Integration Rules for Scattering Equations

TL;DR

This paper introduces a combinatorial rule with a diagrammatic interpretation for efficiently evaluating scattering amplitude integrals localized on solutions to scattering equations, simplifying calculations in quantum field theory and string theory.

Contribution

It provides a new, simple combinatorial rule with a diagrammatic interpretation for evaluating integrals involving scattering equations, applicable to M"obius-invariant integrands with simple poles.

Findings

The rule simplifies the evaluation of scattering amplitude integrals.

It has a clear diagrammatic interpretation.

The method connects to string theory amplitude computations.

Abstract

As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints for any M\"obius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.