Note on symmetric BCJ numerator

TL;DR

This work presents a systematic, KLT-based method to construct BCJ numerators that realize color-kinematics duality, ensuring Jacobi consistency, an amplitude-encoded representation, and manifest relabeling symmetry. By averaging the KLT expression over all external-leg permutations and expanding in the KK basis, the authors identify the BCJ numerators with the corresponding KK-derived objects and verify these properties explicitly up to 5 points (with 6-point results provided in an appendix). They demonstrate that relabeling symmetry is satisfied for the DDM-chain topology by checking two key permutations and ensuring consistency with KK-relations and antisymmetry. The approach highlights how permutation averaging and KK/B basis techniques yield symmetric, amplitude-encoded BCJ numerators, and discusses potential generalizations to higher points and loop-level extensions.

Abstract

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.