Expansion of Einstein-Yang-Mills amplitude

Journal of High Energy Physics, Sep 2017

In this paper, we study from various perspectives the expansion of tree level single trace Einstein-Yang-Mills amplitudes into linear combination of color-ordered Yang-Mills 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 color-order Yang-Mills amplitude. As a byproduct, an algorithm for constructing a polynomial form of the BCJ numerator for Yang-Mills 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 on-shell perspective. And we examine the EYM expansion using KLT relations and show how to evaluate the expansion coefficients efficiently.

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Expansion of Einstein-Yang-Mills amplitude

Received: March Expansion of Einstein-Yang-Mills amplitude Chih-Hao Fu 0 1 3 7 8 Yi-Jian 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 Einstein-Yang-Mills amplitudes into linear combination of color-ordered 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 color-order Yang-Mills amplitude. As a byproduct, an algorithm for constructing a polynomial form of the BCJ numerator for Yang-Mills 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 on-shell 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 Yang-Mills amplitudes: ordered splitting formula 3.6 Expanding to pure Yang-Mills amplitudes: KK basis formula 4 The BCJ numerator of Yang-Mills 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 four-point gluon-scalar amplitude involving two gluons The ve-point YMs amplitudes with two gluons 6.4 The ve and higher point amplitude involving two gravitons 7 Conclusion A Scalar Yang-Mills Feynman rules B Graphical proof of the two-gluon fundamental BCJ relation between cur 1 Introduction 2 Amplitude relations from the perspective of CHY-formulation 3 The gauge invariance determines the amplitude relations A fairly non-trivial relation between Einstein-Yang-Mills (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 n-gluon mgraviton amplitudes by (n + 2m)-gluon amplitudes [2{6]. It is now widely known that, the non-trivial relations among amplitudes are important both in the practical evaluation and { 1 { the analytical study, while the U(1)-relation, Kleiss-Kuijf(KK)relation [7] and especially the Bern-Carrasco-Johansson(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 Yang-Mills amplitudes [9, 10]. As an analogous scenario, where amplitudes of two originally seemingly unrelated theories take part in novel identity, recall that the famous Kawai-Lewellen-Tye (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 bi-linear sum of Yang-Mills 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 Cachazo-He-Yuan (CHY) formulation [21{ 25], by genuinely reformulating the CHY-integrand 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 cross-ratio identity and other o -shell identities of integrands [27, 28] are crucial tools for deforming CHY-integrands 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 CHY-integrand without referring to the scattering equations. The idea of integration rule and cross-ratio identity method was to decompose arbitrary CHY-integrand using cross-ratio identities into those corresponding to cubicscalar Feynman diagrams dressed with kinematic factors. By carefully organizing terms one can identify the resulting CHY-integrands 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 CHY-framework. Starting from CHY-integrand of a theory, it is always possible to reformulate it to another form by cross-ratio and other o -shell relations, for instance the Yang-Mills-scalar (YMs) amplitude can be expanded as linear sum of bi-adjoint cubic-scalar amplitudes. We shall discuss this later in this paper. As it is very often, on-shell technique can prove to be a powerful tool for the purpose of understanding non-trivial amplitude relations within eld theory framework. One such example is the on-shell 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 Britto-Cachazo-Feng-Witten(BCFW) on-shell 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 on-shell 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 Yang-Mills amplitudes. Furthermore, with the philosophy of decomposing CHY-integrands, the same argument applies to the Yang-Mills amplitudes which would be expanded as pure bi-adjoint cubic-scalar amplitudes. This is exactly the cubic-graph expansion of Yang-Mills amplitude which makes the color-kinematics duality manifest [26]. The EYM amplitude relation combined with CHY-integrand, more speci cally the Pfa an expansion, would produce the non-trivial expansion for Yang-Mills amplitude as cubic-scalar graphs, as well as expansion for gravity amplitude as pure YangMills amplitudes and eventually the cubic-scalar 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 CHY-formulation, BCFW on-shell 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 non-trivial amplitude relations from the CHY-formulation, and explain the expansion of amplitudes as the expansion of Pfa an of CHY-integrand. 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 Yang-Mills amplitudes and apply the non-trivial relation to the computation of BCJ numerators. In section 5, we provide the on-shell 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 CHY-formulation The non-trivial relation revealed recently between EYM amplitudes and pure Yang-Mills amplitudes [1, 14, 15] has an intuitive interpretation in the CHY-framework. In fact, the CHY-formulation tells more beyond the EYM amplitudes. In the CHY-formula, it is the so called CHY-integrand ICHY that describes speci c eld theories. The CHY-integrand is an uniform weight-4 rational function of n complex variables zi for n-point scattering system, i.e., with the 1=zi4 scaling behavior in the zi ! 1 limit. For almost all known theories, the weight-4 CHY-integrand can be factorized as two weight-2 ingredients, formally written as ICHY = IL IR : Let us then de ne two new weight-4 CHY integrands as follows where PT( ) is the Parke-Taylor 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 CHY-integrands ILCHY; IRCHY also describe certain physical meaningful eld theories and produce the corresponding color-ordered amplitudes AL( ); AR( ) after CHY-evaluation, then by CHY-construction [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 CHY-integrand (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 color-ordered 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 n-point 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 weight-2 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 cross-ratio identities to the CHY-integrands, where in each step a gauge choice should be taken in the cross-ratio 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 on-shell 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 CHY-integrand is given by products of two weight-2 ingredients as (2.1), for some theories the CHY-integrand is de ned by the product of four weight-1 ingredients. So there are various combinations of them to form weight-2 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 CHY-integrand 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 CHY-integrands as = f 1; : : : ; rg. ILCHY( je) := PTr( )Pf s PTn(e) ; IRCHY( ) := Pf0 n the amplitude AL is the color-ordered YMs amplitude ArY;Mss with r scalars and s gluons, which has two trace structures associated with the two PT-factors, while the amplitude AR is color-ordered Yang-Mills amplitude AnYM. One thing to emphasize is that the scalar carries two groups (one gauge group and one avor group) and has bi-adjoint scalar-cubic 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 Yang-Mills 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 weight-2 ingredient in the CHY-integrand in (2.1). This means that while keeping the same weight-2 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 Yang-Mills amplitudes, we can change the Pf0 n in (2.9) as PTn( ). The resulting CHY-integrand IrY;sMs( j ) = PTr( )Pf s PTn( ) (2.15) { 6 { (2.11) (2.12) (2.13) (2.14) describes a Yang-Mills-scalar amplitude with r scalars and s gluons, and the weight-2 ingredients are now IL = PTr( )Pf s and IR = PTn( ). With the same philosophy as in (2.10), we can de ne two new CHY-integrands as ILCHY := PTr( )Pf s with the same expansion coe cients as in (2.12). This non-trivial relation expresses any single trace color-ordered amplitude of Yang-Mills-scalar theory as linear combination of color-ordered amplitude of bi-adjoint 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 CHY-integrand 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 cross-ratio 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 non-trivial 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 on-shell 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 Yang-Mills 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 color-ordering 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 color-ordering. 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 color-ordering of color-ordered YM amplitudes. A clari cation of the de nition Yp is needed here for the future usage. In the (n + 1)point pure Yang-Mills 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 Yang-Mills 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 Yang-Mills 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 non-trivial 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 Yang-Mills 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 Yang-Mills 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 Yang-Mills 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 multi-graviton 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 non-trivial relations for EYM amplitudes with generic number of gravitons, we start with the following two ansatz, These two ansatz come from lower-point 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 BCJ-like 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 n-gluon two-graviton EYM amplitudes as Yang-Mills 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 color-ordering 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 non-trivial 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 length-m1 set in the gluon side and is a length-m2 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 n-gluon three-graviton EYM amplitudes as Yang-Mills 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 well-de 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 Yang-Mills 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 n-point Yang-Mills 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 n-th 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 non-trivial 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 Yang-Mills amplitudes: ordered splitting formula The recursive construction given in (3.39) is easy to implement and one can eventually get an expansion with pure Yang-Mills 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 one-to-one 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 color-ordered Yang-Mills 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 lower-point 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 color-ordered i amplitudes with color-ordering de ned by the corresponding ordered splitting. Fej ij 1 we re-express 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 on-shell condition of the right-most 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 Yang-Mills-scalar 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 re-write 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 non-trivial relations between YMs amplitude and pure scalar amplitudes (hence the EYM amplitude and Yang-Mills 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 multi-graviton 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 perturbation-friendly 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 Duca-Dixon-Maltoni (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 well-suited 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 self-contained we list the color-ordered Feynman rules for gluonscalar interaction presented in [72] in the appendix A. Consider rst the scalar Yang-Mills amplitudes when there is only one gluon. Note that a (color-stripped) 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 anti-symmetry of the three-vertices, It is very important to notice that the color-kinematics 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 n-point DDM half-ladder 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 Yang-Mills-scalar 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 DDM-chain. 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 four-point gluon-scalar amplitude involving two gluons Next we consider Yang-Mills-scalar amplitudes involving two gluons. For this case, since Feynman diagrams will involve the four-point 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 color-ordered 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 color-kinematics 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 on-shell 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 half-ladder 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 two-gluon partial amplitude can only contain graphs derivable from those appearing at four-point 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 pair-wise 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 ve-point YMs amplitudes with two gluons Having presented the example of four points with two gluons, we further show an example of ve-point amplitude with two gluons. Again, we will use the method of summing over color-ordered 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 Yang-Mills 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 ve-point 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 n-point 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 self-contained, 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 numerator-current 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 ve-point 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 fewer-point sub-currents, 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) ve-point scalar Yang-Mills 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 two-gluon 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 ve-point 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 Berends-Giele decomposed ve-point current in the BCJ sum. The result after cancelation is then collected and identi ed to be the Berends-Giele 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 sub-currents 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 Einstein-Yang-Mills amplitude into Yang-Mills amplitudes shares the same kinematic coe cient as the expansion of YangMills-scalar amplitude into cubic-scalar amplitudes, we copy the EYM amplitude relation to YMs amplitude relation, and generalize the later one to the expansion of pure YangMills amplitude into cubic-scalar amplitudes by the help of Pfa an expansion. With the Yang-Mills amplitude expanded recursively into the cubic graphs, we further outline the strategy of rewriting the scalar amplitudes into KK basis, manifesting the color-kinematics duality and computing the BCJ numerators of Yang-Mills amplitude. We also study the EYM amplitude relations in the S-matrix 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 non-trivial 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 CHY-integrand 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 Yang-Mills 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 CHY-integrand 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 CHY-integrands, 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 four-point 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 Yang-Mills amplitudes. This construction, when generalized to loop-level, 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 Yang-Mills Feynman rules For reference purposes we list below the color-ordered Feynman rules for constructing scalar Yang-Mills 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 two-gluon 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 sub-currents 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 non-adjacent 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 Berends-Giele 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 sub-currents 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 sub-currents, we rewrite the coe cient it carries using the kinematic identity s2;134 = k 52 s134 and then further Berends-Giele 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 sub-currents, 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 Berends-Giele 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 sub-currents. Graphs (a1) and (b1) make up a BCJ sum of the sub-currents 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 Berends-Giele 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. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] S. Stieberger and T.R. Taylor, New relations for Einstein-Yang-Mills 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 higher-order 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, Multi-gluon cross-sections and ve jet production at hadron colliders, [8] Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory 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. Bjerrum-Bohr, 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, Einstein-Yang-Mills from pure Yang-Mills amplitudes, JHEP 10 (2016) 070 [arXiv:1607.05701] [INSPIRE]. [15] L. de la Cruz, A. Kniss and S. Weinzierl, Relations for Einstein-Yang-Mills 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 Einstein-Yang-Mills, JHEP 11 (2016) 074 [arXiv:1608.00130] [INSPIRE]. [17] Y.-J. Du, F. Teng and Y.-S. Wu, Direct evaluation of n-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills 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 gauge-theory and gravity amplitudes at loop level, [20] T. Adamo, E. Casali, K.A. Roehrig and D. Skinner, On tree amplitudes of supersymmetric Einstein-Yang-Mills theory, JHEP 12 (2015) 177 [arXiv:1507.02207] [INSPIRE]. [21] F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye 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, Einstein-Yang-Mills 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 Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE]. [26] N.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Manifesting color-kinematics duality in the scattering equation formalism, JHEP 09 (2016) 094 [arXiv:1608.00006] [INSPIRE]. [27] N.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Analytic representations of Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 964 [arXiv:1605.06501] [INSPIRE]. [28] C. Cardona, B. Feng, H. Gomez and R. Huang, Cross-ratio identities and higher-order poles of CHY-integrand, JHEP 09 (2016) 133 [arXiv:1606.00670] [INSPIRE]. [29] C. Baadsgaard, N.E.J. Bjerrum-Bohr, 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. Bjerrum-Bohr, 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. Bjerrum-Bohr, 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 higher-order poles in CHY construction, JHEP 06 (2016) 013 [arXiv:1604.07314] [INSPIRE]. [33] B. Feng, R. Huang and Y. Jia, Gauge amplitude identities by on-shell recursion relation in S-matrix 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 minimal-basis 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 [hep-th/0412308] [INSPIRE]. [36] R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE]. [37] P. Benincasa and E. Conde, On the tree-level 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, On-shell 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 S-matrix 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, String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry, JHEP 07 (2017) 138 [arXiv:1611.03459] [INSPIRE]. [INSPIRE]. [47] N. Arkani-Hamed, 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 self-dual 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 Yang-Mills and gravity, Phys. Rev. D 84 (2011) 085009 [arXiv:1107.4802] [INSPIRE]. [53] N.E.J. Bjerrum-Bohr, 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 color-kinematic 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]. Maxwell-Einstein and Yang-Mills/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 multi-step 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 , Semi-Abelian Z -theory: NLSM+ 3 from the

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Chih-Hao Fu, Yi-Jian Du, Rijun Huang, Bo Feng. Expansion of Einstein-Yang-Mills amplitude, Journal of High Energy Physics, 2017, 21, DOI: 10.1007/JHEP09(2017)021