Analytic Representations of Yang-Mills Amplitudes

TL;DR

This paper develops a systematic method to analytically compute Yang-Mills scattering amplitudes directly from the CHY formalism, avoiding complex solutions of scattering equations, and provides explicit formulas for up to six gluons.

Contribution

It introduces a new approach using monodromy relations to directly integrate CHY integrals for Yang-Mills amplitudes, enabling explicit formulas for any number of external legs.

Findings

Derived compact analytic expressions for up to six-gluon amplitudes.

Established a systematic procedure for covariant Yang-Mills amplitude calculation.

Demonstrated the method's applicability in any spacetime dimension.

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.