PHYSICAL REVIEW D 68, 054015 ~2003!

Antikaon-kaon condensate in color-flavor locked matter and its implicationfor quark-hadron continuity

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

~Received 19 February 2003; published 18 September 2003!

We study the role of the strange quark mass in color-flavor locked matter. An effective interaction betweenkaons and particle-hole excitations near the Fermi surface is important and should be included in the effectiveLagrangian. The new picture is based on the reasonable assumption of quark-hadron continuity. For a nonzerostrange-quark mass, there is the possibility that both antikaon and kaon condensates may occur at differentdensities. A novel phase diagram of high-density QCD is given, which is different from that given in previousliterature. The implications of the configuration are discussed briefly.

DOI: 10.1103/PhysRevD.68.054015 PACS number~s!: 12.39.Fe, 11.30.Rd

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I. INTRODUCTION

Investigations of strongly interacting matter under etreme conditions have attracted much attention for yeThese interactions are expected to exist in many physsystems of interest from supernovas and neutron starheavy ion collisions. A recent hot topic involves the studyquark matter at high baryon density and low temperatuThere has been renewed interest in this issue since a palar symmetry breaking pattern is proposed for the case wthree massless quarks at high baryon density. As pointedby the authors of Ref.@1#, the original color and flavorSU(3)color3SU(3)L3SU(3)R symmetries are brokendown to the diagonal subgroupSU(3)color1L1R via the for-mation of Cooper pairs of quarks near the Fermi surfaThis suggests the existence of a particularly symmeground state, where color and flavor rotations are lockedgether in the dense quark matter. In the so-called color-flalocked~CFL! phase, the main low-lying spectrum is an ocof pseudoscalar Goldstone bosons that are responsible fobroken chiral symmetry. There exists a gap in the vicinitythe Fermi surface, where quarks and hole excitationsmuch more active than those residing far away fromFermi surface. Their quantum behavior becomes the reledegrees of freedom of the CFL phase of matter. All massgluons in the broken symmetry scenario may be integraout in the low-energy region. Hence the CFL phase candescribed by a nonlinear effective Lagrangian for low engies, namely, energies smaller than the gapD near the Fermisurface.

More recently, the effect of a nonzero strange quark mon high-density quark matter has been investigated. Thethors of Refs.@2–4# find that aK0 condensate occurs at somdensities and thus the CFL matter with an additional mecondensate, namely, the CFLK0 phase, may become enegetically favorable. Similarly, a nonzero lepton chemical ptential results in the formation of charged meson condens@3–5#. These studies imply that the CFL matter with adtional condensates is probably the ground state of hdensity QCD with three flavors. Moreover, these studiesimportant for better understanding the phase structurestrongly interacting matter in the high-density and lo

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temperature region of the QCD phase diagram.Let us first briefly discuss the essential feature of the p

vious treatment of the charge-neutral meson conden@2–4#. In the situation that the strange quark is much heavthan light quarks, the Fermi surface for the strange quobviously shifts with respect to that for the light quarks. Bintroducing a self-interacting term containing the quamasses@Eq. ~4! below#, Bedaque and Scha¨fer proposed thatthe existence of the strange hole excitations shifts the Fesurfaces bydpF.ms

2/(2pF), wherepF is the Fermi momen-tum of the quarks@3#. As a consequence, aK0 condensateoccurs in the CFL vacuum via a process like 0→ds→K0,where ‘‘0’’ means the physical vacuum in the mediumhigh density. On the other hand, a different but complemtary mechanism for condensation in hadronic matter wproposed in Ref.@6# some years ago. There, the meson codensate is caused by an attractive kaon-nucleon interacwhich is mainly proportional to;KKNN. At the mean-fieldlevel, the meson-baryon interaction term causes the kmass to deviate from its vacuum value, and the effectmass decreases with increasing nuclear density. As a rean antikaon condensate becomes possible in a nucleonenvironment. The question arises, then, whether theresimilar mechanism in the color-flavor locking environmenIn other words, the question is whether or not the kaon cdensate in the CFL matter is driven by the Bedaque-Sch¨ferterm @Eq. ~4!# exclusively. We will discuss what kind of ad-ditional information is required to settle the issue, whichthe first purpose of the present note.

Moreover, a remarkable picture has been proposed@7,8#that quark matter and hadronic matter should be contiously connected since the low-lying spectrum of CFL matexhibits a resemblance to that of hadronic matter. If this cjecture is correct, the dense hadronic matter and CFL maprobably possess a coexisting phase at some densities. Incase, it is plausible to assume that in the CFL phase thexist the same meson condensates as in the hadronic phathe aspect of strangeness. This inspires us to comparetwo different condensates in hadronic and CFL matter.the hadron-matter side, most calculations based on the ceffective Lagrangian suggest that a condensate of antika(K2,K0) rather than kaons (K1,K0) is favored@6,9#. On the

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ZHANG, LUO, AND LI PHYSICAL REVIEW D 68, 054015 ~2003!

side of CFL matter, nevertheless, Refs.@2,4# predicted that itis not antikaons but kaons that condense. Qualitatively, thcalculations for CFL matter do not predict the same condsation of strangeness as that obtained on the hadron-mside, being then in some sense inconsistent with the picof quark-hadron continuity. Resolving this problem is aother purpose of the present note.

We will reexamine neutral-meson condensates in Cmatter by introducing an additional effective interaction btween kaons and quasiparticle excitations in the effectivegrangian. The possibility is raised that both antikaons akaons condense in high-density strongly interacting mawith two light quarks and one middle-weight strange quaIt can help to overcome the aforementioned difficulty andof importance in studying the quark-hadron continuity.

II. LOW-ENERGY EFFECTIVE LAGRANGIANOF GOLDSTONE MODES

Let us begin with the leading-order effective Lagrangiof three-flavor QCD in CFL matter. For energies below tgapD, the effective Lagrangian for the Goldstone modes@10,11#

LG5f p

2

4Tr~]0S]0S12vp

2 ] iS] iS1!

2c@det~M !Tr~M 21S!1H.c.#, ~1!

where S5exp(ifala/fp) is the chiral field, f p is the piondecay constant,vp is the velocity of the Goldstone bosonandM is the quark mass matrix. Under a chiral rotation, tGoldstone boson field and the quark mass matrix transfasS→LSR1 andM→LMR1 with L,RPSU(3)L,R . Fromthe perturbative calculations of high-density QCD, the loenergy coefficients in Eq.~1! are @11,12#

f p2 5

2128 ln 2

18

pF2

2p2, c5

3D2

2p2, ~2!

wherepF is the Fermi momentum that is associated withbaryon density of quark matter. The Lagrangian~1! gives themasses of Goldstone modes as

mp2 5

4c

f p2

mqms , mK2 5

2c

f p2

mq~mq1ms!, ~3!

wheremq5mu5md is the light quark mass in the limit oisospin symmetry, whilems is the strange quark mass. Thabove results are different from those obtained with the relar chiral effective Lagrangian at zero density. It is remarkthat Eq.~3! shows that the kaon is lighter than the pionCFL matter. The inverted mass spectrum is related tospecific structure det(M )Tr(M 21S) of the O(M2) massterm in Eq.~1!. Thus, the condensation of the~lighter! kaonsshould be easier than pion condensation in CFL matter. Inpresent work we will consider only the kaon-antikaon codensate. If there exist some unknown physical mechanibeyond the present model Lagrangian Eq.~1!, otherO(M2)

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At high baryon density, the quasiparticle excitations nethe Fermi surface need to be included in the effectivegrangian. Using the high-density effective theory of QCBedaque and Scha¨fer decompose the quark fieldc into twopiecesc and c8, then describe these excitations in termthe field c5 1

2 (11aW • vF)c, wherevF is the Fermi velocity@3#. In the limit of high density, one can eliminate the fiec85c2c and reduce the three-flavor QCD to a simplifieversion which is described by a Lagrangian with only thecfield. To the leading order in 1/pF , a so-called BedaqueSchafer term such as

21

pF~c L

1MM 1c L1c R1M 1Mc R! ~4!

is found@2,3# to play a key role in the formation of the kaocondensate. The Lagrangian~4! is invariant under thesymmetry transformations c L→Lc L and c R→Rc R.MM 1/2pF andM 1M /2pF in Eq. ~4! transform in a similarway as the left- and right-handed flavor gauge fields, resptively. To impose this approximate gauge symmetry onCFL effective Lagrangian, the derivative of the chiral fieldEq. ~1! is replaced by the covariant one@2,3#:

¹0S5]0S1 i S MM 1

2pFDS2 iSS M 1M

2pFD . ~5!

In the present study, we will takems@mqÞ0. In thiscase, Eq.~4! can be further simplified as

Lq52ms

2

2pF~cL

s 1cLs1L→R!. ~6!

It is noted thatc s andc s1 in Eq. ~6! denote the quasiparticleexcitations near the Fermi surface of CFL matter, rather tthe ordinary strange quark with the current ma;150 MeV. To guarantee the existence of the CFL phathe value of ms is constrained by the condition 0<ms

<ADpF @2,3#. This constraint sets a rather wide range ofmsas long asDpF is not small. Thereforems is treated as a freeparameter all through our calculation. Presumably, by assing thatms varies within the allowed region, we are ablereexamine the meson condensates in CFL matter.

It is obvious thatms2/2pF in Eq. ~6! plays the role of an

effective chemical potential. In the situation where all extnal chemical potentials, say the electron chemical potenmay be neglected,ms

2/2pF is the only chemical potentiaconcerned in the present scenario, and we identifyms

2/2pF as

m[ms

2

2pF. ~7!

Since there is a large gapD for quarks near the Fermsurfaces, the meson-quark interaction is very weak in Cmatter@5,13#. So the interaction and its medium effect on t

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ANTIKAON-KAON CONDENSATE IN COLOR-FLAVOR . . . PHYSICAL REVIEW D68, 054015 ~2003!

meson condensate were ignored in existing calculati@2–5#. However, this treatment might be somewhat oversplified in a realistic situation where the strange quarkmuch heavier than the light quarks. As long as Eq.~4! istaken into account, the interaction between the Goldstbosons and strange particle or hole excitations needs tconsidered seriously. In view of the fact that Eq.~4! emergesat the leading order of chiral expansion as well as in 1/pF ,there is no reason to exclude an effective interaction tesuch as

1

pF~c L

1SM 1SM 1c L1c R1S1MS1Mc R! ~8!

from the CFL effective Lagrangian. Also, the Lagrangian~8!

is invariant under the flavor symmetry transformationsc L

→Lc L andc R→Rc R.Based on the effective Lagrangian Eq.~1!, kaon conden-

sation is always more favorable than that of pions in Cmatter. Thus, only the neutral kaon-antikaon condensatecurs in the case ofms@mqÞ0 as long as the electron chemcal potential is ignored. It is practical to retain only the kafields in the Goldstone meson matrixS. In this case, theeffective meson-quark interaction term Eq.~8! can be re-duced to a simpler form. Here we are concerned withleading term that is caused by a nonzero mass of the strquark and we obtain

LKq52ms

2

pF f p2KK~cL

s 1cLs1L→R!. ~9!

Equation~9! bears a formal analogy with the kaon-nucleinteraction ;KKNN derived in Refs.@6,9#. It is the in-medium effect of the kaon-nucleon interaction term thmakes kaon condensation possible in a nucleon rich enviment. We expect that Eq.~9! causes a similar effect in CFLmatter.

III. EFFECT OF THE KAON-QUARK INTERACTION

In analogy with the treatment in hadronic matter, weplace the operator (cL

s 1cLs1L→R) in Eq. ~9! by its in-

medium expectation value in the mean-field approximatiIt seems that the expectation value can be interpreted adensity of strangeness. This is not correct, however. InCFL context, we define the so-called vacuum, whichcludes three flavors of equal number densities. But when~6! is taken into account, there is an extra amount of straquarks with momentadpF.ms

2/(2pF) and the extra part othe baryon density can be treated as the effective mediumfact, it is obvious that Eq.~9! and the medium expectatiovalue of (cL

s 1cLs1L→R) could not exist in the case withou

including Eq.~6!. As long as Eq.~6! is taken into accountthe in-medium expectation value of (cL

s 1cLs1L→R) is non-

zero and can be interpreted as an extra density with resto the CFL vacuum, namely,

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^cLs 1cL

s1L→R&[drs . ~10!

Sincedrs originates from the shiftdpF which is caused essentially by a nonzero mass of the strange quark, its vashould be proportional to the variation in the quark densi.e., drs;(pF

2/2p2)dpF . The explicit form ofdrs dependson the physical assumptions about the Fermi surface comsition in CFL matter. If it is assumed that the strange hexcitations gain the shiftdpF , the value ofdrs should bepositive, as in the picture of superfluid quark matter@13#. Wemight as well assume that the strange particles instead oholes near the CFL Fermi surface are rather active. Incase, the value ofdrs should be negative as in the ordinapicture of quark matter. In principle, the best way to resothis problem is to employ some well-established microscomodels of strongly interacting matter at high density. Hoever, we have no such model in hand at present, soalternatively we may set a simple parametrization onvalue ofdrs without losing generality@see Eq.~15! for de-tails#.

When combined with the free mass term, Eq.~9! gives theeffective mass of the kaon mode as

~mK* !25mK2 1

ms2

pF f p2

drs , ~11!

where the piece containingdrs provides an in-mediummodification to the kaon mass given in Eq.~3!. AlthoughdpF is small with respect topF , Eq. ~11! shows a similardeviation of the kaon mass as was shown in hadronic ma@6,9#.

On the other hand, it is necessary to investigate the Cvacuum carefully. Strictly speaking, the CFL vacuum canmaintained even ifmsÞ0. As pointed out by the authors oRef. @7# there is a common Fermi momentumpF85pF

2ms2/(4pF) for three flavors and thus the CFL vacuum st

exists in the case ofmsÞ0. Although the CFL vacuum itseldoes not affect the in-medium expectation valuedrs , thecontribution from the CFL vacuum cannot be ignored copletely. In order to describe the CFL vacuum withmsÞ0, weconsider the Lagrangian for the kaon such as

LK5~]0K !~]0K !2vp2 ~] i K !~] iK !1mK

2 KK2mK

2

f p2 ~KK !2

1•••. ~12!

At the classical level, the mean field value^KK& satisfies theequation of motion

S mK2 2

2mK2

f p2 ^KK& D uK&50 ~13!

approximately. From Eq.~13!, we obtain the CFL vacuumcontribution ^KK&. f p

2 /2. Inserting it into Eq.~9!, we findthat the system gains energy density such as

dE.mdrs . ~14!

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ZHANG, LUO, AND LI PHYSICAL REVIEW D 68, 054015 ~2003!

Actually, a term like Eq.~14! was introduced in Refs.@6,9#when considering the meson condensates in hadronic maIn addition to the energy density derived from the Lagranian in the usual manner, it is necessary to introduce a tthat is equal to the chemical potential of the condensedson multiplied by the corresponding number density. Indition, adding Eq.~14! in the kaon’s energy density is undestood as another effect of the kaon-quark interaction term

IV. GROUND STATE WITH NEUTRAL KAONCONDENSATES

In the above discussion, the value ofdrs seems to play acrucial role for the neutral kaon condensate at the mean-level. By simply setting a parametrization in the naive fasion, we can include a structure function to denote the inaction and then expressdrs in the form

drs5ApF

2

2p2dpF , ~15!

where so farA is an unknown function ofpF and ms . Al-though this kind of treatment may be rather naive, it domaintain generality regardless of what physical assumpis adopted.

If the strange hole excitations gain a positive shiftdpF ,the in-medium expectation valuedrs and thus the functionAin Eq. ~15! should be positive. This is reasonable for CFmatter at sufficiently high densities, as the phenomenonbeen explained in Ref.@13#. In this case,m is the effectivechemical potential for positive strangeness. Chemical elibrium requires the chemical potentials for kaons and akaons to be

mK05m, m K052m, ~16!

respectively. Considering Eq.~7!, the value ofm varies sincems is a free parameter. Therefore,m is equivalent to an ‘‘ex-ternal’’ chemical potential in the following treatment.

We characterize the condensates by an amplitudeu andrewrite the neutral kaon fields as@2–4#

K05f p

A2u exp~2 imt !, K05

f p

A2u exp~ imt !. ~17!

Substituting Eq.~17! into Eq. ~1! and considering the effecof Eq. ~9!, it is easy to find that the energy density associawith the neutral kaon condensates is

E~u,m!5mdrs21

2f p

2 m2sin2u2 f p2 ~mK* !2~cosu21!,

~18!

where the terms of order higher thanu2 have been neglectesinceu is small @2,9#.

The ground state is determined by minimizing the enedensity that is obtained from the effective Lagrangian.extremizingE with respect tom andu,

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i-i-

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yy

]E]m

50,]E]u

50, ~19!

we have the following set of equations:

3A22A cosu2~cosu21!12D2

ms2 S mq

msD5sin2u

f p2 p2

pF2

~20!

and

cosu5~mK* !2

m2. ~21!

Instead ofm'mK given in Refs.@2–4#, Eq. ~21! indicatesthat atm'mK* theK0 condensation starts. The value ofA canbe obtained from Eq.~20! and Eq.~21! together.

So far we have concentrated on the case ofA.0, whichfavors aK0 condensate. Theoretically, there are no principor symmetries to determine the sign ofA for the relativelylow-density regime, where the CFL description is believedbe valid. As long as hadron-quark continuity is assumwhich is obviously reasonable, we can allow for the posbility of A,0, and this corresponds to the situation whethe in-medium expectation value is negative. In this casenot the strange hole excitations but the strange particle etations that gain an effective chemical potentialm. Thus themeson condensate with negative strangeness is energetfavorable instead:K0 condensation occurs probably viaprocess like 0→ds→K0. To discuss this possibility selfconsistently, we rewrite Eq.~16! as

mK052m, m K05m. ~22!

Correspondingly, Eq.~17! and Eq.~18! are changed throughthe exchange ofm→2m. As a consequence, Eq.~20! isreplaced by

A22A cosu2~cosu21!12D2

ms2 S mq

msD5sin2u

f p2 p2

pF2

,

~23!

while Eq. ~21! remains unchanged. By using Eq.~21! andEq. ~23! the solution forA,0 can be obtained and it corresponds to the CFLK0 phase in these conditions.

V. RESULTS AND DISCUSSION

At a sufficiently high density, a perturbative QCcalculation predicts that the gap behaves asD;pFg25 exp2(3p2/A2g) @13,14#, where g is the weakgauge coupling constant. For smallerpF we extrapolate theabove relation to meet the current estimate:D;80 MeV forpF;500 MeV @2,3#. Presently, the relation betweenms andpF is not well understood from the color superconductigap equation. For the present work we takems as a param-eter, and for the light quark mass we use the quark mass

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ANTIKAON-KAON CONDENSATE IN COLOR-FLAVOR . . . PHYSICAL REVIEW D68, 054015 ~2003!

ms /mq;20 that is derived from the ordinary chiral effectivtheory @15#.

Whether or not a condensate occurs is closely linkedthe comparison between the values ofm andmK* . Whenm>mK* , the corresponding kaon or antikaon condensate cooccur. For illustrative purposes, we consider the maximvalue ofA first. When kaon or antikaon condensation occuwe have

Amax5f p

2 p2

pF2

23D2~mq

21msmq!

pF2m2

. ~24!

This expression is obtained directly fromm5mK* . Only thesolution that satisfiesA<Amax makes sense for the CFphases with kaon or antikaon condensates since one hm>mK* in this case.

If quark matter exists in neutron stars, the range ofFermi momentum ispF;0.4–1 GeV. Let us discuss thcase at a typical Fermi momentumpF50.5 GeV, about fivetimes the normal nuclear density. Depending on the signEq. ~15!, we can obtain the solutions ofA from Eq. ~20! andEq. ~23!, respectively. Figure 1 shows the values ofA andAmax as a function of the strange quark mass. Correspoingly, the variation of the effective kaon massmK* and thechemical potentialm with respect to the strange quark mams is given in Fig. 2. For comparison, we also plot the valof mK that is obtained by Eq.~3! in the case without includ-ing Eq.~9!. There are two phases according to the value oA.

(i) A.0. The critical massmsc.80 MeV is shown in Fig.

2 for startingK0 condensation. Sincemsc is obtained forA

.0, the location of the critical quark mass is not obviousaffected by including Eq.~9!. This is easily understood sincmeson condensation is assumed to be a smoothly infinimal phase transition@2,9#. With ms increasing, we notice thathe effective massmK* increases much faster than the mamK . Including the contribution of Eq.~9!, our calculationshows a tendency for aK0 condensate to disappear at a sficiently large value ofms . This result is more reasonab

40 80 120 160-2.0

-1.5

-1.0

-0.5

0.0

0.5

ms (MeV)

FIG. 1. Solutions forA ~the solid line! and Amax ~the dashedline! with respect toms at pF50.5 GeV.

05401

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s

-

than that without including Eq.~9! in view of the fact thatCFL matter breaks down whenms

2/pF;D @2,3#.(ii) A ,0. Whenms is less than the critical value, Fig.

shows that the value ofAmax is negative. Thus the solutionfor A.0 no longer corresponds to the CFLK0 phase sinceA.Amax in this case. Nevertheless, Eq.~23! yields a solutionfor A,0 as shown in Fig. 1. From Eq.~11! we notice thatthe medium effect of Eq.~9! provides an effective attractionfor the antikaon mode. Similar to the case in hadronic matFig. 2 shows that there is a significant change in the antikmass. So aK0 condensed phase becomes energetically favable in CFL matter, which can be called the CFLK0 phase.

To demonstrate the results for larger Fermi momenta, F3 gives a schematic phase diagram for three-flavor QCDthe pF-ms plane. The solid line in Fig. 3 divides the phadiagram of the CFL phases where strange meson condenexist into two parts, and they correspond to opposite stranness. Also, we give the boundary of CFL matter qualitativusingms

2/pF;D @2,3#. As the strange quark mass increas

40 80 120 1600

5

10

15

ms (MeV)

(MeV

)

FIG. 2. Dependence of the effective kaon massmK* ~the solidline!, the kaon massmK obtained by Eq.~3! ~the dotted line!, andthe chemical potentialm ~the dashed line! on the strange quarkmassms at pF50.5 GeV.

0.5 1.0 1.5 2.0 2.5

50

100

150

200

250

2SC phase

CFLK0 phase

CFLK0 phase

ms (

MeV

)

pF (GeV)

FIG. 3. The second-order phase transition from the CFLK0 tothe CFLK0 phase is shown by the solid line, while the boundaryCFL matter is shown by the dashed line.

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ZHANG, LUO, AND LI PHYSICAL REVIEW D 68, 054015 ~2003!

beyond the values of the dashed line, the CFL descriptiono longer valid and the so-called two-flavor superconducity ~2SC! phase may occur@16,17#.

For very high densities, our result shows that the CFLK0

phase is favorable as long asms is not very small. In thiscase, an additional meson-quark interaction does not oously affect the meson condensate. The kaon condensastill mainly driven by the Bedaque-Scha¨fer term @Eq. ~4!#.However, at a relatively low density, the situation woulddifferent. For the density regime where quark-hadron conuity becomes possible, we find that the antikaon condenis dominant as long asms is small enough. This result idifferent from that given in Refs.@2,4#, but in agreementwith that obtained in Ref.@5#, which found an antikaon (K2)condensed phase in the presence of electrons by usingCFL effective Lagrangian. Note that antikaon condensawere predicted in hadronic matter for these densities@9#; it isstrongly suggested that if quark-hadron continuity exists,antikaon condensate should be observed in a coexisphase of hadronic and quark matter. Possibly, the appearof strangeness in strongly interacting matter of finite dencan be regarded as a unique signature for quark-hadrontinuity. Nevertheless, there still are some unanswered aspto this complex and difficult problem, including how threflavor deconfined quark matter is produced, how stramatter reaches a phase equilibrium with hadronic matter,Until now the questions are far from being satisfactorsolved, and they are beyond the scope of the present w

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In conclusion, we study the phase diagram of dense qumatter in the case ofms@mqÞ0. Our main intention is toexamine the role of an effective interaction term betweenGoldstone bosons and quasiparticles with strange flavodiscussing antikaon-kaon condensation. In addition toBedaque-Scha¨fer term, a self-consistently determined mdium effect based on Eq.~9! allows for color-flavor lockedphases with various condensates. A novelK0 condensedphase is predicted for a moderate density regime, whicessentially in agreement with quark-hadron continuity. Italso noted that theoretically there are some uncertaintiethe present work. First, we have extrapolated the effecLagrangian for describing CFL matter into a relatively lowdensity regime. Even if color superconductivity existsthese densities, it is still controversial whether there are ophases, say the 2SC phase, between the CFL and hadphases in the QCD phase diagram. Therefore, our qualitaconclusion is valid only ifms

2/pF<D is satisfied. Secondlythe chemical potentials for charge or lepton number in natmight be nonzero, so the realistic ground state of three-flaQCD at high density might be a locally charge-neutral qumatter in some conditions@18#. A thorough analysis of thisissue is related to answering the question of how~indeed,whether! CFL matter with additional condensates existsastrophysical situations like neutron stars. Generally, ourcussion is valid when the electron chemical potential isnored. Although our treatment is still schematic, it gives ipetus for further studies of quark-hadron continuity. Workthis direction is worth further pursuit.

v.

@1# M. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys.B537,443 ~1999!.

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