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, e). We present a method to compute the modified E(z, p, e). 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.
Highlights
SYM and QCD amplitudes with one quark-antiquark pair and an arbitrary number of gluons
In this paper we discussed the extension of the representation of a cyclic ordered tree amplitude in the framework of the scattering equations from the pure gluonic case to the case, where the external particles are allowed to have spin ≤ 1
The modified function is given as a linear combination of (n − 3)! basis amplitudes
Summary
Amplitudes corresponding to theories with gauge groups may be decomposed into group-theoretical factors multiplied by kinematic functions called partial amplitudes. The decomposition into gauge-invariant cyclic ordered objects, which are called primitive tree amplitudes, is more involved and consists of three step: (i) elimination of identical flavours, (ii) stripping of colour factors, (iii) elimination of U(1)-gluons. The partial amplitudes Apnartial are gauge-invariant, but are in general for nq > 2 not cyclic ordered This is related to the fact that for nq ≥ 2 there can be so-called U(1)-gluons, corresponding to the second term of the Fierz identity in eq (2.6). Primitive tree amplitudes are purely kinematic objects, which are gauge-invariant and which have a fixed cyclic ordering of the external legs. They are calculated from planar diagrams with the colour-ordered Feynman rules given in appendix A. The partial amplitude Sn(φσ1, . . . , φσn|φσ 1, . . . , φσn) corresponds to Feynman diagrams compatible with the cyclic orders σ and σ [4]
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