Phenomenological QCD equation of state for massive neutron stars
PPhenomenological QCD equation of state for massive neutron stars
Toru Kojo, Philip D. Powell,
1, 2
Yifan Song, and Gordon Baym
1, 3 Department of Physics, University of Illinois at Urbana-Champaign,1110 W. Green Street, Urbana, Illinois 61801, USA Lawrence Livermore National Laboratory, 7000 East Ave., Livermore, CA 94550 Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan (Dated: April 19, 2018)We construct an equation of state for massive neutron stars based on quantum chromodynam-ics phenomenology. Our primary purpose is to delineate the relevant ingredients of equations ofstate that simultaneously have the required stiffness and satisfy constraints from thermodynamicsand causality. These ingredients are: (i) a repulsive density-density interaction, universal for allflavors; (ii) the color-magnetic interaction active from low to high densities; (iii) confining effects,which become increasingly important as the baryon density decreases; (iv) nonperturbative gluons,which are not very sensitive to changes of the quark density. We use the following ”3-window”description: At baryon densities below about twice normal nuclear density, 2 n , we use the Akmal-Pandharipande-Ravenhall (APR) equation of state, and at high densities, ≥ (4 − n , we use thethree-flavor Nambu-Jona-Lasinio (NJL) model supplemented by vector and diquark interactions. Inthe transition density region, we smoothly interpolate the hadronic and quark equations of state inthe chemical potential-pressure plane. Requiring that the equation of state approach APR at lowdensities, we find that the quark pressure in nonconfining models can be larger than the hadronicpressure, unlike in conventional equations of state. We show that consistent equations of state ofstiffness sufficient to allow massive neutron stars are reasonably tightly constrained, suggesting thatgluon dynamics remains nonperturbative even at baryon densities ∼ n . I. INTRODUCTION
As the Relativistic Heavy Ion Collider and the LargeHadron Collider continue to push the limits of our ex-perimental knowledge of hot dense quantum chromody-namics (QCD), neutron stars are the only cosmic lab-oratories in which we can study the structure of colddense QCD [1–5]. The recent discoveries of neutron starswith masses M (cid:39) M (cid:12) ( M (cid:12) is the solar mass), includ-ing the binary millisecond pulsar J1614-2230 with mass(1 . ± . M (cid:12) [6] and the pulsar J0348+0432 with mass(2 . ± . M (cid:12) [7] (also PSR J1311-3430 [8]), togetherwith recent simultaneous determinations of neutron starmasses and radii [9, 10] pose particular challenges to thetheoretical construction of the neutron star equation ofstate.On the one hand, the existence of these massive starssuggests that the equation of state must be stiffer thanconventional hadronic descriptions of matter includinghyperons. Furthermore, the central baryon density inneutron stars with masses ∼ M (cid:12) well exceeds twice nu-clear matter density n , and may reach as high as ∼ n .To understand why such high mass stars are stable re-quires a knowledge of the equation of state at baryondensities n B over a range ∼ − n . However we can-not at present reliably calculate the equation of state oversuch a range; the densities are too high to apply reliablyconventional hadronic equations of state, and too low toapply perturbative QCD.This situation motivates us to investigate the prop-erties of strongly correlated quark matter, intermediatebetween the hadronic and perturbative QCD phases, andask how the properties of such matter is constrained by neutron star observations. Using a schematic quarkmodel, we manifestly take into account quark degrees offreedom, while including interaction effects such as vec-tor repulsion between quarks, known from hadron spec-troscopy, color-magnetic diquark interactions, and six-quark interactions arising from the axial anomaly. Weexamine the roles of these interactions and find that itis possible, within a reasonable parameter range, to con-struct an equation of state that (i) is sufficiently stiffto include stable stars with M ∼ M (cid:12) ; (ii) satisfiesthe thermodynamic constraint that the baryon numberdensity be an increasing function of the baryon chemi-cal potential, ∂n B /∂µ B >
0; and (iii) is consistent withthe (suggestive) causality constraint that the speed ofsound (at zero frequency) not exceed the speed of light[11, 12]. While these conditions provide relatively tightconstraints on the quark matter equation of state, it isnonetheless possible to construct the desired equation ofstate using quark model parameters compatible with thehadron spectroscopy.To further motivate the picture of strongly correlatedquark matter, we briefly review the domain of applicabil-ity of hadronic and perturbative QCD equations of state.Conventional hadronic equations of state are constrainedby experimental data at low energy and density, e.g.,two-body hadronic scattering below the pion productionthreshold, the masses and level structure of light nuclei,and nuclear matter around nuclear saturation density n .While hadronic equations of state include the relevantphysics in the low density regime in which their param-eters are fit, with increasing density multiple meson ex-changes, many-baryon interactions, and virtual baryonicexcitations become increasingly important. (The system- a r X i v : . [ h e p - ph ] J a n FIG. 1:
Pressure vs. quark chemical potential for several equations of state. The black line is the APR result [16] (A18+ δv +UIX*without pion condensation)): the bold line for n B < n and thin dotted line for n B > n . Various effects are successively added tothe standard NJL model; (a) a repulsive density-density interaction, which stiffens the NJL equation of state ; (b) the color-magneticinteraction (diquark correlation), which reduces the average quark energy at all densities; and (c) confining effects, which suppress theartificially large pressure in NJL models at low density down to the APR pressure, discussed in the text. atics can be most clearly seen in the chiral effective theoryapproach [13–15].) In nucleonic potential models [16], thethree-body nucleon interaction is crucial to reproducingnuclear matter properties at n B (cid:39) n , and its contribu-tion to the energy density can be even comparable (and ofopposite sign) to that of the two-body force at n B ∼ n .Beyond baryon densities n B (cid:29) n a well defined expan-sion in terms of static two-, three-, or more, body forcesno longer exists.The equation of state of perturbative QCD [17–19] re-lies on a picture of weakly coupled quarks and gluons.A current state-of-the-art calculation in this regime, tosecond order in the strong interaction fine structure con-stant α s , with strange quark mass corrections [18], findsa relatively strong dependence of the QCD equation ofstate on the renormalization scale below the quark chem-ical potential µ ∼ ∼ n . Such dependence indicates that non-perturbative effects remain quite important in the lowerdensity range relevant to neutron stars.In constructing a phenomenological QCD equation ofstate here, we follow the spirit of the “3-window” ap-proach of Masuda, Hatsuda, and Takatsuka [20] that in-terpolates between a nuclear equation of state at lowdensity and a quark equation of state at high density.At densities below 2 n we adopt the hadronic Akmal-Pandaripande-Ravenhall (APR) equation of state [16](denoted in their paper as A18+ δv +UIX*). At densitiesabove 4-7 n , where a gas of baryons of radius 0.4-0.5fm would begin to percolate [21], we employ a three-flavor Nambu–Jona-Lasinio (NJL) quark model includ-ing vector and diquark interactions. While we use theNJL model to be specific, our discussions are more gen-eral. In the intermediate region, where purely hadronicor purely quark descriptions are not appropriate, we con-struct an equation of state using a smooth polynomialinterpolation in the baryon chemical potential–pressure( µ , P ) plane. In this plane the pressure must be a con-tinuous and monotonically increasing function of µ . Onecannot rule out the possibility of a first order transition as the baryon density increases; such a transition wouldappear as a discontinuity in the first derivative, ∂P/∂µ .The 3-window approach is quite different from the con-ventional hybrid one in which one regards the quark andhadronic phases as distinct. In the latter, the quark pres-sure at given µ must, with increasing µ , intersect thehadronic pressure from below, and moreover must re-main larger than the hadronic pressure at larger µ . Byregarding the hadronic equation of state at density largerthan ∼ n as a valid description of matter, such a hybridconstruction implicitly selects out possible forms of quarkequations of state; in order to have an intersection, thequark pressure must be larger than that of the hadronicphase as large µ . Such quark equations of state are typi-cally soft, and as a consequence hybrid stars with largerquark cores tend to have smaller masses. In contrast, inthe 3-window approach, we construct high density quarkequations of state independently of assuming a trustwor-thy high density hadronic pressure; at high density, theresulting quark pressure at given µ does not have to growfast and may remain smaller than the pressure extrapo-lated from the hadronic phase. Such quark equations ofstate tend to be stiff, and a star with large quark corecan have a large mass.Within this schematic 3-window description, we aimto incorporate the following effects known from ob-served hadronic spectroscopy: (i) A repulsive flavor-independent density-density interaction [22], which stiff-ens the equation of state (Fig.1a). (ii) The attrac-tive color-magnetic interaction, relevant at all densi-ties, which reduces the average single quark energy(Fig.1.(b)). This effect is similar to that observed in theconstituent quark model [23], in which the average quarkenergy in a nucleon is reduced from the constituent quarkmass, ∼
340 MeV, to one-third of the nucleon mass, ∼
313 MeV. As we show, this effect plays an importantrole in ensuring that the interpolated equation of statesatisfies thermodynamic constraints. (iii) Confinement,which, at low densities, traps quarks into baryons andforbids quarks to contribute significantly to the pressure.As we describe, the NJL model, which does not includeconfinement, has a higher pressure at low density thannuclear models. The requirement that the interpolatedpressure merges smoothly into APR at low densities (Fig.1c) effectively suppresses such excess pressure.The present approach of interpolating between ahadronic and an NJL based quark picture makes the tacitassumption that the behavior of the gluon sector does notchange appreciably over the range 0 (cid:46) n B (cid:46) n ; and inparticular, that the gluons do not add a bag constant, B g ,to the energy (and subtracted from the pressure) whenthe nonperturbative gluons become perturbative. Thebag constant measures the energy difference between thetrivial (perturbative vacuum) and the nonperturbativevacuum. With the zero-point of the energy set to makethe QCD (nonperturbative) vacuum energy zero, the per-turbative vacuum has positive energy. Thus, wheneverwe consider the extreme conditions under which nonper-turbative effects disappear, we must include the bag con-stant in addition to the contributions of the perturba-tive effects. However, the stability of massive neutronstars does not permit gluon condensation at the QCDscale Λ QCD ∼ . B g ∼ Λ , since such a term would, as we argue,too greatly soften the equation of state; we thus excludethe possibility of such a term. On the other hand, aquark bag constant, B q of order Λ associated withrestoration of chiral symmetry, is unavoidable in the NJLmodel [24].We extrapolate NJL parameters obtained via hadronphenomenology at n B ∼ n to high density quark mat-ter ( n B ∼ n ) [25, 26], an approach that is consistentwith the observation from analyses for a large number ofcolors, N c , that gluon dynamics is insensitive to quarkloop effects [27].This paper is organized as follows. In Sec. II, we brieflydescribe the hadronic and quark models adopted in thisstudy. In Sec. III A, we examine interaction effects on thequark equation of state. In Sec. IV, we construct the in-terpolated equation of state. In particular, we explain thedifference between the present 3-window description andconventional equations of state which introduce a first or-der phase transition between hadronic and quark matterin [1, 28–31]. As we see the constraints from thermody-namics and causality are quite important. In Sec. V, wesolve the Tolman-Oppenheimer-Volkoff (TOV) equationand examine the resulting mass-radius ( M - R ) relation ofneutron stars. Section VI is devoted to a summary andoutlook.We use the following conventions: g µν =diag(1 , − , − , − γ = γ † , and the charge conju-gation operator C is iγ γ . The flavor and color U (3)generator matrices τ i ( i = 0 , · · · ,
8) and λ a ( a = 0 , · · · , τ i τ j ] = 2 δ ij and tr[ λ a λ b ] = 2 δ ab . We work in units in which c and (cid:126) = 1. II. MODELS
In this section we briefly summarize the features ofthe hadronic (APR) and quark (NJL) matter equationsof state employed in this paper. APR will be used todescribe low density matter, n B < n , while the NJLmodel will be used at high densities, n B > (4 − n .The precise density beyond which we adopt a fully quarkdescription of matter will depend upon details of the in-terpolation, as discussed in section IV. A. The APR equation of state
In this work we adopt the A18+ δv +UIX ∗ version of theAPR equation of state to describe low density hadronicmatter [16]. This equation of state, based on the Argonne v two-body potential, which fits hadronic scatteringdata very well, and the Urbana IX three-body interac-tion, which is important to explain nuclear saturationproperties, includes charge neutrality and β -equilibrium.The δv indicates the inclusion of relativistic corrections.For simplicity, we adopt the APR version excluding neu-tral pion condensate, which emerges at n B ∼ . n . TheAPR model includes only nucleonic degrees of freedom,and does not take into account hyperons, whose interac-tions with nucleons and among themselves are not welldetermined. Typical models of nucleon-hyperon interac-tions predict hyperon onset at a density n B ∼ − n .We restrict our application of APR to n B < n . B. The NJL equation of state
1. The Lagrangian
In descriptions of quark matter, we adopt a three flavorNambu–Jona-Lasinio model with Lagrangian density L = q (i /∂ − ˆ m ) q + L (4) + L (6) , (1)where q is a quark field with color, flavor, and Dirac in-dices, ˆ m is the quark current mass matrix, and L (4) = L (4) σ + L (4) V + L (4) d and L (6) = L (6) σ + L (6) σd are four- andsix-quark interaction terms, respectively, chosen to re-flect the symmetries of QCD. The four-quark interac-tions possess U L (3) × U R (3) symmetry for flavors, whilethe six-quark interactions reflect the U A (1) anomaly.The first of the four-quark interactions describes spon-taneous chiral symmetry breaking: L (4) σ = G (cid:88) i =0 (cid:2) ( qτ i q ) + ( q i γ τ i q ) (cid:3) = 8 G tr( φ † φ ) , (2)where G > φ ij = ( q R ) ja ( q L ) ia is thechiral operator with flavor indices i, j (summed over thecolor index a ).The second of the four-quark terms [32], L (4) V = − g V ( qγ µ q ) , ( g V >
0) (3)describes the repulsive density-density interaction, anal-ogous to ω -meson exchange in nuclear matter.The third of the four-quark terms, L (4) d = H (cid:88) A,A (cid:48) =2 , , (cid:2) (cid:0) q i γ τ A λ A (cid:48) Cq T (cid:1) (cid:0) q T C i γ τ A λ A (cid:48) q (cid:1) + (cid:0) qτ A λ A (cid:48) Cq T (cid:1) (cid:0) q T Cτ A λ A (cid:48) q (cid:1) (cid:3) , = 2 H tr( d † L d L + d † R d R ) , ( H > , (4)describes attractive diquark pairing, where τ A and λ A (cid:48) ( A, A (cid:48) = 2 , ,
7) are the antisymmetric generators of U(3)flavor and SU(3) color, respectively. The structure ofthe interaction can be understood as the color-magneticinteraction in the 2 → s -wave,spin-singlet, flavor- and color- anti-triplet channel. Theoperators ( d L,R ) ai = (cid:15) abc (cid:15) ijk ( q L,R ) jb C ( q L,R ) kc are diquarkoperators of left- and right-handed chirality.Next we discuss the six-quark interactions responsiblefor the U A (1) anomaly [33]. The first term involves theproduct of the chiral condensates of different flavors: L (6) σ = − K (det f φ + h.c.) , ( K >
0) (5)where det f denotes the determinant with respect to flavorindices. The second term couples the chiral and diquarkcondensates [34], L (6) σd = K (cid:48) (tr[( d † R d L ) φ ] + h.c.) , ( K (cid:48) > . (6)At tree level, these two interactions may be related viaa Fierz transformation, which leads to the conclusion K (cid:48) = K . However, renormalization effects will, in gen-eral, destroy this equality so that at the mean field levelwe may treat K and K (cid:48) as independent parameters, butwith K (cid:48) ∼ K .
2. Electric and color charge neutrality constraints
In order to avoid energetically expensive static long-range electric Coulomb interactions and color flux tubeconfigurations in stable homogeneous quark matter, weimpose the local electric and color charge neutrality con-straints n Q ( x ) = n a ( x ) = 0 . ( a = 1 , · · · ,
8) (7)where n Q is the local electric charge density, and the n a are the local color densities. These conditions areenforced via standard Lagrange multipliers – with theappropriate chemical potentials coupled to the electricand color charge densities, respectively [35].Introducing the charge chemical potential µ Q , we addto the Lagrangian the terms L Q = µ Q ( q † Qq − l † i l i ) , ( i : summed) (8) where Q = diag(2 / , − / , − /
3) is the quark chargeoperator for ( u, d, s ) quarks, in units of the proton charge e . The l i = ( e, µ, τ ) are lepton fields and m i the leptonmasses. We may safely omit contributions from µ - and τ - leptons since their populations are vanishingly smallin the density range of interest [20].Colors and flavors in dense matter are coupled throughthe diquark interactions of L (4) d . Thus, an asymmetry inquark flavor densities (e.g., a 2SC phase) leads to a cor-responding net quark color density. Most generally case,we should introduce eight independent color chemicalpotentials [36]. However, for the diquark pairing struc-tures considered in this paper, all color densities except n = (cid:104) q † λ q (cid:105) and n = (cid:104) q † λ q (cid:105) automatically vanish [37].Thus, we only need to add the terms L , = µ q † λ q + µ q † λ q , (9)to constrain the system. The values of ( µ Q , µ , µ ) willbe tuned to satisfy the neutrality conditions.
3. Mean field equation of state
The mean fields for the chiral condensate and quarkdensities are σ i = (cid:104) q i q i (cid:105) , n = (cid:88) i =1 (cid:104) q † i q i (cid:105) . (10)Below we write ( σ , σ , σ ) = ( σ u , σ d , σ s ) for later conve-nience. For the diquark mean fields, we write d i = (cid:104) q T C i γ R i q (cid:105) , (11)where ( R , R , R ) ≡ ( τ λ , τ λ , τ λ ) . (12)With these definitions, the diquark condensates( d , d , d ) correspond to ( ds, su, ud ) quark pairings, re-spectively.The thermodynamic potential may be computed fromthe mean field particle propagators in terms of thesemean fields; the inverse of the propagator S ( k ), can beread off from the mean field Lagrangian [34], S − ( k ) = (cid:18) /k − ˆ M + ˆ µγ i γ ∆ i R i − i γ ∆ ∗ i R i /k − ˆ M − ˆ µγ (cid:19) , (13)where the effective mass matrix has diagonal elements M i = m i − Gσ i + K | (cid:15) ijk | σ j σ k + K (cid:48) | d i | , (14)while the three diquark pairing amplitudes,∆ i = − d i (cid:18) H − K (cid:48) σ i (cid:19) , (15)and the effective chemical potential matrix,ˆ µ = µ − g V n + µ λ + µ Q Q , (16)are color- and flavor-dependent.For each momentum, the inverse propagator is a 72 × (cid:15), − (cid:15) ). The single par-ticle contribution to the thermodynamic potential isΩ single = − (cid:88) j =1 (cid:90) Λ d k (2 π ) (cid:20) T ln (cid:16) e −| (cid:15) j | /T (cid:17) + ∆ (cid:15) j (cid:21) , (17)where ∆ (cid:15) j = (cid:15) j − (cid:15) free j ; here Λ is an ultraviolet cutoff. The µ -dependence is hidden in the eigenvalue (cid:15) j . Because Eq.(13) cannot be inverted analytically, the eigenvalues mustbe computed numerically for each momentum [38].In order to remove the double-counting of interactionstypical of mean-field treatments, we must also include inthe thermodynamic potential the termsΩ cond = (cid:88) i =1 (cid:20) Gσ i + (cid:18) H − K (cid:48) σ i (cid:19) | d i | (cid:21) − Kσ σ σ − g V n . (18)These terms are positive ( σ i < bare q =Ω single + Ω cond . . However, there still remains the nontriv-ial choice of the “zero” of the thermodyanmic potential.For discussions of neutron star masses, this procedure isextremely important because in general relativitity, theabsolute energy density, as in the TOV equation, andnot simply its deviation from the QCD vacuum, is phys-ically relevant. Given that the cosmological constant isextremely small compared to the QCD scale, we set theorigin of the thermodynamic potential to zero at zeroquark density and temperature. Thus in constructing thequark matter equation of state, we will use the renormal-ized thermodynamic potentialΩ q (ˆ µ, T ) ≡ Ω bare q (ˆ µ, T ) − Ω bare q (ˆ µ = T = 0) , (19)which vanishes at T = ˆ µ = 0.Finally, the electron contribution to the thermody-namic potential is the standardΩ e = − T (cid:88) λ = ± (cid:90) d k (2 π ) ln (cid:16) e − ( E e + λµ Q ) /T (cid:17) , (20)with E e = (cid:112) k + m e , and where we recall that the elec-tron chemical potential is µ e − = − µ Q .Writing the total thermodynamic potential as Ω =Ω q + Ω e , the thermodynamic state of the system is de-termined minimizing the free energy with respect to theseven condensates { σ i , d i , n } under the neutrality condi-tions n Q, , = − ∂ Ω ∂µ Q, , = 0 , (21) TABLE I: Three common parameter sets for the three-flavorNJL model: the average up and down bare quark mass m u,d ,strange bare quark mass m s , coupling constants G and K (cid:48) , and3-momentum cutoff Λ [25, 39, 40], with Λ, m u,d , and m s in MeV. Λ m u,d m s G Λ K Λ HK [25] 631.4 5.5 135.7 1.835 9.29RHK [39] 602.3 5.5 140.7 1.835 12.36LKW [40] 750.0 3.6 87.0 1.820 8.90 which yields the “gap equations.”0 = − ∂ Ω ∂σ i = − ∂ Ω ∂d i , n = − ∂ Ω ∂µ . (22)Below we solve these self-consistent equations using themethod outlined in [37]. Whenever we encounter regionsin the solution suggestive of first order phase transitions,we explicitly compare Ω in the relevant phases to deter-mine which local minima is gives the lower free energy.For the ground state we calculate at a nonzero but verysmall temperature ∼ . m e , which makes the numericalcalculations faster and more stable.
4. The NJL parameters
For the model outlined above, we identify two distinctsets of parameters: (Λ , m u,d , m s , G, K ) and ( g V , H, K (cid:48) ).The first set is fixed by matching to QCD vacuum phe-nomenology. In this work we will use the set by Hatsudaand Kunihiro (HK)[25] (Table I), which gives the vac-uum effective masses for light flavors, M u,d (cid:39)
336 MeV,and strange quark, M s (cid:39)
528 MeV. The second set ofparameters does not manifestly affect the quantities inQCD vacuum at the mean field level; we therefore treatthem as free parameters, but of the same order of mag-nitude as the first set, based upon Fierz transformationsconnecting the associated interaction vertices in the ab-sence of any known anomalous suppression. Briefly, wewill investigate g V = 1 − G , H = 1 − . G , and K (cid:48) = 0and K (cid:48) = K . The choice of these values will be explainedin Sec.III. III. QUALITATIVE EFFECTS ON THE QUARKEQUATION OF STATE
In this section we examine a number of qualitative ef-fects related to the quark matter equation of state.
A. When do the quark equations of state becomestiff ?
We begin our discussion of stiffening of the equation ofstate by considering a schematic expression for the quarksector equation of state (see also [29]). For simplicity, wepresently consider only matter in a single phase, ignor-ing any complications arising from phase transitions. Inthis context, the energy density may be parameterized interms of the quark density as ε ( n ) = c n / + c n / + c − n + B , (23)where n ∼ p F with p F the quark Fermi momentum. Thefirst term is the kinetic energy contribution. The secondterm contains contributions from both diquark pairing onthe Fermi surface ( ∼ − p F ∆( p F ) ) and mass corrections( ∼ + p F M ( p F ) ), where we assume that ∆ , M (cid:28) p F .(The limit M (cid:28) p F is applicable for all quarks, evenstrange, at high density, where chiral restoration occurs,and is applicable for u and d quarks at intermediate den-sity.) The third term represents the density-density in-teraction. The last term is the bag constant B ( > p F , the density dependence of thepairing gaps, as well as that of the bag constant.Differentiating (23) yields the chemical potential µ = ∂ε/∂n , from which the pressure P = µn − ε is: P = ε − c n / + 23 c − n − B . (24)The first term is the kinetic pressure, while the remainingterms correspond to corrections arising from the mecha-nisms discussed above. For given ε , the pressure becomeslarge when c < c − >
0. The former condition ismet when the quarks interact attractively near the Fermisurface. The latter condition simply expresses the re-quirement that the density-density interaction should berepulsive in order to stiffen the equation of state. Moregenerally, for stiff equations of state, the coefficients c m ≥ should be negative, while c m< should be positive. Fi-nally, a smaller value of the bag constant also tends tostiffen the equation of state. B. Quark and gluon bag constants: B q and B g To consider the impact of the quark and gluon bag con-stants, we begin by supposing that both the quark andgluon sectors are weakly interacting and that all quarkand gluon condensates are vanishingly small. In the ab-sence of perturbative corrections, the equation of state isthen P ( µ ) = c µ − B , ε ( µ ) = 3 c µ + B , (25)where c = N c N f / π is a function of the number ofquark colors N c and flavors N f , and the net bag constantis sum of quark and gluon contributions, B = B q + B g .The existence of the bag constant changes the energy-pressure relation from ε = 3 P to ε − B = 3 P . Therefore,a smaller B enhances P at given ε , stiffening the equationof state. In fact, for a three flavor ideal quark gas witha bag constant, the maximum neutron star mass scales as [41, 42] M max (cid:39) . M (cid:12) (cid:18)
155 MeV B / (cid:19) , (26)while the corresponding radius scales as R (cid:39) . (cid:18)
155 MeV B / (cid:19) km . (27)Thus, smaller values of B give rise to more massive, largerneutron stars.In the NJL model, the quark bag constant at largedensity appears automatically when the gap in the Diracsea is closed through chiral restoration. As a result, itsvalue be computed explicitly as B NJL q ≡ [ Ω( M eff = m ) − Ω( M eff = M ) ] T = µ =0 , (28)where M eff is the effective mass in the quark energy; M eff becomes the current quark mass ( m ) in the perturbativevacuum and the dynamically generated mass ( M ) in thechiral symmetry-broken vacuum. In the HK parameterset with vacuum effective masses M u,d = 336 MeV and M s = 528 MeV, the bag constant is B NJL q (cid:39)
284 MeV / fm = (219 MeV) . (29)Naively substituting this value into Eq. (26), we obtain amaximum neutron star mass ∼ . M (cid:12) , which less than1/2 the mass of observed massive stars. This low maxi-mum mass indicates the importance of interaction effectsin order to sustain massive neutron stars.Typical values of the bag constant used previously[16, 42] are in the range B / ∼ −
200 MeV. The bagconstant used in [42] to construct strange quark stars, B sq (cid:39) (155MeV) , is one-fourth of the NJL bag con-stant, B sq (cid:39) B NJL q /
4. For a three-flavor free quark gaswith B sq , the star mass is relatively large, (cid:39) . M (cid:12) ,but still does not reach ∼ M (cid:12) ; to do so requires includ-ing interactions. Since the value of the bag constant isnot precisely known, we employ the NJL value (29) forconsistency. As noted, we should also consider a gluonbag constant, B g ∼ Λ when gluons become pertur-bative at some large quark density. However, becausethe quark bag constant in the NJL model alone alreadyprovides considerable softening of the equation of state,a significant contribution from the gluonic bag constantis unlikely in our equation of state, as we show later inSec. IV C.One might argue that considering a gluonic bag con-stant is unnecessary because the gluons are integratedout in determining the interactions in the NJL model,and thus the quark bag constant already contains thegluonic contributions. This is not quite correct. In theNJL model, the long-range components of the gluonssuch as those producing confinement are certainly nottaken into account; integrating out the long-range com-ponents generally produces nonlocal interactions amongquarks, which are not present in the NJL model. There-fore, we must consider the contributions from long-rangegluons separately, and not simply ignore the gluonic bagconstant. P ( G e V /f m ) e (GeV/fm )H=K’=0, g V /G = 0.0g V /G = 1.0g V /G = 2.0APR (n B < 2n ) FIG. 2:
Pressure vs. energy density for several NJL parametersets and APR. The bold lines for the NJL equations of state indi-cate that n B > n . As g V increases, the NJL equation of statebecomes noticeably stiffer. APR is also plotted for comparison, inbold for n B < n , and as double dots in the region above, wherewe do not use APR. C. Repulsive density-density interaction: g V The repulsive quark vector interaction is inspired bythe repulsive density-density interaction in nuclear mat-ter, described, e.g., by omega meson exchange [43]. Ex-trapolating the picture of nuclear matter to the stronglycorrelated quark matter domain, we anticipate that thequark vector coupling is of similar magnitude to thehadronic coupling scale g V ∼ G .Reference [44] demonstrated that the vector couplingshould be g V ∼ G in order to explain the lattice re-sults on the curvature of the chiral restoration line nearzero density [45]. (Note that the coupling constant G here corresponds to half that used in Ref. [44].) In thefollowing, we focus mainly on the value g V = 2 G .The inclusion of a vector coupling smooths out chiralsymmetry restoration [44] because the density-density re-pulsion forbids a rapid increase of the baryon density,and as a result the chiral transition also does not occurrapidly. Indeed, beyond a particular critical coupling,a first order chiral transition is turned into a smoothcrossover.Intuitively, an increasing repulsive vector force stiff-ens the equation of state, P vs. ε , as shown in Fig. 2.While the NJL equation of state is considerably softerthan APR for small vector couplings, when g V is suf-ficiently large the NJL equation of state can achieve astiffness on par with APR across a wide range of densi-ties. By increasing g V sufficiently, we can obtain an equa-tion of state within the present framework stiff enoughto support neutron star masses ∼ M (cid:12) .On the other hand, increasing g V makes it more diffi-cult to interpolate between the APR and NJL regimes.This challenge is seen in the plots of n B ( µ ) and P ( µ ) inFig. 3, where for both P and n B , the APR and NJLcurves become more widely separated in µ as g V in-creases. One might imagine that the matching could be n B / n m (GeV)H=K’=0, g V /G = 0.0g V /G = 1.0g V /G = 2.0APR (n B < 2n ) P ( G e V /f m ) m (GeV) H=K’=0, g V /G = 0.0g V /G = 1.0g V /G = 2.0APR (n B < 2n ) FIG. 3:
Normalized baryon density n B /n (top panel) and pres-sure P (bottom panel) as a function of quark chemical potential µ for several NJL parameter sets and APR. Typically, the baryondensity and pressure in the APR rise faster at lower chemicalpotential than in NJL which has a larger effective quark mass, M (cid:39)
336 MeV, than 1/3 of the nucleon mass. performed rather simply by allowing a first order phasetransition in the interpolated region; however a first or-der transition, a sudden increase in n B is simply a kinkin P vs. µ , which does not help the interpolation.A part of the difficulty of interpolating between APRand NJL is that the constituent quark mass for light fla-vors is M u,d (cid:39)
336 MeV, larger than the one-third ofthe nucleon mass. Accordingly, the P ( µ ) curve in theNJL model tends be below that of APR. We next discusstwo-body correlations mediated by the color magnetic in-teraction, which tend to shift the P ( µ ) curves to the left,rendering the interpolation procedure more feasible. D. Two-body correlations: the color magneticinteraction H At high density, quarks undergo BCS pairing (diquarkcondensation) as a consequence of the color magneticinteraction. Pairing reduces the energy density by anamount δε ∼ − p F ∆ , or equivalently, enhances the pres-sure by δP ∼ + p F ∆ .We expect correlation effects among the quarks to in-crease with decreasing density. Eventually three-quarkcorrelations must be dominant in the hadronic phase.One path to three-quark correlations is increasing di- G ap s ( G e V ) m (GeV) M u M d M s D ds D su D ud FIG. 4:
Chiral and diquark gaps as a function of quark chemicalpotential for K (cid:48) = 0, g V = 2 . G , H = 1 . G . With increasingdensity, the system undergoes a first order phase transition fromthe 2SC to the CFL phase. The gaps are shown as bold lines for n B > n . n B / n m (GeV)K’=0, g V = 2.0G, H=0.0H=1.0 GH=1.25 GH=1.5 GAPR (n B < 2n ) P ( G e V /f m ) m (GeV)K’=0, g V = 2.0G, H=0.0H=1.0 GH=1.25 GH=1.5 GAPR (n B < 2n ) FIG. 5:
Normalized baryon density n B /n (top panel) and pres-sure P (bottom panel) as functions of quark chemical potential µ . We take the vector and axial anomaly couplings in NJL to be g V /G = 2 . K (cid:48) = 0, and allow the diquark coupling to takeon the values H/G = 0, 1.0, 1.25, and 1.5. As H increases thecurves are shifted toward lower chemical potential. quark correlations plus diquark-single quark correlationsbeyond that described by the standard choice of the di-quark coupling H = 0.75-1.0 G , based on Fierz trans-formation of the one-gluon exchange type vertex. In thisrange, we do not find significant effects of H on the equa-tion of state in the density range of interest. A diquarkmean field appears only when the Fermi surface becomessufficiently large to overcome chiral symmetry breakingeffects. However, diquark correlations, which can existeven without a large Fermi sea, reduce the energy of apair to less than twice the effective quark mass. To simu-late such effects within the present mean field approach,we allow the diquark mean field to reduce the single quarkenergy at all densities by exploring somewhat larger val-ues H = 1.0-1.5 G than the standard.As shown in Sec. III A, pairing tends to stiffen theequation of state in the high density regime. Figure 4shows the development of constituent quark masses andmean field pairing gaps; as µ increases, quark matterfirst appears in a 2SC phase in which only up and downquarks are paired (∆ ud (cid:54) = 0, ∆ us = ∆ ds = 0), and laterevolves into a CFL phase in which all three quark flavorspair (∆ ud , ∆ ds , ∆ su (cid:54) = 0). At T = 0 the 2SC-CFL transi-tion appears to be first order for all NJL parameter sets.However, given that this transition occurs at relativelylow density ( n B < n ), the quark model results must betreated with caution.Two-body correlations are also important at low densi-ties. For example, in the constituent quark model, colormagnetic interactions between quarks, in the presence ofconfinement, reduces the nucleon mass from three timesthe constituent quark mass, (cid:39) ×
340 = 1020 MeV bysome 80 MeV to its physical value. Since confinement,by localizing the quarks into a spatial region ∼ Λ − ,increases the quark kinetic energies, as well as addingthe energy of color flux tubes – of typical length ∼ Λ − and energy ∼ σ Λ − , where σ is the string tension – theenergy gain from the color magnetic interaction must ex-ceed ∼
30 MeV.The diquark interaction, H , treated in mean field,qualitatively simulates the reduction in the average quarkenergy at low density that results from pairing effects.As the magnitude of the diquark interaction increases,the curves of the thermodynamic variables as functionsof µ are shifted leftwards to lower chemical potential, asshown in Fig. 5. Thus, by including effects of pairing,one is able to maintain the stiff equation of state pro-duced by a relatively large vector coupling, while at thesame time enabling a smooth interpolation between theNJL and APR equations of state for all thermodynamicvariables.Figure 6 demonstrates the impact of pairing on thestiffness of the NJL equation of state. The discontinuouschange of ε at fixed P reflects the first order 2SC-CFLphase transition. While for 0 < H < . G , the equationsof state exhibit softening immediately following the 2SC-CFL transition, as the quark density increases further,pairing effects stiffen the equation of state for all NJL P ( G e V /f m ) e (GeV/fm )K’=0, g V = 2.0G, H=0.0H=1.0 GH=1.25 GH=1.5 GAPR (n B < 2n ) FIG. 6:
Pressure as a function of energy density for the sameparameter sets as in Fig. 5. The discontinuous change of ε atfixed P reflects the first order 2SC-CFL phase transition. For0 < H < . G , the equation of state is softened immediatelyfollowing the 2SC-CFL phase transition, but as density increases,the equation of state eventually stiffens relative to the H = 0case. For H = 1 . G , the equation of state is stiffer than theunpaired case for all densities. parameter sets, relative to the no-pairing case. Moreover,when the pairing is sufficiently strong ( H > . G ), theequation of state is stiffer than without pairing ( H = 0)across the entire density range.We note that for large H , the quark pressure at given µ exceeds the APR pressure even at very low densities.Taken at face value, this would suggest that even at verylow densities the ground state of QCD matter is quarkrather than hadronic matter. However, as we discuss inSec. IV, this high pressure is an unphysical consequenceof the NJL model not being confining at low densities. E. Chiral-diquark coupling: K (cid:48) We now turn to the axial anomaly-induced couplingbetween the chiral and diquark condensates. The im-portance of the anomalous coupling K (cid:48) depends on thesize of the chiral and diquark condensates. For K (cid:48) > K (cid:48) increases from zero thediquark condensate emerges at lower chemical potentialand the chiral condensate persists to higher chemical po-tential.Figure 7 shows the impact of K (cid:48) on the NJL equationof state. We note that while increasing K (cid:48) from 0 to K slightly stiffens the equation of state, its impact ismuch smaller than that of g V or H . Since the K (cid:48) termin the Lagrangian can be read as a diquark interactionwith an interaction strength proportional to the chiralcondensate, the effect of K (cid:48) can be largely absorbed bythe variation of g V and H ; thus in the following we donot study the variation of K (cid:48) in detail but take K (cid:48) = K P ( G e V /f m ) e (GeV/fm )g V = 2.0G, H=1.5G, K’=0K’=KAPR (n B < 2n ) FIG. 7:
Pressure as a function of energy density in the presenceof the chiral-diquark coupling, K (cid:48) = K . Larger K (cid:48) stiffens theequation of state, but its impact is significantly smaller than thatof variations of g V or H . as a canonical value. IV. INTERPOLATED EOS
We now discuss constructing an interpolated equationof state, P ( µ ) that smoothly joins the equations of stateof the low density APR model to the high density NJLmodel. The first step in defining an interpolation methodfor joining the hadronic and quark sectors is to determinethe “overlap region” in which the two equations of statewill be merged. Beginning in the low density hadronicregime described by APR, we expect that as density in-creases, corrections from many-body forces, hyperon de-grees of freedom, multiple meson exchanges and the like,will become important above n B ∼ n . Thus, we fix thelower boundary of the interpolation to n < ≡ n .As the density decreases in the quark regime , confiningeffects, which trap quarks into baryons, become increas-ingly important. Assuming that the radius of a typicalbaryon is r B ∼ . − . n B ∼ − n . Thus, we set the upperboundary of the interpolation to n > ≡ − n , with theprecise location determined by the details of the givenNJL parameter set being used.In the intermediate density regime at a given chemi-cal potential, the pressure of the interpolated equation ofstate should be lower than the NJL pressure extrapolatedto lower densities (Fig.8). This follows from the fact thatnonconfining models yield excess quark populations atgiven chemical potential, due to the unphysically smallenergy cost of having quarks present. In other words,were the confining effects of QCD incorporated in thedescription of the quark phase, the pressure, especiallyin dilute region, would be significantly suppressed. Thissituation is quite analogous to the “semi”-QGP picturefor finite temperature QCD [46] in which an “overpop-0FIG. 8: Schematic illustration of confining effects on the hy-brid quark-hadron equation of state, P ( µ ), here normalized bythe Stefan-Boltzman gas for N f = 3. Effects of confinementare strong at lower density, suppressing the excess NJL pres-sure. The boundaries of the interpolated equation of state are( n < , n > ) = (2 . , . n . ulated” quark pressure is suppressed by Polyakov loopeffects, until thermal quarks and gluons exhibit quasipar-ticle behavior at temperatures beyond ∼ − T c , where T c is the pseudocritical temperature for deconfinement[47].The present 3-window description is quite differentfrom the conventional one involving a first order hadron-quark phase transition [1]. In the latter case, the quarkpressure at low density of nonconfining models must besmaller than the hadronic one, in order to ensure theintersection of the quark and hadronic pressure at rea-sonable density. This is achieved either by restricting thequark model parameters or by introducing a bag constantto lower the quark pressure. Such choices, however, gen-erally affect the quark matter equation of state not onlyin the (presumably unreliable) low density limit, but alsoin the high density regime in which it should be reliable. A. Thermodynamic constraints
Having discussed the qualitative aspects of a hadron-quark interpolation, we now briefly review the thermo-dynamic constraints imposed on this interpolation whichare necessary to ensure that the interpolated equationof state is physical. These constraints are as follows:(i) the pressure P ( µ ) must be continuous everywhere;(ii) n B ( µ ) = ( ∂P/∂µ ) /N c must be a monotonicallyincreasing function in order to ensure stability of thesystem with respect to phase separation: ∂ P/∂µ = N c ∂n B /∂µ >
0. (iii) In addition one physically expectsthat the speed of sound must be less than the speed oflight: c s = ∂P/∂ε < B. Interpolation method
We now describe a particular method for construct-ing a phenomenological quark-hadron equation of state.To interpolate in the variables µ - P , we employ a simplepolynomial interpolation function, defining an N th orderpolynomial interpolant for the pressure: P ( µ ) = N (cid:88) m =0 b m µ m , for µ < < µ < µ > , (30)where µ < and µ > are defined as the points where n B ( µ < ) ≡ n < = 2 n and n B ( µ > ) ≡ n > = 4 − n . Thecoefficients b m are chosen to satisfy the matching condi-tions at the boundaries of the interpolating interval. At µ = µ < : P APR ( µ < ) = P ( µ < ) , ∂P APR ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ < = ∂ P ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ < , · · · (31)and at µ = µ > : P NJL ( µ > ) = P ( µ > ) , ∂P NJL ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ > = ∂ P ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ > , · · · (32)The number of derivatives that one matches at eachboundary is a matter of choice. In general, matchingmore derivatives results in a smoother interpolation, butat the same time increases the probability of producingunphysical artifacts in the interpolating region (e.g., in-flection points in P ( µ ) which violate thermodynamic con-straints). Here we match up to second order derivativesat each boundary, which ensures that the pressure, num-ber density, and number susceptibility are continuous.Correspondingly, these six boundary conditions requirethat Eq. (30) has six terms ( N = 5). C. Interpolated EOS
Figure 9 shows the interpolated equation of state P ( µ ), with the interpolation boundaries ( n < , n > ) =(2 . , . n . For illustration, we consider two NJL pa-rameter sets:( g V , H ) = (cid:26) (2 . , . G (Set I)(2 . , . G (Set II) (33)with K (cid:48) = K in both cases. For n > (cid:39) n B , Set I satis-fies all the conditions demanded by the thermodynamicconstraints, as we verify shortly. One cannot, however,within the present polynomial interpolation, construct asensible interpolation for Set II, because at the interpo-lation boundaries the APR and NJL pressures are ratherwidely separated in µ , a possibility noted in Sec. III D.This wide separation in µ requires a small slope of theinterpolated pressure, but at the same time the slopemust be larger than the slope of the APR pressure at thelower boundary, because of the compressibility condition ∂ P/∂µ >
0. The interpolated equation of state has a1 P ( G e V /f m ) m (GeV)APR(g V , H)=(2.0, 1.5)Ginterpolated(g V , H)=(2.0, 0.0)Ginterpolatedfixed slope FIG. 9:
Interpolated equations of state, P ( µ ), for two param-eter sets, with up to second order derivatives matched at theinterpolation boundaries. As a guide to the eye, we also plot thepressure with constant slope fixed at n < . Thermodynamic stabil-ity requires that the slopes in the interpolated equation of stateincrease monotonically: ∂ P/∂µ = N c ∂n B /∂µ >
0, which isviolated for the equation of state constructed in the simple inter-polation with parameter Set II. n B / n m (GeV)APR(g V , H)=(2.0, 1.5)Ginterpolated(g V , H)=(2.0, 0.0)Ginterpolated FIG. 10:
Baryon density as a function of chemical potential forthe interpolated equations of state. The parameter set ( g V , H ) =(2 . , . G has an unstable region ∂n B /∂µ < region of ∂n B /∂µ <
0, as is clearly seen in the plot of n B ( µ ) in Fig. 10.The result presented in Fig. 10 does not preclude con-structing a sensible interpolated pressure for Set II. Asone sees in Fig. 10, it is possible to join the low andhigh baryon density curves with a nondecreasing func-tion of µ . The present exercise shows that the class ofinterpolating functions for Set II is much more restric-tive than for Set I. For example, if there is a first orderphase transition between the hadronic and quark regions,the possible density discontinuities are smaller for Set IIthan Set I. More detailed treatments of the interpolationregion are beyond the scope of the present work and wehenceforth restrict our consideration to the simple poly-nomial interpolation, rejecting NJL parameter sets, such P ( G e V /f m ) e (GeV/fm )APR(g V , H)=(2.0, 1.5)Ginterpolated FIG. 11:
Pressure vs. energy density for the interpolatedequation of state for ( g V , H ) = (2 . , . G , K (cid:48) = K , and( n < , n > ) = (2 . , . n . c s e (GeV/fm )n > / n = 4.05.06.07.0 FIG. 12:
The speed of sound c s = ∂P/∂ε as a function of ε .The NJL parameter set is the same as Fig. 11, for different valuesof the high density boundary: n > /n = 4.0, 5.0, 6.0, and 7.0. Foreach of these choices, the causality condition is satisfied. as II, incapable of being joined with APR in a thermo-dynamically consistent manner.Figure 11 shows the pressure vs. energy density forparameter set I. In this case, the high density equationof state is as stiff as APR extrapolated into the regionbeyond n B = 2 n . From Fig. 12, we observe that thecausality constraint is satisfied when the high densityboundary, n > , is varied from 4 n to 7 n . If we take n > (cid:46) n , however, we find that c s >
1, a putative vio-lation of causality.Finally, we consider the possible impact of a gluonicbag constant, B g ∼ (200MeV) (cid:39) . − , whichshould be included when the gluon sector becomes per-turbative. This contribution reduces the pressure in thequark matter region by 30 − ∂n B /∂µ > B g , so the resulting equa-tion of state becomes significantly softer. Strictly speak-ing, even in this situation it would be possible to con-2struct equations of state by increasing g V and H sig-nificantly from our current choices; however the currentchoices for these couplings are already relatively largeand it is difficult to identify a mechanism that would sig-nificantly increase either coupling in the dense regime.Thus, we conclude that B g should be very small in thequark matter equation of state, even at n B ∼ n ; thegluons remain condensed and nonperturbative. V. MASS-RADIUS RELATION
By solving the TOV equation for a given value of thebaryon density at the center, we construct a family ofstars whose masses and radii are functions of n cB . The M - n cB relation for the equation of state with NJL param-eters ( g V , H, K (cid:48) ) = (2 G, . G, K ) is shown in Fig. 13,where we have taken n > = 5 n as in the previous sec-tion. To examine effects of the quark bag constant as-sociated with chiral restoration, we also show resultsfor the three-flavor free quark gas with bag constants B q = B NJL = (219MeV) and B q = (155MeV) . In theNJL model by itself, the quark bag constant is so largethat neutron star masses are restricted to M < M (cid:12) .However, we note that as the bag constant decreases,neutron star masses rise, so that models yielding smallerbag constants allow more massive stars.We see in Fig. 13 that the M ( n cB ) curves for our in-terpolated equation of state and APR are quite similar,a not too surprising result given that our chosen NJLparameter set yields an equation of state quite similarto APR. However, while the thermodynamic propertiesof the two systems are similar, the underlying effectivedegrees of freedom are quite different. Indeed, APR iswell known for its extreme stiffness at high densities, butthe effect of hyperonic degrees of freedom in the hadronicsector are expected to reduce this stiffness. On the otherhand, our NJL treatment of the quark sector includesstrange quarks from the beginning, and is capable ofproducing a sufficiently stiff equation of state to supportstars whose masses exceed 2 M (cid:12) [20].Figure 14 shows the M - R relation for the interpolatedequation of state with parameter set I. For most centraldensities the stellar radius is ∼ −
12 km, which iscompatible with observational data [9, 10]. However,the radius at which the mass starts to rise in the M - R plot is sensitive to the properties of the hadronic equationof state for n ∼ . − . n , with different hadronicequations of state yielding radii differing by up to ∼ VI. SUMMARY
In this paper we have constructed a phenomenologi-cal equation of state over the range of baryon densities n B ∼ − n by interpolating between the low densityhadronic APR equation of state and the high density NJL FIG. 13: Neutron star mass as a function of central baryondensity, n cB . We display curves corresponding to our interpo-lated equation of state for ( g V , H, K (cid:48) ) = (2 G, . G, K ) and n > = 5 n , as well as APR with and without the three-nucleoninteraction, and the three-flavor free quark gas with bag con-stants B q = B NJL = (219MeV) and B q = (155MeV) . Thevertical arrows indicate the boundaries of the interpolation regionfor the interpolated equation of state. FIG. 14:
Mass-radius relation for neutron stars with severalequations of state (same as Fig. 13). quark model. In so doing, we explored a number of rele-vant ingredients of the equation of state needed to realizeneutron stars of mass ∼ M (cid:12) , while satisfying necessarythermodynamic and causality constraints. These require-ments constrain the form of the interpolated equation ofstate and allow one to infer qualitative effects regardingthe intermediate density region between the hadronic andquark regimes. The repulsive density-density interaction,color-magnetic interaction, and confining effects play avital role in determining the structure of the equation ofstate. A crucial result is that the gluonic bag constantmust be small, i.e., the gluons must remain strongly cou-pled throughout the density region of interest in massiveneutron stars; the gluon sector remains nonperturbativeeven at ∼ n . One reaches a similar conclusion from3studies of quarkyonic matter [27, 48, 49], in which non-perturbative gluons in quark matter play a crucial role.We have also emphasized why the three window modelof interpolation [20] is capable of producing stiffer equa-tions of state than conventional hybrid equations of stateinvolving first order phase transitions. As opposed toconventional models, which require the intersection ofquark and hadronic P ( µ ) curves, we propose that thequark pressure P ( µ ) based on nonconfining models neednot (and even should not) intersect the hadronic equationof state before the inclusion of confining effects. Whilethis observation is not directly applicable at low density(where we did not use the NJL model), it results in amuch wider range of possible high density quark equa-tions of state. As a result, we are able to explore a re-gion of parameter space that has been omitted from priorstudies, while still producing a stiff equation of state re-quired to support massive neutron stars.In this work we have not taken into account possiblemeson condensed phases, by which we mean condensatesin which the order parameter has the quantum numberof a mesonic field. Such condensates have been studiedin a nuclear context [50–52] as well in quark matter, e.g.,inhomogeneous diquark [4] and chiral [5] condensates. Ifextant, such exotic phases would likely occur in the theneither purely hadronic nor purely quark density regionin which we have interpolated. Thus one cannot directlytake over previous results for meson condensates, includ-ing the strength, or density discontinuity, of the first or-der phase transition to the condensed phase. For a givenhadronic equation of state at low density and quark equa-tion of state at high density, the strength of such a phasetransition is bounded. Although we have, for simplic-ity, considered only a smooth interpolation scheme, oneshould more generally allow for such exotic phases; thenthe smooth P ( µ ) curve used in this work would be re-placed by one with a small kink, keeping the positivecurvature of P ( µ ) in the interpolation. We anticipatethat even with condensates at intermediate density, itwill still be possible to find a reasonable parameter set forthe color-magnetic and vector interactions that is com-patible with the existence of massive stars. The issue ofexotic phases remains open until we can reliably estimatethe high density quark equations of state. Further stud- ies are needed to understand better the impact of exoticphases on the in-medium NJL parameters and in turn,their implications for the description of massive neutronstars.In order to improve the description of the intermediatedensity matter in neutron stars it is important to fur-ther refine our understanding of the hadronic equationof state near n B ∼ (1 − n . In particular, a more care-ful assessment of the importance of many-body interac-tions and the emergence of hyperons is required. Furtherconstraints may be obtained from heavy-ion collisions,including strangeness production [53], and lattice QCDcalculations of hyperon-nucleon interactions [54]. Stud-ies of the density dependence of nuclear forces in termsof quarks and gluons play a crucial role in determiningwhen (and why) quarks emerge as the proper degrees offreedom at high density. It would be desirable in the fu-ture to extend to finite temperature the present approachto the equation of state to enable us to address dynamicalquestions such as applications to heavy ion collisions, andneutron star cooling. It is also important to obtain animproved estimate of the quark bag constant, for were itmuch smaller than the NJL estimate used here, the soft-ening associated with chiral restoration would be signif-icantly reduced and large vector and diquark couplingswould not be necessary to obtain a stiff quark matterequation of state within the present context. VII. ACKNOWLEDGEMENTS
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