Note on symmetric BCJ numerator

We present an algorithm that leads to BCJ numerators satisfying manifestly the three properties proposed by Broedel and Carrasco in [35]. We explicitly calculate the numerators at 4, 5 and 6-points and show that the relabeling property is generically satisfied.


Introduction
In an inspiring paper [1], Bern, Carrasco and Johansson (BCJ) have made a remarkable observation that the Yang-Mills scattering amplitude can be rearranged into a symmetrical form where its kinematic dependent numerators satisfy the same algebraic identities as the color factors. This duality between color and kinematic factors was later found to be present in a variety of Yang-Mills theories [2,3,4,5,6,7,8,9,10] and, perhaps most surprisingly, was shown to be valid at least for the first few loop levels [11,12,13,3,14,15,16,17,18,5,6,7,8,9,19]. The apparently symmetrical structure also suggests mirror versions of the existing formulations. In particular that studies of the BCJ duals of the original color-ordered and the Del Duca-Dixon-Maltoni formulations [20] can be found in [21,22,23,24,25,26].
Ever since the discovery of the duality, a considerable amount of endeavor has been devoted to the systematic construction of the kinematic numerators. An explicit construction was given by Mafra, Schlotterer and Stieberger using the pure spinor language [27]. Alternatively, it was shown that the kinematic factors can be interpreted in terms of diffeomorphism algebra [28,29,23,30]. In a series of recent papers [31,32,33] another interesting construction was provided by Cachazo, He and Yuan (CHY) using the solutions to the scattering equations. At the moment of writing it is not yet clear how to construct kinematic numerators for the most generic configuration and to arbitrary loop order.
Instead of attempting to decipher the analytic structure responsible for the possible algebraic behavior, another line of thoughts is to solve the kinematic numerators reversely in terms of scattering amplitudes, and indeed, it was discussed in [29,34] that such expression for the numerators can always be derived in suitable basis. A technical issue lies with this approach is that because of the complexity involved, along with the ambiguity introduced by generalized gauge invariance, it is practically difficult to write down an analytic expression for generic numerator. Nevertheless at tree level, explicit numerators were worked out by Broedel and Carrasco [35] at 4 and 5-points, and 6-points in the case of four dimensions, which satisfy the following three properties: 1. The numerators satisfy Jacobi identity, and this property is referred as BCJ representation.
2. All external state information such as particle species and helicity is coded inside the color-ordered partial amplitudes. In other words, we have n i = α c iα A α , where c iα is helicity blind. This property is called amplitude-encoded representation.
3. Expressions for numerators sharing the same topology are relabeling related. In other words, for each topology, if we know the BCJ numerator for a particular ordering of external legs, we know all others simply through relabelings. This property is called symmetric representation.
In this paper we present a systematic construction of the kinematic numerators at generic n-points via Kawai-Lewellen-Tye (KLT) relation [36,37], which satisfy the above three properties of Broedel and Carrasco. As examples we derive the explicit formulas for 4, 5 and 6 points. In particular that the 6-point expression here does not assume spinor identities. The paper is organized as follows. In section 2 we discuss the basic idea used to determine the kinematic numerators. In sections 3 and 4 we present explicit expressions for numerators at 4 and 5-points. Due to complexity we present the 6-point result in appendix A. We show that the numerators produced by the algorithm discussed in this paper satisfy all three properties of Broedel and Carrasco in section 5. Finally, a conclusion is given in section 6.

General framework
Our starting point is the KLT expression of the total Yang-Mills amplitude [36,38,37,39], σ, σ∈S n−3 A n (1, σ(2, n − 2), n − 1, n)S[ σ(2, n − 2))|σ(2, n − 2))] p 1 A n (n − 1, n, σ(2, n − 2), 1) where A n is the color ordered partial amplitude of Yang-Mills theory and A n is the color ordered partial amplitude of scalar theory appearing in the dual DDM-form. Notice that although the total amplitude is totally symmetric under permutations of all n legs, the expressions in the second and the third lines are not manifestly so. In particular that legs 1, n − 1, n are kept fixed, while the ordering of the rest (n − 3) legs appear in A KLT n (1, 2, ..., n − 1, n) has no effect on the expression. To emphasize this feature, we write the parameter of amplitude A KLT n as (1, {2, ..., n − 2}, n − 1, n). The momentum kernel S appears in (2.1) is defined as [38,39] where θ(i t , i q ) is zero when pair (i t , i q ) has same ordering at both set I = In this definition momentum p 1 plays a distinct role, in the sense that for each leg i there is always one term s i1 . In the case when different choices of p 1 are encountered, we should write S[I|J ] p 1 to avoid confusions. Since our goal is to obtain a symmetric representation for BCJ numerators, as a second step we symmetrize expression (2.1) 1 , However note that since KLT expression is already manifestly (n − 3)!-symmetric, equation (2.3) reduces to averaging over n-choices for a 1 , then (n − 1)-choices for a n−1 and (n − 2)-choices for a n , As a third step, we expand A n using KK-basis, with , for example, 1, n fixed at the first and the last position. Thus A S n becomes where the n 1σ(2,...,n−1)n here is a collective factor of color-ordered amplitude A n and kinematic factors.
Comparing with the dual DDM-form [11] of the total Yang-Mills amplitude, we propose that the desired BCJ numerator n S α satisfying the three properties of [35] to be n s 1σ(2,...,n−1)n ≡ n 1σ(2,...,n−1)n . [ns-def] (2.7) To summarize, the basic idea is as follows: 1 Consider the average of KLT relations was firstly suggested in the work [40]. However, the numerator suggested in [40] by taking (n − 2)! cannot satisfy the relabeling property.
• We construct only the KK-basis of BCJ numerator n i in (2.6). The rest numeators are given by Jacobi identity. By this construction, the expression is automatically BCJ representation.
• We consider constructing the numerators using KLT relation, where the helicity information is automatically coded inside the partial amplitude.
• The A S n (2.3) is averaged over all permutations of n external legs, thus the relabeling property is manifestly constructed.

Four-point construction
Having discussed the general framework, let us present a few examples explicitly. We start with averaging KLT relation over all 4! permutations, Translating allÃ into KK-basis, and comparing equation ( The expression above can be further translated into BCJ basis, which is simply the four-point result obtained by Broedel and Carrasco in [35]. The other numerator n 1324 , obtained by collecting the coefficient of A(1, 3, 2, 4), gives the same expression as (3.3) with legs (2, 3) swapped.

The 5-point case
The calculation at 5-points follows in a straightforward manner. Here we list the result for n 12345 in KK-basis. and we do find that relabeling symmetry is satisfied at 5-points. This result is also same with the one obtained by Broedel and Carrasco in [35] The explicit expression for 6-point numerator is considerably more complicated and we leave the result to appendix A.

Verifying symmetry properties of the numerators
In this section let us check whether the BCJ numerators constructed following the algorithm outlined at the beginning of this paper indeed satisfy the three properties proposed in [35]. As remarked at the end of section 2, the n s α 's defined by this algorithm satisfy the BCJ-representation automatically since n s 1σ(2,...,n−1)n works as a basis and other numerators are constructed through antisymmetry and Jacobi identity. The n s α 's are also amplitude-encoded representation since n S α is of the form A n K, where K are kinematic factors constructed by s ij and all helicity information is included in A n .
The last property, i.e., the symmetric representation, is however not trivial. Note that since all other topologies can be constructed using DDM-chains (linear trees), if we can show the relabeling symmetry is true for this topology, it must be true for other topologies. For the topology of DDM-chain, there are n! different labelings. Among them (n − 2)! of the numerators are directly given by the algorithm (2.7) and others can be constructed using the Jacobi-identity and anti-symmetry. In our third step, we have fixed only two legs 1, n, the relabeling property is manifestly true among (2, 3, ..., n − 1) by construction. Since all permutations can be generated by successive permutations between the (n − 1) consequtive pairs, we can reduce our checking to the following two permutations: (12) and (n − 1)n. Now let us consider the DDM-chains n 213...(n−1)n and n 123...n(n−1) . We have two ways to get the same n α from basis numerators: One is by relabeling n 123...(n−1)n and another one, by using Jacobi relation and antisymmetry. If the expressions obtained by these two ways are the same, the relabeling property is fully checked. (12) Now we consider the relabeling property under the permutation (12)

Permutation ((n − 1)n)
The proof of ((n − 1)n) invariance is similar. Using relabeling we find as required by anti-symmetry of the BCJ numerator.

Conclusion
In this paper we discussed a systematic construction of the BCJ numerator based on matching KLT and dual DDM-form of the full Yang-Mills amplitude. Using this method we explicitly calculated the numerators at 4, 5 and 6 points and verified the three symmetry properties proposed by Broedel and Carrasco [35] hold generically. Note the similarity between the expressions discussed and the prescription proposed by Cachazo, He and Yuan [33] despite the method used in this paper does not rely on the existence of the solutions to scattering equations.
There are a few things one can proceed. First, although we have the general algorithm, its computation takes a long time when the number of external leg increases. Thus it will be nice if we can have a general patten of n α expanded into the KK-basis (or BCJ-basis). Secondly, it will be natural to generalize above results to loop level. At loop-level the current method does not seem to straightforwardly generalize because of the lack of support from KLT relation, yet naively it might be possible to solve numerators by comparing integrands in suitable basis. We leave this part of the discussion to future works.

A. 6-point numerator
In this appendix we present the explicit formula of the 6-point BCJ numerator in KK basis. Note however, despite its complixity, this expression holds in all dimensions greater or equal to three and does not depend on helicity except through color-ordered amplitudes.