Hadron-quark phase transition at nonzero isospin density: The effect of quark pairing

G. Pagliara and J. Schaffner-Bielich

Institut fur Theoretische Physik, Ruprecht-Karls-Universitat, Philosophenweg 16, D-69120, Heidelberg, Germany(Received 8 March 2010; published 20 May 2010)

We compute the mixed phase of nuclear matter and 2SC matter for different temperatures and proton

fractions. After showing that the symmetry energy of the 2SC phase is, to a good approximation, 3 times

larger than the one of the normal quark phase, we discuss and compare all the properties of the mixed

phase with a 2SC component or a normal quark matter component. In particular, the local isospin densities

of the nuclear and the quark component and the stiffness of the mixed phase are significantly different

whether the 2SC phase or the normal quark phase are considered. If a strong diquark pairing is adopted for

the 2SC phase, there is a possibility to eventually enter in the nuclear matter 2SC matter mixed phase in

low energy heavy ions collisions experiments. Possible observables able to discern between the formation

of the 2SC phase or the normal quark phase are finally discussed.

DOI: 10.1103/PhysRevD.81.094024 PACS numbers: 12.38.Mh, 25.75.Nq

I. INTRODUCTION

The phase transition from nuclear (hadronic) matter toquark matter at high density may depend strongly on theisospin asymmetry of matter. While the value of the nu-clear symmetry energy is known at saturation, its behaviorat densities larger than nuclear density is still under studyboth theoretically and experimentally by means of heavyions collisions experiments at low energy [1,2]. On theother hand, very little is known about the dependence onthe isospin asymmetry of the energy of the quark phasewhich is believed to take place at large density. There havebeen studies on the phase transition from nuclear matter toquark matter at large isospin densities mainly focused onthe role of the nuclear matter symmetry energy while theinteractions between quarks have been neglected or limitedto the perturbative QCD corrections [3–7]. The main resultof these studies is a steep reduction of the critical densityfor the phase transition as the isospin asymmetry increases.This is due to the large value of the symmetry energy ofnuclear matter and the small value of the symmetry energyof quark matter which is provided only by the Fermikinetic contribution (eventually corrected by the perturba-tive QCD interactions). Moreover, it was argued in [4] thatthe large difference between the symmetry energy of thenuclear and the quark phase could induce, within the mixedphase, the so-called neutron distillation effect: the quarkcomponent of the mixed phase is much more isospinasymmetric with respect to the nuclear phase. In turn,this effect could modify the particles yield ratios and couldbe eventually detected in heavy ions experiments at lowenergies as the ones planned at FAIR and NICA.

An important effect was not considered in the above-mentioned studies: at large densities quark matter is likelyto be in a color superconducting state [8] with supercon-ducting gaps ranging from a few tens of MeV up to200 MeV if a strong diquark pairing is adopted [9,10].Thus a color superconducting state could potentially sur-

vive even at temperatures of the order of 50 MeVor moreand it is therefore interesting to investigate its possibleformation in low energy heavy ions experiments [11].Also in the case in which the temperature reached in thecollision is higher than the critical temperature for colorsuperconductivity, interesting precursor phenomena of thediquark formation might take place [14,15].Among the many possible color superconducting

phases, the 2SC phase is the relevant candidate for heavyions collisions, since it is likely to appear at lower densitieswith respect to the three flavor color-flavor-locked (CFL)phase and because, at low collision energy, a small amountof strange quarks is produced. Based on general arguments,one could expect a different value of the symmetry energyof the 2SC phase with respect to the normal quark phasesince the Cooper pairs are in an isospin symmetric state,i.e. the density of up and down quarks are forced to beequal to allow the formation of the diquark condensate[16]. On the other hand at large isospin asymmetries the2SC pairing pattern is broken and the normal quark phaseis obtained as shown in previous papers [17–22].The aim of this paper is to extend previous calculations

on the effect of isospin on the nuclear matter—quarkmatter phase transition at high density and finite tempera-ture by including the nonperturbative interactions betweenquarks responsible for the phenomenon of color super-conductivity in the 2SC phase. We also discuss underwhich conditions this state of matter might be created inthe laboratory and its possible observational signatures inthe framework of the search for the nuclear matter—quarkmatter mixed phase in heavy ions experiments [23].The paper is organized as follows: in Sec. II we describe

the model adopted to calculate the 2SC equation of stateand we compute its symmetry energy and compare it withthe one of normal quark matter. In Sec. III we compute thecritical densities for the onset of the phase transition atdifferent asymmetries and temperatures and we explain thefeatures of the mixed phases. In Sec. IV we present the

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phase diagrams. Further discussions and conclusions aregiven in Sec. V.

II. QUARK MATTER AND NUCLEAR MATTEREQUATIONS OF STATE

The region of the QCD phase diagram we want toinvestigate, baryon density nB from nuclear matter density,n0 ¼ 0:16 fm�3, to 3–4n0 and temperature T from 0 to�100 MeV has a very rich structure: at these conditionsthe chiral phase transition is believed to occur and also a‘‘deconfinement’’ phase transition, here just meant to be achange from hadronic degrees of freedom to quarks de-grees of freedom. These two transitions are not necessarilycoincident; for instance it has been proven, in the large Nc

limit, the existence of a phase in which chiral symmetry isrestored but quarks are confined, the so-called Quarkyonicphase [24,25]. Moreover the phase transition could involvenormal quark matter (NQ) or color superconducting mat-ter, the 2SC phase, depending on the temperature, theisospin asymmetry, and the value of the superconductinggap. Needless to say, at these regimes QCD is stronglynonperturbative and one has to resort to models to obtainsome qualitative results. One possible approach is to con-sider quark chiral models, like the Nambu–Jona-Lasinio(NJL) or the Polyakov loop NJL model (PNJL) models,and to compute the structure of the phase diagram bymeans of order parameters as the chiral condensates, thediquark condensates [9,10,26–29], and the Polyakov loop[12,13]. One important feature of QCD missing in thesecalculations is quark confinement and therefore they can-not describe the nucleon and nuclear matter. Thus, one canonly predict the line of the chiral phase transition (or thedeconfinement phase transition, but only for the gluonicsector, within the PNJL model) in the temperature chemi-cal potential plane but one has no estimates for the value ofthe baryon density or energy density of the onset of thephase transition which are crucial to make comparisonswith the experiments.

Another approach is to consider two models, one for thelow density low temperature hadronic phase and one forthe high density high temperature quark phase and then tocompute the binodal boundaries by using a Maxwell or aGibbs construction. Although this approach is certainly notsatisfying because of the use of two different Lagrangiansto describe the same matter, it has the advantages ofproviding some numerical estimates of the critical den-sities and therefore we will adopt it here. Some promisingstudies were already performed trying to describe thenucleons within a NJL-type model with quark degrees offreedom but unfortunately the properties of nuclear matterat saturation cannot be correctly reproduced [30–32].Recently, a new approach has been proposed in which aunique Lagrangian is considered having both hadronic andquark degrees of freedom; the phase transition betweennuclear matter and quark matter is regulated by the

Polyakov loop [33]. While interesting, this model neglectsthe formation of diquark condensates.Let us start by describing the quark model we adopt in

our work. In the same spirit of Refs. [34,35], we start withthe thermodynamic potential of normal quark matterwithin the MIT bag model for two massless flavors, upand down quarks, and we correct it with the contributionfrom the quark pairing as calculated in Refs. [19,20]:

� ¼ �NQ þ �2SC (1)

where:

�NQ ¼ � �4u

4�2� �4

d

4�2� 1

2T2�2

u � 1

2T2�2

d �7

30�2T4 þ B

(2)

�u and �d are the chemical potentials of up and downquarks and B is the bag constant.We treat now separately �2SC by using the formalism of

Ref. [19,20] in which an NJL-like model is proposed totreat the diquark pairing. Let us start by introducing thequark chemical potentials �i;� where i ¼ up, down is the

flavor index and � ¼ r, g, b, is the color index for red,green and blue quarks. The relations of chemical equilib-rium read:

�ur ¼ �ug ¼ �þ 23�c þ 1

3�8 (3)

�dr ¼ �dg ¼ �� 13�c þ 1

3�8 (4)

�ub ¼ �þ 23�c � 2

3�8 (5)

�db ¼ �� 13�c � 2

3�8 (6)

where �, �c, and �8 are the quark chemical potential, thecharge chemical potential, and the chemical potential as-sociated with the color generator T8 of SUð3Þc. The differ-ence between the thermodynamic potential of quarks in the2SC phase and of quarks in the normal phase, in mean fieldapproximation [19,20], reads:

�2SC ¼ �2Xa

Z d3p

ð2�Þ3�E2SCa þ 2T ln

�1þ exp

��E2SCa

T

��

� ENQa � 2T ln

�1þ exp

��ENQa

T

���þ �2

4GD

(7)

GD is the diquark coupling, � is the superconducting gap,E2SCa are the (six quark and six antiquark) quasiparticles

dispersion relations in the 2SC phase as calculated in[19,20]

E�ub ¼ p��ub½�1� (8)

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E�db ¼ p��db½�1� (9)

E��� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðp� ��Þ2 þ �2

q� ��½�2� (10)

the numbers in the square brackets represent the degener-

acy, �� ¼ ð�ur þ�dgÞ=2, �� ¼ ð�dg ��urÞ=2, and ENQa

the free quarks (six) and antiquarks (six) dispersion rela-tions in the normal quark phase (obtained by setting � ¼0, �8 ¼ 0 in E2SC

a ). Notice that here we adopt the approxi-mation of massless quarks and we do not take into accountthe effective mass of quarks because, as we will see in thenext section, the phase transition from nuclear matter toquark matter occurs at values of the baryon chemicalpotential larger then the chemical potential of the restora-tion of chiral symmetry as obtained in the model of[19,20]. The integrals, as usual, are regularized by intro-ducing a cutoff �. The values of � and �8 are obtained byminimizing �2SC with respect to � and by solving thecolor-charge neutrality condition: �@�2SC=@�8 ¼ 0. Theequation of state can then be computed as a function of �,�c, and T [36].

Let us discuss now our choice of the parameters: as in[19,20] � ¼ 0:6533 GeV and we consider two values ofthe diquark coupling GD ¼ 3=4GS and GD ¼ GS, whereGS ¼ 5:0163 GeV�2 is the quark-antiquark pairing, withcorresponding gaps �� 130 MeV and �� 200 MeV forsymmetric matter, at zero temperature and at � ¼500 MeV. We will discuss the values of B in the nextsection.

Concerning the nuclear matter (NM) equation of state,we adopt the widely used relativistic mean field model withthe parametrization TM1 [37]. In our calculation we willnot include the pion contribution to the nuclear equation ofstate. The are two reasons for which we neglect pions: theircontribution to the pressure is important for large tempera-tures �100 MeV and small chemical potentials. The sec-ond point is that within the relativistic mean field model weare using, the effective mass of the pion as a function of thetemperature and the density cannot be computed and, byusing its vacuum mass, it is well known that such kind ofmodels for nuclear matter predicts pion condensation atdensities not far from saturation. The possibility of pioncondensation seems, on the other hand, to be ruled out inmore sophisticated chiral models in which the effectivemass of the pion can be computed and it turns out to belarge enough to prevent the condensation [6].

We want to give now some arguments concerning thebehavior of the quark equation of state as a function of theisospin asymmetry. First, we define the asymmetry t for thequark phase as

t ¼ 3nd � nund þ nu

¼ nd � nunB

(11)

where nd and nu are the densities of down and up quarks

and nB is the baryon density. The proton fraction is relatedto t by the relation Z=A ¼ ð1� tÞ=2. In the normal quarkphase, since interactions are neglected, only the Fermikinetic terms contribute to the symmetry energy which

can be easily shown to be aNQsym ¼ �=6 ¼ �B=18. In the

2SC phase, the formation of Cooper pairs forces the den-sities of the paired quarks to be the same. Only the blue upand down quarks are unpaired and therefore can eventuallyhave different densities. Thus, to a good approximation, atfixed values of � and �c the asymmetry of the 2SC phaseis 1=3 of the asymmetry of the normal quark phase t2SC ’tNQ=3. The approximation consists in neglecting the cor-rection to the paired quark densities given by the gap whichin fact scales as ð�=�Þ2 [16]. At fixed � and �c the totalquark densities of the 2SC phase and the normal quarkphase are quite similar. On the other hand, at fixed densityand isospin asymmetry, we expect the mismatch betweenthe chemical potentials of up and down quarks to be largerin the 2SC phase than in the normal quark phase (becauseonly the two blue unpaired quarks contribute to the asym-metry). We can write the energy per baryon of the 2SCphase as:

ðE=NÞ2SC ’ ðE=NÞNQ3

þ paired quarks contribution (12)

the first contribution corresponds to the blue quarks and itis ’ 1=3 of the energy per baryon of the normal quarkphase (again, by neglecting the correction to the pairedquark densities given by the gap). The symmetry energyturns out to be:

a2SCsym ¼ 1

2

d2ðE=NÞ2SCðdt2SCÞ2 ’ 1

2

d2ðE=NÞNQ=3ððdtNQ=3Þ2 ¼ 3aNQsym: (13)

To check this result, we calculate numerically the sym-metry energy of the 2SC phase by using the general defi-nition of symmetry energy:

ðE=NÞt ¼ ðE=NÞt¼0 þ asymt2 (14)

from which its value is easily obtained by computing theequations of state of the 2SC phase for t ¼ 0 and foranother value of t, t ¼ 0:2 in our calculation.In Fig. 1 we show the numerical and the approximated

results for the symmetry energy of the 2SC phase and thesymmetry energy of the normal quark phase as functions ofthe baryon density. For comparison we also show thesymmetry energy of nuclear matter as obtained in therelativistic mean field model TM1 (given by the sum ofthe Kinetic term and the isovector term [4]). Our approxi-mation for the calculation of the symmetry energy of the2SC phase works pretty well, within an error of a fewpercent it is 3 times larger than the one of the normal quarkphase. The symmetry energy of nuclear matter is of course

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larger than the ones of the quark phases but at densities ofthe order of 2 times saturation density it is actually com-parable with the one of the 2SC phase. As we will show inthe next sections, the larger value of the symmetry energyof the 2SC phase with respect to the normal quark phase isresponsible for the behavior of the critical density of theonset of the phase transition as a function of the asymmetryand it affects also the properties of the mixed phase.

III. MIXED PHASES AT DIFFERENT Z=A

The mixed phase between quark matter and nuclearmatter is computed by solving the Gibbs conditions for amulticomponent system with two globally conservedcharges [38,39], the baryonic charge, and the isospincharge associated with the chemical potentials �B ¼�n ¼ 3� and�I ¼ ð�p ��nÞ=2 ¼ ð�u ��dÞ=2 respec-tively, where �n and �p are the chemical potentials of

neutrons and protons. The system can analogously bedescribed by imposing the global conservation of the bar-yonic charge and the electric charge, the electric chargechemical potential being �c ¼ 2�I. We prefer to work byusing the second description therefore we will use theelectric charge ratio or proton fraction Z=A, and the elec-tric charge chemical potential as the second conservedquantity in addition to the baryonic charge. The Gibbsconditions read

PNMð�B;�c; TÞ ¼ P2SCð�B;�c; TÞ (15)

Z=AnB ¼ ð1� �ÞnNMc þ �n2SCc (16)

where PNM, P2SC are the pressures of the nuclear and thequark phase, nNMc , n2SCc are the charge densities of the

nuclear and the quark phase, � is the volume fraction ofthe quark phase, and nB ¼ ð1� �ÞnNMB þ �n2SCB is the

baryon density.As a first step, we compute the value of the critical

density for the onset of the nuclear matter—2SC mixedphase at fixed temperature and by varying the protonfraction. Concerning the choice of the bag parameter, we

select two values B1=4 ¼ 165 MeV and B1=4 ¼ 190 MeVwhich together with the choice of intermediate and strongdiquark pairing, GD ¼ 3=4GS and GD ¼ GS, allow toobtain the onset of the mixed phase a T ¼ 0 and forsymmetric matter at densities larger than �3n0, in agree-ment, as argued in [40], with the constraint put on it by theSIS data. Results are shown in Figs. 2 and 3, where also thecase of normal quark matter is shown for comparison. Atzero temperature and for symmetric matter, the onset of themixed phase is strongly reduced in the 2SC phase withrespect to the normal phase: this is clearly due to the softer2SC equation of state in which the formation of Cooperpairs allows the system to lower its energy with respect tothe case of a system of unpaired quarks. As the protonfraction decreases, or the asymmetry increases, a steepreduction of the critical density is obtained for normalquark matter as noticed in Refs. [3,4,7] due to the strongstiffening of the nuclear equation of state. This effect is dueto the large value of the nuclear symmetry energy. On theother hand, as the proton fraction is reduced, the normalquark equation of state has a mild dependence on theasymmetry due to the small value of its symmetry energy.In the case of 2SC phase instead, the critical density staysalmost constant as the proton fraction is reduced, because

FIG. 1. Symmetry energy of the nuclear phase TM1 (thindashed lines), 2SC phase (numerical calculation, thick solidline), 2SC phase (approximation, thick dashed line), and normalquark phase (thin solid line) as functions of the density. Thesymmetry energy of the 2SC phase is roughly 3 times larger thanthe one of the normal quark phase. The nuclear phase has thelargest value of the symmetry energy. Here, the intermediatevalue of the diquark coupling has been used.

FIG. 2. Density for the onset of the phase transition as afunction of the proton fraction Z=A and for two values oftemperature T ¼ 0 and T ¼ 50 MeV, the intermediate valueof the diquark pairing is adopted. The onset of the phasetransition is smaller in the case of the 2SC phase (solid thinand thick lines) with respect to the case of normal quark matter(dashed thin and thick lines). Before the unpairing transition atZ=A� 0:35 for T ¼ 0 and Z=A� 0:42 for T ¼ 50 MeV, thecritical density for the onset of the 2SC phase has a milddependence on the proton fraction while a strong dependenceis evident in the case of normal quark matter.

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of the larger value of the symmetry energy of the 2SCphase: both the nuclear and the 2SC equations of statebecome substantially stiffer when the proton fraction isreduced and therefore the critical density is almost inde-pendent of the proton fraction. As the proton fraction isfurther decreased, at some point the stress caused on the upand down quark’s Fermi surfaces becomes too large, the2SC pairing is broken and only the normal quark phase canbe formed in the phase transition. Notice the effect of thedifferent values of the diquark pairing we are adopting:while in Fig. 2, for intermediate pairing, the unpairingtransition is obtained for Z=A� 0:35 at zero temperature,for the strong pairing case, Fig. 3, this occurs at Z=A�0:25. For extremely low values of Z=A, as the ones reachedin neutron star matter, by computing the mixed phase ofnuclear matter and color superconducting matter, an inter-esting effect was obtained in Ref. [41]: the conditions ofbeta stability and charge neutrality render the effectivemass of up quarks larger than the one of down quarksand one obtains first a mixed phase between nuclear matterand down quarks and only at larger densities also the upquarks are deconfined. We do not consider here thispossibility.

Finally, the dependence on the temperature is also inter-esting: at finite temperature, T ¼ 50 MeV in Fig. 2 and 3,the value of the superconducting gap is smaller and there-fore the values of the critical densities for the 2SC case arecloser to the ones of normal quark matter and the unpairingtransition occurs at larger values of the proton fraction.This effect is evident in Fig. 2, for intermediate diquarkpairing, but is not so pronounced in Fig. 3 where the strongdiquark pairing is considered and the critical temperature ishigher than 50 MeV.

Let us have a closer look now into the mixed phase itself.Notice that for symmetric matter, the system is actually a

one component system and therefore the Gibbs construc-tion coincides with the Maxwell construction. Thus, withinthe mixed phase the pressure is constant and the two phasesare both symmetric. The situation is different for asym-metric matter: as a result of the Gibbs conditions the twophases in the mixed phase have different isospin asymme-tries, the nuclear phase being the most symmetric, and thepressure increases with the density. In Fig. 4, we show thelocal isospin asymmetry t of the nuclear phase and thenormal quark and 2SC phase within the mixed phase as

functions of the volume fraction �. Parameters are: B1=4 ¼190 MeV, T ¼ 50 MeV, Z=A ¼ 0:4, which implies t ¼0:2, and for the 2SC phase we consider the case of strongcoupling. An interesting qualitative difference is evidentwhether the quark component is in the unpaired quark stateor in the 2SC state. In the case of normal quark matter, dueto its low value of the symmetry energy, close to the onsetof the mixed phase the quark component is very asymmet-ric, t� 1. Notice that an asymmetry larger than 1, whichwould imply a negative value for Z=A, is possible in purequark matter by considering the fact that the density ofprotons in quark matter is given by np ¼ ð2nu � ndÞ=3 as

can be easily verified. On the other hand, the nuclearcomponent, which at the onset of the mixed phase hasasymmetry t� 0:2, becomes more and more symmetricas � increases. This is the so-called neutron distillationeffect [4,7]: there is an excess of isospin density in thequark drops with respect to the nuclear phase which, due toits high value of the symmetry energy, lowers its energyapproaching the symmetric state. As the volume fractionincreases, the asymmetry of the quark phase rapidly de-creases and reaches the value t� 0:2 at � ¼ 1 as it mustbe. Notice that at � ¼ 1, the end of the mixed phase, theasymmetry of the nuclear phase is almost zero.

FIG. 4. Local isospin asymmetries of the nuclear and quarkcomponents of the mixed phase. In the case of normal quarkmatter, close to the onset of the mixed phase, the isospinasymmetry is distillated into the quark component for whicht� 1. In the 2SC case, due to its large symmetry energy, nostrong asymmetries are reached. The neutron distillation effect issuppressed.

FIG. 3. Density for the onset of the phase transition as afunction of the proton fraction Z=A and for two values oftemperature T ¼ 0 and T ¼ 50 MeV, the strong diquark pairingis adopted (the lines denote the same as in Fig. 1). Because of thelarge value of the gap, the unpairing transition occurs at Z=A�0:25 for T ¼ 0 and Z=A� 0:3 for T ¼ 50 MeV.

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The situation is different in the case of the 2SC phase.Let us consider first the onset of the mixed phase: theasymmetry in the chemical potentials of the nuclear phaseaffects only the unpaired blue quarks: due to the pairing,the other quarks do not contribute to the asymmetry. Thisexplains why close to the onset of the mixed phase the 2SCphase is actually much less asymmetric than the normalquark phase. As � increases, slowly the 2SC phase reducesits asymmetry until the value t ¼ 0:2 is reached at the endof the mixed phase; at the same time the nuclear phasereduces its asymmetry and reaches, at � ¼ 1, a value t�0:1, thus larger with respect to the case of normal quarkmatter. This is again explained by considering that for afixed value of t in quark matter, the mismatch of chemicalpotentials of up and down quarks is larger in the 2SC phasethan the normal quark phase, therefore at the end of themixed phase this produces a larger asymmetry of thenuclear phase when the 2SC phase is considered. In con-clusion, the isospin distillation effect in presence of the2SC phase is strongly reduced with respect to the case ofnormal quark matter. We will discuss in the last sectionpossible effects for heavy ion collisions experiments.

In Fig. 5, we show a comparison between the equationsof state when the normal quark phase or the 2SC phase areconsidered, for the same choice of parameters. As alreadynoticed, the onset of the mixed phase occurs at lowerdensities when the 2SC phase is considered because it issofter than the normal quark phase. One can notice that theextension of the NM-2SC mixed phase is reduced withrespect to the case of the NM-NQ mixed phase. At thesame time the variation of the pressure in the NM-2SCmixed phase is much smaller than the one in the NM-NQmixed phase. Indeed the Gibbs construction in the case ofthe 2SC phase provides a result which is quite similar to thesimpler Maxwell construction. Again, this is clear if weconsider that the 2SC phase has a larger symmetry energy

with respect to the normal quark phase and therefore thetwo components of the mixed phase both prefer to be in astate as more symmetric as possible (similarly in neutronstar matter a phase transition from nuclear matter to theCFL phase is treated with a Maxwell construction since thepairing in the CFL phase already enforces its charge neu-trality [42]). In conclusion, the NM-2SC mixed phase ismore compressible than the NM-NQ mixed phase and, aswe will discuss in the last section, this might also beimportant in heavy ions collisions experiments.

IV. PHASE DIAGRAMS

In the last section we have shown how, as the protonfraction decreases, one goes from the phase transition tothe 2SC phase to the phase transition to the normal quarkphase due to the breaking of the pairing pattern. We showin this section the effect of the temperature: in general asecond order phase transition is obtained from the 2SC tothe normal phase as the temperature increases and at fixedchemical potential, with critical temperature Tcrit ¼0:57�0, where �0 is the gap as obtained at zero tempera-ture. Let us consider now the phase diagrams at fixedvalues of Z=A. In Fig. 6, we show the phase diagram inthe temperature baryon chemical potential plane, for strongcoupling and symmetric matter. A first order phase tran-sition line separates nuclear matter from quark matter bothin the normal state and the 2SC state. A second order phasetransition line separates the normal quark phase from the2SC phase. The two lines intersect in a point at T �95 MeV and �B � 1000 MeV, thus potentially interestingfor heavy ions collision experiments as we will discuss inthe following [43]. Interestingly, the overall structure of thephase diagram is reminiscent of the phase diagram of 4Hefor which at small temperatures a first order phase transi-tion line separates the solid phase and the two liquid phasesHeI and HeII. The two liquid phase are among them

FIG. 5. Equations of state, pressure as a function of the density,for nuclear matter TM1, normal quark matter, and 2SC phase.The dots indicate the onset and the end of the mixed phases. Thesoftening of the equation of state due to the formation of 2SCmatter is more pronounced then the one of the normal quarkphase.

FIG. 6. Phase diagram for symmetric matter in the temperaturechemical potential plane. Solid lines indicate first order phasetransitions, the dashed line corresponds to the second orderphase transition between the 2SC phase and normal quarkmatter.

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separated by a second order phase transition line, the so-called � line [44].

Let us look now at the phase diagrams in the temperaturedensity plane. In Fig. 7, we show the case of symmetricmatter and strong diquark coupling. Notice that the phasediagram is divided in five regions: depending on the den-sity and temperature, three pure phases and two mixedphases can be formed, separated by first order transitionlines (thick dashed and solid lines) and second order tran-sition lines (thin solid). To make contact with heavy ionsphysics, where nuclei with Z=A� 0:4 are used, we com-pute the phase diagram for asymmetric matter. We want todiscuss our phase diagram, in comparison with the resultsobtained in Ref. [4]. In that paper a transport model is usedto investigate the conditions reached in semicentral heavyion collisions of 238U (Z=A ¼ 0:387) nuclei at 1A GeV.Interestingly, it was found that rather exotic nuclear matteris formed in a transient time of 10 fm=c having densitiesaround 3n0, T � 50–60 MeV, and Z=A� 0:35� 0:4. InFig. 8 we show the phase diagram for Z=A ¼ 0:35 in thecase of strong diquark pairing. As before, three pure phasesand two mixed phases are obtained separated by first orderand second order transition lines. Notice that within theNM-2SC mixed phase, the line of second order phasetransition to the NM-NQ mixed phase is not constant any-more because for asymmetric matter the chemical potentialvaries in the mixed phase and therefore also the super-conducting gap. As we explained before, in the case ofasymmetric matter the onset of the phase transition occursat lower densities with respect to the case of symmetricmatter and the window of mixed phase is larger. Finally,the two arrows in Fig. 8 indicate the region of the phasediagram which can be reached in experiments as proposedin Ref. [4]. Clearly, under the assumption of strong diquarkpairing, which in our model implies a critical temperaturefor color superconductivity of �80–100 MeV, it would be

possible to reach the NM-2SC mixed phase in heavy ionscollisions.One should however remind that in more sophisticated

calculations within the Dyson-Schwinger approach [45] avalue of the 2SC gap of the order of 70–100 GeV wasobtained which implies, by using the BCS relation Tcrit ¼0:57�0, a critical temperature of the order of 40–60 MeV.If this is indeed the case the 2SC window falls below theregion of the phase diagram which can be reached inexperiments as proposed in Ref. [4] and therefore nosignatures of the formation of the 2SC phase are expected.On the other hand, within the PNJL model the standardBCS relation between the critical temperature and the gapdoes not hold and the quark pairing could survive attemperatures as high as 150 MeV [12,13] (see also thevery recent [46]). We find therefore interesting to inves-tigate the possible signatures of the formation of the 2SCphase in heavy ions collisions.

V. DISCUSSION AND CONCLUSIONS

We have computed the mixed phase of nuclear matterand 2SC phase under different conditions of temperatureand isospin asymmetry (or proton fraction). First, we haveprovided a clear argument by which the symmetry energyof the 2SC phase is, to a good approximation, 3 timeslarger than the one of normal quark matter: due to theCooper pairing of red and green quarks, only the blueunpaired up and down quarks in the 2SC phase can con-tribute to the isospin asymmetry. The argument is con-firmed by a numerical calculation. This fact allows us tounderstand the main differences between the nuclear mat-ter 2SC mixed phase and the one with the normal quark

FIG. 7. Phase diagram for symmetric matter in the temperaturedensity plane. Solid thick lines indicate the onset of the mixedphase ncrit 1, dashed thick lines correspond to the end of themixed phase ncrit 2. The solid thin line is a second order phasetransition between the 2SC phase and normal quark matter.

FIG. 8. Phase diagram for asymmetric matter, Z=A ¼ 0:35, inthe temperature density plane. Solid thick lines indicate the onsetof the mixed phase ncrit 1, dashed thick lines correspond to theend of the mixed phase ncrit 2. The solid thin line is a secondorder phase transition between the 2SC phase and normal quarkmatter. The two arrows indicate the region of the phase diagramwhich might be reached in semicentral low energy heavy ionscollisions [4].

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phase: the onset of the mixed phase shows a mild depen-dence on Z=A because both the nuclear and the 2SCequations of state become substantially stiffer when theproton fraction is reduced. The isospin distillation effectproposed in [4] in the case of normal quark matter isreduced in the nuclear matter 2SC mixed phase becauseof the larger value of its symmetry energy. Finally, thesoftening of the equation of state due to the appearance ofthe nuclear matter quark matter mixed phase is morepronounced in the case of the 2SC phase with respect tothe case of normal quark matter. A further interesting pointis that the phase transition can occur at lower densities thanpreviously thought when the effects from color supercon-ductivity are taken into account. The crucial question iswhether these differences can be probed in heavy ionscollisions experiments. By referring to the results of thetransport calculation of [4] it might be possible indeed toenter in the nuclear matter 2SC mixed phase in low energysemicentral collisions. Concerning the possible observ-ables which allow to distinguish whether normal quarkphase or the 2SC phase are formed, here we want to limitourself to discuss qualitatively a few ideas.

The fact that in a mixed phase between two differentcomponents, nuclear matter and quark matter, there can bea separation or distillation of globally conserved quantitiesfrom one phase to the other is rather old. It was indeedproposed in Ref. [47], that in high energy heavy ionscollisions strange and antistrange quarks are abundantlyproduced (with net strangeness being zero) and thatstrangeness would be much more abundant in the quarkcomponent. This could be a mechanism to producestrangelets (stable or metastable) in the laboratories.Similarly, the isospin distillation effect proposed in [4] isthe migration of the isospin density into the phase with thelowest value of the symmetry energy. It has been proposedthat this phenomenon could invert the trend in the produc-tion of neutron rich fragments and it could affect the��=�þ multiplicities ratio. In the first studies of colorsuperconductivity it was argued that due to the formationof Cooper pairs in the densest region of the system, thequark phase would expel the excess of down quarks and upantiquarks, which then, in the hadronic phase, would even-tually form ��. Moreover, when the diquark condensatebreaks up late in the collision a number of protons largerthen the initial one in the colliding nuclei would be emit-ted. The signature would then be an increase of the��=�þratio and at the same time an increase of the number ofprotons within selected events with an anomalously largedensity and small temperature [18,48]. This effect could beregarded as the opposite of the neutron distillation effect,the quark phase is completely symmetric and expels itsexcess of isospin into the nuclear phase. This would becorrect if all the quarks pair, but in the 2SC phase the

unpaired up and down blue quarks can, as in the normalquark phase, have different densities. Actually the symme-try energy of the 2SC phase, while larger then the one ofthe normal phase, is still smaller than the nuclear mattersymmetry energy (at least within the model of nuclearmatter we consider here). Therefore as we have shown,also when considering the 2SC phase a neutron distillationinto the quark component of the mixed phase occurs but itis simply suppressed with respect to the normal quarkphase. One could then expect that, going from low tem-peratures at which the 2SC phase is formed to highertemperatures (by increasing the energy of the ions) wherethe normal phase is formed, the neutron distillation effectsis gradually enhanced and therefore also its specific sig-natures [4]. At the same time, the mixed phase becomesstiffer passing from the 2SC phase to the normal quarkphase what can be for instance ‘‘detected’’ by using theKþyields which were shown to represent a good probe for thestiffness and the isospin dependence of the equation ofstate [49–55]. Other possible signatures of the formation of

quark matter are associated with the enhancement of the ��to �p ratio as shown in Refs. [56–58] although alternativeexplanations based on multihadron reactions have beenproposed [59,60]. A detailed quantitative study of thesequantities within a transport model would be of course veryimportant. In addition to the particles yields one can cal-culate also the susceptibilites as done in Refs. [61,62]which are important for the charge (baryonic, electric,isospin) fluctuations. In particular one could expect thatthe off-diagonal susceptibility �ud could be different in the2SC phase due to the Cooper pair correlations. Finally, theresults of this paper might be relevant also for neutron starsphysics: in protoneutron stars, where the initial protonfraction is�0:3, during deleptonization the proton fractiondecreases and at some point it might be possible that aphase transition from the 2SC phase to the normal quarkphase can take place due to the increasing stress on thequark’s Fermi surfaces [63–65]. Moreover, the local protonfraction of the nuclear phase within the mixed phase is alsoimportant for the late cooling of neutron stars since itregulates the threshold of the direct Urca processes [66].

ACKNOWLEDGMENTS

The work of G. P. is supported by the DeutscheForschungsgemeinschaft (DFG) under Grant No. PA1780/2-1. J. S. B. is supported by the DFG through theHeidelberg Graduate School of Fundamental Physics. Wethank A. Drago and M. Hempel for many fruitful discus-sions. This work was also supported by CompStar, aResearch Networking Programme of the EuropeanScience Foundation.

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