Polyakov-loop Extended Nambu-Jona-Lasinio Model Research Articles

Effects of tensor spin polarization on the chiral restoration and deconfinement phase transitions

Effects of tensor spin polarization (TSP) on the chiral restoration and deconfinement phase transitions are studied in Polyakov loop extended Nambu–Jona-Lasinio (PNJL) model. For chiral phase transition, the higher the polarized degree of quark-antiquark pairs under the strong magnetic field, the higher the phase transition temperature. The TSP corrects the position of the critical end point. The small impact of TSP on the phase transition temperature is found for the deconfinement phase transition. On the other hand, we divide the phase space into three ranges based on the phase diagram obtained from the PNJL model: the confinement phase with chiral symmetry broken, the deconfinement phase with restored chiral symmetry, and the confinement phase with restored chiral symmetry (quarkyonic phase). It is found that TSP has only a very small effect on the anisotropic pressure in the deconfined phase with chiral symmetry restored and the quarkyonic phase, but it has a very strong effect on the anisotropic pressure in the confined phase with chiral symmetry broken. This is because TSP is closely related to chiral symmetry. The restoration of chiral symmetry means the dissociation of spin polarization condensate. Published by the American Physical Society 2024

Dilepton production from magnetized quark matter with an anomalous magnetic moment of the quarks using a three-flavor PNJL model

Dilepton production from hot, dense and magnetized quark matter is studied using the three-flavor Polyakov loop extended Nambu--Jona-Lasinio (PNJL) model in which the anomalous magnetic moment (AMM) of the quarks is also taken into consideration. This is done by first evaluating the thermo-magnetic spectral function of the vector current correlator employing the real time formalism of finite temperature field theory and the Schwinger proper time formalism. The constituent quark mass which goes as an input in the expression of the dilepton production rate (DPR), has been calculated using the three-flavor PNJL model employing Pauli-Villiars (PV) regularization. The obtained constituent quark mass being strongly dependent on the temperature, density, magnetic field and AMM of the quarks, captures the effect of `strong' interactions specifically around the (pseudo) chiral and confinement-deconfinement phase transition regions. The analytic structure of the spectral function in the complex energy plane has been analyzed in detail and a non-trivial Landau cut is found in the physical kinematic domains resulting from the scattering of the Landau quantized quark/antiquark with the photon which is purely a finite magnetic field effect. Due to the emergence of the Landau cut along with the usual unitary cut, the DPR is found to be largely enhanced in the low invariant mass region. Owing to the magnetic field and AMM dependence of the thresholds of these cuts, we find that the kinematically forbidden gap between the Unitary and Landau cuts vanishes at sufficiently high temperature, density and magnetic field leading to the generation of a continuous spectrum of dilepton emission over the whole invariant mass region. In order to see the effects of strangeness and confinement-deconfinement, the rates are compared with the three-flavor NJL and the two-flavor NJL and PNJL models.

Magnetized baryons and the QCD phase diagram: NJL model meets the lattice

We determine the baryon spectrum of 1 + 1 + 1-flavor QCD in the presence of strong background magnetic fields using lattice simulations at physical quark masses for the first time. Our results show a splitting within multiplets according to the electric charge of the baryons and reveal, in particular, a reduction of the nucleon masses for strong magnetic fields. This first-principles input is used to define constituent quark masses and is employed to set the free parameters of the Polyakov loop-extended Nambu-Jona-Lasinio (PNJL) model in a magnetic field-dependent manner. The so constructed model is shown to exhibit inverse magnetic catalysis at high temperatures and a reduction of the transition temperature as the magnetic field grows — in line with non-perturbative lattice results. This is contrary to the naive variant of this model, which gives incorrect results for this fundamental phase diagram. Our findings demonstrate that the magnetic field dependence of the PNJL model can be reconciled with the lattice findings in a systematic way, employing solely zero-temperature first-principles input.

Strange matter and kaon to pion ratio in the SU(3) Polyakov–Nambu–Jona-Lasinio model

The behavior of strange matter in the frame of the SU(3) Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model including the ${U}_{A}(1)$ anomaly is considered. We discuss the appearance of a peak in the ratio of the number of strange mesons to nonstrange mesons known as the ``horn.'' The PNJL model gives a schematic description of the chiral phase transition and meson properties at finite temperature and density. By using the model, we can show that the splitting of kaon and antikaon masses appears as a result of the introduction of density. This may explain the difference in the ${K}^{+}/{\ensuremath{\pi}}^{+}$ ratio and the ${K}^{\ensuremath{-}}/{\ensuremath{\pi}}^{\ensuremath{-}}$ ratio at low $\sqrt{{s}_{NN}}$ and their tendency to the same value at high $\sqrt{{s}_{NN}}$. We also show that the rise in the ratio ${K}^{+}/{\ensuremath{\pi}}^{+}$ appears near the critical endpoint when we build the ${K}^{+}/{\ensuremath{\pi}}^{+}$ ratio along the phase-transition diagram and it can be considered as a critical region signal.

Discussion of thermodynamic features within the PNJL model**Supported by National Natural Science Foundation of China (11475085, 11535005, 11690030, 11805097), National Major state Basic Research and Development of China (2016YFE0129300) and Jiangsu Provincial Natural Science Foundation of China (BK20180323)

We discuss some thermodynamical features of a QCD system within the two-flavor Polyakov loop extended Nambu–Jona-Lasinio (PNJL) model. Several thermodynamical quantities of interest (pressure, energy density, specific heat, speed of sound, etc.) are investigated and discussed in detail with two different forms of Polyakov loop potential. The effective coupling strength G incorporating a quark feedback (quark condensate) through operator product expansion is also discussed, as well as the relationship between color deconfinement and chiral phase crossover. We find that some thermodynamical quantities have quite different behavior for different Polyakov loop potentials. By changing the characteristic temperature T0 of the pure Yang-Mills field, we find that when T0 becomes small, color deconfinement might happen earlier than chiral phase crossover, while their relationship can be determined via some thermodynamical quantities. Furthermore, the behavior of the thermodynamical quantities is quite different in the two different forms of Polyakov loop potential studied. Especially, one of the potentials, specific heat, has two peaks, which correspond to color deconfinement and chiral phase crossover respectively. This interesting phenomenon may shed some light on whether the inflection points of the chiral condensate and deconfinement transitions happen at the same temperature or not for lattice QCD and experimental studies.

Strangeness Fluctuation Using PNJL Model

The properties of net kaon fluctuations in nuclear matter has been studied within Polyakov loop extended Nambu–Jona-Lasinio model. The ratio of fourth order moment to second order moment (kurtosis) and the third order moment to the second order moment (skewness) of strangeness fluctuations have been calculated and compared with the recent experimental results.

Baryon number fluctuations and the phase structure in the PNJL model

We investigate the kurtosis and skewness of net-baryon number fluctuations in the Polyakov loop extended Nambu–Jona-Lasinio (PNJL) model, and discuss the relations between fluctuation distributions and the phase structure of quark-gluon matter. The calculation shows that the traces of chiral and deconfinement transitions can be effectively reflected by the kurtosis and skewness of net-baryon number fluctuations not only in the critical region but also in the crossover region. The contour plot of baryon number kurtosis derived in the PNJL model can qualitatively explain the behavior of net-proton number kurtosis in the STAR beam energy scan experiments. Moreover, the three-dimensional presentations of the kurtosis and skewness in this study are helpful to understand the relations between baryon number fluctuations and QCD phase structure.

Properties of 2+1 -flavor QCD in the imaginary chemical potential region: A model approach

We study properties of 2+1-flavor QCD in the imaginary chemical potential region by using two approaches. One is a theoretical approach based on QCD partition function, and the other is a qualitative one based on the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. In the theoretical approach, we clarify conditions imposed on the imaginary chemical potentials $\mu_{f}=i\theta_{f}T$ to realize the Roberge-Weiss (RW) periodicity. We also show that the RW periodicity is broken if anyone of $\theta_{f}$ is fixed to a constant value. In order to visualize the condition, we use the PNJL model as a model possessing the RW periodicity, and draw the phase diagram as a function of $\theta_{u}=\theta_{d}\equiv \theta_{l}$ for two conditions of $\theta_{s}=\theta_{l}$ and $\theta_{s}=0$. We also consider two cases, $(\mu_{u},\mu_{d},\mu_{s}) =(i\theta_{u}T,iC_{1}T,0)$ and $(\mu_{u},\mu_{d},\mu_{s})=(iC_{2}T,iC_{2}T,i\theta_{s}T)$; here $C_{1}$ and $C_{2}$ are dimensionless constants, whereas $\theta_{u}$ and $\theta_{s}$ are treated as variables. For some choice of $C_{1}$ ($C_{2}$), the number density of up (strange) quark becomes smooth in the entire region of $\theta_{u}$ ($\theta_{s}$) even in high $T$ region.

Finite-volume effects on phase transition in the Polyakov-loop extended Nambu–Jona-Lasinio model with a chiral chemical potential

To investigate the finite-volume effects on the chiral symmetry restoration and the deconfinement transition for a quantum chromodynamics (QCD) system with [Formula: see text] (two quark flavors), we apply the Polyakov-loop extended Nambu–Jona-Lasinio model by introducing a chiral chemical potential [Formula: see text] artificially. The final numerical results indicate that the introduced chiral chemical potential does not change the critical exponents, but shifts the location of critical end point (CEP) significantly; the ratios for the chiral chemical potentials and temperatures at CEP, [Formula: see text] and [Formula: see text], are significantly affected by the system size [Formula: see text]. The behavior is that [Formula: see text] increases slowly with [Formula: see text] when [Formula: see text] is “large” and [Formula: see text] decreases first and then increases with [Formula: see text] when [Formula: see text] is “small.” It is also found that for a fixed [Formula: see text], there is a [Formula: see text], where the critical end point vanishes and the whole phase diagram becomes a crossover when [Formula: see text]. Therefore, we suggest that for the heavy-ion collision experiments, which is to study the possible location of CEP, the finite-volume behavior should be taken into account.

Phase structure of two-color QCD at real and imaginary chemical potentials: Lattice simulations and model analyses

We investigate the phase structure of two-color QCD at both real and imaginary chemical potentials mu, performing lattice simulations and analyzing the data with the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. Lattice QCD simulations are done on an 8^3 times 4 lattice with the clover-improved two-flavor Wilson fermion action and the renormalization-group improved Iwasaki gauge action. We test the analytic continuation of physical quantities from imaginary mu to real mu by comparing lattice QCD results calculated at real mu with the result of analytic function the coefficients of which are determined from lattice QCD results at imaginary mu. We also test the validity of the PNJL model by comparing model results with lattice QCD ones. The PNJL model is good in the deconfinement region, but less accurate in the transition and confinement regions. This problem is improved by introducing the baryon degree of freedom to the model. It is also found that the vector-type four-quark interaction is necessary to explain lattice data on the quark number density.

Effective model approach to meson screening masses at finite temperature

Temperature dependence of pion and sigma-meson screening masses is evaluated by the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model with the entanglement vertex. We propose a practical way of calculating meson screening masses in the NJL-type effective models. The method based on the Pauli-Villars regularization solves the well-known difficulty that the evaluation of screening masses is not easy in the NJL-type effective models. The PNJL model with the entanglement vertex and the Pauli-Villars regularization well reproduces lattice QCD results on temperature dependence of the chiral condensate and the Polyakov loop. The method is applied to analyze temperature dependence of pion screening masses calculated with state-of-the-art lattice simulations with success in reproducing the lattice QCD results.

The Wigner solution and QCD phase transitions in a modified PNJL model

By employing some modification to the widely used two-flavor Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model, we discuss the Wigner solution of the quark gap equation at finite temperature and zero quark chemical potential beyond the chiral limit, and then try to explore its influences on the chiral and deconfinement phase transitions of QCD at finite temperature and zero chemical potential. The discovery of the coexistence of the Nambu and the Wigner solutions of the quark gap equation with nonzero current quark mass at zero temperature and zero chemical potential, as well as their evolutions with temperature is very interesting for the studies of the phase transitions of QCD. According to our results, the chiral phase transition might be of first order (while the deconfinement phase transition is still a crossover, as in the normal PNJL model), and the corresponding phase transition temperature is lower than that of the deconfinement phase transition, instead of coinciding with each other, which are not the same as the conclusions obtained from the normal PNJL model. In addition, we also discuss the sensibility of our final results on the choice of model parameters.

Differences and similarities between fundamental and adjoint matters inSU(N)gauge theories

We investigate differences and similarities between fundamental fermions and adjoint fermions in SU(N) gauge theories. The gauge theory with fundamental fermions possesses ZN symmetry only in the limit of infinite fermion mass, whereas the gauge theory with adjoint fermions does have the symmetry for any fermion mass. The flavor-dependent twisted boundary condition (FTBC) is then imposed on fundamental fermions so that the theory with fundamental fermions can possess ZN symmetry for any fermion mass. We show similarities between FTBC fundamental fermions and adjoint fermions, using the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. In the mean-field level, the PNJL model with FTBC fundamental fermions has dynamics similar to the PNJL model with adjoint fermions for the confinement/deconfinement transition related to ZN symmetry. The chiral property is somewhat different between the two models, but there is a simple relation between chiral condensates in the two models. As an interesting high-energy phenomenon, a possibility of the gauge symmetry breaking is studied for FTBC fundamental fermions.

Two-color QCD at imaginary chemical potential and its impact on real chemical potential

We study properties of two-color QCD at imaginary chemical potential ($\mu$) from the viewpoint of the Roberge-Weiss (RW) periodicity, the charge conjugation and the pseudo-reality. At $\mu=\pm i\pi T/2$, where $T$ is temperature, the system is symmetric under the combination of the charge conjugation ${\cal C}$ and the ${\mathbb Z}_{2}$ transformation. The symmetry, called ${\cal C} {\mathbb Z}_{2}$ symmetry, is preserved at lower $T$ but broken at higher $T$. The Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model has the same properties as two-color QCD for ${\cal C} {\mathbb Z}_{2}$ symmetry and the pseudo-reality. The nontrivial correlation between the chiral restoration and the deconfinement are investigated by introducing the entanglement vertex in the PNJL model. The order of ${\cal C} {\mathbb Z}_{2}$ symmetry breaking at the RW endpoint is second-order when the correlation is weak, but becomes first-order when the correlation is strong. We also investigate the impact of the correlation on the phase diagram at real $\mu$.

The quarkyonic phase and the [formula omitted] symmetry

We investigate the interplay between the ZNc symmetry and the emergence of the quarkyonic phase, adding the flavor-dependent complex chemical potentials μf=μ+iTθf with (θf)=(0,θ,−θ) to the Polyakov-loop extended Nambu–Jona-Lasinio (PNJL) model. When θ=0, the PNJL model with the μf agrees with the standard PNJL model with the real chemical potential μ. When θ=2π/3, meanwhile, the PNJL model with the μf has the ZNc symmetry exactly for any real μ, so that the quarkyonic phase exists at small T and large μ. Once θ varies from 2π/3, the quarkyonic phase exists only on a line of T=0 and μ larger than the dynamical quark mass, and the region at small T and large μ is dominated by the quarkyonic-like phase in which the Polyakov loop is small but finite.

Quark–gluon thermodynamics with the symmetry

We investigate quark–gluon thermodynamics with the symmetry. The flavor-dependent twist boundary condition (TBC) is imposed on Nc degenerate flavor quarks in the SU(Nc) gauge theory. This model is useful to understand the mechanism of color confinement. Thermodynamics of this quark–gluon system is studied by imposing the TBC on the Polyakov-loop extended Nambu–Jona–Lasinio (PNJL) model. The TBC model is applied to two- and three-color cases. The symmetry is preserved below some temperature Tc, but spontaneously broken above Tc. The color confinement below Tc preserves the flavor symmetry. Above Tc, the flavor symmetry is broken, but the breaking is suppressed by the entanglement between the Polyakov loop and the chiral condensate. Particularly at low temperature, dynamics of the TBC model is similar to that of the PNJL model with the standard fermion boundary condition, indicating that the symmetry is a good approximate concept in the latter model even if the current quark mass is small. The present prediction can be tested in future by lattice QCD, since the quark–gluon dynamics with a flavor-dependent imaginary chemical potential has no sign problem.

Equation of state in the PNJL model with the entanglement interaction

The equation of state and the phase diagram in two-flavor quantum chromodynamics (QCD) are investigated by the Polyakov-loop extended Nambu–Jona-Lasinio (PNJL) model with an entanglement vertex between the chiral condensate and the Polyakov loop. The entanglement-PNJL (EPNJL) model reproduces lattice QCD (LQCD) data at μ ⩾ 0 better than the PNJL model. Hadronic degrees of freedom are taken into account by the free-hadron-gas approximation with the volume-exclusion effect due to the hadron generation. The hadron degrees of freedom improve agreement of the EPNJL model with LQCD data particularly at low temperature. The quarkyonic phase survives, even if the correlation between the chiral condensate and the Polyakov loop is strong and hadron degrees of freedom are taken into account. However, the location of the quarkyonic phase is sensitive to the strength of the volume exclusion.

Theta vacuum effects on QCD phase diagram

Theta vacuum effects on the QCD phase structure in the μ–T plane are studied by using the Polyakov-loop extended Nambu–Jona-Lasinio model and its extension, where μ is the quark chemical potential and T is temperature, respectively. As the parameter θ of the theta vacuum increases, the chiral transition becomes stronger. For large θ, it eventually becomes first order even at zero μ.

Entanglement between chiral and deconfinement transitions under strong uniform magnetic background field

Effects of a uniform (electro-) magnetic background field are investigated in the two-flavor nonlocal Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. Temperature-dependences of the chiral order-parameter and the Polyakov-loop are investigated under the magnetic field. In the nonlocal PNJL model with four-dimensional momentum dependent distribution function, pseudo-critical temperatures of the chiral and deconfinement transitions are coincident each other even if the magnetic field is strong. It means that we can check the reliability of the distribution function under the strong magnetic field by comparing with lattice QCD data, particularly from the viewpoint of the entanglement between the chiral and deconfinement transitions.

Extensions and further applications of the nonlocal Polyakov–Nambu–Jona-Lasinio model

The nonlocal Polyakov-loop-extended Nambu--Jona-Lasinio (PNJL) model is further improved by including momentum-dependent wave-function renormalization in the quark quasiparticle propagator. Both two- and three-flavor versions of this improved PNJL model are discussed, the latter with inclusion of the (nonlocal) 't Hooft-Kobayashi-Maskawa determinant interaction in order to account for the axial U(1) anomaly. Thermodynamics and phases are investigated and compared with recent lattice-QCD results.