# Tree and $1$-loop fundamental BCJ relations from soft theorems

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

###### Abstract

We provide a new derivation of the fundamental BCJ relation among double color
ordered tree amplitudes of bi-adjoint scalar theory, based on the leading soft
theorem for external scalars. Then, we generalize the fundamental BCJ relation
to $1$-loop Feynman integrands. We also use the fundamental BCJ relation to
understand the Adler’s zero for tree amplitudes of non-linear Sigma model and
Born-Infeld theories.

##  I Introduction

As well known, soft theorems describe the universal behavior of scattering
amplitudes when one or more external massless momenta are taken to near zero.
Historically, soft theorems at the leading order for tree amplitudes were
derived using Feynman rules Low ; Weinberg . In 2014, new soft theorems at
higher orders were proposed for gravity (GR) Cachazo:2014fwa  and Yang-Mills
(YM) theory Casali:2014xpa  at tree level, by applying Britto-Cachazo-Feng-
Witten (BCFW) recursion relation Britto:2004ap ; Britto:2005fq . Subsequently,
these new results were generalized to arbitrary space-time dimensions
Schwab:2014xua ; Afkhami-Jeddi:2014fia , by using Cachazo-He-Yuan (CHY)
formulas Cachazo:2013gna ; Cachazo:2013hca ; Cachazo:2013iea ; Cachazo:2014nsa
; Cachazo:2014xea . Quite interestingly, the soft theorems can be understood
as the consequence of asymptotic symmetries, and are related to memory effects
Strominger:2013jfa ; Strominger:2013lka ; He:2014laa ; Kapec:2014opa ;
Strominger:2014pwa ; Pasterski:2015tva ; Barnich1 ; Barnich2 ; Barnich3 .
Furthermore, the soft theorems and asymptotic symmetries for a wide range of
other theories including string theory, and the soft theorems at the loop
level, have been investigated in ZviScott ; HHW ; FreddyEllis ;
Bianchi:2014gla ; Campiglia:2014yka ; Campiglia:2016efb ; Elvang:2016qvq ;
Guerrieri:2017ujb ; Hamada:2017atr ; Mao:2017tey ; Li:2017fsb ;
DiVecchia:2017gfi ; Bianchi:2016viy ; Chakrabarti:2017ltl ; Sen:2017nim ;
Hamada:2018vrw .

On the other hand, the soft theorems were exploited in the construction of
tree level amplitudes, such as using another type of soft behavior called the
Adler’s zero to construct amplitudes of various effective theories, and the
inverse soft theorem program, and so on Cheung:2014dqa ; Luo:2015tat ;
Elvang:2018dco ; Cachazo:2016njl ; Rodina:2018pcb ; Boucher-Veronneau:2011rwd
; Nguyen:2009jk . Using methods in above literatures, one can bootstrap
amplitudes by also assuming the explicit form of soft factors or soft
operators. Recently, in Zhou:2022orv , it was shown that the tree amplitudes
can be fixed by imposing only the factorization property of soft theorems and
the universality of soft operators, without knowing the explicit form of such
operators. The results mentioned above indicate that the tree amplitudes are
completely determined by soft theorems. Thus, it is natural to expect that the
relations among tree amplitudes can also be determined and understood via soft
theorems. This is the main motivation for the current note.

In this note, we use the soft theorems to derive the well known fundamental
Bern-Carrasco-Johansson (BCJ) relation among color ordered amplitudes
Bern:2008qj ; Chiodaroli:2014xia ; Johansson:2015oia ; Johansson:2019dnu .
More explicitly, we use the leading order soft theorem for external scalars to
get the fundamental BCJ relation among double color ordered bi-adjoint scalar
(BAS) amplitudes at tree level. Furthermore, we generalize the fundamental BCJ
relation to the $1$-loop level, via the forward limit method. By expanding
tree amplitudes to BAS ones, we also use the fundamental BCJ relation to
understand the Adler’s zero which describe the soft behavior of non-linear
Sigma model (NLSM) and Born-Infeld (BI) amplitudes at tree level.

The remainder of this note is organized as follows. In section.II, we rapidly
introduce the tree BAS amplitudes, as well as their soft behavior at the
leading order. In section.III, we derive the fundamental BCJ relation by using
the leading soft theorem for external scalars. Then, we generalize the tree
level fundamental BCJ relation to the $1$-loop level in section.IV. In
section.V, we use the fundamental BCJ relation, as well as the expanded
formula of tree NLSM and BI amplitudes, to understand the Adler’s zero.
Finally, we end with a summery in section.VI.

##  II Tree BAS amplitudes

For readers’ convenience, in this section we give a brief review to the
necessary background. In subsection.II.1, we introduce the tree level
amplitudes of bi-adjoint scalar (BAS) theory. Some notations and technics
which will be used in subsequent sections are also mentioned. In
subsection.II.2, we discuss the soft behavior of tree BAS amplitudes,
including the leading soft factor, as well as the statement that the leading
soft behaviors of all external scalars fully determine the tree BAS
amplitudes.

###  II.1 Tree level BAS amplitudes

The BAS theory describes the bi-adjoint scalar field $\phi_{a\bar{a}}$ with
the Lagrangian

$\displaystyle{\cal L}_{BAS}={1\over
2}\,\partial_{\mu}\phi^{a\bar{a}}\,\partial^{\mu}\phi^{a\bar{a}}-{\lambda\over
3!}\,f^{abc}f^{\bar{a}\bar{b}\bar{c}}\,\phi^{a\bar{a}}\phi^{b\bar{b}}\phi^{c\bar{c}}\,,$
(1)

where the structure constant $f^{abc}$ and generator $T^{a}$ satisfy

$\displaystyle[T^{a},T^{b}]=if^{abc}T^{c}\,,$ (2)

and the dual color algebra encoded by $f^{\bar{a}\bar{b}\bar{c}}$ and
$T^{\bar{a}}$ is analogous. The tree level amplitudes of this theory contain
only propagators, and can be decomposed into double color ordered partial
amplitudes via the standard technic. Each double color ordered partial
amplitude is simultaneously planar with respect to two color orderings, arise
from expanding the full $n$-point amplitude to $\mathop{\rm
Tr}(T^{a_{\sigma_{1}}}\cdots T^{a_{\sigma_{n}}})$ and $\mathop{\rm
Tr}(T^{\bar{a}_{\bar{\sigma}_{1}}}\cdots T^{\bar{a}_{\bar{\sigma}_{n}}})$
respectively, where $\sigma_{i}$ and $\bar{\sigma}_{i}$ denote permutations
among all external scalars.

To calculate double color ordered partial amplitudes, it is convenient to
employ the diagrammatical method proposed by Cachazo, He and Yuan in
Cachazo:2013iea . For a BAS amplitude whose double color-orderings are given,
this method provides the corresponding Feynman diagrams as well as the overall
sign directly. Let us consider the $5$-point example ${\cal
A}_{S}(1,2,3,4,5|1,4,2,3,5)$. In Figure.1, the first diagram satisfies both
two color orderings $(1,2,3,4,5)$ and $(1,4,2,3,5)$, while the second one
satisfies the ordering $(1,2,3,4,5)$ but not $(1,4,2,3,5)$. Thus, the first
diagram is allowed by the double color orderings $(1,2,3,4,5|1,4,2,3,5)$,
while the second one is not. It is easy to see other diagrams are also
forbidden by the ordering $(1,4,2,3,5)$, thus the first diagram in Figure.1 is
the only diagram contributes to the amplitude ${\cal
A}_{S}(1,2,3,4,5|1,4,2,3,5)$.

Figure 1: Two $5$-point diagrams

The Feynman diagrams contribute to a given BAS amplitude can be obtained via
systematic diagrammatical rules. For the above example, one can draw a disk
diagram as follows.

  * •

Draw points on the boundary of the disk according to the first ordering
$(1,2,3,4,5)$.

  * •

Draw a loop of line segments which connecting the points according to the
second ordering $(1,4,2,3,5)$.

The obtained disk diagram is shown in the first diagram in Figure.2. From the
diagram, one can see that two orderings share the boundaries $\\{1,5\\}$ and
$\\{2,3\\}$. These co-boundaries indicate channels ${1/s_{15}}$ and
${1/s_{23}}$, therefore the first Feynman diagram in Figure.1. Then the BAS
amplitude ${\cal A}_{S}(1,2,3,4,5|1,4,2,3,5)$ can be computed as

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

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

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

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

Figure 2: Diagram for ${\cal A}_{S}(1,2,3,4,5|1,4,2,3,5)$ and ${\cal
A}_{S}(1,2,3,4,5|1,2,4,3,5)$.

As another example, let us consider the BAS amplitude ${\cal
A}_{S}(1,2,3,4,5|1,2,4,3,5)$. The corresponding disk diagram is shown in the
second configuration in Figure.2, and one can see two orderings have co-
boundaries $\\{3,4\\}$ and $\\{5,1,2\\}$. The co-boundary $\\{3,4\\}$
indicates the channel ${1/s_{34}}$. The co-boundary $\\{5,1,2\\}$ indicates
the channel ${1/s_{512}}$ which is equivalent to $1/s_{34}$, as well as sub-
channels ${1/s_{12}}$ and ${1/s_{51}}$. Using the above decomposition, one can
calculate ${\cal A}_{S}(1,2,3,4,5|1,2,4,3,5)$ as

$\displaystyle{\cal A}_{S}(1,2,3,4,5|1,2,4,3,5)={1\over s_{34}}\Big{(}{1\over
s_{12}}+{1\over s_{51}}\Big{)}\,,$ (5)

up to an overall sign.

The overall sign, determined by the color algebra, can be fixed by following
rules.

  * •

Each polygon with odd number of vertices contributes a plus sign if its
orientation is the same as that of the disk and a minus sign if opposite.

  * •

Each polygon with even number of vertices always contributes a minus sign.

  * •

Each intersection point contributes a minus sign.

We now apply these rules to previous examples. In the first diagram in
Figure.2, the polygons are three triangles, namely $51A$, $A4B$ and $B23$,
which contribute $+$, $-$, $+$ respectively, while two intersection points $A$
and $B$ contribute two $-$. In the second one in Figure.2, the polygons are
$512A$ and $A43$, which contribute two $-$, while the intersection point $A$
contributes $-$. Then we arrive at the full results

$\displaystyle{\cal A}_{S}(1,2,3,4,5|1,4,2,3,5)$ $\displaystyle=$
$\displaystyle-{1\over s_{23}}{1\over s_{51}}\,,$ $\displaystyle{\cal
A}_{S}(1,2,3,4,5|1,2,4,3,5)$ $\displaystyle=$ $\displaystyle-{1\over
s_{34}}\Big{(}{1\over s_{12}}+{1\over s_{51}}\Big{)}\,.$ (6)

In the reminder of this paper, we adopt another convention for the overall
sign. If the line segments form a convex polygon, and the orientation of the
convex polygon is the same as that of the disk, then the overall sign is $+$.
For instance, the disk diagram in Figure.3 indicates the overall sign $+$
under the new convention, while the old convention gives $-$ according to the
square formed by four line segments. Notice that the diagrammatical rules
described previously still give the related sign between different disk
diagrams. For example, two disk diagrams in Figure.2 shows that the relative
sign between ${\cal A}_{S}(1,2,3,4,5|1,4,2,3,5)$ and ${\cal
A}_{S}(1,2,3,4,5|1,2,4,3,5)$ is $+$.

Figure 3: The overall sign $+$ under the new convention.

When considering the soft limit, the $2$-point channels play the central role.
Since the partial BAS amplitude carries two color orderings, if the $2$-point
channel contributes $1/s_{ab}$ to the amplitude, external legs $a$ and $b$
must be adjacent to each other in both two orderings. Suppose the first color
ordering is $(\cdots,a,b,\cdots)$, then $1/s_{ab}$ is allowed by this
ordering. To denote if it is allowed by another one, we introduce the symbol
$\delta_{ab}$ whose ordering of two subscripts $a$ and $b$ is determined by
the first color ordering111The Kronecker symbol will not appear in this paper,
thus we hope the notation $\delta_{ab}$ will not confuse the readers.. The
value of $\delta_{ab}$ is $\delta_{ab}=1$ if another color ordering is
$(\cdots,a,b,\cdots)$, $\delta_{ab}=-1$ if another color ordering is
$(\cdots,b,a,\cdots)$, due to the ani-symmetry of the structure constant,
i.e., $f^{abc}=-f^{bac}$, and $\delta_{ab}=0$ otherwise. From the definition,
it is straightforward to see $\delta_{ab}=-\delta_{ba}$. The symbol
$\delta_{ab}$ will be used frequently latter.

###  II.2 Leading soft behavior of tree BAS amplitudes

In this subsection, we first derive the leading order soft factor for the BAS
scalar, then explain that the leading soft behaviors of all external scalars
uniquely determine the tree BAS amplitudes.

The soft limit can be achieved by re-scaling the massless momenta via a soft
parameter as $k^{\mu}\to\tau k^{\mu}$, and taking the limit $\tau\to 0$.
Consider the double color ordered BAS amplitude ${\cal
A}_{S}(1,\cdots,n|\sigma)$, which carries two color orderings $(1,\cdots,n)$
and $\sigma$. We re-scale $k_{i}$ as $k_{i}\to\tau k_{i}$, and expand the
amplitude in $\tau$. The leading order contribution manifestly aries from
$2$-point channels $1/s_{1(i+1)}$ and $1/s_{(i-1)i}$ which provide the
$1/\tau$ order contributions, namely,

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

where $\not{i}$ stands for removing the leg $i$, $\sigma\setminus 1$ means the
color ordering generated from $\sigma$ by eliminating $i$. The leading soft
factor $S^{(0)}_{s}(i)$ for the scalar $i$ is extracted as

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

which acts on external scalars which are adjacent to $i$ in two color
orderings. Notice that in our notation the parameter $\tau$ is absorbed into
the soft factor $S^{(0)}_{s}(i)$.

If the factorization in (7) is satisfied for arbitrary external leg
$i\in\\{1,\cdots,n\\}$ when $k_{i}\to\tau k_{i}$, the tree BAS amplitude is
completely determined, as can be understood as follows. The tree BAS
amplitudes consist only propagators, thus are determined by correct channels.
Since the soft factor (8) determines the $2$-point channels, factors
$S^{(0)}_{s}(i)$ with $i\in\\{1,2,3,4\\}$ manifestly fix the $4$-point BAS
amplitudes. Assume that all $(n-1)$-point BAS amplitudes are already obtained,
which means all amplitudes ${\cal
A}_{S}(1,\cdots,i-1,\not{i},i+1,\cdots,n|\sigma\setminus i)$ at the r.h.s of
(7) are provided. The soft theorem in (7) provides all correct $2$-point poles
of each $n$-point amplitude when $i$ running around $\\{1,\cdots,n\\}$. For
higher-point channels, we observe that
$1/s_{\boldsymbol{{\alpha}}}=1/s_{\bar{\boldsymbol{{\alpha}}}}$, due to the
momentum conservation. Here $\boldsymbol{{\alpha}}$ is a subset of
$\\{1,\cdots,n\\}$ and $\bar{\boldsymbol{{\alpha}}}$ is the complement, and we
assume that $i\in\boldsymbol{{\alpha}}$ where $i$ is the soft external leg.
Since the pole $1/s_{\bar{\boldsymbol{{\alpha}}}}$ is already included in the
$(n-1)$-point sub-amplitude, it also contributes to the $n$-point amplitude.
Thus, correct poles of $n$-point amplitude for all channels are determined,
and these poles fix the $n$-point BAS amplitude uniquely. Thus, one can start
from the $4$-point amplitudes which are determined by the leading soft
theorem, and use the leading soft theorem to generate higher-point amplitudes
recursively. Therefore, we conclude that the leading soft behaviors of all
external scalars fix the tree BAS amplitudes. This conclusion is equivalent to
the following statement, if a relation among tree BAS amplitudes is satisfied
when taking arbitrary external leg $i\in\\{1,\cdots,n\\}$ to be soft, then
this relation is satisfied by BAS amplitudes themselves. This inference will
play the important role when deriving the fundamental BCJ relation in the next
section.

##  III Fundamental BCJ relation at tree level

In this section, we use the soft theorem in (7) to derive the fundamental BCJ
relation among double color ordered tree BAS amplitudes. In subsection.III.1,
we derive the fundamental BCJ relation at the leading order when one of
external scalar is taken to be soft. In subsection.III.2, we show that such
relation is satisfied at any order, by verifying the soft behaviors of other
external scalars.

###  III.1 Derivation

Consider the color ordered $(n+1)$-point BAS amplitudes whose external legs
are denoted as $i\in\\{1,\cdots,n\\}$ and $s$. The definition of $\delta_{is}$
in subsection.II.1 indicates

$\displaystyle 0=\sum_{i=1}^{n}\,\delta_{is}\,,~{}~{}$ (9)

therefore

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\,(k_{s}\cdot
k_{i})\,{\delta_{is}\over s_{is}}$ (10) $\displaystyle=$
$\displaystyle-(k_{s}\cdot K_{1(n-1)})\,{\delta_{ns}\over
s_{ns}}+\sum_{i=1}^{n-1}\,(k_{s}\cdot k_{i})\,{\delta_{is}\over s_{is}}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n-1}\,(k_{s}\cdot
k_{i})\,\Big{(}{\delta_{is}\over s_{is}}+{\delta_{sn}\over s_{sn}}\Big{)}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n-1}\,\sum_{j=i}^{n-1}\,(k_{s}\cdot
k_{i})\,\Big{(}{\delta_{js}\over s_{js}}+{\delta_{s(j+1)}\over
s_{s(j+1)}}\Big{)}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n-1}\,(k_{s}\cdot K_{1j})\,\Big{(}{\delta_{js}\over
s_{js}}+{\delta_{s(j+1)}\over s_{s(j+1)}}\Big{)}\,,~{}~{}~{}~{}$

where the combinatory momentum $K_{ab}$ is defined as
$K_{ab}\equiv\sum_{i=a}^{b}\,k_{i}$, and the Mandelstam variable $s_{ij}$ is
defined as $s_{ij}=2k_{i}\cdot k_{j}$. The second equality uses the momentum
conservation. The third and fourth equalities are obtained by employing
$\delta_{ab}=-\delta_{ba}$. Using (10) and the leading order soft theorem (7)
for the scalar $s$, we find

$\displaystyle 0=(k_{s}\cdot X_{s})\,{\cal A}^{(0)}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle s,n|\sigma)\,,~{}~{}~{}$ (11)

which hints the full fundamental BCJ relation

$\displaystyle 0=(k_{s}\cdot X_{s})\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle s,n|\sigma)\,.~{}~{}~{}$ (12)

Here the combinatory momentum $X_{s}$ is defined by summing over momenta
carried by external scalars at the l.h.s of $s$ in the color ordering. The
symbol $\shuffle$ means summing over all possible shuffles of two ordered sets
$\boldsymbol{{\beta}}_{1}$ and $\boldsymbol{{\beta}}_{2}$, i.e., all
permutations in the set $\boldsymbol{{\beta}}_{1}\cup\boldsymbol{{\beta}}_{2}$
while preserving the orderings of $\boldsymbol{{\beta}}_{1}$ and
$\boldsymbol{{\beta}}_{2}$. For instance, suppose
$\boldsymbol{{\beta}}_{1}=\\{1,2\\}$ and $\boldsymbol{{\beta}}_{2}=\\{3,4\\}$,
then

$\displaystyle\boldsymbol{{\beta}}_{1}\shuffle\boldsymbol{{\beta}}_{2}=\\{1,2,3,4\\}+\\{1,3,2,4\\}+\\{1,3,4,2\\}+\\{3,1,2,4\\}+\\{3,1,4,2\\}+\\{3,4,1,2\\}\,.~{}~{}~{}~{}$
(13)

The derivation in this subsection only gives (11) which is the fundamental BCJ
relation at the leading order, and the full one (12) should be regarded as a
conjecture at the current step. To make this work self-contained, it seems
that we need to verify the relation (12) at all orders. However, we have
another more efficient choice. In subsection.II.2, we pointed out that the
tree BAS amplitudes are fully determined by leading order soft behaviors of
all external scalars. The derivation of (11) only uses the leading soft
theorem for the scalar $s$. Thus, we can verify the conjectured relation (12)
by checking leading soft behaviors for other external particles, as we will do
in the next subsection. As will be seen, all these soft behaviors are finally
reduced to the algebraic relation in (9).

The fundamental BCJ relation (12) can be generalized to color ordered YM
amplitudes via the well known double copy structure. More explicitly, the
double copy structure indicates the following expansion Fu:2017uzt ;
Teng:2017tbo ; Du:2017kpo ; Du:2017gnh ; Feng:2019tvb ; Zhou:2019mbe

$\displaystyle{\cal A}_{\rm YM}(1,\cdots,n)=\sum_{\sigma^{\prime}\in
S_{n-2}}\,C(\sigma^{\prime})\,{\cal A}_{\rm
S}(1,\cdots,n|1,\sigma^{\prime},n)\,.~{}~{}$ (14)

Here $S_{n-2}$ stands for permutations among $(n-2)$ legs in
$\\{2,\cdots,n-1\\}$. The coefficients $C(\sigma^{\prime})$ serve as the BCJ
numerators, and depend on permutations $\sigma^{\prime}$, external momenta, as
well as the polarizations of external gluons. Combining (12) and (14)
together, we arrive at the relation

$\displaystyle 0=(k_{s}\cdot X_{s})\,{\cal A}_{\rm
YM}(1,\\{2,\cdots,n-1\\}\shuffle s,n)\,.~{}~{}~{}$ (15)

This is the standard BCJ relation in literatures Bern:2008qj . One can also
generate the general BCJ relations from the fundamental one, see in Ma:2011um
.

###  III.2 Verification

In this subsection, we prove the fundamental BCJ relation (12) by considering
the leading order soft behavior of each external scalar. More explicitly, we
will show that under the re-scaling $k_{i}\to\tau k_{i}$ with
$i\in\\{1,\cdots,n\\}$, the leading order contribution of (12) gives the
fundamental BCJ relation for BAS amplitudes with less external legs. Such
reduction procedure is terminated at the simplest fundamental BCJ relation for
$4$-point BAS amplitudes, whose correct soft behaviors are ensured by the
definition of $\delta_{ab}$.

Under the re-scaling $k_{1}\to\tau k_{1}$, the leading order contribution of
(12) is given by

$\displaystyle 0={1\over\tau}\,\Big{(}{\delta_{12}\over
s_{12}}+{\delta_{n1}\over s_{n1}}\Big{)}\,(k_{s}\cdot X_{s})\,{\cal A}_{\rm
S}(2,\\{3,\cdots,n-1\\}\shuffle s,n|\sigma\setminus 1)\,,~{}~{}~{}$ (16)

where the amplitude ${\cal A}(s,2,\cdots,n||\sigma\setminus 1)$ does not
contribute, since $k_{s}\cdot X_{s}$ is proportional to $\tau$ when
$X_{s}=k_{1}$. The relation (16) requires

$\displaystyle 0=(k_{s}\cdot X_{s})\,{\cal A}_{\rm
S}(2,\\{3,\cdots,n-1\\}\shuffle s,n|\sigma\setminus 1)\,,~{}~{}~{}$ (17)

which is nothing but the fundamental BCJ relation among $n$-point BAS
amplitudes ${\cal A}_{\rm S}(2,\\{3,\cdots,n-1\\}\shuffle s,n|\sigma\setminus
1)$.

We can use the momentum conservation to rewrite the fundamental BCJ relation
(12) as

$\displaystyle 0=(k_{s}\cdot X^{\rm R}_{s})\,{\cal A}_{\rm
S}(1,\\{2,\cdots,n-1\\}\shuffle s,n)\,,~{}~{}~{}$ (18)

where $X^{\rm R}_{s}$ is defined as the summation of momenta of legs at the
r.h.s of $s$ in the color ordering. Comparing (18) with (12), we see the
manifest symmetry among external legs $1$ and $n$. This symmetry, together
with the result (16), indicate that for $k_{n}\to\tau k_{n}$ we have

$\displaystyle 0={1\over\tau}\,\Big{(}{\delta_{(n-1)n}\over
s_{(n-1)n}}+{\delta_{n1}\over s_{n1}}\Big{)}\,(k_{s}\cdot X_{s})\,{\cal
A}_{\rm S}(1,\\{2,\cdots,n-2\\}\shuffle s,n-1|\sigma\setminus n)\,,~{}~{}~{}$
(19)

which includes the fundamental BCJ relation among $n$-point BAS amplitudes
${\cal A}_{\rm S}(1,\\{2,\cdots,n-2\\}\shuffle s,n-1|\sigma\setminus n)$.

For $k_{i}\to\tau k_{i}$ with $i\in\\{2,\cdots,n-1\\}$, the leading order
contribution of (12) can be separated as

$\displaystyle 0$ $\displaystyle=$
$\displaystyle{1\over\tau}\,\Big{(}{\delta_{(i-1)i}\over
s_{(i-1)i}}+{\delta_{i(i+1)}\over s_{i(i+1)}}\Big{)}\,(k_{s}\cdot
X_{s})\,{\cal A}_{\rm S}(1,\\{2,\cdots,i-2\\}\shuffle
s,i-1,\not{i},i+1,\cdots,n|\sigma\setminus i)$ (20)
$\displaystyle+{1\over\tau}\,\Big{(}{\delta_{(i-1)i}\over
s_{(i-1)i}}+{\delta_{i(i+1)}\over s_{i(i+1)}}\Big{)}\,(k_{s}\cdot
X_{s})\,{\cal A}_{\rm S}(1,\cdots,i-1,\not{i},i+1,\\{i+2,\cdots,n-1\\}\shuffle
s,n|\sigma\setminus i)$ $\displaystyle+{1\over\tau}\,\Big{(}{\delta_{si}\over
s_{si}}+{\delta_{i(i+1)}\over s_{i(i+1)}}\Big{)}\,(k_{s}\cdot X_{s})\,{\cal
A}_{\rm S}(1,\cdots,i-1,s,\not{i},i+1,\cdots,n|\sigma\setminus i)$
$\displaystyle+{1\over\tau}\,\Big{(}{\delta_{(i-1)i}\over
s_{(i-1)i}}+{\delta_{is}\over s_{is}}\Big{)}\,(k_{s}\cdot X_{s})\,{\cal
A}_{\rm S}(1,\cdots,i-1,\not{i},s,i+1,\cdots,n|\sigma\setminus i)\,.~{}~{}~{}$

One can adding the last two lines at the r.h.s of (20) to get

$\displaystyle{1\over\tau}\,\Big{(}{\delta_{si}\over
s_{si}}+{\delta_{i(i+1)}\over s_{i(i+1)}}\Big{)}\,(k_{s}\cdot X_{s})\,{\cal
A}_{\rm S}(1,\cdots,i-1,s,\not{i},i+1,\cdots,n|\sigma\setminus i)$ (21)
$\displaystyle+{1\over\tau}\,\Big{(}{\delta_{(i-1)i}\over
s_{(i-1)i}}+{\delta_{is}\over s_{is}}\Big{)}\,(k_{s}\cdot X_{s})\,{\cal
A}_{\rm S}(1,\cdots,i-1,\not{i},s,i+1,\cdots,n|\sigma\setminus i)$
$\displaystyle=$ $\displaystyle{1\over\tau}\,\Big{(}{\delta_{(i-1)i}\over
s_{(i-1)i}}+{\delta_{i(i+1)}\over s_{i(i+1)}}\Big{)}\,(k_{s}\cdot
X_{s})\,{\cal A}_{\rm S}(1,\cdots,i-1,s,i+1,\cdots,n|\sigma\setminus
i)\,.~{}~{}~{}$

Substituting (21) into (20), we arrive at

$\displaystyle 0={1\over\tau}\,\Big{(}{\delta_{(i-1)i}\over
s_{(i-1)i}}+{\delta_{i(i+1)}\over s_{i(i+1)}}\Big{)}\,(k_{s}\cdot
X_{s})\,{\cal A}_{\rm S}(1,\\{\\{2,\cdots,n-1\\}\setminus i\\}\shuffle
s,n|\sigma\setminus i)\,,$ (22)

which can be recognized as the fundamental BCJ relations among $n$-point BAS
amplitudes ${\cal A}_{\rm S}(1,\\{\\{2,\cdots,n-1\\}\setminus i\\}\shuffle
s,n|\sigma\setminus i)$.

From calculations mentioned above, we see that taking $k_{i}\to\tau k_{i}$ for
each $i\in\\{1,\cdots,n\\}$ reduces the fundamental BCJ relation (12) among
$(n+1)$-point BAS amplitudes to the same relation among $n$-point BAS ones.
This reduction can be repeated recursively. Thus, to prove the leading order
soft behavior for the conjectured relation (12), we only need to check this
relation among $4$-point BAS amplitudes ${\cal A}(1,2\shuffle s,3|\sigma)$,
namely,

$\displaystyle 0=(k_{s}\cdot X_{s})\,{\cal A}_{\rm S}(1,2\shuffle
s,3|\sigma)\,.~{}~{}~{}$ (23)

The $4$-point case is quiet special, since all Mandelstam variables vanish at
the $\tau^{0}$ order when one of external legs being soft. In other words, the
leading order contribution of (23) is at the $\tau^{0}$ order rather than the
$\tau^{-1}$ order. To see this, we use the momentum conservation to rewrite
(23) as

$\displaystyle 0=(k_{s}\cdot k_{1})\,{\cal A}_{\rm
S}(1,s,2,3|\sigma)-(k_{2}\cdot k_{1})\,{\cal A}_{\rm S}(1,2,s,3|\sigma)\,,$
(24)

then re-scale $k_{1}$ as $k_{1}\to\tau k_{1}$, and get the leading order
contribution as

$\displaystyle 0$ $\displaystyle=$ $\displaystyle(k_{s}\cdot
k_{1})\,\Big{(}{\delta_{31}\over s_{31}}+{\delta_{1s}\over
s_{1s}}\Big{)}\,{\cal A}_{\rm S}(s,2,3|\sigma\setminus 1)-(k_{1}\cdot
k_{2})\,\Big{(}{\delta_{31}\over s_{31}}+{\delta_{12}\over
s_{12}}\Big{)}\,{\cal A}_{\rm S}(2,s,3|\sigma\setminus 1)\,.~{}~{}~{}$ (25)

To verify (25), we observe that ${\cal A}_{\rm S}(s,2,3|\sigma\setminus
1)=-{\cal A}_{\rm S}(2,s,3|\sigma\setminus 1)$, which can be understood via
either the antisymmetry of the structure constant of Lie group, or the well
known Kleiss-Kuijf relation Kleiss:1988ne . This observation turns (25) to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\Big{(}\delta_{12}+\delta_{13}+\delta_{1s}\Big{)}\,{\cal A}_{\rm
S}(s,2,3|\sigma\setminus 1)\,,~{}~{}~{}$ (26)

where we have used the momentum conservation to get the coefficient of
$\delta_{13}$. The relation (26) is guaranteed by
$0=\delta_{12}+\delta_{13}+\delta_{1s}$, due to the definition of
$\delta_{ab}$. The correct soft behavior for $k_{3}\to\tau k_{3}$ is ensured
by the symmetry between legs $1$ and $3$, as discussed around (18).

Next, we use the momentum conservation to rewrite (23) as

$\displaystyle 0=(k_{2}\cdot k_{3})\,{\cal A}_{\rm
S}(1,s,2,3|\sigma)-(k_{2}\cdot k_{1})\,{\cal A}_{\rm S}(1,2,s,3|\sigma)\,,$
(27)

and consider $k_{2}\to\tau k_{2}$. The leading order contribution is

$\displaystyle 0$ $\displaystyle=$ $\displaystyle(k_{2}\cdot
k_{3})\,\Big{(}{\delta_{s2}\over s_{s2}}+{\delta_{23}\over
s_{23}}\Big{)}\,{\cal A}_{\rm S}(1,s,3|\sigma\setminus 2)-(k_{2}\cdot
k_{1})\,\Big{(}{\delta_{12}\over s_{12}}+{\delta_{2s}\over
s_{2s}}\Big{)}\,{\cal A}_{\rm S}(1,s,3|\sigma\setminus 2)$ (28)
$\displaystyle=$
$\displaystyle\Big{(}\delta_{21}+\delta_{23}+\delta_{2s}\Big{)}\,{\cal A}_{\rm
S}(1,s,3|\sigma\setminus 2)\,,~{}~{}~{}$

where the momentum conservation is used to obtain the coefficient of
$\delta_{2s}$. Again, the relation (28) is ensured by the definition of
$\delta_{ab}$.

Thus, we conclude that the relation (12) among double color ordered BAS
amplitudes is satisfied when arbitrary external scalar is taken to be soft,
therefore is correct.

##  IV BCJ relation at $1$-loop level

The purpose of this section is to generalize the tree level fundamental BCJ
relation to the $1$-loop level. To realize the goal, we first generalize the
fundamental BCJ relation to tree BAS amplitudes with two massive external
scalars in subsection.IV.1. In subsection.IV.2, we review the forward limit
method which generates the $1$-loop Feynman integrands from tree amplitudes.
Then, in subsection.IV.3, we use the result obtained in subsection.IV.1,
together with the forward limit method, to obtain the fundamental BCJ relation
among Feynman integrands at $1$-loop level.

###  IV.1 Tree level BCJ relation with two massive external states

In the manipulation (10), suppose all external momenta except $k_{s}$ are
massive, with equal mass $k_{i}^{2}=m^{2}$ where $i\in\\{1,\cdots,n\\}$, and
all $s_{is}$ are replaced by $s_{is}-m^{2}=2k_{i}\cdot k_{s}$, we see that the
result still holds. Furthermore, the soft factor (8) are also valid under the
replacement $s_{is}\to s_{is}-m^{2}$. This observation yields the relation
(11) at the leading order, thus leads us to guess the fundamental BCJ relation
holds for the above massive case. However, in the current situation, one can
not conclude that the tree BAS amplitude is completely fixed by soft behaviors
of external scalars. First, there is only one massless scalar $s$ can be taken
to be soft thus is not sufficient to fix poles for all channels. Secondly,
since we have not chose the explicit structure of interaction vertices for the
new theory which includes massive scalars, in principle each internal scalar
can be either massless or massive, thus it is impossible to determine
corresponding poles. Consequently, the fundamental BCJ relation does not hold
for this case, as can be verified directly.

However, we can restrict ourselves to a quiet special case, the external legs
$1$ and $n$ are massive with $k_{1}^{2}=k_{n}^{2}=m^{2}$, while all other
external legs are massless. Furthermore, we assume that an massive scalar
propagates through the amplitude from leg $1$ to leg $n$. Let us call this
path from $1$ to $n$ the massive scalar line. One can think the full amplitude
as a variety of massless scalars coupled to each other and finally coupled to
the massive scalar line. For this special case, the following simple argument
can convince us that the fundamental BCJ relation (12) still holds. Let us use
the Mandelstam variable $s_{\boldsymbol{{\alpha}}}$ to denote the
corresponding channel, where $\boldsymbol{{\alpha}}$ is a set of external legs
satisfying $\boldsymbol{{\alpha}}\subset\\{1,\cdots,n\\}\cup s$. We separate
all channels into two classes, one is $1\in\boldsymbol{{\alpha}}$,
$n\in\bar{\boldsymbol{{\alpha}}}$, and another one is
$1,n\in\boldsymbol{{\alpha}}$. For the first case, $s_{\boldsymbol{{\alpha}}}$
corresponds to the internal line which belong to the massive line, thus the
propagator takes the form $1/(s_{\boldsymbol{{\alpha}}}-m^{2})$. Since there
is only one massive scalar $1$ in $\boldsymbol{{\alpha}}$, we have
$s_{\boldsymbol{{\alpha}}}-m^{2}=\sum_{i,j\in\boldsymbol{{\alpha}}}2k_{i}\cdot
k_{j}$, this form is the same as that for the massless case. For the second
case, we can use $s_{\bar{\boldsymbol{{\alpha}}}}$ to instead of
$s_{\boldsymbol{{\alpha}}}$, due to the momentum conservation. In other words,
for the second case, one can always chose an expression that can not ”see” the
massive legs $1$ and $2$. To summarize, we find that the expression of the BAS
amplitude is not affected when legs $1$ and $n$ are turned to be massive.
Furthermore, in the fundamental BCJ relation (12), it seems that there is no
chance for $k_{1}^{2}$ or $k_{n}^{2}$ to play any role, even if we use the
momentum conservation to rewrite any $k_{i}\cdot k_{s}$. The above argument
provides the strong evidence for the validity of fundamental BCJ relation for
the case $k_{1}^{2}=k_{n}^{2}=m^{2}$, and we numerically verified this
statement until $8$-point.

Starting from the BAS amplitudes with $k_{1}^{2}=k_{n}^{2}=m^{2}$, one can
generate the $1$-loop Feynman integrand by sewing two massive legs together
via the so called forward limit method. In this procedure, one need to allow
$m^{2}<0$, thus it is more suitable to interpret $1$ and $n$ as off-shell legs
with $k_{1}^{2}=k_{2}^{2}$. Applying this sewing manipulation, one can
generalize the tree level fundamental BCJ relation to the $1$-loop level, as
will be seen in subsection.IV.3.

###  IV.2 Forward limit method

The $1$-loop Feynman integrands can be generated from the corresponding tree
amplitudes, via the so called forward limit procedure. For instance, the
$1$-loop CHY formulas can be obtained by applying this operation, as studied
in He:2015yua ; Cachazo:2015aol ; Feng:2016nrf ; Feng:2019xiq . For readers’
convenience, in this subsection we introduce the general idea and features of
the forward limit.

The forward limit is reached as follows:

  * •

Consider a $(n+2)$-point tree amplitude ${\cal A}_{n+2}(k_{+},k_{-})$
including $n$ massless legs with momenta in $\\{k_{1},\cdots,k_{n}\\}$ and two
off-shell legs with $k_{+}^{2}=k_{-}^{2}\neq 0$.

  * •

Take the limit $k_{\pm}\to\pm\ell$, and glue the two corresponding legs
together. we denote this manipulation as ${\cal L}$. Performing ${\cal L}$ on
the tree amplitude leads to a special tree amplitude with $k_{+}=-k_{-}=\ell$,
rather than a $1$-loop level object.

  * •

Sum over all allowed internal states of the internal particle with loop
momentum $\ell$, such as polarization vectors or tensors, colors, flavors, and
so on222For theories include gauge or flavor groups, we only discuss the color
ordered partial amplitudes in this paper, thus the summations over colors or
flavors are hidden. , we denote this manipulation as ${\cal E}$.

Roughly speaking, the obtained object, times the factor $1/\ell^{2}$ as

$\displaystyle{1\over\ell^{2}}\,{\cal F}\,{\cal
A}_{n+2}(k_{+}^{h_{+}},k_{-}^{h_{-}})={1\over\ell^{2}}\,\sum_{h}\,{\cal
A}_{n+2}(\ell^{h},-\ell^{\bar{h}})\,,~{}~{}~{}~{}$ (29)

contributes to the $n$-point $1$-loop Feynman integrand ${\bf I}_{n}$. Here we
introduced the forward limit operator

$\displaystyle{\cal F}\equiv{\cal E}\,{\cal L}\,,$ (30)

to denote the operation of taking forward limit. In this paper, we denote the
$1$-loop Feynman integrands by ${\bf I}$. From now on, we will neglect the
subscript $n$ of ${\bf I}$, since we will use other manners to denote the
number of external legs.

For the individual Feynman diagram, the manipulation in (29) obviously turns
the tree diagram to the $1$-loop one. However, the full $1$-loop Feynman
integrand is obtained by summing over all appropriate diagrams. Thus, let us
consider what requirement should be satisfied if the resulting object of the
manipulation in (29) can be interpreted as the correct $1$-loop Feynman
integrand. It is easy to observe that after summing over all allowed tree
level diagrams, each $1$-loop diagram receives contributions from tree
diagrams correspond to cutting each propagator in the loop once (cutting is
understood as the inverse operation of gluing legs $+$ and $-$ together, where
$+$, $-$ denote external legs carry $k_{+}$ and $k_{-}$ respectively), as can
be seen in Fig. 4. Thus the statement that the operation in (29) generates the
correct Feynman integrand holds if and only if the term for an individual
$1$-loop diagram can be decomposed to terms for related tree diagrams, as
shown in Fig. 4. Such decomposition can be realized via the so called partial
fraction identity He:2015yua ; Baadsgaard:2015twa :

$\displaystyle{1\over D_{1}\cdots D_{m}}=\sum_{i=1}^{m}\,{1\over
D_{i}}\Big{[}\prod_{j\neq i}\,{1\over D_{j}-D_{i}}\Big{]}\,,$ (31)

which implies

$\displaystyle{1\over\ell^{2}(\ell+K_{1})^{2}(\ell+K_{12})^{2}\cdots(\ell+K_{1(m-1)})^{2}}\simeq{1\over\ell^{2}}\,\sum_{i=1}^{m}\,\Big{[}\prod_{j=i}^{i+m-2}\,{1\over(\ell+K_{ij})^{2}-\ell^{2}}\Big{]}\,.~{}~{}~{}~{}$
(32)

For each individual term at the r.h.s of the above relation, the loop momentum
is shifted while result of Feynman integral is not altered. Here $\simeq$
means the l.h.s and r.h.s are not equivalent to each other at the integrand
level, but are equivalent at the integration level. At the r.h.s of (32), we
have seen the propagators with the denominates $(\ell+K_{ij})^{2}-\ell^{2}$,
which are different from the standard ones $(\ell+K_{ij})^{2}$. This feature
of propagators is the condition which should be satisfied if the manipulation
in (29) provides the correct $1$-loop Feynman integrand. In CHY formulas, this
requirement is satisfied via the $1$-loop level scattering equations. From the
Feynman diagrams point of view, one can assume each propagator in the loop is
massive with $m^{2}=\ell^{2}$. Thus, one can assume that the $1$-loop Feynman
integrand ${\bf I}_{\circ}$ is obtained from the manipulation in (29). To
distinguish the full Feynman integrands and partial Feynman integrands
obtained by decomposing the full ones via the partial fraction identity, from
now on, we use ${\bf I}_{\circ}$ to denote the former ones, and ${\bf I}$ to
denote the latter ones.

Figure 4: Decomposition of $1$-loop Feynman integrand.

The above discussion is for the full Feynman integrands without any color
ordering, and now we turn to the color ordered Feynman integrands. Since we
have made sure that the full $1$-loop Feynman integrand can be generated from
the full tree amplitude via the forward limit operation, let us start with the
full tree amplitude. Consider a theory that external particles are in the
adjoint representation of the $U(N)$ group, the full tree amplitude can be
expanded using the standard color decomposition as a sum over $(n+1)!$ terms

$\displaystyle{\cal A}_{n+2}=\sum_{\sigma_{1}\in
S_{n+2}/\mathbb{Z}_{n+2}}\,{\rm Tr}(T^{a_{\sigma_{+}}}T^{a_{\sigma_{1}}}\cdots
T^{a_{\sigma_{n}}}T^{a_{\sigma_{-}}}){\cal
A}(\sigma_{+},\sigma_{1},\cdots,\sigma_{n},\sigma_{-})\,.~{}~{}~{}~{}$ (33)

Notice that at the r.h.s it is not necessary to add the subscript $n+2$ to
${\cal A}$, since the color ordering
$\sigma_{+},\sigma_{1},\cdots,\sigma_{n},\sigma_{-}$ already reflects the
number of external legs. Taking the forward limit of external legs requires
summing over the $U(N)$ degrees of freedom of the two internal particles. This
gives rise to two kinds of terms. The first comes from permutations such that
legs $+$ and $-$ are adjacent, the corresponding color factors are given as

$\displaystyle\sum_{a_{+}=a_{-}=1}^{N^{2}}\,\delta_{a_{+}a_{-}}\,{\rm
Tr}(T^{a_{+}}T^{a_{\sigma_{1}}}\cdots T^{a_{\sigma_{n}}}T^{a_{-}})=N{\rm
Tr}(T^{a_{\sigma_{1}}}\cdots T^{a_{\sigma_{n}}})\,,$ (34)

thus contributes to the $n$-point color ordered Feynman integrand ${\bf
I}_{\circ}(\sigma_{1},\cdots,\sigma_{n})$. The second case that $+$ and $-$
are not adjacent gives rise to double-trace terms. In this paper, we only
consider the single-trace terms, since the double-trace terms are determined
by the single-trace ones Bern:1996je , as can be proved by employing the tree
level Kleiss-Kuijf relation together with the forward limit operation
Kleiss:1988ne . For the single trace case, the above discussion shows that the
partial integrand obtained form taking the forward limit for ${\cal
A}(+,\sigma_{1},\cdots,\sigma_{n},-)$ contributes to ${\bf
I}_{\circ}(\sigma_{1},\cdots,\sigma_{n})$. To find the full decomposition of
${\bf I}_{\circ}(\sigma_{1},\cdots,\sigma_{n})$, we use the clear observation
that several original color orderings give rise to the same trace factor after
summing over $a_{+}$ and $a_{-}$, due to the cyclic symmetry of the trace
factors. Collecting theses color orderings together, one finds that after
taking the forward limit, the decomposition (33) can be organized as

$\displaystyle{\cal F}\,{\cal A}_{n+2}=\sum_{\sigma_{1}\in
S_{n+2}/\mathbb{Z}_{n+2}}\,{\rm Tr}(T^{a_{\sigma_{1}}}\cdots
T^{a_{\sigma_{n}}})\,\sum_{j=0}^{n-1}\,{\cal F}\,{\cal
A}(+,\sigma_{1+j},\cdots,\sigma_{n+j},-)+({\rm double-trace})\,.$ (35)

Consequently, the full color ordered Feynman integrand can be expanded as the
following cyclic summation

$\displaystyle{\bf
I}_{\circ}(\sigma_{1},\cdots,\sigma_{n})=\sum_{j=0}^{n-1}\,{\bf
I}(+,\sigma_{1+j},\cdots,\sigma_{n+j},-)\,,~{}~{}~{}~{}$ (36)

where the partial color ordered integrands ${\bf
I}(+,\sigma_{1+j},\cdots,\sigma_{n+j},-)$ are obtained from the color ordered
tree amplitudes via the standard forward limit procedure in (29), namely,

$\displaystyle{\bf
I}(+,\sigma_{1+j},\cdots,\sigma_{n+j},-)={1\over\ell^{2}}\,{\cal F}\,{\cal
A}(+,\sigma_{1+j},\cdots,\sigma_{n+j},-)\,.~{}~{}$ (37)

The cyclic summation in (36) indicates that each propagator in the loop has
been cut once, thus ${\bf I}_{\circ}(\sigma_{1},\cdots,\sigma_{n})$ and ${\bf
I}(+,\sigma_{1+j},\cdots,\sigma_{n+j},-)$ are also related via the partial
fraction identity.

The forward limit is well defined for the ${\cal N}=4$ SYM theory. For other
theories, a quite general feature is, the obtained Feynman integrand suffer
from divergence in the forward limit. Fortunately, the singular parts is found
to be physically irrelevant, at least for theories under consideration in this
paper. From the Feynman diagrams point of view, the singular parts generated
by the forward limit correspond to tadpole diagrams, as well as babble
diagrams for external legs, which do not contribute to the $S$-matrix. From
the CHY point of view, the singular parts can be bypassed by employing the
following observation Cachazo:2015aol : as long as the CHY integrand is
homogeneous in $\ell^{\mu}$, the singular solutions contribute to the
scaleless integrals which vanish under the dimensional regularization. The
homogeneity in $\ell^{\mu}$ are satisfied by the BAS Feynman integrands under
consideration in this note. Thus, in this section, we just assume that the
singular parts generated by the forward limit are excluded by an appropriate
way.

###  IV.3 Generalizing BCJ relation to $1$-loop level

With the forward limit method introduced in subsection.IV.2, now we are ready
to generalize the fundamental BCJ relation to the $1$-loop level. Using (36)
and (37), we see that the single trace BAS Feynman integrand can be obtained
through

$\displaystyle{\bf I}_{\circ{\rm S}}(\\{1,\cdots,n\\}\shuffle
s|\sigma)=\sum_{j=1}^{n}\,{\bf I}_{\rm S}(+,\\{j,j+1,\cdots,j-2,j-1\\}\shuffle
s,-|+,\sigma,-)\,,~{}~{}~{}~{}$ (38)

with

$\displaystyle{\bf I}_{\rm S}(+,\\{j,j+1,\cdots,j-2,j-1\\}\shuffle
s,-|+,\sigma,-)$ (39) $\displaystyle=$ $\displaystyle{1\over\ell^{2}}\,{\cal
F}\,{\cal A}_{\rm S}(+,\\{j,j+1,\cdots,j-2,j-1\\}\shuffle
s,-|+,\sigma,-)\,,~{}~{}$

where $+$ and $-$ are two off-shell external legs for tree amplitudes ${\cal
A}_{\rm S}(+,\\{j,j+1,\cdots,j-2,j-1\\}\shuffle s,-|+,\sigma,-)$, satisfying
$k_{+}=-k_{-}=\ell$. Using the tree level fundamental BCJ relation (12), we
know that

$\displaystyle 0=(k_{s}\cdot X^{\prime}_{s})\,{\cal A}_{\rm
S}(+,\\{j,j+1,\cdots,j-2,j-1\\}\shuffle s,-|+,\sigma,-)\,,$ (40)

where $X^{\prime}_{s}$ is defined as the summation over the loop momentum
$\ell$ and momenta carried by external legs at the l.h.s of $s$ in the color
ordering. Since $k_{s}\cdot X^{\prime}_{s}$ is commutable with the forward
limit operator ${\cal F}$, the partial integrands ${\bf I}_{\rm
S}(+,\\{j,j+1,\cdots,j-2,j-1\\}\shuffle s,-|+,\sigma,-)$ satisfies

$\displaystyle 0=(k_{s}\cdot X^{\prime}_{s})\,{\bf I}_{\rm
S}(+,\\{j,j+1,\cdots,j-2,j-1\\}\shuffle s,-|+,\sigma,-)\,.~{}~{}$ (41)

Substituting (41) into (38), we arrive at

$\displaystyle 0=(k_{s}\cdot X^{\prime}_{s})\,{\bf I}_{\circ{\rm
S}}(\\{1,\cdots,n\\}\shuffle s|\sigma)\,,~{}~{}$ (42)

which serves as the fundamental BCJ relation among BAS Feynman integrands at
$1$-loop level.

Using the double copy structure, the relation (42) can be generalized to YM
Feynman integrands straightforwardly,

$\displaystyle 0=(k_{s}\cdot X^{\prime}_{s})\,{\bf I}_{\circ{\rm
YM}}(\\{1,\cdots,n\\}\shuffle s)\,,~{}~{}$ (43)

which is the same as the result found in Du:2012mt .

##  V Understanding Adler’s zero

This section devotes to understand the Adler’s zero for tree NLSM and BI
amplitudes. In subsection.V.1, we introduce the expanded formulas of tree NLSM
and BI amplitudes. Then, in subsection.V.2, we show that the Adler’s zero can
be manifested by applying the fundamental BCJ relation to the expanded
formulas in subsection.V.1.

###  V.1 Expanded NLSM and BI amplitudes

Among recent investigations of scattering amplitudes, one of the remarkable
progresses is the expansions of amplitudes, which implies that the amplitudes
of different theories can be unified, and provides a new tool for calculating
amplitudes. Such expansions have been studied from various angles Fu:2017uzt ;
Teng:2017tbo ; Du:2017kpo ; Du:2017gnh ; Feng:2019tvb ; Zhou:2019mbe ;
Zhou:2022orv . Among all of these angles, the double copy structure plays the
central role. In this section, we will use the following two expansions.

The $n$-point tree NLSM amplitude ${\cal A}_{\rm N}(\sigma)$ with arbitrary
color ordering $\sigma$ can be expanded to BAS amplitudes as follows
Feng:2019tvb ; Zhou:2019mbe ,

$\displaystyle{\cal A}_{\rm N}(\sigma)=\Big{(}\prod_{i=2}^{n-1}\,k_{i}\cdot
X_{i}\Big{)}\,{\cal A}_{\rm S}(1,2\shuffle\cdots\shuffle
n-1,n|\sigma)\,.~{}~{}~{}$ (44)

The analogous expansion holds for the BI amplitudes, i.e., the $n$-point BI
amplitude ${\cal A}_{\rm B}(\\{1,\cdots,n\\})$ can be expanded to YM
amplitudes as Feng:2019tvb ; Zhou:2019mbe

$\displaystyle{\cal A}_{\rm
B}(\\{1,\cdots,n\\})=\Big{(}\prod_{i=2}^{n-1}\,k_{i}\cdot X_{i}\Big{)}\,{\cal
A}_{\rm Y}(1,2\shuffle\cdots\shuffle n-1,n)\,.~{}~{}~{}$ (45)

Obviously, these two expressions of NLSM and BI amplitudes allow us to apply
the fundamental BCJ relation directly.

###  V.2 Adler’s zero

The NLSM and BI amplitudes satisfy the so called Adler’s zero condition, i.e.,
amplitudes vanish when one of external legs being soft. Such phenomenon can be
easily understood via the BCJ relation and expanded formulas in (44) and (45).

We first show that the NLSM amplitude ${\cal A}_{\rm N}(\sigma)$ vanishes when
the external leg $2$ is taken to be soft. The generalization to other leg $i$
with $i\in\\{2,\cdots,n-1\\}$ is straightforward. Let us consider a subset of
the r.h.s of (44),

$\displaystyle(k_{2}\cdot
X_{2})\,\Big{(}\prod_{i=3}^{n-1}\,k_{{\alpha}_{i}}\cdot
X_{{\alpha}_{i}}\Big{)}\,{\cal A}_{\rm
S}(1,2\shuffle\\{{\alpha}_{3},\cdots,{\alpha}_{n-1}\\},n|\sigma)\,.~{}~{}~{}$
(46)

We define $X^{(2)}_{{\alpha}_{i}}$ as

$\displaystyle X^{(2)}_{{\alpha}_{i}}=\lim_{k_{2}\to
0}\,X_{{\alpha}_{i}}\,,~{}~{}$ (47)

which is a constant for ${\cal A}_{\rm
S}(1,2\shuffle\\{{\alpha}_{3},\cdots,{\alpha}_{n-1}\\},n|\sigma)$. Then, the
BCJ relation (12) imposes

$\displaystyle 0=(k_{2}\cdot
X_{2})\,\Big{(}\prod_{i=3}^{n-1}\,k_{{\alpha}_{i}}\cdot
X^{(2)}_{{\alpha}_{i}}\Big{)}\,{\cal A}_{\rm
S}(1,2\shuffle\\{{\alpha}_{3},\cdots,{\alpha}_{n-1}\\},n|\sigma)\,,$ (48)

since all $k_{{\alpha}_{i}}\cdot X^{(2)}_{{\alpha}_{i}}$ are constants. Thus,
the effective part of (46) is

$\displaystyle(k_{2}\cdot
X_{2})\,\Big{[}\Big{(}\prod_{i=3}^{n-1}\,k_{{\alpha}_{i}}\cdot
X_{{\alpha}_{i}}\Big{)}-\Big{(}\prod_{i=3}^{n-1}\,k_{{\alpha}_{i}}\cdot
X^{(2)}_{{\alpha}_{i}}\Big{)}\Big{]}\,{\cal A}_{\rm
S}(1,2\shuffle\\{{\alpha}_{3},\cdots,{\alpha}_{n-1}\\},n|\sigma)\,.$ (49)

Since $X_{{\alpha}_{i}}-X^{(2)}_{{\alpha}_{i}}=0~{}{\rm or}~{}k_{2}$, we have

$\displaystyle\Big{(}\prod_{i=3}^{n-1}\,k_{{\alpha}_{i}}\cdot
X_{{\alpha}_{i}}\Big{)}-\Big{(}\prod_{i=3}^{n-1}\,k_{{\alpha}_{i}}\cdot
X^{(2)}_{{\alpha}_{i}}\Big{)}$ $\displaystyle=$ $\displaystyle
c_{1}\,(k_{2}\cdot R_{11})+c_{2}\,(k_{2}\cdot R_{21})\,(k_{2}\cdot
R_{22})+\cdots\,,~{}~{}~{}$ (50)

where $R_{ij}$ denotes proper Lorentz vectors.

Now we re-scale $k_{2}$ as $k_{2}\to\tau k_{2}$, and expand (46) by $\tau$.
The leading order behavior of ${\cal A}_{\rm
S}(1,2\shuffle\\{{\alpha}_{3},\cdots,{\alpha}_{n-1}\\},n|\sigma)$ is at the
$\tau^{-1}$ order. On the other hand, the observation (50) shows that the
effective coefficient $(k_{2}\cdot
X_{2})\,\Big{[}\Big{(}\prod_{i=3}^{n-1}\,k_{{\alpha}_{i}}\cdot
X_{{\alpha}_{i}}\Big{)}-\Big{(}\prod_{i=3}^{n-1}\,k_{{\alpha}_{i}}\cdot
X^{(2)}_{{\alpha}_{i}}\Big{)}\Big{]}$ is vanishing or at the $\tau^{q}$ order,
where $q$ is an integer satisfying $q\geq 2$. Consequently, we have

$\displaystyle 0=\lim_{\tau\to 0}\,(k_{2}\cdot
X_{2})\,\Big{(}\prod_{i=3}^{n-1}\,k_{{\alpha}_{i}}\cdot
X_{{\alpha}_{i}}\Big{)}\,{\cal A}_{\rm
S}(1,2\shuffle\\{{\alpha}_{3},\cdots,{\alpha}_{n-1}\\},n|\sigma)\,,$ (51)

therefore

$\displaystyle 0=\lim_{\tau\to 0}\,(k_{2}\cdot
X_{2})\,\Big{(}\prod_{i=3}^{n-1}\,k_{{\alpha}_{i}}\cdot
X_{{\alpha}_{i}}\Big{)}\,{\cal A}_{\rm S}(1,2\shuffle\cdots\shuffle
n-1,n|\sigma)\,.$ (52)

The consideration for $k_{2}\to\tau k_{2}$ can be generalized to $k_{i}\to\tau
k_{i}$ directly, with $i\in\\{2,\cdots,n-1\\}$. Thus all these legs satisfy
Adler’s zero condition. The remaining task is to understand Adler’s zero for
external legs $1$ and $n$. To deal with the case $k_{1}\to\tau k_{1}$, we
consider the following subset of the r.h.s those the leg at the r.h.s of $1$
in the color ordering $(1,2\shuffle\cdots\shuffle n-1,n)$ is fixed, for
instance

$\displaystyle\Big{(}\prod_{i=2}^{n-1}\,k_{i}\cdot X_{i}\Big{)}\,{\cal A}_{\rm
S}(1,2,3\shuffle\cdots\shuffle n-1,n|\sigma)\,.~{}~{}~{}$ (53)

Taking $k_{1}\to\tau k_{1}$ and expanding (53) by $\tau$, we get the leading
order contribution

$\displaystyle(k_{2}\cdot k_{1})\,\Big{(}\prod_{i=3}^{n-1}\,k_{i}\cdot
X^{(1)}_{i}\Big{)}\,\Big{(}{\delta_{n1}\over s_{n1}}+{\delta_{12}\over
s_{12}}\Big{)}\,{\cal A}_{\rm S}(2,3\shuffle\cdots\shuffle n-1,n|\sigma)\,,$
(54)

which vanishes due to the BCJ relation

$\displaystyle 0=\Big{(}\prod_{i=3}^{n-1}\,k_{i}\cdot X_{i}\Big{)}\,{\cal
A}_{\rm S}(2,3\shuffle\cdots\shuffle n-1,n|\sigma)\,.$ (55)

Here the definition of $X^{(1)}_{i}$ is analogous to the definition of
$X^{(2)}_{{\alpha}_{i}}$ in (47), i.e., $X^{(1)}_{i}=\lim_{k_{1}\to 0}X_{i}$.
Thus the combination (53) satisfies the Adler’s zero condition. In the above
discussion, we fixed the leg at the r.h.s of $1$ to be $2$. The analogous
argument holds when replacing $2$ by any $i\in\\{2,\cdots,n-1\\}$. Thus we
conclude that the expanded formula (44) vanishes when the external leg $1$
being soft. The Adler’s zero for the leg $n$ being soft can be understood via
the similar manipulation, by employing the equivalent representation (18) for
the fundamental BCJ relation.

The Adler’s zero for BI amplitudes can be understood through the paralleled
procedure, by using the fundamental BCJ relation for YM amplitudes (15).

##  VI Summary

In this note, we used the soft theorem for external scalars to derive the
fundamental BCJ relation among double color ordered tree BAS amplitudes. Then,
we generalized this relation to the case that two external legs are massive.
Using the fundamental BCJ relation for such special tree BAS amplitudes with
massive external legs, as well as the forward limit method, we obtained the
fundamental BCJ relation among BAS Feynman integrands at $1$-loop level. We
also used the fundamental BCJ relation to understand the Adler’s zero, which
describe the soft behavior of tree NLSM and BI amplitudes.

In Ma:2011um , one can see that the general BCJ relations can be generated
from the fundamental one. It is interesting to ask whether such generalization
can be realized via the soft theorems. In this note, all soft limits under
consideration are those only one external leg being soft, which are called the
single soft behavior. In order to obtain the general BCJ relations, it seems
that we need to consider the multiple soft behaviors. Thus it is an
interesting future direction.

In this work, we used the expanded formulas, as well as the fundamental BCJ
relation, to understand the soft behavior of NLSM and BI amplitudes. In
Zhou:2022orv , it was shown that the expansions of single trace tree Yang-
Mills-scalar and Yang-Mills amplitudes can be uniquely determined by assuming
only the existence of soft theorems and the universality of soft factors,
without knowing the details of soft factors. It is natural to ask, can the
expansions of tree NLSM and BI amplitudes be determined by the general
consideration of soft behavior? This question will be answered in our next
work Zhou:20233 .

## Acknowledgments

The authors would thank Prof. Yijian Du for helpful discussions and valuable
suggestions.

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