Search for Bound States in ${\Xi^{-}nn}$, ${\Xi^{-}pn}$, and ${\Xi^{-}pp}$ Systems

Abstract

Search for bound states in $\Xi^{-}nn$, $\Xi^{-}pn$, and $\Xi^{-}pp$ systems is performed by employing coupled homogeneous integral Faddeev equations written in terms of $T$-matrix components. Instead of the traditional partial-wave expansion, a direct integration with respect to angular variables is used in these equations, and three-body coupling in the phase space of each of the $\Xi^{-}nn$–$\Lambda\Sigma^{-}n$–$\Sigma^{-}\Sigma^{0}n$, $\Xi^{-}np$–$\Lambda\Lambda n$–$\Lambda\Sigma^{0}n$, and $\Xi^{-}pp$–$\Lambda\Lambda p$–$\Lambda\Sigma^{0}p$ systems is taken precisely into account within this approach. Two-body $t$ matrices are the only ingredient of the proposed method. In the case of two-body $\Xi^{-}N$ interaction, they are found by solving the coupled Lippmann–Schwinger integral equations for the $\Xi N$–$\Lambda\Lambda$–$\Sigma\Sigma$ system in the ($I=0$, ${}^{1}S_{0}$) state, the $\Xi N$ system in the ($I=0$, ${}^{3}S_{1}$) state, the $\Xi N$–$\Lambda\Sigma$ system in the ($I=1$, ${}^{1}S_{0}$) state, and the $\Xi N$–$\Lambda\Sigma$–$\Sigma\Sigma$ system in the ($I=1$, ${}^{3}S_{1}$) state. An updated version of the ESC16 microscopic model is used to obtain two-body $\Xi^{-}N$, YY, and YN interactions generating $t$ matrices. Two-body NN interaction is reconstructed on the basis of the charge-dependent Bonn model. Direct numerical calculations of the binding energy for the systems being considered clearly indicate that either of the $\Xi^{-}nn$ and $\Xi^{-}np$ systems has one bound state with binding energies of 4.5 and 5.5 MeV, respectively, and that the $\Xi^{-}pp$ system has two bound states with binding energies of 2.7 and 4.4 MeV.