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

TL;DR

The paper extends BCJ relations to 1-loop planar Yang-Mills integrands, proposing a general BCJ formula and validating it through $D$-dimensional unitary-cut proofs. By analyzing 4-point and 5-point cases and employing loop-momentum translations and cyclic symmetry, it shows that the proposed relations hold up to integration-vanishing terms, with rational parts canceling in $D$ dimensions. The results generalize the known fundamental 1-loop BCJ relation and hint at deeper connections with tree-level monodromy and KK-BCJ structures, potentially simplifying loop integrand constructions and master-equation coefficients. The work lays groundwork for extensions to higher points and connections to string-theoretic formulations, highlighting practical impact on simplifying loop amplitude computations in Yang-Mills theory.

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]].

Figures (6)

• Figure 1: The definition of loop momentum, if the leg $n$ is attached to a loop propagator.
• Figure 2: The definition of loop momentum, if the leg $n$ is attached to a tree propagator. Here $K=k_{i_1}+...+k_{i_j}$
• Figure 3: The leg $1$($1\equiv\alpha_1$) is attached to a loop propagator while the leg $n$($n\equiv\alpha_r$) is attached to either a loop propagator(the left diagram) or a tree propagator(the right diagram). In both case, when we replace the reference leg $n$ by $1$, we perform a translation on loop momentum, thus on $l$,$l\rightarrow l'=l+K_{(n,1)}+k_1$. Here $(n,1)$ stands for all the legs(here are the possible $\beta$s) between the leg $n$($\alpha_r$) and the leg $1$($\alpha_1$). $K_{(n,1)}$ denotes the sum of the momenta of the legs in $(n,1)$.
• Figure 4: The leg $1$($1\equiv\alpha_1$) is attached to a tree propagator while the leg $n$($n\equiv\alpha_r$) is attached to either a loop propagator(the left diagram) or a tree propagator(the right diagram). In both cases, a replacement of the reference leg from $n$($\alpha_r$) to $1$($\alpha_1$) demands a replacement of $l\rightarrow l'=l+K_{(n,1)}+k_1$.
• Figure 5: Unitary cut in the channel $K^2=\left(\sum\limits_{a=0}^{j}k_{\alpha_{i+a}}+\sum\limits_{b=0}^vk_{\beta_{u+b}}\right)^2$.