Cooling of neutron stars with color superconducting quark cores

PHYSICAL REVIEW C 71, 045801 (2005)

Cooling of neutron stars with color superconducting quark cores

Hovik Grigorian,1,2 David Blaschke,3,4 and Dmitri Voskresensky5,6

1Institut fur Physik, Universitat Rostock, D-18051 Rostock, Germany2Department of Physics, Yerevan State University, Alex Manoogian Street 1, 375025 Yerevan, Armenia

3Fakultat fur Physik, Universitat Bielefeld, D-33615 Bielefeld, Germany4Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia

5Theory Division, GSI mbH, D–64291 Darmstadt, Germany6Moscow Institute for Physics and Engineering, 115409 Moscow, Russia

(Received 22 November 2004; published 5 April 2005)

We show that within a recently developed nonlocal, chiral quark model the critical density for a phasetransition to color superconducting quark matter under neutron star conditions can be low enough for thesephases to occur in compact star configurations with masses below 1.3 M�. We study the cooling of these objectsin isolation for different values of the gravitational mass. Our equation of state (EoS) allows for two-flavor colorsuperconductivity (2SC) quark matter with a large quark gap (∼100 MeV) for u and d quarks of two colorsthat coexists with normal quark matter within a mixed phase in the hybrid star interior. We argue that, if thephases with unpaired quarks were allowed, the corresponding hybrid stars would cool too fast. If they occurredfor M < 1.3 M�, as follows from our EoS, one could not appropriately describe the neutron star cooling dataexisting today. We discuss a “2SC + X” phase as a possibility for having all quarks paired in two-flavor quarkmatter under neutron star constraints, where the X gap is of the order of 10 keV–1 MeV. Density-independentgaps do not allow us to fit the cooling data. Only the presence of an X gap that decreases with increasing densitywould allow us to appropriately fit the data in a similar compact star mass interval to that following from a purelyhadronic model. This scenario is suggested as an alternative explanation of the cooling data in the framework ofa hybrid star model.

DOI: 10.1103/PhysRevC.71.045801 PACS number(s): 26.60.+c, 12.38.−t, 74.90.+n, 97.60.Jd

I. INTRODUCTION

Recently, we have reinvestigated the cooling of neutronstars within a purely hadronic model [1], that is, suppressingthe possibility of quark cores in neutron star interiors. Wehave demonstrated that the neutron star cooling data availabletoday can be well explained within the nuclear mediumcooling scenario (cf. [2,3]), that is, if one takes mediumeffects into consideration. This scenario puts a tight constrainton the density dependence of the asymmetry energy in thenuclear equation of state (EoS). The latter dependence isan important issue for the analysis of heavy-ion collisions,persued especially within the new CBM (compressed baryonmatter) program to be pursued at the future acceleratorfacility FAIR at GSI Darmstadt. The density dependenceof the asymmetry energy determines the proton fraction inneutron star matter and thus governs the onset of the veryefficient direct Urca (DU) process. This process, once initiated,would lead to very fast cooling of neutron stars. Thereby, ifneutron stars with M < MDU

crit cooled slowly, stars with massM only slightly above MDU

crit already would be characterized byvery rapid cooling, yielding surface temperatures significantlybelow those given by the x-ray data points. Unfortunately,the mass spectrum of cooling neutron stars is not known. Onecannot rely on neutron star mass measurements in binary radiopulsars (BRPs), which have average neutron star masses ofMBRP = (1.35 ± 0.04) M� [4,5], because these objects comefrom a twice-selected population: probably not all neutronstars go through the active radio-pulsar stage [6], and theproperties of neutron stars in binaries may be different from

those of isolated objects (see, e.g., [7]). However, it seemsan unfair assumption that most cooling neutron stars shouldbe essentially lighter than the lower limit for BRP massesof MBRP,min = 1.31 M�. Therefore, the threshold for the DUprocess should occur at high enough densities such that typicalneutron stars are unaffected by it. This assumption about themass distribution can be developed into a more quantitative testof cooling scenarios when these are combined with populationsynthesis models. The latter allow us to obtain log N–log Sdistributions for nearby cooling neutron stars, which can betested with data from the ROSAT catalog [8]. In addition themicroscopically based Argonne model for the EoS, which issupposed to be the most realistic one [9], yields a critical massfor the DU process ∼2 M�. At these high star masses thecentral baryon density already exceeds the nuclear saturationdensity by a factor of 5 such that exotic states of matter(e.g., hyperonic matter or quark matter) are more appropriate.Reference [10] argued that the presence of quark matter inmassive compact star cores is a most reliable hypothesis sincewithin their model hyperons appear at sufficiently low densitiesand the EoS becomes too soft to carry neutron star masses inthe range of MBRP [11].

For quark matter in neutron stars, the interaction (αs �= 0)permits the quark DU process. Previous work [12–15] hasshown that this result does not exclude the possibility thatneutron stars might possess large quark matter cores. Thesuppression of the emissivity of the DU process may arise fromthe occurrence of diquark pairing and thus color superconduc-tivity in low-temperature quark matter. To demonstrate this

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GRIGORIAN, BLASCHKE, AND VOSKRESENSKY PHYSICAL REVIEW C 71, 045801 (2005)

possibility one typically uses the simplifying assumption thatall quarks are gapped with the same gap, which is not valid forsuperconducting phases at low densities. In the present paperwe abandon that restriction and suggest a more subtle pairingpattern that is constrained by comparison with neutron starcooling phenomenology.

II. COLOR SUPERCONDUCTIVITY

The quark-quark interaction in the color antitriplet channelis attractive, driving the pairing with a large zero-temperaturepairing gap � ∼ 100 MeV for the quark chemical potentialµq ∼ 300–500 MeV (cf. [16,17]); for a review see [18]and references therein. The attraction comes either from theone-gluon exchange, or from a nonperturbative four-pointinteraction motivated by instantons [19], or from nonpertur-bative gluon propagators [20]. Various phases are possible.The so-called two-flavor color superconductivity (2SC) phaseallows for unpaired quarks of one color, say blue. There mayalso exist a color-flavor locked (CFL) phase [21] for valuesof the dymanical strange quark mass that are not too large orin other words for very large values of the baryon chemicalpotential [22], where the color superconductivity is completein the sense that the diquark condensation produces a gap forquarks of all three colors and flavors. The value of the gapis of the same order of magnitude as that in the two-flavorcase. The critical density for the occurrence of a CFL phaseis high enough that such a phase might not be realized ina large enough volume of a hybrid star to entail observableeffects. Therefore, in the present paper we will focus on thediscussion of color superconducting quark matter in the 2SCphase for which it was shown earlier that stable hybrid starconfigurations with large quark matter cores can exist [23,24](see also Sec. III).

For the cooling evolution, however, it is important not tohave any unpaired quark species such as, for example, theblue quarks of the 2SC phase with pairing in the (ur-dg)antitriplet channel. Since there exist other attractive quarkpairing channels with a rather weak attraction (X channels), wewill introduce here a hypothetical 2SC + X phase of two-flavorquark matter where all the quarks get paired, some strongly inthe 2SC channel and some weakly in the X channel. Typicalvalues of the X-channel pairing gaps we will expect are in therange �X ∼ 10 keV–1 MeV, as for example in the case ofthe color spin locking (CSL) pairing channel (see [25,26]).Nevertheless, we do not identify the X channel with a spin-1pairing channel like the CSL one simply because this phasedoes not coexist with the 2SC phase, which we assume as thedominant pairing structure here. Furthermore, we suppose thatthere are no Goldstone bosons in the 2SC + X phase.

III. HYBRID STARS

In describing the hadronic part of the hybrid star, asin [1], we exploit the Argonne V 18 + δv + UIX∗ modelof the EoS given in [9], which is based on the mostrecent models for the nucleon-nucleon interaction with theinclusion of a parametrized three-body force and relativistic

boost corrections. Actually we continue to adopt an analyticparametrization of this model by Heiselberg and Hjorth-Jensen[27] (hereafter HHJ) that smoothly incorporates causality athigh densities. The HHJ EoS fits the symmetry energy to theoriginal Argonne V 18 + δv + UIX∗ model in the densityinterval we consider, yielding the threshold density for the DUprocess nDU

c � 5.19 n0 (MDUc � 1.839 M�).

The 2SC phase occurs at lower baryon densities than theCFL phase (see [28,29]). For applications to compact stars theomission of the strange quark flavor is justified by the fact thatchemical potentials in central parts of the stars barely reachthe threshold value at which the mass gap for strange quarksbreaks down and they appear in the system [30].

We will employ the EoS of a nonlocal chiral quark modeldeveloped in [31] for the case of neutron star constraints witha 2SC phase. It has been shown in that work that the Gaussianform-factor ansatz leads to an early onset of the deconfinementtransition and such a model is therefore suitable to discusshybrid stars with large quark matter cores [24].

In some density interval above the onset of the first-orderphase transition, there may appear a mixed phase region(see [32]). Reference [32] disregarded finite-size effects, suchas surface tension and charge screening. For the hadron-quarkmixed phase it was demonstrated [33] that finite-size effectsmight play a crucial role, substantially narrowing the regionof the mixed phase or even forbidding its appearance in ahybrid star configuration. Therefore we omit the possibilityof the hadron-quark mixed phase in our model, assumingthat the quark phase arises via the Maxwell construction. Inthis case we found a tiny density jump on the boundary fromnhadr

c � 0.44 fm−3 to nquarkc � 0.46 fm−3.

A large difference between chemical potentials of u and dquarks forbids the pure 2SC phase (cf. [31]). The CFL phaseis still not permitted at such densities. Ignoring weak pairingpatterns leads to two possibilities: Either the quark matter is inthe normal phase, or there appears a region of the 2SC–normalquark mixed phase. Disregarding finite-size effects, as in [32],Refs. [24,29] have argued for a broad region of the 2SC–normalquark mixed phase instead of a pure 2SC phase. In ourmodel the mixed phase continues up to the central densityncentr. Thus we have the mixed phase in the density intervalnhadr

c < n(mix,1)c < n

quarkc < n < n(mix,2)

c = ncentr. In the givencase the arguments of [33] might be partially relaxed since thesurface tension for the 2SC–normal quark boundary shouldbe significantly smaller than for the quark–hadron boundary.Indeed, as has been indicated in Ref. [15], the surface tensionis proportional to the difference of the energies of the twophases, being proportional to (�/µq)2 � 1 in the given case.Recently, Ref. [34] estimated the surface tension to be typicallysmaller than 5 MeV/fm2 for densities of our interest. For thechoice of the Gaussian form factor the mixed phase appearsfor sufficiently low density n > n(mix,1)

c � nhadrc = 0.44 fm−3

(M > Mquarkc = 1.214 M�). The presence or absence of the

2SC–normal quark mixed phase instead of only one ofthose phases is not as important for the hybrid star coolingproblem since the latter is governed by processes involvingeither normal excitations or excitations with the smallestgap.

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COOLING OF NEUTRON STARS WITH COLOR . . . PHYSICAL REVIEW C 71, 045801 (2005)

0.2 0.4 0.6 0.8 1 1.2

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FIG. 1. (Color online) Mass–central density relations for compactstar configurations with different EoS: purely hadronic star withHHJ EoS (dashed line) and stable hybrid stars with HHJ-Gaussian,nonlocal, chiral quark separable model (SM) with 2SC phase (solidline) and without 2SC phase (dash-dotted line).

In Fig. 1 we present the mass–central density relation forhybrid stars with HHJ EoS versus the Gaussian, nonlocal,chiral quark separable model (SM) EoS. Configurations withthe 2SC–normal quark mixed phase, “HHJ-SM with 2SC”(HHJ hadronic EoS together with the Gaussian, nonlocal,chiral quark model SM permitting a 2SC–normal quark mixedphase) are stable, whereas for those without two-flavor large-gap color superconducting matter (“HHJ-SM without 2SC”)

no stable hybrid star configuration is possible. In the case“HHJ-SM with 2SC” the maximum neutron star mass provesto be 1.793 M�.

IV. COOLING

For the calculation of the cooling of the hadron part ofthe hybrid star we use the same model as in [1]. The mainprocesses are the medium modified Urca (MMU) and the pairbreaking and formation (PBF) processes. The HHJ EoS isadopted. In the following we choose two cooling models usedin [1] distinguished by different sets of nucleon gaps. The firstchoice (model I) is shown in Fig. 2 (Fig. 12 of Ref. [1]). We usea fit for the relation between surface and crust temperatures(called “our fit” in [1]). We also made calculations for theother two crust models used in Ref. [1]. Although the resultsare affected by the choice of crust model, our conclusionsremain untouched. The possibilities of pion condensation andof other so-called exotic processes are suppressed since inthe model [1] these processes may occur only for neutronstar masses exceeding M

quarkc = 1.214 M�. The DU process

is irrelevant in this model up to very large neutron star massM > 1.839 M�. The 1S0 neutron and proton gaps are taken tobe the same as those shown by thick lines in Fig. 5 of Ref. [1].We pay particular attention to the fact that the 3P2 neutron gapis additionally suppressed by the factor 0.1 compared to thatshown in Fig. 5 of [1]. The latter suppresion is motivated bythe result of recent work [35] and is required to fit the coolingdata.

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FIG. 2. (Color online) Model I. Cooling curves according to the nuclear medium cooling scenario (see Fig. 12 of [1]). The labels correspondto the gravitational masses of the configurations in units of the solar mass.

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FIG. 3. (Color online) Model I. Cooling curves for hybrid star configurations with Gaussian quark matter core in the 2SC phase. The labelscorrespond to the gravitational masses of the configurations in units of the solar mass.

For the calculation of the cooling of the quark core in thehybrid star we use the model of [14]. We incorporate the mostefficient processes: the quark direct Urca (QDU) processes onunpaired quarks, the quark modified Urca (QMU), the quarkbremsstrahlung (QB), the electron bremsstrahlung (EB), andthe massive gluon-photon decay (see [12]). Following [36] weinclude the emissivity of the quark pair formation and breaking(QPFB) processes. The specific heat incorporates the quarkcontribution, the electron contribution, and the massless andmassive gluon-photon contributions. The heat conductivitycontains quark, electron, and gluon terms.

We have based our calculation on the picture presentedin Fig. 2 and add the contribution of the quark core. Forthe Gaussian form factor the quark core occurs already forM > 1.214 M� according to the model in [31] (see Fig. 1).Most of the relevant neutron star configurations (see Fig. 2)are then affected by the presence of the quark core. First wecheck for the possibility of 2SC–normal quark phases.

Figure 3 shows the cooling curves calculated with thegiven Gaussian ansatz. The variation of the gaps for thestrong pairing of quarks within the 2SC phase and the gluon-photon mass in the interval �,mg−γ ∼ 20–200 MeV, haveonly slight effects on the results. The main cooling processis the QDU process on normal quarks. We see that thepresence of normal quarks entails cooling that is too fast.The data could be explained only if all the masses lie ina very narrow interval (1.21 < M/M� < 1.22 in our case).For the other two crust models the resulting picture issimilar. The existence of only a very narrow mass interval inwhich the data can be fitted seems unrealistic from the point ofview of observations of neutron stars in binary systems with

different masses [e.g., MB1913+16 � (1.4408 ± 0.0003) M�and MJ0737−3039B � (1.250 ± 0.005 M�); cf. [5]]. Thus thedata cannot be satisfactorily explained.

In Fig. 4 we permit the weak pairing pattern for all quarks(which in Fig. 3 were assumed to be unpaired). We use �X �1 MeV for the corresponding quark gap. Figure 4 demonstratescooling that is too slow. Even slow cooling data are notexplained. The same statement holds for all three crust models.Thus the gaps for formerly unpaired quarks should be evensmaller to obtain a satisfactory description of the cooling data.

In Fig. 5 we again allow for the weak pairing pattern of thosequarks that were assumed to be unpaired in Fig. 3, but now weuse �X = 30 keV for the corresponding residual quark pairinggap (X ). We see that all data points, except the Vela, CTA 1,and Geminga, correspond to hybrid stars with masses in thenarrow interval M = 1.21–1.22 M�, very similar to that inFig. 3. This also seems to be unrealistic by the same argumentused for the presence of unpaired quarks (see Fig. 3). We haveobtained a similar picture for the other two crust models. Wehave varied the constant � in wide limits but the qualitativepicture does not change.

Therefore we would like to explore whether a density-dependent X gap could allow a description of the coolingdata within a larger interval of compact star masses. We donot choose the X gap as a rising function of the chemicalpotential since the feature of such a model will not spread theaccessible mass interval satisfying the cooling data constraintrelative to the density-independent X gap case. We employ theansatz

�X(µ) = �c exp [−α(µ − µc)/µc], (1)

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FIG. 4. (Color online) Model I. Cooling curves for hybrid star configurations with Gaussian quark matter core in the 2SC + X phase. Theweak pairing gap is �X � 1 MeV. The labels correspond to the gravitational masses of the configurations in units of the solar mass.

where the parameters are chosen such that, at the criticalquark chemical potential µc = 330 MeV for the onset of thedeconfinement phase transition, the X gap has its maximalvalue of �c = 1.0 MeV, and at the highest attainable chemical

potential µmax = 507 MeV, that is, in the center of themaximum mass hybrid star configuration, it falls to a valueof the order of 10 keV. We choose the value α = 10 for which�X(µmax) = 4.6 keV. In Fig. 6 we show the resulting cooling

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FIG. 5. (Color online) Same as Fig. 4 but with a weak pairing gap of 30 keV.

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FIG. 6. (Color online) Same as Fig. 4 but with a density-dependent pairing gap according to Eq. (1).

curves for the ansatz (1). We observe that the mass interval forcompact stars that obey the cooling data constraint ranges nowfrom M = 1.32 M� for slow coolers to M = 1.75 M� for fast

coolers such as Vela (cf. found with the purely hadronic model[1] with different parameter choices). Note that, according toa recently suggested independent test of cooling models [8],

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FIG. 7. (Color online) Model II cooling curves. Same as Fig. 20 of Ref. [1]. The labels correspond to the gravitational masses of theconfigurations in units of the solar mass.

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FIG. 8. (Color online) Same as Fig. 6 but for model II.

by comparing results of a corresponding population synthesismodel with the log N–log S distribution of nearby isolatedx-ray sources, cooling model I did not pass the test. Therebyit would be interesting to see whether our quark model withinthe gap ansatz (1) may pass the log N–log S test.

Reference [1] presented different hadronic cooling modelsthat can explain the available cooling data by varying theneutron star mass in some limits. The question arises whetherone of these alternatives could be preferred. In Ref. [1] weconjectured that the model based on Fig. 20 of [1] (Fig. 7here) that has used another choice of gaps demonstrates thebest fit since it shows a more regular dependence on neutronstar masses. We call it model II. In this model II the 1S0 neutrongap is the same as in model I used here, whereas the proton1S0 gap is calculated by [37] (see thin lines in Fig. 5 of [1]).The neutron 3P2 gap is taken from [37] but then is additionallysuppressed by a factor of 0.1 as required by the result of [35]and to appropriately fit the cooling data. According to [8] thismodel II successfully passed the log N–log S test, supportingthe conjecture of [8] and an earlier conjecture of [38] thatneutron star masses should be essentially different.

We have performed the same hybrid cooling calculation asdescribed here for model II. The results are qualitatively thesame as those presented in Figs. 3–6. The difference is thatnow the curves related to low-mass objects (M < MDU

crit ) arepushed downward, describing the Crab data point. However,as in model I, neither scenario—that there are unpaired quarksor that the X gaps are density independent—allows us toappropriately fit the data.

The results for the density-dependent X gap according toEq. (1) now using the hadronic cooling model II (demonstrated

by Fig. 7) are shown in Fig. 8. In comparison with the resultof Fig. 6 the variation of the cooling data between slow andfast coolers is obtained by choosing hybrid star masses in awider range 1.1 � M/M� � 1.7. Next, it would be interestingto check whether this model passes the log N–log S test.

V. CONCLUSION

We have demonstrated that within a recently developednonlocal, chiral quark model the critical densities for a phasetransition to color superconducting quark matter can be lowenough for these phases to occur in compact star configurationswith masses below 1.3 M�. For our choice of Gaussian formfactor the 2SC–normal quark matter mixed phase arises atM � 1.21 M�. We have shown that without a residual pairingthe 2SC quark matter phase could describe the cooling dataonly if compact stars had masses in a very narrow bandaround the critical mass for which the quark core can occur.Since there are observations of neutron stars with higherand essentially different masses such a scenario should bedisfavored.

Then we assumed that formally unpaired quarks can bepaired with small gaps �X < 1 MeV (2SC + X pairing),the values of which we varied in wide limits. With density-independent gaps we failed to appropriately fit the coolingdata.

We showed that the present-day cooling data could beexplained by hybrid stars, however, when assuming a complexpairing pattern, where quarks are partly strongly pairedwithin the 2SC channel, and partly weakly paired with gaps�X < 1 MeV, the size of which rapidly decreases with

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increasing density. Within the given hypothesis and in theframefork of our quark model for the equation of state,the observational data are explained in a broad mass inter-val M = 1.32–1.75 M� for model I and M = 1.1–1.7 M�for model II, as is favored by experimental data. How-ever, we point out that the result is model dependent anddemonstrates only a possibility that the presence of the2SC + X phase might not contradict neutron star coolingdata.

We have found rather strong constraints for the density-dependent X gap pairing. It remains to be investigatedwhich microscopic pairing pattern could fulfill the constraints

obtained in this work. Another indirect check of the modelcould be the log N–log S test.

ACKNOWLEDGMENTS

The authors acknowledge the hospitality and support ofRostock University, where most of this work has beenperformed. The work of H.G. has been supported in part by theVirtual Institute of the Helmholtz Association under Grant No.VH-VI-041 and by the DAAD University partnership program;that of D.V. has been supported in part by DFG Grant No. 436RUS 17/117/03.

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Cooling of neutron stars with color superconducting quark cores