Fermions and the scattering equations

TL;DR

The paper extends the scattering equation framework to include fermions and scalars in tree-level amplitudes with a single gauge group. It proves that a representation in terms of a modified function $hatE$ exists if and only if the amplitudes satisfy cyclic invariance, KK and BCJ relations, and it provides an explicit formula for $hatE$ as a linear combination of $(n-3)!$ basis amplitudes. For amplitudes that do not satisfy BCJ, such as QCD with two or more quark-antiquark pairs, it introduces a generalized Parke-Taylor factor and a pseudo-inverse condition to determine the generalized color structure, with a detailed treatment of the simplest four-point case. The results strengthen the connection between scattering equations and double-copy structures and yield practical methods to compute hatE and generalized color factors, thereby broadening the applicability of the scattering-equation program to fermionic amplitudes and multi-quark processes. This work covers tree-level amplitudes in $N=4$ SYM and QCD with one quark pair and many gluons, and outlines directions for fully general multi-quark amplitudes.

Abstract

This paper investigates how tree-level amplitudes with massless quarks, gluons and/or massless scalars transforming under a single copy of the gauge group can be expressed in the context of the scattering equations as a sum over the inequivalent solutions of the scattering equations. In the case where the amplitudes satisfy cyclic invariance, KK- and BCJ-relations the only modification is the generalisation of the permutation invariant function $E(z,p,\varepsilon)$. We present a method to compute the modified $\hat{E}(z,p,\varepsilon)$. The most important examples are tree amplitudes in ${\mathcal N}=4$ SYM and QCD amplitudes with one quark-antiquark pair and an arbitrary number of gluons. QCD amplitudes with two or more quark-antiquark pairs do not satisfy the BCJ-relations and require in addition a generalisation of the Parke-Taylor factors $C_σ(z)$. The simplest case of the QCD tree-level four-point amplitude with two quark-antiquark pairs is discussed explicitly.