Quark matter in neutron stars within the field correlator method

S. Plumari,1,2 G. F. Burgio,3 V. Greco,1,2 and D. Zappala3

1Dipartimento di Fisica e Astronomia, Universita di Catania, Via Santa Sofia 64, I-95123 Catania, Italia2INFN—Laboratori Nazionali del Sud, Via Santa Sofia 62, I-95125 Catania, Italia

3INFN Sezione di Catania, Via Santa Sofia 64, I-95123 Catania, Italia(Received 11 July 2013; published 15 October 2013)

We discuss the appearance of quark matter in neutron star cores, focusing on the possibility that the

recent observation of a very heavy neutron star could constrain free parameters of quark matter models.

For that, we use the equation of state derived with the field correlator method, extended to the zero

temperature limit, whereas for the hadronic phase we use the equation of state obtained within both the

nonrelativistic and the relativistic Brueckner-Hartree-Fock many-body theory. We find a strong depen-

dence of the maximum mass both on the value of the q �q interaction V1 and on the gluon condensate G2,

for which we introduce a dependence on the baryon chemical potential �B. We find that the maximum

masses are consistent with the observational limit for not too small values of V1.

DOI: 10.1103/PhysRevD.88.083005 PACS numbers: 21.65.Qr, 12.38.Aw, 26.60.Kp, 97.60.Jd

I. INTRODUCTION

The appearance of quark matter in the interior of mas-sive neutron stars (NS) is one of the most debated issues inthe physics of these compact objects. Many equations ofstate (EoS) have been used to describe the interior of NS. Ifwe consider only purely nucleonic degrees of freedom andthe EoS is derived within microscopic approaches [1], itturns out that for the heaviest NS, close to the maximummass (about two solar masses), the central particle densityreaches values larger than 1=fm3. In this density range thenucleon cores (dimension � 0:5 fm) start to touch eachother, and it is hard to imagine that only nucleonic degreesof freedom can play a role. On the contrary, it can beexpected that even before reaching these density values,the nucleons start to lose their identity, and quark degreesof freedom are excited at a macroscopic level.

Unfortunately, it is not straightforward to predict therelevance of quark degrees of freedom in the interior ofNS for the various physical observables, like coolingevolution, glitch characteristics, neutrino emissivity, andso on. The value of the maximum mass of NS is probablyone of the physical quantities that is most sensitive to thepresence of quark matter in NS. If the quark matter EoS isquite soft, the quark component is expected to appear in NSand to appreciably affect the maximum mass value. Therecent observation of a large NS mass in PSR J0348þ0432 with mass M ¼ 2:01� 0:04M� [2] implies that theEoS of NS matter is stiff enough to keep the maximummass at these large values. Purely nucleonic EoS are ableto accommodate such large masses [1]. Since the presenceof non-nucleonic degrees of freedom, like hyperons andquarks, usually tends to considerably soften the EoS withrespect to purely nucleonic matter, thus lowering the massvalue, their appearance would in this case be incompatiblewith observations. The large value of the mass could thenbe explained only if both hyperonic and quark matter EoS

are much stiffer than expected. Unfortunately, while themicroscopic theory of the nucleonic EoS has reached ahigh degree of sophistication, the quark matter EoS ispoorly known at zero temperature and at the high baryonicdensity appropriate for NS. One, therefore, has to relyon models of quark matter, which contain a high degreeof uncertainty. The best one can do is to compare thepredictions of different models and to estimate the uncer-tainty of the results for the NS matter as well as for the NSstructure and mass. In this paper we will use two definitenucleonic EoS, which have been developed on the basis ofthe Brueckner-Hartree-Fock (BHF) many-body theory fornuclear matter, and the field correlator model (FCM) for thequark EoS [3], which in principle is able to cover the fulltemperature-chemical potential plane. The FCM EoScontains ab initio the property of confinement, which isexpected to play a role as far as the stability of a neutron staris concerned [4], at variancewith othermodels like, e.g., theNambu-Jona-Lasinio model. In a previous paper [5] weanalyzed the EoS of the quark matter in the FCM (for areview see [3]) and found that the model could be testedagainst NS observations, and these could seriously con-strain the parameters used in the model. It was shown thatthis approach admits stable NS with gravitational massesslightly larger than 1:44M�, thus providing numerical in-dications on some relevant physical quantities, such as thegluon condensate. In the present paper, we elaborate furtheron this idea and explore the dependence of the model on theq �q potential V1 and, moreover, the dependence of thegluon condensate G2 on the baryon chemical potential�B. The observation of a very large neutron star mass [2]can be used to put constraints on these two parameters.This paper is organized as follows: in the next section

the FCM at finite temperature and density is brieflyrecalled, with an extensive discussion of the modelparameters, while Sec. III contains some details of theEoS for the hadronic phase. In Sec. IV the hadron-quark

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phase transition is illustrated, and the results of our analy-sis are presented in Sec. V. Finally, Sec. VI is devoted to theconclusions.

II. QUARK MATTER: EOS IN THE FIELDCORRELATOR METHOD

The approach based on the FCM provides a naturaltreatment of the dynamics of confinement in terms of thecolor electric (DE and DE

1 ) and color magnetic (DH andDH

1 ) Gaussian correlators, the former one being directlyrelated to confinement, so that its vanishing above thecritical temperature implies deconfinement [3]. The exten-sion of the FCM to finite temperature T and chemicalpotential �q ¼ 0 gives analytical results in reasonable

agreement with lattice data, thus allowing us to correctlydescribe the deconfinement phase transition [6–11]. In thiswork, we are interested in the physics of neutron stars, andtherefore the extension of the FCM to finite values of thechemical potential [7,8] allows us to obtain the equation ofstate of the quark-gluon matter in the range of baryondensity typical of the neutron star interiors.

Within the FCM, the quark pressure for a single flavor issimply given by [7,8,11]

Pq=T4 ¼ 1

�2

���

��q � V1=2

T

�þ��

���q þ V1=2

T

��

(1)

where

��ðaÞ ¼Z 1

0du

u4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ �2

p 1

ðexp ½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ �2

p� a� þ 1Þ

(2)

with � ¼ mq=T, and V1 the large distance static q �q

potential:

V1 ¼Z 1=T

0d�ð1� �TÞ

Z 1

0d��DE

1

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 þ �2

q �: (3)

The potential V1 in Eq. (3) is assumed to be independentof the chemical potential, and this is partially supportedby lattice simulations at very small chemical potential[8,12]. We elaborate more on this point in the followingsubsection.

The EoS is completely specified once the gluon contri-bution is added to the quark pressure, i.e.

Pg=T4 ¼ 8

3�2

Z 1

0d��3 1

exp ð�þ 9V1

8T Þ � 1(4)

and therefore

Pqg ¼ Pg þX

j¼u;d;s

Pjq þ��vac (5)

where Pq and Pjg are, respectively, given in Eqs. (1) and

(4), and

��vac � �ð11� 23NfÞ

32

G2

2(6)

corresponds to the difference of the vacuum energy densityin the two phases, with Nf the flavor number. G2 is the

gluon condensate whose numerical value, determined bythe QCD sum rules, is known with large uncertainty [13],

G2 ¼ 0:012� 0:006 GeV4: (7)

Therefore, the EoS in Eq. (5) essentially depends on twoparameters, namely, the quark-antiquark potential V1 andthe gluon condensate G2. In addition, at finite temperatureand vanishing baryon density, a comparison with the avail-able lattice calculations of the Wuppertal-Budapest [14,15]and hotQCD collaborations [16–18] provides clear indica-tions about the specific values of these parameters and, inparticular, their values at the critical temperature Tc. Theseestimates are related to the corresponding values of theparameters at T ¼ �B ¼ 0 which, in turn, can be used asan input to study the EoS at T ¼ 0 and finite �B.

A. The V1 and G2 parameters

In Ref. [7] the EoS at zero baryon density has beenderived, by explicitly assuming a temperature dependenceof the gluon condensate G2 as found in lattice simulations[19,20], namely, an almost constant G2ðTÞ for 0< T < Tc,with a sudden drop around Tc to one-half of its value,followed by the constant behavior G2ðTÞ¼G2ðT¼0Þ=2,for T > Tc. In addition, an indication on the value ofV1ðTcÞ has been extracted in [21], starting from theexpression of the critical temperature obtained in [7,8],

Tc ¼ a0G1=42

2

0@1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ V1ðTcÞ

2a0G1=42

vuut1A; (8)

where a0 ¼ ð3�2=768Þ1=4. In fact, once the values of G2

and Tc are fixed, one immediately gets V1ðTcÞ from Eq. (8),and in Ref. [21] it has been shown that, for G2ðT ¼ 0Þ ¼0:012 GeV4, the critical temperatures found in [14,17],respectively, Tc ¼ 147� 5 MeV and Tc¼154�9MeV,correspond to rather small values of V1 [V1ðTcÞ &0:15 GeV], while the optimum value indicated in [8],V1ðTcÞ ¼ 0:5 GeV, reproduces those temperatures forsmall values of G2, i.e. G2 ’ 0:004 GeV4.However, one should recall that Eq. (8) is not extremely

accurate, as it is obtained by neglecting the hadron pressureat the transition, which in [8] is estimated as a 10%uncertainty. Hence, a check of the EoS focused on thecritical point T ¼ Tc only could be too restrictive, as thenumerical data on the lattice cover a large temperaturerange above Tc. For that, we compare in Fig. 1 the pre-dictions of the FCM with the available lattice data aroundand above the critical temperature. In Fig. 1 we concentrateon the interaction measure ð�� 3pÞ=T4, which is particu-larly significant because it depends both on the energy

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density and on the pressure of the system and shows,around the critical temperature, large deviations fromzero, i.e. the value of the interaction measure of a freegas of massless particles. The predictions of the FCM arechecked against the lattice data for different parametriza-tions of V1ðTÞ, and also for constant V1 ¼ 0:01 GeV andV1 ¼ 0:1 GeV. From Fig. 1 it is evident that these twoconstant values are too small, and higher values of V1 mustbe considered. As suggested in Ref. [8], we take

V1ðTÞ ¼ c0 þ V1

�T

Tc

�¼ c0 þ 0:175

�1:35

T

Tc

� 1

��1GeV

(9)

with, respectively, c0 ¼ 0:07, 0, �0:05 GeV [correspond-ing to V1ðTcÞ ¼ 0:57, 0.5, 0.45 GeV]. The results, dis-played by the dashed (red), solid (green) and dot-dashed(brown) curves, show a much better agreement with thedata, and in particular, the dashed (red) curve withV1ðTcÞ ¼ 0:57 GeV gives a good fit to the data of [15],represented by full circles (orange), whereas V1ðTcÞ ¼0:45 is preferable for the data in [16] from the hotQCDCollaboration, shown as full diamonds (black). However,we note that, more recently, the hotQCD Collaboration isalso converging toward a smaller peak for ð�� 3PÞ=T4

close to the Wuppertal-Budapest one as presented in [18].This check shows that the analysis of the whole set

of lattice data points toward a value of V1ðTcÞ around0.5–0.6 GeV, while as noticed above, the simple determi-nation of the critical temperature Tc with reasonable valuesof G2 would suggest smaller V1ðTcÞ. The value of thepotential at T ¼ 0, which is essential for the study of NSstructure, has been computed in [21] as a function ofV1ðTcÞ, by making use of Eq. (3), under the assumption

of a temperature-independentDE1 in the region 0< T < TC

[20], obtaining V1ðT ¼ 0Þ ’ 0:8–0:9 GeV in correspon-dence with V1ðTcÞ ¼ 0:5 GeV.It is important to notice that there is no direct relation of

these values with the potential at finite �B. One wouldexpect that an increasing baryon density could produce ascreening effect that reduces the intensity of the quark-antiquark potential, and at large density the quark-quarkinteraction should become more and more relevant. In ouranalysis we choose to keep V1 as a free parameter, andcheck what kind of indications on V1 can be extracted fromthe determination of the maximum mass of neutron stars.Let us now turn to the other parameter of the FCM

model, namely, the gluon condensate G2. As mentionedabove, G2ðTÞ at zero baryon density has been computed onthe lattice [19,20] but, due to technical difficulties, analo-gous calculations in full QCD at large �B are precluded.Therefore, we have to resort to different approaches to getsome indications on the gluon condensate at �B � 0. Inparticular, the QCD sum rules technique has been used tostudy some hadronic properties within a nuclear matterenvironment at T ¼ 0 [22], and it has been found thatthe gluon condensate decreases linearly with the baryondensity �B (mN indicates the neutron mass),

G2ð�BÞ �G2ð�B ¼ 0Þ ¼ �mN�B þOð�2BÞ: (10)

Further analysis [23,24] shows that the corrections toEq. (10), even when including nonlinear effects, are sub-stantially small and can be neglected for our purposes.According to this decreasing trend, the gluon condensatevanishes at some value of the baryon density and, as noticedin [24], one expects that a transition to the deconfined stateshould occur before reaching this point.Once the behavior of the condensate below the transition

is given in Eq. (10), we still need to establish G2ð�BÞ athigher values of the baryon chemical potential to proceedin our analysis. Rather than following the simplest choiceof taking an effective �B-independent G2, which wasadopted in [5,21], we prefer to retain Eq. (10) at lowerdensities and, at the same time, to follow the indicationsproposed in [25,26] at higher densities. In fact, in [25,26] itis suggested that G2ð�BÞ in full three-color (Nc ¼ 3) QCDhas the same qualitative behavior of the correspondingvariable in two-color (Nc ¼ 2) QCD. In this case manytechnical problems that affect the theory with Nc ¼ 3 areabsent and, in particular, the modification of the gluoncondensate at finite chemical potential, namely, the differ-ence fCSð�Þ ¼ G2ð�Þ �G2ð0Þ, is computed from theenergy-momentum tensor of an effective chiral Lagrangianwith the following result:

fCSð�Þ ¼ 4f2�ð�2 �M2Þ�1�M2

�2

�(11)

where M is identified with the pion mass. Equation (11)shows an initial decrease which, after reaching a minimum,

0.2 0.3 0.4 0.5T (GeV)

0

1

2

3

4

5

6

7(ε

-3p)

/T4

Wuppertal-Budapest [14]hotQCD [17]V

1=0.1 GeV

V1=0.01 GeV

V1(T/T

C)+c

0, c

0=0.07 GeV

V1(T/T

C)

V1(T/T

C)+c

0, c

0=-0.05 GeV

FIG. 1 (color online). The interaction measure ð�� 3pÞ=T4 asa function of the temperature as obtained in the FCM for threevalues of c0 in Eq. (9): c0 ¼ 0:07, 0, �0:05 GeV [respectively:dashed (red), solid (green) and dot-dashed (brown) curves],compared with the lattice data of Ref. [15] (orange circles)and Ref. [16] (black diamonds).

QUARK MATTER IN NEUTRON STARS WITHIN THE . . . PHYSICAL REVIEW D 88, 083005 (2013)

083005-3

is followed by a continuous growth. This trend is under-stood with the appearance of a weakly interacting gas ofdiquarks, whose pressure is negligible if compared to itsenergy density, which mostly comes from diquark restmass. Accordingly, the gluon condensate, which is relatedto minus the trace energy-momentum tensor, decreaseswith �B. Only at sufficiently large chemical potentialdoes the contribution of the diquarks on the pressurebecome approximately equal to the energy density, andthe growth of the gluon condensate is observed. This resultis also consistent with lattice calculations [27,28], whichcan be carried out for Nc ¼ 2. Finally, in [25,26] it isclaimed that the color superconducting (CS) phase in theNc ¼ 3 theory should qualitatively reproduce the picturedescribed above, and Eq. (11) it does still hold, providedthat in this case one takes M� 2�QCD and � is identified

with the quark chemical potential: � ¼ ð1=3Þ�B.Therefore, we have put together the two curves of the

gluon condensate at low and at high values of �B, given,respectively, in Eqs. (10) and (11), and give G2ð�BÞ as thesolid (red) line displayed in Fig. 2. More precisely, in Fig. 2the dashed (green) and the dot-dashed (blue) lines, respec-tively, correspond to Eq. (10) [where G2ð�BÞ is reparame-trized in terms of G2ð�BÞ] and Eq. (11). Then, by crudelyassuming that the transition point lies close to the inter-section point of these two curves, we parametrizedG2ð�BÞthrough an effective analytic expression, the solid (red)line, which approximates Eq. (10) at low �B and Eq. (11)at higher �B. This analytic form avoids a discontinuity inthe derivative of G2ð�BÞ at the intersection of the twocurves which could produce unphysical features whencomputing the pressure or the energy density of the system.In Fig. 2 the value of the condensate at zero chemicalpotential is taken, G2ð�B ¼ 0Þ ¼ 0:012 GeV4.

III. HADRONIC PHASE: EOS IN THEBRUECKNER-BETHE-GOLDSTONE THEORY

In this section we remind the reader briefly about theBHF method for the nuclear matter EoS. This theoreticalscheme is based on the Brueckner-Bethe-Goldstone (BBG)many-body theory, which is the linked cluster expansion ofthe energy per nucleon of nuclear matter (see Ref. [29],chapter 1, and references therein). In this many-bodyapproach one systematically replaces the bare nucleon-nucleon (NN) interaction V by the Brueckner reactionmatrix G, which is the solution of the Bethe-Goldstoneequation

Gð�;!Þ ¼ V þ VXkakb

jkakbiQhkakbj!� eðkaÞ � eðkbÞGð�;!Þ; (12)

where � is the nucleon number density, ! is the starting

energy, and jkakbiQhkakbj is the Pauli operator. eðkÞ ¼eðk;�Þ ¼ ℏ2

2m k2 þUðk;�Þ is the single-particle energy,

and U is the single-particle potential,

Uðk;�Þ ¼ Xk0�kF

hkk0jGð�; eðkÞ þ eðk0ÞÞjkk0ia: (13)

The subscript ‘‘a’’ indicates antisymmetrization of thematrix element. In the BHF approximation the energy pernucleon is

E

Að�Þ ¼ 3

5

ℏ2k2F2m

þD2; (14)

D2 ¼ 1

2A

Xk;k0�kF

hkk0jGð�; eðkÞ þ eðk0ÞÞjkk0ia: (15)

The nuclear EoS can be calculated with good accuracyin the Brueckner two hole-line approximation with thecontinuous choice for the single-particle potential, sincethe results in this scheme are quite close to the calculationswhich include also the three hole-line contribution.However, as it is well known, the nonrelativistic calcula-tions, based on purely two-body interactions, fail to repro-duce the correct saturation point of symmetric nuclearmatter, and one needs to introduce three-body forces(TBFs). In our approach the TBFs are reduced to adensity-dependent two-body force by averaging over theposition of the third particle [30].In this work we choose the Argonne v18 nucleon-

nucleon potential [31], supplemented by the so-calledUrbana model [32], as the three-body force. This allowsus to correctly reproduce the nuclear matter saturationpoint �0 � 0:17 fm�3, E=A � �16 MeV, and it givesvalues of incompressibility and symmetry energy at satu-ration compatible with those extracted from phenomenol-ogy [33]. For completeness we will show results obtainedwith the relativistic counterpart, i.e. the Dirac-Brueckner-Hartree-Fock (DBHF) scheme [34], where the Bonn Apotential is used as the NN interaction. In the low density

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2µ

B (GeV)

-0.01

-0.005

0

0.005

0.01

0.015

0.02

G2 (

GeV

4 )

G2(µ

B)=G

2(0)-m

Nρ

BG

2(µ

B)=G

2(0)+f

CS(µ

B)

G2(µ

B): full

FIG. 2 (color online). G2ð�BÞ as obtained from Eq. (10)(green dashed line) and Eq. (11) (blue dot-dashed line) withG2ð�B ¼ 0Þ ¼ 0:012 GeV4. The solid (red) line is the effectiveapproximation used in our analysis (see text).

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region (� < 0:3 fm�3), both BHFþ TBF binding energiesand DBHF calculations are very similar, whereas at higherdensities the DBHF is slightly stiffer [1]. The discrepancybetween the nonrelativistic and relativistic calculations canbe easily understood by noticing that the DBHF treatmentis equivalent to introducing in the nonrelativistic BHF thethree-body force corresponding to the excitation of anucleon-antinucleon pair, the so-called Z diagram [35],which is repulsive at all densities. On the contrary, in theBHF treatment both attractive and repulsive three-bodyforces are introduced, and therefore a softer EoS is expected.

We remind the reader that the BBG approach has beenextended to the hyperonic sector in a fully self-consistentway [36,37], by including the �� and � hyperons, but inthis paper we consider stellar matter as composed byneutrons, protons, and leptons in beta equilibrium [30].The chemical potentials of each species are the fundamen-tal input for solving the equations of chemical equilibrium,charge neutrality and baryon number conservation, yield-ing the equilibrium fractions of all species. Once thecomposition of -stable, charge neutral stellar matter isknown, one can calculate the equation of state, i.e., therelation between pressure P and energy density � as afunction of the baryon density �. It can be easily obtainedfrom the thermodynamical relations

P ¼ � dE

dV¼ PB þ Pl; (16)

PB ¼ �2 dð�B=�Þd�

; Pl ¼ �2 dð�l=�Þd�

(17)

with E the total energy and V the total volume. The totalnucleonic energy density is obtained by adding the energydensities of each species �i. As far as leptons are con-cerned, at those high densities electrons are a free ultra-relativistic gas, whereas muons are relativistic. Hence,their energy densities �l are well known from textbooks[38]. The numerical procedure has often been illustrated inpapers and textbooks [38], and therefore it will not berepeated here.

IV. THE HADRON-QUARK PHASE TRANSITION

We are now able to compare the pressure of the twophases, namely, the pressure in the hadronic phase given inEqs. (16) and (17), and the quark pressure shown in Eq. (5).We adopt the simple Maxwell construction, by assuming afirst order hadron-quark phase transition [39] in beta-stablematter. The more general Gibbs construction [40] is stillaffected by many theoretical uncertainties [41], and inany case the final mass-radius relation of massive neutronstars [42] is slightly affected.

We impose thermal, chemical, and mechanical equilib-rium between the two phases. This implies that the phasecoexistence is determined by a crossing point in the pressurevs chemical potential plot, as shown in Fig. 3. There we

display the pressure P (upper panel) and the baryon density(lower panel) as a function of the baryon chemical potential�B for the baryonic and quark matter phases. The hadronicEoS are plotted as solid (brown, BHF) and dashed (orange,DBHF) curves, whereas symbols are the results for quarkmatter EoS in the FCM and different choices of V1. Weobserve that the crossing points are significantly affected bythe chosen value of the potential V1. Moreover, with increas-ing V1, the onset of the phase transition is shifted to largerchemical potentials. Hence, we expect that the neutron starwill possess a thicker hadronic layer with increasing V1.In Fig. 4 we display the total EoS, i.e. the pressure as a

function of the baryon density for the several cases

-200

0

200

400

600

800

1000

1200

1400

P (M

eV/f

m3 )

BHFDBHFV

1=0.01 GeV

V1=0.1 GeV

V1=0.2 GeV

V1=0.3 GeV

1200 1600 2000 2400µ

B (MeV)

0

0.5

1

1.5

ρ B (

fm-3

)

FIG. 3 (color online). The pressure P is displayed vs thebaryon chemical potential in the upper panel, whereas in thelower panel the baryon density is shown for different values ofV1 in the FCM model. The EoS for the hadronic phase arerepresented by the solid line (brown color) for the BHF, and bythe dashed yellow curve for the DBHF EoS.

0 2 4 6 8 10ρ

B/ρ

0

0

200

400

600

800

1000

1200

1400

P (M

eV/f

m3)

FHBD : PH (b)FHB : PH (a)

V1 = 0.01 GeV

V1 = 0.1 GeV

V1 = 0.2 GeV

V1 = 0.3 GeV

0 2 4 6 8 10 12ρ

B/ρ

0

FIG. 4 (color online). The pressure P is displayed vs thebaryon density for different values of V1 in the FCM model.In the left (right) panel calculations are shown when the BHF(DBHF) approach is used for the hadronic phase.

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discussed above. In particular, we plot in the left panel theEoS obtained by using the BHF approach for the hadronicphase, whereas in the right panel calculations are shown forthe case when the DBHF EoS is adopted. The severalcurves represent different choices of the q �q potential V1.The plateaus are consequence of the Maxwell construction.Below the plateau, -stable and charge neutral stellarmatter is in the purely hadronic phase, whereas for den-sities above the ones characterizing the plateau, the systemis in the pure quark phase. The main features of the phasetransition are displayed in Table I, where we report, for afixed hadronic EoS and several values of the q �q potentialV1, the baryonic chemical potential at the transition �tr

B,the corresponding baryon density �tr in units of the satu-ration density �0, and the gluon condensate Gtr

2 . We seethat, whenever the transition takes place at�B & 1:2 GeV,i.e. below the minimum shown in Fig. 2, the pressure Pshows a kink as a function of the baryon density, and this isdue to the particular parametric form of the gluon conden-sate G2 shown in Fig. 2. The presence of a kink givesunstable neutron star configurations, as will be shown inthe next section.

V. RESULTS AND DISCUSSION

The EoS is the fundamental input for solving the well-known hydrostatic equilibrium equations of Tolman,Oppenheimer, and Volkoff [38] for the pressure P andthe enclosed mass m,

dPðrÞdr

¼ �GmðrÞ�ðrÞr2

½1þ PðrÞ�ðrÞ�½1þ 4�r3PðrÞ

mðrÞ �1� 2GmðrÞ

r

; (18)

dmðrÞdr

¼ 4�r2�ðrÞ; (19)

with � the total energy density (G is the gravitationalconstant). For a chosen central value of the energy density,the numerical integration of Eqs. (18) and (19) provides themass-radius relation. For the description of the neutron starcrust, we have joined the equations of state described abovewith the ones by Negele and Vautherin [43] in the medium-density regime; the ones by Baym, Pethick, and Sutherland

[44] for the outer crust (� < 0:001 fm�3); and by Feynman,Metropolis, and Teller [45]. In Fig. 5 we display the gravi-tational mass (in units of solar massM� ¼ 2� 1033 g) as afunction of the radius R (left panel) and the correspondingcentral baryon density, normalized with respect to the satu-ration value (right panel). Stellar configurations have beenobtained using the BHF EoS for the hadronic phase. Theorange band represents the recently observed neutron starPSR J0348þ 0432 with mass M ¼ 2:01� 0:04M� [2].We have marked the stable configurations by thick lines,whereas full symbols denote the maximum mass. Unstableconfigurations are displayed by thin lines. Among theunstable configurations, we signal those characterized byincreasing mass and decreasing central density, which arerelated to the appearance of the kink in the EoS, as antici-pated in Sec. IV. In Table II we display the values character-izing the maximummass, i.e. the central density (in units of�0) and the corresponding value of the gluon condensateG2

TABLE I. Properties of the hadron-quark phase transition.

V1 (GeV) �trB (GeV) �tr=�0 Gtr

2 (GeV4)

BHF

0.01 0.986 1.22 0.0048

0.1 1.606 5.51 0.0046

0.2 2.09 7.33 0.0109

0.3 2.408 8.4 0.0165

DBHF

0.01 0.987 1.17 0.0047

0.1 1.14 2.26 0.0023

0.2 1.89 5.29 0.0079

0.3 2.24 6.31 0.0165

6 8 10 12 14R (Km)

0

0.5

1

1.5

2

M/M

0

2 4 6 8 10 12ρ

c/ρ

0

V1=0.01 GeV

V1=0.1 GeV

V1=0.2 GeV

V1=0.3 GeV

BHF

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FIG. 5 (color online). The mass-radius (left panel) and themass-central density relation (right panel) are displayed fordifferent values of V1 and the BHF hadronic EoS. The fullsymbols denote the value of the maximum mass. Stableconfigurations are displayed by thick lines, whereas thin linesindicate unstable configurations.G2 is dependent on�B (see textfor details).

TABLE II. Properties of maximum mass configurations.Asterisks denote stable hybrid stars with a pure quark mattercore.

V1 (GeV) Mmax =M� �c=�0 Gc2 (GeV4)

BHF

0.01 1.69* 9.23 0.0042

0.1 1.95 5.47 0.2 2.03 7.24 0.3 2.03 7.24

DBHF

0.01 1.69* 9.23 0.0042

0.1 1.95* 5.88 0.0023

0.2 2.31 5.29 0.3 2.31 5.59

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whenever the core contains quark matter. Those configura-tions are stable, and the mass values are denoted by anasterisk. We see that the maximum value spans over a rangebetween 1.69 and 2.03 solar masses, depending on the valueof the q �q potential V1. However, the observational dataindicate that values ofV1 as small as 0.01 GeVare excluded,and that values of about 0.1 GeV are only marginallycompatible with the observational data. Larger values ofV1 produce increasing maximum masses, but the stablestars are in the purely hadronic phase. We found thatV1 � 0:095 GeV is the largest value which produces stableneutron stars with a quark core.

Similar results are displayed in Fig. 6, where the DBHFEoS is used for the hadronic phase, with the same notationsand coding adopted in Fig. 5. Also in this case the obser-vational data indicate that values of V1 as small as0.01 GeV are excluded, and that values of about 0.1 GeVare only marginally compatible with the observationaldata. This is due to the fact that for V1 ¼ 0:1 GeV thephase transition takes place over a range of densities whereboth BHF and DBHF EoS show a similar behavior, but inthe latter case a stable quark matter phase is produced, ascan also be deduced from the asterisks reported in Table II.

By increasing the value of V1, the value of the maximummass increases, but the stability of the pure quark phase islost, as shown in both Figs. 5 and 6 by the cuspids in themass-radius relation, and the maximum mass contains, inits interior, at most a mixed quark-hadron phase. Themaximum mass can increase well above the observationallimit by increasing the value of V1, but hybrid stars willbe mainly in the hadronic phase. We found that V1 �0:12 GeV is the largest value which produces stable neu-tron stars with a quark core if the DBHF EoS is used for thehadronic phase.

Therefore, generally speaking we can conclude that thismodel gives values of the maximum mass in agreementwith the current observational data if V1 * 0:1 GeV.

We remind the reader that those calculations have beenperformed assuming a dependence of the gluon condensateG2 on the baryon chemical potential �B. For comparison,we show in Fig. 7 the dependence of the maximummass onthe value of a constant G2, which was discussed in ourprevious paper [5]. There are some differences, mainlywhen a small value of V1 ¼ 0:01 GeV is used. In fact, inthis case the value of the maximum mass is compatiblewith observational data only if a stiff EoS for the hadronicphase is adopted and, at the same time, sufficiently largevalues of G2 are selected. With increasing V1, the value ofthe maximummass increases, no matter what the EoS is forthe hadronic phase, as already displayed in Figs. 5 and 6,with G2 dependent on �B. Still, the BHF EoS is onlymarginally compatible with the data, whereas the DBHFpoints lie well above the observed NS masses. We alsonotice that, already at V1 ¼ 0:1 GeV and more evidently atV1 ¼ 0:2 GeV, the values of Mmax collected in Fig. 7 areessentially independent of G2, thus indicating that thequark matter appears only after the hadronic branch hasreached its maximum,Mmax , so that the corresponding starhas no quark matter content.

VI. CONCLUSIONS

In this paper we have studied the effects of the appear-ance of a quark matter core in NS, with the correspondingquark-gluon EoS derived in the framework of the FCM andwith a suitable parametrization of the gluon condensate interms of the baryon chemical potential, as suggested by theanalysis of this variable for the theory with Nc ¼ 2, whereG2ð�BÞ turns out to be a decreasing function at small �B

and an increasing function at larger �B. The inclusion of adensity-dependent gluon condensate is motivated by theexpectations of some significant effect related to the onsetof a superconductive phase at large density, which could behidden by the use of a constant G2. Therefore, the absenceof indications coming from lattice simulations at finite �B

6 8 10 12 14R (Km)

0

0.5

1

1.5

2

M/M

0

2 4 6 8 10 12ρ

c/ρ

0

V1=0.01 GeV

V1=0.1 GeV

V1=0.2 GeV

V1=0.3 GeV

DBHF

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FIG. 6 (color online). Same as Fig. 5 but with DBHF EoS usedfor the hadronic phase. The full symbols denote the value of themaximum mass (see text for details).

0.008 0.012 0.016

G2 (GeV

4)

1.2

1.4

1.6

1.8

2

2.2

2.4

Mm

ax/M

0

DBHF, V1 = 0.01 GeV

DBHF, V1 = 0.1 GeV

DBHF, V1 = 0.2 GeV

BHF, V1 = 0.01 GeV

BHF, V1 = 0.1 GeV

BHF, V1 = 0.2 GeV

PSR J1614-2230

PSR J1903+0327

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FIG. 7 (color online). The maximum mass, in units of the solarmass M�, is displayed vs the gluon condensate G2. G2 isindependent of �B.

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in the theory withNc ¼ 3 has forced us to resort to the onlysuggestion available on the behavior of G2ð�BÞ. Clearlythese comments put some limits on the quantitative accu-racy of the gluon condensate parametrization adopted here,which, however, should be regarded just as a qualitativedescription that could signal potential flaws of the simplerpicture obtained by retaining constant G2.

The same kind of problem shows up for the otherparameter entering the quark matter EoS, namely, V1,which is expected to decrease when �B grows because ofthe screening of the quark-antiquark interaction due to theincreasing density. In this case we have no quantitativeindication about its �B dependence, except for the sugges-tion, given in [11], that V1 is substantially � independent,at least for� � 0. Therefore, we performed our analysis atvarious constant values of the potential V1.

Our results are collected in Figs. 5 and 6 and Table II, andthey indicate that, with the adopted parametrization ofG2ð�BÞ, it is necessary to have V1 * 0:1 GeV in order toachieveMmax � 2M�. For larger V1,Mmax increases but nopure quark phase is present in the core of the NS already forV1 * 0:12 GeV and, beyond 0.3 GeV,Mmax does not growany more. Its value only depends on the particular hadronicEoS chosen. This picture is a substantial refinement of theresults found in [5] where only vanishing or very small V1

were considered. In fact it was observed that Mmax growsboth with V1 and G2, and in particular, it was found thatMmax ¼ 1:78M� at constant G2 ¼ 0:012 GeV4 and V1 ¼0:01 GeV, which was consistent with the NS massesmeasured until then, but cannot explain the more recentmeasurements. In addition, in [5] the disappearance of thepure quark phase when increasing G2 was noticed as well.

Analogous results are also obtained in [21] for constantG2, but in this case larger values of Mmax are obtained

because the chosen hadronic EoS, based on the relativisticmean field model, is considerably stiffer than the oneadopted in this paper.A final comment on the potential V1 is in order. As

already noticed, our analysis shows that the observed NSmasses require values larger than 0.1 GeV, while above0.3 GeV the quark phase is no longer relevant in thedetermination of the maximum NS mass. These character-istic values of V1 are much smaller than those comingfrom the analysis shown in Sec. II A on the lattice data at�B ¼ 0 around and above the critical temperature, which,when extrapolated to T ¼ 0, yield V1ðT ¼ �B ¼ 0Þ ¼0:8–0:9 GeV. This is to be taken as a clear indicationthat the long distance potential is indeed sensitive to theincrease of density that induces a screening effect.However, we also expect that when �B grows, the quarkpopulation increases with respect to the antiquark so thatthe quark-quark interaction becomes more and more rele-vant; eventually, it is conceivable that it becomes predomi-nant with respect to the quark-antiquark interaction.According to this point one could imagine replacing thesinglet V1 with the antitriplet V3 and sextet V6 interactions,whose relative weights with respect to V1 are, respectively,(1=2) and (� 1=4). This leads to an effective interactionwhose strength is about (1=4) of the original one, V1ðT ¼�B ¼ 0Þ, so that this extremely simplified picture never-theless predicts a value of the effective interaction at largebaryon density (around 0.2 GeV), which is compatible withthe relevant range of V1 suggested by our analysis.

ACKNOWLEDGMENTS

The authors warmly thankM. P. Lombardo (INFN-LNF)for enlightening discussions concerning the gluon conden-sate G2.

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