GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s53> GLYPH<s54> GLYPH<s45>GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s51>GLYPH<s48>GLYPH<s48> GLYPH<s52>GLYPH<s48>GLYPH<s48> GLYPH<s97>GLYPH<s116>GLYPH<s116>GLYPH<s114>GLYPH<s97>GLYPH<s99>GLYPH<s116>GLYPH<s105>GLYPH<s118>GLYPH<s101>GLYPH<s32> GLYPH<s78> GLYPH<s32> GLYPH<s105>GLYPH<s110>GLYPH<s116>GLYPH<s101>GLYPH<s114>GLYPH<s97>GLYPH<s99>GLYPH<s116>GLYPH<s105>GLYPH<s111>GLYPH<s110> GLYPH<s114>GLYPH<s101>GLYPH<s112>GLYPH<s117>GLYPH<s108>GLYPH<s115>GLYPH<s105>GLYPH<s118>GLYPH<s101>GLYPH<s32> GLYPH<s78> GLYPH<s32> GLYPH<s105>GLYPH<s110>GLYPH<s116>GLYPH<s101>GLYPH<s114>GLYPH<s97>GLYPH<s99>GLYPH<s116>GLYPH<s105>GLYPH<s111>GLYPH<s110> GLYPH<s40> GLYPH<s41> GLYPH<s78> GLYPH<s77>GLYPH<s68>GLYPH<s73> GLYPH<s45> GLYPH<s72>GLYPH<s121> GLYPH<s112>GLYPH<s32> GLYPH<s105>GLYPH<s110>GLYPH<s116>GLYPH<s101>GLYPH<s114> GLYPH<s97>GLYPH<s99>GLYPH<s116>GLYPH<s105>GLYPH<s111>GLYPH<s110> GLYPH<s102>GLYPH<s111>GLYPH<s114> GLYPH<s32> GLYPH<s115>GLYPH<s121> GLYPH<s109>GLYPH<s109>GLYPH<s101>GLYPH<s116>GLYPH<s114> GLYPH<s105>GLYPH<s99>GLYPH<s32> GLYPH<s110>GLYPH<s117>GLYPH<s99>GLYPH<s108>GLYPH<s101>GLYPH<s97>GLYPH<s114> GLYPH<s32> GLYPH<s109>GLYPH<s97>GLYPH<s116>GLYPH<s116>GLYPH<s101>GLYPH<s114> GLYPH<s32> GLYPH<s32> GLYPH<s85> GLYPH<s40> GLYPH<s112> GLYPH<s61> GLYPH<s48> GLYPH<s41> GLYPH<s32> GLYPH<s40> GLYPH<s77> GLYPH<s101> GLYPH<s86> GLYPH<s41> GLYPH<s47> GLYPH<s48> FIG. 1: (Color online) Single-particle potentials for particles at rest in symmetric nuclear matter as functions of density. An important quantity related to the interaction of a particle in nuclear medium is its single-particle potential as given by Eq. (3), which is also needed later in our study of the properties of neutron stars. The single-particle po- tential of a particle depends not only on the density of the medium but also on the momentum of the particle. In this section, we show and discuss the single-particle po- tentials of both nucleons and hyperons in nuclear matter obtained from the extended MDI interaction. 3 B. Single-particle potentials hyperon interactions. We therefore assume in the present study that A$_{bb}$ ′ , A ′ bb ′ , B$_{bb}$ ′ and C$_{τ}$$_{b}$$_{,τ}$ ′ b ′ are all proportional to corresponding ones in the nucleon-nucleon interaction. Specifically, for A$_{bb}$ ′ , A ′ bb ′ and B$_{bb}$ ′ , we assume and for C$_{τ}$$_{b}$$_{,τ}$ ′ b ′ we assume We determine the values of f$_{bb}$ ′ by fitting the empirical potential U ( b $^{′}$) b of baryon b at rest in a medium consisting of baryon species b $^{′}$. For hyperons in symmetric nuclear matter at saturation density, their potentials are for the Λ potential from the analysis of ( π $^{+}$, K$^{+}$) and ( K $^{−}$, π $^{−}$) reactions on nuclei [51, 52] and for the Ξ potential from the analysis of (Ξ , 4 $_{Λ}$H ) [53] and ( K $^{−}$, K $^{+}$) [54, 55] reactions. This leads to the values f$_{N}$$_{Λ}$ = 1 and f$_{N}$$_{Ξ}$ = 0 . 73. For the Σ hyperon, its po- tential was taken to be attractive in earlier studies [45], but more recent analysis indicate that it is repulsive in the nuclear medium [56–60]. In the present work, we consider both the attractive and repulsive cases By setting f$_{N}$$_{Σ}$ = 1 we get an attractive ΣN interac- tion, and this is called MDI-Hyp-A in the following. To obtain a repulsive ΣN interaction, called MDI-Hyp-R in the following, we adjust the values of positive and negative terms in the single-particle potential by setting B$_{N}$$_{Σ}$ = 2 . 33 B$_{NN}$ and A ′ N Σ = 2 . 33 A ′ NN without chang- ing other parameters as in the case of MDI-Hyp-A in- teraction. A similar method of changing an attractive ΣN interaction to a repulsive one was used in the RMF calculation [31] by changing the coupling constants of ω and ρ mesons. For the hyperon-hyperon interaction, the parameters are fitted according to [61] We first show in Fig. 1 the single-particle potential of a particle at rest in symmetric nuclear matter as a function of density. Although the nucleon potential is more attrac- tive at normal density than those of hyperons, it becomes more repulsive than the hyperon potentials above about 5 times normal nuclear density, including the Σ potential that is attractive at normal density. For the Σ poten- tial that is repulsive at normal density, it becomes more repulsive as the density increases and becomes slightly attractive only at very low densities. Comparing our results with those from other models given in Ref. [62] (and references therein), the single-particle potentials of Λ and Σ are close to those from the chiral effective field theory [63], but more repulsive than those based on the G-matrix calculations using the soft core Nijmegen model or the J¨ ulich meson-exchange model for the free hyperon- nucleon interactions, particularly at high densities. The single-particle potential of a particle from the ex- tended MDI interaction also depends on its momentum. Figure 2 shows the momentum dependence of the single- particle potentials for both nucleons and hyperons in symmetric nuclear matter at saturation density. Again, results using both attractive and repulsive ΣN interac- tions (i.e., MDI-Hyp-A and MDI-Hyp-R) are shown for comparison. Also indicated in the figure is the Fermi which gives the strength of the hyperon-hyperon interac- tions as f$_{ΛΛ}$ = 0 . 63, f$_{ΛΣ}$ = 1 . 25, f$_{ΛΞ}$ = 1 . 25, f$_{ΣΣ}$ = 0 . 49, f$_{ΣΞ}$ = 1, and f$_{ΞΞ}$ = 0 . 53. Detailed values of the param- eters are listed in Table. I. These parameterizations can be viewed as a baseline for studying the properties of the hypernuclear matter, and more sophisticated treatments are left for future work after the in-medium properties of hyperons are better understood. It will be shown in the following that many interesting results can already be obtained even with these simple parameterizations. U ( Y $^{′}$) Y ( ρ$_{Y}$ ′ = ρ$_{0}$ ) ∼ − 40 MeV , (8) U ( N ) Σ ( ρ$_{N}$ = ρ$_{0}$ ) = ± 30 MeV . (7) U ( N ) Ξ ( ρ$_{N}$ = ρ$_{0}$ ) = − 18 MeV (6) U ( N ) Λ ( ρ$_{N}$ = ρ$_{0}$ ) = − 30 MeV (5) A$_{bb}$ ′ = f$_{bb}$ ′ A$_{NN}$, A ′ bb ′ = f$_{bb}$ ′ A ′ $_{NN}$, B$_{bb}$ ′ = f$_{bb}$ ′ B$_{NN}$, (4)