aainstitutetext: School of Physics and Technology, Wuhan University, No.299
Bayi Road, Wuhan 430072, Chinabbinstitutetext: Department of Physics and
Astronomy, Uppsala University, 75108 Uppsala, Swedenccinstitutetext: Suzhou
Institute of Wuhan University, No.377 Linquan Street, Suzhou, 215123, China

# Evaluating EYM amplitudes in four dimensions by refined graphic expansion

Hongxiang Tian 111The unusual ordering of authors is to let authors get proper
recognition of contributions under the practice in China. a,b Enze Gong a
Chongsi Xie a,c Yi-Jian Du222Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM)
amplitudes induces a refined graphic expansion, by which any tree-level EYM
amplitude can be expressed as a summation over all possible refined graphs.
Each graph contributes a unique coefficient as well as a proper combination of
color-ordered Yang-Mills (YM) amplitudes. This expansion allows one to
evaluate EYM amplitudes through YM amplitudes, the latter have much simpler
structures in four dimensions than the former. In this paper, we classify the
refined graphs for the expansion of EYM amplitudes into $\text{N}^{\,k}$MHV
sectors. Amplitudes in four dimensions, which involve $k+2$ negative-helicity
particles, at most get non-vanishing contribution from graphs in
$\text{N}^{\,k^{\prime}(k^{\prime}\leq k)}$MHV sectors. By the help of this
classification, we evaluate the non-vanishing amplitudes with two negative-
helicity particles in four dimensions. We establish a correspondence between
the refined graphs for single-trace amplitudes with $(g^{-}_{i},g^{-}_{j})$ or
$(h^{-}_{i},g^{-}_{j})$ configuration and the spanning forests of the known
Hodges determinant form. Inspired by this correspondence, we further propose a
symmetric formula of double-trace amplitudes with $(g^{-}_{i},g^{-}_{j})$
configuration. By analyzing the cancellation between refined graphs in four
dimensions, we prove that any other tree amplitude with two negative-helicity
particles has to vanish.

###### Keywords:

Amplitude relation, Gauge invariance

##  1 Introduction

The expansion of Einstein-Yang-Mills (EYM) 333In this paper, EYM theory always
means the theory which also involves antisymmetric B field and dilation, while
the external particles of amplitudes can only be gravitons and gluons.
amplitudes in terms of pure Yang-Mills (YM) amplitudes was firstly suggested
in Stieberger:2016lng , where a tree-level single-trace EYM amplitude with one
graviton and $n$ gluons was written as a combination of $n+1$ gluon color-
ordered YM amplitudes. This observation was then extended to amplitudes with a
few gravitons and/or gluon traces in many literatures, see e.g.,
Nandan:2016pya ; delaCruz:2016gnm ; Schlotterer:2016cxa . A general study of
the expansion of single-trace amplitudes can be found in Fu:2017uzt ;
Chiodaroli:2017ngp ; Teng:2017tbo . As pointed in Fu:2017uzt , a single-trace
EYM amplitude could be expanded recursively in terms of EYM amplitudes with
fewer gravitons. Thus, the pure-YM expansion can be obtained by applying the
recursive expansion iteratively. Along this line, the general formulas for
recursive expansion of an arbitrary tree-level multitrace EYM amplitude were
established in Du:2017gnh . Further applications and generalizations of this
recursive expansion have been investigated, including the symmetry-induced
identities Fu:2017uzt ; Du:2017gnh ; Hou:2018bwm ; Du:2019vzf , the proof
Du:2018khm  of the equivalence between distinct approaches to amplitudes in
nonlinear sigma model Du:2016tbc ; Carrasco:2016ldy ; Du:2017kpo , the
construction of polynomial Bern-Carrasco-Johansson (BCJ) Bern:2008qj
numerators Fu:2017uzt ; Du:2017kpo , the expansion into BCJ basis Feng:2020jck
; Feng:2019tvb  as well as the generalization to amplitudes in other theories
Plefka:2018zwm ; Zhou:2020mvz .

Among these progresses, a refined graphic rule which conveniently expands an
EYM amplitude by summing over a set of so-called _refined graphs_ has been
invented Hou:2018bwm ; Du:2019vzf . Each tree graph in this expansion provides
a coefficient (expressed by a product of the Lorentz contractions of external
polarizations and/or momenta) as well as a proper combination of color-ordered
YM amplitudes. This refined graphic expansion was already shown to be a
powerful tool in the study of the relationship between symmetry-induced
identities and BCJ relations Hou:2018bwm ; Du:2019vzf .

On the other hand, EYM amplitudes with particular helicity configurations in
four dimensions was also studied in many work. By substituting solutions to
scattering equations directly into the Cachazo-He-Yuan formula Cachazo:2013gna
; Cachazo:2013hca ; Cachazo:2013iea ; Cachazo:2014nsa ; Cachazo:2014xea , the
single-trace EYM amplitudes with the maximally-helicity-violating (MHV)
configuration, which involves only two negative-helicity particles
$(i^{-},j^{-})$ ($i,j$ can be either (i) two gluons or (ii) one graviton and
one gluon), have been shown to satisfy a compact formula Du:2016wkt  that is
expressed by the well known Hodges determinant Hodges:2012ym . Through a
spanning forest expansion Feng:2012sy , this compact formula was shown (see
Du:2016wkt ) to be equivalent to a generating functional formula which was
proposed in earlier work Selivanov:1997ts ; Selivanov:1997aq ; Bern:1999bx .
At double-trace level, a symmetric formula for the MHV amplitudes with only
external gluons was founded Cachazo:2014nsa . In addition, explicit results of
five- and six-point examples of double-trace MHV amplitudes with one and two
gravitons were also provided in Cachazo:2014xea . However, it is still lack of
a general symmetric formula of double-trace MHV amplitudes with an arbitrary
number of gravitons.

Since YM amplitudes have much simpler forms than EYM ones, the refined graphic
expansion may provide a new approach to the EYM amplitudes in four dimensions.
In this paper, we evaluate the EYM amplitudes in four dimensions by the
refined graphic rule. To achieve this, we classify refined graphs into
$\text{N}^{\,k}$MHV sectors and demonstrate that the $\text{N}^{\,k}$MHV
amplitudes with $(k+2)$ negative-helicity particles at most get nonvanishing
contributions from graphs in the $\text{N}^{\,k^{\prime}(k^{\prime}\leq
k)}$MHV sectors. When $k=0$, the nonvanishing MHV amplitudes can only get
contribution from the refined graphs in the MHV sector and the corresponding
YM amplitudes in the expansion satisfy Parke-Taylor formula Parke:1986gb . For
single-trace MHV amplitudes with two negative-helicity gluons or those with
one-negative helicity graviton and one negative-helicity gluon, a
correspondence between the refined graphs and the spanning forests of the
Hodges determinant is established. Hence this approach precisely reproduces
the known results Du:2016wkt . Inspired by this correspondence, we further
propose a spanning forest formula of double-trace MHV amplitudes with two
negative-helicity gluons, in which the following symmetries are manifest:
_(i)._ invariance under permutations of gravitons, _(ii)._ invariance under
exchanging the two traces, _(iii)._ the cyclic symmetry of each trace. By an
analysis on the cancellation between graphs, we show that all other tree
amplitudes with two negative-helicity particles have to vanish.

The structure of this paper is organized as follows. In section 2, we provide
the construction rule for the MHV sector of tree level single- and double-
trace EYM amplitudes. By the help of the spinor-helicity formalism Xu:1986xb
in four dimensions and helpful identities between Parke-Taylor factors, we
establish a correspondence between the refined graphs in the MHV sector for
single-trace amplitudes and the spanning forests of Hodges determinant in
section 3. In section 4, a symmetric formula of double-trace MHV amplitudes
with two negative-helicity gluons is derived. Other tree amplitudes with two
negative helicity particles are proven to vanish in section 5. This work is
summarized in section 6. The refined graphic rule for the $\text{N}^{\,k}$MHV
sector of an $m$-trace amplitude is provided in appendix A. Spinor-helicity
formalism and helpful identities are reviewed in appendix B.

#### Convention of notations

Convention of notations is displayed as follows444Most notations follow from
the paper Du:2019vzf ..

Permutations: We use boldface Greek letters $\boldsymbol{\beta}$,
$\boldsymbol{\gamma}$, $\boldsymbol{\sigma}$, or boldface uppercase Latin
letters $\boldsymbol{X}$ and $\boldsymbol{Y}$ to denote permutations (or in
other words ordered sets). The $i$-th element in a permutation
$\boldsymbol{\beta}$ is usually denoted by $\beta_{i}$, while the position of
an element $a\in\boldsymbol{\beta}$ is expressed by $\beta^{-1}(a)$. The
condition $\beta^{-1}(a)<\beta^{-1}(b)$ is written as $a\prec b$ in some
places for short. The inverse permutation of $\boldsymbol{\beta}$ is denoted
by $\boldsymbol{\beta}^{T}$. Shuffle permutations of two ordered sets
$\boldsymbol{X}$ and $\boldsymbol{Y}$ are written as
$\boldsymbol{X}\shuffle\boldsymbol{Y}$. Number of elements in $\boldsymbol{X}$
is denoted as $|\boldsymbol{X}|$.

Gravitons, gluon traces and amplitudes: We denote gluon traces by boldface
numbers $\boldsymbol{1}$, $\boldsymbol{2}$, …, or boldface lowercase Latin
letters $\boldsymbol{t}$, $\boldsymbol{i}$ … If a trace $\boldsymbol{i}$ can
be written as
$\boldsymbol{i}=\\{a_{i},\boldsymbol{X}_{i},b_{i},\boldsymbol{Y}_{i}\\}$, we
define
$\mathsf{KK}[\boldsymbol{i},a_{i},b_{i}]\equiv\boldsymbol{X}_{i}\shuffle\boldsymbol{Y}_{i}^{T}$
and $(-1)^{|\boldsymbol{i},a_{i},b_{i}|}\equiv(-1)^{|\boldsymbol{Y}_{i}|}$. A
multitrace EYM amplitude is generally denoted by
$A(1,2,\ldots,r|\boldsymbol{2}|\ldots|\boldsymbol{m}\|\mathsf{H})$, where
$\mathsf{H}$ stands for the graviton set with gravitons $h_{1}$, $h_{2}$, …,
while $\boldsymbol{1},\boldsymbol{2},\dots,\boldsymbol{m}$ are the gluon
traces. Gluons in the trace $\boldsymbol{1}$ are explicitly denoted by
$1,\dots,r$. When we study double trace amplitudes, we use $x_{1},\dots,x_{r}$
and $y_{1},\dots,y_{s}$ to denote the gluons in traces $\boldsymbol{1}$ and
$\boldsymbol{2}$ respectively. The set of all gravitons and gluon traces is
denoted by ${\boldsymbol{\mathcal{H}}}$.

Refined graphs and spanning forests: In this paper, graphs constructed by
refined graphic rule are mentioned as _refined graphs_ and denoted by
$\mathcal{F}$, while graphs that describe helicity amplitudes in the final
formula are mentioned as _spanning forests_ and denoted by $\mathcal{G}$. When
we fix $1$ and $r$ as the first and the last elements, the set of permutations
(of other nodes) established by a refined graph $\mathcal{F}$ is denoted by
${\mathcal{F}}|_{1}\setminus\\{1,r\\}$. Components of a graph $\mathcal{F}$
are denoted by $\mathscr{C}_{i}$, while a chain of components is denoted by
$\mathbb{CH}$. We use $\mathsf{R}$ to stands for the reference order of
gravitons and traces in the refined graphic rule and use
$\mathsf{R}_{\mathscr{C}}$ to denote the reference order of components. The
set of refined graphs which involve $m$ traces and belong to the
$\text{N}^{\,k}$MHV sector is denoted by
$\mathcal{F}\in{\mathcal{F}{[k^{\prime},m]}}$. Disjoint union of graphs are
denoted by $\oplus$.

##  2 Sectors of tree-level EYM amplitudes

Figure 1: Four types of lines in the refined graphic rule for multitrace EYM
amplitudes

As pointed out in Du:2017gnh ; Du:2019vzf , an arbitrary tree level EYM
amplitude $A(1,2,\ldots,r|\boldsymbol{2}|\ldots|\boldsymbol{m}\|\mathsf{H})$
with the graviton set $\mathsf{H}$ and the gluon traces
$\boldsymbol{1}=\\{1,2,\ldots,r\\},\boldsymbol{2},\dots,\boldsymbol{m}$ can be
generally expressed as a combination of tree level color-ordered pure YM
amplitudes

$\displaystyle
A\left(1,2,\ldots,r|\boldsymbol{2}|\ldots|\boldsymbol{m}\|\mathsf{H}\right)=\sum\limits_{\mathcal{F}}(-)^{\mathcal{F}}\mathcal{C}^{\mathcal{F}}\biggl{[}\,\sum\limits_{\boldsymbol{\sigma}\in{\mathcal{F}|_{1}\setminus\\{1,r\\}}}A\big{(}1,\boldsymbol{\sigma},r\big{)}\,\biggr{]},$
(2.1)

where we have summed over all possible connected tree graphs $\mathcal{F}$
which are constructed according to the refined graphic rule Du:2017gnh ;
Du:2019vzf . A tree graph $\mathcal{F}$ defines a coefficient
$\mathcal{C}^{\mathcal{F}}$ as well as a combination of color-ordered YM
amplitudes $A\big{(}1,\boldsymbol{\sigma},r\big{)}$. The polarization tensor
$\epsilon_{h}^{\mu\nu}$ of a graviton splits into two polarization vectors
$\epsilon_{h}^{\mu}\epsilon_{h}^{\nu}$. One of them is absorbed into the
coefficient $\mathcal{C}^{\mathcal{F}}$, the other is considered as the
polarization vector of a gluon which is involved in the pure YM amplitude
$A\big{(}1,\boldsymbol{\sigma},r\big{)}$. In each graph $\mathcal{F}$, all
gluons and gravitons are expressed by nodes. As shown by Fig. 1, distinct
types of lines between nodes are introduced: (a). A type-1 line represents an
$\epsilon\cdot\epsilon$ factor between two gravitons; (b). A type-2 line
denotes an $\epsilon\cdot k$ factor ($k^{\mu}$ denotes the momentum of a node)
between a graviton and a graviton/gluon; (c). A type-3 line stands for a
$k\cdot k$ factor between any two nodes; (d). A type-4 line is introduced to
record the relative order of two adjacent gluons in a gluon trace but does not
contribute any nontrivial kinematic factor. The expansion coefficient
$\mathcal{C}^{\mathcal{F}}$ is then given by the product of factors defined by
all lines in $\mathcal{F}$, while a proper sign $(-)^{\mathcal{F}}$ is also
defined (see appendix A). Permutations
$\boldsymbol{\sigma}\in{\mathcal{F}|_{1}\setminus\\{1,r\\}}$ established by
the graph $\mathcal{F}$ are given by the following two steps: (i). If $a$ and
$b$ are two adjacent nodes which satisfy the condition that $a$ is nearer to
the gluon $1$ than $b$, we have $\sigma^{-1}(a)<\sigma^{-1}(b)$ where
$\sigma^{-1}(a)$ and $\sigma^{-1}(b)$ respectively denote the positions of $a$
and $b$ in $\boldsymbol{\sigma}$.555Supposing that the position of $a$ in
$\boldsymbol{\sigma}$ is $j$, we have $a=\sigma_{j}\equiv{\sigma}(j)$, hence
it is reasonable to define $j={\sigma}^{-1}(a)$. (ii). If two subtree
structures are attached to a same node, we shuffle the corresponding
permutations together such that the permutation in each subtree is preserved.

In order to analyze helicity amplitudes with $m$ traces in four dimensions, we
denote the set of all graphs involving $k$ type-1 lines (i.e.
$\epsilon\cdot\epsilon$ lines) and $m$ gluon traces by $\mathcal{F}{[k,m]}$.
We further define the $\text{N}^{\,k}$MHV sector $I[k,m]$ ($k\geq 0$) of the
expansion (2.1) by the total contribution of graphs in $\mathcal{F}{[k,m]}$ as
follows:

$\displaystyle
I\left[k,m\right]\equiv\sum\limits_{\mathcal{F}\in\mathcal{F}{\left[k,m\right]}}(-)^{\mathcal{F}}\mathcal{C}^{\mathcal{F}}\,\sum\limits_{\boldsymbol{\sigma}\in{\mathcal{F}|_{1}\setminus\\{1,r\\}}}A\bigl{(}1,\boldsymbol{\sigma},r\bigr{)}.$
(2.2)

The expansion (2.1) is then given by summing over all possible $k$

$\displaystyle
A\left(1,2,\ldots,r|\boldsymbol{2}|\ldots|\boldsymbol{m}\|\mathsf{H}\right)=\sum\limits_{k}\,I\left[k,m\right].$
(2.3)

The general refined graphic rule for constructing the $\text{N}^{\,k}$MHV
sector is provided in appendix A. _In the following, we illustrate that an
$\text{N}^{\,k}$MHV amplitude in four dimensions can at most get nonzero
contribution from graphs in the $\text{N}^{\,k^{\prime}(k^{\prime}\leq k)}$MHV
sector._

In four dimensions, both the momentum of a graviton/gluon and the (half)
polarization of a graviton can be expressed according to the spinor-helicity
formalism Xu:1986xb  (see appendix B). The Lorentz contractions between
momenta and/or polarizations are then expressed by spinor products, see eq.
(B.3). In this paper, the reference momenta of all those gravitons with the
same helicity are chosen as the same one. With this choice of gauge, the
contractions between the (half) polarizations of any two gravitons with the
same helicity must vanish. The only possible remaining contractions between
‘half’ polarizations must have the form
$\epsilon_{h_{i}}^{+}\cdot\epsilon_{h_{j}}^{-}$. For an $m$-trace
$\text{N}^{\,k}$MHV amplitude with $l$ negative-helicity gravitons and $k-l+2$
negative-helicity gluons, the number of factors
$\epsilon_{h_{i}}^{+}\cdot\epsilon_{h_{j}}^{-}$ in the coefficient
$\mathcal{C}^{\mathcal{F}}$ is at most $l$. This implies only those graphs
$\mathcal{F}\in{\mathcal{F}{[k^{\prime},m]}}$, which contain $k^{\prime}$
($k^{\prime}\leq l$) type-1 lines, may provide nonvanishing contributions to
an $\text{N}^{\,k}$MHV amplitude in four dimensions. The number of type-1
lines can be further reduced when the following two facts are considered:

  * (i).

It has been known that an EYM amplitude where all the negative-helicity
particles are gravitons (in other words all gluons carry the same helicity)
must vanish. Thus amplitudes in the case of $l=k+2$, for which a graph at most
involves $l=k+2$ nontrivial type-1 lines, have to vanish.

  * (ii).

When the reference momentum of all positive-helicity gravitons are chosen as
the momentum of one of the negative-helicity gravitons, say $h_{i}$, the
maximal number of type-1 lines in a graph is further reduced by one because
$\epsilon^{-}_{h_{i}}(p)\cdot\epsilon^{+}_{h_{j}}(k_{h_{i}})=0$. Thus
amplitudes with $l=k+1$ can only get nonzero contributions from graphs
$\mathcal{F}\in{\mathcal{F}{[k^{\prime},m]}}$ ($k^{\prime}\leq k$).

Therefore, the nonvanishing EYM amplitudes with $k+2$ negative-helicity
particles can at most get nonzero contributions from the graphs in the
$\text{N}^{\,k^{\prime}(k^{\prime}\leq k)}$MHV sector.

In the current paper, we study the single- and double-trace amplitudes with
only two negative helicity particles. The helicity configurations can be
classified into three categories $(g_{i}^{-},g_{j}^{-})$,
$(h_{i}^{-},g_{j}^{-})$ and $(h_{i}^{-},h_{j}^{-})$ which correspond to EYM
amplitudes with (i). two negative-helicity gluons, (ii). one negative-helicity
gluon and one negative-helicity graviton and (iii). two negative-helicity
gravitons. As stated before, the last configuration has to vanish, while the
first two configurations only get contributions from graphs in the MHV sector.
In the coming sections, we show that the MHV sectors of single-trace
amplitudes with the $(g_{i}^{-},g_{j}^{-})$ and $(h_{i}^{-},g_{j}^{-})$
configurations precisely match with the corresponding spanning forests of the
Hodges determinant form in four dimensions. By generalizing this discussion to
double-trace amplitudes, we establish a spanning forest formula for double-
trace amplitudes with the $(g_{i}^{-},g_{j}^{-})$ configuration. Single-trace
amplitudes with the $(h_{i}^{-},h_{j}^{-})$ configuration, double-trace
amplitudes with $(h_{i}^{-},g_{j}^{-})$ and $(h_{i}^{-},h_{j}^{-})$
configurations as well as all $m(m>3)$-trace amplitudes with two negative-
helicity particles will be proven to vanish.

Figure 2: (a): A typical graph for the MHV sector of single-trace EYM
amplitudes. (b): A typical graph for the MHV sector of double-trace EYM
amplitudes.

Before proceeding, we explicitly display the expansions of the single- and
double-trace MHV amplitudes by graphs in the corresponding MHV sectors. The
general construction rule of the $\text{N}^{\,k}$MHV sector can be found in
appendix A.

###  2.1 The expansion of single-trace MHV amplitudes

Following the above discussion, a single-trace MHV amplitude
$A^{(i,j)}(1,2,\ldots,r\|\mathsf{H})$ (where the two negative helicity
particles $(i,j)$ can be either $(g_{i},g_{j})$ or $(h_{i},g_{j})$) is given
by

$\displaystyle
A^{(i,j)}(1,2,\ldots,r\|\mathsf{H})=\sum\limits_{\mathcal{F}\in\mathcal{F}{[0,1]}}(-)^{\mathcal{F}}\mathcal{C}^{\mathcal{F}}A^{(i,j)}(1,\boldsymbol{\sigma},r)=\sum\limits_{\mathcal{F}\in\mathcal{F}{[0,1]}}\mathcal{C}^{\mathcal{F}}\Biggl{[}\,\sum\limits_{\boldsymbol{\sigma}\in{\mathcal{F}\,|_{1}\setminus\\{1,r\\}}}\frac{\left\langle
i,j\right\rangle^{4}}{(1,\boldsymbol{\sigma},r)}\Biggr{]},$ (2.4)

in which all graphs $\mathcal{F}$ in the MHV sector for the single-trace
amplitude have been summed over, while each pure-YM amplitude
$A^{(i,j)}(1,\boldsymbol{\sigma},r)$ involving two negative-helicity particles
$(i,j)$ is further expressed by the Parke-Taylor formula. Here,
$(1,2,\dots,r)$ denotes $\left\langle 1,2\right\rangle\left\langle
2,3\right\rangle\dots\left\langle r,1\right\rangle$ for short.

Graphs $\mathcal{F}\in\mathcal{F}{[0,1]}$ in eq. (2.4) are obtained by the
general construction rule in appendix A. To be specific, the general pattern
of a graph $\mathcal{F}\in\mathcal{F}[0,1]$ is given by connecting gravitons
to gluons in $\boldsymbol{1}\setminus\\{r\\}$ via type-2 lines (i.e.,
$\epsilon\cdot k$ lines) whose arrows point towards the trace
$\boldsymbol{1}$, as shown by Fig. 2 (a). According to appendix A, the sign
$(-)^{\mathcal{F}}$ associating to any graph $\mathcal{F}$ of this pattern is
$+1$.

###  2.2 The expansion of double-trace MHV amplitudes

Suppose the two gluon traces are $\boldsymbol{1}=\\{x_{1},\dots,x_{r}\\}$,
$\boldsymbol{2}=\\{y_{1},\dots,y_{s}\\}$666Here, we slightly change the labels
of gluons so that the two traces are on an equal footing in the final result.,
the graviton set is $\mathsf{H}$ and the two negative-helicity particles are
$(i,j)$ ($(i,j)$ can be either $(g_{i},g_{j})$ or $(h_{i},g_{j})$). A double-
trace amplitude $A^{(i,j)}(x_{1},\ldots,x_{r}|y_{1},\ldots,y_{s}\|\mathsf{H})$
is expanded as

$\displaystyle
A^{(i,j)}(x_{1},\ldots,x_{r}|y_{1},\ldots,y_{s}\|\mathsf{H})=\sum\limits_{\mathcal{F}\in\mathcal{F}{[0,2]}}(-)^{\mathcal{F}}\mathcal{C}^{\mathcal{F}}\Biggl{[}\,\sum\limits_{\boldsymbol{\sigma}\in{\mathcal{F}\,|_{x_{1}}\setminus\\{x_{1},x_{r}\\}}}\frac{\left\langle
i,j\right\rangle^{4}}{(x_{1},\boldsymbol{\sigma},x_{r})}\Biggr{]},$ (2.5)

where we have summed over all possible graphs $\mathcal{F}$ in the MHV sector
of double-trace amplitudes.

As stated in appendix A, graphs $\mathcal{F}\in\mathcal{F}{[0,2]}$ have the
general pattern Fig. 2 (b) where two components $\mathscr{C}_{1}$ and
$\mathscr{C}_{2}$, which respectively contains the two traces $\boldsymbol{1}$
and $\boldsymbol{2}$, are connected by a type-3 line (i.e. the $k\cdot k$
line). Such a graph has the following pattern:

  * •

Gluons of the trace $\boldsymbol{1}$ is arranged in the normal order
$x_{1},\dots,x_{r}$. Assuming the trace $\boldsymbol{2}$ can be written as
$\boldsymbol{2}=\\{a,\boldsymbol{X},b,\boldsymbol{Y}\\}$ where
$a,b\in\boldsymbol{2}$, all gluons in $\boldsymbol{2}$ are arranged in an
order $a,{\beta}_{1},\dots,\beta_{|\boldsymbol{2}|-2},b$ where
$\\{{\beta}_{1},\dots,\beta_{|\boldsymbol{2}|-2}\\}\in\boldsymbol{X}\shuffle\boldsymbol{Y}^{T}$.

  * •

Each of $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ in Fig. 2 (b) is constructed
by connecting gravitons therein to gluons in the corresponding trace (except
the last element $x_{r}$ in the trace $\boldsymbol{1}$) via type-2 lines. The
arrow of any type-2 line should point towards the direction of the
corresponding trace.

  * •

The two end nodes $x$ and $y$ of the type-3 line $k_{x}\cdot k_{y}$
respectively belong to $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$. The node $x$
satisfies $x\in\mathscr{C}_{1}\setminus\\{x_{r}\\}$ and the node $y$ is either
the node $a$ or a node which is connected to $a$ via only type-3 lines.

  * •

The sign $(-)^{\mathcal{F}}$ associating to a graph $\mathcal{F}$ is given by
$(-)^{\mathcal{F}}\equiv(-1)^{\mathcal{N}\left(\mathcal{F}\right)+|\boldsymbol{2},a,b|}$,
where $\mathcal{N}\left(\mathcal{F}\right)$ is the total number of arrows
pointing away from $x_{1}$, $|\boldsymbol{2},a,b|$ denotes the number of
elements in the $\boldsymbol{Y}$ set if the trace $\boldsymbol{2}$ can be
written as $\boldsymbol{2}=\\{a,\boldsymbol{X},b,\boldsymbol{Y}\\}$ for a
given $a\neq b$ ($a\in\boldsymbol{2}$).

The summation over all graphs $\mathcal{F}\in\mathcal{F}{[0,2]}$ in eq. (2.5)
then means that we should sum over (i). all possible graphs constructed by the
above method for a given $a\neq b$ ($b\in\boldsymbol{2}$ is always fixed) and
a given
$\boldsymbol{\beta}=\\{{\beta}_{1},\dots,\beta_{|\boldsymbol{2}|-2}\\}$, (ii).
all possible $\boldsymbol{\beta}$ for a given $a$, (iii). all possible
$a\in\boldsymbol{2}$, ($a\neq b$).

Figure 3: A typical spanning forest for single-trace MHV amplitude. All
positive-helicity gravitons and gluons are treated as nodes. All trees in the
forest are rooted at gluons.

##  3 Single-trace MHV amplitudes with $(g_{i}^{-},g_{j}^{-})$ and
$(h_{i}^{-},g_{j}^{-})$

In this section, we evaluate the single-trace MHV amplitudes
$A^{(i,j)}(1,2,\ldots,r\|\mathsf{H})$ with the helicity configurations
$(g^{-}_{i},g^{-}_{j})$ and $(h^{-}_{i},g_{j}^{-})$ by the expansion (2.4). We
prove that eq. (2.4) precisely gives the spanning forest form

$\displaystyle\boxed{A^{(i,j)}(1,2,\ldots,r\|\mathsf{H})=(-1)^{N_{h^{-}}}\frac{\left\langle
i,j\right\rangle^{4}}{\big{(}1,2,\dots,r\bigr{)}}\left[\,\sum\limits_{\mathcal{G}}\,\prod\limits_{e(x,y)\in\mathcal{E}(\mathcal{G})}\frac{\left\langle
y,\xi\right\rangle\left\langle
y,\eta\right\rangle\left[y,x\right]}{\left\langle
x,\xi\right\rangle\left\langle x,\eta\right\rangle\left\langle
y,x\right\rangle}\,\right]},$ (3.1)

which is the spanning forest expansion Feng:2012sy  of the known formula
(B.15) (see Du:2016wkt ) that is expressed by Hodges determinant Hodges:2012ym
. In the above expression, we summed over all possible spanning forests
$\mathcal{G}$ (see Fig. 3) where all elements in
$\\{1,\dots,r\\}\cup\mathsf{H}^{+}$ ($\mathsf{H}^{+}$ is the set of positive-
helicity gravitons) were treated as nodes, while the gluons $1,\dots,r$ were
considered as roots. The prefactor $(-1)^{N_{h^{-}}}$ is $+1$ for the
$(g^{-}_{i},g^{-}_{j})$ configuration and $-1$ for the $(h^{-}_{i},g_{j}^{-})$
configuration. For a given forest $\mathcal{G}$, $\mathcal{E}(\mathcal{G})$
denotes the set of edges and $e(x,y)$ denotes an edge between nodes $x$ and
$y$. For an edge $e(x,y)$, $y$ is always supposed to be nearer to root than
$x$ and the edge is dressed by an arrow pointing towards $y$. Each $e(x,y)$ in
eq. (3.1) is accompanied by a factor

$\displaystyle\frac{\left\langle y,\xi\right\rangle\left\langle
y,\eta\right\rangle\left[y,x\right]}{\left\langle
x,\xi\right\rangle\left\langle x,\eta\right\rangle\left\langle
y,x\right\rangle},$ (3.2)

where $\xi$ and $\eta$ are two arbitrarily chosen reference spinors which
cannot make the expression divergent. In the following, we prove eq. (3.1)
with $(g^{-}_{i},g^{-}_{j})$ and $(h^{-}_{i},g_{j}^{-})$ configurations in
turn.

###  3.1 Single-trace amplitudes with $(g^{-}_{i},g^{-}_{j})$ configuration

For a single-trace MHV amplitude with the $(g^{-}_{i},g^{-}_{j})$
configuration, all gravitons carry positive helicity. As shown in section 2,
the reference momenta of all gravitons are taken as the same one, say
$\xi^{\mu}$. According to spinor helicity formalism, the Lorentz contraction
$\epsilon^{+}_{h}\cdot k_{l}$ between the half polarization
$\epsilon^{+\mu}_{h}$ of a graviton $h$ and an arbitrary momentum
$k^{\mu}_{l}$ becomes

$\displaystyle\epsilon^{+}_{h_{a}}(\xi)\cdot
k_{l}=\frac{\left\langle\xi,l\right\rangle\left[l,h_{a}\right]}{\left\langle\xi,h_{a}\right\rangle}.$
(3.3)

Therefore, the coefficient $\mathcal{C}^{\mathcal{F}}$ in eq. (2.4) can be
conveniently expressed as

$\displaystyle\mathcal{C}^{\mathcal{F}}=\mathcal{C}^{\mathcal{F}\setminus\\{h_{a}\\}}\,\frac{\left\langle\xi,l\right\rangle\left[l,h_{a}\right]}{\left\langle\xi,h_{a}\right\rangle}$
(3.4)

for a leaf (i.e. an outmost graviton) $h_{a}\in\mathcal{F}$. Here
$\mathcal{F}\setminus\\{h_{a}\\}$ denotes the subgraph that is obtained from
$\mathcal{F}$, by removing $h_{a}$ and the edge attached to it. The node $l$
in eq. (3.4) is supposed to be the neighbour of $h_{a}$.

Figure 4: Evaluating a single-trace amplitude with the $(g_{i}^{-},g_{j}^{-})$
configuration.

On another hand, the summation over all permutations
$\boldsymbol{\sigma}\in{\mathcal{F}\,|_{1}\setminus\\{1,r\\}}$ which are
established by the graph $\mathcal{F}$ in eq. (2.4) can be achieved by the
following steps: (1). summing over all possible permutations
$\boldsymbol{\gamma}\in\mathcal{F}\big{|}_{1}\setminus\\{1,r,h_{a}\\}$
established by the graph $\mathcal{F}\setminus\\{h_{a}\\}$; (2). summing over
all possible permutations $\boldsymbol{\gamma}\shuffle\\{h_{a}\\}|_{l\prec
h_{a}}$ 777Here we write the constraint condition
$\sigma^{-1}(l)<\sigma^{-1}(h_{a})$ by $l\prec h_{a}$ for short. for a given
$\boldsymbol{\gamma}$ in the previous step. Thus the summation in square
brackets of eq. (2.4) is rearranged into

$\displaystyle\sum\limits_{\boldsymbol{\gamma}\in\mathcal{F}|_{1}\setminus\\{1,r,h_{a}\\}}\,\sum\limits_{\boldsymbol{\sigma}\in\boldsymbol{\gamma}\,\shuffle\\{h_{a}\\}|_{l\prec
h_{a}}}\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\big{(}1,\boldsymbol{\sigma},r\big{)}},$ (3.5)

which can be further simplified by the help of the identity (B.8), as follows

$\displaystyle\Biggl{[}\,\sum\limits_{\boldsymbol{\gamma}\in\mathcal{F}|_{1}\setminus\\{1,r,h_{a}\\}}\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\big{(}1,\boldsymbol{\gamma},r\big{)}}\,\Biggr{]}\,\frac{\left\langle
l,r\right\rangle}{\left\langle l,h_{a}\right\rangle\left\langle
h_{a},r\right\rangle}.$ (3.6)

When substituting eq. (3.6) and eq. (3.4) into eq. (2.4) for a given graph
$\mathcal{F}$, we arrive

$\displaystyle T[\mathcal{F}]$ $\displaystyle\equiv$
$\displaystyle\mathcal{C}^{\mathcal{F}}\Biggl{[}\,\sum\limits_{\boldsymbol{\sigma}\in{\mathcal{F}\,|_{1}\setminus\\{1,r\\}}}\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{(1,\boldsymbol{\sigma},r)}\,\Biggr{]}$ (3.7)
$\displaystyle=$
$\displaystyle\Biggl{[}\,\mathcal{C}^{\mathcal{F}\setminus\\{h_{a}\\}}\sum\limits_{\boldsymbol{\gamma}\in\mathcal{F}|_{1}\setminus\\{1,r,h_{a}\\}}\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\big{(}1,\boldsymbol{\gamma},r\big{)}}\,\Biggr{]}\,\frac{\left\langle
l,r\right\rangle\left\langle
l,\xi\right\rangle\left[l,h_{a}\right]}{\left\langle
h_{a},r\right\rangle\left\langle h_{a},\xi\right\rangle\left\langle
l,h_{a}\right\rangle}$ $\displaystyle=$ $\displaystyle
T[\mathcal{F}\setminus\\{h_{a}\\}]\,\frac{\left\langle
l,\xi\right\rangle\left\langle
l,r\right\rangle\left[l,h_{a}\right]}{\left\langle
h_{a},\xi\right\rangle\left\langle h_{a},r\right\rangle\left\langle
l,h_{a}\right\rangle},$

where the expression in the square brackets on the second line is just the
contribution of the graph $\mathcal{F}\setminus\\{h_{a}\\}$. Eq. (3.7) is
shown by the first equality of Fig. 4, while we introduce an arrowed edge
$e(h_{a},l)$ pointing towards $l$ to denote the factor

$\displaystyle\frac{\left\langle l,\xi\right\rangle\left\langle
l,r\right\rangle\left[l,h_{a}\right]}{\left\langle
h_{a},\xi\right\rangle\left\langle h_{a},r\right\rangle\left\langle
l,h_{a}\right\rangle}.$ (3.8)

Apparently, the removed line between $h_{a}$ and $l$ in the graph
$\mathcal{F}$ is transformed into the edge $e(h_{a},l)$ with the new meaning
eq. (3.8) in this step.

The equation (3.7) establishes a relation between contributions of a graph
$\mathcal{F}$ and the subgraph with one gravion extracted out, hence can be
applied iteratively. To be specific, we can further pick out a leaf, say
$h_{b}$, from the graph $\mathcal{F}\setminus\\{h_{a}\\}$. According to eq.
(3.7), $T[\mathcal{F}\setminus\\{h_{a}\\}]$ can be written as

$\displaystyle
T[\mathcal{F}\setminus\\{h_{a}\\}]=T\left[\mathcal{F}\setminus\\{h_{a},h_{b}\\}\right]\frac{\left\langle
l^{\prime},\xi\right\rangle\left\langle
l^{\prime},r\right\rangle\left[l^{\prime},h_{b}\right]}{\left\langle
h_{b},\xi\right\rangle\left\langle h_{b},r\right\rangle\left\langle
l^{\prime},h_{b}\right\rangle},$ (3.9)

where $h_{b}$ is supposed to be connected to the node $l^{\prime}$ via a
type-2 line. Repeating the above steps until all gravitons have been extracted
out, we get the final expression which consists of the following two factors
(see the last term in Fig. 4): (i). the Parke-Taylor factor corresponding to
the gluon trace $\boldsymbol{1}$, (ii). the coefficient characterized by the
forest which has the same structure with that in the original graph
$\mathcal{F}$ but with the new meaning of the edges, i.e. (3.8). When all
possible graphs $\mathcal{F}\in\mathcal{F}[0,1]$ are summed over, the full
single-trace MHV amplitude eq. (2.4) with the $(g_{i}^{-},g_{j}^{-})$
configuration is then written as

$\displaystyle\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\big{(}1,2,\dots,r\bigr{)}}\sum\limits_{\mathcal{G}^{\prime}}\,\prod\limits_{e(x,y)\in\mathcal{E}(\mathcal{G}^{\prime})}\frac{\left\langle
y,\xi\right\rangle\left\langle y,r\right\rangle\left[y,x\right]}{\left\langle
x,\xi\right\rangle\left\langle x,r\right\rangle\left\langle
y,x\right\rangle}.$ (3.10)

In the above expression, we have summed over all possible spanning forests
$\mathcal{G}^{\prime}$ where leaves and internal nodes are gravitons, gluons
$1,2,\dots,r-1$ are roots (recalling that in a graph $\mathcal{F}$ (see eq.
(2.4)) the last gluon $r$ cannot be connected by a type-2 line). The summation
over $\mathcal{G}^{\prime}$ can be further extended to a summation over all
forests $\mathcal{G}$ rooted at gluons $1,2,\dots,r$ because

$\displaystyle\frac{\left\langle r,\xi\right\rangle\left\langle
r,r\right\rangle\left[r,h_{a}\right]}{\left\langle
h_{a},\xi\right\rangle\left\langle h_{a},r\right\rangle\left\langle
l,h_{a}\right\rangle}=0.$ (3.11)

Therefore, we get the expression (3.1) with the choice of reference spinor
$\eta=r$. Since the RHS of eq. (3.1) can be expressed by the Hodges
determinant which is independent of the choice of the reference spinors $\xi$
and $\eta$ Feng:2012sy ; Du:2016wkt ), the proof of eq. (3.1) has been
completed.

Figure 5: Typical graphs for the $(h^{-}_{i},g^{-}_{j})$ configuration. In (a)
the negative-helicity graviton $h_{i}$ is the end node of a type-2 line. In
(b), $h_{i}$ plays as a leaf (i.e. an outmost node).

###  3.2 Single-trace amplitudes with $(h^{-}_{i},g^{-}_{j})$ configuration

For the $(h^{-}_{i},g^{-}_{j})$ configuration, there is one negative-helicity
graviton $h_{i}$ and one negative-helicity gluon $g_{j}$. As stated in section
2, the reference momenta of all positive-helicity gravitons are chosen as
$k^{\mu}_{h_{i}}$. Consequently,

$\displaystyle\epsilon_{h_{l}}^{+}(k_{h_{i}})\cdot k_{h_{i}}=0.$ (3.12)

This indicates those graphs $\mathcal{F}$, where the negative-helicity
graviton $h_{i}$ is the ending node of a type-2 line (as shown by Fig. 5 (a)),
have to vanish. In other words, $h_{i}$ in a nonvanishing graph can only be a
leaf, as shown by Fig. 5 (b). Thus the summation over graphs
$\mathcal{F}\in\mathcal{F}{[0,1]}$ in eq. (2.4) becomes a summation
$\sum_{\mathcal{F}\,^{\prime}}$ over all graphs $\mathcal{F}\,^{\prime}$ with
the structure Fig. 5 (b). This summation can be further arranged as
$\sum_{\mathcal{F}\,^{\prime}}\to\sum_{\mathcal{F}\,^{\prime\prime}}\,\sum_{l\in\mathcal{F}\,^{\prime\prime}\setminus\\{r\\}}$,
where $\mathcal{F}\,^{\prime\prime}$ denote those subgraphs (of graphs
$\mathcal{F}^{\prime}$) which involve only positive-helicity gravitons and all
gluons, while $l$ denotes the neighbour of $h_{i}$ in the original graph
$\mathcal{F}^{\prime}$ (see Fig. 5 (b)). Meanwhile, the coefficient
$\mathcal{C}^{\mathcal{F}\,^{\prime}}$ is factorized as

$\displaystyle\mathcal{C}^{\mathcal{F}\,^{\prime}}=\mathcal{C}^{\mathcal{F}\,^{\prime\prime}}\big{(}\epsilon^{-}_{h_{i}}\cdot
k_{l}\big{)}=\mathcal{C}^{\mathcal{F}\,^{\prime\prime}}\frac{\left[\eta,l\right]\left\langle
l,h_{i}\right\rangle}{\left[\eta,h_{i}\right]},$ (3.13)

where $\eta^{\mu}$ is the reference momentum of the negative-helicity graviton
$h_{i}$. The permutations
$\boldsymbol{\sigma}\in\mathcal{F}\,^{\prime}|_{1}\setminus\\{1,r\\}$ can be
reexpressed as

$\displaystyle\boldsymbol{\sigma}\in\boldsymbol{\gamma}\shuffle\\{h_{i}\\}|_{l\prec
h_{i}},\text{~{}where~{}}\boldsymbol{\gamma}\in\mathcal{F}\,^{\prime\prime}|_{1}\setminus\\{1,r\\}.$
(3.14)

Altogether, the single-trace MHV amplitude
$A^{(h_{i},g_{j})}(1,2,\ldots,r\|\mathsf{H})$ in eq. (2.4) becomes

$\displaystyle\sum\limits_{\mathcal{F}\,^{\prime\prime}}\mathcal{C}^{\mathcal{F}\,^{\prime\prime}}\sum\limits_{l\in\mathcal{F}\,^{\prime\prime}\setminus\\{r\\}}\,\frac{\left[\eta,l\right]\left\langle
l,h_{i}\right\rangle}{\left[\eta,h_{i}\right]}\Biggl{[}\,\sum\limits_{\boldsymbol{\gamma}\in\mathcal{F}\,^{\prime\prime}|_{1}\setminus\\{1,r\\}}\,\sum\limits_{\boldsymbol{\sigma}\in\boldsymbol{\gamma}\,\shuffle\\{h_{i}\\}|_{l\prec
h_{i}}}\,\frac{\left\langle
h_{i},g_{j}\right\rangle^{4}}{(1,\boldsymbol{\sigma},r)}\Biggr{]}.$ (3.15)

When the identity (B.8) is applied, the last summation in the square brackets
gives

$\displaystyle\sum\limits_{\boldsymbol{\sigma}\in\boldsymbol{\gamma}\,\shuffle\\{h_{i}\\}|_{l\prec
h_{i}}}\,\frac{\left\langle
h_{i},g_{j}\right\rangle^{4}}{(1,\boldsymbol{\sigma},r)}=\frac{\left\langle
h_{i},g_{j}\right\rangle^{4}}{\big{(}1,\boldsymbol{\gamma},r\big{)}}\frac{\left\langle
l,r\right\rangle}{\left\langle l,h_{i}\right\rangle\left\langle
h_{i},r\right\rangle},$ (3.16)

Here, the first factor is a Parke-Taylor factor which does not involve the
graviton $h_{i}$ in the denominator, while the second factor,
$\frac{\left\langle l,r\right\rangle}{\left\langle
l,h_{i}\right\rangle\left\langle h_{i},r\right\rangle}$, depends on the choice
of $l$ and is independent of $\boldsymbol{\gamma}$. Thus the summation of the
factors depending on $l$ in eq. (3.15) is given by

$\displaystyle\sum\limits_{l\in\mathcal{F}\,^{\prime\prime}\setminus\\{r\\}}\frac{\left[\eta,l\right]\left\langle
l,h_{i}\right\rangle}{\left[\eta,h_{i}\right]}\cdot\frac{\left\langle
l,r\right\rangle}{\left\langle l,h_{i}\right\rangle\left\langle
h_{i},r\right\rangle}=\sum\limits_{l\in\mathcal{F}\,^{\prime\prime}\setminus\\{r\\}}\frac{\left\langle
l,r\right\rangle\left[\eta,\,l\right]}{\left\langle
h_{i},r\right\rangle\left[\eta,h_{i}\right]}=-\frac{\left\langle
r,r\right\rangle\left[\eta,\,r\right]}{\left\langle
h_{i},r\right\rangle\left[\eta,h_{i}\right]}-\frac{\left\langle
h_{i},r\right\rangle\left[\eta,h_{i}\right]}{\left\langle
h_{i},r\right\rangle\left[\eta,h_{i}\right]}=-1,$ (3.17)

where momentum conservation and the antisymmetry of spinor products have been
applied. After this simplification, the amplitude (3.15) turns into

$\displaystyle
A^{(h_{i},g_{j})}(1,2,\ldots,r\|\mathsf{H})=(-1)\sum\limits_{\mathcal{F}\,^{\prime\prime}}\mathcal{C}^{\mathcal{F}\,^{\prime\prime}}\Biggl{[}\,\sum\limits_{\boldsymbol{\gamma}\in\mathcal{F}\,^{\prime\prime}|_{1}\setminus\\{1,r\\}}\,\frac{\left\langle
h_{i},g_{j}\right\rangle^{4}}{(1,\boldsymbol{\gamma},r)}\Biggr{]},$ (3.18)

in which, $\mathcal{F}\,^{\prime\prime}$ are those tree graphs in the MHV
sector with the negative-helicity graviton $h_{i}$ deleted. Up to an overall
factor, the above expression exactly has the same pattern with the
$(g^{-}_{i},g^{-}_{j})$ case (i.e. eq. (2.4) with the $(g^{-}_{i},g^{-}_{j})$
configuration) in which all gravitons have positive helicity. Thus, following
a parallel discussion with the $(g^{-}_{i},g^{-}_{j})$ case, we immediately
arrive

$\displaystyle
A^{(h_{i},g_{j})}(1,2,\ldots,r\|\mathsf{H})=(-1)\frac{\left\langle
h_{i},g_{j}\right\rangle^{4}}{\big{(}1,2,\dots,r\bigr{)}}\left[\,\sum\limits_{\mathcal{G}}\,\prod\limits_{e(x,y)\in\mathcal{E}(\mathcal{G})}\frac{\left\langle
y,h_{i}\right\rangle\left\langle
y,r\right\rangle\left[y,x\right]}{\left\langle
x,h_{i}\right\rangle\left\langle x,r\right\rangle\left\langle
y,x\right\rangle}\,\right],$ (3.19)

where we have summed over all spanning forests $\mathcal{G}$ with the node set
$\\{1,\dots,r\\}\cup\mathsf{H}^{+}$ and roots $1$, …, $r$. This expression is
identical with eq. (3.1) when choosing $\xi=h_{i}$ and $\eta=r$. Since eq.
(3.1) is independent of the choice of $\xi$ and $\eta$ Du:2016wkt , the proof
for the $(h^{-}_{i},g^{-}_{j})$ configuration has been completed.

###  3.3 Comments

Now we summarize some critical features of the above evaluations, which will
inspire a symmetric formula of double-trace amplitude with the
$(g_{i}^{-},g_{j}^{-})$ configuration in the coming section:

  * •

(i). There is a one-to-one correspondence between the refined graphs which
contain positive-helicity gravitons and gluons as nodes, and the spanning
forests in four dimensions.

  * •

(ii). Although, gravitons are involved in the color-ordered YM amplitudes in
the expansion (2.1), they do not appear in the Parke-Taylor factor (except the
numerator $\left\langle h_{i},g_{j}\right\rangle^{4}$ for the
$(h^{-}_{i},g^{-}_{j})$ configuration) in the spanning forest formula (3.1).
Thus the cyclic symmetry of gluons in eq. (3.1) is manifest.

  * •

(iii). The spanning forest formula is independent of the choice of gauge $\xi$
and $\eta$.

In the coming section, we generalize the spanning forest formula (3.1) to
double-trace MHV amplitudes with the $(g_{i}^{-},g_{j}^{-})$ configuration.

##  4 Double-trace MHV amplitudes with $(g^{-}_{i},g^{-}_{j})$ configuration

Figure 6:  A typical spanning forest for double-trace amplitude with the
$(g^{-}_{i},g^{-}_{j})$ configuration Figure 7:  A bridge between the two
traces in a spanning forest for a double-trace amplitude with the
$(g^{-}_{i},g^{-}_{j})$ configuration

In this section, we generalize the pattern of single-trace MHV amplitudes
(3.1) to double-trace ones with the $(g^{-}_{i},g^{-}_{j})$ configuration. We
prove that the double-trace MHV amplitude
$A^{(g_{i},g_{j})}(x_{1},\ldots,x_{r}|y_{1},\ldots,y_{s}\|\mathsf{H})$ (where
the two gluon traces are $\boldsymbol{1}=\\{x_{1},\dots,x_{r}\\}$,
$\boldsymbol{2}=\\{y_{1},\dots,y_{s}\\}$ and the graviton set is $\mathsf{H}$)
satisfies the following spanning forest formula

$\displaystyle\boxed{\text{\begin{tabular}[]{c}$A^{(g_{i},g_{j})}(x_{1},\ldots,x_{r}|y_{1},\ldots,y_{s}\|\mathsf{H})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$\\\
\\\ $=\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)\left(y_{1},\dots,y_{s}\right)}\Bigg{[}\,\sum\limits_{\mathcal{G}=\mathcal{G}_{1}\oplus\mathcal{G}_{2}}\,\sum\limits_{\begin{subarray}{c}c\in\mathcal{G}_{1}\\\
d\in\mathcal{G}_{2}\end{subarray}}\left(-k_{c}\cdot
k_{d}\right)\frac{\left\langle c,\zeta\right\rangle\left\langle
d,\chi\right\rangle}{\left\langle
c,d\right\rangle\left\langle\zeta,\chi\right\rangle}\prod\limits_{{e(x,y)\in\,\mathcal{E}(\mathcal{G})}}\frac{\left\langle\xi,y\right\rangle\left\langle\lambda_{e},y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\lambda_{e},x\right\rangle\left\langle
y,x\right\rangle}\Bigg{]}$\\\ \end{tabular}}}\,.$ (4.4)

In the above expression, all possible spanning forests $\mathcal{G}$ with the
pattern Fig. 6 are summed over. In each forest $\mathcal{G}$, the node set is
given by $\boldsymbol{1}\cup\boldsymbol{2}\cup\mathsf{H}$ and all gluons in
$\boldsymbol{1}\cup\boldsymbol{2}$ are considered as roots. Edges in a forest
$\mathcal{G}$ are presented by arrow lines pointing towards roots. Each forest
$\mathcal{G}$ is composed of two sub-forests $\mathcal{G}_{1}$ and
$\mathcal{G}_{2}$ whose roots live in $\boldsymbol{1}$ and $\boldsymbol{2}$
correspondingly. We further sum over all possible choices of
$c\in\mathcal{G}_{1}$ and $d\in\mathcal{G}_{2}$. For a given
$\mathcal{G}=\mathcal{G}_{1}\oplus\mathcal{G}_{2}$ and a given choice of
$c\in\mathcal{G}_{1}$ and $d\in\mathcal{G}_{2}$, a crossing factor

$\displaystyle\left(-k_{c}\cdot k_{d}\right)\frac{\left\langle
c,\zeta\right\rangle\left\langle d,\chi\right\rangle}{\left\langle
c,d\right\rangle\left\langle\zeta,\chi\right\rangle}$ (4.5)

is implied where $\zeta$ and $\chi$ are two reference spinors which
respectively reflect the cyclic symmetries of the traces $\boldsymbol{1}$ and
$\boldsymbol{2}$. For convenience, we express this crossing factor by
connecting $c$ and $d$ via a double-arrow line, as shown by Fig. 6. Any other
edge in the graph is either an _inner edge_ or an _outer edge_ , which depends
on whether it lives on the bridge (see Fig. 7) between traces $\boldsymbol{1}$
and $\boldsymbol{2}$ or not. Each edge $e(x,y)$ where the arrow points from
$x$ to $y$ is associated with a factor

$\displaystyle\frac{\left\langle\xi,y\right\rangle\left\langle\lambda_{e},y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\lambda_{e},x\right\rangle\left\langle
y,x\right\rangle},$ (4.6)

where $\xi$ is the common reference spinor of all $e(x,y)$ which reflects the
choice of gauge of the half-polarizations in the refined graphs. The
$\lambda_{e}$ for each edge $e(x,y)$ is a reference spinor reflecting the
cyclic symmetries of the gluon traces. In particular, it is defined by

$\displaystyle\lambda_{e}=\Bigg{\\{}\begin{array}[]{cc}\eta&\text{(if $e(x,y)$
is an outer edge )}\\\ \zeta&~{}~{}~{}~{}\text{(if $e(x,y)$ is an inner edge
and $e(x,y)\in\mathcal{G}_{1}$)}\\\ \chi&~{}~{}~{}~{}\text{(if $e(x,y)$ is an
inner edge and $e(x,y)\in\mathcal{G}_{2}$)}\\\ \end{array},$ (4.10)

where the reference spinors $\zeta$ and $\chi$ were already introduced before,
while $\eta$ is a new reference spinor. _It is worth pointing that the
summations in the expression ( 4.4) can be rearranged by (i). first summing
over all splits of the graviton set
$\mathsf{H}\to\mathsf{H}_{A},\mathsf{H}_{B}$; (ii). for a given split, summing
over all possible bridges $\mathcal{B}\left(\mathsf{H}_{A}\right)$, which
involve all the gravitons in $\mathsf{H}_{A}$ as nodes, between the two traces
$\boldsymbol{1}$ and $\boldsymbol{2}$, (iii). for a given split and a given
bridge, summing over all possible spanning forests $\mathcal{G}^{\prime}$, in
which all gravitons and gluons are considered as nodes and all elements in
$\boldsymbol{1}\cup\boldsymbol{2}\cup\mathsf{H}_{A}$ are considered as roots._
The formula (4.4) is then rewritten as an equivalent formula

$\displaystyle\boxed{\text{\begin{tabular}[]{c}$A^{(g^{-}_{i},g^{-}_{j})}(x_{1},\ldots,x_{r}|y_{1},\ldots,y_{s}\|\mathsf{H})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$\\\
\\\ $=\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)\left(y_{1},\dots,y_{s}\right)}\Bigg{[}\,\sum\limits_{\mathsf{H}\to\mathsf{H}_{A},\mathsf{H}_{B}}\,\sum\limits_{\mathcal{B}\,\left(\mathsf{H}_{A}\right)}\left(-k_{c}\cdot
k_{d}\right)\frac{\left\langle c,\zeta\right\rangle\left\langle
d,\chi\right\rangle}{\left\langle
c,d\right\rangle\left\langle\zeta,\chi\right\rangle}\,\prod\limits_{{e(x,y)\in\,\mathcal{E}_{1}(\mathcal{B}(\mathsf{H}_{A}))}}\frac{\left\langle\xi,y\right\rangle\left\langle\zeta,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\zeta,x\right\rangle\left\langle
y,x\right\rangle}$\\\
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\prod\limits_{{e(x,y)\in\,\mathcal{E}_{2}(\mathcal{B}(\mathsf{H}_{A}))}}\frac{\left\langle\xi,y\right\rangle\left\langle\chi,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\chi,x\right\rangle\left\langle
y,x\right\rangle}\,\sum\limits_{\mathcal{G}^{\prime}}\,\prod\limits_{{e(x,y)\in\,\mathcal{E}(\mathcal{G}^{\prime})}}\frac{\left\langle\xi,y\right\rangle\left\langle\eta,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\eta,x\right\rangle\left\langle
y,x\right\rangle}\Bigg{]}$\\\ \end{tabular}}}\,,$ (4.15) (4.16)

where $c$ and $d$ are the two gravitons connected by the double arrow line on
the bridge $\mathcal{B}\left(\mathsf{H}_{A}\right)$. The
$\mathcal{E}_{1}(\mathcal{B}(\mathsf{H}_{A}))$ and
$\mathcal{E}_{2}(\mathcal{B}(\mathsf{H}_{A}))$ are the set of single-arrowed
edges whose arrows points towards the traces $\boldsymbol{1}$ and
$\boldsymbol{2}$, respectively. Graphs $\mathcal{G}^{\prime}$ denote those
spanning forests rooted at elements in
$\boldsymbol{1}\cup\boldsymbol{2}\cup\mathsf{H}_{A}$. Thus all edges of
$\mathcal{G}^{\prime}$ are apparently those outer edges in eq. (4.4) and eq.
(4.10).

In the following, we first investigate the symmetries of eq. (4.4). After
that, we provide an example for eq. (4.4) and then the general proof by the
refined graphic rule.

###  4.1 Symmetries of the formula

The spanning forest formula (4.4) is much more symmetric than the expansion
(2.5) given by refined graphic rule.

First, permutations in the Parke-Taylor factors of (2.5) involve all external
gluons and gravitons. On the contrary, gravitons and gluon traces in (4.4) are
disentangled from one another: Each trace is expressed by a Parke-Taylor
factor with its own gluons in the original permutation, while gravitons and
the other trace do not participate in. Therefore, eq. (4.4) is explicitly
symmetric under the exchanging of the two gluon traces. Moreover, the
invariance of eq. (4.4) under exchanging any two gravitons seems more
transparent because gravitons are already extracted out from the Parke-Taylor
factors and only involved in the summation over spanning forests.

Second, there are “gauge symmetries” corresponding to the arbitrariness of the
reference spinors $\xi$, $\zeta$, $\chi$ and $\eta$. In the coming
subsections, we will see $\xi$ comes from the reference spinor of all ‘half’
polarizations, thus its arbitrariness is essentially the gauge symmetry of
amplitudes. Nevertheless, the arbitrariness of the choices of $\lambda_{e}$
(i.e., $\zeta$, $\chi$ and $\eta$) that encode the cyclic symmetries of the
two traces is not so clear. Now let us understand these symmetries of the
expression (4.4).

(i). _The invariance of ( 4.4) under the change of $\eta$_ This symmetry is
easily understood when the amplitude is expressed by eq. (4.16): For a given
split $\mathsf{H}\to\mathsf{H}_{A},\mathsf{H}_{B}$ and a given bridge between
the two traces, the summation over all spanning forests rooted at elements in
$\boldsymbol{1}\cup\boldsymbol{2}\cup\mathsf{H}_{A}$ exactly produces the
determinant of the Hodges matrix (B.16) Feng:2012sy ; Du:2016wkt

$\displaystyle\sum\limits_{\mathcal{G}^{\prime}}\prod\limits_{{e(x,y)\in\,\mathcal{E}(\mathcal{G}^{\prime})}}\frac{\left\langle\xi,y\right\rangle\left\langle\eta,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\eta,x\right\rangle\left\langle
y,x\right\rangle}=\det\left[\phi_{\mathsf{H}_{B}}\right],$ (4.17)

which is independent of the choice of $\eta$. Since each bridge is associated
by a Hodges determinant that is independent of $\eta$, $\eta$ can even be
chosen differently for distinct configurations of bridge. Nevertheless, in
this paper, we choose all $\eta$’s corresponding to different bridges as the
same one for convenience.

(ii). _The invariance of ( 4.4) under the change of $\zeta$_ Since $\eta$ can
be chosen arbitrarily, we just set $\eta=\chi$, which does not bring any
divergency to eq. (4.4). Now we proceed our discussion by classifying the
terms inside the square brackets of eq. (4.4) according to _whether
$c\in\mathcal{G}_{1}$ is a gluon or a graviton_ (i.e. $c\in\boldsymbol{1}$ or
$c\in\mathcal{G}_{1}\setminus\boldsymbol{1}$), for given $\mathcal{G}_{1}$ and
$\mathcal{G}_{2}$. In the latter case, there exist single-arrowed edges
pointing towards $\boldsymbol{1}$ on the bridge between $\boldsymbol{1}$ and
$\boldsymbol{2}$ (see Fig. 7), while in the former there is no such line on
the bridge. The contribution of terms with $c\in\boldsymbol{1}$ is given by

$\displaystyle T_{a}=\sum\limits_{\begin{subarray}{c}c\in\boldsymbol{1}\\\
d\in\mathcal{G}_{2}\end{subarray}}\left(-k_{c}\cdot
k_{d}\right)\frac{\left\langle c,\zeta\right\rangle\left\langle
d,\chi\right\rangle}{\left\langle
c,d\right\rangle\left\langle\zeta,\chi\right\rangle}\Biggl{[}\,\prod\limits_{{e(x,y)\in\,\mathcal{E}\left(\mathcal{G}=\mathcal{G}_{1}\oplus\mathcal{G}_{2}\right)}}\frac{\left\langle\xi,y\right\rangle\left\langle\chi,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\chi,x\right\rangle\left\langle
y,x\right\rangle}\Biggr{]},$ (4.18)

and the difference between $T^{\prime}_{a}$ and $T_{a}$ corresponding to
$\zeta^{\prime}$ and $\zeta$ is evaluated as

$\displaystyle\Delta
T_{a}=\sum\limits_{\begin{subarray}{c}c\in\boldsymbol{1}\\\
d\in\mathcal{G}_{2}\end{subarray}}\frac{1}{2}\left[c,d\right]\left\langle
d,\chi\right\rangle\left\langle\chi,c\right\rangle\frac{\left\langle\zeta^{\prime},\zeta\right\rangle}{\left\langle\zeta^{\prime},\chi\right\rangle\left\langle\zeta,\chi\right\rangle}\Biggl{[}\,\prod\limits_{{e(x,y)\in\,\mathcal{E}\left(\mathcal{G}=\mathcal{G}_{1}\oplus\mathcal{G}_{2}\right)}}\frac{\left\langle\xi,y\right\rangle\left\langle\chi,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\chi,x\right\rangle\left\langle
y,x\right\rangle}\Biggr{]},$ (4.19)

where Schouten identity (B.5) has been applied. On another hand, the
expression inside the square brackets of eq. (4.4) when
$c\in\mathcal{G}_{1}\setminus\boldsymbol{1}$ reads

$\displaystyle T_{b}$ $\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}c\in\mathcal{G}_{1}\setminus\boldsymbol{1}\\\
d\in\mathcal{G}_{2}\end{subarray}}\left(-k_{c}\cdot
k_{d}\right)\frac{\left\langle c,\zeta\right\rangle\left\langle
d,\chi\right\rangle}{\left\langle
c,d\right\rangle\left\langle\zeta,\chi\right\rangle}$ (4.20)
$\displaystyle\times$
$\displaystyle\prod\limits_{e(x,y){\in}\,\mathcal{E}_{1}(c)}\frac{\left\langle\xi,y\right\rangle\left\langle\zeta,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\zeta,x\right\rangle\left\langle
y,x\right\rangle}\prod\limits_{\begin{subarray}{c}e(x,y)\in\,\mathcal{E}(\mathcal{G}_{1})\\\
e(x,y)\cancel{\in}\,\mathcal{E}_{1}(c)\end{subarray}}\frac{\left\langle\xi,y\right\rangle\left\langle\chi,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\chi,x\right\rangle\left\langle
y,x\right\rangle}\prod\limits_{{e(x,y)\in\,\mathcal{E}(\mathcal{G}_{2})}}\frac{\left\langle\xi,y\right\rangle\left\langle\chi,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\chi,x\right\rangle\left\langle
y,x\right\rangle},$

where $\mathcal{E}_{1}(c)$ denotes the set of _single-arrowed inner edges_
that are pointing towards the trace $\boldsymbol{1}$ for a given $c$.
Supposing the bridge between the two traces $\boldsymbol{1}$ and
$\boldsymbol{2}$ is given by Fig. 7, the first product in eq. (4.20) for a
given $c\in\mathcal{G}_{1}\setminus\boldsymbol{1}$ and $d\in\mathcal{G}_{2}$
reads

$\displaystyle\prod\limits_{e(x,y){\in}\,\mathcal{E}_{1}(c)}\frac{\left\langle\xi,y\right\rangle\left\langle\zeta,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\zeta,x\right\rangle\left\langle
y,x\right\rangle}$ (4.21) $\displaystyle=$
$\displaystyle\frac{\left\langle\xi,u_{1}\right\rangle\cancel{\left\langle\zeta,u_{1}\right\rangle}\left[u_{1},c\right]}{\left\langle\xi,c\right\rangle\left\langle\zeta,c\right\rangle\left\langle
u_{1},c\right\rangle}\,\frac{\left\langle\xi,u_{2}\right\rangle\cancel{\left\langle\zeta,u_{2}\right\rangle}\left[u_{2},u_{1}\right]}{\left\langle\xi,u_{1}\right\rangle\cancel{\left\langle\zeta,u_{1}\right\rangle}\left\langle
u_{2},u_{1}\right\rangle}\,\dots\,\frac{\left\langle\xi,l_{1}\right\rangle\left\langle\zeta,l_{1}\right\rangle\left[l_{1},u_{i}\right]}{\left\langle\xi,u_{i}\right\rangle\cancel{\left\langle\zeta,u_{i}\right\rangle}\left\langle
l_{1},u_{i}\right\rangle}$ $\displaystyle=$
$\displaystyle\Biggl{[}\frac{\left\langle\xi,u_{1}\right\rangle\left\langle\chi,u_{1}\right\rangle\left[u_{1},c\right]}{\left\langle\xi,c\right\rangle\left\langle\chi,c\right\rangle\left\langle
u_{1},c\right\rangle}\,\frac{\left\langle\xi,u_{2}\right\rangle\left\langle\chi,u_{2}\right\rangle\left[u_{2},u_{1}\right]}{\left\langle\xi,u_{1}\right\rangle\left\langle\chi,u_{1}\right\rangle\left\langle
u_{2},u_{1}\right\rangle}\,\dots\,\frac{\left\langle\xi,l_{1}\right\rangle\left\langle\chi,l_{1}\right\rangle\left[l_{1},u_{i}\right]}{\left\langle\xi,u_{i}\right\rangle\left\langle\chi,u_{i}\right\rangle\left\langle
l_{1},u_{i}\right\rangle}\Biggr{]}\,\frac{\left\langle\chi,c\right\rangle}{\left\langle\chi,l_{1}\right\rangle}\frac{\left\langle\zeta,l_{1}\right\rangle}{\left\langle\zeta,c\right\rangle}$
$\displaystyle=$
$\displaystyle\Biggl{[}\,\prod\limits_{e(x,y){\in}\,\mathcal{E}_{1}(c)}\frac{\left\langle\xi,y\right\rangle\left\langle\chi,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\chi,x\right\rangle\left\langle
y,x\right\rangle}\Biggr{]}\,\frac{\left\langle\chi,c\right\rangle}{\left\langle\chi,l_{1}\right\rangle}\frac{\left\langle\zeta,l_{1}\right\rangle}{\left\langle\zeta,c\right\rangle}.$

The expression (4.20) is therefore arranged as

$\displaystyle T_{b}$ $\displaystyle=$
$\displaystyle\Biggl{[}\sum\limits_{\begin{subarray}{c}c\in\mathcal{G}_{1}\setminus{\boldsymbol{1}}\\\
d\in\mathcal{G}_{2}\end{subarray}}\frac{1}{2}\left[c,d\right]\left\langle\chi,d\right\rangle\frac{\left\langle\chi,c\right\rangle}{\left\langle\chi,l_{1}\right\rangle}\frac{\left\langle\zeta,l_{1}\right\rangle}{\left\langle\zeta,\chi\right\rangle}\Biggr{]}\Biggl{[}\,\prod\limits_{{e(x,y)\in\,\mathcal{E}\left(\mathcal{G}=\mathcal{G}_{1}\oplus\mathcal{G}_{2}\right)}}\frac{\left\langle\xi,y\right\rangle\left\langle\chi,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\chi,x\right\rangle\left\langle
y,x\right\rangle}\Biggr{]}.$ (4.22)

Figure 8: Typical refined graphs for the double-trace amplitude
$A_{m=2}^{\text{MHV}}(x_{1},\ldots,x_{r}|y_{1},\ldots,y_{s}\|h_{1})$

Apparently, only the first factor involves $\zeta$. When $\zeta$ is replaced
by $\zeta^{\prime}$, the first factor differs from the original one (the one
with $\zeta$) by

$\displaystyle\sum\limits_{\begin{subarray}{c}c\in\mathcal{G}_{1}\setminus{\boldsymbol{1}}\\\
d\in\mathcal{G}_{2}\end{subarray}}\frac{1}{2}\left[c,d\right]\left\langle\chi,d\right\rangle\frac{\left\langle\chi,c\right\rangle}{\left\langle\chi,l_{1}\right\rangle}\left(\frac{\left\langle\zeta^{\prime},l_{1}\right\rangle}{\left\langle\zeta^{\prime},\chi\right\rangle}-\frac{\left\langle\zeta,l_{1}\right\rangle}{\left\langle\zeta,\chi\right\rangle}\right)$
$\displaystyle=$
$\displaystyle\Biggl{[}\sum\limits_{\begin{subarray}{c}c\in\mathcal{G}_{1}\setminus\boldsymbol{1}\\\
d\in\mathcal{G}_{2}\end{subarray}}\frac{1}{2}\left[c,d\right]\left\langle\chi,d\right\rangle\left\langle\chi,c\right\rangle\Biggr{]}\frac{\left\langle\zeta^{\prime},\zeta\right\rangle}{\left\langle\chi,\zeta^{\prime}\right\rangle\left\langle\chi,\zeta\right\rangle}$
(4.23)

where we have applied Schouten identity (B.5). Thus

$\displaystyle\Delta
T_{b}=\sum\limits_{\begin{subarray}{c}c\in\mathcal{G}_{1}\setminus\boldsymbol{1}\\\
d\in\mathcal{G}_{2}\end{subarray}}\frac{1}{2}\left[c,d\right]\left\langle
d,\chi\right\rangle\left\langle\chi,c\right\rangle\frac{\left\langle\zeta^{\prime},\zeta\right\rangle}{\left\langle\chi,\zeta^{\prime}\right\rangle\left\langle\chi,\zeta\right\rangle}\Biggl{[}\,\prod\limits_{{e(x,y)\in\,\mathcal{E}\left(\mathcal{G}=\mathcal{G}_{1}\oplus\mathcal{G}_{2}\right)}}\frac{\left\langle\xi,y\right\rangle\left\langle\chi,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\chi,x\right\rangle\left\langle
y,x\right\rangle}\Biggr{]}.$ (4.24)

The sum of $\Delta T_{a}$ and $\Delta T_{b}$ is then given by

$\displaystyle\Delta T_{a}+\Delta T_{b}$ $\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}c\in\mathcal{G}_{1}\\\
d\in\mathcal{G}_{2}\end{subarray}}\frac{1}{2}\left[c,d\right]\left\langle
d,\chi\right\rangle\left\langle\chi,c\right\rangle\frac{\left\langle\zeta^{\prime},\zeta\right\rangle}{\left\langle\chi,\zeta^{\prime}\right\rangle\left\langle\chi,\zeta\right\rangle}\Biggl{[}\,\prod\limits_{{e(x,y)\in\,\mathcal{E}\left(\mathcal{G}=\mathcal{G}_{1}\oplus\mathcal{G}_{2}\right)}}\frac{\left\langle\xi,y\right\rangle\left\langle\chi,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\chi,x\right\rangle\left\langle
y,x\right\rangle}\Biggr{]}$ (4.25) $\displaystyle=$
$\displaystyle-\sum\limits_{c,d\in\mathcal{G}_{1}}\frac{1}{2}\left[c,d\right]\left\langle
d,\chi\right\rangle\left\langle\chi,c\right\rangle\frac{\left\langle\zeta^{\prime},\zeta\right\rangle}{\left\langle\chi,\zeta^{\prime}\right\rangle\left\langle\chi,\zeta\right\rangle}\Biggl{[}\,\prod\limits_{{e(x,y)\in\,\mathcal{E}\left(\mathcal{G}=\mathcal{G}_{1}\oplus\mathcal{G}_{2}\right)}}\frac{\left\langle\xi,y\right\rangle\left\langle\chi,y\right\rangle\left[y,x\right]}{\left\langle\xi,x\right\rangle\left\langle\chi,x\right\rangle\left\langle
y,x\right\rangle}\Biggr{]},$

where momentum conservation has been applied. For given
$c,d\in\mathcal{G}_{1}$, the last line of the above equation is antisymmetric
about $c$ and $d$, thus has to vanish when all nodes $c,d\in\mathcal{G}_{1}$
are summed over. We then conclude that eq. (4.16) (thus eq. (4.4)) is
invariant under the change of $\zeta$.

(iii). _The invariance of ( 4.4) under the change of $\chi$_ When we choose
$\eta=\zeta$, this symmetry naturally follows from a discussion parallel with
(ii) because (4.4) has a symmetric form under the exchange of the roles of
$\boldsymbol{1}$ and $\boldsymbol{2}$.

###  4.2 An example study

Before showing the general proof of the formula (4.4), we explicitly evaluate
the double-trace amplitude
$A^{(g_{i}^{-},g_{j}^{-})}(x_{1},\ldots,x_{r}|y_{1},\ldots,y_{s}\|h_{1})$
involving only one graviton. According to the refined graphic rule given in
section 2.2, this amplitude can be written as eq. (2.5), in which all typical
refined graphs $\mathcal{F}\in\mathcal{F}[0,2]$ are presented by Fig. 8.

We first evaluate the graphs with the structures Fig. 8 (a) and (b), where the
graviton $h_{1}$ plays as an outer node. The contribution of all graphs with
the structure Fig. 8 (a) is given by

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\gamma}}\,\sum\limits_{\boldsymbol{\sigma}\,\in\,\boldsymbol{\gamma}\,\shuffle\,\\{h_{1}\\}}(-k_{a}\cdot
k_{l_{1}})\left(\epsilon_{h_{1}}\cdot
k_{l_{2}}\right)\,A\left(x_{1},\boldsymbol{\sigma},x_{r}\right)$ (4.26)
$\displaystyle=$ $\displaystyle\sum\limits_{\boldsymbol{\gamma}}(-k_{a}\cdot
k_{l_{1}})\,\sum\limits_{\boldsymbol{\sigma}\,\in\,\boldsymbol{\gamma}\,\shuffle\,\\{h_{1}\\}}\frac{\left\langle\xi,l_{2}\right\rangle\left[l_{2},h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle}\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\boldsymbol{\sigma},x_{r}\right)}$
$\displaystyle=$ $\displaystyle\sum\limits_{\boldsymbol{\gamma}}(-k_{a}\cdot
k_{l_{1}})\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\boldsymbol{\gamma},x_{r}\right)}\,\frac{\left\langle\xi,l_{2}\right\rangle\left\langle
x_{r},l_{2}\right\rangle\left[l_{2},h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
x_{r},h_{1}\right\rangle\left\langle l_{2},h_{1}\right\rangle}.$

where $l_{2}\in\boldsymbol{1}\setminus\\{x_{r}\\}=\\{x_{1},\dots,x_{r-1}\\}$
and the permutations $\boldsymbol{\gamma}$ obeys

$\displaystyle\boldsymbol{\gamma}\in\left\\{x_{2},\dots,l_{1}=x_{a},\\{x_{a+1},\dots,x_{r-1}\\}\shuffle\\{a,\beta_{1},\dots,\beta_{|\boldsymbol{2}|-2},b\\}\right\\}.$
(4.27)

On the second line of eq. (4.26), we have written $\epsilon_{h_{1}}(\xi)\cdot
k_{l_{2}}$ by spinor products and the MHV Yang-Mills amplitudes
$A\left(x_{1},\boldsymbol{\sigma},x_{r}\right)$ by Parke-Taylor formula. On
the third line of eq. (4.26), the identity (B.8) was applied. Following a
similar discussion, the contribution of Fig. 8 (b) reads

$\displaystyle T_{2}$ $\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\gamma}}(-k_{a}\cdot
k_{l_{1}})\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\boldsymbol{\gamma},x_{r}\right)}\,\frac{\left\langle\xi,l_{2}\right\rangle\left\langle
x_{r},l_{2}\right\rangle\left[l_{2},h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
x_{r},h_{1}\right\rangle\left\langle
l_{2},h_{1}\right\rangle}.~{}~{}(l_{2}\in\boldsymbol{2}).$ (4.28)

When summed over all possible $l_{2}$ in $T_{1}$ and $T_{2}$, we arrive the
total contribution of graphs with structures Fig. 8 (a) and (b):

$\displaystyle
I_{1}=\sum\limits_{l_{2}\in\boldsymbol{1}\setminus\\{x_{r}\\}}T_{1}+\sum\limits_{l_{2}\in\boldsymbol{2}}T_{2}=\left[\sum\limits_{\boldsymbol{\gamma}}(-k_{a}\cdot
k_{l_{1}})\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\boldsymbol{\gamma},x_{r}\right)}\right]\,\left[\sum\limits_{l_{2}\in\boldsymbol{1}\cup\boldsymbol{2}}\frac{\left\langle\xi,l_{2}\right\rangle\left\langle
x_{r},l_{2}\right\rangle\left[l_{2},h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
x_{r},h_{1}\right\rangle\left\langle l_{2},h_{1}\right\rangle}\right],$ (4.29)

where the case $l_{2}=x_{r}$ is already involved in the second equality since
$\left\langle x_{r},x_{r}\right\rangle=0$. The first factor in the above
expression is explicitly written as

$\displaystyle\sum\limits_{\boldsymbol{\gamma}}(-k_{a}\cdot
k_{l_{1}})\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,l_{1}=x_{a},\boldsymbol{\gamma}\in\\{x_{a+1},\dots,x_{r-1}\\}\shuffle\\{a,\boldsymbol{\beta},b\\},x_{r}\right)}$
(4.30) $\displaystyle=$ $\displaystyle(-k_{a}\cdot
k_{l_{1}})\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)}\,\frac{\left\langle
l_{1},x_{r}\right\rangle}{\left\langle
l_{1},a,\boldsymbol{\beta},b,x_{r}\,\right\rangle}$ $\displaystyle=$
$\displaystyle\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)}\,\frac{1}{\left(a,\boldsymbol{\beta},b\right)}(-k_{a}\cdot
k_{l_{1}})\frac{\left\langle b,a\right\rangle\left\langle
l_{1},x_{r}\right\rangle}{\left\langle l_{1},a\right\rangle\left\langle
b,x_{r}\right\rangle},$

in which, the identity (B.9) has been applied on the second line. When all
possible $\boldsymbol{\beta}$ (with the sign
$(-1)^{|\,\boldsymbol{2},a,b\,|}$) are summed over, the factor
$\frac{1}{\left(a,\boldsymbol{\beta},b\right)}$ turns into
$\frac{1}{\left(y_{1},\dots,y_{s}\right)}$, according to the Kleiss-Kuijf (KK)
relation Kleiss:1988ne  (B.13) between Parke-Taylor factors. Therefore all
contributions of graphs with the structures Fig. 8 (a) and (b) are collected
as

$\displaystyle I_{1}=\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)\left(y_{1},\dots,y_{s}\right)}\left[\,\sum\limits_{l_{1}\in\boldsymbol{1},a\in\boldsymbol{2}}(-k_{a}\cdot
k_{l_{1}})\frac{\left\langle a,b\right\rangle\left\langle
l_{1},x_{r}\right\rangle}{\left\langle l_{1},a\right\rangle\left\langle
x_{r},b\right\rangle}\right]\,\left[\sum\limits_{l_{2}\in\boldsymbol{1}\cup\boldsymbol{2}}\frac{\left\langle\xi,l_{2}\right\rangle\left\langle
x_{r},l_{2}\right\rangle\left[l_{2},h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
x_{r},h_{1}\right\rangle\left\langle l_{2},h_{1}\right\rangle}\right].$ (4.31)

Each graph with the structure Fig. 8 (c) contributes

$\displaystyle T_{3}$ $\displaystyle=$ $\displaystyle(-k_{a}\cdot
k_{h_{1}})(\epsilon_{h_{1}}\cdot
k_{l_{1}})\sum\limits_{\boldsymbol{\sigma}}\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},x_{2},\dots,l_{1}=x_{a},\boldsymbol{\sigma}\in\\{x_{a+1},\dots,x_{r-1}\\}\shuffle\\{h_{1},a,\boldsymbol{\beta},b\\},x_{r}\right)}.$
(4.32)

Noting that permutation $\boldsymbol{\sigma}$ in each Parke-Taylor factor can
be rewritten as follows

$\displaystyle\boldsymbol{\sigma}\in\\{x_{a+1},\dots,x_{r-1}\\}\shuffle\\{h_{1}\\}\shuffle\\{a,\boldsymbol{\beta},b\\}\big{|}_{h_{1}\prec
a},$ (4.33)

we are able to apply the identity (B.9) to extract the trace $\boldsymbol{2}$
and the graviton $h_{1}$ from the Parke-Taylor factor in turn. The $T_{3}$ is
then expressed as

$\displaystyle T_{3}$ $\displaystyle=$ $\displaystyle(-k_{a}\cdot
k_{h_{1}})(\epsilon_{h_{1}}\cdot
k_{l_{1}})\sum\limits_{\boldsymbol{\gamma}}\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,l_{1}=x_{a},\boldsymbol{\gamma}\in\\{x_{a+1},\dots,x_{r-1}\\}\shuffle\\{h_{1}\\},x_{r}\right)}\frac{1}{\left(a,\boldsymbol{\beta},b\right)}\frac{\left\langle
h_{1},x_{r}\right\rangle\left\langle b,a\right\rangle}{\left\langle
h_{1},a\right\rangle\left\langle b,x_{r}\right\rangle}$ (4.34)
$\displaystyle=$ $\displaystyle\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)}\frac{1}{\left(a,\boldsymbol{\beta},b\right)}\left[(-k_{a}\cdot
k_{h_{1}})\frac{\left\langle h_{1},x_{r}\right\rangle\left\langle
b,a\right\rangle}{\left\langle h_{1},a\right\rangle\left\langle
b,x_{r}\right\rangle}\right]\left[\frac{\left\langle\xi,l_{1}\right\rangle\left\langle
x_{r},l_{1}\right\rangle\left[l_{1},h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
x_{r},h_{1}\right\rangle\left\langle l_{1},h_{1}\right\rangle}\right].$

When all possible $\boldsymbol{\beta}$ (with the sign
$(-1)^{|\,\boldsymbol{2},a,b\,|}$), $a\in\boldsymbol{2}\setminus\\{b\\}$,
$l_{1}\in\boldsymbol{1}\setminus\\{x_{r}\\}$ are summed over and the KK
relation Kleiss:1988ne  is applied to the trace $\boldsymbol{2}$, the
contribution of all graphs with the structure Fig. 8 (c) becomes

$\displaystyle I_{2}=\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)\left(y_{1},\dots,y_{s}\right)}\left[\,\sum\limits_{a\in\boldsymbol{2}}(-k_{a}\cdot
k_{h_{1}})\frac{\left\langle h_{1},x_{r}\right\rangle\left\langle
b,a\right\rangle}{\left\langle h_{1},a\right\rangle\left\langle
b,x_{r}\right\rangle}\,\right]\left[\,\sum\limits_{l_{1}\in\boldsymbol{1}}\frac{\left\langle\xi,l_{1}\right\rangle\left\langle
x_{r},l_{1}\right\rangle\left[l_{1},h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
x_{r},h_{1}\right\rangle\left\langle l_{1},h_{1}\right\rangle}\,\right].$
(4.35)

Here, the terms with $a=b$ and $l_{1}=x_{r}$ were also included in the
corresponding summations since $\left\langle b,b\right\rangle=\left\langle
x_{r},x_{r}\right\rangle=0$.

Each graph with the structure Fig. 8 (d) provides a contribution

$\displaystyle T_{4}$ $\displaystyle=$ $\displaystyle(-k_{h_{1}}\cdot
k_{l_{1}})(-\epsilon_{h_{1}}\cdot
k_{a})\sum\limits_{\boldsymbol{\gamma}}\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},x_{2},\dots,l_{1}=x_{a},\boldsymbol{\gamma}\in\\{x_{a+1},\dots,x_{r-1}\\}\shuffle\\{h_{1},a,\boldsymbol{\beta},b\\},x_{r}\right)}.$
(4.36)

When the identity (B.9) is applied, $T_{4}$ turns into

$\displaystyle T_{4}$ $\displaystyle=$ $\displaystyle(-k_{h_{1}}\cdot
k_{l_{1}})(-\epsilon_{h_{1}}\cdot k_{a})\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)}\frac{\left\langle
l_{1},x_{r}\right\rangle}{\left\langle
l_{1},h_{1},a,\boldsymbol{\beta},b,x_{r}\right\rangle}$ (4.37)
$\displaystyle=$ $\displaystyle(-\epsilon_{h_{1}}\cdot
k_{a})\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)}\frac{1}{\left(h_{1},a,\boldsymbol{\beta},b\right)}\left[(-k_{h_{1}}\cdot
k_{l_{1}})\frac{\left\langle l_{1},x_{r}\right\rangle\left\langle
b,h_{1}\right\rangle}{\left\langle l_{1},h_{1}\right\rangle\left\langle
b,x_{r}\right\rangle}\right],$

where $\left\langle a_{1},a_{2},\dots,a_{i}\right\rangle\equiv\left\langle
a_{1},a_{2}\right\rangle\left\langle a_{2},a_{3}\right\rangle\dots\left\langle
a_{i-1},a_{i}\right\rangle$. We further apply the identity (B.8) to extract
$h_{1}$ from the trace $\boldsymbol{2}$ (i.e. the Parke-Taylor factor
${1}/{\left(h_{1},a,\boldsymbol{\beta},b\right)}$) and write
$\epsilon_{h_{1}}\cdot k_{a}$ by spinor products, then obtain

$\displaystyle T_{4}=\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)}\frac{1}{\left(a,\boldsymbol{\beta},b\right)}\left[(-k_{h_{1}}\cdot
k_{l_{1}})\frac{\left\langle l_{1},x_{r}\right\rangle\left\langle
b,h_{1}\right\rangle}{\left\langle l_{1},h_{1}\right\rangle\left\langle
b,x_{r}\right\rangle}\right]\left[\frac{\left\langle\xi,a\right\rangle\left\langle
b,a\right\rangle\left[a,h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
b,h_{1}\right\rangle\left\langle a,h_{1}\right\rangle}\right].$ (4.38)

Summing over all $\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a,b]$ (with
the sign $(-1)^{|\,\boldsymbol{2},a,b\,|}$),
$a\in\boldsymbol{2}\setminus\\{b\\}$,
$l_{1}\in\boldsymbol{1}\setminus\\{x_{r}\\}$ and applying the KK relation on
the Parke-Taylor factors of trace $\boldsymbol{2}$, we get

$\displaystyle I_{3}=\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)\left(y_{1},\dots,y_{s}\right)}\left[\sum\limits_{l_{1}\in\boldsymbol{1}}(-k_{h_{1}}\cdot
k_{l_{1}})\frac{\left\langle l_{1},x_{r}\right\rangle\left\langle
b,h_{1}\right\rangle}{\left\langle l_{1},h_{1}\right\rangle\left\langle
b,x_{r}\right\rangle}\right]\left[\sum\limits_{a\in\boldsymbol{2}}\frac{\left\langle\xi,a\right\rangle\left\langle
b,a\right\rangle\left[a,h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
b,h_{1}\right\rangle\left\langle a,h_{1}\right\rangle}\right],$ (4.39)

where $a=b$ and $l_{1}=x_{r}$ are included again due to the antisymmetry of
the spinor products.

Figure 9: Spanning forests for the double-trace amplitude
$A^{(g_{i},g_{j})}(x_{1},\ldots,x_{r}|y_{1},\ldots,y_{s}\|h_{1})$.

The contribution of all refined graphs with the structures Fig. 8 are obtained
by summing $I_{1}$, $I_{2}$ and $I_{3}$ together:

$\displaystyle
A^{(g_{i},g_{j})}(x_{1},\ldots,x_{r}|y_{1},\ldots,y_{s}\|h_{1})$ (4.40)
$\displaystyle=$ $\displaystyle\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)\left(y_{1},\dots,y_{s}\right)}$
$\displaystyle~{}~{}~{}\times\Biggl{[}\,\sum\limits_{l_{1}\in\boldsymbol{1},a\in\boldsymbol{2}}(-k_{a}\cdot
k_{l_{1}})\frac{\left\langle l_{1},x_{r}\right\rangle\left\langle
a,b\right\rangle}{\left\langle l_{1},a\right\rangle\left\langle
x_{r},b\right\rangle}\,\sum\limits_{l_{2}\in\boldsymbol{1}\cup\boldsymbol{2}}\frac{\left\langle\xi,l_{2}\right\rangle\left\langle
l_{2},x_{r}\right\rangle\left[l_{2},h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
h_{1},x_{r}\right\rangle\left\langle l_{2},h_{1}\right\rangle}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\,\,+\sum\limits_{a\in\boldsymbol{2}}(-k_{a}\cdot
k_{h_{1}})\frac{\left\langle h_{1},x_{r}\right\rangle\left\langle
a,b\right\rangle}{\left\langle h_{1},a\right\rangle\left\langle
x_{r},b\right\rangle}\,\,\,\,\,\sum\limits_{l_{1}\in\boldsymbol{1}}\frac{\left\langle\xi,l_{1}\right\rangle\left\langle
l_{1},x_{r}\right\rangle\left[l_{1},h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
h_{1},x_{r}\right\rangle\left\langle l_{1},h_{1}\right\rangle}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\,\,+\sum\limits_{l_{1}\in\boldsymbol{1}}(-k_{h_{1}}\cdot
k_{l_{1}})\frac{\left\langle l_{1},x_{r}\right\rangle\left\langle
h_{1},b\right\rangle}{\left\langle l_{1},h_{1}\right\rangle\left\langle
x_{r},b\right\rangle}\,\,\sum\limits_{a\in\boldsymbol{2}}\frac{\left\langle\xi,a\right\rangle\left\langle
a,b\right\rangle\left[a,h_{1}\right]}{\left\langle\xi,h_{1}\right\rangle\left\langle
h_{1},b\right\rangle\left\langle a,h_{1}\right\rangle}\,\Biggr{]}.$

which precisely reproduces the formula (4.4) with the choice of gauge
$\zeta=x_{r}$ and $\chi=b$. In this example, all possible graphs expressed by
the spanning forests in eq. (4.4) are given by Fig. 9, where both Fig. 9 (a)
and (b) contribute to the first term, Fig. 9 (c) and (d) correspond to the
second and the third term. _The cyclic symmetry of the traces demand that the
above expression must be preserved when we consider any other gluons
$x_{u}\in\boldsymbol{1}$ and $y_{v}\in\boldsymbol{2}$ as the last ones instead
of $x_{r}$ and $b$ respectively. This has already been guaranteed by the
symmetry under choosing different $\zeta$ and $\chi$ in (4.4)._

###  4.3 The general proof

Inspired by the above example and the study of single-trace amplitudes, we now
prove the general formula (4.4) by three steps: step-1. extracting all outer
gravitons from the Parke-Taylor factors in eq. (2.5), step-2. separating the
two traces, which may be attached by inner gravitons, from one another,
step-3. extracting the inner gravitons from the corresponding Parke-Taylor
factor. The explicit proof is following.

Step-1: For a given refined graph $\mathcal{F}$ in eq. (2.5), we pick a leaf
(i.e. an outermost graviton) $h_{a}$. Permutations established by
$\mathcal{F}$ can be written as
$\\{x_{1},\left(\mathcal{F}|_{x_{1}}\setminus\\{x_{1},h_{a},x_{r}\\}\right)\shuffle\\{h_{a}\\}|_{l\prec
h_{a}},x_{r}\\}$, where $l$ is supposed to be the node adjacent to $h_{a}$.
The coefficient corresponding to the graph $\mathcal{F}$ then reads

$\displaystyle\mathcal{C}^{\,\mathcal{F}}=\mathcal{C}^{\,\mathcal{F}\setminus\\{h_{a}\\}}\,\left(\epsilon_{h_{a}}\cdot
k_{l}\right).$ (4.41)

Figure 10: An example for step-1 in the general proof of the formula (4.4).
Here, the permutation $\boldsymbol{\beta}$ satisfies
$\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a,b]$. The graviton sets
$\\{h_{1},h_{2},h_{3}\\}$, $\\{h_{4},h_{8}\\}$ and
$\\{h_{6},h_{9},h_{10},h_{11}\\}$ are the sets of outer gravitons which belong
to those trees planted at gluons in the trace $\boldsymbol{2}$ (i.e. $l_{3}$
and $l_{4}$), gluons in the trace $\boldsymbol{1}$ (i.e. $l_{1}$ and $l_{2}$)
and the inner gravitons (i.e. $h_{5}$ and $h_{7}$), respectively. After
manipulations in this step, only the two gluon traces as well as the inner
gravitons $h_{5}$, $h_{7}$ remain in the refined graph. All other gravitons
(and the tree structures associated to them) have been extracted out and
described by the spanning forest $\mathcal{G}^{\prime}$.

The YM amplitudes in eq. (2.5) corresponding to the graph $\mathcal{F}$ are
collected as

$\displaystyle\sum\limits_{\boldsymbol{\sigma}\in\mathcal{F}|_{x_{1}}\setminus\\{x_{1},x_{r}\\}}\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\boldsymbol{\sigma},x_{r}\right)}$
$\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\gamma}\in\mathcal{F}|_{x_{1}}\setminus\\{x_{1},h_{a},x_{r}\\}}\,\sum\limits_{\boldsymbol{\sigma}\in(\boldsymbol{\gamma}\shuffle\\{h_{a}\\})|_{l\prec
h_{a}}}\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\boldsymbol{\sigma},x_{r}\right)}$
(4.42) $\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\gamma}\in\mathcal{F}|_{x_{1}}\setminus\\{x_{1},h_{a},x_{r}\\}}\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\boldsymbol{\gamma},x_{r}\right)}\,\frac{\left\langle
l,x_{r}\right\rangle}{\left\langle l,h_{a}\right\rangle\left\langle
h_{a},x_{r}\right\rangle},$

where the identity (B.8) has been applied. Hence the total contribution of the
graph $\mathcal{F}$ is recursively expressed as

$\displaystyle T^{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\Biggl{[}\mathcal{C}^{\,\mathcal{F}\setminus\\{h_{a}\\}}\sum\limits_{\boldsymbol{\gamma}\in\mathcal{F}|_{x_{1}}\setminus\\{x_{1},h_{a},x_{r}\\}}\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\boldsymbol{\gamma},x_{r}\right)}\,\Biggr{]}\Biggl{[}(\epsilon_{h_{a}}\cdot
k_{l})\,\frac{\left\langle l,x_{r}\right\rangle}{\left\langle
l,h_{a}\right\rangle\left\langle h_{a},x_{r}\right\rangle}\Biggr{]}$ (4.43)
$\displaystyle=$
$\displaystyle\Bigg{[}\,\mathcal{C}^{\,\mathcal{F}\setminus\\{h_{a}\\}}\sum\limits_{\boldsymbol{\gamma}\in\mathcal{F}|_{x_{1}}\setminus\\{x_{1},h_{a},x_{r}\\}}\,\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\boldsymbol{\gamma},x_{r}\right)}\Bigg{]}\,\frac{\left\langle\xi,l\right\rangle\left\langle
x_{r},l\right\rangle\left[l,h_{a}\right]}{\left\langle\xi,h_{a}\right\rangle\left\langle
x_{r},h_{a}\right\rangle\left\langle l,h_{a}\right\rangle}$ $\displaystyle=$
$\displaystyle
T^{\mathcal{F}\setminus\\{h_{a}\\}}\frac{\left\langle\xi,l\right\rangle\left\langle
x_{r},l\right\rangle\left[l,h_{a}\right]}{\left\langle\xi,h_{a}\right\rangle\left\langle
x_{r},h_{a}\right\rangle\left\langle l,h_{a}\right\rangle}.$

On the second line, the explicit expression of $\epsilon_{h_{a}}\cdot k_{l}$
has been substituted and the expression in the square brackets is nothing but
the contribution of the graph when the node $h_{a}$ is deleted. Applying the
above relation iteratively until all elements in the outer-graviton set
$\mathsf{H}_{B}$ have been removed from the graph $\mathcal{F}$, we arrive

$\displaystyle
T^{\mathcal{F}}=T^{\mathcal{F}\setminus\mathsf{H}_{B}}\prod\limits_{{e(h_{a},l)\in\,\mathcal{E}(\mathcal{G}^{\prime})}}\frac{\left\langle\xi,l\right\rangle\left\langle
x_{r},l\right\rangle\left[l,h_{a}\right]}{\left\langle\xi,h_{a}\right\rangle\left\langle
x_{r},h_{a}\right\rangle\left\langle l,h_{a}\right\rangle},$ (4.44)

where $\mathcal{E}(\mathcal{G}^{\prime})$ denotes the set of edges belonging
to the spanning forest $\mathcal{G}^{\prime}$ that is the collection of trees
planted at nodes in
$\left(\boldsymbol{1}\setminus\\{x_{r}\\}\right)\cup\boldsymbol{2}\cup\mathsf{H}_{A}$.
These trees in $\mathcal{G}^{\prime}$ have the same structures with those in
the corresponding refined graph $\mathcal{F}$, but the edges have the new
meaning. After this step, the remaining refined graph
$\mathcal{F}\setminus\mathsf{H}_{B}$ involves only the two traces and the
inner gravitons (i.e. gravitons on the bridge between the two traces).
Manipulations in this step is shown by Fig. 10.

Step-2: Now we separate the two traces $\boldsymbol{1}$ and $\boldsymbol{2}$
(and the inner gravitons attached to each) from one another. Assume that the
bridge between $\boldsymbol{1}$ and $\boldsymbol{2}$ in the remaining refined
graph has the following structure

$\displaystyle l_{1}=x_{k}\in\boldsymbol{1}\setminus\\{x_{r}\\}\leftarrow
c_{1}\leftarrow\dots\leftarrow c_{u-1}\leftarrow c(=c_{u})\leftrightarrow
d(=d_{1})\rightarrow\dots\rightarrow d_{v}\rightarrow
a\in\boldsymbol{2}\setminus\\{b\\},$ (4.45)

where each type-2 line is expressed by $\leftarrow$ or $\rightarrow$, the
type-3 line is denoted by $\leftrightarrow$, the gluon
$l_{1}\in\boldsymbol{1}$ is supposed to be $x_{k}$. According to the refined
graphic rule, $T^{\mathcal{F}\setminus\mathsf{H}_{B}}$ in eq. (4.44) is
explicitly written as

$\displaystyle T^{\mathcal{F}\setminus\mathsf{H}_{B}}$ $\displaystyle=$
$\displaystyle\Biggl{[}\,\sum\limits_{\shuffle}\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{k},\\{x_{k+1},\dots,x_{r-1}\\}\shuffle\\{c_{1},\dots,c=c_{u},d=d_{1},\dots,d_{v},a,\boldsymbol{\beta},b\\},x_{r}\right)}\Biggr{]}$
(4.46)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times~{}\left(-k_{c}\cdot
k_{d}\right)\,\prod\limits_{p=1}^{u}\epsilon_{c_{p}}\cdot
k_{c_{p-1}}\,\prod\limits_{q=1}^{v}\left(-\epsilon_{d_{q}}\cdot
k_{d_{q+1}}\right),$

where $c_{0}\equiv x_{k}$ and $d_{v+1}\equiv a$. Note that the permutations
$\\{x_{k+1},\dots,x_{r-1}\\}\shuffle\\{c_{1},\dots,c=c_{u},d=d_{1},\dots,d_{v},a,\boldsymbol{\beta},b\\}$
can be reexpressed as

$\displaystyle\bigl{(}\boldsymbol{\gamma}\in\\{x_{k+1},\dots,x_{r-1}\\}\shuffle\\{c_{1},\dots,c=c_{u}\\}\bigr{)}\shuffle\\{d=d_{1},\dots,d_{v},a,\boldsymbol{\beta},b\\}\big{|}_{c\prec
d}.$ (4.47)

When the identity (B.9) is applied, the expression in the square brackets of
eq. (4.46) turns into

$\displaystyle\sum\limits_{\boldsymbol{\gamma}}\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{k},\boldsymbol{\gamma},x_{r}\right)}\frac{\left\langle
c,x_{r}\right\rangle}{\langle
c,d=d_{1},\dots,d_{v},a,\boldsymbol{\beta},b,x_{r}\rangle}$ (4.48)
$\displaystyle=$
$\displaystyle\sum\limits_{\boldsymbol{\gamma}}\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{k},\boldsymbol{\gamma},x_{r}\right)\left(d=d_{1},\dots,d_{v},a,\boldsymbol{\beta},b\right)}\,\frac{\left\langle
b,d\right\rangle\left\langle c,x_{r}\right\rangle}{\left\langle
c,d\right\rangle\left\langle b,x_{r}\right\rangle}.$

Thus the contribution of two traces (and inner gravitons attached to them) in
eq. (4.46) are separated as follows

$\displaystyle T^{\mathcal{F}\setminus\mathsf{H}_{B}}$ $\displaystyle=$
$\displaystyle\left(-k_{c}\cdot k_{d}\right)\frac{\left\langle
c,x_{r}\right\rangle\left\langle d,b\right\rangle}{\left\langle
c,d\right\rangle\left\langle x_{r},b\right\rangle}$ (4.49)
$\displaystyle\times\Biggl{[}\,\sum\limits_{\boldsymbol{\gamma}}\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{k},\boldsymbol{\gamma},x_{r}\right)}\prod\limits_{p=1}^{u}\epsilon_{c_{p}}\cdot
k_{c_{p-1}}\Biggr{]}\Biggl{[}\,\frac{1}{\left(d=d_{1},\dots,d_{v},a,\boldsymbol{\beta},b\right)}\prod\limits_{q=1}^{v}\left(-\epsilon_{d_{q}}\cdot
k_{d_{q+1}}\right)\Biggr{]}.$

Figure 11: An example for step-2 in the general proof of the formula (4.4). In
this step, the contribution of the remaining refined graph in Fig. 10 is
further decomposed such that the two traces and the inner gravitons attached
to them are separated. After this step, the double-arrow line between $h_{5}$
and $h_{7}$ now stands for the crossing factor $\left(-k_{h_{5}}\cdot
k_{h_{7}}\right)\frac{\left\langle h_{5},b\right\rangle\left\langle
x_{r},h_{7}\right\rangle}{\left\langle h_{7},h_{5}\right\rangle\left\langle
b,x_{r}\right\rangle}$.

Here, without loss of generality, the factor $\left\langle
g_{i},g_{j}\right\rangle^{4}$ is assumed to be absorbed into the Parke-Taylor
factor involving the trace $\boldsymbol{1}$. This step is demonstrated by Fig.
11.

Figure 12: An example for step-3 in the general proof of the formula (4.4). In
this step, the inner gravitons $h_{7}$ and $h_{5}$ in Fig. 12 are further
extracted from the traces $\boldsymbol{1}$ and $\boldsymbol{2}$ respectively.
After summing over all $\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a,b]$
with the proper sign $(-1)^{|\boldsymbol{2},a,b|}$, the second trace becomes a
Parke-Taylor factor with the standard permutation $y_{1},y_{2},\dots,y_{s}$.
Therefore, we get the structure (4.53). Here the gluons in $\boldsymbol{1}$
and $\boldsymbol{2}$ are considered as roots, while gravitons $h_{5}$ and
$h_{7}$ play as the inner gravitons. The bridge between the two traces is
$l_{1}\leftarrow h_{7}\leftrightarrow h_{5}\rightarrow a$.

Step-3: We now extract the gravitons $c_{1},\dots,c_{u}$ and
$d_{1},\dots,d_{v}$ from the traces $\boldsymbol{1}$ and $\boldsymbol{2}$
respectively. The contribution of the trace $\boldsymbol{1}$ (and the inner
gravitons attached to it) is given by the expression inside the first pair of
square brackets in eq. (4.49), which is further simplified as

$\displaystyle\sum\limits_{\boldsymbol{\gamma}}\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{k},\boldsymbol{\gamma},x_{r}\right)}\prod\limits_{p=1}^{u}\epsilon_{c_{p}}\cdot
k_{c_{p-1}}$ (4.50) $\displaystyle=$ $\displaystyle\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)}\left(\prod\limits_{p=1}^{u}\frac{\left\langle
c_{p-1},x_{r}\right\rangle}{\left\langle
c_{p-1},c_{p}\right\rangle\left\langle
c_{p},x_{r}\right\rangle}\right)\left(\prod\limits_{p=1}^{u}\epsilon_{c_{p}}\cdot
k_{c_{p-1}}\right)$ $\displaystyle=$ $\displaystyle\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)}\prod\limits_{p=1}^{u}\frac{\left\langle\xi,c_{p-1}\right\rangle\left\langle
x_{r},c_{p-1}\right\rangle\left[c_{p-1},c_{p}\right]}{\left\langle\xi,c_{p}\right\rangle\left\langle
x_{r},c_{p}\right\rangle\left\langle c_{p-1},c_{p}\right\rangle},$

where permutations $\boldsymbol{\gamma}$ satisfy
$\boldsymbol{\gamma}\in\\{x_{k+1},\dots,x_{r-1}\\}\shuffle\\{c_{1},\dots,c=c_{u}\\}$.
For a given $c_{p}$ in eq. (4.50), the identity (B.8) has been applied and the
explicit expression of $\epsilon_{c_{p}}\cdot k_{c_{p-1}}$ has been inserted.
For the trace $\boldsymbol{2}$ in eq. (4.49), we extract gravitons
$d_{1},\dots,d_{v}$ out from the corresponding Parke-Taylor factor (noting
that the Parke-Taylor factor has cyclic symmetry) as follows

$\displaystyle\frac{1}{\left(d=d_{1},\dots,d_{v},a,\boldsymbol{\beta},b\right)}\prod\limits_{q=1}^{v}\left(-\epsilon_{d_{q}}\cdot
k_{d_{q+1}}\right)$ (4.51) $\displaystyle=$
$\displaystyle\frac{1}{\left(a,\boldsymbol{\beta},b\right)}\left(\prod\limits_{q=1}^{v}\frac{\left\langle
b,d_{q+1}\right\rangle}{\left\langle b,d_{q}\right\rangle\left\langle
d_{q},d_{q+1}\right\rangle}\right)\prod\limits_{q=1}^{v}\left(-\epsilon_{d_{q}}\cdot
k_{d_{q+1}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\left(a,\boldsymbol{\beta},b\right)}\prod\limits_{q=1}^{v}\frac{\left\langle\xi,d_{q+1}\right\rangle\left\langle
b,d_{q+1}\right\rangle\left[d_{q+1},d_{q}\right]}{\left\langle\xi,d_{q}\right\rangle\left\langle
b,d_{q}\right\rangle\left\langle d_{q+1},d_{q}\right\rangle}.$

When we sum over all possible permutations
$\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a,b]$, each of which is
dressed by the sign $(-1)^{|\boldsymbol{2},a,b|}$, the Parke-Taylor factors
${1}\setminus{\left(a,\boldsymbol{\beta},b\right)}$ turn into the one with the
standard permutation $y_{1},\dots,y_{s}$, because of the KK relation (B.13).
We thus get

$\displaystyle\frac{1}{\left(y_{1},\dots,y_{s}\right)}\prod\limits_{q=1}^{v}\frac{\left\langle\xi,d_{q+1}\right\rangle\left\langle
b,d_{q+1}\right\rangle\left[d_{q+1},d_{q}\right]}{\left\langle\xi,d_{q}\right\rangle\left\langle
b,d_{q}\right\rangle\left\langle d_{q+1},d_{q}\right\rangle}.$ (4.52)

Eq. (4.49), eq. (4.50) and eq. (4.52) together imply that eq. (4.44) can be
reexpressed as (shown by Fig. 12)

$\displaystyle\frac{\left\langle
g_{i},g_{j}\right\rangle^{4}}{\left(x_{1},\dots,x_{r}\right)\,\left(y_{1},\dots,y_{s}\right)}\left(-k_{c}\cdot
k_{d}\right)\frac{\left\langle c,x_{r}\right\rangle\left\langle
d,b\right\rangle}{\left\langle c,d\right\rangle\left\langle
x_{r},b\right\rangle}\,\prod\limits_{p=1}^{u}\,\frac{\left\langle\xi,c_{p-1}\right\rangle\left\langle
x_{r},c_{p-1}\right\rangle\left[c_{p-1},c_{p}\right]}{\left\langle\xi,c_{p}\right\rangle\left\langle
x_{r},c_{p}\right\rangle\left\langle c_{p-1},c_{p}\right\rangle}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\,\prod\limits_{q=1}^{v}\,\frac{\left\langle\xi,d_{q+1}\right\rangle\left\langle
b,d_{q+1}\right\rangle\left[d_{q+1},d_{q}\right]}{\left\langle\xi,d_{q}\right\rangle\left\langle
b,d_{q}\right\rangle\left\langle
d_{q+1},d_{q}\right\rangle}\prod\limits_{{e(h_{a},l)\in\,\mathcal{E}(\mathcal{G}^{\prime})}}\,\frac{\left\langle\xi,l\right\rangle\left\langle
x_{r},l\right\rangle\left[l,h_{a}\right]}{\left\langle\xi,h_{a}\right\rangle\left\langle
x_{r},h_{a}\right\rangle\left\langle l,h_{a}\right\rangle}$ (4.53)

when all possible $\boldsymbol{\beta}\in\mathsf{KK}[\boldsymbol{2},a,b]$ are
summed over. This is exactly one term of eq. (4.16), while the inner graviton
set is $\mathsf{H}_{A}=\\{c_{1},\cdots,c_{u},d_{1},\cdots,d_{v}\\}$, the
bridge $\mathcal{B}(H_{A})$ between the two traces is given by (4.45) and the
choice of gauge is

$\displaystyle\chi=b\,,~{}~{}\zeta=\eta=x_{r}.$ (4.54)

After summing over all splits $\mathsf{H}\to\mathsf{H}_{A},\mathsf{H}_{B}$,
all possible bridges (4.45) for a given split
$\mathsf{H}\to\mathsf{H}_{A},\mathsf{H}_{B}$ and all spanning forests rooted
at
$\left(\boldsymbol{1}\setminus\\{x_{r}\\}\right)\cup\boldsymbol{2}\cup\mathsf{H}_{A}$
for a given bridge, we finally get eq. (4.16) (thus eq. (4.4)) with the choice
of gauge (4.54) (noting that the terms in eq. (4.16) which invlove trees
planted on $x_{r}$ and/or the bridge (4.45) with $a=b$ must vanish under this
choice of gauge, due to the antisymmetry of spinor products). Since we have
already proven that eq. (4.16) is independent of the choice of gauge, the
general proof of eq. (4.16) (thus eq. (4.4)) has been completed.

##  5 The vanishing configurations

In this section, we investigate the vanishing amplitudes with two negative-
helicity particles. We first introduce several properties of refined graphs
_when the corresponding YM amplitudes in the expansion ( 2.1) are MHV
amplitudes (which are expressed by Parke-Taylor formula)._ Using these
properties, we prove that the double-trace amplitudes with the
$(h_{i}^{-},g_{j}^{-})$ configuration, the single- and double- trace
amplitudes with the $(h_{i}^{-},h_{j}^{-})$ configuration as well as all
amplitudes with arbitrary two negative-helicity particles and more than two
traces have to vanish.

###  5.1 Helpful properties of refined graphs for amplitudes with two
negative-helicity particles

Now we present several usful properties of the refined graphs when the
amplitude has two negative-helicity particles. _Our choice of gauge is always
the standard one: i.e. all the reference momenta of positive-helicity
gravitons are chosen as $k^{\mu}_{h_{i}}$ ($h_{i}$ is a negative-helicity
graviton), while the reference momenta of all negative-helicity gravitons,
i.e. the $h_{i}$ for the $(h_{i}^{-},g_{j}^{-})$ configuration and $h_{i}$,
$h_{j}$ for the $(h_{i}^{-},h_{j}^{-})$ configuration, are chosen as
$\xi^{\mu}$._ With this choice of gauge, the refined graphs satisfy the
following properties.

Figure 13: Graphs with the structures (a) and (b) cancel with each other if
the corresponding amplitudes are described by Parke-Taylor factors.

Property-1: Suppose $h_{i}$ is a negative-helicity graviton and
$\\{\mathcal{F}^{\prime}\\}$ is the set of graphs satisfying the following two
conditions: (i). The $h_{i}$ is a leaf which starts a type-2 line
$\epsilon_{h_{i}}\cdot k_{l}$, (ii). All graphs in
$\\{\mathcal{F}^{\prime}\\}$ are reduced to an identical graph $\mathcal{F}$
when $h_{i}$ is deleted. _The sum of contributions of all graphs in
$\\{\mathcal{F}^{\prime}\\}$ is reduced to the contribution of
$\mathcal{F}=\mathcal{F}^{\prime}\setminus\\{h_{a}\\}$ with an extra minus._
This property, when the graphs $\mathcal{F}^{\prime}$ involve only one
component (see Fig. 5), has already been proven in section 3.2. Proof for the
cases with more general structure follows from a similar statement.

Property-2: _Graphs of the patterns Fig. 13 (a) and (b) must cancel with each
other._ This is shown as follows. When we extract the gravitons from the
Parke-Taylor factors, the contributions of $\mathscr{C}_{1}$
($\mathscr{C}_{2}$) are the same for distinct graphs Fig. 13 (a) and (b). Thus
we only need to compare the contribution of the substructures colored red. In
Fig. 13 (a), the red part contributes a factor

$\displaystyle(-1)(\epsilon_{h_{j}}^{-}\cdot k_{l})\,\left(\frac{\left\langle
l,x_{r}\right\rangle}{\left\langle l,h_{j}\right\rangle\left\langle
h_{j},x_{r}\right\rangle}\right),$ (5.1)

where the first factor comes from the definition of the type-2 line and the
second factor is obtained when extracting $h_{j}$ out from the Parke-Taylor
factor, $(-1)$ is the overall sign in Fig. 13 (a). When the first factor is
expressed by spinor products, the above expression is reduced into

$\displaystyle-\frac{\left\langle
l,h_{j}\right\rangle\left[\xi,l\right]}{\left[\xi,h_{j}\right]}\frac{\left\langle
l,x_{r}\right\rangle}{\left\langle l,h_{j}\right\rangle\left\langle
h_{j},x_{r}\right\rangle}=-\frac{\left\langle
l,x_{r}\right\rangle\left[\xi,l\right]}{\left\langle
h_{j},x_{r}\right\rangle\left[\xi,h_{j}\right]}.$ (5.2)

The red part of Fig. 13 (b) contributes the following factor

$\displaystyle(\epsilon_{h_{j}}^{-}\cdot k_{h_{i}})\frac{\left\langle
h_{i},x_{r}\right\rangle}{\left\langle h_{i},h_{j}\right\rangle\left\langle
h_{j},x_{r}\right\rangle}(\epsilon_{h_{i}}^{-}\cdot k_{l})\frac{\left\langle
l,x_{r}\right\rangle}{\left\langle l,h_{i}\right\rangle\left\langle
h_{i},x_{r}\right\rangle}$ (5.3) $\displaystyle=$
$\displaystyle\frac{\left\langle
h_{i},h_{j}\right\rangle\left[\xi,h_{i}\right]}{\left[\xi,h_{j}\right]}\frac{\left\langle
h_{i},x_{r}\right\rangle}{\left\langle h_{i},h_{j}\right\rangle\left\langle
h_{j},x_{r}\right\rangle}\frac{\left\langle
l,h_{i}\right\rangle\left[\xi,l\right]}{\left[\xi,h_{i}\right]}\frac{\left\langle
l,x_{r}\right\rangle}{\left\langle l,h_{i}\right\rangle\left\langle
h_{i},x_{r}\right\rangle}$ $\displaystyle=$ $\displaystyle\frac{\left\langle
l,x_{r}\right\rangle\left[\xi,l\right]}{\left\langle
h_{j},x_{r}\right\rangle\left[\xi,h_{j}\right]},$

which is just the corresponding factor in Fig. 13 (a) with an opposite sign.
Thus Fig. 13 (a) and (b) cancel with each other.

Property-3: let $\mathscr{C}$ be a tree structure where all lines are type-2
lines whose arrows point towards the negative-helicity graviton $h_{j}$,
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ be other two tree structures. Suppose
the trace $\boldsymbol{1}=\\{x_{1},\dots,x_{r}\\}$ belongs to
$\mathscr{C}_{1}$. _Graphs with the structures Fig. 14 (a), (b) and (c) must
cancel with each other._ In Fig. 14, a line connected to a boxed region but
not a concrete node always means we sum over graphs where the line is
connected to all nodes (except $x_{r}\in\boldsymbol{1}$) in that region. To
prove the cancellation between these three graphs, we only need to consider
the $\mathscr{C}$ part and the red lines in each graph because either the
$\mathscr{C}_{1}$ or the $\mathscr{C}_{2}$ part provides an identical
expression in all three graphs.

Figure 14: Contributions of graphs with the typical structures (a), (b) and
(c) cancel with each other.

  * •

(a). In Fig. 14 (a), when extracting the $\mathscr{C}$ part from the Parke-
Taylor factors we get a factor of the following form

$\displaystyle\left[(\epsilon_{h_{j}}\cdot k_{l})\frac{\left\langle
l,x_{r}\right\rangle}{\left\langle l,h_{j}\right\rangle\left\langle
h_{j},x_{r}\right\rangle}\right]\left[\prod\limits_{\begin{subarray}{c}e(x,y)\in\mathcal{E}(\mathscr{C})\end{subarray}}(\epsilon_{x}\cdot
k_{y})\frac{\left\langle y,x_{r}\right\rangle}{\left\langle
y,x\right\rangle\left\langle x,x_{r}\right\rangle}\right],$ (5.4)

where $l\in(\mathscr{C}_{1}\setminus\\{x_{r}\\})\cup\mathscr{C}_{2}$ is
supposed to be the end node of the type-2 line which starts at $h_{j}$. We
extract the node $d$ after extracting all other nodes in $\mathscr{C}_{2}$
part from the Parke-Taylor factors. Then the node $d$ is associated with the
following factor

$\displaystyle(-k_{d}\cdot k_{c})\frac{\left\langle
c,x_{r}\right\rangle}{\left\langle c,d\right\rangle\left\langle
d,x_{r}\right\rangle}.$ (5.5)

Putting the above factors together and summing over
$l\in(\mathscr{C}_{1}\setminus\\{x_{r}\\})\cup\mathscr{C}_{2}$ we get

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle\sum\limits_{l\in(\mathscr{C}_{1}\setminus\\{x_{r}\\})\cup\mathscr{C}_{2}}\left[(\epsilon_{h_{j}}\cdot
k_{l})\frac{\left\langle l,x_{r}\right\rangle}{\left\langle
l,h_{j}\right\rangle\left\langle
h_{j},x_{r}\right\rangle}\right]\left[\prod\limits_{\begin{subarray}{c}e(x,y)\in\mathcal{E}(\mathscr{C})\end{subarray}}(\epsilon_{x}\cdot
k_{y})\frac{\left\langle y,x_{r}\right\rangle}{\left\langle
y,x\right\rangle\left\langle x,x_{r}\right\rangle}\right](-k_{d}\cdot
k_{c})\frac{\left\langle c,x_{r}\right\rangle}{\left\langle
c,d\right\rangle\left\langle d,x_{r}\right\rangle}$ (5.6) $\displaystyle=$
$\displaystyle\sum\limits_{l\in\mathscr{C}_{1}\cup\mathscr{C}_{2}}\frac{1}{2}\frac{\left[\xi,l\right]\left\langle
l,x_{r}\right\rangle}{\left[\xi,h_{j}\right]\left\langle
h_{j},x_{r}\right\rangle}\left[\prod\limits_{\begin{subarray}{c}e(x,y)\in\mathcal{E}(\mathscr{C})\end{subarray}}(\epsilon_{x}\cdot
k_{y})\frac{\left\langle y,x_{r}\right\rangle}{\left\langle
y,x\right\rangle\left\langle
x,x_{r}\right\rangle}\right]\frac{\left[c,d\right]\left\langle
c,x_{r}\right\rangle}{\left\langle d,x_{r}\right\rangle}$ $\displaystyle=$
$\displaystyle-\sum\limits_{l\in\mathscr{C}}\frac{1}{2}\frac{\left[\xi,l\right]\left[c,d\right]}{\left[\xi,h_{j}\right]}\left[\frac{\left\langle
l,x_{r}\right\rangle\left\langle c,x_{r}\right\rangle}{\left\langle
h_{j},x_{r}\right\rangle\left\langle
d,x_{r}\right\rangle}\right]\left[\prod\limits_{\begin{subarray}{c}e(x,y)\in\mathcal{E}(\mathscr{C})\end{subarray}}(\epsilon_{x}\cdot
k_{y})\frac{\left\langle y,x_{r}\right\rangle}{\left\langle
y,x\right\rangle\left\langle x,x_{r}\right\rangle}\right],$

in which, $e(x,y)\in\mathcal{E}(\mathscr{C})$ denoted the set of (type-2)
lines in $\mathscr{C}$ and $e(x,y)$ denoted the type-2 line starting at $x$
and ending at $y$, the term with $l=x_{r}$ was included in the second line
because of $\left\langle x_{r},x_{r}\right\rangle=0$, momentum conservation
was applied on the third line.

  * •

(b). In Fig. 14 (b), when nodes in the $\mathscr{C}$ part and the node $d$ are
extracted out from the Parke-Taylor factors, we get the following factor

$\displaystyle T_{2}$ $\displaystyle=$
$\displaystyle\left[\sum\limits_{l\in\mathscr{C}}(-k_{d}\cdot
k_{l})\frac{\left\langle l,x_{r}\right\rangle}{\left\langle
l,d\right\rangle\left\langle
d,x_{r}\right\rangle}\right]\left[(\epsilon_{h_{j}}\cdot
k_{c})\frac{\left\langle c,x_{r}\right\rangle}{\left\langle
c,h_{j}\right\rangle\left\langle
h_{j},x_{r}\right\rangle}\right]\left[\prod\limits_{\begin{subarray}{c}e(x,y)\in\mathcal{E}(\mathscr{C})\end{subarray}}(\epsilon_{x}\cdot
k_{y})\frac{\left\langle y,x_{r}\right\rangle}{\left\langle
y,x\right\rangle\left\langle x,x_{r}\right\rangle}\right]$ (5.7)
$\displaystyle=$
$\displaystyle\sum\limits_{l\in\mathscr{C}}\frac{1}{2}\frac{\left[l,d\right]\left[\xi,c\right]}{\left[\xi,h_{j}\right]}\left[\frac{\left\langle
l,x_{r}\right\rangle\left\langle c,x_{r}\right\rangle}{\left\langle
d,x_{r}\right\rangle\left\langle
h_{j},x_{r}\right\rangle}\right]\left[\prod\limits_{\begin{subarray}{c}e(x,y)\in\mathcal{E}(\mathscr{C})\end{subarray}}(\epsilon_{x}\cdot
k_{y})\frac{\left\langle y,x_{r}\right\rangle}{\left\langle
y,x\right\rangle\left\langle x,x_{r}\right\rangle}\right],$

where all possible $l\in\mathscr{C}$ were summed over.

  * •

(c). To evaluate Fig. 14 (c), we suppose the node $l\in\mathscr{C}$ is
connected to $c$ via the type-3 line. Assuming that nodes on the path from $l$
to $h_{j}$ are $u_{1}$, $u_{2}$, …, $u_{k}$, respectively, all the type-2
lines between adjacent nodes on this path point away from the trace
$\boldsymbol{1}$. When extracting the nodes $u_{1}$, …, $u_{k}$, $h_{j}$, from
the Parke-Taylor factors according to eq. (B.9), we get a factor

$\displaystyle(-\epsilon_{u_{k}}\cdot k_{h_{j}})(-\epsilon_{u_{k-1}}\cdot
k_{u_{k}})\dots(-\epsilon_{u_{1}}\cdot k_{u_{2}})(-\epsilon_{l}\cdot
k_{u_{1}})\frac{\left\langle l,x_{r}\right\rangle}{\left\langle
l,u_{1},\dots,u_{k},h_{j},x_{r}\right\rangle}$ $\displaystyle=$
$\displaystyle(\epsilon_{u_{k}}\cdot k_{h_{j}})\,(\epsilon_{u_{k-1}}\cdot
k_{u_{k}})\,\dots\,(\epsilon_{u_{1}}\cdot k_{u_{2}})\,(\epsilon_{l}\cdot
k_{u_{1}})\,\frac{\left\langle h_{j},x_{r}\right\rangle}{\left\langle
h_{j},u_{k},\dots,u_{1},l,x_{r}\right\rangle}\left(\frac{\left\langle
l,x_{r}\right\rangle}{\left\langle h_{j},x_{r}\right\rangle}\right)^{2}$
$\displaystyle=$ $\displaystyle\biggl{[}(\epsilon_{u_{k}}\cdot
k_{h_{j}})\frac{\left\langle h_{j},x_{r}\right\rangle}{\left\langle
h_{j},u_{k}\right\rangle\left\langle
u_{k},x_{r}\right\rangle}(\epsilon_{u_{k-1}}\cdot k_{u_{k}})\frac{\left\langle
u_{k},x_{r}\right\rangle}{\left\langle u_{k},u_{k-1}\right\rangle\left\langle
u_{k-1},x_{r}\right\rangle}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\dots(\epsilon_{u_{1}}\cdot
k_{u_{2}})\frac{\left\langle u_{2},x_{r}\right\rangle}{\left\langle
u_{2},u_{1}\right\rangle\left\langle
u_{1},x_{r}\right\rangle}(\epsilon_{l}\cdot k_{u_{1}})\frac{\left\langle
u_{1},x_{r}\right\rangle}{\left\langle u_{1},l\right\rangle\left\langle
l,x_{r}\right\rangle}\biggr{]}\left(\frac{\left\langle
l,x_{r}\right\rangle}{\left\langle h_{j},x_{r}\right\rangle}\right)^{2},$

where the antisymmetry of spinor products has been applied. The expression in
the square brackets on the last line would have the form as if arrows on this
path pointed towards the trace $\boldsymbol{1}$. Arrows of lines in
$\mathscr{C}$ which do not on the path from $l$ to $h_{j}$ already point
towards the trace $\boldsymbol{1}$. Hence full contribution of lines in the
$\mathscr{C}$ part reads

$\displaystyle\left[\prod\limits_{\begin{subarray}{c}e(x,y)\in\mathcal{E}(\mathscr{C})\end{subarray}}(\epsilon_{x}\cdot
k_{y})\frac{\left\langle y,x_{r}\right\rangle}{\left\langle
y,x\right\rangle\left\langle
x,x_{r}\right\rangle}\right]\left(\frac{\left\langle
l,x_{r}\right\rangle}{\left\langle h_{j},x_{r}\right\rangle}\right)^{2}.$
(5.8)

The node $d$ and the node $l$ respectively provide the following factors when
they are extracted out

$\displaystyle(-\epsilon_{h_{j}}\cdot k_{d})\frac{\left\langle
h_{j},x_{r}\right\rangle}{\left\langle h_{j},d\right\rangle\left\langle
d,x_{r}\right\rangle},~{}~{}~{}~{}~{}~{}~{}(-k_{c}\cdot
k_{l})\frac{\left\langle c,x_{r}\right\rangle}{\left\langle
c,l\right\rangle\left\langle l,x_{r}\right\rangle}.$ (5.9)

Figure 15: Contributions of graphs with the above typical structures cancel
out. Here, the substructure $\mathscr{C}$ is divided into two parts
$\mathscr{C}_{t}$ and $\mathscr{C}_{b}$ by a dashed line with no arrow.

Therefore, the contributions of lines in $\mathscr{C}$ together with the red
parts in Fig. 14 (c) are collected as

$\displaystyle(-\epsilon_{h_{j}}\cdot k_{d})\frac{\left\langle
h_{j},x_{r}\right\rangle}{\left\langle h_{j},d\right\rangle\left\langle
d,x_{r}\right\rangle}(-k_{c}\cdot k_{l})\frac{\left\langle
c,x_{r}\right\rangle}{\left\langle c,l\right\rangle\left\langle
l,x_{r}\right\rangle}\left[\prod\limits_{\begin{subarray}{c}e(x,y)\in\mathcal{E}(\mathscr{C})\end{subarray}}(\epsilon_{x}\cdot
k_{y})\frac{\left\langle y,x_{r}\right\rangle}{\left\langle
y,x\right\rangle\left\langle
x,x_{r}\right\rangle}\right]\left(\frac{\left\langle
l,x_{r}\right\rangle}{\left\langle h_{j},x_{r}\right\rangle}\right)^{2}.$
(5.10)

When spinor expressions are applied and $l\in\mathscr{C}$ is summed over, the
above expression turns into

$\displaystyle
T_{3}=\sum\limits_{l\in\mathscr{C}}\frac{1}{2}\frac{\left[c,l\right]\left[\xi,d\right]}{\left[\xi,h_{j}\right]}\left[\frac{\left\langle
l,x_{r}\right\rangle\left\langle c,x_{r}\right\rangle}{\left\langle
d,x_{r}\right\rangle\left\langle
h_{j},x_{r}\right\rangle}\right]\left[\prod\limits_{\begin{subarray}{c}e(x,y)\in\mathcal{E}(\mathscr{C})\end{subarray}}(\epsilon_{x}\cdot
k_{y})\frac{\left\langle y,x_{r}\right\rangle}{\left\langle
y,x\right\rangle\left\langle x,x_{r}\right\rangle}\right].$ (5.11)

The sum of $T_{1}$, $T_{2}$ and $T_{3}$ for any $l$ precisely cancel out due
to Schouten identity (B.5).

Property-4: _Graphs with the structures Fig. 15 (a), (b) and (c) cancel with
each other._ Before proving this property, let us first explain the meaning of
dashed lines in the graphs: _(1)_ the dashed arrow lines inherit the
definition of the type-4 line (see Fig. 1 (d)) but the two ends of these lines
can be gravitons and/or gluons, _(2)._ the dashed line with no arrow in Fig.
15 (colored red) also reflects the relative position between its two end
nodes, but it does not bring any sign. The cancellation between graphs Fig. 15
(a), (b) and (c) only relies on the relative positions between nodes and
possible signs caused by the arrows (just as the property-3). Let us now
evaluate the different parts in the three graphs in turn.

Figure 16: (a): the path between $l_{1}$ and $l_{2}$ in eq. (5.13), (b): the
path between $l_{2}$ and $l_{1}$ in Fig. 16 (c)

  * •

(a) Suppose that the two end nodes of the type-3 line between
$\mathscr{C}_{b}$ and $\mathscr{C}_{1}\cup\mathscr{C}_{2}$ in Fig. 15 (a) are
respectively $l_{1}\in\mathscr{C}_{b}$ and
$l_{2}\in(\mathscr{C}_{1}\setminus\\{x_{r}\\})\cup\mathscr{C}_{2}$. Then all
lines in $\mathscr{C}$, the type-3 line between $l_{1}$ , $l_{2}$ and the
type-3 line between $c$, $d$ together produce the following factor

$\displaystyle\sum\limits_{l_{2}\in(\mathscr{C}_{1}\setminus\\{x_{r}\\})\cup\mathscr{C}_{2}}\sum\limits_{l_{1}\in\mathscr{C}_{b}}T[\mathscr{C}(l_{1})]\left[(-k_{l_{1}}\cdot
k_{l_{2}})\frac{\left\langle l_{2},x_{r}\right\rangle}{\left\langle
l_{2},l_{1}\right\rangle\left\langle
l_{1},x_{r}\right\rangle}\right]\left[(-k_{d}\cdot k_{c})\frac{\left\langle
c,x_{r}\right\rangle}{\left\langle c,d\right\rangle\left\langle
d,x_{r}\right\rangle}\right],$ (5.12)

where $T[\mathscr{C}(l_{1})]$ denotes the full contributions of lines in the
region $\mathscr{C}$ when $l_{1}$ is the node nearest to the trace
$\boldsymbol{1}$. Momentum conservation allows one to convert the summation
over $l_{2}\in(\mathscr{C}_{1}\setminus\\{x_{r}\\})\cup\mathscr{C}_{2}$ into a
summation over $l_{2}\in\mathscr{C}_{t}\cup\mathscr{C}_{b}$ with an extra sign
$(-1)$, which further splits into the summations over
$l_{2}\in\mathscr{C}_{b}$ and $l_{2}\in\mathscr{C}_{t}$. The sum of terms with
$l_{2}\in\mathscr{C}_{b}$ is explicitly written as

$\displaystyle-\sum\limits_{l_{1},l_{2}\in\mathscr{C}_{b}}T[\mathscr{C}(l_{1})]\left[(-k_{l_{1}}\cdot
k_{l_{2}})\frac{\left\langle l_{2},x_{r}\right\rangle}{\left\langle
l_{2},l_{1}\right\rangle\left\langle
l_{1},x_{r}\right\rangle}\right]\left[(-k_{d}\cdot k_{c})\frac{\left\langle
c,x_{r}\right\rangle}{\left\langle c,d\right\rangle\left\langle
d,x_{r}\right\rangle}\right].$ (5.13)

For a given $l_{1},l_{2}\in\mathscr{C}_{b}$ in the above expression, one can
always find out another term in the summation by exchanging the roles of
$l_{1}$ and $l_{2}$. The only difference between $T[\mathscr{C}(l_{1})]$ and
$T[\mathscr{C}(l_{2})]$ is the factor corresponding to the path between
$l_{1}$ and $l_{2}$ (see Fig. 16 (a)). Supposing the nodes between $l_{1}$ and
$l_{2}$ are in turn $l_{1},v_{1},\dots,v_{j},b,u_{1},\dots,u_{k},l_{2}$, the
lines between $l_{1}$ and $l_{2}$ contribute the following factor to
$T[\mathscr{C}(l_{1})]$:

$\displaystyle(-1)^{j+1}\frac{\left\langle
l_{1},x_{r}\right\rangle}{\left\langle
l_{1},v_{1},\dots,v_{j},b,u_{1},\dots,u_{k},l_{2},x_{r}\right\rangle}=\frac{\left\langle
l_{1},x_{r}\right\rangle}{\left\langle
b,v_{j},\dots,v_{1},l_{1}\right\rangle\left\langle
b,u_{1},\dots,u_{k},l_{2},x_{r}\right\rangle},$ (5.14)

when eq. (B.9) is been applied. Together with the second factor in eq. (5.13),
the above factor provides

$\displaystyle\frac{\left\langle l_{1},x_{r}\right\rangle}{\left\langle
b,v_{j},\dots,v_{1},l_{1}\right\rangle\left\langle
b,u_{1},\dots,u_{k},l_{2},x_{r}\right\rangle}(-k_{l_{1}}\cdot
k_{l_{2}})\frac{\left\langle l_{2},x_{r}\right\rangle}{\left\langle
l_{2},l_{1}\right\rangle\left\langle l_{1},x_{r}\right\rangle}$ (5.15)
$\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\left[l_{2},l_{1}\right]}{\left\langle
b,v_{j},\dots,v_{1},l_{1}\right\rangle\left\langle
b,u_{1},\dots,u_{k},l_{2}\right\rangle}.$

While exchanging the roles of $l_{1}$ and $l_{2}$, we get the same expression
with an opposite sign. This antisymmetry indicates that eq. (5.13) vanishes
when all $l_{1},l_{2}\in\mathscr{C}_{b}$ are summed over. The expression
(5.12) is then reduced into

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle-\sum\limits_{l_{2}\in\mathscr{C}_{t}}\sum\limits_{l_{1}\in\mathscr{C}_{b}}T[\mathscr{C}(l_{1})]\left[(-k_{l_{1}}\cdot
k_{l_{2}})\frac{\left\langle l_{2},x_{r}\right\rangle}{\left\langle
l_{2},l_{1}\right\rangle\left\langle
l_{1},x_{r}\right\rangle}\right]\left[(-k_{d}\cdot k_{c})\frac{\left\langle
c,x_{r}\right\rangle}{\left\langle c,d\right\rangle\left\langle
d,x_{r}\right\rangle}\right]$ (5.16) $\displaystyle=$
$\displaystyle\sum\limits_{l_{2}\in\mathscr{C}_{t}}\sum\limits_{l_{1}\in\mathscr{C}_{b}}\frac{1}{2}\left[l_{1},l_{2}\right]\left[c,d\right]\frac{\left\langle
l_{2},x_{r}\right\rangle\left\langle c,x_{r}\right\rangle}{\left\langle
l_{1},x_{r}\right\rangle\left\langle
d,x_{r}\right\rangle}T[\mathscr{C}(l_{1})].$

  * •

(b) In Fig. 15 (b), the two type-3 lines and the $\mathscr{C}$ part together
contribute a factor

$\displaystyle T_{2}$ $\displaystyle=$
$\displaystyle\sum\limits_{l_{2}\in\mathscr{C}_{t}}\sum\limits_{l_{1}\in\mathscr{C}_{b}}(-k_{d}\cdot
k_{l_{2}})\frac{\left\langle l_{2},x_{r}\right\rangle}{\left\langle
l_{2},d\right\rangle\left\langle d,x_{r}\right\rangle}(-k_{c}\cdot
k_{l_{1}})\frac{\left\langle c,x_{r}\right\rangle}{\left\langle
c,l_{1}\right\rangle\left\langle
l_{1},x_{r}\right\rangle}T[\mathscr{C}(l_{1})]$ (5.17) $\displaystyle=$
$\displaystyle\sum\limits_{l_{2}\in\mathscr{C}_{t}}\sum\limits_{l_{1}\in\mathscr{C}_{b}}\frac{1}{2}\left[l_{2},d\right]\left[c,l_{1}\right]\frac{\left\langle
l_{2},x_{r}\right\rangle\left\langle c,x_{r}\right\rangle}{\left\langle
l_{1},x_{r}\right\rangle\left\langle
d,x_{r}\right\rangle}T[\mathscr{C}(l_{1})].$

  * •

(c) In Fig. 15 (c), the two type-3 lines and the $\mathscr{C}$ part contribute
the following factor

$\displaystyle T_{3}$ $\displaystyle=$
$\displaystyle\sum\limits_{l_{2}\in\mathscr{C}_{t}}\sum\limits_{l_{1}\in\mathscr{C}_{b}}(-k_{d}\cdot
k_{l_{1}})\frac{\left\langle l_{1},x_{r}\right\rangle}{\left\langle
l_{1},d\right\rangle\left\langle d,x_{r}\right\rangle}(-k_{c}\cdot
k_{l_{2}})\frac{\left\langle c,x_{r}\right\rangle}{\left\langle
c,l_{2}\right\rangle\left\langle
l_{2},x_{r}\right\rangle}T[\mathscr{C}(l_{2})]$ (5.18) $\displaystyle=$
$\displaystyle\sum\limits_{l_{2}\in\mathscr{C}_{t}}\sum\limits_{l_{1}\in\mathscr{C}_{b}}\frac{1}{2}\left[l_{1},d\right]\left[c,l_{2}\right]\frac{\left\langle
l_{1},x_{r}\right\rangle\left\langle c,x_{r}\right\rangle}{\left\langle
l_{2},x_{r}\right\rangle\left\langle
d,x_{r}\right\rangle}T[\mathscr{C}(l_{2})].$

To relate the $T[\mathscr{C}(l_{2})]$, which is the expression of
$\mathscr{C}$ part while $l_{2}$ is the node nearest to the trace
$\boldsymbol{1}$, to $T[\mathscr{C}(l_{1})]$, we note that the only difference
between $T[\mathscr{C}(l_{1})]$ and $T[\mathscr{C}(l_{2})]$ is the path
between the nodes $l_{1}\in\mathscr{C}_{b}$ and $l_{2}\in\mathscr{C}_{t}$ (see
Fig. 16 (b)). In $T[\mathscr{C}(l_{2})]$, this path has the form

$\displaystyle(-1)^{k+1}\frac{\left\langle
l_{2},x_{r}\right\rangle}{\left\langle
l_{2},u_{k},\dots,u_{1},a,b,v_{1},\dots,v_{j},l_{1},x_{r}\right\rangle}=\frac{\left\langle
l_{2},x_{r}\right\rangle}{\left\langle
a,u_{1},\dots,u_{k},l_{2}\right\rangle\left\langle
a,b\right\rangle\left\langle b,v_{1},\dots,v_{j},l_{1},x_{r}\right\rangle},$
(5.19)

where nodes on this path were supposed to be
$l_{2},u_{k},\dots,u_{1},a,b,v_{1},\dots,v_{j},l_{1}$ in turn. Similarly, in
$T[\mathscr{C}(l_{1})]$, the path from $l_{1}$ to $l_{2}$ contributes a factor

$\displaystyle\frac{\left\langle l_{1},x_{r}\right\rangle}{\left\langle
b,v_{1},\dots,v_{j},l_{1}\right\rangle\left\langle
b,a\right\rangle\left\langle a,u_{1},\dots,u_{k},l_{2},x_{r}\right\rangle}.$
(5.20)

Therefore, we arrive $T[\mathscr{C}(l_{2})]=\left(-\frac{\left\langle
l_{2},x_{r}\right\rangle^{2}}{\left\langle
l_{1},x_{r}\right\rangle^{2}}\right)T[\mathscr{C}(l_{1})]$ and $T_{3}$ is
immediately rewritten as

$\displaystyle
T_{3}=\sum\limits_{l_{2}\in\mathscr{C}_{t}}\sum\limits_{l_{1}\in\mathscr{C}_{b}}\frac{1}{2}\left[d,l_{1}\right]\left[c,l_{2}\right]\frac{\left\langle
l_{2},x_{r}\right\rangle\left\langle c,x_{r}\right\rangle}{\left\langle
l_{1},x_{r}\right\rangle\left\langle
d,x_{r}\right\rangle}T[\mathscr{C}(l_{1})].$ (5.21)

When summing $T_{1}$, $T_{2}$ and $T_{3}$ together, we get

$\displaystyle
T_{1}+T_{2}+T_{3}=\sum\limits_{l_{2}\in\mathscr{C}_{t}}\sum\limits_{l_{1}\in\mathscr{C}_{b}}\frac{1}{2}\left(\left[l_{1},l_{2}\right]\left[c,d\right]+\left[l_{2},d\right]\left[c,l_{1}\right]+\left[d,l_{1}\right]\left[c,l_{2}\right]\right)\frac{\left\langle
l_{2},x_{r}\right\rangle\left\langle c,x_{r}\right\rangle}{\left\langle
l_{1},x_{r}\right\rangle\left\langle
d,x_{r}\right\rangle}T[\mathscr{C}(l_{1})],$ (5.22)

which precisely vanishes due to the Schouten identity (B.5). Thus we conclude
that the graphs Fig. 15 (a), (b) and (c) cancel with each other.

Figure 17: Typical graphs involving a type-1 line for a single-trace amplitude
with the $(h_{i}^{-},h_{j}^{-})$ configuration

###  5.2 The vanishing amplitudes

Having the above properties, we are able to prove the vanishing of all the
remaining amplitudes with two negative-helicity particles.

#### Single-trace amplitudes with the $(h_{i}^{-},h_{j}^{-})$ configuration

When evaluating a single-trace amplitude with the $(h_{i}^{-},h_{j}^{-})$
configuration, we have to investigate not only _(i). graphs in the MHV sector
which do not involve the type-3 line_ but also _(ii). graphs with one type-1
line_ (i.e. graphs in the NMHV sector). This comes from our choice of gauge:
those graphs involving the type-1 line $\epsilon^{-}_{h_{j}}\cdot\epsilon^{+}$
do not vanish. We now show cancellations occur in each class of graphs.

(i). The cancellation of graphs in the MHV sector  In a graph with no type-1
component, the graviton $h_{i}$ cannot be the end node of any type-2 line that
starts at a positive-helicity graviton because $\epsilon^{+}\cdot k_{h_{i}}=0$
under our choice of gauge. Hence $h_{i}$ can only be _(1)._ a leaf or _(2)._
an end node of a type-2 line that starts at the other negative-helicity
graviton $h_{j}$. According to property-1, graphs in the former case are
further reduced to graphs with $h_{i}$ removed which have the pattern Fig. 13
(a). Graphs in the latter case have the pattern Fig. 13 (b). Therefore, all
graphs cancel in pairs due to property-2.

(ii). The cancellation of graphs with one type-1 line
$(\epsilon^{-}_{h_{j}}\cdot\epsilon^{+})$ In this case, we have two components
that are separated by a type-3 ($k\cdot k$) line (see appendix A): the
component invloving the type-1 line $(\epsilon^{-}_{h_{j}}\cdot\epsilon^{+})$
and the component involving the trace. According to our choice of gauge, the
graviton $h_{i}$ can be either a leaf (see Fig. 17 (a)) or an end node of the
type-3 line between the two components (see Fig. 17 (b) and (c)). The node
$h_{i}$ cannot be the end node of a type-2 line that starts at $h_{j}$ because
$\epsilon_{h_{j}}$ is already occupied by the type-1 line
$(\epsilon^{-}_{h_{j}}\cdot\epsilon^{+})$ in this case. Graphs Fig. 17 (a),
(b) and (c) agree with the pattern of graphs Fig. 14 (a), (b) and (c) (when
$\mathscr{C}$ in Fig. 14 includes only the graviton $h_{j}$ and
$\mathscr{C}_{2}$ denotes the component involving the type-1 line), thus they
cancel out due to property-3.

#### Double-trace amplitudes with the $(h_{i}^{-},g_{j}^{-})$ configuration

Now we turn to double-trace amplitudes with the $(h_{i}^{-},g_{j}^{-})$
configuration, all graphs for this case are graphs in the MHV sector, as shown
by Fig. 2 (b). According to our choice of gauge, $h_{i}$ cannot be the end
node of a type-2 line which starts at a positive-helicity graviton. Hence
$h_{i}$ can only be a leaf or an end node of the type-3 line. The typical
graphs can be obtained via replacing the $\mathscr{C}_{2}$ in Fig. 17 (a), (b)
and (c) by the component (which is defined by Fig. 18 (b) in appendix A)
involving the trace $\boldsymbol{2}$. Since the patterns of graphs Fig. 17
(a), (b) and (c) are preserved, they must cancel out due to property-3. Thus
we conclude that an amplitude with the $(h_{i}^{-},g_{j}^{-})$ configuration
has to vanish.

#### Double-trace amplitudes with the $(h^{-}_{i},h^{-}_{j})$ configuration

When we evaluate a double-trace amplitude with the $(h_{i}^{-},h_{j}^{-})$
configuration, a graph may or may not involve a type-1 line of the form
$(\epsilon^{-}_{h_{j}}\cdot\epsilon^{+})$. This is analogue to the single-
trace case with the $(h^{-}_{i},h^{-}_{j})$ configuration. We state that
graphs in each case cancel out.

(i). The cancellation of graphs with no type-1 line (i.e. graphs in the MHV
sector, as shown by Fig. 2 (b)) All graphs in this case can be classified
according to positions of the negative-helicity graviton $h_{j}$ (and the tree
structure attached to it). Specifically, $h_{j}$ can be an outer node or an
inner node which lives on the bridge between the two traces. The former case
is described by Fig. 14 (a) while the latter case has the pattern Fig. 14 (b)
or (c). Here, the $\mathscr{C}_{2}$ component is the component which involves
the trace $\boldsymbol{2}$ (see Fig. 18 (b)). According to property-3, all
these graphs cancel with each other.

(ii). The cancellation of graphs involving a type-1 line of the form
$\epsilon_{h_{j}}^{-}\cdot\epsilon^{+}$ (i.e. graphs in the NMHV sector)
According to the refined graphic rule in appendix A, each graph in this case
involves three components with the patterns Fig. 18 (a), (b) and (c) which
respectively contain the type-1 line $\epsilon_{h_{j}}^{-}\cdot\epsilon^{+}$,
the trace $\boldsymbol{2}$ and the trace $\boldsymbol{1}$. These graphs can be
classified according to whether the type-1 line is on the bridges between the
two traces or not. When the common kinematic factors $\epsilon\cdot k$ and
$\epsilon_{h_{j}}^{-}\cdot\epsilon^{+}$ in the component containing the type-1
line are extracted out, graphs in this case precisely match with the patterns
Fig. 15 (a), (b) and (c) (with the correct arrows), while $\mathscr{C}_{1}$
and $\mathscr{C}_{2}$ respectively denote the components containing traces
$\boldsymbol{1}$ and $\boldsymbol{2}$. Then we conclude that these graphs
cancel with each other due to property-4.

#### Amplitudes with more than two traces

For amplitudes with more than two traces and arbitrary two negative-helicity
particles, there exist at least three components which are connected together
by type-3 lines. In order to avoid a tedious discussion, we just sketch the
main pattern of the cancellation of amplitudes in this case. As stated in
Du:2019vzf , an amplitude with more than two traces can be obtained by (i).
constructing so-called the upper and lower blocks, the latter involves the
trace $\boldsymbol{1}$, (ii). attaching a substructure $\mathscr{C}$, which is
divided by the _kernel of either a type-IA or a type-IB component_ (see Fig.
18 (a), (b)), to either the upper or the lower block constructed in the
previous step, (iii). connecting the two disconnected subgraphs obtained in
the previous step via a type-3 line. In a graph constructed by (i)-(iii), the
kernel of the substructure $\mathscr{C}$ may be or may not be on the bridge
between the upper and lower blocks. Correspondingly, when the kinematic factor
of $\mathscr{C}$ is extracted out, such a graph has the pattern Fig. 15 (b),
(c) or (a), accompanied by the correct sign. Therefore, all graphs with the
same $\mathscr{C}$ and the same configuration of upper and lower blocks must
cancel out due to property-4.

##  6 Conclusions

In this paper, we evaluated all tree level EYM amplitudes with two negative-
helicity particles and an arbitrary number of traces in four dimensions, by
the use of refined graphic rule. According to the number of
$\epsilon\cdot\epsilon$ lines in the graphs, all graphs were classified into
$\text{N}^{k}\text{MHV}$ sectors. We pointed that the nonvanishing amplitudes
with $k+2$ negative-helicity particles could at most get contributions from
graphs in the $\text{N}^{k^{\prime}\,(k^{\prime}\leq k)}\text{MHV}$ sectors,
under a proper choice of gauge. For single-trace amplitudes with the
$(g^{-}_{i},g^{-}_{j})$ and the $(h^{-}_{i},g^{-}_{j})$ configurations, we
established the correspondence between the refined graphs and the spanning
forests in four dimensions. A symmetric formula of double-trace amplitudes
with the $(g^{-}_{i},g^{-}_{j})$ configuration was further provided. In this
formula, the two gluon traces are on an equal footing while all gravitons are
disentangled from the Parke-Taylor factors corresponding to gluon traces.
Other amplitudes with two negative-helicity particles were proven to vanish,
due to properties of the graphs. We now provide the following two related
topics that deserve further study to end this paper.

  * (i)

How to extend the discussions to amplitudes with more than two negative-
helicity particles? Since the refined graphic rule is independent of
dimensions, the discussion of this paper can be extended to configurations
with more negative-helicity particles. Nevertheless, the general formula of
$\text{N}^{k\,(k>2)}\text{MHV}$ amplitudes in YM depends on the position of
negative-helicity particles Dixon:2010ik . This may require more hidden
properties of the refined graphs in four dimensions.

  * (ii)

It is worth studying amplitudes with gravitons coupling to matters in other
theories. As pointed in Plefka:2018zwm ; Zhou:2020mvz , the recursive
expansion of EYM amplitudes could be generalized to other theories with
gravitons coupling to matters. Hence we expect that the calculations in the
current paper, by the use of refined graphic rule (which is based on the
recursive expansion of EYM amplitudes), can also be extended to these
theories.

## Acknowledgments

We would like to thank Linghui Hou, Jingpei Liu and Konglong Wu for helpful
discussions. This work is supported by NSFC under Grant No. 11875206, Jiangsu
Ministry of Science and Technology under contract BK20170410 as well as the
“Fundamental Research Funds for the Central Universities”.

##  Appendix A Sectors in the graphic expansion of EYM amplitudes

In this part, we present the refined graphic rule for the $\text{N}^{k}$MHV
sector in the expansion (2.3). This rule follows from a discussion parallel
with those in Du:2017gnh ; Du:2019vzf .

###  A.1 The construction of a graph $\mathcal{F}\in\mathcal{F}{[k,m]}$ in the
$\text{N}^{k}$MHV sector

To construct a graph $\mathcal{F}\in\mathcal{F}{[k,m]}$ in the
$\text{N}^{k}$MHV sector, we introduce the set of all gravitons and the gluon
traces $\boldsymbol{2},\dots,\boldsymbol{m}$,
${\boldsymbol{\mathcal{H}}}=\\{h_{1},h_{2},\dots,h_{s},\boldsymbol{2},\dots,\boldsymbol{m}\\}$.
Each element in this set is denoted by $\mathcal{H}_{i}$ $(i=1,\dots,m+s-1)$.
We further define a reference order $\mathsf{R}$ by the ordered set

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

where each gluon trace is considered as a single object. The position of an
element in the reference order $\mathsf{R}$ is called its _weight_. When all
gravitons and gluons are treated as nodes, a graph $\mathcal{F}{[k,m]}$ is
obtained by connecting lines (see Fig. 1) between these nodes in an
appropriate way. This can be achieved by the following two steps. _(i)._
Construct a _skeleton_ that does not involve any type-3 line and may contain
more than one mutually disjoint maximally connected subgraphs (which are
defined as _components_). _(ii)_. Connect these components of a skeleton via
type-3 lines such that the skeleton becomes a fully connected tree graph
$\mathcal{F}{[k,m]}$. Now we look into details of these two steps
respectively.

Figure 18: Three types of components defined in the refined graphic rule

Step-1 Constructing a skeleton To obtain a skeleton, we first connect type-4
lines between adjacent nodes in each trace. For the trace $\boldsymbol{1}$,
the nodes are arranged in the relative order $1,2,\dots,r$ and all type-4
lines in $\boldsymbol{1}$ point towards the direction of the node $1$. For the
trace $\boldsymbol{i}$ ($\boldsymbol{i}\neq\boldsymbol{1}$), the gluons
therein are arranged in a relative order

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

where
$\mathsf{KK}[\boldsymbol{i},a_{i},b_{i}]=\boldsymbol{X}\shuffle\boldsymbol{Y}^{T}$
when the trace $\boldsymbol{i}$ is supposed to have the form