Renormalization Scheme Dependence with Renormalization Group Summation

TL;DR

This paper explores how renormalization scheme choices affect calculations in quantum field theory, demonstrating methods to sum logarithmic corrections and showing scheme independence in certain physical quantities.

Contribution

It introduces a detailed analysis of scheme dependence in renormalization group summation and relates scheme invariants to Stevenson’s characterization, with applications to the effective potential.

Findings

Renormalization scheme dependence can be absorbed into mass scales.

Effective potential is scheme independent when summed appropriately.

Coupling constants in different schemes can be expressed as power series.

Abstract

We consider all radiative corrections to the total electron-positron cross section showing how the renormalization group equation can be used to sum the logarithmic contributions in two ways. First of all, one can sum leading-log etc. contributions. A second summation shows how all logarithmic corrections can be expressed in terms of log-independent contributions. Next, using Stevenson's characterization of renormalization scheme, we examine scheme dependence when using the second way of summing logarithms. The renormalization scheme invariants that arise are then related to those of Stevenson. We consider two choices of renormalization scheme, one resulting in two powers of a running coupling, the second in an infinite series in the two loop running constant. We then establish how the coupling constant arising in one renormalization scheme can be expressed as a power series of the coupling in any other scheme. Next we establish how by using different mass scale at each order of perturbation theory, all renormalization scheme dependence can be absorbed into these mass scales when one uses the second way of summing logarithmic corrections. We then employ this approach to renormalization scheme dependency to the effective potential in a scalar model, showing the result that it is independent of the background field is scheme independent. The way in which the "principle of minimal sensitivity" can be applied after summation is then discussed.