8
DOI 10.1140/epja/i2007-10497-y Regular Article – Theoretical Physics Eur. Phys. J. A 34, 153–160 (2007) T HE EUROPEAN P HYSICAL JOURNAL A Quark deconfinement in neutron star cores Chang-Qun Ma and Chun-Yuan Gao a School of Physics, Peking University, Beijing 100871, China Received: 29 July 2007 / Revised: 5 October 2007 Published online: 23 November 2007 – c Societ`a Italiana di Fisica / Springer-Verlag2007 Communicated by T. B´ ır´o Abstract. Whether or not the deconfined quark phase exists in neutron star cores is an open question. We use two realistic effective quark models, the three-flavor Nambu-Jona-Lasinio model and the modified quark-meson coupling model, to describe the neutron star matter. We show that the modified quark-meson coupling model, which is fixed by reproducing the saturation properties of nuclear matter, can be consistent with the experimental constraints from nuclear collisions. After constructing possible hybrid equations of state (EOSes) with an unpaired or color superconducting quark phase with the assumption of the sharp hadron-quark phase transition, we discuss the observational constraints from neutron stars on the EOSes. It is found that the neutron star with pure quark matter core is unstable and the hadronic phase with hyperons is denied, while hybrid EOSes with a two-flavor color superconducting phase or unpaired quark matter phase are both allowed by the tight and most reliable constraints from two stars Ter 5 I and EXO 0748-676. And the hybrid EOS with an unpaired quark matter phase is allowed even compared with the tightest constraint from the most massive pulsar star PSR J0751+1807. PACS. 12.38.Mh Quark-gluon plasma – 12.39.-x Phenomenological quark models – 26.60.+c Nuclear matter aspects of neutron stars 1 Introduction Neutron stars are some of the densest objects in the uni- verse and the density in the inner core of a neutron star could be as large as several times the nuclear saturation density ( =0.17 fm 3 ) [1]. The core of a neutron star is so dense that a phase transition from a confined hadronic phase to a deconfined quark phase may exist. The possible emergence of a deconfinement phase in neutron star cores has aroused great interest since it may have a distinct ef- fect on the neutron star structure [2]. Up to now, however, no conclusive observational or experimental evidence sug- gests that the quark matter core conjecture is true and this still remains an open question. In our recent work where the hadronic phase containing octet baryons was considered, it was found that the equation of state (EOS) with deconfinement phase would be ruled out by the ob- servational mass limit of 1.68M of the star Ter 5 I [3]. A similar result was also educed by ¨ Ozel who reported that EOSes with exotic phases would be ruled out by the in- ferred mass and radius for the star EXO 0748-676 and she concluded that the ground state of matter was hadrons and not deconfined quarks [4]. While Alford et al. com- pared ¨ Ozel’s observational limits with predictions based on a more comprehensive set of proposed equations of a e-mail: [email protected] state from the literature, and concluded that the presence of quark matter in EXO 0748-676 was not ruled out [5]. Therefore, the existence of a deconfined quark phase in neutron star cores and the ground state of neutral mat- ter at moderate densities are controversial. Our goal of the present paper is to investigate the observational con- straints on the deconfinement in neutron star cores. It is expected that at extreme conditions chiral symme- try can be restored and quarks and gluons become decon- fined [6]. Consequently, quark matter was frequently dealt with as a non-interacting quark gas, i.e., unpaired quark matter (UQM), and usually described by a phenomenolog- ical bag model [7]. According to the BCS theory [8], any attractive interaction in a cold Fermi sea will cause Cooper instability in the vicinity of the Fermi surface in the mo- mentum space and a superconductor will be formed. Be- cause of the attractive quark-quark interaction in the color antitriple channel [9] it is expected that the color su- perconducting state, with a spontaneous breakdown of the non-Abelian SU (3) color gauge group, would be the ground state of quark matter. The color superconduct- ing state has attracted great interest since it was found that the superconducting gap could be 100 MeV due to non-perturbative features [10]. Depending on quarks par- ticipating in a diquark condensation, one can distinguish several color superconducting phases. The prominent two are the two-flavor color superconducting (2SC) phase con-

Quark deconfinement in neutron star cores

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Page 1: Quark deconfinement in neutron star cores

DOI 10.1140/epja/i2007-10497-y

Regular Article – Theoretical Physics

Eur. Phys. J. A 34, 153–160 (2007) THE EUROPEAN

PHYSICAL JOURNAL A

Quark deconfinement in neutron star cores

Chang-Qun Ma and Chun-Yuan Gaoa

School of Physics, Peking University, Beijing 100871, China

Received: 29 July 2007 / Revised: 5 October 2007Published online: 23 November 2007 – c© Societa Italiana di Fisica / Springer-Verlag 2007Communicated by T. Bıro

Abstract. Whether or not the deconfined quark phase exists in neutron star cores is an open question.We use two realistic effective quark models, the three-flavor Nambu-Jona-Lasinio model and the modifiedquark-meson coupling model, to describe the neutron star matter. We show that the modified quark-mesoncoupling model, which is fixed by reproducing the saturation properties of nuclear matter, can be consistentwith the experimental constraints from nuclear collisions. After constructing possible hybrid equations ofstate (EOSes) with an unpaired or color superconducting quark phase with the assumption of the sharphadron-quark phase transition, we discuss the observational constraints from neutron stars on the EOSes.It is found that the neutron star with pure quark matter core is unstable and the hadronic phase withhyperons is denied, while hybrid EOSes with a two-flavor color superconducting phase or unpaired quarkmatter phase are both allowed by the tight and most reliable constraints from two stars Ter 5 I and EXO0748-676. And the hybrid EOS with an unpaired quark matter phase is allowed even compared with thetightest constraint from the most massive pulsar star PSR J0751+1807.

PACS. 12.38.Mh Quark-gluon plasma – 12.39.-x Phenomenological quark models – 26.60.+c Nuclearmatter aspects of neutron stars

1 Introduction

Neutron stars are some of the densest objects in the uni-verse and the density in the inner core of a neutron starcould be as large as several times the nuclear saturationdensity (∼= 0.17 fm−3) [1]. The core of a neutron star isso dense that a phase transition from a confined hadronicphase to a deconfined quark phase may exist. The possibleemergence of a deconfinement phase in neutron star coreshas aroused great interest since it may have a distinct ef-fect on the neutron star structure [2]. Up to now, however,no conclusive observational or experimental evidence sug-gests that the quark matter core conjecture is true andthis still remains an open question. In our recent workwhere the hadronic phase containing octet baryons wasconsidered, it was found that the equation of state (EOS)with deconfinement phase would be ruled out by the ob-servational mass limit of 1.68M⊙ of the star Ter 5 I [3]. A

similar result was also educed by Ozel who reported thatEOSes with exotic phases would be ruled out by the in-ferred mass and radius for the star EXO 0748-676 and sheconcluded that the ground state of matter was hadronsand not deconfined quarks [4]. While Alford et al. com-

pared Ozel’s observational limits with predictions basedon a more comprehensive set of proposed equations of

a e-mail: [email protected]

state from the literature, and concluded that the presenceof quark matter in EXO 0748-676 was not ruled out [5].Therefore, the existence of a deconfined quark phase inneutron star cores and the ground state of neutral mat-ter at moderate densities are controversial. Our goal ofthe present paper is to investigate the observational con-straints on the deconfinement in neutron star cores.

It is expected that at extreme conditions chiral symme-try can be restored and quarks and gluons become decon-fined [6]. Consequently, quark matter was frequently dealtwith as a non-interacting quark gas, i.e., unpaired quarkmatter (UQM), and usually described by a phenomenolog-ical bag model [7]. According to the BCS theory [8], anyattractive interaction in a cold Fermi sea will cause Cooperinstability in the vicinity of the Fermi surface in the mo-mentum space and a superconductor will be formed. Be-cause of the attractive quark-quark interaction in the colorantitriple channel [9] it is expected that the color su-perconducting state, with a spontaneous breakdown ofthe non-Abelian SU(3) color gauge group, would be theground state of quark matter. The color superconduct-ing state has attracted great interest since it was foundthat the superconducting gap could be ∼ 100MeV due tonon-perturbative features [10]. Depending on quarks par-ticipating in a diquark condensation, one can distinguishseveral color superconducting phases. The prominent twoare the two-flavor color superconducting (2SC) phase con-

Page 2: Quark deconfinement in neutron star cores

154 The European Physical Journal A

taining ud pairs together with unpaired strange quarksand the color-flavor locked (CFL) phase containing ud, dsand us pairs (for recent reviews see ref. [11]).

We adopt two realistic effective quark models for quarkand hadronic phases, respectively, to describe the neu-tron star matter. The extended three-flavor Nambu-Jona-Lasinio (NJL) model with determinant interaction [12],which shares many symmetries with QCD, is used to cal-culate the properties of quark matter in UQM, 2SC andCFL phases. For the hadronic phase, we adopt the EOSesof neutron-proton matter and hypernuclear matter withoctet baryons by the improved modified quark-meson cou-pling (MQMC) model [3,13]. The MQMC model gives asatisfactory description for saturation properties of nu-clear matter [14] and could reproduce the bulk propertiesof finite nuclei well [15]. We then assume a sharp (first or-der) transition from the pure hadronic to the pure quarkphase and consider only the homogeneous quark phaseand not the mixed phase between different quark phases.

The outline of this paper is as follows: In sect. 2, webriefly introduce the three-flavor NJL model for densequark matter and the MQMC model for the hadronicphase. In sect. 3, we construct the possible equations ofstate of neutron star matter and then discuss the obser-vational constraints on the EOSes. Section 4 is devoted tosummary and conclusions.

2 The model

2.1 Quark phase

Strange quark matter at moderate densities can be effec-tively described by the three-flavor NJL model [16,17].The Lagrange density of the extended three-flavor NJLmodel with a six-fermion determinant interaction (t’Hooftterm) is given by

LNJL = q (iγµ∂µ − m0) q + Lqq + Lqq , (1)

where

Lqq = G

8∑

a=0

[

(qτaq)2

+ (qiγ5τaq)2]

−K [detf (q(1 + γ5)q) + detf (q(1 − γ5)q)] (2)

and

Lqq = H∑

A=2,5,7

A′=2,5,7

(qiγ5τAλA′qc) (qciγ5τAλA′q) .

(3)Here, q = (u, d, s)T denotes the quark fields with threecolors. And the current quark mass matrix has the formm0 = diag(m0u,m0d,m0s) in the flavor space, wherem0u = m0d = m0q is assumed throughout this paper.

τ0 =√

231 is proportional to the unit matrix in the flavor

space. τA and λA (A = 1, . . . , 8) are Gell-Mann matrixesin flavor and color spaces, respectively. qc = CqT is thecharge-conjugate spinor.

In the present work, we restrict ourselves to the bulkquark matter in the mean-field approximation and focuson the chiral condensates defined as

φf = 〈qfqf 〉 , f = u, d, s, (4)

and the three-flavor diquark condensates being

∆A = −2H 〈qcγ5τAλAq〉 , A = 2, 5, 7. (5)

After bosonization, one obtains the linearized version ofthe model in the mean-field approximation,

L = q (iγµ∂µ − m) q

+1

2

A

[q (∆Aγ5τAλA) qc + qc (−∆∗Aγ5τAλA) q]

− 1

4H

A

|∆A|2 − 2G∑

f

φ2f + 4Kφuφdφs , (6)

where we have introduced the constituent quark mass

m =

mu 0 00 md 00 0 ms

, (7)

in which

mi = m0i − 4Gφi + 2Kφjφk , (8)

(i, j, k) = any permutation of (u, d, s).

Employing the Nambu-Gorkov formalism, then the ther-modynamic potential per unit volume at temperature Tis obtained via the finite-temperature field theory [18] andtakes the form,

Ω = − T

2V

~P

72∑

i=1

[ |ωi|2T

+ ln(

1 + e−|ωi|/T)

]

+Ωcon +Ωe ,

(9)where

Ωcon = 2G∑

f=u,d,s

φf2−4Kφuφdφs+

1

4H

A

|∆A|2 , (10)

and

Ωe = − 1

12π2

(

µ4e + 2π2T 2µ2

e +7π4

15T 4

)

(11)

is the contribution from the electron gas with chemicalpotential µe.

In eq. (9), ωi is the energy of a quasiparticle and canbe obtained by diagonalizing the inverse propagator inNambu-Gorkov basis. Or, equivalently, ωi can be obtainedby calculating the eigenvalues of the 72 × 72 matrix

M =

~P · ~α + mγ0 − µ −∑

A

∆Aγ0γ5τAλA

A

∆∗Aγ0γ5τAλA

~P · ~α + mγ0 + µ

.

(12)

Page 3: Quark deconfinement in neutron star cores

Chang-Qun Ma and Chun-Yuan Gao: Quark deconfinement in neutron star cores 155

And the chemical potential operator µ is a diagonal 9× 9matrix in flavor and color space. By introducing the quarknumber chemical potential µ, the electrochemical poten-tial µQ and two additional chemical potentials µ3 and µ8

coupled to the color charges λ3 and λ8, respectively, µ canbe expressed as (see ref. [16] for details)

µ = µ + µQ

(

1

2τ3 +

1

2√

3τ8

)

+ µ3λ3 + µ8λ8 . (13)

In beta equilibrium, we have

µe = −µQ . (14)

The order parameters, φf and ∆A, can be obtained byminimizing the thermodynamic potential, and are equiv-alently given by the gap equations,

∂Ω

∂φf= 0, f = u, d, s, (15)

∂Ω

∂∆A= 0, A = 2, 5, 7. (16)

With the diquark condensates, we can explicitly distin-guish different quark phases as

∆2 = ∆5 = ∆7 = 0: UQM;

∆5 = ∆7 = 0,∆2 6= 0: 2SC;

∆2,∆5,∆7 6= 0: CFL.

Five other possible quark phases, i.e. 2SCus, 2SCds, uSC,dSC and sSC, were predicted in ref. [19] with the similarNJL model. However, it was found that at zero tempera-ture these five phases would not appear (see figs. 2 and 4in ref. [19]), so the inclusion of UQM, 2SC and CFL issufficient for neutron stars.

Dense quark matter in neutron star is electrical andcolor neutral, then the following relations should be main-tained:

nQ = − ∂Ω

∂µQ= 0, (17)

n3 = − ∂Ω

∂µ3

= 0, (18)

n8 = − ∂Ω

∂µ8

= 0. (19)

Finally, for a given quark number density

n = −∂Ω

∂µ,

the energy density E and pressure P at zero temperatureare given by

E = Ω + Ωvac + µn, (20)

P = −Ω − Ωvac . (21)

Ωvac is chosen so that P and E vanish in vacuum.For the model parameters, we take the values as fol-

lows. To regularize the divergent integrals we need a sharp

Table 1. The new zero-point motion parameters and radii forthe modified quark-meson coupling model.

M (MeV) Z R (fm)

N 939.0 2.0403 0.6000

Λ 1115.7 1.8099 0.6459

Σ+ 1189.4 1.6318 0.6722

Σ0 1192.6 1.6236 0.6733

Σ− 1197.4 1.6114 0.6749

Ξ0 1314.8 1.4743 0.6922

Ξ− 1321.3 1.4567 0.6942

Table 2. New independent coupling constants, which havebeen defined in ref. [3].

gu,dσ gu,d

ω gu,dρ gbag,N

σ

0.9685 2.7071 7.9288 6.8732

cutoff Λ in the 3-momentum space since the NJL model isnon-renormalizable. Thus, we have a total of 6 parameters,namely, the current masses m0s and m0q for strange andnon-strange quarks, the three couplings G, K and H, andthe cutoff Λ. Following the method adopted in ref. [20],we get Λ = 602.8MeV, GΛ2 = 1.803, KΛ5 = 12.93 andm0s = 140.9MeV by fitting the meson masses [21] mπ =134.98MeV, mK = 497.65MeV and mη′ = 957.78MeVand the π decay constant fπ = 92.2MeV [22] while m0q

is fixed at 5.5MeV. When discussing the superconductingstates, H = G is set as has been used in ref. [16].

2.2 Hadronic phase

For the hadronic phases, we focus on the neutron-protonmatter and the hypernuclear matter consisting of thebaryon octet, i.e., p, n, Λ, Σ+, Σ0, Σ−, Ξ0 and Ξ−. Thehadronic phases are described with the improved MQMCmodel which has been discussed in detail in ref. [3]. Inorder to make the calculation for hadronic phases com-patible with that for quark phases, we should take thesame current quark masses in both models. Therefore thecurrent quark masses are m0u = m0d = 5.5MeV andm0s = 140.9MeV, which are different from those usedin ref. [3], and the other inputs, such as the mass spec-trum and the nucleon’s radius, are unchanged. As a con-sequence, the bag constant in vacuum is changed to be

B1/4

0 = 187.7716MeV. The obtained zero-point motionparameters and bag radii for baryons and coupling con-stants are all changed, their new values are listed in table 1and table 2, respectively.

Nuclear collisions provide the only means to compressnuclear matter to high density within a laboratory envi-ronment. Danielewicz et al. analyzed the flow of matterin nuclear collisions to determine the range of pressure-density relationships for zero-temperature nuclear mat-ter [23]. In fig. 1, the black solid lines represent the equa-tions of state for the symmetric neutron-proton matter

Page 4: Quark deconfinement in neutron star cores

156 The European Physical Journal A

100

101

102

103

0.2 0.3 0.4 0.5 0.6 0.7 0.810

0

101

102

103

P(M

eV

/fm

3)

(fm-3 )

MQMCexperiment

MQMCExp.+Asy_sof tExp.+Asy_stiff

P(M

eV

/fm

3)

symmetric neutron-proton matter

neutron matter

Fig. 1. (Colour on-line) Zero-temperature EOS for symmetricneutron-proton matter and neutron matter. The shaded re-gion in the upper panel corresponds to the region of pressuresfor symmetric neutron-proton matter consistent with the ex-perimental flow data. In the lower panel the upper blue andlower magenta shaded regions correspond to the pressure re-gions for neutron matter consistent with the experimental flowdata after inclusion of the pressures from asymmetry termswith strong and weak density dependences, respectively [23].

and neutron matter in the MQMC model with all pa-rameters fixed as above. We can see that both of thempass through the allowed region. Therefore the modifiedquark-meson coupling model, which is fixed by reproduc-ing the saturation properties of nuclear matter (the sym-metric energy index asym = 32.5MeV, the binding energyEb = −16MeV and the compressibility is 289MeV at thedensity ρ0 = 0.17 fm−3) can be consistent with the exper-imental constraints from nuclear collisions.

3 Results and discussions

3.1 Equation of state

The first-order phase transition takes place at the criticalpoint where baryon chemical potentials and pressuresfor the two phases are equal. The pressures of hadronicand quark phases are shown in fig. 2 as a function ofthe baryon chemical potential and the position of thehadron-quark phase transition can be easily read off asthe point where the lines P(µB) cross. From the figure,we can find four possible hadron-quark phase transitions,i.e., A, B, C and D. EOS of the hypernuclear matter(npH) has no cross with those of quark phases 2SC orUQM, and has only one cross at A with that of CFL.Whereas the neutron-proton matter (np) could changeinto deconfined quark phases in UQM, 2SC or CFL stateat B, C and D. Therefore, we could construct totally

900 1000 1100 1200 1300 1400 1500 16000

100

200

300

400

500

600

B(MeV)

(MeV

/fm

3)

npnpHUQM2SCCFL

A(1459)

B(1456)

C(1341) D(1288)

E(1234)

P

Fig. 2. (Colour on-line) Pressure of the hadronic or quarkphase as a function of the baryon chemical potential. np (blackthick solid line) denotes neutron-proton matter and npH (redthick dashed line) denotes the hypernuclear matter consistingof octet baryons. UQM (green thin dash-dotted line), 2SC (ma-genta thin dashed line) and CFL (blue thin solid line) denotethe three kinds of homogeneous quark phases. Squares orderedby A, B, C and D mark the hadron-quark phase transitionsand E is the phase transition between 2SC and CFL phases.Values in parentheses are the critical chemical potentials.

0 100 200 300 4000

400

800

1200

1600

2000

2400

(MeV/fm3)

(MeV

/fm

3)

npnpHUQM2SCCFL

A

B

C

E

D

P

E

Fig. 3. (Colour on-line) Energy vs. pressure for equations ofstate. The solid lines represent the EOSes of pure phases andthe thin dotted lines indicate the first-order phase transitions.A, B, C, D and E are the same as those defined in fig. 2.

four kinds of hybrid EOSes between the hadronic phaseand the quark deconfinement phase.

EOSes are plotted in fig. 3 and possible phase transi-tions are also shown by dotted lines marked by A, B, C, Dand E. The hybrid EOS of npH+CFL can be constructedby combining the EOS of pure hypernuclear matter npHand the EOS of pure quark matter in the CFL state forlow and high pressures, respectively, with a hadron-quarkphase transition at A. The other EOSes of np+UQM,np+2SC and np+CFL could be similarly constructed.There are some notable features for the hybrid EOSes.First, the hadron-quark phase transitions are first order,which is an inevitable result because we have chosen a

Page 5: Quark deconfinement in neutron star cores

Chang-Qun Ma and Chun-Yuan Gao: Quark deconfinement in neutron star cores 157

0..2 [1.4] [1.6]10

-5

10 -4

10 -3

10 -2

10 -1

100

np+UQM

0 0..2 [1.6] 10

-5

10 -4

10 -3

10 -2

10 -1

100

np+2SC

0 [1.2] [1.4] [1.6]10

-5

10 -4

10 -3

10 -2

10 -1

100

np+CFL

0 1.0 [1.6]10

-5

10 -4

10 -3

10 -2

10 -1

100

npH+CFL

B(fm -3 )

ρi/

B

0.8 [1.25]

u

d

s

e

n p

e

0.67 [0.96]

u

d

s

e

n

p

e

0.61 [1.03]

u/d/s

n

p

e

1.35 [1.46]

u/d/s

n p

e

Λ

(a) (b)

(c) (d)

B(fm -3 ) ρ

B(fm -3)

B(fm -3)

µ

Σ − Ξ-

0

Ξ0

+

0.4 0.60 0.4 0.6 [1.0] [1.2] [1.4]

0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.2

ρρ

i/Bρ

ρi/

Bρρ

i/Bρ

µ

µ

µ

ρ

ρρ

Σ Σ

Fig. 4. Fractions of particles as a function of the baryon number density for different EOSes. The solid lines indicate theparticle compositions in hadronic phases and the dashed lines are for quark compositions. The vertical dotted lines denote thehadron-quark phase transitions, left and right to which are the hadronic and quark phases, respectively. Number on both sidesof the vertical dotted lines denotes the baryon number densities just before and after the hadron-quark transitions. To avoidconfusion, the coordinates of the baryon number density for quark phases are marked in square brackets and those withoutbrackets are for hadronic phases.

sharp interface as claimed in sect. 1. Second, the energygaps of phase transitions are rather large, which is foundto have profound effect on the star structure and is goingto be discussed. Among the hybrid EOSes, the stiffest oneis of np+UQM, and the next are of np+2SC, np+CFLand npH+CFL in turn.

We have treated the three quark phases on an equalfooting, though the relative pressures for different quarkphases are predicted unambiguously by the model, withthe CFL phase having the largest pressure. Because peo-ple’s knowledge about the diquark coupling is very limited,it is possible that diquark coupling might be too small tofulfill the color supercoducting states within the neutralmatter and then UQM is the thermodynamically stablephase. In yet another scenario, the actual density depen-dence of the medium-modified mass of the strange quarkis not well known in QCD. So one could imagine that thestrange quark might be too heavy to participate in Cooperpairing inside the neutron star. Then only a mixture of2SC matter and normal matter of strange quarks couldform, for which we assume that the EOS of 2SC in thepresent model could be approximately right if the strangequark mass is not greatly changed. Or else, the EOS withheavier constituent strange quark mass is stiffer, givingrise to more massive hybrid stars with a given density inthe center. Then, if the present 2SC phase is not excluded,that with more massive strange quark mass should be al-lowed. Taking these into account, the three quark phasesare treated on an equal footing.

3.2 Particle compositions

Fractions of particles as a function of the baryon numberdensity for different EOSes are given in fig. 4. Numbers onboth sides of the vertical dotted lines denotes the baryonnumber densities just before and after the hadron-quarkphase transitions, which reveal that the baryon numberdensities are discontinue at the critical positions. Similarto the energy discontinuities in fig. 3, this is caused byfirst-order phase transitions as well. As to quark phases,the number differences in quarks with different flavors di-minish as the baryon number density increases. Particu-larly, the CFL phase is composed of quarks with three fla-vors in equal numbers. It is seen that electrons are rare inthe quark phases. The electron fraction is lower than 10−4

in UQM and lower than 10−2 in 2SC, while in CFL no elec-tron exists because the numbers of u, d and s quarks areequal. In npH matter, hyperons are very abundant, whichshifts the critical density of the hadron-quark phase tran-sition to 1.35 fm−3, while the critical density is much lowerin np matter.

3.3 Maximum mass

By solving the Tolman-Oppenheimer-Volkoff equa-tions [24], mass-radius relations are obtained for differentequations of state, which are given in fig. 5. The softestEOS of npH+CFL predicts a maximum mass of 1.54M⊙.

Page 6: Quark deconfinement in neutron star cores

158 The European Physical Journal A

9.5 10 10.5 11 11.5 12 12.5 130.6

0.8

1

1.2

1.4

1.6

1.8

2

npnp+UQMnp+2SCnp+CFLnpHnpH+CFL

(10.83, 1.54)

(10.66, 1.99) (11.44, 1.92)

(11.80, 1.82)

(12.00, 1.74)

M/M

R(km)

1.44

1.68

1.90

Fig. 5. (Colour on-line) Neutron star mass-radius relations forpure or hybrid EOSes. Thick lines indicate the pure hadronicstar and thin lines are for hybrid stars. The dots denote themaximum masses and their coordinates are given in the paren-theses. Three horizontal lines represent the observational con-straints from stars PSR 1913+16 (1.44M⊙ [25]), Terzan 5 I(1.68M⊙ [26]) and PSR J0751+1807 (1.90M⊙ [27]), respec-tively.

Compared with the best measured pulsar mass 1.44M⊙

in the binary pulsar PSR 1913+16 [25], which had beentaken as the lower limit of neutron star’s maximum massfor many years, all the EOSes here are allowed. However,very recent measurements strongly indicate that there aremore massive neutron stars. The typical neutron star isreported by Ransom et al. who inferred that at least oneof the stars in Terzan 5, the Ter 5 I, is more massive than1.48, 1.68, or 1.74M⊙ at 99%, 95%, and 90% confidencelevels [26]. Therefore, as indicated in fig. 5, imposing thetighter observational constraint of 1.68M⊙ at 95% con-fidence level, pure hadronic EOS with hyperons and thehybrid EOS of npH+CFL are firmly ruled out, while thehybrid EOSes of np+UQM, np+2SC and np+CFL canbe compatible with this constraint. So far, the most mas-sive pulsar star reported is PSR J0751+1807, which hasan inferred mass of 2.1 ± 0.2M⊙ covering 68% confidenceuncertainties [27]. Therefore, PSR J0751+1807 gives thetightest mass constraint of 1.9M⊙ at about 68% confi-dence level. If this tightest mass limit is confirmed, all thehybrid EOSes are ruled out but that of np+UQM. And, ofcourse, the EOS of pure neutron-proton matter is allowedbecause it is stiffer than the one of np+UQM. However,the measurement uncertainty of PSR J0751+1807 is notyet small enough to draw any firm conclusion and the tightconstraint coming from Ter 5 I is more reliable.

The hybrid stars with quark phases are indicated infig. 5 with thin lines, which reveal that all the stars withquark phase cores are unstable, i.e., those with quark coreswill collapse. The direct reason is that the hybrid EOShas a rather large discontinuity of energy caused by thefirst-order phase transition, as claimed in sect. 3.1. Ourresult confirms those obtained in earlier works, where thehadronic phase has been described by different models,e.g., the relativistic mean-field model or the microscopic

1 1.2 1.4 1.6 1.8 20.1

0.2

0.3

0.4

0.5

0.6

Z

npnp+UQMnp+2SCnp+CFLnpHnpH+CFL

(1.54, 0.31)

(1.99, 0.49)

(1.92, 0.41)

(1.82, 0.36)

(1.75, 0.33)

M/M

0.3325

0.4000

1.44 1.68 1.90

Fig. 6. (Colour on-line) Gravitational redshifts vs. masses fordifferent EOSes. The lower horizontal line is the observationalgravitational redshift of EOX0748-676 [33,35] and the upperone is that of 4U1700+24 [36]. The dots denote the maximummasses and their coordinates are given in the parentheses. Con-straints of observational masses (the same as in fig. 5) are alsoplotted with three vertical lines.

many-body theory [28,29]. Nevertheless, Buballa et al. re-ported recently that by adopting another parameter setwhich was determined by a different method from here,a 2SC phase was possible to exist in the star core withina very tiny window [30]. Baldo et al. suggested that theinstability may be linked to the lack of confinement in thecurrent NJL model [29]. Moreover, parameters obtainedby fitting the vacuum properties might not be suitable athigh densities. Therefore, if the existence of pure quarkcores in neutron stars was confirmed by future observa-tions, the NJL model currently used should be modified.

Besides the assumed sharp transition, there is anothertreatment for the hadron-quark phase transition by takinga hadron-quark mixed phase into account. As has been in-dicated in the recent works by Lawley et al. [31], where thehybrid construction of np and UQM or 2SC is discussedwithin an innovative unified two-flavor NJL model, themixed phases are expected to exist inside a stable neu-tron star. While, as suggested in ref. [32], the existence ofa mixed phase would smooth out the cusps of the mass-radius and mass-redshift relations, but the value of themaximum mass is not expected to be modified by a sub-stantial amount. So the existence of a mixed phase wouldnot change our results if there is no more constraint.

3.4 Gravitational redshift

In principle, the range of gravitational redshift predictedby an EOS should cover all the ever detected redshifts.Cottam et al. have reported a redshift z of 0.35 inferredby identifying three sets of transitions in the spectra of theX-ray binary EXO 0748-676 [33]. The result has been con-firmed by Chang et al. [34]. And it has been shown that thetotal error in this redshift is no more than 5% [35]. There-fore, allowing for the error bar, it imposes a lower limit of

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Chang-Qun Ma and Chun-Yuan Gao: Quark deconfinement in neutron star cores 159

about 0.3325 to the maximum redshift. In fig. 6, gravita-tional redshifts vs. masses of different EOSes are shown.EOSes with hyperons fail to construct the stable neutronstar satisfying the redshift of EXO 0748-676, so they areruled out. Without hyperons, hybrid EOS of np+CFL isnot allowed likely, while those stiffer, namely, of np+2SC,np+UQM and np, are permitted.

Larger observational redshift gives more stringent con-straint. Tiengo et al., inferred that a redshift z = 0.4can be obtained by explaining the emission lines from4U1700+24 with the Ne IX triplet [36]. Then the hybridEOS of np+2SC would be ruled out by this constraint,and only the one of np+UQM is marginally permitted.However, due to the lack of identifications of other spec-tral features at z = 0.4, it seems that the interpretation ofz = 0.012 for 4U1700+24 is more reliable [36]. Thereforethe redshift of EXO 0748-676 is the most reliable con-straint till now.

Combining the constraints of observational masses andredshifts of neutron stars, it can be summarized thatEOSes of np+2SC, np+UQM and np are very likely to befavored. And if either the mass limit of 1.9M⊙ of the mostmassive pulsar star PSR J0751+1807 or the very tentativeinterpretation of the redshift of 4U1700+24 equal to 0.4 isconfirmed, EOSes with CFL or 2SC quark phase are bothdenied, then the allowed hybrid EOS is only of np+UQM.It should be noted that the rotation effect is not consid-ered here because influences of the rotation effect on staticproperties are negligible for the main observations [37] wecurrently discussed.

4 Summary and conclusions

We have used two realistic effective quark models, i.e.,the three-flavor NJL model and the MQMC model,to describe the neutron star matter. For the hadronicphase, EOSes with and without hyperons (npH andnp) are both considered. And we have discussed thequark matter phase in normal and color superconductingstates, namely, UQM, 2SC and CFL. Then four possiblehybrid EOSes between the hadronic phase and thequark deconfinement phase are constructed. For differentEOSes, particle compositions are discussed. Comparingwith observations, we find that EOSes with hyperonsshould be ruled out by the observational constraints fromthe mass of the star Ter 5 I and/or the redshift of thebinary star EXO 0748-676. Moreover, the hybrid EOS ofnp+CFL is also ruled out by the observational redshiftof EXO 0748-676. As a consequence, hybrid EOSes ofnp+2SC and np+UQM as well as pure hadronic EOSof np are most likely to be favored. A tightest but lessreliable constraint can be inferred by the mass of themost massive pulsar star PSR J0751+1807, and if it isconfirmed the permitted hybrid EOS is of np+UQM only.

Therefore, assuming the sharp hadron-quark phasetransition, we conclude that observational constraintsfrom neutron stars could not rule out all the possibleEOSes with a quark phase though the neutron stars with

pure quark cores are found to be unstable in our calcula-tion. Both the normal unpaired quark state and the two-flavor color superconducting state are likely permitted,which indicates that the ground state of neutral matter atmoderate densities could be in a deconfined quark phase.

The authors are grateful to Professor Pawel Danielewicz forproviding the data for the pressure-density relationship con-sistent with the experimental flow data which is indicated bythe shaded region in fig. 1. Financial support by the NationalNatural Science Foundation of China under grants 10305001,10475002 & 10435080 is gratefully acknowledged.

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