Momentum-transfer dependence of the [formula omitted] vertex in a dispersion theoretical model. I

Abstract

We study the dependence of the electropion production vertex on λ, the square of the 4-momentum transferred between electrons, by employing unsubtracted dispersion relations written in terms of the λ-variable itself. Adopting a model in which only spinless pions, nucleons, and photons are considered, we concentrate on complications of the λ-plane singularity structure which result from the instability of the final state π- N system. Physical unitarity in the λ-channel is approximated by the three intermediate states ππ, N N , and πN N . A closed system of form factors and scattering amplitudes which are coupled by unitarity is thus obtained. It is assumed that an s-wave gp- N resonance exists, and the π- N center-of-mass s-wave form factor is singled out for special attention. A dispersion relation is written for this form factor and is found to possess several complex anomalous branch cuts when the s-variable is above thresh-old. We calculate the discontinuities across these anomalous cuts and show that the dispersion relations obtained obey both s- and λ-channel unitarity. A version of the coupled channel N D method is used to rewrite the dispersion relations as an equivalent set of partially decoupled, linear integral equations.