Mass divergences and Callan--Symanzik equations in quantum electrodynamics

We discuss the connection between the theorems on mass divergences of unrenormalized (but regularized) vertex functions and the asymptotic Callan-Symanzik equations in quantum electrodynamics. To illustrate the point, we first use these theorems to give a rather simple derivation of the asymptotic Callan-Symanzik equation for the renormalized photon propagator and later generalize the analysis to the $2n+l$ vertex function. By a natural extension of the argument, one obtains nonasymptotic equations in which the inhomogeneous terms involve mass derivatives of the unrenormalized vertex functions multiplied by suitable products of the renormalization constants ${Z}_{2}$ and ${Z}_{3}$. As a by-product, one obtains partial differential equations for ${Z}_{2}$ and ${Z}_{3}$ regarded as functions of the masses. The method can be used when several internal masses are present in the problem, in either the conventional or the intermediate renormalization schemes.