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. They depend on $(n\ensuremath{-}3)$ quantities ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ which satisfy a set of coupled scattering equations, with momentum dot products as input coefficients. These equations are known to have $(n\ensuremath{-}3)!$ solutions; hence, each ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ is believed to satisfy a single polynomial equation of degree $(n\ensuremath{-}3)!$. In this article, we derive the transformation properties of ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ 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 ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ lead to symmetry relations for the $(n\ensuremath{-}3)!+1$ 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.

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.