Analytic Representations of Yang-Mills Amplitudes

TL;DR

The authors tackle the challenge of obtaining analytic, covariant Yang–Mills amplitudes from the CHY formalism by refining the CHY integrand to be Möbius-invariant term-by-term and employing string-theory monodromy relations to decompose complex terms into those compatible with known integration rules. They generalize monodromy reductions to eliminate problematic k-tuples, enabling systematic analytic evaluation of CHY integrals for YM amplitudes of arbitrary multiplicity, with explicit 5- and 6-point results. The work yields a practical algorithm that produces closed-form, dimensionally agnostic YM tree amplitudes and suggests pathways to gravity and loop extensions, bridging CHY formulations with established field- and string-theory techniques. Overall, the paper provides a robust, scalable method for obtaining analytic Yang–Mills amplitudes directly from CHY representations.

Abstract

Scattering amplitudes in Yang-Mills theory can be represented in the formalism of Cachazo, He and Yuan (CHY) as integrals over an auxiliary projective space---fully localized on the support of the scattering equations. Because solving the scattering equations is difficult and summing over the solutions algebraically complex, a method of directly integrating the terms that appear in this representation has long been sought. We solve this important open problem by first rewriting the terms in a manifestly Mobius-invariant form and then using monodromy relations (inspired by analogy to string theory) to decompose terms into those for which combinatorial rules of integration are known. The result is a systematic procedure to obtain analytic, covariant forms of Yang-Mills tree-amplitudes for any number of external legs and in any number of dimensions. As examples, we provide compact analytic expressions for amplitudes involving up to six gluons of arbitrary helicities.