Benchmark for a quasi-bound state of the system

We present three-body nonrelativistic calculations within the framework of a potential model for the kaonic cluster $K^{-}pp$ using two completely different methods: the method of hyperspherical harmonics in the momentum representation and the method of Faddeev equations in configuration space. To perform a numerical benchmark, different $NN$ and antikaon-nucleon interactions are applied. The results of the calculations for the ground state energy for the $K^{-}pp$ system obtained by both methods are in reasonable agreement. Although the ground state energy is not sensitive to the $NN$ interaction, it shows very strong dependence on the $N$ potential. We show that the dominant clustering of the $K^{-}pp$ system in the configuration $\Lambda (1405)+p$ allows us to calculate the binding energy to good accuracy within a simple cluster approach for the differential Faddeev equations. The theoretical discrepancies in the binding energy and width for the $K^{-}pp$ system related to the different $NN$ and $N$ interactions are addressed.