Permutation Symmetry of the Scattering Equations

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.