Scattering equations and virtuous kinematic numerators and dual-trace functions

TL;DR

The paper presents a constructive framework to obtain permutation-symmetric, amplitude-encoded BCJ numerators and dual-trace functions that satisfy color-kinematic duality for n-point gauge-theory amplitudes. By leveraging permutations of nonsymmetric representations and averaging, it delivers virtuous numerators and symmetric tau for all n, with explicit 4- and 5-point results and manifest gravity via the double-copy. It highlights the non-uniqueness of these constructions and discusses connections to Kleiss-Kuijf relations, leaving open questions about additional constraints that might uniquely fix the representations. The work provides practical tools for cleanly implementing the double-copy and offers insights into the structural freedoms in color-kinematic duality.

Abstract

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 (tau) for n-point amplitudes. These can be used to obtain symmetric kinematic numerators that automatically satisfy color-kinematic duality. The S_n 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.