$Z(3)$ metastable states in a Polyakov quark meson model

We study the existence of $Z(3)$ metastable states in the presence of the dynamical quarks within the ambit of a Polyakov quark meson (PQM) model. Within the parameters of the model, it is seen that for temperatures ${\mathrm{T}}_m$ greater than the chiral transition temperature ${\mathrm{T}}_c$, $Z(3)$ metastable states exist (${\mathrm{T}}_m\sim 310\mathrm{MeV}$ at zero chemical potential). At finite chemical potential ${\mathrm{T}}_m$ is larger than the same at vanishing chemical potential. We also observe a shift of ($\sim 5°$) in the phase of the metastable vacua at zero chemical potential. The energy density difference between true and $Z(3)$ metastable vacua is very large in this model. This indicates a strong explicit symmetry-breaking effect due to quarks in the PQM model. We compare this explicit symmetry breaking in the PQM model with small explicit symmetry breaking as a linear term in a Polyakov loop added to the Polyakov loop potential. We also study the possibility of domain growth in a quenched transition to quark gluon plasma in relativistic heavy ion collisions.

Figure 1

The $Z(3)$ structure of the vacuum in the plot of the Polyakov potential for both parameter sets as a function of $\theta$ at a temperature 400 MeV.

Figure 2

The path of the Polyakov order parameter $\mathrm{\Phi }$ in the complex plane when crossing the $Z(3)$ interface at a temperature of 500 MeV. This path corresponds to a minimum energy configuration as one goes from one vacuum to another $Z(3)$ vacuum.

Figure 3

Effective potential as a function of the phase of the Polyakov loop for $\mathrm{T}=400\mathrm{MeV}$ and $\mu =0$. Set 1 corresponds to the Polyakov potential of Ref. [6], while set 2 corresponds to Ref. [17].

Figure 4

The normalized chiral condensate $\sigma$ and the Polyakov loop $\mathrm{\Phi }$ as a function of temperature.

Figure 5

Free energy density difference between true and metastable vacua as a function of temperature.

Figure 6

Contour plot of effective potential at temperatures (a) 300 MeV and (b) 400 MeV. The legend bar represents energy density in $\mathrm{GeV}{\mathrm{fm}}^{-3}$.

Figure 7

The Polyakov loop potential with explicit symmetry-breaking term ${\mathcal{U}}_e$ [Eq. (27)] and the effective potential $V_{\mathrm{eff}}$ of the PQM model [Eq. (13)] as a function of $\mathrm{\Phi }$ along true and metastable vacua at $\mathrm{T}=400\mathrm{MeV}$.

Figure 8

$V_{\mathrm{eff}}$ as a function of $\theta$ for different numbers of flavors at $\mathrm{T}=400\mathrm{MeV}$.

Figure 9

The temperature ${\mathrm{T}}_m$ beyond which $Z(3)$ metastable vacuum occurs as a function of chemical potential.

Figure 10

Field configurations at different times with explicit symmetry-breaking effect. The shading (color) representing the dominant region in (d) corresponds to the true vacuum with $\theta =0$. (a)–(d) show the growth of domains for $\tau =1.2$, 1.6, 2.2, and 2.4 fm (with corresponding values of temperature $\mathrm{T}=368$, 335, 311, and 298 MeV), respectively.

Figure 11

(a) Polyakov loop. (b) Sigma field (in units of MeV) domain growth at $\tau =3\mathrm{fm}$.

Figure 12

Plot of average Polyakov loop and normalized sigma field value with $\tau$.