Singular Potentials and Peratization. II

Abstract

In a field theory with more than one unrenormalizable interaction (like the W theory with weak and electromagnetic coupling), the important problem arises of resumming multiple series of individually divergent terms. In order to get a first insight in the new questions which arise for multiple as compared to single series, we study the analogous question for a superposition of two singular potentials, using a family of exactly soluble cases. We ask whether one can expand resummed series for the zero‐energy scattering amplitude in powers of one coupling constant with coefficients depending on the other. The answer depends both on the relative magnitude of the coupling constants and on the relative degree of singularity of the interactions. Depending on these two conditions one finds three regimes, one where a convergent power expansion holds, another where an asymptotic expansions obtains, and a third where it is impossible to expand in powers of either single constant separately. It is conjectured that a similar situation will be true in a field theory with leading power singularities only (and no logarithmic ones), if such a theory has meaning.