Comments on the evaluation of massless scattering

TL;DR

This paper introduces a new approach for evaluating massless scattering integrals by transforming polynomial equations into linear systems, and discusses the equivalence of different computational methods for these integrals.

Contribution

It provides a general expression for five-point integrals on M_{0,n} using Chebyshev polynomials and links the companion matrix method to elimination theory.

Findings

Derived a general five-point integral expression on M_{0,n}.

Transformed scattering equations into linear symmetric polynomial systems.

Established equivalence between companion matrices and elimination algorithms.

Abstract

The goal of this work is threefold. First, we give an expression of the most general five point integral on M_{0,n} in terms of Chebyshev polynomials. Second, we choose a special kinematics that transforms the polynomial form of the scattering equations to a linear system of symmetric polynomials. We then explain how this can be used to explicitly evaluate arbitrary point integrals on M_{0,n}. Third, we comment on the recently presented method of companion matrices and we show its equivalence to the elimination theory and an algorithm previously developed by one of the authors.