The CHY representation of tree-level primitive QCD amplitudes

TL;DR

This work extends the CHY formalism to all tree-level primitive QCD amplitudes with massless or massive quarks by introducing a generalized cyclic factor $\hat{C}(w,z)$ and a generalized permutation invariant function $\hat{E}(z,p,\varepsilon)$. The authors prove that amplitudes can be written as a CHY integral or as a sum over inequivalent scattering-equation solutions, with the Jacobian $J(z,p)$, while the ordering and helicity information are cleanly separated into $\hat{C}$ and $\hat{E}$, respectively. A key technical innovation is the construction of a minimal amplitude basis using Dyck words and BCJ/Kleiss-Kuijf relations, plus a rank condition (verified for up to 10 points) ensuring the feasibility of the CHY representation for all cases. An explicit six-point example demonstrates the recursion and the interplay between $\hat{C}$ and $\hat{E}$. The results significantly generalize CHY methods to multi-quark QCD amplitudes and pave the way for efficient, helicity- and ordering-agnostic representations in perturbative QCD.

Abstract

In this paper we construct a CHY representation for all tree-level primitive QCD amplitudes. The quarks may be massless or massive. We define a generalised cyclic factor $\hat{C}(w,z)$ and a generalised permutation invariant function $\hat{E}(z,p,\varepsilon)$. The amplitude is then given as a contour integral encircling the solutions of the scattering equations with the product $\hat{C} \hat{E}$ as integrand. Equivalently, it is given as a sum over the inequivalent solutions of the scattering equations, where the summand consists of a Jacobian times the product $\hat{C} \hat{E}$. This representation separates information: The generalised cyclic factor does not depend on the helicities of the external particles, the generalised permutation invariant function does not depend on the ordering of the external particles.