Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism

TL;DR

This work addresses the problem of efficiently computing tree-level, single-trace, n-point MHV amplitudes in four-dimensional Einstein-Yang-Mills theory. By directly evaluating CHY integrals, the authors derive compact formulas that factorize into a Parke-Taylor denominator and Hodges determinant minors, with the all-graviton negative-helicity amplitudes shown to vanish; they also connect these CHY results to the Selivanov-Bern-De Freitas-Wong (SBDW) generating function via a graph-theoretic matrix-forest theorem. The main contributions include explicit expressions for the $(g^{-}g^{-})$ and $(h^{-}g^{-})$ cases, a proof of vanishing for $(h^{-}h^{-})$, and a rigorous demonstration of equivalence with the SBDW formula through combinatorial interpretations. This work provides analytic evidence for hidden simplicity in quantum field theory and extends CHY methods to a broader class of theories, with potential implications for connections between Einstein-Yang-Mills, Yang-Mills, and gravity amplitudes.

Abstract

In this paper we extend our techniques, developed in a previous paper (Du, etc, JHEP 05(2016)086) for direct evaluation of arbitrary $n$-point tree-level MHV amplitudes in 4d Yang-Mills and gravity theory using the Cachazo-He-Yuan (CHY) formalism, to the 4d Einstein-Yang-Mills (EYM) theory. Any single-trace color-ordered $n$-point tree-level MHV amplitude in EYM theory, obtained by a direct evaluation of the CHY formula, is of an elegant factorized form of a Parke-Taylor factor and a Hodges determinant, much simpler and more compact than the existing formulas in the literature. We prove that our new expression is equivalent to the conjectured Selivanov-Bern-De Freitas-Wong (SBDW) formula, with the help of a new theorem showing that the SBDW generating function has a graph theory interpretation. Together with Ref. (Du, etc, JHEP 05(2016)086), we provide strong analytic evidence for hidden simplicity in quantum field theory.

Figures (1)

• Figure 1: Some spanning forests of $K_5$ rooted in $I=\{1,2\}$.