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

TL;DR

This work shows that explicit solutions to the scattering equations are not required to compute tree-level amplitudes in massless theories. The authors identify a fundamental $SL(2,\mathbb{C})$-invariant quantity depending on cross ratios and construct a generating function for the first nontrivial case $n=5$ to organize amplitudes as linear combinations of this quantity. They develop an algorithm that expresses amplitudes as rational functions of the kinematic invariants using Viète’s relations, and demonstrate this via a detailed Yang–Mills example decomposed into $P_{\vec{\alpha}}$ contributions. The approach provides a compact, algebraic framework that encapsulates the combinatorics of the scattering equations and yields practical, implementable expressions potentially extensible to higher $n$ and other theories within the same formalism.

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.