Isospin Effect in Three-Body Kaonic Clusters

The kaonic clusters $K^{-}K^{-}p$ and $ppK^{-}$ are described based on the configuration space Faddeev equations for $AAB$ system. The $AB$ interaction is given by isospin-dependent potentials. For this isospin model, we show that the relation $\left\vert E_{3}(V_{AA}=0)\right\vert~<~2\left\vert E_{2}\right\vert$ is satisfied when $E_{2}$ is the binding energy of the $AB$ subsystem and $E_{3}(V_{AA}=0)$ is the three-body binding energy when interaction between identical particles is omitted, $V_{AA}=0$. For the $NN{\bar K}$ system, taking into account weak attraction of $NN$ interaction the relation leads to the evaluation $|E_3|\le 2|E_2|$. The "isospinless model" for the kaonic clusters based on the isospin averaged $N{\bar K}$ potential demonstrates the opposite relation $\left\vert E_{3}(V_{AA}=0)\right\vert~>~2\left\vert E_{2}\right\vert$. The isospin "given charge formalism" is presented for $NN{\bar K}$ cluster. This formalism is related to isospin model by unitary transformation of the isospin basis. An interpretation of the "particle representation" for $NN{\bar K}$ system is proposed.