Four-Loop Renormalisation of Chiral Gauge Theories with Non-Anticommuting $\gamma_5$ in the BMHV Scheme

TL;DR

This paper performs the first complete 4-loop renormalisation of an Abelian chiral gauge theory using the BMHV scheme with a non-anticommuting gamma_5, demonstrating the scheme's viability at high loop orders.

Contribution

It provides the highest-order application of the BMHV scheme, including the full set of symmetry restoring counterterms at 4 loops, and details the computational framework used.

Findings

Successfully applied BMHV scheme at 4 loops

Determined symmetry restoring counterterms

Established a foundation for high-precision electroweak calculations

Abstract

We present the complete 4-loop renormalisation of an Abelian chiral gauge theory in the Breitenlohner-Maison / `t Hooft-Veltman (BMHV) scheme. Employing a non-anticommuting $\gamma_5$ in dimensional regularisation, we determine the full set of symmetry restoring counterterms from the quantum action principle. Our calculation represents the highest-order application of the BMHV framework so far, pushing the limits of a self-consistent treatment of $\gamma_5$ at the multi-loop level. We describe the computational setup that we developed to perform the computations and discuss key implementation aspects, such as the BMHV algebra and tensor reduction. Our work demonstrates the feasibility of applying the BMHV scheme at high loop orders and establishes a solid foundation for future studies of high-precision electroweak physics.