Revisiting RGEs for general gauge theories

TL;DR

This paper revises and corrects the renormalisation group equations for general gauge theories at one- and two-loop levels, addressing previous inaccuracies and providing updated formulas with practical implementations.

Contribution

It identifies and corrects errors in existing RGE formulas for gauge theories, especially for scalar mixing and dimensionful parameters, validated through supersymmetric cross-checks.

Findings

Corrected beta-functions for fermion masses and scalar couplings.

Quantified the impact of corrections in toy models with large Yukawa couplings.

Implemented updated formulas in SARAH and PyR@TE packages.

Abstract

We revisit the renormalisation group equations (RGE) for general renormalisable gauge theories at one- and two-loop accuracy. We identify and correct various mistakes in the literature for the $\beta$-functions of the dimensionful Lagrangian parameters (the fermion mass, the bilinear and trilinear scalar couplings) as well as the dimensionless quartic scalar couplings. There are two sources for these discrepancies. Firstly, the known expressions for the scalar couplings assume a diagonal wave-function renormalisation which is not appropriate for models with mixing in the scalar sector. Secondly, the dimensionful parameters have been derived in the literature using a dummy field method which we critically re-examine, obtaining revised expressions for the $\beta$-function of the fermion mass. We perform an independent cross-check using well-tested supersymmetric RGEs which confirms our results. The numerical impact of the changes in the $\beta$-function for the fermion mass terms is illustrated using a toy model with a heavy vector-like fermion pair coupled to a scalar gauge singlet. Unsurprisingly, the correction to the running of the fermion mass becomes sizeable for large Yukawa couplings of the order of O(1). Furthermore, we demonstrate the importance of the correction to the $\beta$-functions of the scalar quartic couplings using a general type-III Two-Higgs-Doublet-Model. All the corrected expressions have been implemented in updated versions of the Mathematica package SARAH and the Python package PyR@TE.