An N/D study of the S 11 channel πN scattering amplitude

Extensive dynamical \begin{document}$N/D$\end{document} calculations are made in the study of \begin{document}$S_{11}$\end{document} channel low energy \begin{document}$\pi$\end{document} N scatterings, based on various phenomenological model inputs of left cuts at tree level. The subtleties of the singular behavior of the partial wave amplitude at the origin of the complex \begin{document}$s$\end{document} plane are carefully analysed. Furthermore, it is found that the dispersion representation for the phase shift, \begin{document}$\delta$\end{document} , has to be modified in the case of \begin{document}$\pi $\end{document} N scatterings. An additional contribution from the dispersion integral exists, which is, however, almost exactly cancelled the contribution from two virtual poles located near the end points of the segment cut induced by \begin{document}$u$\end{document} channel nucleon exchanges. Relying very little on the details of the dynamical inputs, the subthreshold resonance \begin{document}$N^*(890)$\end{document} survives.