CPC(HEP & NP), 2010, 34(9): 1331{1334 Chinese Physics C Vol. 34, No. 9, Sep., 2010 Quark-clustering in cold quark matter * LAI Xiao-Yu(5)1) XU Ren-Xin(M;#)2) School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Abstract Quarks are proposed to be grouped together to make quark-clusters due to the strong interaction 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 pulsar-like compact stars, and could not be ruled by first principles. Key words quark matter, neutron stars, pulsars, nuclear matter, quantum chromodynamics 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 hadrons, 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 color superconductivity, 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-magnitude 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 . Strange quark 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 star 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 quark star 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 = equation of state. 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 state of matter may cause the development mass. Some recent observations have indicated mas- of anisotropic pressure distributed inside solid matter.