text:
\section{Introduction}
\label{sec:intro}

Quantum Chromodynamics (QCD) has a special property called the
asymptotic freedom that implies that the strong coupling constant runs
to a smaller value in parton reactions involving harder momentum
scales.  In this way the perturbative calculation of QCD is a reliable
theoretical description in high-energy nuclear physics.  In reality,
however, a na\"{i}ve perturbative expansion breaks down for
diffractive type reactions in which exchanged momenta are not
necessary hard as compared to the collision energy.  Then, it is
indispensable to make a resummation over large logarithmic enhancement
factors which appear kinematically.  Such a resummation leads to a
picture of increasing parton or gluon density with increasing
scattering energy, and, one would anticipate that the parton density
would eventually enter a new regime where the effect of parton
overlapping is significant.  In such a regime of dense partons, QCD is
still perturbative in a sense that the running coupling constant $g$
is small, but is highly non-linear due to the gluon amplitude $A^\mu$
as large as $\sim 1/g$ [and thus $gA^\mu \sim \mathcal{O}(1)$].  This
non-linear dynamics is a manifestation of the parton or gluon
saturation (see
Refs.~\cite{Mueller:1993rr,Mueller:1994jq,Balitsky:1995ub,Balitsky:2001re,Kovchegov:1999yj}
for pioneering extensions including the non-linearity, see also
Refs.~\cite{Gelis:2012ri,Blaizot:2016qgz} for recent reviews).

The virtue of the gluon saturation is that physical observables
exhibit universal behavior in terms of scaling variables, from which
the saturation momentum, $Q_{\mathrm{s}}$, can be experimentally
fixed~\cite{Stasto:2000er}.  The theoretical framework in the
saturation regime to compute physical observables as functions of
$Q_{\mathrm{s}}$ has been well established and known as the Color Glass
Condensate
(CGC)~\cite{McLerran:1993ni,McLerran:1993ka,McLerran:1994vd}.  The CGC
effective theory is elegantly formulated in the renormalization group
language~\cite{JalilianMarian:1997jx,JalilianMarian:1997gr,JalilianMarian:1997dw},
in which the soft gluons are given by the classical Yang-Mills fields
from the hard parton color source~\cite{Kovchegov:1996ty} whose
transverse density is characterized by $Q_{\mathrm{s}}^2$.  The non-linear
quantum evolution equation for the color source distribution is known
as the JIMWLK equation named after the authors of the pioneering
works~\cite{JalilianMarian:1997jx,JalilianMarian:1997gr,JalilianMarian:1997dw,Iancu:2000hn,Ferreiro:2001qy}.
Although solving the JIMWLK equation demands huge computational
resources (see, for example, Ref.~\cite{Weigert:2000gi} for the
reformulation using the Langevin equation, and
Refs.~\cite{Rummukainen:2003ns,Lappi:2015fma} for numerical
simulations), a Gaussian approximated solution has been
derived~\cite{Iancu:2002aq,Lappi:2012nh}.  The simplest CGC model in
the Gaussian approximation is commonly called the McLerran-Venugopalan
(MV) model, which is often used as an initial input for the JIMWLK
evolution, or, this model itself could provide us with a good
description of the qualitative features of high-energy QCD processes.

One of the most interested and testable quantities calculated in the
CGC framework is the particle
production~\cite{Kovner:2001vi,Kovchegov:2001sc,Blaizot:2004wu,Blaizot:2004wv}
and
correlation~\cite{Baier:2005dv,Marquet:2007vb,Fukushima:2008ya,Albacete:2010pg,Kovner:2016jfp},
including electromagnetic
probes~\cite{Gelis:2002ki,Gelis:2002fw,JalilianMarian:2004er,Benic:2016yqt,Benic:2016uku}.
In the MV model various correlations among gluons and quarks have been
discussed, and these predictions are to be compared to the
experimental data of jets and hadron correlations.  One of the most
well known examples is the CGC picture to understand the ridge
structure seen in the di-jet or di-hadron correlations in rapidity
space~\cite{Dumitru:2008wn,Dumitru:2010iy}.  Thus, it would be a
natural extension to apply the CGC-based calculations to account for
the collective behavior in small size systems.  As the collision
energy grows up, small size systems such as the $p$-$A$
(proton-nucleus) and even the $p$-$p$ collisions may look similar to
the $A$-$A$ collision, yielding systematic patterns of the flow
observables (see Ref.~\cite{Loizides:2016tew} for an experimental
overview).  The final state interaction might be still responsible for
the flow observables, but at the same time, one must also estimate the
initial state effect to quantify which of the initial and the final
state interactions is more important.  Recently, within the dipole
model for the particle production the systematics of the higher order
flow observables, $v_n\{m\}$ (i.e.\ $n$-th harmonics of the
$m$-particle flow) has been quantified in the glasma graph
approximation~\cite{Skokov:2014tka,Lappi:2015vta} and in the full MV
model~\cite{Dusling:2017dqg,Dusling:2017aot}.

Generally, in the CGC model calculations, the most time-consuming part
in numerics is the numerical computation of the expectation value of
the Wilson lines.  In the presence of the CGC background fields
corresponding to the saturated soft gluons, one needs to take account
of multiple scatterings which amount to the Wilson line in the eikonal
approximation.  It has been repeatedly discussed how to compute the
Wilson line correlators efficiently in separate contexts (see
Ref.~\cite{Shi:2017gcq} for a very recent work toward general formulae
for the Wilson line correlators, for instance).  Therefore, it would
be quite useful to establish a prescription to compute the Wilson line
correlators with some generality but in a fairly straightforward way.
There are several preceding works along these lines from an early
attempt in Ref.~\cite{Fukushima:2007dy} to a very recent reformulation
in Ref.~\cite{Dumitru:2017cwt} on top of an explicit calculation in
Ref.~\cite{Shi:2017gcq}.  The purpose of the present work is to
translate the powerful method of Ref.~\cite{Fukushima:2007dy} to a
more specific physics problem in a more handy form.  In particular,
the setup in Ref.~\cite{Fukushima:2007dy} was too general and it was
not clear how to utilize the formulae for phenomenological
applications.  In the present work, hence, we focus on the special
type of correlation function in terms of the
\textit{dipole operators} in the color fundamental representation,
which is the basic building block in the dipole
model~\cite{Kowalski:2006hc}.  For $n$-point dipole correlations the
problem is reduced to the exponentiation of $n!\times n!$ matrices.
The calculation of the exponential of the matrix is numerically
feasible but it is sometimes insightful to perform the large-$N_{\mathrm{c}}$
expansion, especially to identify the $N_{\mathrm{c}}$ scaling of observables.
We emphasize that our formulae take a quite convenient form for the
large-$N_{\mathrm{c}}$ expansion and we will demonstrate the expansion explicitly
up to the order where the first nonzero flow cumulants appear.  We
check the validity of the large-$N_{\mathrm{c}}$ formulae by comparing to the
exact results known for $n=2$ with $N_{\mathrm{c}}=3$.  For $n\ge 4$ we need to
be very careful of the correct large-$N_{\mathrm{c}}$ counting and we will argue
the subtlety in the numerical analysis.

\section{Master formulae}
\label{sec:master}

The Wilson line correlators that we calculate in this paper are
expressed in analogy to quantum mechanics as
follows~\cite{Fukushima:2007dy}:
\begin{equation}
  \label{eq:Wilson}
  \biggl\langle \prod_{i=1}^n U({\boldsymbol{x}_\perp}_i)_{\beta_i\alpha_i}
  U^\ast({\boldsymbol{y}_\perp}_i)_{\bar{\beta}_i\bar{\alpha}_i} \biggr\rangle
  = \exp\bigl[-(H_0 + V)\bigr]_{\beta_1\bar{\beta}_1 \cdots \beta_n\bar{\beta}_n;
    \alpha_1\bar{\alpha}_1 \cdots \alpha_n\bar{\alpha}_n} \,,
\end{equation}
where the \textit{free Hamiltonian}, $H_0$, is defined as
\begin{equation}
  \label{eq:hamiltonian}
  H_0 = Q_{\mathrm{s}}^2 \frac{2N_{\mathrm{c}}}{N_{\mathrm{c}}^2 - 1} L(0,0) \biggl[ \sum_{i=1}^n
    \bigl( {T_{\mathrm{F}}}_i^a - {T_{\mathrm{F}}^\ast}_{\bar{i}}^a \bigr) \biggr]^2\,.
\end{equation}
Here, we take the summation over $a$ in group space (which is always
implicitly assumed).  In this paper we limit our considerations to the
fundamental representation only and ${T_{\mathrm{F}}}_i^a$'s represent the elements
of su($N_{\mathrm{c}}$) algebra in the fundamental representation, while the
formula is valid for any representation.  Here, $L(0,0)$ is
quadratically divergent, so that only color states that have zero
eigenvalues of $H_0$ can make finite contributions to the Wilson line
correlators [and thus, the detailed definition of $L(0,0)$ is not
important here; see Ref.~\cite{Fukushima:2007dy} for the explicit form
of $L(0,0)$].  We note that the definition of $Q_{\mathrm{s}}$ is slightly
different from Refs.~\cite{Dusling:2017dqg,Dusling:2017aot} and we
will come to this point later when we apply our formulae for the flow
observables.  For the moment, Eq.~\eqref{eq:hamiltonian} is understood
as a definition of $Q_{\mathrm{s}}$ in our convention.  The color matrix in the
\textit{interaction part} is
\begin{equation}
  \label{eq:potential}
  \begin{split}
    V = -Q_{\mathrm{s}}^2 \frac{2N_{\mathrm{c}}}{N_{\mathrm{c}}^2 - 1} \biggl\{
    & \sum_{i>j}^n \Bigl[ {T_{\mathrm{F}}}_i^a {T_{\mathrm{F}}}_j^a \Gamma(|{\boldsymbol{x}_\perp}_i-{\boldsymbol{x}_\perp}_j|)
      + {T_{\mathrm{F}}^\ast}_{\bar{i}}^a {T_{\mathrm{F}}^\ast}_{\bar{j}}^a \Gamma(|{\boldsymbol{y}_\perp}_i-{\boldsymbol{y}_\perp}_j|)
      \Bigr] \\
    & - \sum_{i,j=1}^n {T_{\mathrm{F}}}_i^a {T_{\mathrm{F}}^\ast}_{\bar{j}}^a
    \Gamma(|{\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_j|) \biggr\} \,.
  \end{split}
\end{equation}
In the above the common building block, $\Gamma({\boldsymbol{x}_\perp})$, is defined in
the CGC formalism by the following integral:
\begin{equation}
  \Gamma(|{\boldsymbol{x}_\perp}|) := 2g^4 \int_{\Lambda}^\infty
  \frac{\mathrm{d} k}{2\pi}\, \frac{1}{k^3}\, \Bigl[ 1
    - J_0(k|{\boldsymbol{x}_\perp}|) \Bigr]
  \simeq -\frac{g^4}{4\pi} |{\boldsymbol{x}_\perp}|^2
  \ln \bigl(|{\boldsymbol{x}_\perp}| \bar{\Lambda}\bigr)\,,
\end{equation}
where $J_{\alpha}(x)$ is the Bessel function of the first kind,
$\Lambda$ is an infrared (IR) cutoff of order of $\Lambda_{\text{QCD}}$ and we
introduced a shorthand notation as
$\bar{\Lambda}:=\tfrac{1}{2}\Lambda \mathrm{e}^{\gamma-1}$.  This
approximated expression has undesirable behavior in the IR region
especially when $|{\boldsymbol{x}_\perp}|\bar{\Lambda}<1$.  We cure this IR problem,
according to Ref.~\cite{Dusling:2017aot}, by introducing another
regulator as
\begin{equation}
  \Gamma(|{\boldsymbol{x}_\perp}|) \simeq \frac{g^4}{4\pi} |{\boldsymbol{x}_\perp}|^2
  \ln \biggl( \frac{1}{|{\boldsymbol{x}_\perp}| \bar{\Lambda}} + \mathrm{e} \biggr)\,.
\end{equation}
We will adopt this regularized approximation for $\Gamma({\boldsymbol{x}_\perp})$ in
our numerical calculations later in Sec.~\ref{sec:validity}.

Now we shall sketch our strategy to proceed to concrete calculations
of the Wilson line correlators using Eq.~\eqref{eq:Wilson}.  We will first
identify all the color-singlet bases with which $H_0$ vanishes.  For
the $n$-th power product of $U$ and $U^\ast$, there are $2n$ Wilson
lines, and then the number of independent color-singlet bases should
be $n!$, as we will explain in the next section.  We can easily
construct the color index structures by taking the permutations.
After confirming that $H_0$ surely vanishes with such bases, next, we
will consider the matrix elements of $V$.  In general it is not an
easy task to find analytical expressions for the eigenvalues of $V$.
Nevertheless, it is known that the large-$N_{\mathrm{c}}$ approximation works at
good quantitative accuracy, and we will systematically make an
expansion in power of $1/N_{\mathrm{c}}$ to find analytical expressions.

\section{Color singlet bases}
\label{sec:singlet}

Because of the dipole-type structures of the Wilson lines it is easy
to find all the combinations of the color singlet indices.  One
trivial singlet is immediately found as
\begin{equation}
  \langle\{\alpha\};\{\bar{\alpha}\}|s_0\rangle
  := N_{\mathrm{c}}^{-n/2}\,
  \delta_{\alpha_1\bar{\alpha}_1}\cdots\delta_{\alpha_n\bar{\alpha}_n}\,.
\end{equation}
Clearly all the permutations are possible, i.e.\ we introduce
$|s_p\rangle$ as a permutation of $|s_{0}\rangle$.  For this purpose let
us introduce the symmetry group $S_n$ with elements $\pi_p$ that
denotes a permutation as
\begin{equation}
  \pi_p = \begin{pmatrix}
    1 & 2 & \cdots & n \\
    p_1 & p_2 & \cdots & p_n
  \end{pmatrix} \,.
\end{equation}
Then,
\begin{equation}
  \langle\{\alpha\};\{\bar{\alpha}\}|s_p\rangle
  := \langle\{\pi_p \alpha\};\{\bar{\alpha}\}|s_0\rangle
  = N_{\mathrm{c}}^{-n/2}\,
  \delta_{\alpha_{p_1}\bar{\alpha}_1}\cdots\delta_{\alpha_{p_n}\bar{\alpha}_n}\,.
\end{equation}
We note that any $\pi_p$ can be expressed as a product of cycles.
For example, it is easy to confirm the following
relation,
\begin{equation}
  \begin{pmatrix}
    1 & 2 & 3 & 4 & 5 & 6 \\
    5 & 1 & 3 & 6 & 2 & 4
  \end{pmatrix}
    = (152)\cdot (46) \,,
\end{equation}
where one-cycles, i.e.\ $(3)$ in the above case, are trivial and not
explicitly written.  Let us see several useful formulae for later
calculation checks.  Because of the well-known relation,
\begin{equation}
  2{T_{\mathrm{F}}}^a_{\beta_i\alpha_i}{T_{\mathrm{F}}}^a_{\beta_j\alpha_j}
  = \delta_{\beta_i\alpha_j}\delta_{\beta_j\alpha_i}
  - \frac{1}{N_{\mathrm{c}}}\delta_{\beta_i\alpha_i}\delta_{\beta_j\alpha_j} \,,
\end{equation}
we can easily prove that
\begin{equation}
  \langle\{\beta\};\{\bar{\beta}\}|2{T_{\mathrm{F}}}^a_i{T_{\mathrm{F}}}^a_j|s_p\rangle
  = \langle\{(i\,j)\pi_p \beta\};\{\bar{\beta}\}|
  s_0\rangle
  - \frac{1}{N_{\mathrm{c}}}\langle\{\pi_p\beta\};\{\bar{\beta}\}|s_0\rangle\,,
\end{equation}
which is schematically expressed as
\begin{equation}
  \label{eq:formula_tftf}
  2{T_{\mathrm{F}}}_i^a{T_{\mathrm{F}}}_{\bar{j}}^a \,\cdot\,\pi_p
  = \biggl[ (i\,j) - \frac{1}{N_{\mathrm{c}}} \biggr]\,\cdot \pi_p\,.
\end{equation}
For the complex conjugate, because ${T_{\mathrm{F}}}^a$'s are Hermitean, the
following should hold:
\begin{equation}
  2{T_{\mathrm{F}}^\ast}^a_{\bar{\beta}_i\bar{\alpha}_i}{T_{\mathrm{F}}^\ast}^a_{\bar{\beta}_j\bar{\alpha}_j}
  = \delta_{\bar{\alpha}_j\bar{\beta}_i}\delta_{\bar{\alpha}_i\bar{\beta}_j}
  - \frac{1}{N_{\mathrm{c}}}\delta_{\bar{\alpha}_i\bar{\beta}_i}
  \delta_{\bar{\alpha}_j\bar{\beta}_j} \,,
\end{equation}
which implies
\begin{equation}
  \langle\{\beta\};\{\bar{\beta}\}|2{T_{\mathrm{F}}^\ast}^a_{\bar{i}}{T_{\mathrm{F}}^\ast}^a_{\bar{j}}|s_p\rangle
  = \langle\{\pi_p \beta\};\{(i\, j)\bar{\beta}\}|
  s_0\rangle
  - \frac{1}{N_{\mathrm{c}}}\langle\{\pi_p\beta\};\{\bar{\beta}\}|s_0\rangle\,.
\end{equation}
Thanks to the index structure of $|s_0\rangle$, we readily see;
$\langle\{\pi_p \beta\};\{(i\,j)\bar{\beta}\}|s_0\rangle
=\langle\{\pi_p (i\,j)^{-1}\beta\};\{\bar{\beta}\}|s_0\rangle$ and
trivially $(i\,j)^{-1}=(i\,j)$, so that we can again give a schematic
representation as
\begin{equation}
  \label{eq:formula_tfatfa}
  2{T_{\mathrm{F}}^\ast}^a_{\bar{i}} {T_{\mathrm{F}}^\ast}^a_{\bar{j}} \,\cdot \pi_p
  = \pi_p \,\cdot \biggl[ (i\,j) - \frac{1}{N_{\mathrm{c}}} \biggr] \,.
\end{equation}
Also, another useful formula is
\begin{equation}
  2{T_{\mathrm{F}}}^a_{\beta_i\alpha_i} {T_{\mathrm{F}}^\ast}^a_{\bar{\beta}_j\bar{\alpha}_j}
  = \delta_{\beta_i\bar{\beta}_j}\delta_{\alpha_i\bar{\alpha}_j}
  - \frac{1}{N_{\mathrm{c}}}\delta_{\beta_i\alpha_i}
  \delta_{\bar{\beta}_j\bar{\alpha}_j}\,.
\end{equation}
In this case, if $p_j=i$, the contract of
$\delta_{\beta_i\bar{\beta}_j}\delta_{\alpha_i\bar{\alpha}_j}
=\delta_{\beta_{p_j}\bar{\beta}_j}\delta_{\alpha_{p_j}\bar{\alpha}_j}$
in the above and $\delta_{\alpha_{p_j}\bar{\alpha}_j}$ in
$|s_p\rangle$ makes $N_{\mathrm{c}}\delta_{\beta_{p_j}\bar{\beta}_j}$.  For
$p_j\neq i$ the contract of
$\delta_{\beta_i\bar{\beta}_j}\delta_{\alpha_i\bar{\alpha}_j}$ and
$\delta_{\alpha_{p_j}\bar{\alpha}_j}\delta_{\alpha_i\bar{\alpha}_{p^{-1}_i}}$
together with
$\delta_{\beta_{p_j}\alpha_{p_j}}\delta_{\bar{\beta}_{p^{-1}_i}\bar{\alpha}_{p^{-1}_i}}$
(which is always implicitly taken for indices not involved in ${T_{\mathrm{F}}}^a$
nor ${T_{\mathrm{F}}^\ast}^a$) gives
$\delta_{\beta_i\bar{\beta}_j}\delta_{\beta_{p_j}\bar{\beta}_{p^{-1}_i}}
=(i\, p_j)\,\delta_{\beta_{p_j}\bar{\beta}_j}
\delta_{\beta_i\bar{\beta}_{p^{-1}_i}}$,
where $p^{-1}_i$ indicates an index that satisfies
$p_{p^{-1}_i}=i$.  Therefore, we establish the following schematic
relations:
\begin{equation}
  \label{eq:formula_tftfa}
  2{T_{\mathrm{F}}}^a_i {T_{\mathrm{F}}^\ast}^a_j\, \pi_p = \begin{cases}
    \displaystyle \biggl( N_{\mathrm{c}}-\frac{1}{N_{\mathrm{c}}} \biggr)\,\cdot \pi_p
    & \text{for $p_j=i$} \\[1em]
    \displaystyle \biggl[ (i\,p_j) - \frac{1}{N_{\mathrm{c}}} \biggr]\, \cdot \pi_p
    & \text{for $p_j\neq i$}
  \end{cases}
\end{equation}
In what follows below, we will utilize the
formulae~\eqref{eq:formula_tftf}, \eqref{eq:formula_tfatfa}, and
\eqref{eq:formula_tftfa} to perform calculations with a compact
notation.

It would be an instructive check to see how $H_0|s_p\rangle=0$ is
satisfied for all $|s_p\rangle$.  For this purpose we first need to
expand the second-order Casimir operator in $H_0$ as
\begin{equation}
  \biggl[ \sum_{i=1}^n \bigl( {T_{\mathrm{F}}}_i^a - {T_{\mathrm{F}}^\ast}_{\bar{i}}^a \bigr) \biggr]^2
  = \sum_{i=1}^n \bigl( {T_{\mathrm{F}}}_i^{a\,2} + {T_{\mathrm{F}}^\ast}_{\bar{i}}^{a\,2} \bigr)
  + 2\sum_{i>j}^n \bigl( {T_{\mathrm{F}}}_i^a{T_{\mathrm{F}}}_j^a
  + {T_{\mathrm{F}}^\ast}_{\bar{i}}^a{T_{\mathrm{F}}^\ast}_{\bar{j}}^a \bigr)
  - 2\sum_{i,j=1}^n {T_{\mathrm{F}}}_i^a{T_{\mathrm{F}}^\ast}_{\bar{j}}^a \,.
\end{equation}
Here, again, we note that the summation over $a$ is always implicitly
assumed.   The first term is nothing but the Casimir operator, so that
it is simply given by
\begin{equation}
  \label{eq:casimir}
  \sum_{i=1}^n \bigl( {T_{\mathrm{F}}}_i^{a\,2}+{T_{\mathrm{F}}^\ast}_{\bar{i}}^{a\,2} \bigr)
  = n\biggl( N_{\mathrm{c}} - \frac{1}{N_{\mathrm{c}}} \biggr)\,,
\end{equation}
according to the su($N_{\mathrm{c}}$) algebra.  Using the
formulae~\eqref{eq:formula_tftf}, \eqref{eq:formula_tfatfa}, and
\eqref{eq:formula_tftfa}, we can simplify the rest (if applied to
$|s_p\rangle$) as
\begin{align}
  &\biggl[ 2\sum_{i>j}^n \bigl( {T_{\mathrm{F}}}_i^a{T_{\mathrm{F}}}_j^a
    + {T_{\mathrm{F}}^\ast}_{\bar{i}}^a{T_{\mathrm{F}}^\ast}_{\bar{j}}^a \bigr)
    -2 \sum_{i,j=1}^n {T_{\mathrm{F}}}^a_i {T_{\mathrm{F}}^\ast}^a_{\bar{j}} \biggr]\,\cdot \pi_p \notag\\
  &= \sum_{i>j}^n \biggl\{ \biggl[ (i\,j) - \frac{1}{N_{\mathrm{c}}} \biggr] \cdot\pi_p
  + \pi_p \cdot \biggl[ (i\,j) - \frac{1}{N_{\mathrm{c}}} \biggr] \biggr\}
  - \sum_{i,j=1}^n \biggl[ N_{\mathrm{c}}\delta_{i p_j} + (i\,p_j)
    (1-\delta_{i p_j})-\frac{1}{N_{\mathrm{c}}} \biggr] \, \cdot \pi_p \notag\\
  &= -n\biggl(N_{\mathrm{c}}-\frac{1}{N_{\mathrm{c}}}\biggr)\,\cdot \pi_p
  + \sum_{i>j}^n \bigl[ (i\,j)\,\cdot \pi_p + \pi_p \cdot (i\,j)\bigr]
  - \sum_{i,j=1}^n (i\,p_j)(1-\delta_{i p_j})\,\cdot \pi_p \,.
\end{align}
The first term cancels out with Eq.~\eqref{eq:casimir}.
Since $i$ and $j$ run from $1$ to $n$, we can equivalently take a
summation with respect to $j$ instead of using $p_j$, leading to
\begin{equation}
  -\sum_{i,j=1}^n (i\, p_j)(1-\delta_{i p_j})\,\cdot \pi_p
  = -\sum_{i\neq j}^n (i\, j)\,\cdot \pi_p \,.
\end{equation}
Also, we see $\pi_p\cdot (i\, j)=(p_i\,p_j)\cdot \pi_p$, which allows
us to replace $p_i$ and $p_j$ with $i$ and $j$ in the summations.
Finally we arrive at
\begin{equation}
  \sum_{i>j}^n \bigl[ (i\,j)\,\cdot \pi_p + \pi_p \cdot (i\,j)\bigr]
  - \sum_{i,j=1}^n (i\,p_j)(1-\delta_{i p_j})\,\cdot \pi_p = 0\,.
\end{equation}
This completes our confirmation of $H_0|s_p\rangle=0$ for any
$|s_p\rangle$.

For the practical calculation the most important is the evaluation of
the matrix elements of $V$.  In our formalism we can infer the matrix
elements from
\begin{align}
  V\,\cdot \pi_p &= -Q_{\mathrm{s}}^2\,\frac{N_{\mathrm{c}}}{N_{\mathrm{c}}^2-1}\, \Biggl( \sum_{i>j}^n
  \biggl\{
  \Bigl[ (i\,j)-\frac{1}{N_{\mathrm{c}}}\Bigr] \Gamma(|{\boldsymbol{x}_\perp}_i-{\boldsymbol{x}_\perp}_j|)
  +\Bigl[ (p_i\,p_j)-\frac{1}{N_{\mathrm{c}}}\Bigr] \Gamma(|{\boldsymbol{y}_\perp}_i-{\boldsymbol{y}_\perp}_j|)
  \biggr\} \notag\\
  &\quad -\sum_{i,j=1}^n \Bigl[ N_{\mathrm{c}}\delta_{i p_j} + (i\,p_j)
    (1-\delta_{i p_j})-\frac{1}{N_{\mathrm{c}}} \Bigr] \Gamma(|{\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_j|)
  \Biggr)\,\cdot \pi_p \,,
\end{align}
which can be easily verified with Eqs.~\eqref{eq:formula_tftf},
\eqref{eq:formula_tfatfa}, and \eqref{eq:formula_tftfa}.  We can
further simplify the above expression by introducing a notation for a
proper combination of $\Gamma$'s, i.e.,
\begin{equation}
  F({\boldsymbol{x}_\perp},{\boldsymbol{x}_\perp^\prime};{\boldsymbol{y}_\perp},{\boldsymbol{y}_\perp^\prime}) := \Gamma(|{\boldsymbol{x}_\perp^\prime}-{\boldsymbol{y}_\perp}|)
  +\Gamma(|{\boldsymbol{x}_\perp}-{\boldsymbol{y}_\perp^\prime}|) - \Gamma(|{\boldsymbol{x}_\perp}-{\boldsymbol{x}_\perp^\prime}|)
  -\Gamma(|{\boldsymbol{y}_\perp}-{\boldsymbol{y}_\perp^\prime}|) \,.
\end{equation}
We then define the explicit components of the matrix elements as
\begin{equation}
  V|s_p\rangle = \sum_{p'} |s_{p'}\rangle\, V_{p',\, p} \,.
\end{equation}
Here we must be careful of the fact that $|s_p\rangle$'s span a
complete set of bases but they are not orthogonal.  Using these
notations and definitions we can summarize the non-zero components as
follows:
\begin{align}
  \label{eq:Vpp}
  V_{p,\,p} &= Q_{\mathrm{s}}^2 \biggl[ \sum_{i=1}^n \Gamma(|{\boldsymbol{x}_\perp}_{p_i}-{\boldsymbol{y}_\perp}_i|)
    - \frac{1}{N_{\mathrm{c}}^2-1}\sum_{i>j}^n
    F({\boldsymbol{x}_\perp}_{p_i},{\boldsymbol{x}_\perp}_{p_j};{\boldsymbol{y}_\perp}_i,{\boldsymbol{y}_\perp}_j) \biggr] \,,\\
  \label{eq:Vpijp}
  V_{p(i\, j),\, p} &= Q_{\mathrm{s}}^2\, \frac{N_{\mathrm{c}}}{N_{\mathrm{c}}^2-1}\,
  F({\boldsymbol{x}_\perp}_{p_i},{\boldsymbol{x}_\perp}_{p_j};{\boldsymbol{y}_\perp}_i,{\boldsymbol{y}_\perp}_j) \,,
\end{align}
and other matrix elements vanish.
These formulae are our central results in the present paper.  For the
actual application of the formulae, we should compute the exponential
of $V$ as seen in Eq.~\eqref{eq:Wilson}.

To understand how the formulae work, let us consider the simplest
example of $n=2$.  The matrix elements of $2\times 2$ matrix $V$
read
\begin{equation}
  V = \begin{pmatrix}
    V_{0,0} & V_{0,(21)} \\ V_{(21),0} & V_{(21),(21)}
  \end{pmatrix} = Q_{\mathrm{s}}^2 \frac{2N_{\mathrm{c}}}{N_{\mathrm{c}}^2-1} \begin{pmatrix}
    \displaystyle \frac{N_{\mathrm{c}}^2-1}{N_{\mathrm{c}}} \gamma - \frac{1}{N_{\mathrm{c}}}(\alpha-\beta) &
    \gamma-\beta \\
    \alpha-\beta &
    \displaystyle \frac{N_{\mathrm{c}}^2-1}{N_{\mathrm{c}}} \alpha - \frac{1}{N_{\mathrm{c}}}(\gamma-\beta)
  \end{pmatrix} \;,
\end{equation}
where,
\begin{equation}
  \begin{split}
    2\alpha := \Gamma(|{\boldsymbol{x}_\perp}_1-{\boldsymbol{y}_\perp}_2|) + \Gamma(|{\boldsymbol{y}_\perp}_1-{\boldsymbol{x}_\perp}_2|)\,,\\
    2\beta  := \Gamma(|{\boldsymbol{x}_\perp}_1-{\boldsymbol{x}_\perp}_2|) + \Gamma(|{\boldsymbol{y}_\perp}_1-{\boldsymbol{y}_\perp}_2|)\,,\\
    2\gamma := \Gamma(|{\boldsymbol{x}_\perp}_1-{\boldsymbol{y}_\perp}_1|) + \Gamma(|{\boldsymbol{x}_\perp}_2-{\boldsymbol{x}_\perp}_2|)\,.
  \end{split}
\end{equation}
It is a straightforward calculation to obtain two eigenvalues as
$\lambda_\pm=\frac{1}{2}(\mathrm{tr} V\pm\varphi)$ with
\begin{equation}
  \varphi := \sqrt{(\mathrm{tr} V)^2-4\det V} = Q_{\mathrm{s}}^2 \frac{2N_{\mathrm{c}}^2}{N_{\mathrm{c}}^2-1}
  \sqrt{(\alpha-\gamma)^2 + \frac{4}{N_{\mathrm{c}}^2}(\beta-\alpha)(\beta-\gamma)}\;.
\end{equation}
Now we can express the exponential of $V$ in a simple form as
\begin{equation}
  \label{eq:n2}
  \mathrm{e}^{-V} = \mathrm{e}^{-\frac{1}{2}\mathrm{tr} V} \begin{pmatrix}
    \displaystyle \cosh\tfrac{1}{2}\varphi
    - \frac{\sinh\tfrac{1}{2}\varphi}{\varphi} (V_{0,0}-V_{(21),(21)}) &
    \displaystyle -2\frac{\sinh\tfrac{1}{2}\varphi}{\varphi} V_{0,(21)} \\[1em]
    \displaystyle -2\frac{\sinh\tfrac{1}{2}\varphi}{\varphi} V_{(21),0} &
    \displaystyle \cosh\tfrac{1}{2}\varphi
    + \frac{\sinh\tfrac{1}{2}\varphi}{\varphi} (V_{0,0}-V_{(21),(21)})
  \end{pmatrix} \,.
\end{equation}
Although the calculation machinery is rather simple, a larger $n$
would cause a huge computational cost.  Hence, we will seek for an
algorithmic expansion to approximate $\mathrm{e}^{-V}$ without complicated
matrix algebra.

\section{Dipole Wilson line correlators and the large-$N_{\mathrm{c}}$ expansion}
\label{sec:dipole}

For the application for the particle production problem in the
relativistic heavy-ion
collision~\cite{Dusling:2017dqg,Dusling:2017aot}, we are specifically
interested in the correlation functions of the dipole operators.  The
definition of the dipole operator is
\begin{equation}
  D({\boldsymbol{x}_\perp},{\boldsymbol{y}_\perp}) := \frac{1}{N_{\mathrm{c}}}\mathrm{tr}\bigl[ U({\boldsymbol{x}_\perp})\,U^\dag({\boldsymbol{y}_\perp}) \bigr]
  = \frac{1}{N_{\mathrm{c}}}\,\delta_{\beta\bar{\beta}}\delta_{\alpha\bar{\alpha}}
  \, U({\boldsymbol{x}_\perp})_{\beta\alpha}\, U^\ast({\boldsymbol{y}_\perp})_{\bar{\beta}\bar{\alpha}}\,.
\end{equation}
From this form it is obvious that the $n$ dipole expectation value is
given by an matrix element of $\mathrm{e}^{-V}$ evaluated with
$|s_0\rangle$, i.e.,
\begin{equation}
  \biggl\langle \prod_{i=1}^n D({\boldsymbol{x}_\perp}_i,{\boldsymbol{y}_\perp}_i)\biggr\rangle
  = \langle s_0|\, \mathrm{e}^{-V}\, |s_0\rangle \,.
\end{equation}
We may be able to do a direct computation, but we can develop a more
sophisticated method assuming that $N_{\mathrm{c}}$ is large enough.  In view of
Eqs.~\eqref{eq:Vpp} and \eqref{eq:Vpijp}, the off-diagonal components
are suppressed by $1/N_{\mathrm{c}}$, so that we can avoid exponentiating $V$ but
make a systematic expansion in terms of $V_{p(ij),p}$.

For notational brevity we shall denote the diagonal and the
off-diagonal parts of $V$ as $V^{(0)}$ and $V^{(1)}$, respectively.
Then, the starting point for the systematic perturbative expansion is
the \textit{interaction picture} as in quantum mechanics expressed as
\begin{equation}
  \mathrm{e}^{-V} = \mathrm{e}^{-V^{(0)}}\, \mathcal{T}_\tau\, \exp\biggl[
    -\int_0^1 \mathrm{d}\tau\, V^{(1)}(\tau) \biggr]\,,
\end{equation}
where $\mathcal{T}_\tau$ stands for the time-ordered product in terms of
$\tau$ and the time dependent $V^{(1)}(\tau)$ in the interaction
picture is defined as
\begin{equation}
  V^{(1)}(\tau) := \mathrm{e}^{\tau V^{(0)}} V^{(1)}\,
  \mathrm{e}^{-\tau V^{(0)}}\,.
\end{equation}
Thus, up to the second order in $V^{(1)}$ for example, the
perturbative expansion reads
\begin{equation}
  \mathrm{e}^{-V} \simeq \mathrm{e}^{-V^{(0)}} \biggl[
    1-\int_0^1 \mathrm{d}\tau\, V^{(1)}(\tau) +
    \int_0^1 \mathrm{d}\tau_1 \int_0^{\tau_1} \mathrm{d}\tau_2\,
    V^{(1)}(\tau_1)\,V^{(1)}(\tau_2) \biggr] \,.
\end{equation}
Because $V^{(0)}$ is a diagonal matrix, its matrix elements,
$V_{p,p}$, are the eigenvalues of $V$.  Then, let us introduce an
eigenvector $|p\rangle$ with an eigenvalue $E_p$ for $V^{(0)}$.  That
is,
\begin{equation}
  V^{(0)}|p\rangle = E_p|p\rangle = (E_p^{(0)}+E_p^{(2)})|p\rangle\,,
\end{equation}
where we decomposed the eigenvalue according to the $1/N_{\mathrm{c}}$ order as
\begin{align}
  E_p^{(0)} &:= Q_{\mathrm{s}}^2 \sum_{i=1}^n\, \Gamma({\boldsymbol{x}_\perp}_{p_i}-{\boldsymbol{y}_\perp}_i)\,,\\
  E_p^{(2)} &:= -Q_{\mathrm{s}}^2 \frac{1}{N_{\mathrm{c}}^2-1} \sum_{i>j}^n
  F({\boldsymbol{x}_\perp}_{p_i},{\boldsymbol{x}_\perp}_{p_j};{\boldsymbol{y}_\perp}_i,{\boldsymbol{y}_\perp}_j)
  = -\frac{1}{N_{\mathrm{c}}} \sum_{i>j}^n V_{p(ij), p}^{(1)}\,.
\end{align}
It is then easy to re-express the first perturbative correction as
\begin{align}
  -\mathrm{e}^{-V^{(0)}} \int_0^1 \mathrm{d}\tau\, V^{(1)}(\tau)|p\rangle
  &= -\sum_q |q\rangle\, \mathrm{e}^{-E_q} \int_0^1 \mathrm{d}\tau\,
  V_{qp}^{(1)}\, \mathrm{e}^{\tau(E_q-E_p)} \notag\\
  &= \sum_q |q\rangle\, \frac{\mathrm{e}^{-E_q}-\mathrm{e}^{-E_p}}{E_q-E_p}\,
  V_{qp}^{(1)}\,.
\end{align}
Here, $V_{qp}^{(1)}$ is defined as
$V^{(1)}|p\rangle = \sum_q |q\rangle\, V_{qp}^{(1)}$, where we note
that this $V^{(1)}$ is an original matrix, not the one in the
interaction picture.

In the same way, we can proceed to the second perturbative correction
as
\begin{align}
  &\mathrm{e}^{-V^{(0)}} \int_0^1 \mathrm{d}\tau_1 \int_0^{\tau_1} \mathrm{d}\tau_2\,
  V^{(1)}(\tau_1) V^{(1)}(\tau_2)\,|p\rangle \notag\\
  &\qquad = \sum_{q,r}|q\rangle\, \mathrm{e}^{-E_q} \int_0^1\mathrm{d}\tau_1
  \int_0^{\tau_1}\mathrm{d}\tau_2\, \mathrm{e}^{\tau_1(E_q-E_r) + \tau_2(E_r-E_p)}
  \, V_{qr}^{(1)}\, V_{rp}^{(1)} \notag\\
  &\qquad = \sum_{q,r}|q\rangle\, \biggl(
  \frac{\mathrm{e}^{-E_q}-\mathrm{e}^{-E_r}}{E_q-E_r} - \frac{\mathrm{e}^{-E_q}-\mathrm{e}^{-E_p}}
       {E_q-E_p} \biggr)\, \frac{V_{qr}^{(1)}\, V_{rp}^{(1)}}
       {E_r-E_p} \,.
\end{align}
At this point, we can see a general algorithm to go to arbitrary high
orders.  The next order, for example, is generated automatically via
one more iteration as
\begin{align}
  \sum_{q,s,r}|q\rangle\, \biggl[ &
    \biggl(\frac{\mathrm{e}^{-E_q}-\mathrm{e}^{-E_s}}{E_q-E_s}
    -\frac{\mathrm{e}^{-E_q}-\mathrm{e}^{-E_r}}{E_q-E_r}\biggr)
   \frac{1}{E_s-E_r} \notag  \\
    & - \biggl(\frac{\mathrm{e}^{-E_q}-\mathrm{e}^{-E_s}}{E_q-E_s}
    - \frac{\mathrm{e}^{-E_q}-\mathrm{e}^{-E_p}}{E_q-E_p}\biggr)
        \frac{1}{E_s-E_p}
        \biggr]\, \frac{V_{qs}^{(1)}\, V_{sr}^{(1)}\, V_{rp}^{(1)}}
      {E_r-E_p} \,.
      \label{eq:NNLO}
\end{align}

Now, we are ready to compute $\langle s_0|\mathrm{e}^{-V}|s_0\rangle$ up to
the $N_{\mathrm{c}}^{-2}$ order.  Noting that $V^{(1)}$ has a matrix element
between $p$ and $p(ij)$, we can write down
\begin{align}
  \langle s_0|\mathrm{e}^{-V}|s_0\rangle \simeq
  \langle s_0|s_0\rangle \, \mathrm{e}^{-E_0} & + \sum_{i>j}^n
    \langle s_0|(ij)s_0\rangle\, \frac{\mathrm{e}^{-E_{(ij)}}-\mathrm{e}^{-E_0}}
            {E_{(ij)}-E_0}\, V_{(ij), 0}^{(1)} \notag\\
  & + \sum_{i>j}^n \langle s_0|s_0\rangle
    \biggl(\frac{\mathrm{e}^{-E_0}-\mathrm{e}^{-E_{(ij)}}}{E_0-E_{(ij)}}
    + e^{-E_0} \biggr)\,
    \frac{V_{0,(ij)}^{(1)}\, V_{(ij),0}^{(1)}}{E_{(ij)}-E_0} \,.
\end{align}
As we already mentioned, $|s_p\rangle$'s are not orthogonal for
different $p$'s, and a simple calculation leads to the normalization
as $\langle s_0|s_p\rangle = N_{\mathrm{c}}^{-n+n_p}$ where $n_p$ denotes the
number of cycles of $\pi_p$.
Because the second and the
third terms are already suppressed by $N_{\mathrm{c}}^{-2}$, we can replace
$E_{(ij)}$ with $E_{(ij)}^{(0)}$ in the above expansion.  Then, we
notice that $E_{(ij)}^{(0)}$ is always accompanied by $E_0^{(0)}$,
which motivates us to introduce a new notation as
\begin{equation}
  \Delta E_{(ij)}^{(0)} := E_{(ij)}^{(0)} - E_0^{(0)} = Q_{\mathrm{s}}^2 \bigl[
      \Gamma(|{\boldsymbol{x}_\perp}_j-{\boldsymbol{y}_\perp}_i|) + \Gamma(|{\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_j|)
    - \Gamma(|{\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_i|) - \Gamma(|{\boldsymbol{x}_\perp}_j-{\boldsymbol{y}_\perp}_j|) \bigr]\,.
\end{equation}
Now, by expanding $E_0$ and using the above relations, we can reach
the result from the large-$N_{\mathrm{c}}$ expansion up to the second order as
\begin{equation}
  \label{eq:final}
  \biggl\langle \prod_{i=1}^n D({\boldsymbol{x}_\perp}_i,{\boldsymbol{y}_\perp}_i)\biggr\rangle
     = \mathrm{e}^{-E_0^{(0)}} \Biggl\{ 1 + \sum_{i>j}^n \Biggl( 1 -
    \frac{1-\mathrm{e}^{-\Delta E_{(ij)}^{(0)}}}{\Delta E_{(ij)}^{(0)}} \Biggr)\,
    \Biggl(\frac{V_{(ij),0}^{(1)}}{N_{\mathrm{c}}}
    + \frac{V_{0,\,(ij)}^{(1)}\, V_{(ij),\,0}^{(1)}}
    {\Delta E_{(ij)}^{(0)}} \Biggr)\Biggr\}\,.
\end{equation}
This is our final expression expanded up to the $N_{\mathrm{c}}^{-2}$ order in
the MV model.

\section{Comparison to the exact answer}
\label{sec:validity}

In this section let us make a comparison between numerical results
from our expansion~\eqref{eq:final} and the exact answer.  In
particular for the $n=2$ case as we discussed around
Eq.~\eqref{eq:n2}, the full analytical expression for the dipole
Wilson line correlation is known for general $N_{\mathrm{c}}$, which provides us
with a useful benchmark to quantify the validity of the large-$N_{\mathrm{c}}$
approximation in Eq.~\eqref{eq:final}.  Because our present purpose is
to check our formulae~\eqref{eq:final}, concrete values of model
parameters are not relevant.  The coupling $g$ always appears as a
combination of $g^4 Q_{\mathrm{s}}^2$, so we can take $g=1$ without loss of
generality and change $Q_{\mathrm{s}}$.  We measure all variables in the unit of
$\bar{\Lambda}$ here.  This means, for example, $Q_{\mathrm{s}}=5$ in this
section is actually $Q_{\mathrm{s}}=5\bar{\Lambda}$, etc.

\begin{figure}
  \centering
  \includegraphics[width=0.6\textwidth]{ntwo.pdf}
  \caption{Validity check for the large-$N_{\mathrm{c}}$
    approximation~\eqref{eq:final} as compared to the analytically
    exact result~\eqref{eq:n2} for the $n=2$ dipole correlation with
    $Q_{\mathrm{s}}=5$ at ${\boldsymbol{x}_\perp}_1=(0,0)$, ${\boldsymbol{y}_\perp}_1=(0.6,0.6)$, ${\boldsymbol{x}_\perp}_2=(x_2,0)$,
    ${\boldsymbol{y}_\perp}_2=(0.3,0.3)$ (in the unit of $\bar{\Lambda}$).}
  \label{fig:ntwo}
\end{figure}

Figure~\ref{fig:ntwo} shows the validity check between
Eqs.~\eqref{eq:n2} and \eqref{eq:final} for the $n=2$ dipole
correlator with $N_{\mathrm{c}}=3$.  The agreement generally depends on ${\boldsymbol{x}_\perp}_i$
and ${\boldsymbol{y}_\perp}_i$, but our formulae~\eqref{eq:final} work quite well for
almost all ${\boldsymbol{x}_\perp}_i$ and ${\boldsymbol{y}_\perp}_i$ as long as $Q_{\mathrm{s}}$ is not too large
(when $Q_{\mathrm{s}}$ is too large, the outputs are too small, and the errors
become relatively larger).  Here, in Fig.~\ref{fig:ntwo}, we chose
$Q_{\mathrm{s}}=5$ and ${\boldsymbol{x}_\perp}_1=(0,0)$, ${\boldsymbol{y}_\perp}_1=(0.6,0.6)$, ${\boldsymbol{x}_\perp}_2=(x_2,0)$,
${\boldsymbol{y}_\perp}_2=(0.3,0.3)$, which is intentionally chosen to make the
difference as visible as possible for the small $x_2$ region, and so,
the overall agreement is better than shown in Fig.~\ref{fig:ntwo}.

\begin{figure}
  \centering
  \includegraphics[width=0.6\textwidth]{Qs.pdf}
  \caption{Validity check for the large-$N_{\mathrm{c}}$
    approximation~\eqref{eq:final} and the Abelian approximation as
    compared to the analytically exact result~\eqref{eq:n2} for the
    $n=2$ dipole correlation.  The positions are chosen as
    ${\boldsymbol{x}_\perp}_1=(0,0)$, ${\boldsymbol{y}_\perp}_1=(0.6,0.6)$, ${\boldsymbol{x}_\perp}_2=(0.1,0)$,
    ${\boldsymbol{y}_\perp}_2=(0.3,0.3)$.}
  \label{fig:Qs}
\end{figure}

We next see the $Q_{\mathrm{s}}$ dependence of the $n=2$ dipole correlator
together with the Abelian approximation, as depicted in
Fig.~\ref{fig:Qs}.  We introduce the Abelian approximation as employed
in Ref.~\cite{Dusling:2017dqg,Dusling:2017aot} so that the $n=1$
expectation value can reproduce the exact result.  For example of
the $n=2$ case, the Abelian approximation reads
\begin{equation}
  \label{eq:abelian}
  \mathrm{e}^{ -Q_{\mathrm{s}}^2 [\Gamma(|{\boldsymbol{x}_\perp}_1-{\boldsymbol{y}_\perp}_1|)-\Gamma(|{\boldsymbol{x}_\perp}_1-{\boldsymbol{x}_\perp}_2|)
    +\Gamma(|{\boldsymbol{x}_\perp}_1-{\boldsymbol{y}_\perp}_2|)+\Gamma(|{\boldsymbol{x}_\perp}_2-{\boldsymbol{y}_\perp}_1|)-\Gamma(|{\boldsymbol{y}_\perp}_1-{\boldsymbol{y}_\perp}_2|)
    +\Gamma(|{\boldsymbol{x}_\perp}_2-{\boldsymbol{y}_\perp}_2|) ] }\,,
\end{equation}
which would agree with the exact answer in the limit of
${\boldsymbol{x}_\perp}_1={\boldsymbol{y}_\perp}_1$ or ${\boldsymbol{x}_\perp}_2={\boldsymbol{y}_\perp}_2$ (in which the correlator reduces to
the $n=1$ one), but deviates from the exact answer for general
${\boldsymbol{x}_\perp}_i$ and ${\boldsymbol{y}_\perp}_i$.  In this sense, the expression like
Eq.~\eqref{eq:abelian} is to be regarded as an Abelianized
extrapolation from the $n=1$ answer.  Figure~\ref{fig:Qs} clearly
shows that a small disagreement magnified in Fig.~\ref{fig:ntwo} is
actually a negligibly small difference in the whole profile over
various $Q_{\mathrm{s}}$.  The Abelian approximation captures a qualitative
dependence with increasing $Q_{\mathrm{s}}$, while the quantitative values should
be considered as only order estimates.

\section{Flow harmonics and higher order contributions}
\label{sec:flow}

We follow the calculations of the flow observables, $v_n\{m\}$,
according to Refs.~\cite{Dusling:2017dqg,Dusling:2017aot}.  The flows
are characteristic angular distributions defined from the $m$-particle
inclusive spectra, which are in the dipole model given by
\begin{equation}
  \begin{split}
  & \frac{\mathrm{d}^m N}{\mathrm{d}^2{\boldsymbol{p}_\perp}_1 \cdots \mathrm{d}^2{\boldsymbol{p}_\perp}_m} \\
  & = \frac{1}{(4\pi^{3}B)^m} \prod_{i=1}^m \int \mathrm{d}^2{\boldsymbol{x}_\perp}_i
  \,\mathrm{d}^2{\boldsymbol{y}_\perp}_i\,\mathrm{e}^{-\frac{{\boldsymbol{x}_\perp}_i^2+{\boldsymbol{y}_\perp}_i^2}{2B} + \mathrm{i}({\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_i)\cdot{\boldsymbol{p}_\perp}_i}
  \biggl\langle \prod_{j=1}^m D({\boldsymbol{x}_\perp}_j, {\boldsymbol{y}_\perp}_j) \biggr\rangle\,,
  \end{split}
\end{equation}
where $B$ is a dipole model parameter, which is typically of the order
of the nucleon size $\sim 1\;\text{fm}\sim (0.2\;\text{GeV})^{-1}$, and we take
$\sqrt{B}=2\;\text{GeV}^{-1}$.
The general analysis for the flow properties is presented in
Ref.~\cite{Borghini:2001vi} and the $n$-th moment of the $m$-particle
correlation is introduced as
\begin{equation}
  \kappa_n\{m\} := \prod_{i=1}^m \int\frac{\mathrm{d}^2{\boldsymbol{p}_\perp}_i}{(2\pi)^2}
  \, \mathrm{e}^{\mathrm{i} n (-1)^{i+1}\phi_i}\,
  \frac{\mathrm{d}^m N}{\mathrm{d}^2 {\boldsymbol{p}_\perp}_1 \cdots {\boldsymbol{p}_\perp}_m}\,,
\end{equation}
where $\phi_i$ represents the azimuthal angle, i.e.\
${\boldsymbol{p}_\perp}_i=|{\boldsymbol{p}_\perp}_i|(\cos\phi_i,\sin\phi_i)$.  Then, in the dipole model,
we can perform the momentum integrations to find the following
expression,
\begin{equation}
  \label{eq:kappa}
  \begin{split}
    & \kappa_n\{m\} \\
    & = \frac{1}{(4\pi^{3}B)^m}  \prod_{i=1}^m
    \int\mathrm{d}^2{\boldsymbol{x}_\perp}_i\,\mathrm{d}^2{\boldsymbol{y}_\perp}_i\,
    \mathrm{e}^{-\frac{{\boldsymbol{x}_\perp}_i^2+{\boldsymbol{y}_\perp}_i^2}{2B}} K_n^{((-1)^{i+1})}({\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_i)
    \biggl\langle \prod_{j=1}^m D({\boldsymbol{x}_\perp}_j,{\boldsymbol{y}_\perp}_j)\biggr\rangle\,.
  \end{split}
\end{equation}
Here, using the regularized generalized hypergeometric function, we
defined,
\begin{align}
  K_n^{(\pm)}({\boldsymbol{x}_\perp}) &:= \int^{p_{\text{max}}} \frac{\mathrm{d}^2{\boldsymbol{p}_\perp}}{(2\pi)^2}\,
  \mathrm{e}^{\pm\mathrm{i} n \phi + \mathrm{i} {\boldsymbol{x}_\perp}\cdot{\boldsymbol{p}_\perp}} \notag\\
  &= \frac{\mathrm{i}^n p_{\text{max}}^2\,\mathrm{e}^{\pm\mathrm{i} n\theta}}{2\pi (n+2) n!}
  \biggl(\frac{|{\boldsymbol{x}_\perp}| p_{\text{max}}}{2}
  \biggr)^n \!\! {{}_1F_2} \biggl(1+\frac{n}{2};
  \Bigl\{ 1+n,\, 2+\frac{n}{2}\Bigr\};-\frac{1}{4}|{\boldsymbol{x}_\perp}|^2p_{\text{max}}^2\biggr)\,,
\end{align}
where $\theta$ is the azimuthal angle of ${\boldsymbol{x}_\perp}$.  In particular, we
will frequently use the $n=0$ function for the normalization, which is
given by
\begin{equation}
  \label{eq:K0}
  K_0^{(\pm)}({\boldsymbol{x}_\perp}) = \tilde{\delta}_{p_{\text{max}}}^{(2)}({\boldsymbol{x}_\perp})
  := \frac{p_{\text{max}}}{2\pi|{\boldsymbol{x}_\perp}|}J_1(|{\boldsymbol{x}_\perp}|p_{\text{max}})\,.
\end{equation}
It is important to notice that $\pm$ is irrelevant for $n=0$ and there
is no angular dependence any more in $K_0^{(\pm)}({\boldsymbol{x}_\perp})$.  Also, we
must point out that $\tilde{\delta}_{p_{\text{max}}}^{(2)}({\boldsymbol{x}_\perp})$ should
approach $\delta^{(2)}({\boldsymbol{x}_\perp})$ in the $p_{\text{max}}\to\infty$ limit.

Let us first consider the case with $m=2$ using our large-$N_{\mathrm{c}}$
expansion. The $N_{\mathrm{c}}^{0}$ order term in Eq.~\eqref{eq:final} does
not contribute to $\kappa_n\{2\}$ due to the phase factor in
$K_n^{(\pm)}({\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_i)$.
As a result, we can write $\kappa_n\{2\}$ using the $N_{\mathrm{c}}^{-2}$ order
results  as
\begin{equation}
  \kappa_n\{2\}=D_n^{(+-)}\,,
\end{equation}
where
\begin{equation}
  \begin{split}
  D_n^{(+-)} &:= \frac{1}{(4\pi^{3}B)^2}
  \int\mathrm{d}^2{\boldsymbol{x}_\perp}_1\,\mathrm{d}^2{\boldsymbol{y}_\perp}_1\,\mathrm{d}^2{\boldsymbol{x}_\perp}_2\,\mathrm{d}^2{\boldsymbol{y}_\perp}_2\,
  \mathrm{e}^{-\frac{{\boldsymbol{x}_\perp}_1^2+{\boldsymbol{y}_\perp}_1^2+{\boldsymbol{x}_\perp}_2^2+{\boldsymbol{y}_\perp}_2^2}{2B}} \\
  &\quad \times \mathrm{e}^{-Q_{\mathrm{s}}^2 \Gamma({\boldsymbol{x}_\perp}_1-{\boldsymbol{y}_\perp}_1)}K_n^{(+)}({\boldsymbol{x}_\perp}_1-{\boldsymbol{y}_\perp}_1)
  \,\mathrm{e}^{-Q_{\mathrm{s}}^2 \Gamma({\boldsymbol{x}_\perp}_2-{\boldsymbol{y}_\perp}_2)}K_n^{(-)}({\boldsymbol{x}_\perp}_2-{\boldsymbol{y}_\perp}_2) \\
  &\quad \times \biggl( 1-\frac{1-\mathrm{e}^{-\Delta E_{(21)}^{(0)}}}
        {\Delta E_{(21)}^{(0)}}\biggr)\biggl(\frac{V_{(21),0}^{(1)}}{N_{\mathrm{c}}}
        +\frac{V_{0,(21)}^{(1)} V_{(21),0}^{(1)}}{\Delta
          E_{(21)}^{(0)}} \biggr)\,.
  \end{split}
\end{equation}
One could think of $D_n^{(++)}$ and $D_n^{(--)}$ in a similar manner but
they are also vanishing because of the phase factors in $K_n^{(\pm)}$.  The
last part of the integrand is a function of modulus of various
combinations of ${\boldsymbol{x}_\perp}_1$, ${\boldsymbol{x}_\perp}_2$, ${\boldsymbol{y}_\perp}_1$, ${\boldsymbol{y}_\perp}_2$.  Here, it is
crucially important to understand that any term in the
integrand which is factorized into a function of $|{\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_i|$
alone would vanish due to the phase factors in
$K_n^{(\pm)}({\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_i)$ in the factorized integrations.

\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{flow.pdf}
  \caption{2-particle flow harmonics using the analytical
    (unapproximated) expression~\eqref{eq:n2} and our
    formulae~\eqref{eq:final}.  The saturation momentum $Q_{\mathrm{s}}$ is the
    one defined in Eq.~\eqref{eq:newQs} as in
    Ref.~\cite{Dusling:2017aot}.}
    \label{fig:flow}
\end{figure}

Because there is no finite contribution of disconnected parts in the two particle correlation,
we can immediately compute the two particle flow harmonics,
$v_n\{2\}$, from
\begin{equation}
  v_n\{2\} = \sqrt{ \frac{\kappa_n\{2\}}{\kappa_0\{2\}} }\,,
\end{equation}
where the denominator is obtained with Eq.~\eqref{eq:K0}, i.e.\ if we
keep using the expanded expression up to the $N_{\mathrm{c}}^{-2}$ order for
later convenience, we have
\begin{equation}
  \label{eq:k02}
  \kappa_0\{2\} = D_0^2 + D_0^{(+-)}\,.
\end{equation}
Here, we defined
\begin{equation}
  D_0 :=\frac{1}{4\pi^{3}B} \int\mathrm{d}^2{\boldsymbol{x}_\perp}\,\mathrm{d}^2{\boldsymbol{y}_\perp}\,
  \mathrm{e}^{-\frac{{\boldsymbol{x}_\perp}^2+{\boldsymbol{y}_\perp}^2}{2B}}\,
  \mathrm{e}^{-Q_{\mathrm{s}}^2\Gamma({\boldsymbol{x}_\perp}-{\boldsymbol{y}_\perp})}\,\tilde{\delta}_{p_{\text{max}}}^{(2)}({\boldsymbol{x}_\perp}-{\boldsymbol{y}_\perp})\,.
\end{equation}
Of course, up to this order, keeping $D_0^{(+-)}$ in the denominator
is in principle irrelevant since it gives a higher order correction
which we should neglect.

We summarize our numerical results in Fig.~\ref{fig:flow}.  We have
performed the 8 dimensional numerical integration with respect to
$\{{\boldsymbol{x}_\perp}_i,{\boldsymbol{y}_\perp}_i\}$ using the Monte-Carlo method by taking
$10^8$ sampling points.  To draw Fig.~\ref{fig:flow} we
chose $\bar{\Lambda}=0.241\;\text{GeV}$ and $p_{\text{max}}=2\;\text{GeV}$ in accord with
Ref.~\cite{Dusling:2017aot}.  We also note that, only in this section,
we change the definition of the saturation momentum from our original
$Q_{\mathrm{s}}$ in Eqs.~\eqref{eq:hamiltonian} and \eqref{eq:potential} to new
$\bar{Q}_{\text{s}}$ defined by
\begin{equation}
  \label{eq:newQs}
  \bar{Q}_{\text{s}}^2 = \frac{1}{2\Gamma(|\sqrt{2}/Q_{\mathrm{s}}|)}
\end{equation}
according to Refs.~\cite{Dusling:2017dqg,Dusling:2017aot}.
Since there is no confusion, in this section, we will omit bar and
simply denote $Q_{\mathrm{s}}$ to mean $\bar{Q}_{\text{s}}$.  Then, we can make a
direct comparison of our outputs to Fig.~1 of
Ref.~\cite{Dusling:2017dqg}.  The dashed curved in Fig.~\ref{fig:flow}
must precisely reproduce Fig.~1 of Ref.~\cite{Dusling:2017dqg}.  At a
glance of our numerical calculations we see quantitatively good
agreement.  The most interesting question is how useful our
large-$N_{\mathrm{c}}$ formulae~\eqref{eq:final} can be for the $N_{\mathrm{c}}=3$ case.
The comparison between the dashed curves (full analytical results) and
the solid curves (large-$N_{\mathrm{c}}$ approximations) in Fig.~\ref{fig:flow}
concludes that the errors are of only a few (at most $\sim 5$) \%
level except for the $n=3$ case.

Next, it is intriguing to see what happens for $m=4$.  In this case,
$\kappa_n\{4\}$ involves $K_n^{(+)}({\boldsymbol{x}_\perp}_1-{\boldsymbol{y}_\perp}_1)$,
$K_n^{(-)}({\boldsymbol{x}_\perp}_2-{\boldsymbol{y}_\perp}_2)$, $K_n^{(+)}({\boldsymbol{x}_\perp}_3-{\boldsymbol{y}_\perp}_3)$, and
$K_n^{(-)}({\boldsymbol{x}_\perp}_4-{\boldsymbol{y}_\perp}_4)$.  It is then easy to understand that our
formula of the $N_{\mathrm{c}}^{-2}$ order in Eq.~\eqref{eq:final} is
insufficient to get nonvanishing contributions.  Terms of the
formula~\eqref{eq:final} are functions of, say, ${\boldsymbol{x}_\perp}_1$, ${\boldsymbol{x}_\perp}_2$,
${\boldsymbol{y}_\perp}_1$, ${\boldsymbol{y}_\perp}_2$ for $i=2$ and $j=1$, and then the angle
integrations of ${\boldsymbol{x}_\perp}_3-{\boldsymbol{y}_\perp}_3$ and ${\boldsymbol{x}_\perp}_4-{\boldsymbol{y}_\perp}_4$ become zero.
Therefore, one permutation is not enough, but two permutations are
necessary to shuffle all ${\boldsymbol{x}_\perp}_1$, ${\boldsymbol{x}_\perp}_2$, ${\boldsymbol{x}_\perp}_3$, and ${\boldsymbol{x}_\perp}_4$;
we must go to the next $N_{\mathrm{c}}^{-4}$ order to compute a first nonzero
term in $\kappa_n\{4\}$.

It is a straightforward but tedious calculation to pick all the
$N_{\mathrm{c}}^{-4}$ order terms up from the expansion in Eq.~\eqref{eq:NNLO}.
We can slightly simplify the problem by discarding terms which do not
contribute to $\kappa_n\{4\}$.  The $N_{\mathrm{c}}^{-4}$ order terms generally
contain the product of the interaction matrix elements like
\begin{equation}
  \sim V_{0,(pq)}^{(1)} V_{(pq),(ij)(kl)}^{(1)}
  V_{(ij)(kl),(ij)}^{(1)} V_{(ij),0}^{(1)}\,,
\end{equation}
but we already saw that, if an unexchanged pair exists, the angle
integration is vanishing.  For example, if the above matrix elements
are $\sim V_{0,(21)}^{(1)} V_{(21),0}^{(1)} V_{0,(21)}^{(1)}
V_{(21),0}^{(1)}$, which itself is nonzero, the angle integrations of
${\boldsymbol{x}_\perp}_3-{\boldsymbol{y}_\perp}_3$ and ${\boldsymbol{x}_\perp}_4-{\boldsymbol{y}_\perp}_4$ are zero.  Thus, among all
possible combinations of the matrix elements, there are finite
contributions only from
\begin{equation}
  \label{eq:ijkl}
  \begin{split}
  & V_{0,(43)}^{(1)} V_{(43),(0)}^{(1)}
  V_{(0),(21)}^{(1)} V_{(21),0}^{(1)}\,,\qquad
  V_{0,(42)}^{(1)} V_{(42),(0)}^{(1)}
  V_{(0),(31)}^{(1)} V_{(31),0}^{(1)}\,,\\
  & V_{0,(32)}^{(1)} V_{(32),(0)}^{(1)}
  V_{(0),(41)}^{(1)} V_{(41),0}^{(1)}\,.
  \end{split}
\end{equation}
Here, we used relations such as
$V_{(ij)(kl),(ij)}^{(1)}=V_{(kl),0}^{(1)}$,
$V_{(kl),(ij)(kl)}^{(1)}=V_{0,(ij)}^{(1)}$, etc for $(ij)\neq(kl)$.
After long calculations we arrive at a final form which turned out to
be factorized as
\begin{align}
  \kappa_n\{4\} &= \frac{1}{(4\pi^{3}B)^4}
  \prod_{i=1}^4 \int\mathrm{d}^2{\boldsymbol{x}_\perp}_i\,\mathrm{d}^2{\boldsymbol{y}_\perp}_i\,
  \mathrm{e}^{-\frac{{\boldsymbol{x}_\perp}_i^2+{\boldsymbol{y}_\perp}_i^2}{2B}}\,
  \mathrm{e}^{-Q_{\mathrm{s}}^2 \Gamma({\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_i)}K_n^{((-1)^{i+1})}({\boldsymbol{x}_\perp}_i-{\boldsymbol{y}_\perp}_i) \notag\\
  &\quad \times \frac{1}{2!}\sum_{(ij)\neq(kl)}
  \biggl( 1-\frac{1-\mathrm{e}^{-\Delta E_{(ij)}^{(0)}}}
        {\Delta E_{(ij)}^{(0)}}\biggr)\biggl(\frac{V_{(ij),0}^{(1)}}{N_{\mathrm{c}}}
        +\frac{V_{0,(ij)}^{(1)} V_{(ij),0}^{(1)}}{\Delta
          E_{(ij)}^{(0)}} \biggr) \notag\\
  &\qquad\qquad\quad\; \times \biggl( 1-\frac{1-\mathrm{e}^{-\Delta E_{(kl)}^{(0)}}}
        {\Delta E_{(kl)}^{(0)}}\biggr)\biggl(\frac{V_{(kl),0}^{(1)}}{N_{\mathrm{c}}}
        +\frac{V_{0,(kl)}^{(1)} V_{(kl),0}^{(1)}}{\Delta
          E_{(kl)}^{(0)}} \biggr)\,,
\end{align}
which is the full expression of the $N_{\mathrm{c}}^{-4}$ order without any
truncation like the glasma graph approximation.  This result looks
quite reasonable, but we emphasize that the complete cancellation of
$(ij)(kl)$ intermediate states with an energy denominator,
$E_{(ij)}+E_{(kl)}$, is far from trivial.
The sum with respect to $(ij)$ and $(kl)$ should run over
all the permutations of the combinations as listed in
Eq.~\eqref{eq:ijkl}.  Among all the combinations of indices,
$[(ij)=(31), (kl)=(42)]$ and $[(ij)=(42), (kl)=(31)]$ are irrelevant
because $D_n^{(++)}=D_n^{(--)}=0$ (for $n>0$) as we already pointed
out.  Therefore, the remaining four combinations of
$[(ij)=(21), (kl)=(43)]$, $[(ij)=(43), (kl)=(21)]$,
$[(ij)=(41), (kl)=(32)]$, and $[(ij)=(32), (kl)=(41)]$ lead to
\begin{equation}
  \kappa_n\{4\} = 2 \bigl[ D_n^{(+-)} \bigr]^2\,.
\end{equation}
Then, with extra terms corresponding to $D_0^{(++)}$ and $D_0^{(--)}$
which are nonzero, the normalization is written as
\begin{equation}
  \kappa_0\{4\} = D_0^4 + 6D_0^2 D_0^{(+-)}\,,
\end{equation}
up to the $N_{\mathrm{c}}^{-2}$ order in the same way as in Eq.~\eqref{eq:k02}.
Now, the cumulant is then given by
\begin{equation}
  c_n\{4\} = \frac{\kappa_n\{4\}}{\kappa_0\{4\}}
  - 2 \biggl(\frac{\kappa_n\{2\}}{\kappa_0\{2\}}\biggr)^2\,.
\end{equation}
The above quantity itself is zero in the strict order counting for
$c_n\{4\}$ up to $N_{\mathrm{c}}^{-4}$.  Thus, our conclusion is, even in the
full MV model beyond the glasma graph approximation, no connected
cumulant remains for $c_n\{4\}$ at the $N_{\mathrm{c}}^{-4}$ order.

In this way we can understand that the first connected contribution to
cumulants appears from the $N_{\mathrm{c}}^{-2m+2}$ order;  for $m=4$, thus, we
need to go to the $N_{\mathrm{c}}^{-6}$ order and then a completely nested
combination of four indices like $(43), (32), (21)$ is possible.
Therefore, the flow harmonics from the fully nested permutations must
scale as
\begin{equation}
  \label{eq:scaling1}
  v_n\{m\} \sim N_{\mathrm{c}}^{-2+2/m}\,.
\end{equation}
This is our conclusion on the $N_{\mathrm{c}}$ scaling in the full MV model.

\begin{figure}
  \centering
  \includegraphics[width=0.6\textwidth]{flowdelta.pdf}
  \caption{$\delta v_n\{4\}$ estimated from Eq.~\eqref{eq:delta} using
    $v_n\{2\}$ obtained in the large-$N_{\mathrm{c}}$ limit.  Our choice of the
    parameter is $p_{\text{max}}=2\;\text{GeV}$.}
    \label{fig:flowdelta}
\end{figure}

Although our conclusion of $c_n\{4\}=0$ at the $N_{\mathrm{c}}^{-4}$ order is
solid, there may be a subtle point in the large-$N_{\mathrm{c}}$ scaling due to
finiteness of $p_{\text{max}}$.  If
one just uses our formulae up to the $N_{\mathrm{c}}^{-4}$ order to estimate
$\kappa_n\{2\}$, $\kappa_0\{2\}$, $\kappa_n\{4\}$, and
$\kappa_0\{4\}$, one would na\"{i}vely find that the cancellation is
incomplete at the $N_{\mathrm{c}}^{-6}$ order and a finite remainder is given by
\begin{equation}
  \delta c_n\{4\} = \frac{2[D_n^{(+-)}]^2}{D_0^4+6D_0^2D_0^{(+-)}}
  -2\biggl(\frac{D_n^{(+-)}}{D_0^2+D_0^{(+-)}}\biggr)^2
  \simeq -\frac{8[D_n^{(+-)}]^2 D_0^{(+-)}}{D_0^6}
  \simeq -\frac{8D_{0}^{(+-)}}{D_{0}^{2}}
  (v_n\{2\})^4\,.
\end{equation}
This correction is beyond the $N_{\mathrm{c}}^{-4}$ order and should be
identified as a part of the $N_{\mathrm{c}}^{-6}$ terms.  One can easily check
that this $N_{\mathrm{c}}^{-6}$ order correction is vanishing for
$p_{\text{max}}\to\infty$, in which $D_0^{(+-)}\to 0$.  However, for a finite
$p_{\text{max}}$ especially comparable to $Q_{\mathrm{s}}$, the correction could be
sizable and the flow harmonics is modified even from the disconnected
piece.  To see this effect quantitatively, let us compute the flow
harmonics corresponding to $\delta c\{4\}$, which leads to
\begin{equation}
  \label{eq:delta}
  \delta v_n\{4\} = (-\delta c_n\{4\})^{1/4} =
  2^{3/4}  \biggl(\frac{D_{0}^{(+-)}}{D_{0}^{2}} \biggr)^{1/4}
  v_n\{2\} \,.
\end{equation}
We make a plot of $\delta v_n\{4\}$ as a function of $Q_{\mathrm{s}}$ in
Fig.~\ref{fig:flowdelta}.  One must be very careful of the physical
interpretation of this correction by $\delta v_n\{4\}$.  Even though
this is non-negligible as seen in Fig.~\ref{fig:flowdelta}, the
physical origin lies in not the connected correlator but in the
normalization.  Moreover,  this normalization effect makes the $N_{\mathrm{c}}$
counting skewed to become
\begin{equation}
  \label{eq:scaling2}
  \delta v_n\{m\} \sim N_{\mathrm{c}}^{-1-2/m} \qquad (m\ge 4)\,,
\end{equation}
which starts differing from anticipated Eq.~\eqref{eq:scaling1} for
$m>4$.  Thus, to distinguish the connected contribution from the
normalization effect, one can test the $N_{\mathrm{c}}$ scaling properties as in
Eqs.~\eqref{eq:scaling1} and \eqref{eq:scaling2} and also check the
$p_{\text{max}}$ dependence since $\delta v_n\{m\}$ from the normalization
effect is very sensitive to $p_{\text{max}}/Q_{\mathrm{s}}$ as perceived from
Fig.~\ref{fig:flowdelta}.

\section{Conclusions}
\label{sec:conclusions}

We have established general formulae and machineries to compute
$2n$-point Wilson line (or $n$ dipole) correlators with the color
group representation $(N_{\mathrm{c}}\otimes \bar{N_{\mathrm{c}}})^{n}$ in the
McLerran-Venugopalan model.  The color structure accommodates a huge
representation but the nonzero contribution to the Wilson line
correlators reduces to the $n!\times n!$ matrix, whose bases
correspond to the color singlets.  In particular, we have derived the
explicit expression of the matrix elements [see Eqs.~\eqref{eq:Vpp}
  and \eqref{eq:Vpijp}] in the color singlet bases constructed by
permutations.  The formulae are quite powerful not only for the
direct numerical evaluation of the matrix but also for the analytical
large-$N_{\mathrm{c}}$ expansion.  We have developed the systematic large-$N_{\mathrm{c}}$
expansion in a way analogous to time-dependent perturbation theory in
quantum mechanics.  We have then shown the explicit expression up to
$N_{\mathrm{c}}^{-2}$ order for the dipole correlators as given in
Eq.~\eqref{eq:final}.  As a check of the validity, we have compared
results from the exact answers and those in the large-$N_{\mathrm{c}}$ expansion
for the two-particle flow harmonics,
$v_{n}\{2\}$ $(n=2,3,4,5)$, which shows quantitatively good
agreement.  We have continued our large-$N_{\mathrm{c}}$ expansion to higher
orders to discuss the flow harmonics with more particles.  Then, we
have discovered the $N_{\mathrm{c}}$ scaling as $v_n\{m\} \sim N_{\mathrm{c}}^{-2+2/m}$ even
beyond the glasma graph approximation but in the full MV model.  We
have also pointed out that a slightly different $N_{\mathrm{c}}$ scaling could
emerge from the normalization effect at finite cutoff of the
transverse momenta of integrated particles.

Although we focused on only the dipole correlators in the present
paper, our general formulae also provide us with useful approaches to
evaluate Wilson line correlators in channels relevant for the particle
production rate in the $p$-$A$ collision generally.  In fact, not only
fundamental but also adjoint Wilson line correlators appearing in the
multi-gluon production can be derived from our results in
Eq.~\eqref{eq:Wilson} through the relation
$[U_{\text{adj}}]_{ba}({\boldsymbol{x}_\perp}) = 2\mathrm{tr} [U({\boldsymbol{x}_\perp}) {T_{\mathrm{F}}}^{a} U^{\dag}({\boldsymbol{x}_\perp})
{T_{\mathrm{F}}}^{b}] = {T_{\mathrm{F}}}_{\alpha\bar{\alpha}}^{a}{T_{\mathrm{F}}}_{\bar{\beta}\beta}^{b}
U_{\beta\alpha}({\boldsymbol{x}_\perp})U_{\bar{\beta}\bar{\alpha}}^{*}({\boldsymbol{x}_\perp})$.

As we emphasized, our scheme of the large-$N_{\mathrm{c}}$ expansion takes a nice
form which is easily implemented in numerical algorithms to go to
arbitrarily higher orders.  Such higher order numerical evaluations
are left as an intriguing future problem.  It would be also an
important question to think about generalizations beyond the MV
model.  Further systematic considerations on the Wilson line
correlators should deserve more investigations in the future.

\acknowledgments

The authors thank
Kevin~Dusling,
Mark~Mace,
S\"{o}ren~Schlichting,
Vladimir~Skokov,
and Raju~Venugopalan for discussions.
This work was supported by Japan Society for the Promotion of Science
(JSPS) KAKENHI Grant No.\ 15H03652, 15K13479 and 16K17716.

\bibliographystyle{JHEP.bst}

redpajama_set_name: RedPajamaArXiv


text:
Innovation & Change
Government Technology
Digital Customer Service
5G & Mobile
Gigabit & Fibre
People & Processes
Future Workforce
Innovation and change
People and processes
Service deliveryDigital Customer ServiceInterview: How Bristol is leading by example
Interview: How Bristol is leading by example
How has Bristol's use of digital technology helped it become Britain's leading smart city?
Austin Clark
Date published January 31, 2018 Categories
Digital by Default News talks to Barney Smith, Founder of Perform Green, Programme Director for Bristol City Council's Smart Operations Centre through to launch, and Interim CEO for Bristol is Open until November 2017, about the city's use of digital technology and its journey to become Britain's leading smart city.
Last November, Bristol was named as the UK's leading smart city in the second UK Smart Cities Index. The report is based on evaluations of 20 cities and their strategies, key projects and general readiness in using digital technology to improve crucial civil services from transport to healthcare.
Perform Green has been directly involved with the development of a digital change strategy that has helped Bristol take the top spot. By working with Bristol City Council on solutions that will help the people of Bristol, while making cost savings and developing new revenue streams for the city, the partnership has undoubtedly been a success.
At the heart of Bristol's development as a leading smart city is its new Smart Operations Centre, which publicly launched in October 2017. The centre plays an important role in managing the daily safety, accessibility and flow of the city through to its traffic control, incident management, community safety (CCTV) and telecare services.
The centre now offers a world class cooperative and collaborative workspace, bringing council and city organisations together to work in a truly integrated manner for the first time. Perform Green has driven the programme from construction and build of the space, to the delivery and integration of the people, technology and services.
Barney Smith, Founder of Perform Green, was also Programme Director for Bristol City Council's Smart Operations Centre through to launch. He, and is Interim CEO for Bristol is Open, the joint venture between Bristol City Council and University of Bristol that has provided large scale infrastructure for R&D, allowing testing of future technologies at a citywide scale. When asked what's next for Bristol, he told us: "Now that the Operations Centre is up and running – and the council is realising significant financial and operational benefits – we have a platform onto which we can build additional services and, I think, transform the way public services in some of the most difficult and crucial areas councils are dealing with. In fact, we've helped the council develop phase 2 of the Operations Centre, which received approval in December."
There are two elements to this second phase:
CCTV: The largest element of phase two is an upgrade to the council's CCTV estate, moving to an IP-based system that is cheaper to run and requires fewer cameras. It also has a high degree of visual analytics built in, which means a lot more can be processed at the back-end, prior to operator intervention and incident management. Overall operational costs are reduced, and operators can focus on incidents, not just trying to identify which camera saw the incident.
Telehealth: The second element revolves around telehealth, working with the local NHS (Sustainability and Transformation Plan). The Council has telecare and is currently invited into homes. People also have the ability to alert the Operations Centre when they are in difficulty.
"With the predictive analytics platform that we've built in the Operations Centre, we can monitor and respond and build those telecare services into much more of a predictive operation," explains Barney. "We can tell in advance where somebody is likely to have a problem and intervene. That could be a mental health issue, a high likelihood of a fall at some point in the future or some other deviation from the norm. "Clearly this happens with the individual's approval and we wouldn't just do it. They will tell us they want this support and service, so they can stay in the home. This allows the adult social care service to be delivered much more effectively and efficiently. It also allows providers to give targeted intervention that will really help the individual.
This idea also flows through to smart meters – monitoring energy usage to see if there's deviation from the norm and to use that smart meter as an IoT hub for extra sensors in the home, whether that's temperature sensors or humidity measurements.
"If somebody's heating fails and the temperature drops below a certain temperature point, there's a risk to health and we can intervene," explains Barney. "If we can get those sensors integrated with other digital health care monitors and programmes where they might be monitoring characteristics of their health, such as blood sugar levels for diabetes there are obvious benefits. It's about healthcare and adult social care being much more integrated."
Bristol leading the way
Given its ranking at the top of the Smart City Index and its successful ongoing implementation of digital technology, Bristol is clearly leading the way when it comes to digital transformation. Why does Barney think that is the case?
"I think a number of factors have come together at the same time in a rather positive way. The leadership, vision and capability to deliver have been available at the same time.
"There are lots of councils – and we're talking to several – who see the potential, but they're at different points in their journey or have different levels of understanding of what the potential is. Therefore, they're all in slightly different places in their transformation. Yes, Bristol is ahead of other local authorities and that's helped by the fact the city has a very good tech community and people welcome innovation. In many ways it's the ideal scenario. Innovation breeds innovation – because Bristol has invested in other programmes such as the Operations Centre, it's well positioned to take advantage of technology and integration of services."
Barney does say this trailblazing position does mean Bristol has had to learn rapidly.
"Because Bristol has been ploughing the path for everyone else, it has had to learn the lessons that have made it succeed. And that's tough.
"At Perform Green we've been through the journey with Bristol and other authorities and now can see the common pitfalls and opportunities, as well as the ways that step can be leapfrogged. That's critical right now because local government and NHS finances are at the point where they can't afford people to muck around. They can't afford the services they're delivering today and the demands on those services are growing faster than the budgets can support. In many cases the budgets are going in the opposite direction.
"Bristol, because they're further along their journey, should be used as an example of what's possible and where authorities can go. Other councils should be looking to 'borrow' this investment and align what Bristol is doing in their areas."
Overcoming fear of change
Many local authorities are reluctant to invest in digital transformation because they are struggling to justify the spend, can't see a return on investment or can't overcome the fear associated with changing service delivery. Barney, however, says this shouldn't be the case.
"When it comes to return on investment, digital projects really shouldn't be seen in isolation. Quite honestly, Bristol City Council put the investment into the Operations Centre partly on the back of its estate rationalisation programme. If they didn't build the Operations Centre, they wouldn't have been able to vacate one building and sell it for £1m plus. They also wouldn't have been able to vacate another building and release a £1.5m leasehold. That makes the investment easy to justify.
"Other benefits are there too. For example, when you start combining services from across control rooms and integrate them together, you start balancing your peak workload. Different services have different peaks in demand, so by combining services you can flatten your supply, reducing average overall staffing levels while delivering a better service."
Creating opportunities
With most transformation journeys, success often leads to success – something that Barney says can definitely be the case for councils.
"Once you can deliver your service more efficiently you can start to do things you didn't do before," he says. "Take healthcare as an example. If you can start bringing together your adult social care services with NHS services such as GP out of hours, you can start to achieve better outcomes for the person calling the care line or the out of hours service. There are clearly efficiencies to be made by bringing them together. It's worth pointing out here that it's not just cities that can benefit from the Bristol model – all local authorities can achieve efficiencies, regardless of geography.
"A crucial point I would like to make, coming back to the idea of fear, is that when somebody has achieved something it becomes easier for somebody else to come along and achieve the same thing or similar. It's much easier to get buy-in for a project or set up the project in the right way if you can clearly demonstrate what has already been successful. In the current climate it's difficult for local councils to see through the fear of investment."
Barney continues: "Digital leaders need to ask a number of key questions. How can I reimagine how my organisation and service delivery operates? How can I drive efficiencies while protecting citizen outcomes? And, how can I improve citizen outcomes? They important questions, but they're really difficult questions to answer. Particularly when you're on the inside, living it every day. That's where external organisations or other local authorities can help – they can help those responsible for change to make sense of the situation and see where the maximum opportunities are.
"Take Bristol – the Operations Centre is just one element of the outfit. You still need your operators and your sensors. How are you getting your information and data in? What's the connectivity? How are you processing it and analysing the data? And, then, how does the call centre and predictive analytics drive you to the right interventions, at the right time, in the most efficient way? Digital transformation has many tentacles – and that's what Bristol has realised and got right – and will benefit from going forward."
Barney finishes by saying: "The long-term legacy for the people of Bristol will be to improve their day to day life. From the congestion on the roads or to air quality and healthcare provision, this can all be radically improved. The recent Operations Centre launch is the very start of the journey for the city. I am extremely excited about some of the practical ways in which technology will inspire a better living environment for everyone."
Discussing the future of healthcare – part 1
Data Insight Discussing the future of healthcare – part 1
4y Austin Clark
Driving data into the DNA of policing
Data Insight Driving data into the DNA of policing
4y Boyd Mulvey
DASA seeks ideas for open data sharing solutions
Data Insight DASA seeks ideas for open data sharing solutions
4y Jay Ashar
Data on parking spaces set to be standardised
Data Insight Data on parking spaces set to be standardised
Open-source data initiative to help people get active
Data Insight Open-source data initiative to help people get active
Using space technologies and satellite data to deliver world-class services
Data Insight Using space technologies and satellite data to deliver world-class services
Digital innovation for booking fitness and sport activities
Data Insight Digital innovation for booking fitness and sport activities
Data Exploration Licence makes geospatial data more accessible
Data Insight Data Exploration Licence makes geospatial data more accessible
Register to the latest news & insights
Innovation and changekeyboard_arrow_down
Service deliverykeyboard_arrow_down
Data and securitykeyboard_arrow_down
Digital infrastructurekeyboard_arrow_down
People and processeskeyboard_arrow_down
Copyright © 2023 GovTech Leaders

redpajama_set_name: RedPajamaCommonCrawl


text:
Álvaro Iglesias
Eva and the cities DC
A New York story
Ibed Flex
Birthplace of dreams
After completing his university career and after a few years as editor and production assistant on some TV channels (Antena 3, Expansion channel ...) he directed 4 documentaries in the Palestinian refugee camps in Lebanon, Bosnia and Latin America collecting several mentions and Awards at various festivals. During that stage collaborated and worked with the producer Emilio González Martín in several cinema and documentary festivals developing different activities for the company of AVID edition. (Play Doc, Documenta Madrid, Doc Looks ...) In 2010 he began his work as an advertising director with the aim of connecting, seducing and bringing new credibility and personality to brands through film, photography and communication. During these years he has worked for brands and agencies that allowed him to develop his work as a creator with total freedom. Ford, Starwood Hotels & Resorts, Meliá, Iberostar, Lottusse, Vein Magazine, Kaiku, Palma Pictures, Hoss Intropia, Design by Atlas ... In 2010 he founded the international film festival MareMostra, which he directed during his first four editions.
Jordi Estrada
Santi Winer
Back to directors


In this business, time is critical. Ultra-Fab's experience with stainless fabrication means your job goes from blueprints to actual product without delays.
With our machining capabilities we are able to closely monitor the quality of work performed on sub-assembly components. This is critical to the structural integrity of finished products.
Attention to detail and experienced polishers is the key to a proper finish. Our metal finishers set high standards for their work and it shows.
Jet Edge waterjet technology is ideal for cutting heat-sensitive material, and costs due to thermal distortion of machine parts are eliminated.
Risk of fatigue failure resulting from process induced thermal stressing and associated material structural change is minimized.
The practical elimination of airborne dust particles, smoke, fumes and contaminants from cutting materials such as asbestos and fiberglass greatly improves the work environment and reduces problems arising from operator exposure.
Selection of appropriate cutting parameters such as operating pressure, material feed rate, abrasive particle size and flow rate can eliminate the need for secondary operations. Jet Edge systems also reduce machining time and production costs.
USDA approved equipment designs promote the use of ultra-high water pressure jet streams in the sanitary cutting of food products. This results in higher cutting speeds and the elimination of associated downtime incurred for cleaning and resharpening delicate blades.
Jet Edge systems reduce fixturing costs plus avoid excessive stress to finished parts. Also, minimal lateral force simplifies material handling and allows cutting close to material edges.
With typical orifice diameters in the range of .003 - .030 inches, waterjet kerf width is kept to a minimum, thus maximizing material utilization.
Waterjet cut profiles are not confined to straight line geometries. Part complexity is limited only by the motion control system specified and traditional start holes are eliminated.
WaterJet is the most flexible and effective cutting and cleaning solution available for a variety of industrial needs. Abrasive Jet cutting uses a combination of water and garnet to cut through materials considered Unmachinable by conventional cutting methods. Contact Ultra-Fab today and discover how we can put this system to work for you.
Our tradesmen know the importance of a proper weld and more importantly have the know-how to do it right.
Our growing CNC Manufacturing and Engineering Department has 2 Matsuura MC-800 VF vertical machining centers. Let us process, engineer and manufacture your CNC Mill work, whether it is a one piece or 500 piece order, Ultra-Fab, Inc. has got you covered. We design, and build our own fixture and work holding in house and we use Auto Desk along with Feature CAM as our CAD/CAM system to ensure that high quality and perfection is met at all times.
The "Formula 62" process is a resonance based method of vibratory stress relief. Vibratory stress relief is often referred to as VSR or, in it's more generic form, also referred to as vibration conditioning or vibratory weld conditioning (VWC). Workpieces are subjected to low frequency vibrations for a short period of time either during welding or after welding and at various times during machining to promote dimensional stability.
The process is quick, clean and very energy efficient. It is designed as a shop tool for use by your personnel and adaptable to most manufacturing environments.


Julie Fison is the author of eleven books for children and young adults. Her books include the Hazard River series for young adventure lovers and titles in the Choose Your Own Ever After series for girls aged 10 – 14. Julie has also written three books for young adults. The latest, Counterfeit Love, about an ambitious young television reporter trying to make a name for herself in Hong Kong, was inspired by Julie's own time in Asia.
My first foray into fiction was inspired by a family holiday on the beautiful Noosa River in Queensland. My two sons teamed up with friends and spent the summer dodging stingrays, exploring the bush and building secret camps. I had to write about it.
By the end of the summer I had written two books about a gang of kids holidaying on Hazard River, coming up against rogue fishermen, smugglers and dodgy developers. The stories were full of fun and adventure, with a subtle message about threatened wildlife. My kids loved them. I just hoped that I could find a publisher who felt the same way.
I had my fair share of rejections, but I had enough encouraging feedback to continue writing about the gang at Hazard River. A year or so later I met Ford Street Publishing's Paul Collins at a book launch. I used the opportunity to subtly sidle up to Paul and pitch my books. (Paul tells a different version – apparently I wasn't too subtle!) A month later I had a four-book contract. Less than two years later Ford Street Publishing had released six books in the Hazard River series – Shark Frenzy, Tiger Terror, Bat Attack, Snake Surprise, Toads' Revenge and Blood Money.
My most recent books are for girls. I've written two stories for the Choose Your Own Ever After series (Hardie Grant Egmont) – How to Get to Rio and The Call of the Wild. The books let the reader decide how the story goes – with girlfriend dramas, boy troubles, family fun and plenty of decisions along the way.
In How to Get to Rio schoolgirl Kitty McLean has to decide whether to go camping with her old friends or spend the holidays at an exclusive beach resort with her new friend. The big decision for nature-loving Phoebe in The Call of the Wild is whether to go to a party with her besties or help out at the save-the-orangutan fundraiser.
The emphasis is on fun, but the stories present realistic dilemmas for girls to consider. For every choice there's a consequence. But unlike other stories in the choose-your-own-adventure genre, no one dies in this series when they make a bad choice!
I write best when I'm sitting at my desk. I don't like distractions of any kind – which is very strange after working in a noisy, smoky newsroom.
Once I have an idea in my head I write quickly. I settle myself in front of my computer and pound away for as much of the day as I can. I forget to pick up the kids from school, I ignore requests for dinner and Molly, the dog, looks on forlornly waiting for some attention. When I need a break to recharge my ideas I dash around getting chores done and take Molly for a walk. I find walking, bike-riding and napping extremely useful for getting through any kind of block.
I always have an idea of where a story will start and finish before I begin writing. The details of the middle section get worked out as I go along. The Choose Your Own Ever After series was an exception. Each story has eight possible endings and various other choices along the way, so there were a lot of threads to tie in. I couldn't just wing it with those stories; the plots had to be meticulously planned.
I get my ideas from everywhere – my own misadventures, things that happen to my children, stories I read in the paper, conversations I overhear. Everything! Places that I love also feature heavily in my stories. Noosa is the setting for my first YA novel – Tall, Dark and Distant. The magical ruined city of Angkor, in Cambodia, is the backdrop for Lust and Found, while Counterfeit Love takes place in my old stomping ground – Hong Kong.
Deciding what to write can be very difficult. It's very frustrating to have a head full of ideas and no idea of which one is going to work out.
I'm in awe of authors who can resuscitate history – Hilary Mantel, of course, and Geraldine Brooks. I also think Hannah Kent's debut novel Burial Rights is amazing and I loved Toni Jordan's Nine Days.
Thank you for having me on WordMothers, Nicole!


• Where a person proposes to make an investment in an approved business of at least US$ 1,500,000 on their own behalf.
• At least two persons propose to make a joint investment in such an approved business totalling at least US$ 5,000,000, and each of those persons individually propose to contribute at least US$400,000 to the joint investment an application or application for Citizenship by Investment may be submitted on his, her or their behalf through an agent.
• Visit Antigua and Barbuda to collect your passport and to take the oath or affirmation of allegiance.
• Visit an Embassy, High Commission or Consular Office of Antigua and Barbuda to collect your passport and to take the oath or affirmation of allegiance. Link to Embassies/High Commissions/Consular Offices shown on an alternate page.


text:
Posts Tagged 'John Sayles'
Monday Picks: Piranha (1978)
by Douglas Barnett, Aug 6 2012 // 9:00 AM
This week's Monday Pick comes to us from the wonderful world of the king of Independent cinema, Mr. Roger Corman, who produced the low-budget version of Jaws, and that movie was the Joe Dante classic Piranha (1978). Cashing in on the "animals run amok" craze of the late 1970s, Dante and screenwriters John Sayles and Richard Robinson crafted a truly unique horror film that quickly gained a huge cult following.
The film centers around a young woman named Maggie McKeown (Heather Menzies) who is hired to find two missing teenagers who were hiking in the vicinity of Lost River Lake. Maggie enlists the help of a backwoods drunk named Paul Grogan (Bradford Dillman) to be her guide and to help her find the two missing teens. Paul and Maggie's search brings them to an abandoned military installation with a giant pool filled with salt water. Maggie and Paul enter the facility and find a series of mutant specimens and preserved experiments. Maggie finds the master control in order to drain the pool and see whether or not the teens possibly drowned. Maggie pulls the lever and is suddenly attacked by the facility's caretaker who is knocked unconscious, as he was trying to stop the pool from draining.
Maggie and Grogan take the man back to Grogan's cabin where he has sustained further injuries after he crashed Maggie's jeep in an attempt to flee. The man is tied down to a bed and is frantic when he awakens and learns that the pool had been drained. Grogan decides to take the injured man down river in a homemade raft. As the three trek down river, the man introduces himself as Dr. Robert Hoak (played by Dante alumni Kevin McCarthy) a military scientist who explains that the pool was filled with genetically engineered piranhas for a project codenamed Operation: Razorteeth, a plan which was to introduce the deadly strain of fish that could survive in the coldwater river systems of North Vietnam.
Posted in: Blu-Ray · Comedy · Cult Cinema · Directors · DVD · DVD Reviews · Horror · Monday Picks · Movies
Tagged: Barbara Steele, Belinda Balaski, Bradford Dillman, Dick Miller, Heather Menzies, Joe Dante, John Sayles, Keenan Wynn, Kevin McCarthy, Paul Bartel, Peter Kuran, Phil Tippett, Rob Bottin, Roger Corman
Monday Picks: 'Alligator'
by Douglas Barnett, Nov 14 2011 // 1:30 PM
This week's Monday Pick is the classic "animals run amok" monster movie Alligator (1980). The film stars Academy Award nominee Robert Forster, Robin Riker, Michael Gazzo, and Henry Silva.
Alligator is a well written, tongue in cheek horror film centered on the old urban legend of alligators that are flushed down the drain and thrive in the subterranean jungle of the city sewer. The film was penned by writer/director John Sayles who helped write the 1978 Joe Dante Jaws homage Piranha and would also help to write the upcoming release of The Howling. Sayles crafts another horror classic that pits man against a marauding beast with an insatiable appetite.
Robert Forster stars as policeman David Madison who has been investigating some grizzly murders committed in the sewers of a Missouri city. Madison begins investigating further after a pet store owner's remains ended up in the city's sewage treatment facility. One other baffling discovery is a women's small dog that appears to have grown in size substantially after it was reported missing.
Posted in: Academy Awards · Cult Cinema · DVD · Horror · Lionsgate · Monday Picks · Movies · Netflix · Prequels and Sequels · Reviews
Tagged: Henry Silva, James Ingersoll, John Sayles, Lewis Teague, Michael Gazzo, Perry Lang, Robert Forster, Robin Riker


text:
WaterWealth campaign director Ian Stephen wants Kinder Morgan to alter the route of the Trans Mountain oil pipeline twinning project off the city's aquifer. (Jenna Hauck/ The Progress file)
Court's pipeline decision leaves questions for Chilliwack aquifer protection
City councillor Jason Lum agrees with court decision that found consultation is not a one-way street
With the fate of the Trans Mountain oil pipeline expansion (TMX) project at the very least uncertain, the local fight against routing the flow of more oil so close to Chilliwack drinking water source may now be moot.
But for one of the fiercest advocates of protecting the city's aquifer, WaterWealth Project campaign director Ian Stephen, the irony is that if the project is killed, that may make it harder to deal with his hope to move the existing 65-year-old oil pipeline.
"For us here in Chilliwack, we still have this antique pipeline sitting on top of our aquifer," Stephen said Thursday after the announcement of the unanimous decision by the Federal Court of Appeal overturning the Trudeau government's approval of the pipeline twinning project.
"My fear with the old pipeline was they were needing repairs and they were waiting for the new construction. But those repairs will still have to happen."
• READ MORE: NEB approves Trans Mountain pipeline route through Chilliwack residential area, school yard
Stephen said he was actually surprised by the decision Thursday, but maybe because his head was down focusing on presentations to the NEB for the detailed route hearings expected to begin in September. He was unclear immediately if those hearings will still go ahead or be quashed long with Ottawa's approval.
"It's disappointing that we don't get the opportunity to change the route," Stephen said. "That's an old pipeline to be sitting right next to the city wells."
The case decided Thursday involved a number of First Nations opposed to the project, including the Tsleil-Waututh and Squamish, who argued the NEB review was flawed and could not serve as a basis for the federal government to approve it.
In a live-streamed press conference soon after the unanimous decision came down, Dustin Rivers speaking on behalf of the Squamish Nation council, cheered the court victory.
• READ MORE: VIDEO: B.C. First Nations hail court's quash of Kinder Morgan pipeline approval
"Today we won in court against this project," Rivers said. "The process was flawed. The consultation was inadequate. The Trudeau government did not respect our rights as a people and we won today, the courts agreed with every step of that argument."
City of Chilliwack Coun. Jason Lum who has been a vocal advocate for the aquifer, and a critic of the process for TMX, said he didn't want to say "I told you so" but, well, he's said all along that consultation is not a one-way street.
Lum, who spoke to The Progress while on vacation and after reading the court's decision, pointed to a particular section under the subhead "The experience of Sto:lo" that outlined criticism from local First Nations about how consultation means more than simply gathering information on aboriginal interests.
"[A] simple 'what we heard' report is inadequate to this task and the Governor-in-Council must be aware of its obligation to either reject or make changes to the project to protect and preserve the aboriginal rights, title and interests of the Sto:lo Collective," the court stated, quoting from a Sto:lo submission.
• READ MORE: City of Chilliwack 'disappointed' with NEB approval of Trans Mountain pipeline route
Representing local Sto:lo interests, the S'ólh Téméxw Stewardship Alliance, were intervenors in January during the National Energy Board (NEB) hearings regarding detailed routing through the residential area of Sardis where the pipeline runs through Watson elementary schoolyard, dozens of residential backyards and is within the capture zone of the city's drinking water wells.
Stephen of WaterWealth and the City of Chilliwack were also intervenors.
After the NEB hearings, the board approved Trans Mountain's application to build the expansion in the original right-of-way. Critics called the hearings rigged since the alternative approved route, in the BC Hydro right-of-way, was never a possibility according to BC Hydro.
And the city was not happy.
"I was disappointed to learn of the National Energy Board's decision to approve the Kinder Morgan route realignment in Chilliwack, especially given the probability that oil from a leak could make its way to our water wells," Mayor Sharon Gaetz said in April when the routing decision was made.
This week, Lum said the process by which an intervenor must come to the NEB with opposition to a route and then provide a detailed alternative is simply "unacceptable."
"We are not building the project," Lum said. "If we have concerns, very real concerns, about the aquifer and the affect on drinking water then its incumbent on the propent that those concerns are addressed adequately."
Just minutes after quashing the federal government's approval, Kinder Morgan shareholders voted overwhelmingly to agree to see the pipeline for $4.5 billion to Ottawa.
The court decision does not necessarily kill the project, firstly because the government could appeal to the Supreme Court of Canada, but secondly because the decision did not order TMX to start from scratch with consultation, but rather ordered a new review the justices suggested could be kept short and would include restarting consultation with Indigenous groups.
• READ MORE: Trans Mountain pipeline: How we got here – A five-part series on the issues surrounding Kinder Morgan's proposed expansion of the Trans Mountain pipeline
• READ MORE: Chilliwack reacts to the Trans Mountain pipeline buyout
@PeeJayAitch
paul.henderson@theprogress.com
NEB hearing Jan. 15 in Chilliwack into Kinder Morgan's application to change approved routing for the Trans Mountain pipeline expansion project. (Paul Henderson/ The Progress) NEB hearing Jan. 15 in Chilliwack into Kinder Morgan's application to change approved routing for the Trans Mountain pipeline expansion project. (Paul Henderson/ The Progress file)
The originally approved route through Sardis along the BC Hydro right-of-way (in green), and the route of the existing pipeline is the red line. The NEB approved Trans Mountain's application to run the expanded pipeline along the existing route, despite opposition from city hall.
Carol Todd to speak at anti-bullying event in Fraser Valley
Bernier or bust: Mad Max's path to a new political party not all that twisty


text:
zend-escaper

Securely and safely escape HTML, HTML attributes, JavaScript, CSS, and URLs.

$ composer require zendframework/zend-escaper

redpajama_set_name: RedPajamaGithub