Hadron-quark transition and chiral symmetry restoration at high density

Exploration of the QCD phase diagram is an important subject not only in particle and nuclear physics but also in astrophysics and cosmology; for a review, see, e.g., Ref. Fukushima and Hatsuda (2011) and references therein. However, at finite baryon (or quark) chemical potential, the first principle calculation, i.e., the lattice QCD simulation, is not feasible due to the infamous sign problem. To avoid the problem, several methods are proposed and investigated, but, at present, these methods are not complete, and we do not have the established equation of state (EoS) at finite baryon density.

At low temperature, there is nuclear (baryonic) matter at saturation density. As the density increases, other baryons may appear. At the extreme high density, the chiral symmetric quark matter is expected to appear and conformality is restored. However, at present, we do not have a definite information of the EoS in the intermediate region between the saturation density and the extremely high density.

It is known that repulsive effects among baryons are important in the intermediate region. If repulsion is absent, the baryonic matter is realized at sufficiently large baryon density Cleymans et al. (1986a). One of the traditional treatments of such repulsion is to consider the excluded volume effects (EVE) among baryons Cleymans et al. (1986b); Kouno and Takagi (1989); Rischke et al. (1991). The excluded volume effects successfully prevent baryonic matter from realizing at sufficiently large baryon density; for a recent review, see, e.g., Ref. Fujimoto et al. (2022a) and references therein.

At large density, the chiral symmetry restoration is also expected. The Nambu–Jona-Lasinio (NJL) model Nambu and Jona-Lasinio (1961a); *Nambu:1961fr is a simple but very useful model to describe the restoration; as a review, see, e.g., Ref. Hatsuda and Kunihiro (1994) and references therein. However, the NJL model cannot describe the hadron-quark transition.

Furthermore, recently, it has been emphasized that the trace anomaly and the speed of sound are very important to understand the properties of the high density hadron and quark matter; see, e.g., Refs. Fujimoto et al. (2022b); Kojo (2021a) and references therein.

In this paper, we construct a simple phenomenological hybrid hadron-quark model with EVE of baryons and chiral dynamics. The model naturally connects the baryonic matter at low density and the quark matter at high density. It can also describe the chiral restoration. It is found that, in this hybrid model, there is a possibility that the abrupt first-order like transition to pure quark matter induces the strong restoration of chiral symmetry and the speed of sound has a large peak.

This paper is organized as follows. In Sec. II, the hybrid model is formulated. In Sec. III, numerical results are shown for two typical cases. Section IV is devoted to a summary and discussion.

First, we give a sketch of our strategy of calculations. Our main purpose is to know the $\mu_{\rm B}$-dependence of the thermodynamic quantities of hadron or quark matter, where $\mu_{\rm B}$ is the baryon chemical potential. First, we construct the baryon number density $n_{\rm B}$ with the EVE of baryons. Next we require that $n_{\rm B}$ approaches ${n_{\rm q}\over{3}}$ in the high density limit, where $n_{\rm q}$ is the quark number density of pure quark phase. In this procedure, $n_{\rm B}$ depends on the chiral condensates $\sigma_{f}~{}(f=u,d,s)$ which are included in the quark model. Using the thermodynamic equation, the $\sigma_{f}$-dependence of the thermodynamic potential density $\Omega~{}(=-P)$ is obtained, where $P$ is the pressure of the system. The values of $\sigma_{f}$ are determined to minimize $\Omega$ (or maximize $P$). Using the obtained values of $\sigma_{f}$, the other thermodynamic quantities are calculated.

As shown in Fig. 1, we consider $N$ non-pointlike baryons in the system with volume $V$ where $N$ is the number of baryons. We consider this system to be equivalent to the system of $N$ pointlike baryons in the effective volume $V-v_{\rm B}N$ where $v_{\rm B}$ is the volume of a baryon; see Fig. 2. Then, the following equation is satisfied;

where $n_{\rm B}={N\over{V}}$ is the baryon density of non-pointlike baryon and $\tilde{n}_{\rm B}$ is the one of $N$ pointlike baryons. In this paper, we add a tilde to the baryon number density of point-like baryons. Hence, the baryon number density of baryonic matter with EVE is given by

The pressure $P$ of the baryonic matter with the excluded volume effects is determined by the thermodynamic equation

The other thermodynamic quantity, such as the energy density $\varepsilon$, is determined by the thermodynamic relation. Hereafter, we concentrate on EoS at zero temperature and omit the variable $T$ for simplicity. Then, Eq. (3) is represented as ${dP\over{d\mu_{\rm B}}}$.

When a constant $v_{\rm B}$ is used, the speed of sound,

may exceed the speed of light $c~{}(=1)$ and the causality can be easily violated. In fact, when $v_{\rm B}\tilde{n}_{\rm B}\gg 1$, we have

This means that the EoS becomes very hard and the speed of sound diverges. Therefore, the density dependence of $v_{\rm B}$ is very important.

It is natural to assume that the baryon number of the system approaches the one of the pure quark system with the same chemical potential, when the baryon chemical potential is very large. This requirement can be achieved if we assume

where $n_{\rm q}$ is the quark number density of the pure quark matter. In other words, we assume that EoS inside the baryons is described by the pure-quark model. The baryon number density is given by

Hence, the EoS is expected to approach the pure quark EoS at high density. When $3\tilde{n}_{\rm B}\gg n_{\rm q}$, we obtain

When $n_{\rm q}\neq 0$ and $\tilde{n}_{\rm B}\neq 0$, Eq. (9) can be rewritten as

where $v_{\rm q}={1\over{3\tilde{n}_{\rm B}}}$. From Eq. (11), we see that the system can also be regarded as the matter of quarks with finite effective volume $v_{\rm q}$; see Fig. 3. In this sense, this model has a quark-hadron duality; for a general review of quark-hadron duality, see, e.g., Ref. Shifman (2000) and references therein. It is also interesting that Eq. (13) resembles the law of combined resistance of parallel resistances. The large $v_{\rm B}$ causes the strong suppression of baryonic matter. When $3v_{\rm q}<v_{\rm B}$, it is natural to regard that the system is composed of quarks with small volume $v_{\rm q}$ rather than baryons with large volume $v_{\rm B}$.

However, when $n_{\rm q}=0$, $v_{\rm B}$ cannot be defined. On the other side, it is known that, below the saturation density, the nucleon has a finite volume $v_{\rm B0}={4\pi r_{\rm B0}^{3}\over{3}}$ with the nucleon radius $r_{\rm B0}=0.8$ fm. Hence, we interpolate between $v_{\rm B0}$ and ${3\over{n_{\rm q}}}$ with the following smooth function of $n_{\rm q}$;

with

where $a$ is a free parameter and controls how abruptly $v_{\rm B}$ approaches $3/n_{\rm q}$. Note that, as is shown below, $n_{\rm q}$ depends on chiral condensates in our model. Hence, $v_{\rm B}$ and $n_{\rm B}$ also depend on the chiral condensates.

For the pointlike hadron model, we use the hadron resonance gas (HRG) model. The number density is given as

where $g_{{\rm B}i}$ and $m_{{\rm B}i}$ are the spin degeneracy and the mass of $i$-th baryon, respectively. The function $n(g,m,\mu)$ is the number density of ideal fermion gas with the degeneracy factor $g$, the mass $m$ and the chemical potential $\mu$ at zero temperature, and is given by

In this paper, for simplicity, we use the same $v_{\rm B}$ for all baryons.

For pure quark matter, we use the three-flavor NJL model with mean field approximation. The quark number density of the NJL model is given by

where $M_{f}$ and $\mu_{\rm q}(=\mu_{\rm B}/3)$ are the effective mass of $f$-quark and the quark chemical potential, respectively. The effective quark mass is given by

with $f\neq f^{\prime}$, $f\neq f^{\prime\prime}$ and $f^{\prime}\neq f^{\prime\prime}$, where $m_{f}$ is the current quark mass of $f$-quark, $G_{\rm s}$ and $G_{\rm d}$ are coupling constants of four and six-quarks interaction, and $\sigma_{f}$ is the chiral condensate of $f$-quark, respectively. In the pure quark system, the pressure is given by

where $P_{\rm v}$, $U_{\rm m}$ and $P_{\rm D,NJL}$ are the Dirac sea contributions, the mesonic and the density parts of the NJL model, respectively. For each flavor, the density part $P_{\rm D,NJL}$ is given by $P_{\rm F}(M_{f},\mu_{\rm q})$ which is equivalent to the pressure of the free quark gas with the quark mass $M_{f}$ and the quark chemical potential $\mu_{\rm q}$. The Dirac sea contributions are given by

where $E_{{\rm v}f}=\sqrt{\Lambda^{2}+M_{f}^{2}}$ with the three dimensional momentum cutoff $\Lambda$. The Dirac sea contributions have no explicit $\mu_{\rm q}$-dependence. However, $P_{\rm v}$ depends on $\mu_{\rm q}$, since $M_{f}$ depends on $\mu_{\rm q}$. The mesonic part is given by

According to Ref. Rehberg et al. (1996), we set $m_{u,d}=5.5$ MeV, $m_{s}=140.7$ MeV, $G_{\rm s}\Lambda^{2}=1.835$, $G_{\rm d}\lambda^{5}=12.36$ and $\Lambda=602.3$ MeV.

Instead of Eq. (22), in the hybrid model, the total pressure $P$ of the system is given by

Using $n_{\rm B}$ given by Eq. (2) with Eq. (14), the density part $P_{\rm D}$ is given by

where the integration is performed with fixing all $\sigma_{f}$ so as to satisfy the relation

The value of the chiral condensate $\sigma_{f}$ is determined to maximize $P$ and satisfy

Hence, the thermodynamic relation

is satisfied. Therefore, using Eq. (26), once $P_{\rm D}$ is obtained as the function of $\sigma_{f}$ and the solution $\sigma_{f}$ which maximizes the total pressure (25) is determined, the equation of motion (28) and the thermodynamic relation (29) are automatically satisfied. The energy density $\varepsilon$ of the total system is determined using the thermodynamic relation

Following the above procedure, the quark and hadron matters are correlated with each other in the level of the pressure via EVE. This is also true for the thermodynamic potential density $\Omega$ because there is a direct relation to the pressure, namely $\Omega=-P$. The functional form of $\Omega(\sigma_{f})$ deviates largely from the one $\Omega_{\rm NJL}(\sigma_{f})$ in the original NJL model in the intermediate region of $\mu_{\rm B}$. Therefore, our result can provide the crossover behavior even if the NJL model itself has the first-order chiral transition at low $T$ in the moderately high density region; since the thermodynamic potential is deformed by the hadron contributions, they act as the external term that explicitly breaks chiral symmetry.

The hybrid model approaches in the NJL model in the high density limit. As is seen in the next section, when $n_{\rm B}$ approaches ${n_{\rm q}\over{3}}$ gradually, crossover chiral transition takes place. When $n_{\rm B}$ approaches ${n_{\rm q}\over{3}}$ abruptly, first-order like transition happens.

The original NJL model has a cutoff. However, at zero temperature, the chemical potential is the natural cutoff for energy and momentum. Hence, in the numerical calculations, we use the cutoff only in the Dirac Sea contributions. In this procedure, the physical quantities are expected to approach those in the free quark gas model. In the HRG part, we include all baryons listed in the list of Particle Data Group Workman and Others (2022), but they only contribute to the results when their masses are smaller than the baryon chemical potential $\mu_{\rm B}$.

Several hybrid models have been already proposed. For example, in Ref. Masuda et al. (2013), the hyperbolic functions are used as an interpolation function of EoS. In our model, we do not give such an interpolation function but give a more microscopic quantity, namely, the density dependence of baryon volume in EVE. In Ref. Jeong et al. (2020), the hybrid model based on quarkyonic matter and the EVE is investigated. In our model, we do not assume quarkyonic matter, but some kind of more macroscopic quark-hadron duality (11) is assumed. The macroscopic model may be simpler and more convenient than the microscopic one, but it has a less dynamical description. The study of the relation between several hybrid models is an important problem in the future.

Our hybrid model has one free parameter $a$ in Eq. (15). In this section, we show the numerical results in two typical cases, a crossover transition ($a=0.1$) and a first-order like transition ($a=0.8$). Hereafter, we use the quark chemical potential $\mu_{\rm q}={\mu_{\rm B}\over{3}}$ instead of $\mu_{\rm B}$ since we are interested in the chiral dynamics of quarks as well as the hadron-quark transition.

The numerical calculations are done according to the following procedures.

Note that the equation of motion (28) and the thermodynamic relation (29) are automatically satisfied by minimizing (maximizing) $\Omega$ ($P$).

In summary, in this paper, the simple phenomenological hybrid model with the excluded volume effect (EVE) of baryons and the chiral dynamics is investigated. In the model, we combine the Nambu–Jona-Lasinio model and the hadron resonance gas model. The model interpolates between nuclear matter at saturation density and quark matter at high density. The equation of state (EoS) approaches that of pure quark matter as the density increases. There is the intermediate phase, which is difficult to be identified as a pure hadron phase or a pure quark phase. The speed of sound has a double-peak structure. One peak is related to the EVE of baryons, and the other is related to the hadron-quark transition and the chiral symmetry restoration. If the baryon volume $v_{\rm B}$ approaches ${3\over{n_{\rm q}}}$ gradually, the hadron-quark transition is a typical crossover transition, where $n_{\rm q}$ is the quark number density of the pure quark phase. If the baryon volume $v_{\rm B}$ approaches ${3\over{n_{\rm q}}}$ abruptly, the first-order like transition can occur in cooperation with the rapid chiral symmetry restoration, and the speed of sound can be very large.

It seems that the density dependence of the baryon volume is very important for EoS at zero temperature. It is desirable to determine the dependence in the framework of the lattice QCD simulation. Simulations using the imaginary baryon chemical potential de Forcrand and Philipsen (2002); D’Elia and Lombardo (2003); Nagata and Nakamura (2011); Takahashi et al. (2015) may be useful for this purpose; see Refs. Roberge and Weiss (1986); Kashiwa (2019) as an example.

It is interesting to investigate neutron star properties using our hybrid model. However, to extend the model to asymmetric matter, we should know not only the density dependence but also the isospin dependence of the baryon volume. The lattice QCD simulation with imaginary baryon and isospin chemical potentials D’Elia and Sanfilippo (2009) may be useful to determine the dependence.

Recently, repulsion in nuclear matter is discussed McLerran and Reddy (2019); Jeong et al. (2020); Duarte et al. (2020) in the context of quarkyonic matter McLerran and Pisarski (2007). In quarkyonic matter, the onset of the quark Fermi sea suppresses baryonic matter Kojo (2021b). The quarkyonic effective field theory Duarte et al. (2021, 2023) is developed and the relation between the quarkyonic phase and EVE is discussed. Although the existence of quarkyonic matter is not explicitly assumed in our simple phenomenological hybrid model, there is some kind of quark-hadron duality and an intermediate phase which cannot be identified as pure hadron matter or pure quark matter appears. The study of the relation between quarkyonic matter and baryon volume may be important.

On the thermal QCD transition, the possibility of the existence of the partial deconfinement phase has been discussed recently; for a recent review, see, e.g., Ref. Hanada and Watanabe (2023). The partial deconfinement can be regarded as the coexistence of two phases in the internal color space rather than in the coordinate space. The intermediate phase which appears in our model may also be understood as such a phase.