Two-loop RGE of a general renormalizable Yang-Mills theory in a renormalization scheme with an explicit UV cutoff

TL;DR

This paper develops a two-loop renormalization group analysis for a general Yang-Mills theory using a cutoff scheme, providing explicit relations to the MS scheme and exploring implications for hierarchy problem solutions.

Contribution

It introduces a systematic one-loop renormalization in a cutoff scheme and derives two-loop RG equations, connecting them to the MS scheme and analyzing vacuum divergences.

Findings

Derived explicit two-loop RG equations in the cutoff scheme.

Established relations between cutoff and MS renormalized parameters.

Recomputed two-loop scalar potential beta functions.

Abstract

We perform a systematic one-loop renormalization of a general renormalizable Yang-Mills theory coupled to scalars and fermions using a regularization scheme with a smooth momentum cutoff $\Lambda$ (implemented through an exponential damping factor). We construct the necessary finite counterterms restoring the BRST invariance of the effective action by analyzing the relevant Slavnov-Taylor identities. We find the relation between the renormalized parameters in our scheme and in the conventional $\overline{\rm MS}$ scheme which allow us to obtain the explicit two-loop renormalization group equations in our scheme from the known two-loop ones in the $\overline{\rm MS}$ scheme. We calculate in our scheme the divergences of two-loop vacuum graphs in the presence of a constant scalar background field which allow us to rederive the two-loop beta functions for parameters of the scalar potential. We also prove that consistent application of the proposed regularization leads to counterterms which, together with the original action, combine to a bare action expressed in terms of bare parameters. This, together with treating $\Lambda$ as an intrinsic scale of a hypothetical underlying finite theory of all interactions, offers a possibility of an unconventional solution to the hierarchy problem if no intermediate scales between the electroweak scale and the Planck scale exist.