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

This thesis establishes a rigorous, algorithmic framework for restoring gauge and BRST invariance in chiral gauge theories using the BMHV scheme with non-anticommuting $\gamma_5$, successfully demonstrating its feasibility through the complete 4-loop renormalisation of an Abelian theory and the 1-loop renormalisation of the full Standard Model.

The Gist

Here is an explanation of Matthias Weißwange's dissertation, translated from high-level physics jargon into a story about building a perfect, glitch-free video game engine.

The Big Picture: The "Glitch" in the Universe's Code

Imagine the Standard Model of particle physics as the ultimate source code for the universe. It describes how tiny particles (like electrons and quarks) interact with forces (like electromagnetism and the weak nuclear force). This code is incredibly precise, but it has a notorious bug: Chirality.

In the real world, some particles are "left-handed" and some are "right-handed" (like your hands). They behave differently. In the math used to calculate how these particles interact, there is a special symbol called $\gamma_5$ (gamma-five) that acts like a "handedness switch."

The problem? When physicists try to run these calculations on a computer to predict what happens at high energies, they run into a mathematical paradox. It's like trying to fit a 4-dimensional object into a 3-dimensional box. The math breaks down, producing "infinities" (errors that make the calculation impossible).

The Solution: The "BMHV" Patch

For decades, physicists have used a method called Dimensional Regularization to fix these infinities. It's like temporarily stretching the universe into 4.0001 dimensions to smooth out the rough edges of the math.

However, there's a catch. The standard way of stretching the universe (the "Naive" method) breaks a fundamental rule of the game: Symmetry. In physics, if you break symmetry, the laws of physics stop making sense (particles might appear out of nowhere, or probabilities might add up to more than 100%).

The BMHV Scheme (Breitenlohner-Maison/'t Hooft-Veltman) is a specific, mathematically rigorous way of handling the "handedness switch" ($\gamma_5$) in this stretched universe.

• The Good News: It is mathematically perfect. It never crashes.
• The Bad News: It introduces a "spurious" glitch. It breaks the symmetry temporarily during the calculation, even though the final result should be perfect.

Think of it like this: You are building a bridge. The BMHV method is a construction technique that is so precise it never collapses, but it requires you to build the bridge slightly crooked in the middle. You have to know exactly how to bend it back into a straight line at the very end, or the bridge won't work.

The Mission: Fixing the Glitch at 4 Loops

The core achievement of this thesis is solving the "bending back" problem.

In quantum physics, calculations are done in layers called loops.

• 1 Loop: A simple calculation (like a single loop in a road).
• 4 Loops: An incredibly complex calculation involving billions of intermediate steps.

Previous attempts to fix the symmetry breaking in the BMHV scheme usually stopped at 1 or 2 loops. The math got too messy, and the "bending back" (called Symmetry Restoration) became too hard to calculate.

Matthias Weißwange's breakthrough:

1. The 4-Loop Milestone: He successfully performed the complete calculation for an Abelian chiral gauge theory (a simplified version of the Standard Model) all the way up to 4 loops. This is the highest order ever achieved with this specific, rigorous method.
2. The Automated Factory: To do this, he didn't just use a calculator; he built a digital factory. He created a highly optimized computer program (using a language called FORM) that could handle billions of algebraic terms per second. Without this automation, the calculation would have taken a human lifetime.
3. The "Shadow" Analysis: He discovered that there are "shadows" in the math—different ways to set up the initial rules (called Evanescent Shadows). He showed that while these shadows look different, they all lead to the same physical result, provided you fix the symmetry correctly at the end.

The Grand Finale: The Standard Model

The ultimate goal is to apply this rigorous, glitch-free method to the entire Standard Model (the full theory of all known particles).

• The Result: The thesis presents the complete 1-loop renormalization of the full Standard Model using the BMHV scheme.
• Why it matters: This is the "foundation laying" phase. Before you can build a skyscraper (high-precision 2-loop or 3-loop predictions for the Large Hadron Collider), you need a solid, mathematically consistent foundation. This thesis proves that the foundation is solid.

The Analogy: The Perfect Video Game

Imagine you are a game developer trying to simulate a universe where left-handed and right-handed characters have different physics.

1. The Problem: Your physics engine crashes whenever a left-handed character interacts with a right-handed one because of a bug in the code ($\gamma_5$).
2. The Patch (BMHV): You install a patch that stops the crashes, but the patch makes the characters walk slightly sideways during the simulation.
3. The Fix (Renormalization): You need to write a script that automatically corrects the sideways walking at the end of every frame so the characters walk straight again.
4. The Achievement: Previous developers could only write this script for simple scenes (1 loop). Matthias wrote a script that works for the most complex, chaotic scenes imaginable (4 loops), involving billions of moving parts. He proved that no matter how complex the scene gets, the script can always fix the walking so the game remains playable and realistic.

Why Should You Care?

• Precision: As our experiments (like those at CERN) get more precise, our theories must be equally precise. If we use "naive" math that ignores these subtle glitches, our predictions will eventually be wrong.
• New Physics: If the Standard Model is slightly off, it might mean there is "New Physics" (like Dark Matter or extra dimensions). To find it, we need to be 100% sure our current math is perfect. This thesis ensures our math is perfect.
• The Future: This work provides the tools and the confidence to calculate the next generation of particle physics predictions, potentially leading to the discovery of new laws of nature.

In short: Matthias Weißwange built the ultimate mathematical engine to calculate the universe's behavior, proving that even the most complex, "glitchy" parts of the code can be fixed to reveal a perfectly consistent reality.

Technical Summary

Technical Summary: Renormalisation of Chiral Gauge Theories with Non-Anticommuting $\gamma_5$ at the Multi-Loop Level

1. Problem Statement

The renormalisation of chiral gauge theories (such as the Standard Model) in dimensional regularisation (DReg) is obstructed by the $\gamma_5$-problem. The matrix $\gamma_5$ is inherently 4-dimensional, defined via the Levi-Civita tensor $\varepsilon_{\mu\nu\rho\sigma}$. Extending it to $D = 4 - 2\epsilon$ dimensions while preserving its anticommutation relations with all $\gamma_\mu$ matrices leads to mathematical inconsistencies (contradictions in trace identities).

Alternative prescriptions (e.g., naive anticommuting $\gamma_5$, Kreimer's scheme, Larin's prescription) either lack mathematical consistency at multi-loop orders or introduce ambiguities that require external arguments to resolve. The only known mathematically consistent framework is the Breitenlohner-Maison/'t Hooft-Veltman (BMHV) scheme, where $\gamma_5$ is kept strictly 4-dimensional. However, this breaks the anticommutation relation with the $(D-4)$-dimensional components of $\gamma_\mu$, leading to a spurious violation of gauge and BRST invariance at the regularised level. Restoring these symmetries requires the systematic construction of symmetry-restoring counterterms, a task that becomes computationally prohibitive at high loop orders due to the proliferation of terms and the need to compute operator-inserted Green functions.

2. Methodology

The thesis employs a rigorous, algorithmic approach based on algebraic renormalisation and the quantum action principle within the BMHV scheme.

A. Theoretical Framework

• BMHV Scheme: The spacetime is decomposed into a 4-dimensional subspace ($M_4$) and an evanescent subspace ($M_{D-4}$). $\gamma_5$ acts only on $M_4$. Consequently, $\{\gamma_5, \gamma_\mu\} = 0$ for $\mu \in M_4$, but $[\gamma_5, \hat{\gamma}_\mu] = 0$ for $\hat{\gamma}_\mu \in M_{D-4}$.
• Symmetry Breaking: The modified algebra causes the evanescent part of the fermion kinetic term to mix chiralities, breaking BRST invariance. This breaking is encoded in a composite operator $\Delta = \hat{\Delta} + \Delta_{ct}$.
• Restoration Procedure: The Quantum Action Principle is used to relate the breaking of the Slavnov-Taylor identity to the insertion of $\Delta$ into Green functions. The procedure involves:
  1. Computing ordinary 1PI Green functions to determine singular (divergent) counterterms.
  2. Computing operator-inserted 1PI Green functions ($\Delta \cdot \Gamma$) to determine the symmetry breaking.
  3. Constructing finite symmetry-restoring counterterms ($S_{fct}$) such that the Slavnov-Taylor identity is restored: $S(\Gamma_{ren}) = 0$.
• Dimensional Ambiguities: The thesis investigates the non-uniqueness of the $D$-dimensional extension of fermion kinetic terms and gauge interactions. It explores different "Options" for constructing Dirac spinors (combining physical chiralities vs. introducing sterile partners) and the role of evanescent gauge interactions (couplings to the $(D-4)$-dimensional gauge boson).

B. Computational Framework

To handle the extreme complexity of multi-loop calculations (billions of intermediate terms), the author developed a fully automated, high-performance computational framework:

• Software Stack: Primarily based on FORM (for symbolic manipulation of billions of terms) and FIRE/Reduze2/Kira (for Integration-by-Parts (IBP) reduction). A Mathematica interface was used for high-level control.
• Tadpole Decomposition: An improved Taylor expansion method is used to extract UV divergences. By introducing an auxiliary mass scale $M^2$ into all propagators, all multi-scale integrals are mapped to single-scale massive vacuum bubbles (tadpoles). This avoids the complexity of multi-scale master integrals while preserving UV behavior.
• Tensor Reduction: An orbit partition approach (based on group theory) is implemented to efficiently reduce high-rank tensor integrals to scalar master integrals, handling the complex Lorentz structures arising from the BMHV algebra.
• BMHV Algebra Implementation: Dedicated FORM routines handle the non-anticommuting $\gamma_5$ by factorising traces into 4-dimensional and evanescent parts, exploiting the fact that mixed traces vanish.

3. Key Contributions

A. First Complete 4-Loop Renormalisation

The thesis presents the first complete 4-loop renormalisation of a chiral gauge theory (specifically, an Abelian chiral gauge theory with right-handed fermions) within the BMHV scheme. This represents the highest-order application of the BMHV scheme to date, demonstrating that a fully self-consistent treatment of $\gamma_5$ is feasible even at very high loop orders.

B. Systematic Analysis of Dimensional Ambiguities

The work provides a comprehensive analysis of how different $D$-dimensional realisations affect the symmetry breaking:

• Fermion Options: It compares Option 1 (combining physical chiralities into Dirac spinors) with Option 2 (introducing sterile partners). It concludes that while Option 2 preserves global symmetries better, it introduces non-standard propagator denominators for massive fermions, making it impractical for the Standard Model (SM). Option 1 is preferred despite global symmetry violations, which are restored by counterterms.
• Evanescent Interactions: It analyses the impact of evanescent gauge interactions (parameterised by $c_{QED}$ and $c_{QCD}$). The results show that omitting evanescent gauge interactions ($c=0$) yields the most compact and economical counterterm structure, suggesting this is the optimal prescription for practical calculations.

C. Complete 1-Loop Renormalisation of the Standard Model

A major milestone is the complete 1-loop renormalisation of the full Standard Model in the BMHV scheme. This includes:

• Derivation of all singular and finite symmetry-restoring counterterms.
• Analysis of evanescent gauge interactions in the SM context (photon and gluon sectors).
• Establishment of the foundation for future 2-loop and higher-order SM calculations.

D. Validation of Renormalisability

The work explicitly demonstrates that the symmetry breaking in the BMHV scheme can be parametrised by a finite basis of local field monomials (constrained by power counting and ghost number) at every loop order. This confirms the stability and renormalisability of chiral gauge theories in this scheme, proving that no new counterterm structures emerge indefinitely as the loop order increases.

4. Key Results

• Abelian Chiral Theory (4-Loop):

  • Complete determination of divergent and finite counterterms up to 4 loops.
  • Verification that the Slavnov-Taylor identity is restored order-by-order.
  • Explicit coefficients for the breaking of Ward identities (transversality of gauge boson self-energies, relations between fermion self-energies and vertices).
  • Identification of auxiliary counterterms required for the tadpole decomposition method.

• Standard Model (1-Loop):

  • Full set of counterterms for fermions, gauge bosons, Higgs, Yukawa, and ghost sectors.
  • Demonstration that global hypercharge violation induced by the evanescent kinetic term (in Option 1) is cancelled by specific finite counterterms.
  • Confirmation that the choice $c_{QED}=0, c_{QCD}=0$ (purely 4-dimensional interactions for photon and gluon) minimises the complexity of the counterterm action.

• Computational Performance:

  • The framework successfully handled 4-loop diagrams with tensor ranks up to 12 and generated billions of intermediate terms per diagram, validating the efficiency of the FORM-based setup.

5. Significance

• Theoretical Rigor: This work resolves the long-standing challenge of performing high-precision calculations in chiral gauge theories without relying on ad-hoc prescriptions or external consistency arguments. It establishes the BMHV scheme as a viable, mathematically rigorous tool for multi-loop phenomenology.
• Precision Phenomenology: By providing a consistent framework for $\gamma_5$, this research paves the way for next-to-next-to-next-to-leading order (N$^3$LO) and higher precision calculations for electroweak observables (e.g., $W$-boson mass, $Z$-decays, Higgs production). This is crucial for testing the Standard Model against high-precision data from the LHC and future colliders.
• Methodological Advancement: The developed computational tools (tadpole decomposition, orbit partition tensor reduction, automated symmetry restoration) are general and can be applied to other complex quantum field theory problems, including Effective Field Theories (EFTs) like SMEFT.
• Resolution of Ambiguities: The systematic study of evanescent details provides clear guidelines for practitioners on how to implement the regularisation to minimise computational complexity while maintaining physical consistency.

In summary, this thesis represents a landmark achievement in theoretical particle physics, bridging the gap between the mathematical consistency of the BMHV scheme and the practical requirements of high-precision multi-loop calculations in the Standard Model.