Scattering equations, generating functions and all massless five point tree amplitudes

TL;DR

This paper demonstrates that explicit solutions of scattering equations are unnecessary for calculating massless five-point tree amplitudes, using a generating function approach based on SL(2,C) invariance and cross ratios.

Contribution

It introduces a method to evaluate five-point amplitudes without solving scattering equations explicitly, utilizing a generating function approach for SL(2,C) invariant quantities.

Findings

Evaluated the general SL(2,C) invariant quantity for five points.

Constructed a generating function capturing the combinatorial structure.

Provided a new way to compute amplitudes without explicit solutions.

Abstract

We argue that one does not need to know the explicit solutions of the scattering equations in order to evaluate a given amplitude. We consider the most general quantity consistent with SL(2,C) invariance that can appear in an amplitude that admits a scattering equation description. This quantity depends on all cross ratios that can be formed from n points and we evaluate it for the first non-trivial case of n=5. The combinatorial nature of the problem is captured through the construction of an appropriate generating function that depends on five variables.