Pion-nucleon scattering in the P33 channel in the linear sigma model

A consistent study of the NN interaction requires the investigation of the Δ resonance, as the Δ can affect the properties of the effective light σ meson originating from correlated two-pion exchange and vice versa. We study how the Δ appears as an effective degree of freedom in the linear σ model. Since the field-theoretic scattering amplitudes respect certain analytic properties, it is approximate to use partial-wave dispersion relations for πN scattering, which produce non-linear integral equations that are solved by the N/D method. We choose an amplitude that guarantees the correct threshold beaviour and that gives a P33 resonance with a driving term from one-nucleon exchange. Then we investigate the effect of adding the one-sigma-exchange force. With either a very heavy or a light σ meson, we can find a P33 resonance. For intermediate σ masses, we find a πN bound state. We confirm that the calculated resonance mass depends sensitively on the short-range πN attraction. Nevertheless, the existence of a P33 resonance is compatible with the existence of an effective light σ meson in this model. (orig.)