NDCalculation with Inelastic Unitarity ofπNScattering

We calculated the real part $\delta$ of the $\pi N$ phase shifts for partial waves $J\le \frac{3}{2}$ using the $\frac{N}{D}$ equations with inelastic unitarity. The generalized potential is determined by considering single-exchange diagrams for the nucleon, the $N^{*}$ (1238 MeV) and the $\rho$ (760 MeV). The inelastic factor $\eta$ is taken from the recent, extensive complex phase-shift analyses. A straight cutoff $W_c$ on the dispersion integrals in the energy plane $W$ is used to eliminate the high-energy divergences associated with the exchange of particles with spin ≥1. Full numerical solutions of the integral equation for the $N$ function are obtained by the matrix-inversion technique. The single cutoff $W_c$ is separately adjusted for each $J$ to give the best fit to the two coupled waves $l=J-\frac{1}{2} \mathrm{and} J+\frac{1}{2}$. For comparison, we also calculate the $\delta \text{'}\mathrm{s}$ using elastic unitarity, i.e., $\eta \mathrm{}\left(W\right)\mathrm{}\equiv 1$. By including inelastic effects, we obtain better agreement with the phase-shift analyses, except for the $S_{31}$ and $P_{13}$ partial waves. In particular, our calculation of the $D_{13}$ phase shift agrees with the phase-shift analyses for ${\delta }_{D_{13}}$ up to $E_L\approx 450$ MeV (whereas the solution for $\eta \equiv 1$ gives a $\delta$ which is much too small). The $P_{11}$ partial wave is of special importance since (in addition to the nucleon pole) it contains a possible resonance at $E_L\sim 570$ MeV which is very inelastic. We did two different calculations of the $I=\frac{1}{2}$, J = ½ partial wave: (i) $W_c$ was adjusted to yield the nucleon pole as a bound state. The residue (related to ${g_{\overline{N}N\pi }}^2$) is approximately twice what it should be. Both the $S_{11}$ and $P_{11}$ phase shifts are in violent disagreement with the phase-shift analyses. (The calculations with inelastic effects gave only a slight improvement over the $\eta \equiv 1$ calculations.) (ii) The nucleon pole was included in the direct channel at the correct position with the correct residue and $W_c$ was adjusted so that no zero appeared in the $D$ function. We then obtained quantitative fits to the low-energy $S_{11}$ and $P_{11}$ phase shifts.