[en] The modern state of analiticity problem on coupling constant in the models of the quantum field theory (QFT) mainly for spinor electrodynamics, is presented. The main attention is paid to the method of utilization of a renormalization group (RG) in the perturbation theory in the QFT models. The consideration begins with the simplest case of a single-charged spinor electrodynamics, in which the invariant charge (IC) is proportional to the dimensionless photon propagator due to the Ward identity. The link of the spectral functions of various analytical representations of the photon propagator with the Gellman-Low function is considered. The IC analytical properties are combined with the RG invariance requirements. Due to this combination, as well as a combination with analyticity conditions in the pulse representation it is possible to make a number of conclusions about the zero nonanalyticity character on the coupling constant for the renormalizable logarithmic QFT models