Scattering equations and virtuous kinematic numerators and dual-trace functions

Inspired by recent developments on scattering equations, we present a constructive procedure for computing symmetric, amplitude-encoded, BCJ numerators for n-point gauge-theory amplitudes, thus satisfying the three virtues identified by Broedel and Carrasco. We also develop a constructive procedure for computing symmetric, amplitude-encoded dual-trace functions τ for n-point amplitudes. These can be used to obtain symmetric kinematic numerators that automatically satisfy color-kinematic duality. The Sn symmetry of n-point gravity amplitudes formed from these symmetric dual-trace functions is completely manifest. Explicit expressions for four- and five-point amplitudes are presented.

Because the color factors c i in eq. (1.1) satisfy Jacobi identities and hence are not linearly independent, the kinematic numerators n i for a given n-point gauge-theory amplitude are not uniquely determined. This generalized gauge freedom can be used to require that n i satisfy the same Jacobi identities as the color factors [5]. As a consequence all kinematic numerators n i may be expressed in terms of an independent set n 1γn associated with half-ladder diagrams. Defining color-ordered amplitudes A α as the coefficients of the gauge-theory amplitude expressed in the trace basis (1.8) it can then be shown that the half-ladder numerators must satisfy Because there are (n − 2)! independent half-ladder numerators n 1γn and because rank m = (n − 3)!, this equation cannot be uniquely inverted, meaning that n 1γn still possesses some residual generalized gauge freedom even after color-kinematic duality is imposed. Many different representations of the kinematic numerators are possible, e.g., see refs. [26][27][28].

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In an effort to define an economical and natural representation for the numerators, Broedel and Carrasco [29] enumerated three virtues that kinematic numerators would ideally possess: (1) color-kinematic duality (numerators obey the same symmetries as the associated color factors), (2) amplitude-encoding (external-state dependence is expressed in terms of color-ordered amplitudes A α ), and (3) symmetry (numerator functions corresponding to diagrams with the same topology but different labelings of the external legs are all related by permutations of their arguments). They then proceeded to construct virtuous numerators for four-and five-point tree-level amplitudes, and for six-point MHV amplitudes in four dimensions, by assuming a general ansatz for the numerators and imposing functional constraints. Their approach becomes impracticable, however, for larger values of n, and they voiced the hope that a constructive procedure for virtuous numerators for arbitrary n-point amplitudes could be found.
In section 3 of this paper, we present a procedure to produce n-point kinematic numerators satisfying all three virtues of Broedel and Carrasco. We begin with a specific set of nonsymmetric numerators from ref. [4] obtained using the properties of m(α|β). By applying arbitrary permutations to the external legs, we derive other nonsymmetric sets of numerators. Finally, we generate a symmetric numerator by summing over all such representations. 2 Our expression manifestly satisfies all Jacobi identities and diagram symmetries for the numerators without having to invoke the BCJ relations for the color-ordered amplitudes. We give explicit results for four-and five-point amplitudes, but the procedure is completely general.
The double-copy construction relates gauge-theory to gravity amplitudes by replacing the color factor c i in eq. (1.1) with a functionñ i of kinematic variables [5]. Bern and Dennen [31] suggested that, in a similar way, the gravity amplitude could be obtained from the gauge-theory amplitude (1.8) by replacing the color trace factor Tr[α] with a dual-trace function τ α of the kinematic variables (1.10) It can then be shown that Once again, because there are (n − 1)!/2 independent τ α and rank m = (n − 3)!, this equation cannot be uniquely inverted; many choices of dual-trace functions are possible. In ref. [31], Bern and Dennen presented explicit expressions for τ α in terms of n 1γn for four-, five-, and six-point amplitudes, and verified the existence of such expressions through nine points, but did not present a general procedure for arbitrary n. Further progress was achieved by Du et al. [32,33].
In section 4 of this paper, we present a procedure to generate n-point dual-trace functions τ α that possess the three virtues identified by Broedel and Carrasco: (1) they automatically lead to kinematic numerators that obey color-kinematic duality, (2) they are JHEP07(2014)143 expressed in terms of A α , i.e., they are amplitude-encoded, and (3) they are symmetric: a single function τ suffices to determine the full set via permutations of its arguments. We first identify a specific set of nonsymmetric dual-trace functions that satisfy eq. (1.11), apply arbitrary permutations to the external legs to obtain other representations, and then generate a symmetric dual-trace function by summing over all such representations. We present explicit expressions for four-and five-point functions, but again the procedure is completely general.
The expressions for τ α obtained in ref. [31] were observed to obey Kleiss-Kuijf relations, and the authors argued that this was a necessary condition for symmetric dual-trace functions expressed in terms of kinematic numerators. The results for τ α obtained in this paper do not satisfy Kleiss-Kuijf relations; they are expressed directly in terms of color-ordered amplitudes, and so are able to avoid this requirement. Thus, even after the virtues of Broedel and Carrasco are imposed, there remains some freedom in the dual-trace function.
This paper is structured as follows. Section 2 reviews the tree-level amplitudes of gauge, gravity, and double-color scalar theories, and the color-kinematic dualities that relate them. Section 3 describes the criteria of Broedel and Carrasco for virtuous numerators, outlines our procedure for constructing them, and presents explicit results for four-and five-point amplitudes. Section 4 reviews the properties of the dual-trace functions of Bern and Dennen, and then outlines our procedure for constructing them, again presenting explicit results for four-and five-point amplitudes. Section 5 discusses open questions, and an appendix contains the proof of the procedure used to produce symmetric kinematic numerators and dual-trace functions.

Review of color-kinematic duality
In this section, we review tree-level amplitudes of gauge, gravity, and double-color scalar theories, and the color-kinematic dualities that relate these various amplitudes.
We begin with the tree-level n-gluon amplitude A(1, 2, · · · , n), where the arguments denote the momentum p j , polarization ε j , and color a j of the external particles j = 1, · · · , n. The color dependence can be specified in terms of a set of color factors c i obtained by sewing together cubic vertices f abc ≡ Tr(T a [T b , T c ]), where T a denote generators in the fundamental representation of the color group. The n-point amplitude is then expressed as a sum over all cubic diagrams [5] A(1, 2, · · · , n) = i c i n i (1, 2, · · · , n) d i (1, 2, · · · , n) . (2.1) The denominator d i (1, 2, · · · , n) associated with the color diagram c i is a product of the diagram propagators, and is a function of the external momenta p j . The kinematic numerator n i (1, 2, · · · , n) associated with c i is a function of both p j and ε j . (As usual, terms in the Feynman diagram expansion arising from quartic vertices can be parceled into terms involving only cubic vertices.)

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The color factors c i can be expanded in a trace basis The trace basis is independent provided we restrict the sum to permutations of n indices modulo cyclic permutations. The gauge-theory amplitude (2.1) can be decomposed in this basis where the arguments of the color-ordered amplitudes A α (1, 2, · · · , n) denote p j and ε j . Since the trace basis is independent, the color-ordered amplitudes are well-defined and gauge invariant. Because of the invariance of the full amplitude A(1, 2, · · · , n) under permutations of the arguments and the independence of the trace basis, 3 the color-ordered amplitudes are all related to one another via where A(i, j, · · · ) ≡ A 12···n (i, j, · · · ), the coefficient of Tr(T a 1 T a 2 · · · T an ). The color-ordered amplitudes are thus "symmetric" functions, in the sense of Broedel and Carrasco [29]. The color factors c i in eq. (2.1) are not independent but satisfy various Jacobi identities which can be expressed [34][35][36] as i ℓ i c i = 0, where ℓ i are left null vectors of the matrix M iα : i ℓ i M iα = 0. The matrix M iα has rank (n − 2)! so there are (n − 2)! independent color factors. The matrix M iα also possesses a set of right null vectors α M iα r α = 0. By eq. (2.4), these give rise [34][35][36] to a set of constraints α A α r α = 0 on the color-ordered amplitudes, viz., the Kleiss-Kuijf relations [37].
Because of the linear dependence of the color factors c i , the kinematic numerators n i (1, 2, · · · , n) in eq. (2.1) are not uniquely determined but can undergo what are termed generalized gauge transformations [38] without altering the amplitudes. The insight of Bern, Carrasco, and Johannson [5] is that there exists a generalized gauge choice for which the numerators satisfy the same Jacobi identities as the color factors n i (1, 2, · · · , n) + n j (1, 2, · · · , n) + n k (1, 2, · · · , n) = 0 . (2.7) We will refer to a set of kinematic numerators satisfying eq. (2.7) as BCJ numerators. Such a choice is not unique: there remain residual generalized gauge transformations that preserve the Jacobi identities (2.7).

Double-color scalar amplitudes
In their seminal paper [5], Bern, Carrasco, and Johansson showed that, given a set of BCJ numerators n i (1, 2, · · · , n), one can obtain tree-level gravity amplitudes 4 by replacing c i in eq. (2.1) with the kinematic numeratorsñ i (1, 2, · · · , n) of a second gauge theory (the double-copy procedure) where theñ i (1, 2, · · · , n) need not themselves satisfy the Jacobi identities [38]. An alternative double-copy procedure replaces the n i (1, 2, · · · , n) in eq. (2.1) with the color factorsc i of a second color group . (2.14) Such an expression corresponds [4] to the n-point amplitude of a theory of massless scalar particles φ aa' in the adjoint of the color group U(N ) × U(Ñ ) with cubic interactions of the form

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where f abc andf a'b'c' are the structure constants of U(N ) and U(Ñ ). Using together with eq. (2.2), the double-color amplitude (2.14) can be written as  .17), "double-partial amplitudes." They showed that m(α|β) computes the sum of all trivalent scalar diagrams that can be regarded both as α-color-ordered and β-color-ordered, where each diagram's contribution is given by the product of its propagators. The double-partial amplitudes satisfy (on both sides) Kleiss-Kuijf relations Using eq. (2.11), the double-color amplitude (2.14) can also be expressed as The double-partial amplitudes in this equation, m(1γn|1δn), are essentially the entries in the (n − 2)! × (n − 2)! matrix considered in ref. [41]. There it was shown by consideration of low values of n that, as a result of momentum conservation, the rank of this matrix is (n − 3)!. Cachazo et al. showed that the double-partial amplitudes could alternatively be expressed as eq. (1.7), which shows explicitly that m(α|β) has rank (n−3)!. It consequently possesses (n − 3)! − (n − 2)! additional null vectors, dependent on kinematic invariants, which implies that the double-partial amplitudes satisfy BCJ relations [5] in addition to Kleiss-Kuijf relations.

Gauge-theory amplitudes
Just as the Jacobi identities for the color factors (2.6) imply that they can be expressed as a linear combination of independent half-ladder color factors (2.11), so the Jacobi identities (2.7) imply that the kinematic numerators can be written in terms of an independent basis of numerators n 1γn (1, 2, · · · , n) associated with c 1γn

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Then, analogous to eq. (2.19) for double-color scalar amplitude, we can write the gaugetheory amplitude (2.1) as We can also substitute eq. (2.20) into eq. (2.4) to obtain the color-ordered amplitudes As noted above, the matrix m(α|1γn) has rank (n−3)! and has (n−2)!−(n−3)! additional 6 null vectors (dependent on kinematic invariants) which give rise [7,41,42] to the BCJ relations [5] among the color-ordered amplitudes. The kernel of the matrix corresponds to (n − 2)! − (n − 3)! degrees of freedom of residual generalized gauge transformations of the BCJ numerators (i.e. generalized gauge transformations that preserve the Jacobi constraints (2.7)). Because of this residual gauge freedom, eq. (2.22) cannot be inverted to obtain unique expressions for BCJ numerators in terms of color-ordered amplitudes. To invert eq. (2.22), we must first make a choice of gauge. One possible gauge choice is to set which allows us to restrict the sum in eq. (2.22) to S n−3 . Now consider the subset of color-ordered amplitudes (2)···γ(n−2)n−1,n (1, 2, · · · , n) . (2.24) It was shown in ref. [4] by using KLT orthogonality [2] that the (n − 3)! × (n − 3)! submatrix appearing in eq. (2.24) is invertible, with the inverse given by the (negative of the) momentum kernel 7 [27,[43][44][45] Thus the nonzero members of the independent basis of numerators can be expressed as [4] n 1γ (2) The full set of (BCJ) numerators, including the half-ladders not included in the independent set, can then be obtained from eqs.

Gravity amplitudes
By the double-copy construction, we can replace c 1γn in eqs. (2.12) and (2.21) with n 1γn (1, 2, · · · , n) to obtain the gravity amplitude The specific choice of gauge in eqs. (2.23) and (2.26) can then be used to write which is one possible form of the field-theory limit of the KLT relation [27,46,47].

Virtuous kinematic numerators
As described in the previous section, even after requiring the kinematic numerators to satisfy color-kinematic duality, a certain amount of generalized gauge freedom remains. To fix this residual gauge freedom in a natural, economical, and possibly unique way, Broedel and Carrasco [29] identified three desirable features that a set of numerators should possess: 1. color-kinematic duality: BCJ numerators obey the same symmetries as their associated color factors. Such numerators can be used to construct gravity amplitudes using the double-copy procedure.
2. amplitude-encoding: the external state dependence (e.g., helicities) of the numerators is expressed in terms of color-ordered amplitudes. Such a representation would also be independent of the number of space-time dimensions.
3. symmetry: all of the numerators n i for a given diagram topology can be expressed via permutations of the arguments of a single function. This virtue is thus one of economy, and also makes the gravity amplitudes constructed from these numerators manifestly invariant under permutations of the external legs.
Kinematic numerators that satisfy all three features are dubbed "virtuous." As we saw in the previous section, numerators that satisfy the first virtue are expressed in terms of an independent basis of half-ladder numerators n 1γn (1, 2, · · · , n) via eq. (2.20). Color-ordered amplitudes are expressed as in terms of these as (3.1) Were we able to invert this equation, we would possess numerators that also satisfy the second virtue of amplitude-encoding. Because det m = 0, eq. (3.1) has no unique inverse; there JHEP07(2014)143 exists rather a family of generalized inverses [42,48], each corresponding to a particular gauge choice for n 1γn . One such choice is given by [4] n 1γ(2)···γ(n−1)n (1, 2, · · · , n) = 0, This choice, however, does not satisfy the third virtue of symmetry. Next, we describe how one can generate a symmetric representation of numerators starting from a nonsymmetric representation, such as that given in eq. (3.2). 8 We show in the appendix that, given one set of numerators n α (1, 2, · · · , n), one can use an arbitrary permutation β acting on the external legs to generate another set of valid numerators i.e., these also satisfy eq. (3.1). If we average over all permutations β ∈ S n , the resulting function n(1, 2, · · · , n) ≡ n sym (1, 2, · · · , n) = 1 n! β∈Sn n β −1 (β(1), · · · , β(n)) (3.4) will be symmetric; i.e., all half-ladder numerators are given by permutations of the arguments of the single function n(1, 2, · · · , n). Symmetric numerator functions for topologies other than the half-ladder are then obtained from n(1, 2, · · · , n) via eq. (2.20). Since the individual sets of numerators n ′ α (1, 2, · · · , n) satisfy eq. (3.1), so does the average (3.4): A(α(1), · · · , α(n)) = γ∈S n−2 m(α|1γn) n(1, γ(2), · · · , γ(n − 1), n) . (3.5) Hence we have constructed a representation of kinematic numerators satisfying all three virtues of Broedel and Carrasco. We apply this procedure in the following subsections to obtain explicit expressions for virtuous numerators for four-and five-point amplitudes. The expressions become increasingly lengthy for higher-point amplitudes, but the important point is that there exists a constructive proof of the existence of virtuous numerators for all tree-level n-gluon amplitudes. Whether such numerators are unique remains an open question.
In a recent paper [30], Fu, Du, and Feng also gave a prescription for obtaining virtuous numerators based on a similar symmetrization strategy applied to a KLT expression for the gauge-theory amplitude. It seems likely that this gives the same results as eq. (3.4). We will compare specific results below.

Four-point symmetric kinematic numerators
For the four-point amplitude, there is only one topology for color factors, the half ladder. A symmetric half-ladder numerator n(1, 2, 3, 4) must satisfy eq. Our expression (3.8) agrees with that recently obtained in ref. [30]. It is also in agreement with the shorter expression in ref. [29] n(1, 2, 3, 4) = We could further use the BCJ relations to write n(1, 2, 3, 4) in terms of a single color-ordered amplitude, but only at the price of having kinematic invariants in the denominator.

Five-point symmetric numerators
For the five-point amplitude, there is again only one topology for the color factors, the half ladder. A symmetric half-ladder numerator n(1, 2, 3, 4, 5) satisfies eq. (3.5), which after using Jacobi identities is equivalent to as well as all the Jacobi identities. Equation (3.16) agrees with the shorter expression given in Broedel and Carrasco [29] after imposing momentum conservation and BCJ relations on the color-ordered amplitudes. The Broedel-Carrasco expression is virtuous, but to show that it obeys eq. (3.17) requires the imposition of momentum conservation and BCJ relations, whereas for eq. (3.16) the dihedral symmetry is manifest. Both the Broedel-Carrasco expression and eq. (3.16) satisfy eq. (3.14) only after momentum conservation and the BCJ relations are imposed.
We can re-express eq. This expression differs from the result given in ref. [30], but that result apparently contains some typographical errors which will be corrected in a revised version. 9 We emphasize that eq. (3.18) manifestly obeys Jacobi identities and dihedral symmetries using only the Kleiss-Kuijf relations and the cyclic and reversal properties of the color-ordered amplitudes (but not momentum conservation or BCJ relations). Fu et al. [30] also give explicit, rather lengthy, expressions for virtuous numerators for six-point amplitudes.

Virtuous dual-trace functions
The color factors c i and the kinematic numerators n i (1, 2, · · · , n) play dual roles in the gauge-theory amplitude (2.1). Bern and Dennen [31] proposed that a role dual to the

Four-point symmetric dual-trace functions
For the four-gluon amplitude, eq. (4.6) yields a nonsymmetric representation for the dualtrace function with the remaining τ α given by eq. (4.3). We now use eq. The expression for τ given in ref. [32] is equivalent to eq. (4.16). Equation The difference is "pure gauge": it vanishes when substituted into eq. (4.14), and therefore does not contribute to n (1, 2, 3, 4). Hence we see that imposing the three virtues of Broedel and Carrasco on the dual-trace function is not sufficient to determine it uniquely.

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To further elucidate the difference between eqs. (4.13) and (4.16), we rewrite eq. (4.2) as a matrix equation where s = s 12 , t = s 14 , and u = s 13 . As expected, m has rank (4 − 3)! = 1. It therefore does not possess a unique inverse, but rather a family of generalized inverses m +    A generalized inverse must satisfy mm + m = m [42,48], which guarantees that the resulting τ α will satisfy eq. (4.18). Different gauge choices for τ correspond to different generalized inverses. The nonsymmetric representation (4.12) corresponds to The symmetric dual-trace function (4.13) corresponds to whereas the Bern-Dennen dual-trace function corresponds to All three generalized inverses satisfy mm + m = m.

Five-point symmetric dual-trace functions
For the five-gluon amplitude, eq. (4.6) yields a nonsymmetric representation

Discussion
In this paper, we have offered a constructive procedure for computing virtuous kinematic numerators for n-point gauge-theory scattering amplitudes; that is, numerators that simultaneously satisfy color-kinematic duality, are expressed in terms of color-ordered amplitudes, and are symmetric in the sense of Broedel and Carrasco. We have presented explicit expressions for four-and five-point amplitudes, which although somewhat lengthy, have the advantage of manifestly obeying the Jacobi identities and diagram symmetries JHEP07(2014)143 without having to invoke the BCJ relations for the color-ordered amplitudes. Our results are equivalent (upon using the BCJ relations) to other virtuous expressions in the literature, suggesting the possibility that the three virtues of Broedel and Carrasco are sufficient to uniquely determine the kinematic numerators, although we do not have a proof of this.
We have also applied this procedure to compute symmetric, amplitude-encoded dualtrace functions τ for n-point amplitudes, presenting explicit expressions for four-and fivepoint amplitudes. In this case, the results are not uniquely determined by these criteria alone. In particular, symmetric expressions for τ obtained by other authors additionally satisfy (optional) Kleiss-Kuijf relations, whereas our results do not. While it is possible that a constructive procedure could be found to generate virtuous dual-trace functions for n-point amplitudes that also satisfy Kleiss-Kuijf relations, it is also possible that an alternative criterion to single out a unique dual-trace function may be more useful or natural.

Acknowledgments
I am grateful to Johannes Broedel for useful correspondence. I especially wish to thank Freddy Cachazo and Ellis Yuan for crucial conversations at the outset of this project. This research was supported in part by the NSF under grant no. PHY10-67961.
Here we have divided up the color factors c i according to the topology T of the diagram, and arbitrarily assign the color labels a j to the legs of the diagram, calling the result c α denotes the color factor with the same topology but with the color labels on the legs permuted by α ∈ S n . Also, d (T ) (1, 2, · · · , n) is the denominator associated with c (T ) 1 l and d (T ) (α) = d (T ) (α(1), · · · , α(n)).

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Unlike the trace basis, the set of color factors (modulo G (T ) ) does not constitute an independent basis on account of various Jacobi identities of the form where T 1 , T 2 , and T 3 need not all be equal since Jacobi identities may mix color factors with different topologies. Hence the kinematic numerators n (T ) α are not unique and can undergo generalized gauge transformations. Now we apply a permutation β to the external legs of an amplitude expressed in the color basis (eq. (A.7)) to obtain where in the last equation, we used which follows from eq. (A.11). The invariance of the full bosonic amplitude A(1, 2, · · · , n) under permutations of the external legs implies that α (1l), because as stated above the color factors (modulo the symmetries G (T ) ) do not form an independent basis. In other words, the kinematic numerators need not be symmetric, in the sense of Broedel and Carrasco [29]. In particular, the kinematic numerators given in eq. (3.2) are manifestly not symmetric because some of them vanish, which is not possible if they are all related by permutations to a single function.
On the other hand, the permutation symmetry of the amplitude gives us a mechanism for generating new representations for kinematic numerators from existing ones. Specifically, eq. (A.16) demonstrates that, given a valid set of kinematic numerators n and hence are BCJ numerators.
Moreover, normalized linear combinations of valid numerators are also valid numerators. This suggests that, given a set of nonsymmetric kinematic numerators n in terms of a single symmetric function n (T ) (1, 2, · · · , n) for each topology. Furthermore, using eq. (2.20), each n (T ) (1, 2, · · · , n) can be expressed in terms of a single symmetric function for the half-ladder diagram n(1, 2, · · · , n).

(A.27)
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