Expansion formula of one-loop Einstein-Yang-Mills integrand

TL;DR

This work proves that one-loop EYM integrands with a gluon loop can be expanded in terms of conventional one-loop YM integrands with quadratic propagators, provided the expansion coefficients satisfy one-loop consistency conditions. The authors construct the EYM integrand via forward-limit tree amplitudes, decompose YM contributions to BS amplitudes, and show that, with consistent coefficients, the EYM result reduces to a YM-based expansion whose coefficients match those from the YMS case. The key insight is a shared kinematic structure that enables BCJ-numerator construction for both YMS and EYM at one loop, supported by a concrete two-gluon/two-graviton example and a general combinatorial proof. Explicit three-graviton coefficients are given in the appendix, illustrating the practical reach of the expansion. This work clarifies the link between gravity and gauge theories at one loop and strengthens the BCJ framework for mixed YM/EYM theories.

Abstract

Building upon the algebraic consistency construction of one-loop Bern-Carrasco-Johansson (BCJ) numerators for Yang-Mills (YM) and Yang-Mills-scalar (YMS) theories, we explore the expansion formula of one-loop Einstein-Yang-Mills (EYM) integrands (with a gluon loop) in terms of conventional one-loop YM integrands with quadratic propagators. We first express the EYM integrand by tree-level amplitudes according to the forward limit approach. Employing a two-step expansion strategy, the gluon-loop EYM integrand is decomposed into tree-level YM amplitudes under the forward limit, which are subsequently expanded into tree-level bi-adjoint scalar (BS) ones. We then prove that when the kinematic coefficients in both expansion steps satisfy the one-loop consistency conditions, the EYM integrand is finally expanded as a combination of YM integrands with quadratic propagators. The coefficients in this expansion formula coincide exactly with those in the expansion formula for YMS integrands. This correspondence highlights a shared kinematic structure, providing the proper foundation for constructing BCJ numerators in both YMS and EYM theories at one loop.