[Paper Review] Gauge coupling beta functions in the Standard Model

L. Mihaila, J. Salomon+1 more|arXiv (Cornell University)|Sep 25, 2012

Particle physics theoretical and experimental studies4 references3 citations

TL;DR

This paper presents the complete three-loop corrections to the gauge coupling beta functions in the Standard Model using the $\overline{\rm MS}$ renormalization scheme. Employing automated diagram generation and cross-checked calculations via Lorenz gauge and background field gauge approaches, the authors derive exact analytic expressions for $\beta_{\alpha_1}$, $\beta_{\alpha_2}$, and $\beta_{\alpha_3}$, revealing that three-loop terms reduce theoretical uncertainty below experimental precision, particularly for $\alpha_2$. The results are essential for high-precision unification and phenomenological predictions.

ABSTRACT

We report about the computation of three-loop corrections to the gauge coupling beta functions in the Standard Model.

Motivation & Objective

• To compute the three-loop corrections to the gauge coupling beta functions in the Standard Model with high precision.
• To reduce theoretical uncertainty in the running of gauge couplings below the level of experimental error, especially for $\alpha_2$.
• To provide analytic expressions for $\beta_{\alpha_1}$, $\beta_{\alpha_2}$, and $\beta_{\alpha_3}$ in the $\overline{\rm MS}$ scheme, valid for high-energy unification studies.
• To validate results using multiple independent methods—Lorenz gauge and background field gauge—ensuring reliability through cross-checking.
• To incorporate Yukawa and Higgs self-coupling effects, including the possibility of extended fermion generations, in the final beta function expressions.

Proposed method

• The beta functions are computed via the master formula relating renormalization constants $Z_{\alpha_i}$ to the $\mu$-dependence of the couplings, using $\mu^2 \frac{d}{d\mu^2} \frac{\alpha_i}{\pi} = \beta_i$.
• Renormalization constants $Z_{\alpha_i}$ are calculated to three-loop order using vertex and wave function renormalization, with $Z_{\alpha_i} = (Z_{\text{vrtx}})^2 / \prod_k Z_{k,\text{wf}}$.
• The calculation uses the $\overline{\rm MS}$ scheme, which eliminates mass dependence and simplifies loop integrals to massless two-point functions.
• Feynman diagrams (up to $10^6$) are generated and computed automatically using a computational framework that also derives Feynman rules algorithmically.
• Two independent approaches are used: the unbroken phase with Lorenz gauge and the broken phase with background field gauge, enabling cross-validation of results.
• The Higgs self-coupling contribution is treated at leading order in $\epsilon$, and Yukawa couplings are encoded via trace invariants $\text{tr}\hat{L}$, $\text{tr}\hat{T}$, $\text{tr}\hat{B}$.

Experimental results

Research questions

• RQ1 What are the exact three-loop corrections to the beta functions of the hypercharge ($\alpha_1$), weak isospin ($\alpha_2$), and strong ($\alpha_3$) couplings in the Standard Model?
• RQ2 How do three-loop corrections affect the running of $\alpha_1$ and $\alpha_2$ from the $Z$-boson scale to unification scales?
• RQ3 To what extent do three-loop terms reduce theoretical uncertainty compared to experimental uncertainty in gauge coupling unification?
• RQ4 How can the results be generalized to include a fourth generation of fermions with heavy Yukawa couplings?
• RQ5 What is the analytic structure of the three-loop beta functions, including mixed contributions from gauge, Yukawa, and Higgs self-couplings?

Key findings

• The three-loop beta functions for $\alpha_1$, $\alpha_2$, and $\alpha_3$ are derived in closed analytic form, including mixed terms involving $\alpha_1\alpha_2$, $\alpha_1\alpha_3$, $\alpha_2\alpha_3$, $\alpha_i^2\alpha_j$, and $\alpha_i^2\lambda$, where $\lambda$ is the Higgs self-coupling.
• The three-loop corrections significantly reduce theoretical uncertainty in the running of $\alpha_1$ and $\alpha_2$, bringing it below the level of current experimental uncertainty, especially for $\alpha_2$.
• The inclusion of three-loop terms results in a visible shift in the running curves, with the three-loop solid lines deviating notably from one- and two-loop curves, but converging closely to the two-loop band.
• The results are validated through cross-checks using both Lorenz gauge and background field gauge formulations, confirming consistency across different computational frameworks.
• The formulae are extended to include a fourth generation of fermions by modifying the Yukawa trace invariants $\text{tr}\hat{F}^n$ to include a heavy fourth-generation coupling $\alpha_F$, with $\hat{F}_4$ as a $4\times4$ block matrix.
• The contribution of a heavy fourth-generation neutrino is not included, as the formalism assumes only charged and vector-like fermions contribute to the trace invariants.