Renormalisation of Chiral Gauge Theories with Non-Anticommuting $γ_5$ at the Multi-Loop Level

This thesis presents a comprehensive study of the renormalisation of chiral gauge theories in dimensional regularisation (DReg) at the multi-loop level. We employ the mathematically consistent Breitenlohner-Maison/`t~Hooft-Veltman (BMHV) scheme with non-anticommuting γ5, whose modified algebraic relations induce a spurious violation of gauge and BRST invariance. A central focus is the systematic restoration of the broken symmetry, for which we provide a transparent and fully algorithmic procedure based on the quantum action principle. A major achievement of this work is the complete 4-loop renormalisation of an Abelian chiral gauge theory --- the highest-order application of the BMHV scheme to date. This calculation is made possible by an automated, high-performance computational framework incorporating several optimised algorithms. Our results demonstrate that a rigorous, self-consistent treatment of γ5 is feasible even at very high loop orders. We further analyse dimensional ambiguities and evanescent details corresponding to different implementations of the regularisation, and identify practically efficient prescriptions for D-dimensional fermions and gauge interactions. Building on these insights, we present the complete 1-loop renormalisation of the full Standard Model (SM) in the BMHV scheme, providing a first step towards a fully self-consistent multi-loop renormalisation of the SM and establishing a solid foundation for future high-precision electroweak phenomenology.:1. Introduction 2. Gauge Theories 3. Renormalisation and Symmetries 4. Dimensional Regularisation and Renormalisation 5. Multi-Loop Techniques 6. Symmetry Restoration 7. Abelian Chiral Gauge Theories in D Dimensions 8. Non-Anticommuting γ5 at the Multi-Loop Level 9. The Standard Model in the BMHV Scheme 10. Conclusions A. Results --- General Abelian Chiral Gauge Theory B. Results --- Right-Handed Abelian Gauge Theory C. Results --- The Standard Model D. Renormalisation of Quantum Electrodynamics