Comments on the evaluation of massless scattering

TL;DR

This work tackles efficient evaluation of massless CHY amplitudes by presenting three complementary approaches. It first provides a Chebyshev-polynomial expression for the most general five-point integral on $\mathcal{M}_{0,n}$, exposing a recursive structure. It then introduces a two-parameter special kinematics that linearizes the polynomial scattering equations in symmetric polynomials, allowing contour integrals to be computed via linear algebra without solving all roots. Finally, it analyzes the companion-matrix approach, showing its equivalence to elimination theory and to Kalousios' coefficient-based algorithm, with amplitudes expressed as traces of polynomials in a single companion matrix. Together, these results offer new computational tools and a unifying perspective for CHY amplitude evaluation, albeit with remaining computational hurdles for large $n$ due to polynomial-system complexity.

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.