Quark–hadron phase transition in an extended Nambu–Jona-Lasinio model with scalar–vector interaction: Finite temperature and baryon chemical potential case

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The quark–hadron phase transition at ﬁnite temperature and baryon chemical potential is investigated in an extended Nambu–Jona-Lasinio (NJL) model with scalar–vector eight-point interaction by comparing the pressure of symmetric nuclear matter with that of the quark matter. As a result, the extended NJL phase diagram is obtained in the temperature–baryon chemical potential plane and the effects of the scalar–vector coupling constant G qs v on the chiral phase transition are summarized. In our investigation, the quark–hadron phase transition occurs after the chiral phase transition in the nuclear matter. It is shown that a quarkyonic-like phase, in which the chiral symmetry is restored but the elementary excitation modes are nucleonic, appears just before deconﬁnement in this model. . . . .


Brief recapitulation of the extended NJL model for nuclear and quark matters at finite temperature and density
In this section, following Ref. [18], the same NJL-type models including the scalar-vector eightpoint interaction are given for nuclear and quark matters at finite temperature and density, the model parameters and the number of colors being different.

Lagrangian density, gap equation, and pressure
Let us start with the following Lagrangian density for nuclear and quark matters: where the subscript/superscript i denotes nothing but an index that represents the case of nuclear matter (i = N ) or quark matter (i = q). Here, ψ i represents the fermion field, i.e., ψ N is the nucleon field and ψ q is the quark field, and τ are the Pauli matrices in isospin space. The first two terms are the original NJL model Lagrangian density. The third term is a vector-vector repulsive term with G i v . The last term is a scalar-vector and isoscalar-vector coupling term with G i sv . Parameters G i v and G i sv represent a coupling constant of four-point vector-vector interaction and that of eight-point scalar-vector interaction, respectively. As is well known, pure NJL interaction alone is not enough to reproduce the property of nuclear saturation in the original NJL model. Thus, we introduce the last two terms in Eq. (2.1) so as to reproduce the nuclear matter saturation properties [8]. Then, in this paper, this model is called an extended NJL model. This model is nonrenormalizable, so that we adopt a three-momentum cutoff scheme in which the cutoff parameter i is introduced.
Under the mean field approximation 1 , the Lagrangian density L M F i and the Hamiltonian density H M F i are obtained as where 3) for nuclear matter (i = N ) and quark matter (i = q), respectively. Here, the symbol · · · denotes the finite-temperature expectation value that represents the thermal average. In addition, we introduce the chemical potential μ i to deal with a finite density system: 5) 1 We make a replacement such as ψ ψ → ψ ψ + (ψ ψ − ψ ψ ), where ψ ψ are bilinear quantities in the fermion fields and are matrices in Dirac, flavor, and color space. The symbol · · · denotes the expectation value at finite temperature or thermal average. Here, the fluctuation, ψ ψ − ψ ψ , is linearized. Namely, we consider two nonvanishing terms: where μ r i is the effective chemical potential: Here, the expectation values at finite temperature is given as where ν i is the degeneracy factor, in which N i c and N i f represent the numbers of color and flavor, and n i ± are the fermion number distribution functions with β = 1/T , in which T is temperature. Here, we have eliminated the contribution of the occupied negative energy states from the nucleon and/or quark number density itself in Eq. (2.8). As a result, we obtain a self-consistent set of equations, Eqs. (2.3) and (2.6)-(2.8) with (2.10). A self-consistent equation for m i in Eq. (2.3) is nothing but the so-called gap equation in BCS theory.
The thermodynamic potential density ω i for nuclear and quark matters is defined as where and Incidentally, we obtain the gap equation in Eq. (2.3) and the fermion number distribution functions in Eq. (2.10) again by minimizing ω i with respect to m i and n i ± . From this thermodynamic potential density, the pressures of nuclear matter (i = N ) and quark matter (i = q) are given as where we have subtracted the zero-density expectation value of the mean field Hamiltonian in the vacuum. This subtraction method is the same as that used in the expression for energy density, which is given later. We will discuss the determination of the realized phase by comparing the pressures of nuclear matter and quark matter.

Model parameters
For nuclear matter, the extended NJL model has four parameters: G N s , G N v , G N sv , and N . These parameters are determined by four conditions at zero temperature: the nucleon mass in vacuum, m N (ρ N =0) = 939 MeV, the normal nuclear density, ρ 0 N = 0.17/fm 3 , the nucleon mass at normal nuclear density, m N (ρ 0 N ) = 0.6m N (ρ N = 0), and the saturation properties of nuclear matter, W N (ρ 0 N ) = −15 MeV. At zero temperature, the fermion number distribution function n ± i in Eq. (2.10) reduces to the Heaviside step function, n + i = θ(μ r i − p 2 + m 2 i ), and n − i = 1. Thus, in the nuclear matter case, the gap equation at zero temperature is expressed as 20) and ν N = 2N N f N N c with N N f = 2 and N N c = 1. The symbol · · · denotes the expectation value at zero temperature. The energy density per single nucleon at finite baryon density and zero temperature is evaluated as where   As may be seen in this table, the momentum cutoff N is rather small. It is possible, however, that the cutoff similarly increases with density by considering a chemical potential dependent cutoff [19,20]. Under the parameters of Table 1, a rather reasonable value of the incompressibility is obtained numerically as K ≈ 260 MeV, in which the incompressibility of nuclear matter at normal nuclear density is evaluated as Although we fix the value of m N (ρ 0 N ) in this paper, the incompressibility K at normal nuclear density is capable of becoming an input parameter instead of the nucleon mass m N (ρ 0 N ) at normal nuclear density, since m N (ρ 0 N ) has an influence on K .
For the quark matter, there are three parameters, i.e., G q s , G q sv , and q , in the extended NJL model with scalar-vector eight-point interaction. Here, we put G q v = 0 since the effects of the vector coupling G q v are well-known [21]. On the other hand, we introduce the parameter G q sv because it has an influence on the chiral phase transition at finite density and temperature. The scalar-vector attractive interaction with G q sv increases the chiral condensate strength. Namely, the chiral phase transition point is pushed to the higher-density side with increasing G q sv , in which the scalar coupling G q s is effectively regarded as a density-dependent coupling G q s (ρ q ) by introducing G q sv . In this paper, we treat G q sv as a chiral phase transition tuning parameter. The parameters G q s and q are determined by two conditions: the vacuum value for the dynamical quark mass, m q = 313 MeV, and the pion decay constant, f π = 93 MeV. For the value of G q sv , there is no criterion to determine it at this stage. Thus, we treat G q sv as a free parameter. The physical quantities of the quark matter case are obtained from the corresponding ones for the nuclear matter case with The values of the model parameters for quark matter are summarized in Table 2.

Gap solutions and chiral phase transition
In the quark matter case, the coupling constant for the scalar-vector and isoscalar-vector eight-point interaction, G q sv , is a free parameter, as has already been mentioned in the previous section. Hence, we put G q sv 8 q = −68.4 so as to realize the chiral phase transition at a reasonable point. Then, we take the parameter G q sv as [18]. The numerical results that will be presented correspond to this parameter set and to the gap solutions in Eq. The left-hand side of Fig. 1 shows the dynamical quark mass as a function of the quark chemical potential at T = 0, 20, and 40 MeV. For zero temperature, the obtained vacuum value for the dynamical quark mass is μ q = 313 MeV. As may be seen in this figure, the region with multiple solutions for the gap equation shrinks with increasing temperature.
The right-hand side of Fig. 1 shows the quark number density as a function of the quark chemical potential at T = 0, 20, and 40 MeV. Here, the quark number density is given as a multiple of normal nuclear density ρ 0 N in the vertical axis. The relation between the quark number density ρ q and the quark chemical potential μ q is obtained from Eq.  The gap equation has multiple solutions in certain regions, so that there exists an unphysical region with unstable solutions there. Thus, we determine the physically realized solution by calculating the pressure p q in Eq. (2.16) where i = q for quark matter. The physically realized solution corresponds to the largest pressure at each temperature. Figure 2 shows the pressure of quark matter as a function of the quark chemical potential.
In the case of T = 0 MeV, the lower density solution is realized from μ q = 313 MeV to μ q ≈ 326 MeV on the left-hand side of Fig. 1, where the pressure is the largest value. Above μ q ≈ 326 MeV, however, the massless solution becomes physically realized. Namely, the chiral phase transition for T = 0 occurs at μ q ≈ 326 MeV, as seen in Fig. 2. In the quark phase, the phase with dynamical quark mass (chiral broken phase) is realized for the lower quark chemical potential (μ q < 326 MeV) and the phase with massless quark (chiral symmetric phase) for the higher one (μ q > 326 MeV), as may be seen from Fig. 2. In the region from ρ q ∼ 0.38ρ 0 N (ρ B ∼ 0.13ρ 0 N , where ρ B represents the baryon number density) to ρ q ∼ 5.41ρ 0 N (ρ B ∼ 1.80ρ 0 N ) on the right-hand side of Fig. 1, a first-order chiral phase transition is realized and coexistence of quark phases occurs.
In the case of T = 20 MeV, the low density solution is realized up to μ q ≈ 323 MeV and the massless solution becomes physically realized from μ q ≈ 323 MeV on the right-hand side of Fig. 1. From  Fig. 2, it is seen that the chiral phase transition with T = 20 MeV occurs at μ q ≈ 323 MeV. In this case, the chiral broken phase is realized at μ q < 323 MeV and the chiral symmetric phase at μ q > 323 MeV. Here, in the region from ρ q ∼ 1.30ρ 0 N (ρ B ∼ 0.43ρ 0 N ) to ρ q ∼ 5.41ρ 0 N (ρ B ∼ 1.80ρ 0 N ) on the right-hand side of Fig. 1, a first-order chiral phase transition is realized and coexistence of quark phases occurs.
In the case of T = 40 MeV, the density solution with gap solution is realized in all points (from μ q = 0 to μ q ≈ 318 MeV) and the massless solution becomes physically realized from μ q ≈ 318 MeV on the right-hand side of Fig. 1. In Fig. 2 with T = 40 MeV, two branches are smoothly connected and at μ q ≈ 318 MeV a chiral phase transition occurs, or at ρ q ∼ 5.78ρ 0 N (ρ B ∼ 1.93ρ 0 N ) on the right-hand side of Fig. 1. In this case, the phase with chiral symmetry breaking is realized for  or 20 MeV, at T = 40 MeV, there is no jump from the lower density solution to the massless solution since there is no unstable density solution, as the right-hand side of Fig. 1 with T = 40 MeV shows. Hence, coexistence of quark phases does not occur and the order of the chiral phase transition is not a first-order phase transition. In the case in which a jump exists, as in the right-hand side of Fig. 1 with T = 0 or 20 MeV, it is clear that the phase transition is of first order. However, because there is no jump on the right-hand side of Fig. 1 with T = 40 MeV and the slope of the pressure with respect to quark chemical potential is the same, namely, the two branches are smoothly connected as seen in Fig. 2, the phase transition at T = 40 MeV is of second order.
Incidentally, the critical density of the chiral phase transition is lower than that of the quark-hadron phase transition, as is mentioned below. Also, chiral symmetry restoration occurs systematically at the lower density side, so that it has no actual physical consequences and no influence on the curve of the equation of state in the quark phase after the quark-hadron phase transition.

Quark-hadron phase transition
The main object of this paper is to investigate the quark-hadron phase transition in the extended NJL model with scalar-vector eight-point interaction at finite temperature and density. In this subsection, we present a procedure for investigating the phase transition between the nuclear and quark matters and show the numerical results.
For the quark-hadron phase transition, we follow the same approach to determine the physically realized phase, as has already been shown in the previous subsection. Namely, we determine the realized phase by comparing the pressure of nuclear matter with that of quark matter at finite temperature and baryon chemical potential. For this purpose, the condition for chemical equilibrium is required to be μ N (T ) = 3μ q (T ), (3.1) where μ N and 3μ q indicate the chemical potential per baryon. By regarding this condition as the one for thermodynamic equilibrium between the hadron and quark phases, we derive the corresponding condition for the pressure of the hadron and quark phases as  From Eq. (2.16), the pressure p N and p q can be calculated for nuclear matter and for quark matter, respectively. The left-hand side of Fig. 3 shows the pressure of nuclear and quark matter as a function of the baryon chemical potential, which is equivalent to the nuclear chemical potential and the triple of the quark chemical potential at T = 0 in the case of G q sv 8 q = −68.4. As may be seen from this figure, there is a crossing point at a certain chemical potential value. From this crossing point, we can determine the coexistence of nuclear and quark phases. Then, about μ B ≈ 1236 MeV, the quarkhadron phase transition at T = 0 occurs on the left-hand side of Fig. 3. Thus, the nuclear phase (hadron phase) is realized for a smaller baryon chemical potential (μ B = μ N < 1236 MeV) and the quark phase for a larger chemical potential (μ B = 3μ q > 1236 MeV).
On the other hand, the right-hand side of Fig. 3 shows the pressure as a function of the baryon number density at T = 0, where the vertical axis is shown on a logarithmic scale. This figure is depicted by using the relation of the density and chemical potential shown in the right-hand side of Fig. 1. It is seen from this figure that the nuclear and quark phases coexist and a first-order quark-hadron phase transition occurs in the region from ρ N (=ρ B ) ∼ 2.64ρ 0 N to ρ q ∼ 10.9ρ 0 N (ρ B ∼ 3.63ρ 0 N ). In the case of T = 20 MeV, it is seen that the quark-hadron phase transition occurs at μ B ≈ 1190 MeV, as may be seen from the left-hand side of Fig. 4. Thus, the nuclear/hadron phase is realized in the region of μ N < 1190 MeV and the quark phase in the region of 3μ q > 1190 MeV. From the right-hand side of Fig. 4, it is seen that coexistence of nuclear and quark phases occurs from ρ N (=ρ B ) ∼ 2.49ρ 0 N to ρ q ∼ 9.99ρ 0 N (ρ B ∼ 3.33ρ 0 N ) and a first-order chiral phase transition is realized there. As compared with the case of T = 0, it is seen from Fig. 4 that the crossing point (quark-hadron phase transition point) moves toward the lower left (small chemical potential and low pressure side) and the density region of the coexistence of nuclear and quark phases becomes smaller with increasing temperature.
In the process of increasing temperature, at a certain temperature, this crossing point vanishes. This situation is depicted in Fig. 5 with T = 40 MeV. In this figure, there is no crossing point that represents the transition point, so this leads to the situation in which the first-order quark-hadron phase transition has already finished.

Phase diagram
In this section, we present the phase diagram in the extended NJL model with scalar-vector eightpoint interaction at finite temperature and chemical potential. Then, in the quark matter, we also investigate the effects of the scalar-vector coupling constant G q sv , which has an influence on the chiral phase transition in the phase diagram. Namely, we discuss how the chiral phase transition line is affected by varying the strength of the scalar-vector interaction.

Phase diagram with
First, we present the phase diagram without scalar-vector interaction. Thus, for the quark matter, we investigate the chiral and quark-hadron phase transitions by using the original NJL model Lagrangian density without the scalar-vector interaction, i.e., G q sv = 0. The top panel of Fig. 6 shows the phase diagram with G q sv = 0 as a function of temperature and baryon chemical potential. Here, the vertical and horizontal axes represent temperature T and baryon chemical potential μ B (=μ N = 3μ q ), respectively. The solid curve represents the critical line of the first-order chiral phase transition and the dotted one represents that of the second-order chiral phase 10  Then, the endpoint of the first-order quark-hadron phase transition is located on the critical line of the chiral phase transition, as may be seen from the middle panel of Fig. 6.
Here, it should be noted that there is a region where the quark-hadron phase transition occurs after chiral symmetry restoration in the nuclear phase side, in which the nucleon mass is zero. Namely, this suggests that a phase that is chiral symmetric but with nucleonic (hadronic) elementary excitation could exist just before the phase transition from the nuclear phase to the quark one. Recently, McLerran and Pisarski have proposed a new state of matter, the so-called quarkyonic matter [22][23][24], which is a phase characterized by chiral symmetry restoration and confinement based on large N c arguments. Thus, this chiral symmetric nuclear phase predicted by our model may possibly correspond to the quarkyonic phase.

Phase diagram with
Finally, we show the phase diagram with the stronger scalar-vector interaction. We set the value G q sv 8 Fig. 6 shows the phase diagram in the T -μ B plane with G q sv 8 q = −81.9. The critical line of the first-order chiral phase transition, depicted as a solid curve, emerges at T = 0 and μ B ≈ 979 MeV and soon terminates at (μ B , T ) (979, 1) MeV. As for the critical line of the first-order quark-hadron phase transition, depicted as a dash-dotted curve, the same diagram as the previous two is obtained.

The bottom panel of
The reason why the behavior of the first-order quark-hadron phase transition line seen in Fig. 6 does not change is that the critical line of the first-order quark-hadron phase transition exists in the chiral symmetric phase, in which m q = 0 and ψ q ψ q = 0, i.e., the quark-hadron phase transition occurs after the chiral phase transition for the unstable quark phase with G

Summary and concluding remarks
The quark-hadron phase transition at finite temperature and baryon chemical potential has been investigated following Ref. [18] in the extended NJL model with scalar-vector eight-point interaction. In this model, as a first attempt to investigate the quark-hadron phase transition, the hadron side was regarded as symmetric nuclear matter and the quark side as a free quark phase with no quark-pair correlation. Here, the single nucleon in the nuclear matter and the single quark in the quark matter were treated as fundamental fermions with N N c = 1 and N q c = 3, respectively. Then, in this model, 12 the nuclear saturation property has been well reproduced for the nuclear matter side. On the other hand, for the quark matter side, there is one free parameter, G q sv . This model parameter was not fixed in this paper, since there is no criterion to determine the value of G q sv by using physical quantities at this stage. By introducing this parameter G q sv , the effective density-dependent coupling constant was obtained as G q s (ρ q ) = G q s (1 − G q sv /G q s · ρ 2 q ). Hence, the G q sv term plays an important role in pushing the chiral symmetry restoration point to the higher density side for quark matter. Thus, the parameter G q sv controls the chiral symmetry restoration point and/or the strength of the partial chiral symmetry restoration in the nuclear medium. As for the description of the quark-hadron phase transition, we have calculated the pressure of the nuclear matter and the quark matter, and determined the realized phase by comparing their pressures. As a result, a first-order quark-hadron phase transition is obtained at finite temperature and baryon chemical potential. Here, the end point of the first-order quark-hadron phase transition is at (μ B , T ) (955, 39) MeV with G q sv 8 q = −68.4. This phase boundary is not changed even by varying the strength of the scalar-vector interaction because of G q sv -independence. As for the effects of the scalar-vector coupling constant G q sv on the chiral phase transition, the critical line of the first-order chiral phase transition shrinks with increasing G q sv . Namely, G q sv acts to move the endpoint of the first-order chiral phase transition toward larger μ B and lower T .
From the phase diagram in Fig. 6, it should be noted that there is an interesting phase where the quark-hadron phase transition occurs after chiral symmetry restoration in the nuclear matter. This might appear as an exotic phase, i.e., the nuclear phase, not the quark phase, while the chiral symmetry is restored in terms of the quark matter. This phase may possibly correspond to the quarkyonic phase [22][23][24], which is introduced as a chiral symmetric confined matter.
In this paper, we have ignored the color superconducting phase [25]. However, this phase may exist in finite density systems. Thus, the next challenging task may be to investigate the phases of nuclear matter, including nuclear superfluidity and quark matter, and also including the color superconducting state, i.e., nucleon pairing on the nuclear phase side and quark pairing on the quark phase side. Further, it is widely believed that neutron star matter undergoes a phase transition to quark matter at high temperature and/or density. Thus, it is also interesting to investigate the phase transition between neutron star matter and quark matter. This leads to the understanding and development of the physics of neutron stars.