Phase Transition from QMC Hyperonic Matter to Deconfined Quark Matter

# Phase Transition from QMC Hyperonic Matter to Deconfined Quark Matter

J. D. Carroll    D. B. Leinweber    A. G. Williams Centre for the Subatomic Structure of Matter (CSSM), Department of Physics, University of Adelaide, SA 5005, Australia    A. W. Thomas Thomas Jefferson National Accelerator Facility, 12000 Jefferson Ave., Newport News, VA 23606, USA College of William and Mary, Williamsburg, VA 23187, USA. Centre for the Subatomic Structure of Matter (CSSM), Department of Physics, University of Adelaide, SA 5005, Australia
July 19, 2019
###### Abstract

We investigate the possibility and consequences of phase transitions from an equation of state (EoS) describing nucleons and hyperons interacting via mean-fields of , , and mesons in the recently improved Quark-Meson Coupling (QMC) model to an EoS describing a Fermi gas of quarks in an MIT bag. The transition to a mixed phase of baryons and deconfined quarks, and subsequently to a pure deconfined quark phase is described using the method of Glendenning. The overall EoS for the three phases is calculated for various scenarios and these are used to calculate stellar solutions using the Tolman-Oppenheimer-Volkoff equations. The results are compared to recent experimental data and the validity of each case is discussed with consequences for determining the species content of the interior of neutron stars.

###### pacs:
26.60.Kp, 21.65.Qr, 12.39.-x

## I Introduction

The use of hadronic models to describe high density matter enables us to investigate both the microscopic world of atomic nuclei and the macroscopic world of compact stellar objects, encompassing an enormous range of scales. The results of these investigations provide deep fundamental insight into the way that the world around us is constructed.

Experimental data from both extremes of scale aid in constraining such models, from the saturation properties of nuclear matter to the observed properties of neutron stars Podsiadlowski et al. (2005); Grigorian et al. (2006); Klahn et al. (2007). The literature Lattimer and Prakash (2001); Heiselberg and Hjorth-Jensen (2000); Weber (2005); Schaffner-Bielich (2005); Weber and Weigel (1989); Chin and Walecka (1974) provides a plethora of models for the EoS of hadronic matter, at least some of which have been successfully applied to calculate the properties of finite nuclei. There are also important constraints from data involving heavy-ion collisions Danielewicz et al. (2002); Worley et al. (2008). Many of these EoS have also been applied to neutron star features as well. However, the amount of data available for neutron stars (or compact stellar objects) is very limited, with only a single result containing both a mass and radius result simultaneously Ozel (2006) and even that has been recently disputed Alford et al. (2007a).

With such a lack of constraining data, our focus shifts to finding models which better reflect the physics that is expected to be important under the conditions which we are investigating. A prime example of this is that at the densities which we consider to be interesting for this investigation (1–10 times nuclear density, ) it is possible that either hyperonic matter (in which strangeness carrying baryons become energetically favourable as the Fermi sea fills), quark matter (in which it becomes energetically favourable that the quarks inside the baryons become deconfined) or a mixed phase of these is present, rather than the more traditional treatment of nucleons alone.

We construct a model of high density matter which is globally charge neutral, color neutral, rotationally symmetric, and in a state that is energetically favourable. For this purpose we consider hadronic matter modelled by the Quark-Meson Coupling (QMC) model Guichon and Thomas (2004); Guichon et al. (2006), which was recently improved through the self-consistent inclusion of color hyperfine interactions Rikovska-Stone et al. (2007). While this improvement had no significant effect on the binding of nucleons, it led to impressive results for finite hypernuclei Guichon et al. (2007). We follow the method of Glendenning Glendenning (2001) to produce a mixed phase of hyperonic matter and deconfined quark matter under total mechanical stability, then a pure deconfined quark matter phase with relativistic non-interacting quarks.

We begin with a brief presentation of Relativistic Mean-Field Theory in Section II to establish a foundation with which to discuss the general formalism for the QMC model, including new additions, in Section III. Deconfined quark matter is discussed in Section IV. This is followed by Section V providing a summary of the requirements for and method to construct a phase transition from a hadronic phase to a mixed phase and from a mixed phase to a quark phase. Stellar solutions are calculated in Section VI and a summary of our results is presented in Section VII with conclusions in Section VIII.

## Ii Relativistic Mean-Field Theory

We introduce the mean-field description of nuclear matter using the classic example of Quantum Hadrodynamics (QHD) Chin and Walecka (1974); Serot and Walecka (1986); Furnstahl and Serot (2000). Although the Quark-Meson Coupling model (QMC) has a fundamentally different starting point, namely the self-consistent modification of the structure of a hadron immersed in the nuclear medium Guichon (1988); Saito et al. (1997, 2007), in practice the equations for nuclear matter involve only a few key differences. We summarize those in the next section. The original formulation of QHD included only nucleons interacting with scalar-isoscalar, , and vector-isoscalar, , mesons. This was later expanded to include the vector-isovector, , and subsequently the entire octet of baryons, , with global charge neutrality upheld via leptons, .

The Lagrangian density for QHD is

 L = ∑k¯ψk[γμ(i∂μ−gωkωμ−gρ→τ(k)⋅→ρμ)−(Mk−gσkσ)]ψk +12(∂μσ∂μσ−m2σσ2)−14FμνFμν−14RaμνRμνa +12m2ωωμωμ+12m2ρρaμρμa+¯ψℓ[γμi∂μ−mℓ]ψℓ+δL,

where the index represents each isospin group of the baryon states, and corresponds to the Dirac spinors for these

 (2)

The vector field tensors are

 Fμν=∂μων−∂νωμ,Rμνa=∂μρνa−∂νρμa−gρϵabcρμbρνc, (3)

The third components of the isospin matrices are

 (4)

is a spinor for the lepton states, and are renormalisation terms. We do not include pions here as they provide no contribution to the mean-field, because the ground state of nuclear matter is parity-even. We have neglected nonlinear meson terms in this description for comparison purposes, though it has been shown that the inclusion of nonlinear scalar meson terms produces a framework consistent with the QMC model without the added hyperfine interaction Muller and Jennings (1997). The values of the baryon and meson masses in vacuum are summarized in Table 1.

Assuming that the baryon density is sufficiently large, we use a Mean-Field Approximation (MFA) with physical parameters (breaking charge symmetry) in which the meson fields are replaced by their classical vacuum expectation values. With this condition, the renormalisation terms can be neglected.

By enforcing rotational symmetry and working in the frame where the matter as a whole is at rest, we set all of the 3-vector components of the vector meson fields to zero, leaving only the temporal components. Furthermore, by enforcing isospin symmetry we remove all charged meson states. Consequently, because the mean-fields are constant, all meson derivative terms vanish, and thus so do the vector field tensors. The only non-zero components of the vector meson mean fields are then the time components, and . Similarly, only the third component of the meson mean field in iso-space is non-zero, corresponding to the uncharged meson.

The couplings of the mesons to the baryons are found via SU(6) flavor-symmetry Rijken et al. (1999). This produces the following relations for the and couplings to each isospin group (and hence each baryon in that isospin group)

 13gσN=12gσΛ=12gσΣ=gσΞ,13gωN=12gωΛ=12gωΣ=gωΞ. (5)

Using the formalism as above with isospin expressed explicitly in the Lagrangian density, the couplings of the meson to the octet baryons are unified, thus by specifying , , and we are therefore able to determine the couplings to the remaining baryons.

By evaluating the equations of motion from the Euler-Lagrange equations

 ∂L∂ϕi−∂μ∂L∂(∂μϕi)=0, (6)

we find the mean-field equations for each of the mesons, as well as the baryons. The equations for the meson fields are

 ⟨σ⟩ =∑BgσBm2σ⟨¯ψBψB⟩, (7) ⟨ω⟩ =∑BgωBm2ω⟨¯ψBγ0ψB⟩=∑BgωBm2ω⟨ψ†BψB⟩, (8) ⟨ρ⟩ =∑kgρm2ρ⟨¯ψkγ0τ(k)3ψk⟩=∑kgρm2ρ⟨ψ†kτ(k)3ψk⟩=∑Bgρm2ρ⟨ψ†BI3BψB⟩, (9)

where the sum over corresponds to the sum over the octet baryon states, and the sum over corresponds to the sum over isospin groups. is the third component of isospin of baryon , as found in the diagonal elements of . , , and are proportional to the conserved baryon density, isospin density and scalar density respectively, where the scalar density is calculated self-consistently.

The Euler-Lagrange equations also provide a Dirac equation for the baryons

 ∑B[i⧸∂−gωBγ0⟨ω⟩−gργ0I3B⟨ρ⟩−MB+gσB⟨σ⟩]ψB=0. (10)

At this point, we can define the baryon effective mass as

 M∗B=MB−gσB⟨σ⟩, (11)

and the baryon chemical potential (also known as the Fermi energy, the energy associated with the Dirac equation) as

 μB=ϵFB=√k2FB+(M∗B)2+gωB⟨ω⟩+gρI3B⟨ρ⟩. (12)

The chemical potentials for the leptons are found via

 μℓ=√k2Fℓ+m2ℓ. (13)

The energy density, , and pressure, , for the EoS can be obtained using the relations for the energy-momentum tensor (where is the 4-velocity)

 ⟨Tμν⟩=(E+P)uμuν+Pgμν,⇒P=13⟨Tii⟩,E=⟨T00⟩, (14)

since and , where here is the inverse metric tensor having a negative temporal component, and is the energy-momentum tensor. In accordance with Noether’s Theorem, the relation between the energy momentum tensor and the Lagrangian density is

 Tμν=−gμνL+∂μψ∂L∂(∂νψ), (15)

and we find the Hartree-level energy density and pressure for the system as a sum of contributions from baryons, ; leptons, ; and mesons, to be

 E =∑j=B,ℓ,mEj =∑i=B,ℓ(2Ji+1)(2π)3∫θ(kFi−|→k|)√k2+(M∗i)2d3k+∑α=σ,ω,ρ12m2α⟨α⟩2, P =∑j=B,ℓ,mPj =∑i=B,ℓ(2Ji+1)3(2π)3∫k2θ(kFi−|→k|)√k2+(M∗i)2d3k+∑α=ω,ρ12m2α⟨α⟩2−12m2σ⟨σ⟩2,

where is the spin of particle () which in this case accounts for the availability of both up and down spin-states. is the Heaviside Step Function. Note that the pressure arising from the vector mesons is positive, while it is negative for the scalar meson.

The total baryon density, , can be calculated via

 ρ=∑BρB=∑B(2JB+1)(2π)3∫θ(kFB−|→k|)d3k, (18)

where in symmetric matter, the Fermi momenta are related via , and the binding energy per baryon, , is determined via

 E=[1ρ(E−∑BMBρB)]. (19)

The couplings and are determined such that symmetric nuclear matter (in which ) saturates with the appropriate minimum in the binding energy per baryon of at a nuclear density of . The couplings for QHD which provide a fit to saturated nuclear matter are shown in Table 2.

The coupling is fixed such that the nucleon symmetry energy, given by

 asym=g2ρ12π2m2ρk3F+112k2F√k2F+(M∗p)2+112k2F√k2F+(M∗n)2, (20)

is reproduced at saturation as .

The chemical potential for any particle, , can be related to two independent chemical potentials — we choose that of the neutron () and the electron () — and thus we use a general relation

 μi=Biμn−Qiμe ; i∈{p,n,Λ,Σ+,Σ0,Σ−,Ξ0,Ξ−,ℓ}, (21)

where and are the baryon (unitless) and electric (in units of the proton charge) charges respectively. For example, the proton has and , so it must satisfy which is familiar as -equilibrium. Since neutrinos are able to escape the star, we consider . Leptons have , and all baryons have .

The relations between the chemical potentials are therefore derived to be

 μΛ=μΣ0=μΞ0=μn,μΣ−=μΞ−=μn+μe,μp=μΣ+=μn−μe,μμ=μe. (22)

The EoS for QHD can be obtained by finding solutions to Eqs. (79) subject to charge neutrality, conservation of a chosen total baryon number, and equivalence of chemical potentials. These conditions can be summarised as

 0=∑iQiρiρ=∑iBiρiμi=Biμn−Qiμe⎫⎪⎬⎪⎭i∈{p,n,Λ,Σ+,Σ0,Σ−,Ξ0,Ξ−,ℓ}. (23)

With these conditions, we are able to find the EoS for QHD. It should be noted that, as with many relativistic models for baryonic matter, once we include more than one species of baryon this model eventually produces baryons with negative effective masses at sufficiently high densities (). This is a direct result of the linear nature of the effective mass as shown in Eq. (11). As the Fermi energy (see Eq. (12)) approaches zero, the cost associated with producing baryon-anti-baryon pairs is reduced and at this point the model breaks down. From a more physical point of view, as the density rises one would expect that the internal structure of the baryons should play a role in the dynamics. Indeed, within the QMC model, the response of the internal structure of the baryons to the applied mean scalar field ensures that no baryon mass ever becomes negative. We now describe the essential changes associated with the QMC model.

## Iii QMC model

Like QHD, QMC is a relativistic quantum field theory formulated in terms of the exchange of scalar and vector mesons. However, in contrast with QHD these mesons couple not to structureless baryons but to clusters of confined quarks. As the density of the medium grows and the mean scalar and vector fields grow, the structure of the clusters adjusts self-consistently in response to the mean-field coupling. While such a model would be extremely complicated to solve in general, it has been shown by Guichon et al. Guichon et al. (1996) that in finite nuclei one should expect the Born-Oppenheimer approximation to be good at the 3% level. Of course, in nuclear matter it is exact at mean-field level.

Within the Born-Oppenheimer approximation, the major effect of including the structure of the baryon is that the internal quark wave functions respond in a way that opposes the applied scalar field. To a very good approximation this physics is described through the “scalar polarizability,” , which in analogy with the electric polarizability describes the term in the baryon effective mass quadratic in the applied scalar field Guichon (1988); Thomas et al. (2004); Ericson and Chanfray (2008); Massot and Chanfray (2008); Chanfray et al. (2003). Recent explicit calculations of the equivalent energy functional for the QMC model have demonstrated the very natural link between the existence of the scalar polarizability and the many-body forces, or equivalently the density dependence, associated with successful, phenomenological forces of the Skyrme type Guichon and Thomas (2004); Guichon et al. (2006). In nuclear matter the scalar polarizability is the only effect of the internal structure in mean-field approximation. On the other hand, in finite nuclei the variation of the vector field across the hadronic volume also leads to a spin-orbit term in the nucleon energy Guichon et al. (1996).

Once one chooses a quark model for the baryons, and specifies the quark-level meson couplings, there are no new parameters associated with introducing any species of baryon into the nuclear matter. Given the well known lack of experimental constraints on the forces between nucleons and hyperons, let alone hyperons and hyperons, which will be of great practical importance as the nuclear density rises above (2–3), this is a particularly attractive feature of the QMC approach and it is crucial for our current investigation. Indeed, we point to the very exciting recent results of the QMC model, modified to include the effect of the scalar field on the hyperfine interaction, which led to hypernuclei being bound in quite good agreement with experiment and hypernuclei being unbound because of the modification of the hyperfine interaction Guichon et al. (2007) - thus yielding a very natural explanation of this observed fact. We note the success that this description has found for finite nuclei as noted in Guichon et al. (2006).

While we focus on the MIT bag model Chodos et al. (1974) as our approximation to baryon structure, we note that there has been a parallel development Bentz and Thomas (2001) based upon the covariant, chiral symmetric NJL model Nambu and Jona-Lasinio (1961), with quark confinement modelled using the proper time regularization proposed by the Tübingen group Hellstern et al. (1997); Ebert et al. (1996). The latter model has many advantages for the computation of the medium modification of form factors and structure functions, with the results for spin structure functions Cloet et al. (2005, 2006) offering a unique opportunity to test the fundamental idea of the QMC model experimentally. However, in both models it is the effect of quark confinement that leads to a positive polarizability and a natural saturation mechanism.

Although the underlying physics of QHD and QMC is rather different, at the hadronic level the equations to be solved are very similar. We therefore focus on the changes which are required.

1.  Because of the scalar polarizability of the hadrons, which accounts for the self-consistent response of the internal quark structure of the baryon to the applied scalar field Guichon et al. (2006), the effective masses appearing in QMC are non-linear in the mean field. We write them in the general form

 M∗B=MB−wσBgσN⟨σ⟩+d2~wσB(gσN⟨σ⟩)2, (24)

where the weightings, and the scalar polarizability of the nucleon, , must be calculated from the underlying quark model. Note now that only the coupling to the nucleons, , is required to determine all the effective masses.

The most recent calculation of these effective masses, including the in-medium dependence of the spin dependent hyperfine interaction Guichon et al. (2007), yields the explicit expressions:

 MN(⟨σ⟩) = MN−gσN⟨σ⟩ +[0.0022+0.1055RfreeN−0.0178(RfreeN)2](gσN⟨σ⟩)2, MΛ(⟨σ⟩) = MΛ−[0.6672+0.0462RfreeN−0.0021(RfreeN)2]gσN⟨σ⟩ +[0.0016+0.0686RfreeN−0.0084(RfreeN)2](gσN⟨σ⟩)2, MΣ(⟨σ⟩) = MΣ−[0.6706−0.0638RfreeN−0.008(RfreeN)2]gσN⟨σ⟩ +[−0.0007+0.0786RfreeN−0.0181(RfreeN)2](gσN⟨σ⟩)2, MΞ(⟨σ⟩) = MΞ−[0.3395+0.02822RfreeN−0.0128(RfreeN)2]gσN⟨σ⟩ +[−0.0014+0.0416RfreeN−0.0061(RfreeN)2](gσN⟨σ⟩)2.

We take as the preferred value of the free nucleon radius, although in practice the numerical results depend only very weakly on this parameter Guichon et al. (2006).

Given the parameters in Eq. (III), all the effective masses for the baryon octet are entirely determined. They are plotted as functions of in Fig. 1 and we see clearly that they never become negative. (Note that the range of covered here corresponds to densities up to (6–8)).

2.  Since the mean scalar field, , is derived self-consistently by taking the derivative of the energy density with respect to , the scalar field equation

 ⟨σ⟩=∑BgσNm2σC(⟨σ⟩)(2JB+1)(2π)3∫M∗Bθ(kFB−|→k|)√k2+(M∗B)2d3k, (26)

has an extra factor, denoted by

 C(⟨σ⟩)=[wσB−~wσBdgσN⟨σ⟩]. (27)

Note that the term (the scalar polarizability) in does not have the factor of that is found Eq. (24), because of the differentiation.

Given this new term in the equation for the mean scalar field, we can see that this allows feedback of the scalar field which is modelling the internal degrees of freedom of the baryons. This feedback prevents certain values of from being accessed.

3.  The couplings to the proton are re-determined by the fit to saturation properties (minimum binding energy per baryon and saturation density) with the new effective masses for the proton and neutron. The couplings for QMC which provide a fit to saturated nuclear matter are shown in Table 3.

Given these changes alone, QHD is transformed into QMC. When we compare the results of Section VII with those of Ref. Rikovska-Stone et al. (2007) minor differences arise because the QMC calculations in Ref. Rikovska-Stone et al. (2007) are performed at Hartree-Fock level, whereas here they have been performed at Hartree level (mean-field) only.

## Iv Deconfined Quark Matter

We consider two models for a deconfined quark matter phase, both of which model free quarks in -equilibrium. The first model, the MIT bag model Chodos et al. (1974), is commonly used to describe the quark matter phase because of its simplicity.

In this model we consider three quarks with fixed masses to possess chemical potentials related to the independent chemical potentials of Eq. (21) via

 μu=13μn−23μe,μd=13μn+13μe,μs=μd, (28)

where quarks have a baryon charge of since baryons contain 3 quarks. Because the quarks are taken to be free, the chemical potential has no vector interaction terms, and thus

 μq=√k2Fq+m2q ;q∈{u,d,s}. (29)

The EoS can therefore be solved under the conditions of Eq. (23).

As an alternative model for deconfined quark matter, we consider a simplified Nambu-Jona-Lasinio (NJL) model Nambu and Jona-Lasinio (1961), in which the quarks have dynamically generated masses, ranging from constituent quark masses at low densities to current quark masses at high densities. The equation for a quark condensate at a given density (and hence, ) in NJL is similar to the scalar field in QHD/QMC, and is written as

 ⟨¯ψψ⟩=−4 Nc∫1(2π3)M∗qθ(kF−|→k|)θ(Λ−kF)√k2+(M∗q)2d3k, (30)

where denotes the dependent (hence, density dependent) quark mass; is the number of color degrees of freedom of quarks; and is the momentum cutoff. This is self-consistently calculated via

 M∗q=Mcurrent−G⟨¯ψψ⟩, (31)

where is the coupling and the current quark mass.

To solve for the quark mass at each density, we must first find the coupling, , which yields the required constituent quark mass in free space (). The coupling is assumed to remain constant as the density rises. In free space, we can solve the above equations to find the coupling

 G=(M∗q−Mcurrent)4 Nc[∫1(2π)3M∗qθ(|→k|−kF)θ(Λ−kF)√k2+(M∗q)2d3k]−1∣∣∣kF=0. (32)

We solve Eqs. (3032) for to obtain constituent quark masses of using current quark masses of for the light quarks, and to obtain a constituent quark mass of using a current quark mass of for the strange quark, with a momentum cutoff of . At we find the couplings to be

 Gu,d=0.148 fm2,Gs=0.105 fm2. (33)

We can now use these parameters to evaluate the dynamic quark mass , for varying values of , by solving Eq. (30) and Eq. (31) self-consistently. The resulting density dependence of is illustrated in Fig. 2. This shows that the masses of the quarks eventually saturate and are somewhat constant above a certain density.

We can then construct the EoS in the same way as we did for the MIT bag model, but with density-dependent masses, rather than fixed masses.

## V Phase Transitions

### v.1 Equilibrium Conditions

We now have a description of hadronic matter with quark degrees of freedom, but we are still faced with the issue that the baryons are very densely packed. We wish to know if it is more energetically favourable for deconfined quark matter to be the dominant phase at a certain density. To do this, we need to find a point (if it exists) at which stability is achieved between the hadronic phase and the quark phase.

The condition for stability is that chemical, thermal, and mechanical equilibrium between the hadronic () and quark () phases is achieved, and thus that the independent quantities in each phase are separately equal. Thus the two independent chemical potentials, (), are each separately equal to their counterparts in the other phase, i.e. , and (chemical equilibrium); the temperatures are equal () (thermal equilibrium); and the pressures are equal () (mechanical equilibrium). For a discussion of this condition, see Ref. Reif (1965). We consider both phases to be cold on the nuclear scale, and assume , so the temperatures are by construction equal. We must therefore find the point at which, for a given pair of independent chemical potentials, the pressures in both the hadronic phase and the quark phase are the same.

To find the partial pressure of any baryon, quark, or lepton species, , we use

 Pi=(2JB+1)Nc3(2π)3∫k2θ(kFi−|→k|)√k2+(M∗i)2d3k, (34)

where for quarks, and for baryons and leptons. To find the total pressure in each phase we use

 PH=∑BPB+∑ℓPℓ+∑α=ω,ρ12m2α⟨α⟩2−12m2σ⟨σ⟩2, (35)

which is equivalent to Eq. (LABEL:eq:P_H), and

 PQ=∑qPq+∑ℓPℓ−B, (36)

where in the quark pressure is the bag energy density. For the QMC model described in Section III, and a Fermi gas of quarks, both with interactions with leptons for charge neutrality, a point exists at which the condition of stability, as described above, is satisfied.

At this point, it is equally favourable that hadronic matter and quark matter are the dominant phase. Beyond this point, the quark pressure is greater than the hadronic pressure, and so the quark phase has a lower thermodynamic potential (through the relation ) and the quark phase will be more energetically favourable. To determine the EoS beyond this point, we need to consider a mixed phase.

### v.2 Mixed Phase

We can model a mixed phase of hadronic and quark matter — as opposed to modelling a simple direct phase transition between the two, a Maxwell construction, which would have a discontinuity in the density, while retaining a constant pressure between the two phases — using the method of Glendenning. A detailed description of this appears in Ref. Glendenning (2001).

We solve for the hadronic EoS using the independent chemical potentials as inputs for the quark matter EoS, as the order parameter, , the conserved baryon density, increases until we find a point (if it exists) at which the pressure in the quark phase is equal to that of the hadronic phase. Once we have the density and pressure at which the phase transition occurs, we change the order parameter from the conserved baryon density to the quark fraction, . If we consider the mixed phase to be a fraction of the hadronic matter and a fraction of the quark matter, then the mixed phase (MP) of matter will have the following properties; the total density will be

 ρMP=(1−χ)ρHP+χρQP, (37)

where and are the densities in the hadronic and quark phases, respectively. The equivalent baryon density in the quark phase,

 ρQP=∑qρq=3(ρu+ρd+ρs), (38)

arises because of the restriction that a bag must contain 3 quarks.

According to the condition of mechanical equilibrium, the pressure in the mixed phase will be

 PMP=PHP=PQP. (39)

We can step through values and find the density at which equilibrium is achieved, keeping the mechanical stability conditions as they were above. In the mixed phase we need to alter our definition of charge neutrality; it becomes possible now that one phase is (locally) charged, while the other phase carries the opposite charge, making the system globally charge neutral. This is achieved by enforcing

 0=(1−χ)ρcHP+χρcQP+ρcℓ, (40)

where this time we are considering charge densities, which are simply charge proportions of density and is the lepton charge density. For example, the charge density in the quark phase is given by

 ρcQP=∑qQqρq=23ρu−13ρd−13ρs. (41)

We continue to calculate the densities until we reach , at which point the mixed phase is now entirely charge neutral quark matter. After this point, we continue with the EoS for pure charge neutral quark matter, using as the order parameter.

## Vi Stellar Solutions

To test the predictions of these models, we find solutions of the Tolman-Oppenheimer-Volkoff (TOV) Oppenheimer and Volkoff (1939) equation

 dPdR=−G(P+E)(M(R)+4πR3P)R(R−2GM(R)), (42)

where the mass, , contained within a radius is found by integrating the energy density

 M(R)=∫R04πr2Edr, (43)

and and are the energy density and pressure in the EoS, respectively.

Given an EoS and a choice for the central density of the star, this provides static, spherically symmetric, non-rotating, gravitationally stable stellar solutions for the total mass and radius of a star. For studies of the effect of rapid rotation in General Relativity we refer to Refs. Lattimer and Schutz (2005); Owen (2005). This becomes important for comparison to experimental data, as only data for stellar masses exists (with the single, disputed exception from Ozel (2006)), we can use the model to predict the radii of the observed stars.

## Vii Results

To obtain numerical results, we solve the meson field equations, Eqs. (79), with the conditions of charge neutrality, fixed baryon density, and the equivalence of chemical potentials given by Eq. (23), for various models. Having found the EoS by evaluating the energy density, Eq. (II), and pressure, Eq. (LABEL:eq:P_H), we can solve for stellar solutions for an EoS using the TOV equation. The radius of the star is defined as the radius at which the pressure is zero and is calculated using a fourth-order Runge-Kutta integration method.

The EoS for octet QMC hadronic matter is shown in Fig. 4 alongside the same model when including a phase transition to 3-flavor quark matter modelled with the MIT bag model, and the results do not appear to differ much at this scale. The theoretical causality limit of is also shown (corresponding to the limit ) and we can see that these models do not approach this limit at the scale displayed. This is because of the softening of the EoS that occurs with the introduction of hyperons, enlarging the Fermi sea to be filled and reducing the overall pressure.

The species fraction for each particle, , is simply the density fraction of that particle, and is calculated via

 Yi=ρiρ ;i∈{p,n,Λ,Σ+,Σ0,Σ−,Ξ0,Ξ−,ℓ,q}, (44)

where is the total baryon density. The species fractions for octet QMC when a phase transition is neglected are shown in Fig. 5, where we note that the species fraction is enhanced and the species fractions are suppressed with increasing density. From the investigations by Rikovska-Stone et al. Rikovska-Stone et al. (2007) we expect that the would disappear entirely if we were to include Fock terms.

The value of compression modulus and effective nucleon mass at saturation are frequently used as a comparison to experimental evidence. Models which neglect quark-level interactions, such as QHD, typically predict much higher values for the compression modulus than experiments suggest. In the symmetric (nuclear) matter QHD model described in this paper, we find values of and which are in agreement with Serot and Walecka (1986), but as stated in that reference, not with experiment. For QMC we find a significant improvement in the compression modulus; which lies at the upper end of the experimental range. The nucleon effective mass at saturation for QMC is found to be , producing .

When we calculate the EoS including a mixed phase and subsequent pure quark phase, we find that small changes in the parameters can sometimes lead to very significant changes. In particular, the bag energy density, , and the quark masses in the MIT bag model have the ability to both move the phase transition points, and to vary the constituents of the mixed phase. We have investigated the range of parameters which yield a transition to a mixed phase and these are summarised in Table 4. For illustrative purposes we show an example of species fractions for a reasonable set of parameters ( and ) in Fig. 6. Note that in this case the hyperon enters the mixed phase briefly (and at a low species fraction).

Note that the transition density of produced by the combination of the octet QMC and MIT bag models (as shown in Fig. 6) is clearly not physical as it implies the presence of deconfined quarks at densities less than .

With small changes to parameters, such as those used to produce Fig. 7 in which the bag energy density is given a slightly higher value from that used in Fig. 6 ( increased from to , but the quark masses remain the same), it becomes possible for the hyperons to also enter the mixed phase, albeit in that case with small species fractions, .

The TOV solutions for octet QMC with and without a phase transition to a mixed phase are shown in Fig. 8. The stellar masses produced using these methods are similar to observed neutron star masses. Once we have solved the TOV equations, we can examine individual solutions and determine the species content for specific stars. If we examine the solutions with a stellar mass of , where is a solar mass, for the set of parameters used to produce Figs. 5 and 6, we can find the species fraction as a function of stellar radius to obtain a cross-section of the star. This is shown in Fig. 9 for the case of no phase transition, and Fig. 10 for the case where we allow a transition to a mixed phase, and subsequently to a quark matter phase.

If we now examine the stellar solution with mass of the set of parameters ( and used to produce Fig. 7) as shown in Fig. 11, we note that the quark content of this 10.5 km star reaches out to around 8 km, and that the core of the star contains roughly equal proportions of protons, neutrons and hyperons with .

Within a mixed phase, we require that for a given pair of and at any value of the mixing parameter , the quark density is greater than the hadronic density. This condition ensures that the total baryon density increases monotonically within the range , as can be seen in Eq. (37). An example of this is illustrated in Fig. 12 for a mixed phase of octet QMC and 3-flavor quark matter modelled with the MIT bag model.

The use of quark masses corresponding to the NJL model results in a quark density that is lower than the hadronic density, and as a result there are no solutions for a mixed phase in which the proportion of quarks increases with fraction , while at the same time the total baryon density increases. It may be possible that with smaller constituent quark masses at low density, the Fermi momenta would provide sufficiently high quark densities, but we feel that it would be unphysical to use any smaller constituent quark masses. This result implies that, at least for the model we have investigated, dynamical chiral symmetry breaking (in the production of constituent quark masses at low density) prevents a phase transition from a hadronic phase to a mixed phase involving quarks.

We do note, however, that if we restrict consideration to nucleons only within the QMC model (with the same parameters as octet QMC), and represent quark matter with the NJL model, we do in fact find a possible mixed phase. More surprisingly, the phase transition density for this combination is significantly larger than the case where hyperons are present. An example of this is shown in Fig. 13 with parameters found in Table 4. This produces a mixed phase at about () and a pure quark matter phase above about (). We note the coincidence of this phase transition density with the density corresponding to one nucleon per nucleon volume, with the aforementioned assumption of , though we do not draw any conclusions from this. Performing this calculation with quark matter modelled with the MIT bag model produces results similar to those of Fig. 6 except of course lacking the hyperon contribution. Although this example does show a phase transition, the omission of hyperons is certainly unrealistic. This does however illustrate the importance and significance of including hyperons, in that their inclusion alters the chemical potentials which satisfy the equilibrium conditions in such a way that the mixed phase is no longer produced.

For each of the cases where we find a phase transition from baryonic matter to quark matter, the solution consists of negatively charged quark matter, positively charged hadronic matter, and a small proportion of leptons, to produce globally charge neutral matter. The proportions of hadronic, leptonic and quark matter throughout the mixed phase (for example, during a transition from octet QMC matter to 3-flavor quark matter modelled with the MIT bag model) are displayed in Fig. 14. A summary of the results of interest is given in Table 4.

Results for larger quark masses are not shown, as they require a much lower bag energy density to satisfy the equilibrium conditions. For constituent quark masses, we find that no phase transition is possible for any value of the bag energy density, as the quark pressure does not rise sufficiently fast to overcome the hadronic pressure. This is merely because the mass of the quarks does not allow a sufficiently large Fermi momentum at a given chemical potential, according to Eq. (29).

## Viii Conclusions

We have produced several EoS that simulate a phase transition from octet QMC modelled hadronic matter, via a continuous Glendenning style mixed phase to a pure, deconfined quark matter phase. This should correspond to a reasonable description of the relevant degrees of freedom in each density region. The models used here for quark matter provide a framework for exploring the way that this form of matter may behave, in particular under extreme conditions. The success of the QMC model in reproducing a broad range of experimental data gives us considerable confidence in this aspect of these calculations, and provides a reasonable hadronic sector and calculation framework, which then awaits improvement in the quark sector to produce realistic stellar solutions.

We have presented EoS and stellar solutions for octet QMC matter at Hartree level. We have explored several possible phase transitions from this hadronic sector to a mixed phase involving 3-flavor quark matter. The corresponding EoS demonstrate the complexity and intricacy of the solutions as well as the dependence on small changes in parameters. The stellar solutions provide overlap with the lower end of the experimentally acceptable range.

Several investigations were made of the response of the model to a more sophisticated treatment of the quark masses in-medium, namely the NJL model. In that model the quark masses arise from dynamical chiral symmetry breaking and thus take values typical of constituent quarks at low density and drop to current quark masses at higher densities. The result is that no transition to a mixed phase is possible in this case.

The omission of hyperons in the QMC model yields a transition to a mixed phase of either NJL or MIT bag model quark matter, as the hadronic EoS is no longer as soft. This observation makes it clear that hyperons can play a significant role in the EoS. However, we acknowledge that their presence in neutron stars remains speculative.

The models considered here reveal some important things about the possible nature of the dense nuclear matter in a neutron star. It seems that if dynamical chiral symmetry does indeed result in typical constituent quark masses in low density quark matter, then a phase transition from hadronic matter to quark matter is unlikely. This result invites further investigation.

The results presented in Fig. 8 indicate that the model in its current form is unable to reproduce sufficiently massive neutron stars to account for all observations, notably the observed stellar masses of and larger. This is a direct result of the softness of the EoS. This issue will be explored in a future publication via the inclusion of Fock terms, which have been shown to have an effect on the scalar and vector potentials Krein et al. (1999).

Many open questions remain to be investigated in further work, including the effects of Fock terms, and the density dependence of the bag energy density in the quark phase, which can be calculated explicitly within the NJL model. The quark matter models used here are still not the most sophisticated models available, and further work may involve an investigation of the effects of color-superconducting quark matter Alford et al. (2007b); Lawley et al. (2006).

###### Acknowledgements.
This research was supported in part by DOE contract DE-AC05-06OR23177, (under which Jefferson Science Associates, LLC, operates Jefferson Lab) and in part by the Australian Research Council. JDC would like to thank Jefferson Lab for their hospitality and and to thank Ping Wang for helpful discussions.

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