Pion-Nucleon Scattering and Pion-Pion Interactions

Abstract

Low energy $s$- and $p$-wave $\ensuremath{\pi}\ensuremath{-}N$ scattering is analyzed by partial wave dispersion relations. From the experimental $\ensuremath{\pi}\ensuremath{-}N$ phase shifts we derive the in the physical energy region and on the crossed cut. We are able to separate the discrepancies into the short-range (\ensuremath{\lesssim}0.2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm) $\ensuremath{\pi}\ensuremath{-}N$ interactions and the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ contributions to $\ensuremath{\pi}\ensuremath{-}N$ scattering. The $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ contributions found in this way satisfy several stringent tests which show the validity of our method for deriving and separating the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ contributions.The (+) charge combination of $s$- and $p$-wave $\ensuremath{\pi}\ensuremath{-}N$ amplitudes yields considerable information about the $T=0$ $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ interaction at low energies. The $T=0$, $J=0$ $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering is dominant, and we determine possible sets of the corresponding phase shift ${{\ensuremath{\delta}}_{0}}^{0}$. Several of our solutions for ${{\ensuremath{\delta}}_{0}}^{0}$ agree with recent solutions of the Chew-Mandelstam equations for $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering. Comparison with the latter suggests that the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ coupling parameter is $\ensuremath{\lambda}=\ensuremath{-}0.18\ifmmode\pm\else\textpm\fi{}0.05$, and the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering length is ${a}_{0}=1.3\ifmmode\pm\else\textpm\fi{}0.4$ (in units where $\ensuremath{\hbar}=\ensuremath{\mu}=c=1$). Other information, from the $p+d$ and $\ensuremath{\pi}+N\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\pi}+N$ experiments and from $\ensuremath{\tau}$ decay, is consistent with our proposed values of ${{\ensuremath{\delta}}_{0}}^{0}$.The (-) charge combination of $s$- and $p$-wave $\ensuremath{\pi}\ensuremath{-}N$ amplitudes gives information about the $T=1$, $J=1$ $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ interaction which is consistent with the observed $\ensuremath{\rho}$ resonance. However in the $T=1$, case, complete prediction of our $\ensuremath{\pi}\ensuremath{-}N$ results via the helicity amplitudes for $\ensuremath{\pi}+\ensuremath{\pi}\ensuremath{\rightarrow}N+\overline{N}$ is not yet satisfactory. Possible reasons for this are given.The $p$-wave $\ensuremath{\pi}\ensuremath{-}N$ interaction is separated into its constituent parts, and for example, it is seen that any attempt to determine the position of the ($\frac{3}{2}, \frac{3}{2}$) resonance must include the $T=0$, $J=0$ $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ interaction. The extent to which our analysis depends on assuming charge independence is examined. We also discuss how our results can be regarded as a fairly good physical proof of the Mandelstam representation.