Renormalization « groups » extended to masses; application to the structure of the renormalization constantsZ 3 andm o

Abstract

By a precise elucidation of the axioms from which renormalization « groups » result we construct the most general «groups » which include mass renormalization. We discuss in detail the particular case involving only photon field and electron mass renormalizations. An exact integral equation which can be put in a form generalizing that given by Gell-Mann and Low. allows us to show that a dependence of the bare charge α0 on the observed one α is, contrary to an often quoted paradox, consistent with the renormalization group. We demonstrate that in the cases when, the bare or the observed mass of the electron is null or infinite, which should correspond to determined values of α the formal perturbation expansion in α0 of the photon vacuum polarization tensor is no more divergent than a single power of a logarithm. We finally determine wherein lies the failure of the argument leading to the paradox stated above and discuss its implication.