  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.
