Correspondence between Solutions of Scattering Equations and Scattering Amplitudes in Four Dimensions

TL;DR

This paper explores the link between solutions of scattering equations and scattering amplitudes in four-dimensional theories, providing explicit calculations, reproducing known formulas, and proposing new methods for characterizing solutions.

Contribution

It introduces a direct evaluation method for tree-level MHV amplitudes using scattering equations and derives a new compact formula for Einstein-Yang-Mills amplitudes.

Findings

Explicit calculation of MHV amplitudes reproduces known formulas

Other solutions do not contribute to MHV amplitudes in studied theories

Proposes a new method for characterizing solutions of scattering equations

Abstract

In this talk, we review our recent work on direct evaluation of tree-level MHV amplitudes by Cachazo-He-Yuan (CHY) formula. We also investigate the correspondence between solutions to scattering equations and amplitudes in four dimensions along this line. By substituting the MHV solution of scattering equations into the integrated CHY formula, we explicitly calculate the tree-level MHV amplitudes for four dimensional Yang-Mills theory and gravity. These results naturally reproduce the Parke-Taylor and Hodges formulas. In addition, we derive a new compact formula for tree-level single-trace MHV amplitudes in Einstein-Yang-Mills theory, which is equivalent to the known Selivanov-Bern-De Freitas-Wong (SBDW) formula. Other solutions do not contribute to the MHV amplitudes in Yang-Mills theory, gravity and Einstein-Yang-Mills theory. We further investigate the correspondence between solutions of scattering equation and helicity configurations beyond MHV and proposed a method for characterizing solutions of scattering equations.