Khuri-Treiman Representation and Perturbation Theory

Abstract

It is characteristic of decay amplitudes that the spectral functions for their integral representations have branch points overlapping the integration contour. For the amplitude satisfying the Khuri-Treiman dispersion representation, the prescription for passing the branch points is shown to be incomplete. The full prescription is obtained using perturbation theory as a guide. It turns out that the perturbation theory prescription contradicts the "naive" dispersion theory prescription in part, even in the physical decay region of the amplitude. An interpretation is offered for this contradiction. In the perturbation analysis it is found that a non-Landau or second-type singularity appears on the unphysical boundary of the physical sheet of the decay amplitude. In view of this unexpected result, the usual Landau analysis of perturbation amplitudes is extended to include examination of the singularity of the complex non-Landau surface on the physical sheet. Such an extension is valuable when an explicit formula for the spectral function is unavailable. Here the extended Landau analysis facilitates comparison of the present results with previous work on decay amplitudes. One part of the discussion presents and makes use of an analysis in which an internal mass is taken as a complex variable.