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Using separable $\bar{K}N-\pi\Sigma$ potentials in the Faddeev equations, wecalculated the binding energies and widths of the $K^{-}pp$, $K^{-}ppn$ and$K^{-}ppp$ quasi-bound states on the basis of three- and four-bodyAlt-Grassberger-Sandhas equations in the momentum representation. One- andtwo-pole version of $\bar{K}N-\pi\Sigma$ interaction are considered and thedependence of the resulting few-body energy on the two-body$\bar{K}N-\pi\Sigma$ potential was investigated. The $s$-wave [3+1] and [2+2]sub-amplitudes are obtained by using the Hilbert-Schmidt expansion procedurefor the integral kernels. As a result, we found a four-body resonance of the$K^{-}ppn$ and $K^{-}ppp$ quasi-bound states with a binding energy in the range$B_{K^{-}ppn}\sim{55-70}$ and $B_{K^{-}ppp}\sim{90-100}$ MeV, respectively. Thecalculations yielded full width of $\Gamma_{K^{-}ppn}\sim{16-20}$ and$\Gamma_{K^{-}ppp}\sim{7-20}$ MeV.

Coupled-channels Faddeev AGS calculation of $K^{-}ppn$ and $K^{-}ppp$ quasi-bound states