Integrand Oxidation and One-Loop Colour-Dual Numerators in N=4 Gauge Theory

TL;DR

This paper develops a systematic method for constructing BCJ numerators in one-loop amplitudes of N=4 gauge theory, revealing structural insights and simplifying calculations relevant for supergravity and gauge theories.

Contribution

It introduces a new systematic approach to determine BCJ numerators at one loop, connecting integrand reductions, algebraic identities, and gauge theory amplitudes.

Findings

Explicit BCJ numerators up to seven points in N=4 gauge theory

Connection between Jacobi identities and integrand reductions

Simplified formulas for integrand reductions and BCJ box numerators

Abstract

We present a systematic method to determine BCJ numerators for one-loop amplitudes that explores the global constraints on the loop momentum dependence. We apply this method to amplitudes in N=4 gauge theory, working out detailed examples up to seven points in both the MHV and the NMHV sectors. The structure of Jacobi identities between BCJ numerators is seen to be closely connected to that of algebraic integrand reductions. We discuss the consequences for one-loop N=8 supergravity amplitudes obtained through the double copy prescription. Moreover, in the MHV sector, we show how to obtain simple BCJ box numerators using a conjectured relationship to amplitudes in self-dual gauge theory. We also introduce simpler trace-type formulas for integrand reductions.