Two-loop QCD helicity amplitudes for $g\,g \to Z\,g$ and $g\,g \to Z\,γ$

TL;DR

The authors derive the two-loop QCD helicity amplitudes for gg -> Zg and gg -> Zγ in massless QCD, providing essential inputs for NLO predictions of gluon-induced vector-boson processes at the LHC. They adopt a gauge-fixed tensor decomposition to simplify the amplitude structure, perform UV renormalisation and IR subtraction via Catani formalism, and express the finite remainders analytically in terms of harmonic polylogarithms up to weight four. The calculation relies on a full diagrammatic reduction to master integrals, with careful analytic continuation to physical scattering regions for V+jet and V+γ production. Extensive checks against known one-loop results, symmetry properties, and infrared structure corroborate the results, which are complemented by publicly available Mathematica code and expressions. Overall, this work delivers the complete two-loop helicity amplitudes needed for precise gluon-induced Z-boson production phenomenology at the LHC.

Abstract

We compute the helicity amplitudes for the processes $gg\to Zg$ and $gg\to Zγ$ to two loops in massless QCD. The perturbative expansion of these processes starts only at the one-loop level, such that our results are a crucial ingredient to the NLO corrections to $Zγ$ and $Z+$jet production through gluon fusion.