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

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.Comment: 35 pages (1 figure available upon request), Plain TeX, BUTP 93-26 (1 more macro included

Gauge parameter dependence and gauge invariance in the Abelian Higgs model

We analyze gauge parameter dependence by using an algebraic method which relates the gauge parameter dependence of Green functions to an enlarged Slavnov-Taylor identity. In the course of the renormalization it turns out that gauge parameter dependence of physical parameters is already restricted at the level of Green functions. In a first step we consider the on-shell conditions which we find to be in complete agreement with these restrictions to all orders of perturbation theory. The fixing of the coupling, however, is much more involved outside the complete on-shell scheme. In the Abelian Higgs model we prove that this fixing can be properly chosen by requiring the Ward identity of gauge invariance to hold in its tree form to all orders of perturbation theory.Comment: 28 pages, 2 figure