Expansion of Einstein-Yang-Mills Amplitude
Chih-Hao Fu, Yi-Jian Du, Rijun Huang, Bo Feng
TL;DR
This work develops a gauge-invariance–driven, recursive framework to express tree-level single-trace Einstein-Yang-Mills amplitudes as linear combinations of color-ordered Yang-Mills amplitudes, enabling systematic construction for arbitrary numbers of gravitons. It integrates CHY (Pfaffian) and CHY-based KLT relations, BCFW on-shell recursion, and KLT/BCJ-numerator perspectives to derive explicit expansion formulas and to extract BCJ numerators from EYM relations. The authors present constructive algorithms (including ordered-splitting and KK-basis reformulations) to obtain the expansion coefficients and BCJ numerators, with concrete demonstrations for up to three gravitons and two gravitons at higher points, and they connect these relations to YM-scalar and pure YM theories. The results provide a computationally efficient route to evaluate EYM amplitudes and illuminate the color-kinematics duality structure underlying gravity–gauge theory relations, with promising implications for loop-level generalizations and CHY-integrand expansions.
Abstract
In this paper, we provide a thorough study on the expansion of single trace Einstein-Yang-Mills amplitudes into linear combination of color-ordered Yang-Mills amplitudes, from various different perspectives. Using the gauge invariance principle, we propose a recursive construction, where EYM amplitude with any number of gravitons could be expanded into EYM amplitudes with less number of gravitons. Through this construction, we can write down the complete expansion of EYM amplitude in the basis of color-ordered Yang-Mills amplitudes. As a byproduct, we are able to write down the polynomial form of BCJ numerator, i.e., numerators satisfying the color-kinematic duality, for Yang-Mills amplitude. After the discussion of gauge invariance, we move to the BCFW on-shell recursion relation and discuss how the expansion can be understood from the on-shell picture. Finally, we show how to interpret the expansion from the aspect of KLT relation and the way of evaluating the expansion coefficients efficiently.