#  Recursive construction for expansions of tree Yang-Mills amplitudes from
soft theorem

Chang Huab, Kang Zhouc isiahalbert<EMAIL_ADDRESS>aSchool of
Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced
Study, UCAS, Hangzhou 310024, China
b University of Chinese Academy of Sciences, Beijing 100049, China
c Center for Gravitation and Cosmology, College of Physical Science and
Technology, Yangzhou University, Yangzhou, 225009, China

###### Abstract

In this paper, we have introduced a fundamentally different approach, based on
a bottom-up methodology, to expand tree-level Yang-Mills (YM) amplitudes into
Yang-Mills-scalar (YMS) amplitudes and Bi-adjoint-scalar (BAS) amplitudes. Our
method relies solely on the intrinsic soft behavior of external gluons,
eliminating the need for external aids such as Feynman rules or CHY rules. The
recursive procedure consistently preserves explicit gauge invariance at every
step, ultimately resulting in a manifest gauge-invariant outcome when the
initial expression is already framed in a gauge-invariant manner. The
resulting expansion can be directly analogized to the expansions of
gravitational (GR) amplitudes using the double copy structure. When combined
with the expansions of Einstein-Yang-Mills amplitudes obtained using the
covariant double copy method from existing literature, the expansions
presented in this note yield gauge-invariant BCJ numerators.

##  I Introduction

In recent decades, significant progress has been made in understanding
scattering amplitudes, revealing hidden mathematical structures that go beyond
traditional Feynman rules. These advancements have uncovered interconnections
between amplitudes from various theories. For instance, gravitational (GR) and
Yang-Mills (YM) amplitudes at the tree level are linked through the Kawai-
Lewellen-Tye (KLT) relation Kawai:1985xq  and the 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 the a large variety of theories can be
generated from the tree GR amplitudes through the compactifying, squeezing and
the generalized dimensional reduction proceduresCachazo:2013gna ;
Cachazo:2013hca ; Cachazo:2013iea ; Cachazo:2014nsa ; Cachazo:2014xea .
Similar unifying relations have emerged, allowing tree amplitudes from one
theory to be expressed in terms of those from others Cheung:2017ems ;
Zhou:2018wvn ; Bollmann:2018edb ; Fu:2017uzt ; Teng:2017tbo ; Du:2017kpo ;
Du:2017gnh ; Feng:2019tvb ; Zhou:2019gtk ; Zhou:2019mbe .

The expansions of tree-level gravitational (GR) amplitudes into Yang-Mills
(YM) amplitudes have garnered significant attention due to their role in
constructing BCJ numerators Bern:2010yg ; Fu:2017uzt . These expansions
involve recursive techniques, breaking down GR amplitudes into linear
combinations of tree-level Einstein-Yang-Mills (EYM) amplitudes where some
gravitons become gluons. They also expand tree-level EYM amplitudes into
versions with fewer gravitons and more gluons Fu:2017uzt ; Teng:2017tbo ;
Feng:2019tvb . By iteratively applying these recursions, one can derive the
desired GR to YM expansions.

The obtained coefficients or BCJ numerators through the aforementioned process
are local, without any spurious poles in the expansions. However, establishing
gauge invariance is challenging. Recent developments in scattering amplitude
research have shown that amplitudes manifesting gauge invariance at the
expense of locality often lead to innovative insights and unexpected
mathematical structures. For instance, the Britto-Cachazo-Feng-Witten (BCFW)
on-shell recursion relationsBritto:2004ap ; Britto:2005fq , inspiring novel
amplitude constructions like the Grassmannian representation and the
Amplituhedron Arkani-Hamed:2012zlh ; Arkani-Hamed:2013jha ; Arkani-
Hamed:2013kca . Thus, exploring expansion techniques that explicitly showcase
gauge invariance is valuable.

To achieve these new expansions, a natural approach is to modify existing
recursive expansions. Fortunately, we have alternative recursive expansions
for tree Einstein-Yang-Mills (EYM) amplitudes that explicitly exhibit gauge
invariance for each polarization carried by external particles. These new
expansions, discovered by Clifford Cheung and James Mangan using the covariant
color-kinematic duality Cheung:2021zvb , can also be derived using a technique
based on soft theorems, similar to the one presented in this paper Wei:2023yfy
. Another challenge is to discover new expansions that manifest gauge
invariance while expanding tree GR amplitudes into EYM ones. This can be
achieved by altering the old GR expansions through the addition of new terms
that vanish due to the gauge invariance of the expanded GR amplitudes, as
outlined in subsection IV A. Ensuring that the original expansions describe
gauge-invariant amplitudes is essential, making it logical to derive the old
expansions from a gauge-invariant foundational framework, such as the
traditional Lagrangian or CHY formalism. In other words, modifying the old
expansions is a top-down construction.

The central aim of the modern S-matrix program is to construct amplitudes from
a bottom-up approach, free from reliance on Lagrangian techniques. One
prominent example involves bootstrapping 3-point amplitudes within the spinor-
helicity formalism and subsequently generating higher-point amplitudes through
the application of the BCFW recursive method Britto:2004ap ; Britto:2005fq .
The primary objective of this paper is to provide a concise bottom-up approach
for constructing expansions of tree-level gravitational (GR) amplitudes into
tree-level Einstein-Yang-Mills (EYM) amplitudes. This approach ensures the
explicit manifestation of gauge invariance for the polarization of each
external particle.

The method presented in this paper is based on the sub-leading soft behavior
for external gluons. Initially, soft theorems at the tree level were derived
using Feynman rules for photons and gravitons Low ; Weinberg . In 2014, new
soft theorems were discovered for gravity (GR) and Yang-Mills (YM) theory at
the tree level by applying BCFW recursion relations Cachazo:2014fwa ;
Casali:2014xpa . In GR, the soft theorem was extended beyond the leading order
to sub-leading and sub-sub-leading levels, while in YM theory, it was
identified at the leading and sub-leading orders. These new soft theorems were
subsequently generalized to arbitrary spacetime dimensions Schwab:2014xua ;
Afkhami-Jeddi:2014fia  using CHY formulas Cachazo:2013gna ; Cachazo:2013hca ;
Cachazo:2013iea ; Cachazo:2014nsa ; Cachazo:2014xea . These soft theorems have
been instrumental in constructing tree amplitudes, such as through the inverse
soft theorem program, and by utilizing another type of soft behavior known as
the Adler zero to construct amplitudes for various effective theories
Cheung:2014dqa ; Luo:2015tat ; Elvang:2018dco ; Cachazo:2016njl ;
Rodina:2018pcb ; Boucher-Veronneau:2011rwd ; Nguyen:2009jk . In this paper, we
build upon the idea of constructing tree amplitudes from soft theorems, but
our approach differs significantly from the techniques found in existing
literature.

Instead of expanding GR (gravity) amplitudes to EYM (Einstein-Yang-Mills)
ones, this paper focuses on the expansions of color-ordered single-trace tree
YM (Yang-Mills) amplitudes Chiodaroli:2014xia ; Chiodaroli:2017ngp  into tree
Yang-Mills-scalar (YMS) amplitudes, which describe gluons and bi-adjoint
scalars (BAS). This choice is motivated by several factors. Firstly, both
expansions share the same coefficients due to the double copy structure
Kawai:1985xq ; Bern:2008qj ; Chiodaroli:2014xia ; Johansson:2015oia ;
Johansson:2019dnu ; Cachazo:2013iea . Secondly, the pure BAS amplitudes only
contain propagators without numerators Cachazo:2013iea . The approach starts
by bootstrapping the lowest 3-point tree YM amplitudes, while assuming that
higher-point tree amplitudes are uniquely determined by the soft behaviors of
gluons. The advantage lies in the sub-leading soft factor of gluons, which
inserts the soft external leg in a manifestly gauge-invariant manner.
Consequently, if the original 3-point amplitude is expressed in a manifestly
gauge-invariant formula, this gauge invariance is preserved throughout the
process. As demonstrated, relying on the assumption that soft behaviors fully
determine amplitudes, gauge invariance is established for the old expansions
without necessitating a top-down derivation.

The method presented in this paper represents a significant advancement and
improvement compared to the one used in Zhou:2022orv . The new version
constructs general expansions starting from the lowest-point amplitudes, which
can be determined through bootstrapping. As pointed out in Zhou:2022orv , the
soft theorem for gluons can be uniquely fixed by assuming the universality of
soft behaviors. More explicitly, one can find the expansion of tree YMS
amplitudes with one external gluon by imposing the universality of the soft
behaviours of BAS scalars, then derive the soft theorem for gluons from the
resulted expansion of such YMS amplitudes. On the other hand, since the pure
BAS amplitudes only contain propagators, the soft behaviors of BAS scalars are
obvious. Thus, in the whole story, the expansions of YM amplitudes to YMS ones
indeed arise from the assumptions of universality of soft behaviors.

The remainder of this note is organized as follows. In section II, we
introduce necessary background including the expansions of tree amplitudes to
KK BAS basis, the recursive expansions of tree YMS (EYM) amplitudes, as well
as the soft theorems for external BAS scalars and gluons at tree level. In
section III, we introduce our recursive method, and construct the old
expansions of tree YM amplitudes to YMS ones, from the $3$-point amplitudes
fixed by bootstrapping. In section IV, we derive the new expansions which
manifest the gauge invariance, by applying both the direct construction and
the recursive technic. Finally we end with a brief summery in section V.

##  II Background

For readers’ convenience, in this section we give a brief review of necessary
background. In subsection II.1, we introduce the concept of tree level
amplitudes of bi-adjoint scalar (BAS) theory and Yang-Mills scalar (YMS)
theory. Meanwhile, we draw the conclusion that any theory consisting solely of
massless particles can be expanded onto the KK BAS basis. In subsection II.2,
We briefly list the previously established soft theorems for scalars and
gluons in various theories. In subsection II.3, We provide a concise
introduction on how to recursively expand YMS tree-level amplitudes into the
KK basis.

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

The double color-ordered tree amplitudes in bi-adjoint scalar (BAS) theory
exclusively feature propagators for massless scalars. Each amplitude exhibits
simultaneous planarity with respect to two color orderings. We take the
$5$-point amplitude ${\cal A}_{\rm BAS}(1,2,3,4,5|1,4,2,3,5)$ as an example.
In Figure 1, both $(a1)$ and $(a2)$ correspond to the same tree diagram.
Specifically, a1 is associated with color ordering $(1,2,3,4,5)$, while $(a2)$
corresponds to color ordering $(1,4,2,3,5)$. However, in Figure 1, the tree
diagram represented by figure $b$ can only conform to color ordering
$(1,2,3,4,5)$ and does not satisfy color ordering $(1,4,2,3,5)$. Similarly,
one can draw additional tree diagrams that satisfy color ordering
$(1,2,3,4,5)$, but it will be observed that none of them conform to color
ordering $(1,4,2,3,5)$. Then the tree BAS amplitude ${\cal A}_{\rm
BAS}(1,2,3,4,5|1,4,2,3,5)$ can be computed as

$\displaystyle{\cal A}_{\rm BAS}(1,2,3,4,5|1,4,2,3,5)=(-1)^{n_{flip}}{1\over
s_{23}}{1\over s_{51}}\,.$ (1)

The Mandelstam variable $s_{i\cdots j}$ is defined as

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

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

Figure 1: Two $5$-point diagrams. Figure (a1) and (a2) are the same tree
diagram, while (a1) represents the color ordering $12345$ and (a2) represents
the color ordering $14235$.

Each double color-ordered BAS amplitude carries an overall sign
$(-1)^{n_{flip}}$, where $n_{flip}$ is determined by these two color
orderings. Readers should be aware that in this paper, when these two color
orderings are identical, $n_{flip}$ equals zero, implying that the overall
sign is $+$. The systematic method for determining $n_{flip}$ and evaluating
double color ordered partial amplitudes can be referenced in Cachazo:2013iea ,
and as it is not pertinent to the current paper, it will not be further
elaborated upon here. 111In fact, in reference A, the overall sign is defined
as $(-1)^{n-3+n_{flip}}$, where $n$ is the number or external legs. However,
for the convenience of our subsequent work in this paper, we adopt the
notation $(-1)^{n_{flip}}$. The advantage of the new convention stems from the
fact that when we remove a soft external scalar from an $n$-point amplitude,
it changes the number of external legs to $n-1$. Therefore, we desire that
this overall sign contains information solely about the relative ordering
between the two color orderings of the origin $n$-point amplitude, excluding
the influence of changes in the number of external legs on this sign.

The Yang-Mills-scalar (YMS) amplitudes pertain to scalars that are coupled
with gluons. In this context, we are specifically concerned with a subset of
YMS amplitudes known as single-trace YMS amplitudes. The single-trace YMS
amplitude, denoted as ${\cal A}_{\rm
YMS}(1,\cdots,n;\\{p_{1},\cdots,p_{m}\\}|\sigma_{n+m})$, comprises of $n$
external scalars represented by $\\{1,\cdots,n\\}$ and $m$ external gluons
labeled as $\\{p_{1},\cdots,p_{m}\\}$. The ordering $\sigma_{n+m}$ on the
right-hand side of $|$ encompasses all external legs within
$\\{1,\cdots,n\\}\cup\\{p_{1},\cdots,p_{m}\\}$. On the left-hand side of $|$,
$(1,\cdots,n)$ signifies an alternative overall ordering of external scalars,
which is also the reason it is referred to as ’single-trace’, while
$\\{p_{1},\cdots,p_{m}\\}$ constitutes an unordered set. In simpler terms, the
external scalars are BAS scalars, which have a dual ordering, whereas the
external gluons belong to a single ordering, $\sigma_{n+m}$.

Tree level amplitudes for any theory as long as they contain only massless
particles and cubic vertices can be expanded to double color ordered BAS
amplitudes, due to the observation that each Feynman diagram for pure
propagators is included in at least one BAS amplitude. In addition, for tree
diagrams involving higher point vertices, we can decompose them into BAS
amplitudes with solely cubic vertices by multiplying both the numerator and
denominator by the propagator D. An example is shown in Figure 2. Since each
Feynman diagram contributes propagators that can be mapped to BAS amplitudes,
along with a numerator, we can conclude that each tree-level amplitude can be
expanded into double color-ordered BAS amplitudes. These expanded amplitudes
have coefficients that are polynomials depending on Lorentz invariants created
by external kinematic variables.

Figure 2: Turn the $4$-point vertex to $3$-point ones. The bold line
corresponds to the inserted propagator $1/s_{12}$. This manipulation turns the
original numerator $N$ to $s_{12}N$, and split the original coupling constant
$g$ to two $\sqrt{g}$ for two cubic vertices.

In reality, not all BAS amplitudes are independent, therefore, such an
expansion requires the selection of appropriate basis. Such basis can be
determined by the well known Kleiss-Kuijf (KK) relation Kleiss:1988ne

$\displaystyle{\cal A}_{\rm
BAS}(1,\vec{\boldsymbol{{\alpha}}},n,\vec{\boldsymbol{{\beta}}}|\sigma_{n})=(-)^{\vec{|\boldsymbol{{\beta}}}|}\,{\cal
A}_{\rm
BAS}(1,\vec{\boldsymbol{{\alpha}}}\shuffle\vec{\boldsymbol{{\beta}}}^{T},n|\sigma_{n})\,,~{}~{}~{}$
(3)

where $\vec{\boldsymbol{{\alpha}}}$ and $\vec{\boldsymbol{{\beta}}}$ are two
ordered subsets of external scalars, and $\vec{\boldsymbol{{\beta}}}^{T}$
stands for the ordered set generated from $\vec{\boldsymbol{{\beta}}}$ by
reversing the original ordering, the $|\vec{\boldsymbol{{\beta}}}|$ on overall
sign represents the number of elements in subset $\vec{\boldsymbol{{\beta}}}$.
The symbol $\shuffle$ means summing over all possible shuffles of two ordered
sets $\vec{\boldsymbol{{\beta}}}_{1}$ and $\vec{\boldsymbol{{\beta}}}_{2}$,
i.e., all permutations in the set
$\vec{\boldsymbol{{\beta}}}_{1}\cup\vec{\boldsymbol{{\beta}}}_{2}$ while
preserving the orderings of $\vec{\boldsymbol{{\beta}}}_{1}$ and
$\vec{\boldsymbol{{\beta}}}_{2}$. For instance, suppose
$\vec{\boldsymbol{{\beta}}}_{1}=\\{1,2\\}$ and
$\vec{\boldsymbol{{\beta}}}_{2}=\\{3,4\\}$, then

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

One should note that the $n$-point BAS amplitude ${\cal A}_{\rm
BAS}(1,\vec{\boldsymbol{{\alpha}}},n,\vec{\boldsymbol{{\beta}}}|\sigma_{n})$
at the l.h.s of (3) carries two color orderings. We can also have an another
analogous KK relation for the second color ordering $\sigma_{n}$. Therefore,
the KK relation implies that the basis can be chosen as BAS amplitudes ${\cal
A}_{\rm BAS}(1,\sigma_{1},n|1,\sigma_{2},n)$, with $1$ and $n$ are fixed at
two ends in each ordering. Such basis is called the KK BAS basis. Based on the
discussion above, any amplitude which includes only massless particles can be
expanded to this basis222The well known Bern-Carrasco-Johansson (BCJ) relation
Bern:2008qj ; Chiodaroli:2014xia ; Johansson:2015oia ; Johansson:2019dnu
links BAS amplitudes in the KK basis together, and the independent BAS
amplitudes can be obtained by fixing three legs at three particular positions
in the color orderings. However, in the BCJ relation, 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, one can find the expanded formula in which the coefficients contain
no poles. In this paper, we opt for the KK basis, as we aim to ensure that all
poles of tree amplitudes are encompassed within the basis, with coefficients
solely serving as numerators.. In the expansion, the KK basis supplies the
propagators, while the coefficients in the expansions provide the numerators.

In this paper, we will concentrate on the expansion of pure YM amplitudes. The
$n$-point YM amplitude ${\cal A}_{\rm YM}(\sigma_{n})$ with color ordering
$\sigma_{n}$ can be expanded to KK BAS basis as

$\displaystyle{\cal A}_{\rm YM}(\sigma_{n})$ $\displaystyle=$
$\displaystyle\sum_{\alpha_{n-2}}\,{\cal
C}(\alpha_{n-2},\epsilon_{i},k_{i})\,{\cal A}_{\rm
BAS}(1,\alpha_{n-2},n|\sigma_{n})\,,~{}~{}~{}$ (5)

where $\alpha_{n-2}$ denotes color orderings among external legs of
$\\{2,\cdots,n-1\\}$. The double copy structure Kawai:1985xq ; Bern:2008qj ;
Chiodaroli:2014xia ; Johansson:2015oia ; Johansson:2019dnu  indicates that the
coefficient ${\cal C}(\alpha_{n-2},\epsilon_{i},k_{i})$ depends on
polarization vectors $\epsilon_{i}$ and momenta $k_{i}$ of external gluons, as
well as the ordering $\alpha_{n-2}$, but remains independent of ordering
$\sigma_{n}$333Originally, the concept of the double copy implies that the
General Relativity (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 derived by inverting BAS amplitudes. Our assumption that the coefficients
depend on only one color ordering is consistent with the original version, as
discussed in Zhou:2022orv .. Therefore, within this expansion, the
investigation of the structure of YM amplitudes is transformed into the study
of the structure of the coefficients ${\cal
C}(\alpha_{n-2},\epsilon_{i},k_{i})$. These coefficients are known as BCJ
numerators Bern:2010yg ; Fu:2017uzt .

###  II.2 Soft theorems for external scalars and gluons

In this subsection, we provide a brief overview of the soft scattering
theorems for external scalars and gluons, which are crucial for the
discussions in the subsequent sections.

For the double color ordered BAS amplitude ${\cal A}_{\rm
BAS}(1,\cdots,n|\sigma_{n})$, we rescale $k_{i}$ as $k_{i}\to\tau k_{i}$, and
expand the amplitude with respect to $\tau$,

${\cal A}_{\rm BAS}(1,\cdots,n|\sigma_{n})={\cal A}^{(0)_{i}}_{\rm
BAS}(1,\cdots,n|\sigma_{n})+{\cal A}^{(1)_{i}}_{\rm
BAS}(1,\cdots,n|\sigma_{n})+\mathcal{O}(\tau).$ (6)

The leading-order contribution ${\cal A}^{(0)_{i}}_{\rm
BAS}(1,\cdots,n|\sigma_{n})$ arises explicitly from the $2$-point channels
$1/s_{1(i+1)}$ and $1/s_{(i-1)i}$, which are at the $1/\tau$ order. In other
words,

$\displaystyle{\cal A}^{(0)_{i}}_{\rm BAS}(1,\cdots,n|\sigma_{n})$
$\displaystyle=$ $\displaystyle{1\over\tau}\Big{(}{\delta_{i(i+1)}\over
s_{i(i+1)}}+{\delta_{(i-1)i}\over s_{(i-1)i}}\Big{)}\,{\cal A}_{\rm
BAS}(1,\cdots,i-1,\not{i},i+1,\cdots,n|\sigma_{n}\setminus i)$ (7)
$\displaystyle=$ $\displaystyle S^{(0)_{i}}_{s}\,{\cal A}_{\rm
BAS}(1,\cdots,i-1,\not{i},i+1,\cdots,n|\sigma_{n}\setminus i)\,,~{}~{}~{}$

where $\not{i}$ stands for removing the leg $i$, $\sigma_{n}\setminus i$ means
the color ordering generated from $\sigma_{n}$ by eliminating $i$. In Equation
(7), $\delta_{ab}$ is not the well-known Kronecker symbol. Its value is
determined by the second color ordering $\sigma_{n}$. When $a,b$ in
$\sigma_{n}$ are not adjacent, $\delta_{ab}$ takes the value of $0$. When
$a,b$ in $\sigma_{n}$ are adjacent and maintain the same order as in the first
color ordering $(1,\cdots,n)$, $\delta_{ab}$ takes the value of $1$. When
$a,b$ in $\sigma_{n}$ are adjacent but have the reverse order compared to
their order in the first color ordering $(1,\cdots,n)$, it takes the value of
$-1$. The leading soft operator $S^{(0)}_{s}(i)$ for the scalar $i$ is
observed as

$\displaystyle S^{(0)_{i}}_{s}\equiv{1\over\tau}\,\Big{(}{\delta_{i(i+1)}\over
s_{i(i+1)}}+{\delta_{(i-1)i}\over s_{(i-1)i}}\Big{)}\,.~{}~{}~{}~{}$ (8)

For example, for $4$-point amplitude ${\cal A}_{BAS}(1234|1234)$, we rescale
$k_{4}$ as $k_{4}\to\tau k_{4}$, then by our notation, we have

$\displaystyle{\cal A}_{BAS}^{(0)}(1234|1234)=$
$\displaystyle\frac{1}{\tau}(\frac{1}{s_{41}}+\frac{1}{s_{34}})$ (9)
$\displaystyle{\cal A}_{BAS}(123|123)=$ $\displaystyle+1.$

According to our definition of $\delta_{ab}$, we know
$\delta_{34}=\delta_{41}=1$ since $3,4,1$ remains the same in both color
orderings. So we verified

$\displaystyle{\cal A}^{(0)_{4}}_{\rm BAS}(1234|1234)$ $\displaystyle=$
$\displaystyle{1\over\tau}\Big{(}{\delta_{41}\over s_{41}}+{\delta_{34}\over
s_{34}}\Big{)}\,{\cal A}_{\rm BAS}(123|123).$ (10)

For $4$-point amplitude ${\cal A}_{BAS}(1234|1243)$, we also rescale $k_{4}$
as $k_{4}\to\tau k_{4}$, then by our notation, we have

$\displaystyle{\cal A}_{BAS}^{(0)}(1234|1243)=$
$\displaystyle\frac{1}{\tau}(-\frac{1}{s_{34}})$ (11) $\displaystyle{\cal
A}_{BAS}(123|123)=$ $\displaystyle+1.$

Because 3 and 4 have opposite orders in the two color orderings $1234$ and
$1243$, we have $\delta_{34}=-1$. And since 4 and 1 are not adjacent in the
second color ordering $1243$, we have $\delta_{41}=0$. So we verified

$\displaystyle{\cal A}^{(0)_{4}}_{\rm BAS}(1234|1243)$ $\displaystyle=$
$\displaystyle{1\over\tau}\Big{(}{\delta_{41}\over s_{41}}+{\delta_{34}\over
s_{34}}\Big{)}\,{\cal A}_{\rm BAS}(123|123).$ (12)

In our previous work, we introduced an assumption that, the soft operator for
should be universal across different theories. For example, in the YMS theory,
scalar particles, and in the BAS theory, scalar particles, both share the same
form for their respective soft operators. Then the leading contribution of YMS
amplitude is

$\displaystyle{\cal A}^{(0)_{i}}_{\rm
YMS}(1,\cdots,n;p_{1},\cdots,p_{m}|\sigma_{n+m})$ $\displaystyle=$
$\displaystyle S^{(0)_{i}}_{s}\,{\cal A}_{\rm
YMS}(1,\cdots,i-1,\not{i},i+1,\cdots,n;p_{1},\cdots,p_{m}|\sigma_{n+m}\setminus
i)\,.~{}~{}~{}$ (13)

Based on (8), it can be observed that this soft operator $S^{(0)_{i}}_{s}$
does not act on external gluons.

The soft theorems for external gluons at leading and sub-leading orders can be
obtained via various approaches Casali:2014xpa ; Schwab:2014xua ; Afkhami-
Jeddi:2014fia ; Zhou:2022orv . Such soft theorems are given as

$\displaystyle{\cal A}^{(0)_{p_{i}}}_{\rm
YMS}(1,\cdots,n;p_{1},\cdots,p_{m}|\sigma_{n+m})$ $\displaystyle=$
$\displaystyle S^{(0)_{p_{i}}}_{g}\,{\cal A}_{\rm
YMS}(1,\cdots,n;p_{1},\cdots,\not{p_{i}},\cdots,p_{m}|\sigma_{n+m}\setminus
p_{i})\,,~{}~{}~{}$ (14)

and

$\displaystyle{\cal A}^{(1)_{p_{i}}}_{\rm
YMS}(1,\cdots,n;p_{1},\cdots,p_{m}|\sigma_{n+m})$ $\displaystyle=$
$\displaystyle S^{(1)_{p_{i}}}_{g}\,{\cal A}_{\rm
YMS}(1,\cdots,n;p_{1},\cdots,\not{p_{i}},\cdots,p_{m}|\sigma_{n+m}\setminus
p_{i})\,,~{}~{}~{}$ (15)

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

$\displaystyle
S^{(0)_{p_{i}}}_{g}={1\over\tau}\,\sum_{a\in\sigma}\,{\delta_{ap_{i}}\,(\epsilon_{p_{i}}\cdot
k_{a})\over s_{ap_{i}}}\,,~{}~{}$ (16)

and

$\displaystyle
S^{(1)_{p_{i}}}_{g}=\sum_{a\in\sigma}\,{\delta_{ap_{i}}\,\big{(}\epsilon_{p_{i}}\cdot
J_{a}\cdot k_{p_{i}}\big{)}\over s_{ap_{i}}}\,,~{}~{}$ (17)

respectively Casali:2014xpa ; Schwab:2014xua . Here, $\delta_{ab}$ is solely
determined by the second color ordering $\sigma_{n+m}$. When $a,b$ are
adjacent in $\sigma_{n+m}$ (in a cyclically symmetric sense), $\delta_{ab}$
takes the value of $1$ if $a$ precedes $b$, and $-1$ if $a$ follows $b$. When
$a,b$ are not adjacent in $\sigma_{n+m}$, $\delta_{ab}$ takes the value of
$0$. $J_{a}^{\mu\nu}$ is the angular momentum operator, defined as
$J_{a}^{\mu\nu}=k_{a}^{[\mu}\,{\partial\over\partial k_{a,\nu]}}$. In (16) and
(17), 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 (17) for external gluons plays a central role in
the subsequent sections. Here, we present some valuable results regarding the
operation of this operator. The angular momentum operator $J_{a}^{\mu\nu}$
acts on the Lorentz vector $k^{\rho}_{a}$ with the orbital component of the
generator and on $\epsilon^{\rho}_{a}$ with the spin component of the
generator in the vector representation.

$\displaystyle J_{a}^{\mu\nu}\,k_{a}^{\rho}=k_{a}^{[\mu}\,{\partial
k_{a}^{\rho}\over\partial
k_{a,\nu]}}\,,~{}~{}~{}~{}J_{a}^{\mu\nu}\,\epsilon_{a}^{\rho}=\big{(}\eta^{\nu\rho}\,\delta^{\mu}_{\sigma}-\eta^{\mu\rho}\,\delta^{\nu}_{\sigma}\big{)}\,\epsilon^{\sigma}_{a}\,.$
(18)

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

$\displaystyle S^{(1)_{p}}_{g}$ $\displaystyle=$
$\displaystyle-\sum_{V_{a}}\,{\delta_{ap}\over s_{ap}}\,V_{a}\cdot
f_{p}\cdot{\partial\over\partial V_{a}}\,,~{}~{}$ (19)

due to the observation that the amplitude is linear in each polarization
vector. We observe that in (19), the summation over $V_{a}$ includes all
Lorentz vectors, encompassing both momenta and polarizations. The operator
(19) is a differential operator that adheres to Leibniz’s rule. Employing
(19), we promptly obtain

$\displaystyle\big{(}S^{(1)_{p}}_{g}\,k_{a}\big{)}\cdot V=-{\delta_{ap}\over
s_{ap}}\,(k_{a}\cdot f_{p}\cdot
V)\,,~{}~{}~{}~{}\big{(}S^{(1)_{p}}_{g}\,\epsilon_{a}\big{)}\cdot
V=-{\delta_{ap}\over s_{ap}}\,(\epsilon_{a}\cdot f_{p}\cdot V)\,,~{}~{}~{}~{}$
(20)

where $V$ is an arbitrary Lorentz vector, and

$\displaystyle V_{1}\cdot\big{(}S^{(1)_{p}}_{g}\,f_{a}\big{)}\cdot
V_{2}={\delta_{ap}\over s_{ap}}\,V_{1}\cdot(f_{p}\cdot f_{a}-f_{a}\cdot
f_{p})\cdot V_{2}\,,~{}~{}~{}~{}$ (21)

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.

###  II.3 Recursive expansion of single-trace YMS amplitudes

The discussion for the expansion of tree level amplitudes in the previous
subsection.II.1 indicates that the YMS amplitudes can also be expanded to KK
BAS basis. This expansion can be achieved by applying the following recursive
expansion iteratively:

$\displaystyle{\cal A}_{\rm
YMS}(1,\cdots,n;\\{p_{1},\cdots,p_{m}\\}|\sigma_{n+m})$ (22) $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{p}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot
Y_{\vec{\boldsymbol{{\alpha}}}}\big{)}\,{\cal A}_{\rm
YMS}(1,\\{2,\cdots,n-1\\}\shuffle\\{\vec{\boldsymbol{{\alpha}}},p\\},n;\\{p_{1},\cdots,p_{m}\\}\setminus\\{p\cup\boldsymbol{{\alpha}}\\}|\sigma_{n+m})\,,~{}~{}$

where $p$ is the fiducial gluon which can be chosen as any element in
$\\{p_{1},\cdots,p_{m}\\}$, and $\boldsymbol{{\alpha}}$ are subsets of
$\\{p_{1},\cdots,p_{m}\\}\setminus p$ which is allowed to be empty. When
$\boldsymbol{{\alpha}}=\\{p_{1},\cdots,p_{m}\\}\setminus p$, the YMS
amplitudes in the second line of (22) are reduced to 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

$\displaystyle
F^{\mu\nu}_{\vec{\boldsymbol{{\alpha}}}}\equiv\big{(}f_{{\alpha}_{k}}\cdot
f_{{\alpha}_{k-1}}\cdots f_{{\alpha}_{2}}\cdot
f_{{\alpha}_{1}}\big{)}^{\mu\nu}\,,$ (23)

for $\vec{\boldsymbol{{\alpha}}}=\\{{\alpha}_{1},\cdots{\alpha}_{k}\\}$. The
combinatory momentum $Y_{\vec{\boldsymbol{{\alpha}}}}$ is the summation of
momenta carried by external gluons at the l.h.s of ${\alpha}_{1}$ in the color
ordering $\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}}$. The
summation in (38) is over all un-equivalent ordered sets
$\vec{\boldsymbol{{\alpha}}}$. The recursive expansion (22) is found via
various methods Casali:2014xpa ; Schwab:2014xua ; Afkhami-Jeddi:2014fia ;
Zhou:2022orv . It should be noted that this recurrence relation can be
independently derived through the recursive construction based on soft
behaviors. For detailed information, please refer to the reference
Zhou:2022orv . In the recursive expansion (22), the YMS amplitude undergoes
expansion, resulting in YMS amplitudes with fewer gluons and more scalars. By
iteratively applying this expansion, it becomes possible to expand any YMS
amplitude into pure BAS amplitudes.

Furthermore, we can also observe that in the recursive expansion (22), the
gauge invariance for each gluon in $\\{P_{1},\cdots,p_{m}\\}\setminus p$ 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
(22) iteratively, fiducial gluons will be chosen at every step. Consequently,
in the resulting expansion to pure BAS amplitudes, the gauge invariance for
each gluon will be compromised. To achieve a manifestly gauge-invariant
expansion, an alternative recursive expansion should be employed

$\displaystyle{\cal A}_{\rm
YMS}(1,\cdots,n;\\{p_{1},\cdots,p_{m}\\}|\sigma_{n+m})$ (24) $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,{k_{r}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot Y_{\vec{\boldsymbol{{\alpha}}}}\over
k_{r}\cdot k_{p_{1}\cdots p_{m}}}\,{\cal A}_{\rm
YMS}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{1},\cdots,p_{m}\\}\setminus\boldsymbol{{\alpha}}|\sigma_{n+m})\,,~{}~{}$

where $k_{r}$ is a reference massless momentum. In this context, the notations
align with those in (22), and $\boldsymbol{{\alpha}}$ represents subsets of
${p_{1},\cdots,p_{m}}$. The formula (24) was originally discovered by Clifford
Cheung and James Mangan within the framework of the covariant double copy. It
was also independently derived by applying the recursive construction based on
soft theorems[]. Notably, the expansion (24) doesn’t necessitate the use of a
fiducial gluon, and it inherently exhibits gauge invariance for all
polarizations. Through iterative application of (22), one eventually arrives
at the manifestly gauge-invariant expansion of YMS amplitudes into the KK BAS
basis.

##  III Expand YM amplitudes to YMS ones

In this section, we derive the color-ordered Yang-Mills (YM) amplitudes as
part of the formula for expanding YM amplitudes into Yang-Mills-scalar (YMS)
amplitudes. The technique employed in this section is based on the analysis of
the sub-leading-order soft behavior. It represents an improvement over the
method utilized in Zhou:2022orv . In comparison with the previous method from
Zhou:2022orv , the new approach allows us to construct the expanded formula
recursively, starting from the lowest $3$-point amplitudes that can be
uniquely determined through bootstrapping. This is achieved without the need
to rely on other frameworks, such as Feynman rules or CHY formalism. However,
it should be noted that the resulting expansion in this section does not
exhibit gauge invariance for all polarizations. The manifestation of gauge
invariance is the objective of the next section.

###  III.1 $3$-point amplitudes

To ensure our construction is self-contained, without relying on Feynman rules
or the CHY formula, we fix the 3-point color-ordered Yang-Mills (YM)
amplitudes using a bootstrapping method. Our construction is primarily based
on the following ansatz:

1) The amplitude ${\cal A}_{\rm YM}(1,2,3)$ with the color ordering $(1,2,3)$
has a mass dimension of $1$, and due to the absence of factorization channels
for the lowest-point amplitudes, it does not contain any pole structures.

2) This amplitude is linearly dependent on polarization vectors
$\epsilon_{1}$, $\epsilon_{2}$ and $\epsilon_{3}$.

3) Due to cyclic symmetry, ${\cal A}_{\rm YM}(1,2,3)$ remains invariant under
permutation transformations $(1\to 2,2\to 3,3\to 1)$.

In this manner, we establish the foundational elements for our self-contained
construction as

$\displaystyle{\cal A}_{\rm
YM}(1,2,3)=(k_{1}\cdot\epsilon_{2})\,(\epsilon_{3}\cdot\epsilon_{1})+(k_{2}\cdot\epsilon_{3})\,(\epsilon_{1}\cdot\epsilon_{2})+(k_{3}\cdot\epsilon_{1})\,(\epsilon_{2}\cdot\epsilon_{3})\,.~{}~{}$
(25)

It’s worth noting that replacing $k_{1}\cdot\epsilon_{2}$,
$k_{2}\cdot\epsilon_{3}$, and $k_{3}\cdot\epsilon_{1}$ with
$k_{3}\cdot\epsilon_{2}$, $k_{1}\cdot\epsilon_{3}$, and
$k_{2}\cdot\epsilon_{1}$, respectively, results in an overall sign change.
This occurs due to both momentum conservation and the on-shell condition
$k_{i}\cdot\epsilon_{i}=0$.

We can expand ${\cal A}_{\rm YM}(1,2,3)$ onto YMS amplitudes and BAS
amplitudes in the manner of equation (22), by ${\cal A}_{\rm
YMS}(1,3;2|1,2,3)=(\epsilon_{2}\cdot k_{1})\,{\cal A}_{\rm BAS}(1,2,3|1,2,3)$
and ${\cal A}_{\rm BAS}(1,2,3|1,2,3)=1$, we have

$\displaystyle{\cal A}_{\rm YM}(1,2,3)$ $\displaystyle=$
$\displaystyle\big{(}\epsilon_{3}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,3;2|1,2,3)$ (26) $\displaystyle+\big{(}\epsilon_{3}\cdot
f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm BAS}(1,2,3|1,2,3)\,.~{}~{}$

Then the double copy structure indicates the expansion for general $3$-point
color ordered YM amplitude

$\displaystyle{\cal A}_{\rm YM}(\sigma_{3})$ $\displaystyle=$
$\displaystyle\big{(}\epsilon_{3}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,3;2|\sigma_{3})$ (27) $\displaystyle+\big{(}\epsilon_{3}\cdot
f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm BAS}(1,2,3|\sigma_{3})\,,~{}~{}$

where $\sigma_{3}$ is an arbitrary ordering among legs in $\\{1,2,3\\}$.
Equation (27) serves as the initial step for the recursive construction in
this section.

###  III.2 Recursive construction for $4$-point amplitudes

In this subsection, we derive the expansion of 4-point Yang-Mills (YM)
amplitudes based on the previously expanded formula for $3$-point amplitudes
in equation (27). This is accomplished by examining the sub-leading-order soft
behavior of the 4-point amplitude. We label the external legs of the $4$-point
amplitude ${\cal A}{\rm YM}(\sigma_{4})$ as $\sigma_{4}=\\{1,2,3,s\\}$ and
choose the KK basis such that legs $1$ and $3$ are fixed at two ends in the
left color ordering. We focus on the soft behavior of the external particle
$s$ by rescaling $k_{s}$ as $k_{s}\to\tau k_{s}$ and expanding ${\cal A}{\rm
YM}(\sigma_{4})$ with respect to $\tau$. This process of constructing 4-point
amplitudes from 3-point amplitudes provides valuable insights for addressing
more general cases in our subsequent discussions.

According to the soft theorem in (15) and (17), the sub-leading contribution
is given as

$\displaystyle{\cal A}^{(1)_{s}}_{\rm YM}(\sigma_{4})$ $\displaystyle=$
$\displaystyle S^{(1)_{s}}_{g}\,{\cal A}_{\rm YM}(\sigma_{4}\setminus s)$ (28)
$\displaystyle=$ $\displaystyle
S^{(1)_{s}}_{g}\,\Big{[}\big{(}\epsilon_{3}\cdot\epsilon_{1}\big{)}\,{\cal
A}_{\rm YMS}(1,3;2|\sigma_{4}\setminus s)+\big{(}\epsilon_{3}\cdot
f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm BAS}(1,2,3|\sigma_{4}\setminus
s)\Big{]}$ $\displaystyle=$ $\displaystyle P_{1}+P_{2}\,,~{}~{}$

The second line is derived by substituting the expansion (27) into the first
line. The sub-leading contribution is then split into two parts, denoted as
$P_{1}$ and $P_{2}$:

$\displaystyle P_{1}$
$\displaystyle=\left(\epsilon_{3}\cdot\epsilon_{1}\right)\left[S^{(1)_{s}}_{g}{\cal
A}_{\rm YMS}(1,3;2|\sigma_{4}\setminus s)\right]+\left(\epsilon_{3}\cdot
f_{2}\cdot\epsilon_{1}\right)\left[S^{(1)_{s}}_{g}{\cal A}_{\rm
BAS}(1,2,3|\sigma_{4}\setminus s)\right],$ $\displaystyle P_{2}$
$\displaystyle=\left[S^{(1)_{s}}_{g}\left(\epsilon_{3}\cdot\epsilon_{1}\right)\right]{\cal
A}_{\rm YMS}(1,3;2|\sigma_{4}\setminus
s)+\left[S^{(1)_{s}}_{g}\left(\epsilon_{3}\cdot
f_{2}\cdot\epsilon_{1}\right)\right]{\cal A}_{\rm
BAS}(1,2,3|\sigma_{4}\setminus s).$ (29)

Here, $P_{1}$ is obtained by applying the sub-leading soft operator
$S^{(1)_{s}}_{g}$ to YMS or BAS amplitudes, and $P_{2}$ is obtained by acting
$S^{(1)_{s}}_{g}$ on coefficients.

Our objective is to deduce the expanded formula of ${\cal A}_{\rm
YM}(\sigma_{4})$ from the soft behavior of ${\cal A}^{(1)s}_{\rm
YM}(\sigma_{4})$. Hence, it becomes imperative to construe ${\cal
A}^{(1)s}_{\rm YM}(\sigma_{4})$ as a synthesis of the soft behaviors inherent
in the constituent components of the expansion. Here, by ”components,” we
refer to the individual constituents akin to ${\cal A}^{(1)_{s}}_{\rm YMS}$
and ${\cal A}^{(0)_{s}}_{\rm BAS}$ within the expansion. For the $P_{1}$ part,
the soft theorem suggests

$\displaystyle P_{1}$ $\displaystyle=$
$\displaystyle\big{(}\epsilon_{3}\cdot\epsilon_{1}\big{)}\,{\cal
A}^{(1)_{s}}_{\rm YMS}(1,3;s,2|\sigma_{4})+\big{(}\epsilon_{3}\cdot
f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}^{(1)_{s}}_{\rm
YMS}(1,2,3;s|\sigma_{4})\,.~{}~{}$ (30)

The second part $P_{2}$ can be evaluated by applying relations (20) and (21),

$\displaystyle P_{2}$ $\displaystyle=$ $\displaystyle\big{(}\epsilon_{3}\cdot
f_{s}\cdot\epsilon_{1}\big{)}\,\Big{(}{\delta_{1s}\over
s_{1s}}+{\delta_{s3}\over s_{s3}}\Big{)}\,{\cal A}_{\rm
YMS}(1,3;2|\sigma_{4}\setminus s)$ (31)
$\displaystyle+\big{(}\epsilon_{3}\cdot f_{2}\cdot
f_{s}\cdot\epsilon_{1}\big{)}\,\Big{(}{\delta_{1s}\over
s_{1s}}+{\delta_{s2}\over s_{s2}}\Big{)}\,{\cal A}_{\rm
BAS}(1,2,3|\sigma_{4}\setminus s)$ $\displaystyle+\big{(}\epsilon_{3}\cdot
f_{s}\cdot f_{2}\cdot\epsilon_{1}\big{)}\,\Big{(}{\delta_{2s}\over
s_{2s}}+{\delta_{s3}\over s_{s3}}\Big{)}\,{\cal A}_{\rm
BAS}(1,2,3|\sigma_{4}\setminus s)\,.~{}~{}$

Since the symbol $\delta_{ab}$ will not appear in the expansion of ${\cal
A}_{\rm YM}(\sigma_{4})$, it should be absorbed into the soft behaviors of YMS
amplitudes. By applying the soft theorem (7) and (8) for the BAS scalars, we
can recognize that

$\displaystyle\Big{(}{\delta_{1s}\over s_{1s}}+{\delta_{s2}\over
s_{s2}}\Big{)}\,{\cal A}_{\rm BAS}(1,2,3|\sigma_{4}\setminus s)=\tau\,{\cal
A}^{(0)_{s}}_{\rm BAS}(1,s,2,3|\sigma_{4})\,,$
$\displaystyle\Big{(}{\delta_{2s}\over s_{2s}}+{\delta_{s3}\over
s_{s3}}\Big{)}\,{\cal A}_{\rm BAS}(1,2,3|\sigma_{4}\setminus s)=\tau\,{\cal
A}^{(0)_{s}}_{\rm BAS}(1,2,s,3|\sigma_{4})\,.~{}~{}$ (32)

This observation eliminates $\delta_{ab}$ in second and third lines in (31).
Then we turn to the first line in (31). Expanding ${\cal A}_{\rm
YMS}(1,3;2|\sigma_{4}\setminus s)$ as in (22), we get

$\displaystyle\Big{(}{\delta_{1s}\over s_{1s}}+{\delta_{s3}\over
s_{s3}}\Big{)}\,{\cal A}_{\rm YMS}(1,3;2|\sigma_{4}\setminus s)$ (33)
$\displaystyle=$ $\displaystyle(\epsilon_{2}\cdot
k_{1})\,\Big{(}{\delta_{1s}\over s_{1s}}+{\delta_{s3}\over
s_{s3}}\Big{)}\,{\cal A}_{\rm BAS}(1,2,3|\sigma_{4}\setminus s)$
$\displaystyle=$ $\displaystyle(\epsilon_{2}\cdot
k_{1})\,\Big{(}{\delta_{1s}\over s_{1s}}+{\delta_{s2}\over
s_{s2}}+{\delta_{2s}\over s_{2s}}+{\delta_{s3}\over s_{s3}}\Big{)}\,{\cal
A}_{\rm BAS}(1,2,3|\sigma_{4}\setminus s)$ $\displaystyle=$
$\displaystyle\tau\,(\epsilon_{2}\cdot k_{1})\,{\cal A}^{(0)_{s}}_{\rm
BAS}(1,2\shuffle s,3|\sigma_{4})\,.~{}~{}$

The second equality relies on the property $\delta_{ab}=-\delta_{ba}$, while
the third equality is predicated on the soft theorem as given in equation (7).
Adding all of these terms together, we obtain

$\displaystyle{\cal A}^{(1)_{s}}_{\rm YM}(\sigma_{4})$ $\displaystyle=$
$\displaystyle\big{(}\epsilon_{3}\cdot\epsilon_{1}\big{)}\,{\cal
A}^{(1)_{s}}_{\rm YMS}(1,3;s,2|\sigma_{4})+\big{(}\epsilon_{3}\cdot
f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}^{(1)_{s}}_{\rm
YMS}(1,2,3;s|\sigma_{4})$ (34)
$\displaystyle+\tau\,\Big{[}\big{(}\epsilon_{3}\cdot
f_{s}\cdot\epsilon_{1}\big{)}\big{(}\epsilon_{2}\cdot k_{1}\big{)}{\cal
A}^{(0)_{s}}_{\rm BAS}(1,2\shuffle s,3|\sigma_{4})+\big{(}\epsilon_{3}\cdot
f_{2}\cdot f_{s}\cdot\epsilon_{1}\big{)}\,{\cal A}^{(0)_{s}}_{\rm
BAS}(1,s,2,3|\sigma_{4})$ $\displaystyle+\big{(}\epsilon_{3}\cdot f_{s}\cdot
f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}^{(0)_{s}}_{\rm
BAS}(1,2,s,3|\sigma_{4})\Big{]}\,.~{}~{}$

Some discussions are warranted regarding the expansion in (37). This expansion
is derived by examining the soft behavior at the sub-leading order. The reason
for selecting the sub-leading order rather than the leading one is that the
leading-order contributions from the third, fourth, and fifth terms in (37)
are of order $\tau^{0}$, while ${\cal A}^{(0)_{s}}_{\rm YM}(\sigma_{4})$ is of
order $\tau^{-1}$. Consequently, these terms cannot be detected through the
leading-order soft behavior of ${\cal A}^{(0)_{s}}_{\rm YM}(\sigma_{4})$. This
leads to a natural question: does the full expansion include a term whose
leading-order soft behavior is of order $\tau^{1}$, making it undetectable
when examining ${\cal A}^{(1)_{s}}_{\rm YM}(\sigma_{4})$?. We can rule out
this possibility through the following argument. If such a term were to exist,
it must involve a coefficient that is bilinear in $k_{s}$, since the leading-
order soft behavior of each YMS amplitude is of order $\tau^{-1}$.
Consequently, the symmetry between legs $s$ and $2$ would imply the existence
of another term featuring a coefficient that is bilinear in $k_{2}$. However,
the new term with a coefficient bilinear in $k_{2}$ should be detectable when
examining ${\cal A}^{(1)_{s}}_{\rm YM}(\sigma_{4})$ because mass dimension
considerations prohibit the coefficient from being bilinear in both $k_{2}$
and $k_{s}$. Intriguingly, the associated contribution is conspicuously absent
in (34). Consequently, we can confidently assert that the hypothesized
undetectable term with leading-order behavior of $\tau^{1}$ does not exist.

Next, we will go through each term step by step to demonstrate how to
reconstruct the original amplitude ${\cal A}_{\rm YM}(\sigma_{4})$ from this
soft behavior.

  * •

$\big{(}\epsilon_{3}\cdot\epsilon_{1}\big{)}\,{\cal A}^{(1)_{s}}_{\rm
YMS}(1,3;s,2|\sigma_{4})$
It is evident that this term arises from the expansion represented by
$\big{(}\epsilon_{3}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,3;s,2|\sigma_{4})$.

  * •

$\big{(}\epsilon_{3}\cdot f_{2}\cdot\epsilon_{1}\big{)}\,{\cal
A}^{(1)_{s}}_{\rm YMS}(1,2,3;s|\sigma_{4})$
Due to the fact that $k_{s}$ does not contribute to the leading-order terms,
the coefficients resembling $(\epsilon_{3}\cdot
f_{2}^{\prime}\cdot\epsilon_{1})$ are all at leading order
$\big{(}\epsilon_{3}\cdot f_{2}\cdot\epsilon_{1}\big{)}$. Where

$(f^{\prime}_{2})^{\mu\nu}\equiv(k_{2}^{\mu}+xk_{s}^{\mu})\epsilon_{2}^{\nu}-\epsilon_{2}^{\mu}(k_{2}^{\nu}+yk_{s}^{\nu})$
(35)

with $x$, $y$ can take arbitrary constant values. Therefore, this term may
originate from $(\epsilon_{3}\cdot f_{2}^{\prime}\cdot\epsilon_{1}){\cal
A}_{\rm YMS}(1,2,3;s|\sigma_{4})$ in the expansion. On the other hand, besides
$\big{(}\epsilon_{3}\cdot f_{2}\cdot\epsilon_{1}\big{)}\,{\cal
A}^{(1)_{s}}_{\rm YMS}(1,2,3;s|\sigma_{4})$, the sub-leading order of
$(\epsilon_{3}\cdot f_{2}^{\prime}\cdot\epsilon_{1}){\cal A}_{\rm
YMS}(1,2,3;s|\sigma_{4})$ after rescale $k_{s}\to\tau k_{s}$ also contains

$\tau\left(x(\epsilon_{3}\cdot
k_{s})(\epsilon_{2}\cdot\epsilon_{1})-y(\epsilon_{3}\cdot\epsilon_{2})(k_{s}\cdot\epsilon_{1})\right){\cal
A}^{(0)_{s}}_{\rm YMS}(1,2,3;s|\sigma_{4}).$ (36)

Compared with (34), we observe that this kind of sub-leading soft behavior has
not been detected, i,e, $x=y=0$. In summary, the second term arises from
$\big{(}\epsilon_{3}\cdot f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,2,3;s|\sigma_{4})$

  * •

$\big{(}\epsilon_{3}\cdot
f_{s}\cdot\epsilon_{1}\big{)}\big{(}\epsilon_{2}\cdot k_{1}\big{)}{\cal
A}^{(0)_{s}}_{\rm BAS}(1,2\shuffle s,3|\sigma_{4})$
The treatment of the third term, however, can be somewhat intricate. In
addition, We anticipate that the expansion of ${\cal A}_{\rm YM}(\sigma_{4})$
should remain invariant under the interchange of $2$ and $s$ because both $2$
and $s$ represent on-shell massless particles that have not been fixed at any
ends in the color orderings. Therefore, the appearance of
$\big{(}\epsilon_{3}\cdot f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,2,3;s|\sigma_{4})$ in the expansion of ${\cal A}_{\rm YM}(\sigma_{4})$
inevitably leads to the appearance of $\big{(}\epsilon_{3}\cdot
f_{s}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm YMS}(1,s,3;2|\sigma_{4})$. On the
other hand, we can derived the following relationship:

${\cal A}^{(0)_{s}}_{\rm YMS}(1,s,3;2|\sigma_{4})=(\epsilon_{2}\cdot
k_{1})\,{\cal A}^{(0)_{s}}_{\rm BAS}(1,2\shuffle s,3|\sigma_{4}).$

This equality can be confirmed by substituting the expansion (22) and
observing that $k_{s}$ does not contribute to $Y_{2}$ at the leading order.
This confirms that the third term in equation (34) indeed arises from the sub-
leading order soft behavior of $\big{(}\epsilon_{3}\cdot
f_{s}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm YMS}(1,s,3;2|\sigma_{4})$.

  * •

$\big{(}\epsilon_{3}\cdot f_{2}\cdot f_{s}\cdot\epsilon_{1}\big{)}\,{\cal
A}^{(0)_{s}}_{\rm BAS}(1,s,2,3|\sigma_{4})$ and $\big{(}\epsilon_{3}\cdot
f_{s}\cdot f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}^{(0)_{s}}_{\rm
BAS}(1,2,s,3|\sigma_{4})$

Naively, these two terms may come from $\big{(}\epsilon_{3}\cdot
f^{\prime}_{2}\cdot f_{s}\cdot\epsilon_{1}\big{)}\,{\cal A}^{(0)_{s}}_{\rm
BAS}(1,s,2,3|\sigma_{4})$ and $\big{(}\epsilon_{3}\cdot f_{s}\cdot
f^{\prime}_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}^{(0)_{s}}_{\rm
BAS}(1,2,s,3|\sigma_{4})$ respectively, with $f^{\prime}_{2}$ defined in (35).
However, from the discussion above, we realize that the coefficients in the
expansion of ${\cal A}_{YM}(\sigma_{4})$ do not involve terms proportional to
the square of $k_{s}$ or higher-order terms. Therefore, these two terms
exclusively originate from $\big{(}\epsilon_{3}\cdot f_{2}\cdot
f_{s}\cdot\epsilon_{1}\big{)}\,{\cal A}^{(0)_{s}}_{\rm
BAS}(1,s,2,3|\sigma_{4})$ and $\big{(}\epsilon_{3}\cdot f_{s}\cdot
f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}^{(0)_{s}}_{\rm
BAS}(1,2,s,3|\sigma_{4})$.

Finally, we observe the desired expansion of $4$-point amplitude

$\displaystyle{\cal A}^{\rm I}_{\rm YM}(\sigma_{4})$ $\displaystyle=$
$\displaystyle\big{(}\epsilon_{3}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,3;s,2|\sigma_{4})$ (37) $\displaystyle+\big{(}\epsilon_{3}\cdot
f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,2,3;s|\sigma_{4})+\big{(}\epsilon_{3}\cdot
f_{s}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm YMS}(1,s,3;2|\sigma_{4})$
$\displaystyle+\big{(}\epsilon_{3}\cdot f_{2}\cdot
f_{s}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
BAS}(1,s,2,3|\sigma_{4})+\big{(}\epsilon_{3}\cdot f_{s}\cdot
f_{2}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
BAS}(1,2,s,3|\sigma_{4})\,.~{}~{}$

It can be seen that this expression (37) remains invariant under the exchange
of external legs $2$ and $s$.

###  III.3 General case

In this subsection, we construct the expansion of general color ordered YM
amplitudes ${\cal A}_{\rm YM}(\sigma_{n})$, by applying the recursive method
in the previous subsection III.2 iteratively.

The main result of this subsection is the expansion

$\displaystyle{\cal A}^{\rm I}_{\rm
YM}(\sigma_{n})=\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},n;\\{2,\cdots,n-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{n})\,,~{}~{}$
(38)

where $\boldsymbol{{\alpha}}$ denotes a subset of external legs in
$\\{2,\cdots,n-1\\}$ which is allowed to be empty, and 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

$\displaystyle
F^{\mu\nu}_{\vec{\boldsymbol{{\alpha}}}}\equiv\big{(}f_{{\alpha}_{k}}\cdot
f_{{\alpha}_{k-1}}\cdots f_{{\alpha}_{2}}\cdot
f_{{\alpha}_{1}}\big{)}^{\mu\nu}\,,$ (39)

for $\vec{\boldsymbol{{\alpha}}}=\\{{\alpha}_{1},\cdots{\alpha}_{k}\\}$. The
summation in (38) is among all un-equivalent ordered sets
$\vec{\boldsymbol{{\alpha}}}$. In other words, one should sum over all subsets
$\boldsymbol{{\alpha}}$ of $\\{2,\cdots,n-1\\}$, as well as all un-cyclic
permutations of elements in $\boldsymbol{{\alpha}}$. Evidently, the general
formula (38) is satisfied by the expansions of $3$-point and $4$-point
amplitudes, as shown in (27) and (37). In the remainder of this subsection, we
will illustrate that if the formula (38) holds for $m$-point amplitudes, it
also extends to $(m+1)$-point ones. Consequently, we can iteratively ensure
the validity of the general expansion (38). This process shares many
similarities with the one described in the previous subsection III.2, and as
such, we will skip various details.

We can denote the external legs of the $(m+1)$-point amplitude as
$s\cup{1,\cdots,m}$ and investigate the soft behavior associated with the
external leg $s$. The sub-leading order soft behavior of the $(m+1)$-point
amplitude is expressed as follows:

$\displaystyle{\cal A}^{(1)_{s}}_{\rm YM}(\sigma_{m+1})$ $\displaystyle=$
$\displaystyle S^{(1)_{s}}_{g}\,{\cal A}^{\rm I}_{\rm
YM}(\sigma_{m+1}\setminus s)$ (40) $\displaystyle=$ $\displaystyle
S^{(1)_{s}}_{g}\,\Big{[}\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)\Big{]}$ $\displaystyle=$ $\displaystyle P_{1}+P_{2}\,,$

where $P_{1}$ and $P_{2}$ are given as

$\displaystyle
P_{1}=\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\big{)}\,\Big{[}S^{(1)_{s}}_{g}\,{\cal
A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)\Big{]}\,,$ (41)

and

$\displaystyle
P_{2}=\sum_{\vec{\boldsymbol{{\alpha}}}}\,\Big{[}S^{(1)_{s}}_{g}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\big{)}\Big{]}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)\,.$ (42)

The soft theorem (15) leads to

$\displaystyle
P_{1}=\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\big{)}\,{\cal
A}^{(1)_{s}}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;s\cup\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})\,.~{}~{}$
(43)

The block $P_{2}$ can be calculated as

$\displaystyle P_{2}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}\shuffle
s}\cdot\epsilon_{1}\big{)}\,\Big{(}{\delta_{s_{l}s}\over
s_{s_{l}s}}+{\delta_{ss_{r}}\over s_{sr}}\Big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)\,,~{}~{}$ (44)

by using relations in (20) and (21). The notation $s_{l}$ denotes the adjacent
leg of $s$ which is at the l.h.s of $s$ in
$\vec{\boldsymbol{{\alpha}}}\shuffle s$, while $s_{r}$ denotes the r.h.s one.
Using the argument the same as that from previous subsection, we arrive at

$\displaystyle P_{2}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}\shuffle s}\cdot\epsilon_{1}\big{)}\,{\cal
A}^{(0)_{s}}_{\rm YMS}(1,\vec{\boldsymbol{{\alpha}}}\shuffle
s,m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})\,.~{}~{}$
(45)

Combining (43) and (45) together leads to

$\displaystyle{\cal A}^{(1)_{s}}_{\rm YM}(\sigma_{m+1})$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\big{)}^{(0)_{s}}\,{\cal
A}^{(1)_{s}}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;s\cup\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})$
(46)
$\displaystyle+\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}\shuffle
s}\cdot\epsilon_{1}\big{)}^{(0)_{s}}\,{\cal A}^{(0)_{s}}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}}\shuffle
s,m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})\,,$

which indicates the expansion

$\displaystyle{\cal A}_{\rm YM}(\sigma_{m+1})$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;s\cup\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})$
(47)
$\displaystyle+\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}\shuffle s}\cdot\epsilon_{1}\big{)}\,{\cal
A}_{\rm YMS}(1,\vec{\boldsymbol{{\alpha}}}\shuffle
s,m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})$
$\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}^{\prime\prime}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}^{\prime\prime}}\cdot\epsilon_{1}\big{)}\,{\cal
A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}}^{\prime\prime},m;s\cup\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}^{\prime\prime}|\sigma_{m+1})\,.~{}~{}$

The final line in (47) corresponds to the expanded formula (38) applicable to
the $(m+1)$-point case. It’s worth noting that each set
$\boldsymbol{{\alpha}}$ in (47) represents a subset of $\\{2,\ldots,m-1\\}$
excluding the leg $s$, while each $\boldsymbol{{\alpha}}^{\prime\prime}$
denotes a subset of $s\cup\\{2,\ldots,m-1\\}$.

Hence, the general expansion formula (38) is proven to be valid for
$(m+1)$-point amplitudes when it has been established for $m$-point amplitudes
and demonstrated for $3$-point and $4$-point amplitudes. This conclusion is
derived through a mathematical induction argument.

##  IV Manifests the gauge invariance

The expansion presented in (38) doesn’t inherently exhibit gauge invariance
for all polarizations, including $\epsilon_{1}$ and $\epsilon_{n}$. In this
section, we establish an expansion that explicitly maintains gauge invariance
through two distinct approaches. The first approach, detailed in subsection
IV.1, involves a direct modification of (38), incorporating the gauge
invariance condition. While this approach requires the imposition of gauge
invariance rather than its proof, it serves as a valuable starting point. To
address the logical concern associated with the first construction, we
introduce the second approach in subsection IV.2. In this approach, we utilize
the recursive technique established in the previous section III. The key
difference lies in the modification of the starting point of the
recursion,i.e, the $3$-point amplitude, ensuring that the expression at this
initial stage exhibits explicit gauge invariance. Subsequently, at each step
of the recursion, we insert soft particles while maintaining this manifest
gauge invariance. Ultimately, this process allows us to construct an expansion
that ensures gauge invariance without the need for explicit requirements.

###  IV.1 Direct construction

In the expansion (38), gauge invariance is manifest for polarizations
$\epsilon_{i}$ with $i\in\\{2,\cdots,n-1\\}$ because the tensor
$f^{\mu\nu}_{i}$ automatically vanishes under the replacement $\epsilon_{i}\to
k_{i}$. However, ensuring gauge invariance for polarizations $\epsilon_{1}$
and $\epsilon_{n}$ is not as straightforward. To obtain an expansion formula
that also explicitly demonstrates gauge invariance for $\epsilon_{1}$ and
$\epsilon_{n}$, one can impose the following gauge invariance conditions:

$\displaystyle A_{n}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\Big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot k_{1}\Big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},n;\\{2,\cdots,n-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{n})=0\,,$
$\displaystyle B_{n}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\Big{(}k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\Big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},n;\\{2,\cdots,n-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{n})=0\,,$
$\displaystyle C_{n}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,\Big{(}k_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot k_{1}\Big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},n;\\{2,\cdots,n-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{n})=0\,,~{}~{}$
(48)

which are obtained from (38) by replacing $\epsilon_{1}$ or $\epsilon_{n}$
with $k_{1}$ or $k_{n}$, respectively. These conditions lead to the following
new formula,

$\displaystyle{\cal A}^{\rm II}_{\rm
YM}(\sigma_{n})=-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{n}\cdot F_{\vec{\boldsymbol{{\alpha}}}}\cdot f_{1}\big{)}\over
k_{n}\cdot k_{1}}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},n;\\{2,\cdots,n-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{n})\,,~{}~{}$
(49)

since

$\displaystyle{\cal A}^{\rm II}_{\rm YM}(\sigma_{n})$ $\displaystyle=$
$\displaystyle{\cal A}^{\rm I}_{\rm
YM}(\sigma_{n})-{k_{n}\cdot\epsilon_{1}\over k_{n}\cdot
k_{1}}\,A_{n}-{\epsilon_{n}\cdot k_{1}\over k_{n}\cdot
k_{1}}\,B_{n}+{\epsilon_{n}\cdot\epsilon_{1}\over k_{n}\cdot
k_{1}}\,C_{n}\,,~{}~{}$ (50)

where ${\cal A}^{\rm I}_{\rm YM}(\sigma_{n})$ is the expanded formula in (38).

In the above construction, we required the gauge invariance for polarizations
$\epsilon_{1}$ and $\epsilon_{n}$, namely, $A_{n}=B_{n}=C_{n}=0$. However, it
is quite non-trivial to prove this property for general $n$. To ensure this
gauge invariance condition, one need to derive the expansion (38) from a
manifestly gauge invariant framework, such as lagrangian or CHY formalism.
Thus, if we insist the spirt of constructing YM amplitudes recursively from
lowest-point ones, without respecting other frameworks, the above construction
is not so satisfactory.

###  IV.2 Recursive derivation

For the expansion of the simplest $3$-point amplitudes in (27), the gauge
invariance for polarization $\epsilon_{1}$ or $\epsilon_{3}$ is easy to be
observed. For example, replacing $\epsilon_{3}$ by $k_{3}$ in (38) yields

$\displaystyle{\cal A}^{\rm I}_{\rm YM}(\sigma_{3})\Big{|}_{\epsilon_{3}\to
k_{3}}$ $\displaystyle=$
$\displaystyle\big{(}k_{3}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,3;2|\sigma_{3})$ (51)
$\displaystyle-\big{(}k_{3}\cdot\epsilon_{2}\big{)}\,\big{(}k_{2}\cdot\epsilon_{1}\big{)}\,{\cal
A}_{\rm BAS}(1,2,3|\sigma_{3})\,,$

here we have used $k_{3}\cdot k_{2}=0$ due to momentum conservation and on-
shell condition. By utilizing (22) to expand ${\cal A}_{\rm
YMS}(1,3;2|\sigma_{3})$ into the BAS amplitude ${\cal A}_{\rm
BAS}(1,2,3|\sigma_{3})$, we can readily verify that ${\cal A}^{\rm I}_{\rm
YM}(\sigma_{3})\Big{|}{\epsilon_{3}\to k_{3}}=0$. This corresponds to the
gauge invariance condition stated in the second line of (48) for the case of
$3$-point amplitudes. Similar verifications apply to the other two conditions
in (48). Consequently, we can transform the expansion (27) into a new
formulation denoted as ${\cal A}^{\rm II}_{\rm YM}(\sigma_{3})$, which is
expressed as:

$\displaystyle{\cal A}^{\rm II}_{\rm YM}(\sigma_{3})$ $\displaystyle=$
$\displaystyle{{\rm tr}\,\big{(}f_{3}\cdot f_{1}\big{)}\over k_{3}\cdot
k_{1}}\,{\cal A}_{\rm YMS}(1,3;2|\sigma_{3})$ (52) $\displaystyle+{{\rm
tr}\,\big{(}f_{3}\cdot f_{2}\cdot f_{1}\big{)}\over k_{3}\cdot k_{1}}\,{\cal
A}_{\rm BAS}(1,2,3|\sigma_{3})\,.~{}~{}$

Next, we will start from expression (52) and repeat the procedure outlined in
section (III.3) to iteratively construct an alternative expansion form of
${\cal A}^{\rm II}_{\rm YM}(\sigma_{n})$ using lower-point amplitudes.
Simultaneously, we will recursively observe that $A_{n}=B_{n}=C_{n}=0$ in (48)
holds for any $n$, which establishes the gauge invariance for polarizations
$\epsilon_{1}$ and $\epsilon_{n}$ in the expansion (38).

First, let’s make the following inductive assumption: (A) The expansion (49)
that maintains manifestness of all polarization vectors is valid for all
$n\leq m$. (B) The gauge invariance condition (48) $A_{n}=B_{n}=C_{n}=0$ holds
for all $n\leq m$. As shown above, these two conditions hold when $n=3$.
Therefore, we have obtained the starting point for the inductive proof. We
need to prove that these two conditions hold for $n=m+1$ as well. The sub-
leading soft behavior of ${\cal A}^{\rm II}_{\rm YM}(\sigma_{m+1})$ can be
expressed as

$\displaystyle{\cal A}^{(1)_{s}}_{\rm YM}(\sigma_{m+1})$ $\displaystyle=$
$\displaystyle S^{(1)_{s}}_{g}\,{\cal A}^{\rm II}_{\rm
YM}(\sigma_{m+1}\setminus s)$ (53) $\displaystyle=$ $\displaystyle
S^{(1)_{s}}_{g}\,\Big{[}-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}}\cdot f_{1}\big{)}\over
k_{m}\cdot k_{1}}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)\Big{]}$ $\displaystyle=$ $\displaystyle P_{1}+P_{2}\,,~{}~{}$

where

$\displaystyle P_{1}=-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}}\cdot f_{1}\big{)}\over
k_{m}\cdot k_{1}}\,{\cal A}^{(1)_{s}}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;s\cup\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})$
(54)

is obtained by acting the operator $S^{(1)_{s}}_{g}$ on YMS amplitudes, while
$P_{2}$ is obtained by acting $S^{(1)_{s}}_{g}$ on coefficients. The block
$P_{2}$ can be evaluated as

$\displaystyle P_{2}$ $\displaystyle=$
$\displaystyle-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}\shuffle s}\cdot
f_{1}\big{)}\over k_{m}\cdot k_{1}}\,\Big{(}{\delta_{s_{l}s}\over
s_{s_{l}s}}+{\delta_{ss_{r}}\over s_{ss_{r}}}\Big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)$ (55) $\displaystyle-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}}\cdot f_{1}\cdot
f_{s}\big{)}\over k_{m}\cdot k_{1}}\,\Big{(}{\delta_{ms}\over
s_{ms}}+{\delta_{s1}\over s_{s1}}\Big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)$ $\displaystyle+{k_{m}\cdot f_{s}\cdot k_{1}\over k_{m}\cdot
k_{1}}\,\Big{(}{\delta_{1s}\over s_{1s}}+{\delta_{sm}\over
s_{sm}}\Big{)}\,\Big{[}\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}}\cdot f_{1}\big{)}\over
k_{m}\cdot k_{1}}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)\Big{]}\,,~{}~{}$

notice that the last line arises from acting the soft operator on the
denominator $k_{n}\cdot k_{1}$ in (53). The first line in (55) can be
recognized as

$\displaystyle P_{21}=-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}\shuffle s}\cdot
f_{1}\big{)}\over k_{m}\cdot k_{1}}\,{\cal A}^{(0)_{s}}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}}\shuffle
s,m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})\,,$ (56)

via the technic from (44) to (45). Then we combining the second and third
lines in (55) and regroup them as as

$\displaystyle P_{22}$ $\displaystyle=$ $\displaystyle{\cal
C}_{1}\,\Big{[}\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{m}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)\Big{]}$ (57) $\displaystyle+{\cal
C}_{2}\,\Big{[}\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{m}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot k_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)\Big{]}$ $\displaystyle+{\cal
C}_{3}\,\Big{[}\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}k_{m}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)\Big{]}$ $\displaystyle+{\cal
C}_{4}\,\Big{[}\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}k_{m}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot k_{1}\big{)}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1}\setminus
s)\Big{]}\,,~{}~{}$

where coefficients ${\cal C}_{i}$ with $i\in\\{1,\cdots,4\\}$ are independent
of ordered sets $\vec{\boldsymbol{{\alpha}}}$. Using $\delta_{ab}=\delta_{ba}$
and $k_{m}\cdot f_{s}\cdot k_{1}=-k_{1}\cdot f_{s}\cdot k_{m}$, we find ${\cal
C}_{1}=0$. Then remaining three lines in (57) vanish automatically, since the
gauge invariance conditions in (48) for $m$-point amplitudes by the condition
(A) of the induction hypothesis. Thus we obtain

$\displaystyle{\cal A}^{(1)_{s}}_{\rm YM}(\sigma_{m+1})$ $\displaystyle=$
$\displaystyle P_{1}+P_{21}$ (58) $\displaystyle=$
$\displaystyle-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}}\cdot f_{1}\big{)}\over
k_{m}\cdot k_{1}}\,{\cal A}^{(1)_{s}}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;s\cup\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})$
$\displaystyle-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}\shuffle s}\cdot
f_{1}\big{)}\over k_{m}\cdot k_{1}}\,{\cal A}^{(0)_{s}}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}}\shuffle
s,m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})\,,$

which indicates

$\displaystyle{\cal A}^{\rm II}_{\rm YM}(\sigma_{m+1})$ $\displaystyle=$
$\displaystyle-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}}\cdot f_{1}\big{)}\over
k_{m}\cdot k_{1}}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}},m;s\cup\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})$
(59) $\displaystyle-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}\shuffle s}\cdot
f_{1}\big{)}\over k_{m}\cdot k_{1}}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}}\shuffle
s,m;\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}|\sigma_{m+1})$
$\displaystyle=$
$\displaystyle-\sum_{\vec{\boldsymbol{{\alpha}}}^{\prime}}\,{{\rm
tr}\,\big{(}f_{m}\cdot F_{\vec{\boldsymbol{{\alpha}}}^{\prime}}\cdot
f_{1}\big{)}\over k_{m}\cdot k_{1}}\,{\cal A}_{\rm
YMS}(1,\vec{\boldsymbol{{\alpha}}}^{\prime},m;s\cup\\{2,\cdots,m-1\\}\setminus\boldsymbol{{\alpha}}^{\prime}|\sigma_{m+1})\,.~{}~{}$

In the formula (59), $\boldsymbol{{\alpha}}$ denotes subsets of
$\\{2,\cdots,m-1\\}$ that do not include the leg $s$, while
$\boldsymbol{{\alpha}}^{\prime}$ represents subsets of
$s\cup\\{2,\cdots,m-1\\}$. The expansion (59) is essentially the expanded
formula ${\cal A}^{\rm II}_{\rm YM}(\sigma_{n})$ as seen in (49), applied to
the $(m+1)$-point case. It is important to recall that ${\cal A}^{\rm II}_{\rm
YM}(\sigma_{m+1})$ and ${\cal A}^{\rm I}_{\rm YM}(\sigma_{m+1})$ are two
distinct forms of expansion derived using the same method. As a result, we
arrive at the conclusion that ${\cal A}^{\rm II}_{\rm YM}(\sigma_{m+1})={\cal
A}^{\rm I}_{\rm YM}(\sigma_{m+1})$, leading to

$\displaystyle-{k_{m}\cdot\epsilon_{1}\over k_{m}\cdot
k_{1}}\,A_{m+1}-{\epsilon_{m}\cdot k_{1}\over k_{m}\cdot
k_{1}}\,B_{m+1}+{\epsilon_{m}\cdot\epsilon_{1}\over k_{m}\cdot
k_{1}}\,C_{m+1}=0\,,~{}~{}$ (60)

where $A_{m+1}$, $B_{m+1}$, and $C_{m+1}$ are defined in (48) with $(m+1)$
replacing $n$. Consequently, we conclude that the gauge invariance conditions
$A_{m+1}=B_{m+1}=C_{m+1}=0$ hold for the $(m+1)$-point amplitudes, as each
coefficient associated with the independent variables
$k_{m}\cdot\epsilon_{1}$, $\epsilon_{m}\cdot k_{1}$, and
$\epsilon_{m}\cdot\epsilon_{1}$ must vanish individually.

In summary, if the expansion (49) and the gauge invariance conditions (48)
hold for $m$-point amplitudes, they automatically hold for $(m+1)$-point
amplitudes. Given that the $3$-point amplitudes ${\cal A}_{\rm
YM}(\sigma_{3})$ satisfy both the expansion (49) and the gauge invariance
conditions (48), we can apply this result iteratively to conclude that (49)
and (48) are valid for any YM amplitude ${\cal A}_{\rm YM}(\sigma_{n})$. In
other words, we have presented an alternative expression (49) that explicitly
maintains gauge invariance for all polarization vectors constructed from
lower-point amplitudes. Additionally, we’ve verified that the expansions
derived in the previous section, as outlined in (III.3), also preserve gauge
invariance for both $\epsilon_{1}$ and $\epsilon_{n}$.

##  V Summery

In this paper, we have enhanced the recursive method based on the sub-leading
soft theorem for external gluons, as previously employed in Zhou:2022orv .
With our new approach, we have developed two types of expansions from YM
amplitudes to YMS amplitudes. The first type does not exhibit manifest gauge
invariance for each polarization, while the second type does. As detailed in
Section I and Section II, the sub-leading soft theorem for gluons effectively
”grows” a soft gluon in a manifestly gauge-invariant manner. Consequently, as
long as our recursive starting point is manifestly gauge-invariant, any
intermediate result obtained during the recursion process remains explicitly
gauge-invariant.

According to the methodology presented in this paper, we can also make the
following two extensions:

  * •

Based on the double copy structure, one can replace YM (YMS) by GR (EYM) in
(38) and (49) to obtain the expansion of GR amplitudes to EYM ones as follows,

$\displaystyle{\cal A}^{\rm I}_{\rm
GR}(n)=\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\epsilon_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot\epsilon_{1}\big{)}\,{\cal A}_{\rm
EYM}(1,\vec{\boldsymbol{{\alpha}}},n;\\{2,\cdots,n-1\\}\setminus\boldsymbol{{\alpha}})\,,~{}~{}$
(61)

$\displaystyle{\cal A}^{\rm II}_{\rm
GR}(n)=-\sum_{\vec{\boldsymbol{{\alpha}}}}\,{{\rm tr}\,\big{(}f_{n}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot f_{1}\big{)}\over k_{n}\cdot
k_{1}}\,{\cal A}_{\rm
EYM}(1,\vec{\boldsymbol{{\alpha}}},n;\\{2,\cdots,n-1\\}\setminus\boldsymbol{{\alpha}})\,.~{}~{}$
(62)

Then, we perform a similar replacement for (24),

$\displaystyle{\cal A}_{\rm EYM}(1,\cdots,n;\\{p_{1},\cdots,p_{m}\\})$ (63)
$\displaystyle=$ $\displaystyle\sum_{\vec{\boldsymbol{{\alpha}}}}\,{k_{r}\cdot
F_{\vec{\boldsymbol{{\alpha}}}}\cdot Y_{\vec{\boldsymbol{{\alpha}}}}\over
k_{r}\cdot k_{p_{1}\cdots p_{m}}}\,{\cal A}_{\rm
EYM}(1,\\{2,\cdots,n-1\\}\shuffle\vec{\boldsymbol{{\alpha}}},n;\\{p_{1},\cdots,p_{m}\\}\setminus\boldsymbol{{\alpha}})\,.~{}~{}$

Starting from (62), iteratively using (63), one arrives at the expansions of
pure GR amplitudes to pure YM amplitudes, whose coefficients manifest the
gauge invariance for each polarization. Thus the manifestly gauge invariant
BCJ numerators are found.

  * •

The gauge invariant expansion can be extended to the $1$-loop level
straightforwardly. One can first use (38) to expand the $(n+2)$-point tree YM
amplitude as

$\displaystyle{\cal A}_{\rm
YM}(+,\sigma_{n},-)=\sum_{\vec{\boldsymbol{{\alpha}}}}\,(\epsilon_{-}\cdot
F_{\vec{\boldsymbol{a}}}\cdot\epsilon_{+})\,{\cal A}_{\rm
YMS}(+,\vec{\boldsymbol{{\alpha}}},-;\\{1,\cdots,n\\}\setminus\boldsymbol{{\alpha}}|+,\sigma_{n},-)\,,~{}~{}$
(64)

where two fixed legs are encoded as $+$, $-$, and $\boldsymbol{{\alpha}}$
denotes subsets of $\\{1,\cdots,n\\}$. The $1$-loop amplitude can be generated
by taking forward limit of (65) for legs $+$ and $-$, namely, setting
$k_{+}=-k_{-}=\ell$, then gluing two legs $+$ and $-$ together by identifying
$\epsilon_{+}$, $\epsilon_{-}$, and summing over all possible states
He:2015yua ; Cachazo:2015aol ; Dong:2021qai ; Zhou:2021kzv ; Zhou:2022djx .
This manipulation leads to the expansion at amplitude level,

$\displaystyle{\cal A}^{1-{\rm loop}}_{\rm
YM}(\sigma_{n})=\sum_{\vec{\boldsymbol{{\alpha}}}}\,\big{(}\mathop{\rm
Tr}\,F_{\vec{\boldsymbol{a}}}\big{)}\,{\cal A}_{\rm sYMS}^{1-{\rm
loop}}(\vec{\boldsymbol{{\alpha}}};\\{1,\cdots,n\\}\setminus\boldsymbol{{\alpha}}|\sigma_{n})\,,~{}~{}$
(65)

which manifests the gauge invariance. Here the subscript $s$ in ${\cal
A}^{1-{\rm loop}}_{\rm sYMS}$ denotes the special type of $1$-loop YMS
amplitudes those the virtual particle propagating in the loop is a scalar. At
the integrand level, one can employ (24) to expand sYMS integrands
iteratively, end with pure BAS integrands. Since
$Y_{\vec{\boldsymbol{{\alpha}}}}$ should include the loop momentum $\ell$ when
applying (24) to the $1$-loop level, the obtained gauge invariant expansion to
pure BAS ones does not hold at the amplitude level.

Moreover, in section.I, we also pointed out that the explicit formulas of soft
factors can be regarded as the consequence of the universality of soft
behaviors, without respecting any top down derivation. When referring to
universality, we means the soft factor for BAS scalars observed from pure BAS
amplitudes holds for BAS scalars in general YMS amplitudes, and the soft
factors for gluons derived from YMS amplitudes with only one external gluon
also hold for general YMS amplitudes. From the traditional perspective, such
universality is the consequence of the symmetries. For example, the universal
soft behavior of gravitons are ensured by the asymptotic BMS symmetries of
flat space time at null infinity Strominger:2013jfa ; Strominger:2013lka ;
He:2014laa ; Kapec:2014opa ; Barnich1 ; Barnich2 ; Barnich3 . From the bottom
up perspective, universality of soft behaviors can be taken as the basic
principle, and the associated symmetries are hard to be observed. Thus, a
natural question is, can we reproduce the corresponding underlying symmetries
from the bottom up perspective? This is an interesting future direction.

## Acknowledgments

The authors would thank Prof. Ellis Ye Yuan, Prof. Yang Zhang and Prof. Song
He for helpful suggestions.

## 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) Z. Bern, T. Dennen, Y. t. Huang and M. Kiermaier, “Gravity as the Square of Gauge Theory,” Phys. Rev. D 82, 065003 (2010) doi:10.1103/PhysRevD.82.065003 [arXiv:1004.0693 [hep-th]].
  * (22) 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]].
  * (23) 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]].
  * (24) N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov and J. Trnka, “Grassmannian Geometry of Scattering Amplitudes,” Cambridge University Press, 2016, ISBN 978-1-107-08658-6, 978-1-316-57296-2 doi:10.1017/CBO9781316091548 [arXiv:1212.5605 [hep-th]].
  * (25) N. Arkani-Hamed and J. Trnka, “The Amplituhedron,” JHEP 10, 030 (2014) doi:10.1007/JHEP10(2014)030 [arXiv:1312.2007 [hep-th]].
  * (26) N. Arkani-Hamed and J. Trnka, “Into the Amplituhedron,” JHEP 12, 182 (2014) doi:10.1007/JHEP12(2014)182 [arXiv:1312.7878 [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) F. S. Wei and K. Zhou, [arXiv:2306.14774 [hep-th]].
  * (29) F. E. Low, “Bremsstrahlung of very low-energy quanta in elementary particle collisions,” Phys. Rev. 110, 974 (1958).
  * (30) S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140, B516 (1965).
  * (31) F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,” [arXiv:1404.4091 [hep-th]].
  * (32) 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]].
  * (33) 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]].
  * (34) N. Afkhami-Jeddi, “Soft Graviton Theorem in Arbitrary Dimensions,” [arXiv:1405.3533 [hep-th]].
  * (35) C. Cheung, K. Kampf, J. Novotny and J. Trnka, “Effective Field Theories from Soft Limits of Scattering Amplitudes,” Phys. Rev. Lett. 114, no.22, 221602 (2015) doi:10.1103/PhysRevLett.114.221602 [arXiv:1412.4095 [hep-th]].
  * (36) H. Luo and C. Wen, “Recursion relations from soft theorems,” JHEP 03, 088 (2016) doi:10.1007/JHEP03(2016)088 [arXiv:1512.06801 [hep-th]].
  * (37) H. Elvang, M. Hadjiantonis, C. R. T. Jones and S. Paranjape, “Soft Bootstrap and Supersymmetry,” JHEP 01, 195 (2019) doi:10.1007/JHEP01(2019)195 [arXiv:1806.06079 [hep-th]].
  * (38) F. Cachazo, P. Cha and S. Mizera, “Extensions of Theories from Soft Limits,” JHEP 06, 170 (2016) doi:10.1007/JHEP06(2016)170 [arXiv:1604.03893 [hep-th]].
  * (39) L. Rodina, “Scattering Amplitudes from Soft Theorems and Infrared Behavior,” Phys. Rev. Lett. 122, no.7, 071601 (2019) doi:10.1103/PhysRevLett.122.071601 [arXiv:1807.09738 [hep-th]].
  * (40) C. Boucher-Veronneau and A. J. Larkoski, “Constructing Amplitudes from Their Soft Limits,” JHEP 09, 130 (2011) doi:10.1007/JHEP09(2011)130 [arXiv:1108.5385 [hep-th]].
  * (41) D. Nguyen, M. Spradlin, A. Volovich and C. Wen, “The Tree Formula for MHV Graviton Amplitudes,” JHEP 07, 045 (2010) doi:10.1007/JHEP07(2010)045 [arXiv:0907.2276 [hep-th]].
  * (42) 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]].
  * (43) K. Zhou, “Tree level amplitudes from soft theorems,” JHEP 03, 021 (2023) doi:10.1007/JHEP03(2023)021 [arXiv:2212.12892 [hep-th]].
  * (44) R. Kleiss and H. Kuijf, “MULTI - GLUON CROSS-SECTIONS AND FIVE JET PRODUCTION AT HADRON COLLIDERS,” Nucl. Phys. B 312, 616 (1989).
  * (45) S. He and E. Y. Yuan, “One-loop Scattering Equations and Amplitudes from Forward Limit,” Phys. Rev. D 92, no.10, 105004 (2015) doi:10.1103/PhysRevD.92.105004 [arXiv:1508.06027 [hep-th]].
  * (46) F. Cachazo, S. He and E. Y. Yuan, “One-Loop Corrections from Higher Dimensional Tree Amplitudes,” JHEP 08, 008 (2016) doi:10.1007/JHEP08(2016)008 [arXiv:1512.05001 [hep-th]].
  * (47) J. Dong, S. He and L. Hou, “Universal expansions of scattering amplitudes for gravitons, gluons, and Goldstone particles,” Phys. Rev. D 105, no.10, 105007 (2022) doi:10.1103/PhysRevD.105.105007 [arXiv:2111.10525 [hep-th]].
  * (48) K. Zhou, “On differential operators and unifying relations for 1-loop Feynman integrands,” JHEP 10, 150 (2021) doi:10.1007/JHEP10(2021)150 [arXiv:2108.04025 [hep-th]].
  * (49) K. Zhou, “Transmutation operators and expansions for one-loop Feynman integrands,” Phys. Rev. D 106, no.2, 025014 (2022) doi:10.1103/PhysRevD.106.025014 [arXiv:2201.01552 [hep-th]].
  * (50) A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 07, 152 (2014) doi:10.1007/JHEP07(2014)152 [arXiv:1312.2229 [hep-th]].
  * (51) A. Strominger, “Asymptotic Symmetries of Yang-Mills Theory,” JHEP 07, 151 (2014) doi:10.1007/JHEP07(2014)151 [arXiv:1308.0589 [hep-th]].
  * (52) T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP 05, 151 (2015) doi:10.1007/JHEP05(2015)151 [arXiv:1401.7026 [hep-th]].
  * (53) D. Kapec, V. Lysov, S. Pasterski and A. Strominger, “Semiclassical Virasoro symmetry of the quantum gravity $\mathcal{S}$-matrix,” JHEP 08, 058 (2014) doi:10.1007/JHEP08(2014)058 [arXiv:1406.3312 [hep-th]].
  * (54) G. Barnich and C. Troessaert, “Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited,” Phys. Rev. Lett. 105, 111103 (2010) [arXiv:0909.2617 [gr-qc]].
  * (55) G. Barnich and C. Troessaert, “Supertranslations call for superrotations,” PoS CNCFG 2010, 010 (2010) [arXiv:1102.4632 [gr-qc]].
  * (56) G. Barnich and C. Troessaert, “BMS charge algebra,” JHEP 1112, 105 (2011) [arXiv:1106.0213 [hep-th]].