πN S11partial-wave amplitude near theηnproduction threshold

From an analysis of the ${\pi }^{-}$ backward differential cross-section data, which clearly shows a cusplike structure near the $\eta n$ threshold, we deduce, using partial-wave unitarity relations, the size and the orientation of the cusp. The cusp appears because of the strong coupling of the $\eta n$ channel to an $N^{\prime }$ resonance just above its threshold. Using the above information on the cusp, the $\pi N S_{11}$ partial-wave amplitude is fitted through the $\eta n$ threshold and the resonance. The effect of the size of the cusp, acting as a constraint, on the resonance parameters determined in the fit is emphasized. The fit also determines the phase of the $\eta n$ production amplitude at the $\eta n$ threshold, which then using the knowledge of the orientation of the cusp gives information on the phase of the elastic no-spin-flip amplitude at the $\eta n$ threshold. This information could be a useful constraint in $\pi N$ partial-wave analysis.