Prediction ofp-,d-, andf-Wave Pion-Nucleon Scattering

We develop a peripheral method for predicting $\pi -N$ phase shifts up to moderate energies. Precise values are given for the $p$-, $d$-, and $f$-wave phase shifts (with the exception of $p_{11}$, up to 400 MeV, and the general behavior up to around 1 BeV is also predicted. The 600- and 900-MeV ${\pi }^{-}-p$ resonances are clearly identified with the $D_{13}$ and $F_{15}$ amplitudes, respectively, and it is probable that the 1.35 BeV ${\pi }^{+}-p$ resonance is in $F_{37}$. The predictions at 310 MeV select the phase shift set $\mathrm{spdf}$ II of Vik and Rugge. The method consists in evaluating the dispersion relation for $F_{l\pm }\mathrm{}\left(s\right)=\frac{f_{l\pm }\mathrm{}\left(s\right)\mathrm{}}{q^{2l}}$ where $f_{l\pm }\mathrm{}\left(s\right)\mathrm{}$ is the partial-wave amplitude. The factor $q^{-2l}$ suppresses the unknown shorter range parts of the $\pi -N$ interaction. Various means are used to avoid the difficulties arising from lack of knowledge of the inelasticity. The symmetries in spin and isospin of the dispersion relation calculations of the various interactions are examined, together with equivalent model potentials.