On the solutions of the scattering equations

TL;DR

The paper analyzes analytic solutions of the Cachazo-He-Yuan scattering equations in four dimensions using momentum twistors. It proves that there are always two rational solutions, and that for $n\le5$ these two exhaust the inequivalent solutions, while for $n=6$ there exist four additional algebraic solutions constructed explicitly as roots of a quartic in $\sigma_1$. It further demonstrates that, in general, these four algebraic solutions are not rational functions of momentum twistors, highlighting limitations of rational parametrisations. The results clarify the structure of scattering amplitudes and their BCJ representations in low multiplicities and provide explicit formulas up to $n=6$.

Abstract

This paper addresses the question, whether the solutions of the scattering equations in four space-time dimensions can be expressed as rational functions of the momentum twistor variables. This is the case for $n\le5$ external particles. For general $n$ there are always two solutions, which are rational functions of the momentum twistor variables. However, the remaining solutions are in general not rational. In the case $n=6$ the remaining four solutions can be expressed as algebraic functions. These four solutions are constructed explicitly in this paper.