GLYPH<s48> GLYPH<s53>GLYPH<s48>GLYPH<s48> GLYPH<s45>GLYPH<s52>GLYPH<s48> GLYPH<s45>GLYPH<s50>GLYPH<s48> GLYPH<s48> GLYPH<s50>GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s54>GLYPH<s48> GLYPH<s40>GLYPH<s82>GLYPH<s41> GLYPH<s40>GLYPH<s65>GLYPH<s41> GLYPH<s78> GLYPH<s83> GLYPH<s121> GLYPH<s109> GLYPH<s101>GLYPH<s116>GLYPH<s114>GLYPH<s121> GLYPH<s32>GLYPH<s112>GLYPH<s111>GLYPH<s116>GLYPH<s101>GLYPH<s110>GLYPH<s116>GLYPH<s105>GLYPH<s97>GLYPH<s108>GLYPH<s115>GLYPH<s32> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s111>GLYPH<s102>GLYPH<s32>GLYPH<s78> GLYPH<s44>GLYPH<s32> GLYPH<s32>GLYPH<s97>GLYPH<s110>GLYPH<s100>GLYPH<s32> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s97>GLYPH<s116>GLYPH<s32> GLYPH<s32>GLYPH<s61> GLYPH<s32> GLYPH<s48> GLYPH<s32> GLYPH<s32> GLYPH<s85> GLYPH<s115> GLYPH<s121> GLYPH<s109> GLYPH<s32> GLYPH<s40> GLYPH<s77> GLYPH<s101> GLYPH<s86> GLYPH<s41> GLYPH<s112>GLYPH<s32>GLYPH<s40>GLYPH<s77>GLYPH<s101>GLYPH<s86>GLYPH<s41> GLYPH<s120>GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s48> GLYPH<s48> GLYPH<s53>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s51>GLYPH<s48> GLYPH<s54>GLYPH<s48> GLYPH<s57>GLYPH<s48> GLYPH<s49>GLYPH<s50>GLYPH<s48> GLYPH<s49>GLYPH<s53>GLYPH<s48> GLYPH<s49>GLYPH<s56>GLYPH<s48> GLYPH<s40>GLYPH<s82>GLYPH<s41> GLYPH<s40>GLYPH<s65>GLYPH<s41> GLYPH<s120>GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s45>GLYPH<s49> GLYPH<s32> GLYPH<s32> 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> FIG. 4: (Color online) Symmetry potentials of nucleon, Σ and Ξ in asymmetric nuclear matter at density ρ = 3 ρ 0 . Σ( A ) and Σ( R ) are for the MDI-Hyp-A and MDI-Hyp-R interactions, respectively. Note that different scales are used for x = 0 (left panel) and x = − 1 (right panel). A. The hadron phase ∑ i ρ$_{i}$b$_{i}$ = ρ, (9) µ$_{i}$ = µ H b b$_{i}$ − µ H c q$_{i}$. (11) i ρ$_{i}$q$_{i}$ = 0 , (10) 5 interaction. These values are changed to 174 MeV and 63 MeV for the nucleon and the Ξ hyperon, respectively, and for the Σ hyperon to 87 MeV for the MDI-Hyp-A inter- action and 143 MeV for the MDI-Hyp-R interaction for the case of x = − 1. As noted in Ref. [66], the charged Σ baryon ratio in heavy-ion collisions can be used as a probe to constrain the nuclear symmetry energy (potential) at densities larger than 3 ρ$_{0}$ . It will be very interesting to see how the symmetry potentials of Σ and Ξ hyperons in nu- clear matter affect the charged Σ hyperon ratio and the charged Ξ hyperon ratio in heavy-ion collisions induced by neutron-rich nuclei. This is important for determin- ing the high density behavior of the nuclear symmetry energy using these ratios in heavy-ion collisions. The ex- tended MDI interaction with hyperons is therefore useful for studying the nuclear symmetry energy (potential) at ρ > 3 ρ$_{0}$ in the transport model calculation for heavy-ion collisions. metry, however, also holds for Σ and Ξ hyperons. The single-particle potential of a particle in asymmetric nu- clear matter can thus be written in general as U$_{τ}$$_{b}$ ( ρ, δ ) ≈ U$_{τ}$$_{b}$ ( ρ, δ = 0) − τ$_{b}$U b $_{sym}$( ρ ) δ in terms of the symmetry potential U b $_{sym}$( ρ ), defined by U N $_{sym}$( ρ ) = ( U$_{n}$ ( ρ, δ ) − U$_{p}$ ( ρ, δ )) / 2 δ , U Σ $_{sym}$( ρ ) = ( U$_{Σ}$ − ( ρ, δ ) − U$_{Σ}$ + ( ρ, δ )) / 2 δ and U Ξ $_{sym}$( ρ ) = ( U$_{Ξ}$ − ( ρ, δ ) − U$_{Ξ}$ 0 ( ρ, δ )) / 2 δ for the nucleon, Σ and Ξ hyperons, respectively. From the single-particle potentials of nucleons, Σ and Ξ hyperons in asymmet- ric nuclear matter at normal nuclear matter density and of isospin asymmetry δ = 0 . 2, we have calculated their symmetry potentials using above definitions. In Fig. 3, the momentum dependence of these symmetry potentials are compared. All symmetry potentials are seen to de- crease with increasing momentum. At zero momentum, the symmetry potentials of nucleon and Ξ are about 34 MeV and 12 MeV, respectively, and for the Σ hyperon they are 17 MeV for an attractive ΣN interaction and 9 MeV for a repulsive one. The extended MDI interaction is also useful for study- ing the properties of a hybrid star that is expected to have a quark core at high densities, a mixed phase of quarks and hadrons at moderate densities, and a hadron phase at low densities. We review in this section the β - equilibrium, charge neutrality, and baryon number con- servation conditions of such a matter. In the hadron phase of a hybrid star, the matter con- sists of nucleons, hyperons and leptons. For leptons, we include both electrons and muons with their masses taken to be 0 and 106 MeV, respectively. At equilibrium, these particles satisfy following baryon number conservation, charge neutrality and β -equilibrium conditions: Although the symmetry potentials at normal nuclear density are independent of the value of x , which is used in the MDI interaction to model the stiffness of nuclear symmetry energy at densities different from the normal density, this is not the case at other densities. This is demonstrated in Fig. 4 for the symmetry potentials of the nucleon, the Σ hyperon and the Ξ hyperon in asym- metric nuclear matter at density ρ = 3 ρ$_{0}$ for the two sym- metry energy parameters x = 0 and x = − 1. For x = 0, the symmetry potentials at zero momentum are 53 MeV and 19 MeV for the nucleon and the Ξ hyperon, respec- tively, and for the Σ hyperon they are 26 MeV with the MDI-Hyp-A interaction and 2 MeV with the MDI-Hyp-R In the above, q$_{i}$ and b$_{i}$ are, respectively, the charge and baryon numbers of particle species i , where i can be nu- cleons, hyperons or leptons, and their density and chemi- cal potential are denoted, respectively, by ρ$_{i}$ and µ$_{i}$ . The total baryon density of the hadron phase is denoted by ρ , and µ H b and µ H c are, respectively, the baryon and charge chemical potentials of the hadron phase. Taking into account their interactions in the mean-field approximation, the chemical potential of baryon species III. EQUILIBRIUM CONDITIONS AND THERMODYNAMICAL RELATIONS IN DENSE MATTER