Evaluation of the CHY Gauge Amplitude

TL;DR

This paper develops a direct evaluation method for the CHY gauge amplitude integral, overcoming previous indirect proof limitations and enabling explicit calculations for any number of particles.

Contribution

It introduces a novel method to directly evaluate the CHY gauge amplitude integral, handling complex Pfaffian terms and intricate integrations for arbitrary particle numbers.

Findings

Successfully formulated a general evaluation method.

Demonstrated calculations with multiple examples.

Resolved key computational difficulties in CHY amplitude evaluation.

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 $\sigma$-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.