Mass divergences in annihilation processes. I. Origin and nature of divergences in cut vacuum polarization diagrams

The origin of mass divergences in internal loop momentum integrals of cut vacuum polarization diagrams is investigated. It is found that loop momentum configurations which can give rise to mass divergences are of a severely limited form and have a direct interpretation in terms of physically realizable processes. A power-counting procedure suitable for estimating the nature of mass divergences is developed, and it is found that in a large class of field theories, cross sections smeared over a small region of phase space are at worst logarithmically divergent.