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Small, Dense Quark Stars from Perturbative QCD Eduardo S. Fragaa, Robert D. Pisarskia, and Ju¨rgen Schaffner-Bielichb aDepartment of Physics, Brookhaven National Laboratory, Upton, NY 11973-5000, USA bRIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, USA (February 1, 2008) order chiral transition. In addition, however, there is a As a model for nonideal behavior in the equation of state new class of star [7,10], with densities much higher than of QCD at high density, we consider cold quark matter in thatofnuclearmatter. Forthisnewclass,themaximum perturbation theory. To second order in the strong coupling constant, αs, the results depend sensitively on the choice of mass is ≈ 1.M⊙, with a radius ≈ 5 km. Other models with nonideal behavior also generate small, dense quark the renormalization mass scale. Certain choices of this scale 1 correspondtoastronglyfirstorderchiraltransition,andgen- stars [8,9]. 0 erate quark stars with maximum masses and radii approxi- Assume that the chiral phase transition occurs at a 0 mately half that of ordinary neutron stars. At the center of chemicalpotentialµχ [14]. Our perturbativeequationof 2 these stars, quarksare essentially massless. state is applicable only in the chirally symmetric phase, n whenthequarkchemicalpotentialµ>µ . Inthisphase, χ a PACS number(s): 12.38.Bx, 12.38.Mh, 26.60.+c, 97.60.Jd the effects of a strange quark mass, m 100 MeV [16], J s ≈ are small relative to the quark chemical potentials, µ > 4 300 MeV. Thus we take three flavors of massless quarks 1 Strongly interacting matter under extreme conditions with equal chemical potentials [1,17]. can reveal new phenomena in Quantum Chromodynam- 1 The thermodynamic potential of a plasma of massless ics (QCD). Compact stars serve as an excellent obser- v quarksandgluonswascalculatedperturbativelyto α2 3 vatory to probe QCD at large density, as their interior by Freedman and McLerran [2] and by Baluni [3], u∼sings 4 might be dense enough to allow for the presence of chi- the momentum-space subtraction (MOM) scheme. The 1 rally symmetric quark matter, i.e., quark stars [1–11]. MOMcouplingisrelatedtothatinthemodifiedmimimal 1 The usual model used for quark stars is a bag model, 0 substraction scheme, MS, as [3,18–20]: with at most a correction α from perturbative QCD 1 ∼ s [6]. Inthemasslesscase,thefirstordercorrectioncancels 0 αMOM αMS αMS / out in the equation of state, so that one ends up finally s = s 1+ s ; (1) h withafreegasofquarksmodifiedonlybyabagconstant. π π " A π # p If the bag constant is fit from hadronic phenomenology, - α = g2/(4π), with g the QCD coupling constant, and p then the gross features of quark stars are very similar to s he those expected for neutron stars: the maximum mass is Aflav=or1s5.1/(4W8−hil(e5/w1e8)tNafk,ewNith=Nf3t,hweenugmivbeerfoorfmmulaassslefossr v: ≈I2n.Mth⊙is, warittihclae wraedciuosns≈id1e0rqkuma.rkstars,using the equa- arbitaryNf.) In the MS schfeme, to ∼α2s the thermody- i namic potential is then X tion of state for cold, dense QCD in perturbation theory 2 ar ctoon∼trαibsu[t2io,3n].isTthoeuseseremsuoldtseranredewteelrlmkinnoawtino,nasnodfothuerorunnly- Ω(µ)= Nfµ4 1 2 αs − 4π2 − π ning of the QCD coupling constant [12]. At the outset, αn (cid:16) (cid:17)2 Λ¯ α 2 we stress that we do not suggest that the perturbative G+N ln s + 11 N ln s , (2) f f − π − 3 µ π equation of state is a good approximation for the densi- (cid:20) (cid:18) (cid:19) (cid:21)(cid:16) (cid:17) (cid:27) ties of interest in quark stars. Rather, we use it merely where G=G0 0.536Nf +NflnNf, G0 =10.374 .13 as a model for the equation of state of QCD. [21], and Λ¯ is t−he renormalization subtraction poin±t. In To ∼ α2s, there is significant sensitivity to the choice MS scheme, the thermodynamic potential is manifestly of the renormalization mass scale. Under our assump- gauge invariant. We take the scale dependence of the tions,wefindthatthischoiceistightlyconstrainedbythe strong coupling constant, α α (Λ¯) as [12,22]: s s physics. We consider two illustrative values of this pa- ≡ rameter. One choice corresponds to a weakly first order α (Λ¯)= 4π 1 2β1ln(u) chiral transition (or no true phase transition), and gives s β0u(cid:20) − β02 u mtroanximstuarms.mTahsesessecaonnddrcahdoiiicveercyorsriemspiloanrdtsottohaatstorfonneguly- + 4β12 ln(u) 1 2+ β2β0 5 ; (3) first order chiral transition [13], and generates two types β04u2 (cid:18) − 2(cid:19) 8β12 − 4!# of stars. One type has densities a few times that of nuclear matter, and looks like the stars of a weakly first u = ln(Λ¯2/ΛM2 S), β0 = 11−2Nf/3, β1 = 51−19Nf/3, and β2 =2857−5033Nf/9+325Nf2/27. The scale ΛMS 1 isfixedbyrequiringthatα =0.3089atΛ¯ =2GeV[12]; s 1 for Nf =3, (3) gives ΛMS =365 MeV. free gas Allthermodynamicquantitiesfollowconsistentlyfrom Ω(µ). The pressure is givenby p(µ)=−Ω(µ), the quark p00.8 ~αs number density by n(µ)=(∂p/∂µ), and the energy den- p/ sity by ǫ = p + µn. Given our stated assumptions, e 0.6 − ur the only freedom we have in the model is the choice s of the ratio Λ¯/µ. To illustrate this, we take the values es 0.4 ~α2 r s Λ¯/µ=1,2,3. P For reasons which will become clear later, we find the 0.2 choice Λ¯ = 2µ especially interesting. Start with a very largechemicalpotential,suchasµ=100GeV,forwhich 0 αs ∼.095(forthepurposesofdiscussion,assumeNf =3 0.4 C0.h5emic0a.6l pot0e.7ntial µ0. 8(cid:10)(GeV0).9 1 at this scale). The first order term decreases the ideal FIG. 1. The total pressure, relative to the pressure of an gas pressure by 6%; the sum of the first and second order terms decr∼ease the pressure by ∼ 7% of the ideal iadseaalfguansc,tpio0n;ionfclµu;dΛi¯ng=te2rµm.stoorder∼αsandtoorder∼α2s, gas value. Because the strong coupling constant runs relatively slowly with µ at large µ, even at µ = 1 GeV, where αs .31, the first order term decreases the ideal sensitive to nonperturbative effects from static magnetic ∼ gas pressure only by 20%; the first and second order gluons from α3 on, at µ = 0 (and T = 0), the entire ∼ ∼ s 6 terms, by 30%. power series in α is well defined [27]. ∼ s Ascanbeseenfrom(2),theperturbativeexpansionof We now use the perturbative calculation of the ther- the thermodynamic potential is an expansionin a power modynamic potential for µ < 1 GeV. Since both terms series not just in αs, but in αslog(αs). The logarithmof ∼ αs and ∼ α2s have negative coefficients, as αs(µ) in- αs arisesfromthe plasmoneffect, wherethe Debye mass creases with decreasing µ, eventually the pressure van- squaredm2D ∼αsµ2. BecausegluonsatT =0havefour- ishes. While it is clearly invalid using perturbation the- dimensional phase space in loop integrals, however, the ory when p=0, it at least provides a well defined model plasmoneffectisrelativelyinnocuous,andonlyproduces ofdenseQCD.InFig.1weshowthepressureforΛ¯ =2µ; logarithms, log(mD/µ)∼log(αs). it vanishes at µc = 425 MeV, where αs ∼ .65. This cor- This isinstarkcontrasttothe perturbativeexpansion responds to a quark density 4.35ρ0, where ρ0 is the of the free energy atnonzero temperature,T =0. While density of quarks in nuclear m∼atter, 3 .16/fm3. there is again a plasmon effect, m2D ∼ αsT2,6 because in A weakness in our model is how t∼o m×atch the equa- loopintegralsstatic gluonsatT =0haveathree dimen- tion of state for massless quarks, (2), onto that for mas- 6 sional phase space, the perturbative expansion is not in sive quarks and hadrons. The quark chemical potential αs,butin√αs. Theseriesin√αs ismuchworsebehaved must be larger than one third of the nucleon rest mass, than that at µ = 0, T = 0, and does not converge until minus one third the binding energy of nuclear matter, 6 veryhightemperatures[23]. Theconvergenceappearsto µ > µmin 313 5 MeV. While the pressure vanishes ≈ − improve after resummation [24,19,20], or by using Padé atµmin,hadronic(orquark)mattercertainlyexists,with methods [25]. nonzeropressure,forallµ>µmin. Thusweimaginethat Consequently, the perturbative series for the thermo- a very “soft” equation of state for massive quarks (and dynamicpotentialmaybe muchbetterbehavedatµ=0 hadrons) matches onto the equation of state in (2) at 6 (and T = 0) than at T = 0 [26]. This does not imply some µ > µ (see also below). Consequently, µ cannot 6 c c that a given value of αs, which is adequate to compute be much higher than µmin. the thermodynamic potential, works equally well for all It is this which limits the choice of Λ¯/µ in our model. other quantities. In particular, the gaps for color su- For Λ¯ =µ, µ =767 MeV. It is absurd to think that the c perconductivity are nonperturbative, φ exp( 1/√αs) pressure of massive quarks could be small to densities ∼ − [27,15], and much smaller values of αs appear to be re- 33ρ0. Thus we do not consider this case further. quired to reliably compute them [28]. In QCD, effective ∼For Λ¯ = 3µ, µ = 300 MeV when α .6. By the c s ∼ models find that even when µ 400 MeV, these gaps Hugenholtz-van Hove theorem, when the pressure van- ∼ are at most 100 MeV [15]. As the relative change in ishes, the ratioof the totalenergy to the baryonnumber the thermody∼namic potential is only (φ/µ)2, then, for is E/A = 3µ . For iron, E/A = 930 MeV. Thus for ∼ c the equation of state in QCD, color superconductivity is Λ¯ = 3µ, E/A = 3µ = 900 MeV, and, as suggested by c never a large effect. Bodmer and Witten [4,5], strange quark matter is abso- To trulyknow howwellperturbationtheoryconverges lutely stable relative to hadronic matter. atµ=0,itisimperativetocomputethethermodynamic While possible, we prefer an alternate view. Our per- poten6 tial to α3. Unlike the case of T = 0, which is ∼ s 6 2 1 2.4 s ) gas nit Λ = 3µ 3V/fm free ass u 1.8 e m (G Λ=3µ ar 1.2 e ol Λ = 2µ r s su n es Λ=2µ s i 0.6 r s P a M 0 0 0 1 2 3 0 2 4 6 8 10 12 14 Energy density (GeV/fm3) Radius (km) FIG. 3. Mass-radius relation of the quark star for FIG. 2. Equation of state for cold quark matter, for Λ¯/µ=2,3. Λ¯/µ=2,3. turbative equation of state is valid only in the chirally radius satisfy a simple scaling relation, Mmax Rmax 1/B1/2 [5]. ∼ ∼ symmetricphase,for µ>µ . Perhapswhenµ<µ ,the χ χ Forthechemicalpotentialsofrelevancetoaquarkstar, true equationofstate is closeto our perturbativemodel, somewhatsurprisinglywefindnumericallythatthepres- but vanishes smoothly as µ µmin. As discussed later, → surein(2)canbe wellapproximatedbythe effectivebag thisisamodelforaweaklyfirstorder(orno)chiralphase model, (4). This can be seen from Fig. 2, where the transition. relationship between pressure and energy is very nearly The structure of a quark star is determined by the so- linear. When Λ¯ = 3µ, the pressure agrees with a bag lution to the Tolman-Oppenheimer-Volkov(TOV) equations [1]. For the TOV equations, all that matters is the model with Be1ff/4 = 157 MeV and aeff = .626 to within relationship between pressure and energy density. This 2% for µ : 300 470 MeV. This is close to the usual isshowninFig.2forΛ¯/µ=2,3. Thenumericalsolution value in the MI→T bag model, B1/4 = 145 MeV [29]. oftheTOVequationsgivesthemassradiusrelationships When Λ¯ = 2µ, the pressure agrees with a bag model of Fig. 3. For any solution to the TOV equation, the with Be1ff/4 = 223 MeV and aeff = .628 to within 4% for chemicalpotentialreachesitsmaximumvalueatthecen- µ:425 650 MeV. → ter of the star; as one goes out in radius, the chemical Consequently, the mass radius relationships for our potential decreases, and equals µ (where the pressure quark stars agree well with a bag equation of state. To c vanishes) at the edge of the star. For Λ¯ =3µ, the maxi- 5%, the maximum masses and radiiscale accordingto mummassisMmax ≈2.14M⊙. Atthismass,theradiusis ∼∼1/Be1ff/2. Theshapeofthemassradiuscurveisalsothe Rmax 12km;thechemicalpotentialatthecenterofthe sameasfor abag model. Notably,lightquarkstarshave ≈ star is µ 456 MeV, which corresponds to a quark den- small radii. This is because for light stars, M M⊙, sity of ρm≈ax 4.87ρ0. When Λ¯ = 2µ, Mmax 1.05M⊙. the chemical potential at the center of the star≪is near ≈ ≈ Atthismass,theradiusisRmax ≈5.81km;thechemical µc, and the equation of state is controlled by that of potentialatthecenterisµ 649MeV,correspondingto massless fields, minus a bag constant. ≈ a quark density of ρmax 14ρ0. Our results for Λ¯ = 2µ can be compared to other ≈ To help understand these results, it is useful to com- equations of state for dense QCD [8–10]. All of these pare to the equation of state of a nonideal bag model: can be viewed as models in which there is nonideality at a scale significantly higher than nuclear matter den- N Ω(µ)=−4πf2 aeff µ4+Beff ; (4) sities. Ref. [8] uses a Schwinger-Dyson model, and finds Mmax 0.7M⊙ andRmax 9.km. Refs.[9]and[10]use ≈ ≈ Beff is an effective bag constant, and the parameter aeff models with massive quasiparticles, so that the masses measuresdeviationsfromideality. Acommonchoiceisto actasatypeofnonideality. Ref.[9]findsMmax 0.8M⊙ ≈ take aeff from the thermodynamic potential to one loop and Rmax ≈ 4. km; ref. [10] finds Mmax ≈ 1.35M⊙ and order,with a fixedvalue ofthe coupling constant: aeff = Rmax ≈ 10. km. We note that in Nambu-Jona-Lasino 1 2α /π [6]. In a bag model, the relationship between models, stars with a quark core do not arise,even at the s pr−essure and energy density is linear, p = (ǫ 4B)/3, maximum mass [11]. irrespective of the value of aeff. Thus we can u−niformly What about the manifestly nonperturbative phase in scale p, ǫ, and B together, so the maximum mass and whichchiralsymmetryisspontaneouslybroken,µ<µχ? 3 Tounderstandthis,consideranexpansioninalargenum- in which chiral symmetry is broken (assuming µ < χ ber of colors [30]. The usual large N limit is to let 456 MeV). Thus realistically, all of our “quark” stars c N at fixed N . Since quark loops are suppressed havemantleswithmassivequarks,andthenhadrons. For c f → ∞ inthis limit, however,gluonsareonly affectedby quarks stars near the maximum mass, we assume this mantle is when µ N1/4 or larger. This is in contrast to the thin, and does not greatly alter its properties. As the c ∼ transition at a nonzero temperature, which occurs at a mass decreases, however, the portion of the star in the temperature N0 [31]. We then consider a generalized chirally symmetric phase becomes small, until the entire ∼ c largeN limit,inwhichN atfixedN /N [32–34]. stariscomposedentirelyofmassivequarksandhadrons. c f f c The quark thermodynamic→po∞tential is Ω N N µ4, At this point, the relationship between the stars mass f c and the quark number density n N N∼µ3−. In this and radius is no longer like that of Fig. 3. Instead, it f c limit, µ N0, as at nonzero temp∼erature. looks like that of nonrelativistic matter, for which the χ ∼ c For the purposes of our discussion, all that matters radiusincreasesasthemassdecreases. Itisnecessaryfor is that baryon masses m N . The baryon chemical thechiraltransitiontobeweaklyfirstorder,orasmooth B c ∼ potential is related to the quark chemical potential as crossover,for the mass-radius curve to be continuous. µ = N µ N . When the baryons are nonrelativistic, The secondpossibility is thatthe equationofstate for B c c ∼ sotheirFermimomentak 1,the baryonnumber den- massless quarks does not match smoothly onto that for f sity is n d k3. The dege∼neracyof baryonsis at least massive quarks, with a strongly first order chiral transi- B ∼ B f N ,andcouldeasilybelarger, N2. Thebaryonther- tion [13]. As the thermodynamic potential approaches ∼modfynamicpotentialis(naively)∼Ω f d k5/m . For the ideal gas limit at large µ, and vanishes at µmin, B ∼− B f B this requires that the pressure is small at a value of k 1, however, everything is fine: the baryon density f ∼ µ µ . Below µ , by construction the pressure of is N that of quarks, and Ω (1/N )Ω. In terms of χ min χ c B c ≫ ∼ ≤ massive quarks is small, with a “soft” equation of state the quark chemicalpotential, however,µ=µ /N ; with B c m =m /N , µ m +k2/(2m N )+.... That is, for [36]. q B c ≈ q f B c This occurs if Λ¯ = 2µ. At the maximum mass, k 1, the region in µ over which hadrons are a rea- f esonnera∼gbyleodfensuccrilpeatiromnaisttsemraisll,au∼to1m/Natc2ic.aAllylso,1th/Ne 2b.inding µof∼the64st9arMiseVcomatptohseedcoenftmera;ssilfesµsχqu≪ark6s4.9AMsetVhe, mmaossst ∼ c of the star decreases, so does the amount in the chirally From this large N argument, we conclude that a c symmetricphase. Ifthechiralphasetransitionisstrongly hadronic description is applicable only in a very narrow first order, eventually one jumps to a second branch, in region of µ; for larger µ, still of order one, a quark de- which the chemical potentials are always < µ through- scription is appropriate. This need not be a true phase χ out the star. Stars on this second branch are composed transition; rather, simply that the thermodynamic po- only of massive quarks and hadrons, with a maximum tential may be very difficult to compute in terms of mass and radius like that of “ordinary” neutron stars. hadrons, but relatively simple in terms of quarks. For example, when k N1/2, so µ m 1/N , naively A strong first order chiral transition is necessary to en- 3/2 f ∼1/2c − q ∼ c sure that there are two, distinct branches. Using toy Ω d N N Ω. This cannot be right — the B ∼ − B c ∼ c modelsforthe thermodynamicpotential,numericallywe thermodynamic potential of baryons cannot dominate obtained solutions to the TOV equations which display thatofquarks. Theonlyresolutionisthattherearecan- twobranches: wepatchedathermodynamicpotentialfor cellations — analogous to those which occur for baryon- massive quarks, for µ < µ µ , onto that for massless c χ meson couplings [35]— which greatly reduce the baryon ≈ quarks, for µ > µ . In this case, our stars of massless c thermodynamicpotential,sothatitiscomparabletothat quarks constitute a third class of compact stars, after of quarks. In other words, the hadronic thermodynamic white dwarfs and “ordinary” neutron stars [7,10]. potential must “soften” whenever µ m >1/N2. − q c Most pulsars have masses 1.5M⊙ [1]. The MACHO Once one is awayfrom this (narrow)hadronic window ∼ project has also reported micro-lensing events for the inµ,the appropriateequationofstate forµ<µ isthat χ Large Magellanic Cloud with masses M = 0.15–0.9M⊙ for massive quarks. There are then two possibilities. [37]. For a weakly first order chiral phase transition, if Thefirstisthatthethermodynamicpotentialformass- MACHOeventsarehadronicstars,they musthavelarge less quarks matches, more or less smoothly, onto that of radii. For a strongly first order chiral phase transition massive quarks. This requires either a weakly first order [13], MACHO events could be quark stars, with small chiraltransition,orperhapsjustcrossover. Belowµ ,as χ radii,andpulsarsmightrepresentthesecondbranch. We iµn→a faµsmhiinonthteypqiucaarlkofthmearmssoivdeynpaamrtiicclepso;twenitthiainl van1i/shNe2s stress that our numbers for the maximum mass and ra- ∼ c diusaremeantonlytobesuggestive. Evenso,webelieve of µmin, a hadronic description is applicable. thatourconclusionsarequalitativelycorrect;aweak(or This is illustrated by the choice Λ¯ =3µ. For a star at no) chiral phase transition leads to one type of compact its maximum mass, at the center µ 456 MeV; as the ≈ objects, a strongly first order chiral transition, to two. radius increases, µ decreases, until one enters a phase We thank R. Harlander, J. Lenaghan, A. Peshier, K. 4 Rajagopal,A.Rebhan,K.Redlich,D.Rischke,andespe- [20] J. P.Blaizot, E. Iancu and A.Rebhan,hep-ph/0005003. ciallyL.McLerranforfruitful discussions. 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