An $N/D$ study of the $S_{11}$ channel $\pi N$ scattering amplitude

TL;DR

This paper performs detailed N/D calculations of the low-energy $S_{11}$ $\pi N$ scattering amplitude, revealing the necessity to modify the dispersion representation of phase shifts and confirming the survival of the subthreshold resonance $N^*(890)$.

Contribution

It introduces a modified dispersion representation for phase shifts in $\pi N$ scattering and demonstrates the robustness of the $N^*(890)$ resonance against various model inputs.

Findings

The dispersion representation for phase shifts must be modified in $\\pi N$ scattering.

An additional dispersion integral contribution cancels near-endpoint virtual pole effects.

The subthreshold resonance $N^*(890)$ persists across different phenomenological models.

Abstract

Extensive dynamical $N/D$ calculations are made in the study of $S_{11}$ channel low energy $\pi$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 $s$ plane are carefully analysed. Furthermore, { it is found that the dispersion representation for the phase shift, $\delta$, has to be modified in the case of $\pi $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 $u$ channel nucleon exchanges.} Relying very little on the details of the dynamical inputs, the subthreshold resonance $N^*(890)$ survives.