⚛️ phenomenology

The Standard Model with Non-Anticommuting $\gamma_5$ in the BMHV Scheme

This paper presents the complete one-loop renormalization of the full Standard Model within the Breitenlohner-Maison/'t Hooft-Veltman (BMHV) scheme, deriving all necessary symmetry-restoring counterterms and analyzing the impact of evanescent gauge interactions to establish a consistent framework for multi-loop electroweak precision calculations involving $\gamma_5$ .

Original authors: Paul Kühler, Dominik Stöckinger, Matthias Weißwange

Published 2026-11-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Paul Kühler, Dominik Stöckinger, Matthias Weißwange

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Standard Model is the most successful theory physicists have ever built to describe the fundamental particles of nature and the forces that govern them. It explains how matter is constructed from tiny building blocks like quarks and electrons, and how these particles interact through forces such as electromagnetism and the strong nuclear force. A crucial feature of this theory is that it treats left-handed and right-handed versions of these particles differently, a property known as chirality. This difference is essential for explaining why the universe behaves the way it does, including why certain particles decay and others do not. However, when physicists try to calculate the precise behavior of these particles using the tools of quantum mechanics, they encounter a persistent mathematical snag involving a specific object called gamma-five. This object is used to distinguish between left and right, but it behaves in a way that is difficult to reconcile with the standard method used to handle the infinite numbers that appear in quantum calculations. If this problem is not solved correctly, the entire theory can produce contradictory or nonsensical results, undermining our ability to predict what experiments will see.

In a new study, a team of researchers has provided a complete and rigorous solution to this problem for the entire Standard Model. They worked within a specific mathematical framework known as the Breitenlohner-Maison/'t Hooft-Veltman scheme, which is the only method proven to be mathematically consistent at every level of calculation. While this method is reliable, it has a known side effect: it temporarily breaks the fundamental symmetries that keep the theory consistent, much like a temporary glitch in a complex machine. The researchers' task was to identify exactly how this symmetry breaks and to construct a precise set of corrections, called counterterms, to fix it. They performed this calculation for the first time across the full Standard Model, including all types of particles and forces, and in a general setting that allows for different ways of measuring the interactions. Their work produces a complete list of these necessary corrections, ensuring that the theory remains consistent and predictive even when dealing with the most complex quantum effects.

A central part of their investigation involved deciding how to treat the forces that act on particles in a way that looks the same from all directions, such as the force carried by photons and gluons. The researchers explored whether these forces should be treated as strictly four-dimensional or allowed to exist in the extra mathematical dimensions used for the calculations. They found that the choice matters significantly. For the electromagnetic and weak forces, which distinguish between left and right, the best approach is to treat them as strictly four-dimensional. This choice simplifies the corrections needed to fix the symmetry breaking. For the strong force, which binds quarks together, the situation is more nuanced. The team discovered that no matter how the strong force is treated, it always contributes to the symmetry breaking, meaning the strong force cannot be ignored or treated separately from the rest of the model. However, treating the strong force as fully existing in the extra dimensions yields the cleanest and most efficient set of corrections.

The researchers also examined how these corrections behave at the most basic level of interaction, known as the tree level, before any complex quantum loops are added. They showed that even at this simplest stage, the mathematical inconsistencies caused by the gamma-five problem are unavoidable and must be addressed. Their analysis confirmed that the Standard Model must be treated as a single, unified whole; one cannot simply calculate the corrections for the strong force in isolation and expect the rest of the theory to work. The symmetry breaking caused by the strong force is inextricably linked to the behavior of the other forces. By mapping out every single correction needed, the team has created a foundation that allows for future, more complex calculations. This work is a necessary step toward achieving the high-precision predictions required to match the increasingly accurate data coming from modern particle accelerators.

The study was carried out using two completely independent computer systems to ensure the results were correct. One system used a specialized language designed for handling massive algebraic expressions, while the other used a more general mathematical software package. Both systems produced identical results, providing a powerful cross-check that confirms the reliability of the findings. The researchers presented their results in two formats: a compact version that groups all the forces together for efficiency, and a detailed version that breaks them down by the specific type of force, making it easier to see how each part contributes. This dual presentation ensures that the results are both usable for automated calculations and transparent for human understanding.

Ultimately, this work provides the complete toolkit needed to perform multi-loop calculations in the Standard Model with a fully consistent treatment of the gamma-five problem. Multi-loop calculations are essential for predicting the properties of particles with extreme precision, such as the mass of the W boson or the magnetic moment of the muon. Without the corrections identified in this study, these predictions would be flawed. The researchers recommend a specific approach for future work: treating the electroweak forces as strictly four-dimensional and the strong force as fully existing in the extra dimensions. This combination minimizes the number of corrections required and keeps the theory as simple as possible while remaining mathematically sound. By solving this long-standing technical challenge, the team has removed a major obstacle to testing the Standard Model at the highest levels of precision, paving the way for deeper insights into the fundamental laws of the universe.

Technical Summary: The Standard Model with Non-Anticommuting $\gamma_5$ in the BMHV Scheme

Problem Statement
The treatment of the chiral matrix $\gamma_5$ in dimensional regularisation (DReg) presents a fundamental challenge for perturbative calculations in chiral gauge theories like the Standard Model (SM). No extension of $\gamma_5$ to $D = 4 - 2\epsilon$ dimensions can simultaneously preserve the cyclicity of $\gamma$ -traces, anticommutation with all $\gamma_\mu$ matrices, and the correct relation to the $\epsilon$ -tensor. The Breitenlohner-Maison/'t Hooft-Veltman (BMHV) scheme resolves this by treating $\gamma_5$ as a strictly four-dimensional object that anticommutes only with the four-dimensional part of $\gamma_\mu$ and commutes with the evanescent ( $D-4$ ) part. While mathematically consistent to all orders, this algebraic modification spuriously breaks gauge and BRST invariance at the regularised level. Consequently, the Slavnov-Taylor identities are violated, necessitating the introduction of finite, symmetry-restoring counterterms in addition to the standard singular counterterms required for UV renormalisation. Previous applications of the BMHV scheme to the SM were limited to specific sectors, background field gauges, or lower loop orders, leaving the complete 1-loop renormalisation of the full SM in a general $R_\xi$ gauge unresolved.

Methodology
The authors perform a complete 1-loop renormalisation of the full Standard Model within the BMHV scheme using the regularised quantum action principle. The calculation is conducted in a general $R_\xi$ gauge, parametrised by $\xi_F = 1 - \xi$ . The methodology involves:

Lagrangian Formulation: The SM is formulated in $D$ dimensions using a compact "twelveplet" notation for gauge fields ( $V^A_\mu$ ) and generalised generators to handle the $U(1)_Y \times SU(2)_L \times SU(3)_c$ structure efficiently. Fermions are treated as Dirac spinors where possible, combining physical chiral fields, which necessitates the inclusion of fictitious sterile right-handed neutrino fields in the kinetic terms.
Evanescent Interactions: A central feature of the analysis is the parametrisation of evanescent gauge interactions via two parameters, $c_{\text{QED}}$ and $c_{\text{QCD}}$ . These parameters allow the photon and gluon currents to be treated either purely 4-dimensionally ( $c=0$ ) or fully $D$ -dimensionally ( $c=1$ ). The authors analyse the impact of these choices on the tree-level symmetry breaking and the resulting counterterm structure.
Symmetry Restoration: The breaking of the Slavnov-Taylor identity is identified as a local operator insertion $\hat{\Delta}$ . The required counterterm action $S_{\text{ct}} = S_{\text{sct}} + S_{\text{fct}}$ is determined by solving the linearised Slavnov-Taylor equation:
$\lim_{D \to 4} \left( \hat{\Delta} \cdot \Gamma_1 + \hat{D} S_{\text{sct}} + \hat{D} S_{\text{fct}} \right) = 0$
where $\Gamma_1$ is the 1-loop effective action, $S_{\text{sct}}$ contains singular (divergent) counterterms, and $S_{\text{fct}}$ contains finite symmetry-restoring counterterms.
Computational Framework: The calculation employs two independent, automated computational frameworks (one based on FORM and one on Mathematica) to generate Feynman diagrams, perform algebraic manipulations, and reduce integrals. This dual approach serves as a rigorous cross-check.

Key Contributions and Results
The paper provides the first complete 1-loop renormalisation of the full Standard Model in the BMHV scheme, including:

Complete Counterterm Action: The authors derive the full set of singular and finite symmetry-restoring counterterms for all sectors of the SM (fermionic, gauge, Higgs, Yukawa, and ghost/exterior source sectors). These results are presented in two formats: a compact form using twelveplets (suitable for automation) and a decomposed form separating the $U(1)_Y$ , $SU(2)_L$ , and $SU(3)_c$ contributions (suitable for physical interpretation).
Analysis of Evanescent Gauge Interactions: The study demonstrates that the choice of evanescent interactions significantly impacts the structure of the counterterms.
- Electroweak Sector: Setting $c_{\text{QED}} = 0$ (purely 4-dimensional photon current) is shown to be the most economical choice, minimising the number of counterterms and simplifying the structure of symmetry-restoring terms.
- QCD Sector: The analysis reveals that QCD always contributes to regularisation-induced symmetry breaking, regardless of whether the gluon current is treated as 4-dimensional ( $c_{\text{QCD}}=0$ ) or $D$ -dimensional ( $c_{\text{QCD}}=1$ ). A purely 4-dimensional treatment induces pure QCD breaking, while a $D$ -dimensional treatment induces mixed electroweak-QCD breaking. Consequently, the SM cannot be treated as separate sectors; the full theory must be renormalised as a whole.
Recommendation: Based on the minimisation of counterterms and the restoration of BRST invariance in the pure QCD sector, the authors recommend the choice $(c_{\text{QED}}, c_{\text{QCD}}) = (0, 1)$ . This corresponds to a purely 4-dimensional treatment of electroweak gauge interactions and a fully $D$ -dimensional treatment of the gluon interaction.
Consistency Checks: The results are validated by verifying that the counterterms cancel the 1-loop breaking of the Slavnov-Taylor identity and by reproducing known results in specific limits, including the Abelian sector, ordinary QED, and ordinary QCD.

Significance
The authors state that this work constitutes a necessary step towards multi-loop electroweak precision calculations with a fully consistent treatment of $\gamma_5$ . By providing the complete 1-loop counterterm action, the paper establishes the foundation required for higher-order computations in the BMHV scheme. The explicit determination of off-shell Green function counterterms is particularly crucial for the sub-renormalisation procedures needed in multi-loop SM calculations. The results enable the rigorous calculation of electroweak precision observables at higher orders, matching the increasing precision of current and future experiments, while maintaining mathematical consistency in the treatment of chiral anomalies and $\gamma_5$ .

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