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Four Loop Renormalisation Group Equations in general Gross-Neveu-Yukawa Theories

Tom Steudtner

TL;DR

This paper delivers complete four-loop template RGEs for renormalizable theories with fermions and scalars in the MS-bar scheme, focusing on gaugeless Yukawa–quartic interactions and their anomalous dimensions. The computation uses a model-independent template approach, supported by MaRTIn, IR rearrangement, master-integral reduction, and FoRGEr mapping, while addressing the $\gamma_5$-ambiguity and introducing an MS-bar$_3$ scheme to ensure consistent dimensional continuation to $d=3$. The results include extensive four-loop expressions for $\gamma_\phi^{(4)}$, $\gamma_\psi^{(4)}$, $\hat{\beta}_y^{(4)}$, and $\hat{\beta}_\lambda^{(4)}$, validated against SUSY and known scalar templates, and applied to yield new four-loop contributions to the Standard Model top Yukawa and Higgs self-coupling. The work enables high-precision RG analyses in Gross-Neveu–Yukawa theories and provides practical tools for exploring 3D critical phenomena and beyond.

Abstract

We compute template formulae of all four-loop $β$-functions and anomalous dimensions of arbitrary renormalisable quantum field theories with fermions and scalar fields in the $\overline{\text{MS}}$ scheme. Using these results, novel contributions to the $β$-functions of the Higgs and top coupling are extracted. We discuss how four loops present a challenge to dimensional continuation from four to three dimensions and propose a solution.

Four Loop Renormalisation Group Equations in general Gross-Neveu-Yukawa Theories

TL;DR

This paper delivers complete four-loop template RGEs for renormalizable theories with fermions and scalars in the MS-bar scheme, focusing on gaugeless Yukawa–quartic interactions and their anomalous dimensions. The computation uses a model-independent template approach, supported by MaRTIn, IR rearrangement, master-integral reduction, and FoRGEr mapping, while addressing the -ambiguity and introducing an MS-bar scheme to ensure consistent dimensional continuation to . The results include extensive four-loop expressions for , , , and , validated against SUSY and known scalar templates, and applied to yield new four-loop contributions to the Standard Model top Yukawa and Higgs self-coupling. The work enables high-precision RG analyses in Gross-Neveu–Yukawa theories and provides practical tools for exploring 3D critical phenomena and beyond.

Abstract

We compute template formulae of all four-loop -functions and anomalous dimensions of arbitrary renormalisable quantum field theories with fermions and scalar fields in the scheme. Using these results, novel contributions to the -functions of the Higgs and top coupling are extracted. We discuss how four loops present a challenge to dimensional continuation from four to three dimensions and propose a solution.

Paper Structure

This paper contains 9 sections, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Computation
  3. Results
  4. Summary
  5. Scalar Anomalous Dimension
  6. Fermion Anomalous Dimension
  7. Yukawa Vertex
  8. Quartic Vertex
  9. N=1 Supersymmetry Relations

Figures (2)

  • Figure 1: Sample of four-loop diagrams contributing to (a) the scalar and (b) fermion anomalous dimension with external momentum $p$ and loop momenta $k_{1..4}$.
  • Figure 2: (a) Vanishing one-loop subdiagram $\propto \mathrm{tr}(y^3)$. (b) Vanising four-loop contribution to the fermion anomalous dimensions. (c) Non-vanishing Yukawa vertex correction. See main text for further explanations.