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

TL;DR

This paper presents a numerical calculation of the two-loop QCD matrix element for quark-antiquark to Z and Z gamma processes, including axial couplings, directly in four dimensions, avoiding issues with gamma5 in dimensional regularization.

Contribution

It demonstrates a method to include axial couplings in two-loop QCD calculations directly in four dimensions, bypassing gamma5 regularization complications.

Findings

Successfully computed two-loop matrix elements with axial couplings in four dimensions.

Implemented a subtraction scheme for infrared, ultraviolet, and threshold singularities.

Validated the approach within the framework of anomaly cancellation in loop momentum space.

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.