Permutation Symmetry of the Scattering Equations

Understand

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.