Axial triangles in $q\bar{q}\to Z\gamma$ at two loops in QCD directly in four dimensions

Abstract

We numerically evaluate the two-loop QCD squared matrix element for in $q\bar{q}\to Z$ and $q\bar{q}\to Z\gamma$ with heavy top and bottom quarks circulating in a triangular fermion loop, by simultaneously subtracting infrared, ultraviolet, and threshold singularities directly in loop momentum space. This computation serves as an explicit demonstration that axial couplings can be included in the final state within the framework of arXiv:2510.18801. By formulating the entire calculation in four spacetime dimensions, with anomaly cancellation realised locally in loop momentum space, we bypass the complications associated with treating $\gamma^5$ in dimensional regularisation.