Chirally Improved Quark Pauli Blocking in Nuclear Matter and Applications to Quark Deconfinement in Neutron Stars

Abstract

The relativistic mean field (RMF) model of the nuclear matter equation of state was modified by including the effect of Pauli-blocking owing to quark exchange between the baryons. Different schemes of a chiral enhancement of the quark Pauli blocking was suggested according to the adopted density dependence of the dynamical quark mass. The resulting equations of state for the pressure are compared to the RMF model DD2 with excluded volume correction. On the basis of this comparison a density-dependent nucleon volume is extracted which parameterizes the quark Pauli blocking effect in the respective scheme of chiral enhancement. The dependence on the isospin asymmetry is investigated and the corresponding density dependent nuclear symmetry energy is obtained in fair accordance with phenomenological constraints. The deconfinement phase transition is obtained by a Maxwell construction with a quark matter phase described within a higher order NJL model. Solutions for rotating and nonrotating (hybrid) compact star sequences are obtained, which show the effect of high-mass twin compact star solutions for the rotating case.

1. Introduction

The behavior of baryons in a dense, strongly interacting medium and the resulting properties of dense baryonic matter are highly interesting questions because of their relevance for explaining the interior of compact astrophysical objects like pulsars and their mergers as well as heavy-ion collision experiments in the NICA-FAIR energy range. The main problem, which awaits a better theoretical formulation and understanding, is to treat the baryon as a bound state of quarks and to study the effects of this quark substructure as a function of density. In particular, it is expected that at a critical value of the density the many-baryon system will change its character and get transformed to the new state of deconfined quark matter. Already before this transition occurs, the effective interaction between baryons will be strongly modified due to the fact that the effects of quark exchange between different baryons need to be taken into account as a requirement following from the Pauli principle on the quark level of description. These quark substructure effects will eventually dominate over other effects due to, e.g., the meson exchange interaction. The resulting quark exchange contribution to the baryon self-energy entails an increase of the energy per baryon and thus lead to a stiffening of dense baryonic matter. On the other hand, the quark exchange between two baryons involves already a six-quark wave function, which is a partial delocalization of quarks and can be seen as a precursor of the transition to deconfined (i.e., delocalized) quark matter. In this transition from a many-baryon to a many-quark system, the matter is effectively softened, due to the appearance of a mean field emerging from the attractive two-quark interactions. The value of the critical density for deconfinement is crucial for applications in heavy-ion collisions and compact stars.

The aspect we wanted to consider in this work was to investigate the influence of quark exchange on the self energies of baryons and the equation of state of dense baryonic matter on the one hand and on the phase transition to delocalized quark matter on the other. For quantitative estimates we employed a relativistic mean field theory for baryonic matter (the linear Walecka model) as well as for quark matter (the NJL-type model with higher order quark interactions) and superimpose the quark exchange contribution to the baryon self-energy obtained within a nonrelativistic quark potential model of the baryon structure and the six-quark wave function. The quark mass in this calculation has a density dependence (even inside the baryon) and is taken, e.g., from the NJL model calculation.

The effect of Pauli blocking in systems of composite particles can be discussed from the quark and nuclear level to that of atomic clusters. The relationship between Pauli blocking and excluded volume is known from the fact that the hard-sphere model of molecular interactions is based on the electron exchange interaction among atoms (see, e.g., Ebeling et al. [1]) which is captured, e.g., in the Carnahan–Starling EoS [2]. Note that the Carnahan–Starling form of the EoS for multicomponent mixtures [3] has recently been reproduced for a hadron resonance gas model with induced surface tension when the packing fraction is not too large [4]. A recent application of the Pauli blocking effect has been found in [5] where its role for explaining the ionization potential depression accessible in high-pressure experiments with warm dense plasmas has been demonstrated. The temperature, density and momentum dependence of the Pauli blocking depends on the generic form of bound state wave functions and therefore concepts developed for atomic systems could thus be taken over to the case of dense hadronic systems. Detailed parameterizations of the Pauli shift for nuclear clusters in warm, dense nuclear matter are given in [6] (see also references therein). In [7] it has been demonstrated that the repulsive part of effective density-dependent nucleon–nucleon interactions of the Skyrme type (e.g., the one by Vautherin and Brink [8]) can be reproduced by the quark exchange interaction between nucleons.

On the other hand, for the description of repulsive interactions in dense hadronic systems the concept of an excluded volume has been successfully applied [9] and extended to the case of light nuclear clusters [10], but this application requires a medium dependence of the excluded volume parameter [11] and thus hints to a microscopic origin from the composite nature of hadrons and clusters. We therefore use the present approach to quantify such a relationship between quark Pauli blocking in dense nuclear matter and the medium dependence of the excluded volume parameter by comparing the EoS of the present approach to the relativistic mean field approach DD2 with excluded volume [12]. We would like to point out that the inclusion of the Pauli-blocking effect within a quantum statistical description of light cluster formation and dissociation in nuclear matter at subsaturation densities [13,14,15] has important consequences for the equation of state and the composition of matter as seen, e.g., in the nuclear symmetry energy [16] that is successfully compared to experiments and in the description of supernova matter [17,18] where otherwise excluded volume approaches are commonly used [19].

In the present investigation, we also consider the role that the quark exchange interaction can play for the nuclear symmetry energy. Here the interesting question arises inasmuch the quark exchange contribution can make the contribution from isovector meson exchange obsolete. Our present study suggests that the $\rho$-meson mean field may not have any contribution for densities up to the onset of baryon dissociation.

Recently, the question of the softening of dense baryonic matter due to the appearance of strange baryons became very popular and led to the hyperon puzzle: a lowering of the maximum mass of compact stars so that the existence of pulsars with masses as high as $2M_{\odot }$ could not be explained. In principle, the approach can be extended to obtain results on the baryon self-energy shifts also in the case that the strange quark flavor will be included. In that case the presented approach can make a contribution to solving the question: Which effect will dominate when increasing the density: the occurrence of strange baryons or of deconfined strange quark matter? In this present work we want to consider as a first step only the question of nonstrange quark-nucleon matter.

2. Quark Exchange in Nuclear Matter

2.1. Quark Substructure Effect on the Self-Energy of the Nucleons

The quark substructure of nucleons becomes apparent for higher densities, when the nucleon wave functions have a finite overlap so that the effects of quark exchange between nucleons due to the Pauli principle on the quark level are no longer negligible. A quantitative estimate for this effect has been made within a potential model for the nucleons as three-quark bound states [7,20], see the Appendix for details of the derivation, and we employ the resulting contribution to the nucleon self-energy as the basis for our work. The result has been obtained in the form of a Pauli blocking energy shift for a nucleon with momentum P, given spin and the isospin projection $\tau =n,p$ in nuclear matter at $T=0$,

$$\begin{array}{c}\hfill \Delta E_{\tau P}^{\mathrm{Pauli}}(P_{F,n},P_{F,p})={\displaystyle \sum _{{\tau }^{\prime }=n,p}}{\displaystyle \sum _{\alpha =1,2}}{c}_{\tau {\tau }^{\prime }}^{\left(\alpha \right)}{W}_{\alpha }(P,P_{F,{\tau }^{\prime }}),\end{array}$$

(1)

where $P_{F,\tau }$ is the Fermi momentum of a medium nucleon with the isospin projection $\tau =n,p$, which is directly related to the medium density by $P_{F,\tau }={\left(3{\pi }^2{n}_{\tau }\right)}^{1/3}$. The coefficients for the $c_{\tau {\tau }^{\prime }}^{\left(\alpha \right)}$ are given in

Table 1. Quark exchange coefficients $c_{n\tau }^{\left(\alpha \right)}$ in spin-flavor-color space. These coefficients entail the symmetry relation $\Delta E_{pP}^{\mathrm{Pauli}}(P_{F,n},P_{F,p})=\Delta E_{nP}^{\mathrm{Pauli}}(P_{F,p},P_{F,n})$.

$\mathit{\tau }$	${\mathit{c}}_{\mathit{n}\mathit{\tau }}^{\left(1\right)}$	${\mathit{c}}_{\mathit{n}\mathit{\tau }}^{\left(2\right)}$
n	$\frac{15}{81}$	−$\frac{16}{81}$
p	$\frac{12}{81}$	−$\frac{14}{81}$

, and their superscript index $\alpha =1,2$ indicates whether they apply for the one-quark or the two-quark exchange in the two-nucleon system. The functions $W_{\alpha }(P,P_{F{\tau }^{\prime }})$ are the contributions due to the Pauli-shift in the energy spectrum of three quark bound states. Their analytic derivation within a harmonic oscillator confinement model for the ground state nucleons according to the [20] is detailed in the Appendix A. The resulting expression for $W_{\alpha }(P,P_{F{\tau }^{\prime }})$ is

$$\begin{array}{cc}\hfill W_{\alpha }(P,P_{F{\tau }^{\prime }})=\frac{9\sqrt{3}}{64\sqrt{\pi }}\frac{b}{m}\frac{1}{{\lambda }_{\alpha }^3}& \{12\sqrt{\pi }\left[\mathrm{erf}\left({\lambda }_{\alpha }(P_{F{\tau }^{\prime }}-P)\right)+\mathrm{erf}\left({\lambda }_{\alpha }(P_{F{\tau }^{\prime }}+P)\right)\right]\hfill \\ & +\frac{1}{{\lambda }_{\alpha }P}\left\{\left[11-2{\lambda }_{\alpha }^2{P}_{F{\tau }^{\prime }}(P_{F{\tau }^{\prime }}+p)\right]{\mathrm{e}}^{-{\lambda }_{\alpha }^2{(P_{F{\nu }^{\prime }}+P)}^2}\right.\hfill \\ & -\left.\left[11-2{\lambda }_{\alpha }^2{P}_{F{\tau }^{\prime }}(P_{F{\tau }^{\prime }}-P)\right]{\mathrm{e}}^{-{\lambda }_{\alpha }^2{(P_{F{\tau }^{\prime }}-P)}^2}\right\}\}.\hfill \end{array}$$

(2)

Here m is the constituent quark mass and b is the width parameter of the nucleon wave function that describes the quark substructure by a product of two Gaussian functions of the relative (Jacobi) coordinates in the three-quark system with $b^{-2}=\sqrt{3}m\omega$; ${\lambda }_{\alpha }=b\alpha /\left(2\sqrt{3}\right)$ denote the ranges for one- and two-quark exchange processes. Values for b and $\omega$ are given below.

We want to consider as examples the two special cases:

• Symmetric nuclear matter (SNM), for which $P_{F,n}=P_{F,p}=P_F$ and

  $$\begin{array}{ccc}\hfill \Delta E_{n{P}_F}^{\mathrm{Pauli}}(P_F,P_F)& =& c_{nn}^{\left(1\right)}{W}_1\left({\lambda }_1{P}_F\right)+c_{nn}^{\left(2\right)}{W}_2\left({\lambda }_2{P}_F\right)+c_{np}^{\left(1\right)}{W}_1\left({\lambda }_1{P}_F\right)+c_{np}^{\left(2\right)}{W}_2\left({\lambda }_2{P}_F\right).\hfill \end{array}$$

  (3)
  The Pauli shift for protons is obtained using the symmetry relation $\Delta E_{pP}^{\mathrm{Pauli}}(P_{F,n},P_{F,p})=\Delta E_{nP}^{\mathrm{Pauli}}(P_{F,p},P_{F,n})$ that is encoded in the coefficients of Table 1. With these coefficients and the low-density expansion (A29) up to fifth order in the Fermi momentum, we obtain

  $$\begin{array}{ccc}\hfill \Delta E_{n{P}_F}^{\mathrm{Pauli}}(P_F,P_F)& =& \frac{5}{8\sqrt{3\pi }}\frac{b}{m}\left(-P_F^3+\frac{1054}{225}{b}^2{P}_F^5\right).\hfill \end{array}$$

  (4)
  This energy shift can be identified with a shift in the chemical potential and thus be used to derive a contribution to the equation of state, see [7].
  In order to give numerical results for the Pauli shift (4), we adopt the values $m=350$ MeV and $b=0.59$ fm according to [20], which reproduce quite well the single nucleon properties. With the relation $P_F^3=(3{\pi }^2/2)n$, the Pauli blocking shift can be given as a function of the nuclear matter density $\rho$

  $$\Delta E^{\mathrm{Pauli}}\left(n\right)=a_{1}n+a_2{n}^{5/3},$$

  (5)

  with $a_1^{\left(\mathrm{SNM}\right)}=-197.77$ MeV fm³ and $a_2^{\left(\mathrm{SNM}\right)}=1944.45$ MeV fm⁵. As has been discussed already in [7], this density dependent energy shift is in good agreement with the repulsive part of the Skyrme Hartree–Fock shift in nuclear matter obtained by Vautherin and Brink [8].
• Pure neutron matter (PNM), for which $P_{F,p}=0$, $P_{F,n}=P_F$ and

  $$\begin{array}{ccc}\hfill \Delta E_{n{P}_F}^{\mathrm{Pauli}}(P_F,P_F)& =& c_{nn}^{\left(1\right)}{W}_1\left({\lambda }_1{P}_F\right)+c_{nn}^{\left(2\right)}{W}_2\left({\lambda }_2{P}_F\right).\hfill \end{array}$$

  (6)
  Inserting the coefficients from Table 1 and the low-density expansion (A29) up to fifth order in the Fermi momentum, we obtain

  $$\begin{array}{ccc}\hfill \Delta E_{n{P}_F}^{\mathrm{Pauli}}(P_F,P_F)& =& \frac{5}{24\sqrt{3\pi }}\frac{b}{m}\left(-P_F^3+\frac{1666}{225}{b}^2{P}_F^5\right).\hfill \end{array}$$

  (7)
  Inserting the relation $P_F^3=3{\pi }^{2}n$ between Fermi momentum and density for PNM, we obtain the energy shift in the form (5) with the coefficients $a_1^{\left(\mathrm{PNM}\right)}=-131.85$ MeV fm³ and $a_2^{\left(\mathrm{PNM}\right)}=3252.57$ MeV fm⁵.

Figure 1 shows the Pauli blocking shift for the SNM and PNM cases as function of the Fermi momentum.

2.2. Chiral Improvement of the Quark Pauli Blocking

One of the main shortcomings when applying the results for the quark Pauli blocking obtained within the nonrelativistic quark model to the equation of state of dense nuclear and neutron star matter up to the deconfinement phase transition is the fact that the quark mass is a medium-independent constant in this model. From chiral perturbation theory it is known that the chiral condensate $\langle \overline{q}q\rangle$ melts in a dense hadronic matter environment so that the constituent quark mass shall be reduced towards its value in the QCD Lagrangian which obeys approximate chiral symmetry.

Effective chiral quark models for the low-energy sector of QCD are capable of addressing the aspect of dynamical chiral symmetry breaking and its restoration in a hot and dense medium, but have a problem with modeling confinement of quarks in hadrons. Here we suggest a compromise. We adopt a density dependence for the dynamically generated quark mass and thus achieve a chiral improvement of the quark Pauli blocking shift. We discuss in the following three schemes for this density-dependent quark mass: (i) a constant quark mass, (ii) a linear density dependence (called Brown–Rho scaling [21]) and (iii) a density dependence according to the calculation within a higher order Nambu–Jona-Lasinio model [22]. These density dependences of the quark mass are illustrated in Figure 2. In Figure 3 we show the energy shifts ${\Delta }_{\tau }\left(n\right)=\Delta E_{\tau P_F}^{\mathrm{Pauli}}(P_F,P_F)$ resulting from the insertion of the density dependencies for the Fermi momenta and the quark mass as shown in Figure 2 into Equations (4) and (7) for SNM and PNM, respectively.

In order to arrive at a model for dense (asymmetric) nuclear matter with quark substructure effects at supranuclear densities, we adopt a combined approach consisting of a relativistic mean-field (RMF) approach to nuclear matter which in its simplest form is the well-known linear Walecka (LW) model [23,24], to which we add the repulsive quark Pauli blocking interaction, which should then partly replace the vector meson exchange at high densities and play the role of a precursor of the delocalization of the nucleon wave functions in the quark deconfinement transition. Such a combined approach has been very successfully employed before in the description of light nuclear clusters in nuclear matter by Typel et al. [15], where self-energy effects for nucleons were treated within a relativistic mean-field theory while the cluster formation is described within a nonrelativistic quantum statistical approach that allowed to account for the reduction of the binding energy of the clusters due to nucleonic Pauli blocking, leading to the Mott dissociation of the clusters and the formation of uniform nuclear matter around the saturation density.

In the present work, the role of the clusters is played by the nucleons as three-quark bound states, subject to a quark Pauli blocking effect that triggers their Mott dissociation into deconfined quark matter described in a relativistic mean-field model for which we adopt the higher order Nambu–Jona-Lasinio (hNJL) model of [22]. At lower densities, in order to make contact with the phenomenology of nuclear matter saturation properties, the Fermi gas model of nucleons three-quark bound states with a hard core repulsion from quark Pauli blocking has to be augmented with additional attraction and repulsion as described, e.g., by the coupling to scalar and vector mean fields in the $\sigma -\omega$ (LW) model.

3. Equation of State of Cold, Dense Matter with Deconfinement Transition

3.1. Relativistic Mean Field Model with Quark Exchange Contribution

The modification of the LW model to account for quark exchange (Pauli blocking) effects among nucleons is introduced by additional contributions to the pressure ($p_{\mathrm{ex}}$) and to the energy density (${\epsilon }_{\mathrm{ex}}$) as

$$\begin{array}{ccc}\hfill P& =& \frac{1}{8{\pi }^2}{\displaystyle \sum _{\tau =n,p}}\left[-E_{\tau }^{*}{m}_{\tau }^{*2}{P}_{F,\tau }+\frac{2}{3}{E}_{\tau }^{*}{P}_{F,\tau }^3+m_{\tau }^{*4}log\left(\frac{E_{\tau }^{*}+P_{F,\tau }}{m_{\tau }^{*}}\right)\right]+\frac{1}{2}{G}_{\omega }{n}^2-\frac{1}{2}{G}_{\sigma }{n}_s^2+P_{\mathrm{ex}},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill \epsilon & =& \frac{1}{8{\pi }^2}{\displaystyle \sum _{\tau =n,p}}\left[2E_{\tau }^{*3}{P}_{F,\tau }-E_{\tau }^{*}{m}_{\tau }^{*2}{P}_{F,\tau }-m_{\tau }^{*4}log\left(\frac{E_{\tau }^{*}+P_{F,\tau }}{m_{\tau }^{*}}\right)\right]+\frac{1}{2}{G}_{\omega }{n}^2+\frac{1}{2}{G}_{\sigma }{n}_s^2+{\epsilon }_{\mathrm{ex}},\hfill \end{array}$$

(9)

where $n=n_n+n_p$ is the baryon density, $n_s=n_{s,n}+n_{s,p}$ the scalar density, and for each baryon species we have

$$\begin{array}{ccc}\hfill n_{s,\tau }& =& \frac{m_{\tau }^{*}}{2{\pi }^2}\left[E_{\tau }^{*}{P}_{F,\tau }-m_{\tau }^{*2}log\left(\frac{E_{\tau }^{*}+P_{F,\tau }}{m_{\tau }^{*}}\right)\right],\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill E_{\tau }^{*}& =& \sqrt{m_{\tau }^{*2}+P_{F,\tau }^2}\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill n_{\tau }& =& \frac{P_{F,\tau }^3}{3{\pi }^2},\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill m_{\tau }^{*}& =& m_{\tau }-G_{\sigma }{n}_{s,\tau },\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill {\mu }_{\tau }& =& E_{\tau }^{*}+G_{\omega }{n}_{\tau }+{\mu }_{\mathrm{ex},\tau }.\hfill \end{array}$$

(14)

The effective coupling constants $G_{\sigma }={(g_{\sigma }/m_{\sigma })}^2$ and $G_{\omega }={(g_{\omega }/m_{\omega })}^2$ are adjusted in order to fit the saturation point of symmetric nuclear matter with the phenomenological binding energy per nucleon, see

Table 2. Parameter sets for vector ($G_{\omega }={(g_{\omega }/m_{\omega })}^2$) and scalar ($G_{\sigma }={(g_{\sigma }/m_{\sigma })}^2$) meson couplings, the compressibility K and symmetry energy $E_s$ at the nuclear saturation density as well as radius $R_{1.4}$ of the neutron star with mass $1.4M_{\odot }$ for the relativistic mean-field (RMF) linear Walecka (LW) model and for modified LW models with quark exchange contributions for different density-dependences of the quark mass: constant quark mass (LW + Qex), Brown–Rho scaling (LW + MQex) and hNJL model (LW + MhNJL).

	${({\mathit{g}}_{\mathit{\omega }}/{\mathit{m}}_{\mathit{\omega }})}^2$ [fm2]	${({\mathit{g}}_{\mathit{\sigma }}/{\mathit{m}}_{\mathit{\sigma }})}^2$ [fm2]	K [MeV]	${\mathit{E}}_{\mathit{s}}$ [MeV]	${\mathit{R}}_{1.4}$ [km]
RMF (LW)	11.6582	15.2883	608.874	21.58	13.22
LW + Qex	6.11035	9.91197	331.958	32.04	13.70
LW + MQex	8.59170	13.29118	481.713	34.12	14.40
LW + MhNJL	9.25683	13.9474	582.831	31.55	14.29

and the section with the results.

In the relativistic mean-field EoS of Equations (8) and (9) we also introduce the contribution to the thermodynamical quantities that originate from the quark exchange self-energy via

$$\begin{array}{ccc}\hfill {\mu }_{\mathrm{ex},\tau }& =& {\Delta }_{\tau }(n,x)=\Delta E_{\tau P_{F,\tau }}^{\mathrm{Pauli}}(P_{F,n},P_{F,p}),\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\epsilon }_{\mathrm{ex}}& =& {\int }_0^{n}d{n}^{\prime }\{x{\Delta }_p(n^{\prime },x)+(1-x){\Delta }_n(n^{\prime },x)\},\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill P_{\mathrm{ex}}& =& {\displaystyle \sum _{\tau =n,p}}{\mu }_{\mathrm{ex},\tau }{n}_{\tau }-{\epsilon }_{\mathrm{ex}},\hfill \end{array}$$

(17)

where $n_p$ ($n_n$) denotes the proton (neutron) density and $x=n_p/n$ is the proton fraction.

3.2. NJL Model with Higher Order Quark Interactions

In order to describe cold quark matter that is significantly stiffer than the ideal gas, we employ a recently developed generalization of the NJL model, which includes higher order quark interactions in both Dirac scalar and Dirac vector channels (hNJL), see [22] and references therein. The thermodynamic potential density of the 2-flavor hNJL model for a homogeneous quark matter system in the mean-field approximation is given by

$$\begin{array}{ccc}\hfill \Omega & =& -2N_c{\displaystyle \sum _{q=u,d}}\left\{{\int }_0^{\Lambda }\frac{dp{p}^2}{2{\pi }^2}{E}_q-\frac{1}{16{\pi }^2}\left[(\frac{2}{3}{E}_{F,q}{p}_{F,q}^3-M_q^2{E}_{F,q}{p}_{F,q}+M_q^{4}ln\left(\frac{E_{F,q}+p_{F,q}}{M_q}\right)\right]\right\}\hfill \\ & & +U-{\Omega }_0,\hfill \end{array}$$

(18)

where

$$\begin{array}{c}\hfill U=\frac{g_{20}}{{\Lambda }^2}{\sigma }^2+3\frac{g_{40}}{{\Lambda }^8}{\sigma }^4-3\frac{g_{22}}{{\Lambda }^8}{\sigma }^2{\omega }^2-\frac{g_{02}}{{\Lambda }^2}{\omega }^2-3\frac{g_{04}}{{\Lambda }^8}{\omega }^4\end{array}$$

(19)

is the potential energy density and the quark quasiparticle dispersion relation is $E_q=\sqrt{p^2+M_q^2}$, with

$$\begin{array}{ccc}\hfill M_q& =& m_q+2\frac{g_{20}}{{\Lambda }^2}\sigma +4\frac{g_{40}}{{\Lambda }^8}{\sigma }^3-2\frac{g_{22}}{{\Lambda }^8}\sigma {\omega }^2,\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill E_{F,q}& =& {\mu }_q-2\frac{g_{02}}{{\Lambda }^2}\omega -4\frac{g_{04}}{{\Lambda }^8}{\omega }^3-2\frac{g_{22}}{{\Lambda }^8}{\sigma }^2\omega .\hfill \end{array}$$

(21)

The model parameters are the 4-quark scalar and vector couplings $g_{20}$, and $g_{02}$, the 8-quark scalar and vector couplings $g_{40}$ and $g_{04}$ as well as the current quark mass m and the momentum cutoff $\Lambda$ placed on the divergent zero-point energy. Furthermore, the subtraction of the constant ${\Omega }_0$ ensures zero pressure in the vacuum.

The model is solved by minimizing the thermodynamic potential density with respect to the mean-fields $X=\sigma ,\omega$, i.e.,

$$\begin{array}{c}\hfill \frac{\partial \Omega }{\partial X}=0,\end{array}$$

(22)

and the pressure is obtained from the relation $P=-\Omega$.

In this work we use the parameter set of [25] with $g_{20}=2.104$, $g_{40}=3.069$, $m_q=5.5$ MeV, and $\Lambda =631.5$ MeV. The vector channel strengths are quantified by

$$\begin{array}{c}\hfill {\eta }_2=\frac{g_{02}}{g_{20}},{\eta }_4=\frac{g_{04}}{g_{40}}.\end{array}$$

(23)

We concentrate on the parameter space where ${\eta }_2$ is small and use ${\eta }_4$ to control the stiffness of the EoS. With small ${\eta }_2$ we do not delay the onset of quark matter. Additionally, we put $g_{22}=0$ [22].

This approach allows us to calculate partial pressures $P_q$ and partial densities $n_q=\partial P_q/\partial {\mu }_q$ for $q=u,d$. In a cold stellar environment, the processes $d\to u+e^{-}+{\overline{\nu }}_e$ and $u+e^{-}\to d+{\nu }_e$ result in the $\beta$-equilibrium relation for the chemical potentials ${\mu }_d={\mu }_u+{\mu }_e$, since the neutrinos leave the star and do not take part in the chemical equilibration. Local charge neutrality requires

$$\begin{array}{c}\hfill \frac{2}{3}{n}_u-\frac{1}{3}{n}_d-n_e=0.\end{array}$$

(24)

The total pressure in the quark phase is given as $P=P_u+P_d+P_e$, where $P_e$ is the electron pressure given by the relativistic ideal gas formula. The baryon chemical potential in the quark phase can be calculated from

$$\begin{array}{c}\hfill \mu ={\mu }_u+2{\mu }_d.\end{array}$$

(25)

and the respective baryon number density is

$$\begin{array}{c}\hfill n=\frac{\partial P}{\partial \mu }=\frac{n_u+n_d}{3}.\end{array}$$

(26)

3.3. Quark Deconfinement Phase Transition

To construct a thermodynamically consistent hybrid EoS, we use the Maxwell construction, which is tantamount to assuming a large surface tension at the hadron–quark interface. The critical baryon chemical potential is obtained by matching the pressures from the low density and the high density phase. The first order phase transition obtained by the Maxwell construction generates a jump in the density and the energy density. Illustrative examples for this are shown and discussed in the following Section for the parametrization that is introduced there.

4. Results

4.1. Parameterization of the Model

For the calculations, we fixed the parameters of our models on the properties of symmetric nuclear matter at the saturation density $n_0$ = 0.153 fm⁻³, at which the binding energy is $E/A$ = $\epsilon /n-m_n$ = $-15.8$ MeV. The parameters are given in the Table 2.

In Figure 4, we demonstrate the properties of symmetric nuclear matter as a function of the baryon density. As it can be seen from the values of coupling constants of mesons in Table 2 in comparison with those of the LW model the repulsion of the $\omega$-meson is partially replaced by the inclusion of Pauli-blocking via quark exchange mechanism. At low densities, the binding energy per baryon goes to zero since no nuclear cluster formation is included here. For a detailed discussion of this aspect, see [15,16].

4.2. Equation of State

The EoS for the nuclear matter is obtained and in Figure 5 the pressure as a function of the density is shown for symmetric matter (left panel) and for pure neutron matter (right panel). The symmetry energy is shown and discussed below in Section 4.4 for all our models.

4.3. Comparison with Nucleonic Excluded Volume

It is interesting to compare the effect of accounting for the compositeness and finite extension of the nucleon wave function by the Pauli blocking effect with the phenomenological improvement of nuclear matter models by implementing a nucleonic excluded volume like in the van-der-Waals gas. In Figure 6 we show the equations of state for pressure vs. density that results from our LW model with chirally enhanced Pauli blocking and the RMF model DD2 for different values of the nucleonic excluded volume parameter [12].

Also shown is the flow constraint derived from the analysis of heavy ion collision experiments [26]. From comparing the Pauli-blocking improved LW models with excluded-volume corrected DD2 models, we extracted a density-dependent excluded volume parameter and the corresponding hard-core radius for nucleons. We note that these results compare very well with nucleonic hard-core radii obtained within the induced surface tension approach reported in [9]. A thorough analysis of the critical temperature of symmetric nuclear matter, the incompressibility of the normal nuclear matter and the proton flow constraint clearly shows [9,27] that a hard-core radius of nucleons up to 0.45 fm is still consistent with the available experimental data. Therefore, the short dashed curve in Figure 7 is perfectly consistent with the known symmetric nuclear matter properties.

It should be stressed that smaller values of $r\approx 0.35$ fm for the nucleon hard-core radius are obtained from fits to heavy-ion collision data for hadron production at LHC and RHIC energies so that a dependence of the hard-core radius on the chemical freezeout temperature was conjectured in [9]. Extending the present approach to finite temperatures, such a temperature dependence is expected to result from the temperature dependence of the quark Pauli-blocking energy shift.

4.4. Applications for Neutron Stars

In the left panel of Figure 8, we show the symmetry energy as a function of the density for all considered models. In the right panel of Figure 8, we show the proton fraction as a function of the density which results from accounting for the $\beta$-equilibrium with electrons for all considered models.

For all three models we construct the thermodynamics of stellar matter in $\beta$-equilibrium fulfilling the charge neutrality condition with electrons and protons. In Figure 9 we show the EoS in $\beta$-equilibrium for all considered models in comparison with the LW EoS.

We consider the problem of causality in our modeling and show in the lower right panel of Figure 9 the dependence of the squared speed of sound on density. As it is shown for models LW, LW + Qex and LW + MQex for all relevant densities the causality holds since $c_s^2<1$. For the model LW + MhNJL the causality is violated for high densities where already the transition to quark matter has to happen. This fact is consistent with our modeling because as a mass function for the quarks we took the behavior corresponding to hNJL model, see Figure 2.

Having defined the hadronic EoS with three different scenarios of the chiral enhancement of the quark Pauli blocking effect, and four choices for pair of free parameters of the quark matter EoS: $({\eta }_2,{\eta }_4)=(0,14.0),(0,6.0),(0.1,14.0)$ and $(0.1,6.0)$, we perform four Maxwell constructions for each hadronic model, see Figure 10.

In the upper left panel of that figure, we illustrate the Maxwell construction in the pressure-chemical potential plane. Each crossing point of a hadronic EoS $P_H\left(\mu \right)$ with a quark matter one $P_Q\left(\mu \right)$ fulfills the Gibbs conditions for phase equilibrium at $T=0$ because the chemical potentials are equal to the critical value ${\mu }_c$ (chemical equilibrium) where the pressures coincide $P_H\left({\mu }_c\right)=P_Q\left({\mu }_c\right)$ (mechanical equilibrium). According to the principles of equilibrium thermodynamics, the system is at each value of the chemical potential in the phase with the highest pressure. Therefore, at the crossing point ${\mu }_c$ the system switches from the hadronic to the quark matter EoS. Since at ${\mu }_c$ the corresponding pressures have a different slope, this transition is accompanied with a jump in the baryon number density $n=dP/d\mu$ and energy density $\epsilon =-P+\mu n$.

In the remaining three panels of Figure 10 we show the pressure as a function of the energy density for the 12 hybrid EoS models resulting from the combination of the three hadronic EoS: LW + Qex (upper right panel), LW + MQex (lower left panel) and LW + MhNJL (lower right panel) with the four quark matter EoS for the model parameters of the hNJL model: ${\eta }_2=\{0.0,0.1\}$ and ${\eta }_4=\{6.0,14.0\}.$ The other parameters of the hNJL model are fixed to values of [25], see also Section 3.2 above.

In Figure 11 we show the mass-radius relation for compact star configurations considering two models for the density dependence of the quark mass: constant quark mass (LW + Qex, green short-dashed line) and Brown-Rho scaling (LW + MQex, red long-dashed line) without and with the possible phase transition to quark matter. We do not show the M-R curves for LW + MhNJL here because, as we mentioned earlier when discussing Figure 9, this scenario violates causality ($c_s^2>1$) at large densities in the hadronic phase and makes sense only with a phase transition that prevents this problem to occur. The phase transition, however, is the same for LW + MQex and LW + MhNJL, so that the lines for the latter results are indistinguishable from those for the former ones and are not displayed separately. From the calculation we choose the hybrid EoS where quark matter is modeled with the parameters ${\eta }_2=0$ and ${\eta }_4=14$. As it is shown in the figure, the differences between all three models for the masses of stars are small, and all of them satisfying the $2M_{\odot }$ observational constraint from the Shapiro-delay based mass measurement on PSR J0740 + 6220 [28].

Moreover, with this particular hybrid EoS the third family of compact stars [29] is possible because the three conditions are fulfilled [30]: (i) a sufficiently stiff hadronic EoS, (ii) a large jump in energy density at the transition which occurs at a low pressure $P\left({\mu }_c\right)<100$ MeV/fm³, (iii) a sufficiently stiff quark matter EoS to reach a maximum mass of $\sim 2M_{\odot }$. Such a third family of compact stars, if it would be discovered, would signal a strong first-order phase transition and therefore support the existence of a critical endpoint in the QCD phase diagram [30,31]. Recently, it was shown within a Bayesian analysis that the existence of such a class of hybrid EoS is in accordance with modern constraints from multi-messenger astronomy [32].

We like to remark that a similar calculation, with quark Pauli blocking as a repulsive interaction in the nuclear matter phase (for constant quark mass) and with the string-flip model for quark matter has been performed as early as in 1989 with a similar result that stable hybrid stars with quark matter core are possible and have a maximum mass above $2M_{\odot }$ [33]. At that time, measured pulsar masses were below $1.5M_{\odot }$.

In the same plot we also show the relationship between mass and equatorial radius for stars rotating with the maximum possible angular velocity. These calculations have been performed within the slow-rotation approximation described in detail in [34,35].

5. Conclusions

The relativistic mean field model of the nuclear matter equation of state was modified by including the effect of Pauli-blocking owing to quark exchange between the baryons. Different schemes of a chiral enhancement of the quark Pauli blocking due to a density-dependent reduction of the value of the dynamical quark mass were considered. The resulting equations of state for the pressure were compared to the RMF model DD2 with excluded volume correction.

On this basis a density-dependent nucleon excluded volume is extracted which parameterizes the quark Pauli blocking effect in the respective scheme of chiral enhancement. The dependence on the isospin asymmetry of the quark Pauli blocking was investigated and the corresponding density dependent nuclear symmetry energy was obtained in fair accordance with phenomenological constraints.

The deconfinement phase transition was obtained by a Maxwell construction with a quark matter phase described within a higher order NJL model. Solutions for rotating and nonrotating (hybrid) compact star sequences were obtained, which show the effect of high-mass twin compact star solutions for the rotating case. This result is a consequence of the stiffening of the nuclear equation of state due to the quark Pauli blocking effect which at the same time is a precursor of the delocalization of the quark wave function in the deconfinement transition that leads to a strong softening and thus a large enough density jump at the phase transition to induce a gravitational instability as one of the necessary conditions for the occurrence of a third family solution for hybrid star sequences in the neutron star mass-radius diagram. The other one is the sufficient stiffness of deconfined quark matter at high densities which is provided by the 8-quark interactions in the scalar and vector channels of the higher order NJL model.