Fermions and the scattering equations

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.