Effective mass and decay of ${\Theta }^{+}$ in nuclear matter by using the quark-meson coupling models

The in-medium mass of a ${\Theta }^{+},m_{{\Theta }^{+}}^{*}$, in cold symmetric nuclear matter is calculated by use of the quark-meson coupling models. The ${\Theta }^{+}$ is treated as an MIT bag with a quark content $\mathit{uudd}\overline{s}$. The bag parameters for a free ${\Theta }^{+}$ are fixed to reproduce the observed mass of the ${\Theta }^{+}$. In doing so, we use three different values of the s-quark mass as the mass of the s quark is not well known. As usual, the strengths of the u- and d-quark couplings to σ- and ω-meson fields are determined to fit the nuclear saturation properties. However, the coupling constant $g_{\sigma }^s$ between the s quark and the σ meson cannot be fixed from the saturation properties, and thus we treat $g_{\sigma }^s$ as a free parameter and investigate how $m_{{\Theta }^{+}}^{*}$ depends on $g_{\sigma }^s$. We find that $m_{{\Theta }^{+}}^{*}$ depends significantly on the value of $g_{\sigma }^s$ but not on the mass of the s quark. Chemical potentials of the ${\Theta }^{+}$ and the $K+N$ system are calculated to discuss the decay of a ${\Theta }^{+}$ in nuclear matter. We calculate the effective mass of a kaon in nuclear matter in two ways: by using the optical potential of $K^{-}$ in matter and by using a quark model. By comparing the effective masses of K calculated from these two methods, we find that the magnitude of the real part of the optical potential that is consistent with the quark model is about 100 MeV. In that case, we find $g_{\sigma K}\approx 2$.

Figure 1

Ratios of the effective radius R of a ${\Theta }^{+}$ bag to $R_0$. $R/R_0$ from QMC (MQMC) calculations are plotted by thick (thin) curves. $g_{\sigma }^s=0$ and $m_s=150$ MeV are used for the calculations in this figure.

Figure 2

$m_{{\Theta }^{+}}^{*}/m_{{\Theta }^{+}}$ is plotted against density for $g_{\sigma }^s=-2$, 0, 2. Thick (thin) curves denote $m_{{\Theta }^{+}}^{*}/m_{{\Theta }^{+}}$ calculated from QMC (MQMC) for both ${\Theta }^{+}$ and nuclear matter.

Figure 3

The effective mass of a kaon as a meson bag is plotted by the solid curve. The dashed and the dotted curves denote $m_K^{*}$ when we treat a kaon as a point particle with $U_{\overline{K}}({\rho }_0)$ being $-100$ and $-150$ MeV, respectively.

Figure 4

Comparison between ${\mu }_{{\Theta }^{+}}$ and ${\mu }_K+{\mu }_N$ in the MQMC model. At $\rho <{\rho }_1,{\mu }_{{\Theta }^{+}}$ is greater than ${\mu }_K+{\mu }_N$ with $k_K=k_F$, so both ${\Theta }^{+}\to KN$ and $N{\Theta }^{+}\to KNN$ decays are possible. At ${\rho }_1<\rho <{\rho }_2$, only the $N{\Theta }^{+}\to KNN$ channel is open. At ${\rho }_2<\rho ,{\Theta }^{+}$ decay channels to KN are closed.