Renormalization constants and beta functions for the gauge couplings of the Standard Model to three-loop order

TL;DR

This work delivers the three-loop beta functions for the Standard Model gauge couplings in the MSbar scheme, including all Yukawa sectors and the Higgs self-coupling. It performs two independent calculations—Lorenz gauge in the unbroken phase and background field gauge in the broken phase—with a careful γ5 treatment, and it automates diagram generation and evaluation via a FeynArtsToQ2E pipeline feeding MINCER and MATAD. The authors provide explicit three-loop beta functions with full flavor structure and cross-checks for gauge-parameter independence and IR safety, showing small but important corrections that sharpen high-scale extrapolations such as grand unification. The results are verified against prior work and supported by a robust computational framework, underscoring the reliability of perturbative running up to very high energies.

Abstract

We compute the beta functions for the three gauge couplings of the Standard Model in the minimal subtraction scheme to three loops. We take into account contributions from all sectors of the Standard Model. The calculation is performed using both Lorenz gauge in the unbroken phase of the Standard Model and background field gauge in the spontaneously broken phase. Furthermore, we describe in detail the treatment of $γ_5$ and present the automated setup which we use for the calculation of the Feynman diagrams. It starts with the generation of the Feynman rules and leads to the bare result for the Green's function of a given process.

Figures (5)

• Figure 1: Sample Feynman diagrams contributing to the Green's functions which have been used for our calculation of the gauge coupling renormalization constants. Solid, dashed, dotted, curly and wavy lines correspond to fermions, Higgs bosons, ghosts, gluons and electroweak gauge bosons, respectively.
• Figure 2: Overview of our automated setup. Calling up the programs in the uppermost line determines and evaluates a given process in a given model. The vertical workflow leads to the implementation of a new model in the setup. The programs are discussed in more detail in the text.
• Figure 3: The running of the gauge couplings at three loops. The curve with the smallest initial value corresponds to $\alpha_1$, the middle curve to $\alpha_2$, and the curve with the highest initial value to $\alpha_3$.
• Figure 4: The running of the electroweak gauge couplings in the SM. The lines with positive slope correspond to $\alpha_1$, the lines with negative slope to $\alpha_2$. The dotted, dashed and solid lines correspond to one-, two- and three-loop precision, respectively. The bands around the three-loop curves visualize the experimental uncertainty.
• Figure 5: Comparison of the relative experimental and theoretical uncertainty of $\alpha_3$. The theoretical uncertainty is given by the dashed line, the solid curve corresponds to the experimental one.