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

TL;DR

The paper develops a bottom-up integrand-oxidation framework to construct BCJ color-kinematics dual numerators for one-loop amplitudes in N=4 SYM, leveraging global loop-momentum constraints from Jacobi identities. It provides explicit constructions for five-, six-, and seven-point MHV amplitudes, connects box numerators to self-dual Yang-Mills via dimension shifting, and extends to NMHV sectors using unitarity cuts. A key result is that loop-momentum dependence cancels upon integrand reduction to pentagons, enabling clean gravity amplitudes via the BCJ double copy in N=8 supergravity, with trace-based reduction formulas offered as practical tools. The work suggests a robust path to higher-multiplicity and non-MHV amplitudes and highlights deep ties between integrand structure, self-dual theories, and gravity.

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.