Renormalization and scaling behaviour of eikonal perturbation theories

Abstract

In field theory the eikonal approximation has often played an important role, especially in connection with problems concerning infra-red divergences, high-energy scattering and bound states. Several types of approximations have been proposed and their degree of val idi ty has been treated in various contexts. Recently a systematic eikonal per turbat ion theory was discovered by 0RZALESI (1). I t carries some points of similarity to earlier methods due to FRAD~:IN (~), but it is supposed to have a wider range of applicability. Thus, noting tha t this theory in some sense can be considered to be ~ perturbat ion theory around infinite momentum, 0RZALESI suggested tha t approximate Green's functions calculated within this scheme should show the scaling behaviour predicted to a certain order in the coupling constant by, for example, renormalization group techniques. In the present note we shall record some observations on the renormalization and scaling behaviour of the charged-particle propagator in scalar quantum electrodynamics in the ordinary eikonal approximation (EA) as well as in the eikonal perturbation theory (EPT) considered in ref. (1). Our conclusions indicate that scaling behaviour is not realized in the simple sense as anticipated in ref. (z). More precisely the field theory to be considered is that of a charged scalar particle (in the following referred to as an electron) interacting with a quantized radiation field, where we neglect vacuum polarizations. Also we ignore the circumstance that electronelectron scattering subgraphs here are superficially divergent and thus introduce an extra coupling constant. Thus the model is very similar to the Johnson-Baker-Willey (3,4) model of spinor QED.