aainstitutetext: Center for Theoretical Physics, School of Physics and
Technology, Wuhan University,
No.299 Bayi Road, Wuhan 430072, Chinabbinstitutetext: College of Science,
Tibet University,
No.10 Zangda East Road, Lasa, 850000, Chinaccinstitutetext: Suzhou Institute
of Wuhan University,
No.377 Linquan Street, Suzhou, 215123, China

# A graphic approach to identities induced from multi-trace Einstein-Yang-
Mills amplitudes

Yi-Jian Du a Linghui Hou  yijian.du<EMAIL_ADDRESS>
###### Abstract

Symmetries of Einstein-Yang-Mills (EYM) amplitudes, together with the
recursive expansions, induce nontrivial identities for pure Yang-Mills
amplitudes. In the previous work Hou:2018bwm , we have already proven that the
identities induced from tree level single-trace EYM amplitudes can be
precisely expanded in terms of BCJ relations. In this paper, we extend the
discussions to those identities induced from all tree level _multi-trace_ EYM
amplitudes. Particularly, we establish a refined graphic rule for multi-trace
EYM amplitudes and then show that the induced identities can be fully
decomposed in terms of BCJ relations.

###### Keywords:

Amplitude Relation, Gauge invariance

##  1 Introduction

It has been proven that any tree level multi-trace Einstein-Yang-Mills (EYM)
amplitude $A^{(m,s)}$ with $m$ gluon traces and $s$ gravitons can be
recursively expanded in terms of the amplitudes $A^{(m^{\prime},s^{\prime})}$
with $s^{\prime}+m^{\prime}<s+m$ Du:2017gnh . In the special case $m=1$, these
relations give rise to the earlier proposed expansions of single-trace EYM
amplitudes Fu:2017uzt ; Chiodaroli:2017ngp ; Teng:2017tbo . When the recursive
expansions are applied repeatedly until there is no graviton and only one
gluon trace, an arbitrary tree-level EYM amplitude is finally expressed as a
combination of tree level color-ordered Yang-Mills (YM) ones. Such pure-YM
expansions precisely coincide with the earlier studies on amplitudes with only
a few gravitons and/or gluon traces Stieberger:2016lng ; Nandan:2016pya ;
delaCruz:2016gnm ; Schlotterer:2016cxa .

The pure YM expansions of EYM amplitudes, together with the gauge invariance
conditions of gravitons or the cyclic symmetries of gluon traces, induce
nontrivial identities for color-ordered YM amplitudes Du:2017gnh . These
identities guaranteed the localities in the Britto-Cachazo-Feng-Witten (BCFW)
Britto:2004ap ; Britto:2005fq  proof of the recursive expansions Fu:2017uzt ;
Du:2017gnh  and played a crucial role in the proof Du:2018khm  of the
equivalence between distinct approaches Du:2016tbc ; Carrasco:2016ldy ;
Du:2017kpo  to nonlinear sigma model amplitudes.

A prominent feature of the identities induced from multi-trace EYM amplitudes
is that the coefficients therein generally contain factors of all the three
types of Lorentz contractions $\epsilon\cdot\epsilon$, $\epsilon\cdot k$ and
$k\cdot k$ where $\epsilon^{\mu}$ are half polarizations of gravitons and
$k^{\mu}$ are external momenta. This is quite different from the known Kleiss-
Kuijf (KK) Kleiss:1988ne  and Bern-Carrasco-Johansson (BCJ) relations
Bern:2008qj  whose coefficients at most involve $k\cdot k$ factors.
Nevertheless, several clues imply that the identities induced from multi-trace
EYM amplitudes can be related with BCJ relations: (i). First, the relationship
between the identities induced from single-trace EYM amplitudes and BCJ
relations have already been founded Hou:2018bwm , while the multi-trace
amplitudes can be obtained through replacing gravitons by gluon traces in an
appropriate way (as pointed in Du:2017gnh ). (ii). Second, as demonstrated in
Du:2017gnh , examples with a few gluon traces and gravitons provided evidence
of the connection between the induced identities and BCJ relations. (iii).
Third, the fact that color-kinematic duality Bern:2008qj , which is the
underlaid structure of the BCJ relations for YM amplitudes, can be resulted
from gauge invariance Arkani-Hamed:2016rak  also implies that the identities
induced from gauge invariance can be related with BCJ relations.

In the current paper, we extend our discussions in Hou:2018bwm  to multi-trace
cases and show that all these identities induced from multi-trace EYM
amplitudes can be expanded in terms of BCJ relations. The main idea is
sketched as follows:

  * (i).

Refined graphic rule  We first introduce a refined graphic rule111Graphic
approaches to scattering amplitudes can be found in many literatures
including: graphic study on Cachazo-He-Yuan Cachazo:2013gna ; Cachazo:2013hca
; Cachazo:2013iea ; Cachazo:2014xea  formulas (see e.g. Lam:2015sqb ;
Lam:2016tlk ; Bjerrum-Bohr:2016juj ; Huang:2017ydz ; Gao:2017dek ; He:2018pue
; Lam:2018tgm ; Lam:2019mfk ), graphic construction for BCJ numerators
Du:2017kpo  and the refined graphic rule for single-trace EYM amplitudes
Hou:2018bwm . for the coefficients in pure-YM expansions of multi-trace EYM
amplitudes, where the three types of factors $\epsilon\cdot\epsilon$,
$\epsilon\cdot k$ and $k\cdot k$ are presented by distinct types of lines (as
already introduced in the study of single-trace case Hou:2018bwm ) and a new
type of line is invented to record the relative order of gluons in a gluon
trace. The induced identities are then expressed through a summation over
connected tree graphs which are built of the four types of lines, while color-
ordered YM amplitudes corresponding to each graph can be collected in a proper
way.

  * (ii).

Skeletons and components  To relate the induced identities with BCJ relations,
we split gluon traces in an appropriate way and then remove all the $k\cdot k$
lines from the graphs. After that, a _physical graph_ (i.e. a graph defined by
the refined graphic rule) turns to a disconnected one which is called
_skeleton_ and consists of disjoint _components_. The summation over all
physical graphs is thus given by summing over all skeletons and summing over
all the physical graphs corresponding to a given skeleton.

  * (iii).

The final upper and lower blocks  For a given skeleton, the summation over all
possible physical graphs can further be arranged by the following two steps:
(a). Connect components via $k\cdot k$ lines properly such that the skeleton
becomes a graph with only two disjoint maximally connected subgraphs, which
are called _the final upper and lower blocks_ , (b). Connect the final upper
and lower blocks into a physical graph via a $k\cdot k$ line. _Spurious
graphs_ , which are not defined by the refined graphic rule, can also be
introduced for a given configuration of the final upper and lower blocks. When
associated with proper signs, all spurious graphs cancel out. Then the
summation over all physical graphs for a given final upper and lower blocks
can be reexpressed by a summation over all physical and spurious graphs.

  * (iv).

Expressing an induced identity by BCJ relations  We finally find that all
contributions of the physical and spurious graphs, corresponding to a given
configuration of the final upper and lower blocks, together can be written as
a combination of the graph-based BCJ relations Hou:2018bwm  which have been
proven to be combinations of the traditional BCJ relations Bern:2008qj 222In
this paper, we distinguish the earlier proposed forms of BCJ relations (see
Bern:2008qj ; BjerrumBohr:2009rd ; Chen:2011jxa ) from the graph-based BCJ
relations Hou:2018bwm  by the name _traditional BCJ relations_. .

The structure of this paper is the following. In section 2, we introduce a
refined graphic rule for the expansion of multi-trace EYM amplitudes and then
express the induced identities by this rule. We further show two examples in
section 3 which support the fact that the contributions of all those graphs
corresponding to a given skeleton can be written as a combination of graph-
based BCJ relations. In section 4, the general pattern of skeletons and
components are studied. We then provide the general construction of the final
upper and lower blocks for a given skeleton. The pattern of spurious graphs is
also discussed. We finally show how to express the contributions of all
(physical and spurious) graphs for a given final upper and lower blocks in
terms of the graph-based BCJ relations. This work is summarized in section 5.
A review of the background knowledge, the proof of the splitting trace
relation and the pattern of the signs for graphs are included in the appendix.

#### Convention of notations

The notations in this paper are gathered as follows.

Permutations and sets: Permutations are denoted by boldface Greek letters:
$\boldsymbol{\sigma}$, $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$,
$\boldsymbol{\gamma}$, $\boldsymbol{\zeta}$, etc. The $i$-th element in
$\boldsymbol{\sigma}$ is denoted by $\sigma(i)$. The position of an element
$a$ in $\boldsymbol{\sigma}$ is expressed by $\sigma^{-1}(a)$. The inverse
permutation of elements in an ordered set $\boldsymbol{X}$ is denoted by
$\boldsymbol{X}^{T}$. Shuffle permutations of two ordered sets
$\boldsymbol{X}$ and $\boldsymbol{Y}$ are written as
$\boldsymbol{X}\shuffle\boldsymbol{Y}$. The number of elements in set
$\boldsymbol{A}$ is presented by $|\boldsymbol{A}|$. We use
$\boldsymbol{A}\setminus\boldsymbol{B}$ to denote the difference of the sets
$\boldsymbol{A}$ and $\boldsymbol{B}$.

Gravitons and gluon traces: Gluon traces are denoted by boldface numbers
$\boldsymbol{1}$, $\boldsymbol{2}$, …, or boldface lowercase Latin letters
$\boldsymbol{t}$, $\boldsymbol{i}$ … If a trace $\boldsymbol{t}_{i}$ can be
written as
$\boldsymbol{t}_{i}=\\{a_{i},\boldsymbol{X}_{i},b_{i},\boldsymbol{Y}_{i}\\}$ (
$\boldsymbol{X}_{i}$ and $\boldsymbol{Y}_{i}$ are the two ordered sets of
gluons which are separated by the gluons $a_{i}$ and $b_{i}$), we define
$\mathsf{KK}[\boldsymbol{t}_{i},a_{i},b_{i}]\equiv\boldsymbol{X}_{i}\shuffle\boldsymbol{Y}_{i}^{T}$
and
$(-1)^{|\boldsymbol{t}_{i},a_{i},b_{i}|}\equiv(-1)^{|\boldsymbol{Y}_{i}|}$.
The notation $\mathsf{H}$ ( $\mathsf{H}=\\{h_{1},h_{2},\dots,h_{s}\\}$) stands
for a graviton set with gravitons $h_{1}$, …, $h_{s}$. The set of gravitons
$h_{1},\dots,h_{s}$ and gluon traces $\boldsymbol{2},\dots,\boldsymbol{m}$ is
denoted by $\boldsymbol{\mathcal{H}}$
($\boldsymbol{\mathcal{H}}=\\{\boldsymbol{2},\dots,\boldsymbol{m},h_{1},\dots,h_{s}\\}$).
The multi-trace EYM amplitude with traces
$\boldsymbol{1}=\\{1,\dots,r\\},\boldsymbol{2},\dots,\boldsymbol{m}$ and
graviton set $\mathsf{H}$ is written as
$A(1,2,\ldots,r|\boldsymbol{2}|\ldots|\boldsymbol{m}\|\mathsf{H})$.

Graphs: Graphs are denoted by $\mathcal{F}$, $\mathcal{G}$ or $\mathcal{T}$.
The notation $\mathcal{G}^{\prime}$ stands for the skeleton of a graph
$\mathcal{G}$. Reference order and root set are respectively expressed by
$\mathsf{R}$ and $\mathcal{R}$. Components of a skeleton are given by
$\mathscr{A}$, $\mathscr{B}$, $\mathscr{C}$ …, while a chain of components is
denoted by $\mathbb{CH}$. The reference order of components is given by
$\mathsf{R}_{\mathscr{C}}$. The final upper and lower blocks are respectively
presented by $\mathscr{U}$ and $\mathscr{L}$ whose disjoint union is
$\mathscr{U}\oplus\mathscr{L}$.

##  2 Refined graphic rule for multi-trace EYM amplitudes and the induced
identities

In this section, we present a _refined graphic rule_ , by which one expresses
a tree level multi-trace EYM amplitude
$A(1,2,\ldots,r|\boldsymbol{2}|\ldots|\boldsymbol{m}\|\mathsf{H})$ with $m$
gluon traces
$\boldsymbol{1}\equiv\\{1,2,\ldots,r\\},\boldsymbol{2},\ldots,\boldsymbol{m}$
and $s$ gravitons $\mathsf{H}\equiv\\{h_{1},h_{2},\dots,h_{s}\\}$ in terms of
$(m+s)$-point tree level color-ordered YM amplitudes:

$\displaystyle\boxed{A(1,2,\ldots,r|\boldsymbol{2}|\ldots|\boldsymbol{m}\|\mathsf{H})=\sum\limits_{\mathcal{F}}\mathcal{C}^{\mathcal{F}}\left[\sum\limits_{\boldsymbol{\sigma}^{\mathcal{F}}}A(1,\boldsymbol{\sigma}^{\mathcal{F}},r)\right]},$
(2.1)

where we have summed over all possible connected tree graphs $\mathcal{F}$.
Each graph $\mathcal{F}$ defines a coefficient $\mathcal{C}^{\mathcal{F}}$ and
proper permutations $1,\boldsymbol{\sigma}^{\mathcal{F}},r$ (of all elements
in
$\boldsymbol{1}\cup\boldsymbol{2}\cup\ldots\cup\boldsymbol{m}\cup\mathsf{H}$)
according to the refined graphic rule. The expansion (2.1) is obtained by
applying the recursive expansion (A.1) (see Du:2017gnh ) iteratively and it is
essentially equivalent to the graphic expansion given in Du:2017gnh . Two
examples which are helpful for understanding the refined graphic rule are
given in this section. We then provide two identities that are respectively
induced by the gauge invariance condition of a graviton and the cyclic
symmetry of a gluon trace.

###  2.1 Refined graphic rule

To illustrate the refined graphic rule for the expansion (2.1), it is helpful
to consider the gluon trace $\boldsymbol{1}$ in the EYM amplitude
$A(1,2,\ldots,r|\boldsymbol{2}|\ldots|\boldsymbol{m}\|\mathsf{H})$ as a
special one and denote the set of other gluon traces
$\boldsymbol{2},\ldots,\boldsymbol{m}$ and all gravitons $h_{1},\dots,h_{s}$
by
$\boldsymbol{\mathcal{H}}\equiv\\{\boldsymbol{2},\dots,\boldsymbol{m},h_{1},\dots,h_{s}\\}$.
In the set $\boldsymbol{\mathcal{H}}$, each gluon trace is always considered
as a single object. The refined graphic rule is then expressed as follows:

Step-1 Define a reference order of elements in $\boldsymbol{\mathcal{H}}$ as
the following ordered set:

$\displaystyle\mathsf{R}=\left\\{\mathcal{H}_{\rho(1)},\mathcal{H}_{\rho(2)},\dots,\mathcal{H}_{\rho(l=m+s-1)}\right\\},$
(2.2)

where each $\mathcal{H}_{i}$ stands for an element (i.e. a graviton or a gluon
trace) of $\boldsymbol{\mathcal{H}}$ and the position of $\mathcal{H}_{i}$ in
$\mathsf{R}$ is called its _weight_. Apparently, $\mathcal{H}_{\rho(l=m+s-1)}$
is the highest-weight element in the reference order (2.2), while
$\mathcal{H}_{\rho(1)}$ is the lowest-weight one. We also define the root set
$\mathcal{R}$ by collecting elements of the trace $\boldsymbol{1}$:

$\displaystyle\mathcal{R}\equiv\boldsymbol{1}\setminus\\{r\\}=\\{1,2,\dots,r-1\\},$
(2.3)

where the last element $r\in\boldsymbol{1}$ is always excluded333If a set
$\boldsymbol{B}$ is a subset of $\boldsymbol{A}$,
$\boldsymbol{A}\setminus\boldsymbol{B}$ is defined by moving all elements of
$\boldsymbol{B}$ from $\boldsymbol{A}$..

Step-2 Pick out the highest-weight element $\mathcal{H}_{\rho(l)}$ as well as
other elements ${j_{1}},\dots,{j_{u}}$ (not necessary in the same relative
order in $\mathsf{R}$) from the ordered set $\mathsf{R}$, then construct a
chain towards an element $w$ in the root set $\mathcal{R}$:

$\displaystyle\mathcal{H}_{\rho(l)}\to j_{u}\to\dots\to j_{1}\to w.$ (2.4)

In the above chain, the $\mathcal{H}_{\rho(l)}$, ${j_{1}}$, …, ${j_{u}}$ and
$w$ are respectively mentioned as the _starting element_ (graviton or trace),
the _internal elements_ (gravitons and/or traces) and the _ending element_
(also mentioned as the root of the chain). The special case that a chain with
no internal element $\mathcal{H}_{\rho(l)}\to w$ is allowed. The contribution
of the chain (2.4) and the graphic expression are evaluated in the following
way:

Figure 1: The graphs for a starting graviton and an internal graviton are
correspondingly given by (b). Figure 2: The starting trace and an internal
trace of a chain are respectively presented by the graphs (a) and (b). The
trace $\boldsymbol{1}\equiv\\{1,2,\dots,r\\}$ is expressed by the structure
(c). In this paper, the two end nodes of a trace are always colored red.
Figure 3: Four types of lines in the refined graphic rule for multi-trace EYM
amplitudes

  * •

Polarizations and momenta Each half polarization $\epsilon^{\mu}$ of a
graviton is expressed by a solid line connected to the graviton. The momentum
$k^{\mu}$ of any node (graviton or gluon) is presented by a solid arrow line
pointing to the node. Here, if an arrow points away from the direction of
root, an extra minus should be dressed.

  * •

Starting and internal gravitons If the starting element
$\mathcal{H}_{\rho(l)}$ is a graviton $h_{i}$, it contributes a half
polarization $\epsilon^{\mu}_{h_{i}}$ presented by the structure Fig. 1 (a).
If an internal element is a graviton $h_{i}\in\\{j_{1},\dots,j_{u}\\}$, it
contributes a strength tensor $F^{\mu\,\nu}_{h_{i}}\equiv
k^{\mu}_{h_{i}}\epsilon^{\nu}_{h_{i}}-k^{\nu}_{h_{i}}\epsilon^{\mu}_{h_{i}}$
and is expressed by the structure Fig. 1 (b).

  * •

Starting and internal traces To express a gluon trace $\boldsymbol{t}_{i}$, we
select an ordered pair of gluons $\\{a_{i},b_{i}\\}$
$(a_{i},b_{i}\in\boldsymbol{t}_{i})$ where $a_{i}$ and $b_{i}$ play as the
first and the last elements of the trace respectively. Then we arrange other
gluons of $\boldsymbol{t}_{i}$ in a relative order
$\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{t}_{i},a_{i},b_{i}]$. Supposing
that the trace can be written as
$\boldsymbol{t}_{i}=\\{a_{i},\boldsymbol{X},b_{i},\boldsymbol{Y}\\}$ where
$\boldsymbol{X}$ and $\boldsymbol{Y}$ are the two ordered sets of gluons
separated by $a_{i}$ and $b_{i}$, the
$\mathsf{KK}[\boldsymbol{t}_{i},a_{i},b_{i}]$ is defined by permutations
$\boldsymbol{X}\shuffle\boldsymbol{Y}^{T}$ ($\boldsymbol{Y}^{T}$ is the
inverse order of $\boldsymbol{Y}$ and $\boldsymbol{A}\shuffle\boldsymbol{B}$
for two ordered sets $\boldsymbol{A}$, $\boldsymbol{B}$ stands for the set of
all permutations obtained by merging the two sets together with keeping the
relative order of elements in each). Hence all gluons inside the trace are
arranged in the relative order444Here, the $l$-th element $\beta(l)$ in the
permutation $\boldsymbol{\beta}$ is labeled by $\beta_{l}$ for short.

$\displaystyle
a_{i},\beta_{1},\dots,\beta_{|\boldsymbol{t}_{i}|-2},b_{i},~{}~{}(\\{\beta_{1},\dots,\beta_{|\boldsymbol{t}_{i}|-2}\\}\equiv\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{t}_{i},a_{i},b_{i}]).$
(2.5)

Supposing $a_{i}$ is nearer to root than $b_{i}$, we draw a dashed arrow line
between any two adjacent gluons in the permutation (2.5). Each arrow points
towards the direction of the node $a_{i}$ as shown by Fig. 2, thus it also
points towards the root. If the trace $\boldsymbol{t}_{i}$ plays as the
starting element $\mathcal{H}_{\rho(l)}$ of the chain (2.4), it is presented
by the structure Fig. 2 (a) and contributes a $-k_{a_{i}}^{\mu}$ to the
coefficient. If $\boldsymbol{t}_{i}\in\\{j_{1},\dots,j_{u}\\}$ is an internal
trace, it should be expressed by the structure Fig. 2 (b) and contributes a
$-k_{b_{i}}^{\mu}k_{a_{i}}^{\nu}$. As shown by Fig. 2 (c), the special trace
$\boldsymbol{1}=\\{1,2,\dots,r\\}$ is presented via connecting adjacent gluons
by dashed arrow lines whose arrows point towards the direction of the first
gluon $1$.

  * •

Lorentz contractions and Line styles Contracting the Lorentz indices
accompanying with adjacent elements, we get the contribution of the chain
(2.4)

$\displaystyle\mathcal{E}_{\mathcal{H}_{\rho(l)}}\cdot\mathbb{F}_{j_{u}}\cdot...\cdot\mathbb{F}_{j_{1}}\cdot
k_{w},$ (2.6)

in which

$\displaystyle\mathcal{E}^{\mu}_{x}$ $\displaystyle=$
$\displaystyle\biggl{\\{}\begin{array}[]{cc}~{}~{}\epsilon^{\mu}_{h_{i}}&\,(\text{if
$x$ is a graviton $h_{i}$})\\\ ~{}~{}-k^{\mu}_{a_{i}}&\,\,\,\,\,\,\,(\text{if
$x$ is a gluon trace $\boldsymbol{t}_{i}$})\\\ \end{array}$ (2.9)
$\displaystyle\mathbb{F}^{\mu\,\nu}_{y}$ $\displaystyle=$
$\displaystyle\Biggl{\\{}\begin{array}[]{cc}F^{\mu\,\nu}_{h_{i}}\equiv
k^{\mu}_{h_{i}}\epsilon^{\nu}_{h_{i}}-k^{\nu}_{h_{i}}\epsilon^{\nu}_{h_{i}}&(\text{if
$y$ is a graviton $h_{i}$})\\\
-k_{b_{i}}^{\mu}k_{a_{i}}^{\nu}&~{}\,\,\,\,(\text{if $y$ is a gluon trace
$\boldsymbol{t}_{i}$})\\\ \end{array}.$ (2.12)

There are three types of lines Fig. 3 (a), (b) and (c), which are resulted by
the Lorentz contraction (2.6) and correspond to $\epsilon\cdot\epsilon$,
$\epsilon\cdot k$ and $k\cdot k$. Recalling that dashed arrow lines Fig. 3 (d)
between gluons in a same trace have been introduced, we have four types of
lines in all.

Redefine the ordered set $\mathsf{R}$ by removing the elements which have been
used:

$\displaystyle\mathsf{R}\to\mathsf{R}^{\prime}=\mathsf{R}\setminus\\{{j_{1}},\dots,{j_{u}},\mathcal{H}_{\rho(l)}\\}\equiv\left\\{\mathcal{H}_{\rho(1^{\prime})},\mathcal{H}_{\rho(2^{\prime})},\dots,\mathcal{H}_{\rho(l^{\prime})}\right\\}.$
(2.13)

and redefine the root set $\mathcal{R}$ by:

$\displaystyle\mathcal{R}\to\mathcal{R}^{\prime}=\mathcal{R}\cup{j_{1}}\cup\dots\cup{j_{u}}\cup\mathcal{H}_{\rho(l)}.$
(2.14)

Here each element in $\mathcal{R}$ is either a graviton or a gluon (this is
different from the ordered set $\mathsf{R}$ where a trace is considered as a
single element). If $j_{i}$ or $\mathcal{H}_{\rho(l)}$ in eq. (2.14) is a
graviton, it always stands for a single-element set $\\{j_{i}\\}$ or
$\\{\mathcal{H}_{\rho(l)}\\}$.

Step-3 Repeat the above step by using the new defined $\mathsf{R}$ and
$\mathcal{R}$ until the ordered set $\mathsf{R}$ becomes empty. Then a fully
connected tree graph $\mathcal{F}$ which is rooted at the gluon
$1\in\boldsymbol{1}$ is produced.

Step-4 For a given graph $\mathcal{F}$, the coefficient
$\mathcal{C}^{\mathcal{F}}$ in eq. (2.1) can be read off as the product of all
factors corresponding to the type-1,-2 and -3 lines (see Fig. 3). The sign
associated with such a graph gets two distinct contributions (i).
$(-1)^{\mathcal{N}(\mathcal{F})}$, where $\mathcal{N}(\mathcal{F})$ is the
number of arrows pointing away from the gluon $1$; (ii). Each trace
$\boldsymbol{t}_{i}$ contributes a $(-1)^{|\boldsymbol{t}_{i},a_{i},b_{i}|}$
for given $a_{i}$ and $b_{i}$, where $|\boldsymbol{t}_{i},a_{i},b_{i}|$ is the
number of elements in $\boldsymbol{Y}_{i}$ if the trace $\boldsymbol{t}_{i}$
can be written as $a_{i},\boldsymbol{X}_{i},b_{i},\boldsymbol{Y}_{i}$.

Step-5 Collect amplitudes $A(1,\boldsymbol{\sigma}^{\mathcal{F}},r)$ for a
given graph $\mathcal{F}$. In any graph $\mathcal{F}$, the gluons $1$ (i.e.
the root) and $r$ in the trace $\boldsymbol{1}=\\{1,2,\dots,r\\}$ are always
treated as the first and the last elements. Permutations
$\boldsymbol{\sigma}^{\mathcal{F}}$ are determined as follows: (i). Two
adjacent nodes $x$ and $y$ which are connected by a line (of any style) must
live on a path towards the root $1$. If $x$ is nearer to $1$ than $y$ on this
path, we have
$({\sigma}^{\mathcal{F}})^{-1}(x)<({\sigma}^{\mathcal{F}})^{-1}(y)$ where
$({\sigma}^{\mathcal{F}})^{-1}(a)$ denotes the position555This is understood
as follows: assuming that the position of $a$ in
$\boldsymbol{\sigma}^{\mathcal{F}}$ is $j$, we have
$a={\sigma}^{\mathcal{F}}(j)$, hence it is reasonable to define
$j=({\sigma}^{\mathcal{F}})^{-1}(a)$. of $a$. (ii). If there are several
branches attached to a node, the relative order is defined by shuffling the
branches together.

Figure 4: Typical graphs for the pure YM expansion of the amplitude
$A(1,2\ldots r|\boldsymbol{2}\|h_{1})$. Graphs of structures (a), (b) and (c)
contribute to the expansion with the reference order
$\mathsf{R}=\left\\{\boldsymbol{2},h_{1}\right\\}$. Graphs (b), (c), (d) and
(e) contribute to the expansion with the reference order
$\mathsf{R}=\left\\{h_{1},\boldsymbol{2}\right\\}$.

When summing over all possible graphs constructed by the above steps, (i.e.,
(i). summing over all graphs with given $\\{a_{i},b_{i}\\}$ pairs and given
permutations
$\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{t}_{i},a_{i},b_{i}]$ for all
traces, (ii). summing over all possible permutations
$\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{t}_{i},a_{i},b_{i}]$ for given
$\\{a_{i},b_{i}\\}$ pairs in all traces, (iii). summing over all possible
choices of the $\\{a_{i},b_{i}\\}$ pairs for internal traces and all possible
choices of $a_{i}\in\boldsymbol{t}_{i},a_{i}\neq b_{i}$ for starting traces
with fixed $b_{i}$’s666Notice that the refined graphic rule is given by
applying the recursive expansion (A.1) iteratively. The $b_{i}$ of the
starting trace in each step of recursive expansion can be chosen freely (see
section A). This implies that the $b_{i}$’s for the same trace, which plays as
the starting element of different chains, can be chosen differently. In this
paper, we fix the $b_{i}\in\boldsymbol{t}_{i}$ as the same one in all the
graphs where the trace $\boldsymbol{t}_{i}$ plays as a starting element. ), we
finally arrive the expansion (2.1). In the coming subsection, we show a
concrete example to explain this rule.

###  2.2 Examples for the refined graphic rule

Now we take the double-trace amplitude $A(1,2\ldots r|\boldsymbol{2}\|h_{1})$
with one graviton as an example. For this amplitude, the reference order can
be chosen as either $\mathsf{R}=\left\\{\boldsymbol{2},h_{1}\right\\}$ or
$\mathsf{R}=\left\\{h_{1},\boldsymbol{2}\right\\}$ in which the highest-weight
element is respectively the graviton $h_{1}$ or the gluon trace
$\boldsymbol{2}$. We study these two cases separately.

If $\mathsf{R}=\left\\{\boldsymbol{2},h_{1}\right\\}$, the typical graphs are
shown by Fig. 4 (a), (b) and (c). Correspondingly, these graphs contribute

$\displaystyle\mathcal{C}^{\text{(a)}}$ $\displaystyle=$
$\displaystyle-(\epsilon_{h_{1}}\cdot k_{b_{2}})(k_{a_{2}}\cdot k_{l}),$
$\displaystyle\boldsymbol{\sigma}^{\text{(a)}}$ $\displaystyle\in$
$\displaystyle\\{2,\dots,l,\\{l+1,\dots,r-1\\}\shuffle\\{a_{2},\beta_{1},\dots,\beta_{|\boldsymbol{2}|-2},b_{2},h_{1}\\}\\},$
(2.15) $\displaystyle\mathcal{C}^{\text{(b)}}$ $\displaystyle=$
$\displaystyle-(\epsilon_{h_{1}}\cdot k_{l^{\prime}})(k_{a_{2}}\cdot k_{l}),$
$\displaystyle\boldsymbol{\sigma}^{\text{(b)}}$ $\displaystyle\in$
$\displaystyle\bigl{\\{}2,\dots,l^{\prime},\\{h_{1}\\}\shuffle\bigl{\\{}l^{\prime}+1,\dots,l,\\{l+1,\dots,r-1\\}\shuffle\\{a_{2},\beta_{1},\dots,\beta_{|\boldsymbol{2}|-2},c_{2}\\}\bigr{\\}}\bigr{\\}},$
(2.16) $\displaystyle\mathcal{C}^{\text{(c)}}$ $\displaystyle=$
$\displaystyle-(\epsilon_{h_{1}}\cdot k_{l})(k_{a_{2}}\cdot k_{h_{1}}),$
$\displaystyle\boldsymbol{\sigma}^{\text{(c)}}$ $\displaystyle\in$
$\displaystyle\\{2,\dots,l,\\{l+1,\dots,r-1\\}\shuffle\\{h_{1},a_{2},\beta_{1},\dots,\beta_{|\boldsymbol{2}|-2},c_{2}\\}\\},$
(2.17)

where the permutations
$\boldsymbol{\beta}=\\{\beta_{1},\beta_{2},\dots,\beta_{|\boldsymbol{2}|-2}\\}$
in $\boldsymbol{\sigma}^{\text{(a)}}$ and
$\boldsymbol{\sigma}^{\text{(b)},\text{(c)}}$ satisfy
$\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a_{2},b_{2}]$ and
$\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a_{2},c_{2}]$ respectively.
The signs caused by arrows pointing away from the root are absorbed into the
coefficients $\mathcal{C}$ in eqs. (2.15), (2.16) and (2.17), but the signs
$(-1)^{|\boldsymbol{t}_{2},a_{2},b_{2}|}$,
$(-1)^{|\boldsymbol{t}_{2},a_{2},c_{2}|}$ induced by the trace are not. Then
the sum over all graphs in eq. (2.1) for the amplitude $A(1,2\ldots
r|\boldsymbol{2}\|h_{1})$ is given by

$\displaystyle\sum\limits_{l\in\\{1,\dots,r-1\\}}\,\Biggl{[}\,\widetilde{\sum\limits_{\\{a_{2},b_{2}\\}\subset\boldsymbol{2}}}\,\sum\limits_{\boldsymbol{\sigma}^{\text{(a)}}}\,\mathcal{C}^{\text{(a)}}A(1,\boldsymbol{\sigma}^{\text{(a)}},r)$
(2.18)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\widetilde{\sum\limits_{\scriptsize{\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq
c_{2}(c_{2}\in\boldsymbol{2})\end{subarray}}}}\biggl{(}\,\sum\limits_{l^{\prime}\in\\{1,\dots,r-1\\}}\sum\limits_{\boldsymbol{\sigma}^{\text{(b)}}}\mathcal{C}^{\text{(b)}}A(1,\boldsymbol{\sigma}^{\text{(b)}},r)+\sum\limits_{\boldsymbol{\sigma}^{\text{(c)}}}\,\mathcal{C}^{\text{(c)}}\,A(1,\boldsymbol{\sigma}^{\text{(c)}},r)\biggr{)}\Biggr{]}.$

where the summation over
$\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a_{2},b_{2}]$
($\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a_{2},c_{2}]$) and the
factor $(-1)^{|\boldsymbol{t}_{2},a_{2},b_{2}|}$
($(-1)^{|\boldsymbol{t}_{2},a_{2},c_{2}|}$) corresponding to the trace
$\boldsymbol{2}$ have been absorbed into the notation
$\widetilde{\sum}_{\\{a_{2},b_{2}\\}\subset\boldsymbol{2}}$
($\widetilde{\sum}_{\scriptsize{\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq b_{2}(b_{2}\in\boldsymbol{2})\end{subarray}}}$), i.e.,

$\displaystyle\widetilde{\sum\limits_{\\{a_{2},b_{2}\\}\subset\boldsymbol{2}}}$
$\displaystyle\equiv$
$\displaystyle\sum\limits_{\\{a_{2},b_{2}\\}\subset\boldsymbol{2}}(-1)^{|\boldsymbol{t}_{2},a_{2},b_{2}|}\sum\limits_{\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a_{2},b_{2}]},$
$\displaystyle~{}~{}\widetilde{\sum\limits_{\scriptsize{\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq c_{2}(c_{2}\in\boldsymbol{2})\end{subarray}}}}$
$\displaystyle\equiv$
$\displaystyle{\sum\limits_{\scriptsize{\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq
c_{2}(c_{2}\in\boldsymbol{2})\end{subarray}}}}(-1)^{|\boldsymbol{t}_{2},a_{2},c_{2}|}\sum\limits_{\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a_{2},c_{2}]}.$
(2.19)

The $c_{2}\in\boldsymbol{2}$ in the second term of eq. (2.18) is arbitrarily
fixed because the trace $\boldsymbol{2}$ plays as the starting trace in Fig. 4
(b) and (c).

Figure 5: The induced identities (2.23) and (2.25) are obtained respectively
by imposing the replacements (a) and (b) on the graphs in eq. (2.1) where the
corresponding highest-weight elements are the graviton $h_{a}$ and the trace
$\boldsymbol{t}_{0}$. The sum over all
$\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{t}_{0},a_{0},b_{0}]$ (with the
sign $(-1)^{|\boldsymbol{t}_{0},a_{0},b_{0}|}$) on the RHS of the replacement
(b) can be further expressed by (c), where the full trace $\boldsymbol{t}_{0}$
is supposed to be $d_{1},d_{2},\dots
d_{j-1},a_{0}=d_{j},d_{j+1},\dots,d_{|\boldsymbol{t}_{0}|-1},b_{0}=d_{|\boldsymbol{t}_{0}|}$.

If $\mathsf{R}=\left\\{h_{1},\boldsymbol{2}\right\\}$, the typical graphs are
given by Fig. 4 (b), (c), (d) and (e). The expression of Fig. 4 (b) and (c)
are already shown by eq. (2.16) and eq. (2.17). The contributions from Fig. 4
(d) and (e) read

$\displaystyle\mathcal{C}^{\text{(d)}}$ $\displaystyle=$
$\displaystyle(k_{h_{1}}\cdot k_{l})(\epsilon_{h_{1}}\cdot k_{a_{2}}),$
$\displaystyle\boldsymbol{\sigma}^{\text{(d)}}$ $\displaystyle\in$
$\displaystyle\\{2,\dots,l,\\{l+1,\dots,r-1\\}\shuffle\\{h_{1},a_{2},\beta_{1},\dots,\beta_{|\boldsymbol{2}|-2},c_{2}\\}\\}$
(2.20) $\displaystyle\mathcal{C}^{\text{(e)}}$ $\displaystyle=$
$\displaystyle-(\epsilon_{h_{1}}\cdot k_{l^{\prime}})(k_{a_{2}}\cdot k_{l}),$
$\displaystyle\boldsymbol{\sigma}^{\text{(e)}}$ $\displaystyle\in$
$\displaystyle\bigl{\\{}2,\dots,l,\\{l+1,\dots,r-1\\}\shuffle\\{a_{2},\beta_{1},\dots,l^{\prime}=\beta_{j},\\{\beta_{j+1},\dots,\beta_{|\boldsymbol{2}|-2},c_{2}\\}\shuffle\\{h_{1}\\}\\}\bigr{\\}},$
(2.21)

where $l^{\prime}$ can be any element in the trace $\boldsymbol{2}$. The sum
over all graphs in eq. (2.1) for the amplitude $A(1,2\ldots
r|\boldsymbol{2}\|h_{1})$ is then written as

$\displaystyle\sum\limits_{l\in\\{1,\dots,r-1\\}}\widetilde{\sum\limits_{\scriptsize{\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq
c_{2}(c_{2}\in\boldsymbol{2})\end{subarray}}}}\Biggl{[}\,\sum\limits_{l^{\prime}\in\\{1,\dots,r-1\\}}\sum\limits_{\boldsymbol{\sigma}^{\text{(b)}}}\mathcal{C}^{\text{(b)}}A(1,\boldsymbol{\sigma}^{\text{(b)}},r)+\,\sum\limits_{\boldsymbol{\sigma}^{\text{(c)}}}\mathcal{C}^{\text{(c)}}A(1,\boldsymbol{\sigma}^{\text{(c)}},r)$
(2.22)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum\limits_{\boldsymbol{\sigma}^{\text{(d)}}}\mathcal{C}^{\text{(d)}}A(1,\boldsymbol{\sigma}^{\text{(d)}},r)+\sum\limits_{l^{\prime}\in\boldsymbol{2}}\sum\limits_{\boldsymbol{\sigma}^{\text{(e)}}}\mathcal{C}^{\text{(e)}}A(1,\boldsymbol{\sigma}^{\text{(e)}},r)\Biggr{]}.$

###  2.3 Induced identities by refined graphs

As pointed in Du:2017gnh , one can induce a nontrivial identity (see eq.
(A.11) or eq. (A.12)) of EYM amplitudes by imposing each of the following
conditions on the recursive expansion (A.1): (i). the gauge invariance
condition of a graviton, (ii). the cyclic symmetry of a gluon trace. When the
EYM amplitudes on the LHS of eq. (A.11) and eq. (A.12) are further expanded
repeatedly according to eq. (A.1), we finally arrive identities for pure YM
amplitudes. Such an identity can also be obtained by imposing the condition
(i) or (ii) on eq. (2.1) straightforwardly. As shown in the appendix, we only
need to study identities which are induced by conditions of the highest-weight
element. Those identities induced by conditions of elements other than the
highest-weight one are essentially treated as cases with a smaller
$\boldsymbol{\mathcal{H}}$ set.

If the highest-weight element $\mathcal{H}_{\rho(l)}$ in the reference order
(2.2) is a graviton $h_{a}$, the gauge invariance condition of $h_{a}$ states
that the expansion (2.1) vanishes when $\epsilon^{\mu}_{h_{a}}$ is replaced by
$k^{\mu}_{h_{a}}$:

$\displaystyle\boxed{0=\sum\limits_{\mathcal{F}}\mathcal{C}^{\mathcal{F}}\big{|}_{\epsilon_{h_{a}}\to
k_{h_{a}}}\biggl{[}\,\sum\limits_{\boldsymbol{\sigma}^{\mathcal{F}}}A(1,\boldsymbol{\sigma}^{\mathcal{F}},r)\,\biggr{]}}.$
(2.23)

The corresponding graphs can be obtained by the replacement Fig. 5 (a)
777Although, the arrow in Fig. 5 (a) points away from the root, we do not
associate a minus sign with this graph, for an overall sign in the identity
eq. (2.23) can always be neglected..

If the highest-weight element $\mathcal{H}_{\rho(l)}$ in the reference order
(2.2) is a gluon trace $\boldsymbol{t}_{0}$, the following identity is induced
by the replacement

$\displaystyle\\{a_{0},\mathsf{KK}[\boldsymbol{t}_{0},a_{0},b_{0}],b_{0}\\}\to\\{a_{0},\mathsf{KK}[\boldsymbol{t}_{0},a_{0},b_{0}]\\},$
(2.24)

for any $a_{0}\in\boldsymbol{t_{0}}~{}(a_{0}\neq b_{0})$:

$\displaystyle\boxed{0=\sum\limits_{\mathcal{F}}\mathcal{C}^{\mathcal{F}}\biggl{[}\,\sum\limits_{\boldsymbol{\sigma}^{\mathcal{F}}}A(1,\boldsymbol{\sigma}^{\mathcal{F}},r)\,\biggr{]}\bigg{|}_{\begin{subarray}{c}\\{a_{0},\mathsf{KK}[\boldsymbol{t}_{0},a_{0},b_{0}],b_{0}\\}\\\
\to\\{a_{0},\mathsf{KK}[\boldsymbol{t}_{0},a_{0},b_{0}]\\}\end{subarray}}}.$
(2.25)

This identity is understood as follows: when we consider
$\\{a_{0},\mathsf{KK}[\boldsymbol{t}_{0},a_{0},b_{0}]\\}$ (for
$a_{0}\in\boldsymbol{t}_{0}$) (which does not contain the fixed gluon
$b_{0}\in\boldsymbol{t}_{0}$) as the highest-weight element
$\mathcal{H}_{\rho(l)}$ (see Fig. 5 (b)) and then apply the refined graphic
rule, the total contribution of the RHS of eq. (2.1) must vanish. As stated in
Du:2017gnh , this identity is essentially a result of the cyclic symmetry of
the trace $\boldsymbol{t}_{0}$. It is worth pointing out an interesting
property Du:2017gnh : when we sum over
$\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{t}_{0},a_{0},b_{0}]$ with the
sign $(-1)^{|\boldsymbol{t}_{0},a_{0},b_{0}|}$, the trace after the
replacement Fig. 5 (b) becomes the RHS of Fig. 5 (c) (where
$\boldsymbol{t}_{0}$ is supposed to be $d_{1},d_{2},\dots
d_{j-1},a_{0}=d_{j},d_{j+1},\dots,d_{|\boldsymbol{t}_{0}|-1},b_{0}=d_{|\boldsymbol{t}|_{0}}$).
In the coming sections, we always use the RHS of Fig. 5 (c) to stand for the
highest-weight element in the identity (2.25).

In the next section, we show that the identities (2.23) and (2.25) induced
from the double-trace EYM amplitude $A(1,\cdots,r|\boldsymbol{2}\|h_{1})$ can
be expanded in terms of graph-based BCJ relations.

##  3 Identities induced from $A(1,\cdots,r|\boldsymbol{2}\|h_{1})$ as
combinations of BCJ relations

We now take the identities induced from the amplitude
$A(1,\cdots,r|\boldsymbol{2}\|h_{1})$ as explicit examples and show that eq.
(2.23) and eq. (2.25) can be written as combinations of graph-based BCJ
relations (A.15) (thus traditional BCJ relations (A.14)). Critical features of
these examples are further summarized.

Figure 6: Typical graphs for the identity induced from the double-trace
amplitude $A(1,2,\cdots,r|\boldsymbol{2}\|h_{1})$ with the reference order
$\mathsf{R}=\\{\boldsymbol{2},h_{1}\\}$. The trace $\boldsymbol{2}$ plays as
an internal trace in (a) and a starting trace in (b), (c).

###  3.1 Example-1: the identity (2.23) induced from
$A(1,2,\cdots,r|\boldsymbol{2}\|h_{1})$

When the reference order is chosen as $\mathsf{R}=\\{\boldsymbol{2},h_{1}\\}$,
the highest-weight element is the graviton $h_{1}$. Thus the expansion (2.1)
for the double-trace amplitude $A(1,2,\cdots,r|\boldsymbol{2}\|h_{1})$ induces
an identity (2.23) under the replacement $\epsilon^{\mu}_{h_{1}}\to
k^{\mu}_{h_{1}}$. Typical graphs for this identity are given by Fig. 6 (a),
(b) and (c) which are correspondingly obtained from Fig. 4 (a), (b) and (c)
via the replacement Fig. 5 (a). The total contributions of these graphs are
then written as

$\displaystyle T^{\text{(a)}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\sigma}^{\text{(a)}}}(k_{h_{1}}\cdot
k_{b_{2}})(-k_{a_{2}}\cdot
k_{l})A\big{(}1,\,\boldsymbol{\sigma}^{\text{(a)}},\,r\big{)},$ (3.1)
$\displaystyle T^{\text{(b)}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\sigma}^{\text{(b)}}}(k_{h_{1}}\cdot
k_{l^{\prime}})(-k_{a_{2}}\cdot
k_{l})\,A\big{(}1,\,\boldsymbol{\sigma}^{\text{(b)}},\,r\big{)},$ (3.2)
$\displaystyle T^{\text{(c)}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\sigma}^{\text{(c)}}}(k_{h_{1}}\cdot
k_{l})(-k_{b_{2}}\cdot
k_{h_{1}})A\big{(}1,\,\boldsymbol{\sigma}^{\text{(c)}},\,r\big{)},$ (3.3)

where $\boldsymbol{\sigma}^{\text{(a)}}$, $\boldsymbol{\sigma}^{\text{(b)}}$
and $\boldsymbol{\sigma}^{\text{(c)}}$ are already presented in eq. (2.15),
eq. (2.16) and eq. (2.17) respectively. Then the full expression of the RHS of
the identity (2.23) induced from $A(1,\cdots,r|\boldsymbol{2}\|h_{1})$ is
given by

$\displaystyle\Biggl{[}\,\sum\limits_{l\in\\{1,\dots,r-1\\}}\widetilde{\sum\limits_{{\scriptsize\\{a_{2},b_{2}\\}\subset\boldsymbol{2}}}}T^{\text{(a)}}\Biggr{]}+\Biggl{[}\,\sum\limits_{l,l^{\prime}\in\\{1,\dots,r-1\\}}\widetilde{\sum\limits_{\scriptsize\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq
c_{2}\,(c_{2}\in\boldsymbol{2})\end{subarray}}}T^{\text{(b)}}\Biggr{]}+\Biggl{[}\,\sum\limits_{l\in\\{1,\dots,r-1\\}}\widetilde{\sum\limits_{\scriptsize\begin{subarray}{c}b_{2}\in\boldsymbol{2}\\\
b_{2}\neq
c_{2}\,(c_{2}\in\boldsymbol{2})\end{subarray}}}T^{\text{(c)}}\Biggr{]},$ (3.4)

where a summation notation with a tile is already defined by eq. (2.19).

To investigate the relationship between the induced identity and BCJ
relations, we define the _standard basis set_ for a gluon trace
$\boldsymbol{i}$ by the set of permutations
$\bigl{\\{}\\{b_{i},\boldsymbol{\beta}\in\mathsf{KK}[\,\boldsymbol{i},b_{i},c_{i}],c_{i}\\}\big{|}b_{i}\neq
c_{i},b_{i}\in\boldsymbol{i}\bigr{\\}}$ for an arbitrarily fixed end node
$c_{i}\in\boldsymbol{i}$ (i.e. we do not sum over the end node $c_{i}$).
Graphically, any two adjacent nodes for a given permutation in the standard
basis set
$\\{b_{i},\boldsymbol{\beta}\in\mathsf{KK}[\,\boldsymbol{i},b_{i},c_{i}],c_{i}\\}$
are connected by a dashed arrow line pointing towards the node $b_{i}$. A
starting trace defined by the refined graphic rule is already expressed by the
standard basis because an end node of this trace is already fixed. Moreover,
all internal traces can be expanded by the standard basis, according to the
following nontrivial property:

$\displaystyle\boxed{\centering\includegraphics[width=460.36174pt]{SplittingTraces.jpg}},\@add@centering$
(3.5)

where each graph stands for its full contribution (including coefficients and
amplitudes). The LHS of the above equation is an internal trace structure
defined by the refined graphic rule. On the RHS, the trace $\boldsymbol{i}$ is
expressed by the standard basis with one end $c_{i}\in\boldsymbol{i}$ fixed.
The node $a_{i}$ ($b_{i}$) on both sides of eq. (3.5) must be connected to a
same node outside the trace $\boldsymbol{i}$ via the same type of line (i.e.,
type-2 or type-3 line). The summation notations with a tilde in eq. (3.5) was
already defined by eq. (2.19). If a dashed arrow points away from the root, an
extra minus should be dressed. We leave the proof of eq. (3.5) in appendix B.

Figure 7: When the identity (3.5) is applied, the contribution of Fig. 6 (a)
splits into (a1) and (a2) which are expressed by standard basis. The graph
(a1) comes from the second term of eq. (3.5), thus it must be associated with
an extra minus. The graph (a3) is a spurious graph which cannot be directly
obtained by the refined graphic rule and the identity (3.5). It can be
considered as the $a_{i}=b_{i}$ supplement to both terms on the RHS of eq.
(3.5).

Now we apply the relation (3.5) to the first term of eq. (3.4) for a given
$l$. Since the $c_{i}$ in eq. (3.5) can be chosen arbitrarily, we just choose
the $c_{i}\in\boldsymbol{i}$ (in this example $\boldsymbol{i}=\boldsymbol{2}$)
as the fixed gluon $c_{2}$ in the trace $\boldsymbol{2}$ of Fig. 6 (b) and (c)
for convenience. Then the summation
$\widetilde{\sum}_{{\scriptsize\\{a_{2},b_{2}\\}\subset\boldsymbol{2}}}T^{\text{(a)}}$
in the first term of eq. (3.4) splits into

$\displaystyle\widetilde{\sum\limits_{{\scriptsize\\{a_{2},b_{2}\\}\subset\boldsymbol{2}}}}T^{\text{(a)}}=-\widetilde{\sum\limits_{{\scriptsize\begin{subarray}{c}b_{2}\in\boldsymbol{2}\\\
b_{2}\neq
c_{2}\end{subarray}}}}\sum\limits_{\scriptsize\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq
b_{2}\end{subarray}}T^{\text{(a1)}}+\widetilde{\sum\limits_{{\scriptsize\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq
c_{2}\end{subarray}}}}\sum\limits_{\scriptsize\begin{subarray}{c}b_{2}\in\boldsymbol{2}\\\
b_{2}\neq a_{2}\end{subarray}}T^{\text{(a2)}},$ (3.6)

where $T^{\text{(a1)}}$ and $T^{\text{(a2)}}$ are corresponding to Fig. 7 (a1)
and (a2). Substituting eq. (3.6) into eq. (3.4) and introducing the
contribution of _spurious graph_ Fig. 7 (a3) by
$0=T^{\text{(a3)}}-T^{\text{(a3)}}$, we rewrite eq. (3.4) as the sum of
$I_{1}$ and $I_{2}$ which are respectively defined by

$\displaystyle
I_{1}\equiv\widetilde{\sum\limits_{{\scriptsize\begin{subarray}{c}b_{2}\in\boldsymbol{2}\\\
b_{2}\neq
c_{2}\end{subarray}}}}\biggl{[}\,\sum\limits_{l\in\\{1,\dots,r-1\\}}\biggl{(}\,-\sum\limits_{\scriptsize\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq
b_{2}\end{subarray}}T^{\text{(a1)}}-T^{\text{(a3)}}|_{a_{2}=b_{2}}+T^{\text{(c)}}\biggr{)}\biggr{]}$
(3.7)

and

$\displaystyle
I_{2}\equiv\widetilde{\sum\limits_{\scriptsize\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq
c_{2}\end{subarray}}}\biggl{[}\,\sum\limits_{l\in\\{1,\dots,r-1\\}}\biggl{(}\sum\limits_{\scriptsize\begin{subarray}{c}b_{2}\in\boldsymbol{2}\\\
b_{2}\neq
a_{2}\end{subarray}}T^{\text{(a2)}}+T^{\text{(a3)}}+\sum\limits_{l^{\prime}\in\\{1,\dots,r-1\\}}T^{\text{(b)}}\biggr{)}\biggr{]}.$
(3.8)

In the following, we analyze $I_{1}$ and $I_{2}$ in turn and prove that both
of them can be expanded in terms of graph-based BCJ relations.

Figure 8: The graphs Fig. 7 (a1), (a3) and Fig. 6 (c) can be reproduced by
connecting the structure $\mathcal{T}_{1}$ with a gluon
$l\in\\{1,2,\dots,r-1\\}=\boldsymbol{1}\setminus\\{r\\}$ via a type-3 line.
This further implies that the $I_{1}$ in eq. (3.7) can be expanded by graph-
based BCJ relations (A.15) with $\mathcal{T}=\mathcal{T}_{1}$.

(i). It is easy to see the graphs Fig. 7 (a1), (a3) and Fig. 6 (c)
corresponding to the three terms of eq. (3.7) can be reproduced by drawing a
type-3 line between a gluon
$l\in\\{1,2,\dots,r-1\\}=\boldsymbol{1}\setminus\\{r\\}$ and a node
$a\in\mathcal{T}_{1}$ where $\mathcal{T}_{1}$ is the tree structure Fig. 8.
Particularly, $a$ is given by $a_{2}\in\boldsymbol{2}~{}(a_{2}\neq b_{2})$ for
Fig. 7 (a1), $b_{2}$ for Fig. 7 (a3) and $h_{1}$ for Fig. 6 (c). The kinematic
factors of Fig. 7 (a1), (a3) and Fig. 6 (c) can then be uniformly given by
$(k_{h_{1}}\cdot k_{b_{2}})(k_{a}\cdot k_{l})$. Let us count the sign: Given
$b_{2}\in\boldsymbol{2}\,(b_{2}\neq c_{2})$ in eq. (3.7), there is an overall
sign $(-1)^{|\boldsymbol{2},b_{2},c_{2}|}$ which has already been absorbed
into the summation notation with a tilde
$\widetilde{\sum}_{{\scriptsize\begin{subarray}{c}b_{2}\in\boldsymbol{2}\\\
b_{2}\neq c_{2}\end{subarray}}}$. The sign for each term inside the square
brackets is collected as follows: (1). According to the refined graphic rule,
any graph $\mathcal{F}$ with $\mathcal{N}(\mathcal{F})$ arrows (for both solid
and dashed arrow lines) pointing away from the root $1$ is associated with a
sign $(-1)^{\mathcal{N}(\mathcal{F})}$. (2). Each of (a1) and (a3) has an
extra sign $(-1)$. The above observations further lead to the following
pattern:

  * •

Once $a\in\mathcal{T}_{1}$ has been chosen, the sign is independent of the
choice of $l\in\\{1,2,\dots,r-1\\}=\boldsymbol{1}\setminus\\{r\\}$ because
neither $\mathcal{N}(\mathcal{F})$ nor the extra sign in eq. (3.7) relies on
$l$. On another hand, two graphs with adjacent $a\in\mathcal{T}_{1}$ are
associated with opposite signs.

  * •

All permutations established by the graph with any given $a\in\mathcal{T}_{1}$
have the form

$\displaystyle\\{1,\boldsymbol{\sigma}\in\\{2,\dots,r-1\\}\shuffle\mathcal{T}_{1}|_{a},r\\}.$
(3.9)

Here $\mathcal{T}_{1}|_{a}$ is introduced as the relative orders between nodes
of the tree $\mathcal{T}_{1}$ when the node $a$ is considered as the leftmost
one. According to the refined graphic rule, when we connect a type-3 line
between $a\in\mathcal{T}_{1}$ and $l\in\boldsymbol{1}\setminus\\{r\\}$, $l$
must be nearer to the root $1$ than $a$. In other words, for a given
permutation $\boldsymbol{\sigma}$ in eq. (3.9), $l\in\\{1,\dots,r-1\\}$ can be
any node satisfying $\sigma^{-1}(l)<\sigma^{-1}(a)$. Then the total
coefficient (which comes from the type-3 line between $a$ and $l$) for the
permutation $\boldsymbol{\sigma}$ is collected as $-k_{a}\cdot
Y_{a}(\boldsymbol{\sigma})$ where
$Y^{\mu}_{a}(\boldsymbol{\sigma})\equiv\sum_{\scriptsize{\sigma^{-1}(l)<\sigma^{-1}(a)}}k^{\mu}_{l}$
(the momentum of the root $1$ is always included in this summation).

Altogether, $I_{1}$ in eq. (3.7) can be reexpressed by

$\displaystyle
I_{1}=\widetilde{\sum\limits_{{\scriptsize\begin{subarray}{c}b_{2}\in\boldsymbol{2}\\\
b_{2}\neq c_{2}\end{subarray}}}}(k_{h_{1}}\cdot
k_{b_{2}})(-)^{\mathcal{F}({x_{0}})}\biggl{[}\,\sum\limits_{a\in\mathcal{T}_{1}}f^{a}\sum\limits_{\zeta\in\mathcal{T}_{1}|_{a}}\sum\limits_{\boldsymbol{\sigma}\in\zeta\shuffle\\{2,\dots,r-1\\}}(k_{a}\cdot
Y_{a}(\boldsymbol{\sigma}))A(1,\boldsymbol{\sigma},r)\biggr{]},$ (3.10)

where $(-)^{\mathcal{F}({x_{0}})}$ denotes the sign for the graph
$\mathcal{F}$ with $a=x_{0}$ ($x_{0}\in\mathcal{T}_{1}$). The $f^{a}$ for any
$a\in\mathcal{T}_{1}$ is fixed as (i). $f^{x_{0}}=1$, (ii).
$f^{x_{1}}=-f^{x_{2}}$ if $x_{1}$ and $x_{2}$ are two adjacent nodes in
$\mathcal{T}_{1}$. Therefore, the expression in the square brackets is just
the LHS of the graph-based BCJ relation (A.15) which has been proven to be a
combination of traditional BCJ relations (A.14) (see Hou:2018bwm ).

(ii). For $I_{2}$, all the permutations established by the graphs Fig. 6 (b),
Fig. 7 (a2) and (a3) have the form
$\\{1,\boldsymbol{\sigma}\in\\{h_{1}\\}\shuffle\,\boldsymbol{\gamma},r\\}$,
where

$\displaystyle\boldsymbol{\gamma}\in\\{2,\dots,l,\\{l+1,\dots,r-1\\}\shuffle\\{a_{2},\mathsf{KK}[\,\boldsymbol{2},a_{2},c_{2}],c_{2}\\}\\}.$
(3.11)

Coefficient for each permutation
$\boldsymbol{\sigma}\in\\{h_{1}\\}\shuffle\,\boldsymbol{\gamma}$ is collected
as $(-k_{a_{2}}\cdot k_{l})(k_{h_{1}}\cdot Y_{h_{1}}(\boldsymbol{\sigma}))$.
Hence $I_{2}$ turns to

$\displaystyle
I_{2}=\widetilde{\sum\limits_{\scriptsize\begin{subarray}{c}a_{2}\in\boldsymbol{2}\\\
a_{2}\neq
c_{2}\end{subarray}}}\,\sum\limits_{l\in\\{1,\dots,r-1\\}}(-k_{a_{2}}\cdot
k_{l})\sum\limits_{\boldsymbol{\gamma}}\biggl{[}\,\sum\limits_{\boldsymbol{\sigma}\in\\{h_{1}\\}\shuffle\,\boldsymbol{\gamma}}(k_{h_{1}}\cdot
Y_{h_{1}}(\boldsymbol{\sigma}))A(1,\boldsymbol{\sigma},r)\biggr{]},$ (3.12)

in which $\boldsymbol{\gamma}$ satisfy eq. (3.11). Apparently, the expression
in the square brackets is just the LHS of a special case of BCJ relation
(A.14), which can also be understood as the graph-based BCJ relation (A.15)
when the tree graph $\mathcal{T}$ is the single node $h_{1}$.

Figure 9: Typical graphs for the identity (2.25) induced from the double-trace
amplitude $A(1,2,\cdots,r|\boldsymbol{2}\|h_{1})$ with the reference order
$\mathsf{R}=\\{h_{1},\boldsymbol{2}\\}$. Here, gluons of the trace
$\boldsymbol{2}$ are supposed to be in the cyclic order
$d_{1},d_{2},\dots,d_{|\boldsymbol{2}|}$ and the gluon $d_{|\boldsymbol{2}|}$
is the removed gluon $b_{0}$ in the induced identity (2.25).

###  3.2 Example-2: the identity (2.25) induced from
$A(1,2,\cdots,r|\boldsymbol{2}\|h_{1})$

When the reference order for the expansion (2.1) of the double-trace EYM
amplitude $A(1,2,\cdots,r|\boldsymbol{2}\|h_{1})$ is chosen as
$\mathsf{R}=\\{h_{1},\boldsymbol{2}\\}$ (i,e., the trace $\boldsymbol{2}$ is
the highest-weight element), the cyclic symmetry of the trace $\boldsymbol{2}$
induces the identity (2.25) where typical graphs are presented as Fig. 4
(b)-(e). According to the discussions in section 2.3, we do the replacement
Fig. 5 (b) and then Fig. 5 (c) on the graphs Fig. 4 (b)-(e). Thus graphs Fig.
9 (a)-(d) with the following contributions are obtained correspondingly:

$\displaystyle T^{\text{(a)}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\sigma}^{\text{(a)}}}(\epsilon_{h_{1}}\cdot
k_{l})(-k_{a_{2}}\cdot
k_{l^{\prime}})\,A(1,\boldsymbol{\sigma}^{\text{(a)}},r),~{}~{}~{}~{}\,~{}~{}~{}~{}T^{\text{(b)}}=\sum\limits_{\boldsymbol{\sigma}^{\text{(b)}}}(\epsilon_{h_{1}}\cdot
k_{l})(-k_{a_{2}}\cdot k_{h_{1}})A(1,\boldsymbol{\sigma}^{\text{(b)}},r),$
$\displaystyle T^{\text{(c)}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\sigma}^{\text{(c)}}}(-\epsilon_{h_{1}}\cdot
k_{a_{2}})(-k_{h_{1}}\cdot
k_{l})A(1,\boldsymbol{\sigma}^{\text{(c)}},r),~{}~{}~{}~{}~{}\,T^{\text{(d)}}=\sum\limits_{\boldsymbol{\sigma}^{\text{(d)}}}(\epsilon_{h_{1}}\cdot
k_{l^{\prime}})(-k_{a_{2}}\cdot
k_{l})\,A(1,\boldsymbol{\sigma}^{\text{(d)}},r),$ (3.13)

in which,

$\displaystyle\boldsymbol{\sigma}^{\text{(a)}}$ $\displaystyle\in$
$\displaystyle\bigl{\\{}2,\dots,l,\\{h_{1}\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\shuffle\bigl{\\{}l+1,\dots,l^{\prime},\\{l^{\prime}+1,\dots,r-1\\}\shuffle\\{a_{2}=d_{j},\\{d_{j+1},\dots,d_{|\boldsymbol{2}|-1}\\}\shuffle\\{d_{j-1},\dots,d_{1}\\}\\}\bigr{\\}}\bigr{\\}},$
$\displaystyle\boldsymbol{\sigma}^{\text{(b),(c)}}$ $\displaystyle\in$
$\displaystyle\\{2,\dots,l,\\{l+1,\dots,r-1\\}\shuffle\\{h_{1},a_{2}=d_{j},\\{d_{j+1},\dots,d_{|\boldsymbol{2}|-1}\\}\shuffle\\{d_{j-1},\dots,d_{1}\\}\\},$
$\displaystyle\boldsymbol{\sigma}^{\text{(d)}}$ $\displaystyle\in$
$\displaystyle\bigl{\\{}2,\dots,l,\\{l+1,\dots,r-1\\}$ (3.14)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\shuffle\\{a_{2}=d_{k},\\{d_{k+1},\dots,d_{|\boldsymbol{2}|-1}\\}\shuffle\\{d_{k-1},\dots,l^{\prime}=d_{j},\\{h_{1}\\}\shuffle\\{d_{j-1},\dots,d_{1}\\}\\}\\}\bigr{\\}}.$

Once all graphs are summed over, we arrive the RHS of the induced identity eq.
(2.25) for the amplitude $A(1,2,\cdots,r|\boldsymbol{2}\|h_{1})$:

$\displaystyle\underbrace{\sum\limits_{l,l^{\prime}\in\\{1,\dots,r-1\\}}\sum\limits_{j=1}^{|\boldsymbol{2}|-1}T^{\text{(a)}}+\sum\limits_{l\in\\{1,\dots,r-1\\}}\sum\limits_{j=1}^{|\boldsymbol{2}|-1}\Bigl{(}T^{\text{(b)}}}_{I_{1}}+\underbrace{T^{\text{(c)}}\Bigr{)}+\sum\limits_{l\in\\{1,\dots,r-1\\}}\sum\limits_{j,k=1}^{|\boldsymbol{2}|-1}T^{\text{(d)}}}_{I_{2}},$
(3.15)

where contributions of all graphs of the form Fig. 9 (a), (b) and (c), (d)
were collected as $I_{1}$ and $I_{2}$ respectively. Now we prove that both
$I_{1}$ and $I_{2}$ in eq. (3.15) can be expanded in terms of BCJ relations.

Figure 10: The terms $I_{1}$ and $I_{2}$ in eq. (3.15) can be expanded in
terms of graph-based BCJ relations (A.15), where the corresponding tree graphs
$\mathcal{T}$ are chosen as the graph (a) and the graph (b).

For the $I_{1}$ part in eq. (3.15), the summation over
$j=1,\dots,|\boldsymbol{2}|-1$ is nothing but just the summation over all
nodes $a_{2}\in\mathcal{T}_{3}$ where $\mathcal{T}_{3}$ is the tree graph Fig.
10 (a). All permutations established by the graphs Fig. 9 (a), (b) with a
given $a_{2}\in\mathcal{T}_{3}$ and a given $l\in\\{1,\dots,r-1\\}$ have the
form

$\displaystyle\big{\\{}1,\boldsymbol{\sigma}\in(\mathcal{T}_{3}|_{a_{2}})\shuffle\boldsymbol{\gamma},r\big{\\}}.$
(3.16)

Here,
$\boldsymbol{\gamma}\in\\{2,\dots,l,\\{h_{1}\\}\shuffle\\{l+1,\dots,r-1\\}\\}$
and $\mathcal{T}_{3}|_{a_{2}}$ denotes the relative orders of nodes in
$\mathcal{T}_{3}$ when $a_{2}$ is considered as the leftmost one. Based on a
similar discussion with the example-1 in section 3.1, we find the following
patterns: (1). The coefficients $(k_{a_{2}}\cdot k_{l^{\prime}})$
($l^{\prime}\in\\{1,\dots,r-1\\}$ for Fig. 9 (a) and $l^{\prime}=h_{1}$ for
Fig. 9 (b)) corresponding to a same
$\boldsymbol{\sigma}\in(\mathcal{T}_{3}|_{a_{2}})\shuffle\boldsymbol{\gamma}$
with different choices of $l^{\prime}$ are collected as $(-k_{a_{2}}\cdot
Y_{a_{2}}(\boldsymbol{\sigma}))$; (2). Any two graphs, where
$a_{2}\in\mathcal{T}_{3}$ are chosen as adjacent nodes, have opposite signs.
Then the $I_{1}$ part in eq. (3.15) is expressed by

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\sum\limits_{l\in\\{1,\dots,r-1\\}}\sum\limits_{\small\begin{subarray}{c}\boldsymbol{\gamma}\in\\{2,\dots,l,\\{h_{1}\\}\\\
\shuffle\\{l+1,\dots,r-1\\}\\}\end{subarray}}(\epsilon_{h_{1}}\cdot
k_{l})(-)^{\mathcal{F}({x_{0}})}$ (3.17)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\biggl{[}\,\sum\limits_{a_{2}\in\mathcal{T}_{3}}f^{a_{2}}\sum\limits_{\boldsymbol{\sigma}\in\\{\mathcal{T}_{3}|_{a_{2}}\shuffle\,\boldsymbol{\gamma}\\}}(-k_{a_{2}}\cdot
Y_{a_{2}}(\boldsymbol{\sigma}))A(1,\boldsymbol{\sigma},r)\biggr{]}.$

Here $(-)^{\mathcal{F}(x_{0})}$ is the sign for a graph with
$a_{2}=x_{0}\in\mathcal{T}_{3}$. Since $(-)^{\mathcal{F}({x_{0}})}$ has been
extracted as an overall sign, we have $f^{x_{0}}=1$. If
$x_{1},x_{2}\in\mathcal{T}_{3}$ are two adjacent nodes, we have
$f^{x_{1}}=-f^{x_{2}}$. Obviously, the expression in the square brackets is
nothing but (up to a total minus) the LHS of a graph-based BCJ relation
(A.15). As a result, $I_{1}$ is a combination of traditional BCJ relations.

The $I_{2}$ part in eq. (3.15) can be analyzed following a parallel discussion
with $I_{1}$ but replacing the tree graph $\mathcal{T}_{3}$ by
$\mathcal{T}_{4}$ (see Fig. 10 (b)) for
$d_{j}\in\\{d_{1},\dots,d_{|\boldsymbol{2}|-1}\\}$ and replacing
$\boldsymbol{\gamma}$ by $\\{2,\dots,r-1\\}$:

$\displaystyle
I_{2}=\sum\limits_{j=1}^{|\boldsymbol{2}|-1}(\epsilon_{h_{1}}\cdot
k_{d_{j}})(-)^{\mathcal{F}({x_{0}})}\biggl{[}\,\sum\limits_{a_{2}\in\mathcal{T}_{4}}f^{a_{2}}\sum\limits_{\boldsymbol{\sigma}\in\\{\mathcal{T}_{4}|_{a_{2}}\shuffle\,\\{2,\dots,r-1\\}\\}}(-k_{a_{2}}\cdot
Y_{a_{2}}(\boldsymbol{\sigma}))A(1,\boldsymbol{\sigma},r)\biggr{]},$ (3.18)

where $x_{0}\in\mathcal{T}_{4}$ and $f^{x_{0}}=1$. Again, the $f^{a_{2}}$’s
for adjacent choices of $a_{2}$ have the opposite signs. Up to a total sign,
the expression in the square brackets is just the LHS of graph-based BCJ
relation (A.15), where the tree graph $\mathcal{T}_{4}$ for a given $j$ is
Fig. 10 (b). Thus, we conclude that $I_{2}$ is a combination of BCJ relations.

###  3.3 Common features of the examples

Now let us extract some common features from the examples, which will be
extended to general cases in the next section.

Figure 11: A typical skeleton in example-1 is presented as the graph (a) which
consists of three components. Graphs (b) and (c) are possible structures of
skeletons in example-2. The subgraphs $\mathscr{U}$ and $\mathscr{L}$ in each
of (b) and (c) are correspondingly the final upper and lower blocks.

(i). Expressing traces by standard basis In example-1, the trace
$\boldsymbol{2}$ played as an internal trace in Fig. 6 (a) and a starting
trace in either Fig. 6 (b) or (c). In the latter cases, the trace
$\boldsymbol{2}$ was already expressed by the standard basis, i.e., one end of
the trace, the gluon $c_{2}$, was fixed. In the former case, both ends of the
trace $\boldsymbol{2}$ were not fixed (in other words both are summed over).
In order to expand the trace $\boldsymbol{2}$ in Fig. 6 (a) by the standard
basis, we have made used of the splitting trace relation (3.5), in which the
fixed node was conveniently chosen as the same element (i.e. $c_{2}$) with
that in Fig. 6 (b) and (c).

(ii). Skeletons and components We define _skeletons_ by removing all type-3
lines from the graphs where all traces, except the highest-weight element in
$\mathsf{R}$ (if it is a trace), are already expressed by the standard basis.
Since each graph defined by the refined graphic rule is a connected tree
graph, its skeleton must be a disconnected graph. Each maximally connected
subgraph of a skeleton is called a _component_. A typical skeleton in
example-1 is given by Fig. 11 (a) (for $b_{2}\in\boldsymbol{2}$, $b_{2}\neq
c_{2}$) which consists of three components. Skeletons in example-2 have two
distinct structures, Fig. 11 (b) and (c), each of which has two components.
From the examples, we can see _any skeleton must have at least two components
that involve the highest-weight element (graviton or trace) and the trace
$\boldsymbol{1}$ respectively_.

Figure 12: Typical configurations of the final upper and lower blocks
$\mathscr{U}$, $\mathscr{L}$ for the skeleton Fig. 11 (a).

(iii). The final upper and lower blocks Any graph in the examples can be
reproduced by connecting a type-3 line between _the final upper and lower
blocks_ which are two mutually disjoint connected subgraphs and respectively
contain the highest-weight element and the trace $\boldsymbol{1}$. In
example-2, each of the skeletons Fig. 11 (b) and (c) already consists of only
two disjoint connected subgraphs $\mathscr{U}$ and $\mathscr{L}$ which serve
as the final upper and lower blocks. In example-1, there are three components
in the skeleton Fig. 11 (a). A typical configuration of the final upper and
lower blocks is constructed when we connect $b_{2}$ in Fig. 11 (a) to either
(i). $h_{1}$ (see Fig. 12 (a)) or (ii). an element in $\\{1,\dots,r-1\\}$ via
a type-3 line (see Fig. 12 (b)).

(iv). Physical and spurious graphs For a given configuration of the final
upper and lower blocks $\mathscr{U}$ and $\mathscr{L}$, we can connect two
nodes $x\in\mathscr{U}$ and $y\in\mathscr{L}\setminus\\{r\\}$ (recalling that
the gluon $r$ is always excluded) via a type-3 line. Then a fully connected
graph is constructed. In example-2, the graphs Fig. 9 (a), (b) (and Fig. 9
(c), (d)) are reproduced by connecting the final upper and lower blocks in
Fig. 11 (b) (and Fig. 11 (c)) via a type-3 line. Similarly, the graphs Fig. 7
(a1), Fig. 6 (c) (and Fig. 7 (a2), Fig. 6 (b)) in example-1 are constructed
from Fig. 12 (a) (and (b)). All the graphs Fig. 6 (b), (c), Fig. 7 (a1), (a2)
and Fig. 9 (a)-(d) are graphs in standard basis which are directly defined by
the refined graphic rule. These graphs are called _physical graphs_. The
_spurious graph_ Fig. 7 (a3), which is not defined by the refined graphic
rule, can be reproduced from either Fig. 12 (a) (connecting
$b_{2}\in\mathscr{U}$ with $l\in\mathscr{L}\setminus\\{r\\}$) or Fig. 12 (b)
(connecting $h_{1}\in\mathscr{U}$ with $b_{2}\in\mathscr{L}\setminus\\{r\\}$).
In the former case, a minus sign is introduced so that the spurious graph
constructed by distinct ways cancel with one another. Therefore, for any
skeleton, the sum of all physical graphs can be given by (1). summing over all
possible configurations of the final upper and lower blocks $\mathscr{U}$,
$\mathscr{L}$, (2). for a given $\mathscr{U}$ and $\mathscr{L}$, connecting
two nodes $x\in\mathscr{U}$ and $y\in\mathscr{L}\setminus\\{r\\}$ via a type-3
line and summing over all possible choices of $x$, $y$ (in other words summing
over all possible physical and spurious graphs corresponding to $\mathscr{U}$
and $\mathscr{L}$).

(v). Induced identities as combinations of graph-based BCJ relations A crucial
observation is that the sum over all the graphs corresponding to a given
configuration of the final upper and lower blocks $\mathscr{U}$ and
$\mathscr{L}$ (i.e. either $I_{1}$ or $I_{2}$ in each example) is a
combination of graph-based BCJ relations (A.15).

In the next section, we extend these observations to general cases and show
that both identities (2.23) and (2.25) can be expanded in terms of BCJ
relations.

##  4 General study

To investigate the general induced identities (2.23) and (2.25) in a unified
way, we write them as

$\displaystyle
0=\sum\limits_{\mathcal{G}}\mathcal{C}^{\mathcal{G}}\,\sum\limits_{\boldsymbol{\sigma}^{\mathcal{G}}}\,A(1,\boldsymbol{\sigma}^{\mathcal{G}},r),~{}$
(4.1)

where the graphs $\mathcal{G}$ are obtained by imposing the replacement Fig. 5
(a) or (b), which corresponds to eq. (2.23) or eq. (2.25), on the graphs
$\mathcal{F}$ in eq. (2.1).

When we introduce skeletons $\mathcal{G}^{\prime}$ by deleting all type-3
lines from the graphs $\mathcal{G}$ and expressing all traces by standard
basis according to eq. (3.5), eq. (4.1) is rearranged as

$\displaystyle
0=\widetilde{\sum\limits_{\mathcal{G}^{\prime}}}\mathcal{P}^{[\mathcal{G}^{\prime}]}\left[\,\sum\limits_{\mathcal{G}\supset\mathcal{G}^{\prime}}\,\sum\limits_{\boldsymbol{\sigma}^{\mathcal{G}}}(-)^{\mathcal{G}}\mathcal{K}^{[\mathcal{G}\setminus\mathcal{G}^{\prime}]}A(1,\boldsymbol{\sigma}^{\mathcal{G}},r)\right].~{}$
(4.2)

In the above equation, the summation notation
$\widetilde{\sum}_{\mathcal{G}^{\prime}}$ means that all possible skeletons
$\mathcal{G}^{\prime}$ are summed over and the signs
$(-1)^{|\boldsymbol{t}_{i},a_{i},c_{i}|}$ and/or
$(-1)^{|\boldsymbol{t}_{i},b_{i},c_{i}|}$ accompanying to the traces in each
$\mathcal{G}^{\prime}$ are absorbed. The factor
$\mathcal{P}^{[\mathcal{G}^{\prime}]}$ denotes the kinematic factor
corresponding to the skeleton $\mathcal{G}^{\prime}$. Since a skeleton does
not involve any type-3 line, $\mathcal{P}^{[\mathcal{G}^{\prime}]}$ only
consists of $\epsilon\cdot\epsilon$ and $\epsilon\cdot k$ factors. In the
expression inside the square brackets, all possible physical graphs
$\mathcal{G}$ (i.e. graphs generated by refined graphic rule with all internal
traces expressed by the standard basis) containing the skeleton
$\mathcal{G}^{\prime}$ and all permutations
$\boldsymbol{\sigma}^{\mathcal{G}}$ for each graph $\mathcal{G}$ are summed
over. The factor $\mathcal{K}^{[\mathcal{G}\setminus\mathcal{G}^{\prime}]}$ in
eq. (4.2) stands for the product of all $k\cdot k$ factors that are presented
by the type-3 lines in $\mathcal{G}$. Those signs caused by arrows pointing
away from the root and the extra signs caused by the second term of eq. (3.5)
are all collected as $(-)^{\mathcal{G}}$.

Figure 13: The graph (a) is a chain that does not involve the highest-weight
element $\mathcal{H}_{\rho(l)}$ in the reference order $\mathsf{R}$. The graph
(b) is a chain involving the highest-weight element. Each node (except for the
ending node $w$) outside the boxed structures in the chains (a) and (b) is a
graviton and we define $F^{\mu\nu}_{(+)}\equiv k^{\mu}\epsilon^{\nu}$,
$F^{\mu\nu}_{(-)}\equiv-\epsilon^{\nu}k^{\mu}$. The structures shown by the
graph (c) are not allowed. Each boxed structure in the chain (a) and in an
internal sector of the chain (b) can be (1). two gravitons connected together
by a type-1 line (as shown by the graph (d)) or (2). a gluon trace in standard
basis (as shown by (e) and (f) which correspond to the two terms in eq.
(3.5)). The node $c$ in (e), (f) denotes the fixed node of a trace
$\boldsymbol{t}$. The boxed structure in the starting sector of (b) can be
either a graviton (as shown by (g)) or a gluon trace
$\boldsymbol{t}\to\\{d_{1},d_{2},\dots,d_{|\boldsymbol{t}|-1}\\}$ (as shown by
(h)).

As observed in section 3.3, all physical graphs $\mathcal{G}$ involving a
given skeleton $\mathcal{G}^{\prime}$ can be generated by connecting the
components via type-3 lines in a proper way: (i). first generate all possible
configurations of the final upper and lower blocks
$\mathscr{U}\oplus\mathscr{L}$; (ii). then connect a type-3 line between two
nodes $x\in\mathscr{U}$ and $y\in\mathscr{L}\setminus\\{r\\}$ appropriately.
One should take care of the step (ii) because spurious graphs may also be
produced. Nevertheless, the spurious graphs in fact all cancel out in the
examples. Hence we suppose that the expression in the square brackets in eq.
(4.2) can be generally written as

$\displaystyle
I[\mathcal{G}^{\prime}]\equiv\sum\limits_{\mathscr{U}\oplus\mathscr{L}}\mathcal{K}^{[\mathscr{U}\oplus\mathscr{L}\setminus\mathcal{G}^{\prime}]}\Biggl{[}\,\sum\limits_{\begin{subarray}{c}x\in\mathscr{U}\\\
y\in\mathscr{L}\setminus\\{r\\}\end{subarray}}\,\sum\limits_{\boldsymbol{\sigma}^{\mathcal{G}}}(-)^{\mathcal{G}}\,(k_{x}\cdot
k_{y})\,A(1,\boldsymbol{\sigma}^{\mathcal{G}},r)\Biggr{]}.~{}$ (4.3)

Here,
$\mathcal{K}^{[\mathscr{U}\oplus\mathscr{L}\setminus\mathcal{G}^{\prime}]}$ is
the product of all $k\cdot k$ factors corresponding to the given configuration
of the final upper and lower blocks $\mathscr{U}$, $\mathscr{L}$, while
$k_{x}\cdot k_{y}$ is the factor corresponding to the type-3 line between
$\mathscr{U}$ and $\mathscr{L}$. The first summation in eq. (4.3) is taken
over all possible configurations of the final upper and lower blocks
$\mathscr{U}$, $\mathscr{L}$ for the skeleton $\mathcal{G}^{\prime}$. In the
square brackets, all choices of nodes $x\in\mathscr{U}$ and
$y\in\mathscr{L}\setminus\\{r\\}$ as well as the permutations
$\boldsymbol{\sigma}^{\mathcal{G}}$ defined by the (physical or spurious)
graph $\mathcal{G}$ (determined by $\mathscr{U}$, $\mathscr{L}$, $x$ and $y$)
are summed over. The sign for the (physical or spurious) graph $\mathcal{G}$
is denoted by $(-)^{\mathcal{G}}$.

In this section, we study eq. (4.3) schematically. We first classify
components of skeletons, then show how to construct the final upper and lower
blocks from a given skeleton $\mathcal{G}^{\prime}$. After that, we show all
spurious graphs cancel out. Thus the summation over all possible choices of
$x\in\mathscr{U}$ and $y\in\mathscr{L}\setminus\\{r\\}$ is equivalent to
summing over all possible physical graphs for given $\mathscr{U}$ and
$\mathscr{L}$. At last, we demonstrate that the expression inside the square
brackets in eq. (4.3) is a combination of BCJ relations.

Figure 14: There are three types of sectors (a), (b) and (c). The starting and
internal sectors of the chain Fig. 13 (a) and the internal sectors of the
chain Fig. 13 (b) have the same general structure (a). Such sectors are called
type-I sectors. The starting sector of the chain Fig. 13 (b) is called the
type-II sector and has the pattern (b). The ending sector of any chain has the
structure (c) and is called a type-III sector.  Figure 15: Graph (a) is a
type-IA component, while graph (b) is a type-IB component. The kernel of (a)
is defined by the type-1 line between nodes $a$ and $b$. The kernel of (b) is
defined by the type-4 line which is connected to the (unfixed) end node $d$.
The nodes $x$ and $y$ in each graph are supposed to be the highest-weight
nodes in the regions $\mathscr{A}$ and $\mathscr{B}$, respectively. If the
weight of $x$ is higher than that of $y$, $\mathscr{A}$ and $\mathscr{B}$ are
correspondingly the top and the bottom sides. Contrarily, if the weight of $y$
is higher than that of $x$, $\mathscr{A}$ becomes the bottom side while
$\mathscr{B}$ the top.  Figure 16: The component involving the highest-weight
element $\mathcal{H}_{\rho(l)}$ is defined as (a). the type-IIA component if
$\mathcal{H}_{\rho(l)}$ is a graviton $h_{\rho(s)}$, (b). the type-IIB
component if $\mathcal{H}_{\rho(l)}$ is a gluon trace
$\boldsymbol{t}\to\\{d_{1},\dots,d_{|\boldsymbol{t}|-1}\\}$. The component
containing the trace $\boldsymbol{1}=\\{1,2,\dots,r-1,r\\}$, as shown by the
graph (c), is defined as the type-III component.

###  4.1 Skeletons and components

When all type-3 lines (i.e. $k\cdot k$ factors) are removed, a graph
$\mathcal{G}$ becomes a skeleton $\mathcal{G}^{\prime}$. To analyze possible
structures of components which are maximally connected subgraphs of
$\mathcal{G}^{\prime}$, we should look into the inner structure of a chain via
expanding eq. (2.6) by eq. (2.9), eq. (2.12) and eq. (3.5). According to
whether the starting node is the highest-weight element in the reference order
$\mathsf{R}$ (defined in eq. (2.2)) or not, we carry out the discussion as
follows:

  * •

(i). If the starting node of a chain is not the highest-weight element
$\mathcal{H}_{\rho(l)}$, a graph corresponding to this chain can only have the
general pattern Fig. 13 (a). Now we comment on crucial features of Fig. 13
(a): (1). Distinct sectors are separated by type-3 lines. The sector
containing the starting (ending) node is called the _starting (ending)
sector_. Other sectors between the starting and ending sectors are mentioned
as _internal sectors_. (2). The possible structures inside the boxes in Fig.
13 (a) are given by Fig. 13 (d) (where two gravitons are connected by a type-1
line) and Fig. 13 (e), (f) (which involve a gluon trace expressed by standard
basis). (3). If a chain Fig. 13 (a) contains only one sector, the sector must
be the ending sector. (4). If the starting element of a chain is a gluon
trace, there must be no graviton on the left hand side of the box in the
starting sector, as shown by Fig. 13 (f). (5). Other structures of a chain are
forbidden because the substructures in Fig. 13 (c) are not allowed by (eq.
(2.6)) (in other words, $F_{h}^{\mu\nu}\equiv
k_{h}^{\mu}\epsilon_{h}^{\nu}-\epsilon_{h}^{\mu}k_{h}^{\mu}$ for a graviton
$h$ involves neither $k^{\mu}k^{\nu}$ nor
$\epsilon_{h}^{\mu}\epsilon_{h}^{\nu}$).

  * •

(ii). If the starting node of a chain is the highest-weight element
$\mathcal{H}_{\rho(l)}$, its corresponding graph must have the general pattern
Fig. 13 (b), where the highest-weight element (graviton or trace) is already
replaced according to Fig. 5 (a) (for a graviton) or Fig. 5 (b) and (c) (for a
gluon trace). An important feature is _the chain Fig. 13 (b) has at least two
sectors_, which follows from the fact that structures in Fig. 13 (c) are
forbidden.

Having the above discussions, sectors of chains can be easily classified as
Fig. 14. In a full graph $\mathcal{G}$, nodes of any sector may play as the
ending nodes of other chains. When all type-3 lines are removed, each sector
in a skeleton thus can be attached by the type-3 sectors Fig. 14 (c) (or
equivalently ending sectors) of other chains. Consequently, components in a
skeleton can be classified by the following way.

  * •

Type-I component: _A component consisting of a type-I sector Fig. 14 (a) and
possible type-III sectors Fig. 14 (c) whose arrow lines point towards the
type-I sector_ Type-I components can further be classified according to
different structures inside the box of Fig. 14 (a): If the box contains the
structure Fig. 13 (d), as shown by Fig. 15 (a), this component is called a
type-IA component. Else, if the structure in the box is given by Fig. 13 (e)
or (f), as shown by Fig. 15 (b), the component is called a type-IB component.
We define the kernel of a type-I component by (i). the type-1 line of a type-
IA component (see Fig. 15 (a)), (ii). the type-4 line that is attached to the
unfixed end node of the trace (in standard basis) inside a type-IB component
(see Fig. 15 (b)). For a given reference order, any type-IA and -IB component
is divided into two parts by the kernel: the part involving the highest-weight
node (although a trace is considered as a single object in the reference
order, the fixed node $c$ is always considered as the highest-weight node of
this trace and it carries the weight of the full trace in the reference order)
of this component is called the top side, while the opposite part is called
the bottom side.

  * •

Type-II component: _A component consisting of a type-II sector Fig. 14 (a) and
possible type-III sectors whose arrows point towards the type-II sector_ If
the structure in the box of the type-II sector is Fig. 13 (g) (i.e. the
highest-weight element is a graviton), this component is called a type-IIA
component (see Fig. 16 (a)). If the structure in the box is a gluon trace Fig.
13 (h), the component is called a type-IIB component (see Fig. 16 (b)).

  * •

Type-III component: _A component consisting of the trace $\boldsymbol{1}$ and
possible type-III sectors with arrows pointing towards the trace
$\boldsymbol{1}$ (see Fig. 16 (c))_

Since the chain Fig. 13 (b) that is lead by the highest weight node has at
least two sectors, a skeleton must at least contain two components: the type-
II and the type-III components. In general, Type-I components may also be
involved in a skeleton. All those graphs corresponding to a given skeleton are
reproduced by connecting type-3 lines between components of a skeleton
$\mathcal{G}^{\prime}$ in an appropriate way.

#### Multi-trace from single-trace

It is worth pointing out that the refined graphic rule given in section 2 can
be obtained from the rule for identities induced from single-trace amplitudes
(which was presented in Hou:2018bwm ) by an appropriate replacement.
Particularly, we consider a single-trace amplitude
$A(1,2,\dots,r\|\mathsf{H})$ where we have $s+m-1$ gravitons
$\mathsf{H}=\\{h_{1},\dots,h_{s+m-1}\\}$. When we replace $m-1$ gravitons,
e.g., $h_{s+1},h_{s+2},\dots,h_{s+m-1}$ by $m-1$ gluon traces
$\boldsymbol{2},\dots,\boldsymbol{m}$, we get the multi-trace amplitude
$A(1,2,\dots,r|\boldsymbol{2}|\dots|\boldsymbol{m}\|\mathsf{H})$ with
$\mathsf{H}=\\{h_{1},\dots,h_{s}\\}$. Such replacement is reflected in the
refined graphic rule via replacing the reference order and chains with only
gravitons by those with gravitons and/or gluon traces. The corresponding
replacement for coefficients are given by

$\displaystyle\epsilon\cdot F\cdot F\cdot\ldots\cdot F\cdot
k\xrightarrow{h_{s+1},\dots,h_{s+m-1}\to\boldsymbol{2},\dots,\boldsymbol{m}}\mathcal{E}\cdot\mathbb{F}\cdot\mathbb{F}\cdot...\cdot\mathbb{F}\cdot
k,$ (4.4)

where $\epsilon$, $F$ are the half polarizations and strength tensors of
gravitons, while $\mathcal{E}$ and $\mathbb{F}$ are generalized polarizations
and generalized strength tensors for gravitons and/or gluon traces (see eq.
(2.9) and eq. (2.12)). Graphically, this replacement is given by

Fig. 1 (a) $\displaystyle\to$
$\displaystyle\text{Fig.~{}\ref{Fig:RefinedTrace} (a)},(\text{if $h_{i}$ is
replaced by $\boldsymbol{t}_{i}$})$ Fig. 1 (b) $\displaystyle\to$
$\displaystyle\text{Fig.~{}\ref{Fig:RefinedTrace} (b)},(\text{if $h_{i}$ is
replaced by $\boldsymbol{t}_{i}$}).$ (4.5)

For identities induced from a single-trace amplitude, the highest-weight
element (graviton) is further given by $k^{\mu}$ (see Fig. 5 (a)). In the
corresponding multi-trace case, the highest-weight element can either be a
graviton (if it is not replaced by a trace) or be replaced by a gluon trace.
The latter is described via replacing the node (the highest-weight graviton
for identity induced from single-trace amplitude) by the RHS of Fig. 5 (c).

Now we look into the inner structure of a chain by further expanding internal
gravitons on the LHS of eq. (4.4) according to
$F^{\mu\nu}=k^{\mu}\epsilon^{\nu}-\epsilon^{\mu}k^{\nu}\equiv
F_{(+)}^{\mu\nu}-F_{(-)}^{\mu\nu}$. On the RHS, internal gravitons are also
expanded by $F_{(+)}^{\mu\nu}-F_{(-)}^{\mu\nu}$, while internal gluon traces
are expanded according to the relation eq. (3.5). Although this expansion of
gluon trace does not affect the tensor $\mathbb{F}^{\mu\nu}$ for a given
$a_{i}$ and $b_{i}$ (see eq. (2.12)), it splits the graphs corresponding to
the trace into the standard basis. Hence, the replacement
$h_{i}\to\boldsymbol{i}$ (for an internal graviton $h_{i}$) is achieved
graphically through replacing the first (second) graph in Fig. 1 (b) by the
first (second) graph on the RHS of eq. (3.5) (for a given
$\\{a_{i},b_{i}\\}$). It follows that the general chain structures Fig. 13 (a)
and (b) are obtained from those chain structures for the single-trace case
Hou:2018bwm  (where only type-IA, type-IIA and type-III sectors are allowed)
by incorporating more types of sectors: type-IB and type-IIB sectors which
reflect structures of gluon traces. Consequently, the full classification of
components in multi-trace cases can be given by enlarging the families of the
type-I and the type-II components that were defined in Hou:2018bwm :

Type-I components $\displaystyle\to$ $\displaystyle\text{Type-IA or Type-IB
components},$ Type-II components $\displaystyle\to$ $\displaystyle\text{Type-
IIA or Type-IIB components},$ (4.6)

where the type-IIA and type-IIB components are respectively the highest-weight
components of the identities (2.23) and (2.25). All the above discussions
allow us to borrow some crucial conclusions from the single-trace case
Hou:2018bwm :

  * •

(i) When keeping track of chains in the single-trace case Hou:2018bwm , one
can build all possible physical graphs corresponding to a skeleton by
connecting type-3 lines between components properly (see appendix D in
Hou:2018bwm ). This construction can be immediately generalized to multi-trace
cases by the enlargement (4.6).

  * •

(ii) As proved in Hou:2018bwm  (see sections 6.1, 6.2 and appendix D of
Hou:2018bwm ), all the physical graphs in the single-trace case, which are
corresponding to a given skeleton and are constructed by the above step, can
be reproduced by (1). constructing the final upper and lower blocks, (2).
connecting the final upper and lower blocks via a type-3 line appropriately.
In multi-trace cases, we just follow the same construction rule but enlarging
the type-I and type-II classes of components according to (4.6).

  * •

(iii) As pointed in section 6.3 of Hou:2018bwm , in the single-trace case, the
sum over all physical graphs which are produced by (ii) can be further written
as the sum of all physical and spurious graphs (those graphs which are not
directly constructed from the refined graphic rule). The latter all cancel out
after summation. Again, the spurious graphs for single-trace induced
identities can be straightforwardly extended to multi-trace cases by the help
of (4.6) and they all cancel out (we have seen this cancellation by the
examples in section 3).

In the coming two subsections, we display the construction rule of the final
upper and lower blocks as well as the construction of physical and spurious
graphs without a proof. In fact, all the proofs follow from discussions
parallel with those in the single-trace case Hou:2018bwm .

###  4.2 The final upper and lower blocks

Now we provide the general rule for constructing all possible configurations
of the final upper and lower blocks corresponding to a given skeleton
$\mathcal{G}^{\prime}$:

  * •

Step-1 For any skeleton $\mathcal{G}^{\prime}$, we define _the reference order
$\mathsf{R}_{\mathscr{C}}$ of all type-I components (including type-IA and
type-IB components)_ by the relative order of the highest-weight nodes
therein. In other words, the weight (i.e. the position in
$\mathsf{R}_{\mathscr{C}}$) of a component inherits from its highest-weight
node. For example, suppose there are three type-I components (IA and/or IB)
$\mathscr{C}_{1}$, $\mathscr{C}_{2}$, and $\mathscr{C}_{3}$ with the
corresponding highest-weight nodes (graviton or a gluon) $a_{1}$, $a_{2}$ and
$a_{3}$. If the weights $W_{a_{i}}$ have the relation
$W_{a_{2}}<W_{a_{1}}<W_{a_{3}}$, the reference order of these components is
then given by the ordered set
$\mathsf{R}_{\mathscr{C}}=\\{\mathscr{C}_{2},\mathscr{C}_{1},\mathscr{C}_{3}\\}$.
We further define the upper block $\mathscr{U}$ and lower block $\mathscr{L}$
as the components respectively containing the highest-weight element (graviton
or trace) and the trace $\boldsymbol{1}$. At the beginning, the upper and the
lower blocks are nothing but the type-II and the type-III components.

  * •

Step-2 Supposing the reference order of components is
$\mathsf{R}_{\mathscr{C}}=\left\\{\mathscr{C}_{1},\mathscr{C}_{2},\dots,\mathscr{C}_{N}\right\\}$,
pick out the highest-weight component $\mathscr{C}_{N}$ as well as arbitrary
components $\mathscr{C}_{a_{1}}$, $\mathscr{C}_{a_{2}}$, …,
$\mathscr{C}_{a_{i}}$ (the relative order of these components is not necessary
the same relative order in $\mathsf{R}_{\mathscr{C}}$). Construct a chain of
components towards either the upper block or the lower block as follows

$\displaystyle\mathbb{CH}=\left[(\mathscr{C}_{N})_{t},(\mathscr{C}_{N})_{b}\leftrightarrow(\mathscr{C}_{a_{i}})_{t\,(\text{or}\,b)},(\mathscr{C}_{a_{i}})_{b\,(\text{or}\,t)}\leftrightarrow\cdots\leftrightarrow(\mathscr{C}_{a_{1}})_{t\,(\text{or}\,b)},(\mathscr{C}_{a_{1}})_{b\,(\text{or}\,t)}\leftrightarrow\mathscr{U}\,\text{or}\,\mathscr{L}\setminus\\{r\\}\right].$

Here the subscripts $t$ and $b$ respectively denote the top and bottom sides
of a type-I component, which are separated by a comma888Notations here are
slightly different from those in Hou:2018bwm .. The double arrow line
‘$\leftrightarrow$’ between two components stands for the type-3 line (i.e.
$k\cdot k$), which connects any two nodes belonging to the corresponding
regions. For example, if the chain of components has the form
$\left[(\mathscr{C}_{N})_{t},(\mathscr{C}_{N})_{b}\leftrightarrow(\mathscr{C}_{a_{i}})_{t},(\mathscr{C}_{a_{i}})_{b}\leftrightarrow\cdots\right]$,
the two ends $x$ and $y$ of the type-3 line between the components
$\mathscr{C}_{N}$ and $\mathscr{C}_{a_{i}}$ must belong to
$(\mathscr{C}_{N})_{b}$ and $(\mathscr{C}_{a_{i}})_{t}$ respectively. After
this step, we redefine the reference order of components as well as the upper
and lower blocks by:

$\displaystyle\mathsf{R}_{\mathscr{C}}$ $\displaystyle\to$
$\displaystyle\mathsf{R}^{\prime}_{\mathscr{C}}=\mathsf{R}_{\mathscr{C}}\setminus\\{\mathscr{C}_{N},\mathscr{C}_{a_{i}},\dots,\mathscr{C}_{a_{1}}\\}=\left\\{\mathscr{C}^{\prime}_{1},\mathscr{C}^{\prime}_{2},\dots,\mathscr{C}^{\prime}_{N}\right\\},$
$\displaystyle\mathscr{U}$ $\displaystyle\to$
$\displaystyle\mathscr{U}^{\prime}=\mathscr{U}\cup\\{\mathscr{C}_{N},\mathscr{C}_{a_{i}},\dots,\mathscr{C}_{a_{1}}\\},~{}~{}~{}~{}\mathscr{L}\to\mathscr{L}^{\prime}=\mathscr{L}~{}~{}\text{(if
$\mathbb{CH}$ was attached to $\mathscr{U}$)}$ $\displaystyle\mathscr{L}$
$\displaystyle\to$
$\displaystyle\mathscr{L}^{\prime}=\mathscr{L}\cup\\{\mathscr{C}_{N},\mathscr{C}_{a_{i}},\dots,\mathscr{C}_{a_{1}}\\},~{}~{}~{}~{}\mathscr{U}\to\mathscr{U}^{\prime}=\mathscr{U}~{}~{}\text{(if
$\mathbb{CH}$ was attached to $\mathscr{L}$)}.$ (4.8)

  * •

Step-3 Repeating step-2 with the new defined $\mathsf{R}_{\mathscr{C}}$,
$\mathscr{U}$ and $\mathscr{L}$ iteratively until the ordered set $\mathsf{R}$
becomes empty, we get a graph with only two mutually disjoint subgraphs: the
final upper and lower blocks $\mathscr{U}$ and $\mathscr{L}$.

Figure 17:  A typical spurious graph where the path starting from the highest-
weight element and ending at the root $1$ passes through some single sides of
type-I (IA or IB) components
$\mathscr{C}_{a_{1}},\mathscr{C}_{a_{2}},\dots,\mathscr{C}_{a_{m}}$ (which are
called spurious components).

All possible configurations of the final upper and lower blocks are produced
by the above steps. The final upper and lower blocks for the examples in
section 3 are precisely reproduced by this rule (see Fig. 12 (a), (b) for
example-1 and Fig. 11 (b), (c) for example-2).

###  4.3 Physical and spurious graphs

For a given configuration of the final upper and lower blocks $\mathscr{U}$
and $\mathscr{L}$ which are constructed previously, a fully connected graph
$\mathcal{G}$ in eq. (4.3) is produced by connecting arbitrary two nodes
$x\in\mathscr{U}$ and $y\in\mathscr{L}\setminus\\{r\\}$ via a type-3 line. As
pointed in section 3, such a graph can be either a physical graph or a
spurious one. We have already stated that physical and spurious graphs can be
obtained from those in single-trace case Hou:2018bwm  by the enlargement
(4.6). As a result, a spurious graph has the structure Fig. 17, where the
chain starting form the highest-weight element (graviton for the identity
(2.23) and gluon for the identity (2.25)) and ending at the root $1$ passes
through single sides of some type-IA and/or type-IB components
$\mathscr{C}_{a_{1}},\mathscr{C}_{a_{2}},\dots,\mathscr{C}_{a_{m}}$ which
respectively belong to the chains of components
$\mathbb{CH}_{1},\dots,\mathbb{CH}_{m}$.

In order to display more details of spurious graphs, we define the weight
$W_{i}$ of a chain $\mathbb{CH}_{a_{i}}$ by the weight of the starting
component (equivalently the highest-weight component) of
$\mathbb{CH}_{a_{i}}$. As pointed in Hou:2018bwm , if the lowest-weight chain
among $\mathbb{CH}_{1},\dots,\mathbb{CH}_{m}$ in Fig. 17 is $\mathbb{CH}_{l}$,
we must have $W_{1}>\dots>W_{l-1}>W_{l}$ and $W_{l}<W_{l+1}<\dots<W_{m}$.
Following a discussion which is parallel with that in Hou:2018bwm , we
conclude that all the chains $\mathbb{CH}_{1},\dots,\mathbb{CH}_{l-1}$ (and
structures attached to them) belong to the final lower block $\mathscr{L}$,
while $\mathbb{CH}_{l+1},\dots,\mathbb{CH}_{m}$ (and structures attached to
them) belong to the final upper block $\mathscr{U}$. Only the lowest-weight
chain $\mathbb{CH}_{l}$ (among $\mathbb{CH}_{l+1},\dots,\mathbb{CH}_{m}$) can
live in either $\mathscr{L}$ or $\mathscr{U}$. Correspondingly, the type-3
line (colored by red in Fig. 17) on either the LHS or the RHS of
$\mathscr{C}_{a_{l}}$ is considered as the one between the final upper and
lower blocks. Thus a given spurious graph is corresponding to two distinct
configurations of $\mathscr{U}$ and $\mathscr{L}$. In other words, all
spurious graphs must appear in pairs! This fact allows us to associate a pair
of spurious graphs with opposite signs so that all spurious graphs cancel out.

By the help of the above discussion, we now determine the sign
$(-)^{\mathcal{G}}$ in eq. (4.3) for a (physical or spurious) graph
$\mathcal{G}$:

  * •

(i). As proposed in Hou:2018bwm , a graph is accompanied by a sign
$(-1)^{S(\mathscr{U}_{x})}$, where $S(\mathscr{U}_{x})$ is the number of
spurious components living in the final upper block $\mathscr{U}$ for a given
$x\in\mathscr{U}$ in eq. (4.3). For the typical spurious graph Fig. 17,
$S(\mathscr{U}_{x})=(-1)^{m-l+1}$, if $\mathscr{C}_{a_{l}}$ belongs to the
final upper block $\mathscr{U}$, while $S(\mathscr{U}_{x})=(-1)^{m-l}$, if
$\mathscr{C}_{a_{l}}$ belongs to the final lower block $\mathscr{L}$. Hence
spurious graphs cancel in pairs precisely.

  * •

(ii). Another sign which should be taken into account is introduced by the
relation eq. (3.5). Particularly, if the arrow of the kernel in a type-IB
component (see Fig. 15 (b)) is pointing away from the root (i.e. the second
term of eq. (3.5)), this component should be dressed by an extra minus. Since
the choice of $y\in\mathscr{L}\setminus\\{r\\}$ in eq. (4.3) does not affect
the direction of arrows in the final lower block $\mathscr{L}$, the number of
such type-IB components only depends on $\mathscr{U}$, $\mathscr{L}$ and the
choice of $x\in\mathscr{U}$ in eq. (4.3). We use
$Tr(\mathscr{U}_{x},\mathscr{L})$ to denote this number, the resulting sign is
then written as $(-1)^{Tr(\mathscr{U}_{x},\mathscr{L})}$.

  * •

(iii). For any graph, the total number
$\mathcal{N}(\mathscr{U}_{x})+\mathcal{N}(\mathscr{L})+1$ of arrows pointing
away from the root induces the third sign
$(-1)^{\mathcal{N}(\mathscr{U}_{x})+\mathcal{N}(\mathscr{L})+1}$ (as required
by the refined graphic rule), where $\mathcal{N}(\mathscr{U}_{x})$ and
$\mathcal{N}(\mathscr{L})$ count the corresponding numbers in $\mathscr{U}$
(for $x\in\mathscr{U}$) and $\mathscr{L}$. The extra minus is caused by the
type-3 line between $\mathscr{U}$ and $\mathscr{L}$.

To sum up, the sign for any (physical or spurious) graph is given by

$\displaystyle(-)^{\mathcal{G}}=(-1)^{S(\mathscr{U}_{x})+Tr(\mathscr{U}_{x},\mathscr{L})+\mathcal{N}(\mathscr{U}_{x})+\mathcal{N}(\mathscr{L})+1}.$
(4.9)

For a physical graph $\mathcal{G}$, there is no spurious component, thus
$S(\mathscr{U}_{x})=0$. For identities induced from single-trace amplitudes,
$Tr(\mathscr{U}_{x},\mathscr{L})$ vanishes and the sign (4.9) turns into the
one given in Hou:2018bwm .

###  4.4 Expanding induced identities in terms of BCJ relations

Figure 18:  Relative positions of nodes in a graph $\mathcal{G}$ which is
determined by $\mathscr{U}$, $\mathscr{L}$ and $x\in\mathscr{U}$,
$y\in\mathscr{L}\setminus\\{r\\}$ can be characterized by dashed lines with no
arrow.

We are now ready to show the expression in the square brackets in eq. (4.3),
i.e.

$\displaystyle
I[\mathscr{U},\mathscr{L}]\equiv\sum\limits_{\begin{subarray}{c}x\in\mathscr{U}\\\
y\in\mathscr{L}\setminus\\{r\\}\end{subarray}}\,\sum\limits_{\boldsymbol{\sigma}^{\mathcal{G}}}(-)^{\mathcal{G}}\,(k_{x}\cdot
k_{y})\,A\big{(}1,\boldsymbol{\sigma}^{\mathcal{G}},r\big{)}~{}$ (4.10)

for a given configuration of the final upper and lower blocks $\mathscr{U}$,
$\mathscr{L}$ is the LHS of the graph-based BCJ relation (A.15) (hence a
combination of traditional BCJ relations (A.14)). Here a graph $\mathcal{G}$
in the above expression is a physical or a spurious graph which is constructed
by connecting two nodes $x\in\mathscr{U}$, $y\in\mathscr{L}\setminus\\{r\\}$
via a type-3 line. Our discussion is carried out by the follow steps:

  * •

(i). According to the refined graphic rule, permutations
$\boldsymbol{\sigma}^{\mathcal{G}}$ in eq. (4.10) are independent of the line
styles in $\mathcal{G}$. Therefore, all lines in ${\mathcal{G}}$ can be
replaced by dashed lines (with no arrow) which only characterize the relative
positions of nodes in ${\mathcal{G}}$, as shown by Fig. 18. Apparently, for a
given configuration of the final upper and lower blocks $\mathscr{U}$,
$\mathscr{L}$ and a given choice of $x\in\mathscr{L}$, the corresponding
permutations $\boldsymbol{\sigma}^{\mathcal{G}}$ satisfy

$\displaystyle\boldsymbol{\sigma}^{\mathcal{G}}\in\Bigl{[}\boldsymbol{\zeta}\shuffle\boldsymbol{\gamma}\Bigr{]}\Big{|}_{y\prec
x}~{}~{}~{}~{}\big{(}\text{for~{}}\boldsymbol{\zeta}\in{\mathscr{U}}\big{|}_{x}\text{~{}and~{}}\boldsymbol{\gamma}\in\bigl{(}\mathscr{L}\big{|}_{1}\bigr{)}\setminus\\{1,r\\}\big{)},$
(4.11)

where ${\mathscr{U}}\big{|}_{x}$ and $\mathscr{L}\big{|}_{1}$ denote the
permutations established by $\mathscr{U}$ and $\mathscr{L}$ when
$x\in\mathscr{U}$ and the root $1\in\mathscr{L}$ are the leftmost elements
respectively. Since $1$ and $r$ are fixed as the first and the last elements
in eq. (4.10), they should be excluded from
$\boldsymbol{\sigma}^{\mathcal{G}}$. The $y\prec x$ means $y$ situates before
$x$ in the permutation, i.e.
$(\sigma^{\mathcal{G}})^{-1}(y)<(\sigma^{\mathcal{G}})^{-1}(x)$. Noting that
the choice of $y$ is independent of the relative orders $\boldsymbol{\zeta}$
and $\boldsymbol{\gamma}$, we rewrite the summations in eq. (4.10) as follows

$\displaystyle\sum\limits_{\begin{subarray}{c}x\in\mathscr{U}\\\
y\in\mathscr{L}\setminus\\{r\\}\end{subarray}}\,\sum\limits_{\boldsymbol{\sigma}^{\mathcal{G}}}\to\sum\limits_{x\in\mathscr{U}}\,\sum\limits_{\boldsymbol{\zeta}\in\mathscr{U}|_{x}}\,\sum\limits_{\boldsymbol{\gamma}\in\mathscr{L}|_{1}\setminus\\{1,r\\}}\,\sum\limits_{y\in\mathscr{L}\setminus\\{r\\}}\,\sum\limits_{\boldsymbol{\sigma}\in[\boldsymbol{\zeta}\shuffle\boldsymbol{\gamma}]|_{y\prec
x}}.$ (4.12)

  * •

(ii). For a given permutation
$\boldsymbol{\sigma}\in\boldsymbol{\zeta}\shuffle\boldsymbol{\gamma}$, one can
collect together the coefficients $k_{x}\cdot k_{y}$ in eq. (4.11) with
different choices of $y\in\mathscr{L}\setminus\\{r\\}$. Specifically, only
those $y$ satisfying $y\prec x$ in $\boldsymbol{\sigma}$ have nonzero
contributions and the sign (4.9) is totally independent of $y$. Hence, all the
$k_{x}\cdot k_{y}$ factors for a given $\boldsymbol{\sigma}$ are collected as
$k_{x}\cdot Y_{x}(\boldsymbol{\sigma})$ where
$Y^{\mu}_{x}(\boldsymbol{\sigma})\equiv\sum_{y\prec x}k_{y}^{\mu}$ (the gluon
$1$ is always included as the leftmost $y$ in this summation). Meanwhile, the
last two summations in eq. (4.12) turns into

$\displaystyle\sum\limits_{y\in\mathscr{L}\setminus\\{r\\}}\,\sum\limits_{\boldsymbol{\sigma}\in[\boldsymbol{\zeta}\shuffle\boldsymbol{\gamma}]|_{y\prec
x}}\to\sum\limits_{\boldsymbol{\sigma}\in\boldsymbol{\zeta}\shuffle\boldsymbol{\gamma}}.$
(4.13)

  * •

(iii). As illustrated by appendix C, the sign $(-)^{\mathcal{G}}$ in eq.
(4.10) has the following pattern. Two graphs with $x=x_{1}$ and $x=x_{2}$
where $x_{1},x_{2}\in\mathscr{U}$ are adjacent to each other must be
associated with opposite signs. Hence one can extract the sign eq. (4.9) for a
graph with $x=x_{0}\in\mathscr{U}$ as an overall factor and then the sign for
an arbitrary choice of node $x$ is given by

$\displaystyle(-1)^{S(\mathscr{U}_{x_{0}})+Tr(\mathscr{U}_{x_{0}},\mathscr{L})+\mathcal{N}(\mathscr{U}_{x_{0}})+\mathcal{N}(\mathscr{L})+1}f^{x}\equiv(-)^{\mathcal{G}({x_{0}})}f^{x},$
(4.14)

where $f^{x}$ is defined by (i). $f^{x_{0}}=1$, (ii). $f^{x_{1}}=-f^{x_{2}}$
for two adjacent nodes $x_{1}$ and $x_{2}$ ($x_{1},x_{2}\in\mathscr{U}$).

When all the above are taken into account, eq. (4.10) is finally expressed by

$\displaystyle
I[\mathscr{U},\mathscr{L}]=(-)^{\mathcal{G}({x_{0}})}\sum\limits_{\boldsymbol{\gamma}\in\mathscr{L}|_{1}\setminus\\{1,r\\}}\biggl{[}\sum\limits_{x\in\mathscr{U}}f^{x}\sum\limits_{\boldsymbol{\zeta}\in\mathscr{U}|_{x}}\sum\limits_{\boldsymbol{\sigma}\in\boldsymbol{\zeta}\shuffle\boldsymbol{\gamma}}(k_{x}\cdot
Y_{x}(\boldsymbol{\sigma}))A\Bigl{(}1,\boldsymbol{\sigma},r\Bigr{)}\biggr{]},$
(4.15)

where the summation over $\boldsymbol{\gamma}$ was extracted out because it is
independent of the choice of $x\in\mathscr{U}$. The expression in the square
brackets in eq. (4.15) is nothing but the LHS of the graph-based BCJ relation
(A.15).

##  5 Conclusions

In this paper, we provided the refined graphic rule for expanding tree level
multi-trace EYM amplitudes in terms of color-ordered YM amplitudes. When the
gauge invariance condition of a graviton and the cyclic symmetry of a gluon
trace were imposed, this expansion induced two identities (2.23) and (2.25)
respectively. By extending the analysis for the single-trace case Hou:2018bwm
to an arbitrary multi-trace induced identity, we demonstrated that eq. (2.23)
and eq. (2.25) can finally be expressed as a combination of graph-based BCJ
relations (thus traditional BCJ relations).

There are several related topics that deserve further study: (i). First, how
to understand the induced identities from the view of string theory? String
theory studies of the expansions of EYM amplitudes have been established in
Stieberger:2016lng ; Schlotterer:2016cxa ; He:2018pol ; He:2019drm , while BCJ
relations have also been proven in string theory BjerrumBohr:2009rd ;
Stieberger:2009hq . Hence it is reasonable to expect a string-theory approach
to both induced identities and graph-based BCJ relations. (ii). Second, it is
worth investigating the induced identities in various theories systematically.
In Zhou:2019mbe , a unified web of expansions of amplitudes was founded with
the help of the unifying relation Cheung:2017ems , which inspires that the
induced identities may exist in many other theories. (iii). Third, the YM
expansion of EYM amplitudes, which have been used in this paper, is in KK
basis Kleiss:1988ne . As pointed in Feng:2019tvb , this expansion can be
extended to BCJ basis Bern:2008qj . We expect that the refined graphic rule
can also be generalized to expansions in BCJ basis Bern:2008qj . (iv). Last
but not least, a kinematic algebra for constructing BCJ numerators in the MHV
sector was proposed Chen:2019ywi . It seems that distinct sectors of
numerators are corresponding to graphs with different numbers of type-IA
kernels. Thus, one may provide a general rule for constructing all sectors of
BCJ numerators, with the help of refined graphic rule.

## Acknowledgments

The authors are grateful to Chih-Hao Fu, Song He, Xiaodi Li, Hui Luo, Gang
Yang, Yihong Wang and Yong Zhang for helpful discussions or/and valuable
comments. This work is supported by NSFC under Grant Nos. 11875206, 11847309,
Jiangsu Ministry of Science and Technology under contract BK20170410 as well
as the “Fundamental Research Funds for the Central Universities”.

##  Appendix A Recursive expansions and BCJ relations

In this section, we review the recursive expansions of multi-trace EYM
amplitudes and the BCJ relations for YM amplitudes.

###  A.1 Recursive expansions of multi-trace EYM amplitudes

When all gravitons $h_{1},\dots,h_{s}$ and gluon traces
$\boldsymbol{2},\dots,\boldsymbol{m}$ are collected into the set
$\boldsymbol{\mathcal{H}}\equiv\\{h_{1},\dots,h_{s},\boldsymbol{2},\dots,\boldsymbol{m}\\}$,
a tree level multi-trace EYM amplitude
$A(1,2,\ldots,r|\boldsymbol{2}|\ldots|\boldsymbol{m}\|\mathsf{H})$ where
$\mathsf{H}\equiv\\{h_{1},h_{2},\dots,h_{s}\\}$ can be briefly expressed by
$A(1,2,\ldots,r\|\boldsymbol{\mathcal{H}})$. As proven in Du:2017gnh , this
multi-trace amplitude satisfies the following recursive expansion relation:

$\displaystyle
A(1,2,\ldots,r\,\|\,\boldsymbol{\mathcal{H}})=\sum_{\small\begin{subarray}{c}\boldsymbol{\mathcal{H}}\setminus\\{\mathcal{H}_{a}\\}\\\
\to\text{perms}\,\boldsymbol{\mathcal{H}}_{A}|\boldsymbol{\mathcal{H}}_{B}\end{subarray}}\widetilde{\sum\limits_{\mathsf{Tr}}}\,\biggl{[}\sum\limits_{\boldsymbol{\sigma}}C(1,\boldsymbol{\sigma},r)A(1,\boldsymbol{\sigma},r\,\|\,\boldsymbol{\mathcal{H}}_{B})\biggr{]}.$
(A.1)

In the expansion (A.1), we have picked out an arbitrary element
$\mathcal{H}_{a}$ (a graviton or a gluon trace) from
$\boldsymbol{\mathcal{H}}$, which is called the _fiducial element_.
Apparently, $\mathcal{H}_{a}$ can be either a graviton or a gluon trace. On
the RHS of eq. (A.1):

  * •

The first summation is taken over (i). all possible splittings of the set
$\boldsymbol{\mathcal{H}}\setminus\\{\mathcal{H}_{a}\\}$ into two subsets
$\boldsymbol{\mathcal{H}}_{A}$, $\boldsymbol{\mathcal{H}}_{B}$ and (ii). all
permutations of elements in $\boldsymbol{\mathcal{H}}_{A}$ for a given
splitting.

  * •

For a given splitting of
$\boldsymbol{\mathcal{H}}\setminus\\{\mathcal{H}_{a}\\}$ and a given
permutation of elements in $\boldsymbol{\mathcal{H}}_{A}$, the summation
$\widetilde{\sum}_{\mathsf{Tr}}$ is defined as follows

if $\mathcal{H}_{a}$ is a graviton:
$\displaystyle\widetilde{\sum_{\mathsf{Tr}}}\to\sum_{\scriptsize\begin{subarray}{c}\\{a_{i},b_{i}\\}\subset\boldsymbol{t}_{i}\\\
\text{for all
$\boldsymbol{t}_{i}\in\boldsymbol{\mathcal{H}}_{A}$}\end{subarray}}(-1)^{|\boldsymbol{t}_{i},a_{i},b_{i}|}\sum\limits_{\boldsymbol{\beta}_{i}}\,$
(A.2) if $\mathcal{H}_{a}=\boldsymbol{t}_{0}$ is a gluon trace:
$\displaystyle\widetilde{\sum_{\mathsf{Tr}}}\to\sum_{\scriptsize\begin{subarray}{c}a_{0}\in\boldsymbol{t}_{0}\\\
\text{for $a_{0}\neq
b_{0}$($b_{0}\in\boldsymbol{t}_{0}$)}\end{subarray}}(-1)^{|\boldsymbol{t}_{0},a_{0},b_{0}|}\sum\limits_{\boldsymbol{\beta}_{0}}\sum_{\scriptsize\begin{subarray}{c}\\{a_{i},b_{i}\\}\subset\boldsymbol{t}_{i}\\\
\text{for all
$\boldsymbol{t}_{i}\in\boldsymbol{\mathcal{H}}_{A}$}\end{subarray}}(-1)^{|\boldsymbol{t}_{i},a_{i},b_{i}|}\sum\limits_{\boldsymbol{\beta}_{i}}.$
(A.3)

This means we sum over all possible choices of the ordered pair of gluons
$\\{a_{i},b_{i}\\}\subset\boldsymbol{t}_{i}$ for all traces
$\boldsymbol{t}_{i}\in\boldsymbol{\mathcal{H}}_{A}$. If the fiducial element
$\mathcal{H}_{a}$ is also a trace, namely $\boldsymbol{t}_{0}$, we should fix
an arbitrary gluon $b_{0}\in\boldsymbol{t}_{0}$ and then sum over all choices
of $a_{0}\neq b_{0}$ in this trace. For a given choice of $a_{i}$ and $b_{i}$,
a gluon trace (including $\mathcal{H}_{a}$ if it is also a trace) can always
be written into the form $a_{i},\boldsymbol{X}_{i},b_{i},\boldsymbol{Y}_{i}$
where $\boldsymbol{X}_{i}$ and $\boldsymbol{Y}_{i}$ are the two ordered sets
of gluons separated by $a_{i}$ and $b_{i}$. The sign
$(-1)^{|\boldsymbol{t}_{i},a_{i},b_{i}|}$ in eq. (A.2) and eq. (A.3) for each
trace is defined as $(-1)^{|\boldsymbol{Y}_{i}|}$ where $|\boldsymbol{Y}_{i}|$
is the number of elements in $\boldsymbol{Y}_{i}$. Permutations
$\boldsymbol{\beta}_{i}$ (for a given $\\{a_{i},b_{i}\\}$) and
$\boldsymbol{\beta}_{0}$ (for a given $a_{0}\neq b_{0}$) in eq. (A.2) and/or
eq. (A.3) which satisfy

$\displaystyle\boldsymbol{\beta}_{i}$
$\displaystyle\in\mathsf{KK}[\boldsymbol{t}_{i},a_{i},b_{i}]$
$\displaystyle\equiv\boldsymbol{X}_{i}\shuffle\boldsymbol{Y}_{i}^{T},$ (A.4)
$\displaystyle\boldsymbol{\beta}_{0}$
$\displaystyle\in\mathsf{KK}[\boldsymbol{t}_{0},a_{0},b_{0}]$
$\displaystyle\equiv\boldsymbol{X}_{0}\shuffle\boldsymbol{Y}_{0}^{T},$ (A.5)

are also summed over.

  * •

Supposing that the permutation of elements in $\boldsymbol{\mathcal{H}}_{A}$
in the first summation (see eq. (A.1)) is given by $j_{1}$, $j_{2}$,…, $j_{u}$
and the gluon pairs in the second summation are
$\\{a_{i},b_{i}\\}\subset\boldsymbol{t}_{i}$ ($i$ can be $0$ if the fiducial
element $\mathcal{H}_{a}$ is a gluon trace $\boldsymbol{t}_{0}$), we sum over
all permutations $\boldsymbol{\sigma}$ satisfying

$\displaystyle\boldsymbol{\sigma}\in\\{2,\dots,r-1\\}\shuffle\\{j_{1},j_{2},\dots,j_{u},\mathcal{H}_{a}\\}.$
(A.6)

_Here the traces in the ordered set
$\\{j_{1},j_{2},\dots,j_{u},\mathcal{H}_{a}\\}$ are no longer considered as
single elements but considered as proper permutations of all gluons in them. _
Particularly, if $j_{i}$ denotes a gluon trace $\boldsymbol{t}_{i}$ and this
trace can be written as $a_{i},\boldsymbol{X}_{i},b_{i},\boldsymbol{Y}_{i}$
for a given choice of ordered pair
$\\{a_{i},b_{i}\\}\subset\boldsymbol{t}_{i}$, we should replace $j_{i}$ in eq.
(A.6) by a permutation $\\{a_{i},\boldsymbol{\beta}_{i},b_{i}\\}$ where
$\boldsymbol{\beta}_{i}$ satisfies eq. (A.4). Similarly, if the fiducial
element $\mathcal{H}_{a}$ is also a trace, say $\boldsymbol{t}_{0}$, it must
be replaced by a permutation $\\{a_{0},\boldsymbol{\beta}_{0},b_{0}\\}$ where
$\boldsymbol{\beta}_{0}$ satisfies eq. (A.5). Then the summation over
$\boldsymbol{\sigma}$ in eq. (A.1) means summing over all possible shuffle
permutations in eq. (A.6).

  * •

The coefficient $C(1,\boldsymbol{\sigma},r)$ in eq. (A.1) is defined as

$\displaystyle\mathcal{E}_{\mathcal{H}_{a}}\cdot\mathbb{F}_{j_{u}}\cdot\ldots\cdot\mathbb{F}_{j_{i}}\cdot
Y_{j_{1}}(\boldsymbol{\sigma}),$ (A.7)

where $\mathcal{E}^{\mu}$ and $\mathbb{F}^{\mu\,\nu}$ are defined by eq. (2.9)
and eq. (2.12).

$\displaystyle Y^{\mu}_{j_{1}}(\boldsymbol{\sigma})$ $\displaystyle=$
$\displaystyle\Biggl{\\{}\begin{array}[]{cc}\sum\limits_{\tiny\sigma^{-1}(l)<\sigma^{-1}(h_{x})}k^{\mu}_{l}&~{}~{}~{}~{}~{}~{}~{}~{}\,(\text{if
$j_{1}$ is a graviton $h_{x}$})\\\
\sum\limits_{\tiny\sigma^{-1}(l)<\sigma^{-1}(b_{i})}k^{\mu}_{l}&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,(\text{if
$j_{1}$ is a gluon trace $\boldsymbol{t}_{i}$})\end{array}.$ (A.10)

###  A.2 Identities induced from multi-trace EYM amplitudes

Symmetries of multi-trace EYM amplitudes, together with the recursive
expansion (A.1), induce nontrivial identities for EYM amplitudes with fewer
gravitons and/or gluon traces Du:2017gnh . There are two symmetries under
consideration in this paper: the gauge invariance condition for a graviton and
the cyclic symmetry of a gluon trace.

Muti-trace EYM amplitude satisfies gauge invariance condition, which states
that the amplitude has to vanish once half polarization
$\epsilon^{\mu}_{h_{x}}$ of a graviton $h_{x}$ is replaced by the momentum
$k^{\mu}_{h_{x}}$. If this replacement is performed on the RHS of the
recursive expansion (A.1), we should consider two distinct situations:

  * •

If the fiducial element $\mathcal{H}_{a}$ is a graviton $h_{x}$ (i.e. eq.
(A.1) is type-I expansion), the gauge invariance condition for $h_{x}$ induces
a nontrivial relation between EYM amplitudes with fewer gravitons

$\displaystyle\sum_{\small\begin{subarray}{c}\boldsymbol{\mathcal{H}}\setminus
h_{x}\\\
\to\text{perms}\,\boldsymbol{\mathcal{H}}_{A}|\boldsymbol{\mathcal{H}}_{B}\end{subarray}}\widetilde{\sum\limits_{\mathsf{Tr}}}\bigg{[}\sum\limits_{\boldsymbol{\sigma}}C(1,\boldsymbol{\sigma},r)\Big{|}_{\epsilon_{h_{x}}\to
k_{h_{x}}}A(1,\boldsymbol{\sigma},r\,\|\,\boldsymbol{\mathcal{H}}_{B})\bigg{]}=0.$
(A.11)

  * •

If $h_{x}$ is not the fiducial one, it may belong to either
$\boldsymbol{\mathcal{H}}_{A}$ or $\boldsymbol{\mathcal{H}}_{B}$ in eq. (A.1)
. Terms with $h_{x}\in\boldsymbol{\mathcal{H}}_{A}$ have to vanish due to the
antisymmetry of the strength tensor $F_{h_{x}}^{\mu\nu}\equiv
k_{h_{x}}^{\mu}\epsilon_{h_{x}}^{\nu}-k_{h_{x}}^{\nu}\epsilon_{h_{x}}^{\mu}$,
while terms with $h_{x}\in\boldsymbol{\mathcal{H}}_{B}$ have to vanish due to
gauge invariance of amplitudes with fewer gravitons.

Therefore, the only nontrivial identity induced by the gauge invariance
condition of a graviton is eq. (A.11) which is called type-I identity in
Du:2017gnh .

Another identity (called type-II identity in Du:2017gnh ) is induced from the
expansion (A.1) where the fiducial element $\mathcal{H}_{a}$ is a gluon trace
$\boldsymbol{t}_{0}$. In particular, we notice that the end element
$b_{0}\in\boldsymbol{t}_{0}$ can be chosen arbitrarily in the fiducial trace
$\boldsymbol{t}_{0}$. This arbitrariness is essentially caused by the cyclic
symmetry of the trace $\boldsymbol{t}_{0}$ Du:2017gnh  and indicates the
following identity

$\displaystyle\sum_{\small\begin{subarray}{c}\boldsymbol{\mathcal{H}}\setminus\mathcal{H}_{a}\\\
\to\text{perms}\,\boldsymbol{\mathcal{H}}_{A}|\boldsymbol{\mathcal{H}}_{B}\end{subarray}}\widetilde{\sum\limits_{\mathsf{Tr}}}\bigg{[}\sum\limits_{\boldsymbol{\sigma}^{\prime}}C(1,\boldsymbol{\sigma}^{\prime},r)A(1,\boldsymbol{\sigma}^{\prime},r\,\|\,\boldsymbol{\mathcal{H}}_{B})\bigg{]}=0,$
(A.12)

where $\boldsymbol{\sigma}^{\prime}$ is defined by

$\displaystyle\boldsymbol{\sigma}^{\prime}\in\\{2,\dots,r-1\\}\shuffle\\{j_{1},j_{2},\dots,j_{u},a_{0},\boldsymbol{\beta}_{0}\in\mathsf{KK}[\boldsymbol{t}_{0},a_{0},b_{0}],\bcancel{b_{0}}\\}.$
(A.13)

Here we removed $b_{0}$ from the trace $\boldsymbol{t}_{0}$ first, then
shuffled these permutations according to eq. (A.6). The $j_{1}$, $j_{2}$, …,
$j_{u}$ is a permutation of elements in $\boldsymbol{\mathcal{H}}_{A}$.

When the recursive expansion (A.1) for amplitudes with fewer gravitons and/or
gluon traces are applied repeatedly, the two types of relations (A.11) and
(A.12) respectively turn into the induced identities (2.23) and (2.25) for
pure YM amplitudes.

###  A.3 BCJ relations

Tree level color-ordered YM amplitudes satisfy the following _traditional BCJ
relation_ BjerrumBohr:2009rd ; Chen:2011jxa :

$\displaystyle\sum\limits_{\boldsymbol{\sigma}\in\boldsymbol{\beta}\,\shuffle\,\boldsymbol{\alpha}}\,\sum\limits_{l\in\boldsymbol{\beta}}\left(k_{l}\cdot
X_{l}(\boldsymbol{\sigma})\right)A(1,\boldsymbol{\sigma},r)=0,$ (A.14)

where $\boldsymbol{\beta}$ and $\boldsymbol{\alpha}$ are two ordered sets of
external gluons, $X_{l}(\boldsymbol{\sigma})$ denotes the sum of all momenta
of gluons $a\in\\{1,\boldsymbol{\sigma}\\}\cup\boldsymbol{\beta}$ satisfying
$\sigma^{-1}(a)<\sigma^{-1}(l)$.

In Hou:2018bwm , the following _graph-based BCJ relation_ for YM amplitudes
was proposed

$\displaystyle\sum\limits_{a\in\mathcal{T}}f^{a}\sum\limits_{\boldsymbol{\zeta}\in{\mathcal{T}}|_{a}}\sum\limits_{\boldsymbol{\sigma}\in\boldsymbol{\zeta}\shuffle\,\boldsymbol{\gamma}}\left[k_{a}\cdot
Y_{a}(\boldsymbol{\sigma})\right]\,A\left(1,\boldsymbol{\sigma},r\right)=0.~{}~{}$
(A.15)

Here, $\boldsymbol{\gamma}$ is an arbitrary permutation of elements in
$\\{2,\dots,r-1\\}$ and $\mathcal{T}$ is an arbitrary connected tree graph.
When a node $a$ is chosen as the leftmost element, the tree graph
$\mathcal{T}$ establishes permutations $\boldsymbol{\zeta}\in\mathcal{T}|_{a}$
as follows (i). For two adjacent nodes $x$ and $y$, if $x$ is nearer to $a$
than $y$, we have $\zeta^{-1}(x)<\zeta^{-1}(y)$, (ii). If there are subtree
structures attached to a same node, we should shuffle the permutations
established by these subtrees together. The factor $f^{a}$ is a relative sign
depending on the node $a$. This factor is determined by the following steps.
(i). Choose an arbitrary node $c$ and require $f^{c}=1$. (ii). For arbitrary
two adjacent nodes $c_{1}$ and $c_{2}$, we have $f^{c_{1}}=-f^{c_{2}}$. As
already proven in Hou:2018bwm , the graph-based BCJ relation (A.15) can always
be written as a combination of the traditional ones (A.14).

##  Appendix B Proof of eq. (3.5)

To prove eq. (3.5), we focus on a term with a given choice of
$\\{a_{i},b_{i}\\}\subset\boldsymbol{i}$ in eq. (3.5) and prove the following
stronger relation:

$\displaystyle\boxed{\centering\includegraphics[width=480.59395pt]{SplittingTraces2.jpg}},\@add@centering$
(B.1)

where the $\mathcal{T}$ denotes a tree structure which is attached to the node
$b_{i}$. In fact, all these nodes in the trace can be attached by arbitrary
tree structures and the relation (B.1) still holds. Once all
$\\{a_{i},b_{i}\\}\subset\boldsymbol{i}$ are summed over, we arrive the
relation (3.5).

Suppose the gluons in trace $\boldsymbol{i}$ are in the cyclic order
$d_{1},d_{2},\dots,d_{|\boldsymbol{i}|}$. Without loss of generality, $a_{i}$
and $b_{i}$ are respectively chosen as $a_{i}=d_{j}$,
$b_{i}=d_{|\boldsymbol{i}|}$. Then the permutations
$\\{a_{i},\boldsymbol{\beta}\in\mathsf{KK}[\,\boldsymbol{i},a_{i},b_{i}],b_{i}\\}$
on the LHS of eq. (B.1) are explicitly displayed by

$\displaystyle\big{\\{}a_{i}=d_{j},\\{d_{j+1},\dots,d_{|\boldsymbol{i}|-1}\\}\shuffle\\{d_{j-1},\dots,d_{1}\\},b_{i}=d_{|\boldsymbol{i}|},\mathcal{T}|_{x}\big{\\}},$
(B.2) $\displaystyle=$
$\displaystyle\Bigl{\\{}\big{\\{}a_{i}=d_{j},\\{d_{j+1},\dots,d_{|\boldsymbol{i}|-1}\\}\shuffle\\{d_{j-1},\dots,d_{1}\\},b_{i}=d_{|\boldsymbol{i}|}\big{\\}}\shuffle\mathcal{T}|_{x}\Bigr{\\}}|_{b_{i}\prec
x}.$

with a sign $(-1)^{|\boldsymbol{i},a_{i},b_{i}|}=(-1)^{j-1}$ (the overall sign
coming from $\mathcal{T}|_{x}$ has been neglected). Here $x$ denotes the
nearest to $b_{i}$ node in $\mathcal{T}$ and $\mathcal{T}|_{x}$ are the
permutations established by the tree $\mathcal{T}$. The notation $b_{i}\prec
x$ means the position of $b_{i}$ in the permutation is less than that of $x$.
The second line of eq. (B.2), where we shuffled $\mathcal{T}|_{x}$ with the
full trace and required the node $b_{i}$ is always on the left of the node
$x$, is apparently equivalent to the first line.

On the RHS of eq. (B.1), we assume that the $c_{i}$ is chosen as $d_{l}$
($l>j$). The case with $l<j$ follows from a similar discussion. Then
permutations in the first term on the RHS of eq. (B.1) is given by

$\displaystyle\Bigl{\\{}\big{\\{}a_{i}=d_{j},\\{d_{j+1},\dots,d_{l-1}\\}\shuffle\\{d_{j-1},\dots,d_{1},b_{i}=d_{|\boldsymbol{i}|},\dots,d_{l+1}\\},c_{i}=d_{l}\bigr{\\}}\shuffle\mathcal{T}|_{x}\Bigr{\\}}|_{b_{i}\prec
x},$ (B.3)

with the sign
$(-1)^{|\boldsymbol{i},a_{i},c_{i}|}=(-1)^{|\boldsymbol{i}|-l+j-1}$. The
permutations in the second term of the RHS of eq. (3.5) are given by the
following two steps:

  * •

First shuffle the gluons in the trace $\boldsymbol{i}$ so that
$b_{i}=d_{|\boldsymbol{i}|}$ and $c_{i}=d_{l}$ become the two ends of the
trace. Then the permutations
$\boldsymbol{\beta}\in\mathsf{KK}[\,\boldsymbol{i},b_{i},c_{i}]$ are
explicitly given by

$\displaystyle\boldsymbol{\beta}\in\\{d_{|\boldsymbol{i}|-1},\dots,d_{l+1}\\}\shuffle\\{d_{1},\dots,d_{l-1}\\}.$
(B.4)

which is associated with a sign
$(-1)^{|\boldsymbol{i},b_{i},c_{i}|}=(-1)^{|\boldsymbol{i}|-l-1}$. These
permutations can be classified according to the relative orders between the
$a_{i}=d_{j}$ $(d_{j}\in\\{d_{1},\dots,d_{l-1}\\})$ and elements in the set
$\\{d_{|\boldsymbol{i}|-1},\dots,d_{l+1}\\}$:

$\displaystyle\boldsymbol{\beta}^{{(1)}}$ $\displaystyle\in$
$\displaystyle\bigl{\\{}d_{1},\dots,d_{j-1},d_{j},\\{d_{j+1},\dots,d_{l-1}\\}\shuffle\\{d_{|\boldsymbol{i}|-1},\dots,d_{l+1}\\}\bigr{\\}}$
$\displaystyle\boldsymbol{\beta}^{{(2)}}$ $\displaystyle\in$
$\displaystyle\bigl{\\{}\\{d_{1},\dots,d_{j-1}\\}\shuffle\\{d_{|\boldsymbol{i}|-1}\\},d_{j},\\{d_{j+1},\dots,d_{l-1}\\}\shuffle\\{d_{|\boldsymbol{i}|-2},\dots,d_{l+1}\\}\bigr{\\}}$
$\displaystyle\dots$ $\displaystyle\boldsymbol{\beta}^{{(q)}}$
$\displaystyle\in$
$\displaystyle\bigl{\\{}\\{d_{1},\dots,d_{j-1}\\}\shuffle\\{d_{|\boldsymbol{i}|-1},\dots,d_{|\boldsymbol{i}|-q+1}\\},d_{j},\\{d_{j+1},\dots,d_{l-1}\\}\shuffle\\{d_{|\boldsymbol{i}|-q},\dots,d_{l+1}\\}\bigr{\\}}$
$\displaystyle\dots$
$\displaystyle\boldsymbol{\beta}^{(|\boldsymbol{i}|-l-1)}$ $\displaystyle\in$
$\displaystyle\bigl{\\{}\\{d_{1},\dots,d_{j-1}\\}\shuffle\\{d_{|\boldsymbol{i}|-1},\dots,d_{l+2}\\},d_{j},\\{d_{j+1},\dots,d_{l-1}\\}\shuffle\\{d_{l+1}\\}\bigr{\\}}$
$\displaystyle\boldsymbol{\beta}^{(|\boldsymbol{i}|-l)}$ $\displaystyle\in$
$\displaystyle\bigl{\\{}\\{d_{1},\dots,d_{j-1}\\}\shuffle\\{d_{|\boldsymbol{i}|-1},\dots,d_{l+1}\\},d_{j},d_{j+1},\dots,d_{l-1}\bigr{\\}}.$
(B.5)

  * •

For any given permutation $\boldsymbol{\beta}^{(q)}$ in eq. (B.5), the
permutations $\boldsymbol{\sigma}^{(q)}$ established by the second term on the
RHS of eq. (B.1) are obtained by shuffling the two branches attached to the
node $a_{i}=d_{i}$ together:

$\displaystyle\boldsymbol{\sigma}^{(q)}$ $\displaystyle\in$
$\displaystyle\biggl{\\{}\Bigl{\\{}a_{i}=d_{j},\bigl{\\{}\\{d_{j+1},\dots,d_{l-1}\\}\shuffle\\{d_{|\boldsymbol{i}|-q},\dots,d_{l+1}\\},c_{i}=d_{l}\bigr{\\}}$
(B.6)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\shuffle\bigl{\\{}\\{d_{j-1},\dots,d_{1}\\}\shuffle\\{d_{|\boldsymbol{i}|-q+1},\dots,d_{|\boldsymbol{i}|-1}\\},b_{i}=d_{|\boldsymbol{i}|}\bigr{\\}}\Bigr{\\}}\shuffle\mathcal{T}|_{x}\biggr{\\}}\Big{|}_{b_{i}\prec
x}$

with a sign $(-1)^{|\boldsymbol{i}|-l-1}(-1)^{\mathcal{N}^{(q)}_{a_{i}}}$.
Here $\mathcal{N}^{(q)}_{a_{i}}$ denotes the number of arrows pointing away
from root in the trace $\boldsymbol{i}$.

According to the relative orders between $d_{|\boldsymbol{i}|-q}$ and
$d_{|\boldsymbol{i}|-q+1}$, permutations $\boldsymbol{\sigma}^{(q)}$ in eq.
(B.6) splits into
$\boldsymbol{\sigma}^{(q)}_{\text{A}}\equiv\boldsymbol{\sigma}^{(q)}|_{d_{|\boldsymbol{i}|-q}\prec
d_{|\boldsymbol{i}|-q+1}}$ and
$\boldsymbol{\sigma}^{(q)}_{\text{B}}\equiv\boldsymbol{\sigma}^{(q)}|_{d_{|\boldsymbol{i}|-q}\succ
d_{|\boldsymbol{i}|-q+1}}$. It is easy to see

$\displaystyle\boldsymbol{\sigma}^{(q)}_{\text{A}}$ $\displaystyle=$
$\displaystyle\boldsymbol{\sigma}^{(q+1)}_{\text{B}},~{}~{}~{}~{}\mathcal{N}^{(q)}_{a_{i}}=\mathcal{N}^{(q+1)}_{a_{i}}+1.$
(B.7)

Hence the contributions from $\boldsymbol{\sigma}^{(q)}_{\text{A}}$ and
$\boldsymbol{\sigma}^{(q+1)}_{\text{B}}$ for all
$q=1,\dots,|\boldsymbol{i}|-l-1$ must cancel one another. The remaining
nonzero terms are the two boundaries

$\displaystyle\boldsymbol{\sigma}^{(1)}_{\text{B}}$ $\displaystyle\in$
$\displaystyle\biggl{\\{}\Bigl{\\{}a_{i}=d_{j},\bigl{\\{}\\{d_{j+1},\dots,d_{l-1}\\}\shuffle\\{d_{|\boldsymbol{i}|-1},\dots,d_{l+1}\\},c_{i}=d_{l}\bigr{\\}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\shuffle\bigl{\\{}d_{j-1},\dots,d_{1},b=d_{|\boldsymbol{i}|}\bigr{\\}}\Bigr{\\}}\big{|}_{d_{|\boldsymbol{i}|-1}\succ
d_{|\boldsymbol{i}|}}\shuffle\mathcal{T}|_{x}\biggr{\\}}\Big{|}_{b_{i}\prec
x}$ $\displaystyle=$
$\displaystyle\Bigl{\\{}\bigl{\\{}a_{i}=d_{j},\\{d_{j+1},\dots,d_{l-1}\\}\shuffle\\{d_{j-1},\dots,d_{1},b_{i}=d_{|\boldsymbol{i}|},d_{|\boldsymbol{i}|-1},\dots,d_{l+1}\\},c_{i}=d_{l}\bigr{\\}}\shuffle\mathcal{T}|_{x}\Bigr{\\}}\big{|}_{b_{i}\prec
x}.$ $\displaystyle\boldsymbol{\sigma}^{(|\boldsymbol{i}|-l)}_{\text{A}}$
$\displaystyle\in$
$\displaystyle\biggl{\\{}\Bigl{\\{}a_{i}=d_{j},\\{d_{j+1},\dots,d_{l-1},c_{i}=d_{l}\bigr{\\}}$
(B.8)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\shuffle\bigl{\\{}\\{d_{j-1},\dots,d_{1}\\}\shuffle\\{d_{l+1},\dots,d_{|\boldsymbol{i}|-1}\\},b_{i}=d_{|\boldsymbol{i}|}\bigr{\\}}\Bigr{\\}}\big{|}_{d_{l}\prec
d_{l+1}}\shuffle\mathcal{T}|_{x}\biggr{\\}}\Big{|}_{b_{i}\prec x}$
$\displaystyle=$
$\displaystyle\Bigl{\\{}\bigl{\\{}a_{i}=d_{j},\\{d_{j-1},\dots,d_{1}\\}\shuffle\\{d_{j+1},\dots,d_{|\boldsymbol{i}|-1}\\},b_{i}=d_{|\boldsymbol{i}|}\bigr{\\}}\shuffle\mathcal{T}|_{x}\Bigr{\\}}\big{|}_{b_{i}\prec
x},$

with the signs
$(-1)^{|\boldsymbol{i}|-l-1}(-1)^{\mathcal{N}^{(1)}_{a_{i}}}=(-1)^{|\boldsymbol{i}|-l-1+j}$
and
$(-1)^{|\boldsymbol{i}|-l-1}(-1)^{\mathcal{N}^{(|\boldsymbol{i}|-l)}_{a_{i}}}=(-1)^{j}$
respectively. An extra minus which is introduced from the second term of eq.
(B.1) must also be taken into account. Hence the permutations
$\boldsymbol{\sigma}^{(1)}_{\text{B}}$ are same with the permutations in eq.
(B.3), with an opposite sign. As a result, they cancel with each other. The
remaining permutations $\boldsymbol{\sigma}^{(|\boldsymbol{i}|-l)}_{\text{A}}$
are nothing but the permutations (B.2) with the corrected sign. Therefore, we
have proven the relation (B.1) for a given $\\{a_{i},b_{i}\\}$. After summing
over all possible choices of the $\\{a_{i},b_{i}\\}$ pairs, the proof of eq.
(3.5) is completed.

Figure 19: All possible situations when choosing $x\in\mathscr{U}$ as two
adjacent nodes $x_{1}$ and $x_{2}$.

_Comments on the proof:_ In the above proof, the tree structure
$\mathcal{T}|_{x}$ is always treated separately from the trace in each step
(i.e. it is shuffled with the full trace with a proper constraint, as shown in
eqs. (B.2), (B.3), (B.6), (B.8)). Thus manipulations on nodes inside the trace
are independent of $\mathcal{T}|_{x}$. This observation allows us to
generalize eq. (3.5) to cases where more tree structures are attached to nodes
in the trace $\boldsymbol{i}$ straightforwardly.

##  Appendix C The sign in eq. (4.10)

In eq. (4.10), the graph $\mathcal{G}$ is constructed by connecting
$x\in\mathscr{U}$ and $y\in\mathscr{L}$ via a type-3 line. As mentioned
before, the sign for such a graph is dependent of the choice of $x$ but
independent of the choice of $y$. Now we show that two graphs with choosing
adjacent $x\in\mathscr{U}$ must have opposite signs. To see this, we study all
possible structures presented by Fig. 19 (a)-(e), where $x=x_{1}$ is adjacent
to $x=x_{2}$ ($x_{1},x_{2}\in\mathscr{U}$), as follows.

(i). As shown by Fig. 19 (a), nodes $x_{1}$ and $x_{2}$ connected by a type-3
line must belong to a same (top or bottom) side of a type-IA or type-IB
component. In this case, each one of $S(\mathscr{U}_{x})$,
$Tr(\mathscr{U}_{x},\mathscr{L})$ and $\mathcal{N}(\mathscr{L})$ is the same
for choosing two adjacent $x$’s. But the numbers
$\mathcal{N}(\mathscr{U}_{x})$ in eq. (4.9) for $x=x_{1}$ and $x=x_{2}$ differ
by one because the type-2 line between $x_{1}$ and $x_{2}$ has opposite
directions for these two cases.

(ii). If $x_{1}$ and $x_{2}$ are connected by a type-4 line, they must belong
to a same type-IB component. If the type-4 line is not the kernel of the type-
IB component, the structure is given by Fig. 19 (b). In this case, $x_{1}$ and
$x_{2}$ must belong to a same side of the component. The numbers
$\mathcal{N}(\mathscr{U}_{x})$ for choosing $x=x_{1}$ and $x=x_{2}$ should
differ by one because the type-4 line between $x_{1}$ and $x_{2}$ has opposite
directions. Each of $S(\mathscr{U}_{x})$, $Tr(\mathscr{U}_{x},\mathscr{L})$
and $\mathcal{N}(\mathscr{L})$ in eq. (4.9) is the same for both choices of
$x$.

(iii). If the two adjacent nodes $x_{1}$ and $x_{2}$ connected by a type-4
line belong to opposite sides of a type-IB component, the type-4 line must be
the kernel of the type-IB component (as shown by Fig. 19 (c)). In this case,
each of $\mathcal{N}(\mathscr{U}_{x})$, $Tr(\mathscr{U}_{x},\mathscr{L})$ and
$S(\mathscr{U}_{x})$ for $x=x_{1}$ and $x=x_{2}$ differ by one but
$\mathcal{N}(\mathscr{L})$ is the same.

(iv). If $x_{1}$ and $x_{2}$ are connected by a type-1 line which must be the
kernel of a type-IA component (as shown by Fig. 19 (d)), only the number
$S(\mathscr{U}_{x})$ for choosing $x=x_{1}$ and $x=x_{2}$ differ by one. Any
of $Tr(\mathscr{U}_{x},\mathscr{L})$, $\mathcal{N}(\mathscr{U}_{x})$ and
$\mathcal{N}(\mathscr{L})$ is the same for the two choices of $x$.

(v). If $x_{1}$ and $x_{2}$ are connected by a type-3 line as shown by Fig. 19
(e), only $S(\mathscr{U}_{x})$ for choosing $x=x_{1}$ and $x=x_{2}$ differ by
one. Any other number is the same for the two choices.

Therefore, in all the above cases,
$S(\mathscr{U}_{x})+Tr(\mathscr{U}_{x},\mathscr{L})+\mathcal{N}(\mathscr{U}_{x})+\mathcal{N}(\mathscr{L})+1$
for choosing $x=x_{1}$ and $x=x_{2}$, where $x_{1},x_{2}\in\mathscr{U}$ are
two adjacent nodes, must differ by an odd number. In other words, two graphs
with adjacent $x$’s have opposite signs.

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