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Two-loop effective potential for a general renormalizable theory and softly broken supersymmetry

Stephen P. Martin

TL;DR

The paper delivers a comprehensive calculation of the two-loop effective potential in the Landau gauge for a general renormalizable theory, providing explicit results in the MSbar, DR, and DR' renormalization schemes. It introduces a DR to DR' parameter redefinition to decouple evanescent epsilon scalars in softly broken supersymmetry, and proves renormalization-group invariance by deriving the scalar anomalous dimensions and the two-loop vacuum-energy beta function. The work validates the formalism with multiple consistency checks across Wess-Zumino models, SUSY QED, SU($N_c$) gauge theories, and softly broken SUSY QED, demonstrating precise cancellations in supersymmetric vacua. These results enable more accurate, RG-scale-stable predictions for vacuum structure and electroweak symmetry breaking in MSSM-like theories, with planned applications to MSSM phenomenology and beyond.

Abstract

I compute the two-loop effective potential in the Landau gauge for a general renormalizable field theory in four dimensions. Results are presented for the \bar{MS} renormalization scheme based on dimensional regularization, and for the \bar{DR} and \bar{DR}' schemes based on regularization by dimensional reduction. The last of these is appropriate for models with softly broken supersymmetry, such as the Minimal Supersymmetric Standard Model. I find the parameter redefinition which relates the \bar{DR} and \bar{DR}' schemes at two-loop order. I also discuss the renormalization group invariance of the two-loop effective potential, and compute the anomalous dimensions for scalars and the beta function for the vacuum energy at two-loop order in softly broken supersymmetry. Several illustrative examples and consistency checks are included.

Two-loop effective potential for a general renormalizable theory and softly broken supersymmetry

TL;DR

The paper delivers a comprehensive calculation of the two-loop effective potential in the Landau gauge for a general renormalizable theory, providing explicit results in the MSbar, DR, and DR' renormalization schemes. It introduces a DR to DR' parameter redefinition to decouple evanescent epsilon scalars in softly broken supersymmetry, and proves renormalization-group invariance by deriving the scalar anomalous dimensions and the two-loop vacuum-energy beta function. The work validates the formalism with multiple consistency checks across Wess-Zumino models, SUSY QED, SU() gauge theories, and softly broken SUSY QED, demonstrating precise cancellations in supersymmetric vacua. These results enable more accurate, RG-scale-stable predictions for vacuum structure and electroweak symmetry breaking in MSSM-like theories, with planned applications to MSSM phenomenology and beyond.

Abstract

I compute the two-loop effective potential in the Landau gauge for a general renormalizable field theory in four dimensions. Results are presented for the \bar{MS} renormalization scheme based on dimensional regularization, and for the \bar{DR} and \bar{DR}' schemes based on regularization by dimensional reduction. The last of these is appropriate for models with softly broken supersymmetry, such as the Minimal Supersymmetric Standard Model. I find the parameter redefinition which relates the \bar{DR} and \bar{DR}' schemes at two-loop order. I also discuss the renormalization group invariance of the two-loop effective potential, and compute the anomalous dimensions for scalars and the beta function for the vacuum energy at two-loop order in softly broken supersymmetry. Several illustrative examples and consistency checks are included.

Paper Structure

This paper contains 17 sections, 100 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Conventions and setup
  3. Field-dependent masses and couplings
  4. The Feynman diagrams
  5. Two-loop integral functions needed for vacuum graphs
  6. Conventions for softly-broken supersymmetric models
  7. One-loop effective potential in the $\overline{\rm MS}$, $\overline{\rm DR}$, and $\overline{\rm DR}'$ schemes
  8. Two-loop effective potential in the $\overline{\rm MS}$ scheme
  9. Two-loop effective potential in the $\overline{\rm DR}$ scheme
  10. Two-loop effective potential in the $\overline{\rm DR}'$ scheme
  11. Renormalization group invariance of the two-loop effective potential in softly broken supersymmetry
  12. Examples and consistency checks
  13. The Wess-Zumino Model
  14. Supersymmetric QED in supersymmetric vacua
  15. Supersymmetric $SU(N_{c})$ gauge theory with one flavor in supersymmetric vacua
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Topologies of one-particle-irreducible connected vacuum Feynman diagrams for the one-loop and two-loop contributions to the effective potential.
  • Figure 2: The one-particle-irreducible connected Feynman diagrams contributing to the two-loop effective potential. Dashed lines denote real scalars, solid lines denote Weyl fermions carrying helicity along the arrow direction, wavy lines are for vector bosons, and dotted lines are for ghosts. The large dots between opposing arrows on the fermion lines in the $\overline{FF}S$ and $\overline{FF}V$ diagrams denote mass insertions. The $\overline{FF}S$ diagram is accompanied by its complex conjugate (the same diagram with all arrows reversed).