New Phase-Shift Parametrization Based on a Phase-Shift Dispersion Relation

Abstract

The dispersion relation for the phase shift used by Ball and Frazer is generalized to include the effects of zeros and poles in the $S$ matrix. For complex $S$-matrix zeros near the physical cut, one obtains a new parametrization for a resonant state. For $l=0$, the usual Breit-Wigner phase shift is found. The method is easily extended to resonances in higher partial waves (in terms of two parameters for the case of pure elastic unitarity). The results are used to describe three partial waves in $\ensuremath{\pi}N$ scattering: ${P}_{33}$, ${D}_{13}$, and ${P}_{11}$. Excellent agreement with the published ${P}_{33}$ phase shifts is obtained by using our resonant form plus the integral over the inelastic cut. The inelastic contribution to the phase shift is small at low energies but becomes important for ${E}_{L}>300$ MeV (${E}_{L}$ is the incident-pion kinetic energy). Inelastic effects in the ${D}_{13}$ partial wave can account for almost all of the phase shift up to ${E}_{L}=500$ MeV. A narrow resonance plus inelasticity gives a good fit to the phase shifts up to ${E}_{L}=900$ MeV. Our fit to the ${P}_{11}$ phase shift depends on the nucleon pole, the Roper resonance, a zero in the $S$ matrix below the elastic threshold, and inelastic effects. A good fit to the low-energy phase shifts can be obtained only if one assumes large inelasticity above 1 BeV; however, the model does account for the qualitative behavior of the ${P}_{11}$ phase shift.