Coupling Constant andP-Wave Phase Shifts in Low-Energy Pion-Nucleon Scattering

By means of a model for low-energy pion-nucleon scattering, it is shown that analyticity, unitarity, and the crossing properties are sufficiently strong conditions to determine the $\pi N$ coupling constant, the $P$-wave phase shifts [including the position and width of the ($\frac{3}{2}, \frac{3}{2}$) resonance] and to give restrictions on the $\pi \pi$ cut. In addition, the solution for the (½,½) scattering amplitude turns out to be of the form describing the nucleon as a bound state of the $\pi N$ system in close analogy to a bound state in potential theory. In the model, the $\pi \pi$ cut is represented by a pair of conjugate poles, and crossing is approximated by the static crossing relations. No cutoff parameter has to be introduced, and the crossing relations are fulfilled to a high degree.