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Quark description of the Nambu-Goldstone bosons in the color-flavor locked phase



MIT-CTP 3479

Quark description of the Nambu-Goldstone bosons in the color-?avor locked phase
Kenji Fukushima1, 2, ?
1

Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
2

arXiv:hep-ph/0403091v3 5 Sep 2004

Department of Physics, University of Tokyo,

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Abstract
We investigate the color-singlet order parameters and the quark description of the NambuGoldstone (NG) bosons in the color-?avor locked (CFL) phase. We put emphasis on the NG boson (phason) called “H” associated with the UB (1) symmetry breaking. We qualitatively argue the nature of H as the second sound in the hydrodynamic regime. We articulate, based on a diquark picture, how the structural change of the condensates and the associated NG bosons occurs continuously from hadronic to CFL quark matter if the quark-hadron continuity is realized. We sharpen the qualitative di?erence between the ?avor octet pions and the singlet phason. We propose a conjecture that super?uid H matter undergoes a crossover to a superconductor with tightly-bound diquarks, and then a crossover to superconducting matter with diquarks dissociated.
PACS numbers: 12.38.Aw, 11.30.Rd

?

Electronic address: kenji@lns.mit.edu

1

I.

INTRODUCTION

Quantum Chromodynamics (QCD) describing the dynamics of quarks and gluons has the rich phase structure depending on the temperature and the baryon density. In particular, the region at low temperature and high density is the arena of a number of possibilities competing. Our knowledge in condensed matter physics provides us with useful information on dense QCD matter. For example, we know that the attractive force between electrons mediated by phonons brings about superconductivity of metals, and that a fermionic 3 He liquid has the super?uid (B, A, and A1 ) phases originating from pair condensation of 3 He atoms [1, 2]. It is natural to pursue similar possibilities for nucleons [3] and quarks [4]. Actually, the pairing force in the NN interaction leads to a ?nite 1 S0 gap in nuclear matter. The super?uid properties of neutron matter and neutron-rich nuclei have been paid much attention in both nuclear and neutron star physics. When the baryon density is large enough, a triplet-state 3 P2 super?uidity of neutron matter is possibly favored [3, 5]. At extremely high density, quarks are expected to participate in the dynamics directly [6] and eventually the perturbative QCD calculation at weak coupling is feasible owing to asymptotic freedom. As a matter of fact, the one gluon exchange interaction between quarks is attractive in itself in the color anti-triplet channel, which results in the color-superconducting phase [7, 8]. In the case of three massless ?avors, the color-?avor locked (CFL) phase [9] is well established at su?ciently high density. The CFL phase preserving a symmetry under the color-?avor rotation is similar to the B phase of 3 He where a symmetry remain when the orbit and spin rotations are locked together. Interestingly enough, it has been discussed that the spontaneous breaking of global symmetries in the CFL phase has the same pattern as that in hadronic matter, as brie?y reviewed below. It leads us to a conjecture that super?uid (hyper-nuclear) hadronic matter can be smoothly connected to CFL quark matter (i.e., quark-hadron continuity [10]). If this happens, the physical content of the Nambu-Goldstone (NG) bosons must continuously change from one to the other phase. In order to investigate the low energy dynamics governed by the NG bosons in the CFL phase, the non-linear representation of the chiral e?ective Lagrangian has been studied [11, 12, 13, 14]. There are, on the other hand, some attempts to describe the NG bosons in terms of quarks directly [15, 16]. An apparent problem thus is that the e?ective Lagrangian approach suggests the (pionic) NG bosons consisting of four quarks, i.e., two quarks and 2

two quark-holes. Due to this structure, as discussed in [11, 12, 13], the inverse mass ordering of the NG bosons occurs, which leads to the possible meson condensation in the presence of ?nite quark masses or an electron chemical potential. The fact that the NG bosons are made of four quarks has not been fully taken into account in quark based calculations. [This has been considered with respect to not the NG bosons but the sigma meson [16].] Hence, it is important to formulate the quark description of the NG bosons in the CFL phase clearly. The purpose of this paper is to give plain expressions for the NG bosons in the CFL phase. Such a description enables us to have a clear speculation on the quark-hadron continuity from the point of view of the quark content of the condensates and the associated NG bosons. We would emphasize that the structural change of the quark content can be intuitively understood, even though it is hard to imagine how the correspondence is realized, as originally conjectured in [10], between the color-?avor octet quarks and the ?avor octet baryons, or between the massive gauge bosons and the vector mesons.

II.

NAMBU-GOLDSTONE BOSONS

We shall limit our discussion to the chiral limit with three massless ?avors since we are interested in the ideal realization of the NG bosons at ?rst. Although a ?nite ms might change the phase structure qualitatively [17], the chiral limit would be a well-de?ned starting point and an optimal underpinning to consider more complicated situations. In the super?uid phase of diquarks the relevant degrees of freedom are given by diquarks rather than quarks because quarks are all gapped. This is a dynamical assumption meaning that we are working in the CFL phase. The statement that the NG bosons in the CFL phase consist of four quarks is generic not depending on the assumption. The picture, as we argue below, that the four-quark boson is regarded as a diquark-dihole state is derived from the assumption. In other words we specify our problem by taking a diquark model picture. We de?ne the left- and right-handed diquark ?elds as follows; φ? = ?ijk ?αβγ qLjβ qLkγ , ?C Liα φ? = ?ijk ?αβγ qRjβ qRkγ , ?C Riα (1)

where the complex conjugation (?) is attached in order that φL,Riα transforms as a triplet under the color and ?avor rotations as in [11]. The Latin and Greek indices represent ?avor and color, respectively. It is assumed that dominant diquarks are anti-symmetric in 3

spin, color, and therefore ?avor. Since the condensate has positive parity, which is favored energetically, we have φLiα = ? φRiα ∝ δiα ? in the CFL phase if we choose a certain gauge. The left-handed diquark condensate breaks the symmetries as SUC (3) × SUL (3) × UL (1) → SUC+L (3) × ZL (2) (φL is invariant under qL → ?qL ). The right-handed diquark condensate breaks the symmetries likewise [9]. The residual symmetry is in a sense similar to the custodial symmetry in the Higgs-Kibble model [18]. After all, the chiral symmetry breaking takes place in the CFL phase as SUC (3) × SUL (3) × SUR (3) × UV (1) × UA (1) → SUC+L+R (3) × ZL (2) × ZR (2), (2)

as argued ?rst in [9]. We should remark that the UA (1) symmetry is explicitly broken due to the axial anomaly and thus the residual discrete symmetry is only Z(2) in fact. Nevertheless we leave both ZL,R (2) symmetries for later convenience. In the subsequent paragraphs we will clarify the nature of the NG bosons associated with (2). Let us ?rst consider about the pions in the CFL phase, that is, the mesonic excitations forming an octet of SUC+L+R (3). As stated above, a vectorial symmetry is preserved when the color rotation is locked with both the left- and right-handed ?avor (opposite) rotations. Only the axial part of chiral symmetry (i.e., coset space SUL (3) × SUR (3)/SUL+R (3)) is broken actually and the NG bosons appear corresponding to the axial ?uctuations. The important point is that the pions should be colorless because the local color symmetry is never broken spontaneously [19]. In other words, gapless ?uctuations with non-trivial color charge are to be absorbed in the longitudinal polarization of the gauge ?elds (Anderson-Higgs mechanism). As the resulting NG bosons must be all colorless, it is rather preferable to begin with color-singlet order parameters characterizing the symmetry breaking pattern (2). The simplest choice is to form a singlet order parameter from a triplet φL,Riα and an antitriplet φ? L,Riα in color space. Then we have 6 × 6 = 36 combinations (left- and right-handed triplet and anti-triplet) for the ?avor indices. The condition that it should have positive parity and break chiral symmetry reduces those into 17 combinations as follows; σ = φ? φRiα + φ? φLiα , ? Liα Riα
a CV = φ ? Liα a CA = φ ? Liα

(3) λ 2 λa 2
a ij

λ 2 λa 2

a

ij

φLjα + φ? Riα φRjα + φ? Riα

φRjα , φLjα ,

(4) (5)

ij

ij

4

where λa /2 (a = 1, . . . , 8) form an algebra of su(3) in ?avor space. σ is a singlet under the ?
a a ?avor SUV (3) transformation, but neither CV nor CA are. The essential point is that only

σ serves as a proper order parameter for the CFL phase. This can be seen algebraically ?
a from trλa = 0. Otherwise ( CV,A = 0), no vectorial subgroup of chiral symmetry would

survive. One might wonder where the color and ?avor are locked in the order parameter (3) as they should be. The answer is that the choice of σ already re?ects the fact that the ? system is in the CFL phase. This is because the only way to preserve the chiral SUV (3) symmetry with a ?nite diquark condensate is the color-?avor locking and thus the presence of the unbroken SUV (3) symmetry signi?es the CFL phase (see also the argument in [20]). In the construction of the chiral e?ective Lagrangian, the choice of the ?avor-singlet currents (X ? ?X and Y ? ?Y in [11]) corresponds to the choice of (3) in the present formulation. As we stated before, the four-quark nature is generically derived from the symmetry breaking pattern (2) and the color-singletness of the order parameter. The diquark description like (3) is, however, based on the diquark model picture. Actually the four-quark order parameter can be given by other expressions once some Fierz transformation is applied to (3). In principle one cannot distinguish the CFL phase from the hadronic phase with exotic chiral symmetry breaking [21] only from the symmetry breaking pattern. Apparently σ is a counterpart of the ordinary sigma meson, σ ? qR qL + qL qR , which ? ? ? is regarded as a four-quark object as discussed in [16]. We must be cautious about this statement, however. It does not mean that σ is a bound state of two quarks and two holes ?
1

. We should rather understand that one diquark (dihole) only supplies the color charge

as a background compensating for the other dihole (diquark). Also it should be noted here that σ has an essential di?erence from σ. Under either ZL (2) or ZR (2) transformation, σ is ? ? invariant but σ is not. It should be noted that σ breaks the UA (1) symmetry up to Z(2) is a larger symmetry than σ leaves 2 . This di?erence is responsible for the inverse mass that amounts to only a part of unbroken UV (1). Therefore ZL(2) × ZR (2) in the CFL phase ordering of the NG bosons [11] and, in general, the attribute of higher dimensionful order
1

2

For convenience, we will simply call a state excited by the operator q a hole. It is actually mixed with an anti-quark whose contribution is negligible at high density and low temperature. When super?uidity occurs, the UV (1) symmetry is also broken up to Z(2) in addition to UA (1) → Z(2) due to the chiral condensate. The residual symmetry in this case is not Z(2) × Z(2) but only Z(2) and certainly smaller than ZL (2) × ZR (2) realized in the CFL phase.

5

parameters as discussed in [21]. Now that the order parameter is given by (3), the NG bosons can be read from the operator identity to de?ne the spontaneous symmetry breaking as formulated ?rst in [22]. Let Qa A be generators of the axial part of chiral symmetry, i.e., Qa = A
a dx jA?=0 =

dx q γ5 γ0 (λa /2)q. ?

By using the anti-commutation relation, {λa /2, λb/2} = δ ab /3+dabc λc /2, we can readily show i c iQa , π b = δ ab σ + idabc CA ? A ? 3 with π a = φ? ? Liα λa 2 φRjα ? φ? Riα λa 2 φLjα . (6)

ij

ij

(7)

In general, as explicitly seen in the Umezawa-Kamefuchi–Lehmann-K¨ll?n representation, a e we can prove that π a couples to a massless state if the expectation value of the right?
a jA? → (fπ ?0 χa , fπ v 2 ?i χa )+· · · asymptotically, where v is the velocity, χa is the (asymptotic) ? ?

hand-side of (6) is non-vanishing [22]. As a result, it follows that π a → Z 1/2 χa + · · · and ?

NG boson operator, and fπ Z 1/2 = σ/3 . This non-perturbative relation may be made use ? ? of to determine a non-perturbative value of fπ . [c.f. fπ = σ in the mean-?eld analysis of ? the linear sigma model.] The four-quark operators, π a , are thus the interpolating ?elds of ? the pions in the CFL phase. Equivalently, by using the Jacobi identity, one can intuitively regard the pions as ?uctuations around the order parameter, i.e., π a ? [iQa , σ]. ? A ? A similar description in terms of four-quark states in terms of diquarks (diquark–dihole or diquark–anti-diquark) has been argued in diquark models [23]. In contrast to the normal hadronic phase, however, we would emphasize that the four-quark nature in the CFL phase is rigid and there is no mixing with the quark–hole nor quark–anti-quark component. This is because of the unbroken ZL (2) × ZR (2) symmetry in the CFL phase. The “parity” under either ZL (2) or ZR (2) transformation becomes a good quantum number. The quark–hole or quark–anti-quark object has odd “parity”, while the diquark–dihole or diquark–anti-diquark object has even “parity”. Therefore, the dominant Fock-state of the pionic excitation in the CFL phase is considered as a four-quark state, apart from the instanton e?ect. Next let us consider about the phason, that is, the NG boson in connection with the super?uidity. We can construct color-singlet order parameters with a non-vanishing baryon number. The simplest choice is to form a color-singlet from three triplets. In this case the ?avor indices are more complicated than in the pion case, namely, 6 × 6 × 6 = 216 6

combinations. They can be reduced to 10 combinations by parity and anti-symmetric nature in color; H = HL ? HR = ?ijk ?αβγ φLiα φLjβ φLkγ ? φRiα φRjβ φRkγ , λ ?a H8 = ?αβγ ?ijk φLjα φLkβ 2 ? H1 = ?ijk ?αβγ φLiα φLjβ φRkγ ? φRiα φRjβ φLkγ ,
a il

(8) (9)

φRlγ ? ?ijk φRjα φRkβ

λ 2

a il

φLlγ .

(10)

In our notation the above expression of H gives positive parity since φL,R → ?φR,L under ? the parity transformation. In the CFL phase both H and H1 are SUV (3)-singlets and can ?a take a ?nite expectation value, while H8 = 0. A non-vanishing expectation value of H would break only the UV (1) and UA (1) symmetries up to ZL (2) × ZR (2). On the other hand, ? H1 is variant also under the axial part of chiral symmetry as well as the U(1) symmetries. A ? natural interpretation on H1 is a composite state made of σ and H. Therefore we will give ? ? ? little thought about H1 hereafter though H1 could be quali?ed as an order parameter. We would mention that one can directly prove that (TrX ? ?X)2 + (TrY ? ?Y )2 in [11] originates from (?HL )2 + (?HR )2 in the present formulation. The interpolating ?eld of the NG boson can be deduced in the same way as in the pion case. The generator of the UV (1) rotation is given by QV = (phason) [25]. We can easily show [iQV , H] = ?6iH. (11) dx j?=0 = dx q γ0 q. Then, ?

as in the familiar Goldstone model [24], the phase of the order parameter is the NG boson

Interestingly enough, the H ?eld de?ned in (8) has the same quark content as the Hdibaryon [26] as already pointed out in [10, 11, 27]. Let us clarify physical interpretation on the H ?eld. The H ?eld has two degrees of freedom. In the UV (1) broken phase the vacuum state is described by the coherent state of H, which is the superposition of many H and H ? states. [H is a complex scalar ?eld and H ? is its conjugate.] If we choose the real part, (H + H ? )/2, as a condensate, the phason, χH , is given by the imaginary part, i(H ? H ? )/2. The relation between the real and imaginary parts of H under the UV (1) transformation can be understood in analogy with the UA (1) rotation for σ ? qL qR + qR qL ? ? and η0 ? qL qR ? qR qL . It would be legitimate to regard both parts of H as particles just like ? ? there is no clear distinction for the H ?eld. 7 σ and η0 (which eventually becomes η ′ due to mixing with η8 when ms > mu ? md ), though

The velocity of χH has been calculated at zero temperature [11] and the zero temperature value, v 2 = 1/3, has been prevailing in model studies [14]. When it comes to the velocity of χH at T = 0, it is important to distinguish the collisionless regime from the hydrodynamic one. Let ω and τ be the frequency of χH and the characteristic collision time which is roughly of order 1/T . The zero temperature estimate is expected to work in the collisionless regime, ωτ ? 1, meaning H with energies much higher than T . The collisionless regime shrinks with increasing temperature. Then, the hydrodynamic description becomes more relevant in the hydrodynamic regime, ωτ ? 1. One might consider that χH is not a propagating (particle) mode but a di?usive mode in the hydrodynamic regime, for it looks like a density ?uctuation. This is not the case in the super?uid phase since the number density is no longer conserved. Actually the order parameter ?eld and the entropy density are mixed to lead to a propagating mode (second sound) in a super?uid [28, 29]. In a dilute super?uid system the nature of hydrodynamic modes have been investigated [29, 30, 31]. Then in the hydrodynamic regime it has been actually shown that the phason mainly corresponds to the second sound in the hydrodynamic regime. This consequence makes sense; the second sound only exists in the super?uid (symmetry-broken) phase, while the ?rst sound is present in both the super?uid and normal phases. As to the QCD context, due to the asymptotic freedom at high density, we can expect that the analyses in a dilute system are relevant to our problem. Once the χH propagation is identi?ed as the second sound, the velocity of χH in the hydrodynamic regime would be quite di?erent from the zero temperature value. In general a super?uid at ?nite temperature has a non-vanishing normal component whose density is ρn as well as a super?uid one whose density is ρs . The two-?uid dynamics gives the general expression for the second sound speed as c2 = ρs s2 T , ρn ρcv (12)

where ρ = ρn + ρs . The entropy density and the speci?c heat are denoted by s and cv √ respectively. The velocity, which is proportional to ρs , approaches zero as the system goes closer to the critical temperature. This observation is analogous to the pion velocity vanishing at the critical temperature [32] though the physics is di?erent. Whether we should consider in the collisionless or hydrodynamic regime depends on the energy of H and the temperature. The energy and temperature dependence of the velocity 8

of χH will be of importance, for example, in χH -involved processes in the proto-neutron star at temperatures of tens MeV [14].

III.

STRUCTURAL CHANGE

Now we shall consider how the quark-hadron continuity can be viewed from the quark content of the color-singlet condensates and the NG bosons. We will ?nd it important to recognize that the meaning of continuity is twofold; one is with respect to chiral symmetry and the other is related to the con?nement-Higgs crossover [33]. Their physical meanings are distinct. For the moment we will adopt a working de?nition; “hadronic matter” is used for the phase with σ = 0, and “CFL phase” for the phase with σ = 0. This is in accordance with the prevailing convention in the chiral Lagrangian approach [11, 27]. In hadronic matter pions are mainly composed of a quark (q) and an anti-quark (?) in the constituent quark q picture. As discussed some time ago [34] there can be a signi?cant mixture of q 2 q 2 even in the ? hadronic phase and we have not only σ = 0 but also σ = 0. In the CFL phase, i.e., in ? the phase having the ZL (2) × ZR (2) symmetry (apart from the axial anomaly), σ becomes zero and any q q component in the pions vanishes. The quark-hadron continuity is realized ? by a non-vanishing common ingredient of the pions, π a , made of a diquark and a dihole. ? The instanton e?ect, of course, breaks the UA (1) symmetry explicitly and blurs a sharp phase separation. The important point is, however, that we can have a de?nite limit where hadronic matter is distinguished from the CFL phase with respect to the pions 3 . In contrast, there remain more or less controversial points in the discussion about super?uidity. This is because hyper-nuclear matter has not been completely understood. It is expected at least that hyper-nuclear matter has a ?nite gap leading to a super?uid phase where UV (1) is broken into Z(2) 4 . There are 8 baryons forming an octet in the SUV (3)
3

4

The limit where ZL (2) × ZR (2) is present does not corresponds to the absence of instantons. Otherwise, not only UA (1) but also chiral symmetry is not broken. This limit should be understood in the same sense as the e?ective restoration of the UA (1) symmetry discussed in [35]. As stated in [36], the UB (1) symmetry generated by the baryon charge is di?erent from UV (1) discussed here. Since we always deal with color-singlet objects, however, the UV (1) rotation is equivalent with three (six, nine, etc) times UB (1) rotation, apart from the discrete symmetries. In this sense, though it is not

9

symmetry. The super?uid condensate is an SUV (3) singlet, which can be constructed by two baryon octets as ?N N ? (Σ0 )2 + (Λ0 )2 + Σ+ Σ? + Σ? Σ+ + pΞ? + Ξ? p + nΞ0 + Ξ0 n . (13)

Moreover we can also expect H = 0 (H is given by (8), i.e., a three diquark-like object) because it is not prohibited by the symmetry. As a matter of fact, the H-dibaryon state could be an admixture of two-nucleon-like and three-diquark-like con?gurations. This can be typically seen in the di?erent treatments, namely, the resonating group method (RGM) [37] and the MIT bag model [26]. Unlike the pion case, we have no de?nite limit where we can give a rigorous statement about the structural change of the super?uid component. We can only draw a qualitative picture in an intuitive picture. In the phase where the ZL (2) × ZR (2) symmetry is manifested, any nucleon made of 3 quarks cannot be present, though a nucleon pair can be. Thus it is likely that the nucleon-like condensate, ?N N , becomes less signi?cant in the CFL phase. It means that the overlap of the wave-function in the condensate becomes larger for H (probably decon?ned three-diquark condensate) then. This can be interpreted as a crossover from a con?ned phase to a Higgs phase. [Actually the major di?erence between the RGM and the MIT bag model comes from the way to impose the con?nement condition. The RGM is stricter in con?nement than the MIT bag model [37].] We would note that a quite similar picture (continuity between hyper-nuclear matter and partially decon?ned quark matter) has already been proposed some time ago in the context of H-matter in the neutron star [38]. In contrast to the pion case, the important message here is that the continuity with respect to the phason is inherently indistinguishable. We summarized our point of view in Table I and Fig. 1 schematically. In principle the color-superconductivity can be a di?erent phenomenon from the chiral phase transition, though we have considered chiral symmetry to characterize the phase. The situation is even similar to the relation between the decon?nement and chiral phase transitions at ?nite temperature [39]. There, the decon?nement transition is a crossover in the presence of dynamical quarks in the fundamental representation, while the chiral phase transition is well-de?ned in the chiral limit. In the present case, the liberation of color degrees of freedom corresponds to a crossover from hadronic matter to the color-superconducting
rigorously correct, we will use a sloppy notation and sometimes write UV (1) and ZV (2) to mean UB (1) and ZB (2) respectively.

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Hyper Nuclear Matter Chiral Symmetry σ ? qq = 0 ? σ ? q 2 q 2 ? small ? ? π ? q q + (?q )(qq) + · · · ? q? Super-?uidity ?N N = Σ2 + Λ2 + N Ξ = 0 H =0

CFL σ =0 σ =0 ? π ? (?q )(qq) + · · · ? q? ?N N ? small H =0

Z(2) ?→ ZV (2) (instanton) ←? ZL (2) × ZR (2)

χH ? 2 nucleons + 3 diquarks χH ? 3 diquarks con?nement–Higgs crossover TABLE I: Summary of qualitative changes from nuclear matter to CFL quark matter.

Hyper Nuclear Matter CFL _ none (apart from UA(1) breaking) qq pion __ __ q q qq qq qq qq _ _ q q qqq small qqq phason q q qq qq q q qq qq qq qq q q ?
BEC of colorless H BEC of colored qq BCS

FIG. 1: Schematic picture of the structural change and the two-step crossover.

phase where colored diquarks play an essential role. From the point of view of the NG bosons this can be seen as dissociation of the hadrons into constituent diquarks. As far as H is concerned, we can say in the following way; the hadronic phase has a Bose-Einstein condensate (BEC) of the color-singlet H-dibaryon, while the dissociated colored diquarks lead to a superconducting state at higher baryon density, and yet they compensate for their color charge to be a color-singlet in the CFL phase. In this sense, the attractive force between diquarks controls the state of matter. If the interaction is strong enough, the state is BEC-like, and otherwise, it is BCS-like. This BEC-BCS crossover looks quite di?erent 11

from that demonstrated in the 3 He super?uid [40], discussed in the system at ?nite isospin chemical potential [41], and investigated in the color-superconductivity [42]. In the present case, the BEC-BCS crossover is seen, not in the Cooper pair itself, but in the combination of two (for the pion) or three (for the phason) Cooper pairs. This is actually a crossover from the BEC of H to the BEC of diquarks. So far in our discussion we have implicitly assumed the existence of compact (tightly bound) diquarks. In the weak coupling regime diquarks become spatially spreading, which would cause another (and more familiar) crossover from the BEC of diquarks to the BCS. This type of the BEC-BCS crossover is well-known and have been intensely studied in condensed matter physics. Thus, as the baryon density increases, the system should undergo the two-step crossover; BEC of colorless H → color-superconductivity, and then BEC of colored diquarks → BCS. This scenario in sketched in Fig. 1.
IV. CONCLUSIONS AND AN OUTLOOK

In this paper, we discussed the quark description of the NG bosons in the CFL phase. We emphasized the four-quark nature of the pions stemming from the residual Z(2) symmetry and the color-singletness. We wrote down the plain expressions for the interpolating ?elds of the NG bosons motivated by the diquark picture. Then we discussed the H particle, that is the NG boson associated with the UB (1) symmetry breaking, in the hydrodynamic regime. We point out the velocity of the phason is given by the second sound speed which vanishes at the critical temperature. Finally we proposed a conjecture about the quarkhadron continuity. If the quark-hadron continuity is realized, the NG boson must change its nature continuously from one to the other phase. Based on the constituent diquark picture and the quark-hadron continuity hypothesis, we drew the two-step crossover scenario; BEC of H → BEC of diquarks, and then BEC of diquarks → BCS. This scenario can be con?rmed by developing the H matter description discussed in [38] or the tightly-bound diquark picture as discussed in [23]. Although the diquark model leading to too light H dibaryons may not be realistic in the vacuum, we can expect that it is a proper description at high baryon density where the color-superconductivity occurs. A three-body problem in terms of diquarks at ?nite density would be the next step to go further into quantitative investigations. As shown in Fig. 1 in 12

[23] the diquark interaction through the ’t Hooft term is a?ected by the chiral condensate. This may cause an entanglement between the chiral condensate and the diquark, in other words, between the pions and the phason. It would be intriguing to study how far the phase transition with respect to the pions can bring about the structural change of the phason. In this work the explicit breaking of the UA (1) symmetry has been regarded as an external perturbation smearing a sharp distinction based on chiral symmetry. It would be a challenging problem to study its e?ect not only on the pions but also on the super?uid structure. Since it is known that a strong three-body repulsion induced by the instanton e?ect makes the H-dibaryon weakly bound or unbound theoretically [43], the instanton-induced interaction will a?ect the content of the super?uid component. Our speculation implies that diquarks would become important for the hadronic phase if it is close to CFL quark matter at high baryon density. This must be a robust consequence even in the presence of ?nite ms as long as there is no ?rst order phase transition. The nature of diquarks in the hadronic phase deserves further investigation not only in the vacuum [44] but rather at high baryon density. This is partially because the importance of diquarks would be seen once the chiral condensate is vanishingly small. A possible suggestion is that, near the chiral phase transition at high temperature where the chiral condensate melts, the diquark correlation can be relevant to thermodynamic quantities, as has been already pointed out [45]. In order to make our argument applicable, for instance, to neutron star physics, it is necessary to take account of ?nite ms and the neutrality conditions. The CFL phase is robust as long as m2 /? < 2?CFL [46]. As discussed in [17], if ms is large enough to suppress s the hyperon number density, the transition from ordinary nuclear matter to the CFL phase is discontinuous. Detailed calculations, however, suggest the presence of H matter in the neutron star [38], though further analyses are needed to reveal the actual properties of H matter in the cores of compact stellar objects. Since the diquark picture tends to give rise to tightly-bound H-dibaryon, we may well think that the calculation with diquarks taken into account would result in the existence of H matter and our discussion here is not altered qualitatively. If our scenario is realized, then it would be quite interesting to see how the nature of H (or χH ) a?ects the internal structure of the vortices in a super?uid along the lines of [47]. We believe that our pictorial understanding could shed light on the non-perturbative 13

region between hadronic and quark matter and cast novel and challenging problems.

Acknowledgments

The author thanks K. Rajagopal for critical comments.

He is also grateful to

M. M. Forbes, E. Gubankova and O. Schr¨der for conversation. He would like to acknowledge o T. Kunihiro and M. A. Stephanov for useful comments. This work was partially supported by Japan Society for the Promotion of Science for Young Scientists and the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818.

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