GLYPH<s57> GLYPH<s49>GLYPH<s49> GLYPH<s49>GLYPH<s51> GLYPH<s49>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s50>GLYPH<s46>GLYPH<s48> GLYPH<s120>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s45>GLYPH<s49> GLYPH<s32> GLYPH<s32> GLYPH<s77> GLYPH<s47> GLYPH<s77> GLYPH<s82>GLYPH<s32>GLYPH<s40>GLYPH<s107>GLYPH<s109> GLYPH<s41> GLYPH<s77> GLYPH<s68> GLYPH<s73>GLYPH<s45>GLYPH<s72> GLYPH<s121> GLYPH<s112>GLYPH<s45>GLYPH<s82> GLYPH<s105>GLYPH<s110>GLYPH<s116>GLYPH<s101>GLYPH<s114>GLYPH<s97>GLYPH<s99>GLYPH<s116>GLYPH<s105>GLYPH<s111>GLYPH<s110> GLYPH<s120>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s50>GLYPH<s46>GLYPH<s48> GLYPH<s110>GLYPH<s117>GLYPH<s99>GLYPH<s108>GLYPH<s101>GLYPH<s111>GLYPH<s110>GLYPH<s105>GLYPH<s99> GLYPH<s66> GLYPH<s49>GLYPH<s47>GLYPH<s52> GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s49>GLYPH<s55>GLYPH<s48>GLYPH<s32>GLYPH<s77> GLYPH<s101>GLYPH<s86> GLYPH<s66> GLYPH<s49>GLYPH<s47>GLYPH<s52> GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s49>GLYPH<s56>GLYPH<s48>GLYPH<s32>GLYPH<s77> GLYPH<s101>GLYPH<s86> GLYPH<s32> GLYPH<s32> GLYPH<s99> GLYPH<s32>GLYPH<s40>GLYPH<s102>GLYPH<s109> GLYPH<s45>GLYPH<s51> GLYPH<s41> GLYPH<s57> GLYPH<s49>GLYPH<s49> GLYPH<s49>GLYPH<s51> GLYPH<s49>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s50>GLYPH<s46>GLYPH<s48> GLYPH<s120>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s48> GLYPH<s99> GLYPH<s32>GLYPH<s40>GLYPH<s102>GLYPH<s109> GLYPH<s45>GLYPH<s51> GLYPH<s41> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s82>GLYPH<s32>GLYPH<s40>GLYPH<s107>GLYPH<s109> GLYPH<s41> 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<s120>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s45>GLYPH<s49> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s50>GLYPH<s46>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<s110>GLYPH<s117>GLYPH<s99>GLYPH<s108>GLYPH<s101>GLYPH<s111>GLYPH<s110>GLYPH<s105>GLYPH<s99> GLYPH<s32> GLYPH<s32> GLYPH<s77> GLYPH<s47> GLYPH<s77> GLYPH<s114>GLYPH<s101>GLYPH<s112>GLYPH<s117>GLYPH<s108>GLYPH<s115>GLYPH<s105>GLYPH<s118>GLYPH<s101>GLYPH<s32> GLYPH<s78> FIG. 12: (Color online) The hybrid star mass as a function of radius (left panel) and central density (right panel) in the presence of the hadron-quark phase transition. Results from the MDI-Hyp-R interaction for the hadron phase with x = 0 and x = − 1 and the MIT bag model for the quark phase with B 1 / 4 = 180 MeV and 170 MeV are shown for comparison. Results from a pure nucleonic approach are also displayed. FIG. 11: (Color online) The hybrid star mass as a function of radius (left panel) and central density (right panel) based on the MDI-Hyp interaction with x = 0 and x = − 1. Results from attractive and repulsive ΣN interactions are shown for comparison. Results from a pure nucleonic approach are also displayed. 11 transition. With the Maxwell construction, the radius of a maximum mass hybrid star would be larger. Within our model, both treatments of the phase transition give, however, reasonable masses and radii for hybrid stars. Finally, we note that the original MDI interaction does not lead to the violation of causality in β -stable npeµ matter up to 10 ρ$_{0}$ as shown in Panel (d) of Fig. 2 in Ref. [13]. Since the EOS of neutron star matter is soft- ened by the presence of hyperons and quarks, the causal- ity condition is still satisfied for the extended MDI inter- action used in the present study, and this is confirmed by explicit calculations. of state. The maximum mass obtained with an attrac- tive ΣN interaction is 1 . 59 M$_{⊙}$ for x = 0 and 1 . 57 M$_{⊙}$ for x = − 1, where M$_{⊙}$ is the solar mass. With a repulsive ΣN interaction the maximum mass increases to 1 . 65 M$_{⊙}$ for x = 0 and 1 . 67 M$_{⊙}$ for x = − 1. Our results thus dif- fer from those of many previous works [38, 39] that the maximum mass of hybrid stars cannot reach the canoni- cal value of 1 . 4 M$_{⊙}$ . In our model, the radius of a hybrid star with a mass of 1 . 4 M$_{⊙}$ is 11 . 2 km for x = 0 and 11 . 9 km for x = − 1 for an attractive ΣN interaction, and 11 . 9 km for x = 0 and 13 . 2 km for x = − 1 for a repulsive ΣN interaction, respectively. In Fig. 12, the M-R and M- ρ$_{c}$ relations are displayed for hybrid stars with the hadron-quark phase transition in their liquid core. Again, results from the hadron phase with x = 0 and x = − 1 and the quark phase with B 1 / 4 = 180 MeV and B 1 / 4 = 170 MeV are shown, and those from a pure nucleonic approach are also shown for comparison. The maximum mass for B 1 / 4 = 180 MeV is 1 . 50 M$_{⊙}$ for x = 0 and 1 . 46 M$_{⊙}$ for x = − 1, and for B 1 / 4 = 170 MeV it is 1 . 46 M$_{⊙}$ for x = 0 and 1 . 45 M$_{⊙}$ for x = − 1, respectively. The radius of a standard neutron star with a mass of 1 . 4 M$_{⊙}$ is 11 . 0 km for x = 0 and 10 . 8 km for x = − 1 for B 1 / 4 = 180 MeV, and 10 . 2 km for x = 0 and 10 . 0 km for x = − 1 for B 1 / 4 = 170 MeV. If we further reduce the value of B , the hadron-quark phase transition would happen at an even lower density, and the maximum mass for the hybrid star would correspond to a larger central density and a smaller radius. Above results are obtained with the Gibbs construction for the hadron-quark phase V. SUMMARY We have extended the MDI interaction for the nucleon- nucleon effective interaction in nuclear medium to include the nucleon-hyperon and hyperon-hyperon interactions by assuming that they have the same density and mo- mentum dependence as that for the nucleon-nucleon in- teraction. The parameters were determined by fitting the empirical hyperon single-particle potentials in symmetric nuclear matter at its saturation density. As an example for the application of the extended MDI interaction, we investigated the properties of hybrid stars that include not only the hyperon degree of freedom but also that of quarks by taking into account the phase transition be- tween the hadron and quark phases. Our results indicate that the extended MDI interaction can give a reasonable