We review our latest calculations of $K^-$ nuclear quasi-bound states. We apply a self-consistent scheme for constructing $K^-$-nuclear potentials $V_K$ from subthreshold chirally inspired in-medium ${\bar K}N$ scattering amplitudes, which was introduced in Ref. [1,2]. We consider two in-medium versions of the scattering amplitudes: the version which takes into account only Pauli blocking in the intermediate states, and the version which adds self-consistently hadron self-energies.
To explore the model dependence of our calculations, we constructed the underlying ${\bar K}N$ amplitudes within chirally motivated meson-baryon coupled-channel interaction models: Prague [3], Kyoto- Munich [4], Murcia [5], and Bonn [6]. They capture the physics of the $\Lambda(1405)$ and reproduce low energy ${\bar K}N$ observables, including the recent $1s$ level shift and width in the $K^-$ hydrogen atom from the SIDDHARTA experiment [7].

Energy dependence of the in-medium scattering amplitudes, particularly in the $K^-N$ subthreshold region, is the decisive mechanism that controls the self-consistent evaluation of corresponding $K^-$ optical potentials. The role of hadron self-energies in the self-consistent calculations of the $K^-$ binding energies $B_K$ is less pronounced than the model dependence of predicted $B_K$.

The widths of low-lying $K^-$ states due to $K^-N \rightarrow \pi Y$ conversions are substantially reduced in the self-consistent calculations, thus reflecting the proximity of the $\pi\Sigma$ threshold. On the contrary, the widths of higher excited $K^-$ states are quite large even if only the pion conversion modes on a single nucleon are considered. After including 2 body $K^-NN \rightarrow YN$ absorption modes, the total decay widths $\Gamma_K$ are comparable with the corresponding binding energies $B_K$ for all $K^-$ nuclear quasi-bound states, exceeding considerably the level spacing.

[1] A. Cieply, E. Friedman, A. Gal, D. Gazda, J. Mares, Phys. Lett. B 702 (2011) 402; Phys. Rev. C 84 (2011) 045206

[2] D. Gazda, J. Mares, Nucl. Phys. A 881 (2012) 159

[3] A. Cieply, J. Smejkal, Nucl. Phys. A 881 (2012) 115

[4] Y. Ikeda, T. Hyodo and W. Weise, Nucl. Phys. A 881 (2012) 98

[5] Z. H. Guo and J. A. Oller, Phys. Rev. C 87, no. 3,(2013) 035202

[6] M. Mai and U.-G. Meißner, Nucl. Phys. A 900, (2013) 51

[7] M. Bazzi et al (SIDDHARTA Collaboration), Phys. Lett. B 704 (2011) 113