QCD phase diagram with the improved Polyakov loop effective potential

Special Section on International Workshop on Nuclear Dynamics in Heavy-Ion Reactions (IWND2016)

QCD phase diagram with the improved Polyakov loop effective potential

Guo-Yun Shao，

Xue-Yan Gao，

Zhan-Duo Tang，

Ning Gao

Nuclear Science and Techniques

Vol.27, No.6

Article number 151

Published in print 20 Dec 2016

Available online 31 Oct 2016

DOI：10.1007/s41365-016-0152-0

51200

We report our recent progress on the QCD phase structure. We explore the properties of quark-gluon matter in the improved Polyakov—Nambu—Jona-Lasinio (PNJL) model by introducing a chemical potential dependent Polyakov loop potential. This treatment effectively reflects the quantum backreaction of matter sector to glue sector at none zero chemical potential. Compared with the original PNJL model, a superiority of the improved PNJL model is that it can effectively describe the confinement-deconfinement transition at low temperature and high density. And the QCD phase diagram will be modified to a certain degree if the strength of the quantum backreaction of matter sector to glue sector is strong. One evident variation is that the region of quarkyonic phase will be greatly reduced in the improved PNJL model. This means that the modification to the Polyakov loop potential with the chemical potential dependence possibly a significant improvement in exploring the full QCD phase structure.

QCD phase diagramChiral symmetryConfinement-deconfinement phase transition

1 Introduction

Exploring the QCD phase diagram of strongly interacting matter, and searching for phase transition signatures from nuclear to quark-gluon matter are subjects of great interest. Recently, intensive searches on high-energy heavy-ion collision (HIC) have been performed in laboratories such as RHIC and LHC, and a near perfect fluid of quark-gluon plasma (QGP) has been created [1]. To look for the critical end point and the boundaries of the phase transition, more experiments are in plan on the next generation facilities such as the second stage of beam energy scan (BES II) project on RHIC and programs on NICA/FAIR/J-PARC. In particular, experiments will be performed in the region of high baryon density where a promising observation of the signatures of the phase transformation is being looked forward to. Ultimately, these phenomena need to be understood in the frame of quantum chromodynamics (QCD). However, in spite of tremendous theoretical and experimental efforts, the QCD phase diagram has not been unveiled yet [2, 3]. Lattice QCD simulation is very successful in investigating the thermodynamics of QCD matter at vanishing and/or small chemical potential [4-9], but the situation is not clear at large chemical potential μ_B since lattice QCD suffers the sign problem of the fermion determinant with three colors at finite baryon chemical potential. Although some approximation methods have been proposed to try to overcome the problem, the region of large chemical potential and low temperature essentially remains inaccessible [10-12].

Complementary to lattice QCD simulation, some quantum field theory approaches and phenomenological models, such as the Dyson-Schwinger equation approach [13, 14], the Nambu–Jona-Lasinio (NJL) model [15-19], the PNJL model [20-22], the entanglement extended PNJL (EPNJL) model [23], the Polyakov-loop extended quark-meson (PQM) model [24-26], the enhanced pQCD model [27], and the equivparticle model [28, 29] have been developed to describe the properties of quark matter.

Among these models, the PNJL model, which takes into account both the chiral dynamics and (de)confinement effect at high temperature, gives a good reproduction of lattice data at vanishing chemical potential. On the other hand, in the original PNJL model, a "quarkyonic phase" in which quarks are confined, but the dynamical chiral symmetry is already restored and appears at high density and finite temperature [30, 31]. In theory, quark deconfinement should also occur at high density. The absence of quark deconfinement at low T and high density in the original PNJL model originates from the Polyakov-loop potential is extracted from pure Yang-Mills lattice simulation at vanishing. In the presence of dynamical quarks, the contribution from matter sector and its quantum backreaction to the glue sector should be included. This was realized by introducing a flavor and chemical potential dependent Polyakov loop potential in the functional renormalization group (FRG) approach [32]. Therefore, with the incorporation of both the matter and glue dynamics, the flavor and chemical potential dependent Polyakov loop potential should be taken in to study the QCD phase transition and thermodynamics [32-38] .

As a further study along this line, we take a chemical potential dependent Polyakov loop potential in the PNJL model to explore the relation between quark condensate and Polyakov loop dynamics, and investigate the full QCD phase diagram. Compared with the results derived in the original PNJL model, the calculation presents that, with the inclusion of the backreaction of matter sector to glue sector, the deconfinement phase transition line moves towards low temperature at large chemical potential. The critical end point of the first-order phase transition moves towards low T and large μ_B in the QCD phase diagram. The transition region near by the critical end point is possibly reached in the planed experiments at the facilities of NICA/FAIR/J-PARC and BES II program at RHIC. The future heavy-ion-collision experiments will provide us some hints on the QCD phase structure.

The paper is organized as follows. In Sect. 2, we describe briefly the improved PNJL model with the inclusion of quantum backreaction of matter sector to glue sector at finite chemical potential, and give the relevant formulas. In Sect. 3, we present the full QCD phase diagram in the improved PNJL model, and analyze the influence of the μ-dependent Polyakov loop potential on the chiral and deconfinement phase transition. Finally, a summary is given in Sect. 4.

2 Theoretical descriptions

To describe the properties of quark matter first, we introduce the standard two-flavor PNJL model, and then consider the μ-dependent Polyakov loop potential which the quantum backreaction of matter sector to glue sector at none zero quark chemical potential. The Lagrangian of the standard two-flavor PNJL model is

(1)

where q denotes the quark fields with two flavors, u and d, and three colors ( ${\hat{m}}_{0} = d i a g (m_{u}, m_{d})$ in flavor space). The covariant derivative in the Lagrangian is defined as $D_{μ} = \partial_{μ} - i A_{μ} - i μ_{q} δ_{μ}^{0}$ . The glue background field $A_{μ} = δ_{μ}^{0} A_{0}$ is supposed to be homogeneous and static, with $A_{0} = g A_{0}^{α} \frac{λ^{α}}{2}$ , where $\frac{λ^{α}}{2}$ is SU(3) color generators.

The effective potential $U (Φ [A], \bar{Φ} [A], T)$ is expressed in terms of the traced Polyakov loop $Φ = ({Tr}_{c} L) / N_{C}$ and its conjugate $\bar{Φ} = ({Tr}_{c} L^{†}) / N_{C}$ . The Polyakov loop, L, is a matrix in color space

L (\vec{x}) = P exp [i \int_{0}^{\frac{1}{T}} d τ A_{4} (\vec{x}, τ)],

(2)

where A₄=iA₀.

The temperature-dependent Polyakov loop effective potential, $U (Φ, \bar{Φ}, T)$ , proposed in Ref. [39], takes the form

\begin{array}{l} \frac{U (Φ, \bar{Φ}, T)}{T^{4}} = - \frac{a (T)}{2} \bar{Φ} Φ + b (T) ln [1 - 6 \bar{Φ} Φ \\ + 4 ({\bar{Φ}}^{3} + Φ^{3}) - 3 (\bar{Φ} Φ)^{2}], \end{array}

(3)

where

a (T) = a_{0} + a_{1} (\frac{T_{0}}{T}) + a_{2} {(\frac{T_{0}}{T})}^{2}, b (T) = b_{3} {(\frac{T_{0}}{T})}^{3} .

(4)

The parameters ai, bi, summarized in Table 1, are precisely fitted according to the result of lattice QCD thermodynamics in pure gauge sector.

Parameters in Polyakov effective potential given in Ref. [39]

a₀	a₁	a₂	b₃
3.51	-2.47	15.2	-1.75

The parameter T₀=270 MeV is the confinement-deconfinement transition temperature in the pure Yang-Mills theory at vanishing chemical potential [40]. In the presence of fermions, the quantum back-reaction of the matter sector to the glue sector should be considered, which leads to a flavor and quark chemical potential dependence of the transition temperature, T₀(N_{f, μ}) [32-36]. By using renormalization group theory in Ref. [32], the form of T₀(N_{f, μ}) is proposed to be

T_{0} (N_{f}, μ) = T_{τ} e^{- 1 / (α_{0} b (N_{f}, μ))},

(5)

where

b (N_{f}, μ) = \frac{11 N_{c} - 2 N_{f}}{6 π} - β \frac{16 N_{f}}{π} \frac{μ^{2}}{T_{τ}^{2}} .

(6)

The running coupling α₀=0.304 is fixed at the τ scale Tτ=1.770 GeV according to the deconfinement transition temperature T₀=270 MeV of pure gauge field with Nf=0 and μ=0. When fermion fields are included, T₀ is rescaled to 208 MeV for 2 flavor and 187 MeV for 2+1 flavor at vanishing chemical potential. The parameter β in Eq. (6) governs the curvature of T₀(μ) as a function of quark chemical potential.

With the consideration of the chemical potential dependence of Polyakov loop potential, this improved PNJL model is named the μPNJL model. We then replace the T₀ with T₀(N_{f, μ}) in the Polyakov loop potential given in Eq. (4). The thermodynamical potential of quark matter in the μPNJL model within the mean field approximation can be derived then as

(7)

where $E_{i} = \sqrt{k^{2} + M_{i}^{2}}$ is energy-momentum dispersion relation of quark flavor, i, and μi is the corresponding quark chemical potential.

The dynamical quark masses and quark condensates are coupled with the following equations

M_{i} = m_{0} - 2 G (ϕ_{u} + ϕ_{d}),

(8)

ϕ_{i} = - 2 N_{c} \int \frac{d^{3} k}{{(2 π)}^{3}} \frac{M_{i}}{E_{i}} (1 - n_{i} (k) - {\bar{n}}_{i} (k)),

(9)

where ni(k) and ${\bar{n}}_{i} (k)$

n_{i} (k) ​ = ​ \frac{Φ e^{​ - ​ (E_{i} ​ - ​ μ_{i}) / T} ​ + ​ 2 \bar{Φ} e^{​ - ​ 2 (E_{i} ​ - ​ μ_{i}) / T} + e^{- 3 (E_{i} - μ_{i}) / T}}{1 ​ + ​ 3 Φ e^{​ - ​ (E_{i} ​ - ​ μ_{i}) / T} ​ + ​ 3 \bar{Φ} e^{​ - ​ 2 (E_{i} ​ - ​ μ_{i}) / T} ​ + ​ e^{​ - ​ 3 (E_{i} ​ - ​ μ_{i}) / T}},

(10)

{\bar{n}}_{i} (k) ​ = ​ \frac{\bar{Φ} e^{​ - ​ (E_{i} ​ + ​ μ_{i}) / T} ​ + ​ 2 Φ e^{​ - ​ 2 (E_{i} ​ + ​ μ_{i}) / T} ​ + ​ e^{- 3 (E_{i} + μ_{i}) / T}}{1 ​ + ​ 3 \bar{Φ} e^{​ - ​ (E_{i} ​ + ​ μ_{i}) / T} ​ + ​ 3 Φ e^{​ - ​ 2 (E_{i} ​ + ​ μ_{i}) / T} ​ ​ + ​ ​ e^{​ - ​ 3 (E_{i} ​ + ​ μ_{i}) / T}}

(11)

are modified Fermion distribution functions of quark and antiquark. The values of ϕu, ϕd, Φ, and $\bar{Φ}$ can be determined by minimizing the thermodynamical potential,

\frac{\partial Ω}{\partial ϕ_{u}} = \frac{\partial Ω}{\partial ϕ_{d}} = \frac{\partial Ω}{\partial Φ} = \frac{\partial Ω}{\partial \bar{Φ}} = 0.

(12)

All the thermodynamic quantities relevant to the bulk properties of quark matter can be obtained from Ω. Particularly, we note that the pressure and energy density should be zero in the vacuum. In the calculation a cut-off, Λ, is implemented in 3-momentum space for divergent integrations. Λ=651 MeV, G=5.04 GeV^-2, and m_u,d=5.5 MeV will be taken by fitting the experimental values of pion decay constant fπ=92.3 MeV and pion mass mπ=139.3 MeV [21].

3 Results and discussion

In this section, we present the properties of quark matter in the improved PNJL model with a chemical potential dependent Polyakov effective potential which effectively reflects the backreaction of matter sector to glue sector at non zero chemical potential. In the original PNJL model, the parameter T₀, describing the confinement-deconfinement transition temperature in the pure Yang-Mills theory at vanishing chemical potential, is a constant. When the backreaction of matter sector to glue sector is included, T₀ will show a feature of chemical potential dependence, as presented in Fig. 1.

Fig. 1.

(color online) T₀(μ) as a function of μ with various β from 0 to 1. The case β=0 corresponds to the standard PNJL model.

To mimic the strength of the quantum backreaction of matter sector to glue sector, different values of β are used in the calculation for a tentative study. In the case of β=0, corresponding to the standard PNJL model, in which only the contribution from gauge field to Polyakov loop potential is considered, T₀(μ)=208 MeV is a constant, as shown with the solid line in Fig. 1. The dotted lines show the results for β 0. These lines manifest that T₀(μ) is sensitive to the value of β which reflects the strength of quantum backreaction of matter sector to glue sector at finite chemical potential. This figure indicates that the deconfinement temperature decreases with the increase of β.

We plot in Fig. 2 the values of Polyakov loop Φ and $\bar{Φ}$ as functions of baryon density for various β at T=20 MeV. In the original PNJL model (the case β=0), Φ and $\bar{Φ}$ always take small values at low temperature. This means that quarks are confined, even in the high density region where the chiral symmetry is restored. This forms the so-called quarkyonic phase at a low T and high density region. However, with the consideration of quark quantum backreaction to glue sector at finite chemical potential, quark confinement-deconfinement phase transition can occur at low T, as shown by the dotted lines with different values of β. If we take the standard that Φ or $\bar{Φ}$ =0.5 marks the happening of deconfinement transition, as adopted in [20, 23], we find the deconfinement transition density moves to a lower one for a larger β. However, we should note that when β>1.2 is taken in the calculation, unphysical results will be derived with too small deconfined baryon density where the chiral symmetry is still breaking. For the relevant study, one can also refer to [35-37].

Fig. 2.

(color online) Polyakov loop Φ and

\bar{Φ}

as functions of baryon density, ρ_B, for different β at T=20 MeV.

We plot the full QCD phase diagram in Fig. 3 for different values of β=0, 0.25, 0.5, and 0.75 to show the influence of quantum backreaction of matter sector to glue sector on the QCD phase structure. In this figure, the red dash-dotted line in each panel describes the confinement-deconfinement phase transition. The dotted and solid lines are the chiral phase transition lines. The dotted line at high temperature in each panel means the chiral phase transition is a smooth crossover, and the chiral phase transition at low temperature is the first order, as shown with the black solid line. The solid dot is the end point of the first order phase transition for each β.

Fig. 3.

(color online) QCD phase diagram with different parameter β. "QY" in the figure means quarkyonic phase. the red dash-dotted line in each panel describes the confinement-deconfinement phase transition, and the dotted and solid lines are the chiral phase transition lines. The dotted line at high temperature means the chiral phase transition is a smooth crossover. The chiral phase transition at low temperature as shown with the black solid line is the first order. The solid dot is the end point of the first order phase transition.

In the upper panel, the case of β=0 is the phase diagram derived in the standard two-flavor PNJL model. We can see that there exist a quite large region of quarkyonic matter (marked QY in Fig. 3), which means a phase where the chiral symmetry restores, but quarks are still confined. The appearance of quarkyonic phase in fact displays the relation between chiral restoration of quark condensate and color deconfinement in the QCD phase diagram at large chemical potential.

As indicated in Fig. 3 with different values of β, we can see that strength of quantum backreaction plays an important role on the QCD phase diagram at large chemical potential. When a larger β is taken, the first-order transition line becomes shorter and correspondingly the critical end point moves towards lower temperature and larger chemical potential. What’s more, compared with the original PNJL model, the confinement-deconfinement transition can take place at lower temperature in the improved μPNJL model, which result in a great shrink of quarkyonic phase when a larger β is taken. Once again, it demonstrates that the quantum backreaction of matter sector to glue sector at finite chemical potential is significant for the QCD phase structure. The future heavy-ion-collision experiments will provide us more hints on the QCD phase structure and provide opportunities to test the quark quantum backreaction effect.

From Fig. 3, we also notice that the critical temperatures of chiral and deconfinement phase transition at μ=0 are different. The difference is about 15 MeV in the two-flavor PNJL model. For the incoincidence of the two kinds of phase transition at zero quark chemical potential, lattice QCD still cannot provide strict limit on this aspect. For example, lattice QCD simulation shows the difference of the critical point of the two phase transition is 25 MeV in Ref. [8], 18 MeV in Ref. [9]. But in Ref. [6], the lattice calculation shows that the two phase transitions happen simultaneously. More lattice QCD results about the difference of the two phase transitions are compiled in Ref. [36]. Besides, Coleman and Witten conjectured that chiral and deconfinement transition should happen coincidently. In theory, this can be realized by introducing the entanglement interaction between chiral condensate and Polyakov loop, as shown in Ref. [23]. Relevant discussions can also be found in our previous study [41]. A further study involving the coincidence problem is also in progress in an improved three-flavor PNJL quark model.

4 Summary

We have studied the properties of quark matter in the improved PNJL model with the chemical potential dependent Polyakov loop effective potential which effectively reflects, to some degree, the quantum backreaction of matter sector to glue sector at none zero chemical potential. Compared with the original PNJL model, a superiority of the improved μPNJL model is that it can effectively describe the confinement-deconfinement transition at low T and high density region. And the QCD phase structure will be effectively modified if the strength of the quantum backreaction of matter sector to glue sector is strong. One evident variation is that the region of the so-called quarkyonic phase will be greatly reduced in the μPNJL model. It indicates that the modification to the Polyakov loop potential with the chemical potential dependence is a significant improvement in exploring the full QCD phase diagram.

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