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CPC(HEP & NP), 2010, 34(9): 1331–1334 Chinese Physics C Vol. 34, No. 9, Sep., 2010

Quark-clustering in cold *

LAI Xiao-Yu(5ˆ)1) XU Ren-Xin(M;#)2) School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China

Abstract are proposed to be grouped together to make quark-clusters due to the in cold quark matter at a few nuclear densities, because a weakly coupling treatment of the interaction between quarks there would be inadequate. Cold quark matter is then conjectured to be in solid state (i.e., forming a crystal structure) if the inter-cluster potential is deep enough to localize clusters in lattice. Such a solid state of cold quark matter would be very necessary for us to understand different manifestations of -like compact , and could not be ruled by first principles. Key words quark matter, stars, , nuclear matter, PACS 21.65.Qr, 97.60.Jd, 97.60.Gb

1 Introduction to the physics of mat- cause of the existence of strange quarks), in contrast ter at supra-nuclear density with normal neutron stars whose dominant degrees of freedom are , and there are possible observa- On one hand, pulsar-like compact stars are unique tional evidences that pulsar-like stars could be quark laboratories for the physics of cold matter at supranu- stars [1]. It depends on the state of cold quark mat- clear densities, which certainly relates to the funda- ter at supra-nuclear densities to model quark stars, mental color interaction between quarks. However, which is unfortunately not certain due to the non- one of the challenges for physicists today is to under- perturbative nature of QCD. stand the strong interaction at low energy scales. Al- Therefore, the study of cold quark matter opens though QCD (quantum chromo-dynamics) is believed an important window to connect three active fields: to be the underlying theory for describing the ele- particle physics, condensed matter physics, and as- mentary strong interaction and has been tested well trophysics. The combination of these fields could be at high energy scale, the non-perturbative nature of helpful and necessary for us to explore the nature of QCD at low energy scale makes it difficult for us to both strong interaction and pulsar-like compact stars. deal with particle systems under strong interaction, especially the case of cold matter at a few nuclear 2 Cold quark matter: color supercon- densities. ducting v.s. quark clustering On the other hand, one of the challenges for as- trophysicist is to understand the nature of pulsar-like compact stars, although it has been more than 40 QGP (quark-gluon plasma) is a state predicted years since the discovery of the first pulsar. Of partic- theoretically in QCD, when the temperature is ex- ular interest is whether the density in such compact tremely high (& 200 MeV). QGP can be also call hot stars could be high enough to result in unconfined quark matter, and could be produced in the experi- quarks (quark matter). Stars composed of deconfined ments of relativistic heavy ion collisions. Cold quark quarks (and possible gluons) as the dominant degrees matter, on the other hand, could only be produced of freedom are called quark stars (or strange stars be- at extremely high chemical potential, and it could

Received 19 January 2010 * Supported by NSFC (10778611, 10973002), National Basic Research Program of China (2009CB824800) and LCWR (LHXZ200602) 1) E-mail: [email protected] 2) E-mail: [email protected] ©2010 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd 1332 Chinese Physics C (HEP & NP) Vol. 34 only exist in rare astrophysical conditions, the pulsar- as the energy scale µ increases, and in pQCD it can like compact stars. In principle, we could explore be written approximately as the properties of cold quark matter by astrophysical 4π α ≈ , (1) observations, sine there are possible evidences that s 2 µ2 11− n ln pulsar-like stars are actually quark stars.  3 f  Λ2 At first glance, the ground state of extremely where n is the number of quark flavors, Λ ∼ 200– dense quark matter seems that of an ideal Fermi gas. f 300 MeV is the renormalization parameter, and µ is of Nevertheless, it has also been found that the highly the order of chemical potential. Therefore, for n = 3, degenerate Fermi surface is unstable against the for- f we can get α ' (0.8 to 3) for ρ ∼ 3ρ although pQCD mation of quark Cooper pairs, which condense near s 0 would not be applicable in this case. We can also see the Fermi surface due to the existence of color attrac- that even if ρ ∼ 106ρ , then α ∼ 0.15, which indicates tive channels between the quarks. In fact, a BCS- 0 s that the interaction is still much stronger than the like , similar to electric super- electromagnetic interaction (with coupling constant conductivity, has been formulated within perturba- α ' 1/137). In a word, one could possibly have tive QCD (pQCD) at ultra-high baryon densities. It em α > 1 in the realistic cold quark matter, and this has further been argued, based on QCD-like effective s strong interaction could render the quark grouped models, that color superconductivity could also occur into clusters, rather than making quarks condensate even at the more realistic baryon densities of pulsar- in momentum space to form color super-conductivity. like compact stars. Therefore a color superconductiv- If Q -like particles are created in cold quark mat- ity state [2] is currently focused on in studying cold α ter [3], the distance between neighbored clusters is quark matter. about 2 fm if the mass density ρ = 3ρ . From the However, is there any other possibility about the 0 Heinsenberg’s uncertainty relation, the quarks inside state of cold quark matter? What kind of cold quark one cluster with length scale l have typically the ki- matter can we expected from observations of compact netic energy of ∼ ~2/(m l2). They are bound by color stars rather than from first principles? We will dis- q interaction with energy of ∼ α ~c/l. The length scale cuss another conjectured state of cold quark matter s l can be estimated by equaling the kinetic energy here, i.e., the state of quark-clusters. and color interaction energy, then l ∼ ~c/(α m c2) ∼ Let’s begin with some of order-of- esti- s q ∼ 2 1 fm/αs, and the interaction energy 300αs MeV. mates. For a typical pulsar, the mass is about 1.4M Then quarks are bound tightly inside one quark- and the radius is about 10 km, then the average den- cluster, and because the length scale of one cluster sity mass density is ρ ' 2.4ρ , where ρ is the nuclear 0 0 (l < 1 fm) is much shorter than the inter-cluster dis- matter density. Let us see some properties of quark tance (∼ 2 fm) in this case, quark-clusters can be matter with mass density ρ ∼ 3ρ . 0 considered as classical particles in cold quark matter. matter is most reliable cold quark matter, which is composed of u, d and s quarks, so their number den- −3 3 To understand observations in solid sities are nu = nd ∼ ns ≈ 0.48 fm , and the to- −3 tal quark number density is nq ≈ 1.4 fm , with quark models −1/3 ≈ the distance between quarks lq = nq 0.9 fm.

The de Broglie wavelength is λq = h/ 3mqkT ≈ Quark-clusters in cold quark matter would crys- 1 1 3 − 2 6 − 2 4×10 fm (mq/300 MeV) (T/10 K) p. Therefore, tallize to form classical solid, if the inter-cluster po- 6 for mq ' 300 MeV and the temperature T ' 10 tential is deep enough. Various astrophysical obser-

K (the typical temperature inside pulsars), λq  lq, vations could be understood in terms of solid quark which means that the quantum Fermi-Dirac statis- star models. tics should be applied. Turning off the interac- High-mass pulsars. The solid models tion between quark, then we can get the degener- are much different from the conventional models (e.g., ate quark chemical potential at zero temperature. MIT bag model) in which quarks are relativistic par- NR ∼ NR NR For the non-relativistic case, µs µu = µd = ticles. For a relativistic system, the pressure is pro- 2 2 2 3 ~ (3π nu) /2mq ≈ 380 MeV  T , and for the portional to the energy density, so it cannot have stiff ER ∼ ER ER extremely relativistic case, µs µu = µd = . In a solid quark star, on the other 1 2 3 ~c(3π nu) ≈ 480 MeV  T . hand, quarks are grouped in clusters and these clus- However, the strong interaction should play an im- ters are non-relativistic particles, which means that it portant role. The QCD coupling constant decreases could have stiffer equation of state. A stiffer equation No. 9 LAI Xiao-Yu et al: Quark-clustering in cold quark matter 1333 of state could have very important astrophysical im- a quark star cools (even spinning constantly), the plications because it can lead to a higher maximum changing may cause the development mass. Some recent observations have indicated mas- of anisotropic pressure distributed inside solid matter. sive (∼ 2M ) pulsars [4], which could be explained (ii) A uniform fluid star would keep approximately its naturally in solid quark star models. One of the solid eccentricity of Maclaurin spheres, and the eccentric- quark star models is that the inter-cluster potential ity decreases as a star spins down. However, for a could be described by the Lennard-Jones form just solid star, the shear stress would prevent the eccen- like the case in a system of inert gas [5] tricity of the star from decreasing during spin-down.

12 6 In this case, even the state of matter does not change, r0 − r0 u(r) = 4U0 , (2) and stress energy could still develop as the solid star  r   r   spins down. where U0 is the depth of the potential well, r0 is During an process, the solid star could the range of the inter-cluster force. In this model, additionally support the accreted matter against the equation of state can be very stiff, because at by these forces, unless the forces become so a small inter-cluster distance, there is a very strong strong that a star-quake occurs. This is the so-called repulsion. Under some reasonable values of parame- AIQ (Accretion-Induced star-Quake) mechanism pro- ters, a quark star with Lennard-Jones matter could posed previously [6], which might be responsible to be very massive (> 2M ). The mass-radius curves the bursts (even the supergiant flares) and glitches and mass-central density curves (the central density observed in soft γ-ray repeaters/anomalous X-ray only includes the rest mass energy density) are shown pulsars [7]. The gravitational energy releases during in Fig. 1. The mass of a star increases and reaches star-quakes for the polytropic quark star model have the maximum value as the increases of the central been calculated [8]. The results for polytropic index density, and after that, the star will become gravita- n = 1 are shown in Fig. 2. Three supergiant flares tionally instable. from soft γ-ray repeaters have been observed, with released photon energy being order of ∼ 1047 ergs. Our numerical results imply that for all the parame- 6 ters we chosen, the released energy could be as high N =3, U =50 MeV q 0 as the observed. 4 N =3, U =100 MeV q 0 M

/ N =18, U =50 MeV q 0 M 53 2 N =18, U =100 MeV 10 q 0 ε = 10−2

52 0 10 0 5 10 15 20 ε = 10−3 R / km 51 10 6 ε = 10−4

50 10

4 = 0) (erg) −5 ε ε = 10 ( g M

/ 49 10 M −6 ) − E

2 ε ε = 10 ( g

E 48 10 0 2 2.5 3 3.5 4 4.5 5 n=1, Λ=0 ρ / ρ 47 −3 c 0 10 n=1, Λ=80 MeV fm

46 Fig. 1. The mass-radius and mass-central den- 10 0 0.5 1 1.5 2 2.5 3 sity (rest-mass energy density) curves, for a M/M given surface density ρs = 2ρ0. Nq is the num- Fig. 2. The gravitational energy difference be- ber of quarks in one quark-cluster, and U0 is tween stars with and without anisotropic pres- the depth of the inter-cluster potential. sures, which may be released during sequen- tial star quakes. Λ is the difference of vac- Energy releases during star-quakes. A solid stel- uum energy between inside and outside of a lar object would inevitably result in star-quakes when quark star, and ε denotes the deviation of strain energy develops to a critical value, and a huge tangential P⊥ from the radial pressure Pr, − of gravitational and elastic energies would then be ε = (P⊥ Pr)/Pr. released. Actually there are two types of stress force inside solid stars, and two factors could result in the Pulsar glitches. Pulsar glitches, which are the development of stress energy in a solid star. (i) As occasional disruptions of regular pulses, provide us 1334 Chinese Physics C (HEP & NP) Vol. 34 with rich source of information concerning the prop- quark star should be in solid state in order to explain erties of pulsars. Glitches can happen during the observational properties of pulsar-like compact stars, star-quakes of solid quark stars, but cannot be ex- as we have discussed in this section. plained in terms of conventional quark star models in which quark stars are considered to be in liquid 4 Conclusions and discussions state. In terms of normal neutron stars, the leading model involves the angular momentum trans- Realistic cold quark matter is suggested in a solid fer in the crust from the superfluid to the normal state, where quark-clustering occurs, and solid quark component; however, the observations of long period stars could also be very necessary to understand var- precession for pulsars are inconsistent with this glitch ious observations. To explore the real QCD phases of model, and in addition, it can hardly explain the re- cold quark matter, it should be necessary to combine cent discovered slow glitches. A solid pulsar with different research fields, such as the Lattice QCD, the crust could solve these problems, and both normal QCD-based effective models (e.g., NJL model and and slow glitches can then be modeled [9, 10]. DSE), and the phenomenological models proposed Others. It is worth mentioning that some observa- from astrophysical side. tions of pulsars favor bare strange stars (i.e., a quark In this paper, the quark stars are supposed to without crust) over normal neutron stars, al- normal or classical solid, in which the quark-clusters though they do not depend on the specific state of are in periodic lattices and the barrier penetration is matter inside quark stars. Clear drifting sub-pulses negligible. However, another possibility of quantum suggest that both positively and negatively charged or super solid can still not be ruled out in effective particles should be strongly bound on the surface of QCD models. The BCS-type quark pairing was pro- a pulsar, which could be explained naturally if the posed to form at a Fermi surface of cold quark mat- pulsar is a bare . Another example is the ter, and the shear moduli of the rigid crystalline color observational properties of dead pulsars that do not super-conducting quark matter could be much larger have radio emission because the potential drop in the than those of crusts [12], and this might open field line region is lower than a critical value: one provide explanations to pulsar glitches too. would expect that both thermal X-ray emission and Cold quark matter could be in normal solid state if atomic lines from atmospheres of neutron stars should interaction of the lattices (quark-clusters) is so strong have been discovered; however, no clear atomic fea- that each lattice can only vibrate deep inside the po- tures has been found, and such featureless spectrum tential well. In contrast, quantum solid state could could be a probe for identifying bare strange stars. exist when the interaction of lattices is relatively Moreover, a one-dimensional calculation weak and the potential wells are not deep enough, so shows that the lepton-dominated fireball supported that the quantum penetration is significant. It could by a bare quark surface do play a significant role in be interesting to observationally distinguish between the explosion dynamics under a photon-driven sce- normal solid and quantum solid state in the future. nario [11]. To distinguish bare strange stars and neu- tron stars is important, which is in fact the first step We would like to acknowledge useful discussions to demonstrate the existence of solid quark stars. A at our pulsar group of PKU.

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