Field renormalization constant for unstable particles

A recently proposed normalization condition for the imaginary part of the self-energy of an unstable particle is shown to lead to closed, exact expressions for the renormalized self-energy and propagator, which are consequently free of ultraviolet divergences to all orders in perturbation theory. In turn, the corresponding closed expressions for the mass and field renormalization counterterms are necessary in some important cases to avoid power-like infrared divergences in high orders of perturbation theory. In the same examples, the width plays the rôle of an infrared cutoff, and, consequently, the field renormalization counterterm is not an analytic function of the coupling constant.