Expansion of EinsteinYangMills amplitude
Received: March
Expansion of EinsteinYangMills amplitude
ChihHao Fu 0 1 3 7 8
YiJian Du 0 1 3 5 8
Rijun Huang 0 1 3 6 8
Bo Feng 0 1 2 3 4 8
0 No. 1 Wenyuan Road, Nanjing 210046 , P.R. China
1 No. 299 Bayi Road, Wuhan 430072 , P.R. China
2 Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University
3 No. 620 West Chang'an Avenue, Xi'an 710119 , P.R. China
4 Center of Mathematical Science, Zhejiang University
5 Center for Theoretical Physics, School of Physics and Technology, Wuhan University
6 Department of Physics and Institute of Theoretical Physics, Nanjing Normal University
7 School of Physics and Information Technology, Shaanxi Normal University
8 No. 38 Zheda Road, Hangzhou 310027 , P.R. China
In this paper, we study from various perspectives the expansion of tree level single trace EinsteinYangMills amplitudes into linear combination of colorordered YangMills amplitudes. By applying the gauge invariance principle, a programable recursive construction is devised to expand EYM amplitude with arbitrary number of gravitons into EYM amplitudes with fewer gravitons. Based on this recursive technique we write down the complete expansion of any single trace EYM amplitude in the basis of colororder YangMills amplitude. As a byproduct, an algorithm for constructing a polynomial form of the BCJ numerator for YangMills amplitudes is also outlined in this paper. In addition, by applying BCFW recursion relation we show how to arrive at the same EYM amplitude expansion from the onshell perspective. And we examine the EYM expansion using KLT relations and show how to evaluate the expansion coe cients e ciently.
Scattering Amplitudes; Gauge Symmetry

3.1
3.2
3.3
3.4
3.5
6.1
6.2
6.3
rents
1
Introduction
A constructive algorithm for producing general EYM amplitude relations
Expanding to pure YangMills amplitudes: ordered splitting formula
3.6 Expanding to pure YangMills amplitudes: KK basis formula
4 The BCJ numerator of YangMills theory
5 Inspecting the amplitude relations through BCFW recursions
5.1
Contributions of nite poles
5.2 The boundary contributions
6 Inspecting the amplitude relations through KLT relation
The case with single gluon
The fourpoint gluonscalar amplitude involving two gluons
The vepoint YMs amplitudes with two gluons
6.4 The ve and higher point amplitude involving two gravitons
7 Conclusion
A Scalar YangMills Feynman rules B Graphical proof of the twogluon fundamental BCJ relation between cur
1 Introduction
2 Amplitude relations from the perspective of CHYformulation
3 The gauge invariance determines the amplitude relations
A fairly nontrivial relation between EinsteinYangMills (EYM) amplitude and pure
YangMills amplitudes was proposed in [1] recently, where the amplitude of n gluons coupled to a
formulation, which is di erent from the earlier proposed relations that express ngluon
mgraviton amplitudes by (n + 2m)gluon amplitudes [2{6]. It is now widely known that, the
nontrivial relations among amplitudes are important both in the practical evaluation and
{ 1 {
the analytical study, while the U(1)relation, KleissKuijf(KK)relation [7] and especially
the BernCarrascoJohansson(BCJ) relations [8] among amplitudes of the same eld theory
have received considerable investigations in the past few years, and inspired the
colorkinematics duality for gravity and YangMills amplitudes [9, 10]. As an analogous scenario,
where amplitudes of two originally seemingly unrelated theories take part in novel identity,
recall that the famous KawaiLewellenTye (KLT) relation [11] was proposed quite a
long time ago, which formulates a closed string amplitude as products of two open string
amplitudes, and in the eld theory limit it expands a pure gravity amplitude as bilinear
sum of YangMills amplitudes. The newly proposed linear EYM amplitude relation was
also inspired by the study of string theory, where monodromy relations for mixed
closedopen string amplitudes, previously been applied to the study of BCJ relations [3, 12, 13],
has been considered.
Because of its compact and simple nature, a substantial research interests has been
drawn to the study of EYM amplitude relations and to its generalizations [14{19].1 In
particular most of the discussions are based on the CachazoHeYuan (CHY) formulation [21{
25], by genuinely reformulating the CHYintegrand in an appropriate form. Notably,
explicit expressions for EYM amplitude relations with arbitrary number of gluons coupled
to up to three gravitons were provided in [14]. The technique for reformulating the
CHYintegrands in these papers developed into a systematic explanation in [26], and it is revealed
therein that the crossratio identity and other o shell identities of integrands [27, 28] are
crucial tools for deforming CHYintegrands into alternative forms corresponding to di
erent eld theories. These powerful tools bene t from the integration rule method [29{32]
developed for the purpose of evaluating CHYintegrand without referring to the scattering
equations. The idea of integration rule and crossratio identity method was to decompose
arbitrary CHYintegrand using crossratio identities into those corresponding to
cubicscalar Feynman diagrams dressed with kinematic factors. By carefully organizing terms
one can identify the resulting CHYintegrands as amplitudes of certain eld theories, hence
the amplitude relations, as was done in [14, 15]. In fact, there is more about EYM
amplitude relations from the perspective of CHYframework. Starting from CHYintegrand of
a theory, it is always possible to reformulate it to another form by crossratio and other
o shell relations, for instance the YangMillsscalar (YMs) amplitude can be expanded as
linear sum of biadjoint cubicscalar amplitudes. We shall discuss this later in this paper.
As it is very often, onshell technique can prove to be a powerful tool for the purpose
of understanding nontrivial amplitude relations within eld theory framework. One such
example is the onshell proof of BCJ relations [33, 34]. The central idea is to deduce physical
identities only from general principles such as locality, unitarity and gauge invariance.
This is particularly true with the advent of BrittoCachazoFengWitten(BCFW) onshell
recursion relation [35, 36], which utilizes the rst two. In most cases, the BCFW recursion
relation computes the amplitude in a way such that only contributions from
nite local
single poles are summed over, which requires a vanishing behavior in the boundary of
1Remark that in paper [20], a formula for single trace EYM amplitudes in four dimension for arbitrary
many gravitons is provided, although not mentioning the amplitude relations.
{ 2 {
BCFW complex parameter plane. This is exactly the case for BCJ relations of
YangMills amplitudes. However, for generic situations, the amplitude as a rational function of
BCFW parameter z is not vanishing in z ! 1 and the boundary contributions can not be
avoid. This is a problem one would meet when applying BCFW recursions to the EYM
amplitude relations, and such subtlety complicates the onshell proof of EYM amplitude
relations. The evaluation of boundary contributions is generically a di cult problem, but
many methods have been proposed to deal with it.
Noteworthily systematic algorithm has also been proposed recently [37{43] so that at least in principle it is indeed possible to systematically study the EYM amplitude relations using BCFW recursion relations. On the other hand it is also known that very often gauge invariance can become a very
handy tool in constraining the speci c analytic form of the scattering amplitude. Recent
progresses have pushed the gauge invariance principle forward and indicate that, the gauge
invariance along with cubic graph expansion are enough to determine the amplitudes [44{
49]. In a less but still quite challenging situation, we claim that the gauge invariance should
uniquely determine the EYM amplitude relations, and from which we can explicitly write
down the expansion for EYM amplitude with arbitrary number of gravitons.
As the number of gravitons increases and that of gluons decreases, in the extremal
limit we would come to the amplitude with pure gravitons. This is the important problem
of expanding gravity amplitude as pure YangMills amplitudes. Furthermore, with the
philosophy of decomposing CHYintegrands, the same argument applies to the YangMills
amplitudes which would be expanded as pure biadjoint cubicscalar amplitudes. This is
exactly the cubicgraph expansion of YangMills amplitude which makes the colorkinematics
duality manifest [26]. The EYM amplitude relation combined with CHYintegrand, more
speci cally the Pfa an expansion, would produce the nontrivial expansion for YangMills
amplitude as cubicscalar graphs, as well as expansion for gravity amplitude as pure
YangMills amplitudes and eventually the cubicscalar graphs. This provides a way of computing
the BCJ numerators, which is usually considered to be very di cult [46, 50{62]. When KLT
relation is in action, the EYM amplitude relation can be connected to the BCJ numerator
problem. We will learn more about this in later sections.
In this paper we examine the EYM amplitude relations from the perspectives of
CHYformulation, BCFW onshell recursion, KLT relation, and through the contruction of
BCJ numerators. This paper is organized as follows. In section 2, we present the general
theoretical playground of nontrivial amplitude relations from the CHYformulation, and
explain the expansion of amplitudes as the expansion of Pfa an of CHYintegrand. In
section 3, we facilitate the principle of gauge invariance to determine the EYM amplitude
relations for gluons coupled to arbitrary number of gravitons. In section 4, we generalize
the EYM amplitude relations to pure YangMills amplitudes and apply the nontrivial
relation to the computation of BCJ numerators. In section 5, we provide the onshell proof
of some EYM amplitude relations by BCFW recursion relations. In section 6 we study
in the language of KLT relations. Conclusion is presented in section 7 and some useful
backgrounds are summarized in the appendix.
{ 3 {
Amplitude relations from the perspective of CHYformulation
The nontrivial relation revealed recently between EYM amplitudes and pure YangMills
amplitudes [1, 14, 15] has an intuitive interpretation in the CHYframework. In fact, the
CHYformulation tells more beyond the EYM amplitudes. In the CHYformula, it is the
so called CHYintegrand ICHY that describes speci c eld theories. The CHYintegrand
is an uniform weight4 rational function of n complex variables zi for npoint scattering
system, i.e., with the 1=zi4 scaling behavior in the zi ! 1 limit.
For almost all known theories, the weight4 CHYintegrand can be factorized as two
weight2 ingredients, formally written as
ICHY = IL
IR :
Let us then de ne two new weight4 CHY integrands as follows where PT( ) is the ParkeTaylor factor
ILCHY( ) := IL
PT( ) ;
IRCHY( ) := IR
PT( ) ;
PT( ) :=
1
(z 1
z 2 )
(z n 1
z n )(z n
z 1 )
:
Supposing the two CHYintegrands ILCHY; IRCHY also describe certain physical meaningful
eld theories and produce the corresponding colorordered amplitudes AL( ); AR( )
after CHYevaluation, then by CHYconstruction [21{23] we could arrive at the following
generalized KLT relation,
where A is the amplitude of speci c eld theory determined by the theories of AL, AR,
while Sn denotes permutations on n elements and S[ je] is some kinematic kernel. The
summation is over Sn 3 permutations of sets f2; : : : ; n
2g, depending on our choice of
legs k1; kn 1; kn being
xed.
The expression (2.4) denotes a general expansion for the original amplitude A de ned
by CHYintegrand (2.1). If for a speci c
ordering, we sum over all Sn 3 permutations of
and de ne the result as
then the original amplitude can be expressed as
sum over all Sn 3 permutations of e
and de ne the summation as
where C(e) serves as the expansion coe cients. Similarly, if for a speci c
ordering we
C(e) :=
X
2Sn 3
then the original amplitude can be expanded as
2Sn 3
Ce( )AL(n
1; n; ; 1) :
(2.8)
The expressions (2.6) and (2.8) have provided two di erent expansions of the original
theory. There are several general remarks regarding the expansion in above,
Firstly, the expansion is into a chosen (n
3)! BCJ basis, and the corresponding
expansion coe cients C(e) and Ce( ) would also be unique. However, as we will
discuss soon, sometimes it is better to expand the original amplitude into the (n
2)!
KK basis. Because of the BCJ relations among colorordered partial amplitudes,
the expansion coe cients in the KK basis will not be unique and depend on the
generalized gauge choice in the BCJ sense.
Secondly, with the amplitude expansion formula in hand, the next is to compute
the expansion coe cients. For this purpose, there are several approaches. The rst
approach is to use the de nitions (2.5) and (2.7) directly. However, in general it
is very hard to evaluate the summation for generic npoint situation, and only in
certain special case a direct evaluation is possible, which we shall explain later. The
second approach seeds back to the expression (2.1), and the major idea is to expand
the weight2 ingredients IL or IR into the PT( ) factor of n elements. In fact, this is
the approach followed in [14, 15]. The expansion can be systematically achieved by
successively applying crossratio identities to the CHYintegrands, where in each step
a gauge choice should be taken in the crossratio identity. In general, such expansion
leads to a result with (n
1)! cyclic basis. Then one can use the KK relation to
rewrite it into the (n
2)! KK basis. As already mentioned, the gauge dependence
remains in the expansion coe cients at each step, and it would disappear only after
using the BCJ relations to rewrite all into (n
3)! BCJ basis.
Besides the above two direct evaluation methods for expansion coe cients, there
are also some indirect ways. For example, one can propose some ansatz for the
expansion coe cients, then prove and generalize it by onshell recursion relations.
One can also use some general considerations, for instance the gauge invariance or
the soft behavior, to determine the coe cients [47{49].
In this paper, we will investigate the expansion from these di erent views.
Thirdly, although in most theories, the CHYintegrand is given by products of two
weight2 ingredients as (2.1), for some theories the CHYintegrand is de ned by the
product of four weight1 ingredients. So there are various combinations of them to
form weight2 parts. In other words, there are possibilities to have more than two
expansions given in (2.6) and (2.8). It would be interesting to survey the consequence
of di erent combinations for these theories.
After above general discussions, now we focus on our major topic in this paper, i.e.,
the single trace part of EYM theory, whose CHYintegrand is de ned as
IrE;YsM( ) = PTr( )Pf s
Pf0 n ;
(2.9)
{ 5 {
for scattering system of r gluons and s gravitons with r + s = n, and
We can de ne two new CHYintegrands as
= f 1; : : : ; rg.
ILCHY( je) := PTr( )Pf s
PTn(e) ;
IRCHY( ) := Pf0 n
the amplitude AL is the colorordered YMs amplitude ArY;Mss with r scalars and s gluons,
which has two trace structures associated with the two PTfactors, while the amplitude
AR is colorordered YangMills amplitude AnYM. One thing to emphasize is that the scalar
carries two groups (one gauge group and one avor group) and has biadjoint scalarcubic
interactions.
An immediate consequence from (2.4) reads
2Sn 3
=
Ce( j )AnYM(n
The expansion (2.11) is into the BCJ basis with (n 3)! independent YangMills amplitudes.
However, as it will be clear soon, an expansion into (n
2)! KK basis is more favorable, and we would present it here as with the expansion coe cients
X
2Sn 2
ArE;YsM( ) =
Ce0( j )AnYM(n; ; 1) ;
Ce0( j ) = lim
1
X
kn2!0 kn2 e2Sn 2
S[ je]ArY;Mss( j1; e; n) :
The expansion coe cient in (2.13) is the desired quantity we want to compute in
this paper. As we have discussed in previous paragraph, these coe cients are determined
by only one weight2 ingredient in the CHYintegrand in (2.1). This means that while
keeping the same weight2 ingredient, we have the freedom to change the other
weight2 ingredient. As an implication of such modi cation, we could work out the expansion
for di erent eld theories but with the same expansion coe cients. This freedom could
simplify our investigation of expansion coe cients. For example, in the context of EYM
amplitude as YangMills amplitudes, we can change the Pf0 n in (2.9) as PTn( ). The
resulting CHYintegrand
IrY;sMs( j ) = PTr( )Pf s
PTn( )
(2.15)
{ 6 {
(2.11)
(2.12)
(2.13)
(2.14)
describes a YangMillsscalar amplitude with r scalars and s gluons, and the weight2
ingredients are now IL = PTr( )Pf
s and IR = PTn( ). With the same philosophy as
in (2.10), we can de ne two new CHYintegrands as
ILCHY := PTr( )Pf s
with the same expansion coe cients as in (2.12). This nontrivial relation expresses any
single trace colorordered amplitude of YangMillsscalar theory as linear combination of
colorordered amplitude of biadjoint scalar 3 theory.
After studying the expansion of single trace part of EYM theory to YM theory, we
will brie y discuss the expansion of gravity theory to YM theory. The CHYintegrand of
gravity theory is
If expanding the reduced Pfa an
IrG;s( ) = Pf0
n
Pf0
n :
X
2Sn 3
ArY;Mss( j ) =
Ce( j )An3 ( jn
by crossratio identities, we will get
X
2Sn 2
X
2Sn 2
Pf0
n =
n(1; ; n) PTn(1; ; n)
AG =
n(1; ; n) AYM(1; ; n)
by (2.18). As already pointed out in papers [23, 54, 57, 63{66], the coe cients n(1; ; n)
in the expansion (2.19)(hence also the one in the expansion (2.20)) is nothing but the
DDM basis for the BCJ numerator of YM amplitude. While in the expansion (2.8), i.e.,
1; n; ; 1), suppose we can rewrite the (n
3)! BCJ basis into
A = P
2Sn 3 e
C( )AL(n
(n
2)! KK basis, then identifying the resulting formula with the one given by (2.20), and
equaling the expansion coe cients of the same KK basis, we will get the BCJ numerator
n(1; ; n) as linear combination of Ce( ). Thus here we provided a new way of computing
the BCJ numerators via the computation of amplitude expansion (2.8). Although in (2.20)
we have taken gravity amplitude as example, the same consideration can be applied to
large number of theories, and the BCJ numerators of those theories can also be identi ed
as the expansion coe cients after rewriting (2.8) into KK basis.
With the above general framework for studying the nontrivial relations among di
erent theories, we will start our exploration from next section. As we will see, these relations
encode many surprising structures and connect many important topics, such as the
construction of BCJ numerators mentioned above, the boundary contribution of amplitudes
under BCFW onshell recursion relation, the DDM chain and BCJ relations, etc,.
{ 7 {
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
The gauge invariance determines the amplitude relations
A physical amplitude should be gauge invariant, i.e., vanishes on the condition i ! ki. If
considering the amplitudes with gravitons and expressing the graviton polarization states as
products of two YangMills polarization states i i , then it should also vanish by setting
one of the polarization vector as ki. The gauge invariance is an important property of
amplitude, and of course it is also valid in the amplitude relations. As we have mentioned
in previous section, there are di erent approaches to study expansion coe cients. In this
section, we will demonstrate how to use the gauge invariance to x coe cients, which is
the same spirit spelled out in [47{49].
3.1
With single graviton
To motivate our discussion, let us start with the single trace EYM amplitude with one
graviton with the known expansion
AEnY;1M(1; 2; : : : ; n; p) =
X( p Yp)AnY+M1(1; f2; : : : ; n
1
g
fpg; n) ;
(3.1)
e, i.e., all permutation sets of
[ e while
where the summation
is over all shu es
e
preserving the ordering of each ; . The colorordering of external legs in AnY+M1 has cyclic
invariance. However if we conventionally x the leg 1 in the rst position, then every
external leg could have a de nite position in the colorordering. In this convention, we can de ne
Yp(also Xp in the following paragraph) as the sum of momenta of all the gluons at the left
hand side (l.h.s.) of leg p, given the de nite colorordering of colorordered YM amplitudes.
A clari cation of the de nition Yp is needed here for the future usage. In the (n +
1)point pure YangMills amplitude, the gluon legs has two di erent correspondents in the
EYM amplitude, i.e., The gluon legs ki, i = 1; : : : ; n are also gluons in EYM amplitude while
gluon leg p is originally graviton leg in EYM amplitude. Yp is speci cally de ned as the sum
of momenta in the gluon subset of EYM amplitude at the l.h.s. of p, while we also de ne
another quantity Xp as the sum of all momenta at the l.h.s. of p no matter it is in the gluon
subset or graviton subset of EYM amplitude. Xp; Yp would be di erent when considering EYM
amplitude with more than one gravitons, but in the current discussion they are the same.
Let us now return to the relation (3.1). In the l.h.s., imposing any gauge condition
i ! ki for gluon legs would vanish the EYM amplitude, while any YangMills amplitudes
in the right hand side (r.h.s.) also vanish under the same gauge condition. For the graviton
polarization p p , setting either p ! p would vanish the EYM amplitude. In the r.h.s.,
the graviton polarization is distributed in two places: one is in the YangMills amplitude
and the other, in the expansion coe cients. The vanish of r.h.s. for the former case is
trivial, while for the latter case, it vanishes due to the fundamental BCJ relations
X(p Yp)AnY+M1(1; f2; : : : ; n
1
g
fpg; n) = 0 :
(3.2)
This consequence is rather interesting. For the nontrivial relation (3.1) to be true and the
gauge invariance be not violated, we eventually end up with BCJ relations. On the other
{ 8 {
Ansatz 1: the single trace EYM amplitude AEnY;mM with m gravitons can always be
expanded into EYM amplitudes AEnY+Mm m0;m0 with m0 < m gravitons.
Ansatz 2: when an EYM amplitude AEnY;mM is expanded into pure YangMills
amplitudes, the terms whose expansion coe cients contains only
k but no
takes the
form,2
X
Xh m )AnY+Mm(1; f1; 2; : : : ; ng
f h1 ; : : : ; hm g; n) :
hand, if we assume AEnY;1M can be expanded as linear combination of YangMills amplitudes
in the KK basis for convenience, and the expansion coe cients should be certain sum of
p ki to compensate the extra graviton polarization,
X
2Sn 1
AEnY;1M(1; : : : ; n; p) =
( p xp)AnY+M1(1; 2; : : : ; n 1; p; n) ;
(3.3)
then P (p xp)AnY+M1(1; 2; : : : ; n 1; p; n) should be in the BCJ relation form!
The lesson learned from the EYM amplitude with one graviton suggests that, while
expressing EYM amplitudes as linear combination of YangMills amplitudes, the gauge
invariance strongly constraints the possible form of expansion coe cients. This property
motivates us to nd the expansion of the single trace multigraviton EYM amplitude with
more than one graviton
gauge condition hi ! hi.
AEnY;mM(1; : : : ; n; h1; : : : ; hm) ;
by gauge invariance, i.e., we want coe cients to make the expression to zero under each
In order to construct the nontrivial relations for EYM amplitudes with generic number
of gravitons, we start with the following two ansatz,
These two ansatz come from lowerpoint known results. The rst ansatz is in fact a general
statement saying that a recursive construction for EYM amplitude expansion exists. While
the second ansatz is presented in an explicit expression which has an obvious BCJlike
relation form. The validation of ansatz 2 in fact can be veri ed by BCFW recursions. In the
expression (3.5), we emphasize again that Xhi is de ned to be the sum of all momenta in the
l.h.s. of leg hi, no matter those legs representing gluons or gravitons in the EYM amplitude.
Bearing in mind that any EYM amplitude expansion relations should follow the above
mentioned two ansatz, we are now ready to extend the introductory one graviton example
to EYM amplitudes with arbitrary number of gravitons. However, before presenting the
general algorithm, let us familiarize ourselves by studying EYM amplitudes with two and
three gravitons.
2We have taken the convention that, for an EYM amplitude we choice the sign of these terms to be (+).
It would be possible that for results in other conventions, for instance the CHY results in recent literatures,
there could be a sign di erence.
{ 9 {
(3.4)
(3.5)
Expressing ngluon twograviton EYM amplitudes as YangMills
amplitudes
The algorithm for producing general EYM amplitude relations is to expand AEnY;mM as
AEnY+M1;m 1 successively until AEnY+Mm;0
AYmM+n. Note that the gravitons are colorless, and
it has no colorordering in EYM amplitudes. But in our construction of EYM amplitude
relation by gauge invariance principle, it is necessary to specify a graviton in each step of
expansion AEnY;mM ! AEnY+M1;m 1, which in the AEnY;mM amplitude it denotes a graviton but in the
AEnY+M1;m 1 amplitude it denotes a gluon, such that we can apply gauge invariance principle
with that graviton. Furthermore, it requires us to select one arbitrary graviton to start with.
Now let us outline the arguments that lead to the correct expansion of EYM amplitude
with two gravitons AEnY;2M(1; 2; : : : ; n; p; q). In the rst step, let us specify the graviton hp,
and following the Ansatz 1 let us propose the following terms that would contribute to
the expansion of AEnY;2M,
T1 =
In fact, these proposed terms (3.6) are reminiscent of the expression (3.1) for expanding
the single trace EYM amplitude with one graviton. This is not yet the complete expansion
expression for AEnY;2M, but we will explain soon after how to recover the remaining terms.
Let us investigate the gauge invariance of gravitons hp and hq for the proposed terms (3.6).
The gauge invariance for hq is obvious since AEnY+M1;1(
; q) vanishes under q ! q gauge
condition. However, T1 is not gauge invariant under p ! p due to the expansion coe cients
p Yp, and there are some missing terms in order to produce the complete expansion for
AEnY;2M. Let us proceed to expand the AEnY+M1;1 in T1 with the known result (3.1), which gives
T1 =
X( p Yp) X( q Xq)AnY+M2(1; (f2; : : : ; n
1
g
1 fpg)
2 fqg; n) :
(3.7)
1
2
Note that in the permutation shu e f2; : : : ; n
1
g
p
f g
fqg, the position of leg q would
be either at the l.h.s. of p or r.h.s. of p. But the leg p denotes a graviton in AEnY;2M. So the
expansion coe cient is q Xq but not q Yq(remind that Xq is the sum of all momenta in
the l.h.s. of q, while Yq is the sum of all momenta in the l.h.s. of q excluded the leg p, which
means that if p is at the r.h.s. of q, Xq = Yq, but if p is at the l.h.s. of q, Xq = Yq + p).
From the Ansatz 2 (3.5), we know that in the AEnY;2M expansion, the correct terms with
coe cients ( q
)( q
) must be
X
X( p Xp)( q Xq)AnY+M2(1; f2; : : : ; n
1
g
1 fpg
2 fqg; n) :
(3.8)
1
2
Comparing T1 (3.7) with the correct result (3.8), it is easy to see that for those terms
with p in the l.h.s. of q, Yp = Xp so that the corresponding terms in T1 and (3.8) are the
same. While for those terms with p in the r.h.s. of q, we have Yp + q = Xp. So in order to
reproduce the correct result (3.8), we should add another contribution
T2 =
X( p q)( q Xq)AnY+M2(1; f2; : : : ; n
1
g
HJEP09(217)
such that
T1 + T2 =
is exactly equivalent to the correct result (3.8). Remind that Ansatz 2 gives correct
answer for contributing terms without (
) coe cients for EYM amplitude expansion,
and up to now, we have reformulated the correct result as (3.10) which guarantees an easy
generalization.
the form
Of course, (3.10) is still not yet the complete expansion for AEnY;2M, since those terms
with coe cient ( p q) are still missing. Let us propose that the complete expansion takes
HJEP09(217)
AEnY;2M(1; 2; : : : ; n; p; q) = T1 + T2 + ( p
q)T3 ;
and the remaining task is to determine T3. It can be determined either by gauge condition
p ! p or by q ! q, however the latter is much more convenient since T1 is already
manifestly gauge invariant for leg q. Setting q ! q, we have
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
0 =
T1 +T2 +( p q)T3
= ( p q)
q!q
which has a solution
T3 =
X(q Xq)AnY+M2(1; f2; : : : ; n
1
g
Hence we get the nontrivial relation for EYM amplitude with two gravitons as
X(q Xq)AnY+M2(1; f2; : : : ; n 1
g fq; pg; n)+T3 ;
+ X( p q)( q Xq)AnY+M2(1; f2; : : : ; n
X( p
q)(q Xq)AnY+M2(1; f2; : : : ; n
1
g
1
g
the above EYM amplitude expansion can be reformulated in a more compact form as
From expression (3.16) we can infer some important features. Firstly, for terms in
the rst line, leg p denotes a gluon and leg q denotes a graviton, while for terms in the
second line, leg q denotes gluon instead of graviton. This di erence leads to the di erence
of expansion coe cient, such that the Yp factor in the rst line has been replaced by the
factor Fq Xq. Or we can say a factor Fq is inserted. As we would see shortly after, this is
a general pattern for EYM amplitudes involving more gravitons.
Secondly, in the expression (3.16), the gauge invariance for leg q is manifest, while
gauge invariance for leg p is not manifest and requires further checking. Although it can
be checked directly, we will follow another approach. Note that the whole result should
be symmetric under switching p $ q. For the terms with kinematic factors (
k)(
k),
this symmetry is manifest since it is given by (3.8). For the terms with kinematic factors
( p
q), the result is not manifestly symmetric. In order to shown the symmetry, we need
to use the generalized BCJ relation. Let us divide the set f2; : : : ; n
subsets
= fa1; : : : ; mg and
= f 1; : : : ; tg such that m + t = n
1g into two ordered
2, then the general
BCJ relation is given by [12, 34]
t
i=1
X
X k i X i
!
A(1;
; n) = 0 ;
where the rst summation is over all shu es, and X i is the summation of all momenta of
legs at the l.h.s. of leg i. Using (3.17) with
= fq; pg it is easy to see that
( q
p) X(q Xq)A(1; f2; : : : ; n
1
g
Next, we use the general BCJ relation (3.17) with the choice
BCJ relation) to reach
= fpg (i.e., the fundamental
( q
p) X(p Xp)A(1; f2; : : : ; n
1
g
fp; qg; n) :
Hence the symmetry of legs q; p for the terms with factor ( p q) is apparent. Since the
gauge invariance for leg q is satis ed, by the symmetry, the gauge invariance for leg p is
also satis ed.
The above discussion allows a systematical generalization to EYM amplitude with any
number of gravitons. Before doing so, let us introduce a new quantity which would be
useful in later discussions. Assuming the gravitons have been split into two subsets ; ,
where
is the ordered lengthm1 set in the gluon side and
is a lengthm2 set in the
graviton side whose ordering is not relevant, we de ne
(3.17)
(3.18)
(3.19)
T [ j ]=X
= fqg, we have m1 = m2 = 1, and
i k i+1 ( m1 Y m1 )AEnY+Mm1;m2 (1;f2;:::;n 1
g f m1 ;:::; 1g;n; ): (3.20)
T [fpgjfqg] =
which is in fact the rst line of (3.14). While if
= fq; pg;
= ;, m1 = 2; m2 = 0, and we
have
T [fq; pgj;] =
X( p q)( q Yq)AnY+M2(1; f2; : : : ; n
1
g
which reproduces the second line of (3.14).
Expressing ngluon threegraviton EYM amplitudes as YangMills
amplitudes
Now let us explore the details by the EYM amplitude with three gravitons
AEnY;3M(1; : : : ; n; p; q; r). Our purpose is to construct a recursive algorithm for EYM
amplitude expansion which is manifestly gauge invariant in each step for gravitons (except
the initial one), and the terms without ( hi
hj ) factor matches the Ansatz 2 (3.5). In
the current case it is
X
Again, the starting point is specifying an arbitrary graviton for expanding AEnY;3M ! AEnY+M1;2
and without lose of generality we choice p. The proposed contributing terms are
T [fpgjfq; rg] =
X( p Yp)AEnY+M1;2(1; f2; : : : ; n
1
g
fpg; n; q; r) :
(3.24)
Note that q; r are manifestly gauge invariant in T [fpgjfq; rg], and legs q; r denote gravitons.
To match the correct result (3.23), we need to add terms where leg q or r is at the
l.h.s. of p. This means that we need to add terms AEnY+M2;1 where leg q or leg r now denotes
gluon and its position is at the l.h.s. of leg p. For AEnY+M2;1 terms where leg p; q are gluons
but leg r is graviton, the added term should be
T [fq; pgjfrg] =
X( p q)( q Yq)AEnY+M2;1(1; f2; : : : ; n
1
g
fq; pg; n; r) :
(3.25)
These terms introduce the missing terms for T [fpgjfq; rg] in order to match the
result (3.23), however the gauge invariance for q is still broken. In order to keep gauge
invariance for legs q; r at every step, we should further modify (3.25) by adding terms with ( p q)
coe cients, and the resulting terms should not alter the matching with result (3.23). From
experiences gained in the previous subsection, we can propose the following modi cation
G[fq; pgjfrg] =
X( p q)( q Yq)AEnY+M2;1(1; f2; : : : ; n
1
g
fq; pg; n; r)
X( p
q)(q Yq)AEnY+M2;1(1; f2; : : : ; n
1
g
fq; pg; n; r)
=
X( p Fq Yq)AEnY+M2;1(1; f2; : : : ; n
1
g
fq; pg; n; r) ;
(3.26)
which are manifestly gauge invariant for q; r.
Similarly, for AEnY+M2;1 terms where legs p; r are gluons but leg q is graviton, the proposed
gauge invariant term should be
G[fr; pgjfqg] =
X( p Fr Yr)AEnY+M2;1(1; f2; : : : ; n
1
g
fr; pg; n; q) :
(3.27)
Emphasize again that the above proposals are based on the gauge invariant principle, the
Ansatz 1 and the Ansatz 2.
Notice that in (3.26) and (3.27), we have ( q Yq) or ( r Yr) instead of ( q;r Xq;r). So
in order to arrive at a complete matching with result (3.23), we should further add AEnY+M3;0
terms where all p; q; r are gluon legs. For G[fq; pgjfrg], the Ansatz 1 guides us to propose
additional terms as
However, these terms are not gauge invariant for leg r, and according to gauge invariance
principle we need to modify (3.28) as
G[fr; q; pgj;] =
X( p Fq r)( r Yr)AnY+M3(1; f2; : : : ; n
1
g
X( p Fq Fr Yr)AnY+M3(1; f2; : : : ; n
1
g
fr; q; pg; n) ;
(3.29)
= X( p Yp)AEnY+M1;2(1;f2;:::;n 1
g fpg;n;q;r)
+X( p Fq Yq)AEnY+M2;1(1;f2;:::;n 1
g fq;pg;n;r)
+X( p Fr Yr)AEnY+M2;1(1;f2;:::;n 1
g fr;pg;n;q)
+X( p Fq Fr Yr)AnY+M3(1;f2;:::;n 1
g fr;q;pg;n)
+X( p Fr Fq Yq)AnY+M3(1;f2;:::;n 1
g fq;r;pg;n):
which reproduces the correct result (3.23) yet is gauge invariant manifestly. The ( p Fq r)
coe cient in the second line of (3.29) is
( p Fq
r) = ( p q)( q
r)
( p
q)(q
r) ;
so we can see clearly that, the second line of (3.29) only introduces terms with ( hi
hj )
factor which will not contribute to the (3.23) terms.
Similarly, for G[fr; pgjfqg], we need to add the following gauge invariant terms
G[fq; r; pgj;] =
X( p Fr Fq Yq)AnY+M3(1; f2; : : : ; n
1
g
Summarizing all parts together, we have
AEnY;3M(1;:::;n;p;q;r) = T [fpgjfq;rg]+G[fq;pgjfrg]+G[fr;pgjfqg]+G[fr;q;pgj;]+G[fq;r;pgj;]
(3.30)
(3.31)
(3.32)
Expression (3.32) has demonstrated the recursive construction pattern more clearly, i.e.,
expanding the EYM amplitude successively and keep the gauge invariance in each step by
introducing additional terms. The starting point is to specify an arbitrary graviton and
propose the contributing terms T [fpgjfq; rg], which are terms of AEnY+M1;2. It reproduces
a part of the correct result (3.23) from Ansatz 2, and another part would come from
AEnY+M2;1 terms. Specifying graviton q, we can propose the contributing terms G[fq; pgjfrg],
deduced from gauge invariance principle, Ansatz 1 and the matching of Ansatz 2. While
specifying graviton r, we can propose G[fr; pgjfqg]. The remaining part could be proposed
by specifying the last graviton, which gives G[fr; q; pgj;], G[fq; r; pgj;]. The correctness
of terms without ( hi
hj ) is guaranteed by construction, while the terms with ( hi
hj )
factor are determined by gauge invariance in each step. The gauge invariance for q; r is then
manifest at each term, except for the leg p. It is also easy to see that, in each step when leg
hi in the amplitude denotes a gluon while in the previous step it denotes a graviton, the
corresponding gauge invariant term is just to insert a Fhi into the kinematic factor in an
appropriate position. It corresponds to replacing khi hi as Fhi . So similar to the de nition
of T [ j ] in (3.20), we can de ne a new quantity
X( 1 F 2 F 3
F m1 Y m1 )AEnY+Mm1;m2 (1; f2; : : : ; n 1
g
f m1 ; : : : ; 1g; n; ) :
Note that when m1 = 1, T [ j ] = G[ j ].
Before presenting the algorithm for general EYM amplitude relations, we give a remark
on the gauge invariance of p. It is not apparent, but one can show the full S3 symmetry
among three gravitons after using various BCJ relations. Hence the gauge invariance of
leg p should be indicated by the symmetry.
3.4
A constructive algorithm for producing general EYM amplitude relations
The basic idea of constructive algorithm for producing general EYM amplitude relations
AEnY;mM is to write down the contributing terms AEnY+M1;m 1; AEnY+M2;m 2; : : : ; AEnY+Mm;0 successively,
and the explicit expression corresponding to AEYM
n+m0;m m0 relies on AEnY+Mm0 1;m
m0+1
recursively. Brie y speaking, provided we have written down the contribution of AEnYM
m2;m2 ,
where the gravitons in this EYM amplitude are labeled as
specifying a graviton, say h i ,
= f 1; : : : ; m2 g. Then by
we can directly write down a gauge invariant contributing term AEnYMm2+1;m2 1 as G[f ig [
j =f ig], whose coe cients are obtained by replacing Y m1 ! F i Y i in the coe cients
G[fh1gjfh2; : : : ; hmg] :
(3.34)
(3.35)
of AEnYM
m2;m2 .
amplitude with m gravitons
Now let us describe the algorithm for generic EYM amplitude relations. For the EYM
AEnY;mM(1; 2; : : : ; n; h1; h2; : : : ; hm) ;
Step 1: specify arbitrary one graviton, say h1, and record the contribution
Step 2: from the previous step, specify one graviton h02 2 fh2; : : : ; hmg, and record
the corresponding contribution
G[fh02; h1gjfh2; : : : ; hmg=fh02g] ;
h02 2 fh2; : : : ; hmg :
(3.36)
Step 3: for each G in the previous step, specify one graviton h03 2 fh2; : : : ; hmg=fh02g
and record the corresponding contribution
G[fh03;h02;h1gjfh2;:::;hmg=fh02;h03g] ; h022fh2;:::;hmg ; h032fh2;:::;hmg=fh02g: (3.37)
HJEP09(217)
Step m:
for each G in the previous step, specify one graviton h0m
=
fh2; : : : ; hmg=fh02; : : : ; h0m 1g and record the contribution
G[fh0m; h0m 1; : : : ; h02; h1gj;] ; h0i 2 fh2; : : : ; hmg=fh02; : : : ; h0i 1g for i = 2; : : : ; m :
Summing over all the recorded contributions, we get the relation for generic EYM amplitude
expansion as
AEnY;mM(1; 2; : : : ; n; h1; h2; : : : ; hm) =
where H is a subset of fh2; : : : ; hmg, and jHj is the length of set H. Explicitly writing
down, we have
(3.38)
(3.39)
(3.40)
Since G[ j ] is wellde ned in (3.33), the explicit expression for (3.39) can be readily written
down. Note that relation (3.39) expands an EYM amplitude with m gravitons as linear
combination of EYM amplitudes with m0 < m gravitons and YangMills amplitudes. In
this expression (3.40), the gauge invariance is manifest for (m
1) gravitons fh2; : : : ; hmg,
since by construction, each contributing term G[ j ] that building up the expansion relation
is gauge invariant for fh2; : : : ; hmg. For the leg h1, the gauge invariance is not manifest.
However, as argued in [47{49], for npoint YangMills amplitudes, manifest gauge invariance
for (n
1) points will be enough to guarantee the correctness of the result, so the gauge
= G[fh1gjfh2; : : : ; hmg] +
X
X
h022fh2;:::;hmg
+
+
+
+
X
2Sm 1
X
fh02;:::;h0kg fh2;:::;hmg 2Sk 1
G[f h02 ; : : : ; h0m ; h1gj;] :
G[fh02; h1gjfh2; : : : ; hmg=fh02g]
G[f h02 ; : : : ; h0k ; h1gjfh2; : : : ; hmg=fh02; : : : ; h0kg]
[fh1gjfh2;h3;h4g]!
< [fh3;h1gjfh2;h4g] !
<
:
invariance of the nth point. We believe the same conclusion can be made for EYM theory
by similar argument. If we buy this argument, result (3.39) must be the right expression.
As a demonstration, let us brie y present the nontrivial relations for EYM
amplitude with four gravitons AEnY;4M(1; : : : ; n; h1; h2; h3; h4). The contributions in each step are
abbreviated as follows,
+
+
2S3
The rst vertical line corresponds to the contributions of AEnY+M1;3, where we have speci ed
leg h1 as the gluon leg in AEnY+M1;3 amplitude. The second vertical line corresponds to the
contributions of AEnY+M2;2, and seen from the rst vertical line, we can specify either h2; h3 or
h4 as gluon leg in AEnY+M2;2 amplitudes. Thus we get three contributions. The third vertical
line corresponds to the contributions of AEnY+M3;1, deduced from the second vertical line by
specifying a graviton leg in AEnY+M2;2 as gluon leg in AEnY+M3;1. And so arrives at the fourth
vertical line. Each one in the above table corresponds to a gauge invariant term G de ned
in (3.33), and summing over all contributions we get the expansion for EYM amplitude
with four gravitons,
AEnY;4M(1;:::;n;h1;h2;h3;h4)
= X( h1 Yh1)AEnY+M1;3(1;f2;:::;n 1
g fh1g;n;h2;h3;h4)
(3.41)
X( h1 Fhi Yhi)AEnY+M2;2(1;f2;:::;n 1
g fhi;h1g;n;fh2;h3;h4g=fhig)
X( h1 F hi F hj Y hj )AEnY+M3;1(1;f2;:::;n 1
g f hj ; hi;h1g;n;fh2;h3;h4g=fhi;hjg)
+ X X( h1 F h2 F h3 F h4 Y h4 )AnY+M4(1;f2;:::;n 1
g f h2; h3; h4;h1g;n):
3.5
Expanding to pure YangMills amplitudes: ordered splitting formula
The recursive construction given in (3.39) is easy to implement and one can eventually
get an expansion with pure YangMills amplitudes. In this subsection, we will present the
related discussion.
To familiarize ourselves with this problem, let us start with some examples. The rst
example is the one with two gravitons. After substituting (3.1) into the rst term of (3.16),
we get
X
1; 2
( p Yp)( q Xq)AnY+M2(1; f2; : : : ; n
1
g
1 fpg
2 fqg; n)
where it is crucial to use Xq instead of Yq in the rst term of the expansion, since to the leg
q, leg p is actually a gluon. Although the expression (3.42) is very suggestive, the pattern
is still not clear, so we go ahead to the examples with three gravitons (3.32). Doing similar
manipulations, we arrive at
AEnY;3M(1; : : : ; n; p; q; r)
(3.43)
f g
p
f g
q
f g
g
1
g
Some explanations for (3.43) are in order. Firstly, when expanding AEnY+M1;2(1; f2; : : : ; n 1
g
fpg; n; q; r) we need to specify a graviton leg which would be the gluon leg in AEnY+M2;1, and
our choice is leg q. Secondly, we have de ned a new notation Zhi . To de ne Zhi , we shall
introduce a new concept, i.e., the ordered splitting of m elements. To de ne the ordered
splitting, we must rst de ne an ordering of m elements, for example, h1
h2
hm
(we will call it ordered gauge ). Once the ordered gauge is xed, the ordered splitting is then
de ned by the following ordered set of subsets f 1; : : : ; tg satisfying following conditions,
Each subset i
fh1; : : : ; hmg is ordered,
Join[f 1; : : : tg] = fh1; : : : ; hmg,
R 1
R 2
subset i in the set fa1; : : : ; atg),
Denoting R i as the last element of the ordered subset i (or named the pivot), then
R t according to the ordered gauge(it de nes the ordering of
In each subset, all other elements must be larger than R i according to the ordered
gauge. However, there is no ordering requirement for all other elements.
To better understand the de nition of ordered splitting, we take the set fp; q; rg with
ordered gauge p
q
r as an example to write down all ordered splitting,
With only one subset, we can have two cases: fr; q; pg and fq; r; pg,
With two subsets, we can have three cases:
ffpg; fr; qgg, ffr; pg; fqgg and
ffq; pg; frgg,
With three subsets, we have only one case ffpg; fqg; frgg.
Now let us de ne the notation Zhi . It is easy to notice that, the six lines in (3.43) are
onetoone mapped to the six ordered splitting of fp; q; rg with ordered gauge p
q
The Zhi is the sum of momenta of legs satisfying the following two conditions: (1) legs at
the l.h.s. of the leg hi in the colorordered YangMills amplitudes, (2) legs at the l.h.s. of the
label chain de ned by the ordered splitting. The label chain for a given ordered splitting
is the ordered set f1; 2; : : : ; n
1; 1; : : : ; t; ng. For instance, for the ordered splitting
ffpg; fqg; frgg in the rst line of (3.43), the label chain is f1; 2; : : : ; n
for ffpg; fr; qgg in the second line of (3.43), the label chain is f1; 2; : : : ; n
1; p; q; r; ng, and
1; p; r; q; ng.
With the understanding of Zhi , it is easy to see that all Yhi appearing in (3.43) is equal
to Zhi , so we can rewrite (3.43) as
AEnY;3M(1; : : : ; n; p; q; r)
(3.44)
=
X( p Zp)( q Zq)( r Zr)AnY+M3(1; f2; : : : ; n
1
g
+ X( p Zp)( q Fr Zr)AnY+M3(1; f2; : : : ; n
+ X( p Fq Zq)( r Zr)AnY+M3(1; f2; : : : ; n
+ X( p Fr Zr)( q Zq)AnY+M3(1; f2; : : : ; n
+ X( p Fq Fr Zr)AnY+M3(1; f2; : : : ; n
+ X( p Fr Fq Zq)AnY+M3(1; f2; : : : ; n
1
g
1
g
1
g
p
f g
p
f g
f g
g
1
g
result proposed in [14].
arbitrary number of gravitons,
We have numerically checked the above relation, by comparing with AEnY;3M directly evaluated
with the CHY de nition and found agreements in the lowerpoint examples up to A3E;Y3M.
In addition, when expanding the amplitude AEnY;3M(1; : : : ; n; p; q; r) into terms of pure
YangMills ones by (3.44), (3.42) and (3.1) and considering BCJ relations, we obtain the same
With the above result (3.44), it is ready to outline the rule for generalizing (3.43) for
Decide an ordered gauge a priori, and write down all possible ordered splitting.
For each ordered splitting f 1; 2; : : : ; tg, write down a factor ( e1 Fe2
Fej ij
Zej ij ) for each subset i = fej ij
; : : : ; e2; e1g, and product the factors for all
0 s in the ordered splitting. This is the desired coe cients for the colorordered
i
amplitudes with colorordering de ned by the corresponding ordered splitting.
Fej ij 1
we reexpress the boundary term in the r.h.s. of (5.2) as
X( p Yp)f a0na01e i
s1n
+ X( p Fq Yq)f a0na01e i
s1n
AenY;M1s(2; f3; : : : ; n
1
g
fpg; P1e;n; q)
3
Aen+1(2; f3; : : : ; n
1
g
fq; pg; P1e;n) :
(5.17)
f a0na01e s1in AeYn M1s;2(2; : : : ; n
the l.h.s. of (5.2).
Remind that our proof of (5.2) does not rely on the onshell condition of the rightmost
scalar kn, hence (5.2) is also valid for amplitudes with o shell kn. Assuming the
validation of (5.2) for YMs amplitude with n0 < n gluons, we simply get the sum (5.17) as
1; P1e;n; p; q), which is identical to the boundary contribution in
The case with n = 2: this case is much more subtle. The boundary contributions
in the l.h.s. of (5.2) come from the diagrams as shown in
gure 1.b, gure 1.c, while the
boundary contributions in the r.h.s. of (5.2) come from the diagrams as shown in gure 1.d,
gure 1.e, gure 1.f.
According to the Feynman rules for YangMillsscalar amplitudes, we can compute the
three terms for the r.h.s. of (5.2) as
On the other hand, we can compute the two terms for l.h.s. of (5.2) as
If we rewrite the second line in the result of gure 1.b by Jacobi identity
f a02a01ef ea0pa0q = f a01a0pef ea0qa02
f a01a0qef ea0pa02 ;
then the matching of boundary contribution in both sides of (5.2) can be easily checked.
With above discussions, we have con rmed the nontrivial relations between YMs
amplitude and pure scalar amplitudes (hence the EYM amplitude and YangMills amplitudes)
by BCFW recursion relations. The proof of relations for YMs amplitude with more than
two gluons requires more labors, but the strategy is similar, which includes comparing the
contributions from
nite poles and boundary contributions. We will not discuss it further.
Inspecting the amplitude relations through KLT relation
In the following discussions we will demonstrate that, at least in the rst few simplest
scenarios, the newly discovered multigraviton relations [1, 14, 15] can be readily understood
from the perspective of KLT relations. It was demonstrated in [72] that the KLT relation
provides a much more perturbationfriendly construction of the EYM amplitudes, which
would be otherwise di cult to calculate in viewing of the in nite vertices that constitute
the linearized gravity Feynman rules. In this setting, EYM amplitude factorizes into a
copy of pure gluon amplitude and a copy that gluon interacts with scalars, through which
the color dependence is introduced. To have simpler expression, we will use the (n
2)!
symmetric KLT relation rst introduced in [73{75],
AEYM(1; 2; : : : ; n) = lim
AYM(n; ; 1)S[ j ]AYMs(1; ; n)
(6.1)
1
X
2Sn 2
AYM(1; ; n) n(1; ; n) :
where the numerator in the expression de ned using gluon scalar currents
X
2Sn 2
n(1; ; n) =
S[ j ]J YMs(1; ; n)
(6.2)
carries both kinematic and color factors. The formula de ned in (6.2) has provided a way
of evaluating the numerator n(1; ; n). However, it is obvious that, directly calculating all
currents and then making the sum is not an e cient method. There are two alternative
methods to compute the coe cients n(1; ; n). The rst is to carry out the summation
step by step as was done in [61, 71]. The idea is to divide the full Sn 2 permutation sums
appearing in (6.2) into (n
2) blocks of Sn 3 permutation sums, such that in each block
we can pull out a format of BCJ sums. Then one can use the Fundamental BCJ relation
for currents to simplify the expression and arrive at a similar sum as the one given in (6.2)
but with only Sn 3 permutation sums. Iterating this procedure several times, we can
nally compute the coe cients. Establishing the Fundamental BCJ relation for currents is
a crucial point for this method, and we will show how to do this in the later sections. The
second method is, however, less straightforward. When expanding the amplitude into KK
basis with the formulation given in the second line of (6.1), it is shown in [23, 54, 57, 66]
that, the coe cients n(1; ; n) are nothing but the numerators of Del DucaDixonMaltoni
(DDM) basis provided we write the whole AYMs(1; ; n) amplitude into BCJ form (i.e.,
numerators satisfying the Jacobi relations). Using this aspect, the problem is translated
to computing the BCJ numerators of DDM basis by any conventional methods.
The purpose of this section is to show that, the newly discovered EYM amplitude
relations can also be tted in the framework of KLT relations. The methods that developed
in the computation of BCJ numerators in various theories [61, 71] are also wellsuited in
the analysis of EYM amplitude expansion, with only a few modi cation. This connects the
problem of EYM amplitude expansion with many other theories. In the following
discussions, we will use both methods developed years ago for computing the BCJ numerators
to address the problem of constructing the expansion coe cients n(1; ; n).
6.1
For the purpose of being selfcontained we list the colorordered Feynman rules for
gluonscalar interaction presented in [72] in the appendix A. Consider rst the scalar YangMills
amplitudes when there is only one gluon. Note that a (colorstripped) gluon propagator
does not transmit the color/ avor of scalars attached to its two ends, so that for single trace
part of the partial amplitude, gluon lines cannot be internal or the color factors carried by
the scalars at its two ends factorize. A consequence is that all single gluon amplitudes are
consisting of cubic graphs. For example at four points when, say leg 3, is the gluon line there
are only three cubic graphs in the KK sector, up to antisymmetry of the threevertices,
It is very important to notice that the colorkinematics duality is ensured by the vanishing
of the sum of their numerators (even at the o shell level)
f a1a2a4 (k1 + k2 + k4)
3
k3
3 = 0 :
(6.3)
This observation (i.e., only cubic vertex is allowed and the gauge invariance), when
generalized to higher points, indicates that the Feynman diagrams provide the desired BCJ
form. In particular, an npoint DDM halfladder numerator n(1; 2; 3;
i; pg; i + 1;
; n)
is therefore given by the corresponding Feynman diagrams as
k122k1223
= hf 1;2; f ;3; : : : f ;i;a
ab
f b;i+1;
f ;n 1;ni
p n 2
2
( 1)n+1 i p (k1 +
+ ki) :
(6.4)
So what are the allowed DDM numerators? The YangMillsscalar theory has a gauge
group and a avor group. The avor ordering xes the ordering of scalars, thus the only
allowed freedom is the location of gluon leg along the DDMchain. In other words, the
desired expansion coe cients in (6.1) are nothing but the one given in (6.4) with all
possible insertions of gluon legs,
AEnY;1M(1; 2;
; n; p) = p2n 2
( 1)n+1 i X( p Xp)AYM(1; f2; : : : ; n 1
g fpg; n) ;
(6.5)
and we nd agreement with the new single graviton relation (up to an overall factor).
6.2
The fourpoint gluonscalar amplitude involving two gluons
Next we consider YangMillsscalar amplitudes involving two gluons. For this case, since
Feynman diagrams will involve the fourpoint vertex, the BCJ form will not be manifest for
Feynman diagrams and the computation will be more complicated. Thus in this subsection,
we will follow the method of summing over the colorordered KK basis.
At four points we have the following two KK basis amplitudes,
+
+
+
+
a1a4
p
2
i 2
i;
A2Y;M2 s(1; 2g; 3g; 4) =
ns
s12
s23
nt + n4 =
A2Y;M2 s(1; 3g; 2g; 4) =
nu +
s13
s23
nt + n4 =
where ns, nt, nu and n4 denote the factors
ns = k122
nt = k223
nu = ( 1)k123
and
= ((k3 +k4) k1) 2
(k4 (k1 +k2)) 3
=
2 3 (k3 k2) (k4 k1) + (k4 k1) 3 ((k1 +k4) k3) 2
a1a4
i
p
2
= ( 1) ((k2 +k4) k1) 3 (k4 (k1 +k3)) 2
p
i
2
a1a4
i;
p
i 2
2
i;
n4 =
=
2
i a1a4 ( 2
3) :
Now that with quartic graph present, the original Jacobi identity inevitably needs to be
modi ed if colorkinematics duality is to remain holding. To better keep track of how this
is done we write the BCJ sum of the two KK basis amplitudes in terms of the factors just
introduced so that every term appears in the sum has a clear graphical interpretation. Also
for future reference we analytically continue one of the scalar legs, say leg 4, and write
s21A2Y;M2 s(1; 2g; 3g; 4) + (s21 + s23)A2Y;M2 s(1; 3g; 2g; 4)
= s12
ns
s12
s23
nt + n4
= (ns + nt + nu + (s12
+ (s21 + s23)
s13)n4) + k42
nu +
s13
s23
nt + n4
s13
nu + n4 :
;
(6.6)
;
(6.7)
(6.8)
(6.9)
(6.10)
(6.11)
(6.12)
The fact that BCJ sum vanishes in the onshell limit suggests that the Jacobi identity is
modi ed as ns + nt + nu + (s12
s13)n4 = 0 up to terms proportional to k42. A careful
inspection shows that these terms actually cancel completely. Plugging equations (6.8)
to (6.11) into the left hand side of this modi ed Jacobi sum, we see that (neglecting an
overall factor ( i) a1a4 =2),
( 2 3) part :
We obtain the numerator by feeding the o shell continued BCJ sum just computed into the
KLT inspired prescription (6.2), taking the modi ed Jacobi identity into account, yielding
1
4
n(13g2g4) =
k2 s31 s21A2Y;M2 s(1; 2g; 3g; 4) + (s21 + s23)A2Y;M2 s(1; 3g; 2g; 4)
=
nu + s13 n4 =
+ s13
;
where by an abuse of notation we neglected factors of inverse propagators so that the graphs
appear in the equation above should be understood as representing the corresponding
numerators rather than the original Feynman graphs. In the following discussions we
shall not distinguish numerators from Feynman graphs unless it is not apparent from the
context. The other two gluon numerator at four points can be readily obtained by swapping
labels (2 $ 3). Inserting the halfladder numerators back into KLT relation and we nd
agreement with the two graviton relation (equation (4) in [14]).
A2E;Y2M(1; 4; h2; h3) = ( 2i) ( 2 X2)( 3 X3)
( 2 3)s21 A4YM(1; 2; 3; 4)+(2 $ 3) : (6.15)
Note that the above relation is not exactly the same as (5.1), but equivalent to it after
using certain BCJ relations, and note particularly that the new 2 3 term came from the
quartic graph contribution.
O shell continued Jacobi identity.
The key point of the above calculation is the
modi ed Jacobi identity when some of the legs becoming o shell, e.g., ns + nt + nu +
(s12
s13)n4 = 0. This modi cation will lead to modi ed fundamental BCJ relations, to
be discussed later. When considering situations for higher points, one note that the color
dependency will factorize when the scalars are connected by an internal gluon line, thus
the single trace part of a twogluon partial amplitude can only contain graphs derivable
from those appearing at fourpoint case by welding pure scalar currents to their two scalar
legs. Therefore we only need to consider analytically continuing the two scalar lines of
the modi ed Jacobi identity when two gluons are present. Careful inspection of (6.14)
1
4
shows that the ( k)( k) part of the Jacobi sum is a pairwise cancelation, up to terms
proportional to ( 2 k2) or ( 3 k3), and therefore remains valid even when scalars become o
shell. The only modi cation comes from the ( 2 3) part. To completely cancel the (k2 k3)
k4) factor produced by nt, we see that the quartic graph needs to be multiplied by the
same factor. The o shell continued identity we need for all two gluon amplitudes is then
+
+ (k2
k3) (k1
k4)
= 0 ;
(6.16)
HJEP09(217)
and we will be using this identity in the following discussions.
The vepoint YMs amplitudes with two gluons
Having presented the example of four points with two gluons, we further show an example
of vepoint amplitude with two gluons. Again, we will use the method of summing over
colorordered KK basis.
At ve points the number of graphs increases considerably. Recall from [8] that there
are 15 di erent graphs in total in the KK sector at ve points if the amplitudes are to be
described by cubic graphs only, 6 of them are independent when Jacobi identities are taken
into account. Similarly we label the cubic graphs as n1, n2, : : :, n15, and we regard the
quartic graphs as additional corrections n16, n17, n18. The amplitudes are given by
A3Y;M2 s(1; 2g; 3g; 4; 5) =
A3Y;M2 s(1; 4; 3g; 2g; 5) =
A3Y;M2 s(1; 3g; 4; 2g; 5) =
A3Y;M2 s(1; 2g; 4; 3g; 5) =
A3Y;M2 s(1; 4; 2g; 3g; 5) =
A3Y;M2 s(1; 3g; 2g; 4; 5) =
n1
n6
n9
s12s45
s14s25
s13s25
n12
s12s35
n14
s14s35
n15
s13s45
n5
s23s15
s34s15
s34s15
n11
s24s15
n11
s24s15
n2
s23s15
n7
s34s12
s23s14
n10
s24s13
n3
n7
s34s12
s23s14
n10
s24s13
n8
n8
s45s23
s25s34
s25s34
n13
s35s24
n13
s35s24
n4
s45s23
n2
s15s34
s15s23
n11
s15s24
n5
n2
s15s34
s15s23
n11
s15s24
2 n17 ;
s15
n18 +
s14
n16 +
s45
n17 ;
s15
n17 ;
s15
n17 ;
s15
n17 :
s15
Together there are 15 cubic graphs and 3 quartic graphs in the two gluon scalar YangMills
amplitudes at ve points, which we list below,
n1 =
n6 =
n2 =
n7 =
;
;
n3 =
n8 =
;
;
n4 =
n9 =
;
n5 =
; n10 =
(6.17)
(6.18)
(6.19)
(6.20)
(6.21)
(6.22)
;
;
n8
As in the case of a cubic theory, not all graphs are independent. Together there are seven
HJEP09(217)
Jacobi identities derived from cyclic permutations involving one gluon, one of the identities
obtainable as a linear combination of the others,
n5 + n8 = 0 ;
n3
n1 + n12 = 0 ;
n4
n2 + n7 = 0 ;
n6 + n9 = 0 ; n10
n9 + n15 = 0 ; n10
n11 + n13 = 0 ;
n13
n12 + n14 = 0 :
In addition there are three more modi ed Jacobi identities where two gluons participate
the permutations, and therefore contains quartic graphs,
n4
n5
n1 + n15
n2 + n11 + (s34
s31)n16 = 0 ;
s24)n17 = 0 ;
n7
n6 + n14 + (s35
s25)n18 = 0 :
The above constraints allows us to trade n7, n8, : : :, n15 in terms of the rst six independent
cubic graphs plus the three quartic graphs,
Furthermore we note that all three quartic graphs actually contribute the same value,
n18 = n17 = n16 = ( 2 3)
1
2
f a1a4a5 :
Bearing all these in mind we calculate the vepoint numerator n(12g3g45) from KLT
relation by summing over KK basis and get
X
2S3
n(12g3g45) =
S[234j ]J YMs(1; ; 5)
n7 = n2
n8 =
n3 + n5 ;
n9 = n3
n5 + n6 ;
n4 ;
n5
n10 =
n1 + n3 + n4
n11 = n2
n13 = n1 + n2
n3
n14 =
n2 + n4 + n6
n5 + n6
s24)n17 ;
n4
n6
(s35
(s21
s31)n16 ;
n12 = n1
n3 ;
(s34
s24)n17 + (s21
s25)n18 ;
n15 = n1
s31)n16 ;
n4 + (s21
s31)n16 : (6.30)
= n1 + s12 n16 =
+ s12
(6.23)
(6.24)
(6.25)
(6.26)
(6.27)
(6.28)
(6.29)
(6.31)
(6.32)
n(12g43g5) =
n(13g2g45) =
n(13g42g5) =
n(142g3g5) =
+ (s21 + s24)
n(143g2g5) =
+ (s31 + s34)
;
All other numerators follow the same derivation, and we obtain
+ s21
+ s31
+ s31
Plugging the above results into DDM expression yields the two graviton EYM amplitude
at ve points,
A3E;Y2M(1; 4; 5; h2; h3) = p 3
2 i ( 2 X2)( 3 X3)
p 3
+ 2 i ( 2 X2)( 3 X3)
1
4
( 2
1
4
( 2
3)s21 A5YM(1; 2; 3; 4; 5)
(6.38)
3)s21 A5YM(1; 2; 4; 3; 5) +
6.4
ve and higher point amplitude involving two gravitons
Having witness that KLT relation successfully explains the new EYM amplitude expression
for two graviton scattering at four and
ve points, perhaps it is not much of a surprise
that the explanation generalizes to higher points. Indeed, one can actually read o the
npoint two gluon numerator, and the two graviton EYM amplitude is determined by the
corresponding DDM expression. We shall use the algorithm introduced originally for the
pure scalar scenario in [71] to systematically calculate the numerator (i.e., to systematically
sum over KK basis). As we shall see, in the case when only two gluons (p; q) are involved,
the numerators remain fairly simple,
n(12
i pg
j qg
n) =
+ 2(p Yp)
(6.39)
= p2n 2
( 1)n+1 i ( p xp)( q xq)
( p
q) (p Yp) :
;
;
1
2
(6.33)
(6.34)
(6.35)
(6.36)
(6.37)
HJEP09(217)
A brief review of the algorithm for numerators in the scalar scenario: for the
purpose of being selfcontained, we brie y review the algorithm used by the authors in [71]
and [61] to calculate numerators. The idea is to divide the full Sn 2 permutation sum
appears in the numeratorcurrent relation n(1
n) = P
2Sn 2 S[ T j ] J YMs(1; ; n) into
BCJ sums, and proceed repeatedly if the Fundamental BCJ relation between currents
admits further simpli cations. For example, it was shown in [71] that the Fundamental
BCJ relation between
3 currents yields another current, with the leg running through all
insertions in the BCJ sum
xed at the o shell continued line,
s21
+ (s21 + s31)
+
;
(6.40)
HJEP09(217)
X
2S3
so that if we divide the full permutation sum S3 in the vepoint numerator calculation into
BCJ sums, after substituting these summations using Fundamental BCJ relation (6.40),
the collected result is yet another BCJ sum, but only performed over permutations of the
legs of fewerpoint subcurrents,
S[432j 2 3 4]J 3 (1; 2; 3; 4;5) = s21s31 s41J 3 (14325)+(s41 +s43)J 3 (13425)+:::
(6.41)
+s21(s31 +s32) s41J (14235)+(s41 +s42)J (12435)+:::
3
3
= k52 s21 s31
+(s31 +s32)
!
;
where we used (6.40) to replace the rst and the second line of the equation above with the
two graphs in (6.41). The result is another BCJ sum over currents. Repeat the substitution
using Fundamental BCJ relation, and we obtain the numerator
2 2 2
k1234 k123 k12
:
(6.42)
vepoint scalar YangMills numerators involving two gluons.
The
calculation explained above only complicates slightly when few gluons are present. As far as
single trace contributions are concerned, all amplitudes are consisted of Jacobi satisfying
cubic graphs when only one gluon participates the scattering, and the same algorithm
applies. It is straightforward to see that the numerator is given by n(12
pg
n) =
f 1;2; f ;3;
f ;n 1;n p xp, which when plugged into the summing expression readily
reproduces the new EYM formula. In other words, (6.4) can also be understood from this
point of view.
Things will become a little bit more complicated when two and more gluons are
involved, since quartic vertices start to come into play, although they still remain quite
manageable, in the sense that the modi ed Fundamental BCJ relations brought by the
quartic term also permit repeated use of the relation when we carry out the summation.
Explicitly, at ve points the twogluon Fundamental BCJ relations are modi ed as
s21JYMs(12g3g45)+s2;13JYMs(13g2g45)+s2;134JYMs(13g42g5) = k52
s21JYMs(12g34g5)+s2;13JYMs(132g4g5)+s2;134JYMs(134g2g5) = k52
;
+
(6.43)
!
;
(6.44)
HJEP09(217)
.
.
.
The rules to modi cation is as follows. Generically one only needs to replace the appropriate
scalar by gluon lines in the original Fundamental BCJ relation between currents (6.40), and
the right hand side of the equation is a current with the running leg
xed at the o shell
line. The only exception is when the running leg is gluonic, also that either leg 1 or leg n 1
(legs adjacent to the o shell line) is a gluon line. In these cases an additional current needs
to be added, where a quartic vertex resides on the o shell line connects the two gluons.
We leave the details of a proof to these relations at vepoint to appendix B because of
its complicated nature. The principles are however not much di erent from the pure scalar
scenario and is conceptually straightforward. Basically we cancel graphs related by Jacobi
identities among BerendsGiele decomposed vepoint current in the BCJ sum. The result
after cancelation is then collected and identi ed to be the BerendsGiele decomposition
of the right hand side of the equation.
The proof for generic n points follows rather
trivially from the structure of the proof, since adding more scalar lines into subcurrents
at peripherals does not change Jacobi identities.
Assuming the Fundamental BCJ relations above, it is not di cult to see that the
numerator is genuinely given by the formula (6.39) we claimed earlier. Consider for example
the derivation that leads to numerator n(123g4g5),
n(123g4g5) = s21s31 s41J YMs(14g3g25) + (s41 + s43)J YMs(13g4g25) + : : :
(6.45)
+s21(s31 + s32) s41J YMs(14g23g5) + (s41 + s42)J YMs(124g3g5) + : : :
= k52 s21 s31
+ (s31 + s32)
+
!!
:
As was explained earlier we obtain the numerator by rst dividing the full Sn 2
permutation sum appears in the KLT inspired prescription (6.2) into BCJ sums, and then use the
Fundamental BCJ relation between currents to x the n
2 legs one by one in descending
order. For the most part, this procedure is not di erent from the derivation of a pure scalar
numerator, and the result does contain a cubic half ladder graph. The only modi cation
occurs whenever the leg we attempt to x is gluonic, in which case an additional graph is
included, where a quartic vertex connecting both gluon lines emerges. The derivation
afterwards again follows that of a pure scalar numerator. In the n(123g4g5) example this leads to
2 2 2
n(123g4g5) = k1234 k123 k12
+k12234 k122(s31+s32)
:
(6.46)
Note that the Mandelstam variables associated with the quartic graph was furnished by
momentum kernel. Careful inspection of the derivation that leads to (6.45) shows that
they should contain the inner products between gluon line carrying the smaller label and
all the scalar lines which precede it. As another illustration we consider n(12g34g5),
n(12g34g5) = k52 s21 s31
+ (s31 + s32)
!
:
(6.47)
HJEP09(217)
A repeated use of the Fundamental BCJ relation yields
2 2 2
n(12g34g5) = k1234 k123 k12
+ k12234 k123 s21
:
(6.48)
As a veri cation, note that applying the same rules to derive numerators with all possible
combinations of gluon positions yields the same results as those listed from equation (6.32)
to (6.37) previously obtained exclusively for ve points.
7
Conclusion
In this paper, we studied the newly discovered EYM amplitude relation by gauge invariance
principle, the BCFW recursion relation as well as the KLT relation respectively. It turns
out that the problem of EYM amplitude expansion is also closely related to the problem
of computing BCJ numerators and the boundary contribution of BCFW terms.
The major context of this paper is devoted to the principle of gauge invariance applied
to the determination of EYM amplitude relations. We propose a constructive algorithm
by expanding any EYM amplitude AEnY;mM as a linear sum of AEnY+Mi;m i for i = 1; 2; : : : ; m
with given expansion coe cients, and the contributing terms of AEnY+Mi;m i are determined
by AEnY+Mi 1;m i+1. This means that any contributing terms can be recursively determined
by the very
rst one AEnY+M1;m 1
, while keeping the gauge invariance in each step. This
leads to a compact formula (3.39) for general EYM amplitude relations with arbitrary
number of gravitons. Realizing that the expansion of EinsteinYangMills amplitude into
YangMills amplitudes shares the same kinematic coe cient as the expansion of
YangMillsscalar amplitude into cubicscalar amplitudes, we copy the EYM amplitude relation
to YMs amplitude relation, and generalize the later one to the expansion of pure
YangMills amplitude into cubicscalar amplitudes by the help of Pfa an expansion. With the
YangMills amplitude expanded recursively into the cubic graphs, we further outline the
strategy of rewriting the scalar amplitudes into KK basis, manifesting the colorkinematics
duality and computing the BCJ numerators of YangMills amplitude.
We also study the EYM amplitude relations in the Smatrix framework, and present the
proof of EYM amplitude relations with two gravitons by BCFW recursion relations. In this
case, any choice of deformed momenta is not possible to avoid the boundary contributions,
so we need to compare the contributions of both sides in the relations from
nite poles and
also the boundary. The matching of both contributions also constraints the possible form
of the nontrivial relations. Besides, we examine the problem again from the perspective
of KLT relations. The expansion coe cients of EYM amplitude relations are identical to
the BCJ numerators of DDM basis, and by computing the BCJ relations for currents we
con rm the validation of EYM amplitude relations.
Following our results, there are many interesting directions to explore further. In our
paper, one of the most important results is the recursive construction (3.39) of EYM
amplitude relation. We have claimed this expression by a few explicit examples plus the guidance
of gauge invariance principle. For the con rmation of the claim, a rigorous derivation by
other methods is favorable. In an upcoming paper, we would explore the recursive
construction directly from operations on the CHYintegrand level. Furthermore, we believe
that, such recursive pattern can also nd its hints in the BCFW recursion relation or KLT
relation investigation of EYM amplitude expansion, which worth to work on with.
Another possible work would be that, in our recursive construction, the gauge
invariance is manifest for all gravitons at each step except the rst one that started the recursive
algorithm. As shown in [45, 47{49], for YangMills theory, the requirement of gange
invariance for (n
1) points is su cient to guarantee the correctness of the full amplitude.
This observation seems to be also true in the EYM theory, thus nding an explicit proof
along the same line as in [47{49] would be a thing worth to do.
A most interesting and important future direction would be the systematic study of
the CHYintegrand expansion. In section 2, we have laid down the general framework for
the expansion, while in the whole paper we are focus only on the expansion of (reduced)
Pfa an. However, many CHYintegrands, such as (Pf 0(A))2 can be obtained from Pfa an
with proper reduction. Thus our results could be easily generalized to many other theories.
Especially by similar calculations, we can check if the soft theorem can be used to uniquely
determine the amplitude for some theories, such as NLSM as advertised in [47{49].
Finally, as a byproduct of the EYM expansion, we have outlined the strategy of
computing BCJ numerators4 from the expansion relation for general EYM amplitudes. The
fourpoint example shows the procedure of computing the BCJ numerators as polynomial
of (
), (
k) and (k
k), constructed neatly from the expansion coe cients of EYM
amplitudes into YangMills amplitudes. This construction, when generalized to looplevel,
would fascinate many important calculations involving gravitons.
Acknowledgments
We would like to thank Fei Teng for valuable discussions. BF is supported by
QiuShi Funding and the National Natural Science Foundation of China (NSFC) with Grant
No.11575156, No.11135006, and No.11125523. YD would like to acknowledge NSFC under
Grant Nos.11105118, 111547310, as well as the support from 351 program of Wuhan
University. RH would like to acknowledge the supporting from NSFC No.11575156 and the
Chinese Postdoctoral Administrative Committee.
A
Scalar YangMills Feynman rules
For reference purposes we list below the colorordered Feynman rules for constructing scalar
YangMills amplitudes [72]. The scalars and gluons are understood to be represented by
straight lines and wavy lines respectively,
(k1 k2) +cyclic;
= p
i
2
= i
=
= i ab
:
i
2
ab(k1 k2) ;
(A.2)
(A.3)
B
Graphical proof of the twogluon fundamental BCJ relation between
currents
As a demonstration of the general idea, in this appendix we prove two of the Fundamental
BCJ relations at ve points involving two gluons, equations (6.43) and (6.44), following the
4The polynomial expression of BCJ numerator of (reduced) Pfa an has been applied to the proof of
vanishing double poles in a recent work [76].
method used in [71] (which was also brie y outlined earlier in section 6.4). We shall neglect
repeating a similar proof for generic n points, as it can be readily derived by induction
and by attaching more external legs on the subcurrents represented by blank circles in the
graphs below.
Relations with no gluon adjacent to the o shell leg.
Consider rst the con gura
tion where the leg running over all possible insertions in the BCJ sum is a gluon, and the
other gluon is nonadjacent to the o shell leg. We would like to prove that
s21J YMs(12g3g45)+(s2;13)J YMs(13g2g45)+(s2;134)J YMs(13g42g5) =
:
(B.1)
HJEP09(217)
For this purpose we BerendsGiele decompose all three currents appear in the BCJ sum,
yielding altogether nine graphs,
s21J YMs(12g3g45) = s21
+s21
+s21
;
(B.2)
(a1)
s2;13J YMs(13g2g45) = s2;13
+s2;13
+s2;13
;
(B.3)
(b1)
(c1)
(b2)
(b3)
(c2)
(c3)
s2;134J YMs(13g42g5) = s2;134
+s2;134
+s2;134
; (B.4)
and notice that, aside from (c1), rest of the graphs can be regrouped as BCJ sums of
subcurrents. Indeed, graphs (a1)and (b1) together make up a BCJ sum of the subcurrents
involving legs 1, 2 and 3,
s21
+ s2;13
+ k1223
;
(B.5)
(a1)
(b1)
and graph (a2) is by itself a (trivial) BCJ sum of three point current. The combination of
(b2) and (c2) is also a BCJ sum of the three point current, after eliminating part of the
sum that carries an s21 using U(1) decoupling identity,
s21
= s23
and similarly (a3), (b3) and (c3) combine to give, up to terms vanishing under U(1)
decoupling identity,
s21
+s2;13
+s2;134
=k2234
(b3)
(c3)
In the equations above we have assumed the BCJ relations between currents at four points.
As for the remaining graph (c1) that does not regrouped with the others into a BCJ sum
of subcurrents, we rewrite the coe cient it carries using the kinematic identity s2;134 =
k
52 s134 and then further BerendsGiele decompose the part that carries a factor s134, giving
s134
s134
(B.7)
Because of the regrouping and the application of lower point BCJ relation on subcurrents,
the full BCJ sum (B.1) is now translated into the right hand side of equations (B.5), graph
(a2), (B.6), (B.7) and (B.8) combined. To see that this combination is indeed identical to
the right hand side of equation (B.1) we must show that all other graphs cancel, and this
is true because of the Jacobi identities
(c1)
2
k123
+ s24
s134
= 0 ;
(B.9)
+ k2234
s134
= 0 ;
(B.10)
and the fact that the following two graphs contribute the same, up to a relative minus sign,
and
therefore nishing our proof.
Relations with one gluon adjacent to the o shell leg.
The proof when one gluon
is adjacent to the o shell leg follows exactly the same derivation, except that now we have
a few additional quartic graphs. The relation we are aiming to prove is
s21J YMs(12g34g5)+s2;13J YMs(132g4g5)+s2;134J YMs(134g2g5)=k52
+k52
Currents that appear in the BCJ sum BerendsGiele decompose as
:
(B.12)
s21J YMs(12g34g5) = s21
+ s21
+ s21
; (B.13)
(a1)
(a2)
(a3)
s2;13J YMs(132g4g5) = s2;13
+ s2;13
i
(b4)
+s2;13
+ s2;13
;
(B.14)
+ s2;134
:
(B.15)
(c3)
(c4)
Note the presence of two new quartic graphs (b4) and (c4). As in the previous example we
regroup graphs into BCJ sums of subcurrents. Graphs (a1) and (b1) make up a BCJ sum
of the subcurrents involving legs 1, 2 and 3,
s21
s21
(a3)
= k2234
(b3)
+ s2;13
+ s2;13
= s23
s21
+ s2;13
+ s2;134
graph (a2) forms a trivial BCJ sum of the three point current by itself, graphs (b2) and
(c2) add up to another BCJ sum after eliminating terms using U(1) decoupling,
(b2)
(c2)
and similarly for the sum of graphs (a3), (b3) and (c3),
(B.16)
(B.17)
(c3)
(B.18)
As for (c1), we BerendsGiele decompose it as in the previous example to give The full BCJ sum (B.12) is now translated into the right hand side of equation (B.16), graph (b2), (B.17), (B.18), (B.19) plus the additional graphs (b4) and (c4). To nish the
proof we need to further translate these graphs into one of those on the right hand side of
equation (B.12), and this is done by using Jacobi identities
s21
+k2234
s134
s134
s123
s134
= (s4;13 s2;13)
; (B.20)
= 0;
(B.21)
and the fact that the following two graphs contribute the same.
2
k234
= s13
4 :
(B.22)
Collecting terms gives
which completes our proof.
k
s2;13) + s2;13 + s2;134 + s13
2
= k5
+ k52
Open Access.
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[1] S. Stieberger and T.R. Taylor, New relations for EinsteinYangMills amplitudes, Nucl. Phys.
B 913 (2016) 151 [arXiv:1606.09616] [INSPIRE].
[2] Y.X. Chen, Y.J. Du and Q. Ma, Relations between closed string amplitudes at higherorder
tree level and open string amplitudes, Nucl. Phys. B 824 (2010) 314 [arXiv:0901.1163]
[4] Y.X. Chen, Y.J. Du and Q. Ma, Disk relations for tree amplitudes in minimal coupling
theory of gauge eld and gravity, Nucl. Phys. B 833 (2010) 28 [arXiv:1001.0060] [INSPIRE].
[5] Y.X. Chen, Y.J. Du and B. Feng, On tree amplitudes with gluons coupled to gravitons,
JHEP 01 (2011) 081 [arXiv:1011.1953] [INSPIRE].
[6] S. Stieberger and T.R. Taylor, Graviton as a pair of collinear gauge bosons, Phys. Lett. B
739 (2014) 457 [arXiv:1409.4771] [INSPIRE].
Nucl. Phys. B 312 (1989) 616 [INSPIRE].
[7] R. Kleiss and H. Kuijf, Multigluon crosssections and ve jet production at hadron colliders,
[8] Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gaugetheory amplitudes, Phys.
Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].
[9] Z. Bern, T. Dennen, Y.T. Huang and M. Kiermaier, Gravity as the square of gauge theory,
[10] Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy
of gauge theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].
[11] H. Kawai, D.C. Lewellen and S.H. Henry Tye, A relation between tree amplitudes of closed
and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
[12] N.E.J. BjerrumBohr, P.H. Damgaard and P. Vanhove, Minimal basis for gauge theory
amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].
[13] S. Stieberger and T.R. Taylor, Disk scattering of open and closed strings (I), Nucl. Phys. B
903 (2016) 104 [arXiv:1510.01774] [INSPIRE].
[14] D. Nandan, J. Plefka, O. Schlotterer and C. Wen, EinsteinYangMills from pure YangMills
amplitudes, JHEP 10 (2016) 070 [arXiv:1607.05701] [INSPIRE].
[15] L. de la Cruz, A. Kniss and S. Weinzierl, Relations for EinsteinYangMills amplitudes from
the CHY representation, Phys. Lett. B 767 (2017) 86 [arXiv:1607.06036] [INSPIRE].
[16] O. Schlotterer, Amplitude relations in heterotic string theory and EinsteinYangMills, JHEP
11 (2016) 074 [arXiv:1608.00130] [INSPIRE].
[17] Y.J. Du, F. Teng and Y.S. Wu, Direct evaluation of npoint singletrace MHV amplitudes
in 4d EinsteinYangMills theory using the CHY formalism, JHEP 09 (2016) 171
[arXiv:1608.00883] [INSPIRE].
[18] D. Nandan, J. Plefka and W. Wormsbecher, Collinear limits beyond the leading order from
the scattering equations, JHEP 02 (2017) 038 [arXiv:1608.04730] [INSPIRE].
[19] S. He and O. Schlotterer, New relations for gaugetheory and gravity amplitudes at loop level,
[20] T. Adamo, E. Casali, K.A. Roehrig and D. Skinner, On tree amplitudes of supersymmetric
EinsteinYangMills theory, JHEP 12 (2015) 177 [arXiv:1507.02207] [INSPIRE].
[21] F. Cachazo, S. He and E.Y. Yuan, Scattering equations and KawaiLewellenTye
orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
[22] F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimensions,
Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
[23] F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and
[24] F. Cachazo, S. He and E.Y. Yuan, EinsteinYangMills scattering amplitudes from scattering
equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].
[25] F. Cachazo, S. He and E.Y. Yuan, Scattering equations and matrices: from Einstein to
YangMills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
[26] N.E.J. BjerrumBohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Manifesting
colorkinematics duality in the scattering equation formalism, JHEP 09 (2016) 094
[arXiv:1608.00006] [INSPIRE].
[27] N.E.J. BjerrumBohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Analytic representations
of YangMills amplitudes, Nucl. Phys. B 913 (2016) 964 [arXiv:1605.06501] [INSPIRE].
[28] C. Cardona, B. Feng, H. Gomez and R. Huang, Crossratio identities and higherorder poles
of CHYintegrand, JHEP 09 (2016) 133 [arXiv:1606.00670] [INSPIRE].
[29] C. Baadsgaard, N.E.J. BjerrumBohr, J.L. Bourjaily and P.H. Damgaard, Integration rules
for scattering equations, JHEP 09 (2015) 129 [arXiv:1506.06137] [INSPIRE].
[30] C. Baadsgaard, N.E.J. BjerrumBohr, J.L. Bourjaily and P.H. Damgaard, Scattering
equations and Feynman diagrams, JHEP 09 (2015) 136 [arXiv:1507.00997] [INSPIRE].
[31] C. Baadsgaard, N.E.J. BjerrumBohr, J.L. Bourjaily, P.H. Damgaard and B. Feng,
Integration rules for loop scattering equations, JHEP 11 (2015) 080 [arXiv:1508.03627]
[INSPIRE].
[32] R. Huang, B. Feng, M.X. Luo and C.J. Zhu, Feynman rules of higherorder poles in CHY
construction, JHEP 06 (2016) 013 [arXiv:1604.07314] [INSPIRE].
[33] B. Feng, R. Huang and Y. Jia, Gauge amplitude identities by onshell recursion relation in
Smatrix program, Phys. Lett. B 695 (2011) 350 [arXiv:1004.3417] [INSPIRE].
[34] Y.X. Chen, Y.J. Du and B. Feng, A proof of the explicit minimalbasis expansion of tree
amplitudes in gauge eld theory, JHEP 02 (2011) 112 [arXiv:1101.0009] [INSPIRE].
[35] R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons,
Nucl. Phys. B 715 (2005) 499 [hepth/0412308] [INSPIRE].
[36] R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of treelevel recursion relation in
YangMills theory, Phys. Rev. Lett. 94 (2005) 181602 [hepth/0501052] [INSPIRE].
[37] P. Benincasa and E. Conde, On the treelevel structure of scattering amplitudes of massless
particles, JHEP 11 (2011) 074 [arXiv:1106.0166] [INSPIRE].
relation, JHEP 03 (2015) 023 [arXiv:1411.0452] [INSPIRE].
[39] Q. Jin and B. Feng, Recursion relation for boundary contribution, JHEP 06 (2015) 018
[42] C. Cheung, C.H. Shen and J. Trnka, Simple recursion relations for general eld theories,
HJEP09(217)
JHEP 06 (2015) 118 [arXiv:1502.05057] [INSPIRE].
[43] C. Cheung, K. Kampf, J. Novotny, C.H. Shen and J. Trnka, Onshell recursion relations for
e ective eld theories, Phys. Rev. Lett. 116 (2016) 041601 [arXiv:1509.03309] [INSPIRE].
[44] L.A. Barreiro and R. Medina, RNS derivation of N point disk amplitudes from the revisited
Smatrix approach, Nucl. Phys. B 886 (2014) 870 [arXiv:1310.5942] [INSPIRE].
[45] R.H. Boels and R. Medina, Graviton and gluon scattering from rst principles, Phys. Rev.
Lett. 118 (2017) 061602 [arXiv:1607.08246] [INSPIRE].
[46] M. Berg, I. Buchberger and O. Schlotterer, Stringmotivated oneloop amplitudes in gauge
theories with halfmaximal supersymmetry, JHEP 07 (2017) 138 [arXiv:1611.03459]
[INSPIRE].
[INSPIRE].
[47] N. ArkaniHamed, L. Rodina and J. Trnka, Locality and unitarity from singularities and
gauge invariance, arXiv:1612.02797 [INSPIRE].
[48] L. Rodina, Uniqueness from locality and BCFW shifts, arXiv:1612.03885 [INSPIRE].
[49] L. Rodina, Uniqueness from gauge invariance and the Adler zero, arXiv:1612.06342
[50] C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors,
JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].
(2011) 007 [arXiv:1105.2565] [INSPIRE].
[51] R. Monteiro and D. O'Connell, The kinematic algebra from the selfdual sector, JHEP 07
03 (2014) 110 [arXiv:1311.1151] [INSPIRE].
[arXiv:1403.6262] [INSPIRE].
[52] J. Broedel and J.J.M. Carrasco, Virtuous trees at ve and six points for YangMills and
gravity, Phys. Rev. D 84 (2011) 085009 [arXiv:1107.4802] [INSPIRE].
[53] N.E.J. BjerrumBohr, P.H. Damgaard, R. Monteiro and D. O'Connell, Algebras for
amplitudes, JHEP 06 (2012) 061 [arXiv:1203.0944] [INSPIRE].
[54] C.H. Fu, Y.J. Du and B. Feng, An algebraic approach to BCJ numerators, JHEP 03 (2013)
050 [arXiv:1212.6168] [INSPIRE].
[55] R.H. Boels and R.S. Isermann, On powercounting in perturbative quantum gravity theories
through colorkinematic duality, JHEP 06 (2013) 017 [arXiv:1212.3473] [INSPIRE].
[56] R. Monteiro and D. O'Connell, The kinematic algebras from the scattering equations, JHEP
[57] C.H. Fu, Y.J. Du and B. Feng, Note on symmetric BCJ numerator, JHEP 08 (2014) 098
functions, JHEP 07 (2014) 143 [arXiv:1404.7141] [INSPIRE].
MaxwellEinstein and YangMills/Einstein supergravity, JHEP 01 (2015) 081
[arXiv:1511.01740] [INSPIRE].
174 [arXiv:1606.05846] [INSPIRE].
open string, JHEP 08 (2017) 135 [arXiv:1612.06446] [INSPIRE].
Mary University, London U.K., May 2010.
kernel of gauge and gravity theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [INSPIRE].
theory, JHEP 10 (2013) 069 [arXiv:1305.2996] [INSPIRE].
JHEP 07 (2013) 057 [arXiv:1304.2978] [INSPIRE].
93 (2016) 105008 [arXiv:1602.06419] [INSPIRE].
theory, JHEP 07 (2010) 093 [arXiv:1004.1282] [INSPIRE].
[40] B. Feng , J. Rao and K. Zhou , On multistep BCFW recursion relations , JHEP 07 ( 2015 ) 058 [41] Q. Jin and B. Feng , Boundary operators of BCFW recursion relation , JHEP 04 ( 2016 ) 123 [59] M. Chiodaroli , M. Gunaydin, H. Johansson and R. Roiban , Scattering amplitudes in N = 2 [60] M. Chiodaroli , M. Gunaydin, H. Johansson and R. Roiban , Spontaneously broken [61] Y.J. Du and C.H. Fu , Explicit BCJ numerators of nonlinear sigma model , JHEP 09 ( 2016 ) [62] J.J.M. Carrasco , C.R. Mafra and O. Schlotterer , SemiAbelian Z theory: NLSM+ 3 from the