$\bar{K}NN$ quasi-bound state and the $\bar{K}N$ interaction: coupled-channel Faddeev calculations of the $\bar{K}NN-\pi \mathrm{\Sigma }N$ system

Coupled-channel three-body calculations of an $I=1\mathrm{/}2$, $J^{\pi }=0^{-}$ $\bar{K}NN$ quasi-bound state in the $\bar{K}NN-\pi \mathrm{\Sigma }N$ system were performed and the dependence of the resulting three-body energy on the two-body $\bar{K}N-\pi \mathrm{\Sigma }$ interaction was investigated. Earlier results of binding energy $B_{K^{-}pp}\sim 50-70$ MeV and width ${\mathrm{\Gamma }}_{K^{-}pp}\sim 100$ MeV are confirmed [N.V. Shevchenko {\it et al.}, Phys. Rev. Lett. {\bf 98}, 082301 (2007)]. It is shown that a suitably constructed energy-independent complex $\bar{K}N$ potential gives a considerably shallower and narrower three-body quasi-bound state than the full coupled-channel calculation. Comparison with other calculations is made.