Multiple Couplings and Renormalization Scheme Ambiguities

TL;DR

This paper explores the ambiguities in renormalization schemes for massless theories with multiple couplings, analyzing how scheme choices affect beta-functions and physical predictions, and demonstrating methods to sum logarithms for improved accuracy.

Contribution

It introduces a scheme-independent way to characterize renormalization schemes and examines how summing logarithms cancels scale dependence in physical quantities.

Findings

Beta-functions cannot be made to vanish beyond first order.

Summation of logarithms cancels explicit and implicit scale dependencies.

A scheme exists where all higher-order effects are absorbed into running couplings.

Abstract

The ambiguities inherent in renormalization are considered when using mass-independent renormalization in massless theories that involve two coupling coupling constants. We review how there is no renormalization scheme in which the beta-functions can be chosen to vanish beyond a certain order in perturbation theory, but that the beta-functions always contain ambiguities beyond first order. We examine how the coupling constants depend on the coefficients of the beta-function beyond one loop order. A way of characterizing renormalization schemes that doesn't use coefficients of the beta-function is considered for models with either one or two couplings. The renormalization scheme ambiguities of physical quantities computed to finite order in perturbation theory are also examined. We demonstrate how summation of the logarithms that have explicit dependence on the renormalization scale parameter mu in a physical quantity R leads to a cancellation with the implicit dependence of R on mu through the running couplings. It is also shown that there exists a renormalization scheme in which all radiative effects beyond lowest order are incorporated into the behaviour of the running couplings.