EPJ manuscript No. (will be inserted by the editor) Quark Stars: Features and Findings Prashanth Jaikumar Department of Physics and Astronomy,OhioUniversity,Athens,Ohio 45701 USA 7 0 Received: date/ Revised version: date 0 2 Abstract. Under extreme conditions of temperature and/or density, quarks and gluons are expected to n undergoadeconfinementphasetransition.Whilethisisanephemeralphenomenonattheultra-relativistic a heavy-ion collider (BNL-RHIC), quark matter may exist naturally in the dense interior of neutron stars. J Herein,wepresentanappraisalofthepossiblephasestructureofdensequarkmatterinsideneutronstars, 5 and the likelihood of its existence given the current status of neutron star observations. We conclude 2 that quarkmatterinsideneutron starscannot bedismissed asapossibility, although recent observational evidence rules out most soft equations of state. 1 v 4 PACS. 97.60.Jd Neutron stars – 26.60.+c Nuclear matter aspects of Neutron stars 3 7 1 1 Introduction of stable strange matter and focus on the consequence 0 for the surface structure of quark stars. We emphasize 7 BNL-RHIC is engagedina voyageofexplorationanddis- recentdevelopmentsinneutronstarobservationsthatcan 0 coveryinthehigh-temperature,lowbaryondensityregime shedlightonthepossibleexistenceofquarkmatterinside / of QCD’s phase diagram[1,2]. One of the central aims of h neutron stars. p thisprogramistocharacterizethedeconfinementandchi- - ral phase transition in QCD at temperatures reminiscent o of the hot Big Bang. It is widely believed that a strongly 2 Strange quark matter in heavy-ion collisions r t interactingmediumofquarksandgluons,displayingideal s liquid-like behaviour, has been created in the most ener- The rationale behind stable strange quark matter is the a : getic central Au-Au collisions at RHIC[3,4,5]. However, Wittenhypothesis[10],whicharguesthattheintroduction v the lifetime of this phase is of the order of 10fm/c, re- ofstrangenessinupanddownquarkmatterreducesPauli i X quiring penetrating experimental probes that carry the repulsionbyincreasingtheflavordegeneracyfromupand imprint of the early hot partonic phase, and which are downtoup,downandstrangequarks,ensuringinthepro- r a not washed out by the hadronization process[6,7,8]. cess also a lower charge-to-baryonratio for strange quark matter compared to nuclear matter. While this hypoth- Thedenseinteriorofneutronstarsprovidesacomplemen- esis is clearly not borne out for small baryon numbers, tary testing ground for quark deconfinement. The central where strange baryons are definitely heavier than their densities inside neutron stars can be as high as 5-10ρ0 (ρ0 = 2.5 1014 g/cc is the nuclear saturation density), non-strange counterparts, there is no observational evi- × dence to suggest that this is also the case in bulk strange and nucleons overlap to an extent that quarks and glu- quark matter. Nuclei would not spontaneously decay to ons become the effective degrees of freedom. Under such strange matter even if the latter was more stable, since extreme conditions of density, it is possible that strange that would require A weak reactions to occur simulta- quark matter is energetically more stable than nuclear ∼ neously in a nuclear volume containing A nucleons. This matter[9,10]. If this is the case, there is a critical pres- conversionwouldhappenmuchmoreeasilyinthe interior sure at which a first order phase transition from nuclear of neutron stars,where pressures and densities are supra- to quark matter will occur. Quark matter can then com- nuclear. The critical question regarding quark matter in priseanarbitraryfractionofthestar,fromzeroforapure neutron stars is then whether the central density of neu- neutron star to one for a pure quark star, depending on tron stars is large enough so that strange quark matter the equation of state of matter at high density. becomes the ground state of strongly interacting matter. In these proceedings, we outline the rationale behind the This question evades a precise answer because QCD is possible stability of three flavor (up, down and strange) stillnotsufficientlywellunderstoodatneutronstardensi- quark matter at high baryon density and review results ties.Latticemethodsfailatsuchhighdensitiesduetothe fromstrangeletsearchesin heavy-ioncollisions fromAGS complexion of the measure involved in importance sam- toCERN-SPSandRHIC.Weconsiderthe effectsoffinite pling.Inthe absence ofconcreteresultsfromlattice stud- sizeandinterfaceenergycorrectionstothephasestructure ies of QCDat finite density and zerotemperature, simple 2 Prashanth Jaikumar: Quark Stars: Features and Findings model-dependent studies[11] admit a parameter window abatic expansion rather than by evaporation[18]. Alto- (the parametersbeing the strangequark mass,the strong gether, it is very unlikely that stable strangelets would coupling constant and a phenomenological Bag constant) be produced in a heavy-ion collision. within which bulk strange quark matter is stable, evenat zero pressure. If true, this implies that, if central densi- ties inside neutron stars are large enough to create two- 3 Strange quark matter in neutron stars flavor(upanddown)quarkmatter,orifasmallnuggetof cosmological/cosmic-ray origin (”strangelet”) enters the Ifstrangequarkmatter isstable onlyatveryhighbaryon star, the entire neutron matter inside the star will con- number (A 107), a neutron star with 1057 baryons is vert to strange quark matter by absorbing neutrons and a natural ca≫ndidate where such matter can exist. Two equilibrating strangeness. possibilities then arise: (i) quark matter is stable in bulk at some large value of the pressure. In this case, a first- Strangeletsearchesinterrestrialmaterials,cosmicraysor ordertransition is likely to occur at some depth (density) asby-productsofneutronstar-neutronstarcollisionshave insidethe neutronstarandquarkmatter is admixedwith thus far yielded negative results[12,13]. Even if strange hadronicmatterinamixedphasewhosestructuraldetails quark matter is stable in bulk, it may be destabilized aredeterminedbysurfaceandCoulombeffects.(ii)quark by prohibitive surface and Coulomb energy costs so that matterisstableinbulkevenatzeropressure(stillatfinite strangelets do not survive until the present day although density, since it is self-bound). This would imply that all they may have existed in the hot and dense epoch of neutronstarsarereallystrangequarkstars,withpossibly the early universe. If so, conditions in the forward rapid- athinlayerofhadronicmatteratsurface.Letusexamine ity regime of ultra-relativistic nucleus-nucleus collisions these possibilities in more detail below: (and mid-rapidity at fixed target experiments) may be (i) In the event that a first order phase transition oc- able to create strangelets for a short while before they curs inside the star, a mixed phase of nuclear and quark evaporate[14,15]. The experimental signal searched for is matter can occupy a significant portion of the star’s inte- a particle with a large mass-to-charge ratio that, owing rior. This conclusion follows from the fact that there are to its large rigidity, would not be deflected by magnetic twoconservedquantities,electricchargeandbaryonnum- fields,andwouldbeabletoreachthezero-degreecalorime- ber, which can be arranged differently in the two phases, ter (ZDC). There, it would produce a shower originating quarkandnuclear,atdifferentequilibriumpressures.Thus, from a single point, unlike spectator neutrons which are weexpectagraduallyincreasingproportionofquarkmat- dispersed in the transverse plane due to Fermi motion. ter with increasing depth inside the neutron star. The While the strangelet search at NA52[14] at the CERN- structureandsizeoftherarerphase(droplets/rods/slabs) SPS was sensitive to long lived strangelets(τ µs), the atagivendensitydependsonthesurfacetensionbetween ∼ corresponding experiments at AGS[16] and at RHIC[15] thetwophases,thecurvatureenergyandthesmallestDe- were sensitive down to lifetimes of τ 50ns. From these bye screening length. While positive surface tension and ∼ experiments, the production rate of strangelets was lim- curvature energy tend to disfavor small sizes, Coulomb ited to less than one in 107 109 central collisions for energy disfavors large sizes, leading to deformed struc- strangelets exceeding a mass−of 30 GeV/c2 and lifetime tures when Debye screening effects are included. If sur- greater than a few nanoseconds. face tension and Coulomb costs are prohibitively large in themixedphase,thestandardpictureofasharpinterface Suchalowprobabilityforproducingstrangeletsinheavy- with a density discontinuity between hadronic and quark ion collisions is expected on theoretical grounds as well. matterinduced bygravity,is applicable,eventhoughitis Various models have been examined as a mechanism to not the minimum energy configuration in bulk matter. produce stable strangelets (see Ref[17] for a review). The (ii) If strange quark matter is stable in bulk even at coalescencemechanisminvolvesformingaclumpofstrange zero pressure,what we call neutron stars are really quark matter by overlap of a sufficient number of baryons (of stars that contain quark matter almost upto the surface. appropriate strangeness) with small relative momentum. 2 At scales where the strange quark mass m /4µ 1, Thisishighlyimprobableatcolliderenergies.Thethermal s Q ≪ with µ the quark chemical potential, quark matter is Q model, which has its parameters, temperature and bary- effectively neutral with equal numbers of up, down and ochemical potential, tuned to reproduce observed parti- strange quarks. Near the surface of the star, however, cle ratios at chemical freeze-out, predicts less than one wherem2/4µ isnotsmall,electronsarerequiredtomake strangelet for every 1027 collisions for a strangelet with s Q up the deficit of strange quarks in order to form a neu- Z/A 0.1 and mass 20GeV/c2. This production mech- tral object. Microscopically, the electron distribution at ∼ anism falls off rapidly with increasing collider energies. the surface is governed by electrostatics on the length The QGP distillation method relies on the enhancement of strange quarks in a QGP followed by evaporation of scale le 1/qαρ1e/3 1000fm (ρe is the number den- ∼ ∼ the baryonrich QGP through nucleons, thereby distilling sity of electrons and α is the fine structure constant), strangeness and enabling a cool and stable strangelet to while quarks are bound by the strong force (QCD scale emerge.However,baryonrichandstrangeness-richregimes 1fm). Consequently, charge neutrality at the surface ∼ are well separated in rapidity, and there is clear evidence is impossible at scales smaller than a 1000fm in a pic- that the QGP cools through rapid, approximately adi- ture where quark matter is assumed to be homogeneous. Prashanth Jaikumar: Quark Stars: Features and Findings 3 The electrons in this case distribute themselves accord- 48 r=8.25 fm ing to the laws of electrostatics and mechanical equilib- 0 rium. They form a thin chargedskin atop the star, which 1 ) 46 is held to the surface by enormous electric fields E 3 1016V/cm[19].Suchanenormouselectricfieldisexpecte∼d fm 0.8 / toemitelectron-positronpairsatsufficientlyhightemper- V 0.6 φ(r)/φ(0) aturesT 1010K,therebyproducingadramaticsignalof Me 44 nu(r)/nu(0) htiomteqoufarak≥fsetwards.ayTsh,issinsicgenathleissatalsroctoroalnssireanpti,dwlyitthoalolwifeer- -40 0.4 nnnds(((rrr)))///nnnsd(((000))) 1 42 0.2 e e temperatures,shuttingoffthepairemission[20].Thus,ob- ( servation of this signal is highly improbable from an as- C 0 + 0.1 1 10 100 trophysical viewpoint. εs40 r (fm) Thereisamoreappealingalternativeforthesurfacestruc- ture which takes into account the fact that a two-phase 38 systemcanbegloballyratherthanlocallychargeneutral[21]. 4 5 6 7 8 9 10 11 12 r (fm) Relaxingtheconditionoflocalchargeneutralityallowsto 0 reduce the strangeness fraction in quark matter at small Fig. 1. Surface plus Coulomb energy cost as a function of µ , thereby lowering its free energy. Global charge neu- Q nugget size. The optimum size of the quark nugget for the trality is achieved in a mixed phase by having the phase choice (ii) of parameters described in the text is 8.25fm.The fractions vary as a function of the pressure. This hetero- inset shows the quark and electron number densities, as well geneous phase is favored when surface and Coulomb en- as theelectrostatic potential φ inside the nugget. ergies are negligible, as shown in the model independent approach followed in ref[22]. Since µ µ for all rea- e Q ≪ sonable equations of state describing dense quark matter, In the context of the Bag model for dense quark matter, the quark pressure may be expanded in powers of µe/µQ. the condition for forming a mixed phase becomes To second order in µ /µ , it is given by e Q ms 3 ms 2 σ 12 MeV/fm . (4) ≤ (cid:16)150 MeV(cid:17) µ Q 1 2 P(µQ,µe)=P0(µQ)−nQ(µQ) µe+ 2 χQ(µQ) µe, (1) Using estimates of the surface energy of strangelets[24, 25]: (i) σ 8 MeV/fm2 for m =150 MeV and µ 300 s Q MeV; and≃(ii) σ 5 MeV/fm2 for m = 200 M≃eV at where n (µ ) = ∂P/∂µ is the positive charge den- s Q Q e ≃ sity, χ (µ ) = ∂2P−/∂µ2 is the charge susceptibility and µQ 300MeV. The condition in Eq. 4 implies that a ho- Q Q e mog≃eneousphaseismarginallyfavoredform =150MeV P0 is the pressure of the electron-free quark phase. They s while the structured mixed phase is favoredfor m =200 depend on µ , m , and strong interactions. Typical val- s Q s ues in the Bag model description are n = m2µ /2π2, MeV. The sensitivity to ms in Eq. 4 and uncertainty in χQ = 2µ2Q/π2 and P0 = 3(µ4Q −m2sµ2Q)/Q4π2 −sB,Qwhere omtahteers.fiInfittehesiszteruecffteucrtesdcpanhaaselteisr ftahveosreedq,uaitntwitilaltbiveeceosmti-- B is the bagconstant.At fixedµ ,fromEq.1,the quark Q posedofquarknuggetsimmersedinaseaofelectrons.The pressureP is zeroandquarkmatter ispositivelycharged q size of the quark nuggets in this phase is determined by when µ takes on the value e minimizingthesurface,Coulombandotherfinitesizecon- tributions to the energy (see Fig.1). At low temperature, nQ 2P0χQ µ˜e = χ (1−p1−ξ) where ξ = n2 . (2) this mixed phase will be a solid with electrons contribut- Q Q ing to the pressure while quarks contribute to the energy density - much like the mixed phase with electrons and Amixedphaseis possiblewhen0<ξ <1.Inthis regime, nuclei in the crust of a conventional neutron star. This the mixed phase has lower free energy (larger pressure) modified picture of the strange star surface has a much than homogeneous matter. The mixed phase is however reduceddensity gradientat surfaceand negligible electric penalized by Coulomb, surface, and other finite size con- field unlike the old paradigmfor quark stars.In the mod- tributions to the energy. At zero temperature and pres- ern viewpoint, there is no need for the electron skin or sure, the magnitude of the change in Gibbs free energy associatedlargeelectric fields, since matter atthe surface per quark in going from a homogeneous to a mixed phase is globally neutral. The observed luminosity from such a should be more than the surface tension, i.e. the energy surface will be very different than from a charged surface cost of creating a droplet surface, Then, the mixed phase with an electron skin[26]. is preferredoverhomogeneousquark matter.This critical It is also of interest to estimate the radial extent ∆R of surface tension is[23] themixedphasecrust.Thisisbecausesomeneutronstars 0.8n2 exhibit ”glitches” in their rotation, when they suddenly σcrit = 12√παχQ3/2 . (3) ssptainrs-uhpavbeefaorlearggreadcruuasltly, wrehseurme iangsusppeirnfludiodwann.dIfastlarattnigcee Q 4 Prashanth Jaikumar: Quark Stars: Features and Findings 1014 ing pattern can be quite complex involving phases with gapless modes for certain quark quasiparticles. Of partic- 13 10 ular interest are the crystalline phases[29], where quarks 1012 withdifferentFermisurfacespairatnon-zeromomentum, resulting in an inhomogenous but spatially periodic or- 11 10 der parameter. The crystalline structure may also serve ) 3m1010 as sites for pinning rotational vortices formed in the su- c perfluid as a result of stellar rotation, and could generate g/ 9 ( 10 theobservedglitchphenomenainneutronstarspin-down. ρ 8 Thegaplessandcrystallinephasescanalsoleadtotemper- 10 aturedependencesthataremodifiedfromtheusualforms 107 inungappedquarkmatter.Thesephasesalsohaveunique dispersion relations for certain quark quasiparticles, and 6 10 consequently, a specific heat per unit volume that is also 5 different from ungapped quark matter. These two factors 10 0 10 20 30 40 50 imply a change in the stellar cooling curve that can be ∆R (m) confronted by observations[30]. The question as to which Fig.2.Densityprofileofthecrustforastrangestarwithmass is the preferredstable state ofquarkmatter atintermedi- M=1.4 solar masses and radius R=10 km[22]. ate densities and physical strange quark mass remains an open one at this time. structure can co-exist, they could explain this glitching mechanism. Using Newtonian approximations to Hydro- statics, the size of the crust is given by[22] 4 Constraints from neutron star observations R n2 ∆R= Q R, (5) Neutron star observations can help in constraining the Rs χQǫ0 equationofstate ofdense matter,andalsoin distinguish- ing between different models for the crust as discussed 2 whereRs =2GM/c 3(M/M⊙)kmistheSchwarzschild above. Individual neutron star masses are most precisely ≃ radius of the star, R is the radius of the star and ǫ0 is determined by measuring post-Keplerian orbital param- the energy density inside a quark nugget. For m = 150 s eters in close binary systems. Neutron star masses thus MeV and µ 300 MeV, the Bag model with B = 65 MeV/fm3 yiQeld≃sn 0.045fm−3,χ 46MeV/fm,and determined lie in the range 1.18-1.44M⊙ and have errors ǫ0 ≃283 MeV/fm3Q.≃From Eq.(5), ∆RQ ≃≃100 meters for a oafscleerstsatihnaedntaotheingthhaocfcuarpaceyrc,etnhte.eIqfutahteiornadoifusstactaenoafldseonbsee star with mass M =1.4M⊙. The Newtonian estimate for neutronstarmattercanbepinneddown.Inpractice,there ∆R is close to a more accurate value for ∆R obtained by areseveralcomplications thatmake radius measurements solving generalrelativistic equations for hydrostatic equi- a challenge so that it is only possible to infer the radius libriumnumerically[22].Fig.2showsthedensityprofileof at infinity which is related to the true radius of the star the crust thus obtained. through the relation R∞ = (1+z) R with the red-shift The interior structure of a quark star can be quite com- factor(1+z)−1 = 1 2GM/Rc2.Consequently,instead p − plicated. Recently, a lot of progresshas been made in un- ofmeasuringaradiuswecanonlyinferarelationbetween derstandingQCDatasymptoticallyhighdensities.Inthat mass and radius. It has recently been realized that com- regime,wherepertubativestudiesarereliable,quarkmat- pact objects in low-mass X-ray binaries (LMXBs) which ter is believed to be in a color-flavor-locked(CFL) phase, exhibit X-ray bursting behavior may provide a promising characterized by quark pairing with a completely gapped new avenue to determine, simultaneously, both the mass spectrum.Such a phaseis anelectromagneticinsulatorin andradiusofaneutronstar[31],therebysetting firmcon- bulkandadmitsnoelectrons,evenwhenstressedbysmall straintsonthe equationof state ofdense matter.In these quark masses[27]. If dense quark matter indeed exists in- objects, there is the potential to observe, in addition to side neutron stars, where densities are well above nuclear the quiescent luminosity that can be used to infer R , ∞ matter density but below the density where perturbative the Eddington luminosity during the burst and the grav- QCD is expected to be valid, the ground state of quark itational red-shift through direct observation of the shift matterisuncertain.Nevertheless,insucha“hybridstar”, in identifiable atomic absorption lines in the atmosphere. attractiveinteractionsbetweenquarkswillleadtothefor- Thesimultaneousdeterminationofmass(2.10 0.28M⊙) ± mation of a color superconducting state,characterizedby andradius(13.8 1.8km)[31]oftheburstingLMXBEXO ± quark pairing and superfluidity. The singlet pairing gaps 0748-676 eliminates most soft equations of state and is canbeaslargeas100MeV[28]andtheymodifytransport compatible only with the stiffest neutronor quarkmatter propertiesduetothepresenceofcollectiveexcitationsbe- equations of state. It does not rule out all quark matter low the scale of the gap. At densities relevant to neutron equations ofstate, therefore,hybrid starsor strange stars stars, with µ 500 MeV or less, and with the physi- remainviable[32]. However,this andother heavy neutron Q ∼ cal requirements of charge and color neutrality, the pair- star candidates and the rather large inferred radius for Prashanth Jaikumar: Quark Stars: Features and Findings 5 EXO 0748-676 disfavors the scenario in which significant 16. G.Van Buren,J. Phys.G25, (1999) 411. softeningduetoaphasetransitionathighdensityoccurs. 17. C. Greiner, J. Phys.G25, (1999) 389. These recent developments are reviewed in [33,34]. 18. P. F. Kolb, Heavy Ion Phys. 21, (2004) 243. 19. V. 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