Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame

TL;DR

The authors establish a CHY-frame recursive expansion for tree-level single-trace EYM amplitudes with any number of gluons and gravitons. By performing a Pfaffian Laplace expansion along graviton rows and defining a level-by-level current T^μ, they prove that the EYM integrand can be expressed in terms of lower-graviton EYM and YM integrands, ultimately yielding a pure YM expansion in the KK basis. They further introduce graphic spanning-tree rules that directly compute the expansion coefficients, revealing an elegant combinatorial structure behind the coefficients. The results unify prior conjectures with a rigorous CHY-based proof and point to practical avenues for constructing BCJ numerators and exploring double-copy relations in YM and related theories, with potential extensions to multitrace EYM and loop-level CHY formalisms.

Abstract

Using the Cachazo-He-Yuan (CHY) formalism, we prove a recursive expansion of tree level single trace Einstein-Yang-Mills (EYM) amplitudes with arbitrary number of gluons and gravitons, which is valid for general spacetime dimensions and any helicity configurations. The recursion is written in terms of fewer-graviton EYM amplitudes and pure Yang-Mills (YM) amplitudes, which can be further carried out until we reach an expansion in terms of pure YM amplitudes in Kleiss-Kuijf (KK) basis. Our expansion then generates naturally a spanning tree structure rooted on gluons whose vertices are gravitons. We further propose a set of graph theoretical rules based on spanning trees that evaluate directly the pure YM expansion coefficients.

Figures (3)

• Figure 1: The rooted tree with three levels.
• Figure 2: A tree whose first level contains $N$ branches.
• Figure 3: Schematic expansion of the EYM integrand