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We start from the following Lagrangian density of the three-flavor pNJL model[50-52]
$$ \begin{split} {\cal{L}}_\mathrm{pNJL}^{} =\, & \bar{\psi}\left({\rm{i}}\gamma^\mu D_\mu^{}+\hat{\mu}\gamma^0-\hat{m}\right)\psi+ {\cal{L}}_\mathrm{S}^{} + {\cal{L}}_\mathrm{V}^{}+ \\ & {\cal{L}}_{\mathrm{KMT}}^{} + {\cal{L}}_\mathrm{IS}^{} + {\cal{L}}_\mathrm{IV}^{}-{\cal{U}}\left(\varPhi,\bar{\varPhi},T\right), \end{split} $$ (1) where
$$ {\cal{L}}_\mathrm{S}^{} = \frac{G_\mathrm{S}^{}}{2}\sum\limits_{a = 0}^8 \left[\left(\bar{\psi}\lambda^a\psi\right)^2 + \left(\bar{\psi}{\rm{i}} \gamma^5 \lambda^a \psi\right)^2\right], $$ (2) $$ {\cal{L}}_\mathrm{V}^{} = - \frac{G_\mathrm{V}^{}}{2}\sum\limits_{a = 0}^8 \left[\left(\bar{\psi}\gamma^{\mu}\lambda^a\psi\right)^2 + \left(\bar{\psi} \gamma^5 \gamma^{\mu} \lambda^a \psi\right)^2\right], $$ (3) $$ {\cal{L}}_{\mathrm{KMT}}^{} = -K\left[\mathrm{det}\bar{\psi}\left(1+\gamma^5\right)\psi + \mathrm{det}\bar{\psi}\left(1-\gamma^5\right)\psi\right], $$ (4) $$ {\cal{L}}_\mathrm{IS}^{} = G_\mathrm{IS}^{} \sum\limits_{a = 1}^3\left[\left(\bar{\psi}\lambda^a\psi\right)^2 + \left(\bar{\psi}{\rm{i}} \gamma^5 \lambda^a \psi\right)^2\right], $$ (5) $$ {\cal{L}}_\mathrm{IV}^{} = - G_\mathrm{IV}^{} \sum\limits_{a = 1}^3\left[\left(\bar{\psi}\gamma^{\mu}\lambda^a\psi\right)^2 + \left(\bar{\psi} \gamma^5 \gamma^{\mu} \lambda^a \psi\right)^2\right], $$ (6) are the scalar-isoscalar term, the vector-isoscalar term, the Kobayashi-Maskawa-t'Hooft (KMT) term, the scalar-isovector term, and the vector-isovector term, respectively. In the above,
$\psi = (u, d, s)^T$ represents the three-flavor quark fields with each flavor containing quark fields of three colors;$\hat{\mu} = \text{diag}(\mu_u^{}, \mu_d^{}, \mu_s^{})$ and$\hat{m} = \text{diag}(m_u^{},m_d^{},m_s^{})$ are the matrices of the chemical potential and the current quark mass for$ u $ ,$ d $ , and$ s $ quarks;$D_\mu^{} = \partial_\mu^{} - {\rm{i}} A_\mu^{}$ is the covariant derivative with$A_\mu^{} = \delta_{\mu}^0 A_0^{}$ , where$A_0^{} = {g} A_0^a \lambda^a/2 = -{\rm{i}}A_4^{}$ is the non-Abelian$ \mathrm{SU}(3) $ gauge field with the gauge coupling$ {g} $ conveniently absorbed in the definition of$ A_\mu^{} $ ;$ \lambda^a $ ($a = 1, \cdots ,8$ ) are the Gell-Mann matrices in$ \mathrm{SU}(3) $ flavor space with$\lambda^0 = \sqrt{2/3} {\mathbb{1}}_3^{}$ ;$ G_\mathrm{S}^{} $ and$ G_\mathrm{V}^{} $ are respectively the scalar-isoscalar and the vector-isoscalar coupling constant;$ G_\mathrm{IS}^{} $ and$ G_\mathrm{IV}^{} $ are respectively the scalar-isovector and the vector-isovector coupling constants. Since the Gell-Mann matrices with$a = 1,\, 2,\, 3$ are identical to the Pauli matrices in$ u $ and$ d $ space, the isovector couplings break the$ \mathrm{SU}(3) $ symmetry while keeping the isospin symmetry.$ K $ denotes the strength of the six-point KMT interaction[53] that breaks the axial$ U(1)_\mathrm{A}^{} $ symmetry, where ‘det’ denotes the determinant in flavor space. In the present study, we employ the parameters$m_u^{} = m_d^{} = 3.6$ MeV,$m_s^{} = 87$ MeV,$G_\mathrm{S}^{}\varLambda^2 = 3.6$ ,$K\varLambda^5 = 8.9$ , and the cutoff value in the momentum integral$\varLambda = 750$ MeV/c given in Refs. [19, 54-55]. In our previous study[31],$G_\mathrm{IS}^{} = -0.002 G_\mathrm{S}^{}$ and$G_\mathrm{IV}^{} = 0.25G_\mathrm{S}^{}$ are determined by fitting the physical pion mass$m_\pi^{} \approx 140.9$ MeV and the reduced isospin density from LQCD calculations at zero temperature[56], at which the pNJL model reduces to the NJL model. We set$G_\mathrm{V}^{} = 0$ and$\mu_s^{} = 0$ throughout the present study.We take the temperature-dependent effective potential
$ {\cal{U}}(\varPhi,\bar{\varPhi},T) $ from Ref. [21], i.e.,$$ \begin{split} {\cal{U}}(\varPhi,\bar{\varPhi},T) =\, & -b \cdot T\left\{54{\rm e}^{-a/T}\varPhi\bar{\varPhi} +\ln\left[1-6\varPhi\bar{\varPhi} - \right.\right. \\ &3\left.\left. \left(\varPhi\bar{\varPhi}\right)^2+4\left(\varPhi^3+\bar{\varPhi}^3\right)\right]\right\}. \end{split}$$ (7) The parameters
$a = 664$ MeV and$b = 0.028\varLambda^3$ are determined by the condition that the first-order phase transition in the pure gluodynamics takes place at$T = 270$ MeV[21], and the simultaneous crossover of the chiral restoration and the deconfinement phase transition occurs around$T \approx 212$ MeV. The Polyakov loop$ \varPhi $ and its (charge) conjugate$ \bar{\varPhi} $ are expressed as[48, 57]$$ \varPhi = \frac{1}{N_{\rm{c}}^{}}\mathrm{Tr}_{\rm{c}}^{} L,\; \; \; \; \; \; \bar{\varPhi} = \frac{1}{N_{\rm{c}}^{}}\mathrm{Tr}_{\rm{c}}^{} L^{\dagger}, $$ (8) where
$N_{\rm{c}}^{} = 3$ is the color degeneracy, and the matrix$ L $ in color space is explicitly given by$$ L({{\boldsymbol{x}}}) = {\cal{P}}\mathrm{exp}\left[{\rm{i}}\int\nolimits_0^{\beta}{\rm{d}}\tau A_4^{}(\tau, {{\boldsymbol{x}}}) \right] = \mathrm{exp}\left(\frac{{\rm{i}}A_4^{}}{T}\right), $$ (9) with
$ {\cal{P}} $ being the path ordering and$ \beta = 1/T $ being the inverse of temperature. The coupling between the Polyakov loop and quarks is uniquely determined by the covariant derivative$ D_\mu^{} $ in the pNJL Lagrangian [Eq. (1)][48]. The second equal sign in the above equation is valid by treating the temporal component of the Euclidean gauge field$ A_4^{} $ as a constant in the pNJL model. In this way, the Polyakov loop$ \varPhi $ and its conjugate$ \bar{\varPhi} $ can be treated as classical field variables.Based on the mean-field approximation, the Lagrangian density of the pNJL model can be written as
$$ {\cal{L}}_\mathrm{MF}^{} = \bar{\psi}{\cal{S}}^{-1}\psi -{\cal{V}}-{\cal{U}}(\varPhi,\bar{\varPhi},T), $$ (10) where
$$ {\cal{S}}^{-1} (p) = \left( {\begin{array}{*{20}{c}} {{\cal{S}}_{uu}^{ - 1}(p)}&{{\rm{i}}\varDelta {\gamma ^5}}&0\\ {{\rm{i}}\varDelta {\gamma ^5}}&{{\cal{S}}_{dd}^{ - 1}(p)}&0\\ 0&0&{{\cal{S}}_{ss}^{ - 1}(p)} \end{array}} \right) $$ (11) is the inverse of the quark propagator
$ {\cal{S}}(p) $ as a function of quark momentum$ p $ , with$$ \begin{array}{l} {\cal{S}}_{uu}^{-1} (p) = \gamma^\mu p_\mu^{} + \tilde{\mu}_\mathrm{u}^\star \gamma^0-M_u^{}, \\ {\cal{S}}_{dd}^{-1} (p) = \gamma^\mu p_\mu^{} +\tilde{\mu}_\mathrm{d}^\star \gamma^0 -M_d^{}, \\ {\cal{S}}_{ss}^{-1} (p) = \gamma^\mu p_\mu^{} + \tilde{\mu}_\mathrm{s}^\star \gamma^0 -M_s^{} \end{array} $$ being the inverse of the
$ u $ ,$ d $ , and$ s $ quark propagators, respectively,$$ \varDelta = \left(G_\mathrm{S}^{} + 2G_\mathrm{IS}^{} -K \sigma_s^{}\right)\pi $$ (12) being the gap parameter, and
$$\begin{split} {\cal{V}} =\, & G_\mathrm{S}^{} \left( \sigma_u^2 + \sigma_d^2 + \sigma_s^2 \right) + \frac{G_\mathrm{S}^{}}{2}\pi^2 + G_\mathrm{IS}^{} \left(\sigma_u^{}-\sigma_d^{}\right)^2 + \\ & G_\mathrm{IS}^{}\pi^2 - 4K \sigma_u^{} \sigma_d^{} \sigma_s^{} - K\sigma_s^{} \pi^2 - \\ & \frac{1}{3}G_\mathrm{V}^{} \left( \rho_u^{}+\rho_d^{}+\rho_s^{} \right)^2 -G_\mathrm{IV}^{}\left(\rho_{u}^{}-\rho_{d}^{}\right)^2 \end{split} $$ (13) being the condensation energy independent of the quark fields. In the above,
$ \rho_q^{} = \langle \bar{q} \gamma^0 q \rangle $ and$ \sigma_q^{} = \langle \bar{q} q \rangle $ are the net-quark density and the chiral condensate, respectively, with$ q = u,\,d,\,s $ being the quark flavor, and$\pi = \langle \bar{\psi} {\rm i} \gamma^5 \lambda^1 \psi \rangle$ is the pion condensate. The constituent mass of quarks can be expressed as$$ \begin{array}{l} M_u^{} = m_u^{}-2G_\mathrm{S}^{}\sigma_u^{} - 2G_\mathrm{IS}^{}(\sigma_u^{}-\sigma_d^{}) + 2K\sigma_d^{} \sigma_s^{} ,\\ M_d^{} = m_d^{}-2G_\mathrm{S}^{}\sigma_d^{} + 2G_\mathrm{IS}^{}(\sigma_u^{}-\sigma_d^{}) + 2K\sigma_u^{} \sigma_s^{} ,\\ M_s^{} = m_s^{}-2G_\mathrm{S}^{}\sigma_s^{} + 2K\sigma_u^{} \sigma_d^{} + \frac{K}{2} \pi^2. \end{array} $$ The effective chemical potentials for
$ u $ ,$ d $ , and$ s $ quarks in the propagator are defined as$$ \tilde{\mu}_\mathrm{u}^\star = \frac{\tilde{\mu}_\mathrm{B}^{}}{3}- {\rm{i}}A_4^{} + \frac{\tilde{\mu}_\mathrm{I}^{}}{2}, $$ (14) $$ \tilde{\mu}_\mathrm{d}^\star = \frac{\tilde{\mu}_\mathrm{B}^{}}{3}- {\rm{i}}A_4^{} - \frac{\tilde{\mu}_\mathrm{I}^{}}{2}, $$ (15) $$ \tilde{\mu}_\mathrm{s}^\star = \frac{\tilde{\mu}_\mathrm{B}^{}}{3}- {\rm{i}}A_4^{} - \tilde{\mu}_\mathrm{S}^{}, $$ (16) with the effective baryon, isospin, and strangeness chemical potentials expressed as
$$ \begin{array}{l} \tilde{\mu}_\mathrm{B}^{} = \mu_\mathrm{B}^{} - 2G_\mathrm{V}^{}\rho, \\ \tilde{\mu}_\mathrm{I}^{} = \mu_\mathrm{I}^{} - 4G_\mathrm{IV}^{}\left(\rho_u^{}-\rho_d^{}\right) , \\ \tilde{\mu}_\mathrm{S}^{} = \mu_\mathrm{S}^{}, \end{array}$$ (17) and
$$ \begin{split} &\mu_\mathrm{B}^{} = \frac{3(\mu_u^{} + \mu_d^{})}{2} , \\ &\mu_\mathrm{I}^{} = \mu_u^{} - \mu_d^{}, \\ & \mu_\mathrm{S}^{} = \frac{\mu_u^{} + \mu_d^{}}{2} - \mu_s^{} \end{split}$$ (18) are the real baryon, isospin, and strangeness chemical potentials.
The thermodynamic potential of the quark system can be obtained through
$$ \varOmega = -T\sum\limits_n^{}\int \frac{{\rm{d}}^3 p}{\left(2\pi\right)^3} \text{Tr ln } {\cal{S}}({\rm{i}}\omega_n^{}, {{\boldsymbol{p}}})^{-1} +{\cal{V}}+{\cal{U}}\left(\varPhi,\bar{\varPhi},T\right). \\ $$ (19) In the above, the four-momentum
$p = \left(p_0^{}, {{\boldsymbol{p}}}\right)$ becomes$p = \left({\rm{i}}\omega_n^{},{{\boldsymbol{p}}}\right)$ with$\omega_n^{} = (2n+1)\pi T$ being the Matsubara frequency for a Fermi system. In order to evaluate$ \varOmega $ for each momentum$ p $ numerically, we need to find the zeros of$ {\cal{S}}^{-1} (p) $ . Similar to the method in Refs. [58-60], it can be proved that the eigenvalues$ \lambda_k^{} $ $ (k = 1,\,2,\,3,\,4) $ of the following “Dirac Hamiltonian density”$$ {\cal{H}} (\boldsymbol{p}) = - \left( {\begin{array}{*{20}{c}} {\dfrac{{\tilde \mu _{\rm{I}}^{}}}{2} - M_u^{}}&{|\boldsymbol p|}&0&{ - \varDelta }\\ {|\boldsymbol p|}&{\dfrac{{\tilde \mu _{\rm{I}}^{}}}{2} + M_u^{}}&\varDelta &0\\ 0&\varDelta &{ - \dfrac{{\tilde \mu _{\rm{I}}^{}}}{2} - M_d^{}}&{|\boldsymbol p|}\\ { - \varDelta }&0&{|\boldsymbol p|}&{ - \dfrac{{\tilde \mu _{\rm{I}}^{}}}{2} + M_d^{}} \end{array}} \right) $$ are zeros of
$ {\cal{S}}^{-1} (p) $ . Using the relationship$\mathrm{Trln} = \mathrm{lnDet}$ , one can get the following expression of the thermodynamic potential$$ \begin{split} \varOmega =\, & \varOmega^+(\lambda_1'^{})+\varOmega^+(\lambda_2'^{})+\varOmega^-(-\lambda_3'^{})+\varOmega^-(-\lambda_4'^{}) +\\ & \varOmega^+\left(E_s^-\right)+\varOmega^-\left(E_s^+\right)+{\cal{V}}+{\cal{U}}\left(\varPhi,\bar{\varPhi},T\right) \end{split} $$ (20) with
$$ \varOmega^{\pm}(\lambda) = -2N_{\rm{c}}^{}\,\int\nolimits_0^{\varLambda} \frac{{\rm{d}}^3 p}{(2\pi)^3}\frac{\lambda}{2} -2T\int\nolimits_0^{\varLambda} \frac{{\rm{d}}^3 p}{(2\pi)^3}Z^{\pm}(-\lambda), \\ $$ (21) where the integrands in the second integral are
$$ \begin{array}{l} Z^-(\lambda) = \mathrm{Tr}_{\rm{c}}^{}\text{ln}\left( 1+L \xi_{\lambda}^{}\right) = \text{ln} \left\{ 1+N_{\rm{c}}^{}\varPhi \xi_{\lambda}^{}+N_{\rm{c}}^{}\bar{\varPhi}\xi_{\lambda}^2 + \xi_{\lambda}^3 \right\}, \\ Z^+(\lambda) = \mathrm{Tr}_{\rm{c}}^{}\text{ln}\left( 1+L^\dagger \xi_{\lambda}^{}\right) = \text{ln} \left\{ 1+N_{\rm{c}}^{}\bar{\varPhi} \xi_{\lambda}^{}+N_{\rm{c}}^{}{\varPhi}\xi_{\lambda}^2 + \xi_{\lambda}^3 \right\}, \end{array} $$ with
$ \xi_{\lambda}^{} = e^{\beta \lambda} $ . In Eq. (20),$ \lambda_k'^{} $ and$ E_s^{\pm} $ are defined respectively as$\lambda_k'^{} = \lambda_k^{}-\frac{\tilde{\mu}_\mathrm{B}^{}}{3}$ and$ E_s^{\pm} = E_s^{} \pm \tilde{\mu}_s^{} $ , with$ E_s^{} = \sqrt{M_s^2 + \boldsymbol{p}^2} $ being the single$ s $ quark energy. Throughout this paper,$ \mathrm{Tr} $ and$ \mathrm{Det} $ represent respectively the trace and determinant over Dirac, flavor, and color space, while$ \mathrm{Tr}_c^{} $ and$ \mathrm{Det}_c^{} $ represent those only taken over color space. It should be pointed out that we introduce a momentum cutoff in the two integrals in Eq. (21) as in Ref. [51], otherwise the integrals will be divergent at large baryon and isospin chemical potentials. This is, however, slightly different from our previous studies[61-63].By taking the trace of the corresponding component of the propagator[35], the chiral condensates
$ \sigma_q^{} $ , the net-quark densities$ \rho_q^{} $ , and the pion condensate$ \pi $ can be expressed as$$ \sigma_u^{} = 4\, N_{\rm{c}}^{}\sum\limits_{k = 1}^4\int \frac{{\rm{d}}^3 p}{(2\pi)^3}g_{\sigma u}^{}\left(\lambda_k^{}\right) \left[-\frac{1}{2}+F^+\left(\lambda_k'^{}\right)\right], $$ (22) $$ \sigma_d^{} = 4\, N_{\rm{c}}^{}\sum\limits_{k = 1}^4\int \frac{{\rm{d}}^3 p}{(2\pi)^3}g_{\sigma d}^{}\left(\lambda_k^{}\right) \left[-\frac{1}{2}+F^+\left(\lambda_k'^{}\right)\right], $$ (23) $$ \sigma_{s}^{} = 2\,N_{\rm{c}}^{}\int \frac{{\rm{d}}^3 p}{(2\pi)^3}\frac{M_s^{}}{E_s^{}} \left[F^+\left(E_s^-\right)+F^-\left(E_s^+\right)-1\right], $$ (24) $$ \rho_u^{} = 4\, N_{\rm{c}}^{}\sum\limits_{k = 1}^4\int \frac{{\rm{d}}^3 p}{(2\pi)^3}g_{\rho u}^{}\left(\lambda_k^{}\right) \left[-\frac{1}{2}+F^+\left(\lambda_k'^{}\right)\right], $$ (25) $$ \rho_{{d}}^{} = 4\, N_{\rm{c}}^{}\sum\limits_{k = 1}^4\int \frac{{\rm{d}}^3 p}{(2\pi)^3}g_{\rho d}^{}\left(\lambda_k^{}\right) \left[-\frac{1}{2}+F^+\left(\lambda_k'^{}\right)\right], $$ (26) $$ \rho_{s}^{} = 2\,N_{\rm{c}}^{}\int \frac{{\rm{d}}^3 p}{(2\pi)^3}\left[F^+\left(E_s^-\right)-F^-\left(E_s^+\right)\right], $$ (27) $$ \pi = 4\,N_{\rm{c}}^{} \sum\limits_{k = 1}^4\int \frac{{\rm{d}}^3 p}{(2\pi)^3}g_{\pi}^{}\left(\lambda_k^{}\right) \left[-\frac{1}{2}+F^+\left(\lambda_k'^{}\right)\right], $$ (28) where the
$ g $ functions have the same form as those in Ref. [31], since the$ g $ functions are actually independent of$ \tilde{\mu}_\mathrm{B}^{} $ , and the$ {\rm{i}}A_4^{} $ terms are always combined with$ {\tilde{\mu}_\mathrm{B}^{}}/{3} $ in the quark propagator. In the above,$ F^+(\lambda) $ and$ F^-(\lambda) $ are, respectively, the effective phase-space distribution for quarks and antiquarks, and they are expressed as$$\begin{array}{l} F^+(\lambda) = \dfrac{1}{N_{\rm{c}}^{}}\mathrm{Tr}_c^{}\left( \dfrac{1}{1+L \xi_{\lambda}^{}}\right) = \dfrac{1+2\varPhi \xi_{\lambda}^{}+ \bar{\varPhi}\xi_{\lambda}^2} { 1+N_c^{}\varPhi \xi_{\lambda}^{}+N_{\rm{c}}^{}\bar{\varPhi}\xi_{\lambda}^2 + \xi_{\lambda}^3}, \\ F^-(\lambda) = \dfrac{1}{N_{\rm{c}}^{}}\mathrm{Tr}_{\rm{c}}^{}\left( \dfrac{1}{1+L^\dagger \xi_{\lambda}^{}}\right) = \dfrac{1+2\bar{\varPhi} \xi_{\lambda}^{}+ \varPhi \xi_{\lambda}^2} { 1+N_c^{}\bar{\varPhi} \xi_{\lambda}^{}+N_c^{}\varPhi \xi_{\lambda}^2 + \xi_{\lambda}^3}. \end{array} $$ It is seen that the above distributions reduce to the normal Fermi-Dirac form at high temperatures when the Polyakov loops are approaching 1, while they become the Fermi-Dirac form with a reduced temperature of
$ T/3 $ at low temperatures when the Polyakov loops are almost zero. This leads to a CEP at a higher temperature in the pNJL model than in the NJL model. Eqs. (22)~(28) can be also obtained equivalently from$$ \frac{\partial \varOmega}{\partial \sigma_q^{}} = \frac{\partial \varOmega}{\partial \rho_q^{}} = \frac{\partial \varOmega}{\partial \pi} = 0, $$ (29) with
$ q = u,d,s $ being the quark flavor, leading to the relations$$ \sigma_q^{} = \frac{\partial \varOmega}{\partial M_q^{}},\; \rho_q^{} = -\frac{\partial \varOmega}{\partial \mu_q^{}},\; \pi = -\frac{\partial \varOmega}{\partial \varDelta}. $$ (30) The values of
$ \varPhi $ and$ \bar{\varPhi} $ can be similarly determined by minimizing the grand potential with respect to the Polyakov loops, i.e.,$$ \frac{\partial \varOmega}{\partial \varPhi} = \frac{\partial \varOmega}{\partial \bar{\varPhi}} = 0. $$ (31)
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摘要: 基于三味Polyakov-looped Nambu−Jona-Lasinio(pNJL)模型,通过研究手征凝聚、pion凝聚和Polyakov圈之间的相互作用,研究了温度、重子化学势和同位旋化学势依赖的三维QCD相图结构。虽然pNJL模型得到的正常夸克物质相、pion超流相和Sarma相的结构以及相边界与NJL模型定性上相似,但Polyakov 圈的引入大大扩展了pion超流相和Sarma相的存在区域,并导致临界点出现在较高的温度。由于有效地引入了胶子动力学的贡献,与 NJL模型相比,该研究有望对三维QCD相图给出更可靠的预测。Abstract: Based on the three-flavor Polyakov-looped Nambu−Jona-Lasinio(pNJL) model, we have studied the structure of the three-dimensional QCD phase diagram with respect to the temperature, the baryon chemical potential, and the isospin chemical potential, by investigating the interplay among the chiral quark condensate, the pion condensate, and the Polyakov loop. While the pNJL model leads to qualitatively similar structure of the normal quark phase, the pion superfluid phase, and the Sarma phase as well as their phase boundaries, when compared to the NJL model, the inclusion of the Polyakov loop enlarges considerably the areas of the pion superfluid phase and the Sarma phase, and leads to critical end points at higher temperatures. With the contribution of the gluon dynamics effectively included, the present study is expected to give a more reliable prediction of the three-dimensional QCD phase diagram compared to that in the NJL model.
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Key words:
- Nambu-Jona-Lasinio /
- QCD phase diagram /
- pion condensate
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Figure 1. Reduced pion condensate
$\pi/2\sigma_0^{}$ , Sarma phase solution$\pi_2^{}/2\sigma_0^{}$ , and chiral condensate$\sigma_u^{}/\sigma_0^{}$ as well as the Polyakov loop$\varPhi$ as a function of the isospin chemical potential$\mu_\mathrm{I}^{}$ in hot [$T = 50$ (left) and$100$ (right) MeV] and baryon-rich [$\mu_\mathrm{B}^{} = 400$ (a),(e), 500 (b),(f), 800 (c),(g), and 1000 (d),(h) MeV] quark matter. Results are compared with those obtained from the NJL model. (color online)Figure 2. Reduced pion condensate
$\pi/2\sigma_0^{}$ , Sarma phase solution$\pi_2^{}/2\sigma_0^{}$ , and chiral condensate$\sigma_u^{}/\sigma_0^{}$ as well as the Polyakov loop$\varPhi$ as a function of the temperature$T$ in quark matter of different baryon chemical potentials$\mu_\mathrm{B}^{}$ and isospin chemical potentials$\mu_\mathrm{I}^{}$ . Results are compared with those obtained from the NJL model. (color online)Figure 3. Phase diagrams in the
$T-\mu_\mathrm{B}^{}$ plane at different isospin chemical potentials$\mu_\mathrm{I}^{} = 200$ (a), 400 (b), 600 (c), and 800 (d) MeV in the pNJL model compared with those in NJL model. Solid lines represent the first-order phase transition (PT) between Phase I and Phase III, dashed lines represent the second-order phase transition between Phase I and Phase II, and dash-dotted lines represent the second-order phase transition between Phase II and Phase III. Blue solid (dotted) lines represent the deconfinement phase transition with (without) the pion condensate. (color online)Figure 4. Similar to Fig. 3 but in the
$T-\mu_\mathrm{I}^{}$ plane at different baryon chemical potentials$\mu_\mathrm{B}^{} = 400$ (a), 600 (b), 800 (c), and 1 000 (d) MeV. (color online)Figure 5. Similar to Fig. 3 but in the
$\mu_\mathrm{B}^{}-\mu_\mathrm{I}^{}$ plane at different temperatures$T = 50$ (a), 100 (b), 150 (c), and 200 (d) MeV. (color online) -
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