$NK$ and $\Delta K$ states in the chiral $\mathrm{SU}\left(3\right)$ quark model

The isospin $I=0$ and $I=1$ kaon-nucleon $S$, $P$, $D$, $F$ wave phase shifts are studied in the chiral $\mathrm{SU}\left(3\right)$ quark model by solving the resonating group method (RGM) equation. The calculated phase shifts for different partial waves are in agreement with the experimental data. Furthermore, the structures of the $\Delta K$ states with $L=0$, $I=1$, and $L=0$, $I=2$ are investigated. We find that the interaction between $\Delta$ and $K$ in the case of $L=0$, $I=1$ is attractive, which is not like the situation of the $NK$ system, where the $S$-wave interactions between $N$ and $K$ for both $I=0$ and $I=1$ are repulsive. Our numerical results also show that when the model parameters are taken to be the same as in our previous $NN$ and $YN$ scattering calculations, the $\Delta K$ state with $L=0$ and $I=1$ is a weakly bound state with about $2\mathrm{MeV}$ binding energy, while the one with $I=2$ is unbound in the present one-channel calculation.

Figure 1

$KN$ $S$-wave phase shifts as a function of the laboratory momentum of kaon meson. The hole circles and the triangles correspond respectively to the phase shifts analysis of Hyslop et al. [20] and Hashimoto [21].

Figure 2

$KN$ $P$-wave phase shifts. Same notation as in Fig. 1.

Figure 3

$KN$ $D$-wave phase shifts. Same notation as in Fig. 1.

Figure 4

$KN$ $F$-wave phase shifts. Same notation as in Fig. 1.

Figure 5

$\Delta K$ $S$-wave phase shifts as a function of the energy of the center of mass motion. The solid lines represent the results obtained by considering ${\theta }^S=35.264°$ while the dotted lines $-18°$.

Figure 6

The GCM matrix elements of OGE.