Relativistic propagators forππresonances and theNNinteraction

Abstract

A relativistic resonance propagator is presented whose form is guided by an approximation to the full $S$ matrix, utilizing $K$-matrix unitarization of the lowest-order quantum-field-theory perturbation term. This propagator is designed to fit the full $S$ matrix and therefore the corresponding phase-shift data, and to also satisfy a threshold scattering-length relationship. The formalism is generalized to include the propagation of a system of resonances on an individual phase shift, such as occurs on the $\ensuremath{\pi}\ensuremath{\pi}$ $S$ wave. A dispersion relation is then used to obtain the corresponding spectral function (or mass-squared distribution). The influence of inelasticity is considered. This model is fitted to several $\ensuremath{\pi}\ensuremath{\pi}$ $S$- and $P$-wave phase-shift solutions. The spectral functions so obtained can be used in calculations of the $\ensuremath{\pi}\ensuremath{\pi}$-system exchange contributions to the $\mathrm{NN}$ interaction. The implications of these spectral functions upon the nonrelativistic configuration-space one-boson exchange potential for the $\mathrm{NN}$ interaction as well as upon the corresponding relativistic momentum-space potential for use with the Bethe-Salpeter equation are discussed.