PHYSICAL REVIEW D, VOLUME 70, 054010

K�-meson condensation and phase transition from nuclear matterto color-flavor-locked matter

Xiao-Bing Zhang and Xue-Qian LiDepartment of Physics, Nankai University, Tianjin 300071, China

(Received 23 June 2004; published 10 September 2004)

0556-2821=20

We investigate effects of discontinuous K�-meson condensation on the phase transition from nuclearmatter to color-flavor-locked quark matter. Two specific scenarios (one is a first-order transition viamixing of opposite-charged phases, whereas the other is a coexisting phase of the nuclear matter withK�-meson condensation and the color-flavor-locked matter) are examined. It is found that the presenceof electrically neutral color-flavor-locked matter is preceded by the coexisting phase in the sense of thehadron-quark continuity, once K�-meson condensation occurs in a neutron star environment.

DOI: 10.1103/PhysRevD.70.054010 PACS numbers: 12.39.Fe, 11.30.Rd

1Although both the kaon mode in CFL matter and the kaonmeson in nuclear/hadronic matter are associated with sponta-neous chiral symmetry breaking, the former has a much lightermass of order of 10 MeV [2], compared with the original massof the latter. So we must distinguish the two kinds of kaonicbosons in expression of the present paper.

Nuclear matter is undoubtedly one of the phases ofQCD, but not the most stable phase especially underextreme conditions. For high-density QCD with threeflavors, its original color and flavor symmetriesSU�3�color � SU�3�L � SU�3�R might be broken down toa diagonal subgroup SU�3�color�L�R via the Bardeen-Cooper-Schrieffer-like pairing of the quarks near theFermi surface [1]. The quark matter with a particularsymmetry pattern, namely, color and flavor rotationsare locked together, is called color-flavor-locked (CFL)matter. The CFL matter is much more stable than normalquark matter due to the existence of a pairing energy gap� of order of tens to 100 MeV in the CFL matter [1,2]. Atsufficiently high density and low temperature, the CFLmatter will become the ground state of three-flavor QCD.In general, quark matter at high density has a rich phasestructure, consisting of different color-superconductingphases. The nuclear-matter phase and the CFL phasemight be separated by other color-superconductingphases, e.g., one- or two-flavor superconducting phases,gapless CFL phases, crystalline phases, etc. [3]. Thesimplest possibility, however, is the transition(s) betweennuclear matter and CFL quark matter, which is expectedto occur in the cores of neutron stars [3,4]. Investigationon this issue is necessary for understanding the QCDphase diagram.

Recently, the nuclear-CFL phase transition has beenstudied in a realistic situation, i.e., the neutron star, whichis made of matter with nonzero electron chemical poten-tial �e [4,5]. On the nuclear side, the system is usuallycomposed of neutrons, protons, and electrons and mightbe electrically non-neutral. On the CFL side, the quarksystem has equal numbers of u, d, and s quarks and thusthe CFL phase keeps electrically neutral itself. Noticingthat the pseudo Nambu-Goldstone modes associated withspontaneous chiral symmetry breaking are the low-energy degrees of freedom in the CFL spectrum, thenegative-charged kaon mode (K� mode) condenses easilyin the case of �e � 0 [6–8]. Thus, the K�-mode con-densed phase of the CFL matter, named CFLK�, be-

04=70(5)=054010(5)$22.50 70 0540

comes more energetically stable on the CFL side.Although both the nuclear and CFLK� phases may becharged, a mixed phase of them is required to be globallyneutral in neutron stars. In general, the pressures andcharge densities for nuclear matter and quark matter insuch a mixed phase satisfy [4,9]

P N��;�e� � PQ��;�e�; (1)

�qQ��;�e� � �1� ��qN��;�e� � 0; (2)

respectively, where � is the quark chemical potential orone-third of the baryon chemical potential and � 2 �0; 1�denotes the volume fraction of the quark matter in themixed phase. Equation (1) reflects the pressure equilib-rium, while Eq. (2) is introduced to guarantee the globalneutrality of the nuclear-quark mixed phase. In this way,a first-order nuclear-CFL transition was proposed to takeplace in a finite mixed phase region [4].

In Ref. [4], the typical value of the quark chemicalpotential where CFL matter is favored in the nuclear-CFLmixed phase region was suggested to be �� 400 MeV,equivalently about 5 times of the saturant density ofnuclear matter. In such a density regime, however, ithas been well known that condensations of the ordinarykaon-meson1 might occur in hadronic matter [10–12].When the K� meson condenses in the nucleon-rich envi-ronment with �e � 0, the nuclear side involved in thenuclear-CFL transition is not a normal nuclear-matter(NNM) phase but the nuclear phase with K�-mesoncondensation, which is named NMK� in the following.In this case, effects of K�-meson condensation on thephase structure of nuclear-CFL transition must be inves-

10-1 2004 The American Physical Society

2In Refs. [11,12] !K was named as the zero-momentumenergy of the kaon meson. For s-wave condensation, the energyis just the effective kaon-meson mass in nuclear medium.

3In the vicinity of �c, this scenario of NNM-NMK� tran-sition has been further corrected [12]. By using the Gibbscondition analogous with Eqs. (1) and (2), a phase transitionvia mixing NNM and NMK� was obtained. There, K�-mesoncondensation is still of first order, but it occurs in a finite mixedphase region. Although the ‘‘finite’’ transition is more thermo-dynamically favorable than the above ‘‘single’’ transition, thetwo scenarios obtain the identical description of nuclear matterexcept for in the mixed region (see Ref. [12] for details).Moreover, such a NNM-NMK� mixed phase dissolves whenthe CFL matter is introduced in strongly interacting matter. Soit is reasonable to adopt a discontinuous ‘‘jump’’ for describingK�-meson condensation in this work.

XIAO-BING ZHANG AND XUE-QIAN LI PHYSICAL REVIEW D 70 054010

tigated seriously, which is just the purpose of the presentwork.

Assuming that the transition takes place via mixing ofnuclear and CFL matter, the two phases in the mixedregion must have opposite charges. The CFL side iselectrically negative due to the K�-mode condensation,so that the nuclear side has to be electrically positiveduring this transition. Once NMK� becomes the energeti-cally stable phase of nuclear matter, questions arise natu-rally whether and how the NMK� phase becomes elec-trically positive and how to reconstruct a mixed phaseconsisting of both NMK� and CFLK�. Answering thesequestions is important for understanding the detailedphase structure of nuclear-CFL transition. On the otherhand, in view of the fact that the Nambu-Goldstonemodes in CFL quark matter exhibit a resemblance tothe ordinary Nambu-Goldstone mesons, a remarkableconjecture has been proposed [13] that the hadronic mat-ter and the CFL quark matter should be continuouslyconnected. Generally, the continuity between normal nu-clear matter and CFL quark matter is impossible, becausethe latter is strange matter whereas the former is not. Asthe NNM is replaced by the NMK� on the nuclear side,however, a continuous transition from the nuclear matterwith additional K�-meson condensation to CFL matterbecomes very likely, since it is consistent with the pictureof hadron-quark continuity. This is different essentiallyfrom the scenario that a first-order transition takes placefrom nuclear matter to CFL matter directly, and thusinvestigation on it is needed also.

As discussed above, the concept of the nuclear matterwhich is considered in the nuclear-CFL transition needsto be generalized; namely, the matter can be NNM andNMK�. The description of such generalized matter oftendepends on the mechanism of K�-meson condensation.By taking an attractive kaon-nucleon interacting terminto account explicitly, Kaplan and Nelson suggestedthat kaon-meson condensation exists in a nuclear environ-ment [10]. In recent years, a relativistic mean-field modelfor neutron star matter has been developed, where theinteractions of kaon meson with the sigma, omega, andrho mean fields are incorporated in a similar way to thatfor the nucleon interaction [14]. Using this model, thepressure for the nuclear matter takes the form [4]

PN �

Pi � p; n

1

3�2

Z piF

0

p4dp��������������������p2 �m2

N

q �1

2m2

��2 �U���

�1

2m2

!!2 �1

2m2

��2 ��4

e

12�2 ; (3)

where mN � mN � g�N� is the nucleon effective mass,

which is smaller than the free nucleon mass mN due to thescalar coupling. The contribution from the kaon-nucleoninteraction is included in the mean fields of �, !, and �self-consistently [14]. In this way Eq. (3) is valid regard-

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less of whether the K� meson condenses. Similar to themechanism proposed in Ref. [10], kaon-meson condensa-tion in the nuclear matter described by Eq. (3) is stillcaused by the kaon-nucleon interaction essentially. At themean-field level, the interaction enforces the kaon-mesonmass to deviate from its original value of about 500 MeVobserved at zero density. The modification in kaon-mesonmass makes kaon-meson condensation possible since themass of an antikaon meson was widely predicted todecrease with increasing matter density [11,12]. Themechanism of meson condensation proposed inRef. [12] is adopted in the present work. In the neutronstar matter with �e � 0, the critical condition ofK�-meson condensation reads [12]

!K � �e; (4)

where !K denotes the effective kaon-meson mass2 innuclear medium. As Eq. (4) is satisfied, K�-meson con-densation occurs. In this case, the density of the con-densed K� meson �K has a nonzero value and the chargedensity on the nuclear side is �p � �e � �K. Here �p and�e are the densities of proton and electron, respectively.Otherwise, without K�-meson condensation, the chargedensity of a normal nuclear system is given by �p � �e.

By using a Maxwell construction, the phase structureof the generalized nuclear matter on the basis of Eqs. (3)and (4) is shown in Fig. 1. A discontinuous K�-mesoncondensation occurs at � � �c; equivalently, the baryonchemical potential is 3�c. Thus, the vertical dotted line inFig. 1 separates NNM and NMK� simply3. The solid linesin the region of �<�c and �>�c are the neutralitylines for NNM and NMK�, respectively. In the casewithout considering K�-meson condensation, the neutral-ity line for �>�c is shown by the dashed line for acomparison. Introduction of K�-meson condensation re-sults in a falling trend of �e with increasing matterdensity, whereas it rises monotonously if neglecting themeson condensation. Another remarkable difference isthat the NMK� phase is significantly more stable thanthe corresponding phase without K�-meson condensa-

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50

100

150

n. l. of N

MK

n. l. of CFL

MIXNNM

CONTINUITY

NMK --C

FL

µc

NNM-CFL

µ e (MeV)

350 400 450 500

50

100

150

200

250

300

µc

NNM-phase NMK--phase

µ e (M

eV)

µ (MeV)

FIG. 1. The electron chemical potential and the quark chemi-cal potential (i.e., one-third of the baryon chemical potential)in the generalized nuclear matter using a Maxwell construc-tion, where the nucleon-meson and kaon-meson coupling pa-rameters are chose as done in Refs. [12,14], respectively.

K�-MESON CONDENSATION AND PHASE TRANSITION. . . PHYSICAL REVIEW D 70 054010

tion. Because of these facts, nuclear-CFL transition isexpected to have a more complicated phase structureonce K�-meson condensation occurs.

On the CFL side, the pressure for the pure CFL mattertakes the form [1,4]

P CFL � �

Pi � u; d; s

Z pF

0

3

�2 p2�

������������������p2 �m2

i

q���dp

�3�2�2

�2 � B; (5)

where pF is the common Fermi momentum for threeflavors and is given by pF � ��m2

s=�6�� approximately[4], B corresponds to the bag constant reflecting thephysics of confinement, and � is the color-superconducting gap which determines the CFL pairingenergy. As for the quark masses, we adopt a finite strange-quark mass ms � 150 MeV and a small light-quark massmu;d � ms=20 all through this work. According to theleading-order effective Lagrangian for the Goldstonemodes in the CFL matter [2,7], the mass of a kaonmode is obtained by

m2K �

3�2

�2f2�mu;d�mu;d �ms�; (6)

where f� is the decay constant of the Goldstone modes inthe context of the CFL matter [2].

Noticing that the K�-mode condensation occurs in aninfinitesimal way4, the contribution from the condensed

4Different from K�-meson condensation in nuclear matter,the K� mode condenses infinitesimally in CFL matter; namely,the phase transition from CFL to CFLK� is of second order[4,6,8,15]. The main reason lies on the fact that one requires noelectrons to make CFL matter electrically neutral, and thus thedescription of CFL matter is independent of �e approximately.

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K�-modes to the pressure is

P K �f2�2�2

e

�1� 2

m2K

�2e�

m4K

�4e

�; (7)

where the chemical potential for the K� mode has beenassumed to be �e approximately [4]. Equations (7) and(9) (in the following) are valid only for �e mK.Otherwise, the K�-mode condensation does not exist atall. The total pressure of the quark matter is

P Q � P CFL � PK; (8)

as long as the K� mode condenses in the CFL matter.Meanwhile, the density of the condensed K� modescontributes to the charge density of the CFLK� phase.The charge density on the quark side is [4,6,8]

qQ � �f2��e

�1�

m4K

�4e

�; (9)

when the K�-mode condensate exists.In view of the fact that the pressure of quark matter is

sensitive to the bag constant, we choose B1=4 �200 MeV, a middle value within the physically allowedregion for the bag constant, to investigate the detailedstructure of nuclear-CFL transition. As for the nuclearside, we use the same set of parameters as done inRef. [12], where the critical value for a discontinuousK�-meson condensation is obtained to be �c ’410 MeV. Figure 2 shows the phase structure of thenuclear-CFL transition in the ��;�e� plane by using theGibbs condition Eqs. (1) and (2). For an illustrative pur-pose, let us first concentrate on the physical picture ofstrongly interacting matter for �< 410 MeV. Startingwith the normal nuclear matter, a first-order phase tran-sition via the mixing of NNM and CFLK� takes place in

350 400 450 500

µ (MeV)

FIG. 2. Schematic phase structure of the nuclear-CFL tran-sition in the ��;�e� plane, using the parameter set in Fig. 1 onthe nuclear side, whereas adopting B1=4 � 200 MeV and � �100 MeV on the CFL side. The label of ‘‘n.l.’’ is used in themeaning of ‘‘neutrality line.’’

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XIAO-BING ZHANG AND XUE-QIAN LI PHYSICAL REVIEW D 70 054010

the region of 367 � � � 410 MeV. At � � 367 MeV,the CFL matter starts to exist as a droplet phase, whichis immersed in nuclear background. Thus, NNM andCFLK� are mixed in this region, where the correspond-ing solid line denoted as ‘‘NNM-CFL’’ is a globallyneutral phase.

With �e decreasing and � increasing, the end point ofthe NNM-CFLK� mixed phase is close to the point of�� � �c;�e � 0� gradually. So the pure and electricallyneutral CFL matter seems to become a stable state forstrongly interacting matter as long as � �c.Nevertheless, there exists a discontinuous K�-meson con-densation in the vicinity of � � �c, so that it is doubtfulthat the CFL matter is energetically favorable with re-spect to the NMK� phase. Comparing the pressures ofthree phases, NMK�, NNM, and CFL at � � �c, we findthat for small �e the CFL matter cannot remain stablesince NMK� is favored over it. To examine the stronglyinteracting matter for � �c, another mixed phase con-sisting of NMK� and CFLK� is constructed by usingEqs. (1) and (2), as shown in the region of 410 � � �450 MeV in Fig. 2. In the vicinity of � � �c, the neutral-ity line of the NMK�-CFL mixed phase moves up incomparison to that of the NNM-CFL mixed phase. Thereason is mainly that the volume ratio of the nuclearmatter [i.e., 1� �; see Eq. (2)] in the mixed phase ischanged in favor of the NMK� phase rather than theNNM phase.

It is not the whole story yet. The numerical calculationshows that the NMK� and CFL phases reach a pressureequilibrium easily as long as � is large enough and �e issmall enough. A shaded region is shown in the window of450 � � � 500 MeV in Fig. 2 and it is surrounded by thefollowing lines: the neutrality line of CFL matter, whichis obtained from �e � mK; the neutrality line of NMK�

matter, which was given in Fig. 1; and the line of aNMK�-CFLK� mixed phase. In such a window, notonly is the pressure equilibrium satisfied, but also boththe nuclear matter with K�-meson condensation and theCFL matter are approximately electrically neutral. In thisway the strongly interacting matter in this window, as acoexisting phase of NMK� and CFL, becomes energeti-cally stable. It must be stressed that this is not a simplenuclear-CFL mixed phase, but a coexisting phase madeup of high-density nuclear matter and CFL quark matter.Therefore, there is not a regular nuclear-CFL transition insuch a shaded region. More important, the effective kaon-meson mass !K has the same order of magnitude as thekaon-mode mass mK in this window. The reason is that!K is required to be equal to �e [see Eq. (4)] for theNMK� matter while mK is equal to (or a little larger than)�e for the neutral CFL matter. Thus, the kaon meson inhadronic context is related to the kaon mode in CFLcontext, which is in complete agreement with the pictureof hadron-quark continuity. The appearance of this coex-

054010

isting phase in such a window should be understood as therealization of the hadron-quark continuity.

Instead of a direct transition from the normal nuclearmatter to the CFL matter, the presence of electricallyneutral CFL matter is preceded by the coexisting phasein the sense of the hadron-quark continuity. Thus, the pureCFL matter becomes absolutely stable only for very largequark chemical potential, say, �> 500 MeV, as shown inFig. 2. Also, the present work points out that the nuclear-CFL transition in neutron star matter takes place in a‘‘hybrid’’ way, containing both a first-order transition andthe hadron-quark continuity at the same time. Dependingon the parameters at the nuclear and CFL sides, thelocation of the window of hadron-quark continuity mightbe changed quantitatively. Even if so, the above physicalpicture is generally valid as long as K�-meson condensa-tion occurs in a nuclear environment. If the bag constantis large enough so that the nuclear matter is alwaysfavored over CFL matter in a relatively low densityregion, say, �<�c, then a NNM-CFLK� mixed phaseregion would not be constructed. In that case, onlyNMK� takes part in the nuclear-CFL transition and itis very likely that the hadron-quark continuity becomesdominant for the formation of a pure CFL matter. Furtherinvestigation on this issue needs to consider other effectson the CFL side, such as the in-medium dependent pa-rametrization of the bag ‘‘constant’’ [16], the variation ofthe color-superconducting gap, and the perturbative in-teraction among quarks, all of which are beyond the scopeof this work.

In the present paper, we investigate effects ofK�-meson condensation on the nuclear -CFL transitionin the neutron star matter with �e � 0. By taking theK�-meson-condensed phase of nuclear matter (NMK�)into account, we reexamine a first-order phase transitionvia mixing of opposite-charged phases. More impor-tantly, introducing the NMK� phase provides a possibil-ity that high-density nuclear/hadronic matter isconnected with the CFL quark matter via an irregular‘‘transition.’’ It is found that, for large � and small �e, thehadron-quark continuity may be realized by a coexistingphase of NMK� and CFL, where the mass differencebetween kaon mode and kaon meson becomes negligible.This result is expected to be important for finding signa-tures of the presence of color-superconducting quark mat-ter in neutron stars. Nevertheless, there still are someunanswered aspects, including how the kaon meson in anuclear environment is transposed to the kaon mode inCFL matter, whether the other color-superconductingphases of quark matter exist between the high-densitynuclear/hadronic matter and CFL matter, and whether theconnection in masses of two kinds of kaon is a uniquesignature of hadron-quark continuity. Until now, thesequestions are far from being satisfactorily solved andsome of the problems are being investigated.

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K�-MESON CONDENSATION AND PHASE TRANSITION. . . PHYSICAL REVIEW D 70 054010

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