Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories

TL;DR

This paper investigates the algebraic structure of one-loop BCJ numerators in Yang-Mills and related theories, providing explicit construction methods and clarifying their relation to tree-level numerators and double-copy structures.

Contribution

It introduces a systematic approach to construct one-loop BCJ numerators satisfying Jacobi identities and provides explicit results for low-point cases, extending to Einstein-Yang-Mills and gravity.

Findings

Explicit one-loop numerators for up to three gluons derived.

Consistency conditions uniquely determine numerator coefficients.

Method extends to Einstein-Yang-Mills and gravity amplitudes.

Abstract

We study the algebraic structure of one-loop BCJ numerators in Yang-Mills and related theories. Starting from the propagator matrix that connects colour-ordered integrands to numerators, we identify the consistency conditions that ensure the existence of Jacobi-satisfying numerator solutions and determine the unique construction. The relation between one-loop numerators and forward-limit tree numerators is clarified, together with the additional physical conditions required for a consistent double-copy interpretation. We propose a two-step expansion strategy for obtaining explicit one-loop numerators. The Yang-Mills integrand is first decomposed into scalar-loop Yang-Mills-scalar building blocks, which are then expanded into bi-adjoint scalar integrands. We derive explicit results for up to three external gluons, showing how the kinematic consistency conditions uniquely determine the coefficients in each case. Similar results for Einstein-Yang-Mills and gravity amplitudes are also presented.