Isospin- and momentum-dependent effective interactions for the baryon octet and the properties of hybrid stars

# Isospin- and momentum-dependent effective interactions for the baryon octet and the properties of hybrid stars

## Abstract

The isospin- and momentum-dependent MDI interaction, which has been extensively used in intermediate-energy heavy-ion reactions to study the properties of asymmetric nuclear matter, is extended to include the nucleon-hyperon and hyperon-hyperon interactions by assuming same density, momentum and isospin dependence as for the nucleon-nucleon interaction. The parameters in these interactions are determined from the empirical hyperon single-particle potentials in symmetric nuclear matter at saturation density. The extended MDI interaction is then used to study in the mean-field approximation the equation of state of hypernuclear matter and also the properties of hybrid stars by including the phase transition from the hypernuclear matter to the quark matter at high densities. In particular, the effects of attractive and repulsive N interactions and different values of symmetry energies on the hybrid star properties are investigated.

###### pacs:
24.10.-i, 21.65.-f, 26.60.-c, 97.60.Jd, 12.39.Ba

## I Introduction

The study of the in-medium baryon-baryon effective interactions is one of the fundamental problems in nuclear physics. While significant progress has been made in understanding the nucleon-nucleon interaction in nuclear medium, our knowledge on the hyperon-nucleon and hyperon-hyperon interactions in nuclear medium is very limited. The latter is important for understanding the properties of hypernuclei, the equation of state (EOS) of dense baryonic matter, the properties of compact stars in which hyperons are expected to appear abundantly, and strangeness production in heavy-ion collisions. Therefore, it is of great interest to develop an effective model for the hyperon-nucleon and hyperon-hyperon interactions in nuclear medium.

The in-medium single-particle potential of a nucleon generally depends not only on the nuclear density and its momentum but also on its isospin. The isospin- and momentum-dependent MDI interaction is such an effective nuclear interaction based on the finite-range Gogny interaction (1). This interaction has recently been extensively used in studying the isospin and momentum-dependent effects in nuclear reactions and the properties of neutron stars (for a recent review see Ref. (2)). In particular, using this interaction in the Isospin-dependent Boltzmann-Uheling-Ulenback (IBUU) transport model with the isospin-dependent in-medium nucleon-nucleon scattering cross sections, a relatively stringent constraint on the density dependence of the nuclear symmetry energy at subsaturation densities was obtained from the isospin diffusion data in intermediate-energy heavy-ion collisions  (3); (4); (5). The resulting symmetry energy has been used to constrain the neutron skin thickness of heavy nuclei (6) and the properties of neutron stars (7); (8); (9); (10); (11) including the transition density, which separates the liquid core from the inner crust of a neutron star, and other properties of neutron stars by assuming that they only consist of nucleons, electrons, and muons (12); (13).

In this paper, we extend the MDI interaction to include nucleon-hyperon and hyperon-hyperon interactions. This is achieved by assuming that the nucleon-hyperon and hyperon-hyperon interactions have the same density and momentum dependence as the nucleon-nucleon interaction with the interaction parameters fitted to known experimental data at normal nuclear matter density. With this extended MDI interaction, we then study, as an example, the properties of hybrid stars that not only consist of appreciable fraction of hyperons but also a possible quark matter in their dense core (14); (15); (16); (17); (18); (19); (20); (21); (22); (23); (24); (25); (26). For the hadron-quark phase transition, we use the Gibbs construction (27); (28) with the quark phase described by a simple MIT bag model (29); (16). In particular, we study the effects of attractive and repulsive N interactions and different symmetry energies on the properties of hybrid stars. We note that there have been extensive studies on hybrid stars based on various approaches, such as the relativistic mean field (RMF) model (30); (31); (32); (34); (33), the Brueckner-Hartree-Fock theroy (35); (36); (37); (38); (39), and phenomenological models (40); (41).

This paper is organized as follows. In Sec. II, we describe the method used in extending the MDI interaction to include the hyperon-nucleon and hyperon-hyperon interactions. The extended MDI interaction is then used in Sec. III.1 to study the thermodynamical properties of hypernuclear matter and its equilibrium conditions. The MIT bag model is used in Sec. III.2 to describe the properties of the quark matter and in Sec. III.3 to study the hadron-quark phase transition in dense matter. In Sec. IV, we show and discuss the results on the particle fractions in dense matter and its equation of state as well as the mass-radius relation of hybrid stars in different scenarios. A summary is given in Sec. V.

## Ii The MDI interaction with hyperons

The MDI interaction is an effective nuclear interaction with its density and momentum dependence deduced from the phenomenological finite-range Gogny interaction (1). In the mean-field approximation, the potential energy density in a nuclear matter of density and isospin asymmetry with and being, respectively, the neutron and proton densities, is given by

 V(ρ,δ) = Au(x)ρnρpρ0+Al(x)2ρ0(ρ2n+ρ2p)+Bσ+1ρσ+1ρσ0 (1) × (1−xδ2)+1ρ0∑τ,τ′Cτ,τ′ × ∫∫d3pd3p′fτ(→r,→p)fτ′(→r,→p′)1+(→p−→p′)2/Λ2,

where is the nucleon isospin, is the nucleon phase distribution function, and fm is the saturation density of normal nuclear matter. Values of the parameters , , , , , and can be found in Refs. (1); (4). The parameter , related to the coefficient of the spin-exchange operator in the Gogny/Skyrme interactions, is used to adjust the density dependence of the symmetry energy away from the saturation density without changing the properties of symmetric nuclear matter. At subsaturation densities, its value has been constrained between and from the analysis of the isospin-diffusion and neutron skin thickness data (3); (4); (5); (42).

### ii.1 The extended MDI interaction

To extend the MDI interaction for the nucleon-nucleon () interaction to include the nucleon-hyperon () and hyperon-hyperon () interactions, we assume that the latter have the same density and momentum dependence as that between two nucleons. The potential energy density of a hypernuclear matter due to interactions between any two baryons then has the following general form:

 Vbb′ = ∑τb,τ′b′[Abb′2ρ0ρτbρτ′b′+A′bb′2ρ0τbτb′ρτbρτ′b′ (2) + Bbb′σ+1ρσ−1ρσ0(ρτbρτ′b′−xτbτb′ρτbρτ′b′) + Cτb,τ′b′ρ0∫∫d3pd3p′fτb(→r,→p)fτ′b′(→r,→p′)1+(→p−→p′)2/Λ2⎤⎥⎦,

where () denotes the baryon octet included in the present study, i.e., , , , and . We use the conventions that for neutron and for proton (43), for , for , for and for , and for and for . In the above, the total baryon density is now given by , and is the phase-space distribution function of particle species . The interaction parameters are denoted by , , , and . If there are only nucleons, we can rewrite , , , and for and for , which then reduce to the original parameters in the MDI interaction for nucleons (1); (4). The parameter is again used to model the isospin effect on the interaction between two baryons, and its value is taken to be or in the present study for all baryon pairs.

The single-particle potential for a baryon of species in a hypernuclear matter can then be obtained from the total potential energy density of the hypernuclear matter, given by , as

 Uτb(p) = δδρτbVHP (3) = ∑b′(b′≠b)∑τ′b′[Abb′2ρ0ρτ′b′+A′bb′2ρ0τbτ′b′ρτ′b′ + Bbb′σ+1ρσ−1ρσ0(ρτ′b′−xτbτ′b′ρτ′b′)+Cτb,τ′b′ρ0 × Missing or unrecognized delimiter for \right + A′bbρ0τbτ′bρτ′b+2Bbbσ+1ρσ−1ρσ0(ρτ′b−xτbτ′bρτ′b) + 2Cτb,τ′bρ0∫d3p′fτ′b(→r,→p′)1+(→p−→p′)2/Λ2⎤⎦ + ∑b′,b′′[Bb′b′′σ−1σ+1ρσ−2ρσ0 × ∑τb′∑τ′b′′(ρτb′ρτ′b′′−xτb′τ′b′′ρτb′ρτ′b′′)⎤⎥⎦.

For the interaction parameters , , and that involve hyperons, they can in principle be determined from the nucleon-hyperon and hyperon-hyperon interactions in free space. Because of the lack of hyperon-nucleon scattering experiments, information on the hyperon-nucleon interactions has been mainly obtained from the hyperon single-particle potentials extracted empirically from studying  (44) as well as  (45); (46) and  (47) production in nuclear reactions. Although this has led to extensive studies of the hyperon-nucleon interaction in the past (48); (50); (49), the isospin and momentum dependence of the hyperon-nucleon in-medium interactions is still not very well determined, and the situation is even worse for hyperon-hyperon interactions. We therefore assume in the present study that , , and are all proportional to corresponding ones in the nucleon-nucleon interaction. Specifically, for , and , we assume

 Abb′ = fbb′ANN, A′bb′ = fbb′A′NN, Bbb′ = fbb′BNN, (4)

and for we assume

 Unknown environment '%

with and treated differently.

We determine the values of by fitting the empirical potential of baryon at rest in a medium consisting of baryon species . For hyperons in symmetric nuclear matter at saturation density, their potentials are

 U(N)Λ(ρN=ρ0)=−30 MeV (5)

for the potential from the analysis of and reactions on nuclei (51); (52) and

 U(N)Ξ(ρN=ρ0)=−18 MeV (6)

for the potential from the analysis of  (53) and  (54); (55) reactions. This leads to the values and . For the hyperon, its potential was taken to be attractive in earlier studies (45), but more recent analysis indicate that it is repulsive in the nuclear medium (58); (56); (59); (60); (57). In the present work, we consider both the attractive and repulsive cases

 U(N)Σ(ρN=ρ0) = ±30 MeV. (7)

By setting we get an attractive N interaction, and this is called MDI-Hyp-A in the following. To obtain a repulsive N interaction, called MDI-Hyp-R in the following, we adjust the values of positive and negative terms in the single-particle potential by setting and without changing other parameters as in the case of MDI-Hyp-A interaction. A similar method of changing an attractive N interaction to a repulsive one was used in the RMF calculation (31) by changing the coupling constants of and mesons. For the hyperon-hyperon interaction, the parameters are fitted according to (61)

 U(Y′)Y(ρY′=ρ0)∼−40 MeV, (8)

which gives the strength of the hyperon-hyperon interactions as , , , , , and . Detailed values of the parameters are listed in Table. 1. These parameterizations can be viewed as a baseline for studying the properties of the hypernuclear matter, and more sophisticated treatments are left for future work after the in-medium properties of hyperons are better understood. It will be shown in the following that many interesting results can already be obtained even with these simple parameterizations.

### ii.2 Single-particle potentials

An important quantity related to the interaction of a particle in nuclear medium is its single-particle potential as given by Eq. (3), which is also needed later in our study of the properties of neutron stars. The single-particle potential of a particle depends not only on the density of the medium but also on the momentum of the particle. In this section, we show and discuss the single-particle potentials of both nucleons and hyperons in nuclear matter obtained from the extended MDI interaction.

We first show in Fig. 1 the single-particle potential of a particle at rest in symmetric nuclear matter as a function of density. Although the nucleon potential is more attractive at normal density than those of hyperons, it becomes more repulsive than the hyperon potentials above about 5 times normal nuclear density, including the potential that is attractive at normal density. For the potential that is repulsive at normal density, it becomes more repulsive as the density increases and becomes slightly attractive only at very low densities. Comparing our results with those from other models given in Ref. (62) (and references therein), the single-particle potentials of and are close to those from the chiral effective field theory (63), but more repulsive than those based on the G-matrix calculations using the soft core Nijmegen model or the Jlich meson-exchange model for the free hyperon-nucleon interactions, particularly at high densities.

The single-particle potential of a particle from the extended MDI interaction also depends on its momentum. Figure 2 shows the momentum dependence of the single-particle potentials for both nucleons and hyperons in symmetric nuclear matter at saturation density. Again, results using both attractive and repulsive N interactions (i.e., MDI-Hyp-A and MDI-Hyp-R) are shown for comparison. Also indicated in the figure is the Fermi momentum of nucleons. For nucleons, the single-particle potential from the MDI interaction is consistent with the Schrdinger equivalent potential obtained by Hama et al. from the nucleon-nucleus scattering data (64); (65) up to the nucleon momentum of MeV. For hyperons, the momentum dependence of their single-particle potentials from the extended MDI interaction is similar to that obtained from the G-matrix calculations based on the free Nijmegen interaction (37). Both show an increase with increasing momentum and are similar at low momenta as both are constrained by available experimental data. They are, however, slightly different at high momenta. The momentum dependence of and interactions thus remains an open question, especially at high momenta. Also, the density dependence of the high-momentum behavior of the mean-field potential is poorly known. Since the maximum Fermi momenta of nucleons and hyperons reached in hybrid stars are not very high, the incorrect nucleon potential and the uncertainty of hyperon potentials at high momenta given by the extended MDI interaction are thus not expected to affect significantly the properties of cold hypernuclear matter considered in the present study.

It is known that both proton and neutron single-particle potentials in asymmetric nuclear matter of unequal proton and neutron densities are approximately linear in the isospin asymmetry of the matter. Whether this is also the case for hyperons is not clear in the literature. For the extended MDI interaction introduced in the present study, such a linear dependence in isospin asymmetry, however, also holds for and hyperons. The single-particle potential of a particle in asymmetric nuclear matter can thus be written in general as in terms of the symmetry potential , defined by , and for the nucleon, and hyperons, respectively. From the single-particle potentials of nucleons, and hyperons in asymmetric nuclear matter at normal nuclear matter density and of isospin asymmetry , we have calculated their symmetry potentials using above definitions. In Fig. 3, the momentum dependence of these symmetry potentials are compared. All symmetry potentials are seen to decrease with increasing momentum. At zero momentum, the symmetry potentials of nucleon and are about MeV and MeV, respectively, and for the hyperon they are MeV for an attractive N interaction and MeV for a repulsive one.

Although the symmetry potentials at normal nuclear density are independent of the value of , which is used in the MDI interaction to model the stiffness of nuclear symmetry energy at densities different from the normal density, this is not the case at other densities. This is demonstrated in Fig. 4 for the symmetry potentials of the nucleon, the hyperon and the hyperon in asymmetric nuclear matter at density for the two symmetry energy parameters and . For , the symmetry potentials at zero momentum are MeV and MeV for the nucleon and the hyperon, respectively, and for the hyperon they are MeV with the MDI-Hyp-A interaction and MeV with the MDI-Hyp-R interaction. These values are changed to MeV and MeV for the nucleon and the hyperon, respectively, and for the hyperon to MeV for the MDI-Hyp-A interaction and MeV for the MDI-Hyp-R interaction for the case of . As noted in Ref. (66), the charged baryon ratio in heavy-ion collisions can be used as a probe to constrain the nuclear symmetry energy (potential) at densities larger than . It will be very interesting to see how the symmetry potentials of and hyperons in nuclear matter affect the charged hyperon ratio and the charged hyperon ratio in heavy-ion collisions induced by neutron-rich nuclei. This is important for determining the high density behavior of the nuclear symmetry energy using these ratios in heavy-ion collisions. The extended MDI interaction with hyperons is therefore useful for studying the nuclear symmetry energy (potential) at in the transport model calculation for heavy-ion collisions.

## Iii equilibrium conditions and thermodynamical relations in dense matter

The extended MDI interaction is also useful for studying the properties of a hybrid star that is expected to have a quark core at high densities, a mixed phase of quarks and hadrons at moderate densities, and a hadron phase at low densities. We review in this section the -equilibrium, charge neutrality, and baryon number conservation conditions of such a matter.

In the hadron phase of a hybrid star, the matter consists of nucleons, hyperons and leptons. For leptons, we include both electrons and muons with their masses taken to be and MeV, respectively. At equilibrium, these particles satisfy following baryon number conservation, charge neutrality and -equilibrium conditions:

 ∑iρibi = ρ, (9) ∑iρiqi = 0, (10) μi = μHbbi−μHcqi. (11)

In the above, and are, respectively, the charge and baryon numbers of particle species , where can be nucleons, hyperons or leptons, and their density and chemical potential are denoted, respectively, by and . The total baryon density of the hadron phase is denoted by , and and are, respectively, the baryon and charge chemical potentials of the hadron phase.

Taking into account their interactions in the mean-field approximation, the chemical potential of baryon species is given by

 μτb(pFτb)=mτb+p2Fτb2mτb+Uτb(pFτb), (12)

where is their Fermi momentum, and and are, respectively, the single-particle potential and mass of the baryon. For leptons (, ) their chemical potential is given by with being their Fermi momentum. The relative abundances of various hadrons and leptons for a given total baryon density are then obtained by solving above equations.

In terms of the densities of various particles, the total energy density of the hadron phase can be written as

 ϵH=VH+VL, (13)

where and are the contributions from baryons and leptons, respectively. The former can be written as

 VH=VHP+VHK+VHM, (14)

where is the potential energy density of baryons with calculated from Eq. (2), and and are, respectively, the kinetic energy and mass contributions given by

 VHK = ∑b∑τbp5Fτb10π2mτb, (15) VHM = ∑b∑τbρτbmτb. (16)

The contribution from leptons is calculated by treating them as a free Fermi gas, i.e.,

 VL = Ve+Vμ, Ve = p4Fe4π2, Vμ = 14π2[pFμμ3μ−12m2μpFμμμ − 12m4μln(pFμ+μμmμ)].

The pressure of the hadron phase is obtained from the thermodynamical relation

 PH=PH+PL=(∑b∑τbμτbρτb−VH)+(∑lμlρl−VL), (18)

where and run over all species of baryons and leptons, respectively. We note that the thermodynamical consistency condition

 PH=ρ2d(ϵH/ρ)dρ (19)

is satisfied.

### iii.2 The quark phase

As the nuclear matter density increases, such as in the core of neutron stars, not only hyperons appear but also a quark matter could exist (15). To take into consideration possible transition between the hadronic matter and the quark matter, we follow many previous studies to use in the present study the MIT bag model (29); (16) to describe the cold quark matter that might exist in the dense core of neutron stars.

For the quark phase consisting of quarks and leptons, the baryon number conservation and charge neutrality conditions are given by expressions similar to Eqs. (9) and (10) with denoting now quarks and leptons. For the -equilibrium condition in the quark phase, it is given by with and being the baryon and charge chemical potentials of the quark phase, respectively.

The total energy density and pressure of the quark phase can be calculated from

 ϵQ=VQ+VL, (20) PQ=PQ+PL, (21)

where and are the energy density and pressure of quarks, which can be calculated from the MIT bag model as described in the following, and and are the energy density and pressure of leptons given by same expressions as those in the hadron phase.

At zero temperature, the density , chemical potential , and energy density of quarks of flavor in the quark matter are given, respectively, by

 ρf = p3Ffπ2, μf = √m2f+p2Ff, Vf = 34π2[pFfμ3f−12m2fpFfμf (22) − 12m4fln(pFf+μfmf)],

where is the Fermi momentum of quarks of flavor . For the quark masses, they are taken to be and MeV. In the bag model, the energy density is modified by a bag constant , resulting in an energy density given by

 VQ=∑fVf+B. (23)

This leads to the following pressure for the quark matter:

 PQ=∑fμfρf−VQ=ρ2d(VQ/ρ)dρ, (24)

where is the total baryon density of the quark phase

 ρ=13∑fρf. (25)

### iii.3 The hadron-quark phase transition

The hadron-quark phase transition leads to a mixed phase of hadronic and quark matters, which is usually described by the Gibbs conditions  (27); (28)

 TH = TQ,PH=PQ, μb=μHb = μQb,μc=μHc=μQc. (26)

The Gibbs conditions for the chemical potentials can also be expressed as

 μu=13μn−23μe, μd=μs=13μn+13μe. (27)

Since only the case of zero temperature is considered in this paper, the first condition in Eq. (III.3) is always satisfied. To solve above equations, we follow the method of Ref. (28). In this method, one first calculates the pressure of the hadron phase at a series of baryon densities. From the known chemical potentials and of the hadronic matter at these densities, one then calculates the pressure of the quark matter that is in chemical equilibrium with corresponding hadronic matter, i.e., with the quark chemical potentials , and determined by the two relations in Eq. (III.3). The hadronic baryon density at which the two pressures are the same is then the low-density boundary of the hadron-quark phase transition. As an illustration, we show in Fig. 5 the results for the case of the symmetry energy parameter and the bag constant . The solid line is the density dependence of the pressure of the hadron phase calculated using the model described in Sec. III.1, while the dash-dotted line is the pressure of the quark matter that is in chemical equilibrium with the hadronic matter as a function of the baryon density of the hadronic matter. The low-density boundary of the hadron-quark phase transition is indicated by the intersection of the solid line and the dash-dotted line.

Above the low-density boundary determined above, the dense matter enters the mixed phase, in which the hadron phase and the quark phase satisfy following chemical equilibrium, mechanical equilibrium, baryon number conservation, and charge neutrality conditions:

 μbbi−μcqi=μi, PH=PQ, (1−Y)∑b∑τbρτb+Y3∑fρf=ρ, (1−Y)∑b∑τbρτbqτb+Y3∑fρfqf+∑lρlql=0,

where runs over baryons, leptons and quarks, and is the baryon number fraction of the quark phase. We note that in the mixed phase the total charge is zero and the leptons play an important role in maintaining the charge neutrality and -equilibrium conditions. The total energy density and pressure of the mixed phase are calculated according to

 ϵM = (1−Y)VH+YVQ+VL, PM = (1−Y)PH+YPQ+PL.

It is obvious that we have a pure hadron phase for and a pure quark phase for . For the case of the symmetry energy parameter and the bag constant considered above, the pressure of the mixed phase as a function of the baryon density obtained from solving Eq. (III.3) is shown in Fig. 5 by the dotted line. The mixed phase starts at the low-density boundary of the hadron-quark phase transition and ends at the high-density boundary of the hadron-quark phase transition when the matter is a pure quark matter, whose pressure is calculated using the model described in Sec. III.2 and is shown by the dashed line in Fig. 5 as a function of the baryon density.

There is a recent study based on the Relativistic Mean-Field model for the hadron phase and the MIT bag model for the quark phase to compare the behavior of the mixed phase in the Gibbs construction with that in the Maxwell construction, which does not require same charge chemical potential for the hadronic matter and quark matter in the mixed phase (67). The pressure of the mixed phase in the Maxwell construction is found to be constant with respect to its baryon density, which is in contrast with the increasing pressure of the mixed phase with increasing baryon density in the Gibbs construction as seen in Fig. 5. It is worthwhile to point out that although the charge neutrality condition in Eq. (LABEL:cnhq) is satisfied globally, it is violated in each phase. A more realistic equation of state of the mixed phase can be obtained from the Wigner-Seitz cell calculation by taking into account the Coulomb and surface effects. The equation of state of the mixed phase obtained from this approach lies between those from the Gibbs and Maxwell constructions. The latter can thus be viewed as two extreme cases corresponding to certain values of the surface tension parameter in the Wigner-Seitz cell calculation (68); (69).

## Iv Results and discussions

In this section, we use the extended MDI (MDI-Hyp) interaction to study the equation of state and the relative particle fractions in the charge neutral and -stable hypernuclear matter. Including the hadron-quark phase transition in the hypernuclear matter, we further study the mass-radius relation of hybrid stars. Results from different values of the symmetry energy parameter , the bag constant, and attractive and repulsive N interactions are compared.

### iv.1 The hypernuclear matter

Using the extended MDI interaction with hyperons (MDI-Hyp), we have studied the particle fractions in a hypernuclear matter as functions of the baryon density. The results are shown in Fig. 6 for MDI-Hyp-A ((a) with and (b) with ) and MDI-Hyp-R ((c) with and (d) with ) interactions. The density dependence of corresponding particle chemical potentials is shown in Fig. 7. It is seen that the effect of hyperons is more pronounced at higher densities, and the appearance of hyperons prevents the neutron chemical potential from increasing too fast. Similar to the results from other works, the hyperon appears at baryon density of about . For the hyperon, the density at which it appears depends on the sign of the N interaction. For the attractive MDI-Hyp-A interaction, it appears at about , while for the repulsive MDI-Hyp-R interaction, it does not appear until very high densities due to the rapid increase of the chemical potentials of hyperons with increasing total baryon density as shown in Fig. 7, making it difficult to satisfies the -equilibrium condition at small baryon densities. This result is similar to that reported in the literature (31); (40). Also, with the value , corresponding to a stiffer symmetry energy, the fractions of leptons are lower at subsaturation densities but higher at higher densities, which results in a larger charge chemical potential at higher densities and the appearance of negatively (positively) charged hyperons at a lower (higher) density. A stiffer symmetry energy also increases the neutron chemical potential and thus leads to a larger total hyperon fractions at higher densities. Since the total charge of the hypernuclear matter is conserved, an increase in the fraction of negatively charged hyperon results in an increasing fraction of protons and decreasing fraction of leptons. We note that for the MDI-Hyp-R interaction and at the density , about of the particle fraction is made up by hyperons, and the fraction of particles is larger than that of protons at high densities.

In Fig. 8, we show the equation of state of nuclear matter from the MDI interaction with and and of hypernuclear matter from the MDI-Hyp-A and MDI-Hyp-R interactions. It is seen that the EOS is softened when hyperons are present as compared to the EOSs of pure nuclear matter. We note that the EOS at moderate densities plays an important role in determining the maximum mass of neutron stars. Without hyperons, the EOS at moderate densities is stiffer for smaller value of , while it becomes stiffer for larger value of when hyperons are included, especially for a repulsive N interaction. As shown in Fig. 6, a stiffer symmetry energy leads to a larger number of hyperons, which results in a softer EOS as a result of the lower pressure due to the presence of more degrees of freedom, and the effect of hyperons on the EOS is smaller for a soft symmetry energy, as fewer hyperons are then present in the hypernuclear matter. One the other hand, the symmetry energy contribution to the pressure of the hypernuclear matter is larger for than . As a result, the EOS of hypernuclear matter shows similar stiffness for different symmetry energy parameters. Furthermore, the EOS from the MDI-Hyp-R interaction is stiffer than that from the MDI-Hyp-A interaction, as particles do not appear in the former case as shown in Fig. 6. Our results thus show that the hypernuclear matter has a stiffer (softer) EOS at lower densities but a softer (stiffer) EOS at moderate densities with a stiffer (softer) nuclear symmetry energy. We note that the results from both and are consistent with the constraints on the nuclear equation of state obtained from the analysis of the collective flow data in heavy-ion collisions (70).

### iv.2 The hadron-quark phase transition

At higher densities in the core of a neutron star, a transition from hadron matter to quark matter is expected to occur. Here we only consider the hadron-quark phase transition for the repulsive N interaction as it is more consistent with the latest empirical information (58); (56); (59); (60); (57). In Fig. 9, we display the particle fractions of each species in the presence of a hadron-quark phase transition, with the hadron phase from the MDI-Hyp-R interaction with and and the quark phase from the MIT bag model with bag constants MeV and MeV. The particle fractions are weighted by the baryon number and corresponding phase fraction. It is seen that for MeV, the phase transition begins at the density fm and ends at the density fm for , and begins at the density fm and ends at the density fm for , while for MeV it begins at the density fm and ends at the density fm for , and begins at the density fm and ends at the density fm for . The hadron-quark phase transition thus happens at lower baryon number density for a stiffer symmetry energy and for a smaller value of , while the density at which the hadron-quark phase transition ends seems to depend only on the value of but not much on the value of the symmetry energy parameter . With a smaller value of the bag constant, the hadron-quark phase transition both begins and ends earlier. It is also seen that and quarks occupy a larger fraction than quarks in the mixed phase because of their negative charges, so the lepton fraction decreases while the proton fraction increases when the hadron-quark phase transition occurs. Furthermore, only hyperons (no other hyperons) appear in the mixed phase in our model. The fraction of hyperons is, however, sensitive to the value of the bag constant , and with a smaller value of its fraction becomes smaller.

For the EOSs in the presence of the hadron-quark phase transition, they are shown in Fig. 10 with MeV and 170 MeV. Our results show that the EOSs of the mixed phase and the quark phase are both softened in comparison with a pure hadron phase. The difference between the results from and is not large except for the different starting density of the hadron-quark phase transition. Since the EOS is more sensitive to the value of bag constant than to the value of when the quark degrees of freedom are introduced, the bag constant is thus the main parameter in determining the EOS of dense matter as reported in other work (22). The energy density at the end of the hadron-quark phase transition is about GeV/fm for MeV and about GeV/fm for MeV, with the former closer to the value of about GeV/fm obtained from the lattice QCD calculation (71) and extracted from heavy-ion collision experiments (72). Compared with the equation of state constrained by the collective flow data in heavy-ion collisions (70), our results from both values of bag constants satisfy the empirical constraint.

### iv.3 Hybrid stars

In this subsection, we use the MDI-Hyp interaction to study the properties of static hybrid stars with spherically symmetric mass distributions. In particular, we calculate the mass-radius relation of a hybrid stars using Tolman-Oppenheimer-Volkoff (TOV) equation (73); (74)

 dPdr=−(ϵ+P)(mg+4πr3P)r(r−2mg), (30)

where is the gravitational mass inside the radius of the hybrid star given by

 dmgdr=4πr2ϵ(r), (31)

with being the energy density. Above equations are solved by starting from a central energy density and integrating outward until the pressure on the surface of the hybrid star defined by vanishes, i.e, . This gives the radius , and the total gravitational mass of a hybrid star is then given by .

In our calculations, the hybrid star is divided into three parts from the center to the surface: the liquid core, the inner crust and the outer crust as in our previous work (12); (13). In the liquid core it is assumed to be the hypernuclear matter or that with the hadron-quark phase transition, and the resulting EOSs shown in previous subsections are used. In the inner crust, a parameterized EOS of is used as in the previous treatment (12); (13). The outer crust usually consists of heavy nuclei and the electron gas, where we use the BPS EOS (75). The transition density between the liquid core and the inner crust has been consistently determined in our previous work (12); (13), and for the density which distinguishes the edge of inner crust and outer crust, we take it to be fm. The parameters and can then be determined by the pressures and energy densities at and .

In Fig. 11, we show the mass-radius (M-R) and mass-central density (M-) relations of hybrid stars from the EOS of the hypernuclear matter without the hadron-quark phase transition as obtained above based on the MDI-Hyp interaction. Results using both attractive and repulsive N interactions are shown, and the results from a pure nucleonic approach without hyperons are also shown for comparison. The maximum mass obtained with hyperons in hybrid stars is seen to be smaller as the EOS of the hypernuclear matter is softer than that of the nuclear matter. The EOSs at moderate densities play an important role in determining the maximum mass, as the M- relation shows that the neutron star mass can not increase even the central density reaches a very high value. The mass is larger when a repulsive N interaction is used as a result of the smaller number of degrees of freedom and stiffer EOS, as discussed in the previous subsection. For smaller values of a stiffer symmetry energy gives a larger mass as the nuclear matter dominates this density range, and the results are thus the same as those in our previous work (12); (13). With larger , the effect due to the nuclear symmetry energy is small, and the mass from is similar to that from because of the opposite effects from the symmetry pressure and the fraction of hyperons on the equation of state. The maximum mass obtained with an attractive N interaction is for and for , where is the solar mass. With a repulsive N interaction the maximum mass increases to for and for . Our results thus differ from those of many previous works (38); (39) that the maximum mass of hybrid stars cannot reach the canonical value of . In our model, the radius of a hybrid star with a mass of is km for and km for for an attractive N interaction, and km for and km for for a repulsive N interaction, respectively.

In Fig. 12, the M-R and M- relations are displayed for hybrid stars with the hadron-quark phase transition in their liquid core. Again, results from the hadron phase with and and the quark phase with MeV and MeV are shown, and those from a pure nucleonic approach are also shown for comparison. The maximum mass for MeV is for and for , and for MeV it is for and for , respectively. The radius of a standard neutron star with a mass of is km for and km for for MeV, and km for and km for for MeV. If we further reduce the value of , the hadron-quark phase transition would happen at an even lower density, and the maximum mass for the hybrid star would correspond to a larger central density and a smaller radius. Above results are obtained with the Gibbs construction for the hadron-quark phase transition. With the Maxwell construction, the radius of a maximum mass hybrid star would be larger. Within our model, both treatments of the phase transition give, however, reasonable masses and radii for hybrid stars.

Finally, we note that the original MDI interaction does not lead to the violation of causality in -stable matter up to as shown in Panel (d) of Fig. 2 in Ref. (13). Since the EOS of neutron star matter is softened by the presence of hyperons and quarks, the causality condition is still satisfied for the extended MDI interaction used in the present study, and this is confirmed by explicit calculations.

## V Summary

We have extended the MDI interaction for the nucleon-nucleon effective interaction in nuclear medium to include the nucleon-hyperon and hyperon-hyperon interactions by assuming that they have the same density and momentum dependence as that for the nucleon-nucleon interaction. The parameters were determined by fitting the empirical hyperon single-particle potentials in symmetric nuclear matter at its saturation density. As an example for the application of the extended MDI interaction, we investigated the properties of hybrid stars that include not only the hyperon degree of freedom but also that of quarks by taking into account the phase transition between the hadron and quark phases. Our results indicate that the extended MDI interaction can give a reasonable description of the hypernuclear matter. We found that the EOS of the hypernuclear matter is much softer than that of the pure nuclear matter, and that it becomes even softer if the hadron-quark phase transition is included. The masses and radii of hybrid stars were also studied with these EOSs, and they were found to remain reasonable after including hyperons and the hadron-quark phase transition. We have also studied the effects of attractive and repulsive N interactions and different symmetry energies on the hybrid star properties. The results show that the appearance of the hyperon in hybrid stars depends sensitively on the sign of the N interaction with a repulsive N interaction giving a higher critical density for the appearance of hyperons. In addition, a stiffer symmetry energy usually leads to a larger fraction of hyperons in the hypernuclear matter. We further found that both the low-density boundary of the hadron-quark phase transition and the EOS at high densities in hybrid stars are more sensitive to the bag constant than to the stiffness of the nuclear symmetry energy at high densities. This extended MDI interaction, which gives an isospin- and momentum-dependent in-medium effective interactions for the baryon octet, will also be very useful in transport models that simulate heavy-ion reactions in future radioactive beam facilities, particularly at the FAIR/GSI energies.

###### Acknowledgements.
We thank David Blaschke for critical comments on an earlier version of this paper. This work was supported in part by U.S. National Science Foundation under Grant No. PHY-0758115, PHY-0652548 and PHY-0757839, the Welch Foundation under Grant No. A-1358, the Research Corporation under Award No. 7123, the Texas Coordinating Board of Higher Education Award No. 003565-0004-2007, the National Natural Science Foundation of China under Grant Nos. 10675082 and 10975097, MOE of China under project NCET-05-0392, Shanghai Rising-Star Program under Grant No. 06QA14024, the SRF for ROCS, SEM of China, the National Basic Research Program of China (973 Program) under Contract No. 2007CB815004.

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