Kinematic Numerators and a Double-Copy Formula for N = 4 Super-Yang-Mills Residues

TL;DR

The paper introduces residue numerators as a residue-level analogue of BCJ kinematic numerators, showing that RSVW residues in $ N=4$ SYM satisfy BCJ identities and admit a double-copy construction that yields gravity residues. By formulating a linear-algebra framework with a generalized inverse, it proves the BCJ identities are equivalent to a consistency condition, and extends these structures to RSVW residues and gravity via KLT orthogonality. The central result is a concrete residue-double-copy formula $R_r^{G} = (rac{ }{2})^{n-2} N_r^{T} F ilde{N}_r$, which mirrors the amplitude-level relation and suggests pathways to loop-level generalizations. Although demonstrated explicitly at $n=6$, the framework provides a systematic route to residue-level color-kinematic duality and could inform loop integrand construction and broader theories such as ABJM. This work thus ties BCJ/KLT structures to a residue-based perspective, with potential impact on loop calculations and the understanding of gravity as a double copy of gauge theory at the level of residues.

Abstract

Recent work by Cachazo, He, and Yuan shows that connected prescription residues obey the global identities of $\mathcal{N} = 4$ super-Yang-Mills amplitudes. In particular, they obey the Bern-Carrasco-Johansson (BCJ) amplitude identities. Here we offer a new way of interpreting this result via objects that we call residue numerators. These objects behave like the kinematic numerators introduced by BCJ except that they are associated with individual residues. In particular, these new objects satisfy a double-copy formula relating them to the residues appearing in recently-discovered analogs of the connected prescription integrals for $\mathcal{N} = 8$ supergravity. Along the way, we show that the BCJ amplitude identities are equivalent to the consistency condition that allows kinematic numerators to be expressed as amplitudes using a generalized inverse.

Figures (2)

• Figure 1: Reduction of the Kleiss-Kuijf amplitude basis to the BCJ amplitude basis for $n=4$. In the figure, $A\left(1,2,3,4\right)\equiv A_{23}$ and $A\left(1,3,2,4\right)\equiv A_{32}$. Because the BCJ basis is the minimal basis, any Kleiss-Kuijf amplitude vector actually lies on the "BCJ line". Both $A_{23}$ and $A_{32}$ are complex numbers, indicated by the $\mathbb{C}$ labels on the axes. The "rescaling by $Q$" arrows indicate the $GL\left(1\right)$ freedom that rescales the point $A$ along the BCJ line.
• Figure 2: The integral for the $n=6$, $k=3$ Yang-Mills amplitude has one integration variable not fixed by gauge choice or momentum conservation, and so may be calculated as a standard contour integral of a complex variable $c\in\mathbb{C}$. The four poles $c_{1}$, $c_{2}$, $c_{3}$, and $c_{4}$ correspond to the four roots of $S\left(c\right)$, and the three remaining poles $\tilde{c}_{1}$, $\tilde{c}_{2}$, and $\tilde{c}_{3}$ correspond to the poles of the function $H\left(c\right)$. This figure is meant only as a guide; the actual location of the poles changes for different external momenta.