Gradient flows for beta functions via multi-scale renormalization group equations

Ulrich Ellwanger

doi:10.1140/epjc/s10052-019-6720-9

Abstract

Renormalization schemes and cutoff schemes allow for the introduction of various distinct renormalization scales for distinct couplings. We consider the coupled renormalization group flow of several marginal couplings which depend on just as many renormalization scales. The usual beta functions describing the flow with respect to a common global scale are assumed to be given. Within this framework one can always construct a metric and a potential in the space of couplings such that the beta functions can be expressed as gradients of the potential. Moreover the potential itself can be derived explicitely from a prepotential which, in turn, determines the metric. Some examples of renormalization group flows are considered, and the metric and the potential are compared to expressions obtained elsewhere.

Highlights

Multi-scale renormalization group (RG) flows were introduced to deal with physical problems involving distinct energy scales [1]
On the other hand it is plausible to consider multi-scale RG flows motivated by purely formal arguments: In dimensional regularization marginal couplings acquire a dimension d − 4 which requires the introduction of a scale μ, and in perturbation theory the corresponding renormalized couplings depend on t ≡ log(μ2/μ20) where μ0 serves to define initial conditions for the running couplings
Distinct momentum space cutoffs can be introduced in the form of distinct form factors attached to the vertices corresponding to marginal couplings, as it happens automatically in the case of compositeness

Summary

Introduction

In the presence of an ultraviolet (UV) cutoff the renormalization group can be used to describe the running of bare couplings with keeping the renormalized couplings fixed. In principle such a prepotential can always be constructed if one solves the system of coupled RG equations for ga(t), inserts the solutions into the potential (g(t)), integrates with respect to t and re-expresses t in terms of ga(t) In practice these steps are hardly feasable, whereas within the present approach the prepotential is related to the metric ηab (see the section) which allows for its construction. A candidate ηaZb for a metric is the correlation function of two composite operators l2d Oa(x)Ob(0) ||x|=l (l denotes an UV cutoff) where the composite operators Oa, Ob are dual to the couplings ga, gb respectively Such a metric was introduced by Zamolodchikov [10] in order to show the irreversibility of the RG flux in d = 2 dimensional field theory where the positivity of ηaZb can be shown. Our results for gradient flows in some simple field theory models to those obtained elsewhere

Gradient flow from multiple scales

Examples

Conclusions