Fermions and the scattering equations

TL;DR

This paper extends the scattering equations framework to include tree-level amplitudes with massless quarks, gluons, and scalars, providing methods to compute these amplitudes in gauge theories like QCD and ${\mathcal N}=4$ SYM.

Contribution

It generalizes the scattering equations approach to accommodate amplitudes with quarks and scalars, introducing a modified permutation invariant function and addressing cases with multiple quark pairs.

Findings

Derived a method to compute the modified $\hat{E}(z,p,\varepsilon)$ function.

Applied the framework to QCD amplitudes with one and two quark-antiquark pairs.

Explicitly discussed the four-point QCD amplitude with two quark pairs.

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 $C_\sigma(z)$. The simplest case of the QCD tree-level four-point amplitude with two quark-antiquark pairs is discussed explicitly.