On General BCJ Relation at One-loop Level in Yang-Mills Theory

TL;DR

This paper extends the Bern-Carrasco-Johansson (BCJ) relations to one-loop level in Yang-Mills theory, providing a proof for a general formula using unitarity cut methods, thus advancing understanding of amplitude relations at loop level.

Contribution

It introduces a general BCJ relation for one-loop integrands in Yang-Mills theory and proves it using D-dimensional unitarity cut techniques, generalizing previous tree-level results.

Findings

Proved a general BCJ relation at one-loop level.

Demonstrated the relation using unitarity cut method.

Extended BCJ relations beyond tree level.

Abstract

BCJ relation reveals a dual between color structures and kinematic structure and can be used to reduce the number of independent color-ordered amplitudes at tree level. Refer to the loop-level in Yang-Mills theory, we investigate the similar BCJ relation in this paper. Four-point 1-loop example in N = 4 SYM can hint about the relation of integrands. Five-point example implies that the general formula can be proven by unitary- cut method. We will then prove a 'general' BCJ relation for 1-loop integrands by D-dimension unitary cut, which can be regarded as a non-trivial generalization of the (fundamental)BCJ relation given by Boels and Isermann in [arXiv:1109.5888 [hep-th]] and [arXiv:1110.4462 [hep-th]].