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arXiv · 2511.10963

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

Yi-Jian Du·Chih-Hao Fu·Yihong Wang·Chongsi Xie
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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.

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Yi-Jian Du, Chih-Hao Fu, Yihong Wang, Chongsi Xie. 2026-03-02. Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories. https://doi.org/10.1103/nf97-wmfd

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