The Kinematic Algebras from the Scattering Equations
EEdinburgh 2013/29
Prepared for submission to JHEP
The Kinematic Algebras from the Scattering Equations
Ricardo Monteiro a , Donal O’Connell b a Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK b Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh,Edinburgh EH9 3JZ, Scotland, UK
Abstract:
We study kinematic algebras associated to the recently proposed scattering equations,which arise in the description of the scattering of massless particles. In particular, we describe therole that these algebras play in the BCJ duality between colour and kinematics in gauge theory,and its relation to gravity. We find that the scattering equations are a consistency condition for aself-dual-type vertex which is associated to each solution of those equations. We also identify anextension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Bothvertices correspond to the structure constants of Lie algebras. We give a prescription for the use of thegenerators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. Inparticular, we write BCJ numerators for each contribution to the amplitude associated to a solutionof the scattering equations. This leads to a decomposition of the determinant of a certain kinematicmatrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present thekinematic analogues of colour traces, according to these algebras, and the associated decompositionof that determinant. a r X i v : . [ h e p - t h ] N ov ontents X -amplitudes 12 Contents1 Introduction
Cachazo, He and Yuan [1, 2] have recently proposed remarkably compact formulas for the tree-levelscattering amplitudes of gluons and gravitons, valid in any number of spacetime dimensions. Theseare expressed in terms of an integral over points on a sphere, which localises to a sum over ( n − n -particle scattering. The localisation of the integral is associated to the ( n − Review of the scattering equations
The scattering equations at n points are a deceptively simple set of equations, to be solved for n complex variables σ a : (cid:88) b (cid:54) = a k a · k b σ a − σ b = 0 , (2.1)where the k a are the momenta of the external particles. Thus, there are n equations. However, only n − SL (2 , C ) when the k a satisfy conservation of momentum. That is, given aparticular solution for the σ a , an equally good solution is given by σ (cid:48) a = Aσ a + BCσ a + D , (2.2)when AD − BC = 1. Thus we may interpret the σ a as points on S .Up to this redundancy, there are always ( n − n = 3 is trivial: one may exploit the SL (2 , C ) to fix the locations of the threepoints; meanwhile, all the kinematic invariants vanish. For n = 4, after fixing SL (2 , C ), there isonly one variable. Meanwhile, there is one equation to be solved, which is linear in the remainingvariable. Solutions of the scattering equations at n points may be obtained from the solutions at n − n − n − n -point system, for a total of ( n − D dimensions are easily writtendown using the solutions of the scattering equations. The colour-ordered n point Yang-Mills amplitudes A n , and the n -point gravity amplitudes M n are simply [1, 2] A n = (cid:90) d n σ vol SL(2 , C ) (cid:89) a (cid:48) δ (cid:88) b (cid:54) = a k a · k b σ a − σ b E n ( { k, (cid:15), σ } ) σ σ · · · σ n , (2.3) M n = (cid:90) d n σ vol SL(2 , C ) (cid:89) a (cid:48) δ (cid:88) b (cid:54) = a k a · k b σ a − σ b E n ( { k, (cid:15), σ } ) , (2.4)where σ ab = σ a − σ b , and (cid:89) a (cid:48) δ (cid:88) b (cid:54) = a k a · k b σ a − σ b = σ ij σ jk σ ki (cid:89) a (cid:54) = i,j,k δ (cid:88) b (cid:54) = a k a · k b σ a − σ b . (2.5)This is independent of the choice of i, j and k ; therefore, it is permutation symmetric. Meanwhile, theobject E n ( { k, (cid:15), σ } ) is a gauge-invariant function of the polarisations (cid:15) a of the particles. Moreover, it issymmetric under permutations of the particles. It is most easily described in terms of an antisymmetric2 n × n matrix Ψ, which is given in block form asΨ = (cid:18) A − C T C B (cid:19) . (2.6)– 3 –he n × n blocks of Ψ are defined by A ab = k a · k b σ ab a (cid:54) = b, a = b, (2.7) B ab = (cid:15) a · (cid:15) b σ ab a (cid:54) = b, a = b, (2.8) C ab = (cid:15) a · k b σ ab a (cid:54) = b, − (cid:88) c (cid:54) = a (cid:15) a · k c σ ac a = b. (2.9)The Pfaffian of Ψ vanishes; however, leaving out two rows i and j as well as the corresponding columns i and j yields a matrix Ψ ijij which has a non-vanishing Pfaffian. Indeed, the object E n ( { k, (cid:15), σ } )appearing in the amplitudes (2.3) and (2.4) is E n ( { k, (cid:15), σ } ) ≡ Pf (cid:48) (Ψ ijij ) ≡ − i + j σ ij Pf (Ψ ijij ) . (2.10)The delta functions appearing under the integral sign in the amplitudes (2.3) and (2.4) completelylocalise the integrals. So, in fact, there are no integrations to do, and the amplitudes can be expressedas a sum over the ( n − ab = k a · k b σ ab a (cid:54) = b, − (cid:88) c (cid:54) = a k a · k c σ ac a = b. (2.11)This matrix, introduced in [5], is closely connected to the Hodges formula for MHV graviton amplitudes[16]. Because the delta functions (2.5) instruct us to omit rows i, j and k , we omit these from theJacobian determinant. In addition, to gauge fix the SL (2 , C ) we may fix the position of three points,say σ r , σ s and σ t . As usual, this gauge-fixing procedure introduces a factor σ rs σ st σ ts into the integral.The result is that the Jacobian determinant is the minor determinant of Φ, omitting rows i, j and k and columns r, s and t . It is convenient to introduce the simple notationdet (cid:48) Φ = | Φ | ijkrst σ rs σ st σ ts σ ij σ jk σ ki . (2.12)The amplitudes are then written as sums over contributions from distinct solutions of the scatteringequations, A n = (cid:88) solutions σ σ · · · σ n Pf (cid:48) Ψdet (cid:48) Φ , (2.13) M n = (cid:88) solutions (Pf (cid:48) Ψ) det (cid:48) Φ . (2.14)– 4 –he scattering equations are relevant not just for the scattering of spin 1 and spin 2 particles, butalso for the scattering of certain massless scalar theories. These are theories of scalar fields φ aa (cid:48) witha double-coloured cubic vertex, where a and a (cid:48) transform under the adjoint of two groups G and G (cid:48) .Ref. [28] presents an expression for the amplitudes of particles of spin s , valid for these scalar theoriesas well as for gluons and gravitons, A ( s ) n = (cid:88) solutions (cid:18) Tr( T a T a · · · T a n ) σ σ · · · σ n + non-cyclic permutations (cid:19) − s (Pf (cid:48) Ψ) s det (cid:48) Φ . (2.15)In the scalar case, we can have distinct groups G (cid:54) = G (cid:48) , or simply distinct algebra indices a r (cid:54) = a (cid:48) r , sothat we should substitute (cid:18) Tr( T a T a · · · T a n ) σ σ · · · σ n + . . . (cid:19) → (cid:18) Tr( T a T a · · · T a n ) σ σ · · · σ n + . . . (cid:19) (cid:32) Tr( T a (cid:48) T a (cid:48) · · · T a (cid:48) n ) σ σ · · · σ n + . . . (cid:33) . (2.16)The most remarkable property of the solutions of the scattering equations, which leads to thesimplicity of the formulas above, is the so-called KLT orthogonality, discovered in [5] and proven in[1]. The KLT relations [17, 34], proven in [35], give a graviton amplitude as a product of two sets ofgauge theory colour-ordered amplitudes, M n = (cid:88) P,P (cid:48) ∈ S n A n ( P ) S KLT n ( P, P (cid:48) ) A n ( P (cid:48) ) , (2.17)mediated by a momentum kernel S KLT n dependent on the Mandelstam variables [31]. This defines anatural inner product (equivalent to the BCJ double-copy to be reviewed later). The statement ofKLT orthogonality is that the Parke-Taylor amplitudes constructed from solutions of the scatteringequations, e.g. 1 σ σ · · · σ n , (2.18)are orthogonal with respect to the KLT inner product when they arise from two different solutions.To be more specific, in the KLT product above, let A n ( P ) denote the permutations of a Parke-Taylorfactor with solution I of the scattering equations, and let A n ( P (cid:48) ) denote the permutations of a Parke-Taylor factor with solution J (cid:54) = I ; then, the product vanishes. The consequence of this fact is thatthe expression for gravity amplitudes (2.14) contains only one sum over solutions of the scatteringequations, and not two sums with mixed contributions, as would in principle arise from the KLTrelations. There are two main aspects to the BCJ story: colour-kinematics duality, and the double-copy relationbetween gauge and gravity amplitudes. The duality and the double-copy were first noticed by Bern,Carrasco and Johansson at tree level [18], and were later generalised by the same authors to loops [19].Our focus in this paper will be on tree level, and therefore we will restrict our review to this case forsimplicity.We begin by discussing colour-kinematics duality. This is a property of amplitudes in gauge theory(with or without supersymmetry). It is always possible to express gauge amplitudes as a sum over cubicdiagrams. One way to achieve this is to begin with Feynman diagrams, and then to systematically– 5 –ssign diagrams with four point vertices to cubic diagrams by introducing the missing propagatordenominators with a compensating factor in the numerator. For example,
12 3 4 56 =
12 3 4 56 s =
12 3 4 56 s . (3.1)We see that this is not unique. In gauge theory, diagrams are associated not only with a set ofpropagators and a kinematic numerator, but also with a colour factor. Since the colour factors arebuilt from the structure constants ˜ f abc of the gauge group , they are cubic in nature. Thus, we assigncontact terms to cubic diagrams by inspecting their colour structure, and introducing the missingpropagators with compensating factors in the numerators.We may therefore express any (colour-dressed) n -point amplitude in a gauge theory as A n = (cid:88) α ∈ cubic n α c α D α , (3.2)where the sum runs over the set of distinct cubic n -point diagrams. Meanwhile, the objects n α , c α and D α are the kinematic numerators, colour factors and (total) propagator denominators associated withthe diagram α . The statement of colour-kinematics duality is now simple. Take any triple ( α, β, γ ) ofdiagrams such that their colour factors are related by a Jacobi identity: c α + c β + c γ = 0 . (3.3)It is possible to find a set of valid numerators such that n α + n β + n γ = 0 . (3.4)Moreover, these kinematic numerators have the same antisymmetry properties as the colour factors.If, under interchanging two legs, a colour factor changes sign, then so does the corresponding kinematicfactor: c α → − c α ⇒ n α → − n α . (3.5)Thus, the kinematic structure mirrors the algebraic structure of the colour factors. In general, suchchoice of numerators is non-unique, but non-trivial to find. By now, various algorithms have beendescribed for finding BCJ numerators; see, for example, [23, 31–33, 36]. We shall discuss below howthe scattering equations are naturally associated with a certain set of numerators.Before we move on, let us dwell a little on one previous algorithm for finding BCJ numerators. Infact, the algorithm proposed in [32] is particularly relevant for understanding the scattering equations,as we shall see. The idea of [32] is simple. There are ( n − n points [18, 37–39]. Thus, we can view the colour-ordered amplitudes as a vector (cid:126)A in an ( n − φ aa (cid:48) where a and a (cid:48) transform underthe adjoint of two groups G and G (cid:48) . We suppose that these scalar fields interact only through a cubicvertex, with Feynman rule ˜ f abc ˜ f a (cid:48) b (cid:48) c (cid:48) . These are exactly the scalar theories considered in the previoussection. We may then construct colour-ordered amplitudes (cid:126)θ for this theory, with respect to the group G , say. Because the analogues of the kinematic numerators in this scalar theory are, in fact, directly Following [18, 19], we use a tilde to specify the normalisation ˜ f abc = Tr([ T a , T b ] T c ). – 6 –ade of the structure constants of the group G (cid:48) , they automatically satisfy colour-kinematics duality,so that there are ( n − (cid:126)θ is an element of the same vector space asthe gauge amplitudes (cid:126)A . Since there are infinitely many groups to choose from (and different particlelabellings within each group), we can find many vectors living in this space. Indeed we may find a basisof this space using the scalar amplitudes. Therefore, the gauge amplitudes (cid:126)A are a linear combinationof the scalar amplitudes: (cid:126)A = ( n − (cid:88) I =1 α ( I ) (cid:126)θ ( I ) , (3.6)where (cid:126)θ ( I ) is the vector of independent colour-ordered amplitudes of the I th scalar theory. Now,consider the k th colour-ordered gauge amplitude; denoting the numerator of the scalar theory by c (cid:48) ,it is given by A k = ( n − (cid:88) I =1 α ( I ) (cid:88) α ∈ k th ordered cubic c (cid:48) ( I ) α D α = (cid:88) α ∈ k th ordered cubic D α ( n − (cid:88) I =1 α ( I ) c (cid:48) ( I ) α . (3.7)The α -sum runs over the cubic diagrams in the k th colour order. Notice that the gauge numeratorshave been expressed as a linear combination of the c (cid:48) ( I ) ; these automatically satisfy Jacobi identitiesbecause they are built from the structure constants ˜ f a (cid:48) b (cid:48) c (cid:48) of the group G (cid:48) . In this way, we can alwaysfind colour-dual kinematic numerators. However, to do so we must invert the linear system (3.6) tocompute the expansion coefficients α ( I ) . We will see that the work of Cachazo, He and Yuan [1, 2]provides a canonical choice for the basis amplitudes, and an explicit formula for the α ( I ) . In particular,the basis is orthogonal with respect to the natural inner product, the double-copy formula, which webriefly review below.One intriguing aspect of colour-kinematics duality is that it suggests that there exists an algebraicstructure which can be used to directly compute BCJ numerators in gauge theories, including at looplevel, using a procedure analogous to Feynman rules. However, to date an understanding of thisalgebraic structure remains elusive, except in the special case of self-dual Yang-Mills theory in fourdimensions [20]. In the self-dual theory, it is known that the kinematic algebra is an area-preservingdiffeomorphism algebra. It was also shown in [20] that colour-dual kinematic numerators for MHVamplitudes in (full) Yang-Mills theory can be computed directly from a knowledge of the self-dualtheory, using a particular choice of gauge. As we shall see below, the scattering equations allow us toextend essentially the same structure to the full theory, in any dimension and for any polarisationsof the external particles. Since this material is central to this paper, we will review it in detail inSection 4.The second major aspect of the BCJ story [18, 19] is the double-copy formula which relates gaugeamplitudes and gravitational amplitudes. Given a set of colour-dual kinematic numerators n α , validfor an n -point gauge amplitude, there is an associated gravitational amplitude ( − n − M n = (cid:88) α n α n α D α . (3.8)That is, the gravitational amplitude is obtained from the gauge amplitude by replacing the colourfactors of the gauge amplitude by another copy of the kinematic numerators. More generally, we maybuild a gravitational amplitude using n -point numerators of different gauge theories. For example, we We are fixing the normalisation of gravitational amplitudes according to (2.14). – 7 –ay construct a gravitational amplitude from one set of pure Yang-Mills numerators n α and a set of N = 4 super-Yang-Mills numerators ˜ n α . The result is an amplitude in N = 4 supergravity, given by( − n − M n = (cid:88) α ∈ cubic n α ˜ n α D α . (3.9)In fact, only one of the numerators, n α or ˜ n α , needs to satisfy the colour-kinematics duality. Thedouble-copy formula has been proven at tree level [28, 40]. This procedure is therefore equivalent tothe KLT relations.At loop level, the colour-kinematics duality remains conjectural. However, it has been verified invarious examples [21, 22, 41–48]. We will start by reviewing the kinematic algebras arising in (anti)-self-dual gauge theory, and theirrelation to the colour-kinematics duality [20]. Then we will see that an extension of these algebrasapplies more generally in the context of the scattering equations.
Consider four-dimensional gauge theory in light-cone gauge, with light-cone coordinates ( u, v, w, ¯ w ).We will follow the discussion in [20]. The spacetime metric is defined such that, for any two four-vectors A and B , 2 A · B = A u B v + A v B u − A w B ¯ w − A ¯ w B w . (4.1)We take the polarisation vectors to be ε + a = (0 , k aw , , k au ) , ε − a = (0 , k a ¯ w , k au , , (4.2)so that they satisfy ε ± a · ε ± b = 0 , ε + a · ε − b = − k au k bu . (4.3)The contraction of the polarisation vectors with the momenta determines two antisymmetric bilinearforms of interest, 2 ε + a · k b = k aw k bu − k au k bw ≡ X a,b , (4.4)2 ε − a · k b = k a ¯ w k bu − k au k b ¯ w ≡ ¯ X a,b , (4.5)and we have s ab = X a,b ¯ X a,b k au k bu . (4.6)While the latter equation is valid only if k a and k b are on-shell, we can extend the definitions (4.4)-(4.5) to allow for momenta k which are not massless; for instance X , = X , + X , . For clarity,we will use capital Greek letters A, B, . . . to denote off-shell momenta later on.Let us connect these quantities to the scattering equations. We write a four-vector with spinorialindices as k α ˙ α = (cid:18) k u k w k ¯ w k v (cid:19) . (4.7)– 8 –f the four-vector is on-shell, then this matrix is singular, and it can be written in terms of two spinorsas k α ˙ α = λ α ˜ λ ˙ α . Let us choose these spinors to be λ α = (cid:18) σ (cid:19) , ˜ λ ˙ α = (cid:18) k u k w (cid:19) , where σ = k ¯ w k u = k v k w . (4.8)We can now define the spinor products ( (cid:15) = − (cid:104) ab (cid:105) = (cid:15) αβ λ ( a ) α λ ( b ) β = σ a − σ b = ¯ X a,b k au k bu , (4.9)[ ab ] = − (cid:15) ˙ α ˙ β ˜ λ ( a )˙ α ˜ λ ( b )˙ β = − X a,b . (4.10)From the relation (4.6), or equivalently s ab = (cid:104) ab (cid:105) [ ba ], we obtain X a,b = s ab σ a − σ b . (4.11)Notice that, from the definitions (4.4)-(4.5) and from momentum conservation, we have (cid:88) b (cid:54) = a X a,b = 0 , (cid:88) b (cid:54) = a ¯ X a,b = 0 . (4.12)These identities take the form of the scattering equations (2.1). The identity on the left correspondsto σ a = k a ¯ w /k au , as considered above, while the identity on the right can be seen as the equivalentstatement for its conjugate ¯ σ a = k aw /k au . So, in four dimensions and for n >
4, these are alwaystwo of the solutions to the scattering equations. In particular, these two solutions give the singlenon-vanishing contribution to the MHV and the MHV amplitudes, respectively; the factor Pf (cid:48)
Ψ inthe expression (2.13) vanishes for the other solutions with such a choice of polarisations. The n = 4case, where MHV= MHV, is special: there is a single solution to the scattering equations, and indeedthe two solutions discussed above are the same, up to an SL (2 , C ) transformation. To see this, let usfix the SL (2 , C ) freedom by performing a transformation such that ( σ , σ , σ ) → ( ∞ , , σ → σ (cid:48) = ( σ − σ )( σ − σ )( σ − σ )( σ − σ ) . (4.13)The cross ratio on the right-hand-side is equal to its conjugate (¯ σ replacing σ ), since( σ − σ )( σ − σ )( σ − σ )( σ − σ ) = s s X , ( σ − σ ) X , ( σ − σ ) = − s s , (4.14)where we used momentum conservation for the second equality, X , ( σ − σ ) = − X , ( σ − σ ).The action of SL (2 , C ) on this type of solution to the scattering equations, σ → σ (cid:48) = Aσ + BCσ + D , AD − BC = 1 , (4.15)corresponds to the standard action of SL (2 , C ) on the spinor λ , λ α → λ (cid:48) α = (cid:18) D CB A (cid:19) (cid:18) σ (cid:19) = t (cid:18) σ (cid:48) (cid:19) , t = Cσ + D, (4.16)followed by a rescaling of the other spinor, ˜ λ ˙ α → ˜ λ (cid:48) ˙ α = t − ˜ λ ˙ α .– 9 –ur interest in the quantity X A,B arises from the fact that it is the kinematic part of the vertexof self-dual gauge theory [49, 50]: (cid:64)(cid:64)(cid:64)(cid:64) (cid:0)(cid:0)(cid:0)(cid:0) = (2 π ) δ (4) ( K A + K B + K C ) X A,B f a A a B a C .AB C (4.17)Seeing the vertex as part of a trivalent graph, K A , K B , K C correspond to sums of momenta of theexternal particles; for instance, if K A = k + k and K B = k , then X A,B = X , = X , + X , . Abasic condition for X A,B to represent a vertex is that we can read it with any two of the three legsmeeting at that vertex, e.g. X A,B = X B,C = X C,A in the figure above. This is ensured by momentumconservation, and by the fact that X is an antisymmetric bilinear form.Next, we review how this vertex is related to the colour-kinematics duality, as shown in [20]. Themain point is that the vertex X A,B is the structure constant of a Lie algebra. In particular, it is theLie algebra of area-preserving diffeomorphisms in the plane w − u , generated by the vectors V + A = − e − iK A · x (cid:15) + A · ∂ such that [ V + A , V + B ] = iX A,B V + A + B . (4.18)We take the definition (4.2) of the polarisation vectors to extend to off-shell momenta. The Jacobiidentity involving the generators leads to X A,B X A + B,C + X B,C X B + C,A + X C,A X C + A,B = 0 . (4.19)Therefore, in self-dual gauge theory, we have a single cubic vertex whose kinematic part satisfies Jacobiidentities, exactly like the colour part. This is the simplest manifestation of the colour-kinematicsduality, since the duality is satisfied already by Feynman diagrams.Now we can write BCJ numerators for self-dual gauge theory. Consider, for instance, the four-pointcase. The numerators corresponding to the three channels, associated to the colour factors˜ f a a b ˜ f ba a , ˜ f a a b ˜ f ba a , ˜ f a a b ˜ f ba a , (4.20)are given by n , = αX , X , , n , = αX , X , , n , = αX , X , , (4.21)respectively. We introduced a factor α which is independent of the particle ordering, and which takesinto account overall factors coming from polarisations and their normalisation; we are not interestedin this factor for now. The construction of numerators in general is straightforward. To give a moreelaborate example, consider the seven-point trivalent graph˜ f a a b ˜ f ba c ˜ f cde ˜ f da a ˜ f ea a , (4.22)whose numerator is n = αX , X , X , X , X , . (4.23)We described how to find BCJ numerators for self-dual gauge theory. However, the tree-levelscattering amplitudes of self-dual gauge theory vanish. They correspond to helicity configurationswhere there is a single external particle with negative helicity, and it is well known that such amplitudes– 10 –anish by Ward identities (for n > In this section, we will extend the concept of the self-dual vertex discussed above so that it applies toa generic solution of the scattering equations, independently of the number of spacetime dimensions.A natural observation is that solutions still come in pairs in a certain sense. If our choice of externalmomenta is such that the Mandelstam variables s ab are real-valued, then the complex conjugate of asolution to the scattering equations is clearly also a solution. However, it is not clear that one canobtain a relation analogous to (4.6) from those two conjugate solutions; indeed, generically it is notpossible. To be more precise, let us define these quantities for a given solution σ a of the scatteringequations as X a,b ≡ s ab σ a − σ b , ¯ X a,b ≡ ( σ a − σ b ) h a h b , (4.24)and we also set X a,a ≡
0. We introduced the quantities h a , which are the analogues of the p au fromthe previous section, so that the definition of ¯ X a,b generalises the relation (4.9). The h a must obeythe constraints n (cid:88) a =1 h a = n (cid:88) a =1 σ a h a = 0 , (4.25)so that we still have (cid:88) b (cid:54) = a X a,b = 0 , (cid:88) b (cid:54) = a ¯ X a,b = 0 . (4.26)We shall impose no other constraints on the h a . The question of whether X and ¯ X are obtained fromconjugate solutions is very simple: can ¯ X a,b , as defined in (4.24), be given by ¯ X a,b = s ab / (¯ σ a − ¯ σ b ) fora solution to the scattering equations denoted by ¯ σ a ? To address this question, one can try to solvethe equations X a,b ¯ X a,b = s ab h a h b (4.27)using ¯ X a,b = s ab / (¯ σ a − ¯ σ b ). We find numerically that there is no solution for h a in general, unless thepair of solutions σ a and ¯ σ a admits a four-dimensional interpretation in terms of projective holomorphicand anti-holomorphic spinors, as in the previous section. Henceforth, we will consider ¯ X a,b withoutassuming that it is in any way related to solutions of the scattering equations other than the one usedin its definition (4.24). Notice that this is always the case at five points, since the scattering occurs in a four-dimensional subspace (due tomomentum conservation). Beyond five points, it is only possible generically in four spacetime dimensions, and there isa single pair of such solutions. – 11 –he generalisation (4.24) of the vertices X and ¯ X breaks the symmetry between them encounteredin the previous section, but maintains a crucial property: the equations (4.26). We shall now see thatthese equations give the consistency condition for X (or ¯ X ) to behave like a three-point vertex (4.17).Let us define the “off-shell” version of X as X A,B = (cid:88) a ∈{ A } X a,B = (cid:88) b ∈{ B } X A,b = (cid:88) a ∈{ A } (cid:88) b ∈{ B } X a,b , (4.28)where { A } and { B } are two sets of external particles. The picture is that, for a vertex like (4.17) ina tree-level graph, we have a partition of the external particles into the three sets { A } , { B } and { C } .These sets contain the particles connected through the graph to the lines A , B and C of the vertex,respectively. Then we must have, as a consistency condition, that the vertex can be read with anytwo of the three lines, X A,B = X B,C = X C,A . (4.29)This is precisely what the scattering equations guarantee. Let us see this in detail, X A,B = (cid:88) a ∈{ A } (cid:88) b ∈{ B } X a,b = − (cid:88) a ∈{ A } (cid:88) c/ ∈{ B } X a,c == − (cid:88) a ∈{ A } (cid:88) c ∈{ A } X a,c − (cid:88) a ∈{ A } (cid:88) c ∈{ C } X a,c = − X A,C = X C,A . (4.30)The scattering equations where used in the second equality, first line. From the first to the secondline, we used that the complement of { B } is { A } ∪ { C } . In the second line, the first term vanishesbecause X A,A = 0.Consider now a partition of the external particles into four sets, { A } , { B } , { C } , { D } , correspondingto the four lines involved in a Jacobi identity. The Jacobi identity for the vertices X (or ¯ X ) followsdirectly from (4.29), and therefore from the scattering equations: X A,B X C,D + X B,C X A,D + X C,A X B,D == − X A,B ( X A,D + X B,D ) − X B,C ( X B,D + X C,D ) − X C,A ( X C,D + X A,D ) = 0 . (4.31)We have defined objects X and ¯ X which can be interpreted as the kinematic part of cubic vertices,and which obey Jacobi identities. These Jacobi identities imply that we are dealing with Lie algebras,[ ˆ V + A , ˆ V + B ] = iX A,B ˆ V + A + B , [ ˆ V − A , ˆ V − B ] = i ¯ X A,B ˆ V − A + B . (4.32)The na¨ıve extension of the representation (4.18) doesn’t work; there is in general no set of vectors ˆ ε + a such that 2ˆ ε + a · k b = X a,b , mirroring (4.4). Nevertheless, there are Lie algebras whose generators ˆ V ± A can be associated to lines in a trivalent graph. We will see in later sections how the elements ˆ V ± A canbe used to write BCJ numerators for gauge theory amplitudes. X -amplitudes It is now natural to address the following question. The vertices X from the previous section, corre-sponding to anti-holomorphic spinor brackets in four dimensions (and their off-shell extension) led toBCJ numerators for tree-level self-dual amplitudes. These amplitudes vanish, although the individualnumerators don’t. So we would like to know what happens if we consider a certain solution to thescattering equations, σ a , and compute the “amplitudes” constructed uniquely from the vertices X – 12 –ssociated to σ a , which we can call the X -amplitudes. We find that they vanish for n ≥
4, just likein the self-dual case. On the other hand, we find that the analogous ¯ X -amplitudes don’t vanish. (Forthe special four-dimensional solutions to the scattering equations, the vanishing of the ¯ X -amplitudesrequires specific values for the h a , corresponding to p au .)The proof that the X-amplitudes vanish closely follows an argument given by Cangemi [49]. First,notice that the scattering equations for n particles imply that, for m ≤ n , m − (cid:88) j =1 j (cid:88) s =1 m (cid:88) t = j +1 X s,t σ j,j +1 = ( p + · · · + p m ) . (4.33)We may prove this result by a simple rearrangement of the summation: m − (cid:88) j =1 j (cid:88) s =1 m (cid:88) t = j +1 X s,t σ j,j +1 = (cid:88) ≤ s ≤ j 4) = A ( X )4 = X , X , s + X , X , s = σ ( X , + X , ) + σ ( X , + X , ) σ σ . (4.38)By Eq. (4.33), we see that we get the simple result A ( X )4 = p σ σ . (4.39)Thus, when p is on shell, the amplitude vanishes. We can also consider the case p (cid:54) = 0; for example,this could be a four-point subamplitude of a larger amplitude. Indeed, let us consider the five-point X -amplitude as a further illustration. The amplitude is A ( X )5 = X , X , X , s s + (cid:18) X , X , s + X , X , s (cid:19) X , s + (cid:18) X , X , s + X , X , s (cid:19) X , s . (4.40)The last two terms in this expression have the structure of off-shell colour-ordered four-point subdi-agrams, connected to a three-point tree involving particle 5. Meanwhile, the first term includes twothree-point vertices, which we view as off-shell colour-ordered three-point subdiagrams, connected toa three-point tree involving particle 5. Thus, A ( X )5 = 1 σ σ X , + ( p + p + p ) σ σ X , s + ( p + p + p ) σ σ X , s (4.41)= 1 σ σ σ ( σ X , + σ X , + σ X , ) (4.42)= p σ σ σ . (4.43)– 13 – + 112 j + 1 KQjmj + 2 JJ Figure 1 . Construction of X -amplitudes using Berends-Giele recursion. Once again, we see that the amplitude vanishes.We will prove that the X -amplitudes always vanish by induction. These amplitudes can beconstructed recursively in terms of graphs with one leg off shell, using Berends-Giele recursion [51];see Figure 1. The concept is simple; let us consider a subgraph with m external legs and a singleinternal leg (from the point of view of the complete graph), which we can call m + 1. This leg isoff-shell until the last stage of the recursion, when we get the complete graph, in which case it is thelast remaining external leg. The leg m + 1 must connect to a three-point vertex. The other legs of thisvertex, call them K and Q , connect to colour-ordered subdiagrams with a single off-shell leg. Let usdefine this single off-shell graph, with j on-shell legs and a final leg K off shell to be J (1 , , . . . , j ; K ).By recursion, the object J satisfies J (1 , , . . . , m ; m + 1) = m − (cid:88) j =1 J (1 , , . . . , j ; K ) J ( j + 1 , j + 2 , . . . , m ; Q ) 1 K Q X K,Q . (4.44)The colour-ordered amplitude may simply be obtained as A ( X ) (1 , , . . . , n ) = J (1 , , . . . , n − n ) (cid:12)(cid:12) p n =0 . (4.45)We adopt the inductive hypothesis that J (1 , , . . . , m ; m + 1) = 1 σ σ · · · σ m − ,m (cid:32) m (cid:88) s =1 p s (cid:33) (4.46)holds for m + 1 ≤ n ; this is easily checked for small n , as in the four and five-point examples above.Now, consider the m + 2 point case, obtained from the recursion formula: J (1 , , . . . , m + 1; m + 2) = 1 σ σ · · · σ m,m +1 m (cid:88) j =1 X ... + j, ( j +1)+ ... +( m +1) σ j,j +1 , (4.47)= 1 σ σ · · · σ m,m +1 (cid:32) m +1 (cid:88) s =1 p s (cid:33) , (4.48)where we have used the identity Eq. (4.33). Thus, our inductive hypothesis is proven, and consequentlyall the X -amplitudes (with the exception of the three-point case) vanish identically.– 14 – BCJ numerators from the scattering equations In this section, we will show how to construct BCJ numerators based on elements of the Lie algebrasassociated to solutions of the scattering equations. We will start by giving the general idea, and thenwe will present the complete BCJ numerators. Finally, we will consider a special SL (2 , C ) frame inwhich the numerators simplify, allowing for a simple proof of their validity. The starting point is a very simple observation. Consider three-point scattering, and take vectors ofthe type V a = e − ik a · x ε a · ∂, a = 1 , , , (5.1)as an example. The most natural Lorentz invariant quantity constructed from the vectors is V · [ V , V ] + V · [ V , V ] + V · [ V , V ] == − ie − i ( k + k + k ) · x (( ε · ε )( k − k ) · ε + ( ε · ε )( k − k ) · ε + ( ε · ε )( k − k ) · ε ) . (5.2)Indeed, this is the three-point gluon amplitude. The natural generalisation to four points, in the caseof the s -channel, is n , = V · [ V , [ V , V ]] + V · [[ V , V ] , V ] + V · [ V , [ V , V ]] + V · [[ V , V ] , V ] , (5.3)which corresponds to the colour factor ˜ f a a b ˜ f ba a . Notice how the structure of the commutatorsreflects the orientation of the vertices. Viewing this quantity as a numerator, it turns out that theBCJ Jacobi identities follow from the standard Jacobi identities of the algebra of spacetime vectors, n , + n , + n , == V · ([ V , [ V , V ]] + [ V , [ V , V ]] + [ V , [ V , V ]])+ V · ([ V , [ V , V ]] + [ V , [ V , V ]] + [ V , [ V , V ]])+ V · ([ V , [ V , V ]] + [ V , [ V , V ]] + [ V , [ V , V ]])+ V · ([[ V , V ] , V ] + [[ V , V ] , V ] + [[ V , V ] , V ]) = 0 . (5.4)It is straightforward to generalise this construction for any number of external particles, so that theJacobi identities hold. Take the example (5.3). If we substitute V r by T a r and the · product by standardmatrix multiplication, and then take the trace, we find that each term reproduces the correspondingcolour factor. This will be the general rule. So, the numerator of the trivalent graph α is n α = n (cid:88) a =1 V a · G ( α ) a , (5.5)where G ( α ) a is the commutator structure of graph α as read from particle a . Let us give anotherexample, the seven-point graph with colour factor˜ f a a b ˜ f ba c ˜ f cde ˜ f da a ˜ f ea a . (5.6)Its numerator is n = V · [ V , [ V , [[ V , V ] , [ V , V ]]]] + V · [[ V , [[ V , V ] , [ V , V ]]] , V ]+ V · [[[ V , V ] , [ V , V ]] , [ V , V ]]+ V · [ V , [[ V , V ] , [[ V , V ] , V ]]] + V · [[[ V , V ] , [[ V , V ] , V ]] , V ]+ V · [ V , [[[ V , V ] , V ] , [ V , V ]]] + V · [[[[ V , V ] , V ] , [ V , V ]] , V ] . (5.7)– 15 –ur BCJ numerators for gluon amplitudes will be based on the same idea, although the algebrasdo not admit this representation in general. The idea described here follows from the self-dual storythat we saw before, and it was presented in [52], where it was claimed to apply directly to MHVamplitudes, which is a particular case of the analysis to be presented next. The idea was also presentedindependently in [36], where it was used to obtain basis amplitudes in the spirit of [32]. We will construct BCJ numerators for each contribution to the gauge theory amplitude correspondingto a solution of the scattering equations. According to (2.13), each of these contributions is of thetype Parke-Taylor factor × permutation-invariant factor . (5.8)The permutation-invariant factor can be pulled out. Therefore, we are really looking for BCJ numer-ators reproducing the Parke-Taylor amplitudes, A ( I )PT = Tr( T a T a · · · T a n ) σ ( I )12 · · · σ ( I ) n + non-cyclic permutations , (5.9)where I denotes the particular solution to the scattering equations. Below, we will omit the label I inorder to simplify the notation.The Lie algebras (4.32) associated to each solution of the scattering equations are not of the sametype as the Lie algebra discussed in the subsection above. However, the latter will serve as inspiration.First, it will be convenient to define the action of an element of the Lie algebra on another as, in thecase of the X -algebra, ˆ V + A ˆ V + B = iX A,B ˆ V + A | B . (5.10)Here, ˆ V + A | B is not itself an element of the X -algebra, but satisfies the propertyˆ V + A | B + ˆ V + B | A = ˆ V + A + B . (5.11)This is consistent with [ ˆ V + A , ˆ V + B ] = iX A,B ˆ V + A + B , and it clearly mirrors the vector algebra seen above,where (5.10) corresponds to the first part of the vector commutator. Based on this idea, and furtherspecifying ˆ V ±∅| A = ˆ V ± A , we define ˆ V + A | B ˆ V ± C | D = iX B,C + D ˆ V ± A + B + C | D , (5.12)ˆ V − A | B ˆ V ± C | D = i ¯ X B,C + D ˆ V ± A + B + C | D . (5.13)Moreover, we define a symmetric product ∗ such thatˆ V ± A | B ∗ ˆ V ± C | D = 0 , ˆ V + A | B ∗ ˆ V − C | D = (cid:26) − h B h D if K A + K B + K C + K D = 0 , h A = (cid:80) a ∈{ A } h a .The sole goal of these definitions, for our purposes, is to determine objects such as, at four points,ˆ V − ∗ ˆ V +2 ˆ V +3 ˆ V +4 = 2 X , X , h h = 2 X , X , h h , (5.15)where we used the scattering equations. If we consider commutators, and recall the conditions (4.25)on the h a , we getˆ V − ∗ [ ˆ V +2 , [ ˆ V +3 , ˆ V +4 ]] = 2 X , X , h ( h + h + h ) = − h X , X , . (5.16)– 16 –e can now consider complete numerators such as n − + , + + = ˆ V − ∗ [ ˆ V +2 , [ ˆ V +3 , ˆ V +4 ]] + ˆ V +2 ∗ [[ ˆ V +3 , ˆ V +4 ] , ˆ V − ] + ˆ V +3 ∗ [ ˆ V +4 , [ ˆ V − , ˆ V +2 ]] + ˆ V +4 ∗ [[ ˆ V − , ˆ V +2 ] , ˆ V +3 ] , (5.17)which satisfy Jacobi identities exactly for the same reason as in (5.4). The extension of this constructionof numerators to n points is straightforward, as pointed out in the previous subsection.Regarding numerators where only one of the ˆ V ’s is of the ( − ) type, as in the example above, thereare two important observations to make, which extend to any number of particles. The first is thatwe do not need to consider a sum over all the particles when we take ˆ V a ∗ [commutators]. For instance,in our example, n − + , + + = − h X , X , = 2 ˆ V − ∗ [ ˆ V +2 , [ ˆ V +3 , ˆ V +4 ]] . (5.18)In general, the sum over all the particles is equivalent to considering only the ( − ) particle, up to afactor of two; i.e. for graph α , if r labels the ( − ) particle, we have n α,r = n (cid:88) a =1 ˆ V a ∗ ˆ G ( α ) a = 2 ˆ V − r ∗ ˆ G ( α ) r . (5.19)We have verified this numerically, but it would be nice to have a proof.The second observation is that numerators with a single ( − ) particle contain only X vertices,apart from an overall factor proportional to h r . Therefore, these are the numerators of what we called X -amplitudes. We have proven previously that these amplitudes vanish, even though the numeratorsdo not. Notice also that, if we only had (+) particles, the numerators themselves would vanish due tothe product ∗ .We now proceed to the main result of this paper. The Parke-Taylor amplitude (5.9) is obtainedwith a very simple prescription: we consider numerators of the type described above but with two ( − )particles. Let these particles be r and s . We obtain A PT = α rs (cid:88) α ∈ cubic n α, rs c α D α , (5.20)where the factor α rs , which is independent of the particle ordering, is given by α rs = − (cid:34) i n ( σ r − σ s ) h r h s ( h r + h s ) n (cid:88) a =1 σ a h a (cid:35) − . (5.21)The special dependence of α rs on particles r and s compensates that of the numerators n α, rs , sothat the choice of these particles is irrelevant for A PT . We view this fact as an analogue of choosingarbitrarily two columns/rows to be eliminated in the Pf (cid:48) Ψ defined in (2.10). For two ( − ) particles,we find the property analogous to (5.19) that n α,rs = n (cid:88) a =1 ˆ V a ∗ ˆ G ( α ) a = 2 (cid:16) ˆ V − r ∗ ˆ G ( α ) r + ˆ V − s ∗ ˆ G ( α ) s (cid:17) . (5.22)We emphasise that these results do not depend on the values of the quantities h a , as long as they obeythe conditions (4.25). Notice that ( ± ) does not refer to the polarisation of the external particles. That information is included in thepermutation-invariant factor, as mentioned in (5.8). The labels ( ± ) will be used only in reproducing the Parke-Taylorfactor. – 17 –e conclude that a natural choice of BCJ numerator for graph α in a gauge theory amplitude is n α = ( n − (cid:88) I =1 α ( I ) rs n ( I ) α, rs Υ ( I ) , with Υ ( I ) = Pf (cid:48) Ψ ( I ) det (cid:48) Φ ( I ) , (5.23)where we reintroduced the label I of each solution to the scattering equations. Since the factors α ( I ) rs and Υ ( I ) are independent of the particle ordering, the complete BCJ numerators n α satisfy the sameJacobi identities as the numerators n ( I ) α, rs . We have presented above the complete BCJ numerators based on the kinematic algebras. Their validityhas been verified numerically up to eight points. In the following, we will use a certain SL (2 , C ) framein order to obtain a simpler form of the numerators, which allows us to prove their validity for anymultiplicity. This form of the numerators reduces to the one described in [20] for MHV amplitudes.We mentioned above that the numerators obtained with the elements ˆ V ± a are, in the case of one( − ) particle, the numerators of an X -amplitude (which vanishes), and, in the case of two ( − ) particles,the numerators of a Parke-Taylor amplitude (up to a permutation-invariant factor). In the case of one( − ) particle, there are only X vertices, while in the case of two ( − ) particles there is a single ¯ X vertex,the rest being X vertices. This is easily checked by direct inspection. We are interested in the case oftwo ( − ) particles, and we want to choose an SL (2 , C ) frame such that we force the single vertex ¯ X tobe attached to a certain external particle. Let us say that this reference particle is particle n . This isachieved by taking the limit σ n → ∞ . (5.24)In this limit, due to the second condition in (4.25), we must have also h n → 0, so that σ n h n is finite.Therefore, we have X n,A → , ¯ X n,A → ( σ n h n ) h A , (5.25)which leads to the vanishing of all the contributions for which particle n is attached to an X vertex,rather than the ¯ X vertex. It is then trivial to write down the BCJ numerators: they are the same as forthe X -amplitudes, except that one of the vertices – the one attached to particle n – is ¯ X . Therefore,all the Jacobi identities involving only X vertices are satisfied. The only additional requirement isthat the Jacobi identities involving propagators connected to particle n are also satisfied. If the otherthree (generically off-shell) lines connected to such a propagator are A , B and C , we have¯ X n,A X B,C + ¯ X n,B X C,A + ¯ X n,C X A,B → ( σ n h n ) ( h A X B,C + h B X C,A + h C X A,B )= ( σ n h n ) ( h A ( X B,C + X B,A ) + h B ( X C,A + X B,A ) − h n X A,B )= ( σ n h n ) ( h A X n,B + h B X A,n − h n X A,B ) → , (5.26)where we used the condition (cid:80) a h a = 0 in the third line and the scattering equations in the fourthline.We now proceed to prove the validity of these BCJ numerators using Berends-Giele recursion.The procedure is essentially the same as in Section 4.3, except that the final vertex in the recursion is– 18 –ot X , but ¯ X . Therefore, A PT (1 , , . . . , n ) = α n n − (cid:88) j =1 J (1 , , . . . , j ; K ) J ( j + 1 , j + 2 , . . . , n − Q ) 1 K Q ¯ X n,K (5.27)= α n σ σ · · · σ n − ,n − n − (cid:88) j =1 ¯ X n, ...j σ j,j +1 , (5.28)where α n is a proportionality coefficient independent of particle ordering. It is useful to rearrange thesummation on the last line: n − (cid:88) j =1 ¯ X n, ...j σ j,j +1 = (cid:88) ≤ s ≤ j We thank Lionel Mason, Gustav Mogull, V. Parameswaran Nair and David Skinner for helpful discus-sions. RM is supported by the European Commission through a Marie Curie Fellowship, while DOCis supported in part by the STFC grant Particle Physics at the Tait Institute. References [1] F. Cachazo, S. He and E. Y. Yuan, arXiv:1306.6575 [hep-th].[2] F. Cachazo, S. He and E. Y. Yuan, arXiv:1307.2199 [hep-th].[3] R. Roiban, M. Spradlin and A. Volovich, Phys. Rev. D (2004) 026009 [hep-th/0403190]. – 21 – 4] E. Witten, Commun. Math. Phys. (2004) 189 [hep-th/0312171].[5] F. Cachazo and Y. 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