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 β-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 β-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 β-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 β-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.
Highlights
Renormalisation Group Equations (RGEs) are important as they provide the necessary link between the physics at different energy scales
The procedure is very similar to the case of the cubic scalar coupling, and we find the following relations for the wavefunction corrections to the terms appearing for the quartic scalar coupling:
Making use of the results of Ref. [8], which are implemented in the package SARAH, we find the following expressions for the one- and two-loop RGEs for the different parts of the model: 1. Gauge Couplings βg(1)
Summary
Renormalisation Group Equations (RGEs) are important as they provide the necessary link between the physics at different energy scales. The two-loop RGEs for all dimensionless parameters in general gauge theories have been derived already more than 30 years ago [1,2,3,4,5,6]. These results have been re-derived by Luo et al [7] including the β-functions for dimensionful parameters. We provide an independent cross-check using well tested supersymmetric RGEs which confirms our results We believe that these corrections and validations are non-trivial and important in view of the wide use of the RGEs. Still, an independent direct calculation of the dimensionful β-functions would be useful. The overarching purpose is to present the current state-of-the art of the two-loop β-functions and to collect the corrected expressions such that all the relevant information is at hand in one place
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