Permutation Symmetry of the Scattering Equations

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 ${\text{s}}_{\text{a}}$ 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 ${\text{s}}_{\text{a}}$ is believed to satisfy a single polynomial equation of degree $(n-3)!$. In this article, we derive the transformation properties of ${\text{s}}_{\text{a}}$ 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 ${\text{s}}_{\text{a}}$ 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.