Few-body calculations of $\eta$-nuclear quasibound states

We report on precise hyperspherical-basis calculations of $\eta NN$ and $\eta NNN$ quasibound states, using energy dependent $\eta N$ interaction potentials derived from coupled-channel models of the $S_{11}$ $N^{\ast}(1535)$ nucleon resonance. The $\eta N$ attraction generated in these models is too weak to generate a two-body bound state. No $\eta NN$ bound-state solution was found in our calculations in models where Re $a_{\eta N}\lesssim 1$ fm, with $a_{\eta N}$ the $\eta N$ scattering length, covering thereby the majority of $N^{\ast}(1535)$ resonance models. A near-threshold $\eta NNN$ bound-state solution, with $\eta$ separation energy of less than 1 MeV and width of about 15 MeV, was obtained in the 2005 Green-Wycech model where Re $a_{\eta N}\approx 1$ fm. The role of handling self consistently the subthreshold $\eta N$ interaction is carefully studied.