B. The hadron-quark phase transition 9 GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s117> GLYPH<s115> GLYPH<s100> GLYPH<s101> GLYPH<s112> GLYPH<s32> GLYPH<s32> GLYPH<s120>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s48> GLYPH<s77>GLYPH<s68>GLYPH<s73> GLYPH<s45>GLYPH<s72>GLYPH<s121> GLYPH<s112>GLYPH<s45>GLYPH<s82>GLYPH<s32> GLYPH<s110> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s117> GLYPH<s115> GLYPH<s100> GLYPH<s101> GLYPH<s112> GLYPH<s110> GLYPH<s120>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s45>GLYPH<s49> GLYPH<s32> 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<s48> GLYPH<s50> GLYPH<s52> GLYPH<s54> GLYPH<s56> GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s40>GLYPH<s99>GLYPH<s41> GLYPH<s117> GLYPH<s115> GLYPH<s100> GLYPH<s101> GLYPH<s112> GLYPH<s110> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s112> GLYPH<s97> GLYPH<s114> GLYPH<s116> GLYPH<s105> GLYPH<s99> GLYPH<s108> GLYPH<s101> GLYPH<s32> GLYPH<s102> GLYPH<s114> GLYPH<s97> GLYPH<s99> GLYPH<s116> GLYPH<s105> GLYPH<s111> GLYPH<s110> GLYPH<s32> GLYPH<s40> GLYPH<s37> GLYPH<s41> GLYPH<s47> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s54> GLYPH<s56> GLYPH<s49>GLYPH<s48> GLYPH<s40>GLYPH<s100>GLYPH<s41> GLYPH<s117> GLYPH<s115> GLYPH<s100> GLYPH<s101> GLYPH<s112> GLYPH<s110> 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<s32> GLYPH<s32> GLYPH<s48> GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s52>GLYPH<s48>GLYPH<s48> GLYPH<s54>GLYPH<s48>GLYPH<s48> GLYPH<s56>GLYPH<s48>GLYPH<s48> GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s56>GLYPH<s48> GLYPH<s49>GLYPH<s50>GLYPH<s48> GLYPH<s77>GLYPH<s68>GLYPH<s73>GLYPH<s45>GLYPH<s72>GLYPH<s121>GLYPH<s112>GLYPH<s45>GLYPH<s82>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<s32> GLYPH<s80> GLYPH<s32> GLYPH<s40> GLYPH<s77> GLYPH<s101> GLYPH<s86> GLYPH<s47> GLYPH<s102> GLYPH<s109> GLYPH<s51> GLYPH<s41> GLYPH<s40>GLYPH<s77>GLYPH<s101>GLYPH<s86>GLYPH<s47>GLYPH<s102>GLYPH<s109> GLYPH<s51> GLYPH<s41> GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s56>GLYPH<s48> GLYPH<s49>GLYPH<s50>GLYPH<s48> GLYPH<s32> GLYPH<s120>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s32> GLYPH<s48>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<s120>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s45>GLYPH<s49>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<s120>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s32> GLYPH<s48>GLYPH<s32> GLYPH<s104>GLYPH<s121> GLYPH<s112>GLYPH<s101>GLYPH<s114>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<s120>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s45>GLYPH<s49>GLYPH<s32> GLYPH<s104>GLYPH<s121> GLYPH<s112>GLYPH<s101>GLYPH<s114>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<s32> GLYPH<s77>GLYPH<s68>GLYPH<s73>GLYPH<s45>GLYPH<s72>GLYPH<s121>GLYPH<s112>GLYPH<s45>GLYPH<s65>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. 9: (Color online) Particle fractions in a hypernuclear matter with the presence of hadron-quark phase transition from the MDI-Hyp-R interaction with x = 0 ((a) and (c)) and x = − 1 ((b) and (d)) for the hadron phase and the MIT bag model for the quark phase. Results from B 1 / 4 = 180 MeV ((a) and (b)) and 170 MeV ((c) and (d)) are shown for comparison. FIG. 8: (Color online) Equation of state of pure nuclear mat- ter and hypernuclear matter from the MDI-Hyp interaction with x = 0 and x = − 1 as well as with attractive (upper panel) and repulsive (lower panel) ΣN interactions. ter from the MDI interaction with x = 0 and x = − 1 and of hypernuclear matter from the MDI-Hyp-A and MDI- Hyp-R interactions. It is seen that the EOS is softened when hyperons are present as compared to the EOSs of pure nuclear matter. We note that the EOS at mod- erate densities plays an important role in determining the maximum mass of neutron stars. Without hyperons, the EOS at moderate densities is stiffer for smaller value of x , while it becomes stiffer for larger value of x when hyperons are included, especially for a repulsive ΣN in- teraction. As shown in Fig. 6, a stiffer symmetry energy leads to a larger number of hyperons, which results in a softer EOS as a result of the lower pressure due to the presence of more degrees of freedom, and the effect of hy- perons on the EOS is smaller for a soft symmetry energy, as fewer hyperons are then present in the hypernuclear matter. One the other hand, the symmetry energy con- tribution to the pressure of the hypernuclear matter is larger for x = − 1 than x = 0. As a result, the EOS of hypernuclear matter shows similar stiffness for differ- ent symmetry energy parameters. Furthermore, the EOS from the MDI-Hyp-R interaction is stiffer than that from the MDI-Hyp-A interaction, as Σ particles do not appear in the former case as shown in Fig. 6. Our results thus show that the hypernuclear matter has a stiffer (softer) EOS at lower densities but a softer (stiffer) EOS at mod- erate densities with a stiffer (softer) nuclear symmetry energy. We note that the results from both x = 0 and x = − 1 are consistent with the constraints on the nu- clear equation of state obtained from the analysis of the collective flow data in heavy-ion collisions [70]. At higher densities in the core of a neutron star, a tran- sition from hadron matter to quark matter is expected to occur. Here we only consider the hadron-quark phase transition for the repulsive ΣN interaction as it is more consistent with the latest empirical information [56–60]. In Fig. 9, we display the particle fractions of each species in the presence of a hadron-quark phase transition, with the hadron phase from the MDI-Hyp-R interaction with x = 0 and x = − 1 and the quark phase from the MIT bag model with bag constants B 1 / 4 = 180 MeV and 170 MeV. The particle fractions are weighted by the baryon number and corresponding phase fraction. It is seen that for B 1 / 4 = 180 MeV, the phase transition begins at the density 0 . 35 fm − 3 and ends at the density 1 . 38 fm − 3 for x = 0, and begins at the density 0 . 26 fm − 3 and ends at the density 1 . 38 fm − 3 for x = − 1, while for B 1 / 4 = 170 MeV it begins at the density 0 . 21 fm − 3 and ends at the density 1 . 05 fm − 3 for x = 0, and begins at the den- sity 0 . 18 fm − 3 and ends at the density 1 . 05 fm − 3 for x = − 1. The hadron-quark phase transition thus hap- pens at lower baryon number density for a stiffer symme- try energy and for a smaller value of B , while the density at which the hadron-quark phase transition ends seems to depend only on the value of B but not much on the value of the symmetry energy parameter x . With a smaller value of the bag constant, the hadron-quark phase tran- sition both begins and ends earlier. It is also seen that d and s quarks occupy a larger fraction than u quarks in the mixed phase because of their negative charges, so