aainstitutetext: Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University,
No.180, Siwangting Road, Yangzhou, 225009, P.R. China.

# On soft factors and transmutation operators

Fang-Stars Wei a Kang Zhou<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The well known soft theorems state the specific factorizations of tree level
gravitational (GR) amplitudes at leading, sub-leading and sub-sub-leading
orders, with universal soft factors. For Yang-Mills (YM) amplitudes, similar
factorizations and universal soft factors are found at leading and sub-leading
orders. Then it is natural to ask if the similar factorizations and soft
factors exist at higher orders. In this note, by using transformation
operators proposed by Cheung, Shen and Wen, we reconstruct the known soft
factors of YM and GR amplitudes, and prove the nonexistence of higher order
soft factor of YM or GR amplitude which satisfies the universality.

###### Keywords:

Scattering Amplitude, Soft Theorem, Transmutation Operator

##  1 Introduction

In recent years, the investigation on soft theorems of scattering amplitudes
has been an active area of research, leading to remarkably insights and
applications ranging from gauge theory and gravity, to various effective field
theories (EFTs). Soft theorems describe the universal behaviors of amplitudes
when one or more external momenta are taken to near zero. Historically, they
were originally discovered for photons and gravitons, at tree level Low:1958sn
; Weinberg:1965nx . In 2014, the soft behaviors of tree gravitational (GR) and
Yang-Mills (YM) amplitudes were extended to higher-orders Cachazo:2014fwa ;
Casali:2014xpa ; Schwab:2014xua ; Afkhami-Jeddi:2014fia ; Zlotnikov:2014sva ,
by using modern technics beyond Feynman diagrams, like Britto-Cachazo-Feng-
Witten (BCFW) Britto:2004ap ; Britto:2005fq  recursion relation and Cachazo-
He-Yuan (CHY) formula Cachazo:2013gna ; Cachazo:2013hca ; Cachazo:2013iea ;
Cachazo:2014nsa ; Cachazo:2014xea . Subsequently, soft theorems were further
studied in a wider range including string theory and the loop level
Bern:2014oka ; He:2014bga ; Cachazo:2014dia ; Bianchi:2014gla ; Sen:2017nim .
Meanwhile, it turns out that tree amplitudes can be constructed by solely
exploiting soft behaviors, with out the aid of a Lagrangian or Feynman rules,
see in progresses in Nguyen:2009jk ; Boucher-Veronneau:2011rwd ;
Rodina:2018pcb ; Ma:2022qja ; Cheung:2014dqa ; Cheung:2015ota ; Luo:2015tat ;
Elvang:2018dco ; Zhou:2022orv ; Wei:2023yfy ; Hu:2023lso ; Du:2024dwm .

The soft limit can be achieved by rescaling the external momentum $k_{i}$
carried by particle $i$ as $k_{i}\to\tau k_{i}$, then take the limit $\tau\to
0$. For gravity and gauge theory, soft theorems state that in the soft limit
the full $n$-point amplitude factorizes into a soft factor, as well as a
$(n-1)$-point sub-amplitude. For instance, the $n$-point GR amplitude
factorizes as

$\displaystyle{\cal
A}_{n}\,\to\,\Big{(}\tau^{-1}\,S^{(0)_{i}}_{h}+\tau^{0}\,S^{(1)_{i}}_{h}+\tau\,S^{(2)_{i}}_{h}\Big{)}\,{\cal
A}_{n-1}+{\cal O}(\tau^{2})\,,~{}~{}~{}~{}$ (1)

where ${\cal A}_{n-1}$ is the sub-amplitude of ${\cal A}_{n}$, generated from
${\cal A}_{n}$ by removing the soft external graviton. The operators
$S^{(0)_{i}}_{h}$, $S^{(1)_{i}}_{h}$, $S^{(2)_{i}}_{h}$ are called soft
factors, at leading, sub-leading, and sub-sub-leading orders respectively,
their precise forms can be seen in Cachazo:2014fwa ; Schwab:2014xua ; Afkhami-
Jeddi:2014fia ; Zlotnikov:2014sva . These factors are universal, namely, their
forms are valid for arbitrary $n\geq 4$. For GR amplitudes, $S^{(0)_{i}}_{h}$,
$S^{(1)_{i}}_{h}$, $S^{(2)_{i}}_{h}$ in (1) are all already known soft
factors. For YM amplitudes, we have two known soft factors $S^{(0)_{i}}_{g}$
and $S^{(1)_{i}}_{g}$, at leading and sub-leading orders. Since each amplitude
can always be expanded to a series with respect to $\tau$,

$\displaystyle{\cal A}_{n}(\tau)=\sum_{a=0}^{+\infty}\,\tau^{a-1}\,{\cal
A}_{n}^{(a)_{i}}\,,~{}~{}$ (2)

it is natural to ask, can higher order terms ${\cal A}_{n}^{(a)_{i}}$
factorize as ${\cal A}^{(a)_{i}}_{n}=S^{(a)_{i}}\,{\cal A}_{n-1}$ in the soft
limit?

Formally, the answer is extremely trivial, since one can define
$S^{(a)_{i}}\equiv{\cal A}_{n}^{(a)_{i}}/{\cal A}_{n-1}$ in (2), then the
factorization behavior ${\cal A}^{(a)_{i}}_{n}=S^{(a)_{i}}\,{\cal A}_{n-1}$
holds at any order. However, such formal factorization does not lead to any
physical insight. Thus, it is necessary to impose further physical criteria as
the constraint on soft factors. The most natural candidate of such criteria is
the universality, which is satisfied by known soft factors for GR and YM
amplitudes, as well as the distinct soft behavior called Adler zero for
various EFTs. Thus, we are interested in the existence of universal soft
factors $S^{(a)_{i}}_{h}$ and $S^{(a)_{i}}_{g}$ at higher orders, with $a\geq
3$ for GR and $a\geq 2$ for YM, respectively.

In this note, with the help of transmutation operators which connect
amplitudes of different theories together Cheung:2017ems ; Zhou:2018wvn ;
Bollmann:2018edb , we prove that there is no universal soft factor can be
found at higher order. In other words, for GR amplitudes, the only soft
factors satisfy the universality are those $S^{(a)_{i}}_{h}$ with $a=0,1,2$.
Meanwhile, soft factors for YM amplitudes are $S^{(a)_{i}}_{g}$ with $a=0,1$.
Our method is as follows. The GR, YM, bi-adjoint scalar (BAS) amplitudes are
linked by the transmutation operator ${\cal T}[1,\cdots,n]$. On the other
hand, as will be explained in section 2.1, it is straightforward to figure out
the leading soft factor of BAS amplitudes, and observe that no soft factor
compatible with universality can be found at higher order. Based on
transmutation relations and the known leading soft behavior of BAS amplitudes,
we can establish equations which allow us to solve potential $S^{(a)_{i}}_{g}$
and $S^{(a)_{i}}_{h}$. We then find all solutions of $S^{(a)_{i}}_{g}$ and
$S^{(a)_{i}}_{h}$, coincide with those in literatures Cachazo:2014fwa ;
Casali:2014xpa ; Schwab:2014xua ; Afkhami-Jeddi:2014fia ; Zlotnikov:2014sva ,
and prove the nonexistence of solution at higher order. As a byproduct, we
also clarify that the consistent soft factors $S^{(a)_{i}}_{h}$ with $a=1,2$
found in literatures Cachazo:2014fwa ; Schwab:2014xua ; Afkhami-Jeddi:2014fia
; Zlotnikov:2014sva  only hold for amplitudes of standard Einstein gravity.
For the extended theory that Einstein gravity couples to $2$-form and dilaton
field, they are spoiled.

The note is organized as follows. In section 2, we give a brief review for
soft behavior of tree BAS amplitudes, as well as transmutation operators.
Then, in section 3 we rederive soft factors of YM amplitudes at leading and
sub-leading orders, and prove the nonexistence of higher order soft factors.
Subsequently, in section 4, we rederive soft factors of GR amplitudes at
leading, sub-leading, sub-sub-leading orders, and prove the nonexistence of
higher order ones. Finally, a brief summary will be presented in section 5.

##  2 Back ground

In this section, we give a rapid review for necessary background, including
the soft behavior of BAS amplitudes, as well as transmutation operators
proposed in Cheung:2017ems .

###  2.1 Soft behavior of BAS amplitudes

The bi-adjoint scalar (BAS) amplitudes describe scattering of massless
scalars, with cubic interactions. In this paper, we are interested in the
double ordered partial BAS amplitudes at the tree level. Each $n$-point double
ordered amplitude ${\cal A}_{\rm
BAS}(\vec{\boldsymbol{\sigma}}|\vec{\boldsymbol{\sigma}}^{\prime}_{n})$
carries two orderings encoded as $\vec{\boldsymbol{\sigma}}$ and
$\vec{\boldsymbol{\sigma}}^{\prime}_{n}$, and is simultaneously planar with
respect to both two orderings. Here we give a $5$-point example. The first
diagram in Figure.1 contributes to the amplitude ${\cal A}_{\rm
BAS}(1,2,3,4,5|1,4,2,3,5)$, since it is compatible with both two orderings
$1,2,3,4,5$ and $1,4,2,3,5$. Meanwhile, the second diagram violates the
ordering $1,4,2,3,5$, thus is forbidden. One can verify that the first diagram
in Figure.1 is the only candidate satisfies two orderings simultaneously, thus
the amplitude ${\cal A}_{\rm BAS}(1,2,3,4,5|1,4,2,3,5)$ reads

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

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

$\displaystyle s_{i\cdots j}\equiv k_{i\cdots j}^{2}\,,~{}~{}~{}{\rm
with}~{}k_{i\cdots j}\equiv\sum_{a=i}^{j}\,k_{a}\,,~{}~{}~{}~{}$ (4)

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

Figure 1: Two $5$-point diagrams

In double ordered BAS amplitudes, each interaction vertex features
antisymmetry for lines attached to it. In other words, swamping two lines $a$
and $b$ at a vertex creates a $-$ sign, therefore BAS amplitudes with
different orderings carry different overall $\pm$ sign. To simplify the
description of soft behavior, we choose the overall sign to be $+$ if two
orderings carried by the BAS amplitude are the same. For example, the
amplitude ${\cal A}_{\rm BAS}(1,2,3,4|1,2,3,4)$ carries the overall sign $+$
under our convention. The sign for amplitudes with two different orderings can
be generated from the above reference one by counting flippings, via the
diagrammatic technic proposed in Cachazo:2013iea .

Due to the definition of tree BAS amplitudes introduced above, it is direct to
observe the leading soft behavior of BAS amplitude ${\cal A}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})$. Take the external scalar $i$
to be the soft particle, with $k_{i}^{\mu}\sim\tau k_{i}^{\mu}$ and $\tau\to
0$, then the propagators $1/s_{i(i-1)}$ and $1/s_{i(i+1)}$ become divergent in
such soft limit, therefore

$\displaystyle{\cal A}^{(0)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})$ $\displaystyle=$
$\displaystyle{1\over\tau}\,S^{(0)_{i}}_{s}\,{\cal A}_{\rm
BAS}(1,\cdots,i-1,i+1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n}\setminus
i)\,.~{}~{}~{}$ (5)

Throughout this paper, we use the superscript $(a)_{i}$ to denote the
contribution at the $a^{\rm th}$ order, with the particle $i$ taken to be
soft. In the above, the leading soft factor $S^{(0)_{i}}_{s}$ is given by

$\displaystyle S^{(0)_{i}}_{s}=\left(\,{\delta_{i(i+1)}\over
s_{i(i+1)}}+{\delta_{(i-1)i}\over s_{(i-1)i}}\,\right)\,,~{}~{}~{}~{}$ (6)

where the symbol $\delta_{ij}$ is determined by positions of $i$ and $j$ in
the ordering $\vec{\boldsymbol{\sigma}}_{n}$. If $i$, $j$ are not adjacent to
each other, $\delta_{ij}=0$. If $i$ and $j$ are two adjacent elements, we have
$\delta_{ij}=1$ for $i\prec j$, $\delta_{ij}=-1$ for $j\prec i$. The notation
$a\prec b$ for a given permutation including elements $a$, $b$ means the
position of $a$ is on the left side of the position of $b$.

The higher order soft behaviors can also be analysed by considering
contributions from allowed Feynman diagrams. However, suppose we formally
express them as

$\displaystyle{\cal A}^{(a)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})$ $\displaystyle=$
$\displaystyle\tau^{(a-1)}\,S^{(a)_{i}}_{s}\,{\cal A}_{\rm
BAS}(1,\cdots,i-1,i+1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,,$
(7)

the soft factors $S^{(a)_{i}}_{s}$ with $a\geq 1$ do not satisfy the
universality. For example, the $4$-point BAS amplitudes only have the leading
order term, while the higher-point amplitudes receives contribution from
higher orders. In this sense, for tree BAS amplitudes under consideration, the
expected universal soft factor only exist at the leading order.

###  2.2 Transmutation operators

The transmutation operators proposed by Cheung, Shen and Wen connects tree
amplitudes of different theories together, by transmuting amplitudes of one
theory to those of another Cheung:2017ems . In this paper, we only use the
combinatorial operator ${\cal T}[1,\cdots,n]$, which turns GR amplitudes to
ordered YM amplitudes, and also transmutes YM amplitudes to double ordered BAS
amplitudes, namely,

$\displaystyle{\cal A}_{\rm YM}(1,\cdots,n)$ $\displaystyle=$
$\displaystyle\widetilde{\cal T}[1,\cdots,n]\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n})\,,$ $\displaystyle{\cal A}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})$ $\displaystyle=$
$\displaystyle{\cal T}[1,\cdots,n]\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})\,.~{}~{}$ (8)

In the above, ${\boldsymbol{h}}_{n}$ stands for the unordered set of $n$
external gravitons, and $\vec{\boldsymbol{\sigma}}_{n}$ labels the ordered set
of $n$ external gluons. The polarization tensor of each graviton is decomposed
as
$\varepsilon_{p}^{\mu\nu}=\epsilon_{p}^{\mu}\widetilde{\epsilon}_{p}^{\nu}$,
where $\epsilon_{p}$ and $\widetilde{\epsilon}_{p}$ with
$p\in\\{1,\cdots,n\\}$ are two sectors of polarization vectors. The operator
$\widetilde{\cal T}[1,\cdots,n]$ is defined for polarization vectors
$\widetilde{\epsilon}_{p}$ (as will be explicitly explained soon), transmutes
the unordered GR amplitude ${\cal A}_{\rm GR}(\boldsymbol{h}_{n})$ to the YM
amplitude ${\cal A}_{\rm YM}(1,\cdots,n)$ with ordering $1,\cdots,n$, and the
external gluons carry polarization vectors $\epsilon_{p}$. The operator ${\cal
T}[1,\cdots,n]$ is defined for polarization vectors $\epsilon_{p}$, transmutes
the ordered YM amplitude ${\cal A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})$ to
the double ordered BAS amplitude ${\cal A}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})$ for scalars without any
polarization. Notice that the relation in (8) which turns GR amplitude to YM
one only makes sense for the extended gravity theory that Einstein gravity is
coupled to a $2$-form and dilaton field.

One of the explicit forms for the combinatorial operator ${\cal
T}[1,\cdots,n]$ is given by

$\displaystyle{\cal T}[1,\cdots,n]=\Big{(}\prod_{a=2}^{n-1}\,{\cal
I}_{(a-1)an}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,,~{}~{}~{}~{}{\rm
with}~{}\partial_{x}\equiv{\partial\over\partial x}\,,~{}~{}$ (9)

where the insertion operator ${\cal I}_{bac}$ is defined as

$\displaystyle{\cal I}_{bac}\equiv\partial_{\epsilon_{a}\cdot
k_{b}}-\partial_{\epsilon_{a}\cdot k_{c}}\,.$ (10)

The above ${\cal T}[1,\cdots,n]$ is defined for polarization vectors
$\epsilon_{p}$. The dual operator $\widetilde{\cal T}[1,\cdots,n]$, defined
for polarization vectors $\widetilde{\epsilon}_{p}$, can be obtained from
${\cal T}[1,\cdots,n]$ by replacing all $\epsilon_{p}$ with
$\widetilde{\epsilon}_{p}$. As proved in Zhou:2018wvn ; Bollmann:2018edb , the
effect of insertion operator ${\cal I}_{bac}$ is to reduce the spin of
particle $a$ by $1$, and insert it between $b$ and $c$ in the ordering.
According to the above interpretation for insertion operator, the effect of
the operator ${\cal T}[1,\cdots,n]$ in (9) can be understood as,

  * •

Generating two endpoints $1$ and $n$ of the ordering.

  * •

Inserting the leg $2$ between $1$ and $n$.

  * •

Inserting the leg $3$ between $2$ and $n$.

  * •

Repeating the above procedure to insert other legs in turn, until the full
ordering $1,\cdots,n$ is completed.

The above steps exhibit the process for generating the full ordering. Notice
that we assume the order of performing differentials in (9) to be from right
to left. However, since all insertion operators in (9) are algebraically
commutative, we can rearrange the order as ${\cal I}_{(n-2)(n-1)n}{\cal
I}_{(n-3)(n-2)n}\cdots{\cal I}_{23n}{\cal I}_{12n}$, and interpret the effects
of them in the above way.

To create any desired ordering $\vec{\boldsymbol{\sigma}}_{n}$, the
corresponding formula of combinatorial operator ${\cal
T}[\vec{\boldsymbol{\sigma}}_{n}]$ is not unique, due to the interpretation
for the insertion operator ${\cal I}_{bac}$. For instance, to generate the
ordering $1,2,3,4$ one can chose

$\displaystyle{\cal T}[1,2,3,4]={\cal I}_{234}\,{\cal
I}_{124}\,\partial_{\epsilon_{1}\cdot\epsilon_{4}}\,,$ (11)

which realizes the goal as

  * •

Generating two endpoints $1$ and $4$.

  * •

Inserting the leg $2$ between $1$ and $4$.

  * •

Inserting the leg $3$ between $2$ and $4$.

However, the different choice

$\displaystyle{\cal T}[1,2,3,4]={\cal I}_{123}\,{\cal
I}_{134}\,\partial_{\epsilon_{1}\cdot\epsilon_{4}}$ (12)

is also correct, this operator generates the ordering $1,2,3,4$ as

  * •

Generating two endpoints $1$ and $4$.

  * •

Inserting the leg $3$ between $1$ and $4$.

  * •

Inserting the leg $2$ between $1$ and $3$.

Such freedom for choosing the combinatorial operator ${\cal T}[1,\cdots,n]$
will play the crucial role in subsequent sections.

The analogous operator ${\cal T}[\vec{\boldsymbol{\sigma}}_{m}]$, where
$\vec{\boldsymbol{\sigma}}_{m}$ is the ordering among $m$ elements in the
complete set $\\{1,\cdots,n\\}$, with $m<n$, also leads to the meaningful
interpretation. For instance, one can define ${\cal T}[2,3,4]={\cal
I}_{234}\partial_{\epsilon_{2}\cdot\epsilon_{4}}$. When acting on the
$4$-point YM amplitude ${\cal A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{4})$ with
external gluons encoded as $\\{1,2,3,4\\}$, the above operator ${\cal
T}[2,3,4]$ turns gluons in the set $\\{2,3,4\\}$ to scalars, and generates the
ordering $2,3,4$. The resulted amplitude ${\cal A}_{\rm
YMS}(2,3,4;1|\vec{\boldsymbol{\sigma}}_{4})$ is known as the Yang-Mills-scalar
(YMS) amplitude that the gluon $1$ interacts with BAS scalars in
$\\{2,3,4\\}$. Similarly, the operators $\widetilde{\cal
T}[\vec{\boldsymbol{\sigma}}_{m}]$ with $m<n$ transmutes GR amplitudes to
Einstein-Yang-Mills (EYM) amplitudes those gravitons couple to gluons.

##  3 Soft behavior of YM amplitudes

In this section, we use the transmutation operator introduced previously to
reconstruct known soft factors of YM amplitudes at leading and sub-leading
orders, and prove the nonexistence of higher order soft factor which satisfies
universality. To avoid the treatment for complexity induced by momentum
conservation, in this and next sections, we use the momentum conservation to
eliminate all $k_{n}$ in each amplitude.

###  3.1 Constraints on soft factors

Consider the soft behavior of any $n$-point amplitude ${\cal A}_{n}$, with
$k_{i}\to\tau k_{i}$, $\tau\to 0$. One can always expand the full amplitude as
in (2), to acquire the formal factorization behavior ${\cal
A}^{(a)_{i}}_{n}=S^{(a)_{i}}\,{\cal A}_{n-1}$ at any order. Since such formula
is not the physically expected factorization, we need to impose appropriate
constraints on soft factors.

The first important constraint is the universality, i.e., the expression of
the soft factor is independent on the number of external legs. For $4$-point
amplitudes, all allowed propagators have the form $1/s_{ij}$ where $i$ is the
soft particle under consideration, thus the denominator of each term behaves
as $\tau s_{ij}$ in the soft limit. Therefore, for the $4$-point case, the
formula of soft factor at $a^{\rm th}$ order is restricted to

$\displaystyle S^{(a)_{i}}_{g}=\sum_{j}\,{N^{(a)}_{ij}\over s_{ij}}\,,~{}~{}$
(13)

where the summation is over all $j$ those $1/s_{ij}$ contribute to ${\cal
A}_{n}$. Then the universality requires the formula of soft factors in (13)
should also be satisfied for higher-point cases. It is impossible to extended
the formula (13) to incorporate propagators like $1/s_{ijk}$ from multi-
particle channels without breaking the universality, since these propagators
do not proportional to $1/\tau$ thus can not be on an equal footing with
$1/s_{ij}$. If the formula (13) for arbitrary number of external particles
does not hold at the $a^{\rm th}$ order, then we say the universal soft factor
does not exist at this order, since the independence on the number of external
legs is violated.

For the YM case under consideration in this section, the numerator
$N^{(a)}_{ij}(\epsilon_{i},k_{i},k_{\ell})$ depends on $\epsilon_{i}$, $k_{i}$
for the soft gluon, and $k_{\ell}$ carried by other external gluons, but is
independent of any $\epsilon_{\ell}$ with $\ell\neq i$, since both ${\cal
A}^{(a)_{i}}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})$ and ${\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$ are linear on each
$\epsilon_{\ell}$. We allow each soft factor to be an operator, acts on
$(n-1)$-point amplitudes, then the independence on $\epsilon_{\ell}$ should be
understood as that the effect of acting operator
$N^{(a)}_{ij}(\epsilon_{i},k_{i},k_{\ell})$ doe not break the linearity on
$\epsilon_{\ell}$.

Furthermore, for the gluons of YM theory, we can extended the universality of
soft behavior to a stronger version: the soft factors of pure YM amplitudes
also holds for Yang-Mills-scalar (YMS) amplitudes in which gluons interact
with BAS scalars. The reason is, the $3$-point YMS amplitude with two external
scalars $1,3$ and one external gluon $2$, is related to the $3$-point YM
amplitude via the operator $\partial_{\epsilon_{1}\cdot\epsilon_{3}}$
introduced in section 2.2. The differential
$\partial_{\epsilon_{1}\cdot\epsilon_{3}}$ is equivalent to the dimensional
reduction, namely, consider the $(d+1)$-dimensional space-time, let the
nonzero components of $\epsilon_{1}$ and $\epsilon_{3}$ to be in the extra
dimension, and keep $\epsilon_{2}$ and all $k_{p}$ with $p\in\\{1,2,3\\}$ to
be in the ordinary $d$-dimensional space-time. From $d$-dimensional point of
view, $1$ and $3$ behaves as two scalars while $2$ behaves as a gluon.
However, from the $(d+1)$-dimensional perspective, the $d$-dimensional scalar-
scalar-gluon and gluon-gluon-gluon vertices are exactly the same interaction.
Based on the above reason, it is nature to generalize the universality of
gluon soft behaviors to the YMS case. Such extended universality will be
useful in subsequent works.

Now we discuss the constraints from gauge invariance. To make the discussion
clear, let us define the Ward identity operator as

$\displaystyle{\cal W}_{q}\equiv\sum_{v}\,(k_{q}\cdot
v)\,\partial_{\epsilon_{q}\cdot v}\,,$ (14)

where the summation is over all Lorentz vectors $v^{\mu}$. Each numerator
$N^{(a)}_{ij}(\epsilon_{i},k_{i},k_{\ell})$ in (13) should be consistent with
gauge invariance for polarizations carried by external gluons, i.e.,

$\displaystyle 0=S^{(a)_{i}}_{g}\,{\cal W}_{q}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)={\cal
W}_{q}\,S^{(a)_{i}}_{g}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,.~{}~{}$ (15)

In the above, the first equality holds because the $(n-1)$-point amplitude
${\cal A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$ in independent
on $\epsilon_{i}$ when $q=i$, and states the gauge invariance of the amplitude
${\cal A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$ when $q\neq i$.
The second is based on the gauge invariance of ${\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ and the definition ${\cal
A}^{(a)_{i}}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})=S^{(a)_{i}}_{g}\,{\cal
A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$. Thus, we conclude the
commutativity $[{\cal W}_{q},S^{(a)_{i}}_{g}]=0$, due to the relation in (15).

Such commutativity requires the soft operator $S^{(a)_{i}}_{g}$ to have the
form

$\displaystyle
S^{(a)_{i}}_{g}=\sum_{j}\,\sum_{r=1}^{k}\,{P^{(a)}_{ij;r}(\epsilon_{i},k_{i},k_{\ell})\over
s_{ij}}\,{\cal O}^{(a)}_{ij;r}\,,~{}~{}$ (16)

where each $P^{(a)}_{ij;r}(\epsilon_{i},k_{i},k_{\ell})$ is a polynomial of
Lorentz invariants, while each ${\cal O}^{(a)}_{ij;r}$ is an operator. The
summation over integers $r$ means we allow more than one operators contribute
to the numerator $N^{(a)}_{ij}(\epsilon_{i},k_{i},k_{\ell})$, albeit all cases
which will be encountered in this note are those $k=1$. The operator ${\cal
O}^{(a)}_{ij;r}$ satisfies that when acting on any Lorentz invariant
$k_{p}\cdot v$, the linearity on $k_{p}$ is kept, due to the following reason.
The operator $S^{(a)_{i}}_{g}$ transmutes the Lorentz invariant
$\epsilon_{p}\cdot v$ with $p\neq i$ as follows

$\displaystyle S^{(a)_{i}}_{g}\,\Big{(}\epsilon_{p}\cdot
v\Big{)}=\sum_{j}\,\sum_{r=1}^{k}\,{P^{(a)}_{ij;r}(\epsilon_{i},k_{i},k_{\ell})\over
s_{ij}}\,\epsilon_{p}\cdot v^{\prime}_{j;r}\,,$ (17)

without breaking the linearity on $\epsilon_{p}$, as discussed previously.
Here we have replaced the Lorentz vector $v^{\mu}$ by new ones
$(v^{\prime}_{j;r})^{\mu}$, to reflect the action of ${\cal O}^{(a)}_{ij;r}$.
Therefore, we have

$\displaystyle{\cal W}_{p}\,S^{(a)_{i}}_{g}\,\Big{(}\epsilon_{p}\cdot
v\Big{)}=\sum_{j}\,\sum_{r=1}^{k}\,{P^{(a)}_{ij}(\epsilon_{i},k_{i},k_{\ell})\over
s_{ij}}\,k_{p}\cdot v^{\prime}_{j;r}\,.$ (18)

Then the commutation relation $[{\cal W}_{p},S^{(a)_{i}}_{g}]=0$ leads to

$\displaystyle S^{(a)_{i}}_{g}\,\Big{(}k_{p}\cdot
v\Big{)}=S^{(a)_{i}}_{g}\,{\cal W}_{p}\,\Big{(}\epsilon_{p}\cdot
v\Big{)}=\sum_{j}\,\sum_{r=1}^{k}\,{P^{(a)}_{ij}(\epsilon_{i},k_{i},k_{\ell})\over
s_{ij}}\,k_{p}\cdot v^{\prime}_{j;r}\,,$ (19)

which means the operators ${\cal O}^{(a)}_{ij;r}$ do not affect the linearity
on $k_{p}$.

With the expected formula (16) and the property of ${\cal O}^{(a)}_{ij;r}$
displayed above, now we are ready to study the existence of soft factors at
each order.

###  3.2 Leading order

In this subsection, we use transmutation operators to investigate whether the
leading order soft behavior of the YM amplitude ${\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ can be represented as the factorized
formula

$\displaystyle{\cal A}^{(0)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})=S^{(0)_{i}}_{g}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,,~{}~{}$ (20)

where the leading soft factor $S^{(0)_{i}}_{g}$ satisfies the required form in
(16).

We chose the combinatorial transmutation operator ${\cal T}[1,\cdots,n]$ to be

$\displaystyle{\cal T}_{0}[1,\cdots,n]$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\epsilon_{i}\cdot
k_{i-1}}-\partial_{\epsilon_{i}\cdot
k_{i+1}}\Big{)}\,\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{i+1}\cdot
k_{i-1}}\,\Big{(}\prod_{a=2}^{i-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,,~{}~{}$ (21)

which creates the ordering $1,\cdots,n$ as follows:

  * •

Generating two endpoints $1$ and $n$ of the ordering.

  * •

Inserting legs $a$ with $a\in\\{2,\cdots,i-1\\}$ between $1$ and $n$.

  * •

Inserting the leg $(i+1)$ between $(i-1)$ and $n$.

  * •

Inserting legs $a$ with $a\in\\{i+2,\cdots,n-1\\}$ between $(i+1)$ and $n$.

  * •

Inserting the leg $i$ between $(i-1)$ and $(i+1)$.

In the operator (21), all $\partial_{\epsilon_{a}\cdot k_{n}}$ in ${\cal
I}_{(a-1)an}$ are removed, since all $k_{n}$ in the amplitude are eliminated
by using momentum conservation. It is straightforward to recognize that

$\displaystyle{\cal T}_{0}[1,\cdots,n]={\cal I}_{(i-1)i(i+1)}\,{\cal
T}[1,\cdots,i-1,i+1,\cdots,n]\,,~{}~{}$ (22)

where the operator ${\cal T}[1,\cdots,i-1,i+1,\cdots,n]$ transmutes the
$n$-point YM amplitude to YMS one as follows

$\displaystyle{\cal T}[1,\cdots,i-1,i+1,\cdots,n]\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})={\cal A}_{\rm
YMS}(1,\cdots,i-1,i+1,\cdots,n;i|\vec{\boldsymbol{\sigma}}_{n})\,.~{}~{}$ (23)

Now we use the operator chosen in (21) to link the soft behaviors of gluons
and BAS scalars together. We can expand the BAS and YM amplitudes by $\tau$,
then the transmutation relation in (8) reads

$\displaystyle{1\over\tau}\,{\cal A}^{(0)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})+{\cal A}^{(1)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})+\tau\,{\cal A}^{(2)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})+\cdots$ (24) $\displaystyle=$
$\displaystyle{\cal T}_{0}[1,\cdots,n]\,\Big{(}{1\over\tau}\,{\cal
A}^{(0)_{i}}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})+{\cal A}^{(1)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})+\tau\,{\cal A}^{(2)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})+\cdots\Big{)}\,.$

Since the operator ${\cal T}_{0}[1,\cdots,n]$ in (21) does not include any
$\partial_{\epsilon_{p}\cdot k_{i}}$, it is independent of the soft parameter
$\tau$. Consequently, we have

$\displaystyle{\cal A}^{(a)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})={\cal T}_{0}[1,\cdots,n]\,{\cal
A}^{(a)_{i}}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})\,,~{}~{}$ (25)

holds at any order.

At the leading order, one can substitute the soft behavior of BAS amplitude
given in (5) and (6), to obtain

$\displaystyle{\cal T}_{0}[1,\cdots,n]\,{\cal A}^{(0)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ $\displaystyle=$
$\displaystyle\Big{(}{\delta_{(i-1)i}\over s_{(i-1)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,{\cal A}_{\rm
BAS}(1,\cdots,i-1,i+1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n}\setminus
i)\,.~{}~{}$ (26)

Suppose ${\cal A}^{(0)_{i}}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})$ satisfies
the factorized formula (20), then we have

$\displaystyle{\cal T}_{0}[1,\cdots,n]\,{\cal A}^{(0)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ (27) $\displaystyle=$ $\displaystyle{\cal
I}_{(i-1)i(i+1)}\,{\cal T}[1,\cdots,i-1,i+1,\cdots,n]\,{\cal A}^{(0)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ $\displaystyle=$ $\displaystyle{\cal
I}_{(i-1)i(i+1)}\,{\cal A}^{(0)_{i}}_{\rm
YMS}(1,\cdots,i-1,i+1,\cdots,n;i|\vec{\boldsymbol{\sigma}}_{n})$
$\displaystyle=$ $\displaystyle{\cal I}_{(i-1)i(i+1)}\,S^{(0)_{i}}_{g}\,{\cal
A}_{\rm BAS}(1,\cdots,i-1,i+1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n}\setminus
i)\,,~{}~{}$

where the second equality uses the observation

$\displaystyle{\cal T}[1,\cdots,i-1,i+1,\cdots,n]\,{\cal A}^{(0)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})={\cal A}^{(0)_{i}}_{\rm
YMS}(1,\cdots,i-1,i+1,\cdots,n;i|\vec{\boldsymbol{\sigma}}_{n})\,,$ (28)

based on the transmutation relation (23). The third uses

$\displaystyle{\cal A}^{(0)_{i}}_{\rm
YMS}(1,\cdots,i-1,i+1,\cdots,n;i|\vec{\boldsymbol{\sigma}}_{n})=S^{(0)_{i}}_{g}\,{\cal
A}_{\rm BAS}(1,\cdots,i-1,i+1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n}\setminus
i)\,,~{}~{}$ (29)

which is indicated by the universality of soft behavior, namely, the formula
(29) holds as long as (20) holds, with exactly the same soft factor.
Substituting (27) into (26), we find the equations

$\displaystyle{\cal I}_{(i-1)i(i+1)}\,S^{(0)_{i}}_{g}$ $\displaystyle=$
$\displaystyle{\delta_{(i-1)i}\over s_{(i-1)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\,,~{}~{}$ (30)

hold for any $i\in\\{2,\cdots,n-1\\}$ (for $i=n-1$, the operator
$\partial_{\epsilon_{i}\cdot k_{i+1}}$ in ${\cal I}_{(i-1)i(i+1)}$ should be
removed, since we have eliminated all $k_{n}$ in the amplitude via momentum
conservation).

The unique solution to equations (30) for all $i\in\\{2,\cdots,n-1\\}$ is
found to be

$\displaystyle S^{(0)_{i}}_{g}=\sum_{j\neq
i}\,{\delta_{ji}\,(\epsilon_{i}\cdot k_{j})\over s_{ij}}\,.~{}~{}$ (31)

Notice that for $i=n-1$, the corresponding equation

$\displaystyle\partial_{\epsilon_{n-1}\cdot k_{n-2}}\,S^{(0)_{i}}_{g}$
$\displaystyle=$ $\displaystyle{\delta_{(n-2)(n-1)}\over
s_{(n-2)(n-1)}}+{\delta_{(n-1)n}\over s_{(n-1)n}}$ (32)

is not sufficient to determine the term $\delta_{ni}(\epsilon_{i}\cdot
k_{n})/s_{in}$ in the solution (31). This term is fixed by considering the
gauge invariance for the polarization $\epsilon_{i}$. Start from the the
ansatz

$\displaystyle S^{(0)_{i}}_{g}={X\over s_{in}}+\sum_{j\neq
i,n}\,{\delta_{ji}\,(\epsilon_{i}\cdot k_{j})\over s_{ij}}\,,$ (33)

the gauge invariance indicates

$\displaystyle 0={X\big{|}_{\epsilon_{i}\to k_{i}}\over s_{in}}+\sum_{j\neq
i,n}\,{\delta_{ji}\over 2}\,,$ (34)

then the relation $\sum_{k\neq i}\delta_{ki}=0$ together with the linearity on
$\epsilon_{i}$ fix $X$ to be $\delta_{ni}(\epsilon_{i}\cdot n)$. Comparing
with the expected formula (16), we see that the polynomial is
$P^{(0)}_{ij;1}(\epsilon_{i},k_{i},k_{\ell})=\delta_{ji}(\epsilon_{i}\cdot
k_{j})$, the operator is the identity operator ${\cal
O}^{(0)}_{ij;1}=\boldsymbol{1}$. The symbol $\delta_{ji}$ requires the
effective legs $j$ to be those adjacent to $i$ in the ordering
$\vec{\boldsymbol{\sigma}}_{n}$, thus ensured that the summation is for all
$j$ those the corresponding $1/s_{ij}$ contribute to the amplitude.

It is worth to point out that the result in (27) also implies the
commutativity

$\displaystyle\Big{[}S^{(0)_{i}}_{g},{\cal
T}[1,\cdots,i-1,i+1,\cdots,n]\Big{]}=0\,,~{}~{}$ (35)

due to the soft behavior (20) and the transmutation relation

$\displaystyle{\cal T}[1,\cdots,i-1,i+1,\cdots,n]\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)={\cal A}_{\rm
BAS}(1,\cdots,i-1,i+1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n}\setminus
i)\,.~{}~{}$ (36)

In general, the above interpretation is not correct for the operator ${\cal
T}[1,\cdots,i-1,i+1,\cdots,n]$ defined in (22), since momenta carried by
gluons in the set $\\{1,\cdots,n\\}\setminus i$ violate momentum conservation.
However, such interpretation makes sense in the soft limit $\tau\to 0$. The
commutativity in (35) can be generalized to arbitrary order as

$\displaystyle{\cal T}[1,\cdots,i-1,i+1,\cdots,n]\,S^{(a)_{i}}_{g}\,{\cal
A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$ (37) $\displaystyle=$
$\displaystyle{\cal T}[1,\cdots,i-1,i+1,\cdots,n]\,{\cal A}^{(a)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ $\displaystyle=$ $\displaystyle{\cal
A}^{(a)_{i}}_{\rm
YMS}(1,\cdots,i-1,i+1,\cdots,n;i|\vec{\boldsymbol{\sigma}}_{n})$
$\displaystyle=$ $\displaystyle S^{(a)_{i}}_{g}\,{\cal A}_{\rm
BAS}(1,\cdots,i-1,i+1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n}\setminus i)$
$\displaystyle=$ $\displaystyle S^{(a)_{i}}_{g}\,{\cal
T}[1,\cdots,i-1,i+1,\cdots,n]\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,,$

if the soft factor satisfies the requirement in (16) exisit at the $a^{\rm
th}$ order. The general commutation relation

$\displaystyle\Big{[}S^{(a)_{i}}_{g},{\cal
T}[1,\cdots,i-1,i+1,\cdots,n]\Big{]}=0\,,~{}~{}$ (38)

will be useful in subsequent subsections.

###  3.3 Sub-leading order

In this subsection, we continue to study the soft behavior of YM amplitudes at
the sub-leading order. The relation (72) also links the soft behaviors of YM
and BAS amplitudes at the sub-leading order. However, since the factorized
formula for BAS amplitudes is lacked, one can not repeat the manipulation in
the previous subsection 3.2, to solve the sub-leading YM soft factor from the
BAS one. Therefore, the operator ${\cal T}_{0}[1,\cdots,n]$ chosen in (21) and
the relation (72) are not effective for the current case. The above obstacle
motivates us to chose new operator ${\cal T}_{1}[1,\cdots,n]$ which connects
the sub-leading term of the YM side and the leading term of the BAS side
together.

The new operator ${\cal T}_{1}[1,\cdots,n]$ is chosen to be that in (9). Based
on the assumption that all $k_{n}$ in amplitudes are removed via momentum
conservation, we can remove all $\partial_{\epsilon_{p}\cdot k_{n}}$ in the
insertions operators ${\cal I}_{(a-1)an}$ to obtain

$\displaystyle{\cal
T}_{1}[1,\cdots,n]=\Big{(}\prod_{a=2}^{n-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,.~{}~{}$ (39)

In the soft limit, the operator (39) transmutes the expanded YM amplitude to
expanded BAS amplitude as follows

$\displaystyle{1\over\tau}\,{\cal A}^{(0)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})+{\cal A}^{(1)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})+\tau\,{\cal A}^{(2)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})+\cdots$ (40) $\displaystyle=$
$\displaystyle{\cal T}_{1}[1,\cdots,n]\,\Big{(}{1\over\tau}\,{\cal
A}^{(0)_{i}}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})+{\cal A}^{(1)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})+\tau\,{\cal A}^{(2)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})+\cdots\Big{)}\,.$

Using the leading soft factor (31), it is direct to see that the operator
${\cal T}_{1}[1,\cdots,n]$ annihilates the leading order YM term ${\cal
A}^{(0)_{i}}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})$, since ${\cal
T}_{1}[1,\cdots,n]$ involves the differential operator
$\partial_{\epsilon_{i+1}\cdot k_{i}}$, while ${\cal A}^{(0)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ is independent of $k_{i}$. The operator
$\partial_{\epsilon_{i+1}\cdot k_{i}}$ carries the parameter $1/\tau$ when
$k_{i}\to\tau k_{i}$, therefore, the operator ${\cal T}_{1}[1,\cdots,n]$
transmutes the sub-leading YM term ${\cal A}^{(1)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ to the leading BAS term ${\cal
A}^{(0)_{i}}_{\rm BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})$. Thus,
suppose the sub-leading soft behavior of YM amplitude satisfies the
factorization

$\displaystyle{\cal A}^{(1)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})=S^{(1)_{i}}_{g}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,,~{}~{}$ (41)

one can use the transmutation relation based on ${\cal T}_{1}[1,\cdots,n]$, to
solve the soft factor $S^{(1)_{i}}_{g}$ from the leading soft behavior of BAS
amplitude.

Based on above discussions, we can find the following relation for the assumed
sub-leading soft factor $S^{(1)_{i}}_{g}$,

$\displaystyle{\cal A}^{(0)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})$ $\displaystyle=$
$\displaystyle{\cal T}_{1}[1,\cdots,n]\,{\cal A}^{(1)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ (42) $\displaystyle=$ $\displaystyle{\cal
T}_{1}[1,\cdots,n]\,S^{(1)_{i}}_{g}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$ $\displaystyle=$
$\displaystyle\partial_{\epsilon_{i+1}\cdot
k_{i}}\,\partial_{\epsilon_{i}\cdot k_{i-1}}\,S^{(1)_{i}}_{g}\,{\cal
P}_{1}\,{\cal A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,,~{}~{}$

where the commutation relation

$\displaystyle\Big{[}S^{(1)_{i}}_{g},{\cal P}_{1}\Big{]}=0~{}~{}$ (43)

with the operator ${\cal P}_{1}$ defined as

$\displaystyle{\cal
P}_{1}=\Big{(}\prod_{a=2}^{i-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,,$ (44)

is ensured by the commutativity in (38), since ${\cal P}_{1}$ is a subpart
involved in ${\cal T}[1,\cdots,i-1,i+1,\cdots,n]$. Substituting the leading
soft factor of BAS amplitude (6) and the transmutation relation (36) into
(42), we arrive at the equations for $S^{(1)_{i}}_{g}$,

$\displaystyle\partial_{\epsilon_{i+1}\cdot
k_{i}}\,\partial_{\epsilon_{i}\cdot k_{i-1}}\,S^{(1)_{i}}_{g}\,{\cal
P}_{1}\,{\cal A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus
i)=\Big{(}{\delta_{(i-1)i}\over s_{(i-1)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,\partial_{\epsilon_{i+1}\cdot k_{i-1}}\,{\cal
P}_{1}\,{\cal A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,,~{}~{}$
(45)

hold for any $i\in\\{2,\cdots,n-1\\}$. Notice that ${\cal P}_{1}\,{\cal
A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})$ involves only one polarization
$\epsilon_{i+1}$, thus the above equations are convenient for analysing the
effect of operator $S^{(1)_{i}}_{g}$.

To solve equations (45), we observe that the effect of differential
$\partial_{\epsilon_{i+1}\cdot k_{i-1}}$ is turning $\epsilon_{i+1}\cdot
k_{i-1}$ to $1$ and annihilating all terms without $\epsilon_{i+1}\cdot
k_{i-1}$, due to the linear dependence on polarization $\epsilon_{i+1}$ of
each physical amplitude. Similarly, the operator
$\partial_{\epsilon_{i+1}\cdot k_{i}}\,\partial_{\epsilon_{i}\cdot k_{i-1}}$
turns $(\epsilon_{i+1}\cdot k_{i})(\epsilon_{i}\cdot k_{i-1})$ to $1$ and
annihilates all terms do not contain $(\epsilon_{i+1}\cdot
k_{i})(\epsilon_{i}\cdot k_{i-1})$. The Lorentz invariant
$(\epsilon_{i+1}\cdot k_{i})(\epsilon_{i}\cdot k_{i-1})$ under the action of
$\partial_{\epsilon_{i+1}\cdot k_{i}}\,\partial_{\epsilon_{i}\cdot k_{i-1}}$
must be created by acting $S^{(1)_{i}}_{g}$ on ${\cal P}_{1}\,{\cal A}{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$, since ${\cal P}_{1}\,{\cal
A}{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$ is independent of
$\epsilon_{i}$ and $k_{i}$. Thus, the operator $S^{(1)_{i}}_{g}$ should turn
$\epsilon_{i+1}\cdot k_{i-1}$ to a new Lorentz invariant which involves
$(\epsilon_{i+1}\cdot k_{i})(\epsilon_{i}\cdot k_{i-1})$, namely,

$\displaystyle S^{(1)_{i}}_{g}\,\epsilon_{i+1}\cdot
k_{i-1}=\Big{(}{\delta_{(i-1)i}\over s_{(i-1)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,(\epsilon_{i+1}\cdot k_{i})(\epsilon_{i}\cdot
k_{i-1})+C\,,$ (46)

where $C$ is the potential part which is annihilated by
$\partial_{\epsilon_{i+1}\cdot k_{i}}\partial_{\epsilon_{i}\cdot k_{i-1}}$.
Using the gauge invariance requirement $\big{[}{\cal
W}_{i},S^{(1)_{i}}_{g}\big{]}=0$, one can fix the undetected part $C$, and
obtain

$\displaystyle S^{(1)_{i}}_{g}\,\epsilon_{i+1}\cdot
k_{i-1}=\Big{(}{\delta_{(i-1)i}\over s_{(i-1)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,(\epsilon_{i+1}\cdot f_{i}\cdot k_{i-1})\,,~{}~{}$ (47)

where the antisymmetric strength tensor is defined as $f_{a}^{\mu\nu}\equiv
k^{\mu}_{a}\epsilon^{\nu}_{a}-\epsilon^{\mu}_{a}k^{\nu}_{a}$. Furthermore, the
commutativity in(43) implies that the Lorentz invariants
$\epsilon_{1}\cdot\epsilon_{n}$ and $\epsilon_{a}\cdot k_{a-1}$ with
$a\in\\{1,\cdots,i-1\\}\cup\\{i+2,\cdots,n-1\\}$ are unaffected while
$\epsilon_{i+1}\cdot k_{i-1}$ is transmuted as in (47). It means the operator
$S^{(1)_{i}}_{g}$ should satisfy the Leibnitz rule. Consequently, the operator
$S^{(1)_{i}}_{g}$ is found to be

$\displaystyle S^{(1)_{i}}_{g}=\sum_{j\neq i}\,{\delta_{ij}\over
s_{ij}}\,\Big{(}k_{j}\cdot f_{i}\cdot\partial_{k_{j}}+\epsilon_{j}\cdot
f_{i}\cdot\partial_{\epsilon_{j}}\Big{)}\,,~{}~{}$ (48)

which is equivalent to

$\displaystyle S^{(1)_{i}}_{g}=\sum_{j\neq
i}\,{\delta_{ji}\,(\epsilon_{i}\cdot J_{j}\cdot k_{i})\over s_{ij}}\,,~{}~{}$
(49)

where $J_{j}$ serves as the angular momentum carried by the external particle
$j$.

Comparing (49) with the desired formula in (16), we see that the polynomial
$P^{(1)}_{ij;1}(\epsilon_{i},k_{i},k_{\ell})$ is trivially $\delta_{ji}$,
while the operator is ${\cal O}^{(1)}_{ij;1}=\epsilon_{i}\cdot J_{j}\cdot
k_{i}$. Again, the symbol $\delta_{ji}$ ensures that the effective summation
is for external legs $j$, which contribute $1/s_{ij}$ to the amplitude.

###  3.4 Higher order

The leading and sub-leading soft factors in (31) and (49) are standard soft
factors of tree YM amplitudes, found in literatures Casali:2014xpa ;
Schwab:2014xua . In this subsection, we argue that the sub-sub-leading soft
factor $S^{(2)_{i}}_{g}$ satisfies the expectation in (16) does not exist.

Similar as in the previous subsection, our method is to choose an operator
${\cal T}_{2}[1,\cdots,n]$ which relates the sub-sub-leading YM term ${\cal
A}_{\rm YM}^{(2)_{i}}(\vec{\boldsymbol{\sigma}}_{n})$ to the leading BAS term
${\cal A}^{(0)_{i}}_{\rm BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})$, then
try to solve $S^{(2)_{i}}_{g}$ from $S^{(0)_{i}}_{s}$. Such operator is chosen
as

$\displaystyle{\cal T}_{2}[1,\cdots,n]=\Big{(}\partial_{\epsilon_{i-1}\cdot
k_{i-2}}-\partial_{\epsilon_{i-1}\cdot
k_{i}}\Big{)}\,\Big{(}\prod_{a=i+1}^{n-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{i}\cdot
k_{i-2}}\,\Big{(}\prod_{a=2}^{i-2}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,,~{}~{}$ (50)

which is similar to the operator ${\cal T}_{0}[1,\cdots,n]$ in (21), but with
$(i+1)$ replaced by $i$. The operator ${\cal T}_{2}[1,\cdots,n]$ in (50) can
be separated as

$\displaystyle{\cal T}_{2}[1,\cdots,n]={\cal T}_{21}[1,\cdots,n]+{\cal
T}_{21}[1,\cdots,n]\,,$ (51)

where

$\displaystyle{\cal
T}_{21}[1,\cdots,n]=\Big{(}\prod_{a=i+1}^{n-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{i}\cdot
k_{i-2}}\,\Big{(}\prod_{a=2}^{i-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,,~{}~{}$ (52)

and

$\displaystyle{\cal T}_{22}[1,\cdots,n]=-\partial_{\epsilon_{i-1}\cdot
k_{i}}\,\Big{(}\prod_{a=i+1}^{n-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{i}\cdot
k_{i-2}}\,\Big{(}\prod_{a=2}^{i-2}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,.~{}~{}$ (53)

It is straightforward to verify that the operator ${\cal T}_{21}[1,\cdots,n]$
annihilates ${\cal A}^{(0)_{i}}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})$,
since ${\cal T}_{21}[1,\cdots,n]$ includes the differential
$\partial_{\epsilon_{i+1}\cdot k_{i}}$ but ${\cal A}^{(0)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ is independent of $k_{i}$. Meanwhile, the
operator ${\cal T}_{22}[1,\cdots,n]$ annihilates both ${\cal A}^{(0)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ and ${\cal A}^{(1)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$, since differentials
$\partial_{\epsilon_{i+1}\cdot k_{i}}$ and $\partial_{\epsilon_{i-1}\cdot
k_{i}}$ in ${\cal T}_{22}[1,\cdots,n]$ requires the bilinearity on $k_{i}$,
but ${\cal A}^{(0)_{i}}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})$ is
independent of $k_{i}$ and ${\cal A}^{(1)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ is linear on $k_{i}$. Two operators ${\cal
T}_{21}[1,\cdots,n]$ carry scale parameters $1/\tau$ and $1/\tau^{2}$, arise
from $\partial_{\epsilon_{i+1}\cdot k_{i}}$ and $\partial_{\epsilon_{i+1}\cdot
k_{i}}\partial_{\epsilon_{i-1}\cdot k_{i}}$, respectively. Therefore, at the
$\tau^{-1}$ order we have

$\displaystyle{\cal A}^{(0)_{i}}_{\rm
BAS}(1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n})={\cal
T}_{21}[1,\cdots,n]\,{\cal A}^{(1)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})+{\cal T}_{22}[1,\cdots,n]\,{\cal
A}^{(2)_{i}}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})\,.~{}~{}$ (54)

By employing the sub-leading soft behavior of YM amplitudes in (41) and (49),
one can find

$\displaystyle{\cal T}_{21}[1,\cdots,n]\,{\cal A}^{(1)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ (55) $\displaystyle=$
$\displaystyle\partial_{\epsilon_{i+1}\cdot
k_{i}}\,\partial_{\epsilon_{i}\cdot
k_{i-2}}\,S^{(1)_{i}}_{g}\,\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\Big{(}\prod_{a=2}^{i-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$ $\displaystyle=$
$\displaystyle\Big{(}{\delta_{(i-2)i}\over s_{(i-2)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,\partial_{\epsilon_{i+1}\cdot
k_{i-2}}\,\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\Big{(}\prod_{a=2}^{i-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,,~{}~{}$

where the first equality uses the commutation relation (38), and the second
uses the following property

$\displaystyle S^{(1)_{i}}_{g}\,(\epsilon_{i+1}\cdot
k_{i-2})=\Big{(}{\delta_{(i-2)i}\over s_{(i-2)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,(\epsilon_{i+1}\cdot f_{i}\cdot k_{i-2})\,,$ (56)

as can be directly verified by using the definition of $S^{(1)_{i}}_{g}$ in
(49) and (48). Meanwhile, we also have

$\displaystyle{\cal T}_{22}[1,\cdots,n]\,{\cal A}^{(2)_{i}}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n})$ (57) $\displaystyle=$
$\displaystyle-\partial_{\epsilon_{i+1}\cdot
k_{i}}\,\partial_{\epsilon_{i}\cdot k_{i-2}}\,\partial_{\epsilon_{i-1}\cdot
k_{i}}\,S^{(2)_{i}}_{g}\,\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\Big{(}\prod_{a=2}^{i-2}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,,~{}~{}$

if the desired $S^{(2)_{i}}_{g}$ exist. In the above derivation, the
commutativity in (38) is used again.

Combining results in (55), (57) and the leading soft behavior of BAS amplitude
together, we get the equations

$\displaystyle\partial_{\epsilon_{i+1}\cdot
k_{i}}\,\partial_{\epsilon_{i}\cdot k_{i-2}}\,\partial_{\epsilon_{i-1}\cdot
k_{i}}\,S^{(2)_{i}}_{g}\,{\cal P}_{2}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$ (58) $\displaystyle=$
$\displaystyle\Big{[}\Big{(}{\delta_{(i-2)i}\over
s_{(i-2)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,\partial_{\epsilon_{i+1}\cdot
k_{i-2}}\,\partial_{\epsilon_{i-1}\cdot k_{i-2}}-\Big{(}{\delta_{(i-1)i}\over
s_{(i-1)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,\partial_{\epsilon_{i+1}\cdot
k_{i-1}}\,\partial_{\epsilon_{i-1}\cdot k_{i-2}}\Big{]}\,{\cal P}_{2}\,{\cal
A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,,~{}~{}$

hold for any $i\in\\{2,\cdots,n-1\\}$, with

$\displaystyle{\cal
P}_{2}=\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\Big{(}\prod_{a=2}^{i-2}\,\partial_{\epsilon_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\epsilon_{1}\cdot\epsilon_{n}}\,.$ (59)

The above equations imply that the operator $S^{(2)_{i}}_{g}$ should transmute
the Lorentz invariant $(\epsilon_{i+1}\cdot k_{i-2})(\epsilon_{i-1}\cdot
k_{i-2})$ or $(\epsilon_{i+1}\cdot k_{i-1})(\epsilon_{i-1}\cdot k_{i-2})$ to a
new Lorentz invariant, which contains a part proportional to
$(\epsilon_{i+1}\cdot k_{i})(\epsilon_{i}\cdot k_{i-2})(\epsilon_{i-1}\cdot
k_{i})$. For the first case, the bilinearity on $k_{i-2}$ is turned to the
linearity. For the second case, the linearity on $k_{i-1}$ is turned to the
independence. Both two situations can not be realized via a polynomial
$P^{(a)}_{ij}(\epsilon_{i},k_{i},k_{\ell})$ and an operator ${\cal O}_{ij}$
which keeps the linearity on any $k_{p}$, as required in (16). Thus, we
conclude that the YM soft factor satisfies our expectation can not be found at
the sub-sub-leading order.

In equations (58), we have used the transmutation relation

$\displaystyle{\cal A}_{\rm
BAS}(1,\cdots,i-1,i+1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n}\setminus
i)=\partial_{\epsilon_{i+1}\cdot k_{i-1}}\,\partial_{\epsilon_{i-1}\cdot
k_{i-2}}\,{\cal P}_{2}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,.~{}~{}$ (60)

Instead of the above one, we can also consider the equivalent relation

$\displaystyle{\cal A}_{\rm
BAS}(1,\cdots,i-1,i+1,\cdots,n|\vec{\boldsymbol{\sigma}}_{n}\setminus
i)=\Big{(}\partial_{\epsilon_{i-1}\cdot k_{i-2}}-\partial_{\epsilon_{i-1}\cdot
k_{i+1}}\Big{)}\,\partial_{\epsilon_{i+1}\cdot k_{i-2}}\,{\cal P}_{2}\,{\cal
A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,,~{}~{}$ (61)

where the operator $\partial_{\epsilon_{i+1}\cdot k_{i-2}}$ serves as the
insertion operator ${\cal I}_{(i-2)(i+1)n}$ which inserts the leg $(i+1)$
between $(i-2)$ and $n$, while the operator $(\partial_{\epsilon_{i-1}\cdot
k_{i-2}}-\partial_{\epsilon_{i-1}\cdot k_{i+1}})$ is interpreted as ${\cal
I}_{(i-2)(i-1)(i+1)}$, which inserts $(i-1)$ between $(i-2)$ and $(i+1)$.
Replacing (60) by (61), the equations (58) are modified to

$\displaystyle\partial_{\epsilon_{i+1}\cdot
k_{i}}\,\partial_{\epsilon_{i}\cdot k_{i-2}}\,\partial_{\epsilon_{i-1}\cdot
k_{i}}\,S^{(2)_{i}}_{g}\,{\cal P}_{2}\,{\cal A}_{\rm
YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)$ (62) $\displaystyle=$
$\displaystyle\Big{[}\Big{(}{\delta_{(i-2)i}\over
s_{(i-2)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,\partial_{\epsilon_{i+1}\cdot
k_{i-2}}\,\partial_{\epsilon_{i-1}\cdot k_{i-2}}-\Big{(}{\delta_{(i-1)i}\over
s_{(i-1)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,\partial_{\epsilon_{i-1}\cdot
k_{i-2}}\,\partial_{\epsilon_{i+1}\cdot k_{i-2}}$
$\displaystyle+\Big{(}{\delta_{(i-1)i}\over s_{(i-1)i}}+{\delta_{i(i+1)}\over
s_{i(i+1)}}\Big{)}\,\partial_{\epsilon_{i-1}\cdot
k_{i+1}}\,\partial_{\epsilon_{i+1}\cdot k_{i-2}}\Big{]}\,{\cal P}_{2}\,{\cal
A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n}\setminus i)\,.~{}~{}$

For the above equations, the solution $S^{(2)_{i}}_{g}$ is also forbidden,
since the similar analysis indicates the effect of turning the bilinearity on
$k_{i-2}$ to the linearity, or turning the linearity on $k_{i+1}$ to the
independence.

The similar argument can also be applied to exclude the solution of
$S^{(a)_{i}}_{g}$ satisfying the requirement (16), with $a\geq 3$. For
instance, one can choose the differentials those create the ordering
$1,\cdots,i-3,i,\cdots,n$ first, then insert $(i-2)$ between $(i-3)$ and $i$,
and subsequently insert $(i-1)$ between $(i-2)$ and $i$. Such combinatorial
operator can be decomposed into three parts which carry scale parameters
$1/\tau$, $1/\tau^{2}$ and $1/\tau^{3}$ respectively, therefore transmutes the
summation of leading, sub-leading and sub-sub-leading terms of YM amplitude to
the leading contribution of BAS amplitude. Then one can observe the similar
phenomenon, the assumed soft operators $S^{(2)_{i}}_{g}$ and $S^{(3)_{i}}_{g}$
decrease the power of some external momenta thus are forbidden. It is easy to
see that when $k_{i}$ appears more than once in at lest one part of the
combinatorial operator, then the above phenomenon, which excludes the solution
under the constraint (16), always happen.

##  4 Soft behavior of GR amplitudes

In this section, we study the factorization of GR amplitudes in the soft
limit,

$\displaystyle{\cal A}^{(a)_{i}}_{\rm
GR}({\boldsymbol{h}}_{n})=S^{(a)_{i}}_{h}\,{\cal A}_{\rm
GR}({\boldsymbol{h}}_{n}\setminus i)\,,~{}~{}$ (63)

with assumed soft factors $S^{(a)_{i}}_{h}$, at $a^{\rm th}$ order. Through
the argument paralleled to that in section 3.1, we expect soft factors of GR
amplitudes to take the form

$\displaystyle S^{(a)_{i}}_{h}=\sum_{j}\,{N_{ij}\over
s_{ij}}=\sum_{j}\,\sum_{r=1}^{k}\,{P^{(a)}_{ij;r}(\varepsilon_{i},k_{i},k_{\ell})\over
s_{ij}}\,{\cal O}^{(a)}_{ij;r}\,,~{}~{}$ (64)

where the operators ${\cal O}^{(a)}_{ij;r}$ maintain the linearity on $k_{p}$
when acting on $k_{p}\cdot v$, similar as its YM counterpart in (16). Here
$\varepsilon_{i}^{\mu\nu}$ is the polarization tensor carried by the soft
graviton $i$. Meanwhile, the commutation relation in (38) is now extended to

$\displaystyle\Big{[}S^{(a)_{i}}_{h},\widetilde{\cal
T}[1,\cdots,i-1,i+1,\cdots,n]\Big{]}=0\,,~{}~{}$ (65)

According to the transmutation in (8), the combinatorial operator
$\widetilde{\cal T}[1,\cdots,n]$ turns the GR amplitudes to the YM ones with
specific ordering $1,\cdots,n$. For such YM amplitudes ${\cal A}_{\rm
YM}(1,\cdots,n)$ , the soft factors in (31) and (49) are reduced to

$\displaystyle S^{(0)_{i}}_{g}={\epsilon_{i}\cdot k_{i-1}\over
s_{i(i-1)}}-{\epsilon_{i}\cdot k_{i+1}\over s_{i(i+1)}}\,,~{}~{}$ (66)

and

$\displaystyle S^{(1)_{i}}_{g}={\epsilon_{i}\cdot J_{i-1}\cdot k_{i}\over
s_{i(i-1)}}-{\epsilon_{i}\cdot J_{i+1}\cdot k_{i}\over s_{i(i+1)}}\,.~{}~{}$
(67)

We will find that the consistent sub-leading and sub-sub-leading soft factors
in literatures Cachazo:2014fwa ; Schwab:2014xua ; Afkhami-Jeddi:2014fia ;
Zlotnikov:2014sva  should be defined for pure Einstein gravity. On the other
hand, the transmutation operators make sense for extended gravity that
Einstein gravity couples to $2$-form and dilaton field, whose amplitudes
manifest the double copy structure. This gap complicates the discussion. Thus,
it is worth to explain the double copy structure and its implications in more
detail. We do this in subsection 4.1 by employing the CHY formula. Then, in
subsequent subsections, we rederive leading, sub-leading and sub-sub-leading
soft factors for GR amplitudes, and prove the nonexistence of higher order
soft factor which satisfies the expectation (64).

###  4.1 CHY formula and double copy structure

The well known Cachazo-He-Yuan (CHY) formula manifests the double copy
structure of GR amplitudes, which will play the important role in this
section. In CHY formula, the integrands for GR, YM, and BAS theories are
Cachazo:2013hca ; Cachazo:2013iea

$\displaystyle{\cal I}_{\rm GR}$ $\displaystyle=$ $\displaystyle{\rm
Pf}^{\prime}\Psi(\epsilon_{p},k_{p})\,{\rm
Pf}^{\prime}\widetilde{\Psi}(\widetilde{\epsilon}_{p},k_{p})\,,$
$\displaystyle{\cal I}_{\rm YM}$ $\displaystyle=$ $\displaystyle{\rm
Pf}^{\prime}\Psi(\epsilon_{p},k_{p})\,{\rm
PT}(\vec{\boldsymbol{\sigma}}_{n})\,,$ $\displaystyle{\cal I}_{\rm BAS}$
$\displaystyle=$ $\displaystyle{\rm PT}(\vec{\boldsymbol{\sigma}}_{n})\,{\rm
PT}(\vec{\boldsymbol{\sigma}}^{\prime}_{n})\,.~{}~{}$ (68)

In the above, ${\rm Pf}^{\prime}\Psi(\epsilon_{p},k_{p})$ encodes the reduced
Pffafian of the matrix $\Psi(\epsilon_{p},k_{p})$ which depends on external
polarizations $\epsilon_{p}$ and momenta $k_{p}$, while ${\rm
PT}(\vec{\boldsymbol{\sigma}}_{n})$ denotes the Parke-Taylor factor with the
ordering $\vec{\boldsymbol{\sigma}}_{n}$ which is independent of any external
kinematic variable. The polarization tensor carried by a graviton is
decomposed as
$\varepsilon_{p}^{\mu\nu}=\epsilon_{p}^{\mu}\widetilde{\epsilon}_{p}^{\nu}$.
For the extended gravity, $\epsilon_{p}$ and $\widetilde{\epsilon}_{p}$ are
independent of each other. For Einstein gravity, $\epsilon_{p}$ and
$\widetilde{\epsilon}_{p}$ are equivalent. The tree amplitudes for above three
theories can be obtained by doing the contour integration for integrands given
in (68), with the poles determined by so called scattering equations.

As demonstrated in Zhou:2018wvn ; Bollmann:2018edb , the transmutation
operator ${\cal T}[\vec{\boldsymbol{\sigma}}_{n}]$ transmutes ${\rm
Pf}^{\prime}\Psi(\epsilon_{p},k_{p})$ to ${\rm
PT}(\vec{\boldsymbol{\sigma}}_{n})$, and analogously $\widetilde{\cal
T}[\vec{\boldsymbol{\sigma}}^{\prime}_{n}]$ transmutes ${\rm
Pf}^{\prime}\widetilde{\Psi}(\widetilde{\epsilon}_{p},k_{p})$ to ${\rm
PT}(\vec{\boldsymbol{\sigma}}^{\prime}_{n})$. In other words, they connect CHY
integrands of GR, YM and BAS theories together.

Each ${\rm Pf}^{\prime}\Psi(\epsilon_{p},k_{p})$ (or ${\rm
Pf}^{\prime}\widetilde{\Psi}(\widetilde{\epsilon}_{p},k_{p})$) can be expanded
to Parke-Taylor factors as

$\displaystyle{\rm
Pf}^{\prime}\Psi(\epsilon_{p},k_{p})=\sum_{\vec{\boldsymbol{\sigma}}_{n}}\,C(\epsilon_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})\,{\rm
PT}(\vec{\boldsymbol{\sigma}}_{n})\,,$ (69)

where the coefficients $C(\epsilon_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})$
are polynomials of Lorentz invariants arise from external polarizations and
momenta. Consequently, the GR and YM amplitudes can be expanded to BAS
amplitudes, namely,

$\displaystyle{\cal A}_{\rm GR}(\boldsymbol{h}_{n})$ $\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{\sigma}}_{n}}\,\sum_{\vec{\boldsymbol{\sigma}}^{\prime}_{n}}\,C(\epsilon_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})\,{\cal
A}_{\rm
BAS}(\vec{\boldsymbol{\sigma}}_{n}|\vec{\boldsymbol{\sigma}}^{\prime}_{n})\,\widetilde{C}(\widetilde{\epsilon}_{p},k_{p},\vec{\boldsymbol{\sigma}}^{\prime}_{n})\,,$
$\displaystyle{\cal A}_{\rm YM}(\vec{\boldsymbol{\sigma}}_{n})$
$\displaystyle=$
$\displaystyle\sum_{\vec{\boldsymbol{\sigma}}^{\prime}_{n}}\,C(\epsilon_{p},k_{p},\vec{\boldsymbol{\sigma}}^{\prime}_{n})\,{\cal
A}_{\rm
BAS}(\vec{\boldsymbol{\sigma}}_{n}|\vec{\boldsymbol{\sigma}}^{\prime}_{n})\,.~{}~{}$
(70)

This structure indicates that the transmutation operator ${\cal
T}[1,\cdots,n]$ turns $C(\epsilon_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})$
with $\vec{\boldsymbol{\sigma}}_{n}=\\{1,\cdots,n\\}$ to $1$, and annihilates
all other $C(\epsilon_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})$. The analogous
statement holds for $\widetilde{\cal T}[1,\cdots,n]$ and
$\widetilde{C}(\widetilde{\epsilon}_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})$.

The CHY integrands in (68) and the expansions in (70) indicate that the
Lorentz invariants in the form $\epsilon_{p}\cdot\widetilde{\epsilon}_{k}$
never occur in amplitudes of extended gravity. In subsequent subsections, we
will maintain such character carefully when deriving soft operators.

Another new situation indicated by the double copy structure is as follows.
Since transmutation operators act on only one of two reduced Pfaffians in CHY
integrands (68) (or equivalently one of two coefficients in expansions (70)),
while another one also contributes to soft behaviors, when performing such
operators to solve soft factors, new undetectable terms which can not be
determined by imposing gauge invariance will occur. We will see the examples
when considering the sub-leading and sub-sub-leading soft behaviors of GR
amplitudes.

###  4.2 Leading and sub-leading orders

In this subsection, we derive the soft factors of GR amplitudes at leading and
sub-leading orders. The method in this subsection is similar as that in
section 3.2. We choose the combinatorial operator $\widetilde{\cal
T}_{0}[1,\cdots,n]$ as

$\displaystyle\widetilde{\cal T}_{0}[1,\cdots,n]$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-1}}-\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i+1}}\Big{)}\,\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-1}}\,\Big{(}\prod_{a=2}^{i-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{1}\cdot\widetilde{\epsilon}_{n}}$
(71) $\displaystyle=$ $\displaystyle\widetilde{\cal
I}_{(i-1)i(i+1)}\,\widetilde{\cal T}[1,\cdots,i-1,i+1,\cdots,n]\,,~{}~{}$

which is paralleled to the definition of ${\cal T}_{0}[1,\cdots,n]$ in (21),
with each $\epsilon_{p}$ replaced by $\widetilde{\epsilon}_{p}$. The operator
$\widetilde{\cal T}_{0}[1,\cdots,n]$ connects soft behaviors of GR and YM
amplitudes as

$\displaystyle{\cal A}^{(a)_{i}}_{\rm YM}(1,\cdots,n)=\widetilde{\cal
T}_{0}[1,\cdots,n]\,{\cal A}^{(a)_{i}}_{\rm
GR}({\boldsymbol{h}}_{n})\,,~{}~{}$ (72)

allows us to solve $S^{(0)_{i}}_{h}$ and a part of $S^{(1)_{i}}_{h}$ from
$S^{(0)_{i}}_{g}$ and $S^{(1)_{i}}_{g}$, respectively.

At the leading order $a=0$, we have

$\displaystyle{\cal A}^{(0)_{i}}_{\rm YM}(1,\cdots,n)$ $\displaystyle=$
$\displaystyle\widetilde{\cal T}_{0}[1,\cdots,n]\,{\cal A}^{(0)_{i}}_{\rm
GR}({\boldsymbol{h}}_{n})$ (73) $\displaystyle=$ $\displaystyle\widetilde{\cal
I}_{(i-1)i(i+1)}\,S^{(0)_{i}}_{h}\,\widetilde{\cal
T}[1,\cdots,i-1,i+1,\cdots,n]\,{\cal A}_{\rm GR}({\boldsymbol{h}}_{n}\setminus
i)$ $\displaystyle=$ $\displaystyle\widetilde{\cal
I}_{(i-1)i(i+1)}\,S^{(0)_{i}}_{h}\,{\cal A}_{\rm
YM}(1,\cdots,i-1,i+1,\cdots,n)\,,$

where the second equality uses the leading order factorization (63), as well
as the commutativity (65). Substituting the leading soft factor of YM
amplitude in (66), we get the equations

$\displaystyle\Big{(}\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-1}}-\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i+1}}\Big{)}\,S^{(0)_{i}}_{h}={\epsilon_{i}\cdot k_{i-1}\over
s_{i(i-1)}}-{\epsilon_{i}\cdot k_{i+1}\over s_{i(i+1)}}\,,~{}~{}$ (74)

hold for any $i\in\\{2,\cdots,n-1\\}$. The solution to the above equations
(74) is

$\displaystyle S^{(0)_{i}}_{h}=\sum_{j\neq i}\,{(\epsilon_{i}\cdot
k_{j})\,(\widetilde{\epsilon}_{i}\cdot k_{j})\over s_{ij}}=\sum_{j\neq
i}\,{k_{j}\cdot\varepsilon_{i}\cdot k_{j}\over s_{ij}}\,.~{}~{}$ (75)

coincides with the leading soft factor given in Cachazo:2014fwa ;
Schwab:2014xua ; Afkhami-Jeddi:2014fia . The gauge invariance of the above
soft factor $S^{(0)_{i}}_{h}$ is ensured by momentum conservation and on-shell
condition. For instance, replacing $\epsilon_{i}$ by $k_{i}$ yields

$\displaystyle\sum_{j\neq i}\,{\widetilde{\epsilon}_{i}\cdot k_{j}\over
2}=-{\widetilde{\epsilon}_{i}\cdot k_{i}\over 2}=0\,.$ (76)

The gauge invariance for polarization $\widetilde{\epsilon}_{i}$ is analogous.

The paralleled procedure leads to equations at the sub-leading order

$\displaystyle\Big{(}\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-1}}-\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i+1}}\Big{)}\,S^{(1)_{i}}_{h}\,{\cal A}_{\rm
YM}(1,\cdots,i-1,i+1,\cdots,n)$ (77) $\displaystyle=$
$\displaystyle\Big{(}{\epsilon_{i}\cdot J_{i-1}\cdot k_{i}\over
s_{i(i-1)}}-{\epsilon_{i}\cdot J_{i+1}\cdot k_{i}\over
s_{i(i+1)}}\Big{)}\,{\cal A}_{\rm YM}(1,\cdots,i-1,i+1,\cdots,n)\,,~{}~{}$

where the soft factor in (67) is used. The solution to above equations (77) is
found to be

$\displaystyle S_{1}$ $\displaystyle=$ $\displaystyle\sum_{j\neq
i}\,{(\epsilon_{i}\cdot{\cal J}_{j}\cdot
k_{i})\,(\widetilde{\epsilon}_{i}\cdot k_{j})\over s_{ij}}\,,~{}~{}$ (78)

where ${\cal J}_{j}$ are operators

$\displaystyle{\cal J}_{j}^{\mu\nu}\,k_{j}^{\rho}=k_{j}^{\mu}\,{\partial
k_{j}^{\rho}\over\partial k_{j,\nu}}-k_{j}^{\nu}\,{\partial
k_{j}^{\rho}\over\partial k_{j,\mu}}\,,~{}~{}~{}~{}{\cal
J}_{j}^{\mu\nu}\,\epsilon_{j}^{\rho}=\big{(}\eta^{\nu\rho}\,\delta^{\mu}_{\sigma}-\eta^{\mu\rho}\,\delta^{\nu}_{\sigma}\big{)}\,\epsilon^{\sigma}_{j}\,,~{}~{}$
(79)

which do not act on Lorentz invariants contributed by
$\widetilde{C}(\widetilde{\epsilon}_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})$
in (70). When transmuted to YM amplitudes with only one reduced Pfaffian in
the corresponding CHY integrand, these operators with the effects in (79)
restore the standard angular momentum operators. For the extended gravity,
they can not be interpreted as angular momentum operators, since they do not
act on all orbital and spin parts of an external graviton. The reason for
introducing the above ${\cal J}_{j}$ is to maintain the correct sub-leading
soft factor for YM amplitudes and the double copy structure in (68) and (70)
simultaneously. Suppose we allow the operator ${\cal J}_{j}$ to act on
$\widetilde{C}(\widetilde{\epsilon}_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})$,
then $\epsilon_{\ell}\cdot\widetilde{\epsilon}_{k}$ will occur.

Based on symmetry, it is natural to expect another block

$\displaystyle S_{2}$ $\displaystyle=$ $\displaystyle\sum_{j\neq
i}\,{(\epsilon_{i}\cdot k_{j})\,(\widetilde{\epsilon}_{i}\cdot\widetilde{\cal
J}_{j}\cdot k_{i})\over s_{ij}}\,,~{}~{}$ (80)

which do not act on Lorentz invariants from
$C(\epsilon_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})$, and is connected to
$S_{1}$ by exchanging $\epsilon$ and $\widetilde{\epsilon}$. This part can not
be detected by the operator $\widetilde{\cal T}_{0}[1,\cdots,n]$, and will be
found in next subsection 4.3 by using the operator $\widetilde{\cal
T}_{1}[1,\cdots,n]$. The full soft factor at the sub-leading order is the
summation of two parts $S_{1}$ and $S_{2}$, namely,

$\displaystyle S^{(1)_{i}}_{h}=S_{1}+S_{2}=\sum_{j\neq i}\,{(\epsilon_{i}\cdot
k_{j})\,(\widetilde{\epsilon}_{i}\cdot\widetilde{\cal J}_{j}\cdot
k_{i})+(\widetilde{\epsilon}_{i}\cdot k_{j})\,(\epsilon_{i}\cdot{\cal
J}_{j}\cdot k_{i})\over s_{ij}}\,.~{}~{}$ (81)

In practice, the formula (81) does not make sense, due to the following
reason. By definition, the operators ${\cal J}_{j}$ do not act on
$k_{\ell}\cdot k_{k}$ from
$\widetilde{C}(\widetilde{\epsilon}_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})$,
while operators $\widetilde{\cal J}_{j}$ do not act on $k_{\ell}\cdot k_{k}$
from $C(\epsilon_{p},k_{p},\vec{\boldsymbol{\sigma}}_{n})$. However, it is
impossible to distinguish the origins of these $k_{\ell}\cdot k_{k}$ in each
amplitude. The situation has changed dramatically if we restrict ourselves to
standard Einstein gravity. For Einstein gravity, in which
$\epsilon^{\mu}_{p}=\widetilde{\epsilon}^{\mu}_{p}$, the solution (81) is
reduced to

$\displaystyle S^{(1)_{i}}_{h}=\sum_{j\neq i}\,{(\epsilon_{i}\cdot
k_{j})\,(\epsilon_{i}\cdot J_{j}\cdot k_{i})\over s_{ij}}=\sum_{j\neq
i}\,{k_{j}\cdot\varepsilon_{i}\cdot J_{j}\cdot k_{i}\over s_{ij}}\,,~{}~{}$
(82)

since in this case ${\rm Pf}^{\prime}\Psi(\epsilon_{p},k_{p})$ and ${\rm
Pf}^{\prime}\widetilde{\Psi}(\widetilde{\epsilon}_{p},k_{p})$ are equivalent
to each other. In (82), the operators $J_{j}$ are angular momentum operators
which act on all $k_{j}$, $\epsilon_{j}$ and $\widetilde{\epsilon}_{j}$,
therefore affect orbital and spin parts of the $j^{\rm th}$ graviton in the
correct manner. The equivalence between ${\rm
Pf}^{\prime}\Psi(\epsilon_{p},k_{p})$ and ${\rm
Pf}^{\prime}\widetilde{\Psi}(\widetilde{\epsilon}_{p},k_{p})$ implies that it
is not necessary to distinguish the origins of $k_{\ell}\cdot k_{k}$ for the
current case. The sub-leading soft factor in (82) is also coincide with the
result in Cachazo:2014fwa ; Schwab:2014xua ; Afkhami-Jeddi:2014fia .

###  4.3 Sub-leading and sub-sub-leading orders

In this subsection, we derive the $S_{2}$ part of the soft factor
$S^{(1)_{i}}_{h}$, as well as a part of $S^{(2)_{i}}_{h}$, by using the
operator

$\displaystyle\widetilde{\cal
T}_{1}[1,\cdots,n]=\Big{(}\prod_{a=2}^{n-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{1}\cdot\widetilde{\epsilon}_{n}}\,,~{}~{}$
(83)

obtained by replacing $\epsilon_{p}$ with $\widetilde{\epsilon}_{p}$ in (39).
This operator annihilates the leading GR term ${\cal A}^{(0)_{i}}_{\rm
GR}(\boldsymbol{h}_{n})$, and links GR terms at $(a+1)^{\rm th}$ order to YM
terms at $a^{\rm th}$ order,

$\displaystyle{\cal A}^{(a)_{i}}_{\rm YM}(1,\cdots,n)=\widetilde{\cal
T}_{1}[1,\cdots,n]\,{\cal A}^{(a+1)_{i}}_{\rm GR}(\boldsymbol{h}_{n})\,.$ (84)

Such connection allows us to detect $S^{(a+1)_{i}}_{h}$ by substituting known
$S^{(a)_{i}}_{g}$.

At the sub-leading order, we have

$\displaystyle{\cal A}^{(0)_{i}}_{\rm YM}(1,\cdots,n)$ $\displaystyle=$
$\displaystyle\widetilde{\cal T}_{1}[1,\cdots,n]\,{\cal A}^{(1)_{i}}_{\rm
GR}(\boldsymbol{h}_{n})$ (85) $\displaystyle=$
$\displaystyle\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-1}}\,S^{(1)_{i}}_{h}\,\Big{[}\Big{(}\prod_{a=2}^{i-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{1}\cdot\widetilde{\epsilon}_{n}}\Big{]}\,{\cal
A}_{\rm GR}(\boldsymbol{h}_{n}\setminus i)\,,~{}~{}$

where the commutativity in (65) is used again. Substituting the soft factor in
(66), as well as the transmutation relation

$\displaystyle{\cal A}_{\rm YM}(1,\cdots,i-1,i+1,\cdots,n)=\widetilde{\cal
T}[1,\cdots,i-1,i+1,\cdots,n]\,{\cal A}_{\rm GR}(\boldsymbol{h}_{n}\setminus
i)\,,~{}~{}$ (86)

into the relation (85), we find the equations

$\displaystyle\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-1}}\,S^{(1)_{i}}_{h}\,\widetilde{\cal P}_{1}\,{\cal A}_{\rm
GR}({\boldsymbol{h}}_{n}\setminus i)=\Big{(}{\epsilon_{i}\cdot k_{i-1}\over
s_{(i-1)i}}-{\epsilon_{i}\cdot k_{i+1}\over
s_{i(i+1)}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-1}}\,\widetilde{\cal P}_{1}\,{\cal A}_{\rm
GR}({\boldsymbol{h}}_{n}\setminus i)\,,~{}~{}$ (87)

with

$\displaystyle\widetilde{\cal
P}_{1}=\Big{(}\prod_{a=2}^{i-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{1}\cdot\widetilde{\epsilon}_{n}}\,.$
(88)

Through the technic extremely similar to that for solving equation (45), we
find the solution to equation (87) is $S_{2}$ in (80). As discussed in the
previous subsection 4.2, the combination of two parts $S_{1}$ and $S_{2}$
leads to the full sub-leading soft factor, which can be reduced to the
standard formula in Einstein gravity.

At the sub-sub-leading order, the same manipulation gives the similar
equations

$\displaystyle\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-1}}\,S^{(2)_{i}}_{h}\,\widetilde{\cal P}_{1}\,{\cal A}_{\rm
GR}({\boldsymbol{h}}_{n}\setminus i)=\Big{(}{\epsilon_{i}\cdot{\cal
J}_{i-1}\cdot k_{i}\over s_{(i-1)i}}-{\epsilon_{i}\cdot{\cal J}_{i+1}\cdot
k_{i}\over s_{i(i+1)}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-1}}\,\widetilde{\cal P}_{1}\,{\cal A}_{\rm
GR}({\boldsymbol{h}}_{n}\setminus i)\,,~{}~{}$ (89)

and the solution is found to be

$\displaystyle S=\sum_{j\neq i}\,{(\epsilon_{i}\cdot{\cal J}_{j}\cdot
k_{i})\,(\widetilde{\epsilon}_{i}\cdot\widetilde{\cal J}_{j}\cdot k_{i})\over
s_{ij}}\,,~{}~{}$ (90)

by using the analogous technic. For Einstein gravity with ${\rm
Pf}^{\prime}\Psi(\epsilon_{p},k_{p})={\rm
Pf}^{\prime}\widetilde{\Psi}(\widetilde{\epsilon}_{p},k_{p})$, the above
formula is reduced to

$\displaystyle S^{(2)_{i}}_{h}={1\over 2}\,\sum_{j\neq i}\,{(\epsilon_{i}\cdot
J_{j}\cdot k_{i})^{2}\over s_{ij}}=-{1\over 2}\,\sum_{j\neq i}\,{k_{i}\cdot
J_{j}\cdot\varepsilon_{i}\cdot J_{j}\cdot k_{i}\over s_{ij}}\,,~{}~{}$ (91)

coincide with the result in Cachazo:2014fwa ; Zlotnikov:2014sva , where the
factor $1/2$ is introduced to cancel the over-counting. In the next
subsection, we will explain that the solution in (90) only corresponds to a
part of the sub-sub-leading soft behavior of extended gravity which can be
detected by the operator ${\cal T}[1,\cdots,n]$, rather than the full one. On
the other hand, the formula in (91) serves as the complete sub-sub-leading
soft factor for Einstein gravity.

###  4.4 Higher order

This subsection aims to argue that the factorized formula (63) of GR
amplitudes, with the expected soft factor in (64), can not be found at the
$3^{\rm th}$ order. The argument is similar as that for excluding the expected
YM soft factor $S^{(2)_{i}}_{g}$, in section 3.4. Paralleled to ${\cal
T}_{2}[1,\cdots,n]$ in (50), we now choose the operator

$\displaystyle\widetilde{\cal T}_{2}[1,\cdots,n]=\widetilde{\cal
T}_{21}[1,\cdots,n]+\widetilde{\cal T}_{22}[1,\cdots,n]\,,$ (92)

where

$\displaystyle\widetilde{\cal
T}_{21}[1,\cdots,n]=\Big{(}\prod_{a=i+1}^{n-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-2}}\,\Big{(}\prod_{a=2}^{i-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{1}\cdot\widetilde{\epsilon}_{n}}\,,~{}~{}$
(93)

and

$\displaystyle\widetilde{\cal
T}_{22}[1,\cdots,n]=-\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i}}\,\Big{(}\prod_{a=i+1}^{n-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-2}}\,\Big{(}\prod_{a=2}^{i-2}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{1}\cdot\widetilde{\epsilon}_{n}}\,.~{}~{}$
(94)

It is straightforward to verify that the operator $\widetilde{\cal
T}_{21}[1,\cdots,n]$ annihilates ${\cal A}^{(0)_{i}}_{\rm
GR}(\boldsymbol{h}_{n})$, while $\widetilde{\cal T}_{22}[1,\cdots,n]$
annihilates both ${\cal A}^{(0)_{i}}_{\rm GR}(\boldsymbol{h}_{n})$ and ${\cal
A}^{(1)_{i}}_{\rm GR}(\boldsymbol{h}_{n})$. The operator $\widetilde{\cal
T}_{2}[1,\cdots,n]$ links the soft behaviors of GR and YM amplitudes as
follows

$\displaystyle{\cal A}^{(a)_{i}}_{\rm YM}(1,\cdots,n)=\widetilde{\cal
T}_{21}[1,\cdots,n]\,{\cal A}^{(a+1)_{i}}_{\rm
GR}(\boldsymbol{h}_{n})+\widetilde{\cal T}_{22}[1,\cdots,n]\,{\cal
A}^{(a+2)_{i}}_{\rm GR}(\boldsymbol{h}_{n})\,.~{}~{}$ (95)

Before studying the soft behavior of GR amplitudes at the $3^{\rm th}$ order,
let us verify that the transmutation relation (95) holds for $a=0$, namely,

$\displaystyle{\cal A}^{(0)_{i}}_{\rm YM}(1,\cdots,n)=\widetilde{\cal
T}_{21}[1,\cdots,n]\,{\cal A}^{(1)_{i}}_{\rm
GR}(\boldsymbol{h}_{n})+\widetilde{\cal T}_{22}[1,\cdots,n]\,{\cal
A}^{(2)_{i}}_{\rm GR}(\boldsymbol{h}_{n})\,.~{}~{}$ (96)

The above statement only holds for Einstein gravity, with $S^{(1)_{i}}_{h}$
and $S^{(2)_{i}}_{h}$ given in (82) and (91). For the general extended
gravity, the transmutation in (96) does not hold if we naively regard the
solution in (90) as $S^{(2)_{i}}_{h}$. This observation means the sub-sub-
leading soft behavior with soft factor in (90) is not the complete one.
Therefore, let us restrict our selves to the Einstein gravity.

Since the transmutation operator makes sense for amplitudes of extended
gravity, we need to keep notations $\epsilon_{p}$ and
$\widetilde{\epsilon}_{p}$ to manifest the double copy structure, and assume
that each differential in the combinatorial operator $\widetilde{\cal
T}_{2}[1,\cdots,n]$ only acts on Lorentz invariants which carry
$\widetilde{\epsilon}_{p}$. For the $\widetilde{\cal T}_{21}[1,\cdots,n]$
part, we realize this goal by going back to the formula of sub-leading soft
factor in (81), which is equivalent to (82) when setting
$\epsilon_{p}^{\mu}=\widetilde{\epsilon}_{p}^{\mu}$. Using the sub-leading
soft factor in (81), we get

$\displaystyle\widetilde{\cal T}_{21}[1,\cdots,n]\,{\cal A}^{(1)_{i}}_{\rm
GR}(\boldsymbol{h}_{n})$ $\displaystyle=$
$\displaystyle\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-2}}\,S^{(1)_{i}}_{h}\,\widetilde{\cal P}_{21}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)$ (97) $\displaystyle=$
$\displaystyle\Big{(}{2\,\epsilon_{i}\cdot k_{i-2}\over
s_{i(i-2)}}-{\epsilon_{i}\cdot k_{i+1}\over
s_{i(i+1)}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-2}}\,\widetilde{\cal P}_{21}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)\,,~{}~{}$

where the commutativity in (65) is used again. In the above, the operator
$\widetilde{\cal P}_{21}$ is given as

$\displaystyle\widetilde{\cal
P}_{21}=\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\Big{(}\prod_{a=2}^{i-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{1}\cdot\widetilde{\epsilon}_{n}}\,.$
(98)

The second equality of (97) is obtained as follows. The operator $S_{2}$ in
(81) acts on the Lorentz invariant $\widetilde{\epsilon}_{i+1}\cdot k_{i-2}$
as

$\displaystyle S_{2}\,(\widetilde{\epsilon}_{i+1}\cdot k_{i-2})$
$\displaystyle=$ $\displaystyle\Big{(}{\epsilon_{i}\cdot k_{i-2}\over
s_{i(i-2)}}-{\epsilon_{i}\cdot k_{i+1}\over
s_{i(i+1)}}\Big{)}\,\widetilde{\epsilon}_{i+1}\cdot\widetilde{f}_{i}\cdot
k_{i-2}\,,$ (99)

which includes $(\widetilde{\epsilon}_{i+1}\cdot
k_{i})(\widetilde{\epsilon}_{i}\cdot k_{i-2})$ in the factor
$\widetilde{\epsilon}_{i+1}\cdot\widetilde{f}_{i}\cdot k_{i-2}$, while the
operator $S_{1}$ in (81) acts on the Lorentz invariant
$\widetilde{\epsilon}_{i+1}\cdot k_{i-2}$ as

$\displaystyle S_{1}\,(\widetilde{\epsilon}_{i+1}\cdot k_{i-2})$
$\displaystyle=$ $\displaystyle\Big{(}{\widetilde{\epsilon}_{i}\cdot
k_{i-2}\over s_{i(i-2)}}-{\widetilde{\epsilon}_{i}\cdot k_{i+1}\over
s_{i(i+1)}}\Big{)}\,\widetilde{\epsilon}_{i+1}\cdot f_{i}\cdot k_{i-2}\,,$
(100)

Then the second equality is indicated by the observation that the effect of
applying $\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\partial_{\widetilde{\epsilon}_{i}\cdot k_{i-2}}$ is turning
$(\widetilde{\epsilon}_{i+1}\cdot k_{i})(\widetilde{\epsilon}_{i}\cdot
k_{i-2})$ to $1$ while eliminating all terms without
$(\widetilde{\epsilon}_{i+1}\cdot k_{i})(\widetilde{\epsilon}_{i}\cdot
k_{i-2})$, and the effect of performing
$\partial_{\widetilde{\epsilon}_{i+1}\cdot k_{i-2}}$ is turning
$\widetilde{\epsilon}_{i+1}\cdot k_{i-2}$ to $1$ while annihilating all terms
without $\widetilde{\epsilon}_{i+1}\cdot k_{i-2}$.

Meanwhile, using the sub-sub-leading soft factor in (91), we find

$\displaystyle\widetilde{\cal T}_{22}[1,\cdots,n]\,{\cal A}^{(2)_{i}}_{\rm
GR}(\boldsymbol{h}_{n})$ (101) $\displaystyle=$
$\displaystyle-\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i}}\,S^{(2)_{i}}_{h}\,\widetilde{\cal P}_{22}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)$ $\displaystyle=$
$\displaystyle\Big{(}{\epsilon_{i}\cdot k_{i+1}\over
s_{i(i+1)}}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i+1}}+{\epsilon_{i}\cdot k_{i-1}\over
s_{i(i-1)}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-1}}-{2\,\epsilon_{i}\cdot k_{i-2}\over
s_{i(i-2)}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot k_{i-2}}\Big{)}$
$\displaystyle\widetilde{\cal P}_{22}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)\,,~{}~{}$

where

$\displaystyle\widetilde{\cal
P}_{22}=\Big{(}\prod_{a=i+2}^{n-1}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\Big{(}\prod_{a=2}^{i-2}\,\partial_{\widetilde{\epsilon}_{a}\cdot
k_{a-1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{1}\cdot\widetilde{\epsilon}_{n}}\,.$
(102)

The second equality in (101) is based on the observations

$\displaystyle S^{(2)_{i}}_{h}\,(\epsilon_{i+1}\cdot
k_{i-2})\,(\epsilon_{i-1}\cdot k_{i+1})$ $\displaystyle=$
$\displaystyle-{(\epsilon_{i+1}\cdot f_{i}\cdot k_{i-2})\,(\epsilon_{i-1}\cdot
f_{i}\cdot k_{i+1})\over s_{i(i+1)}}$ $\displaystyle\sim$
$\displaystyle-{(\widetilde{\epsilon}_{i+1}\cdot
k_{i})\,(\widetilde{\epsilon}_{i-1}\cdot\widetilde{k}_{i})\,(\widetilde{\epsilon}_{i}\cdot
k_{i-2})\,(\epsilon_{i}\cdot k_{i+1})\over s_{i(i+1)}}\,,$ $\displaystyle
S^{(2)_{i}}_{h}\,(\epsilon_{i-1}\cdot k_{i-2})\,(\epsilon_{i+1}\cdot k_{i-1})$
$\displaystyle=$ $\displaystyle-{(\epsilon_{i-1}\cdot f_{i}\cdot
k_{i-2})\,(\epsilon_{i+1}\cdot f_{i}\cdot k_{i-1})\over s_{i(i-1)}}$
$\displaystyle\sim$ $\displaystyle-{(\widetilde{\epsilon}_{i+1}\cdot
k_{i})\,(\widetilde{\epsilon}_{i-1}\cdot\widetilde{k}_{i})\,(\widetilde{\epsilon}_{i}\cdot
k_{i-2})\,(\epsilon_{i}\cdot k_{i-1})\over s_{i(i-1)}}\,,$ $\displaystyle
S^{(2)_{i}}_{h}\,(\epsilon_{i-1}\cdot k_{i-2})\,(\epsilon_{i+1}\cdot k_{i-2})$
$\displaystyle=$ $\displaystyle{(\epsilon_{i-1}\cdot f_{i}\cdot
k_{i-2})\,(\epsilon_{i+1}\cdot f_{i}\cdot k_{i-2})\over s_{i(i-2)}}$ (103)
$\displaystyle\sim$ $\displaystyle{2\,(\widetilde{\epsilon}_{i+1}\cdot
k_{i})\,(\widetilde{\epsilon}_{i-1}\cdot\widetilde{k}_{i})\,(\widetilde{\epsilon}_{i}\cdot
k_{i-2})\,(\epsilon_{i}\cdot k_{i-2})\over s_{i(i-2)}}\,.$

Here $\sim$ means collecting effective terms survive under the action of
$\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot k_{i}}$. We turned
$\epsilon_{i+1}$ and $\epsilon_{i-1}$ to $\widetilde{\epsilon}_{i+1}$ and
$\widetilde{\epsilon}_{i-1}$, based on the fact that $\widetilde{\cal
T}_{22}[1,\cdots,n]$ only acts on ${\rm
Pf}^{\prime}\widetilde{\Psi}(\widetilde{\epsilon}_{p},k_{p})$, and kept the
structure
$\varepsilon^{\mu\nu}_{i}=\epsilon_{i}^{\mu}\widetilde{\epsilon}_{i}^{\nu}$.
The factor $2$ in the last line comes from two alternative choices of turning
one of two $\epsilon_{i}\cdot k_{i-2}$ to $\widetilde{\epsilon}_{i}\cdot
k_{i-2}$.

Combining (97) and (101) together, we arrive at

$\displaystyle\widetilde{\cal T}_{21}[1,\cdots,n]\,{\cal A}^{(1)_{i}}_{\rm
GR}(\boldsymbol{h}_{n})+\widetilde{\cal T}_{22}[1,\cdots,n]\,{\cal
A}^{(2)_{i}}_{\rm GR}(\boldsymbol{h}_{n})$ (104) $\displaystyle=$
$\displaystyle\Big{(}{\epsilon_{i}\cdot k_{i+1}\over
s_{i(i+1)}}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i+1}}+{\epsilon_{i}\cdot k_{i-1}\over
s_{i(i-1)}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-1}}-{\epsilon_{i}\cdot k_{i+1}\over
s_{i(i+1)}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot k_{i-2}}\Big{)}$
$\displaystyle\widetilde{\cal P}_{22}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)$ $\displaystyle=$
$\displaystyle\Big{(}{\epsilon_{i}\cdot k_{i-1}\over
s_{i(i-1)}}-{\epsilon_{i}\cdot k_{i+1}\over s_{i(i+1)}}\Big{)}\,{\cal A}_{\rm
YM}(1,\cdots,i-1,i+1,\cdots,n)\,.$

In the last step, we have used the observation that both
$\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i-2}}\partial_{\widetilde{\epsilon}_{i+1}\cdot k_{i-1}}$ and
$(\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i+1}}-\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i-2}})\partial_{\widetilde{\epsilon}_{i+1}\cdot k_{i-2}}$ transmutes the
object $\widetilde{\cal P}_{22}{\cal A}_{\rm GR}(\boldsymbol{h}_{n}\setminus
i)$ to the YM amplitude ${\cal A}_{\rm YM}(1,\cdots,i-1,i+1,\cdots,n)$, since
the latter one can be interpreted as $\widetilde{\cal
I}_{(i-2)(i-1)(i+1)}\widetilde{\cal I}_{(i-2)(i+1)n}$, which inserts the leg
$(i+1)$ between $(i-2)$ and $n$, and subsequently insert $(i-1)$ between
$(i-2)$ and $(i+1)$. Consequently, the expected transmutation relation (96) is
valid.

Then we turn to study the soft factor $S^{(3)_{i}}_{h}$ at the $3^{\rm th}$
order. We use the transmutation relation (95) with $a=1$, i.e.,

$\displaystyle{\cal A}^{(1)_{i}}_{\rm YM}(1,\cdots,n)=\widetilde{\cal
T}_{21}[1,\cdots,n]\,{\cal A}^{(2)_{i}}_{\rm
GR}(\boldsymbol{h}_{n})+\widetilde{\cal T}_{22}[1,\cdots,n]\,{\cal
A}^{(3)_{i}}_{\rm GR}(\boldsymbol{h}_{n})\,,~{}~{}$ (105)

to solve $S^{(3)_{i}}_{h}$ by substituting already known $S^{(1)_{i}}_{g}$ and
$S^{(2)_{i}}_{h}$. Since the complete sub-sub-leading soft factor
$S^{(2)_{i}}_{h}$ in (91) only makes sense for Einstein gravity, we restrict
ourselves to Einstein gravity when considering $S^{(3)_{i}}_{h}$. Meanwhile,
we still keep the notation $\epsilon_{p}$ and $\widetilde{\epsilon}_{p}$, to
manifest the effect of applying operators $\widetilde{\cal
T}_{21}[1,\cdots,n]$ and $\widetilde{\cal T}_{22}[1,\cdots,n]$.

Now let us figure out the expression for each block in (105). We can represent
${\cal A}^{(1)_{i}}_{\rm YM}(1,\cdots,n)$ as

$\displaystyle{\cal A}^{(1)_{i}}_{\rm YM}(1,\cdots,n)$ $\displaystyle=$
$\displaystyle\Big{(}{\epsilon_{i}\cdot J_{i-1}\cdot k_{i}\over
s_{i(i-1)}}-{\epsilon_{i}\cdot J_{i+1}\cdot k_{i}\over
s_{i(i+1)}}\Big{)}\,{\cal A}_{\rm YM}(1,\cdots,i-1,i+1,\cdots,n)$ (106)
$\displaystyle=$ $\displaystyle\Big{(}{\epsilon_{i}\cdot{\cal J}_{i-1}\cdot
k_{i}\over s_{i(i-1)}}-{\epsilon_{i}\cdot{\cal J}_{i+1}\cdot k_{i}\over
s_{i(i+1)}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-1}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot k_{i-2}}\,\widetilde{\cal
P}_{22}\,{\cal A}_{\rm GR}(\boldsymbol{h}_{n}\setminus i)$ $\displaystyle=$
$\displaystyle\Big{(}{\epsilon_{i}\cdot{\cal J}_{i-1}\cdot k_{i}\over
s_{i(i-1)}}-{\epsilon_{i}\cdot{\cal J}_{i+1}\cdot k_{i}\over
s_{i(i+1)}}\Big{)}\,\Big{(}\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i-2}}-\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i+1}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-2}}\,\widetilde{\cal P}_{22}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)\,,~{}~{}$

where the second and third lines choose two different schemes of insertions,
which are $\widetilde{I}_{(i-1)(i+1)n}\widetilde{I}_{(i-2)(i-1)n}$ and
$\widetilde{I}_{(i-2)(i-1)(i+1)}\widetilde{I}_{(i-2)(i+1)n}$ respectively.
Both two choices will be considered. On the other hand, by using the sub-sub-
leading soft factor in (91), we find

$\displaystyle\widetilde{\cal T}_{21}[1,\cdots,n]\,{\cal A}^{(2)_{i}}_{\rm
GR}(\boldsymbol{h}_{n})$ $\displaystyle=$
$\displaystyle\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-2}}\,S^{(2)_{i}}_{h}\,\widetilde{\cal P}_{21}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)$ (107) $\displaystyle=$
$\displaystyle\Big{(}{\epsilon_{i}\cdot{\cal J}_{i-2}\cdot k_{i}\over
s_{i(i-2)}}-{\epsilon_{i}\cdot{\cal J}_{i+1}\cdot k_{i}\over
s_{i(i+1)}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot k_{i-2}}\,\widetilde{\cal
P}_{22}\,{\cal A}_{\rm GR}(\boldsymbol{h}_{n}\setminus i)\,,~{}~{}$

since the effective terms survive under the action of
$\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\partial_{\widetilde{\epsilon}_{i}\cdot k_{i-2}}$ is
$\widetilde{\epsilon}_{i+1}\cdot\widetilde{f}_{i}\cdot k_{i+2}$, arises from
acting operators $\widetilde{\epsilon}_{i}\cdot\widetilde{\cal J}_{i-2}\cdot
k_{i}$ or $\widetilde{\epsilon}_{i}\cdot\widetilde{\cal J}_{i+1}\cdot k_{i}$
on $\widetilde{\epsilon}_{i+1}\cdot k_{i+2}$. This observation selects
$(\epsilon_{i}\cdot{\cal J}_{i-2}\cdot
k_{i})(\widetilde{\epsilon}_{i}\cdot\widetilde{\cal J}_{i-2}\cdot
k_{i})/s_{i(i-2)}$ and $(\epsilon_{i}\cdot{\cal J}_{i+1}\cdot
k_{i})(\widetilde{\epsilon}_{i}\cdot\widetilde{\cal J}_{i+1}\cdot
k_{i})/s_{i(i+1)}$ in (91), ultimately yields (107).

Substituting the second line of (106) and the relation (107) into (105) leads
to the equations

$\displaystyle\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i}}\,S^{(3)_{i}}_{h}\,\widetilde{\cal P}_{22}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)$ (108) $\displaystyle=$
$\displaystyle\Big{[}\Big{(}{\epsilon_{i}\cdot{\cal J}_{i-2}\cdot k_{i}\over
s_{i(i-2)}}-{\epsilon_{i}\cdot{\cal J}_{i+1}\cdot k_{i}\over
s_{i(i+1)}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i-2}}-\Big{(}{\epsilon_{i}\cdot{\cal J}_{i-1}\cdot k_{i}\over
s_{i(i-1)}}-{\epsilon_{i}\cdot{\cal J}_{i+1}\cdot k_{i}\over
s_{i(i+1)}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-1}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot k_{i-2}}\Big{]}$
$\displaystyle\widetilde{\cal P}_{22}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)\,.$

Then, similar to the argument at the end of section 3.4, the solution
$S^{(3)_{i}}_{h}$ which satisfies the expectation in (64) is forbidden by the
effect of turning the bilinearity on $k_{2}$ to linearity, or turning the
linearity on $k_{i-1}$ to independence. On the other hand, substituting the
third line of (106) yields

$\displaystyle\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i}}\,\partial_{\widetilde{\epsilon}_{i}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i}}\,S^{(3)_{i}}_{h}\,\widetilde{\cal P}_{22}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)$ (109) $\displaystyle=$
$\displaystyle\Big{[}\Big{(}{\epsilon_{i}\cdot{\cal J}_{i-2}\cdot k_{i}\over
s_{i(i-2)}}-{\epsilon_{i}\cdot{\cal J}_{i+1}\cdot k_{i}\over
s_{i(i+1)}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i-2}}-\Big{(}{\epsilon_{i}\cdot{\cal J}_{i-1}\cdot k_{i}\over
s_{i(i-1)}}-{\epsilon_{i}\cdot{\cal J}_{i+1}\cdot k_{i}\over
s_{i(i+1)}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i-2}}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot k_{i-2}}$
$\displaystyle+\Big{(}{\epsilon_{i}\cdot{\cal J}_{i-1}\cdot k_{i}\over
s_{i(i-1)}}-{\epsilon_{i}\cdot{\cal J}_{i+1}\cdot k_{i}\over
s_{i(i+1)}}\Big{)}\,\partial_{\widetilde{\epsilon}_{i-1}\cdot
k_{i+1}}\,\partial_{\widetilde{\epsilon}_{i+1}\cdot
k_{i-2}}\Big{]}\,\widetilde{\cal P}_{22}\,{\cal A}_{\rm
GR}(\boldsymbol{h}_{n}\setminus i)\,,$

then the existence of desired $S^{(3)_{i}}_{h}$ is forbidden by the effect of
turning the bilinearity on $k_{2}$ to linearity, or turning the linearity on
$k_{i+1}$ to independence. The above argument excludes the existence of
expected universal soft factor $S^{(3)_{i}}_{h}$.

##  5 Summary

In this note, with the help of transmutation operators, we reconstruct known
soft factors of YM and GR amplitudes, and prove the nonexistence of higher
order soft factor under the constraint of universality. We also clarify that
the consistent soft factors $S^{(1)_{i}}_{h}$ and $S^{(2)_{i}}_{h}$ of GR
amplitudes in literatures should be defined for pure Einstein gravity, rather
than for the extended one. This phenomenon is quite natural, since the
asymptotic BMS symmetry which predicts the soft behavior of GR amplitudes at
leading, and sub-leading orders only makes sense for pure Einstein gravity
Strominger:2013jfa ; He:2014laa . Our method is purely bottom-up thus can not
reveal the underlying symmetry, but also leads to the conclusion coincide with
the prediction of symmetry.

It is also interesting to relax the requirement of universality, and study
soft behavior at higher orders. For example, one can find universal soft
factor of BAS amplitudes with the number of external legs $n\geq 5$, while the
soft behavior of $4$-point amplitudes is distinct. It means one can still talk
about the universal soft behavior at a ”weaker” level. It is natural to expect
the existence of similar feature in YM and GR cases. An interesting future
direction is to figure out such ”weaker” universal soft factors, and
investigate their physical applications.

## Acknowledgments

This work is supported by NSFC under Grant No. 11805163.

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