Dependence of calculated binding energies and widths of $eta$-mesic nuclei on treatment of subthreshold $eta$-nucleon interaction

We demonstrate that the binding energies and widths of eta-mesic nuclei depend strongly on subthreshold eta-N interaction. This strong dependence is made evident from comparing three different eta-nucleus optical potentials: (1) a microscopic optical potential taking into account the full effects of off-shell eta-nucleon interactions; (2) a factorization approximation to the microscopic optical potential where a downward energy shift parameter is introduced to approximate the subthreshold eta-nucleon interaction; and (3) an optical potential using on-shell eta-nucleon scattering length as the interaction input. Our analysis indicates that the in-medium $eta$N interaction for bound-state formation is about 30 MeV below the free-space $eta$N threshold, which causes a substantial reduction of the attractive force between the $eta$ and nucleon with respect to that implied by the scattering length. Consequently, the scattering-length approach overpredicts the binding energies and caution must be exercised when these latter predictions are used as guide in searching for $eta$-nucleus bound states. We also show that final-state-interaction analysis cannot provide an unequivocal determination of the existence of $eta$-nucleus bound state. More direct measurements are, therefore, necessary.