Energy-Dependent Pion-Nucleon Phase-Shift Analysis

Energy-dependent pion-nucleon phase-shift analyses have been carried out in the energy ranges 0-50, 0-100, 0-350, 0-700, and 300-800 MeV. An outline of the procedure is given. The results obtained with all available data are presented. The best solution in the 0-350-MeV range is obtained by starting from Donnachie, Hamilton, and Lea's predicted phases, although the final results differ widely from the input. The best solution obtained in the 0-700-MeV range contains three resonances: in the ${P}_{33}$ state at 193 MeV, the ${P}_{11}$ state at approximately 585 MeV, and the ${D}_{13}$ state at 638 MeV. The last two are strongly absorptive. A good solution has been obtained without the ${P}_{11}$ phase shift actually going through 90\ifmmode^\circ\else\textdegree\fi{}, but it must be large (about 80\ifmmode^\circ\else\textdegree\fi{}) at 700 MeV. The 0-100-, the 0-350-, and the 0-700-MeV solutions are consistent with each other in the large phases. The ${P}_{11}$, ${P}_{13}$, and ${P}_{31}$ phases cannot be precisely determined by using only the 0-50-MeV data.