Medium Effect in S-shell Double-$\Lambda$ Hypernuclei and Hyperon Superfluidity in Neutron Star Cores

Abstract

The effective $\Lambda$-$\Lambda$ interaction energies in s-shell double-$\Lambda$ hypernuclei, $^{6}_{\Lambda \Lambda }\text {He}$, $^{5}_{\Lambda \Lambda }\text {He}$ and $^{5}_{\Lambda \Lambda }\text {H}$ have been investigated by solving three-body system ($\Lambda$ + $\Lambda$ + core) within the framework of coupled rearrangement channel Gaussian basis treatment. The Nijmegen soft-core potential models, NSC97d, NSC97e and NSC97f, have been applied for free-space $\Lambda$-$\Lambda$ interaction. By taking into account the medium effects, it was adjusted to fit the experimental $\Lambda$-$\Lambda$ interaction energies of $^{6}_{\Lambda \Lambda }\text {He}$ (0.67 ± 0.17 MeV). The effective $\Lambda$-$\Lambda$ interaction energies of $^{5}_{\Lambda \Lambda }\text {He}$ are 0.92 MeV, 0.88 MeV, 0.86 MeV for each potential model and that of $^{5}_{\Lambda \Lambda }\text {H}$ is 0.6 MeV for all potential modes. Moreover, the $\Lambda$-superfludity in the neutron star cores which is related to cooling process has also been investigated by applying our constructed effective $\Lambda$-$\Lambda$ NSC97e potential of $^{6}_{\Lambda \Lambda }\text {He}$. It is found that $\Lambda$ superfludity begins to appear when the interaction strength is 1.5 times more attractive than our constructed interaction. The $\Lambda$-superfluidity in the neutron star cores might occur at the hyperon percentage $Y_\Lambda$ (1%–5%), and vanish at $Y_\Lambda$ (15%– 20%).