Isospin and particle representations for quasi-bound state of kaonic clusters

In the framework of the method of the Faddeev equations in configuration space, the $NN\overline K (I=0)$ (and $\overline K \overline K N)$ kaonic cluster system including two identical particles is considered. We use the formalism of isospin and particle representations [1] to describe the systems. The treatment of $I=1$ and $I=0$ isospin $\overline K N$ channels is discussed. The presence of the Coulomb force in $ppK^{-}$ channel violates the isospin symmetry of the $NN\overline K (I=0)$ system. According to the particle representation, $NN\overline K $ is a two-level system of coupled $ppK^{-}$ and $pn\overline K^{0}$ channels with and without the Coulomb energy, respectively. The results of calculations for the bound states with the phenomenological and chiral motivated $\overline K N$ potentials are given for different representations. In particular, new single channel calculations for the $ppK^{-}$ (and $K^{-}K^{-}p)$ cluster are presented. It is shown that the exchange of identical particles plays an important role in the formation of a bound state of the systems. The relation of the exchange and the three-body mass rearrangement effects is discussed. [1] J. Revai, arXiv: 1608.01802v1 (2016).