$D^{*}\mathrm{\Xi }N$ bound state in strange three-body systems

The recent update of the strangeness $-2$ ESC08c Nijmegen potential incorporating the NAGARA and KISO events predicts a $\mathrm{\Xi }N$ bound state, $D^{*}$, in the ${}^{3}S_1(I=1)$ channel. We study if the existence of this two-body bound state could give rise to stable three-body systems. For this purpose we solve the bound state problem of three-body systems where the $\mathrm{\Xi }N$ state is merged with $N\text{’s}$, $\mathrm{\Lambda }\text{’s}$, $\mathrm{\Sigma }\text{’s}$, or $\mathrm{\Xi }\text{’s}$, making use of the most recent updates of the two-body ESC08c Nijmegen potentials. We found that there appear stable states in the $\mathrm{\Xi }NN$ and $\mathrm{\Xi }\mathrm{\Xi }N$ systems, the $\mathrm{\Xi }\mathrm{\Lambda }N$ and $\mathrm{\Xi }\mathrm{\Sigma }N$ systems being unbound.

Figure 1

(a) $V_{\mathrm{\Lambda }N}\left(r\right)$ potential as given by Eq. (19) with the parameters of Table 1. (b) Same as (a) for the $V_{\mathrm{\Sigma }N}\left(r\right)$ potential. (c) Same as (a) for the $V_{\mathrm{\Xi }N}\left(r\right)$ potential. (d) Same as (a) for the $V_{\mathrm{\Lambda }\mathrm{\Xi }}\left(r\right)$ potential. (e) Same as (a) for the $V_{\mathrm{\Sigma }\mathrm{\Xi }}\left(r\right)$ potential. (f) Same as (a) for the $V_{\mathrm{\Xi }\mathrm{\Xi }}\left(r\right)$ potential.

Figure 2

(a) Fredholm determinant for the $J=1/2$ and $J=3/2I=1/2\mathrm{\Xi }NN$ channels. (b) Fredholm determinant for the $J=1/2I=3/2\mathrm{\Xi }NN$ channel.

Figure 3

(a) Fredholm determinant for the $J=1/2$ and $J=3/2I=0\mathrm{\Xi }\mathrm{\Lambda }N$ channels. (b) Fredholm determinant for the $J=1/2$ and $J=3/2I=1\mathrm{\Xi }\mathrm{\Lambda }N$ channels.

Figure 4

Fredholm determinant for the $J=1/2$ and $J=3/2I=2\mathrm{\Xi }\mathrm{\Sigma }N$ channels.

Figure 5

(a) Fredholm determinant for the $J=1/2$ and $J=3/2I=1/2\mathrm{\Xi }\mathrm{\Xi }N$ channels. (b) Fredholm determinant for the $J=1/2I=3/2\mathrm{\Xi }\mathrm{\Xi }N$ channel.