Multi-scale renormalization

Abstract

The standard MS renormalization prescription is inadequate for dealing with multi-scale problems. To illustrate this we consider the computation of the effective potential in the Higgs-Yukawa model. It is argued that it is natural to employ a two-scale renormalization group. We give a modified version of a two-scale scheme introduced by Einhorn and Jones. In such schemes the beta functions necessarily contain potentially large logarithms of the RG scale ratios. For credible perturbation theory one must implement a large logarithms resummation on the beta functions themselves. We show how the integrability condition for the two RG equations allows one to perform this resummation.

Unique determination of the effective potential in terms of renormalization group functions

“…Similarly, conversion of the RG functions from the MS scheme to the CW scheme in massless scalar QED is also more elaborate, and requires the use of multiscale renormalization group methods. Although multiscale RG techniques are not widely known, the necessary multiscale RG functions can either be calculated directly by introducing a renormalization scale for each kinetic term (and hence propagator) in the theory and exploiting the usual relation between the RG functions and the 1= terms in the renormalization constants, or they may be reconstructed from the single-scale MS-scheme RG functions in conjunction with integrability conditions related to the commutator of the RG operator associated with each renormalization scale [11,12].…”

“…As suggested by (12) and (13), the explicit solutions to Eqs. (10) and (11) take the form of polynomials in w and logw:…”

“…To circumvent these problems, the concept of multiple renormalization scales, one scale for each appearance of the traditional MS RG scale in the Lagrangian, was first considered in [11]. This method was refined in [12] by associating a renormalization scale with each kinetic term in the Lagrangian, which, in the case of massless scalar QED, will introduce two renormalization scales ( g and ) resulting in a MSscheme perturbation series for the effective potential containing two logarithms:…”

“…The results are quite similar to the 4 scenario; iterative methods are developed to determine the scalar-field effective potential in terms of the RG functions in the CW scheme. The presence of multiple couplings requires the use of multiscale RG methods [11,12] to convert the RG coefficients to the CW scheme.…”

Vacuum stability of Froggatt-Nielsen models

“…In the current study, we have the additional complication that we have to deal with two scalar VEVs. There are several proposals in the literature how to obtain a RG-improved effective potential in this circumstance [10,11,12,13,14]. We will see that we do not require these techniques here and discuss in detail how to resum the leading logs in our context.…”