Single-${\mathrm{\Xi }}^{-}$ hypernuclei within a quark mean-field model

The single-${\mathrm{\Xi }}^{-}$ hypernuclei are studied using a quark mean-field model. At the quark level, the ${\mathrm{\Xi }}^{-}$ hyperon is composed of one $d$ quark and two $s$ quarks, which are confined by the harmonic oscillator potentials. In the case of hadrons, the baryons interact with each other by exchanging $\sigma ,\omega$, and $\rho$ mesons between quarks in different baryons. The single-${\mathrm{\Xi }}^{-}$ binding energies of $\mathrm{\Xi }$ hypernuclei are investigated from $\mathrm{Be}$ to $\mathrm{Tl}$ using different parameter sets, which are determined by the ground-state properties of several stable nuclei and the empirical values of the single-$\mathrm{\Lambda }$ and single-$\mathrm{\Xi }$ potentials at the nuclear saturation density. For the bound states of ${\mathrm{\Xi }}^{-}+{}^{14}\mathrm{N}$ (i.e., $\mathrm{C}$) system named as KISO event in the KEK-E373 experiment, it is found that the ${\mathrm{\Xi }}^{-}$ binding energies are around $5.61–5.89$ MeV for $1s$ state and $0.94–1.21$ MeV for $1p$ state with QMF-NK1S, QMF-NK2S, and QMF-NK3S parameter sets, whose single-$\mathrm{\Xi }$ potentials are $-12$ MeV. These results and those from cluster models with the Gaussian expansion method concerning on the ${\mathrm{\Xi }}^{-}+{}^{12}\mathrm{Be}$ show that in the KISO event, the ${\mathrm{\Xi }}^{-}$ hyperon may occupy the $1p$ state. Furthermore, the ${\mathrm{\Xi }}^{-}$ binding energies are achieved around $27\mathrm{MeV}$ for the ${\mathrm{\Xi }}^{-}+{}^{207}\mathrm{Pb}$ (i.e., $\mathrm{Tl}$) system. The energies were nearly comparable to the single-$\mathrm{\Lambda }$ binding energy of $\mathrm{Pb}$ observed by experiments. It demonstrates that the $\mathrm{\Lambda }$ and ${\mathrm{\Xi }}^{-}$ hyperons seem to appear simultaneously in a neutron star.

Figure 1

Systematic calculations of the binding energies of ${\mathrm{\Xi }}^{-}$ hypernuclei within (a) QMF-NK2S and (b) ${\mathrm{QMF}-\mathrm{NK}2\mathrm{S}}^{\prime }$ parameter sets from $\mathrm{Be}$ to $\mathrm{Tl}$.

Figure 2

The potentials from various fields in $\mathrm{C}$ within QMF-NK2S and ${\mathrm{QMF}-\mathrm{NK}2\mathrm{S}}^{\prime }$. The dot-dashed ones are from the spurious contributions of $\rho$ meson and dashed curves for its real effects. Dotted lines represent Coulomb field. The dash-dot-dotted curves denote the potentials from $\sigma$ and $\omega$ mesons. The potentials from all mesons are given as solid curves, where $V_{\mathrm{all}}=V_{\sigma }+V_{\omega }+V_{\rho }+V_A$.

Figure 3

The potentials from various fields in $\mathrm{Be}$ within QMF-NK2S and ${\mathrm{QMF}-\mathrm{NK}2\mathrm{S}}^{\prime }$. The WS potential used in the analysis of Nakazawa et al. [7] is also plotted for comparison as long dashed curves.