Recent calculations of $\eta$-nuclear bound states in few-body, as well as many-body systems are reviewed [1,2,3]. Underlying energy-dependent $\eta N$ interactions are derived from coupled-channel models that incorporate the $S_{11}$ $N^*(1535)$ nucleon resonance.

The role of self-consistent handling of the subthreshold, strongly energy-dependent $\eta N$ interactions is thoroughly discussed. Due to relatively large downward energy shift and rapid decrease of the $\eta N$ amplitudes, our calculations impose stronger constraints than ever on the onset of $\eta$-nuclear binding.

Binding energies and widths of $\eta$-nuclear bound states were calculated within several $\eta N$ interaction models for nuclei across the periodic table. No $\eta NN$ bound states were found in models where Re$a_{\eta N} \leq 1$ fm, with $a_{\eta N}$ the $\eta N$ scattering length, i.e., in the majority of coupled-channel models of the $N^*(1535)$ resonance. For $\eta NNN$, a weakly bound and relatively broad state was found within the GW model [4] where Re$a_{\eta N} \approx 1$ fm. Bound states of the $\eta$ meson in $^{12}$C are unlikely in models with Re$a_{\eta N} \leq 0.5$ fm, and Re$a_{\eta N} \approx 0.9$ fm is required to reproduce the $\eta$ bound-state candidate in $^{25}$Mg from the COSY-GEM experiment [5].

[1] N. Barnea, E. Friedman, A. Gal, Phys. Lett. B 747 (2015) 345

[2] E. Friedman, A. Gal, J. Mares, Phys. Lett. B 725 (2013) 334

[3] A. Cieply, E. Friedman, A. Gal, J. Mares, Nucl. Phys. A 925 (2014) 126

[4] A. M. Green, S. Wycech, Phys. Rev. C 71 (2005) 014001

[5] A. Budzanowski et al (COSY-GEM Collaboration), Phys. Rev. C 79 (2009) 012201(R)