Complex Propagators in Perturbation Theory

Abstract

A method is given for studying the analytic properties of an arbitrary Feynman graph $F$, in which a full two-particle propagator $G$ is inserted between one pair of points. Three special graphs are treated in detail: the two-particle amplitude itself, with two- and three-particle intermediate states, and the "triangle" graph. When $G$ has a resonance, a possible approximation for $F$ is to replace $G$ by a complex pole, obtaining thereby a new graph $f$ in which one internal particle has a complex mass. We show that, although the singularities of $F$ and $f$ are in general different, this approximation is appropriate for calculating "enhancement" effects due to singularities of $F$, near the physical region, associated with the resonance. For the cases considered, we predict the ranges of the external variables for which such effects will occur, and show how to calculate them explicitly.