All tree-level amplitudes in massless QCD

Abstract

We derive compact analytical formulae for all tree-level color-ordered gauge theory amplitudes involving any number of external gluons and up to four massless quarkanti-quark pairs. A general formula is presented based on the combinatorics of paths along a rooted tree and associated determinants. Explicit expressions are displayed for the next-to-maximally helicity violating (NMHV) and next-to-next-to-maximally helicity violating (NNMHV) gauge theory amplitudes. Our results are obtained by projecting the previously-found expressions for the super-amplitudes of the maximally supersymmetric super Yang-Mills theory ( $$ \mathcal{N} = 4 $$ SYM) onto the relevant components yielding all gluon-gluino tree amplitudes in $$ \mathcal{N} = 4 $$ SYM. We show how these results carry over to the corresponding QCD amplitudes, including massless quarks of different avors as well as a single electroweak vector boson. The public Mathematica package GGT is described, which encodes the results of this work and yields analytical formulae for all $$ \mathcal{N} = 4 $$ SYM gluon-gluino trees. These in turn yield all QCD trees with up to four external arbitrary-flavored massless quark-anti-quark pairs.