Pion-Nucleon Elastic Scattering at 310 MeV: Phase-Shift Analysis

A phase-shift analysis of ${\pi }^{+}-p$ and ${\pi }^{-}-p$ elastic scattering at 310 MeV has been performed. The data include differential and total cross section and recoil-proton polarization data for both ${\pi }^{+}-p$ and ${\pi }^{-}-p$ elastic scattering, as well as differential cross-section data for charge-exchange scattering. Inclusion of $d$ waves was necessary to attain an adequate fit to the data; in the case of ${\pi }^{-}-p$ differential cross section, the best fit included $f$ waves. A general phase-shift search using $s$, $p$, and $d$ waves was carried out; a single solution was obtained that adequately fit all the available data. The most notable characteristics of this solution are isotopic-spin 3/2 phase shifts similar to those obtained in a previous analysis of the ${\pi }^{+}-p$ data and a relatively large $D_{1,5}$ phase shift equal to approximately 15 deg. Errors on the isotopic-spin 1/2 phase shifts of this solution range from 0.3 to 0.9 deg. The isotopic-spin 3/2 phase-shift errors are similar to those obtained previously. Because the ${\pi }^{-}-p$ differential cross section data indicated a possible need for $f$ waves, and since the only satisfactory $\mathrm{spd}$ solution displayed a large $d$-wave phase shift in the isotopic-spin 1/2 state, the analysis was extended to include $f$ waves. The result of allowing $f$ waves was to increase the errors on each of the phase shifts (up to about 2 deg), and also to introduce two new solutions, neither of which can be ruled out statistically. These new solutions are similar to the $d$-wave solution in the isotopic-spin 3/2 phase shifts, but vary rather widely in isotopic-spin 1/2 phase shifts. Inelastic scattering processes were neglected throughout most of the analysis; however, a study of their effects on the final solutions was made and these effects were seen to be unimportant.