. Article .

SCIENCE CHINAPhysics, Mechanics & Astronomy

doi: 10.1007/s11433-013-5160-z

c© Science China Press and Springer-Verlag Berlin Heidelberg 2013 phys.scichina.com www.springerlink.com

Properties of color-flavor locked strange quark matter and strangestars in a new quark mass scaling

CHANG Qian1, CHEN ShiWu1, PENG GuangXiong1,2 & XU JianFeng1*

1College of Physics, University of Chinese Academy of Sciences, Beijing 100049, China;2Theoretical Physics Center for Science Facilities, Institute of High Energy Physics, Beijing 100049, China

Received September 18, 2012; accepted December 14, 2012

Considering the effect of one-gluon-exchange interaction between quarks, the color-flavor locked strange quark matter and strangestars are investigated in a new quark mass density-dependent model. It is found that the color-flavor locked strange quark mattercan be more stable if the one-gluon-exchange effect is included. The lower density behavior of the sound velocity in this model isdifferent from the previous results. Moreover, the new equation of state leads to a heavier acceptable maximum mass, supportingthe recent observation of a compact star mass as large as about 2 times the solar mass.

color-flavor locked, strange quark matter, one-gluon-exchange effect

PACS number(s): 24.85+p, 25.75-q, 12.38.Mh

Citation: Chang Q, Chen S W, Peng G X, et al. Properties of color-flavor locked strange quark matter and strange stars in a new quark mass scaling. SciChina-Phys Mech Astron, 2013, doi: 10.1007/s11433-013-5160-z

1 Introduction

Quantum chromodynamics (QCD) is popularly accepted asthe fundamental theory of strong interactions. Accordingto QCD, nuclear matter of high enough density is expectedto undergo phase transition to a deconfined phase in whichquarks and gluons roam around in the medium and chiralsymmetry is nearly restored. Therefore, researchers have be-come increasingly interested in whether a deconfined phaseof matters consisting merely of quarks is possible. Strangequark matter (SQM) has been investigated for decades be-cause of its great theoretical and experimental significance.It has long been suggested that SQM could be the groundstate of strongly interacting matter [1,2].

Because of the well-known difficulties in working outQCD in a direct and strict way, researchers have to rely

*Corresponding author (email: xjfil@126.com)

on phenomenological models to investigate the properties ofSQM. Recent decades have witnessed lots of papers investi-gating this subject. Many models, such as the Nambu-Jona-Lasinio model [3] and various versions of the bag model withan effective bag constant and effective masses [4], have beenemployed to study the properties of SQM. Most of the re-sults indicated that SQM is probably more stable than thehadronic matter [1,2]. For example, Jaffe and Farhi had ex-plicitly shown that SQM could be stable near the normal nu-clear saturation density for a wide range of parameters [5].

Because SQM might be absolutely stable, many scientistsbelieve that the so-called pulsars might be strange stars [6]. In1969, the possible existence of a quark core in a neutron starwas discussed [7]. In 1970, the structure of a quark star wasstudied with a Fermi gas model of u, d, s quarks [8]. Here-after the analysis of strange stars consisting of quark matterhas been carried out in a number of investigations. For ex-ample, Cheng et al. [9] pointed out that strange stars are most

2

likely formed in the collapse of the core of a massive star af-ter a supernova explosion. Therefore, the core of a neutronstar is a reasonable place for the phase transition of hadronicmatter to quark matter.

It was believed that the BCS mechanism in metals islikely to operate even more effectively in a dense quark mat-ter [10,11]. Bailin and Love pointed out almost 30 years agothat asymptotic freedom and the presence of a Fermi surfaceimply a color superconductor at very high density [12]. Un-fortunately, this result has been largely forgotten until it wasrealized that the super-conducting gap is quite large, on theorder of Δ ≈ 100 MeV at density ρ ∼ 5ρ0 via using the inter-actions that reproduce the strength of chiral symmetry at zerodensity [13,14]. Quark matter and/or its lumps, the so-calledstrangelets, may form color-superconductor at high densityby the chiral-symmetry violating condensate [15]. Currently,color-flavor locked (CFL) SQM is a topic of interest for re-searchers in nuclear physics. Recently, the properties of CFLquark matter have been investigated by lots of authors withvarious models [16–20].

In recent years, the quark mass density-dependent modelhas been applied to study the properties of CFL phase ofstrange quark matter [21] and strangelets [22], in whichmerely the confinement interaction is considered. Lately,it has been shown that the Coulomb-like interaction has astrong effect on the quark mass scaling, and accordingly af-fects the stability of quark matter in ordinary phase [23]. Inthis paper we study the color-flavor locked SQM with one-gluon-exchange interaction included. It is found that the one-gluon-exchange effect can make the CFL quark matter morestable, and the lower density behavior of the sound velocityis different from the previous result. This new approach pro-duces the maximum acceptable mass as big as two times thesolar mass.

This paper is organized as follows. In sect. II, we brieflyreview the thermodynamic formulas, and then study the prop-erties of CFL SQM in the newly obtained quark mass scal-ing. In sect. III, we investigate the mass-radius relation ofstrange stars with the effect of one-gluon-exchange interac-tion included. And we end with a short summary in sect. IV.

2 Properties of color-flavor locked strangequark matter

Let us consider the quasiparticle contribution of the thermo-dynamic potential density

ΩCFL =∑

i

Ωi + Ωpair,V + B, (1)

where Ωi is the usual Fermi-gas result, Ωpair,V is the pairingenergy term, and the bag constant B is appended as usually

done. Only the volume term of pairing contribution is con-sidered here, because the finite-size contribution to Ωpair isassumed small compared with the corrections toΩi [17]. Thevolume term of pairing contribution is

Ωpair,V ≈ −3Δ2μ2

π2, (2)

where μ =(μu + μd + μs

)/3 is the average effective chemical

potential of quarks, and Δ is the super-conducting gap. Thethermodynamic potential density of unpaired strange quarkmatter is

Ωi = − giT2π2

∫ ∞

0

{ln[1 + e−(εi−μi)/T

]

+ ln[1 + e−(εi+μi)/T

]}p2 dp, (3)

where gi is the degeneracy factor (it equals 6 for quarks),εi =

√p2 + mi

2 is the dispersion relation with mi being theparticle mass.

At zero temperature, eq. (3) is simplified to

Ωi =gi

2π2

∫ pF

0(εi − μi)p2 dp, (4)

where pF is the common Fermi momentum. It is a fictionalintermediate parameter in CFL matter and is determined by

∑

i

√pF

2 + mi2 = 3μ. (5)

The number density of flavor type i is

ni = −∂ΩCFL

∂μi=

pF3 + 2Δ2μ

π2. (6)

Accordingly, the baryon number density is nb = (nu + nd +

ns)/3 = nu = nd = ns. Therefore, the CFL strange quarkmatter is always naturally neutral, without requiring any elec-trons. Thus we have μe = 0.

The energy density is

E = ΩCFL +∑

i

μini = ΩCFL + 3nbμ. (7)

And the pressure can be expressed as

P = −ΩCFL + nb

∑

i

∂ΩCFL

∂mi

∂mi

∂nb. (8)

The additional term is because of the density dependence ofthe quark masses. It is necessary to maintain thermodynamicconsistency [24]. The weak chemical equilibrium in SQM isalways maintained by the weak interaction processes such asd(s) ↔ u + e− + νe, u + d ↔ u + s. The relevant chemicalpotentials satisfy

μd = μs, μd + μν = μu + μe. (9)

Because neutrinos enter and leave the system freely, one hasμν = 0. We know that the ordinary quark matter is charge

3

neutral because it can support electrons within it. The CFLstrange quark matter, however, is automatically neutral be-cause quarks of different flavors have the same Fermi mo-mentum. Thus in studying the properties of CFL strangequark matters, the contribution from electrons can be safelyneglected. We therefore have the simple relation μu = μd =

μs.For a given baryon number density nb, we first work out

the common Fermi momentum pF and chemical potential μby solving eqs. (5) and (6). Then the energy density and pres-sure can be easily calculated from eqs. (7) and (8), respec-tively.

In a recent work, we have obtained a new quark mass scal-ing, which reads [23]

mq = mq0 +D

n1/3b

−Cn1/3b , (10)

where mq0 is the current mass of the corresponding quark fla-vor, and nb is baryon number density. The current massesof light quarks are very small compared to that of strangequarks, and can thus be neglected, though the question isstill under active investigations. The current mass of strangequarks ms0 is (95 ± 25) MeV according to the Particle DataGroup. Generally speaking, SQM becomes less stable withthe value of ms0 increasing. The parameters D and C can belinked to known quantities in conventional nuclear physics,such as pion mass mπ, decay constant of pion fπ, strong cou-pling constant αs, string tension σ, vacuum chiral conden-sate 〈qq〉0 etc. Because there are large uncertainties in someof these parameters, they are not used to definitely determinethe D value. A reasonable strategy is to use them for an rangeestimation. One can find that the reasonable value of

√D

should be in the range of (147, 270) MeV [25], and the valueof C varies in the range of (0, 0.918) [23].

The last two terms on the right hand side of eq. (10) are,respectively, the contributions of the confinement and one-gluon-exchange interactions. We can specially denote themby

mI =D

n1/3b

−Cn1/3b . (11)

They are the same for all flavors, indicating the interactioneffect.

One can note that mI approaches zero as the value of nb

goes high enough. By solving mI = 0, the correspondingcritical density can then be easily obtained as

nc =

(DC

)3/2. (12)

When the density is lower than this value, the interaction partof quark mass is given by eq. (11). If nb > nc, the quarkmass has only the corresponding current mass, and it does

not depend on density any more. In this case, the quark mat-ter reaches the asymptotically-free state. Therefore, nc is thecritical density for SQM to become completely free.

In the following numeric calculations, mi should be re-placed by the expression in eq. (10). Throughout this paper,we take C = 0.1, 0.3 and D1/2 = 120, 160 MeV as exam-ples. Because the current mass of u and d quarks is verysmall, we simply take mu0 = md0 = 0. For the current massof s quarks, we take ms0 = 100 MeV. The super-conductinggap varies from several tens of MeVs to several hundreds ofMeVs [13,14]. In this paper, we take Δ = 100 MeV. In thecalculation for the pure bag model, the bag constant is takento be (150 MeV)4.

In Figure 1, we show the energy per baryon as a func-tion of density in three groups of parameter sets. We can seefrom the figure that the lowest energy per baryon correspondsexactly to the zero pressure (showed by full dots). For com-parison, we have also shown the result given by C = 0 atD1/2 = 120 Mev (C = 0 indicates the one-gluon-exchangeinteraction between quarks has not been considered) usingdashed line. It can be seen that if one-gluon-exchange effectis added, the energy per baryon is lower in comparison withabsence of the effect, which indicates that SQM can be eas-ier to maintain stable states in this case. The correspondingequation of state is given in Figure 2.

In Figure 3, we plot the velocity of sound as a function ofdensity in quark mass density-dependent (QMDD) model. Athigher densities, the results approach to the ultra-relativisticcase as expected. At lower densities, the density behaviordiffers. This probably indicates that QMDD model is morereasonable according to the special relativity.

Ene

rgy

per b

aryo

n (M

eV)

1050

1000

950

900

850

800

7500.0 0.2 0.4 0.6 0.8 1.0

Density (fm–3)

D1/2=120 MeV, C=0.3D1/2=120 MeV, C=0D1/2=160 MeV, C=0.1Zero pressure

Figure 1 Energy per baryon vs baryon number density in the cases of dif-ferent parameters. The zero pressure points marked with “•” are coincidentwith the lowest energy density points exactly.

4

400

300

200

100

0

Pre

ssur

e (M

eV fm

–3)

D1/2=120 MeV, C=0.3D1/2=120 MeV, C=0D1/2=160 MeV, C=0.1

200 400 600 800 1000 1200

Energy density (MeV fm–3)

Figure 2 EoS of SQM for different parameter groups. They approach tothe free gas EoS at higher density.

Velo

city

of s

ound

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00.0 0.2 0.4 0.6 0.8 1.0

Density (fm–3)

D1/2=120 MeV, C=0.3D1/2=120 MeV, C=0D1/2=160 MeV, C=0.1Ultra relativistic case

Figure 3 The velocity of sound as a function of density in the quark massdensity-dependent model. With inclusion of the one-gluon-exchange inter-action, the velocity of sound becomes larger than without considering it.

3 Mass-radius relation of strange stars

Strange stars have been investigated intensely in recent years,thus we will investigate the mass-radius relation of CFLquark stars. As usually done, we assume the strange star tobe a spherically symmetric object. Its stability is determined

by the Tolman-Oppenheimer-Volkoff (TOV) equation

dPdr=−GmE

r2

(1+

PE

)(1+

4πr3Pm

)(1−2Gm

r

)−1

, (13)

which is a general relativistic equilibrium equation of idealspherically symmetric hydrostatic object under the action ofgravitational field. In eq. (13),

dmdr= 4πr2E, (14)

and r is the distance from the core of the star, G = 6.707 ×10−45 MeV−2 is the gravitational constant, m = m(r) is themass within the radius r, E = E(r) is the energy density andP = P(r) is the pressure.

In Figure 4, we plot the mass-radius relation of CFLstrange stars. It is seen that the maximum mass in the quarkmass density-dependent model with one-gluon-exchange in-teractions can reach 1.8 m (m is the solar mass). If usinga larger value of the parameter C, one will produce a largermaximum mass, e.g. as large as about two times the solarmass for C = 0.6.

4 Summary

We have investigated the color-flavor locked strange quarkmatter and strange stars in a new quark mass density-dependent model, and compared the results with those inprevious work. It is found that if one-gluon-exchange effectbetween quarks is considered, the color-flavor locked strange

Mas

s (S

olar

mas

s)

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

D1/2=120 MeV, C=0.3D1/2=120 MeV, C=0D1/2=160 MeV, C=0.1Maximum mass

0 2 4 6 8 10 12

Radius (km)

Figure 4 Mass-Radius relation of strange stars. The points marked withcircles (“◦”) represent the maximum acceptable masses. The maximummasses can nearly reach to 1.8 m in the quark mass density-dependentmodel.

5

quark matter can be more stable, and the velocity of soundbecomes larger than without considering this effect, whichmakes the possible maximum mass of a strange star larger,e.g., as large as 1.8–2 m.

The authors would like to thank support from the National Nat-ural Science Foundation of China (Grant Nos. 11135011 and11045006) and the Key Project of Chinese Academy of Sciences (No.Y12A0A0012).

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