text:
\section{Introduction}
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Recent exciting work by Cachazo, He, and
Yuan \cite{Cachazo:2013iaa,Cachazo:2013gna,Cachazo:2013hca,Cachazo:2013iea}
has shed new light on the color-kinematic duality of gauge-theory amplitudes
and the double-copy construction that relates gauge-theory and gravity amplitudes \cite{Bern:2008qj,Bern:2010ue}.
Bern, Carrasco, and Johansson showed that tree-level gauge-theory
and gravity amplitudes may be expressed as sums over cubic diagrams
\begin{equation}
 {\cal A} ~=~ \sum_i {c_i ~ n_i \over d_i }, \qquad\qquad
 {\cal M} ~=~ \sum_i { \tilde{n} _i ~ n_i \over d_i }
\label{amps}
\end{equation}
where the kinematic numerators $n_i$ are chosen to possess the same symmetries as
the color factors $c_i$.
Cachazo et al. \cite{Cachazo:2013iea}
extended the gauge-gravity nexus to include
amplitudes of the theory of massless scalars in the adjoint representation
of $U(N) \times U(\tilde{N})$ with cubic interactions
\begin{equation}
 {\cal A}^{{\rm scalar}}  ~=~ \sum_i { \tilde{c} _i ~ c_i \over d_i }
\end{equation}
and they showed that tree-level $n$-point scattering amplitudes for all three theories may be
expressed in terms of an integral over the space of $n$ marked points on a sphere, $\{ \sigma_j ~|~ j=1, \cdots n\}$.
For example, the double-color scalar theory amplitudes may be expressed as\footnote{The definition of $m(\alpha|\beta)$
here differs from that of ref.~\cite{Cachazo:2013iea} by an overall sign when $n$ is even.}
\begin{eqnarray}
 {\cal A}^{{\rm scalar}}
&=&
\sum_{\alpha, \beta}
\mathop{\rm Tr}\nolimits[\alpha]
~m(\alpha|\beta) ~
\widetilde{\mathop{\rm Tr}\nolimits}[\beta]
\\
\mathop{\rm Tr}\nolimits [\alpha]
&=&
{\rm Tr}({T}^{\textsf{a}_{\alpha(1)}}{T}^{\textsf{a}_{\alpha(2)}}\cdots {T}^{\textsf{a}_{\alpha(n)}})
\\
m(\alpha|\beta)\ &=& (-1)^{n-1} ~\int \frac{d\,^n\sigma}{\textrm{vol}\,\mathrm{SL}(2,\mathbb{C})}
\frac{\prod'_j
~ \delta \left(\sum_{k\neq j} \frac{s_{jk}}{\sigma_{j,k}} \right)
}{(\sigma_{\alpha(1),\alpha(2)}\cdots\sigma_{\alpha(n),\alpha(1)})
(\sigma_{\beta(1),\beta(2)}\cdots\sigma_{\beta(n),\beta(1)})}
\label{local}
\end{eqnarray}
where $\sigma_{j,k}= \sigma_j - \sigma_k$
and $s_{jk}=( p_j + p_k)^2$ with $p_j$ the momenta of the external particles.
The delta function localizes the integral (\ref{local}) on the solutions of the scattering
equations \cite{Cachazo:2013iaa,Cachazo:2013gna}
\begin{equation}
\sum_{k\neq j} \frac{s_{jk}}{\sigma_{j,k}} ~=~ 0 \quad {\rm for} \quad j=1, \cdots n \,.
\label{scatt}
\end{equation}
As these equations have, up to $\mathrm{SL}(2,\mathbb{C})$ transformations,
$(n-3)!$ solutions $\{\sigma_j^{(I)}\}$,
the double-partial amplitude $m(\alpha|\beta)$ may
expressed as a sum over solutions
\begin{equation}
m(\alpha|\beta)=
~(-1)^{n-1}~\sum_{I=1}^{(n-3)!} \frac{1}{(\sigma^{(I)}_{\alpha(1),\alpha(2)}\cdots\sigma^{(I)}_{\alpha(n),\alpha(1)})(\sigma^{(I)}_{\beta(1),\beta(2)}\cdots\sigma^{(I)}_{\beta(n),\beta(1)}){\det}'\Phi(\sigma^{(I)})}
\label{sumoversolutions}
\end{equation}
demonstrating that $m(\alpha|\beta)$ has rank $(n-3)!$.
The amplitudes (\ref{amps}) for gauge theory and gravity
may also be succinctly expressed in terms of sums over
solutions\cite{Cachazo:2013hca,Cachazo:2013iea}.
Related work on the scattering equations includes
refs.~\cite{Litsey:2013jfa,Adamo:2013tca,Monteiro:2013rya,Mason:2013sva,Chiodaroli:2013upa,Dolan:2013isa,Adamo:2013tsa,Gomez:2013wza,Kalousios:2013eca,Stieberger:2014hba,Yuan:2014gva,Weinzierl:2014vwa,Dolan:2014ega,Bjerrum-Bohr:2014qwa,He:2014wua,Kol:2014yua,Kol:2014zca,Geyer:2014fka,Schwab:2014xua}.
\\ [-2mm]

Because the color factors $c_i$ in \eqn{amps} satisfy Jacobi identities
and hence are not linearly independent, the kinematic numerators $n_i$
for a given $n$-point gauge-theory amplitude are not uniquely determined.
This generalized gauge freedom can be used to require that $n_i$ satisfy
the same Jacobi identities as the color factors \cite{Bern:2008qj}.
As a consequence all kinematic numerators $n_i$ may be expressed in terms of
an independent set ${\bf n}_{1\gamma n}$ associated with half-ladder diagrams.
Defining color-ordered amplitudes $ A_\alpha$ as
the coefficients of the
gauge-theory amplitude expressed in the trace basis
\begin{equation}
 {\cal A}
= \sum_{\alpha} \mathop{\rm Tr}\nolimits[ \alpha] ~  A_{\alpha}
\label{trace}
\end{equation}
it can then be shown that the  half-ladder numerators must satisfy
\begin{equation}
 A_\alpha   ~=~  \sum_{\gamma} m( \alpha| 1 \gamma n) ~{\bf n}_{1 \gamma n }  \,.
\end{equation}
Because there are $(n-2)!$ independent half-ladder numerators
${\bf n}_{1 \gamma n}$ and because rank $m = (n-3)!$,
this equation cannot be uniquely inverted, meaning that
${\bf n}_{1 \gamma n }$ still possesses some residual generalized gauge freedom
even after color-kinematic duality is imposed.
Many different representations of the kinematic numerators are possible,
{e.g.}, see refs.~\cite{Kiermaier,BjerrumBohr:2010hn,Mafra:2011kj}.
\\ [-2mm]

In an effort to define an economical and natural representation for the numerators,
Broedel and Carrasco \cite{Broedel:2011pd} enumerated
three virtues that kinematic numerators would ideally possess:
(1) color-kinematic duality (numerators obey the same symmetries
as the associated color factors),
(2) amplitude-encoding (external-state dependence is expressed in terms of
color-ordered amplitudes $A_\alpha$), and
(3) symmetry (numerator functions corresponding to diagrams with the same topology
but different labelings of the external legs
are all related by permutations of their arguments).
They then proceeded to construct virtuous numerators for four- and five-point
tree-level amplitudes, and for six-point MHV amplitudes in four dimensions,
by assuming a general ansatz for the numerators and imposing functional constraints.
Their approach becomes impracticable, however, for larger values of $n$,
and they voiced the hope that a constructive procedure for virtuous numerators
for arbitrary $n$-point amplitudes could be found.
\\ [-2mm]

In sec.~3 of this paper, we present a procedure to produce $n$-point kinematic numerators
satisfying all three virtues of Broedel and Carrasco.
We begin with a specific set of nonsymmetric numerators from
ref.~\cite{Cachazo:2013iea} obtained using the properties of $m(\alpha|\beta)$.
By applying arbitrary permutations to the external legs,
we derive other nonsymmetric sets of numerators.
Finally, we generate a symmetric numerator by summing over all such representations.\footnote{Recently,
Fu, Du, and Feng \cite{Fu:2014pya}
presented a different but apparently equivalent algorithm by obtaining symmetric numerators
from a KLT expression for the gauge-theory amplitude.}
Our expression manifestly satisfies all Jacobi identities and diagram symmetries
for the numerators without having to invoke the BCJ relations for the color-ordered amplitudes.
We give explicit results for four- and five-point amplitudes, but the procedure is
completely general.
\\ [-2mm]

The double-copy construction relates gauge-theory to gravity amplitudes by replacing
the color factor $c_i$ in \eqn{amps} with a function $ \tilde{n} _i$ of kinematic variables \cite{Bern:2008qj}.
Bern and Dennen \cite{Bern:2011ia}
suggested that, in a similar way, the gravity amplitude could be
obtained from the gauge-theory amplitude (\ref{trace})
by replacing the color trace factor $\mathop{\rm Tr}\nolimits[\alpha]$
with a dual-trace function $\tau_\alpha$ of the kinematic variables
\begin{equation}
 {\cal M}
= \sum_{\alpha} \tau_\alpha ~  A_{\alpha}  \,.
\end{equation}
It can then be shown that
\begin{equation}
 A_\alpha   ~=~  \sum_{\beta} m( \alpha| \beta) \tau_\beta  \,.
\label{AYMmtau}
\end{equation}
Once again, because there are $(n-1)!/2$ independent $\tau_\alpha$ and rank $m = (n-3)!$, this
equation cannot be uniquely inverted;  many choices of dual-trace functions are possible.
\\ [-2mm]

In ref.~\cite{Bern:2011ia}, Bern and Dennen
presented explicit expressions for $\tau_\alpha$
in terms of ${\bf n}_{1 \gamma n }$
for four-, five-, and six-point amplitudes,
and verified the existence of such expressions through nine points,
but did not present a general procedure for arbitrary $n$.
Further progress was achieved by Du et al. \cite{Du:2013sha,Fu:2013qna}.
\\ [-2mm]

In sec.~4 of this paper, we present a procedure to generate
$n$-point dual-trace functions $\tau_\alpha$ that possess
the three virtues identified by Broedel and Carrasco:
(1) they automatically lead to kinematic numerators that obey color-kinematic duality,
(2) they are expressed in terms of $A_\alpha$, {i.e.}, they are amplitude-encoded, and
(3) they are symmetric:  a single function $\tau$ suffices to determine the
full set via permutations of its arguments.
We first identify a specific set of nonsymmetric dual-trace functions
that satisfy \eqn{AYMmtau},
apply arbitrary permutations to the external legs to obtain other representations,
and then generate a symmetric dual-trace function by summing over all such representations.
We present explicit expressions for four- and five-point functions,
but again the procedure is completely general.
\\ [-2mm]

The expressions for $\tau_\alpha$ obtained in ref.~\cite{Bern:2011ia}
were observed to obey Kleiss-Kuijf relations,
and the authors argued that this was a necessary condition
for symmetric dual-trace functions expressed in terms of kinematic numerators.
The results for $\tau_\alpha$ obtained in this paper do not satisfy Kleiss-Kuijf relations;
they are expressed directly in terms of color-ordered amplitudes,
and so are able to avoid this requirement.
Thus, even after the virtues of Broedel and Carrasco are imposed,
there remains some freedom in the dual-trace function.
\\ [-2mm]

This paper is structured as follows.
Section 2 reviews the tree-level amplitudes of
gauge, gravity, and double-color scalar theories,
and the color-kinematic dualities that relate them.
Section 3 describes the criteria of Broedel and Carrasco
for virtuous numerators, outlines our procedure for constructing them,
and presents explicit results for four- and five-point amplitudes.
Section 4 reviews the properties of the dual-trace functions of
Bern and Dennen, and then outlines our procedure for constructing them,
again presenting explicit results for four- and five-point amplitudes.
Section 5 discusses open questions,
and an appendix contains the proof of the procedure used
to produce symmetric kinematic numerators and dual-trace functions.

\section{Review of color-kinematic duality}
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In this section, we review tree-level
amplitudes of gauge, gravity, and double-color scalar theories,
and the color-kinematic dualities that relate these various
amplitudes. \\ [-2mm]

We begin with the tree-level $n$-gluon amplitude
$ {\cal A} ( 1,2, \cdots, n )$, where the arguments denote
the momentum $p_j$, polarization $\varepsilon_j$, and color $\textsf{a}_j$
of the external particles $j=1, \cdots, n$.
The color dependence can be specified in terms of a set of
color factors
$c_i$ obtained by sewing together cubic
vertices $f_{\textsf{abc}} \equiv \mathop{\rm Tr}\nolimits(T^\textsf{a} [ T^\textsf{b}, T^\textsf{c}] )$,
where $T^\textsf{a}$ denote generators in the fundamental representation of the
color group.
The $n$-point amplitude is then expressed as a sum over all cubic diagrams
\cite{Bern:2008qj}
\begin{equation}
 {\cal A} ( 1,2, \cdots, n )
~=~ \sum_i
{c_i ~ n_i ( 1,2, \cdots, n )
\over d_i ( 1,2, \cdots, n )} \,.
\label{gluonamp}
\end{equation}
The denominator $d_i ( 1,2, \cdots, n )$
associated with the color diagram $c_i$
is a product of the diagram propagators,
and is a function of the external momenta $p_j$.
The kinematic numerator $n_i ( 1,2, \cdots, n )$ associated with $c_i$
is a function of both $p_j$ and $\varepsilon_j$.
(As usual, terms in the Feynman diagram expansion arising
from quartic vertices can be parceled into terms involving only cubic vertices.) \\ [-2mm]

The color factors $c_i$ can be expanded in a trace basis
\begin{equation}
c_i  ~=~
\sum_{\alpha \in   {S_n /\mathbb{Z} } }  M_{i \alpha} \mathop{\rm Tr}\nolimits[\alpha] , \qquad \qquad
\mathop{\rm Tr}\nolimits [\alpha] ~\equiv~
{\rm Tr}({T}^{\textsf{a}_{\alpha(1)}}{T}^{\textsf{a}_{\alpha(2)}}\cdots {T}^{\textsf{a}_{\alpha(n)}})  \,.
\label{colortrace}
\end{equation}
The trace basis is independent provided we restrict the sum to permutations
of $n$ indices modulo cyclic permutations.
The gauge-theory amplitude (\ref{gluonamp}) can be decomposed in this basis
\begin{eqnarray}
 {\cal A}  ( 1,2, \cdots, n )
&=& \sum_{\alpha \in   {S_n /\mathbb{Z} } } \mathop{\rm Tr}\nolimits[ \alpha] ~  A_{\alpha} ( 1,2, \cdots, n )  ,
\label{tracedecomp}
\\
 A_{\alpha} ( 1,2, \cdots, n )
&=& \sum_i
{M_{i \alpha}  ~ n_i ( 1,2, \cdots, n )
\over d_i ( 1,2, \cdots, n )}
\label{colorordered}
\end{eqnarray}
where the arguments of the color-ordered amplitudes
$ A_{\alpha} ( 1,2, \cdots, n )$ denote $p_j$ and $\varepsilon_j$.
Since the trace basis is independent, the color-ordered amplitudes
are well-defined and gauge invariant.
Because of the invariance of the full amplitude $ {\cal A} ( 1,2, \cdots, n )$ under
permutations of the arguments and the independence of the
trace basis,\footnote{See discussion in the appendix.}
the color-ordered amplitudes are all related to one another
via
\begin{equation}
 A_\alpha ( 1,2, \cdots, n ) ~=~  A ( \alpha(1), \cdots, \alpha(n))
\end{equation}
where $ A  (i,j,\cdots) \equiv  A_{12\cdots n}(i, j, \cdots) $,
the coefficient of
${\rm Tr}(T^{\textsf{a}_{1}}T^{\textsf{a}_{2}}\cdots T^{\textsf{a}_{n}})$.
The color-ordered amplitudes are thus ``symmetric'' functions,
in the sense of Broedel and Carrasco\cite{Broedel:2011pd}. \\ [-2mm]

The color factors $c_i$ in \eqn{gluonamp}
are not independent but satisfy various Jacobi identities
\begin{equation}
c_i + c_j + c_k ~=~ 0
\label{cjacobi}
\end{equation}
which can be expressed
\cite{Naculich:2011ep,Edison:2011ta,Edison:2012fn}
as $\sum_i \ell_i c_i  = 0$,
where $\ell_i$ are left null vectors of the matrix $M_{i\alpha}$:
  $\sum_i  \ell_i M_{i\alpha}=0$.
The matrix $M_{i\alpha}$ has rank $(n-2)!$ so there are $(n-2)!$ independent
color factors.
The matrix $M_{i\alpha}$ also possesses a set of right null vectors
$\sum_\alpha  M_{i\alpha} r_\alpha=0$.
By \eqn{colorordered}, these give
rise\cite{Naculich:2011ep,Edison:2011ta,Edison:2012fn}
to a set of constraints
$\sum_\alpha  A_{\alpha} r_\alpha=0$
on the color-ordered amplitudes, {viz.},
the Kleiss-Kuijf relations \cite{Kleiss:1988ne}. \\ [-2mm]

Because of the linear dependence of the color factors $c_i$,
the kinematic numerators $n_i ( 1,2, \cdots, n )$ in \eqn{gluonamp}
are not uniquely determined but can undergo
what are termed generalized gauge transformations \cite{Bern:2010yg}
without altering the amplitudes.
The insight of Bern, Carrasco, and Johannson \cite{Bern:2008qj}
is that there exists a generalized gauge choice
for which the numerators satisfy the same Jacobi identities as the color factors
\begin{equation}
n_i ( 1,2, \cdots, n ) ~+~ n_j ( 1,2, \cdots, n ) ~+~ n_k ( 1,2, \cdots, n ) ~=~ 0  \,.
\label{njacobi}
\end{equation}
We will refer to a set of kinematic numerators satisfying \eqn{njacobi} as BCJ numerators.
Such a choice is not unique: there remain residual generalized
gauge transformations that preserve the Jacobi identities (\ref{njacobi}).
\\ [-2mm]

A subset of the color factors $c_i$ are the half-ladder diagrams,
labeled by $ \alpha \in S_n$:
\begin{equation}
{\bf c}_{\alpha} ~=~  \sum_{\textsf{b}_1,\ldots,\textsf{b}_{n{-}3}}
f_{\textsf{a}_{\alpha(1)} \textsf{a}_{\alpha(2)} \textsf{b}_1}\cdots f_{\textsf{b}_{n{-}3} \textsf{a}_{\alpha(n{-}1)} \textsf{a}_{\alpha(n)}}
  ~=~ \mathop{\rm Tr}\nolimits(
T^{\textsf{a}_{\alpha(1)}}
[T^{\textsf{a}_{\alpha(2)}} ,
[ \cdots [
T^{\textsf{a}_{\alpha(n-1)}} ,
T^{\textsf{a}_{\alpha(n)}} ] \cdots ]] ) \,.
\label{c-basis}
\end{equation}
An independent set of color factors \cite{DelDuca:1999ha,DelDuca:1999rs}
consists of those half ladders with $\alpha(1)=1$ and $\alpha(n)=n$:
\begin{equation}
{\bf c}_{1 \gamma n } ~\equiv~
{\bf c}_{1 \gamma(2) \cdots \gamma(n-1) n }
, \qquad\qquad \gamma \in S_{n-2}
\end{equation}
whose expansion in the trace basis begins
\begin{equation}
{\bf c}_{1 \gamma n }
~=~ \mathop{\rm Tr}\nolimits[ 1 \gamma(2) \cdots \gamma(n-1) n]
~+~ (-1)^n \mathop{\rm Tr}\nolimits[ n \gamma(n-1) \cdots \gamma(1) 1]  ~+~ \cdots
\label{halfladderexpand}
\end{equation}
where $+ \cdots$ denotes traces without $1$ and $n$ adjacent.
That the set ${\bf c}_{1 \gamma n }$ is independent
follows from \eqn{halfladderexpand} together with the independence of the trace basis
(modulo cyclic permutations).
That the set is complete was shown in ref.~\cite{DelDuca:1999rs}
where an arbitrary color diagram was reduced to a linear combination
$c_i = \sum_i \lambda_{i,\gamma} {\bf c}_{1 \gamma n }$
using the Jacobi identities.
By matching the coefficients of $\mathop{\rm Tr}\nolimits[ 1 \gamma(2) \cdots \gamma(n-1) n] $
on both sides of this equation,
one then establishes that
\begin{equation}
c_i ~=~ \sum_{\gamma \in S_{n-2}}  M_{i, 1\gamma n} {\bf c}_{1 \gamma n } \,.
\label{cMc}
\end{equation}
Then \eqns{gluonamp}{colorordered} imply \cite{DelDuca:1999ha,DelDuca:1999rs}
\begin{equation}
 {\cal A} ( 1,2, \cdots, n ) ~=~ \sum_{\gamma \in S_{n-2}}
{\bf c}_{1 \gamma n } A(1, \gamma(2), \cdots, \gamma(n-1), n) \,.
\label{cgammaA}
\end{equation}

\subsection{Double-color scalar amplitudes}

In their seminal paper \cite{Bern:2008qj},
Bern, Carrasco, and Johansson showed that,
given a set of BCJ numerators $n_i( 1,2, \cdots, n )$,
one can obtain tree-level gravity amplitudes\footnote{
Up to an overall factor depending on coupling strengths
that we suppress throughout this paper}
by replacing $c_i$ in \eqn{gluonamp} with the kinematic
numerators $ \tilde{n} _i( 1,2, \cdots, n )$ of a second gauge theory (the double-copy procedure)
\begin{equation}
 {\cal M} ( 1,2, \cdots, n )
~=~ \sum_i
{  \tilde{n} _i ( 1,2, \cdots, n ) ~ n_i ( 1,2, \cdots, n )
\over d_i ( 1,2, \cdots, n )}
\label{gravamp}
\end{equation}
where the $ \tilde{n} _i( 1,2, \cdots, n )$ need not themselves satisfy the Jacobi
identities \cite{Bern:2010yg}.
\\ [-2mm]

An alternative double-copy procedure replaces the $n_i( 1,2, \cdots, n )$ in
\eqn{gluonamp} with the color factors $ \tilde{c} _i$ of a second color group
\begin{equation}
 {\cal A}^{{\rm scalar}}  ( 1,2, \cdots, n )
~=~ \sum_i
{ \tilde{c} _i ~ c_i \over d_i ( 1,2, \cdots, n )} \,.
\label{scalaramp}
\end{equation}
Such an expression corresponds \cite{Cachazo:2013iea}
to the $n$-point amplitude of a theory of massless scalar
particles $ \phi^{\textsf{a}\textsf{a'}}$
in the adjoint of the color group $U(N) \times U(\tilde{N})$
with cubic interactions of the form
\begin{equation}
f_{\textsf{abc}}\tilde f_{\textsf{a'b'c'}}\phi^{\textsf{a}\textsf{a'}}\phi^{\textsf{b}\textsf{b'}}\phi^{\textsf{c}\textsf{c'}}
\end{equation}
where
$f_{\textsf{abc}}$ and $\tilde f_{\textsf{a'b'c'}}$
are the structure constants of $U(N)$ and $U(\tilde{N})$.
Using
\begin{equation}
 \tilde{c} _i  ~=~
\sum_{\alpha \in   {S_n /\mathbb{Z} } }  M_{i \alpha} \widetilde{\mathop{\rm Tr}\nolimits}[\alpha] , \qquad \qquad
\widetilde{\mathop{\rm Tr}\nolimits} [\alpha] ~\equiv~
{\rm Tr}({\tilde T}^{\textsf{a}_{\alpha(1)}}{\tilde T}^{\textsf{a}_{\alpha(2)}}\cdots {\tilde T}^{\textsf{a}_{\alpha(n)}})
\end{equation}
together with \eqn{colortrace}, the double-color amplitude (\ref{scalaramp}) can be written as
\begin{equation}
 {\cal A}^{{\rm scalar}}  ( 1,2, \cdots, n )
~=~
\sum_{\alpha \in   {S_n /\mathbb{Z} } }  \sum_{\beta \in   {S_n /\mathbb{Z} } }
\mathop{\rm Tr}\nolimits[\alpha]
~m(\alpha|\beta) ~
\widetilde{\mathop{\rm Tr}\nolimits}[\beta]
\label{TmT}
\end{equation}
where\footnote{This definition of $m(\alpha|\beta)$ differs by a
sign from that of ref.~\cite{Cachazo:2013iea} when $n$ is even.}
\begin{equation}
m(\alpha| \beta)
~=~ \sum_i
{M_{i\alpha} M_{i\beta} \over d_i( 1,2, \cdots, n ) },
\qquad \alpha, \beta \in S_n \,.
\label{doublepartial}
\end{equation}
The coefficients $A_\alpha$ of $\mathop{\rm Tr}\nolimits[\alpha]$ of the gauge-theory amplitude (\ref{tracedecomp})
are sometimes termed ``partial amplitudes.''
In ref.~\cite{Cachazo:2013iea},
Cachazo, He, and Yuan dubbed $m(\alpha|\beta)$,
the coefficients of
$\mathop{\rm Tr}\nolimits[\alpha] \widetilde{\mathop{\rm Tr}\nolimits}[\beta]$ in \eqn{TmT},
``double-partial amplitudes.''
They showed that $m(\alpha|\beta)$ computes the sum of all trivalent scalar diagrams
that can be regarded both as $\alpha$-color-ordered and $\beta$-color-ordered,
where each diagram's contribution is given by the product of its propagators.
The double-partial amplitudes satisfy (on both sides)
Kleiss-Kuijf relations
$ \sum_\alpha r_\alpha m(\alpha|\beta) = \sum_\alpha m(\alpha|\beta)  r_\beta = 0$
where $r_\alpha$ are right null vectors of the rank $(n-2)!$ matrix $M_{i\alpha}$.
\\ [-2mm]

Using \eqn{cMc}, the double-color amplitude (\ref{scalaramp}) can also be expressed as
\begin{equation}
 {\cal A}^{{\rm scalar}}  ( 1,2, \cdots, n )
~=~  \sum_{\gamma \in S_{n-2}}  \sum_{\delta \in S_{n-2}}
{\bf  \tilde{c} }_{1 \gamma n } ~m( 1 \gamma n| 1 \delta n) ~{\bf c}_{1 \delta n }  \,.
\label{cmc}
\end{equation}
The double-partial amplitudes in this equation,
$m( 1 \gamma n| 1 \delta n)$,
are essentially the entries in the $(n-2)! \times (n-2)!$ matrix
considered in ref.~\cite{Vaman:2010ez}.
There it was shown by consideration of low values of $n$ that,
as a result of momentum conservation,
the rank of this matrix is $(n-3)!$.
Cachazo et al. showed that the double-partial amplitudes could alternatively be expressed
as \eqn{sumoversolutions}, which shows explicitly that
$m(\alpha|\beta)$ has rank $(n-3)!$.
It consequently possesses $(n-3)!-(n-2)!$ additional null vectors,
dependent on kinematic invariants,
which implies that the double-partial amplitudes satisfy
BCJ relations \cite{Bern:2008qj}  in addition to Kleiss-Kuijf relations.

\subsection{Gauge-theory amplitudes}

Just as the Jacobi identities for the color factors (\ref{cjacobi})
imply that they can be expressed as a linear combination of
independent half-ladder color factors (\ref{cMc}),
so the Jacobi identities (\ref{njacobi})
imply that the kinematic numerators can be written
in terms of an independent basis of numerators
${\bf n}_{1 \gamma n } ( 1,2, \cdots, n ) $ associated with ${\bf c}_{1 \gamma n } $
\begin{equation}
n_i ( 1,2, \cdots, n ) ~=~ \sum_{\gamma \in S_{n-2}}  M_{i, 1\gamma n}
~{\bf n}_{1 \gamma n } ( 1,2, \cdots, n ) \,.
\label{nMn}
\end{equation}
Then, analogous to  \eqn{cmc} for double-color scalar amplitude,
we can write the gauge-theory amplitude (\ref{gluonamp})
as
\begin{equation}
 {\cal A} ( 1,2, \cdots, n )
~=~  \sum_{\gamma \in S_{n-2}}  \sum_{\delta \in S_{n-2}}
{\bf c}_{1 \gamma n } ~m( 1 \gamma n| 1 \delta n) ~{\bf n}_{1 \delta n }  \,.
\label{cmn}
\end{equation}
We can also substitute \eqn{nMn} into \eqn{colorordered}
to obtain the color-ordered amplitudes
\begin{equation}
 A_\alpha   ( 1,2, \cdots, n )
~=~  \sum_{\gamma \in S_{n-2}}
m( \alpha| 1 \gamma n)
~{\bf n}_{1 \gamma n } ( 1,2, \cdots, n ) \,.
\label{Amn}
\end{equation}
As noted above,
the matrix $m( \alpha| 1 \gamma n) $ has rank $(n-3)!$
and has $(n-2)! - (n-3)!$ additional\footnote{In addition,
that is, to the null vectors of $M_{i\alpha}$
which give rise to the Kleiss-Kuijf relations.}
null vectors (dependent on kinematic invariants)
which give rise \cite{Vaman:2010ez,Boels:2012sy,Litsey:2013jfa}
to the BCJ relations \cite{Bern:2008qj} among the color-ordered amplitudes.
\\ [-2mm]

The kernel of the matrix corresponds to
$(n-2)! - (n-3)!$ degrees of freedom of
residual generalized gauge transformations of the BCJ numerators
(i.e. generalized gauge transformations  that preserve the Jacobi constraints (\ref{njacobi})).
Because of this residual gauge freedom,
\eqn{Amn} cannot be inverted
to obtain unique expressions for BCJ numerators
in terms of color-ordered amplitudes.
To invert \eqn{Amn}, we must first make a choice of gauge.
One possible gauge choice is to set
\begin{equation}
{\bf n}_{1\gamma(2)\cdots\gamma(n-1)n} = 0, \qquad \qquad \gamma(n-1)\neq n-1
\label{zeronumerators}
\end{equation}
which allows us to restrict the sum in \eqn{Amn} to $S_{n-3}$.
\\ [-2mm]

Now consider the subset of color-ordered amplitudes
\begin{equation}
 A (1, \beta(2), \cdots, \beta(n-2), n, n-1)
~=~  \sum_{\gamma \in S_{n-3}}
m( 1 \beta n, n-1 | 1 \gamma n-1,n)
~ {\bf n}_{1\gamma(2)\cdots\gamma(n-2)n-1,n} ( 1,2, \cdots, n )   \,.
\label{mbetagamma}
\end{equation}
It was shown in ref.~\cite{Cachazo:2013iea}
by using KLT orthogonality \cite{Cachazo:2013gna} that
the  $(n-3)! \times (n-3)!$ submatrix
appearing in \eqn{mbetagamma} is invertible,
with the inverse given by
the (negative of the) momentum kernel\footnote{Here $\gamma,\beta\in S_{n-3}$ are permutations acting on
labels $2,3,\ldots,n{-}2$; $\theta(r,s)_\beta=1$ if the ordering of $r,s$
is the same in both sequences of labels, $\gamma(2),\ldots,\gamma(n{-}2)$
and $\beta(2),\ldots,\beta(n{-}2)$, and zero otherwise.} \cite{BjerrumBohr:2010ta,BjerrumBohr:2010zb,BjerrumBohr:2010yc,BjerrumBohr:2010hn}
\begin{equation}
S[\gamma|\beta]~=~
\prod^{n{-}2}_{i=2}\left[s_{1, \gamma(i)}+\sum^{i{-}1}_{j=2} \theta(\gamma(j), \gamma(i))_{\beta} s_{\gamma(j),\gamma(i)}\right].
\end{equation}
Thus the nonzero members of the independent basis of numerators
can be expressed as \cite{Cachazo:2013iea}
\begin{equation}
{\bf n}_{1\gamma(2)\cdots\gamma(n-2)n-1,n} ( 1,2, \cdots, n )
~=~ -~\sum_{\beta\in S_{n-3}}S[\gamma|\beta]
 A (1,\beta(2), \cdots, \beta(n-2) ,n,n-1) \,.
\label{BCJnumerator}
\end{equation}
The full set of (BCJ) numerators,
including the half-ladders not included in the independent set,
can then be obtained from \eqns{zeronumerators}{BCJnumerator} via \eqn{nMn}.
\\ [-2mm]

\subsection{Gravity amplitudes}

By the double-copy construction, we can replace
${\bf c}_{1 \gamma n }$ in \eqns{cgammaA}{cmn} with
${\bf  n}_{1 \gamma n } ( 1,2, \cdots, n )$
to obtain the  gravity amplitude
\begin{eqnarray}
 {\cal M} ( 1,2, \cdots, n )
&=& \sum_{\gamma \in S_{n-2}}
{\bf n}_{1 \gamma n } ( 1,2, \cdots, n )
 A (1, \gamma(2), \cdots, \gamma(n-1),n)
\nonumber\\
&=&  \sum_{\gamma \in S_{n-2}}  \sum_{\delta \in S_{n-2}}
{\bf n}_{1 \gamma n } ( 1,2, \cdots, n ) ~m( 1 \gamma n| 1 \delta n) ~{\bf n}_{1 \delta n } ( 1,2, \cdots, n ) \,.
\end{eqnarray}
The specific choice of gauge in \eqns{zeronumerators}{BCJnumerator} can then
be used to write
\begin{eqnarray}
&&  {\cal M} ( 1,2, \cdots, n )
\nonumber\\
&& ~~~
=
 \sum_{\gamma \in S_{n-3}}
   \sum_{\delta \in S_{n-3}}
 A (1, \gamma(2), \cdots, \gamma(n-2),n-1,n)
S[\gamma|\delta]
 A (1,\delta(2), \cdots, \delta(n-2) ,n,n-1)
\nonumber\\
\end{eqnarray}
which is one possible form of the field-theory limit of the KLT relation \cite{Kawai:1985xq,Bern:1998sv,BjerrumBohr:2010hn}.
\\ [-2mm]

\section{Virtuous kinematic numerators}
\setcounter{equation}{0}

As described in the previous section,
even after requiring the kinematic numerators to satisfy color-kinematic duality,
a certain amount of generalized gauge freedom remains.
To fix this residual gauge freedom in a natural, economical,
and possibly unique way,
Broedel and Carrasco \cite{Broedel:2011pd} identified three
desirable features that a set of numerators should possess:

\begin{enumerate}
\item
color-kinematic duality:
BCJ numerators obey the same symmetries as their associated color factors.
Such numerators can be used to construct gravity amplitudes using the double-copy procedure.
\item
amplitude-encoding: the external state dependence ({e.g.}, helicities) of
the numerators is expressed in terms of color-ordered amplitudes.
Such a representation would also be independent of the number of space-time dimensions.
\item
symmetry:
all of the numerators $n_i$ for a given diagram topology can
be expressed via permutations of the arguments of a single function.
This virtue is thus one of economy, and also makes the gravity amplitudes
constructed from these numerators manifestly invariant under permutations
of the external legs.
\end{enumerate}
Kinematic numerators that satisfy all three features are dubbed ``virtuous.''
\\ [-2mm]

As we saw in the previous section, numerators that satisfy the
first virtue are expressed in terms
of an independent basis of half-ladder numerators
${\bf n}_{1 \gamma n } ( 1,2, \cdots, n )$ via \eqn{nMn}.
Color-ordered amplitudes are expressed as in terms of these as
\begin{equation}
 A_\alpha ( 1,2, \cdots, n )
~=~  \sum_{\gamma \in S_{n-2}}
m( \alpha| 1 \gamma n)
~{\bf n}_{1 \gamma n } ( 1,2, \cdots, n ) \,.
\label{Amnrepeat}
\end{equation}
Were we able to invert this equation, we would possess numerators
that also satisfy the second virtue of amplitude-encoding.
Because $\det m= 0$, \eqn{Amnrepeat} has no unique inverse;
there exists rather a family of generalized inverses \cite{BenIsrael,Boels:2012sy},
each corresponding to a particular gauge choice for ${\bf n}_{1 \gamma n }$.
One such choice is given by \cite{Cachazo:2013iea}
\begin{eqnarray}
{\bf n}_{1\gamma(2)\cdots\gamma(n-1)n}( 1,2, \cdots, n ) &=& 0, \qquad \qquad \gamma(n-1)\neq n-1
\nonumber\\
{\bf n}_{1\gamma(2)\cdots\gamma(n-2)n-1,n} ( 1,2, \cdots, n )  &=&
-~\sum_{\beta\in S_{n-3}}S[\gamma|\beta]
 A (1,\beta(2), \cdots, \beta(n-2) ,n,n-1)\,.
\label{gaugechoice}
\end{eqnarray}
This choice, however, does not satisfy the third virtue of symmetry.
\\ [-2mm]

Next, we describe how one can generate a symmetric representation of
numerators starting from a nonsymmetric representation,
such as that given in \eqn{gaugechoice}.\footnote{We thank Freddy Cachazo for
suggesting this approach.}
We show in the appendix that,
given one set of numerators
${\bf n}_{\alpha} ( 1,2, \cdots, n )$,
one can use an arbitrary permutation $\beta$ acting on the
external legs to generate another set of valid numerators
\begin{equation}
{\bf n}'_{\alpha} ( 1,2, \cdots, n )
~=~ {\bf n}_{\beta^{-1} \alpha} (\beta(1), \cdots, \beta(n) )
\end{equation}
{i.e.}, these also satisfy \eqn{Amnrepeat}.
If  we average over all permutations $\beta \in S_n$,
the resulting function
\begin{equation}
{\bf n} ( 1,2, \cdots, n )
~\equiv~
 {\bf n}^{\rm sym} ( 1,2, \cdots, n )
~=~ {1 \over n!} \sum_{\beta \in S_n}  {\bf n}_{\beta^{-1} } (\beta(1), \cdots, \beta(n) )
\label{symmetrized}
\end{equation}
will be symmetric; {i.e.}, all half-ladder numerators are given by permutations
of the arguments of the single function ${\bf n} ( 1,2, \cdots, n ) $.
Symmetric numerator functions for topologies other than the half-ladder are then
obtained from ${\bf n} ( 1,2, \cdots, n ) $ via \eqn{nMn}.
Since the individual sets of numerators
${\bf n}'_{\alpha} ( 1,2, \cdots, n )$ satisfy \eqn{Amnrepeat},
so does the average (\ref{symmetrized}):
\begin{equation}
 A (\alpha(1), \cdots, \alpha(n))
 ~=~
\sum_{\gamma \in S_{n-2}}
~m( \alpha| 1 \gamma n)
~{\bf n} (1, \gamma(2), \cdots, \gamma(n-1), n )\,.
\label{validvirtue}
\end{equation}
Hence we have constructed a representation of kinematic numerators satisfying all three virtues of Broedel and Carrasco.
\\ [-2mm]

We apply this procedure in the following subsections to obtain
explicit expressions for virtuous numerators for four- and five-point amplitudes.
The expressions become increasingly lengthy for higher-point amplitudes, but
the important point is that there exists a constructive proof of the existence of
virtuous numerators for all tree-level $n$-gluon amplitudes.
Whether such numerators are unique remains an open question.
\\ [-2mm]

In a recent paper \cite{Fu:2014pya}, Fu, Du, and Feng
also gave a prescription for obtaining virtuous numerators
based on a similar symmetrization strategy applied to
a KLT expression for the gauge-theory amplitude.
It seems likely that this gives the same results as \eqn{symmetrized}.
We will compare specific results below.

\subsection{Four-point symmetric kinematic numerators}

For the four-point amplitude, there is only one topology
for color factors, the half ladder.
A symmetric half-ladder numerator ${\bf n}(1,2,3,4)$  must satisfy \eqn{validvirtue}
which in this case becomes
\begin{equation}
A(1,2,3,4) ~=~   \left( {1 \over s_{12}} + {1 \over s_{14}} \right) {\bf n}(1,2,3,4)
		~+~ {1 \over s_{14}} {\bf n}(1,3,4,2)   \,.
\label{fourpointcolorordered}
\end{equation}
\Eqn{gaugechoice} yields
a nonsymmetric amplitude-encoded BCJ representation
for independent half-ladder numerators
\begin{eqnarray}
{\bf n}_{1234}(i,j,k,l) &=& - s_{ij}  A (i,j,l,k)\,,  \nonumber\\
{\bf n}_{1324}(i,j,k,l) &=& 0
\label{fourpointnonsym}
\end{eqnarray}
with the remaining numerators given by \eqn{nMn}, {i.e.}, by Jacobi identities and numerator symmetries.
We obtain a virtuous representation by summing over all permutations (\ref{symmetrized})
\begin{equation}
{\bf n}(1,2,3,4) ~=~
{1 \over 12} {(s_{12} + s_{34})} \left[
A(1,2,3,4) -A(1,3,4,2)
\right]
+{1 \over 12} {(s_{13} +s_{24} -s_{14} - s_{23})}
A(1,4,2,3)  \,.
\label{fourpointvirtue}
\end{equation}
We have reduced the number of terms by using
the cyclic invariance of $A(1,2,3,4)$
as well as the reversal symmetry $A(1,2,3,4)= A(4,3,2,1)$.
This expression manifestly obeys the Jacobi identities
\begin{equation}
{\bf n}(1,2,3,4) + {\bf n}(1,3,4,2) + {\bf n}(1,4,2,3)  ~=~ 0
\end{equation}
as well as the dihedral symmetries of the half-ladder diagram
\begin{equation}
{\bf n}(1,2,3,4) ~=~  - {\bf n}(2,1,3,4) ~=~  - {\bf n}(1,2,4,3) ~=~  {\bf n}(4,3,2,1)  \,.
\end{equation}
\Eqn{fourpointvirtue} also satisfies \eqn{fourpointcolorordered}
provided that  the color-ordered amplitudes satisfy the four-point BCJ relations
\begin{equation}
s_{ij} A(i,j,k,l) ~=~  s_{ik} A(i,l,j,k) \,.
\label{fourpointBCJ}
\end{equation}
Our expression (\ref{fourpointvirtue}) agrees with
that recently obtained in ref.~\cite{Fu:2014pya}.
It is also in agreement with the shorter expression in ref.~\cite{Broedel:2011pd}
\begin{equation}
{\bf n}(1,2,3,4) ~=~   {1 \over 3} \left[ s_{12}  A (1, 2, 3, 4)  -  s_{14}  A(1, 4, 2, 3) \right]
\label{fourpointBC}
\end{equation}
once momentum conservation and the Kleiss-Kuijf (subcyclic) identity
$ A(1,2,3,4) + A(1,3,4,2) + A(1, 4, 2, 3) =  0 $
are imposed.
In fact, the three virtues listed above are sufficient to uniquely determine
the four-point numerator.
\\ [-2mm]

The BCJ relations (\ref{fourpointBCJ}) can be used to rewrite \eqn{fourpointBC} as
\begin{equation}
{\bf n}(1,2,3,4) ~=~   {1 \over 3} s_{12} \left[ A(1,2,3,4) - A(1, 3, 4, 2) \right]  \,.
\end{equation}
We could further use the BCJ relations to write
${\bf n}(1,2,3,4)$
in terms of a single color-ordered amplitude,
but only at the price of having kinematic invariants in the denominator.

\subsection{Five-point symmetric numerators}

For the five-point amplitude, there is again only one topology
for the color factors, the half ladder.
A symmetric half-ladder numerator ${\bf n}(1,2,3,4,5)$  satisfies \eqn{validvirtue},
which after using Jacobi identities is equivalent to
\begin{equation}
A(1,2,3,4,5) ~=~
 {{\bf n}(1,2,3,4,5)    \over s_{12}s_{45}}
~+~{{\bf n}(2,3,4,5,1)    \over s_{23}s_{51}}
~+~{{\bf n}(3,4,5,1,2)    \over s_{34}s_{12}}
~+~{{\bf n}(4,5,1,2,3)    \over s_{45}s_{23}}
~+~{{\bf n}(5,1,2,3,4)    \over s_{51}s_{34}}   \,.
\label{fivepointcolorordered}
\end{equation}
\Eqn{gaugechoice} yields
a nonsymmetric amplitude-encoded BCJ representation
for independent half-ladder numerators
\begin{eqnarray}
{\bf n}_{12345}(i,j,k,l,m) &=&  - s_{ij} (s_{ik}+s_{jk} )  A(i,j,k,m,l) - s_{ij} s_{ik} A (i,k,j,m,l)\,, \nonumber\\
{\bf n}_{13245}(i,j,k,l,m) &=&  - s_{ij} s_{ik}           A(i,j,k,m,l)  - s_{ik} (s_{ij}+s_{jk} )   A(i,k,j,m,l)\,,  \\
{\bf n}_{14235}(i,j,k,l,m) &=&   0\,, \nonumber\\
{\bf n}_{12435}(i,j,k,l,m) &=& 0\,, \nonumber\\
{\bf n}_{14235}(i,j,k,l,m) &=& 0\,, \nonumber\\
{\bf n}_{14325}(i,j,k,l,m) &=&0
\nonumber
\end{eqnarray}
where the other half-ladder numerators are obtained using \eqn{nMn}.
Again, we obtain a virtuous representation by summing over all permutations (\ref{symmetrized})
\begin{eqnarray}
{\bf n}(1,2,3,4,5) &=&
{1 \over 60} \Big[
\left(s_{1 2} s_{1 3} + 2 s_{1 2} s_{2 3} + 2 s_{3 4} s_{4 5} + s_{3 5} s_{4 5}
\right) A(1, 2, 3, 4, 5) \nonumber\\
&+&\left(s_{1 3} s_{1 4} + s_{1 2} s_{1 5} + s_{2 3} s_{2 4} + 2 s_{2 3} s_{3 4} + s_{2 4} s_{3 4} + s_{2 5} s_{3 5} + s_{1 5} s_{4 5}
\right) A(1, 4, 3, 2, 5) \nonumber\\
&+&\left(s_{1 3} s_{1 4} + s_{2 3} s_{2 4} + s_{1 2} s_{2 5} + 2 s_{1 3} s_{3 4} + s_{1 4} s_{3 4} + s_{1 5} s_{3 5} + s_{2 5} s_{4 5}
\right) A(1, 3, 4, 2, 5) \nonumber\\
&+&\left(s_{1 2} s_{1 4} - 2 s_{1 2} s_{1 5} + 2 s_{1 2} s_{2 4} - s_{1 2} s_{2 5} - s_{3 4} s_{4 5} + s_{3 5} s_{4 5}
\right) A(1, 2, 4, 3, 5) \nonumber\\
&+&\left(s_{1 2} s_{1 4} + s_{1 3} s_{1 5} + s_{2 3} s_{2 5} + s_{2 4} s_{3 4} + 2 s_{2 3} s_{3 5} + s_{2 5} s_{3 5} + s_{1 4} s_{4 5}
\right) A(1, 4, 2, 3, 5) \nonumber\\
&+&\left(s_{1 2} s_{1 3} - s_{1 2} s_{2 3} - s_{1 4} s_{4 5} - 2 s_{1 5} s_{4 5} + 2 s_{2 4} s_{4 5} + s_{2 5} s_{4 5}
\right) A(1, 3, 2, 4, 5) \nonumber\\
&+&\left(2 s_{1 2} s_{1 3} + s_{1 2} s_{2 3} + 2 s_{3 4} s_{4 5} + s_{3 5} s_{4 5}
\right) A(1, 2, 5, 4, 3) \nonumber\\
&+&\left(s_{1 2} s_{1 3} - s_{1 2} s_{2 3} - 2 s_{1 4} s_{4 5} - s_{1 5} s_{4 5} + s_{2 4} s_{4 5} + 2 s_{2 5} s_{4 5}
\right) A(1, 4, 5, 2, 3) \nonumber\\
&+&\left(-s_{1 3} s_{1 5} - s_{1 2} s_{2 4} - s_{2 3} s_{2 5} - s_{1 4} s_{3 4} - 2 s_{1 3} s_{3 5} - s_{1 5} s_{3 5} - s_{2 4} s_{4 5}
\right) A(1, 3, 5, 2, 4) \nonumber\\
&+&\left(2 s_{1 2} s_{1 4} - s_{1 2} s_{1 5} + s_{1 2} s_{2 4} - 2 s_{1 2} s_{2 5} - s_{3 4} s_{4 5} + s_{3 5} s_{4 5}
\right) A(1, 2, 5, 3, 4) \nonumber\\
&+&\left(s_{1 2} s_{1 3} + 2 s_{1 2} s_{2 3} + s_{3 4} s_{4 5} + 2 s_{3 5} s_{4 5}
\right) A(1, 4, 5, 3, 2) \nonumber\\
&+&\left(2 s_{1 2} s_{1 3} + s_{1 2} s_{2 3} + s_{3 4} s_{4 5} + 2 s_{3 5} s_{4 5}
\right) A(1, 3, 5, 4, 2)
\Big]
\label{fivepointvirtue}
\end{eqnarray}
where we have used the reversal property
$ A(1,2,3,4,5)= -A(5,4,3,2,1) $
together with cyclic invariance of $A(1,2,3,4,5)$.
This expression automatically obeys the dihedral symmetry of the half-ladder diagram
\begin{equation}
{\bf n}(1,2,3,4,5) ~=~  - {\bf n}(2,1,3,4,5) ~=~  - {\bf n}(1,2,3,5,4) ~=~ - {\bf n}(5,4,3,2,1)
\label{fivepointdihedral}
\end{equation}
as well as all the Jacobi identities.
\\ [-2mm]

\Eqn{fivepointvirtue} agrees with the shorter expression
given in Broedel and Carrasco \cite{Broedel:2011pd}
after imposing momentum conservation and BCJ relations
on the color-ordered amplitudes.
The Broedel-Carrasco expression is virtuous,
but to show that it obeys \eqn{fivepointdihedral}
requires the imposition of momentum conservation and BCJ relations,
whereas for \eqn{fivepointvirtue} the dihedral symmetry is manifest.
Both the Broedel-Carrasco expression and \eqn{fivepointvirtue}
satisfy \eqn{fivepointcolorordered} only after momentum conservation and the BCJ relations
are imposed.
\\ [-2mm]

We can re-express \eqn{fivepointvirtue} in the Kleiss-Kuijf basis
to find
\begin{eqnarray}
{\bf n}(1,2,3,4,5) &=&
{1 \over 10} \Big[
 s_{12} \left(s_{13}+s_{23}\right)+s_{45}\left(s_{34}+s_{35}\right)
\Big] A(1, 2, 3, 4, 5)
\nonumber\\ &+&
{1 \over 60} \Big[
s_{12} \left(s_{13}+2 s_{14}-s_{23}-2 s_{25}\right)
+s_{13} \left(s_{14}-s_{15}-2 s_{35}\right)
+s_{23} \left(s_{24}-s_{25}+2 s_{34}\right)
\nonumber\\ &&
+s_{34} \left( s_{24}-s_{14} \right)
+s_{35} \left(s_{25}-s_{15}\right)
+s_{45} \left( -2 s_{14}+2 s_{25}-s_{34}+s_{35} \right)
\Big]
A(1, 4, 3, 2, 5)
\nonumber\\ &+&
{1 \over 60} \Big[
s_{12} \left(3 s_{13}-s_{24}+s_{25}\right)+s_{13} \left(s_{14}-s_{15}+2 s_{34}-2 s_{35}\right)
+ s_{23} \left( s_{24}-s_{25} \right)
\nonumber\\ &&
+ s_{45} \left(-2 s_{14}-s_{15}+3 s_{25}+2 s_{34}+s_{35}\right)
\Big]
A(1, 3, 4, 2, 5)
\nonumber\\ &+&
{1 \over 20} \Big[
s_{12} \left(s_{13}+s_{14}-s_{15}+s_{23}+s_{24}-s_{25}\right)+2 s_{35} s_{45}
\Big]
A(1, 2, 4, 3, 5)
\nonumber\\ &+&
{1 \over 60} \Big[
 s_{12} \left(s_{13}+3 s_{14}-s_{15}+2 s_{23}-2 s_{25}\right)
+s_{14} \left(s_{45}-s_{34}\right)
\nonumber\\ &&
+ s_{24} \left(s_{34}-s_{45}\right)+s_{35} \left(-2 s_{13}-s_{15}+2 s_{23}+s_{25}+3 s_{45}\right)
\Big]
A(1, 4, 2, 3, 5)
\nonumber\\ &+&
{1 \over 20} \Big[
 2 s_{12} s_{13}+
s_{45}\left(-s_{14}-s_{15}+s_{24}+s_{25}+s_{34}+s_{35}\right)
\Big]
A(1, 3, 2, 4, 5) \,.
\label{kkbasis}
\end{eqnarray}
This expression differs from the result given in ref.~\cite{Fu:2014pya},
but that result apparently contains some typographical errors which will be corrected
in a revised version\footnote{Private communication}.
We emphasize that \eqn{kkbasis} manifestly obeys Jacobi identities and dihedral symmetries
using only the Kleiss-Kuijf relations and the cyclic and reversal properties of the color-ordered
amplitudes (but not momentum conservation or BCJ relations).
\\ [-2mm]

Fu et al. \cite{Fu:2014pya} also give explicit, rather lengthy, expressions
for virtuous numerators for six-point amplitudes.

\section{Virtuous dual-trace functions}
\setcounter{equation}{0}

The color factors $c_i$ and the kinematic numerators $n_i( 1,2, \cdots, n )$
play dual roles in the gauge-theory amplitude (\ref{gluonamp}).
Bern and Dennen \cite{Bern:2011ia} proposed that a role dual
to the traces of generators $\mathop{\rm Tr}\nolimits[\alpha]$ in \eqn{tracedecomp} could be played
by a function $\tau_\alpha( 1,2, \cdots, n )$ of the variables $p_j$ and $\varepsilon_j$
that is related to $n_i( 1,2, \cdots, n )$
in the same way (\ref{colortrace}) that $\mathop{\rm Tr}\nolimits[\alpha]$ is related to $c_i$:
\begin{equation}
n_i ( 1,2, \cdots, n )  ~=~
\sum_{\alpha  \in   {S_n /\mathbb{Z} } }
M_{i \alpha} \, \tau_\alpha ( 1,2, \cdots, n )   \,.
\label{nMtau}
\end{equation}
Just as the Jacobi identities for the color factors (\ref{cjacobi})
can be expressed as $\sum_i \ell_i c_i  = 0$,
where $\ell_i$ are left null vectors of the rank $(n-2)!$ matrix $M_{i\alpha}$,
so the Jacobi identities for the kinematic numerators (\ref{njacobi})
are expressed as $\sum_i \ell_i n_i  = 0$,
and will therefore be automatically satisfied by \eqn{nMtau}.
Using \eqn{nMtau}, together with \eqns{colorordered}{doublepartial}, the color-ordered amplitudes can be written
in terms of the dual-trace functions as
\begin{equation}
 A_\alpha ( 1,2, \cdots, n ) ~=~
\sum_{\beta \in   {S_n /\mathbb{Z} } }
m(\alpha|\beta)
~ \tau_{\beta} ( 1,2, \cdots, n ) \,.
\label{Amtau}
\end{equation}
If we could invert this equation, we would have
a prescription for an amplitude-encoded dual-trace function,
i.e. in which the dependence on the external states is expressed
through the color-ordered amplitudes.
Since $m(\alpha|\beta)$ has rank $(n-3)!$,
there is no unique inverse, but rather a family of generalized inverses,
each corresponding to a particular
gauge choice imposed on the dual-trace functions.
\\ [-2mm]

First we consider the symmetries of the dual-trace functions $\tau_\alpha$.
Because $M_{i\alpha} = M_{i\alpha'}$
when $\alpha$ and $\alpha'$ are related by cyclic permutations,
and $ M_{i\alpha} = (-1)^n M_{i\alpha'} $
when $\alpha'$ is the reverse of $\alpha'$,
we can impose these properties on $\tau_\alpha$ without loss of generality
\begin{eqnarray}
\tau_{\alpha(1)\alpha(2)\cdots \alpha(n)} ( 1,2, \cdots, n )
&= &\tau_{\alpha(2)\alpha(3)\cdots \alpha(1)} ( 1,2, \cdots, n ) \,,
\label{taucyclic}
\\[2 mm]
\tau_{\alpha(1) \alpha(2)\cdots \alpha(n)} ( 1,2, \cdots, n )
&=& (-1)^n \tau_{\alpha(n) \cdots \alpha(2) \alpha(1)} ( 1,2, \cdots, n )
\label{taureverse}
\end{eqnarray}
leaving $(n-1)!/2$ independent dual-trace functions to be determined.
These must satisfy \eqn{nMtau}, which for the independent half-ladder numerators
takes the form
\begin{eqnarray}
{\bf n}_{1 \gamma(2) \cdots \gamma(n-1) n} ( 1,2, \cdots, n )  &=&  \tau_{ 1[ \gamma(2), [ \cdots, [\gamma(n-1), n] \cdots ]]  } ( 1,2, \cdots, n )
\label{ntau}
\\ [2mm]
&=& \tau_{1 \gamma(2) \cdots \gamma(n-1) n}  ( 1,2, \cdots, n ) + (-1)^n
                 \tau_{1 n \gamma(n-1) \cdots \gamma(2)} ( 1,2, \cdots, n )  + \cdots
\nonumber
\end{eqnarray}
where $+\cdots$ consists of terms
$\tau_{1\gamma(2)\cdots\gamma(n)}$
for which $\gamma(2)\neq n$ and $\gamma(n)\neq n$.
Since there are only  $(n-2)!$ kinematic numerators
${\bf n}_{1 \gamma n} $
and $(n-1)!/2$ dual-trace functions $\tau_\alpha$,
there remains a great deal of (gauge) freedom in choosing $\tau_\alpha$.
\\ [-2mm]

In refs.~\cite{Bern:2011ia,Du:2013sha}, Kleiss-Kuijf relations $\sum_\alpha r_\alpha \tau_\alpha = 0$
were imposed on $\tau_\alpha$, reducing the number of independent dual-trace functions to $(n-2)!$.
This is an optional\footnote{
Recall that Kleiss-Kuijf relations follow from the existence of right null
vectors
$\sum_\alpha M_{i\alpha}r_\alpha = 0$
of the matrix $M_{i\alpha}$.
By virtue of \eqn{colorordered},
the color-ordered amplitudes $A_\alpha$ satisfy the
relations $\sum_\alpha r_\alpha A_\alpha =0$.
Kleiss-Kuijf relations do not apply to the trace basis $\mathop{\rm Tr}\nolimits[\alpha]$
and therefore one is not required to impose them on $\tau_\alpha.$
For example, see ref.~\cite{Fu:2013qna}.}
gauge choice, however,
and we choose instead to set to zero all terms of the form
$\tau_{1\gamma(2)\cdots\gamma(n)}  ( 1,2, \cdots, n )$
except
\begin{eqnarray}
\tau_{1\gamma(2)\cdots\gamma(n-2)n-1,n}  ( 1,2, \cdots, n )
&=& -{1\over 2}\sum_{\beta\in S_{n-3}}S[\gamma|\beta]A(1,\beta(2), \cdots, \beta(n-2) ,n,n-1),
\nonumber\\
\tau_{1,n,n-1\gamma(n-2)\cdots\gamma(2)}  ( 1,2, \cdots, n )
&=& ( -1)^n  \tau_{1\gamma(2)\cdots\gamma(n-2)n-1,n}  ( 1,2, \cdots, n ) \,.
\label{tauchoice}
\end{eqnarray}
All remaining dual-trace functions follow from cyclic invariance (\ref{taucyclic}).
Note that our gauge choice for $\tau_\alpha$ also implies a particular gauge choice for
${\bf n}_{1 \gamma n} $, namely, \eqn{gaugechoice}.
According to ref.~\cite{Bern:2011ia}, the ability to express the dual-trace functions in
terms of kinematic numerators requires us to impose Kleiss-Kuijf relations on $\tau_\alpha$.
If, however, our goal is to write amplitude-encoded dual-trace functions, then this restriction
is not necessary, as we will see explicitly below.
\\ [-2mm]

The dual-trace functions defined in \eqn{tauchoice}
are amplitude-encoded and satisfy \eqn{Amtau},
but they are not symmetric functions in the sense of Broedel and Carrasco.\footnote{In the language of
ref.~\cite{Du:2013sha}, they do not have a natural relabeling property.}
In the appendix, it is shown that we can
follow the same procedure as in the previous section
to generate from \eqn{tauchoice} a symmetric dual-trace function
\begin{equation}
\tau ( 1,2, \cdots, n  ) ~\equiv~
{1 \over n!} \sum_{\beta \in S_n}  \tau_{\beta^{-1} } (\beta(1), \cdots, \beta(n))  \,.
\label{symmetrictau}
\end{equation}
This then provides a constructive definition
for a virtuous dual-trace function for tree-level $n$-point amplitudes,
proving that such a representation exists for all $n$.
The symmetric dual-trace function can be used to express the $n$-gluon amplitude as
\begin{equation}
 {\cal A} ( 1,2, \cdots, n )
~=~
\sum_{\alpha \in   {S_n /\mathbb{Z} } }  \sum_{\beta \in   {S_n /\mathbb{Z} } }
\mathop{\rm Tr}\nolimits[\alpha]
~m(\alpha|\beta)
~\tau (\beta)
\label{Tmtau}
\end{equation}
where $\tau(\beta) \equiv \tau(\beta(1), \beta(2), \cdots, \beta(n))$.
\\ [-2mm]

By substituting \eqn{nMtau} into (\ref{gravamp}),
the scattering amplitude for gravitons may be written
\begin{equation}
 {\cal M} ( 1,2, \cdots, n )
~=~
\sum_{\alpha \in   {S_n /\mathbb{Z} } }  \sum_{\beta \in   {S_n /\mathbb{Z} } }
\tau (\alpha) ~ m(\alpha|\beta) ~ \tau (\beta) \,.
\end{equation}
This may equivalently be obtained by replacing $\mathop{\rm Tr}\nolimits[\alpha]$ with $\tau_\alpha$ in \eqn{Tmtau}.
The gravity amplitude may also be written \cite{Bern:2011ia}
\begin{equation}
 {\cal M} ( 1,2, \cdots, n )
~=~
\sum_{\alpha \in   {S_n /\mathbb{Z} } }
\tau (\alpha) ~ A (\alpha)
\end{equation}
which has the nice feature of demonstrating that
the gravity amplitude is {\it manifestly} invariant under an arbitrary
permutation $\beta \in S_n$:
\begin{equation}
 {\cal M} (\beta )
~=~
\sum_{\alpha \in   {S_n /\mathbb{Z} } }
\tau (\beta \alpha)
~ A (\beta \alpha)
~=~
\sum_{\alpha' \in   {S_n /\mathbb{Z} } }
\tau (\alpha')
~ A (\alpha')
~=~
 {\cal M} (  {\mathsf{1}\kern -3pt \mathsf{l} } )\,.
\end{equation}
\\ [-2mm]

In the following subsections, we compute the symmetric dual-trace function (\ref{symmetrictau})
explicitly for four- and five-point amplitudes.
Despite being symmetric, our expressions do not satisfy the Kleiss-Kuijf relations,
illustrating that the virtues of Broedel and Carrasco
are not sufficient to single out a unique dual-trace function.
\\ [-2mm]

\subsection{Four-point symmetric dual-trace functions}

For the four-gluon amplitude, \eqn{tauchoice} yields a nonsymmetric representation
for the dual-trace function
\begin{eqnarray}
\tau_{1234}(i,j,k,l) &=& - {1 \over 2} s_{ij}  A (i,j,l,k)   \,,
\nonumber\\
\tau_{1342}(i,j,k,l) &=& 0 \,,
\nonumber\\
\tau_{1423}(i,j,k,l) &=& 0
\label{fournonsymtau}
\end{eqnarray}
with the remaining $\tau_\alpha$ given by \eqn{taucyclic}.
We now use \eqn{symmetrictau} to generate a symmetric dual-trace function, obtaining
\begin{equation}
\tau(1,2,3,4) ~=~
- \frac{1}{24}
\Big[
\left(s_{12}+s_{34}\right) A(1,3,4,2) + \left(s_{14}+s_{23}\right)
A(1,4,2,3)
\Big]
\label{fourpointtau}
\end{equation}
where we have used
$A(1,2,3,4)= A(4,3,2,1)$
and the cyclic invariance of $A(1,2,3,4)$.
\Eqn{fourpointtau} manifestly obeys
$\tau(1,2,3,4)= \tau(2,3,4,1)$ and $\tau(1,2,3,4)= \tau(4,3,2,1)$;
however,
$\tau(1,2,3,4) + \tau(1,3,4,2) + \tau(1,4,2,3)$ does not vanish;
{i.e.},
\eqn{fourpointtau} does not satisfy the Kleiss-Kuijf relations.
When substituted into \eqn{ntau}, which takes the form
\begin{eqnarray}
{\bf n}(1,2,3,4) &=&  2 \left[ \tau(1,2,3,4)  -   \tau(1,2,4,3)  \right] \,,
\label{fourntau}
\end{eqnarray}
\eqn{fourpointtau} yields precisely the symmetric kinematic numerator found in \eqn{fourpointvirtue}.
\Eqn {fourpointtau} can be written more briefly as
\begin{equation}
\tau (1,2,3,4)
~=~  - {1 \over 6} s_{12}  A(1, 3, 4,2)
\end{equation}
by using momentum conservation and the BCJ relations (\ref{fourpointBCJ}).
\\ [-2mm]

Bern and Dennen proposed that the four-point dual-trace function takes the form \cite{Bern:2011ia}
\begin{equation}
\tau_{BD} (1,2,3,4) ~=~
{1 \over 6} \left[ {\bf n}(1,2,3,4) + {\bf n}(2,3,4,1)  \right] \,.
\label{BernDennentau}
\end{equation}
The expression for $\tau$ given in ref.~\cite{Du:2013sha} is equivalent to \eqn{BernDennentau}.
\Eqn{BernDennentau}
manifestly obeys
$\tau(1,2,3,4)= \tau(2,3,4,1)$ and $\tau(1,2,3,4)= \tau(4,3,2,1)$,
and in addition satisfies the Kleiss-Kuijf relation.
To compare our expression with that of Bern and Dennen,
we substitute the symmetric numerator (\ref{fourpointvirtue}) into \eqn{BernDennentau} to find
\begin{eqnarray}
&&\tau_{BD} (1,2,3,4) ~-~ \tau(1,2,3,4)
\\
&&~~~~ = ~\frac{1}{36}
\Big[
\left(s_{12}+s_{14}\right) A(1,2,3,4) +
\left(s_{12}+s_{13}\right) A(1,3,4,2) +
\left(s_{13}+s_{14}\right) A(1,4,2,3)
\Big] \,.
\nonumber
\end{eqnarray}
The difference is ``pure gauge'': it vanishes when substituted into \eqn{fourntau},
and therefore does not contribute  to ${\bf n}(1,2,3,4)$.
Hence we see that imposing the three virtues of Broedel and Carrasco
on the dual-trace function is not sufficient to
determine it uniquely.
\\ [-2mm]

To further elucidate the difference between \eqns{fourpointtau}{BernDennentau},
we rewrite \eqn{Amtau}
as a matrix equation
\begin{equation}
\begin{pmatrix}
A_{1234} \\
A_{1342} \\
A_{1423}
\end{pmatrix}
~=~ m
\begin{pmatrix}
\tau_{1234} \\
\tau_{1342} \\
\tau_{1423}
\end{pmatrix},
\qquad\qquad
m ~=~-~{2\over s t u} \begin{pmatrix}   u \\ t \\ s \end{pmatrix} \begin{pmatrix}   u &t & s\end{pmatrix}
\label{matrix}
\end{equation}
where $s=s_{12}$, $t=s_{14}$, and $u=s_{13}$.
As expected, $m$ has rank $(4-3)! = 1$.
It therefore does not possess a unique inverse, but rather a
family of generalized inverses $m^+$
\begin{equation}
\begin{pmatrix}
\tau_{1234} \\
\tau_{1342} \\
\tau_{1423}
\end{pmatrix}
~=~ m ^+
\begin{pmatrix}
A_{1234} \\
A_{1342} \\
A_{1423}
\end{pmatrix},
\qquad\qquad m m^+ m= m \,.
\end{equation}
A generalized inverse must satisfy $m m^+ m= m$  \cite{BenIsrael,Boels:2012sy},
which guarantees that the resulting $\tau_\alpha$ will satisfy \eqn{matrix}.
\\ [-2mm]

Different gauge choices for $\tau$ correspond to different generalized inverses.
The nonsymmetric representation (\ref{fournonsymtau})
corresponds to
\begin{equation}
m_{\rm nonsym}^+ ~=~-~{1\over 2} \begin{pmatrix}   0 & s & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}  \,.
\end{equation}
The symmetric dual-trace function (\ref{fourpointtau}) corresponds to
\begin{equation}
m_{\rm sym}^+ ~=~-~{1\over 12} \begin{pmatrix}   0 & s & t \\ s & 0 & u \\ t & u & 0 \end{pmatrix}
\end{equation}
whereas  the Bern-Dennen dual-trace function corresponds to
\begin{equation}
m_{BD}^+ ~=~{1\over 36}
\begin{pmatrix}
-u & -3 s-t & -3t-s \\
 -3 s-u & -t & -3u-s \\
 -3 t-u & -3u-t u & -s \\
\end{pmatrix} \,.
\end{equation}
All three generalized inverses satisfy $m m^+ m= m$.

\subsection{Five-point symmetric dual-trace functions}

For the five-gluon amplitude, \eqn{tauchoice} yields a nonsymmetric representation
\begin{eqnarray}
\tau_{12345}(i,j,k,l,m) &=&
- {1\over 2} \left[ s_{ij} (s_{ik}+s_{jk} )  A(i,j,k,m,l) + s_{ij} s_{ik} A (i,k,j,m,l)\right]\,, \nonumber\\
\tau_{13245}(i,j,k,l,m) &=&
- {1\over 2} \left[ s_{ij} s_{ik}           A(i,j,k,m,l)  + s_{ik} (s_{ij}+s_{jk} )   A(i,k,j,m,l) \right]
\label{fivepointnonsymtau}
\end{eqnarray}
with all remaining dual-trace functions set to zero, except for those related to \eqn{fivepointnonsymtau} by \eqn{taucyclic}.
We now use \eqn{symmetrictau} to generate a symmetric dual-trace function, obtaining
\begin{eqnarray}
\tau(1,2,3,4,5) &=&
\frac{1}{120}
\Big[
(s_{23} s_{24}+s_{34} s_{24}+2 s_{23} s_{34}) A(1, 4, 3, 2, 5)
+(s_{23} s_{24}+s_{12} s_{25}) A(1, 3, 4, 2, 5)
\nonumber\\ &-&
(2 s_{12} s_{15}+s_{25} s_{15}+s_{12} s_{25}) A(1, 2, 4, 3, 5)
+(s_{24} s_{34}+s_{14} s_{45}) A(1, 4, 2, 3, 5)
\nonumber\\ &-&
(s_{14} s_{15}+2 s_{45} s_{15}+s_{14} s_{45}) A(1, 3, 2, 4, 5)
+(s_{34} s_{35}+s_{45} s_{35}+2 s_{34} s_{45}) A(1, 2, 5, 4, 3)
\nonumber\\ &+&
(s_{12} s_{13}+s_{14} s_{15}) A(1, 4, 5, 2, 3)
+(s_{15} s_{25}+s_{35} s_{45}) A(1, 2, 5, 3, 4)
\nonumber\\ &+&
(s_{12} s_{13}+s_{23} s_{13}+2 s_{12} s_{23}) A(1, 4, 5, 3, 2)
-(s_{13} s_{23}+s_{34} s_{35}) A(1, 3, 5, 4, 2)
\Big]
\label{fivepointtau}
\end{eqnarray}
where we have used
$ A(1,2,3,4,5)= -A(5,4,3,2,1) $
together with cyclic invariance of $A(1,2,3,4,5)$.
This expression manifestly satisfies $\tau(1,2,3,4,5)=\tau(2,3,4,5,1)$
and $\tau(1,2,3,4,5) = - \tau(5,4,3,2,1)$, but not the Kleiss-Kuijf relations.
It therefore differs from the expressions given
in refs. \cite{Bern:2011ia,Du:2013sha},
which do satisfy the Kleiss-Kuijf relations.
\\ [-2mm]

\Eqn{fivepointtau} yields precisely the symmetric kinematic numerator \eqn{fivepointvirtue}
when substituted into \eqn{fiventau}, which takes the form
\begin{eqnarray}
{\bf n}(1,2,3,4,5) &=&
2 \left[\tau(1,2,3,4,5)  + \tau(1,2,5,4,3) + \tau(1,4,5,3,2) + \tau(1,3,5,4,2) \right] \,.
\label{fiventau}
\end{eqnarray}
Finally, the Kleiss-Kuijf relations for the color-ordered amplitudes can be used to rewrite \eqn{fivepointtau}
as
\begin{eqnarray}
{\bf \tau}(1,2,3,4,5) &=&
{1 \over 120} \Big[
(
 s_{12} s_{13}+2 s_{12} s_{23}+2 s_{34} s_{45}+s_{35} s_{45}
) A(1, 2, 3, 4, 5)
\nonumber\\ &+&
(
 s_{12} s_{13}+s_{14} s_{15}+s_{23} s_{24}+s_{15} s_{25}+2 s_{23} s_{34}+s_{24} s_{34}+s_{35} s_{45}
)A(1, 4, 3, 2, 5)
\nonumber\\ &+&
(
 s_{12} s_{13}+s_{14} s_{15}+s_{23} s_{24}+s_{12} s_{25}+s_{34} s_{35}+2 s_{34} s_{45}+s_{35} s_{45}
)A(1, 3, 4, 2, 5)
\nonumber\\ &+&
(
 s_{12} s_{13}-2 s_{12} s_{15}+2 s_{12} s_{23}-s_{12} s_{25}-s_{34} s_{35}+s_{35} s_{45}
) A(1, 2, 4, 3, 5)
\nonumber\\ &+&
(
 s_{12} s_{13}+s_{23} s_{13}+2 s_{12} s_{23}+s_{15} s_{25}+s_{24} s_{34}+s_{14} s_{45}+s_{35} s_{45}
)A(1, 4, 2, 3, 5)
\nonumber\\ &+&
(
 s_{12} s_{13}-s_{23} s_{13}-s_{14} s_{45}-2 s_{15} s_{45}+2 s_{34} s_{45}+s_{35} s_{45}
)A(1, 3, 2, 4, 5)
\Big]\,.
\nonumber\\
\end{eqnarray}

\section{Discussion}
\setcounter{equation}{0}

In this paper, we have offered a constructive procedure for computing
virtuous kinematic numerators for $n$-point gauge-theory scattering amplitudes;
that is, numerators that simultaneously satisfy color-kinematic duality,
are expressed in terms of color-ordered amplitudes,
and are symmetric in the sense of Broedel and Carrasco.
We have presented explicit expressions
for four- and five-point amplitudes,
which although somewhat lengthy,
have the advantage of manifestly obeying the Jacobi identities and
diagram symmetries without having to invoke the BCJ relations
for the color-ordered amplitudes.
Our results are equivalent (upon using the BCJ relations)
to other virtuous expressions in the literature,
suggesting the possibility that the three virtues of Broedel and Carrasco
are sufficient to uniquely determine the kinematic numerators,
although we do not have a proof of this.
\\ [-2mm]

We have also applied this procedure to
compute symmetric, amplitude-encoded dual-trace functions $\tau$ for $n$-point amplitudes,
presenting explicit expressions for four- and five-point amplitudes.
In this case, the results are not uniquely determined by these criteria alone.
In particular, symmetric expressions for $\tau$ obtained by other authors
additionally satisfy (optional) Kleiss-Kuijf relations, whereas our results do not.
While it is possible that a constructive procedure could be found to
generate virtuous dual-trace functions
for $n$-point amplitudes that also satisfy Kleiss-Kuijf relations,
it is also possible that an alternative criterion
to single out a unique dual-trace function
may be more useful or natural.

\section*{Acknowledgments}
I am grateful to Johannes Broedel for useful correspondence.
I especially wish to thank Freddy Cachazo
and Ellis Yuan for crucial conversations
at the outset of this project.
This research was supported in part by the NSF under grant no. PHY10-67961.

redpajama_set_name: RedPajamaArXiv


text:
Trump Family Values
Donald Trump Jr. Killed an Endangered Sheep in Mongolia, Got Government Approval Days Later
The First Son, who claims to hate nepotism, also met one-on-one with the president of Mongolia, in addition to other special treatment.
By Bess Levin
By Sue Ogrocki/AP/REX/Shutterstock.
Like his namesake, Donald Trump Jr. frequently complains about how unfair life has been since his father moved into the White House, claiming that his family has had to make so many sacrifices since January 2017. Recalling a visit to Arlington National Cemetery the day before Trump Senior's inauguration, Junior wrote in his book that while he'd never actually served and wasn't six feet under, he felt a connection with those who'd died on the battlefield. "I had a deep sense of the importance of the presidency and a love of our country," he wrote. "I also thought of all the attacks we'd already suffered as a family, and about all the sacrifices we'd have to make to help my father succeed—voluntarily giving up a huge chunk of our business and all international deals to avoid the appearance that we were 'profiting off of the office. Frankly, it was a big sacrifice, costing us millions and millions of dollars annually."
Recently, Donny and his brother Eric—who, by the by, actually haven't stopped doing international deals—have taken to comparing their situation to that of Hunter Biden and insisting that while the former V.P.'s son corruptly and outrageously traded on his father's name for access and profit, they would never do such a thing. "When you're the father, and your son's entire career is dependent on that, they own you," Eric whined to Fox News' Jeanine Pirro in a televised fit of clinical un-self-awareness. "If I was doing the same thing that that family was doing, I'd be in jail. Why is it that every family goes into politics and enriches themselves?" Similarly piqued, Junior moaned to Sean Hannity one night in October, "I wish my name was Hunter Biden. I could go abroad, make millions off of my father's presidency—I'd be a really rich guy. It would be incredible."
While it may be news to him, Donny, along with Eric, runs his family's business, which has gone abroad and made millions off his father's presidency. As of August, Don Jr. was reportedly worth more than $150 million, the bulk of which was presumably acquired by virtue of being a Trump. In addition to continuing to do foreign deals, the company from which he profits has also received a nice chunk of change from various GOP lawmakers, corporate executives with business before the Justice Department, visiting foreign government officials, and anyone looking to kiss the ring who knows their money will go far at the family's Washington hotel. Oh, and he also apparently had the red carpet rolled out for him on one of his recent hunting trips abroad, where he—a supposedly private citizen—met one-on-one with the country's president and received special treatment vis-à-vis killing endangered species.
ProPublica has the lowdown:
The rocky highlands of Central Asia, in a remote region of Western Mongolia, are home to a plummeting population of the largest sheep in the world, the argali. The endangered species is beloved for its giant curving horns, which can run over 6 feet in length. On a hunting trip this August, Donald Trump Jr. shot and killed one. His adventure was supported by government resources from both the U.S. and Mongolia, which each sent security services to accompany the president's eldest son and grandson on the multiday trip. It also thrust Trump Jr. directly into the controversial world of Mongolian trophy hunting—a polarizing practice in a country that views the big-horned rams as a national treasure. The right to kill an argali is controlled by an opaque permitting system that experts say is mostly based on money, connections, and politics.
The Mongolian government granted Trump Jr. a coveted and rare permit to slay the animal retroactively on September 2, after he'd left the region following his trip. It's unusual for permits to be issued after a hunter's stay. It was one of only three permits to be issued in that hunting region, local records show. Afterward, Trump Jr. met privately with the country's president, Khaltmaagiin Battulga, before departing the capital of Ulaanbaatar back to the U.S., according to Khuantai Khafezyn, a local government official in the region where Trump Jr. hunted the argali and a former government official with knowledge of the meeting. It isn't clear what was discussed. Trump Jr. wouldn't answer questions about the meeting. Representatives for Battulga haven't responded to requests for comment.
According to reporters Jake Pearson and Anand Tumurtogoo, the hunt and presidential sit-down came "just weeks after high-level government discussions—including a White House meeting—between officials from the U.S. and Mongolia," which relies on America for economic and security support. (If only it could offer up some dirt on Joe Biden!) "What are the chances the Mongolian government would've done any of that to someone who wasn't the son of the United States' president?" wondered Kathleen Clark, a professor specializing in legal ethics at Washington University in St. Louis School of Law. She noted that while Trump Jr. is not an employee of the U.S. government, his last name incentivizes foreign officials like the ones in Mongolia to give him special treatment out of a "desire on the part of a foreign government to curry favor with the president's family." Which is an academic way of saying, and hopefully the Trump Boys will hold their ears for this, that Donny only received the full-court press on his sheep-killing expedition because his father is president.
In a statement, a spokesman for Don Jr., who posted photos of his vacation—including a historically embarrassing shot of him on a horse, and another of himself in traditional Mongolian garb despite the fact that local next to him is in regular clothing—said the trip was strictly a personal expedition, and that he purchased the seven-day package at a National Rifle Association charity auction before his father announced his candidacy for president in 2015.
More Great Stories From Vanity Fair
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— Is Rudy Giuliani truly in trouble?
— The secret life and strange death of Quadriga cofounder, Gerald Cotten
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Looking for more? Sign up for our daily Hive newsletter and never miss a story.
Get the Hive Newsletter
The freshest-and most essential-updates from Washington, Wall Street, and Silicon Valley.
Of Course Trump's Idiot Son Outed the Alleged Whistle-Blower
Would you expect anything less from Don Jr.?
Don Jr. Thinks Not Doing Foreign Deals is the Same as Dying in Battle
A visit to the Arlington National Cemetery made the first son tear up thinking about all the "sacrifices" he's been forced to make.
Don Jr.'s Girlfriend Thinks Online Dating Is for Losers
Kimberly Guilfoyle had some harsh words for the conservative protesters who booed the hell out of her and Don Jr. on Sunday.
Daddy's Boys: Eric and Donald Trump Jr.
A peek into the lives of the future First Sons.
By Vanity Fair

redpajama_set_name: RedPajamaCommonCrawl


Meizu 16 and Meizu 16 Plus were launched in China on Wednesday as the successors to the Meizu 15 and Meizu 15 Plus that debuted in April. With a starting price of CNY 2,698 (roughly Rs. 27,100), the new models sport 18:9 display panels and are powered by Qualcomm Snapdragon 845 SoC, along with up to 8GB of RAM. Meizu has also offered an in-display fingerprint sensor that is touted to have a recognition rate of 99.12 percent. The Meizu 16 and Meizu 16 Plus additionally include mEngine 2.0 that is highly inspired by Apple's Taptic Engine. There is a built-in thermal management system that uses a custom copper cooling tube to maintain the on-body temperature. The Meizu 16 and Meizu 16 Plus also come with up to 256GB of onboard storage.
The Meizu 16 price in China has been set at CNY 2,698 (roughly Rs. 27,100) for the 6GB RAM and 64GB storage option, while its 6GB RAM and 128GB storage variant comes with a price tag of CNY 2,998 (roughly Rs. 30,100), and the 8GB RAM variant with the same 128GB storage can be purchased at CNY 3,298 (roughly Rs. 33,100). In contrast, the Meizu 16 Plus price has been set at CNY 3,198 (roughly Rs. 31,400) for 6GB RAM and 128GB storage configuration, whereas its 8GB RAM and 128GB storage model is priced at CNY 3,498 (roughly Rs. 35,100), and the 8GB RAM and 256GB storage variant can be bought at CNY 3,998 (roughly Rs. 40,100). All these variants are available for pre-orders in China starting Wednesday. However, the sale of the Meizu 16 models will begin from August 16, while the Meizu 16 Plus will go on sale starting August 20.
The dual-SIM (Nano) Meizu 16 runs Flyme OS based on Android 8.0 Oreo and features a 6-inch full-HD+ (1080×2160 pixels) Super AMOLED display with an 18:9 aspect ratio, 430cd/m2 brightness, and a 402ppi pixel density. Under the hood, the smartphone is powered by a Qualcomm Snapdragon 845 SoC, coupled with Adreno 630 GPU and 6GB or 8GB RAM options. The smartphone sports a dual rear camera setup that has a 12-megapixel primary 1/2.3-inch Sony IMX380 sensor along with a wide-angle lens that has an f/1.8 aperture and phase detection autofocus (PDAF) and a 20-megapixel Sony IMX350 sensor along with a telephoto lens that has an f/2.6 aperture. The camera setup is powered by a four-axis optical image stabilisation (OIS) and paired with a 6-LED ring flash. For selfies, there is a 20-megapixel sensor at the front – along with an f/2.0 aperture. The frontal sensor also supports an artificial intelligence (AI) powered face unlock. Further, there is an ArcSoft beauty algorithm adaptive skin technology as well as Face AE facial brightness enhancement. The rear camera also works with a preloaded portrait background blur mode and 3x lossless zoom mode.
On the part of storage, the Meizu 16 has 64GB and 128GB storage options that are backed by UFS 2.1 but not expandable via microSD card. The smartphone has 4G VoLTE, Wi-Fi 802.11ac, Bluetooth v5.0, GPS/ A-GPS, and USB Type-C. Onboard sensors include an ambient light sensor, electronic compass, gravity sensor, gyroscope, hall magnetic, and pressure sensing. There is also an in-display fingerprint sensor. Also, the handset has linear dual speakers with an immersive surround sound technology. There is a 3010mAh battery that supports fast charging through the proprietary mCharge. Besides, the model measures 160.4×78.2×7.3mm and weighs 182 grams.
Similar to the Meizu 16, the dual-SIM (Nano) Meizu 16 Plus runs Android 8.0 Oreo-based Flyme OS. But unlike the smaller variant, there is a 6.5-inch Super AMOLED panel that comes in (1080×2160 pixels) full-HD+ resolution along with an 18:9 aspect ratio and a pixel density of 374ppi. There is a Qualcomm Snapdragon 845 SoC, paired with Adreno 630 GPU and 6GB or 8GB RAM options. It bears the same camera setup as the Meizu 16.
Meizu has provided 128GB and 256GB UFS 2.1 storage options that both are not expandable through microSD card. The Meizu 16 Plus has 4G VoLTE, Wi-Fi 802.11ac, Bluetooth v5.0, GPS/ A-GPS, and USB Type-C. Sensors on board include an ambient light sensor, electronic compass, gravity sensor, gyroscope, hall magnetic, and pressure sensing. There is also an in-display fingerprint sensor. The handset packs a 3640mAh battery that supports mCharge technology. Lastly, it measures 160.4×78.2×7.3mm and weighs 182 grams.


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Budget allocation for Cork and Shannon Airports is "utterly inadequate" – says TD
Clare Independent T.D. Michael McNamara says the €10m allocated to Cork and Shannon Airports in Budget 2021 is "utterly inadequate" to compensate the airports for the Government's decision to shut down the Irish aviation sector.
He said, "Aviation is one of Ireland's largest sectors, yet the Government has chosen to shut it down in a way that no other European country has done."
Deputy McNamara added, "Clarity is required around the exact breakdown and conditionality of this funding given that it is to be shared between two airports. Shannon Group alone is facing weekly revenue losses of €1.3m and is expecting annual revenue for 2020 to be down by 80%, and as Group CEO Mary Considine told me last week, the airport's costs have not substantially reduced."
"The Government's track record on providing money to Shannon Group doesn't inspire confidence. We saw great fanfare around the announcement of €2.6m in funding for Shannon Heritage but it has since transpired that only €550,000 is being provided," concluded Deputy McNamara.
AREA: SOUTH CORK, NEWS
Ireland tourism industry is really suffering from COVID19 shutdowns
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I broke off a tooth while eating a piece of candy that had a foreign object in it. What can I do?
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San Francisco is a wonderful and beautiful city by itself, but viewing this city from out at sea is an experience on its own. Take this year's celebrations aboard the San Francisco Spirit Yacht and allow yourself to be treated to an unforgettable night out on the seas. Whether you want to celebrate a special occasion or holidays such as New Year's Eve, Halloween, or July 4th, this yacht will have you covered!
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The San Francisco Spirit will transform any celebration to an experience on the seas. When you sail on this yacht, there will be fully stocked bars available for you to order up drinks from. This beautiful yacht also has multiple decks for you to enjoy and outside observation decks. Enjoy the beautiful San Francisco weather as you sail on your cruise and celebrate a night out like you never have. Check out all of our events aboard the San Francisco Spirit and book your tickets for an unforgettable celebration!


"[T]he idea of a single state was not a plan for the future — to be sought or averted — but an accurate description of the reality on the ground… Legally, commercially and administratively, the Jewish and Palestinian populations were interlaced.
• The Israeli Jews will never willingly give up their state in order to dissolve, unarmed and unprotected, into an Arab-majority state along with a majority population with who they have been in bitter conflict with for many decades.
This is a policy which has started with its hoped-for endpoint — a single (Arab) state — and been worked backwards to produce a justification. Our policy starts with the reality for Palestinian and Israeli workers, on the ground, and worked forward to find the most democratic, feasible arrangement that can begin to mend the antagonism between the two peoples.
Of course it is understandable why such a set of ideas might be attractive to a British activist willing to ignore the obvious problems with this version of a single state "solution". Firstly, it avoids the need to advocate the "smashing" — the bloody destruction — of Israel by some outside force (as the SWP have said: "The road to Jerusalem lies through [an aggressive government in] Cairo"). In this version of the single state "solution" the Israeli Jews will voluntarily see the error of their ways, disarm, dismantle their state — their armed protection — and dissolve into an Arab majority state. Whoever thinks this is a possible is a fantasist — or a cynic, who has a second agenda in the background, and is using this policy to dupe the unquestioning British middle-class liberals who sincerely advocate the policy.
In contrast, the two states solution is possible because it asks less of the Israeli Jews, stopping short of attacking their democratic rights to self-determination (meaning, the right to a state). Any policy which denies the Israeli Jews the right to self-determination is unreasonable — and right now, given the balance of power, utterly unfeasible.
The 1960 South African census showed that three million of the population were white (19%), a little less than one-in-five of the population. The white South Africans were a caste and even white workers benefited from the exploitation of black labour.
By 1960 black South Africans had been stripped of almost all forms of political representation. Pass books and Bantustans were part of an intricate system which regulated the use of black labour on behalf of racist South African capitalism. The solution was to abolish distinctions of race and deliver political equality.
The reason that calling Israel an apartheid state is so important for the single state activists is what they want to take from the South African experience is the political programme — one person, one vote in a single state (with the abolition of Bantustans etc).
The problem is that the structure of Israel is nothing like that of apartheid era South Africa. In Israel — inside the 1967 borders — Israeli Jews are the big majority, with Arab Israelis a little more than one-in-five of the population. Israeli capitalism relies — mainly — on the exploitation of Jewish Israeli workers.
Arab Israelis are a discriminated-against minority. However Arab Israelis have not had their rights crushed. They are able to vote, to build their own parties, to protest. Arab parties have MKs in the Knesset.
The answer to the discrimination is to end the discrimination, not to attempt to destroy Israel!
And Gaza and the West Bank are not Bantustans containing cheap labour for the Israeli economy. The time when many thousands of Palestinians crossed the Green Line each day to work in Israel — before the first intifada — has long gone. The Arabs that work in Israel are Israeli citizens. Gaza is currently sealed off and strangled by the Israeli (and Egyptian) state(s), not integrated into the Israeli economy as a subordinate part.
The plight of working-class Palestinians is very much our concern. But before we know what to say, we must actually understand the full reality of the situation.
Still more ridiculous is to describe the Palestinian diaspora living in other Arab states as existing under Israeli apartheid. The descendants of those Palestinians that fled in 1948 living in camps in, say, Lebanon or Syria, have been denied the right to integrate into their host societies and denied political rights by the Arab governments of those states.
For example, the Lebanese government has denied Palestinians civil rights and they have no automatic right to work, join a union, or get social security. Palestinians are banned from 25 areas of work, including medicine, law and engineering. Troops guard Palestinian camps in order — for example — to stop Palestinians building and improving their housing. They are deliberately kept in squalor.
This is not the Israeli state's action, it is the action of an Arab state. If Palestinians in Lebanon are living under apartheid it is Lebanese apartheid.
The attempt to claim that a single state already exists — across Israel, the West Bank, Gaza and the scattered Palestinians in a dozen Arab countries — is a piece of not very clever political cheating available only to those who are determined that a single (Arab) state should be advocated and are not troubled by the facts of the real world.
Take Gaza. Are we supposed to believe that the Hamas-run Islamic police state, is really a part of Israel? Despite the withdrawal of the Israeli state, and the dismantling of all Israeli settlements in Gaza, in 2005?
How does this make any sense? The precise purpose of Israeli withdrawal was that Israel wanted nothing more to do with Gaza.
Any solution to the Israeli-Palestinian conflict must not only focus on the reality of Palestinian suffering (the realities of checkpoints in the West Bank, the discrimination, the unfair detentions of children etc), but also — honestly — note the reality of the social structures in these societies.
Many Israelis already accept the idea of a two states solution; almost no Israelis will ever accept the voluntary abolition of Israel in favour of dissolving themselves, unprotected, into an Arab state. The experience of the Holocaust and the extreme tension in the region for decades, the wars and hostility to Israel, guarantee this. The only honest way to advocate a single state is to advocate extreme violence against Israel —that way leads to a war against the Israeli Jews, not peace and reconciliation.
The left would not expect peoples who have been at peace for decades to dissolve themselves into a unitary state. The left does not expect Argentina and Chile to merge, nor Poland and Germany, so why expect it in a much more fraught, difficult situation in Israel-Palestine?
Any solution we advocate should be consistent with other, similar disputes. No one on the left advocates that Turkey, which is as brutal an occupying power towards the Kurds as Israel is to the Palestinians, be replaced with a unitary Kurdish state. None of the left advocates a comprehensive campaign of boycotting Turkish produce or culture.
This singling out of Israel, and treating Israel as an illegitimate and essentially, inevitably and eternally racist state gets to the heart of much of the left's agitation on the question. It begs the question: why has the left one rule for Turkey and one for Israel? To avoid this type of antisemitism the left must be willing to grant equal rights to the Jewish Israeli nation.
Workers' unity through mutual recognition of national rights is the only democratic policy for socialists to adopt. We are for an independent Palestinian state alongside Israel, with the same rights as Israel.
Solidarity 477, 29 August 2018Israel/Palestine Boycott Israel?


The Boardman girls swim team took first place against Canfield and Massillon Perry with a score of 158 on Dec. 22. Mikel Flatley was the only double winner for the Spartans taking the 50 freestyle (28.31) and the 100 butterfly (1:11.04). Single event winners included Madison Ricciuti in 1 meter diving (186), Jennah Markovitch in the 200 individual medley (2:47.85), Alyssa DiNapoli in the 100 backstroke (1:12.51) and Hayley Ho in the 100 breaststroke (1:22.30). All three of the Boardman girls relays swept first place finishes. This included the 200 free relay that touched out the Canfield Lady Cardinals by 13 hundredths of a second. Members of this relay included Hayley Ho, Sarah Varga, Mikel Flatley and Jennah Markovitch. The 200 medley relay team of DiNapoli, Ho, Flatley and Kuras placed first (2:12.18), as did the 400 freestyle relay team of Casey Wert, Sarah Varga, Allison Kuras and Alyssa DiNapoli with a time of 4:37.21.
The girls varsity basketball team lost to Struthers on Dec. 22 with a score of 56-52. The leading scorer for the Spartans was Maria Torres who finished with a game high 23 points, which included five 3-pointers. Raegan Burkey hit for 15 points and led the team in steals with four. Also scoring in double digits was Katie Stamp, who had 10 points. She led the team in rebounding with nine. Cate Green and Serene Khatib each had two points to round out the scoring for Boardman.
The boys varsity basketball team suffered a loss to Hudson with a score of 65-47 on Dec. 22. Che Trevena and Derrick Anderson each had 11 points. Daeone Martin had nine points, Ethan Andersen had six points, Cam Kreps had four points, and Zach Ryan and Charlie Davis each had three points.
The boys freshman basketball team defeated Hudson 47-27 on Dec. 22. The game was back and forth for much of the first three quarters with both teams exchanging scoring runs that resulted in a 23-23 tie heading into the fourth. The Spartan defense was the difference in the fourth quarter, forcing eight Hudson turnovers. Cam Thompson and Terence Thomas led the way on defense, allowing Boardman to capitalize in transition. Seth Cervello had a big fourth quarter on offense, scoring all eight of his points in the period. The leading scorer for Boardman was Terence Thomas with 14 points. Also scoring for the Spartans were DJ Evans and Reshaud Biggs with six each, Cam Thompson with four, Tommy Nawrocki with three and Luke Ryan, Anthony Hightower and Ryder Kreps all adding two points.
The Boardman junior varsity basketball team fell to Struthers with a score of 44-40 on Dec. 22. Boardman was led in scoring by Jenna Olexa with 12. Tini Zheng and Serene Khatib each tallied six for the maroon and white. Emma Mills scored four, while Josie Porter and Emma Tokarsky had three each. Madi Larch, Priya Shah and Tanna Ortega each added two for Boardman.
The freshman boys basketball team continued their winning ways on Dec. 20 by defeating Youngstown East by a score of 45-32. After a sluggish first quarter, the Spartans found a nice rhythm with their second unit, leading to a 19 point second quarter outburst. Seth Cervello led the charge with three triples in the period. Boardman was able to sustain their advantage the rest of the game, resulting in their sixth straight win to start the season. Leading scorers for the Spartans were DJ Evans with 10 points, Seth Cervello with nine and Luke Ryan with seven points. Ben Alvarico added four points. Cam Thompson, Ryder Kreps and Max Rassega all chipped in with three points. Also getting in the book were Terence Thomas, Anthony Hightower and Tommy Nawrocki, each with two points.
The seventh-grade boys white team beat Warren gold with a score of 29-19 on Dec. 20. Jaden Payne led the way with 14 points, while Jack Ericson and Landon Mitchell each finished with four. DeMayne Earvin, Noah Ziak and Jack Neill each scored two points and Ryan Conti finished the scoring with one point.
Boardman maroon eighth-grade boys posted a convincing 47-32 win over Austintown Middle School red at Glenwood on Dec. 20. The win improves the Spartans to a 4-3 record going into the Christmas break. In this one, Boardman led wire-to-wire and were up as many as 21 points. Brady Depietro led the way with 22 points, Nate Christmas added eight, and five other players contributed into the scoring column. The defensive effort was outstanding, shutting down the top two threats for the Falcons.
The Boardman girls seventh-grade maroon basketball team beat Austintown red with a score of 33-29 on Dec. 20. Leading scorers for Boardman were Ava Bosnjak - 16, CiCi Rivera - 7, Dana Qutail - 4, Leah Evans - 2, Gabby Goddard - 2 and Brooke Yoder - 2. Solid defense from Camryn Lyda, along with the rest of the girls, keyed the come from behind win.