Evaluation of the CHY Gauge Amplitude

TL;DR

This work provides a direct, general method to evaluate the CHY gauge amplitude for tree-level $n$-gluon scattering by systematically organizing the reduced Pfaffian into cycles and employing a crystal-based residue calculus. The authors develop shift invariance and helicity-gauge techniques to reduce the combinatorial complexity, and introduce crystal graphs, a $σ$-table, and a $C$-table to extract propagator structures and assemble numerators. They demonstrate explicit consistency with Feynman amplitudes for $n=3,4,5$, including recovery of the triple- and four-gluon vertices, and argue the general applicability of their approach to arbitrary $n$ with methods to handle potential double-pole contributions. The results illuminate how CHY amplitudes, though devoid of explicit vertices and propagators, reproduce standard gauge-theory dynamics and provide a unified, global perspective on tree-level gluon scattering. The techniques have potential to streamline high-$n$ calculations and deepen understanding of the CHY formulation's relation to conventional Feynman theory.

Abstract

The Cachazo-He-Yuan (CHY) formula for $n$-gluon scattering is known to give the same amplitude as the one obtained from Feynman diagrams, though the former contains neither vertices nor propagators explicitly. The equivalence was shown by indirect means, not by a direct evaluation of the $(n\! - \!3)$-dimensional integral in the CHY formula. The purpose of this paper is to discuss how such a direct evaluation can be carried out. There are two basic difficulties in the calculation: how to handle the large number of terms in the reduced Pfaffian, and how to carry out the integrations in the presence of a $σ$-dependence much more complicated than the Parke-Taylor form found in a CHY double-color scalar amplitude. We have solved both of these problems, and have formulated a method that can be applied to any $n$. Many examples are provided to illustrate these calculations.