Chiral and deconfinement transitions in strong coupling lattice QCD with finite coupling and Polyakov loop effects

We investigate chiral and deconfinement transitions in the framework of the strong coupling lattice QCD for color SU(3) with one species of unrooted staggered fermion at finite temperature and quark chemical potential. We take account of the leading order Polyakov loop terms as well as the next-to-next-to-leading order ($1/g^4$) fermionic terms of the strong coupling expansion in the effective action. We investigate the Polyakov loop effects by comparing two approximation schemes, a Haar measure method (no fluctuation from the mean field) and a Weiss mean-field method (with fluctuations). The effective potential is obtained in both cases, and we analytically clarify the Polyakov loop contributions to the effective potential. The Polyakov loop is found to suppress the chiral condensate and to reduce the chiral transition temperature at $\mu =0$, and the chiral transition temperature roughly reproduces the Monte Carlo results at $\beta =2N_c/g^2\lesssim 4$. The deconfinement transition is found to be the crossover and first order for light ($a{m}_0\lesssim 4$ at $\beta =4$) and heavy quark masses, respectively.

Figure 1

The leading order of the Polyakov loop effects in the strong coupling expansion. The squares in the left and loops in the right represent the temporal plaquettes and the Polyakov loops, respectively.

Figure 2

The quark excitation on the lattice. We show the one- and two-quark excitation in the first line, while the temporal baryon hopping in the third line.

Figure 3

The panels indicate meson hopping and quark excitation in the W method: (b) temporal meson hopping; (c) one-quark and one-antiquark excitations, two-quark and two-antiquark excitations; (d) one-antiquark and two-quark excitations. The labels (b), (c), and (d) correspond to Eqs. (22b), (22c), and (22d), respectively.

Figure 4

The Polyakov loop potential in the W (upper panel) and H (lower panel) methods. The filled circles show the equilibrium points. We show the results in the lattice unit.

Figure 5

The temperature and quark-mass dependence of the chiral condensate (upper panel) and the Polyakov loop (lower panel) in the W method. We show both of the results in the lattice unit.

Figure 6

The chiral condensate and Polyakov loop in P-SC-LQCD (solid lines), and chiral condensate in SC-LQCD without the Polyakov loop effects (dashed line) as functions of $T$ at $\mu =0$ in the W method. We show the results in the lattice unit.

Figure 7

Upper panel: The temperature dependence of $-d\sigma /dT$ (dashed line) and $d\ell /dT$ (solid line). Lower panel: $\beta$ dependence of the critical temperatures for the chiral (dashed line) and deconfinement (solid line) phase transitions. All of them are obtained in the W method. We show both of the results in the lattice unit.

Figure 8

A comparison of P-SC-LQCD (solid line) and SC-LQCD without the Polyakov loop (dashed line) in the lattice unit. We show the results of Weiss mean-filed method (bold solid line) and Haar measure method (thin solid line) in P-SC-LQCD at $m_0=0.0$. The triangles represent the results of the critical temperature ($T_{c,\mu =0}$, open triangle) and the critical coupling (${\beta }_c$, filled triangles) obtained in Monte-Carlo simulations with one species of unrooted staggered fermion: From the left, $T_{c,\mu =0}$ in the SCL with monomer-dimer-polymer simulations [19], ${\beta }_c$ at $(N_{\tau },m_0)=(2,0.025)$ [44], (2, 0.05) [44], (4, 0.1) [45], (4, 0.05) [46, 47], and (8, 0.0) [48].

Figure 9

A comparison of the chiral transition temperature $T_{\chi ,c}$ in P-SC-LQCD with NNLO (solid line) and NLO (dashed line) finite coupling corrections for quarks. The triangles represent the same MC results in Fig. 8. We show the results in the lattice unit.

Figure 10

The chiral condensate and Polyakov loop as functions of $T$ at $\mu =0$ in the W and H methods. The notation is the same as Fig. 6. We show the results in the lattice unit.

Figure 11

Th temperature dependence of $-d\sigma /dT$ (dashed line) and $d\ell /dT$ (solid line) in the Haar measure method (H method). The notation is the same as Fig. 7. We show the results in the lattice unit.

Figure 12

Upper panel: The Polyakov-quark coupling $L_1$ and $N_c\ell$ as functions of $\ell$ at $\mu =0$ ($T=0.5,\dots ,1.0$). The dots represent the value of $L_1$ under the stationary condition ($m_0=0.05$). Lower panel: Stationary value of the Polyakov-quark coupling $L_1$ and $N_c\ell$ as functions of the normalized temperature $T/T_c$. We show both of the results in the lattice unit.