Two-Loop Renormalization in the Standard Model Part II: Renormalization Procedures and Computational Techniques

TL;DR

The paper develops a complete two-loop renormalization framework for the Standard Model, extending the MS-bar scheme and the 't Hooft–Veltman approach to ensure local, polynomial counterterms remove all ultraviolet divergences. It delivers a systematic UV decomposition for one- and two-loop integrals, and provides explicit two-loop self-energies and counterterms, showing that two-point functions suffice to renormalize the theory while preserving Ward-Slavnov-Taylor identities. The methodology combines automated diagram generation with analytic treatment of collinear logs, enabling precise, numerically stable predictions. The work lays the groundwork for finite renormalization and physical observable calculations in Part III and demonstrates the consistency of the renormalization procedure across photon, Z, W, and Higgs sectors, including mixed transitions.

Abstract

In part I general aspects of the renormalization of a spontaneously broken gauge theory have been introduced. Here, in part II, two-loop renormalization is introduced and discussed within the context of the minimal Standard Model. Therefore, this paper deals with the transition between bare parameters and fields to renormalized ones. The full list of one- and two-loop counterterms is shown and it is proven that, by a suitable extension of the formalism already introduced at the one-loop level, two-point functions suffice in renormalizing the model. The problem of overlapping ultraviolet divergencies is analyzed and it is shown that all counterterms are local and of polynomial nature. The original program of 't Hooft and Veltman is at work. Finite parts are written in a way that allows for a fast and reliable numerical integration with all collinear logarithms extracted analytically. Finite renormalization, the transition between renormalized parameters and physical (pseudo-)observables, will be discussed in part III where numerical results, e.g. for the complex poles of the unstable gauge bosons, will be shown. An attempt will be made to define the running of the electromagnetic coupling constant at the two-loop level.

Figures (15)

• Figure 1: Sources related to the gauge-fixing functions ${\cal{C}}^{\pm}$ defined in Eq.(\ref{['eq:one:GAUGEREN']}). The momentum $p$ is flowing inwards.
• Figure 2: Doubly-contracted WST identity with two external ${\cal{C}}^{\pm}$ sources. Gray circles contain all the irreducible and reducible Feynman diagrams contributing to the needed Green functions. Black dotted circles represent sources. Their expression can be read in Fig. \ref{['fig:one:sources']}.
• Figure 3: Source vertices related to the gauge-fixing functions ${\cal{C}}^{\pm}$. We start from Fig. \ref{['fig:one:sources']} and expand the renormalization constants up to ${\cal O}\left(g^2\right)$. The black squares denote source vertices at ${\cal O}\left(1\right)$, whereas the white squares denote source vertices at ${\cal O}\left(g^2\right)$.
• Figure 4: Doubly-contracted WST identity with two external ${\cal{C}}^{\pm}$ sources at ${\cal O}\left(g^2\right)$. Gray circles contain all the irreducible and reducible Feynman diagrams contributing to the Green functions at ${\cal O}\left(g^2\right)$. Black squares and white squares represent source vertices. Their expression can be read in Fig. \ref{['fig:one:specsources']}.
• Figure 5: Doubly-contracted WST identities with two external gauge bosons at ${\cal O}\left(g^2\right)$. Gray circles denote the sum of the needed Feynman diagrams at ${\cal O}\left(g^2\right)$. Source vertices are given in Fig. \ref{['fig:one:Fsources']}.