Study of double-antikaonic $K^{-} K^{-} p$ cluster: numerical benchmarks

A double-antikaonic cluster $K^{-} K^{-} p$ is studied using two different methods: the method of hyperspherical functions in momentum representation and method of the Faddeev equations in configuration space. Binding energy and width of the system $K^{-} K^{-} p$ is calculated by employing the energy dependent chiral \textit{KN} interaction, as well as a phenomenological KN potential. The ground state energy shows very strong dependence on the antikaon-nucleon potential, as well as the extreme sensitivity of the width to the $K^{-} p$ potential. The energy of the ground state calculated for the energy independent \textit{KN} interaction is more than three times bigger than one obtained for the energy dependent chiral \textit{KN} potential. The energies of the ground state obtained in both methods are in a reasonable agreement. Cluster approach for the Faddeev equations was applied to demonstrate contributions of the configurations ($K^{-} K^{-}$) $+ p$ and ($K^{-}p$)$^{singlet} + K^-$ , ($K^{-} p$)$^{triplet} + K^{-}$ to total wave function of the system. The comparison of our results with calculations within variational methods and the Faddeev equations in momentum representation is presented.