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3D Physics and the Electroweak Phase Transition: Perturbation Theory

K. Farakos, K. Kajantie, K. Rummukainen, M. Shaposhnikov

TL;DR

This paper develops a 3d effective field theory framework for the high-temperature electroweak sector, enabling renormalization-group based summation of leading logarithms across all perturbative orders. By dimensional reduction and careful matching to 4d finite-temperature theory, it derives a renormalization-group improved 3d effective potential up to two loops and investigates the convergence of perturbation theory, including the impact of integrating out the heavy $A_0$ field. The authors quantify the electroweak phase transition parameters using the 2-loop potential, assess perturbative uncertainties (including possible 3-loop logs and a hypothetical 3-loop linear term), and demonstrate that the 3d EFT approach remains well-behaved and suitable for non-perturbative lattice studies. Overall, the work provides a practical, RG-resummed perturbative backbone for studying hot electroweak matter and guides lattice investigations of the phase transition.

Abstract

We develop a method for the construction of the effective potential at high temperatures based on the effective field theory approach and renormalization group. It allows one to sum up the leading logarithms in all orders of perturbation theory. The method reproduces the known one-loop and two-loop results in a very simple and economic way and clarifies the issue of the convergence of the perturbation theory. We also discuss the assumptions being made for the determination of the critical temperature of the electroweak phase transition, and analyse different perturbative uncertainties in its determination. These results are then used for the non-perturbative lattice Monte Carlo simulations of the EW phase transition in forthcoming paper.

3D Physics and the Electroweak Phase Transition: Perturbation Theory

TL;DR

This paper develops a 3d effective field theory framework for the high-temperature electroweak sector, enabling renormalization-group based summation of leading logarithms across all perturbative orders. By dimensional reduction and careful matching to 4d finite-temperature theory, it derives a renormalization-group improved 3d effective potential up to two loops and investigates the convergence of perturbation theory, including the impact of integrating out the heavy field. The authors quantify the electroweak phase transition parameters using the 2-loop potential, assess perturbative uncertainties (including possible 3-loop logs and a hypothetical 3-loop linear term), and demonstrate that the 3d EFT approach remains well-behaved and suitable for non-perturbative lattice studies. Overall, the work provides a practical, RG-resummed perturbative backbone for studying hot electroweak matter and guides lattice investigations of the phase transition.

Abstract

We develop a method for the construction of the effective potential at high temperatures based on the effective field theory approach and renormalization group. It allows one to sum up the leading logarithms in all orders of perturbation theory. The method reproduces the known one-loop and two-loop results in a very simple and economic way and clarifies the issue of the convergence of the perturbation theory. We also discuss the assumptions being made for the determination of the critical temperature of the electroweak phase transition, and analyse different perturbative uncertainties in its determination. These results are then used for the non-perturbative lattice Monte Carlo simulations of the EW phase transition in forthcoming paper.

Paper Structure

This paper contains 23 sections, 156 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Dimensional reduction, dimensional regularization and effective field theories
  3. Peculiarities of renormalization of 3d theory
  4. The effective potential of the 3d theory to 2 loops
  5. Renormalization group and convergence of perturbation theory
  6. Preliminaries
  7. Example: The scalar theory
  8. SU(2) theory
  9. Integrating out the $A_0$ field
  10. The relation between 3d and 4d computations on the 2-loop level
  11. Parameters of the EW phase transition from perturbation theory
  12. The 2-loop results
  13. Perturbative uncertainties in predictions
  14. The renormalization group corrections
  15. The 3-loop linear term
  16. ...and 8 more sections

Figures (6)

  • Figure 1: The one-loop linear-divergent diagrams contributing to the mass of the doublet and the triplet of scalar fields.
  • Figure 2: The two-loop log-divergent diagrams contributing to the mass of the doublet of scalar fields.
  • Figure 3: The two-loop log-divergent diagrams contributing to the mass of the triplet of scalar fields.
  • Figure 4: The two-loop contributions to the effective potential.
  • Figure 5: The two-loop (upper curve) and the one-loop (lower curve) rg improved effective potentials at the 2-loop critical temperature for $m_H = 35$ GeV. The x-axis is 4d scalar field, in GeV, the y axis is a dimensionless 3d potential $\frac{12}{g_3^6}V(\phi)$.
  • ...and 1 more figures