The structure of the invariant charge in massive theories with one coupling

TL;DR

This paper investigates the structure of the invariant charge in massive theories with one coupling, analyzing how renormalization group invariants behave in different models and the impact of mass and symmetry breaking on high-energy limits.

Contribution

It provides a detailed analysis of RG-invariants in massive models, highlighting differences from massless theories and the effects of spontaneous symmetry breaking on RG behavior.

Findings

RG-invariants are power series in logarithms in massless models

In massive models, RG-invariants are not necessarily logarithmic series

Spontaneously broken models with fermions show dependence on normalization point at high energies

Abstract

Invariance under finite renormalization group (RG) transformations is used to structure the invariant charge in models with one coupling in the 4 lowest orders of perturbation theory. In every order there starts a RG-invariant, which is uniquely continued to higher orders. Whereas in massless models the RG-invariants are power series in logarithms, there is no such requirement in a massive model. Only, when one applies the Callan-Symanzik (CS) equation of the respective theories, the high-energy behavior of the RG-invariants is restricted. In models, where the CS-equation has the same form as the RG-equation, the massless limit is reached smoothly, i.e. the beta-functions are constants in the asymptotic limit and the RG-functions starting the new invariant tend to logarithms. On the other hand in the spontaneously broken models with fermions the CS-equation contains a beta-function of a physical mass. As a consequence the beta-functions depend on the normalization point also in the asymptotic region and a mass independent limit does not exist anymore.