{Speedup over Grover search with }1/\delta\text{ qubits}} \ar@<1ex>[l]^{\text{Theorem } \ref{searchimpliesdelta} \hspace{3em}} \\
{\text{Superluminal signaling with }1/\delta\text{ qubits}}\ar@<1ex>[u]^{\text{Theorem }\ref{signalingimpliesborn}}}$
Relationship between theorems connecting signaling and search.
In short, we find that a violation of the Born rule by $\delta$ is
equivalent to allowing a super-Grover speedup and an instantaneous
signaling protocol using $1/\delta$ qubits. Hence in theories in which
$\delta$ is only polynomially suppressed (as a function e.g. of the
number of fields $N$ in Super-Yang-Mills), then such theories allow
for superluminal signaling and violations of the Grover lower bound
with reasonable overheads. On the other hand, our results do not rule
out violations of the Born rule in which $1/\delta$ is unphysically
§.§ Power law violations are unique
We now show that, given some reasonable assumptions about the function
$f(\alpha)$, the only possible violation of the Born rule is
given by $f(\alpha)=|\alpha|^p$. In particular we will demand the
following properties of $f$:
* Well-behaved derivative: $f(\alpha)$ is continuous and
differentiable, and $f'(\alpha)$ changes sign at most a finite
number of times on $[0,1]$
* Scale invariance: for any $k\in\mathbb{C}$, we have that
$\frac{f(k\alpha)}{\sum_x f(k\alpha_x)}=\frac{f(\alpha)}{\sum_x
f(\alpha_x)}$. In other words the calculation of the probability
$p_x$ of seeing outcome $x$ is independent of the norm or phase of
the input state; it only depends on the state of the projective
Hilbert space.
There are a number of other reasonable constraints one could impose;
for instance one could demand that the modified Born rule has to
behave well under tensor products. Suppose you have a state
$\ket{\psi}=\sum_x \alpha_x \ket{y}$ and a state $\ket{\phi}=\sum_y
\beta_y \ket{y}$. A reasonable assumption would be to impose that in
the state $\ket{\psi}\otimes \ket{\phi}$, the probability $p_{xy}$ of
measuring outcome $xy$ should be equal to $p_xp_y$, i.e. a tensor
product state is equivalent to independent copies of each state. More
formally this would state that
\begin{equation}
\frac{f(\alpha_{x}\beta_y)}{\sum_{x'y'}f(\alpha_{x'}\beta_{y'})} =
\frac{f(\alpha_{x})}{\sum_{x'}f(\alpha_{x'})}
\frac{f(\beta_{y})}{\sum_{y'}f(\beta_{y'})}.
\end{equation}
Let us call this the Tensor product property. It will turn out that
the Tensor product property is implied by the Scale invariance
property, which we will show in our proof.
We now show that the Well-behaved derivative and Scale invariance
properties imply $f(\alpha)=|\alpha|^p$ for some $p$.
Suppose that $f$ satisfies the Well-behaved derivative
and Scale invariance properties. Then $f(\alpha)=|\alpha|^p$ for
some $p\in\mathbb{R}$.
First, note that the functions $f(\alpha)$ and $cf(\alpha)$ give the
same measurement statistics for any scalar $c\in\mathbb{R}$. To
eliminate this redundancy in our description of $f$, we'll choose $c$
such that $f(1)=1$.
For any $\alpha\in\mathbb{C}$, consider the (non-normalized) state
$\alpha\ket{0}+\ket{1}$. By scale invariance, for any
$\beta\in\mathbb{C}$, we must have that
\begin{equation}
\frac{f(\alpha)}{f(\alpha)+f(1)}=\frac{f(\alpha\beta)}{f(\alpha\beta) + f(\beta)}
\end{equation}
which implies that
$f(\alpha)f(\beta)=f(\alpha\beta)f(1)=f(\alpha\beta)$ for all
$\alpha,\beta \in\mathbb{C}$. One can easily check that this implies
the tensor product property.
In particular this holds for any phase, so if
$\alpha=|\alpha|e^{i\theta}$, we must have that
$f(\alpha)=\hat{f}(|\alpha|)g(\theta)$ for some functions
$\hat{f}:\mathbb{R}^{\geq0}\rightarrow \mathbb{R}^+$ and
$g:[0,2\pi)\rightarrow \mathbb{R}^+$. Note that taking $g\rightarrow
cg$ and $\hat{f}\rightarrow \hat{f}/c$ leaves $f$ invariant for any
scalar $c\in\mathbb{R}^+$. So without loss of generality, since
$f(1)=1$, we can set $\hat{f}(1)=g(0)=1$ as well by an appropriate
choice of scalar $c$. Now, for any phases $e^{i\theta}$ and
$e^{i\phi}$, we have
$f(e^{i\theta})f(e^{i\phi})=f(e^{i(\theta+\phi)})$. Since
$\hat{f}(1)=1$ this implies $g(\theta)g(\phi)=g(\theta+\phi)$,
i.e. $g$ must be a real one-dimensional representation of $U(1)$. The
only such representation is $g=1$, hence $f(\alpha)=f(|\alpha|)$.
Now we will show that $f(x)=x^p$ for some $p$. Consider any $0<x<1$
and $0<x'<1$ where $x\neq x'$. Since
$f(\alpha)f(\beta)=f(\alpha\beta)$, we must have that $f(x^k)=f(x)^k$
and $f(x'^k)=f(x')^k$ for any $k\in\mathbb{N}$. Let
$p=\log(f(x))/\log(x)$ and $p'=\log(f(x'))/\log(x')$. Then the above
equations imply that $f(x^k)=x^{kp}$ and $f(x'^k)=x'^{kp'}$ for all
Now suppose by way of contradiction that there exist $x,x'$ such that
$p\neq p'$. Since both $x<1$ and $x'<1$, as $k\rightarrow\infty$ we
have that $f(x^{kp})\rightarrow0$ and $f(x'^{kp})\rightarrow
0$. However, the sequence of points $f(x),f(x^2),f(x^3),\ldots$
approaches zero along the curve $h(x)=x^p$ while the sequence of
points $f(x'),f(x'^2),f(x'^3),\ldots$ approaches zero along the curve
$h'(x)=x^{p'}$. This implies $f$ must oscillate infinitely many times
between the curves $h$ and $h'$, which implies $f'$ must change signs
infinitely many times by the intermediate value theorem. This
contradicts the Well-behaved derivative assumption.
Hence we have for all $0<x<1$, $f(x)=x^p$ for some $p$. Now if $x>1$,
we have $f(x)f(1/x)=f(1)=1$. Since $1/x <1$, then we have
$f(1/x)=1/x^p$, so $f(x)=x^p$ as well. Also $f(1)=1^p=1$, and by
continuity $f(0)=0$. Hence for all $x\geq0$ we must have $f(x)=x^p$
for some $p$, as claimed.
§.§ Born rule violations imply signaling and super-Grover speedup
We first show that large violations of the Born rule imply a large
speedup to Grover search and allow for large amounts of superluminal
signaling. This was previously shown by Aaronson
<cit.>, but for completeness we will summarize the
proof here.
Suppose that the Born rule is modified such
that $f(\alpha)=|\alpha|^{2+\delta}$ where $\delta\neq 0$. Then one
can solve problems on instances of size $n$ in time
$O(\frac{n^2}{|\delta|})$. In particular one can search an unordered
list of $2^n$ indices in $O(\frac{n^2}{|\delta|})$ time.
We will use the modified Born rule to simulate postselection. Suppose
one has a state $\ket{\Psi}=\sum_x (\alpha_x \ket{0} + \beta_x
\ket{1}) \ket{x}$ and wishes to simulate postselection of the first
qubit on the state $\ket{0}$. Suppose $\delta>0$; the case $\delta<0$
follows analogously. To simulate postselection on zero, simply append
$k$ ancilla qubits in the $\ket{0}$ state. Then apply a Hadamard to
each of the ancilla qubits controlled on the first qubit being a
1. The state now evolves to
\begin{equation}
\sum_x \left( \alpha_x \ket{0} \ket{x} \ket{0}^{n/\delta} + \beta_x
\ket{1} \ket{x} \sum_{y} 2^{-k/2} \ket{y} \right)
\end{equation}
When measuring this state in the computational basis, the probability
of measuring a $0$ on the first qubit is proportional to $\sum_x
|\alpha_x|^{2+\delta}$, while the probability of getting a $1$ on the
first qubit is proportional to $2^{-k\delta/2} \sum_x
|\beta_x|^{2+\delta}$. Hence setting $k=n/\delta$, the probability of
getting a $1$ on the first qubit is exponentially suppressed by a
factor of $2^{-n}$. This effectively postselects the first qubit to
have value $0$ as desired. The rest of the proof follows from the fact
that Aaronson's algorithm to solve -hard
problems on instances of size $n$ runs in time $O(n)$ and involves
$O(n)$ postselections; hence using this algorithm to solve
-hard problems when the Born rule is violated takes time
$O(\frac{n^2}{\delta})$ as claimed.
Aaronson's result also implies that large violations of the Born rule imply one can send
superluminal signals with small numbers of qubits.
Suppose that the Born rule is
modified such that $f(\alpha)=|\alpha|^{2+\delta}$ where $\delta\neq
0$. Then one can transmit a bit superluminally in a protocol
involving a state on $O(n/|\delta|)$ qubits which succeeds with
probability $1-2^{-n}$. Note one can use this protocol to send
either classical bits or quantum bits.
The proof follows almost immediately from the proof of Theorem
<ref>. Suppose that Alice wishes to send a bit 0 or 1 to
Bob. Alice and Bob can perform the standard teleportation protocol
<cit.>, but instead of Alice sending her classical
measurement outcomes to Bob, Alice simply postselects her measurement
outcome to be 00 (i.e. no corrections are necessary to Bob's state)
using the trick in Theorem <ref>. If Alice uses
$O(n/|\delta|)$ qubits to simulate the postselection, and then measures
her qubits, she will obtain outcome 00 with probability $1-2^{-n}$ and
the bit will be correctly transmitted as desired.
§.§ Signaling implies large power law violation
We now show that if one can send a superluminal signal with bias
$\epsilon$ using a shared state on $n$ qubits, then the violation of
the Born rule $\delta$ must satisfy $|\delta| \geq O(\epsilon/n)$. Hence
$\delta$ and $\epsilon$ must be polynomially related. Put less
precisely, if a physically reasonable experiment can send a
superluminal signal with a nontrivial probability, then there must be
a nontrivial (and hence observable) violation of the Born rule. This
in turn, implies by Theorem <ref> that one can solve
-hard problems with a reasonable multiplicative overhead.
Suppose that the Born rule is modified such that
$f(\alpha)=|\alpha|^{2+\delta}$, and suppose there is a signaling
protocol using an entangled state on $n$ qubits signaling with
probability $\epsilon$. Then $|\delta| \geq O(\frac{\epsilon}{n})$.
Consider the most general signaling protocol to send a bit of
information. Suppose that Alice and Bob share an entangled state
$\ket{\Phi}$ on $n$ qubits, $m$ of which are held by Bob and $n-m$ of
which are held by Alice. To send a zero, Alice performs some unitary
$U_0$ on her half of the state, and to send a one, Alice performs some
unitary $U_1$ on her half of the state. Bob then measures in some
fixed basis $B$. This is equivalent to the following protocol: Alice
and Bob share the state $\ket{\Psi} = U_0 \ket{\Phi}$ ahead of time,
and Alice does nothing to send a $0$, and applies $U=U_1U_0^\dagger$
to obtain $\ket{\Psi'}=U\ket{\Psi}$ send a $1$. Then Bob measures in
basis $B$. We say the protocol succeeds with probability $\epsilon$ if
the distributions seen by Bob in the case Alice is sending a $0$ vs. a
$1$ differ by $\epsilon$ in total variation distance. As shown in
section <ref> of Appendix B, the total variation distance is polynomially
related to the capacity of the resulting classical communication
Let $\alpha_{xy}$ be the amplitude of the state $\ket{x}\ket{y}$ in
$\ket{\Psi}$, where the $\ket{x}$ is an arbitrary basis for Alice's
qubits and $\ket{y}$ are given by the basis $B$ in which Bob measures
his qubits. Let $\alpha'_{xy}$ be the amplitude of $\ket{x}\ket{y}$
in the state $\ket{\Psi'}$, so we have $\alpha'_{xy}=\sum_{x'} U_{xx'}
\alpha_{x'y}$. In short
\begin{align}
\ket{\Psi} = \sum_{xy} \alpha_{xy}\ket{x}\ket{y} & &
\ket{\Psi'}=\sum_{xy}\alpha_{xy}' \ket{x} \ket{y}
\end{align}
Assume that $\sum_{x,y} |\alpha_{xy}|^2 =1$, i.e. the state is
normalized in the $\ell_2$ norm. Since $U$ is unitary this implies the
state $U\ket{\psi'}$ is normalized in the $\ell_2$ norm as well.
Now suppose that the protocol has an $\epsilon$ probability of
success. Let $D_0$ be the distribution on outcomes $y\in \{0,1\}^m$
when Alice is sending a zero, and $D_1$ be the distribution when Alice
is sending a $1$. Let $D_b(y)$ denote the probability of obtaining
outcome $y$ under $D_b$. Then the total variation distance between
$D_0$ and $D_1$, given by $\frac{1}{2} \sum_y|D_0(y)-D_1(y)|$, must be
at least $\epsilon$. Equivalently, there must be some event $S \subset
\{0,1\}^m$ for which
\begin{equation}
\sum_{y\in S} D_0(y)-D_1(y) \geq \epsilon
\end{equation}
and for which, for all $y\in S$, we have $D_0(y) > D_1(y)$.
Assume for the moment that $\delta>0$; an analogous proof will hold in
the case $\delta<0$. Let $N=2^n$ be the dimension of the Hilbert space
of $\ket{\Psi}$. Plugging in the probabilities $D_0(y)$ and $D_1(y)$
given by the modified Born rule, we obtain
\begin{align}
\epsilon &\leq \sum_{x\in \{0,1\}^{n-m}, y\in S} \frac{|\alpha_{xy}|^{2+\delta}}{\sum_{x'y'} |\alpha_{x'y'}|^{2+\delta}} -\frac{|\alpha'_{xy}|^{2+\delta}}{\sum_{x'y'} |\alpha'_{x'y'}|^{2+\delta}}\\
&\leq \sum_{x\in \{0,1\}^{n-m}, y\in S} N^{\delta/2}|\alpha_{xy}|^{2+\delta} - |\alpha'_{xy}|^{2+\delta} \label{eq:usingnorms}\\
&= \sum_{x\in \{0,1\}^{n-m}, y\in S} \left(1+\frac{\delta}{2}\log(N)\right) |\alpha_{xy}|^{2}\left(1+\delta \log|\alpha_{xy}|\right) - |\alpha'_{xy}|^{2} (1+\delta \log|\alpha'_{xy}|) + O(\delta^2) \label{eq:taylordelta} \\
&= \sum_{x\in \{0,1\}^{n-m}, y\in S} \left( |\alpha_{xy}|^2-|\alpha'_{xy}|^2\right) + \frac{\delta}{2} \log(N) |\alpha_{xy}|^{2} + \delta \left(|\alpha_{xy}|^2\log|\alpha_{xy}| - |\alpha'_{xy}|^2\log|\alpha'_{xy}|\right)
+ O(\delta^2) \\
&\leq \frac{\delta}{2} \log(N) + \frac{\delta}{2} \sum_{x\in \{0,1\}^{n-m}, y\in S}\left(|\alpha_{xy}|^2\log|\alpha_{xy}|^2 - |\alpha'_{xy}|^2\log|\alpha'_{xy}|^2\right) + O(\delta^2) \label{eq:deltasimplify} \\
&\leq \frac{\delta}{2} \log(N) + \frac{\delta}{2} \log(N) + O(\delta^2) = \delta n + O(\delta^2) \label{eq:deltaentropy}
\end{align}
On line (<ref>) we used the fact that for any vector
$\ket{\phi}=\sum_y \beta_y \ket{y}$ of $\ell_2$ norm 1 over a Hilbert
space of dimension $N$, we have $N^{-\delta/2} \leq \sum_y
|\beta_y|^{2+\delta}\leq 1$ when $\delta>0$. On line
(<ref>) we expanded to first order in $\delta$. On line
(<ref>) we used the fact that the first term is zero
because applying a unitary to one half of a system does not affect
measurement outcomes on the other half of the system and the second
sum is upper bounded by 1. On line (<ref>) we used the
fact that the sum is given by a difference of entropies of (possibly
subnormalized) probability distributions, each of which is between
zero and $\log(N)$.
Hence we have that $\delta n + O(\delta^2 )\geq \epsilon$, so to first
order in $\delta$ we must have $\delta \geq \epsilon/n$ as claimed.
The following corollary follows from Theorem <ref>, and
hence we've shown that superluminal signaling implies a super-Grover
Suppose that the Born rule is modified such that
$f(\alpha)=|\alpha|^{2+\delta}$, and that there is a signaling
protocol using an entangled state on $n$ qubits which signals with
probability $\epsilon$. Then there is an algorithm to solve
-hard and -hard instances of size $m$
(e.g. ) in time
§.§ Super-Grover speedup implies signaling
We now show that even a mild super-Grover speedup implies that
$\delta$ is large, and hence one can send superluminal signals. Our
proof uses the hybrid argument of Bennett, Bernstein, Brassard and
Vazirani <cit.> combined with the proof techniques of Theorem
Suppose that the Born rule is modified such that
$f(\alpha)=|\alpha|^{2+\delta}$, and there is an algorithm to search
an unordered list of $N$ items with $Q$ queries using an algorithm
over a Hilbert space of dimension $M$. Then
\begin{equation}
\frac{1}{6} \leq \frac{2 Q}{\sqrt{N}} + |\delta| \log (M) +O(\delta^2).
\end{equation}
Suppose that such an algorithm exists. It must consist of a series of
unitaries and oracle calls followed by a measurement in the
computational basis.
Let $\ket{\psi^0} = \sum_y \alpha^0_y \ket{y}$ be the state of the
algorithm just before the final measurement when there is no marked
item, and let $\ket{\psi^x} = \sum_y \alpha^x_y \ket{y}$ be the state
if there is a marked item. Let $D_0$ be the distribution on $y$
obtained by measuring $\ket{\psi^0}$ in the computational basis, and
$D_x$ be the distribution obtained by measuring $\ket{\psi^x}$. We
know that $\ket{\psi^0}$ and $\ket{\psi^x}$ must be distinguishable
with 2/3 probability for every $x$. Hence we must have that the total
variation distance between $D_0$ and $D_x$ must be at least $1/6$ for
every $x$ (otherwise one could not decide the problem with bias
1/6). This implies there must exist some event $S_x$ for which
\begin{equation}
\frac{1}{6} \leq \sum_{y\in S_x} D_0(y)-D_1(y)
\end{equation}
Assume $\delta>0$; an analogous proof holds for $\delta<0$. Plugging
in the expressions for $D_0$ and $D_1$ and averaging over $x$ we
\begin{align}
\frac{1}{6} &\leq \frac{1}{N} \sum_x \sum_{y\in S_x} \frac{|\alpha^0_y|^{2+\delta}}{\sum_{y
} |\alpha^0_{y'}|^{2+\delta}}- \frac{|\alpha^x_y|^{2+\delta}}{\sum_{y
} |\alpha^x_{y'}|^{2+\delta}}\\
&\leq \frac{1}{N} \sum_x \sum_{y\in S_x} M^{\delta/2}|\alpha^0_y|^{2+\delta}- |\alpha^x_y|^{2+\delta} \label{eq:normchange}\\
&= \frac{1}{N} \sum_x \sum_{y\in S_x} \left(1+\frac{\delta}{2}\log(M)\right)|\alpha^0_y|^2\left(1+\delta \log|\alpha^0_y|\right) - |\alpha^x_y|^2\left(1+\delta\log|\alpha^x_y|\right) + O\left(\delta^2\right) \label{eq:deltataylor} \\
&= \frac{1}{N} \sum_x \sum_{y\in S_x} \left(|\alpha^0_y|^2 - |\alpha^x_y|^2\right)+ \frac{\delta}{2} \log(M) |\alpha^0_y|^2 + \frac{\delta}{2} \left(|\alpha^0_y|^2\log|\alpha^0_y|^2 - |\alpha^x_y|^2\log|\alpha^x_y|^2\right) + O\left(\delta^2\right) \\
&\leq \delta \log(M) + O(\delta^2) +\frac{1}{N} \sum_x \sum_{y\in S_x}
\left(|\alpha^0_y|^2 - |\alpha^x_y|^2\right) \label{eq:simplify}.
\end{align}
In line (<ref>) we used the fact that $M^{-\delta/2}
\leq \sum_{y} |\alpha_y|^{2+\delta} \leq 1$ for any state $\alpha_y$
normalized in the $\ell_2$ norm, in line (<ref>) we
Taylor expanded to first order in $\delta$, and in line
(<ref>) we used the fact that the sum in the second term
is upper bounded by one and the sum on third term is a difference of
entropies of (subnormalized) probability distributions which is at
most $\log(M)$.
We next consider the final term
\begin{equation}
R \equiv \frac{1}{N} \sum_x \sum_{y \in S_x} \left( |\alpha_y^0|^2 -
|\alpha_y^x|^2 \right).
\end{equation}
Let $\hat{S}_x$ be the observable
\begin{equation}
\hat{S}_x = \sum_{y \in S_x} \ket{y} \bra{y}.
\end{equation}
\begin{eqnarray}
R & = & \frac{1}{N} \sum_x \left[ \bra{\psi^0} \hat{S}_x \ket{\psi^0}
- \bra{\psi^x} \hat{S}_x \ket{\psi^x} \right] \\
& = & \frac{1}{N} \sum_x \left[ \left( \bra{\psi^0} - \bra{\psi^x}
\right) \hat{S}_x \ket{\psi^0} + \bra{\psi^x} \hat{S}_x \left(
\ket{\psi^0} - \ket{\psi^x} \right) \right] \\
& \leq & \frac{2}{N} \sum_x \left\| \ket{\psi^0} - \ket{\psi^x} \right\|,
\end{eqnarray}
where the last inequality uses the fact that $\| \hat{S}_x \| =
1$. Next we note that $\sum_x \left\| \ket{\psi^0} - \ket{\psi^x}
\right\|$ is the $\ell_1$ norm of the $N$-dimensional vector whose
$x^{\mathrm{th}}$ component is
$\left\| \ket{\psi^0} - \ket{\psi^x}\right\|$. For any $N$-dimensional
vector $\vec{v}$, $\|v\|_1 \leq \sqrt{N} \| \vec{v} \|_2$. Thus,
\begin{equation}
\label{rfin}
R \leq \frac{2}{\sqrt{N}} \sqrt{ \sum_x \left\| \ket{\psi^0} - \ket{\psi^x}
\right\|^2 }.
\end{equation}
As shown in <cit.>,
a unitary search algorithm using $Q$ oracle queries yields
\begin{equation}
\label{maxdiff}
\sum_x \left\| \ket{\psi^0} - \ket{\psi^x} \right\|^2 \leq 4 Q^2.
\end{equation}
Together, rfin and maxdiff imply
\begin{equation}
\label{rineq}
R \leq \frac{2Q}{\sqrt{N}}.
\end{equation}
Now, rineq bounds the last term in eq:simplify yielding our
final result.
\begin{equation}
\frac{1}{6} \leq \delta \log(M) + O(\delta^2) + \frac{2Q}{\sqrt{N}}.
\end{equation}
The following Corollary follows immediately from Theorem
<ref> and Theorem <ref>.
Suppose that the Born rule is
modified such that $f(\alpha)=|\alpha|^{2+\delta}$, and one can
search a list of $N=2^n$ items using $m$ qubits and $Q$
queries. Then to first order in $\delta$, we have
\[
|\delta| \geq \frac{1}{m}\left(\frac{1}{6} - \frac{2Q}{\sqrt{N}}\right).
\]
In particular, if one can search an $N$ element list with $Q\leq
\sqrt{N}/24$ queries on a state of $m$ qubits, then $|\delta| \geq
\frac{1}{12m}$, and hence by Theorem <ref> one can
send superluminal signals with probability 2/3 using $O(m)$
In contrast, Grover's algorithm uses $\frac{\pi}{4} \sqrt{N}$ queries
to solve search, which is optimal <cit.>. So Corollary
<ref> shows that if one can achieve even a modest
factor of ($6 \pi \approx 19$) speedup over Grover search using $m$
qubits, then one can send superluminal signals using $O(m)$
§ APPENDIX C: CLONING OF QUANTUM STATES
One way to modify quantum mechanics is to allow perfect copying of
quantum information, or “cloning”. As a minimal example, we will
here introduce the ability to do perfect single-qubit cloning. As with
nonlinear dynamics, care must be taken to formulate a version of
quantum cloning that is actually well defined. It is clear that
perfect single qubit cloning should take $\ket{\psi} \mapsto
\ket{\psi} \otimes \ket{\psi}$ for any single-qubit pure state. The
nontrivial task is to define the behavior of the cloner on qubits that
are entangled. It is tempting to simply define cloning in terms of the
Schmidt decomposition of the entangled state. That is, applying the
cloner to qubit $B$ induces the map $\sum_i \lambda_i \ket{i_A}
\ket{i_B} \mapsto \sum_i \lambda_i \ket{i_A} \ket{i_B} \ket{i_B}$. However, this
prescription is ill-defined due to the non-uniqueness of Schmidt
decompositions. The two decompositions of the EPR pair given in
11 and ++ provide an example of the inconsistency of the
above definition.
Instead, we define our single-qubit cloner as follows.
Let $\rho_{AB}$ be a state on a bipartite system $AB$. Let
$\rho_B$ be the reduced density matrix of $B$. Then applying the
cloner to $B$ yields
\[
\rho_{AB} \mapsto \rho_{AB} \otimes \rho_B.
\]
In particular, for pure input, we have $\ket{\psi_{AB}}\bra{\psi_{AB}}
\mapsto \ket{\psi_{AB}}\bra{\psi_{AB}}\otimes \rho_B$. Thus, this
version of cloning maps pure states to mixed states in
general. Furthermore, the clones are asymmetric. The cloner takes one
qubit as input and produces two qubits as output. The two output
qubits have identical reduced density matrices. However, one of the
output qubits retains all the entanglement that the input qubit had
with other systems, whereas the other qubit is unentangled with
anything else. By monogamy of entanglement it is impossible for both
outputs to retain the entanglement that the input qubit had.
It is worth noting that the addition of nonlinear dynamics, and
cloning in particular, breaks the equivalence between density matrices
and probabilistic ensembles of pure states. Here, we take density
matrices as the fundamental objects in terms of which our generalized
quantum mechanics is defined.
In analyzing a model of computation involving cloning, we will treat
the cloning operation as an additional gate, with the same “cost” as
any other. In circuit diagrams, we denote the cloning gate as follows.
\[
\includegraphics[width=0.6in]{clonegate.eps}
\]
This notation reflects the asymmetric nature of our cloning gate; the
arrow indicates the output qubit that retains the entanglement of the
input qubit.
§.§ Grover Search using Quantum Cloning
Cloning is a nonlinear map on quantum states. As argued by Abrams and
Lloyd <cit.>, one can solve Grover search on a database
of size $N$ using $O(1)$ oracle queries and $O(\log N)$ applications
of $S$, for any nonlinear map $S$ from pure states to pure states,
except perhaps some pathological cases. Here, with theorem
<ref>, we have formalized this further, showing that this
holds as long as $S$ is differentiable. However, the cloning gate
considered here maps pure states to mixed states. Therefore, this gate
requires a separate analysis. We cannot simply invoke theorem
<ref>. Instead we specifically analyze the cloning gate
given above and arrive at the following result.
Suppose we have access to a standard Grover bit-flip oracle, which
acts as $U_f \ket{y} \ket{z} = \ket{y} \ket{z \oplus f(y)}$ where
$f:\{0,1\}^n \to \{0,1\}$. Using one query to this oracle, followed by
a circuit using $\mathrm{poly}(n)$ conventional quantum gates and
$O(n)$ of the single-qubit cloning gates described in definition
<ref>, one can distinguish between the cases that
$|f^{-1}(1)| = 0$ and $|f^{-1}(1)| = 1$ with high probability.
For the design of nonlinear Grover search algorithms it is helpful to
have a nonlinear map from a fixed state space to itself. To this end,
we consider circuits of the following form, which implement nonlinear
maps from the space of possible density matrices of a qubit to itself.
\[
\includegraphics[width=2in]{genmap.eps}
\]
Here, one clones the input qubit, performs some unitary $U$ between
the two resulting copies, and lastly discards one of the qubits.
With a small amount of trial and error one can find a choice of $U$
which enables single-query Grover search using an analogue of the
Abrams-Lloyd algorithm. Specifically, we choose $U$ to be the
controlled-not gate. That is, let
\[
\includegraphics[width=2.2in]{cnotmap.eps}.
\]
$\mathcal{M}$ is a quadratic map on density matrices. By direct
\begin{equation}
\mathcal{M}\left( \left[ \begin{array}{cc}
r_{00} & r_{01} \\
r_{10} & r_{11} \end{array} \right] \right)
= \left[ \begin{array}{cc}
r_{00}^2 + r_{00} r_{11} & r_{01}^2 + r_{01} r_{10} \\
r_{10}^2 + r_{10} r_{01} & r_{11}^2 + r_{11} r_{00} \end{array} \right].
\end{equation}
One can find the fixed points of $\mathcal{M}$ by solving the system
of four quadratic equations implied by $\mathcal{M}(\rho) = \rho$. The
solutions are as follows.
\begin{eqnarray}
r_{10} & = & 1 - r_{01}\textrm{,}\quad r_{11} = 1-r_{00} \label{sol1} \\
r_{00} & = & 0\textrm{,}\quad r_{10}=1-r_{01}\textrm{,}\quad r_{11} = 0 \label{sol2}\\
r_{01} & = & 1\textrm{,}\quad r_{10} = 0\textrm{,}\quad r_{11} = 1-r_{00} \label{sol3}\\
r_{00} & = & r_{01} = r_{10} = r_{11} = 0 \label{sol4}
\end{eqnarray}
The solutions sol2 and sol4 are traceless and therefore
unphysical. Solution sol3 is an arbitrary mixture of $\ket{0}$
and $\ket{1}$. That is,
\begin{equation}
\rho_r = \left[ \begin{array}{cc} r & 0 \\ 0 & 1-r \end{array} \right]
\quad \textrm{is a fixed point for all $r \in [0,1]$.}
\end{equation}
As a matrix, the solution sol1 is
\begin{equation}
\rho_{a,b} = \left[ \begin{array}{cc} a & b \\ 1-b & 1-a \end{array} \right].
\end{equation}
This is only Hermitian if $b = (1-b)^*$, which implies that $b =
\frac{1}{2} + \alpha i$ for some $\alpha \in \mathbb{R}$. However, if
$\alpha \neq 0$ then $\rho_{a,b}$ fails to be positive semidefinite,
which is unphysical. Thus, $b = \frac{1}{2}$. The eigenvalues of
$\rho_{a,1/2}$ are
\begin{equation}
\frac{1}{2} \pm \sqrt{ \frac{1 + 2a(a-1)}{2}}.
\end{equation}
Thus, unless $a = \frac{1}{2}$, the largest eigenvalue of
$\rho_{a,1/2}$ exceeds one, which is unphysical. So, the only physical
fixed point other than $\rho_r$ is
\begin{equation}
\rho_+ = \left[ \begin{array}{cc} \frac{1}{2} & \frac{1}{2} \vspace{4pt}\\
\frac{1}{2} & \frac{1}{2} \end{array} \right] = \ket{+} \bra{+}.
\end{equation}
Numerically, one finds that $\rho_r$ is an attractive fixed point and
$\rho_+$ is a repulsive fixed point. Let
\begin{equation}
\rho_\epsilon = \left[ \begin{array}{cc} \frac{1}{2} & \frac{1}{2} -
\epsilon \\ \frac{1}{2} - \epsilon & \frac{1}{2} \end{array}
\right] = (1-\epsilon) \ket{+}\bra{+} + \epsilon \ket{-} \bra{-}.
\end{equation}
\begin{eqnarray}
\mathcal{M}(\rho_{\epsilon}) = \rho_{2 \epsilon + O(\epsilon^2)}.
\end{eqnarray}
Consequently, $\mathcal{M}^r(\rho_{\epsilon})$ is easily
distinguishable from $\mathcal{M}^r(\rho_+) = \rho_+$ after $r = O(
\log(1/\epsilon) )$ iterations of $\mathcal{M}$.
Let $U_f$ be the standard Grover bit-flip oracle, which acts as $U_f
\ket{y}\ket{z} = \ket{y} \ket{z \oplus f(y)}$ where $f:\{0,1\}^n \to
\{0,1\}$.
Now, consider the following circuit.
\[
\Qcircuit @C=1em @R=1em {
\lstick{\frac{\id}{2^n}} & {/} \qw & \multigate{1}{U_x} & \qw & \qw & \\
\lstick{\ket{0}} & \qw & \ghost{U_x} & \gate{H} & \qw &
\]
One sees that the bottom qubit emerges in the state $\rho_+$ if $f$
has no solution and emerges in the state $\rho_{\epsilon}$ with
$\epsilon = \frac{1}{2^n}$ if $f$ has one solution. By making one such
query and then applying the map $\mathcal{M}$ a total of $O(n)$ times
to the resulting state, one obtains single-qubit states in the
no-solution and one-solution cases that are easily distinguished with
high confidence using conventional quantum measurements.
For simplicity, in theorem <ref>, we have restricted our
attention to search problems which are promised to have exactly one
solution or no solutions and our task is to determine which of these
is the case. Note that can be reduced to
in randomized polynomial time
<cit.>. Hence solving the Grover problem in poly(n)
time when there is either exactly one solution or no solutions
suffices to solve -hard problems in randomized polynomial
It is interesting to note that probability distributions also cannot
be cloned. The map $\vec{p} \mapsto \vec{p} \otimes \vec{p}$ on
vectors of probabilities is nonlinear and hence does not correspond to
any realizable stochastic process. Furthermore, one finds by a
construction similar to the above that cloning of classical
probability distributions also formally implies polynomial-time
solution to -hard problems via logarithmic-complexity single-query
Grover search. However, nonlinear maps on probabilities do not appear
to be genuinely well-defined. Suppose we have probability $p_1$ of
drawing from distribution $\vec{p}_1$ and probability $p_2$ of drawing
from distribution $\vec{p}_2$. Normally this is equivalent to
drawing from $p_1 \vec{p_1} + p_2 \vec{p_2}$. However, if we apply a
nonlinear map $\mathcal{M}$ then $\mathcal{M}(p_1 \vec{p_1} + p_2
\vec{p_2})$ is in general not equal to $p_1 \mathcal{M}(p_1) + p_2
\mathcal{M}(\vec{p_2})$. It is not clear that a well-defined
self-consistent principle can be devised for resolving such
§.§ Superluminal Signaling using Quantum Cloning
Suppose Alice and Bob share an EPR pair $\frac{1}{\sqrt{2}} \left(
\ket{00} + \ket{11} \right)$. If Alice wishes to transmit a zero she
does nothing. If she wishes to transmit a one she measures her qubit
in the computational basis. If Alice doesn't measure then Bob's
reduced density matrix is maximally mixed. Hence if he makes several
clones and measures them all in the computational basis he will obtain
a uniformly random string of ones and zeros. If Alice does measure
then Bob's reduced density matrix is either $\ket{0}\bra{0}$ or
$\ket{1} \bra{1}$, with equal probability. If he makes several clones
and measures them all in the computational basis he will get
$000\ldots$ or $111\ldots$, with equal probability. Thus, by making
logarithmically many clones, Bob can achieve polynomial certainty
about the bit that Alice wished to transmit.
§ APPENDIX D: POSTSELECTION
In <cit.> it was shown that adding the ability to
postselect a single qubit onto the state $\ket{0}$ to the quantum
circuit model yields a model of computation whose power is equal to
the classical complexity class PP. Furthermore, postselection onto
$\ket{0}$ allows perfect superluminal signaling by postselected quantum
teleportation. Here we consider a more general question: suppose we
have the ability to postselect on some arbitrary but fixed $n$-qubit
state $\ket{\psi}$. Does this still yield efficient means of solving
problems in PP and sending superluminal signals? It is clear that one
can use postselection onto $\ket{\psi}$ to simulate postselection onto
$\ket{0}$ given a quantum circuit for a unitary $U$ such that $U
\ket{00\ldots0} = \ket{\psi}$. However, for a generic $n$-qubit state
$\ket{\psi}$, no polynomial-size quantum circuit for this tasks
exists. Nevertheless, in this appendix we show that, for Haar random
(but fixed) $\ket{\psi}$, postselection onto $\ket{\psi}$ can with
high probability be used to simulate postselection onto $\ket{0}$ with
exponential precision.
We first note that the maximally entangled state of $2n$ qubits:
\begin{equation}
\ket{\Phi_{2n}} = \sum_{x \in \{0,1\}^n} \ket{x} \otimes \ket{x}
\end{equation}
can be prepared using $n$ Hadamard gates followed by $n$ CNOT
gates. Postselecting the second tensor factor of $\ket{\Phi_{2n}}$
onto $\ket{\psi}$ yields $\ket{\psi}$ on the first tensor factor. In
this manner, one may extract a copy of $\ket{\psi}$. We assume that
$\ket{\psi}$ is Haar random but fixed. That is, each time one uses the
postselection “gate,” one postselects onto the same state
$\ket{\psi}$. Hence, using the above procedure twice yields two copies
of $\ket{\psi}$. Applying $\sigma_x$ to one qubit of one of the copies
of $\ket{\psi}$ yields a state $\ket{\psi'} = \sigma_x \ket{\psi}$. As
shown below, the root-mean-square inner product between $\ket{\psi}$
and $\ket{\psi'}$ is of order $1/\sqrt{2^n}$. That is, they are nearly
orthogonal. Thus, one can simulate postselection onto $\ket{0}$ with
the following circuit.
\[
\Qcircuit @C=1em @R=1em {
& \qw & \ctrl{1} & \qw \\
\lstick{\ket{\psi'}} & {/} \qw & \multigate{1}{\mathrm{SWAP}} & \qw \\
\lstick{\ket{\psi}} & {/} \qw & \ghost{\mathrm{SWAP}} & \gate{\ket{\psi}}
\]
Here, the top qubit gets postselected onto $\ket{0}$ with fidelity
$1-O(1/\sqrt{2^n})$, the middle register is discarded, and the bottom
register is postselected onto $\ket{\psi}$, an operation we denote by
$\Qcircuit @C=1em @R=1em { & \gate{\ket{\psi}} & \qw }$.
Lastly, we prove the claim that the root-mean-square inner
product between a Haar random $n$-qubit state $\ket{\psi}$ and
$\ket{\psi'} = \sigma_x \ket{\psi}$ is of order $1/\sqrt{2^n}$. This
mean-square inner product can be written as
\begin{eqnarray}
\bar{I} & = & \int dU | \bra{0\ldots0} U^\dag \sigma_x U
\ket{0\ldots0}|^2 \\
& = & \sum_{a,b \in \{0,1\}^n} \int dU U^\dag_{0a} U_{\bar{a} 0} U^\dag_{0b}
U_{\bar{b} 0} \label{barb}
\end{eqnarray}
where $\bar{a}$ indicates the result of flipping the first bit of $a$,
$\bar{b}$ indicates the result of flipping the first bit of
$b$, and $0$ in the subscripts is shorthand for the bit string $0
\ldots 0$. (We arbitrarily choose the $\sigma_x$ to act on the first
Next we recall the following identity regarding integrals on the Haar
measure over $U(N)$. (See <cit.> or appendix D of <cit.>.)
\begin{equation}
\begin{array}{rcl}
\int dU \ U_{ij} U_{kl} U^\dag_{mn} U^\dag_{op} & = & \frac{1}{N^2-1}
\left( \delta_{in} \delta_{kp} \delta_{jm} \delta_{lo} + \delta_{ip}
\delta_{kn} \delta_{jo} \delta_{lm} \right) \\
& & - \frac{1}{N (N^2-1)} \left( \delta_{ij} \delta_{kp} \delta_{jo}
\delta_{lm} + \delta_{ip} \delta_{kn} \delta_{jm} \delta_{lo}
\right) \label{ident}
\end{array}
\end{equation}
Applying ident to barb shows that the only nonzero terms
come from $a = \bar{b}$ and consequently
\begin{eqnarray}
\bar{I} & = & \sum_{a \in \{0,1\}^n} \int dU \ U_{\bar{a} 0} U_{a0} U^\dag_{0a}
U^{\dag}_{0 \bar{a}} \\
& = & \frac{N}{N^2-1} - \frac{1}{N^2-1}.
\end{eqnarray}
Consequently, the RMS inner product for large $N$ is
\begin{equation}
\sqrt{\bar{I}} \simeq \frac{1}{\sqrt{N}}.
\end{equation}
Recalling that $N = 2^n$ completes the argument.
§ APPENDIX E: GENERAL NONLINEARITIES
Our discussion of final-state projection models can be thought of as
falling within a larger tradition of studying the
information-theoretic and computational complexity implications of
nonlinear quantum mechanics, as exemplified by
<cit.>. A question within this subject
that has been raised multiple times <cit.> is
whether all nonlinearities necessarily imply that Grover search can be
solved with a single query. In this note we shed some light on this
question. However, note that the setting differs from that of section
<ref> of Appendix B in that (following <cit.>) we
assume the nonlinear map is the same each time, and we can apply it
polynomially many times. In section <ref> of Appendix B we have included the possibility that black holes (and the nonlinear maps that they generate) are scarce and that they may differ from one another.
We first note that, for dynamics that map normalized pure states to
normalized pure states, the terms nonunitary and nonlinear are
essentially interchangeable. Let $V$ be the manifold of normalized
vectors on a complex Hilbert space $\mathcal{H}$, which could be
finite-dimensional or infinite-dimensional. Let $S:V \to V$ be a
general map, not necessarily linear or even continuous. We'll call $S$
a unitary map if it preserves the magnitude of inner
products. That is, $|\langle S \psi|S \phi \rangle| = | \langle
\psi|\phi \rangle |$ for all $\ket{\phi},\ket{\psi} \in
\mathcal{H}$. Wigner's theorem <cit.> states that all unitary
maps are either unitary linear transformations, or antiunitary
antilinear transformations. (Antiunitary transformations are equivalent
to unitary transformations followed by complex conjugation of all
amplitudes in some basis.) Extending quantum mechanics by allowing
antiunitary dynamics does not affect computational complexity, as can
be deduced from <cit.>. Thus, without loss of generality, we
may ignore antiunitary maps. Hence, within the present context, if a
map is unitary then it is linear. Conversely, by linear algebra, if
map $S$ is linear, and maps $V \to V$, i.e. is norm-preserving,
then it is also inner-product preserving, i.e. unitary.
A standard version of the Grover problem is, for some function
$f:\{0,1\}^n \to \{0,1\}$, to decide whether the number of solutions
to $f(y) = 1$ is zero or one, given that one of these is the case. The
search problem of finding a solution is reducible to this decision
problem with logarithmic overhead via binary search. In
<cit.> Abrams and Lloyd show how to solve the decision
version of Grover search using a single quantum query to $f$ and
$O(n)$ applications of a single-qubit nonlinear map. This suffices to
solve in polynomial time. We now briefly describe their
algorithm. In contrast to section <ref> of Appendix B, it is more convenient
here to assume a bit-flip oracle rather than a phase-flip oracle. That
is, for $y \in \{0,1\}^n$ and $z \in \{0,1\}$ the oracle $O_f$
acts as
\begin{equation}
O_f \ket{y}\ket{z} = \ket{y} \ket{z \oplus f(y)}.
\end{equation}
Querying the oracle with the state
$\frac{1}{\sqrt{2^n}} \sum_{y \in \{0,1\}^n} \ket{y} \ket{0}$ yields
$\frac{1}{\sqrt{2^n}} \sum_{y \in \{0,1\}^n} \ket{y} \ket{f(y)}$.
Applying a Hadamard gate to each qubit of the first register and
measuring the first register in the computational basis yields the
outcome $00\ldots0$ with probability at least $\frac{1}{4}$. Given
that this occurs, the post-measurement state of the second register is
\begin{equation}
\ket{\psi_s} = \frac{(2^n-s) \ket{0} + s \ket{1}}{\sqrt{(2^n-s)^2+s^2}},
\end{equation}
where $s$ is the number of solutions, i.e. $s =
|f^{-1}(1)|$. Thus, we can solve the Grover search problem by
distinguishing two exponentially-close states, namely $\ket{\psi_0}$
and $\ket{\psi_1}$. For the particular nonlinear map on the manifold
of normalized pure single-qubit states considered in
<cit.>, a pair of states $\epsilon$-close together
can be separated to constant distance by iterating the map
$O(\log(1/\epsilon))$ times.
We now show that any differentiable nonlinear map from pure states to
pure states on any finite-dimensional Hilbert space can achieve
this. (See theorem <ref>.) Let $V^{(n)}$ be the manifold of
normalized pure states on $\mathbb{C}^n$. Thus, $V^{(n)}$ is a $2n-1$
dimensional real closed compact manifold. For points $a,b$ on
$V^{(n)}$ let $|a-b|$ denote their distance. (Our choice of distance
metric is not important to the argument, but for concreteness, we
could choose the angle between quantum states, that is, $|a-b|=
\cos^{-1} |\langle a | b \rangle|$. That this is a metric is proven in
section 9.2.2 of <cit.>.)
Let $S:V^{(n)} \to V^{(n)}$ be a differentiable map,
that is, a self-diffeomorphism of $V^{(n)}$. Let $r = \max_{a,b \in
V^{(n)}} \frac{|S(a)-S(b)|}{|a-b|}$. Then there exists some
sufficiently short geodesic $l$ in $V^{(n)}$ such that for all $x,y
\in l$, $\frac{|S(x)-S(y)|}{|x-y|} \geq r$.
Choose two points $x,y$ on $V^{(n)}$ that maximize the ratio $r =
\frac{|S(x) - S(y)|}{|x-y|}$. By assumption, $S$ is not unitary, so
not all distances are preserved. Because $S$ is a map from $V^{(n)}$
to another manifold of equal volume (namely $V^{(n)}$ itself) it
cannot be that all distances are decreased. Thus, this maximum ratio
must be larger than one. The extent that this ratio exceeds one
quantifies the deviation from unitarity.
Now, consider the geodesic $g$ on $V^{(n)}$ from $x$ to $y$. Because
it is a geodesic, $g$ has length $|x-y|$. Now consider the image of
$g$ under the map $S$. Because $S$ is a continuous map, $S(g)$ will
also be a line segment. By the construction, the endpoints of $S(g)$
are distance $r|x-y|$ apart. Therefore, the length of $S(g)$, which we
denote $|S(g)|$, satisfies $|S(g)| \geq r|x-y|$, with equality
if $S(g)$ happens to also be a geodesic. Thus, $S$ induces a
diffeomorphism $S_g$ from the line segment $g$ to the line segment
$S(g)$, where $|S(g)|/|g| \geq r$. Because $S_g$ is a diffeomorphism
it follows that on any sufficiently small subsegment of $g$ it acts by
linearly magnifying or shrinking the subsegment and translating to
some location on $S(g)$. Because $|S(g)|/|g| \geq r$ it follows that
there exists some subsegment $l$ such that this linear magnification is
by a factor of at least $r$. (There could be some subsegments that
grow less than this or even shrink, but if so, others have to make up
for it by growing by a factor of more $|S(g)|/|g|$.)
We now argue that the existence of $l$ suffices to ensure success for
the Abrams-Lloyd algorithm. Let $f$ denote the “magnification
factor” that $S$ induces on $l$. According to theorem
<ref>, $f \geq r$. We are interested in asymptotic
complexity, so the distance $\epsilon$ between $\ket{\psi_0}$ and
$\ket{\psi_1}$ is asymptotically small. Therefore, we assume $\epsilon$
is smaller than the length of $l$. So, we can append ancilla qubits
and apply a unitary transformation such that the resulting isometry
maps $\ket{\psi_0}$ and $\ket{\psi_1}$ to two points
$\ket{\phi_0^{(0)}}$ and $\ket{\phi_1^{(0)}}$ that lie on $l$. We then
apply $S$, resulting in the states $\ket{\phi_0^{(1)}}$ or
$\ket{\phi_1^{(1)}}$, which have distance $f \epsilon$. If $f
\epsilon$ is larger than the length $l$ then we terminate. Because we
have a fixed nonunitary map, the distance between our states is now a
constant (independent of $\epsilon$ and hence of the size of the
search space). If $f \epsilon$ is smaller than the length of $l$, then
we apply a unitary map that takes $\ket{\phi_0^{(1)}}$ and
$\ket{\phi_1^{(1)}}$ back onto $l$ and apply $S$ again. We then have
states $\ket{\phi_0^{(2)}}$ and $\ket{\phi_1^{(2)}}$ separated by
distance $f^2 \epsilon$. We then iterate this process until we exceed
the size $l$, which separates the states to a constant distance and
uses $\log_f(1/\epsilon)$ of the nonunitary operations. States with
constant separation can be distinguished within standard quantum
mechanics by preparing a constant number of copies and collecting
statistics on the outcomes of ordinary projective measurements.
§ APPENDIX F: A CAUTIONARY NOTE ON NONLINEAR QUANTUM MECHANICS
The Horowitz-Maldecena final-state projection model, cloning of
quantum states, and the Gross-Pitaevsky equation (if interpreted as a
quantum wave equation) all involve nonlinear dynamics of the
wavefunction. In such cases, one must be very careful to ensure that
subsystem structure, which is captured by tensor product structure in
conventional quantum mechanics, is well-defined. Indeed, subsystem
structure is lost by introducing generic nonlinearities, and in
particular by the nonlinearity of the Gross-Pitaevsky equation. This
makes the question about superluminal signaling in the Gross-Pitaevsky
model ill-posed. The Horowitz-Maldecena model does have a natural
notion of subsystem structure, which is one of the features that makes
it appealing. Furthermore, the model of cloning that we formulate in Appendix D preserves subsystem structure by virtue of being
phrased in terms of reduced density matrices.
More formally, let $V$ be the manifold of normalized vectors in the
Hilbert space $\mathbb{C}^{d}$. We will model nonlinear quantum
dynamics by some map $S:V \to V$ which may not be a linear map on
$\mathbb{C}^d$. In general, specifying a map $S$ on $V$ does not
uniquely determine the action of $S$ when applied to a subsystem of a
larger Hilbert space. For example, consider the map $S_0$ on the
normalized pure states of one qubit given by
\begin{equation}
S_0 \ket{\psi} = \ket{0} \quad \forall \ket{\psi}
\end{equation}
Now, consider what happens if we apply $S_0$ to half of an EPR
pair $\ket{\Psi_{\mathrm{EPR}}}$. We can write the EPR state in two
equivalent ways
\begin{eqnarray}
\ket{\Psi_{\mathrm{EPR}}} & = & \frac{1}{\sqrt{2}} \left( \ket{0}
\ket{0} + \ket{1} \ket{1} \right) \label{11} \\
& = & \frac{1}{\sqrt{2}} \left( \ket{+} \ket{+} +
\ket{-} \ket{-} \right) \label{++}
\end{eqnarray}
\begin{equation}
\ket{\pm} = \frac{1}{\sqrt{2}} \left( \ket{0} \pm \ket{1} \right)
\end{equation}
Symbolically applying the rule $S_0 \ket{\psi} \mapsto \ket{0}$ to the
first tensor factor of 11 yields $\ket{0} \ket{+}$, whereas
applying this rule to the first tensor factor of $\eq{++}$ yields
$\ket{0} \ket{0}$.
This example illustrates that one must specify additional information
beyond the action of a nonlinear map on a fixed Hilbert space in order
to obtain a well-defined extension to quantum theory incorporating the
notion of subsystems.
§ APPENDIX G: OPEN PROBLEMS
We have shown that in several domains of modifications of
quantum mechanics, the resources required to observe
superluminal signaling or a speedup over Grover's algorithm are polynomially related.
We extrapolate that this
relationship holds more generally, that is, in any quantum-like
theory, the Grover lower bound is derivable from the no-signaling
principle and vice-versa.
A further hint in this direction is that, as shown in <cit.>, the limit on distinguishing non-orthogonal states in quantum mechanics is dictated by the no-signaling principle. Thus, any improvement over the Grover lower bound based on beyond-quantum state discrimination can be expected to imply some nonzero capacity for superluminal signaling.
There is a substantial literature on
generalizations of quantum mechanics which could be drawn upon to
address this question. In particular, one could consider the
generalized probabilistic theories framework of Barrett
<cit.>, the category-theoretic framework of Abramsky and
Coecke <cit.>, the Newton-Schrödinger equation <cit.>, quaternionic quantum mechanics <cit.>, or the Papadodimas-Raju
state-dependence model of black hole dynamics <cit.>.
In these cases the investigation of computational and
communication properties is inseparably tied with the fundamental
questions about the physical interpretations of these
models. Possibly, such investigation could help shed light on these
fundamental questions.
Our finding can be regarded as evidence
against the possibility of using black hole dynamics to efficiently
solve -complete problems, at least for problem instances of
reasonable size.
Note however that there are other
independent questions regarding the feasibility of computational
advantage through final-state projection and other forms of
non-unitary quantum mechanics. In particular, the issue of
fault-tolerance in modified quantum mechanics remains largely open,
although some discussion of this issue appears in <cit.>.
Also, while our results focus on the query complexity of search, in practice one also is interested
in the time complexity. Harlow and Hayden <cit.> have argued that decoding the Hawking radiation emitted by a black hole may require exponential time on a quantum computer.
If the Harlow-Hayden argument is
correct, then exponential improvement in query complexity for search
does not imply exponential improvement in time-complexity. We
emphasize however that query complexity sets a lower bound on time
complexity, and therefore the reverse implication still holds, namely
exponential improvement in time complexity implies exponential
improvement in query complexity, which in the models we considered
implies superluminal signaling.
Hence an operational version of the Grover lower bound can be derived from an operation version of the no-signaling principle.
|
1511.00679
|
On intra-regular and left regular and left duo
$\Gamma$-semigroupsNiovi Kehayopulu, Michael TsingelisAbstract. For an intra-regular or a left regular and
left duo ordered $\Gamma$-semigroup $M$, we describe the principal
filter of $M$ which plays an essential role in the structure of this
type of $po$-$\Gamma$-semigroups. We also prove that an ordered
$\Gamma$-semigroup $M$ is intra-regular if and only if the ideals of
$M$ are semiprime and it is left (right) regular and left (right) duo
if and only if the left (right) ideals of $M$ are semiprime.AMS Subject Classification: 20M99 (06F99)Keywords: ordered $\Gamma$-semigroup; filter;
intra-regular; left regular
§ INTRODUCTION AND PREREQUISITES
Croisot, who used the term
“inversive" instead of “regular", connects the matter of
decomposition of a semigroup with the regularity and semiprime
conditions [1]. A semigroup $S$ is said to be left (resp. right)
regular if for every $a\in S$ there exists $x\in S$ such that
$a=xa^2$ (resp. $a=a^2x$). That is, if $a\in Sa^2$ (resp. $a\in
a^2S)$ for every $a\in S$ which is equivalent to saying that
$A\subseteq A^2S$ (resp. $A\subseteq SA^2$) for every $A\subseteq S$.
A semigroup $S$ is said to be intra-regular if for every $a\in S$
there exist $x,y\in S$ such that $a=xa^2 y$. In other words, if $a\in
Sa^2 S$ for every $a\in S$ or $A\subseteq SA^2S$ for every
$A\subseteq S$. For decompositions of an intra-regular, left regular
or both left regular and right regular semigroup we refer to [2, 10].
The concepts of intra-regular ordered semigroup and of right regular
ordered semigroups have been introduced in [3, 4] in which the
decomposition of an intra-regular ordered semigroup into simple
components and the decomposition of a right regular and right duo
ordered semigroup into right simple components has been studied. The
principal filter of $S$ has a very simple form, both for ordered and
non-ordered case of semigroups, and it plays an essential role in the
For two nonempty sets $M$ and $\Gamma$, we denote by $A\Gamma B$ the
set containing the elements of the form $a\gamma b$ where $a\in A$,
$\gamma\in\Gamma$ and $b\in B$. That is, we define
B:=\{a\gamma b \mid a\in A, b\in B, \gamma\in\Gamma\}.$$
Then $M$ is
called a $\Gamma$-semigroup if the following assertions are
(1) $M\Gamma M\subseteq M$;
(2) $a\gamma (b\mu c)=(a\gamma b)\mu c$ for all $a,b,c\in M$
and all $\gamma,\mu\in\Gamma$;
(3) if $a,b,c,d\in M$ and $\gamma,\mu\in\Gamma$ such that
$a=c$, $\gamma=\mu$ and $b=d$, then $a\gamma b=c\mu
An ordered $\Gamma$-semigroup (shortly, $po$-$\Gamma$-semigroup) is a $\Gamma$-semigroup $M$ with an order
relation “$\le$" on $M$ such that $a\le b$ implies $ac\le bc$ and
$ca\le cb$ for every $c\in M$. A nonempty subset $A$ of $M$ is called
a subsemigroup of $M$ if, for every $a,b\in A$ and every
$\gamma\in\Gamma$, we have $a\gamma b\in A$. A subsemigroup $F$ of
$M$ is called a filter of $M$ if
(1) for every $a,b\in F$ and every $\gamma\in\Gamma$ such that
$a\gamma b\in F$, we have $a\in F$ and $b\in F$ and (2) if $a\in F$
and $M\ni b\ge a$, then $b\in F$. For an element $x$ of $M$, we
denote by $N(x)$ the filter of $M$ generated by $x$ (that is,
the least with respect to the inclusion relation filter of $M$
containing $x$). A nonempty subset $A$ of $M$ is called a left
(resp. right) ideal of $M$ if (1) $M\Gamma A\subseteq A$
(resp. $A\Gamma M\subseteq A$) and (2) if $a\in A$ and $M\ni b\le a$,
then $b\in A$. It is called an ideal or (two-sided ideal)
of $M$ if it is both a left and right ideal of $M$. A
$po$-$\Gamma$-semigroup $M$ is called left (resp. right)
duo if the left (resp. right) ideals of $M$ are two-sided. A
subset $T$ of $M$ is called semiprime if for every $x\in M$ and
every $\gamma\in\Gamma$ such that $x\gamma x\in T$, we have $x\in T$.
For a subset $H$ of $M$ we denote by $(H]$ the subset of $M$ defined
by $(H]=\{t\in M \mid t\le a \mbox { for some } t\in H\}$. We clearly
have $M=(M]$, and for any subsets $A, B, C, D$ of $M$, we have
$A\subseteq (A]=((A]]$; if $A\subseteq B$, then $(A]\subseteq (B]$;
if $A\subseteq B$ and $C\subseteq D$, then $(A\Gamma C]\subseteq
(B\Gamma D]$; $(A]\Gamma (B]\subseteq (A\Gamma B]$; and $((A]\Gamma
(B]]=((A]\Gamma B]=(A\Gamma (B]]=(A\Gamma B]$. As we know, some
results on semigroups (ordered semigroups) can be transferred into
$\Gamma$-semigroups ($po$-$\Gamma$-semigroups) just putting a Gamma
in the appropriate place, while for some other results the transfer
is not easy. A $\Gamma$-semigroup $M$ is called intra-regular
if $a\in M\Gamma a\Gamma a\Gamma M$ for every $a\in M$, equivalently
if $A\subseteq M\Gamma A\Gamma A\Gamma M$ for every $A\subseteq M$.
It is called left (resp. right) regular if $a\in
M\Gamma a\Gamma a$ (resp. $a\in \Gamma a\Gamma M$) for every $a\in M$
or $A\subseteq M\Gamma A\Gamma A$ (resp. $A\subseteq \Gamma A\Gamma
M$) for every $A\subseteq M$. An ordered $\Gamma$-semigroup $M$ is
called intra-regular if for every $a\in M$ we have $a\in
(M\Gamma a\Gamma a\Gamma M]$, equivalently if for every $A\subseteq
M$ we have $A\subseteq (M\Gamma A\Gamma A\Gamma M]$. An ordered
$\Gamma$-semigroup $M$ is called left (resp. right) regular if $a\in (M\Gamma a\Gamma a]$ (resp. $(a\in a\Gamma a\Gamma
M]$) for every $a\in M$, equivalently if $A\subseteq (M\Gamma A\Gamma
A]$ (resp. $A\subseteq (A\Gamma A\Gamma M]$) for every $A\subseteq
M$. Although some interesting results on $\Gamma$-semigroups are
obtained with these definitions, these definitions fail to describe
the principal filter of intra-regular, left regular and right regular
$\Gamma$-semigroups (ordered $\Gamma$-semigroups) which play an
essential role in the investigation. To overcome this difficulty, in
[8] a new definition of intra-regular and of left regular
$\Gamma$-semigroups has been introduced. The intra-regular
$\Gamma$-semigroup has been defined as a $\Gamma$-semigroup such that
$a\in M\Gamma a\gamma a\Gamma M$ for each $a\in M$ and each
$\gamma\in\Gamma$ and the left (resp. right) regular
$\Gamma$-semigroup as a $\Gamma$-semigroup in which $a\in M\Gamma
a\gamma a$ (resp. $a\in a\gamma a\Gamma M$) for each $a\in M$ and
each $\gamma\in\Gamma$ and it is proved that a $\Gamma$-semigroup $M$
is left regular (in that new sense) if and only if it is a union of a
family of left simple subsemigroups on $M$. And in [9] we gave some
further structure theorems of this type of $\Gamma$-semigroups using
that new definition and the form of principal filters. But what
happens in case of intra-regular and left or right regular
$po$-$\Gamma$-semigroups? Can we describe the form of the principal
filters using some new definitions like in the non-ordered case? The
present paper gives the related answer. For more information on
$\Gamma$ (or $po$-$\Gamma$)-semigroups cf., for example, the papers
in [5–7] of the References, and the papers in which these papers
refer. Examples on $\Gamma$-semigroups are also given in these
§ ON INTRA-REGULAR ORDERED $PO$-$\GAMMA$-SEMIGROUPS
We characterize here the intra-regular $po$-$\Gamma$-semigroups in
terms of filters, and we prove that a $po$-$\Gamma$-semigroup $M$ is
intra-regular if and only if the ideals of $M$ are
semiprime.Definition 1. An ordered $\Gamma$-semigroup $M$ is
called intra-regular if
$$x\in (M\Gamma x\gamma x\Gamma M]$$
every $x\in M$ and every $\gamma\in\Gamma$.Theorem 2. An ordered $\Gamma$-semigroup $M$ is
intra-regular if and only if, for every $x\in M$, we have
\{y\in M \mid x\in (M\Gamma y\Gamma M]\}.$$
$\Longrightarrow$. Let $x\in M$ and $T:=
\{y\in M \mid x\in (M\Gamma y\Gamma M]\}$. Then we have the
(1) $T$ is a nonempty subset of $M$. Indeed: Take an element
$\gamma\in\Gamma$ ($\Gamma\not=\emptyset$). Since $M$ is
intra-regular, we have
$$x\in (M\Gamma x\gamma x\Gamma M]={\Big(}(M\Gamma x)\gamma x\Gamma
M{\Big]}\subseteq {\Big(}(M\Gamma M)\Gamma x\Gamma M{\Big]}\subseteq
(M\Gamma x\Gamma M],$$
so $x\in T$.
(2) Let $a,b\in T$ and $\gamma\in\Gamma$. Then $a\gamma b\in T$.
Indeed: Since $a\in T$, we have $x\in (M\Gamma a\Gamma M]$. Since
$b\in T$, we have $x\in (M\Gamma b\Gamma M]$. Since $M$ is
intra-regular, $x\in M$ and $\gamma\in\Gamma$, we have $x\in (M\Gamma
x\gamma x\Gamma M]$. Then we have
\begin{eqnarray*}x\in (M\Gamma
x\gamma x\Gamma M]&\subseteq&{\Big(}M\Gamma (M\Gamma b\Gamma M]\gamma
(M\Gamma a\Gamma M]\Gamma M{\Big]}\\&=&{\Big(}M\Gamma (M\Gamma
b\Gamma M)\gamma (M\Gamma a\Gamma M)\Gamma
M{\Big]}\\&=&{\Big(}(M\Gamma M)\Gamma (b\Gamma M\gamma M\Gamma
a)\Gamma (M\Gamma M){\Big]}\\&\subseteq&{\Big(}M\Gamma (b\Gamma
M\gamma M\Gamma a)\Gamma M{\Big]}.\end{eqnarray*}
We prove that
$b\Gamma M\gamma M\Gamma a\subseteq {\Big(}M\Gamma (a\gamma b)\Gamma
M{\Big]}$. Then we have
\begin{eqnarray*}x&\in& {\bigg(}M\Gamma
{\Big(}M\Gamma (a\gamma b)\Gamma M{\Big]}\Gamma
M{\bigg]}={\bigg(}M\Gamma {\Big(}M\Gamma (a\gamma b)\Gamma
M{\Big)}\Gamma M{\bigg]}\\&=&{\Big(}(M\Gamma M)\Gamma (a\gamma
b)\Gamma (M\Gamma M){\Big]}\subseteq {\Big(}M\Gamma (a\gamma b)\Gamma
so $a\gamma b\in T$. Let now $b\lambda
u\gamma v\delta a\in b\Gamma M\gamma M\Gamma a$ for some $u,v\in M$,
$\lambda,\delta\in\Gamma$. Since $M$ is intra-regular, for the
elements $b\lambda u\gamma v\delta a\in M$ and $\gamma\in\Gamma$, we
\begin{eqnarray*}b\lambda u\gamma v\delta a&\in& {\Big(}M\Gamma
(b\lambda u\gamma v\delta a)\gamma (b\lambda u\gamma v\delta a)\Gamma
M{\Big]}\\&=&{\Big(}(M\Gamma b\lambda u\gamma v)\delta (a\gamma
b)\lambda (u\gamma v\delta a\Gamma
M){\Big]}\\&\subseteq&{\Big(}M\Gamma (a\gamma b)\Gamma M{\Big]}.
\end{eqnarray*}
(3) Let $a,b\in M$ and $\gamma\in\Gamma$ such that $a\gamma b\in T$.
Then $a,b\in T$. Indeed: Since $a\gamma b\in T$, we have $x\in
{\Big(}M\Gamma (a\gamma b)\Gamma M{\Big]}\subseteq {\Big(}M\Gamma
a\gamma (M\Gamma M){\Big]}\subseteq (M\Gamma a\Gamma M]$, so $a\in
T$. Since $x\in {\Big(}M\Gamma (a\gamma b)\Gamma M{\Big]}\subseteq
{\Big(}(M\Gamma M)\gamma b\Gamma M{\Big]}\subseteq (M\Gamma b\Gamma
M]$, we have $b\in T$.
(4) Let $a\in T$ and $M\ni b\ge a$. Then $b\in T$. Indeed: Since
$a\in T$, we have $x\in (M\Gamma a\Gamma M]$. Since $a\le b$, we have
$(M\Gamma a\Gamma M]\subseteq (M\Gamma b\Gamma M]$. Then we have
$x\in (M\Gamma b\Gamma M]$, and $b\in T$.
(5) Let $F$ be a filter of $M$ such that $x\in F$. Then $T\subseteq
F$. Indeed: Let $a\in T$. Then $x\in (M\Gamma a\Gamma M]$, so $F\ni
x\le u\lambda (a\mu v)$ for some $u,v\in M$, $\lambda,\mu\in \Gamma$.
Since $F$ is a filter of $M$, $x\in F$ and $M\ni u\lambda (a\mu v)\ge
x$, we have $u\lambda (a\mu v)\in F$. Since $F$ is a filter of $M$,
$u, a\mu v\in M$, $\lambda\in \Gamma$ and $u\lambda (a\mu v)\in F$,
we have $a\mu v\in F$, again since $F$ is a filter of $M$, $a,v\in M$
and $\mu\in \Gamma$, we have $a\in F$.
$\Longleftarrow$. Let $x\in M$ and $\gamma\in\Gamma$. Then
$x\in (M\Gamma x\gamma x\Gamma M]$. Indeed: Since $N(x)$ is a
subsemigroup of $M$, $x\in N(x)$ and $\gamma\in\Gamma$, we have
$x\gamma x\in N(x)$. By hypothesis, we get
$x\in{\Big(}M\Gamma(x\gamma x)\Gamma M{\Big]}=(M\Gamma x\gamma
x\Gamma M]$, thus $M$ is intra-regular.$\hfill\Box$Theorem 3. An ordered $\Gamma$-semigroup $M$ is
intra-regular if and only if the ideals of $M$ are
semiprime.Proof. $\Longrightarrow$. Let $A$ be an ideal of $M$,
$x\in M$ and $\gamma\in\Gamma$ such that $x\gamma x\in A$. Since $M$
is intra-regular, we have $x\in {\Big(}M\Gamma (x\gamma x)\Gamma
M{\Big]}\subseteq {\Big(}(M\Gamma A)\Gamma M{\Big]}\subseteq (A\Gamma
M]\subseteq (A]=A$, then $x\in A$, and $A$ is semiprime.
$\Longleftarrow$. Let $x\in M$ and $\gamma\in\Gamma$. Then $x\in
(M\Gamma x\gamma x\Gamma M]$. In fact: The set $(M\Gamma x\gamma
x\Gamma M]$ is an ideal of $M$. This is because it is a nonempty
subset of $M$, $M\Gamma (M\Gamma x\gamma x\Gamma M]\subseteq
{\Big(}M\Gamma(M\Gamma x\gamma x\Gamma
M]{\Big]}={\Big(}M\Gamma(M\Gamma x\gamma x\Gamma M){\Big]}\subseteq
(M\Gamma x\gamma x\Gamma M]$, $(M\Gamma x\gamma x\Gamma M]\Gamma
M\subseteq (M\Gamma x\gamma x\Gamma M]$, and ${\Big(}(M\Gamma x\gamma
x\Gamma M]{\Big]}=(M\Gamma x\gamma x\Gamma M]$. Since $(M\Gamma
x\gamma x\Gamma M]$ is semiprime and
$(x\gamma x)\gamma (x\gamma x)=x\gamma (x\gamma x)\gamma x\in M\Gamma
x\gamma x\Gamma M\subseteq (M\Gamma x\gamma x\Gamma M]$, we have
$x\gamma x\in (M\Gamma x\gamma x\Gamma M]$. Again since $(M\Gamma
x\gamma x\Gamma M]$ is semiprime, we have $x\in (M\Gamma x\gamma
x\Gamma M]$, so $M$ is intra-regular.$\hfill\Box$
§ ON LEFT REGULAR AND LEFT DUO $PO$-$\GAMMA$-SEMIGROUPS
we notice that the left (and the right) $po$-$\Gamma$-semigroups are
intra-regular. Then we characterize the $po$-$\Gamma$-semigroups
which are both left regular and left duo in terms of filters and we
prove that a $po$-$\Gamma$-semigroup $M$ is left (resp. right)
regular if and only if the left (resp. right) ideals of $M$ are
semiprime.Definition 4. An ordered $\Gamma$-semigroup $M$ is
called left regular (resp. right regular) if
(M\Gamma x\gamma x] \mbox { (resp. } x\in (x\gamma x\Gamma M])$$
every $x\in M$ and every $\gamma\in\Gamma$.Proposition 5. Let M be an ordered
$\Gamma$-semigroup. If M is left (resp. right) regular, then M is
intra-regular.Proof. Let $M$ be left regular, $x\in M$ and
$\gamma\in\Gamma$. Then we have
\begin{eqnarray*}x\in (M\Gamma x\gamma
x]&\subseteq&{\Big(}M\Gamma (M\Gamma x\gamma x]\gamma x{\Big
]}={\Big(}M\Gamma (M\Gamma x\gamma x)\gamma x{\Big ]}\\&\subseteq
&{\Big(}(M\Gamma M)\Gamma (x\gamma x)\Gamma M{\Big]}\subseteq
{\Big(}M\Gamma x\gamma x\Gamma M{\Big]},\end{eqnarray*}
thus $M$ is
intra-regular.$\hfill\Box$Theorem 6. An ordered $\Gamma$-semigroup $M$ is
left regular and left duo if and only if, for every $x\in M$, we have
\{y\in M \mid x\in (M\Gamma y]\}.$$
$\Longrightarrow$. Let $x\in M$ and $T:=
\{y\in M \mid x\in (M\Gamma y]\}$. Since $M$ is left regular, we
$x\in (M\Gamma x\gamma x]\subseteq {\Big(}(M\Gamma M)\Gamma
x{\Big]}\subseteq (M\Gamma x]$, so $x\in T$, and $T$ is a nonempty
subset of $M$. Let $a,b\in T$ and $\gamma\in\Gamma$. Since $x\in
(M\Gamma a]$, $x\in (M\Gamma b]$ and $M$ is left regular, we
\begin{eqnarray*}x\in (M\Gamma x\gamma
x]&\subseteq&{\Big(}M\Gamma (M\Gamma b]\gamma (M\Gamma
a]{\Big]}={\Big(}M\Gamma (M\Gamma b)\gamma (M\Gamma
a){\Big]}\\&\subseteq&{\Big(}M\Gamma (b\gamma M\Gamma
In addition, $b\gamma M\Gamma a\subseteq
(M\Gamma a\gamma b]$. Indeed: Let $b\gamma u\mu a\in b\gamma M\Gamma
a$, where $u\in M$ and $\mu\in\Gamma$. Since $M$ is left regular, we
$$b\gamma u\mu a\in {\Big(}M\Gamma (b\gamma u\mu a)\gamma
(b\gamma u\mu a){\Big]}\subseteq {\Big(}M\Gamma (a\gamma b) \Gamma
M{\Big]}={\Big(}(M\Gamma a\gamma b]\Gamma M{\Big]}.$$
Since $(M\Gamma
a\gamma b]$ is a left ideal, it is a right ideal of $M$ as well, so
$(M\Gamma a\gamma b]\Gamma M\subseteq (M\Gamma a\gamma b]$, then
$b\gamma u\mu a\in{\Big(}(M\Gamma a\gamma b]{\Big]}=(M\Gamma a\gamma
b]$. Hence we obtain
$$x\in {\Big(}M\Gamma(M\Gamma a\gamma
b]{\Big]}={\Big(}M\Gamma(M\Gamma a\gamma b){\Big]}\subseteq
{\Big(}M\Gamma(a\gamma b){\Big]},$$
from which $a\gamma b\in T$.
$a,b\in M$ and $\gamma\in\Gamma$ such that $a\gamma b\in T$. Since
$x\in (M\Gamma a\gamma b]\subseteq (M\Gamma b]$, we have $b\in T$.
Besides, $x\in (M\Gamma a\gamma b]\subseteq {\Big(}(M\Gamma a]\Gamma
M{\Big]}$. The set $(M\Gamma a]$ as a left ideal, is a right ideal of
$M$ as well, so $(M\Gamma a]\Gamma M\subseteq (M\Gamma a]$. Thus we
have $x\in {\Big(}(M\Gamma a]{\Big]}=(M\Gamma a]$, and $a\in T$.
$a\in T$ and $M\ni b\ge a$. Since $M$ is left regular, we have
(M\Gamma a\gamma a]\subseteq (M\Gamma b\gamma b]\subseteq
{\Big(}(M\Gamma b]\Gamma M{\Big]}.$$
$(M\Gamma b]$ as a left ideal is
a right ideal of $M$, so $(M\Gamma b]\Gamma M\subseteq (M\Gamma b]$.
Hence we have $x\in {\Big(}(M\Gamma b]{\Big]}=(M\Gamma b]$, and $b
\in T$.
Let $F$ be a filter of $M$ such that $x\in F$ and let $a\in
T$. Since $x\in (M\Gamma a]$, we have $F\ni x\le u\mu a$ for some
$u\in M$, $\mu\in\Gamma$. Since $F$ is a filter of $M$, we have $u\mu
a\in F$, and $a\in F$.
$\Longleftarrow$. Let $x\in M$ and
$\gamma\in\Gamma$. Since $x\in N(x)$ and $N(x)$ is a subsemigroup of
$M$, we have $x\gamma x\in N(x)$. By hypothesis, we get $x\in
(M\Gamma x\gamma x]$, so $M$ is left regular. Let now $A$ be a left
ideal of $M$, $a\in A$, $\gamma\in\Gamma$ and $u\in M$. Since
$a\gamma u\in N(a\gamma u)$ and $N(a\gamma u)$ is a filter of $M$, we
have $a\in N(a\gamma u)$. By hypothesis, we have $a\gamma u\in
(M\Gamma a]\subseteq (M\Gamma A]\subseteq (A]=A$. Thus $A$ is right
ideal of $M$. $\hfill\Box$
The right analogue of Theorem 6 also
holds, and we haveTheorem 7. An ordered $\Gamma$-semigroup $M$ is
right regular and right duo if and only if, for every $x\in M$, we
$$N(x)=\{y\in M \mid x\in (y\Gamma M]\}.$$
Theorem 8. An ordered $\Gamma$-semigroup $M$ is
left (resp. right) regular if and only if the left (resp. right)
ideals of $M$ are semiprime.Proof. $\Longrightarrow$. Let $M$ be left regular, $A$
a left ideal of $M$, $x\in M$ and $\gamma\in\Gamma$ such that
$x\gamma x\in A$. Then we have $x\in {\Big(}M\Gamma (x\gamma
x){\Big]}\subseteq (M\Gamma A]\subseteq (A]=A$, so $M$ is
$\Longleftarrow$. Suppose the left ideals of $M$ are
semiprime and let $x\in M$ and $\gamma\in\Gamma$. Since $(M\Gamma
x\gamma x]$ is a left ideal of $M$ and $(x\gamma x)\gamma (x\gamma
x)\in (M\Gamma x\gamma x]$, we have $x\gamma x\in (M\Gamma x\gamma
x]$, and $x\in (M\Gamma x\gamma x]$, so $M$ is left
1R. Croisot, Demi-groupes inversifs et demi-groupes
reunions de demi-groupes simples, Ann. Sci. Ecole Norm. Sup.
3, No. 70 (1953), 205–208.
2A.H. Clifford, G.B. Preston, The Algebraic Theory of
Semigroups, Vol. I, Amer. Math. Soc. Math. Surveys 7,
Providence, Rhode Island, 1961.
3N. Kehayopulu, On right regular and right duo ordered
semigroups, Math. Japon. 36, No. 2 (1991), 201–206.
4N. Kehayopulu, On intra-regular ordered semigroups,
Semigroup Forum 46 (1993), 271–278.
5N. Kehayopulu, On prime, weakly prime ideals in
$po$-$\Gamma$-semigroups, Lobachevskii J. Math. 30, No. 4
(2009), 257–262.
6N. Kehayopulu, On ordered $\Gamma$-semigroups, Sci.
Math. Jpn. 71, No. 2 (2010), 179–185.
7N. Kehayopulu, On regular duo
$po$-$\Gamma$-semigroups, Math. Slovaca 61, no. 6 (2011),
8N. Kehayopulu, On left regular $\Gamma$-semigroups,
Int. J. Algebra 8, no. 8 (2014), 389–394.
9N. Kehayopulu, M. Tsingelis, On intra-regular and
some left regular $\Gamma$-semigroups, Quasigroups and Related
Systems, 23 no. 2 (2015), to appear.
10M. Petrich, Introduction to Semigroups, Charles E.
Merrill Publ. Comp., A Bell $\&$ Howell Comp. Columbus, Ohio
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1511.00796
|
In this article, we propose a control law for almost-global asymptotic tracking (AGAT) of a smooth reference trajectory for a fully actuated simple mechanical system (SMS) evolving
on a Riemannian manifold which can be embedded in a Euclidean space. The existing results on tracking for an SMS are either local, or almost-global, only in the case the manifold is a Lie group. In the latter case, the notion of a configuration error is naturally defined by the group operation and facilitates a global analysis. However, such a notion is not intrinsic to a Riemannian manifold. In this paper, we define a configuration error followed by
error dynamics on a Riemannian manifold, and then prove AGAT. The results are demonstrated for a spherical pendulum which is an SMS on $S^2$ and for a particle moving on a Lissajous curve in $\mathbb{R}^3$.
§ INTRODUCTION
The problem of stabilization of an equilibrium point of an SMS on a Riemannian manifold has been well studied in the literature in a geometric framework <cit.>, <cit.>, <cit.>, <cit.>, <cit.>. Further extensions of these results to the problem of locally tracking a smooth and bounded trajectory can be found in <cit.>. An SMS is completely specified by a manifold, the kinetic energy of the mechanical system, which defines the Riemannian metric on the manifold, the potential forces, and the external forces or one forms on the manifold. If the Riemannian manifold is embedded in a Euclidean space, the metric on the manifold is induced from the Euclidean metric. In <cit.>, a proportional and derivative plus feed forward (PD+FF) feedback control law is proposed for tracking a trajectory on a Riemannian manifold using error functions. This controller achieves asymptotic tracking only when the initial configuration of the SMS is in a neighborhood of the initial reference configuration. Therefore, such a tracking law achieves local convergence. As pointed out in <cit.> and <cit.>, global stabilization and global tracking is guaranteed only when the configuration manifold is diffeomorphic to $\mathbb{R}^n$. This leads us to the question: Is almost-global asymptotic stabilization (AGAS) of an equilibrium point and, almost-global asymptotic tracking (AGAT) of a suitable class of reference trajectories possible on a Riemannian manifold?
AGAS problems on a compact Riemannian manifold trace their origin to an early work by Koditschek. In <cit.>, a potential function called as “navigation function" is introduced, which is a Morse function on the manifold with a unique minimum. It is shown that there exists a dense set from which the trajectories of a negative gradient vector field generated by the navigation function converge to the minimum. A class of simple mechanical systems can be generated by the “lifting" of the gradient vector field to the tangent bundle of the manifold. It is shown that the integral curves of such an SMS on the tangent bundle behave similar to integral curves of the gradient vector field of the navigation function on the manifold. In particular, the integral curve of such an SMS originating from a dense set in the tangent bundle converges asymptotically to the minimum of the navigation function in the zero section of the tangent bundle.
Koditschek's idea can be extended to almost-global tracking of a smooth and bounded trajectory on a Lie group. In a tracking problem, it is essential to define the notion of both configuration and velocity errors between the reference and the system trajectory. For an SMS on a lie group, a configuration error is defined by the left or right group operation and the velocity error is defined on the Lie algebra. This defines the error dynamics on the tangent bundle of the Lie group. It is shown in <cit.> that the error dynamics is an SMS on the tangent bundle of the Lie group and, is generated by the tangent lift of a navigation function. Therefore, Koditschek's theorem can be applied to achieve AGAS of the error dynamics. Specific problems of almost-global tracking and stabilization in Lie groups have been studied in the literature as well. In <cit.>, <cit.>, control laws for AGAT are proposed on $SE(3)$ using Morse functions.
Our contribution extends the existing results on AGAT to compact manifolds embedded in an Euclidean space. It is shown in <cit.> that a navigation function exists on any compact manifold. We choose a configuration error map on the manifold subject to certain requirements imposed by the navigation function. The velocity error between the reference and system trajectory is defined along the error trajectory on the manifold with the help of two “transport maps” defined by the configuration error map. This construction gives rise to error dynamics on the tangent space of the error trajectory. To the best of our knowledge, this approach of AGAT for an SMS has not been explored before. In <cit.>, a transport map is introduced to compare velocities at two configurations. However, in such a construction the error dynamics is not the “lift" of a gradient vector field of a navigation function. Therefore, Koditschek's theorem is not applicable to the error dynamics. In this paper, the error dynamics we introduce is an SMS on a compact Riemannian manifold and hence Koditschek's theorem can be applied for AGAS of the error dynamics. This leads to AGAT of the reference trajectory.
The paper is organised as follows. The second section is a brief introduction to relevant terminology in associated literature. In the third section we elaborate on the “lift" of a gradient vector field and state the main result on AGAS from <cit.>. The following section is on AGAT for a fully actuated SMS on a compact manifold. We first define the allowable configuration error map and navigation function for the tracking problem. Subsequently in the main theorem, we state our proposed control for AGAT on a Riemannian manifold. In the next section we demonstrate the idea of two transport maps for AGAT on a Lie group by choosing a configuration error defined by the group operation. The last section shows simulation results for a spherical pendulum which is an SMS on $S^2$ and for a particle moving on a Lissajous curve in $\mathbb{R}^3$.
§ PRELIMINARIES
A Riemannian manifold is denoted by the 2-tuple $(M, \mathbb{G})$, where $M$ is a smooth connected manifold and $\mathbb{G}$ is a smooth, symmetric, positive definite $(0,2)$ tensor field (or a metric) on $M$. $\stackrel{\mathbb{G}}{\nabla}$ denotes the Riemannian connection on $(M, \mathbb{G})$ (see <cit.>,<cit.> for more details). Let $\Psi: M \to \mathbb{R}$ be a twice differentiable function on $(M, \mathbb{G})$.
The Hessian of $\Psi$ is the symmetric $(0,2)$ tensor field denoted by $Hess \Psi$ and defined as $ Hess \Psi(q)(v_q, w_q) = \langle \langle v_q, \stackrel{\mathbb{G}}{\nabla}_{w_q} grad \Psi \rangle \rangle,$ where $v_q$, $w_q \in T_q M$ and $\langle \langle v_q, w_q\rangle \rangle \coloneq \mathbb{G}(q)(v_q, w_q)$. Let $x_0$ be a critical point of $\Psi$ and $\{ x^1, \dotsc , x^n\}$ are local coordinates at $x_0$. The Hessian at $x_0$ is given in coordinates as ${(Hess \Psi(x_0))}_{ij} = \frac{\partial ^2 \Psi}{\partial x^i \partial x^j}(x_0)$ (see chapter 13 in <cit.> for details). The map $\mathbb{G}^{\flat}(q) : T_q M \to T_q ^* M$ is defined by $\langle \mathbb{G}(q)^{\flat} (v_1), v_2 \rangle \coloneq \mathbb{G}(q)(v_1, v_2)$ for $v_1$,$v_2 \in T_q M$. Therefore, if $\{ e^i\}$ is a basis for $T^*_q M$ in a coordinate system then $\mathbb{G}^{\flat}$ is expressed in coordinates as $ \mathbb{G}^{\flat}(q) (v_q) = \mathbb{G}_{ij} {v_q}^j e^i$, where $\mathbb{G}_{ij}$ is the matrix representation of $\mathbb{G}(q)$ in the chosen basis. The map $ \mathbb{G}^\sharp: T_q ^* M \to T_q M$ is dual of $\mathbb{G}^\flat$. It is expressed in coordinates as $ \mathbb{G}^\sharp (w) = \mathbb{G}^{ij} w_j e_i$ for $w \in T_q^*M$.
§.§ SMS on a Riemannian manifold
A fully actuated simple mechanical system (or an SMS) on a smooth, connected Riemannian manifold $(M, \mathbb{G})$ is denoted by the 3-tuple $(M,\mathbb{G}, F)$, where $F$ is an external force. The governing equations are
\begin{equation}\label{eq:3}
\stackrel{\mathbb{G}}{\nabla}_{\dot{\gamma}(t)} \dot{\gamma}(t) = \mathbb{G}^{\sharp} (F(\dot{\gamma}(t)))
\end{equation}
where $\gamma(t)$ is the system trajectory.
§.§ SMS on a Riemannian manifold embedded in $\mathbb{R}^m$
Consider a Riemannian manifold $(M,\mathbb{G})$. By Nash embedding theorem (see <cit.>), there exists an isometric embedding $f : M \to \mathbb{R}^m$ for some $m$ depending on the dimension of $M$. The Euclidean metric $\mathbb{G}_{id}$ on $\mathbb{R}^m$ is the Riemannian metric such that in Cartesian coordinates
\begin{equation}\label{gid}
G_{id} = \delta_{ij} = \begin{cases}
1 & \text{if} \quad i =j\\
0 & \text{if} \quad i \neq j.
\end{cases}
\end{equation}
Therefore, the metric $\mathbb{G}$ on $M$ is induced by the Riemannian metric as follows
\begin{equation} \label{metricind}
\mathbb{G}= f^{*}\mathbb{G}_{id}
\end{equation}
where $f^{*}\mathbb{G}_{id}$ is the pull back of $\mathbb{G}_{id}$ (see Definition 3.81 in <cit.>). The equations of motion for the SMS $(M, \mathbb{G},F)$ in (<ref>) can be simplified by embedding $M$ in $\mathbb{R}^m$. The idea behind this approach is that we consider the SMS to evolve on $(\mathbb{R}^m, \mathbb{G}_{id})$ subject to a distribution $\mathcal{D}$ (the velocity constraint) whose integral manifold is $M$ (see section 4.5 in <cit.> for more details). The subspace $\mathcal{D}_{x} = T_{f^{-1}(x)} M$. Let $P_{\mathcal{D}_y}$ and $P_{\mathcal{D}_y}^{\perp}$ be projection bundle maps from
$T_y \mathbb{R}^m$ to $\mathcal{D}_y$ and $\mathcal{D}^\perp_y$ respectively so that for $v_y \in T_y \mathbb{R}^m$, $P_{\mathcal{D}_y}(v_y) \in \mathcal{D}_y$ and $P_{\mathcal{D}_y}^\perp(v_y) \in \mathcal{D}^\perp_y$ respectively.
The constrained affine connection on $M$ is denoted by $\stackrel{\mathcal{D}}{\nabla}$ and is defined for $X$, $Y \in \Gamma^{\infty} (TM)$ as
\begin{equation}\label{eq:111}
\stackrel{\mathcal{D}}{\nabla}_X Y = \stackrel{\mathbb{G}_{id}}{\nabla}_X Y + (\stackrel{\mathbb{G}_{id}}{\nabla}_X P_{\mathcal{D}}^\perp) Y
\end{equation}
The equations of motion for the SMS $(\mathbb{R}, \mathbb{G}_{id}, \mathcal{D})$ are (see Theorem 4.87 in <cit.>)
\begin{align}\label{123}
\stackrel{\mathcal{D}}{\nabla} _{\dot{\boldsymbol{\gamma}}} \dot{\boldsymbol{\gamma}} &=0,\\
P_{\mathcal{D}_{\boldsymbol{\gamma}}}^\perp (\dot{\boldsymbol{\gamma}}(t_0))&=0 \quad \text{for some} \quad t_0 \in \mathbb{R}^{+},\nonumber
\end{align}
where $\boldsymbol{\gamma}:\mathbb{R}^{+} \to \mathbb{R}^m$ denotes the system trajectory, $\dot{\boldsymbol{\gamma}}(t) \in \mathcal{D}_{\boldsymbol{\gamma}(t)}$ for all $t \in \mathbb{R}^{+}$ and $\ddot{\boldsymbol{\gamma}}(t) \in {\mathcal{D}}^\perp_{\gamma(t)}$. (<ref>) can be simplified by substituting for $\stackrel{\mathcal{D}}{\nabla}$ from (<ref>) as follows
\begin{equation}\label{112}
\ddot{\boldsymbol{\gamma}} +(\stackrel{\mathbb{G}_{id}}{\nabla}_{\dot{\boldsymbol{\gamma}}} P_{\mathcal{D}}^\perp)\dot{\boldsymbol{\gamma}}=0.
\end{equation}
Now consider the SMS $(\mathbb{R}, \mathbb{G}_{id},F, \mathcal{D})$. The equations are
\begin{align}\label{113}
\stackrel{\mathcal{D}}{\nabla} _{\dot{\boldsymbol{\gamma}}}\dot{\boldsymbol{\gamma}}&= u,\\
P_{\mathcal{D}_{\boldsymbol{\gamma}}}^\perp (\dot{\boldsymbol{\gamma}}(t_0))&=0 \quad \text{for some} \quad t_0 \in \mathbb{R}^{+} \nonumber
\end{align}
where $\boldsymbol{\gamma}:\mathbb{R} \to \mathbb{R}^m$ denotes the system trajectory and, $u = P_{\mathcal{D}_\gamma} \{\mathbb{G}_{id}^\sharp (F)\}$. Further, substituting from (<ref>), (<ref>) can be written as
\begin{equation}\label{134}
\ddot{\boldsymbol{\gamma}} +(\stackrel{\mathbb{G}_{id}}{\nabla}_{\dot{\boldsymbol{\gamma}}} P_{\mathcal{D}}^\perp)\dot{\boldsymbol{\gamma}}= u.
\end{equation}
Remark 1: For a fully actuated SMS on $M$, the number of independent controls available is the dimension of $M$. However, as (<ref>) and (<ref>) are expressed in Euclidean coordinates, the control vector field $u$ is in $\mathbb{R}^m$.
Remark 2: Equation (<ref>) and (<ref>) both represent the dynamics of the SMS $(M, \mathbb{G},F)$. The system trajectory $\boldsymbol{\gamma}(t) \in \mathbb{R}^m$ (in (<ref>)) is the push forward of $\gamma(t) \in M$ (in (<ref>)) by the embedding $f: M \to \mathbb{R}^m$. Therefore, $\boldsymbol{\gamma}(t)= f_* \gamma(t)$ (see Remark 4.98 in <cit.>).
§ AGAS OF ERROR DYNAMICS
The main objective is almost global asympototic tracking (AGAT) of a given reference trajectory on the Riemannian manifold $(M, \mathbb{G})$ which is embedded in $\mathbb{R}^m$ (described in section II B.). The tracking problem is reduced to a stabilization problem by introducing the notion of a configuration error on the manifold. Almost global asymptotic stabilization (AGAS) of the error dynamics about a desired "zero error" configuration leads to AGAT of the reference trajectory. In this section, we (a) introduce this configuration error map between two configurations on a Riemmanian manifold to express the error between the reference and system trajectories and, (b) explicitly obtain the error dynamics for the tracking problem.
A function $\psi: M \to \mathbb{R}$ on $(M, \mathbb{G})$ is a navigation function iff
* $\psi$ attains a unique minimum.
* $Det(Hess \psi(q)) \neq 0$ whenever $\mathrm{d}\psi(q) = 0$ for $q \in M$.
Let $\gamma:\mathbb{R}^+ \to M$ denote the system trajectory of $(M, \mathbb{G})$ and $\gamma_{ref}:\mathbb{R}^+ \to M$ denote a reference trajectory on the $M$. We define a $\mathcal{C}^2$ map $E:M \times M \to M$ between any two configurations on the manifold called the configuration error map. The error trajectory on $M$ is $E(\gamma(t), \gamma_{ref}(t))$. The following condition characterizes the class of error maps for the AGAT on $M$.
Consider a navigation function $\psi$ on $M$ (as in Definition <ref>). A configuration error map $E$ is compatible with $\psi$ iff
* $\psi\circ E(\gamma(t),\gamma_{ref}(t))= \psi \circ E(\gamma_{ref}(t),\gamma(t))$ for all $t \in \mathbb{R}^+$ and,
* $E(q,q) = q_0$, where $q_0$ is the minimum of $\psi$.
The following equation describes the controlled error dynamics for the tracking problem on $(M, \mathbb{G})$
\begin{align}\label{errdyn}
&\stackrel{\mathcal{D}}{\nabla}_{\dot{\textbf{E}}(\boldsymbol{\gamma}(t), \boldsymbol{\gamma}_{ref}(t))}{\dot{\textbf{E}}(\boldsymbol{\gamma}(t), \boldsymbol{\gamma}_{ref}(t))} =\\
&P_{\mathcal{D}_{\textbf{E}(t)}}\{\mathbb{G}_{id}^\sharp (- K_p \mathrm{d}\boldsymbol{\psi}(\textbf{E})+F_{diss}(\dot{\textbf{E}} ))\} \nonumber
\end{align}
where $\boldsymbol{\gamma} = f_* \gamma$, $\boldsymbol{\gamma}_{ref} = f_* \gamma_{ref}$, $\boldsymbol{\psi} \coloneq f_* \psi$, $\textbf{E} \coloneq f_* E$, $E$ is a compatible with the navigation function $\psi$, $K_p$ is a positive definite matrix and, $F_{diss}: T\mathbb{R}^m \to T^* \mathbb{R}^m$ is a dissipative force, which means $\langle F_{diss}(v), v \rangle \leq 0$ for all $v \in \mathbb{R}^m$.
From (<ref>), we conclude that the error dynamics (in (<ref>)) is an SMS. In the following Lemma, we apply the main results from <cit.> to conclude AGAS of the error dynamics about the minimum of $\psi$ lifted to the zero section of $TM$.
Note: From henceforth we shall denote the push forward of entities by $f$ in bold font.
Consider the SMS $(M, \mathbb{G})$ whose dynamics is given by (<ref>), and a smooth reference trajectory $\gamma_{ref}: \mathbb{R}^+ \to M$. The error dynamics in (<ref>) is AGAS about $(q_m,0)$ where $q_m$ is the unique minimum of the navigation function $\psi$.
We first rewrite (<ref>) so that the flow $\dot{E}(t)$ evolves on $TM$. From the equivalence in representation of Riemmannian connection $ \stackrel{\mathbb{G}}{\nabla}$ and the constrained connection $ \stackrel{\mathcal{D}}{\nabla}$ noted in Remark 2, (<ref>) can be expressed as
\begin{align}\label{eq:75}
\dot{E} &= v_e \\ \nonumber
\stackrel{\mathbb{G}}{\nabla}_{\dot{E}}{v_e} &= - \mathbb{G}^\sharp K_p \mathrm{d}\psi(E) + \mathbb{G}^\sharp F_{diss}(v_e) \nonumber
\end{align}
We define an energy like function $E_{cl}$ on $TM$ as $E_{cl}(E,v_E) \coloneq K_p \psi(E) + \frac{1}{2} {||v_e ||}^2$. Then, $E_{cl}(q_m,0)=0$ and $E_{cl}(q,0)>0$ for all $(q,0)$ in a neighborhood of $(q_m,0)$. Also,
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t} E_{cl} &= \langle K_p \mathrm{d} \psi(E),v_e \rangle + \ll v_e, \stackrel{\mathbb{G}}{\nabla}_{\dot{E}} v_e \gg \\
&=\langle K_p \mathrm{d} \psi(E),v_e \rangle + \mathbb{G}(v_{e}, -\mathbb{G}^{\sharp} (K_p \mathrm{d} \psi(E) - F_{diss}(v_{e}) )\\
&= K_p \langle \mathrm{d} \psi, v_{e} \rangle- K_p \langle \mathrm{d}\psi, v_{e} \rangle + \langle F_{diss}(v_{e}), v_{e} \rangle \leq 0
\end{align*}
as $F_{diss}$ is dissipative. Therefore $E_{cl}$ is a Lyapunov function and the error dynamics in (<ref>) is locally stable around $(q_m,0)$. In what follows, we obtain all parts of from proposition 3.6 in <cit.> for the error dynamics.
In proposition 3.2 in <cit.>, as $M$ is a manifold without boundary, $b_1 = +\infty$ and $b_0 = \psi(q_m)$. Therefore, $\epsilon^b = TM$ and by Propostion 3.2, $TM$ is a positively invariant set.
Observe that $(q*,0)$ is an equilibrium state of (<ref>) iff $q*$ is a critical point of $\psi$. By proposition 3.3 in <cit.>, the positive limit set of all solution trajectories of (<ref>) originating in the positive invariant set $TM$ is $\{(p,0)\in TM: \mathrm{d}\psi(p)=0\}$.
In order to study the behavior of the flow of the vector field in (<ref>), we linearize the equations around $(q*,0)$ to obtain
\begin{align}\label{linerr}
\dot{E}\\
\dot{v}_e
\end{pmatrix}=\\ \nonumber
0 & I_n\\
-G^{ij}(q*) {Hess \psi}(q*) & G^{ij}(q*) \circ \frac{\partial F_{diss}}{\partial v_e}(q*,0)
\end{pmatrix}\begin{pmatrix}
E\\ v_e\\
\end{pmatrix}.
\end{align}
By Lemma 3.5 in <cit.> the origin of the LIT system (<ref>) in $\mathbb{R}^{2n}$, $n$ being the dimension of $M$, is (a) asymptotically stable, (b) stable but not attractive or (c) unstable if the origin of the following LTI system in $\mathbb{R}^n$ has the corresponding property
\[\dot{E}= -Hess \psi(E)
\]
The local behavior of the error dynamics around $(q*,0)$ is therefore, determined by the nature of Hessian at $q^*$.
The trajectories of $-\mathrm{d}\psi(E(t))$ converge to $q_m$ from all but the stable manifolds of the maxima and saddle points. As $\psi$ is a navigation function, the stable manifolds of the maxima and saddle points constitutes a nowhere dense set. Therefore there is a dense set in $TM$ for which all trajectories of (<ref>) converge to $(q_m,0)$.
§ AGAT FOR AN SMS
In this section we propose a control law for AGAT of a reference trajectory for a fully actuated SMS $(M, \mathbb{G})$ for which the equations of motion are given in (<ref>). We first obtain a simplified expression for the constrained covariant derivative of two vector fields on $M$.
(Proposition 4.85 in <cit.>)
Let $X \in \Gamma^{\infty}(TM)$ and $Y \in \Gamma^{\infty}(\mathcal{D})$ be vector fields on $M$. The constrained covariant derivative of $Y$ along $X$ is given as
\begin{equation}\label{116}
\stackrel{\mathcal{D}}{\nabla}_X Y = P_{\mathcal{D}}(\stackrel{\mathbb{G}_{id}}{\nabla}_X Y ).
\end{equation}
Consider the SMS $(M,\mathbb{G})$ given by (<ref>) and a smooth trajectory $\gamma_{ref}:\mathbb{R} \to M$ with bounded velocity. Let $\psi: M \to \mathbb{R}$ be a navigation function and $E:M \times M \to M$ be a compatible error map on the manifold. Then there exists an open dense set $S \in TM$ such that AGAT of $\gamma_{ref}$ is achieved for all $(\gamma(0),\dot{\gamma}(0))\in S$ with $u$ in (<ref>) given by the solution to the following equations
\begin{align}\label{thm11}
\mathrm{d}_1 \mathbf{E}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})(u) &= P_{\mathcal{D}_{\mathbf{E}}}(- K_p \mathrm{d}\boldsymbol{\psi}(\mathbf{E}) + F_{diss}(\dot{\mathbf{E}})\\ \nonumber
&- \mathrm{d}_1 (\mathrm{d}_1 \mathbf{E} )(\dot{\boldsymbol{\gamma}},\dot{\boldsymbol{\gamma}}) \\ \nonumber
&+ \mathrm{d}_2 (\mathrm{d}_1 \mathbf{E}) (\dot{\boldsymbol{\gamma}}_{ref},\dot{\boldsymbol{\gamma}})+ \frac{\mathrm{d}}{\mathrm{d}t}(\mathrm{d}_2 \mathbf{E} \dot{\boldsymbol{\gamma}}_{ref})) ,\\ \nonumber
\text{and}, \quad P_{\mathcal{D}_{\boldsymbol{\gamma}}}^\perp (u)&=0.\nonumber
\end{align}
where $F_{diss}: T\mathbb{R}^m \to T^*\mathbb{R}^m $ is a dissipative force and $K_p \in \mathbb{R}^+$.
Let $\textbf{E}(\boldsymbol{\gamma}(t), \boldsymbol{\gamma}_{ref}(t))$ be error trajectory and the closed loop error dynamics be given by (<ref>). The velocity vector of the error trajectory is given by
\begin{equation}\label{eq:73}
\dot{\textbf{E}}(\boldsymbol{\gamma}(t), \boldsymbol{\gamma}_{ref}(t)) = \mathrm{d}_1\textbf{E}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref} ) \dot{\boldsymbol{\gamma}} + \mathrm{d}_2\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref}) \dot{\boldsymbol{\gamma}}_{ref}
\end{equation}
where $\mathrm{d}_1\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref}): T_{\boldsymbol{\gamma}} M \to T_{\textbf{E}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})}M$ is the partial derivative of $\textbf{E}$ with respect to the first argument and, $\mathrm{d}_2\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref}): T_{{\boldsymbol{\gamma}}_{ref}} M \to T_{\textbf{E}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})}M$ is the partial derivative of $\textbf{E}$ with respect to the second argument and, $\dot{\textbf{E}}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref}) \in T_{\textbf{E}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})}M $.
$\mathrm{d}_1\textbf{E}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref}) $ and $\mathrm{d}_2\textbf{E}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})$ are similar to “transport maps" in <cit.> as they transport vectors along the system and reference trajectory respectively to vectors along the error trajectory.
Two transport maps from controlled trajectory $\boldsymbol{\gamma}(t)$ and reference trajectory $\boldsymbol{\gamma}_{ref}(t)$ to the error trajectory $\textbf{E}(\boldsymbol{\gamma},\boldsymbol{\gamma}_{ref})$
\begin{align}\label{118}
\stackrel{\mathcal{D}}{\nabla}_{\dot{\textbf{E}}}{\dot{\textbf{E}}}&= \stackrel{\mathcal{D}}{\nabla}_{\dot{\textbf{E}}}(\mathrm{d}_1\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})\dot{\boldsymbol{\gamma}} + \mathrm{d}_2\textbf{E}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref}) \dot{\boldsymbol{\gamma}}_{ref})
\end{align}
As $\mathrm{d}_1\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})\dot{\boldsymbol{\gamma}}$ and $\mathrm{d}_1\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})\dot{\boldsymbol{\gamma}}_{ref}$ are vector fields along $E
(t)$ on $Q$, therefore, by Lemma <ref>,
\begin{equation}
\stackrel{\mathcal{D}}{\nabla}_{\dot{\textbf{E}}}(\mathrm{d}_1\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})\dot{\boldsymbol{\gamma}}) = P_{\mathcal{D}_\textbf{E}}(\stackrel{\mathbb{G}_{id}}{\nabla}_ {\dot{\textbf{E}}}\mathrm{d}_1\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})\dot{\boldsymbol{\gamma}} )
\end{equation}
\begin{equation}
\stackrel{\mathcal{D}}{\nabla}_{\dot{\textbf{E}}}(\mathrm{d}_2\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})\dot{\boldsymbol{\gamma}}_{ref}) = P_{\mathcal{D}_\textbf{E}}(\stackrel{\mathbb{G}_{id}}{\nabla}_ {\dot{\textbf{E}}}\mathrm{d}_2\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})\dot{\boldsymbol{\gamma}}_{ref})
\end{equation}
Note: We shall drop arguments and refer to $\textbf{E}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})$ as $\textbf{E}$ and similarly refer to $\mathrm{d}_1\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref}) $ and $\mathrm{d}_2\textbf{E} (\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})$ as $\mathrm{d}_1\textbf{E}$, $\mathrm{d}_2\textbf{E}$ respectively.
From (<ref>) and (<ref>),
\begin{align}\label{119}
\stackrel{\mathcal{D}}{\nabla}_{\dot{\textbf{E}}}{\dot{\textbf{E}}} &= P_{\mathcal{D}_\textbf{E}}(\stackrel{\mathbb{G}_{id}}{\nabla}_ {\dot{\textbf{E}}}\mathrm{d}_1\textbf{E} \dot{\boldsymbol{\gamma}} + \stackrel{\mathbb{G}_{id}}{\nabla}_ {\dot{\textbf{E}}}\mathrm{d}_2\textbf{E} \dot{\boldsymbol{\gamma}}_{ref}) \\ \nonumber
&= P_{\mathcal{D}_\textbf{E}}(\frac{\mathrm{d}}{\mathrm{d}t}(\mathrm{d}_1\textbf{E} \dot{\boldsymbol{\gamma}}) + \stackrel{\mathbb{G}_{id}}{\nabla}_ {\dot{\textbf{E}}}\mathrm{d}_2\textbf{E} \dot{\boldsymbol{\gamma}}_{ref})
\end{align}
$\mathrm{d}_1 \textbf{E}(\boldsymbol{\gamma}, \boldsymbol{\gamma}_{ref})$ is a $(1,1)$ tensor which depends on two configurations at which error is defined. Therefore $\mathrm{d}_1 (\mathrm{d}_1 \textbf{E}): TM \times TM \to TM$ and $\mathrm{d}_2 (\mathrm{d}_1 \textbf{E}):TM \times TM \to TM$ are $(2,1)$ tensors. The first term in the bracket can be expressed in terms of these $(2,1)$ tensors as follows.
\begin{align}\label{120}
\frac{\mathrm{d}}{\mathrm{d}t} (\mathrm{d}_1\textbf{E} \dot{\boldsymbol{\gamma}}) &= (\mathrm{d}_1 (\mathrm{d}_1 \textbf{E}))(\dot{\boldsymbol{\gamma}}, \dot{\boldsymbol{\gamma}}) + (\mathrm{d}_2 (\mathrm{d}_1 \textbf{E}))(\dot{\boldsymbol{\gamma}}_{ref}, \dot{\boldsymbol{\gamma}})\\ \nonumber
&+ \mathrm{d}_1\textbf{E}(\stackrel{\mathcal{D}}{\nabla}_{\dot{\boldsymbol{\gamma}}} \dot{\boldsymbol{\gamma}})
\end{align}
In the last term in (<ref>) we consider the constrained covariant derivative to differentiate $\dot{\boldsymbol{\gamma}}$ as $\mathrm{d}_1\textbf{E}$ is a transport map from $T_{\boldsymbol{\gamma}}M $ to $T_\textbf{E} M$. From equations (<ref>) and (<ref>),
\begin{align}\label{121}
\stackrel{\mathcal{D}}{\nabla}_{\dot{\textbf{E}}}{\dot{\textbf{E}}} &= P_{\mathcal{D}_\textbf{E}}(
(\mathrm{d}_1 (\mathrm{d}_1 \textbf{E}))(\dot{\boldsymbol{\gamma}}, \dot{\boldsymbol{\gamma}}) + (\mathrm{d}_2 (\mathrm{d}_1 \textbf{E}))(\dot{\boldsymbol{\gamma}}_{ref}, \dot{\boldsymbol{\gamma}})\\ \nonumber
&+ \stackrel{\mathbb{G}_{id}}{\nabla}_ {\dot{\textbf{E}}}\mathrm{d}_2\textbf{E} \dot{\boldsymbol{\gamma}}_{ref}+ \mathrm{d}_1\textbf{E}(\stackrel{\mathcal{D}}{\nabla}_{\dot{\boldsymbol{\gamma}}}\dot{\boldsymbol{\gamma}})).
\end{align}
From (<ref>), (<ref>), and using $P_{\mathcal{D}_{\textbf{E}}}(\mathrm{d}_1\textbf{E} (u))=\mathrm{d}_1\textbf{E} (u)$,
\begin{align}\label{122}
\mathrm{d}_1\textbf{E} (u)&= \stackrel{\mathcal{D}}{\nabla}_{\dot{\textbf{E}}}\dot{\textbf{E}} - P_{\mathcal{D}(\textbf{E})}(
(\mathrm{d}_1 (\mathrm{d}_1 \textbf{E}))(\dot{\boldsymbol{\gamma}}, \dot{\boldsymbol{\gamma}}) \\ \nonumber
&+ (\mathrm{d}_2 (\mathrm{d}_1 \textbf{E}))(\dot{\boldsymbol{\gamma}}_{ref}, \dot{\boldsymbol{\gamma}})
+ \stackrel{\mathbb{G}_{id}}{\nabla}_ {\dot{\textbf{E}}} \mathrm{d}_2\textbf{E} \dot{\boldsymbol{\gamma}}_{ref}).
\end{align}
Substituting for $\stackrel{\mathcal{D}}{\nabla}_{\dot{\textbf{E}}}\dot{\textbf{E}}$ from (<ref>) in Lemma <ref> we get (<ref>). As the error dynamics is AGAS, therefore, $u$ in (<ref>) leads to AGAT of $\gamma_{ref}(t)$.
Remark 1: (<ref>) is the solution to an underdetermined system of $m-$equations. Recall that $u \in \mathbb{R}^m$ is the representation of available independent controls in the $m-$ dimensional Euclidean space.
Remark 2: The Nash embedding theorem ensures that a Riemannian manifold can be embedded isometrically in some Euclidean space. Therefore, Theorem <ref> reduces the problem of almost-global tracking of a given reference trajectory on a Riemannian manifold to finding a navigation function $\psi$ and a compatible configuration error map $E$ on the manifold. It is well known that a navigation function exists on a compact manifold (<cit.>, <cit.>). The compatible error map is obtained from the embedding as will be seen in examples.
Remark 3: The control law given in <cit.> uses a single transport map $\mathcal{T}(\gamma, \gamma_{ref})$ to transport $\dot{\gamma}_{ref}$ to $T_{\gamma} M$. Instead of this, we have two transport maps $\mathrm{d}_1 E $ and $\mathrm{d}_2 E$ to transport $\dot{\gamma}$ and $\dot{\gamma}_{ref}$ respectively to $T_{E(\gamma, \gamma_{ref})} M$ along the error trajectory $\dot{E}$. The tracking error function $\Psi: M \times M \to \mathbb{R}$ in <cit.> is similar to $\psi(E)$. However, instead of the velocity error along $\dot{\gamma}(t)$, we consider the velocity error $\dot{E}$ along $E(t)$. This is the essential difference in our approach to tracking a trajectory for an SMS.
Remark 4: In this remark we follow the procedure in <cit.> to obtain error dynamics and show that the theorem in <cit.> cannot be applied to conclude AGAT even if a navigation function is chosen as a potential function. Let us consider the tracking error function $\Psi: M \times M \to \mathbb{R}$ defined as $\Psi = \psi \circ E$ for a navigation function $\psi$ and a compatible error map $E$. The control law for local tracking of $\gamma_{ref}$ in <cit.> is
\begin{align*}
F(\gamma, \dot{\gamma}) &= -d_1 \Psi(\gamma, \gamma_{ref}) - F_{diss}(\dot{\gamma} -\mathcal{T}(\gamma, \gamma_{ref}) \dot{\gamma}_{ref})\\
&+{\mathbb{G}}^\flat(\stackrel{\mathbb{G}}{\nabla}_{\dot{\gamma}} \mathcal{T}(\gamma, \gamma_{ref}) \dot{\gamma}_{ref}(t)+ \frac{\mathrm{d}}{\mathrm{d}t} (\mathcal{T} \dot{\gamma}_{ref}) )
\end{align*}
where $\mathcal{T}(\gamma, \gamma_{ref}):T_{\gamma_{ref}}M \to T_{\gamma}M $ is a transport map compatible with the error function $\Psi$ (<cit.>). The velocity error is defined along $\gamma(t)$ as $v'_e \coloneqq \dot{\gamma} - \mathcal{T} \dot{\gamma}_{ref}$. The error dynamics in this case is given by
\begin{equation}\label{eq:23}
\dot{E} = \mathrm{d}_1E . v'_{e}
\end{equation}
\begin{equation} \label{eq:54}
\dot{v'}_e = \mathbb{G}^{\sharp} (-\mathrm{d}\psi (E) \mathrm{d}_1 E + F_{diss}(v'_{e})) - I(v'_e,\mathcal{T}\dot{\gamma}_{ref} ) -C(v'_{e})
\end{equation}
where $I(v'_e,\mathcal{T}\dot{\gamma}_{ref} ) =\Gamma_{ij}^k {v'_e}^i{(\mathcal{T}\dot{\gamma}_{ref})}^j$ and $C(v'_e) = \Gamma_{ij}^k {v'_e}^i {v'_e}^j$. (<ref>) results from the following equivalent compatibility condition
\begin{equation}\label{eq:93}
\mathrm{d}_2 E(\gamma, \gamma_{ref}) = - \mathrm{d}_1 E(\gamma, \gamma_{ref}) \mathcal{T}(\gamma, \gamma_{ref})
\end{equation}
and (<ref>) is given by the following identity
\begin{align}\label{eq:5}
\stackrel{\mathbb{G}}{\nabla}_{\dot{\gamma}(t)} v'_{e} &= \stackrel{\mathbb{G}}{\nabla}_{\dot{\gamma}} (\dot{\gamma}- \mathcal{T}(\gamma, \gamma_{ref}). \dot{\gamma}_{ref}(t))\\ \nonumber
&= \mathbb{G}^{\sharp} (F_{PD} + F_{FF}) - \mathbb{G}^{\sharp}F_{FF}\\ \nonumber
&= -\mathbb{G}^{\sharp} (d_1 \Psi (\gamma, \gamma_{ref})) +\mathbb{G}^{\sharp} (F_{diss}(v'_{e})). \nonumber
\end{align}
Linearizing (<ref>)-(<ref>) about an equilibrium state $(E*,0)$,
\begin{align*}
\dot{E}\\
\dot{v}'_e
\end{pmatrix}= \\
0 & \mathrm{d}_1E\\
-G^{ij} {\mathrm{d}^2 \psi}(E^*)\mathrm{d}_1E & G^{ij} \circ F_{diss}(E^*,0) - I^{\flat} (\mathcal{T} \dot{\gamma}_{ref})
\end{pmatrix} \begin{pmatrix}
E\\ v'_e\\
\end{pmatrix}.
\end{align*}
As $I^{\flat} (\mathcal{T} \dot{\gamma}_{ref})$ is a time dependent term, the flow of error dynamics around $(E^*,0)$ cannot be determined by the flow of $- \mathrm{d}\psi$. As a result, the lifting property of dissipative systems cannot be used to establish AGAS of the error dynamics for the PD+FF tracking control law in <cit.>.
§ AGAT FOR AN SMS ON A LIE GROUP
In this section, we utilize the idea of two transport maps originating from the configuration error map to study AGAT for an SMS on a compact Lie group. The configuration error map is defined using the group operation. Therefore, the problem of AGAT for an SMS on a compact Lie group is reduced to choosing a compatible navigation function.
§.§ Preliminaries
Let $G$ be a Lie group and let $\mathfrak{g}$ denote its Lie algebra. Let $\phi: G \times G \to G$ be the left group action in the first argument defined as $\phi(g,h) \coloneq L_{g} (h)= gh $ for all $g$, $h \in G$. The Lie bracket on $\mathfrak{g}$ is $[,]$. The adjoint map $ad_\xi : \mathfrak{g} \to \mathfrak{g}$ for $\xi \in \mathfrak{g}$ and defined as $ad_\xi \eta \coloneq [\xi,\eta]$. Let $\mathbb{I} :\mathfrak{g} \to \mathfrak{g}^*$ be an isomorphism on the Lie algebra to its dual and the inverse is denoted by $\mathbb{I}^\sharp: \mathfrak{g}^* \to \mathfrak{g}$. $\mathbb{I}$ induces a
left invariant metric on $G$ (see section 5.3 in <cit.>). This metric on $G$ is denoted by $\mathbb{G}_{\mathbb{I}}$ and defined as $\mathbb{G}_{\mathbb{I}}(g).(X_g,Y_g) \coloneq \langle \mathbb{I}(T_gL_{g^{-1}} (X_g)),T_gL_{g^{-1}} (Y_g)\rangle$ for all $g \in G$ and $X_g$, $Y_g \in T_g G$. The equations of motion for the SMS $(G, {\mathbb{I}},F)$ where $F \in \mathfrak{g}^*$ are given by
\begin{align}\label{dynliegrp}
\xi &= T_g L_{g^{-1}} \dot{g},\\ \nonumber
\dot{\xi} - I^\sharp ad^*_\xi I \xi &= \mathbb{I}^{\sharp} F
\end{align}
where $g(t)$ describes the system trajectory. $\xi(t)$ is called the body velocity of $g(t)$.
Given a differentiable parameterized curve $\gamma: \mathbb{R} \to G$ and a vector field $X$ along $\gamma(t)$ we have the following equality
\[ \stackrel{\mathbb{G}_{\mathbb{I}}}{\nabla}_{\dot{\gamma}} X = T_eL_{\gamma}( \frac{\mathrm{d}}{\mathrm{d}t}
(T_{\gamma}L_ {\gamma^{-1}} X(t) ) +\stackrel{\mathfrak{g}}{\nabla}_{\xi(t)} T_{\gamma}L_ {\gamma^{-1}}X(t))
\]
where $\stackrel{\mathfrak{g}}{\nabla}$ is the bilinear map defined as
\begin{equation}\label{eq:60}
\stackrel{\mathfrak{g}}{\nabla}_\eta \nu = \frac{1}{2} [\eta, \nu] - \frac{1}{2} \mathbb{I}^ \sharp ( ad^*_{\eta} \mathbb{I}\nu + ad^*_{\nu} \mathbb{I}\eta )
\end{equation}
for $\nu$, $\eta \in \mathfrak{g}$.
Let $\{ e_1, \dotsc , e_n\}$ be a basis for $\mathfrak{g}$. Let,
\begin{align*} X(\gamma(t)) &= T_eL_{\gamma} \big(\sum_{i=1}^{n} v_{X(\gamma(t))}^i(t) e_i \big) \\
&= \sum_{i=1}^{n} v_{X(\gamma(t))}^i(t) (e_i)_L (\gamma(t))
\end{align*}
where $v_X= T_{\gamma}L_ {\gamma^{-1}}X(\gamma(t))$ and $(e_i)_L(g) = T_eL_g e_i $ and therefore $(e_i)_L(g)$
is a basis for $T_gG$ for all $g \in G$.
Similarly let
\[ \dot{\gamma}(t) = T_eL_{\gamma} \big(\sum_{j=1}^{n} v_{\gamma}^j(t) e_j \big)
= \big(\sum_{j=1}^{n} v_{\gamma}^j(t) (e_j)_L (\gamma(t)) \big)
\]
where $v_{\gamma}(t)= T_{\gamma}L_ {\gamma^{-1}}\dot{\gamma}(t)$.
Using properties of affine connection we have,
\begin{align*}
\stackrel{\mathbb{G}_{\mathbb{I}}}{\nabla}_{\dot{\gamma}} X
&= \stackrel{\mathbb{G}_{\mathbb{I}}}{\nabla}_{ v_{\gamma}^j(t) (e_j)_L (\gamma(t))} v_{X}^i(t) (e_i)_L (\gamma(t)) \\
&= \frac{\mathrm{d}}{\mathrm{d}t}(v_X^i)(e_i)_L(\gamma(t))
+ v_X^k v_\gamma^j (\stackrel{\mathfrak{g}}{\nabla}_{(e_j)_L}(e_k)_L ) (\gamma(t)) \\
&= T_eL_\gamma \big( \frac{\mathrm{d}}{\mathrm{d}t}(v_X^i) \big)e_i + v_X^k v_\gamma^j \stackrel{\mathfrak{g}}{\nabla}_{(e_j)}(e_k) \\
&= T_eL_{\gamma}( \frac{\mathrm{d}}{\mathrm{d}t} (T_{\gamma}L_ {\gamma^{-1}} X(t))
+\stackrel{\mathfrak{g}}{\nabla}_{\xi(t)} T_{\gamma}L_ {\gamma^{-1}}X(t))
\end{align*}
The configuration error $E: G \times G \to G$ map is defined as
\begin{equation} \label{errmap}
E(g, h) = L_{h}g^{-1}.
\end{equation}
§.§ AGAT for an SMS on a compact Lie group
(AGAT for Lie group) Let $G$ be a compact Lie group and $\mathbb{I}: \mathfrak{g} \to \mathfrak{g}^*$ be an isomorphism on the Lie algebra. Consider the SMS on the Riemannian manifold $(G, {\mathbb{I}})$ given by (<ref>) and a smooth reference trajectory $g_r :\mathbb{R} \to G$ on the Lie group which has bounded velocity. Let $\psi : G \to \mathbb{R}$ be a navigation function compatible with the error map in (<ref>). Then there exists an open dense set $S \in G \times \mathfrak{g}$ such that AGAT of $g_r$ is achieved for all $(g(0), \xi(0)) \in S$ with $u = \mathbb{I}^\sharp(F)$ in (<ref>) given by the following equation
\begin{align}\label{thm2eqn}
u&= - g^{-1}_r \mathbb{G}_{\mathbb{I}}^\sharp (-K_p \mathrm{d}\psi (E) + F_{diss}\dot{E}) g + g^{-1} (\stackrel{\mathfrak{g}}{\nabla}_\eta \eta \\ \nonumber
&+ \frac{\mathrm{d}}{\mathrm{d}t}{E^{-1}}\mathrm{d}_2E(\dot{g}_r)) g - I^\sharp ad^*_\xi I \xi
\end{align}
where $\eta$ is the body velocity of the error trajectory and $K_p >0$.
$\mathrm{d}_1E(\dot{g})$ is a vector field along $E(t)$
The error trajectory is $E(g(t), g_r(t))$ where $E$ is defined in (<ref>). The error dynamics on $(G, {\mathbb{I}})$ is similar to (<ref>) with the appropriate Riemannian connection as follows
\begin{equation}\label{errdynlie}
\stackrel{\mathbb{G}_{\mathbb{I}}}{\nabla}_{\dot{E}} \dot{E} = \mathbb{G}_{\mathbb{I}}^\sharp (-K_p \mathrm{d}\psi (E) + F_{diss}\dot{E})
\end{equation}
As $(E, \psi)$ is a compatible pair, by Lemma <ref>, the error dynamics is AGAS about the minimum of $\psi$. The derivative of the error trajectory is given by (<ref>). Therefore, $\mathrm{d}_1E(\dot{g})= -g_r g^{-1} \dot{g}g^{-1} = T_g L_{g_r g^{-1}}R_{g^{-1}}\dot{g}$ and $\mathrm{d}_2 E(\dot{g}_r)= \dot{g}_r g^{-1} = T_{g_r}R_{g^{-1}} \dot{g}_r$ and,
\begin{align}\label{eq:55}
\stackrel{\mathbb{G}_{\mathbb{I}}}{\nabla}_{\dot{E}} \dot{E} &=
\stackrel{\mathbb{G}_{\mathbb{I}}}{\nabla}_{\dot{E} }(\mathrm{d}_1E(\dot{g}) + \mathrm{d}_2E(\dot{g}_r) ).\\ \nonumber
\end{align}
As $\mathrm{d}_1E(\dot{g})$ and $\mathrm{d}_2E(\dot{g}_r)$ are vector fields along $E(t)$, we use Lemma <ref> to expand $ \nabla_{\dot{E} }(\mathrm{d}_1E(\dot{g}))$ as
\begin{align*}
\stackrel{\mathbb{G}_{\mathbb{I}}}{\nabla}_{\dot{E}} \mathrm{d}_1E(\dot{g}) &= T_eL_{E}( \frac{\mathrm{d}}{\mathrm{d}t}
(T_{E}L_ {E^{-1}} \mathrm{d}_1E(\dot{g})) \\ &+\stackrel{\mathfrak{g}}{\nabla}_{T_{E}L_ {E^{-1}} \dot{E}} T_{E}L_ {E^{-1}}\mathrm{d}_1E(\dot{g}) )\\
&= E( \frac{\mathrm{d}}{\mathrm{d}t}
(E^{-1}\mathrm{d}_1E(\dot{g})) +\stackrel{\mathfrak{g}}{\nabla}_ { E^{-1}\dot{E}} {E^{-1}}\mathrm{d}_1E(\dot{g}) )\\
&= E( g \dot{\xi} g^{-1} + \stackrel{\mathfrak{g}}{\nabla}_ { E^{-1}\dot{E}} {E^{-1}}\mathrm{d}_1E(\dot{g}))
\end{align*}
Similarly, the second term in (<ref>) is
\begin{align*}
\stackrel{\mathbb{G}_{\mathbb{I}}}{\nabla}_{\dot{E} }\mathrm{d}_2E(\dot{g}_r) &= \frac{\mathrm{d}}{\mathrm{d}t}
(E^{-1}\mathrm{d}_2E(\dot{g}_r))+ \stackrel{\mathfrak{g}}{\nabla}_ { E^{-1}\dot{E}} {E^{-1}}\mathrm{d}_2E(\dot{g}_r)
\end{align*}
Hence from (<ref>),
\begin{align}\label{eq:51}
\stackrel{\mathbb{G}_{\mathbb{I}}}{\nabla}_{\dot{E}} \dot{E} &= E( g \dot{\xi} g^{-1} + \stackrel{\mathfrak{g}}{\nabla}_ { E^{-1}\dot{E}} {E^{-1}}\mathrm{d}_1E(\dot{g})\\ \nonumber
&+ \frac{\mathrm{d}}{\mathrm{d}t} (E^{-1}\mathrm{d}_2E(\dot{g}_r))+ \stackrel{\mathfrak{g}}{\nabla}_ { E^{-1}\dot{E}} {E^{-1}}\mathrm{d}_2E(\dot{g}_r))\\ \nonumber
&= E( g \dot{\xi} g^{-1} + \stackrel{\mathfrak{g}}{\nabla}_ { E^{-1}\dot{E}} E^{-1}\dot{E} + \frac{\mathrm{d}}{\mathrm{d}t}{E^{-1}}\mathrm{d}_2E(\dot{g}_r)) \nonumber
\end{align}
Let $u_1=\mathbb{G}_{\mathbb{I}}^\sharp (- \mathrm{d}\psi (E) + F_{diss}\dot{E})$. From (<ref>), $ \dot{\xi} = u + I^\sharp ad^*_\xi I \xi$ and $\eta= E^{-1} \dot{E}$, therefore, (<ref>) is
\begin{align}
u_1 &= E(-g(u+I^\sharp ad^*_\xi I \xi ) g^{-1} +\stackrel{\mathfrak{g}}{\nabla}_\eta \eta + \frac{\mathrm{d}}{\mathrm{d}t}({E^{-1}}\mathrm{d}_2E(\dot{g}_r))) \nonumber
\end{align}
which gives
\begin{align}\label{eq:53}
u &= g^{-1} (- E^{-1}u_1+ \stackrel{\mathfrak{g}}{\nabla}_\eta \eta + \frac{\mathrm{d}}{\mathrm{d}t}{E^{-1}}\mathrm{d}_2E(\dot{g}_r)) g - I^\sharp ad^*_\xi I \xi \\ \nonumber
&= - g^{-1}_r u_1 g + g^{-1} (\stackrel{\mathfrak{g}}{\nabla}_\eta \eta + \frac{\mathrm{d}}{\mathrm{d}t}{E^{-1}}\mathrm{d}_2E(\dot{g}_r)) g - I^\sharp ad^*_\xi I \xi \nonumber
\end{align}
A comparison of tracking results
Comparison of control effort for initial conditions (i) and (ii)
Remark 1:
The control law in <cit.> for tracking the reference trajectory $t \to g_r(t) \in G$ by a fully actuated SMS given by $(G, \mathbb{I}, \mathbb{R}^n)$ is
\begin{align} \label{132}
&F(t,g,\xi) = - T_eL_{g^{-1}_{r} g}^* (\mathrm{d} \Psi (g^{-1}_{r} g)) +F_{diss} (\xi - Ad_{g^{-1} {g^{-1}_{r}}}v_r ) \\ \nonumber
&+ \mathbb{I}^\sharp (\stackrel{\mathfrak{g}}{\nabla}_{\xi} Ad_{g^{-1} {g_r}}v_r + [Ad_{g^{-1}g_r} v_r, \xi ] + Ad_{g^{-1}g_r} \dot{v}_{r} )
\end{align}
where $v_r(t)$ is the body velocity of the reference trajectory defined by $v_r(t) = T_{g_r} L_{g_r^{-1}} \dot{g}_{r}(t)$ and $\Psi : G \times G \to \mathbb{R}$ is a tracking error function. It is shown in <cit.> that by choosing $\Psi = \psi \circ E$, where $E$ is defined as in (<ref>) and $\psi$ is a navigation function, the control law in (<ref>) achieves AGAT of $g_r (t)$. On comparing (<ref>) and (<ref>) with $\Psi = \psi \circ E$, it is observed that in (<ref>) the acceleration of the error trajectory on the Lie algebra given by $\stackrel{\mathfrak{g}}{\nabla}_\eta \eta$ appears as an additional term.
In order to observe the effect of this term on the controlled trajectory we compare the tracking results for an externally actuated rigid body obtained by the existing the control law with the proposed control law. The rigid body is an SMS on $SO(3)$ and $\psi(E)= tr(P(I_3-E))$, where $P$ is a symmetric positive definite matrix is chosen as the compatible navigation function which has a unique minimum at $I_3$. We consider a rigid body with an inertia matrix given by $\mathbb{I} =\begin{pmatrix} 4&1 &1\\ 1 5 & 0.2& 2\\ 1& 2 & 6.3\\ \end{pmatrix}$ and initial conditions $R(0) = \begin{pmatrix} 0.36& 0.48 &-0.8 \\ -0.8 &0.6& 0\\ 0.48 & 0.64 & 0.60\\ \end{pmatrix}$ and $\mathbb{I}\Omega = \begin{pmatrix} 1 &2.2 &5.1\\ \end{pmatrix}$. and the reference is generated by a dummy rigid body with inertia matrix $\mathbb{I}_d=\begin{pmatrix}
1 & 0& 0\\ 0 &1.2& 0\\ 0& 0 &2 \\
\end{pmatrix}$, initial conditions, $R_d(0)=I_3 $ and $\mathbb{I}_d{\Omega_d} = \begin{pmatrix}-0.8& -0.3 &-0.5\end{pmatrix}$.
In the Morse function, $P = diag(4 ,4.5 ,4.2)$ and in the intermediate control $F_{diss} = -diag( 3.5 ,3.5, 3.7)$. In figures <ref> and <ref>, the reference and two controlled trajectories obtained by the existing and proposed control law are plotted together.
In order to compare the control effort, we compute the $2$ norm of $\breve{u}(t) \in \mathfrak{so}(3)$ for (i) the rigid body with the above initial conditions and (ii) with initial conditions given by $R(0) = \begin{pmatrix} 0.7071& 0.7071 & 0 \\ -0.7071 & 0.7071 & 0\\ 0 & 0 & 1\\ \end{pmatrix}$ and $\mathbb{I}\Omega = \begin{pmatrix} 1 &2.2 &5.1\\ \end{pmatrix}$ and compare it with the $2$ norm of existing control in figures <ref> and <ref> respectively.
§ SIMULATION RESULTS
§.§ AGAT on $S^2$
[xyplane] (-2*,-2*) rectangle (2.2*,2.8*);
[ball color=white] (0,0) circle ();
(0,0) circle ();
(O) at (0,0);
[mark coordinate](N) at (0,)̋;
[mark coordinate](S) at (0,-)̋;
[pzplane] (,0) coordinate (PE);
[xzplane] (,0) coordinate (XE);
(PE) ++(0,-)̋ coordinate (Paux);
second line=(S)–(Paux));
[xyplane,<->] (1.8*,0) node[below] $y$ – (0,0) – (0,2.4*)
node[right] $x$;
[->] (0,-)̋ – (0,1.6*) node[above] $z$;
A 2-sphere in $\mathbb{R}^3$
§.§.§ Navigation function
We consider the restriction of the height function in $\mathbb{R}^3$ to $S^2$ given by $\boldsymbol{\psi}(x,y,z)=z $ for $(x,y,z)\in \mathbb{R}^3$. It is a navigation function on $S^2$ with $(0,0,-1)^T$ as the unique minimum and $(0,0,1)^T$ as maximum. It can be verified that the $Hess \boldsymbol{\psi}$ is non-degenerate at both extremal points. The projection map $P_{\mathcal{D}_q}:\mathbb{R}^3 \to T_qS^2$ is defined as
\begin{equation}\label{125}
P_{\mathcal{D}_q}v= - \{\hat{q}\}^2v
\end{equation}
§.§.§ Configuration Error map
The configuration error map $\textbf{E}: S^2 \times S^2 \to S^2$ is chosen as
\begin{equation}\label{126}
\textbf{E}(q_1,q_2)= \begin{pmatrix}
\sqrt{1- {\langle q_1,q_2\rangle_{\mathbb{R}^3}}^2} & 0 & {-\langle q_1,q_2\rangle_{\mathbb{R}^3}}^2
\end{pmatrix}^T
\end{equation}
for $q_1$, $q_2 \in S^2$. As $E$ is symmetric, therefore, $\boldsymbol{\psi} \circ \textbf{E}$ is also symmetric. $\textbf{E}(q,q)= \begin{pmatrix}
0 & 0 & -1
\end{pmatrix}^T$ hence, $E(q,q)$ is the minimum of $\boldsymbol{\psi}$. Therefore $(\boldsymbol{\psi},\textbf{E})$ is a compatible pair according to Definition 1.
The $(1,1)$ tensors are
\begin{align*}
\mathrm{d}_1\textbf{E}(q_1,q_2)&= \begin{pmatrix}
\frac{\langle q_1,q_2\rangle_{\mathbb{R}^3}q_2^T}{ \sqrt{1- {\langle q_1,q_2\rangle_{\mathbb{R}^3}}}} & 0_{1 \times 3} & -q_2^T
\end{pmatrix}^T \quad \text{and,}\\
\mathrm{d}_2\textbf{E}(q_1,q_2)&= \begin{pmatrix}
\frac{\langle q_1,q_2\rangle_{\mathbb{R}^3}q_1^T}{ \sqrt{1- {\langle q_1,q_2\rangle_{\mathbb{R}^3}}^2}} & 0_{1 \times 3} & -q_1^T
\end{pmatrix}^T.
\end{align*}
The $(2,1)$ tensors are $3 \times 3$ arrays given by
\begin{align*}
\mathrm{d}_1\mathrm{d}_1\textbf{E}(q_1,q_2)&= \begin{pmatrix}
\frac{q_2q_2^T}{ {\sqrt{1- {\langle q_1,q_2\rangle_{\mathbb{R}^3}}^2}^3}} & 0_{3\times 3} & 0_{3 \times 3}
\end{pmatrix}^T,\\
\mathrm{d}_2\mathrm{d}_1\textbf{E}(q_1,q_2)&= \begin{pmatrix}
\frac{q_1q_2^T -{\langle q_1,q_2\rangle}_{\mathbb{R}^3}^3 I_3 + {\langle q_1,q_2\rangle}_{\mathbb{R}^3} I_3}{ {\sqrt{1- {\langle q_1,q_2\rangle_{\mathbb{R}^3}}^2}^3}} & 0_{3\times 3} & -I_3
\end{pmatrix}^T,\\
\mathrm{d}_2\mathrm{d}_2\textbf{E}(q_1,q_2)&= \begin{pmatrix}
\frac{q_1q_1^T}{ {\sqrt{1- {\langle q_1,q_2\rangle_{\mathbb{R}^3}}^2}^3}} & 0_{3\times 3} & 0_{3 \times 3}
\end{pmatrix}^T \quad \text{and,}\\
\mathrm{d}_1\mathrm{d}_2\textbf{E}(q_1,q_2)&= \begin{pmatrix}
\frac{q_2q_1^T -{\langle q_1,q_2\rangle}_{\mathbb{R}^3}^3 I_3 + {\langle q_1,q_2\rangle}_{\mathbb{R}^3} I_3}{ {\sqrt{1- {\langle q_1,q_2\rangle_{\mathbb{R}^3}}^2}^3}} & 0_{3\times 3} & -I_3
\end{pmatrix}^T.
\end{align*}
§.§.§ AGAT results
x coordinate
y coordinate
z coordinate
Tracking results for first set of initial conditions
x coordinate
y coordinate
z coordinate
Tracking results for second set of initial conditions
3D visualization of tracking problem for first set of initial conditions
3D visualization of tracking problem for second set of initial conditions
The constrained affine connection on $S^2$ is given by
\[ \stackrel{\mathcal{D}}{\nabla} _{\dot{\boldsymbol{\gamma}}} \dot{\boldsymbol{\gamma}} = \ddot{\boldsymbol{\gamma}}(t) + || \dot{\boldsymbol{\gamma}}||_2^2\boldsymbol{\gamma}.
\]
Therefore, from (<ref>), the system trajectory $\boldsymbol{\gamma}(t)$ for any spherical pendulum satisfies the following equation
\begin{equation}\label{130}
\ddot{\boldsymbol{\gamma}}(t) + || \dot{\boldsymbol{\gamma}}||_2^2\boldsymbol{\gamma} = P_{\mathcal{D}_{\boldsymbol{\gamma}}}(u).
\end{equation}
The reference trajectory is generated by a dummy spherical pendulum with the following initial conditions $(\boldsymbol{\gamma}_{ref}(0),\dot{\boldsymbol{\gamma}}_{ref}(0))=\begin{pmatrix}
\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} & 3 & 0 & -3
\end{pmatrix}^T$ and $u = \begin{pmatrix}
1 & 2 & -1
\end{pmatrix}^T$.
The initial conditions for the system trajectory is given by $
-1 & 0 & 0& 0 & 1 & 0
\end{pmatrix}^T
$. Theorem 1 is applied to compute the tracking control given in (<ref>) with $F_{diss}=-4$, $K_p = 3.7$. The system trajectory is generated using ODE45 solver of MATLAB. The reference (in blue) and system trajectory (in red) are compared in all $3$ coordinates in figures <ref>, <ref> and <ref>. We consider another set of initial conditions as follows.
\frac{1}{\sqrt(3)} & \frac{1}{\sqrt(3)} & \frac{1}{\sqrt(3)}& 3 & 0 & -3
\end{pmatrix}^T$ and $u = \begin{pmatrix}
1 & 2 & 1
\end{pmatrix}^T$ for the dummy spherical pendulum. The initial conditions for the system trajectory are $
0 & -1 & 0& 1 & 2 & 2
\end{pmatrix}^T.$ Theorem 1 is applied to compute the tracking control with $F_{diss} =-5.7$ and $K_p= 4$. The reference (in blue) and system trajectory (in red) are compared in all $3$ coordinates in figures <ref>, <ref> and <ref>.
§.§ AGAT on Lissajous curve
The space curve L
A Lissajous curve in $3$ dimensions (shown in figure <ref>) is a $1-$ dimensional smooth, connected, compact manifold in $\mathbb{R}^3$. It is denoted by $L$ and defined as $L = h^{-1}\{\begin{pmatrix}
0 & 0
\end{pmatrix}^T\}$
where $h : \mathbb{R}^3 \to \mathbb{R}^2$ is given by $h(x,y,z) = \begin{pmatrix}
x^2+y^2+z^2-2xyz -1 & 4x^2y -2xz-y
\end{pmatrix}^T$.Therefore $T_\textbf{x} L= \{v \in \mathbb{R}^3: \mathrm{D}h(\textbf{x}) v=0\}$ for $\textbf{x} \in L$.
§.§ Navigation function:
We consider $\boldsymbol{\psi}: L \to \mathbb{R}$ given as $\boldsymbol{\psi}(x,y,z) = x$. It is observed that $\boldsymbol{\psi}$ has a unique minimum at $\begin{pmatrix}
-1 & 0 & 0
\end{pmatrix}^T$ and a unique maximum at $\begin{pmatrix}
1 & 0 & 0
\end{pmatrix}^T$. Using parameterizations $\psi_1(t)= \begin{pmatrix}
\cos(t) & \sin(2t) & \sin(3t)
\end{pmatrix}^T, t \in (-\pi,\pi)$ around $\begin{pmatrix} -1 & 0& 0 \end{pmatrix}^T$ and the parameterization $\psi_2(t)= \begin{pmatrix}
\cos(t) & \sin(2t) & -\sin(3t)
\end{pmatrix}^T, t \in (-\pi,\pi)$ around $\begin{pmatrix} 1 &0 &0 \end{pmatrix}^T$, it is verified that $Hess ({\psi_i(t)})|_{t=0} \neq 0$, $i=1,2$. Therefore $\psi$ is a navigation function.
§.§.§ Configuration error map
The configuration error map $\textbf{E}:L \times L \to L$ is chosen as
\begin{equation}\label{confige}
\textbf{E}(q_1,q_2)= \begin{pmatrix}
-\frac{\langle q_1,q_2\rangle}{|q_1||q_2|}_{\mathbb{R}^3} \\
-2\frac{|q_1 \times q_2|{\langle q_1,q_2\rangle}_{\mathbb{R}^3}}{|q_1|^2|q_2|^2} \\ 4\frac{|q_1 \times q_2|{{\langle q_1,q_2\rangle}_{\mathbb{R}^3}}^2}{|q_1|^3|q_2|^3 }-\frac{|q_1 \times q_2|}{|q_1||q_2| }
\end{pmatrix}.
\end{equation}
It is observed that $\textbf{E}(q_1,q_2) \in L$ and that $(\boldsymbol{\psi}\circ \textbf{E})(q_1,q_2)$ is symmetric. As $\textbf{E}(q,q)= \begin{pmatrix}
-1 & 0 & 0
\end{pmatrix}$ hence, $\textbf{E}(q,q)$ is the minimum of $\boldsymbol{\psi}$. Therefore, $(\boldsymbol{\psi}, \textbf{E})$ is a compatible pair according to Definition 1. We define $\beta = \frac{|q_1 \times q_2|}{|q_1||q_2|}$ and $\textbf{E} = \begin{pmatrix}
E_1& E_2&E_3
\end{pmatrix}^T$. Observe that $\beta = \sqrt{1- E_1^2}$. Therefore, $\mathrm{d}_1\textbf{E}(q_1,q_2) =\begin{pmatrix} \frac{\partial E_1 }{\partial q_1} & \frac{\partial E_2 }{\partial q_1}& \frac{\partial E_3 }{\partial q_1} \end{pmatrix}^T$ and, $\frac{\partial \beta }{\partial q_1} = -\frac{E_1}{\beta}\frac{\partial E_1}{ \partial q_1}$, $\frac{\partial \beta }{\partial q_2} = -\frac{E_1}{\beta}\frac{\partial E_1}{ \partial q_2}$.The $(1,1)$ tensors are $3 \times 3$ matrices given as
\begin{align}\label{onet}
\mathrm{d}_1\textbf{E}(q_1,q_2)&= \begin{pmatrix}
-E_1\frac{q_1^T}{2|q_1|^2}- \frac{q_2^T}{|q_1||q_2|} \\
2E_1 \frac{\partial \beta }{\partial q_1}+2\beta \frac{\partial E_1 }{\partial q_1} \\
(4E_1^2 - 1)\frac{\partial \beta }{\partial q_1} + 8 \beta E_1 \frac{\partial E_1 }{\partial q_1} \end{pmatrix} \quad \text{and},\\ \nonumber
\mathrm{d}_2\textbf{E}(q_1,q_2)&=\begin{pmatrix}
-E_1\frac{q_2^T}{2|q_2|^2}- \frac{q_1^T}{|q_1||q_2|} \\
2E_1 \frac{\partial \beta }{\partial q_2}+2\beta \frac{\partial E_1 }{\partial q_2} \\
(4E_1^2 - 1)\frac{\partial \beta }{\partial q_2} + 8 \beta E_1 \frac{\partial E_1 }{\partial q_2}
\end{pmatrix}.
\end{align}
The $(2,1)$ tensors are $3 \times 3 \times 3$ arrays given as
\begin{align}\label{twot}
&\mathrm{d}_1(\mathrm{d}_1\textbf{E})(q_1,q_2)=\\ \nonumber
\frac{q_1 q_2^T}{2|q_1|^3|q_2|}- I_3\frac{E_1}{2|q_1|^2} + E_1\frac{q_1 q_1^T}{2|q_1|^4} -\{\frac{\partial E_1 }{\partial q_1}\}^T\frac{q_1^T}{2|q_1|^2} \\ \nonumber
4 sym\big (\{\frac{\partial E_1}{\partial q_1}\}^T \frac{\partial \beta }{\partial q_1} \big)+2E_1 \frac{\partial^2 \beta }{\partial q_1^2}+2\beta \frac{\partial^2 E_1 }{\partial q_1^2}\\ \nonumber
16E_1 sym\big ({\frac{\partial E_1}{\partial q_1}}^T \frac{\partial \beta }{\partial q_1} \big)+\alpha \frac{\partial^2 \beta }{\partial q_1^2}+8\beta E_1 \frac{\partial^2 E_1 }{\partial q_1^2}+ 8 \beta {\frac{\partial E_1}{\partial q_1}}^T \frac{\partial E_1}{\partial q_1}
\end{pmatrix}\\ \nonumber
&\mathrm{d}_2(\mathrm{d}_1\textbf{E})(q_1,q_2)=\\ \nonumber
-\frac{I_3}{|q_1||q_2|} + \frac{q_2 q_2^T}{2 |q_2|^3|q_1|} - \frac{q_1^T}{2|q_1|^2}\frac{\partial E_1 }{\partial q_2}\\
2{\frac{\partial E_1}{\partial q_2}}^T \frac{\partial \beta }{\partial q_1}
+2{\frac{\partial \beta}{\partial q_2}}^T \frac{\partial E_1 }{\partial q_1}
+2E_1 \frac{\partial^2 \beta }{\partial q_2 \partial q_1}+2\beta \frac{\partial^2 E_1 }{\partial q_2 \partial q_1}\\
8E_1\big({\frac{\partial E_1}{\partial q_2}}^T \frac{\partial \beta }{\partial q_1}
+{\frac{\partial \beta}{\partial q_2}}^T \frac{\partial E_1 }{\partial q_1}\big)
+ \alpha \frac{\partial^2 \beta }{\partial q_2\partial q_1}+8\beta\big( E_1\frac{\partial^2 E_1 }{\partial q_2\partial q_1}+ {\frac{\partial E_1}{\partial q_2}}^T \frac{\partial E_1}{\partial q_1}\big)
\end{pmatrix} \nonumber
\end{align}
The tensors $\mathrm{d}_2(\mathrm{d}_2\textbf{E})$ and $\mathrm{d}_1(\mathrm{d}_2\textbf{E})$ are obtained similarly as $\textbf{E}(q_1,q_2)$ is symmetric in $q_1$ and $q_2$.
§.§.§ AGAT results
We consider a particle moving on the curve $L$. The equations of motion of the particle are given by the geodesic $\boldsymbol{\gamma}(t)$ on $L$ for $t \in \mathbb{R}^+$. Therefore,
\begin{equation}\label{geod1}
\ddot{\boldsymbol{\gamma}}(t)= \lambda_1 \mathrm{d}h(1) + \lambda_2 \mathrm{d}h(2)
\end{equation}
as $\ddot{\boldsymbol{\gamma}}(t) \in \mathcal{D}^\perp_\gamma$ for all $t$. Since $\gamma(t) \in L$ implies $h \circ \boldsymbol{\gamma}(t)=0$, therefore,
\begin{equation}\label{geod2}
\dot{\boldsymbol{\gamma}}^T \mathrm{D}^2h \dot{\boldsymbol{\gamma}} + \mathrm{D}h \ddot{\boldsymbol{\gamma}}=0.
\end{equation}
From (<ref>) and (<ref>) we obtain $\lambda_1$, $\lambda_2$ and hence the geodesic curve $\boldsymbol{\gamma}$. Therefore the affine connection on $L$ is given as
\begin{equation}\label{geodL}
\stackrel{\mathcal{D}}{\nabla}_{\dot{\boldsymbol{\gamma}}}\dot{\boldsymbol{\gamma}} =\ddot{\boldsymbol{\gamma}}(t)- \lambda_1 \mathrm{d}h(1) - \lambda_2 \mathrm{d}h(2)
\end{equation}
We consider the reference trajectory $\boldsymbol{\gamma_{ref}}(t)=\begin{pmatrix} \cos(\sin(t))& \sin(2\sin(t)) &\sin(3\sin(t)) \end{pmatrix}^T, t \geq 0$ with $(\boldsymbol{\gamma_{ref}}(0),\dot{\boldsymbol{\gamma}}_{ref}(0))= \begin{pmatrix}
1& 0 &0 & 0 & 2 & 3
\end{pmatrix}^T$. The initial conditions for system trajectory are $ (\boldsymbol{\gamma}(0),\dot{\boldsymbol{\gamma}}(0))= \begin{pmatrix}
-0.82& 0.9386 & 0.9672 & -1.197 & 1.1346 & 2.0798
\end{pmatrix}^T$. Theorem 1 is applied to compute the tracking control in (<ref>) with $F_{diss}=-1.2$, $K_p = 5.4$. The system trajectory is generated using ODE45 solver of MATLAB. The reference (in blue) and system trajectory (in red) are compared in all $3$ coordinates in figures <ref>, <ref>, <ref>. Another simulation is performed with the reference trajectory $\boldsymbol{\gamma_{ref}}(t)=\begin{pmatrix} \cos(t)& \sin(2t) &\sin(3t) \end{pmatrix}^T, t \geq 0$ with $(\boldsymbol{\gamma_{ref}}(0),\dot{\boldsymbol{\gamma}}_{ref}(0))= \begin{pmatrix}
1& 0 &0 & 0 & 2 & 3
\end{pmatrix}^T$. The initial conditions for system trajectory are $ (\boldsymbol{\gamma}(0),\dot{\boldsymbol{\gamma}}(0))= \begin{pmatrix}
0.54 & 0.9093 & 0.1411 & -0.8415 & -0.8323 & -2.97
\end{pmatrix}^T$. Theorem 1 is applied to compute the tracking control in (<ref>) with $F_{diss}=-1.6$, $K_p = 5.3$.The reference (in blue) and system trajectory (in red) are compared in all $3$ coordinates in figures <ref>, <ref>, <ref>.
x coordinate
y coordinate
z coordinate
Tracking results for first set of initial conditions
x coordinate
y coordinate
z coordinate
Tracking results for second set of initial conditions
|
1511.00667
|
$^1$ Max-Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1, 85748 Garching, Germany
$^2$ Space Telescope Science Institute, 3700 San Martin Drive, Baltimore MD 21218
$^3$ University of Colorado at Boulder, UCB 391, Boulder, CO 80309
$^4$ Australian National University, Mount Stromlo Observatory, Cotter Road, Weston Creek, ACT 2611, Australia
$^5$ Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH
$^6$ Department of Physics, University of Warwick, Coventry CV4 7AL
$^7$ National Space Science & Technology Center, 320 Sparkman Drive, Huntsville AL 35805
$^8$ Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital-263002, Uttarakhand, India
$^9$ University of California Davis, 1 Shields Avenue, Davis, CA, 95616
$^{10}$ Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599
We present our imaging and spectroscopic observations of the host galaxies of two dark long bursts with anomalously high metallicities, LGRB 051022 and LGRB 020819B, which in conjunction with another LGRB event with an optical afterglow <cit.> comprise the three LGRBs with high metallicity host galaxies in the <cit.> sample. In <cit.>, we showed that LGRBs exhibit a strong and apparently intrinsic preference for low metallicity environments (12+log(O/H) $<$ 8.4 in the KK04 scale) in spite of these three cases with abundances of about solar and above. Not only do these exceptions not share the typical low metallicities of LGRB hosts, they are consistent with the general star-forming galaxy population of comparable brightness & redshift. This result is intrinsically surprising: even among a preselected sample of the high metallicity LGRBs, were the metal aversion to remain in effect for these objects, we would expect the metallicity these LGRBs to still be lower than the typical metallicity for the galaxies at that luminosity and redshift (i.e., either a outlier of said population, or among the lowest galaxies available within it). Therefore we deduce that it is likely possible to form an LGRB in a high metallicity environment although with greater rarity.
From this we conclude that there are three possible explanations for the presence of the LGRBs observed in high metallicity hosts as seen to date: (1) LGRBs do not occur in high metallicity environments and those seen in high metallicity hosts are in fact occurring in low metallicity environments that have become associated with otherwise high metallicity hosts but remain unenriched. (2) The LGRB formation mechanism while preferring low metallicity environments does not strictly require it resulting in a gradual decline in burst formation with increasing metallicity. (3) The typical low metallicity LGRBs and the few high metallicity cases are the result of physically different burst formation pathways with only the former affected by the metallicity and the later occurring much more infrequently.
§ INTRODUCTION
Long soft gamma-ray bursts (LGRBs) are frequently found in a particular type of host galaxy: blue irregulars <cit.>. LGRBs have shown a strong preference for occurring in star-forming galaxies <cit.>, which often exhibit bright emission lines <cit.> indicative of substantial populations of young, massive stars. LGRBs have also frequently been associated with broad-lined Type Ic (Ic-bl) supernovae (SNe) <cit.> so named because their spectral lines show broadening from high-velocity ($\sim$15,000 km s$^{-1}$) ejecta.
<cit.> performed a detailed study of the LGRB host galaxy population, using the GOODS core-collapse supernovae sample as a comparative group, and shows a surprising bias against hosts being grand design spiral galaxies. Only one out of 42 LGRB host galaxies was a grand design spiral (as compared to the GOODS supernovae sample where about half the host galaxies were spirals). If one constrains the LGRB host population to a redshift of 1.2 or less so as to match the redshift distribution of the supernovae sample this drops to one out of 18, still a rather surprising result. The remainder of the GRB host population are composed of generally faint, irregular galaxies, whereas by contrast, half of all the core collapse supernovae hosts were spirals. Additionally, the <cit.> sample showed a strong preference for LGRB's occurring in the brightest, and hence likely the most star-forming regions of their hosts.
One of the conclusions of <cit.> was that LGRB formation requires a low metallicity progenitor and the bias toward irregular galaxies is a result of their low metallicity as expected by the mass-metallicity relation. This conclusion was supported by subsequent works of Stanek et al.(2006), where a comparison of LGRB hosts and galaxies in the Sloan sample (of similar magnitude) shows that the very nearest hosts have low metallicity. Most convincingly, <cit.> compared the host metallicity luminosity relation of LGRBs with known Ic supernovae counterparts to nearby Ic Supernovae without LGRB associations (see Figure 5 in ). They found a dramatic difference in metallicity between the two samples, even when host luminosity is accounted for, showing a profound metallicity avoidance for the LGRBs. <cit.> compared the mass distribution of LGRB hosts to the general star-forming galaxy population as a function of redshift, finding that the populations remain dissimilar out to z $\sim$ 1 with an upper limit on the stellar mass of LGRB hosts evolving with redshift. This favors a smooth decrease in the LGRB formation rate with increasing metallicity above 12+log(O/H) = 8.7 over an extremely low metallicity cut-off thus limiting LGRB hosts to low mass spirals and dwarfs at low redshift and suggesting that LGRBs remain metallicity biased tracers of star-formation out to intermediate redshifts. On the other hand <cit.> argue that LGRB hosts lie on the same mass-metallicity relation as regular galaxies. <cit.> have used this claim to argue that because the host of GRB 020127 is unusually bright it must also be metal-rich. <cit.> however show the existence of low metallicity outliers on the luminosity-metallicity relation, and argue against the assignment of metallicities (to individual galaxies) based only on their luminosities. In particular they highlight the morphologically similarities of their bright outliers to the brighter hosts in the <cit.> sample. <cit.> claimed that the observed low metallicity LGRB bias is not an intrinsic preference but only a byproduct of the higher star-formation rates observed in LGRB hosts and the <cit.> anti-correlation between mass, star-formation rate, and metallicity.
In <cit.>, we showed that LGRBs exhibit a strong and apparently intrinsic preference for environments with metallicities below 12+log(O/H) $<$ 8.4 on the KK04 <cit.> scale. However we note therein that some exceptions do exist to this trend — three of the 14 LGRB in the sample possess abundances of about solar and above. While the majority of the LGRB population is constrained to low metallicities of about a third solar and below these exceptions probably show that is it still possible to form an LGRB in a high metallicity environment although with greater rarity. For us to use LGRBs to trace the star-formation of the Universe (even at low metallicities), we must understand the conditions required for their production and thus selection effects that take place even before we see the LGRB. The implications of these high metallicity bursts are important not only for understanding the formation of LGRBs but also for any hope of being ever able to use them as cosmological probes.
Here, we specifically address these exceptions by beginning with a more detailed examination of the individual cases, LGRBs 051022, 020819B, and 050826, followed by an analysis of these objects as a population. Initial metallicity results for the LGRB 051022 and 020819B host galaxies were published in our conference proceedings <cit.> and a short letter <cit.> respectively and we perform a more detailed metallicity analysis (particularly with regard to error), study additional object parameters, and give greater consideration of the scientific impact of these results herein. We will also briefly discuss LGRB 050826, particularly with regard to its host properties. Both LGRB 051022 and 020819B are among the brightest LGRB host galaxies yet seen (L* or above in the Schechter luminosity function), and via the luminosity-mass-metallicity relationship, such galaxies would be expected to possess a comparatively higher metallicity than that yet seen in long burst hosts. Both are at a low enough redshift that the emission lines necessary to study their metallicity are visible in either the optical or near-IR. Also both lack an optical transient or SNe contribution, classifying them as
dark bursts (See <cit.> for more detail on dark bursts and their criteria for classification).
<cit.> performed a detailed study of 23 dark bursts and identified their host galaxies to be more massive, more star-forming, and more dust obscured than LGRB hosts with optical afterglows. While this likely implies that dark bursts also are more metal rich than their optically visible counterparts both <cit.> and <cit.> conclude that, even after accounting for the contribution of dark bursts, a strong preference for LGRB to occur in low metallicity host galaxies remains. Furthermore <cit.> suggest that per unit underlying star-formation the outer parts of submillimeter galaxies overproduce LGRBs while the inner parts underproduce them consistent with a metallicity or IMF gradient therein.
Our third object however, LGRB 050826, did have an optical transient establishing that such bursts do also exist in high metallicity environments. This furthers the argument that dark bursts are just normal LGRBs with heavy dust extinction attenuating the viable afterglow. LGRB 050826 was also the first case where the a high metallicity LGRB was found without without previous suspicion that this was likely the case before obtaining the metallicity measurement. After examining these cases individually and as a population, we will propose and discuss explanations for the existence of high metallicity LGRBs in light of the now established intrinsic preference of LGRBs for low metallicity environments. Such an explanation must reconcile a formation mechanism that is biased against a metal rich environment with one that seems to still permit some formation against that bias.
§ OBSERVATIONS
§.§ GRB 051022
§.§.§ Burst Detection and Localization
LGRB 051022 was initially detected by all three instruments on the High Energy Transient Explorer 2 (HETE-2) satellite, the FREnch GAmma TElescope (Fregate), Wide-field X-ray Monitor (WXM), and the Soft X-ray Camera (SXC) ( GCN 4131). The burst was sufficiently bright in soft X-rays that the SXC position was determined independently of the WXM location ( GCN 4137). HETE-2 observations indicated a burst duration in excess of 2 to 4 minutes securely identifying this as a long gamma ray burst ( GCN 4137). Within the HETE-2 error circles a fading afterglow was detected in the X-ray, mm, and radio bands. In spite of this, no variable optical source was found at the afterglow position (, GCN 4134, GCN 4143) thus placing this LGRB in a special class of bursts called the “dark” bursts. The afterglow position identifies a host galaxy at a redshift of z = 0.8 ( GCN 4156). See Table <ref> for details of prompt and afterglow localizations.
$\Delta$T (days) RA Dec Error Instrument Band Reference
0 23$^{h}$55$^{m}$53$^{s}$ +19$^{\circ}$37$^{m}$43$^{s}$ 14' HETE-2 WXM 2-25 keV GCN 4131
0 23$^{h}$56$^{m}$00$^{s}$ +19$^{\circ}$35$^{m}$51$^{s}$ 2.5' HETE-2 SXC 0.5-10 keV GCN 4131; GCN 4137
0.15 23$^{h}$56$^{m}$04.1$^{s}$ +19$^{\circ}$36$^{m}$25.1$^{s}$ 4" Swift XRT X-ray <cit.> GCN 4141
1.35 23$^{h}$56$^{m}$04.1$^{s}$ +19$^{\circ}$36$^{m}$25.4$^{s}$ 0.5" Plateau de Bure Interferometer 3 mm <cit.> GCN 4157, <cit.>
1.54 23$^{h}$56$^{m}$04.1$^{s}$ +19$^{\circ}$36$^{m}$24.1$^{s}$ 1" VLA 8.4, 4.9 and 1.4GHz <cit.> GCN 4154
3.46 23$^{h}$56$^{m}$04.1$^{s}$ +19$^{\circ}$36$^{m}$23.9$^{s}$ 0.7" Chandra ACIS-S X-ray <cit.> GCN 4163, <cit.>
Prompt and afterglow localizations of LGRB 051022.
§.§.§ Burst Host Imaging
We obtained imaging on the LGRB 051022 field in r, i, and z bands with the GMOS instrument on Gemini North on October 22nd, 2005. A later image, for image subtraction, was acquired on October 25th, 2005. More details on observations and data analysis are presented in <cit.>. The resultant host magnitudes in r, i, and z bands are 22.04 $\pm$ 0.01, 21.77 $\pm$ 0.01, and 21.30 $\pm$ 0.04 respectively (). The r band image is shown in figure <ref> with the astrometric solution determined in section <ref>.
GRB051022 host galaxy (r band) with our astrometrically matched 0.17" 1 sigma radius Chandra X-ray error circle overplotted. Note that the morphology of this host is unresolved in ground-based images.
Approximating the redshift as a straight i to B band central wavelength conversion yields a shift in central wavelength from 780 nm (for the i band filter) to 432 nm vs. 438 nm for a B band filter. Preserving the flux values gives a rest frame absolute B band magnitude of -21.60 $\pm$ 0.01 for the host galaxy of GRB 051022. The estimated absolute magnitude is unusually bright for an LGRB host and belongs at about 1.4 $\times$ L* on the Schechter luminosity function (we adopt blue galaxy M$^{*}_{B}$ values from Table 3 of for all L* comparisons and as is typical for L* we do not attempt to correct for extinction).
ACS/WFC F606W (left) and WFC3/IR F160W (right) images of the LGRB 051022 host galaxy. The images were astrometrically aligned to our Gemini GMOS imaging and the astrometric fit from section <ref> was transformed to the HST images giving the 0.17" 1 sigma radius Chandra X-ray error circle shown. The images are also shown aligned with each other in the figure.
To discern the host morphology, the field of GRB 051022 was imaged with the Hubble Space Telescope (HST) using repaired ACS/WFC in F606W on August 21$^{st}$ 2009 and one orbit with WFC3/IR in F160W on October 12$^{th}$ 2009 (see figure <ref> left and right respectively). Four dithered 520 second exposures were obtained in the F606W filter of the ACS WFC, and four dithered 600 second exposures were obtained in the F160W filter of the WFC3 IR channel. The two data sets were separately combined using the drizzle/multidrizzle packages and process outlined in <cit.>. To allow a localized color comparison the combined ACS field was also mapped to the scale of the WFC image using “blot” <cit.>. Using a consistent aperture across both bands (for optimal color term accuracy) we determine a host magnitude of 21.903 in F606W and 20.608 in F160W relative to the instrument zero points. Adopting the noise correlation ratio of <cit.> equation 9 we calculate a statistical error of $\pm$ 0.006 magnitudes on both measurements. For absolute photometry we estimate out of aperture flux lost at 0.05 mags or less.
The ACS/WFC F606W image is just blue of u band in the host galaxy rest frame and thus provides a map of the active star-formation and likely LGRB progenitor regions.
The bimodal (perhaps trimodal) nature of the F606W image and the presence of additional regions in the F160W imaging (approximately equivalent to rest frame z band) suggests a clumpy system with varying degrees of star-formation and potentially variant metallicities. More interesting from a morphological perspective is the presence of a tidal tail emanating south west from the host galaxy (Figure <ref> left). Our Gemini GMOS r band imaging also shows faint evidence of a similar extended emission (Figure <ref> right) which is less apparent in the ACS/WFC F606W image (Figure <ref> center) due to the latter's lower surface brightness sensitivity.
Astrometrically matched WFC3/IR F160W (left), ACS/WFC F606W (center), and Gemini GMOS r band images of the LGRB 051022 host galaxy showing the tidal tail emanating south west from the host galaxy. The 0.17" 1 sigma radius Chandra X-ray error circle from from section <ref> is shown as before. The presence of a tidal tail identifies the system as a merger as speculated in <cit.> and explains the system's high star-formation rate and starburst history.
§.§.§ Astrometric Refinement of GRB 051022
We obtained 20 ks of Chandra ACIS observation of the burst afterglow as described in <cit.>. In addition to the afterglow, our observations detected 6 X-ray background sources. In order to better constrain the location of the burst we identified counterparts for all 6 sources in our Gemini GMOS r band imaging and generated an astrometric solution between the X-ray source fits and their optical counterpart positions. Note the astrometric solution determined and used here is (aside from the X-ray source fitting) independent of the similar analysis presented in <cit.>. (r band observations are typically optimal for astrometry as the objects are brighter than in the bluer bands and the GMOS i and redder bands have significant fringing which disrupts object centroiding). We observe that the residual in the astrometric fit of the background sources is proportional to almost exactly the inverse square root of the number of counts. Given that the brightest of the background sources is only about half the flux of the observed afterglow the error in the X-ray source fitting is dominated by the inaccuracy in centering the background X-ray objects. The deviation from fitting the residual as the inverse square root of the number of counts we observe to be bounded by a monotonically increasing function of the distance from the image center. As the LGRB afterglow is centered on the X-ray image this increasing error with distance from the image center is negligible for the X-ray afterglow source. Therefore, the error on the object position is dominated by the error in the astrometric solution obtained on the background objects. Thus we adopt the RMS fit accuracy as the error in the our astrometrically matched source position locating the X-ray afterglow to our GMOS image with a one sigma accuracy of 0.17" (or 1.2 binned 2 $\times$ 2 GMOS pixels). This places the burst at the center of the host galaxy in the ground based image (See Figure <ref>).
Using the coordinates of ten 2MASS catalogue objects as an absolute reference we determined astrometric calibration for our Gemini optical images using the starlink astrom package with an error of 0.057" RMS. Applied in concert with the Chandra X-ray afterglow to Gemini GMOS astrometric solution from the preceding paragraph, this places GRB 051022 at an absolute astrometric position of RA: 23$^{h}$56$^{m}$04.110$^{s}$ Dec: +19$^{\circ}$36$^{m}$24.03$^{s}$ (J2000) with an accuracy of 0.18" statistical and 0.05" systematic.
Astrometric alignment of the ACS/WFC F606W and WFC3/IR F160W reduced drizzled images to the Gemini GMOS r band imaging yielded an accuracy (RMS error) of 0.010 and 0.022 arc-seconds respectively. The RMS error is essentially insignificant when added in quadrature with the refined Chandra X-ray afterglow error. This likely locates the burst to the southern pair of star-forming regions and excludes the brighter northern region beyond the two sigma limit (see figure <ref>).
§.§.§ Host Optical Spectroscopy
Initial spectroscopic observations were obtained with the GMOS instrument on Gemini North on
November 25$^{th}$ 2005 for a total exposure time of 1 hour. The R400 grating offers a reasonable compromise between spectral resolution (1.37 Å/pixel) and width of coverage (about 4000 Å) giving a spectral range of 5500 to 9500 Å for a central wavelength of 7500 Å. A 50 Å dither in wavelength was also added to ensure continuous spectral coverage across chip gaps and allow for easy removal of other chip based effects. Due to the abundance of skylines in the spectral range the Nod & Shuffle method was used offering a dramatic improvement in sky subtractions over conventional spectroscopy due to its more coincident and technically consistent sampling of object and sky spectra. A brief introduction to the Nod & Shuffle process is provided in <cit.>, for a more detailed description of the Nod & Shuffle process see <cit.> and <cit.>, and for its use on Gemini see <cit.> and <cit.>.
In order to determine whether nearby galaxies were in association with the host, Nod & Shuffle Multi-Object Spectroscopy (MOS) was employed to obtain spectra of several objects with no increase in observing time. Observations consisted of two 30 min Nod & Shuffle integrations, each containing two 15 min spectra.
The individual spectroscopic exposures were reduced using the standard IRAF “Gemini.GMOS" Nod & Shuffle packages. This is essentially the same as a conventional spectroscopic reduction except the two shuffled images on each exposure are subtracted from each other after bias subtraction and before flat fielding. Due to the small number of Nod & Shuffle spectra observed, the spectra were combined in 2d to optimize cosmic ray rejection as described in <cit.>. A custom Nod & Shuffle dark was also used.
Spectral extraction was performed with IRAF task “apall" along with a matched extraction on an arc spectrum subsequently used for wavelength calibration. The process yielded a spectrum with a spectral resolution of 1.37 Å per pixel and a spatial resolution of 0.15 arc seconds per pixel. The host galaxy spectrum contains several bright emission lines placing it at a redshift of $z=0.806$ consistent with the previous redshift measurements given by <cit.> and <cit.>.
The equivalent widths of several lines identified in the host galaxy spectrum are shown in Figure <ref> and listed in Table <ref>. None of the additional MOS objects showed features placing them at a similar enough redshift for a cluster association. The additional multi-object spectra collected are shown in Figure <ref>.
Gemini GMOS Nod & Shuffle optical spectrum of GRB 051022. The [O II], [O III], and H$\beta$ lines are used for the R$_{23}$ method.
The Gemini GMOS Nod & Shuffle multi-object spectra aside from the host of LGRB 051022. (The LGRB 051022 host location is indicated on the large image as the circle without a line connecting with a spectrum). An eight slit mask was employed in MOS configuration, one slit being the object, another slit failed to yield a detectable spectrum and the other six lack any features of a redshift similar to the host of LGRB 051022. On three of the spectra the annotated lines were identified and place the objects at the labeled redshifts.
§.§.§ Host Near Infrared Spectroscopy
Near infrared spectroscopic observations of the LGRB 051022 host were obtained with NIRSPEC on the Keck II telescope on October 23$^{th}$ 2007.
Our observations consisted of four 900 second exposures in the NIRSPEC-2 filter, using a 0.76 arc-second slit, and giving a spectral coverage from 1.09 to 1.29 $\mu$m.
Keck NIRSPEC near infrared spectrum of GRB 051022 shifted to the rest frame. The [N II] to H$\alpha$ ratio is used to break the degeneracy of the R$_{23}$ method.
Individual NIRSPEC exposures were reduced using the standard procedure described in the online documentation from the NIRSPEC manual.[<http://www2.keck.hawaii.edu/inst/nirspec/redux.html>] The object's placement on the slit was dithered between two locations with two exposures each, so that the combined image from each placement could be subtracted from the other to remove sky features.
Due to the subtraction between the two dither placements, this image (like those described in the Nod & Shuffle process perviously) contains two spectra, one of which is inverted. Similarly, an inverted copy of the image is created, shifted to align the now positive spectrum with the positive spectrum on the original and then coadded, to yield a single combined spectrum.
Spectral extraction was performed with IRAF task “apall" along with another extraction on non dither subtracted data to generate a sky spectrum that was used for wavelength calibration. The process yielded a spectrum with a spectral resolution of 1.92 Å per pixel and a spatial resolution of 0.16 arc seconds per pixel. The resulting spectrum with identified H$\alpha$ and 6583 Å [N II] lines is shown in Figure <ref>.
The continuum was fit with a high order polynomial and the H$\alpha$ line with a Gaussian (with the continuum parameter set to the value returned from the polynomial fitting). Similar Gaussian fitting on the 6583 Å [N II] line was suboptimal and returned an inconsistent line width with a high fractional error. In order to to obtain a more accurate result, the width of the H$\alpha$ line (with a much lower fractional error) was assumed for the 6583 Å [N II] line and the line fit only for height (fitting also for center did not produce a statically different result in flux). The fitting results are shown in Table <ref>. Due do the proximity of the two lines the continuum is effectively unchanged between them thus, a direct flux ratio of the two lines can be calculated with lower error than would be naively expected from propagating the errors for the equivalent widths though the ratio and yields a value of 6.57 $\pm$ 0.52 for the H$\alpha$ to 6583 Å [N II] line flux ratio.
Line Rest Wavelength (Å) Equivalent width
[O II] 3727 -67.00 $\pm$ 4.80
[Ne III] 3869 -8.00 $\pm$ 1.56
H$\gamma$ 4340 -10.00 $\pm$ 3.34
He II 4686 -4.86 $\pm$ 0.98
H$\beta$ 4861 -25.29 $\pm$ 4.85
[O III] 4959 -22.24 $\pm$ 4.55
[O III] 5007 -59.57 $\pm$ 6.36
H$\alpha$ 6563 -104.99 $\pm$ 4.09
[N II] 6583 -15.97 $\pm$ 1.37
Spectral line rest frame equivalent widths for the LGRB 051022 host.
§.§ GRB 020819B
§.§.§ Burst Detection and Localization
LGRB 020819B was initially detected by all three instruments on the HETE-2 satellite as well as the Gamma Ray Burst instrument in the Ulysses spacecraft. Ulysses observations indicated a moderately long burst duration of approximately 20 seconds securely identifying this as a long gamma ray burst ( GCN 1507). <cit.> (GCN 1842) discovered a fading radio afterglow in VLA observations of the region. <cit.> (GCN 1844) found a clearly resolved galaxy with R $\sim$ 19.8 mag coincident with the radio afterglow position. <cit.> measured a redshift of z=0.41 for the GRB 020319B host and observed that the radio afterglow is superposed on a faint structure they term “the blob" located around 3 arcsecs from the galaxy center. No optical or NIR afterglow was detected to a limiting magnitude of R $>$ 22 ( GCN 1844) and K' $>$ 19.5 mag <cit.>, thus classifying LGRB 020819B as a “dark" burst. See Table <ref> for details of prompt and afterglow localizations.
$\Delta$T (days) RA Dec Error Instrument Band Reference
0 23$^{h}$27$^{m}$07$^{s}$ +6$^{\circ}$21$^{m}$50$^{s}$ 7' HETE-2 WXM 2-25 keV GCN 1508
0 23$^{h}$27$^{m}$24$^{s}$ +6$^{\circ}$16$^{m}$08$^{s}$ 2.5' HETE-2 SXC 0.5-10 keV GCN 1508
0 23$^{h}$27$^{m}$19.5$^{s}$ +6$^{\circ}$17$^{m}$05$^{s}$ 82" HETE-2 SXC refined 0.5-10 keV <cit.>
1.75 23$^{h}$27$^{m}$19.475$^{s}$ +6$^{\circ}$15$^{m}$55.95$^{s}$ 1" VLA 8.46 GHz <cit.> GCN 1842
Prompt and afterglow localizations of LGRB 020819B. The SXC burst position was subsequently refined in <cit.>. This burst was also detected by the Gamma Ray Burst instrument in the Ulysses spacecraft. Using differential time of arrival from the two spacecraft the InterPlanetary Network (IPN) located the burst to a 4' wide annulus centered at RA: 345.048$^{\circ}$ Dec: -43.013$^{\circ}$ radius: 49.664$^{\circ}$ ( GCN 1507).
Gemini North GMOS image of the host of GRB 020819. The 1" diameter radio afterglow error circle is shown. Unfortunately the resolution of the image and the diameter of “the blob" are both about 0.6 arc-seconds. While our spectroscopy places “the blob" and the rest of the host at the same redshift, greater resolution than that obtainable from the ground is needed to determine whether “the blob" is part of the spiral, an accreting galaxy like the Sagittarius Dwarf, or a satellite galaxy like the LMC.
§.§.§ Burst Host Imaging
We acquired three dithered 600 second Gemini GMOS r band exposures of the LGRB 020819B host on August 30th 2008 under photometric conditions to obtain high resolution imaging (program GN-2008B-Q-99). High resolution imaging is necessary for an accurate slit placement and it also provides a better understanding of the dynamics of the system. The photometric data was reduced with the standard Gemini GMOS IRAF[IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.] packages and combined with a simple cosmic ray rejecting sum. The Gemini GMOS image of the neighborhood of the radio afterglow of GRB 020819B along with the faint “blob" is shown in Figure <ref>. Aperture photometry yields an r band magnitude of 24.20 $\pm$ 0.15 for “the blob" and 19.829 $\pm$ 0.010 for the entire host galaxy system using the Gemini GMOS photometric zero points. The host magnitudes agree well with the SDSS galaxy magnitude of 19.83 $\pm$ 0.04 and is reasonably consistent with the R band value of 19.5 mag after uncorrecting for the galactic extinction of E(B-V) = 0.070 and converting to SDSS magnitude system.
For GRB 020819B, we estimate an absolute B band magnitude of -21.92 $\pm$ 0.01 for the host galaxy and -17.55 for “the blob" using a similar approach as discussed in section <ref>.
The galaxy is unusually luminous for an LGRB host and belongs at about 3 $\times$ L* on the Schechter luminosity function. "The blob" has an absolute magnitude approximately equal to the Large Magellanic Cloud (LMC -17.86 $\pm$ 0.05 using <cit.> and <cit.> for the LMC magnitude and distance values respectively) and about half the LMC's linear size.
Our imaging resolution and the diameter of “the blob", both being about 0.6 arc-seconds, makes it almost impossible to determine whether “the blob" is an outlying part of the spiral, an accreting galaxy like the Sagittarius Dwarf, or a satellite galaxy like the LMC. The asymmetry apparent in the disk of the spiral galaxy suggests that the systems are indeed interacting.
While HST imaging is required to definitively resolve the morphological nature of the LGRB progenitor region, a merging system was the operating assumption in selecting this burst as an analogous system to that of LGRB 051022 and “the blob's" high star-formation rate supports (but does not prove) this.
§.§.§ Burst Host Spectroscopy
Spectroscopic observations of the spiral host galaxy of GRB 020819B were obtained at two separate epochs. The first epoch observations of the host galaxy nucleus were obtained with the Low Resolution Imaging Spectrograph (LRIS; ) on the Keck I telescope on November 2nd, 2008. We acquired a second epoch spectrum of the star-forming region associated with the radio afterglow of GRB 020819B on November 19th, 2009. The results of the spectroscopic observations are presented in <cit.>. As noted in <cit.> technical problems prevented the detection of 4861 Å H$\beta$ and 4959 & 5007 Å [O III] lines on “the blob" spectrum.
§ ANALYSIS OF HOST GALAXY METALLICITIES
§.§ Methodology
Since oxygen is the most abundant contributor to metallicity and metallicity is dominated by the combined abundance of oxygen, carbon and nitrogen the latter two of which share an approximately similar formation history to oxygen, oxygen is an ideal proxy of the overall metal abundance. Oxygen is also typically the brightest contributor of metal lines in a spectrum and has no shortage of lines in the optical range making it a useful tracer. Emission line diagnostics are specifically measuring the metallicity of H II regions, when applied to other galaxies this is used extensively as a proxy for the overall metallicity. In specific application to the metallicity of bright stars, such as LGRB and core collapse supernovae progenitors, this is even more apt since a high degree of ionizing radiation (from the progenitor and similar neighboring stars) is virtually unavoidable and thus it is to be expected that the surrounding environment is an H II region. This is doubly true in the former case as LGRB's occur in the brightest, most star-forming regions of their hosts <cit.>. Thus, any potential bias between the host galaxy's overall metallicity and the overall metallicity of its H II regions works in favor of a more accurate estimate of the LGRB progenitor's metallicity. However, as galaxies are not uniform objects, the remaining issue is then how well the overall metallicity of the host galaxy's H II regions serves as a proxy for the specific metallicity of the LGRB progenitor's H II region (and thus the metallicity of the progenitor itself).
§.§.§ $R_{23}$ Diagnostic
The R$_{23}$ method is a commonly used metallicity diagnostic based on the electron temperature sensitivity of the oxygen spectral lines, achieved via using the ratio of Oxygen line strength to a spectral feature independent of metallicity. In particular [O III] has lines at 4959 and 5007 Å and [O II] has a particularly strong, typically unresolved, doublet at 3727 Å. The metallicity independent 4861 Å H$\beta$ line's convenient placement between the aforementioned Oxygen lines, gives the equation for R$_{23}$ used in the classical application of this diagnostic, see equation <ref>. First proposed by Bernard Pagel in 1979 <cit.>, R$_{23}$ has become the primary metallicity diagnostic for galaxies at $z>0.3$ (especially those where the faint [O III] 4363 Å line is not measurable).
\begin{equation}
\label{eqn}
R_{23} = {I_{3727} + I_{4959} + I_{5007} \over I_{H\beta}}
\end{equation}
In order to correlate the flux of a line belonging to an individual atomic ionization level with the total abundance of that element it is necessary to know what fraction of the element is ionized to the level in question. This is achieved by measuring the flux ratio between the [O II] and [O III] line strength, giving the relative population in the O II and O III ionization states, and fitting the metallicity for that specific ionization state ratio.
Thus in the classical application of this diagnostic the R$_{23}$ value would be calculated from the measured line ratios and then compared along an [O III] to [O II] line ratio contour against the best calibration data available. This classical application however treats ionization as a parameter independent of metallicity and ignores the feedback the latter has on the former. <cit.> solved this issue by using iterative fitting to dynamically factor the effects of the metallicity on the ionization parameter (without calculating the R$_{23}$ value shown above). <cit.> established that the R$_{23}$ method can be directly used on equivalent width values (instead of reddening corrected flux values) and is found to be more accurate than flux ratios when reddening information is not available.
The R$_{23}$ method diagnostics suffer a degeneracy due to different effects being dominant at different regimes. In the low metallicity regime the effects of the metals on the electron temperature of the system can be ignored due to their low relative abundance. Thus the more metals in the electron gas, the more collisional excitations and more resultant flux in the metal lines. As the metallicity rises however, emission from infrared fine-structure lines becomes significant and serves as a cooling mechanism, lowering the electron temperature, the electron velocity, the number of collisional excitations, and the metal line flux.
This temperature dependence causes two metallicity values (one high, one low) to generate the same metal line flux. Unless one of the values can be obviously excluded or the two values are within the error range of the R$_{23}$ calibration, new empirical data is the only accurate way to break the degeneracy. This is usually facilitated, as we have done here, by application of the [N II]/H$\alpha$ diagnostic described in section <ref>. This has the obvious disadvantage of requiring additional measurements considerably outside the R$_{23}$ wavelength range often requiring a separate observation and for higher redshift objects a separate, near-infrared, instrument.
§.§.§ [N II]/[O II] Diagnostic
The [N II]/[O II] ratio provides a diagnostic characterized in <cit.>. Notably this diagnostic is relatively insensitive to differences in the ionization parameter and lacks the degeneracy issues plaguing R$_{23}$. Given the wide separation between the 3727 Å [O II] and 6584 Å [N II] lines accurate flux calibration and measurement of the reddening are obvious issues and tend to limit this application. Also the diagnostic's low variability to metallicity below half solar constrains its use to abundances above that, however as the [N II] line flux decreases with metallicity, for all but the most well measured spectra the low signal to noise ratio on the [N II] line flux will likely be the limiting factor.
Unlike R$_{23}$, [N II]/[O II] has not been tested using equivalent width values in place of fluxes. However as the spectral energy distribution of the bluest galaxies almost never exceeds that of a flat continuum (in erg cm$^{-2}$ s$^{-1}$ Å$^{-1}$), and extinction will preferentially suppress the 3727 Å [O II] flux over the 6584 Å [N II] flux, we can assume that the equivalent width ratio will function as an upper limit on the metallicity. While we experiment with using this upper limit here, with encouraging results, our usage should be considered only anecdotal and a detailed study with statistical rigor and full characterization of errors is needed before its adoption is considered as a hard limit.
§.§.§ [N II]/H$\alpha$ Diagnostic
The [N II]/H$\alpha$ ratio (also characterized in ) provides a crude metallicity indicator however due to its strong dependence on the ionization parameter it provides only a gross estimate of abundance unless the ionization in known. Since measure of the ionization parameter requires measurement of lines which themselves constitute better metallicity diagnostics application of ionization parameter correction to this diagnostic is of limited utility. Also the diagnostic is easily distorted by contamination from shock excitation and AGN rendering it comparatively inaccurate. Thus its primary application is selecting between the degenerate upper and lower branch values provided by the R$_{23}$ diagnostic (as stated in section <ref>).
§.§.§ Scale & Code
Due to differences between metallicity diagnostics and their various calibrations, a true comparison of metallicity requires using a common scale and if possible a consistent methodology and diagnostic. Here we adopt the <cit.> scale. Based on the <cit.> 6300 Å [O I] line measurements of the Sun, solar metallicity is estimated to be log(O/H)+12 = 8.69 $\pm$ 0.05 in this scale. It should be noted that while the <cit.> diagnostics listed above can be cross calibrated to a high degree of accuracy due to the large number of H II regions where all the diagnostics can be applied this is not true of the density-sensitive 6300 Å [O I] line measurement where the line strength is insufficient for widespread application. Thus the <cit.> scale is more accurate internally than to any absolute reference to solar value and such comparisons should be limited to broad generalizations. For conversion to other scales and discussion of associated issues we refer the reader to the transforms in <cit.>.
To ensure consistency of method, we employ an improved version of the idl code outlined in <cit.> (updated to the scale) to calculate metallicity values, including recalculation of comparison values from published fluxes. Since the Kewley code applies ionization across all diagnostics and was used to cross calibrate the various diagnostics with each other, we expect this methodology to give the best cross diagnostic agreement. Still, when possible and unless otherwise specifically described, the R$_{23}$ diagnostic output is the metallicity value adopted.
§.§ LGRB 051022
Using the [O II], [O III], and H$\beta$ host galaxy line equivalent widths from our Gemini GMOS South spectroscopy yields degenerate R$_{23}$ metallicity values of log(O/H)+12 = 8.18 and log(O/H)+12 = 8.77 (about 1/4 solar and slightly super-solar respectively).
The 6583 Å [N II] and 6563 Å H$\alpha$ line values (obtained with NIRSPEC on Keck II due to their being red-shifted into the infrared) constrains the R$_{23}$ degeneracy to the upper branch yielding a host galaxy metallicity of log(O/H)+12 = 8.77 $\pm$ 0.07 in the <cit.> scale
which was the highest measured metallicity of any long burst host galaxy then seen. We adopt this metallicity value for the system subsequently in the paper.
Interestingly however application of the [N II]/H$\alpha$ diagnostic yields a value of log(O/H)+12 = 8.90 notably higher than with R$_{23}$. Application of the [N II]/[O II] diagnostic with equivalent widths assuming the flat continuum blue constraint outlined section <ref> gives a soft upper limit of log(O/H)+12 = 8.84 on the metallicity. The [N II]/H$\alpha$ diagnostic is known to be comparatively inaccurate <cit.>, even when it is corrected with the ionization parameter determined from the [O II] to [O III] line ratio. Still the higher [N II]/[O II] diagnostic value may be indicative of either a nitrogen over abundance in the system (respective to oxygen), ionization parameter differences between the slit placement in the GMOS and NIRSPEC observations, or a metallicity variation across the merging host system.
Additionally <cit.> obtained independent optical spectroscopy of the GRB 051022 host with Keck LRIS and determined an R$_{23}$ metallicity of log(O/H)+12 = 8.62 using flux values. <cit.> use the H$\beta$ to H$\gamma$ ratio to determine a reddening of E(B-V) = 0.50 and our [N II]/H$\alpha$ ratio to resolve degeneracy. Based on their published flux values we recalculate an R$_{23}$ value of log(O/H)+12 = 8.64 (via the methodology in section <ref>). Applying that ionization parameter to the [N II]/H$\alpha$ ratio & diagnostic yields a value of log(O/H)+12 = 8.83 again notably higher than with R$_{23}$. We adopt the reddening value from <cit.> to attempt a crude flux calibration of the 6583 Å [N II] line via setting the H$\alpha$ flux at its expected value with respect to the H$\beta$ line from the Balmer decrement. Following this by applying the observed [N II]/H$\alpha$ ratio gives an [N II]/[O II] value of log(O/H)+12 = 8.67 for the system. The values or limits available for the various diagnostics in GRB 051022 host system using both fluxes and equivalent widths are listed in Table <ref>. For LGRB 051022 the fluxes are the published values from <cit.> which were obtained via separate optical spectroscopy and suspected to be of a separate region of the host.
While a difference between the two independent R$_{23}$ observations of only 0.13 dex might seem minor it is almost twice the intrinsic $\pm$0.07 dex error of the <cit.> diagnostic. The surprisingly good agreement between the R$_{23}$ and [N II]/[O II] diagnostics applied to the <cit.> LRIS spectrum suggests that our Keck NIRSPEC spectroscopy might more closely match the LRIS spectrum than our GMOS spectroscopy and highlights that the difference may lie in the position angles used. Our multi-object spectroscopy of the host was constrained to a slit position angle of 90$^{\circ}$ so as to match our GMOS imaging from which the slit mask was derived (which itself was oriented to a position angle to obtain a bright On Instrument Wave Front Sensor [OIWFS] star while still orienting the image with respect to a cardinal direction). This places the slit axis almost exactly perpendicular to the axis between the northern and southern regions in Figure <ref> and maximizes the likelihood of preferentially obtaining one of the regions over the other. Similarly, the Keck LRIS spectroscopy of <cit.> used a slit a position angle of 105$^{\circ}$ (private communication) also placing their slit axis roughly perpendicular to the extended direction of the host. Our Keck NIRSPEC observations however used slit a position angle of 210$^{\circ}$ to orient the slit along the apparent length of the galaxy (in the ground based imaging). While the separation between the two regions is only about 0.6" and thus a large amount of contamination is to be expected even if the slit were perfectly covering only one of the regions this only implies that any apparent metallicity differences seen between the two spectra are likely a significant underestimate of those in the actual system.
§.§ LGRB 020819B
§.§.§ Host Galaxy Nucleus
<cit.> give metallicity values of log(O/H)+12 = 9.0 and 8.8 for the [N II]/[O II] and [N II]/H$\alpha$ diagnostics respectively. For maximum possible consistency with other objects we recalculate these values using the code and procedure outlined in section <ref>. This yields values of log(O/H)+12 = 8.97, 8.98, and 8.99 for the R$_{23}$, [N II]/[O II] and [N II]/H$\alpha$ metallicity diagnostics respectively. (The primary difference in methodology being the correction here of galactic extinction based on the dust maps before correcting for internal extinction via the Balmer decrement. In this as most cases this difference is quite minor but is employed for consistency). The R$_{23}$ value is adopted as the metallicity of the LGRB 020819B host galaxy.
§.§.§ “The Blob" outlying region hosting the LGRB
Given the localization of the burst to an outlying structure the properties of the host galaxy nucleus are likely irrelevant to understand the GRB progenitor environment (but useful in understanding sources of potential error on host studies where such positioning is not available or where the system geometry does not allow for such a clean identification of a progenitor region, i.e. an edge on system). Thus direct metallicity measurement of “the blob", outlying the galaxy where the burst occurred, is necessary for inferring the GRB progenitor properties.
This process is greatly complicated by the failure to obtain the H$\beta$ and [O III] lines <cit.>. The preferred R$_{23}$ metallicity diagnostic obviously cannot be applied at all. The [N II]/[O II] diagnostic while itself unaffected is essentially useless without knowledge of the extinction (obtained by fitting the observed Balmer lines to expected ratios) required to correct the observed flux values for reddening. The [N II]/H$\alpha$ diagnostic is virtually unaffected by reddening (due to the close proximity of the two lines) however it is highly dependent on the ionization parameter (typically determined from the [O III] to [O II] line ratio). Still the [N II]/H$\alpha$ diagnostic is the best estimate possible from the flux values available and yields a metallicity of log(O/H)+12 = 8.95 for “the blob." This value and those for the host galaxy nucleus are consistent with the metallicity values given in <cit.> shifted to the <cit.> scale using the transformations in <cit.>. However without knowledge of the ionization parameter the [N II]/H$\alpha$ diagnostic at best a crude estimate of the metallicity.
In order to quantize constraints on [N II]/H$\alpha$ diagnostic error we adopt a more involved approach. Given that the H$\alpha$ flux is known and the intrinsic H$\alpha$ to H$\beta$ ratio set by the Balmer decrement, the observed H$\beta$ flux is a function of the extinction. Similarly as the 3727 Å [O II] flux is known and the [O II] to [O III] line ratios are a function of the ionization, the flux of the [O III] lines is a function of the ionization parameter. Thus the problem can be decomposed into three metallicity diagnostics ([N II]/[O II], R$_{23}$, and [N II]/H$\alpha$) and three unknown parameters (extinction, [O III] flux, and metallicity). The H$\beta$ flux is estimated from the H$\alpha$ flux and extinction using the Balmer decrement. The system can thus be modeled by iterating though the reasonable extinction and [O III] flux range and seeking concordance between the three metallicity diagnostics. The iteration yields three solutions of E(B – V), 5007 Å [O III] flux and metallicity as listed in Table <ref>.
Extinction Extinction derived Estimated 5007 Å Metallicity
E(B – V) H$\beta$ flux [O III] flux log(O/H)+12
(magnitudes) 10$^{-17}$ erg cm$^{-2}$ s$^{-1}$ 10$^{-17}$ erg cm$^{-2}$ s$^{-1}$ <cit.> scale
0.31 9.3 4.7 9.08
0.67 6.2 4.3 8.93
1.09 3.9 1.5 8.70
Iterated multi-metallicity diagnostic solutions. The flux values are given as they would have been measured (i.e. prior to correcting for extinction).
Allowed multi-metallicity diagnostic solutions for given extinction and [O III] flux iterated values for “the blob" region of the LGRB 020819B host. Fill color shows the R$_{23}$ metallicity of the given solution and the contours show the standard deviation between the different metallicity diagnostics. The three metallicity diagnosis ($R_{23}$, [N II]/[O II], & [N II]/H$\alpha$) and the three unknowns (i.e. the two missing lines 4861 Å H$\beta$ & 5007 Å [O III] and the metallicity) effectively form a three equations and three variables problem. Thus we iterate across reasonable extinction and 5007 Å line flus values discarding those solutions where the $R_{23}$ and [N II]/[O II] diagnosis differ by more than 0.05 dex or the mean of those differ from the [N II]/H$\alpha$ by more than 0.15 dex (approximately the optimal cross diagnostic agreement observed). Though not plotted we iterated out to an extinction of E(B – V) of 2 without any additional allowed values. Standard deviation contours of 0.005, 0.01, 0.02, 0.04, & 0.08 are plotted with a + marking the point of exact agreement (i.e. where all three metallicity diagnostics give the same value). Note that the allowed solution (the area in the figure with a shown metallicity color) cutoff does not follow any of these contours due to its being determined exactly as described with different criteria on agreement between the different diagnostics. Additionally, a vertical line shows the extinction lower limit value of E(B – V) = 0.48 discussed in section <ref>. Given the wide range of allowed metallicities even with optimal cross diagnostic agreement we conclude that the metallicity of the LGRB 020819B hosts “the blob" region cannot be constrained much beyond the upper branch of the R$_{23}$ diagnostic. Thus while the specific region termed “the blob" hosting LGRB 020819B is at much higher metallicity than typical LGRB hosts, comparison of its metallicity to the center of its host galaxy cannot be determined from the data yet obtained.
The three cases in Table <ref> represent the zero error cases only. In reality disagreement between the metallicity values given by the different diagnostics must be considered. In figure <ref> the diagnostics disagreement is shown with the overplotted contours or representing the degree of difference between the different metallicity diagnostics (the + marks the three cases where the diagnostics agree exactly given in Table <ref>). Assuming a conservative $\sim$0.1 dex error on the values gives an approximate metallicity range for the [N II]/H$\alpha$ ratio and diagnostic at hand of about log(O/H)+12 = 8.67 to 9.18 or a value of log(O/H)+12 = 8.95$^{+0.22}_{-0.28}$ on the [N II]/H$\alpha$ diagnostic.
While the lack of an extinction value for “the blob" makes it impractical to apply the [N II]/[O II] diagnostic to determine metallicity it can however be used to set some limits on the system. The metallicity value decreases as one increases the extinction estimate. Thus for the zero extinction case “the blob" metallicity of log(O/H)+12 = 9.18 is a hard upper limit. Similarly adopting the flat continuum blue constraint (outlined in section <ref>) on the equivalent width values with the [N II]/[O II] diagnostic gives a soft upper limit of log(O/H)+12 = 9.01 for “the blob" region. This added [N II]/[O II] upper limit constraint removes the upper metallicity zero error case in Table <ref> and reduces the metallicity range from that allowed under the [N II]/H$\alpha$ diagnostic. Additionally, adopting the equivalent width soft upper limit value and applying the [N II]/[O II] diagnostic with fluxes yields, where its metallicity converges at, an E(B – V) = 0.48 lower limit on the extinction.
This corresponds to an extinction of A$_V$ = 1.3 magnitudes or greater which is the criteria for modest extinction in <cit.> model 1 and thus potentially explains the “dark" nature of this burst. The values or limits available for the various diagnostics in the GRB 020819B host nucleus and “the blob" region using both fluxes and equivalent widths are given in Table <ref>.
Input R$_{23}$ [N II]/[O II] [N II]/H$\alpha$
LGRB 020819B host “blob" fluxes ... $\leq$ 9.18 8.95
LGRB 020819B host “blob" equivalent widths ... $\leq$ 9.01 9.02
LGRB 020819B host nucleus fluxes 8.97 8.98 8.99
LGRB 020819B host nucleus equivalent widths 9.08 $\leq$ 9.10 8.97
LGRB 051022 host system equivalent widths 8.77 $\leq$ 8.84 8.90
LGRB 051022 host system fluxes 8.64 8.67 8.83
Metallicity values for various diagnostics and inputs. (The LGRB 051022 host system fluxes are from the <cit.> spectrum which is distinct from the spectrum used for the LGRB 051022 equivalent widths).
§.§ LGRB 050826
In addition to LGRBs 020819B and 051022, a third case, LGRB 050826, was identified as being at high metallicity by <cit.>. Although we did not observe this object, a discussion of LGRB 050826 is in order due to its being the remaining high metallicity LGRB in the <cit.> sample. This object, with a host at log(O/H)+12 = 8.84 (recalculated using the published fluxes via the methodology described in section <ref> and consistent with value), is unique both in having an optical counterpart ( GCN 4749), and in being the first high metallicity LGRB host found without some prior expectation of this result (remember than both LGRBs 020819B and 051022 were suspected to be at high metallicity due to their unusually bright host galaxies). Due to its discovery in such an untargeted manner and otherwise being a burst of no particular importance to the community, it is as yet quite poorly studied.
Unfortunately the host of LGRB 050826 is lacking any HST observations so it is difficult to study its properties in great detail — see Figure <ref> for an image of the host galaxy. Morphologically it appears as the confluence of the properties of the other two high metallicity LGRB host galaxies and the larger (presumably predominately low metallicity) LGRB host sample of <cit.>. Like the other high metallicity LGRB hosts this galaxy is brighter than the typical host but consistent with the general luminosity metallicity relation for its redshift. However the as with most LGRB hosts the galaxy is small (0.3 L*) and clearly irregular. It is difficult to determine with the obtained image quality if the host is a merging system however this possibility certainly can not be excluded. The burst itself is sub luminous consistent with the majority of LGRB events detected in the local population <cit.>. Not only does LGRB 050826 establish that a burst in a high metallicity environment can have an optical transient but also this is the first case where the metallicity was determined to be high without previous suspicion that this was likely the case before obtaining the metallicity measurement.
Image of the LGRB 050826 host galaxy taken with RETROCAM on the MDM 2.4m telescope. Inset is a heavily smoothed cutout of the same image at the same scale. Note the irregular and likely dynamical host galaxy morphology. Image from .
§ HIGH METALLICITY LGRBS AS A POPULATION
The individual characteristics of the high metallicity LGRBs and their hosts warrant some consideration. Given that these initial two high metallicity LGRBs (051022 & 020819B) were both dark bursts, this led to early speculation of whether it was the dark nature of the burst that allowed it to occur in such a high metallicity environment (i.e. dark bursts were a distinct phenomenon without the metal aversion seen in typical bursts with an optical afterglow) or whether the absence of an optical transient was simply a product of this burst's environment (i.e. larger degree of extinction due to geometry and greater prevalence of dust with metallicity). The addition of LGRB 050826 with an optical transient strongly favors the latter argument and strongly suggests that further dark bursts will also be high metallicity. Our observations of LGRBs 051022 & 020819B are also consistent with the <cit.>
observation that dark burst host galaxies to be more massive, star-forming, dust obscured, and thus presumably metal rich than than LGRB hosts with optical afterglows. This has obvious potential implications on the LGRB metal aversion debate. However as current estimates are that no more than 20 to 30 % of LGRBs are dark <cit.>, even assuming all dark LGRBs are high metallicity, there is still an insufficient fraction of such bursts to ascribe LGRB formation as a metallicity independent function of star-formation (even using the most generous estimates of as much as half the star-formation occurring in small metal-poor galaxies), thus there is still a strong selection effect toward low metallicity environments <cit.>.
The high metallicity LGRB host morphology is noteworthy. Our HST imaging conclusively shows that LGRB 051022's host is a merging system and our ground based Gemini imaging of LGRB 020819B is also shown a clearly disturbed morphology consistent with (but not conclusive of) a small galaxy (hosting the burst) being absorbed into the grand design spiral. LGRB 050826 lacks deep resolved imaging and is described as possessing a "bright core and irregular extension" <cit.>.
While the association between high metallicity environments and bursts being dark could be as simple as the metallicity being necessary for the formation of sufficient dust to provide the required extinction the effect of morphology is more complicated. Given that two of the three known high metallicity LGRBs are dark, and assuming that both cases are mergers, this suggests that this is likely more than a chance occurrence. Combined with the fact that these two bursts have the highest star-formation rate of any LGRB hosts z $<$ 0.9 (approximately the redshift range of measured emission line host metallicity) suggests that extremely rapid self enrichment is occurring.
Central galaxy metallicity versus B band absolute magnitude of high metallicity LGRB hosts (squares), TKRS galaxies (diamonds), and SDSS galaxies (small dots) with color used to index redshift. For clarity, the SDSS galaxy population has been restricted to galaxies with a similar redshift to GRB 050826 (the other plotted LGRBs exceed the redshift range of the SDSS population). Note that the three high metallicity LGRB hosts are consistent with the general galaxy population (SDSS & TKRS galaxies combined) luminosity-metallicity trends for their redshift. Thus the metallicities of the high metallicity LGRBs are fully consistent with typical star-forming galaxies of comparable brightness and redshift.
In <cit.>, we analyzed the general metallicity trends of the LGRB population, compiling data from the two objects studied here as well as all other suiTable LGRB host spectra and other object classes for comparison and found that LGRBs have a strong intrinsic preference for low metallicity environments. Of particular note is that the high metallicity LGRB population (of these three objects out of 14 total LGRBs) appears consistent with the general star-forming galaxy population of comparable brightness and redshift.
§ DISCUSSION
While the majority of the LGRB population is found in host galaxies far more metal-poor than typical galaxies of comparable luminosity and redshift, thus reflecting an intrinsic preference of LGRBs to occur preferentially in low metallicity environments, high metallicity LGRB hosts (while rare) do exist. Specifically, we identify the cases of LGRBs 020819B, 050826, & 051022. What is most notable about the three cases observed to date is that each of the LGRB host galaxies are consistent with the TKRS sample with no visible offset toward a lower metallicity (see figure <ref>). Given that there is some intrinsic distribution in galaxy metallicities, this result is surprising — where there an ongoing preference for low metallicity environments then even after selecting a set of LGRBs with high metallicity host galaxies one would still expect those LGRBs to have as low a metallicity environment as that selection allowed. Even if an individual LGRB containing galaxy is at a high metallicity we would still expect it to be at a lower metallicity than typical for a galaxy with the same brightness and redshift. Within a high metallicity galaxy we would expect to find the LGRB in a region of lower than typical metallicity. However we find that all three of our high galaxies LGRB host galaxies to be at a brightness and redshift typical for the general galaxy population of <cit.>. Furthermore, we don't see any obvious signs that high metallicity LGRB are in the lowest metallicity parts of there hosts. The location of LGRB 020819B looked prosing in this respect however metallicity measurement of the specific host galaxy region where it occurred excludes a typical low metallicity environment. Without either sign of ongoing metal aversion in environments of higher metallicity than in which LGRBs are typically observed we therefore deduce that it is likely still possible to form an LGRB in a high metallicity environment although with greater rarity.
In <cit.>, we analyze LGRB population metallically distribution in detail and find that aside from our sample's 3 high metallicity hosts (out of a total population of 14) the remaining LGRB population is constrained to the lowest metallicity tenth of available star-formation. This simple counting statistic is complicated by a number of factors. First of all, 2 of the 3 high metallicity LGRB hosts were selected for metallicity measurement in a biased manner, deliberately searching based on prior information for cases that would likely violate the low metallicity LGRB host trend observed to date. Secondly, the general galaxy mass metallicity relationship in concert with the greater observational practicality of measuring the metallicity of brighter galaxies and the greater range and thus sample size available therein combines to skew general LGRB host metallicity surveys (those without efforts to cherry pick anomalous cases) toward the higher metallicity end of the host population. These selection biases are addressed in <cit.> although we suspect the total effect of these biases is likely a factor of a few, at most.
Thus while the likelihood of forming an LGRB is decreasing with increasing metallicity, either asymptotically or linearly, the preference appears to be truncated at high metallicity such that the probability of forming an LGRB, instead of approaching zero approaches a small nearly (or perhaps actually) constant value. This LGRB metallicity dependance function leads to three potential interpretations.
(1) LGRBs do not occur in high metallicity environments and those seen in high metallicity hosts are in fact occurring in low metallicity environments that have become associated with otherwise high metallicity hosts but remain unenriched. The spatially resolved metallicity observations of LGRB 020918B seem to suggest that this is not the case. However due to the limits on the precision of the site metallicity measurement (due to detector problems), the limited spatial resolution of the dark burst, the lack of definitive morphology information on the burst progenitor region and the presence of only a single such spatially resolved high metallicity LGRB observation we find this evidence insufficient to be fully exclusionary. Thus while this explanation cannot be completely ruled out, based on our work to date we believe it to be rather improbable.
(2) The LGRB formation mechanism while preferring low metallicity environments does not absolutely require it resulting in a continuous (though probably nonlinear) decline in burst formation with increasing metallicity with an observed sharp drop off around half solar metallicity. The primary physical implication of this being that there is no physical limit on the metallicity allowed for LGRB formation only a degradation of the formation process with increasing metallicity. Both the absence of a correlation between LGRB energetics and metallicity <cit.> as well as the absence of any apparent bias in the high metallicity LGRB population toward lower than average metallicity (for galaxies of the same brightness and redshift) both suggest that this is not the case. However the small number of such high metallicity LGRBs and the intrinsic noise in the metallicity measurements makes this evidence again insufficient to be exclusionary.
(3) The typical low metallicity LGRBs and the few high metallicity cases are the result of physically differentiable burst formation pathways with only the former affected by the metallicity of the burst environment and the later simply occurring much more infrequently. Presumably such a metallicity insensitive pathway would also produce LGRBs at low metallicity (as an infrequent addition to those produced through more common metallicity sensitive route) in numbers predicable from extending the high metallicity LGRB rate per unit star-formation down to smaller metal-poor galaxies.
The binary explosive common-envelope ejection mechanism of <cit.> provides a credible production scenario for short-period black-hole binaries with the bare carbon oxygen core progenitor likely needed for LGRB and type Ic supernovae production. As noted therein this mechanism likely incorporates a low metallicity bias as the required red-supergiant branch ( case C) mass transfer is more common at lower metallicity. Whether this mechanism best explains only the high metallicity LGRBs (whose short period systems are not produced through typical pathways ) or all LGRBs is debatable however as only a small fraction of Ic supernovae become LGRBs we suspect in general that the introduction of a binary companion to LGRB formation, assuming it is not a prerequisite, would probably prove quite disruptive. The metallicity bias in this formation mechanism would likely favor a continuing preference for low metallicity, even in the high metallicity range, which conflicts with the exceedingly tentative evidence (that at high metallicity marginal differences in galaxy metallicity do not have an appreciable effect) presented in section <ref> and favors the case 2 scenario in section <ref>.
Overall, a binary formation model with a very particular and rare mass transfer process and bounds for LGRB production is well-suited to provide the necessary scarcity of LGRB events required to match the observed disparity between LGRB formation rates and those of non-LGRB broad-lined type Ic supernovae <cit.>. Considering the much less total coverage of present supernovae searches (compared with GRB detectors) the number of broad-lined type Ic supernovae found at their respective low redshifts likely significantly out paces LGRB losses due to non-detection of off axis bursts and the lower total star-formation in requisite low metallicity environments. The additional requirement of a binary system close enough to allow mass transfer interaction of a specific rare type would thus introduce such an additional lowering of the LGRB formation rate. Whether SNe progenitor rotation can provide the required one LGRB per $\sim$40 Type Ic-bl SNe under optimal low metallicity conditions <cit.> degrading to one LGRB per $\sim$120 Type Ic-bl SNe at higher metallicities remains an open question <cit.>.
§ SUMMARY
We present our imaging and spectroscopic observations of the host galaxies of two dark long bursts with anomalously high metallicities, LGRB 051022 and LGRB 020819B, which in conjunction with another LGRB event, LGRB 050826 (shown in figure <ref>), with an optical afterglow <cit.> comprise the three LGRBs with high metallicity host galaxies in the <cit.> sample. For both LGRB 051022 and LGRB 020819B no optical counterpart was detected however radio and X-ray source was detected classifying these as “dark" LGRBs but allowing us to locate the bursts upon their host galaxies. LGRB 051022's host is a z=0.8 extended object with two separated star-forming regions and a tidal tail (visible in figures <ref> and <ref> respectivly). LGRB 020819B occurred in a small outlying structure termed “the blob" on the outskirts of a z=0.4 grand design spiral galaxy (shown in figure <ref>).
We used the $R_{23}$, [N II]/[O II] and [N II]/H$\alpha$ diagnostics to calculate the metallicity in LGRB 051022 and LGRB 020819B. The metallicity estimates for LGRB 051022 using different diagnostics are in reasonable agreement with each other, and slightly different than that given in <cit.> from independent optical spectroscopy however our discovery of a merging morphology for this host raises the issue of internal metallicity disparity and makes such ensemble metallicity measurements less than ideal. For LGRB 020819B in addition to the host galaxy core we were also able to measure “the blob" metallicity independently and do not find its metallicity to differ from the galaxy center within the error of our estimate though only with the considerably less sensitive [N II]/H$\alpha$ diagnostic In an attempt to improve on this result (despite less than complete spectroscopy due to instrument failure) we use the lines we do have to create essentially a 3 equations (3 different metallicity diagnostics), with 3 variables problem (metallicity, extinction, and ionization), and naturally find three solutions (shown in figure <ref>). One of which does allow for “the blob" region to be at a significantly lower metallicity than the galaxy core but all solutions are still considerably higher in metallicity than that seen in typical GRB hosts.
In <cit.>, we showed that LGRBs exhibit a strong and apparently intrinsic preference for low metallicity environments. However, as we note therein, some exceptions do exist to this trend — three of the 14 LGRBs in the sample possess abundances of about solar and above. Not only do the three high metallicity LGRB hosts (051022, 020819B, and 050826) not share the typical low metallicities of LGRB hosts, they are consistent with the general star-forming galaxy population of comparable brightness & redshift. The result is intrinsically surprising: were the metal aversion to remain in effect for these objects, we would expect their occurrence (if still in the high metallicity range) to be far lower than the typical metallicity for the population at that luminosity and redshift (i.e., either a outlier of said population, or among the lowest galaxies available within it). While the majority of the LGRB population is constrained to low metallicities of about a third solar and below these exceptions probably show that is it still possible to still form an LGRB in a high metallicity environment although with greater rarity.
From this we conclude that there are three possible explanations for the presence of the LGRBs observed in high metallicity hosts as seen to date: (1) LGRBs do not occur in high metallicity environments and those seen in high metallicity hosts are in fact occurring in low metallicity environments that have become associated with otherwise high metallicity hosts but remain unenriched. (2) The LGRB formation mechanism while preferring low metallicity environments does not strictly require it resulting in a gradual decline in burst formation with increasing metallicity. (3) The typical low metallicity LGRBs and the few high metallicity cases are the result of physically different burst formation pathways with only the former affected by the metallicity and the later occurring much more infrequently. To discriminate between these possibilities we recommend scrutiny of the metallicity distribution of high metallicity LGRBs within their host galaxies (i.e. Do they favor low metallicity regions, or do they track the brightest stars, as seen in the general LGRB population <cit.>).
§ CONCLUSIONS
Here we present our imaging and spectroscopic observations of the host galaxies of two dark long bursts with anomalously high metallicities, LGRB 051022 and LGRB 020819B, which, in conjunction with another LGRB event with an optical afterglow <cit.>, comprise the known three LGRBs with high metallicity host galaxies. In <cit.> we analyze the metallicity distribution of the LGRB population at large as well as comparing and contrasting with the general star-forming galaxies and supernovae populations to conclude that the LGRB hosts are significantly depressed in metallically due to an intrinsic metal aversion preference in LGRB formation. Here, we focused on the few exceptions to this trend: high metallicity LGRB events, whose occurrence is astonishingly rare compared to the much greater volume of star-formation available at high metallicity.
Most notably, aside form their existence, is that these high metallicity LGRBs lack any apparent preference for a low metallically environment either with regard to other galaxies of similar luminosity and redshift or the location of the burst occurrence within their host galaxies. We thus conclude that despite a massive preference for low metallicity in LGRB formation, once that threshold in grossly exceeded, there remains no marginal preference for a lower metallically. This result is intrinsically surprising, as were the metal aversion effect to remain in effect for these objects we would expect their occurrence, if still in the high metallicity range, to be far lower than the typical metallicity for the population at that luminosity and redshift (i.e. either a outlier of said population or among the lowest galaxies available within it).
We do however find this result to be consistent with that of another paper we are publishing concurrently. In <cit.> we extended the <cit.> analysis by normalizing the LGRB rate to the rate of underlying star-formation across different metallicities to directly probe and quantize how much more likely is an LGRB to form at one metallicity as compared with another. We find that the gradient in LGRBs per unit star-formation is comparably flat at high metallicities after undergoing a sharp decline at log(O/H)+12 $\sim$ 8.3. Understanding this metallicity cutoff is essential to efforts to compare the LGRB rate with that of star-formation as a function of redshift <cit.>.
From this we conclude that there are three possible explanations for the presence of the LGRBs observed in high metallicity hosts as seen to date: (1) LGRBs do not occur in high metallicity environments and those seen in high metallicity hosts are in fact occurring in low metallicity environments that have become associated with otherwise high metallicity hosts but remain unenriched. (2) The LGRB formation mechanism while preferring low metallicity environments does not strictly require it resulting in a gradual decline in burst formation with increasing metallicity. (3) The typical low metallicity LGRBs and the few high metallicity cases are the result of physically different burst formation pathways with only the former affected by the metallicity and the later occurring much more infrequently.
To discriminate between these possibilities we recommend scrutiny of the metallicity distribution of high metallicity LGRBs within their host galaxies (i.e. Do they favor low metallicity regions, or do they track the brightest stars, as seen in the general LGRB population a la ). Such spatially resolved host spectroscopy have been conducted via IFU survey on one of the closest bursts, LGRB 980425 <cit.> and could be extended to slightly higher redshifts (especially for brighter & bigger galaxies). For the higher redshift targets z $\gtrsim$ 0.8 we propose a combination of laser guide star adaptive optics infrared integral field unit spectroscopy and tunable narrow band ACS ramp filter observations to overcome the limits of ground based seeing.
Finally we hypothesize on binaries as the possible source of a second lesser traveled and metallicity independent LGRB formation pathway. Assuming that the specific star-forming regions hosting the bursts and the burst progenitors themselves are high metallicity systems this significantly complicates current theories of LGRB formation. These theories nearly universally require a rapidly spinning progenitor to collimate the formation of jets required for GRB emission. The higher mass loss rates of high metallicity stars would thus bleed off the requisite angular momentum and prevent the formation of jets capable of escaping their host star. The presence of a binary companion could, under the right circumstances, spin up the progenitor star allowing for the jet formation even in high metallicity environments.
Based on observations obtained at the Gemini Observatory acquired through the Gemini Science Archive and processed using the Gemini IRAF package, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), MinistÄrio da CiÉncia, Tecnologia e InovaŹŗo (Brazil) and Ministerio de Ciencia, TecnologÆa e InnovaciŚn Productiva (Argentina).
The W.M. Keck Observatory is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program $\#$ 11343.
Support for this work was provided by NASA through grant number $\_\_\_\_\_\_\_$ from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.
Facilities: Gemini:Gillett (GMOS), Keck:II (NIRSPEC), HST (ACS/WFC, WFC3/IR), MDM:Hiltner (RETROCAM)
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1511.01032
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Which song will Smith listen to next?
Which restaurant will Alice go to tomorrow?
Which product will John click next?
These applications have in common the prediction of user trajectories that are in a constant state of flux over a hidden network (e.g. website links, geographic location).
But what users are doing now may be unrelated to what they will be doing in an hour from now.
Mindful of these challenges we propose , a method designed to cope with the complex challenges of learning personalized predictive models of non-stationary, transient, and time-heterogeneous user trajectories.
is a general method that can perform next product recommendation, next song recommendation, next location prediction, and general arbitrary-length user trajectory prediction without domain-specific knowledge.
is more accurate and up to ${\bf 413\times}$ faster than top competitors.
§ INTRODUCTION
Web users are in a constant state of flux in their interactions with products, places, and services.
User preferences and the environment that they navigate determine the sequence of items that users visit (links they click, songs they listen, businesses they visit).
In this work we refer to the sequence of items visited by a user as the user's trajectory.
Both the environment and user preferences affect such trajectories.
The underlying navigation environment may change or vary over time: a website updates its design, a suburban user spends a weekend in the city.
Similarly, user preferences may also vary or change over time: a user has different music preferences at work and at home, a user prefers ethnic food on weekdays but will hit all pizza places while in Chicago for the weekend.
The above facts result in user trajectories that over multiple time scales can be non-stationary (depend on wall clock times), transient (some visits are never repeated), and time-heterogeneous (user behavior changes over time); please refer to Section <ref> for examples.
Unfortunately, mining non-stationary, transient, and time-heterogeneous stochastic processes is a challenging task.
It would be easier if trajectories were stationary (behavior is independent of wall clock times), ergodic (visits are infinitely repeated), and time-homogeneous (behavior does not change over time).
is at least an order of magnitude faster than state-of-the-art methods for next-item predictions.
In this work we propose to tackle the problem of mining and predicting user trajectories.
takes as input a set of users and a sequence items they visit (user trajectories),
including the timestamps of these visits if available, and outputs a model for personalized next-item prediction (or next $n > 1$ items).
can be readily applied to personalized trajectories from next check-in recommendations, to next song recommendations, to product recommendations.
is highly parallel and nearly two orders of magnitude faster than the top state-of-the-art competitors.
In order to be application-agnostic we ignore application-specific user and item features, including time-of-day effects, but these can be trivially incorporated into .
To illustrate the performance of consider Figure <ref>,
where we seek to compare the Mean Reciprocal Rank (MRR) of over datasets with up to 1.6 million items and 86 million item visits (further details about this dataset is given in Section <ref>) against that of state-of-the-art methods such as Multi-core Latent Markov Embedding (MultiLME) <cit.>, personalized ranking LME (PRLME) <cit.>, and Context-aware Ranking with Factorizing Personalized Markov Chains <cit.> (FPMC).
Unfortunately, MultiLME, PRLME, and FPMC cannot finish any of these tasks in less than 10 days while for it takes between one and thirteen hours.
In significantly sub-sampled versions of the same datasets we find that is at least 23% more accurate than its competitors.
works by decomposing potentially non-stationary, transient, time-heterogeneous user trajectories into very short sequences of random walks on latent environments that are stationary, ergodic, and time-homogeneous.
An intuitive way to understand is as follows.
Random walks have been widely used for ranking items on observed graph topologies (e.g. PageRank-inspired approaches <cit.>); meanwhile,
overlapping community detection algorithms <cit.> also use observed graphs to infer latent weighted subgraphs.
But what if we were not given the environments (weighted graphs & time scales) but could see the output of a random surfer over them?
sees user trajectories and infers a set of latent environments (weighted item-item graphs and their time scales) that best describe user trajectories through short random walks over these environments; after the short walk users perform a weighted jump between environments; the jump allows to infer user preference of latent environments.
Once the model infers the relationships between short trajectories and latent environments,
we can give it any user history and current trajectory to infer a posterior over the latent environment that the user is currently surfing and, this way, perform accurate personalized next-item prediction using a random walk.
Our main contributions can be summarized as:
Comparison of Properties of State-of-art Methods.
MC MLE <cit.> Gravity Model <cit.> LDA/TM-LDA <cit.> LME/MultiLME <cit.> P(R)LME <cit.> FPMC <cit.> Temporal Tensors
(our method)
General Approach
Trajectory Model
Multiple Time Scales
Trajectory Memory
Sense Making
* (Accuracy). In our datasets predictions are always more accurate than state-of-the-art methods.
The state-of-the-art methods include Latent Markov Embedding (LME) of Chen et al. <cit.>, Multi-LME of Moore et al. <cit.>, PRLME of Feng et al. <cit.>,
and FPMC of Rendle et al. <cit.>.
is also more accurate than an application of the time-varying latent factorization method (TM-LDA) of Wang et al. <cit.>.
We also see why can better capture the latent space of user trajectories than state-of-the-art tensor decomposition methods <cit.>.
* (Parameter-free). In all our results is used without parameter tuning. Because is a nonparametric hierarchical Bayesian method, it has a few parameters that are set as small constants and do not seem significantly affect the performance if there is enough data.
The only parameter that affects performance ($B \in \mathbb{Z}^+$ explained in the model description) is safely set to a small constant ($B=1$).
* (Scalability). uses a scalable parallel algorithm to infer model parameters from the data.
is between 48$\times$ to 413$\times$ faster than our top competitors, including LME, PLME, PRLME, and FPMC.
When available, we evaluated the performance of over the datasets originally used to develop the competing methods.
* (Novelty). provides a general framework (random surfer over infinite latent environments) to build upon for application-specific recommendation systems.
Reproducibility. Of separate interest is the reproducibility of our work. Datasets, the source code and extra source code of competing methods that were not publicly available (implemented by us) can be found on our website[<http://flaviovdf.github.io/tribeflow>].
() Alice randomly chooses a (latent) environment (weighted item-item graph $G_1$ and associated time scales) according to her preferences and surfs for a short time (two steps) before randomly jumping environments.
We now present the outline of this work.
Section <ref> reviews the related work.
Section <ref> describes the model.
Section <ref> presents our results on both small and reasonably-sized datasets of real-world user trajectories.
Section <ref> also compares both in terms of accuracy and speed against state-of-the-art and naive methods.
Section <ref> shows that has excellent sense-making capabilities.
Finally, Section <ref> presents our conclusions.
§ RELATED WORK
How do we find a compact representation of the state space of network trajectories that allows easy-to-infer and accurate models? In this section, we present an overview of previous efforts related to that were also motivated by this question.
Random Walks over Observed Networks
The naive solution to the above question would be to simplify the state space of trajectories by merging nodes into communities via community detection algorithms <cit.>.
However, we do not have the underlying network, only the user trajectories.
is able to infer latent environments (weighted graphs and inter-event times) from user trajectories without knowledge of the underlying network.
Latent Markov Embedding
Latent Markov Embedding (LME) <cit.> was recently proposed to tractably learn trajectories through a Markov chain whose states are projected user trajectories into Euclidean space.
However, even with parallel optimizations <cit.> the method does not yet scaled beyond hundreds of thousands of users and items (as shown in Section <ref>).
The LME method can be seen as one of the first practical approaches in the literature to jointly learn user memory & item preferences in trajectories.
Wu et al. <cit.> present a factorization embedding adding user personalization to the LME predictions, called PLME, which is not parallel and suffers from a quadratic runtime. Very recently Feng et al. <cit.> relaxed some of the assumptions of PLME in order to focus only on rankings (thus, PRLME), making the quadratic learning algorithm become linear but not multi-core as Multi-LME.
Factorizing Personalized Markov Chains
Rendle et al. <cit.> proposes Factorizing Personalized Markov Chains (FPMC) to learn the stochastic process of user purchasing trajectories.
FPMC predicts which item a customer would add next in his or her basket by reducing the state space of trajectories of each user into an unordered set (i.e., trajectory sequence is neglected).
The resulting Markov model of each user transition in the space of items forms the slice of a tri-mode tensor which then is embedded into a lower dimensional space via a tensor decomposition.
Similarly, Aizenberg et al. <cit.> performs a matrix factorization for embedding.
Wang et al. <cit.> and Feng et al. <cit.> also make personalized factorization-style projections.
FPMC seems to be the most widely used method of this class that predicts next-items from unordered sets.
Collaborative Filtering Methods
Collaborative filtering methods can be broadly classified into two general approaches: memory-based (e.g. <cit.>) and item-based (e.g. <cit.>).
In memory-based models next item predictions are derived from the trajectory each user independently of other users.
In item-based models, next item predictions are based on the unordered trajectories of all users, which disregards the sequence of events in the trajectory.
More general unordered set predictions use Collective Matrix Factorization <cit.>.
Recently, Hierarchical Poisson Factorization of Gopalan et al. <cit.> deal with a different problem: item recommendation based on user ratings rather than user trajectories.
Chaney et al. <cit.> extends the Gopalan et al. model for recommendations when network side information exists but also does not consider trajectories.
The work of Wang et al. <cit.> uses a Latent Dirichlet Allocation-type embedding to capture latent topic transitions which we adapt to model trajectories in our evaluation (TM-LDA).
Naive Methods
Naive methods such as Gravity Model (GM) <cit.> are used to measure the average flow of users between items. Recently, Smith et al. <cit.> employs GMs to understand the flow of humans within a city. Galivanes et al. <cit.> employs GMs to understand Twitter user interactions. Note that GMs are application-dependent as they rely on a pre-defined distance function between items.
Section <ref> shows that is significantly more accurate than GM while retaining fast running times.
Other Markov Chain Approaches
A naive Markov Chain (MC) of the item-item transitions can be inferred via Maximum Likelihood Estimation (MLE) <cit.> but it does not provide enough flexibility to build good predictive models.
Fox et al. <cit.> and Matsubara et al. <cit.> propose Bayesian generalizations of Markov Switching Models <cit.> (MSMs) for a different problem: to segment video and audio.
Liebman et al. <cit.> uses reinforcement learning (RL) to predict song playlists but the computational complexity of the method is prohibitive even for our smallest datasets.
Hidden Markov Models (HMMs) can also be used to model trajectories. However, even in recent approaches <cit.>, fitting HMMs requires quadratic time in the number of latent spaces. The fitting algorithm is, in contrast, linear. HMMs are nevertheless interesting since inference is conditioned on the full sequence of a trajectory. can mimick this behavior with the $B$ parameter. We also considered novel HMM based model (called Stages) that has been proposed by Yang et al. <cit.>. Stages has sub-quadratic runtimes in the number of visited items (transitions) but the author-supplied source code did not converge to usable parameter values in our larger datasets. It is unclear why Stages is unable to converge over large datasets. In smaller datasets, where convergence did occur, Stages is less accurate than . The lower accuracy is likely because Stages is more focused on explicit trajectory commonalities and does not model personalized user transitions explicitly as does.
Table <ref> compares with the strongest competitors in the literature for trajectory prediction and sense-making. is the only method that meets all criteria: general, personalized, multiple time scales, and scalable. Our inference algorithm is sub-quadratic in asymptotic runtime, as well as fully parallel. is the only approach that is accurate, general, and scalable.
§ THE MODEL
models each user as a random surfer over latent environments.
User trajectories are the outcome of a combination of latent user preferences and the latent environment that users are exposed to in their navigation.
We use a nonparametric model of short user trajectory sequences as steps of semi-Markov random walks over latent environments composed of random graphs and associated inter-event time distributions.
Inter-event times are defined as the time difference between two consecutive items visited by a user.
The model is learned via Bayesian inference.
Our model is illustrated in Figure <ref>; in our illustration user Alice jumps to a latent environment ($\cM=1$) according to her environment preferences and performs two random walk steps on the graph $G_\cM$ with inter-event times associated to environment $\cM$.
After the two steps Alice randomly jumps to another latent environment of her preference.
Random walks on our latent environments are not to be confused with random walks on dynamic graphs.
In the former the underlying graph topology and associated environment characteristics do not change once they are drawn from an unknown probability distribution while in the latter the graph structure is redrawn at each time step.
In our applications probabilistic static environments seem like a better framework: a user with a given latent intent in mind (listen to heavy metal, eat spicy Indian food) at a given location (a webpage, a neighborhood) has item preferences (edge weights to songs, restaurants) similar to other like-minded users in the same network location.
A random graph framework would be more accurate if webpages and restaurants randomly changed every time users made a choice.
Our probabilistic environments are different from random graphs as the environment distribution is unknown and defines other characteristics such as the inter-event time distribution (simpler unidimensional examples of Markovian random walks on random environments with known distributions are given in Alexander et al. <cit.> and Hughes <cit.>.).
By construction, 's semi-Markov random walk generates ergodic, stationary, and time-homogeneous (E-S-Ho) trajectories.
models potentially non-ergodic, transient, and time-heterogeneous user trajectories (Ne-T-He) as short sequences of E-S-Ho trajectories.
By its nature E-S-Ho processes are generally easier to predict than Ne-T-He processes.
And while a single user may not spend much time performing E-S-Ho transitions, other users will use the same E-S-Ho process which should allow us to infer well its characteristics.
§.§ Detailed Description
The set of users ($\cU$) in can be agents such as people, bots, and cars.
The set of items ($\Omega$) can be anything: products, places, services, or websites.
The latent environment $\cM = 1,2,\ldots$ is a latent weighted clique $G_\cM = (\Omega,E_\cM)$ over the set of items $\Omega$.
Edge weights in $E_\cM$ have gamma distribution priors, $w_{(\cdot,v)} = w_v \sim \text{Gamma}(\beta,1)$, $\forall v \in \Omega$.
In what follows we define the operator $|\cdot|$ to be the size of a set and $\otimes$ denotes the outer-product.
Each user $u \in \cU$ generates a “sequence trajectory” of length $B+1$ at the $t$-th visit, $t \geq 1$,
$$(x_{u,t},\ldots, x_{u,t+B}) \in \Omega^{B+1} \, , B \geq 1 \, , $$
before jumping to another environment $\cM'$ according to user preference distribution $\pi_{\cM' | u}$.
The entire trajectory $x_{u,1}, x_{u,2},\ldots$ of user $u \in U$ is the concatenation of such sequences.
The time between observations $x_{u,t+k}$ and $x_{u,t+k+1}$ is the $k$-th inter-event time $\tau_{u,t+k}$, $k = 0,\ldots,B$.
Special care must be taken with the last event of a user, which does not have an inter-event time.
The random walk over $G_\cM$ is modeled as a semi-Markov process with inter-event time $\tau_{u,t} \sim \lambda(\cM)$ (a.k.a. holding or residence times).
Note again that inter-event times depend on the current latent environment.
The model of the user and the semi-Markov transition probability matrix mixture.
We now define the transition probability matrix of our random walk over the random graph $G_\cM$.
The random walk over graph $G_\cM$ of environment $\cM$ is a semi-Markov chain with a random $|\Omega| \times |\Omega|$ transition
probability matrix distributed as
\begin{align}\label{e:P}
{\bf P}_\cM \sim (I - \text{diag}(\phi_\cM))^{-1} (\phi_\cM \otimes \phi_\cM - \text{diag}(\phi_\cM^2)) \, ,
\end{align}
where $\text{diag}(\cdot)$ is a diagonal matrix and
$\phi_{\cM} \sim Dirichlet(\cdot \mid \beta)$.
Semi-Markov chain $\cM$ is stationary, ergodic, and time-homogeneous with high probability if $|\Omega| > 2$.
Without loss of generality we assume that the walker starts at $o \in \Omega$.
The semi-Markov random walk with transition probability matrix ${\bf P}_\cM$ over $G_\cM$, $\cM \geq 1$, sees edge weights $w_v \sim \text{Gamma}(\beta,1)$, $\forall v \in \Omega \backslash \{o\}$.
The probability that the walk moves to $v \neq o$ is $w_v / S_{\neq u}$, where $S_{\neq o} = \sum_{j \in \Omega \backslash \{o\}} w_j$.
Let ${\bf P}_\cM(o,\Omega \backslash\{ o\})$ denote the off-diagonal elements of the random walk transition probability matrix ${\bf P}_\cM$.
Because $\{w_j\}_{j \in \Omega}$ are independent and Gamma distributed then
$${\bf P}_\cM(o,\Omega \backslash\{ o\}) = (w_v / S_{\neq u})_{v \in \Omega \backslash \{o\}}$$
follows a Dirichlet distribution $\text{Dirichlet}(\beta,\ldots, \beta)$ <cit.>.
Note that ${\bf P}_\cM(o,o) = 0$.
A little algebra gives Eq. (<ref>).
The chain is trivially stationary and time-homogeneous as transition probabilities do not change over time.
We now show the chain is ergodic.
Any state $j \in \Omega$ is reachable from any state $i \in \Omega$ as $({\bf P}_\cM)^n(i,j) > 0$ for some $n \geq 1$ implying that the chain is recurrent <cit.>. As $|\Omega| < \infty$ the chain is positive recurrent.
By construction $G_1$ is connected and thus ${\bf P}_\cM$ is irreducible.
For $|\Omega| > 2$ the graph $G_1$ is not bipartite and making the chain aperiodic.
If a chain is irreducible, aperiodic, and positive recurrent, then it is ergodic <cit.>.
Note that the random walk does not have revisits, $x_t \neq x_{t+1}$, $\forall t > 0$.
In the datasets we remove all revisits because re-consumption (repeated accesses to the same item) tends to be easy to predict <cit.>, highly application-specific, and can be decoupled entirely from our problem via stochastic complementation using phase-type Markov chains with a single entry state such as the ones in Neuts <cit.>, Robert and Le Boudec <cit.> and Kleinberg <cit.>.
Gathering all elements together we obtain the model illustrated in Figure <ref>, which can be seen as a random surfer taking $B$ steps over a latent graph $G_\cM$ and then randomly moving to a new environment according to the following generative model:
* Draw $\zeta \sim \text{GEM}(\gamma)$ according to a stick-breaking process.
* For each user $u \in \cU$ sample $\pi_{\cM|u} \sim \text{Dirichlet}(\cdot \mid \alpha \zeta)$.
* Draw a semi-Markov random walk transition probability matrix ${\bf P}_\cM \sim (I - \text{diag}(\phi_\cM))^{-1} (\phi_\cM \otimes \phi_\cM - \text{diag}(\phi_\cM^2)) \, ,$ where $\text{diag}(\cdot)$ is a diagonal matrix and $\phi_{\cM} \sim \text{Dirichlet}(\cdot \mid \beta)$, $\cM = 1,2,\ldots$.
* For a given user $u$ each sequence burst $(x_t,\ldots,x_{t+B})_{u}$ with inter-event times $(\tau_{t},\ldots,\tau_{t+B-1})$ is generated as follows:
* Draw a latent semi-Markov chain
$\cM \sim \text{Multinomial}(\pi_{\cM|u})$.
* For $k=0,\ldots,(B-1)$ select item $x_{t+k}$ according to probability ${\bf P}_\cM(x_{t+k},x_{t+k+1})$ and inter-event time $\tau_{t+k'} \sim \lambda(\cM)$, $k'=0,\ldots,B$, where $\lambda(\cM)$ is the inter-event time distribution of environment $\cM$. Item $x_1$ is drawn uniformly from $\Omega$.
§.§ Inferring Model from Data
In what follows we describe how we learn from data.
Given a set of user trajectories $\{(x_{u,1},x_{u,2},\ldots) : \forall u \in \cU\}$ from a set of items $x_{u,t} \in \Omega$, $t \geq 1$, we infer:
* The number of environments $K > 1$ from the data.
* $K$ semi-Markov transition probability matrices $\{{\bf P}_\cM : \cM = 1,\dots,K\}$ corresponding random walks over a finite set of graphs $\{G_\cM$ : $\cM=1,\dots,K\}$.
* A distribution of user environment preferences $\{\pi_{\cM|u} : u \in \cU\}$.
If the inter-event time distribution $\lambda(\cM)$ comes of a known family we can also get a distribution of inter-event times for each environment.
The probability that a user sees a sequence $x_{u,t},\ldots,x_{u,t+B}$ with inter-arrival times $\tau_{u,t},\ldots,\tau_{u,t+B}$ at environment $\cM$ is
\begin{align*}
P[&x_{u,t},\ldots,x_{u,t+B},\tau_{u,t},\ldots,\tau_{u,t+B} | \cM] = \nonumber\\
&\prod_{k=0}^{B-1} {\bf P}_\cM(x_{u,t+k},x_{u,t+k+1}) P[\tau_{u,t+k} | \cM] P[\tau_{u,t+B} | \cM] \, \nonumber,
\end{align*}
with ${\bf P}_\cM$ as given in Eq. (<ref>).
The probability we observe such burst for user $u \in \cU$ is then
\begin{equation}
\label{eq:Bm}
\begin{aligned}
&P[x_{u,t},\ldots,x_{u,t+B},\tau_{u,t},\ldots,\tau_{u,t+B} | u] = \\
&\sum_{K =1}^\infty P[ \zeta ] P[\tau_{u,t+B} | \cM] \pi_{\cM|\alpha\zeta,u} \\
&\qquad \times \prod_{k=0}^{B-1} {\bf P}_\cM(x_{u,t+k},x_{u,t+k+1}) P[\tau_{u,t+k} | \cM] \, \, ,
\end{aligned}
\end{equation}
where $P[\zeta]$ is the stick-breaking prior over $\cM$.
Unrolling Eq. (<ref>) into Eq. (<ref>) we obtain the equation that describes the trajectory:
\begin{equation}
\label{eq:full}
\begin{aligned}
P[&x_{u,t},\ldots,x_{u,t+B},\tau_{u,t},\ldots,\tau_{u,t+B-1} | u] \propto \\
&\sum_{K =1}^\infty P[ \zeta ] P[\tau_{u,t+B} | \cM]\, \pi_{\cM|\alpha\zeta,u} \\
&\qquad \times \prod_{k=0}^{B-1} \frac{\phi_\cM(x_{u,t+k+1})}{1 - \phi_\cM(x_{u,t+k})} P[\tau_{u,t+k} | \cM] \, .
\end{aligned}
\end{equation}
We use collapsed Gibbs sampling to estimate the model parameters. Initially, given a sequence of size $B + 1$, we transform user trajectories into a set $\cD$ of tuples using a sliding window over the trajectories of each user.
To exemplify, for $B=2$ each entry is:
$$(u, x_{u,t},x_{u,t+1},x_{u,t+2},\tau_{u,t},\tau_{u,t+1}) \in \cD \:, t \geq 1 .$$
This tuple represents the user $u$, the trajectory $x_{u,t},\ldots,x_{u,t+B}$ and the inter-event times $\tau_{u,t},\ldots,\tau_{u,t+B}$ for every time $t \geq 1$.
The use of sliding windows, while not theoretically justified by our model, tremendously simplify our inference problem by not forcing us to decide how
to segment the data into blocks of $B$ events or forcing us to make $B$ a random variable.
Adding random jumps would entail a costly forward-backward inference procedure needed to decide when users jump between environments.
To infer , our heuristic starts with an initial estimate of the number of environments $K$ and randomly assign each tuple in $\mathcal{D}$ to one environment.
After the initial assignment we count the number of tuples of each user $u$: $n_u = \sum_{\forall (u', \ldots) \in \mathcal{D}} {\bf 1}(u' = u)$, where ${\bf 1}$ is the indicator function.
We also count the number of times environment $\cM$ is assigned to a tuple from user $u$: $e_{\cM,u}$, as well as the joint count of items, at any position, and environments: $c_{i,\cM}$, and count the number of tuples assigned to an environment $\cM$: $a_{\cM}$.
Assuming, for now, that $\zeta$ is given, we can infer <cit.>
\begin{align} \label{eq:dirichlet}
\pi_{\cM|\alpha\zeta,u} = \frac{e_{\cM,u} + \alpha \zeta(\cM)}{n_{u} + K\alpha\zeta(\cM)} \quad , \quad
\phi_{\cM}(i) = \frac{c_{i,\cM} + \beta}{a_{\cM} + |\Omega|\beta}.
\end{align}
We then employ ECME <cit.> inference where: (1) the e-step consists of one pass over the entire dataset performing a Gibbs sampling update (in other terms, one iteration of the collapsed Gibbs sampler); (2) an m-step where the algorithm considers the probability of inter-event times according to the following procedure.
Summary of our datasets.
# Transitions # # Inter-event Times Timestamp Span
Last.FM-1k User Artist 10,132,959 992 348,156 Yes Feb. 2005 to May 2009
Last.FM-Groups User Artist 86,798,741 15,235 1,672,735 Yes Feb. 2005 to Aug. 2014
BrightKite User Venue 2,034,085 37,357 1,514,460 Yes Apr. 2008 to Oct. 2010
FourSQ User Venue 453,429 191,061 87,345 Yes Dec. 2012 to April 2014
YooChoose Session Product 19,721,515 6,756,575 96,094 Yes Apr. 2014 to Sept. 2014
Yes Playlist Song 1,542,372 11,139 75,262 No (No timestamps)
If the inter-event times of environments $\cM=1,2,\ldots$ have a known probability law $\lambda(\cM)$ we can include this law in our model with the appropriate priors (e.g. a Normal distribution with fixed variance can have a Normal prior), updating the distribution parameters in each m-step. If the law of $\lambda(\cM)$ is unknown but inter-event times are observed, we use the empirical complementary cumulative distribution function (ECCDF) in the the following heuristic: We estimate the ECCDF of each $\cM$ based on the entries assigned to $\cM$ on the last e-step.
If $T_\cM$ is the random variable that defines the inter-arrival at environment $\cM = 1,\ldots$.
For inter-event time $\tau_{u,t}$ the probability $P[T_\cM > \tau_{u,t}]$ given the current ECCDF is the number of entries whose observed inter-event times in $\cM$ are greater than $\tau_{u,t}$.
Thus, ${\bf 1}(T_\cM > \tau_{u,t})$ is a Bernoulli random variable <cit.> with parameter $p = P[T_\cM > \tau_{u,t}]$.
Adding a conjugate prior $Beta(1, K-1)$ to the Bernoulli gives the following predictive posterior <cit.>:
\begin{align} \label{eq:ptau}
P[\tau_{u,t} | \cM] \propto \frac{b_{>\tau_{u,t},\cM} + 1}{n_{\cM} + K} \, ,
\end{align}
where $b_{>\tau_{u,t},\cM} $ is the number of tuples currently assigned to $\cM$ that have inter-arrival times greater than $\tau_{u,t}$. It is easy to see from Eq. (<ref>) that transitions with large inter-event times w.r.t. other inter-event times in $\cM$ are less likely to have been generated by $\cM$. This captures the intuition that inter-event times in an environment should be similar and is an integral part of how we learn $\zeta$ as detailed below.
In our experiments the ECCDF heuristic consistently provides better results than assuming Normal inter-event time distributions. Testing other application-specific inter-event time posteriors is of interest in more application-specialized future work.
We now infer $K$, the number of environments, by adapting the partial expectation (partial-e) and partial maximization (partial-m) heuristics of Bryant and Sudderth <cit.> for hierarchical Dirichlet processes (HDPs) as follows: (a) merge pairs of redundant environments when there is gain in the joint posterior probability of model & data; and (b) split environments based on their inter-event times by separating the 5% entries with highest inter-event time if there is a marginal gain in the posterior. The merge step (partial-e) is equivalent to the one described in <cit.>. Our split step (partial-m) splits a environment with high variance inter-event times into two environments with lower variance if the splitting improves the joint posterior probability of model & data.
Parallel Learning: Finally, the model can be fully trained in parallel using the approach described in Asuncion et al. <cit.>. Each processor learns the model on a subset of the dataset. After a processor performs an “E” and an “M” steps, another processor is chosen at random to synchronize the model state (merge count matrices of the Gibbs sampler and update ECCDF estimates), leaving them with the same count matrices and ECCDF estimates. After a fixed number of iterations, which we fix as 200, every processor meets a barrier and the partial-e and partial-m steps (merges and splits) are performed on a single master processor. After these partial-e and partial-m steps, the master-processor updates all slave processors and the parallel learning continues as previously described. The learning ends after a fixed number of iterations, which we set as $2,\!000$ for the results in Section <ref>.
Inference without Timestamps: In a few of our datasets timestamps are not available.
In these cases we only need to infer ${\bf P}_\cM$ and $\pi_{\cM|\alpha \zeta, u}$ without the need to take into account $P[\tau_{t+k} | \cM] $.
In our results we also take the opportunity to assume that the number of environments is fixed ($\zeta(\cM) = 1$) and infer the model posteriors using collapsed Gibbs sampling <cit.> employing Eqs. (<ref>) and (<ref>), updating the counts at each iteration (updating the posterior probabilities). We denote the latter simpler approach -NT (-NoTimestamps).
Prior Parameters: For all of our experiments, we fix prior parameters $\alpha = 50/K$ and $\beta = 0.001$. Constant $\gamma$ does not need to be stated explicitly. In the presence of large amounts of data these priors are expected to have little impact on the outcome of Bayesian nonparametric models <cit.>.
§.§ Predictions
In this work has two prediction tasks: next-item likelihood predictions and ranking.
The personalized predictive likelihood of user $u \in \cU$ for candidate next-item $\tilde{x}_{u,t+1} \in \Omega$ is based on the last $B$ items and inter-event times of the user.
Observing that user $u$ has chosen items $x_{u,t-1},\ldots,x_{u,t-B}$ with inter-event times $\tau_{u,t-1},\ldots,\tau_{u,t-B}$ gives a posterior probability that $u$ is performing a random walk at latent environment $\cM \in \{1,\ldots,K\}$, where $K$ is the learned number of latent environments, typically $K < 10^3$ in our experiments.
More precisely, the posterior probability that $u\in \cU$ is in environment $\cM$ after choosing items $x_{u,t-1},$ $\ldots,x_{u,t-B}$ with inter-event times $\tau_{u,t-1},\ldots,\tau_{u,t-B}$ is
$$P[\cM | u, x_{u,t-1},\ldots,x_{u,t-B},\tau_{u,t-1},\ldots,\tau_{u,t-B}],$$
which yields the full likelihood
\begin{align} \label{eq:ll}
&P[\tilde{x}_{u,t+1} | u, x_{u,t-1},\ldots,x_{u,t-B},\tau_{u,t-1},\ldots,\tau_{u,t-B}] = \\
% & \sum_{\cM = 1}^K {\bf P}_\cM(x_{u,t},\tilde{x}_{u,t+1}) P[\cM | u, x_{u,t-1},\ldots,x_{u,t-B}, \tau_{u,t-1},\ldots,\tau_{u,t-B}] = \nonumber\\
& \left( \prod_{k=1}^{B} {\bf P}_\cM(x_{u,t-k},x_{u,t-k+1}) P[\tau_{u,t-k} | \cM] \pi_{\cM|\alpha \zeta,u} \right) \nonumber \\
& \times \frac{\sum_{\cM = 1}^K {\bf P}_\cM(x_{u,t},\tilde{x}_{u,t+1})}{\sum_{\cM = 1}^K \prod_{h=1}^{B} {\bf P}_\cM(x_{u,t-h},x_{u,t-h+1}) P[\tau_{u,t-h} | \cM] \pi_{\cM|\alpha \zeta,u}} \nonumber .
\end{align}
The task of ranking the next items is easier and faster than predicting their likelihood.
This is because we can speed up the predictions by not computing the denominator in eq. (<ref>), as the denominator is the same for all values of $\tilde{x}_{u,t+1} \in \Omega$.
Note that our rankings are personalized and consider the inter-event times.
§ RESULTS
Previous sections introduce and explain how to infer its posteriors from data.
We now turn our attention to compare against state-of-the-art approaches, solving the very same problems over some of the very same datasets (if publicly available) as their original papers.
We also use larger publicly available dataset and one ever larger dataset that we collected for this study (available for download at our website).
In what follows Section <ref> contrasts against state-of-the-art methods for next-item ranking.
Section <ref> compares against methods that learn latent Markov chains and predict the likelihood of next items.
Finally, Section <ref> contrasts ability to predict average mean-field user flows against that of Gravity Model.
Although is able to predict not just the next-item but also the next $n \geq 1$ items, our evaluations are based at next-item predictions because previous efforts mostly focused their evaluations on this task.
We also consider the reconsumption problem (consecutive visits to the same item) treated in Figueiredo et al. <cit.> as a separate, often easier, problem that can be dealt with via stochastic complementation as discussed in Section <ref>.
§.§ Datasets and Evaluation Setup
Our datasets encompass three broad range of applications: (a) location-based social networks (check-in datasets), (b) music streaming applications, and (c) user clicks on e-commerce websites.
It is important to point our that recommendation engines and user interfaces influence user navigation and their trajectories. Such effects are considered to be an integral part of our predictive task.
But can as easily learn pure user preferences if given a dataset with no environment bias (when that is possible).
Table <ref> summarizes our datasets showing the number of users and items, the total number of $x_{u,t}, x_{u,t+1}$ pairs visited all users (or transitions/trajectories when $B=1$) as well as the time span covered by the dataset. The set of items, $\Omega$, can be songs or artists on music datasets, venues on check-in data, and products on e-commerce data. The set of users $\cU$ are individuals, a “playlist”, or a “browser session” as described next.
Last.FM is a music streaming service that aggregates data from various forms of digital music consumption, ranging from desktop/mobile media players to other streaming services. This dataset was crawled in August 2014, using the user groups[Pages in which the user discusses musical artists] feature from Last.FM. We manually selected 15 groups of pop artists and two general interest groups[Active Users, Music Statistics, Britney Spears, The Strokes, Arctic Monkeys, Miley Cyrus, LMFAO, Katy Perry, Jay-Z, Kanye West, Lana Del Rey, Snoop Dogg, Madonna, Rihanna, Taylor Swift, Adelle, and The Beatles]. For each group, we crawled the listening history of a subset of the users (the first users listed in the group).
Last.FM-1k. The second Last.FM dataset was collected in 2009 using snowball sampling by Celma et al. <cit.>.
BrightKite. Brightkite is a location based social network (LBSN) where users share their current locations by check-ins. In this publicly available dataset, each is a location where are checks-in. Collected by Cho et al. <cit.>.
FourSQ Our second LBSN dataset was gathered from FourSquare by Sarwt et al. <cit.> in 2014.
YooChoose. This dataset is comprised of user clicks on a large e-commerce business. Each is captured by a session and the trajectories capture clicks on different products within the session.
As “users” are actually browser sessions, there is an upper-limit of 12 hours on recorded inter-event times of a single “user” (session).
Yes. Finally, the Yes dataset consists of song transitions (playlists) of popular broadcast (offline) radios in the United States. This dataset does not provide explicit user information or timestamps. However, we use -NT by defining the playlists as users and each song as an item. The Yes dataset was collected by Chen et al. <cit.> to develop LME.
No filtering or trimming is done over the original data for the results shown in Figure <ref>.
In our evaluation of trajectory predictions we divide the datasets into “past” ($\cD_\text{past}$) and “future” ($\cD_\text{future}$) by selecting a timestamp that splits the dataset into “the first 70% transitions” for “past” (training set) and the remaining 30% transitions in the data in “future” (test set), with the exception of Yes that has no timestamps.
This training and testing scenarios best represent real-life situations where the training is performed in batches over existing data.
For instance, in this realistic training and test setting we may need to predict the next transitions of a user that belongs to the test set but not to the training set.
Note that some users will have trajectories confined in the “past” dataset while the trajectory of “new users” may be entirely placed in the “future” dataset.
For the Yes data the training and testing sets are the ones pre-defined by Chen et al. <cit.>.
Due to limitations in scalability of state-of-the-art methods we also exploit subsamples of our larger datasets when necessary.
These subsamples have the first 1000, 2000, 5000, 10,000, 20,000 and 100,000 transitions ordered by timestamps.
We test 's robustness through these dataset subsamples.
Setup: Our tests run on a server with 2$\times$10-core Intel Xeon-E5 processors and 256 GB of RAM.
We tested and present results of with $B = 1,2,3,4,5$, fixing other hyper-parameters as discussed in Section <ref>.
Our results show that regardless of the choice of $B$, outperforms competitors.
§.§ for Next-item Ranking
Before we compare against competing approaches it is important to note only can handle our larger datasets as evidenced by Figure <ref>.
All of our comparison against competing state-of-the-art methods are performed over small or sub sampled datasets due to the large execution times.
We start our discussion on ranking methods by focusing on the evaluation metric: the mean reciprocal rank.
The reciprocal rank, $RR$, is the inverse of the position of the destination ${x}_{u,t+1}$ on the ranking of all potential candidates in decreasing order. That is, if candidate, $\tilde{x}_{u,t+1}$, destinations Big Brewery, Pizza Place, and Sandwich Shop are ranked with probabilities $0.4$, $0.5$, and $0.1$ respectively, and the true destination was Big Brewery, the reciprocal rank has value $1/2$. Using , the RR can be computed using Eq. (<ref>) as follows (with $B=1$):
\begin{align}RR({x}_{u,t+1}, x_{u,t}, \tau_t, u) = \frac{1}{rank(P[{x}_{u,t+1} \mid x_{u,t}, \tau_{u,t}, u])}.\end{align}
Based on the reciprocal rank we define a single metric measured over the entire test set. This metric is simply the mean of reciprocal rank values over every transition in $\mathcal{D}_{test}$ (MRR).
with HDP heuristics Before comparing with competing approaches, we test if our heuristic of HDP expansion and contraction of latent environments & inter-event time ECCDF inference improves the quality of the predictions. That is, we run with and without the partial e- and m-steps described in Section <ref>.
We train with an initial guess of $K=100$ environments and -NT (without partial e- and m-steps) with $100$ environments.
We evaluate the MRR at our largest datasets: BrightKite, LastFM-Groups, LastFM-1k, and YooChoose, and one small dataset (FourSQ).
At FourSQ and YooChoose both and -NT have the same accuracy, possibly because of lower quality timestamps: FourSQ user data is sub-sampled in time and YooChoose browser sessions timeout at 12 hours.
For the remaining datasets the MRR gains of over -NT are 7%, 45%, and 53%, for LastFM-1k, LastFM-Groups, and BrightKite, respectively.
This shows that our partial e- and m-step heuristics can significantly improve the results.
again outperforms in ranking task (subsampled datasets with only $\bf 10^4$ transitions).
v.s. state-of-the-art We now turn to our comparison of against state-of-the-art competing ranking approaches.
Our first competitor is Factorizing Personalized Markov Chain (FPMC) of Rendle et al. <cit.>.
FPMC was initially proposed to predict the next object a user will insert into an online shopping basket.
If we consider each source $x_t$ as a size one shopping basket, FPMC can be used to rank the candidate items, or destinations $\tilde{x}_{t+1}$, that a user will consume next.
Our second competitor is the best-performing Latent Markov Embedding (LME) method in our datasets: Personalized Ranking by Latent Markov Embedding (PRLME) of Feng et al. <cit.>.
Inspired by Personalized Latent Markov Embedding <cit.> (PLME), the PRLME approach focuses on rankings and not on extracting Markov chains.
Figure <ref> presents the results of against FPMC and PRLME over datasets Bkite, FourSQ, LFM-Groups, LFM-1k, YooChoose subsampled to the first 10,000 transitions only due to scalability issues of FPMC and PRLME (even when limiting inference with 1000 stochastic gradient descent iterations).
The figure shows the MRR scores (y-axis) against running times in log-scale (x-axis).
Each point in the figure represents one method-dataset pair, while different datasets are represented by different colors. We label each point in order to help readability.
The top-left corner of the figure indicates the best accuracy and shorter runtime.
In these tests, as with all our tests, we use 70% of the initial user transitions to perform inference and the last 30% to test accuracy.
In Figure <ref> we see that is 23% more accurate and 46$\times$ times faster than the best result of PRLME.
In all cases is more accurate and faster than PRLME and FPMC, oftentimes is two orders of magnitude faster.
This is surprising as runs parameter-free while PRLME and FPMC both optimize over a large parameter space to obtain their best results[We perform a grid-search over parameters, testing 3 to 5 different values for each parameter of each method.].
Our evaluation also considers a range of subsampled transitions in the datasets, from $10^3$ to $10^5$ transitions.
The accuracy results are similar in all cases, with showing consistently more accurate results.
Interestingly, using the results from the LastFM-Groups subsamples ($1000, 5000, 10000, 20000,10^5)$ using a simple linear regression reveals that it would take over six years to run PRMLE and FPMC in the full LastFM-Groups dataset using our server.
The minimum expected running time of PRMLE/FPMC on a complete dataset is 14 days for the FourSQ dataset (the smallest dataset).
The parameter search, as well as the lack of parallelism, greatly impact the runtime of PRLME and FPMC. Nevertheless, both methods are sub-quadratic as is . In the next sub-section we shall compare with another fully parallelized baseline.
Finally, we point that we also compare with the Stages method proposed by Yang et al. <cit.>. Our simulations used an author-supplied source code[<http://infolab.stanford.edu/ crucis>] that, unfortunately, did not converge to usable parameter values except over a few of the sub-sampled datasets. In the cases where the model converged, best results of Stages over were observed in the YooChoose data (sub-sampled to 10k transitions) where Stage's MRR value is 0.15 while 's MRR is 0.25. That is, in its best-performing dataset Stages is 40% less accurate than . Moreover, Stages achieved MRR values of 0.05 for Brighkite (against 0.57 on ) and of 0.10 on FourSQ (against 0.14 for ).
The results discussed so far consider only average measures of the effectiveness of the rankings produced by the methods. Going a step further, we also performed a Kolmogorov-Smirnov test between the distributions of reciprocal rank values obtained by and the competing methods.
The results again clearly indicate that obtains larger RR values over all datasets ($p < 0.001$), which is consistent the MRR results.
Another aspect is the impact of the walker sequence length $B$ on performance.
We test with $B = 1,2,3,4,5$ finding that larger $B$ improves accuracy at FourSQ, LFM-1k, LFM-Groups and reduces accuracy at Bkite and YooChoose as shown in Figure <ref>.
As the best choice of $B$ is application dependent, we recommend fixing $B=1$.
Impact of walker sequence length $B$.
§.§ 's Execution Time
In this section we compare the performance of against Multi-LME <cit.> in our smallest datasets, namely Yes and FourSQ. These experiments validate the performance (in training time) of against another fully parallelized method.
For the sake of fairness, we also compare with MultiLME using the next-item likelihood measure: $P[x_{u,t+1} \mid x_{u,t}]$ as in Chen et al. <cit.> and other naive approaches (see below).
The predictive likelihood using can be computed as:
\begin{align} \label{eq:ourll}
P[x_{u,t+1} \mid x_{u,t}] = \sum_{\cM} \mathbf{P}_{\cM}(x_{u,t+1}) \mathbf{P}_{\cM}(x_{u,t}) P[\cM]
%{\bf P}^{(\text{TF})} = \sum_{\cM} {\bf P}_\cM \sum_{u} \pi_{\cM|\alpha\zeta,u} P[u] \, ,
\end{align}
where $P[\cM] \propto$ the number of tuples assigned to $\cM$ in the inference.
Accuracy on the test set is measure using the log likelihood:
\begin{align} \label{eq:predll}
\text{PredLL}(\cD_\text{future}) = \sum_{x_{u,t+1}, x_{u,t} \in \cD_\text{future}} log(P[x_{u,t+1} \mid x_{u,t}]).
\end{align}
Naive Approaches
To compute $P[x_{u,t+1} \mid x_{u,t}]$ one can also trivially adapt both Latent Dirichlet Allocation (LDA) <cit.> and Transition Matrix LDA (TM-LDA) <cit.> for the same task.
In LDA each we define users as “documents”, environments are “topics”, and $x_{u,t}$ is the $t$-th word.
TM-LDA follows the same definitions.
LDA is trained with $K$ latent factors and TM-LDA creates a $(K,K)$ matrix capturing the probability of transitioning between a latent “topics”.
Note that in these models $P[x_{u,t+1} \mid x_{u,t}] = P[x_{u,t+1}]$.
We refer to these adaptations as LDA' and TM-LDA' for simplicity as they do not reflect the original applications of LDA and TM-LDA.
Inference Procedure
In our evaluations we use the fully parallelized Multi-LME implementation <cit.>[<http://www.cs.cornell.edu/People/tj/playlists/>]. The LDA' and TM-LDA' adaptations to our problem are trained using <cit.> which includes a fast/online <cit.> and parallelized implementation of these methods. While faster LDA' training methods do exist <cit.>, we preferred to make use of a mature software package.
The Yes dataset of Chen et al. <cit.> has no timestamps, thus we use -NT for this comparison instead of the more accurate general method.
Each method was trained using different values of $K$, the number of ($K \in \{10, 50, 25, 100\}$) on the Yes dataset and with $K=10$ on FourSQ because the original Multi-LME code has difficulties scaling to more factors on FourSQ.
outperforms competition in next-item predictive log-likelihood task (small dataset due to competitors).
[“Classical Music”]
[“Movie Composers”]
[“Heavy Metal”]
[“Electro House”]
Examples of popular items in latent environments in a dataset dominated by pop music fans (LFM-G).
Figure <ref> presents our results in the next-item predictive log-likelihood task using $K=100$ and $K=10$ on the Yes and FourSQ datasets.
We get similar results for Yes with $K \in \{10, 25, 50\}$.
For the sake of interpretability we normalize the predictive log likelihood of each method by the best result (always ).
Each point in the figure represents one method-dataset pair, while different datasets are represented by different colors. We labelled each point in order to help the interpretation of the results. With these settings, it is expected that the method that performs the best embedding is placed in the top-left corner of the figure (higher accuracy, lower runtime).
The x-axis represents the execution time, whereas the y-axis represents the normalized likelihood.
In Figure <ref> we clearly see that -NT is the best approach on both datasets.
Compared to MultiLME -NT achieves a higher log-likelihood at a fraction of the runtime (speedups are up to 413$\times$).
As expected, -NT and MultiLME usually outperforms the LDA-based baselines since -NT and MultiLME were built to explicitly capture user trajectories.
Interestingly, comparing the speed of and -NT in Figures <ref> and <ref> we see that for FourSQ -NT is one order of magnitude faster than but less accurate than . This occurs because -NT does not infer inter-event time distributions using the ECCDF heuristic.
§.§ Flows of Users Between Locations
outperforms the state-of-the-art methods in sophisticated tasks such as ranking and predictive next-item likelihoods.
But what about a simpler task? Can outperform simple application-specific methods?
In this section we compare against the Gravity Model (GM) for uncovering the average flows of users between two locations.
The widely popular Gravity Model (GM) uses GPS coordinates and requires a pre-defined distance function, $dist$, capturing the proximity of locations around the globe.
As in Smith et al. <cit.> and Garcia-Gavilanes et al. <cit.> we employ the distance on a sphere from the latitude and longitude coordinates of the venues in our LBSN datasets.
The three parameters, $\theta_1$, $\theta_2$ and $\theta_3$ of the distance are fitted using a Poisson regression that is known to lead to better results <cit.>.
Our goal is to estimate $f_{ds}$, the flow of users going from location $s \in \Omega$ to location $d \in \Omega$.
Let $n_d = \sum_{x_{u,t+1}, x_{u,t} \in \cD_{present}} {\bf 1}(x_{u,t+1} = d)$ denote the number of visits of all users to a destination $d$ and $r_s$ be the equivalent number of visits of all users to a source location $s$.
GM captures the flows of users between the two locations $f_{ds}$ as
$$\hat{f}^{(\text{GM})}_{ds} = \frac{r_s^{\theta_1} n_d^{\theta_2}}{dist(d, s)^{\theta_3}}.$$
can trivially estimate the flow $\hat{f}^{(\text{TF})}_{ds}$ from Eq. (<ref>).
Since gravity models are limited to geolocated datasets (need GPS coordinates), we compare with GM on our FourSQ and Brightkite.
Note that unlike GM, is application-agnostic and does not use GPS coordinates, albeit it would be straightforward to incorporate such application-specific features in the latent environments.
Further, we opt to use -NT (instead of the more accurate method) because of its slightly faster running time, trading-off accuracy for speed.
We infer the posteriors from -NT with $K=10$ environments.
In this setting, training both models takes less than 5 minutes.
Methods are evaluated using the mean absolute error (MAE).
-NT significantly outperforms GM for geolocation flows with just a few ($K=10$) random environments and, unlike GM, without GPS coordinates.
Specifically, GM achieves MAE results of 10.48 and 9.81 on the Bkite and FourSQ datasets, respectively.
In contrast, -NT achieves 1.606 and 1.41 on Bkite and FourSQ, respectively.
The improvements of range from roughly 800% to 900% in mean absolute error.
Again, validating our results using the Kolmogorv-Smirnov test showed that -NT is statistically more accurate than GM ($p < 0.001$).
§ EXPLORATORY ANALYSIS
Latent flow environments inferred by without GPS information.
[User 1]
[User 2]
Example of Two Users From Last.FM-Groups
In this section we consider how can be used for sense-making in our datasets.
Section <ref> discusses the semantics of inferred by in our LastFM-Groups dataset, specially in the presence of non-stationary, transient, and time-heterogeneous user trajectories.
Section <ref> introduces latent environments inferred in the FourSQ dataset without GPS data.
Finally, Section <ref> discusses how compares with tensor decomposition approaches when uncovering meaningful patterns of user behavior from stationary user trajectory data.
§.§ Artist-to-Artist Transitions
Figure <ref> shows five latent environments inferred by from the LastFM-Groups dataset.
Each latent environment is represented as a word-cloud of the names of the top 15 artists in the environment ranked by the random walk steady-state probability at each environment.
We cross-reference the top artists at each latent environment in Figures <ref>(a-e) with the AllMusic guide[<http://www.allmusic.com/>], finding that the environments discovered are semantically meaningful.
It is worth noting that users in the LastFM-Groups dataset are overwhelmingly declared fans of pop music (as described in Section <ref>), but even then, is able to extract the user interests of very diverse musical themes such as those depicted in Figure <ref>.
Taking a more in-depth look at the latent environment represented in Figure <ref>(a) shows a sequential user trajectory preference for songs related to “Classical Music”.
Some of the main artists/composers in this environment are: Chopin, Bach, Beethoven and Mozart.
The latent environment in Figure <ref>(b) shows composers of motion picture sound tracks.
John Williams, for instance, compose soundtracks for popular movies such as Star Wars, Jaws, ET and Superman.
The environment in Figure <ref>(c) represents popular rock bands such as Nirvana and Green Day, whereas the environment in Figure <ref>(d) represents heavy metal bands such as Iron Maiden and Slayer.
Finally, the environment in Figure <ref>(e) represents electro house music being composed of groups such as Skrillex, Pendulum, deadmau5, and Nero.
§.§ Flow Semantics in Check-in Data
We now turn our attention to the Foursquare dataset. Environments in this dataset capture user trajectories between businesses.
We want see whether infers semantically meaningful latent environments for check-in trajectory data.
Interestingly, because does not use GPS features, can identify latent environments of “nearby” locations in any geometry.
Figure <ref> shows the top-20 locations for two different latent environments inferred by .
Each point in the figure is a latitude-longitude coordinate plotted on the world map from its GPS coordinate.
Two latent environments best exemplify the sense-making abilities of .
The first latent environment is represented by U.S. airports and seems to capture flows of U.S. domestic flights (including Hawaii).
The second latent environment captures check-in trajectories of connections to/from Pacific-Asia-based airports. Also in this environment are major U.S./U.K. major hubs (JFK in New York City (NYC), LAX in Los Angeles and Heathrow in London) that connect to Pacific-Asia.
Although omitted from the figure, also extracted environments based on user trajectories within cities including NYC, Miami and Atlanta. In these settings, check-ins are related to different places in these cities.
This illustrates that latent environments can be a powerful tool for sense-making in user trajectory datasets.
§.§ Transient User Trajectories
In this section we illustrate how can help us identify transient user trajectories (also showing that can cope well with the transience).
To illustrate this, Figures <ref>(a,b) present the number of song plays (y-axis) over multiple years (x-axis) of two users from Last.FM-Groups broken down into the user's four preferred latent environments.
More precisely, the y-axis shows the cumulative sum of the song plays at a given month color-coded by the song's most likely latent environment for that user, $P[\cM | u, x_{t}]$.
As shown in the Figure <ref>(a), from 2010 until mid 2011 the user goes through a strong Pop phase – most representative (top) artists in the environment labeled “U.S. Pop (1)” are Madonna, Nelly Furtado, and Alicia Keys, and top artists in the “U.S. Pop (2)” environment are Britney Spears, Leona Lewis, and Kelly Clarkson.
We also note some interest in 70-80's Rock overtones – top artists being Queen, Michael Jackson, and The Beatles.
After mid 2011 the user moves away from Pop artists towards a “Classic Rock” environment, with The Beatles, Pink Floyd, and Nirvana as top artists.
The user represented in Figure <ref>(b) also changes interest over time, most markably from “Classic Rock” & “Heavy Metal” to “Korean Pop”.
The three major take-aways are: (a) user trajectories are indeed transient (Figure <ref>); (b) users can show interests in the same latent environment at different points in time: User 1 Figure <ref>(a) shows strong “Classic Rock” environment preference between 2011-2012 while User 2 Figure <ref>(b) shows strong preference for the same environment between 2013-2014; and (c) can cope well with transient trajectories.
To provide further evidence in support 's ability to extract stationary, ergodic, and time-homogeneous behavior from user trajectories consider the following synthetic dataset.
For comparison, we contrast with a state-of-the-art temporal tensor approach in the same scenario.
This comparison sheds light into the reasons why tensor-factorization-based methods should have difficulty in extracting stationary, ergodic, and time-homogeneous behavior from user trajectory data.
Our synthetic dataset has 50 users event assigned to one of five Markov chains.
The Markov chains are used to create user trajectories.
We model the popularity of items at each chain as a Lognormal distribution.
Also, each environment has exponentially distributed inter-event times but timestamps are not recorded.
We simulate a total of 5 days where and each user selects in average 100 items per day.
We simulate users joining the system 1 or 2 days apart.
Comparison with Temporal Decomposition.
Applying -NT to this synthetic data almost perfectly recovers user preferences of transition matrices as presented in Figure <ref>. should work even better if timestamps were available, but that would be an unfair comparison because of the extra feature.
A state-of-the-art temporal decomposition method <cit.> (tensor with time mode) on the same data has trouble finding the ground truth (i.e., user ids $0-9$ using Markov chain 0, user ids $10-19$ using Markov chain 1, etc).
Note that only is able to recover the different user trajectory preferences despite the asynchronous behavior of users.
Clearly this type of tensor decomposition is meant to uncover only synchronized behavior as originally proposed by Matsubara et al. <cit.>.
§ CONCLUSIONS
In this work we introduced , a general method to mine and predict user trajectories.
decomposes non-stationary, transient, time-heterogeneous user trajectories into a small number of short random walks on latent random environments.
The decomposed trajectories are stationary, erratic, and time-homogeneous in short time scales.
User activity (e.g., listening to music, shopping for products online or checking-in different places in a city) is then captured by different latent environments inferred solely from observed user navigation patterns (e.g., listening to “classical music”, listening to “Brazilian Pop”, shopping for shoes, shopping for electronics, checking-in into airport fast-food venues, etc.).
We summarize our major contributions as follows:
* Accurate: outperforms various state-of-the-art baseline methods in three different tasks: extracting Markov embedding of user behavior, next-item prediction and capturing the flows of users between locations. Gains are up-to 900% depending on the dataset and task analyzed.
* Scalable: is at least tens and up to hundreds of times faster than state-of-the-art competitors even in relatively small datasets. If we consider the only other fully parallel competitor, MultiLME <cit.>, is 413x faster and still more accurate.
* Novelty: provides a general framework to build upon (random surfer over infinite latent random environments) and make application-specific personalized recommendation systems.
§ ACKNOWLEDGMENTS
Research was funded by Brazil's National Institute of Science and Technology for Web Research (MCT/CNPq/INCT Web 573871/2008-6). Research was also sponsored by the Defense Threat Reduction Agency and was accomplished under contract No. HDTRA1-10-1-0120, as well as by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-09-2-0053. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation here on.
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1511.00967
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§ INTRODUCTION
The details of the QCD phase diagram are still largely unknown because of the sign problem of Monte Carlo simulations of lattice
QCD. Approximate methods are able to circumvent this problem only for small quark chemical potentials $\mu=\mu_B/3\losim T$
<cit.>. So far, no sign of a critical point or a first order phase transition has been found in this controlled region.
Complex Langevin algorithms do not suffer from the sign problem, but in some models converge to incorrect answers or in QCD are
restricted to certain regions in parameter space. A lot of progress has been made recently, but no true phase transition has been
reported in this approach either <cit.>.
These difficulties have motivated the development of effective lattice theories, which can either be handled analytically or whose
sign problem is mild enough to simulate the cold and dense region of QCD. Here we update an ongoing project to describe the cold
and dense regime of QCD by means of a 3d effective theory, which is derived by combined strong coupling and hopping expansions.
The resulting effective theory is valid for the description of very heavy quarks for sufficiently weak couplings, such that a
continuum limit of thermodynamic functions can be attempted. Its sign problem is mild enough for complex Langevin or even Monte
Carlo simulations to describe cold nuclear matter. The heavy dense region is also studied by complex Langevin simulations in 4d
<cit.>, for which the effective theory results can serve as a benchmark. Furthermore, the effective theory approach can be
tested against full simulations in two-colour QCD <cit.>. In a new development, we show how the thermodynamic
functions of the effective theory can also be computed fully analytically by means of linked cluster expansion.
§ DERIVATION OF THE EFFECTIVE THEORY
We start on a $(3+1)$-dimensional lattice with Wilson's gauge and fermion actions on infinite spatial volume and at finite
temperature $T=(aN_\tau)^{-1}$. After Grassmann integration the partition function is
\begin{eqnarray}
S_g=\frac{\beta}{2N_c}\sum_p\left[\tr\, U_p+\tr\, U_p^\dagger\right]\;,
\end{eqnarray}
with the quark hopping matrix
\begin{eqnarray}
\kappa\sum_{\nu=0}^3\left[e^{a\mu\delta_{\nu0}}(1+\gamma_\nu)_{\alpha\beta}U_\nu^{ab}(x)
\delta_{x,y-\hat{\nu}}+e^{-a\mu\delta_{\nu0}}(1-\gamma_\nu)_{\alpha\beta}U_{-\nu}^{ab}(x)
\delta_{x,y+\hat{\nu}}\right]
\;.\nonumber
\end{eqnarray}
The lattice gauge coupling and hopping parameter are defined as
\begin{equation}
\beta=\frac{2N_C}{g^2},\quad \kappa=\frac{1}{2am+8}\;.
\end{equation}
A 3d effective action is then defined by integrating out the spatial link variables
\begin{eqnarray}
\label{eq_defeffth}
\end{eqnarray}
The resulting effective theory depends on temporal link variables closing through the temporal boundary, i.e. on Polyakov loops
\begin{eqnarray}
W_i\equiv\prod_{\tau=1}^{N_\tau}U_0\left(\vec{x}_i;\tau\right),\quad L_i = \tr W_i\:.
\end{eqnarray}
Without truncations, the effective action is unique and exact. Since all spatial links are integrated over, the resulting
effective action has long-range interactions of Polyakov loops at all distances and to all powers so that in practice truncations
are necessary. For non-perturbative ways to determine truncated theories, see <cit.>. Here we expand
eq. (<ref>) in a combined strong coupling and hopping expansion, with interaction terms ordered according to their
leading powers in $\beta, \kappa$.
For the gauge action it is advantageous to perform a character expansion
\begin{eqnarray}
\exp\left[\frac{\beta}{2N_c}\Big(\tr\, U+\tr\, U^\dagger\Big)\right]
\end{eqnarray}
where the factor $c_0(\beta)$ can be neglected as it is independent of gauge links and cancels in expectation values. In earlier
publications <cit.>, we have shown how to compute the effective gauge theory up to rather high
orders in the fundamental character expansion coefficient $u(\beta)\equiv a_f(\beta)$. Note that $u(\beta)=\beta/18+O(\beta^2)<1$,
thus improving the convergence compared to straightforward expansion in $\beta$. To leading order the effective gauge action
\begin{eqnarray}
e^{S_{\mathrm{eff}}^{(1)}}=\lambda(u,N_\tau)\sum_{<ij>}\left(L_iL_j^\ast+L_i^\ast L_j\right)\;,
\qquad\lambda(u,N_\tau)=u^{N_\tau}\Big[1+\ldots\Big]\;,
\label{eq_seffgauge}
\end{eqnarray}
where higher order corrections of $\lambda(u,N_\tau)$ as well as a discussion of higher order interaction terms can be found in
<cit.>. Going via an effective action results in a resummation with better convergence properties than a direct
series expansion of the thermodynamic observables as in <cit.>. We observe that $\lambda(u,N_\tau)$ is suppressed
for large $N_\tau$. In this presentation our main interest is in the cold and dense region, for which all $\lambda_i<10^{ -18}$
and the pure gauge contribution can be safely neglected.
To compute the quark determinant in a spatial hopping expansion, we split the quark matrix in positive and negative temporal and
spatial parts,
\begin{eqnarray}
\end{eqnarray}
The static determinant is given by neglecting the spatial parts and can be computed exactly <cit.>
\begin{eqnarray}
\det[Q_{\mathrm{stat}}]&=& \prod_{\vec{x}} \det \Big[1+h_1W({\vec{x}})\Big]^2
\det \Big[1+\bar{h}_1W^{\dagger}({\vec{x}})\Big]^2 \nonumber\\
&=&\prod_{\vec{x}} \left[1 + h_1 L_{\vec{x}} + h_1^2 L^{\dagger}_{\vec{x}}+h_1^3\right]^2
\left[1 + \bar{h}_1 L^{\dagger}_{\vec{x}} + \bar{h}_1^2 L_{\vec{x}}+\bar{h}_1^3\right]^2,
\label{q_static}
\end{eqnarray}
with the leading order couplings
\begin{equation}
h_1=(2 \kappa e^{a \mu})^{N_{\tau}},\quad \bar{h}_1=(2 \kappa e^{-a \mu})^{N_{\tau}}\;.
\end{equation}
In order to compute a systematic hopping expansion, we define the kinetic quark determinant
\begin{eqnarray}
\det[Q]&\equiv&\det[Q_{\mathrm{stat}}]\det[Q_{\mathrm{kin}}]\;,\\
\det[Q_{\mathrm{kin}}]&=&[1-(1-T)^{-1}(S^++S^-)]\equiv\det[1-P-M]=\exp\left[\tr\, \ln (1-P-M)\right]\;.
\label{eq_detqkin}
\end{eqnarray}
The static propagator $(1-T)^{-1}$ is also known exactly. It contains summation of all temporal windings to produce the basic
building blocks of the effective action, for details see <cit.>,
\begin{equation}
W_{n,m}(\vec{x}) = \tr \, \frac{\big(h_1 W(\vec{x})\big)^m}{\big(1+h_1 W(\vec{x})\big)^n} \, .
\end{equation}
We have calculated the effective action through order $\kappa^8 u^5$ in the low temperature limit, i.e. the leading power of
$N_\tau$. However, because of its length we will give the result only in a compact, graphical representation. We symbolise
factors of $W_{n,m}(\vec{x})$ by vertices, where $n$ is the number of bonds entering a vertex, and $m$ is the number indicated on
the node. Furthermore, vertices which are connected by one or more bonds are nearest neighbours on the lattice.
\begin{align} \label{eq:effective_action}
S_{\mathrm{eff}} &=
h_2 N_f \sum_{\mathrm{dof}} \; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(270:.75)}] n1) {1}
edge[thick] (n1);
\end{midtikzpicture}
\; - h_2^2 N_f\sum_{\mathrm{dof}} \: \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:{sqrt(2/3)})}] n1) {1}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:{sqrt(2/3)})}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture}
- h_2^2 N_f^2\sum_{\mathrm{dof}} \: \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(270:.75)}] n1) {1}
edge[thick,bend left=45] (n1)
edge[thick,bend right=45] (n1);
\end{midtikzpicture}
+ h_2^3 N_f\sum_{\mathrm{dof}} \: \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:{sqrt(2/3)})}] n1) {1}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:{sqrt(2/3)})}] n2) {1}
edge[thick] (n2);
\node[circ] (n4) at ([shift={(60:{sqrt(2/3)})}] n3) {1}
edge[thick] (n3);
\end{midtikzpicture} \nonumber\\
&+\frac{1}{3} h_2^3 N_f \sum_{\mathrm{dof}} \Bigg(\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] at ([shift={(-30:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(90:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(210:.5)}] n1) {1}
edge[thick] (n1);
\end{midtikzpicture}
\;-\;\begin{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] at ([shift={(-30:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(90:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(210:.5)}] n1) {1}
edge[thick] (n1);
\end{midtikzpicture} \;\Bigg)
+2 h_2^3 N_f^2 \sum_{\mathrm{dof}} \Bigg(\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {1}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1);
\node[circ] at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture}
\;-\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {2}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1);
\node[circ] at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture} \Bigg) \nonumber\\
&+\frac{1}{6} h_2^3 N_f \sum_{\mathrm{dof}} \Bigg(\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] at ([shift={(270:.75)}] n1) {1}
edge[thick,bend left=45] (n1)
edge[thick,bend right=45] (n1)
edge[thick] (n1);
\end{midtikzpicture}
\;-\; \begin{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] at ([shift={(270:.75)}] n1) {2}
edge[thick,bend left=45] (n1)
edge[thick,bend right=45] (n1)
edge[thick] (n1);
\end{midtikzpicture} \Bigg)
- \frac{4}{3} h_2^3 N_f^3 \sum_{\mathrm{dof}} \; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] at ([shift={(270:.75)}] n1) {2}
edge[thick,bend left=45] (n1)
edge[thick,bend right=45] (n1)
edge[thick] (n1);
\end{midtikzpicture}
- h_2^4 N_f\sum_{\mathrm{dof}} \: \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:{sqrt(2/3)})}] n1) {1}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:{sqrt(2/3)})}] n2) {1}
edge[thick] (n2);
\node[circ] (n4) at ([shift={(60:{sqrt(2/3)})}] n3) {1}
edge[thick] (n3);
\node[circ] (n5) at ([shift={(300:{sqrt(2/3)})}] n4) {1}
edge[thick] (n4);
\end{midtikzpicture} \nonumber\\
&- \frac{1}{12} h_2^4 N_f\sum_{\mathrm{dof}} \Bigg(\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] at ([shift={(45:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(135:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(225:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(315:.5)}] n1) {1}
edge[thick] (n1);
\end{midtikzpicture}
\;-2\; \begin{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] at ([shift={(45:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(135:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(225:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(315:.5)}] n1) {1}
edge[thick] (n1);
\end{midtikzpicture}
\;+\; \begin{midtikzpicture}
\node[circ] (n1) {3};
\node[circ] at ([shift={(45:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(135:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(225:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(315:.5)}] n1) {1}
edge[thick] (n1);
\end{midtikzpicture} \Bigg)
- h_2^4 N_f\sum_{\mathrm{dof}} \Bigg(\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:{sqrt(1/3)})}] n1) {1}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:{sqrt(1/3)})}] n2) {1}
edge[thick] (n2);
\node[circ] (n4) at ([shift={(60:{sqrt(1/3)})}] n3) {1}
edge[thick] (n3);
\node[circ] (n5) at ([shift={(0:.5)}] n3) {1}
edge[thick] (n3);
\end{midtikzpicture}
\;-\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:{sqrt(1/3)})}] n1) {1}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:{sqrt(1/3)})}] n2) {2}
edge[thick] (n2);
\node[circ] (n4) at ([shift={(60:{sqrt(1/3)})}] n3) {1}
edge[thick] (n3);
\node[circ] (n5) at ([shift={(0:.5)}] n3) {1}
edge[thick] (n3);
\end{midtikzpicture} \Bigg) \nonumber\\
&- h_2^4 N_f^2 \sum_{\mathrm{dof}} \Bigg(\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] at ([shift={(-30:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(90:.65)}] n1) {1}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1);
\node[circ] at ([shift={(210:.5)}] n1) {1}
edge[thick] (n1);
\end{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] at ([shift={(-30:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(90:.65)}] n1) {1}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1);
\node[circ] at ([shift={(210:.5)}] n1) {1}
edge[thick] (n1);
\end{midtikzpicture}
\node[circ] (n1) {3};
\node[circ] at ([shift={(-30:.5)}] n1) {1}
edge[thick] (n1);
\node[circ] at ([shift={(90:.65)}] n1) {1}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1);
\node[circ] at ([shift={(210:.5)}] n1) {1}
edge[thick] (n1);
\end{midtikzpicture} \;\Bigg)
- h_2^4 N_f^2 \sum_{\mathrm{dof}} \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at (0.65,0) {1}
edge[thick] (n1);
\node[circ] (n3) at (0.65,0.65) {1}
edge[thick] (n2);
\node[circ] (n4) at (0,0.65) {1}
edge[thick] (n3)
edge[thick] (n1);
\end{midtikzpicture} \nonumber\\
&-2 h_2^4 N_f^2 \sum_{\mathrm{dof}} \Bigg( \; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {1}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\node[circ] at ([shift={(60:.75)}] n3) {1}
edge[thick] (n3);
\end{midtikzpicture}
\;-\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {2}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\node[circ] at ([shift={(60:.75)}] n3) {1}
edge[thick] (n3);
\end{midtikzpicture} \Bigg)
- h_2^4 N_f^2 \sum_{\mathrm{dof}} \Bigg( \; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(65:{sqrt(2/3)})}] n1) {1}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(-65:{sqrt(2/3)})}] n2) {1}
edge[thick,bend left=30] (n2)
edge[thick,bend right=30] (n2);
\node[circ] (n4) at ([shift={(65:{sqrt(2/3)})}] n3) {1}
edge[thick] (n3);
\end{midtikzpicture}
\;-2\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(65:{sqrt(2/3)})}] n1) {1}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(-65:{sqrt(2/3)})}] n2) {2}
edge[thick,bend left=30] (n2)
edge[thick,bend right=30] (n2);
\node[circ] (n4) at ([shift={(65:{sqrt(2/3)})}] n3) {1}
edge[thick] (n3);
\end{midtikzpicture}
\;+\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(65:{sqrt(2/3)})}] n1) {2}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(-65:{sqrt(2/3)})}] n2) {2}
edge[thick,bend left=30] (n2)
edge[thick,bend right=30] (n2);
\node[circ] (n4) at ([shift={(65:{sqrt(2/3)})}] n3) {1}
edge[thick] (n3);
\end{midtikzpicture} \Bigg) \nonumber\\
&-\frac{1}{3} h_2^4 N_f \sum_{\mathrm{dof}} \Bigg( \; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {1}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1)
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture}
\;-2\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {2}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1)
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture}
\;+2\; \begin{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {2}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1)
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture}
\;-\; \begin{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {3}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1)
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture} \Bigg) \nonumber\\
&+\frac{4}{3} h_2^4 N_f^3 \sum_{\mathrm{dof}} \Bigg( \; \begin{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {1}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1)
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture}
\;-2\; \begin{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {2}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1)
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture}
\;+2\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {2}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1)
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture}
\;-\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {3}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1)
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture} \Bigg) \nonumber\\
&-\bigg(\frac{1}{12} N_f + \frac{2}{3} N_f^3 \bigg) h_2^4 \sum_{\mathrm{dof}} \Bigg( \; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {1}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick,bend left=30] (n2)
edge[thick,bend right=30] (n2);
\end{midtikzpicture}
\;-4\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {2}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick,bend left=30] (n2)
edge[thick,bend right=30] (n2);
\end{midtikzpicture}
\;+\; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:.75)}] n1) {3}
edge[thick,bend left=30] (n1)
edge[thick,bend right=30] (n1);
\node[circ] (n3) at ([shift={(300:.75)}] n2) {1}
edge[thick,bend left=30] (n2)
edge[thick,bend right=30] (n2);
\end{midtikzpicture} \Bigg)
-\frac{2}{3} h_2^4 N_f^4 \sum_{\mathrm{dof}} \Bigg( \; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at (0,-.75) {3}
edge[thick, bend right=45] (n1)
edge[thick, bend left=45] (n1)
edge[thick, bend right=15] (n1)
edge[thick, bend left=15] (n1);
\end{midtikzpicture}
\;+2\; \begin{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] (n2) at (0,-.75) {2}
edge[thick, bend right=45] (n1)
edge[thick, bend left=45] (n1)
edge[thick, bend right=15] (n1)
edge[thick, bend left=15] (n1);
\end{midtikzpicture} \Bigg) \nonumber \\
&-\frac{1}{12} h_2^4 N_f^2 \sum_{\mathrm{dof}} \Bigg( \; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at (0,-.75) {1}
edge[thick, bend right=45] (n1)
edge[thick, bend left=45] (n1)
edge[thick, bend right=15] (n1)
edge[thick, bend left=15] (n1);
\end{midtikzpicture}
\;+12\; \begin{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] (n2) at (0,-.75) {2}
edge[thick, bend right=45] (n1)
edge[thick, bend left=45] (n1)
edge[thick, bend right=15] (n1)
edge[thick, bend left=15] (n1);
\end{midtikzpicture}
\;+\; \begin{midtikzpicture}
\node[circ] (n1) {3};
\node[circ] (n2) at (0,-.75) {3}
edge[thick, bend right=45] (n1)
edge[thick, bend left=45] (n1)
edge[thick, bend right=15] (n1)
edge[thick, bend left=15] (n1);
\end{midtikzpicture} \Bigg)
+\frac{2}{3} h_2^4 N_f^2 \sum_{\mathrm{dof}} \Bigg( \; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at (0,-.75) {2}
edge[thick, bend right=45] (n1)
edge[thick, bend left=45] (n1)
edge[thick, bend right=15] (n1)
edge[thick, bend left=15] (n1);
\end{midtikzpicture}
\;+\; \begin{midtikzpicture}
\node[circ] (n1) {2};
\node[circ] (n2) at (0,-.75) {3}
edge[thick, bend right=45] (n1)
edge[thick, bend left=45] (n1)
edge[thick, bend right=15] (n1)
edge[thick, bend left=15] (n1);
\end{midtikzpicture} \Bigg) +\mathcal{O} \big(\kappa^{10}\big)
\end{align}
The effective couplings to this order are
\begin{eqnarray}
h_1&=& e^{N_{\tau} (a \mu + \log(2 \kappa))} e^{6 N_{\tau} \kappa^2 u (\frac{1}{1-u}
+4u^4-12\kappa^2+9\kappa^2 u +4\kappa^2 u^2-4\kappa^4)}, \\
h_2&=& \frac{\kappa^2 N_{\tau}}{N_c} \Big[1 + \frac{2 u}{1-u} + 8 u^5 \Big] \;.
\end{eqnarray}
The sign problem of this effective theory is weak and it can be simulated either by complex Langevin without any convergence or
runaway problems, or even by standard Metropolis using reweighting. For details and tests of the simulation algorithm see
§ THE QCD PHASE DIAGRAM FOR HEAVY QUARKS
[]Left: The deconfinement critical surface with heavy quarks for real and imaginary chemical
potential. From <cit.>. Right: The qualitative phase diagram for QCD with heavy quarks. The location of the lines
and critical endpoints depends on $N_f$ and the quark mass.
The effective theory has been used to study the QCD phase diagram for QCD with heavy quarks. In the static limit, the finite
temperature deconfinement transition is of first order, corresponding to the spontaneous breaking of centre symmetry. With
dynamical quarks, the fermion determinant breaks the symmetry explicitly and the transition weakens until it changes to a
crossover at a critical quark mass. Similarly, real chemical potential weakens the transition. This behaviour is reflected in the
heavy mass corner of the Columbia plot. <ref> (left) shows the deconfinement critical surface separating the
first order from the crossover region as calculated with a $\kappa^2$-action but full $N_\tau$- dependence in <cit.>.
[]Left: Baryon density, Polyakov loop and conjugate Polyakov loop
obtained from Monte Carlo $N_s=3$, complex Langevin ($N_s=6$) and the static strong coupling limit, respectively. From
<cit.>. Middle: Continuum extrapolation for two densities. Right: Baryon density in the continuum. From
The cold and dense regime was considered in <cit.>. <ref> (left) shows results at a fixed lattice
spacing. We set the scale using the $r_0$ parameter and compute the corresponding pion mass from strong coupling formulae
<cit.>. To keep our truncated series in full control, we choose $\beta=5.7, \kappa = 0.0000887, N_\tau=116$ corresponding to
$m_\pi=20$ GeV, $T=10$ MeV, $a=0.17$ fm. The silver blaze property as well as lattice saturation are clearly observed. Note that
the Polyakov loop as well as its conjugate get screened in the presence of a baryonic medium, and hence rise. The ensuing
decrease is due to the artefact of lattice saturation which forces all $Z(3)$ states to be occupied. Note that this decrease
happens before saturation near the point of half filling. There is an approximate particle anti-particle symmetry about this point
and one expects artefacts to be dominant <cit.>. <ref> (right) shows the continuum extrapolated results for
the baryon density as a function of chemical potential, each point being extrapolated from results for 4-7 lattice spacings
<cit.>. We observe the silver blaze property followed by the onset of nuclear matter, which is steeper for $N_f=2$ than for
$N_f=1$ as expected. Note that the onset transition happens at $\mu_c\losim m_B$, due to the binding energy between the nucleons.
Note also that the onset transition here is a smooth crossover, in contrast to the first-order phase transition for physical QCD
in nature. This is due to the fact that the binding energy decreases with growing quark mass and for the heavy quarks studied here
is smaller than the temperature realized in the plot. It was confirmed in <cit.> that the first-order behaviour indeed results
for sufficiently large hopping parameters/small quark masses, but in that mass range the effective theory is not yet converging
and terms of higher order in $\kappa$ are necessary to quantitatively reproduce QCD.
§ SYSTEMATICS OF THE EFFECTIVE THEORY TO ORDER $\KAPPA^8$
An important question is for which parameter regions the effective theory is valid. Since both the character as well as the
hopping expansion provide convergent series within their radius of convergence, we are able to self-consistently check this by
comparing physical observables at different orders of the effective action. Here we are interested in the cold and dense region
around the onset of nuclear matter and our observable of choice is the baryon number. <ref> (left) tests the
hopping expansion in the strong coupling limit by comparing results obtained with effective actions of increasing order in
$\kappa$. One observes clearly how adjacent orders stay together for larger values of the coupling $h_2$ as the order is
increased, thus extending the range where our effective action is reliable. <ref> (right) shows the same exercise
for the largest $h_2\sim 0.1$ considered here, this time increasing the orders of the character expansion. We observe good
convergence up to $\beta\sim 7$, which is a sufficiently weak coupling to allow for continuum extrapolations. It is interesting to
note that the convergence properties are not determined by the size of the expansion parameters alone. Even though the
$u(\beta)$-values far exceed the $\kappa$-values employed in the figures, convergence in $u(\beta)$ appears to be faster.
[]Left: Convergence of the baryon density as a function of $h_2 = \frac{\kappa^2 N_t}{N_c}$, computed with
effective actions of different orders in the hopping expansion. Right: Convergence in $u$.
The gain in convergence region can be exploited in two ways. Firstly, at fixed temperature and quark masses it allows for the use
of finer lattices, which can be employed in a continuum extrapolation. <ref> (middle) shows results from our
previous simulations obtained with the $\kappa^4$-action as well as new ones with the $\kappa^8$-action at two values of
$\mu>\mu_c$. The baryon density just about reaches the domain with leading cut-off effects linear in $a$, as is expected for
Wilson fermions. The break-off from this behaviour for finer lattices is due to truncation errors and indicates the limit of
validity of the effective action. The new data generated with the $\kappa^8$-action indeed smoothly extends the linear section
towards the continuum limit. We conclude that our hopping expansion is systematic and controlled, with additional orders in the
action allowing for simulations on finer lattices. For sufficiently heavy masses a continuum extrapolation appears possible.
A second way to benefit from the additional orders in the hopping expansion is to keep the lattice spacing fixed and study smaller
masses. This is shown in <ref> for two different lattice spacings. The error bars in these plots are systematic and
give the difference between results obtained by the action to the highest two orders in the hopping expansion. Growing error bars
thus indicate the loss of good convergence. As expected, increasing orders allow for smaller quark masses, and so do coarser
lattices. However, the gain in mass range per additional order in the hopping expansion is too small to envisage an extension to
the physical quark masses of QCD at the present stage.
[]Baryon number density as a function of pion mass.
§ ANALYTIC EVALUATION OF THE EFFECTIVE THEORY
So far we have used Monte-Carlo and Complex Langevin simulations to simulate the effective theory on a 3d lattice and numerically
calculate the remaining integral over Polyakov loops. However, restricted to some finite order in the expansion parameters
$\kappa$ and $u(\beta)$, these integrals can also be carried out analytically. In particular, since the effective couplings are
small and correspond to power series in the original couplings, a perturbative evaluation should have good convergence behaviour,
as noted before <cit.>. In order to get the correct thermodynamic limit we now employ a linked cluster expansion, and
thus calculate the free energy directly rather than going through the partition function. A review of the standard linked cluster
expansion can be found in <cit.>. One should note however that the standard linked cluster expansion tailored to spin
models deals with nearest neighbour interactions. Our action on the other hand contains $n$-point interactions at all distances
and with all possible geometries. It can then be mapped into a spin model, whose partition function thus takes the more
generalised form
\begin{align}
\mathcal{Z} = \int \prod_{x,i} \mathrm{d} &\phi_i(x) \, \exp \bigg\{ -S_0[\phi_i] %
- \frac{1}{2!} \sum_{x,y} v_{ij}(x,y) \phi_i(x) \phi_j(x)\nonumber\\
&-\frac{1}{3!}\sum_{x,y,z} u_{ijk}(x,y,z) \phi_i(x) \phi_j(y) \phi_k(z) + \dots \bigg\}.
\end{align}
In our case the effective theory is fully encoded in the coupling constants $v_{ij}(x,y), u_{ijk}(x,y,z), \dots$, which themselves are
series in the expansion parameters. The free energy can hence be rewritten as
\begin{align}
\mathcal{W}[v,u,\dots] = \Bigg[&\exp\bigg(\frac{1}{2!}\sum_{x,y}\sum_{i,j}v_{ij}(x,y)
\frac{\delta}{\delta\tilde{v}_{ij}(x,y)}\bigg) \nonumber\\
\frac{\delta}{\delta\tilde{u}_{i,j,k}(x,y,z)}\bigg) \nonumber\\
& \hspace{3.4cm}\cdots\hspace{3.4cm} \Bigg] \mathcal{W}[\tilde{v},\tilde{u},\dots]
\Bigg|_{\substack{\tilde{v} = 0\\\tilde{u} = 0\\\dots}},
\end{align}
and with the expressions for the couplings at hand, the 3d theory can be evaluated analytically in a systematic, well defined way
by carrying out the higher order derivatives and summing over the set of topologically invariant terms. Although this approach is
valid, it lacks the elegance of the graphical techniques of the linked cluster expansion. However, because all of the terms
given in (<ref>) can be embedded on a square lattice, one can use another embedding scheme to simplify the
calculations. This scheme will be introduced next.
§ GRAPHICAL METHODS
The linked cluster expansion graphs constitute all contributions to the free energy in the thermodynamic limit up to a certain
order. Therefore by embedding the terms from the effective action onto all connected skeleton graphs from the linked cluster
expansion with two-point interactions, we can calculate the free energy of the effective theory. This procedure is most easily
demonstrated with an example.
Consider for now an effective theory consisting of two terms, a nearest neighbour term, and a "wedge"
\begin{equation}
S =
\;a \sum_{\mathrm{dof}} \; \begin{midtikzpicture}
\node[dot] (n1) {};
\node[dot] (n2) at ([shift={(270:.75)}] n1) {}
edge[thick] (n1);
\end{midtikzpicture}
\;+b\; \sum_{\mathrm{dof}} \: \begin{midtikzpicture}
\node[dot,blue] (n1) {};
\node[dot,blue] (n2) at ([shift={(60:{sqrt(2/3)})}] n1) {}
edge[thick,blue] (n1);
\node[dot,blue] (n3) at ([shift={(300:{sqrt(2/3)})}] n2) {}
edge[thick,blue] (n2);
\end{midtikzpicture} .
\end{equation}
To calculate the free energy, we can embed these terms onto the linked cluster graphs, calculating symmetry factors and lattice
embeddings for every embedding on every graph. For example embedding onto the fourth order graph
\begin{equation}
g_8 = \begin{midtikzpicture}
\node[inner] (n1) {};
\node[inner] (n2) at ([shift={(60:.75)}] n1) {}
edge[bend right=30] (n1);
\node[inner] (n3) at ([shift={(300:.75)}] n2) {}
edge[bend left=45] (n2)
edge[bend right=45] (n2);
\node[inner] (n4) at ([shift={(60:.75)}] n3) {}
edge[bend left=30] (n3);
\end{midtikzpicture}, \hspace{1cm} \text{ symmetry: } 4
\end{equation}
we get the following contribution to $\mathcal{W}$
\begin{equation}
\mathcal{W}_{g_8} =
\;\frac{a^4 \Lambda}{4} \; \begin{midtikzpicture}
\node[dot] (n1) {};
\node[dot] (n2) at ([shift={(60:.75)}] n1) {}
edge[thick,bend right=30] (n1);
\node[dot] (n3) at ([shift={(300:.75)}] n2) {}
edge[thick, bend left=45] (n2)
edge[thick, bend right=45] (n2);
\node[dot] (n4) at ([shift={(60:.75)}] n3) {}
edge[thick, bend left=30] (n3);
\end{midtikzpicture}
+a^2 b \Lambda \; \begin{midtikzpicture}
\node[dot,blue] (n1) {};
\node[inner] (n2i) at ([shift={(60:.75)}] n1) {};
\node[outer,blue] (n2o) at ([shift={(60:.75)}] n1) {}
edge[thick, blue, bend right=30] (n1);
\node[inner] (n3i) at ([shift={(300:.75)}] n2) {};
\node[outer,blue] (n3o) at ([shift={(300:.75)}] n2) {}
edge[thick, bend left=45] (n2o)
edge[thick, bend right=45,blue] (n2o);
\node[dot] (n4) at ([shift={(60:.75)}] n3) {}
edge[thick, bend left=30] (n3);
\end{midtikzpicture}
+\frac{a^2 b \Lambda}{2} \; \begin{midtikzpicture}
\node[dot] (n1) {};
\node[inner] (n2i) at ([shift={(60:.75)}] n1) {};
\node[outer,blue] (n2o) at ([shift={(60:.75)}] n1) {}
edge[thick,bend right=30] (n1);
\node[inner] (n3i) at ([shift={(300:.75)}] n2) {};
\node[outer,blue] (n3o) at ([shift={(300:.75)}] n2) {}
edge[thick, bend left=45,blue] (n2o)
edge[thick, bend right=45,blue] (n2o);
\node[dot] (n4) at ([shift={(60:.75)}] n3) {}
edge[thick, bend left=30] (n3);
\end{midtikzpicture}
+\frac{b^2 \Lambda}{2} \; \begin{midtikzpicture}
\node[dot,blue] (n1) {};
\node[inner,blue] (n2i) at ([shift={(60:.75)}] n1) {};
\node[outer,blue] (n2o) at ([shift={(60:.75)}] n1) {}
edge[thick, blue, bend right=30] (n1);
\node[inner,blue] (n3i) at ([shift={(300:.75)}] n2) {};
\node[outer,blue] (n3o) at ([shift={(300:.75)}] n2) {}
edge[thick, bend left=45,blue] (n2o)
edge[thick, bend right=45,blue] (n2o);
\node[dot,blue] (n4) at ([shift={(60:.75)}] n3) {}
edge[thick, blue, bend left=30] (n3);
\end{midtikzpicture} .
\end{equation}
The denominators give the overall symmetry factor of the embedding and $\Lambda$ is the lattice embedding of the graph itself. For
$d$ dimensional square lattices $\Lambda=(2d)^3$. After the embeddings we are left with overlapping nodes at individual
lattice points which gives integrals on the form
\begin{equation}
\int \mathrm{d} W \; \det \big[Q_{\mathrm{stat}}\big]^{2 N_f} W_{n_1,m_1}^{k_1} \cdots
W_{n_N, m_N}^{k_N}\;,
\end{equation}
which are analytically calculable for $\sum_i n_i k_i \leq 2 N_f$ using for example Polyakov loop integral lookup tables. This means
that for a system with $2$ degenerate quark flavours we can carry out the calculation up to order $\kappa^8$. The full result for
the free energy is too lengthy to be included here, and will be published elsewhere in the near future.
§ COMPARISON WITH NUMERICS
With an analytic expression for the free energy at hand, various bulk thermodynamic quantities can be calculated and the results
compared to those obtained with numerics. It should be stressed that the analytical results emerged from a two stage expansion, first the
effective action is calculated to a fixed order. Second, the expanded action is used to compute e.g. the free energy to a specific
order. This implies that we have for example a $\mathcal{O}\big(\kappa^8\big)$ linked cluster expansion for the
$\mathcal{O}\big(\kappa^2\big)$ action, which we expect to reproduce the $\mathcal{O}\big(\kappa^2\big)$ numerical results. A
comparison between the various orders of the linked cluster expansion is presented in <ref>. Expanding the
$\kappa^2$ action is shown to the left, while the $\kappa^8$ action is used in the right plot. From the figure one can see that
the $\kappa^2$ action converges much quicker than the $\kappa^8$ action. This is only natural as the cluster expansion to order
$\kappa^4$ only contains effective action terms to the same order, and the expansion thus have to "catch up" with the order at
Linked cluster expansion of the $\kappa^2$ (left) and $\kappa^8$ (right) actions for
$\beta=0$ and $h_1=0.8$.
A linked cluster expansion is done for a fixed lattice spacing. By applying the same procedure to different lattices it is also
possible to carry out continuum extrapolations. In <ref> (left) it is demonstrated that this gives remarkably
similar results to the numerical ones. In both cases error bars include a systematic error, measured as the difference between the
two highest order effective actions along with an estimate of the continuum extrapolation error. <ref> (right)
shows a continuum extrapolated equation of state for the dense nuclear matter. It is interesting to note that when the curve is
fitted to a power law, the exponent suggests that heavy dense matter behaves as a non-relativistic free gas of fermions. The
origin of this behaviour will be the subject of further study and we believe that analytical equations will be helpful in this
endeavour as one can toggle the interconnected degrees of freedom individually to obtain insight into the underlying processes at
Continuum extrapolated results from the analytic calculation. Left: Baryon density as a function of baryon chemical
potential, compare to <ref>. Right: The equation of state in the cold and dense regime with a fitted curve.
§ CHAIN RESUMMATION
So far we have managed to reproduce most of the simulated results with analytical calculations. In this section we will present a
resummation scheme for the analytic approach which even extends its reach beyond that of the numerical methods. First let us
demonstrate the pattern with a small example before moving on to the full resummation. Consider for now the following four terms
from (<ref>),
\begin{equation}
h_2 N_f \sum_{\mathrm{dof}} \; \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(270:.75)}] n1) {1}
edge[thick] (n1);
\end{midtikzpicture}, \;
- h_2^2 N_f\sum_{\mathrm{dof}} \: \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:{sqrt(2/3)})}] n1) {1}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:{sqrt(2/3)})}] n2) {1}
edge[thick] (n2);
\end{midtikzpicture}, \;
h_2^3 N_f\sum_{\mathrm{dof}} \: \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:{sqrt(2/3)})}] n1) {1}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:{sqrt(2/3)})}] n2) {1}
edge[thick] (n2);
\node[circ] (n4) at ([shift={(60:{sqrt(2/3)})}] n3) {1}
edge[thick] (n3);
\end{midtikzpicture}, \;
- h_2^4 N_f\sum_{\mathrm{dof}} \: \begin{midtikzpicture}
\node[circ] (n1) {1};
\node[circ] (n2) at ([shift={(60:{sqrt(2/3)})}] n1) {1}
edge[thick] (n1);
\node[circ] (n3) at ([shift={(300:{sqrt(2/3)})}] n2) {1}
edge[thick] (n2);
\node[circ] (n4) at ([shift={(60:{sqrt(2/3)})}] n3) {1}
edge[thick] (n3);
\node[circ] (n5) at ([shift={(300:{sqrt(2/3)})}] n4) {1}
edge[thick] (n4);
\end{midtikzpicture} .
\end{equation}
It is easy to see a common pattern appearing from these terms. Each term extends the length of the chain by one node while
maintaining a common prefactor. Looking at the equations we see that every link in the chain adds a factor $h_2 W_{2,1}$ to the
term, along with the necessary spatial geometry. One can check with the terms up to order $\kappa^8$ in
(<ref>) that this holds not only for the simple chain shown above, but for all terms in the action with a
singly connected node, meaning nodes that correspond to a factor of $W_{1,1}$. The "chain" resummation scheme thus corresponds to
the substitution
\begin{equation}
W_{1,1}(x) \to W_{1,1}(x) \sum_{n=0}^{\infty} \mathcal{G}(\{x_n\}) \prod_{i=1}^{n} (-h_2) W_{2,1}(x_i),
\end{equation}
where $\mathcal{G}(\{x_n\})$ contains the geometry of the chain. We will not go into more detail about the correctness of the
chain resummation here as it is quite involved. A more satisfactory argument based on the specifics of the effective 3d theory
will be given in a separate publication.
While we now have a resummation to all orders in $\kappa$ for every order in the effective theory, evaluating the final gauge
integral is still impossible. This is because we need to sum over all geometries for all the terms in
the resummation, which is inaccessible. To proceed, an additional constraint must be introduced, namely that only embeddings with
the same basic geometry as the chain itself is included. This implies that all nodes of the chain will be at separate
lattice points, and an $n+1$ long chain will result in the following integral
\begin{align}
\bigg( (2d) h_2 \int \mathrm{d} W \; \det\big[Q_{\mathrm{stat}}\big]^{2 N_f} W_{2,1} \bigg)^n
(2d) h_2 \int \mathrm{d} W\; \det&\big[Q_{\mathrm{stat}}\big]^{2 N_f} W_{1,1} \nonumber\\
&\equiv (2d h_2)^{n+1} I_{11} I_{21}^n.
\end{align}
and thus the full chain will give
\begin{equation}
(2d) h_2 I_{11} \sum_{n=0}^{\infty} (-(2d) h_2)^n I_{2,1}^n = \frac{(2d) h_2 I_{11}}{1 + (2d)h_2 I_{21}}.
\end{equation}
Carrying out this resummation also introduces a small error. This arises because the embedding factor of $(2d)$ is too large, and
the linked cluster expansion relies on cancellations from other same order graphs to correct this. These would be graphs that
overlap with itself as it spans the lattice. One could replace the embedding factor with a more natural one from e.g. studies of
the self-avoiding walk. For a 3d square lattice the factor would be somewhere between 4 and 5. However, in
<ref> one can clearly see that the contributions from self-overlapping terms are subleading, especially as we
increase the baryon chemical potential.
Value of the integrated nodes where the red curve comes from three separate nodes, the green curve from
the case where two of the three nodes overlap and finally the blue curve from the case where all three nodes overlap.
§ RESUMMED RESULTS
With the improved analytical results at hand we redo the convergence plot. In <ref> (left) the
effects of the resummation are clearly visible, more than doubling the convergence region in $h_2$. On the right of the same
figure we have also provided the result of carrying out a Padé approximation in $h_2$, choosing the close to diagonal
approximants. It is not surprising that the Padé approximation gives similar improvements, although slightly superior. Both
correspond to resummations giving rational expressions. The Padé is not restricted to a particular class of diagrams and
might therefore predict more of the higher order behaviour. Nonetheless, the similarity between the two results is reassuring as
to the validity of the chain resummation, which originates from the effective theory itself.
Convergence of the analytically improved 3d theory results. Left: Using the resummation scheme introduced in
Section <ref>. Right: Using the near diagonal Padé approximants.
Finally, we use this improved convergence behaviour to study another physics observable. In <ref> we plot
the binding energy per nucleon, defined by the energy density minus the mass density in the zero temperature limit,
\begin{equation}
\epsilon \equiv \frac{e - n_B m_B}{n_B m_B}\;.
\end{equation}
This is an important quantity that characterises nuclear matter. In previous work we have shown numerically and to leading order
in the hopping expansion, that it displays the silver blaze property until the onset transition, where it becomes negative
<cit.>. We can now extend this study to slightly larger densities. <ref> shows the binding
energy extracted from both the resummation scheme as well as the Padé approximants to the partition function at various orders.
While quantitative convergence breaks down shortly after the onset transition near $3\mu\sim m_B$, we obtain a new qualitative
result: in higher orders we see the binding energy becoming positive again with growing chemical potential, as is expected from
nuclear physics. A minimum characterising nuclear density appears, which however is not yet settled quantitatively at the
available orders.
Plots of the binding energy as a function of baryon chemical potential, $\beta=0$, $\kappa=0.08$, $N_t=50$. Left: Using
the resummation scheme. Right: Using the Padé approximants.
§ CONCLUSION
In a continuation of a long term project, we have derived a 3d effective lattice action for the description of thermodynamics of
QCD with heavy quarks through the orders $u^5\kappa^8$ in a combined character and hopping parameter expansion. The effective
action can be simulated by either complex Langevin or, due to the mildness of its sign problem, Metropolis with reweighting. By
comparing results for observables obtained with effective actions of neighbouring orders, full control over the convergence
properties of the expansion is obtained. The additional orders in the effective action allow for finer lattices and somewhat
lighter quark masses, though an extension to the physically interesting mass region is not in reach at present. By generalising
linked cluster expansion methods from spin models to our effective lattice theory with long distance and many-point couplings, a
fully analytic calculation of thermodynamic functions was achieved that quantitatively agrees with the simulations of the
effective theory. Finally, we have devised a resummation scheme that sums up infinite chains of graphs analytically, which in
addition includes higher order terms in the effective theory.
This project was supported by the German BMBF, No. 06FY7100 as well as by the Helmholtz International Center for FAIR within the
LOEWE program of the State of Hesse.
P. de Forcrand,
PoS LAT 2009 (2009) 010
[arXiv:1005.0539 [hep-lat]].
G. Aarts, F. Attanasio, B. Jäger, E. Seiler, D. Sexty and I. O. Stamatescu,
arXiv:1412.0847 [hep-lat].
G. Aarts, F. Attanasio, B. Jäger, E. Seiler, D. Sexty and I. O. Stamatescu,
arXiv:1510.09100 [hep-lat].
P. Scior, D. Scheffler, D. Smith and L. von Smekal,
PoS LATTICE 2014 (2015) 173
[arXiv:1412.7089 [hep-lat]].
P. Scior and L. von Smekal,
arXiv:1508.00431 [hep-lat].
C. Wozar, T. Kaestner, A. Wipf and T. Heinzl,
Phys. Rev. D 76 (2007) 085004
[arXiv:0704.2570 [hep-lat]].
J. Greensite and K. Langfeld,
Phys. Rev. D 88 (2013) 074503
[arXiv:1305.0048 [hep-lat]].
J. Greensite and K. Langfeld,
Phys. Rev. D 90 (2014) 014507
[arXiv:1403.5844 [hep-lat]].
G. Bergner, J. Langelage and O. Philipsen,
arXiv:1505.01021 [hep-lat].
J. Langelage, S. Lottini and O. Philipsen,
JHEP 1102 (2011) 057
[Erratum-ibid. 1107 (2011) 014]
[arXiv:1010.0951 [hep-lat]].
J. Langelage, S. Lottini and O. Philipsen,
PoS LATTICE 2010 (2010) 196
[arXiv:1011.0095 [hep-lat]].
J. Langelage and O. Philipsen,
JHEP 1001 (2010) 089
[arXiv:0911.2577 [hep-lat]].
J. Langelage and O. Philipsen,
JHEP 1004 (2010) 055
[arXiv:1002.1507 [hep-lat]].
T. C. Blum, J. E. Hetrick and D. Toussaint,
Phys. Rev. Lett. 76 (1996) 1019
M. Fromm, J. Langelage, S. Lottini and O. Philipsen,
JHEP 1201 (2012) 042
[arXiv:1111.4953 [hep-lat]].
M. Fromm, J. Langelage, S. Lottini, M. Neuman and O. Philipsen,
Phys. Rev. Lett. 110 (2013) 122001
[arXiv:1207.3005 [hep-lat]].
J. Langelage, M. Neuman and O. Philipsen,
JHEP 1409 (2014) 131
[arXiv:1403.4162 [hep-lat]].
T. Rindlisbacher and P. de Forcrand,
arXiv:1509.00087 [hep-lat].
M. Wortis,
In *Domb, C., Green, M.S.: Phase Transitions and Critical Phenomena, Vol.3*, 113-180
|
1511.00675
|
We develop an approach for simulating acousto-elastic wave phenomena,
including scattering from fluid-solid boundaries, where the solid is
allowed to be anisotropic, with the Discontinuous Galerkin method. We
use a coupled first-order elastic strain-velocity, acoustic
velocity-pressure formulation,
and append penalty terms based on
interior boundary continuity conditions to the numerical (central)
flux so that the consistency condition holds for the discretized
Discontinuous Galerkin weak formulation. We incorporate the
fluid-solid boundaries through these penalty terms and obtain a stable
Our approach avoids the diagonalization into
polarized wave constituents such as in the approach based on solving
elementwise Riemann problems.
Discontinuous Galerkin method – penalty flux – fluid-solid
boundaries – anisotropy
§ INTRODUCTION
The accurate computation of waves in realistic three-dimensional Earth
models represents an ongoing challenge in local, regional, and global
seismology. Here, we focus on simulating coupled acousto-elastic wave
phenomena including scattering from fluid-solid boundaries, where the
solid is allowed to be anisotropic, with the Discontinuous Galerkin
method. Of particular interest are applications in geophysics, namely,
marine seismic exploration and global Earth inverse problems using
earthquake-generated seismic waves as the probing field. In the first
application, we are concerned with the presence of the ocean bottom
and in the second one with
the core-mantle-boundary (CMB)
and inner-core-boundary (ICB).
Our formulation closely follows the
analysis of existence of (weak) solutions of hyperbolic first-order
systems of equations by
<cit.>. We use an
unstructured tetrahedral mesh with local refinement to accommodate
highly heterogeneous media and complex geometries, which is also an
underlying motivation for employing the Discontinuous Galerkin method
from a computational point of view.
In the past three decades, a wide variety of numerical techniques has
been employed in the development of computational methods for
simulating seismic waves.
The most widely used one is based on the finite difference method
<cit.> and
<cit.>]. This method has been applied to
computing the wavefield in three-dimensional local and regional models
<cit.> and
<cit.>]. The use of optimal or compact
finite-difference operators has provided a certain improvement [e.g.,
<cit.> and
<cit.>]. Methods
that resort to spectral and pseudospectral techniques based on global
gridding of the model have also been used both in regional [e.g.,
<cit.>] and global [e.g.,
<cit.> and
<cit.>] seismic wave
propagation and scattering problems. However, because of the use of
global basis functions (polynomial: Chebyshev or Legendre, or
harmonic: Fourier), these techniques are limited to coefficients which
are (piecewise) sufficiently smooth. The finite difference method
suffers from a limited accuracy in the presence of
a free surface or surface discontinuities with topography
within the model [e.g.,
<cit.> and
A procedure for
the stable imposition of free-surface boundary conditions for a
second-order formulation can be found in
<cit.>. Another approach, belonging to a broader family of
interface methods, handles both free surfaces [e.g.,
<cit.>] and fluid-solid interfaces [e.g.,
<cit.>] in such a way, conjectured by the authors, that
enables higher-order accuracy to be obtained.
<cit.> use summation-by-parts finite difference
operators along with a weak enforcement of boundary conditions to
develop a multi-block finite difference scheme which achieves
higher-order accuracy for complex geometries.
A key development in the computation of seismic waves has been based
on the spectral element method (SEM)
[<cit.>]. In its
original formulation, in terms of displacement
[<cit.>], continuity of displacement and velocity is
enforced everywhere within the model. In the case of a boundary
between an inviscid fluid and a solid, however, the kinematic boundary
condition is perfect slip; therefore, only the normal component of
velocity is continuous across such a boundary, and thus this
formulation is not applicable. Some classical finite-element
methods (FEMs) alternatively introduce coupling conditions on fluid-solid
interfaces between
displacement in the solid and pressure in the fluid
[e.g. <cit.>].
The FEM and SEM are commonly (but not exclusively)
based on the second-order form of the system
of equations describing acousto-elastic waves. In this case, the
acousto-elastic interaction is effected by coupling the respective
wave equations through appropriate interface conditions. To resolve
the coupling, a predictor-multicorrector iteration at each time step
has been used
<cit.>]. A computationally more efficient time
stepping method for global seismic wave propagation accommodating the
effects of fluid-solid boundaries, as well as transverse isotropy with
a radial symmetry axis and radial models of attenuation, was proposed
It uses a velocity potential
formulation and a second-order accurate Newmark time integration, in
which a time step is first performed in the acoustic fluid and then in
the elastic solid using interface values based on the fluid
solution. Currently the SEM is used in a variety of
implementations in global and regional seismic simulation, with the
effects of variations in elastic parameters, density, ellipticity,
topography and bathymetry, fluid-solid interfaces, anisotropy, and
self-gravitation included [e.g. <cit.>].
In contrast to classical finite element discretizations, the
Discontinous Galerkin (DG) method imposes continuity of approximate
solutions between elements only weakly through a numerical flux. The
Discontinuous Galerkin method has been employed for solving
second-order wave equations in both the acoustic and elastodynamic
settings [e.g.
<cit.> and
<cit.> employ a central numerical flux in a
DG scheme combined with a leap-frog time integration for the
velocity-stress elastic-wave formulation.
<cit.> developed a non-conservative
formulation with an upwind numerical flux using
only the material properties from the side of the interface
that is opposite to the outer normal direction.
<cit.> derived an upwind numerical flux by solving
the exact Riemann problem on interior boundaries of each element with
material discontinuities based on a velocity-strain formulation of the
coupled acousto-elastic equations.
In this work, we essentially extend the upwind flux, given by
<cit.> for hyperbolic systems, to a penalty flux
based on the boundary continuity condition for general
fluid-solid interfaces.
The novelties of our approach are the following: we
* use a coupled first-order elastic strain-velocity, acoustic
velocity-pressure formulation,
* obtain a self-consistent Discontinuous Galerkin weak formulation
without diagonalization into polarized wave constituents,
* append penalty terms, derived from interior boundary
continuity conditions, with an appropriate weight to the
numerical (central) flux so that the consistency condition holds
for the discretized Discontinuous Galerkin weak formulation,
* incorporate fluid-solid boundaries through the mentioned
penalty terms.
We note that the DG method is naturally adapted to well-posedness,
in the sense that it makes use of coercivity of the operator
defining the part of the system containing the spatial derivatives
separately in the solid and fluid regions.
§ THE SYSTEM OF EQUATIONS DESCRIBING ACOUSTO-ELASTIC WAVES
We consider a bounded domain $\Omega \subset \mathbb{R}^3$ which is
divided into solid and fluid regions, $\Os$ and $\Of$,
The interior boundaries include solid-solid interface $\Sss$,
fluid-fluid interface $\Sff$, and fluid-solid interface $\Sfs$, $\Ssf$
(where we distinguish whether the fluid or solid is on a particular side).
We present the weak form of the coupled acousto-elastic system of equations.
Hooke's law in an elastodynamical system is expressed by relating
stress, $S_{ij}$, and strain, $E_{kl}$. Assuming small deformations
gives a linear relationship, that is, $S_{ij} = c_{ijkl} E_{kl}$,
where $c_{ijkl}$ is the stiffness tensor. Through the relevant
symmetries, this tensor only contains 21 independent components.
We use the Voigt notation which simplifies the writing of tensors
while introducing
$\mathbi{S} = (S_{11}, S_{22}, S_{33}, S_{23} ,
S_{12}, S_{13})^T$ and $\mathbi{E} = (E_{11}, E_{22}, E_{33}, E_{23} ,
E_{12}, E_{13})^T$.
In this notation the stiffness tensor takes the form of
a 6 by 6 matrix, $\mathbi{C}$, defined by,
\begin{equation}
\mathbi{S} = \mathbi{C} \mathbi{E} ,\quad
\tsC = \begin{bmatrix}
C_{11} & C_{12} & C_{13} & 2 C_{14} & 2 C_{15} & 2 C_{16} \\[-1mm]
C_{12} & C_{22} & C_{13} & 2 C_{24} & 2 C_{25} & 2 C_{26} \\[-1mm]
C_{13} & C_{23} & C_{33} & 2 C_{34} & 2 C_{35} & 2 C_{36} \\[-1mm]
C_{14} & C_{24} & C_{34} & 2 C_{44} & 2 C_{45} & 2 C_{46} \\[-1mm]
C_{15} & C_{25} & C_{35} & 2 C_{45} & 2 C_{55} & 2 C_{56} \\[-1mm]
C_{16} & C_{26} & C_{36} & 2 C_{46} & 2 C_{56} & 2 C_{66}
\end{bmatrix} .
\label{eqn:stiffness}
\end{equation}
The isotropic case is obtained by setting all of the $C_{ij}$
components to zero except for $C_{11} = \lambda+2\mu$, $C_{12} =
C_{13} = C_{23} = \lambda$, $C_{44} = \mu$, $C_{55} = \mu$, and
$C_{66} = \mu$; $(\lambda,\mu)$ are the Lamé
Furthermore, $\rho$ denotes the density. The anisotropic
elastodynamical equations are written in terms of the strain,
$\mathbi{E}$, and the particle velocity, $\mathbi{v}$,
\begin{equation}\label{eq:elasticPDE}
\Dt{\tsE}
= \tfrac12 \left( \nabla \vev + \nabla \vev^T \right) ,
\quad
\rho \, \Dt{\vev}
= \nabla \cdot (\tsC \tsE) + \vef
\end{equation}
in $\Os$. In fluid regions, $\Of$, we use the pressure-velocity
\begin{equation}\label{eq:acousticPDE}
\Dt{\tE}
= \nabla \cdot \tvev
-\frac{\tf}{\tlambda} ,
\quad
\widetilde{\rho} \, \Dt{\tvev}
= \nabla (\tlambda \tE) .
\end{equation}
Here, $\widetilde P=-\tP$ is the pressure,
while we use $\ \widetilde{ }\ $ to
distinguish acoustic field quantities and material parameters from the
elastic ones.
In the above, $\tf$ denotes a volume source density of injection and
$\vef$ denotes a volume source density of force.
The solid-solid, fluid-solid and fluid-fluid
boundary conditions are given by
\begin{align}
\vev^+ - \vev^- = 0 &
\quad\quad \mbox{ and } &
\ven \cdot \tsS^+ - \ven \cdot \tsS^- =0 &
\quad\quad\text{ on } \Sss ,
\label{eq:BC1}\\
\ven \cdot (\vev^\pm - \tvev^\mp) = 0 &
\quad\quad \text{ and } &
\ven \cdot \tsS^\pm - (\tP)^\mp \ven = 0 &
\quad\quad\text{ on } \Ssf \mbox{ and } \Sfs,
\label{eq:BC2}\\
\ven \cdot (\tvev^+ - \tvev^-) = 0 &
\quad\quad \text{ and } &
(\tP)^+ -(\tP)^- =0 &
\quad\quad\mbox{ on } \Sff .
\label{eq:BC3}
\end{align}
The $\pm$ convention is determined by the direction of the interface
normal, $\ven$.
The outer normal vector points in the direction of the “$+$” side of the
We introduce test functions (tensors)
$\tsH, \vew$ in the solid regions and
$\tvew, \tH$ in the fluid regions, which are assumed to be contained
in the same spaces and satisfy the same boundary conditions
as $\tsE, \vev, \tvev$ and $\tE$. Using
(<ref>) and (<ref>), we find that
\begin{align}
\int_{\Os} \ddt{\tsE} : (\tsC \tsH\,) \dd\Omega
&= \int_{\Os} \tfrac12 (\nabla \vev + \nabla \vev^T) :
(\tsC \tsH\,) \dd\Omega ,
\label{eq:Weakform1}\\
\int_{\Os} \rho\, \ddt{\vev} \cdot \vew \dd\Omega
&= \int_{\Os} (\nabla \cdot (\tsC \tsE\,))
\cdot \vew \dd\Omega + \int_{\Os} \vef \cdot \vew \dd\Omega ,
\label{eq:Weakform2}\\
\int_{\Of} \ddt{\tE} \tlambda\, \tH \dd\Omega
&= \int_{\Of} (\nabla \cdot \tvev) \tlambda\, \tH \dd\Omega
- \int_{\Of} \tf \, \tH \dd\Omega ,
\label{eq:Weakform3}\\
\int_{\Of} \trho\, \ddt{\tvev} \cdot \tvew \dd\Omega
&= \int_{\Of} \nabla (\tP) \cdot \tvew \dd\Omega .
\label{eq:Weakform4}
\end{align}
Assuming an outer traction-free boundary condition in (<ref>)
and an outer pressure-free boundary condition in (<ref>),
and applying an integration by parts, we obtain
\begin{align}
\int_{\Os} \rho\, \ddt{\vev} \cdot \vew \dd\Omega
=& -\int_{\Os} \tsS : \nabla \vew \dd\Omega
+ \int_{\Sfs} (\ven \cdot \tsS^\mm) \cdot \vew^\mm \dd\Sigma
+ \int_{\Os} \vef \cdot \vew \dd\Omega ,
\label{eq:elasticWeak0}\\
\int_{\Of} \ddt{\tE} \tlambda\, \tH \dd\Omega
=& -\int_{\Of} \tvev\cdot\nabla (\tlambda\tH) \dd\Omega
+ \int_{\Sfs} (\ven \cdot \tvev^\mm)(\tlambda\tH)^\mm \dd\Sigma
- \int_{\Of} \tf \, \tH \dd\Omega .
\label{eq:acousticWeak0}
\end{align}
We use the fluid-solid boundary conditions (<ref>),
replacing the fluid-solid surface integrals in
(<ref>) and (<ref>)
by taking the average of both sides
consistent with a central flux scheme, and obtain
\begin{align}
\int_{\Os} \rho\, \ddt{\vev} \cdot \vew \dd\Omega
&= -\int_{\Os} \tsS : \nabla\vew \dd\Omega
\nonumber\\ &
+ \int_{\Sfs} \tfrac12((\tP)^\pp \ven+\ven \cdot \tsS^\mm)
\cdot \vew^\mm \dd\Sigma
+ \int_{\Os} \vef \cdot \vew \dd\Omega ,
\label{eq:WeakCouple1}\\
\int_{\Of} \ddt{\tE} \tlambda\tH \dd\Omega
&=-\int_{\Of} \tvev \cdot\nabla(\tlambda\tH) \dd\Omega
\nonumber\\ &
+ \int_{\Sfs} {\tfrac12(\ven \cdot \vev^\mm+\ven \cdot \tvev^\pp)}\,
(\tlambda\tH)^\mm \dd\Sigma
- \int_{\Of} \tf \, \tH \dd\Omega .
\label{eq:WeakCouple2}
\end{align}
This form of the equations is analogous to the one used in the spectral
element method, see
<cit.>. Applying
an integration by parts, again, in (<ref>), we recover
the coupled strong formulation,
\begin{align}
\int_{\Os} \rho\, \ddt{\vev} \cdot \vew \dd\Omega
&= \int_{\Os} (\nabla \cdot \tsS) \cdot \vew \dd\Omega
\nonumber\\ &
+ \int_{\Sfs} {\tfrac12}((\tP)^\pp \ven - \ven \cdot \tsS^\mm)
\cdot \vew^\mm \dd\Sigma
+ \int_{\Os} \vef \cdot \vew \dd\Omega ,
\label{eq:StrongCouple1}\\
\int_{\Of} \ddt{\tE} \tlambda\tH \dd\Omega
&= \int_{\Of} (\nabla \cdot \tvev) \tlambda\tH \dd\Omega
\nonumber\\ &
+ \int_{\Sfs} {\tfrac12}(\ven \cdot (\vev^\pp - \tvev^\mm))\,
(\tlambda\tH)^\mm \dd\Sigma
- \int_{\Of} \tf \, \tH \dd\Omega .
\label{eq:StrongCouple2}
\end{align}
We use this system of equations together with (<ref>) and
(<ref>) to develop our Discontinuous Galerkin method
based approach.
§ DISCONTINUOUS GALERKIN METHOD WITH FLUID-SOLID BOUNDARIES
The domain is partitioned into elements, $D^e$. We
distinguish elements, $\DeS$, in the solid regions from elements,
$\DeF$, in the fluid regions.
Correspondingly, we distinguish
fluid-fluid ($\Tff$), solid-solid ($\Tss$) and
fluid-solid ($\Tfs, \Tsf$) faces for each element;
thus the interior boundaries are decomposed as
\[
\Sigma_{\ast\,\bullet}=\cup\Sigma_{\ast\,\bullet}^\mathrm{e},
\quad \ast,\,\bullet\in\{\mathrm{S},\mathrm{F}\},
\]
and so are the elements'
boundaries: $\partial\DeS = \Tss \cup \Tsf$ and
$\partial\DeF = \Tff \cup \Tfs$.
The mesh size, $h$, is defined as the
maximum radius of each tetrahedral's inscribed sphere.
We introduce the broken polynomial space $ V_h = \bigoplus_{\De}V_h^{\De}$
where the local space is defined elementwise as
$V_h^{\De}=\mathrm{span}\{\phi_n(\De)\}_{n=1}^{\Ndof}$, with $\phi_n$
a set of polynomial basis further discussed in Section <ref>.
The subscript “$h$” indicates the refinement of $V_h$ with decrease in
mesh size.
The semi-discrete time-domain, discontinuous Galerkin formulation
using a central flux yields: Find $\tsE_h, \vev_h, \tvev_h, \tE_h $,
with each component for each one of them in $ V_h $
such that
\begin{equation}\label{eq:DG1}
\begin{split}
& \int_{\DeS} \ddt{\tsE_h} : (\tsC \tsH_h) \dd\Omega
+ \int_{\DeS} \rho\, \ddt{\vev_h} \cdot \vew_h \dd\Omega
\\
- \int_{\DeS} \tfrac12 (\nabla \vev_h + \nabla \vev_h^T) :
(\tsC \tsH_h) \dd\Omega
- \int_{\DeS} (\nabla \cdot (\tsC \tsE_h)) \cdot \vew_h \dd\Omega
\\
- \int_{\Tss} \tfrac12 \jmp{\vev_h}_\rSS \cdot
(\ven \cdot (\tsC \tsH_h)^\mm) \dd\Sigma
- \int_{\Tsf} {\color{black}\tfrac12} \jmp{\vev_h}_\rSF
\cdot (\ven \cdot (\tsC \tsH_h)^\mm) \dd\Sigma
\\
- \int_{\Tss} \tfrac12 \ven\cdot(\jmp{\tsC\tsE_h}_\rSS)
\cdot \vew_h^\mm \dd\Sigma
- \int_{\Tsf} {\tfrac12} \ven\cdot(\jmp{\tsC\tsE_h}_\rSF)
\cdot \vew_h^\mm \dd\Sigma
= \int_{\DeS} \vef_h \cdot \vew_h \dd\Omega ,
\end{split}
\end{equation}
\begin{equation}\label{eq:DG2}
\begin{split}
& \int_{\DeF} \ddt{\tE_h} \tlambda\tH_h \dd\Omega
+ \int_{\DeF} \trho\, \ddt{\tvev_h} \cdot \tvew_h \dd\Omega
\\
- \int_{\DeF} (\nabla \cdot \tvev_h) \, \tQ_h \dd\Omega
- \int_{\DeF} \nabla (\tP_h) \cdot \tvew_h \dd\Omega
\\
- \int_{\Tff} \tfrac12 (\ven\cdot\jmp{\tvev_h}_\rFF) \,
(\tQ_h)^\mm \dd\Sigma
- \int_{\Tfs} {\tfrac12} (\ven \cdot \jmp{\tvev_h}_\rFS) \,
(\tQ_h)^\mm \dd\Sigma
\\
- \int_{\Tff} \tfrac12 \jmp{\tP_h}_\rFF
(\ven \cdot \vew_h^\mm) \dd\Sigma
- \int_{\Tfs} {\tfrac12} \jmp{\tP_h}_\rFS
(\ven \cdot \vew_h^\mm) \dd\Sigma
= -\int_{\DeF} \tf_h \, \tH_h \dd\Omega ,
\end{split}
\end{equation}
hold for each element $\DeS$ or $\DeF$,
for all test functions $\tsH_h, \vew_h, \tvew_h, \tH_h
{\in V_h}$. The notations $\vef_h$ and $\tf_h$ indicate
polynomial approximation of $\vef$ and $\tf$.
\begin{align}
\jmp{\vev}_\rSS =& \vev^+ - \vev^- \\
\jmp{\tsC\tsE}_\rSS =& \ven\,(\ven \cdot (\tsC \tsE)^+
- \ven \cdot (\tsC \tsE)^-)
\end{array}
&\quad \mbox{ on } \Tss , \\
\jmp{\vev}_\rSF=&(\ven\cdot(\tvev^+ - \vev^-))\ven \\
\jmp{\tsC\tsE}_\rSF=&\ven\,((\tP)^+ \ven
- \ven \cdot (\tsC \tsE)^-)
\end{array}
&\quad \mbox{ on } \Tsf
\end{align}
in the solid regions, while
\begin{align}
\jmp{\tvev}_\rFF=& (\ven \cdot (\tvev^+ - \tvev^-))\ven \\
\jmp{\tP}_\rFF=& (\tP)^+ - (\tP)^-
\end{array}
&\quad \mbox{ on } \Tff , \\
\jmp{\tvev}_\rFS=& (\ven \cdot (\vev^+ - \tvev^-))\ven \\
\jmp{\tP}_\rFS=&\ven \cdot (\tsC \tsE)^+ \cdot \ven - (\tP)^-
\end{array}
&\quad \mbox{ on } \Tfs
\end{align}
in the fluid regions, using interior boundary continuity conditions. A
similar formulation for Maxwell's equations, using the central flux,
can be found in
§.§ Energy function of central flux
We consider a time-dependent energy function comprising both the solid
and fluid regions, $\Energy_h = \Energy_{\mathrm S,h} +
\Energy_{\mathrm F,h}$, with
\begin{equation}\label{eq:energy}
\begin{split}
\Energy_{\mathrm S,h}
&= \Half\SumS\int_\DeS
(\tsE_h : (\tsC \tsE_h) + \rho \, \vev_h
\cdot \vev_h) \dd\Omega ,
\\
\Energy_{\mathrm F,h}
&= \Half\SumF\int_\DeF
\left( \tP_h^2 + \trho \, \tvev_h \cdot
\tvev_h \right) \dd\Omega.
\end{split}
\end{equation}
The functions in (<ref>) define a norm both in the solid and
in the fluid regions.
Taking the time derivative
and noting that $\tsC$ is symmetric, we have
\begin{align}
\DDt{\Energy_{\mathrm S,h}}
&= \SumS\int_\DeS
\left( \ddt{\tsE_h} : (\tsC \tsE_h) + \rho \, \ddt{\vev_h}
\cdot \vev_h \right) \dd\Omega ,
\label{eq:energyS}
\\
\DDt{\Energy_{\mathrm F,h}}
&= \SumF\int_\DeF
\left( \ddt{\tE_h} \tP_h + \trho \, \ddt{\tvev_h}
\cdot \tvev_h \right) \dd\Omega .
\label{eq:energyF}
\end{align}
Starting from (<ref>) and (<ref>)
and carrying out the summation over all
the elements yields
\begin{equation}
\label{eq:central energy}
\DDt{\Energy_h} = \SumS\int_\DeS \vef_h \cdot \vev_h \dd\Omega
- \SumF\int_\DeF \tf_h \, \tE_h \dd\Omega .
\end{equation}
This property is obtained as follows:
In (<ref>) and (<ref>) we let $\tsH_h=\tsE_h, \vew_h=\vev_h,
\tH_h=\tE_h, \tvew_h=\tvev_h$, and obtain elementwise
\begin{equation}\label{eq:solid VolE to SurfE}
\begin{split}
& \int_{\DeS} \tfrac12 (\nabla \vev_h + \nabla \vev_h^T) :
(\tsC \tsE_h) \dd\Omega
+ \int_{\DeS} (\nabla \cdot (\tsC \tsE_h)) \cdot \vev_h \dd\Omega
\\
% = & -\left(
% \int_{\DeS} (\nabla \cdot (\tsC \tsE_h)) \cdot \vev_h \dd\Omega
% + \int_{\DeS} \tfrac12 (\nabla \vev_h + \nabla \vev_h^T) :
% (\tsC \tsE_h) \dd\Omega
% \right)
% &+ \int_{\Tss\cup\Tsf} \vev_h^\mm \cdot
% (\ven \cdot (\tsC \tsE_h)^\mm) \dd\Sigma
% + \int_{\Tss\cup\Tsf} \ven\cdot(\tsC\tsE_h)^\mm
% \cdot \vev_h^\mm \dd\Sigma
% = &\frac12 \left(
= & \int_{\Tss\cup\Tsf} \vev_h^\mm \cdot
(\ven \cdot (\tsC \tsE_h)^\mm) \dd\Sigma,
% + \int_{\Tss\cup\Tsf} \ven\cdot(\tsC\tsE_h)^\mm
% \cdot \vev_h^\mm \dd\Sigma \right),
\end{split}
\end{equation}
and similarily
\begin{equation}\label{eq:fluid VolE to SurfE}
\begin{split}
& \int_{\DeF} (\nabla \cdot \tvev_h) \, \tP_h \dd\Omega
+ \int_{\DeF} \nabla (\tP_h) \cdot \tvev_h \dd\Omega
\\
% = &\frac12\left(
= & \int_{\Tff\cup\Tfs} \ven\cdot\tvev_h^\mm \,
(\tP_h)^\mm \dd\Sigma.
% + \int_{\Tff\cup\Tfs} (\tP_h)^\mm
% (\ven \cdot \tvev_h^\mm) \dd\Sigma \right).
\end{split}
\end{equation}
From (<ref>), (<ref>), (<ref>) and
\begin{align}
\DDt{\Energy_{\mathrm S,h}}
\SumS\int_\DeS \vef_h \cdot \vev_h \dd\Omega
\nonumber\\
\Sumsf\tfrac12\int_{\Tsf}\left(
(\jmp{\vev_h}_\rSF+\vev_h^\mm) \cdot (\ven \cdot (\tsC \tsE_h)^\mm) +
\ven\cdot(\jmp{\tsC\tsE_h}_\rSF+(\tsC\tsE_h)^\mm)\cdot \vev_h^\mm
\right)\dd\Sigma
\tag{$\Theta_1$}\label{Theta 1} \\
\Sumss\tfrac12\int_{\Tss}\left(
(\jmp{\vev_h}_\rSS+\vev_h^\mm) \cdot (\ven \cdot (\tsC \tsE_h)^\mm) +
\ven\cdot(\jmp{\tsC\tsE_h}_\rSS+(\tsC\tsE_h)^\mm)\cdot \vev_h^\mm
\right)\dd\Sigma.
\tag{$\Theta_2$}\label{Theta 2}
\end{align}
In the above,
\begin{equation}
\Theta_2=
\Sumss\tfrac12\int_{\Tss}\left(
\vev_h^\pp \cdot(\ven \cdot (\tsC \tsE_h)^\mm) +
\ven\cdot(\tsC\tsE_h)^\pp \cdot \vev_h^\mm
\right)\dd\Sigma =0.
\end{equation}
The surface integration terms cancel out when summed from both sides
of the solid-solid interfaces because of the continuity condition
(<ref>) and the opposite outer normal directions.
We are left with the contributions from solid-fluid inner faces,
\begin{equation}
\DDt{\Energy_{\mathrm S,h}}=
\Sumsf\tfrac12\int_{\Tsf}\left(
\tvev_h^\pp \cdot(\ven \cdot (\tsC \tsE_h)^\mm)+
(\tP)^\pp \ven\cdot \vev_h^\mm
\right)\dd\Sigma
+ \SumS\int_\DeS \vef_h \cdot \vev_h \dd\Omega.
\label{eq:summation1}
\end{equation}
A similar result in the fluid region obtained from (<ref>),
(<ref>), (<ref>) and (<ref>)
\begin{equation}
\DDt{\Energy_{\mathrm F,h}}=
\Sumfs\tfrac12\int_{\Tfs}\left(
(\ven \cdot \vev_h^\pp)(\tP)^\mm +
\ven \cdot (\tsC \tsE_h)^\pp \cdot\tvev_h^\mm
\right)\dd\Sigma
-\SumF\int_\DeF \tf_h \, \tE_h \dd\Omega,
\label{eq:summation2}
\end{equation}
and the surface integration terms on the solid-fluid and fluid-solid
interfaces in (<ref>) and (<ref>) cancel out
due to (<ref>).
Therefore (<ref>) is obtained. We note that the surface
integration along solid-fluid interfaces
$\int_{\Tsf} {\tfrac12} \ven\cdot(\jmp{\tsC\tsE_h}_\rSF)
\cdot \vew_h^\mm \dd\Sigma$ and
$ \int_{\Tfs} {\tfrac12} (\ven \cdot \jmp{\tvev_h}_\rFS) \,
(\tQ_h)^\mm \dd\Sigma$
are essential to guarantee energy conservation.
§.§ Nodal basis functions
The discretized solution follows an expansion, componentwise,
into $\Ndof = \Ndof(N_p)$ nodal trial basis functions of order $N_p$,
as is in
\begin{equation}\label{eq:expand-E}
\begin{split}
(\tsE_{h,n}^{\De})_{ij}(t)\phi_n(\mathbi{x}) ,\\
\mbox{with } (\tsE_{h,n}^{\De})_{ij}(t)=&(\tsE_h)_{ij}
(\mathbi{x}_n,t) , n=1,2,\cdots,\Ndof,
\end{split}
\end{equation}
and similarly for the other fields, $\vev_h, \tvev_h, \tE_h$.
The superscript, ${\centerdot}^{D^e}$,
indicates a local expansion within element
$D^e$. In the above, ${\{ \phi_n(\mathbi{x}) \}}_{n = 1}^{\Ndof}$
is a set of three-dimensional Lagrange polynomials
associated with the nodal points, $ {\{ \mathbi{x}_n \}}_{n = 1}^{\Ndof} $
(see Figure <ref>), with each polynomial defined as
\[
\phi_k(\mathbi{x})=\prod_{j=1,j\neq k}^{\Ndof}
\frac{\mathbi{x}-\mathbi{x}_j}{\mathbi{x}_k-\mathbi{x}_j}.
\]
We use the warp & blend method [<cit.>] to determine the
coordinates of nodal points in the tetrahedron by numerically minimizing
the Lebesgue constant of interpolation.
For an order $N_p$ interpolation there are
$\Ndof = \frac16(N_p + 1)(N_p + 2)(N_p + 3)$ nodal points.
$ N_p $ = 1 $ N_p $ = 3 $ N_p $ = 8
Warp & blend tetrahedral nodal point distribution for $ N_p
$ = 1, 3, 8. For clarity only facial nodes are illustrated.
The medium coefficients are expanded in a likewise manner
\begin{equation}\label{eq:expand-c}
\begin{split}
(\tsC_h )_{ij}(\mathbi{x}) = &\bigoplus_{\De}\sum_{n=1}^{N_p}
(\tsC_{h,n}^{\DeS})_{ij} \phi_n(\mathbi{x}) ,\\
\mbox{with }(\tsC_{h,n}^{\DeS})_{ij} =& (\tsC_h)_{ij}
(\mathbi{x}_n) , n=1,2,\cdots,\Ndof,
\end{split}
\end{equation}
and similarly for $\rho, \trho, \tlambda$.
When refining a mesh, we expect an increase in number of elements $\De$
with decreased size.
§.§ The system of equations in matrix form
To simplify the notation in the further development of a numerical
scheme, we introduce a joint matrix form of the system of equations.
We map the components of $\tsE, \vev$ and $\tE, \tvev$ to $9
\times 1$ and $4 \times 1$ matrices, respectively,
\begin{equation}\label{eq:unknown vec1}
\veq=(E_{11},E_{22},E_{33},E_{23},E_{13},E_{12},v_1,v_2,v_3)^T
\quad\text{and}\quad
\tveq=(\tE,\widetilde{v}_1,\widetilde{v}_2,\widetilde{v}_3)^T ,
\end{equation}
and, correspondingly, the components of body forces $ \vef $
and $ \tf $ to the matrix
\[
\veg = (0,0,0,0,0,0,f_1,f_2,f_3)^T
\quad\text{and}\quad
\tveg = \left(-\frac{\tf}{\tlambda},0,0,0\right)^T
\quad.
\]
Equations (<ref>) and (<ref>) attain the form
\begin{equation}
\tsQ \, \ddt{\veq} - \nabla \cdot (\tsA \veq) = \veg
\quad\mbox{and}\quad
\ttsQ \, \ddt{\tveq} - \nabla \cdot (\ttsA \tveq)
\end{equation}
\[ \tsQ=\left(
\begin{array}{ccc}I_{6\times 6} & \vline & 0 \\ \hline 0 & \vline &
\rho I_{3\times 3}\end{array}\right) \quad\text{and}\quad
\ttsQ=\left(\begin{array}{ccc} {1} & \vline & 0 \\ \hline 0 &
\vline & \trho I_{3\times 3}\end{array}\right)
\]
\[ \tsA=(A_1,A_2,A_3) \quad\text{and}\quad \ttsA=(\widetilde{A}_1,
\widetilde{A}_2,\widetilde{A}_3) ,
\]
that is,
\[
\begin{split}
(\nabla \cdot (\tsA \veq))_l
= \partial_{x_k} ( (\tsA_{k})_{lm} \veq_m)
\quad\text{and}\quad
(\nabla \cdot (\ttsA \tveq))_l
= \partial_{x_k} ( (\ttsA_{k})_{lm} \tveq_m)
k=1,2,3,\quad l,m=1,\cdots,9 \mbox{ or } 1,\cdots,4
\end{split}
\]
\[
\begin{array}{ccc}
{\displaystyle \mathbf{0}} & \hspace*{-4mm} \vline \hspace*{-4mm} &
\begin{array}{ccc}
1 & 0 & 0\\[-2mm]
0 & 0 & 0\\[-2mm]
0 & 0 & 0\\[-2mm]
0 & 0 & 0\\[-2mm]
0 & 0 & \frac{1}{2}\\[-2mm]
0 & \frac{1}{2} & 0
\end{array}\\
\hline
\begin{array}{cccccc}
C_{11} & C_{12} & C_{13} & 2C_{14} & 2C_{15} & 2C_{16} \\[-2mm]
C_{16} & C_{26} & C_{36} & 2C_{46} & 2C_{56} & 2C_{66} \\[-2mm]
C_{15} & C_{25} & C_{35} & 2C_{45} & 2C_{55} & 2C_{56}
\end{array} & \hspace*{-4mm} \vline \hspace*{-4mm} &
\end{array}
\hspace*{-2mm}\right)
\quad \text{and}\quad
\widetilde{A}_1=\left(\hspace*{-2mm}
\begin{array}{ccc}
{\displaystyle \mathbf{0}} & \hspace*{-2mm} \vline \hspace*{-4mm} &
\begin{array}{ccc}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\\
\hline
& \hspace*{-2mm} \vline \hspace*{-4mm} &
\end{array}
\hspace*{-2mm}\right) ,
\]
\[
\begin{array}{ccc}
{\displaystyle \mathbf{0}} & \hspace*{-4mm} \vline \hspace*{-4mm} &
\begin{array}{ccc}
0 & 0 & 0\\[-2mm]
0 & 1 & 0\\[-2mm]
0 & 0 & 0\\[-2mm]
0 & 0 & \frac{1}{2}\\[-2mm]
0 & 0 & 0\\[-2mm]
\frac{1}{2} & 0 & 0
\end{array}\\
\hline
\begin{array}{cccccc}
C_{16} & C_{26} & C_{36} & 2C_{46} & 2C_{56} & 2C_{66} \\[-2mm]
C_{12} & C_{22} & C_{23} & 2C_{24} & 2C_{25} & 2C_{26} \\[-2mm]
C_{14} & C_{24} & C_{34} & 2C_{44} & 2C_{45} & 2C_{46}
\end{array} & \hspace*{-4mm} \vline \hspace*{-4mm} &
\end{array}
\hspace*{-2mm}\right)
\quad \text{and}\quad
\widetilde{A}_2=\left(\hspace*{-2mm}
\begin{array}{ccc}
{\displaystyle \mathbf{0}} & \hspace*{-2mm} \vline \hspace*{-4mm} &
\begin{array}{ccc}
0 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{array}\\
\hline
& \hspace*{-2mm} \vline \hspace*{-4mm} &
\end{array}
\hspace*{-2mm}\right) ,
\]
\[
\begin{array}{ccc}
{\displaystyle \mathbf{0}} & \hspace*{-4mm} \vline \hspace*{-4mm} &
\begin{array}{ccc}
0 & 0 & 0\\[-2mm]
0 & 0 & 0\\[-2mm]
0 & 0 & 1\\[-2mm]
0 & \frac{1}{2} & 0\\[-2mm]
\frac{1}{2} & 0 & 0\\[-2mm]
0 & 0 & 0
\end{array}\\
\hline
\begin{array}{cccccc}
C_{15} & C_{25} & C_{35} & 2C_{45} & 2C_{55} & 2C_{56} \\[-2mm]
C_{14} & C_{24} & C_{34} & 2C_{44} & 2C_{45} & 2C_{46} \\[-2mm]
C_{13} & C_{23} & C_{33} & 2C_{34} & 2C_{35} & 2C_{36}
\end{array}&
\hspace*{-4mm} \vline \hspace*{-4mm} &
\end{array}
\hspace*{-2mm}\right)
\quad \text{and}\quad
\widetilde{A}_3=\left(\hspace*{-2mm}
\begin{array}{ccc}
{\displaystyle \mathbf{0}} & \hspace*{-2mm} \vline \hspace*{-4mm} &
\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1
\end{array}\\
\hline
& \hspace*{-2mm} \vline \hspace*{-4mm} &
\end{array}
\hspace*{-2mm}\right) .
\]
We define the coefficient matrices $ \tsA_n $ in the normal
directions $ \ven=(\,n_1,n_2,n_3) $ as $ \tsA_n=n_1A_1+n_2A_2+n_3A_3
$, thus $ \tsA_n\veq \equiv \ven\cdot(\tsA\veq) $; similarly, $
\ttsA_n=n_1\tilde A_1+n_2\tilde A_2+n_3\tilde A_3 $. We can also give
them in the matrix form,
\[
\tsA_n =\left(\begin{array}{cc}0&T_{12}\\T_{21}\cdot \tsC&0\end{array}\right)
\quad\text{and}\quad
\ttsA_n = \left(\begin{array}{cc} 0 & \ven^T \\
{\tlambda}\ven & 0 \end{array}\right)
\]
\[
n_1 & 0 & 0 & 0 & \tfrac12n_3 & \tfrac12n_2 \\
0 & n_2 & 0 & \tfrac12n_3 & 0 & \tfrac12n_1 \\
0 & 0 & n_3 & \tfrac12n_2 & \tfrac12n_1 & 0
\end{array}\right)^T,
\quad
n_1 & 0 & 0 & 0 & n_3 & n_2 \\
0 & n_2 & 0 & n_3 & 0 & n_1 \\
0 & 0 & n_3 & n_2 & n_1 & 0
\end{array}\right).
\]
We introduce
\[
\tsLa = \left(
\begin{array}{ccc} \tsC & \vline & 0 \\ \hline 0 & \vline &
I_{3\times 3}\end{array} \right)
\quad\text{and}\quad
\ttsLa = \left(
\begin{array}{ccc} \tlambda & \vline & 0 \\ \hline 0 & \vline &
I_{3\times 3}\end{array} \right)
\]
In the solid regions, we write $ \vep = {
\left(
% \begin{array}{c}\tsH \\ \vew \end{array}
\right)}^T $, and in the
fluid regions, we write $ \tvep =
% \left(
% \begin{array}{c}\tH \\ \tvew \end{array}
(\tH,\tilde w_1,\tilde w_2,\tilde w_3)^T
% \right)^T
The inner product $\innprod{\veq,\vep}_\Omega$ indicates the dot
product of vectors $\veq$ and $\vep$ followed by integration over
the domain $\Omega$.
Equation (<ref>) is then rewritten, regarding the supports of basis
functions $\vep_h$ localized to an element $\De_{\mathrm{S},\mathrm{F}}$,
\begin{equation}
\label{eq:semi_disc_DG_S}
\begin{split}
\innprodDeS{\tsQ_h\ddt{\veq_h},{\tsLa_h}\vep_h}
-&\innprodDeS{\nabla \cdot (\tsA_h\veq_h),{\tsLa_h} \vep_h}
\jmp{\tsA_{n,h}\veq_h}_\rSS,
\jmp{\tsA_{n,h}\veq_h}_\rSF,
= \innprodDeS{\veg,{\tsLa_h}\vep_h} ,
\end{split}
\end{equation}
\begin{equation}
\label{eq:semi_disc_DG_F}
\begin{split}
\innprodDeF{\ttsQ_h\ddt{\tveq_h},{\ttsLa_h}\tvep_h}
-&\innprodDeF{\nabla \cdot (\ttsA_h\tveq_h),{\ttsLa_h} \tvep_h}
\jmp{\ttsA_{n,h}\tveq_h}_\rFF,
\jmp{\ttsA_{n,h}\tveq_h}_\rFS,
= \innprodDeF{\tveg,{\ttsLa_h}\tvep_h}.
\end{split}
\end{equation}
In the above we identify the central flux as
\begin{equation}\label{eq:central flux}
\Fluxc_{\mathrm S*}=\tfrac12\innprod{\jmp{\tsA_n \veq}_{\mathrm S*},
(\tsLa \vep)^-}_{\Sigma_{\mathrm S*}^\mathrm{e}},
\quad
\tFluxc_{\mathrm F*}=\tfrac12\innprod{\jmp{\ttsA_n \tveq}_{\mathrm F*},
(\ttsLa \tvep)^-}_{\Sigma_{\mathrm F*}^\mathrm{e}},
\quad
\end{equation}
in which we redefine
\begin{equation}\label{eq:fluid-solid projection}
\begin{split}
& \jmp{\tsA_n\veq}_\rSS=(\tsA_n\veq)^\pp-(\tsA_n\veq)^\mm,\quad
\jmp{\tsA_n\veq}_\rSF=O^T(\ttsA_n\tveq)^\pp-(\tsA_n\veq)^\mm,\\
& \jmp{\ttsA_n\tveq}_\rFF=(\ttsA_n\tveq)^\pp-(\ttsA_n\tveq)^\mm,\quad
\jmp{\ttsA_n\tveq}_\rFS=O^{\phantom{*}}(\tsA_n\veq)^\pp-(\ttsA_n\tveq)^\mm,
\end{split}
\end{equation}
with the map $O: \mathbb{R}^9 \to \mathbb{R}^4$ given by
\[
O \veq = \left(\begin{array}{c}
\ven \cdot \tsE \cdot \ven \\
(\ven \cdot \vev)\ven
\end{array}\right),
\quad \mbox{ and its adjoint } \quad
O^T \tveq = \left(\begin{array}{c}
( \ven\ven )\tE \\
(\ven \cdot \tvev)\ven
\end{array}\right),
\]
which can also be explicitly given in the matrix form
\[
\begin{array}{ccc}
\begin{array}{cccccc}
n_1n_1 & n_2n_2 & n_3n_3 & n_2n_3 & n_1n_3 & n_1n_2
\end{array} & \hspace*{-4mm}
\vline \hspace*{-4mm} & {\displaystyle \mathbf{0}} \\
\hline
{\displaystyle \mathbf{0}} & \hspace*{-4mm} \vline \hspace*{-4mm} &
\begin{array}{ccc}
n_1n_1 & n_1n_2 & n_1n_3 \\[-2mm]
n_1n_2 & n_2n_2 & n_2n_3 \\[-2mm]
n_1n_3 & n_2n_3 & n_3n_3
\end{array}
\end{array}
\hspace*{-2mm}\right).
\]
THE BOUNDARY CONDITION PENALIZED NUMERICAL FLUX AND STABILITY
Here, we construct our penalized numerical flux. The flux is designed
such that the penalized discrete counterpart of the
weak form (<ref>) and (<ref>)
the condition of non-increasing energy
and guarantees a proper error estimate.
We replace the central fluxes, $ \Fluxc $ and $\tFluxc$, in
(<ref>), by penalized fluxes, $ \Fluxp $ and $\tFluxp$,
by adding penalty terms, that is:
\begin{equation}\label{eq:penalty flux}
\begin{split}
\Fluxp_{\mathrm S*}=&
\tfrac12\innprod{\jmp{\tsA_n \veq}_{\mathrm S*},
(\tsLa \vep)^-}_{\Sigma_{\mathrm S*}^\mathrm{e}}
+\alpha\innprod{\tsA_n^{T,\mm}\jmp{\tsA_n \veq}_{\mathrm S*},
\vep^-}_{\Sigma_{\mathrm S*}^\mathrm{e}}
\\
&=\tfrac12\innprod{\jmp{\tsA_n \veq}_{\mathrm S*},
(\tsLa \vep)^-}_{\Sigma_{\mathrm S*}^\mathrm{e}}
+\alpha\innprod{\jmp{\tsA_n \veq}_{\mathrm S*},
(\tsA_n\vep)^-}_{\Sigma_{\mathrm S*}^\mathrm{e}},
\\
\tFluxp_{\mathrm F*}=&
\tfrac12\innprod{\jmp{\ttsA_n \tveq}_{\mathrm F*},
(\ttsLa \tvep)^-}_{\Sigma_{\mathrm F*}^\mathrm{e}}
+\alpha\innprod{\ttsA_n^{T,\mm}\jmp{\ttsA_n \tveq}_{\mathrm F*},
\tvep^-}_{\Sigma_{\mathrm F*}^\mathrm{e}}
\\
&=\tfrac12\innprod{\jmp{\ttsA_n \tveq}_{\mathrm F*},
(\ttsLa \tvep)^-}_{\Sigma_{\mathrm F*}^\mathrm{e}}
+\alpha\innprod{\jmp{\ttsA_n \tveq}_{\mathrm F*},
(\ttsA_n\tvep)^-}_{\Sigma_{\mathrm F*}^\mathrm{e}},
\quad *\in\{\mathrm S,\mathrm F\}
\end{split}
\end{equation}
with $\alpha $ some positive constant scalar.
With this modification, (<ref>) and
(<ref>) becomes
\begin{align}
\begin{aligned}
\innprodDeS{\tsQ_h\ddt{\veq_h},{\tsLa_h}\vep_h}
-&\innprodDeS{\nabla \cdot (\tsA_h\veq_h),{\tsLa_h} \vep_h}
\jmp{\tsA_{n,h}\veq_h}_{\mathrm S*},
(\tsLa_h\vep_h)^\mm}_{\Sigma_{\mathrm S*}^\mathrm{e}}
% \jmp{\tsA_{n,h}\veq_h}_\rSS,
% (\tsLa_h\vep_h)^\mm}_\Tss
% \jmp{\tsA_{n,h}\veq_h}_\rSF,
% (\tsLa_h\vep_h)^\mm}_\Tsf
\\
\jmp{\tsA_{n,h}\veq_h}_{\mathrm S*},
(\tsA_{n,h}\vep_h)^\mm}_{\Sigma_{\mathrm S*}^\mathrm{e}}
% \jmp{\tsA_{n,h}\veq_h}_\rSS,
% (\tsA_{n,h}\vep_h)^\mm}_\Tss
% \jmp{\tsA_{n,h}\veq_h}_\rSF,
% (\tsA_{n,h}\vep_h)^\mm}_\Tsf
= \innprodDeS{\veg,{\tsLa_h}\vep_h} ,
\end{aligned}
\label{eq:penalty_DG_S}
\\
\begin{aligned}
\innprodDeF{\ttsQ_h\ddt{\tveq_h},{\ttsLa_h}\tvep_h}
-&\innprodDeF{\nabla \cdot (\ttsA_h\tveq_h),{\ttsLa_h} \tvep_h}
\jmp{\ttsA_{n,h}\tveq_h}_{\mathrm F*},
(\ttsLa_h\tvep_h)^\mm}_{\Sigma_{\mathrm F*}^\mathrm{e}}
% \jmp{\ttsA_{n,h}\tveq_h}_\rFF,
% (\ttsLa_h\tvep_h)^\mm}_\Tff
% \jmp{\ttsA_{n,h}\tveq_h}_\rFS,
% (\ttsLa_h\tvep_h)^\mm}_\Tfs
\\
\jmp{\ttsA_{n,h}\tveq_h}_{\mathrm F*},
(\ttsA_{n,h}\tvep_h)^\mm}_{\Sigma_{\mathrm F*}^\mathrm{e}}
% \jmp{\ttsA_{n,h}\tveq_h}_\rFF,
% (\ttsA_{n,h}\tvep_h)^\mm}_\Tff
% \jmp{\ttsA_{n,h}\tveq_h}_\rFS,
% (\ttsA_{n,h}\tvep_h)^\mm}_\Tfs
= \innprodDeF{\tveg,{\ttsLa_h}\tvep_h},
\quad *\in\{\mathrm S,\mathrm F\}.
\end{aligned}
\label{eq:penalty_DG_F}
\end{align}
In Appendix <ref> we provide a guideline how to choose an
$\alpha$ based on an error analysis. We set $\alpha=1/2$, in which case
the energy function with the penalty terms coincides with the one
using an upwind flux <cit.>.
For the convergence analysis, we follow <cit.>
while obtaining an error estimate.
Following the matrix form in Subsection <ref>, we immediately
rewrite the definition of energy functions (<ref>) in solid
and fluid region as
\begin{equation}
\begin{split}
\Energy_{\mathrm S,h}=&
\Half\SumS \innprod{\tsQ_h\veq_h,\tsLa_h\veq_h}_\DeS
= \Half\SumS \norm{\veq}_{L^2(\DeS;\tsQ_h,\tsLa_h)}\\
\Energy_{\mathrm F,h}=&
\Half\SumF \innprod{\ttsQ_h\tveq_h,\ttsLa_h\tveq_h}_\DeF
= \Half\SumF \norm{\tveq}_{L^2(\DeF;\ttsQ_h,\ttsLa_h)}.
\end{split}
\label{eq:Energy matrixform}
\end{equation}
Here $\Norm{\centerdot}_{L^2(\DeS; \tsQ,\tsLa)}$ and
$\Norm{\centerdot}_{L^2(\DeF; \ttsQ,\ttsLa)}$ are the energy norms in solid
and fluid regions, and we simplify the notification
without causing ambiguity by
$\normVS{\centerdot}$ and $\normVF{\centerdot}$, respectively.
We also define the energy norms in solid-solid, fluid-fluid and
solid-fluid interfaces similarly as $\normSS{\centerdot}$,
$\normFF{\centerdot}$ and $\normSF{\centerdot}$, $\normFS{\centerdot}$.
Upon taking the penalty terms into consideration,
equation (<ref>) is replaced by
\begin{equation}\label{eq:DG-penalty-energy}
\begin{split}
+\frac\alpha 2
\Bigl (
\Sumss \Norm{
\jmp{\tsA_{n,h}\veq_h}_\rSS
+ \Sumff \Norm{
\jmp{\ttsA_{n,h}\tveq_h}_\rFF
}_{L^2(\Tff)}^2 %\right.
\\
+ 2\Sumsf \Norm{
\jmp{\tsA_{n,h}\veq_h}_\rSF
}_{L^2(\Tsf)}^2 \Bigr )
% \right)
= \SumS\int_\DeS \veg_h \cdot \tsLa_h\veq_h \dd\Omega
+ \SumF\int_\DeF \tveg_h \cdot \ttsLa_h \tveq_h \dd\Omega .
\end{split}
\end{equation}
To obtain this result, in (<ref>) – (<ref>),
we let $\vep=\veq, \tvep=\tveq$.
Taking the summation over all penalty terms on solid-solid interfaces
\begin{equation}\label{eq:penaltyE1}
\begin{split}
\Sumss\innprod{
\jmp{\tsA_{n,h}\veq_h}_\rSS,
\\&%\hspace{1cm}
% (\tsA_{n,h}\veq_h)^\pp-(\tsA_{n,h}\veq_h)^\mm
\jmp{\tsA_{n,h}\veq_h}_\rSS
%% (\tsA_{n,h}\veq_h)^\pp-(\tsA_{n,h}\veq_h)^\mm
% \jmp{\tsA_{n,h}\veq_h}_\rSS
% }_{L^2(\Sss)}^2 .
\end{split}
\end{equation}
Taking the summation over all penalty terms on fluid-fluid interfaces yields
\begin{equation}\label{eq:penaltyE2}
\Sumff\innprod{
\jmp{\ttsA_{n,h}\tveq_h}_\rFF,
% (\ttsA_{n,h}\tveq_h)^\pp-(\ttsA_{n,h}\tveq_h)^\mm,
% (\ttsA_{n,h}\tveq_h)^\mm}_\Tff
\jmp{\ttsA_{n,h}\tveq_h}_\rFF
% \jmp{\ttsA_{n,h}\tveq_h}_\rFF
% }_{L^2(\Sff)}^2 .
\end{equation}
We rewrite the penalty terms on fluid-solid interface from the solid side as
\begin{equation}
\begin{split}
\jmp{\tsA_{n,h}\veq_h}_\rSF,
(\tsA_{n,h}\veq_h)^\mm}_\Tsf \\
(\tsA_{n,h}\veq_h)^\mm}_\Tsf ,
\end{split}
\label{eq:sf energy}
\end{equation}
and from the fluid side as
\begin{equation}
\begin{split}
\jmp{\ttsA_{n,h}\tveq_h}_\rFS,
(\ttsA_{n,h}\tveq_h)^\mm}_\Tfs \\
(\ttsA_{n,h}\tveq_h)^\mm}_\Tfs \\
\end{split}
\label{eq:fs energy}
\end{equation}
in which the property $O O^T=I_{4\times 4}$ is used.
Changing from the fluid to the solid sides yields
\begin{equation}
\begin{split}
\jmp{\ttsA_{n,h}\tveq_h}_\rFS,
(\ttsA_{n,h}\tveq_h)^\mm}_\Tfs \\
\end{split}
\label{eq:SF energy}
\end{equation}
Summation over all fluid-solid interfaces with
(<ref>) and (<ref>),
\begin{equation}\label{eq:penaltyE3}
\begin{split}
\jmp{\tsA_{n,h}\veq_h}_\rSF,
\jmp{\ttsA_{n,h}\tveq_h}_\rFS,
\\
\\
\jmp{\tsA_{n,h}\veq_h}_\rSF
\end{split}
\end{equation}
Thus we obtain (<ref>).
Our approach is reminiscent of earlier work, in which an upwind flux
is defined by the Riemann solutions which are obtained by
diagonalizing $\tsA_n$, that is, $\tsA_n = R D R^T$, on the faces of
each element [<cit.>], and $D$ is the diagonal matrix of
eigenvalues of $\tsA_n$.
The upwind flux takes the form,
\begin{equation}\label{eq:upwind flux}
\begin{split}
&\Flux^{\mathrm U}_{\mathrm S*}=
\innprod{\jmp{\tsA_n \veq}_{\mathrm S*},
(\tsLa \vep)^-}_{\Sigma_{\mathrm S*}^\mathrm{e}}
+\innprod{\jmp{(R |D| R^T)
\veq}_{\mathrm S*},
(\tsLa \vep)^-}_{\Sigma_{\mathrm S*}^\mathrm{e}},
\\
&\tFlux^{\mathrm U}_{\mathrm F*}=
\innprod{\jmp{\ttsA_n \tveq}_{\mathrm F*},
(\ttsLa \tvep)^-}_{\Sigma_{\mathrm F*}^\mathrm{e}}
+\innprod{\jmp{(\widetilde R |\widetilde D| \widetilde R^T)
\tveq}_{\mathrm F*},
(\ttsLa \tvep)^-}_{\Sigma_{\mathrm F*}^\mathrm{e}},
\quad *\in\{\mathrm{S},\mathrm{F}\},
\end{split}
\end{equation}
where $|\centerdot|$ stands for the operaton of taking the absolute
value of each entry of the diagonal matrix, that is, $|D|_{ij}=|D_{ij}|$.
Our approach avoids this diagonalization, allowing general
heterogeneous media with anisotropy.
§ TIME DISCRETIZATION
In this section, we discuss a time discretization that is
computationally efficient for complex domains.
Often, the computational meshes used to model the subsurface
must contain regions where the characteristic lengths of the elements
drop far below that of a wavelength because the subsurface contains
very complex geometries and discontinuities. As a result, the time
steps must be equally reduced to produce a stable solution. We follow
two different time discretization schemes: (1) for non-complex
domains, it is advantageous to use a traditional Runge–Kutta (RK)
method and (2) for complex domains, a semi implicit–explicit (IMEX)
method is used. The IMEX method enables the solver to perform implicit
time integration in areas of oversampling, while keeping the
computational efficiency of RK in regions of proper sampling.
§.§ Explicit Runge–Kutta
We use an explicit time integration method when the variation in
element size is small. There are a variety of time-stepping methods
available, however, we employ the five stage low-storage explicit
Runge–Kutta (LSERK) method from <cit.>.
LSERK is an explicit method the time-step of which is dictated by the
Courant–Friedrichs–Lewy (CFL) condition.
Efforts to define, quantitatively, a stable CFL condition depending on
polynomial order $N_p$, can be found in <cit.>.
The LSERK method is preferred over other methods because it saves
memory at the cost of computation time.
§.§ Explicit–Implicit Runge–Kutta
When the domain in question contains complex geometries within large
domains, such as rough surfaces,
the resulting mesh will contain regions of oversampling relative to
the relevant wavelengths. This hinders the use of an
implicit time-stepping method
because its accuracy depends on the size of the time step, which in
turn is dependent on the region of highest spatial sampling.
A natural approach is
the IMEX method, (e.g. <cit.>),
which allows the regions of oversampling to be integrated in time with
an L-stable third-order and 3-stage Diagonally Implicit Runge–Kutta
(DIRK) method, while using a fast and simple 4-stage third-order ERK
method in the regions of more reasonable sampling (8–10 nodes per
The system can be solved without requiring an
interpolation at the boundary of the implicit–explicit regions.
The intermediate abscissaes of each time step
for implicit Runge–Kutta stages and
for explicit ones
are selected to equal one another
so as to synchronize the explicit and
implicit schemes, and the so-called Butcher matrix
is calculated correspondingly.
The implicit stages
are solved using a multifrontal factorization.
§ COMPUTATIONAL EXPERIMENTS
Here, we illustrate our DG method by verifying its convergence rate and
carrying out computational experiments.
We use the fourth-order LSERK algorithm for time integration.
For visualization of wavefields or model parameters,
we write the value in the Visualization Toolkit (VTK)
unstructured mesh format and visualize the result using
Paraview <cit.>.
§.§ Convergence tests at (interior) boundaries
We carry out computational tests using wave propagation and
scattering problems in 3-dimensional cubic subdomains.
We first test the propagation of a plane wave
in a homogeneous isotropic elastic medium,
in which periodic boundary conditions are applied.
We also test the free-surface boundary condition with
a homogeneous isotropic elastic solid,
in which both Rayleigh and Love waves are generated.
We focus on the Rayleigh wave,
the particle motion of which is
in the plane perpendicular to the free surface.
A Stoneley wave, generated at a solid-elastic
interface <cit.> in an unbounded domain
composed of two half spaces with different material properties,
is also simulated and
compared with the closed-form solution in
For the test of our DG method at an acousto-elastic interface,
we generate a Scholte wave.
We refer to <cit.>
for the closed-form solution.
The external boundary conditions, beside those mentioned above,
are imposed
by using the traction of the exact solution as boundary “forces”
(A) (B)
(C) (D)
$L^2$ error of partical velocity $\vev$ as a function of
mesh size $h$,
for the simulation of
(A) a plane wave,
(B) a Rayleigh wave,
(C) a Stoneley wave, and
(D) a Scholte wave,
for different orders $N_p = 2,3,\cdots,6 $ .
The computational domains are discretized as regular tetrahedral meshes.
A sufficiently small constant, $K_\mathrm{CFL}=0.05$,
was selected during the tests for time stepping,
and a large simulation time (10 s) is choosen for the error computation.
The domain geometry and boundary conditions for each test are given in
Table <ref>.
The relevant material parameters, that is, the Lamé
parameters $\lambda$ and $\mu$, and density $\rho$, are given in
Table <ref>.
We calculate the $L^2$ errors for the particle velocity of the
numerical solutions,
which are discretized by $N_p$ order polynomials.
The magnitudes of the numerical errors at time $t =$ 10 s are shown in
Figure <ref>, as a function of mesh size $h$ for different
values of $N_p$, and least-squares fits to lines,
with the estimated convergence order for each line shown in the legend.
We observe that the $L^2$ error of our numerical scheme
achieves a convergence rate higher than $N_p+\frac12$.
We also show a comparison of accuracies and convergence rates
tested with the wave types described in this section for the
upwind flux, the central flux and our penalty flux in Appendix
wave type domain range (in km)
boundary conditions
plane wave $[-1,1]\times[-1,1]\times[-1,1]$
periodic boundaries
Rayleigh wave $[-1,1] \times [-1,1] \times [0,2]$
free surface boundary at $x_3=0$,
exact boundary “force” at $x_3=2$,
periodic boundaries otherwise
Stoneley wave $[-2,2] \times [-2,2] \times [-2,2]$
exact boundary “force” at $x_3=\pm 2$,
periodic boundaries otherwise
Scholte wave $[-2,2] \times [-2,2] \times [-2,2]$
exact boundary “force” at $x_3=\pm 2$,
periodic boundaries otherwise
Geometry and boundary conditions
for the four wave types in the convergence tests.
wave type
material properties
plane wave
$\lambda=2.00$ GPa, $\mu=1.00$ GPa,
$\rho=1.00$ g/cm3
Rayleigh wave
$\lambda=2.00$ GPa, $\mu=1.00$ GPa,
$\rho=1.00$ g/cm3
Stoneley wave
$\lambda=1.20$ Gpa, $\mu=1.20$ GPa,
$\rho=1.20$ g/cm3,
for $x_3>0$
$\lambda=3.00$ Gpa, $\mu=1.20$ GPa,
$\rho=4.00$ g/cm3,
for $x_3<0$
Scholte wave
$\lambda=1.20$ Gpa, $\mu=1.30$ GPa,
$\rho=1.10$ g/cm3,
for $x_3>0$
$\lambda=1.11$ Gpa, $\mu=0.00$ GPa,
$\rho=1.32$ g/cm3,
for $x_3<0$
Material parameters
for the four wave types in the convergence tests.
§.§ Homogeneous orthorhombic solid: Caustics
Here, we simulate a band-limited fundamental solution in an anisotropic
elastic medium, forming caustics. The medium is orthorhombic and
Several minerals in Earth's mantle
have orthorhombic symmetry; this symmetry also appears in regions of
sedimentary basins where fracture sets are commonly found in sandstone
beds, shales, and granites.
The material properties are selected as follows,
$\rho$ $C_{11}$ $C_{22}$ $C_{33}$ $C_{44}$ $C_{55}$ $C_{66}$
$C_{23}$ $C_{13}$ $C_{12}$
1.0 (g/cm3) 30.40 19.20 16.00
4.67 10.86 12.82 4.80 4.00 6.24 (GPa)
which produce a medium
whose P phase velocities are 5.51 km/s, 4.38 km/s, and 4.00 km/s and
S phase velocities are 2.16 km/s, 3.26 km/s, and 3.58 km/s in the
principal directions (perpendicular to the symmetry planes).
The computational domain is a $5 \times 5
\times 5$ (in km) cube. We place an explosive Gaussian source
at the center of the cube, using a Ricker wavelet with a
center frequency of 5Hz.
Images of isosurfaces of the different components of the particle velocity
are shown in Figure <ref>.
We note the presence of caustics in one of the shear polarizations.
(A) (B) (C)
Snapshots of the contours for the particle velocity
(A) $v_1$, (B) $v_2$, and (C) $v_3$ at $t=0.45 s$.
The black arrow in (C) indicates
the shear wave front forming caustics.
§.§ Flat isotropic fluid-solid interface:
Propagation of Scholte wave
Fluid-solid configuration visualized in the
$x_1$–$x_3$ plane at $x_2=15.0$, with source and receiver located
in the fluid.
A snapshot at $ t=12s $ is shown in (a), and a snapshot at $ t=26s $
is shown in (b).
Seismic trace from a hydrophone located at $ (40.0,\, 15.0,\,
6.0) $km in the fluid side. Arrival times of head wave Pn, direct P waves
and Scholte waves are indicated by vertical lines.
We present a model with dimensions $[0,50]\times[0,30]\times[0,15]$km
with a flat fluid-solid interface located at $ x_3=7.5 $km. The fluid
side is homogeneous isotropic with an acoustic wave speed $ 1.5 $km/s
and density $ 1.0 $g/cm3. The solid side is homogeneous
isotropic with a P-wave speed $ 3.0 $km/s and S-wave speed $ 1.5
$km/s, and density $ 2.5 $g/cm3. The Scholte wave speed
is computed numerically as $1.2455 $km/s [e.g.,
We place an explosive source in the fluid at location $(5.0,\, 15.0,\,
6.5)$km, using a Ricker wavelet as the source-time series with a central
frequency of $ 2.0 $Hz. A
receiver is located at $ (45.0,\, 15.0,\, 6.5) $km and records the
synthetic phases for 40 seconds. We apply convolutional perfect matching
layers (CPMLs)
[e.g., <cit.>]
for all external boundaries of the model,
highlighting the effects of a fluid-solid internal boundary.
Two snapshots are shown in Figure <ref>,
one for the solution at $ t=12 $s
and the other for the solution
at $ t=26 $s, in which we observe
the occurence of a Scholte wave which is well seperated from the body wave
phases at long times.
The amplitude of the Scholte wave decays exponentially with the
distance from fluid-solid interface [<cit.>].
Figure <ref> shows
the seismogram as well as the arrival times of the head wave Pn,
the direct P wave and
Scholte wave. The modelled phase arrivals agree well with the travel times
marked by perpendicular lines.
§.§ Seismic waves in a geological structure: SEAM model
In this application, the DG method's ability to model the propagation
and scattering of seismic waves in a field-scale domain with complex
geological structures is demonstrated. The 3D SEAM (SEG Advanced
Modeling) Phase I acoustic model is used that has heterogeneous structures
and represents the sea-bed of the Gulf of Mexico [<cit.>].
It spans a 35 km
by 40 km region of the earth's surface and has a depth of 15 km,
and is discretized as a regular grid
with 20m $ \times$ 20m $\times$ 10m
sample interval. The model has several geological features that we
will use to test the robustness of the DG method. It contains a
high-velocity salt body that extends through the center of the model
(Figure <ref>). The rapid contrast in velocity makes the
model, in the language of partial differential equations, a stiff
domain. Another geometric feature is the sedimentary layering at
approximately 10 km under the surface. These layers will cause multiple
scattering that will lead to constructive and destructive
A tetrahedral meshing for the 3D SEAM generated by
segmentation and mesh deformation techniques.
The color map shows the P wavespeed $v_p$ interpolation.
Slices of the 3D SEAM acoustic velocity model and snapshot of
pressure wave field at $t=5.0$s, with the same viewpoint as in
Figure <ref>.
Slices of the isotropic extension of 3D SEAM Phase I shear
wavespeed model and snapshot of 3-component of particle velocity at $t=5.0$s, with the same viewpoint as in
Figure <ref> and <ref>.
A tetrahedral mesh with 863,973 elements of order 3 is generated adaptively
starting from the contours of the wave speed model, including the
rough boundary of the salt body (Figure <ref>)
and selected smooth interfaces associated with the sedimentary layers.
We generate triangular isosurfaces based on domain partitioning of the
wavespeed model into four primary subdomains:
the ocean layer, the salt body, a high-contrast sediment layer and
the sediment background. We also adaptively add
vertices by tracking the contrasts of wavespeed inside each subregion.
Using these, a tetrahedral mesh was created using TetGen
A point source is located at the ocean bottom $(x_1,x_2,x_3)=(17.5 , 15.0 ,
1.45)$km and the source function was a Ricker wavelet with a center
frequency of 10.0 Hz.
A snapshot of the acoustic pressure wave field
solution is shown in Figure <ref>.
We also consider an extension of the SEAM Phase I model to
isotropic elasiticity as is presented by
<cit.>. We represent, via interpolation,
the S wave speed and density on the unstructured mesh based on
the four distinct subdomains,
and place a point source inside the ocean layer at
$(x_1,x_2,x_3)=(17.5 , 15.0 , 0.10)$km
using a Ricker wavelet with a center frequency of 5.0 Hz.
We apply a pressure-free surface boundary condition
on the ocean surface, and CPMLs elsewhere.
The S wavespeed and $3$-component of the particle velocity are shown
in Figure <ref>,
in which the shear wave front can be clearly observed after the P arrivals.
§.§ Scattering from a rough surface: Fractured carbonate
Here, we model the reflection generated by an explosive point source
from a rough surface embedded in a transversely
isotropic medium. This type of medium closely resembles fractured
samples of carbonate rocks [<cit.>]. Carbonates are
abundantly found in nature. They pose many complications when working
with them in the field because the physical properties vary from site
to site and are strongly heterogeneous within the bulk rock. A
homogeneous transversely isotropic medium can be used to model a
carbonate because a variation in velocity amongst layers is the most
common form of heterogeneity [<cit.>].
(A) (B)
(A) Domain of the digitized rough surface. (B) Zoomed in of the mesh.
The unit of the axises are in meters.
(A) (B)
Slices of the $V_3$ wave field after (A) 21 $\mu s$, (B) 31 $\mu s$,
and (C) 41 $\mu s$ from a 3D rough surface.
Laser profilometry was used to measure the surface roughness of an induced
fracture in Austin Chalk, a carbonate rock sample.
From these measurements, a profile of the surface was extracted to provide
a rough boundary in an otherwise cubic domain
with edge length of 0.1 m. The rough surface was placed
on the top plane of the box, i.e.$x_3=0.1$m (Figure <ref>). The material properties were
chosen such that the symmetry axis was in the
$(\hat{x}_1,\hat{x}_2,\hat{x}_3)=(0,1,0)$ direction.
P- and S-phase velocities
along the axis of symmetry are 4000 m/s and 2280 m/s
respectively, and are 4900 m/s and 2000 m/s respectively along the
other two directions.
The following table provides a list of the specific
elastic constants used:
$\rho$ $C_{11}$ $C_{22}$ $C_{33}$ $C_{44}$ $C_{55}$ $C_{66}$
$C_{23}$ $C_{13}$ $C_{12}$
1.5 (g/cm3) 24.00 16.00 24.00
4.00 5.20 4.00 8.00 13.60 8.00 (GPa)
The tetrahedral mesh contains 686,444 elements, with $N_p=4$.
We place an explosive source at $(x_1,x_2,x_3) = (.05,.05,0)$,
using a Ricker wavelet with a central frequency of 1 MHz.
Two snapshots of the wave field were taken of the
$3$-component of the particle velocity (Figure
<ref>) that display the formation of shear-wave
caustics due to anisotropy at $t = 21 \mu s$,
and the solutions
of scattering at $t = 31 \mu s$ and $t= 41 \mu s$, respectively.
§.§ Heterogeneous anisotropic solid-fluid boundary with
Here, we use our DG method to simulate the wave propagation and
scattering in a heterogeneous anisotropic solid-fluid
configuration. The solid-fluid boundary has topography, which is well
described by adaptively fitting an unstructured mesh (see Figure
<ref>(a)). The model has
dimensions $[0,50] \times [0,30] \times [0,15]$km. The fluid side is
homogeneous isotropic with an acoustic wave speed $ 1.5 $km/s and
density $ 1.0 $g/cm3. The solid side consists of a reference
HTI medium component with elastic parameters given by $
C_{11}=33.75,\,C_{22}=22.50,\, C_{33}=13.85,\,
$ (GPa), and $ \rho=2.5 $g/cm3. A low-velocity lens is
superimposed with its center located at $ (25,\,15,\,9) $km. We place an
explosive source in the fluid at location $(8.0,\, 15.0,\,6.5)$km,
using a Ricker wavelet as source-time series with a central frequency of
$ 1.0 $Hz. We apply convolutional perfect matching layers (CPMLs) for all
external boundaries of the model with the thickness of approximately two
central wavelengths.
Heterogeneous HTI solid-fluid boundary with topography.
(a) 3D model setting, with color indicating quasi-P wavespeed; (b) snapshot at t=4.0s; (c) snapshot at t=18.0s.
The waves are propagated for 40 seconds. Two snapshots in time
of the wave field are shown; the solution at $ t=4 $s (Figure
<ref> (b)) and the solution at $ t=14 $s (Figure
<ref> (c)), with the occurence of a Scholte
wave and seperation from body waves while propagating.
We note the fomation of caustics in the solid region,
caused by the anisotropy and the low-velocity lens.
§ DISCUSSION
We develop a DG-method based numerical approach to simulate
acousto-elastic wave phenomena. We demonstrate
its ability to generate accurate solutions in domains with
heterogeneous and complex geometries for long-time simulation. We
briefly discuss the specifics of and differences between our and
earlier developed DG methods for general acousto-elastic wave
Most of the existing DG
discretizations for solving the acousto-elastic system of equations in
the first-order formulation make use of an upwind numerical flux
derived from the elementwise solution of a Riemann problem.
In <cit.>, a Godunov upwind flux is applied upon
diagonalizing the coefficient matrix in the stress-velocity
formulation at element-element interfaces. Specifically, they use a
“one-sided” upwind numerical flux and, to avoid elementwise
numerical integration and make use of pre-calculated matrices instead,
restrict the coefficients to be
constant in each element.
Steger-Warming flux-vector splitting in <cit.>
is another way to obtain an exact Riemann solution for
the linear system
with flexibly parameterized isotropic elastic media,
allowing variable coefficient within elements.
The velocity-strain formulation introduced by <cit.> uses
the Rankine-Hugoniot jump condition to obtain an upwind flux for isotropic
solid-fluid interfaces while designing a uniform conservative
formulation for coupled elasto-acoustic systems.
Meanwhile, there are penalty based DG schemes designed to solve numerically
the second-order system of equations for the displacement.
The interior penalty Galerkin method is used by
<cit.> to solve a nonlinear parabolic system, and a symmetric
interior penalty term was employed by
to make the stiffness matrix symmetric positive definite.
studies the dispersion and convergence of these interior penalty
DG-method based schemes for the second-order elliptic
Lamé system.
<cit.> defines for a general hyperbolic system a flux that
penalizes the fields based on their continuity.
In our DG-method based scheme, we introduce a penalized numerical flux
the form of which is motivated by the interior boundary continuity
conditions. The fluid-solid boundary conditions are accounted for in
the coupling of elements through the fluxes.
Our penalty weight
does not depend on the normal direction of the interior faces of the
elements, and moreover, unlike the interior penalty scheme in the
second-order displacement formulation, does not depend on the mesh
size either.
§ ACKNOWLEDGMENT
This research was supported in part by the members, BGP, ExxonMobil, PGS,
Statoil and Total, of the Geo-Mathematical Imaging Group.
§ CONVERGENCE ANALYSIS
In this section we consider the $L^2$ error of numerical solutions $\veq_h$
and $\tveq_h$, which satisfy (<ref>)–(<ref>)
for any $\vep_h$ and $\tvep_h \in V_h^{N_p}$.
We denote by $\pi_h^{N_p}:L^2\mapsto V_h^{N_p}$ the $L^2$ projection onto
the polynomial space of order $N_p$.
We assume that $\vef-\vef_h=0$ and $\tf-\tf_h=0$, and no
error occurs for $L^2$ projection of coefficient matrices, that is,
$\tsA-\tsA_h=0, \tsQ-\tsQ_h=0$ and $ \tsLa-\tsLa_h=0$.
We define $\verr:=\veq-\veq_h$ and $\tverr:=\tveq-\tveq_h$,
where $\veq$ and $\tveq$ are the exact solutions.
We also denote $\vxi:=\veq_h-\pi_h^{N_p}\veq\, , \,\,
\tvxi:=\tveq_h-\pi_h^{N_p}\tveq \,$,
and $\veta:=(1-\pi_h^{N_p})\veq\, , \,\, \tveta:=(1-\pi_h^{N_p})\tveq \,$;
thus $\verr=\veta-\vxi, \, \tverr=\tveta-\tvxi$.
We define the volume residuals
\begin{equation}\label{eq:vol_res}
\resv(\veq_h):=\tsLa^T\left(\tsQ\ddt{\veq_h}
\quad
\tresv(\tveq_h):=\ttsLa^T\left(\ttsQ\ddt{\tveq_h}
\end{equation}
and surface residuals
\begin{equation}\label{eq:surf_res}
\begin{split}
\ressS(\veq_h):=\tfrac12(\tsLa^\mm)^T\jmp{\tsA_n\veq_h}_\rSS
\\&
\tressS(\tveq_h):=\tfrac12(\ttsLa^\mm)^T\jmp{\ttsA_n\tveq_h}_\rFS
\\&
\ressF(\veq_h):=\tfrac12(\tsLa^\mm)^T\jmp{\tsA_n\veq_h}_\rSF
\\&
\tressF(\tveq_h):=\tfrac12(\ttsLa^\mm)^T\jmp{\ttsA_n\tveq_h}_\rFF
\end{split}
\end{equation}
Using (<ref>)–(<ref>),
it follows that $(\verr,\tverr)$ satisfy
\begin{equation}\label{eq:numerical solution}
\begin{split}
\SumS \int_{\DeS}\resv(\verr)\cdot\vep_h\dd \Omega
- \Sumss\int_{\Tss}\ressS(\verr)\cdot\vep_h^\mm \dd \Sigma
- \Sumsf\int_{\Tsf}\ressF(\verr)\cdot\vep_h^\mm \dd \Sigma =&0,
\\
\SumF \int_{\DeF}\tresv(\tverr)\cdot\tvep_h\dd \Omega
- \Sumfs\int_{\Tfs}\tressS(\tverr)\cdot\tvep_h^\mm \dd \Sigma
- \Sumff\int_{\Tff}\tressF(\tverr)\cdot\tvep_h^\mm \dd \Sigma =&0,
\end{split}
\end{equation}
upon setting $\tsQ_h=\tsQ$ and $\tsA_h=\tsA\,$.
We take inner products of (<ref>) and (<ref>) with
corresponding test functions, and immediately get,
after summing up all the terms,
\begin{align}
&\SumS\int_{\DeS} \tsQ\ddt{\veq_h} \cdot \tsLa\vep_h\dd \Omega
- \SumS\int_{\DeS} (\nabla\cdot(\tsA\veq_h))\cdot\tsLa\vep_h\dd \Omega
\\&\hspace{1cm}
- \Half\Sumss\int_{\Tss} \JmpRd\jmp{\tsA_n\veq_h}_{\rSS}
\cdot (\tsLa\vep_h)^\mm \dd \Sigma
- \Half\Sumsf\int_{\Tsf} \JmpRd\jmp{\tsA_n\veq_h}_{\rSF}
\cdot (\tsLa\vep_h)^\mm \dd \Sigma
\\&\hspace{1cm}
- \alpha\Sumss\int_{\Tss} \JmpRd\jmp{\tsA_n\veq_h}_{\rSS}
\cdot (\tsA_n\vep_h)^\mm \dd \Sigma
- \alpha\Sumsf\int_{\Tsf} \JmpRd\jmp{\tsA_n\veq_h}_{\rSF}
\cdot (\tsA_n\vep_h)^\mm \dd \Sigma \\
=&\SumS\int_{\DeS}\resv(\veq_h)\cdot\vep_h\dd \Omega
- \Sumss\int_{\Tss}\ressS(\veq_h)\cdot\vep_h^\mm \dd \Sigma
- \Sumsf\int_{\Tsf}\ressF(\veq_h)\cdot\vep_h^\mm \dd \Sigma, \\
\end{aligned}
\label{eq:numerical residual elastic}
\\[5mm]&
\begin{aligned}
&\SumF\int_{\DeF} \ttsQ\ddt{\tveq_h} \cdot \ttsLa\tvep_h\dd \Omega
- \SumF\int_{\DeF} (\nabla\cdot(\ttsA\tveq_h))\cdot\ttsLa\tvep_h\dd \Omega
\\&\hspace{1cm}
- \Half\Sumff\int_{\Tff}\JmpRd\jmp{\ttsA_n\tveq_h}_{\rFF}
\cdot (\ttsLa\tvep_h)^\mm \dd \Sigma
- \Half\Sumfs\int_{\Tfs}\JmpRd\jmp{\ttsA_n\tveq_h}_{\rFS}
\cdot (\ttsLa\tvep_h)^\mm \dd \Sigma
\\&\hspace{1cm}
- \alpha\Sumff\int_{\Tff}\JmpRd\jmp{\ttsA_n\tveq_h}_{\rFF}
\cdot (\ttsA_n\tvep_h)^\mm \dd \Sigma
- \alpha\Sumfs\int_{\Tfs}\JmpRd\jmp{\ttsA_n\tveq_h}_{\rFS}
\cdot (\ttsA_n\tvep_h)^\mm \dd \Sigma \\
=&\SumF\int_{\DeF}\tresv(\tveq_h)\cdot\tvep_h\dd \Omega
- \Sumff\int_{\Tff}\tressF(\tveq_h)\cdot\tvep_h^\mm \dd \Sigma
- \Sumfs\int_{\Tfs}\tressS(\tveq_h)\cdot\tvep_h^\mm \dd \Sigma.\\
\label{eq:numerical residual acoustic}
\end{aligned}
\end{align}
We let $\veq_h=\vep_h=\vxi$, $\tveq_h=\tvep_h=\tvxi$, when equations
(<ref>) and
(<ref>) become
\begin{align}
\begin{aligned}
- \alpha\Sumss\int_{\Tss}\JmpRd\jmp{\tsA_n\vxi}_{\rSS}
\cdot (\tsA_n\vxi)^\mm \dd \Sigma
- \alpha\Sumsf\int_{\Tsf}\JmpRd\jmp{\tsA_n\vxi}_{\rSF}
\cdot (\tsA_n\vxi)^\mm \dd \Sigma \\[-2mm]
&\hspace{5mm} - \left(
\SumS\int_{\DeS} (\nabla\cdot(\tsA\vxi)) \cdot \tsLa\vxi\dd \Omega
+ \Half\Sumss\int_{\Tss}\JmpRd\jmp{\tsA_n\vxi}_{\rSS}
\cdot (\tsLa\vxi)^\mm \dd \Sigma
\right.\\[-4mm]
& \hspace{8cm}\left.
+ \Half\Sumsf\int_{\Tsf}\JmpRd\jmp{\tsA_n\vxi}_{\rSF}
\cdot (\tsLa\vxi)^\mm \dd \Sigma \right)\\
& =\SumS\int_{\DeS}\resv(\vxi)\cdot\vxi \dd \Omega
- \Sumss\int_{\Tss}\ressS(\vxi)\cdot\vxi^\mm \dd \Sigma
- \Sumsf\int_{\Tsf}\ressF(\vxi)\cdot\vxi^\mm \dd \Sigma ,
\end{aligned}
\label{eq:error elastic}
\\[5mm]&
\begin{aligned}
- \alpha\Sumff\int_{\Tff}\JmpRd\jmp{\ttsA_n\tvxi}_{\rFF}
\cdot (\ttsA_n\tvxi)^\mm \dd \Sigma
- \alpha\Sumfs\int_{\Tfs}\JmpRd\jmp{\ttsA_n\tvxi}_{\rFS}
\cdot (\ttsA_n\tvxi)^\mm \dd \Sigma \\[-2mm]
&\hspace{5mm} - \left(
\SumF\int_\DeF ( \nabla\cdot(\ttsA\tvxi) ) \cdot \ttsLa\tvxi\dd \Omega
+ \Half\Sumff\int_{\Tff}\JmpRd\jmp{\ttsA_n\tvxi}_{\rFF}
\cdot (\ttsLa\tvxi)^\mm \dd \Sigma
\right. \\[-4mm]
& \hspace{8cm}\left.
+ \Half\Sumfs\int_{\Tfs}\JmpRd\jmp{\ttsA_n\tvxi}_{\rFS}
\cdot (\ttsLa\tvxi)^\mm \dd \Sigma \right)\\
& =\SumF\int_\DeF\tresv(\tvxi)\cdot\tvxi \dd \Omega
- \Sumff\int_{\Tff}\tressF(\tvxi)\cdot\tvxi^\mm \dd \Sigma
- \Sumfs\int_{\Tfs}\tressS(\tvxi)\cdot\tvxi^\mm \dd \Sigma .
\end{aligned}
\label{eq:error acoustic}
\end{align}
Adding (<ref>) and (<ref>), and using
the energy result in Section <ref>,
the terms in between parentheses on
the left-hand sides of both equations cancel one another, and the penalty
terms turn into quadratic forms, that is,
\begin{equation}\label{eq:energy estimate}
\begin{split}
& \hspace{-3mm}
\frac12\DDt{}\left(\SumS\normVS{\vxi}^2
+\SumF\normVF{\tvxi}^2 \right) \\[-2mm]
& \hspace{2mm} +\frac\alpha2\left(
\Sumss\normSS{\jmp{\tsA_n\vxi}_{\rSS}}^2
+2\Sumsf\normSF{\jmp{\tsA_n\vxi}_{\rSF}}^2 \right)\\
=\SumS\int_{\DeS}\resv(\vxi)\cdot\vxi \dd \Omega
- \Sumss\int_{\Tss}\ressS(\vxi)\cdot\vxi^\mm \dd \Sigma
- \Sumsf\int_{\Tsf}\ressF(\vxi)\cdot\vxi^\mm \dd \Sigma \\[-2mm]
& \hspace{2mm} +
\SumF\int_{\DeF}\tresv(\tvxi)\cdot\tvxi \dd \Omega
- \Sumff\int_{\Tff}\tressF(\tvxi)\cdot\tvxi^\mm \dd \Sigma
- \Sumfs\int_{\Tfs}\tressS(\tvxi)\cdot\tvxi^\mm \dd \Sigma .
\end{split}
\end{equation}
Let $\vep_h=\vxi$ in (<ref>), and subtract it from the
right-hand side of (<ref>). We note that
$\verr=\veta-\vxi, \, \tverr=\tveta-\tvxi$, and obtain
\begin{equation}\label{eq:energy estimate1}
\begin{split}
& \hspace{-3mm}
\frac12\DDt{}\left(\SumS\normVS{\vxi}^2
+\SumF\normVF{\tvxi}^2 \right) \\[-1mm]
& \hspace{2mm} +\frac\alpha2\left(
\Sumss\normSS{\jmp{\tsA_n\vxi}_{\rSS}}^2
+2\Sumsf\normSF{\jmp{\tsA_n\vxi}_{\rSF}}^2 \right) \\
=\SumS\int_{\DeS}\resv(\veta)\cdot\vxi \dd \Omega
- \Sumss\int_{\Tss}\ressS(\veta)\cdot\vxi^\mm \dd \Sigma
- \Sumsf\int_{\Tsf}\ressF(\veta)\cdot\vxi^\mm \dd \Sigma \\[-2mm]
& \hspace{2mm} + \SumF\int_{\DeF}\tresv(\tveta)\cdot\tvxi \dd \Omega
- \Sumff\int_{\Tff}\tressF(\tveta)\cdot\tvxi^\mm \dd \Sigma
- \Sumfs\int_{\Tfs}\tressS(\tveta)\cdot\tvxi^\mm \dd \Sigma .
\end{split}
\end{equation}
Having the energy result (<ref>),
which corresponds with Equation (5.10) in
<cit.>, we follow the same process
as described in the reference.
We apply integration by parts:
\begin{equation}
\begin{split}
\int_\DeS (\nabla\cdot(\tsA\veq))\cdot(\tsLa\vep)\dd\Omega
\int_\DeS
\dd\Omega
\\ &\hspace{1cm} =
\tsS:\tfrac12(\nabla\vew+\nabla\vew^T)
\dd\Omega
\\[-3mm] &\hspace{2.4cm}
\dd\Sigma
\\ &\hspace{1cm} =
\boxed{
- \int_\DeS (\nabla\cdot(\tsA\vep))\cdot(\tsLa\veq)\dd\Omega
+ \int_{\Tss\cup\Tsf}(\tsA_n\veq)^\mm\cdot(\tsLa\vep)^\mm
\dd\Sigma,
\end{split}
\label{eq:matform intbyparts 1}
\end{equation}
and similarly
\begin{equation}
\hspace{-5mm}
\int_\DeF (\nabla\cdot(\ttsA\tveq))\cdot(\ttsLa\tvep)\dd\Omega
= \boxed{
- \int_\DeF (\nabla\cdot(\ttsA\tvep))\cdot(\ttsLa\veq)\dd\Omega
+ \int_{\Tfs\cup\Tff}(\ttsA_n\tveq)^\mm\cdot(\ttsLa\tvep)^\mm
\dd\Sigma.
\label{eq:matform intbyparts 2}
\end{equation}
We set $\veq=\veta,\,\vep=\vxi$ in (<ref>)
and $\tveq=\tveta,\,\tvep=\tvxi$ in (<ref>).
The boxed terms in (<ref>) and
(<ref>) vanish as the projection errors $\veta$
and $\tveta$ are orthogonal to the spatial derivatives of the polynomial
solutions $\veq_h$ and $\tveq_h$
by Galerkin approximation,
and then the right-hand side of (<ref>) becomes
\begin{align}
& \hspace{-5mm}
\SumS\int_{\DeS}\resv(\veta)\cdot\vxi \dd \Omega
- \Sumss\int_{\Tss}\ressS(\veta)\cdot\vxi^\mm \dd \Sigma
- \Sumsf\int_{\Tsf}\ressF(\veta)\cdot\vxi^\mm \dd \Sigma
\\[-2mm] & \hspace{2mm}
+ \SumF\int_{\DeF}\tresv(\tveta)\cdot\tvxi \dd \Omega
- \Sumff\int_{\Tff}\tressF(\tveta)\cdot\tvxi^\mm \dd \Sigma
- \Sumfs\int_{\Tfs}\tressS(\tveta)\cdot\tvxi^\mm \dd \Sigma
\end{aligned}
\nonumber\\
\hspace{-8mm}
= \hspace{4mm}
\SumS\int_{\DeS}\tsQ\ddt{\veta}\cdot(\tsLa\vxi) \dd\Omega
+ \SumF\int_{\DeF}\ttsQ\ddt{\tveta}\cdot(\ttsLa\tvxi) \dd\Omega
}\tag{$\Xi_1$}\label{Xi 1}\\
\begin{aligned}
& \hspace{0mm}
- \Sumss\int_{\Tss}\avg{\tsA_n\veta}_\rSS\cdot(\tsLa\vxi)^\mm \dd \Sigma
- \Sumsf\int_{\Tsf}\avg{\tsA_n\veta}_\rSF\cdot(\tsLa\vxi)^\mm \dd \Sigma
\\ & \hspace{0mm}
- \Sumff\int_{\Tff}\avg{\ttsA_n\tveta}_\rFF\cdot(\ttsLa\tvxi)^\mm \dd \Sigma
- \Sumfs\int_{\Tfs}\avg{\ttsA_n\tveta}_\rFS\cdot(\ttsLa\tvxi)^\mm \dd \Sigma
\\ & \hspace{0mm}
- \alpha\Sumss\int_{\Tss}\jmp{\tsA_n\veta}_\rSS\cdot(\tsA_n\vxi)^\mm
\dd \Sigma
- \alpha\Sumsf\int_{\Tsf}\jmp{\tsA_n\veta}_\rSF\cdot(\tsA_n\vxi)^\mm
\dd \Sigma
\\ & \hspace{0mm}
- \alpha\Sumff\int_{\Tff}\jmp{\ttsA_n\tveta}_\rFF\cdot(\ttsA_n\tvxi)^\mm
\dd \Sigma
- \alpha\Sumfs\int_{\Tfs}\jmp{\ttsA_n\tveta}_\rFS\cdot(\ttsA_n\tvxi)^\mm
\dd \Sigma,
\end{aligned}
\tag{$\Xi_2$}\label{Xi 2} }
\end{align}
in which we use the following simplified notation for averaging:
\[
\begin{split}
& \avg{ \tsA_n \veq}_\rSS=\tfrac12
( ( \tsA_n \veq)^\pp+( \tsA_n \veq)^\mm),\quad
\avg{ \tsA_n \veq}_\rSF=\tfrac12
(O^T (\ttsA_n\tveq)^\pp+( \tsA_n \veq)^\mm),\\
& \avg{\ttsA_n\tveq}_\rFF=\tfrac12
( (\ttsA_n\tveq)^\pp+(\ttsA_n\tveq)^\mm),\quad
\avg{\ttsA_n\tveq}_\rFS=\tfrac12
(O^{\phantom{*}}( \tsA_n \veq)^\pp+(\ttsA_n\tveq)^\mm).
\end{split}
\]
For the volume integration terms (cf. (<ref>)) we obtain the estimate
\begin{equation}
\begin{split}
\SumS \normVS{\vxi} \NormVS{\ddt{\veta}}
+ \SumF \normVF{\tvxi}\NormVF{\ddt{\tveta}}
\\
& \hspace{1cm}\leq
\sqrt{ \SumS \normVS{\vxi}^2+ \SumF \normVF{\tvxi}^2}
\\&\hspace{3cm}
\sqrt{
\SumS \NormVS{\ddt{\veta}}^2
+\SumF \NormVF{\ddt{\tveta}}^2
}\quad .
\end{split}
\label{eq:err estimate vol}
\end{equation}
For the surface integration terms (cf. (<ref>)),
we use the symmetry in $\tsA$ and $\tsLa$ to find that
\begin{equation}
(\tsA_n\veq)^\pm \cdot (\tsLa\vep)^\mm =
\ven \cdot \tsS^\pm \cdot \vew^\mm +
\ven \cdot (\tsC\tsH\,)^\mm \cdot \vev^\pm =
(\tsA_n\vep)^\mm \cdot (\tsLa\veq)^\pm .
\label{eq:rule 1}
\end{equation}
\begin{equation}
\begin{split}
&\hspace{-4mm} \Sumss \int_\Tss
\avg{\tsA_n\veta}_\rSS \cdot(\tsLa\vxi)^\mm \dd \Sigma
\\& \hspace{1cm} =
\Half\Sumss \int_\Tss
\cdot(\tsLa\vxi)^\mm \dd \Sigma
\Half\Sumss \int_\Tss
\cdot(\tsLa\vxi)^\mm \dd \Sigma
\\& \hspace{1cm} =
\Half\Sumss \int_\Tss
\cdot(\tsLa\veta)^\pp \dd \Sigma
\Half\Sumss \int_\Tss
\cdot(\tsLa\veta)^\mm \dd \Sigma
\\& \hspace{1cm} =
\Half\Sumss \int_\Tss
\hspace{-3mm}-
(\tsA_n\vxi)^\pp\cdot(\tsLa\veta)^\mm \dd \Sigma
\Half\Sumss \int_\Tss
\cdot(\tsLa\veta)^\mm \dd \Sigma
\\& \hspace{1cm} =
\Half\Sumss \int_\Tss
\hspace{-3mm}-
\jmp{\tsA_n\vxi}_\rSS \cdot(\tsLa\veta)^\mm \dd \Sigma
\quad=\,
-\Half\Sumss \int_\Tss \JmpRd
\jmp{\tsA_n\vxi}_\rSS \cdot\avg{\tsLa\veta}_\rSS \dd \Sigma,
\end{split}
\label{eq:Tss}
\end{equation}
in which the second equality uses (<ref>),
and the third equality is obtained by exchanging the summation order
of elements between solid-solid interfaces. Similarly, we have
\begin{equation}
(\ttsA_n\tveq)^\pm \cdot (\ttsLa\tvep)^\mm =
(\tlambda\tE\,)^\pm \ven\cdot \tvew^\mm +
(\tlambda\tH\,)^\mm \ven\cdot \tvev^\pm =
(\ttsA_n\tvep)^\mm \cdot (\ttsLa\tveq)^\pm ,
\label{eq:rule 2}
\end{equation}
\begin{equation}
\Sumff \int_\Tff
\avg{\ttsA_n\tveta}_\rFF \cdot(\ttsLa\tvxi)^\mm \dd \Sigma
-\Half\Sumff \int_\Tff \JmpRd
\jmp{\ttsA_n\tvxi}_\rFF \cdot\avg{\ttsLa\tveta}_\rFF
\dd \Sigma.
\label{eq:Tff}
\end{equation}
For fluid-solid interfaces we also have the symmetry
\begin{equation}
\begin{split}
% \hspace{-3mm}
O^T(\ttsA_n\tveq)^\pp \cdot (\tsLa\vep)^\mm =\, &
(\tlambda\tE)^\pp \ven \cdot \vew^\mm +
(\ven \cdot (\tsC\tsH\,)^\mm \cdot\ven)(\ven\cdot \tvev^\pp) =
(\tsA_n\vep)^\mm \cdot O^T(\ttsLa\tveq)^\pp, \\[5mm]
% & \hspace{-3mm}
O^{\phantom{*}}\,(\tsA_n\veq)^\pp \cdot (\ttsLa\tvep)^\mm =\, &
(\ven \cdot \tsS^\pp \cdot\ven) (\tvew^\mm\cdot\ven) +
(\tlambda\tH)^\mm \ven \cdot \vev^\pp =
O^T(\ttsA_n\tvep)^\mm \cdot (\tsLa\veq)^\pp ,
\end{split}
\label{eq:rule 3}
\end{equation}
and using (<ref>), (<ref>) and (<ref>),
\begin{equation}
\begin{split}
& \hspace{-3mm}
\Sumsf \int_\Tsf
\avg{\tsA_n\veta}_\rSF \cdot(\tsLa\vxi)^\mm \dd \Sigma \,
\Sumfs \int_\Tfs
\avg{\ttsA_n\tveta}_\rFS \cdot(\ttsLa\tvxi)^\mm \dd \Sigma
\\ & \hspace{1cm} =
\Sumsf \int_\Tsf \tfrac12 (
O^T(\ttsA_n\tveta)^\pp \cdot(\tsLa\vxi)^\mm
+(\tsA_n\veta)^\mm \cdot (\tsLa\vxi)^\mm
) \dd \Sigma \,
\\[-3mm] & \hspace{3cm} +
\Sumfs \int_\Tfs \tfrac12 (
O^{\phantom{*}}(\tsA_n\veta)^\pp \cdot(\ttsLa\tvxi)^\mm
+(\ttsA_n\tveta)^\mm \cdot (\ttsLa\tvxi)^\mm
) \dd \Sigma \,
\\ & \hspace{1cm} =
\Sumsf \int_\Tsf \tfrac12 (
(\tsA_n\vxi)^\mm\cdot O^T(\ttsLa\tveta)^\pp
+(\tsA_n\vxi)^\mm \cdot (\tsLa\veta)^\mm
) \dd \Sigma \,
\\[-3mm] & \hspace{3cm} +
\Sumfs \int_\Tfs \tfrac12 (
O^T(\ttsA_n\tvxi)^\mm \cdot(\tsLa\veta)^\pp
+(\ttsA_n\tvxi)^\mm \cdot (\ttsLa\tveta)^\mm
) \dd \Sigma \,
\\ & \hspace{1cm} =
\Sumsf \int_\Tsf \tfrac12 (
(\tsA_n\vxi)^\mm\cdot O^T(\ttsLa\tveta)^\pp
+(\tsA_n\vxi)^\mm \cdot (\tsLa\veta)^\mm
\\[-3mm] & \hspace{3cm} -
O^T(\ttsA_n\tvxi)^\pp \cdot(\tsLa\veta)^\mm
-O^T(\ttsA_n\tvxi)^\pp \cdot O^T(\ttsLa\tveta)^\pp
) \dd \Sigma \,
\\ & \hspace{1cm} =
-\Sumsf \int_\Tsf
\jmp{\tsA_n\vxi}_\rSF \cdot\avg{\tsLa\veta}_\rSF
\dd \Sigma \,.
\end{split}
\label{eq:Tsf}
\end{equation}
For the penalty terms in (<ref>), it is straightforward to check that
\begin{align}
\Sumss\int_{\Tss}\jmp{\tsA_n\veta}_\rSS\cdot
\Sumss\int_{\Tss}\jmp{\tsA_n\veta}_\rSS\cdot
\jmp{\tsA_n\vxi}_\rSS\dd\Sigma,
\label{eq:alfa Tss}
\\
\Sumff\int_{\Tff}\jmp{\ttsA_n\tveta}_\rFF\cdot
\Sumff\int_{\Tff}\jmp{\ttsA_n\tveta}_\rFF\cdot
\jmp{\ttsA_n\tvxi}_\rFF\dd\Sigma,
\label{eq:alfa Tff}
\end{align}
\begin{equation}
\begin{split}
\Sumsf\int_{\Tsf}\jmp{\tsA_n\veta}_\rSF\cdot
\Sumfs\int_{\Tfs}\jmp{\ttsA_n\tveta}_\rFS\cdot
\\ & \hspace{1cm}
\Sumsf\int_{\Tsf} (
) \dd\Sigma
\\[-3mm] & \hspace{3.2cm}
\Sumfs\int_{\Tfs} (
) \dd\Sigma
\\ & \hspace{1cm}
\Sumsf\int_{\Tsf} (
\\[-3mm] & \hspace{3.2cm}
(\tsA_n\veta)^\mm\cdot O^T(\ttsA_n\tvxi)^\pp
O^T(\ttsA_n\tveta)^\pp\cdot O^T(\ttsA_n\tvxi)^\pp
) \dd\Sigma
\\ & \hspace{1cm}
\Sumsf\int_{\Tsf}\jmp{\tsA_n\veta}_\rSF\cdot
\jmp{\tsA_n\vxi}_\rSF\dd\Sigma.
\end{split}
\label{eq:alfa Tsf}
\end{equation}
Using (<ref>), (<ref>), (<ref>), (<ref>),
(<ref>) and (<ref>) in (<ref>) yields
the estimate for (<ref>)
\begin{equation}
\begin{split}
\Half\Sumss \int_\Tss
\jmp{\tsA_n\vxi}_\rSS \cdot
\Bigl(
% \jmp{\tsA_n\vxi}_\rSS \cdot
\avg{\tsLa\veta}_\rSS
+ \alpha
% \jmp{\tsA_n\vxi}_\rSS \cdot
\jmp{\tsA_n\veta}_\rSS \Bigr)
\dd \Sigma
\\[-0mm] &
\phantom{=} \hspace{5mm} +
\Half\Sumff \int_\Tff
\jmp{\ttsA_n\tvxi}_\rFF \cdot
\Bigl(
% \jmp{\ttsA_n\tvxi}_\rFF \cdot
\avg{\ttsLa\tveta}_\rFF
+ \alpha
% \jmp{\ttsA_n\tvxi}_\rFF \cdot
\jmp{\ttsA_n\tveta}_\rFF \Bigr)
\dd \Sigma
\\[-0mm] &
\phantom{=} \hspace{5mm} + \phantom{\tfrac12}
\Sumsf \int_\Tsf
\jmp{\tsA_n\vxi}_\rSF \cdot
\,\Bigl(
% \jmp{\tsA_n\vxi}_\rSF \cdot
\avg{\tsLa\veta}_\rSF
+ \alpha
% \jmp{\tsA_n\vxi}_\rSF \cdot
\jmp{\tsA_n\veta}_\rSF \,\Bigr)
\dd \Sigma
\\ &
\leq \hspace{5mm}\phantom{+}
\Half\Sumss %\int_\Tss
\left(
\normSS{\jmp{\tsA_n\vxi}_\rSS} \normSS{\avg{\tsLa\veta}_\rSS}
\right.
\\[-4mm] & \hspace{4cm}
\left.
+ \alpha
\normSS{\jmp{\tsA_n\vxi}_\rSS} \normSS{\jmp{\tsA_n\veta}_\rSS}
\right)
% \dd \Sigma
\\[-0mm] &
\phantom{=} \hspace{5mm} +
\Half\Sumff %\int_\Tff
\left(
\normFF{\jmp{\ttsA_n\tvxi}_\rFF} \normFF{\avg{\ttsLa\tveta}_\rFF}
\right.
\\[-4mm] & \hspace{4cm}
\left.
+ \alpha
\normFF{\jmp{\ttsA_n\tvxi}_\rFF} \normFF{\jmp{\ttsA_n\tveta}_\rFF}
\right)
% \dd \Sigma
\\[-0mm] &
\phantom{=} \hspace{5mm} + \phantom{\tfrac12}
\Sumsf %\int_\Tsf
\left(
\normSF{\jmp{\tsA_n\vxi}_\rSF} \normSF{\avg{\tsLa\veta}_\rSF}
\right.
\\[-4mm] & \hspace{4cm}
\left.
+ \alpha
\normSF{\jmp{\tsA_n\vxi}_\rSF} \normSF{\jmp{\tsA_n\veta}_\rSF}
\right)
% \dd \Sigma
\\ &
\leq \frac1{2\beta} \Bigl(
\Sumss\normSS{\jmp{\tsA_n\vxi}_{\rSS}}^2
+2\Sumsf\normSF{\jmp{\tsA_n\vxi}_{\rSF}}^2 \Bigr)
\\&\hspace{1cm}
+\frac\beta4 \Bigl(
\hspace{6mm}\,\phantom{+2}\Sumss
\left(\normSS{\avg{\tsLa\veta}_{\rSS}}^2
\,+\alpha^2\normSS{\jmp{\tsA_n\veta}_{\rSS}}^2
\,\right)
% \right.
\\[-1mm]&\hspace{2cm}
\hspace{5mm}+\phantom{2}\Sumff
\left(\normFF{\avg{\ttsLa\tveta}_{\rFF}}^2
\right)
\\[-1mm]&\hspace{2cm} %\left.
\hspace{5mm}+2\,\Sumsf
\left(\normSF{\avg{\tsLa\veta}_{\rSF}}^2
\,\right)
\quad \Bigr).
\end{split}
\label{eq:err estimate surf0}
\end{equation}
The first inequality is obtained by Cauchy–Schwarz,
and the second one is based on Young's inequality with factor $\beta$
(or so-called “Peter–Paul inequality”).
\[
\begin{split}
\Sumss \normSS{\avg{\tsLa\veta}_{\rSS}}^2
= \Sumss \left( \tfrac12
\normSS{(\tsLa\veta)^\pp+(\tsLa\veta)^\mm}
\right)^2
\\ &\hspace{1cm}
\leq \Sumss \tfrac14
\left(\normSS{(\tsLa\veta)^\pp}^2+\normSS{(\tsLa\veta)^\mm}^2
\\ &\hspace{1cm}
\leq \Sumss \tfrac12\left(
\normSS{(\tsLa\veta)^\pp}^2 + \normSS{(\tsLa\veta)^\mm}^2
\right)
= \Sumss \normSS{(\tsLa\veta)^\mm}^2,
\\ &
\Sumss \normSS{\jmp{\tsA_n\veta}_{\rSS}}^2
= 4\Sumss \left( \tfrac12
\normSS{(\tsA_n\veta)^\pp+(\tsA_n\veta)^\mm}
\right)^2
\\ &\hspace{1cm}
\leq 4\Sumss \tfrac14
\left(\normSS{(\tsA_n\veta)^\pp}^2+\normSS{(\tsA_n\veta)^\mm}^2
\right)
\\ &\hspace{1cm}
\leq 4\Sumss \tfrac12\left(
\normSS{(\tsA_n\veta)^\pp}^2 + \normSS{(\tsA_n\veta)^\mm}^2
\right)
= 4\Sumss \normSS{(\tsA_n\veta)^\mm}^2,
\end{split}
\]
due to Cauchy–Schwarz followed by Young's inequality,
\[
\begin{split}
\Sumff \normFF{\avg{\ttsLa\tveta}_{\rFF}}^2
\leq \hspace{1.5mm}\Sumff \normFF{(\ttsLa\tveta)^\mm}^2,
\\ &
\Sumff \normFF{\jmp{\ttsA_n\tveta}_{\rFF}}^2
\leq 4\Sumff \normFF{(\ttsA_n\tveta)^\mm}^2,
\\ &
\Sumsf \normSF{\avg{\tsLa\veta}_{\rSF}}^2
\leq \Half\Sumsf \normSF{(\tsLa\veta)^\mm}^2
+ \Half\Sumfs \normFS{(\ttsLa\tveta)^\mm}^2,
\\ &
\Sumsf \normSF{\jmp{\tsA_n\veta}_{\rSF}}^2
\,\leq 2\Sumsf \normSF{(\tsA_n\veta)^\mm}^2
+ 2\Sumfs \normFS{(\ttsA_n\tveta)^\mm}^2,
\end{split}
\]
we get the estimate for (<ref>),
\begin{equation}
\begin{split}
\Xi_2 &
\leq \frac1{2\beta} \left(
\Sumss\normSS{\jmp{\tsA_n\vxi}_{\rSS}}^2
+2\Sumsf\normSF{\jmp{\tsA_n\vxi}_{\rSF}}^2 \right)
\\&\hspace{1cm}
+\frac\beta4 \left(
\hspace{5mm}\phantom{+2}\SumS
\left(\Norm{(\tsLa\veta)^\mm}_{L^2(\Tss\cup\Tsf)}^2
\right)
\right.
\\[-3mm]&\hspace{2cm} \left.
\hspace{5mm}+\phantom{2}\SumF
\left(\Norm{(\ttsLa\tveta)^\mm}_{L^2(\Tff\cup\Tfs)}^2
\right)
\quad \right).
\end{split}
\label{eq:err estimate surf}
\end{equation}
Using (<ref>) and (<ref>)
in (<ref>) yields
\begin{equation}
\begin{split}
& \frac12\DDt{}\left(
\SumS\normVS{\vxi}^2 +\SumF\normVF{\tvxi}^2
\right)
\left(
\Sumss\Norm{\jmp{\tsA_n\vxi}_\rSS}^2_{L^2(\Sss)}
\right.
\\[-0mm]& \hspace{3cm} \left.
\Sumff\Norm{\jmp{\ttsA_n\tvxi}_\rFF}^2_{L^2(\Sff)}
+ 2\Sumsf\Norm{\jmp{\tsA_n\vxi}_\rSF}^2_{L^2(\Ssf)}
\right)
\\&
\leq
\sqrt{ \SumS \normVS{\vxi}^2+ \SumF \normVF{\tvxi}^2}
\sqrt{
\SumS \NormVS{\ddt{\veta}}^2
+\SumF \NormVF{\ddt{\tveta}}^2
\\[-0mm]&\hspace{2cm}
+\frac\beta4 \left(
\hspace{0mm}\phantom{+2}\SumS
\left(\Norm{(\tsLa\veta)^\mm}_{L^2(\Tss\cup\Tsf)}^2
\right)
\right.
\\[-0mm]&\hspace{2cm} \left.
\hspace{11mm}+\phantom{2}\SumF
\left(\Norm{(\ttsLa\tveta)^\mm}_{L^2(\Tff\cup\Tfs)}^2
\right)
\quad \right).
\end{split}
\label{eq:err est final}
\end{equation}
Following <cit.>, we can finally obtain the
required error estimate from (<ref>).
We take $\alpha = 1/2$; by choosing $\beta$ sufficiently large in
Young's inequality, we control the error by applying a modified
Gronwall's lemma <cit.>.
§ COMPUTATIONAL BENCHMARK AMONG TYPES OF FLUX
In this appendix, we compare three types of numerical fluxes:
the central flux, the upwind flux proposed by
<cit.>, and the boundary condition penalized flux
proposed in our DG method. The comparisons are conducted using plane
waves, Rayleigh waves, Stoneley waves and Scholte waves,
with the parameter settings as in <ref>.
(A) (B)
(C) (D)
comparison of the accuracies and convergence rates between
the penalized numerical fluxes and the upwind flux when simulating
(A) a plane wave,
(B) a Rayleigh wave,
(C) a Stoneley wave, and
(D) a Scholte wave,
for different orders $N_p = 3$ and $6$.
Accuracy as a function for propagation time of three types of
numerical fluxes for the simulation of a Scholte wave.
The maximum propagation time is $25$ s, which
corresponds with a distance of $5.4$ wavelengths; $h = 0.25$km
($0.125$ wavelengths). When the numerical method starts to blow up,
the error function increases exponentially and the curve becomes jittery.
Figure <ref> compares the accuracies and convergence rates
of the penalized numerical fluxes with the upwind flux when simulating
a plane wave (A), a Rayleigh
wave (B), a Stoneley wave (C) and a Scholte wave (D),
for both the lower-order case ($ N_p=3 $) and
the relatively higher-order case ($ N_p=6 $).
We observe that the measured errors for the penalized flux
are sometimes slightly larger than those for the upwind flux,
while the orders of convergence are essentially the same
(and both better than $\mathcal{O}(h^{N_p+\frac12})$).
Figure <ref> shows the time dependent error measure
for all three types of numerical fluxes.
We simulate the Scholte wave for 25 seconds, with uniform mesh size
$h=0.125$km and order $N_p=2,4,6$ for each numerical flux.
The central flux turns out to be unstable with
growing oscillatory numerical errors during time evolution.
This instability may be caused by the inaccuracy of numerical integration,
in which case the integration by parts does not necessary hold,
and the energy-conserving property no longer holds.
In comparison, the upwind and penalty fluxes
extend the tolerance to inexact numerical integration.
|
1511.01015
|
Dipartimento di Fisica e Astronomia
dell'Università di Catania,
INFN Sezione di Catania,
Via S.Sofia 64, I-95123 Catania, Italy
Pure Yang-Mills SU(N) theory is studied in the Landau gauge and four dimensional space.
While leaving the original Lagrangian unmodified, a double perturbative expansion is
devised, based on a massive free-particle propagator. In dimensional
regularization, all diverging mass terms cancel exactly in the double
expansion, without the need to include mass counterterms that would spoil the symmetry
of the Lagrangian. No free parameters are included that were not in
the original theory, yielding a fully analytical approach from first principles.
The expansion is safe in the infrared and is equivalent to the standard perturbation theory in the UV.
At one-loop, explicit analytical expressions are given for the propagators and the running coupling and
are found in excellent agreement with the data of lattice simulations.
A universal scaling property is predicted for the inverse propagators and shown to be satisfied by
the lattice data. Higher loops are found to be negligible in the infrared below $300$ MeV where
the coupling becomes small and the one-loop approximation is under full control.
12.38.Bx, 12.38.Lg, 12.38.Aw, 14.70.Dj
§ INTRODUCTION
In modern textbooks on QCD, the infrared domain is usually called non-perturbative
just because standard perturbation theory breaks down at
the low-energy scale $\Lambda_{QCD}\approx$ 200 MeV.
While the high energy behaviour of the theory is under control and an analytical study of
non-Abelian gauge theories is usually achieved by a loop expansion in the UV, no analytical
first-principle description of the infrared can be found in books where the subject is usually
discussed by phenomenological models that rely on numerical lattice simulations.
In the last years, important progresses have been achieved by non-perturbative approaches
based on Schwinger-Dyson equations (SDE)
variational methods<cit.>, Gribov copies<cit.>
and by simulating
larger and larger lattices<cit.> of course.
While we still miss a full analytical description, the numerical solution of truncated
sets of SDE integral equations together with the measures that come from the lattice
yield a more clear picture of the infrared behaviour of QCD and Yang-Mills theory.
It is now widely believed that in the Landau gauge the gluon propagator is finite and
an effective coupling can be defined that is infrared safe and relatively small.
As discussed by Cornwall<cit.> in 1982, the gluon may acquire a dynamical mass
in the infrared without breaking the gauge invariance of the theory. The effect cannot
be described by the standard perturbation theory at any finite order because of gauge
invariance that makes the polarization transverse and prohibits any shift of the pole in the
gluon propagator. That is one of the reasons why the standard perturbation theory cannot
predict the correct phenomenology in the infrared.
An other reason is the occurrence of a Landau pole in the running of the coupling that makes
evident the failure of the perturbative expansion below $\Lambda_{QCD}$.
However, in the Landau gauge the ghost-gluon vertex function can be shown to be finite<cit.>
and a running coupling can be defined by the product of two-point correlators. Being massive,
the gluon propagator is finite and its dressing function vanishes in the infrared yielding a finite
running coupling that reaches a maximum and decreases in the low-energy limit<cit.>.
On the other hand, if the Landau pole is an artifact of the perturbative expansion, the relatively
small value of the real coupling suggests that we could manage to set up a different perturbative scheme
in the infrared. Actually, in order to make physical sense, a perturbative expansion requires that
the lowest order term should approximately describe the exact result. While that condition is fulfilled by
the standard perturbation theory in the UV, where the propagator is not massive, the dynamical
mass of the gluon makes the free propagator unsuitable for describing the low energy limit.
Thus we would expect that, by a change of the expansion point, a perturbative approach to QCD in the
infrared could be viable.
There is some evidence that inclusion of a mass by hand
in the Lagrangian gives a phenomenolgical model that describes very well the lattice data
in the infrared at one loop<cit.>.
However that model could be hardly justified by first principles because of the mass that breaks the
gauge invariance of the Lagrangian. Even in a fixed gauge, BRST symmetry is broken and a mass counterterm
must be included for renormalizing the theory, thus introducing spurious free parameters in the model.
A change of the expansion point can be achieved by first principles without changing the original
Lagrangian. Variational calculations have been proposed<cit.>
where the zeroth order propagator is a trial
unknown function to be determined by some set of stationary conditions. The added propagator is
subtracted in the interaction, leaving the total action unchanged. The idea is not new and goes
back to the works on the Gaussian effective potential<cit.>
where an unknown mass parameter was inserted
in the zeroth order propagator and subtracted from the interaction, yielding a pure variational
approximation with the mass that acts as a variational parameter.
Some recent variational calculations on Yang-Mills theory<cit.> have shown that,
provided that we change the expansion point, a fair agreement with the lattice data can be
achieved without too much numerical effort. Thus we expect that it is not very important the actual
choice of the zeroth order propagator provided that it is massive. A simple free-particle
Yukawa-type massive propagator would be enough and the corrections due to the interaction would then be
manageable by perturbation theory.
A first attempt along these lines was reported in Ref.<cit.> where, by a second order massive
expansion, the gluon and ghost propagators are evaluated and found in fair agreement with the lattice data.
The integrals were regularized by a simple cutoff that breaks the BRST symmetry and gives rise to several
drawbacks like quadratic divergences and the need of a mass counterterm. However, a fine tuning of the
mass parameter seems to cure the drawbacks yielding an optimized expansion that reproduces the lattice data.
In this paper, the difficulties of dealing with a cutoff are avoided by the use of a more robust
dimensional regularization scheme, yielding a more rigorous perturbative study of
pure $SU(N)$ Yang-Mills theory from first principles. While the original Lagrangian is not changed in any way,
the outcome is a one-loop analytical description that is infrared safe and in striking agreement with the
data of lattice simulations. Moreover the result can be improved by including higher-order terms and by use of
standard Renormalization Group (RG) techniques for reducing the effect of higher order terms.
A very interesting property of the massive expansion is the cancellation of all diverging mass terms without
including any spurious mass counterterm. Only wave function renormalization constants
are required and, in the minimal subtraction scheme, these constants are the same of the standard perturbative
expansion, thus ensuring that the correct UV behaviour is recovered.
The massive expansion is discussed in the Landau gauge in the present paper. The Landau
gauge is probably the optimal choice for the expansion, because of the transversality of the propagator
that makes the longitudinal polarization irrelevant.
In the Landau gauge the problem decouples and a fully analytical result can be found for the propagators at one-loop.
While massive models have been studied before and found in good agreement with the
data of lattice simulations<cit.>,
the present calculation is very different because the Lagrangian is not modified, overall BRST symmetry is not
broken and no free parameters are added to the exact Yang-Mills theory,
yielding a description that is based on first principles and can be improved order by order.
Thus, at variance with previous massive models, the present method would not give a mass to the photon.
The paper is organized as follows:
in Section II the massive expansion is developed for pure $SU(N)$ Yang-Mills theory in a generic
covariant gauge; in Section III the double expansion is set up in the Landau gauge; in Section IV
the explicit cancellation of the diverging mass terms is discussed in detail; in Section V explicit
analytical expressions are derived for the propagators at one-loop; in Section VI the one-loop propagators
and their scaling properties are compared with the available lattice data; in Section VII the running
coupling is evaluated and its sensitivity to the renormalization conditions is discussed, showing that
the approximation is under full control below $300$ MeV; finally, in Section VIII the main results are discussed.
Explicit analytical expressions for the propagators are given in the Appendix.
§ SET UP OF THE MASSIVE EXPANSION IN A GENERIC GAUGE
Let us consider pure Yang-Mills $SU(N)$ gauge theory without
external fermions in a $d$-dimensional space.
The Lagrangian can be written as
where ${\cal L}_{YM}$ is the Yang-Mills term
L_YM=-1/2 ( F̂_μνF̂^μν)
${\cal L}_{fix}$ is a gauge fixing term and ${\cal L}_{FP}$ is the ghost term
arising from the Faddev-Popov determinant.
In terms of the gauge fields, the tensor operator $\hat F_{\mu\nu}$ is
F̂_μν=∂_μÂ_ν-∂_νÂ_μ-i g [Â_μ, Â_ν]
Â^μ=∑_a X̂_a A_a^μand the generators of $SU(N)$ satisfy the algebra
[ X̂_a, X̂_b]= i f_abc X̂_c
with the structure constants normalized according to
f_abc f_dbc= Nδ_ad.
If a generic covariant gauge-fixing term is chosen
L_fix=-1/ξ [(∂_μÂ^μ)(∂_νÂ^ν)]
the total action can be written as $S_{tot}=S_0+S_I$ where the free-particle term is
\begin{align}
S_0&=\frac{1}{2}\int A_{a\mu}(x)\delta_{ab} {\Delta_0^{-1}}^{\mu\nu}(x,y) A_{b\nu}(y) {\rm d}^dx{\rm d}^dy \nn \\
&+\int \omega^\star_a(x) \delta_{ab}{{\cal G}_0^{-1}}(x,y) \omega_b (y) {\rm d}^dx{\rm d}^dy
\label{S0}
\end{align}
and the interaction is
S_I=∫d^dx [ L_gh + L_3 + L_4].
with the three local interaction terms that read
\begin{align}
{\cal L}_3&=-g f_{abc} (\partial_\mu A_{a\nu}) A_b^\mu A_c^\nu\nn\\
{\cal L}_4&=-\frac{1}{4}g^2 f_{abc} f_{ade} A_{b\mu} A_{c\nu} A_d^\mu A_e^\nu\nn\\
{\cal L}_{gh}&=-g f_{abc} (\partial_\mu \omega^\star_a)\omega_b A_c^\mu.
\label{Lint}
\end{align}
In Eq.(<ref>), $\Delta_0$ and ${\cal G}_0$ are the standard free-particle propagators for
gluons and ghosts and their Fourier transforms are
\begin{align}
{\Delta_0}^{\mu\nu} (p)&=\Delta_0(p)\left[t^{\mu\nu}(p)
+\xi \ell^{\mu\nu}(p) \right]\nn\\
\Delta_0(p)&=\frac{1}{-p^2}, \qquad {{\cal G}_0} (p)=\frac{1}{p^2}.
\label{D0}
\end{align}
Here the transverse and longitudinal projectors are defined as
t_μν (p)=η_μν - p_μp_ν/p^2; ℓ_μν (p)=p_μp_ν/p^2
where $\eta_{\mu\nu}$ is the metric tensor.
A shift of the pole in the gluon propagator can be introduced by an unconventional
splitting of the total action. Since we have the freedom of adding and subtracting the same
arbitrary term $\delta S$ to the total action
\begin{align}
S_0&\rightarrow S_0+\delta S\nn\\
S_I&\rightarrow S_I-\delta S
\label{shift}
\end{align}
we can take
δS= 1/2∫A_aμ(x)δ_ab δΓ^μν(x,y)
A_bν(y) d^dxd^dy
where the vertex function $\delta\Gamma$ is a shift of the inverse propagator
[Δ_m^-1^μν(x,y)- Δ_0^-1^μν(x,y)]
and ${\Delta_m}^{\mu\nu}$ is a massive free-particle propagator
\begin{align}
{\Delta_m^{-1}}^{\mu\nu} (p)&={\Delta_m}(p)^{-1} t^{\mu\nu}(p)
+\left[\frac{-p^2}{\xi}+{\cal A}(p)\right]\ell^{\mu\nu}(p)\nn\\
{\Delta_m}(p)^{-1}&=-p^2+{\cal M}(p)^2.
\label{Deltam}
\end{align}
Here the dynamical mass ${\cal M}(p)$ and the longitudinal function ${\cal A}(p)$ are left
totally arbitrary. While the total action cannot depend on them, just because $\delta S$ is added
and subtracted again, any expansion in powers of the new shifted interaction $S_I\to S_I-\delta S$ is going
to depend on the choice of $\delta S$ because of the approximation.
Thus, it is the approximation
that changes but we are not changing the content of the exact theory.
The shift $\delta S$ has two effects: the free-particle propagator ${\Delta_0}^{\mu\nu}$ is replaced by
the massive propagator ${\Delta_m}^{\mu\nu}$ in $S_0$; a counterterm $-\delta S$ is added to the interaction
From now on, let us drop all color indices in the diagonal matrices.
Inserting Eq.(<ref>) and (<ref>) in Eq.(<ref>) the counterterm reads
δΓ^μν (p)=M(p)^2 t^μν(p) +A(p)ℓ^μν(p)
and must be added to the standard vertices arising from Eq.(<ref>).
The proper gluon polarization $\Pi$ and ghost self energy $\Sigma$ can be evaluated, order by order,
summing up Feynman graphs where ${\Delta_m}^{\mu\nu}$ is the zeroth order gluon propagator.
By Lorentz invariance we can write
Π^μν(p)=Π^T(p) t^μν(p)+Π^L(p) ℓ^μν(p)
and the dressed propagators are
\begin{align}
\Delta_{\mu\nu}(p)&=\Delta^T (p)t_{\mu\nu}(p)+\Delta^L (p)\ell^{\mu\nu}(p)\nn\\
{\cal G}^{-1}(p)&=p^2-\Sigma (p)
\end{align}
where the transverse and longitudinal parts read
\begin{align}
{\Delta^T}^{-1} (p)&=\left[-p^2+{\cal M}(p)^2-\Pi^T(p)\right]\nn\\
{\Delta^L}^{-1} (p)&=\left[\frac{-p^2}{\xi}+{\cal A}(p)-\Pi^L(p)\right].
\label{DTL}
\end{align}
At tree level, the polarization is just given by the counterterm $\delta \Gamma$ of Eq.(<ref>),
so that the tree-terms $\Pi^T_{tree}={\cal M}^2$, $\Pi^L_{tree}={\cal A}$ just cancel the shifts in the
dressed propagator $\Delta$ of Eq.(<ref>), giving back the standard free-particle propagator
of Eq.(<ref>).
In fact, at tree-level, nothing really changes.
Summing up all loops, the exact dressed propagator
can be written as
\begin{align}
\label{DTLloop}
\end{align}
As a consequence of gauge invariance, the exact longitudinal polarization $\Pi^L_{loops}$ must be zero
and the longitudinal part of the exact propagator must be equal to its tree-level value
$\Delta^L=-\xi/p^2$, just because the loop-terms cannot change it, as recently confirmed
by lattice simulations<cit.>.
Since $\Pi^L_{loops}$ and $\Pi^T_{loops}$ are evaluated by insertion of the modified propagator
${\Delta_m}^{\mu\nu}$ in the loops, they can be considered as functionals of the arbitrary functions
${\cal M}$, ${\cal A}$. Thus, summing up all loops, the following constraints must hold for the
exact polarization functions:
\begin{align}
&\frac{\delta \Pi^T_{loops}}{\delta {\cal A}}=\frac{\delta \Pi^T_{loops}}{\delta {\cal M}}=0\nn\\
&\Pi^L_{loops}[{\cal A},{\cal M}]=0.
\label{constraint}
\end{align}
Expanding in powers of the total interaction, including the counterterm $\delta \Gamma$ among the vertices
and writing down the Feynman graphs, we can truncate
the expansion at a finite order yielding approximate functionals that may not satisfy the constraints
of Eq.(<ref>) exactly. For instance, the exact vanishing of the transverse polarization would be
lost unless BRST symmetry is mantained order by order. Actually, while the total Lagrangian has not been
changed and mantains its symmetry, the two parts $S_0$ and $S_I$ might be not BRST invariant because
of the arbitrary shift $\delta S$. Then the exact symmetry is lost in the expansion at any finite order
and the constraints are expected to hold only approximately unless all the graphs are summed up.
The outcome of the truncated expansion becomes sensitive to the choice of the functions ${\cal A}$, ${\cal M}$
thus suggesting the use of Eq.(<ref>) as variational stationary conditions.
Since we expect that the approximation should work better if the zeroth order propagator $\Delta_m^{\mu\nu}$
is a good approximation of the exact one, then comparing Eq.(<ref>) and Eq.(<ref>),
a self-consistent method could be set up by requiring that
M(p)^2=Π^T_loops[M]; A=Π^L_loops=0.
Summing up all the loops these equations would be equivalent to SDE. Variational methods of this kind
have been investigated in several works<cit.> and require the solution
of integral equations that can hardly be treated analytically.
In the Landau gauge the problem decouples and a fully analytical result can be found for the propagators.
The Landau gauge is very special as the gluon propagator is transverse and does not depend on the choice of the
function ${\cal A}$. In the limit $\xi\to 0$ the longitudinal part $\Delta^L$ is exactly zero in
Eq.(<ref>) and is decoupled from the longitudinal polarization that becomes irrelevant for the
calculation of the propagator. In fact, in the same limit, the zeroth order propagator ${\Delta_m}^{\mu\nu}$
in Eq.(<ref>) becomes transverse for any choice of ${\cal A}$ and the longitudinal part of
the counterterm $\delta \Gamma$ does not give any contribution in the loops when sandwiched by two transverse
propagators. Thus the only action of ${\cal A}$ is at tree level where it cancels itself in the gluon propagator
and disappears. Then, in the Landau gauge,
we can safely drop all longitudinal parts and set ${\cal A}=0$ without affecting the calculation.
Because of the decoupling, the calculation of the longitudinal and of the transverse parts can be seen as
two separate problems that may even require different orders of approximation. That simplifies things
considerably, since a poor approximation for the longitudinal polarization would not affect the accuracy
of the propagator.
That also explains why reliable results for the propagator can be achieved even when the BRST symmetry is
broken<cit.> in the Landau gauge.
Moreover, since the total Lagrangian has not been modified, the overall BRST symmetry is unbroken and the
constraints in Eq.(<ref>) must be satisfied asymptotically. Thus, if required, a better approximation
for the longitudinal polarization can always be achieved in a separate calculation
by adding more terms to the expansion.
§ DOUBLE EXPANSION IN THE LANDAU GAUGE
The Landau gauge is probably the optimal choice for the massive expansion, as discussed in the previous section.
In the limit $\xi\to 0$ the gluon propagators are transverse exactly and, having set ${\cal A}=0$,
we can simplify the notation and drop the projectors $t^{\mu\nu}$ everywhere whenever each term is
Moreover, we make the minimal assumption of taking the arbitrary function ${\cal M}$ equal
to a constant mass scale ${\cal M}=m$. In fact, variational calculations seem to suggest<cit.>
that the actual form of the zeroth order propagator is not important provided that it is massive.
A constant mass simplifies the calculation and allows the use of dimensional regularization.
Graphical illustration of Eq.(<ref>). The cross is the counterterm of Eq.(<ref>)
that gives a factor $m^2$.
We can use the standard formalism of Feynman graphs with a massive zeroth
order propagator that reads
and a counterterm
that must be added to the standard three-particle ghost-gluon and gluon-gluon vertices
of order ${\cal O} (g)$ and to the four-particle gluon-gluon vertex of order ${\cal O} (g^2)$
according to Eq.(<ref>). Thus, the total interaction is a mixture of terms that depend on
the coupling strength $g$ and a counterterm that does not vanish in the limit $g\to 0$.
A perturbative expansion in powers of the total interaction would contain at any order different
powers of $g$ but the same number of vertices (including the counterterm among vertices) and
we may define the order of a term as the number of vertices in the graph.
Of course, we could easily sum up some infinite set of graphs, like the chain graphs in Fig.1.
Formally, summing up all graphs with $n$ insertions
of the counterterm in the internal gluon lines, would cancel the pole shift and would give back the standard
perturbation theory
1/-p^2+m^2∑_n=0^∞[ m^2 1/-p^2+m^2]^n=1/-p^2.
In fact, this is what we would get exactly at tree level, as discussed in the previous section.
That just says that the massive and the standard expansions are equivalent if we sum up all graphs.
On the other hand, at any finite order, the massive expansion is not
equivalent to the standard perturbation theory, but the two expansions differ by an infinite class of graphs
that amounts to some non-perturbative content. For instance, the massive zeroth order
propagator $\Delta_m$ cannot be obtained by the standard perturbation theory at any finite order because
of the gauge invariance of the theory that does not allow any shift of the pole.
We may reverse the argument and observe that, while in the UV the geometric expansion in Eq.(<ref>)
is convergent and the two perturbation theories must give the same result,
when $p^2\to m^2$ each single term in Eq.(<ref>) diverges and the formal sum of infinite poles
amounts to some non-perturbative content that makes the theories different at any finite order.
We can predict that the scale $m$ should be close to the Landau pole $\Lambda$ where the
standard perturbation theory breaks down.
Two-point graphs with no more than three vertices and no more than one loop.
In the next sections, the ghost self energy and the gluon polarization
are obtained by the sum of all the graphs in the figure.
Since we know that the gluon develops a dynamical mass in the infrared, we do not want to sum the chain
graphs in Fig.1 but prefer to truncate the power expansion at some finite order. An
expansion in powers of the total interaction $S_I$ is more efficient than the
standard expansion in powers of the coupling $g$. The counterterm $\delta \Gamma$ has the important
effect of reducing the weight of the total interaction since, in principle, if the zeroth order
propagator were exact, the total polarization would be exactly zero. For that reason, we define the order
of a graph as
the number of vertices that are included, reflecting the power of $S_I$ rather than the number of
loops. Thus the tree-level graphs must be regarded as first order. As shown in Fig.2 the one-loop
tadpole $(1b)$ is first order while the gluon loop $(2b)$ is second order. Any insertion of the
counterterm $\delta \Gamma$ increases the order by one.
If the effective coupling is small, as it turns out to be according to non-perturbative calculations,
not all the graphs have the same weight in the expansion. Since the number of loops is equal to the power
of $g^2$ in a graph, two-loop graphs must be much smaller than one-loop and tree graphs.
We can consider a double expansion in powers of the total
interaction and in powers of the coupling: we expand up to the $n$th order, retaining graphs with $n$ vertices
at most, and then neglect all graphs with more than $\ell$ loops. In Fig.2 the lower order graphs are
shown up to $n=3$ and $\ell=1$.
A very important feature of the double expansion is that there is no need to include mass counterterms
for regularizing the divergences. All diverging mass terms cancel exactly in the expansion.
Thus we avoid to insert spurious mass parameters that were not in the original Lagrangian.
The cancellation can be easily explained by the following argument. Since we did not change the Lagrangian
and it was renormalizable (without mass counterterms because of BRST symmetry), all the diverging mass terms
must cancel if we sum up all graphs. In fact, no diverging mass term arises in the standard perturbative
expansion. The cancellation must be given by the sum of infinite graphs with counterterm insertions
$\delta \Gamma$ in the loops that, according to Eq.(<ref>) and Fig.1, restore the pole
of the propagator and cancel the mass. However, if we inspect the graphs in Fig.2, we can easily see that
any insertion of $\delta \Gamma$ in a loop reduces the degree of divergence of the graph so that they
become finite after a finite number of insertions.
Thus, if the divergences must cancel, they will cancel at a finite order of the expansion
provided that we retain more counterterm insertions than loops.
If $n$ is large enough, then all divergences in the mass terms are cancelled by the counterterms in the loops.
For instance, at one loop we only need $n=3$ as shown in Fig.2.
While in this paper we report the results for a one-loop (third-order) approximation,
the extension to higher loops is straightforward and the regularization follows the same path
of standard perturbation theory, with all the divergences that can be cancelled by the usual
wave function renormalization constants.
§ CANCELLATION OF MASS DIVERGENCES IN DIMENSIONAL REGULARIZATION
The exact cancellation of the diverging mass-terms can be carried out explicitely in
dimensional regularization expanding in powers of $\epsilon=4-d$.
The insertion of one counterterm in a loop can be seen as the replacement
1/-p^2+m^2→1/-p^2+m^2 m^21/-p^2+m^2=-m^2∂/∂m^2Δ_m
in the internal gluon line. If there are no other counterterm insertions in the same graph,
then the dependence on $m^2$ must come from the massive propagators and a derivative of the whole
$n$th-order $\ell$-loop graph gives the sum of all $(n+1)$-order $\ell$-loop graphs
that can be written by a single insertion of $\delta\Gamma$ in any position.
In dimensional regularization, any diverging mass term that arises from a loop can be expressed
as a pole $c\> m^2/\epsilon$ where $c$ is a factor. Inserting this term in Eq.(<ref>), we see
that a counterterm in the loop gives a crossed-loop graph with the opposite diverging
term $-c\> m^2/\epsilon $. The argument also suggests a simple
way to evaluate the crossed-loop graphs by Eq.(<ref>).
At one-loop, we must truncate the expansion at the order $n=3$ for a full cancellation of
all the diverging mass terms. While higher-order terms could be included without introducing any
further divergence at one-loop, in this paper we explore the minimal approximation and sum up all
graphs up to $n=3$ as shown in Fig.2.
It is not difficult to show that in the limit $p\to 0$ the gluon polarization is finite but not zero.
The existence of a finite limit $\Pi(0)\not=0$ is crucial for the existence of a finite gluon
propagator in the infrared.
First of all, let us evaluate the constant graphs at the order $n=3$.
At the lowest order ($n=1$, $\ell=0$) the counterterm $\delta \Gamma$ gives the
constant graph $\Pi_{1a}=m^2$ that cancels the shift of the pole in the propagator.
Exact integral expressions for the loop graphs have been reported by other authors in the Landau gauge.
In Ref.<cit.> all one-loop graphs are reported for any gauge, any space dimension and
any choice of the zeroth order propagator. In Landau gauge and Euclidean space,
the constant tadpole $\Pi_{1b}$ can be written as
Π_1b=-N g^2 (d-1)^2/d ∫1/k^2+m^2.
Expanding around $d=4$, in the $\overline{MS}$ scheme,
Π_1b=3/4 αm^2(2/ϵ+logμ^2/m^2+1/6)
having hided the factor $N$ inside an effective coupling $\alpha$ defined as
α=3N/4π α_s; α_s=g^2/4π.
The crossed tadpole $\Pi_{1c}$ follows by a derivative according to Eq.(<ref>)
Π_1c=-m^2∂Π_1b/∂ m^2=
-3/4 αm^2(2/ϵ+logμ^2/m^2-5/6).
As expected, the diverging terms cancel in the sum $\Pi_{1b}+\Pi_{1c}$. In fact,
the double-crossed tadpole $\Pi_{1d}$ is finite and including its symmetry factor it reads
Π_1d= 1/2 m^4∂^2Π_1b/∂ (m^2)^2= -3/8 αm^2
so that the sum of the constant graphs is
Π_1b+Π_1c+Π_1d=3/8 αm^2.
While the ghost loop vanishes in the limit $p\to 0$, a finite $\Pi(0)\not=0$ can also
arise from the gluon loop $\Pi_{2b}$ that in the Landau gauge (in Euclidean space)
can be written as<cit.>
Π_2b (p)=2Ng^2∫k_⊥^2F(k,p)/(k^2+m^2)[(k+p)^2+m^2]
where $k_\perp^2=[k^2-(k\cdot p)^2/p^2]$ and the kernel ${\cal F}$ can be decomposed as
F (k,p)=k^2+p^2/k^2+p^2/(k+p)^2-p^2k_⊥^2/(d-1)(k+p)^2 k^2
The calculation of this graph is straightforward but
tedious. The integral can be evaluated analytically and the result is reported in the next section.
If we take the limit $p\to 0$ before integrating, we find that ${\cal F}\to 1$ and a mass term arises
Π_2b (0)=2Ng^2(d-1)/d∫k^2/(k^2+m^2)^2.
The integral is trivial and expanding around $d=4$ in the $\overline{MS}$ scheme we can write it as
Adding the crossed loop $\Pi_{2c}$ with its symmetry factor, the divergences cancel
and adding the constant graphs in Eq.(<ref>),
the one-loop dressed propagators can be written as
\begin{align}
\Delta(p)^{-1}&=-p^2+\frac{5}{8}\alpha m^2-\left[ \Pi(p)-\Pi(0)\right]\nn\\
{\cal G} (p)^{-1}&=p^2-\Sigma (p).
\label{dressed}
\end{align}
While the explicit calculation requires the evaluation of the gluon and ghost loop
and of the ghost self-energy,
we observe that a finite mass-term has survived in the cancellation, so that
the dressed propagator $\Delta(0)^{-1}=5\alpha m^2/8$ is finite and of order $\alpha$.
Actually, we checked that the full propagators in Eq.(<ref>), when renormalized, do
not depend on the precise value of the factor $5/8$ that arises by truncating
the expansion at the third order. A minor change of that coefficient is absorbed by a change
of the mass parameter and of the renormalization constants without affecting the final result.
That is an important feature since otherwise the whole calculation would depend on the somehow
arbitrary truncation of the expansion. In fact, while higher-order terms
would add very small corrections in the UV because of the factor $(-p^2+m^2)^{n+1}\sim(-p^2)^{n+1}$ in
the denominators of Eq.(<ref>), in the limit $p\to 0$ the corrections might not be
negligible. In that limit we find a hierarchy in the significance of the crossed terms.
The most important effect arises at tree-level since the tree-graph $\Pi_{(1a)}$ in Fig.2 cancels
the entire shift of the pole in the propagator, as discussed in Section II. Thus, a finite
$\Pi(0)\not=0$ can only arise from loops and the massive expansion would not predict any mass
for the photon. At one-loop, a first insertion of the counterterm gives diverging crossed
graphs that cancel the divergence of the loops entirely. Inclusion of those terms is crucial
for the renormalization of the theory. On the other hand, the insertion of $n$ counterterms in a loop,
with $n\ge 2$, gives finite terms that only add some fractions of $\alpha m^2$ to $\Pi (0)$.
These terms decrease as $\sim 1/n^2$ and subtract each other, with a positive series of terms coming
from the tadpole graph and a negative series arising from the gluon loop.
Thus, the inclusion of higher order terms would only give a slight decrease of the coefficient of
$\Pi(0)=-5\alpha m^2/8$ in Eq.(<ref>).
That change is compensated by an increase of the mass parameter and by a change of the
renormalization constants, without making any real difference in the renormalized propagators.
In that sense, the minimal choice of a third order expansion has nothing special in itself and no dramatic effect is expected if higher-order terms are included.
§ ONE-LOOP PROPAGATORS
The explicit evaluation of the propagators at one-loop and order $n=3$ requires the sum of the
gluon loop $\Pi_{2b}$, the crossed loop $\Pi_{2c}$ and the ghost loop $\Pi_{2a}$ for the gluon
propagator and the sum of the one-loop and crossed-loop self energy graphs for the ghost
propagator, as shown in Fig.2.
From now on, we switch to Euclidean space, expand the graphs around $d=4$ in the
$\overline{MS}$ scheme and use the adimensional variable $s=p^2/m^2$.
The gluon loop $\Pi_{2b}$ is given by the integral in Eq.(<ref>) that gives
a diverging part
Π^ϵ_2b(p)=-α( m^2-25/36 p^2)
and a finite part
Π^f_2b=αm^2/72[2/s-135+226/3s+s^3logs -s L_A(s)-s L_B(s)]
where $L_A$, $L_B$ are the logarithmic functions
\begin{align}
\log\left(\frac{\sqrt{4+s}-\sqrt{s}}{\sqrt{4+s}+\sqrt{s}}\right)\nn\\
\label{logs}
\end{align}
The ghost loop $\Pi_{2a}$ is a standard graph and in the Landau gauge it is given by the
The integral is straightforward and the diverging part is
Π^ϵ_2a (p)=αp^2/36
while the finite part reads
Π^f_2a (p)=αm^2/36
The first of self-energy graphs in Fig.2, the standard one-loop graph,
is given by the integral<cit.>
Σ_1 (p)=-Ng^2
∫ p^2 k_⊥^2/k^2 (k-p)^2 (k^2+m^2)
that yields a diverging term
and a finite part
Σ_1^f (p)=αp^2/12[g(s)-5]
where the function $g(s)$ is
If we do not add the crossed loops and take the sum of the finite parts, Eqs.(<ref>) and (<ref>), then
we recover the finite part of the one-loop polarization function
Π_1^f (p)-Π_1^f (0)=-αp^2/72[f(s)-238/3+111/s]
where the function $f(s)$ is
f(s)=L_A(s)+L_B(s)+(2-s^2)logs-2 s^-2.
We observe that in the limit $s\to 0$ the logarithmic functions have the limits $sL_A(s)\to -96$ and
$s(L_B(s)-2s^{-2})\to -15$, so that $s f(s)\to -111$.
The sum of the one-loop diverging parts, Eqs.(<ref>) and (<ref>),
yields the one-loop diverging term
Up to irrelevant terms that depend on the renormalization scheme, the finite and diverging parts
$\Sigma_1^f$, $\Pi_1^f$,$\Sigma_1^\epsilon$, $\Pi_1^\epsilon$,
as given by Eqs.(<ref>), (<ref>), (<ref>) and (<ref>),
coincide with previous results<cit.> for the one-loop functions with a massive propagator.
The crossed loops can be included very easily by a derivative with respect to $m^2$, as discussed in the
previous section below Eq.(<ref>)
\begin{align}
\Sigma_{tot}&=\left(1-m^2\frac{\partial}{\partial m^2}\right)\Sigma_1=
\left(1+s\frac{\partial}{\partial s}\right)\Sigma_1\nn\\
\Pi_{tot}&=\Pi_{2a}+\Pi_{2b}+\Pi_{2c}=\left(1+s\frac{\partial}{\partial s}\right)\Pi_1
\label{crossed}
\end{align}
where we include all finite and diverging parts in the derivative.
The derivative of the diverging parts gives the finite terms $-\alpha p^2/4$ and
$13\alpha p^2/18$ that must be added to the finite parts of self-energy and polarization
function, respectively. Thus the diverging parts do not change and are given by the
one-loop terms, Eqs.(<ref>) and (<ref>).
Performing the derivative of the finite parts we obtain
\begin{align}
\Sigma_{tot}^f (p)&=\frac{\alpha p^2}{12}\left[sg^\prime(s)+g(s)-8\right]\nn\\
\Pi_{tot}^f (p)-\Pi_{tot}^f (0)&=-\frac{\alpha p^2}{72} \left[sf^\prime(s)+f(s)-\frac{394}{3}\right]
\label{totf}
\end{align}
where $f^\prime$ and $g^\prime$ are the derivatives of $f$ and $g$, respectively.
The bare propagators follow by the insertion of finite and diverging parts in Eq.(<ref>).
The propagators can be made finite by the standard wave function renormalization.
At one loop, the only residual
mass term is finite and of order $\alpha$,
so that the divergences in Eq.(<ref>) are absorbed by the wave function
renormalization constants $Z_A$, $Z_\omega$.
In the $\overline{MS}$ scheme we find by Eqs.(<ref>) and (<ref>)
\begin{align}
Z_A&=1+\frac{13\alpha}{9\epsilon}=1+\frac{13}{3}\frac{g^2 N}{16\pi^2}\frac{1}{\epsilon}\nn\\
Z_\omega&=1+\frac{\alpha}{2\epsilon}=1+\frac{3}{2}\frac{g^2 N}{16\pi^2}\frac{1}{\epsilon},
\label{z}
\end{align}
thus reproducing the same UV behaviour of the standard one-loop approximation.
It is useful to introduce the adimensional ghost and gluon dressing functions
χ(p)=-p^2G(p); J(p)=p^2 Δ(p)
that, once renormalized by the constants $Z_A$, $Z_\omega$, are finite and read
\begin{align}
\chi (s)^{-1}&=1+\alpha\left[G(s)-\frac{2}{3}-\frac{1}{4}\log{\frac{\mu^2}{m^2}}\right]\nn\\
J (s)^{-1}&=1+\alpha\left[F(s)-\frac{394}{216}-\frac{13}{18}\log{\frac{\mu^2}{m^2}}\right]
\label{dress2}
\end{align}
\begin{align}
F(s)&=\frac{5}{8s}+\frac{1}{72}\left[ s f^\prime (s)+f(s)\right]\nn\\
G(s)&=\frac{1}{12}\left[ s g^\prime (s)+g(s)\right].
\label{FG}
\end{align}
Explicit expressions for the universal functions $F(s)$, $G(s)$ are given in the Appendix.
Here we give the asymptotic behaviour. In the UV, for $s\gg 1$ we have
F(s)≈17/18 +13/18log(s), G(s)≈1/3 +1/4log(s)
while in the infrared, for $s\to 0$, we find that $G(s)$ tends to a constant and $F(s)\approx 5/(8s)$,
so that $\chi(0)$ is finite and $J(s)\approx 8s/(5\alpha)$, yielding
$\Delta (0)^{-1}=5\alpha m^2/8$ as expected from Eq.(<ref>).
We observe that in the UV, the asymptotic behaviour of Eq.(<ref>) is precisely what we need
for canceling the dependence on $m$ in the dressing functions. In fact, in the UV, Eq.(<ref>)
can be written as
\begin{align}
\chi (p)^{-1}&=\chi (\mu)^{-1}+\frac{\alpha}{4}\log\frac{p^2}{\mu^2}\nn\\
J (p)^{-1}&=J (\mu)^{-1}+\frac{13\>\alpha}{18}\log\frac{p^2}{\mu^2}
\label{dressUV}
\end{align}
which is the standard UV behaviour that we expected by inspection of the renormalization constants Eq.(<ref>).
The constants in Eq.(<ref>) have no direct physical meaning and depend on the special choice of
renormalization constants in the $\overline{MS}$ scheme. We can subtract the dressing functions at a generic
point $s_0$ and, without fixing any special renormalization condition, we can write them in the more
general form
\begin{align}
\left[\alpha\> \chi (s)\right]^{-1}&=\left[\alpha\>\chi (s_0)\right]^{-1}+\left[G(s)-G(s_0)\right]\nn\\
\left[\alpha\> J (s)\right]^{-1}&=\left[\alpha\> J (s_0)\right]^{-1}+\left[F(s)-F(s_0)\right]
\label{dress3}
\end{align}
that extends the standard UV one-loop behaviour of the Eqs.(<ref>), sharing with them
the same asymptotic behaviour for $s,s_0\gg 1$ according to Eq.(<ref>).
We observe that in general, we might not have the freedom of setting $J(s_0)=\chi(s_0)=1$
in Eq.(<ref>). Actually, $F(s)$ is not a monotonic function, it has a minimum and is
bounded from below, so that $J(s)^{-1}$ must also be bounded in Eq.(<ref>).
Of course, that is just a limit of the
one-loop approximation and the dressing functions can be renormalized at will by a different
choice of the renormalization constants. The point is that if the dressing functions are multiplied
by the arbitrary factor $Z=1+\alpha \delta Z$ then, at one-loop, that is equivalent to the subtraction
of $\alpha \delta Z$ on the right-hand sides of the Eqs.(<ref>). That only makes sense
if $\delta Z$ is small and $Z\approx 1$. While in principle $Z$ can take any value, even much larger or
smaller than $1$, the one-loop subtraction can only compensate a small value of $\delta Z$. That is not
a problem in Eq.(<ref>) provided that we take account of any large renormalization factor by
direct multiplicative renormalization of $\chi(s_0)$ and $J(s_0)$. Then, if the energy $s$ is not too
far from the subtraction point $s_0$, the one-loop correction is small as it must be.
An important consequence is that by Eq.(<ref>) we can predict that at one-loop, up to an arbitrary
multiplicative renormalization constant, the inverse dressing functions are given by
the universal functions $F(s)$ and $G(s)$ up to an additive renormalization constant.
Such scaling property is satisfied quite
well by the lattice data, thus enforcing the idea that perturbation theory can provide important
insights on QCD in the infrared.
§ SCALING PROPERTIES ON THE LATTICE
The predictive content of the theory can be tested by a direct comparison with the lattice
data. First of all, we would like to explore the scaling properties that emerge from
Eq.(<ref>) and that seem to be satisfied by the available lattice data for $SU(2)$ and
$SU(3)$. In fact, in Eq.(<ref>) any dependence on $\alpha$
is absorbed by the multiplicative renormalization constants of $\chi$ and $J$.
By such renormalization, the inverse dressing functions are entirely determined by
the universal functions $F(s)$ and $G(s)$ up to an additive constant.
In other words, by a special choice of the renormalization constants, all dressing functions can
be translated on top of the same curve by a vertical shift. In order to make that more explicit,
we can write Eq.(<ref>) as
\begin{align}
\left[Z_G\>\chi (s)\right]^{-1}&=G(s)+G_0\nn\\
\left[Z_F \>J (s)\right]^{-1}&=F(s)+F_0
\label{dress4}
\end{align}
where $F_0$ and $G_0$ are a pair of constants depending on the subtraction point $s_0$, on the bare coupling
and on the normalization of the dressing functions $\chi(s_0)$, $J(s_0)$, while
$Z_G$, $Z_F$ are arbitrary renormalization constants that also absorb the dependence on $\alpha$.
While these equations predict a scaling property that is a stringent test for the one-loop approximation,
the predictive content is remarkable: the derivatives of the inverse dressing functions must be equal to
the derivatives of the universal functions $F(s)$, $G(s)$ up to an irrelevant multiplicative factor,
while the additive constants $F_0$, $G_0$ emerge as unknown integration constants.
The function $F(p^2/m^2)+F_0$ (solid line) is plotted together with the
lattice data for the
inverse gluon dressing function $1/J(p)=p^2\Delta(p)$ renormalized by the factors in Table I according
to Eq.(<ref>).
The points are $SU(3)$ data extracted from a figure of Ref.<cit.> ($N=3$, $\beta=5.7$, $L=96$).
The broken line is a fit of $SU(2)$ data by the empirical function of Ref.<cit.> ($\beta=2.40$, $L=42$),
only valid in the range $0.7-3$ GeV.
The energy scale is set by taking $m=0.73$ GeV for $N=3$ and $m=0.77$ GeV for $N=2$ (for the $SU(2)$ data the energy
is scaled by the ratio of the masses in order to superimpose the curves).
Inverse gluon dressing function. An enlargement of the minimum area of Fig.3 is shown by
a linear scale.
The function $G(p^2/m^2)+G_0$ (solid line) is plotted together with the
lattice data for the
inverse ghost dressing function $1/\chi(p)$ renormalized by the factors in Table I according
to Eq.(<ref>).
The points are $SU(3)$ data extracted from a figure of Ref.<cit.> ($N=3$, $\beta=5.7$, $L=80$).
The broken line is a fit of $SU(2)$ data by the empirical function of Ref.<cit.> ($\beta=2.40$, $L=42$),
only valid in the range $0.2-3$ GeV.
The energy scale is set by taking $m=0.73$ GeV for $N=3$ and $m=0.77$ GeV for $N=2$ (for the $SU(2)$ data the energy
is scaled by the ratio of the masses in order to superimpose the curves).
The mass parameter $m$ provides the natural energy units that cannot be predicted by the theory and
can only be fixed by comparison with physical observables or lattice data. In fact, the total
Lagrangian does not contain any energy scale and, as for lattice calculations, the natural scale must
be regarded as a phenomenological quantity. However, once the mass $m$ is fixed, the original arbitrariness
of its choice is reflected in a spurious dependence on the subtraction point $s_0$ which is the only scale that
remains free in the theory. We expect that the residual dependence on $s_0$, which is implicit in the constants
$F_0$, $G_0$, should decrease if the approximation is improved by the inclusion of higher loops.
$N$ $m$ (GeV) $Z_G$ $G_0$ $Z_F$
2 0.77 0.888 0.285 0.62 -0.98
3 0.73 0.637 0.24 0.30 -1.05
Multiplicative and additive renormalization constants in Eq.(<ref>) and mass scales used
in the figures.
The function $F(s)+F_0$ is shown in Fig.3 together with the lattice data for the gluon inverse dressing function.
For $SU(3)$ the data points are extracted from a figure of Ref.<cit.> while for $SU(2)$ the
interpolation function of Ref.<cit.> is used, valid in the range 0.7-3.0 GeV. The data are scaled by
the renormalization constants in Table I and shown to collapse on the one-loop function $F(s)$ by a vertical
translation. Eq.(<ref>) is satisfied very well in the whole range of the lattice data. There is
a pronounced minimum that fixes the energy scale at $m=0.73$ GeV for $SU(3)$ and $m=0.77$ GeV for $SU(2)$.
In Fig.3 and Fig.4, the energy units of the data for $SU(2)$ have been scaled by the ratio of
the masses in order to superimpose them on the data for $SU(3)$.
An enlargement of the area of the minimum is shown in Fig.4 where the deviations
between the curves are amplified but found to be smaller than the fluctuations of the lattice data.
The function $G(s)+G_0$ is shown in Fig.5 together with the lattice data for the ghost inverse dressing function.
As in Fig.3, the lattice data for $SU(3)$ are extracted from a figure of Ref.<cit.> while the
data for $SU(2)$ are given by the interpolation function of Ref.<cit.>, valid in the range 0.2-3.5 GeV.
Again, the data are scaled by the renormalization constants in Table I and collapse on the one-loop function
$G(s)$ by a vertical translation. The energy units are the same of Fig.3 and Fig.4, i.e. the same values of $m$
are required for ghost and gluon dressing functions. We can see that the scaling properties
predicted by Eq.(<ref>) are also satisfied very well by the ghost dressing function.
The one-loop gluon propagator $\Delta (p)$ (line)
is plotted together with the lattice data (points) extracted from
a figure of Ref.<cit.> ($N=3$, $g=1.02$, L=96) and scaled
by the same renormalization constants of Table I. The energy scale is set by taking $m=0.73$ GeV.
The ghost dressing function $\chi (p)$ (line) is plotted together with the
Lattice data (points) extracted from
a figure of Ref.<cit.> ($N=3$, $g=1.02$, L=80) and scaled
by the same renormalization constants of Table I. The energy scale is set by taking $m=0.73$ GeV.
Gluon propagator. Enlarged detail of Fig.6 deep in the infrared.
Overall, we find a very satisfactory description of the lattice data if the renormalization constants, the
multiplicative factors $Z_F$, $Z_G$ and the additive constants $F_0$, $G_0$ are fixed as in Table I.
While the multiplicative factors are not relevant anyway, we find a slight dependence on the additive constants
that cannot be compensated by a change of the factors, because of the one-loop approximation.
Once the energy scale $m$ is fixed, no other free parameters are left besides the renormalization constants,
so that the agreement with the lattice data is remarkable and really encouraging.
The gluon propagator and the ghost dressing function seem to show an even better accuracy than their inverse,
because of the scale.
For instance, for $SU(3)$ the gluon propagator and the ghost dressing function are reported in Fig.6
and Fig.7, respectively, together with the lattice data of Ref.<cit.>. The renormalization constants
are set at the same values of Table I as discussed above.
We observe that the gluon propagator is not convex. Actually, it is not even a monotonic function of $p$, as shown
in Fig.8 where an enlargement of the deep infrared area is displayed in more detail. That property is usually assumed
to be a sign of confinement. A comparison of the gluon propagator with the lattice data of Ref.<cit.>
for $SU(2)$ is given in Fig.9.
The one-loop gluon propagator $\Delta (p)$ (solid line)
is plotted together with the interpolation function of Ref.<cit.> (points)
that fits the lattice data for
$SU(2)$ in the range $0.7-3$ GeV ($N=2$, $\beta=2.40$, $L=42$).
The renormalization constants of Table I are used. The energy scale is set by taking $m=0.77$ GeV.
The running coupling $\alpha_s(\mu)$ by Eq.(<ref>) (solid line) is
compared with the lattice data of Ref.<cit.> for $N=3$,
$\beta=5.7$, $L=64$ (triangles) and $L=80$ (circles). The constants $F_0$, $G_0$ and
the mass $m$ are set at the values of Table I for $N=3$. The coupling is
renormalized at the point $\mu=2$ GeV where we set $\alpha_s=0.37$. The broken line is
obtained by the function $S(s)$, according to Eq.(<ref>), renormalized at the maximum
$\mu=0.67$ where $\alpha_s=1.21$.
The filled grey pattern is the area spanned by the
coupling $\alpha_s(\mu)$ when the constants $F_0$, $G_0$ are changed
by $\pm 25\%$ with respect to the values in Table I for $N=3$. All couplings are
renormalized at $\mu=2$ GeV where $\alpha_s=0.37$.
The lattice data points and the solid line of Fig. 10 are superimposed for
The same as Fig. 11, but with all couplings renormalized in the infrared at $\mu=0.15$ GeV where
The same as Fig. 11, but with all couplings renormalized at the maximum point $\mu=0.67$ GeV where
§ RUNNING COUPLING
In the Landau gauge the ghost-gluon vertex
is regular<cit.> and the vertex renormalization constant can be set to one in a momentum-subtraction
scheme, so that a runnig coupling is usually defined by the RG invariant product of the dressing
α_s(μ)=α_s(μ_0) J(μ) χ(μ)^2/J(μ_0) χ(μ_0)^2.
Having reproduced the dressing functions very well, we expect a very good agreement with the lattice for
the running coupling. Inserting Eq.(<ref>), the coupling reads
α_s(μ)=α_s(μ_0) [F(μ_0^2/m^2)+F_0] [G(μ_0^2/m^2)+G_0]^2/[F(μ^2/m^2)+F_0] [G(μ^2/m^2)+G_0]^2
and depends on the renormalization point $\mu=\mu_0$ where we set $\alpha_s(\mu_0)$ at a given
phenomenological value. We can renormalize the coupling at the point $\mu=2$ GeV where the
lattice data of Ref.<cit.> give $\alpha_s=0.37$ for $SU(3)$. That is a good compromise
as the coupling is still quite small while the energy is not too large, so that we can still neglect the RG
effects that become important in the UV limit<cit.>. We will refer to this point as the large energy
renormalization point. Using the values of Table I for $F_0$, $G_0$ and $m$ at $N=3$, the running coupling of
Eq.(<ref>) is displayed in Fig. 10, together with the lattice data of Ref.<cit.>.
The agreement is very good in the whole infrared range for $\mu<2.5$ GeV. In the UV, when $\mu> 2.5$ GeV,
we observe that Eq.(<ref>) starts to deviate from the lattice data. That is a known problem
that can be cured by a consistent running of the coupling
in the one-loop calculation according to the RG equations, as shown in Ref.<cit.>.
On the other hand, in the infrared the agreement is impressive for a one-loop calculation.
It is instructive to explore how sensitive the result is to the choice of the additive renormalization constants
$F_0$, $G_0$, which are the only free parameters of the calculation. From a physical point of view, we would expect
that if the running coupling $\alpha_s(\mu)$ is the true effective coupling at the scale $\mu$, then the one-loop
approximation should be working very well deep in the infrared where $\alpha_s\to 0$. That would be very interesting
for future perturbative work. A test of the one-loop approximation comes from the sensitivity to changes of the
additive constants. If the approximation is under full control, then any small change of $F_0$ and $G_0$
should be compensated by the multiplicative renormalization constants, thus canceling in the normalized
ratio of Eq.(<ref>). In Fig.11, the grey pattern shows the area spanned by the running coupling $\alpha_s(\mu)$ of
Eq.(<ref>) when the additive renormalization constants $F_0$, $G_0$ are changed by $\pm25\%$ around the
values of Table I. Ignoring RG effects in the UV and comparing with the best running coupling of Fig.10,
which is also shown in the figure, we see that the deviations are very small in the UV and start growing up
when $\alpha_s\approx 0.6$. They increase until $\alpha_s$ reaches its maximum and then decrease getting smaller
and smaller in the infrared limit $\mu\to 0$. That enforces the idea that, deep in the infrared, the one-loop
approximation could be under full control. Moreover, the sensitivity to the additive constants seems to be
even smaller in the infrared if the renormalization point is taken at a very low energy.
In Fig. 12 the deviations are evaluated as before, by Eq.(<ref>), but renormalizing the coupling
at $\mu=0.15$ GeV where $\alpha_s=0.2$. We can see that the running coupling seems to be not
sensitive at all to the choice of the additive constants until $\alpha_s\approx 0.6$, and the approximation
seems to be under full control below $300$ MeV. In other words, regardless of the actual value of the
renormalization constants, all curves evaluated by Eq.(<ref>) collapse on the lattice data below $300$ MeV.
That is a remarkable feature as, by a proper choice of the renormalization point, the present one-loop
approximation provides a very accurate description of the running coupling below $300$ MeV (Fig. 12) or above
$1.5$ GeV (Fig. 11), without adjusting any free parameter, from first principles.
In the range between $0.3$ and $1.3$ GeV, where $\alpha_s>0.6$, Eq.(<ref>) can be still tuned on the lattice data,
as shown in Fig. 10, but the increased sensitivity to the additive renormalization constants is a sign of the
limits of the one-loop approximation. However, since the calculation is from first principles, we expect that
the sensitivity to the additive constants should decrease when higher loops are included in the expansion.
From a technical point of view, Eq.(<ref>) provides a very good interpolation of the lattice data
and is not sensitive to the choice of the renormalization constants below $300$ MeV and above
$1.5$ GeV. Then, it could make sense to introduce a third fixed point by just renormalizing at the scale
where the deviations are larger and pinpoint $\alpha_s$ at its maximum. If we renormalize at the maximum
point $\mu=0.67$ GeV setting $\alpha_s=1.21$, the deviations are quite small over the whole range of energies,
as shown in Fig.13. That suggests that we can get rid somehow of the additive constants and write some
universal function for the running coupling, free of any parameter, albeit slightly approximate.
Let us pretend that we can set $\chi(s_0)=J(s_0)=1$ in Eq.(<ref>) and insert it in Eq.(<ref>).
Then, neglecting higher powers of $\alpha$, the running coupling takes the simple shape
where $\alpha(p^2/m^2)=3N\alpha_s(p)/(4\pi)$ and the universal function $S(s)$ is defined as
and does not contain any free parameter.
In the UV, the running coupling $\alpha(s)$ incorporates the standard one-loop leading behaviour.
In fact, by Eq.(<ref>), for $s,s_0\gg 1$
which does not depend on the scale $m$.
In the infrared, the function $S(s)$ replaces the standard log, yielding a finite running coupling
without encountering any Landau pole.
In the limit $s\to 0$ the function diverges as $S(s)\sim (1/s)$ and the coupling goes
to zero as a power $\alpha(s)\sim s$.
A maximum is found at the point where ${\rm d}S(s)/{\rm d}s=0$, which occurs at $s_M= 1.044$.
Of course, this point does not depend on any parameter and provides an independent way to fix the scale $m$ by
a comparison with the lattice. From the data of Ref.<cit.> in Fig.10 the maximum occurs at
$p\approx0.6-0.7$ GeV yielding a scale $m\approx 0.6$ GeV, not too far from the values in Table I.
Taking $m=0.6$ GeV and the maximum as renormalization point, namely $p=0.67$ GeV and $\alpha_s=1.21$ as in Fig. 13,
the plot of Eq.(<ref>) is shown in Fig.10 as a broken line.
The running coupling $\alpha(s)$ in Eq.(<ref>)
provides a nice qualitative description from first-principles, incorporates the standard leading UV behaviour
at one-loop and can be used for extending the standard one-loop running coupling deep in the infrared.
§ DISCUSSION
Let us summarize the main findings of the paper.
It has been shown that, from first principles,
without changing the original Lagrangian, Yang-Mills theory can be studied
by a perturbative expansion by just taking a massive propagator as the expansion point.
Without the need to include spurious parameters or mass counterterms, the expansion can be renormalized
and all the divergences are canceled by the standard wave function renormalization of the fields.
At one-loop, the derivatives of the inverse propagators are determined, up to irrelevant multiplicative
factors, by the derivatives of the universal functions $F(s)$, $G(s)$, that do not depend on any
parameter. Thus, once a scale is fixed (the theory does not contain a scale that must come from the phenomenology),
the inverse dressing functions are determined up to an integration constant.
The relevant features of the dressing functions are contained in the universal functions $F$, $G$ regardless
of the specific value of the bare coupling and of $N$. That scaling property has been shown to be satisfied very well
by the lattice data, enforcing the idea that the infrared range of QCD can be studied by perturbation theory.
While the derivatives of the dressing functions are derived exactly, the propagators depend on the integration
constants $F_0$, $G_0$. If the coupling is small and the one-loop approximation is under full control
we would expect that a slight change of the additive constants could be compensated by a change of the
irrelevant multiplicative factors. Actually, that only occurs in the UV and deep in the infrared where the
effective running coupling is small. In the range $0.5-1$ GeV, where the coupling reaches its maximum, the
propagators are sensitive to the choice of the additive constants. That seems to be a sign that higher
loops might be relevant when the effective coupling is larger. Thus, we expect that the sensitivity
should decrease when higher loops are included in the calculation.
Even where the coupling $\alpha_s$ is not very small and two-loop corrections seem to be relevant, the one-loop calculation may acquire a variational meaning. The dependence on the renormalization constants is a consequence of an overall dependence on the ratio of the two energy scales: the mass parameter $m$ and the renormalization point $\mu$. Since the exact result should not depend on that ratio, the dependence is expected
to decrease when higher loops are included in the calculation. Thus a best choice for that ratio could be
obtained by some stationary condition on the observables, requiring that the sensitivity should be minimal in the
predicted phenomenology. However, there is no proof that a best choice of the renormalization constants does
exist, mimimizing two-loop corrections everywhere.
Thus, it is encouraging to know that, by tuning the additive constants, the one-loop calculation already
provides an excellent description of the lattice data for the propagators and the running coupling. We conclude
that, while not anomalously small in general, two-loop corrections can be minimized by a best choice of
the constants.
Moreover, the sensitivity to the additive constants $F_0$, $G_0$ seems to be really negligible below $300$ MeV
and above $1.5$ GeV, namely when $\alpha_s<0.6$. In those ranges the running coupling collapses on the lattice
data without the need to tune any constant or parameter, from first principles and by a fully analytical
The existence of an energy range, deep in the infrared, where the one-loop approximation seems to be under
full control, could open the way for a more general analytical study of QCD below $\Lambda_{QCD}$ where
many interesting phenomena still suffer the lack of a full description from first principles.
§ EXPLICIT FUNCTIONS $F$ AND $G$
The functions $F(x)$ and $G(x)$ are defined in Eq.(<ref>) in terms of the functions $f(x)$, $g(x)$ and
their derivatives. Here we give the explicit expressions. The functions $f$, $g$ are given in Eqs.(<ref>)
and (<ref>), respectively. They are encountered in the calculation of the standard one-loop polarization
and self energy with a massive propagator and coincide with the result of other authors<cit.>.
The derivatives are straightforward and have been checked by a software package. The result is
\begin{align}
\label{FGx}
\end{align}
where the logarithmic functions $L_x$ are
\begin{align}
\log\left(\frac{\sqrt{4+x}-\sqrt{x}}{\sqrt{4+x}+\sqrt{x}}\right)\nn\\
\label{logsA}
\end{align}
and the rational parts $R_x$ are
\begin{align}
\label{rational}
\end{align}
aguilar8 A. C. Aguilar, D. Binosi, J. Papavassiliou, Phys. Rev. D78, 025010 (2008).
aguilar10 A. C. Aguilar, J. Papavassiliou, Phys. Rev. D81, 034003 (2010).
aguilar14 A. C. Aguilar, D. Binosi, J. Papavassiliou, Phys. Rev. D 89, 085032 (2014).
aguilar14b A. C. Aguilar, D. Binosi, D. Ibanez, J. Papavassiliou, arXiv:1405.3506.
papa15 A. C. Aguilar, D. Binosi, J. Papavassiliou, Phys. Rev. D 91, 085014 (2015).
papa15b D. Binosi, L. Chang, J. Papavassiliou, C. D. Roberts, Phys. Lett. B 742, 183 (2015).
huber14 A. L. Blum, M. Q. Huber, M. Mitter, L. von Smekal, Phys. Rev. D 89, 061703 (2014).
huber15g M. Q. Huber, Phys. Rev. D 91, 085018 (2015).
huber15b A. K. Cyrol, M. Q. Huber, L. von Smekal, Eur.Phys.J. C75 (2015) 102.
reinhardt04 C. Feuchter and H. Reinhardt, Phys. Rev. D 70, 105021 (2004).
reinhardt05 H. Reinhardt and C. Feuchter, Phys.Rev. D 71, 105002, (2005).
reinhardt08 D. Epple, H. Reinhardt, W. Schleifenbaum, A.P. Szczepaniak, Phys. Rev. D77, 085007,(2008).
reinhardt14 M. Quandt, H. Reinhardt, J. Heffner, Phys. Rev. D 89, 065037 (2014).
sigma F. Siringo and L. Marotta, Eur. Phys. J. C 44, 293 (2005).
sigma2 F. Siringo, Mod. Phys. Lett. A 29, 1450026 (2014), arXiv:1308.4037
gep2 F. Siringo, Phys. Rev. D 88, 056020 (2013), arXiv:1308.1836.
varqed F. Siringo, Phys. Rev. D 89, 025005 (2014), arXiv:1308.2913.
varqcd F. Siringo, Phys. Rev. D 90, 094021 (2014), arXiv:1408.5313.
genself F. Siringo, Phys. Rev. D 92, 074034 (2015), arXiv:1507.00122.
ptqcd F. Siringo, arXiv:1507.05543.
dudal08 D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel, H. Verschelde, Phys. Rev. D 78,
065047 (2008).
sorella15 M.A.L. Capri, A.D. Pereira, R.F. Sobreiro, S.P. Sorella, Eur. Phys. J. C 75, 479 (2015).
dudal15 M.A.L. Capri, D. Dudal, D. Fiorentini, M.S. Guimaraes, I.F. Justo, A.D. Pereira,
B.W. Mintz, L.F. Palhares, R.F. Sobreiro, S.P. Sorella, Phys. Rev. D 92, 045039 (2015).
bogolubsky I.L. Bogolubsky, E.M. Ilgenfritz, M. Muller-Preussker, A. Sternbeckc,
Phys. Lett. B 676, 69 (2009).
twoloop E.-M. Ilgenfritz, M. Müller-Preussker, A. Sternbeck, A. Schiller, arXiv:hep-lat/0601027.
dudal D. Dudal, O. Oliveira, N. Vandersickel, Phys. Rev. D 81, 074505 (2010).
binosi15 P. Bicudo, D. Binosi, N. Cardoso, O. Oliveira, P. J. Silva, arXiv:1505.05897.
burgio15 G. Burgio, M. Quandt, H. Reinhardt, H. Vogt, Phys. Rev. D 92, 034518 (2015).
taylor J. C. Taylor, Nucl. Phys. B 33, 436 (1971).
cornwall J. M. Cornwall, Phys. Rev. D 26, 1453 (1982)
tissier10 M. Tissier, N. Wschebor, Phys. Rev. D 82, 101701(R) (2010).
tissier11 M. Tissier, N. Wschebor, Phys. Rev. D 84, 045018 (2011).
tissier14 M. Pelaez, M. Tissier, N. Wschebor, Phys. Rev. D 90, 065031 (2014).
stevenson P.M. Stevenson, Phys. Rev. D 32, 1389 (1985).
var F. Siringo, Phys. Rev. D 62, 116009 (2000).
light F. Siringo, Europhys. Lett. 59, 820 (2002).
bubble F. Siringo and L. Marotta, Int. J. Mod. Phys. A25, 5865 (2010), arXiv:0901.2418v2.
ibanez R. Ibañez-Meier, I. Stancu, P.M. Stevenson, Z. Phys. C 70, 307 (1996).
su2 F. Siringo, L. Marotta, Phys. Rev. D 78, 016003 (2008).
LR F. Siringo and L. Marotta, Phys. Rev. D 74, 115001 (2006).
HT F. Siringo, Phys. Rev. D 86, 076016 (2012), arXiv: 1208.3592v2.
superc1 M. Camarda, G.G.N. Angilella, R. Pucci, F. Siringo,
Eur. Phys. J. B 33, 273 (2003).
superc2 L. Marotta, M. Camarda, G.G.N. Angilella and F. Siringo,
Phys. Rev. B 73, 104517 (2006).
AF L. Marotta and F. Siringo, Mod. Phys. Lett. B, 26, 1250130 (2012), arXiv:0806.4569v3.
stancu2 I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1990).
stancu I. Stancu, Phys. Rev. D 43, 1283 (1991).
su2glu V. G. Bornyakov, V. K. Mitrjushkin, M. Ml̈ler-Preussker, Phys. Rev. D 81, 054503 (2010).
su2gho V. G. Bornyakov, E.-M. Ilgenfritz, C. Litwinski, V. K. Mitrjushkin, M. Muller-Preussker, arXiv:1302.5943.
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1511.00630
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Lorentz and CPT Violation in Top-Quark Production
Zhi Liu
Signals for Lorentz and CPT violation can appear in a wide range of experiments including hadron colliders like the LHC. We present a calculation of the Lorentz-violating cross section for top-quark pair production via gluon fusion. This process dominates at the LHC, and analysis of LHC data should permit sharpening the constraints on top-quark Lorentz violation obtained recently by the D0 Collaboration. We also present a separate calculation of single-top production, which is sensitive to CPT violation. Data from the LHC can be used to measure coefficients for CPT violation in the top-quark sector for the first time.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
Lorentz and CPT symmetries are fundamental symmetries that both the Standard Model and general relativity are based on. Up to now, a great many experiments have confirmed these symmetries. However, in the history of science, many symmetries that people had taken for granted turned out to be violated, including parity and CP. Therefore, it is possible to think about tiny violations of Lorentz and CPT symmetries. In fact, Lorentz violation may occur as a spontaneous symmetry breaking in string theories <cit.>. Thus it is becoming an important task for physicists to test these symmetries.
To study Lorentz and CPT violation, one approach is to forget about the underlying theories that cause the violation but to come up with an effective field theory. This is more universal because it does not depend on the model we use, which is convenient for experimental searches. The framework that implement this idea is the Standard-Model Extension (SME) <cit.>, which includes all known physics and all possible Lorentz violating effects. It also has the covariance under observer Lorentz transformations, which means physics does not depend on the coordinates we choose. Under these conditions, we can write down all possible Lorentz violating terms in the Lagrange density. Each term consists of a Lorentz violating operator contracted with a coefficient. These coefficients, which control the size of the violation, are expected to be quite small, for no deviations from Lorentz symmetry have been found. This means we can treat the coefficients perturbatively. On the other hand, in effective field theories, CPT violation leads to Lorentz violation <cit.>. As a result, about half of the Lorentz violating operators in the SME also violate CPT. Therefore, the SME is also a framework to study CPT violation.
The framework of the SME allows us to predict what would happen in experiments. In turn, experiments are able to set bounds on the coefficients. One of the most useful signals in Lorentz violation searches is the sidereal signal. As the Earth rotates on its axis, the background fields, i.e. the coefficients for Lorentz violation, which are taken to be constant in the Sun-centered frame <cit.>, will change over time in the laboratories fixed on Earth. If the experimental systems are coupled to these fields, the results may depend on sidereal time and give sidereal signals. The sidereal day is the time it takes the Earth to complete one rotation, which is nearly 4 minutes shorter than the 24-hour mean solar day.
Various kinds of experiments have been performed in order to test Lorentz and CPT symmetries under the framework of the SME, including experiments with neutral-meson oscillations, neutrino oscillations, electroweak physics, photon, gravity, etc <cit.>. Many of them have achieved excellent sensitivities. In contrast, in the top-quark sector, only one measurement searching for Lorentz violation <cit.> and no measurement searching for CPT violation have been performed. So this work explores the possibility of increasing the sensitivity of Lorentz violation measurement in the top quark and the prospect of testing CPT symmetry in the same context.
§ TOP PAIR PRODUCTION
Being the heaviest quark discovered by now, the top quark plays a special role in particle physics, from the Standard Model to its extensions, including the SME <cit.>. So it's of special interest to test Lorentz and CPT symmetries using the top quark <cit.>. Up to now, the only places in the world that can produce the top quark are the Tevatron and the LHC. The top pair production, through quark fusion ($q\bar{q} \rightarrow t\bar{t}$) and gluon fusion ($gg\rightarrow t\bar{t}$), is the main production mechanism in these hadron colliders. At the Tevatron, about 85% of the production is from quark fusion. At the LHC, gluon fusion dominates. To take advantage of the much larger cross sections of the top pair production at the LHC, it's essential to analyze how the SME modifies the cross sections of gluon fusion.
After a top-antitop pair is produced, the top and the antitop quickly decay. Therefore, the squared matrix element ($|{\cal M}|^2$) for both the production and the decay is needed for experimental analysis. Using the narrow-width approximation, $|{\cal M}|^2 $ can be written in terms of the product of three factors, which come from the production, the top decay and the antitop decay, respectively. From now on, we concentrate on the production part. The whole process is treated in Ref. <cit.>.
The theoretical framework we use is the SME in the top-quark sector with some minor assumptions. We assume the Lorentz and CPT violation occurs only in the third-generation quarks. By field redefinitions, the dimensionless coefficient $c_{\mu\nu}$ is the only observable coefficient for the cross sections we compute. So $c_{\mu\nu}$ produces general Lorentz violation in this sense. The field redefinitions also show the antisymmetric part of $c_{\mu\nu}$ is not observable. In addition, no leading-order CPT violation can appear in top pair production <cit.>.
The path from the theory to the experiments is like this. Starting with the SME Lagrange density, we derive the Feynman rules, which are used to calculate the squared matrix element of the production part. Combining it with the decay part gives the $|{\cal M}|^2 $ of the whole process. These results can then be used as input for experimental analysis to run simulations and to compare with experimental signals like cross sections and sidereal variations.
The D0 Collaboration has used the squared matrix element for the quark fusion to obtain the first measurement of Lorentz violation in the top-quark sector <cit.>. The measured $c_{\mu\nu}$ coefficient is consistent with zero with $\sim$10% sensitivity. Given the much greater statistical power at the LHC, we expect the $c_{\mu\nu}$ coefficient can be measured to about 1%.
Now we outline the calculations of the SME corrections to $|{\cal M}|^2$ for the gluon fusion. The Standard Model diagrams at tree level consist of three diagrams: $s$, $t$ and $u$ channels. The modified Feynman rules have insertions on vertices and propagators, which means Lorentz violating effects come in as insertions on the Standard Model diagrams. Since the Lorentz violating effects are expected to be small, we only consider the leading-order corrections. So only one insertion is needed at a time. The corrections from the vertices can be found by adding the diagrams with vertex insertions to the Standard Model diagrams, taking the modulus square and averaging/summing over spins and colors.
The calculations of the corrections from the propagators are slightly different. Instead of using insertions, we compute the full propagator. We first obtain the modified Dirac equation from the Lagrange density. The full propagator is $i$ times the factor in front of the spinor in the corresponding momentum-space equation. The modified spin sums are also needed for this calculation. They are $-i$ times the numerators of the full propagators after we eliminate the gamma matrices in the denominators of these propagators.
The explicit expressions for the SME corrections to the squared matrix element are given in Ref. <cit.> but are omitted here due to space limitations. When $c_{\mu\nu}$ is factored out in the sum of the SME corrections, the remaining factor is symmetric under the interchange of $\mu$ and $\nu$, which is compatible with the antisymmetric part of $c_{\mu\nu}$ being unobservable.
§ SINGLE-TOP PRODUCTION
Unlike top pair production, CPT violation is observable in single-top production, which makes it a more interesting process to consider. Since the cross sections of the single-top production are much larger at the LHC, we concentrate on the production processes at the LHC in this section.
As before, our goal is to calculate the squared matrix element for the whole process and we still use narrow-width approximation. Besides the theoretical assumption we make in Sec. 2, we further assume the only nonzero coefficient is $b_{\mu}$, which is for CPT violation. This is reasonable, because in practice it is easier to consider one coefficient at a time and set a bound on this coefficient. If we find a nonzero result, we can reconsider the interference between coefficients <cit.>. In addition, according to Greenberg's theorem <cit.>, top and antitop have the same invariant mass in our framework.
We use the same idea to calculate the SME corrections to the squared matrix element in the production part. In the Standard Model, the single-top production include the following processes: $q\ol{q'}\rightarrow t \ol{b}$ ($s$-channel); $bq \rightarrow t q'$ and $b\ol{q} \rightarrow t\ol{q'}$ ($t$-channel); $bg\rightarrow t W^-$ ($tW$ mode). At tree level, each of the first three processes has one diagram, the $t W$ mode has two diagrams. One complication in this calculation is the modified spin sums, which can be computed using the approximate solutions to the modified Dirac equation. These solutions are given in Appendix A of Ref. <cit.>. To simplify the calculation, we first compute the spin sums in the zero-momentum frame of the particle, then we transform the results to the frame in which the particle has momentum $\vec{p}$. The transformation is an observer Lorentz transformation, so both the particle and the background fields $b_{\mu}$ are transformed.
The explicit expressions for the SME corrections to $|{\cal M}|^2 $ are still given in Ref. <cit.>. One interesting result is that, if we consider single-antitop production by reversing the fermion lines in all the tree-level diagrams for single-top production, the resulting SME corrections for each process obtain an overall minus sign. This is useful, because in the Standard Model the theoretical cross sections for the $tW$ mode and for the $\ol{t}W$ mode are equal at the LHC <cit.> but the SME corrections are different. Therefore, a difference in these two cross sections is another signal for CPT violation comparing to sidereal variations.
§ SUMMARY
The SME is a general framework to study Lorentz and CPT violation. Under this framework, we calculate the corrections to the squared matrix element for top pair production via gluon fusion, which allows the LHC Collaboration, including ATLAS and CMS, to increase the sensitivity of the coefficient for Lorentz violation to about 1% comparing with the previous $\sim$10% sensitivity. The CPT violating corrections to the squared matrix element for single-top production can be used by the LHC Collaboration to obtain the first measurement in CPT violation in the top-quark sector with an estimated sensitivity of about 5%. The data taken with 8 TeV at the LHC can already be used to make these measurements, and using the full dataset will increase the sensitivity.
I am grateful to Micheal S. Berger and V. Alan Kostelecký for their collaboration in this work. This work is supported partly by Department of Energy under grant number DE-SC0010120 and by the
Indiana University Center for Spacetime Symmetries
V. A. Kostelecký and S. Samuel,
Phys. Rev. D 39, 683 (1989).
V. A. Kostelecký and R. Potting,
Nucl. Phys. B 359, 545 (1991).
D. Colladay and V. A. Kostelecký,
Phys. Rev. D 55, 6760 (1997).
D. Colladay and V. A. Kostelecký,
Phys. Rev. D 58, 116002 (1998).
O. W. Greenberg,
Phys. Rev. Lett. 89, 231602 (2002).
R. Bluhm, V. A. Kostelecký, C. D. Lane and N. Russell,
Phys. Rev. D 68, 125008 (2003).
R. Bluhm, V. A. Kostelecký, C. D. Lane and N. Russell,
Phys. Rev. Lett. 88, 090801 (2002).
V. A. Kostelecký and M. Mewes,
Phys. Rev. D 66, 056005 (2002).
Data Tables for Lorentz and CPT Violation, 2015 edition
, V. A. Kostelecký and N. Russell, arXiv:0801.0287v8 [hep-ph].
V. M. Abazov et al. [D0 Collaboration],
Phys. Rev. Lett. 108, 261603 (2012).
K. A. Olive et al. [Particle Data Group Collaboration],
Chin. Phys. C 38, 090001 (2014).
M. S. Berger, V. A. Kostelecký and Z. Liu,
arXiv:1509.08929 [hep-ph].
W. Bernreuther,
J. Phys. G 35, 083001 (2008).
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1511.00864
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[Contact email address: ][email protected]
School of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland
School of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland
School of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland
The ultra-fast demagnetization of small iron clusters initiated by an intense optical excitation is studied with time-dependent spin density functional theory (TDSDFT). In particular we investigate the effect of the spin-orbit interaction on the onset of the demagnetization process. It is found that the initial rate of coherent spin loss is proportional to the square of the atomic spin-orbit coupling constant, $\lambda$. A simplified quantum spin model comprising spin-orbit interaction and a local time-dependent magnetic field is found to be the minimal model able to reproduce our ab initio results. The model predicts the $\lambda^2$ dependence of the onset rate of demagnetization when it is solved either numerically or analytically in the linear response limit. Our findings are supported by additional TDSDFT simulations of clusters made of Co and Ni.
75.78.Jp, 75.75.-c, 75.70.Tj, 75.10.Jm
Achieving control over the magnetization dynamics at the femtosecond timescale is a desirable
asset for new magnetic data storage technologies. The ultrafast optical demagnetization (UOD) phenomenon,
discovered by Beaurepaire et al <cit.>, in which an intense femtosecond laser pulse
induces an abrupt and dramatic loss of magnetization in a metallic film, initiated what is now
the highly active field of femto-magnetism.
Typical UOD experiments are based on the pump-probe method, where a
femtosecond laser pulse in the optical range (pump) is shed onto the magnetic sample and
then a delayed short electromagnetic pulse (probe) is used to detect the magnetic response through
possible linear or non-linear magneto-optical effects <cit.>. By varying the time delay
between the pump and the probe the magnetization dynamics can be reconstructed in the time domain
over a typical range spanning from a few femtoseconds to a few picoseconds. The rapid demagnetization process
that develops over this time can be characterized by two distinct
stages: (i) a coherent stage in the first few tens of fs when the light interacts with the electrons and (ii) a relaxation stage when
hot electrons and spins interact with each other and with the lattice so to thermalize. Although the role
of the particular microscopic spin-flip mechanisms is often unclear and dependent on the details of the
magnet investigated, the thermalization process is in general tractable through empirical three-temperature
models <cit.>, which establish rate equations between the spin, electron and phonon systems.
In contrast, theory for the coherent stage is rather unsettled and spans a range of different views (not necessarily mutually exclusive), from relativistic accounts of the direct photon-spin coupling <cit.> to
semi-classical transport models <cit.>.
Experimental works, focussed on the coherent regime, have described a strong dependence of the rate of UOD on the material and, particularly, its spin-orbit coupling (SOC) properties. For instance, it has been
reported <cit.> that materials exhibiting stronger SOC demagnetize significantly faster than
lighter ones. The SOC has been identified as a key component enabling UOD also in earlier theoretical works based on model Hamiltonian <cit.>. Very recently time-dependent spin-density functional theory (TDSDFT) calculations <cit.> have provided another confirmation of its essential role for the ultrafast laser-induced loss of spin in bulk transition metals. In this letter we seek to gain further understanding of the microscopic mechanisms responsible for the very initiation of the UOD. We employ the only practically-applicable first-principles theoretical framework, the TD(S)DFT <cit.>, that allows to simulate the UOD process for real atomic clusters directly in the time domain and for experimentally-relevant times. We demonstrate that the very onset of the demagnetization is triggered by the electronic charge response to the electric field of the pulse. The charge and spin currents generated give rise to a magnetic field which in combination with the SOC facilitates spin flips and global spin decay. We also establish that the initial, coherent demagnetization rate is proportional to the square of the ionic SOC strength for a range of small transition metal clusters.
0.80.8trueFig01(Color online) (a) Typical electric field pulses used to excite the Fe$_6$ cluster
(cartooned as inset) and time evolution of the total TDSDFT energy (b), total spin (c) and total KS angular momentum (d)
of the cluster when subjected to each of the pulses in panel (a) with corresponding color code on each of the other
panels. fig01
In particular, we focus on Fe$_6$, the geometry and ground state (GS) spin $2S=20\,\hbar$ of which have been previously predicted
by the LSDA<cit.>. It is well-known that open $d$-shell systems are problematic to local
approximations of the exchange and correlation (XC) functional. Here, however, our intention is to study the
generality of the spin dynamics so that possible quantitative features are not important at this time.
We consider the adiabatic temporal extension <cit.> of the local spin-density approximation (ALSDA),
parameterized by Perdew and Wang <cit.> and implemented in the Octopus code <cit.>.
In all our calculations spin dynamics is initiated by a single intense electric field pulse (we neglect the
magnetic field component). We solve the time-dependent Kohn-Sham (KS) equations
\begin{equation}
i\hbar\frac{\partial}{\partial t}\psi_{j}(\vec{r},t) = H_\mathrm{KS}(\vec{r},t)\psi_{j}(\vec{r},t)\:,
\end{equation}
where $H_\mathrm{KS}(\vec{r},t)$ is $2\times 2$ matrix in spin space to account for non-collinearity,
$\psi_{j}$ are two-dimensional spinors and we consider the time-dependent KS Hamiltonian
with electric field, $\vec{E}(t)$, introduced in the length gauge
\begin{eqnarray}
H_\mathrm{KS}(\vec{r},t) & = & -\frac{\hbar^{2}\nabla^{2}}{2m} + v_\mathrm{s}(\vec{r},t) - \frac{2\mu_\mathrm{B}}{\hbar}\hat{\vec{S}}\cdot\vec{B}_\mathrm{xc}(\vec{r},t) \,\,\,\,\,\, \\
v_\mathrm{s}(\vec{r},t) & = & \sum_{\vec{R}_{I}} V_\mathrm{PP}(|\vec{r}-\vec{R}_{I}|) + \int d^{3}r' \frac{n(\vec{r'})}{|\vec{r}-\vec{r'}|} + \nonumber \\ & & v_\mathrm{xc}(\vec{r},t) + \vec{r}\cdot \vec{E}(t).
\end{eqnarray}
We substitute the ionic potentials, centered at each site $\vec{R}_{I}$, with soft norm-conserving fully relativistic pseudopotentials that
reproduce correctly the semi-core and valence wavefunctions beyond a certain core
radius<cit.> [Our fully-relativistic norm-conserving pseudopotentials
have been generated with the multi-reference pseudopotential method as implemented in the APE
code <cit.>. This includes, in addition to valence states, also the semi-core ones ($3s$ and
$3p$), since semi-core states play an important role in describing the chemical bond and the
magnetic properties of transition metal clusters <cit.>.]
\begin{eqnarray} \label{eq:Vpp}
V_\mathrm{PP}(r) & = & \sum_{l}\sum_{m=-l}^{l}\bigg( \bar{V}_{l}^\mathrm{ion}(r) + \frac{1}{4} V_{l}^\mathrm{SO}(r) + \nonumber \\
& & V_{l}^\mathrm{SO}(r) \hat{\vec{L}} \cdot \hat{\vec{S}}\bigg)\ket{l, m}\bra{l, m}\:.
\end{eqnarray}
Here, $\hat{\vec{S}}$ is the spin operator, $\hat{\vec{L}}$ is the angular momentum operator
associated to the atomic center, while the scalar
part of the pseudopotential $\bar{V}_{l}^\mathrm{ion}(r)$ includes the effect of the mass shift and the
Darwin term. $V_{l}^\mathrm{SO}(r)$ defines the range of the SOC term
[For values of the radius greater than the range of the SOC component the ionic
pseudopotential reduces to a purely local object.]. Within the adiabatic LSDA, at every
time $t$ the XC potentials are calculated through the ground state LSDA exchange-correlation
energy functional corresponding to the instantaneous electron charge density
$n(\vec{r},t)=\sum_{j\in \mathrm{occ.}} \psi_j^*(\vec{r},t) \psi_j(\vec{r},t)$ and spin density
$\vec{s}(\vec{r},t)=\frac{\hbar}{2}\sum_{j\in \mathrm{occ.}} \psi_j^*(\vec{r},t) \vec{S} \psi_j(\vec{r},t)$ , i.e.
\begin{equation}
v_\mathrm{xc}(\vec{r},t) \!=\!\left. \frac{\delta E_{xc}}{\delta n}\right|_{\!\begin{array}[b]{c} n(\vec{r},t)\\ \vec{s}(\vec{r},t) \end{array}}, \,\, \vec{B}_\mathrm{xc}(\vec{r},t) \!=\!\frac{\hbar}{2\mu_\mathrm{B}}\left. \frac{\delta E_{xc}}{\delta \vec{s}}\right|_{\!\begin{array}[b]{c} n(\vec{r},t)\\ \vec{s}(\vec{r},t) \end{array}} .
\end{equation}
For the XC magnetic field the zero-torque theorem holds <cit.>, i.e.
$\int d^{3}r \vec{B}_\mathrm{xc}(\vec{r}) \times \vec{s}(\vec{r}) = 0$,
where the integral is taken over the entire space. In other words, $\vec{B}_\mathrm{xc}$ cannot produce
a global spin torque over the system. The Heisenberg equation of motion for the spin operator leads to a spin
continuity equation in the form
\begin{equation} \label{eq:spincont}
\frac{d\vec{s}(\vec{r},t)}{dt} = -\nabla\cdot \mathrm{J}_\mathrm{KS}(\vec{r},t) + 2\frac{\mu_\mathrm{B}}{\hbar}\vec{s}(\vec{r},t)\times\vec{B}_\mathrm{xc}(\vec{r},t) + \vec{\Gamma}(\vec{r},t)
\end{equation}
\begin{eqnarray}
\vec{\Gamma}_{n}(\vec{r},t) & = & \sum_{\vec{R}_{I}}\sum_{\alpha,\beta}\sum_{m,m',l} \epsilon_{njk}V_{l}^\mathrm{SO}(|\vec{r}-\vec{R}_{I}|)\cdot \nonumber \\
& & \cdot \braket{\Psi^\mathrm{KS}_{\alpha}|l, m', \vec{R}_{I}} \hat{S}_{\alpha,\beta}^{k} \label{eq:3} \\
& & \bra{l, m', \vec{R}_{I}}\hat{L}^{j}_{I}\ket{l, m, \vec{R}_{I}} \braket{l, m, \vec{R}_{I}|\Psi^\mathrm{KS}_{\beta}}. \nonumber
\end{eqnarray}
The first term in Eq. (<ref>) can be written as a two components tensor,
$ \mathrm{J}^{ij}_\mathrm{KS}(\vec{r},t)= \Tr{\hat{S}^i \sum_k (\psi_k^* \partial_{j} \psi_k - \psi_k \partial_{j} \psi^*_k)}$
is the KS spin-current operator. The second term is the torque exerted locally by $\vec{B}_\mathrm{xc}(\vec{r},t)$,
which vanishes within the adiabatic LSDA ($\vec{B}^{ALDA}_\mathrm{xc}(\vec{r})\parallel \vec{s}(\vec{r})$).
The first term, however, can be re-written in a form $ \frac{2\mu_\mathrm{B}}{\hbar}\vec{s} \times \vec{B}_\mathrm{kin}$,
where the kinetic
magnetic field<cit.> is defined as $\vec{B}_\mathrm{kin} = \frac{1}{2en} \sum_k \nabla_k (n \nabla_k \vec{s})$.
An effective local magnetic field can be defined as
$\vec{B}_\mathrm{eff} (\vec{r},t)=\vec{B}_\mathrm{kin} (\vec{r},t) + \vec{B}_\mathrm{xc} (\vec{r},t)$.
This field is not necessary locally parallel to $\vec{s} (\vec{r},t)$ and produces a local torque.
The last term in Eq. (<ref>) $\vec{\Gamma}(\vec{r},t)$ is the only source of global spin relaxation
in the temporal evolution. Note that in this description the global spin change is only determined by the orbital
dependent scattering properties of the non-local component of the atomic pseudopotential as a result of the
SOC and it is, therefore, not dependent directly on the Kohn-Sham orbital momentum defined
as $\vec{L}_\mathrm{KS}=\vec{r}\times\vec{J}_\mathrm{KS}$. Later in this letter we will revisit
Eq. (<ref>) and use it as a base for a simplified quantum model for the spin operator.
11trueFig02(Color online) Contour plots of the time-averaged and $z$-averaged observables
evaluated only in spheres of radius 0.85 Å around each atom: (a) the averaged temporal variation of the
spin density $S^z (\vec{r})$ with respect to the ground state; (b) difference of the latter and its counterpart in
the case of no SOC; (c) the effective magnetic field $B^z_\mathrm{eff} (\vec{r})$ in the ground state
($t=0$) and (d) the variation of $B^z_\mathrm{eff}(t)$ with respect to the ground state.
See text for details.fig02
Figure <ref> shows representative results from the full time-dependent simulations of the dynamics
of the Fe$_6$ cluster. For a range of pulse shapes and amplitudes the magnetic response is a decay in the
global spin expectation value, $S^z (t) = \int d^3 r s^z(\vec{r},t)$ . The pulse excites the cluster and the rate of
the spin decay triggered by the excitation is correlated with the total variation of the TDSDFT energy
before and after the pulse (the larger the energy deposited in the cluster, the larger the induced spin-decay
rate). Note, that after the pulse the total energy is conserved, Fig. <ref>(b).
The spatial distribution of the calculated demagnetization is visualized in Fig. <ref>, where we plot
the time and space averaged (along the direction of the symmetry axis of the cluster, $z$) planar distributions of the
temporal variations (with respect to the GS) of the spin-density and the effective magnetic field
[The definitions of the quantities plotted in Fig. <ref> are:
$\overline{\Delta S^z(x,y)} = \frac{1}{T} \sum_i \int_T dt \int_{z_{i,1}}^{z_{i,2}} dz $ $ \SqB{s^z \RnB{x,y,z,t} - s^z \RnB{x,y,z,0}} / \RnB{z_{i,2}-z_{i,1}}$
and $\overline{\Delta B_\mathrm{eff}^z(x,y)}\, =\, \frac{1}{T} \sum_i \int_T dt \int_{z_{i,1}}^{z_{i,2}} dz $ $ \SqB{B_\mathrm{eff}^z \RnB{x,y,z,t} - B_\mathrm{eff}^z \RnB{x,y,z,0}}\,/\,\RnB{z_{i,2}-z_{i,1}}$, where $T$ is the total simulation time and $z_{i,1}, z_{i,2}$ are functions of $(x,y)$ and belong to one of the
non-overlapping atom-centered spheres $\Xi_i$.].
It is notable that the negative variation of $s^z(\vec{r})$ is predominantly localized around the atomic
centers [panel (a)]. Furthermore, the averaged spin-variation distribution difference between analogous
simulations with and without SOC [panel (b)], which approximately represents the global spin loss
in the presence of the SOC, is also localized[The apparent symmetry breaking is due
to the finiteness of the time domain simulation.]. In particular it is more pronounced along the direction
of the bond with the apex atoms, the shortest bond length in the system along the direction of the laser
electric field. The effective magnetic field is very inhomogeneous in the GS. Its temporal variation,
however, shows a spatial correlation with the variation of the spin density. The regions of decrease
of spin density exhibit an increase in $B^z_\mathrm{eff}(\vec{r})$ (note that in the same regions
$B^z_\mathrm{eff}(\vec{r})$ is mostly negative in the GS). Hence, regions of spin decay are associated
with decrease in the absolute value of $B^z_\mathrm{eff}(\vec{r})$.
0.90.9trueFig03(Color online) (a) Evolution of global spin expectation value for different
factors $\alpha$ in front of the SO term in Eq. (<ref>). (b) Same as in panel (a) but having the
GS spin subtracted and $\Delta S^z_\mathrm{tot}$ values multiplied by $(1/\alpha^2)$. Panels (c)
and (d) show the corresponding trajectories of the averaged over the non-overlapping atom-centered
spheres $B^z_\mathrm{eff}(t)$ [see Eq (<ref>)].fig03
As suggested by Eq. (<ref>), the SOC is expected to have a major role in the
spin-decay process. In order to extract such effect we have introduced an artificial scaling factor, $\alpha$,
in front of the SOC term of Eq. (<ref>). Depicted in Fig. <ref> is the effect on the global
spin-variation trajectory of the variation of $\alpha$ from 0 to 4. The rate of spin loss, both pulse-coherent
and post-pulse, is strongly affected by the SOC strength with the limit of $\alpha=0$ (no SOC) resulting in global
spin conservation. In panel (b) we have plotted the same spin trajectories after removing their GS offset and
scalling them by a factor of $1/\alpha^2$. The overlap of the curves demonstrates that in the initial coherent stage the
spin-decay rate scales as the square of the SOC strength.
We now compare the global spin trajectories to those of the $B_\mathrm{eff}^z$ in the vicinity of the
atomic centers. We define a measure of the local variation of the latter as
\begin{equation} \label{eq:Bxcloc}
\Avr{B^z_\mathrm{eff}(t)} = \sum_i \frac{1}{V_{\Xi_i}} \int_{\Xi_i} B_\mathrm{eff}^z (\vec{r},t) d^3 r \,,
\end{equation}
where $\Xi_i$ are non-overlapping atom-centered spheres of radius 0.85 Å for all the quantities plotted
in <ref>(c,d). Although such defined $\Avr{B^z_\mathrm{eff}(t)}$ appears noisy due to spatial grid effects, it does show
a coherent response to the external field pulse and during this stage it is practically independent of the SOC strength.
After the pulse dies out the decrease
in the absolute value of $\Avr{B^z_\mathrm{eff}(t)}$ correlates to the global spin decay in panel (a),
and that is especially notable for higher $\alpha$.
This is related to the fact that practically all the spin loss takes place in the same atomic vicinity regions
where $\Avr{B^z_\mathrm{eff}(t)}$ is defined.
0.90.9trueFig04(Color online) Trajectories for $S^z$ corresponding to the model Hamiltonian of
Eq. (<ref>) in the case of (a) having an initial state with spin up or spin down or (b) different values of the SOC (factor $\alpha$) for an initial spin up. (c) Similarly to Fig. <ref>(b), the latter
trajectories re-scaled by $1/\alpha$. The shaded area is a reference for the temporal profile of $\vec{B}=[0,0,B(t)]$.
The initial orbital momentum state is a linear combination of $l^z=0,1,2$ states.fig04
0.90.9trueFig05(Color online) (a) TDSDFT trajectories of $S^z_{tot}$ with respect of the GS for
three different clusters: Fe$_6$, Ni$_6$ and Co$_6$, all sharing the same Fe$_6$ geometry <cit.>.
(b) Closer view into the coherent part of the trajectories where a parabolic decay [$y=A(t-t_0)^2 +B$] is fitted
for each trajectory (dashed curves). (c) The coefficient in the panel (b) fit $A$ versus an effective atomic
SOC [Eq. (<ref>)] for each material.fig05
The insights drawn from Fig. <ref> for Fe$_6$ suggest that the SOC is key in the demagnetization
process, which in turn takes place in the vicinity of the atomic sites, i.e. where the SOC is the strongest.
Furthermore, we have observed that in the same regions the $B_\mathrm{eff}^z$ also decays rapidly in time,
coherently with the laser field.
As a minimal model for understanding the demagnetization process we propose the following spatially
homogeneous and time-dependent spin Hamiltonian
\begin{equation}\label{eq:modH}
\hat{H}(t) = \lambda \hat{L}\cdot\hat{S} + \vec{B}(t)\cdot\hat{S}\:,
\end{equation}
where $\lambda$ defines the SOC strength and $\vec{B}(t)$ is a time-dependent magnetic field.
This model effectively mimics the local effective spin dynamics at a given point in space arising from the TDSDFT
calculation as described by Eq. (<ref>). The basis set used to expand the wavefunction, solution of
the corresponding time-dependent Schrödinger equation, is given by the eigenstates of $\hat{L}^{z}$ and
$\hat{S}^{z}$, $\ket{l^z, s^z}$. For instance, considering $l=1$ for the orbital momentum quantum number we
can write
\begin{equation}
\ket{\Psi(t)} = \sum_{l^z=-1}^{1}\sum_{s^z=-1/2}^{1/2} c_{l^z,s^z} \ket{l^z, s^z}\:,
\end{equation}
and solve numerically the 6-dimensional Schrödinger equation to obtain the evolution of the spin observables.
The calculated dynamics show that in the absence of SOC there is no spin dynamics regardless
of whether or not the initial state is collinear to $\vec{B}(t)$. This is because the spin and orbital angular momenta
are decoupled.
In contrast when $\lambda\neq 0$, an initial state with $l^z \ne \pm l$ and a step-like variation of $\vec{B}(t)$
(similarly to $\Avr{B^z_\mathrm{eff}(t)}$ in Fig. <ref>) produce a sharp change in the expectation value
of $\hat{S}^z$ (see Fig. <ref>). In particular, for an initial spin-up state we find a decrease of the $S^z$
expectation value, while an initially down spin-state shows an increase in $S^z$. In other words, any change
of the local magnetic field, combined with the SOC, leads to a decrease in the modulus of the
expectation value of the spin of even initially collinear with the field (assumed along the quantization axis) spin-states with $l^z \ne \pm l$
(like the states in the open $d$-shell of the Fe atoms).
In addition, the model reproduces the $\lambda^2$ dependence of the coherent demagnetization rate
observed in the TDSDFT calculations [Fig. <ref>(b)]. In Fig. <ref>(b) and (c) we show that
if the SOC strength is rescaled by a factor $\alpha$ the demagnetization curves get steeper
by a factor $\alpha^2$.
This property can also be demonstrated analytically by looking at the first order term in the perturbative
expansion of the solution $\myket{\Psi(t)}$. For instance, in the case of
$\myket{\Psi(0)} = c_{1}(0) \myket{1,-1/2} + c_{2}(0)\myket{0, 1/2}$, the variation of
expectation value of $\hat{S}^z$ with respect to the ground state reads
\begin{equation}\label{eq:deltaSz}
\Braket{\hat{S}^{z}(t)} - \Braket{\hat{S}^{z}}_{0} = \frac{\lambda^{2}t^{2}}{2}\big( -\frac{\sqrt{2}}{2}c_{1}c_{2} - c_{1}^{2} + c_{2}^{2}\big)\:,
\end{equation}
i.e. it scales as $\lambda^2$.
As a final proof for the $\lambda^2$ dependence of the demagnetization speed, we look at the laser-induced
response of clusters analogous to Fe$_6$ but composed of Co and Ni (we keep the same geometry of Fe$_6$).
We quantify their ionic SOC strength through the following definition
\begin{equation}\label{eq:lambdaeff}
\lambda_\mathrm{eff} = \sum_{l \in \mathrm{occ.}} \frac{n_l}{n_\mathrm{tot}} \int V^\mathrm{SO}_l (\vec{r}) R_l^2(\vec{r}) d^3 r \:,
\end{equation}
where $n_l$ are the KS state occupations with $l$ spanning the valence states (in this case $3s, 3p, 3d$ and $4s$),
$n_\mathrm{tot}= \sum_{l \in \mathrm{occ.}}n_l$, $R_l(\vec{r})$ are the radial pseudo-atomic wavefunctions and
$V^\mathrm{SO}_l(\vec{r}) = \frac{2l}{2l+1}\SqB{V_\mathrm{PP}^{l+1/2}(\vec{r})-V_\mathrm{PP}^{l-1/2}(\vec{r})}$
is the same object as in Eq. <ref> also calculated in the LSDA. By fitting the first few femtoseconds of the
demagnetization curve to a quadratic time decay we extract the demagnetization rate of the three different clusters.
These show a systematic dependence on $\lambda_\mathrm{eff}^2$ [see Fig. <ref>(c)].
In conclusion TD-SDFT calculations obtained with fully relativistic pseudopotentials for transition metal clusters
display ultrafast demagnetization with a leading quadratic time dependence and rates ranging between 4 and 10
$\hbar/$fs$^2$. The phenomenon is then explained in terms of the resulting laser-induced coherent drop of
the effective magnetic field. The latter, combined with the SOC which is the strongest in the vicinity of the atomic centers,
leads to a local decrease of the expectation
value of $\hat{S}^z$. The external electric pulse is therefore only indirectly involved in the demagnetization process,
that must be ascribed to the large variations of the effective magnetic field due to excited spin-polarized currents.
Furthermore, the onset of the demagnetization shows a clear direct dependence
on the ionic SOC properties of the material, scaling quadratically with the SOC strength. Because of the localised nature
of this ultrafast demagnetization mechanism, we believe our findings are valid beyond the cluster systems and may provide a formal
backing to experimental observations like the comparison of Ni and CoPt$_3$ demagnetization rates in Ref. Bigot09.
This work has been funded by the EU FP7 project CRONOS (grant no. 280879). We gratefully acknowledge the
DJEI/DES/SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities
and support and the Trinity Centre for High Performance Computing for technical support.
|
1511.01034
|
Instructions for authors
T.H.E. Editor & G.H. Ostwriter
Department of Astronomy and Space Sciences, Faculty of Arts and Sciences,
Erciyes University, Talas Yolu, 38039, Kayseri, Turkey
Istanbul University, Science Faculty, Department of Astronomy and
Space Sciences, 34119, Üniversite-Istanbul, Turkey, [email protected]
Universidade Federal do Rio Grande do Sul,
Departamento de Astronomia, CP 15051, RS, Porto Alegre 91501-970,
Brazil, [email protected]
We investigate the dynamical evolution of 40 open
clusters (OCs) by means of their astrophysical parameters derived
from field-decontaminated 2MASS photometry. We find a bifurcation
in the planes core radius vs. age and cluster radius vs. age, in
which part of the clusters appear to expand with time probably due
to the presence of stellar black holes while others seem to shrink
due to dynamical relaxation.
Mass functions (MFs) are built for 3$/$4 of the sample (31 OCs),
which are used to search for indications of mass segregation and
external dynamical processes by means of relations among
astrophysical, structural and evolutionary parameters.
We detect a flattening of MF slopes ocurring at the evolutionary
parameters $\tau_{core}\leq 32$ and $\tau_{overall}\leq 30$,
Within the uncertainties involved, the overall MF slopes of 14
out of 31 OCs with $m_{overall} > 500~M_{\odot}$ are consistent with
Kroupa's initial mass function, implying little or no dynamical
evolution for these clusters. The remaining 17 OCs with MF slopes
departing from that of Kroupa show mild/large scale mass
segregation due to dynamical evolution.
§ INTRODUCTION
The internal dynamical processes of open clusters (OCs) are
mass loss during stellar evolution, mass segregation and evaporation
of its stellar content with time. Tidal interactions with the Galaxy's disc and bulge,
as well as collisions with Giant Molecular clouds (hereafter GMCs) are
the main external dynamical effects upon OCs. Because of these dynamical interactions, as
clusters age, their structures are subject to considerable
changes, and may even be dissolved in the Galactic field.
A massive cluster can be dissolved by central
tidal effects in $\approx$ 50 Myr <cit.>. This time
is much shorter than $\sim 1\; Gyr$ found for most OCs
within the Solar circle <cit.>.
Interactions with the galactic disc, the tidal pull of the
Galactic bulge and collisions with GMCs destroy more easily
the poorly-populated OCs, on a time-scale of $10^{8}$ yr ,
particularly inside the Solar circle <cit.>.
A cluster loses low-mass stars from its outer regions into the field by stellar
evaporation. As a result of this mass segregation, low-mass stars are transferred from its
core to the cluster's outskirts while massive stars accumulate in
the core <cit.>. This results in a flat mass function (MF hereafter) in
the core and steep one in the halo.
These external and internal dynamical processes play different
roles, depending on the location of an OC with respect to the
Solar circle: old OCs with Age $>1$ Gyr tend to be concentrated in
the anti-centre, a region with a low density of GMCs <cit.>. Tidal shocks from the Galaxy and from GMCs and observational
incompleteness or biases are responsible for the scarcity of OCs in direction
to the Galactic center <cit.>. Due to absorption and crowding
in regions dominated by disc and bulge stars, the OCs' observational completeness
is decreased. With the effect of tidal interaction, an OC heats and its stars
gain kinetic energy, which leads to an increase in the evaporation rate.
In this paper we have considered 40 OCs with 2MASS JH${K_{s}}$ photometric data, which are
selected in respective to the cluster location and the age (Age$\ge100$ Myr) from WEBDA OC and Dias et al. catalogues <cit.>.
These OCs have been considered to study their dynamical evolution, particularly in dependence of their
location in the Galaxy. We state that our sample is relatively small but our work have the advantage of being
based on a uniform database, in the sense that we determine the parameters following
the same methods, based on the same kind of photometry.
The robust structural parameters have been
derived from high-contrast stellar radial density profiles
following the method of <cit.>, and the ages were derived
from a fit of isochrones to decontaminated colour-magnitude
diagrams of the 2MASS JH${K_{s}}$ photometric data.
As can be seen from the WEBDA database, CCD-based CMDs of these 40 OCs
are also available. We stress that the CMDs presented here go fainter than is available there.
From our sample of young, intermediate and old OCs ($100\; Myr \leq Age \leq 5\; Gyr$),
the relations between the dynamical evolution indicators and cluster radius
(R$_{RDP}$, hereafter), core radius (R$_{core}$, hereafter), mass, mass function slope $\chi$,
mass density $\rho$, evolutionary parameter $\tau$-, and the parameters $(Age, d, R_{GC}, z)$
have been derived and compared with the values given in the
literature. Here, d, R$_{GC}$, and z denote heliocentric,
galactocentric (hereafter R$_{GC}$), and Galactic plane distances,
respectively. Such relations have been studied by <cit.>,
<cit.>,<cit.>, <cit.>, <cit.>,
<cit.>, <cit.>, <cit.>, <cit.>,
<cit.>, and <cit.>. In this paper,
R$_{\odot}$$=7.2\pm0.3$ kpc which is based on the
updated distances of Galactic globular clusters <cit.>, is taken through this paper.
This paper is organised as follows: the selection of the OCs is
presented in Section 2. In Section 3 the 2MASS JH${K_{s}}$
photometry and the field star decontamination algorithm (employed
in the CMD analyses) are given. The derivations of astrophysical
and structural parameters, mass and mass functions, relaxation
time and evolutionary parameter are presented in Sections 4 to 6.
Section 7 is devoted to Results, which
contain the following subsections: 7.1 the relation
between $R_{RDP}$ and $R_{core}$, 7.2 relations of cluster dimensions
with distance and age, 7.3 the relations between $R_{RDP}$ and $Age$
and $R_{core}$ with $Age$, 7.4 the relations $R_{RDP}$ with
$R_{GC}$ and $R_{core}$ with $R_{GC}$, 7.5 the spatial
distribution of the 40 OCs in the Galaxy, 7.6 relations between
the overall mass with ($R_{RDP}$, $R_{core}$) and with $(Age,
R_{GC})$, 7.7 the relations between the mass density with $MF$
slopes, $Age$, $R_{RDP}$ and with $R_{GC}$, 7.8 the relation
between the $MF$ slopes and the evolutionary parameter, and a
comparison with Kroupa's IMF. Conclusions are presented in Section 8.
§ OPEN CLUSTER SAMPLE AND SPATIAL DISTRIBUTION
We applied two criteria to select the OCs for our work from WEBDA
OC and Dias et al. catalogues <cit.>. Namely, the
cluster location in the Galaxy and their ages, see Fig. 1.
In order to study dynamical evolution of middle- and older-age
OCs, 40 OCs with $100\; Myr \leq Age \leq 5\; Gyr$ as a
function of the Galactic location (see Fig. 1, slices I-IV) from
the 2MASS data base are considered.
The location criteria is important because
the longevity/survival rate of the OCs are related to the Galactic
slices inside/outside the Solar circle. Over 40 OCs in WEBDA OC
<cit.> and <cit.> catalogue have been considered. The
OCs which were not appropriate to the decontamination technique
of field stars were eliminated by examining their decontamination
surface density distributions (see sect. 3). Thus the sample size
resulted to be 40 OCs. We are aware that the sample is not large but we
intended that the sample with robust parameters
would be significant to address the dynamical problems mentioned
earlier. From the 40 OCs, we have also studied the relations
between the parameters (Age, d, R$_{GC}$, $z$) and dynamical
indicators (R$_{core}$, R$_{RDP}$, $m$, $\chi$, $\tau$).
[width = 7cm, height = 7cm]Figure1.jpg
Spatial distribution (X,Y) of 40 OCs.
Open triangles and filled circles represent OCs with ages younger than 1 Gyr
and older than 1 Gyr, respectively. The schematic projection of the Galaxy is seen
from the North pole. The Sun's distance to the
Galactic center is taken to be 7.2 kpc of <cit.>.
1148 OCs out of 2000 OCs in <cit.> catalogue, which can be
considered the most representative for our purpose have
age determinations. As can be seen from Table 1, there are 13 OCs
with $200\; Myr \leq Age < 1\; Gyr$ (3.5 %), 26 OCs with $1\;
Gyr \leq Age <5 Gyr$ (14.5 %) and one with $Age \le 200\; Myr$ in
our sample.
Comparison of our cluster sample to the catalogue of
<cit.> for the age data.
Age (Myr) N (This work) N(Dias) Percentage (%)
Age $<$ 200 1 579 0.17
200 $\leq$ Age $<$ 1000 13 373 3
1000 $\leq$ Age $<$ 5000 26 179 15
5000 $\leq$ Age $\leq$10000 - 17 -
Total 40 1148 3
The spatial distribution in (X, Y) plane together spiral arms [(X, Y) is a right handed
Cartesian coordinate system with the Sun on its center,
with the X axis pointing towards the Galactic centre and the Y axis pointing in
the disc rotation direction.] of the 40 OCs is displayed in Fig. 1.
As seen from the Fig. 1, our sample comprises the OCs of four Galactic slices (I-IV).
Note that the number of OCs towards the anti-center in Fig. 1 is larger than the ones toward the Galactic center directions.
Six out of eight OCs with Age $ < 1\, $ Gyr fall in the Galactic anticenter directions,
whereas the remaining two occupy the Galactic center direction.
This is because the OCs in Galactic center directions cannot be observed due to strong absorption,
crowding or were systematically dissolved by different tidal effects such as high frequency of collisions with GMCs <cit.>.
The majority of OCs with Age $\ge$ 1 Gyr lies outside the Solar circle.
From Fig. 1, one readily sees that the number of OCs inside
the Solar radius is biased in direction of the Galactic center. The
reason is that the inner Galaxy clusters cannot be observed because of
strong absorption and crowding, or because they have been dissolved by
a combination of tidal effects. In a good measure the latter is caused
by the expected higher frequency of collisions with GMCs in that direction
<cit.>. From an inspection of Fig. 1, there
are more OCs in the anticentre direction than in the opposite direction,
in agreement with <cit.>, who find that the OCs with Age $\ge$
1.0 Gyr tend to be concentrated in the anticentre, which is a region with
lower density of GMCs. Our sample has small statistics to draw
significant conclusions in that respect. However, statistically, working with a representative
sub-population of the Galactic OCs minimizes the occurrence of biases in the analyses.
Finally, to put the present OC sample in context, in Fig. 2 we compare some observational data
together with fundamental parameters (derived in subsequent sections) with the corresponding
ones found in OC databases. This analysis is also important for checking for the presence of
systematic biases in our sample. For this analysis we use the parameters derived by
<cit.> for 3006 OCs. The advantage of their work is that the parameters
follow from a systematic and uniform analysis. Since <cit.> do not
provide cluster mass, we take such values from <cit.>, although for a
smaller number of OCs, 236.
[width = 7cm, height = 11cm]Figure2.eps
Normalized distribution functions of our OC sample (circles)
compared to those of <cit.> and <cit.> (solid
Our analysis compares distribution functions of the several parameters between both
sets, as is seen from Fig. 2. Uncertainties in the parameters have been incorporated
into the respective distribution function. And, since the samples differ significantly in the number of OCs, the distribution
functions have been scaled to provide the best visual comparison between both.
The top panels of Fig. 2 show how the OCs distribute with respect to the Galactic longitude
(left) and latitude (right). Clearly, most of our sample corresponds to clusters directed
towards the 2nd and 3rd Galactic quadrants. Regarding Galactic latitude, our sample tend
to avoid the plane. In terms of distance from the Sun (middle-left), our sample is
somewhat consistent with that of <cit.>, particularly for distances
in excess of 2 kpc. The same applies to the core radius (middle-right) for R$_{core}$ $>$ 1pc;
below this threshold, our sample appears to contain a lower fraction of OCs than that
in <cit.>. Regarding mass, both distributions have a similar shape,
but with a shift of $\approx$ 0.7 dex between the peaks, which suggests that our sample
occupies the high-mass wing observed in <cit.> distribution. The age
distributions also have similar shapes, with our sample consisting essentially of clusters
older than 100 Myrs. Thus, we can conclude that the 40 OCs dealt with here are a
representative sub-sample of the Galactic OC population, with no systematic biases.
§ THE 2MASS PHOTOMETRY AND THE FIELD-STAR DECONTAMINATION
We have used JH${K_{s}}$ photometry of 2MASS[The Two Micron All Sky Survey Catalogue,
available at http://www.ipac.caltech.edu/2mass/releases/allsky/]
to find the apparently cluster members of 40 OCs <cit.>. We used
to extract the near infrared (NIR) (J, H, and ${K_{s}}$ 2MASS) photometry for a large-area centered
on each cluster, which is essential to build the RDPs with a high contrast relative to
the background, and for a better field star decontamination. 2MASS provides an all-sky
coverage with the spatial and photometric uniformity required for high star count statistics.
For the photometric constraint, the 2MASS magnitude extractions have been restricted
to stars with errors smaller than 0.2 mag in JH${K_{s}}$ magnitudes.
The extraction radii of 40 OCs have been chosen by visual inspection on the DSS-I
image[Extracted from the Canadian Astronomy Data Centre (CADC),
at http://ledas-www.star.le.ac.uk/DSSimage/], and taking into account the RDP,
in the sense that the profile must become relatively stable in the outer region.
As an example we show only the DSS-I image of Pismis 19 in Fig. 3.
The technique used here for determining the cluster members of the 40
OCs is known as the field star decontamination procedure coupled to
the 2MASS JH${K_{s}}$ photometry, and it was succesfully used by <cit.> and more recently by <cit.>, This decontamination procedure was
applied to the 40 OCs discussed here. This technique samples photometric
properties of the stars in a neighbour comparison field considered free of
cluster stars to (statistically) remove the contamining field stars from
the cluster stars with help of the colour magnitude diagrams (CMD).
[width = 7cm, height = 7cm]Figure3.jpg
The image of Pismis 19 in DSS-I $25\,^\prime$x $25\,^\prime$.
Firstly, the stellar surface densities $\sigma(stars\,\rm arcmin^{-2})$
and the surface isopleths of both the raw and decontamination data of
40 OCs, computed for a mesh size of $3^{\prime}\times3^{\prime}$ and
centred on the galactic coordinates of Table 2 (see Supplementary
material section) If necessary, we have re-determined them in this work
(see below). Here, isopleth is star density map. These maps have been used to maximise the contrast of the
cluster against the background. In Figs. 4 and 5 we show the result for
Pismis 19 as an example. The central stellar density excesses are
significant in the decontamination surface-density distributions, as is
seen in Fig. 5 for Pismis 19.
Literature (left columns) and presently optimised (right columns) Equatorial and Galactic coordinates of 40 OCs.
Cluster ${\alpha}$(2000) ${\delta}$(2000) l b ${\alpha}$(2000) ${\delta}$(2000) l b
(h m s) ($^{o}$ $'$ $''$) ( $^{o}$ ) ( $^{o}$ ) (h m s) ($^{o}$ $'$ $''$) ( $^{o}$ ) ( $^{o}$ )
NGC 436 01 15 58 58 48 42 126.11 -3.91 01 15 58 58 48 42 126.11 -3.91
King 5 03 14 45 52 41 12 143.78 -4.29 03 14 45 52 41 12 143.78 -4.29
NGC 1513 04 09 57 49 30 54 152.59 -1.57 04 09 50 49 31 17 152.57 -1.58
Be 15 05 02 06 44 30 43 162.26 1.62 05 02 06 44 30 43 162.26 1.62
NGC 1798 05 11 39 47 41 30 160.70 4.85 05 11 39 47 41 30 160.70 4.85
Be 17 05 20 36 30 36 00 175.65 -3.65 05 20 38 30 34 28 175.67 -3.66
NGC 1907 05 28 05 35 19 30 172.62 0.31 05 28 09 35 18 20 172.64 0.31
NGC 2112 05 53 45 00 24 36 205.87 -12.62 05 53 51 00 25 44 205.87 -12.58
Koposov 12 06 00 56 35 16 36 176.16 6.00 06 00 56 35 16 36 176.16 6.00
NGC 2158 06 07 25 24 05 48 186.63 1.78 06 07 30 24 05 50 186.64 1.80
Koposov 53 06 08 56 26 15 49 184.90 3.13 06 08 56 26 15 49 184.90 3.13
NGC 2194 06 13 45 12 48 24 197.25 -2.35 06 13 45 12 48 24 197.25 -2.35
NGC 2192 06 15 17 39 51 18 173.42 10.65 06 15 22 39 51 06 173.42 10.67
NGC 2243 06 29 34 -31 17 00 239.48 -18.01 06 29 34 -31 17 00 239.48 -18.01
Trumpler 5 06 36 42 09 26 00 202.86 1.05 06 36 36 09 25 21 202.86 1.02
Col 110 06 38 24 02 01 00 209.65 -1.98 06 38 35 02 01 30 209.66 -1.93
NGC 2262 06 39 38 01 08 36 210.57 -2.10 06 39 38 01 08 36 210.57 -2.10
NGC 2286 06 47 40 -03 08 54 215.31 -2.27 06 47 43 -03 10 20 215.33 -2.27
NGC 2309 06 56 03 -07 10 30 219.84 -2.24 06 56 02 -07 11 05 219.85 -2.25
Tombaugh 2 07 03 05 -20 49 00 232.83 -6.88 07 03 05 -20 49 00 232.83 -6.88
Be 36 07 16 06 -13 06 00 227.38 -0.59 07 16 24 -13 11 23 227.49 -0.56
Haffner 8 07 23 24 -12 20 00 227.53 1.34 07 23 09 -12 16 12 227.45 1.32
Mel 71 07 37 30 -12 04 00 228.95 4.50 07 37 30 -12 04 00 228.95 4.50
NGC 2425 07 38 22 -14 52 54 231.52 3.31 07 38 22 -14 52 54 231.52 3.31
NGC 2506 08 00 01 -10 46 12 230.56 9.93 07 59 59 -10 45 28 230.55 9.93
Pismis 3 08 31 22 -38 39 00 257.86 0.50 08 31 16 -38 39 02 257.85 0.48
NGC 2660 08 42 38 -47 12 00 265.93 -3.01 08 42 38 -47 12 00 265.93 -3.01
NGC 3680 11 25 38 -43 14 36 286.76 16.92 11 25 35 -43 15 11 286.76 16.91
Ru 96 11 50 38 -62 08 23 295.89 -0.10 11 50 37 -62 09 04 295.89 -0.11
Ru 105 12 34 15 -61 34 11 300.88 1.24 12 34 12 -61 33 00 300.88 1.25
Trumpler 20 12 39 34 -60 37 00 301.48 2.22 12 39 34 -60 37 00 301.48 2.22
Pismis 19 14 30 40 -60 53 00 314.71 -0.30 14 30 40 -60 53 00 314.71 -0.30
NGC 6134 16 27 46 -49 09 06 334.92 -0.20 16 27 46 -49 09 06 334.92 -0.20
IC 4651 17 24 49 -49 56 00 340.09 -7.91 17 24 46 -49 55 06 340.10 -7.89
NGC 6802 19 30 35 20 15 42 55.33 0.92 19 30 33 20 15 48 55.32 0.92
NGC 6819 19 41 18 40 11 12 73.98 8.48 19 41 18 40 11 12 73.98 8.48
Be 89 20 24 36 46 03 00 83.16 4.82 20 24 30 46 02 53 83.15 4.84
NGC 6939 20 31 30 60 39 42 95.90 12.30 20 31 30 60 39 42 95.90 12.30
NGC 7142 21 45 09 65 46 30 105.35 9.48 21 45 12 65 47 43 105.36 9.50
NGC 7789 23 57 24 56 42 30 115.53 -5.39 23 57 24 56 42 30 115.53 -5.39
The stellar radial density profiles (RDP) were derived from the
isopleth surfaces of each cluster, the coordinates were checked and
the cluster radii were determined (e.g. Table 4). The residual
background level of each RDP corresponds to the average number of
CM-filtered stars measured in the comparison field. A wide external
ring $(\Delta R=13'-70')$ centered in the cluster (Col. 11 of
Tables 4 and S4) has been considered to eliminate field stars of the
40 OCs. Stars within the cluster radii have been considered to be
probable members.
The stellar radial density profile (RDP) of each cluster,
built based on the JH${K_{s}}$ photometry extracted with the
WEBDA[www.univie.ac.at/WEBDA-Mermilliod & Paunzen
(2003)] coordinates are displayed in Table 2 and have been
computed to check cluster centering. In some cases the RDP
built with the original cluster coordinates presented a dip
at the center. Then, new central coordinates are searched
after field star decontamination to maximise the star counts
in the innermost RDP bin. From these RDPs, the cluster radii
of 40 OCs are determined (Table 4). The stellar RDP is the
projected number of stars per area around the cluster centre.
To avoid oversampling near the centre and undersampling for
large radii, the RDPs are built by counting stars in concentric
rings of increasing width with distance to the centre. The number
and width of rings are optimised so that the resulting RDPs have
adequate spatial resolution with moderate $1\sigma$ Poission
errors. The residual background level of each RDP corresponds
to the average number of CM-filtered stars measured in the
comparison field.
[width = 7cm, height = 7cm]Figure4.jpg
[width = 7cm, height = 7cm]Figure5.jpg
For observed (raw) photometry, top panel: stellar surface$-$density
$\sigma (stars\,\rm arcmin^{-2}$) of Pismis 19, computed for a mesh size of
$3^\prime\times3^\prime$, centred on the coordinates in Table 2.
Bottom panel : The corresponding isopleth surface.
[width = 6.5cm, height = 7cm]Figure6.jpg
[width = 6.5cm, height = 7cm]Figure7.jpg
For decontaminated photometry, top panel: stellar surface$-$density
$\sigma (stars\,\rm arcmin^{-2}$) of Pismis 19, computed for a mesh size of
$3^\prime\times3^\prime$, centred on the coordinates in Table 2.
Bottom panel : The corresponding isopleth surface.
As <cit.> noted, RDPs of OCs built based on
the WEBDA coordinates usually show a dip in the inner RDP
region when a mismatch between the ”true” and catalogue
coordinates exists. For this reason, new central coordinates
of these clusters have been searched to maximise the star
counts at the innermost RDP bin. Then, the 2MASS photometry
was extracted again, but now centered on the optimized cluster
coordinates. As a representative, the optimised central
coordinate of Pismis 19 is displayed in Fig. 3 as small circle,
and given in the right section of Table 2.
To have the intrinsic morphology of the clusters in the CMD,
as explained above, the statistical field star decontamination
procedure of <cit.> is used. This procedure is based on
the relative stars densities per sky area in a cluster region and
on a neighboring offset field. It divides the full range of magnitudes
and colours of a CMD into the cell dimensions of $\Delta{J}=1.0$,
and $\Delta(J-H)={\Delta(J-K_{s})}=0.15$. These dimensions are
adequate to allow for sufficient star counts in individual cells
and preserve the intrinsic morphology of the evolutionary sequences.
<cit.> showed that the field star decontamination procedure
with 2MASS JH${K_{s}}$ photometry is efficient isolating those
stars with a high probability of being cluster members. More
details on the algorithm can be found in <cit.>,
<cit.>, and <cit.>.
By following the field decontamination technique which is briefly
explained above,the probable cluster members of the 40 OCs have been
identified for further analysis.
Derived fundamental astrophysical parameters from 2MASS JH${K_{s}}$ photometry of 40 OCs.
Cluster Z E(J-H) E(B-V) Age(Gyr) (m-M)j d(kpc) R$_{GC}$(kpc)
NGC 436 0.019 0.13$\pm$0.03 0.42$\pm$0.10 0.4$\pm$0.1 12.54$\pm$0.31 3.22$\pm$0.46 9.48$\pm$0.28
King 5 0.0105 0.26$\pm$0.05 0.83$\pm$0.16 1.0$\pm$0.2 11.53$\pm$0.24 2.03$\pm$0.23 8.93$\pm$0.18
NGC 1513 0.019 0.23$\pm$0.02 0.74$\pm$0.06 0.1$\pm$0.02 10.37$\pm$0.28 1.18$\pm$0.15 8.29$\pm$0.13
Be 15 0.019 0.27$\pm$0.03 0.86$\pm$0.10 0.5$\pm$0.1 12.45$\pm$0.31 3.10$\pm$0.44 10.21$\pm$0.42
NGC 1798 0.0105 0.16$\pm$0.04 0.51$\pm$0.13 1.5$\pm$0.3 13.51$\pm$0.26 5.03$\pm$0.59 12.07$\pm$0.55
Be 17 0.006 0.26$\pm$0.04 0.83$\pm$0.13 5.0$\pm$0.5 11.93$\pm$0.29 2.43$\pm$0.33 9.65$\pm$0.33
NGC 1907 0.019 0.18$\pm$0.03 0.58$\pm$0.10 0.4$\pm$0.1 11.45$\pm$0.26 1.95$\pm$0.24 9.16$\pm$0.23
NGC 2112 0.019 0.20$\pm$0.04 0.64$\pm$0.13 2.0$\pm$0.3 10.15$\pm$0.23 1.07$\pm$0.11 8.18$\pm$0.10
Koposov 12 0.0105 0.07$\pm$0.02 0.22$\pm$0.06 1.8$\pm$0.2 11.56$\pm$0.18 2.05$\pm$0.17 9.26$\pm$0.17
NGC 2158 0.019 0.05$\pm$0.01 0.16$\pm$0.03 2.5$\pm$0.3 13.21$\pm$0.10 4.39$\pm$0.21 11.59$\pm$0.21
Koposov 53 0.019 0.01$\pm$0.00 0.03$\pm$0.02 1.0$\pm$0.1 13.05$\pm$0.18 4.08$\pm$0.34 11.28$\pm$0.34
NGC 2194 0.019 0.13$\pm$0.04 0.42$\pm$0.13 0.8$\pm$0.2 11.87$\pm$0.27 2.37$\pm$0.30 9.51$\pm$0.28
NGC 2192 0.019 0.01$\pm$0.00 0.03$\pm$0.00 1.3$\pm$0.1 13.12$\pm$0.15 4.21$\pm$0.29 11.37$\pm$0.28
NGC 2243 0.0105 0.01$\pm$0.00 0.03$\pm$0.00 2.0$\pm$0.2 13.37$\pm$0.12 4.73$\pm$0.26 10.36$\pm$0.14
Trumpler 5 0.006 0.24$\pm$0.05 0.77$\pm$0.16 3.0$\pm$0.3 12.19$\pm$0.29 2.74$\pm$0.36 9.80$\pm$0.33
Col 110 0.019 0.06$\pm$0.01 0.19$\pm$0.03 3.0$\pm$0.2 11.93$\pm$0.15 2.44$\pm$0.17 9.41$\pm$0.15
NGC 2262 0.0105 0.11$\pm$0.01 0.35$\pm$0.03 1.3$\pm$0.1 12.36$\pm$0.30 2.96$\pm$0.41 9.88$\pm$0.35
NGC 2286 0.019 0.03$\pm$0.00 0.10$\pm$0.02 1.0$\pm$0.2 11.82$\pm$0.30 2.31$\pm$0.32 9.20$\pm$0.26
NGC 2309 0.019 0.16$\pm$0.02 0.51$\pm$0.06 0.5$\pm$0.1 12.41$\pm$0.21 3.03$\pm$0.29 9.74$\pm$0.22
Tombaugh 2 0.019 0.35$\pm$0.05 1.12$\pm$0.16 3.0$\pm$0.3 10.43$\pm$0.24 1.22$\pm$0.14 8.01$\pm$0.08
Be 36 0.019 0.12$\pm$0.02 0.38$\pm$0.06 3.0$\pm$1.0 13.67$\pm$0.16 5.42$\pm$0.40 11.59$\pm$0.27
Haffner 8 0.006 0.06$\pm$0.02 0.19$\pm$0.06 1.0$\pm$0.1 11.98$\pm$0.16 2.49$\pm$0.18 9.09$\pm$0.12
Mel 71 0.019 0.01$\pm$0.00 0.03$\pm$0.02 1.5$\pm$0.2 11.54$\pm$0.15 2.03$\pm$0.14 8.69$\pm$0.09
NGC 2425 0.019 0.10$\pm$0.02 0.32$\pm$0.06 3.2$\pm$0.5 12.27$\pm$0.26 2.85$\pm$0.34 9.26$\pm$0.21
NGC 2506 0.006 0.03$\pm$0.01 0.10$\pm$0.03 2.0$\pm$0.3 12.27$\pm$0.20 2.84$\pm$0.26 9.27$\pm$0.17
Pismis 3 0.006 0.33$\pm$0.02 1.06$\pm$0.06 3.2$\pm$0.2 11.19$\pm$0.11 1.73$\pm$0.09 7.77$\pm$0.03
NGC 2660 0.019 0.13$\pm$0.03 0.42$\pm$0.10 1.5$\pm$0.3 11.89$\pm$0.17 2.39$\pm$0.19 7.76$\pm$0.06
NGC 3680 0.019 0.05$\pm$0.01 0.16$\pm$0.03 1.5$\pm$0.2 10.16$\pm$0.10 1.08$\pm$0.05 7.00$\pm$0.02
Ru 96 0.019 0.07$\pm$0.01 0.22$\pm$0.03 1.0$\pm$0.1 12.01$\pm$0.25 2.52$\pm$0.29 6.53$\pm$0.15
Ru 105 0.019 0.05$\pm$0.01 0.16$\pm$0.03 1.0$\pm$0.4 11.56$\pm$0.20 2.05$\pm$0.19 6.41$\pm$0.10
Trumpler 20 0.019 0.10$\pm$0.03 0.32$\pm$0.10 1.5$\pm$0.5 12.52$\pm$0.31 3.20$\pm$0.46 6.19$\pm$0.27
Pismis 19 0.019 0.41$\pm$0.03 1.31$\pm$0.10 0.8$\pm$0.1 11.42$\pm$0.38 1.92$\pm$0.34 6.02$\pm$0.24
NGC 6134 0.019 0.10$\pm$0.01 0.32$\pm$0.03 1.5$\pm$0.1 10.22$\pm$0.12 1.11$\pm$0.06 6.23$\pm$0.06
IC 4651 0.019 0.02$\pm$0.00 0.06$\pm$0.02 2.5$\pm$0.3 9.64$\pm$0.20 0.85$\pm$0.08 6.44$\pm$0.07
NGC 6802 0.019 0.23$\pm$0.03 0.74$\pm$0.10 0.9$\pm$0.1 11.77$\pm$0.31 2.25$\pm$0.32 6.22$\pm$0.19
NGC 6819 0.019 0.02$\pm$0.00 0.06$\pm$0.02 2.5$\pm$0.5 11.84$\pm$0.15 2.34$\pm$0.16 6.96$\pm$0.06
Be 89 0.019 0.23$\pm$0.02 0.74$\pm$0.06 2.0$\pm$0.5 12.37$\pm$0.21 2.97$\pm$0.28 7.47$\pm$0.11
NGC 6939 0.019 0.12$\pm$0.03 0.38$\pm$0.10 2.0$\pm$0.3 11.27$\pm$0.31 1.79$\pm$0.26 7.61$\pm$0.06
NGC 7142 0.019 0.13$\pm$0.03 0.42$\pm$0.10 2.5$\pm$0.3 12.04$\pm$0.22 2.56$\pm$0.25 8.27$\pm$0.10
NGC 7789 0.0105 0.08$\pm$0.02 0.26$\pm$0.06 1.8$\pm$0.2 11.23$\pm$0.21 1.76$\pm$0.17 8.13$\pm$0.08
§ ASTROPHYSICAL PARAMETERS
We have derived the fundamental parameters of 40 OCs using the
decontaminated $(J, J-H)$ CMDs (see Figs. S5-S9 in the
supplementary material) eye-fitted with Padova isochrones
Since the spectroscopic metal abundances [Fe/H]$_{spec}$ are only available for 21 (Col. 7, Table 4) out of 40 OCs,
we have considered the abundances of $Z= +0.019$ ([Fe/H]=0), $Z= +0.0105$ ([Fe/H]=$-$0.25), and $Z=
+0.006$ ([Fe/H]=$-$0.50), respectively. In the sense OCs need to be uniformly and homogeneously analysed.
M08 isochrones for three Z abundances were fitted to the $(J, J-H)$ CMDs of
each of the 40 OCs. The most appropriate $Z$ fit solution on the CMDs
has been made by eye. Accordingly, the M08 isochrones of $Z=
+0.019$ for 29 OCs, $Z= +0.0105$ for six OCs, and $Z= +0.006$
for five OCs, respectively, have provided us good fits for
reddening, age and distance modulus.
As an example, such $(J, J-H)$ CMDs have been
displayed in Figs. 6(a)-(c) for Pismis 19, for three Z abundances.
The shaded areas in the panels are the
colour-magnitude filters which follow the distribution of the
decontaminated star sequences in the CMDs, or stars comprised
in the shaded area are considered probable members.
These filters are wide enough to accommodate the colour distributions of main sequence and evolved stars
of the clusters, allowing 1 $\sigma$ photometric uncertainties.
The fitted 0.8 Gyr isochrone of $Z= +0.019$ for Pismis 19 in
panel (a) provides a good solution. As can be seen from Fig. 6(a), the
M08 isochrone fits well the main sequence (MS), turn$-$off
(TO) and Red Giant/Red Clump (RG/RC) regions on the CMD of Pismis 19. Due to the
presence of binaries, the M08 isochrones have been shifted to
the left and below of the main sequence in Figs. 6(a)$-$(c),
and all CMDs of the 40 OCs are presented in Figs. S5$-$S9 as
supplementary material.
The reddening, distance modulus (i.e. distance), age and the appropriate $Z$
abundances were derived this way for all 40 OCs
of our sample. These astrophysical parameters together with their uncertainties are presented in Table 3.
However, the reddening is degenerate with the metallicity. For this, we have determined E(B-V),
d (pc), Age (Gyr) of 21 OCs (Table 4) for three Z abundances.
The E(B-V) and d (pc) values (Table 4) of three Z abundances are reasonably
close to our original ones (Table 3) within the uncertainties.
The age values (Col. 6, Table 4) derived from three Z values are the same.
As stated by <cit.>, any metallicity for the range of $+0.006\leq Z
\leq+0.019$ would produce acceptable solutions for the
astrophysical parameters, due to the filters of 2MASS.
Our derived ages here are almost robust enough to allow inferences about cluster evolution.
For this, NGC 2286 (Fig. 7) is presented as an example. The 0.8 Gyr (blue line), 1 Gyr (solid black line),
and 1.2 Gyr (red line) isochrones of $Z= +0.019$ for NGC 2286
are fitted to CMD of the cluster. As is seen from Fig. 7, 1$\pm$0.2 Gyr isochrone (solid line) fit well the main sequence (MS), turn$-$off
(TO) and Red Giant/Red Clump (RG/RC) regions on the CMD of the cluster.
The uncertainties in our derived ages of 40 OCs are in the level of $\pm$0.02$-$0.5 Gyr (Table 3), except for Be 36 ($\pm$1 Gyr).
JHK photometry is unsensitive to metallicity, in opposition
to optical photometry, where the blue (B) and principally the
ultraviolet (U) are sensitive to the photospheric metal lines,
reaching its maximum blanketing effect by SpT F5. For later
than SpT = G2 it becomes too fuzzy to disentagle it from the
molecular lines. On the other hand, metallicity affects
significantly the distance and the age of a cluster, i.e. the
less Z is, the shorter the distance and larger the age.
[width = 4.5cm, height = 5.5cm]Figure8.jpg
[width = 4.5cm, height = 5.5cm]Figure9.jpg
[width = 4.5cm, height = 5.5cm]Figure10.jpg
Observed decontaminated $J\times(J-H)$ CMDs
extracted from the region of $R=11'.03$ for Pismis 19.
The solid lines in the panels represent
the fitted 0.8 Gyr Padova isochrones
for Z$=$+0.019 (solar), Z$=$+0.0105, and Z$=$+0.006, respectively.
The CMD filter used to isolate cluster
MS/evolved stars is shown with the shaded area.
[width = 4.5cm, height = 5.5cm]isochrone.jpg
Observed decontaminated $J\times(J-H)$ CMD
of NGC 2286. The solid lines in the panels represent
the fitted 0.8 Gyr (blue line), 1 Gyr (solid black line),
and 1.2 Gyr (red line) isochrones of $Z= +0.019$.
The CMD filter used to isolate cluster
MS/evolved stars is shown with the shaded area.
Z, E(B-V), d (pc), Age (Gyr) values of 21 OCs with [Fe/H]$_{spec}$. E(B-V) values are listed in Col. 3 for three Z abundances of 21 OCs in our sample.
[Fe/H]$_{iso}$ values in Col. 4 are converted from the expression $Z = Z_\odot \cdot 10^{[Fe/H]}$.
The solar abundance value is taken as $Z_\odot = +0.019$. Ages are given in Col. 6.
[Fe/H]$_{spec}$ values together with literature are listed in Col. 7-8.
Cluster Z E(B-V) [Fe/H]$_{iso}$ d (kpc) Age(Gyr) [Fe/H]$_{spec}$ Reference
1cTrumpler 5 0.019 0.54$\pm$0.13 3.13$\pm$0.39 3
1c 0.0105 0.70$\pm$0.16 2.87$\pm$0.38 3
1c 0.006 0.77$\pm$0.16 -0.50 2.74$\pm$0.36 3 -0.36 Carrera et al. 2007
1cNGC 2158 0.019 0.16$\pm$0.03 0 4.39$\pm$0.21 2.5 -0.28 Jacobson et al. 2011
1c 0.0105 0.29$\pm$0.03 3.98$\pm$0.19 2.5
1c 0.006 0.38$\pm$0.03 3.75$\pm$0.18 2.5
1cCol 110 0.019 0.19$\pm$0.03 0 2.44$\pm$0.17 3 -0.01 Carrera et al. 2007
1c 0.0105 0.29$\pm$0.06 2.29$\pm$0.17 3
1c 0.006 0.42$\pm$0.06 2.03$\pm$0.15 3
1cNGC6134 0.019 0.32$\pm$0.03 0 1.11$\pm$0.06 1.5 0.12 Smiljanic et al.2009
1c 0.0105 0.48$\pm$0.06 0.97$\pm$0.07 1.5
1c 0.006 0.64$\pm$0.10 0.85$\pm$0.07 1.5
1cNGC2425 0.019 0.32$\pm$0.06 0 2.85$\pm$0.34 3.2 -0.15 Jacobson et al. 2011
1c 0.0105 0.42$\pm$0.10 2.74$\pm$0.33 3.2
1c 0.006 0.51$\pm$0.16 2.64$\pm$0.35 3.2
1cTrumpler 20 0.019 0.32$\pm$0.10 0 3.20$\pm$0.46 1.5 0.09 Carraro et al. 2014
1c 0.0105 0.48$\pm$0.10 2.80$\pm$0.40 1.5
1c 0.006 0.64$\pm$0.10 2.45$\pm$0.35 1.5
1cNGC 2112 0.019 0.64$\pm$0.13 0 1.07$\pm$0.11 2 -0.10 Brown et al. 1996
1c 0.0105 0.77$\pm$0.06 0.87$\pm$0.12 2
1c 0.006 0.90$\pm$0.06 0.77$\pm$0.09 2
1cMel 71 0.019 0.03$\pm$0.02 0 2.03$\pm$0.14 1.5 -0.30 Brown et al. 1996
1c 0.0105 0.16$\pm$0.06 1.93$\pm$0.14 1.5
1c 0.006 0.29$\pm$0.06 1.92$\pm$0.17 1.5
1cNGC 7789 0.019 0.19$\pm$0.03 1.81$\pm$0.17 1.8
1c 0.0105 0.26$\pm$0.06 -0.25 1.76$\pm$0.17 1.8 0.02 Jacobson et al. 2011
1c 0.006 0.32$\pm$0.06 1.72$\pm$0.19 1.8
1cNGC 3680 0.019 0.16$\pm$0.03 0 1.08$\pm$0.05 1.5 0.04 Smiljanic et al.2009
1c 0.0105 0.32$\pm$0.06 0.95$\pm$0.06 1.5
1c 0.006 0.48$\pm$0.10 0.85$\pm$0.07 1.5
1cIC 4651 0.019 0.06$\pm$0.02 0 0.85$\pm$0.08 2.5 0.10 Pasquini et al. 2004
1c 0.0105 0.16$\pm$0.03 0.78$\pm$0.08 2.5
1c 0.006 0.26$\pm$0.06 0.72$\pm$0.08 2.5
1cNGC 6819 0.019 0.06$\pm$0.02 0 2.34$\pm$0.16 2.5 0.09 Bragaglia et al.2001
1c 0.0105 0.16$\pm$0.03 2.15$\pm$0.18 2.5
1c 0.006 0.32$\pm$0.10 1.84$\pm$0.15 2.5
1cNGC 1798 0.019 0.32$\pm$0.06 5.69$\pm$0.54 1.5
1c 0.0105 0.51$\pm$0.13 -0.25 5.03$\pm$0.59 1.5 -0.12 Carrera 2012
1c 0.006 0.7$\pm$0.16 4.45$\pm$0.58 1.5
1cNGC 2243 0.019 0.005$\pm$0.005 5.01$\pm$0.23 2
1c 0.0105 0.03$\pm$0.005 -0.25 4.73$\pm$0.26 2 -0.48 Gratton et al. 1994
1c 0.006 0.16$\pm$0.03 4.49$\pm$0.25 2
1cNGC 6939 0.019 0.38$\pm$0.10 0 1.79$\pm$0.26 2 0 Jacobson et al. 2007
1c 0.0105 0.45$\pm$0.13 1.64$\pm$0.26 2
1c 0.006 0.58$\pm$0.19 1.50$\pm$0.26 2
1cNGC 7142 0.019 0.42$\pm$0.10 0 2.56$\pm$0.25 2.5 0.08 Jacobson et al. 2008
1c 0.0105 0.54$\pm$0.13 2.32$\pm$0.26 2.5
1c 0.006 0.67$\pm$0.16 2.11$\pm$0.25 2.5
1cNGC 2194 0.019 0.42$\pm$0.13 0 2.37$\pm$0.30 0.8 -0.08 Jacobson et al. 2011
1c 0.0105 0.51$\pm$0.16 2.15$\pm$0.26 0.8
1c 0.006 0.61$\pm$0.16 1.97$\pm$0.24 0.8
1cNGC 2660 0.019 0.42$\pm$0.10 0 2.39$\pm$0.19 1.5 0.04 Bragaglia et al.2008
1c 0.0105 0.51$\pm$0.13 2.20$\pm$0.19 1.5
1c 0.006 0.67$\pm$0.16 1.97$\pm$0.21 1.5
1cBe 17 0.019 0.64$\pm$0.13 3.02$\pm$0.38 5
1c 0.0105 0.74$\pm$0.13 2.71$\pm$0.40 5
1c 0.006 0.83$\pm$0.13 -0.50 2.43$\pm$0.33 5 -0.10 Friel et al. 2005
1cTombaugh 2 0.019 1.12$\pm$0.16 0 1.22$\pm$0.14 3 -0.45 Brown et al. 1996
1c 0.0105 1.22$\pm$0.19 1.18$\pm$0.14 3
1c 0.006 1.31$\pm$0.19 1.13$\pm$0.18 3
1cNGC 2506 0.019 0.02$\pm$0.005 3.07$\pm$0.28 2
1c 0.0105 0.06$\pm$0.03 2.94$\pm$0.27 2
1c 0.006 0.10$\pm$0.03 -0.50 2.84$\pm$0.26 2 -0.20 Carretta et al. 2004
The reddenings $E(J-H)$ (Col. 3 in Table 3) of the 40 OCs
were derived from the CMD diagrams. These are converted to $E(B-V)$ (Col. 4 in Table 3) with
the extinction law $A_{J}/{A_{V}}=0.276,\, A_{H}/{A_{V}}=0.176,\,
A_{K_{s}}/{A_{V}}=0.118, A_{J}=2.76\times{E(J-H)}$, and $E(J-H)=0.33
\times{ E(B-V)}$ <cit.>, assuming a constant total-to-selective
absorption ratio $R_{V}=3.1$. The distance
moduli of the clusters have been derived and listed in Col. 6 of
Table 3. The estimated heliocentric $d~(kpc)$ and its corresponding
galactocentric $R_{GC}$ (kpc) distances are given in
Cols. 7$-$8, respectively. When estimating the $R_{GC}$ distances,
we adopted the galactocentric distance of the Sun as $R_{\odot}=7.2\pm
0.3$ kpc of <cit.>.
The errors in E(J-H), hence in colour excess E(B-V), distance moduli
and ages, given in Table 3 have been estimated as follows:
* The uncertainties of E(J-H) were estimated moving the M08
isochrones up and down, back and forward and in direction of the
reddening vector in the colour-magnitude diagram $(J,J-H)$ until a
good fit with the observed MS, TO, the subgiant branch (SG), RG/RC sequences were
* The uncertainties of distance moduli in Table 3 stem to a lesser
degree from the photometric errors and fitting the appropriate isochrone
to the observational data points in the CMDs. A larger uncertainty, up
to 2 mag in the distance moduli, originates by the assumption of the
metallicity: for a larger Z the OCs are more distant and metal poor
stars are nearer.
* For the uncertainties in the age estimates, see those of the
distance moduli. Again, metal-rich stars are younger than the metal-
poor ones.
The precision of the parameters depends on the scatter
of the data points in the CMDs. The uncertainties of distance moduli
in Table 3 stem from fitting the appropriate isochrone to the observation
in the CMDs, by taking into the uncertainties of the photometric data.
The uncertainties of distance moduli of 40 OCs are at the level of
0.10$-$0.31. The uncertainty of age is obtained from fitting the M08
isochrone with appropriate heavy element to the CMDs. In this regard,
the uncertainty of the age depends on the uncertainties of E(J-H) and
distance moduli of 40 OCs. The uncertainties of the ages of 40 OCs in
Table 3 fall in the range of 0.02$-$1.0 Gyr.
The relations of $E(B$–$V)$ versus Galactic longitude $\textit
l^{\circ}$ and $E(B$–$V)$ versus Galactic latitude $\it b^{\circ}$
as a function of the cluster distances, are displayed in Figs. 8(a)
and (b), respectively. In Figs. 8 open and filled circles show the
$d=[0,~2.1]$ kpc and $d=(2.1,~5.42]$ kpc subsets, respectively.
The reddenings of the OCs in the anticentre directions have
$0.03 \leq E(B-V) \leq 1.31$. From panel (a), the bulk of the 40
clusters lies within $|b|\leq 5^{\circ}$ and $0.03\leq E(B-V) \leq 1.31$.
There are two OCs with $E(B-V)>0.50$ in the Galactic centre directions.
[width = 7cm, height = 10cm]Figure11.eps
$E(B$-$V)$ versus $\textit l^{\circ}$ (panel a) and versus
$b^{\circ}$ (panel b) for the 40 OCs. Open and filled circles
show clusters with $d=[0, 2.1]$ kpc and $d=(2.1, 5.42]$
kpc, respectively.
The reddenings of 40 OCs have been compared to those of the dust maps
of <cit.>, which are based on the COBE/DIRBE and
IRAS/ISSA maps. These maps take into account the dust absorption
$E(B-V)_{\infty}$ all the way to infinity.
The relations of $E(B$–$V)_{\rm SFD,\infty}$ versus $E(B$–$V)$, and
$E(B$–$V)_{\rm SFD}$versus $E(B$–$V)$ of the 40 OCs are displayed in Figs. 9(a)
and (b), respectively. As is seen from Fig. 9(a), the values of $E(B$–$V)_{\rm
SFD,\infty}$ are at the level of $0.07 \leq E(B$–$V)_{\rm SFD,\infty}\leq25.81$.
For seven clusters, differences in between both reddenings are $\Delta E(B-V)\leq0.10$,
while the differences of 33 OCs are larger than 0.10 mag. The equation given by
<cit.> has been adopted to correct the SFD reddening estimates. Then the final
reddening, $E(B$–$V)_{\rm SFD}$, for a given star is reduced compared to the total
reddening $E(B$–$V)(\ell, b)_\infty$by a factor $\lbrace1-\exp[-d \sin |b|/H]\rbrace$,
given by <cit.>, where $b$, $d$, and $H$ are the Galactic latitude (Col. 9 of
Table 2), the distance from the observer to the object (Col. 7 of Table 3), and the
scale height of the dust layer in the Galaxy, respectively. The value of $H=125$ pc is
adopted <cit.>. The reduced final reddenings have been compared with the ones of 40
OCs in Fig. 9(b). The reduced $E(B$–$V)$ values fall in the range of $0.07 \leq E(B-V)
\leq 1.261$.
[width = 7cm, height = 10cm]Figure12.eps
Relations of E(B-V)$_{cluster}$-E(B-V)$_{SFD, \infty}$ (panel a),
E(B-V)$_{cluster}$-E(B-V)$_{SFD, d}$ (panel b), respectively.
There are significant differences for 27 OCs between both $E(B$–$V)$ color excess
values. For the rest, the $E(B$–$V)$ values of 13 OCs are quite close to the ones of
SFD. Note that SFD maps are not reliable at regions $|b|<5^{\circ}$ due to
contaminating sources and uncertainties in the dust temperatures <cit.>.
Therefore, the SFD values resulted from line-of-sight integral through the Milky Way and
with low spatial resolution, it is quite a normal to have different reddening values for
these relatively close ($\sim 1$ kpc) star clusters.
Structural parameters of 40 OCs. Col. 2 represents arcmin to parsec scale.
$\sigma_0K$ in Col. 3 and 7 is the central density of stars.
$\sigma_{bg}$ in Col. 4 and 8 is the residual background density.
R$_{core}$ in Col. 5 and 8 and R$_{RDP}$ in Col. 6 and 10 are the core and cluster radii, respectively.
The symbols $* pc^{-2}$ and $*^{-2}$ in cols. 3, 4, 7 and 8 mean $stars~pc^{-2}$ and $stars~arcmin^{-2}$, respectively.
$\Delta$ R($'$) in Col. 11 denotes comparison field ring. Col. 12 represents the correlation coefficient.
Cluster (1$'$) pc $\sigma_{0K}$ (*pc$^{-2}$) $\sigma_{bg}$ (*pc$^{-2}$) R$_{core}$(pc)
R$_{RDP}$ (pc) $\sigma_{0K}$ (*'$^{-2}$) $\sigma_{bg}$ (*'$^{-2}$) R$_{core}$ ($'$)
R$_{RDP}$ ($'$) $\Delta$ R($'$) C.C.
NGC 436 0.94 10.94$\pm$2.97 0.71$\pm$0.03 1.04$\pm$0.20 6.97$\pm$0.26 9.60$\pm$2.60 0.62$\pm$0.03 1.11$\pm$0.22 7.44$\pm$0.27 22-32 0.93
King 5 0.59 24.79$\pm$5.28 2.70$\pm$0.06 0.95$\pm$0.15 5.62$\pm$0.18 8.65$\pm$1.84 0.94$\pm$0.02 1.60$\pm$0.25 9.52$\pm$0.30 20-30 0.94
NGC 1513 0.34 30.60$\pm$4.73 17.44$\pm$0.44 1.65$\pm$0.26 6.51$\pm$0.20 3.61$\pm$0.55 2.05$\pm$0.05 4.80$\pm$0.75 16.99$\pm$0.58 42-57 0.94
Be 15 0.90 25.70$\pm$11.33 0.97$\pm$0.02 0.35$\pm$0.10 5.04$\pm$0.30 20.89$\pm$9.20 0.79$\pm$0.02 0.39$\pm$0.11 5.59$\pm$0.33 30-40 0.86
NGC 1798 1.46 6.64$\pm$1.90 0.58$\pm$0.01 1.10$\pm$0.22 9.11$\pm$0.48 18.20$\pm$5.22 1.59$\pm$0.03 0.67$\pm$0.13 5.51$\pm$0.29 50-60 0.92
Be 17 0.71 7.25$\pm$1.38 3.26$\pm$0.10 2.10$\pm$0.39 5.29$\pm$0.20 3.62$\pm$0.69 1.63$\pm$0.05 2.98$\pm$0.55 7.48$\pm$0.29 42-52 0.94
NGC 1907 0.57 17.03$\pm$4.16 4.32$\pm$0.19 1.28$\pm$0.27 4.26$\pm$0.16 5.47$\pm$1.34 1.39$\pm$0.06 2.26$\pm$0.47 7.50$\pm$0.28 50-60 0.91
NGC 2112 0.31 24.69$\pm$3.51 6.96$\pm$0.21 1.64$\pm$0.21 5.92$\pm$0.19 2.39$\pm$0.34 0.67$\pm$0.02 5.28$\pm$0.68 19.01$\pm$0.61 50-60 0.95
Koposov 12 0.60 8.96$\pm$3.74 1.27$\pm$0.05 0.87$\pm$0.27 3.82$\pm$0.20 3.18$\pm$1.33 0.45$\pm$0.02 1.46$\pm$0.46 6.41$\pm$0.33 15-25 0.83
NGC 2158 1.27 28.85$\pm$4.67 1.45$\pm$0.06 1.74$\pm$0.20 14.03$\pm$0.71 47.05$\pm$7.61 2.37$\pm$0.10 1.36$\pm$0.16 10.99$\pm$0.56 45-60 0.97
Koposov 53 1.18 7.12$\pm$0.23 0.48$\pm$0.05 0.66$\pm$0.04 4.18$\pm$0.33 10.04$\pm$0.33 0.67$\pm$0.08 0.56$\pm$0.03 3.52$\pm$0.28 25-35 0.99
NGC 2194 0.69 21.00$\pm$3.31 3.30$\pm$0.15 1.66$\pm$0.22 6.55$\pm$0.21 9.98$\pm$1.58 1.57$\pm$0.07 2.41$\pm$0.32 9.5$\pm$0.31 40-50 0.96
NGC 2192 1.22 5.26$\pm$1.59 0.41$\pm$0.01 1.11$\pm$0.24 5.47$\pm$0.34 7.87$\pm$2.42 0.62$\pm$0.02 0.91$\pm$0.19 4.47$\pm$0.28 45-55 0.90
NGC 2243 1.38 13.35$\pm$4.18 0.22$\pm$0.01 0.89$\pm$0.18 12.94$\pm$0.37 25.28$\pm$7.95 0.42$\pm$0.02 0.65$\pm$0.13 9.40$\pm$0.27 20-30 0.93
Trumpler 5 0.79 13.62$\pm$1.72 3.37$\pm$0.09 3.86$\pm$0.43 15.18$\pm$0.47 8.65$\pm$1.09 2.14$\pm$0.06 4.85$\pm$0.54 19.05$\pm$0.58 27-37 0.97
Col 110 0.71 5.32$\pm$0.51 2.68$\pm$0.05 6.25$\pm$0.63 12.12$\pm$0.40 2.68$\pm$0.26 1.35$\pm$0.02 8.79$\pm$0.88 17.07$\pm$0.57 40-50 0.97
NGC 2262 0.86 22.03$\pm$5.17 1.93$\pm$0.05 0.85$\pm$0.14 6.37$\pm$0.24 16.32$\pm$3.81 1.43$\pm$0.04 0.99$\pm$0.17 7.40$\pm$0.28 30-45 0.94
NGC 2286 0.67 6.81$\pm$2.29 2.66$\pm$0.12 1.59$\pm$0.48 6.39$\pm$0.19 3.07$\pm$1.03 1.20$\pm$0.05 2.37$\pm$0.72 9.51$\pm$0.29 27-37 0.85
NGC 2309 0.88 12.41$\pm$5.96 0.64$\pm$0.05 0.84$\pm$0.28 7.50$\pm$0.25 9.64$\pm$4.63 0.50$\pm$0.04 0.95$\pm$0.32 8.51$\pm$0.29 50-60 0.84
Tombaugh 2 0.35 134.29$\pm$83.62 3.67$\pm$0.25 0.17$\pm$0.07 1.92$\pm$0.11 16.91$\pm$10.5 0.46$\pm$0.03 0.47$\pm$0.18 5.42$\pm$0.31 20-25 0.98
Be 36 1.57 3.30$\pm$1.76 0.51$\pm$0.02 1.32$\pm$0.51 10.23$\pm$0.40 8.21$\pm$4.39 1.27$\pm$0.04 0.83$\pm$0.32 6.50$\pm$0.25 25-40 0.79
Haffner 8 0.72 5.31$\pm$2.85 3.89$\pm$0.08 1.47$\pm$0.68 6.93$\pm$0.21 2.79$\pm$1.50 2.04$\pm$0.04 2.03$\pm$0.94 9.56$\pm$0.29 45-60 0.69
Mel 71 0.59 22.61$\pm$3.97 4.39$\pm$0.09 1.27$\pm$0.18 5.00$\pm$0.17 7.89$\pm$1.39 1.53$\pm$0.03 2.16$\pm$0.30 8.46$\pm$0.29 45-60 0.95
NGC 2425 0.83 11.00$\pm$2.04 2.28$\pm$0.04 1.16$\pm$0.17 5.43$\pm$0.22 7.55$\pm$1.41 1.57$\pm$0.03 1.40$\pm$0.20 6.54$\pm$0.26 42-47 0.95
NGC 2506 0.82 18.56$\pm$3.13 1.07$\pm$0.04 1.65$\pm$0.20 10.76$\pm$0.48 12.67$\pm$2.14 0.73$\pm$0.03 2.00$\pm$0.24 13.02$\pm$0.58 40-50 0.96
Pismis 3 0.50 26.09$\pm$2.91 6.53$\pm$0.11 2.10$\pm$0.20 8.58$\pm$0.29 6.61$\pm$0.74 1.65$\pm$0.03 4.17$\pm$0.40 17.05$\pm$0.57 37-47 0.98
NGC 2660 0.69 90.44$\pm$24.32 3.84$\pm$0.09 0.39$\pm$0.07 5.27$\pm$0.17 43.71$\pm$11.74 1.86$\pm$0.04 0.55$\pm$0.10 7.58$\pm$0.25 25-35 0.94
NGC 3680 0.31 19.37$\pm$7.61 1.84$\pm$0.05 0.47$\pm$0.13 2.98$\pm$0.10 1.90$\pm$0.75 0.18$\pm$0.005 1.49$\pm$0.41 9.49$\pm$0.32 40-50 0.85
Ru 96 0.73 6.52$\pm$2.75 5.95$\pm$0.13 1.43$\pm$0.53 2.60$\pm$0.21 3.54$\pm$1.45 3.20$\pm$0.07 1.94$\pm$0.73 3.54$\pm$0.29 50-60 0.77
Ru 105 0.56 2.63$\pm$1.81 1.52$\pm$0.05 1.35$\pm$0.78 3.83$\pm$0.21 0.93$\pm$0.64 0.54$\pm$0.02 2.27$\pm$1.30 6.42$\pm$0.34 47-57 0.64
Trumpler 20 0.93 10.22$\pm$1.22 4.73$\pm$0.12 3.12$\pm$0.38 13.98$\pm$0.55 8.86$\pm$1.06 4.10$\pm$0.10 3.36$\pm$0.41 15.02$\pm$0.59 40-50 0.97
Pismis 19 0.56 104.06$\pm$15.47 13.68$\pm$0.18 0.54$\pm$0.06 6.16$\pm$0.33 32.46$\pm$4.82 4.27$\pm$0.05 0.96$\pm$0.10 11.03$\pm$0.59 47-57 0.97
NGC 6134 0.32 72.12$\pm$24.74 7.82$\pm$0.18 0.45$\pm$0.11 3.08$\pm$0.10 7.52$\pm$2.59 0.81$\pm$0.02 1.39$\pm$0.34 9.53$\pm$0.30 13-22 0.86
IC 4651 0.25 38.57$\pm$7.88 12.50$\pm$0.99 1.02$\pm$0.22 2.35$\pm$0.08 2.36$\pm$0.48 0.76$\pm$0.06 4.13$\pm$0.91 9.49$\pm$0.31 50-60 0.92
NGC 6802 0.65 34.48$\pm$9.18 7.90$\pm$0.27 1.03$\pm$0.18 4.24$\pm$0.18 14.72$\pm$3.63 3.38$\pm$0.12 1.58$\pm$0.28 6.49$\pm$0.38 45-60 0.92
NGC 6819 0.68 38.22$\pm$4.18 3.97$\pm$0.08 1.50$\pm$0.12 12.92$\pm$0.40 17.15$\pm$1.95 1.84$\pm$0.04 2.20$\pm$0.18 18.98$\pm$0.59 40-50 0.98
Be 89 0.88 7.02$\pm$1.35 3.80$\pm$0.12 2.75$\pm$0.53 7.48$\pm$0.26 5.42$\pm$1.04 2.90$\pm$0.09 3.10$\pm$0.60 8.50$\pm$0.30 15-20 0.93
NGC 6939 0.52 33.06$\pm$5.19 3.35$\pm$0.17 1.16$\pm$0.15 4.92$\pm$0.18 8.96$\pm$1.41 0.91$\pm$0.05 2.24$\pm$0.28 9.46$\pm$0.28 35-45 0.97
NGC 7142 0.74 10.15$\pm$1.87 1.72$\pm$0.10 1.98$\pm$0.32 11.19$\pm$0.44 5.63$\pm$1.04 0.95$\pm$0.05 2.65$\pm$0.43 15.02$\pm$0.59 50-60 0.94
NGC 7789 0.51 31.23$\pm$2.22 3.88$\pm$0.04 2.32$\pm$0.13 26.88$\pm$0.74 8.18$\pm$0.58 1.02$\pm$0.01 4.52$\pm$0.25 52.5$\pm$1.44 55-70 0.99
§ STRUCTURAL PARAMETERS
We derived the structural parameters of 40 OCs from the stellar radial density profiles (RDPs).
Usually, the RDPs of star clusters can be described by an analytical profile, like the
empirical, single mass, modified isothermal spheres of
<cit.> and <cit.>, and the power law with a core
of <cit.>. These functions are characterized by different
sets of parameters that are related to the cluster structure.
Here we adopted the two-parameter function $\sigma(R) =
\sigma_{bg} + \sigma_0/(1+(R/R_c)^2)$, where $\sigma_{bg}$
is the residual background density, $\sigma_0$ the central
density of stars, and R$_{core}$ the core radius. Applied to
star counts, this function is similar to that used by
<cit.> to describe the surface brightness profiles in
the central parts of globular clusters. To minimize degrees of
freedom in RDP fits with the King-like profile, $\sigma_{bg}$
was kept fixed (measured in the respective comparison fields)
while $\sigma_{0}$ and $R_{core}$ were determined by the best
profile fit to the data. As a representative of the OCs sample,
the RDP of Pismis 19 fitted with King's profile is shown in
Fig. 10, where the solid line shows the best profile fit. The
horizontal red bar in the figure denotes the stellar background
level measured in the comparison field, and the $1\sigma$ profile
fit uncertainty is shown by the
shaded domain. The stellar RDPs fitted profiles of the 40 OCs have
been given in Figs. S10$-$S13 as supplementary material. The cluster
radius (R$_{RDP}$) is also obtained from the measured distance from
the cluster centre where the RDP and residual background are
statistically indistinguishable <cit.>. The R$_{RDP}$ can
be taken as an observational truncation radius, whose value depends
on the radial distribution of member stars and the stellar field
density. $\Delta R$ means the wide external ring of the stellar
comparison field (see also Sect. 3). These structural parameters and
their meaning are listed in Table 5.
[width = 7cm, height = 7cm]Figure13.jpg
Stellar RDP (open circles) of Pismis 19 built with
CMD filtered photometry. Solid line shows the best-fit King profile.
Horizontal red bar: stellar background level measured
in the comparison field. Shaded region: $1\sigma$ King fit uncertainty.
From the distributions of R$_{core}$ and R$_{RDP}$, given in
Fig. 11(a) and (b), there seems to be two groupings at R$_{RDP}
$$=$7 pc and R$_{core}$$=$1.5 pc, respectively, which are
close to the values of 10 pc and 1.5 pc of <cit.>.
[width = 7cm, height = 11cm]Figure14.jpg
Distributions of R$_{RDP}$
(panel a) and R$_{core}$ (panel b) of 40 OCs, respectively.
§ MASS AND MASS FUNCTIONS
The stellar masses stored in the OCs of our sample have been
determined by means of their mass functions (MFs), built for
the observed MS mass range, according to <cit.>.
By following the algorithm, which is basically defined by
<cit.>, luminosity functions from the decontaminated
$(J, J-H)$ diagrams of the OCs have been transformed into MFs
through the corresponding mass-luminosity relations derived
from the M08 isochrones which correspond to the ages in Col. 5
of Table 3. We determined the overall masses of 26 OCs and the
core masses of 24 OCs in our sample. The total mass locked up
in stars of these OCs was obtained by considering all stars from
the turnoff to the H-burning mass limit. We do this by directly
extrapolating the low-mass MFs down to $0.08M_{\odot}$. Here we
have based our results on the CMD filtered photometry of open
cluster and offset field stars. The filtering process contemplates
most of the background, leaving a residual contamination. Due
to the relatively large sizes of the OCs and the brightness
limitation of the 2MASS photometry, we do not have access to
the whole stellar mass range of the OCs.
Here, we stress that the values we derive should be taken as approximations.
[width = 7cm, height = 7cm]Figure15.jpg
$\phi(m)(stars ~m_\odot^{-1})$ versus $m_\odot$ of Pismis 19 cluster,
as a function of distance from the core.
The relation of $\phi(m)(stars ~m_{\odot}^{-1})$ versus $m_{\odot}$
of our representative open cluster Pismis 19 is shown in
Figs. 12(a)$-$(c) for different cluster regions.
The main sequence mass functions (MFs) in the panels (a)$-$(c) of
Fig. 12 are fitted with the function $\phi(m)\propto{m}^{-(1+\chi)}$,
and the MF slopes ($\chi$) have been determined for the different
segments of the mass function MF in Col. 1 of Table 6. More details
of this approach are given in Table 6, where we also show the number
and mass of the evolved stars (m$_{evol}$). The MF slopes of the core
(29 OCs) and the overall (31 OCs) regions of OCs are presented in
Cols. 2 and 5 of Table 7. Since the lower MS is not accessible on the
$(J, J-H)$ diagrams of the OCs sample, we assumed that the low-mass
content is still present, and use Kroupa's MF[$\chi=0.3\pm0.5$
<cit.> for $0.08<M_{\odot}<0.5$, $\chi=1.3\pm0.3$ for $0.5 <
M_{\odot}<1.0$, and $\chi=1.3\pm 0.7$ for $1.0<M_{\odot}$] to estimate
the total stellar mass, down to the H-burning mass limit. The results:
number of stars, MS and evolved star contents (m$_{obs}$), MF
slope ($\chi$), and mass extrapolated (m$_{tot}$) to 0.08 $M_{\odot}$)
for each cluster region are given in Table 6. The mass densities of
$\rho$ in unit of $M_{\odot}\: pc^{-3}$ are also estimated and given
in Cols. 8 and 11 of Table 6 (See also sect. 7.7).
When deriving the mass functions, the part of the steep, that is
observed in the core may come from crowding and completeness. 2MASS
is not very photometrically deep and has just a moderate spatial
resolution. So, in crowded regions (such as the core of most
clusters) many stars are not detected, especially the faint ones.
This, in turn may mimic mass segregation.
The relaxation time $t_{rlx}$ (Myr) is the characteristic
time-scale for a cluster to reach some level of energy
equipartition <cit.>. As discussed in <cit.>,
<cit.>, and <cit.>, the evolutionary parameter
($\tau = Age/t_{rlx}$) appears to be a good indicator of dynamical
state. Following <cit.>, we parameterize $t_{rlx}$ as
where N is the number of stars located inside the region of radius R.
The relaxation time and evolutionary parameter for both core and the
overall regions are listed in Table 7. The uncertainties in the
evolutionary parameters ($\tau$) of OCs have been estimated by
propagating the errors in Age (Table 3), Radii (Table 5) and N
(Table 6) into $t_{rlx}$ and $\tau$. When propagated, the latter
two errors produce a large uncertainty in $t_{rlx}$ (Table 7) and,
consequently, a large uncertainty in the evolutionary parameter.
In this sense, both $t_{rlx}$ and $\tau$ should be taken simply as
an order of magnitude estimate.
From the overall mass distribution ($m_{overall}$) of 26 OCs
displayed in Fig. 13, 2000 $M_\odot$ value is considered as a
criteria in classifying the clusters as less massive and
[width = 6.5cm, height = 7cm]Figure16.eps
The overall mass distribution of 26 OCs.
The number of stars, mass information, mass function slope, mass density,
which correspond to cluster regions of available clusters for the cases of
Evolved, Observed+Evolved, and Extrapolated+Evolved.
The full version is available in the online version of this manuscript in the supplementary material section (Table S6).
11cNGC 436
2c $\chi$
Region N* m$_{evol}$ 1.38-2.78 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$ $M_{\odot} pc^{-3}$
0.0-1.04 1$\pm$1 0.4$\pm$0.4 -1.46$\pm$0.47 - 0.25$\pm$0.03 0.56$\pm$0.28 11.9$\pm$5.97 0.4$\pm$0.1 0.7$\pm$0.03 15.2$\pm$6.11
1.04-6.97 12$\pm$6 3.5$\pm$1.8 1.74$\pm$0.36 - 1.01$\pm$0.1 2$\pm$0.57 0.14$\pm$0.04 25.6$\pm$19.6 9.8$\pm$3.8 0.69$\pm$0.27
0.0-6.97 14$\pm$6 3.9$\pm$1.9 0.86$\pm$0.29 - 1.12$\pm$0.09 2.55$\pm$0.63 0.18$\pm$0.04 17.1$\pm$11.9 7.9$\pm$2.4 0.56$\pm$0.17
$\cdot \cdot \cdot$ $\cdot \cdot \cdot$ $\cdot \cdot \cdot$ $\cdot \cdot \cdot$ $\cdot \cdot \cdot$
$\cdot \cdot \cdot$ $\cdot \cdot \cdot$ $\cdot \cdot \cdot$ $\cdot \cdot \cdot$ $\cdot \cdot \cdot$ $\cdot \cdot \cdot$
Col. 1: the distance from the core. Cols. 2,6,9 : cluster stars for the regions in Col. 1.
Col. 4 gives the MF slopes ($\chi$), derived for the low-mass and high-mass ranges.
The masses of $m_{evol}$, $m_{obs}$, and $m_{tot}$ are listed in Cols. 3, 7 and 10, respectively.
The mass densities are given in Cols. 8 and 11.
Mass function slopes ($\chi$), relaxation time (t$_{rlx}$(Myr))
and evolutionary parameter ($\tau$) of core and overall regions of the available clusters.
3cCore 3cOverall
Cluster $\chi$ t$_{rlx}$(Myr) $\tau$$_{core}$ $\chi$ t$_{rlx}$(Myr) $\tau$$_{overall}$
NGC 436 $-$1.46$\pm$0.47 0.46$\pm$0.12 869.57$\pm$314.19 0.86$\pm$0.29 63.95$\pm$38.60 6.25$\pm$4.08
King 5 $-$3.06$\pm$0.96 0.58$\pm$0.11 1724.14$\pm$475.22 1.80$\pm$0.49 215.00$\pm$148.09 4.65$\pm$3.34
NGC 1513 1.12$\pm$0.24 10.21$\pm$6.40 9.79$\pm$6.44 1.90$\pm$0.12 175.88$\pm$118.13 0.57$\pm$0.40
Be 15 - - - $-$1.54$\pm$1.15 5.05$\pm$2.78 99.01$\pm$57.99
NGC 1907 $-$0.76$\pm$0.40 1.71$\pm$0.43 233.92$\pm$82.95 0.00$\pm$0.23 - -
NGC 2112 $-$1.28$\pm$0.51 3.28$\pm$3.59 609.76$\pm$673.63 0.50$\pm$0.42 126.22$\pm$78.13 15.85$\pm$10.10
NGC 2158 $-$4.24$\pm$1.00 5.05$\pm$0.16 495.05$\pm$61.44 $-$1.55$\pm$0.71 - -
Koposov 53 $-$3.96$\pm$3.40 - - 0.93$\pm$0.81 15.51$\pm$11.41 64.47$\pm$47.86
NGC 2194 0.38$\pm$0.42 13.15$\pm$7.87 60.84$\pm$39.46 2.52$\pm$0.37 456.55$\pm$311.22 1.75$\pm$1.27
NGC 2192 $-$2.78$\pm$0.96 0.36$\pm$0.10 3611.11$\pm$1040.84 $-$3.12$\pm$0.43 7.23$\pm$0.85 179.81$\pm$25.26
NGC 2243 - - - 2.09$\pm$1.01 826.05$\pm$654.27 2.42$\pm$1.93
Trumpler 5 0.42$\pm$0.75 1195.27$\pm$869.82 2.51$\pm$1.84 1.32$\pm$1.18 3804.93$\pm$2817.13 0.79$\pm$0.59
Col 110 $-$2.58$\pm$0.21 29.78$\pm$4.37 100.74$\pm$16.24 $-$2.84$\pm$0.57 100.01$\pm$22.83 30.00$\pm$7.13
NGC 2262 $-$1.49$\pm$0.80 0.43$\pm$0.15 3023.26$\pm$1079.96 1.01$\pm$0.44 186.81$\pm$126.13 6.96$\pm$4.73
NGC 2286 1.30$\pm$0.50 5.97$\pm$4.26 167.50$\pm$124.13 1.45$\pm$0.30 99.95$\pm$65.65 10.01$\pm$6.87
NGC 2309 $-$1.52$\pm$1.03 0.35$\pm$0.07 1428.57$\pm$404.06 $-$0.89$\pm$0.60 - -
Haffner 8 1.28$\pm$0.77 5.50$\pm$4.79 181.82$\pm$159.39 1.82$\pm$0.59 101.14$\pm$68.47 9.89$\pm$6.77
Mel 71 0.30$\pm$1.04 - - 1.29$\pm$0.40 146.55$\pm$99.37 10.24$\pm$7.08
NGC 2506 4.11$\pm$1.63 56.00$\pm$49.28 35.71$\pm$31.88 0.97$\pm$0.63 822.94$\pm$594.95 2.43$\pm$1.79
Pismis 3 $-$1.60$\pm$0.84 10.42$\pm$2.03 307.10$\pm$62.83 1.71$\pm$0.49 1348.20$\pm$939.15 2.37$\pm$1.66
Ru 96 4.58$\pm$0.98 17.70$\pm$15.06 56.50$\pm$48.40 4.55$\pm$0.65 57.49$\pm$40.92 17.39$\pm$12.50
Trumpler 20 $-$1.07$\pm$0.50 12.85$\pm$17.60 116.73$\pm$164.55 2.06$\pm$0.68 2272.78$\pm$1663.64 0.66$\pm$0.53
Pismis 19 $-$2.42$\pm$1.07 0.45$\pm$0.10 1777.78$\pm$453.27 1.18$\pm$0.38 402.04$\pm$272.33 1.99$\pm$1.37
NGC 6134 $-$0.95$\pm$0.90 - - $-$0.83$\pm$1.11 - -
IC 4651 $-$2.78$\pm$0.75 1.09$\pm$0.06 2293.58$\pm$302.81 $-$0.60$\pm$0.41 - -
NGC 6802 $-$0.46$\pm$0.84 - - 1.66$\pm$0.24 232.20$\pm$156.40 3.88$\pm$2.65
NGC 6819 $-$1.07$\pm$0.55 - - 0.47$\pm$0.40 680.42$\pm$429.71 3.67$\pm$2.43
Be 89 0.18$\pm$0.66 - - 1.64$\pm$0.93 312.83$\pm$220.34 6.39$\pm$4.78
NGC 6939 $-$2.17$\pm$0.66 1.42$\pm$0.07 1408.45$\pm$222.38 0.84$\pm$0.46 - -
NGC 7142 $-$1.97$\pm$1.44 - - $-$1.06$\pm$0.56 - -
NGC 7789 $-$0.42$\pm$0.45 - - 0.79$\pm$0.65 5471.49$\pm$3972.27 0.33$\pm$0.24
§ RESULTS
§.§ Relation of R$_{RDP}$–R$_{core}$
The cluster and core radii (R$_{RDP}$, R$_{core}$) of 40 OCs, given
in Fig. 14 are related by the following relation,
with a mild correlation coefficient (CC, hereafter) of 0.61.
This relation of Fig. 14 is almost linear between $log\: (R_{RDP})$ and
$log\: (R_{core})$, where the axes are in a log-log scale.
Their core and cluster sizes are $0.17\leq R_{core}~(pc) \leq 6.25$
and $1.92\leq R_{RDP}~(pc)\leq 26.88$, respectively. The OCs in our
sample which do not follow the relation above are either intrinsically
small or have been suffering significant evaporation effects. Our
coefficient value (4.69) of Fig. 14 falls in the
range of $3.1-8.9$ of the literature (Table 8). However the coefficients
in Table 8 are affected by the sample size. The relation between R$_{RDP}$
and R$_{core}$ found by us is reasonably similar to that given by <cit.>.
However analogue functions were found by other authors, <cit.>.
[width = 7cm, height = 7cm]Figure17.eps
Relation of R$_{RDP}$ - R$_{core}$ of 40 OCs.
Empty circles show 40 OCs. Solid line and shaded area show
the best fit and $1\sigma$ uncertainty, respectively.
The coefficients of the relation, R$_{RDP}$$=$a+bR$_{core}$, given
in the literature between R$_{core}$ and R$_{RDP}$.
The form of the relation of Camargo et al. (2010) is $R_{RDP}=bR_{core}^{a}$.
CC and N in last two columns mean the correlation coefficient and data number, respectively.
Author a b CC N
<cit.> - 6 - 38
<cit.> 1.05 7.73 0.95 16
<cit.> - 3.1 - 9
<cit.> - 3.1 0.74 42
<cit.> 0.3 8.9 - 50
<cit.> 0.58 6.98 0.93 140
§.§ Relations of Cluster Dimensions to the Distance and Age
The relations of R$_{RDP}$ and R$_{core}$ with d(kpc) are apparently
linear, and are displayed in Figs. 15(a)$-$(b). The linear best fit to the
data (solid lines) are the following, R$_{RDP}=(2.67\pm0.27)\: d(kpc)$
(CC$=$0.84) and R$_{core}= (0.50 \pm 0.07)$ d(kpc)
(CC$=$0.76), respectively. Given a couple of deviants, the sizes (R$_{RDP}$ and
R$_{core}$) increase on the average with the distance from the Sun.
Similar trends were also obtained by <cit.>,
<cit.>, <cit.>, <cit.> and <cit.>.
[width = 6.5cm, height = 11cm]Figure18.jpg
Relations of R$_{RDP}$ - d(kpc)(panel a),
R$_{core}$ - d(kpc)(panel b), respectively. Solid and dashed lines
show the best fit and $1\sigma$ uncertainty, respectively.
The relations of $|z|$ and R$_{GC}$ as a function of Age and R$_{RDP}$,
respectively, are presented in Figs. 16(a)$-$(b). Younger and older
clusters than 1 Gyr in panels (a)$-$(b) lie inside/outside the Solar
circle. No cluster with R$_{RDP} > 8$ pc is seen in panel (a).
The OCs with Age$\geq$1 Gyr in panel (b)
do not show any dependence of R$_{GC}$ and R$_{RDP}$. The OCs, NGC 2243
and NGC 2192 with $|z| > 800$ pc outside the Solar circle in panels (b)$-$(c),
where GMCs are scarce, might have been moved to outer
parts of the Galactic radii via tidal interactions with the disc and the
Galactic bulge, and collisions with GMCs. Alternatively, they may have been
formed from molecular clouds at these distances. Note that
<cit.> have also detected large and small sized clusters outside
the Solar circle. From panel (b) we note that most of large/small sized
OCs inside or outside the Solar circle are located near the galactic
plane ($|z| < 300$ pc) and the OCs inside the Solar circle seem to
survive four or more rotations around the Galactic centre. Their
survival can be explained by which they survived against external shocks
[width = 7cm, height = 13cm]Figure19.eps
Relations of $|z|$- R$_{GC}$ in terms of
R$_{RDP}$ (panel a) and Age (Myr) (panel b).
Filled squares and empty circles show the OCs with
R$_{RDP}<8$ pc and R$_{RDP}\ge8$ pc, respectively.
Relation of $|z|$-Age as function of R$_{\odot}$ (panel c).
Old clusters with large dimensions inside the Solar circle in panel (b)
may have a primordial origin, or their sizes may have been increased
via expansion due to stellar mass black hole couples.
For the relation, $|z|$- Age as a function of R$_{\odot}$
in Fig. 16(c), the OCs with Age $\geq 1$ Gyr reach higher z distances,
whereas those with Age $<1 $ Gyr have $|z|<300$ pc.
§.§ Relations of R$_{RDP}$-Age and R$_{core}$-Age
The relations of R$_{core}$$-$Age and R$_{RDP}$$-$Age have been
displayed in Figs. 17(a)$-$(b). In Fig. 17, filled circles and
empty triangles show 16 OCs with $m_{overall} \ge 2000~M_{\odot}$
and 10 OCs with $m_{overall}<2000~M_{\odot}$, respectively. 14
OCs which have no mass determinations are marked by open squares.
The relation in Figs. 17(a)$-$(b) suggests a bifurcation which is seen
at an age $\approx$ 1 Gyr. In the sense, in Fig. 17 some clusters
appear to expand (`A' arrow), while others contract (`B' arrow)
with a bifurcation occuring at about 1 Gyr. <cit.> observed
the bifurcation at $\approx$ 500-600 Myr (shown with `C' in the
panels of Fig. 17). This kind of relations in the panels were
also observed by <cit.>, <cit.>, and <cit.>
from their OC samples.
[width = 7cm, height = 9.5cm]Figure20.eps
Relations of Age–R$_{RDP}$ (panel a) and
Age–R$_{core}$ (panel b), respectively.
Filled circles and empty triangles show
16 OCs with $m_{overall}\ge20~M_{\odot}$ and 10 OCs with $m_{overall}<20~M_{\odot}$, respectively.
14 OCs which have no mass determinations are marked by open squares. R1, R2, R3 and R4 mean the regions.
<cit.> argue that some clusters show the expanded cores due to
stellar mass black holes (hereafter BHs), and others contract due to
dynamical relaxation and core collapse. To be able to see the effect
of BHs in our core radius-age relation, the information of
the OCs in regions of R2 and R4 in Fig. 17(a) is given in Tables 9$-$10.
We call the regions in Fig. 17 as R1, R2, R3 and R4.
N$_{bh}$ in Tables 9$-$10 means the estimated number of stellar mass
black holes (BHs). This value is estimated from a relation N$_{bh}=6
\times 10^{-4}N_{star}$, given by <cit.>. Here, $N_{star}$ is the
extrapolated number of stars in the OCs, and is given in Col. 9 of
Table 6 for the overall regions of OCs. Because the extrapolated stellar number for NGC 2158 is
not available (Col. 9 of Table 6; supplementary material), the number
of BHs could be estimated from the relation of $N_{bh} \approx 0.002 \:
M_{cluster}$, given by <cit.>. The BH numbers of seven OCs in the regions
R2 and R4 in Fig. 17(a) cannot be estimated, because their extrapolated
star numbers or overall masses are not available (see Col. 9 of
Table 6; supplementary material).
Age, dimensions and mass (Cols. 2$-$5) for OCs,
which show the core expansion in Fig. 17(a).
The number of black holes (N$_{bh}$) is listed in last column.
Cluster Age R$_{RDP}$ R$_{core}$
m$_{overall}$ N$_{bh}$
(Myr) (pc) (pc) (100m$_{\odot})$
NGC 2112 2000 5.92 1.64 18.10 4
NGC 2158 2500 14.03 1.74 33.40 7
Trumpler 5 3000 15.18 3.86 223.0 45
Col 110 3000 12.12 6.25 16.50 3
NGC 2286 1000 6.39 1.59 11.10 2
NGC 2506 2000 10.76 1.65 65.80 13
Pismis 3 3200 8.58 2.10 133.00 27
Trumpler 20 1500 13.98 3.12 150.00 30
NGC 6819 2500 12.92 1.50 49.10 10
Be 89 2000 7.48 2.75 32.00 6
NGC 7789 1800 26.88 2.32 194.00 39
Age, dimensions and mass (Cols. 2$-$5) for OCs,
which show the core shrinkage in Fig. 17(a).
The number of black holes (N$_{bh}$) is listed in last column.
Cluster Age R$_{RDP}$ R$_{core}$
m$_{overall}$ N$_{bh}$
(Myr) (pc) (pc) (100m$_{\odot})$
King 5 1000 5.62 0.95 30.00 6
Koposov 53 1000 4.18 0.66 2.37 0
NGC 2192 1300 5.47 1.11 2.27 0
NGC 2243 2000 12.94 0.89 51.70 10
NGC 2262 1300 6.37 0.85 24.10 5
Haffner 8 1000 6.93 1.47 9.85 2
Mel 71 1500 5.00 1.27 21.80 4
Ru 96 1000 2.60 1.43 15.80 3
§.§ Relations of R$_{RDP}$ and R$_{core}$ with R$_{GC}$
The dependence of the structural parameters (R$_{RDP}$ & R$_{core}$)
with their galactocentric distance R$_{GC}$ of the 40 OCs and as a
function of the ages are plotted in Figs. 18(a)$-$(b). The large and
small-sized clusters in Fig. 18(a) occupy the inner- and the outer-
Galactic radii. Two OCs with R$_{RDP}$ $ < $ 7 pc and Age $ < $ 1 Gyr
in Fig. 18(a) are locate inner Galactic radius. Such OCs with these
sizes and ages are also seen in <cit.>.
The relation between R$_{RDP}$ and R$_{GC}$ is the following,
R$_{RDP}$$=(0.98\pm0.25)R_{GC}+(-3.07\pm2.03)$, with a
correlation coefficient of 0.53 (see Fig. 18a), Our result shows that
there is no strong dependence of R$_{RDP}$ on R$_{GC}$. However,
<cit.>, <cit.> and <cit.>
mention a correlation from their OC samples.
[width = 7cm, height = 7cm]Figure22.jpg
[width = 7cm, height = 7cm]Figure23.eps
Relations of R$_{GC}$–R$_{RDP}$ (panel a) and
R$_{GC}$–R$_{core}$ (panel b), respectively.
Filled squares and open circles denote the OCs with Age $<$ 1 Gyr
and Age $\ge$ 1 Gyr, respectively.
§.§ Relations of m$_{overall}$ with R$_{RDP}$, R$_{core}$, Age and R$_{GC}$
Figs. 19(a) and (b) show the relations of m$_{overall}$ versus R$_{RDP}$
and m$_{overall}$ versus R$_{core}$ as a function of Age of 26
of our 40 OCs. The relations to fit m$_{overall}$ with R$_{RDP}$
and with R$_{core}$ are $\ln m_{overall} = (1.57\pm0.42)\: \ln R_{RDP}+(0.01\pm0.02)$
(CC$=$0.60) and $\ln m_{overall} = (1.14\pm0.37)\\ln R_{core}+(2.81\pm0.26)$
(CC$=$0.53), respectively. These correlations between size and mass of the clusters
are in concordance with the mass-radius relation for massive OCs with
Age $ > $ 100 Myr <cit.>.
In Figs. 20(a) and (b) the relations of m$_{overall}$ with R$_{GC}$ and of
m$_{overall}$ with Age of 26 of our 40 OCs are shown.
As is seen from Fig. 20(a),
massive and less massive OCs than m$_{overall} =2000\:
M_{\odot}$ are located indistinctly in- or outwards of the Solar circle.
[width = 7cm, height = 7cm]Figure25.eps
[width = 7cm, height = 7cm]Figure26.eps
Relations of R$_{RDP}$ - m$_{overall}$
(panel a) and R$_{core}$ - m$_{overall}$(panel b) of 26 OCs.
Filled squares and open circles represent the OCs with Age $<$ 1 Gyr
and Age $\ge$ 1 Gyr. Dashed lines denote the best fits.
[width = 7cm, height = 7cm]Figure27.eps
[width = 7cm, height = 7cm]Figure28.eps
Relations of R$_{GC}$ - m$_{overall}$ (panel a)
and Age - m$_{overall}$ (panel b) of 26 OCs.
§.§ Relations between MF slopes, Age, R$_{RDP}$, R$_{GC}$, and the mass density
The relation of $\chi_{overall}$ with $\chi_{core}$ of 29 OCs is
presented in Fig. 21. The fit which is applied to the data is given as
following, $\chi_{overall}=(0.47\pm0.12)\chi_{core}+(1.10\pm0.26)$,
with a moderate CC$=$0.60. The OCs with flat/steep
positive overall MF slopes for $\chi_{core}<0$ in Fig. 21 show signs of
a mild to large scale mass segregation, whereas the OCs with negative
overall MF slopes for $\chi_{core} < 0$ indicate an advanced dynamical
evolution. These MF slopes of $\chi_{core}<0$ in Fig. 21 can be explained
by the external dynamical effects such as tidal stripping by tidal
interactions (in the form of shocks) due to disc and bulge
crossings, as well as encounters with GMCs.
[width = 7cm, height = 7cm]Figure29.eps
Relation of $\chi_{core}$ - $\chi_{overall}$ of 29 OCs.
Dashed line shows the best fit.
The relations of Age vs. $\chi_{overall}$ of 31 OCs and Age vs. $\chi_{core}$
of 29 OCs of our sample are displayed in Figs. 22(a) and (b).
The age dependence of the overall and core MF slopes
has been parameterised by the linear-decay function
(shown as dashed curve) $\chi (t) = \chi_\circ - t/t_{f}$,
where $\chi_\circ$ represents the MF slope in the early phases and
$t_{f}$ is the flattening time scale. For the overall MF we derive
$\chi_\circ = 1.68\pm0.30$ and $t_{f}=1569\pm600$ Myr (CC$=$0.44);
the core values $\chi_\circ = 0.74\pm0.39$ and
$t_{f}=1006\pm206$ Myr (CC$=$0.68). Within the
expected uncertainties the overall MF values are quite close to
$\chi_\circ=1.30\pm0.30$ of <cit.> and $\chi_\circ=1.35$ of
[width = 7cm, height = 7cm]Figure30.eps
[width = 7cm, height = 7cm]Figure31.eps
Relations of Age - $\chi_{overall}$ (31 OCs, panel a) and
Age - $\chi_{core}$ (29 OCs, panel b). Dashed and solid lines show
the best fit and $1\sigma$ uncertainty, respectively.
The relations of m$_{overall}$ with the slope $\chi_{overall}$ of 26
OCs and m$_{core}$ with $\chi_{core}$ of 24 OCs have been presented
in Figs. 23(a) and (b). In Fig. 23(a), most of the OCs of
with positive overall slopes are mass-rich and present little or no
signs of mass segregation.
For the relation of m$_{core}$ with $\chi_{core}$ dispayed in Fig. 23(b),
most of OCs with m$_{core}<1000\: m_{\odot}$ have negative core MF
slopes implicating mass segregation effects at a larger scale.
[width = 7cm, height = 7cm]Figure32.eps
[width = 7cm, height = 7cm]Figure33.eps
Relations of m$_{overall}$ - $\chi_{overall}$
(panel a) of 26 OCs and m$_{core}$ - $\chi_{core}$ (panel b) of 24 OCs, respectively.
In the relations of R$_{RDP}$ with $\chi_{overall}$ and R$_{RDP}$ with
$\chi_{core}$ for 31 and 29 OCs of our sample, respectively, given in
Figs. 24(a) and (b), the OCs with larger or smaller dimensions than R$_{RDP} =
7\:pc$ have a positive or negative sloped overall- and core-MFs,
[width = 7cm, height = 7cm]Figure34.eps
[width = 7cm, height = 7cm]Figure35.eps
Relations of $\chi_{overall}$ - R$_{RDP}$
(panel a) of 31 OCs and $\chi_{core}$ - R$_{RDP}$ (panel b) of 29 OCs.
Filled squares and open circles denote clusters with Age $<$ 1 Gyr
and Age $\ge$ 1Gyr, respectively.
From the relations between R$_{GC}$ and $\chi_{overall}$ of 31 OCs, and
between R$_{GC}$ and $\chi_{core}$ of 29 OCs shown in Figs. 25(a) and (b),
apparently MF slopes are not correlated with R$_{GC}$ or Age.
[width = 7cm, height = 7cm]Figure36.eps
[width = 7cm, height = 7cm]Figure37.eps
Relations of $\chi_{overall}$ - R$_{GC}$ (panel a) of 31 OCs
and $\chi_{core}$ - R$_{GC}$ (panel b) of 29 OCs, respectively.
The cluster mass density $\rho(m_\odot\: pc^{-3})$ is plotted in
Fig. 26(a) and (b) as a function of $\chi_{overall}$ for 26 OCs and
$\chi_{core}$ for 24 OCs, respectively. In panel (a) the mass
densities of the OCs having $\chi_{overall} < 0$ are low, as
compared to the ones of the OCs with $\chi_{overall} > 0$. This
indicates that low mass stars of OCs with negative MF slopes are
significantly lost due to external dynamical processes. From
panel (b) one can see that $\chi_{core}$ and $\rho_{core}$ are not
[width = 7cm, height = 7cm]Figure38.eps
[width = 7cm, height = 7cm]Figure39.eps
Relations of $\chi_{overall}$ - $\rho_{overall}$ (panel a) of 26 OCs
and $\chi_{core}$ - $\rho_{core}$ (panel b) of 24 OCs, respectively.
§.§ Relation between the MF slope and the evolutionary
parameter and a comparison to Kroupa's IMF
From MF slopes and evolutionary parameters of the overall and the
core of the OCs given in Table 7, the relations of $\tau_{overall}$
with $ \chi_{overall} $ of 24 OCs and similarly, of $\tau_{core}$
with $\chi_{core}$ of 21 OCs have been plotted in Figs. 27(a)$-$(b).
The dashed curve on the figure shows the fit, $\chi (\tau) = \chi_\circ
- \chi_{1}e^{-(\frac{\tau_{o}}{\tau})}$. As seen in panels (a) and
(b) of Figure 27, the overall- and core-MF slopes undergo an exponential
decay with $\tau$. Here, $\chi_\circ$ and $\chi_{1}$ mean MF slopes at
birth and in the advanced stage, respectively. For the overall MF slope,
we derive $\chi_\circ = 1.67\pm 0.18$ and $\tau_{o}=29.92\pm 12.29$
(CC$=$0.77); the core values $\chi_\circ =
1.19\pm 0.89$ and $\tau_{o} = 31.62\pm 34.79$ (CC$=$0.64).
Similar relations were obtained by <cit.>
and <cit.>.
[width = 7cm, height = 7cm]Figure40.eps
[width = 7cm, height = 7cm]Figure41.eps
Relations of $\tau_{overall}$ - $\chi_{overall}$ (panel a) of 24 OCs and
$\tau_{core}$ - $\chi_{core}$ (panel b) of 21 OCs. Dashed and solid lines show the best fit
and $1\sigma$ uncertainty, respectively. In panel (b), $1\sigma$ uncertainty is
not shown due to too large error of $\tau_{core}$.
§ CONCLUSIONS
Our main conclusions are summarized as follows:
* The astrophysical and structural parameteres of 40 OCs
have been derived from the filtered 2MASS $(J, J-H)$ CMDs, and
the stellar RDPs. The field star decontamination technique is utilised
for separating the cluster members. The astrophysical parameters (Age, d, E(B-V))
of 40 OCs comprise with ages in the range of 0.1 Gyr to 5.0 Gyr,
at heliocentric distances, 0.85 kpc to 5.42 kpc, and with reddenings, $0.03 \leq E(B-V)
\leq 1.31$ (Table 3).
Having combined the derived structural, mass and mass functions, relaxation and evolutionary
parameters with astrophysical parameters of 40 OCs, dynamical evolution of these OCs have been studied.
The reduced final reddenings from the dust maps of SFD have been compared with the ones of 40
OCs (Fig. 9(b)). There are significant differences for 27 OCs between both $E(B$–$V)$ color excess
values. For the rest, the $E(B$–$V)$ values of 13 OCs are quite close to the ones of
SFD. Note that SFD maps are not reliable at regions $|b|<5^{\circ}$ due to
contaminating sources and uncertainties in the dust temperatures <cit.>.
Therefore, the SFD values resulted from line-of-sight integral through the Milky Way and
with low spatial resolution, it is quite a normal to have different reddening values for
these relatively close ($\sim 1$ kpc) star clusters.
* The relation between R$_{RDP}$ and R$_{core}$ in Fig. 14 found by us is reasonably
similar to that given by <cit.>. The OCs in our sample which do not follow the relation
are either intrinsically small or have been suffering significant evaporation effects.
The dimensions (R$_{RDP}$ and R$_{core}$) in Figs. 15(a) and (b) increase on
the average with the distance from the Sun.
* From Fig. 17(a) and Tables 9-10, apparently the sizes of core radii
of the OCs are related with their respective BH numbers. The black
hole numbers of the OCs in region R2 of Fig. 17(a) are
generally larger than the ones of the OCs in the region of IV. Note
that the BH numbers of six, out of 10 OCs in Table 10 are almost
close to the ones of the OCs in Table 9. However, with almost similar BH numbers, these six OCs show
shrinkage, whereas those in Table 9 indicate the expanded cores. For
example, If the statement of <cit.> is correct, NGC 2243 with
its 10 BHs would develope a large core. However, NGC 2243 has small
core value, R$_{core}$ = 0.89 pc. Col. 110 with
a few BHs shows an expanding core with R$_{core}$ = 6.25 pc. The core
sizes of NGC 7789 (R$_{core}$ = 2.32 pc) and Be 89 (R$_{core}$ = 2.75
pc), respectively are quite close together. But their BH numbers of
the two OCs are very different. Therefore, the presence of BHs is not the only possible
explanation for the bifurcation seen in Figs. 17(a) and (b).
Alternatively, one should also consider the effect of the mass
range of OCs. In other words, for clusters older than 1 Gyr,
Fig. 17 shows that massive OCs (filled circles) can be found in
both regions R2 and R4. There are two low-mass OCs (open triangles)
in the region R4, (see Fig. 17(a)). In this sense, the distribution
of OCs in Fig. 17 can be partly attributed to clusters with large radii
retaining larger masses.
<cit.> also argue that the expanded cores are the cause of
growth of the limiting radii and the shrinking cores lead to the
contraction of the limiting radii. There are 32 OCs of our sample in the
regions R2 and R4 in Fig. 17(a); 16 out of 19 OCs with R$_{core} < 1.5$
pc in the region R4 of Fig. 17(a) have R$_{RDP} < 7$ pc; three of 19 OCs
have R$_{RDP} > 7$ pc. Similarly, 10 out of 13 OCs in the region R2 of
Fig. 17(a) with R$_{core} > 1.5$ pc have R$_{RDP} > 7$ pc; three of them
have R$_{RDP} < 7$ pc. Here, R$_{RDP} = 7$ pc means the separation into
two groups of our sample (Fig.11a). These findings imply that the OCs
with their core expanding could have small cluster limiting radii, in a
similar manner, the OCs with shrinking cores could have large limiting
cluster radii. Note that there are six OCs with incompatible cores and
limiting radii in the regions R2 and R4 of Fig. 17(b). These six OCs are
inconsistent with the arguments of <cit.>.
* For this paper, we do not make an effort to determine the binary fractions of our sample OCs.
However, the OCs Binaries widen the main sequence of the OCs by as much as 0.75 mag, so
theoretical isochrones are fitted to the mid-points of CMDs of the OCs,
rather than the faint or blue sides, as emphasized by <cit.>.
Binaries are indeed an effective way of storing energy in a cluster. Non-primordial
binary formation, especially the close ones, requires many encounters of at least 3
stars, 2 of which end up having orbits around each other and the 3rd one gets
"ejected". So, depending on the binary fraction, a cluster can get dynamically
swollen. As a consequence of the dynamical evolution in OCs, multiple systems
tend to concentrate in central regions <cit.>. As indicated by <cit.>, the main effect of
a significant fraction of binaries in central parts of OCs
is that the number of low-mass stars is underestimated with respect to the higher mass stars.
<cit.> give the complete fractions of binaries as 35 $\%$ to 70 $\%$.
They also give a minimum binary fraction, which is larger than 11 $\%$ within the core of OCs.
* It is seen from Figs. 18(a) and (b) that the OCs with R$_{RDP} < 3$ pc and
R$_{core} < 0.6$ pc inside the Solar circle are older than 1 Gyr.
As they lost their stellar content, they shrunk in size and mass with
time. Nevertheless, they seem to survive against external shocks for a
longer time, according to the simulations of <cit.>. As one can see
from Fig. 18(b), there is no strong dependence of R$_{core}$ with
* As can be seen in Figs. 19(a) and (b), OCs with large dimensions are on
the average more massive. There does not seem to be an age dependence
for the relations in the panels (a)$-$(b) of Fig 19. As can be seen from Fig. 20(a),
massive and less massive OCs than m$_{overall} =2000\:
M_{\odot}$ are located indistinctly in- or outwards of the Solar circle.
Less massive OCs which are located outside the Solar circle appear to
survive, because they are subject to less external dynamical processes:
The OCs inside the Solar radius survive against the combined dynamical
effects such as interactions with GMCs, tidal effects with the spiral
arm and the Galactic disc, which are quite efficient in the Galactic
center directions. As is seen in Fig. 20(b), less massive OCs older than
1 Gyr are scarcer since they are dissolved into the field, i.e. the more
massive and older OCs ($ > 1$ Gyr) survive.
* The OCs with flat/steep positive overall MF slopes for $\chi_{core}<0$ in Fig. 21 show signs of
a mild to large scale mass segregation, whereas the OCs with negative
overall MF slopes for $\chi_{core} < 0$ indicate an advanced dynamical
evolution. These MF slopes of $\chi_{core}<0$ in Fig. 21 can be explained
by the external dynamical effects such as tidal stripping by tidal
interactions (in the form of shocks) due to disc and bulge
crossings, as well as encounters with GMCs.
* As considered these MF slopes in Figs. 22(a) and (b), OCs are formed with
flat core and Kroupa and Salpeter-like overall MFs, as stated by
<cit.>. As is seen from Fig. 22, at cluster birth the core MF
seems to be much flatter than the overall MF. Early core flattening may
be partly linked to primordial processes associated to molecular-cloud
Within the expected uncertainties the overall MF values are quite close to
$\chi_\circ=1.30\pm0.30$ of <cit.> and $\chi_\circ=1.35$ of
<cit.>. However, our core MF value are smaller than the ones of
Kroupa and Salpeter.
As is seen from panels (a)$-$(b), except for few MF slopes, the overall
and core MF slopes tend to be negative values towards older ages,
because of mild/large scale mass segregation, the presence of GMCs and
tidal effects from disk and Bulge crossings as external processes.
* Most of the OCs of with positive overall slopes in Fig. 23(a) are
mass-rich and present little or no signs of mass segregation.
Apparently they retain their low-mass stars
because they are strongly bounded to the clusters. The OCs with negative
overall MF slopes in Fig. 23(a) seem to be in the phase of more advanced
dynamical evolution.
In panels (a)$-$(b) of Fig. 23 there is no indication of age dependence
that is seen between the MF slopes and the core- or overall-masses.
In panels (a)$-$(b) of Fig. 23, the OCs with steep overall and
core MF slopes present signs of larger scale mass segregation in the core
or halo region. As expected, there are no indications of age dependence
among the positive/negative MF slopes.
* From Fig. 27(a), one sees that for $\tau > 30$, the overall MF slopes of the
OCs are negative, with one exception. For $\tau < 30$, the overall MF
slopes of the remaining OCs fall in the range of $ +0.5< \chi_{overall}
< +2.5$. For $\tau>30$, as a result of the loss of low mass stars, $\chi_{overall}$
tends to negative values. As can be seen from panel (b), the core MF slopes
for the majority of OCs tend to be negative values after $\tau\approx 32$,
with two exceptions. In panel (b) there are two
OCs with flat slopes for $\tau< 32$.
For $\tau>32$, it is seen from panel (b) that the OCs with dynamically
evolved cores reveal a sign of strong mass segregation.
From eleven OCs, <cit.> detected the significant flattening in
MF slopes for $\tau_{core} \leq 100$ and $\tau_{overall} \leq 7$,
respectively. From their OCs, <cit.> give for these values
$\tau_{core} \le 1000$ and $\tau_{overall}\le 450$, respectively.
Here we detect the flattening of MF slopes at $\tau_{core} \leq 32$
and $\tau_{overall} \leq 30$, respectively. However, these values
are affected by the sample size with young and old OCs.
Note that our sample also contains the OCs with intermediate
and old ages.
The overall MF slopes of 31 OCs with $m > 0.5$, M$_{\odot}$ could
have been compared with the one given by <cit.>.
As compared to the uncertainties of our MF slopes (Col. 5; Table 7)
and the one ($\pm0.3$) of Kroupa, the overall MF slopes of 14 out of
31 OCs are consistent with the one of <cit.>,
which implies little or no dynamical evolution for these clusters. The
remaining 17 OCs with MF slopes that depart from that of <cit.>
show mild to large scale mass segregation, due to the dynamical evolution.
* We do not make an effort to determine the binary fractions of our sample OCs.
<cit.> give the complete fractions of binaries as 35 $\%$ to 70 $\%$.
They also give a minimum binary fraction, which is larger than 11 $\%$ within the core of OCs.
However, binaries of the OCs widen the main sequence of the OCs by as much as 0.75 mag, so
theoretical isochrones are fitted to the mid-points of CMDs of the OCs,
rather than the faint or blue sides, as emphasized by <cit.>. We have considered this issue
for isochrone fitting to CMDs (see Sect.4).
Binaries are indeed an effective way of storing energy in a cluster. Non-primordial
binary formation, especially the close ones, requires many encounters of at least 3
stars, 2 of which end up having orbits around each other and the 3rd one gets
"ejected". So, depending on the binary fraction, a cluster can get dynamically
swollen. As a consequence of the dynamical evolution in OCs, multiple systems
tend to concentrate in central regions <cit.>. As indicated by <cit.>, the main effect of
a significant fraction of binaries in central parts of OCs
is that the number of low-mass stars is underestimated with respect to the higher mass stars.
We thank the anonymous referee for her/his comments and suggestions on the manuscript.
We thank C. Chavarria for the correction of English in the text.
This publication makes use of data products from the Two Micron All Sky Survey,
which is a joint project of the University of Massachusetts and the
Infrared Processing and Analysis Centre/California Institute of Technology,
funded by the National Aeronautics and Space Administration and the National
Science Foundation. This research has made use of the WEBDA database, operated
at the Institute for Astronomy of the University of Vienna.
[Arce & Goodman (1999)]arc99
Arce, H. G., & Goodman, A. A. 1999., ApJ, 512, L135
[Bahcall & Soneira (1980)]bs80
Bahcall, J.N., & Soneira, R.M., 1980, ApJ, 238, 17
[Bergond et al.(2001)]ber01
Bergond, G., Leon, S., Guibert, J., 2001, A&A, 377, 462
[Bica & Bonatto(2005)]bic05
Bica, E., Bonatto, Ch., 2005, A&A, 443, 465
[Bica et al.(2006a)]bic06a
Bica, E., Bonatto, Ch., Blumberg, R., 2006a, A&A, 460, 83
[Bica et al.(2006b)]bic06b
Bica, E., Bonatto, Ch., Barbuy, B., Ortolani, S., 2006b, A&A, 450, 105
[Bica et al.(2008a)]bic08a
Bica, E., Bonatto, Ch., Camargo, D., 2008, MNRAS, 385, 349
[Bica et al.(2008b)]bic08b
Bica, E., Bonatto, Ch., Dutra, C., 2008, A&A, 489, 1129
[Binney & Merrifield (1998)]bin98
Binney, J. & Merrifield, M. 1998, in Galactic Astronomy, Princeton,
NJ: Princeton University Press. (Princeton series in astrophysics)
[Bonatto et al.(2004)]bon04
Bonatto, Ch., Bica, E., Girardi, L., 2004, A&A, 415, 571
[Bonatto & Bica (2005)]bon05
Bonatto, Ch., Bica, E., 2005, A&A, 437, 483
[Bonatto et al.(2005)]bont05
Bonatto, Ch., Bica, E., Santos Jr., J.F.C., 2005, A&A, 433, 917
[Bonatto & Bica (2006a)]bon06a
Bonatto, Ch., Bica, E., 2006a, A&A, 455, 931
[Bonatto et al. (2006b)]bon06b
Bonatto, Ch., Kerber, L.O., Bica, E., Santiago, B.X., 2006b, A&A, 446, 121
[Bonatto & Bica (2007a)]bon07a
Bonatto, Ch., Bica, E., 2007a, A&A, 473, 445
[Bonatto & Bica (2007b)]bon07b
Bonatto, Ch., Bica, E., 2007b, MNRAS, 377, 1301
[Bonatto & Bica (2008)]bon08
Bonatto, C., Bica, E, 2008, A&A, 485, 81
[Bonatto & Bica (2009a)]bon09a
Bonatto, Ch., Bica, E., 2009a, MNRAS, 392, 483
[Bonatto & Bica (2009b)]bon09b
Bonatto, Ch., Bica, E., 2009b, MNRAS, 394, 2127
[Bonatto & Bica (2009c)]bon09c
Bonatto, Ch., Bica, E., 2009c, MNRAS, 397, 1915
[Bonatto & Bica (2010)]bon10
Bonatto, Ch., Bica, E., 2010, A&A, 521, 74
[Bonatto & Bica (2011)]bon11
Bonatto, Ch., Bica, E., 2011, MNRAS, 415, 2827
[Bonifacio et al.(2000)]bon
Bonifacio, P., Monai, S., Beers, T.C., 2000, AJ, 120, 2065
[Bragaglia et al.(2001)]bra01
Bragaglia, A., Carretta, E., Gratton, R. G., Tosi, M. et al., 2001, AJ, 121, 327
[Bragaglia et al.(2008)]bra08
Bragaglia, A., Sestito, P., Villanova, S., Carretta, E., Randich, S., Tosi, M., 2008 A&A, 480, 79
[Brown et al.(2008)]bro96
Brown, J. A., Wallerstein, G., Geisler, D., Oke, J. B., 1996, AJ, 112, 1551
[Bukowiecki et al.(2011)]buk11
Bukowiecki, L., Maciejewski, G., Konorski, P., Strobel, A., 2011, AcA, 61, 231
[Caetano et al.(2015)]cae15
Caetano T.C., Dias, W.S., Lepine, J.R.D., Monteiro, H.S., Moitinho, A., Hickel, G.R., Oliveira, A.F., 2015, New A, 38, 31
[Camargo et al.(2009)]cam09
Camargo, D., Bonatto, C., Bica, E., 2009, A&A, 508, 211
[Camargo et al.(2010)]cam10
Camargo, D., Bonatto, C., Bica, E, 2010, A&A, 521, 42
[Cambresy et al. (2005)]camb05
Cambresy, L., Jarett, T.H., Beichman, C.A., 2005, A&A, 435, 131
[Carney (2001)]car01
Carney, B. 2001, Star Clusters, Saas-Fee Advanced Course 28, Lecture Notes 1998,
Swiss Society for Astrophysics and Astronomy, eds. L. Labhardt and B. Binggeli
(Berlin: Springer-Verlag) pp. 1-222
[Carraro et al. (2014)]car14
Carraro, G., Villanova, S., Monaco, L., Beccari, G., Ahumada, J. A., Boffin, H. M. J., 2014, A&A, 562, A39
[Carrera et al. (2007)]carr07
Carrera, R., Gallart, C., Pancino, E., Zinn, R., 2007, AJ, 134, 1298
[Carrera (2007)]carr12
Carrera, R., 2012, A&A, 544, A109
[Carretta et al. (2004)]cart04
Carretta, E., Bragaglia, A., Gratton, R. G., Tosi, M. 2004, A&A, 422, 951
[Chen et al. (1999)]chen99
Chen, B., Figueras, F., Torra, J., Jordi, C, Luri, X., Galadi-Enriquez, D., 1999, A&A, 352, 459
[Dias et al. (2002)]dia02
Dias, W. S., Alessi, B. S., Moitinho, A., & Lépine, J. R. D., 2002, A&A, 389, 871
[Dias et al. (2012)]dia12
Dias, W. S., Alessi, B. S., Moitinho, A., & Lépine, J. R. D., 2012, 2012yCat, VizieR On-line Data Catalog: B/ocl.
[Dutra et al.(2002)]dut02a
Dutra, C.M., Santiago, B.X., Bica, E., 2002, A&A, 383, 219
[Dutra et al.(2002)]dut02b
Dutra, C.M., Bica, E., 2002, A&A, 383, 631
[Elson et al.(1987)]els87
Elson, R.A.W., Fall, S.M., & Freeman, K.C., 1987, ApJ, 323, 54
[Friel et al.(2005)]fri05
Friel, E. D., Jacobson, H.R., Pilachowski, C.A., 2005, AJ, 129, 272
[Gieles et al.(2006)]gie06
Gieles, M., Portegies-Zwart, S., Athanassoula, E., Baumardt, H., Lamers,
H.J.G.L.M., Sipior, M, Leenaarts, J., 2006, MNRAS, 371, 793
[Gieles et al.(2007)]gie07
Gieles, M., Athanassoula, E, Portegies-Zwart, S. 2007, MNRAS, 376, 809
[Gonzalez et al.(2012)]gon12
Gonzalez, O.A., Rejkuba, M., Zoccali, M., Valenti, E., Minniti, D., Schultheis, M., Tobar, R., Chen, B., 2012, A&A, 543, 13
[Gratton et al.(1994)]gra94
Gratton, R. G., Contarini, G. 1994, A&A, 283, 911
[Güneş et al.(2012)]gun12
Güneş, O., Karataş, Y., Bonatto, C., 2012, New A, 17, 720
[Jacobson et al.(2007)]jac07
Jacobson, H. R., Friel, E. D., Pilachowski, C. A., 2007, AJ, 134, 1216
[Jacobson et al.(2008)]jac08
Jacobson, H. R., Friel, E. D., Pilachowski, C. A., 2008, AJ, 135, 2341
[Jacobson et al.(2011)]jac11
Jacobson, H. R., Pilachowski, C. A., Friel, E. D., 2011, AJ, 142, 59
[Janes & Phelps (1994)]jan94
Janes, K.A., Phelps, R.L., 1994, AJ, 108, 1773
[Joshii (2005)]josh05
Joshii, Y.C., 2005, MNRAS, 362, 1259
[Kharchenko et al.(2013)]kha13
Kharchenko, N. V., Piskunov, E. A., Schilbach, E., Roeser, S., Scholz, R. -D, 2013, A&A, 558A, 53
[King (1962)]kin62
King, I., 1962, AJ, 67, 471
[King (1966)]kin66
King, I., 1966, AJ, 71, 64
[Kroupa (2001)]kro01
Kroupa, P., 2001, MNRAS, 322, 231
[Lamers & Gieles (2006)]lam06
Lamers, H.J.G.L.M., Gieles, M., 2006, A&A, 455, 17
[Lynga (1982)]lyn82
Lynga, G., 1982, A&A, 109, 213
[Lynga (1988)]lyn88
Lynga, G., 1988, ESO Conf. Workshop Proc., 28, 379
[Mackey et al.(2003)]mac03
Mackey, A.D., Gilmore, G.F., 2003, MNRAS, 338, 120
[Mackey et al.(2008)]mac08
Mackey, A. D., Wilkinson, M. I., Davies, M. B., Gilmore, G. F., 2008, MNRAS, 386, 65
[Maciejewski & Niedzielski(2007)]mn07
Maciejewski, G., Niedzielski, A., 2007, A&A, 467, 1065
[Marigo et al. (2008)]mar08
Marigo, P., Girardi, L., Bressan, A., Groenewegen, M. A. T., Silva, L., & Granato, G. L., 2008, A&A, 482, 883 (M08)
[Mermilliod (1992)]mer92
Mermilliod, J.C., 1992, Bull. Inform, CDS, 40, 115
[Momany et al. (2006)]mom06
Momany, Y., Zaggia, S., Gilmore, G.F., Piotto, G., Carraro, G., Bedin, L.R., de Angeli, F., 2006, A&A, 451, 515
[Nilakshi et al.(2002)]nil02
Nilakshi, S. R., Pandey, A. K., Mohan, V., 2002, A&A, 383, 153
[Nishiyama et al.(2006)]nis06
Nishiyama, S., Nagata, T., Sato, S., Kato, D., Nagayama, T., Kusakabe, N., Matsunaga, N., Naoi, T., Sugitani, K., Tamura, M., 2006, ApJ, 647, 1093
[Pasquini et al.(2004)]pas04
Pasquini, L., Randich, S., Zoccali, M., Hill, V., Charbonnel, C., Nordstrom, B., 2004, A&A, 424, 951
[Piskunov et al.(2008)]pis08
Piskunov, E. A., Schilbach, E., Kharchenko, N. V., Roeser, S., Scholz, R. -D, 2008, A&A, 477, 165
[Portegies Zwart & McMillan (2000)]pm00
Portegies Zwart, S., McMillan, S., 2000, ApJ, 17, 2000
[Portegies Zwart et al. (2002)]por02
Portegies Zwart, S., Makino, J., McMillan, S., Hut, P., 2002, ApJ, 565, 265
[Portegies Zwart et al. (2010)]por10
Portegies Zwart, S., McMillan, S., Gieles, M., 2010, ARA&A, 48, 431
[Salpeter (1955)]sal55
Salpeter, E. 1955, ApJ, 121, 161
[Schilbach et al.(2006)]sch06
Schilbach, E., Kharchenko, N. V., Piskunov, A.E., R�ser, S., Scholz, R.-D., 2006, A&A, 456, 523
[Schuster et al. (2004)]sch04
Schuster, W. J., Beers, T. C., Michel, R., Nissen, P. E., & García G.,
2004, A&A, 422, 527
[Schlegel et al.(1998)]sch98
Schlegel, D.J., Finkbeiner, D. P., & Davis, M., 1998, ApJ, 500, 525
[Sharma et al.(2006)]sha06
Sharma, S., Pandey, A.K., Ogura, K., Mrro, H., Tarusawa, K., Sagar, R., AJ, 132, 1669,
[Skrutskie et al.(2006)]skr06
Skrutskie, M.F., Cutri, R., Stiening, R., Weinberg, M.D., Schneider, S.E., Carpenter, J.M., Beichman, C., Capps, R., 2006, AJ, 131, 1163
[Smiljanic et al.(2008)]smi08
Smiljanic, R., Gauderon, R., North, P., Barbuy, B., Charbonnel, C., Mowlavi, N., 2008, A&A, 502, 267
[Sollima et al.(2010)]sol10
Sollima, A., Carballo-Bello, J.A., Beccari, G., Ferraro, F.R., Fusi Pecci, F., Lanzoni, A., 2010, MNRAS, 401, 577
[Spitzer&Chevalier (1973)]sc73
Spitzer, L., Chevalier, R.A., 1973, ApJ, 183, 565
[Tadross et al.(2002)]tad02
Tadross, A.L., Werner, P., Osman, A., Marie, M., 2002, New A, 7, 553
[Takahasi& Portegies Zwart(2000)]tak00
Takahasi, K., & Portegies Zwart, S. F. 2000, ApJ, 535, 759
[Trumpler (1930)]tru30
Trumpler, R.J., 1930, Lick Obs. Bull., 14, 154
[Wilson (1975)]wil75
Wilson, C.P., 1975, AJ, 80, 175
[van den Bergh & McClure (1980)]van80
van den Bergh, S., McClure, R.D., 1980, A&A, 88, 360
[van den Berg et al. (1991)]van91
van den Berg, S., Morbey, C., Pazder, J., 1991, ApJ, 375, 594
§ SUPPLEMENTARY MATERIAL
Figures S5-S9: Observed decontaminated J, (J-H) CMDs of 40 OCs. The CMD filter shown
with the shaded area is used to isolate cluster MS/evolved stars.
Figures S10-S13: Stellar RDPs (open circles) of 40 Ocs built with CMD filtered photometry.
Solid line shows the best-fit King profile. Horizontal red bar: stellar background level measured
in the comparison field. Shaded region: $1\sigma$ King fit uncertainty.
Table S6: The number of stars, mass information, mass function slope, mass density, which correspond to cluster regions of available clusters
for the cases of Evolved, Observed+Evolved, and Extrapolated+Evolved.
[width = 14cm, height = 15cm]FigS5.jpg
Observed decontaminated J, (J-H) CMDs of 10 OCs.
The CMD filter shown with the shaded area is used to isolate cluster MS/evolved stars.
The OCs in the panels are NGC 436, King 5, NGC 1513, Be 15, Be 17, NGC 1907, NGC 2112, and Koposov 12, respectively.
[width = 14cm, height = 15cm]FigS6.jpg
Observed decontaminated J, (J-H) CMDs of NGC 1798, Koposov 53, NGC 2194, NGC 2192, NGC 2158, Col 110, NGC 2262,
NGC 2286, respectively.
The symbols are the same as Fig. S5.
[width = 14cm, height = 15cm]FigS7.jpg
Observed decontaminated J, (J-H) CMDs of NGC 2243, Trumpler 5, Be 36, Haffner 8, NGC 2309, To 2, Pismis 3, NGC 2660, respectively.
The symbols are the same as Fig. S5.
[width = 14cm, height = 15cm]FigS8.jpg
Observed decontaminated J, (J-H) CMDs of Mel 71, NGC 2425, NGC 2506, Trumpler 20, NGC 3680, Ru 96, Ru 105, NGC 6819, respectively.
The symbols are the same as Fig. S5.
[width = 14cm, height = 15cm]FigS9.jpg
Observed decontaminated J, (J-H) CMDs of Pismis 19, NGC 6134, IC 4651, NGC 6802, Be 89, NGC 6939, NGC 7142, NGC 7789, respectively.
The symbols are the same as Fig. S5.
[width = 14cm, height = 15cm]FigS10.eps
Stellar RDPs (open circles) of 10 Ocs built with CMD filtered photometry.
Solid line shows the best-fit King profile. Horizontal red bar: stellar background level measured
in the comparison field. Shaded region: $1\sigma$ King fit uncertainty.
The OCs in the panels are NGC 436, Ki 05, NGC 1513, Be 15, NGC 1798, NGC 2112, Koposov 12, and NGC 2158, respectively.
[width = 14cm, height = 15cm]FigS11.eps
Stellar RDPs (open circles) of Koposov 53, NGC 2194, NGC 2192, NGC 2243, Trumpler 5, Col 110, NGC 2262, NGC 2286, NGC 2309, To 2, respectively.
The symbols are the same as Fig. S10.
[width = 14cm, height = 15cm]FigS12.eps
Stellar RDPs (open circles) of Be 36, Haffner 8, Mel 71, NGC 2425, NGC 2506, Pismis 3, NGC 2660, NGC 3680, Ru 96, Ru 105, respectively.
The symbols are the same as Fig. S10.
[width = 14cm, height = 15cm]FigS13.eps
Stellar RDPs (open circles) of Trumpler 20, Pismis 19, NGC 6134, IC 4651, NGC 6802, NGC 6819, Be 89, NGC 6939, NGC 7142, NGC 7789, respectively.
The symbols are the same as Fig. S10.
The number of stars, mass information, mass function slope, mass density, which correspond to cluster regions of available clusters
for the cases of Evolved, Observed+Evolved, and Extrapolated+Evolved.
11cNGC 436
2c $\chi$
Region N* m$_{evol}$ 1.38-2.78 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.04 1$\pm$1 0.4$\pm$0.4 -1.46$\pm$0.47 - 0.25$\pm$0.03 0.56$\pm$0.28 11.9$\pm$5.97 0.4$\pm$0.1 0.7$\pm$0.03 15.2$\pm$6.11
1.04-6.97 12$\pm$6 3.5$\pm$1.8 1.74$\pm$0.36 - 1.01$\pm$0.1 2$\pm$0.57 0.14$\pm$0.04 25.6$\pm$19.6 9.8$\pm$3.8 0.69$\pm$0.27
0.0-6.97 14$\pm$6 3.9$\pm$1.9 0.86$\pm$0.29 - 1.12$\pm$0.09 2.55$\pm$0.63 0.18$\pm$0.04 17.1$\pm$11.9 7.9$\pm$2.4 0.56$\pm$0.17
11cKing 5
2c $\chi$
Region N* m$_{evol}$ 1.28-1.78 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-0.95 9$\pm$3 1.8$\pm$0.6 -3.06$\pm$0.96 - 0.48$\pm$0.05 0.77$\pm$0.34 21.5$\pm$9.6 0.63$\pm$0.09 0.91$\pm$0.35 25.3$\pm$9.73
0.95-5.62 27$\pm$8 5.5$\pm$1.7 2.65$\pm$0.41 - 1.98$\pm$0.14 3.07$\pm$0.62 0.41$\pm$.0.08 80.2$\pm$61.7 27.3$\pm$11.7 3.69$\pm$1.58
0.0-5.62 35$\pm$9 7.3$\pm$1.8 1.8$\pm$0.49 - 2.44$\pm$0.16 3.84$\pm$0.89 0.52$\pm$0.12 86.9$\pm$67.2 30.0$\pm$12.8 4.04$\pm$1.73
11cNGC 1513
2c $\chi$
Region N* m$_{evol}$ 0.68-0.98 0.98-3.33 N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.65 - - -3.37$\pm$0.52 1.12$\pm$0.24 1.77$\pm$0.11 2.31$\pm$0.27 12.3$\pm$1.41 10.8$\pm$7.65 4.54$\pm$1.43 24.1$\pm$7.6
1.65-6.51 - - - 2.08$\pm$0.17 4.25$\pm$0.3 5$\pm$0.42 0.44$\pm$0.04 45.2$\pm$34.6 15.4$\pm$6.42 1.35$\pm$0.57
0.0-6.51 - - - 1.9$\pm$0.12 5.9$\pm$0.32 7.13$\pm$0.46 0.62$\pm$0.04 58.6$\pm$44.4 20.4$\pm$8.23 1.77$\pm$0.71
11cBe 15
2c $\chi$
Region N* m$_{evol}$ 2.03-2.68 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.16 - - - - - - - - - -
1.16-5.73 - - -3.07$\pm$1.07 - 0.35$\pm$0.09 0.83$\pm$1.11 0.15$\pm$0.21 0.61$\pm$0.28 1.23$\pm$1.19 0.23$\pm$0.22
0.0-5.73 - - -1.54$\pm$1.15 - 0.42$\pm$0.09 1$\pm$1.4 0.19$\pm$0.26 1.2$\pm$0.83 1.96$\pm$1.73 0.36$\pm$0.32
11cNGC 1907
2c $\chi$
Region N* m$_{evol}$ 1.18-2.83 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.28 9$\pm$3 2.5$\pm$0.9 -0.76$\pm$0.4 - 0.78$\pm$0.07 1.68$\pm$0.61 19.2$\pm$6.99 1.62$\pm$2.02 2.24$\pm$0.84 25.5$\pm$9.55
1.28-4.26 3$\pm$3 0.9$\pm$1 -0.73$\pm$0.43 - 1.6$\pm$0.11 2.81$\pm$0.93 0.89$\pm$0.3 3.69$\pm$8.12 4$\pm$2.42 1.27$\pm$0.77
0.0-4.26 12$\pm$5 3.5$\pm$1.3 0.0$\pm$0.23 - 2.36$\pm$0.14 4.56$\pm$0.91 1.41$\pm$0.28 - - -
11cNGC 2112
2c $\chi$
Region N* m$_{evol}$ 0.93-1.63 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.64 19$\pm$5 3$\pm$0.8 -1.28$\pm$0.51 - 1.56$\pm$0.09 2.06$\pm$0.34 11.1$\pm$1.86 2.8$\pm$3.7 2.73$\pm$1.17 14.8$\pm$6.36
1.64-5.92 10$\pm$8 1.6$\pm$1.4 0.26$\pm$0.39 - 3.59$\pm$0.23 4.33$\pm$0.5 0.51$\pm$0.06 - - -
0.0-5.92 28$\pm$10 4.6$\pm$1.6 0.5$\pm$0.42 - 5.11$\pm$0.24 6.38$\pm$0.78 0.73$\pm$0.09 44.9$\pm$31.5 18.1$\pm$5.96 2.08$\pm$0.69
11cNGC 2158
2c $\chi$
Region N* m$_{evol}$ 1.28-1.53 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.74 57$\pm$8 8.7$\pm$1.2 -4.24$\pm$1 - 1.9$\pm$0.11 2.74$\pm$0.91 12.4$\pm$4.12 4.21$\pm$0.82 5.21$\pm$1.27 23.6$\pm$5.74
1.74-14.03 203$\pm$22 30.8$\pm$0.3.3 -0.08$\pm$0.6 - 8.39$\pm$0.31 12$\pm$2.56 0.1$\pm$.0.02 - - -
0.0-14.03 260$\pm$23 39.6$\pm$3.5 -1.55$\pm$0.71 - 10.3$\pm$0.33 14.7$\pm$3.69 1.27$\pm$0.04 - 33.4$\pm$26.6 0.29$\pm$0.23
11cKoposov 53
2c $\chi$
Region N* m$_{evol}$ 1.28-2.08 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-0.66 2$\pm$1 0.4.$\pm$0.3 -3.96 $\pm$3.4 - - - - - - -
0.66-4.18 1$\pm$3 0.3$\pm$0.6 1.36$\pm$0.92 - 0.26$\pm$0.05 0.43$\pm$0.26 0.14$\pm$0.08 7.33$\pm$6.22 2.62$\pm$1.3 0.86$\pm$0.43
0.0-4.18 3$\pm$3 0.5$\pm$0.7 0.93$\pm$0.81 - 0.33$\pm$0.05 0.54$\pm$0.28 0.18$\pm$0.09 5.94$\pm$5.15 2.37$\pm$1.1 0.78$\pm$0.36
11cNGC 2194
2c $\chi$
Region N* m$_{evol}$ 1.23-1.98 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.66 18$\pm$5 4.1$\pm$1.0 0.38$\pm$0.42 - 1.39$\pm$0.09 2.27$\pm$0.5 11.9$\pm$2.6 14.5$\pm$9.81 6.84$\pm$1.96 35.7$\pm$10.2
1.66-6.55 19$\pm$7 4.3$\pm$1.6 3.65$\pm$0.48 - 3.8$\pm$0.17 5.69$\pm$1.38 0.49$\pm$.0.12 159$\pm$123 53.2$\pm$23.3 4.6$\pm$2.01
0.0-6.55 38$\pm$9 8.4$\pm$1.9 2.52$\pm$0.37 - 5.18$\pm$0.2 8.03$\pm$1.56 0.68$\pm$0.13 170$\pm$129 58.4$\pm$24.3 4.96$\pm$2.07
11cNGC 2192
2c $\chi$
Region N* m$_{evol}$ 1.43-1.68 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.11 2$\pm$1 0.3$\pm$0.3 -2.78$\pm$0.96 - 0.17$\pm$0.03 0.27$\pm$0.16 4.71$\pm$2.77 0.27$\pm$0.06 0.37$\pm$0.17 6.49$\pm$2.89
1.11-5.47 17$\pm$5 3.3$\pm$1 -3.07$\pm$0.76 - 0.62$\pm$0.07 1.03$\pm$0.36 0.15$\pm$0.05 1.32$\pm$0.26 1.78$\pm$0.45 0.26$\pm$0.07
0.0-5.47 19$\pm$6 3.5$\pm$1 -3.12$\pm$0.43 - 0.79$\pm$0.08 1.28$\pm$0.28 0.19$\pm$0.04 1.7$\pm$0.21 2.27$\pm$0.35 0.33$\pm$0.05
11cNGC 2243
2c $\chi$
Region N* m$_{evol}$ 1.43-1.63 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-0.89 6$\pm$3 1.$\pm$0.4 - - - - - - - -
0.89-12.94 29$\pm$10 4.8$\pm$1.6 4.81$\pm$1.16 - 1.33$\pm$0.08 1.99$\pm$1.31 0.03$\pm$0.01 198$\pm$182 65.1$\pm$39.5 0.72$\pm$0.43
0.0-12.94 36$\pm$10 5.8.1$\pm$1.7 2.09$\pm$1.01 - 1.49$\pm$0.08 2.26$\pm$1.35 0.03$\pm$0.01 154$\pm$136 51.7$\pm$29.0 0.57$\pm$0.32
11cTrumpler 5
2c $\chi$
Region N* m$_{evol}$ 1.18-1.43 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-3.86 93$\pm$12 13.2$\pm$1.7 0.42$\pm$0.75 - 4.35$\pm$0.18 5.75$\pm$1.23 2.39$\pm$0.51 879$\pm$693 340$\pm$138 14.1$\pm$5.72
3.86-15.18 169$\pm$34 24$\pm$4.9 1.64$\pm$1.43 - 11.2$\pm$0.5 14.7$\pm$6.37 0.1$\pm$0.04 530$\pm$444 169$\pm$88.8 1.19$\pm$0.62
0.0-15.18 262$\pm$37 37.2$\pm$5.2 1.32$\pm$1.18 - 15.6$\pm$0.54 20.4$\pm$7.15 0.14$\pm$0.05 699$\pm$568 223$\pm$111 1.55$\pm$0.76
11cCol 110
2c $\chi$
Region N* m$_{evol}$ 1.08-1.48 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-6.25 93$\pm$12 13.1$\pm$1.7 -2.58$\pm$0.21 - 4.57$\pm$0.19 5.91$\pm$0.41 0.58$\pm$0.04 7.97$\pm$1 8.54$\pm$0.56 0.83$\pm$0.05
6.25-12.12 70$\pm$16 10$\pm$2.3 -1.84$\pm$0.51 - 4.5$\pm$0.26 5.78$\pm$0.86 0.09$\pm$0.01 10$\pm$6.74 9.67$\pm$2.45 0.15$\pm$0.04
0.0-12.12 163$\pm$21 23.1$\pm$3 -2.84$\pm$0.57 - 9.06$\pm$0.34 11.7$\pm$1.84 0.16$\pm$0.02 15.1$\pm$3.95 16.5$\pm$2.44 0.22$\pm$0.03
11cNGC 2262
2c $\chi$
Region N* m$_{evol}$ 1.33-1.83 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-0.85 9$\pm$3 0. 16$\pm$0.06 -1.49$\pm$0.80 - 0.26$\pm$0.04 0.43$\pm$0.16 21.8$\pm$8.3 0.5$\pm$0.2 0.6$\pm$0.2 31.8$\pm$9.59
0.85-6.37 22$\pm$8 0.42$\pm$0.15 1.32$\pm$0.61 - 2.12$\pm$0.14 3.35$\pm$1.12 0.31$\pm$0.1 79.5$\pm$63 27.6$\pm$12.4 2.6$\pm$1.1
0.0-6.37 31$\pm$8 0.59$\pm$0.16 1.01$\pm$0.44 - 2.4$\pm$0.14 3.8$\pm$0.9 0.35$\pm$0.08 64.4$\pm$49 24.1$\pm$9.4 2.2$\pm$0.87
11cNGC 2286
2c $\chi$
Region N* m$_{evol}$ 0.98-2.03 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.59 2$\pm$2 0.4$\pm$0.4 1.3$\pm$0.5 - 0.46$\pm$0.05 0.62$\pm$0.17 3.93$\pm$1.02 6$\pm$4.6 2.3$\pm$0.9 13.7$\pm$5.31
1.59-6.39 3$\pm$5 0.6$\pm$1 1.76$\pm$0.33 - 1.55$\pm$0.14 2.08$\pm$0.34 0.19$\pm$0.03 27$\pm$20.4 9.3$\pm$3.8 0.86$\pm$0.35
0.0-6.39 5$\pm$5 1$\pm$1 1.45$\pm$0.3 - 1.98$\pm$0.15 2.7$\pm$0.39 0.25$\pm$0.04 31.5$\pm$23.6 11.1$\pm$4.4 1.01$\pm$0.4
11cNGC 2309
2c $\chi$
Region N* m$_{evol}$ 1.68-2.53 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-0.84 - - -1.52$\pm$1.03 - 0.19$\pm$0.03 0.41$\pm$0.46 - 0.36$\pm$0.18 0.58$\pm$0.48 23.4$\pm$19.3
0.84-7.5 - - -0.84$\pm$0.93 - 0.98$\pm$0.08 2.05$\pm$2 0.12$\pm$0.11 - 3.87$\pm$3.46 0.22$\pm$0.2
0.0-7.5 - - -0.89$\pm$0.6 - 1.16$\pm$0.09 2.43$\pm$1.54 0.14$\pm$0.09 - 4.5$\pm$2.51 0.25$\pm$0.14
11cBe 36
2c $\chi$
Region N* m$_{evol}$ - - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.32 15$\pm$4 2.1$\pm$0.6 - - - - - - - -
1.32-13.39 83$\pm$19 11.8$\pm$2.7 - - - - - - - -
0.0-13.39 98$\pm$20 13.8$\pm$2.8 - - - - - - - -
11cHaffner 8
2c $\chi$
Region N* m$_{evol}$ 1.08-1.88 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.47 - - 1.28$\pm$0.77 - 0.29$\pm$0.05 0.4$\pm$0.16 2.27$\pm$0.9 5.98$\pm$5.23 2.07$\pm$1.05 11.6$\pm$5.88
1.47-6.93 - - 1.99$\pm$0.71 - 1.02$\pm$0.15 1.39$\pm$0.48 0.1$\pm$0.03 25$\pm$19.6 8.43$\pm$3.75 0.61$\pm$0.27
0.0-6.93 - - 1.82$\pm$0.59 - 1.23$\pm$0.16 1.69$\pm$0.5 0.12$\pm$0.04 29.1$\pm$22.5 9.85$\pm$4.27 0.71$\pm$0.31
11cMel 71
2c $\chi$
Region N* m$_{evol}$ 1.08-1.78 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.27 13$\pm$4 2.4$\pm$0.7 0.3$\pm$1.04 - 0.77$\pm$0.06 1.13$\pm$0.44 13.1$\pm$5.11 - - -
1.27-5.0 18$\pm$7 3.2$\pm$1.3 1.82$\pm$0.42 - 2.4$\pm$0.15 3.39$\pm$0.63 0.66$\pm$0.12 62.9$\pm$47.8 21.4$\pm$8.95 4.16$\pm$1.74
0.0-5.0 31$\pm$8 5.5$\pm$1.4 1.29$\pm$0.4 - 3.17$\pm$0.16 4.44$\pm$0.72 0.85$\pm$0.14 64.1$\pm$49.0 21.8$\pm$9.21 4.26$\pm$1.76
11cNGC 2506
2c $\chi$
Region N* m$_{evol}$ 1.23-1.53 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.65 17$\pm$4 2.8$\pm$0.7 4.11$\pm$1.63 - 1.14$\pm$0.07 1.69$\pm$1.21 8.96$\pm$6.42 75.8$\pm$74.4 25.1$\pm$16.5 133$\pm$87.9
1.65-10.76 45$\pm$13 7.4$\pm$2.2 1.48$\pm$0.6 - 4.43$\pm$0.21 6.16$\pm$1.45 0.12$\pm$.0.03 207$\pm$161 68.2$\pm$30.6 1.31$\pm$0.59
0.0-10.76 63$\pm$14 10.2$\pm$2.3 0.97$\pm$0.63 - 5.58$\pm$0.22 7.77$\pm$1.9 0.15$\pm$0.04 188$\pm$151 65.8$\pm$29.4 1.26$\pm$0.56
11cPismis 3
2c $\chi$
Region N* m$_{evol}$ 1.13-1.38 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-2.1 63$\pm$9 8.6$\pm$1.3 -1.6$\pm$0.84 - 2.77$\pm$0.14 3.54$\pm$0.74 6.77$\pm$1.41 8.34$\pm$16.6 7.45$\pm$5.43 14.2$\pm$10.4
2.1-8.58 29$\pm$21 4$\pm$0.2.8 2.94$\pm$0.56 - 6.37$\pm$0.34 7.9$\pm$1.32 0.31$\pm$.0.05 345$\pm$267 109$\pm$49.8 4.19$\pm$1.92
0.0-8.58 92$\pm$23 12.5$\pm$3.1 1.71$\pm$0.49 - 9.14$\pm$0.38 11.4$\pm$1.59 0.43$\pm$0.06 418$\pm$321 133$\pm$59.8 5.03$\pm$2.26
11cNGC 3680
2c $\chi$
Region N* m$_{evol}$ 1.08-1.48 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-0.47 4$\pm$2 0.7$\pm$0.4 - - - - - - - -
0.47-2.98 11$\pm$4 1.9$\pm$0.8 0.24$\pm$1.04 - 0.36$\pm$0.06 0.44$\pm$0.16 0.57$\pm$0.16 - - -
0.0-1.72 14$\pm$5 2.6$\pm$0.8 -1.2$\pm$2.03 - 0.55$\pm$0.08 0.77$\pm$0.35 0.7$\pm$0.32 - - -
11cRu 96
2c $\chi$
Region N* m$_{evol}$ 1.33-1.93 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.43 7$\pm$3 1.4$\pm$0.7 4.58$\pm$0.98 - 0.36$\pm$0.05 0.71$\pm$0.37 5.75$\pm$2.99 23.9$\pm$21 8.15$\pm$4.46 66.5$\pm$36.4
1.43-2.6 9$\pm$4 1.8$\pm$0.9 6.18$\pm$0.72 - 0.52$\pm$0.07 0.82$\pm$0.29 1.34$\pm$0.48 32.5$\pm$26.6 10.9$\pm$5.3 17.7$\pm$8.63
0.0-2.6 15$\pm$6 3.1$\pm$1.1 4.55$\pm$0.65 - 0.92$\pm$0.09 1.49$\pm$0.48 2.02$\pm$0.66 46.5$\pm$37.3 15.8$\pm$7.36 21.4$\pm$10
11cRu 105
2c $\chi$
Region N* m$_{evol}$ - - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.79 10$\pm$4 2.2$\pm$0.8 - - - - - - - -
1.79-3.83 9$\pm$5 1.8$\pm$1.1 - - - - - - - -
0.0-3.83 19$\pm$7 4$\pm$1.4 - - - - - - - -
11cTrumpler 20
2c $\chi$
Region N* m$_{evol}$ 1.38-1.78 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-3.12 69$\pm$10 12.3$\pm$1.8 -1.07$\pm$0.5 - 2.18$\pm$1.35 3.58$\pm$0.79 2.81$\pm$0.62 6.7$\pm$10.8 6.94$\pm$3.32 5.45$\pm$2.61
3.12-13.98 188$\pm$32 33.6$\pm$5.7 2.85$\pm$0.95 - 7.9$\pm$0.45 12.6$\pm$5.46 0.11$\pm$0.05 379$\pm$326 130$\pm$68.2 1.15$\pm$0.6
0.0-13.98 256$\pm$34 45.9$\pm$6 2.06$\pm$0.68 - 10.1$\pm$0.47 16.2$\pm$4.96 0.14$\pm$0.04 434$\pm$350 150$\pm$69.9 1.31$\pm$0.61
11cPismis 19
2c $\chi$
Region N* m$_{evol}$ 1.38-2.18 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-0.54 14$\pm$4 3.1$\pm$0.9 -2.42$\pm$1.07 - 0.63$\pm$0.06 1.16$\pm$0.71 175$\pm$108 0.95$\pm$0.23 1.47$\pm$0.74 222$\pm$112
0.54-6.16 26$\pm$12 5.9$\pm$2.7 1.57$\pm$0.4 - 5.16$\pm$0.3 8.91$\pm$2.55 0.91$\pm$0.26 173$\pm$133 61.5$\pm$25.8 6.35$\pm$2.64
0.0-6.16 40$\pm$13 9$\pm$2.9 1.18$\pm$0.38 - 5.38$\pm$0.28 9.19$\pm$2.69 1.03$\pm$0.28 158$\pm$119 59.1$\pm$22.9 6.04$\pm$2.34
11cNGC 6134
2c $\chi$
Region N* m$_{evol}$ 1.28-1.78 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-0.45 9$\pm$3 1.5$\pm$0.5 -0.95$\pm$0.9 - 0.34$\pm$0.04 0.52$\pm$0.20 138$\pm$53.6 - 0.99$\pm$0.64 259$\pm$167
0.45-3.08 42$\pm$21 7.6$\pm$1.5 -1.3$\pm$1.33 - 1.54$\pm$0.14 2.47$\pm$1.39 2.03$\pm$.1.14 - 3.97$\pm$2.41 3.26$\pm$1.97
0.0-3.08 51$\pm$9 9.1$\pm$1.6 -0.83$\pm$1.11 - 1.87$\pm$0.15 2.98$\pm$1.39 2.44$\pm$1.13 - - -
11cIC 4651
2c $\chi$
Region N* m$_{evol}$ 0.93-1.53 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.02 16$\pm$4 2.4$\pm$0.7 -2.78$\pm$0.75 - 0.97$\pm$0.07 1.27$\pm$0.28 28.5$\pm$6.27 1.24$\pm$0.21 1.45$\pm$0.29 32.6$\pm$6.53
1.02-2.35 19$\pm$6 2.9$\pm$0.9 0.26$\pm$0.7 - 1.59$\pm$0.12 1.93$\pm$0.3 3.87$\pm$0.6 - - -
0.0-2.35 35$\pm$7 5.4$\pm$1.1 -0.6$\pm$0.41 - 2.55$\pm$0.14 3.21$\pm$0.35 5.91$\pm$0.65 - - -
11cNGC 6802
2c $\chi$
Region N* m$_{evol}$ 1.28-2.08 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.03 18$\pm$4 3.9$\pm$0.9 -0.46$\pm$0.84 - 1.02$\pm$0.07 1.8$\pm$0.88 39.4$\pm$19.2 - - -
1.03-4.24 25$\pm$7 5.3$\pm$1.4 2.4$\pm$0.24 - 3.59$\pm$0.17 5.77$\pm$0.85 1.83$\pm$0.27 121$\pm$91.2 42.1$\pm$17 13.4$\pm$5.42
0.0-4.24 43$\pm$8 9.2$\pm$1.7 1.66$\pm$0.24 - 4.59$\pm$0.18 7.53$\pm$1.09 2.36$\pm$0.34 130$\pm$97.7 46.4$\pm$18.3 14.5$\pm$5.72
11cNGC 6819
2c $\chi$
Region N* m$_{evol}$ 1.03-1.53 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.5 32$\pm$6 4.9$\pm$0.9 -1.07$\pm$0.55 - 1.94$\pm$0.1 2.6$\pm$0.45 18.4$\pm$3.2 - 4.57$\pm$4.19 32.3$\pm$29.6
1.5-12.92 68$\pm$23 10.3$\pm$3.5 0.96$\pm$0.42 - 8.71$\pm$0.41 11$\pm$1.4 0.12$\pm$0.01 152$\pm$114 53.6$\pm$21.4 0.59$\pm$0.24
0.0-12.92 100$\pm$24 15.1$\pm$3.7 0.47$\pm$0.4 - 10.6$\pm$0.42 13.6$\pm$1.63 0.15$\pm$0.02 124$\pm$87.5 49.1$\pm$16.5 0.54$\pm$0.18
11cBe 89
2c $\chi$
Region N* m$_{evol}$ 1.13-1.63 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-2.75 11$\pm$5 1.9$\pm$0.8 0.18$\pm$0.66 - 1.15$\pm$0.09 1.56$\pm$0.37 1.79$\pm$0.43 - - -
2.75-7.48 12$\pm$10 2$\pm$1.7 1.27$\pm$1.05 - 2.12$\pm$0.21 2.84$\pm$1.13 0.17$\pm$0.07 62.5$\pm$60.5 20.9$\pm$12.3 1.25$\pm$0.74
0.0-7.48 23$\pm$12 3.8$\pm$1.9 1.64$\pm$0.93 - 3.26$\pm$0.23 4.42$\pm$1.57 0.25$\pm$0.09 96$\pm$75.8 32$\pm$14.6 1.83$\pm$0.83
11cNGC 6939
2c $\chi$
Region N* m$_{evol}$ 1.03-1.63 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.16 25$\pm$5 4.$\pm$0.8 -2.17 $\pm$0.66 - 1.01$\pm$0.07 1.44$\pm$0.31 22$\pm$4.7 1.45$\pm$0.44 1.75$\pm$0.35 26.7$\pm$5.53
1.16-4.92 43$\pm$9 7$\pm$1.5 -0.51$\pm$0.46 - 3.3$\pm$0.16 4.47$\pm$0.71 1.41$\pm$0.22 - - -
0.0-4.92 68$\pm$10 11.1$\pm$1.7 -0.84$\pm$0.46 - 4.3$\pm$0.17 5.89$\pm$0.91 1.82$\pm$0.28 - - -
11cNGC 7142
2c $\chi$
Region N* m$_{evol}$ 1.18-1.53 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-1.98 22$\pm$5 3.4$\pm$0.8 -1.97 $\pm$1.44 - 0.96$\pm$0.07 1.36$\pm$0.68 4.19$\pm$2.09 - 2.48$\pm$1.19 7.61$\pm$3.67
1.98-11.19 85$\pm$18 12.9$\pm$2.7 -0.58$\pm$0.68 - 4.24$\pm$0.25 5.86$\pm$1.35 0.1$\pm$0.02 - - -
0.0-11.19 107$\pm$19 16.3$\pm$2.9 -1.06$\pm$0.56 - 5.21$\pm$0.25 7.22$\pm$1.38 0.12$\pm$0.02 - - -
11cNGC 7789
2c $\chi$
Region N* m$_{evol}$ 1.08-1.38 - N* m$_{obs}$ $\rho$ N* m$_{tot}$ $\rho$
(pc) (Stars) ($10^1 M_{\odot}$) ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$ ($10^2 Stars$) ($10^2 M_{\odot}$) $M_{\odot} pc^{-3}$
0.0-2.32 74$\pm$9 12$\pm$1.5 -0.42$\pm$0.45 - 3.87$\pm$0.14 5.4$\pm$0.82 10.3$\pm$1.56 - - -
2.32-26.88 147$\pm$60 23.6$\pm$9.7 1.03$\pm$0.66 - 19.9$\pm$1.02 24.7$\pm$4.25 0.03$\pm$0.005 574$\pm$471 192$\pm$90.5 0.24$\pm$0.11
0.0-26.88 221$\pm$61 35.6$\pm$9.8 0.79$\pm$0.65 - 23.8$\pm$1.03 29.7$\pm$4.87 0.04$\pm$0.006 556$\pm$444 194$\pm$85.5 0.24$\pm$0.1
Col. 1: the distance from the core. Cols. 2,6,9 : cluster stars for the regions in Col. 1.
Col. 4 gives the MF slopes ($\chi$), derived for the low-mass and high-mass ranges.
The masses of $m_{evol}$, $m_{obs}$, and $m_{tot}$ are listed in Cols. 3, 7 and 10, respectively.
The mass densities are given in Cols. 8 and 11.
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1511.00897
|
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
Laser Physics and Nonlinear Optics (LPNO), MESA+ Institute for Nanotechnology, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
NanoBioInterface, Saxion University of Applied Sciences,
P.O. Box 70000, 7500 KB Enschede, The Netherlands
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
Laser Physics and Nonlinear Optics (LPNO), MESA+ Institute for Nanotechnology, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
We investigate two-photon quantum interference in an opaque scattering medium that intrinsically supports $10^6$ transmission channels. By adaptive spatial phase-modulation of the incident wavefronts, the photons are directed at targeted speckle spots or output channels. From $10^3$ experimentally available coupled channels, we select two channels and enhance their transmission, to realize the equivalent of a fully programmable $2\times2$ beam splitter. By sending pairs of single photons from a parametric down-conversion source through the opaque scattering medium, we observe two-photon quantum interference. The programmed beam splitter need not fulfill energy conservation over the two selected output channels and hence could be non-unitary. Consequently, we have the freedom to tune the quantum interference from bunching (Hong-Ou-Mandel-like) to antibunching. Our results establish opaque scattering media as a platform for high-dimensional quantum interference that is notably relevant for boson sampling and physical-key-based authentication.
42.50.Dv, 42.25.Dd, 42.50.Ex
Light waves propagating through an opaque scattering medium exhibit a random walk inside the medium, which is caused by multiple scattering from spatial inhomogeneities <cit.>. An alternative description describes this by a transmission matrix <cit.>. The transmission matrix describes how a large amount of input channels is coupled to a similarly large amount of output channels, see Fig. <ref>. The number of these channels can be controlled, and easily made to exceed millions, by increasing the illuminated area on the medium. Recent advances in control of light propagation through complex wavefront shaping allow for complete control over these channels in multiple-scattering media <cit.>. Because of their large number of controllable channels, we explore the use of multiple-scattering media to study quantum interference between multiple photons. Employed as a platform for high-dimensional quantum interference, over a large number of channels, multiple-scattering media are of relevance to boson sampling <cit.>, quantum information processing <cit.>, and physical-key-based authentication <cit.>.
It has previously been observed that quantum states are robust against multiple scattering. Correlations in two-photon speckle patterns in single-scattering media have been studied <cit.>. Further, propagation of quantum noise <cit.> and propagation of single-photon Fock states through multiple-scattering media <cit.> have also been explored. So far it has remained an open question if quantum interference of multiple photons could be demonstrated inside a multiple-scattering medium. A hurdle one might expect in an experimental implementation is the low transmission of almost all channels in the multiple-scattering medium. Remarklably, the transmission per channel is not necessarily low since complex wavefront shaping allows funneling of light into selected output modes <cit.>.
A multiple-scattering medium couples millions of input and output channels. Light incident in the input channels results through multiple scattering in a complex interference pattern (speckle, see right panel) at the output, which can be imaged by a CCD camera. Each speckle spot in the image corresponds to an independent output channel. In this work we have programmed their interaction to create a network with 2 inputs ($k'$, $l'$) and 2 outputs ($m$, $n$).
Here we report on an experiment in which we study quantum interference in a multiple-scattering medium. We observe quantum interference of pairs of single photons in a programmable $2\times2$ beam splitter <cit.>, made of a multiple-scattering medium and a spatial light modulator (SLM). The spatial light modulator controls $10^3$ optical channels that are coupled in a reproducible yet unpredictable way in the multiple-scattering medium. The complex-wavefront-shaping technique using the SLM allows us to select two output channels out of $10^3$, by enhancing the amplitude of light transmitted in these channels. Programmability in this beam splitter is achieved by controlling the relative phase between the input and output arms, unlike a conventional beam splitter. We exploit this property to demonstrate not only the well-known Hong-Ou-Mandel-like bunching <cit.>, but also the antibunching of the outgoing photon pairs, as well as any intermediate situation. Our result establishes opaque scattering media as a platform for high-dimensional quantum interference experiments as needed in, e.g., boson sampling. At present we control about $10^3$ channels, but this can readily be scaled up to a number comparable to the number of pixels in modern spatial light modulators of order $10^6$.
Our quantum light source. (a) Photon pairs generated in a PPKTP crystal are separated by a polarizing beam splitter and coupled into single-mode fibers. The photons can be frequency-filtered by inserting a bandpass filter. (b) Measured Hong-Ou-Mandel interference without bandpass filter (cyan squares) and with bandpass filter (magenta circles). Error bars indicate the standard deviation in the measurements.
Pairs of single photons in our experiment are generated using collinear Type-II spontaneous parametric down-conversion (SPDC) with a mode-locked pump at wavelength centered at 395 nm <cit.> as shown in Fig. <ref>(a) <cit.>. The two orthogonally-polarized single photons ($\lambda_{\rm c}$ = 790 nm) are separated using a polarizing beam splitter (PBS) and coupled into single-mode fibers (SMF). The temporal delay between the two photons can be controlled with the help of a linear delay stage in one of the single-photon channels. To measure the degree of indistinguishability of the two photons, we observe Hong-Ou-Mandel (HOM) interference at a conventional 50:50 beam splitter as shown in Fig. <ref>(b). At a pump power of 10 mW, a HOM dip with a visibility of 64% is observed (cyan squares), with the usual definition of the visibility of $V\equiv\left(R_{\rm indist}-R_{\rm dist}\right)/R_{\rm dist}$, where $R_{\rm dist}$ and $R_{\rm indist}$ are the coincidence rates of distinguishable and indistinguishable photons, respectively. The visibility is less than 100% because of some residual spectral distinguishability of the two photons that was present within the broad spectrum of the Type-II SPDC. To improve the spectral indistinguishability, a bandpass filter with a bandwidth of 1.5 nm was used that resulted in a HOM dip with an increased visibility of 86% (magenta circles). The decrease in the spectral width also increases the width of the HOM dip as is evident in the figure.
To demonstrate programmable quantum interference in a multiple-scattering medium, we direct the light from the quantum-light source to a complex-wavefront-shaping setup with two single-mode fibers as shown in Fig. <ref>(a). The two fiber outputs have identical polarization and beam waist and form the input modes $k$ and $l$ for the quantum interference experiment. Both modes are phase modulated with a liquid-crystal spatial light modulator, and afterwards spatially overlapped using a half-wave plate and a polarizing beam splitter, resulting in a collinear propagation of the two modes with orthogonal polarizations. An objective (NA=0.95) is used to focus the light onto the scattering medium slab. The scattering medium consists of a layer of polytetrafluoroethylene (PTFE, Teflon) with a thickness of approximately 500 $\mu$m and has a scattering mean free path of 150 $\pm$ 10 $\mu$m, that we determined by a coherent backscattering cone measurement. The scattering medium was found to be more stable over extended periods of time in our laboratory environment than powder-based samples, which we attribute to its impermeability and therefore insensitivity to humidity changes. The experimental setup, including multiple-scattering medium, is interferometrically stable for a duration longer than a month. The transmitted light is collected by a second objective (NA=0.6) and after transmission through a PBS it is coupled into two multimode fibers (output modes $m$ and $n$) connected to single-photon counting modules (SPCM). The multimode fibers have a core diameter of 200 $\mu$m, which is smaller than the size of a single speckle in the transmitted light, to ensure that they each collect light of only a single mode. The total number of contributing modes (speckles) is approximately $4\times10^3$. By rotating a half-wave plate this light can also be reflected off the PBS and directly projected onto a CCD camera.
Complex-wavefront-shaping setup for programming quantum interference. (a) Two input modes ($k$, $l$) are modulated with a SLM, spatially overlapped with orthogonal polarizations, and focused on a layer of Teflon. The transmitted light is either projected onto a CCD camera, or onto two fibers connected to SPCMs which are in this way selectively observing two output modes ($m$, $n$). (b) Interference between the input modes onto the output modes $m$ (cyan) and $n$ (magenta). The solid curves indicate sine fits to the data, resulting in $\alpha=1.02\pi$.
Realization of a programmable 2$\times$2 beam splitter inside a scattering medium follows a complex-wavefront-shaping process. To ease this process, we couple classical laser light ($\lambda_{\rm c}$ = 790 nm) into the single-mode fibers and monitor the light in the output modes $m$ and $n$ using amplified photodiodes. In short, the process starts with a single input mode $k$ incident on the scattering medium. We optimize the output mode $m$ by fitting the optimal phase for each SLM segment that results in maximum constructive interference in the multimode fiber <cit.>, using direct (classical) light from the laser. Each input mode is controlled by approximately 960 segments on the SLM. The intensity of the single enhanced output spot coupled to the multimode fiber is about $200$ times greater than the average intensity of the background speckles. Next we optimize the second output mode $n$ in the same way. A camera image of two optimized output modes is shown in Fig. <ref>(a). This optimization procedure is repeated for the input mode $l$, resulting in four phase patterns. By combining these four interference patterns we program any $2\times2$ circuit <cit.>. The transmission through the circuit considered in this experiment is described by the following equation that relates the electric field of the two output modes, $E_{m}$ and $E_{n}$, to the electric fields of the two input modes, $E_{k}$ and $E_{l}$, by means of the transmission matrix $\bf{T}$:
\begin{equation}\label{eq:matrix}
\left[\begin{array}{c}
E_{m} \\
E_{n} \\
\end{array}\right]
= {\bf T} \left[\begin{array}{c}
E_{k} \\
E_{l} \\
\end{array}\right]
= t \mat{1}{1}{1}{\exp{(\rm{i} \alpha)}}
\left[\begin{array}{c}
E_{k} \\
E_{l} \\
\end{array}\right],
\end{equation}
where the parameter $\alpha$ is set in the algorithm when combining the interference patterns. The amplitude transmission coefficient $t$ has $\lvert{t\rvert}<\frac{1}{\sqrt{2}}$, emphasizing that the circuit is inherently lossy. Since the two selected channels stem from a manifold of $10^3$ channels, the programmed $2\times2$ circuit need not fulfill energy conservation and could thus be non-unitary. Consequently, we have the freedom to tune the quantum interference from bunching (Hong-Ou-Mandel-like) to antibunching. Only for $\alpha=\pi$ and $\lvert{t\rvert}=\frac{1}{\sqrt{2}}$ we do have a unitary matrix with ${\bf T}^\dagger {\bf T}=1$, and this transmission matrix represents an ideal 50:50 beam splitter. To confirm the functionality of each circuit, classical light is injected into both input modes $k$ and $l$ and the intensities of the output modes $m$ and $n$ are monitored while applying a phase difference $\Delta\theta$ between the input modes. An example of such an interference measurement is shown in Fig. <ref>(b) for $\alpha=\pi$. A similar measurement at $\alpha=0$ shows two overlapping $\left(1+\sin{\Delta\theta}\right)$-shaped curves, indicating the inherent non-unitary behavior of this $2\times2$ circuit. After programming the functionality, we switch back from using the direct classical light of the laser to the down-converted photons from the quantum light source.
Writing the input of the circuit described by Eq. (<ref>) as $\Psi_{\rm in}=\ket{1}_{k} \ket{1}_{l}$, one can find the probabilities for all possible outputs in a straightforward manner <cit.> for indistinguishable photons:
\begin{equation}\label{eq:probindist}
\begin{split}
P\left(2_{m},0_{n}\right) &= P\left(0_{m},2_{n}\right) = 2{\lvert t\rvert}^4 \\
P\left(1_{m},1_{n}\right) &= 2{\lvert t\rvert}^4 \left(1+\cos{\alpha}\right) \\
P\left(1_{m},0_{n}\right) &= P\left(0_{m},1_{n}\right) = 2{\lvert t\rvert}^2 - 2{\lvert t\rvert}^4 \left(3+\cos{\alpha}\right) \\
P\left(0_{m},0_{n}\right) &= 1 - 4{\lvert t\rvert}^2 + 2{\lvert t\rvert}^4 \left(3+\cos{\alpha}\right)
\end{split}
\end{equation}
Of interest are the first three probabilities $P\left(2_{m},0_{n}\right)$, $P\left(0_{m},2_{n}\right)$, and $P\left(1_{m},1_{n}\right)$, for which both photons arrive in output modes $m$ and $n$. Using the circuit of Fig. <ref>(b) programmed with $\alpha=\pi$, representing a 50:50 beam splitter, should result in Hong-Ou-Mandel interference in the scattering medium. Figure <ref>(a) shows the measured quantum interference for this circuit which indeed shows a HOM dip (squares). These measurements were done using a pump power of 100 mW and without a bandpass filter. While this higher pump power (in comparison to Fig. 1) generates more photon pairs ($> 4\times10^6$ s$^{-1}$), it also increases the production rate of higher photon-number states. These states reduce the visibility of the HOM dip. The measured HOM dip with a conventional beam splitter at this pump power indeed shows a reduced visibility of 24% (solid curve). To consolidate the quantum nature of the interference <cit.>, we repeated the measurements at a reduced pump power of 40 mW with a bandpass filter in place, which results in a visibility of 59% as shown in Fig <ref>(b) (squares). Indeed the data match the prediction and therefore confirms the quantum nature of the interference.
We now explore the programmability of the quantum interference. For indistinguishable single photons and for arbitrary $\alpha$, we expect the probability to detect coincidences between output modes $m$ and $n$ to scale as $\left(1+\cos{\alpha}\right)$. For instance, setting $\alpha=0$ in the patterns on the SLM would double the rate of coincidences compared to the rate obtained for distinguishable photons. This appears as a peak instead of a dip in the measured coincidences as shown in Fig. <ref> (circles). Although the case with $\alpha$ = 0 can not be realized in a conventional beam splitter, we show a flipped trace of the measured HOM dip with a conventional beam splitter as a guide to the eye (dashed curve). Note that the probabilities for bunched photons in the outputs is independent of $\alpha$, as is evident from Eq. (<ref>), which makes the output state different from the states typically found when recombining the two outputs from a traditional HOM experiment in a Mach-Zehnder type interferometer <cit.>.
Programmable quantum interference. Setting $\alpha=\pi$ gives rise to a dip in coincidence counts (squares), while $\alpha=0$ gives a peak (circles). Error bars indicate the standard deviations in the measurements. (a) depicts measurements performed at a pump power of 100 mW without bandpass filter, and (b) at a pump power of 40 mW with the filter in place. The solid curves in the plots correspond to the measured coincidence dips using a conventional beam splitter. The flipped traces of these are shown by the dashed curves as a guide to the eye.
Fully programmable quantum interference is demonstrated in our complex scattering medium in Fig. <ref>. Here the visibility of the quantum interference $V_0$ is plotted as a function of phase $\alpha$. Negative visibility corresponds to a dip in the coincidence counts and postive visilibty to a peak. At $\alpha=0$ a coincidence peak is observed. The visibility of this peak vanishes at phase $\alpha=\pi/2$, after which the visibility increases again as a coincidence dip. The well-known HOM dip with a high visibility occurs at phase $\alpha=\pi$. The solid curves indicate $V_0 \cos \alpha$ fits to these data with the prefactor $V_0$ as the only free parameter .
Programmable quantum interference as a function of programmed phase $\alpha$ at a pump power of 100 mW without filter (cyan), and at 40 mW with bandpass filter (magenta). Error bars indicate the standard deviation in the measurements. The solid curves indicate $V_0\cos{\alpha}$ fits to these data with as only free parameter the prefactor $V_0$.
For the measurements at a pump power of 100 mW the average coincidence detection rate is only 1240 h$^{-1}$, corresponding to a single photon detection rate of 2500 s$^{-1}$. This detection rate is small in comparison to the generation rate of $17 \times 10^6$ s$^{-1}$ as a result of our choice to be deep in the multiple-scattering regime. In this way, we approach the assumption of maximal entropy of the matrix in random-matrix theory <cit.>. Multimode fibers that were recently exploited for two-photon quantum interference <cit.>, do not provide such a high entropy, since the transmission matrix of a multimode fiber can be transformed into a block-diagonal matrix by a suitable basis <cit.>. We note that having maximal entropy is essential for the application of a scattering medium in quantum-secure authentication <cit.>, which requires a physical unclonable function with a transmission matrix that cannot predictably be approximated by a near-diagonal matrix <cit.>.
In summary, we demonstrated two-photon quantum interference in a high-dimensional linear optical network consisting of an opaque scattering medium. Out of the approximately $10^6$ available channels, we control about $10^3$ channels in this linear network from which we construct a programmable $2\times2$ circuit in which we have studied quantum interference between two single photons. We have demonstrated that by programming the functionality of this circuit, the well-known Hong-Ou-Mandel bunching can be made to vanish, or be transformed into antibunching. Our results demonstrate the feasibility of using complex scattering media as high-dimensional linear optical network for quantum information processing <cit.> and boson sampling <cit.>. By using complex wavefront shaping one can achieve a programmable functionality to realize universal quantum gates <cit.>.
We thank Simon Huisman, Thomas Huisman, Ad Lagendijk, Allard Mosk, Boris S̆korić, and Tristan Tentrup for discussions and support. This work is financially supported by FOM, NWO (Vici), and STW.
Akkermans2007 E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, New York, 2007).
Beenakker1997 C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731 (1997).
Mosk2012 A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nature Photon. 6, 283 (2012).
Vellekoop2007 I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309 (2007).
Popoff2014 S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent Control of Total Transmission of Light through Disordered Media,” Phys. Rev. Lett. 112, 133903 (2014).
Aaronson2011 S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” in Proceedings of the ACM Symposium on Theory of Computing `11 (ACM, New York, 2011), p. 333.
Peruzzo2010 A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. -Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O'Brien, “Quantum Walks of Correlated Photons,” Science 329, 1500 (2010).
Childs2013 A. M. Childs, D. Gosset, and Z. Webb, “Universal Computation by Multiparticle Quantum Walk,” Science 339, 791 (2013).
Broome2013 M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic Boson Sampling in a Tunable Circuit,” Science 339, 794 (2013).
Spring2013 J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson Sampling on a Photonic Chip,” Science 339, 798 (2013).
Tillmann2013 M. Tillmann, B. Daki, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nature Photon. 7, 540 (2013).
Crespi2013 A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvo, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Integrated multimode interferometers with arbitrary designs for photonic boson sampling,” Nature Photon. 7, 545 (2013).
Spagnolo2014 N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nature Photon. 8, 615 (2014).
Carolan2014 J. Carolan, J. D. A. Meinecke, P. J. Shadbolt, N. J. Russell, N. Ismael, K. Wörhoff, T. Rudolph, M. G. Thompson, J. L. O'Brien, J. C. F. Matthews, and A. Laing, “On the experimental verification of quantum complexity in linear optics,” Nature Photon. 8, 621 (2014).
Knill2001 E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46 (2001).
Kok2007 P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135 (2007).
Carolan2015 J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O'Brien, and A. Laing, “Universal linear optics,” Science 349, 711 (2015).
Harris2015 N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” arXiv:1507.03406 (2015).
Goorden2014 S. A. Goorden, M. Horstmann, A. P. Mosk, B. S̆korić, and P. W. H. Pinkse, “Quantum-secure authentication of a physical unclonable key,” Optica 1, 421 (2014).
Beenakker2009 C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
Peeters2010 W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
Lodahl2005a P. Lodahl, A. P. Mosk, and A. Lagendijk, “Spatial Quantum Correlations in Multiple Scattered Light,” Phys. Rev. Lett. 95, 173901 (2005).
Lodahl2005b P. Lodahl and A. Lagendijk, “Transport of Quantum Noise through Random Media,” Phys. Rev. Lett. 94, 153905 (2005).
Smolka2009 S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of Spatial Quantum Correlations Induced by Multiple Scattering of Nonclassical Light,” Phys. Rev. Lett. 102, 193901 (2009).
TJHuisman2014 T. J. Huisman, S. R. Huisman, A. P. Mosk, and P. W. H. Pinkse, “Controlling single-photon Fock-state propagation through opaque scattering media,” Appl. Phys. B 116, 603 (2014).
Defienne2014 H. Defienne, M. Barbieri, B. Chalopin, B. Chatel, I. A. Walmsley, B. J. Smith, and S. Gigan, “Nonclassical light manipulation in a multiple-scattering medium,” Opt. Lett. 39, 6090 (2014).
Huisman2014 S. R. Huisman, T. J. Huisman, S. A. Goorden, A. P. Mosk, and P. W. H. Pinkse, “Programming balanced optical beam splitters in white paint,” Opt. Express 22, 8320 (2014).
Huisman2015 S. R. Huisman, T. J. Huisman, T. A. W. Wolterink, A. P. Mosk, and P. W. H. Pinkse, “Programmable multiport optical circuits in opaque scattering materials,” Opt. Express 23, 3102 (2015).
Hong1987 C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
Huisman2009 S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon Fock state tomography,” Opt. Lett. 34, 2739 (2009).
Bimbard2010 E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nature Photon. 4, 243 (2010).
SupplementalMaterial See Supplemental Material at [URL will be inserted by publisher] for details about the spontaneous parametric down-conversion source.
Barnett1998 S. M. Barnett, J. Jeffers, A. Gatti, and R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134 (1998).
Ghosh1987 R. Ghosh and L. Mandel, “Observation of Nonclassical Effects in the Interference of Two Photons,” Phys. Rev. Lett. 59, 1903 (1987).
Rarity1990 J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348 (1990).
Chen2007 J. Chen, K. F. Lee, and P. Kumar, “Deterministic quantum splitter based on time-reversed Hong-Ou-Mandel interference,” Phys. Rev. A 76, 031804(R) (2007).
Silverstone2014 J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O'Brien and M. G. Thompson, “On-chip quantum interference between silicon photon-pair sources, ” Nature Photon. 8, 104 (2014).
Defienne2015 H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Control of two-photon quantum walk in a complex multimode system by wavefront shaping,” arXiv:1504.03178 (2015).
Carpenter2014 J. Carpenter, B. J. Eggleton, and J. Schröder, “110x110 optical mode transfer matrix inversion,” Opt. Express 22, 96 (2014).
Cismar2015 M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nature Photon. 9, 529 (2015).
Pappu2002 R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical One-Way Functions,” Science 297, 2026 (2002).
|
1511.00955
|
Tief Belichtete Galaxien group of Vereinigung der
Sternfreunde e.V., Fachgruppe Astrofotografie,
D-64629 Germany
June 22, 2015 July 31, 2015
We conduct a survey of low surface brightness (LSB) satellite
galaxies around the Local Volume massive spirals using long
exposures with small amateur telescopes. We identified 27 low and
very low surface brightness objects around the galaxies NGC 672,
891, 1156, 2683, 3344, 4258, 4618, 4631, and 5457 situated within
10 Mpc from us, and found nothing new around NGC 2903, 3239,
4214, and 5585. Assuming that the dwarf candidates are the
satellites of the neighboring luminous galaxies, their absolute
magnitudes are in the range of , their
effective diameters are 0.4–4.7 kpc, and the average surface
brightness is $26\fm1/\sq\arcsec$. The mean linear projected
separation of the satellite candidates from the host galaxies is
73 kpc. Our spectroscopic observations of two LSB dwarfs with the
Russian 6-meter telescope confirm their physical connection to the
host galaxies NGC 891 and NGC 2683.
§ INTRODUCTION
The two past decades witnessed the rapid formation of a new
direction in extragalactic astronomy, called the cosmology of the
nearby universe. This is facilitated by the ongoing wide-field
surveys of the northern and southern sky in the optical,
infrared, and radio
ranges ,
which allowed astronomers to detect a lot of dwarf galaxies and to
measure their radial
velocities .
Accurate bulk measurements of distances to nearby galaxies carried
out with the
Hubble Space Telescope are also a significant advancement factor.
According to the latest observational data report, titled the
Updated Nearby Galaxy Catalog
(UNGC) <cit.>, the surrounding volume
around the Milky Way with a radius of about 10 Mpc contains
about nine hundred galaxies, the majority of which have their
distances and radial velocities measured, and stellar masses, star
formation rates, and other basic characteristics determined. The
generally accessible database of observational data on the
galaxies of the Local Volume <cit.> (http://www.sao.ru/lv/lvgdb/) is regularly updated with new
objects. About 85% of the UNGC <cit.>
sample are dwarf galaxies whose integrated luminosity is lower
than that of the Magellanic Clouds.
The numerical simulations of the large-scale structure of the
Universe performed on supercomputers within the standard
$\Lambda$CDM cosmological
model <cit.>
revealed a huge discrepancy in the observed number of dwarf
galaxies relative to their number expected within the standard
model. The observed number of dwarf satellites around nearby high
luminosity galaxies turned out to be ten times smaller than
expected. This as yet unexplained situation was called the
“missing satellites” paradox. An ad hoc search for dwarf
galaxies around the nearest massive galaxies
M 31 <cit.>
and M 81 <cit.> has
only partially alleviated this paradox. Therefore, a further
in-depth search for yet fainter dwarf systems
continues to be a topical task of observational cosmology of the nearby universe.
According to theoretical and observational data, the integrated
luminosity of a galaxy $L$ is proportional to the cube of its
effective linear diameter $A$. This implies that the average
luminosity density of galaxies $L/A^3$ is about the same
regardless of their size, however, their average surface
brightness $L/A^2$ drops with decreasing linear diameter of the
galaxy. For this reason the smallest dwarf satellites are to be
searched for among LSB objects. Medium-sized amateur telescopes
(about 0.3 m in diameter) with a focal ratio
and equipped with CCD detectors can be
used for this task. The images obtained on such telescopes with
exposures of about 10 hours reveal quite well objects with the
surface brightness and angular sizes exceeding $0\farcm2$, which
roughly corresponds to the typical parameters of dwarf galaxies
located within the Local Volume.
According to preliminary estimates, the UNGC is about 50%
complete for the galaxies with the absolute magnitude $M_B$
brighter than $-11\fm0$ and the surface brightness ${\rm
SB}\leq26^{\rm m}/\sq\arcsec$. The characteristic linear diameter
of dwarf galaxies near the 50% limit is about 1 kpc, which
corresponds to an angular size
$a\sim0\farcm3$ at the far edge of the Local Volume.
Therefore, hours-long exposures of the vicinity of nearby bright
galaxies with modern amateur telescopes allow successful detection
of new dwarf satellites around them. Such a systematic
observational program provides an independent opportunity to
clarify the extent of completeness of the UNGC in terms of the
luminosity of dwarf galaxies, their linear dimensions and surface
§ TBG GROUP OBSERVATIONAL PROJECT
The TBG (Tief Belichtete Galaxien) group deals with very long
exposure images of galaxies made using amateur telescopes of
medium caliber. The group was organized by P. Riepe in January
2012 in the Astrophotography department of the German
association VdS (Vereinigung der Sternfreunde e.V.). Now the
group consists of about 30 astrophotographers from Germany,
Austria, and Switzerland, equipped with 10 to 110 cm diameter telescopes. Some TBG telescopes are located in the United States
and Spain and are remotely operated. The entire project is
coordinated by P. Riepe and T. Zilch. For deep-sky images,
reaching , the group
uses high-quality CCD detectors and data reduction packages which
include the dark frame subtraction, flat field correction, and
calibration procedures.
One of the main tasks of the TBG group is photographing the
neighborhoods of nearby bright galaxies to look for LSB dwarf
satellites around them. The program includes more than 50 galaxies
of sufficiently high luminosity located within a distance of
10 Mpc. Exposure time with broadband filters varied depending on
weather conditions. The typical exposure lasted from 10 to
15 hours, though in some cases it reached up to 50 hours.
After the necessary image processing steps and contrasting of the
images, we conducted a visual search for low and very low surface
brightness objects, focusing on the average characteristics of the
known satellites of the Milky Way and Andromeda (M 31).
Below we present the results of the satellite candidate search
around thirteen nearby spiral galaxies, obtained in
implementing the initial phase of this program. The following
articles of this series will present the results of monitoring the
neighborhoods of still about 40 massive galaxies in the Local
§ FIRST-SEASON OBSERVATIONS
§.§ NGC 4631
This late-type spiral galaxy, seen edge-on, is the brightest
representative of a scattered group, which includes about
30 members. The distance to it, $D=7.38$ Mpc, was measured by
Radburn-Smith et al. <cit.> from the
luminosity of red giant branch stars. Another bright spiral
galaxy, NGC 4656, is located at an angular distance of
$32\arcmin$ south of NGC 4631. Both of them have structural
distortions which are obviously caused by mutual gravitational
In spring 2013, several TBG group members have obtained the
images of NGC 4631 and its environs with a total exposure of
$24^{\rm h}$. Methodological details of these observations were
given earlier <cit.>. We marked three
very low surface brightness dwarf galaxies in the combined image,
dw1, dw2, dw3, as well as a tidal strip that stretches from
NGC 4631 to the northwest through dw1 and to the southeast
towards the galaxy NGC 4656. The presence of this tidal strip was
later confirmed by Martinez-Delgado
et al. <cit.>. In February 2013 and March
2014, F. Neyer obtained new, deeper images of the NGC 4531/56
pair with an exposure of $49\fh5$
(Fig. 1, see also http://tbg.vdsastro.de). This
image reveals different signs of interaction between the
components of this pair with a characteristic surface brightness
of $29^{\rm m}$–$30^{\rm m}/\sq\arcsec$. Some of them are
indistinguishable from faint reflection nebulae. All the three
objects dw1/2/3, according to their texture and location, are
probably the physical satellites of NGC 4631. However, radial
velocity measurements of these dwarfs are required to confirm this
obvious assumption, which is a difficult observational task.
§.§ M 101 $=$ NGC 5457
The spiral galaxy M 101, viewed face-on, is one of the most
prominent representatives of the Local Volume population. The
distance to it by the Cepheids is estimated as
7.38 Mpc <cit.>,
which happens to coincide with the distance to NGC 4631. According to <cit.>,
M 101 has six satellites, including an intergalactic H I cloud GBT 1355+5439 <cit.>.
In 2009 and 2010 Mihos et al. <cit.>
obtained deep images of the neighborhood of M 101 with the
Burrell Schmidt telescope. The image mosaics covered a field sized
with the surface brightness limit
. The authors noted the
presence in M 101 of two structural disturbances on the
periphery: the northeastern spot and the eastern protrusion.
However, they did not report a discovery of any new nearby
satellites. In March 2012, a deep image of M 101 with an
exposure of $40^{\rm h}$ was obtained by F. Neyer using a
telescope with a diameter of 15 cm and a focal ratio of $f/7.2$.
The image size is $121\arcmin\times 80\arcmin$. The reproduction
of the fragments of this image is shown in
three panels of Fig. 2. In addition to the above-mentioned distortions of the periphery of
M 101, on the
northern and northeastern side the image reveals ten small LSB objects.
All of them are located in the eastern half of the image with
respect to M 101, which looks pretty mysterious.
In May–June 2013, van Dokkum et al. <cit.>
and Merritt et al. <cit.> performed a
survey of the neighborhood of M 101 using the Dragonfly
Telephoto Array. This telescope is a robotic system of eight
lenses, each one with a focal length of 40 cm and a focal ratio of
$f/2.8$. This results in an effective aperture of 403 mm and the
summed up focal ratio of about $f/1$. The field of view is
with a resolution of $2\farcs8$/px.
Given a total exposure of $ 35^{\rm h}$, the Dragonfly team
reached a surface brightness limit .
As a result, the authors found seven M 101 dwarf
satellite candidates and designated them as DF1, DF2, ..., DF7.
Six of them proved to be common with the objects in the image of
F. Neyer, and one, DF5, is located outside the image field. At the
same time, two LSB objects that we have designated as M 101 dwA
and M 101 dwC were not noted by the Dragonfly team. The
Dragonfly deep image limit allowed Merritt
et al. <cit.> to conduct surface photometry
of the detected objects and to determine their integrated $g$ and
$r$ magnitudes, effective diameter, central surface brightness,
and the Sérsic structural index. Recently, M. Elvov obtained a
broader image of the vicinity of M 101, where we found two more
satellite candidates: M 101 dwB and M 101 dwD.
5mm normal
The galaxy NGC 4631 and three of its supposed
companions, located along the diagonal tidal stripe. A
$45\arcmin\times 33\arcmin$ fragment of an image obtained by
F. Neyer with an exposure time of $49\fh5$. North is at the top,
east is to the left.
5mm normal
Fragments of the image displaying the neighborhood of M 101, obtained by
F. Neyer with an exposure of $45\fh6$. (a) The region of an M 101
satellite, the galaxy NGC 5477, and a new satellite candidate
dwA; the frame size is $36\arcmin\times 24\arcmin$. (b) The area
to the northeast of M 101 sized $39\arcmin\times 31\arcmin$, in
which the Dragonfly DF 6 and DF 7 objects and two new M 101
satellite candidates dwB and dwC are located.
Down from the center, a far background galaxy NGC 5485 is
located (a distance of 28 Mpc), around which diffuse elliptical
shells, shown by the arrows were detected for the first time. (c)
A fragment of an image obtained by M. Elvov with an exposure of
$11\fh3$. In the area sized $34\arcmin\times 34\arcmin$, the
NGC 5477 galaxy (the upper left corner), the
Dragonfly DF 3 object, and a new M 101 satellite candidate dwD
are visible. In all the images north is at the top, and east is to
the left.
5mm normal
An image of the spiral galaxy NGC 2683 (in the upper
right corner) and its two prospective satellites. The frame size
is $29\arcmin\times 21\arcmin$. The image was obtained by
G. Kerschhuber with an exposure of $12^{\rm h}$. Based on the
measured radial velocity, the object “1” is a true satellite of
NGC 2683.
§.§ NGC 2683
This massive spiral galaxy at a distance of
has two close and one distant satellite, KK 69, KK 70, and AGC 182595 respectively, the distances to which were measured with the
Hubble Space
Telescope <cit.>.
This poor group is located in the region of low galaxy number
on the front edge of the Void.
Two deep images centered on NGC 2683 were obtained by M. Elvov
and G. Kerschhuber. The field sizes and exposures were
$t=15^{\rm h}$ and
$t=12^{\rm h}$ respectively.
The reproduction of the second image fragment (Fig. 3) reveals two
LSB dwarf objects, designated as dw1 and dw2. The brighter of them
is identified with a UV source of the GALEX
survey <cit.>. Its image in the H$\alpha$
line <cit.>, taken with the 6-meter
telescope of the Special Astrophysical Observatory of the Russian
Academy of Sciences (SAO RAS) with the focal
reducer <cit.>, detects weak emission.
The [O III], H$\beta$, and H$\gamma$ <cit.>
emission lines are
present in the spectrum of the object obtained with the same
telescope <cit.>. Based on them, the
heliocentric radial velocity of the object is
, which is close to the radial
velocity of NGC 2683 itself, .
The agreement of radial velocities indicates that the new dwarf
galaxy is a real companion of NGC 2683. Low surface brightness of
the second (dSph) dwarf galaxy makes
its radial velocity measurement quite difficult.
5mm normal
(a) The spiral galaxy NGC 891 and its two
dwarf satellites: dwA $=$ [TT 2009] 25 and dwB $=$
[TT 2009] 30. A $33\arcmin\times 32\arcmin$ fragment of an image
obtained by M. Elvov with an exposure of $12^{\rm h}$.
(b, c) Images of the dwarf galaxies [TT 2009] 25 and
[TT 2009] 30, obtained by S. S. Kaisin on the SAO RAS 6-meter
telescope with the SED607 filter with 1650-s and 2400-s exposures respectively.
In all the images north is at the top, east is to the left.
§.§ NGC 891
According to <cit.>, the entourage of the
spiral galaxy NGC 891, seen edge-on, consists of four Sm and
Irr galaxies: DDO 22, DDO 24, UGC 1807, and UGC 2172. The
distance to the main galaxy, 9.77 Mpc, was determined from the
surface brightness fluctuations <cit.>.
The group itself is associated with the spiral galaxies NGC 925
and NGC 1023, forming a more extended and
non-virialized complex.
5mm normal
An image of the spiral galaxy NGC 3344 and its suspected
dwarf companion NGC 3444 dw1, marked by arrows. The image was
obtained by G. Kerschhuber and M. Blauensteiner based on
simultaneous data from four telescopes with a total exposure of
$28\fh8$. The size of the fragment is $22\arcmin\times 16\arcmin$.
North is at the top, east is to the left.
Faint satellites of NGC 891 were searched for by Trentham and
Tully <cit.> and
Schulz <cit.>. The former authors used
broadband exposures with the MegaCam detector of the 3.6-m CFHT
telescope. In the immediate surroundings of NGC 891, the authors
found only two dwarf companion candidates: [TT 2009] 25 and
[TT 2009] 30. Schulz <cit.>
searched for new satellites of NGC 891 based on the data from
various available surveys of the sky from the ultraviolet (GALEX)
to the infrared (2MASS, WISE) range. Two out of seven suspected
new satellites turned out to be already known, and the remaining
five are likely to be the objects of distant background. A CCD
image of the vicinity of NGC 891 obtained by M. Elvov on the
refractor $(f/5$)
with an exposure of $12^{\rm h}$ (Fig. 4a) detects the presence of
two LSB dwarf galaxies dwA and dwB, which are identified with the objects [TT 2009] 25 and [TT 2009] 30.
We have obtained large-scale images of both galaxies in the
H$\alpha$ line and in the continuum (filter
SED607) <cit.> with the SAO RAS 6-m
telescope (Figs. 4b and 4c). They did not reveal any H$\alpha$
even though in the GALEX survey they show weak FUV fluxes.
For a brighter spheroidal galaxy [TT 2009] 25,
a spectrum was obtained with the SAO RAS 6-m
telescope <cit.>. Based on three absorption
lines, the radial velocity of the galaxy is
$V_h=692\pm58$ km s$^{-1}$, which is close to the heliocentric velocity
of NGC 861 itself:
According to the texture of the object [TT 2009] 30, it too,
with a high probability, is
a satellite of NGC 891.
§.§ NGC 3344 $=$ KIG 435
This is an isolated galaxy seen face-on, with a regular spiral
pattern. None of its physical satellites were known to date. An
image of the surroundings of NGC 3344 made by Kerschhuber and
Blauensteiner with a total exposure of $28\fh8$ and field of
reveals a suspected LSB
(Fig. 5) $12\arcmin$ northwest of NGC 3344. In this place, the
SDSS survey <cit.> reveals a faint bluish
spot with an angular size of $0\farcm3$. The GALEX survey did not
detect this object.
5mm normal
The spiral galaxy NGC 4258 $=$ M 106 and its new
suspected satellite NGC 4258 dwA. The image was obtained by
M. Elvov with a 20-h exposure. The reproduced part of the image is $ 65\arcmin\times 48\arcmin$. North is
at the top, east is to the left.
§.§ NGC 4258 $=$ M 106
A careful search for faint satellites around the spiral galaxy
NGC 4258 was complied by Kim
et al. <cit.>. For this purpose, the
authors used images taken with the CFHT telescope MegaCam
detector, covering a field of $1\fdg7\times 2\fdg0$. They found
sixteen M 106 satellite candidates of and presented for them the
surface photometry in the $g$ and $r$ bands. Two objects, S 11
and S 16, proved to be new compared to the previous
investigations of the M 106
neighborhood <cit.>. Later, Spencer
et al. <cit.> made
radial-velocity measurements of the suspected satellites
of NGC 4258. This way they were trying to identify the physical
satellites of NGC 4258 among the background galaxies. However,
the M 106 group is located in a complicated region at the
equator of the Local Supercluster. Next to it there are two more
groups around the galaxies NGC 4346 and NGC 4157 with radial
velocities close to that of M 106. The groups have average
distance estimates of 16.4 Mpc and 17.5 Mpc respectively, forming
the distant background for the NGC 4258 group ($D=7.83$ Mpc by
the Cepheid method).
The TBG group members obtained a few images of the neighborhood of
M 106 with exposures ranging from $6^{\rm h}$ to $20^{\rm h}$.
In the area common with the field of view
of <cit.>, all the objects identified
in <cit.> as suspected companions of
M 106 were detected.
Moreover, we have found a new LSB dwarf galaxy, indicated in
Fig. 6 as dwA. As Dr. H. Ann reported to us, in their image made
with MegaCam this object has not been noted, since it fell in the
gap between two CCD chips.
In addition, outside the MegaCam field of view we found two more LSB objects dwB and dwC, which look like very
probable companions of M 106 (see http://tbg.vdsastro.de).[Near the southern edge of
Fig. 6, a distant background galaxy NGC 4217 is located. Its
radial velocity , and the
distance, according to NED, is 19.3 Mpc. On the northeastern edge
of NGC 4217, a round LSB feature is visible, which may be a tidal
bulge in the disk of the spiral. If, however, this is a dwarf
companion of NGC 4258, projected on the outskirts of NGC 4217,
then its absolute magnitude $M_B=-11.5$, its linear diameter is
2.5 kpc, and the mean surface brightness ${\rm
§.§ NGC 672/IC 1727
This tight pair of late-type spiral galaxies at a distance of
7.16 Mpc has three dwarf systems as its close satellites: KK 13,
KK 14, and KK 15 <cit.>. In three images
of the neighborhood of the pair obtained by the TBG members, we
have selected four probable satellite candidates. They are marked
on the three panels of Fig. 7 by the letters A, B, and C. Another
candidate with the coordinates 014738.4+272620 is located in
contact with NGC 672, and is the brightest
and the most compact. We have obtained its spectrum with the SAO
RAS 6-meter telescope. The velocity of the object proved to be
very large,
km s$^{-1}$ <cit.>,
and we excluded it from the group. Three other LSB galaxies look
more like close dwarfs. Radial-velocity measurements are required
to confirm their membership in the NGC 672 group.
5mm normal
(a) The galaxy IC 1727 and the dwarf galaxy
N 672 dwA (a $14\arcmin\times 14\arcmin$ part of an image
obtained by R. Pöelz with an exposure of $21^{\rm h}$).
(b) The galaxy NGC 672 and its assumed satellite dwB (a
$19\arcmin\times 19\arcmin$ fragment of the same image by
R. Pölzl). (c) The area to the north of NGC 672 with its two
supposed satellites NGC 672 dwB and NGC 672 dwC (a
$10\arcmin\times 10\arcmin$ fragment of an image obtained by
S. Küppers with an exposure of $5\fh3$). In all the images
north is at the top, east is to the left.
§.§ NGC 4618/NGC 4625
This is a pair of dwarf spirals at a distance of 7.9 Mpc with a
radial velocity difference of70 km s$^{-1}$. The
spiral structure is distorted in both galaxies, which indicates
their interaction. An dwarf galaxy UGC 7751 is
associated with this pair, and possibly another, fainter irregular
dwarf LV J1243+4127 too <cit.>.
Investigating an extended ultraviolet disk around NGC 4625,
Gil de Pazet al. <cit.> noted
the presence $4\arcmin$ east of it of an LSB object, which they
named NGC 4625 A. The reproduction of an image of the
NGC 4618/25 pair and the dwarf galaxy NGC 4625 A obtained by
R. Sparenberg is shown in Fig. 8. We have included NGC 4625 A in
the list of objects for measuring radial velocities, assuming that
it can prove to be a physical companion of the NGC 4618/25 pair.
5mm normal
An interacting pair of galaxies NGC 4618 (left) and
NGC 4625, and a dwarf galaxy NGC 4626 A, marked by the arrows.
A $23\arcmin\times 16\arcmin$ fragment of an image obtained by
R. Sparenberg on a 44-inch telescope with an exposure of $3\fh5$.
The arrow in the upper right corner indicates the north
§.§ NGC 1156 $=$ KIG 121
This isolated Magellanic type dwarf galaxy with an apparent
magnitude $B=12\fm3$ and radial velocity $V_h=373$ km s$^{-1}$
is located in the zone of significant ($A_b=0\fm97$) Galactic
extinction. The image of its surroundings with a $46\arcmin\times
34\arcmin$ field of view
was obtained by P. Hochleitner on a
36 cm diameter telescope with an exposure of $11^{\rm h}$.
Reflection nebulae (cirri), the presence of which imposes a limit
on the detection of LSB galaxies, are seen almost over the entire
area of the image. Nevertheless, two LSB objects, dw1 and dw2,
which can be classified as satellites of NGC 1156, are present in
the image (Fig. 9). The first one of them is located in the halo
of a bright star SAO 75679. Both objects are barely discernible
in the photographic sky survey DSS2.
Note that the vicinity of NGC 1156 was investigated by Minchin
et al. <cit.> within the deep H I AGES
survey using the Arecibo radio telescope. These observations led
to the discovery of an irregular dwarf galaxy AGES J030039+254656
with a radial velocity
$V_h=308$ km s$^{-1}$ and apparent magnitude ,
which is located to the north, beyond the scope of our image.
Two NGC 1156 satellite candidates that we have discovered look
about two magnitudes fainter than the AGES satellite. Obviously,
their H I fluxes could be below the AGES survey limit.
§.§ NGC 2903, NGC 3239, NGC 4214, NGC 5585
Apart from the nine galaxies mentioned above, we have obtained
long-exposure images for four other galaxies in the Local Volume.
However, the search for new candidate satellites of these galaxies proved to be unsuccessful.
The massive spiral galaxy NGC 2903 is listed in the catalog of isolated galaxies <cit.>.
Although it has four tiny satellites, due to their small size this
is not inconsistent with the catalog isolation criterion. Just
like in the case of NGC 2683, this galaxy is on the front edge of
the nearby Gemini–Leo Void, whose center lies about 18.4 Mpc from
us, and whose radius is 7.5 Mpc. Deep images of the vicinity of
NGC 2903 made by M. Blauensteiner, M. Elvov, and S.Küppers
did not reveal any new companion candidates of the massive spiral.
An interactive pair of merging galaxies contains a scattering
of star formation centers. Two curved tails extending to the south
of the main body of the galaxy make it similar to the Greek letter
“$\pi$.” Around this violently interacting system one would
expect to find a multitude of small irregular satellites formed by
the fragmentation of tidal tails. However, they have not been
found in the image made by B. Hubl with an exposure of $32^{\rm
A magellanic-type dwarf galaxy NGC 4214 at a distance of
2.94 Mpc <cit.> is in the stage of violent
star formation. Next to it there is a dwarf spheroidal galaxy
KDG 90, the distance to which,
2.86 Mpc <cit.>, indicates the physical
connection with NGC 4214. However, the images obtained by
G. Willems and G. Kerschhuber with an exposure of $19\fh1$ do not
show any obvious signs of mutual perturbations in these galaxies,
apart from a faint diffuse protrusion on the southwestern
periphery of NGC 4214. No new NGC 4214 satellite candidates have
either been found.
A dwarf Sm spiral NGC 5585 at a distance of 5.7 Mpc is a
distant satellite of a giant spiral M 101. The periphery of the
galaxy looks quite regular, without disturbances. The image
obtained by B. Hubl with an exposure of $15^{\rm h}$ did not
reveal any satellite candidates of this galaxy in the $30\arcmin\times 30\arcmin$ field.
The reproductions of the said images and the comments to them can
be found on .
§ DISCUSSION
The main features of the thirteen nearby spiral galaxies that we
have observed with long exposures are presented in Table 1. Its
columns contain: (1) the names of the galaxies, ranked by right
ascension; (2) the coordinates of the galaxy for the epoch
(3) the morphological type according to de Vaucouleurs' scale;
(4) the radial velocity in km s$^{-1}$ relative to the centroid of the Local Group;
(5) the distance to the galaxies in Mpc and the applied method of
distance determination: “cep”—the Cepheid luminosity method,
“rgb”—by the luminosity of red giant branch stars,
“sbf”—by surface brightness fluctuations, “tf”—by the
Tully–Fisher relation between the amplitude of rotation of a galaxy and its luminosity, “bs”—by the luminosity of the brightest stars;
(6) the absolute magnitude of the galaxy in the $B$ band adjusted
for the Galactic <cit.> and internal
extinction; (7) the linear Holmberg diameter of the galaxy in kpc;
(8, 9) the logarithm of the
stellar mass and the mass of neutral hydrogen in solar mass units;
(10) the logarithm of the stellar mass density within the 1 Mpc
radius sphere around the galaxy, taken in relation to the average
cosmic stellar mass density; (11) the number of known satellites
of the galaxy within the “zero-velocity radius” sphere around it
plus the number of new satellite candidates that we found and
discussed above; (12) the signs of disturbances on the periphery
of the galaxy if any of them can be seen in our deep images.
5mm normal
An isolated galaxy NGC 1156 in the lower right corner
and two of its suspected satellites dw1 and dw2. A
$19\arcmin\times 15\arcmin$ fragment of an image obtained by
P. Hochleitner with an exposure of $11^{\rm h}$. North is at the
top, east is to the left.
Nearby massive galaxies observed by the TBG group
1c|Name RA (2000.0) Dec $T$ $V_{\rm LG}$, 1c|$D$, Mpc; $M_B$, 1c|$A_H$, 1c|$\log M^*$, $\log M_{\rm HI}$, 1c|$\Theta_j$ 2c|$N_{\rm sat}$ 1cPeriphery
km s$^{-1}$ 1c|method mag 1c|kpc 1c|$[M_{\odot}]$ $[M_{\odot}]$ 2c| 1cshape
1c|(1) (2) (3) (4) 1c|(5) (6) 1c|(7) 1c|(8) (9) 1c|(10) 2c|(11) 1c(12)
NGC 672 014753.2+272601 6 626 7.16 rgb $-18.76$ 15.7 10.22 9.23 $0.2$ 3 3 regular
NGC 891 022232.8+422048 3 736 9.97 sbf $-20.58$ 38.7 10.98 9.66 $-0.1$ 4 2 regular
NGC 1156 025942.4+251415 8 507 7.80 bs $-18.14$ 9.2 9.31 8.82 $-1.5$ 1 2 NE extention
NGC 2683 085240.9+332502 3 365 9.36 rgb $-20.78$ 35.7 10.76 9.10 $-1.4$ 2 2 regular
NGC 2903 093209.6+213002 4 443 8.87 bs $-20.89$ 32.4 10.82 9.44 $-0.8$ 4 0 regular
NGC 3239 102504.9+170949 8 623 7.90 tf $-18.09$ 11.6 9.52 8.89 $-0.9$ 0 0 faint halo
NGC 3344 104330.2+245525 4 500 9.83 rgb $-19.72$ 22.2 10.33 9.44 $-0.8$ 0 1 regular?
NGC 4214 121538.9+361939 8 295 2.94 rgb $-17.20$ 7.3 9.00 8.48 $-0.7$ 3 0 SSW protrusion?
NGC 4258 121857.5+471814 4 506 7.83 cep $-21.20$ 41.5 10.94 9.64 $1.0$ 19 3 twisted
NGC 4618 124132.8+410903 6 576 7.90 tf $-18.33$ 9.7 9.65 8.90 $0.5$ 2 1 N, E protrusions
NGC 4631 124208.0+323229 7 581 7.38 rgb $-20.28$ 33.7 10.49 9.72 $1.0$ 5 3 tails
M 101 140312.8+542102 6 378 7.38 cep $-21.12$ 65.2 10.85 9.91 $0.2$ 6 10 asymmetric
NGC 5585 141948.3+564349 7 457 5.70 bs $-17.81$ 9.7 9.03 8.82 $-3.0$ 0 0 regular
Most of the Table 1 data are taken from the
UNGC <cit.>, which gives evaluations of the
parameters used.
Some distance estimates have been updated with recent observations of the Hubble Space Telescope.
Despite the small statistics, the Table 1 data show the presence
of a positive correlation between the luminosity of a galaxy (its
linear diameter, mass) and the number of satellites around it. The
average density of the environment also affects the amount of
physical satellites. For example, two galaxies in the region of a
cosmic void, NGC 2683 and NGC 2903, have six satellites and two
new candidates, while the two galaxies in the dense environment,
NGC 4258 and NGC 4631 with approximately the same luminosity,
have twenty-four physical satellites and six new candidates. The
noted trends suggest near which galaxies there is a chance to find
the greatest number of new satellites during a further search.
The distant periphery of Andromeda-type massive galaxies preserves
in itself in the form of faint stellar streams relic signs of
interaction with neighbors or traces of the capture of satellites
which happened a few
billion years ago.
The last column of Table 1 provides indications on the possible
events of this kind
in the history of thirteen nearby high luminosity galaxies.
New LSB satellite candidates around nearby massive
1c|Name RA (2000.0) Dec $T$ $B$, $a$, ${\rm SB}$ 1c|$r_p$, $R_p$, 1c|$M_B$, $A$, 1cNote
mag arcmin 1c|arcmin kpc 1c|mag kpc
1c|(1) (2) (3) (4) (5) (6) 1c|(7) (8) 1c|(9) (10) 1c(11)
NGC 672 dwB 014711.1+274100 Ir-VL 21.0 0.20 25.8 17.7 37 $-8.6$ 0.42
NGC 672 dwA 014719.1+271516 Ir-L 19.8 0.26 25.2 13.1 27 $-9.8$ 0.54 GALEX
NGC 672 dwC 014720.4+274324 Sph-L 18.7 0.40 25.0 18.9 39 $-10.9$ 0.83
NGC 891 dwA 022112.4+422150 Tr-L 17.9 0.76 25.7 14.9 43 $-12.3$ 2.20 [TT09]25
NGC 891 dwB 022254.7+424245 Ir-VL 18.9 1.16 27.6 22.4 65 $-11.3$ 3.36 [TT09]30
NGC 1156 dw1 030018.2+251456 Ir-L 19.6 0.38 25.2 8.1 18 $-10.8$ 0.86
NGC 1156 dw2 030028.0+251817 Ir-VL 20.0 0.38 25.6 11.1 25 $-10.4$ 0.86 GALEX
NGC 2683 dw1 085326.8+331820 Ir-L 19.0 0.40 25.5 11.7 32 $-11.0$ 1.09 GALEX
NGC 2683 dw2 085420.5+331458 Sph-VL 19.6 0.40 26.1 23.1 63 $-10.4$ 1.09
NGC 3344 dw1 104244.0+250130 Ir-VL 20.0 0.30 26.0 11.9 34 $-10.1$ 0.86
NGC 4258 dwC 121026.8+464449 Sph-L 19.0 0.27 24.7 93.3 212 $-10.5$ 0.61
NGC 4258 dwA 121551.0+473256 Ir-L 19.0 0.43 25.7 34.8 79 $-10.5$ 0.98
NGC 4258 dwB 122410.9+470723 Sph-L 18.3 0.45 25.1 54.6 124 $-11.2$ 1.02 BTS134
NGC 4631 dw1 124057.0+324733 Ir-VL 16.1 2.20 26.4 21.3 46 $-13.3$ 4.72 GALEX
NGC 4631 dw2 124206.8+323715 Ir-VL 18.5 0.90 26.8 4.8 10 $-10.9$ 1.93 GALEX
NGC 4625 A 124211.0+411510 Tr-L 18.6 0.45 25.4 9.4 22 $-11.0$ 1.03
NGC 4631 dw3 124252.5+322735 Sph-VL 19.7 0.60 27.1 10.6 23 $-9.7$ 1.29
M 101 DF3 140305.7+533656 Sph-VL 17.9 1.00 26.5 44.1 95 $-11.5$ 2.15
M 101 DF1 140345.0+535640 Ir-L 18.9 0.47 25.8 23.9 51 $-10.5$ 1.01
M 101 dwD 140424.6+531619 Sph-VL 19.2 0.38 25.7 65.6 141 $-10.2$ 0.81
M 101 dwC 140518.0+545356 Tr-VL 20.2 0.30 26.2 37.6 81 $-9.2$ 0.64
M 101 DF7 140548.3+550758 Sph-XL 20.4 0.67 28.1 52.0 117 $-9.0$ 1.44
M 101 dwA 140650.2+534432 Sph-L 19.2 0.36 25.6 45.3 97 $-10.2$ 0.77
M 101 DF4 140733.4+544236 Ir-XL 18.8 0.93 27.2 43.5 93 $-10.6$ 1.99
M 101 DF6 140819.0+551124 Ir-XL 20.1 0.73 28.0 67.2 144 $-9.3$ 1.57
M 101 DF2 140837.5+541931 Sph-L 19.8 0.33 26.0 47.1 101 $-9.6$ 0.71
M 101 dwB 140843.1+550957 Sph-VL 20.1 0.30 26.1 68.0 146 $-9.3$ 0.64
Mean 19.2 0.57 26.1 32.4 73 -10.4 1.31
A summary of the data on twenty-seven detected satellite
candidates of the thirteen nearby spiral galaxies is given in
Table 2. Its columns contain: (1) the name of the dwarf galaxy;
(2) the galaxy coordinates for the epoch 2000.0; (3) the
morphological type of the dwarf: irregular (Ir), spheroidal (Sph),
or transition (Tr), with a visual estimation of surface
brightness: low (L), very low (VL), or extremely low (XL);
(4) the apparent $B$ magnitude estimated by eye via comparing with
other dwarf objects of similar structure and known photometry; in
some of the brightest objects the $B$ magnitude is determined
from the SDSS survey $g$ and $r$ magnitudes, the measure of
inaccuracy of our estimates is about
$0\fm5$; (5) the maximum apparent angular diameter $a$ in arcminutes;
(6) the surface brightness
in magnitudes per square arcsecond, where the
$B$ magnitude is corrected for the Galactic extinction;
(7, 8) the projected distance of the satellite from the main galaxy in
arcminutes and in kpc; (9, 10) the absolute magnitude and linear
diameter of the satellite; (11) whether the object has a FUV flux
from the GALEX survey or an alias. The last line of the table
shows the average values of the above parameters.
We can see from these data that the absolute magnitudes of the
suspected new satellites around nearby spirals are confined in the
range from
to $-13\fm3$ with an average value of $-10\fm4$.
Linear diameters of the satellites lie in the range from 0.4 kpc
to 4.7 kpc with an average of 1.3 kpc. The range of both
parameters is typical for the dwarf satellites of M 31 and M 81.
The average surface brightness of the discovered dwarfs
exceeds the Holmberg isophote brightness of $26\fm5/\sq\arcsec$,
which is the detection limit for low-contrast objects observed
with photographic emulsions.
The average projected separation of new satellites from their
host galaxies is 73 kpc. This value is smaller
than the characteristic radius of an entourage of dwarfs around
a Milky Way–type massive galaxy. It is obvious that such a
difference is due to
the small field of view of the used telescopes, which is comparable to the average angular separation of the satellites
. It hence follows that a
significant number of low and very low surface brightness dwarf
satellites can still be found around massive Local Volume galaxies
in the field of view with a radius of .
However, their search would require a lot more observing time of
the telescopes in use or attraction of new astronomical
imaging enthusiasts to this program.
§ CLOSING REMARKS
Detection of ultra-faint dwarf satellites around nearby massive
galaxies is of important cosmological significance.
Successful search for such objects in the Local Group around the Milky Way and the Andromeda Nebula (M 31)
attracted great interest to the features of their spatial distribution and
kinematics <cit.>.
Many believe that ultra-faint dwarf galaxies are among the
“darkest” objects in the Universe and in this sense they can
serve as a natural laboratory to study the nature of dark matter.
The program of detection of faint dwarf galaxies with small
telescopes should obviously be accompanied by systematic radial
velocity measurements of new LSB objects to confirm their
physical connection with massive galaxies. Radial velocity
measurements as well as the studies of the photometric structure
of new dwarf galaxies require the capabilities of large
telescopes. It remains to add that extending the program to the
southern sky objects is a quite evident and topical task.
The authors are grateful to S. S. Kaisin,D. I. Makarov,
M. E. Sharina, Yu A. Perepelitsyna, and E. S. Safonova, who took
part in the observations at the 6-m telescope and data processing.
This work was supported by Russian Science Foundation grant
No. 14-12-00695. Observations on the telescope of the
Special Astrophysical Observatory are held with financial support
from the Ministry of Education and Science of the Russian
Federation (contract No. 14.619.21.0004, project identifier
K. N. Abazajian, J. K. Adelman-McCarthy, M. A. Agueros, et al.,
182, 543 (2009).
R. Giovanelli, M. P. Haynes, B. R. Kent, et al., 130,
2598 (2005).
M. P. Haynes, R. Giovanelli, A. M. Martin, et al., 142,
170 (2011).
T. H. Jarrett, T. Chester, R. Cutri, et al., 119, 2498
T. H. Jarrett, T. Chester, R. Cutri, et al., 125, 525
B. S. Koribalski, L. Staveley-Smith, V. A. Kilborn, et al., 128, 16 (2004).
M. A. Zwaan, L. Staveley-Smith, B. S. Koribalski, et al., 125, 2842 (2003).
K. Kovač, T. A. Oosterloo, and J. M. van der Hulst, 400, 743 (2009).
J. L. Tonry, C. W. Stubbs, K. R. Lykke, et al., 750,
99 (2012).
V. E. Karachentseva and I. D. Karachentsev, 127, 409
V. E. Karachentseva and I. D. Karachentsev, 146, 359
W. K. Huchtmeier, I. D. Karachentsev, and V. E. Karachentseva, 506, 677 (2009).
I. D. Karachentsev, D. I. Makarov, and E. I. Kaisina, 145, 101 (2013).
E. I. Kaisina, D. I. Makarov, I. D. Karachentsev, and
S. S. Kaisin, 67, 115 (2012).
B. Moore, S. Ghigna, F. Governato, et al., 524, L19 (1999).
A. Klypin, A. V. Kravtsov, O. Valenzuela, and F. Prada, 522, 82 (1999).
R. Ibata, N. F. Martin, M. Irwin, et al., 671, 1591
R. A. Ibata, G. F. Lewis, A. R. Conn, et al., Nature, 493,
62 (2013).
N. F. Martin, A. W. McConnachie, M. Irwin, et al., 705, 758 (2009).
K. Chiboucas, I. D. Karachentsev, and R. B. Tully, 137,
3009 (2009).
K. Chiboucas, B. A. Jacobs, R. B Tully, and I. D. Karachentsev,
146, 126 (2013).
D. J. Radburn-Smith, R. S. de Jong, A. C. Seth, et al., 195, 18 (2011).
I. D. Karachentsev, D. Bautzmann, F. Neyer, et al.,
D. Martinez-Delgado, E. D'Onghia, T. S. Chonis, et al., arXiv:1410.6368.
L. Ferrarese, H. C. Ford, J. Huchra, et al. 128, 431
I. D. Karachentsev, E. I. Kaisina, and D. I. Makarov, 147, 13 (2014).
J. C. Mihos, K. M. Keating, K. Holley-Bockelmann, et al., 761, 186 (2012).
J. C. Mihos, P. Harding, C. E. Spengler, et al., 762,
82 (2013).
P. G. van Dokkum, R. Abraham, and A. Merritt, 782, 24
A. Merritt, P. van Dokkum, and R. Abraham, 787, L37
I. D. Karachentsev, R. B. Tully, L. N. Makarova, et al., 805, 144 (2015).
K. B. W. McQuinn, J. M. Cannon, A. E. Dolphin, et al., 785, 3 (2014).
A. Gil de Paz, S. Boissier, B. F. Madore, et al., 173, 185 (2007).
I. D. Karachentsev, S. S. Kaisin, and E. I. Kaisina, Astrophysics,
58, 331 (2015).
V. L. Afanasiev and A. V. Moiseev, Astron. Lett 31, 194 (2005).
I. D. Karachentsev, M. E. Sharina, D. I. Makarov, et al.,
Astrophysics, 58 (2015) (in press).
L. Ferrarese, J. R. Mould, R. C. Kennicutt, et al., 529, 745 (2000).
N. Trentham N. and R. B. Tully, 398, 722 (2009).
E. Schulz, 790, 76 (2014).
E. Kim, M. Kim, N. Hwang, et al., 412, 1881 (2011).
I. D. Karachentsev, V. E. Karachentseva, and W. K. Huchtmeier,
Astron. Lett. 33, 512 (2007).
M. Spencer, S. Loebman, and P. Yoachim, 788, 146 (2014).
A. Gil de Paz, B. F. Madore, S. Boissier, et al., 627,
L29 (2005).
R. F. Minchin, E. Momjian, R. Auld, et al., 140, 1093
V. E. Karachentseva, Soobschenia SAO, No. 8, 3 (1973).
D. J. Schlegel, D. P. Finkbeiner, and M. Davis, 500,
525 (1998).
P. Kroupa, C. Theis, and C. M. Boily, 431, 517 (2005).
R. A. Ibata, G. F. Lewis, A. R. Conn, et al., Nature, 493,
62 (2013).
E. J. Shaya and R. B. Tully, 436, 2096 (2013).
Translated by A. Zyazeva
|
1511.00851
|
Laboratoire Aimé Cotton, CNRS, Univ. Paris-Sud, ENS Cachan, Université Paris-Saclay, 91405 Orsay Cedex, France
GEMAC, CNRS, UVSQ, Université Paris-Saclay, 45 avenue des États Unis 78035 Versailles cedex, France.
Laboratoire Aimé Cotton, CNRS, Univ. Paris-Sud, ENS Cachan, Université Paris-Saclay, 91405 Orsay Cedex, France
Due to their high potentiality for photovoltaic applications or coherent light sources, a renewed interest in hybrid organic perovskites has emerged for few years. When they are arranged in two dimensions, these materials can be considered as hybrid quantum wells. One consequence of the unique structure of 2D hybrid organic perovskites is a huge exciton binding energy that can be tailored through chemical engineering. We present experimental investigations of the exciton non-linearities by means of femtosecond pump-probe spectroscopy. The exciton dynamics is fitted with a bi-exponential decay with a free exciton life-time of $\sim$100 ps. Moreover, an ultrafast intraband relaxation ($<150$ fs) is also reported. Finally, the transient modification of the excitonic line is analyzed through the moment analysis and described in terms of reduction of the oscillator strength and linewidth broadening. We show that excitonic non-linearities in 2D hybrid organic perovskites share some behaviours of inorganic semiconductors despite their high exciton binding energy.
§ INTRODUCTION
Hybrid organic-inorganic materials have attracted much attention for few years because of their high potentialities for optoelectronic and photovoltaic applications. The idea is to combine the advantages of organic materials such as the tunability of most of their physical properties, with the one of inorganic materials such as band engineering or electrical injection. The understanding of the electronic properties of these materials is a key issue to optimize their potential in view of applications. From a fundamental point of view, studying hybrid materials leads to new physics where features of organic and inorganic materials are observed at the same time <cit.>.
(top) Chemical structure of $(C_{6}H_{5}C_{2}H_{4}–NH_{3})_{2}PbI_{4}$. (bottom) Optical density of a 50 nm thin film of PEPI, and spectra of pump pulses at 2.397 eV (green dash line) and at 2.818 eV (blue dash line).
Hybrid organic perovskites (HOPs) have been known for a long time, but a new craze for these materials has emerged for about three years. This renewed interest is due to the exceptional performance of photovoltaic cells with HOPs as active materials <cit.>. Bulk $(CH_{3}NH_{3})PbI_{3}$ and its derivatives are the most widely used HOPs for applications of solar energy conversion. Very recently, the report of amplified spontaneous emission (ASE) highlights also the potential of these materials for the building of on-chip coherent light sources <cit.>. One of their 2D counterparts is $(C_{6}H_{5}C_{2}H_{4}–NH_{3})_{2}PbI_{4}$ (PEPI). Indeed, this structure has been shown to behave as a quantum well <cit.>. The inorganic lead-iodine octahedra layer (see figure 1) acts as a quantum well (QW) where the electronic excitations are confined, and the organic part behaves as a potential barrier <cit.>. In addition to the standard 2D confinement of a QW, carriers are exposed to a large dielectric confinement due to the difference of dielectric constant between the organic and inorganic parts <cit.>. The combination of both confinements leads to exciton binding energy as high as 220 meV, leading to stable excitons at room temperature <cit.>. One consequence is an excitonic absorption around 2.397 eV which is overhung by an absorption continuum at higher energy. The 2D nature of these hybrid QWs has been used, for instance, to reach strong coupling regime at room temperature in cavities and with plasmons <cit.>. Finally, one interesting property of 2D-HOPs is the possibility of chemical engineering that leads to a control and an improvement of their electronic properties <cit.>.
Relative variation of transmission as a function of the pump/probe time delay in a degenerate configuration (pump and probe at 2.397 eV) at exciton density n$\sim 2.2\times10^{13}$cm$^{-2}$.
The knowledge of the carrier dynamics is of high interest for the use of any materials in optoelectronic devices. A lot of studies have been devoted to the optical properties of 2D-HOPs, some about the exciton structure <cit.>, or about the influence of the dielectric confinement <cit.>. These studies often compare 2D-HOPs with GaAs QWs and show that excitons share common behaviours. For instance, 2D-HOPs show 2D-Wannier exciton series <cit.>. Nevertheless, there is a lack of information on their excitonic non linearities. One question is to know whether or not they can be described with models developed for GaAs QWs. Only few works have been devoted to these aspects. Some studies, performed at low temperature, have reported on the exciton relaxation or on the existence of the excitonic molecule <cit.>. Finally, a very recent study has reported excitonic many-body interaction in 2D-HOPs showing the influence of the well thickness on exciton-exciton interaction <cit.>.
In this paper, we report on the exciton non-linearities in PEPI, i.e. modifications of the excitonic resonance, such as broadening or loss of oscillator strength, due to the optical injection of carriers, by means of femtosecond pump/probe spectroscopy. In this study we focus on the so-called linear regime where all the probed quantities depend linearly on the excitonic population. The exciton relaxation is well described by a bi-exponential decay involving free excitons ($\tau_{F}\sim 100$ ps) and exciton trapped on dark states ($\tau_{D}\geqslant 5$ ns). Intraband relaxation is also investigated and turned to be ultrafast ($\tau_{intra}\leq$ 150 fs). Finally, moment analysis of transient absorption spectra allows to disentangle the modification of the exciton linewidth, oscillator strength and energy in presence of carriers.
Time resolved photoluminescence of PEPI excited at 3.1 eV and detected at 2.384 eV (black circles). Impulse Response Function (grey circles). Inset: comparison with the pump/probe measurement.
§ EXPERIMENTS
The sample consists in a 50 nm film of PEPI perovskite deposited on a quartz substrate, that corresponds to approximately 31 QWs. The 50 nm thin film is obtained by spin coating a 10 % wt solution of $C_{6}H_{5}C_{2}H_{4}–NH_{3}I$ and $PbI_{2}$ dissolved in stoichiometric amounts in N,N-dimethylformamide. The exciton dynamics is studied thanks to femtosecond pump/probe non-linear spectroscopy. The pump pulse corresponds to the fourth harmonic of an optical parametric amplifier (OPA) at a 1kHz repetition rate. Its wavelength can be tuned from 2.818 eV to 2.397 eV. The pump spectra are plotted as blue (2.818 eV) and green (2.397 eV) dash lines (figure 1). The probe consists in a spectrally broad pulse obtained by self-phase modulation in the sapphire crystal of the OPA. This white probe pulse allows to perform transient absorption spectroscopy. The chirp of the probe pulse was measured separately and compensated in the measurements of transient spectra. The cross correlation of the pump and probe pulses was measured by stimulated Raman diffusion in water, which gives a $\sim$ 150 fs time resolution of the experiment. The pump fluence is kept in the range of 5-10 $\mu J.cm^{-2}$. In order to control that no sample degradation occurs during the experiment, several acquisitions are performed successively at the same point of the sample and the superimposition of all of them is checked. Finally, time correlated single photon counting was used to perform the luminescence life-time measurements (TR-PL).
§ RESULTS AND DISCUSSION
§.§ Exciton dynamics
Figure 1 (bottom) displays the optical density spectrum of such a multiple QW structure. It shows an excitonic absorption peak at 2.397 eV. Figure 2 displays the relative variation of transmission of the sample as a function of the time delay measured by pump/probe spectroscopy. In this experiment, both pump and probe pulses are tuned in resonance with the excitonic transition. A positive variation of the transmission is observed, that corresponds to a transient photobleaching of the excitonic transition due to the presence of an excitonic population. The relaxation dynamics can not be fitted with a mono-exponential decay. The red curve is a fit with the convolution of a gaussian (instrumental response function) and a bi-exponential decay. The main contribution to the relaxation is related to a relaxation time of $\sim$ 100 ps. The tail at long time delay observed in figure 2 is not resolved in our experiment and is arbitrarily fixed at 5 ns in the fit. In order to get more insight into the exciton dynamics, we performed time-resolved photoluminescence. Figure 3 shows a PL decay that is slightly slower than the time resolution of the experiment ($\sim$70 ps). Here again, the relaxation can not be fitted with a mono-exponential decay. Inset of figure 3 displays a comparison of the dynamics measured both in time-resolved photoluminescence and pump/probe spectroscopy. It shows a much faster decay of the PL signal than of the relative variation of transmission. A precise comparison of the fast decays in TR-PL and pump/probe spectroscopy is not straight forward since the time resolutions of both experiments are quite different ($\sim$ 70 ps for TR-PL and $\sim$ 150 fs for pum/probe spectroscopy). Nevertheless, both time constants are of the order of $\tau_{F}\sim 100$ ps and are attributed to the relaxation of free excitons. Moreover, the lower decay rate of pump/probe experiment indicates that an excitonic population is trapped on dark states before relaxing to the ground state. These results can be compared with the one obtained by Wu et al on $(C_{4}H_{9}NH_{3})_{2}PbI_{4}$ <cit.>. In this paper the authors reported a decay dynamics fitted with three exponential with time constants of $0.36$ ps, $24.3$ ps and $910$ ps. The shortest time is attributed to hot exciton cooling time, the second time is attributed to radiative recombination whereas the longest time is related to trapped states <cit.>. The shortest time is absent in our experiments as the pump power is not increased (see figure 4, grey curve) supporting that our experiments are indeed performed in a low carrier density regime. Moreover, the main relaxation time and the longest one are of the same order of magnitude in both studies. The lack of calculations of the excitonic structure of PEPI prevents from drawing any conclusions on the nature of dark states. It could be either intrinsic (triplet states for instance) or extrinsic (traps on inhomogeneities of the film). Concerning the sources of disorder in PEPI films obtained by spin coating, they are known to be well organized in the direction perpendicular to the octahedra plans, whereas the roughness in the parallel direction is large <cit.>. Moreover, a difference between PEPI and common GaAs QWs is that, here, the well is composed of a single layer of [PbI$_6$]$^{4-}$ octahedra. This configuration prevents any fluctuation of the width of the well. On the contrary, variations in the relative position between the R-NH$_{3}^{+}$ barrier and [PbI$_6$]$^{4-}$ octahedrons are a source of disorder that lead to fluctuations of the image charge effects <cit.>. These fluctuations may also affect the exciton dynamics. Therefore, the differences observed between relaxation times in this study in comparison with the one reported by Wu et al may be related to the different nature of the cation in the two materials. In order to disentangle the influence of the different sources of disorder, experiments on micron scale exfoliated nanosheets that exhibits better crystallinity would be very helpful <cit.>.
(top) Relative variation of transmission as a function of the pump/probe time delay for a probe at 2.397 eV and a pump at 2.397 eV (green) or at 2.818 eV (blue). The $\frac{\Delta T}{T}$ amplitudes are normalized by the injected carrier density n. (bottom) Normalized variation of transmission and zoom at short time delays for both pump energies. The grey curve displays the zoom at short delays for a pump at 2.818 eV for a fluence twice the one of the blue curve (i.e $\sim 20 \mu J.cm^{-2}$).
§.§ Intraband relaxation
Figure 4 displays the relative variation of transmission of the sample as a function of the time delay for pump energies in resonance with the exciton (2.397 eV) and in the continuum (2.818 eV). It shows similar dynamics whatever the pump energy within the measurement uncertainties. Moreover, the bottom of figure 4 shows a zoom at short delays. One can notice that the two curves are superimposed and that no sizeable rise time is observed. Moreover, when the excitation power is increased, one observes the appearance of a short decay time just after the pump pulse (see grey curve in the bottom of figure 4). This is a signature of the appearance of exciton-exciton annihilation processes. In order to work in the so-called low carrier density regime a particular care is taken on the control of the pump fluence so as to get the plateau in the dynamics at short delays (see blue and green curves at the bottom of figure 4). In the following, we will discuss the possible origins of this instantaneous rise time. First, this kind of instantaneous rise time could be attributed to a ground state bleaching effect. That means that the absorption probability on the exciton state is reduced due to a depletion of its ground state when carriers are created at higher energy. This effect can not account by itself for our observation. Indeed, the transient signals share the same dynamics at the different time scales whether carriers are injected at high energy or directly on the excitonic state. If the instantaneous rise time was only due to a ground state bleaching-like effect, one should see a signature of the building of the exciton population at longer time delays, which is not the case. Secondly, in a two color pump/probe experiment, the signature of the intraband relaxation leading to the excitonic population is a rise time on the relative variation of transmission experiment. When the intraband relaxation occurs on a timescale faster than the time resolution of the experiment, it leads to an instantaneous rise time. The time resolution of our experiment is i.e. $\sim$ 150 fs. Therefore, we conclude that the intraband relaxation in PEPI is faster than $\sim$ 150 fs ($\tau_{intra}\leq 150$ fs). Such an ultrafast intraband relaxation time has to be compared to the energy difference between the initial and final states i.e. $\sim$ 421 meV (2.818 - 2.397 eV). In common semiconductors QWs, where excitons have binding energies of few meV, the intraband relaxation is phonon-assisted with, for instance, a 130 fs relaxation time per emitted LO-phonon in CdZnTe QWs <cit.>. In PEPI, the lack of calculations both of the band structure and of the phonon dispersion makes it difficult to definitely conclude about the mechanism at the origin of this ultrafast intraband relaxation. Nevertheless, this seems difficult to invoke standard phonon assisted mechanisms (in the Born-Oppenheimer approximation) alone to account for the relaxation of 421 meV in less than 150 fs. Ultrafast intraband relaxation has also been reported in other systems such as CdSe quantum dots <cit.> or 3D perovskite <cit.>. In CdSe, the relaxation of hot carriers depends on the size of the dot and can vary from picoseconds to few hundreds of femtoseconds. In 3D-HOP, a 400 fs hot hole cooling has been reported. In order to account for these ultrafast relaxation times, two main mechanisms are invoked. The first mechanism is related to non-adiabatic relaxation pathways. It is based on the breakdown of the Born-Oppenheimer approximation where vibrational motions and electronic states are strongly bound together <cit.>. The second mechanism consists in Auger mediated recombination. For instance, in CdSe quantum dots, a hot exciton cools down through the interaction with a hole in an Auger process <cit.>. These processes have attracted much attention for few years since they are of high interest for optoelectronic and photovoltaic applications <cit.>. Therefore, the observation of such an ultrafast relaxation in 2D-HOPs should bring attention to these materials, and highlights the need for further studies, and in particular calculations, in order to investigate both processes.
Finally, note that here, the relative variation of transmission is only normalized by the number of injected carriers. It is remarkable to observe that the amplitude of the exciton line bleaching is almost identical for a given carrier density whatever the pump energy. To support this observation, a statistics of the amplitude of $\Delta T/T$ normalized by the carrier densities has been performed on $\sim 50$ acquisitions. It shows that within the 8% deviation due to experimental fluctuations, the amplitude of the exciton line bleaching does not depend on the pump energy. Supposing that the amplitude of $\Delta T/T$ is proportional to the excitonic population (see below), this suggests that almost all the carriers injected in the continuum relax down to the excitonic state with almost no loss.
§.§ Physical origin of the transient signal
Let us now discuss the physical origin of the transient variation of the excitonic line. The oscillator strength ($f_{X}$), the exciton energy ($\hbar\omega_{X}$ ) and linewidth ($\Gamma_{X}$) may be modified by the presence of excitons in the system. In order to investigate which of these parameters are affected, some transient spectra have been performed.
a) Results of the moment analysis. $f_X$ stands for the oscillator strength, $\Gamma_X$ for the linewidth, $E_X$ the exciton energy and $E_b$ the exciton binding energy. b) Spectra of relative variation of transmission for different pump/probe time delays for a pump at 2.818 eV for $n=1.3\times10^{13}cm^{-2}$. c) Variation of the linewidth $\Delta\Gamma_X$ as a function of $-\frac{\Delta f_X}{f_X}$.
Figure 5b) displays a relative variation of transmission spectrum at several pump/probe delays. The main feature is a positive variation of transmission (photobleaching) at the energy of the exciton line and a negative signal (induced absorption) on the wings. The line-shape of the exciton of PEPI being not simple, the analysis of $\Delta T/T$ in the spectral domain is not straightforward. In order to disentangle which physical parameter is influenced by the presence of excitons, data have been analyzed by a method using the moments of the transient absorption spectra following the procedure proposed by Litvinenko et al. for GaAs/AlGaAs QWs <cit.>. The three quantities $f_{X}$, $\hbar\omega_{X}$ and $\Gamma_{X}$ are calculated via the following equations:
\begin{equation}
f_{X}\propto \int\alpha(\hbar\omega)d\omega
\end{equation}
\begin{equation}
\end{equation}
\begin{equation}
\Gamma^{2}_{X}\propto\int(\hbar\omega-\hbar\omega_{X})^{2}\alpha(\hbar\omega)d\omega/f_{X}
\end{equation}
where $\alpha(\hbar\omega)$ is the absorption coefficient. Results of the moment analysis are presented in figure 5a). One observes that the presence of excitons in the system modifies both $f_{X}$ and $\Gamma_{X}$. On the contrary, $\hbar\omega_{X}$ is less modified, at least more than one order of magnitude smaller than $|f_{X}|$ and $\Gamma_{X}$. At zero time delay, the presence of excitons ($n=1.3\times10^{13}cm^{-2}$) induces a reduction of oscillator strength $\frac{\Delta f_X}{f_X}$ of the order of -6$\times$10$^{-3}$ and a broadening $\frac{\Delta\Gamma_X}{\Gamma_X}\sim4\times10^{-3}$. Note that the same behaviour is observed with a pump in resonance with the exciton line.
Here, we show that the exciton energy $\hbar\omega_{X}$ in PEPI is not modified by the presence of excitons in the system. This is in strong contrast with the work of Wu et al on $(C_{4}H_{9}NH_{3})_{2}PbI_{4}$ <cit.> where a blue shift of several meV is reported with no observation of bleaching and broadening. This observation is interpreted by the authors as a signature of the localization of excitation in 2D perovskites <cit.>. Therefore, our observation may account for a larger exciton delocalization in PEPI than in $(C_{4}H_{9}NH_{3})_{2}PbI_{4}$ highlighting the importance of the organic barrier in the optoelectronic properties of 2D-HOPs.
Let's now discuss about the physical origin of the transient broadening and loss of oscillator strength of the exciton line. Concerning the broadening, it may originate, as in classical inorganic QWs, from collision broadening <cit.>. This process should lead to a linear dependance of the linewidth with the exciton density: $\Gamma_X=\Gamma_{X}^{0}+\gamma E_ba_{0}^{2}n_1$ (4), with $E_b$ the exciton binding energy, $a_0$ the 2D exciton Bohr radius, $n_1$ the carrier density per quantum well and $\gamma$ the scattering efficiency <cit.>. The loss of oscillator strength in common QWs and in the limit of low densities depends also linearly on the exciton density and follows the equation: $-\frac{\Delta f_X}{f_X}\sim \frac{n_1}{N_s}$, with $N_s$ a saturation parameter inversely proportional to the square of the exciton Bohr radius <cit.>. In this context, both the broadening and the loss of oscillator strength are linear with the exciton density. Figure 5c) displays $\Delta\Gamma_{X}$ as a function of $-\frac{\Delta f_{X}}{f_{X}}$. This representation allows to get rid of both the carrier density and the exciton Bohr radius. The overall shape of the curve is linear with a slope of $19\pm0.5$ meV. This shows that $\Delta\Gamma_{X}$ and $-\frac{\Delta f_{X}}{f_{X}}$ share the same dynamics and the same dependency on the exciton density, suggesting that these experiments are indeed performed in the low density regime where both quantities depend linearly on the exciton population. In order to support this assertion, it is possible to compare the excitation regime with the one for which the linear behaviour is observed in GaAs QWs. In reference <cit.>, the linear dependence is observed up to $5.10^9$ cm$^{-2}$ per QW, assuming a homogeneous repartition of the photocreated carriers between wells in the multilayer structure[In our experiments, with a pump at 2.818 eV, the 31$^{th}$ QW contains 8 times less carriers than the first one]. In our work, the carrier density is $\sim4.10^{11}$cm$^{-2}$ per QW, approximately 80 times higher. Nevertheless, these values have to be compared taking into account the difference in the values of exciton Bohr radius between the two materials. Indeed, both $\Delta\Gamma_{X}$ and $-\frac{\Delta f_X}{f_X}$ are proportional to the exciton density and to the square of the exciton Bohr radius <cit.>. In GaAs QWs, the exciton Bohr radius is $\sim$6.4 nm whereas it is $\sim$0.75nm in PEPI, leading to a factor $\sim73$ between the square of both Bohr radii <cit.>. Therefore, by normalizing the carriers densities by the square of exciton Bohr radii, it is possible to say that the experimental conditions are relatively close, supporting the idea that these experiments are performed in the low density regime.
The theoretical models cited above have been developed to describe quantitatively the transient optical nonlinearities in GaAs QWs <cit.>. For instance, Schmitt-Rink et al reported on the excitonic nonlinearities that arise when the oscillator strength, the exciton energy or linewidth are modified by optically injected carriers. In particular, they focus on variations in the low density regime where quantities depends linearly with the exciton density. They calculated a relative variation of oscillator strength $\frac{\Delta f_{X}}{f_{X}}=-\frac{n}{N_s}$ with $\frac{1}{N_s}=\frac{1}{N_s}|_{_{PSF}}+\frac{1}{N_s}|_{_{EXCH}}$, where $\frac{1}{N_s}|_{_{PSF}}$ corresponds to the effect of phase space filling and $\frac{1}{N_s}|_{_{EXCH}}$ to the effect of the wave function renormalization <cit.>. The calculation of both terms gives an expression depending on the square of the Bohr radius and on a numerical prefactor: $\frac{1}{N_s}= 8.51\times\pi a_{0}^{2}$. In the same manner, the broadening calculated by Manzke et al is given by equation (4) where the value of $\gamma$ is equal to 0.4169 for the case of a QW <cit.>. Taking the expressions of $\Delta\Gamma_{X}$ and $-\frac{\Delta f_{X}}{f_{X}}$ reported in these references <cit.> the theoretical value of the slope in figure 5c) depends only on the numerical prefactors and on the exciton binding energy that has been reported to be 220 meV in PEPI <cit.>. Then, the theoretical value of the slope is 3.3 meV, approximately 6 times lower than our experimental finding. The gap between the two values sounds reasonable. It is therefore possible to conclude that exciton non-linearities in 2D-HOPs also share common behaviors with classical quantum wells. Nevertheless, one major difference between the two materials is the gap of dielectric constant between the well and the barrier that exists in 2D-HOPs unlike GaAs QWs. This gap leads to the dielectric confinement effect at the origin of the huge exciton binding energy. This effect should at least influence the values of the numerical prefactors in the expressions of $\Delta\Gamma_{X}$ and $-\frac{\Delta f_{X}}{f_{X}}$ as a function of the exciton density. For instance, the contribution of wave function renormalization term in the relative variation of oscillator strength involves a sum over all the excited states of the exciton <cit.>. In the case of 2D-HOPs, the situation is more complicated than in common semiconductors due to the dielectric confinement effect. Indeed, the exciton states (1S, 2S, 3S ...) are not influenced in the same way by the screening effects, leading to a deviation from the hydrogenoid series used to describe excitons in QWs <cit.>. Therefore, screening effects have to be included in calculations to describe quantitatively the transient excitonic nonlinearities in 2D-HOPs.
§ CONCLUSION
In conclusion, we have reported time-resolved measurements performed on $(C_{6}H_{5}C_{2}H_{4}–NH_{3})_{2}PbI_{4}$, a 2D-HOP, at room temperature. A non-exponential relaxation has been reported with a characteristic decay $\tau_{F}\sim100$ ps interpreted as the life time of free excitons. Furthermore, the presence of a long tail in the pump/probe signal is attributed to trapped excitons on dark states. Moreover, an ultrafast intraband relaxation ($\tau_{intra}\leq150$ fs) has been reported. The transient modification of the excitonic line has been analyzed through the moment analysis. It shows both a reduction of oscillator strength and a broadening of the line. The analysis of the transient broadening and loss of oscillator strength demonstrates that 2D-HOPs share common behaviours with standard semiconductors QWs despite their large exciton binding energy that is closer to the one reported for organic semiconductors. Moreover, by comparison with a recent study <cit.>, the importance of the nature of the organic part on the electronic properties of 2D-HOPs has been highlighted. Finally, the quantitative differences in the amplitudes of the transient phenomena highlight the need to take into account screening effects in the description of excitonic nonlinearities of 2D-HOPs. This could open the way to excitonic engineering of 2D-HOPs in the view of their use, for instance, as amplification medium for coherent light sources.
The authors are grateful to G. Cassabois, Ph. Roussignol, P. Gilliot, J. Bloch, P. Voisin and J. Even for helpful discussions. The authors are grateful to L. Galmiche and P. Audebert for their help in chemistry. This work has been supported by ANR 'PEROCAI' and C'Nano Ile de France grant 'Perovolt'.
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1511.00616
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§ INTRODUCTION
Studying the response of quantum field theories to the presence of external probes is an interesting way to understand these theories better. However, for generic quantum field theories it is prohibitively hard to obtain exact results. The situation improves for theories with additional symmetries, like conformal invariance and/or supersymmetry. In particular, in conformal field theories, for simple enough questions, the additional symmetry disentangles the space-time dependence from the coupling dependence, and the full answer is given in terms of some unknown coefficients that possibly depend on the marginal couplings of the CFT <cit.>. In these cases, to actually compute these coefficients, one must resort to other techniques to determine them, like perturbation theory, the AdS/CFT correspondence, integrability or supersymmetric localization.
Besides their intrinsic interest, a comparatively less explored but potentially far-reaching application of the study of probes in conformal field theories is as useful diagnostics to characterize their holographic duals <cit.>.
In this note we are going to focus on the study of probes in the fundamental representation of four dimensional ${\cal N}=2$ conformal gauge theories. The main reason to limit ourselves to this small family of conformal field theories is to take full advantage of the technique of supersymmetric localization <cit.>. There have been already many works devoted to the use of supersymmetric localization to study probes of these theories (see <cit.> for a review). The main novelties of the present work are the derivation of a single integral equation that governs the eigenvalue density of all these SCFT, in the saddle-point approximation, and the study of correlators of Wilson loops with local operators. Let's comment on these two points in some detail.
The matrix models that compute the partition functions of all these superconformal theories can't be solved exactly at finite N (except for the case of ${\cal N}=4$ theories). We resort to study their partition functions in the saddle-point approximation, by introducing eigenvalue densities $\rho(x)$ for each of them. As a first result, we notice that we can write the integral equation for the eigenvalue densities of all these theories in a unified way
∫_-μ^μdy ρ(y)(1/x-y-νK(x-y))=8π^2/ λx-νK(x) ,
where $K(x)$ is a function to be defined below and the parameter $\nu$ counts what fraction of the matter multiplets transforms in the fundamental representation of the gauge group. Roughly speaking, the large N limit washes out many finite N details about the gauge groups and the relevant representations, and the only possible contributions of hypermultiplets in the fundamental representation to the matter content of the theory are $\nu=0,\frac{1}{2},1$.
Even before we attempt to solve equation (<ref>), it is apparent that the resulting eigenvalue density presents two qualitatively different behaviors, for $\nu=0$ and $\nu>0$. This was already realized in <cit.>, where the cases $\nu=0,1$ were compared.
In <cit.> it was argued as well that the physical reason for the qualitatively different behavior are the screening properties of matter in the fundamental representation.
It is also worth pointing out that this sharply different behavior has a reflection on the possible holographic duals of these field theories. It was argued in <cit.> that a necessary condition for a 4d CFT to have a holographic dual described by a gravitational action with a sensible higher derivative expansion is that at large N their central charges satisfy
\begin{equation}
c,a\gg 1, \hspace{1cm}
\frac{|c-a|}{c}\ll 1 \,.
\label{bucheletal}
\end{equation}
As it turns out, among the ${\cal N}=2$ SCFTs considered here, only theories with $\nu=0$ (i.e. the number of hypermultiplets in the fundamental representation does not scale with $N$) satisfy this constraint. So we observe a correlation between having a Wigner eigenvalue density and potentially having a holographic dual with a sensible derivative expansion.
Turning to the solution of eq. (<ref>), in the limit of strictly infinite 't Hooft coupling, $\lambda=g^2_{YM}N$, we find an analytic expression for the eigenvalue density, slightly generalizing the result in <cit.>. For strong but finite 't Hooft coupling, we can't solve analytically the saddle point equation, so we must resort to some approximation. We do so by following a couple of methods already present in the literature <cit.>.
Once we have found the eigenvalue density for generic $\nu$, we put this result to use by computing various correlation functions involving circular Wilson loops in the fundamental representation. The qualitatively different behavior mentioned above is manifested here as follows: for $\nu=0$ theories like ${\cal N}=4$ SYM, the vev grows exponentially in $\sqrt{\lambda}$ <cit.>, while for $\nu =1$ theories, like ${\cal N}=2$ SQCD, it grows with a power law, $\vev{W}\sim \lambda^3$ <cit.> . One can then anticipate that for theories with $\nu=1/2$, the vev of the circular Wilson loop should present a growth in between $\nu=0$ and $\nu=1$. Indeed, we obtain
W_ν=1/2∼λ^5 .
We then compute the two-point function of this Wilson loop with local operators. More specifically, we compute the normalized two-point function of the straight Wilson line with the Lagrangian density, and then with the stress-energy tensor. Conformal invariance fixes these normalized two-point functions up to a single coefficient each. In ${\cal N}=4$ SYM, the coefficients are essentially the same <cit.>, since the Lagrangian density and the stress-energy tensor belong to the same supermultiplet. This is no longer the case for ${\cal N}=2$ SCFTs, so it is interesting to obtain and compare the behavior of these coefficients. We find that for these two-point functions the $\nu\neq 0$ dependence enters through an angle $\theta$ defined by
cosθ=1-ν .
For instance, the coefficient in the two-point function of the straight Wilson line with the Lagrangian density is given by a constant in the large N, large $\lambda$ limit
L(x) W/W= f_W/|x⃗|^4 , f_W=1/8π^2(2π/θ-1 ) .
Recently, it has been conjectured in <cit.> that for ${\cal N}=2$ SCFTs, one can compute the normalized two-point function of the straight Wilson line with the stress-energy tensor from the vev of the Wilson loop in a squashed four-sphere, $\mathbb{S}^4_b$. Granting that this conjecture is correct, this two-point function displays a logarithmic dependence on the coupling
T_00(x) W/W=h_W/|x⃗|^4 , h_W=1/6πθ lnλ .
Furthermore, based on <cit.> it was also conjectured in <cit.> that the Bremsstrahlung function <cit.> of this Wilson loop for any ${\cal N}=2$ SCFT is given by essentially the same coefficient above,
B= 3h_W=1/2πθ lnλ .
Finally, using the general expression derived in <cit.>, we also compute the change in entanglement entropy of a spherical region of the vacuum state, due the presence of these probes. It is given by
S=(2π/3θ-1) lnλ .
These results are to be contrasted with the well-known corresponding results for ${\cal N}=4$ SYM. In this case, all these coefficients are essentially the same due to the extra amount of supersymmetry, $B=3 h_W=4 f_W$, and can be computed exactly <cit.> for various gauge groups and representations <cit.>. In the large N, large $\lambda$ regime, they scale as $\sqrt{\lambda}$. Our results further exemplify to what extent the properties of probes of ${\cal N}=4$ SYM are not generic among ${\cal N}=2$ theories.
As possible extensions of this work, our results could be generalized to Wilson loops in higher rank representations. It might be also interesting to compute subleading corrections to the results obtained here. Finally, while we carry out this analysis for Lagrangian theories for which $\nu$ can only take the values $\nu=0,\frac{1}{2},1$, an interesting question is whether there are non-Lagrangian ${\cal N}=2$ SCFTs whose correlators are captured by the expressions presented here, for other values of $\nu$.
The present paper is organized as follows: in section 2 we introduce the superconformal theories that we are going to study, we recall the matrix model that computes their partition function, and derive their eigenvalue density in the large N, large $\lambda$ regime. In section 3 we use this eigenvalue density to compute various correlation functions related to heavy probes coupled to these theories.
Note added: As this paper was being typed, we learned of upcoming work <cit.> that studies similar matters for quiver ${\cal N}=2$ SCFTs. In that work, the regime when one of the gauge couplings is strong while the other ones tend to zero is not considered, so there is no immediate overlap with the present paper.
§ SADDLE-POINT EQUATION FOR 4D ${\CAL N}=2$ SCFTS
In this section we present the ${\cal N}=2$ superconformal field theories (SCFTs) that we are going to study and recall the matrix model that computes their partition functions. We then derive the saddle-point equation for these matrix models, and solve them in the large N, large $\lambda$ limit, to obtain their eigenvalue densities.
Let's start by recalling how to obtain all 4d ${\cal N}=2$ SCFTs theories, with a single gauge group, and a marginal coupling. With ${\cal N}=2$ supersymmetry the $\beta$-function is exactly zero if and only if the one-loop contribution is zero <cit.>. Since we are interested in SCFTs that admit a large N limit, we restrict to classical (i.e. non-exceptional) gauge groups and matter content in representations with up to two indices: fundamental, 2-symmetric, 2-antisymmetric and adjoint. The complete list of such theories is well-known <cit.>, and we present it in table <ref>, together with their central charges.
A quantity that will turn out to be relevant in what follows is
ν≡lim_N→∞ n_f/2N
which counts what fraction of the matter in these theories belongs to the fundamental representation in the large N limit. For these theories, we observe in table <ref> that $\nu$ can only take the values $\nu=0,1/2,1$.
$(n_{adj},n_f,n_{S_2},n_{A_2})$ $c$ $a$ $\delta\equiv(c-a)/c$ $\nu$
$(1,0,0,0)$ $\frac{1}{4}N^2-\frac{1}{4}$ $\frac{1}{4}N^2-\frac{1}{4}$ $0$ 0
$(0,0,1,1)$ $\frac{1}{4}N^2-\frac{1}{6}$ $\frac{1}{4}N^2-\frac{5}{24}$ $\frac{1}{6N^2}+\mathcal{O}(N^{-4})$ 0
$(0,4,0,2)$ $\frac{1}{4}N(N+1)-\frac{1}{6}$ $\frac{1}{4}N(N+\frac{1}{2})-\frac{5}{24}$ $\frac{1}{2N}+\mathcal{O}(N^{-2})$ 0
$(0,2N,0,0)$ $\frac{1}{3}N^2-\frac{1}{6}$ $\frac{7}{24}N^2-\frac{5}{24}$ $\frac{1}{8}+\mathcal{O}(N^{-2})$ 1
$(0,N+2,0,1)$ $\frac{7}{24}N^2+\frac{1}{8}N-\frac{1}{6}$ $\frac{13}{48}N^2+\frac{1}{16}N-\frac{5}{24}$ $\frac{1}{14}+\mathcal{O}(N^{-1})$ $\frac{1}{2}$
$(0,N-2,1,0)$ $\frac{7}{24}N^2-\frac{1}{8}N-\frac{1}{6}$ $\frac{13}{48}N^2-\frac{1}{16}N-\frac{5}{24}$ $\frac{1}{14}+\mathcal{O}(N^{-1})$ $\frac{1}{2}$
$(n_{adj},n_f,n_{S_2})$ $c$ $a$ $\delta\equiv(c-a)/c$ $\nu$
$(1,0,0)$ $\frac{1}{2}N^2-\frac{1}{4}N$ $\frac{1}{2}N^2-\frac{1}{4}N$ 0 0
$(0,2N-2,0)$ $\frac{2}{3}N^2-\frac{1}{2}N$ $\frac{7}{12}N^2-\frac{3}{8}N$ $\frac{1}{8}-\frac{3}{32 N}+\mathcal{O}(N^{-2})$ 1
$(n_{adj},n_f,n_{S_2})$ $c$ $a$ $\delta\equiv(c-a)/c$ $\nu$
$(1,0,0)$ $\frac{1}{2}N^2+\frac{1}{4}N$ $\frac{1}{2}N^2+\frac{1}{4}N$ 0 0
$(0,2N-1,0)$ $\frac{2}{3}N^2+\frac{1}{6}N-\frac{1}{12}$ $\frac{7}{12}N^2+\frac{5}{24}N-\frac{1}{24}$ $\frac{1}{8}-\frac{3}{32 N}+\mathcal{O}(N^{-2})$ 1
$(n_{adj},n_f,n_{A_2})$ $c$ $a$ $\delta\equiv(c-a)/c$ $\nu$
$(1,0,0)$ $\frac{1}{2}N^2+\frac{1}{4}N$ $\frac{1}{2}N^2+\frac{1}{4}N$ 0 0
$(0,4,1)$ $\frac{1}{2}N^2+\frac{3}{4}N-\frac{1}{12}$ $\frac{1}{2}N^2+\frac{1}{2}N-\frac{1}{24}$ $\frac{1}{2N}+\mathcal{O}(N^{-2})$ 0
$(0,2N+2,0)$ $\frac{2}{3}N^2+\frac{1}{2}N$ $\frac{7}{12}N^2+\frac{3}{8}N$ $\frac{1}{8}+\frac{3}{32 N}+\mathcal{O}(N^{-2})$ 1
List of 4d ${\cal N}=2$ SCFT families admitting a large N limit for each classical simple Lie algebra
§.§ Partition function
Due to supersymmetric localization, the partition function of these theories on $\mathbb{S}^4$ reduces to an integral over the Lie algebra $\mathfrak{g}$ of the gauge group
Z_𝕊^4=1/(G)∫_[da]e^-8π^2r^2/g^2_YM(a,a)Z_1-loop(ra)|Z_inst(ia,r^-1,r^-1,q)|^2 ,
where $(\, , \, )$ denotes the bilinear form obtained from tracing the product in the fundamental representation and $r$ is the radius of $\mathbb{S}^4$.
This formula can be rewritten in terms of an integral over the Cartan subalgebra whose integration measure is given by a Faddeev-Popov determinant of the form
\Delta^{2}(a) = \prod_{\alpha\in roots({\galg})}(\alpha\cdot a)^{2}\, .
In this gauge the factor $Z_{1-loop}(ra)$ is a certain infinite dimensional product, which appears as a 1-loop determinant in the localization computation.
For an $\mathcal{N}=2$ theory with massless hypermultiplets in any $G$-representation $\mathcal{R}$, the 1-loop determinant is <cit.>
Z^𝒩=2,W_1-loop(ra)=∏_α∈roots()H( α·a r)/∏_w∈weights(ℛ)H(w ·a r) ,
where $H(x)$ is given by
H\left(x\right)=\prod_{n=1}^{\infty}\left(\left(1+\frac{x^{2}}{n^{2}}\right)^{n}e^{-x^{2}/n}\right) \, .
Formula (<ref>) literally holds if the divergent factors are the same in the one-loop determinants for the vector and hypermultiplets. This happens for representations $W$ such that
∑_α(α·a)^2=∑_w(w ·a)^2 ; a ∈ that is if the $\beta$-function vanishes and the $\mathcal{N}=2$ theory is superconformal.
The factor $Z_{inst}(ia,\epsilon_1,\epsilon_2,q)$ is the Nekrasov's instanton partition function of the gauge theory in the $\Omega$-background on $\bR^4$ <cit.>. For $\mathcal{N}=4$ all instanton corrections vanish ($Z_{inst}=1$). As is customary, we will assume that their contribution is negligible in the large N limit.
We now proceed to derive the saddle-point equation for these matrix models.
Following the standard procedure, we bring the Faddeev-Popov and one-loop factors to the exponent. In the large N limit, we can pass to a continuum version. To do so, introduce the eigenvalue density
ρ(x)=1/N∑_iδ(x-ra_i) ,
defined in the interval $\Gamma =[-\mu,\mu]$ and unit normalized. It is convenient to introduce
K(x)=-d logH(x)/d x .
Since $H(x)$ is an even function under $x\rightarrow -x$, $K(x)$ is odd. It is straightforward to write down an integral equation for the eigenvalue density for each ${\cal N}$=2 SCFT. We are now going to argue that all these integral equations can be written in a unified fashion. Let's first consider SCFTs with gauge group $SU(N)$, and for concreteness let's illustrate the argument with the specific example of the SCFT with a hypermultiplet in the antisymmetric representation and $N+2$ hypermultiplets in the fundamental one. The integral equation for the eigenvalue density is
∫_-μ^μdyρ(y)(1/x-y-K(x-y)+1/2K(x+y))=8π^2/λx-1/2K(x) .
The terms inside the parenthesis in the integral come respectively from the Faddeev-Popov determinant, the vector multiplet contribution and the hypermultiplet in the antisymmetric representation. The $K(x)$ term on the RHS corresponds to the hypermultiplets in the fundamental representation. Combining this equation with the one that we obtain by changing $x\rightarrow -x, y\rightarrow -y$, we learn that the eigenvalue density is even. Then, by combining this equation with the one we obtain by changing $x\rightarrow -x$ we learn that under the integral $K(x+y)$ can be replaced by $K(x-y)$. The same argument goes through for all the other SCFTs with gauge group SU(N), and we learn that their integral equations can be written in a compact way in terms of $\nu$
∫_-μ^μdy ρ(y)(1/x-y-νK(x-y))=8π^2/λx-νK(x) .
The discussion can be easily generalized to SCFTs with other classical gauge groups. For $G=SO(2N)$ the Faddeev-Popov determinant is
Δ^2(a)=∏_i<j^N|a_i^2-a_j^2|^2 ,
so its contribution to the integral equation is naively different from the case of $SU(N)$. However
∫_-μ^μdy ρ(y)2x/x^2-y^2=∫_-μ^μdy ρ(y) 2/x-y-0∫_-μ^μdy ρ(y)2y/(x-y)(x+y) ,
so it turns out to give the same kernel as $SU(N)$, except for a factor of two. A factor of two will be generated as well in the term with $\lambda^{-1}$ because the trace in the fundamental representation includes both $\pm a_i$ weights for $SO(2N)$. Finally, the Faddeev-Popov determinants of $SO(2N+1),Sp(2N)$ present further additional terms, but their contribution is subleading in the large N limit. The upshot of this analysis is that for all these SCFTs, the singular integral equation that determines the eigenvalue distribution is
\begin{equation}
\int_{-\mu}^\mu dy \rho(y)\left(\frac{1}{x-y}-\nu K(x-y)\right)=\frac{8\pi^2}{
\lambda}x-\nu K(x) \, ,
\label{thesaddleeq}
\end{equation}
For $\nu=0$ this is of course the integral equation for the Wigner distribution, while for $\nu=1$ this equation was derived in <cit.> for the particular case of ${\cal N}=2$ SQCD.
Before we proceed, let's pause to comment on the holographic implications of this result. A very interesting question is what 4d CFTs admit a holographic dual with a sensible gravitational description in at least some regime of parameters. In this regard, it is possible to find necessary conditions in terms of the central charges of the 4d CFT. If one requires that the gravitational dual is described by an action with two derivatives (i.e. Einstein -Hilbert in the gravitational sector) then in the large N limit the central charges must satisfy <cit.>
\begin{equation}
c,a \gg1 \, , \hspace{1cm} c-a=0+{\cal O}(1/N) \, .
\label{hensken}
\end{equation}
If one relaxes the requirement that the gravitational action involves just two derivatives, and requires only a sensible higher derivative expansion, the constraint on the large N value of the central charges is weakened to <cit.>
\begin{equation}
c,a \gg1 \, , \hspace{1cm} \frac{|c-a|}{c}\ll 1 \,.
\label{buchelagain}
\end{equation}
Going through the list of theories considered here, we observe that this condition is satisfied precisely by the $\nu=0$ theories. It seems that having a Wigner eigenvalue density is necessary to have a gravitational description with a sensible higher derivative expansion.
After this holographic interlude, we come back to the task of solving the saddle-point equation (<ref>).
§.§ Infinite coupling limit
In the strict limit $\frac{1}{\lambda}=0$, $\mu\rightarrow\infty$ and equation (<ref>) reduces to
\begin{equation}
\int_{-\infty}^\infty dy \rho(y)\left(\frac{1}{x-y}-\nu K(x-y)\right)=-\nu K(x) \, .
\end{equation}
This equation can be solved analytically for $\nu \neq 0$. Taking its Fourier transform we arrive at
\hat \rho_\infty(p)=\frac{1}{1+\frac{2}{\nu}\sinh ^2\frac{p}{2}} \, ,
which implies
\rho_\infty(x)=\frac{1}{\sqrt{\frac{2}{\nu}-1}}\frac{\sinh\left((\pi-\theta)x\right)}{\sinh \pi x} \, ,
θ=cos^-1 (1-ν) .
This result is just a slight generalization of the $\nu=1$ case, already obtained in <cit.>.
§.§ Strong coupling
At finite coupling, we are not aware of a technique that allows to solve the saddle-point equation, (<ref>). For finite but strong 't Hooft coupling, $\lambda \gg 1$, there are a couple of works in the literature using different approximations to solve this equation. We will follow <cit.> and also briefly comment on the approximation used in <cit.>.
The first approach to solve approximately the saddle point equation (<ref>) will closely follow <cit.>, and it is based in the Wiener-Hopf method. Our computations will only differ in the treatment of the zero-momentum mode.
Given the integral equation (<ref>), one might be tempted to solve it via a Fourier transform, after extending the definition of $\rho(x)$ to be zero outside its support, $[-\mu,\mu]$. This idea cannot be implemented to (<ref>) as it stands, since the Fourier transforms of $K(x)$ and $x$ are divergent. To arrive at an equation amenable to be Fourier transformed, we follow <cit.> and
make use of the integral operator
\mathcal{P}^{-1}_{x\rightarrow z}\left[f\left(x\right)\right]=-\frac{1}{\pi^2}\fint_{-\mu}^{\mu}\frac{dx}{z-x}\sqrt{\frac{\mu^{2}-z^{2}}{\mu^{2}-x^{2}}}f\left(x\right)
which inverts the principal part integral operator in the following regard:
\mathcal{P}^{-1}_{x\rightarrow z}\left[\fint_{-\mu}^{\mu}dy\frac{\rho\left(y\right)}{x-y}\right]=
\rho\left(z\right);\;z\in\left[-\mu,\mu\right]
Its action onto (<ref>) leads to
ρ(z)-8π/λ√(μ^2-z^2)-ν∫_-μ^μdyρ(y)(f(y,z)-f(0,z))=0; z∈[-μ,μ]
f\left(y,z\right)\equiv\mathcal{P}^{-1}_{x\rightarrow z}\left[K\left(x-y\right)\right]
The kernel does not only depend on the difference $z-y$ anymore, so the use of Fourier transformation would lead now to more involved integral expressions.
We observe nonetheless that by virtue of the symmetry $y\leftrightarrow-y$ the result (<ref>) will remain valid if we use
\hat{f}\left(y,z\right)\equiv\mathcal{P}^{-1}_{x\rightarrow z}\left[\fint_{-\infty}^{\infty}\frac{\omega\coth\left(\pi\omega\right)}{x-y-\omega}\right]=\left(z-y\right)\coth\left(\pi\left(z-y\right)\right)+\delta \hat{f}\left(y,z\right)
in place of $f\left(y,z\right)$. The advantage in this replacement is that the Fourier transform of the term $\delta \hat{f}\left(y,z\right)$ can be argued to be small, and therefore subdominant in the saddle point equation. This endows us with the possibility of solving the equation iteratively, using at each step the distribution obtained in the previous iteration to improve the estimate on the term that contains $\delta \hat{f}$. For our purposes the first step of the algorithm suffices, where this subleading term is fully neglected.
Once we have reformulated the original equation in this fashion, we are finally ready to apply the Wiener-Hopf method. The first step is to extend the definition of the eigenvalue density $\rho(y)$, outside the interval $[-\mu,\mu]$, by defining $\rho(y)=0$ outside this interval.
This is compatible with analytic methods for $\rho(y)$ as long as it is understood that $\rho(y)$ admits a branch cut outside the domain of integration and we are taking the ill-defined values on it as
\rho\left(\left|x\right|>\mu\right)=\frac{1}{2}\lim_{\epsilon\rightarrow 0} \left(\rho\left(x+i \epsilon\right)+\rho\left(x-i\epsilon)\right)\right)
Provided that we take the Fourier transform of the eigenvalue density with this prescription, we obtain
\int_{-\infty}^{\infty} e^{-i p z}\left(
\hat \rho(p)\left(1+\frac{\nu}{2\sinh^2 \frac{p}{2}}\right)
\right)=0;\;z\in\left[-\mu,\mu\right]
\int_{-\infty}^{\infty} e^{-i p z}\hat \rho\left(p\right)=0;\;z\not\in\left[-\mu,\mu\right]
F(p)\equiv 8\pi^2 \mu\frac{J_1\left(\mu p\right)}{\lambda p}+\frac{\nu}{2\sinh^2 \frac{p}{2}}+\ldots
where the dots make reference to the terms coming from $\delta\hat{f}$ that we are neglecting. The general solution for the Fourier transform of the eigenvalue density should consequently be of the form
ρ̂(p)(1+ν/2sinh^2 p/2)= F(p)-χ_- (p)-χ_+ (p) ,
where the functions $\chi_{\pm}$ in the position space are nonvanishing in the real line only on one side of $\left|x\right|>\mu$ each. Their exact expressions can be determined from analiticity constraints in momentum space.
In order to impose those constraints we should pause our calculation for a moment to focus on the analytic structure of
1+\frac{\nu}{2\sinh^2 \frac{p}{2}} \, .
This function does have double poles at $p=2 \pi n i$ and simple zeroes at $p=2 \pi n i \pm \theta$ with $\theta$ defined in eq. (<ref>). The following splitting will turn out to be very convenient:
1+\frac{\nu}{2\sinh^2 \frac{p}{2}}\equiv\frac{1}{G_+ \left(p\right)G_- \left(p\right)}
G_{+} \left(p\right)\equiv \frac
{p^2 \Gamma \left(1+\frac{\theta-i p}{2 \pi}\right)
\Gamma \left(1-\frac{\theta+i p}{2\pi}\right)}
\left(1-\frac{ip}{2\pi}\right)}
because the constructions
C_+ =\frac{\hat \rho \left(p\right) e^{-i p \mu}}{G_- \left(p\right)};\;
C_- =\frac{p^2 \hat \rho \left(p\right) e^{i p \mu}}{G_+ \left(p\right)};
are either totally annihilated or left invariant by the action of $\int_{-\infty}^{\infty}\left(2\pi i\right)^{-1}\left(p-p_0\pm i\epsilon\right)^{-1}$ operators. We can straightforwardly read expressions for $\chi_\pm$ from these projections. We obtain
(F(p)-e^i p μ/G_+(p)
∑_e^-iαμF(α)R_α/p-α)+𝒪(e^-i p μ)
R_{\alpha}\equiv \hbox{Res}\left(G_{+},\alpha\right)
The expression we have obtained for $\hat{\rho}$ is only useful to obtain $\rho\left(x\right)$ at $x\gg-\mu$, but this covers our needs in this case because of the $x\leftrightarrow-x$ symmetry. The normalization condition can be applied as
1=2\int_{0}^{\infty}dx\rho\left(x\right)=\lim_{\epsilon \rightarrow 0}\frac{1}{i\pi}\int_{-\infty}^{\infty}dp\frac{\hat{\rho}\left(p\right)}{p-i\epsilon}
R_{\alpha}\equiv \hbox{Res}\left(G_{+},\alpha\right);\;\tilde{R}_{\beta}\equiv \hbox{Res}\left(G_{-},\beta\right)
Observe that $F\left(\alpha\right)$ has an exponential contribution that makes all $\alpha$ poles equally important, but the sum in $\beta$ will be dominated by the pole at $\beta=i\theta$. Keeping only this dominant contribution and using asymptotic expressions for the Bessel functions in $F\left(p\right)$ we obtain an equation for the dependence $\mu\left(\lambda\right)$, which at large $\lambda$ can be summarized as
The expression for the eigenvalue density (in momentum space), eq. (<ref>) together with the normalization (<ref>) are the main result of this section.
Before we put these results to work, let's briefly comment on a different approximation to solve the saddle-point eq. (<ref>). In <cit.>, Bourgine solved (<ref>) by truncating the expansion of $K(x)$ and keeping only the first terms in a large $x$ expansion,
K(x) →K_sc(x) =2xln|x|+2γx+1/6x
This truncation simplifies the computation enormously, compared with the method we just described. As explained in <cit.>, when computing the vev of the Wilson loop, it works remarkably well in capturing the exponent, but not so well with the prefactor. For the sake of comparison, the expressions work out to be the same, with the replacement
θ_B= √(2ν/1-ν/6)
Remarkably, this expression differs from $\theta=\cos^{-1}(1-\nu)$ in less than 1,8$\%$ in the range $0\leq \nu\leq 1$. Presumably, keeping further terms in the large $x$ expansion of $K(x)$ would improve the agreement of these two methods. Nevertheless, we will stick to the results obtained by the first method, since they capture exactly the exponent in the power law dependence of $\vev{W}$.
§ RESULTS
In this section we put to use the eigenvalue densities found in the previous section, by computing various quantities that characterize the heavy probe. We first compute the vacuum expectation value of the Wilson loop itself; we then compute the normalized two-point function of the Wilson loop and the Lagrangian density, and similarly the normalized two-point function of the Wilson loop and the stress-energy tensor. From this last result we deduce the Bremsstrahlung function and the entanglement entropy associated to the probe.
§.§ Circular Wilson loop
We start by computing the vev of a 1/2-BPS circular Wilson loop. In his seminal paper <cit.>, Pestun showed that due to localization, the path integral reduces to a matrix model. In the saddle-point approximation, the integral boils down to a rather simple expression in terms of the eigenvalue density,
W=∫_Γe^2πx ρ(x) dx .
When $\nu=0$, the eigenvalue density follows the semi-circle law, and the vev of the Wilson loop displays exponential growth <cit.>
ρ_G(x;λ)= 4/λ√(λ-(2πx)^2) ; ⟨W(C) ⟩=2/√(λ)I_1(√(λ))∼e^√(λ)/(λ)^3/4On the other hand, for any given $\nu \neq0$, we obtain that the vev of the circular Wilson loop displays a power law dependence on $\lambda$
θμ(λ) +1/2lnμ(λ)∼log(λ) ; ⟨W(C)⟩∼√(μ(λ))/λe^2πμ(λ)∼(λ/√(log(λ)) ) ^2π/θ-1For $\nu=1$ we have $\theta =\frac{\pi}{2}$ and we recover the known result, $\vev{W}_{\nu=1}\sim \lambda^3$ <cit.>. For the other value of $\nu$ realized by Lagrangian theories, $\nu=1/2$ we have $\theta=\frac{\pi}{3}$ and we obtain
W_ν=1/2∼λ^5 .
It is amusing that for the two values of $\nu$ realized by large N Lagrangian ${\cal N}=2$ CFTs, $\nu=1/2$ and $\nu=1$, the exponent in the power law dependence of $\vev{W}$ happens to be given by integers. We don't know if there is any deeper reason behind this observation.
§.§ Two-point function of the Lagrangian density and the Wilson loop
We now want to compute the normalized two-point function of the 1/2 BPS Wilson loop and the Lagrangian density. We will first derive a general expression for such two-point function, valid for any Lagrangian CFT, and then evaluate it for the theories at hand.
Consider any CFT that can be written in terms of a Lagrangian density. The Lagrangian density is a scalar operator with scaling dimension $\Delta = 4$. Conformal invariance fixes the normalized two-point function with a straight Wilson line to be
where the coefficient $f_W(g^i)$ is a function of the possible marginal couplings of the theory.
For any Euclidean CFT, a conformal transformation maps the straight Wilson line to a circular one. It is well-known that there is a conformal anomaly associated with this mapping, and the vacuum expectation values of these two operators do not coincide <cit.>. Nevertheless, the contribution of this anomaly is localized on the Wilson line, so it is reasonable to expect that it cancels in a normaliized two-point function like the one above, and the same coefficient $f$ also appears in a similarly normalized two-point function with the circular Wilson loop. This expectation is borne out by explicit computations <cit.>.
We are going to write an expression for $f_W$ in terms of the vev of the circular Wilson loop. To do so, we are going to assume that by field redefinitions we can write the action in such a way that the gauge coupling appears only as an overall factor. The vev of the Wilson loop is
W=∫DϕW e^-1/g^2 ∫d^4 x L/∫Dϕe^-1/g^2 ∫d^4 x L ,
and we have
g^2∂_g^2lnW=-1/g^2∫d^4x L(x)W/W .
This gives us a relation in terms of the integrated two-point function. To proceed we have to do the integral in the numerator, which is divergent. A convenient regularization was used in <cit.>. It consists of mapping the space to $S^1\times H_3$,
ds^2=dτ^2+dρ^2+sinh ^2 ρ(dθ^2+sin^2θdϕ^2)
and introduce a short distance cut-off $\rho_c$ for the coordinate $\rho$. The divergence appears then as a pole $1/\rho_c$, which is discarded. Following this procedure we arrive at
f_W=1/8 π^2 g^2∂_g^2 lnW
This expression is valid for any Lagrangian 4d CFT, supersymmetric or not. As a check, for ${\cal N}=4$ SYM, this relation coincides, up to a number, with the expression found for the Bremsstrahlung coefficient in <cit.>,
4 f_W^N=4 = B^N=4 ,
and these coefficients must indeed be related in this way, since on the one hand, in ${\cal N}=4$, the lagrangian density and the stress-energy tensor are in the same supermultiplet, and on the other hand, for ${\cal N}=4$ theories, the Bremsstrahlung function is related to the two-point function of the stress-energy tensor and the Wilson loop <cit.>.
Having derived a general formula for this coefficient, we can now use the results just derived for $\vev{W}$ to obtain this coefficient for ${\cal N}=2$ SCFTs, in the large N, large $\lambda$ regime. For theories with $\nu=0$, we reproduce the known result <cit.>,
f_W=√( λ)/16 π^2 .
For theories with $\nu \neq 0$, we find that at large $\lambda$ and large N, the leading term in $f(\lambda)$ is independent of $\lambda$
f_W=1/8 π^2(2 π/θ-1) .
In our derivation, this result follows immediately from the fact that the Wilson loop grows only as a power law for large $\lambda$. Nevertheless, we find it quite remarkable. From its definition (<ref>) we can interpret this coefficient as giving the strength of the fields sourced by a static probe; our computation implies that for superconformal theories with matter in the fundamental representation, this strength reaches a limiting value in the large N, large $\lambda$ limit.
§.§ Two-point function of the stress-energy tensor and the Wilson loop
We move now to the computation of a similarly normalized two-point function, that of the stress-energy tensor and the Wilson loop. Again, for a straight Wilson line, conformal invariance fixes this two-point function up to a coefficient <cit.>,
T_00W/W=h_W(g^i)/|x⃗|^4 .
It was recently conjectured <cit.> that for ${\cal N}=2$ SCFTs, this coefficient can be related to the vev of a circular Wilson loop in a squashed four-sphere $\mathbb{S}^4_b$ <cit.>, since varying the squashing parameter will insert the stress-energy tensor,
h_W=1/12 π^2 ∂_b lnW_b|_b=1 .
Furthermore, it was argued in <cit.> that this computation can be carried out by just inserting $W_b$ in the matrix model for $\mathbb{S}^4$,
W_b=∫ dx e^2πb x ρ(x) .
which is a computation we can readily perform using the results derived in the previous section. For $\nu=0$ we have $\vev{W_b}=e^{b\sqrt{ \lambda}}$ so applying eq. (<ref>) we arrive at
h_W=√( λ)/12 π^2 ,
a result that can be alternatively obtained by a supergravity computation <cit.>.
For $\nu\neq 0$ theories, it is more convenient to compute $\vev{W_b}$ directly in momentum space using
\vev{W_{b}}=\hat{\rho}\left(-2\pi b i\right) \, .
Keeping the relevant term in the asymptotic $\lambda\gg 1$ limit and plugging (<ref>) into the result we conclude that
ln⟨W⟩_b∼lnλ(2πb/θ-1)+𝒪((1-b)^2,lnμ/2 )
so we arrive at
h_W=1/6πθlnλ .
Again, the result for any $\nu\neq 0$ differs parametrically from the known result of $\nu=0$ theories, which displays the ubiquitous $\sqrt{\lambda}$ dependence, as in eq. (<ref>). Notice also that for generic ${\cal N}=2$ theories, the $\lambda$ dependence of the two coefficients just considered is different. This should not come as a surprise, since for ${\cal N}=2$ theories (unlike what happens in ${\cal N}=4$ SYM) the Lagrangian density and the stress-energy tensor don't belong to the same supermultiplet.
As a bonus, the computation of $h_W$ immediately gives us two other interesting quantities. The first one is the Bremsstrahlung function of the corresponding probes. For any 4d CFT, the Bremsstrahlung coefficient can be defined <cit.> as the coefficient that appears in the formula for the energy loss of an accelerated probe,
E=2πB∫ dt a^2 .
It also captures the momentum fluctuations of the accelerated probe <cit.>. Intuitively, it seems natural that the two-point function of the stress-energy tensor would capture the energy loss of the probe. However, the details are subtle and there is no simple universal relation for B and $h_W$ <cit.>, valid for all four-dimensional CFTs. Nevertheless, for probes of ${\cal N}=2$ SCFTs it is conjectured <cit.> that
B=3h_W .
and granting that this conjectured relation is true, we conclude that
B=1/2πθlnλ .
One lesson of this result is the following. It has been argued in <cit.> that a certain class of observables of planar ${\cal N}=2$ superconformal gauge theories can be obtained from the corresponding result of planar ${\cal N}=4$ SYM, by means of replacing the ${\cal N}=4$ coupling by a single function, universal for a given ${\cal N}$=2 SCFT. Comparing the results we have obtained for $\vev{W}$ and $B$ for ${\cal N}$=4 and ${\cal N}$=2 theories, we conclude that this substitution rule does not apply to the computation of $B$, for theories with a single gauge group.
Finally, we can use our result for $h_W$ to compute the additional entanglement entropy of a spherical region when we add a external probe to the vacuum of the theory. According to <cit.> it is given by
S=lnW-8π^2h_W ,
so for the probes we are considering we have
S=(2π/3θ-1)lnλ .
We would like to thank Efrat Gerchkovitz, Zohar Komargodski, Vladimir Mitev and Elli Pomoni for discussions. The research of BF and GT is partially supported by the Spanish MINECO under projects FPA2013-46570-C2-2-P and MDM-2014-0369 of ICCUB (Unidad de Excelencia “María de Maeztu"), and by AGAUR, grant 2014-SGR-1474. GT is further supported by an FI scholarship by the Generalitat de Catalunya.
|
1511.00660
|
WIPAC]J. Vandenbrouckecorauths
Rochester]S. BenZvi
WIPAC]S. Bravo
WIPAC]P. Karn
WIPAC]M. Meehan
Sensorcast]J. Peacock
WIPAC]M. Plewa
Physics]T. Ruggles
Columbia]M. Santander
WIPAC]D. Schultz
Loyola]A. L. Simons
WIPAC]D. Tosi
[WIPAC]Physics Department and Wisconsin IceCube Particle Astrophysics Center, University of Wisconsin, Madison, WI 53706, USA
[Physics]Physics Department, University of Wisconsin, Madison, WI 53706, USA
[Rochester]Department of Physics and Astronomy, University of Rochester, Rochester, NY, USA
[Columbia]Department of Physics and Astronomy, Barnard College, Columbia University, New York, NY, USA
[Loyola]Seaver College of Science and Engineering, Loyola Marymount University, Los Angeles, CA 90045, USA
Camera image sensors can be used to detect ionizing radiation in addition to optical photons. In particular, cosmic-ray muons are detected as long, straight tracks passing through multiple pixels. The distribution of track lengths can be related to the thickness of the active (depleted) region of the camera image sensor through the known angular distribution of muons at sea level. We use a sample of cosmic-ray muon tracks recorded by the Distributed Electronic Cosmic-ray Observatory to measure the thickness of the depletion region of the camera image sensor in a commercial smart phone, the HTC Wildfire S. The track length distribution prefers a cosmic-ray muon angular distribution over an isotropic distribution. Allowing either distribution, we measure the depletion thickness to be between 13.9 $\mu$m and 27.7 $\mu$m. The same method can be applied to additional models of image sensor. Once measured, the thickness can be used to convert track length to incident polar angle on a per-event basis. Combined with a determination of the incident azimuthal angle directly from the track orientation in the sensor plane, this enables direction reconstruction of individual cosmic-ray events.
[corauths]Corresponding author: [email protected]
cosmic rays, muons, cell phones, CMOS, depletion region
§ INTRODUCTION
The Distributed Electronic Cosmic-ray Observatory (DECO) <cit.> is a network of mobile devices in which camera image sensors are used to detect ionizing radiation including cosmic rays and ambient radioactivity. Although designed to detect optical photons, cell phone camera image sensors (predominantly using CMOS technology) are also sensitive to ionizing radiation incident on the depletion region. Consumer technology can therefore be used for purposes similar to those of custom-built trackers used for particle physics and astro-particle physics. For an overview of DECO, see <cit.>.
Although the area of each sensor is small (typically $\sim$0.15 cm$^2$, varying from model to model), many sensors can be harnessed together. The area of silicon installed in camera image sensors of cell phones and other consumer devices throughout the world[Worldwide there are $\sim$10$^9$ cell phones with cameras, each with a sensor area $\sim$10$^1$ mm$^2$.] is at least $\sim$10$^4$ m$^2$. This is two orders of magnitude larger than the largest professional silicon trackers (installed in the Compact Muon Solenoid <cit.> and the Fermi Large Area Telescope <cit.>). Each image sensor provides megapixels of resolution for track direction determination and particle identification. Finally, we benefit from many years of experience of astronomers in identifying, classifying, and removing cosmic-ray tracks from CCD images <cit.>. CCDs and CMOS sensors are established detectors of cosmic rays and background radioactivity (which are usually a background to optical or X-ray photon detection) and can potentially detect dark matter as well <cit.>.
The DECO mobile app[Available at http://wipac.wisc.edu/deco] records camera images continuously and applies an online filter, implemented in the phone CPU, to select particle candidates. These data, along with associated metadata, are automatically synchronized to a central server for analysis by scientists and members of the public. DECO features a web-based data browser[Available at http://wipac.wisc.edu/deco/data] where users can query the data including quantities such as device ID, model ID, timestamp, and geolocation. Users can click individual events to view the event image or the event location in Google Maps. For privacy, latitude and longitude are degraded to 0.01$^\circ$ resolution and image data are provided only in a zoomed, cropped, and false colored version.
There are several challenges to identifying and calibrating images obtained from a heterogeneous network of consumer devices. Each model, and potentially each device of an individual model, has different noise characteristics. Furthermore, the noise varies with environmental conditions such as temperature. Cosmic rays must be discriminated from background events including those due to sensor artifacts, thermal noise fluctuations, and low-energy particle events induced by radioactivity within or near the device. Finally, cameras in advanced mobile devices include post-processing hardware, firmware, and software designed to remove noise, which can also remove particle events. There is evidence in the DECO data that particular advanced models have low rats of particle detection because of this noise removal feature.
Using the DECO dataset, we demonstrate discrimination between GeV cosmic-ray muon tracks and MeV electron tracks caused by radioactive decay using topological cuts based on track images. Using the muon event sample, we fit the track length distribution with a model based on the known cosmic-ray muon angular distribution at sea level and with a single degree of freedom corresponding to the thickness of the depletion region of the camera image sensor. We also fit the distribution with an analogous functional form corresponding to an isotropic distribution. The results simultaneously validate the use of cell phone camera image sensors as cosmic-ray muon detectors and provide a measurement of a parameter of camera image sensor performance which is not otherwise publicly available. We focus on a particular model, the HTC Wildfire S.
Map of DECO data collection locations. Data have been collected on all seven continents.
§ DATA COLLECTION
DECO has been collecting data since September 2011, with the app released publicly in September 2014. To search for particle events, images from the camera are continuously examined by an online filter that requires a certain number of pixels to reach a certain threshold in the sum of red, green, and blue pixel amplitudes. The minimum pixel multiplicity has been varied during development but is typically five.
A DECO run begins automatically when the user opens the app and continues until the app is closed. The amplitude threshold is automatically determined at the beginning of each run by a calibration routine that measures the sensor noise, which varies from device to device and with temperature. In addition to the filtered data stream of particle candidates, the app saves a single image every five minutes (regardless of whether it passed the filter criteria) as a minimum bias stream for sensor noise and other performance characterization.
Each image contains RGB colors with 8 bits of depth per color. Complete images are saved in order to characterize background noise fluctuations and resolve the full transition from brightly hit pixels through sub-threshold neighboring pixels down to the thermal noise background. Users are advised to collect data with the phone plugged in, connected to wifi, and with the camera lens covered in order to minimize ambient light reaching the sensor.
A map of DECO acquisition locations is shown in Figure <ref>. DECO users have taken data on all seven continents with more than different models of Android devices.
§ METHOD OF DEPLETION THICKNESS MEASUREMENT
At sea level, the cosmic-ray muon flux is dominated by minimum ionizing particles, with a mean energy of $\sim$4 GeV <cit.>. The zenith angle ($\theta$) distribution is proportional to $\cos^2 \theta$ and the total flux integrated over energy and solid angle is $\sim$1 particle per minute per cm$^2$ <cit.>.
We approximate the depletion region as uniformly sensitive to ionizing radiation. Under this assumption, the depletion thickness determines the distribution of the measured track length (projected on the image plane). Given a depletion thickness ($H$) and track length component in the sensor plane ($L$), the distribution of measured track lengths, in the case of an isotropic particle flux (for example, dominated by alpha particles from radioactive decay within or near the phone), is given by
\begin{equation}
\frac{dN}{dL} = A \frac{L H^2}{(L^2 + H^2)^{2}} {\rm~~~(isotropic)}
\end{equation}
where $A$ is a normalization constant proportional to the absolute particle flux, sensor detection efficiency, and livetime.
Alternatively, if the tracks are distributed according to the sea-level cosmic-ray angular distribution ($\cos^2\theta$), a different length distribution is expected:
\begin{equation}
\frac{dN}{dL} = B \frac{L H^4}{(L^2 + H^2)^{3}} {\rm~~~(cosmic~rays)}
\end{equation}
where B is analogous to A. See the appendix (Section <ref>) for derivations of these distributions.
The two distributions have slightly different shapes that can be distinguished from one another with sufficient statistics.
§ EVENT CLASSIFICATION AND SELECTION
Events representative of the three topologies commonly detected by DECO are shown in Figures <ref>, <ref>, and <ref>. These event classes detected by DECO in cell phone CMOS image sensors are the same as those identified in astronomical CCDs. Following the terminology of <cit.>, we designate these event classes as “tracks”, “worms”, and “spots”. Tracks are straight, likely due to high energy (GeV scale) cosmic-ray muons that exhibit little Coulomb scattering. Worms show significant deflection along the trajectory, curving and/or kinking. They are most likely due to low-energy (MeV scale) electrons that exhibit multiple Coulomb scattering. Such electrons are a product of radioactive decay in the materials of the phone itself or in the surroundings. They could be produced directly in a beta decay event or by a Compton scatter from an incident gamma ray produced in a decay. Spots could also be produced by short-range alpha particles <cit.>.
Alpha particles produced by nuclear decay could also deposit detectable tracks. Due to their short range, they would need to be produced within the phone itself or within a few cm of it. The range of alpha particles between 1 and 10 MeV in silicon is dozens of microns, comparable to the expected sensor depletion thickness. Future studies may enable identification of alpha tracks through their ionization energy loss rate ($dE/dx$), which is four times larger than that of muons and electrons due to their twice larger charge, or through their Bragg peak as discussed below.
Some tracks are brighter at one end than the other. In the case of stopping particles, this could indicate a Bragg peak. While the probability of cosmic-ray muons stopping within the sensor is negligible due to their long range, alpha particles have a good likelihood of stopping within the depletion region. It is also possible that the bright spots on one end are due to front-back asymmetry in the response of the depletion region to ionizing radiation <cit.>. If confirmed, this effect could be used to break the bilateral degeneracy in direction reconstruction along the track.
In addition to the depletion thickness measurement presented here using the muon candidate sample, a similar measurement could be performed using the worm events by quantifying their multiple scattering within the depletion region as well as their trajectory into and out of the depletion region slab.
Example “track” event likely due to a GeV cosmic-ray muon.
Example “worm” event likely due to a low-energy (MeV) electron, potentially produced by a Compton scatter from an incident gamma-ray produced by radioactive decay in the phone or surroundings.
Example “spot” event. The spot topology is likely due to a Compton scatter event that produces a low energy (MeV) electron which is quickly absorbed.
To classify events, we first convert the RGB pixel values to a single gray-scale amplitude (luminance) using the ITU-R 601-2 luma transform <cit.>. We next calculate a contour using the “marching squares” algorithm <cit.>, implemented in the scikit-image python module <cit.>, to delimit the pattern of pixels that detected ionization above a particular threshold. This algorithm interpolates within pixels to determine the best iso-luminance contour through each pixel based on the luminance of it and its neighbors. The algorithm requires a single parameter which is the luminance value at which the iso-luminance contour should be drawn. The value found to perform best in containing the hit pixels and not being susceptible to noise was 20.
Several metrics are calculated for each event. The first is the integrated area within the iso-luminance contour. To remove background events caused by noise fluctuations, we require a minimum area of 10 pixels. We also determine the maximum and minimum $x$ and $y$ values spanned by the contour and define the distance between these to be the track length.
Next we calculate the principal moments of the luminance image. From these, we determine the image eccentricity $\epsilon$. We find this eccentricity parameter to to be an excellent discriminant for separating the long, straight tracks from worms and spots. To select a sample rich in muon candidate events, we require $\epsilon > 0.99$.
Measured length distribution of tracks from muon candidates detected by the HTC Wildfire S. In each panel the distribution is fit with a single-parameter function whose parameter is the sensor depletion thickness in units of the pixel width. In each case there are $8-1=7$ degrees of freedom. wildfire_isotropic Fit with functional form according to an isotropic particle flux. The $p$ value for the fit is 0.008. wildfire_cosmic Fit with functional form according to a sea-level cosmic-ray angular distribution ($\cos^2\theta$). The $p$ value for the fit is 0.48.
§ RESULTS
In this analysis we focus on 5829 events recorded by the HTC Wildfire S between January 28, 2012 and January 21, 2014.
After requiring a minimum area of 10 pixels, 2018 events remain. We next apply a cut designed to remove worms, noise fluctuations, and multi-site depositions: We require that the marching squares algorithm determines only one closed contour in the entire camera image. This reduces the sample to 1043 events. Finally, requiring the eccentricity to be at least 0.99 reduces the number of events to 198.
Using these events we determined the length distribution (in pixels) and fit both the isotropic and cosmic-ray parameterizations to the measured distribution, as shown in Figure <ref>. In each case (isotropic and cosmic-ray), a single-parameter fit provides a depletion thickness estimate. The estimate is 29.2 $\pm$ 1.5 pixels in the cosmic-ray case and 16.7 $\pm$ 1.2 in the isotropic case. According to <cit.>, this camera image sensor is 2400 $\mu$m $\times$ 1800 $\mu$m for its 2592 $\times$ 1944 pixels, corresponding to a 0.9 $\mu$m square pixel size. This means that the DECO estimated depletion thickness is 26.3 $\pm$ 1.4 $\mu$m (cosmic-ray case) or 15.0 $\pm$ 1.1 $\mu$m (isotropic case).
The $p$ value of the cosmic-ray fit is 0.48 and the $p$ value of the isotropic fit is 0.008. The present dataset is therefore fit moderately better by a cosmic-ray distribution than an isotropic distribution. This gives us additional confidence that we are detecting and accurately selecting cosmic-ray muons in the DECO data. With future datasets it will be possible to more significantly distinguish between a cosmic-ray and isotropic distribution. This will provide additional confidence in the cosmic-ray origin of the track-shaped signals and will also enable a quantification of a possible isotropic background component produced for example by alpha decays.
Although the event selection is developed to obtain a muon-rich sample and the cosmic-ray fit is preferred over the isotropic fit, there could be an isotropic particle population that contaminates the cosmic-ray muons. The isotropic result therefore provides an estimate of the systematic uncertainty of the measurement. Conservatively, the depletion thickness could be within the full range spanned by the two fits: $(13.9, 27.7)$ $\mu$m.
§ CONCLUSION
We have demonstrated a particular capability of a cell phone app and central database, DECO, to detect cosmic-ray muons and other ionizing radiation. We identified several classes of commonly occurring events consistent with those found in professional astronomical CCDs, and developed algorithms to classify the events. Using a muon track sample, we used DECO to measure the thickness of the depletion region in camera image sensors of an example cell phone model, the HTC Wildfire S, under the approximation that the depletion region responds uniformly to ionizing radiation.
We note that the track length distribution is different for cosmic rays at sea level than for isotropic particles. The cosmic-ray fit is preferred over the isotropic fit with moderate significance. Larger data sets available in the future will more powerfully discriminate between the two cases.
The quality of the directional information of muon tracks in cell phones indicates that additional future measurements are feasible. Because most modern cell phones have accurate orientation sensors, the absolute direction of each cosmic-ray muon can be determined by fitting the azimuthal angle within the sensor plane and using the track length (with the depletion thickness, which can eventually be measured for each device) to determine the polar angle relative to the sensor plane. The depletion thickness relative to the pixel width, not the absolute depletion thickness, is necessary for such direction reconstruction and is determined by the method presented here.
We also note that bright spots at one end of many tracks may be due to front-back asymmetry within the active region and may therefore enable breaking the bilateral degeneracy in determining the cosmic-ray incident direction along the track. It may be possible in future studies to use mobile devices to detect the East-West effect <cit.> in cosmic rays, an effect which historically provided the first indication that cosmic rays are positively charged. DECO provides a large cosmic-ray monitoring network that could enable correlation studies with other data sets and events such as solar storms. Although exceedingly difficult <cit.>, it may also be possible to detect extensive air showers produced by ultra-high-energy cosmic rays through detection with sufficiently many devices in coincidence <cit.>. In any case, DECO is a powerful tool for students and members of the public to carry a particle detector in their pocket and use it to engage with astronomy and particle physics.
§ APPENDIX: DERIVATION OF PARAMETERIZATIONS
To derive the functional form of the parameterization in the two cases, we consider the dependence on $H$ and $L$ and do not include overall normalization constants.
§.§ Isotropic case
In the isotropic case, we have
\begin{equation}
dN \propto A_\textit{eff}(\theta) d\Omega
\end{equation}
where $A_\textit{eff}$ is the effective area that the sensor presents to the particle flux. It is a function of $\theta$ ($A_\textit{eff} = a \cos \theta$, where $a$ is the sensor geometrical area) because the sensor presents a greater cross sectional area to the flux at normal incidence than the flux at oblique incidence. Furthermore, $d\Omega = \sin \theta d\theta d\phi$. Therefore,
\begin{equation}
dN \propto \cos \theta \sin \theta d\theta
\end{equation}
\begin{equation}
dN \propto d ( \sin^2 \theta)
\end{equation}
Now, $\sin \theta = L/\sqrt{L^2 + H^2} $, so
\begin{equation}
\sin^2 \theta = L^2 / (L^2 + H^2)
\end{equation}
\begin{equation}
d(\sin^2\theta) = \frac{2L H^2 dL}{(L^2+H^2)^2}
\end{equation}
\begin{equation}
\frac{dN}{dL} = A \frac{L H^2}{(L^2 + H^2)^{2}} {\rm~~~(isotropic)}
\end{equation}
§.§ Cosmic-ray case
In the cosmic-ray case, we have an additional two factors of $\cos\theta$:
\begin{equation}
dN \propto \cos^3 \theta \sin \theta d\theta
\end{equation}
\begin{equation}
dN \propto d ( \cos^4 \theta)
\end{equation}
\begin{equation}
d(\cos^4\theta) = \frac{-2(L^2+H^2) 2L H^4 dL}{(L^2+H^2)^4}
\end{equation}
\begin{equation}
\frac{dN}{dL} = B \frac{L H^4}{(L^2 + H^2)^{3}} {\rm~~~(cosmic~rays)}
\end{equation}
§ ACKNOWLEDGMENTS
DECO is supported by the American Physical Society, the Knight Foundation, and the Simon-Strauss Foundation. We are grateful for valuable conversations with Keith Bechtol, Andy Biewer, Patricia Burchat, Duncan Carlsmith, Alex Drlica-Wagner, Lucy Fortson, Stefan Funk, Mandeep Gill, Giorgio Gratta, David Kirkby, Carsten Rott, David Saltzberg, Ignacio Taboada, and Ian Wisher. We appreciate beta testing, data collection, and data studies by Colin Adams, Ilhan Bok, Paul Brink, Felipe Campos, Alex Diebold, Mike DuVernois, Laura Gladstone, Jim Haugen, Kyle Jero, and Heather Levy.
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1511.00705
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We propose a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry. The list includes both local questions as well as issues related to the compact complex manifold setting.
§ INTRODUCTION
Pluripotential theory is the several complex variables analogue of classical potential theory in the complex plane (or in Riemann surfaces). While the latter is a linear theory associated to the Laplacian
of a Kähler metric, the former is highly non-linear and associated to
complex Monge-Ampère operators
\f \mapsto MA(\f):=c \, (\omega+dd^c \f)^n,
where $\omega$ is a Kähler metric (or merely a semi-positive and big form), $n$ is the complex dimension
of the ambient manifold
and $c>0$ a normalizing constant.
When $\f$ is smooth and the ambient manifold is an open subset of
$\C^n$ equipped with the euclidean Kähler metric
$\omega=dd^c ||z||^2=\sum_{j=1}^n \frac{i}{\pi} dz_j \wedge d \overline{z}_j$, a change of unknown
function $\f \mapsto \f+||z||^2$ reduces the previous expression to
MA(\f)=c' \, \det \left( \frac{\partial^2 \f}{\partial z_i \partial \overline{z}_j} \right) \, dV(z),
where $dV$ denotes the Lebesgue volume form and $c'>0$ is a normalizing constant.
Complex Monge-Ampère equations
$MA(\f)=F(\f) \mu$
have been one of the most powerful tools in Kähler geometry since Aubin and Yau's classical works
<cit.>, culminating in Yau's solution to the Calabi conjecture. A notable application is the construction of Kähler-Einstein metrics on compact Kähler manifolds.
In recent years, following Tsuji's pioneering work <cit.>, degenerate complex Monge-Ampère equations have been intensively studied by many authors. In relation to the Minimal Model Program, they led to the construction of singular Kähler-Einstein metrics
<cit.> or, more generally, of canonical volume forms on compact Kähler manifolds with nonnegative Kodaira dimension [ST12].
Making sense of and constructing singular Kähler-Einstein metrics on these mildly singular varieties has required advanced tools in the study of degenerate complex Monge-Ampère equations and led
to many open problems.
We have been asked by several colleagues to propose a list of such open questions.
There is no pretention to be exhaustive,
the selection we propose reflects our own taste and limitations.
The questions are of various nature, some would lead to breakthrough and require new ideas,
others are probably reasonably easy.
Although there are many interesting subjects that we haven't covered, we hope there is nevertheless
enough interesting questions for the curious reader to work on.
§.§ Structure of the article
The article is organized as follows:
* In the first section we list open questions related to the domain of definition of the complex Monge-Ampère operator and quasi-plurisubharmonic Green functions.
* We collect in the second section problems related to the description of the range of the complex Monge-Ampère operator acting on various classes of (quasi-)psh functions.
* The third section focuses on regularity issues, in the local and the compact settings.
* The fourth section is devoted to pluripotential theory on non Kähler manifolds.
* We mention in the fifth section some problems about the extension of quasi-psh functions,
as well as how to control their singularities.
* The sixth and last section is devoted to various open questions that arise in the viscosity approach,
for degenerate elliptic and parabolic Monge-Ampère equations.
The authors would like to express their gratitude to Z.Blocki, D.Coman, U.Cegrell C.Favre, S.Kołodziej, M.Păun and A.Rahskovskii for several useful conversations.
§ GREEN FUNCTIONS
We let $MA(\f)$ denote the complex Monge-Ampère measure of a (quasi-)plurisubharmonic function.
When $\f$ is smooth and plurisubharmonic in a domain $\Omega \subset \C^n$, then
MA(\f)=(dd^c \f)^n=c \, \det \left( \frac{\partial^2 \f}{\partial z_i \partial \overline{z}_j} \right) \, dV(z),
where $dV$ denotes the Lebesgue volume form and $c>0$ is a normalizing constant.
When $\f$ is smooth and $\omega$-plurisubharmonic on a compact Kähler manifold $(X,\omega)$, then
MA(\f)=c \, (\omega+dd^c \f)^n,
where $c=(\int_X \omega^n)^{-1}$ is chosen so that $MA(\f)$ is a probability measure.
§.§ Domains of definition
§.§.§ The local context
When $n=1$, the complex Monge-Ampère operator reduces to the linear Laplace operator $\Delta$, whose action on all (p)sh functions is well defined in the sense of distributions. In higher dimensions one cannot usually define the product of two distributions, so the definition of the complex Monge-Ampère operator is more involved.
Let $\f$ be a plurisubharmonic function in some domain $\Omega \subset \C^n$. We say that
the complex Monge-Ampère measure $(dd^c \f)^n$ is well defined if there exists a Radon measure
$\mu$ such that for any sequence
$(\f_j)$ of smooth plurisubharmonic functions decreasing to $\f$ in some relatively compact
subdomain $V \subset \subset \Omega$, the Monge-Ampère measures
$(dd^c \f_j)^n$ converge weakly to $\mu$ in $V$.
If such is the case, one sets $(dd^c \f)^n:=\mu$.
Bedford and Taylor have shown that the complex Monge-Ampère operator $\f \mapsto (dd^c \f)^n$ is well defined for all locally bounded
psh functions <cit.>, however it can not be defined for all
psh functions when $n \geq 2$ (see <cit.>), motivating the following questions <cit.>:
Given $\f$ an arbitrary plurisubharmonic function in some domain $\Omega \subset \C^n$, what can be said about the set of cluster points
of $(dd^c \f_j)^n$, where $\f_j$ are bounded plurisubharmonic functions which decrease to $\f$ ?
By definition this set is reduced to a single Radon measure when $\f$ belongs to the domain of definition
$DMA(\Omega)$ of the complex Monge-Ampère operator. One further difficulty
is that this operator is not continuous for the $L^1_{loc}$-topology <cit.>:
Given $\f$ a bounded plurisubharmonic function in some domain $\Omega \subset \C^n$, what can be said about the set of cluster points of
$(dd^c \f_j)^n$, where $\f_j$ are bounded plurisubharmonic functions which converge to $\f$ in $L_{loc}^1$ ?
When $n=2$, a psh function $\f$ belongs to the domain of definition $DMA(\Omega)$,
$\Omega$ a bounded domain,
if and only if its gradient
$\nabla \f$ belongs to $L^2_{loc}(\Omega)$,
DMA(\Omega)=PSH(\Omega) \cap W^{1,2}_{loc}.
Indeed if such is the case, a simple integration by parts shows that $\f$ is locally integrable with respect to the trace measure of the positive current $dd^c \f$,
hence the current $\f \, dd^c \f$ is well defined and so is
(dd^c \f)^2:=dd^c (\f dd^c \f).
This has been observed by Bedford and Taylor in <cit.>.
To show the converse, Cegrell provides in <cit.> explicit computations showing that
$-\sqrt{-\log|z_2|}$ does not belong to $DMA(\B)$, as it can be approximated by a decreasing sequence of bounded psh functions whose Monge-Ampère masses blow up.
This is extended by Blocki in <cit.> who shows that any psh function
$\f \notin W^{1,2}_{loc}(\Omega)$ is the decreasing limit of smooth psh functions whose Monge-Ampère masses blow up.
A simpler argument has been further provided by Cegrell in <cit.>.
Let us recall that psh functions have gradient in $L^q_{loc}$ for all $q<2$. The function
$\f(z)=\log|z_1|$ does not have gradient in $L_{loc}^2$ while any locally bounded psh function $\f$
does: indeed we can assume $\f \geq 0$ and observe that
dd^c \f^2 =2 \f dd^c \f +2 d\f \wedge d^c \f,
hence the $L^2$-norm of
$\nabla \f$ is bounded from above by the mass of the positive current $dd^c \f^2$.
In higher dimension $n \geq 3$, the domain of definition $DMA(\Omega)$ has been fully characterized by Cegrell <cit.> and Blocki <cit.>, but the characterization is more involved.
We only stress, for further use, the following important subclass: plurisubharmonic functions with
compact singularities belong to $DMA(\Omega)$. We say here that a function $\f$ has compact singularities if it is locally bounded near the boundary of $\Omega$.
§.§.§ Maximal plurisubharmonic functions
A plurisubharmonic function $\f$ is called maximal in a domain $\Omega \subset \C^n$
if for any other plurisubharmonic function $v$ in $\Omega$ such that $v \leq \f$ near $\partial \Omega$,
then $v \leq \f$ in $\Omega$.
When $n=1$, maximal plurisubharmonic function are precisely harmonic ones, in particular they are smooth, but in higher dimension they may be quite irregular: a plurisubharmonic function that only depends on
$n-1$ variables is maximal. When $\f \in DMA(\Omega)$ then
\f \text{ is maximal } \Longleftrightarrow MA(\f)=0,
as explained by Blocki in <cit.>. In particular maximality is a local notion in $DMA(\Omega)$.
Is maximality a local notion for arbitrary plurisubharmonic functions?
If there exists a holomorphic foliation by complex discs $\Delta_t$ along which a plurisubharmonic function
$\f$ is harmonic, then $\f$ is maximal (see e.g. <cit.>).
Conversely assume $n=2$ (for simplicity): if a maximal plurisubharmonic function $\f$ is ${\mathcal C}^3$-smooth, it follows from Frobenius theorem that
the support of the differential form $dd^c \f$ is foliated by holomorphic discs $\Delta_t$ such that $\f_{|\Delta_t}$
is harmonic (see <cit.>). Dujardin has provided in <cit.> several examples of maximal
plurisubharmonic functions that are ${\mathcal C}^{1,\alpha}$-smooth and such that
there is no such holomorphic disc.
Assume $0 \leq \alpha <1$ and $n=2$.
Assume $\f$ is ${\mathcal C}^{2,\alpha}$-smooth and maximal. Is the suppport of $dd^c \f$ foliated (or laminated)
by holomorphic disc along which $\f$ is harmonic ?
§.§.§ Compact Kähler manifolds
The situation turns out to be quite different on compact Kähler manifold.
Let $(X,\omega)$ be a compact Kähler manifold and let $PSH(X,\omega)$ denote the set of all
$\omega$-psh functions. By mimicking the definition in the local case, we propose:
Fix $\f \in PSH(X,\omega)$. We say that
the complex Monge-Ampère measure $(\omega+dd^c \f)^n$ is well defined and write
$\f \in DMA(X,\omega)$, if
there exists a Radon measure $\mu$ such that for any sequence
$(\f_j)$ of smooth $\omega$-plurisubharmonic functions decreasing to $\f$
on $X$, the Monge-Ampère measures
$(\omega+dd^c \f_j)^n$ converge weakly to $\mu$.
If such is the case, one sets $(\omega+dd^c \f)^n:=\mu$.
The existence of smooth $\omega$-psh approximants $\f_j \in PSH(X,\omega) \cap {\mathcal C}^{\infty}(X)$
is not an obvious fact, see <cit.>. One can replace in the definition smooth by
bounded approximants by using Bedford-Taylor's result: bounded $\omega$-psh approximants are easy to construct, for example $\f_j=\max(\f,-j) \in PSH(X,\omega) \cap L^{\infty}(X)$
would do.
We say that $\f$ belongs to $DMA_{loc}(X,\omega)$ if for any
covering of $X$ by bounded strictly pseudonvex domains $U_{\alpha}$, and any local potential
$\rho_{\alpha}$ of $\omega$ in $U_{\alpha}$, the function
$\rho_{\alpha}+\f$ belongs to $DMA(U_{\alpha})$.
The (global) domain of definition $DMA(X,\omega)$ of the complex Monge-Ampère operator
$\f \mapsto (\omega+dd^c \f)^n$ is different than its local counterpart
The construction of Blocki <cit.> breaks down here as the total mass of
the Monge-Ampère measures of any approximating sequence is fixed, equal to the volume
$Vol_\om(X)=\int_X \omega^n$.
It turns out indeed that $DMA(X,\omega)$ contains functions
which do not have gradient in $L^2$.
Characterize $DMA(X,\omega)$.
Again $DMA(X,\omega)$ is a strict subclass of $PSH(X,\omega)$, Lelong numbers provide for instance an obstruction to being in $DMA(X,\omega)$ (see <cit.>).
An important subclass ${\mathcal E}(X,\omega) \subset DMA(X,\omega)$ of functions with full
Monge-Ampère mass has been introduced in <cit.> (see section <ref>).
Functions in ${\mathcal E}(X,\omega)$ have zero Lelong numbers. Thus $DMA(X,\omega)$ contains
the two distinct subsets ${\mathcal E}(X,\omega)$ and $DMA_{loc}(X,\omega)$.
Apart from this not much is known:
Is $DMA(X,\omega)$ convex ?
If $\f \in DMA(X,\omega)$ and $\p \in PSH(X,\omega)$, does it imply that $\max(\f,\p) \in DMA(X,\omega)$ ?
We refer the reader to <cit.> for more open questions, notably a tentative candidate for
describing $DMA(X,\omega)$ in complex dimension $n=2$.
§.§ The Dirac mass question
Let $\f$ be a plurisubharmonic function in the unit ball $\B$ of $\C^n$. Assume $\f$ has a well-defined
complex Monge-Ampère measure $(dd^c \f)^n$, we could for instance
assume that $\f$ is locally bounded outside the origin.
[Guedj, Rashkovskii]
Assume that $(dd^c \f)^n $ has a Dirac mass at the origin. Does it imply that
$\f$ has a positive Lelong number at $0 \in \C^n$ ?
The reverse implication is well-known to hold (see <cit.>). Note however
\f(z,w)=\max( \e \log |z|, \e^{-1} \log |w|)
yields an example of a psh function such that $(dd^c \f)^2$ has Dirac mass $1$ at the origin, but
an arbitrarily small Lelong number $\nu(\f,0)=\e$.
We list below
some partial information and a few situations where the answer is known:
* one necessarily has $\f(0)=-\infty$ (an observation due to Cegrell, see <cit.>);
* if $\f$ is toric, i.e. $\f(e^{i\theta_1} z_1,\ldots,e^{i\theta_n} z_n)=\f(z_1,\ldots,z_n)$
for all $\theta=(\theta_1,\ldots,\theta_n)$ and $z=(z_1,\ldots,z_n)$, then the answer is positive (see <cit.>);
* if $e^{\f}$ is Hölder continuous, in particular if $\f$ has analytic singularities, then the answer is positive (see <cit.>);
* if $\f$ is bounded from below by $\gamma \log \| z\|-C$, the answer is positive <cit.>;
* when $\f$ is invariant under a rational mapping, the answer is positive
(see <cit.>).
Demailly has proposed <cit.> a stronger question. Consider the Bergman spaces
\cH(j \f):=\left\{ f \in \cO(\B) \, | \, \int_\B |f|^2 e^{-2j\f} dV <+\infty \right\}
of holomorphic functions that are $L^2$ in the ball with respect to the measure $e^{-2j\f} dV$, where
$dV$ denotes the Lebesgue measure. Let $(f_{k,j})$ denote an orthonormal basis of $\cH(j \f)$ and set
\f_j:=\frac{1}{2j} \log \left[ \sum_{k=0}^{+\infty} |f_{k,j}|^2 \right].
As observed by Demailly in <cit.>, it follows from the Ohsawa-Takegoshi theorem that
the $\f_j$'s approximate $\f$ in a very accurate manner, hence motivating the following:
Does one have $(dd^c \f_j)^n(0) \longrightarrow (dd^c \f)^n(0)$ ?
Since the $\f_j$'s have analytic singularities and $\nu(\f_j,0) \rightarrow \nu(\f,0)$, a positive answer to Demailly's question would also imply a positive answer to Question <ref>.
The interested reader will get more information on these questions in Rahskovskii's papers (see
§.§ Quasi-plurisubharmonic Green functions
Cegrell and Wiklund have made some contributions to Question <ref>
showing in particular that it suffices to answer it for functions whose Monge-Ampère measure
is concentrated at the origin (see <cit.>).
Such functions are called Green functions whenever they have zero boundary values.
Classical Green functions are fundamental solutions to the Laplace equations. As such they play a crucial role in linear potential theory.
Appropriate notions of plurisubharmonic Green functions have been defined and studied in domains of $\C^n$, we refer the reader to <cit.> for a survey of their properties and applications, see also section 5.6.
One can similarly study $\omega$-psh Green functions on a compact Kähler manifold
$(X,\omega)$: for $a \in X$, these are the functions $g_a \in DMA(X,\omega)$ such that
MA(g_a)=V^{-1}(\omega+dd^c g_a)^n=\delta_a
is the Dirac mass at point $a$.
Such functions have been used in complex dynamics to construct invariant measures of maximal entropy
(see <cit.>).
The article <cit.> studies quasi-plurisubharmonic Green functions for themselves, exhibits
some examples and several questions, for instance:
Assume $(X,\omega)$ is the complex projective space $\C\PP^n$ equipped with the Fubini-Study Kähler form. Can one find examples of a $\omega$-psh function $G_a \in DMA(X,\omega)$, whose Monge-Ampère measure is concentrated at some point $a \in \C\PP^n$ and having an irrational Lelong number at $a$ ?
The answer is negative in dimension $1$, as the Lelong number coincides in this case with the Dirac mass.
The situation is more subtle in higher dimension, as explained in section <ref>.
The reader will find in <cit.> various constructions of such quasi-psh Green functions with given rational Lelong number at $a$.
§ RANGE OF THE COMPLEX MONGE-AMPÈRE OPERATOR
In this section we try and describe the range of the complex Monge-Ampère operator on various classes
of ($\omega$-)plurisubharmonic functions.
§.§ The non-pluripolar product
Let $(X,\omega)$ be a compact Kähler manifold. Given $\f \in PSH(X,\omega)$ we consider the bounded approximants
\f_j:=\max(\f,-j) \in PSH(X,\omega) \cap L^{\infty}(X).
It has been realized in <cit.> that the truncated Monge-Ampère measures
1_{\{ \f>-j\}} (\omega+dd^c \f_j)^n
form an increasing sequence of Borel measures with mass uniformly bounded by $V=\int_X \omega^n$.
A similar local construction had been previously proposed by Bedford and Taylor <cit.>.
Set $T=\omega+dd^c \f \geq 0$. The measure
\langle T^n \rangle=\lim \nearrow 1_{\{ \f>-j\}} (\omega+dd^c \f_j)^n
is called the non-pluripolar measure product of $T^n$.
It has been shown in <cit.> that two cohomologous positive closed currents
$T \sim S$ satisfy
\langle T^n \rangle \leq \langle S^n \rangle
if $T$ is more singular than $S$ and both currents have small unbounded locus, i.e. their local potentials are locally bounded outside a closed complete pluripolar set.
This latter condition seems purely technical and the question makes sense in full generality:
Assume $T \sim S$ with $T$ more singular than $S$. Is it true that
\langle T^n \rangle \leq \langle S^n \rangle \; \; ?
The construction of the non-pluripolar product makes sense when the cohomology class of $T,S$ is not necessarily Kähler, but it uses the fact that the manifold is compact and Kähler (or at least
belongs to the Fujiki class). It would be interesting to understand whether non-pluripolar products are well-defined on arbitrary compact complex manifolds.
In the sequel we set
MA(\f):=V^{-1} \langle (\omega+dd^c \f)^n \rangle,
\text{ with }
V=\int_X \langle \omega^n \rangle.
This is a positive Radon measure of total mass at most one.
We let ${\mathcal E}(X,\omega)$ denote the set of those functions $\f \in PSH(X,\omega)$ such that
the measure $MA(\f)$ is a probability measure.
The "finite energy class" ${\mathcal E}(X,\omega)$ has been introduced in <cit.> and further studied in <cit.>. Clearly $PSH(X,\omega) \cap L^{\infty}(X) \subset {\mathcal E}(X,\omega)$, but the finite energy class also contains many unbounded functions: for example if $\f$ is an arbitrary
$\omega$-psh function then $-(-\f-\sup_X \f-1)^\a \in {\mathcal E}(X,\omega)$ for any $0 \leq \a <1$.
The range of the complex Monge-Ampère operator is well understood in this context:
A probability measure belongs to the range of the complex Monge-Ampère operator acting
on ${\mathcal E}(X,\omega)$
if and only if it does not charge pluripolar sets.
A subtle point is to understand precisely when a function $\f \in DMA(X,\omega)$ belongs to
${\mathcal E}(X,\omega)$. This has been investigated in <cit.> with poor success:
Assume $\f \in DMA(X,\omega)$ is such that $MA(\f)$ does not charge $(\f=-\infty)$.
Does it imply that $\f$ belongs to ${\mathcal E}(X,\omega)$ ?
The answer is positive when $n=1$. The problem seems more subtle when
$n \geq 2$, it is again positive when $e^{\f}$ is continuous <cit.>.
§.§ Pluripolar measures
Thanks to the works of Cegrell and the authors, there is a fairly good understanding of the non-pluripolar range of the complex Monge-Ampère operator both in domains
of $\C^n$ <cit.> and on compact Kähler manifolds <cit.>.
Despite recent results by Cegrell and his coauthors <cit.>, the corresponding problem
for measures which do charge pluripolar sets is fully open.
We mention the following particular issue:
Let $X \stackrel{\pi}{\rightarrow} \C\PP^2$ be the blow up of the complex projective plane at one point.
This is a DelPezzo surface, hence its first Chern class is positive.
Pick $\omega \in c_1(X)$ a Kähler form,
$D$ a smooth anticanonical divisor, and let $\mu_D=\omega \wedge [D] /V$ denote a normalized
Lebesgue measure along $D$. Can one find $\f_D \in DMA(X,\omega)$ such that
MA(\f_D)=\mu_D \; \; ?
The measure $\mu_D$ is pluripolar as it is supported on $D$.
The question is motivated by the fact that $X$ does not admit any Kähler-Einstein metric, hence
the Kähler-Ricci flow diverges on $X$.
Such a Monge-Ampère potential $\f_D$ could be a cluster point of renormalized
Kähler potentials evolving along the Kähler-Ricci flow on $X$.
§.§ The subsolution problems
We now focus on the case of bounded and continuous plurisubharmonic functions.
Although it is the setting of the original breakthrough of Bedford and Taylor <cit.>,
the range of the complex Monge-Ampère operator acting on bounded/continuous (quasi-)psh functions
is yet not characterized (despite rather sharp sufficient conditions <cit.>).
We recall the following question from <cit.>:
Is the range of the complex Monge-Ampère operator acting on $PSH \cap L_{loc}^{\infty}$
characterized by a uniform integrability condition ?
When $n=1$, the answer is positive: a probability measure $\mu$ can be written
$\mu=\Delta \f$ for some locally bounded subharmonic function $\f$ if and only if
it satisfies
SH(\Omega) \subset L^1_{loc}(\mu),
as the reader will check. The integrability condition $PSH(\Omega) \subset L^1_{loc}(\mu)$ is no longer sufficient if $n \geq 2$.
In a bounded hyperconvex domain $\Omega$ of $\C^n$,
a probability measure $\mu$ can be written
\mu=MA(\f)
\; \; \text{ with } \; \;
\f_{|\partial \Omega} \equiv 0
if and only if $\mu$ is dominated by the Monge-Ampère measure of a bounded
plurisubharmonic function. This follows from the comparison principle
and the homogeneity of $MA$ in the local context,
$MA(\lambda \f)=\lambda^n \MA(\f)$ <cit.>. In other words: if there is a bounded subsolution, then there is a bounded solution to the corresponding Dirichlet problem.
Assume there is a continuous subsolution to the Dirichlet problem, i.e. there is a continuous
psh function $u$ in $\Omega$ such that $u_{|\partial \Omega} =0$ and
$\mu \leq (dd^c u)^n$ in $\Omega$. Does it imply that there is a continuous solution, i.e.
a continuous psh function $\f$ such that $\f_{|\partial \Omega} =0$ and
$\mu = (dd^c \f)^n$ in $\Omega$ ?
The answer is positive in dimension $n=1$ as follows from the following characterization: a probability measure $\mu$ can be written
$\mu=\Delta \f$ for some continuous subharmonic function $\f$ if and only if
the inclusion
SH(\Omega) \hookrightarrow L^1_{loc}(\mu)
is continuous,
as the reader will check.
The situation is even more delicate on compact Kähler manifolds:
Let $(X,\omega)$ be a compact Kähler manifold. Let $\mu$ be a probability measure which is
locally dominated by the Monge-Ampère measures of bounded plurisubharmonic functions.
Can one find $\f \in PSH(X,\omega) \cap L^{\infty}(X)$ such that
$\mu=MA(\f)$ ?
The homogeneity property does not hold in the compact setting. Kołodziej has proposed an approach in
<cit.>, but the proof contains a gap.
One can show that the hypothesis on $\mu$ is equivalent to the existence of
$u \in PSH(X,\omega) \cap L^{\infty}(X)$ and $C>0$ such that
\mu \leq C \, MA(u).
Note in particular that $\mu$ does not charge pluripolar sets. It follows therefore from
<cit.> that there exists a unique finite energy function
$\f \in {\mathcal E}(X,\omega)$ such that $\mu=MA(\f)$, and the problem is to show that
$\f$ is bounded below.
A positive answer to this question would imply that
the set $MA(PSH(X,\omega) \cap L^{\infty}(X))$ is convex since,
for $u,v \in PSH(X,\omega) \cap L^{\infty}(X)$,
MA(u)+MA(v) \leq 2^n \, MA \left( \frac{u+v}{2} \right).
§.§ Hölder continuous solutions
Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$. We let
{\mathcal Hol}(X,\omega):=PSH(X,\omega) \cap \rm{Holder}(X)
denote the set of Hölder continuous $\omega$-psh functions on $X$.
Recall that the Lelong numbers of $\omega$-psh functions are uniformly bounded
by a constant that only depends on $X,n,\{\omega\}$.
It follows therefore from Skoda's integrability theorem <cit.> (see <cit.> for the uniform version)
that there exists $\e>0$ such that
\exp(-\e PSH(X,\omega) ) \subset L^1(\mu),
for any smooth volume form $\mu$.
Dinh-Nguyen-Sibony <cit.> have shown that
the same integrability property holds true for any Monge-Ampère measure
$\mu=MA(\f)$ of a Hölder continuous $\omega$-psh function $\f$.
We wonder whether this property actually characterizes $MA({\mathcal Hol}(X,\omega))$:
Are the following properties equivalent ?
(i) There exists $\e_{\mu}>0$ such that $\exp(-\e_\mu PSH(X,\omega) ) \subset L^1(\mu)$.
(ii) There exists $\f \in {\mathcal Hol}(X,\omega)$ sucht that $\mu=MA(\f)$.
These properties are indeed equivalent when $n=1$, or when the singularities have toric symmetries
(see <cit.>).
A (technical) characterization of $MA({\mathcal Hol}(X,\omega))$ is established in
<cit.>, which solves in particular the subsolution problem in the Hölder case:
a measure belongs to $MA({\mathcal Hol}(X,\omega))$ if and only if it admits a Hölder continuous
subsolution. This property is therefore local. The local Hölder subsolution problem is however
Let $\mu$ be a probability measure in a bounded pseudoconvex domain $\Omega \subset \C^n$.
Assume there is a Hölder continuous subsolution to the Dirichlet problem, i.e. there is a
Hölder continuous psh function $u$ in $\Omega$ such that $u_{|\partial \Omega} =0$ and
$\mu \leq (dd^c u)^n$ in $\Omega$.
Is there a Hölder continuous psh function $\f$ such that $\f_{|\partial \Omega} =0$ and
$\mu = (dd^c \f)^n$ in $\Omega$ ?
§ REGULARITY ISSUES
§.§ Uniform a priori estimates
The foundations of an existence and regularity theory for complex Monge-Ampère
equations with smooth data were laid by Yau <cit.> and Caffarelli-Kohn-Nirenberg-Spruck <cit.>.
Yau notably provided in <cit.> a crucial $L^{\infty}$- a priori estimate in the context of
compact Kähler manifolds. His method, based on Moser iteration technique can be generalized to work for right hand side data in $L^p$ with $p>n=dim_{\mathbb C} X$. Later on Cheng and Yau suggested that the Alexandrov-Bakelman-Pucci
estimate can be applied to the complex Monge-Ampère equation. A detailed account of their work
has been provided by Bedford <cit.> and Cegrell-Persson <cit.>. This approach works for $L^p$- right hand side with $p\geq 2$.
This has been since then extended by Kołodziej who showed in <cit.> the following:
Let $\B$ be the unit ball of $\C^n$ and fix $p>1$. If $\f$ is a smooth plurisubharmonic function
in $\B$ such that $\f_{| \partial \B} =0$, then
|| \f ||_{L^{\infty}(\B)} \leq C ||f||_{L^p(\B)}^{1/n},
where $\det(\f_{j\overline{k}})=f$ and $C$ only depends on $n,p$.
It is interesting that no PDE proof of the above fact is known.
Find a proof of the above estimate that relies on PDE techniques.
§.§ Hölder continuity
Let $X$ be a $n$-dimensional compact Kähler manifold. Let $\omega$ be a semi-positive closed $(1,1)$-form with $V_{\omega}:=\int_X \omega^n>0$, and fix $0 \leq f \in L^p(dV)$, where
$dV$ is a volume form and $p>1$.
When $\omega$ is Kähler it has been shown by Kołodziej in <cit.> that the equation
(\omega+dd^c \f)^n=f \, dV
admits a unique normalized $\omega$-psh function $\f$ which is Hölder continuous.
This result has been refined in <cit.> (better exponent, $L^p$-property and control in families). The following local $1$-dimensional example shows that one cannot expect
more than Hölder regularity, and how the exponent is linked to the integrability properties of the density:
The function $\f:z \longmapsto |z|^{2\a} $ is subharmonic in the unit disk of $\C$ with
dd^c \f=\frac{c}{|z|^{2-2\a}} dV=f dV,
with $f \in L^p$ if and only if $p<1/(1-\a)$.
If the form $\omega$ is merely semi-positive and big (i.e. $\int_X\omega^n>0$) then solutions to the Monge-Ampère equations with $L^p$- right hand sides are continuous as shown in <cit.> by using viscosity techniques (solutions are then unique up to an additive constant). Further regularity is then largely open. In particular the following is an important problem:
Assume $\omega$ is merely semi-positive and big and
(\omega+dd^c \f)^n=f \, dV,\ f\in L^p,\ p>1.
Is $\f$ Hölder continuous on $X$ ?
It is known that the solution is Hölder continuous on the ample locus of $\{\omega\}$
<cit.>. The problem is thus to understand the regularity at the boundary of the ample locus.
Such equations show up naturally in Kähler geometry, when constructing singular Kähler-Einstein metrics
on mildly singular varieties (see <cit.>). The problem is to understand the asymptotic behavior of these metrics near the singularities; the function $\f$ above occurs as the potential of such a metric.
The orbifold setting (quotient singularities) is well understood and shows that one cannot expect more than global Hölder continuity of the potentials near the singularities. One can hope to be able to give precise asymptotics, but this remains a major open problem so far.
§.§ Totally real submanifolds
Let $(X, \omega)$ be a compact $n$- dimensional Kähler manifold. Recall that a real submanifold $S \subset X$ is called totally real
if for every point $x\in S$ the real tangent plane $T_xS$ does not contain complex lines.
Let $\mu_S$ denote the normalized $n$-dimensional Hausdorff measure on a smooth totally real manifold $S \subset X$ of maximal real dimension $n$.
It has been shown by Sadullaev in <cit.> that $S$ is non pluripolar, hence $\mu_S$ is the Monge-Ampère measure
$\mu_S=(\omega+dd^c \f_S)^n$ of a unique function $\f_S \in {\mathcal E}(X,\omega)$ normalized by $\sup_X \f_S=0$.
What is the optimal regularity of $\f_S$ ?
One expects $\f$ to be Lipschitz in this case: the torus $S=S^1 \times \cdots S^1 \subset \C\PP^n$ is such that
\f_S=\max_{0 \leq i \leq n} \log |z_i| -\log ||z||.
The potential $\f_S$ is Lipschitz continuous for any real analytic totally real submanifold $S \subset \C\PP^n$.
It has been shown by Sadullaev and Zeriahi <cit.> that extremal functions of such manifolds
are Lipschitz-continuous.
§.§ Smooth currents in big cohomology classes
Let $X$ be a compact manifold in the Fujiki class (i.e. $X$ is bimeromorphic to a compact Kähler manifold).
It follows from the work of Demailly
that $X$ admits one (hence many)
Kähler current(s), i.e. a positive closed $(1,1)$-current $T$ which dominates a positive hermitian form.
A cohomology class $\a \in H^{1,1}(X,\R)$ is big if it can be represented by a Kähler current.
The ample locus $Amp(\a)$ is the set of points $x \in X$ such that there exists a Kähler current
representing $\a$ which is smooth near $x$.
The ample locus is a non empty Zariski open subset, as follows from Demailly's regularization result
<cit.>. The latter insures that any big cohomology class can be represented by a Kähler current with
analytic singularities:
A positive closed current $T$ has analytic singularities if it can be locally written $T=dd^c u$, with
u=\frac{c}{2}\log \left[ \sum_{j=1}^s |f_j|^2 \right]+v,
where $c>0$, $v$ is smooth and the $f_j$'s are holomorphic functions.
The theory of complex Monge-Ampère equations has been extended to big cohomology classes in <cit.>.
Theorem <ref> holds in this context (the boundary of the ample locus is pluripolar hence negligible for non-pluripolar measures): given $\mu$ a non-pluripolar probability measure,
there exists a unique finite energy current $T \in \a$ such that
\langle T \rangle^n=\mu,
where $\langle T \rangle^n$ denotes the non-pluripolar product.
The regularity theory is more involved:
Let $\a \in H^{1,1}(X,\R)$ be a big cohomology class, $\mu$ a smooth volume form and let $T$ be the unique finite energy current in $\a$ such that
$\langle T \rangle^n=\mu$.
Is $T$ smooth in $Amp(\a)$ ?
The answer is positive under the extra assumption that $\a$ is nef <cit.>.
It is also known to hold when $\a=c_1(K_X)$, i.e. when $X$ is a manifold of general type,
but this relies on important recent progresses in birational geometry <cit.>.
A similar problem concerns Monge-Ampère equations on quasi-projective manifolds.
In geometrical applications, singularities of the right hand side are often of divisorial type i.e. $f\approx ||\sigma||^a$ for some defining section $\sigma$ of a fixed divisor and $a\in\mathbb R$. It is natural to ask whether the singular metric associated to the solution of the Monge-Ampère equation is in fact smooth off the divisor:
Let $(X,\omega)$ be a compact $n$-dimensional Kähler manifold.
Assume $\f \in {\mathcal E}(X,\omega)$ solves the equation $(\omega+dd^c \f)^n=f\omega^n$, with $f\in L^1$ being
smooth off a divisor $D\subset X$. Is the function $\f$ smooth on $X\setminus D$?
Since the measure $\mu=f\omega^n$ is non-pluripolar, it is known <cit.> that there exists a unique
normalized solution $\f \in {\mathcal E}(X,\omega)$, so the point is to study its regularity in $X \setminus D$.
Under some mild additional conditions the answer is affirmative <cit.>. The general case however remains open.
§.§ Pogorelov interior estimate
A bounded domain $\Omega \subset \C^n$ is hyperconvex if there exists a negative plurisubharmonic function
$\rho$ in ${\Omega}$ such that the sublevel sets $\{ \rho<c\}$ are relatively compact in $\Omega$ for any $c<0$.
Note that all smooth and strongly pseudoconvex domains are hyperconvex, but they are plenty of hyperconvex non-smooth (and/or not strongly pseudoconvex) domains.
Let $\Omega \subset \C^n$ be a bounded hyperconvex domain.
Let $u$ be the unique continuous plurisubharmonic function in
$\Omega$ such that $u_{| \partial \Omega}=0$ and
(dd^c u)^n=f dV,
with $f$ smooth and bounded away from zero $f \geq \e >0$. Is $u$ automatically smooth ?
Note that the main result of <cit.> implies that any bounded hyperconvex domain admits one exhaustion function $u$ with the above properties.
The problem is open even if one assumes $f$ is constant.
The answer is positive when $\Omega$ is smooth and strongly pseudoconvex <cit.>.
It is however crucial for applications to avoid any a priori regularity assumption on $\partial \Omega$,
so as to be able to apply such a result to the open sublevel sets $\Omega_c=\{u<c\}$.
The answer is also positive for the real Monge-Ampère equation <cit.>, the main ingredient being an interior ${\mathcal C}^2$-estimate of Pogorelov <cit.>.
A complex version of this estimate is not known.
In a similar direction we mention the following:
Let $u$ be a plurisubharmonic function in a smooth strictly pseudoconvex bounded domain
$\Omega \subset \C^n$. Assume $u \in {\mathcal C}^{1,\a}(\Omega),\a>1-\frac2n$ and
(dd^c u)^n=f dV,
with $f$ smooth and bounded away from zero $f \geq \e >0$. Is $u$ automatically smooth ?
Example <ref> below shows that the condition $\a>1-\frac 2n$ cannot be dropped.
The answer is affirmative for the real Monge-Ampère equation <cit.>. In the complex setting the best result is due to Wang <cit.>,
who showed that $u$ is smooth assuming $\Delta u\in L^{\infty}(\Omega)$ instead of $u \in {\mathcal C}^{1,\a}(\Omega),\a>1-\frac2n$.
§.§ Krylov's result
It is an interesting and largely open question to determine
when pluripotential solutions are actually strong solutions, i.e. when the Monge-Ampère operator can be defined pointwise almost everywhere. A special case of the above situation is when the solution is $\mathcal C^{1,1}$ smooth. Hence it is interesting to know when such a regularity holds.
Fix $\Omega \subset \C^n$ a smooth strictly pseudoconvex bounded domain, and
$\phi \in {\mathcal C}^{1,1}(\partial \Omega)$.
Let $u$ be the unique plurisubharmonic function in $\Omega$, continuous up to the boundary,
such that $u=\phi$ on $\partial \Omega$ and
(dd^c u)^n=0 \; \;
\text{ in } \; \; \Omega.
Show, by using complex analytic methods, that $u$ is ${\mathcal C}^{1,1}$-smooth in $\Omega$.
This result has been established by Bedford and Taylor when $\Omega=\B$ is the unit ball <cit.>.
The general case has been established by Krylov in a series of papers culminating in <cit.>, by using probabilistic methods.
We refer the reader to <cit.> for a comprehensive presentation of Krylov's approach which remains quite involved. Demailly has suggested on several occasions that one should be able to produce a proof of Krylov's result by using more standard tools from complex analysis.
We refer the reader to <cit.> for a brief historical account of related questions.
A related problem concerns ${\mathcal C}^{1,1}$ regularity up to the boundary:
Assume that $\Omega \subset \C^n$ is a ${\mathcal C}^{3,1}$ smooth strictly pseudoconvex bounded domain, and
$\phi \in {\mathcal C}^{3,1}(\partial \Omega)$.
Let $u$ be the unique plurisubharmonic function in $\Omega$, continuous up to the boundary,
such that $u=\phi$ on $\partial \Omega$ and
(dd^c u)^n=f \; \;
\text{ in } \; \; \Omega
with $f\geq 0, f^{1/n}\in\mathcal C^{1,1}(\bar{\Omega})$. Then $u\in {\mathcal C}^{1,1}(\bar{\Omega})$.
Bedford and Fornaess <cit.> have shown that $u$ need not belong to $\mathcal C^2(\bar{\Omega})$ under the assumptions above. Their construction relies on the boundary behavior of the $\mathcal C^2$ norm.
Under what minimial assumptions does $u$ belong to $\mathcal C^2(\Omega)$ ?
The problem requires to better understand how singularites form in the interior of $\Omega$.
§.§ Regularity of the pluricomplex Green function
Let $\Omega$ be a bounded hyperconvex domain. The pluricomplex Green function with pole at $w\in\Omega$ is defined by
$$g_{\Omega}(z,w)=sup\lbrace u(z)|\ u\in\psh(\Omega),\ u\leq 0,\ u(s)\leq \log ||s-w||+C\rbrace,$$
where the constant $C$ may depend on $u$.
It is easy to prove that for a hyperconvex domain $\Omega$ and a fixed pole $w$ one has
Is it true that for a bounded hyperconvex domain $g_{\Omega}(z,w)$ converges locally uniformly to zero, as the pole $w$ converges to the boundary of $\Omega$?
The question is only interesting if $\Omega$ has very rough boundary.
[Blocki] Let $\Omega$ be smoothly bounded strictly pseudoconvex domain. It is known that $g_{\Omega}\in\mathcal C^{1,1}(\Omega\setminus\lbrace w\rbrace)$. Is there an example where $g$ fails to be in $\mathcal C^2(\Omega\setminus\lbrace w\rbrace)$?
Coman <cit.> and independently Edigarian and Zwonek <cit.> computed explicitly the pluricomplex Green function with two poles for the unit ball in $\mathbb C^n$ and the function is not $\mathcal C^2$. No such example with one pole is known. On the other hand Bedford and Demailly <cit.> have constructed an example which need not be $\mathcal C^2$ smooth up to the boundary but the interior regularity is open.
§ HERMITIAN SETTING
In this section we work on a compact complex manifold $X$ that is not necessarily Kähler.
The Kähler condition imposes topological and geometrical restrictions.
The reference form $\omega$ is now the $(1,1)$-form of a hermitian metric, it is no longer closed.
There always exist such a hermitian metric $\omega>0$, as can be seen by patching together local metrics by partition of unity. The plurisubharmonicity with respect to such a hermitian metric $\omega$ can be defined in an analogous way as in the Kähler setting by the inequality $\omega+dd^c\f\geq 0$. Note also that $MA\f)=(\omega+dd^c\f)^n$ makes sense if $\omega$ is merely hermitian positive, even though it is not true that $\omega$ is locally the $dd^c$ of a plurisubharmonic potential.
The foundations of pluripotential theory with respect to a general hermitian metric have been developed in <cit.>.
There are numerous extra technical difficulties in this hermitian (non Kähler) setting.
The lack of local potentials makes it more difficult to localize arguments in coordinate patches since the outcome is not the usual local Monge-Ampère equation.
Moreover the Hermitian form $\omega$ being not closed, integrating by parts $(\omega+dd^c\f)^n$ over $X$
yields additional terms involving $d\omega$ which have to be handled separately.
§.§ The total volume
A substantial difference with the Kähler setting is that the total volume $\int_X(\omega+dd^c\f)^n$ depends on $\f$ in general. Such a property is important in various comparison principle arguments <cit.>.
Let $(X,\omega)$ be a compact Hermitian manifold of dimension greater or equal to $3$.
Characterize the forms $\omega$ for which $\int_X(\omega+dd^c\f)^n=\int_X\omega^n$ for any bounded $\omega$-plurisubharmonic function
A good candidate for such a condition is the property studied by Guan and Li <cit.>: namely one assumes that
dd^c\omega=0,\ d\omega\wedge d^c\omega=0.
The same phenomenon leads to the following questions:
Let $(X,\omega)$ be a compact Hermitian manifold. Define $T(X,\omega)$ by
T(X,\omega)=\inf \left\lbrace\int_X(\omega+dd^c\f)^n|\ \f\in\mathcal C^{\infty}(X)\cap PSH(X,\omega) \right\rbrace.
Is it possible that $T(X,\omega)=0$ for some manifold?
Let $(X,\omega)$ be a compact Hermitian manifold. Define $cap(X,\omega)$ by
cap(X,\omega)=\inf \left\lbrace\int_X(\omega+dd^c\f)^n|\ \f\in PSH(X,\omega),\ 0\leq\f\leq 1\right\rbrace.
Relate the quantity $cap(X,\omega)$ to the geometry of $(X,\omega)$. In particular is this quantity computable in terms of geometric
data? How does it behave under deformations?
§.§ Monge-Ampère equations
Just like in the Kähler case one can pose the problem of solving the equation
(\omega+dd^c \f)^n=e^cf\omega^n
for a given nonnegative function $f$. The additional constant factor $e^c$ stems from the lack of invariance of the total volumes.
This equation has been solved in the smooth setting by Tosatti and Weinkove <cit.>:
Let $(X,\omega)$ be a compact Hermitian manifold. Given any smooth strictly positive function $f$
there is a unique constant $c$ and a unique function $\f\in\mathcal C^{\infty}(X)\cap PSH(X,\omega)$ satisfying
sup_X\f=0,\ \ \ (\omega+dd^c \f)^n=e^cf\omega^n.
In the singular setting the following result is due to Kołodziej and Nguyen <cit.>:
Let $(X,\omega)$ be a compact Hermitian manifold. The problem
sup_X\f=0,\ \ \ (\omega+dd^c \f)^n=e^cf\omega^n
admits a unique continuous solution $\f$ and a unique constant $c$ for any nonnegative function $0\neq f\in L^p(X,\omega),\ p>1$.
The higher regularity of solutions was investigated in <cit.>. It is based on the following stability result:
Let $0\neq f, g$ be two nonnegative functions, such that $f,g\in L^p(\omega^n)$ for some $p>1$.
Assume $u$ and $v$ solve the problems
u\in PSH(X,\omega)\cap\mathcal C(X),\ sup_Xu=0,\ (\omega+dd^cu)^n=f\omega^n$$
$$v\in PSH(X,\omega)\cap\mathcal C(X),\ sup_Xv=0,\ (\omega+dd^cv)^n=g\omega^n
(the additonal constants have been incorporated in $f$ and $g$).
Assume moreover that $f\geq c_0$ for some $c_0>0$. Then
$$||u-v||_{\infty}\leq C||f-g||_p^{\alpha}$$
for any $0<\alpha<\frac 1{n+1}$ and some constant $C=C(X,\omega,c_0,\alpha,p,||f||_p,||g||_p).$
Can one remove the condition $f\geq c_0 >0$ ?
This would significantly improve the regularity theory in the hermitian setting.
§.§ Prescribed singularities
Just as in the Kähler case solutions of sequences of complex Monge-Ampère equations with right hand sides clustering along analytic subvarieties should converge weakly to very singular $\omega$-plurisubharmonic functions (compare <cit.>).
It is expected that the limits should have analytic singularities and hence can serve as singular weights in various $\bar{\partial}$-type problems. The easiest case is when clustering occurs at discrete set of points <cit.>. Motivated by the Kähler setting Tosatti and Weinkove studied the following problem:
Let $z_1,\cdots,z_k$ be a collection of points on a Hermitian manifold $X$ and let $\tau_1,\cdots,\tau_k$ be a collection of positive numbers satisfying $\sum_{j=1}^k\tau_j^n<\int_X\omega^n$. Construct an $\omega$-plurisubharmonic function, so that $\phi(z)\leq \tau_j \log(||z-z_j||)+O(1)$ near $z_j$.
Tosatti and Weinkowe were able to apply Demailly's method in complex dimension 2 and (under some assumptions) in dimension 3 <cit.>. The general case remains open.
§ INTEGRABILITY PROPERTIES
§.§ Desingularization of quasi-plurisubharmonic functions
Let $\f$ be a quasi-psh function on a compact complex hermitian manifold $(X,\omega)$.
We can assume without loss of generality that $\omega+dd^c \f \geq 0$ is a positive current.
When $n=1$, the Lelong number $\nu(\f,x)$ of $\f$ at $x$ equals the Dirac mass that
the measure $\omega+dd^c \f$ puts at $x$, thus
$E^+(\f):=\{ x \in X \, | \, \nu(\f,x) >0 \}$ is at most countable and
\omega+dd^c \f=\sum_{x \in E^+(\f)} \nu(\f,x) \delta_x +\mu,
where $\mu$ is a positive Radon measure with no atom. Note that for all $\e>0$, the
set $E_\e(\f):=\{ x \in X \, | \, \nu(\f,x) \geq \e \}$ is finite and $\omega+dd^c \f$
can be equally decomposed as
\omega+dd^c \f=\sum_{x \in E_\e(\f)} \nu(\f,x) \delta_x +\mu_\e,
where the sum is finite and $\mu_\e$ is a positive Radon measure with atoms of size less than $\e$.
It follows from Siu's celebrated result <cit.> that in any dimension $n \geq 2$,
the sets $E_\e(\f):=\{ x \in X \, | \, \nu(\f,x) \geq \e \}$ are closed analytic subsets.
One can similarly decompose
\omega+dd^c \f=\sum_{j \geq 0} m_j [D_j] +R
where $m_j \in \R^+$, $D_j$ is a divisor and $R$ is a closed positive current of bidegree
$(1,1)$ such that $E_+(R)$ does not contain any divisor. The set $E_+(R)$ can however contain infinitely
many analytic subsets of codimension $\geq 2$.
When $n=\dim_\C \leq 2$,
it follows from the work of Blel-Mimouni <cit.>, the second author <cit.> and
Favre-Jonsson <cit.> that $\f$ can be further decomposed as follows:
for any $\e>0$, there exists $\pi_\e:X_\e \rightarrow X$ a finite composition of blow-ups
such that
\pi_\e^* (\omega+dd^c \f)=[D_\e]+R_\e,
where $D_\e$ is a simple normal crossing divisor with coefficients in $\R^+$ and
$R_\e$ is a positive current with Lelong numbers smaller than $\e$ at all points.
Can one similarly desingularize quasi-plurisubharmonic functions in any dimension $n \geq 3$ ?
The methods of proofs so far make essential use of the $2$-dimensional setting.
Such a result, if true, should have many applications.
§.§ Uniform integrability bounds
Let $\Omega$ be a bounded hyperconvex domain. The volume of sublevel sets of plurisubharmonic functions in $\Omega$ is exponentially decreasing. It is important to have more precise asymptotics in the case when the Monge-Ampère operator is well defined for such a function. The following is a fundamental result <cit.> which deals with functions living in the class $\mathcal F(\Omega)$ (we refer to <cit.> for its definition):
There is a uniform constant $c_n$, such that for any $s>0$ and any $\f\in\mathcal F(\Omega)$ one has
{\rm Vol}(\lbrace\f<-s\rbrace)\leq c_n\delta_{\Omega}^{2n}(1+s\lambda^{-1})^{n-1}exp(-2ns\lambda^{-1}),
with $\delta_{\Omega}$ denoting the diameter of $\Omega$ whereas $\lambda$ is defined by $\lambda^n=\int_{\Omega}(dd^c\f)^n$.
While giving the optimal exponential integrability the result is still not sharp because of the polynomial term in front.
Is it true that (for a different constant $c_n$)
{\rm Vol}(\lbrace\f<-s\rbrace)\leq c_n\delta_{\Omega}^{2n}exp(-2ns\mu^{-1})?
As explained by Berman and Berndtsson <cit.> this problem is linked to the optimal Moser-Trudinger and Brezis-Merle type inequalities in $\Omega$.
§.§ Lelong numbers of finite energy currents
Let $X$ be a compact Kähler manifold (or a manifold from the Fujiki class).
Let $\omega$ be a smooth semi-positive $(1,1)$ form which is big, i.e.
such that $\int_X \omega^n >0$, where $n=\dim_\C X$.
Assume $\f \in {\mathcal E}(X,\omega)$ is a $\omega$-psh function with finite energy.
Is it true that $\f$ has zero Lelong number at all points ?
This is shown to be the case when $\omega$ is a Kähler form <cit.>. The argument
to show that $\nu(\f,a)=0$ requires
the existence of a function $g_a \in PSH(X,\omega) \cap L^{\infty}_{loc}(X \setminus \{x\})$
such that
g(x)=\e \log \rm{dist}(x,a)+O(1),
where $\e>0$. When $\omega$ is merely semi-positive and big, the same argument applies at any point of the ample locus of $\{\omega\}$
(see section <ref> for the definition), so the delicate
point is to check that this remains true at the boundary of the ample locus.
A positive answer is known when the situation corresponds to the desingularization of a finite energy
current on a compact normal space <cit.>:
Let $(V,\omega_V)$ be a compact normal space and let $\p \in {\mathcal E}(V,\omega_V)$
be a finite energy $\omega_V$-psh function. Let $\pi:X \rightarrow V$ be any resolution of singularities
and set $\omega=\pi^* \omega$, $\f=\p\circ \pi$.
Then $\p \in {\mathcal E}(X,\omega)$ has finite energy
and zero Lelong number at all points.
The ample locus of $\{ \omega\}$ is here $X \setminus \pi^{-1}(V_{sing})$ and the point is to show that
the Lelong number of $\f$ at $x \in \pi^{-1}(V_{sing})$ is controlled by the slope of
$\p$ at $y=\pi(x)$ (see <cit.>).
This is of course a very particular setting (although already quite useful for geometric applications), and the general case is largely open.
§.§ Extension problems
§.§.§ Extension from a subvariety
Let $(X,\omega)$ be a compact Kähler manifold and $Y \subset X$ a complex submanifold.
It is natural to wonder whether all $\omega$-plurisubharmonic functions on $Y$ are induced by global $\omega$-plurisubharmonic functions:
Is it true that
PSH \left(Y,\omega_{|Y} \right)=PSH(X,\omega)_{|Y} \; \; ?
The answer is positive when $\omega$ is a Hodge form (i.e. it is the curvature form of an ample line bundle),
as follows from the the main result of <cit.>. A result of Matsumura <cit.> shows that, conversely, such an extension property forces the associated line bundle to be ample.
An alternative approach to the case of Hodge forms has been proposed by Hisamoto <cit.>.
It requires a better understanding of
the Ohsawa-Takegoshi extension theorem with prescription of jets of high order, a problem of independent interest.
For transcendental classes
the answer is easy and positive for smooth strongly $\omega$-psh functions
(see <cit.>). A recent result of Collins and Tosatti <cit.>
establishes such an extension result for functions with analytic singularities.
The general case is however largely open.
§.§.§ Families
We consider now a holomorphic family of compact Kähler manifolds.
Let $\pi:{\mathcal X} \rightarrow \D$ be a smooth family of compact Kähler manifolds (i.e. $\pi$ is a proper holomorphic submersion)
and let ${\mathcal L} \rightarrow {\mathcal X}$ be a pseudoeffective holomorphic line bundle.
We let $X_t=\pi^{-1}(t)$ denote the fibers of $\pi$ and set $L_t={\mathcal L}_{| X_t}$.
Assume $h_0=e^{-\f_0}$ is a singular hermitian metric of $L_0$
on $X_0$ which is "positive", i.e. such that $dd^c \f_0 \geq 0$ in the weak sense of currents.
Can one find $h=e^{-\f}$ a hermitian metric of ${\mathcal L}$ on ${\mathcal X}$ which is a semi-positive extension of $\f_0$, i.e.
such that $dd^c \f \geq 0$ and $\f_{|X_0}=\f_0$ ?
This question is related to the problem of the invariance of plurigenera, a fundamental result due to Siu in the case of projective families (see <cit.>).
A related problem is to obtain an effective version of Demailly's extension theorem <cit.>.
We thank M.Păun for emphasizing this.
§ VISCOSITY PROBLEMS
A viscosity approach complementary to the pluripotential one has been proposed in
<cit.>. It raises a number of interesting open questions.
§.§ Differential tests
In this section we fix $\Omega$ a bounded pseudoconvex domain (for example the unit ball) of $\C^n$
and study local properties of plurisubharmonic functions from a viscosity point of view.
Let $\f$ be an upper-semi continuous function in $\Omega$ and pick $z \in \Omega$.
A function is called a differential test from above
at $z$ if it is ${\mathcal C}^2$-smooth in a neighborhood $V$ of $z$, and such that
(\f-q)(z)=\sup_V (\f-q).
Let $\f$ be a plurisubharmonic function which belongs to the domain of definition $DMA(\Omega)$.
It is strictly plurisubharmonic in the classical sense if there exists $\e>0$ such that
$\f-\e ||z||^2$ is still a plurisubharmonic function. It follows in particular that
(dd^c \f)^n \geq \e^n dV(z)>0.
It is natural to wonder whether the converse implication holds true: if $(dd^c \f)^n>0$, does it imply that
$\f$ is strictly plurisubharmonic ? It is obviously true when $\f$ is ${\mathcal C}^2$-smooth.
The answer is also positive in a dynamical context, as shown by Berteloot and Dupont in <cit.>. It is however false in general as the following example shows:
The function
\f(z,w)=(1+|z|^2) |w|
is Lipschitz continuous and plurisubharmonic in $\C^2$. It satisfies
(dd^c \f)^2=c dV,
where $dV$ is the Lebesgue volume form and $c>0$ is a normalizing constant, while
dd^c \f \wedge \frac{i}{2} dw \wedge d\overline{w}=c' |w| \, dV,
hence $\f$ is not strictly plurisubharmonic along the line $(w=0)$.
One can nevertheless show that $\f$ is strictly plurisubharmonic in the viscosity sense:
We say that $\f$ is strictly plurisubharmonic in the viscosity sense if for all $z \in \Omega$ and
for all differential test from above $q$ at $z$, one has $(dd^c q)_+^n(z)>0$.
The point in the example above is that there is no differential test from above for $\f$ along the line
$(w=0)$. A simpler example in any dimension is provided by $\f(z)=||z||$: there is no differential test from above at the origin; the function is strictly plurisubharmonic in the viscosity sense, but not in the classical sense.
Let $\f$ be a plurisubharmonic function. What can be said about the set of points at which $\f$ does not admit a differential test from above ?
It is known that this set has empty interior (see <cit.>). Is it of zero Lebesgue measure (or capacity) ?
§.§ Supersolutions
We fix here $f \geq 0$ a continuous density and let $dV$ denote the Lebesgue euclidean measure.
For a smooth hermitian $(1,1)$-form $\a$, we let $ \a_+(x)$ denote
$\a(x)$ if it is non-negative, or zero if $\a(x)$ has at least one negative eigenvalue.
A supersolution of the equation
-(dd^c V)^n+fdV=0
is a lower semi-continuous function $v:\Omega \rightarrow \R$ such that
for all $x \in \Omega$ and for every differential test $q$ from below at $x$,
-(dd^c q)(x)^n/dV+f(x)\geq 0.
A subsolution of the equation
-(dd^c V)^n+fdV=0
is an upper semi-continuous function $u:\Omega \rightarrow \R$ such that
for all $x \in \Omega$ and for every differential test $q$ from above at $x$,
-(dd^c q) (x)^n/dV+f(x)\leq 0.
A viscosity solution of the equation $-(dd^c V)^n+fdV=0$ is a continuous function which is both a
super- and a sub-solution.
It follows from <cit.> that a bounded upper semi-continuous function
$u$ satisfies $-(dd^c u)^n+fdV \leq 0$ in the viscosity sense if and only if it is plurisubharmonic
and satisfies $-(dd^c u)^n+fdV \leq 0$ in the pluripotential sense of Bedford-Taylor <cit.>.
It follows from <cit.> that a continuous function
satisfies $-(dd^c u)^n+fdV= 0$ in the viscosity sense if and only if it is plurisubharmonic
and satisfies $-(dd^c u)^n+fdV =0$ in the pluripotential sense of Bedford-Taylor.
The situation is non symmetric and more delicate to interpret for supersolutions.
Observe that a viscosity supersolution is not necessarily plurisubharmonic.
We therefore consider its plurisubharmonic envelope,
\sup \{ \p \, | \, \p \in PSH(\Omega) \; \& \; \p \leq v \}
\right)^*.
This is the greatest plurisubharmonic function that lies below $v$.
Let $v$ be a continuous viscosity supersolution of the equation
-(dd^c V)^n+fdV=0.
Is $P(v)$ a pluripotential supersolution of the same equation ?
The answer is positive when $v$ is ${\mathcal C}^2$-smooth, as follows from <cit.>.
While the concept of viscosity subsolution makes perfect sense in pluripotential theory, the dual concept of
viscosity supersolution is more delicate to interpret.
§.§ Approximation of quasi-plurisubharmonic functions
Let $\omega$ be a smooth semi-positive and big form on a compact Kähler manifold $X$.
Is any function $\f \in PSH(X,\omega)$ the decreasing limit of smooth $\omega$-psh functions ?
The answer is positive when $\omega$ is Kähler, as follows from the work of Demailly <cit.>
(see also <cit.> for an elementary proof). It is also positive when
$\omega=\pi^* \omega_V$ is the pull-back of a Hodge form $\omega_V$ on a singular variety $V$
by a resolution $\pi=X \rightarrow V$ <cit.> (see also <cit.>).
In general, it follows from <cit.> that
one can find a sequence of continuous $\omega$-psh functions decreasing to $\f$, but
the smooth approximation is largely open.
The viscosity approach proposed in <cit.> has been extended to the case of general big cohomology classes in <cit.>. Let $\a \in H^{1,1}(X,\R)$ be a big cohomology class and fix
$\theta \in \a$ a smooth closed $(1,1)$-form.
The function
V_{\theta}:=\sup \{ v \, | \, v \in PSH(X,\theta) \text{ and } \sup_X v\le 0 \}
is an example of $\theta$-psh function with minimal singularities. It satisfies $\sup_X V_{\theta}=0$. We let
P(\a):=\{ x \in X \, | \, V_{\theta}(x) =-\infty \}
\text{ and }
NB(\a):=\{ x \in X\, | \, V_{\theta} \notin L_{loc}^{\infty}(\{x\}) \}
denote respectively the polar locus and the non bounded locus of $\a$.
The definitions clearly do not depend on the
choice of $\theta$ and coincide with the polar (resp. non bounded locus) of any $\theta$-psh function with minimal singularities. Note that $P(\a)$ may a priori not be closed, while $NB(\a)$ is always closed.
Does $P(\a)$ always coincide with $NB(\a)$ ?
This condition is both necessary and sufficient in order to approximate any $\theta$-psh function $\f$
by a decreasing sequence of exponentially continuous $\theta$-psh functions $\f_j$,
i.e. such that $e^{\f_j}$ is continuous. While it is easy to construct examples of
quasi-psh functions $\p$ such that $P(\p)$ is strictly smaller than $NB(\p)$, we do not know a single example of a big cohomology class $\a$ for which $P(\a)$ is smaller than $NB(\a)$.
§.§ The Kähler-Ricci flow
The Ricci flow introduced by R.Hamilton is an evolution equation of Riemannian metrics
$(g_t)_{t>0}$ on a fixed riemannian manifold $X$,
\frac{\partial g_t}{\partial t}=-\Ric(g_t),
starting from an initial riemannian metric $g_0$. When $(X,g_0)$ is actually a Kähler manifold,
the Kähler condition is preserved along the flow which is thus called the Kähler-Ricci flow.
§.§.§ Smoothing properties: the compact setting
In this section we fix $X$ a compact Kähler manifold and $\a_0 \in H^{1,1}(X,\R)$ a Kähler cohomology class.
Pick $S_0$ a positive closed current representing $\a_0$.
The Kähler-Ricci flow starting at $S_0$ is the evolution equation
\frac{\partial \omega_t}{\partial t}=-\Ric(\omega_t),
where $(\omega_t)_{0<<T_{max}}$ is a family of Kähler forms and $\omega_t$ weakly converges to $S_0$
as $t \rightarrow 0$.
It is classical <cit.> that when $S_0$ is a Kähler form, such a flow admits a unique solution on a maximal interval of time $[0,T_{max}[$, where
T_{max}:=\sup \left\{ t \geq 0 \, | \, t K_X+\alpha_0 \text{ is K\"ahler}\right\}.
This result has been extended by many authors (see <cit.> and references therein).
The best result so far is due to DiNezza and Lu who showed in <cit.> that there exists
a unique such family of Kähler forms if and only if $S_0$ has zero Lelong numbers at all points.
When $S_0$ has some non-zero Lelong numbers, DiNezza and Lu construct a maximal family of positive currents
$(\omega_t)_{0<t<T_{max}}$ which are honest Kähler forms on a time-increasing family of Zariski open subsets $\Omega_t$ where they satisfy the Kähler-Ricci flow equation.
By analogy with Demailly regularization results <cit.>, it is natural to expect that this family
has analytic singularities.
Does the Kähler-Ricci flow produce approximants with analytic singularities ?
§.§.§ Monge-Ampère flows in pseudoconvex domains
The study of complex Monge-Ampère flows in pseudoconvex domains
has been initiated in <cit.>. A theory of weak solutions has been proposed in <cit.>, using a viscosity approach,
and the smoothing properties have been investigated by Do in <cit.>
who showed in particular the following:
Let $\Omega$ be a bounded smooth strictly pseudoconvex domain of $\C^n$
and $T>0$. Let $u_0$ be a bounded plurisubharmonic function defined on a neighborhood
$\tilde{\Omega}$ of $\overline{\Omega}$.
Assume that $\f \in {\mathcal C}^{\infty} (\overline{\Omega} \times [0,T))$
satisfies $\f(z,t)=u_0(z)$ for $z \in \partial \Omega$.
Then there exists a unique function $u \in {\mathcal C}^{\infty} (\overline{\Omega} \times (0,T))$
such that
(i) for all $t \in (0,T)$, $z \mapsto u_t(z)=u(z,t)$ is a strictly plurisubharmonic function in $\Omega$;
(ii) $\dot{u}=\log \det (u_{i\overline{j} })$ on $\Omega \times (0,T)$;
(iii) $u=\f$ on $\partial \Omega \times (0,T)$;
(iv) $\lim_{t \rightarrow 0} u(z,t)=u_0(z)$, for all $z \in \overline{\Omega}$.
Moreover $u$ is uniformly bounded and
$u_t$ converges to $u_0$ in capacity when $t \rightarrow 0$.
Do actually deals with more general flows which contain, as a particular case, the local version of the Kähler-Ricci flow. These flows can thus be run from a bounded plurisubharmonic initial data.
The following question is natural, in view of the corresponding global results from <cit.>:
Can one run the above complex Monge-Ampère flow from an arbitrary plurisubharmonic function ?
§.§.§ Viscosity comparison principles
Let $X$ be a compact Kähler space.
The Kähler-Ricci flow on $X$ can be reduced to a parabolic scalar equation.
When $X$ is a compact mildly singular variety, this equation is
a complex degenerate Monge-Ampère flow, as studied for example in <cit.>.
We consider here the flows
(\omega_t +dd^c \f_t)^n = e^{\dot{\f}_t + F (t,x,\f) } \mu (t,x)
on $[0,T] \times X$, where
* $T \in ]0,+\infty]$;
* $\omega=\omega(t,x)$ is a continuous family of semi-positive and big $(1,1)$-forms on $X$,
* $F (t,z,r)$ is continuous in $[0,T[ \times \Omega \times \R$ and non decreasing in $r$,
* $\mu (t,z)\geq 0$ is a bounded continuous volume form on $X$,
* $\f : X_T := [0,T[ \times X \rightarrow \R$ is the unknown function, with $\f_t: = \f (t,\cdot)$.
We assume that the forms $\omega_t \geq \theta$ are uniformly bounded below by a fixed
semi-positive and big $(1,1)$-form $\theta$.
A central problem in the viscosity approach is to establish the comparison principle:
recall that a bounded upper semi-continuous function $u$ is a viscosity subsolution if it
(\omega_t +dd^c \f_t)^n \geq e^{\dot{\f}_t + F (t,x,\f) } \mu (t,x)
in the viscosity sense. One defines, by duality, supersolutions as those lower semi-continuous functions
that verify the reverse inequality $\leq$ in the sense of viscosity.
Assume $u$ (resp. $v$) is a viscosity subsolution (resp. supersolution) to the above degenerate complex
Monge-Ampère flow such that
$u \leq v$ on the parabolic boundary $\{0\} \times X$.
Is it true that that $u \leq v$ on $[0,T] \times X$ ?
Such a comparison principle has been established in <cit.> in several particular cases of interest.
The above general case however remains open
(note that a general comparison principle has been established in <cit.> in the context of
pseudoconvex domains of $\C^n$).
[ACCH09]ACCH09 P.Ahag, U.Cegrell, R.Czyz, P.H.Hiep:
Monge-Ampère measures on pluripolar sets. J. Math. Pures Appl. (9) 92 (2009), no. 6, 613-627.
[ACH15]ACH15 P.Ahag, U.Cegrell, P.H.Hiep:
Monge-Ampère measures on subvarieties.
J. Math. Anal. Appl. 423 (2015), no 1, 94-105.
[ACHKZ09]ACKHZ P.Ahag, U.Cegrell, P.H.Hiep, S.Kołodziej, A.Zeriahi: Partial pluricomplex energy and integrability exponents of plurisubharmonic functions. Adv. Math. 222 (2009), no. 6, 2036-2058.
[Aub78]Aub78 T. Aubin: Equation de type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. 102 (1978), 63–95.
[Bed93]Bed93 Bedford: Survey of pluri-potential theory. Several complex variables (Stockholm, 1987/1988), 48-97, Math. Notes, 38, Princeton Univ. Press, Princeton, NJ, 1993.
[BeDe88]BeDe E. Bedford, J.P. Demailly: Two counterexamples concerning the pluri-complex Green function in $\mathbb C^n$. Indiana Univ. Math. J. 37 (1988), no. 4, 865-867.
[BeFo79]BeFo E. Bedford, J. E. Fornaess: Counterexamples to regularity for the complex Monge-Ampère equation.
Invent. Math. 50 (1978/79), no. 2, 129-134.
[BK77]BK77 E. Bedford, M.Kalka: Foliations and complex Monge-Ampère equations.
Comm. Pure Appl. Math. 30 (1977), no. 5, 543–571.
[BT76] BT76 E. Bedford, B. A. Taylor: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37 (1976), no. 1, 1-44.
[BT78] BT78 E. Bedford, B. A. Taylor: Variational properties of the complex Monge-Ampère equation. I. Dirichlet principle. Duke Math. J. 45 (1978), no. 2, 375-403.
[BT82] BT82 E. Bedford, B. A. Taylor: A new capacity for plurisubharmonic functions. Acta Math. 149 (1982), no. 1-2, 1–40.
[BT87] BT87 E. Bedford, B. A. Taylor: Fine topology, Šilov boundary, and $(dd^c)^n$. J.F.A. 72 (1987), 225–251.
[BB11]BB R. Berman, B. Berndtsson: Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin's “hypothèse fondamentalle”. Preprint arXiv 1109.1263.
[BBGZ13] BBGZ13 R.Berman, S.Boucksom, V. Guedj, A. Zeriahi:
A variational approach to complex Monge-Ampère equations.
Publ. Math. I.H.E.S. 117 (2013), 179-245.
[BBEGZ11] BBEGZ R. Berman, S. Boucksom, P.Eyssidieux, V. Guedj, A. Zeriahi: Kähler-Ricci flow and Ricci iteration on log-Fano varieties.
To appear in Journal für die reine und angewandte Mathematik.
[BD05] BD05 F.Berteloot, C.Dupont:
Une caractérisation des endomorphismes de Lattès par leur mesure de Green.
Comment. Math. Helv. 80 (2005), no. 2, 433-454.
[BCHM10]BCHM10 C. Birkar, P. Cascini, C. Hacon, J. McKernan: Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), no. 2, 405-468.
[BM05] BM05 M.Blel, S.Mimouni: Singularités et intégrabilité des fonctions plurisousharmoniques. Ann. Inst. Fourier (Grenoble) 55 (2005), no. 2, 319-351.
[Bło96]Blo96 Z. Błocki: The complex Monge-Ampère operator in hyperconvex domains.
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 4, 721-747
[Bło04]Blo04 Z. Błocki: On the definition of the Monge-Ampère operator in $\mathbb C^2$. Math. Ann. 328 (2004), no. 3, 415-423.
[Bło06]Blo06 Z. Błocki: The domain of definition of the complex Monge-Ampère operator. Amer. J. Math. 128 (2006), no. 2, 519-530.
[BK07] BK07 Z. Błocki, S. Kołodziej: On regularization of plurisubharmonic functions on manifolds. Proc. Amer. Math. Soc. 135 (2007), no. 7, 2089–2093.
[BEGZ10]BEGZ10 S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi: Monge-Ampère equations in big cohomology classes. Acta Math. 205 (2010), 199–262.
[BFJ07] FJ07 S.Boucksom, C.Favre, M.Jonsson:
Valuations and plurisubharmonic singularities. Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 449-494.
[Caf90]Caf90 L. Caffarelli:
Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation.
Ann. of Math. (2) 131 (1990), no. 1, 135-150.
[CKNS85]CKNS85 L. Caffarelli, J. J. Kohn, L. Nirenberg, J. Spruck: The Dirichlet problem for nonlinear second-order
elliptic equations II. Complex Monge-Ampère, and uniformly elliptic, equations. Comm. Pure Appl. Math. 38 (1985), no.
2, 209–252.
[Cao85]Cao85 H.D. Cao: Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math. 81 (1985), no. 2, 359-372.
[Ceg83]Ceg83 U. Cegrell: Discontinuité de l'opérateur de Monge-Ampère complexe.
C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 21, 869-871.
[Ceg98]Ceg98 U. Cegrell: Pluricomplex energy. Acta Math. 180 (1998), no. 2, 187–217.
[Ceg02]Ceg02 U. Cegrell:
Explicit calculation of a Monge-Ampère measure. Actes des Rencontres d'Analyse Complexe (Poitiers-Futuroscope, 1999), 39-42, Atlantique, Poitiers, 2002.
[Ceg04]Ceg04 U. Cegrell: The general definition of the complex Monge-Ampère operator. Ann. Inst. Fourier (Grenoble) 54 (2004), no. 1, 159-179.
[Ceg07]Ceg07 U. Cegrell: The gradient lemma.
Ann. Polon. Math. 91 (2007), no. 2-3, 143-146.
[CP92]CP92 U. Cegrell, L.Persson: The Dirichlet problem for the complex Monge-Ampère operator: stability in $L^2$.
Michigan Math. J. 39 (1992), no. 1, 145-151.
[CT14]CT14 T.Collins, V.Tosatti: An extension theorem for Kähler currents with analytic singularities. Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), no. 4, 893-905.
[Com00]Co D. Coman: The pluricomplex Green function with two poles of the unit ball of $\mathbb C^n$. Pacific J. Math.
194 (2000), no. 2, 257-283.
[CG04]CG04 D. Coman, V. Guedj: Invariant currents and generalized Lelong numbers.
J. Geom. Anal. 14 (2004), no. 2, 199–213.
[CG09]CG09 D. Coman, V. Guedj: Quasiplurisubharmonic Green functions. Journal
de Math.Pures et Appliquées 92 (2009), 456-475.
[CGZ08]CGZ08 D. Coman, V. Guedj, A. Zeriahi: Domains of definition of complex Monge-Ampère operators
on compact Kähler manifolds. Math. Z. 259 (2008), no. 2, 393–418.
[CGZ13]CGZ13 D. Coman, V. Guedj, A. Zeriahi:
Extension of plurisubharmonic functions with growth control.
Journal für die reine und angewandte Mathematik 676 (2013), 33-49.
[Del12]Del12 F.Delarue: Probabilistic approach to regularity. Complex Monge-Ampère equations and geodesics in the space of Kähler metrics, 55-198, L.N.M. 2038, Springer, Heidelberg, 2012.
[Dem85]DemSMF J.P. Demailly: Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines. Mém. Soc. Math. France 19 (1985).
[Dem92]Dem92 J.P. Demailly: Regularization of closed positive currents and intersection theory. J. Alg. Geom. 1
(1992), no. 3, 361–409.
[Dem93]Dem93 J.P. Demailly: Monge-Ampère operators, Lelong numbers and intersection theory. Complex analysis and geometry, 115–193, Univ. Ser. Math., Plenum, New York, 1993.
[Dem14]Dem14 J.P. Demailly: lecture series at KAWA-NORDAN, http://euclides.imub.ub.es/kawa14/
[Dem15]Dem15 J.P. Demailly: Extension of holomorphic functions defined on non reduced analytic subvarieties. Preprint arXiv:1510.05230
[DP04]DP04 J.P. Demailly, M. Păun: Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. of Math. (2) 159 (2004), no. 3, 1247–1274.
[D&all14]DDGHKZ14 J.P. Demailly, S.Dinew, V.Guedj, P.H.Hiep, S.Kołodziej, A.Zeriahi:
Hölder continuous solutions to Monge-Ampère equations.
J. Eur. Math. Soc. 16 (2014), 619-647.
[Din09] Din09 S. Dinew: Uniqueness and stability in $PSH(X,\omega)$, J.F.A. 256, vol 7 (2009), 2113-2122.
[Din10] Din10 S. Dinew: Hölder continuous potentials on manifolds with partially positive curvature. J. Inst. Math.
Jussieu 9 (2010), no. 4, 705-718.
[DiKo12]DiKo S Dinew, S. Kołodziej: Pluripotential estimates on compact Hermitian manifolds. Advances in geometric analysis, 69-86, Adv. Lect. Math. (ALM), 21, Int. Press, Somerville, MA, 2012.
[DiNL14]DiNL14 E.DiNezza, H.C.Lu:
Complex Monge-Ampère equations on quasi-projective varieties.
Preprint arXiv:1401.6398. To appear in Journal für die reine und angewandte Mathematik.
[DiNL14b]DiNL14b E.DiNezza, H.C.Lu:
Uniqueness and short time regularity of the weak Kähler-Ricci flow.
Preprint arXiv:1411.7958
[DNS10]DNS10 T.C. Dinh, V.A. Nguyên, N. Sibony: Exponential estimates for plurisubharmonic functions. J. Diff. Geom. 84, no. 3 (2010), 465-488.
[Do15]Do15 Hoang Son Do: Degenerate complex Monge-Ampère flows on strictly pseudoconvex domains. Preprint (2015) arXiv 1501.07167.
[Duj10]Duj10 R. Dujardin: Wermer examples and currents.
Geom. Funct. Anal. 20 (2010), no. 2, 398-415.
[DG12]DG12 R.Dujardin, V.Guedj: Geometric properties of maximal psh functions.
Complex Monge-Ampère equations and geodesics in the space of Kähler metrics, 33-52, Lecture Notes in Math., 2038, Springer, Heidelberg, 2012.
[EZ98]EdZwo A. Edigarian, W. Zwonek: Invariance of the pluricomplex Green function under proper mappings with applications. Complex Variables Theory Appl. 35 (1998), no. 4, 367-380.
[EGZ09] EGZ09 P. Eyssidieux, V. Guedj, A. Zeriahi: Singular Kähler-Einstein metrics. J. Amer. Math. Soc. 22 (2009), 607-639.
[EGZ11] EGZ11 P. Eyssidieux, V. Guedj, A. Zeriahi: Viscosity solutions to degenerate Complex Monge-Ampère equations. Comm.Pure & Appl.Math 64 (2011), 1059–1094.
[EGZ15] EGZ13 P. Eyssidieux, V. Guedj, A. Zeriahi:
Continuous approximation of quasiplurisubharmonic functions.
Cont. Math. Vol 644 (2015), 67-78.
[EGZ15b] EGZ14 P. Eyssidieux, V. Guedj, A. Zeriahi: Weak solutions to degenerate complex
Monge-Ampère Flows I. Math. Ann. 362 (2015), no. 3-4, 931-963.
[EGZ14] EGZ14b P. Eyssidieux, V. Guedj, A. Zeriahi: Weak solutions to degenerate complex
Monge-Ampère Flows II. Preprint arXiv 2014.
[FJ05] FJ05 C.Favre, M.Jonsson: Valuative analysis of planar plurisubharmonic functions.
Invent. Math. 162 (2005), no. 2, 271-311.
[GL10]GuLi B. Guan, Q. Li: Complex Monge-Ampère equations and totally real submanifolds. Adv. Math. 225 (2010), no. 3, 1185-1223.
[G05] G05 V. Guedj: Courants extrémaux et dynamique complexe.
Ann. Sc.E.N.S. 38 (2005), 407-426.
[G05b] G05b V. Guedj
Desingularization of quasiplurisubharmonic functions.
Int.J.M. 16 (2005), 555–560.
[G10] G10 V. Guedj: Propriétés ergodiques des applications rationnelles.
Quelques aspects des systèmes dynamiques polynomiaux
S.Cantat, A.Chambert-Loir, V.Guedj
Panoramas et Synth. 30 (2010).
[GZ07]GZ07 V. Guedj, A. Zeriahi: The weighted Monge-Ampère energy of quasiplurisubharmonic functions. J. Funct. An. 250 (2007), 442-482.
[GZ12]GZ12 V. Guedj, A. Zeriahi: Dirichlet problem in domains of $\mathbb C^n$.
Complex Monge-Ampère equations and geodesics in the space of Kähler metrics, 13-32,
L.N.M. 2038, Springer, Heidelberg, 2012.
[GZ13]GZ13 V. Guedj, A. Zeriahi: Regularizing properties of the Kähler-Ricci flow.
Journal für die reine und angewandte Mathematik, to appear.
[HL09]HL09 Harvey, F. R. Lawson, H. B.
Dirichlet Duality and the non linear Dirichlet problem on Riemannian manifolds.
Comm. Pure Appl. Math. 62 (2009), 396-443.
[Hiep10]Hiep10 P.H. Hiep: Hölder continuity of solutions to the complex Monge-Ampere equations on compact Kähler manifolds. Ann. Inst. Fourier 60 (2010), no. 5, 1857-1869.
[His12]His12 T. Hisamoto: Remarks on $L^2$-jet extension and extension of singular Hermitian metric with semipositive curvature. Preprint arXiv:1205.1953
[HLi10]HLi10 Z. Hou, Q. Li: Energy functionals and complex Monge-Ampère equations.
J. Inst. Math. Jussieu 9 (2010), no. 3, 463-476.
[Kis83] Kis83 C. Kiselman: Sur la définition de l'opérateur de Monge-Ampère complexe. Analyse Complexe (Toulouse, 1983), 139–150, Lecture Notes in Math. 1094, Springer, Berlin, 1984.
[Kis00] Kis00 C. Kiselman: Plurisubharmonic functions and potential theory in several complex variables. Development of mathematics 1950-2000, 655-714, Birkhäuser, Basel, 2000.
[Koł95] Kol95 S. Kołodziej:
The range of the complex Monge-Ampère operator. II.
Indiana Univ. Math. J. 44 (1995), no. 3, 765-782.
[Koł96] Kol96 S. Kołodziej: Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge-Ampère operator. Ann. Polon. Math. 65 (1996), no. 1, 11-21.
[Koł98] Kol98 S. Kołodziej: The complex Monge-Ampère equation. Acta Math. 180 (1998), no. 1, 69–117.
[Koł05] Kol05 S.Kołodziej: The set of measures given by bounded solutions of the complex Monge-Ampère equation on compact Kähler manifolds.
J. London Math. Soc. (2) 72 (2005), no. 1, 225-238.
[Koł05b] KolBook S.Kołodziej: The complex Monge-Ampère equation and pluripotential theory. Mem. Amer. Math. Soc. 178 (2005), no. 840, 64 pp.
[Koł08] Kol08 S. Kołodziej: Hölder continuity of solutions to the complex Monge-Ampère equation with the right-hand side in $L^p$: the case of compact Kähler manifolds. Math. Ann. 342 (2008), 379–386.
[KN15]KoNg1 S. Kołodziej, N. C. Nguyen: Weak solutions to the complex Monge-Ampère equation on Hermitian manifolds. Contemp. Math. 644 (2015) 141-158.
[KN15b]KoNg2 S. Kołodziej, N. C. Nguyen: Stability and regularity of solutions of the Monge-Ampère equation on Hermitian manifolds, preprint arXiv: 1501.05749.
[Kry89]Kry89 N. V. Krylov: Smoothness of the payoff function for a controllable diffusion process in a domain. Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 66-96.
Math. USSR-Izv. 34 (1990), 65-95.
[Lel83]Lel83 P.Lelong: Discontinuité et annulation de l'opérateur de Monge-Ampère complexe. P. Lelong-P. Dolbeault-H. Skoda analysis seminar, 1981/1983, 219-224, L.N.M. 1028, Springer, 1983.
[Mat13]Mats13 S.Matsumura: An ampleness criterion with the extendability of singular positive metrics. Math. Z. 273 (2013), no. 1-2, 43-54.
[Pau07]Pau07 M. Păun: Siu's invariance of plurigenera: a one-tower proof.
J. Differential Geom. 76 (2007), no. 3, 485-493.
[PSS12]PSS12 D. H. Phong, J. Song, J. Sturm: Complex Monge-Ampère equations. Surveys in differential geometry. Vol. XVII, 327-410, Surv. Differ. Geom., 17, Int. Press, Boston, MA, 2012.
[Pog71]Pog71 A.V. Pogorelov: A regular solution of the $n$-dimensional Minkowski problem. Dokl. Akad. Nauk SSSR 199 785–788 (Russian); translated in Soviet Math. Dokl. 12 1971 1192-1196.
[Ra01]Ra01 A.Rashkovskii: Lelong numbers with respect to regular plurisubharmonic weights, Results Math. 39 (2001), 320-332.
[Ra06]Ra06 A.Rashkovskii: Relative types and extremal problems for plurisubharmonic functions. Int. Math. Res. Not. 2006, Art. ID 76283, 26 pp.
[Ra13]Ra13 A.Rashkovskii: Analytic approximations of plurisubharmonic singularities.
Math. Z. 275 (2013), no. 3-4, 1217-1238.
[Sad76]Sad A. Sadullaev: A boundary uniqueness theorem in $\C^n$. Math. Sbor. 101 (1976), no4, 568-583.
[SZ15]SadZer A. Sadullaev, A.Zeriahi: Hölder regularity of generic manifolds. To appear in Ann.Sc.Nor.Pisa.
[Siu74]Siu74 Y. T. Siu: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974) 53-156.
[Siu98]Siu98 Y. T. Siu: Invariance of Plurigenera, Inv. Math. 134 (1998), 661-673.
[Siu02]Siu02 Y. T. Siu: Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. Complex geometry (Göttingen, 2000), 223-277, Springer, Berlin, 2002.
[Sko72]Sko72 H. Skoda: Sous-ensembles analytiques d'ordre fini ou infini dans $\C^{n}$. Bull. Soc. Math. France 100 (1972), 353-408.
[ST12]ST12 J. Song, G. Tian: Canonical measures and Kähler-Ricci flow.
J.A.M.S. 25 (2012), no. 2, 303-353.
[ST09]ST09 J. Song, G. Tian: The Kähler-Ricci flow through singularities. Preprint (2009) arXiv:0909.4898.
[Ti00]Tian G. Tian: Canonical metrics in Kähler geometry. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2000).
[TW10]ToWe V. Tosatti, B. Weinkove: The complex Monge-Ampère equation on compact Hermitian manifolds. J. Amer. Math. Soc. 23 (2010), no. 4, 1187-1195.
[TW12]ToWe2 V. Tosatti, B. Weinkove: Plurisubharmonic functions and nef classes on complex manifolds.
Proc. Amer. Math. Soc. 140 (2012), no. 11, 4003-4010.
[Tsu88]Tsu88 H. Tsuji: Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281 (1988), no. 1, 123–133.
[Tsu02]Tsu02 H. Tsuji: Deformation invariance of plurigenera.
Nagoya Math. J. 166 (2002), 117-134.
[Yau78]Yau78 S. T. Yau: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.
[Wan12a]Wa Y. Wang: On the $\mathcal C^{2,\alpha}$-regularity of the complex Monge-Ampère equation. Math. Res. Lett. 19
(2012), no. 4, 939-946.
[Wan12b]Wan12 Y. Wang: A viscosity approach to the Dirichlet problem for complex
Monge-Ampère equations. Math. Z. 272 (2012), no. 1-2, 497-513.
[Wik05]Wik05 J. Wiklund: Pluricomplex charge at weak singularities.
Preprint arXiv:math/0510671.
[Zer01]Zer01 A. Zeriahi: Volume and capacity of sublevel sets of a Lelong class of psh functions. Indiana Univ. Math. J. 50 (2001), no. 1, 671–703.
[Zer13]Zer14 A. Zeriahi: A viscosity approach to degenerate complex Monge-Ampère equations. Ann. Fac. Sc. Toulouse 22 (2013), no4, 843-913.
[BGZ08a] BGZ08a S. Benelkourchi, V. Guedj, A. Zeriahi: A priori estimates for weak solutions of complex Monge-Ampère equations. Ann. Scuola Norm. Sup. Pisa C1. Sci. (5), Vol VII (2008), 1-16.
[BGZ08b] BGZ08b S. Benelkourchi, V. Guedj, A. Zeriahi:
Plurisubharmonic functions with weak singularities.
Complex analysis and digital geometry, 57-74, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 86, Uppsala Universitet, Uppsala (2009).
[Bło99]Blo99 Z. Błocki: On the regularity of the complex Monge-Ampè�re operator. Complex geometric analysis in Pohang (1997) 181–189, Contemp. Math. 222, Amer. Math. Soc., Providence, RI, 1999.
[Bło]Blo Z. Błocki: The Calabi-Yau theorem. In
"Complex Monge-Ampère equations and geodesics in the space of Kähler metrics",
Lecture Notes in Mathematics 2038, ed. V. Guedj, Springer, 2012.
[Bou02]Bou02 S. Boucksom: On the volume of a line bundle. Internat. J. Math. 13 (2002), no. 10, 1043–1063.
[Bou04]Bou04 S. Boucksom: Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. Ecole Norm. Sup. (4) 37 (2004), no. 1, 45–76.
[Cal57] Cal57 E. Calabi: On Kähler manifolds with vanishing canonical class. Algebraic geometry and topology. A symposium in honor of S. Lefschetz, pp. 78–89. Princeton Univ. Press, Princeton, 1957.
[CKZ05]CKZ05 U. Cegrell, S.Kołodziej, A.Zeriahi: Subextension of plurisubharmonic functions with weak singularities. Math.
Z. 250 (2005), no. 1, 7-22.
[CY80]CY80 S.Y.Cheng, S.-T.Yau: On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman's equation.
Comm. Pure Appl. Math. 33 (1980), no. 4, 507-544.
[CY76]CY76 S. Y. Cheng, S. T. Yau: On the regularity of the solution of the $n$-dimensional Minkowski problem. Comm. Pure Appl. Math. 29 (1976), no. 5, 495–516.
[DK01]DK J.P. Demailly, J. Kollár: Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 4, 525–556.
[Gua98]Gua98 B. Guan: The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function. Comm. Anal. Geom. 6 (1998), no. 4, 687–703.
[GZ05] GZ05 V. Guedj, A. Zeriahi: Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15 (2005), no. 4, 607-639.
[Mab87]Mab87 T. Mabuchi: Some symplectic geometry on compact Kähler manifolds. Osaka J. Math. 24 (1987), no. 2, 227-252.
[Pau08]Pau08 M. Păun: Regularity properties of the degenerate Monge-Ampère equations on compact Kähler manifolds. Chin. Ann. Math. Ser. B 29 (2008), no. 6, 623-630.
[PSeS07]PSS D. H. Phong, N.Sesum, J. Sturm: Multiplier ideal sheaves and the Kähler-Ricci flow. Comm. Anal. Geom. 15 (2007), no. 3, 613-632.
[PSSW08]PSSW D. H. Phong, J. Song, J. Sturm, B. Weinkove: The Moser-Trudinger inequality on Kähler-Einstein manifolds. Amer. J. Math. 130 (2008), no. 4, 1067–1085.
[PS10]PS10 D. H. Phong, J. Sturm: Lectures on stability and constant scalar curvature. Handbook of geometric analysis, No. 3, 357-436, Adv. Lect. Math., 14, Int. Press, Somerville, MA (2010).
[Sic81]Sic81 J.Siciak: Extremal plurisubharmonic functions in $\C^n$. Ann. Polon. Math. 39 (1981), 175–211.
[Siu87]Siu87 Y. T. Siu: Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics. DMV Seminar, 8. Birkhäuser Verlag, Basel, 1987.
What is the optimal constant $C=C(n,p)$ ?
§ MISCELLANEA
In this section we collect questions which do not fall in the previous categories but have strong pluripotential flavor:
Lelong numbers are well-defined not only for positive closed currents- in particular it is enough to assume that the current is positive and harmonic i.e. $dd^c T=0$. The next question deals with Siu- type problem for harmonic currents:
Let $X$ be a compact Kähler manifold and $T$ be a positive harmonic current on $X$. Is it true that the sets
$$E_c(T)=\lbrace z\in X|\ v(T,z)\geq c\rbrace$$
are analytic?
The question has obviously a negative answer locally: if $(z_1,z')$ are coordinates in $\mathbb C^n$ and $h(z')$ is any positive pluriharmonic function then
with $[A]$ being the current of integration over $A$ is a counterexample.
The next problem is an Ancona type result for the relative capacity. Recall that for a Borel subset $K$ of the unit ball the relative capacity is defined by
$$cap(K)=cap(K,\mathbb B^n):=sup\lbrace\int_K(dd^cu)^n|\ u\in\psh(\mathbb B^n) 0\leq u\leq 1\rbrace.$$
The supremum is achieved by the relative extremal function
$$h_K(z)=\limsup_{\zeta\rightarrow z}sup\lbrace u(\zeta)|\ u\in \psh(\mathbb B^n),\ u\leq 1,\ u|_K\leq 0\rbrace.$$
The set $K$ is called regular if the relative extremal function is continuous.
Given any subset $K$ of positive capacity and $\varepsilon>0$ is there a regular set $V\subset K$ such that
$$cap(V)\geq cap(K)-\varepsilon?$$
For the logarithmic capacity in the complex plane this result was proven by Ancona <cit.>. His methods essentially rely on the linearity of the Laplacian operator.
|
1511.00613
|
When partitioning workflows in realistic scenarios, the knowledge of the processing units is often vague or unknown. A naive approach to addressing this issue is to perform many controlled experiments for different workloads, each consisting of multiple number of trials in order to estimate the mean and variance of the specific workload. Since this controlled experimental approach can be quite costly in terms of time and resources, we propose a variant of the Gibbs Sampling algorithm that uses a sequence of Bayesian inference updates to estimate the processing characteristics of the processing units. Using the inferred characteristics of the processing units, we are able to determine the best way to split a workflow for processing it in parallel with the lowest expected completion time and least variance.
§ INTRODUCTION
Many large and time consuming tasks can be broken down into independent components, as for example, $i$ and $j$, with proportions of $f$ and $1 - f$ for processing in parallel <cit.>. The task is considered complete when both independent components complete, with the processing time taken as the maximum of the two components. Given that each component operates on distinct processing unit with different configurations and capabilities, each component has a completion time that follows a different statistical distribution. If we let $\Theta_i$ represent the parameters of $i$'s completion time $t_i$, and $\Theta_j$ to represent the parameters of $j$'s completion time $t_j$,
* The probability that the task has a completion time $t$ before $\epsilon$ is given by,
\begin{align*}
P(t \leq \epsilon | f, \Theta) &= P(t_i \leq \epsilon | f, \Theta_i) P(t_j \leq \epsilon | f, \Theta_j) \\
\Theta &= \{ \Theta_i, \Theta_j \}
\end{align*}
* The expected completion time of the task $t$ is given by,
\[ E(t | f, \Theta) = \int_0^{\infty} 1 - P(t \leq \epsilon | f, \Theta) ~ d\epsilon \]
* While the variance of completion time $t$ is given by,
\[ Var(t | f, \Theta) = \left\{ 2 \int_0^{\infty} \epsilon \Big[ 1 - P(t \leq \epsilon | f, \Theta) \Big] ~ d\epsilon \right\} - \Big[ E(t | f, \Theta) \Big]^2 \]
For brevity, we shall denote the expected completion time $E(t|f,\Theta)$ as $\mu(f)$ and the variance of completion time $Var(t|f,\Theta)$ as $\sigma^2(f)$.
For the purpose of illustration on how $\mu(f)$ and $\sigma^2(f)$ vary as a function of $f$, we shall assume that the completion times of each processing unit is Gaussian in nature, with known values of the parameter $\Theta$, which governs the processing capabilities of the two processing units. By using the hypothetical values $\mu_i = 30$, $\sigma_i = 2$, $\mu_j = 20$, $\sigma_j = 6$, we obtain the numerical results as shown in Figures <ref>, <ref> and <ref>.
$\mu(f)$ and $\sigma^2(f)$ for each value of $f$
[$\mu(f)$ with respect to $f$]
[$\sigma^2(f)$ with respect to $f$]
$\mu(f)$ and $\sigma^2(f)$ as a function of $f$
In Figure <ref>, each point gives the respective value of $\mu(f)$ and $\sigma^2(f)$ for a specific $f$. The curve formed from these points is parabolic which indicates that for some values of $\mu(f)$, there are two possible choices of $\sigma^2(f)$. The converse is true as well, i.e. some values of $\sigma^2(f)$ has two choices of $\mu(f)$. If our assumptions on the statistical distribution of completion times for the two parallel machines hold, then the theoretical results derived in Figure <ref> allows us to decide the appropriate values of $f$ which,
* Minimizes the expected time $\mu(f)$ for a desired amount of uncertainty $\sigma^2(f)$.
* Minimizes the amount of uncertainty $\sigma^2(f)$ for a desired expected time $\mu(f)$.
The appropriate choice of $f$ that provides the optimal values of $\mu(f)$ and $\sigma^2(f)$ is given by the efficient frontier in the lower left portion of the curve which is highlighted in bolded red $\bullet$. $f$ thus denote the amount of appropriate parallelism necessary to achieve a desired level of Quality-of-Service (QoS).
The scenarios for which QoS is important are supply chain management, computer networking, parallel and distributed systems, or even military strategies that often require the achievement of a common objective orchestrated by several teams working in parallel.
§.§ Problem Description
But in many realistic scenarios, the knowledge of the processing unit is often vague or unknown. A naive approach to addressing this issue is to perform many controlled experiments of different workloads by varying $f$, with each $f$ having multiple number of trials in order to estimate the mean and variance at each value of $f$. However, in realistic or deployed systems, such controlled experiments represent an opportunity cost and the resources used to conduct such experiments would reap more benefits by running actual workloads.
It is therefore necessary to have an algorithm which learns the (processing) system parameters quickly based on several trials of using the processing units without deliberate selection of $f$. To fulfill this requirement, we propose to use a Bayesian approach to infer the parameters $\Theta$. Using a Bayesian approach allows several benefits as follows,
* The current understanding of the systems' performance can be given as input to the algorithm using prior beliefs expressed using statistical distributions.
* Based on an observed batch of data, such as completion time with respect to the amount of parallelism $f$, the likelihood of the observations can be combined with the prior beliefs to obtain a posterior belief of the systems' performance.
* The posterior belief obtained from the previous batch of observations can become the prior belief for the next batch of observations. By chaining a sequence of prior and posterior updates, the algorithm can adjust the systems' parameters for a dynamically fast changing environment.
§ THE SPLITTING WORKFLOW MODEL BASED ON THE NORMAL DISTRIBUTION
For simplification (but without loss of generality) of the learning algorithm description, we shall assume that the completion time $t_i, t_j$ for each of the processing unit $i,j$ follows that of a Normal distribution.
\begin{gather*}
p(t_i | f, \mu_i, \sigma_i, \alpha_i, \beta_i) \sim \mathcal{N} \left( f^{\alpha_i} \mu_i, \left[ f^{\beta_i} \sigma_i \right]^2 \right) \\
p(t_j | f, \mu_j, \sigma_j, \alpha_j, \beta_j) \sim \mathcal{N} \left( [1-f]^{\alpha_j} \mu_j, \left[ (1-f)^{\beta_j} \sigma_j \right]^2 \right)
% f = \frac{|D_i|}{|D_i| + |D_j|}
\end{gather*}
where $\alpha$ and $\beta$ are scaling exponents that affects the completion time for varying size of the workload. In efficient and ideal parallel systems, $\alpha$ and $\beta$ would have values of $1.0$. But due to coordination costs and communication overheads in parallel processing, the values of $\alpha$ and $\beta$ are unlikely to have an exact value of $1.0$. Since the $\alpha$ and $\beta$ have an inter-dependency with the values of $\mu$ and $\sigma$, that implies estimating the parameters of the model cannot be easily reduce to estimating the parameters of a Normal distribution.
Let's simplify the notation so that,
\begin{align*}
f_i &= f \\
f_j &= 1 - f_i
\end{align*}
Then we can see that the analysis for $i$ and $j$ is identical,
\begin{align*}
p(t_i | f_i, \mu_i, \sigma_i) &\sim \mathcal{N} \left( f_i^{\alpha_i} \mu_i, f_i^{2\beta_i} \sigma_i^2 \right) \\
p(t_j | f_j, \mu_j, \sigma_j) &\sim \mathcal{N} \left( f_j^{\alpha_j} \mu_j, f_j^{2\beta_j} \sigma_j^2 \right)
\end{align*}
The purpose of simplifying for $i$ and $j$ is to show that if we can derive the bayesian updates of $i$, then we can similarly apply the same equations to $j$. With that, we can reduce the clutter in the equations by dropping the subscripts $i$ and $j$ so that we only have to work on the following,
\begin{align}
\label{eqn:model_t}
p(t|f, \mu, \sigma, \alpha, \beta) \sim \mathcal{N} \left( f^{\alpha} \mu, f^{2\beta} \sigma^2 \right)
\end{align}
As stated in Equation <ref>, the completion time $t$ can be predicted conditioned on the assumption that $\mu,\sigma, \alpha, \beta$ are known. The original motivation of our discussion does not assume knowledge of these values.
In the next few sections, we will derive the Bayesian inference equations that allow us to obtain estimations for the values of $\mu, \sigma, \alpha, \beta$.
§ BAYESIAN INFERENCE FOR $\MU$ AND $\SIGMA$
Since these values are unknowns, we can assume that they are drawn from some statistical distributions. For notational convenience, let's replace the variance $\sigma^2$ with the precision $\lambda$ using the following relationship,
\[ \lambda = \frac{1}{\sigma^2} \]
An appropriate choice of prior distribution for $\mu$ is the following Normal distribution,
\begin{align*}
\mu | \mu_0, \kappa_0, \lambda &\sim \mathcal{N} \left( \mu_0, ( \kappa_0 \lambda )^{-1} \right) \\
p(\mu | \mu_0, \kappa_0, \lambda) &\propto \lambda^{\frac{1}{2}} \exp\left( -\frac{\kappa_0 \lambda}{2} (\mu-\mu_0)^2 \right)
\end{align*}
While the prior distribution of $\lambda$ is the Gamma distribution,
\begin{align*}
\lambda | \nu_0, \psi_0 &\sim \text{Gamma}(\nu_0, \text{rate}=\psi_0) \\
p(\lambda | \nu_0, \psi_0) &\propto \lambda^{\nu_0 - 1} \exp \left( -\psi_0 \lambda \right)
\end{align*}
where $\mu_0, \kappa_0, \nu_0$ and $\psi_0$ are parameters for the prior distributions of $\mu$ and $\lambda$, which are constants that can be set based on subjective prior knowledge.
Then expressing the pdf as a multiplication of the two distribution,
\begin{align}
p(\mu, \lambda | \mu_0, \kappa_0, \nu_0, \psi_0) &\propto \lambda^{\frac{1}{2}} \exp \left( -\frac{\kappa_0 \lambda}{2} (\mu - \mu_0)^2 - \psi_0 \lambda \right) \lambda^{\nu_0 - 1} \nonumber \\
\label{eqn:mu_sigma_prior}
&\propto \lambda^{\nu_0 - \frac{1}{2}} \exp \left( -\frac{\lambda}{2} \left[ \kappa_0 (\mu - \mu_0)^2 + 2 \psi_0 \right] \right)
\end{align}
The next step is to merge the prior distribution with the likelihood of some observed data to obtain the posterior distribution. In statistical notation, we would like to obtain the posterior distribution conditioned on the observations of some completion time $T = \{t_1, t_2, \ldots, t_N\}$ for a given set of workload $F = \{f_1, f_2, \ldots, f_N \}$. And assuming that the values of $\alpha$ and $\beta$ is known. i.e.
\begin{gather}
\label{eqn:mu_sigma_posterior}
p(\mu, \lambda | T, F, \alpha, \beta, \mu_0, \kappa_0, \nu_0, \psi_0) \propto p(T | F, \mu, \lambda, \alpha, \beta) p(\mu, \lambda | \mu_0, \kappa_0, \nu_0, \psi_0)
\end{gather}
The likelihood is then given by,
\begin{align}
p(T | F, \mu, \lambda, \alpha, \beta) = &\prod_n p(t_n | f_n, \mu, \lambda, \alpha, \beta) \nonumber \\
p(t_n | f_n, \mu, \lambda, \alpha, \beta) &\propto \frac{\sqrt{\lambda}}{f_n^\beta} \exp \left( -\frac{\lambda}{2} \left[ \frac{t - f_n^\alpha \mu}{f_n^\beta} \right]^2 \right) \nonumber \\
p(T | F, \mu, \lambda, \alpha, \beta) &\propto \prod_n \frac{\sqrt{\lambda}}{f_n^\beta} \exp \left(-\frac{\lambda}{2} \left[ \frac{t - f_n^\alpha \mu}{f_n^\beta} \right]^2 \right) \nonumber \\
\label{eqn:mu_sigma_likelihood_0}
&\propto \frac{\lambda^{\frac{N}{2}}}{\prod_n f_n^\beta} \exp \left( - \frac{\lambda}{2} \sum_n \left[ \frac{t - f_n^\alpha \mu}{f_n^\beta} \right]^2 \right) \\
\label{eqn:mu_sigma_likelihood}
&\propto \lambda^{\frac{N}{2}} \exp \left( - \frac{\lambda}{2} \sum_n \left[ \frac{t - f_n^\alpha \mu}{f_n^\beta} \right]^2 \right)
\end{align}
Substitute Equations <ref> and <ref> into <ref>. Then through some algebraic manipulations (expansion, completing the square, factorization and simplification), we can obtain the posterior distribution given by,
\begin{align*}
&p(\mu, \lambda|T,F,\alpha,\beta,\mu_0,\kappa_0,\nu_0,\psi_0) \\
&\propto \lambda^{\nu_N - \frac{1}{2}} \exp \left( -\frac{\lambda}{2} \left[ \kappa_N (\mu - \mu_N)^2 + 2 \psi_N \right] \right)
\end{align*}
With $\mu_N, \kappa_N, \nu_N$ and $\psi_N$ given by,
\begin{align}
\label{eqn:mu_N}
\mu_N &= \frac{\mu_0 \kappa_0 + \sum_n f_n^{\alpha - 2\beta} t_n}{\kappa_0 + \sum_n f_n^{2\alpha - 2\beta}} \\
\label{eqn:kappa_N}
\kappa_N &= \kappa_0 + \sum_n f_n^{2\alpha - 2\beta} \\
\label{eqn:nu_N}
\nu_N &= \nu_0 + \frac{N}{2} \\
\label{eqn:psi_N}
\psi_N &= \psi_0 + \frac{1}{2} \left[ -\mu_N^2 \kappa_N + \mu_0^2 \kappa_0 + \sum_n \left( \frac{t_n}{f_n^{\beta}} \right)^2 \right]
\end{align}
§ BAYESIAN INFERENCE FOR $\ALPHA$ AND $\BETA$
$\alpha$ and $\beta$ represents the scalability of the processing unit when given different workloads governed by $f$. A perfect system would have $\alpha = 1.0$ and $\beta=1.0$ indicating that the expected completion time and variance scales linearly with the size of the workload. $\alpha > 1.0$ and $\beta > 1.0$ represents an impossible scenario since this suggests that the system takes less time and have less uncertainty when given more workload. Since $\alpha$ and $\beta$ could only take values between $0$ and $1$, it would be appropriate to use the Beta distribution as the prior of $\alpha$ and $\beta$.
\begin{align*}
p(\alpha | \theta_0, \phi_0) &\propto \alpha^{\theta_0 - 1} (1 - \alpha)^{\phi_0 - 1} \\
p(\beta | \delta_0, \eta_0) &\propto \beta^{\delta_0 - 1} (1-\beta)^{\eta_0 - 1}
\end{align*}
Using the likelihood given by Equation <ref>, the posterior distribution of $\alpha$ conditioned on a set of observations $T$ for a given set of $F$ is,
\begin{align}
&p(\alpha | T, F, \mu, \lambda, \theta_0, \phi_0, \beta) \propto p(T | F, \mu, \lambda, \beta, \alpha) ~ p(\alpha | \theta_0, \phi_0) \nonumber \\
\label{eqn:alpha_posterior}
&\propto \lambda^{\frac{N}{2}} \exp \left( - \frac{\lambda}{2} \sum_n \left[ \frac{t - f_n^\alpha \mu}{f_n^\beta} \right]^2 \right) \alpha^{\theta_0 - 1} (1 - \alpha)^{\phi_0 - 1}
\end{align}
For the posterior distribution of $\beta$, we would have to use the likelihood given by Equation <ref> which gives us the following,
\begin{align}
&p(\beta | T, F, \mu, \lambda, \delta_0, \eta_0, \alpha) \propto p(T | F, \mu, \lambda, \alpha, \beta) ~ p(\beta | \delta_0, \eta_0) \nonumber \\
\label{eqn:beta_posterior}
&\propto \frac{\lambda^{\frac{N}{2}}}{\prod_n f_n^\beta} \exp \left( - \frac{\lambda}{2} \sum_n \left[ \frac{t - f_n^\alpha \mu}{f_n^\beta} \right]^2 \right) \beta^{\delta_0 - 1} (1-\beta)^{\eta_0 - 1}
\end{align}
[True posterior distribution of $\alpha$ as given by Equation <ref>]
[Approximate posterior distribution of $\alpha$ using the method of moments]
Comparison between the true and approximate posterior distribution of $\alpha$
[True posterior distribution of $\beta$ as given by Equation <ref>]
[Approximate posterior distribution of $\beta$ using the method of moments]
Comparison between the true and approximate posterior distribution of $\beta$
Unfortunately, there is no algebraic solution to manipulate the posterior distribution given by Equations <ref> and <ref> into Beta distributions. In fact, there is no analytical proof that the posterior distributions remains as Beta distributions.
We could continue to assume that the posterior distribution can be approximated by a Beta distribution with parameters $\theta_N, \phi_N$ for $\alpha$ and $\delta_N, \eta_N$ for $\beta$. Using the method of moments,
\begin{align}
\label{eqn:theta_N}
\theta_N &= E(\alpha) \left[ \frac{E(\alpha)[1 - E(\alpha)]}{Var(\alpha)} - 1 \right] \\
\label{eqn:phi_N}
\phi_N &= [1 - E(\alpha)] \left[ \frac{E(\alpha)[1 - E(\alpha)]}{Var(\alpha)} - 1 \right] \\
\label{eqn:delta_N}
\delta_N &= E(\beta) \left[ \frac{E(\beta)[1 - E(\beta)]}{Var(\beta)} - 1 \right] \\
\label{eqn:eta_N}
\eta_N &= [1 - E(\beta)] \left[ \frac{E(\beta)[1 - E(\beta)]}{Var(\beta)} - 1 \right]
\end{align}
Then using the standard definitions for $E(\alpha)$ and $Var(\alpha)$ to derive their specific values,
\begin{align}
\label{eqn:expected_alpha}
E(\alpha) &= \int_0^1 \alpha \cdot p(\alpha | T, F, \mu, \lambda, \theta_0, \phi_0) ~ d\alpha \\
\label{eqn:expected_alpha2}
E(\alpha^2) &= \int_0^1 \alpha^2 \cdot p(\alpha | T, F, \mu, \lambda, \theta_0, \phi_0) ~ d\alpha \\
\label{eqn:variance_alpha}
Var(\alpha) &= E(\alpha^2) - \left[ E(\alpha) \right]^2
\end{align}
Although unproven, it is unlikely that the integrals due to the PDF given by Equation <ref> and <ref> have closed form solutions. In our solver, we employ the use of numerical integration to obtain an approximate value for the expectations and variances. Similar procedure applies for $\delta_N$ and $\eta_N$ of $\beta$.
Figures <ref> and <ref> show an example of the differences between the true and approximate posterior distribution of $\alpha$. Figures <ref> and <ref> show an example of the differences between the true and approximate posterior distribution of $\beta$. The green line in Figures <ref> and <ref> shows that the mean of the distribution is also close to the mode of the distribution, which has important implications for the Gibbs Sampling algorithm which we will describe in the next section.
§ GIBBS SAMPLING ALGORITHM
Algorithm <ref> summarizes the use of the Bayesian inference equations for estimating the parameters of the processing system. After updating the parameters of the prior distributions, we sample from the distributions instead of taking their mode or mean so as to avoid getting trapped in a local maxima of the log likelihood. Due to the fact that the mean is also closed to the mode as shown in Figures <ref> and <ref>, it suggest that sampling from their distributions will have the desired side effect of increasing the log likelihood of the overall system.
Gibbs Sampling of $\mu, \sigma, \alpha, \beta$
$\{ \mu_0, \kappa_0, \nu_0, \psi_0 \}, \{ \theta_0, \phi_0 \}, \{ \delta_0, \eta_0 \}$
Sample $\alpha$ from Beta distribution using $\theta_0$ and $\phi_0$.
Sample $\beta$ from Beta distribution using $\delta_0$ and $\eta_0$.
$T \leftarrow$ [], $F \leftarrow$ []
$n \leftarrow 1$ to $N$
Add $t_n$ to $T$, add $f_n$ to $F$
some number of iterations
$\mu_N \leftarrow$ using Equation <ref>.
$\kappa_N \leftarrow$ using Equation <ref>.
$\nu_N \leftarrow$ using Equation <ref>.
$\psi_N \leftarrow$ using Equation <ref>.
Sample $\lambda$ from Gamma distribution using $\nu_N$ and $\psi_N$.
Sample $\mu$ from Normal distribution using $\mu_N$ and $(\kappa_N \lambda)^{-1}$.
$\theta_N \leftarrow$ using Equation <ref>.
$\phi_N \leftarrow$ using Equation <ref>.
Sample $\alpha$ from Beta distribution using $\theta_N$ and $\phi_N$.
$\delta_N \leftarrow$ using Equation <ref>.
$\eta_N \leftarrow$ using Equation <ref>.
Sample $\beta$ from Beta distribution using $\delta_N$ and $\eta_N$.
$\mu_0 \leftarrow \mu_N$, $\kappa_0 \leftarrow \kappa_N$, $\nu_0 \leftarrow \nu_N$, $\phi_0 \leftarrow \phi_N$
$\theta_0 \leftarrow \theta_N$, $\phi_0 \leftarrow \phi_N$, $\delta_0 \leftarrow \delta_N$, $\eta_0 \leftarrow \eta_N$
$\sigma \leftarrow \sqrt{1/\lambda}$
$\mu, \sigma, \alpha, \beta$
Figure <ref> shows the convergence of the Gibbs Sampling algorithm presented in Algorithm <ref>. The fast increase of the log likelihood (y-axis) using relatively low number of data points (x-axis) shows that the Bayesian inference equations and the Gibbs Sampling algorithm is able to estimate the system parameters.
Results for estimating the network characteristics for a file transfer. Each point represents the logarithm likelihood of Equation <ref>.
|
1511.00849
|
[footnoteinfo]This work was supported by the COMPANION EU project, the Knut and Alice Wallenberg Foundation, and the Swedish Research Council.
First]Sebastian van de Hoef
First]Karl H. Johansson
First]Dimos V. Dimarogonas
[First]ACCESS Linnaeus Center and the School of Electrical Engineering, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden (e-mail: {shvdh, kallej, dimos}@kth.se).
Vehicle platooning facilitates the partial automation of vehicles and can significantly reduce fuel consumption. Mobile communication infrastructure makes it possible to dynamically coordinate the formation of platoons en route. We consider a centralized system that provides trucks with routes and speed profiles allowing them to dynamically form platoons during their journeys. For this to work, all possible pairs of vehicles that can platoon based on their location, destination, and other constraints have to be identified. The presented approach scales well to large vehicle fleets and realistic road networks by extracting features from the transport assignments of the vehicles and rules out a majority of possible pairs based on these features only. Merely a small number of remaining pairs are considered in depth by a complete and computationally expensive algorithm. This algorithm conclusively decides if platooning is possible for a pair based on the complete data associated with the two vehicles. We derive appropriate features for the problem and demonstrate the effectiveness of the approach in a simulation example.
Transportation, Agents, Platooning, Spatial Networks
§ INTRODUCTION
Platooning is one of the fundamental building blocks for controlling connected vehicles. Vehicles are arranged in a convoy and the longitudinal spacing in the convoy is maintained with the help of automatic control. This simplifies the automation of the trailing vehicles and reduces their fuel consumption due to the slipstream effect (<cit.>). While the low-level platoon control is well developed (<cit.>), the dynamic formation of platoons has only recently attracted the interest of researchers (<cit.>).
We envision an integrated system that centrally coordinates the formation of platoons. Trucks would connect to such a system via vehicle-to-infrastructure communication. The system continuously provides updated fuel-efficient routes and speed profiles to the connected trucks. These routes and speed profiles allow vehicles to meet on their journeys in order to form platoons. The computation of these routes and speed profiles takes various constraints such as arrival deadlines, speed-limits, and rest periods into account.
The contribution of this paper is a method to efficiently rule out the majority of transport assignment pairs that cannot form a platoon due to their geographic or temporal separation.
The elements of the significantly smaller set of candidate pairs is then treated one by one.
The computation of routes and coordinated speed profiles happens in several stages. The first stage is the route calculation that produces a route to the destination for each transport assignment based on the current position of the vehicle or the planned start point of the transport assignment in the future. It also computes a set of possible trajectories along that route. The next stage identifies for which pairs of transport assignments the associated vehicles can meet in the future in order to platoon based on the output of the route calculation. This is the input to an optimization routine that determines the platoons formed and the corresponding trajectories. These trajectories are sent to the individual vehicles which execute them. This process is frequently repeated in order to account for new or changed assignments, vehicles deviating from the planned trajectories, and updated traffic information.
Comparing all pairs individually in order to find out which pairs can platoon is only feasible for a small number of transport assignments.
On the other hand, the number of transport assignments increases when larger geographic regions or longer time horizons are considered in the planning process. In 2011 over 1.7 million heavy trucks were in use (<cit.>) in the European Union, a number so large that even a fraction of these can be challenging to coordinate.
A related problem to the one considered in this paper is to compute the collision of a large number of geometric objects, which has been considered in the field of computer graphics (<cit.>) and to some extent in the area of interest management for distributed virtual environments (<cit.>).
The way this problem is tackled might also be relevant for other spatial, large-scale multi-agent systems where the possible interactions need to be identified in real time and where the number of actual interactions is small compared to the number of agent pairs. Examples of such systems are collision avoidance of (autonomous) mobile agents (<cit.>) and ride-sharing systems (<cit.>). Similar to the application considered in this paper, these are large-scale spatially-distributed systems (<cit.>) that are enabled by the rapid development of the communication infrastructure.
This paper is organized as follows. After having introduced the problem and related notation (Section <ref>), we first abstract approaches that have been developed in computer graphics (Section <ref>). We introduce the concept of features and classifiers that can indicate a pair of vehicles not being able to platoon on their routes. In Section <ref> we derive a family of features for the problem setting introduced in Section <ref>. We demonstrate the method with a simulation study in Section <ref>.
§ PROBLEM FORMULATION
We proceed with introducing the problem setup considered in this paper. We model the road network as a directed graph $\set{G}_\op{r} = (\set{N}_\op{r},\set{E}_\op{r})$ with nodes $\set{N}_\op{r}$ and edges $\set{E}_\op{r} \in \set{N}_\op{r} \times \set{N}_\op{r}$. Nodes correspond to intersections or endpoints in the road network and links correspond to road segments connecting these intersections. Each node in $\set{N}_\op{r}$ can be associated with a 2-D coordinate $\ve{P}: \set{N}_\op{r} \rightarrow \mathbb{R}^2$.
Furthermore, we assume that the length of a road segment modeled by an edge equals the euclidean distance between the positions of the two nodes that comprise the edge.
We have $K$ transport assignments and let $\set{N}_\op{c} = \{1, \dots, K\}$ be an index set of all assignments.
Each transport assignment consists of a start node $n^\op{S} \in \set{N}_\op{r}$ and a destination node $n^\op{D} \in \set{N}_\op{r}$. Furthermore, for each transport assignment, there is an earliest start time $t^\op{S}$ and a latest arrival time $t^\op{D}$.
The input to the vehicles are paths and speed profiles, one for each vehicle, that implement the transport assignments. To that end, we define a trajectory (route and speed-profile) of a vehicle and the requirements for a trajectory to implement a transport assignment.
A trajectory is a pair $(\ve{n},\ve{t})$. The route $\ve{n} = n[1], \dots, n[N]$ is a sequence of nodes in $\set{N}_\op{r}$ that describe a path in $\set{G}_\op{r}$. We refer to $\ve{n}$ as a route. The second element $\ve{t} = t[1], \dots, t[N]$ is a sequence of time instances such that $t[a+1] - t[a] \geq \|\ve{P}(n[a+1]) - \ve{P}(n[a])\|_2/v_\op{max}$ for $a = 1, \dots, N-1$, where $v_\op{max}$ is the maximum speed. The number of nodes $N$ may be different for different trajectories.
We neglect that the speed on a link is restricted by the speed on the adjacent links and that the maximum speed $v_\op{max}$ depends on the link and on time. This is mainly for the ease of presentation but can as well be a reasonable simplification in the culling phase and can be accounted for in the routine that calculates the actual speed profiles.
A trajectory $(\ve{n}, \ve{t})$ implements a transport assignment $(n^\op{S}, n^\op{D}, t^\op{S}, t^\op{D})$ if it starts after the start time at the start node of the transport assignment and arrives before the deadline at destination node of the transport assignment, i.e., if $n[1] = n^\op{S}$, $n[N] = n^\op{D}$, $t[1] \geq t^\op{S}$, $t[N] \leq t^\op{D}$.
We assume that the route $\ve{n}$ of the trajectory is given, typically the shortest path. The coordination is restricted to adapting the speed profile, i.e., the sequence $\ve{t}$.
We need to be able to test whether there are trajectories that platoon, i.e., partially coincide. To this end we calculate for each node in $\ve{n}$ a time interval.
The element in $\ve{t}$ that corresponds to the node in $\ve{n}$ lies in this interval if $\ve{t}$ belongs to a trajectory that implements the transport assignment. Only when the intervals for two transport assignments overlap at a common node, the possibility of implementing trajectories that platoon exists. We denote the sequence of lower bounds on the elements of $\ve{t}$ as $\underline{\ve{t}} = \underline{t}[1], \dots, \underline{t}[N]$ and the upper bounds as $\bar{\ve{t}} = \bar{t}[1], \dots, \bar{t}[N]$. They are computed for $a = 1, \dots, N$ as
\begin{align}
\underline{t}[a] &= t^\op{S} + \sum\limits_{m = 1}^{a-1} \frac{\|\ve{P}(n[m+1]) - \ve{P}(n[m])\|_2}{v_\op{max}} \label{eq:lower_bound}\\
\bar{t}[a] &= t^\op{D} - \sum\limits_{m = a}^{N-1} \frac{\|\ve{P}(n[m+1]) - \ve{P}(n[m])\|_2}{v_\op{max}}\label{eq:upper_bound}.
\end{align}
Next, we define a function that indicates whether platooning between two transport assignments is possible or not. This is the case if there is at least one common edge in the routes of the transport assignments where the time bounds of the two assignments overlap.
The coordination function $\coordf: \set{N}_\op{c} \times \set{N}_\op{c} \rightarrow \{0, 1\}$ has the following properties. Let $\underline{\ve{t}}_i, \underline{\ve{t}}_j$ be lower bounds and $\bar{\ve{t}}_i, \bar{\ve{t}}_j$ be upper bounds on the node arrival times of transport assignments $i$ and $j$ according to (<ref>), (<ref>). Then it holds that $\coordf(i,j) = 1$, if there are indices $a, b$ such that $\ve{P}(n_i[a]) = \ve{P}(n_j[b])$ and $\ve{P}(n_i[a+1]) = \ve{P}(n_j[b+1])$, and $[\underline{t}_i[a], \bar{t}_i[a]] \cap [\underline{t}_j[b], \bar{t}_j[b]] \neq \emptyset$ and $[\underline{t}_i[a+1], \bar{t}_i[a+1]] \cap [\underline{t}_j[b+1], \bar{t}_j[b+1]] \neq \emptyset$. Otherwise $\coordf(i,j) = 0$.
Comparing the routes and the time bounds in order to evaluate $\coordf$, is straightforward but computationally expensive. We refer to this as the exact algorithm.
The goal of this work is to find a scalable method for computing the set of all possible platoon pairs $\set{C} = \{(i,j) \in \set{N}_\op{c} \times \set{N}_\op{c}: \coordf(i,j) = 1\}$. Instead of iterating over all elements in $\set{N}_\op{c} \times \set{N}_\op{c}$ and using the exact algorithm, we propose to first efficiently compute an overapproximation $\hat{\set{C}} \supset \set{C}$ and then applying the exact algorithm.
§ CULLING
The key idea of our approach is to extract features from the routes and time bounds ($\ve{n},\underline{\ve{t}}, \bar{\ve{t}}$) of the transport assignments to compute $\hat{\set{C}}$. These features can be more efficiently processed than $\ve{n}, \underline{\ve{t}}, \bar{\ve{t}}$. Features are designed in a way that no platooning opportunity in $\set{C}$ will be excluded from $\hat{\set{C}}$, so that $\set{C}$ can be computed from $\hat{\set{C}}$ using the exact algorithm. However, there might be some additional elements in $\hat{\set{C}}$ that do not actually correspond to platooning opportunities. We call these additional elements false-positives. The less false-positives there are in $\hat{\set{C}}$, the faster is the computation of $\set{C}$ from $\hat{\set{C}}$. This approach is inspired by a related problem of detecting which pairs of a large number of geometric objects intersect or collide.
We consider two types of features. These are interval features and binary features. Interval features map each object to an interval. The corresponding classifier indicates an intersection between two objects if the intervals generated by the objects overlap. There are algorithms (<cit.>)
that can compute this classifier for all object pairs more efficiently than checking each pair individually, if the number of reported intersecting pairs is small.
Binary features map each object to a boolean value. The corresponding classifier indicates an intersection between two objects if the feature holds true for both objects. In Section <ref>, we derive appropriate features for the problem stated in Section <ref>.
The classifiers are aggregated using boolean connectives. We formalize this in the remainder of the section. Let $\set{N}$ be a set of objects. We define a classifier as a function $c: \set{N} \times \set{N} \rightarrow \{0, 1\}$. If $c(i,j) = 0$, we call the combination of $c$ and $(i,j)$ a negative, and if $c(i,j) = 1$, we call it a positive.
Let $g: \set{N} \times \set{N} \rightarrow \{0, 1\}$ be the ground truth which can be computed by the exact algorithm. If for a pair $(i,j)$ we have $g(i,j) = 0$ and $c(i,j) = 1$, we call it a false-positive, and if $g(i,j) = 1$ and $c(i,j) = 0$, we call it a false-negative. Our aim is to design classifiers which yield no false negatives for all elements of $\set{N} \times \set{N}$ and few false-positives that have to be processed by the exact algorithm in addition to the true-positives.
We can identify two types of basic classifiers that are combined in a specific way in order to achieve the above objective.
A classifier $c$ is required if $\neg c(i,j) \Rightarrow \neg g(i,j)$ for all $i, j \in \set{N} \times \set{N}$.
In some cases, we have to take into account a set of classifiers to conclude that $g$ does not hold.
A set of classifiers $\set{D}$ is required if $\neg \bigvee\limits_{c \in \set{D}} c(i,j) \Rightarrow \neg g(i,j)$ for all $i, j \in \set{N} \times \set{N}$.
It is straightforward to construct a required classifier from a required set of classifiers.
If a set $\set{D}$ of classifiers is required, then $\bigvee\limits_{c \in \set{D}} c$ is a required classifier.
We can combine two required classifiers into one required classifier that performs no worse than any of the required classifiers it is combined of.
If $c_1$ and $c_2$ are required classifiers, then $c_{12} := c_1 \wedge c_2$ is a required classifier. Let $\bar{\set{E}}_{12} = \{(i,j) \in \set{N} \times \set{N}: c_{12}(i,j) = 0\}$ be the set of negatives of $c_{12}$ and let $\bar{\set{E}}_1$, $\bar{\set{E}}_2$ be the set of negatives for $c_1$ and $c_2$ respectively. Then $\bar{\set{E}}_1 \subseteq \bar{\set{E}}_{12}$ and $\bar{\set{E}}_2 \subseteq \bar{\set{E}}_{12}$.
For $c_{12}$ to be required, we need to show that $\neg c_{12}(i,j) \Rightarrow \neg g(i,j)$ for all $i, j \in \set{N} \times \set{N}$.
We have
(\neg c_1 \Rightarrow \neg g) \wedge (\neg c_2 \Rightarrow \neg g)
= (c_1 \vee \neg c_1 \wedge \neg g) \wedge (c_2 \vee \neg c_2 \wedge \neg g)
= c_1 \wedge c_2 \vee \neg g \wedge(\neg c_1 \wedge \neg c_2 \vee \neg c_1 \wedge c_2 \vee c_1 \wedge \neg c_2)
= c_1 \wedge c_2 \vee \neg g \wedge(\neg c_1 \vee \neg c_2)
= c_1 \wedge c_2 \vee \neg g \wedge \neg (c_1 \wedge c_2)
= \neg (c_1 \wedge c_2) \Rightarrow \neg g
= \neg c_{12} \Rightarrow \neg g
Let $(i,j) \in \bar{\set{E}}_1$. Then from the definition of $\bar{\set{E}}_1$ we have that $c_1(i,j) = 0$. We have that $c_{12}(i,j) = c_1(i,j) \wedge c_2(i,j) = 0 \wedge c_2(i,j) = 0$. It follows from the definition of $\bar{\set{E}}_{12}$ that $(i,j) \in \bar{\set{E}}_{12}$. Similarly, we see that any element of $\bar{\set{E}}_1$ is an element of $\bar{\set{E}}_{12}$.
In this manner, we can combine as many required classifiers as we want and have at our disposal. With each classifier we add, we potentially decrease the set of remaining candidates that need to be checked by the exact algorithm. There is a trade-off between doing more work to evaluate more classifiers and less instances which have to be processed by the exact algorithm (<cit.>).
§ FEATURES AND CLASSIFIERS
In order to apply the results from Section <ref>, we need to specify appropriate features and classifiers based on these features for the problem stated in Section <ref>. Once we know how to compute appropriate features that yield required classifiers or required sets of classifiers, we can use the results from Section <ref> to execute the culling phase. The remaining candidate pairs are passed on to the exact algorithm to compute $\set{C}$. Hence, we will derive a selection of features and corresponding classifiers in this section. In Section <ref>, we will demonstrate these classifiers and combinations of them in a simulation example.
The first feature projects the possible trajectories on a line which yields an interval.
Formally, we define this feature as follows.
Let $\ve{p} \in \mathbb{R}^3$ be a three dimensional vector which defines the orientation of the line which the trajectories are projected onto. Then the associated interval feature is defined as
\begin{equation}
\set{I} = [\min\limits_{\ve{v} \in \set{R}} (\ve{p}^\transp \ve{v}), \max\limits_{\ve{v} \in \set{R}} (\ve{p}^\transp \ve{v})]
\label{eq:projection_interval}
\end{equation}
\begin{equation}
\begin{split}
\set{R} = \left\{
\begin{bmatrix}\ve{P}(n[1])\\ \underline{t}[1]\end{bmatrix}, \dots, \begin{bmatrix}\ve{P}(n[N])\\ \underline{t}[N]\end{bmatrix},\right. \\ \left.
\begin{bmatrix}\ve{P}(n[1])\\ \bar{t}[1]\end{bmatrix}, \dots, \begin{bmatrix}\ve{P}(n[N])\\ \bar{t}[N]\end{bmatrix}
\right\}.\label{eq:def_R}
\end{split}
\end{equation}
This feature is illustrated in Figure <ref>. The projection vector $\ve{p}$ is a design choice.
Proposition <ref> allows us to combine arbitrarily many classifiers based on this kind of feature with different $\ve{p}$.
Illustration of the projection feature. It shows how the two routes (solid lines) are projected onto a line in the direction of the vector $\ve{p}$. The borders of the intervals are indicated with dashed lines. For illustration purposes the third dimension is omitted here.
Next, we establish that if for a pair of transport assignments the intervals do not overlap the coordination function is equal to zero. This allows us to define a required feature based on the overlap between these intervals.
Let $(i,j)$ refer to a pair of transport assignments. Let $\set{I}_i, \set{I}_j$ be the interval features according to (<ref>) for the two transport assignments.
Then $\set{I}_i \cap \set{I}_j = \emptyset \Rightarrow \coordf(i,j) = 0$.
According to Definition <ref>, $\coordf(i,j) = 1$ implies that there must be indices $a, b$ such that $\ve{P}(n_i[a]) = \ve{P}(n_j[b])$ and $[\underline{t}_i[a], \bar{t}_i[a]] \cap [\underline{t}_j[b], \bar{t}_j[b]] \neq \emptyset$, where $\ve{n}_i, \underline{\ve{t}}_i, \bar{\ve{t}}_i$ and $\ve{n}_j,\underline{\ve{t}}_j, \bar{\ve{t}}_j$ are the node sequences and time bounds of transport assignment $i, j$ respectively. We have
\begin{equation*}
[\underline{t}_i[a], \bar{t}_i[a]] \cap [\underline{t}_j[b], \bar{t}_j[b]] \neq \emptyset
\Leftrightarrow
\underline{t}_i[a] \leq \bar{t}_j[b] \wedge \underline{t}_j[b] \leq \bar{t}_i[a].
\end{equation*}
Let $\ve{p} = [p_1, p_2, p_3]^\transp$, $\ve{P} = \ve{P}(n_i[a]) = \ve{P}(n_j[b])$, and $P^0 = [p_1, p_2] \ve{P}$. We have
\begin{align*}
&\underline{t}_i[a] \leq \bar{t}_j[b] \wedge \underline{t}_j[b] \leq \bar{t}_i[a]
% \wedge
% \underline{t}_i[a] \leq \bar{t}_i[a] \wedge \underline{t}_j[b] \leq \bar{t}_j[b]
\\
\Rightarrow
&\min ( p_3 \underline{t}_i[a], p_3 \bar{t}_i[a]) \leq \max (p_3 \underline{t}_j[b], p_3 \bar{t}_j[b])
\\
\Rightarrow
&\min ( p_3 \underline{t}_i[a] + P^0, p_3 \bar{t}_i[a] + P^0)
\leq \max (p_3 \underline{t}_j[b] + P^0, p_3 \bar{t}_j[b] + P^0)
\\
\Rightarrow
&\min \left(\ve{p}^\transp \begin{bmatrix}\ve{P}\\ \underline{t}_i[a]\end{bmatrix},
\ve{p}^\transp \begin{bmatrix}\ve{P}\\ \bar{t}_i[a]\end{bmatrix}\right)
\leq
\max \left(\ve{p}^\transp \begin{bmatrix}\ve{P}\\ \underline{t}_j[b]\end{bmatrix},
\ve{p}^\transp \begin{bmatrix}\ve{P}\\ \bar{t}_j[b]\end{bmatrix}\right)
\\
\Rightarrow
&\min\limits_{\ve{v} \in \set{R}_i} (\ve{p}^\transp \ve{v})
\leq
\max\limits_{\ve{v} \in \set{R}_j} (\ve{p}^\transp \ve{v})
\end{align*}
with $\set{R}_i, \set{R}_j$ as in (<ref>) for transport assignment $i, j$, respectively.
Similarly, by swapping $i$ and $j$, we can show that the conditions of the proposition imply that
\begin{align*}
\min\limits_{\ve{v} \in \set{R}_j} (\ve{p}^\transp \ve{v})
\leq
\max\limits_{\ve{v} \in \set{R}_i} (\ve{p}^\transp \ve{v}).
\end{align*}
The above two conditions combined imply that $\set{I}_i \cap \set{I}_j \neq \emptyset$. Thus $\coordf = 1 \Rightarrow \set{I}_i \cap \set{I}_j \neq \emptyset$ or equivalently $\set{I}_i \cap \set{I}_j = \emptyset \Rightarrow \coordf = 0$.
Next, we will introduce a binary feature that leads to a required classifier. This feature is based on the orientations of the individual links in a route. It will only be useful if all segments in a route point approximately from start to goal location. Later on, we will address the problem of outliers. Here, we derive a set of required classifiers each based on a binary feature from the orientation.
The orientation $\Theta(n_1,n_2) \in [0, 2\pi]$ of an edge $(n_1, n_2) \in \set{E}_\op{r}$ is the angle in polar coordinates of the vector $\ve{P}(n_2) - \ve{P}(n_1)$.
We choose a partition of the interval $[0, 2\pi]$. Each element of the partition is related to one binary feature which holds true if the orientation of at least one edge in the route falls in the range of that element. When two routes overlap there must be at least one edge that has the same orientation.
Let $(i,j)$ refer to the pair of transport assignments. Let $\set{P}$ be a partition of $[0, 2\pi]$. If there is no element $I \in \set{P}$ and edges in the routes of the transport assignments $(n_i[a], n_i[a+1])$, $(n_j[b], n_j[b+1])$ such that $\Theta(n_i[a], n_i[a+1]) \in I$ and $\Theta(n_j[b], n_j[b+1]) \in I$, then $\coordf(i,j) = 0$.
According to Definition <ref>, $\coordf(i,j) = 1$ implies that there must be indices $a, b$ such that $\ve{P}(n_i[a]) = \ve{P}(n_j[b])$ and $\ve{P}(n_i[a+1]) = \ve{P}(n_j[b+1])$, where $n_i, n_j$ are the node sequences of transport assignment $i, j$ respectively. For these it holds that $\Theta(n_i[a],n_i[a+1]) = \Theta(n_j[b],n_j[b+1])$.
Since $\set{P}$ is a partition of $[0, 2\pi]$ and $\Theta(n_i[a],n_i[a+1]) \in [0, 2\pi]$, there must be $I \in \set{P}$ with $\Theta(n_i[a],n_i[a+1]) \in \set{I}$. Since $\Theta(n_j[b],n_j[b+1]) = \Theta(n_i[a],n_i[a+1])$, it follows that also $\Theta(n_j[b],n_j[b+1]) \in \set{I}$. The proof follows from contradiction.
Next, we discuss how we can make this classifier based on orientation more efficient if we can disregard routes that overlap only over a short distance.
Apart from the direct reduction in true positives, this approach will also reduce the false-positive rate of the classifiers, since some outlier route edges can be disregarded.
In order to cover the notion that there must be a minimum overlap in routes to be considered, we extend the definition of the coordination function (Definition <ref>).
A coordination function $\coordf: \set{N}_\op{c} \times \set{N}_\op{c} \rightarrow \{0,1\}$ according to Definition <ref> requires minimum distance $l_{\min}$ if the following properties hold: if for a pair $(i,j)$ we have $\coordf(i,j) = 1$, there must be a set of pairs of indices $\set{A}$ such that for all $(a,b) \in \set{A}$ it holds that $\ve{P}(n_i[a]) = \ve{P}(n_j[b])$ and $\ve{P}(n_i[a+1]) = \ve{P}(n_j[b+1])$, and $[\underline{t}_i[a], \bar{t}_i[a]] \cap [\underline{t}_j[b], \bar{t}_j[b]] \neq \emptyset$ and $[\underline{t}_i[a+1], \bar{t}_i[a+1]] \cap [\underline{t}_j[b+1], \bar{t}_j[b+1]] \neq \emptyset$. Furthermore we require $\sum\limits_{(a,b) \in \set{A}} \|\ve{P}(n_i[a]) - \ve{P}(n_i[a+1])\|_2 \geq l_{\min}$.
We adapt the orientation-based classifier (Proposition <ref>) to exclude links of a total length less than $l_{\min}$.
The approach is to calculate the fraction of route length that lies in each element of the partition. We can ignore the intersection with some elements of the partition as long as the lengths of the links whose orientation is contained in these elements sums up to a value less than $l_{\min}/2$.
Let $(i,j)$ refer to a pair of transport assignments. Let $\set{P}$ be a partition of $[0, 2\pi]$. Let $\set{I}_i \subseteq \set{P}$ and let $\bar{\set{E}}_i \subseteq \set{E}_i$, where $\set{E}_i = \{(n_i[a],n_i[a+1]): a \in \{1, \dots, N_i\}\}$, such that for all $e \in \bar{\set{E}}_i$, it holds that there exists $I \in \set{I}_i$ with $\Theta(e) \in I$ and we have $\sum\limits_{(n_1,n_2) \in \set{E}_i \setminus \bar{\set{E}}_i} \|\ve{P}(n_1) - \ve{P}(n_2)\|_2 < l_{\min}/2$. Similarly, by replacing $i$ by $j$, we define $\set{I}_j$ for transport assignment $j$. If $\set{I}_i \cap \set{I}_j = \emptyset$, then $\coordf(i,j) = 0$ with $\coordf$ according to Definition <ref>.
If $\coordf(i,j) = 1$, then we have a set of pairs of indices $\set{A}$ such that for all $(a,b) \in \set{A}$ it holds that $\ve{P}(n_i[a]) = \ve{P}(n_j[b])$ and $\ve{P}(n_i[a+1]) = \ve{P}(n_j[b+1])$. Thus it also holds that $\Theta(n_i[a], n_i[a+1]) = \Theta(n_j[b], n_j[b+1])$. Since $\set{P}$ is a partition of the image of $\Theta(\cdot)$, there is exactly one element $I \in \set{P}$ with $\Theta(n_i[a], n_i[a+1]) \in I$ and since $\Theta(n_i[a], n_i[a+1]) = \Theta(n_j[b], n_j[b+1])$, we have $\Theta(n_i[a], n_i[a+1]) \in I \Leftrightarrow \Theta(n_j[b], n_j[b+1]) \in I$.
Furthermore, we have from Definition <ref> that $\sum\limits_{(a,b) \in \set{A}} \|\ve{P}(n_i[a]) - \ve{P}(n_i[a+1])\|_2 \geq l_{\min}$.
Let $\bar{\set{A}}_i$ be a set of the indices of the head nodes of edges in $(\set{E}_i \cap \set{E}_j) \setminus \bar{\set{E}}_i$ paired with the corresponding indices in route $j$, with $\set{E}_i, \set{E}_j, \bar{\set{E}}_i$ as defined in the proposition. These are the pairs of indices of the edges in the common part of the route that are ignored in transport assignment $i$. Similarly, let $\bar{\set{A}}_j$ be the index pairs that are excluded due to transport assignment $j$. We need to show now that $\set{A}$ is not empty without the pairs in $\bar{\set{A}}_i$ and $\bar{\set{A}}_j$, or in other words, that even if the features for either route ignore up to $l_{\min}/2$ of the common part of the route, there are still edges left that let the set of classifiers indicate that the routes intersect.
We have from the assumptions made in the proposition $\sum\limits_{(a,b) \in \bar{\set{A}}_i} \|\ve{P}(n_i[a]) - \ve{P}(n_i[a+1])\|_2 < l_{\min}/2$, $\sum\limits_{(a,b) \in \bar{\set{A}}_j} \|\ve{P}(n_i[a]) - \ve{P}(n_i[a+1])\|_2 < l_{\min}/2$, and from Definition <ref> that $\sum\limits_{(a,b) \in \set{A}} \|\ve{P}(n_i[a]) - \ve{P}(n_i[a+1])\|_2 \geq l_{\min}$. Thus $\sum\limits_{(a,b) \in \set{A} \setminus (\bar{\set{A}}_j \cup \bar{\set{A}}_j)} \|\ve{P}(n_i[a]) - \ve{P}(n_i[a+1])\|_2 > 0$ and since this is a sum over positive elements, we deduce that $\set{A} \setminus (\bar{\set{A}}_j \cup \bar{\set{A}}_j) \neq \emptyset$. But then there is $I \in \set{P}$ and $(a,b) \in \set{A} \setminus (\bar{\set{A}}_j \cup \bar{\set{A}}_j)$ such that $\Theta(n_i[a], n_i[a+1]) = \Theta(n_j[b], n_j[b+1]) \in I$ and thus $\set{I}_i \cap \set{I}_j \neq \emptyset$. By contraposition it follows that $\set{I}_i \cap \set{I}_j = \emptyset \implies \coordf(i,j) = 0$.
It is possible to combine various classifiers as defined in Propositions <ref> and <ref> in various ways according to Propositions <ref> and <ref> in Section <ref>.
§ SIMULATIONS
In this section, we demonstrate in a realistic scenario the method derived in this paper. We demonstrate that the application of 6 classifiers can rule out 99 % of the transport assignment pairs, leaving only 1 % for the computationally expensive exact algorithm.
The simulation setup is as follows. The start and goal locations are sampled randomly with probability proportional to an estimate of the population density in the year 2000 (<cit.>). We limit the area to a large part of Europe.
We calculate shortest routes with the Open Source Routing Machine (<cit.>). If the route is longer than 400 kilometers, a subsection of 400 kilometers of the route is randomly selected.
The maximum speed is $v_{\max} = 80\,$km/h. We set the start times $t^\op{S}$ of half the assignments to 0 and sample the start times of the remaining assignments uniformly in an interval of 0 to 24 h. The first half is to account for assignments that are currently on the road while the other half is to account for assignments that are scheduled to depart later. The deadlines $t^\op{D}$ are set in such a way that the interval $\bar{t}[a] - \underline{t}[a] = 0.5\,h$ where $a$ is any valid index. We consider the minimum length that two assignments have to overlap to be considered for platooning, $l_{\min}$, to be 20 km.
We implemented all features and corresponding classifiers that are described in Section <ref>, i.e., interval projection (Proposition <ref>) and minimum distance orientation partition (Proposition <ref>). Note that Proposition <ref> is a special case of Proposition <ref> with $l_{\min} = 0$. For interval projection we tested vectors of the form
$\begin{bmatrix} 1 , 0 , 0 \end{bmatrix}^\transp$,
$\begin{bmatrix} 0 , 1 , 0 \end{bmatrix}^\transp$,
$\begin{bmatrix} 0 , 0 , 1 \end{bmatrix}^\transp$,
$\begin{bmatrix} 1 , 1 , 0 \end{bmatrix}^\transp$,
$\begin{bmatrix} -1 , 1 , 0 \end{bmatrix}^\transp$,
$\begin{bmatrix} -\cos(\alpha), \frac{-\sin(\alpha)}{\cos(0.278\pi )} , \frac{v_{\max} 180^\circ}{6371\pi} \end{bmatrix}^\transp$,
with $\alpha = 0, \pi/4, \dots, 7\pi/4$. The position $\ve{P}$ is expressed here as latitude and longitude and measured in degrees. The vectors parametrized by $\alpha$ are approximately orthogonal to a trajectory at maximum speed at the latitude of 50 degrees with heading angle $\alpha$ and should work well for trajectory pairs that have similar orientation, that cover the same area, and that are only separated by a small time margin. We will refer to the corresponding classifiers in the following discussion as $c_{100}, c_{010}, c_{001}, c_{110}, c_{-110}, c_{\alpha 0}, \dots, c_{\alpha 7}$ respectively. For the orientation-based classifier we use 100 equally sized cells to partition $[0, 2\pi]$. For each cell the fraction of the route distance that falls in this cell is computed. Matches up to $l_{\min}/2$ starting in ascending order of route distance contained in the cells are excluded. We will refer to this classifier as $c_{\op{o}}$.
We test $K = 1000$ transport assignments. All classifiers are evaluated in parallel. Then the sequence of classifiers that filters the most assignments at every stage is computed. The number of positives for each classifier is listed in Table <ref>. Figure <ref> shows the number of remaining pairs at each stage, the ground truth, and the sequence of classifiers for this sample. The optimization of the classifier order would typically be done when the system is designed and is to some extent specific for the exact problem setting. In a running platoon coordination system the order in which classifiers are applied would remain fixed.
We can see in Figure <ref> that two classifiers, $c_{110}$ and $c_{\alpha 7}$, combined are able to reduce the number of pairs by one order of magnitude. The first classifier, $c_{110}$, only takes into account longitude and latitude of the routes. The second one, $c_{\alpha 7}$ is orthogonal to the first one, $c_{110}$, in the plane but also takes into account timing. The third classifier, $c_{\alpha 3}$, is also of the projection type that is able to identify that a pair of assignments cannot platoon if they are geographically close but differ in timing, and it covers the opposite orientation compared to the previous classifier. The fourth classifier, $c_{100}$, covers a third direction in the plane. It is interesting to see that the fifth classifier, $c_{\op{0}}$, is the orientation-based classifier. Alone it performs much worse than the other classifiers as can be seen in Table <ref>. Two transport assignments that take the same route in opposite directions and that “meet” on the way are impossible to identify as a negative with the projection based classifiers. The orientation-based classifier might be able to achieve that. The classifier that only takes into account start and arrival time, $c_{001}$, is selected last, since most cases it rules out are already covered by the classifiers $c_{\alpha 0}, \dots, c_{\alpha 7}$ and since half the assignments start at the same time. We see that the benefit from adding more classifiers diminishes quickly as classifiers are added. All classifiers combined can reduce the number of pairs by two orders of magnitude and get within one order of magnitude from the ground truth. The false-positives are mostly very curvy routes that intersect geographically and are separated little in time in the area of the intersection. To be able to correctly identify such pairs as negatives is often not possible with the features presented in this paper. We get fairly consistent results for different runs of the simulation that are omitted here owing to space constraints.
None 499,500 $c_{-110}$ 108,403 $c_{\alpha 4}$ 134,019
$c_{100}$ 104,380 $c_{\alpha 0}$ 129,282 $c_{\alpha 5}$ 107,287
$c_{010}$ 101,542 $c_{\alpha 1}$ 103,240 $c_{\alpha 6}$ 105,883
$c_{001}$ 208,896 $c_{\alpha 2}$ 103,453 $c_{\alpha 7}$ 109,934
$c_{110}$ 98,343 $c_{\alpha 3}$ 109,626 $c_{\op{o}}$ 453,246
Number of positives for different classifiers.
This plot shows the number of remaining pairs when the classifiers are consecutively applied from left to right. The order the classifiers are chosen in a way that each stage removes as many pairs as possible. The classifier applied at each stage is indicated on the horizontal axis. The dashed line shows the ground truth from the exact algorithm.
§ CONCLUSIONS AND FUTURE WORK
We presented a method that can significantly reduce the computational effort of centrally coordinating truck platooning over large geographic areas and time intervals. With this framework we can combine different classifiers to efficiently cull the pairs of transport assignments before they are passed on to an exact algorithm that checks which transport assignments can platoon. We developed three types of classifiers and demonstrated their potential in simulation. This approach might also be useful for dynamically computing the interaction network of spatially distributed multi-agent systems.
There are various directions for future work. While this work focuses on algorithmic efficiency, it would be interesting to tune the implementation for best performance.
Furthermore, we would like to gain more insight from simulations and from theoretical analysis on how different parameters affect the effectiveness of the approach. It would be, for instance, desirable to better understand which classifiers should be chosen in which order based on the way transport assignments are generated.
|
1511.00881
|
We present detailed analytic calculations of finite-volume energy spectra,
mean field theory, as well as a systematic low-energy effective field theory
for the square lattice quantum dimer model. The analytic considerations
explain why a string connecting two external static charges in the confining
columnar phase fractionalizes into eight distinct strands with electric flux
$\frac{1}{4}$. An emergent approximate spontaneously broken $SO(2)$ symmetry
gives rise to a pseudo-Goldstone boson. Remarkably, this soft phonon-like
excitation, which is massless at the Rokhsar-Kivelson (RK) point, exists far
beyond this point. The Goldstone physics is captured by a systematic low-energy
effective field theory. We determine its low-energy parameters by matching the
analytic effective field theory with exact diagonalization results and Monte
Carlo data. This confirms that the model exists in the columnar (and not in a
plaquette or mixed) phase all the way to the RK point.
§ INTRODUCTION
Despite the extensive work on high-temperature superconductivity during the
past decades since their discovery <cit.>, understanding the mechanism
of electron or hole pairing still represents a major unsolved problem in
condensed matter physics. One of the various proposed scenarios is related to
the quantum dimer model that was introduced by Rokhsar and Kivelson in
Ref. <cit.>. It represents a simple realization of the resonating
valence bond (RVB) state, proposed by Anderson in his pioneering paper
<cit.>, and provides a possible route towards understanding
high-temperature superconductivity. Quantum dimer models have attracted a lot
of attention over the years, as they are also relevant beyond high-temperature
superconductivity, e.g., in connection with deconfined quantum criticality or
topological order. Unraveling the phase structure of both the classical and
the quantum dimer model has been the subject of many publications
These studies include dimer models on both bipartite and non-bipartite
lattices, which are defined in spatial dimensions $d \ge 2$. Quite
surprisingly, even in the case of the simple square lattice the question of
which phases are realized as a function of the Rokhsar-Kivelson (RK) parameter
$\lambda$ has been controversial. This may even be more surprising in view of
the fact that Monte Carlo simulations of quantum dimer models on the square
lattice are not affected by the sign problem. While some authors claimed that
a plaquette phase arises from a columnar phase in a first order phase
transition around $\lambda \approx 0.6$ <cit.>, other studies found
evidence for a mixed phase for $\lambda \gtrsim 0$, exhibiting features of both
the columnar and the plaquette phase <cit.>.
In a recent study <cit.>, using quantum Monte Carlo applied to dual height
variables as well as exact diagonalization, we have challenged these various views.
In particular, we pointed out that there is no evidence for a plaquette or
mixed phase in the square lattice quantum dimer model — rather the columnar
phase extends all the way to the RK point at $\lambda = 1$. Moreover, we showed
that two external static charges $\pm 2$ are confined by an electric flux
string that fractionalizes into eight strands carrying fractionalized flux
$\frac{1}{4}$. Inside these strands, which represent interfaces separating
different columnar orders, we found plaquette phase. However, the plaquette
phase only exists inside the strands and not in the bulk. Finally, as a
consequence of an approximate emergent $SO(2)$ symmetry, we found evidence for
a soft pseudo-Goldstone boson that exists in the parameter regime
$0 \lesssim \lambda < 1$, i.e., even far beyond the RK point.
In the present article we complement our previous Monte Carlo and exact
diagonalzation results with detailed analytic calculations of finite-volume
energy spectra, mean field theory, as well as a systematic low-energy
effective field theory for the pseudo-Goldstone boson. Overall, we consolidate
our previous findings that contradict the earlier views on the phase structure
of the square lattice quantum dimer model.
The paper is organized as follows. In section <ref> we define
the quantum dimer model and discuss its symmetries on the square lattice. We
then introduce height variables on the dual lattice. On the one hand, these
allow us to define order parameters that distinguish the various candidate
phases. On the other hand, the dual height variables are the basic degrees of
freedom on which our Monte Carlo simulations operate. We also perform a
systematic mean field analysis of the quantum dimer model. In Section
<ref> we investigate the finite-volume energy spectrum as a diagnostic of
the phase structure. Section <ref> is dedicated to the low-energy
effective field theory for the soft pseudo-Goldstone mode and the
corresponding rotor spectrum. In section <ref> we present
exact diagonalization results and use them to estimate some low-energy
parameters of the effective field theory. In Section <ref> we present new
Monte Carlo data for the confining strings in the columnar phase. Finally, in
section <ref> we present our conclusions. Appendix A summarizes
the symmetry properties of the relevant order parameters.
§ MODEL AND OBSERVABLES
In this section, we consider the quantum dimer model and discuss its
symmetries on the square lattice. We then define height variables on the dual
lattice, which are the basic degrees of freedom in our Monte Carlo
simulations. They also serve to construct order parameters that signal which
phase is realized. Finally, we perform a systematic mean-field analysis with the
intention to gain qualitative insight into this question.
§.§ Model
The Hamiltonian of the quantum dimer model coincides with the Hamiltonian of
the $(2+1)$-d $U(1)$ quantum link model <cit.>. However, the
corresponding Gauss law is realized differently. The Hamiltonian of both the
$U(1)$ quantum link model and the quantum dimer model takes the form
\begin{equation}
\label{Hamiltonian}
H = - J \sum_{\Box} \left[U_\Box + U_\Box^\dagger - \lambda
(U_\Box + U_\Box^\dagger)^2\right].
\end{equation}
In the above Hamiltonian, the quantity
$U_\Box = U_{wx} U_{xy} U_{zy}^\dagger U_{wz}^\dagger$ represents a plaquette
operator expressed in terms of quantum links $U_{xy}$ that connect the
nearest-neighbor sites $x$ and $y$ on the square lattice. A $U(1)$ quantum link
$U_{xy} = S_{xy}^+$ is a raising operator of the electric flux
$E_{xy} = S_{xy}^3$, which is built from a quantum spin $\frac{1}{2}$ associated
with the link $xy$. In the $U(1)$ quantum link model, each link has two
possible states characterized by electric flux $\pm \frac{1}{2}$, represented
pictorially by arrows as shown in Fig. <ref>. A
typical flux configuration of the $U(1)$ quantum link model is depicted in the
same figure.
Definition of flux states (left) and typical flux configuration on
the square lattice U(1) quantum link model (right).
The results of applying the Hamiltonian of Eq. (<ref>) to
some plaquette flux states. Here $H_1$ and $H_{2}$ represent the terms
in Eq. (<ref>) proportional to $J$ and $J \lambda$, respectively.
When the Hamiltonian acts on other plaquette configurations (which are not
shown explicitly) the result vanishes.
Applying the Hamiltonian of Eq. (<ref>) to a plaquette flux state
leads to the results shown in Fig. <ref>. In summary, the first
contribution to the Hamiltonian (<ref>), proportional to the
parameter $J$, flips a loop of flux that winds around a plaquette. Flux states
that do not correspond to closed flux loops are referred to as non-flippable
plaquettes, which are annihilated by the Hamiltonian. On the other hand, the
second contribution to the Hamiltonian (<ref>), proportional to the
RK parameter $\lambda$, counts the plaquettes that are flippable.
Notice that the configurations of the
square lattice quantum dimer model are characterized in terms of variables
$D_{xy} \in\{0,1\}$ which signal whether a dimer is present or absent on the
link that connects two neighboring sites $x$ and $y$. In addition, the
electric flux variables $E_{xy}$ can be expressed through the dimer variables
$D_{xy}$ as
\begin{equation}
E_{xy} = (-1)^{x_1+x_2}(D_{xy}-\frac{1}{2}) .
\end{equation}
This mapping between a dimer and a flux configuration of the quantum dimer
model is illustrated in Fig. <ref>.
Mapping between dimer and flux configurations.
Notice that, in the $U(1)$ quantum link model, the physical state
$|\psi\rangle$ satisfies
\begin{equation}
\label{GaussQLM}
G_x |\psi \rangle = 0,
\end{equation}
where the quantity
\begin{equation}
\label{generators}
G_x = \sum_{i}(E_{x,x+\hat{i}}-E_{x-\hat{i},x} )
\end{equation}
commutes with the Hamiltonian and describes an infinitesimal $U(1)$ gauge
transformation. Here $\hat i$ is the unit-vector in the $i$-direction.
Eq. (<ref>) represents the Gauss law for the $U(1)$ quantum link
model. In the quantum dimer model, using the connection between the electric
flux and the dimer variables, one has
\begin{equation}
G_x = (-1)^{x_1+x_2} \sum_i (D_{x,x+\hat i} + D_{x-\hat i,x}) = (-1)^{x_1+x_2}.
\end{equation}
In other words, the dimer covering constraint implies that the quantum dimer
model is characterized by background electric charges $\pm 1$ that are
arranged in a staggered pattern. Accordingly, physical states in the quantum
dimer model satisfy
\begin{equation}
G_x |\Psi\rangle = (-1)^{x_1+x_2} |\Psi\rangle .
\end{equation}
§.§ Symmetries
The quantum dimer model on the square lattice exhibits various symmetries. We
first have a continuous $U(1)$ gauge symmetry and a global $U(1)^2$ center
symmetry. The latter is associated with “large" gauge transformations
<cit.>. The model also has various discrete global symmetries. These
include translations by one lattice vector followed by charge conjugation
($CT_x$ and $CT_y$), which are equivalent to ordinary translations of the
dimers $D_{xy}$. Note that charge conjugation changes the sign of all electric
flux variables. It is important to point out that, in contrast to the quantum
link model, the transformations $T_x, T_y$ and $C$ individually are not
symmetries of the quantum dimer model because they are explicitly violated by
the Gauss law. Furthermore we have 90 degrees rotations around a plaquette
corner ($O$), 90 degrees rotations around a plaquette center followed by
charge conjugation ($CO'$), and finally, reflections on the $x$- and $y$-axes
($R_x$ and $R_y$). Below we will construct order parameters that will help us
to determine which phases are realized in the square lattice quantum dimer
model. It is then crucial to know how these different order parameters
transform under the various symmetries (see Sec. <ref> and
Appendix <ref>).
§.§ Dual Height Variables
In this subsection we introduce height variables that reside on the dual
lattice. This height representation of the quantum dimer model is essential in
our approach. It allows us, on the one hand, to design cluster and Metropolis
algorithms that operate in the space of these height variables and, on the
other hand, to construct order parameters to unambiguously distinguish the
various phases.
The four dual sublattices $A, B, C, D$ used in the construction of
the height variables $h^{A,B,C,D}$.
As illustrated in Fig. <ref>, in the case of the square
lattice quantum dimer model, we define four dual sublattices $A, B, C$, and
$D$, which consist of the points
\begin{equation}
\tilde{x} = (x_1+\frac{1}{2},x_2+\frac{1}{2}) .
\end{equation}
Each of the dual sublattices $X$ carries dual height variables $h^X$ that take
the values
\begin{equation}
h^{A,D}_{\widetilde x} = 0,1 , \qquad h^{B,C}_{\widetilde x} = \pm \frac{1}{2} .
\end{equation}
When defining the height variables on the dual lattice, we will encounter an additional
complication compared to the U(1) quantum link model, which is due to the fact
that the Gauss law is realized differently in these two models. This further
complication requires the introduction of so-called Dirac-strings, in order to
consistently relate the HV $h^X_{\widetilde x}$ with the electric fluxes
$E_{x,y}$. These Dirac-strings are located in a staggered fashion on the
vertical links and are denoted by empty squares on the links (see
Fig. <ref>).
The quantities $h^{A,B,C,D}$, residing at the sites of a dual sublattice, are
related to the electric flux variables on the links by
\begin{eqnarray}
E_{x,x+\hat 1} &=& [h^X_{\widetilde x} - h^{X'}_{\widetilde x - \hat 2}] \,
\mbox{mod} 2 = \pm \frac{1}{2},\nonumber \\
E_{x,x+\hat 2} &=& (-1)^{x_1+x_2} [h^X_{\widetilde x} - h^{X'}_{\widetilde x - \hat 1}]
\, \mbox{mod} 2 = \pm \frac{1}{2}, \\
&& X,X' \in \{A,B,C,D\}. \nonumber
\end{eqnarray}
Note that whenever $(-1)^{x_1+x_2}= -1$, it indicates the presence
of a Dirac-string on the relevant vertical link. The corresponding height
representation and the flux
representation for a columnar dimer configuration is illustrated in
Fig. <ref>. Beside the height and flux variables, we
have also marked positive and negative background charges (filled and empty
circles) as well as the Dirac-strings (empty squares).
The mapping between the dimer configuration and the corresponding
flux and height representation for a columnar quantum dimer configuration.
It should be noted that this construction of height variables is new and fundamentally
different from other height variables definitions that have been introduced in the
literature, in particular, the one described in the review paper of Moessner
and Raman <cit.>. For instance, in our construction the
height variables take only four values: 0, 1, $\pm 1/2$. On the other hand,
with the conventional method they are always labelled with integers and can
take many more values. Our height variables take much fewer values than the
conventional ones, because they provide an exact representation of the dimer
model Hilbert space.
§.§ Order Parameters and Candidate Phases
In this subsection we review four order parameters in terms of the height
variables that we introduced in <cit.>. Each of the four candidate
phases — staggered, columnar, plaquette or mixed — can then be
unambiguously identified by the specific values these order parameters take in
the different phases.
Let us first discuss the various phases that have been established or
conjectured for the square lattice quantum dimer model. Intuitively, in the
limit $\lambda \rightarrow -\infty$, the system maximizes the number of
flippable plaquettes. On the square lattice, such a state can be obtained by
arranging the dimers in a columnar pattern as depicted in
Fig. <ref>(a). Note that we are dealing with four-fold degeneracy:
the four columnar configurations are related by translations or rotations.
On the other hand, in the opposite limit $\lambda \rightarrow +\infty$, the
system minimizes the number of flippable plaquettes. On the square lattice,
such a state can be obtained by arranging the dimers in a staggered pattern,
shown in Fig. <ref>(c). The four degenerate staggered phases are
related by discrete transformations.
Another candidate phase is the so-called plaquette arrangement of dimers,
which is also four-fold degenerate, and is illustrated in
Fig. <ref>(b). In this phase, pairs of parallel dimers oriented in
both possible directions resonate on the plaquettes belonging to one of the
four sublattices A, B, C, D.
Finally, on the square lattice, another conjectured phase is the so-called
mixed phase which is eight-fold degenerate and corresponds to a superposition
of quantum dimer states, sharing features of both the columnar and the
plaquette phase.
Apart from the established columnar and staggered phases in the limits
$\lambda \rightarrow -\infty$ and $\lambda \rightarrow +\infty$, respectively,
the question of which phases are realized between these two points of
reference — and what type of associated phase transitions might exist —
has remained controversial.
Established and conjectured phases for the square
lattice quantum dimer model: (a) Columnar, (b) plaquette, (c) staggered order
on the dual sublattices $A$, $B$, $C$, and $D$. (d) Phase diagram for the
square lattice quantum dimer model as a function of the RK parameter
$\lambda$: three different scenarios <cit.>.
In <cit.> we have challenged the various conflicting scenarios that
have been proposed in earlier studies <cit.> and are depicted in
Fig. <ref>d. An important point of reference is the RK point
($\lambda = 1$) where the model is exactly solvable. Away from the RK point, the
situation becomes less clear. Using Green's function Monte Carlo simulations,
the author of Ref. <cit.> concludes that on the square lattice there is
a phase transition between the columnar and plaquette phase around
$\lambda \approx 0.6$ (scenario 2 in Fig. <ref>d). However,
this view is not shared by Ref. <cit.> which favors a mixed phase for
$\lambda \gtrsim 0$ according to their Green's function Monte Carlo analysis
(scenario 3 in Fig. <ref>d). Based on our new order
parameters and a novel Monte Carlo technique we concluded that the system
exists in a columnar phase all the way up to the RK point (scenario 1 in
Fig. <ref>d). In particular, we found no evidence for plaquette or
mixed phases.
For completeness, we now review the four order parameters whose construction
is based on the dual height representation. Remember that we have two sets of
height variables, the first one associated with the even sublattices A and D,
the second one related to the odd sublattices B and C (see
Fig. <ref>).
We first define four auxiliary order parameters $M_A, M_B, M_C, M_D$ as
\begin{equation}
M_X = \sum_{\widetilde x \in X} s^X_{\widetilde x} h^X_{\widetilde x},
\end{equation}
\begin{eqnarray}
s^A_{\widetilde x} & = & s^C_{\widetilde x} = (-1)^{({\widetilde x}_1 + \frac{1}{2})/2} ,
\qquad \text{if} \quad {\widetilde x}_1 + \frac{1}{2}
\quad \text{even} ,
\nonumber \\
s^B_{\widetilde x} & = & s^D_{\widetilde x} = (-1)^{({\widetilde x}_1 - \frac{1}{2})/2} ,
\qquad \text{if} \quad {\widetilde x}_1 + \frac{1}{2}
\quad \text{odd} .
\end{eqnarray}
Remember that the height variables on the various sublattices take the values
\begin{equation}
h^{A,D}_{\widetilde x} = 0,1 , \qquad h^{B,C}_{\widetilde x} = \pm \frac{1}{2} .
\end{equation}
We then form the linear combinations,
\begin{eqnarray}
\label{OrderParametersMij}
M_{11} & = & M_A - M_B - M_C + M_D = M_1 \cos\varphi_1, \nonumber \\
M_{22} & = & M_A + M_B - M_C - M_D = M_1 \sin\varphi_1, \nonumber \\
M_{12} & = & M_A - M_B - M_C - M_D = M_2 \cos\varphi_2, \nonumber \\
M_{21} & = & - M_A + M_B - M_C - M_D = M_2 \sin\varphi_2,
\end{eqnarray}
which define the order parameters $M_{11}, M_{12}, M_{21}, M_{22}$ that are more
appropriate to distinguish the phases. The two angles $\varphi_1$ and
$\varphi_2$ define the angle
\begin{equation}
\varphi = \frac{1}{2}(\varphi_1 + \varphi_2 + \frac{\pi}{4}) .
\end{equation}
In the columnar phase this angle amounts to
$\varphi = 0 \, \mbox{mod} \frac{\pi}{4}$, while in the plaquette phase it
takes the value $\varphi = \frac{\pi}{8} \, \mbox{mod} \frac{\pi}{4}$. Note
that the order parameter values $\pm (M_A,M_B,M_C,M_D)$, and therefore
$\varphi$ and $\varphi + \pi$, represent the same physical dimer
configuration, because a dimer configuration is invariant under a shift of the
height variables,
\begin{equation}
h^X_{\widetilde x}(t)' = [h^X_{\widetilde x}(t) + 1] \, \mbox{mod} 2 .
\end{equation}
As illustrated in Fig. <ref>, each of the four phases —
columnar, plaquette, mixed, staggered — is characterized by its specific
order parameter pattern. While there are four columnar phases (1,2,3,4) and
four plaquette phases (A,B,C,D), there are eight realizations of the mixed
phase (A1,A2,B2,B3,C3,C4,D4,D1). The mixed phases share features of both the
columnar and the plaquette phases. For instance, in a hypothetical phase
transition between a columnar and a mixed phase, a peak in the order parameter
distribution of the columnar phase would split into two individual peaks: the
columnar peak 1 would split into the peaks D1 and A1 referring to the mixed
phase, etc. On the other hand, in a hypothetical phase transition between a
mixed and a plaquette phase, two peaks in the order parameter distribution of
the mixed phase would merge pairwise into one peak referring to the plaquette
phase: the mixed peaks A1 and A2 would merge into the plaquette peak A, etc.
As we will elaborate in more detail below, in our numerical simulations no
such splitting or merging of peaks is detected.
The four candidate phases — columnar, plaquette,
mixed, staggered — can unambiguously be distinguished by their
characteristic order parameter distributions <cit.>.
For completeness, in Appendix <ref> we show how the four order
parameters transform under the symmetries $CT_x, CT_y, O, CO', R_x$, and
$R_y$ of the square lattice quantum dimer model and how the different
symmetries $CT_x, CT_y, O, CO', R_x, R_y$ act on the columnar, plaquette,
and mixed phases, respectively.
It should be pointed out that the specific phases that are actually realized,
depend on both the lattice geometry and the spatial dimension. While we
restrict ourselves to the (two-dimensional) square lattice, more complicated
phases are indeed possible on other lattices. On non-bipartite lattices, in
two or higher spatial dimensions, a $Z_2$ resonance valence bond liquid phase
is formed <cit.>. This is a phase with $Z_2$ topological
order, characterized by four-fold degenerate gapped ground states in the case
of two-dimensional lattices with periodic boundary conditions. In particular,
it has nontrivial excitations and represents a liquid phase because all dimer
correlations decay exponentially. Another example is the U(1) resonating
valence bond liquid phase that is possible on bipartite lattices and in
spatial dimensions three or higher <cit.>. Even more
complex phases include e.g. the $\sqrt{12} \times \sqrt{12}$ phase, which
appears to be realized on the triangular lattice <cit.>.
§.§ Mean Field Theory
Using mean field theory, in this subsection we address the question which
phases in the square lattice quantum dimer model may be realized in the
vicinity of the RK point. Following the Ginsburg-Landau-Wilson paradigm, we
formulate an effective action for the system in terms of the order parameters
$M_{11}, M_{12}, M_{21}$, and $M_{22}$, defined in Eq. (<ref>).
The most general expression up to quartic order that respects all the
symmetries of the underlying quantum dimer model, is given by
\begin{eqnarray}
\label{fullPotential}
&&V = \mu_1 O_1 + \mu_2 O_2 + \nu_0 O_1 O_2 +
\sum_{i=1}^5 \nu_i |O_i|^2, \nonumber \\
&&O_1 = M_{11}^2 + M_{22}^2 + M_{12}^2 + M_{21}^2, \nonumber \\
&&O_2 = M_{11} M_{12} - M_{11} M_{21} + M_{22} M_{12} + M_{22} M_{21}, \nonumber \\
&&O_3 = M_{11}^2 + M_{22}^2 - M_{12}^2 - M_{21}^2, \nonumber \\
&&O_4 = M_{11} M_{12} + M_{11} M_{21} - M_{22} M_{12} + M_{22} M_{21}, \nonumber \\
&&O_5 = M_{11} M_{22} + i M_{12} M_{21}.
\end{eqnarray}
We have two quadratic and six quartic operators, i.e., a total of eight
parameters $\mu_1, \mu_2, \nu_0, \dots, \nu_5$. Each of the terms in the
effective potential is invariant under the discrete symmetries, i.e., under
$O, CO', R_x, R_y, CT_x$ and $CT_y$.
We perform a systematic analysis of the minima of the potential $V$ in the
infinitesimal neighborhood of the staggered phase which begins at the RK
point, and is characterized by $M_{11}$ = $M_{12}$ = $M_{21}$ = $M_{22}$ = 0.
Since the staggered phase corresponds to a stable
minimum, we first diagonalize the mass squared matrix $\mathbf{M}$
\begin{equation}
\mu_1 O_1 + \mu_2 O_2 = (M_{11}, M_{12}, M_{21}, M_{22}) \, \mathbf{M} \,
{(M_{11}, M_{12}, M_{21}, M_{22})}^T
\end{equation}
near this point. All eigenvalues turn out to be positive if the two conditions
\begin{equation}
\mu_1 + \frac{\mu_2}{\sqrt{2}} > 0 , \quad \mu_1 - \frac{\mu_2}{\sqrt{2}} > 0 ,
\end{equation}
are satisfied. Assuming $\mu_1, \mu_2 > 0$, we obtain two zero eigenvalues
if $\mu_2=\sqrt{2}\mu_1$. The corresponding eigenvectors $v_1$ and $v_2$
define the $xy$-plane of vectors $v$ parametrized by
\begin{eqnarray}
\label{flatDirections}
\left(\begin{array}{c}
\frac{x}{\sqrt{2}} \\
\mbox{$\frac{1}{2}$} (y-x) \\
\mbox{$\frac{1}{2}$} (y+x) \\
-\frac{y}{\sqrt{2}} \end{array} \right) .
\end{eqnarray}
This plane corresponds to the flat directions in which the staggered phase is
about to become unstable. Let us therefore evaluate the quartic potential
along these flat directions. The calculation shows that the potential can be
reduced to the simple form
\begin{eqnarray}
V(x,y) & = & (\mu_1 - \frac{\mu_2}{\sqrt{2}}) (x^2 + y^2)
+ \Big( -\frac{\nu_0}{\sqrt{2}} + \nu_1 + \frac{\nu_2}{2}
+ \frac{\nu_5}{16} \Big) {(x^2 + y^2)}^2 \nonumber \\
& = & \mu (x^2 + y^2) + \nu {(x^2 + y^2)}^2 .
\end{eqnarray}
The minima of the potential $V$ form a circle of radius $r=\sqrt{x^2 + y^2}$
with $r^2=-\mu/2\nu$. Parametrizing the vacuum circle by an angle $\phi$
as $x=r \cos \phi,y= r \sin \phi$, we get
\begin{eqnarray}
\label{minCircle}
\left(\begin{array}{c}
\frac{r}{\sqrt{2}} \cos\phi \\
\frac{r}{2} (\sin\phi - \cos\phi) \\
\frac{r}{2} (\sin\phi + \cos\phi) \\
- \frac{r}{\sqrt{2}} \sin\phi \end{array} \right).
\end{eqnarray}
We now derive general conditions for the minima of the full potential $V$
displayed in Eq. (<ref>). After a lenghty, but
otherwise trivial calculation, for the columnar phase, characterized
by $M_{22}=0$ and $M_{21} = -M_{12}$, the potential at a columnar minimum amounts
\begin{equation}
\label{minColumnar}
V(M_{11},M_{12},-M_{12},0) = \frac{\mu_1}{2} (M_{11}^2 + 2 M_{12}^2)
+ \mu_2 M_{11} M_{12} .
\end{equation}
An analogous calculation for the plaquette phase, characterized by
$M_{12} = M_{11}$ and $M_{21} = -M_{22}$, shows that the potential at a plaquette
minimum corresponds to
\begin{equation}
\label{minPlaquette}
V(M_{11},M_{11},-M_{22},M_{22}) = \mu_1 (M_{11}^2 + M_{22}^2) + \frac{\mu_2}{2}
(M_{11}^2 - M_{22}^2 + 2 M_{11} M_{22}) .
\end{equation}
However, it turns out that these two points, along with the six additional
points that correspond to the other columnar and plaquette phases, all have
the same energy on the circle of minima, Eq. (<ref>). Hence, in
order to decide which phase — columnar or plaquette — is in fact favored,
we have to perturb around these minima.
A stability analysis shows that there are indeed unstable directions,
associated with negative eigenvalues of the mass squared matrix of the second
derivatives. In fact, both the columnar and the plaquette phase can be
associated with negative eigenvalues and the energy of both phases can be
lowered by proceeding into the unstable directions. However, the relative
energies of the phases reached in this way are very sensitive to the parameters
$\nu_i$ that are unknown. The mean-field analysis hence does not lead to a
conclusive answer of which phase — columnar or plaquette — is preferred
near the RK point. Symmetries alone do not favor one of these two candidate
phases over the other. We are thus dealing with a truly dynamical question
which has to be explored with more elaborate methods such as Monte Carlo
simulations (see below).
§ LOW-ENERGY SPECTRUM IN FINITE VOLUME
In this section, we consider the lowest states in the finite-volume energy
spectrum associated with the columnar, plaquette, and mixed phases,
respectively. This will be useful for identifying the phase structure based on
numerical results obtained by exact diagonalization studies.
§.§ Low-Energy Spectrum in the Columnar Phase
Let us first consider the finite-volume energy spectrum in the columnar phase.
The four columnar phases give rise to four almost degenerate eigenstates,
which can be chosen as simultaneous eigenstates of the 90 degrees rotation $O$
with eigenvalues $+1, -i, +i, -1$ as
\begin{eqnarray}
|+1 \rangle & = & \mbox{$\frac{1}{2}$} (|1\rangle + |2\rangle + |3\rangle
+ |4\rangle) , \nonumber \\
|-i \rangle & = & \mbox{$\frac{1}{2}$} (|1\rangle + i |2\rangle - |3\rangle
- i |4\rangle) , \nonumber \\
|+i \rangle & = & \mbox{$\frac{1}{2}$} (|1\rangle - i |2\rangle - |3\rangle
+ i |4\rangle) , \nonumber \\
|-1 \rangle & = & \mbox{$\frac{1}{2}$} (|1\rangle - |2\rangle + |3\rangle
- |4\rangle) ,
\end{eqnarray}
\begin{eqnarray}
O \, |+1 \rangle & = & |+1 \rangle , \quad O \, |-i \rangle = -i |-i \rangle ,
\nonumber \\
O \, |+i \rangle & = & i |+i \rangle , \quad O \, |-1 \rangle = - |-1 \rangle .
\end{eqnarray}
Under the other discrete symmetries, these states transform as
\begin{eqnarray}
CO' \, |+1 \rangle & = & |+1 \rangle , \quad CO' \, |-i \rangle = -i |+i
\rangle , \nonumber \\
CO' \, |+i \rangle & = & i |-i \rangle , \quad CO' \, |-1 \rangle = - |-1
\rangle , \nonumber \\
CT_x \, |+1 \rangle & = & |+1 \rangle , \quad CT_x \, |-i \rangle = |+i
\rangle , \nonumber \\
CT_x \, |+i \rangle & = & |-i \rangle , \quad CT_x \, |-1 \rangle = |-1
\rangle , \nonumber \\
CT_y \, |+1 \rangle & = & |+1 \rangle , \quad CT_y \, |-i \rangle = - |+i
\rangle , \nonumber \\
CT_y \, |+i \rangle & = & - |-i \rangle , \quad CT_y \, |-1 \rangle = |-1
\rangle.
\end{eqnarray}
Besides $|\pm 1 \rangle$, we can also construct linear combinations of
$|\pm i \rangle$ which are eigenstates of $CT_x$ and $CT_y$ such that
\begin{eqnarray}
&&CT_x |+1 \rangle = |+1 \rangle, \nonumber \\
&&CT_x \frac{1}{\sqrt{2}}(|+i \rangle \pm |-i \rangle) = \pm
\frac{1}{\sqrt{2}}(|+i \rangle \pm |-i \rangle), \nonumber \\
&&CT_x |-1 \rangle = |-1 \rangle, \nonumber \\
&&CT_y |+1 \rangle = |+1 \rangle, \nonumber \\
&&CT_y \frac{1}{\sqrt{2}}(|+i \rangle \pm |-i \rangle) = \mp
\frac{1}{\sqrt{2}}(|+i \rangle \pm |-i \rangle), \nonumber \\
&&CT_y |-1 \rangle = |-1 \rangle.
\end{eqnarray}
This implies that in the columnar phase, in a finite volume there are four
almost degenerate ground states with $(CT_x,CT_y)$ quantum numbers
$(+,+)$, $(+,-)$, $(-,+)$, $(+,+)$.
These four states are eigenstates of a reduced transfer matrix
\begin{eqnarray}
\label{transition}
T = \exp(- \beta H)&=&\left(\begin{array}{cccc}
A & B & C & B \\
B & A & B & C \\
C & B & A & B \\
B & C & B & A \end{array} \right) .
\end{eqnarray}
Here, $A, B, C$ are transition amplitudes connecting the various phases. The
corresponding transfer matrix eigenvalues are
\begin{eqnarray}
\exp(-\beta E_{+1}) & = & A + 2B +C , \nonumber \\
\exp(-\beta E_{\pm i}) & = & A - C , \nonumber \\
\exp(-\beta E_{-1}) & = & A - 2B +C .
\end{eqnarray}
Notice only three transition amplitudes appear in
Eq. (<ref>). This is because the transitions from $|+1 \rangle$ to
$|i \rangle$ and $|+1 \rangle$ to $|-i \rangle$ are of the same type.
A pictorial representation for each transition amplitude is depicted in
Fig. <ref>. Using a dilute instanton gas approximation,
one can derive analytic
expressions for the transfer matrix elements $A, B, C$. There are instantons
that represent tunneling events between the phases 1 or 3 to 2 or 4. These
instantons have a Boltzmann weight
$\delta_{\perp} \exp(- \alpha_{\perp} L_x L_y)$. In addition, there are instantons
connecting the phases 1 with 3, as well as 2 with 4. These have a Boltzmann
weight $\delta_{\parallel} \exp(- \alpha_{\parallel} L_x L_y)$. The factors
$\delta_{\perp}$ and $\delta_{\parallel}$ describe capillary wave fluctuations of
the instantons. Denoting the free energy in a bulk phase by $f$, an additional
Boltzmann factor $\exp(-\beta f L_x L_y)$ arises as well. The explicit
calculation for $A, B, C$ then yields the following expressions for the
exponentially small energy gaps,
\begin{eqnarray}
E_{\pm i} - E_{+1} & = & 2 \delta_{\perp} \exp(- \alpha_{\perp} L_x L_y)
+ 2 \delta_{\parallel} \exp(- \alpha_{\parallel} L_x L_y) , \nonumber \\
E_{-1} - E_{+1} & = & 4 \delta_{\perp} \exp(- \alpha_{\perp} L_x L_y) .
\end{eqnarray}
In this calculation we have assumed that the interfaces that correspond to a
1-3 (or 2-4) instanton with an interface tension $\alpha_{\parallel}$ are not
completely wet by the other phases 2, 4 (or 1, 3). This assumption implies
$\alpha_{\parallel} < 2 \alpha_{\perp}$. Antonov's rule <cit.> excludes
$\alpha_{\parallel} > 2 \alpha_{\perp}$, because interfaces with tension
$\alpha_{\parallel}$ would then be unstable against the decay into two
interfaces with tension $\alpha_{\perp}$. This is the situation of complete
wetting. Interfaces with tension $\alpha_{\parallel}$ then simply do not exist
and the corresponding equations turn into
\begin{eqnarray}
E_{\pm i} - E_{+1} & = & 2 \delta_{\perp} \exp(- \alpha_{\perp} L_x L_y) ,
\nonumber \\
E_{-1} - E_{+1} & = & 4 \delta_{\perp} \exp(- \alpha_{\perp} L_x L_y)
= 2 (E_{\pm i} - E_{+1}) .
\end{eqnarray}
Equidistant level spacings are characteristic for complete wetting. At least
for $\lambda \to -\infty$ one indeed expects complete wetting.
A pictorial representation of the transition amplitudes
appearing in Eq. (<ref>). The numbers 1, 2, 3, 4 represent
the four columnar phases.
§.§ Low-Energy Spectrum in the Plaquette Phase
We now consider the finite-volume energy spectrum in the plaquette phase.
Similar to the analysis in the columnar phase, we define the four plaquette
eigenstates as
\begin{eqnarray}
|+1 \rangle' & = & \mbox{$\frac{1}{2}$} (|A\rangle + |B\rangle + |C\rangle
+ |D\rangle) , \nonumber \\
|-i \rangle' & = & \mbox{$\frac{1}{2}$} (|A\rangle + i |B\rangle - |C\rangle
- i |D\rangle) , \nonumber \\
|+i \rangle' & = & \mbox{$\frac{1}{2}$} (|A\rangle - i |B\rangle - |C\rangle
+ i |D\rangle) , \nonumber \\
|-1 \rangle' & = & \mbox{$\frac{1}{2}$} (|A\rangle - |B\rangle + |C\rangle
- |D\rangle) .
\end{eqnarray}
Under the discrete symmetries they transform as
\begin{eqnarray}
O \, |+1 \rangle' & = & |+1 \rangle' , \quad O \, |-i \rangle'
= -i |-i \rangle' , \nonumber \\
O \, |+i \rangle' & = & i |+i \rangle' , \quad O \, |-1 \rangle'
= - |-1 \rangle', \nonumber \\
CO' \, |+1 \rangle' & = & |+1 \rangle' , \quad CO' \, |-i \rangle'
= - |+i \rangle' , \nonumber \\
CO' \, |+i \rangle' & = & - |-i \rangle' , \quad CO' \, |-1 \rangle'
= |-1 \rangle' , \nonumber \\
CT_x \, |+1 \rangle' & = & |+1 \rangle' , \quad CT_x \, |-i \rangle'
= -i |+i \rangle' , \nonumber \\
CT_x \, |+i \rangle' & = & i |-i \rangle' , \quad CT_x \, |-1 \rangle'
= - |-1 \rangle' , \nonumber \\
CT_y \, |+1 \rangle' & = & |+1 \rangle' , \quad CT_y \, |-i \rangle'
= i |+i \rangle' , \nonumber \\
CT_y \, |+i \rangle' & = & -i |-i \rangle' , \quad CT_y \, |-1 \rangle'
= - |-1 \rangle'.
\end{eqnarray}
Besides $|\pm 1 \rangle'$, we can also construct linear combinations of
$|\pm i \rangle'$ which are eigenstates of $CT_x$ and $CT_y$ such that
\begin{eqnarray}
&&CT_x |+1 \rangle' = |+1 \rangle', \nonumber \\
&&CT_x \frac{1}{\sqrt{2}}(|+i \rangle' \pm i |-i \rangle') = \pm
\frac{1}{\sqrt{2}}(|+i \rangle' \pm i |-i \rangle'), \nonumber \\
&&CT_x |-1 \rangle' = - |-1 \rangle', \nonumber \\
&&CT_y |+1 \rangle' = |+1 \rangle', \nonumber \\
&&CT_y \frac{1}{\sqrt{2}}(|+i \rangle' \pm i |-i \rangle') = \mp
\frac{1}{\sqrt{2}}(|+i \rangle' \pm i |-i \rangle'), \nonumber \\
&&CT_y |-1 \rangle' = - |-1 \rangle'.
\end{eqnarray}
Like in the columnar phase, in the plaquette phase there are four almost
degenerate ground states. However, in contrast to the columnar phase, their
$(CT_x,CT_y)$ quantum numbers are $(+,+)$, $(+,-)$, $(-,+)$, $(-,-)$. In
particular, the quantum numbers of the third excited state are different in
the two cases.
The calculation of the energy spectrum in the plaquette phase is the same as
in the columnar phase and shall not be repeated here.
§.§ Low-Energy Spectrum in the Mixed Phase
Finally we discuss the lowest states in the finite-volume energy spectrum of
the mixed phase. Here we have a total of eight states that become degenerate
in the infinite volume limit, with exponentially small gaps at finite volume.
Let us construct the states as eigenstates of an explicitly broken and thus
only approximate $\Z(8)$ symmetry. Defining $z = \exp(2 \pi i/8)$, we obtain
\begin{eqnarray}
|+1 \rangle'' & = & \frac{1}{\sqrt{8}} \Big( |A_1\rangle + |A_2\rangle
+ |B_2\rangle + |B_3\rangle
+ |C_3\rangle + |C_4\rangle + |D_4\rangle + |D_1\rangle \Big) , \nonumber \\
|z \rangle'' & = & \frac{1}{\sqrt{8}} \Big( |A_1\rangle + z |A_2\rangle
+ i |B_2\rangle + z^3 |B_3\rangle
- |C_3\rangle + z^5 |C_4\rangle - i |D_4\rangle + z^7 |D_1\rangle \Big) ,
\nonumber \\
|+i \rangle'' & = & \frac{1}{\sqrt{8}} \Big( |A_1\rangle + i |A_2\rangle
- |B_2\rangle -i |B_3\rangle
+ |C_3\rangle + i |C_4\rangle - |D_4\rangle -i |D_1\rangle \Big) , \nonumber \\
|z^3 \rangle'' & = & \frac{1}{\sqrt{8}} \Big( |A_1\rangle + z^3 |A_2\rangle
- i |B_2\rangle + z |B_3\rangle
- |C_3\rangle + z^7 |C_4\rangle + i |D_4\rangle + z^5 |D_1\rangle \Big) ,
\nonumber \\
|-1 \rangle'' & = & \frac{1}{\sqrt{8}} \Big( |A_1\rangle - |A_2\rangle
+ |B_2\rangle - |B_3\rangle
+ |C_3\rangle - |C_4\rangle + |D_4\rangle - |D_1\rangle \Big) , \nonumber \\
|z^5 \rangle'' & = & \frac{1}{\sqrt{8}} \Big( |A_1\rangle + z^5 |A_2\rangle
+ i |B_2\rangle + z^7 |B_3\rangle
- |C_3\rangle + z |C_4\rangle - i |D_4\rangle + z^3 |D_1\rangle \Big) ,
\nonumber \\
|-i \rangle'' & = & \frac{1}{\sqrt{8}} \Big( |A_1\rangle -i |A_2\rangle
- |B_2\rangle +i |B_3\rangle
+ |C_3\rangle -i |C_4\rangle - |D_4\rangle + i |D_1\rangle \Big) , \nonumber \\
|z^7 \rangle'' & = & \frac{1}{\sqrt{8}} \Big( |A_1\rangle + z^7 |A_2\rangle
- i |B_2\rangle + z^5 |B_3\rangle
- |C_3\rangle + z^3 |C_4\rangle + i |D_4\rangle + z |D_1\rangle \Big) .\nonumber \\
\end{eqnarray}
This gives rise to the following transformation rules,
\begin{eqnarray}
O \, |+1 \rangle'' & = & |+1 \rangle'' , \quad O \, |z \rangle''
= -i |z \rangle'' , \nonumber \\
O \, |+i \rangle'' & = & - |+i \rangle'' , \quad O \, |z^3 \rangle''
= i |z^3 \rangle'', \nonumber \\
O \, |-1 \rangle'' & = & |-1 \rangle'' , \quad O \, |z^5 \rangle''
= -i |z^5 \rangle'' , \nonumber \\
O \, |-i \rangle'' & = & - |-i \rangle'' , \quad O \, |z^7 \rangle''
= i |z^7 \rangle'', \nonumber \\
CO' \, |+1 \rangle'' & = & |+1 \rangle'' , \quad CO' \, |z \rangle''
= z^5 |z^7 \rangle'' , \nonumber \\
CO' \, |+i \rangle'' & = & i |-i \rangle'' , \quad CO' \, |z^3 \rangle''
= z^7 |z^5 \rangle'', \nonumber \\
CO' \, |-1 \rangle'' & = & - |-1 \rangle'' , \quad CO' \, |z^5 \rangle''
= z |z^3 \rangle'' , \nonumber \\
CO' \, |-i \rangle'' & = & -i |+i \rangle'' , \quad CO' \, |z^7 \rangle''
= z^3 |z \rangle'', \nonumber \\
CT_x \, |+1 \rangle'' & = & |+1 \rangle'' , \quad CT_x \, |z \rangle''
= z^7 |z^7 \rangle'' , \nonumber \\
CT_x \, |+i \rangle'' & = & -i |-i \rangle'' , \quad CT_x \, |z^3 \rangle''
= z^5 |z^5 \rangle'', \nonumber \\
CT_x \, |-1 \rangle'' & = & - |-1 \rangle'' , \quad CT_x \, |z^5 \rangle''
= z^3 |z^3 \rangle'' , \nonumber \\
CT_x \, |-i \rangle'' & = & i |+i \rangle'' , \quad CT_x \, |z^7 \rangle''
= z |z \rangle'', \nonumber \\
CT_y \, |+1 \rangle'' & = & |+1 \rangle'' , \quad CT_y \, |z \rangle''
= z^3 |z^7 \rangle'' , \nonumber \\
CT_y \, |+i \rangle'' & = & -i |-i \rangle'' , \quad CT_y \, |z^3 \rangle''
= z |z^5 \rangle'', \nonumber \\
CT_y \, |-1 \rangle'' & = & - |-1 \rangle'' , \quad CT_y \, |z^5 \rangle''
= z^7 |z^3 \rangle'' , \nonumber \\
CT_y \, |-i \rangle'' & = & i |+i \rangle'' , \quad CT_y \, |z^7 \rangle''
= z^5 |z \rangle'' .
\end{eqnarray}
By construction, the eight states are eigenstates of the approximate continuous
$U(1)$ symmetry $U$ restricted to $\Z(8)$,
\begin{eqnarray}
U \, |+1 \rangle'' & = & |+1 \rangle'' , \quad U \, |z \rangle''
= z |z \rangle'' , \nonumber \\
U \, |+i \rangle'' & = & i |+i \rangle'' , \quad U \, |z^3 \rangle''
= z^3 |z^3 \rangle'', \nonumber \\
U \, |-1 \rangle'' & = & - |-1 \rangle'' , \quad U \, |z^5 \rangle''
= z^5 |z^5 \rangle'' , \nonumber \\
U \, |-i \rangle'' & = & -i |-i \rangle'' , \quad U \, |z^7 \rangle''
= z^7 |z^7 \rangle''.
\end{eqnarray}
The symmetries $CT_x, CT_y, CO'$ have $|+1 \rangle''$ and $|-1 \rangle''$
unmixed, and they mix $|+i \rangle''$ with $|-i \rangle''$, $|z \rangle''$
with $|z^7 \rangle'' = |z^* \rangle''$, and $|z^3 \rangle''$ with
$|z^5 \rangle'' = |z^{3*} \rangle''$. The energy spectrum will thus contain two
non-degenerate states, as well as three pairs of two-fold degenerate states.
We do not explicitly work out the energy spectrum, but point out that in a
mixed phase eight finite-volume states become degenerate in the infinite
volume limit, while for the columnar or plaquette phase only four states
become degenerate.
§ LOW-ENERGY EFFECTIVE THEORY
In <cit.> we found strong numerical evidence for an emergent soft
pseudo-Goldstone mode at and below the RK point. In this chapter we discuss
the theoretical concepts underlying our numerical analysis that we present in
the next section. These include the effective field description of the
pseudo-Goldstone boson mode and the energy spectrum in a finite volume.
§.§ Goldstone Boson Fields, Symmetries, Lagrangian
The basic degree of freedom in the effective theory — the soft
pseudo-Goldstone mode — is parametrized by the angle
$\varphi = \frac{1}{2}(\varphi_1 + \varphi_2 + \frac{\pi}{4})$.
Note that the angles $\varphi_1$ and $\varphi_2$ have been defined in
Eq. (<ref>). Under the various symmetries, the angle
$\varphi$ transforms as
\begin{eqnarray}
&&^{CT_x}\varphi = \pi - \varphi, \quad
^{CT_y}\varphi = \frac{\pi}{2} - \varphi, \nonumber \\
&&^O\varphi = \frac{\pi}{4} + \varphi, \quad
^{CO'}\varphi = - \frac{\pi}{4} - \varphi.
\end{eqnarray}
The leading Euclidean effective Lagrangian takes the form
\begin{equation}
{\cal L} = \frac{\rho}{2} \Big( \frac{1}{c^2} \partial_t \varphi \partial_t
\varphi + \partial_i \varphi \partial_i \varphi \Big)
+ \kappa (\partial_i \partial_i \varphi)^2 + \delta \cos^2(4 \varphi) ,
\end{equation}
which is identical with the effective Lagrangian of the $(2+1)$-dimensional
$\RP(1)$ model. Note that the angles $\varphi$ and $\varphi + \pi$ are
indistinguishable, such that the physical Hilbert space only contains states
that are invariant under this shift.
While $\rho$ is the spin stiffness, the quantity $c$ is the limiting velocity
of the pseudo-Goldstone boson. The term proportional to the low-energy
effective constant $\delta$ explicitly breaks the emergent $SO(2)$ symmetry to
the discrete subgroup $\Z(8)$. Accordingly, we are dealing with a light
pseudo-Goldstone mode with mass
\begin{equation}
M c = 4 \sqrt{2 |\delta|/ \rho } .
\end{equation}
At the RK point ($\lambda=1$) all flux configurations cost zero energy in
their ground state. This implies that the individual effective couplings
$\delta$ and $\rho$ (but not the ratio $\rho/c^2)$ are zero. Note that the
condition $\rho=0$ at the RK point is an analytic result which does not
require any fine-tuning. In this case, the quartic kinetic term proportional
to $\kappa$ becomes the dominant contribution. Remarkably,
$\partial_i \partial_i \varphi = 0$ for all configurations of static external
charges in their ground state, such that this term indeed does not contribute
any ground state energy. This is not true for the term
$\sum_{i=1,2}\partial_i \partial_i \varphi\partial_i \partial_i \varphi$. Hence,
such a term cannot arise at the RK point, but it can arise away from it. Also
all terms in the potential energy vanish at the RK point.
§.§ Rotor Spectrum
Let us first consider the spectrum of vacuum states in a periodic volume
$L_1 \times L_2$ at $\delta=0$. At zero temperature, to lowest order, we may
assume $\varphi(x,t) = \varphi(t)$, i.e. the low-energy dynamics reduces to the
one of the spatial zero-mode, which represents a single quantum mechanical
degree of freedom. The action then reduces to
\begin{equation}
S[\varphi] = \int dt \, \Bigg[ \frac{\rho L_1 L_2}{2 c^2} \, \partial_t
\varphi \partial_t \varphi + \delta L_1 L_2 \cos^2(4 \varphi) \Bigg] \, ,
\end{equation}
and the corresponding quantum mechanical Hamilton operator is given by
\begin{equation}
H_{\rm eff} = -\frac{c^2}{2 \rho L_1 L_2} \, \partial^2_{\varphi} + \delta L_1 L_2
\cos^2(4 \varphi) \, .
\end{equation}
At $\delta=0$, this describes a free “particle” on a circle. The
corresponding energy eigenstates and eigenvalues are
\begin{equation}
\psi_m(\varphi) = \frac{1}{\sqrt{2 \pi}} \exp(i m \varphi) \, ,
\qquad E_m = \frac{m^2 c^2}{2 \rho L_1 L_2}.
\end{equation}
Since $\varphi$ and $\varphi + \pi$ are physically equivalent, $m$ is
restricted to even integers.
Let us consider the effects of small $\delta$ in perturbation theory. The
ground state with $m=0$ is non-degenerate and has a constant wavefunction,
\begin{equation}
\psi_0(\varphi) = \frac{1}{\sqrt{2 \pi}} \, .
\end{equation}
Its energy shift is
\begin{equation}
E^{(1)}_0 = \delta L_1 L_2 \langle \psi_0 | \cos^2(4 \varphi) | \psi_0 \rangle
= \frac{\delta L_1 L_2}{2} \, .
\end{equation}
The excited states with $m = \pm 2, \pm 4, \dots $ are 2-fold degenerate.
Their energy shifts result from
\begin{eqnarray}
V_{m,m} & = & V_{-m,-m} = \frac{\delta L_1 L_2}{2} \, , \\
V_{m,-m} & = & V_{-m,m} = \frac{\delta L_1 L_2}{16 \pi}
{\Bigg[ \frac{\sin(2(m-4)\varphi)}{m-4} + \frac{2 \sin(2 m \varphi)}{m}
+ \frac{\sin(2(m+4)\varphi)}{m+4} \Bigg]}_0^{2\pi} \, . \nonumber
\end{eqnarray}
The case $m = \pm 4$ thus needs to be considered separately. Since
\begin{equation}
\lim_{m\to\pm4} V_{m,-m} = \frac{\delta L_1 L_2}{4} \, ,
\end{equation}
the corresponding energy shift takes the form
\begin{equation}
E^{(1)}_{4 \pm} = \frac{\delta L_1 L_2}{2} \pm \frac{\delta L_1 L_2}{4} \, ,
\end{equation}
and the previously degenerate energy levels split,
\begin{equation}
E_{\pm 4} - E_0 = \frac{8 c^2}{\rho L_1 L_2} \pm
\frac{\delta L_1 L_2}{4} \, .
\end{equation}
This formula is only valid in the regime
\begin{equation}
\delta L_1 L_2 \ll \frac{c^2}{\rho L_1 L_2} \, .
\end{equation}
For $m = \pm 2, \pm 6, \pm 8, \dots$, there is no such effect and we simply
\begin{equation}
E_{\pm m} - E_0 = \frac{m^2 c^2}{2\rho L_1 L_2} \, .
\end{equation}
We now turn to second order perturbation theory in $\delta$. Note that the
leading order correction to the higher excited states $m=\pm 2, \pm 4, \dots$
only arises at order $\delta^2$. To avoid degenerate perturbation theory, we
separately consider even and odd wave functions. We begin with the even wave
\begin{equation}
\psi^e_m(\varphi) = \frac{1}{\sqrt{\pi}} \cos(m \varphi) \, , \qquad E_m
= \frac{m^2 c^2}{2 \rho L_1 L_2} \, , \qquad
\psi^e_0(\varphi) = \frac{1}{\sqrt{2\pi}} \, .
\end{equation}
With the matrix elements, $m,n > 0$,
\begin{equation}
\langle m | V(\varphi) | n \rangle = \frac{\delta L_1 L_2}{\pi} \,
\int_0^{2 \pi} d \varphi \cos(m \varphi) \cos^2(4 \varphi)
\cos(n \varphi) \, ,
\end{equation}
we obtain
\begin{eqnarray}
E^{even,(2)}_0 & = & - \sum_{n\neq0} \frac{{| \langle 0 | V(\varphi) | n
\rangle |}^2}{E_n -E_0}
= - \frac{{| \langle 0 | V(\varphi) | 8 \rangle |}^2}{E_8 -E_0}
= - \frac{\delta^2 \rho L^3_1 L^3_2}{256 c^2}
\, , \nonumber \\
E^{even,(2)}_2 & = & - \sum_{n\neq2} \frac{{| \langle 2 | V(\varphi) | n
\rangle |}^2}{E_n -E_2}
= - \frac{{| \langle 2 | V(\varphi) | 6 \rangle |}^2}{E_6 -E_2}\
- \frac{{| \langle 2 | V(\varphi) | 10 \rangle |}^2}{E_{10} -E_2}
= - \frac{\delta^2 \rho L^3_1 L^3_2}{192 c^2}
\, , \nonumber \\
E^{even,(2)}_4 & = & - \sum_{n\neq4} \frac{{| \langle 4 | V(\varphi) | n
\rangle |}^2}{E_n -E_4}
= - \frac{{| \langle 4 | V(\varphi) | 12 \rangle |}^2}{E_{12} -E_4}
= - \frac{\delta^2 \rho L^3_1 L^3_2}{1024 c^2}
\, .
\end{eqnarray}
Analogously, for the odd wave functions ($m,n > 0$),
\begin{equation}
\psi^o_m(\varphi) = \frac{1}{\sqrt{\pi}} \sin(m \varphi) \, ,
\qquad E_m = \frac{m^2 c^2}{2 \rho L_1 L_2} \, ,
\end{equation}
and with
\begin{equation}
\langle m | V(\varphi) | n \rangle = \frac{\delta L_1 L_2}{\pi} \,
\int_0^{2 \pi} d \varphi \sin(m \varphi) \cos^2(4 \varphi)
\sin(n \varphi) \, ,
\end{equation}
second order perturbation theory leads to
\begin{eqnarray}
E^{odd,(2)}_2 & = & - \sum_{n\neq2} \frac{{| \langle 2 | V(\varphi) | n
\rangle |}^2}{E_n -E_2}
= - \frac{{| \langle 2 | V(\varphi) | 6 \rangle |}^2}{E_6 -E_2}
- \frac{{| \langle 2 | V(\varphi) | 10 \rangle |}^2}{E_{10} -E_2}
= - \frac{\delta^2 \rho L^3_1 L^3_2}{192 c^2}
\, , \nonumber \\
E^{odd,(2)}_4 & = & - \sum_{n\neq4} \frac{{| \langle 4 | V(\varphi) | n
\rangle |}^2}{E_n -E_4}
= - \frac{{| \langle 4 | V(\varphi) | 12 \rangle |}^2}{E_{12} -E_4}
= - \frac{\delta^2 \rho L^3_1 L^3_2}{1024 c^2}
\, .
\end{eqnarray}
We now proceed with a nonperturbative treatment of $\delta$ and consider the
nonperturbative Schrödinger equation that takes the form of a Hill equation,
\begin{eqnarray}
& & -\frac{1}{2} \delta^2_{\varphi} \psi(\varphi) + V_0 \cos^2(4 \varphi)
\psi(\varphi) = \varepsilon \psi(\varphi) \, , \nonumber \\
& & V_0=\frac{\delta \rho L^2_1 L^2_2}{c^2} \, , \qquad \varepsilon
= \frac{\rho L_1 L_2}{c^2} \, E \, .
\end{eqnarray}
Since $\varphi$ and $\varphi + \pi$ are to be identified, we introduce a new
angle as $\varphi' = 4 \varphi$, with $\varphi'$ having period $4 \pi$. The
above equation is then converted into the following Mathieu equation,
\begin{equation}
- \delta^2_{\varphi'} \psi(\varphi') + \frac{V_0}{16} \, \cos(2 \varphi')
\psi(\varphi') = \Bigg( \frac{\varepsilon}{8} -\frac{V_0}{16}
\Bigg) \psi(\varphi') \, .
\end{equation}
The corresponding solutions are even and odd Mathieu functions
\begin{equation}
\psi_{2m}(\varphi') = \frac{1}{\sqrt{\pi}} \, ce_m (\varphi') \, ,
\quad \psi_{2m+1}(\varphi') = \frac{1}{\sqrt{\pi}} \, se_m (\varphi')
\, ,
\end{equation}
with eigenvalue $\lambda_0$ given to lowest order by
\begin{equation}
\lambda_0 = \frac{\varepsilon_0}{8} - \frac{V_0}{16}
= -\frac{1}{2} {\Bigg( \frac{V_0}{32} \Bigg)}^2 + {\cal O}(V^4_0) \, .
\end{equation}
Accordingly, the ground state energy reads
\begin{equation}
E_0 = \frac{\delta L_1 L_2}{2} - \frac{\delta^2 \rho L^3_1 L^3_2}{256 c^2}
+ {\cal O}(\delta^4) \, ,
\end{equation}
in agreement with the leading perturbative results. In the nonperturbative
regime, for the excited states we obtain
\begin{equation}
E_m = \frac{c^2}{\rho L_1 L_2} \varepsilon_m = \frac{\delta L_1 L_2}{2}
+ \frac{8c^2}{\rho L_1 L_2} \, \lambda_m\Big(
\frac{\delta \rho L^2_1 L^2_2}{16c^2} \Big) \, .
\end{equation}
The energy splittings in the rotor spectrum are thus given by the eigenvalues
\begin{equation}
\lambda_m = \lambda_m\Big(\frac{\delta \rho L^2_1 L^2_2}{16c^2}\Big)
\end{equation}
of the Mathieu equation,
\begin{equation}
E_m - E_0 = \frac{8 c^2}{\rho L_1 L_2} \, \Bigg[ \lambda_m\Big( \frac{\delta
\rho L_1^2 L_2^2}{16c^2} \Big)
- \lambda_0\Big( \frac{\delta \rho L_1^2 L_2^2}{16c^2}\Big) \Bigg] \, .
\end{equation}
As a consistency check we also consider the eigenvalue $\lambda_1$ that
corresponds to the odd Mathieu function $se_1(\varphi')$,
\begin{equation}
\lambda_1 = \frac{\varepsilon_1}{8} - \frac{V_0}{16} = 1 - \frac{V_0}{32}
- \frac{1}{8} {\Bigg( \frac{V_0}{32} \Bigg)}^2
+ \frac{1}{64} {\Bigg( \frac{V_0}{32} \Bigg)}^3 + {\cal O}(V^4_0) \, .
\end{equation}
Accordingly, the energy $E_1$ is given by
\begin{equation}
E_1 = \frac{8c^2}{\rho L_1 L_2} + \frac{\delta L_1 L_2}{4}
- \frac{\delta^2 \rho L^3_1 L^3_2}{1024 c^2}
+ \frac{\delta^3 \rho^2 L^5_1 L^5_2}{262144 c^4} + {\cal O}(\delta^4) \, ,
\end{equation}
such that
\begin{equation}
E_1 - E_0 = \frac{8c^2}{\rho L_1 L_2} - \frac{\delta L_1 L_2}{4}
+ \frac{3 \delta^2 \rho L^3_1 L^3_2}{1024 c^2}
+ \frac{\delta^3 \rho^2 L^5_1 L^5_2}{262144 c^4} + {\cal O}(\delta^4) \, .
\end{equation}
This is consistent with the leading order perturbative calculation, when one
identifies the state corresponding to $se_1(\varphi')$ with the state $m=-4$.
Note that the results of second order perturbation theory in $\delta$ are also
consistent with the expansion of the Mathieu function eigenvalues.
The theoretical results derived in this section will be compared with exact
diagonalization and Monte Carlo simulation results in sections
<ref> and <ref>, respectively.
§ EXACT DIAGONALIZATION RESULTS
In this section we discuss exact diagonalization results for $L_1 \times L_2$
lattices with $L_1,L_2 \in \{ 4,6,8 \}$, which allow us to determine some
low-energy parameters of the effective field theory discussed in the previous
[Color online] Energy spectrum on an $8^2$ lattice as a
function of the RK coupling $\lambda$ <cit.>.
[Color online] Logarithmic energy gaps as a function of the
volume at $\lambda=-1$.
Using exact diagonalization we were able to calculate the low-lying energy
spectrum on lattices up to $8 \times 8$. Fig. <ref> shows the
energy gaps on the largest lattice. For $\lambda < 1$, the ground state is
non-degenerate and transforms trivially under the symmetry operations, i.e.
it has quantum numbers ($CT_x,CT_y$) = (+,+). The first two excited states
with energy gap $E_1 = E_2$ are degenerate and have quantum numbers $(+,-)$
and $(-,+)$, while the next excited state with energy gap $E_3$ has again
quantum numbers $(+,+)$. As Fig. <ref> shows for $\lambda=-1$, the
energy gaps of these three excited states decrease exponentially with the
volume $L_1 L_2$, i.e. $E_{1,2}, E_3 \sim \exp(- \alpha L_1 L_2)$ for
$-0.2 \lesssim \lambda \lesssim 0.8$. The fact that the gap between the
finite-volume ground state and the three first excited states is exponentially
small indicates that four phases coexist at zero temperature. The
$(CT_x,CT_y)$ quantum numbers $(+,+)$, $(+,-)$, $(-,+)$, $(+,+)$ indicate that
we are in a columnar and not in a plaquette phase.
If the columnar phase were replaced by the plaquette phase for larger values
of $\lambda$, one would expect a level crossing of the excited $(+,+)$ state
with the lowest $(-,-)$ state. Interestingly, no such level crossing arises in
our exact diagonalization study. Notably, the next excited state with energy
gap $E_4$, does not decrease exponentially with the volume. It has quantum
numbers $(-,-)$ and almost degenerates with the $(+,+)$ state with energy
$E_3$ for $-0.2 \lesssim \lambda \lesssim 0.8$. Furthermore, the next two
states with energy $E_5 = E_6$ are exactly degenerate and again have quantum
numbers $(+,-)$ and $(-,+)$. The next states, with energies $E_7$ and $E_8$
are once more almost degenerate and transform as $(+,+)$ and $(-,-)$. The
energy ratios of these states are given by
$E_{1,2}:E_{3,4}:E_{5,6}:E_{7,8} \approx 1:4:9:16$, which is indicated by the
dashed lines in Fig. <ref>. This hints at an approximate rotor
spectrum. Indeed in <cit.> we presented numerical evidence for an
emergent approximate spontaneously broken $SO(2)$ symmetry with an associated
pseudo-Goldstone boson. Since the Goldstone boson has a small mass, it does
not qualify as a dual photon and the theory remains confining before one
reaches the RK point. While the exact diagonalization study alone is not
sufficient to come to this conclusion, it is fully consistent with it.
[Color online] Results for the parameter $\delta$ (top) and the
combination $\frac{c^2}{\rho}$ (bottom) of the effective theory from a fit to
the exact diagonalization data for different $\lambda$ for the lattice sizes
$6\times 6$ and $8\times8$. The error bar that appears in the figures for each
value of $\lambda$ is the uncertainty of the calculated quantity from the fit.
Using the analytic results of the effective theory obtained in the previous
section, we now estimate some low-energy parameters by comparison with the
exact diagonalization results for the rotor spectrum. Fig. <ref>
shows the results for the symmetry breaking parameter $\delta$ (top) and the
combination $\frac{c^2}{\rho}$ (bottom). These results have been obtained
from a global fit using data from $6\times 6$ and $8 \times 8$ lattices for
different values of $\lambda$. Note that $\frac{c^2}{\rho}$ is positive for
all values of $\lambda$, while $\delta \geq 0$ approaches zero near the RK
point. Remarkably, the fit works rather well up to values of
$\lambda \approx 0.6$. Even though the errors are increasing near the RK
point, the results are still consistent with positive values of $\delta$, thus
indicating the absence of a phase transition before the RK point. This
suggests that the columnar phase extends all the way up to $\lambda=1$.
However, the precision reachable with the moderate volumes accessible to exact
diagonalization is not sufficient to definitively settle this issue. In
<cit.> we have provided numerical evidence based on Monte Carlo data
obtained on much larger systems, which implies that $\delta$ remains positive
until one reaches the RK point, thus excluding a transition into the plaquette
One may wonder whether a mixed phase, sharing features of both the columnar
and the plaquette phase, would give rise to a similar finite-volume spectrum.
As we have pointed out in the previous section, in the mixed phase eight
vacuum states, separated by exponentially small energy gaps, are almost
degenerate in a finite volume. This is qualitatively different from the rotor
spectrum observed in our exact diagonalization studies. First of all, the
energy of the rotor states decreases inversely proportional to and not
exponentially with the volume. In addition, on the moderate volumes accessible
to exact diagonalization, the observed spectrum contains at least nine rotor
states, while the mixed phase would be characterized by eight low-energy
states separated from the rest of the spectrum by a gap.
Finally, Fig. <ref> shows two sets of states with energies $E_1'$
and $E_2'$. These states have the quantum numbers ($CT_x,CT_y$) = (+,+), and
represent strings of non-zero electric flux
\begin{equation}
{\cal E}_i = \frac{1}{L_i} \sum_x E_{x,x+\hat i},
\end{equation}
wrapping around the periodic spatial volume. There are four states with energy
$E'_1$ with electric fluxes $({\cal E}_1,{\cal E}_2) = (\pm 1,0),
(0,\pm 1)$, while there are two states with energy $E'_2$ with electric fluxes
$(\pm 2,0), (0,\pm 2)$. The energy gaps of these states vanish at the RK point
$\lambda=1$. This implies that at this point, flux strings cost zero energy
thus signaling deconfinement and the spontaneous breakdown of the $U(1)$
center symmetry.
§ MONTE CARLO RESULTS
Green's function Monte Carlo simulations have been applied earlier to the
square lattice quantum dimer model <cit.>, with lattice
sizes $L^2$ up to $L=48$. In our previous study <cit.> we have used
a more efficient Monte Carlo algorithm that enabled us to reach volumes up to
$L^2 = 144 \times 144$ and temperatures down to $T = J/500$. Our algorithm is
based on the height variable representation of the quantum dimer model. More
details about the algorithm have been presented in <cit.>.
Some Monte Carlo data, in particular those which provide convincing numerical
evidence that the columnar phase is realized in the square lattice quantum
dimer model all the way to the RK point, are already shown in <cit.>.
Here we present new results and we give a more detailed explanation of the
results obtained earlier.
[Color online] The presence of two external static charges violates
the dimer covering constraint <cit.>.
[Color online] (Top to bottom) Energy density $-J \langle U_\Box
+ U_\Box^\dagger\rangle$ on a $120 \times 120$ lattice in the presence of two
charges $\pm 2$, separated by 49 lattice spacings for $\beta J = 64$ and
$\lambda= -5, -2, -1$.
By putting two external static charges $\pm 2$ (relative to the staggered
charge background) into the system, one violates the dimer covering
constraint. As depicted in Fig. <ref>, this leads to two
defects, associated with three dimers that overlap at the same lattice point.
The two static charges, separated by an odd number of lattice spacings, are
connected by an electric flux string and are thus confined. In addition, the
flux string fractionalizes into eight individual strands — displaying the
four plaquette phases — which each carry electric flux $\frac{1}{4}$, thus
adding up to the total flux 2. The energy density
$-J \langle U_\Box + U_\Box^\dagger\rangle$ for $\lambda = -5, -2, -1$ is
shown in Fig. <ref>: one notices
that, as one moves from large negative values of $\lambda$ towards
$\lambda \approx 0$, the individual strands emerge around $\lambda \approx -2$.
Furthermore, inside the different strands plaquette order is present. These
regions of plaquette order are interfaces separating the various columnar
phases. In fact, Fig. <ref> implies that as one
moves from bottom to top, each of the four possible columnar phases is visited
once. The same is true for the four degenerate plaquette orders.
[Color online] The appearance of plaquette order, which results from
the interface between different columnar phases, in the eight strands. The
result is obtained on a $120 \times 120$ lattice with $\beta J = 64$ and
In the presence of two external charges $\pm 2$ separated along the $x$-axis,
both translation and rotation invariance are explicitly broken, while the
reflection on the $x$-axis remains an exact symmetry. As a result, one of the
columnar phases, with the columns oriented in the $y$-direction, is
energetically favored. Interestingly, Fig. <ref> shows that, for
$\lambda=-0.5$ an asymmetric distribution of the eight strands is observed,
thus indicating the spontaneous breakdown of the reflection symmetry. Strictly
speaking, in an infinite volume spontaneous breaking of the reflection symmetry
only arises when the distance between the charges also approaches infinity.
At finite distances, the two
asymmetric flux patterns, which are related to one another by reflection,
coexist with each other through quantum tunneling. As the charges are
separated further and further, still assuming an infinite volume,
tunneling is exponentially suppressed. When we consider a finite volume, the
asymmetry in the flux distribution disappears when the distance between the
charges becomes compatible with the lattice size (Fig. <ref>).
Squeezing the flux distribution into a small volume leads to a restoration
of the spontaneously broken reflection symmetry
due to finite-size effects. This scenario also arises for other negative
values of $\lambda$.
[Color online] Energy density
$-J \langle U_\Box + U_\Box^\dagger\rangle$ on a $120 \times 120$ lattice in the
presence of two charges $\pm 2$, separated by 43 lattice spacings for
$\beta J = 64$ and $\lambda = -0.5$.
[Color online] Energy density
$-J \langle U_\Box + U_\Box^\dagger\rangle$ on a $120 \times 120$ lattice in the
presence of two charges $\pm 2$, separated by 53 lattice spacings for
$\beta J = 64$ and $\lambda=-0.5$.
§ CONCLUSIONS
We have investigated the finite-volume energy spectrum of the square lattice
quantum dimer model using both exact diagonalization and quantum Monte Carlo.
By comparison with analytic predictions we have consolidated our previous
evidence that the columnar phase extends all the way to the RK point, without
any intervening plaquette or mixed phases. In addition, we have studied a soft
pseudo-Goldstone mode that becomes massless at the RK point but still dominates
a large region in parameter space away from it. This mode is described by a
systematic low-energy effective field theory whose parameters we have extracted
by comparison of numerical data with analytic predictions of the effective
theory. It will be an interesting topic for future studies to investigate the
possible role of the soft mode for the preformation of pairs in the pseudogap
regime of high-temperature superconductors. This could be done in the context
of hole-doped quantum dimer models
We have also studied the internal structure of the strings connecting external
charges embedded in the confining columnar phase. For topological reasons, the
string fractionalizes into strands, each carrying electric flux $\frac{1}{4}$.
The flux strands play the role of interfaces separating the four realizations
of the columnar phase. As we noted earlier <cit.>, the interior of
the flux strands shows plaquette order, despite the fact that the plaquette
phase is not stable in the bulk. The interfaces that separate two columnar
phases 1 and 3 or 2 and 4, with the columns oriented in the same direction,
show the universal phenomenon of complete wetting. This manifests itself by
the appearance of a third columnar phase at the interface, with its columns
oriented in an orthogonal direction. Hence, a 1-3 interface splits into two
1-2-3 or 1-4-3 interfaces. Remarkably, reflections on the lattice axis
connecting charges $\pm 2$ are spontaneously broken, which gives rise to
asymmetric electric flux profiles.
As we have seen, the simple square lattice quantum dimer model has a rich
confining dynamics, characterized by strings with an intriguing anatomy.
Understanding these dynamics required the interplay between numerical
simulations and analytic effective field theory calculations. It will be
interesting and promising to apply this strategy to quantum dimer models with
other lattice geometries <cit.>.
§ ACKNOWLEDGMENTS
DB acknowledges interesting discussions with A. Läuchli. CPH thanks the
members of the ITP at Bern University for their hospitality, and acknowledges
support through the project Redes Temáticas de Colaboración
Académica 2013, UCOL-CA-56. The research leading to these results has
received funding from the Schweizerischer Nationalfonds and from the
European Research Council under the European Union's Seventh Framework
Programme (FP7/2007-2013)/ ERC grant agreement 339220.
§ SYMMETRIES AND CANDIDATE PHASES
It is important to know how the four order parameters transform under the
symmetries $CT_x, CT_y, O, CO', R_x$, and $R_y$ of the square lattice
quantum dimer model. This is illustrated in Table <ref>.
$S$ CT$_x$ CT$_y$ O CO' R$_x$ R$_y$
$M_{11}[^S{\cal C}]$ -$M_{11}[{\cal C}]$ -$M_{22}[{\cal C}]$ $M_{21}[{\cal C}]$
-$M_{21}[{\cal C}]$ -$M_{22}[{\cal C}]$ -$M_{11}[{\cal C}]$
$M_{12}[^S{\cal C}]$ $M_{21}[{\cal C}]$ -$M_{12}[{\cal C}]$ -$M_{11}[{\cal C}]$
-$M_{22}[{\cal C}]$ -$M_{12}[{\cal C}]$ $M_{21}[{\cal C}]$
$M_{21}[^S{\cal C}]$ $M_{12}[{\cal C}]$ $M_{21}[{\cal C}]$ -$M_{22}[{\cal C}]$
-$M_{11}[{\cal C}]$ $M_{21}[{\cal C}]$ $M_{12}[{\cal C}]$
$M_{22}[^S{\cal C}]$ $M_{22}[{\cal C}]$ -$M_{11}[{\cal C}]$ -$M_{12}[{\cal C}]$
-$M_{12}[{\cal C}]$ -$M_{11}[{\cal C}]$ $M_{22}[{\cal C}]$
Transformation properties of the order parameters $M_{ij}$ under
the symmetries $S =$ CT$_x$, CT$_y$, O, CO', R$_x$, R$_y$ . The order
parameter $M_{ij}[^S{\cal C}]$ evaluated in the transformed configuration
$^S{\cal C}$ as a function of the order parameters $M_{ij}[{\cal C}]$ evaluated
in the original configuration ${\cal C}$.
$S$ CT$_x$ CT$_y$ O CO' R$_x$ R$_y$
$^S$1 1 3 2 4 3 1
$^S$2 4 2 3 3 2 4
$^S$3 3 1 4 2 1 3
$^S$4 2 4 1 1 4 2
Transformation properties of the four columnar phases 1, 2, 3, 4
under the symmetries $S =$ CT$_x$, CT$_y$, O, CO', R$_x$, R$_y$.
$S$ CT$_x$ CT$_y$ O CO' R$_x$ R$_y$
$^S$A D B B C B D
$^S$B C A C B A C
$^S$C B D D A D B
$^S$D A C A D C A
Transformation properties of the four plaquette phases A, B, C, D
under the symmetries $S =$ CT$_x$, CT$_y$, O, CO', R$_x$, R$_y$.
$S$ CT$_x$ CT$_y$ O CO' R$_x$ R$_y$
$^S$A1 D1 B3 B2 C4 B3 D1
$^S$A2 D4 B2 B3 C3 B2 D4
$^S$B2 C4 A2 C3 B3 A2 C4
$^S$B3 C3 A1 C4 B2 A1 C3
$^S$C3 B3 D1 D4 A2 D1 B3
$^S$C4 B2 D4 D1 A1 D4 B2
$^S$D4 A2 C4 A1 D1 C4 A2
$^S$D1 A1 C3 A2 D4 C3 A1
Transformation properties of the eight mixed phases A1, A2, B2,
B3, C3, C4, D4, D1 under the symmetries $S =$ CT$_x$, CT$_y$, O, CO', R$_x$,
In tables <ref>, <ref>, and <ref>, we show
how the different symmetries $CT_x, CT_y, O, CO', R_x, R_y$ act on the
columnar, plaquette, and mixed phases, respectively.
J. C. Bednorz and K. A. Müller, Z. Phys. B: Condens. Matter
64, 189 (1986).
[Rokhsar and Kivelson (1988)]Rok88
D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376
P. W. Anderson, Science 235, 1196 (1987).
S. Sachdev, Phys. Rev. B 40, 5204 (1989).
L. S. Levitov, Phys. Rev. Lett. 64, 92 (1990).
P. W. Leung, K. C. Chiu, and K. J. Runge, Phys. Rev. B 54,
12938 (1996).
R. Moessner, S. L. Sondhi, and E. Fradkin, Phys. Rev. B 65,
024504 (2002).
C. L. Henley, J. Phys.: Condens. Matter 16, S891 (2004).
F. Alet, J. L. Jacobsen, G. Misguich, V. Pasquier, F. Mila, and
M. Troyer, Phys. Rev. Lett. 94, 235702 (2005).
F. Alet, Y. Ikhlef, J. L. Jacobsen, G. Misguich, and V. Pasquier,
Phys. Rev. E 74, 041124 (2006).
D. Charrier and F. Alet, Phys. Rev. B 82, 014429 (2010).
J. Cano and P. Fendley, Phys. Rev. Lett. 105, 067205 (2010).
A. F. Albuquerque and F. Alet, Phys. Rev. B 82, 180408(R)
Y. Tang, A. W. Sandvik, and C. L. Henley, Phys. Rev. B 84,
174427 (2011).
C. A. Lamas, A. Ralko, M. Oshikawa, D. Poilblanc, and P. Pujol,
Phys. Rev. B 87, 104512 (2013).
O. F. Syljuasen, Phys. Rev. B 71, 020401(R) (2005).
O. F. Syljuasen, Phys. Rev. B 73, 245105 (2006).
A. Ralko, D. Poilblanc, and R. Moessner, Phys. Rev. Lett. 100,
037201 (2008).
D. Banerjee, M. Bögli, C. P. Hofmann, F.-J. Jiang, P. Widmer, and
U.-J. Wiese, Phys. Rev. B 90, 245143 (2014).
D. Horn, Phys. Lett. 100B, 149 (1981).
P. Orland and D. Rohrlich, Nucl. Phys. B 338, 647 (1990).
S. Chandrasekharan and U.-J. Wiese, Nucl. Phys. B 492, 455
G. 't Hooft, Nuci. Phys. B 153, 141 (1979).
[Moessner and Raman (2008)]MR08
R. Moessner and K. S. Raman, arXiv:0809.3051.
[Moessner and Sondi (2001)]MS01
R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001).
[Ivanov (2004)]Iva04
D. A. Ivanov, Phys. Rev. B 70, 094430 (2004).
[Ralko et al.(2005)]RFBIM05
A. Ralko, M. Ferrero, F. Becca, D. Ivanov, and F. Mila, Phys. Rev. B
71, 224109 (2005).
[Huse et al (2003)]HKMS03
D. A. Huse, W. Kraut, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett.
91, 167004 (2003).
[Moessner and Sondi (2003)]MS03
R. Moessner and S. L. Sondhi, Phys. Rev. B 68, 184512 (2003).
M. Hermele, M. P. A. Fisher, and L. Balents, Phys. Rev. B 69,
[Moessner et al. (2000)]MSC00
R. Moessner, S. L. Sondhi, and P. Chandra, Phys. Rev. Lett. 84,
4457 (2000).
[Zeng and Elser (1995)]ZE95
C. Zeng and V. Elser, Phys. Rev. B 51, 8318 (1995).
[Nikolic and Senthil (2003)]NS03
P. Nikolic and T. Senthil, Phys. Rev. B 68, 214415 (2003).
[Moessner et al. (2001)]MSC01
R. Moessner, S. L. Sondhi, and P. Chandra, Phys. Rev. B 64,
144416 (2001).
[Antonov (1907)]Ant07
G. Antonov, J. Chem. Phys. 5, 372 (1907).
L. Balents, L. Bartosch, A. Burkov, S. Sachdev, and K. Sengupta,
Phys. Rev. B 71, 144509 (2005).
D. Poilblanc, F. Alet, F. Becca, A. Ralko, F. Trousselet, and F. Mila,
Phys. Rev. B 74, 014437 (2006).
A. Ralko, F. Mila, and D. Poilblanc,
Phys. Rev. Lett. 99, 127202 (2007).
S. Papanikolaou, E. Luijten, and E. Fradkin,
Phys. Rev. B 76, 134514 (2007).
H. Ribeiro, S. Bieri, and D. Ivanov,
Phys. Rev. B 76, 172301 (2007).
D. Poilblanc,
Phys. Rev. Lett. 100, 157206 (2008).
C. A. Lamas, A. Ralko, D. C. Cabra, D. Poilblanc, and P. Pujol, Phys.
Rev. Lett.109, 016403 (2012).
[Schlittler et al.(2015)]SBMVM15
T. M. Schlittler, T. Barthel, G. Misguich, J. Vidal, and R. Mosseri,
[Schlittler et al.(2015)]SMB15
T. M. Schlittler, R. Mosseri, and T. Barthel,
|
1511.00718
|
Brain connectivity analysis is now at the foreground of neuroscience research. A connectivity network is characterized by a graph, where nodes represent neural elements such as neurons and brain regions, and links represent statistical dependences that are often encoded in terms of partial correlations. Such a graph is inferred from matrix-valued neuroimaging data such as electroencephalography and functional magnetic resonance imaging. There have been a good number of successful proposals for sparse precision matrix estimation under normal or matrix normal distribution; however, this family of solutions do not offer a statistical significance quantification for the estimated links. In this article, we adopt a matrix normal distribution framework and formulate the brain connectivity analysis as a precision matrix hypothesis testing problem. Based on the separable spatial-temporal dependence structure, we develop oracle and data-driven procedures to test the global hypothesis that all spatial locations are conditionally independent, which are shown to be particularly powerful against the sparse alternatives. In addition, simultaneous tests for identifying conditional dependent spatial locations with false discovery rate control are proposed in both oracle and data-driven settings. Theoretical results show that the data-driven procedures perform asymptotically as well as the oracle procedures and enjoy certain optimality properties. The empirical finite-sample performance of the proposed tests is studied via simulations, and the new tests are applied on a real electroencephalography data analysis.
Key Words: Connectivity analysis; False discovery rate; Gaussian graphical model; Matrix-variate normal distribution; Multiple testing.
§ INTRODUCTION
In recent years, matrix-valued data are becoming ubiquitous in a wide range of scientific and business applications, including bioinformatics <cit.>, brain imaging analysis <cit.>, finance <cit.>, among many others. Accordingly, the matrix normal distribution is becoming increasingly popular in modeling the matrix-variate observations <cit.>. Our motivating example is an electroencephalography (EEG) data, which measures voltage value from electrodes placed at various brain locations over a period of time for a group of alcoholic subjects and normal controls. One scientific goal is to infer the connectivity patterns among those spatial locations. More generally, accurate and informative mapping of the human connectivity network is now at the center stage of neuroscience research. The objective is to infer brain connectivity network, which is commonly characterized as a graph consisting of nodes and links. Here nodes represent neural elements, from micriscopic neurons to macroscopic brain regions, and links represent statistical dependencies between neural components <cit.>. Partial correlations, reported by a precision matrix, are frequently employed to describe such statistical dependencies <cit.>. This precision matrix, in turn, is to be derived from imaging modalities, such as EEG, magnetoencephalography, and functional magnetic resonance imaging. The data of those imaging modalities are in the common form of a two-dimensional matrix, with one spatial dimension and the other temporal dimension.
Adopting a matrix normal distribution framework, we formulate the brain connectivity network analysis as a precision matrix inference problem. Specifically, let $\X \in \real{p \times q}$ denote the spatial-temporal matrix data from an image modality, e.g., EEG. It is assumed to follow a matrix normal distribution with the Kronecker product covariance structure,
\begin{eqnarray*}
\Cov\{\text{vec}(\X)\} = \S_{L} \otimes \S_{T},
\end{eqnarray*}
where the operator vec$(\X)$ stacks the columns of the matrix $\X$ to a vector, $\otimes$ is the Kronecker product, $\S_{L} \in \real{p \times p}$ denotes the covariance matrix of $p$ spatial locations, $\S_{T} \in \real{q \times q}$ denotes the covariance matrix for $q$ time points. Correspondingly,
\begin{eqnarray*}
\Cov^{-1}\{\text{vec}(\X)\} = \S_{L}^{-1} \otimes \S_{T}^{-1} = \O_{L} \otimes \O_{T},
\end{eqnarray*}
where $\O_{L} \in \real{p \times p}$ is the spatial precision matrix, $\O_{T} \in \real{q \times q}$ is the temporal precision matrix. In brain connectivity analysis, our primary interest is to infer the the connectivity network characterized by the spatial precision matrix $\O_L$. By contrast, the temporal precision matrix $\O_T$ is of little interest here and is to be treated as a nuisance parameter in our analysis. We also make some remarks regarding the assumptions of our adopted framework. First, the matrix normal assumption has been frequently adopted in various applications <cit.>, and is scientifically plausible in neuroimaging analysis. For instance, the majority standard neuroimaging processing software, such as SPM <cit.> and FSL <cit.>, adopt a framework that assumes the data are normally distributed per voxel (location) with a noise factor and an autoregressive structure, which shares a similar spirit as the matrix normal formulation. Second, it is commonly assumed that the precision matrix is sparse, which we adopt for our inferential procedure as well. Again, this sparsity assumption is scientifically justifiable, as it is known that brain region connections are energy consuming <cit.>, and biological units tend to minimize energy-consuming activities <cit.>.
In this article, we aim to address the following two hypothesis testing questions. The first is to test if all spatial locations are conditionally independent, namely, we test the null hypothesis
H_0: Ø_L is diagonal versus H_1: Ø_L is not diagonal.
The second is to identify those conditionally dependent pairs of locations with false discovery rate and false discovery proportion control; i.e., we simultaneously test
H_0,i,j: ω_L,i,j= 0 H_1,i,j: ω_L,i,j ≠0, 1≤i<j≤p,
where $\omega_{L,i,j}$ is the $(i,j)$th element of $\O_L$.
In the literature, there have been a good number of methods proposed to estimate a sparse precision matrix under normal distribution <cit.>. There are extensions of this line of work from a single precision matrix to multiple precision matrices <cit.>, and from a Gaussian distribution to a more flexible class of nonparanormal distribution <cit.>. Extension of sparse precision matrix estimation has also emerged for matrix-valued data under matrix normal distribution <cit.>. However, all those methods tackle the estimation aspect of the problem and induce a connectivity network from an estimated precision matrix. Only recently, hypothesis testing procedures have been developed under Gaussian graphical model. In particular, <cit.> proposed a testing procedure to recover a network in the one-sample case, whereas <cit.> proposed a method to test the equality of two precision matrices, so to infer the differential network structures in genomics. Both papers, however, worked with vector normal data instead of matrix normal data.
We aim at hypothesis testing for the spatial precision matrix, under the matrix normal framework, to induce the connectivity network of brain regions. We separate the spatial and temporal dependence structures, then infer the precision matrix $\O_L$ through inverse regression models by relating its elements to the regression coefficients. Two procedures are considered. One is to assume the temporal covariance matrix is known, and we term the resulting method as an oracle procedure. The other is to use a data-driven approach to estimate and plug in the temporal covariance matrix, and accordingly we term it a data-driven procedure. We construct test statistics based on the covariances between the residuals from the inverse regression models. We first develop a global test for (<ref>), and show it is particularly powerful against a sparse alternative, then develop a multiple testing procedure for simultaneously testing the hypotheses (<ref>) with false discovery rate and false discovery proportion control. We study the theoretical properties of the two testing procedures, in both the oracle and data-driven scenarios. We show that the data-driven procedure performs asymptotically as well as the oracle procedure, and enjoys certain optimality under the regularity conditions. Our numerical analysis also supports such findings.
Our contributions are multi-fold. First, brain connectivity analysis is now becoming a central goal of neuroscience research <cit.>, and it constantly calls for statistical significance quantification of the inferred connection between neural elements. However, there is a paucity of systematic hypothesis testing solutions developed for this type of problems in the literature, and our proposal offers a timely response. Second, although various network sparse estimation methods have been successfully applied in neural connectivity analyses, network hypothesis testing is an utterly different problem than estimation. The key of estimation is to seek a bias-variance tradeoff, and many common sparse estimation solutions such as graphical lasso <cit.> and constrained $\ell_1$-minimization for inverse matrix estimation <cit.> are biased estimators. Such estimation methods do not produce a direct quantification of statistical significance for the network edges. By contrast, hypothesis testing starts with an nearly unbiased estimator, and produces an explicit significance quantification. Third, among the few network hypothesis testing solutions, <cit.> and <cit.> focused on a vector-valued $\X$ following a normal distribution rather than a matrix-valued data. Directly applying their methods to test the spatial conditional dependence, with no regard for the separable structure of $\O_L$ and $\O_T$, is equivalent to assuming the columns of the matrix $\X$ are independent. This is clearly not true as the measurements at different time points can be highly correlated. Thus it is important to separate the spatial-temporal dependence structure before testing.
The following notations are used throughout this article. For any matrix normal data $\{\X_k\}_{k=1}^n$, with $\X_k\in \real{p \times q}$, $X_{k,i,l}$ denotes the $i$-th spatial location at the $l$-th time point for the $k$-th sample, and $\bar{X}_{i,l}=\frac{1}{n}\sum_{k=1}^nX_{k,i,l}$; $\X_{k,-i,l}$ denotes the column spatial vector with the $i$-th entry removed, and $\bar{\X}_{\cdot,-i,l}=\frac{1}{n}\sum_{k=1}^n\X_{k,-i,l} \in \real{1\times (p-1)}$. For a $p \times 1$ vector $\a$, $\a_{-i}$ denotes the $(p-1)\times 1$ vector by removing the $i$th entry $a_i$ from $\a$. For an $nq \times p$ data matrix $\A=(\A_1,\ldots,\A_{nq})^{\T}$ , $\A_{\cdot,-i}$ denotes an $nq\times (p-1)$ matrix $(\A^{\T} _{1,-i},\ldots,\A^{\T} _{nq,-i})^{\T}$, $\bar{\A}_{\cdot,-i}=\frac{1}{nq}\sum_{k=1}^n\A_{k,-i} \in \real{1\times (p-1)}$, $\A_{(i)}=(A_{1,i},\ldots,A_{nq,i})^{\T} \in \real{nq\times 1}$, $\bar{\A}_{(i)}=(\bar{A}_i,\ldots,\bar{A}_i)^{\T} \in \real{nq\times 1}$, where $\bar{A}_i=\frac{1}{nq}\sum_{k=1}^{nq} A_{k,i}$, and $\bar{\A}_{(\cdot,-i)}=(\bar{\A}_{\cdot,-i}^{\T} ,\ldots,\bar{\A}_{\cdot,-i}^{\T} )^{\T} \in \real{nq\times (p-1)}$.
Furthermore, for any $\A\in \real{p \times q}$, $\A_{i,-j}$ denotes the $i$th row of $\A$ with its $j$th entry removed, and $\A_{-i,j}$ denotes the $j$th column of $\A$ with its $i$th entry removed. $\A_{-i,-j}$ denotes the submatrix of $\A$ with its $i$th row and $j$th column removed. If $\A$ is symmetric, then $\lambda_{\max}(\A)$ and $\lambda_{\min}(\A)$ denote the largest and smallest eigenvalues of $\A$. We also use the following norms. For a vector $\a \in \real{p \times 1}$, define its $\ell_{s}$ norm as $|\a|_{s}=(\sum_{i=1}^{p}|a_{i}|^{s})^{1/s}$ for $1\le s \le \infty$. For a matrix $\A \in \real{p \times p}$, the matrix $1$-norm is the maximum absolute column sum, $\|\A\|_{L_{1}}=\max_{1\leq j\leq p}\sum_{i=1}^{p}|a_{i,j}|$. The matrix element-wise infinity norm is $\|\A\|_{\infty}=\max_{1\leq i,j\leq p}|a_{i,j}|$ and the element-wise $\ell_1$ norm is $\|\A\|_1=\sum_{i=1}^p\sum_{j=1}^p|a_{i,j}|$. Finally, for two sequences of real numbers $\{a_{n}\}$ and $\{b_{n}\}$, $a_{n} = O(b_{n})$ if there exists a constant $c$ such that $|a_{n}| \leq c|b_{n}|$ holds for all $n$, $a_{n} = o(b_{n})$ if $\lim_{n\rightarrow\infty}a_{n}/b_{n} = 0$, and $a_{n}\asymp b_{n}$ if there are positive constants $c_0$ and $c_1$ such that $c_0\leq a_{n}/b_{n}\leq c_1$ for all $n$.
§ METHODOLOGY
In this section, we derive test statistics and testing procedures for the global hypothesis (<ref>), and the entry-wise hypothesis (<ref>) with false discovery rate control.
§.§ Separation of spatial-temporal dependency
Let $\{\X_1,...,\X_{n}\}$, each with dimension $p \times q$, denote $n$ i.i.d. random samples from a matrix normal distribution. The mean, without loss of generality, is assumed to be zero, and the covariance is of the form $\S = \S_L \otimes \S_T$. Our interest is to infer about $\O_L = \S_L^{-1}$, while treating $\O_T=\S_T^{-1}$ as a nuisance. To separate the spatial and temporal dependence structures, we build hypothesis testing procedures for (<ref>) and (<ref>) based upon the linear transformation of the original samples $\{\X_{k}{\S}_{T}^{-1/2}, k=1,\ldots, n\}$. Specifically, we consider two scenarios. We first treat $\S_T$ as known, and term the resulting testing procedure an oracle procedure. In practice, however, $\S_T$ is rarely known, and as such we plug in an estimator of $\S_T$. When trace$(\S_L)=p$ holds true, the sample covariance matrix $\hat{\S}_{T}=\frac{1}{np}\sum_{k=1}^n\X_k^{\T} \X_k$ is an unbiased estimator of $\S_{T}$. However, when trace$(\S_L)\neq p$, $\hat{\S}_{T}$ is biased and we have $\ep(\hat{\S}_T)=\{$trace$(\S_L)/p\} \S_T$. As we will show in Remark <ref> in the next section, trace$(\S_L)$ does not affect our proposed test statistics, and thus we can assume without loss of generality that trace$(\S_L)=p$ to simplify the notations. Subsequently, we plug in the sample covariance matrix $\hat{\S}_T$, develop the hypothesis testing based on the transformed samples, $\{\X_{k}\hat{\S}_{T}^{-1/2}, k=1,\ldots, n\}$, and term it a data-driven procedure. One may also use other estimators of $\S_T$ or $\O_T$, and we will briefly discuss those alternatives in Section <ref>.
§.§ Test statistics
We first develop test statistics for the two hypotheses in the oracle case. The development of the data-driven statistics is very similar, so we omit the details and will remark clearly the difference between the oracle and data-driven cases. For simplicity, we also use the same set of notations for the two scenarios, and will only differentiate them when we study their respective asymptotic properties in Section <ref>.
It is well established that, under the normal distribution, the precision matrix can be described in terms of regression models <cit.>. Specifically, letting $\Y_k = \X_{k}{\S}_{T}^{-1/2}$, $k=1,\ldots,n$, denote the transformed samples, we have,
\begin{eqnarray} \label{regmdl}
Y_{k,i,l} = \Y_{k,-i,l}^{\T}\be_{i}+\epsilon_{k,i,l}, \quad 1\leq i \leq p, 1\leq l \leq q,
\end{eqnarray}
where $\epsilon_{k,i,l}\sim N(0,\sigma_{L,i,i}-\S_{L,i,-i}\S_{L,-i,-i}^{-1}\S_{L,-i,i})$ is independent of $\Y_{k,-i,l}$. The regression coefficient vector $\be_{i}$ and the error term $\epsilon_{k,i}$ satisfy that
\[
\be_{i} = -\omega_{L,i,i}^{-1}\O_{L,-i,i}, \quad \text{and} \quad
r_{i,j} = \Cov(\epsilon_{k,i,l},\epsilon_{k,j,l})=\frac{\omega_{L,i,j}}{\omega_{L,i,i}\omega_{L,j,j}}.
\]
As such, the elements $\omega_{L,i,j}$ of $\O_L$ can be represented in terms of $r_{i,j}$. Next, we construct an estimator of $r_{i,j}$ and its bias-corrected version. We then build on this estimator to obtain an estimator of $\omega_{L,i,j}$, upon which we further develop our test statistics.
A natural estimator of $r_{i,j}$ is the sample covariance between the residuals $\hat{\epsilon}_{k,i,l}=Y_{k,i,l}-\bar{Y}_{i,l}-(\Y_{k,-i,l}-\bar{\Y}_{\cdot,-i,l})^{\T}\hat{\be}_{i}$,
\[ %\label{residual}
\tilde{r}_{i,j}=\frac{1}{nq}\sum_{k=1}^{n}\sum_{l=1}^q\hat{\epsilon}_{k,i,l}\hat{\epsilon}_{k,j,l},
\]
where $\hat{\be}_i$, $i=1,\ldots,p$, are estimators of $\be_i$ that satisfy Condition (C1) in the oracle case and satisfies Condition (C1$^\prime$) in the data-driven case, and these estimators can be obtained via standard estimation methods such as the Lasso and Dantzig selector, as will be discussed in Section <ref>. When $i \ne j$, however, $\tilde{r}_{i,j}$ tends to be biased due to the correlation induced by the estimated parameters. We thus consider a bias-corrected estimator of $r_{i,j}$,
\[
\hat{r}_{i,j}=-(\tilde{r}_{i,j}+\tilde{r}_{i,i}\hat{\beta}_{i,j}+\tilde{r}_{j,j}\hat{\beta}_{j-1,i}), \mbox{ for }1\leq i<j\leq p.
\]
For $i=j$, we let $\hat{r}_{i,i}=\tilde{r}_{i,i}$, which is a nearly unbiased estimator of $r_{i,i}$. An estimator of the entry $\omega_{L,i,j}$ of the spatial precision matrix $\O_L$ can then be constructed as,
\[
T_{i,j}=\frac{\hat{r}_{i,j}}{\hat{r}_{i,i}\cdot \hat{r}_{j,j}}, \;\; 1\leq i< j\leq p.
\]
To further estimate the variance of $T_{i,j}$, note that
θ_i,j = {ϵ_k,iϵ_k,j/(r_i,ir_j,j)}/(nq) = (1+ρ^2_i,j)/(nqr_i,ir_j,j), where $\rho_{i,j}^2={\beta_{i,j}^2r_{i,i}}/{r_{j,j}}$. Then $\theta_{i,j}$ can be estimated by
\[
\hat{\theta}_{i,j} = (1+\hat{\beta}_{i,j}^2\hat{r}_{i,i}/\hat{r}_{j,j})/(nq\hat{r}_{i,i}\hat{r}_{j,j}).
\]
Given $\{T_{i,j}, 1\leq i<j\leq p\}$ are heteroscedastic and can possibly have a wide variability, we standardize $T_{i,j}$ by its standard error, which leads to the standardized statistics,
\[
W_{i,j}=\frac{T_{i,j}}{\sqrt{\hat{\theta}_{i,j}}}, \;\; 1\leq i< j\leq p.
\]
In the next section, we test hypotheses (<ref>) and (<ref>) based on $\{W_{i,j}\}_{i,j=1}^{p}$.
Construction of the test statistics for the data-driven procedure is almost the same as that for the oracle procedure, except that the oracle procedure starts with transformed sample $\Y_k = \X_{k}{\S}_{T}^{-1/2}$ in (<ref>), whereas the data-driven one replaces it with $\Y_k = \X_{k}\hat{\S}_{T}^{-1/2}$. Furthermore, the regression coefficients slightly vary at different time points in the data-driven scenario, and we shall replace (<ref>) by $Y_{k,i,l} = \Y_{k,-i,l}^{\T}\be_{i,l}+\epsilon_{k,i,l}$, for $1\leq i \leq p, 1\leq l \leq q$.
When $\S_T$ is unknown, $\ep(\hat{\S}_T)=\{$trace$(\S_L)/p\} \S_T$. If trace$(\S_L)=cp$, with $c\neq 1$, an unbiased estimator of $\S_T$ becomes $\hat{\S}_T/c$. Accordingly, we shall define the transformed data $\Y_k = \sqrt{c} \X_{k}\hat{\S}_{T}^{-1/2}$, for $k=1,\ldots,n$. Then we have the bias-corrected estimator $\hat{r}_{i,j}=c\hat{r}_{i,j}$, which in turn leads to $T_{i,j} = T_{i,j}/c$, and $\hat{\theta}_{i,j}=\hat{\theta}_{i,j}/c^2$. Thus, the standardized statistic $W_{i,j}$ remains the same, as the constant $c$ is cancelled. Therefore, $c$ does not affect our final test statistics, and thus for notational simplicity, we set $c=1$ from the beginning, without loss of any generality.
§.§ Global testing procedure
We propose the following test statistic for testing the global null hypothesis $H_0: \O_L$ is diagonal,
\[
M_{nq}=\max_{1\leq i< j\leq p}W_{i,j}^2.
\]
Furthermore, we define the global test $\Psi_\alpha$ by
\[
\Psi_{\alpha}= I( M_{nq}\geq q_{\alpha}+4\log p-\log\log p)
\]
where $q_{\alpha}$ is the $1-\alpha$ quantile of the type I extreme value distribution with the cumulative distribution function $\exp\{{({8\pi})^{-1/2}}e^{-t/2}\}$, i.e.,
\[
q_\alpha = -\log (8\pi) - 2 \log\log(1-\alpha)^{-1}.
\]
The hypothesis $H_0$ is rejected whenever $\Psi_\alpha =1$.
The above test is developed based on the asymptotic properties of $M_{nq}$, which will be studied in detail in Section <ref>. Intuitively, $\{W_{i,j}\}_{i,j=1}^{p}$ are approximately standard normal variables under the null distribution, and are only weakly dependent under suitable conditions. Thus $M_{nq}$ is the maximum of the squares of $p(p-1)/{2}$ such random variables, and its value should be close to $2\log \{p(p-1)/{2}\}\approx 4\log p$ under $H_0$. We will later show that, under certain regularity conditions, $M_{nq}-4\log p-\log \log p$ converges to a type I extreme value distribution under $H_0$.
§.§ Multiple testing procedure
Next we develop a multiple testing procedure for $H_{0,i,j}: \omega_{L,i,j}= 0$, so to identify spatial locations that are conditionally dependent. The test statistic $W_{i,j}$ defined in Section <ref> is employed. Since there are $(p^2-p)/2$ simultaneous hypotheses to test, it is important to control the false discovery rate. Let $t$ be the threshold level such that $H_{0,i,j}$ is rejected if $|W_{i,j}|\geq t$. Let $\mathcal{H}_0=\{(i,j): \O_{L,i,j}=0,1\leq i<j\leq p\}$ be the set of true nulls. Denote by $R_{0}(t) = \sum_{(i,j)\in \mathcal{H}_0}I(|W_{i,j}|\geq t)$ the total number of false positives, and by $R(t)= \sum_{1\leq i<j\leq p}I(|W_{i,j}|\geq t)$ the total number of rejections. The false discovery proportion and false discovery rate are then defined as
\[
\text{FDP}(t)=\frac{R_{0}(t)}{R(t)\vee 1}, \quad \text{FDR}(t)=\ep\{\text{FDP}(t)\}.
\]
An ideal choice of $t$ would reject as many true positives as possible while controlling the false discovery rate and false discovery proportion at the pre-specified level $\alpha$. That is, we select
\[
t_0=\inf\left\{0\leq t\leq 2(\log p)^{1/2}: \; \text{FDP}(t)\leq \alpha\right\}.
\]
We shall estimate $\sum_{(i,j)\in \mathcal{H}_0}I\{|W_{i,j}|\geq t\}$ by $2\{1-\Phi(t)\}|\mathcal{H}_0|$, where $\Phi(t)$ is the standard normal cumulative distribution function. Note that $|\mathcal{H}_0|$ can be estimated by $(p^2-p)/2$ due to the sparsity of $\O_L$. This leads to the following multiple testing procedure.
* Calculate the test statistics $W_{i,j}$.
* For given $0\leq \alpha\leq 1$, calculate
\[
\hat{t}=\inf\left\{0\leq t\leq 2(\log p)^{1/ 2}: \; \frac{2\{1-\Phi(t)\}(p^2-p)/2}{R(t)\vee 1}\leq \alpha\right\}.
\]
If $\hat{t}$ does not exist, set $\hat{t}=2(\log p)^{1/ 2}$.
* For $1\leq i<j\leq p$, reject $H_{0,i,j}$ if and only if $|W_{i,j}|\geq \hat{t}$.
§ THEORY
In this section, we analyze the theoretical properties of the global and multiple testing procedures for both the oracle and data-driven scenarios. We show that the data-driven procedures perform asymptotically as well as the oracle procedures and enjoy certain optimality under the regularity conditions. For separate treatment of the oracle and data-driven procedures, we now distinguish the notations of the two, and add the superscript “$o$" to denote the statistics and tests of the oracle procedures, e.g., $\hat{\be}_i^o, M_{nq}^o, \Psi_\alpha^o, \hat{t}^o$, and the superscript “$d$" to denote those of the data-driven procedures, e.g., $\hat{\be}_i^d, M_{nq}^d, \Psi_\alpha^d$, and $\hat{t}^d$.
§.§ Regularity conditions
For the oracle procedure, we require the following set of regularity conditions.
* Assume that $\max_{1\leq i\leq p}|\hat{\be}_{i}^{o} - \be_{i}|_1=o_{\rm p}[\{\log \max(p,q,n)\}^{-1}\}]$, and $\max_{1\leq i\leq p}|\hat{\be}_{i}^{o} - \be_{i}|_2=o_{\rm p}\left\{(nq\log p)^{-1/4}\right\}$.
* Assume that $\log p=o\{(nq)^{1/5}\}$, and there are constants $c_0, c_1>0$ such that, $c_0^{-1}\leq \lambda_{\min}(\O_L)\leq \lambda_{\max}(\O_L)\leq c_0$, and $c_1^{-1}\leq \lambda_{\min}(\O_T)\leq \lambda_{\max}(\O_T)\leq c_1$.
* Let $\D_{L}$ be the diagonal of $\O_L$ and let $\R_{L}=\D_L^{-1/2}\O_L\D_{L}^{-1/2}$ with elements $\eta_{L,i,j}, 1\leq i, j \leq p$. Assume that $\max_{1\leq i< j\leq p} |\eta_{L,i,j} | \leq \eta_L < 1$, for some constant $0 < \eta_L < 1$.
For the data-driven procedure, we replace the above condition (C1) with a slightly different one (C1$^\prime$), then introduce a new condition (C4).
(C1$^\prime$) Assume that $\max_{1\leq i\leq p,1\leq l\leq q}|\hat{\be}_{i}^{d} - \be_{i,l}|_1=o_{\rm p}[\{\log \max(p,q,n)\}^{-1}\}]$,
and $\max_{1\leq i\leq p,1\leq l\leq q}|\hat{\be}_{i}^{d} - \be_{i,l}|_2=o_{\rm p}\left\{(nq\log p)^{-1/4}\right\}$.
(C4) Define $s_p=\max_{1\leq l\leq q}\max_{1\leq i\leq p}\sum_{j=1}^p\max\{I(\omega_{L,i,j}\neq 0),I(\omega^d_{l,i,j}\neq 0)\}$, where $(\omega^d_{l,i,j})_{p\times p}$ $=\O^d_l=\Cov^{-1}\{(\X_k\hat{\S}_T^{-1/2})_{\cdot,l}\}$. Assume that $\|\O_T\|_{L_1}^2\|\O_L\|_{L_1}^2=$
$o\{\min(r_{1,n,p,q}, r_{2,n,p,q})\}$, where $r_{1,n,p,q}=[np/\{s_p^2q^3\log q\log^3 \max(p,q,n)\}]^{1/2}$, and $r_{2,n,p,q}=(np^2/[s_p^2q^7\{\log q\log\max(p,q,n)\}^2\log p])^{1/4}$.
A few remarks are in order. The estimator $\hat{\be}_{i}^{o}$ satisfying (C1) can be easily obtained via standard estimation methods such as Lasso and Dantzig selector. For instance, if one uses the Lasso estimator, then (C1) is satisfied under (C2) and the sparsity condition $\max_{1\leq i\leq p}|\be_i|_0=o[(nq)^{1/2}/\{\log \max(p,q,n)\}^{3/2}]$. Similarly, $\hat{\be}_{i}^{d}$ satisfying (C1$^\prime$) can be obtained by Lasso if (C4) holds and the data-driven regression coefficients ${\be}_{i,l}$ satisfy the similar sparsity condition. Conditions (C2) and (C3) are regularity conditions that are commonly used in the high-dimensional hypothesis testing setting <cit.>. (C4) is a mild technical condition. If $\O_T$, $\O_L$ and $\O_l^d$ satisfy $\max_{1\leq i\leq q}\sum_{j=1}^qI(\omega_{T,i,j}\neq 0)\leq s$ and $s_p\leq s$, for some constant $s>0$, then the conditions on matrix 1-norms can be relaxed to the conditions only related to $n, p$ and $q$, namely, $q^3\log q\log^3\max(p,q,n)=o(np)$ and $q^7\{\log q\log\max(p,q,n)\}^2\log p=o(np^2)$.
§.§ Oracle global testing procedure
We next analyze the limiting null distribution of the oracle global test statistic $M_{nq}^o$ and the power of the corresponding test $\Psi_{\alpha}^o$. We are particularly interested in the power of the test under the alternative when $\O_L$ is sparse, and show that the power is minimax rate optimal.
The following theorem states the asymptotic null distribution for $M_{nq}^o$, and indicates that, under $H_0$, $M_{nq}^o - 4\log p + \log\log p$ converges weakly to a Gumbel random variable with distribution function $\exp\{-({8\pi})^{-1/2}e^{-t/2}\}$.
Assume (C1), (C2) and (C3). Then under $H_{0}$, for any $t \in \real{}$,
\begin{eqnarray*} %\label{th1}
\pr(M_{nq}^o - 4\log p+\log\log p\leq t )\rightarrow \exp\{-({8\pi})^{-{1/2}}\exp(-{t/ 2})\}, \; \textrm{ as } nq,p\rightarrow\infty.
\end{eqnarray*}
Under $H_0$, the above convergence is uniform for all $\{\X_k\}_{k=1}^{n}$ satisfying (C1)-(C3).
We next study the power of the corresponding test $\Psi_{\alpha}^o$. We define the following class of precision matrices for spatial locations:
𝒰(c)={Ø_L: max_1≤i<j≤p|ω_L,i,j|/θ_i,j^1/2≥c(logp)^1/ 2}.
This class of matrices include all precision matrices such that there exists one standardized off-diagonal entry having the magnitude exceeding $c(\log p)^{1/2}$. By the definition in (<ref>), $\theta_{i,j}$ is of the order $1/(nq)$, and thus we only require one of the off-diagonal entries to have size larger than $C \{\log p/(nq)\}^{1/2}$ for some constant $C>0$, where $C$ is fully determined by $c_0$ and $c_1$ in Condition (C2). Then if we choose the constant $c=4$, that is, if there exists one standardized off-diagonal entry having the magnitude larger or equal than $4(\log p)^{1/2}$, the next theorem shows that the null parameter set in which $\O_L$ is diagonal is asymptotically distinguishable from $\mathcal{U}(4)$ by the test $\Psi_\alpha^o$. That is, $H_{0}$ is rejected by the test $\Psi_{\alpha}^o$ with overwhelming probability if $\O_L \in \mathcal{U}(4)$.
Assume (C1) and (C2). Then,
\[
\inf_{\O_L\in\mathcal{U}(4)}\pr(\Psi_{\alpha}^o=1)\rightarrow 1, \; \textrm{ as } nq,p \rightarrow\infty.
\]
The next theorem further shows that this lower bound $4(\log p)^{1/2}$ is rate-optimal. Let ${\cal T}_\alpha$ be the set of all $\alpha$-level tests, i.e., $\pr(T_{\alpha}=1)\leq \alpha$ under $H_{0}$ for all $T_{\alpha}\in{\cal T}_\alpha$.
Suppose that $\log p=o(nq)$. Let $\alpha,\beta>0$
and $\alpha+\beta<1$. Then there exists a constant $c_2>0$ such that for all sufficiently large $nq$ and $p$,
\begin{eqnarray*}
\inf_{\O_L\in\mathcal{U}(c_2)}\sup_{T_{\alpha}\in{\cal T}_\alpha}\pr(T_{\alpha}=1)\leq 1-\beta.
\end{eqnarray*}
As Theorem <ref> indicates, if $c_2$ is sufficiently small, then any $\alpha$ level test is unable to reject the null hypothesis correctly uniformly over $\O_L\in\mathcal{U}(c_2)$ with probability tending to one. So the order $(\log p)^{1/2}$ in the lower bound of $ \max_{1\leq i< j\leq p}\{ |\omega_{L,i,j}| \theta_{i,j} ^{-1/2}\}$ in (<ref>) cannot be further improved.
§.§ Oracle multiple testing procedure
We next investigate the properties of the oracle multiple testing procedure. The following theorem shows that the oracle procedure controls the false discovery proportion and false discovery rate at the pre-specified level $\alpha$ asymptotically.
Assume (C1) and (C2), and let
\[
\mathcal{S}_\rho= \left\{(i,j): 1\leq i<j\leq p, \frac{|\omega_{L,i,j}|}{\theta_{i,j}^{1/2}} \geq (\log p)^{{1/ 2}+\rho}\right\}.
\]
Suppose for some $\rho, \delta>0$, $|\mathcal{S}_\rho| \geq [{1}/\{({8\pi})^{1/2}\alpha\}+\delta]({\log \log p})^{1/2}$. Suppose $l_0= |\mathcal{H}_0|\geq c_0 p^2$ for some $c_0>0$, and $p\leq c(nq)^r$ for some $c, r > 0$. Letting $l=(p^2-p)/2$, then,
\[
\lim_{(nq,p)\rightarrow \infty}\frac{\text{FDR}(\hat{t}^o)}{\alpha l_0/l}=1,
\quad
\frac{\text{FDP}(\hat{t}^o)}{\alpha l_0/l} \rightarrow 1
\]
in probability, as $(nq,p)\rightarrow \infty$.
We comment that the condition $|\mathcal{S}_\rho| \geq [{1}/\{({8\pi})^{1/2}\alpha\}+\delta]({\log \log p})^{1/2}$ in Theorem <ref> is mild, because we have $(p^2-p)/2$ hypotheses in total and this condition only requires a few entries of $\O_L$ having standardized magnitude exceeding $\{(\log p)^{{1/ 2}+\rho}/(nq)\}^{1/2}$ for some constant $\rho>0$.
§.§ Data-driven procedures
We next turn to data-driven procedures for both the global testing and the multiple testing. We show that they perform as well as the oracle testing procedures asymptotically.
Assume (C1$^\prime$) , (C2)-(C4).
* Under $H_{0}$, for any $t \in \real{}$,
\begin{eqnarray*} %\label{th1.data}
\pr(M_{nq}^{d}-4\log p+\log\log p\leq t )\rightarrow \exp\{-({8\pi})^{-{1/2}}\exp(-{t/ 2})\}, \; \textrm{ as } nq,p\rightarrow\infty.
\end{eqnarray*}
Under $H_0$, the above convergence is uniform for all $\{\X_k\}_{k=1}^{n}$ satisfying (C1$^\prime$), (C2)-(C4).
* Furthermore,
\inf_{\O_L\in\mathcal{U}(4)}\pr(\Psi_{\alpha}^{d}=1)\rightarrow 1, \; \textrm{ as } nq,p\rightarrow\infty.
This theorem shows that $M_{nq}^d$ has the same limiting null distribution as the oracle test statistics $M_{nq}^o$, and the power of the corresponding test $\Psi_\alpha^{d}$ performs as well as the oracle test and is thus minimax rate optimal. The same observation applies to Theorem <ref> below, which shows that the data-driven multiple procedure also performs as well as the oracle case, and controls the false discovery proportion and false discovery rate at the pre-specified level $\alpha$ asymptotically.
Assume (C1$^\prime$) and (C4). Then under the same conditions as in Theorem <ref>,
\[
\lim_{(nq,p)\rightarrow \infty}\frac{\text{FDR}(\hat{t}^{d})}{\alpha l_0/l}=1,
\quad
\frac{\text{FDP}(\hat{t}^{d})}{\alpha l_0/l} \rightarrow 1
\]
in probability, as $(nq,p)\rightarrow \infty$.
§ SIMULATIONS
We study in this section the finite-sample performance of the proposed testing procedures. For the global testing of (<ref>), we measure the size and power of the oracle test $\Psi_{\alpha}^{o}$ and the data-driven version $\Psi_{\alpha}^{d}$, and for the multiple testing of (<ref>), we measure the empirical FDR and power. We compare the oracle and data-driven testing procedures, as well as a simple alternative that was developed by <cit.> under normal instead of matrix normal distribution, which ignores the separable spatial-temporal structure. The temporal covariance matrix $\S_T$ is constructed with elements $\sigma_{T,i,j}=0.4^{|i-j|}$, $1\leq i,j\leq p$. The sample size and the number of time points is set at $n=20$, $q=20$ and $n=50$, $q=30$, respectively, whereas the spatial dimension $p$ varies among $\{50, 200, 400, 800\}$. We have chosen this setting, since our primary interest is on inferring about spatial connectivity networks with different spatial dimensions. We keep the temporal dimension small, since it is a nuisance in our setup, and choose a relatively small sample size to reflect the fact that there is usually only a limited sample size in many neuroimaging studies.
For each generated dataset below, we use Lasso to estimate $\be_i$ as
where $\Y$ is the $nq\times p$ data matrix by stacking the transformed samples $\{(\Y_{k,\cdot,l}, k=1,\dots, n, l=1,\dots, q\}$, where $\Y_k = \X_{k}{\S}_{T}^{-1/2}$ for the oracle procedure and $\Y_k = \X_{k}\hat{\S}_{T}^{-1/2}$ for the data-driven procedure, $k=1,\ldots,n$, $\D_{i}=\text{diag}(\hat{\S}_{L,-i,-i})$, and $\hat{\S}_L$ is the sample covariance matrix of $\S_L$ with $nq$ transformed samples, and $\lambda_{n,i}=\kappa \{\hat{\S}_{L,i,i}\log p/(nq) \}^{1/2}$.
§.§ Global testing simulation
For the global testing, the data $\{\X_1, \dots, \X_n\}$ are generated from a matrix normal distribution with mean zero and precision matrix $\I\otimes \O_T$ under the null. To evaluate the power, let $\U$ be a matrix with eight random nonzero entries. The locations of four nonzero entries are selected randomly from the upper triangle of $\U$, each with a magnitude generated randomly and uniformly from the set $[-4\{{\log p/(nq)}\}^{1/2},-2\{{\log p/(nq)}\}^{1/2}] \cup [2\{{\log p/(nq)}\}^{1/2},4\{{\log p/(nq)}\}^{1/2}]$. The other four nonzero entries in the lower triangle are determined by symmetry. We set $\O_L=(\I+\U+\delta\I)/(1+\delta)$, with $\delta=|\lambda_{\min} (\I+\U)|+0.05$, and choose the tuning parameter $\kappa=2$ in (<ref>).
The size and power, in percentage, of the global testing are reported in Table <ref>, based on 1000 data replications and the significance level $\alpha_1=0.05$. We see from Table <ref> that the empirical sizes of the proposed oracle and data-driven procedures are well controlled under the significance level $\alpha_1=0.05$. However, for the vector normal based procedure that ignores the spatial-temporal dependence structure, there is a serious size distortion across all settings. The empirical sizes for the new procedures are slightly below the nominal level for high dimensions, due to the correlation among the variables. Similar phenomenon has also been observed and justified in <cit.>. We also see from the table that the new procedures are powerful in all settings, even though the two spatial precision matrices differ only in eight entries with the magnitude of difference of the order $\{\log p/(nq)\}^{1/2}$. For both the empirical sizes and powers, the data-driven procedure is seen to perform similarly as the oracle procedure.
method 1|c|$p=50$ 1|c|$p=200$ 1|c|$p=400$ 1|c|$p=800$
oracle 3.6 3.5 2.8 2.9
data-driven 3.8 3.8 2.9 2.9
vector normal 36.4 56.7 64.7 75.3
oracle 3.7 3.5 5.2 4.3
data-driven 3.5 3.5 5.1 4.1
vector normal 39.7 64.8 73.5 88.4
oracle 77.9 85.6 87.7 90.9
data-driven 83.1 88.1 87.8 90.7
vector normal 86.2 94.1 95.0 99.1
oracle 62.4 74.3 68.2 75.2
data-driven 66.1 74.7 68.3 75.5
vector normal 76.2 92.1 93.1 90.1
Empirical sizes and powers (%) for global testing. Three methods are compared: the proposed oracle and data-driven procedures based on the matrix normal distribution, and the simple alternative based on the vector normal distribution that ignores the spatial-temporal dependency. The results are based on 1000 data replications.
§.§ Multiple testing simulation
For the multiple testing, the data $\{\X_1, \dots, \X_n\}$ are generated from a matrix normal distribution with mean zero and precision matrix $\O_L\otimes \O_T$. Three choices of $\O_L$ are considered:
* $\O_L^{(1)}=(\omega^{(1)}_{L,i,j})$ where $\omega^{(1)}_{L,i,i}=1$, $\omega^{(1)}_{L,i,i+1}=\omega^{(1)}_{L,i+1,i}=0.6$, $\omega^{(1)}_{L,i,i+2}=\omega^{(1)}_{L,i+2,i}=0.3$ and $\omega^{(1)}_{L,i,j}=0$ otherwise.
* $\O_L^{*(2)}=(\omega^{*(2)}_{L,i,j})$ where $\omega^{*(2)}_{L,i,j}=\omega^{*(2)}_{L,j,i}=0.5$ for $i=10(k-1)+1$ and $10(k-1)+2\leq j\leq 10(k-1)+10$, $1\leq k\leq p/10$. $\omega^{*(2)}_{L,i,j}=0$ otherwise. $\O_L^{(2)}=(\O_L^{*(2)}+\delta\I)/(1+\delta)$ with $\delta=|\lambda_{\min}(\O_L^{*(2)})|+0.05$.
* $\O_L^{*(3)}=(\omega^{*(3)}_{L,i,j})$ where $\omega^{*(3)}_{L,i,i}=1$, $\omega^{*(3)}_{L,i,j}= 0.8\times\text{Bernoulli}(1,2/p)$ for $i < j$ and $\omega^{*(3)}_{L,j,i}=\omega^{*(3)}_{L,i,j}$. $\O_L^{(3)}=(\O_L^{*(3)}+\delta\I)/(1+\delta)$ with $\delta=|\lambda_{\min}(\O_L^{*(3)})|+0.05$.
We select the tuning parameters $\lambda_{n,i}$ in (<ref>) in the Lasso estimation adaptively given the data, following the general principle of making $\sum_{(i,j)\in \mathcal{H}_0}I(|W_{i,j}|\geq t)$ and $\{2-2\Phi(t)\}(p^2-p)/2$ close. The steps of parameter tuning are summarized as follows.
* Let $\lambda_{n,i}=b/20\sqrt{\hat{\S}_{L,i,i}\log p/(nq)}$, for $b=1,\cdots,40$. For each $b$, calculate $\hat{\be}^{(b)}_{i}$, $i=1,\cdots,p$, and construct the corresponding standardized statistics $W_{i,j}^{(b)}$.
* Choose $\hat{b}$ as the minimizer of
\[
\sum_{s=1}^{10} \left( \frac{\sum_{(i,j)\in \cH}I(|W_{i,j}|^{(b)}\geq \Phi^{-1}[1-s\{1-\Phi(\sqrt{\log p})\}/10])}{s\{1-\Phi(\sqrt{\log p})\}/10\cdot p(p-1)}-1 \right)^2.
\]
* The tuning parameters $\lambda_{n,i}$ are then set as,
\[
\lambda_{n,i}=\hat{b}/20\sqrt{\hat{\S}_{L,i,i}\log p/(nq)}.
\]
For comparison, we also carry out the alternative procedure that ignores the Kronecker product structure by using the stacked original data samples $\{\X_{k,\cdot,l}, k=1,\dots, n, l=1,\dots, q\}$.
The empirical FDR and the empirical power of FDR control, in percentage, are summarized in Tables <ref> and <ref>, based on 100 data replications and the FDR level set at $\alpha_2=0.1$ and $\alpha_3=0.01$, respectively. In particular, the power is calculated as
\[
\frac{1}{100}\sum_{l=1}^{100}\frac{\sum_{(i,j)\in \mathcal{H}_1}I(|W_{i,j,l}|\geq \hat{t})}{|\mathcal{H}_1|},
\]
where $W_{i,j,l}$ denotes standardized statistic for the $l$-th replication and $\mathcal{H}_1$ denotes the nonzero locations. We observe from Tables <ref> and <ref> a similar pattern as that from Table <ref>. That is, the empirical FDRs of the proposed oracle and data-driven procedures are both close to the significance levels across all settings, whereas the vector normal based procedure ignoring the spatial-temporal dependence structure yields empirical FDRs much larger than the significance levels. We also see from the table that the new procedures achieve a high power, and the data-driven procedure again performs similarly as the oracle procedure.
4|c|Model 1 4|c|Model 2 4|c|Model 3
$\alpha$ $p$ 1|c| 50 1|c| 200 1|c| 400 1|c| 800 1|c| 50 1|c| 200 1|c| 400 1|c| 800 1|c| 50 1|c| 200 1|c| 400 1|c| 800
12|c|Empirical FDR (in %)
oracle 7.4 6.7 6.4 6.0 8.9 8.5 8.2 7.8 8.9 8.4 8.0 7.8
data-driven 8.0 6.9 6.5 6.0 11.4 9.9 9.0 8.2 11.4 9.3 8.4 7.9
vector normal 18.9 23.1 22.7 20.7 24.5 34.1 36.5 36.5 26.1 32.0 33.5 33.8
oracle 0.6 0.5 0.4 0.7 0.9 0.8 0.6 0.5 0.8 0.7 0.6 0.7
data-driven 0.6 0.5 0.4 0.7 1.2 0.9 0.7 0.5 1.2 0.9 0.7 0.7
vector normal 2.5 3.2 2.6 5.6 4.6 5.6 5.4 2.2 4.4 5.3 0.5 5.3
12|c|Empirical Power (in %)
oracle 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.9 99.9 99.7
data-driven 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.9 99.9 99.7
vector normal 100.0 99.9 99.9 99.9 100.0 100.0 100.0 100.0 100.0 99.8 99.6 99.2
oracle 99.9 99.9 99.8 100.0 100.0 100.0 100.0 99.8 100.0 99.7 99.2 98.5
data-driven 99.9 99.9 99.8 100.0 99.9 99.9 100.0 99.8 100.0 99.6 99.1 98.5
vector normal 99.8 99.7 99.6 99.7 99.9 99.7 99.7 99.4 99.9 99.0 97.9 96.7
Empirical FDRs and powers (%) for multiple testing with $n=20$ and $q=20$. Three methods are compared: the proposed oracle and data-driven procedures based on the matrix normal distribution, and the simple alternative based on the vector normal distribution that ignores the spatial-temporal dependency. The results are based on 100 data replications.
4|c|Model 1 4|c|Model 2 4|c|Model 3
$\alpha$ $p$ 1|c| 50 1|c| 200 1|c| 400 1|c| 800 1|c| 50 1|c| 200 1|c| 400 1|c| 800 1|c| 50 1|c| 200 1|c| 400 1|c| 800
12|c|Empirical FDR (in %)
oracle 8.1 7.7 8.0 7.7 8.9 9.1 8.8 8.6 8.9 8.7 8.4 8.2
data-driven 8.4 7.9 8.1 7.8 10.3 10.0 9.2 8.9 11.7 9.3 8.8 8.2
vector normal 23.3 29.8 33.6 36.1 28.2 39.8 44.4 49.2 29.0 37.5 42.2 45.8
oracle 0.7 0.6 0.6 0.6 1.1 0.8 0.7 0.8 0.7 0.8 0.8 0.7
data-driven 0.7 0.6 0.6 0.6 1.2 0.9 0.8 0.8 0.9 0.9 0.8 0.7
vector normal 3.8 4.8 5.5 5.6 5.9 7.1 7.3 9.1 5.1 6.5 7.3 8.3
12|c|Empirical Power (in %)
oracle 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
data-driven 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
vector normal 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
oracle 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
data-driven 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
vector normal 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Empirical FDRs and powers (%) for multiple testing with $n=50$ and $q=30$. Three methods are compared: the proposed oracle and data-driven procedures based on the matrix normal distribution, and the simple alternative based on the vector normal distribution that ignores the spatial-temporal dependency. The results are based on 100 data replications.
§ REAL DATA ANALYSIS
We illustrate our testing method on an electroencephalography (EEG) data. The data was collected in a study examining EEG correlates of genetic predisposition to alcoholism and is available at . It consists of 77 alcoholic individuals and 45 controls, and each subject was fitted with a 61-lead electrode cap and was recorded 256 samples per second for one second. There were in addition a ground and two bipolar deviation electrodes, which are excluded from the analysis. The electrode positions were located at standard sites (Standard Electrode Position Nomenclature, American Electroencephalographic Association 1990), and were organized into frontal, central, parietal, occipital, left temporal and right temporal regions. Each subject performed 120 trials under three types of stimuli. More details of data collection can be found in <cit.>. We preprocessed the data in the following ways. Similarly as <cit.>, we focused only on the average of all trials under a single stimulus condition for each subject. We then performed an $\alpha$ band filtering on the signals following <cit.>. Finally, we downsized the temporal dimension by averaging signals at eight consecutive times points. This is to facilitate estimation of the temporal covariance in our testing procedure since it is treated as a nuisance. More discussion of temporal covariance estimation is given in Section <ref>. The resulting data is a $61 \times 32$ matrix for each subject, and our goal is to infer the $61 \times 61$ connectivity network of the brain spatial locations.
We applied our testing procedures for the alcoholic and control groups separately. We first applied the global test and obtained the $p$-value $0$ for the alcoholic group, and $1.89e^{-15}$ for the control, clearly indicating that some brain regions are connected in the two groups. We then applied the data-driven multiple testing procedure, with a pre-specified FDR significance level $\alpha=0.01$. There were totally $61 \times 60 / 2=1,830$ pairs of spatial locations, among which we identified 208 significant pairs for the alcoholic group, and 155 pairs for the control group. For graphical illustration, we report the top 30 most significant pairs of spatial locations, ordered by their p-values, in Figure <ref>. Examining the connection patterns among those electrodes in the frontal region (denoted by symbols FP, AF, and F), we noted a slight decrease in connections and some asymmetry between the left and right frontal regions in the alcoholic group compared to the control. A similar phenomenon has been observed in <cit.>. We observed similar connection decrease in the central region (FC and C) for the alcoholic group, but connection increase for the parietal region (CP and P). Such findings require additional scientific validation. We also repeated the multiple testing procedure for the downsized EEG data with temporal dimension equal to 16, and observed similar but less clear patterns. For the sake of space, we omit the plots here. In summary, our testing procedure produces a useful list of connections warranted for further examination.
Alcoholic group Control group
Connectivity network inferred by the multiple testing procedure for the EEG data. The left panel is for the alcoholic group, and the right panel for the control. Top 30 significant links are shown in this graph.
§ DISCUSSION
We have proposed in this article global and multiple testing procedures under the matrix normal distribution for detecting the conditional dependence between spatial locations. It has been shown that the procedures perform well both theoretically and numerically. In this section, we discuss the strength and some potential limitations of our methods. We also explore some alternatives and point out possible future extension of our work.
Our work is one of the few to tackle hypothesis testing aspect of connectivity analysis, and is a useful complement to the general literature of statistical estimation and inference regarding high-dimensional precision matrix. The tests and associated theory have been built upon some of the early works on precision matrix inference such as <cit.> and <cit.>. However, our work is the first that explicitly exploits the special covariance (precision) structure of a matrix-variate normal distribution, and our simulations suggest using such information would improve the test. A strength of our method is that it works for an adequately large network; in our simulations, the number of nodes varied from 50 to 800, which encompasses the typical region/location-based connectivity analysis in neuroscience. A potential limitation is that our method treats the temporal component as nuisance, and the developed tests hinge on the accuracy of the temporal covariance estimation. In this article, we have simply used the usual sample covariance matrix $\hat{\S}_T$ to estimate $\S_T$. As such the method usually prefers a relatively small temporal dimension.
Then a potential extension of our proposal is to employ some alternative estimators of $\S_T$. We note that, many such estimators can be used in conjunction with our proposed testing procedures, as long as the estimator $\tilde{\S}_T$ satisfies the condition,
\[
\|\tilde{\S}_T-c\S_T\|_{\infty}=O_{\rm p}[\{\log q/(np)\}^{1/2}],
\]
for arbitrary constant $c>0$. As an example, if the temporal covariance matrix is sparse, in the sense that $\max_{1\leq i\leq q}\sum_{j=1}^qI(\sigma_{T,i,j}\neq 0)\leq c'$ for some constant $c'>0$, then the adaptive thresholding estimator proposed in <cit.> satisfies the above condition and can be used in our testing procedures. Another alternative is that one can directly estimate the precision matrix $\O_T$, and base the testing procedures on $\{\X_k\hat{\O}_T^{1/2}\}_{k=1}^{n}$, as long as the estimator $\tilde{\O}_T$ satisfies that,
\[
\|{\tilde{\O}}_T-c\O_T\|_{\infty}=O_{\rm p}\Big{[}\|\O_T\|_{L_1}^2\{\log q/(np)\}^{1/2}\Big{]},
\]
for any constant $c>0$. For instance, if the precision matrix $\O_T$ belongs to the following uniformity class of matrices
\[
\mathcal{U}=\Big{\{}\O_T:\max_{1\leq i\leq q}\sum_{j=1}^q|\omega_{T,i,j}|^l\leq s_0(q)\Big{\}},
\]
for $0\leq l<1$, then the clime estimator of <cit.> can be employed.
In this article, we have primarily focused on the one-sample inference scenario, where we assume the matrix-valued observations came from a single population. In the EEG data analysis, we simply carried out the tests for the two populations separately. Practically, it is of equal interest to study the two-sample scenario, where the aim is to identify changes of the conditional dependence structures for spatial locations across two or more populations. Specifically, let $\X, \Y \in \real{p \times q}$ follow two matrix normal distributions with the Kronecker product covariance structures, $\S\{\text{vec}(\X)\}=\S_{L1} \otimes \S_{T1}, \mbox{ and }\S\{\text{vec}(\Y)\}=\S_{L2} \otimes \S_{T2}$, respectively. The goal becomes the global testing of
\[
H_0:~ \O_{L1} = \O_{L2} \; \text{ versus } \; H_1:~ \O_{L1}\neq \O_{L2},
\]
and simultaneous testing of
\[
H_{0,i,j}: \omega_{L1,i,j}= \omega_{L2,i,j} \; \mbox{ versus } \; H_{1,i,j}: \omega_{L1,i,j}\neq \omega_{L2,i,j}, \quad\mbox{ $1\leq i<j\leq p$,}
\]
where $\omega_{L1,i,j}$ is the $(i,j)$th element of $\S_{L1}^{-1}=\O_{L1}=(\omega_{L1,i,j})$, and $\omega_{L2,i,j}$ is the $(i,j)$th element of $\S_{L2}^{-1}=\O_{L2}=(\omega_{L2,i,j})$. In the one-sample scenario, we have used the sample covariance matrix $\hat\S_T$ to estimate temporal covariance matrix, because it can be turned into an unbiased estimator without affecting the test statistics, as shown in Remark <ref>. However, in the two-sample scenario, Remark <ref> can no longer apply, and thus trace$(\S_{L1})$ and trace$(\S_{L2})$ need to be carefully estimated in order to get good estimators for temporal covariance matrices. Consequently, the two-sample comparison is technically much more challenging, and we leave it as our future research.
§ APPENDIX: PROOFS
§.§ Technical Lemmas
We prove the main results in this section. We begin by collecting some technical lemmas.
Let $B=\cup_{t=1}^{p} B_{t}$. For any $k<[p/2]$, we have
\begin{equation*}
\sum_{t=1}^{2k}(-1)^{t-1}F_{t}\leq\pr(B)\leq
\sum_{t=1}^{2k-1}(-1)^{t-1}F_{t},
\end{equation*}
where $ F_{t}=\sum_{1\leq i_{1}<\cdots<i_{t}\leq
p}\pr(B_{i_{1}}\cap\cdots\cap B_{i_{t}}).$
Define $U_{i,j}=\frac{1}{nq}\sum_{k=1}^{n}\sum_{l=1}^q\{\epsilon_{k,i,l}\epsilon_{k,j,l}/(r_{i,i}r_{j,j})-\ep\epsilon_{k,i,l}\epsilon_{k,j,l}/(r_{i,i}r_{j,j})\}$. The follow lemma states the results in the oracle case.
Suppose that (C1) and (C2) hold. Then we have
\[
\max_{1\leq i\leq p}|\hat{r}_{i,i}-r_{i,i}|=o_{\rm p}[\{{\log p/(nq)}\}^{1/2}],
\]
\[
\tilde{r}_{i,j}=\tilde{R}_{i,j}-\tilde{r}_{i,i}(\hat{\beta}_{i,j}-\beta_{i,j})-\tilde{r}_{j,j}(\hat{\beta}_{j-1,i}-\beta_{j-1,i})+o_{\rm p}\{(nq\log p)^{-1/2}\},
\]
for $1\leq i<j\leq p$, where $\tilde{R}_{i,j}$ is the empirical covariance between $\{\epsilon_{k,i,l}, k=1,\dots,n, l=1,\dots,q\}$ and $\{\epsilon_{k,j,l}, k = 1, \dots, n, l=1,\dots,q\}$. Consequently, uniformly in $1\leq i<j\leq p$,
\[
\hat{r}_{i,j}-(\omega_{L,i,i}\hat{\sigma}_{i,i,\epsilon}+\omega_{L,j,j}\hat{\sigma}_{j,j,\epsilon}-1)r_{i,j}=-U_{i,j}+o_{\rm p}\{(nq\log p)^{-1/2}\},
\]
where $(\hat{\sigma}_{i,j,\epsilon})=\frac{1}{nq}\sum_{k=1}^{n}\sum_{l=1}^q(\eps_{k,,l}-\bar{\eps}_l)(\eps_{k,,l}-\bar{\eps}_l)'$,
$\eps_{k,,l}=(\epsilon_{k,1,l},\dots,\epsilon_{k,p,l})$ and $\bar{\eps}_l=\frac{1}{n}\sum_{k=1}^{n}\eps_{k,,l}$.
This lemma is essentially proved in <cit.> with $nq$ inverse regression models instead.
§.§ Proof of Theorem <ref>
Without loss of generality, throughout this section, we assume that $\omega_{L,i,i}=1$ for $i=1,\ldots,p$.
\[
\]
By Lemma <ref>, we have
\[
\max_{1\leq i\leq p}|\hat{r}_{i,i}-\tilde{R}_{i,i}|=o_{\rm p}\{(nq\log p)^{-1/2}\}.
\]
Note that
max_1≤i< j≤p(β̂_i,j^2r̂_i,i/r̂_j,j-ρ_i,j^2)=o_p(1/logp),
Then by Lemma <ref>,
it is easy to see that, under conditions (C1) and (C2), we have under $H_0$
\[
\max_{1\leq i<j\leq p}||W_{i,j}|-|V_{i,j}||=o_{\rm p}\{(\log p)^{-1/2}\}.
\]
Thus it suffices to prove that
\[
\pr(\max_{1\leq i< j\leq p} V^2_{i,j}-4\log p+\log\log p\leq t )\rightarrow \exp\{-({8\pi})^{-{1/ 2}}\exp(-{t/ 2})\}.
\]
We arrange the indices $\{1\leq i< j\leq p\}$ in any ordering and set them as $\{(i_{m},j_{m}): m=1,\dots,s\}$ with $s=p(p-1)/2$. Let $\theta_{m}=\Var(\epsilon_{k,i_m,l}\epsilon_{k,j_m,l})$, and define
$Z_{k,m,l} =\epsilon_{k,i_{m},l}\epsilon_{k,j_{m},l}$ for $1\leq k\leq n$ and $1\leq l\leq q$,
$V_{m} = {(nq\theta_m)^{-1/2}}\sum_{k=1}^{n}\sum_{l=1}^qZ_{k,m,l}$, and
$\hat{V}_{m} ={(nq\theta_m)^{-1/2}}\sum_{k=1}^{n}\sum_{l=1}^q\hat{Z}_{k,m,l}$,
$ \hat{Z}_{k,m,l}=Z_{k,m,l}I(|Z_{k,m,l}|\leq \tau_{n})-\ep \{Z_{k,m,l}I(|Z_{k,m,l}|\leq \tau_{n})\}$,
and $\tau_{n}=32\log (p+nq)$.
Note that
$\max_{1\leq i<j\leq p}V^{2}_{i,j}=\max_{1\leq m\leq s}V_{m}^{2}$, and that
\begin{eqnarray*}
\max_{1\leq m\leq s}(nq)^{-{1/ 2}}\sum_{k=1}^{n}\sum_{l=1}^q&&\ep[|Z_{k,m,l}|I\{|Z_{k,m,l}|\geq 32\log (p+nq)\}]\cr
&&\leq C(nq)^{1/ 2}\max_{1\leq k\leq n}\max_{1\leq l\leq q}\max_{1\leq m\leq s}\ep[|Z_{k,m,l}|I\{|Z_{k,m,l}|\geq 32\log (p+nq)\}]\cr
&&\leq C(nq)^{1/ 2}(p+nq)^{-4}\max_{1\leq k\leq n}\max_{1\leq l\leq q}\max_{1\leq m\leq s}\ep[|Z_{k,m,l}|\exp\{|Z_{k,m,l}|/8\}]\cr
&&\leq C(nq)^{1/ 2}(p+nq)^{-4}.
\end{eqnarray*}
This yields to
\[
\pr\Big{\{}\max_{1\leq m\leq s}|V_{m}-\hat{V}_{m}|\geq (\log p)^{-1}\Big{\}}\leq \pr\Big{(}\max_{1\leq m\leq s}\max_{1\leq k\leq n}\max_{1\leq l\leq q}|Z_{k,m,l}|\geq
\tau_{n}\Big{)}=O(p^{-1}).
\]
Note that
\[
\Big{|}\max_{1\leq m\leq s}V_{m}^{2}-\max_{1\leq m\leq s}\hat{V}_{m}^{2}\Big{|}\leq 2\max_{1\leq m\leq s}|\hat{V}_{m}|\max_{1\leq m\leq s}|V_{m}-\hat{V}_{m}|+\max_{1\leq m\leq s}|V_{m}-\hat{V}_{m}|^{2},
\]
it suffices to prove that for any $t\in \RR$, as $nq,p\rightarrow\infty$,
( max_1≤m≤sV̂^2_m-4logp+loglogp≤t)→exp{-(8π)^-1/ 2exp(-t/ 2)}.
By Lemma <ref>, for any integer $l$ with $0<h<s/2$,
\begin{eqnarray}\label{p9}
\sum_{d=1}^{2h}(-1)^{d-1}\sum_{1\leq m_{1}<\cdots<m_{d}\leq q}
\pr\Bigg{(}\bigcap_{j=1}^{d}F_{m_{j}}\Bigg{)}
&\le& \pr\Big{(}\max_{1\leq m\leq s}\hat{V}^{2}_{m}\geq y_{p}\Big{)}\nonumber \\
&\le& \sum_{d=1}^{2h-1}(-1)^{d-1}\sum_{1\leq m_{1}<\cdots<m_{d}\leq q} \pr\Bigg{(}\bigcap_{j=1}^{d}F_{m_{j}}\Bigg{)}, \quad\quad
\end{eqnarray}
where $y_p=4\log p-\log\log p+t$ and
F_{m_{j}}=(\hat{V}^{2}_{m_{j}}\geq y_{p}).
Let $\tilde{Z}_{k,m,l}=\hat{Z}_{k,m,l}/(\theta_{m})^{1/2}$ for $m=1,\dots,s$ and
\W_{k,l}=(\tilde{Z}_{k,m_{1},l},\ldots,\tilde{Z}_{k,m_{d},l}),
$ for $1\leq k\leq n$ and $1\leq l\leq q$. Define $|\a|_{\min}=\min_{1\leq i\leq d}|a_{i}|$ for any vector $\a\in R^{d}$. Then we have
\[
\pr\Bigg{(}\bigcap_{j=1}^{d}F_{m_{j}}\Bigg{)}=\pr\Bigg{(}\Bigg{|}(nq)^{-{1\over 2}}\sum_{k=1}^{n}\sum_{l=1}^q\W_{k,l}\Bigg{|}_{\min}\geq y^{1\over 2}_{p}\Bigg{)}.
\]
Then it follows from Theorem 1 in Zaïtsev (1987) that
\begin{eqnarray}\label{p10}
\pr\Bigg{(}\Bigg{|}(nq)^{-1/2}\sum_{k=1}^{n}\sum_{l=1}^q\W_{k,l}\Bigg{|}_{\min}\geq y^{1/2}_{p}\Bigg{)}&\leq& \pr
\Big{\{}|\N_{d}|_{\min}\geq y^{1/ 2}_{p}-\epsilon_{n}(\log p)^{-{1/ 2}}\Big{\}}\cr
& &+ c_{1}d^{5\over 2}\exp\Bigg{\{}-\frac{(nq)^{1/2}\epsilon_{n}}{c_{2}d^{3}\tau_{n}(\log p)^{1/2}}\Bigg{\}},
\end{eqnarray}
where $c_{1}>0$ and $c_{2}>0$ are constants, $\epsilon_{n}\rightarrow 0$ which will be specified later and
$\N_{d}=(N_{m_{1}},\ldots,N_{m_{d}})$ is a normal random vector with $\ep (\N_{d})=0$ and $\Cov(\N_{d})=\Cov(\W_{1,1})$.
Recall that $d$ is a fixed integer which does not depend on $n,p,q$. Because $\log p=o((nq)^{1/5})$, we can let $\epsilon_{n}\rightarrow 0$ sufficiently slowly that, for any large $M>0$
c_1d^5/ 2exp{-(nq)^1/2ϵ_n/c_2d^3τ_n(logp)^1/2}=O(p^-M).
Combining (<ref>), (<ref>) and (<ref>) we have
(max_1≤m≤sV̂^2_m≥y_p)≤∑_d=1^2h-1(-1)^d-1∑_1≤m_1<⋯<m_d≤s{|_d|_min≥y^1/ 2_p-ϵ_n(logp)^-1/ 2}+o(1).
Similarly, using Theorem 1 in <cit.> again, we can get
(max_1≤m≤sV̂^2_m≥y_p)≥∑_d=1^2h(-1)^d-1∑_1≤m_1<⋯<m_d≤s{|_d|_min≥y^1/ 2_p+ϵ_n(logp)^-1/ 2}-o(1).
We recall the following lemma, which is shown in the supplementary material of <cit.>.
For any fixed integer $d\geq 1$ and real number $t\in \RR$,
∑_1≤m_1<⋯<m_d≤q{|_d|_min≥y^1/2_p±ϵ_n(logp)^-1/2}=1d!{(8π)^-1/2exp(-t/ 2)}^d{1+o(1)}.
Then Lemma <ref>, (<ref>) and (<ref>) yield that
lim sup_nq,p→∞(max_1≤m≤sV̂^2_m≥y_p) ≤ ∑_d=1^2h(-1)^d-11d!{(8π)^-1/ 2exp(-t/ 2)}^d
lim inf_nq,p→∞(max_1≤m≤sV̂^2_m≥y_p) ≥ ∑_d=1^2h-1(-1)^d-11d!{(8π)^-1/ 2exp(-t/2)}^d
for any positive integer $h$. By letting $h\rightarrow\infty$, we obtain (<ref>) and Theorem <ref> is proved.
§.§ Proof of Theorem <ref>
By Lemma <ref>, we have
\[
\max_{1\leq i<j\leq p}\Big{|}\frac{T_{i,j}-\{1+o(1)\}\ep T_{i,j}}{\hat{\theta}_{i,j}^{1/2}}-V_{i,j}\Big{|}=o_{\rm p}\{(\log p)^{-1/2}\}.
\]
\[
M_{nq}^{1}=\max_{1\leq i<j\leq p}\frac{T_{i,j}-\{1+o(1)\}\ep T_{i,j}}{\hat{\theta}_{i,j}^{1/2}}.
\]
Then the proof of Theorem <ref> yields
\[
\pr(M_{nq}^1\leq 4\log p- 2^{-1}\log \log p)\rightarrow 1,
\]
as $nq,p\rightarrow \infty$. By (<ref>), (<ref>) and the fact that
\[
\max_{1\leq i< j\leq p}\omega_{L,i,j}^2/\hat{\theta}_{i,j}=\max_{1\leq i< j\leq p}[\{1+o(1)\}\ep T_{i,j}]^2/\hat{\theta}_{i,j}\leq 2M_{nq}^{1}+2M_{nq},
\]
the fact that
\[
\max_{1\leq i< j\leq p} |\omega_{L,i,j}|/\theta_{i,j}^{1/2}\geq 4(\log p)^{1/2},
\]
we have
\[
\pr(M_{nq}\geq q_{\alpha}+4\log p-\log\log p)\rightarrow 1
\]
as $nq,p\rightarrow \infty$.
§.§ Proof of Theorem <ref>
This theorem is essentially proved in <cit.>, we skip the proof here.
§.§ Proof of Theorem <ref>
By separation of spatial-temporal dependence structure, we have the following $nq$ inverse regression models
\[
(\X_{k}{\S}_{T}^{-1/2})_{i,l} = \alpha_{i,l} + (\X_{k}{\S}_{T}^{-1/2})_{-i,l}\be_{i}+\epsilon_{k,i,l}, \quad 1\leq k\leq n, 1\leq l\leq q.
\]
Then Theorem <ref> is proved by applying Theorem 3.1 in <cit.> in $nq$ regression models.
§.§ Proof of Theorems <ref> and <ref>
We use the superscript “$d$” to denote the corresponding statistics we derived from the the data-driven inverse regression models
\[
Y^{d}_{k,i,l} = (\Y_{k,-i,l}^{d})^{\T}\be^{d}_{i,l}+\epsilon^{d}_{k,i,l}, \quad 1\leq i \leq p, 1\leq l \leq q,
\]
where $\Y^{d}_k = \X_{k}\hat{\S}_{T}^{-1/2}$, for $k=1,\ldots,n$.
By the proofs of Theorem <ref> and Theorem <ref>, it suffices to prove that uniformly in $1\leq i\leq j\leq p$,
where $\hat{r}_{i,j}^{d}=-(\tilde{r}_{i,j}^{d}+\tilde{r}_{i,i}^{d}\hat{\beta}_{i,j}^d+\tilde{r}_{j,j}^{d}\hat{\beta}_{j-1,i}^d)$ with $\tilde{r}_{i,j}^{d}=1/(nq)\sum_{k=1}^n\sum_{l=1}^q\hat{\epsilon}_{k,i,l}^d\hat{\epsilon}_{k,j,l}^d$ and $\hat{\epsilon}_{k,i,l}^d=Y_k^d-(\Y_{k,-i,l}^{d})^{\T}\hat{\be}^{d}_{i,l}$.
Let $\tilde{\epsilon}_{k,i,l}=\epsilon_{k,i,l}-\bar{\epsilon}_{i,l}$. Then we have
ϵ̂_k,i,l^dϵ̂_k,j,l^d = ϵ̃_k,i,lϵ̃_k,j,l-ϵ̃_k,i,l(_k,-j,l-_-j,l)^(_j^d-_j)-ϵ̃_k,j,l(_k,-i,l-_-i,l)^(_i^d-_i) +(_i^d-_i)^(_k,-i,l-_-i,l)(_k,-j,l-_-j,l)^(_j^d-_j) +[{Y_k,i,l^d-Y̅_i,l^d-(_k,-i,l^d-_-i,l^d)^_i^d}{Y_k,j,l^d-Y̅_j,l^d-(_k,-j,l^d-_-j,l^d)^_j^d}- {Y_k,i,l-Y̅_i,l-(_k,-i,l-_-i,l)^_i^d}{Y_k,j,l-Y̅_j,l-(_k,-j,l-_-j,l)^_j^d}].
Note that
\[
\hat{\S}_T=\frac{1}{np}\sum_{k=1}^n\X_k^{\T}\X_k=\frac{1}{np}\sum_{k=1}^n\Z_k^{\T}\S_L\Z_k,
\]
where $\Z_k=\S_L^{-1/2}\X_k$. Let $\S_L=U^{\T}\Lambda U$, to be the eigen-decomposition of $\S_L$, then we have
\[
\hat{\S}_T=\frac{1}{np}\sum_{k=1}^n(U\Z_k)^{\T}\Lambda (U\Z_k),
\]
where rows of $U\Z_k$ are independent.
Thus it is easy to show that
\[
\|{\hat{\S}}_T-\S_T\|_{\infty}=O_{\rm p}\Big{[}\{\log q/(np)\}^{1/2}\Big{]}.
\]
Thus, we have
\[
\|{\hat{\S}}_T^{-1}-\S_T^{-1}\|_{\infty}=O_{\rm p}\Big{[}\|\O_T\|_{L_1}^2\{\log q/(np)\}^{1/2}\Big{]}.
\]
This yields that
\[
\|{\hat{\S}}_T^{-1/2}-\S_T^{-1/2}\|_{\infty}=\|({\hat{\S}}_T^{-1/2}+\S_T^{-1/2})^{-1}({\hat{\S}}_T^{-1}-\S_T^{-1})\|_{\infty}=O_{\rm p}[\{q\log q/(np)\}^{1/2}\|\O_T\|_{L_1}^2],
\]
which implies, uniformly for $l=1,\ldots,q$, $k=1,\ldots,n$,
\[
|(\X_k\hat{\S}_T^{-1/2})_{\cdot,l}-(\X_k{\S}_T^{-1/2})_{\cdot,l}|_{\infty}=O_{\rm p}[\{q^3\log q\log \max(p,q,n)/(np)\}^{1/2}\|\O_T\|_{L_1}^2].
\]
We shall focus on the event $\{\|{\hat{\S}}_T-\S_T\|_{\infty}=O[\{\log q/(np)\}^{1/2}]\}$ in the following analysis. We have, uniformly for $l=1,\ldots,q$, $k=1,\ldots,n$,
\[
\|\Cov^{-1}[(\X_k\hat{\S}_T^{-1/2})_{\cdot,l}]-\O_L\|_{\infty}=O[\{q^3\log q\log \max(p,q,n)/(np)\}^{1/2}\|\O_T\|_{L_1}^2\|\O_L\|_{L_1}^2].
\]
Thus, by (C4), it is easy to show that
\[
\max_{1\leq i\leq p,1\leq l\leq q}|{\be}_{i,l}^{d}-\be_{i}|_1=o[\{\log \max(p,q,n)\}^{-1}\}],\text{ and }
\max_{1\leq i\leq p,1\leq l\leq q}|{\be}_{i,l}^{d}-\be_{i}|_2=o\left\{(nq\log p)^{-1/4}\right\}.
\]
Thus we have
\[
\max_{1\leq i\leq p}|\hat{\be}_{i}^{d}-\be_{i}|_1=o_{\rm p}[\{\log \max(p,q,n)\}^{-1}\}],\text{ and }
\max_{1\leq i\leq p}|\hat{\be}_{i}^{d}-\be_{i}|_2=o_{\rm p}\left\{(nq\log p)^{-1/4}\right\}.
\]
Hence, by the proof of Lemma <ref>, we have
1/nq∑_k=1^n∑_l=1^q{ϵ̃_k,i,lϵ̃_k,j,l-ϵ̃_k,i,l(_k,-j,l-_-j,l)^(_j^d-_j)-ϵ̃_k,j,l(_k,-i,l-_-i,l)^(_i^d-_i) +(_i^d-_i)^(_k,-i,l-_-i,l)(_k,-j,l-_-j,l)^(_j^d-_j)} =-1/nq∑_k=1^n∑_l=1^q(ϵ̂_k,i,l^d)^2β̂_i,j^d-1/nq∑_k=1^n∑_l=1^q(ϵ̂_k,j,l^d)^2β̂_j-1,i^d- (ω_L,i,iσ̂_i,i,ϵ+ω_L,j,jσ̂_j,j,ϵ-1)r_i,j+U_i,j+o_p((nqlogp)^-1/2),
uniformly in $1\leq i\leq j\leq p$.
Note that
1/nq∑_k=1^n∑_l=1^q[{Y_k,i,l^d-Y̅_i,l^d-(_k,-i,l^d-_-i,l^d)^_i^d}{Y_k,j,l^d-Y̅_j,l^d-(_k,-j,l^d-_-j,l^d)^_j^d}- {Y_k,i,l-Y̅_i,l-(_k,-i,l-_-i,l)^_i^d}{Y_k,j,l-Y̅_j,l-(_k,-j,l-_-j,l)^_j^d}] =1/nq∑_k=1^n∑_l=1^q({Y_k,i,l^d-Y̅_i,l^d-(_k,-i,l^d-_-i,l^d)^_i^d}[{Y_k,j,l^d-Y̅_j,l^d-(Y_k,j,l-Y̅_j,l)} -{_k,-j,l^d-_-j,l^d-(_k,-j,l-_-j,l)}^_j^d]+{Y_k,j,l-Y̅_j,l-(_k,-j,l-_-j,l)^_j^d} [{Y_k,i,l^d-Y̅_i,l^d-(Y_k,i,l-Y̅_i,l)}-{_k,-i,l^d-_-i,l^d-(_k,-i,l-_-i,l)}^_i^d])
It suffices to show that, uniformly in $1\leq i\leq j\leq p$,
s_1,i,j = 1/nq∑_k=1^n∑_l=1^q{Y_k,i,l^d-Y̅_i,l^d-(_k,-i,l^d-_-i,l^d)^_i^d}[{Y_k,j,l^d-Y̅_j,l^d-(Y_k,j,l-Y̅_j,l)} -{_k,-j,l^d-_-j,l^d-(_k,-j,l-_-j,l)}^_j^d]=o_p{(nqlogp)^-1/2}.
Recall that $\hat{\epsilon}_{k,i,l}^{d}=Y_{k,i,l}^{d}-\bar{Y}_{i,l}^{d}-(\Y_{k,-i,l}^{d}-\bar{\Y}_{-i,l}^{d})^{\T}\hat{\be}_{i}^{d}$, then we have
Note that $\hat{\epsilon}_{k,i,l}^{d}={\epsilon}_{k,i,l}^{d}+(\Y_{k,-i,l}^{d}-\bar{\Y}_{-i,l}^{d})^{\T}(\be_{i,l}^{d}-\hat{\be}_{i}^{d})$. Let event $A=(\max_{1\leq i\leq p,1\leq l\leq q}|\hat{\be}_{i}^{d}-\be_{i}^{d}|_1=o[\{\log \max(p,q,n)\}^{-1}])$.
Then we have
max_1≤i≤p,1≤l≤q|(ϵ̂_k,i,l^d|A)| ≤ max_1≤i≤p,1≤l≤q|ϵ_k,i,l^d|+max_1≤i≤p,1≤l≤q|{(_k,-i,l^d-_-i,l^d)^(_i,l^d-_i^d)|A}| = o[{logmax(p,q,n)}^-1/2].
It is easy to check that, for arbitrarily small $\gamma>0$,
max_1≤i≤p,1≤l≤q,1≤k≤n|{Y_k,i,l-Y̅_i,l-(Y_k,i,l^d-Y̅_i,l^d)}| =max_1≤i≤p,1≤l≤q,1≤k≤n|{(_k-1/n∑_k=1^n_k)_T^-1/2}_i,l-{(_k-1/n∑_k=1^n_k)_T^-1/2}_i,l|
Let event
B={max_1≤i≤p,1≤l≤q,1≤k≤n|{Y_k,i,l-Y̅_i,l-(Y_k,i,l^d-Y̅_i,l^d)}| =o[{q^3logqlogmax(p,q,n)log^γp/(np)}^1/2Ø_T_L_1^2]}.
Thus, by the fact that $\max_{1\leq i\leq p,1\leq l\leq q,1\leq k\leq n}|\hat{\epsilon}_{k,i,l}^{d}-\epsilon_{k,i,l}|=o_{\rm p}[\{\log \max(p,q,n)\}^{-1/2}]$, it can be shown that, for arbitrarily small $\gamma>0$,
(max_1≤i≤pmax_1≤h≤p|1/nq∑_k=1^n∑_l=1^qϵ̂_k,i,l^d{Y_k,h,l^d-Y̅_h,l^d-(Y_k,h,l-Y̅_h,l)}| ≥C√(q^2logqlogmax(p,q,n)log^1+γ p/n^2p)Ø_T_L_1^2) ≤(max_1≤i≤pmax_1≤h≤p|1/nq∑_k=1^n∑_l=1^qϵ̂_k,i,l^d{Y_k,h,l^d-Y̅_h,l^d-(Y_k,h,l-Y̅_h,l)}| ≥C√(q^2logqlogmax(p,q,n)log^1+γ p/n^2p)Ø_T_L_1^2, A∩B)+(A^c)+(B^c) ≤(max_1≤i≤pmax_1≤h≤p|1/√(nq)∑_k=1^n∑_l=1^qϵ̂_k,i,l^d{Y_k,h,l^d-Y̅_h,l^d-(Y_k,h,l-Y̅_h,l)}/√(q^3logqlogmax(p,q,n)log^γp/np)Ø_T_L_1^2| ≥C√(logp),A∩B)+o(1)=o(1).
This, together with (C4), implies that
max_1≤i≤j≤p|∑_h≠j1/nq∑_k=1^n∑_l=1^qϵ̂_k,i,l^d{Y_k,h,l^d-Y̅_h,l^d-(Y_k,h,l-Y̅_h,l)}β̂_h,j^d| =O_p(√(q^2logqlogmax(p,q,n)log^1+γ p/n^2p))Ø_T_L_1^2Ø_L_L_1=o_p{(nqlogp)^-1/2}.
Thus (<ref>) is proved.
Thus we have
|1/nq∑_k=1^n∑_l=1^q[{Y_k,i,l^d-Y̅_i,l^d-(_k,-i,l^d-_-i,l^d)^_i^d}{Y_k,j,l^d-Y̅_j,l^d-(_k,-j,l^d-_-j,l^d)^_j^d}- {Y_k,i,l-Y̅_i,l-(_k,-i,l-_-i,l)^_i^d}{Y_k,j,l-Y̅_j,l-(_k,-j,l-_-j,l)^_j^d}]| =o_p{(nqlogp)^-1/2}
uniformly in $1\leq i\leq j\leq p$.
Hence equation (<ref>) is proved and Theorems <ref> and <ref> thus follow.
|
1511.00956
|
§ INTRODUCTION
Dicke <cit.> predicted that if an ensemble of two-level atoms is prepared in a collective state where half of the atoms are in the excited state and half of the atoms are in the ground state the spontaneous emission
is proportional to the square of the number of atoms as if the particles would radiate coherently in phase like synchronized antennas <cit.>. To analyze the phenomenon Dicke introduced the concept of collective spins where $N$ two level atoms are described by the collective spin eigenstates $\ket{N/2, M}$, with
$M$ running from $M = -N/2, \ldots, +N/2$ in steps of unity.
Among these states the state $\ket{N/2, 0}$ radiates with an intensity $N^2$ times as strong as that of a single atom. The origin of superradiance is difficult to see since all states $\ket{N/2, M}$ exhibit no macroscopic dipole moment whereas such a dipole moment is commonly assumed to be required for a radiation rate proportional to $N^2$. The reason is that the Dicke states display strong quantum entanglement. The entangled character of the states is particularly apparent for the case of two two-level atoms where the individual atomic states are labeled by $\ket{e_l}$ and $\ket{g_l}$, $l = 1, 2$, for the excited and ground state, respectively. In this case the Dicke state $\ket{1,0} = 1/\sqrt{2} \left( \ket{e_1, g_2} + \ket{g_1, e_2} \right)$, also known as the Bell state or the EPR state, is clearly maximally entangled. For three atoms one of the Dicke states is denoted by $\ket{3/2,-1/2}$, which in current language would be the $W$ state $1/\sqrt{3} \left( \ket{e_1, g_2, g_3} + \ket{g_1, e_2, g_3} + \ket{g_1, g_2, e_3} \right)$ <cit.>. The single excited generalized $W$ state, where only one atom is excited and $N-1$ atoms are in the ground state, is also known to be fully entangled and plays a particularly important role for single-photon superradiance <cit.>.
In fact, it has been recognized that most of the important aspects of superradiance <cit.> can be studied by examining samples in single excited generalized $W$ states
<cit.> as the emission from these states possesses all the features of superradiance that originally were calculated for samples with an arbitrary number of excitations <cit.>. The spatial features of one photon superradiance have been extensively studied for example from the perspective of timed Dicke states <cit.> and also the spectral and temporal aspects have been investigated in a large variety of systems <cit.>.
Note that a number of recent works <cit.> have also discussed how
single excited generalized $W$ states for a small number of atoms can be prepared in the laboratory.
The single excited generalized $W$ state does however not allow one to study the quantum statistical properties of superradiance. In order to explore these aspects the system must emit at least two photons. Only then one has access to the photon-photon correlations which display amongst others the particular quantum characteristics of the spontaneously scattered radiation <cit.>. To this end it is required to investigate what we will term two-photon superradiance from generalized $W$ states with $n_{e}\geq2$ excitations.
In the present paper we show that in two-photon superradiance the emitted radiation can exhibit both bunched as well as nonclassical and antibunched light depending on the angle of observation, i.e., the position of the detectors collecting the scattered photons, and on the particular $W$ state, i.e., the number of atoms $N$ and the number of excitations $n_{e}$, considered. In particular, in certain cases it is also possible to observe the phenomenon of superbunching, i.e., photon-photon correlations larger than those maximally measurable for classical light sources. In all the cases the mean intensity displays the familiar features of superradiance produced by the corresponding $W$ state. While we derive our results for two-photon superradiance for arbitrary generalized $W$ states we focus in this paper on systems in doubly excited $W$ states; the outcomes for arbitrary $W$ states with more than two excitations are presented in the Supplementary Information section.
Note that bunching in the radiation of generalized $W$ states can be explained semi-classically. However, the phenomenon of superbunching as well as the emission of nonclassical light, first demonstrated in 1977 <cit.>, can only be understood in a quantum mechanical description <cit.>. The latter is a feature arising from the light's particle nature where photon fluctuations become smaller than for coherent light. Demonstrating nonclassicality in the light of arbitrary $W$ states thus directly leads to a manifestation of the particular quantum mechanical characteristics of these superradiant states. We finally discuss also the spatial cross correlations of photons in two-photon superradiance. Here, likewise, superbunching and nonclassicality can be observed.
We note that recent experiments by Jahnke et al. with quantum dots in a cavity have already reported the observation of superbunching <cit.>.
In this paper we bring out for a simple model system in free space the reasons for the appearance of this phenomenon,
a curio which does not commonly occur, the squeezed vacuum being one of the rare examples <cit.>.
§ RESULTS
To focus on the key aspects of two-photon superradiance we consider a linear system of $N$ equidistantly aligned identical emitters, e.g., atoms or ions
with upper state $\ket{e_{l}}$ and ground state $\ket{g_{l}}$, $l = 1, \ldots , N$, trapped in a linear arrangement <cit.> at positions ${\vec R}_{l}$
with spacing $d \gg \lambda$ such that the dipole-dipole coupling between the particles can be neglected (see Fig. <ref>). The atoms are assumed to be prepared initially in a generalized $W$ state with $n_{e}$ excitations, i.e., in the state $\ket{W_{n_{e},N}}=\binom{N}{n_{e}}^{-\frac{1}{2}}\sum_{\{\alpha_{l}\}=\mathcal{P} {\{l\}}} \prod_{i=1}^{n_{e}}\ket{e_{\alpha_{i}}}\prod_{i=n_{e}+1}^{N}\ket{g_{\alpha_{i}}} $,
where $\mathcal{P} {\{l\}}$ denotes all permutations of the set of atoms $\{l\}=\{1,2,\hdots,N\}$. In what follows we study the second order correlation functions at equal times emitted by the atoms in the described $W$ state.
To this end two detectors are placed at positions $\vec{r}_{1}$ and $\vec{r}_{2}$ in the far field each measuring a single photon coincidentally, i.e., within a small time window much smaller than the lifetime of the upper state.
To simplify the calculations we suppose that the emitters and the detectors are in one plane and that the atomic dipole moments of the transition $\ket{e_{l}} \rightarrow \ket{g_{l}}$ are oriented perpendicular to this plane (see Fig. <ref>).
Scheme of considered setup: $N$ two-level atoms are aligned on the x-axis, whereby neighboring atoms are separated by a distance $d$. The intensity in the far field $I(\vec{r}_{1})$ is measured by a single detector at position $\vec{r}_{1}$, whereas the second order correlation function $G^{(2)}(\vec{r}_{1},\vec{r}_{2})$ is measured by two detectors at $\vec{r}_{1}$ and $\vec{r}_{2}$.
Due to the far field condition and therefore the inability to identify the individual photon sources, the electric field operator at ${\vec r}_j$ takes the form <cit.>
$\left[ \hat{E}^{(-)}(\vec{r}_j) \right]^\dagger \! = \hat{E}^{(+)}(\vec{r}_j) \sim \sum_{l=1}^{N} e^{-i\,\varphi_{lj}} \;\hat{s}^{-}_l = \sum_{l=1}^{N} e^{-i\,l\,\delta_{j}} \;\hat{s}^{-}_l$,
where $\hat{s}_{l}^{-}=\ket{g_{l}}\!\bra{e_{l}}$ is the atomic lowering operator for atom $l$, and $\varphi_{lj} = - k \, \frac{\vec{r}_j\cdot\vec{R}_l}{r_j} = l\, k\, d \, \sin\theta_{j} \cos\phi_j =l\,\delta_{j}$ the relative optical phase accumulated by a photon emitted by source $l$
and recorded by detector $j$
with respect to a photon emitted at the origin.
Note that the field operators have been chosen dimensionless as all dimension defining prefactors cancel out in the normalized correlation functions.
The first and second order spatial correlation functions at equal times are defined as <cit.> $G^{(1)}(\vec{r}_{1})=\left<\hat{E}^{(-)}(\vec{r}_{1})\hat{E}^{(+)}(\vec{r}_{1})\right>$ and $G^{(2)}(\vec{r}_{1},\vec{r}_{2})=\left<\hat{E}^{(-)}(\vec{r}_{1})\hat{E}^{(-)}(\vec{r}_{2})\hat{E}^{(+)}(\vec{r}_{2})\hat{E}^{(+)}(\vec{r}_{1})\right> $, respectively,
where $G^{(1)}(\vec{r}_{1})$ is proportional to the mean intensity of the emitted radiation, i.e., $G^{(1)}(\vec{r}_{1})\sim I(\vec{r}_{1}) $. To compare the photon statistics of various systems radiating with different intensities we further introduce the normalized second order correlation function <cit.> $g^{(2)}(\vec{r}_{1},\vec{r}_{2})=G^{(2)}(\vec{r}_{1},\vec{r}_{2})/(G^{(1)}(\vec{r}_{1})\,G^{(1)}(\vec{r}_{2}))$.
For the state $\ket{W_{n_{e},N}}$ the first order correlation function in the configuration of Fig. <ref> has been calculated <cit.> to
\begin{equation}
\begin{aligned}
G^{(1)}_{n_{e},N}(\delta_{1})&= \frac{n_{e}(n_{e}-1)}{N-1} + \frac{N\, n_{e}(N-n_{e})}{N-1} \chi^{2}(\delta_{1}) \ ,
\label{eq:intdefinition}
\end{aligned}
\end{equation}
where $\chi(x)=\frac{\sin(\frac{Nx}{2})}{N\sin(\frac{x}{2})}$ corresponds to the normalized far-field intensity distribution of a coherently illuminated $N$-slit grating <cit.>.
As it is well-known from classical optics, the distribution $\chi^{2}(x)$ is strongly peaked in particular directions. The fact that $\chi^{2}(x)$ appears in the context of spontaneous emission of atoms in
generalized $W$ states as in Eq. (<ref>) has been coined by Dicke spontaneous emission of coherent radiation or simply superradiance <cit.>. Note that even though $\chi^{2}(x)$ displays pronounced maxima in certain directions the distribution assumes also very small values and even vanishes in other directions what has been interpreted as subradiance of the states $\ket{W_{n_{e},N}}$ <cit.>.
From Eq. (<ref>) we find that the intensity distribution of the state $\ket{W_{1,N}}$ with only one excitation ($n_{e}=1$) simplifies to $G^{(1)}_{1,N}(\delta_{1})=N\chi^{2}(\delta_{1})$, with $G^{(1)}_{1,N}(\delta_{1}) \rightarrow N \ \ \text{for} \ \delta_{1}=0$,
displaying a maximal visibility $\mathcal{V}=1$ and a peak value $N$ times the intensity of a single atom.
This kind of single photon superradiance has been extensively studied in the past <cit.>. Its particular superradiant characteristics have been shown to result from quantum path interferences occurring due to the particular interatomic correlations of the collective state $\ket{W_{1,N}}$ <cit.>.
As discussed below the strong correlations of the states $\ket{W_{n_{e},N}}$ may lead to photon-photon correlations with $g_{n_{e},N}^{(2)}(\delta_{1},\delta_{1})> 2$ as well as $g_{n_{e},N}^{(2)}(\delta_{1},\delta_{1})<1$, corresponding to superbunched as well as nonclassical light, respectively.
Note that from the form of normalized second order correlation function $g_{n_{e},N}^{(2)}(\delta_{1},\delta_{2})$ it is obvious that bunching necessitates small intensities, i.e., $G^{(1)}(\delta_{1}) \ll G^{(2)}(\delta_{1},\delta_{1})$, whereas nonclassical light requires small values of the two-photon correlation function, i.e., $G^{(2)}(\delta_{1},\delta_{1})\ll G^{(1)}(\delta_{1})$.
In what follows we study two-photon superradiance for the simplest form of generalized $W$ states, i.e., $W$ states with only two excitations, as the main features of two-photon superradiance can already be observed for this configuration; two-photon superradiance for arbitrary generalized $W$ states $\ket{W_{n_{e},N}}$ with $n_{e} > 2$ is discussed in the Supplementary Information section.
Note that an atomic ensemble in the state $\ket{W_{n_{e}=2,N}}$ can be prepared for example by using photon pairs generated in a down conversion process. When sent on a $50:50$ beam splitter the photon pair would then produce two photons at either of the two output ports of the beam splitter, i.e., if one port delivers zero photons then the other one has two photons <cit.>. Assuming perfect detection efficiency, a photon pair not registered at one output port of the beam splitter and not registered at the other one after having passed the atomic ensemble would then herald the absorption of two photons by the atomic system.
For $n_{e}=2$ the normalized second order correlation function for the considered configuration of Fig. <ref> takes the form
\begin{equation}
\begin{aligned}
& g_{2,N}^{(2)}(\delta_{1},\delta_{2})= \frac{N(N-1)}{2} \frac{\left( N \chi(\delta_{1})\chi(\delta_{2}) - \chi(\delta_{1}+\delta_{2}) \right)^{2}}{\left(1 + N(N-2)\chi^2(\delta_{1})\right) \left(1 + N(N-2)\chi^2(\delta_{2})\right)} \ .
\label{eq:Gcompletene2}
\end{aligned}
\end{equation}
This expression will be investigated in detail in the following subsections.
§.§ Superbunching in Two-photon Superradiance
In this section we investigate whether bunching in two-photon superradiance, in particular the phenomenon of superbunching with $g_{n_{e},N}^{(2)}(\delta_{1},\delta_{2})> 2$, can be observed in the radiation produced by states of the form $\ket{W_{n_{e}=2,N}}$.
We start to explore the photon-photon correlations with the two detectors placed at equal positions, i.e., with the two spontaneously emitted photons recorded in the same mode; photon-photon cross correlations are studied thereafter.
Second order correlation function $g^{(2)}_{2,N}(\delta_{1},\delta_{1})$ (left) and first order correlation function, i.e., the intensity distribution, $G^{(1)}(\delta_{1})$ (right) of the radiation emitted by $N$ two-level atoms in the doubly excited $W$ state $\ket{W_{2,N}}$ for $N=3$ (dotted), $N=4$ (dashed), $N=6$ (solid). Superbunching is observed at $\delta_{1}=\pi$ for $N \geq 4$, while antibunching occurs at detector positions $\delta_{1}$ fulfilling $N \chi^{2}(\delta_{1})=\chi(2\delta_{1})$. For increasing $N$ superbunching becomes stronger, similar to single photon superradiance of the state $\ket{W_{1,N}}$. Comparing the two plots one can see that for high values of $G^{(1)}(\delta_{1})$ small values of $g^{(2)}_{2,N}(\delta_{1},\delta_{1})$ are obtained and vice versa.
According to Eq. (<ref>) the second order correlation function for the state $\ket{W_{2,N}}$ with two detectors placed at the same position takes the form
\begin{equation}
\begin{aligned}
g^{(2)}_{2,N}(\delta_{1},\delta_{1})= \frac{ N(N-1)\, \left( N\,\chi^{2}(\delta_{1})-\chi(2\delta_{1}) \right)^{2}}{2\, \left( 1 + N(N-2)\, \chi^{2}(\delta_{1}) \right)^{2}} \ .
\label{eq:g2ne2d1d1}
\end{aligned}
\end{equation}
In order to access whether the system displays bunching for this configuration we have to search for values $g^{(2)}_{2,N}(\delta_{1},\delta_{1}) > 1$. Hitherto,
we choose detector positions for which the values of the first order correlation function $G^{(1)}(\delta_{1})$ remain smaller than the unnormalized second order correlation function $G^{(2)}_{2,N}(\delta_{1},\delta_{1})$, i.e., locations where the intensity is low. This occurs for example at $\delta_{1}=\pi$, where $g^{(2)}_{2,N}(\delta_{1},\delta_{1})$ attains its maximal value (see Fig. <ref>).
To investigate this outcome quantitatively we have to study the case of even and odd $N$ separately since $\chi(\pi)$ and $\chi(2\pi)$ yield different results in these two cases <cit.>.
For an even number $N$ of atoms one obtains the following two identities $\chi(\pi)=0$, $\chi(2\pi)=-1$,
from which we deduce $g^{(2)}_{2,N_{even}}(\pi,\pi)=N(N-1)/2$,
leading to bunched light in case that $N_{even}\geq 4$ (see Fig. <ref>). In fact, for $N_{even}\geq 4$, we even obtain superbunching, i.e., $g^{(2)}_{2,N_{even}}(\pi,\pi) > 2$, what surpasses the maximum value achievable with classical light.
Note that
$g^{(2)}_{2,N_{even}}(\pi,\pi)$ as a function of $N$ has in principal no upper limit as it increases $\sim N^{2}$ for $N\gg1$. This means that we can produce principally unlimited values of superbunching if we add more and more atoms in the ground state to the system <cit.>.
In case of odd $N$ the above identities read $\chi(\pi)=\pm 1/N$, $\chi(2\pi)= +1$.
what leads to the a maximal value of the second oder correlation function of $g^{(2)}_{2,N_{odd}}(\pi,\pi)=N(N-1)/8$.
Here we obtain bunched and superbunched light in case that $N_{odd} \geq 5$.
Again, as in the case of even $N$, there is no upper limit for the maximum value of the two-photon correlation function,
as once more we have $g^{(2)}_{2,N_{odd}}(\pi,\pi) \sim N^{2}$.
§.§ Nonclassicality in Two-photon Superradiance
In this section we investigate whether for the initial state $\ket{W_{2,N}}$ and two detectors at equal positions we can obtain nonclassical light, a sub-Poissonian photon statistics and antibunching in two-photon superradiance.
As known from the radiation of a single atom <cit.> a sub-Poissonian photon statistics derives from the discrete nature of the scattered radiation and can be explained only in a quantum mechanical treatment of resonance fluorescence. A complete vanishing of the second order correlation function at equal times for the state $\ket{W_{2,N}}$ indicates that $g^{(2)}_{2,N}(\delta_{1},\delta_{1}) = 0$ at $\tau = 0$ what proves true antibunching, whereas values $g^{(2)}_{2,N}(\delta_{1},\delta_{1}) < 1$ result from the nonclassical nature of the radiation scattered by the state $\ket{W_{2,N}}$.
Noting that the second order correlation function $g^{(2)}_{2,N}(\delta_{1},\delta_{1})$ displays a visibility of $\mathcal{V}=1$ (see Fig. <ref>) there must be indeed detector positions where $g^{(2)}_{2,N}(\delta_{1},\delta_{1}) = 0$.
More specifically, according to Eq. ($\ref{eq:g2ne2d1d1}$), the photon-photon correlation function vanishes independently of the atom number $N$ in case that the detectors are located at positions $\tilde{\delta}_{1}=a\frac{2\pi}{N}$, with $ a = 1,2, \ldots < N/2$, as in this case $\chi(\tilde{\delta}_{1})=\chi(2 \tilde{\delta}_{1})=0$. Towards these positions the atomic system thus radiates photons which display complete antibunching <cit.>.
If the two detectors are located at $\delta_{1}=0$ we have $g^{(2)}_{2,N}(\delta_{1},\delta_{1}) < 1$ as long as $N\geq 3$, as in this case the photon-photon correlation function takes the form $g^{(2)}_{2,N}(0,0)=N/(2(N-1))$ (cf. Eq. (<ref>)).
Hence, in those directions the atomic system emits nonclassical light with photon number fluctuations smaller than those for coherent light.
Left: Second order correlation function $g_{n_{e},N}^{(2)}(\delta_{1},\delta_{2})$ for $n_{e}=2$, $\delta_{2}=-\delta_{1}$ and $N=3$ (dotted), $N=4$ (dashed), $N=6$ (solid). The function displays both superbunching and antibunching with a maximal visibilities of $\mathcal{V}=1$. It can be seen that the superbunching effect increases with increasing $N$, while for $\delta_{1}=0$ the second order correlation function converges to $1/2$. Right: First order correlation function $G^{(1)}_{2,N}(\delta_{1})$ for $N=3$ (dotted), $N=4$ (dashed), $N=6$ (solid). From the figure it can be seen that for small values of $G^{(1)}_{2,N}(\delta_{1})$ high values of $g_{n_{e},N}^{(2)}(\delta_{1},\delta_{2})$ are obtained. In contrast to the case $\delta_{1}=\delta_{2}$ several superbunching peaks occur.
§.§ Two-photon Cross Correlations
Finally we study the spatial cross correlations in two-photon superradiance for atoms in the state $\ket{W_{2,N}}$, i.e, the behavior of the second order correlation function $g^{(2)}_{2,N}(\delta_{1}, \delta_{2})$ in case that the scattered photons are recorded at different positions $\delta_{1} \neq \delta_{2}$. We start to explore the particular configuration of counter-propagating detectors, i.e., detectors at positions $\delta_{1}= - \delta_{2}$, followed by the case where $\delta_{2}$ is fixed and only $\delta_{1}$ is varied.
For $\delta_{1}= -\delta_{2}$ (i.e., where we choose $\theta_1 = \theta_2$, $\phi_2 = 0$ and $\phi_1 = \pi$),
the second order correlation function reads (cf. Eq. (<ref>))
\begin{equation}
g^{(2)}_{2,N}(\delta_{1},-\delta_{1})= \frac{N (N-1)\, \left(N\, \chi^{2}(\delta_{1}) - 1 \right)^{2} }{ 2\, \left( 1 + N(N-2)\, \chi^{2}(\delta_{1}) \right)^{2} } \ .
\label{eq:g2ne2d1minusd1}
\end{equation}
To determine the possibilities for superbunching in cross correlations of the scattered photons we look for the maximum of Eq. (<ref>). This is attained at $\bar{\delta}_{1}=a\frac{2\pi}{N}$, with $a = 1, \ldots, N/2$, in which case $\chi(\delta_{1})$ vanishes and the second order correlation function reads $g^{(2)}_{2,N}(\bar{\delta},-\bar{\delta})=N(N-1)/2$.
This result is principally identical to $g^{(2)}_{2,N_{even}}(\pi,\pi)$. However, in the case of counter-propagating detectors it is valid for arbitrary $N$, i.e., for $N$ even or odd. Here, the threshold for superbunching is exceeded if $N\geq 3$ (see Fig. <ref>).
In the same configuration also
complete antibunching
can be obtained. This occurs for $\hat{\delta}_{1}$ such that $\chi^{2}(\hat{\delta}_{1}) = 1/N$ (cf. Eq. (<ref>)), in which case the photon correlation function vanishes identically, i.e., $g^{(2)}_{2,N}(\hat{\delta}_{1},-\hat{\delta}_{1})= 0$.
Another interesting configuration is the case when $\delta_{2}$ is fixed and only $\delta_{1}$ is varied. If we fix $\delta_{2}=0$ the photon-photon cross correlation function takes the form
(cf. Eq. (<ref>))
\begin{equation}
g^{(2)}_{2,N}(\delta_{1},0) = \frac{N(N-1)\chi^{2}(\delta_{1})}{2 + 2N(N-2)\chi^{2}(\delta_{1})} \ ,
\label{eq:g2ne2delta10}
\end{equation}
which can maximally take a value of $1$ (for $N=2$ and $\delta_{1} = 0$), whereas for $N > 2$ the cross correlation function remains always $< 1$. In Fig. <ref> the corresponding photon-photon cross correlations are shown for $N = 2, 4, 6$ for the entire range $\delta_{1} \in [0, \pi]$. It can be seen that by increasing $N$ the second order correlation function $g^{(2)}_{2,N}(\delta_{1},0)$ decreases in absolute values. The reason is the following: since $G^{(2)}_{2,N}(\delta_{1},0)$ and $G^{(1)}_{2,N}(\delta_{1})$ depend on $\chi(\delta_{1})$ in a similar way the overall behavior of $g^{(2)}_{2,N}(\delta_{1},0)$ is determined by the prefactors of $\chi(\delta_{1})$ in the numerator and denominator of $g^{(2)}_{2,N}(\delta_{1},0)$, the ratio of which decreases with increasing $N$ and converges to $1/2$ for $N \gg 1$.
Left: Second order correlation function $g_{n_{e},N}^{(2)}(\delta_{1},\delta_{2})$ for $n_{e}=2$ where one detector is fixed at $\delta_{2}=0$. For all $N$ (in the plot: $N=2$ (solid), $N=4$ (dashed), $N=6$ (dotted)) the two-photon correlation function remains smaller than one for any $\delta_{1}\neq 0$. For $\delta_{1}=0$ the maximal value of one is attained only in the case $N=2$. Right: First order correlation function $G^{(1)}_{2,N}(\delta_{1})$ for $N=2$ (solid), $N=4$ (dashed), $N=6$ (dotted).
§ DISCUSSION
In conclusion we investigated for a prototype ensemble of $N$ identical non-interacting
two-level atoms prepared in collective
superradiant generalized $W$-states with $n_{e}$ excitations the particular quantum statistical properties of the emitted radiation. Such investigations require the collective system to have at least two excitations as we explore the photon-photon correlations of the scattered light. We derived conditions for which the atomic system emits bunched and even superbunched light, as well as nonclassial and antibunched radiation. Here, superbunching refers to values of the normalized second order correlation function $g^{(2)}_{2,N}(\delta_{1}, \delta_{1}) > 2$ and antibunching to values $g^{(2)}_{2,N}(\delta_{1}, \delta_{1}) = 0$. In some cases the results were obtained under the condition that the number of atoms in the ensemble exceed a certain threshold. For example, the smallest number of atoms producing superbunching in the state $\ket{W_{2,N}}$ is $N=3$;
similar results were derived for $N$ atoms with arbitrary number of excitations,
as shown in the Supplementary Information section.
Note that in coherently driven atomic systems superbunching can be observed already for $N = 2$. For example, in <cit.> it was demonstrated that arbitrarily high values of $g^{(2)}_{N = 2}(\delta_{1}, \delta_{1})$ can be produced for $\delta_{1} = \pi$ in case that the two atoms are very weakly excited, as in this case $g^{(2)}_{N = 2}(\pi, \pi) \sim 1/\Omega^4$, where $\Omega$ is the Rabi frequency (in units of the spontaneous decay rate $\gamma$). The effect is even stronger if the two atoms are subject to a strong dipole-dipole interaction as in dipole blockade systems; here $g^{(2)}_{N = 2}(\pi, \pi) \sim \delta_0^2/\Omega^4$ where $\delta_0$ is the level shift of the doubly excited state (again in units of $\gamma$) <cit.>.
In the last part of the paper we finally investigated the spatial cross correlations in two-photon superradiance, i.e., the second order correlation functions $g^{(2)}_{2,N}(\delta_{1}, \delta_{2})$ for detector positions $\delta_{1} \neq \delta_{2}$. Here, again, positions were found where $g^{(2)}_{2,N}(\delta_{1}, \delta_{2}) > 2$ and $g^{(2)}_{2,N}(\delta_{1}, \delta_{2}) = 0$, corresponding in this case to superbunching and antibunching of the cross correlations of the scattered photons.
authorDicke, R. H.
titleCoherence in spontaneous radiation processes.
journalPhys. Rev. volume93,
pages99–110 (year1954).
authorAllen, L. & authorEberly, J. H.
titleOptical Resonance and Two-Level Atoms
(publisherJohn Wiley & Sons, New York, year1975).
authorDür, W., authorVidal, G. &
authorCirac, J. I.
titleThree qubits can be entangled in two inequivalent
journalPhys. Rev. A volume62,
pages062314 (year2000).
authorScully, M. O., authorFry, E. S.,
authorOoi, C. H. R. & authorWódkiewicz, K.
titleDirected spontaneous emission from an extended
ensemble of ${N}$ atoms: Timing is everything.
journalPhys. Rev. Lett.
volume96, pages010501
authorScully, M.
titleCorrelated spontaneous emission on the Volga.
journalLaser Physics
volume17, pages635–646
authorSvidzinsky, A. A., authorChang, J.-T. &
authorScully, M. O.
titleDynamical evolution of correlated spontaneous
emission of a single photon from a uniformly excited cloud of ${N}$ atoms.
journalPhys. Rev. Lett.
volume100, pages160504
authorScully, M. O.
titleCollective lamb shift in single photon Dicke
journalPhys. Rev. Lett.
volume102, pages143601
authorScully, M. O. & authorSvidzinsky, A. A.
titleThe super of superradiance.
journalScience volume325,
pages1510–1511 (year2009).
authorEberly, J. H. & authorRehler, N. E.
titleSuperradiant intensity fluctuations.
journalPhys. Rev. A volume2,
pages1607–1610 (year1970).
authorRehler, N. E. & authorEberly, J. H.
journalPhys. Rev. A volume3,
pages1735–1751 (year1971).
authorAgarwal, G.
titleQuantum statistical theories of spontaneous emission
and their relation to other approaches.
In booktitleQuantum Optics,
vol. volume70 of seriesSpringer Tracts in Modern
Physics, pages1–128 (publisherSpringer Berlin
Heidelberg, year1974).
authorGross, M. & authorHaroche, S.
titleSuperradiance: An essay on the theory of collective
spontaneous emission.
journalPhysics Reports
volume93, pages301 – 396
authorNienhuis, G. & authorSchuller, F.
titleSpontaneous emission and light scattering by atomic
lattice models.
journalJ. Phys. B volume20,
pages23 (year1987).
authorSvidzinsky, A. A., authorChang, J.-T. &
authorScully, M. O.
titleCooperative spontaneous emission of ${N}$ atoms:
Many-body eigenstates, the effect of virtual lamb shift processes, and
analogy with radiation of ${N}$ classical oscillators.
journalPhys. Rev. A volume81,
pages053821 (year2010).
authorWiegner, R., authorvon Zanthier, J. &
authorAgarwal, G. S.
titleQuantum-interference-initiated superradiant and
subradiant emission from entangled atoms.
journalPhys. Rev. A volume84,
pages023805 (year2011).
authorBienaimé, T., authorPiovella, N. &
authorKaiser, R.
titleControlled Dicke subradiance from a large cloud of
two-level systems.
journalPhys. Rev. Lett.
volume108, pages123602
authorBienaimé, T., authorBachelard, R.,
authorPiovella, N. & authorKaiser, R.
titleCooperativity in light scattering by cold atoms.
journalFortschritte der Physik
volume61, pages377–392
authorRöhlsberger, R.
titleCooperative emission from nuclei: The collective lamb
shift and electromagnetically induced transparency.
journalFortschritte der Physik
volume61, pages360–376
authorO’Connell, A. D. et al.
titleQuantum ground state and single-phonon control of a
mechanical resonator.
journalNature volume464,
pages697–703 (year2010).
authorTeufel, J. D. et al.
titleCircuit cavity electromechanics in the
strong-coupling regime.
journalNature volume471,
pages204–208 (year2011).
authorLambert, N., authorJohansson, R. &
authorNori, F.
titleMacrorealism inequality for optoelectromechanical systems.
journalPhys. Rev. B volume84,
pages245421 (year2011).
authorChen, G.-Y., authorLambert, N.,
authorLi, C.-M., authorChen, Y.-N. &
authorNori, F.
titleDelocalized single-photon Dicke states and the
Leggett-Garg inequality in solid state systems.
journalScient. Rep. volume2,
pages869 (year2012).
authorThiel, C., authorvon Zanthier, J.,
authorBastin, T., authorSolano, E. &
authorAgarwal, G. S.
titleGeneration of symmetric Dicke states of remote
qubits with linear optics.
journalPhys. Rev. Lett.
volume99, pages193602
authorLinington, I. E. & authorVitanov, N. V.
titleDecoherence-free preparation of Dicke states of
trapped ions by collective stimulated raman adiabatic passage.
journalPhys. Rev. A volume77,
pages062327 (year2008).
authorLinington, I. E. & authorVitanov, N. V.
titleRobust creation of arbitrary-sized Dicke states of
trapped ions by global addressing.
journalPhys. Rev. A volume77,
pages010302 (year2008).
authorPrevedel, R. et al.
titleExperimental realization of Dicke states of up to
six qubits for multiparty quantum networking.
journalPhys. Rev. Lett.
volume103, pages020503
authorWieczorek, W. et al.
titleExperimental entanglement of a six-photon symmetric
Dicke state.
journalPhys. Rev. Lett.
volume103, pages020504
authorHume, D. B., authorChou, C. W.,
authorRosenband, T. & authorWineland, D. J.
titlePreparation of Dicke states in an ion chain.
journalPhys. Rev. A volume80,
pages052302 (year2009).
authorLemr, K. & authorFiurá ššek, J.
titleConditional preparation of arbitrary superpositions
of atomic Dicke states.
journalPhys. Rev. A volume79,
pages043808 (year2009).
authorGlauber, R. J.
titleThe quantum theory of optical coherence.
journalPhysical Review
volume130, pages2529–2539
authorKimble, H. J., authorDagenais, M. &
authorMandel, L.
titlePhoton antibunching in resonance fluorescence.
journalPhys. Rev. Lett.
volume39, pages691–695
authorWalls, D. F. & authorMilburn, G. J.
titleQuantum Optics (publisherSpringer
Berlin Heidelberg, year1994).
authorPaul, H.
titlePhoton antibunching.
journalRev. Mod. Phys.
volume54, pages1061–1102
authorAuffèves, A., authorGerace, D.,
authorPortolan, S., authorDrezet, A. &
authorSantos, M. F.
titleFew emitters in a cavity: from cooperative emission
to individualization.
journalNew Journal of Physics
volume13, pages093020
authorJahnke, F. et al.
(yearto be published).
authorBoitier, F. et al.
titlePhoton extrabunching in ultrabright twin beams
measured by two-photon counting in a semiconductor.
journalNat. Commun. volume2,
pages425 (year2011).
authorMiroshnychenko, Y. et al.
titlePrecision preparation of strings of trapped neutral
journalNew Journal of Physics
volume8, pages191 (year2006).
authorBlatt, R. & authorWineland, D.
titleEntangled states of trapped atomic ions.
journalNature volume453,
pages1008 – 1015 (year2008).
authorMonz, T. et al.
title14-qubit entanglement: Creation and coherence.
journalPhys. Rev. Lett.
volume106, pages130506
authorMitsch, R., authorSayrin, C.,
authorAlbrecht, B., authorSchneeweiss, P. &
authorRauschenbeutel, A.
titleQuantum state-controlled directional spontaneous
emission of photons into a nanophotonic waveguide.
journalNat. Commun. volume5,
pages5713 (year2014).
authorThiel, C. et al.
titleQuantum imaging with incoherent photons.
journalPhys. Rev. Lett.
volume99, pages133603
authorBorn, M. & authorWolf, E.
titlePrinciples of Optics: Electromagnetic Theory of
Propagation, Interference and Diffraction of Light
(publisherCambridge University Press, year1999).
authorHong, C. K., authorOu, Z. Y. &
authorMandel, L.
titleMeasurement of subpicosecond time intervals between
two photons by interference.
journalPhys. Rev. Lett.
volume59, pages2044–2046
titleNote that in the regime $k d \gg 1$ of widely spaced atoms we
are investigating, it is simple to obtain $\delta_i = \pi$ or $\delta_i = 2
\, \pi$, $i = 1, 2$, as these values can be achieved already with observation
angles $\theta \ll 1$ (see Fig. <ref>).
titleNote that generally for all detector positions $\delta_{1}$
fulfilling ${N} \chi^2(\delta_{1})=\chi(2\delta_{1})$ complete antibunching
is obtained (cf. Eq. (<ref>)).
authorSkornia, C., authorvon Zanthier, J.,
authorAgarwal, G. S., authorWerner, E. &
authorWalther, H.
titleDirected spontaneous emission from an extended
ensemble of ${N}$ atoms: Timing is everything.
journalPhys. Rev. A volume64,
pages063801 (year2001).
authorGillet, J., authorAgarwal, G. S. &
authorBastin, T.
titleTunable entanglement, antibunching, and saturation
effects in dipole blockade.
journalPhys. Rev. A volume81,
pages013837 (year2010).
§ ACKNOWLEDGEMENTS
GSA and JvZ thank Frank Jahnke for fruitful discussions and the disclosure of documents prior to publication. DB and JvZ warmly thank the hospitality of GSA at various stays at the Oklahoma State University. DB thanks the Friedrich-Alexander Universität Erlangen-Nürnberg for a travel grant “Hochschule International” and the German Academic Exchange Service for a PROMOS travel grant. The authors gratefully acknowledge funding by the Erlangen Graduate School in Advanced Optical Technologies (SAOT) by the German Research Foundation (DFG) in the framework of the German excellence initiative.
§ AUTHOR CONTRIBUTIONS STATEMENT
DB did all the calculations while the ideas were formulated by GSA. JvZ helped with thoughts in possible experimental implementation. JvZ primarily wrote the manuscript for publication which was reviewed by both DB and GSA.
§ ADDITIONAL INFORMATION
The authors declare no competing financial interests.
|
1511.00992
|
Department of Physics, Brock University, St. Catharines, Ontario, Canada L2S 3A1
We calculate the energy spectrum of quasiparticles trapped by a domain wall separating different time reversal symmetry-breaking ground states in
a hexagonal superconductor, such as UPt$_3$. The bound state energy is found to be strongly dependent on the gap symmetry, the domain wall orientation, the quasiparticle's direction of semiclassical propagation,
and the phase difference between the domains. We calculate the corresponding density of states and show how one can use its prominent features, in particular, the zero-energy singularity,
to distinguish between different pairing symmetries.
74.20.-z, 74.55.+v
§ INTRODUCTION
The presence of domain walls (DWs) in a superconductor is a direct evidence of an unconventional nature of the pairing, because the DWs can only appear if there are two or more
distinct degenerate ground states, which transform one into another by some discrete symmetry operations, e.g., by time reversal. This is possible if the
superconducting order parameter has more than one component, i.e., corresponds to either a multidimensional irreducible representation (IREP) of the crystal point
group or to a mixture of different one-dimensional (1D) representations. The chiral $p$-wave state, which is realized, for example, in Sr$_{2}$RuO$_{4}$ (Ref. Mackenzie),
is a well-known example of a system in which DWs are believed to play a prominent role. Strong evidence of the superconducting states with broken time reversal symmetry (TRS) has also been reported
in URu$_2$Si$_2$ (Ref. Schemm), UPt$_3$ (Ref. UPt3-review), SrPtAs (Refs. SrPtAs-exp and Fischer), and PrOs$_4$Sb$_{12}$ (Refs. PrOsSb-exp and AC07).
Various TRS-breaking states have been proposed theoretically in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ (Ref. spis), doped graphene (Ref. NLC), undoped bilayer silicene (Ref. Liu2),
Na$_x$CoO$_2\cdot y$H$_2$O (Ref. KPHT), and other materials.
Superconducting DWs can be created in these systems, e.g., due to the nucleation of the order parameters of opposite chirality in different
parts of an inhomogeneous sample.
It is well known that a superconducting DW can trap quasiparticles in its vicinity, creating the Andreev bound states (ABS), see, e.g., Ref. p-wave-ABS.
The energy of these states is inside the bulk gap and their very existence can be explained by topological arguments, see Refs. Volovik and Bernevig.
The ABS contribution to the tunneling density of states (DOS) can be easily separated from that of the bulk quasiparticles and can, therefore, be used to prove the DW presence. Moreover, the ABS spectrum is sensitive to the gap
structure in the bulk of the domains, which allows one to confirm or rule out certain pairing symmetries.
We focus on the case of a three-dimensional (3D) hexagonal superconductor with the crystallographic point group $D_{6h}$, which describes UPt$_3$.
The quasi-two-dimensional tetragonal case, which is applicable to Sr$_2$RuO$_4$ and the iron-based superconductors, was previously studied in Ref. Samokhin1.
The heavy-fermion superconductor UPt$_3$ has a complicated phase diagram, with two distinct phases (called $A$ and $B$ phases) even in the absence of external magnetic field, see Ref. UPt3-review for a review.
A variety of thermodynamic and transport measurements have revealed an unconventional superconducting state with nodal excitations.
Although there is still no general consensus on the pairing symmetry in UPt$_3$, the most promising candidate model, which has recently received further
support from the Josephson interferometry<cit.> and the polar Kerr effect<cit.> experiments, is based on the two-dimensional (2D) IREP $E_{2u}$ of the point group $D_{6h}$. The corresponding order parameter is real
in the high-temperature $A$ phase (at 0.45K $<T<$ 0.5K), and complex, i.e. TRS-breaking, in the low-temperature $B$ phase (at $T<$ 0.45K).
Our goal is to study the quasiparticle tunneling features which are uniquely associated with the DWs and can be used to probe the symmetry of the superconducting order parameter, e.g., in the $B$ phase of UPt$_3$.
We analyze the TRS-breaking states corresponding to all IREPs of $D_{6h}$ that support the formation of DWs.
The paper is organized as follows. In Sec. II, we derive a general expression for the ABS energy in the semiclassical (Andreev) approximation. In Sec. III, we calculate the ABS spectrum and the corresponding contribution to the DOS separately
for each of the four possible TRS-breaking states. The summary of our results is presented in Sec. IV. Throughout the paper we use the units in which $\hbar=e=c=1$.
§ ANDREEV BOUND STATES
We consider a hexagonal superconductor described by the point group $D_{6h}$, in zero magnetic field. The $z$ axis is along the sixfold symmetry axis and the $xy$ plane coincides with the basal plane.
The electron band dispersion is assumed to be $\xi(\bk)=(k^2-k_F^2)/2m^*$, where $m^*$ is the effective mass,
with generalization to a more general, e.g., ellipsoidal, case being straightforward. The superconductor is divided into two semi-infinite superconducting domains by a planar DW.
Since the scale $\xi_d$ of the order parameter variation in
the DW is much greater than the inverse Fermi wavevector, we can use the Andreev approximation,<cit.> in which the Bogoliubov quasiparticles propagate along the semiclassical trajectories characterized by the Fermi-surface wavevectors
$\bk_F=k_{F}(\sin\theta \cos \phi,\sin\theta \sin \phi,\cos\theta)$ ($\theta$ and $\phi$ are the spherical angles, with the polar axis directed along the positive $z$ axis).
In the semiclassical approximation, the gap function is given by a $2\times 2$ spin matrix, which depends on the position $\br$ and the wavevector $\bk_F$:
$\hat\Delta(\bk_F,\br)=i\hat\sigma_2\psi(\bk_F,\br)$ for singlet pairing, and $\hat\Delta(\bk_F,\br)=i\hat{\bm{\sigma}}\hat\sigma_2\bm{d}(\bk_F,\br)$ for triplet pairing. We consider only 2D IREPs of $D_{6h}$,
therefore the gap functions can be written in the following form (Ref. Mineev): $\psi(\bk_F,\br)=\eta_1(\br)\phi_1(\hat{\bk}_F)+\eta_2(\br)\phi_2(\hat{\bk}_F)$ and
$\bm{d}(\bk_F,\br)=\eta_1(\br)\bm{\phi}_1(\hat{\bk}_F)+\eta_2(\br)\bm{\phi}_2(\hat{\bk}_F)$, where $\hat{\bk}_F=\bk_F/k_F$,
$\eta_{1,2}$ are the order parameter components, and the basis functions satisfy $\phi_{1,2}(\bk)=\phi_{1,2}(-\bk)$, $\bm{\phi}_{1,2}(\bk)=-\bm{\phi}_{1,2}(-\bk)$.
The direction of the spin vector $\bm{d}$ is assumed to be fixed along $\hat{\bm{z}}$ by the strong spin-orbit coupling of electrons with the crystal lattice.<cit.>
In general, the quasiparticle wave function has four components, corresponding to the Nambu (electron-hole) and spin degrees of freedom, but in the models considered in this paper the spin channels are decoupled and the equations
are reduced to a two-component form. For each spin projection, the wave function is a product of a rapidly oscillating plane wave $e^{i\bk_F\br}$ and a slowly varying Andreev envelope function $\Psi(\br)$, which satisfies the equation
-iv_F∇_ Δ__F()
Δ^*__F() iv_F∇_ Ψ() = EΨ().
Here $\bm{v}_F=\bk_F/m^*$ is the Fermi velocity and $\Delta_{\bk_F}(\br)$ is the gap function sensed by the quasiparticles as they propagate along the semiclassical trajectory defined by $\bk_F$:
$\Delta_{\bk_F}(\br)=\psi(\bk_F,\br)$ for singlet pairing, and $\Delta_{\bk_F}(\br)=d_z(\bk_F,\br)$ for triplet pairing.
We use the sharp DW model, in which the gap function is described by two different complex constants in the two domains along each semiclassical trajectory:
\begin{equation}
\label{Delta-pm-definition}
\left.\begin{array}{ll}
\Delta_{\bk_F}(\br)=\Delta_{+}(\theta, \phi),\quad (\br\cdot\hat{\bf{n}})>0\medskip \\
\Delta_{\bk_F}(\br)=\Delta_{-}(\theta, \phi),\quad (\br\cdot\hat{\bf{n}})<0,
\end{array}\right.
\end{equation}
where $\hat{\bn}$ is normal to the DW plane. The angular dependence of the gap function is different for different IREPs of the point group, see Sec. <ref> below.
The solution of Eq. (<ref>) which is exponentially localized near the DW has the form $\Psi_\pm(\br)\sim e^{\mp\kappa_{\pm}(\br\cdot\hat{\bf{n}})}$, where
\kappa_{\pm}=\frac{\sqrt{|\Delta_{\pm}|^2-E^2}}{|(\bm{v}_F\cdot\hat{\bf{n}})|}>0.
From the continuity of the wave function across the DW, we obtain the following equation for the Andreev bound state (ABS) energy:
\begin{equation}
\label{ABS-energy-eq}
\frac{E + i(\bm{v}_{F}\cdot\hat{\bn})\kappa_{-}}{E - i(\bm{v}_{F}\cdot\hat{\bn})\kappa_{+}} =\gamma,
\end{equation}
where $\gamma=\Delta_-/\Delta_+=\gamma_R+i\gamma_I$.
The solution of Eq. (<ref>) is straightforward, see, e.g., Ref. Samokhin1, and we find that for each direction of semiclassical propagation satisfying the condition
sgn(|γ|^2-γ_R) sgn(1-γ_R)=1,
there is exactly one ABS, whose energy is given by
E_b(θ,ϕ)=|Δ_+(θ,ϕ)|1/√(1+β^2(θ,ϕ)) sgnβ(θ,ϕ)(v̂_F·n̂),
where $\hat{\bm{v}}_F=\bm{v}_F/v_F$ and
β(θ,ϕ)= 1-γ_R/γ_I.
In all cases studied in this work, we have $|\Delta_+(\theta,\phi)|=|\Delta_-(\theta,\phi)|$, therefore $|\gamma|^2=1$ and the condition (<ref>) is satisfied for every direction.
It is straightforward to show that the ABS energy (<ref>) is inside the bulk gap,
i.e., $|E_b(\theta,\phi)|\leq|\Delta_\pm(\theta,\phi)|$. The dependence of the ABS energy on the direction of semiclassical propagation is not continuous, showing abrupt changes when either $\beta$ or the Fermi
velocity projection on the DW normal change their signs, see Appendix <ref>.
The sharp DW model (<ref>) can be justified by the following argument. The ABS wave function is exponentially localized on both sides of the DW, with the characteristic scales given by $\kappa_{\pm}^{-1}$.
The sharp DW approximation is legitimate for those directions of semiclassical propagation for which the DW width $\xi_d$ is smaller than $\kappa_{\pm}^{-1}$.
This condition is strongly angle-dependent and, in particular, fails for the trajectories corresponding to $(\bm{v}_F\cdot\hat{\bn})\to 0$. However, for such trajectories the Andreev approximation itself is not applicable.
For most directions of $\bm{k}_F$, one can use the following estimate: $\kappa_{\pm}^{-1}\gtrsim v_F/\Delta_0\sim\xi\sim\xi_d$, where $\Delta_0$ is a characteristic value of the gap and $\xi$ is the superconducting correlation length.
The quantity of interest is the DOS of the ABS's, which can be measured in tunneling experiments.
We consider two orientations of the DW, with the normal vector either parallel or perpendicular to the basal plane. In the first case, assuming $\hat{\bm{n}}\parallel\hat{\bm{x}}$, the order parameter depends only on $x$,
the momentum components parallel to the DW are conserved, and the DOS per unit DW area for both spin projections has the following form:
\begin{equation}
\label{DOS-x-general}
\end{equation}
where $L_y$ and $L_z$ are the system's dimensions in the $yz$ plane. Derivation of this expression is outlined in Appendix <ref>. Taking the thermodynamic limit and changing the integration variables
to the spherical angles, we obtain:
N_b(E) = 1/2N_Fv_F∫_0^2π dϕ |cosϕ|∫_0^π dθ sin^2θ δ[E-E_b(θ, ϕ)],
where $N_F=m^*k_F/2\pi^2$ is the normal-state DOS in 3D at the Fermi surface per one spin projection. The ABS energy, see Eq. (<ref>), has the following form:
E_b(θ,ϕ)=|Δ_+(θ,ϕ)|1/√(1+β^2(θ,ϕ)) sgnβ(θ,ϕ)sinθcosϕ.
In the case of $\hat{\bm{n}}\parallel\hat{\bm{z}}$, the order parameter depends only on $z$, and we obtain the following expressions for the DOS per unit DW area for both spin projections:
N_b(E) = 1/4N_Fv_F∫_0^2π dϕ∫_0^π dθ |sin2θ| δ[E-E_b(θ, ϕ)],
and for the ABS energy:
E_b(θ,ϕ)=|Δ_+(θ,ϕ)|1/√(1+β^2(θ,ϕ)) sgnβ(θ,ϕ) cosθ.
Due to the electron-hole symmetry, $N_{b}(E)=N_{b}(-E)$, so that below we calculate the DOS only for $E>0$.
§ ANDREEV BOUND STATES SPECTRA
The point group $D_{6h}$ has twelve IREPs, six even and six odd, of which eight are 1D and four are 2D. The formation of DWs is possible only for those
superconducting classes which are degenerate with respect to some discrete symmetry.<cit.> Since the 1D IREPs cannot support DWs, we focus
on the 2D IREPs. We consider only the TRS-breaking chiral states, with the order parameters given by $\bm{\eta}=(\eta_1,\eta_2)\propto\Delta_0(1,\pm i)$.
The momentum dependence of the corresponding gap functions is listed in Table <ref>. Note that, due to the similarity of the basis functions, our results for the IREPs $E_{1u}$ and $E_{1g}$ are also applicable to
a tetragonal superconductor with the point group $D_{4h}$.
The momentum dependence of the singlet ($\psi$) and triplet ($d_z$) gap functions of the chiral states corresponding to the 2D IREPs of $D_{6h}$, for a strong spin-orbit coupling. The singlet gap functions
correspond to the even IREPs $E_{1g}$ and $E_{2g}$, while the triplet gap functions correspond to the odd IREPs $E_{1u}$ and $E_{2u}$.
IREP $\psi(\bk)$, $d_z(\bk)$
$E_{1u}$ $k_x\pm ik_y$
$E_{1g}$ $k_{z}(k_x \pm i k_y)$
$E_{2u}$ $k_z(k^{2}_x-k^{2}_y\pm 2ik_xk_y)$
$E_{2g}$ $k^{2}_x-k^{2}_y\pm 2ik_xk_y$
§.§ $E_{1u}$ representation
For the chiral $p$-wave state corresponding to the IREP $E_{1u}$, we find from Table <ref> the following expressions for the gap functions in the two domains: $\Delta_+=\Delta_0e^{i\chi}(\hat k_{F,x}-i\hat k_{F,y})$
and $\Delta_-=\Delta_0(\hat k_{F,x}+i\hat k_{F,y})$. Here $0\leq\chi\leq\pi$ is the Josephson phase difference between the domains, which has to be included in order to satisfy the current conservation across the DW,
see Ref. Sam12 and also below. Its value depends on the microscopic details of the system, but here we regard it just as an additional phenomenological parameter.
In terms of the spherical angles the gap functions become
\begin{equation}
\left.\begin{array}{l}
\Delta_{+}(\theta, \phi)=\Delta_0 \sin\theta\, e^{i(\chi-\phi)},\\
\Delta_{-}(\theta, \phi)=\Delta_0 \sin\theta\, e^{i\phi}.
\end{array}\right.
\end{equation}
It follows from Eq. (<ref>) that $\beta=\tan(\phi-\chi/2)$. Below we calculate the ABS energies and the DOS for both orientations of the DW.
$\bm{\hat{n}}||\bm{\hat{x}}$. We obtain from Eq. (<ref>) the following expression for the ABS energy:
\begin{equation}
\label{Eb-E1u-chi}
\end{equation}
It is shown, for $\chi=0$ and $\chi=\pi$, in the upper panels of Fig. <ref>. For general $\chi$, the energy is discontinuous at $\phi-\chi/2=0,\pi$ and also at $\phi=\pi/2,3\pi/2$, see Appendix <ref>.
The ABS energy has two lines of zeros at $\phi=(\chi\pm\pi)/2$, which correspond to the quasiparticle trajectories with
$\hat k_{F,x}/\hat k_{F,y}=-\tan(\chi/2)$. These zeros in the ABS dispersion have a topological origin, see Sec. <ref> below. There are also two point zeros in the ABS energy at the poles of the Fermi surface, i.e. at $\theta=0$ and $\pi$.
However, for these directions (corresponding to the “grazing” trajectories, parallel to the DW) one has $\hat{\bm{v}}_{F}\cdot\hat{\bm{n}}=0$, and the Andreev calculation resulting in Eq. (<ref>) is not applicable.
The quasiparticle DOS is given by Eq. (<ref>) and can be found analytically for $\chi=0$ and $\pi$. Since the calculation is similar in both cases, here we outline it only for $\chi=0$,
when we have
N_b(E)=N_Fv_F\int_{0}^{\pi} d\theta \sin^2 \theta \int_{0}^{\pi/2} d\phi\cos\phi \lb \delta \lp E-\Delta_{0} \sin\theta \cos \phi \rp + \delta \lp E + \Delta_{0} \sin\theta \cos \phi \rp \rb.
Since we consider only positive energies, the second delta function does not contribute to the integral and we obtain:
\begin{eqnarray*}
N_b(E) &=& 2N_Fv_F\int_{0}^{\pi/2} d\theta\sin^2\theta \int_{0}^{\pi/2} d\phi\cos\phi\,\delta(E-\Delta_{0} \sin\theta \cos \phi)\\
&=& \frac{2N_Fv_FE}{\Delta^{2}_{0}} \int_{E/\Delta_{0}}^{1} \frac{x dx}{\sqrt{1-x^2}\sqrt{x^{2}-(E/\Delta_0)^2}},
\end{eqnarray*}
where $x=\sin\theta$. Evaluating the last integral,<cit.> we finally arrive at the following expression:
In a similar fashion, we obtain:
N_b(E)= 2N_Fv_F/Δ_0√(1-E^2/Δ^2_0)
for $\chi=\pi$. The DOS curves corresponding to Eqs. (<ref>) and (<ref>), normalized by $N_Fv_F/2\Delta_0$, are shown in the bottom panels of Fig. <ref>. We can see that the overall magnitude of the ABS
contribution to the DOS (per unit area) is of the same order as in the normal state, since $N_Fv_F/\Delta_0\sim N_F\xi\sim N_F\xi_d$.
$\bm{\hat{n}}||\bm{\hat{z}}$. Using Eq. (<ref>) we obtain for the ABS energy:
\begin{equation}
\label{E-E1u-nz}
E_b(\theta,\phi)=\Delta_{0}|\sin\theta|\cos\left(\phi-\frac{\chi}{2}\right)\,\mathrm{sgn}\lb \sin\left(\phi-\frac{\chi}{2}\right)\cos\theta \rb.
\end{equation}
It is discontinuous at $\phi-\chi/2=0,\pi$ and also at $\theta=\pi/2$, see Appendix <ref>.
One can show that the current conservation requires that $\chi=\pi/2$ in the lowest order of the Ginzburg-Landau (GL) gradient expansion. The superconducting current can
be obtained in the standard fashion from the gradient terms in the GL free energy density. Since the order parameter components depend only on $z$, the gradient energy has the form
$F_{grad}=K_4|\nabla_z\bm{\eta}|^2$, in the notations of Ref. Mineev. Replacing the gradients by the covariant derivatives,
$\bm{\nabla}\to\bm{\nabla}+2i\bm{A}$, and varying with respect to the vector potential $\bm{A}$, we obtain for the superconducting current:
$\bm{j}=2K_4\,\mathrm{Im}\,(\bm{\eta}^*\nabla_{z}\bm{\eta})\bm{\hat{z}}$. We use the constant-amplitude approximation for the order parameter:
\eta_1(z)=\Delta_0 e^{i\varphi(z)},\quad \eta_2(z)=\Delta_0 e^{i\varphi(z)-i\gamma(z)},
where $\varphi$ is the common (or Josephson) phase of the order parameter components and $\gamma$ is the relative phase, satisfying $\gamma(\pm\infty)=\pm\pi/2$. Then the supercurrent becomes $j_z=2K_4\Delta_0^2(2\nabla_{z}\varphi-\nabla_{z}\gamma)$.
It follows from the current conservation that $j_z$ has a constant value, which is fixed by external sources. Setting $j_z=0$, one obtains $\nabla_z\varphi=\nabla_z\gamma/2$, and, therefore,
χ≡φ(+∞)-φ(-∞) = γ(+∞)-γ(-∞)/2=π/2.
It is easy to see that this result holds for the chiral states corresponding to all four 2D IREPs.
In contrast to the case of $\bm{\hat{n}}||\bm{\hat{x}}$ (Refs. Sam12 and Samokhin1), the Josephson phase difference between the domains for $\bm{\hat{n}}||\bm{\hat{z}}$ takes a universal value $\pi/2$,
i.e., does not depend on the coefficients in the GL expansion. This last conclusion can be invalidated by the inclusion of higher-order gradient terms and going beyond the constant-amplitude approximation.
However, one can see from the way the angle $\phi$ enters Eq. (<ref>) that the ABS dispersion for different $\chi$ is obtained from that for $\chi=\pi/2$ by simply translating the latter along the $\phi$
axis by $\chi/2$. Therefore, the DOS, see Eq. (<ref>), does not actually depend on $\chi$. The calculation is similar to the $\bm{\hat{n}}||\bm{\hat{x}}$ case and the final
result has the following form:
N_b(E)= 2N_Fv_F/Δ_0√(1-E^2/Δ^2_0).
The ABS energy for $\chi=\pi/2$ and the DOS for any $\chi$ are shown in Fig. <ref>.
(Color online) The ABS energy as a function of the direction of semiclassical propagation (upper panels) and the corresponding DOS (lower panels),
for $\chi=0$ and $\chi=\pi$, in the case of the chiral $p$-wave state ($E_{1u}$) and $\bm{\hat{n}}||\bm{\hat{x}}$. The grazing trajectories (for which $v_{F,x}=0$)
are shown at $\phi=\pi/2$ and $\phi=3\pi/2$ by horizontal dotted lines.
(Color online) The ABS energy as a function of the direction of semiclassical propagation for $\chi=\pi/2$ (upper panel) and the corresponding DOS (lower panel), in the case of the chiral $p$-wave state ($E_{1u}$)
and $\bm{\hat{n}}||\bm{\hat{z}}$. The vertical dotted line at $\theta=\pi/2$ corresponds to the grazing trajectory (for which $v_{F,z}=0$).
§.§ $E_{1g}$ representation
For the chiral $d$-wave state corresponding to the IREP $E_{1g}$, we obtain from Table <ref> the following expressions for the gap functions in the two domains:
$\Delta_+=2\Delta_0e^{i\chi}\hat k_{F,z}(\hat k_{F,x}-i\hat k_{F,y})$ and $\Delta_-=2\Delta_0\hat k_{F,z}(\hat k_{F,x}+i\hat k_{F,y})$. Therefore,
Δ_+(θ, ϕ)=Δ_0 sin2θ e^i(χ-ϕ),
Δ_-(θ, ϕ)=Δ_0 sin2θ e^iϕ,
and $\beta=\tan(\phi-\chi/2)$.
$\bm{\hat{n}}||\bm{\hat{x}}$. In this case, Eq. (<ref>) takes the following form:
\begin{equation}
\label{Eb-E1g-chi}
E_b(\theta,\phi)=\Delta_{0}|\sin 2\theta|\cos\left(\phi-\frac{\chi}{2}\right)\,\mathrm{sgn}\lb\sin\left(\phi-\frac{\chi}{2}\right)\cos\phi\rb.
\end{equation}
The ABS energy is discontinuous at $\phi-\chi/2=0,\pi$ and also at $\phi=\pi/2,3\pi/2$, see Appendix <ref>.
It has two lines of zeros at $\phi=(\chi\pm\pi)/2$ and another one in the basal plane, i.e., at $\hat k_{F,z}=0$. The point zeros at $\theta=0$ and $\pi$ correspond to the trajectories parallel to the DW,
for which the Andreev approximation is not applicable. In the upper panels of Fig. <ref>, we show the ABS energy for $\chi=0$ and $\pi$.
The DOS is given by Eq. (<ref>). Following the same steps as in the previous subsection, we obtain a constant DOS:
for $\chi=0$, see the bottom panel of Fig. <ref>. For $\chi=\pi$, we have from Eq. (<ref>): $E_b(\theta,\phi)=\Delta_{0}|\sin 2\theta|\sin\phi$, and Eq. (<ref>) can be reduced to the form
N_b(E)=2N_Fv_F\int_{0}^{\pi/2} d\theta \sin^2\theta \int_{0}^{\pi/2} d\phi\cos\phi\,\delta\lp E-\Delta_{0}\sin2\theta\sin\phi\rp
=\frac{N_Fv_F}{\Delta_0}\int_{\alpha}^{\pi/2-\alpha} d\theta\tan\theta,
where $\alpha=(1/2)\arcsin(E/\Delta_0)$. The last integral can be easily evaluated and we arrive at the following final expression for the DOS:
N_b(E)= N_Fv_F/Δ_0ln(1/2arcsinE/Δ_0),
which diverges logarithmically at $E\rightarrow 0$, as shown Fig. <ref>. This divergence is nothing but the van Hove singularity due to the saddle points in the ABS dispersion
at $\theta=\pi/2$ and $\phi=0,\pi$, i.e., for $\hat{\bk}_F$ perpendicular to the DW.
$\bm{\hat{n}}||\bm{\hat{z}}$. We obtain from Eq. (<ref>):
\begin{equation}
E_b(\theta,\phi)=\Delta_{0}\sin 2\theta\cos\left(\phi-\frac{\chi}{2}\right)\,\mathrm{sgn}\lb\sin\left(\phi-\frac{\chi}{2}\right)\rb.
\end{equation}
The energy is discontinuous at $\phi-\chi/2=0$ and $\pi$, see Appendix <ref>.
As for the $E_{1u}$ IREP, the ABS dispersion as a function of $\phi$ simply shifts upon changing $\chi$, therefore the DOS does not depend on $\chi$. A straightforward calculation yields the following result:
The ABS energy for $\chi=\pi/2$ and the DOS for any $\chi$ are shown in Fig. <ref>.
(Color online) The ABS energy as a function of the direction of semiclassical propagation (upper panels) and the corresponding DOS (lower panels),
for $\chi=0$ and $\chi=\pi$, in the case of the chiral $d$-wave state ($E_{1g}$) and $\bm{\hat{n}}||\bm{\hat{x}}$. The grazing trajectories (for which $v_{F,x}=0$)
are shown at $\phi=\pi/2$ and $\phi=3\pi/2$ by horizontal dotted lines.
(Color online) The ABS energy as a function of the direction of semiclassical propagation for $\chi=\pi/2$ (upper panel) and the corresponding DOS (lower panel), in the case of the chiral $d$-wave state ($E_{1g}$)
and $\bm{\hat{n}}||\bm{\hat{z}}$. The vertical dotted line at $\theta=\pi/2$ corresponds to the grazing trajectory (for which $v_{F,z}=0$).
§.§ $E_{2u}$ representation
For the chiral $f$-wave state corresponding to the IREP $E_{2u}$, we obtain from Table <ref> the following expressions for the gap functions in the two domains:
$\Delta_+=\Delta_0e^{i\chi}\hat k_{F,z}(\hat k^2_{F,x}-\hat k^2_{F,y}-2i\hat k_{F,x}\hat k_{F,y})$ and $\Delta_-=\Delta_0\hat k_{F,z}(\hat k^2_{F,x}-\hat k^2_{F,y}+2i\hat k_{F,x}\hat k_{F,y})$. Therefore,
Δ_+(θ, ϕ)=Δ_0 sin^2θcosθ e^i(χ-2ϕ),
Δ_-(θ, ϕ)=Δ_0 sin^2θcosθ e^2iϕ,
and $\beta=\tan(2\phi-\chi/2)$.
$\bm{\hat{n}}||\bm{\hat{x}}$. In this case, Eq. (<ref>) takes the following form:
\begin{equation}
\label{Eb-E2u-chi}
\end{equation}
This expression has discontinuities at $2\phi-\chi/2=0,\pi$ and also at $\phi=\pi/2,3\pi/2$, see Appendix <ref>.
It has four lines of zeros at $\phi=(\chi\pm\pi)/4$ and $\phi=(\chi\pm 3\pi)/4$, and another one at $\hat k_{F,z}=0$. The isolated second-order point zeros at $\theta=0$ and $\pi$ correspond to the trajectories parallel to the DW,
for which the Andreev approximation is not applicable.
The ABS energy for $\chi=0$ and $\pi$ is shown in the upper panels of Fig. <ref>.
The DOS for $\chi=0$, see Eq. (<ref>), can be reduced to the following form:
N_b(E)= N_Fv_F/√(2)Δ_0 ∫_0^1 dx/√(x)[1/√(x(1-x^2)-(E/Δ_0))+1/√(x(1-x^2)+(E/Δ_0))] Θ[x(1-x^2)-E/Δ_0],
where $\Theta(x)$ is the Heaviside step function and $x=\cos\theta$.
Since the function $x(1-x^2)$ attains its maximum at $x=1/\sqrt{3}$, the DOS vanishes at $E>2\Delta_{0}/3\sqrt{3}$. The integral in Eq. (<ref>) is evaluated numerically.
The logarithmic van Hove singularity in the DOS at $E\to 0$ is due to the saddle points in the ABS dispersion in the basal plane, at $\theta=\pi/2$ and $\cos(2\phi)=0$.
One can easily show that the DOS has a zero-energy singularity at all values of $\chi$. Indeed, we have
away from the spectrum discontinuities.
This last expression has the following zeros: (i) $\theta=0,\pi$, whose contribution to the DOS is nonsingular, due to the factor in front of the $\delta$-function in Eq. (<ref>); (ii) $\cos^2\theta=1/3$ and $\sin(2\phi-\chi/2)=0$,
which corresponds to a maximum (minimum) of $E_b$; and (iii) $\theta=\pi/2$ and $\cos(2\phi-\chi/2)=0$, which corresponds to the saddle points of $E_b$. It is the saddle points, which are located at the four perpendicular directions in the basal plane
where the lines of zeros of $E_b$ intersect, that produce the van Hove singularity at $E\to 0$. The DOS for $\chi=0$ and $\chi=\pi$ are shown in Fig. <ref>.
$\bm{\hat{n}}||\bm{\hat{z}}$. It follows from Eq. (<ref>) that
\begin{equation}
\end{equation}
which is discontinuous at $2\phi-\chi/2=0$ and $\pi$, see Appendix <ref>. The ABS dispersion as a function of $\phi$ shifts upon changing $\chi$, therefore,
the DOS does not depend on $\chi$ and we obtain from Eq. (<ref>):
N_b(E)=2N_Fv_F/Δ_0∫_0^1 x dx/√(x^2(1-x^2)^2-(E/Δ_0)^2)Θ[x(1-x^2)-E/Δ_0],
where $x=\cos\theta$.
The integral here is calculated numerically. The ABS energy for $\chi=\pi/2$ and the DOS for any $\chi$ are shown in Fig. <ref>. The logarithmic singularity in the DOS at $E\to 0$ comes from the saddle points in the
ABS dispersion at $\theta=0,\pi$, i.e., for $\hat{\bk}_F$ perpendicular to the DW.
The ABS energy as a function of the direction of semiclassical propagation (upper panels) and the corresponding DOS (lower panels),
for $\chi=0$ and $\chi=\pi$, in the case of the chiral $f$-wave state ($E_{2u}$) and $\bm{\hat{n}}||\bm{\hat{x}}$. The grazing trajectories (for which $v_{F,x}=0$)
are shown at $\phi=\pi/2$ and $\phi=3\pi/2$ by horizontal dotted lines.
(Color online) The ABS energy as a function of the direction of semiclassical propagation for $\chi=\pi/2$ (upper panel) and the corresponding DOS (lower panel), in the case of the chiral $f$-wave state ($E_{2u}$)
and $\bm{\hat{n}}||\bm{\hat{z}}$. The vertical dotted line at $\theta=\pi/2$ corresponds to the grazing trajectory (for which $v_{F,z}=0$).
§.§ $E_{2g}$ representation
Finally, we consider the chiral $d$-wave state corresponding to the IREP $E_{2g}$, in which case $\Delta_+=\Delta_0e^{i\chi}(\hat k^2_{F,x}-\hat k^2_{F,y}-2i\hat k_{F,x}\hat k_{F,y})$ and
$\Delta_-=\Delta_0(\hat k^2_{F,x}-\hat k^2_{F,y}+2i\hat k_{F,x}\hat k_{F,y})$. Therefore,
Δ_+(θ, ϕ)=Δ_0 sin^2θ e^i(χ-2ϕ),
Δ_-(θ, ϕ)=Δ_0 sin^2θ e^2iϕ,
and $\beta=\tan(2\phi-\chi/2)$.
$\bm{\hat{n}}||\bm{\hat{x}}$. In this case, Eq. (<ref>) takes the following form:
\begin{equation}
\label{Eb-E2g-chi}
\end{equation}
which is discontinuous at $2\phi-\chi/2=0,\pi$ and also at $\phi=\pi/2,3\pi/2$, see Appendix <ref>.
It has four lines of zeros at $\phi=(\chi\pm\pi)/4$ and $\phi=(\chi\pm 3\pi)/4$. The isolated second-order point zeros at $\theta=0$ and $\pi$ correspond to the trajectories parallel to the DW,
for which the Andreev approximation is not applicable.
The ABS energy for $\chi=0$ and $\pi$ is shown in the upper panels of Fig. <ref>.
The DOS, see Eq. (<ref>), can be calculated analytically for $\chi=0$ and $\chi=\pi$. Following the same steps as in the previous subsections, we obtain:
N_b(E) = N_Fv_F/√(2)Δ_0 π/2 + arcsin√(1-E/Δ_0)/√(1+E/Δ_0)for $\chi=0$, and
for $\chi=\pi$, see Fig. <ref>.
$\bm{\hat{n}}||\bm{\hat{z}}$. We obtain from Eq. (<ref>):
\begin{equation}
\end{equation}
The discontinuities of the ABS energy are located at $2\phi-\chi/2=0,\pi$ and also at $\theta=\pi/2$, see Appendix <ref>.
As in the previous subsections, the ABS dispersion as a function of $\phi$ shifts upon changing $\chi$, the DOS does not depend on $\chi$, and we obtain:
N_b(E)=N_Fv_F/Δ_0∫_E/Δ_0^1dx/√(x^2-(E/Δ_0)^2)=N_Fv_F/Δ_0 ln√(1-(E/Δ_0)^2)+1/E/Δ_0,
where $x=\sin^2\theta$. The ABS energy for $\chi=\pi/2$ and the DOS for any $\chi$ are shown in Fig. <ref>. The logarithmic singularity in the DOS at $E\to 0$ comes from the saddle points in the
ABS dispersion at $\theta=0,\pi$, i.e., for $\hat{\bk}_F$ perpendicular to the DW.
The ABS energy as a function of the direction of semiclassical propagation (upper panels) and the corresponding DOS (lower panels),
for $\chi=0$ and $\chi=\pi$, in the case of the chiral $d$-wave state ($E_{2g}$) and $\bm{\hat{n}}||\bm{\hat{x}}$. The grazing trajectories (for which $v_{F,x}=0$)
are shown at $\phi=\pi/2$ and $\phi=3\pi/2$ by horizontal dotted lines.
(Color online) The ABS energy as a function of the direction of semiclassical propagation for $\chi=\pi/2$ (upper panel) and the corresponding DOS (lower panel), in the case of the chiral $d$-wave state ($E_{2g}$)
and $\bm{\hat{n}}||\bm{\hat{z}}$. The vertical dotted line at $\theta=\pi/2$ corresponds to the grazing trajectory (for which $v_{F,z}=0$).
§.§ Topological origin of the ABS zero modes
The number of zero-energy ABS localized at the DW separating degenerate chiral states is determined by the difference between topological invariants characterizing the
superconducting states in the bulk of the domains, which is known as the bulk-boundary correspondence.<cit.> As an illustration of this statement, we focus on the case of $\bm{\hat{n}}||\bm{\hat{x}}$.
To define the appropriate topological invariant, we introduce the Matsubara-like Green's function of the Bogoliubov quasiparticles in the bulk:
\begin{equation}
\label{BdG-GF}
\hat G^{-1}(k_0,\bk)=ik_0-\hat H(\bk),
\end{equation}
where $ik_0$ is imaginary “frequency”, $\bk=(k_x,k_y,k_z)$ takes values in the 3D Brillouin zone, and
\hat H(\bk)=\lp\begin{array}{cc} \xi(\bk) & \Delta(\bk)\\ \Delta^*(\bk) & -\xi(\bk)\end{array}\rp
is the Bogoliubov-de Gennes (BdG) Hamiltonian, with the gap function $\Delta(\bk)$.
Since we consider only the singlet pairing and the triplet pairing with $\bm{d}\parallel\hat{\bm{z}}$, see Sec. <ref>, the spin channels are decoupled and the $4\times 4$ BdG equations
are reduced to a two-component (electron-hole, or Nambu) form for each spin. The BdG Hamiltonian can be written in the form $\hat H(\bk)=\bm{\nu}(\bk)\hat{\bm{\tau}}$, where $\hat{\bm{\tau}}$ are the Pauli matrices in the Nambu space and
\bm{\nu}(\bk)=\left(\begin{array}{c}
\re\Delta(\bk)\\
\xi(\bk)
\end{array}\right).
The eigenvalues of $\hat H(\bk)$ are given by $\pm E(\bk)$, where $E(\bk)=\sqrt{\xi^2(\bk)+|\Delta(\bk)|^2}$ the energy of the Bogoliubov fermionic excitations.
At given $k_z$, regarded as a parameter, one can define the following topological invariant:<cit.>
\begin{equation}
\label{N-kz-def}
N(k_z)=-\frac{1}{24\pi^2}\int\mathrm{tr}(\hat Gd\hat G^{-1})^3.
\end{equation}
Here “tr” stands for the Nambu matrix trace, the powers of the 1-form $\hat Gd\hat G^{-1}$ should be understood in the sense of combined exterior and matrix multiplication,
and the integration is performed over $k_0$ and $\bk_\perp=(k_x,k_y)$, with $\bk_\perp$ taking values in the 2D cross-section of the Brillouin zone by the constant $k_z$ plane.
After some algebra, we obtain:
\begin{equation}
\label{N-kz-nu}
N(k_z)=\frac{1}{4\pi^2}\int\frac{\bm{\nu}(d\bm{\nu}\times d\bm{\nu})dk_0}{(k_0^2+E^2)^2}=\frac{1}{8\pi}\int_{k_z=\mathrm{const}}\hat{\bm{\nu}}(d\hat{\bm{\nu}}\times d\hat{\bm{\nu}}),
\end{equation}
where $\hat{\bm{\nu}}=\bm{\nu}/|\bm{\nu}|$. We assume that the superconducting pairing is BCS-like and effective only near the Fermi surface. At given $k_z$, this results in the gap function being nonzero only near the Fermi line
$\mathrm{FL}(k_z)$, which is the intersection of the Fermi surface and the constant $k_z$ plane. We represent the gap function in the form $\Delta(\bk)=|\Delta(\bk)|e^{i\varphi(\bk)}$, and assume
that there are no gap nodes, i.e. the gap magnitude does not vanish anywhere on the Fermi line. Then it follows from Eq. (<ref>) that
\begin{equation}
\label{N-kz-final}
\end{equation}
therefore the topological invariant (<ref>) is nothing but the phase winding number of the gap function around the cross-section of the Fermi surface at given $k_z$. Assuming a spherical Fermi surface, the cross-section is a circle of
radius $k_{F,\perp}=\sqrt{k_F^2-k_z^2}$.
For the superconducting states considered above, the topological invariant (<ref>) takes opposite nonzero values for the states of opposite chirality, see Table <ref>. The topological invariants are not
defined at the bulk gap nodes, i.e. at $k_z=\pm k_F$ for all four IREPs and additionally at $k_z=0$ for the IREPs $E_{1g}$ and $E_{2u}$.
According to Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), the ABS energy for all four IREPs can be written in the following form:
\begin{equation}
\label{Eb-common}
\end{equation}
where $n=1$ for $\Gamma=E_{1u},E_{1g}$ and $n=2$ for $\Gamma=E_{2u},E_{2g}$, and the function $f_\Gamma(\theta)$ depends on the IREP. At fixed $k_z=k_F\cos\theta$, the last expression vanishes at some values of $\phi$, corresponding to
the ABS zero modes. It is easy to see that there are $2n$ zero modes: for $\Gamma=E_{1u},E_{1g}$ they correspond to $\phi=(\chi\pm\pi)/2$, while for $\Gamma=E_{2u},E_{2g}$ they correspond to
$\phi=(\chi\pm\pi)/4$ and $\phi=(\chi\pm 3\pi)/4$.
One can define the algebraic number $\nu(k_z)$ of the ABS zero modes as the number of positive-velocity modes minus the number of negative-velocity modes. According to the bulk-boundary correspondence (Ref. Volovik),
$\nu$ is equal to the difference between the topological invariants in the bulk of the two domains:
\begin{equation}
\label{bulk-boundary}
\nu(k_z)=N(k_z)|_{x>0}-N(k_z)|_{x<0}.
\end{equation}
Expressing the ABS energy, see Eq. (<ref>), in terms of $k_y=k_{F,\perp}\sin\phi$, one can show that the ABS zero modes propagate along the DW in the same direction:
$\mathrm{sgn}(\partial E_b/\partial k_y)=\mathrm{sgn}(\partial E_b/\partial\phi)\,\mathrm{sgn}(\cos\phi)=-1$, therefore $\nu=-2n$.
On the other hand, it follows from Table <ref> that $N(k_z)|_{x>0}=-n$ and $N(k_z)|_{x<0}=n$, which means that Eq. (<ref>) is indeed satisfied. Taking into account the doubling of the degrees
of freedom due to spin, the total number of the ABS zero modes localized near the DW is equal to $4n$, at given $k_z$. Note that the same topological argument can be used to
prove the existence of zero-energy ABS near the surface of an unconventional superconductor, see Ref. ZBCP. For UPt$_3$, it was done recently in Ref. GN15.
Topological invariant, Eq. (<ref>), for the chiral states corresponding to the 2D IREPs of $D_{6h}$.
IREP gap function $N(k_z)$
$E_{1u}$ $k_x\pm ik_y$ $\pm 1$
$E_{1g}$ $k_{z}(k_x \pm i k_y)$ $\pm 1$
$E_{2u}$ $k_z(k^{2}_x-k^{2}_y\pm 2ik_xk_y)$ $\pm 2$
$E_{2g}$ $k^{2}_x-k^{2}_y\pm 2ik_xk_y$ $\pm 2$
§ CONCLUSION
We have found that the DWs separating degenerate TRS-breaking superconducting states in a 3D hexagonal crystal always create the quasiparticle ABS, for all directions of the semiclassical propagation.
We have considered all four 2D IREPs of the point group $D_{6h}$ (two singlet and two triplet cases) and two orientations of the DW, parallel and perpendicular to the $z$ axis.
The ABS spectrum strongly depends on the order parameter symmetry and the DW orientation. Additionally, it is affected by the Josephson phase difference $\chi$ between the
domains, which is determined by the microscopic parameters. If the DW is parallel to the $z$ axis, then there is a significant difference between
the chiral states $(1,\pm i)$ (corresponding to $\chi=0$) and $(\pm 1,i)$ (corresponding to $\chi=\pi$), which can be treated analytically.
The spectrum of the DW ABS's can be probed in tunneling experiments by measuring their DOS, which has very different energy dependence from that of the bulk quasiparticles.
We have calculated the DOS per unit area of the DW and found a widely varying behaviour, the most prominent feature being the logarithmic van Hove singularity at zero energy, which is present
in several cases.
Despite the qualitative sensitivity of the DOS to the microscopic parameters that cannot be easily controlled in experiment, we can still make some firm predictions for the DW effects on the tunneling measurements in UPt$_3$.
First, there is strong evidence that the gap symmetry in the $B$ phase of UPt$_3$ is described by the chiral $f$-wave state corresponding to the IREP $E_{2u}$. If this is the case, then our results in Sec. <ref>
indicate that the zero-energy singularity in the DOS is a universal feature, which, in contrast to the other three IREPs, is present for both orientations of the DW and for all values of $\chi$.
Second, if the DW is perpendicular to the $z$ axis, then the DOS does not actually depend on $\chi$, showing different behaviour for the four IREPs: a broad dome-like maximum for $E_{1u}$, a constant for $E_{1g}$, the zero-energy singularity with
two sharp edges for $E_{2u}$, and the zero-energy singularity without sharp edges for $E_{2g}$. We hope that these features can be directly probed in tunneling experiments, thus shedding light on the presence of the DWs as well as
the underlying pairing symmetry in UPt$_3$ and other hexagonal superconductors.
This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
§ DISCONTINUITIES OF THE ABS SPECTRUM
In all cases studied in this paper the gap function has the same magnitude on both sides of the DW: $\Delta_\pm=\Delta e^{i\varphi_\pm}$.
Therefore, $\gamma=\Delta_-/\Delta_+=e^{i\Phi}$, where $\Phi=\varphi_--\varphi_+$. We obtain from Eqs. (<ref>) and (<ref>):
\beta=\tan\left(\frac{\Phi}{2}\right)
\begin{equation}
\label{app-E_b}
\end{equation}
Both $\Delta$ and $\Phi$ depend on the direction of semiclassical propagation, characterized by the Fermi-surface wavevector $\bk_F$. At each $\bk_F$, we have $|E_b|\leq\Delta$, therefore the ABS are disconnected from the bulk states.
It follows from Eq. (<ref>) that the ABS energy is not defined for some $\hat{\bk}_F$.
It is discontinuous at $\Phi=0\mod 2\pi$, which corresponds to $\Delta_+=\Delta_-$. For such semiclassical trajectories, the DW is “invisible” to the quasiparticles. The ABS energy is also discontinuous at $\hat{\bm{v}}_{F}\perp\hat{\bm{n}}$,
i.e. when the quasiparticles move parallel to the DW. For such trajectories, the Andreev approximation itself is not applicable.
§ QUASIPARTICLE DOS NEAR A DOMAIN WALL
The local quasiparticle DOS for both spin projections is given by the following expression:
N(\br,E)=-\frac{1}{\pi}\sum_{\alpha=\uparrow,\downarrow}\im G^R_{\alpha\alpha}(\br,\br;E).
Here $G^R$ is the retarded Green's function, which is obtained in the standard fashion, by analytically continuing the Fourier transform of the Matsubara Green's function
$G_{\alpha}(\br_1,\br_2;\tau)=-\langle T_\tau\psi_\alpha(\br_1,\tau)\psi^\dagger_\beta(\br_2,0)\rangle$ to real frequencies.<cit.>
Next, we represent the field operators as
\begin{eqnarray*}
\psi_\alpha(\br)=\sum_a[u_a(\br,\alpha)\gamma_a+v_a^*(\br,\alpha)\gamma_a^\dagger],\\
\psi^\dagger_\alpha(\br)=\sum_a[v_a(\br,\alpha)\gamma_a+u_a^*(\br,\alpha)\gamma_a^\dagger],
\end{eqnarray*}
where $\gamma^\dagger_a,\gamma_a$ are the creation and annihilation operators of the Bogoliubov quasiparticles and the quantum numbers $a$ label the upper half of the spectrum ($E_a\geq 0$) of the $4\times 4$ Bogoliubov-de Gennes Hamiltonian.
Both in the singlet case and in the triplet case with $\bm{d}\parallel\hat{\bm{z}}$, the spin channels decouple and the local DOS becomes
\begin{equation}
\label{app: DOS-2by2}
\end{equation}
where the two-component Nambu spinor satisfies the following equation:
\begin{equation}
\label{app: BdG-2by2}
\left(\begin{array}{cc}
\hat\xi & \hat\Delta\\
\hat\Delta^\dagger & -\hat\xi
\end{array}\right)
\left(\begin{array}{c}
\end{array}\right)=E_a
\left(\begin{array}{c}
\end{array}\right).
\end{equation}
Due to the electron-hole symmetry of the BdG spectrum, one can focus only on the electron-like branch with $E\geq 0$.
If the order parameter depends only on $x$, then the normalized solutions of Eq. (<ref>) have the form
\begin{eqnarray*}
&& \left(\begin{array}{c}
\end{array}\right)=\frac{1}{\sqrt{L_yL_z}}
\left(\begin{array}{c}
\end{array}\right)e^{ik_yy}e^{ik_zz},\\
&& \int_{-\infty}^\infty dx\;(|u|^2+|v|^2)=1.
\end{eqnarray*}
One can define the quasiparticle DOS per unit area in the $yz$ plane as follows:
\begin{equation}
\label{app: N-area}
N(E)=\int_{-\infty}^\infty dx\; N(\br,E).
\end{equation}
For the ABS, we have $|u|^2=|v|^2$, therefore $\int_{-\infty}^\infty|u|^2dx=1/2$. Inserting this last expression in Eqs. (<ref>) and (<ref>), we arrive at Eq. (<ref>).
Mackenzie A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003).
Schemm E. R. Schemm, R. E. Baumbach, P. H. Tobash, F. Ronning, E. D. Bauer, and A. Kapitulnik, Phys. Rev. B 91, 140506(R) (2015).
UPt3-review R. Joynt and L. Taillefer, Rev. Mod. Phys. 74, 235 (2002).
SrPtAs-exp P. K. Biswas, H. Luetkens, T. Neupert, T. Stürzer, C. Baines, G. Pascua, A. P. Schnyder, M. H. Fischer, J. Goryo, M. R. Lees, H. Maeter, F. Brückner, H.-H. Klauss, M. Nicklas, P. J. Baker, A. D. Hillier, M. Sigrist, A. Amato,
and D. Johrendt, Phys. Rev. B 87, 180503(R) (2013).
Fischer M. H. Fischer, T. Neupert, C. Platt, A. P. Schnyder, W. Hanke, J. Goryo, R. Thomale, and M. Sigrist, 89, 020509(R) (2014).
PrOsSb-exp Y. Aoki, A. Tsuchiya, T. Kanayama, S. R. Saha, H. Sugawara, H. Sato, W. Higemoto, A. Koda, K. Ohishi, K. Nishiyama, and R. Kadono, Phys. Rev. Lett. 91, 067003 (2003).
AC07 T. R. Abu Alrub and S. H. Curnoe, Phys. Rev. B 76, 054514 (2007).
spis S. Maiti and A. V. Chubukov, Phys. Rev. B 87, 144511 (2013).
NLC R. Nandkishore, L. S. Levitov, and A. V. Chubukov, Nature Phys. 8, 158 (2012).
Liu2 F. Liu, C.-C. Liu, K. Wu, F. Yang, and Y. Yao, 111, 066804 (2013).
KPHT M. L. Kiesel, C. Platt, W. Hanke, and R. Thomale, Phys. Rev. Lett. 111, 097001 (2013).
p-wave-ABS T. L. Ho, J. R. Fulco, J. R. Schrieffer, and F. Wilczek, 52, 1524 (1984); M. Nakahara, J. Phys. C 19, L195 (1986); L. I. Burlachkov and N. B. Kopnin, JETP Lett. 47, 48 (1988);
M. Matsumoto and M. Sigrist, 68, 994 (1999).
Volovik M. A. Silaev and G. E. Volovik, JETP 119, 1042 (2014); G. E. Volovik, The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003).
Bernevig B. A. Bernevig, Topological Insulators and Topological Superconductors (Princeton University Press, USA, 2013).
Samokhin1 S. P. Mukherjee and K. V. Samokhin, Phys. Rev. B 91, 104521 (2015).
UPt3-phase J. D. Strand, D. J. Van Harlingen, J. B. Kycia, and W. P. Halperin, Phys. Rev. Lett. 103, 197002 (2009).
UPt3-Kerr E. R. Schemm, W. J. Gannon, C. M. Wishne, W. P. Halperin, and A. Kapitulnik, Science 345, 190 (2014).
And64 A. F. Andreev, Sov. Phys. JETP 19, 1228 (1964); Ch. Bruder, Phys. Rev. B 41, 4017 (1990); C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994);
I. Adagideli, P. M. Goldbart, A. Shnirman, and A. Yazdani, Phys. Rev. Lett. 83, 5571 (1999).
Mineev V. P. Mineev and K. V. Samokhin, Introduction to Unconventional Superconductivity (Gordon and Breach Science Publishers, London, 1999).
z-axis Another possibility is that the $\bm{d}$ vector is “pinned” to the basal plane, e.g. $\bm{d}\propto k_y\hat{\bm{x}}+k_x\hat{\bm{y}}$, see Ref. Mineev.
We do not consider this possibility here, because it does not seem to be supported by the experimental data for UPt$_3$ (Refs. UPt3-review,UPt3-phase, and UPt3-Kerr).
VG85 G. E. Volovik and L. P. Gor'kov, Sov. Phys. JETP 61, 843 (1985).
Sam12 K. V. Samokhin, Phys. Rev. B 85, 014515 (2012).
Prudnikov A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions (Gordon and Breach, London, 1986).
ZBCP S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641 (2000).
GN15 P. Goswami and A. Nevidomskyy, Phys. Rev. B 92, 214504 (2015).
AGD A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1975).
|
1511.00911
|
x]Erik Sjöqvist
[x]Department of Physics and Astronomy, Uppsala University, Box 516,
SE-751 20 Uppsala, Sweden
We generalize nonadiabatic holonomic quantum computation in a resonant $\Lambda$
configuration proposed in [New J. Phys. 14 (2012) 103035] to the case of off-resonant
driving lasers. We show that any single-qubit holonomic gate can be realized by
separately varying the detuning, amplitude, and phase of the lasers.
Geometric phase; Quantum gates
03.65.Vf, 03.67.Lx
Holonomic quantum computation (HQC) is the idea to use non-Abelian geometric phases to
implement a universal set of quantum gates. It was first proposed in the context of adiabatic
evolution by Zanardi and Rasetti <cit.> based on Wilczek-Zee geometric phases
<cit.> associated with degenerate energy subspaces driven by slowly varying
parameters. More recently, a scheme for fast holonomic quantum gates has been proposed
<cit.>. This scheme has subsequently been implemented experimentally
The nonadiabatic HQC scheme in Ref. <cit.> is based on a $\Lambda$ system
driven by short resonant laser pulses. The high-speed feature makes the resulting gates
potentially easier to implement as it implies a shorter exposure to detrimental
decoherence effects. Here, we modify the original setup in Ref. <cit.> by allowing
nonvanishing detuning of the two lasers driving off-resonant transitions between the excited
state and the computational levels. This modified scheme can be used to implement
any single-qubit gate by separately varying the detuning, amplitude, and phase of the
lasers, at the expense of restricting to square-shaped pulses.
Consider a quantum system exhibiting a three-level $\Lambda$-type configuration, in which
two energy levels $\ket{0}$ and $\ket{1}$, spanning the computational state space, are coupled
to an excited state $\ket{e}$ by two pulsed laser beams with detuning $\delta$ and associated
with Rabi frequencies $f_0 (t)$ and $f_1 (t)$, see Fig. <ref>. The Hamiltonian in a frame that
rotates with the laser fields reads ($\hbar = 1$ from now on)
\begin{eqnarray}
H(t) & = & F (t) \left( e^{-i\varphi} \sin \frac{\theta}{2} \ket{e} \bra{0} -
\cos \frac{\theta}{2} \ket{e} \bra{1} + \textrm{H.c.} \right) + \delta \ket{e} \bra{e}
\nonumber \\
& \equiv & F(t) H_0 + \delta \ket{e} \bra{e} ,
\label{eq:hamiltonian}
\end{eqnarray}
where rapidly oscillating counter-rotating terms have been neglected (rotating wave
approximation) and we have put $f_0 (t) = F (t) e^{-i\varphi} \sin \frac{\theta}{2}$ and
$f_1 (t) = -F (t) \cos \frac{\theta}{2}$. Here, $\theta$ and $\varphi$ are time independent
over the duration of the pulse pair, which is controlled by the real-valued pulse envelope $F (t)$.
Off-resonant $\Lambda$ system consisting of two energy levels $\ket{0}$ and
$\ket{1}$ coupled to an excited state $\ket{e}$ by two pulsed laser beams with detuning
$\delta$ and associated with Rabi frequencies $f_0 (t)$ and $f_1 (t)$.
The standard form of nonadiabatic HQC in the $\Lambda$ configuration <cit.>
assumes that the lasers are on resonance with the transition frequencies, i.e., that the detuning
$\delta$ vanishes. In this case, the evolution of the computational subspace $\mathcal{M} =
\textrm{Span} \{ \ket{0},\ket{1} \}$ becomes purely geometric and cyclic with period $\tau$
such that $\int_0^{\tau} F (t) dt = \pi$, irrespective of the detailed form of $F (t)$. The path
$C_{\bf n}$ traversed by the computational subspace in the Grassmannian
$G(3;2)$[That is, the space of two-dimensional subspaces of
a three-dimensional complex vector space.] is parametrized by the fixed laser parameters
$\theta$ and $\varphi$, as captured by the unit vector ${\bf n} = (\sin \theta \cos \varphi,
\sin \theta \sin \varphi, \cos \theta)$. Since the evolution is purely geometric, the holonomic
one-qubit gate $U(C_{\bf n})$ associated with $C_{\bf n}$ coincides with the action of the
time evolution operator $U (\tau,0)$ on $\mathcal{M}$, i.e., $P_{\mathcal{M}}
U(\tau,0) P_{\mathcal{M}}$ with $P_{\mathcal{M}} = \ket{0} \bra{0} +
\ket{1} \bra{1}$. Explicitly, we find <cit.>
\begin{eqnarray}
U (C_{\bf n}) = \ket{d} \bra{d} - \ket{b} \bra{b}
\label{eq:idealgateoperator}
\end{eqnarray}
with the dark and bright states $\ket{d} = \cos \frac{\theta}{2} \ket{0} + e^{-i\varphi}
\sin \frac{\theta}{2} \ket{1}$ and $\ket{b} = e^{i\varphi} \sin \frac{\theta}{2} \ket{0} -
\cos \frac{\theta}{2} \ket{1}$, respectively, which constitute another orthonormal frame
spanning $\mathcal{M}$. We may write
\begin{eqnarray}
U(C_{\bf n}) = i e^{- i\frac{1}{2} \pi {\bf n} \cdot \boldsymbol{\sigma}} =
{\bf n} \cdot \boldsymbol{\sigma}
\label{eq:idealgatematrix}
\end{eqnarray}
with $\boldsymbol{\sigma} = (\sigma_x,\sigma_y,\sigma_z)$ the Pauli operators
$\sigma_x = \ket{0} \bra{1} + \ket{1} \bra{0}$, $\sigma_y = -i \ket{0} \bra{1} +
i\ket{1} \bra{0}$, and $\sigma_z = \ket{0} \bra{0} - \ket{1} \bra{1}$. Here, $U(C_{\bf n})$
is the nonadiabatic non-Abelian geometric phase <cit.> associated with the
path $C_{\bf n}$.
$U(C_{\bf n})$ corresponds to a $\pi$ rotation of the qubit around the direction ${\bf n}$.
To obtain an arbitrary SU(2) operation, this gate must therefore be combined with another
holonomic gate produced by a second pulse pair. To see this, assume that the laser
parameters of two sequentially applied pulse pairs define unit vectors ${\bf n}$ and
${\bf m}$. The combined gate reads
\begin{eqnarray}
U(C_{\bf m}) U(C_{\bf n}) = {\bf m} \cdot {\bf n} - i \boldsymbol{\sigma} \cdot
({\bf n} \times {\bf m}) .
\end{eqnarray}
This corresponds to a rotation angle $2\arccos \left( {\bf n} \cdot {\bf m} \right)$
around the rotation axis ${\bf n} \times {\bf m}$, i.e., an arbitrary SU(2).
Let us now turn to the off-resonant case where $\delta \neq 0$. Here, the geometric nature
of the evolution of $\mathcal{M}$ depends on the detailed form of the pulse envelope $F(t)$
since $H(t)$ may no longer commute with the time evolution operator[The geometric
nature depends on the pulse form since the extra term $\delta \ket{e} \bra{e}$ in the Hamiltonian
$H(t)$ makes it in general necessary to use time ordering to compute the time evolution
operator. As a consequence, $H(t)$ does not necessarily commute with the associated time
evolution operator, which in general implies that $U^{\dagger} (t,0) H(t) U(t,0)$ would
not vanish on $\mathcal{M}$ and thereby creating a nontrivial non-Abelian dynamical phase
acting on the qubit. The square pulse takes care of this since the time evolution operator acts
trivially on $\mathcal{M}$ before and after the pulse and can be computed without time
ordering during the pulse.]. However, there is one physically justified choice where the
evolution of $\mathcal{M}$ is purely geometric, viz., when $F (t)$ is a square pulse, i.e.,
$F (t) = F_0$ for $0 \leq t \leq \tau$ and zero otherwise. For such a pulse, consider evolution
between $t_0$ and $t_1$, where $t_0 \leq 0$ and $t_1 \geq \tau$. The corresponding
time evolution operator reads:
\begin{eqnarray}
U(t_1,t_0) & = & U(t_1,\tau) U(\tau,0) U(0,t_0) =
e^{-i(t_1-\tau)\delta \ket{e} \bra{e}} U(\tau,0) e^{it_0 \delta \ket{e} \bra{e}}
\end{eqnarray}
\begin{eqnarray}
U(\tau,0) = e^{-i\tau (F_0 H_0 + \delta \ket{e} \bra{e})} .
\end{eqnarray}
The action of $U(t_1,t_0)$ is trivial on $\mathcal{M}$ on $t_0 < t < 0$ and $\tau < t < t_1$,
provided $\tau$ is chosen such that $\mathcal{M}$ undergoes cyclic evolution. Thus, it is
sufficient to consider $U(\tau,0)$ in the following.
Since $\bra{k} U^{\dagger} (t,0) \left( F_0 H_0 + \delta \ket{e} \bra{e} \right)
U(t,0) \ket{l} = \bra{k} \left( F_0 H_0 + \delta \ket{e} \bra{e} \right) \ket{l} = 0$, $k,l=0,1$,
on $0 \leq t\leq \tau$, the nontrivial part $U(\tau,0)$ of the time evolution operator is purely
geometric on the single-qubit subspace $\mathcal{M}$. It further corresponds to cyclic
evolution with period $\tau = 2\pi / \sqrt{\delta^2 + 4F_0^2}$ for which we find holonomic
\begin{eqnarray}
U({\bf n},\chi) = \ket{d} \bra{d} - e^{-i\chi} \ket{b} \bra{b} =
e^{i\frac{1}{2} (\pi - \chi)} e^{-i\frac{1}{2} (\pi - \chi) {\bf n} \cdot \boldsymbol{\sigma}}
\label{eq:nonidealgateoperator}
\end{eqnarray}
\begin{eqnarray}
\chi = \frac{\pi \delta}{\sqrt{\delta^2 + 4F_0^2}} .
\label{eq:chi}
\end{eqnarray}
Up to the unimportant overall phase factor $e^{i\frac{1}{2} (\pi - \chi)}$, we
see that $U({\bf n},\chi)$ corresponds to a single-qubit rotation with angle $\pi - \chi$
around the direction ${\bf n}$. Thus, an arbitrary single-qubit operation can be reached
by independently varying the detuning and laser parameters $\theta$ and $\varphi$. In
particular, $U({\bf n},\chi)$ connects to the identity in the $\delta /(2F_0) \rightarrow \infty$
limit[In the small rotation angle limit, $\delta$ should thus be very large compared
to the pulse strength $F_0$. By comparing with Eq. (<ref>), we see that the duration
$\tau$ becomes very short in this limit, which may invalidate the rotating wave approximation.
This in turn implies that the geometric nature as well as the stability of the gate would be
lost <cit.>. Thus, gates corresponding to small qubit rotations are probably
difficult to realize in practise by using our scheme.], i.e., $U({\bf n},\chi \rightarrow \pi) =
\hat{1}$, and it coincides with the nonadiabatic holonomic gate proposed in Ref. <cit.>
in the resonant case, i.e., $U({\bf n},0) = U(C_{\bf n})$.
Further insight into the geometry of $U({\bf n},\chi)$ can be obtained by calculating
the connection 1-form associated with the time-evolved computational subspace, i.e.,
$\mathcal{M} (t) = \textrm{Span} \{ U(t,0) \ket{d}, U(t,0) \ket{b} \}$ with $\mathcal{M} (\tau) =
\mathcal{M} (0) = \mathcal{M}$. By solving the Schrödinger equation, we find
\begin{eqnarray}
\ket{d(t)} & = & U(t,0) \ket{d} = \ket{d} ,
\nonumber \\
\ket{b(t)} & = & U(t,0) \ket{b}
\nonumber \\
& = & e^{-i\frac{1}{2}\delta t} \left( e^{-i\frac{1}{2}\sqrt{\delta^2 + 4F_0^2}t}
\sin \nu \ket{+} + e^{i\frac{1}{2}\sqrt{\delta^2 + 4F_0^2}t} \cos \nu \ket{-} \right) ,
\end{eqnarray}
\begin{eqnarray}
\tan \nu = \frac{\delta + \sqrt{\delta^2 + 4F_0^2}}{2F_0}
\label{eq:nu}
\end{eqnarray}
and $\ket{\pm}$ are the bright eigenstates of $H_0 + \delta \ket{e} \bra{e}$. Clearly,
the only nonzero component of the vector potential
\begin{eqnarray}
A(t) = i\sum_{k,l=d,b} \bra{k(t)} \dot{l}(t) \rangle \ket{k(t)} \bra{l(t)} =
\sum_{k,l=d,b} A_{kl} (t) \rangle \ket{k(t)} \bra{l(t)}
\end{eqnarray}
is $A_{bb} (t)$. We find,
\begin{eqnarray}
A_{bb} (t) = - \sqrt{\delta^2 + 4F_0^2} \sin^2 \nu .
\end{eqnarray}
The holonomy can be obtained as $\ket{d} \bra{d} + e^{i\gamma} \ket{b} \bra{b}$,
where $\gamma$ is the Aharonov-Anandan geometric phase <cit.> associated
with $\ket{b(t)}$, i.e.,
\begin{eqnarray}
\gamma & = & \int_0^{\tau} A_{bb} (t) dt = -2\pi \sin^2 \nu =
-2\pi \frac{1}{2} \left( 1 + \frac{\delta}{\sqrt{\delta^2 + 4F_0^2}} \right)
\nonumber \\
& = & \pi - \chi , \ \textrm{mod} (2\pi) ,
\end{eqnarray}
where we have used Eq. (<ref>). Thus, $\ket{d} \bra{d} + e^{i\gamma} \ket{b} \bra{b}$
coincides with $U({\bf n},\chi)$.
Note that $A(t)$ commutes with itself within each pulse pair. This is the underlying reason
why the holonomy $U({\bf n},\chi)$ can be understood in terms of the Abelian geometric
phase factor $e^{i\gamma}$ of $\ket{b(t)}$. However, $[A(t),\tilde{A}(t')]$ can be nonvanishing
for $A$ and $\tilde{A}$ evaluated for two subsequent pulse pairs with different laser
parameters. To see this, consider $A(t) = A_{bb} (t) \ket{b(t)} \bra{b(t)}$ and $\tilde{A}(t) =
A_{\tilde{b}\tilde{b}} (t) \ket{\tilde{b} (t)} \bra{\tilde{b} (t)}$ with $\ket{b(t)}$ and
$\ket{\tilde{b} (t)}$ corresponding to two different sets of laser parameters
$\theta,\varphi,\delta$ and $\tilde{\theta},\tilde{\varphi},\tilde{\delta}$, respectively.
We obtain
\begin{eqnarray}
[A(t),\tilde{A}(t')] & = & A_{bb} (t) A_{\tilde{b}\tilde{b}} (t')
\nonumber \\
& & \times \left( \ket{b(t)} \bra{b(t)} \tilde{b}(t) \rangle
\langle \tilde{b}(t)| - |\tilde{b}(t) \rangle \langle \tilde{b}(t) \ket{b(t)} \bra{b(t)} \right) \neq 0 ,
\end{eqnarray}
where $t$ and $t'$ belong to the support of the respective two laser pulse pairs.
This proves the non-Abelian nature of the gate.
Before concluding, let us briefly comment on how the restriction to square-shaped
pulses may influence the flexibility of our scheme. The key point here is that it should
be possible to vary freely the detuning, amplitude, and phase of the pulses, despite this
restriction. To see explicitly what this means, let us consider a possible implementation of
our scheme in which transitions between two atomic levels $j=0,1$ and an excited state
$e$ are induced by pulsed electric fields ${\bf E}_j (t) = g_j (t) \cos (\omega_j t) \boldsymbol{\epsilon}_j$,
where $\boldsymbol{\epsilon}_j$ are the polarizations. Here, the time dependent
part consists of two factors: $g_j(t)$ determining the pulse shape and $\cos (\omega_j t)$
determining the detuning $\delta_j = \omega_{je} - \omega_j$, $\omega_{je}$ being
the energy difference (in units where $\hbar = 1$) between the bare energy eigenstates
$\ket{e}$ and $\ket{j}$. The scheme requires that $g_0 (t) = g_1 (t) \propto F(t)$ being
square-shaped and $\delta_0 = \delta_1 = \delta$. The amplitude and phase parameters
$\theta$ and $\varphi$ are determined by the ratio $\bra{e} \boldsymbol{\mu}
\cdot \boldsymbol{\epsilon}_0 \ket{0} / \bra{e} \boldsymbol{\mu} \cdot
\boldsymbol{\epsilon}_1 \ket{1} = - e^{-i \varphi} \tan \frac{\theta}{2}$, $\boldsymbol{\mu}$
being the electric dipole operator. We thus see that the required flexibility is obtained in this
particular setting if the polarization $\boldsymbol{\epsilon}_j$ and oscillation frequency
$\omega_j$ of each electric field pulse can be varied independently, although $g_j (t)$ is
restricted to having square shape.
In conclusion, we have demonstrated holonomic single-qubit gates in off-resonant
$\Lambda$ system. These gates can be used to implement any single-qubit gate and
would, together with an entangling holonomic two-qubit gate, constitute an all-geometric
universal set of gates. The off-resonant holonomic gates require square-shaped pulses
in order to preserve the purely geometric nature. Our finding implies that the assumption
of zero detuning in the original scheme <cit.> is not a necessary requirement
to perform nonadiabatic holonomic quantum computation in $\Lambda$ systems.
We further note that the additional flexibility associated with the detuning makes it
possible to perform arbitrary single-qubit operations by a single pulse pair. This latter
feature may help experimental realizations of holonomic quantum computation as it
reduces the number of pulses needed to implement arbitrary single-qubit operations.
The scheme can be implemented experimentally in various systems, such as trapped
atoms or ions, superconducting qubits, or NV-centers in diamond.
0.3 cm
Financial support from the Swedish Research Council (VR) through Grant No. D0413201
is acknowledged.
zanardi99 P. Zanardi, M. Rasetti,
Phys. Lett. A 264 (1999) 94.
wilczek84 F. Wilczek, A. Zee,
Phys. Rev. Lett. 52 (1984) 2111.
sjoqvist12 E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo,
M. Johansson, K. Singh,
New J. Phys. 14 (2012) 103035.
abdumalikov13 A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger,
A. Wallraff, S. Filipp,
Nature (London) 496 (2013) 482.
feng13 G. Feng, G. Xu, G. Long,
Phys. Rev. Lett. 110 (2013) 190501.
arroyo-camejo14 S. Arroyo-Camejo, A. Lazariev, S. W. Hell, G. Balasubramanian,
Nature Comm. 5 (2014) 4870.
zu14 C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, L.-M. Duan,
Nature (London) 512 (2014) 72.
anandan88 J. Anandan,
Phys. Lett. A 133 (1988) 171.
spiegelberg13 J. Spiegelberg, E. Sjöqvist,
Phys. Rev. A 88 (2013) 054301.
aharonov87 Y. Aharonov, J. Anandan,
Phys. Rev. Lett. 58 (1987) 1593.
|
1511.00936
|
Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, USA
School of Mathematics and Statistics, Southwest University, Chongqing, China
The characteristic is a simple yet important invariant of an algebra. In this paper, we study the characteristic of a Rota-Baxter algebra, called the Rota-Baxter characteristic. We introduce an invariant, called the ascent set, of a Rota-Baxter characteristic. By studying its properties, we classify Rota-Baxter characteristics in the homogenous case and relate Rota-Baxter characteristics in general to the homogeneous case through initial ideals. We show that
the Rota-Baxter quotients of Rota-Baxter characteristics have the same underlying sets as those in the homogeneous case.
We also give a more detailed study of Rota-Baxter characteristics with special base rings. In particular, we determine the prime characteristics of Rota-Baxter rings.
[2010]13C05, 13E05, 16W99
§ INTRODUCTION
A Rota-Baxter algebra is an associative algebra together
with a linear endomorphism that is an algebraic analogue of the integral operator.
This concept has it origin from a work of G. Baxter <cit.> in probability theory.
In the late 1960s, Rota <cit.> studied the subject from
an algebraic and combinatorial perspective and suggested that they are closely related to
hypergeometric functions, incidence algebras and symmetric functions <cit.>.
Since then, these algebras have been investigated by mathematicians and mathematical physicists with various motivations. For example, a Rota-Baxter operator on a Lie algebra is closely related to the operator form of the classical Yang-Baxter equation. Here the Baxter is the physicist R. Baxter <cit.>. In recent years Rota-Baxter operators have found applications to many areas, such as number theory <cit.>, combinatorics <cit.>, operads <cit.> and
quantum field theory <cit.>. See <cit.> and
the references therein for further details.
For the theoretic study of this important algebraic structure, it is useful to generalize the study of characteristics of algebras to Rota-Baxter algebras. The characteristic of a (unitary) ring $R$ is (the nonnegative generator of) the kernel of the structure map $\ZZ\to R$ sending $1$ to the identity element of $R$. More generally, the characteristic of an algebra $R$ over a commutative ring $\bfk$ is the kernel of the structure map $\bfk\to R$. To generalize this concept to the context of Rota-Baxter algebras, we note that the structure map $\bfk\to R$ comes from the fact that $\bfk$ is the initial object in the category of $\bfk$-algebras, as the free $\bfk$-algebra on the empty set. Then to study the characteristics of Rota-Baxter $\bfk$-algebras, we consider the initial object in the category of Rota-Baxter $\bfk$-algebras, as the free Rota-Baxter algebra on the empty set. Thus every Rota-Baxter $\bfk$-algebra comes with a (unique) structure map from the initial object to this Rota-Baxter algebra. Then the kernel of the structure map should be the characteristic of the Rota-Baxter algebra.
In this paper, we study the characteristics of Rota-Baxter algebras. More precisely, we study the Rota-Baxter ideals, and their corresponding quotients, of the initial Rota-Baxter $\bfk$-algebra. In particular, we study the Rota-Baxter ideals of the initial Rota-Baxter ring. The initial Rota-Baxter algebra is a generalization of the divided power algebra, and the polynomial algebra $\bfk[x]$ when $\bfk$ is taken to contain $\QQ$. So some our results are naturally related to results of these algebras.
The construction of the initial object will be reviewed at the beginning of Section <ref> and will be applied to give the concept of Rota-Baxter characteristics. In Section <ref>, we first classify homogeneous Rota-Baxter characteristics and their quotients (Theorem <ref>). We then show that the Rota-Baxter quotients of general Rota-Baxter characteristics have the same underlying sets as those in the homogeneous case (Theorem <ref>).
Finally specializing to the case of $\bfk=\ZZ$ in Section <ref>,
we give the structures of Rota-Baxter characteristics of Rota-Baxter rings, and determine the prime Rota-Baxter characteristics (Theorem <ref>).
§ CHARACTERISTICS OF ROTA-BAXTER ALGEBRAS
Notations. Unless otherwise specified, an algebra in this paper is assumed to be unitary associative defined over a unitary commutative ring $\bfk$.
Let $\NN$ and $\PP$ denote the set of nonnegative and positive integers respectively. For $n\in \NN$, denote $[n]:=\{1,\cdots,n\}$. For notational convenience, we also denote $[\infty]=\PP$. Let $\ZZ_n$ denote the set of integers modulo $n$.
Before giving the definition of the characteristic of a Rota-Baxter algebra, we first provide some background and preliminary results on Rota-Baxter algebras. See EG,guo2012,guokeigher20001 for details.
Let $\lambda\in \bfk$ be given. A Rota-Baxter $\bfk$-algebra of weight $\lambda$ is a
$\bfk$-algebra $R$ paired with a linear operator $P$ on $R$ that satisfies the identity
\begin{equation}\label{rtequ}
P(x)P(y)=P(xP(y))+P(P(x)y)+\lambda P(xy)
\end{equation}
for all $x,y\in R$. When $\bfk$ is $\ZZ$, then the pair $(R,P)$ is called a Rota-Baxter ring.
A Rota-Baxter ideal of a Rota-Baxter algebra $(R,P)$ is an ideal $I$ of $R$ such that $P(I)\subseteq I$. Then we denote $I\leq R$.
The concepts of Rota-Baxter subalgebras, quotient Rota-Baxter algebras and homomorphisms
of Rota-Baxter algebras can be similarly defined.
We recall that the free Rota-Baxter algebra on a set $X$ is a Rota-Baxter algebra $(F_{RB}(X),P_X)$ together with a set map $i_X:X\to F_{RB}(X)$ characterized by the universal property that, for any Rota-Baxter algebra $(R,P)$ and set map $f:X\to R$, there is a unique Rota-Baxter algebra homomorphism $\tilde{f}:F_{RB}(X)\to R$ such that $i_X\circ \tilde{f}=f$.
The free Rota-Baxter algebra on the empty set is also the free Rota-Baxter $\mathbf{k}$-algebra $F_{RB}(\bfk)$ on the $\bfk$-algebra $\mathbf{k}$ guokeigher20001, characterized by the universal property that, for any Rota-Baxter algebra $(R,P)$ and $\bfk$-algebra homomorphism $f:\bfk\to R$, there is a unique Rota-Baxter algebra homomorphism $\tilde{f}:F_{RB}(\bfk)\to R$ such that $i_\bfk\circ \tilde{f}=f$, where $i_\bfk$ is the structure map $i_\bfk:\bfk\to F_{RB}(\bfk)$. Since there is only one $\bfk$-algebra homomorphism $\bfk\to R$, this universal property shows that $F_{RB}(\bfk)$ is the initial object in the category of Rota-Baxter $\bfk$-algebras.
We refer the reader to <cit.>, as well as <cit.>, for the general constructions of free Rota-Baxter algebras, but will focus on a simple construction of $F_{RB}(\bfk)$ following <cit.>, where it is denoted by $\sha_\bfk (\bfk)$.
This free Rota-Baxter $\mathbf{k}$-algebra not only provides the simplest example of free Rota-Baxter algebras but also establishes the connection between Rota-Baxter algebra and some well-known concepts such as divided powers and
Stirling numbers AGKO,Gust.
As a $\bfk$-module, $\sha_{\mathbf{k} }(\mathbf{k})$ is given by the free $\bfk$-module
\begin{align*}
\sha_{\mathbf{k} }(\mathbf{k})=\bigoplus_{m=0}^{\infty}\mathbf{k} \fraka_{m}
\end{align*}
on the basis $\left\{\fraka_m\,|\, m\geq 0\right\}$.
For a given $\lambda\in\bfk$, the product $\diamond=\diamond_\lambda$ on $\sha_{\mathbf{k}}(\mathbf{k})$ is defined by
\begin{align}\label{productformula1}
\fraka_{m}\diamond_{\lambda}\fraka_{n}
=\sum_{i=0}^{{\rm min}(m,n)}\binom{m+n-i}{m}\binom{m}{i}\lambda^i\fraka_{m+n-i}, \qquad m,n\in \mathbb{N}.
\end{align}
Thus when $\lambda=0$, $\sha_\bfk(\bfk)$ is the divided power algebra.
Note that $\diamond$ is an extension of the product on $\bfk$ viewed as $\bfk \fraka_0$. Thus there should no confusion if the notation $\diamond$ is suppressed, as we often do.
Define the $\bfk$-linear operator $P=P_{\mathbf{k}}$ on $\sha_{\mathbf{k}}(\mathbf{k})$ by assigning
$P_{\mathbf{k}}(\fraka_m)=\fraka_{m+1}$, $m\geq0$
and extending by additivity.
By guokeigher20001, when $\bfk$ contains $\QQ$ and $\lambda=0$, we have $\sha_\bfk(\bfk)\cong\bfk[x]$ as a $\bfk$-algebra.
The pair $(\sha_{\bfk}(\bfk),P_\bfk)$ is the initial object in the category of Rota-Baxter $\bfk$-algebras.
More precisely, for any Rota-Baxter $\bfk$-algebra $(R, P)$, there is a unique
Rota-Baxter $\bfk$-algebra homomorphism $\varphi: (\sha_{\bfk}(\bfk), P_{\bfk})\rightarrow(R, P)$.
Suppose a $\bfk$-algebra $A$ has a linear basis $X$. By <cit.>, the free Rota-Baxter algebra $F_{RB}(A)$ on $A$ (denoted by $\sha^{\mathrm{NC}}_\bfk(A)$) has a linear basis consisting of Rota-Baxter words in the alphabet set $X$. By definition, a Rota-Baxter word in $X$ is a bracketed word in the alphabet set $X$ in which there are no adjacent pairs of brackets. Taking $A=\bfk$, then $X=\{1\}$. Thus a Rota-Baxter word in $X$ can only be of the form
$$u_r:=\underbrace{\lc\cdots\lc}_{r\ \text{iterations}} 1\underbrace{\rc\cdots \rc}_{r\ \text{iterations}}$$
(namely applying the bracket operator $\lc\ \rc$ to $1$ $r$ times), for $r\geq 1$, together with $u_0:=1$. Thus
$$F_{RB}(\bfk)=\sum_{r\geq 0} \bfk u_r,$$
on which the Rota-Baxter operator $Q_\bfk$ is given by $Q_\bfk(u_r)=\lc u_r\rc=u_{r+1}$. By the universal property of $F_{RB}(\bfk)$, the natural inclusion map $f:\bfk \to \sha_\bfk(\bfk)$ sending $1\to \fraka_0$ (that is, the structure map) extends to a Rota-Baxter algebra homomorphism
$$\tilde{f}: F_{RB}(\bfk) \to \sha_\bfk(\bfk).$$
As such, we obtain $\tilde{f}(u_0)=\fraka_0$ and recursively,
$$ \tilde{f}(u_{r+1})=\tilde{f}(Q_\bfk(u_r))=P_\bfk(\tilde{f}(u_r)) =P_\bfk(\fraka_r)=\fraka_{r+1}, \quad r\geq 0.$$
Therefore $\tilde{f}$ is a linear isomorphism and thus a Rota-Baxter algebra isomorphism, showing that $\sha_\bfk(\bfk)$ is the free Rota-Baxter algebra on $\bfk$ and hence the initial object in the category of Rota-Baxter $\bfk$-algebras.
Thus $\sha_{\mathbf{k}}(\bfk)$ plays the same role in the category of Rota-Baxter $\mathbf{k}$-algebras as the role played by $\bfk$ in the category of $\bfk$-algebras.
Let $(R,P)$ be a Rota-Baxter $\bfk$-algebra and let $\varphi=\varphi_{(R,P)}:(\sha_\bfk(\bfk),P_\bfk)\to (R,P)$ be the unique Rota-Baxter algebra homomorphism from the initial object $(\sha_\bfk(\bfk),P_\bfk)$ in the category of Rota-Baxter $\bfk$-algebras to $(R,P)$.
The kernel of the structure map $\varphi$ is called the Rota-Baxter characteristic of $(R,P)$.
In view of Theorem <ref>, the characteristic of a Rota-Baxter $\bfk$-algebra $R$ must be a Rota-Baxter ideal of $\sha_{\bfk}(\bfk)$. Conversely, any Rota-Baxter ideal $I$ of $\sha_\bfk(\bfk)$ is the characteristic of some Rota-Baxter algebra, for example of $\sha_\bfk(\bfk)/I$.
Based on the above observation, we will use the terminology Rota-Baxter characteristics in exchange with Rota-Baxter ideals of $\sha_\bfk(\bfk)$ in the rest of the paper.
§ CLASSIFICATION OF THE CHARACTERISTICS OF ROTA-BAXTER ALGEBRAS
In this section, we study Rota-Baxter characteristics and their quotients. First in Section ss:ascend, we introduce an invariant, called the ascent set, of a Rota-Baxter characteristic. We then apply the invariants to classify all homogeneous Rota-Baxter characteristics in Section ss:homog. Finally in Section ss:general we relate a Rota-Baxter characteristic to a homogeneous Rota-Baxter characteristic by taking the initial terms and show that the quotients of the two Rota-Baxter characteristics share the same underlying sets, but not the same $\bfk$-modules. This approach is motivated from the study in the polynomial algebra $\bfk[x_1,\cdots,x_n]$ when $\bfk$ is a field. There the quotient modulo an ideal and the quotient modulo the corresponding initial ideal are known to share the same basis <cit.>.
§.§ The ascent set of a Rota-Baxter characteristic
As recalled in the last section, the Rota-Baxter algebra $\sha_\bfk(\bfk)$, as the initial object in the category of Rota-Baxter algebras, is the direct sum
$$ \sha_\bfk(\bfk)=\bigoplus_{m\geq 0} \bfk \fraka_m$$
on the basis $\{\fraka_m\,|\,m\geq 0\}$ and hence is an $\NN$-graded $\bfk$–module. Any nonzero element $f$ of $\sha_\bfk(\bfk)$ can be uniquely written as $f=\sum\limits_{i=0}^n c_i\fraka_{i}$ with $c_n\neq 0$. Then $n$ is called the degree of $f$ and $c_n\fraka_n$ is called the initial term of $f$, denoted by $\deg f$ and ${\rm in}(f)$ respectively. In addition, we define $\deg 0=-\infty$. We call the set ${\rm supp}(f):=\{c_i\fraka_{i}|c_i\neq 0\}$ the support of $f$.
Let $I$ be a Rota-Baxter characteristic, namely an ideal of $\sha_\bfk(\bfk)$. We introduce an invariant of $I$.
For each $j\in \mathbb{N}$, we denote
\Omega_{j}:=\Omega_{j}(I):=\left\{b_j\in\mathbf{k}\,\left|\,(\exists f\in I)\ f=\sum\limits_{i=0}^jb_i\fraka_{i}\right.\right\}.
\Omega_{j}=\{0\}\cup\left\{b_j\in\mathbf{k}\,\left|\,(\exists f\in I)\ {\rm{in}}(f)=b_j\fraka_{j}\right.\right\}.
The smallest index $j$ such that $\Omega_j\neq 0$ is called the starting point of $I$, denoted by $\td(I)$, that is,
\begin{align*}
\td(I)=\min\left\{j\in\NN\,\left|\,\Omega_j\neq0\right.\right\}.
\end{align*}
Let $I$ be a Rota-Baxter ideal of $\sha_{\bfk}(\bfk)$. Then for each $j\in \mathbb{N}$, $\Omega_j$ is an ideal of $\bfk$ and $\Omega_j\subseteq\Omega_{j+1}$.
For any $b_j,c_j\in\Omega_j$, there exist $f$ and $g\in I$ such that $f=\sum\limits_{i=0}^jb_i\fraka_{i}$
$g=\sum\limits_{i=0}^jc_i\fraka_{i}$. Hence $f+g=\sum\limits_{i=0}^j(b_i+c_i)\fraka_{i}$ is in $I$.
So $b_j+c_j\in \Omega_j$. On the other hand, for any $c\in \mathbf{k}$,
$cf=\sum\limits_{i=0}^jcb_i\fraka_{i}\in I$. This yields $cb_j\in \Omega_j$. Thus $\Omega_j$ is an ideal of $\mathbf{k}$.
Since $I$ is a Rota-Baxter ideal of $\sha_{\bfk}(\bfk)$, we have $P(f)=\sum\limits_{i=0}^jb_i\fraka_{i+1}\in I$. So $b_j\in\Omega_{j+1}$. Hence $\Omega_j\subseteq\Omega_{j+1}$, as required.
For a given Rota-Baxter ideal $I$ of $\sha_{\bfk}(\bfk)$, Lemma <ref> shows that the ideals $\Omega_j\subseteq \bfk, j\in \NN$, form a non-decreasing sequence of ideals of $\bfk$. Thus the sequence is controlled by the locations and extents where the increases occur. This motivates us to introduce the following notions.
$s_1< s_2<\cdots$ be the integers $t\in \NN$ such that $\Omega_{t-1}\subsetneq \Omega_{t}$ with the convention that $\Omega_{-1}=\left\{0\right\}$.
We will use the notation $s_i, i\in [N_I]$ where $N=N_I$ is either in $\NN$ or $\infty$ with the convention adopted at the beginning of the last section.
The integers $s_i, i\in [N_I],$ are called the ascending points of $I$ and the ideals $\Omega_{s_i}, i\in [N_I]$ are called the ascending levels of $I$. In view of Lemma <ref>, we have $s_1=\td(I)$.
Thus for a given Rota-Baxter characteristic $I$, the set of pairs
\begin{equation}
A(I):=\left\{\left . (s_j,\Omega_{s_j})\,\right|\, j\in [N_I]\right\},
\label{eq:elev}
\end{equation}
called the ascent set, is uniquely determined by $I$.
By the definition of $\Omega_{s_j}$, we have
${s_j}<{s_{j+1}}$ and
\begin{align}\label{idealasscend}
\Omega_{{s_j}}=\Omega_{{s_j}+1}=\cdots=\Omega_{s_{j+1}-1}\subsetneq \Omega_{{s_{j+1}}}
\end{align}
for all $j\in [N_I]$, as illustrated by Figure $1$.
(47,-8)Figure 1.
We next study how the information from $A(I)$ can be used to recover $I$.
\begin{equation}
\cala:=\left\{ \left.\{(s_j,\Omega_{s_j})\}_{j\in [N]} \,\right|\, s_j\in \NN, \Omega_{s_j}\leq \bfk, s_j<s_{j+1}, \Omega_{s_j}\subsetneq \Omega_{s_{j+1}}, j\in [N], 1\leq N\leq \infty\right\},
\label{eq:ascpair}
\end{equation}
called the set of ascending pairs. So $\cala$ consists of pairs of sequences of the same lengths with one sequence of strictly increasing nonnegative integers and a second sequence of strictly increasing ideals of $\bfk$. Let $\cali=\cali(\bfk)$ denote the set of Rota-Baxter characteristics, namely the set of Rota-Baxter ideals of $\sha_\bfk(\bfk)$. Then taking the ascent set of a Rota-Baxter characteristic defines a map
\begin{equation}
\Phi: \cali \to \cala, \quad I\mapsto A(I), I\leq \sha_\bfk(\bfk).
\mlabel{eq:phi}
\end{equation}
In the rest of the paper, we study the property of this map, including its surjectivity and fibers, that is, inverse images. In Theorem <ref>, we show that the restriction of $\Phi$ to the subset of homogeneous Rota-Baxter characteristics gives a bijection to $\cala$, proving the surjectivity of $\Phi$. In Proposition <ref>, we show that two Rota-Baxter characteristics are in the same fiber if and only if they have the same initial ideal.
Let $I$ be a Rota-Baxter ideal of $\sha_\bfk(\bfk)$ with $A(I)=\left\{ (s_j,\Omega_{s_j})\,|\, j\in [N_I]\right\}$.
For each $j\in[N_I]$, we let $\Theta_j:=\left\{\omega_{s_j,\ell}\,\left|\,\ell\in [N_j]\right.\right\}$ be a set of generators of the ideal
$\Omega_{s_j}$. Here $N_j$ is either a positive integer or $\infty$. For each $\omega_{s_j,\ell}\in \Theta_j$, let $f_{s_j,\ell}$ be an element of $I$ whose initial term is $\omega_{s_j,\ell}\fraka_{s_j}$.
Then the set
$$\bigcup_{j\in [N_I]}\left\{\left. f_{s_j,\ell}\,\right|\, \ell\in [N_j]\right\}$$
is a generating set of the Rota-Baxter ideal $I$.
We call the above-mentioned set $\bigcup\limits_{j\in [N_I]}\left\{\left. f_{s_j,\ell}\,\right|\, \ell\in [N_j]\right\}$ an ascent generating set
of the Rota-Baxter ideal $I$.
Let $I'$ be the Rota-Baxter ideal generated by the set $\bigcup\limits_{j\in [N_I]}\left\{\left. f_{s_j,\ell}\,\right|\, \ell\in [N_j]\right\}$.
Since $I'\subseteq I$ is trivial, it suffices to show that each element $f$ of $I$ belongs to $I'$.
So take an $f\in I$. Then $f$ is in $I'$ if $f=0$.
We next assume that $f\neq 0$. Then we have ${\rm in}(f)=b\fraka_{\deg f}$ for some $0\neq b\in \bfk$.
We proceed by induction on $\deg f$. Clearly, $\deg f\geq \td(I)=s_1$.
If $\deg f=s_1$, then $b\in \Omega_{s_1}$. So there exist $c_{s_1,\ell}\in \bfk, \ell\in [N_1],$ all but finitely many of which being zero, such that $b=\sum\limits_{\ell\in [N_1]}c_{s_1,\ell}\omega_{s_1,\ell}$.
Thus the element
$$g=f-\sum\limits_{\ell\in [N_1]}c_{s_1,\ell}f_{s_1,\ell}$$
is in $I$. But now the degree of $g$ is less than $\deg f$. It follows from $\deg f=\td(I)$ that $g$ must be $0$,
which means that $f=\sum\limits_{\ell\in[N_1]}c_{s_1,\ell}f_{s_1,\ell}$ is in $I'$ and we are done. For a given $n\geq \td(I)$,
assume that all $f\in I$ with $\deg f\leq n$ are in $I'$ and take $f\in I$ with $\deg f=n+1$.
Then there exists $r\in [N_I]$ such that $s_{r}\leq n+1<s_{r+1}$ with the convention that $s_{N_I+1}=\infty$ if $N_I$ is finite.
By Eq. (<ref>), we have $\Omega_{n+1}=\Omega_{s_r}$. So $b=\sum\limits_{\ell\in [N_r]}c_{s_r,\ell}\omega_{s_r,\ell}$ where $c_{s_r,\ell}\in \bfk, \ell\in [N_r]$,
with all but a finite number of $c_{s_r,\ell}$ being zero.
h=f-\sum\limits_{\ell\in[N_r]} c_{s_r,\ell}P^{n+1-s_r}\left(f_{s_r,\ell}\right).
Then $h\in I$ with $\deg h<\deg f$. So we can apply the induction hypothesis to obtain that $h$ is in $I'$ and hence $f$ is in $I'$, as required.
§.§ Classification of homogeneous Rota-Baxter characteristics
We now apply ascent sets to classify homogeneous Rota-Baxter characteristics.
A Rota-Baxter ideal $I$ of $\sha_{\bfk}(\bfk)$ is called a homogeneous Rota-Baxter ideal if $I$ is a Rota-Baxter ideal generated by a set of homogeneous elements. Then the Rota-Baxter characteristic $I$ is called homogeneous.
We next show that, for a Rota-Baxter ideal of $\sha_\bfk(\bfk)$, its homogeneity as a Rota-Baxter ideal is equivalent to its homogeneity as an ideal. For this purpose, we first give a general relation between generating a Rota-Baxter ideal and generating an ideal in a Rota-Baxter algebra.
Let $(R,P)$ be a Rota-Baxter $\mathbf{k}$-algebra of weight $\lambda$, and $S$ a subset of $R$.
Then the Rota-Baxter ideal $(S)_{RB}$ generated by $S$ is the ideal of $R$ generated by the set
\begin{align}\label{rtgeseteq}
S_{RB}:=\bigcup_{m\in\mathbb{N}}\left\{(\circ_{i=1}^mP_{x_{i}})(a)|a\in S, x_{i}\in R, 1\leq i\leq m\right\},
\end{align}
where $(\circ_{i=1}^mP_{x_{i}})(a):=P(x_{m}P(x_{m-1}P(\cdots P(x_{1}a))))$ with the convention that $(\circ_{i=1}^0P_{x_{i}})(a):=a$.
If $R=\sha_{\bfk}(\bfk)$, then it suffices to take $x_{i}$ to be the homogeneous elements in Eq. (<ref>).
Let $I$ be the ideal of $R$ generated by $S_{RB}$. Then an element of $I$ is a sum of elements of the form $r(\circ_{i=1}^mP_{x_{i}})(a)$ with $r\in R$. Then we have
\begin{align*}
\end{align*}
where $x_{m+1}=r$. So, by the additivity of $P$, we obtain $P(I)\subseteq I$, and hence $I$ is the Rota-Baxter ideal of $R$ generated by $S_{RB}$,
that is, $I=( S_{RB})_{RB}$.
Notice that $S\subseteq S_{RB}\subseteq (S)_{RB}$, so $(S)_{RB}=( S_{RB})_{RB}$
whence $I=(S)_{RB}$, as required. The second statement follows since any element in $\sha_\bfk(\bfk)$ is a linear combination of homogeneous elements.
Now we can give the following characterization of homogeneous Rota-Baxter ideals in $\sha_\bfk(\bfk)$.
Let $I\subseteq \sha_{\bfk}(\bfk)$ be a Rota-Baxter ideal. Then $I$ is a homogeneous Rota-Baxter ideal if only if $I$ is a homogeneous ideal.
($\Longleftarrow$) If a set of homogeneous elements generates $I$ as an ideal, then it does so as a Rota-Baxter ideal.
($\Longrightarrow$) Suppose that $I$ is a homogeneous Rota-Baxter ideal. Then we can assume that there are a subset $\Lambda$ of $\NN$ and numbers $n_i\in \PP$ ($i\in \Lambda$) or $n_i=\infty$ such that $I$ is generated by the following set of homogeneous elements:
$$\mathcal{G}:=\{b_{ij}\fraka_i\in\bfk\fraka_i\,|\,j\in [n_i], i\in \Lambda\}.$$
Then, by Lemma <ref>, $I$ is the ideal generated by the set
\mathcal{G}':=\left\{(\circ_{k=1}^mP_{x_{k}})(b_{ij}\fraka_i)\,|\, b_{ij}\fraka_i\in \mathcal{G}, x_{k}\in \sha_{\bfk}(\bfk)\ {\rm\ homogeneous}, 1\leq k\leq m, m\geq 0\right\}
with the convention that $(\circ_{i=1}^0P_{x_{i}})(a):=a$.
Hence $I$ is contained in the homogeneous ideal generated by the supports of elements in $\mathcal{G}'$.
Thus to complete the proof, we just need to prove that for each $u\in \mathcal{G}'$, we have $\mathrm{supp}(u)\subset I$. We will prove this for $u$ in the form $(\circ_{k=1}^mP_{x_{k}})(b_{ij}\fraka_i)$, where $b_{ij}\fraka_i\in \mathcal{G}$, by induction on $m\geq 0$. When $m=0$, we have $u=b_{ij}a_i\in \mathcal{G}$ which is assumed to be in $I$. Suppose that the statement is true for $m=k\geq 0$ and consider
By the induction hypothesis, the support of $(\circ_{k=1}^{m}P_{x_{k}})(b_{ij}\fraka_i)$, that is, the set of homogenous components of it, is contained in $I$. Let $b\fraka_p\in I, b\in \bfk,$ be such a homogenous component and denote $x_{m+1}=c \fraka_n, c\in \bfk.$
Then by Eq. (<ref>), we have
P_{x_{m+1}}(b\fraka_p)=P\big((c\fraka_n) (b\fraka_p)\big)=P\left(\sum_{i=0}^{{\rm min}(n,p)}\binom{n+p-i}{p}\binom{p}{i}\lambda^ibc\fraka_{n+p-i}\right)
=\!\!\!\sum_{i=0}^{{\rm min}(n,p)}\binom{n+p-i}{p}\binom{p}{i}\lambda^icP^{n-i+1}(b\fraka_{p}).
Since $P^{n-i+1}(b\fraka_{p})$ is in $I$, the homogeneous components of $P_{x_{m+1}}(b\fraka_p)$ are in $I$. Hence the homogeneous components of $u$ are in $I$. This completes the induction.
* Let $I$ be a homogeneous Rota-Baxter characteristic, that is, a homogeneous Rota-Baxter ideal of $\sha_\bfk(\bfk)$, with $A(I)=\left\{\left. (s_j,\Omega_{s_j})\,\right|\, j\in [N_I]\right\}$.
\begin{equation}
I=\bigoplus_{i=s_1}^\infty \Omega_i \fraka_i =\bigoplus_{j=1}^{N_I}\left(\bigoplus_{i=s_{j}}^{s_{j+1}-1}\Omega_{s_{j}} \fraka_{i}\right)
\bigoplus\limits_{j=1}^\infty \bigoplus\limits_{i=s_j}^{s_{j+1}-1} \Omega_{s_j}\fraka_i, & N_I=\infty, \\
\left(\bigoplus\limits_{j=1}^{N_I-1} \bigoplus\limits_{i=s_j}^{s_{j+1}-1} \Omega_{s_j}\fraka_i\right)\bigoplus \left(\bigoplus\limits_{i=s_{N_I}}^\infty \Omega_{s_{N_I}}\fraka_i\right), & N_I<\infty,
\end{array}\right.
\label{eq:i}
\end{equation}
as the direct sum of the $\bfk$-modules $\Omega_{s_j} \fraka_{i}$.
* The quotient $\sha_{\bfk}(\bfk)/I$ is the direct sum of the $\bfk$-modules $(\bfk/\Omega_{s_j})\fraka_i$, that is,
\begin{eqnarray}
\sha_{\bfk}(\bfk)/I&\cong& \bigoplus_{j=0}^{N_I}
\left(\bigoplus_{i=s_{j}}^{s_{j+1}-1}({\mathbf{k}}/\Omega_{s_{j}}) \fraka_{i}\right) \notag\\
\left(\bigoplus\limits_{i=0}^{s_1-1} \bfk\fraka_i\right) \bigoplus
\left(\bigoplus\limits_{j=1}^\infty \bigoplus\limits_{i=s_j}^{s_{j+1}-1} (\bfk/\Omega_{s_j})\fraka_i\right), & N_I=\infty, \\
\left(\bigoplus\limits_{i=0}^{s_1-1} \bfk\fraka_i\right) \bigoplus
\left(\bigoplus\limits_{j=1}^{N_I-1} \bigoplus\limits_{i=s_j}^{s_{j+1}-1} (\bfk/\Omega_{s_j})\fraka_i\right)\bigoplus \left(\bigoplus\limits_{i=s_{N_I}}^\infty (\bfk/\Omega_{s_{N_I}})\fraka_i\right), & N_I<\infty,
\end{array}\right.
\label{inhomdecompzzash1}
\label{rbida5}
\end{eqnarray}
with the convention that $s_{0}=0$, $\Omega_{s_0}=0$ and $s_{N_I+1}=\infty$ if $N_I$ is finite.
* For any ascent pair $\{(s_j,\Omega_{s_j})\}_{j\in [N]}$ in $\cala$ defined in Eq. (<ref>), consisting of a positive integer $N$ or $N=\infty$, strictly increasing nonnegative integers $s_i$ and strictly increasing ideals
$\Omega_{s_j}, j\in [N]$, there is a unique homogeneous Rota-Baxter ideal $I$ of $\sha_{\bfk}(\bfk)$ such that $A(I)=\{(s_j,\Omega_{s_j})\}_{j \in [N]}$. In other words,
the restriction of the map $\Phi:\cali\to \cala$ in Eq. (<ref>) to the set of homogeneous Rota-Baxter characteristics is a bijection.
By Proposition <ref>, the $\bfk$-module $I$ is the direct sum of the $\bfk$-modules $\Omega_{i} \fraka_{i}$.
(<ref>) Since $I$ is a graded submodule of $\sha_{\bfk}(\bfk)$, Item (<ref>) is proved.
For a given ascending pair $\{(s_j,\Omega_{s_j})\}_{j\in [N]}\in \cala$, define $I\subseteq \sha_\bfk(\bfk)$ as in Eq. (<ref>) with $N_I$ replaced by $N$.
We next show that $I$ is a homogeneous Rota-Baxter ideal.
It follows from $s_1<s_2<\cdots$ and
$\Omega_{s_1}\subsetneq \Omega_{s_2}\subsetneq\cdots$ that $P(I)\subseteq I$. $I$ is a $\bfk$-submodule since $\Omega_{s_j}$ is an ideal of $\bfk$ for each
$j\in [N]$. Further by its definition, $I$ is homogeneous as a $\bfk$-module. We next prove that
$I(\sha_{\bfk}(\bfk))\subseteq I$.
Since each element of $\sha_{\bfk}(\bfk)$ can be written as the summand of finitely many homogeneous components,
it suffices to show that $(b\fraka_{n})(c\fraka_{p})\in I$, where $b\fraka_{n}\in I$ and $c\fraka_{p}\in\sha_{\bfk}(\bfk)$.
By Eq. (<ref>), we have
\begin{align*}
\left(b\fraka_{n}\right)\left( c\fraka_{p}\right)
=\sum\limits_{i=0}^{{\rm min}(n,p)}\binom{n+p-i}{p}\binom{p}{i}bc\lambda^i\fraka_{n+p-i}.
\end{align*}
Since $b\fraka_{n}\in I$, we have $b\in\Omega_{s_r}$ where $r\in[N]$ is such that $s_r\leq n<s_{r+1}$.
For any given $i$ with $0\leq i\leq {\rm min}(n,p)$, let $t$ be the integer in $[N]$ such that $s_t\leq n+p-i<s_{t+1}$.
Note that we always have $n+p-i\geq n\geq s_r$, so $s_r\leq s_t$. Thus $\Omega_{s_r}\subseteq \Omega_{s_t}$ whence $b\in \Omega_{s_t}$.
Then for each $i$ with $0\leq i\leq {\rm min}(n,p)$, the element
$\binom{n+p-i}{p}\binom{p}{i}bc\lambda^i\fraka_{n+p-i}$ is in $\Omega_{s_t}\fraka_{n+p-i}.$
This shows that $\left(b\fraka_{n}\right)\left( c\fraka_{p}\right)\in I$. Thus $I$ is a homogeneous Rota-Baxter ideal of $\sha_{\bfk}(\bfk)$.
Thus we obtain a map $\Psi$ from $\cala$ to the set of homogeneous Rota-Baxter ideals of $\sha_\bfk(\bfk)$. It is direct to check that the left and right compositions of $\Psi$ with the restriction of $\Phi$ to homogeneous Rota-Baxter ideals of $\sha_\bfk(\bfk)$ are the identities. Thus the restriction of $\Phi$ to the set of homogeneous Rota-Baxter ideals is bijective.
§.§ Rota-Baxter characteristics and their quotients
Now we relate a Rota-Baxter characteristic to the homogeneous case and describe the structure of the quotient of $\sha_\bfk(\bfk)$ modulo a Rota-Baxter characteristic.
We will see that such a quotient has the underlying set of the form defined by Eq. (<ref>).
However, the same underlying set may be shared by different Rota-Baxter characteristics, as can be seen from the following theorem.
As mentioned at the beginning of this section, this theorem is a property of $\sha_{\bfk}(\bfk)$ coming as an analog to polynomial algebras.
We first relate a Rota-Baxter ideal of $\sha_\bfk(\bfk)$ to a suitable homogeneous Rota-Baxter ideal.
The initial Rota-Baxter ideal of a Rota-Baxter ideal $I$ of $\sha_{\bfk}(\bfk)$ is the Rota-Baxter ideal ${\rm in}(I)$ generated by
$\left\{{\rm in}(f)|f\in I\right\}$.
Thus ${\rm in}(I)$ is a homogeneous ideal.
Let $I$ be a Rota-Baxter ideal of $\sha_\bfk(\bfk)$. Then for its ascent set we have $A(I)=A({\rm in}(I))$. In other words, two Rota-Baxter characteristics are in the same fiber under the map $\Phi$ in Eq. (<ref>) if and only if they have the same initial ideal.
Assume that $A(I):=\left\{\left . (s_j,\Omega_{s_j})\,\right|\, j\in [N_I]\right\}$.
We only need to show that $\Omega_{i}(I)=\Omega_{i}({\rm in}(I))$ for all $i\in \NN$.
If $i<\td(I)$, then there is no element $f\in I$ with degree lower than $i$, so $\Omega_i(I)=\Omega_{i}({\rm in}(I))=\emptyset$.
Next we assume that $i\geq \td(I)$.
Take a nonzero element $b\in \bfk$. Then $b\in \Omega_{i}(I)$ if and only if there exists an element $f$ in $I$ such that ${\rm in}(f)=b\fraka_{i}$, which
is equivalent to $b\in \Omega_{i}({\rm in}(I))$. Thus $A(I)=A({\rm in}(I))$, as required.
Now we can determine the underlying set of the quotient of a Rota-Baxter characteristic.
Let $I$ be a Rota-Baxter ideal of $\sha_{\mathbf{k}}(\mathbf{k})$.
Then $\sha_{\mathbf{k}}(\mathbf{k})/I$ has the same underlying set as $\sha_{\mathbf{k}}(\mathbf{k})/{\rm in}(I)$ in Eq. (<ref>).
More precisely, for each $j$ with $j\in\{0\}\cup[N_I]$, fix a complete set $T_j\subseteq \bfk$, such that $0\in T_j$, of representatives of $\bfk$ modulo $\Omega_{s_j}$, with the convention that $s_{0}=0$, $\Omega_{s_0}=0$ and $s_{N_I+1}=\infty$ if $N_I$ is finite.
Then $\sha_{\bfk}(\bfk)/I$ has a complete set of representatives given by the following subset of $\sha_\bfk(\bfk)=\bigoplus\limits_{m=0}^\infty \bfk \fraka_m$:
\begin{equation}
\mathcal{T}:=\bigoplus\limits_{j=0}^{N_I}
\left(\bigoplus\limits_{i=s_{j}}^{s_{j+1}-1}T_j \fraka_{i}\right)
\left(\bigoplus\limits_{i=0}^{s_1-1} \bfk\fraka_i\right) \bigoplus
\left(\bigoplus\limits_{j=1}^\infty \bigoplus\limits_{i=s_j}^{s_{j+1}-1} T_j\fraka_i\right), & N_I=\infty, \\
\left(\bigoplus\limits_{i=0}^{s_1-1} \bfk\fraka_i\right) \bigoplus
\left(\bigoplus\limits_{j=1}^{N_I-1} \bigoplus\limits_{i=s_j}^{s_{j+1}-1} T_j\fraka_i\right)\bigoplus \left(\bigoplus\limits_{i=s_{N_I}}^\infty T_{N_I}\fraka_i\right), & N_I<\infty.
\end{array}\right.
\label{decompzzash2}
\end{equation}
Let $I$ be a Rota-Baxter ideal of $\sha_{\mathbf{k}}(\mathbf{k})$ with its ascent set $ A(I)=\left\{ (s_j,\Omega_{s_j})\,|\,j\in [N_I]\right\}$.
By Proposition <ref>, we have $A(I)=A({\rm in}(I))$.
Thus it suffices to show that the underlying set of
$\sha_{\bfk}(\bfk)/I$ is $\mathcal{T}$ in light of Theorem <ref>(<ref>).
By convention, we take $s_0=0$ and $\Omega_{s_0}=0$.
Then $T_0=\bfk$ and
$$\bigoplus\limits_{i=0}^{s_1-1}{T_0} \fraka_{i}=\bigoplus\limits_{i=s_0}^{s_1-1}\mathbf{k} \fraka_{i}.$$
Note that this term does not exist if $s_1=0$.
Let $f\in\sha_{\mathbf{k}}(\mathbf{k})$ be a nonzero element with ${\rm in}(f)= b_n \fraka_{n}$ for some $b_n\in\bfk$. So $\deg f=n\geq 0$.
We first show that there is a unique $f'$ in $\mathcal{T}$ such that $f-f'$ is in $I$.
Let $\bigcup\limits_{j\in [N_I]}\left\{\left. f_{s_j,\ell}\,\right|\, \ell\in [N_j]\right\}$ be an ascent generating set of $I$,
we may assume that for each $j\in [N_I]$ and $\ell\in [N_j]$ we have ${\rm in}(f_{s_j,\ell})=\omega_{s_j,\ell}\fraka_{s_j}$ and hence $\Omega_{s_j}$ is generated by the set
$\Theta_j:=\left\{\omega_{s_j,\ell}\,\left|\,\ell\in [N_j]\right.\right\}$ by Lemma <ref>.
If $\deg f<s_1$, then $f\in \bigoplus\limits_{i=0}^{s_1-1}\mathbf{k}\fraka_{i}$, which equals to $\bigoplus\limits_{i=s_0}^{s_1-1}T_0 \fraka_{i}$
since $s_0=0$, $\Omega_{s_0}=0$.
Thus, it is enough to take $f'=f$.
Next we assume that $\deg f\geq s_1$. We prove that there exists an element
$f'\in \mathcal{T}$ with $\deg f'\leq \deg f$ such that $f-f'\in I$ by induction on $\deg f$.
If $\deg f=s_1$, then ${\rm in}(f)=b_{s_1}\fraka_{s_1}$ for some $b_{s_1}\in\bfk$.
Since $T_1\subseteq \bfk$ is a complete set of representatives of $\bfk$ modulo $\Omega_{s_1}$,
there exists a unique $b_{s_1}'\in T_1$ such that $b_{s_1}-b_{s_1}'\in \Omega_{s_1}$.
Note that $\Omega_{s_1}$ is generated by $\Theta_1$, so there exist $c_{{s_1},\ell}$ in $\mathbf{k}$,
$\ell\in [N_1]$, with all but a finite number of $c_{{s_1},\ell}$ being zero, such that $b_{s_1}-b_{s_1}'=\sum\limits_{\ell\in [N_1]}c_{{s_1},\ell}\omega_{s_1,\ell}$.
If we take $f'=f-\sum\limits_{\ell\in [N_1]}c_{{s_1},\ell}f_{s_1,\ell}$, then $f-f'\in I$.
It is clear that $f'\in \mathcal{T}$ if $f'=0$. If $f'\neq0$, then $f'=b_{s_1}'\fraka_{s_1}+g'$ for some $g'\in \sha_{\bfk}(\bfk)$ with $\deg g'<\deg f=s_1$. Thus $g'\in \bigoplus\limits_{i=0}^{s_1-1}\mathbf{k}\fraka_{i}=\bigoplus\limits_{i=s_0}^{s_1-1}T_0 \fraka_{i}$ so that $f'$ is also in $\mathcal{T}$.
Now we assume that the claim has been proved for $f$ with $\deg f\leq n-1$ for a given $n\geq1$ and show that the claim is true when $\deg f= n$.
Assume that ${\rm in}(f)=b_n\fraka_n$.
Let $r\in \{0\}\cup [N_I]$ be such that $s_r\leq n<s_{r+1}$.
So $\Omega_{n}=\Omega_{s_r}$ by Eq. (<ref>), and hence $b_n-b_n'\in\Omega_{s_r}$ for some $b_n'\in T_r$.
Then there exist $c_{n,\ell}\in \bfk$, $\ell\in [N_r]$, all but finitely many of which being zero, such that
$b_n-b_n'=\sum\limits_{\ell\in [N_r]}c_{n,\ell}\omega_{s_r,\ell}$.
$g=f-P^{n-s_r}(\sum\limits_{\ell\in [N_r]}c_{n,\ell} f_{s_r,\ell})$,
then we have $g=b_{n}'\fraka_{n}+g'$ for some $g'\in\sha_{\bfk}(\bfk)$ with $\deg g'<\deg f$.
If $g'=0$, then put $f'=b_{n}'\fraka_{n}$ so that $f'\in \mathcal{T}$ and we have
$f-f'\in I$.
If $g'\neq 0$, then $\deg g'<\deg f=n$.
By the induction hypothesis, there exists $g''\in\mathcal{T}$ with $\deg g''\leq \deg g'$ such that
$g'-g''\in I$.
If we take
$f'=b_{n}'\fraka_{n}+g''$, then $f-f'\in I$.
Since $\deg g''\leq \deg g'$, we have $\deg g''<\deg f=n$,
so that $f'$ ˇˇis an element of $\mathcal{T}$.
Suppose that there is another element $h'$ in $\mathcal{T}$ such that $h'\neq f'$ and $f-h'\in I$.
Then $f'-h'\in I$. Let $f'=\sum\limits_{i=0}^kc_i\fraka_i$, $h'=\sum\limits_{i=0}^mb_i\fraka_i$,
where $b_i,c_i \in T_{r}$ if $s_r\leq i<s_{r+1}$ for some
$r\in \{0\}\cup[N_I]$. By symmetry, we may assume that $k\leq m$ and put $c_i=0$ for all $i$ with $k< i\leq m$.
Thus $h'-f'=\sum\limits_{i=0}^m(b_i-c_i)\fraka_i$.
Since $h'\neq f'$, there is a nonnegative integer $p$ with $0\leq p\leq m$ such that $b_p-c_p\neq0$ and ${\rm in}(h'-f')=(b_p-c_p)\fraka_p$.
Let $t\in \{0\}\cup[N_I]$ be such that $s_t\leq p<s_{t+1}$. Then $b_p\in T_t$, $c_p\in T_t$ and $b_p-c_p\in \Omega_{s_t}$.
But $T_t$ is a complete set of representatives of $\bfk$ modulo $\Omega_{s_t}$, so $b_p=c_p$, a contradiction.
It remains to show that every element of $\mathcal{T}$ represents an element of $\sha_{\bfk}(\bfk)/I$.
For any $f'$ in $\mathcal{T}$ we take $f=f'$ in $\sha_{\bfk}(\bfk)$,
then $f-f'\in I$ whence $f'\in \mathcal{T}$ is corresponding to the element $f+I$ in $\sha_{\bfk}(\bfk)/I$. Thus, the underlying set of $\sha_{\mathbf{k}}(\mathbf{k})/I$ is $\mathcal{T}$ and the statement follows at once.
We remark that the underlying $\bfk$-module $\mathcal{T}$ in Eq. (<ref>) is usually not the direct sum of these $\bfk$-modules $({\mathbf{k}}/\Omega_{s_{j}}) \fraka_{i}$ for a nonhomogeneous Rota-Baxter ideal $I$, even though we write it in the form of a direct sum. The scalar multiplication by $\bfk$ should be defined according to the $\bfk$-module
$\sha_{\bfk}(\bfk)/I$. We illustrate this by the following example.
Let $\sha_{\ZZ}(\ZZ)$ be the free Rota-Baxter $\ZZ$-algebra on $\ZZ$ of weight $1$. Let $I$ be the Rota-Baxter ideals of $\sha_{\ZZ}(\ZZ)$ generated by the element $f=2(\fraka_{1}+\fraka_{0})$. Then $\sha_{\ZZ}(\ZZ)/I$ and $\sha_\ZZ(\ZZ)/{\rm{in}}(I)$ are in bijection as sets, but are not isomorphic as $\ZZ$-modules (that is, abelian groups).
Since ${\rm{in}}(f)=2\fraka_1$, we have $2a_{m+1}=P^{m}(2\fraka_1)\in {\rm{in}}(I)$ for
$m\geq0$ whence ${\rm{in}}(I)\supseteq\bigoplus\limits_{i\geq1}2\ZZ\fraka_i$.
We note from Lemma <ref> that all elements of $I$ can be obtained from iterated operations on $f$ by
the Rota-Baxter operator $P$, the scalar product, the multiplication and the addition of the algebra $\sha_{\ZZ}(\ZZ)$.
On one hand, we have
$P^m(f)=2(\fraka_{m+1}+\fraka_{m})$. On the other hand, it follows from Eq. (<ref>) that
\begin{align*}
\fraka_{m}f=2(m+1)(\fraka_{m+1}+\fraka_{m}).
\end{align*}
Thus, any element of $I$ must be of the form
\begin{align}\label{form1I2}
\end{align}
where $n\in\NN$, $c_i\in\ZZ$, $1\leq i\leq n+1$. Consequently, ${\rm{in}}(I)\subseteq\bigoplus\limits_{i\geq1}2\ZZ\fraka_i$
and hence ${\rm{in}}(I)=\bigoplus\limits_{i\geq1}2\ZZ\fraka_i$.
Clearly, $A({\rm in}(I))=\{(1, 2\ZZ)\}$, whence both the underlying sets of $\sha_{\ZZ}(\ZZ)/{\rm in}(I)$ and $\sha_{\ZZ}(\ZZ)/I$ are
\ZZ\fraka_0\bigoplus \left(\bigoplus\limits_{i\geq1}\ZZ_2\fraka_i\right)
by Theorems <ref> and <ref>.
For the $\ZZ$-module $\sha_{\ZZ}(\ZZ)/{\rm in}(I)$, it follows from $2\fraka_1\in {\rm in}(I)$ that $\bar{2\fraka_1}=0$. However,
for the $\ZZ$-module $\sha_{\ZZ}(\ZZ)/I$, we have $2\fraka_1+I=-2\fraka_0+I$ so that $\bar{2\fraka_1}=\bar{-2\fraka_0}$, which is clearly not zero since
$-2\fraka_0$ is not in $I$ by Eq. (<ref>).
Therefore, the Rota-Baxter ideals $I$ and ${\rm in}(I)$ share the same quotient sets, but not $\ZZ$-modules. From the fact that
$\bar{2\fraka_1}\neq 0$ in $\sha_{\ZZ}(\ZZ)/I$ we also see that $\sha_{\ZZ}(\ZZ)/I$ is not the direct sum of the $\ZZ$-modules
$\ZZ\fraka_0$ and $\ZZ_2\fraka_i$, $i\geq1$.
The following example shows that a similar phenomenon can already be found in the polynomial algebra $\bfk[x]$.
Let $I$ be the ideal of $\ZZ[x]$ generated by the polynomial $2x+2$. Then the initial ideal ${\rm in}_{\leq}(I)$ is generated by $2x$. Thus the ideal $I$ and its initial ideal ${\rm in}(I)$ have the same quotient set
\ZZ\bigoplus\left(\bigoplus_{n\geq1}\ZZ_2x^n\right).
However, the two ideals do not have isomorphic quotient groups. For instance, $2x+I=-2+I$ is not zero since $-2$ is not in $I$, but $2x+{\rm in}_{\leq}(I)=0$ since
$2x\in {\rm in}_{\leq}(I)$. Furthermore, $2x+I\neq 0$ shows that $\ZZ[x]/I$ is not the direct sum of the abelian groups $\ZZ$ and $\ZZ_2x^n$, $n\geq1$.
§ CHARACTERISTICS OF ROTA-BAXTER RINGS
We next focus on the case when $\bfk=\ZZ$ and classify the Rota-Baxter ideals and prime Rota-Baxter ideal of $\sha_{\mathbb{Z}}(\mathbb{Z})$. Note that $\sha_{\mathbb{Z}}(\ZZ)$ is the initial object in the category of unitary Rota-Baxter rings (that is, Rota-Baxter $\ZZ$-algebras). So we are talking about characteristics of Rota-Baxter rings.
Let $I$ be a Rota-Baxter ideal of $\sha_{\mathbb{Z}}(\mathbb{Z})$.
Then there exist a positive integer $m$ and $m$ pairs $(s_1,\omega_1)$, $\cdots$, $(s_m,\omega_m)\in \NN\times \PP$ with $s_j<s_{j+1}$, $\omega_{j+1}\neq \omega_{j}$ and $\omega_{j+1}|\omega_{j}$, $1\leq j\leq m-1$, such that $A(I)=\left\{(s_1,\omega_1\ZZ),\cdots,(s_m,\omega_m\ZZ)\right\}$.
Thus if $I$ is homogeneous then
I=\bigoplus_{j=1}^m\left(\bigoplus_{i=s_j}^{s_{j+1}-1} \omega_{s_j}\ZZ\fraka_{i}\right)
and the quotient $\sha_{\mathbb{Z}}(\mathbb{Z})/I$ is isomorphic to
\begin{equation}\label{decompzzash1pz}
\bigoplus\limits_{j=0}^m
\left(\bigoplus\limits_{i=s_{j}}^{s_{j+1}-1}\mathbb{Z}_{\omega_j} \fraka_{i}\right),
\end{equation}
with the convention that $s_0=0$, $\omega_{s_0}=0$ and $s_{m+1}=\infty$. If $I$ is nonhomogeneous then $\sha_{\mathbb{Z}}(\mathbb{Z})/I$ has the same underlying set as $\sha_{\bfk}(\bfk)/{\rm in}(I)$ in Eq. (<ref>).
Conversely, for any positive integer $m$ and $m$ pairs $(s_1,\omega_1)$, $\cdots$, $(s_m,\omega_m)\in \NN\times \PP$ with $s_j<s_{j+1}$, $\omega_{j+1}\neq \omega_{j}$ and $\omega_{j+1}|\omega_{j}$, $1\leq j\leq m-1$,
there is a Rota-Baxter ideal $I$ of $\sha_{\ZZ}(\ZZ)$ such that the underlying set of $\sha_{\ZZ}(\ZZ)/I$ is the construction defined by Eq. (<ref>).
Let $I$ be a Rota-Baxter ideal of $\sha_{\mathbb{Z}}(\mathbb{Z})$.
Since $\ZZ$ satisfies the ascending chain condition on ideals, $I$ has only finite ascend steps, say $s_1,\cdots,s_m$, where $m$ is a positive integer.
Since $\ZZ$ is a PID, we have $\Omega_{s_j}=\omega_j\ZZ$ for some positive integer $\omega_j$, $j=1,\cdots,m$.
Then, by Lemma <ref>, we have $\omega_{j+1}\neq \omega_{j}$ and $\omega_{j+1}|\omega_{j}$, $1\leq j\leq m-1$. It follows from
the definition of $A(I)$ that $A(I)=\left\{(s_1,\omega_1\ZZ),\cdots,(s_m,\omega_m\ZZ)\right\}$. Consequently, in view of Theorems <ref> and <ref>,
the first part of the theorem follows.
Conversely, take $\Omega_{s_j}=\omega_j\ZZ$. Then it follows from $s_j<s_{j+1}$, $\omega_{j+1}\neq \omega_{j}$ and $\omega_{j+1}|\omega_{j}$, $1\leq j\leq m-1$
that $\Omega_{s_1}\subsetneq\Omega_{s_2}\subsetneq\cdots\subsetneq\Omega_{s_m}$.
So by Theorem <ref>, there is a Rota-Baxter ideal $I$ of $\sha_{\ZZ}(\ZZ)$ such that the underlying set of $\sha_{\ZZ}(\ZZ)/I$ is the construction defined by Eq. (<ref>).
Let $\bfk$ be an integral domain with characteristic $0$.
The next lemma tells us that if $\Omega_t=\bfk\omega_t$ is a principal ideal for $t=\td(I)$, then the element $f\in I$ with ${\rm in}(f)=\omega_t\fraka_{t}$ is completely determined by $\omega_t$ and the number of nonzero terms in $f$.
Let $\bfk$ be an integral domain with characteristic $0$ and $\sha_{\bfk}(\bfk)$ the free Rota-Baxter algebra of weight $\lambda$, and let $I\subseteq \sha_{\bfk}(\bfk)$ be a nonzero Rota-Baxter ideal. Suppose that $\td(I)=t$ and
$\Omega_t=\bfk \omega_t$.
* If $f=\sum\limits_{i=r}^{t}c_i\fraka_{i}\in I$ with $c_t=\omega_t$ and $c_r\neq0$, then
\begin{align*}
c_i=\binom{t-r}{t-i}\lambda^{t-i}c_t, \qquad r\leq i\leq t-1.
\end{align*}
* If $\lambda=0$, then $I\subseteq \bigoplus\limits_{i\geq t}\bfk\fraka_{i}$.
* If $\lambda=0$ and $\bfk$ is a field, then $I=\bigoplus\limits_{i\geq t}\bfk\fraka_{i}$.
(<ref>) Since $I$ is a Rota-Baxter ideal and $f=\sum\limits_{i=r}^{t}c_i\fraka_{i}$ is in $I$, we have $g_1:=(t+1)P(f)-\fraka_{1}f\in I$.
Now from Eq. (<ref>) it follows that
\begin{align*}
\fraka_{1}f
=(t+1)c_t\fraka_{t+1}+ \sum\limits_{i=r+1}^{t}i(c_{i-1}+\lambda c_{i})\fraka_{i}
+\lambda r c_{r}\fraka_{r},
\end{align*}
which together with
\begin{align*}
\end{align*}
implies that
\begin{align*}
g_1=\sum\limits_{i=r+1}^{t}\left((t+1-i)c_{i-1}-\lambda ic_{i}\right)\fraka_{i}-\lambda rc_{r}\fraka_{r}.
\end{align*}
Thus, the coefficient of $\fraka_{t}$ is in $\Omega_t$, that is, $c_{t-1}-\lambda tc_{t}$ is in $\bfk c_t$ so that
$c_{t-1}$ is in $\bfk c_t$. Hence there exists $b\in\bfk$ such that $c_{t-1}=bc_t$.
\begin{align*}
g_2:=&g_1-(b-\lambda t)f=\sum\limits_{i=r+1}^{t}\left((t+1-i)c_{i-1}+(\lambda (t-i)-b)c_i\right)\fraka_{i}+(\lambda (t-r)-b)c_r\fraka_{r}
\end{align*}
is in $I$.
But the coefficient of $\fraka_{t}$ in $g_2$ is $0$, so the fact that $\td(I)=t$ yields that
$g_2=0$, which is equivalent to $(\lambda (t-r)-b)c_r=0$ and
\begin{align}\label{eqsf}
(t+1-i)c_{i-1}+(\lambda (t-i)-b)c_i=0,\qquad r+1\leq i\leq t.
\end{align}
By the hypotheses that $\bfk$ is an integral domain and $c_r\neq0$, it follows from $(\lambda (t-r)-b)c_r=0$ that $b=\lambda (t-r)$.
Consequently, Eq. (<ref>) becomes
\begin{align*}
(t+1-i)c_{i-1}=\lambda(i-r)c_i,\qquad r+1\leq i\leq t.
\end{align*}
So for each $r\leq i\leq t-1$, we have
\begin{align*}
\prod_{j=i+1}^t\left((t+1-j)c_{j-1}\right)=\prod_{j=i+1}^t\left(\lambda(j-r)c_j\right),
\end{align*}
which is equivalent to
\begin{align*}
\end{align*}
Since $\bfk$ is an integral domain with characteristic $0$, we obtain
\begin{align*}
c_i=\lambda^{t-i}\frac{(t-r)!}{(t-i)!(i-r)!}c_t=\binom{t-r}{t-i}\lambda^{t-i}c_t,\qquad r\leq i\leq t-1.
\end{align*}
This completes the proof of part (<ref>) of this lemma.
(<ref>) Take arbitrary $f\in I$.
Since $\td(I)=t$, we may assume that
\begin{align*}
f=b_m\fraka_{m}+b_{m-1}\fraka_{m-1}+\cdots+b_t\fraka_{t}+f_0\in I,
\end{align*}
where $m\geq t$ and $\deg\ f_0\leq t-1$.
Note that $\Omega_t=\bfk\omega_t$. So there exists an element $g$ in $I$ with $\omega_t\fraka_t$ as the initial term.
Since $\lambda=0$, by part (<ref>), we must have that $g=\omega_t\fraka_{t}$ is in $I$. So
$\omega_t\fraka_{m}=P^{m-t}(\omega_t\fraka_{t})$ is in $I.$
Then $h:=\omega_tf-b_m\omega_t\fraka_{m}$ is in $I$, that is,
\begin{align}\label{arfx1}
h=b_{m-1}\omega_t\fraka_{m-1}+\cdots+ b_t\omega_t\fraka_{t}+\omega_tf_0
\end{align}
is in $I$. Since $P(I)\subseteq I$, we have $\omega_t\fraka_{\ell}=P^{\ell-t}(\omega_t\fraka_{t})\in I$ for all $t\leq \ell\leq m-1$,
so that $b_{m-1}\omega_t\fraka_{m-1}+\cdots+ b_t\omega_t\fraka_{t}\in I$. Then Eq. (<ref>) implies that
$\omega_tf_0\in I$.
Note that $\deg\ f_0\leq t-1$, which together with $\td(I)=t$ yields $\omega_tf_0=0$. But $\bfk$ is an integral domain, so $f_0=0$.
Therefore, $I\subseteq \bigoplus\limits_{i\geq t}\bfk\fraka_{i}$ and we are done.
(<ref>) Since $\lambda=0$, we know that $\omega_t\fraka_{t}$ is in $I$. But $\bfk$ is a field, so $\Omega_t=\bfk$,
whence $\bigoplus\limits_{i\geq t}\bfk\fraka_{i}\subseteq I$, which together with part (<ref>) gives
$I= \bigoplus\limits_{i\geq t}\bfk\fraka_{i}$.
We finally give a classification for the prime Rota-Baxter ideals of $\sha_{\ZZ}(\ZZ)$ when $\lambda=0$.
Recall that a Rota-Baxter ideal $I$ of a Rota-Baxter algebra $(R,P)$ is said to be prime if it is a prime ideal with $P(I)\subseteq I$.
If $\lambda=0$, then for any $m,n\in \NN$, it follows from Eq. (<ref>) that
\begin{align}\label{productformula0}
\fraka_{m} \fraka_{n}
\end{align}
Let $\sha_{\ZZ}(\ZZ)$ be the free Rota-Baxter $\ZZ$-algebra on $\ZZ$ of weight $0$ and let $I$ be a proper nonzero Rota-Baxter ideal of $\sha_{\ZZ}(\ZZ)$. Then $I$ is prime if and only if
$$I=p\ZZ\fraka_{0} \bigoplus \left(\bigoplus\limits_{i\geq 1}\ZZ\fraka_{i}\right),$$
where $p$ is either $0$ or a prime number of $\ZZ$.
By Theorems <ref> and <ref>, the quotient of a prime characteristic of a Rota-Baxter ring is either $\ZZ$ or $\ZZ/p\ZZ$ for a prime number $p$, as in the case of prime characteristic of a ring.
Let $I$ be a prime Rota-Baxter ideal. Denote $t=\td(I)$ and $\Omega_t=\omega\ZZ$ for some positive integer $\omega$.
Then, by Proposition <ref>(<ref>), the element in $I$ of the form
$$\omega\fraka_{t}+ \text{ lower degree terms}$$
must be
$\omega\fraka_{t}$, that is, $\omega\fraka_{t}\in I$.
If $t\geq 1$, then $\omega \fraka_{t-1}\not\in I$ since $\td(I)=t$. From Eq. (<ref>) we know that $\fraka_{1}(\omega\fraka_{t-1})=t \omega\fraka_{t}\in I$.
Since $I$ is prime, we have $\fraka_{1}\in I$, which means that $t={\rm st}(I)\leq1$ and hence $t=1$. Now $\fraka_{1}\in I$ gives $1\in \Omega_1$ so that
$\Omega_1=\ZZ$. By Lemma <ref>, $\Omega_j=\ZZ$ for all positive integer $j$. Therefore,
$$I=\bigoplus\limits_{i\geq 1}\ZZ\fraka_{i}.$$
If $t=0$, then $\omega \fraka_{0}\in I$.
Since $I$ is a prime ideal and $\fraka_0$ is the identity of $\sha_{\ZZ}(\ZZ)$, $\omega$ must be a prime number. Let $\omega =p$.
Then $p\geq2$ and $p\fraka_{0}\in I$.
$\fraka_1^p=p! \fraka_p=(p-1)!P^p(p\fraka_0)\in I$ which means that $\fraka_1$ is in the prime ideal $I$.
Hence $I$ is generated by $p\fraka_{0}$ and $\fraka_{1}$.
$$I=p\ZZ\fraka_{0} \bigoplus \left(\bigoplus\limits_{i\geq 1}\ZZ\fraka_{i}\right).$$
Conversely, $I$ is a homogeneous Rota-Baxter ideal generated by either $\{\fraka_1\}$ or $\{p\fraka_0,\fraka_1\}$,
where $p$ is a prime number. By Theorem <ref>, if $I$ is generated by $\left\{\fraka_{1}\right\}$, then $\sha_{\ZZ}(\ZZ)/I$ is isomorphic to $\ZZ$. If $I$ is generated by $\left\{p\fraka_{0},\fraka_{1}\right\}$, then $\sha_{\ZZ}(\ZZ)/I$ is isomorphic to $\ZZ_p$.
In either case, $I$ is a prime ideal.
This work was supported by NSFC grant $11426183$, $11501467$,
Chongqing Research Program of Application Foundation and Advanced Technology $($No. cstc2014jcyjA00028$)$.
H. Yu would like to thank the NYU Polytechnic School of Engineering
for hospitality and support.
AL M. Aguiar, On the associative analog of Lie bialgebras. J. Algebra 244 (2001), 492-532.
AGKO G. E. Andrews, L. Guo, W. Keigher and K. Ono, Baxter algebras and Hopf algebras,
Trans. Amer. Math. Soc. 355 (2003), 4639-4656.
Bai C. Bai, A unified algebraic approach to classical
Yang-Baxter equation, J. Phys. A: Math. Theor. 40 (2007), 11073-11082.
C. Bai, O. Bellier, L. Guo and X. Ni, Spliting of operations, Manin products and Rota-Baxter operators,
Int. Math. Res. Not. 3 (2013), 485-524.
G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math.
10 (1960), 731-742.
A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry,
Commun. Math. Phys. 199 (1998), 203-242.
C1972 P. Cartier, On the structure of free Baxter algebras, Adv. Math. 9 (1972), 253-265.
D. Cox, J. Little and D. O'shea, Ideals, Varieties, and Algorithms, Springer, 1992.
EG K. Ebrahimi-Fard and L. Guo, Free Rota-Baxter algebras and dendriform algebras, J. Pure Appl.
Algebra 212 (2008), 320-339.
N. S. Gu and L. Guo, Generating functions from the viewpoint of Rota-Baxter algebras. Discrete Math. 338 (2015), 536–554.
L. Guo, An Introduction to Rota-Baxter Algebra, International Press (US) and Higher Education Press (China), 2012.
Gust L. Guo, Baxter algebras, Stirling numbers and partitions, J. Alg. Appl. 4 (2005), 153-164.
L. Guo, WHAT IS a Rota-Baxter algebra? Notice Amer. Math. Soc. 56 (2009), 1436-1437.
L. Guo and W. Keigher, Baxter algebras and shuffle products, Adv. Math. 150 (2000), 117-149.
L. Guo and B. Zhang, Renormalization of multiple zeta values, J. Algebra. 319 (2008), 3770-3809.
Ro1 G.-C. Rota, Baxter algebras and combinatorial identities I & II, Bull. Amer. Math. Soc. 75
(1969), 325-329, 330-334.
Ro2 G.-C. Rota, Baxter operators, an introduction, In: “Gian-Carlo Rota on Combinatorics, Introductory papers and commentaries", Joseph P.S. Kung, Editor,Birkhäuser, Boston, 1995.
G.-C. Rota, Ten mathematics problems I will never solve, Invited address at the joint meeting
of the American Mathematical Society and the Mexican Mathematical Society, Oaxaca,
Mexico, December 6, 1997, Mitt. Dtsch. Math.-Ver., Heft 2 (1998), 45-52.
STS M. Semonov-Tian-Shansky, What is a classical
R-matrix? Funct. Anal. Appl. 17 (1983), 259-272.
|
1511.00857
|
]Enriched Yoneda lemma
Department of Mathematics, University of Haifa,
Mount Carmel, Haifa 3498838, Israel
We present a version of enriched Yoneda lemma for
conventional (not $\infty$-) categories. We do not require the base monoidal category
$\cM$ to be closed or symmetric monoidal. In the case $\cM$ has colimits and the monoidal structure in $\cM$ preserves colimits in each argument, we prove that
the Yoneda embedding $\cA\to P_\cM(\cA)$ is a universal functor from $\cA$ to a category
with colimits, left-tensored over $\cM$.
§ INTRODUCTION
The principal source on enriched category theory is the classical Max Kelly's book <cit.>. The theory is mostly developed under the assumption that the basic monoidal category $\cM$ is symmetric monoidal, and is closed, that is admits an internal Hom — a functor right adjoint to the tensor product.
The aim of this note is to present an approach which would make both conditions unnecessary.
Throughout the paper we study categories enriched over an arbitrary monoidal category
$\cM$. Note that this means that,
if $\cA$ is enriched over $\cM$, the opposite category $\cA^\op$ is enriched
over the monoidal category $\cM_\op$ having the opposite multiplication.
Also, since we do not require $\cM$ to be closed, $\cM$ may not be enriched over itself.
Our approach is based on the following observation. Even though categories
left-tensored over $\cM$ are not necessarily enriched over $\cM$, it
makes a perfect sense to talk about $\cM$-functors $\cA\to \cB$
where $\cA$ is
$\cM$-enriched, and $\cB$ is left-tensored over $\cM$.
Thus, $\cM$-enriched categories and categories left-tensored over $\cM$ appear in our approach as distinct but interconnected species.
In this note we present two results in the enriched setting.
The first is construction of the category of enriched presheaves
and the Yoneda lemma. The second result, claiming a universal property of the
category of enriched presheaves, requires $\cM$ to have colimits, so that the tensor product in $\cM$ preserves colimits in both arguments.
In this note we adopt the language which allows us not to mention
associativity constraints explicitly. Thus is done as follows.
The small categories are considered belonging to $(2,1)$-category
$\Cat$, with functors as 1-morphisms and isomorphisms of functors as
2-morphisms. Associative algebras in 2-category $\Cat$
are precisely monoidal categories, and left modules over these algebras
are left-tensored categories.
Similarly, we denote $\Cat^L$ the $(2,1)$-category whose objects are
the categories with colimits, 1-morphisms are colimit preserving
functors, and 2-morphisms are isomorphisms of such functors.
This is a symmetric monoidal $(2,1)$-category, with tensor product
defined by the formula
\begin{equation}
\Fun(A\otimes B,C)=\{f:A\times B\to C| f\textrm{ preserves colimits in
both arguments}\}.
\end{equation}
Associative algebras in $\Cat^L$ are monoidal categories with colimits,
such that
tensor product preserves colimits in each argument
[As it is shown in <cit.>, Chapter 2, there is no
necessity of keeping explicit track of various coherences even in the
more general context of quasicategories.].
As it was pointed to us by the referee, enriched Yoneda lemma in the
presented in this note is not a new result. A recent paper <cit.>
contains it (see
Sections 5,7), as well as many other results, in even more general
context of monoidal
The approach of op. cit is close to ours. The authors do not have the notion
of $\cM$-functor $\cA\to\cB$ from $\cM$-enriched category $\cA$ to a category $\cB$
left-tensored over $\cM$; but they construct the category of $\cM$-presheaves
$P_\cM(\cA)$ ad hoc using the same formulas.
We are very grateful to the referee for providing this reference, as well as for indicating that we do not use cocompletness of $\cM$ in Sections 2, 3.
The approach to Yoneda lemma presented in this note is very instrumental
in the theory of enriched infinity categories. We intend to address this in a subsequent publication.
§ TWO TYPES OF ENRICHMENT
Let $\cM$ be a monoidal category.
In this section we define $\cM$-categories and categories
left-tensored over $\cM$.
§.§ $\cM$-enriched categories
Let $\cM$ be a monoidal category. An $\cM$-enriched category $\cA$
(or just $\cM$-category) has a set of objects, an object $\hom_\cA(x,y)\in\cM$
for each pair of objects (“internal Hom”), identity maps
$\one\to\hom(x,x)$ for each $x$ and associative compositions
Let $\cA$ be $\cM$-enriched category. Its opposite $\cA^\op$ is a category
enriched over $\cM_\op$. The latter is the same category as $\cM$, but having the
opposite tensor product structure. The category $\cA^\op$ has the same objects as
$\cA$. Morphisms are defined by the formula
\hom_{\cA^\op}(x^\op,y^\op)=\hom_\cA(y,x),
with the composition defined in the obvious way.
§.§ Left-tensored categories
A left-tensored category $\cA$ over $\cM$ is just a left (unital) module
for the associative algebra $\cM\in\Alg(\Cat)$. Note that unitality is
not an extra structure, but a property saying that the unit
of $\cM$ acts on $\cA$ as an equivalence.
Right-tensored categories over $\cM$ are defined similarly. They are the same
as the categories left-tensored over $\cM_\op$.
In case $\cM\in\Alg(\Cat^L)$, that is, $\cM$ has colimits and the
monoidal operation in $\cM$ preserves colimits in each argument, we will
define left-tensored categories over $\cM$ as left $\cM$-modules over
the associative algebra $\cM\in\Alg(\Cat^L)$.
A left-tensored category so defined has colimits,
and the tensor product preserves colimits in both arguments.
Left-tensored categories over $\cM$ often give rise to an $\cM$-enriched
structure: we can define $\hom(x,y)$ as an object of $\cM$ representing
the functor
\begin{equation}\label{eq:internalhom}
m\mapsto\Hom(m\otimes x,y).
\end{equation}
Even if the above functor is not representable, we will use the notation
$\hom(x,y)$ to define the functor (<ref>).
Note that left-tensored categories are categories (with extra
Enriched categories are not, formally speaking, categories: maps from
one object to another form an object of $\cM$ rather than a set.
§ $\CM$-FUNCTORS
In this section we present two contexts for the definition of
a category of $\cM$-functors: from one category left-tensored over $\cM$
to another, and from an $\cM$-category to a left-tensored category
over $\cM$.
§.§ $\cA$ and $\cB$ are left-tensored
Given two categories $\cA$ and $\cB$, left-tensored over $\cM$, one defines
a category $\Fun_\cM(\cA,\cB)$ of $\cM$-functors as follows.
The objects are functors $f:\cA\to\cB$, together with
a natural equivalence between two compositions in the diagram
\begin{equation}\label{eq:modmap}
\xymatrix{
&{\cM\otimes \cA}\ar[r]\ar[d]^{\id\otimes f} &{\cA}\ar[d]^f \\
\end{equation}
satisfying a compatibility in the diagram
\begin{equation}\label{eq:modmap2}
\xymatrix{
&{\cM\otimes\cM\otimes\cA}\ar@<1ex>[r]\ar@<-1ex>[r]\ar[d]^{\id\otimes\id\otimes f}
&{\cM\otimes \cA}\ar[r]\ar[d]^{\id\otimes f}
&{\cA}\ar[d]^f \\
\end{equation}
The morphisms in $\Fun_\cM(\cA,\cB)$ are morphisms of functors
compatible with natural equivalences (<ref>). Note that we
have no unit condition on $f:\cA\to\cB$ as unitality of left-tensor
categories is a property rather than extra data
[saying that the functor $\one\otimes:\cA\to\cA$ is an equivalence.], so the “unit constraints” $\one\otimes x\to x$ are uniquely reconstructed and automatically preserved by $\cM$-functors.
In case $\cM\in\Alg(\Cat^L)$ and $\cA,\cB$ are left-tensored, we define
$\Fun^L_\cM(\cA,\cB)$ as the category of colimit-preserving functors $f:
\cA\to\cB$, with a natural equivalence (<ref>) satisfying
compatibility (<ref>).
§.§ $\cA$ is $\cM$-category and $\cB$ is left-tensored
Let $\cA$ be $\cM$-enriched category and $\cB$ be left-tensored over $\cM$. We
will define $\Fun_\cM(\cA,\cB)$, the category of $\cM$-functors from $\cA$ to
$\cB$, as follows.
An $\cM$-functor $f:\cA\to\cB$ is given by a map
$f:\Ob(\cA)\to\Ob(\cB)$, together with a compatible collection
of maps
\begin{equation}\label{eq:enr-to-mod}
\hom_\cA(x,y)\otimes f(x)\to f(y),
\end{equation}
given for each pair $x,y\in\Ob(\cA)$.
The compatibility means that, given three objects $x,y,z\in\cA$,
one has a commutative diagram
\begin{equation}\label{eq:enr-to-mod2}
\xymatrix{
&{\hom_\cA(y,z)\otimes\hom_\cA(x,y)\otimes f(x)}\ar[r]\ar[d]
&{\hom_\cA(y,z)\otimes f(y)}\ar[d]\\
&{\hom_\cA(x,z)\otimes f(x)}\ar[r]
\end{equation}
Note that here, once more, we need no special unitality condition:
the map (<ref>) applied to $x=y$ , composed with the
unit $\one\to\hom_\cA(x,x)$, yields automatically the “unit
constraint” $\one\otimes f(x)\to f(x)$: this follows from
(<ref>) and the unitality of $\cB$.
$\cM$-functors from $\cA$ to $\cB$ form a category: a map from $f$ to
$g$ is given by a compatible collection of arrows $f(x)\to g(x)$ in
$\cB$ for any $x\in\Ob(\cA)$.
§.§ $\cM$-presheaves
The category $\cM$ is both left and right-tensored over $\cM$.
Given an $\cM$-category $\cA$, the opposite category $\cA^\op$ is
enriched over $\cM_\op$, so one has a category of $\cM_\op$-functors
$\Fun_{\cM_\op}(\cA^\op,\cM)$. We will call it the category of
$\cM$-presheaves on $\cA$ and we will denote it $P_\cM(\cA)$.
Let us describe explicitly what is an $\cM$-presheaf on $\cA$.
This is a map $f:\Ob(\cA)\to\Ob(\cM)$, together with a compatible
collection of maps
\begin{equation}\label{eq:Mpre}
f(y)\otimes\hom_\cA(x,y)\to f(x).
\end{equation}
Let us show that $P_\cM(\cA)$ is left-tensored over $\cM$. Given
$f\in P_\cM(\cA)=\Fun_{\cM_\op}(\cA^\op,\cM)$ and $m\in\cM$, the
presheaf $m\otimes f$ is defined as follows.
It carries an object $x\in\cA^\op$ to $m\otimes f(x)$.
For a pair $x,y\in\Ob(\cA)$ the map
\begin{equation}
(m\otimes f(y))\otimes\hom_\cA(x,y)\to m\otimes f(x).
\end{equation}
is obtained from (<ref>) by tensoring with $m$ on the left.
The Yoneda embedding $Y:\cA\to P_\cM(\cA)$ is an $\cM$-functor
defined as follows.
For $z\in\cA$ the presheaf $Y(z)$ carries $x\in\cA$ to $\hom_\cA(x,z)\in\cM.$
The map (<ref>)
\begin{equation}\label{eq:Y}
Y(z)(y)\otimes\hom_\cA(x,y)\to Y(z)(x)
\end{equation}
is defined by the composition
The functor $\hom_{P_\cM(\cA)}(Y(x),F)$
is represented by $F(x)\in\cM$.
The map of presheaves
\begin{equation}\label{eq:FYF}
F(x)\otimes Y(x)\to F
\end{equation}
is given by the collection of maps
$F(x)\otimes\hom(z,x)\to F(z)$ which is a part of data for $F$.
We have to verify that (<ref>) is universal. That is, any map
$\alpha:m\otimes Y(x)\to F$ in $P_\cM(\cA)$ comes from a unique map
$\tilde\alpha:m\to F(x)$. The map $\tilde\alpha$ is the composition
$$ m\to m\otimes\hom_\cA(x,x)\to F(x).$$
§.§.§ Yoneda lemma
Lemma <ref> is a version of Yoneda lemma.
Theorem <ref> below saying Yoneda embedding is fully faithful is almost an immediate corollary.
An $\cM$-functor $f:\cA\to\cB$ from an $\cM$-category
to an enriched category is fully faithful if for
any $x,y\in\cA$ the functor $\hom_\cB(f(x),f(y))$ defined by the
formula (<ref>), is represented by $\hom_\cA(x,y)$.
The Yoneda embedding $Y:\cA\to P_\cM(\cA)$ is fully faithful for any small
$\cM$-category $\cA$.
Let $x,y\in\cA$.
We have to prove that the canonical map
$$\hom_\cA(x,y)\otimes Y(x)\to Y(y)$$
induces a bijection
\begin{equation}
\Hom_\cM(m,\hom_\cA(x,y))\to\Hom_{P_\cM(\cA)}(m\otimes Y(x),Y(y)).
\end{equation}
This is a special case of Lemma <ref>.
§ UNIVERSAL PROPERTY OF $\CM$-PRESHEAVES
In this section we assume $\cM\in\Alg(\Cat^L)$.
The Yoneda embedding $Y:\cA\to P_\cM(\cA)$ induces, for each left-tensored category $\cB$ over $\cM$, a natural map
\begin{equation}\label{eq:univP}
\end{equation}
In this section we will show that the above map is an equivalence of categories.
In other words, we will prove that $P_\cM(\cA)$ is the universal left-tensored
category over $\cM$ with colimits generated by $\cA$.
§.§ Weighted colimits
Let, as usual, $\cA$ be $\cM$-category and $\cB$ be left-tensored over $\cM$.
Given $W\in P_\cM(\cA)$ and $F:\cA\to\cB$, we define the weighted colimit
$Z=\colim_W(F)$ as a object of $\cB$ together with a collection of arrows
$\alpha_x:W(x)\otimes F(x)\to Z$ making the diagrams
\begin{equation}
\xymatrix{
&{W(y)\otimes\hom_\cA(x,y)\otimes F(x)}\ar[d]\ar[r]
&{W(y)\otimes F(y)}\ar[d]^{\alpha_y}\\
&{W(x)\otimes F(x)}\ar[r]^{\alpha_x} &{Z}
\end{equation}
commutative for each pair $x,y\in\cA$, and satisfying an obvious universal
It is clear from the above definition that weighted colimits are special
kind of colimits, so they always exist.
Weighted colimit is a functor
P_\cM(\cA)\times\Fun_\cM(\cA,\cB)\to \cB
preserving colimits in both arguments.
Weighted colimits are very convenient in presenting presheaves as colimits of representable presheaves. This can be done in a very canonical way:
any presheaf $F\in P_\cM(\cA)$ is the weighted colimit
$$ F=\colim_F(Y),$$
where $Y:\cA\to P_\cM(\cA)$ is the Yoneda embedding.
The functor (<ref>) is an equivalence of categories.
We will construct a functor $Ext$ in the opposite direction. Given
$F\in\Fun_\cM(\cA,\cB)$, we define $Ext(F)$ by the formula
\begin{equation}
\end{equation}
It is easily verified that the functors $Ext$ and $Res$ form
a pair of equivalences.
[GS]GS R. Garner, M. Shulman, Enriched categories as a free
cocompletion, Adv. Math., 289:1–94, 2016.
[K]K G.M. Kelly, Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press 1982, 245 pp.
[L.HA]L.HA J. Lurie, Higher algebra, available at the author's homepage, <http://www.math.harvard.edu/ lurie/papers/higheralgebra.pdf>
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