The Structure of the Invariant Charge in Massive Theories with One Coupling

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 is the high-energy behavior of the RG-invariants restricted. In models where the CS-equation has the same form as the RG-equation, the massless limit is reached smoothly, i.e., the β-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 β-function of a physical mass. As a consequence the β-functions depend on the normalization point also in the asymptotic region and a mass independent limit does not exist anymore.