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

Matthias Weißwange

Abstract

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 $γ_5$, 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 $γ_5$ 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.

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

Abstract

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 , 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 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 -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.
Paper Structure (260 sections, 24 theorems, 660 equations, 3 figures, 9 tables, 2 algorithms)

This paper contains 260 sections, 24 theorems, 660 equations, 3 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Gauge Theories
  3. Basic Concepts
  4. Chiral Gauge Theories
  5. Quantisation of Gauge Theories and the BRST Formalism
  6. Canonical Quantisation and Faddeev-Popov Approach:
  7. The BRST Formalism:
  8. The physical Hilbert Space and the $S$-Matrix:
  9. Anomalies
  10. Renormalisation and Symmetries
  11. Renormalisation Theory
  12. Causal Perturbation Theory:
  13. Renormalisation on the level of Feynman Diagrams:
  14. Concluding Comments on Renormalisation:
  15. Symmetries at the Quantum Level
  16. ...and 245 more sections

Key Result

Theorem 3.1

Let $S_T(\mathbf{g})$ and $S_{T'}(\mathbf{G})$ be two perturbative constructions of the scattering operator as a formal functional power series in $\mathbf{g}=(g_i)_{i\geq1}$ and $\mathbf{G}=(G_i)_{i\geq1}$, see Eq. Eq:ScatteringOperator-FormalPowerSeries, corresponding to two sequences of time-orde

Figures (3)

  • Figure 1: Representative 4-loop Feynman diagrams contributing to the $\Delta$-operator-inserted 1PI Green function $i{(\Delta\cdot\Gamma)}{}_{B_{\mu}c}^{(4)}$.
  • Figure 2: Representative 4-loop Feynman diagrams contributing to the $\Delta$-operator-inserted 1PI Green function $i{(\Delta\cdot\Gamma)}{}_{B_{\rho}B_{\nu}B_{\mu}c}^{(4)}$.
  • Figure 3: Representative 3-loop Feynman diagrams with counterterm insertions contributing to the 4-loop, subrenormalised, $\Delta$-operator-inserted 1PI Green function $i{(\Delta\cdot\Gamma)}{}_{B_{\rho}B_{\nu}B_{\mu}c}^{(4)}$. The left diagram features the insertion of a new $\Delta$-vertex resulting from a 1-loop correction to the $\Delta$-operator, while the right diagram involves the insertion of a 1-loop, finite, symmetry-restoring counterterm.

Theorems & Definitions (71)

  • Definition 2.1: BRST Operator
  • Theorem 3.1: Main Theorem of Renormalisation
  • Remark : on the physical conditions
  • Remark : on the ambiguity in the renormalisation procedure and physical equivalence
  • Remark : on the formal power series
  • Remark : Epstein-Glaser vs. Bogoliubov-Shirkov
  • Definition 3.1: Overall Degree of Divergence
  • Theorem 3.2: Convergence Theorem for power-counting finite Integrals
  • Definition 3.2: $\mathcal{R}$-Operation
  • Theorem 3.3: Zimmermann's Forest Formula
  • ...and 61 more