Permutation Symmetry of the Scattering Equations

TL;DR

This work analyzes how the CHY scattering equation solutions $\sigma_\alpha$ transform under momentum permutations, deriving explicit symmetry relations and a Klein-group identity that constrain the solutions. It translates these sigma-transformations into corresponding transformations of the coefficients $A_p^{(\alpha)}$ of the single-variable polynomials that encode each $\sigma_\alpha$, and verifies the relations in low-$n$ cases ($n=4,5$) and for special momentum configurations. The authors show that symmetry can determine substantial portions of the polynomial coefficients (e.g., all but an overall normalization for $n=4$, and 36 of 45 for $n=5$), and they discuss the remaining ambiguities and how symmetry might generate new solutions for permuted momenta. The results illuminate the structural constraints on CHY amplitudes, offering a route to generate and relate solutions across permutations and to reduce computational effort in solving scattering equations, with potential implications for efficient amplitude calculations in gauge, gravity, and scalar theories.

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.