Color neutrality effects in the phase diagram of the Polyakov–Nambu–Jona-Lasinio model

D. Gomez Dumm,1,2,* D. B. Blaschke,3,4,† A.G. Grunfeld,2,5,‡,x and N.N. Scoccola2,5,6,k1IFLP, CONICET - Departmento de Fısica, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina

2CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentina3Institute for Theoretical Physics, University of Wrocław, Max Born Place 9, 50204 Wrocław, Poland

4Bogoliubov Laboratory of Theoretical Physics, JINR Dubna, Joliot-Curie Street 6, 141980 Dubna, Russia5Physics Department, Comision Nacional de Energıa Atomica, Avenida Libertador 8250, 1429 Buenos Aires, Argentina

6Universidad Favaloro, Solıs 453, 1078 Buenos Aires, Argentina(Received 18 July 2008; published 18 December 2008)

The phase diagram of a two-flavor Polyakov-loop Nambu–Jona-Lasinio model is analyzed imposing

the constraint of color charge neutrality. The main effect of this constraint is a coexistence of the chiral

symmetry breaking (�SB) and two-flavor superconducting phases. Additional effects are a shrinking of

the �SB domain in the T �� plane and a shift of the end point to lower temperatures, but their

quantitative importance is shadowed by the intrinsic uncertainties of the model. The effects can be

understood in view of the presence of a nonvanishing color chemical potential �8, which is introduced to

compensate the color charge density �8 induced by a background color gauge mean field �3. At low

temperatures and large chemical potentials the model exhibits a quarkyonic phase, which gets additional

support from the diquark condensation.

DOI: 10.1103/PhysRevD.78.114021 PACS numbers: 12.39.Fe, 21.65.Qr, 25.75.Nq

I. INTRODUCTION

Constraints on the phase diagram of QCD under extremeconditions of high excitation (temperature) and compres-sion (density) are of vital interest for large-scale experi-mental programmes with ultrarelativistic heavy-ion beams,searching for signatures of the QCD phase transitions:chiral symmetry restoration and deconfinement.

Under conditions of finite temperature T and smallchemical potentials �, which are probed in heavy-ioncollisions at SPS, RHIC, and in the near future at LHC,Lattice QCD simulations have provided insight into thephase structure since the temperature-dependent change ofthe order parameters of the above phases, namely, thechiral condensate h �qqi and the traced Polyakov-loop �,have been determined [1–4]. The peak positions of thesusceptibilities derived from these calculations for bothtransitions are remarkably coincident at the same pseudo-critical temperature Tc. This might be understood in termsof general effective theory [5]. It is also remarkable thatthese results can be nicely described with an effectivemodel of the Nambu–Jona-Lasinio (NJL)-type, extendedwith the inclusion of a coupling to the Polyakov loop [6–9]with a potentialUð�; TÞ that can be extracted from latticeQCD simulations of the pressure in the pure gauge theory[10,11].

The challenge for experiments as well as for theory is toextend this knowledge into the domain of finite baryondensities, where precursors of color superconductivity(pseudogap phase [12]) or even color-superconductingquark matter phases themselves can occur [13,14], andnew constraints from observations of compact star proper-ties may apply [15,16].Most of the effective model analyses addressing the

chiral phase transition in the QCD phase diagram havebeen performed with NJL-type models which, however,lack quark confinement and lead to an onset of finite quarkdensities already at unphysically low temperatures of about50 MeV [17], far below the deconfinement temperature ofabout 200 MeV [3,4].The straightforward step to implement the effects of

confining forces and to suppress unphysical quark degreesof freedom is to study the phase diagram of quark matter inthe Polyakov-loop NJL (PNJL) model. This has beencarried out, e.g., in Refs. [11,18–22]. In these works, how-ever, the important question of color neutrality of quarkmatter has not been considered. The inclusion of neutralityconstraints in the PNJL model has been taken up recentlyin Ref. [23]. In that paper, however, only the high-densityregion for a quark chemical potential � ¼ 500 MeV hasbeen considered, and light quark masses have been ne-glected. It has been found that the Polyakov-loop formu-lation together with neutrality constraints can give rise toan increase of the critical temperature for the color super-conductivity transition, in deviation from the well-knownBCS relationship. An investigation of the interrelationbetween chiral symmetry breaking (�SB) and color super-conductivity, as well as a study of the effect of color

*dumm@fisica.unlp.edu.ar†blaschke@ift.uni.wroc.pl‡Present address: Institute for Nuclear Physics, TU Darmstadt,

Schloßgartenstr. 9, D-64289 Darmstadt, Germanyxag.grunfeld@gmail.comkscoccola@tandar.cnea.gov.ar

PHYSICAL REVIEW D 78, 114021 (2008)

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neutrality without enforcing electric neutrality, is stillmissing. The aim of the present paper is to address thesesubjects, analyzing the QCD phase diagram in the frame-work of a generalization of the PNJL model that fulfills theconstraint of color neutrality.

The article is organized as follows: In Sec. II, we in-troduce the formalism for enforcing color neutrality in thePNJL model by color chemical potentials. In Sec. III, wediscuss the numerical results for the phase diagram, first inthe nonsuperconducting case, then including color super-conductivity. In the final section, we present our conclu-sions and discuss possible consequences for thephenomenology of next heavy-ion collision experiments.

II. FORMALISM

The Euclidean action for the two-flavor PNJL at tem-perature T is given by

SE ¼Z �

0d�

Zd3x

��c ð�i��D� þ mÞc

�G

2½ð �c c Þ2 þ ð �c i�5 ~�c Þ2�

�H

2ð �c Ci�5�2�ac Þð �c Ci�5�2�ac Þy þUð�; TÞ

�;

(1)

where c is the Nf ¼ 2 fermion doublet c � ðudÞT , m ¼diagðmu;mdÞ stands for the current quark mass matrix, and� ¼ 1=T. For simplicity we consider the isospin symmetrylimit, in which mu ¼ md ¼ �m0. Charge conjugated fieldsin Eq. (1) are defined by c C ¼ �2�4

�c T , while ~� and �a,with a ¼ 2, 5, 7, stand for Pauli and Gell-Mann matricesacting on flavor and color spaces, respectively. ThisEuclidean action leads to local chiral invariant current-current interactions in the quark-antiquark and quark-quark channels. The latter is expected to be responsiblefor the presence of a color-superconducting phase in theregion of low temperatures and moderate chemicalpotentials.

The coupling of fermions to the Polyakov loop has beenimplemented in Eq. (1) through the covariant derivative inthe fermion kinetic term ��D�, where D� � @� � iA�.

Here, the Euclidean operator ��@� is defined as �4@@� þ

~� � ~r, with �4 ¼ i�0. As usual, we assume that the quarksmove in a background gauge field A0 ¼ g�0G

�a �a=2,

where G�a are the SU(3) color gauge fields. Then, at the

mean field level, the traced Polyakov loop is given by� ¼13 Tr expði��Þ, with � ¼ i �A0 ¼ constant. The traced

Polyakov loop can be taken as an order parameter for theconfinement transition. In the pure glue theory it can beassociated with the spontaneous breaking of the global Z3

center symmetry of color SUð3Þ, � ¼ 0 corresponding tothe symmetric, confined phase [24]. Here, we will work inthe so-called Polyakov gauge, in which the matrix � is

given a diagonal representation � ¼ �3�3 þ�8�8, whichleaves only two independent variables, �3 and �8. Finally,the action (1) also includes an effective potential U thataccounts for gauge field self-interactions.In order to obtain the grand canonical thermodynamical

potential of the PNJL model at temperature T and chemicalpotential � ¼ �B=3 we start from Eq. (1), performing astandard bosonization of the theory. This can be done byintroducing fields ð; ~�Þ and �a corresponding to thequark-antiquark and diquark channels, respectively. Thenwe consider the mean field approximation, keeping thenonzero vacuum expectation values � and � and droppingthe respective fluctuations. Concerning the superconduct-ing vacuum, we adopt here the usual ansatz in which one

has ��5 ¼ ��7 ¼ 0, ��2 ¼ ��. In this way, using the standardMatsubara formalism one obtains

�MFA ¼ �T

2

X1n¼�1

Z d3p

ð2�Þ3 lndet½�S�1ðT;�Þ� þ �2

2G

þ��2

2HþUð�; TÞ; (2)

where the inverse propagator S�1 is a 48� 48 matrix inDirac, flavor, color, and Nambu-Gorkov spaces. The de-pendence of S�1 with T and � (� ¼ �B=3) is obtainedfrom the T ¼ � ¼ 0 four-momentum integrals in the fer-mion determinant by replacing p4 ! !n � i ~�, where!n ¼ ð2nþ 1Þ�T are the usual fermionic Matsubara fre-quencies, and ~� stands for a complex ‘‘chemical potential’’matrix in color space,

~� ¼ diagð�r;�g;�bÞ þ ið�3�3 þ�8�8Þ; (3)

with

diag ð�r;�g;�bÞ ¼ �1þ�3�3 þ�8�8: (4)

In general, the inclusion of color chemical potentials �3

and �8 is necessary to ensure color neutrality. Concerningthe mean field effective potentialUð�; TÞ, which accountsfor the Polyakov-loop dynamics, we use here a form thatappears to be consistent with group theory constraints aswell as lattice results (from which one can estimate thetemperature dependence). Following Ref. [11], we take thelogarithmic form of the Polyakov-loop potential, moti-vated by the SU(3) Haar measure

Uð�; TÞT4

¼ � 1

2aðTÞ���

þ bðTÞ ln½1� 6���þ 4ð�3 þ��3Þ� 3ð���Þ2�; (5)

with the corresponding definitions of aðTÞ and bðTÞ from[11].It is worth noting that in the presence of the Polyakov

loop the thermodynamical potential could be in general acomplex quantity. As discussed in Ref. [19], in order to

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properly define the mean field approximation one shouldrequire that the mean field configuration provides themaximal contribution to the partition function. Then, the

mean field values (order parameters) �, ��, �3, and �8

should fulfill the coupled set of ‘‘gap equations’’ [19]

@Re½�MFA�ð@ �; @ ��; @�3; @�8Þ

¼ 0: (6)

If one also imposes color charge neutrality conditions, onehas two additional equations that fix the color chemicalpotentials �3 and �8. The vanishing of color chargesimplies

@Re½�MFA�ð@�3; @�8Þ

¼ 0: (7)

Finally, in order to obtain a physically meaningful solution,it has to be verified that the resulting field configurationleads to a real-valued thermodynamical potential.

At vanishing chemical potential, owing to the chargeconjugation properties of the QCD Lagrangian, the meanfield traced Polyakov loop � is expected to be a realquantity. Since �3 and �8 have to be real-valued [19],this condition implies �8 ¼ 0, and it is easy to see that thethermodynamical potential turns out to be real, as desired.In general, this need not to be the case at finite � [25–27].As in Refs. [11,19,23] we will assume that the potentialUis such that the condition �8 ¼ 0 is well satisfied for therange of values of � and T investigated here. The tracedPolyakov loop is then given by � ¼ �� ¼ ½1þ2 cosð��3Þ�=3. In addition, it can be seen that if �8 ¼ 0then Eqs. (6) and (7) lead to�3 ¼ 0, leaving�8 as the onlypotentially nonvanishing color chemical potential. Thisalso implies that the condition �3 ¼ 0 is trivially satisfied.Notice that the fact that �3 ¼ 0, together with the ansatz

chosen for ��a, allow for a residual r� g color symmetry,leading to �r ¼ �g.

The Matsubara sums can be now explicitly evaluated.One obtains

�MFAðTÞ ¼ �2Z d3p

ð2�Þ3X6j¼1

½2T lnð1þ e��EjÞ þ Ej�

þ �2

2Gþ

��2

2HþUð�; TÞ; (8)

where the quasiparticle energies Ej are given by

E1;2 ¼ "p ��b; (9)

E3;4 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið"p ��rÞ2 þ ��2

q� i�3; (10)

E5;6 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið"p ��gÞ2 þ ��2

qþ i�3: (11)

Here, we have defined "p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij ~pj2 þM2p

, being M ¼�m0 þ � the quark constituent mass.

The momentum integral is of course divergent.Following the usual NJL regularization prescription, wecut the integrals of the zero-point energies Ej at a given

cutoff �, while the logarithmic terms are integrated up toinfinity. In these last integrals, however, we assume thatNJL interactions vanish for energies above the cutoff,

therefore we set � ¼ �� ¼ 0 for j ~pj>�.

III. RESULTS

A. Nonsuperconducting case

As in most of the effective models of QCD interactions,the presence of a 2SC phase shows up at low temperaturesand moderate chemical potentials. In order to study theeffects of imposing the condition of color neutrality, let usstart by considering the NJL model without quark-quark

interactions (H ¼ 0), which means to set �� ¼ 0 in theeffective mean field thermodynamical potential in Eq. (8).Notice that, even in the absence of color superconductivity,one still may require a nonvanishing color chemical poten-tial�8 owing to the presence of the quark interactions withthe Polyakov loop. According to the discussion in theprevious section, we take �3 ¼ 0, �8 ¼ 0, leading to areal-valued mean field thermodynamical potential. Thus,the set of six relations (6) and (7) is reduced to only threenontrivial equations; two for the order parameters

@�MFA

ð@ �; @�3Þ ¼ 0; (12)

and one for the constraint of color charge neutrality

@�MFA

@�8

¼ �8 ¼ 0: (13)

If we do not include quark-quark interactions, the modelincludes only three free parameters, namely, the quarkcurrent mass �m0, the coupling constant G, and the cutoff�. For definiteness, to perform the numerical analysis wewill use a standard set of values for these parameters, takenfrom Refs. [10,11]:

�m 0 ¼ 5:5 MeV; G ¼ 10:1 GeV�2;

� ¼ 650 MeV:(14)

We start by considering the case of T ¼ 0. In this limitthe Polyakov loop decouples, since the mean field value�3

can be removed from the momentum integrals by a shift ofthe integration variable p4. One gets then two coupledequations, namely

M� �m0

G¼ M

�2

Z �

0dp

p2

E½2SgðEþ�rÞ

þ 2SgðE��rÞþ SgðEþ�bÞþ SgðE��bÞ�;(15)

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0 ¼ 2ffiffiffi3

p�2

Z �

0dpp2½SgðEþ�rÞ � SgðE��rÞ

� SgðEþ�bÞ þ SgðE��bÞ�: (16)

For low values of the chemical potential�, it is easy to seethat both equations are trivially satisfied for a wide range ofvalues of�8, while from Eq. (15) the value ofM is fixed atM ¼ M0 ¼ 324:1 MeV. Thus, as expected, the system isin a phase in which the chiral symmetry is strongly broken.These results are valid as long as � � M0. The allowedvalues of�8 are those which satisfy j�rj � M0 and j�bj �M0:

�M0 ��

2� �8ffiffiffi

3p �

� ðM0 þ�Þ=2 if � � M0=3M0 �� if M0=3<�<M0

(17)

Now if � is increased above M0, the value of the effectivemass M slightly decreases, while from Eq. (16) one has�8 ¼ 0. This holds up to a critical value �cðT ¼ 0Þ ¼342 MeV. At this point one finds a first order transitioninto a normal quark matter phase, in which the chiralsymmetry is approximately restored. This behavior isshown by the T ¼ 0 curve in the upper panel of Fig. 1,where we plot the value of � ¼ M� �m0 as function of thechemical potential �.

In order to determine the effect of the Polyakov loop onehas to turn on the temperature, solving Eqs. (12) and (13).Let us first consider the region of low temperatures (T �) and values of � up to M0. In this region, while thevalue of the effective mass is kept atM ’ M0, it is possibleto show that the value of �3 is basically determined by theminimization of the effective potential Uð�; TÞ, whichleads to�3 ’ 2�=3 (or, equivalently,� ’ 0). In particular,it is interesting to consider the range 0 � � � M0=3,where it is possible to perform some analytic calculationsto solve Eq. (13). From this analysis, the value of �8 isfound to be given by the simple expression

�8ffiffiffi3

p ¼ 2�� T ln2: (18)

This result allows us to find the actual value of �8 in the

limit T ¼ 0: one has �8=ffiffiffi3

p ¼ 2�, which leads to �r ¼3�, �b ¼ �3�. Notice that the condition � � M0=3 isnecessary to be in agreement with the constraints for �8

that we established above in the T ¼ 0 case. For lowtemperatures and M0=3 � � � M0, it is not easy to get arelation as simple as that in Eq. (18). However, it can beseen that in the T ¼ 0 limit Eq. (13) implies �r ¼ M0,

which means �8=ffiffiffi3

p ¼ M0 ��. These results are shownby the T ¼ 0 curve in the central panel of Fig. 1, where wequote the value of�8 as function of the chemical potential.

In this way, it has been shown that for nonzero tempera-tures and baryon chemical potentials the interaction be-tween the fermions and the Polyakov loop requires thepresence of a nonzero color chemical potential �8 in order

to keep color neutrality. This still holds in the T ¼ 0 limit ifone demands continuity in the values of�8 and�3. Hence,the fact that in this limit the Polyakov loop decouples does

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0 100 200 300 400 5000.0

0.2

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0.6

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T = 300 MeV

T = 200 MeVT = 100 MeV

T = 0

σ [

MeV

]

T = 300 MeV

T = 200 MeV

T = 100 MeV

T = 0

µ 8 [ M

eV ]

T = 300 MeV

T = 200 MeV

T = 100 MeV

Φ

µ [ MeV ]

FIG. 1 (color online). Behavior of the effective mass �, thecolor chemical potential �8, and the traced Polyakov loop � asfunctions of the chemical potential, for various values of thetemperature, in the nonsuperconducting two-flavor PNJL. Solidand dashed lines correspond to the results with and without theimposition of color neutrality, respectively. Notice that in theupper and lower plots the solid and dashed curves correspondingto T ¼ 300 MeV are almost coincident.

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not mean that the system is blind to the color symmetrybreaking induced by �3.

For finite values of the temperature, the critical values ofthe chemical potential �cðTÞ can be found numerically,defining a first order phase transition line in the T ��plane. This holds up to a given ‘‘end point’’ ð�E; TEÞ,which is found to be located at about (333 MeV,64 MeV) for our parameter set. For T > TE the transitionto the normal quark matter phase proceeds as a smoothcrossover. The corresponding transition line can be estab-lished by looking at the peak in the chiral susceptibility,defined by

� � � 1

2

@2�

@ �m02: (19)

Our numerical results are shown in Figs. 1 and 2. InFig. 2 we represent the T �� phase diagram, denoting bysolid and dashed lines the curves corresponding to firstorder and crossover chiral restoration transitions, respec-tively. For comparison we also include the correspondingcurves for the PNJL model without the imposition of colorneutrality. It can be seen that the first order transition lineextends up to about TE ¼ 100 MeV in this latter case.Notice that at � ¼ 0 the phase transition temperature isfound to be about 255 MeV, somewhat larger than thevalues obtained from lattice QCD [2,3]. This is a typicalfeature of local NJL models, and we do not adopt in ourcalculations the rescaling procedure [28] of the Polyakov-loop potential, which leads to a flavor-dependent loweringof the critical temperature. It has been demonstrated that innonlocal generalizations of the PNJL model [29] the criti-cal temperature at � ¼ 0 is in agreement with the lattice

result. In Fig. 1 we show the behavior of the effective mass�, the color chemical potential �8, and the tracedPolyakov loop � as functions of the chemical potential,for some representative values of the temperature. Thecurves for T ¼ 100 MeV and T ¼ 200 MeV show clearlythe crossover transition, while for T ¼ 300 MeV the sys-tem is always in the normal quark matter phase (see Fig. 2).For relatively low values of�, the central and lower panelsof Fig. 1 show the correlation between the deconfinementtransition and the restoration of color symmetry: when thetemperature is increased, the average mean field value ofthe traced Polyakov loop rises from 0 (confinement) to-ward 1 (deconfinement); this comes together with a de-crease of the color chemical potential �8, which has beenintroduced in order to recover color neutrality. The effect isdiluted in presence of a large baryon chemical potential.Finally, in the upper and lower panels we have also in-cluded with dashed lines the curves corresponding to thePNJL model without color neutrality (in the central panel,these correspond simply to �8 ¼ 0). The results for themodels with and without color neutrality are approxi-mately coincident in the case of T ¼ 300 MeV.

B. Color-superconducting model

Let us now take into account the quark-quark pairing

interaction in Eq. (1), with the ansatz ��5 ¼ ��7 ¼ 0, ��2 ¼��. According to the discussion in Sec. II, we have to solvethe system of Eq. (12), together with a fourth equation

@�MFA=@ �� ¼ 0. To proceed with the corresponding nu-merical calculations one has to fix the value of H, whichcannot be directly obtained from T ¼ � ¼ 0 phenomenol-ogy. For definiteness we will consider values of the ratioH=G around 0.75, which is the value obtained from a Fierzrearrangement of the local quark-quark interaction inEq. (1).Our main results are shown in Figs. 3 and 4. In Fig. 3, we

show the arising T �� phase diagrams. In order to see thequalitative dependence of the results on the value of H, wehave considered the cases H=G ¼ 0:7, 0.75, and 0.8. It isseen that now the phase diagrams include in general a largeregion of two-flavor superconducting (2SC) phases. Forlarge values of the chemical potential and relatively lowtemperatures, the system is in a 2SC phase in which thechiral symmetry is approximately restored. Then, at T ’150 MeV one finds a second order phase transition into anormal quark matter phase. In addition, for intermediatevalues of �, the system gets into a 2SC phase in which thechiral symmetry is still strongly broken (�SB). Here, thequarks acquire large dynamical masses, which are notsubstantially different from those obtained in the nonsu-perconducting model. The size of this phase region is largefor H=G � 0:75, and becomes reduced if the ratio isdecreased. Once again, one finds a second order phasetransition into the non-2SC phase, while the chiral resto-ration is driven by a first order phase transition for low

0 50 100 150 200 250 300 350 4000

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150

200

250

With colorneutrality

T [

MeV

]

µ [ MeV ]

Without colorneutrality

FIG. 2 (color online). Phase diagram for the nonsuperconduct-ing two-flavor PNJL model with and without the imposition ofcolor neutrality. Solid and dashed lines correspond to first orderphase transition and smooth crossover, respectively. The fat dotsdenote the position of the end points.

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temperatures, and by a smooth crossover for temperaturesabove a given endpoint. The shrinking of the �SB-2SCcoexisting phase appears to be the main qualitative effectof the change in H=G within the considered range.

In Fig. 4 we show the behavior of the mean field diquark

condensate �� (left panel) and the traced Polyakov loop �(right panel) as functions of the chemical potential for T ¼0, 50 MeV, 100 MeV, and 150 MeV. The curves correspondto the case H=G ¼ 0:75. Both the first order and the cross-over chiral restoration transitions can be noticed throughtheir effects in the diquark condensate, while the size of thediquark gap is found to be of the order of a hundred MeVfor the bulk of the temperature range (it decreases rapidlyto zero near the phase border, at T 150 MeV). Con-cerning the traced Polyakov loop, it is seen that the behav-ior is similar to that found in the case of the nonsupercon-ducting model, in the sense that� & 0:15 for temperaturesbelow 100 MeV and chemical potentials up to 500 MeV(see lower panel of Fig. 1). Here, the rise with � is evenless pronounced than in the nonsuperconducting case,since one finds a plateau in the �SB-2SC coexistenceregion (see curves corresponding to T ¼ 50 and

100 MeV). For comparison, the behavior of � in the non-superconducting model for T ¼ 100 MeV has also beenplotted in the figure. For T � 150 MeV there is no 2SCphase, thus there is no difference between the behavior of� in both models. Finally, the behavior of the color chemi-cal potential�8 in both superconducting and nonsupercon-ducting models is also found to be qualitatively similar (seecentral panel of Fig. 1).It has been noted earlier by Fukushima [20] that the

phase diagram of the PNJLmodel exhibits at densities�>�c and low temperatures a so-called quarkyonic phasecharacterized by the coexistence of confinement (� 1)and chiral symmetry restoration (see also [22]). The pos-sibility of such a phase was suggested in Ref. [30] for largeNc QCD where also the term was coined. In a differentcontext it has been discussed in Ref. [31]. It is at present anopen question whether a quarkyonic phase shall exist in thereal world withNc ¼ 3. As a result of the present study, wefind that the requirement of color neutrality together withthe occurrence of diquark condensation are consistent withthe existence of a quarkyonic phase in the PNJL model.Moreover, as it is shown in the right panel of Fig. 4, the

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2SCχSB - 2SC

EP

µ [ MeV ]

χSB

NQM

H / G = 0.7 H / G = 0.8H / G = 0.75

2SCχSB - 2SC

EP

µ [ MeV ]

χSB

NQM

2SCχSB - 2SC

EP

T [

MeV

]

µ [ MeV ]

χSB

NQM

FIG. 3. T �� phase diagrams for the neutral superconducting two-flavor PNJL model, for various values of the ratio H=G. Solid,dashed, and dotted lines denote first order, second order, and crossover phase transitions, respectively. The fat dot indicates the endpoint.

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∆ [M

eV]

µ [MeV]

FIG. 4 (color online). Behavior of the color-superconducting condensate �� (left) and the traced Polyakov loop� (right) as functionsof the chemical potential, for various values of the temperature and a coupling ratio H=G ¼ 0:75. For comparison, the dashed-dottedline in the right panel shows the curve corresponding to T ¼ 100 MeV in the nonsuperconducting model (H ¼ 0).

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presence of diquark condensation leads to a reduction ofthe expectation value of the traced Polyakov loop � andthus enforce the confining aspects of the model.

It is important to point out that the presence of a non-vanishing Polyakov-loop variable �3 breaks the colorSU(3) symmetry down to SU(2), so that the rotationalinvariance for the choice of the orientation of the 2SCgap under color neutrality constraints [32] might be lost.

As a consequence, the above ansatz for ��a may not corre-spond to the true minimum of the thermodynamical poten-tial, which instead might point to a different direction in the�a space [33]. While accounting for such a disorientationof the 2SC condensate may lead to quantitative corrections,we believe that the qualitative structure of the phase dia-gram for the PNJL model under color neutrality constraintsshown in the left panel of Fig. 3 remains robust. The 2SCphase might coexist with chiral symmetry breaking andconfinement in the temperature range between about50 MeVand 100 MeVat densities below the onset of chiralrestoration with possible observable consequences inheavy-ion collision experiments, e.g., due to diquark-antidiquark annihilation into lepton pairs [34]. Note thatwith the coexistence of �SB and 2SC phases the conditionfor Bose-Einstein condensation (BEC) of diquark boundstates is fulfilled and the chiral restoration transition in theabove temperature window may therefore be characterizedas a BEC-BCS crossover transition where the diquarkbound states undergo a Mott effect to unbound, but reso-nant diquark Cooper pairs [35,36]

IV. CONCLUSIONS

In this paper we have explored the consequences ofimposing the color neutrality constraint within the frame-work of the standard mean field treatment of the PNJLmodel at finite chemical potential. We have found that thefulfillment of such a constraint leads to some changes inthe phase diagram of the model, as e.g., a shrinking of the�SB domain in the T �� plane and a shift of the endpointto lower temperatures. The most noticeable effect turns outto be the presence of a coexistence region of �SB and 2SCphases, which may have consequences for possible appli-cations of the model in heavy-ion collision experiments.

The effects can be understood due to the nonvanishingcolor chemical potential �8, which is introduced to com-pensate the color charge �8 induced by the color back-ground field �3. For a given baryochemical potential �,�8 > 0 reduces the chemical potential of blue quarks,while simultaneously increasing that of red and greenones. The net effect is a shift of the �SB phase border tolower� values than without the color neutrality constraint,due to the nonlinear dependence of the quark mass gap onthe chemical potential close to its critical value for a given,not too low temperature. It is seen that this effect isrelatively slight, taking into account the theoretical uncer-

tainty that can be in general expected from an effectivequark model. In addition, the increase of chemical poten-tials of red and green quarks, which pair in the 2SCchannel, entails an effective lowering of the critical valueof� ¼ �c for the onset of the 2SC phase, again for not toolow temperatures. One finds in this way a �SB-2SC coex-isting phase, whose size depends on the value of theparameter H that drives the quark-quark interaction. Wenotice that the 2SC border has been obtained after choosing

a given ansatz for the mean field values ��a. The actualorientation of the 2SC gap in color space under the pres-ence of the Polyakov loop is a subject that deserves furtherstudy. Moreover, it is worth pointing out that at finitedensities there is no fundamental need to be restricted toLorentz invariant condensates, thus more general meanfield configurations could be considered.Signatures of the coexisting �SB and 2SC phases might

become apparent in not too energetic heavy-ion collisionexperiments (about 3–5 GeV=nucleon), as they are pos-sible at the future facilities FAIR-CBM (Darmstadt),NICA-MPD (Dubna), and J-PARC (Japan). Such a mixedphase can give rise, e.g., to an enhancement of low-massdilepton production due to diquark-antidiquark annihila-tion. It is a realization of the BEC-BCS crossover in quarkmatter. While such a transition has previously been dis-cussed in the NJL model by artificially increasing thediquark coupling to large values [36–38], which can resultin conflict with the nuclear matter ground state, in thepresent PNJL model the �SB-2SC coexisting phase hasbeen obtained without any artificial tuning of parameters. Itjust occurs as a consequence of color neutrality in a domainof temperatures and chemical potentials around thecritical endpoint of the first order chiral phase transitionsfor the standard Fierz value of the diquark coupling,H=G ¼ 0:75, without affecting the nuclear matter groundstate at T ¼ 0.Finally, it has been shown that the coexistence of con-

finement and chiral symmetry restoration characterizing aquarkyonic phase in the PNJL model at finite densities andlow temperatures gets even reinforced due to color super-conductivity. The occurrence of diquark condensation re-duces the expectation value of the traced Polyakov loopand thus acts toward strengthening the confinement aspectof the model.

ACKNOWLEDGMENTS

This work has been supported in part by CONICET andANPCyT (Argentina), under Grant Nos. PIP 6009, PIP6084, and PICT04-03-25374. D. B. acknowledges supportfrom the Polish Ministry of Science and Higher Educationunder Contract No. N N202 0953 33. He is grateful for thehospitality of the Institute for Nuclear Theory at theUniversity of Washington and for partial support fromthe Department of Energy during the completion of thiswork.

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