9 FIG. 15: The bulk viscosity over entropy density ratio ζ/s in the PLSM for different chemical potentials as functions of T/T c . The figure is taken from Ref.[53]. FIG. 14: The T − µ phase diagram in the Polyakov linear sigma model. The figure is taken from Ref.[53]. GLYPH<s48> GLYPH<s46>GLYPH<s56> GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s56> GLYPH<s53> GLYPH<s48> GLYPH<s46>GLYPH<s57> GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s57> GLYPH<s53> GLYPH<s49> GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s49> GLYPH<s46>GLYPH<s48> GLYPH<s53> GLYPH<s49> GLYPH<s46>GLYPH<s49> GLYPH<s48> GLYPH<s49> GLYPH<s46>GLYPH<s49> GLYPH<s53> GLYPH<s49> GLYPH<s46>GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s50> GLYPH<s48> GLYPH<s46>GLYPH<s52> GLYPH<s48> GLYPH<s46>GLYPH<s54> GLYPH<s48> GLYPH<s46>GLYPH<s56> GLYPH<s49> GLYPH<s46>GLYPH<s48> GLYPH<s32> GLYPH<s32> GLYPH<s47>GLYPH<s115> GLYPH<s84> GLYPH<s47>GLYPH<s84> GLYPH<s99> GLYPH<s32> GLYPH<s61>GLYPH<s48> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s77> GLYPH<s101> GLYPH<s86> GLYPH<s32> GLYPH<s61>GLYPH<s56> GLYPH<s48> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s77> GLYPH<s101> GLYPH<s86> GLYPH<s32> GLYPH<s61>GLYPH<s49> GLYPH<s51> GLYPH<s57> GLYPH<s46>GLYPH<s53> GLYPH<s32>GLYPH<s77> GLYPH<s101> GLYPH<s86> GLYPH<s32> GLYPH<s61>GLYPH<s49> GLYPH<s54> GLYPH<s48> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s77> GLYPH<s101> GLYPH<s86> GLYPH<s48> GLYPH<s53> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s49> GLYPH<s53> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s53> GLYPH<s48> GLYPH<s51> GLYPH<s48> GLYPH<s48> GLYPH<s51> GLYPH<s53> GLYPH<s48> GLYPH<s52> GLYPH<s48> GLYPH<s48> GLYPH<s52> GLYPH<s53> GLYPH<s48> GLYPH<s48> GLYPH<s52> GLYPH<s48> GLYPH<s56> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s48> GLYPH<s49> GLYPH<s54> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s48> GLYPH<s50> GLYPH<s56> GLYPH<s48> GLYPH<s32>GLYPH<s99>GLYPH<s114>GLYPH<s111> GLYPH<s115>GLYPH<s115>GLYPH<s111> GLYPH<s118> GLYPH<s101> GLYPH<s114> GLYPH<s32>GLYPH<s49> GLYPH<s115>GLYPH<s116>GLYPH<s32>GLYPH<s111> GLYPH<s114>GLYPH<s100> GLYPH<s101> GLYPH<s114> GLYPH<s32> GLYPH<s121> GLYPH<s39>GLYPH<s40>GLYPH<s84> GLYPH<s41>GLYPH<s32>GLYPH<s50> GLYPH<s110> GLYPH<s100> GLYPH<s32>GLYPH<s112> GLYPH<s101> GLYPH<s97> GLYPH<s107> GLYPH<s32> GLYPH<s32> GLYPH<s39>GLYPH<s32>GLYPH<s40>GLYPH<s84> GLYPH<s41>GLYPH<s32>GLYPH<s50> GLYPH<s110> GLYPH<s100> GLYPH<s32>GLYPH<s112> GLYPH<s101> GLYPH<s97> GLYPH<s107> GLYPH<s32> GLYPH<s39>GLYPH<s40>GLYPH<s84> GLYPH<s41>GLYPH<s32>GLYPH<s50> GLYPH<s110> GLYPH<s100> GLYPH<s32>GLYPH<s112> GLYPH<s101> GLYPH<s97> GLYPH<s107> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s76>GLYPH<s83>GLYPH<s77>GLYPH<s32>GLYPH<s67>GLYPH<s69>GLYPH<s80> GLYPH<s32>GLYPH<s32>GLYPH<s80>GLYPH<s76>GLYPH<s83>GLYPH<s77>GLYPH<s32>GLYPH<s67>GLYPH<s69>GLYPH<s80> GLYPH<s32>GLYPH<s40>GLYPH<s77> GLYPH<s101> GLYPH<s86>GLYPH<s41> GLYPH<s32> GLYPH<s32> GLYPH<s84>GLYPH<s32>GLYPH<s40>GLYPH<s77>GLYPH<s101> GLYPH<s86>GLYPH<s41>GLYPH<s32> divergent at the critical temperature. As discussed earlier, the sharp rise of the bulk viscosity will lead to the breakdown of the hydrodynamic approx- imation around the critical temperature, and will affect the hadronization and freeze-out processes of QGP cre- ated at heavy ion collisions Refs. [40–43]. For example, in Ref. [40], it is pointed out that a sharp rise of bulk vis- cosity near phase transition might induce an instability in the hydrodynamic flow of the plasma, and this mode will blow up and tear the system into droplets. Another scenario is pointed out in Ref. [29, 42] that the large bulk viscosity near phase transition might induce “soft statis- tical hadronization”, i.e. the expansion of QCD mat- ter close to the phase transition is accompanied by the production of many soft partons, which may be mani- fested through both a decrease of the average transverse momentum of the resulting particles and an increase in the total particle multiplicity. Therefore the critical end point might be located through the observables which are sensitive to the ratio of bulk viscosity over entropy density. However, it is noticed that the results of bulk viscosity in this paper are based on Eq. (19), where the ansatz for the spectral function The chiral phase transition for the strange quark and the deconfinement phase transition in the T − µ plane are shown by the dash-dotted line and dotted line, respec- tively. It is found that with the increase of chemical po- tential, the critical temperature for strange quark to re- store chiral symmetry decreases. However, for the decon- finement phase transition, with the increase of chemical potential, the deconfinement critical temperature keeps almost a constant around 220 MeV. It can be seen that in the Polyakov linear sigma model, there exists two- flavor quarkyonic phase [62] at low density, where the u, d quarks restore chiral symmetry but still in confine- ment, and three-flavor quarkyonic phase at high density, where the u, d, s quarks restore chiral symmetry but still in confinement. For the Polyakov linear sigma model, the result shows that the critical end point is around ( T$_{E}$, µ$_{E}$ ) = (188 MeV , 139 . 5 MeV), which is close to the lattice re- sult ( T$_{E}$, µ$_{E}$ ) = (162 ± 2 MeV , µ$_{E}$ = 120 ± 13 MeV) [59]. For the linear sigma model without the Polyakov loop, the critical end point is located at ( T$_{E}$, µ$_{E}$ ) ≃ (92 . 5MeV , 216 MeV). The critical chemical potential µ$_{E}$ in PLSM is much lower than that in the PNJL model with three quark flavors where the predicted critical end point is µ$_{E}$ > 300 MeV[60, 61]. Because the Polyakov-loop in the PLSM is not intro- duced dynamically, it is difficult to calculate the trans- port properties from the Boltzmann equation. However, we can use Eq.(19) to calculate the bulk viscosity. In Fig.15, we plot the bulk viscosity over entropy density ratio ζ/s as a function of the temperature for different chemical potentials. It shows ζ/s as function of the scaled temperature T /T$_{c}$ for different chemical potentials with µ = 0 , 80 , 139 . 5 , 160 MeV. We can see that when the chemical potential increases up to µ = 80 MeV, there is an upward cusp appearing in ζ/s right at the critical tem- perature T$_{c}$ . With the increase of the chemical potential, the upward cusp becomes sharper, and the height of the cusp increases. At the critical end point µ$_{E}$ and when µ > µ$_{E}$ for the first order phase transition, ζ/s becomes has been used in the small frequency, and ω$_{0}$ is a scale at which the perturbation theory becomes valid. In our calculation, ω$_{0}$ = 1 GeV, its magnitude at T$_{c}$ is in agree- ment with that obtained in ChPT for massive pion gas system in Ref. [39]. Qualitatively, the bulk viscosity corresponds to nonconformality, thus it is reasonable to observe a sharp rising of bulk viscosity near phase tran- sition. Ref. [39] has investigated the correlation between the bulk viscosity and conformal breaking, and supports the results in Ref.[29, 30]. The sharp rising of bulk vis- cosity has also been observed by another lattice result [31] and in the linear sigma model [33]. However, till ρ ( ω, ⃗ 0) ω = 9 ζ π ω 2 0 π ( ω 2 + ω $^{2}$) (35)