Permutation Symmetry of the Scattering Equations

TL;DR

This paper investigates the permutation symmetry properties of the solutions to the scattering equations in particle physics, deriving how these solutions transform under permutations and exploring implications for polynomial coefficients.

Contribution

It derives the transformation properties of scattering equation solutions under momentum permutations and explores their implications for polynomial coefficients.

Findings

Derived permutation transformation properties of scattering solutions

Verified symmetry relations with known solutions at low n

Discussed how coefficient symmetries can determine polynomial coefficients

Abstract

Closed formulas for tree amplitudes of $n$-particle scatterings of gluon, graviton, and massless scalar particles have been proposed by Cachazo, He, and Yuan. It depends on $(n-3)$ quantities $\s_\a$ which satisfy a set of coupled {\it scattering equations}, with momentum dot products as input coefficients. These equations are known to have $(n-3)!$ solutions, hence each $\s_\a$ is believed to satisfy a single polynomial equation of degree $(n-3)!$. In this article, we derive the transformation properties of $\s_\a$ under momentum permutation, and verify them with known solutions at low $n$, and with exact solutions at any $n$ for special momentum configurations. For momentum configurations not invariant under a certain momentum permutation, new solutions can be obtained for the permuted configuration from these symmetry relations. These symmetry relations for $\s_\a$ lead to symmetry relations for the $(n-3)!+1$ coefficients of the single-variable polynomials, whose correctness are checked with the known cases at low $n$. The extent to which the coefficient symmetry relations can determine the coefficients is discussed.