# Expanding single trace YMS amplitudes with gauge invariant coefficients

Fang-Stars Wei<EMAIL_ADDRESS>Kang Zhou<EMAIL_ADDRESS>Center for Gravitation and Cosmology, College of Physical Science and
Technology, Yangzhou University, Yangzhou, 225009, China

###### Abstract

In this note, we use the new bottom up method based on soft theorems to
construct the expansion of single-trace Yang-Mills-scalar amplitudes
recursively. The resulted expansion manifests the gauge invariance for any
polarization carried by external gluons, as well as the permutation symmetry
among external gluons. Our result is equivalent to that found by Clifford
Cheung and James Mangan via the so called covariant color-kinematic duality
approach.

##  I Introduction

The investigations of $S$-matrix in the past decade revealed deep connections
among amplitudes of various different theories, which are invisible upon
inspecting traditional Feynman rules. For instance, tree level gravitational
(GR) and Yang-Mills (YM) amplitudes are related by the so called Kawai-
Lewellen-Tye (KLT) relation Kawai:1985xq , and Bern-Carrasco-Johansson (BCJ)
color-kinematic duality Bern:2008qj ; Chiodaroli:2014xia ; Johansson:2015oia ;
Johansson:2019dnu . In the well known CHY formalism Cachazo:2013gna ;
Cachazo:2013hca ; Cachazo:2013iea ; Cachazo:2014nsa ; Cachazo:2014xea , tree
amplitudes for a large variety of theories can be generated from tree GR
amplitudes through the compactifying, squeezing and the generalized
dimensional reduction procedures. Similar unifying relations were proposed by
introducing appropriate differential operators, which transmute tree GR
amplitudes to others Cheung:2017ems ; Zhou:2018wvn ; Bollmann:2018edb . At the
same time, another type of relations also caused attentions, which can be
called the expansions of amplitudes, namely, tree amplitudes of one theory can
be expanded to tree amplitudes of other theories Fu:2017uzt ; Teng:2017tbo ;
Du:2017kpo ; Du:2017gnh ; Feng:2019tvb ; Zhou:2019gtk ; Zhou:2019mbe . Indeed,
the expansions serve as the dual picture of differential operators, as
interpreted in Zhou:2019mbe . In the dual web, tree amplitudes for a wide
range of theories can be expanded to tree bi-adjoint scalar (BAS) amplitudes.

In this note, we focus on the expansion of the tree single-trace Yang-Mills-
scalar (YMS) amplitudes Stieberger:2016lng ; Schlotterer:2016cxa ;
Chiodaroli:2014xia ; Chiodaroli:2017ngp ; DelDuca:1999rs ; Nandan:2016pya ;
delaCruz:2016gnm , due to the special role of these amplitudes. Both tree YM
and BAS amplitudes can be regarded as special cases of tree single-trace YMS
amplitudes. When performing the squeezing procedure in CHY framework, or
acting differential operators, the YM amplitudes are transmuted to single-
trace YMS amplitudes, then end with pure BAS ones. These connections can be
extended to tree GR amplitudes, tree YM amplitudes, and single-trace tree EYM
amplitudes directly Cachazo:2013iea ; Cachazo:2014nsa ; Cachazo:2014xea ;
Zhou:2018wvn ; Bollmann:2018edb . Furthermore, single-trace tree YMS
amplitudes exhibit double copy structure and color-kinematic duality in an
elegant manner Cachazo:2013iea ; Cachazo:2014nsa ; Cachazo:2014xea ;
Cheung:2021zvb ; Chiodaroli:2014xia ; Chiodaroli:2017ngp . The single-trace
YMS amplitudes which describe the scattering of massless gluons and scalars,
can be expanded to YMS ones with less external gluons and more external
scalars Fu:2017uzt ; Teng:2017tbo ; Du:2017kpo ; Du:2017gnh ; Feng:2019tvb ;
Zhou:2022orv . Such recursive expansion can be applied iteratively, end with
the expansion to pure BAS amplitudes.

In the recursive expansion given in Fu:2017uzt ; Teng:2017tbo ; Feng:2019tvb ,
a fiducial external gluon is required. This special gluon breaks the manifest
permutation symmetry among external gluons: if one re-chose the fiducial
gluon, we expect that the new expansion is equivalent to the old one, but such
equivalence is hard to prove. Another disadvantage of the expansion in
Fu:2017uzt ; Teng:2017tbo ; Feng:2019tvb  is related to the gauge invariance.
In the expansion in Fu:2017uzt ; Teng:2017tbo ; Feng:2019tvb , the gauge
invariance for the polarization vector carried by the fiducial gluon is
obscure, namely, if one replace the polarization by the corresponding
momentum, the vanishing of amplitude is not manifest. suppose one use this
expansion iteratively to expand the YMS amplitude to BAS ones, in the
resulting expansion, non of polarizations has the manifest gauge invariance,
since each external gluon will play the role of the fiducial gluon once in the
iterative process. As well known, the gauge invariance plays the crucial role
in the modern S-matrix program, and always hints new understanding and new
mathematical structure for scattering amplitudes. For instance, tree
amplitudes in general relativity and Yang-Mills theory, turned out to be
completely determined by gauge invariance and singularity structures Arkani-
Hamed:2016rak ; Rodina:2016jyz . Another well known example is that the
Britto-Cachazo-Feng-Witten (BCFW) on-shell recursion relation expresses YM
amplitudes in gauge invariant formulas which include spurious poles
Britto:2004ap ; Britto:2005fq . These new formulas motivated elegant new
constructions of amplitudes, such as Grassmannia representation and
Amplituhedron Arkani-Hamed:2012zlh ; Arkani-Hamed:2013jha ; Arkani-
Hamed:2013kca . These experiences force us to seek the new expansion with the
explicit gauge invariance. After expanding to BAS amplitudes, the BAS basis
contributes only poles, thus the gauge invariance for each polarization is
completely determined by coefficients. Thus, it is natural to expect a formula
of coefficients which manifests the gauge invariance. The above consideration
leads to an important question, how to achieve the new expansion which have
manifest permutation symmetry among external gluons and the gauge invariance
for each polarization of external gluons?

Such new expansion was first found by Clifford Cheung and James Mangan in
Cheung:2021zvb , via the so called covariant color-kinematic duality method.
The approach in Cheung:2021zvb  is based on the traditional Lagrangian and
equations of motion. The very premise of the modern S-matrix program is to
bootstrap scattering dynamics without the aid of an action or equation of
motion (see for reviews in Elvang:2013cua ; Cheung:2017pzi ). Thus, it is
natural to ask whether the new expansion in Cheung:2021zvb  can be obtained
through the bottom up construction which uses only on-shell information? This
question is the main motivation for the current short note.

In this note, we reconstruct the expansion in Cheung:2021zvb , from a totally
different perspective based on universal soft behaviors of massless particles.
We first bootstrap the lowest $3$-point tree single-trace YMS amplitudes with
only one external gluon, by imposing general principles such as the
appropriate mass dimension and Lorentz invariance. Then, we invert the soft
theorem for external scalars to construct the expanded single-trace YMS
amplitudes with more external scalars, while keeping the number of external
gluons to be un-altered. After such construction, we use the well known Bern-
Carrasco-Johansson (BCJ) relation Bern:2008qj ; Chiodaroli:2014xia ;
Johansson:2015oia ; Johansson:2019dnu  to turn the resulted expansion to the
new formula which manifests the gauge invariance. Next, we invert the sub-
leading soft theorem for external gluons to generate the expanded single-trace
YMS amplitudes with more external gluons. Such procedure inserts external
gluons to the original amplitude in a manifestly gauge invariant pattern, thus
the explicit gauge invariance will be kept if the starting point is expressed
in a gauge invariant form. The manifest permutation symmetry among external
gluons are also kept at each step.

The remainder of this note is organized as follows. In section.II, we rapidly
introduce the necessary background including the expansion of tree massless
amplitudes to BAS amplitudes, the recursive expansion of single-trace YMS
amplitudes, as well as soft theorems for external scalars and gluons. In
section.III, we construct the expansion in Cheung:2021zvb  by using our
recursive method based on inverting soft theorems. Then, we end with a brief
summery in section.IV.

##  II Background

For readers’ convenience, in this section we give a brief review of necessary
background. In subsection.II.1, we introduce the tree level amplitudes of bi-
adjoint scalar (BAS) theory, as well as expansions of tree amplitudes to BAS
amplitudes. In subsection.II.2, we give the recursive expansion of Yang-Mills-
scalar (YMS) amplitudes. In subsection.II.3, we review the soft theorems for
external scalars and gluons.

###  II.1 Expanding tree level amplitudes to BAS basis

The bi-adjoint scalar (BAS) theory describes the massless bi-adjoint scalar
fields $\phi^{Aa}$ with cubic interaction, the Lagrangian is given as

$\displaystyle{\cal L}_{\rm BAS}={1\over
2}\,\partial_{\mu}\phi^{Aa}\,\partial^{\mu}\phi^{Aa}+{\lambda\over
3!}\,F^{ABC}f^{abc}\,\phi^{Aa}\phi^{Bb}\phi^{Cc}\,,$ (1)

where the structure constant $F^{ABC}$ and generator $T^{A}$ satisfy

$\displaystyle[T^{A},T^{B}]=iF^{ABC}T^{C}\,,$ (2)

and the dual algebra encoded by $f^{abc}$ and $T^{a}$ is analogous.

Tree amplitudes of this theory only contain propagators for massless scalars.
Decomposing the group factors via the standard procedure gives

$\displaystyle{\cal A}_{n}=\sum_{\sigma\in{\cal
S}_{n}}\,\sum_{\widetilde{\sigma}_{n}\in\widetilde{\cal S}_{n}}\,\mathop{\rm
Tr}[T^{A_{\sigma_{1}}},\cdots T^{A_{\sigma_{n}}}]\,\mathop{\rm
Tr}[T^{a_{\widetilde{\sigma}_{1}}}\cdots
T^{a_{\widetilde{\sigma}_{n}}}]\,{\cal
A}_{S}(\sigma_{1},\cdots,\sigma_{n}|\widetilde{\sigma}_{1},\cdots,\widetilde{\sigma}_{n})\,,$
(3)

where ${\cal A}_{n}$ denotes the kinematic part of $n$-point amplitude in
which the coupling constants are dropped, ${\cal S}_{n}$ and $\widetilde{\cal
S}_{n}$ are non-cyclic permutations of external legs. In this note, partial
amplitudes ${\cal
A}_{S}(\sigma_{1},\cdots,\sigma_{n}|\widetilde{\sigma}_{1},\cdots,\widetilde{\sigma}_{n})$
play the crucial role when considering expansions of amplitudes. Each partial
amplitude is simultaneously planar with respect to two color orderings
$(\sigma_{1},\cdots,\sigma_{n})$ and
$(\widetilde{\sigma}_{1},\cdots,\widetilde{\sigma}_{n})$. In other words, it
is double color ordered. Let us take the $5$-point amplitude ${\cal A}_{\rm
S}(1,2,3,4,5|1,4,2,3,5)$ as the example. In Figure.1, the first diagram is
allowed by both two orderings $(1,2,3,4,5)$ and $(1,4,2,3,5)$, while the
second one violates the ordering $(1,4,2,3,5)$. Other diagrams which satisfy
the ordering $(1,2,3,4,5)$ are also forbidden by the ordering $(1,4,2,3,5)$,
thus the first diagram in Figure.1 is the only candidate for the amplitude
${\cal A}_{\rm S}(1,2,3,4,5|1,4,2,3,5)$. Then the amplitude ${\cal A}_{\rm
S}(1,2,3,4,5|1,4,2,3,5)$ can be evaluated as

$\displaystyle{\cal A}_{\rm S}(1,2,3,4,5|1,4,2,3,5)={1\over s_{23}}{1\over
s_{51}}\,,$ (4)

up to an overall sign. The Mandelstam variable $s_{i\cdots j}$ is defined as

$\displaystyle s_{i\cdots j}\equiv k_{i\cdots j}^{2}\,,~{}~{}~{}~{}k_{i\cdots
j}\equiv\sum_{a=i}^{j}\,k_{a}\,,~{}~{}~{}~{}$ (5)

where $k_{a}$ is the momentum carried by the external leg $a$.

Figure 1: Two $5$-point diagrams

Each double color ordered partial BAS amplitude carries an overall $\pm$ sign,
arises from swapping two lines at the common vertex, due to the anti-symmetry
of structure constants $F^{ABC}$ and $f^{abc}$. In this note, we choose the
convention that the overall sign is $+$ if two orderings carried by the BAS
amplitude are the same. For instance, the amplitude ${\cal A}_{\rm
S}(1,2,3,4|1,2,3,4)$ carries the overall sign $+$ under the above convention.
Notice that this convention is different from that in Cachazo:2013iea . The
overall signs for other BAS amplitudes with general orderings can be
determined by counting the number of flipping Cachazo:2013iea .

The double color ordered partial amplitudes can be systematically evaluated by
applying the diagrammatical rules proposed by Cachazo, He and Yuan in
Cachazo:2013iea . We do not introduce this method in the current note, the
reader can see this interesting and useful approach in Cachazo:2013iea .

Tree level amplitudes for massless particles and cubic interactions can be
expanded to double color ordered partial BAS amplitudes, since each associated
Feynman diagram can be included in at least one partial BAS amplitude. For
higher-point vertices, one can transmute them to cubic ones by inserting the
propagator $1/D$ and the numerator $D$ simultaneously, as shown in Figure.2.
This manipulation expands any tree amplitude to tree Feynman diagrams with
only cubic vertices. Since each Feynman diagram contributes propagators which
can be provided by partial BAS amplitudes, accompanied with a numerator, one
can conclude that each tree amplitude for massless particles can be expanded
to double color ordered partial BAS amplitudes. In such expansions,
coefficients are polynomials depend on Lorentz invariants created by external
kinematical variables, without any pole.

Figure 2: Turn the $4$-point vertex to $3$-point ones. The bold line
corresponds to the inserted propagator $1/D$. This manipulation turns the
original numerator $N$ to $DN$.

The expansion requires appropriate basis. Such basis can be obtained by
employing the well known Kleiss-Kuijf (KK) relation Kleiss:1988ne

$\displaystyle{\cal A}_{\rm
S}(1,\vec{\boldsymbol{{\alpha}}},n,\vec{\boldsymbol{{\beta}}}|\sigma_{n})=(-)^{|\boldsymbol{{\beta}}|}\,{\cal
A}_{\rm
S}(1,\vec{\boldsymbol{{\alpha}}}\shuffle\vec{\boldsymbol{{\beta}}}^{T},n|\sigma_{n})\,,~{}~{}~{}$
(6)

where $\vec{\boldsymbol{{\alpha}}}$ and $\vec{\boldsymbol{{\beta}}}$ are two
ordered subsets of external scalars, and $|\boldsymbol{{\beta}}|$ stands for
the number of elements included in $\vec{\boldsymbol{{\beta}}}$. The ordered
set $\vec{\boldsymbol{{\beta}}}^{T}$ is the inversion of
$\vec{\boldsymbol{{\beta}}}$. For example,
$\vec{\boldsymbol{{\beta}}}^{T}=\\{3,2,1\\}$ if
$\vec{\boldsymbol{{\beta}}}=\\{1,2,3\\}$. The $n$-point BAS amplitude ${\cal
A}_{\rm
S}(1,\vec{\boldsymbol{{\alpha}}},n,\vec{\boldsymbol{{\beta}}}|\sigma_{n})$ at
the l.h.s of (6) carries two orderings, which are encoded by
$(1,\vec{\boldsymbol{{\alpha}}},n,\vec{\boldsymbol{{\beta}}})$ and
$\sigma_{n}$ respectively. The symbol $\shuffle$ means summing over all
possible shuffles of two ordered sets $\vec{\boldsymbol{\gamma}}_{1}$ and
$\vec{\boldsymbol{\gamma}}_{2}$, i.e., all permutations in the set
$\vec{\boldsymbol{\gamma}}_{1}\cup\vec{\boldsymbol{\gamma}}_{2}$ while
preserving the orderings of $\vec{\boldsymbol{\gamma}}_{1}$ and
$\vec{\boldsymbol{\gamma}}_{2}$. For instance, suppose
$\vec{\boldsymbol{\gamma}}_{1}=\\{1,2\\}$ and
$\vec{\boldsymbol{\gamma}}_{2}=\\{3,4\\}$, then

$\displaystyle{\cal
A}(\vec{\boldsymbol{\gamma}}_{1}\shuffle\vec{\boldsymbol{\gamma}}_{2})$
$\displaystyle=$ $\displaystyle{\cal A}(1,2,3,4)+{\cal A}(1,3,2,4)+{\cal
A}(1,3,4,2)$ (7) $\displaystyle+{\cal A}(3,1,2,4)+{\cal A}(3,1,4,2)+{\cal
A}(3,4,1,2)\,.~{}~{}~{}~{}$

The analogous KK relation holds for the ordering $\sigma_{n}$. The KK relation
among partial BAS amplitudes implies that the basis can be chosen as
amplitudes ${\cal A}_{\rm S}(1,\sigma_{1},n|1,\sigma_{2},n)$, with $1$ and $n$
are fixed at two ends in each ordering. Such basis is called the KK BAS basis.
Consequently, any tree amplitude for massless particles can be expanded to
this basis. The basis provides poles, while the coefficients contribute
numerators.

The double color ordered partial BAS amplitudes also satisfy well known Bern-
Carrasco-Johansson (BCJ) relations Bern:2008qj ; Chiodaroli:2014xia ;
Johansson:2015oia ; Johansson:2019dnu . Here we give the explicit formula of
the fundamental BCJ relation,

$\displaystyle(k_{s}\cdot Y_{s})\,{\cal A}_{\rm
S}(1,s\shuffle\\{2,\cdots,n-1\\},n|\sigma_{n+1})=0\,,~{}~{}$ (8)

since it will be used subsequently. The combinatory momentum $Y_{s}$ is
defined as the summation of external momenta carried by legs at the l.h.s of
$s$ in the ordering $(1,s\shuffle\\{2,\cdots,n-1\\},n)$, the ordering
$\sigma_{n+1}$ is defined for $n+1$ legs in $\\{1,\cdots,n\\}\cup s$. BCJ
relations imply the independence of BAS amplitudes in KK basis, thus the so
called BCJ basis can be chosen as BAS amplitudes with three fixed legs in
color orderings. However, in BCJ relations, coefficients of BAS amplitudes
depend on Mandelstam variables, this character leads to poles in coefficients
when expanding to BCJ basis. On the other hand, when expanding to KK basis,
coefficients contain no pole. In this note, we choose the KK basis since we
expect that all poles are included in basis, and coefficients only contribute
numerators.

###  II.2 Recursive expansion of single-trace YMS amplitudes

The YMS theory under consideration is the massless ${\rm YM}\oplus{\rm BAS}$
theory with Lagrangian Chiodaroli:2017ngp

$\displaystyle{\cal L}_{{\rm YM}\oplus{\rm BAS}}$ $\displaystyle=$
$\displaystyle-{1\over 4}\,F^{a}_{\mu\nu}\,F^{a\mu\nu}+{1\over
2}\,D_{\mu}\phi^{Aa}\,D^{\mu}\phi^{Aa}-{g^{2}\over
4}\,f^{abe}\,f^{ecd}\,\phi^{Aa}\phi^{Bb}\phi^{Ac}\phi^{Bd}$ (9)
$\displaystyle+{\lambda\over
3!}\,F^{ABC}f^{abc}\,\phi^{Aa}\phi^{Bb}\phi^{Cc}\,.~{}~{}$

The indices $a$, $b$, $c$ and $d$ run over the adjoint representation of the
gauge group. Scalar fields carry additional flavor indices $A$, $B$, $C$. The
field strength and covariant derivative are defined in the usual way

$\displaystyle
F^{a}_{\mu\nu}=\partial_{\mu}\,A^{a}_{\nu}-\partial_{\nu}\,A^{a}_{\mu}+g\,f^{abc}\,A^{b}_{\mu}A^{c}_{\nu}\,,$
$\displaystyle
D_{\mu}\phi^{Aa}=\partial_{\mu}\phi^{Aa}+g\,f^{abc}\,A^{b}_{\mu}\phi^{Ac}\,.$
(10)

The general tree amplitude of this theory includes both massless external
scalars and massless external gluons. Through the standard technic, one can
decompose the gauge group factors to obtain

$\displaystyle{\cal A}_{n+m}=\sum_{\sigma\in{\cal S}_{n+m}}\,{\rm
Tr}[T^{a_{\sigma_{1}}}\cdots T^{a_{\sigma_{n+m}}}]\,{\cal
A}_{n+m,1}(\sigma_{1},\cdots,\sigma_{n+m})\,,~{}~{}$ (11)

where ${s}_{n+m}$ denotes non-cyclic permutations among $n$ external scalars
and $m$ external gluons, $T^{a_{\sigma_{i}}}$ encodes the generator of the
gauge group. Here ${\cal A}_{n+m}$ is the kinematic part of the amplitude
without coupling constants. Meanwhile, decomposition of the flavor group
factors leads to the structure of tree amplitudes which is similar to that of
loop amplitudes in that, unlike pure gauge theories, it is not restricted to
have only single-trace term Chiodaroli:2014xia ; Chiodaroli:2017ngp :

$\displaystyle{\cal A}_{n+m}$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in{\cal S}_{n}}\,{\rm Tr}[T^{A_{\sigma_{1}}}\cdots
T^{A_{\sigma_{m}}}]\,{\cal A}_{n+m,1}(\sigma_{1},\cdots,\sigma_{n})$ (12)
$\displaystyle+\sum_{n_{1}+n_{2}=n}\,\sum_{{\alpha}\in{\cal
S}_{n_{1}}}\,\sum_{{\beta}\in{\cal S}_{n_{2}}}\,{\rm
Tr}[T^{A_{{\alpha}_{1}}}\cdots T^{A_{{\alpha}_{n_{1}}}}]\,{\rm
Tr}[T^{A_{{\beta}_{1}}}\cdots T^{A_{{\beta}_{n_{2}}}}]\,{\cal
A}_{n+m,2}({\alpha}_{1},\cdots,{\alpha}_{n_{1}}|{\beta}_{1},\cdots,{\beta}_{n_{2}})$
$\displaystyle+\cdots\,,~{}~{}$

where ${\cal S}_{n_{i}}$ again stands for the set of non-cyclic permutations.
The single-trace and double-trace terms are collected in the first and second
lines respectively, and $\cdots$ in the third line denotes the remaining
multi-trace terms. External scalars for the partial amplitudes ${\cal
A}_{n+m,2}({\alpha}_{1},\cdots,{\alpha}_{n_{1}}|{\beta}_{1},\cdots,{\beta}_{n_{2}})$
in the second line are grouped into two orderings which are
$({\alpha}_{1},\cdots,{\alpha}_{n_{1}})$ and
$({\beta}_{1},\cdots,{\beta}_{n_{2}})$. Analogously, partial amplitudes in
multi-trace terms in the third line include more orderings among external
scalars.

In this note, we focus on single-trace partial tree amplitudes ${\cal
A}_{n+m,1}(\sigma_{1},\cdots,\sigma_{n})$ in the first line of (12), due to
the following reasons. First, there are special connections between tree YM
amplitudes, tree BAS amplitudes, and single-trace tree YMS amplitudes
Cachazo:2013iea ; Cachazo:2014xea . These relations can be extended to tree GR
amplitudes, tree YM amplitudes, and single-trace tree EYM amplitudes, via the
well known double copy structure Cachazo:2013iea ; Cachazo:2014nsa ;
Cachazo:2014xea ; Zhou:2018wvn ; Bollmann:2018edb . Secondly, single-trace
tree YMS amplitudes play the crucial role when studying color-kinematic
duality Cachazo:2013iea ; Cachazo:2014nsa ; Cachazo:2014xea ; Cheung:2021zvb ;
Chiodaroli:2014xia ; Chiodaroli:2017ngp . By definition, single-trace YMS
amplitudes under consideration are partial amplitudes ${\cal A}_{\rm
YS}(\sigma_{n};\\{p_{i}\\}_{m}|\sigma_{n+m})$, obtained by decomposing flavor
and gauge group factors simultaneously. Here the color ordering in (11) among
all external legs is denoted as $\sigma_{n+m}$, while the flavor ordering
among external scalars, for single-trace terms in the first line in (12), is
labeled as $\sigma_{n}$. The un-ordered set of $m$ external gluons is denoted
by $\\{p_{i}\\}_{m}\equiv\\{p_{1},\cdots,p_{m}\\}$. In other words, gluons
belong to only one ordering $\sigma_{n+m}$. Notice that the single-trace
sector of YMS theory is equivalent to dropping the $\phi^{4}$ terms in the
Lagrangian (9), see in Cheung:2021zvb ; Chiodaroli:2014xia ;
Chiodaroli:2017ngp .

The discussion for expansions of tree level amplitudes in the previous
subsection.II.1 indicates that the single-trace tree YMS amplitude ${\cal
A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{m}|1,\sigma_{n+m-2},n)$ can be expanded to
KK BAS basis as

$\displaystyle{\cal A}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{m}|1,\sigma_{n+m-2},n)$ $\displaystyle=$
$\displaystyle\sum_{\widetilde{\sigma}_{n+m-2}}\,{\cal
C}(\widetilde{\sigma}_{n+m-2},\epsilon_{i},k_{j})\,{\cal A}_{\rm
S}(1,\widetilde{\sigma}_{n+m-2},n|1,\sigma_{n+m-2},n)\,,~{}~{}~{}$ (13)

where $\sigma_{n+m-2}$ and $\widetilde{\sigma}_{n+m-2}$ denote orderings among
external legs in $\\{2,\cdots,n-1\\}\cup\\{p_{i}\\}_{m}$. The double copy
structure Kawai:1985xq ; Bern:2008qj ; Chiodaroli:2014xia ; Johansson:2015oia
; Johansson:2019dnu ; Cachazo:2013iea  indicates that the coefficient ${\cal
C}(\widetilde{\sigma}_{n+m-2},\epsilon_{i},k_{j})$ depends on polarization
vectors $\epsilon_{i}$ carried by external gluons, momenta $k_{i}$ carried by
either gluons or scalars, and orderings $\widetilde{\sigma}_{n+m-2}$, but is
independent of the ordering $\sigma_{n+m-2}$111Originally, the double copy
means the GR amplitude can be factorized as ${\cal A}_{\rm G}={\cal A}_{\rm
YM}\times{\cal S}\times{\cal A}_{\rm YM}$, where the kernel ${\cal S}$ is
obtained by inverting BAS amplitudes. Our assumption that the coefficients
depend on only one color ordering is equivalent to the original version, see
in Zhou:2022orv .. The independence of the ordering $\sigma_{n+m-2}$ leads to
the more general ansatz

$\displaystyle{\cal A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{m}|\sigma_{n+m})$
$\displaystyle=$ $\displaystyle\sum_{\widetilde{\sigma}_{n+m-2}}\,{\cal
C}(\widetilde{\sigma}_{n+m-2},\epsilon_{i},k_{j})\,{\cal A}_{\rm
S}(1,\widetilde{\sigma}_{n+m-2},n|\sigma_{n+m})\,,~{}~{}~{}$ (14)

where $\sigma_{n+m}$ stands for the general ordering among all external legs,
without fixing any one at any particular position.

The expansion in (14) can be achieved by applying the following recursive
expansion iteratively Fu:2017uzt ; Teng:2017tbo ; Feng:2019tvb ,

$\displaystyle{\cal A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{m}|\sigma_{n+m})$ (15)
$\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{p}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot
Y_{\vec{\boldsymbol{{\alpha}}}}\big{)}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\\{\vec{\boldsymbol{{\alpha}}},p\\},n;\\{p_{i}\\}_{m}\setminus\\{p\cup\boldsymbol{{\alpha}}\\}|\sigma_{n+m})\,,~{}~{}$

where $p$ is the fiducial gluon which can be chosen as any element in
$\\{p_{i}\\}_{m}$, and $\boldsymbol{{\alpha}}$ are subsets of
$\\{p_{i}\\}_{m}\setminus p$ which is allowed to be empty. When
$\boldsymbol{{\alpha}}=\\{p_{i}\\}_{m}\setminus p$, the YMS amplitudes in the
second line of (15) are reduced to pure BAS ones. The ordered set
$\vec{\boldsymbol{{\alpha}}}$ is generated from $\boldsymbol{{\alpha}}$ by
endowing an order among elements in $\boldsymbol{{\alpha}}$. The tensor
$F^{\mu\nu}_{\vec{\boldsymbol{{\alpha}}}}$ is defined as

$\displaystyle
F^{\mu\nu}_{\vec{\boldsymbol{{\alpha}}}}\equiv\big{(}f_{{\alpha}_{k}}\cdot
f_{{\alpha}_{k-1}}\cdots f_{{\alpha}_{2}}\cdot
f_{{\alpha}_{1}}\big{)}^{\mu\nu}\,,$ (16)

for $\vec{\boldsymbol{{\alpha}}}=\\{{\alpha}_{1},\cdots{\alpha}_{k}\\}$, where
each anti-symmetric strength tensor $f_{i}$ is given as $f^{\mu\nu}_{i}\equiv
k^{\mu}_{i}\epsilon^{\nu}_{i}-\epsilon^{\mu}_{i}k^{\nu}_{i}$. The combinatory
momentum $Y_{\vec{\boldsymbol{{\alpha}}}}$ is the summation of momenta carried
by external scalars at the l.h.s of ${\alpha}_{1}$ in the color ordering
$(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n)$, where
${\alpha}_{1}$ is the first element in the ordered set
$\vec{\boldsymbol{{\alpha}}}$. The symbol $\shuffle$ means summing over
permutations of $\\{2,\cdots,n-1\\}\cup\\{\vec{\boldsymbol{{\alpha}}},p\\}$
which preserve the orderings of two ordered sets $\\{2,\cdots,n-1\\}$ and
$\\{\vec{\boldsymbol{{\alpha}}},p\\}$, as explained around (7). The summation
in (15) is over all un-equivalent ordered sets $\vec{\boldsymbol{{\alpha}}}$.
In the recursive expansion (15), the YMS amplitude is expanded to YMS
amplitudes with less gluons and more scalars. Repeating such expansion, one
can finally expand any YMS amplitude to pure BAS ones.

In the recursive expansion (15), the gauge invariance for each gluon in
$\\{p_{i}\\}_{m}\setminus p$ is manifest, since the tensor $f^{\mu\nu}$
vanishes automatically under the replacement $\epsilon_{i}\to k_{i}$, due to
the definition. However, the gauge invariance for the fiducial gluon $p$ has
not been manifested. When applying (15) iteratively, a fiducial gluon will be
required at every step. Consequently, in the resulted expansion to pure BAS
amplitudes, the gauge invariance for each polarization will be spoiled.
Furthermore, the manifest permutation invariance among external gluons are
also broken. Notice that the breaking of manifest permutation invariance among
external scalars can not be avoided, since the KK basis requires fixing two
legs at two special positions in orderings. However, a special external gluon
is not necessary. To obtain the expansion which manifests the gauge invariance
and permutation invariance for external gluons simultaneously, one should
employ another recursive expansion

$\displaystyle{\cal A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{m}|\sigma_{n+m})$ (17)
$\displaystyle=$ $\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,{k_{r}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot Y_{\vec{\boldsymbol{{\alpha}}}}\over
k_{r}\cdot k_{p_{1}\cdots p_{m}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m})\,,~{}~{}$

where $k_{r}$ is a reference massless momentum. Here the notations are
parallel to those for (15). The formula (17) was found by Clifford Cheung and
James Mangan, in the so called covariant color-kinematic duality framework
Cheung:2021zvb . The expansion (17) does not require any fiducial gluon, the
gauge invariance for any polarization and the permutation symmetry among
external gluons are manifested. Using (15) iteratively, one finally obtains
the new expansion of YMS amplitudes to KK BAS basis, with explicit gauge and
permutation invariance for gluons. Reconstruct the expansion (17) through a
bottom up method is the main purpose of this note.

###  II.3 Soft theorems for external scalars and gluons

In this subsection, we rapidly review the soft theorems for external scalars
and gluons, which are crucial for subsequent constructions in next section.

For the double color ordered BAS amplitude ${\cal A}_{\rm
S}(1,\cdots,n|\sigma_{n})$, we re-scale $k_{i}$ as $k_{i}\to\tau k_{i}$, and
expand the amplitude by $\tau$. The leading order contribution aries from
propagators $1/s_{1(i+1)}$ and $1/s_{(i-1)i}$ which are at the $\tau^{-1}$
order,

$\displaystyle{\cal A}^{(0)_{i}}_{\rm S}(1,\cdots,n|\sigma_{n})$
$\displaystyle=$ $\displaystyle{1\over\tau}\Big{(}{\delta_{i(i+1)}\over
s_{i(i+1)}}+{\delta_{(i-1)i}\over s_{(i-1)i}}\Big{)}\,{\cal A}_{\rm
S}(1,\cdots,i-1,\not{i},i+1,\cdots,n|\sigma_{n}\setminus i)$ (18)
$\displaystyle=$ $\displaystyle S^{(0)_{i}}_{s}\,{\cal A}_{\rm
S}(1,\cdots,i-1,\not{i},i+1,\cdots,n|\sigma_{n}\setminus i)\,,~{}~{}~{}$

where $\not{i}$ stands for removing the leg $i$, $\sigma_{n}\setminus i$ means
the color ordering generated from $\sigma_{n}$ by eliminating $i$. The
superscript $(0)_{i}$ is introduced for denoting the leading order when
considering the soft behaviour of $k_{i}$. The symbol $\delta_{ab}$ is defined
as follows222The Kronecker symbol will not appear in this paper, thus we hope
the notation $\delta_{ab}$ will not confuse the readers. Zhou:2022orv . In an
ordering, if two legs $a$ and $b$ are adjacent, then $\delta_{ab}=1$ if $a$
precedes $b$, and $\delta_{ab}=-1$ if $a$ follows $b$. If $a$ and $b$ are not
adjacent, $\delta_{ab}=0$. From the definition, it is straightforward to
observe $\delta_{ab}=-\delta_{ba}$. In (18), $\delta_{i(i+1)}$ and
$\delta_{(i-1)i}$ are defined for the ordering $\sigma_{n}$, and the leading
soft operator $S^{(0)_{i}}_{s}$ for the scalar $i$ is given as

$\displaystyle S^{(0)_{i}}_{s}={1\over\tau}\,\Big{(}{\delta_{i(i+1)}\over
s_{i(i+1)}}+{\delta_{(i-1)i}\over s_{(i-1)i}}\Big{)}\,.~{}~{}~{}~{}$ (19)

By requiring the universality of soft behavior, the above result can be
generated to the YMS amplitude as Zhou:2022orv

$\displaystyle{\cal A}^{(0)_{i}}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{m}|\sigma_{n+m})$ $\displaystyle=$ $\displaystyle
S^{(0)_{i}}_{s}\,{\cal A}_{\rm
YS}(1,\cdots,i-1,\not{i},i+1,\cdots,n;\\{p_{i}\\}_{m}|\sigma_{n+m}\setminus
i)\,,~{}~{}~{}$ (20)

where the soft factor $S^{(0)_{i}}_{s}$ is the same as in (19). In other
words, the soft operator $S^{(0)_{i}}_{s}$ does not act on any external gluon.

The soft theorems for external gluons at leading and sub-leading orders can be
obtained via various approaches Casali:2014xpa ; Schwab:2014xua . Notice that
one of these methods is to use the expanded formula of YMS amplitude in (15),
as can be seen in Zhou:2022orv . Such soft theorems are given as

$\displaystyle{\cal A}^{(0)_{p_{j}}}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{m}|\sigma_{n+m})$ $\displaystyle=$ $\displaystyle
S^{(0)_{p_{j}}}_{g}\,{\cal A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{m}\setminus
p_{j}|\sigma_{n+m}\setminus p_{j})\,,~{}~{}~{}$ (21)

and

$\displaystyle{\cal A}^{(1)_{p_{j}}}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{m}|\sigma_{n+m})$ $\displaystyle=$ $\displaystyle
S^{(1)_{p_{j}}}_{g}\,{\cal A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{m}\setminus
p_{j}|\sigma_{n+m}\setminus p_{i})\,,~{}~{}~{}$ (22)

where the external momentum $k_{p_{j}}$ is re-scaled as $k_{p_{j}}\to\tau
k_{p_{j}}$. The soft factors at leading and sub-leading orders are given by

$\displaystyle
S^{(0)_{p_{j}}}_{g}={1\over\tau}\,\sum_{a\in\\{1,\cdots,n\\}\cup\\{p_{i}\\}_{m}\setminus
p_{j}}\,{\delta_{ap_{j}}\,(\epsilon_{p_{j}}\cdot k_{a})\over
s_{ap_{j}}}\,,~{}~{}$ (23)

and

$\displaystyle
S^{(1)_{p_{j}}}_{g}=\sum_{a\in\\{1,\cdots,n\\}\cup\\{p_{i}\\}_{m}\setminus
p_{j}}\,{\delta_{ap_{j}}\,\big{(}\epsilon_{p_{j}}\cdot J_{a}\cdot
k_{p_{j}}\big{)}\over s_{ap_{j}}}\,,~{}~{}$ (24)

respectively. In (23) and (24), one should sum over all external legs $a$,
i.e., these soft operators for external gluon act on both external scalars and
gluons.

The sub-leading soft operator (24) for external gluon plays the central role
in the next section. Here we list some useful results for the action of this
operator. The angular momentum operator $J_{a}^{\mu\nu}$ acts on Lorentz
vector $k^{\rho}_{a}$ with the orbital part of the generator, and on
$\epsilon^{\rho}_{a}$ with the spin part of the generator in the vector
representation,

$\displaystyle J_{a}^{\mu\nu}\,k_{a}^{\rho}=k_{a}^{[\mu}\,{\partial
k_{a}^{\rho}\over\partial
k_{a,\nu]}}\,,~{}~{}~{}~{}J_{a}^{\mu\nu}\,\epsilon_{a}^{\rho}=\big{(}\eta^{\nu\rho}\,\delta^{\mu}_{\sigma}-\eta^{\mu\rho}\,\delta^{\nu}_{\sigma}\big{)}\,\epsilon^{\sigma}_{a}\,.$
(25)

Then the action of $S^{(1)_{p}}_{g}$ can be re-expressed as

$\displaystyle S^{(1)_{p}}_{g}$ $\displaystyle=$
$\displaystyle-\sum_{V_{a}}\,{\delta_{ap}\over s_{ap}}\,V_{a}\cdot
f_{p}\cdot{\partial\over\partial V_{a}}\,,~{}~{}$ (26)

due to the observation that the amplitude is linear in each polarization
vector. In (26), the summation over $V_{a}$ is among all Lorentz vectors
including both momenta and polarizations. The operator (26) is a differential
operator which satisfies Leibnitz’s rule. Using (26), we immediately get

$\displaystyle\big{(}S^{(1)_{p}}_{g}\,k_{a}\big{)}\cdot V=-{\delta_{ap}\over
s_{ap}}\,(k_{a}\cdot f_{p}\cdot
V)\,,~{}~{}~{}~{}\big{(}S^{(1)_{p}}_{g}\,\epsilon_{a}\big{)}\cdot
V=-{\delta_{ap}\over s_{ap}}\,(\epsilon_{a}\cdot f_{p}\cdot V)\,,~{}~{}~{}~{}$
(27)

where $V$ is an arbitrary Lorentz vector, and

$\displaystyle V_{1}\cdot\big{(}S^{(1)_{p}}_{g}\,f_{a}\big{)}\cdot
V_{2}={\delta_{ap}\over s_{ap}}\,V_{1}\cdot(f_{p}\cdot f_{a}-f_{a}\cdot
f_{p})\cdot V_{2}\,,~{}~{}~{}~{}$ (28)

for two arbitrary Lorentz vectors $V_{1}$ and $V_{2}$, where the anti-
symmetric tensor $f_{i}$ is defined as $f_{i}^{\mu\nu}\equiv
k_{i}^{\mu}\epsilon_{i}^{\nu}-\epsilon_{i}^{\mu}k_{i}^{\nu}$, as introduced
previously.

##  III Expansion of single-trace YMS amplitude

In this section, we study the expansion of single-trace tree YMS amplitudes.
Our purpose is to reconstruct the expansion in (17) through a purely bottom up
approach. For simplicity, we will chose the reference momentum $k_{r}$ in (17)
as $k_{r}=k_{n}$. We first bootstrap the $3$-point YMS amplitudes with only
one external gluon. Then, we invert the soft theorem for BAS scalars to
construct YMS amplitudes with more external scalars, while keeping the number
of external gluon to be fixed. After this construction, we use the BCJ
relation to transmute the resulted expansion of such special BAS amplitudes
with only one external gluon to a manifestly gauge invariant form. Next, we
invert the sub-leading soft theorem for external gluons, to give a recursive
pattern which leads to the general expansion of single-trace YMS amplitudes
with arbitrary number of external gluons, which is manifestly gauge invariant
for any polarization. Through the whole process, the manifest permutation
symmetry among external gluons is kept at each step.

###  III.1 YMS amplitude with one external gluon

In this subsection, we construct the single-trace YMS amplitude ${\cal A}_{\rm
YS}(1,\cdots,n;p|\sigma_{n+1})$ which contains external scalars
$i\in\\{1,\cdots,n\\}$ and only one external gluon $p$, by using the purely
bottom up method. To start, we first bootstrap $3$-point amplitudes ${\cal
A}_{\rm YS}(1,2;p|\sigma_{3})$. In $d$-dimensional space-time, any $n$-point
amplitude has the mass dimension $d-{d-2\over 2}n$, thus the $3$-point one has
the mass dimension $3-{d\over 2}$. On the other hand, the coupling constant
$g$ in Lagrangian (9) has mass dimension $2-{d\over 2}$, thus the kinematic
part ${\cal A}_{\rm YS}(1,2;p|\sigma_{3})$ has mass dimension $1$. Meanwhile,
the amplitude ${\cal A}_{\rm YS}(1,2;p|\sigma_{3})$ with one external gluon
$p$ should be linear in $\epsilon_{p}$, where $\epsilon_{p}$ is the
polarization vector carried by $p$. Finally, the $3$-point amplitude does not
include any pole since it can never be factorized into lower-point amplitudes.
The above constraints uniquely fix ${\cal A}_{\rm YS}(1,2;p|\sigma_{3})$ to be
$\epsilon_{p}\cdot k_{1}$, up to an overall sign333When talking about
$3$-point amplitudes, we allow the components of external momenta to take
complex values, otherwise the momentum conservation and on-shell conditions
can not be satisfied simultaneously. A well known example is the MHV and anti-
MHV YM amplitudes in spinor-helicity representation in $4$-d space-time..
Notice that $\epsilon_{p}\cdot k_{2}$ is equivalent to $\epsilon_{p}\cdot
k_{1}$, due to the momentum conservation and the on-shell condition
$\epsilon_{p}\cdot k_{p}=0$. Using the observation ${\cal A}_{\rm
S}(1,p,2|1,p,2)=1$, we arrive at the following expansion

$\displaystyle{\cal A}_{\rm YS}(1,2;p|\sigma_{3})=(\epsilon_{p}\cdot
k_{1})\,{\cal A}_{\rm S}(1,p,2|\sigma_{3})\,,~{}~{}$ (29)

coincides with the general ansatz in (14). Here the overall sign of ${\cal
A}_{\rm YS}(1,2;p|1,p,2)$ is chosen to be $1$, which forces the overall sign
of ${\cal A}_{\rm YS}(1,2;p|2,p,1)$ to be $-1$. Notice that $\sigma_{3}$ has
only two inequivalent choices which are $(1,p,2)$ and $(2,p,1)$, due to the
cyclic symmetry of ordering.

Then we invert the leading soft theorem for external scalars in (18) and (19),
to insert external scalars into ${\cal A}_{\rm YS}(1,2;p|2,p,1)$. Consider the
$4$-point amplitude ${\cal A}_{\rm YS}(1,2,3;p|\sigma_{4})$ with external
scalars $i\in\\{1,2,3\\}$ and external gluon $p$. The soft theorem in (18) and
(19) indicates the following leading soft behavior when taking $k_{2}\to\tau
k_{2}$,

$\displaystyle{\cal A}_{\rm YS}^{(0)_{2}}(1,2,3;p|\sigma_{4})$
$\displaystyle=$ $\displaystyle S^{(0)_{2}}_{s}\,{\cal A}_{\rm
YS}^{(0)_{2}}(1,3;p|\sigma_{4}\setminus 2)$ (30) $\displaystyle=$
$\displaystyle\Big{(}{\delta_{12}\over s_{12}}+{\delta_{23}\over
s_{23}}\Big{)}\,\Big{(}(\epsilon_{p}\cdot k_{1})\,{\cal A}_{\rm
S}(1,p,3|\sigma_{4}\setminus 2)\Big{)}$ $\displaystyle=$
$\displaystyle\Big{(}{\delta_{12}\over s_{12}}+{\delta_{2p}\over
s_{2p}}+{\delta_{p2}\over s_{p2}}+{\delta_{23}\over
s_{23}}\Big{)}\,\Big{(}(\epsilon_{p}\cdot k_{1})\,{\cal A}_{\rm
S}(1,p,3|\sigma_{4}\setminus 2)\Big{)}$ $\displaystyle=$
$\displaystyle(\epsilon_{p}\cdot k_{1})\,\Big{(}{\cal A}^{(0)_{2}}_{\rm
S}(1,2,p,3|\sigma_{4})+{\cal A}^{(0)_{2}}_{\rm
S}(1,p,2,3|\sigma_{4})\Big{)}\,,~{}~{}$

where the second equality uses the soft factor in (19) and the expansion (29)
simultaneously, the third uses the anti-symmetry of $\delta_{ab}$, while the
third uses the inversion of soft theorem in (18) and (19). The soft behavior
in (30) implies that ${\cal A}_{\rm YS}(1,2,3;p|\sigma_{4})$ can be expanded
as

$\displaystyle{\cal A}_{\rm YS}(1,2,3;p|\sigma_{4})$ $\displaystyle=$
$\displaystyle C(1,2,p,3)\,{\cal A}_{\rm S}(1,2,p,3|\sigma_{4})$ (31)
$\displaystyle+C(1,p,2,3)\,{\cal A}_{\rm S}(1,p,2,3|\sigma_{4})\,,$

where coefficients $C(1,2,p,3)$ and $C(1,p,2,3)$ satisfy

$\displaystyle C^{(0)_{2}}(1,2,p,3)=C^{(0)_{2}}(1,p,2,3)=\epsilon_{p}\cdot
k_{1}\,.~{}~{}$ (32)

The constraint (32) requires $C(1,2,p,3)$ and $C(1,p,2,3)$ to have the form

$\displaystyle C(1,2,p,3)=\epsilon_{p}\cdot
k_{1}+{\alpha}_{1}\,\epsilon_{p}\cdot
k_{2}\,,~{}~{}~{}~{}C(1,p,2,3)=\epsilon_{p}\cdot
k_{1}+{\alpha}_{2}\,\epsilon_{p}\cdot k_{2}\,.$ (33)

To determine ${\alpha}_{1}$ and ${\alpha}_{2}$, one can consider the leading
soft behavior of $k_{1}\to\tau k_{1}$ to obtain

$\displaystyle{\cal A}^{(0)_{1}}_{\rm YS}(1,2,3;p|\sigma_{4})$
$\displaystyle=$ $\displaystyle S^{(0)_{1}}_{s}\,{\cal A}_{\rm
YS}(2,3;p|\sigma_{4}\setminus 1)$ (34) $\displaystyle=$
$\displaystyle\Big{(}{\delta_{31}\over s_{31}}+{\delta_{12}\over
s_{12}}\Big{)}\,\Big{(}(\epsilon_{p}\cdot k_{2})\,{\cal A}_{\rm
S}(2,p,3|\sigma_{4}\setminus 1)\Big{)}$ $\displaystyle=$
$\displaystyle-\Big{(}{\delta_{21}\over s_{21}}+{\delta_{1p}\over
s_{1p}}+{\delta_{p1}\over s_{1p}}+{\delta_{13}\over
s_{13}}\Big{)}\,\Big{(}(\epsilon_{p}\cdot k_{2})\,{\cal A}_{\rm
S}(2,p,3|\sigma_{4}\setminus 1)\Big{)}$ $\displaystyle=$
$\displaystyle-(\epsilon_{p}\cdot k_{2})\,\Big{(}{\cal A}^{(0)_{1}}_{\rm
S}(2,1,p,3|\sigma_{4}\setminus 1)+{\cal A}^{(0)_{1}}_{\rm
S}(2,p,1,3|\sigma_{4}\setminus 1)\Big{)}$ $\displaystyle=$
$\displaystyle(\epsilon_{p}\cdot k_{2})\,{\cal A}^{(0)_{1}}_{\rm
S}(1,2,p,3|\sigma_{4}\setminus 1)\,,~{}~{}$

where the last equality uses the fundamental BCJ relation in (8). The soft
behavior in (34) indicates

$\displaystyle C^{(0)_{1}}(1,2,p,3)=\epsilon_{p}\cdot
k_{2}\,,~{}~{}~{}~{}C^{(0)_{1}}(1,p,2,3)=0\,.~{}~{}$ (35)

Comparing (35) with (32), we find ${\alpha}_{1}=1$, ${\alpha}_{2}=0$, thus the
full expansion of ${\cal A}_{\rm YS}(1,2,3;p|\sigma_{4})$ is given as

$\displaystyle{\cal A}_{\rm YS}(1,2,3;p|\sigma_{4})$ $\displaystyle=$
$\displaystyle(\epsilon_{p}\cdot k_{12})\,{\cal A}_{\rm
S}(1,2,p,3|\sigma_{4})+(\epsilon_{p}\cdot k_{1})\,{\cal A}_{\rm
S}(1,p,2,3|\sigma_{4})$ (36) $\displaystyle=$ $\displaystyle(\epsilon_{p}\cdot
Y_{p})\,{\cal A}_{\rm S}(1,2\shuffle p,3|\sigma_{4})\,,$

where the symbol $\shuffle$ is defined around (7), and the combinatory
momentum $Y_{p}$ is defined as the summation over momenta carried by scalars
at the l.h.s of $p$ in the color ordering $(1,2\shuffle p,3)$.

Repeating the above manipulation, one can determine the general YMS amplitude
${\cal A}_{\rm YS}(1,\cdots,n;p|\sigma_{n+1})$ with arbitrary number of
external scalars and one external gluon $p$, in the expanded formula

$\displaystyle{\cal A}_{\rm YS}(1,\cdots,n;p|\sigma_{n+1})=(\epsilon_{p}\cdot
Y_{p})\,{\cal A}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle
p,n|\sigma_{n+1})\,,~{}~{}~{}~{}$ (37)

where the combinatory momentum $Y_{p}$ was defined in analogously. In the
above expansion, it is direct to observe the gauge invariance for the
polarization $\epsilon_{p}$, since the replacement $\epsilon_{p}^{\mu}\to
k_{p}^{\mu}$ yields the BCJ relation (8),

$\displaystyle 0=(k_{p}\cdot Y_{p})\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle p,n|\sigma_{n+1})\,.$ (38)

However, if one use the formula (37) and the soft theorem for external gluons
to construct YMS amplitudes with more external gluons, the resulting expansion
is in the formula (15) Zhou:2022orv , and the gauge invariance for the
polarization of the fiducial gluon is obscure. The proof of gauge invariance
for such fiducial polarization requires the application of BCJ relations in a
complicated way, and the complexity increases rapidly as the number of
external gluons increases. To avoid this disadvantage, we want to rewrite (37)
so that the manifest gauge invariance is carried by coefficients. To realize
the goal, one can use the BCJ relation to modify (37) as

$\displaystyle{\cal A}_{\rm YS}(1,\cdots,n;p|\sigma_{n+1})$ $\displaystyle=$
$\displaystyle(\epsilon_{p}\cdot Y_{p})\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle p,n|\sigma_{n+1})$ (39)
$\displaystyle-{k_{n}\cdot\epsilon_{p}\over k_{n}\cdot k_{p}}\,(k_{p}\cdot
Y_{p})\,{\cal A}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle p,n|\sigma_{n+1})$
$\displaystyle=$ $\displaystyle{k_{n}\cdot f_{p}\cdot Y_{p}\over k_{n}\cdot
k_{p}}\,{\cal A}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle
p,n|\sigma_{n+1})\,.~{}~{}~{}~{}$

In the last line at the r.h.s of (39), the coefficients vanish under the
replacement $\epsilon_{p}^{\mu}\to k_{p}^{\mu}$, due to the definition
$f^{\mu\nu}_{p}\equiv
k^{\mu}_{p}\epsilon^{\nu}_{p}-\epsilon^{\mu}_{p}k^{\nu}_{p}$. This is the
desired new expansion for the YMS with one external gluon, trivialized the
gauge invariance of polarization $\epsilon^{\mu}_{p}$, with the cost that a
spurious pole $k_{n}\cdot k_{p}$ is introduced. As will be seen, by applying
the recursive technic based on the soft theorem for external gluons, this new
expansion leads to general expansion of YMS amplitudes with arbitrary number
of external gluons, which manifests the gauge invariance for each
polarization.

###  III.2 YMS amplitude with two external gluons

Now we turn to the single-trace YMS amplitude ${\cal A}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{2}|\sigma_{n+2})$ with two external gluons labeled
as $p_{1}$ and $p_{2}$. Let us consider the soft behavior of gluon $p_{2}$,
i.e., we re-scale $k_{p_{2}}$ as $k_{p_{2}}\to\tau k_{p_{2}}$, and expand
${\cal A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{2}|\sigma_{n+2})$ in $\tau$. The
soft theorem (22) indicates that the contribution at sub-leading order should
be

$\displaystyle{\cal A}^{(1)_{p_{2}}}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{2}|\sigma_{n+2})$ $\displaystyle=$ $\displaystyle
S^{(1)_{p_{2}}}_{g}\,{\cal A}_{\rm YS}(1,\cdots,n;p_{1}|\sigma_{n+2}\setminus
p_{2})$ (40) $\displaystyle=$ $\displaystyle
S^{(1)_{p_{2}}}_{g}\,\Big{[}{k_{n}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over
k_{n}\cdot k_{p_{1}}}\,{\cal A}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n|\sigma_{n+2}\setminus p_{2})\Big{]}$ $\displaystyle=$ $\displaystyle
B_{1}+B_{2}+B_{3}\,,~{}~{}~{}$

where

$\displaystyle B_{1}={k_{n}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot
k_{p_{1}}}\,{\cal A}^{(1)_{p_{2}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;p_{2}|\sigma_{n+2})\,,~{}~{}$ (41) $\displaystyle B_{2}$
$\displaystyle=$ $\displaystyle\tau\,{k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot
Y_{p_{1}}\over k_{n}\cdot k_{p_{1}}}\,{\cal A}^{(0)_{p_{2}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+2})$ (42)
$\displaystyle+\tau\,{k_{n}\cdot f_{p_{1}}\cdot f_{p_{2}}\cdot Y_{p_{2}}\over
k_{n}\cdot k_{p_{1}}}\,{\cal A}^{(0)_{p_{2}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{2},p_{1}\\},n|\sigma_{n+2})\,,~{}~{}$

and

$\displaystyle B_{3}=-\tau\,{(k_{n}\cdot f_{p_{1}}\cdot
Y_{p_{1}})\,(k_{n}\cdot f_{p_{2}}\cdot k_{p_{1}})\over(k_{n}\cdot
k_{p_{1}})^{2}}\,{\cal A}^{(0)_{p_{2}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+2})\,.~{}~{}~{}$
(43)

The second equality of (40) is obtained by substituting the expansion (39).
Three parts $B_{1}$, $B_{2}$ and $B_{3}$ arise from acting
$S^{(1)_{p_{2}}}_{g}$ on ${\cal A}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n|\sigma_{n+2}\setminus p_{2})$, $k_{n}\cdot f_{p_{1}}\cdot Y_{p_{1}}$
and $k_{n}\cdot k_{p_{1}}$ respectively, due to Leibnitz’s rule. The
expression for the first part $B_{1}$ in (41) is obtained via the soft theorem

$\displaystyle S^{(1)_{p_{2}}}_{g}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle p_{1},n|\sigma_{n+2}\setminus p_{2})={\cal
A}^{(1)_{p_{2}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;p_{2}|\sigma_{n+2})\,.$ (44)

The second part $B_{2}$ can be calculated as follows. Using relations (27) and
(28), we have

$\displaystyle S^{(1)_{p_{2}}}_{g}\,(k_{n}\cdot f_{p_{1}}\cdot
Y_{p_{1}})=H_{1}+H_{2}\,,$ (45)

where

$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle-{\delta_{np_{2}}\,(k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot
Y_{p_{1}})\over s_{np_{2}}}+{\delta_{p_{1}p_{2}}\,(k_{n}\cdot f_{p_{2}}\cdot
f_{p_{1}}\cdot X_{p_{1}})\over s_{p_{1}p_{2}}}$ $\displaystyle H_{2}$
$\displaystyle=$ $\displaystyle-{\delta_{p_{1}p_{2}}\,(k_{n}\cdot
f_{p_{1}}\cdot f_{p_{2}}\cdot Y_{p_{1}})\over
s_{p_{1}p_{2}}}+\sum_{i=1}^{j}\,{\delta_{ip_{2}}\,(k_{n}\cdot f_{p_{1}}\cdot
f_{p_{2}}\cdot k_{i})\over s_{ip_{2}}}\,,~{}~{}~{}~{}$ (46)

with $Y_{p_{1}}=\sum_{i=1}^{j}\,k_{i}$. We use $\delta_{ab}=-\delta_{ba}$ to
reorganize $H_{1}$ as

$\displaystyle H_{1}=\Big{[}\Big{(}{\delta_{p_{1}p_{2}}\over
s_{p_{1}p_{2}}}+{\delta_{p_{2}(j+1)}\over
s_{p_{2}(j+1)}}\Big{)}+\sum_{k=j+1}^{n-1}\,\Big{(}{\delta_{kp_{2}}\over
s_{kp_{2}}}+{\delta_{p_{2}(k+1)}\over
s_{p_{2}(k+1)}}\Big{)}\Big{]}\,(k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot
Y_{p_{1}})\,.~{}~{}~{}~{}$ (47)

The reason for expressing $H_{1}$ in the above manner is that we want to
interpret $H_{1}\,{\cal A}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n|\sigma_{n+2}\setminus p_{2})$ as coefficients times leading or sub-
leading terms of amplitudes, as can be seen in (42). Combining (47) and the
leading soft factor (19) for the scalar, we find

$\displaystyle H_{1}\,{\cal A}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n|\sigma_{n+2}\setminus p_{2})$ (48) $\displaystyle=$
$\displaystyle\sum_{j=1}^{n-1}\,H_{1}\,{\cal A}_{\rm
S}(1,\cdots,j,p_{1},j+1,\cdots,n|\sigma_{n+2}\setminus p_{2})$
$\displaystyle=$ $\displaystyle\tau\,(k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot
Y_{p_{1}})\,{\cal A}^{(0)_{p_{2}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+2})\,.~{}~{}$

We emphasize that ${\cal A}^{(0)_{p_{2}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+2})$ is the
leading order contribution of ${\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+2})$ which is at
the $\tau^{-1}$ order, while $f_{p_{2}}$ in the coefficients is accompanied
with $\tau$. Combining them together leads to the $\tau^{0}$ order
contribution which satisfies the order of ${\cal A}^{(1)_{p_{2}}}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{2}|\sigma_{n+2})$. Based on the reason similar as
that for rewriting $H_{1}$, we reorganize $H_{2}$ as

$\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{j}\,\Big{(}{\delta_{ip_{2}}\over
s_{ip_{2}}}-{\delta_{p_{1}p_{2}}\over s_{p_{1}p_{2}}}\Big{)}\,(k_{n}\cdot
f_{p_{1}}\cdot f_{p_{2}}\cdot k_{i})$ (49) $\displaystyle=$
$\displaystyle\sum_{i=1}^{j}\,\Big{[}\sum_{l=i}^{j-1}\,\Big{(}{\delta_{lp_{2}}\over
s_{lp_{2}}}+{\delta_{p_{2}(l+1)}\over
s_{p_{2}(l+1)}}\Big{)}+{\delta_{jp_{2}}\over
s_{jp_{2}}}+{\delta_{p_{2}p_{1}}\over s_{p_{2}p_{1}}}\Big{]}\,(k_{n}\cdot
f_{p_{1}}\cdot f_{p_{2}}\cdot k_{i})$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{j}\,\Big{(}{\delta_{lp_{2}}\over
s_{lp_{2}}}+{\delta_{p_{2}(l+1)}\over
s_{p_{2}(l+1)}}\Big{)}\,\Big{[}k_{n}\cdot f_{p_{1}}\cdot
f_{p_{2}}\cdot\Big{(}\sum_{i=1}^{l}\,k_{i}\Big{)}\Big{]}\,,$

where $\delta_{q(j+1)}$ and $s_{q(j+1)}$ in the last line should be understood
as $j+1=p_{1}$. Thus,

$\displaystyle H_{2}\,{\cal A}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n|\sigma_{n+2}\setminus p_{2})$ (50) $\displaystyle=$
$\displaystyle\sum_{j=1}^{n-1}\,\sum_{l=1}^{j}\,\Big{(}{\delta_{lp_{2}}\over
s_{lp_{2}}}+{\delta_{p_{2}(l+1)}\over
s_{p_{2}(l+1)}}\Big{)}\,\Big{[}k_{n}\cdot f_{p_{1}}\cdot
f_{p_{2}}\cdot\Big{(}\sum_{i=1}^{l}\,k_{i}\Big{)}\Big{]}\,{\cal A}_{\rm
S}(1,\cdots,j,p_{1},j+1,\cdots,n|\sigma_{n+2}\setminus p_{2})$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n-1}\,\sum_{l=1}^{j}\,\tau\,\Big{[}k_{n}\cdot
f_{p_{1}}\cdot f_{p_{2}}\cdot\Big{(}\sum_{i=1}^{l}\,k_{i}\Big{)}\Big{]}\,{\cal
A}^{(0)_{p_{2}}}_{\rm
S}(1,\cdots,l,p_{2},l+1,\cdots,j,p_{1},j+1,\cdots,n|\sigma_{n+2})$
$\displaystyle=$ $\displaystyle\tau\,(k_{n}\cdot f_{p_{1}}\cdot f_{p_{2}}\cdot
Y_{p_{2}})\,{\cal A}^{(0)_{p_{2}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{2},p_{1}\\},n|\sigma_{n+2})\,.~{}~{}$

Putting (48) and (50) together yields the expression of $B_{2}$ in (42). The
third part $B_{3}$ can be calculated by using the relation (27),

$\displaystyle S^{(1)_{p_{2}}}_{g}\,{1\over k_{n}\cdot k_{p_{1}}}$
$\displaystyle=$ $\displaystyle-{k_{n}\cdot f_{p_{2}}\cdot
k_{p_{1}}\over(k_{n}\cdot k_{p_{1}})^{2}}\,\Big{(}{\delta_{p_{1}p_{2}}\over
s_{p_{1}p_{2}}}-{\delta_{np_{2}}\over s_{np_{2}}}\Big{)}$ (51)
$\displaystyle=$ $\displaystyle-{k_{n}\cdot f_{p_{2}}\cdot
k_{p_{1}}\over(k_{n}\cdot k_{p_{1}})^{2}}\,\Big{[}{\delta_{p_{1}p_{2}}\over
s_{p_{1}p_{2}}}+{\delta_{p_{2}a}\over
s_{p_{2}a}}+\sum_{k=a}^{n-1}\,\Big{(}{\delta_{kp_{2}}\over
s_{kp_{2}}}+{\delta_{p_{2}(k+1)}\over s_{p_{2}(k+1)}}\Big{)}\Big{]}\,,$

which indicates

$\displaystyle\sum_{i=1}^{n-1}\,\Big{[}S^{(1)_{p_{2}}}_{g}\,{1\over k_{n}\cdot
k_{p_{1}}}\Big{]}\,{\cal A}_{\rm
S}(1,\cdots,i,p_{1},i+1,\cdots,n|\sigma_{n+2}\setminus p_{2})$ (52)
$\displaystyle=$ $\displaystyle-\tau\,{k_{n}\cdot f_{p_{2}}\cdot
k_{p_{1}}\over(k_{n}\cdot k_{p_{1}})^{2}}\,{\cal A}^{(0)_{p_{2}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+2})\,,$

therefore we get the expression of $B_{3}$ in (43).

We will use $B_{1}$, $B_{2}$ and $B_{3}$ to reconstruct the complete expansion
of ${\cal A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{2}|\sigma_{n+2})$.
Mathematically, it is impossible to restore the full amplitude from the sub-
leading contribution, thus further physical constrains are needed. The most
important one is the symmetry under the permutation $p_{1}\leftrightarrow
p_{2}$. The physical amplitude ${\cal A}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{2}|\sigma_{n+2})$ should be invariant under such
re-labeling. In the expansion (15), since a fiducial gluon $p_{1}$ is
required, such symmetry is not manifest. However, one can use the BCJ
relations to prove that the resulting expansions obtained by choosing
different fiducial gluons are equivalent to each other. On the other hand, the
purpose of the current work is to seek the new expansion which manifests the
gauge invariance for all polarizations, which implies the democracy among all
external gluons, thus it is natural to expect the manifested symmetry under
$p_{1}\leftrightarrow p_{2}$.

With the additional condition of symmetry $p_{1}\leftrightarrow p_{2}$, let us
try to find the expansion of the amplitude ${\cal A}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{2}|\sigma_{n+2})$. It is straightforward to
recognize that $B_{2}$ arises as the leading term of

$\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle{k_{n}\cdot
f_{p_{2}}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal
A}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+2})$ (53)
$\displaystyle+{k_{n}\cdot f_{p_{1}}\cdot f_{p_{2}}\cdot Y_{p_{2}}\over
k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{2},p_{1}\\},n|\sigma_{n+2})\,,~{}~{}~{}~{}$

which is at the $\tau^{0}$ order. The reason for turning $k_{n}\cdot
k_{p_{1}}$ in the denominators to $k_{n}\cdot k_{p_{1}p_{2}}$ is to keep the
symmetry $p_{1}\leftrightarrow p_{2}$. There is another way to keep such
symmetry, which is adding terms with $k_{n}\cdot k_{p_{2}}$ in the
denominator. However, ${\cal A}^{(1)_{p_{2}}}_{\rm
YS}(1,\cdots,n;p_{1},p_{2}|\sigma_{n+2})$ shall receive contributions from
such terms, for example the coefficients carry $1/(\tau k_{n}\cdot k_{p_{2}})$
under the re-scaling $k_{p_{2}}\to\tau k_{p_{2}}$, and the amplitudes
contribute $\tau^{1}$, which combine to terms at the $\tau^{0}$ order. Since
such contributions do not exist in ${\cal A}^{(1)_{p_{2}}}_{\rm
YS}(1,\cdots,n;p_{1},p_{2}|\sigma_{n+2})$ in (40), the $1/k_{n}\cdot
k_{p_{2}}$ terms are excluded. Then the symmetry $p_{1}\leftrightarrow p_{2}$
also excludes the $1/k_{n}\cdot k_{p_{1}}$ terms, and the remaining choice is
only $1/k_{n}\cdot k_{p_{1}p_{2}}$.

One can also observe that $B_{1}$ comes from the sub-leading term of

$\displaystyle P_{2}={k_{n}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot
k_{p_{1}p_{2}}}\,{\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;p_{2}|\sigma_{n+2})\,,~{}~{}$ (54)

which is also at the $\tau^{0}$ order. Again, the denominator is set to be
$k_{n}\cdot k_{p_{1}p_{2}}$ due to the symmetry. However, the sub-leading
contribution of $P_{2}$ receives additional contribution from expanding the
denominator in $\tau$. Further more, the symmetry among two gluons requires
the existence of

$\displaystyle P_{3}={k_{n}\cdot f_{p_{2}}\cdot Y_{p_{2}}\over k_{n}\cdot
k_{p_{1}p_{2}}}\,{\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{2},n;p_{1}|\sigma_{n+2})\,,~{}~{}$ (55)

which serves as the counter part of $P_{2}$ under the permutation
$p_{1}\leftrightarrow p_{2}$. These missing terms will be figured out by
considering $B_{3}$. The expression of $B_{3}$ in (43) also breaks the
symmetry among $p_{1}$ and $p_{2}$. Based on such symmetry, the appearance of
${\cal A}^{(0)_{p_{2}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+2})$ in (43)
implies the existence of ${\cal A}^{(0)_{p_{2}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{2},p_{1}\\},n|\sigma_{n+2})$. To realize
this, we observe that

$\displaystyle-(k_{n}\cdot f_{p_{2}}\cdot k_{p_{1}})\,{\cal
A}^{(0)_{p_{2}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+2})$ (56)
$\displaystyle=$ $\displaystyle\Big{(}k_{n}\cdot f_{p_{2}}\cdot
Y_{p_{2}}-k_{n}\cdot f_{p_{2}}\cdot X_{p_{2}}\Big{)}\,{\cal
A}^{(0)_{p_{2}}}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle
p_{2},n|\sigma_{n+2})\,,~{}~{}$

where the combinatory momentum $X_{a}$ is defined as the summation of momenta
carried by external legs at the l.h.s of $a$ in the color ordering, without
distinguishing scalars and gluons (recall that $Y_{a}$ only includes momenta
carried by external scalars). Then, $B_{3}$ can be identified as

$\displaystyle B_{3}=B_{4}+B_{5}\,,$ (57)

where

$\displaystyle B_{4}=\tau\,{k_{n}\cdot f_{p_{2}}\cdot Y_{p_{2}}\over
k_{n}\cdot k_{p_{1}}}\,{\cal A}^{(0)_{p_{2}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{2},n;p_{1}|\sigma_{n+2})\,,~{}~{}$ (58)

and

$\displaystyle B_{5}=-\tau\,{(k_{n}\cdot k_{p_{2}})\,(k_{n}\cdot
f_{p_{1}}\cdot Y_{p_{1}})\over(k_{n}\cdot k_{p_{1}})^{2}}\,{\cal
A}^{(0)_{p_{2}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;p_{2}|\sigma_{n+2})\,.~{}~{}$ (59)

Here we have used expansions

$\displaystyle{\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{2},n;p_{1}|\sigma_{n+2})$ $\displaystyle=$ $\displaystyle{k_{n}\cdot
f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot k_{p_{1}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle p_{2},n|\sigma_{n+2})\,,$
$\displaystyle{\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;p_{2}|\sigma_{n+2})$ $\displaystyle=$ $\displaystyle{k_{n}\cdot
f_{p_{2}}\cdot Y_{p_{2}}\over k_{n}\cdot k_{p_{2}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle p_{2},n|\sigma_{n+2})\,.~{}~{}$
(60)

Notice that $Y_{p_{2}}$ in (60) equals to $Y_{p_{2}}$ in (56) since in the
second line of (60) the leg $p_{1}$ is a scalar. The combinatory momentum
$Y_{p_{1}}$ equals to $Y_{p_{1}}$ in (43) and (60) at the leading order, since
$k_{p_{2}}$ is accompanied with $\tau$ thus is dropped into the sub-leading
term.

Now one can observe that $B_{4}$ is the leading term of $P_{3}$, and $B_{5}$
arises from the term at $\tau^{1}$ order in the expansion of the denominator
of $P_{2}$. Thus, we conclude that

$\displaystyle{\cal A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{2}|\sigma_{n+2})$
$\displaystyle=$ $\displaystyle P_{1}+P_{2}+P_{3}$ (61) $\displaystyle=$
$\displaystyle{k_{n}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot
k_{p_{1}p_{2}}}\,{\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;p_{2}|\sigma_{n+2})$ $\displaystyle+{k_{n}\cdot f_{p_{2}}\cdot
Y_{p_{2}}\over k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{2},n;p_{1}|\sigma_{n+2})$
$\displaystyle+{k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over
k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+2})$
$\displaystyle+{k_{n}\cdot f_{p_{1}}\cdot f_{p_{2}}\cdot Y_{p_{2}}\over
k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{2},p_{1}\\},n|\sigma_{n+2})\,.~{}~{}$

One can verify that taking $k_{p_{1}}\to\tau k_{p_{1}}$ and expanding (61) in
$\tau$ give the correct behavior at sub-leading order, namely,

$\displaystyle{\cal A}^{(1)_{p_{1}}}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{2}|\sigma_{n+2})=S^{(1)_{p_{1}}}_{g}\,{\cal
A}_{\rm YS}(1,\cdots,n;p_{2}|\sigma_{n+2}\setminus p_{1})\,.$ (62)

Notice that in the formula (61), both the symmetry among two gluons and the
gauge invariance for each polarization are manifested.

###  III.3 YMS amplitude with three external gluons

The next example is the single-trace YMS amplitude ${\cal A}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{3}|\sigma_{n+3})$ which includes three external
gluons $p_{1}$, $p_{2}$ and $p_{3}$. The treatment is parallel to that in the
previous subsection for the $2$-gluon case, and such similarity exhibits the
recursive pattern. As in the previous case, we consider $k_{p_{3}}\to\tau
k_{p_{3}}$, and expand ${\cal A}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{3}|\sigma_{n+3})$ in $\tau$. The soft theorem
requires the sub-leading term to be

$\displaystyle{\cal A}^{(1)_{p_{3}}}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{3}|\sigma_{n+3})$ (63) $\displaystyle=$
$\displaystyle S^{(1)_{p_{3}}}_{g}\,{\cal A}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{2}|\sigma_{n+3}\setminus p_{3})$ $\displaystyle=$
$\displaystyle S^{(1)_{p_{3}}}_{g}\,\Big{[}{k_{n}\cdot f_{p_{1}}\cdot
Y_{p_{1}}\over k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1},n;p_{2}|\sigma_{n+3}\setminus
p_{3})+p_{1}\leftrightarrow p_{2}$ $\displaystyle+{k_{n}\cdot f_{p_{2}}\cdot
f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+3}\setminus
p_{3})+p_{1}\leftrightarrow p_{2}\Big{]}\,,$

where the expanded formula (61) for YMS amplitude with two external gluons is
used. Applying the Leibnitz’s rule, we get

$\displaystyle{\cal A}^{(1)_{p_{3}}}_{\rm
YS}(1,\cdots,n;p_{1},p_{2},p_{3}|\sigma_{n+3})=B_{1}+B_{2}+B_{3}+B_{4}+B_{5}+B_{6}\,,$
(64)

where

$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle{k_{n}\cdot
f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot
k_{p_{1}p_{2}}}\,\Big{[}S^{(1)_{p_{3}}}_{g}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1},n;p_{2}|\sigma_{n+3}\setminus
p_{3})\Big{]}+p_{1}\leftrightarrow p_{2}$ (65) $\displaystyle=$
$\displaystyle{k_{n}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot
k_{p_{1}p_{2}}}\,{\cal A}^{(1)_{p_{3}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;\\{p_{i}\\}_{3}\setminus p_{1}|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}\,,~{}~{}$ $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle{k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over
k_{n}\cdot k_{p_{1}p_{2}}}\,\Big{[}S^{(1)_{p_{3}}}_{g}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+3}\setminus
p_{3})\Big{]}+p_{1}\leftrightarrow p_{2}$ (66) $\displaystyle=$
$\displaystyle{k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over
k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}^{(1)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n;p_{3}|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}\,,~{}~{}$ $\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle{\Big{[}S^{(1)_{p_{3}}}_{g}\,k_{n}\cdot f_{p_{1}}\cdot
Y_{p_{1}}\Big{]}\over k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1},n;p_{2}|\sigma_{n+3}\setminus
p_{3})+p_{1}\leftrightarrow p_{2}$ (67) $\displaystyle=$
$\displaystyle\tau\,{k_{n}\cdot f_{p_{3}}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over
k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}^{(0)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{3}\\},n;p_{2}|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}$ $\displaystyle+\tau\,{k_{n}\cdot f_{p_{1}}\cdot f_{p_{3}}\cdot
Y_{p_{3}}\over k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}^{(0)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{3},p_{1}\\},n;p_{2}|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}\,,~{}~{}$ $\displaystyle B_{4}$ $\displaystyle=$
$\displaystyle{\Big{[}S^{(1)_{p_{3}}}_{g}\,k_{n}\cdot f_{p_{2}}\cdot
f_{p_{1}}\cdot Y_{p_{1}}\Big{]}\over k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+3}\setminus
p_{3})+p_{1}\leftrightarrow p_{2}$ (68) $\displaystyle=$
$\displaystyle\tau\,{k_{n}\cdot f_{p_{3}}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot
Y_{p_{1}}\over k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal A}^{(0)_{p_{3}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2},p_{3}\\},n|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}$ $\displaystyle+\tau\,{k_{n}\cdot f_{p_{2}}\cdot f_{p_{3}}\cdot
f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal
A}^{(0)_{p_{3}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{3},p_{2}\\},n|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}$ $\displaystyle+\tau\,{k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot
f_{p_{3}}\cdot Y_{p_{3}}\over k_{n}\cdot k_{p_{1}p_{2}}}\,{\cal
A}^{(0)_{p_{3}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{3},p_{1},p_{2}\\},n|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}\,,~{}~{}$ $\displaystyle B_{5}$ $\displaystyle=$
$\displaystyle\Big{[}S^{(1)_{p_{3}}}_{g}\,{1\over k_{n}\cdot
k_{p_{1}p_{2}}}\Big{]}\,(k_{n}\cdot f_{p_{1}}\cdot Y_{p_{1}})\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1},n;p_{2}|\sigma_{n+3}\setminus
p_{3})+p_{1}\leftrightarrow p_{2}$ (69) $\displaystyle=$
$\displaystyle-\tau\,{(k_{n}\cdot f_{p_{1}}\cdot Y_{p_{1}})\,(k_{n}\cdot
f_{p_{3}}\cdot K_{p_{3}})\over\big{(}k_{n}\cdot
k_{p_{1}p_{2}}\big{)}^{2}}\,\widetilde{\cal A}^{(0)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle
p_{3},n;p_{2}|\sigma_{n+3})+p_{1}\leftrightarrow p_{2}\,,~{}~{}$
$\displaystyle B_{6}$ $\displaystyle=$
$\displaystyle\Big{[}S^{(1)_{p_{3}}}_{g}\,{1\over k_{n}\cdot
k_{p_{1}p_{2}}}\Big{]}\,(k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot
Y_{p_{1}})\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n|\sigma_{n+3}\setminus
p_{3})+p_{1}\leftrightarrow p_{2}$ (70) $\displaystyle=$
$\displaystyle-\tau\,{(k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot
Y_{p_{1}})\,(k_{n}\cdot f_{p_{3}}\cdot K_{p_{3}})\over\big{(}k_{n}\cdot
k_{p_{1}p_{2}}\big{)}^{2}}\,{\cal A}^{(0)_{p_{3}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\}\shuffle
p_{3},n|\sigma_{n+3}\setminus p_{3})$ $\displaystyle+p_{1}\leftrightarrow
p_{2}\,.~{}~{}$

The first and second parts $B_{1}$ and $B_{2}$ in (65) and (66) are analogous
to $B_{1}$ in (41), come from the soft theorem. The computation of $B_{3}$ and
$B_{4}$ in (81) and (68) are parallel to that for obtaining $B_{2}$ in (42).
The calculation of $B_{5}$ and $B_{6}$ in (69) and (70) are parallel to the
process for obtaining $B_{3}$ in (43). The combinatory momentum $K_{p_{3}}$ is
defined as the summation of momenta in $\\{k_{p_{1}},k_{p_{2}}\\}$ whose
corresponding legs are at the l.h.s of $p_{3}$ in the color ordering. The
notation $\widetilde{\cal A}^{(0)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle p_{3},n;p_{2}|\sigma_{n+3})$ in
the last line of (69) means expanding ${\cal A}^{(0)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle p_{3},n;p_{2}|\sigma_{n+3})$ to
pure BAS basis, and $K_{p_{3}}$ are defined for the corresponding color
orderings in these BAS amplitudes. If one use other expressions for ${\cal
A}^{(0)_{p_{3}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle
p_{3},n;p_{2}|\sigma_{n+3})$, (69) does not hold.

Using $K_{p_{3}}=Y_{p_{3}}-X_{p_{3}}$, where for (69) $X_{p_{3}}$ is defined
for BAS amplitudes after expanding ${\cal A}^{(0)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle p_{3},n;p_{2}|\sigma_{n+3})$,
one can separate $B_{5}$ and $B_{6}$ as

$\displaystyle B_{5}=B_{7}+B_{8}\,,~{}~{}~{}~{}B_{6}=B_{9}+B_{10}\,,$ (71)

where

$\displaystyle B_{7}=\tau\,{(k_{n}\cdot f_{p_{1}}\cdot Y_{p_{1}})\,(k_{n}\cdot
f_{p_{3}}\cdot Y_{p_{3}})\over\big{(}k_{n}\cdot
k_{p_{1}p_{2}}\big{)}^{2}}\,{\cal A}^{(0)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle
p_{3},n;p_{2}|\sigma_{n+3})+p_{1}\leftrightarrow p_{2}\,,~{}~{}$ (72)
$\displaystyle B_{8}=-\tau\,{(k_{n}\cdot k_{p_{3}})\,(k_{n}\cdot
f_{p_{1}}\cdot Y_{p_{1}})\over\big{(}k_{n}\cdot
k_{p_{1}p_{2}}\big{)}^{2}}\,{\cal A}^{(0)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1},n;\\{p_{i}\\}_{3}\setminus
p_{1}|\sigma_{n+3})+p_{1}\leftrightarrow p_{2}\,,~{}~{}$ (73) $\displaystyle
B_{9}$ $\displaystyle=$ $\displaystyle\tau\,{(k_{n}\cdot f_{p_{2}}\cdot
f_{p_{1}}\cdot Y_{p_{1}})\,(k_{n}\cdot f_{p_{3}}\cdot
Y_{p_{3}})\over\big{(}k_{n}\cdot k_{p_{1}p_{2}}\big{)}^{2}}\,{\cal
A}^{(0)_{p_{3}}}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\}\shuffle
p_{3},n|\sigma_{n+3})$ (74) $\displaystyle+p_{1}\leftrightarrow p_{2}\,~{}~{}$
$\displaystyle B_{10}=-\tau\,{(k_{n}\cdot k_{p_{3}})\,(k_{n}\cdot
f_{p_{2}}\cdot f_{p_{1}}\cdot Y_{p_{1}})\over\big{(}k_{n}\cdot
k_{p_{1}p_{2}}\big{)}^{2}}\,{\cal A}^{(0)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n;p_{3}|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}\,,~{}~{}$ (75)

The result $B_{8}$ in (73) can be obtained as follows. Using the expansion of
${\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;\\{p_{i}\\}_{3}\setminus p_{1}|\sigma_{n+3})$, we have

$\displaystyle{\cal A}^{(0)_{p_{3}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;\\{p_{i}\\}_{3}\setminus p_{1}|\sigma_{n+3})$ (76) $\displaystyle=$
$\displaystyle{k_{n}\cdot f_{p_{2}}\cdot Y_{p_{2}}\over k_{n}\cdot
k_{p_{2}}}\,{\cal A}^{(0)_{p_{3}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1}\shuffle p_{2},n;p_{3}|\sigma_{n+3})\,,~{}~{}$

since all other terms are at the $\tau^{1}$ order. Then we use the expansion
of ${\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle
p_{2},n;p_{3}|\sigma_{n+3})$ to get

$\displaystyle{\cal A}^{(0)_{p_{3}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1}\shuffle p_{2},n;p_{3}|\sigma_{n+3})$ (77) $\displaystyle=$
$\displaystyle{k_{n}\cdot f_{p_{3}}\cdot X_{p_{3}}\over k_{n}\cdot
k_{p_{3}}}\,{\cal A}^{(0)_{p_{3}}}_{\rm S}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1}\shuffle p_{2}\shuffle p_{3},n|\sigma_{n+3})\,.~{}~{}$

Notice that for pure BAS amplitudes $Y_{p_{3}}$ is equivalent to $X_{p_{3}}$.
Expanding ${\cal A}^{(0)_{p_{3}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1}\shuffle p_{2},n;p_{3}|\sigma_{n+3})$ will not alter $Y_{p_{2}}$ in (76)
at the leading order, since $k_{p_{3}}$ is accompanied with $\tau$ thus does
not contribute. Substituting (77) into (76), we find

$\displaystyle{\cal A}^{(0)_{p_{3}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;\\{p_{i}\\}_{3}\setminus p_{1}|\sigma_{n+3})$ (78) $\displaystyle=$
$\displaystyle{k_{n}\cdot f_{p_{3}}\cdot X_{p_{3}}\over k_{n}\cdot
k_{p_{3}}}\,\widetilde{\cal A}^{(0)_{p_{3}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle
p_{3},n;p_{2}|\sigma_{n+3})\,,~{}~{}$

where the expansion of ${\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1}\shuffle p_{3},n;p_{2}|\sigma_{n+3})$ has been used. The relation (78)
yields the expression of $B_{8}$ in (73). The expression of $B_{9}$ in (75)
can be calculated in the analogous way.

Now one can recognize that $B_{1}$ in (65) together with $B_{8}$ in (73) gives
the sub-leading term of

$\displaystyle P_{1}={k_{n}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot
k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1},n;\\{p_{i}\\}_{3}\setminus p_{1}|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}\,,~{}~{}$ (79)

while $B_{2}$ in (66) and $B_{10}$ in (75) gives the sub-leading term of

$\displaystyle P_{2}={k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over
k_{n}\cdot k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2}\\},n;p_{3}|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}\,.~{}~{}$ (80)

Similar as in the $2$-gluon case, $k_{n}\cdot k_{p_{1}p_{2}}$ in the
denominator is turned to $k_{n}\cdot k_{p_{1}p_{2}p_{3}}$, to keep the
manifest permutation symmetry among external gluons $p_{1}$, $p_{2}$, $p_{3}$.
The block $B_{3}$ in (81) can be interpreted as the leading term of

$\displaystyle P_{3}$ $\displaystyle=$ $\displaystyle{k_{n}\cdot
f_{p_{3}}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot
k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{3}\\},n;p_{2}|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}$ (81) $\displaystyle+{k_{n}\cdot f_{p_{1}}\cdot f_{p_{3}}\cdot
Y_{p_{3}}\over k_{n}\cdot k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{3},p_{1}\\},n;p_{2}|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}\,,~{}~{}$

$B_{4}$ in (68) can be interpreted as the leading term of

$\displaystyle P_{4}$ $\displaystyle=$ $\displaystyle{k_{n}\cdot
f_{p_{3}}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot
k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{2},p_{3}\\},n|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}$ (82) $\displaystyle+{k_{n}\cdot f_{p_{2}}\cdot f_{p_{3}}\cdot
f_{p_{1}}\cdot Y_{p_{1}}\over k_{n}\cdot k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{1},p_{3},p_{2}\\},n|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}$ $\displaystyle+{k_{n}\cdot f_{p_{2}}\cdot f_{p_{1}}\cdot f_{p_{3}}\cdot
Y_{p_{3}}\over k_{n}\cdot k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{p_{3},p_{1},p_{2}\\},n|\sigma_{n+3})+p_{1}\leftrightarrow
p_{2}\,.~{}~{}$

Finally, combining $B_{7}$ in (72) and $B_{9}$ in (74) together gives the
leading contribution of

$\displaystyle P_{5}={k_{n}\cdot f_{p_{3}}\cdot Y_{p_{3}}\over k_{n}\cdot
k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{3},n;\\{p_{i}\\}_{2}|\sigma_{n+3})\,.~{}~{}$ (83)

Thus, we arrive at

$\displaystyle{\cal A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{3}|\sigma_{n+3})$ (84)
$\displaystyle=$ $\displaystyle P_{1}+P_{2}+P_{3}+P_{4}+P_{5}$
$\displaystyle=$
$\displaystyle\sum_{{\alpha}_{1}\in\\{p_{i}\\}_{3}}\,{k_{n}\cdot
f_{{\alpha}_{1}}\cdot Y_{{\alpha}_{1}}\over k_{n}\cdot
k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle{\alpha}_{1},n;\\{p_{i}\\}_{3}\setminus{\alpha}_{1}|\sigma_{n+3})$
$\displaystyle+\sum_{{\beta}_{1},{\beta}_{2}\in\\{p_{i}\\}_{3}}\,{k_{n}\cdot
f_{{\beta}_{2}}\cdot f_{{\beta}_{1}}\cdot Y_{{\beta}_{1}}\over k_{n}\cdot
k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\\{{\beta}_{1},{\beta}_{2}\\},n;\\{p_{i}\\}_{3}\setminus\\{{\beta}_{1},{\beta}_{2}\\}|\sigma_{n+3})$
$\displaystyle+\sum_{\gamma_{j}\in\\{p_{i}\\}_{3}}\,{k_{n}\cdot
f_{\gamma_{3}}\cdot f_{\gamma_{2}}\cdot f_{\gamma_{1}}\cdot
Y_{{\beta}_{1}}\over k_{n}\cdot k_{p_{1}p_{2}p_{3}}}\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle\\{\gamma_{1},\gamma_{2},\gamma_{3}\\},n|\sigma_{n+3})\,,~{}~{}$

where $j\in\\{1,2,3\\}$ in the final line.

###  III.4 General case

Comparing the constructions in subsections.III.2 and III.3, we see the
processes bear strong similarity. This similarity implies the recursive
pattern which yields the general expansion for single trace YMS amplitudes
with arbitrary number of external gluons. Such general expansion is the goal
of this subsection.

In general, the single-trace YMS amplitude ${\cal A}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{m}|\sigma_{n+m})$ with $m$ external gluons can be
expanded as in (17). When $\boldsymbol{{\alpha}}=\\{p_{i}\\}_{m}$, the YMS
amplitudes at the second line of (17) are reduced to pure BAS ones. As
mentioned previously, in this note we restrict ourselves to the special choice
of the reference momentum $k_{r}$ in (17), which is $k_{r}=k_{n}$.

Obviously, expansions in (39), (61), (84) satisfy the general formula in (17)
with $k_{r}=k_{n}$. To achieve the general formula, let us prove that suppose
(17) is satisfied for a particular $m$, then it is also satisfied for $m+1$.
Consider $k_{p_{m+1}}\to\tau k_{p_{m+1}}$, and expand ${\cal A}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{m+1}|\sigma_{n+m+1})$ in $\tau$ to get

$\displaystyle{\cal A}^{(1)_{p_{m+1}}}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{m+1}|\sigma_{n+m+1})$ $\displaystyle=$
$\displaystyle S^{(1)_{p_{m+1}}}_{g}\,{\cal A}_{\rm
YS}(1,\cdots,n;\\{p_{i}\\}_{m}|\sigma_{n+m+1}\setminus p_{m+1})$ (85)
$\displaystyle=$ $\displaystyle B_{1}+B_{2}+B_{3}\,,~{}~{}$

where

$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,{k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot Y_{\vec{\boldsymbol{{\alpha}}}}\over
k_{n}\cdot k_{p_{1}\cdots p_{m}}}$ (86)
$\displaystyle\Big{[}S^{(1)_{p_{m+1}}}_{g}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1}\setminus
p_{m+1})\Big{]}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,{k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot Y_{\vec{\boldsymbol{{\alpha}}}}\over
k_{n}\cdot k_{p_{1}\cdots p_{m}}}$ $\displaystyle{\cal A}^{(1)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m+1}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})\,,~{}~{}$
$\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,{\Big{[}S^{(1)_{p_{m+1}}}_{g}\,k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot
Y_{\vec{\boldsymbol{{\alpha}}}}\Big{]}\over k_{n}\cdot k_{p_{1}\cdots p_{m}}}$
(87) $\displaystyle{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1}\setminus
p_{m+1})$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}^{\prime}}\,\tau\,{k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}^{\prime}}\cdot
Y_{\vec{\boldsymbol{{\alpha}}}^{\prime}}\over k_{n}\cdot k_{p_{1}\cdots
p_{m}}}$ $\displaystyle{\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}}^{\prime},n;\\{p_{i}\\}_{m+1}\setminus\boldsymbol{{\alpha}^{\prime}}|\sigma_{n+m+1})\,,~{}~{}$

with $\vec{\boldsymbol{{\alpha}}}^{\prime}=\vec{\boldsymbol{{\alpha}}}\shuffle
p_{m+1}$, and

$\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\Big{[}S^{(1)_{p_{m+1}}}_{g}\,{1\over
k_{n}\cdot k_{p_{1}\cdots p_{m}}}\Big{]}\,(k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot Y_{\vec{\boldsymbol{{\alpha}}}})$ (88)
$\displaystyle{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1}\setminus
p_{m+1})$ $\displaystyle=$
$\displaystyle-\sum_{\vec{\boldsymbol{{\alpha}}}}\,\tau\,{(k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot
Y_{\vec{\boldsymbol{{\alpha}}}})\,(k_{n}\cdot f_{p_{m+1}}\cdot
K_{p_{m+1}})\over\big{(}k_{n}\cdot k_{p_{1}\cdots p_{m}}\big{)}^{2}}$
$\displaystyle\widetilde{\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}}\shuffle
p_{m+1},n;\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})\,.~{}~{}$

The calculation are paralleled to those in subsections III.2 and III.3. Again,
the notation $\widetilde{\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}}\shuffle
p_{m+1},n;\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})$ is
understood as expanding ${\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}}\shuffle
p_{m+1},n;\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})$ to
BAS basis. To obtain $B_{2}$ in the second line of (87), we used the property

$\displaystyle\vec{\boldsymbol{a}}\shuffle\vec{\boldsymbol{b}}\shuffle\vec{\boldsymbol{c}}=\vec{\boldsymbol{a}}\shuffle\\{\vec{\boldsymbol{b}}\shuffle\vec{\boldsymbol{c}}\\}\,,$
(89)

where $\vec{\boldsymbol{a}}$, $\vec{\boldsymbol{b}}$ and
$\vec{\boldsymbol{c}}$ are three ordered sets. In (88), $K_{p_{m+1}}$ stands
for the summation of momenta carried by external gluons $p_{i}$ at the l.h.s
of $p_{m+1}$ in the color ordering. We emphasize that the external leg
$p_{m+1}$ is not included in the set $\boldsymbol{{\alpha}}$ appear in
$B_{1}$, $B_{2}$ and $B_{3}$.

Using $K_{p_{m+1}}=Y_{p_{m+1}}-X_{p_{m+1}}$, one can split $B_{3}$ as

$\displaystyle B_{3}=B_{4}+B_{5}\,,$ (90)

where

$\displaystyle B_{4}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\tau\,{(k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot
Y_{\vec{\boldsymbol{{\alpha}}}})\,(k_{n}\cdot f_{p_{m+1}}\cdot
Y_{p_{m+1}})\over\big{(}k_{n}\cdot k_{p_{1}\cdots p_{m}}\big{)}^{2}}$ (91)
$\displaystyle{\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}}\shuffle
p_{m+1},n;\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})$
$\displaystyle=$ $\displaystyle\tau\,{k_{n}\cdot f_{p_{m+1}}\cdot
Y_{p_{m+1}}\over k_{n}\cdot k_{p_{1}\cdots p_{m}}}\,{\cal
A}^{(0)_{p_{m+1}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{m+1},n;\\{p_{i}\\}_{m}|\sigma_{n+m+1})\,,~{}~{}$ $\displaystyle B_{5}$
$\displaystyle=$
$\displaystyle-\sum_{\vec{\boldsymbol{{\alpha}}}}\,\tau\,{(k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot
Y_{\vec{\boldsymbol{{\alpha}}}})\,(k_{n}\cdot f_{p_{m+1}}\cdot
X_{p_{m+1}})\over\big{(}k_{n}\cdot k_{p_{1}\cdots p_{m}}\big{)}^{2}}$ (92)
$\displaystyle\widetilde{\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}}\shuffle
p_{m+1},n;\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})$
$\displaystyle=$
$\displaystyle-\sum_{\vec{\boldsymbol{{\alpha}}}}\,\tau\,{(k_{n}\cdot
k_{p_{m+1}})\,(k_{n}\cdot F_{\vec{\boldsymbol{{\alpha}}}}\cdot
Y_{\vec{\boldsymbol{{\alpha}}}})\over\big{(}k_{n}\cdot k_{p_{1}\cdots
p_{m}}\big{)}^{2}}$ $\displaystyle{\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m+1}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})\,.~{}~{}$

The computations of $B_{4}$ and $B_{5}$ are analogous to those for obtaining
$B_{7}$, $B_{8}$, $B_{9}$ and $B_{10}$ in subsection. III.3. It is worth to
give a brief discussion for how to generalize the manipulation from (76) to
(78) to the current general case. We first observe that

$\displaystyle{\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m+1}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})$
(93) $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\beta}}}}\,{k_{n}\cdot
F_{\vec{\boldsymbol{{\beta}}}}\cdot Y_{\vec{\boldsymbol{{\beta}}}}\over
k_{n}\cdot k_{\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}}}\,{\cal
A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}}\shuffle\vec{\boldsymbol{{\beta}}},n;\\{p_{i}\\}_{m+1}\setminus\\{\boldsymbol{{\alpha}}\cup\boldsymbol{{\beta}}\\}|\sigma_{n+m+1})\,.~{}~{}$

Here $\boldsymbol{{\beta}}$ is a subset of
$\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}$ which does not include
$p_{m+1}$, since otherwise $k_{p_{m+1}}$ will occur in
$F_{\vec{\boldsymbol{{\beta}}}}$ and contributes $\tau$ under the re-scaling
$k_{p_{m+1}}\to\tau k_{p_{m+1}}$. The combinatory momentum
$k_{\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}}$ is the summation of
momenta for external legs in the set
$\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}$. Applying the above procedure
iteratively, one can expand the leading terms of YMS amplitudes in the second
line of (93) further, until there is only one remaining gluon $p_{m+1}$. In
other words, ${\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{\\{p_{i}\\}_{m+1}\setminus\boldsymbol{{\alpha}}\\}|\sigma_{n+m+1})$
can be expanded to ${\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle\cdots\shuffle
p_{m},n;p_{m+1}|\sigma_{n+m+1})$. Then, we expand them further by employing

$\displaystyle{\cal A}^{(0)_{p_{m+1}}}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{1}\shuffle\cdots\shuffle p_{m},n;p_{m+1}|\sigma_{n+m+1})$ (94)
$\displaystyle=$ $\displaystyle{k_{n}\cdot f_{p_{m+1}}\cdot X_{p_{m+1}}\over
k_{n}\cdot k_{p_{m+1}}}\,{\cal A}^{(0)_{p_{m+1}}}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle p_{1}\shuffle\cdots\shuffle p_{m}\shuffle
p_{m+1},n|\sigma_{n+m+1})\,,~{}~{}$

and reorganize the full expansion as

$\displaystyle{\cal A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m+1}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})$
(95) $\displaystyle=$ $\displaystyle{k_{n}\cdot f_{p_{m+1}}\cdot
X_{p_{m+1}}\over k_{n}\cdot k_{p_{m+1}}}\,\widetilde{\cal
A}^{(0)_{p_{m+1}}}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}}\shuffle
p_{m+1},n;\\{p_{i}\\}_{m}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})\,,~{}~{}$

based on the observation that $k_{p_{m+1}}$ which is accompanied with $\tau$
does not contribute to any $Y$ at the leading order. The relation (95) is the
generalization of the previous result (78). Using (95), we arrive at the
expression of $B_{5}$ in (92).

Now we observe that $B_{1}$ in (86) together with $B_{5}$ in (92) give the
sub-leading term of

$\displaystyle P_{1}=\sum_{\vec{\boldsymbol{{\alpha}}}}\,{k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot Y_{\vec{\boldsymbol{{\alpha}}}}\over
k_{n}\cdot k_{p_{1}\cdots p_{m+1}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m+1}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m+1})\,,~{}~{}$
(96)

$B_{2}$ in (87) can be interpreted as the leading term of

$\displaystyle P_{2}=\sum_{\vec{\boldsymbol{{\alpha}}}^{\prime}}\,{k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}^{\prime}}\cdot
Y_{\vec{\boldsymbol{{\alpha}}}^{\prime}}\over k_{n}\cdot k_{p_{1}\cdots
p_{m+1}}}\,{\cal A}_{\rm
YS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}}^{\prime},n;\\{p_{i}\\}_{m+1}\setminus\boldsymbol{{\alpha}^{\prime}}|\sigma_{n+m+1})\,,~{}~{}$
(97)

where $\boldsymbol{{\alpha}}^{\prime}$ includes the external leg $p_{m+1}$ and
$\boldsymbol{{\alpha}}^{\prime}\setminus p_{m+1}\neq\emptyset$, and $B_{4}$ in
(91) corresponds to the leading terms of

$\displaystyle P_{3}={k_{n}\cdot f_{p_{m+1}}\cdot Y_{p_{m+1}}\over k_{n}\cdot
k_{p_{1}\cdots p_{m+1}}}\,{\cal A}_{\rm YS}(1,\\{2,\cdots,n-1\\}\shuffle
p_{m+1},n;\\{p_{i}\\}_{m}|\sigma_{n+m+1})\,.~{}~{}$ (98)

Combining $P_{1}$, $P_{2}$ and $P_{3}$ together, we see that the general
formula (17) is correct for single-trace YMS amplitude with $(m+1)$ external
gluons, thus the proof is completed.

##  IV Summery

In this note, we reconstructed the expansion of single-trace YMS amplitudes
proposed by Clifford Cheung and James Mangan in Cheung:2021zvb , via the
bottom up method based on soft theorems, with out the aid of a Lagrangian or
equations of motion. The whole process is on-shell, without using any off-
shell ansatz. The new recursive method inserts gluons into the original
amplitude in the manifestly gauge invariant manner, leads to the resulting
expansion which manifests the gauge invariance for all polarizations carried
by external gluons, and the permutation symmetry among external gluons, with
the cost of sacrificing the manifest locality. One can use such expansion
iteratively to expand the single-trace YMS amplitudes to pure BAS ones without
breaking the manifest gauge invariance for any polarization, but a verity of
spurious poles will be created.

Through the double copy structure, our result can be extended to the expansion
of single-trace Einstein-Yang-Mills amplitudes directly,

$\displaystyle{\cal A}_{\rm EY}(1,\cdots,n;\\{p_{i}\\}_{m})$ (99)
$\displaystyle=$ $\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,{k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot Y_{\vec{\boldsymbol{{\alpha}}}}\over
k_{n}\cdot k_{p_{1}\cdots p_{m}}}\,{\cal A}_{\rm
EY}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m}\setminus{\boldsymbol{{\alpha}}})\,,~{}~{}$

where $1,\cdots,n$ are color ordered gluons, and elements in $\\{p_{i}\\}_{m}$
are gravitons without any ordering. It is straightforward to verify the
expansion (99) is equivalent to that found by Clifford Cheung and James Mangan
in Cheung:2021zvb . After some modifications of notations, the expansion in
Cheung:2021zvb  with the choice of reference momentum $k_{r}=k_{n}$ is given
as

$\displaystyle{\cal A}_{\rm YS}(1,\cdots,n;\\{p_{i}\\}_{m})$ (100)
$\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\sum_{i=1}^{n-1}\,{k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot k_{i}\over k_{n}\cdot k_{1\cdots
n}}\,{\cal A}_{\rm
YS}(1,\cdots,i,\\{i+1,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{i}\\}_{m}\setminus{\boldsymbol{{\alpha}}})\,,~{}~{}$

up to an overall sign. Here $k_{i}$ contributes when $i$ is at the l.h.s of
the first element in $\vec{\boldsymbol{{\alpha}}}$ in the ordering, therefore
the definition of $Y_{\vec{\boldsymbol{{\alpha}}}}$ is satisfied. On the other
hand, the momentum conservation gives $k_{1\cdots n}=-k_{p_{1}\cdots p_{m}}$.
Thus the equivalence between (99) and (100) is proved.

The sub-leading soft theorem for external gluon plays the crucial role in our
recursive method. One may ask the reason for choosing sub-leading soft theorem
rather than the leading order one. The answer is, the leading order soft
theorem can not detect all terms in the expansion. Suppose we re-scale
$k_{p_{j}}$ as $k_{p_{j}}\to\tau k_{p_{j}}$, as can be seen in the general
expanded formula (17), various terms include $f_{p_{j}}^{\mu\nu}$ in
corresponding coefficients therefore are accompanied with $\tau$, these terms
have no contribution to the leading order. Another question is whether the
sub-leading soft theorem is sufficient to detect all terms in the complete
expansion? For example, if the coefficient of one term behaviors as
$\tau^{2}$, then this coefficient can not be detected at both leading and sub-
leading orders. This question was answered in Zhou:2022orv , by employing the
universality of soft factor for external scalars, as well as the counting of
mass dimensions. The argument in Zhou:2022orv  at least shows that the old
expansion (15) can be fully detected by the sub-leading soft theorem for
external gluon. Since any correct expansion must be equivalent to (15), we
claim that the new expansion, which manifests gauge invariance for all
polarizations, can also be detected completely.

A related interesting future direction is seeking the expanded formula for
pure YM amplitudes which manifests the gauge invariance for each polarization.
Such expansion leads to the manifestly gauge invariant BCJ numerators. We
expect the recursive method in this note can be applied to solve this
challenge.

## Acknowledgments

The authors would thank Prof.Yijian Du for stimulating discussions. We also
thank Dr.Linghui Hou who told us the expansion found by Clifford Cheung and
James Mangan.

## References

  * (1) H. Kawai, D. C. Lewellen and S. H. Tye, “A Relation Between Tree Amplitudes of Closed and Open Strings,” Nucl. Phys. B 269, 1 (1986).
  * (2) Z. Bern, J. J. M. Carrasco and H. Johansson, “New Relations for Gauge-Theory Amplitudes,” Phys. Rev. D 78, 085011 (2008) [arXiv:0805.3993 [hep-ph]].
  * (3) M. Chiodaroli, M. Gnaydin, H. Johansson and R. Roiban, “Scattering amplitudes in $\mathcal{N}=2$ Maxwell-Einstein and Yang-Mills/Einstein supergravity,” JHEP 1501, 081 (2015) doi:10.1007/JHEP01(2015)081 [arXiv:1408.0764 [hep-th]].
  * (4) H. Johansson and A. Ochirov, “Color-Kinematics Duality for QCD Amplitudes,” JHEP 1601, 170 (2016) doi:10.1007/JHEP01(2016)170 [arXiv:1507.00332 [hep-ph]].
  * (5) H. Johansson and A. Ochirov, “Double copy for massive quantum particles with spin,” JHEP 1909, 040 (2019) doi:10.1007/JHEP09(2019)040 [arXiv:1906.12292 [hep-th]].
  * (6) F. Cachazo, S. He, and E. Y. Yuan, “Scattering Equations and Kawai-Lewellen-Tye Orthogonality,” Phys. Rev. D90 (2014) no. 6, 065001, arXiv:1306.6575 [hep-th].
  * (7) F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles in Arbitrary Dimensions,” Phys. Rev. Lett. 113 (2014) no. 17, 171601, arXiv:1307.2199 [hep-th].
  * (8) F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles: Scalars, Gluons and Gravitons,” JHEP 1407 (2014) 033, arXiv:1309.0885 [hep-th].
  * (9) F. Cachazo, S. He and E. Y. Yuan, “Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations,” JHEP 1501, 121 (2015) [arXiv:1409.8256 [hep-th]].
  * (10) F. Cachazo, S. He and E. Y. Yuan, “Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM,” JHEP 1507, 149 (2015) [arXiv:1412.3479 [hep-th]].
  * (11) C. Cheung, C. H. Shen and C. Wen, “Unifying Relations for Scattering Amplitudes,” JHEP 1802, 095 (2018) [arXiv:1705.03025 [hep-th]].
  * (12) K. Zhou and B. Feng, “Note on differential operators, CHY integrands, and unifying relations for amplitudes,” JHEP 1809, 160 (2018) [arXiv:1808.06835 [hep-th]].
  * (13) M. Bollmann and L. Ferro, “Transmuting CHY formulae,” JHEP 1901, 180 (2019) [arXiv:1808.07451 [hep-th]].
  * (14) C. H. Fu, Y. J. Du, R. Huang and B. Feng, “Expansion of Einstein-Yang-Mills Amplitude,” JHEP 1709, 021 (2017) doi:10.1007/JHEP09(2017)021 [arXiv:1702.08158 [hep-th]].
  * (15) F. Teng and B. Feng, “Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame,” JHEP 1705, 075 (2017) [arXiv:1703.01269 [hep-th]].
  * (16) Y. J. Du and F. Teng, “BCJ numerators from reduced Pfaffian,” JHEP 1704, 033 (2017) [arXiv:1703.05717 [hep-th]].
  * (17) Y. J. Du, B. Feng and F. Teng, “Expansion of All Multitrace Tree Level EYM Amplitudes,” JHEP 1712, 038 (2017) [arXiv:1708.04514 [hep-th]].
  * (18) B. Feng, X. Li and K. Zhou, “Expansion of EYM theory by Differential Operators,” arXiv:1904.05997 [hep-th].
  * (19) K. Zhou and S. Q. Hu, “Expansions of tree amplitudes for Einstein–Maxwell and other theories,” PTEP 2020, no.7, 073B10 (2020) doi:10.1093/ptep/ptaa095 [arXiv:1907.07857 [hep-th]].
  * (20) K. Zhou, “Unified web for expansions of amplitudes,” JHEP 10, 195 (2019) doi:10.1007/JHEP10(2019)195 [arXiv:1908.10272 [hep-th]].
  * (21) S. Stieberger and T. R. Taylor, “New relations for Einstein-Yang-Mills amplitudes,” Nucl. Phys. B 913, 151 (2016) [arXiv:1606.09616 [hep-th]].
  * (22) O. Schlotterer, “Amplitude relations in heterotic string theory and Einstein-Yang-Mills,” JHEP 1611, 074 (2016) [arXiv:1608.00130 [hep-th]].
  * (23) M. Chiodaroli, M. Gunaydin, H. Johansson and R. Roiban, “Explicit Formulae for Yang-Mills-Einstein Amplitudes from the Double Copy,” JHEP 1707, 002 (2017) doi:10.1007/JHEP07(2017)002 [arXiv:1703.00421 [hep-th]].
  * (24) V. Del Duca, L. J. Dixon and F. Maltoni, “New color decompositions for gauge amplitudes at tree and loop level,” Nucl. Phys. B 571, 51 (2000) doi:10.1016/S0550-3213(99)00809-3 [hep-ph/9910563].
  * (25) D. Nandan, J. Plefka, O. Schlotterer and C. Wen, “Einstein-Yang-Mills from pure Yang-Mills amplitudes,” JHEP 1610, 070 (2016) [arXiv:1607.05701 [hep-th]].
  * (26) L. de la Cruz, A. Kniss and S. Weinzierl, “Relations for Einstein-Yang-Mills amplitudes from the CHY representation,” Phys. Lett. B 767, 86 (2017) [arXiv:1607.06036 [hep-th]].
  * (27) C. Cheung and J. Mangan, “Covariant color-kinematics duality,” JHEP 11, 069 (2021) doi:10.1007/JHEP11(2021)069 [arXiv:2108.02276 [hep-th]].
  * (28) K. Zhou, “Tree level amplitudes from soft theorems,” JHEP 03, 021 (2023) doi:10.1007/JHEP03(2023)021 [arXiv:2212.12892 [hep-th]].
  * (29) N. Arkani-Hamed, L. Rodina and J. Trnka, “Locality and Unitarity of Scattering Amplitudes from Singularities and Gauge Invariance,” Phys. Rev. Lett. 120, no.23, 231602 (2018) doi:10.1103/PhysRevLett.120.231602 [arXiv:1612.02797 [hep-th]].
  * (30) L. Rodina, “Uniqueness from gauge invariance and the Adler zero,” JHEP 09, 084 (2019) doi:10.1007/JHEP09(2019)084 [arXiv:1612.06342 [hep-th]].
  * (31) R. Britto, F. Cachazo and B. Feng, “New recursion relations for tree amplitudes of gluons,” Nucl. Phys. B 715, 499-522 (2005) doi:10.1016/j.nuclphysb.2005.02.030 [arXiv:hep-th/0412308 [hep-th]].
  * (32) R. Britto, F. Cachazo, B. Feng and E. Witten, “Direct proof of tree-level recursion relation in Yang-Mills theory,” Phys. Rev. Lett. 94, 181602 (2005) doi:10.1103/PhysRevLett.94.181602 [arXiv:hep-th/0501052 [hep-th]].
  * (33) N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov and J. Trnka, Cambridge University Press, 2016, ISBN 978-1-107-08658-6, 978-1-316-57296-2 doi:10.1017/CBO9781316091548 [arXiv:1212.5605 [hep-th]].
  * (34) N. Arkani-Hamed and J. Trnka, “The Amplituhedron,” JHEP 10, 030 (2014) doi:10.1007/JHEP10(2014)030 [arXiv:1312.2007 [hep-th]].
  * (35) N. Arkani-Hamed and J. Trnka, “Into the Amplituhedron,” JHEP 12, 182 (2014) doi:10.1007/JHEP12(2014)182 [arXiv:1312.7878 [hep-th]].
  * (36) H. Elvang and Y. t. Huang, “Scattering Amplitudes,” [arXiv:1308.1697 [hep-th]].
  * (37) C. Cheung, “TASI Lectures on Scattering Amplitudes,” doi:10.1142/9789813233348_0008 [arXiv:1708.03872 [hep-ph]].
  * (38) R. Kleiss and H. Kuijf, “MULTI - GLUON CROSS-SECTIONS AND FIVE JET PRODUCTION AT HADRON COLLIDERS,” Nucl. Phys. B 312, 616 (1989).
  * (39) E. Casali, “Soft sub-leading divergences in Yang-Mills amplitudes,” JHEP 08, 077 (2014) doi:10.1007/JHEP08(2014)077 [arXiv:1404.5551 [hep-th]].
  * (40) B. U. W. Schwab and A. Volovich, “Subleading Soft Theorem in Arbitrary Dimensions from Scattering Equations,” Phys. Rev. Lett. 113, no.10, 101601 (2014) doi:10.1103/PhysRevLett.113.101601 [arXiv:1404.7749 [hep-th]].