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Two-Loop Renormalization in the Standard Model Part III: Renormalization Equations and their Solutions

S. Actis, G. Passarino

TL;DR

Actis and Passarino develop a comprehensive two-loop renormalization framework for the Standard Model using a complex-pole input scheme. They derive and solve coupled renormalization equations that connect renormalized Lagrangian parameters to an IPS consisting of α, G_F, and complex pole masses, while separating QED, weak, and hadronic corrections. The work emphasizes gauge-parameter independence, unitarity, and Ward–Takahashi identities, and discusses dressed propagators and the complex-mass scheme as practical tools with clear limitations. They provide detailed methodological steps, notations, and numerical illustrations to enable precision electroweak predictions at current and future colliders.

Abstract

In part I and II of this series of papers all elements have been introduced to extend, to two loops, the set of renormalization procedures which are needed in describing the properties of a spontaneously broken gauge theory. In this paper, the final step is undertaken and finite renormalization is discussed. Two-loop renormalization equations are introduced and their solutions discussed within the context of the minimal standard model of fundamental interactions. These equations relate renormalized Lagrangian parameters (couplings and masses) to some input parameter set containing physical (pseudo-)observables. Complex poles for unstable gauge and Higgs bosons are used and a consistent setup is constructed for extending the predictivity of the theory from the Lep1 Z-boson scale (or the Lep2 WW scale) to regions of interest for LHC and ILC physics.

Two-Loop Renormalization in the Standard Model Part III: Renormalization Equations and their Solutions

TL;DR

Actis and Passarino develop a comprehensive two-loop renormalization framework for the Standard Model using a complex-pole input scheme. They derive and solve coupled renormalization equations that connect renormalized Lagrangian parameters to an IPS consisting of α, G_F, and complex pole masses, while separating QED, weak, and hadronic corrections. The work emphasizes gauge-parameter independence, unitarity, and Ward–Takahashi identities, and discusses dressed propagators and the complex-mass scheme as practical tools with clear limitations. They provide detailed methodological steps, notations, and numerical illustrations to enable precision electroweak predictions at current and future colliders.

Abstract

In part I and II of this series of papers all elements have been introduced to extend, to two loops, the set of renormalization procedures which are needed in describing the properties of a spontaneously broken gauge theory. In this paper, the final step is undertaken and finite renormalization is discussed. Two-loop renormalization equations are introduced and their solutions discussed within the context of the minimal standard model of fundamental interactions. These equations relate renormalized Lagrangian parameters (couplings and masses) to some input parameter set containing physical (pseudo-)observables. Complex poles for unstable gauge and Higgs bosons are used and a consistent setup is constructed for extending the predictivity of the theory from the Lep1 Z-boson scale (or the Lep2 WW scale) to regions of interest for LHC and ILC physics.

Paper Structure

This paper contains 27 sections, 238 equations, 24 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Outline of the calculation
  3. The Fermi-coupling constant
  4. Extraction of the electromagnetic components
  5. Wave-function renormalization factors
  6. Two-Loop Corrections to $\Delta g$
  7. Process independent, resummed, Fermi constant
  8. The $\gamma^5$ problem
  9. The fine structure constant
  10. The QED case
  11. The standard model case
  12. Running of $\alpha$ beyond one-loop: issues in perspective
  13. Complex poles: the paradigm
  14. Renormalization equations and their solutions
  15. Running parameters
  16. ...and 12 more sections

Figures (24)

  • Figure 1: Renormalization - flowchart. Feynman rules define the theory, renormalizability guarantees that ultraviolet poles have polynomial residues and can be subtracted. Any input parameter set allows us to replace renormalized quantities with experimental data. A prediction follows.
  • Figure 2: Diagrammatic interpretation of Eq.(\ref{['sdec']}). The first graph is a box diagram in the full SM context. The one-photon vertex diagram is a QED correction in the Fermi-contact interaction, denoted by the black circle.
  • Figure 3: Two-loop box diagrams for muon decay.
  • Figure 4: Two-loop soft $\otimes$ soft splitting.
  • Figure 5: Two-loop soft $\otimes$ hard splitting.
  • ...and 19 more figures