s-Wave Pion-Nucleon Scattering

Abstract

The Mandelstam relations for pion-nucleon scattering are used to obtain equations for the $s$-wave partial wave amplitudes in the two isotopic spin states. The solutions of these equations are investigated in the approximation where only the one-nucleon contributions and the unitarity integral are kept. It is found that there are no solutions of the form $\frac{N}{D}$ without complex zeros, and that this is a consequence of the large size of the one-nucleon terms. A comparison with experiment is made which suggests that the dominant contribution to the $T=\frac{3}{2}$ $s$-wave amplitude (other than the one-nucleon contribution) comes from a region of the complex energy plane that is outside the physical region for the related processes ($\pi -\pi$ into $\mathrm{NN}$ and "crossed" $\pi -N$ scattering). An Appendix is devoted to discussing the available experimental data and they are found to be consistent with a scattering length ($\frac{\delta }{k}$ at threshold) of 0.098±0.004 in the $T=\frac{3}{2}$ state.