Pion-Nucleon Scattering and Pion-Pion Interactions

Low energy $s$- and $p$-wave $\pi -N$ scattering is analyzed by partial wave dispersion relations. From the experimental $\pi -N$ phase shifts we derive the "discrepancies" in the physical energy region and on the crossed cut. We are able to separate the discrepancies into the short-range (≲0.2×${10}^{-13\mathrm{}}$ cm) $\pi -N$ interactions and the $\pi -\pi$ contributions to $\pi -N$ scattering. The $\pi -\pi$ contributions found in this way satisfy several stringent tests which show the validity of our method for deriving and separating the $\pi -\pi$ contributions.