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

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Main Authors: Du, Yi-Jian, Fu, Chih-Hao, Wang, Yihong, Xie, Chongsi
Format: Preprint
Published: 2025
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author Du, Yi-Jian
Fu, Chih-Hao
Wang, Yihong
Xie, Chongsi
author_facet Du, Yi-Jian
Fu, Chih-Hao
Wang, Yihong
Xie, Chongsi
contents 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.
format Preprint
id arxiv_https___arxiv_org_abs_2511_10963
institution arXiv
publishDate 2025
record_format arxiv
spellingShingle Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories
Du, Yi-Jian
Fu, Chih-Hao
Wang, Yihong
Xie, Chongsi
High Energy Physics - Theory
High Energy Physics - Phenomenology
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.
title Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories
topic High Energy Physics - Theory
High Energy Physics - Phenomenology
url https://arxiv.org/abs/2511.10963