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Abnormal number of Nambu-Goldstone bosons in the color-asymmetric 2SC phase of an NJL-type model

D. Blaschke?

Fachbereich Physik, Universit¨ at Rostock, D-18051 Rostock, Germany Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia

D. Ebert?

Institut f¨ ur Physik, Humboldt-Universit¨ at zu Berlin, D-12489 Berlin, Germany

K. G. Klimenko?

Institute of High Energy Physics, 142281 Protvino, Moscow Region, Russia

arXiv:hep-ph/0403151v1 14 Mar 2004

M. K. Volkov, and V. L. Yudichev§

Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia We consider an extended Nambu–Jona-Lasinio model including both (q q ?)- and (qq )-interactions with two light-quark ?avors in the presence of a single (quark density) chemical potential. In the color superconducting phase of the quark matter the color SUc (3) symmetry is spontaneously broken down to SUc (2). If the usual counting of Goldstone bosons would apply, ?ve Nambu–Goldstone (NG) bosons corresponding to the ?ve broken color generators should appear in the mass spectrum. Unlike that expectation, we ?nd only three gapless diquark excitations of quark matter. One of them is an SUc (2)-singlet, the remaining two form an SUc (2)-(anti)doublet and have a quadratic dispersion law in the small momentum limit. These results are in agreement with the Nielsen–Chadha theorem, according to which NG-bosons in Lorentz-noninvariant systems, having a quadratic dispersion law, must be counted di?erently. The origin of the abnormal number of NG-bosons is shown to be related to a nonvanishing expectation value of the color charge operator Q8 re?ecting the lack of color neutrality of the ground state. Finally, by requiring color neutrality, two massive diquarks are argued to become massless, resulting in a normal number of ?ve NG-bosons with usual linear dispersion laws.

PACS numbers: 11.30.Qc, 12.39.-x, 21.65.+f Keywords: Nambu–Jona-Lasinio model; Color superconductivity; Nambu–Goldstone bosons

I.

INTRODUCTION

It is well known that, in accordance with the Goldstone theorem [1, 2], N Nambu–Goldstone (NG) bosons appear in Lorentz-invariant systems if an internal continuous symmetry group G is spontaneously broken down to a subgroup H (here N is the number of generators in the coset space G /H), i. e. the number of NG-modes is equal to the number of broken generators. However, in Lorentz-noninvariant systems the number of NG-bosons can be less than N . In this case, the counting of NG-bosons is regulated by the Nielsen–Chadha (NC) theorem [3]: Let n1 and n2 be the numbers of gapless excitations that in the limit of long wavelengths have the dispersion laws E ? |p| and E ? |p|2 , respectively, then N ≤ n1 + 2n2 . (Here, E is the energy and p is the three-momentum of the particle.) In particular, this theorem is valid for relativistically covariant theories as well, since in this case: i) the total number of NG-bosons equals N , the number of broken symmetry generators [2]; ii) evidently, the dispersion law for these N massless excitations looks like E ? |p|, thus N = n1 . Recently, in some relativistic models describing the dynamics of the kaon condensate in the color-?avor-locked phase of dense quark matter, an abnormal number of NG-bosons has been revealed [4, 5]. Since the Lorentz invariance is broken in this case and some of the gapless excitations have a quadratic dispersion law, there are no contradictions with neither Goldstone nor NC theorems. The super?uid 3 He in the A-phase [6] and ferromagnets [7, 8] are other known examples of condensed-matter systems with an abnormal number of NG-bosons. In the present paper, we demonstrate the abnormal number of NG-bosons in the dense color superconducting phase (2SC) of quark matter for a simple version of the Nambu–Jona-Lasinio (NJL) model with two light quarks and a single (quark number) chemical potential. In this phase, which can be realized naturally only at su?ciently large

address: address: ? Electronic address: § Electronic address:

? Electronic

? Electronic

david.blaschke@physik.uni-rostock.de debert@physik.hu-berlin.de kklim@mx.ihep.su yudichev@thsun1.jinr.ru

2 values of the chemical potential (300 MeV≤ ? <1 GeV) , the initial color SUc (3) symmetry is spontaneously broken down to the SUc (2) group. Hence, in accordance with the usual counting of the Goldstone theorem, one might expect ?ve NG-bosons, corresponding to the ?ve broken symmetry generators, to appear in the NJL model. We shall prove that in the 2SC phase only three gapless excitations that can be identi?ed with NG-bosons appear. Two of them have a quadratic dispersion law, thus yielding no contradiction with the NC theorem. For further applications, notice also the following important criterion which is su?cient for the equality between the number of NG-bosons and the number of broken generators [5]: If Qi , i = 1, ..., N is the full set of broken generators and if [Qi , Qj ] = 0 for any pair (i, j ), then the number of NG-bosons is equal to the number N of broken generators. Recall, that there is an alternative approach to study color superconductivity, which is based on the weak coupling perturbative QCD [9]. In this case, using the Schwinger-Dyson equation with one gluon exchange, the color superconducting phase was proved to exist at asymptotically high densities. In the two-?avored QCD investigations of color superconductivity ?ve scalar NG-bosons are shown to exist, which are mixing with gluons and required as longitudinal components for ?ve massive gluons (color Meissner e?ect) [10]. Hence, the results concerning the number of NG-bosons of both the QCD and the NJL model approaches seem to be in contradiction1 . The origin of the above discrepancy is related to the fact that the ground state of the considered simple NJL–type model is not color-neutral. Indeed, since the expectation value of the color charge operator Q8 is non-vanishing, Q8 = 0, the criterion [5] is not applicable, and an abnormal number of NG-bosons must arise. On the other hand, in quark models with a color-neutral ground state, according to the above criterion, there should arise a normal number of ?ve NG-bosons all having normal (linear) dispersion laws. Most interestingly, we ?nd from the analysis of the mass spectrum in our model that the masses of two scalar diquarks are proportional to the ground state expectation value of the color charge Q8 . Thus, when color neutrality, i. e. Q8 = 0, is imposed in NJL–type models as an additional condition (which can be realized by introducing a color-chemical potential, ?8 , ”by hand” [11]), then two additional massless particles should appear, and the total number of NG-bosons would be equal to ?ve. Recently, it was shown that the ground state of the 2SC phase of QCD is automatically color-neutral [12] due to a dynamical generation of the color chemical potential ?8 by gluon condensation. So, we see that the original discrepancy in the number of NG-bosons between the QCD approach and an extended NJL approach would disappear. The paper is organized as follows. In Section II we investigate the considered NJL–type model in the framework of the Nambu–Gor’kov formalism and derive an e?ective meson–diquark action for a ?nite chemical potential. Moreover, the gap equations for the quark and diquark condensates are numerically studied. Sections III and IV contain a detailed analysis of the massless and massive excitations in the diquark sectors and explicitly show the in?uence of the color properties of the ground state on the diquark mass spectrum. Section V contains a summary and discussions. Finally, dispersion laws for gapless diquarks are derived in Appendix.

II. THE MODEL AND ITS EFFECTIVE ACTION

It is well known that perturbative methods are not applicable in the low-energy/density QCD regions. Instead, e?ective ?eld theories are usually considered. The most popular e?ective theories are based on Lagrangians with four-fermion interactions, like the NJL-type models. Let us ?rst give several (very approximate) arguments somehow justifying the chosen structure of our QCD-motivated NJL model introduced below. For this aim, consider two-?avor QCD with a nonzero chemical potential and the color group SUc (3). By integrating in the generating functional of QCD over gluons and further “approximating” the nonperturbative gluon propagator by a δ ?function, one arrives at an e?ective local chiral four-quark interaction of the NJL type describing low-energy hadron physics. Finally, by performing a Fierz transformation of the interaction term and taking into account only scalar and pseudo-scalar (? q q )as well as scalar (qq )-interaction channels, one obtains a four-fermionic model given by the following Lagrangian (in Minkowski space-time notation)2 L=q ?[γ ν i?ν + ?γ 0 ? mo ]q + G1 [(? q q )2 + (? q iγ 5 τ q )2 ] + G2 [? q C ?δ γ 5 τ 2 q ][? q ?δ γ 5 τ 2 q C ]. (1)

In (1), ? ≥ 0 is the quark chemical potential, which in isotopically symmetric quark matter is the same for both quark ?avors, q C = C q ?t , q ?C = q t C are charge-conjugated spinors, and C = iγ 2 γ 0 is the charge conjugation matrix (the symbol t denotes the transposition operation). The quark ?eld q ≡ qiα is a ?avor doublet and color triplet as well as a four-component Dirac spinor, where i = 1, 2; α = 1, 2, 3. (Latin and Greek indices refer to ?avor and color indices, respectively; spinor indices are omitted.) Furthermore, we use the notations τ ≡ (τ 1 , τ 2 , τ 3 ) for Pauli matrices in ?avor space; (?δ )αβ ≡ ?αβδ is the totally antisymmetric tensor in color space, respectively. Clearly, Lagrangian

1

2

Of course, it is necessary to point out that the QCD weak coupling considerations are valid at asymptotic values of the chemical potential, i. e. at ? > 1 GeV, whereas NJL model results are correct for su?ciently lower values ? < 1 GeV. Thus, a direct comparison of results seems to be problematic. The most general Fierz transformed four-fermion interaction includes additional vector and axial-vector (? q q ) as well as pseudo-scalar, vector and axial-vector-like (qq )-interactions. However, these terms are omitted here for simplicity.

3 (1) is invariant under the chiral SU (2)L × SU (2)R (at mo = 0) and color SUc (3) symmetry groups. The physics of light mesons [13, 14, 15, 16], diquarks [17, 18] and meson-baryon interactions [19, 20] was successfully described in the framework of di?erent NJL models. These e?ective theories were involved for the investigation of both ordinary [21, 22, 23], hot and dilute [15], and color superconducting dense quark matter [24, 25, 26, 27]. Usually, on the basis of light-meson and baryon phenomenology, the following restrictions on the coupling constants are considered G1 > π 2 /(6Λ2 ), G2 < G1 , (2)

where Λ is the cuto? parameter in the three-dimensional momentum space, necessary to eliminate the ultraviolet divergences appearing when quantum e?ects (loops) are taken into account (usually Λ < 1 GeV). Since (1) is a low-energy e?ective theory for QCD, the external parameter ? is restricted from the above: ? < Λ. In the region of coupling constants (2) and at small values of ? (the case of low baryon densities), only the operator q ?q has a nonzero vacuum expectation value. Thus, the chiral symmetry of the model is spontaneously broken in this case, while SUc (3) remains intact. The behavior of quark and meson masses is established rather well in quark matter of low density [21, 22]. In particular, the number of π -mesons, which are the three NG-bosons in this phase, is equal to the number of broken symmetry generators. Hence, this example is a good demonstration of the fact that Lorentz noninvariance is a necessary but not su?cient condition for the abnormal number of NG-bosons to appear in the system. At su?ciently high values of the chemical potential ? ? 300 ÷ 350 MeV, the two-?avor color superconductivity phase occurs in model (1) [26]. So, a nonzero diquark condensate ( qq = 0) is formed in this case, and, as a consequence, color symmetry is spontaneously broken down to the SUc (2) subgroup. A naive counting gives us ?ve NG-bosons in this case (it is just the number of broken generators of the SUc (3) group). However, as it will be shown further, there are only three gapless bosonic excitations of the 2SC quark-matter ground state. Two of them satisfy the quadratic dispersion law, in agreement with the NC theorem. The linearized version of Lagrangian (1) that contains auxiliary bosonic ?elds has the following form σ2 + π2 ? = q L ?[γ ν i?ν + ?γ 0 ? σ ? mo ? iγ 5 τ π ]q ? ? 4G1 i??δ C δ 5 2 i?δ δ 5 2 C ??δ ?δ + [? q ? γ τ q] ? [? q ? γ τ q ]. ? 4G2 2 2 Lagrangians (1) and (3) are equivalent on the equations of motion for bosonic ?elds, from which it follows that ?δ ? q ?C ?δ γ 5 q, σ?q ?q, π ? iq ?γ 5 τ q. (4)

(3)

However, it is more convenient to start our consideration from Lagrangian (3). Clearly, the σ and π ?elds are color singlets. Besides, the (bosonic) diquark ?eld ?δ is a color antitriplet and a (isoscalar) singlet under the chiral SU (2)L × SU (2)R group. Note further that σ and ?δ are (Lorentz) scalars, but π are pseudoscalar ?elds. Hence, if σ = 0, then the chiral symmetry of the model is spontaneously broken (at mo = 0), whereas ?δ = 0 indicates the dynamical breaking of color symmetry. In the framework of the Nambu–Gor’kov formalism (for a recent relativistic . Integrating in the generating treatment see, e.g. ref. [28]) quark ?elds are represented by a bispinor Ψ = qq C functional based on the Lagrangian (3) over the quark ?elds, one obtains an e?ective meson-diquark action of the original model (1) Se? (σ, π, ?δ , ??δ ) = ? d4 x σ2 + π2 i ?δ ??δ ? Trsf cx ln + 4G1 4G2 2 D+ K ? K + D? . (5)

Besides of an evident trace over the two-dimensional Nambu–Gor’kov matrix, the Tr-operation in (5) stands for calculating the trace in spinor- (s), ?avor- (f ), color- (c) as well as four-dimensional coordinate- (x) spaces, correspondingly. We have used also the following notations: K + = i??δ ?δ γ 5 , D = iγ ?ν ± ?γ 0 ? mo ? Σ± ,

± ν

K ? = ?i?δ ?δ γ 5 , Σ± = σ ± iγ 5 πτ .

(6)

Let us introduce notations for the ground state expectation values of the meson and diquark ?elds: σ = σo , π = πo , δ δ ??δ = ?? = ?δ o , ? o and

± ± ± ± ± (Ko , Do , Σ± o ) = (K , D , Σ ) σ=σo ,π =πo ,?δ =?δ o ,... k ?δ The quantities σo , πo , ?δ o , ?o (k, δ = 1, 2, 3) correspond to the global minimum of the e?ective potential Ve? and can be found as a solution of the system of equations

?Ve? = 0, ?π k

?Ve? = 0, ?σ

?Ve? = 0, ? ?δ

?Ve? = 0, ? ??δ

(7)

4 usually called gap equations. In (7) we used the following de?nition of the e?ective potential: Se? = ?Ve? (σ, π, ?δ , ?δ ) d4 x, (8)

σ,π,?δ ,?δ =const

where, in the spirit of the mean-?eld approximation, all ?elds are considered to be independent of x. Without loosing 2 3 any generality of the consideration, one can search for a solution of equations (7) in the form: (?1 o , ?o , ?o )≡ (0, 0, ?). Moreover, we suppose also that πo = 0. (The last assumption is maintained by the observation that in the theory of strong interactions P-parity is a conserved quantity.) If ? = 0, the color symmetry of the model remains intact. If ? = 0, then SUc (3) symmetry is spontaneously broken down to SUc (2), and the 2SC phase is realized in the model. In this case, the system of gap equations (7) is reduced to the following one: d4 q E+ E? 2E + 2E ? σo = 4iM + + + 2 ? (E + )2 2 ? (E ? )2 2G1 (2π )4 E q0 q0 D+ (q0 ) D? (q0 ) 4 ? d q 1 1 , = 4 i? + 4G2 (2π )4 D+ (q0 ) D? (q0 ) , (9) (10)

2 where D± (q0 ) = q0 ? (E ± )2 ? |?|2 , E ± = E ± ?, E = q 2 + M 2 , and M = mo + σo is the constituent quark mass. In these and other similar expressions, q0 is a shorthand notation for q0 + iε · sgn(q0 ), where the limit ε → 0+ must be taken at the end of all calculations. This prescription correctly implements the role of ? as chemical potential and preserves the causality of the theory (see, e. g. [29]). δ δ δ δ Let us now make a ?eld shift σ (x) → σ (x) + σo , ??δ (x) → ??δ (x) + ?? o , ? (x) → ? (x) + ?o and then expand (2) the expression (5) into a power series of the meson and diquark ?elds. The second-order term Se? of this expansion is responsible for the mass spectrum of mesons and diquarks. It looks like

Se? (σ, π, ?δ , ??δ ) = ?

(2)

d4 x

σ2 + π2 ?δ ??δ + + 4G1 4G2 Σ+ , ? K ? ? K + , Σ?

?1

i + Trsf cx S0 4

S0

Σ+ , ? K ? ? K + , Σ?

,

(11)

where S0 is the quark propagator, which is an evident 2×2 matrix in the Nambu–Gor’kov space S0 =

+ ? Do Ko + ? K o Do

≡

a b c d

,

(12)

whose matrix elements can be found by means of the projection operator technique [30]: a= d4 q ?iq(x?y) q0 ? E + q0 + E ? 0? 0? e γ Λ + + 2 ? (E + )2 + |?|2 ?3 ?3 2 ? (E ? )2 + |?|2 ?3 ?3 γ Λ? , (2π )4 q0 q0 1 1 d4 q ?iq(x?y) ?? + ?+ , e γ5Λ γ5Λ b = ?i??3 2 2 4 + 2 2 3 3 ? 2 (2π ) q0 ? (E ) + |?| ? ? q0 ? (E ) + |?|2 ?3 ?3 d4 q ?iq(x?y) 1 1 ?+ + ?? , c = i?? ?3 e γ5Λ γ5Λ 2 2 4 + 2 2 3 3 ? 2 (2π ) q0 ? (E ) + |?| ? ? q0 ? (E ) + |?|2 ?3 ?3 d4 q ?iq(x?y) q0 + E + q0 ? E ? 0? 0? d= e γ Λ + ? 2 ? (E + )2 + |?|2 ?3 ?3 2 ? (E ? )2 + |?|2 ?3 ?3 γ Λ+ , (2π )4 q0 q0 1±

γ 0 (γp?M ) E

(13) (14) (15) (16)

?± = where Λ

3 3

1 2

are projectors on the solutions of the Dirac equation with positive/negative energy,

? ? = diag(?1, ?1, 0) is the projector (up to a sign) in the color space. Quantities (13)–(16) are nontrivial operators in the coordinate, spinor, and color spaces, but they are unit operators in ?avor space. (2) A tedious but straightforward analysis of Se? shows that there arises a mixing between the σ and ?3 ?elds. We have

(2) Se? (σ, π, ?δ , ??δ ) = Sππ (π ) + Sσ3 (σ, ?3 , ??3 ) + S1 (?1 , ??1 ) + S2 (?2 , ??2 ), (2) (2) (2) (2)

(17)

where (ρ = 1, 2)

(2) Sρ (?ρ , ??ρ ) = ? d4 x

i ?ρ ??ρ + Trsf cx a?ρ ?ρ γ 5 d??ρ ?ρ γ 5 . 4G2 2

(18)

5 In the following we are going to search for gapless bosonic excitations of the 2SC quark-matter ground state of the model only. Since we are mainly interested in the diquark sector, the expression for the π -meson e?ective action (2) (2) Sππ (π ) is omitted here. Moreover, Sσ3 (σ, ?3 , ??3 ) has a rather cumbersome form, so we also do not show it. (If needed, both these expressions can be obtained immediately from (11).) Before proceeding further, we should ?x the parameters of our model. Here, we choose the parametrization procedure given in [14]. We simplify that model by ignoring the corrections to the pion-quark coupling constant that come from the mixing between the pion and axial-vector. To ?x the model parameters, we require the model to reproduce the Goldberger-Treiman relation M gπ = Fπ , where Fπ ≈ 93 MeV is the pion weak-decay constant, and gπ is the pion-quark coupling constant determined in the local NJL model as follows

?2 gπ =

?iNc (2π )4

(q 2

d4 q . ? M 2 )2

(19)

(This integral is divergent, and a regularization is supposed to be implemented here. In our work we use a threedimensional cuto? to make such integrals meaningful.) Then, we require the observed pion mass (? 140 MeV) to be reproduced by the local NJL model with our parameters. After this, one more condition is needed to complete the parameter ?xing procedure. We consider two possibilities to ?nalize it: i) ?x the model parameters so that the QCD sum rules estimate for the chiral condensate q ?q ≈ (?245 MeV)3 is reproduced in our model, ii) choose the model parameters in a way that allows one to obtain the correct description of the ρ → ππ decay, as it was done in [14]. In the ?rst case (let us refer to this set of parameters as set-A), the cuto? Λ is about 618 MeV, the four-quark interaction constant G1 is equal to 5.86 GeV?2 . The remaining constant, G2 , is chosen according to the Fierz-transformation as G2 = 3G1 /4 = 4.40 GeV ?2 . The current quark mass mo is ?xed via the gap equations to the value: 5.7 MeV. The constituent quark mass in the vacuum (at zeroth temperature and chemical potential) amounts to 350 MeV. In the second case, we obtain a di?erent set of model parameters, which will be referred to, throughout the paper, as the set-B. Imposing the condition that an extension of this model to the vector channel would give the experimental value for the ρ-meson width (the ρ-meson decays mostly into a couple of charged pions, and the decay is described by the constant gρ ≈ 6.1.) and keeping in mind that in √the local NJL model one gets a natural connection between gρ and gπ (discarding pion–axial-vector transitions), 6gρ = gπ , we obtain Λ = 856 MeV, G1 = 2.49 GeV?2 , G2 = 3G1 /4 = 1.87 GeV?2 , and m0 = 3.6 MeV. In the vacuum the constituent quark mass is 233 MeV, which is small, compared to the case, where the chiral condensate is used to ?x the model parameters3. A similar procedure has been implemented in [27] in the chiral limit. Here, we keep the current quark mass non-zero in order to reproduce the pion mass in the vacuum. Solving the gap equations at a ?xed chemical potential, we obtain the constituent quark mass M and the color gap ? that satisfy the global minimum of the e?ective potential. In Figs. 1 and 2, one can see the quark mass M and the color gap ? vs. the chemical potential for parameter sets A and B. For the parameter set A, one observes a ?rst-order phase transition from hadron matter to the 2SC phase. The gaps (constituent quark mass and color gap) change abruptly when the chemical potential reaches the vacuum value of M (see Fig. 1). On the contrary, if the parameter set-B is chosen, the gaps reveal a rather smooth character near the phase transition, which is typical for a phase transition of the second-order (see Fig. 2).

III. GAPLESS EXCITATIONS IN THE DIQUARK SECTORS

Since ?1 in (17) is not mixed with other ?elds, let us, ?rst of all, focus on the diquark ?1 -sector. If ?1 (x) = √ √ (2) (?1 (x)+ i?2 (x))/ 2, ??1 (x) = (?1 (x) ? i?2 (x))/ 2, then the e?ective action S1 (?1 , ??1 ) from (17) can be presented in the form: S1 (?1 , ??1 ) ≡ S1 (?1 , ?2 ) = ?

(2) (2)

1 2

d4 x d4 y?k (x)Γkl (x ? y )?l (y ),

(20)

where the 2 × 2 matrix Γ(x ? y ) is the inverse propagator of the ?1 , ?2 -?elds (summation over k, l = 1, 2 is implied (2) in (20)). Its matrix elements can be found via a second variation of S1 : Γkl (x ? y ) = ? δ 2 S1 . δ?l (y )δ?k (x)

(2)

(21)

3

A larger quark mass can be obtained if one takes into account the mixing between the pion and axial-vector. However, we omit it in our paper for simplicity and for the reason that this mixing is negligible in the 2SC phase.

6

350 300 250

M, ? [MeV]

200 150 100 50 0 0 100 200 300 400

? [MeV]

FIG. 1: The constituent quark mass M (solid line) and the color gap ? (dashed line) as functions of the chemical potential (parameter set-A).

250

200

M, ? [MeV]

150

100

50

0 0 100 200 300 400

? [MeV]

FIG. 2: The constituent quark mass M (solid line) and the color gap ? (dashed line) as functions of the chemical potential (parameter set-B).

7 In momentum space, the Fourier-transformed components of the Γ-matrix have the following structure Γ11 (p) = Γ22 (p) = 1 Γ?? ? (p) + Γ?? ? (?p) , 2 Γ12 (p) = ?Γ21 (p) = i Γ?? ? (p) ? Γ?? ? (?p) . 2 (22)

The derivation of these relations is given in Appendix, where the quantity Γ?? ? (p) is also presented (see eq. (A5)). The particle dispersion laws are de?ned by zeros of detΓ(p) in the p0 -plane, i. e. by the equation: detΓ(p) = Γ11 (p)Γ22 (p) ? Γ12 (p)Γ21 (p) = Γ?? ? (p)Γ?? ? (?p) = 0. (23)

Clearly, the solutions of (23) with positive/negative p0 correspond to particles/antiparticles. Let us ?rst put p = (p0 , 0, 0, 0). In this case, detΓ(p0 ) is an even function of p0 , so the solutions of eq. (23) in the p2 0 -plane might be searched for. They are the squared masses of excited states in the ?1 -sector of the model. Moreover, all these formulae are then greatly simpli?ed. Indeed, it follows from (A5) at p 2 = 0 that Γ?? ? (p0 ) = 4ip0 d4 q (2π )4 1 1 + + (p0 + q0 + E )D+ (q0 ) (p0 + q0 ? E ? )D? (q0 ) ≡ ?4p0 H (p0 ), (24)

where it is possible to integrate over q0 , using the following prescription: (p0 + q0 ) → (p0 + q0 ) + iε · sgn(p0 + q0 ) and q0 → q0 + iε · sgn(q0 ) with ε → 0+ (see also comments after formula (10)), and H (p0 ) = ? 1 2 d3 q (2π )3 1 θ(E ? ) θ(?E ? ) + + + + ? ? ? ? (p0 + E + + E? )E? (p0 ? E ? ? E? )E? (p0 ? E ? + E? )E? . (25)

± In (25) E? = (E ± )2 + |?|2 . Since the integral in the RHS of this equation is ultraviolet divergent, we regularize it, as the other divergent integrals, by using a three-dimensional cuto? Λ, i. e. q 2 ≤ Λ2 . Then, eq. (23) is transformed to

p2 0 H (p0 )H (?p0 ) = 0,

(26)

from which it is evident that there is a solution p2 0 = 0, corresponding to a gapless excitation of the 2SC ground state. Since the function H (p0 ) does not vanish at p0 = 0 and ? = 0, this solution may be identi?ed in the case of ?nite p2 with a NG-boson with the quadratic dispersion law: p0 ? p 2 /H (0) (see Appendix). Now, let us suppose that at some nonzero value p0 = ?m1 the function H (p0 ) has a zero, i. e. H (?m1 ) = 0. Then, detΓ(p0 ) has two zeroes p0 = ±m1 , 2 i. e. the point p2 0 = m1 is a solution of eq. (26), and the second bosonic excitation of this sector has the nonzero mass m1 . In the low-p0 expansion we have: H (p0 ) = H (0) + p0 H ′ (0) + · · ·, where H (0) = H ′ (0) = 1 2 1 2 d3 q (2π )3 d3 q (2π )3 θ(E ? ) θ(?E ? ) 1 + ? ? ? ? ? + + (E ? + E? )E? (E ? ? E? )E? (E + + E? )E? , . (27)

θ(E ? ) θ(?E ? ) 1 + ? ? 2 ? 2 ? + + + 2 ? ? + E? (E + E? ) (E ? E? ) E? E? (E + E? )

Thus, in this approximation m1 = H (0)/H ′ (0). The m1 vs. ? plots for particular values of the coupling constants G1 , G2 and the cuto? parameter Λ discussed at the end of section 2 (or, equivalently, for the values ? and M from Figs. 1 and 2) are presented in Figs. 3 and 4. In Fig. 3 the values of m1 are calculated for ? = 350 ÷ 400 MeV when the parameter set-A is involved. The leftmost point corresponds to the condition of the phase transition (? ≈ 350 MeV). At smaller ? the color gap equals zero and eqs. (27) cannot be applied, since they have been derived by using eq. (10) with ? = 0. Similarly, in Fig. 4 the axis ? begins with 250 MeV. At smaller ?, (but grater than 233 MeV), the value of m1 is almost zero. Common to both parameter sets A and B is that the values of m1 are small and thus justify the series expansion of the function H (p0 ) around p0 = 0.4 Analogous results are obtained for the ?2 -sector of the model. There, as in the ?1 -sector, only one massless boson with the quadratic dispersion law as well as a massive one with the mass m2 ≡ m1 are found. Evidently, the two massless and two massive bosons in the ?1 , ?2 -sectors form antidoublets with respect to the unbroken SUc (2) symmetry. Now let us consider the sector with mixing of the complex diquark ?3 -?eld and the scalar σ -meson. Introducing √ √ (2) new real ?elds: φ(x) = (?3 (x) + ??3 (x))/ 2, ψ (x) = i(??3 (x) ? ?3 (x))/ 2, we rewrite the e?ective action Sσ3 (here we omit the cumbersome expression for this quantity, however it can be easily reproduced from (11)) in the following form: Sσ3 (σ, ?3 , ??3 ) = ?

(2)

1 2

d4 x d4 y (σ (x), φ(x), ψ (x)) Π(x ? y ) (σ (y ), φ(y ), ψ (y ))t ,

(28)

4

The numerical investigation shows that the mass value m1 found in the low-p0 approximation (see Figs. 3 and 4) di?ers from the exact solution of the equation H (m1 ) = 0 by no more than 10%.

8

30

m 1 [MeV]

20

10

0 350 375 400

? [MeV]

FIG. 3: The diquark mass m1 = H (0)/H ′ (0) as a function of the chemical potential (parameter set-A).

35

30

25

[MeV] m1

20

15

10

5

0 250 300 350 400

? [MeV]

FIG. 4: The diquark mass m1 = H (0)/H ′ (0) as a function of the chemical potential (parameter set-B).

9 where Π(x ? y ) is the inverse propagator of the σ -meson and the diquark ?elds φ, ψ . Clearly, it is a 3×3 matrix, (2) whose elements can be obtained from (28) by taking all second variational derivatives of Sσ3 over the ?elds σ, φ, and ψ . We have calculated the momentum-space matrix Π(p) (corresponding explicit formulae and relations are omitted), and then, putting p = (p0 , 0, 0, 0), found that det(Π(p0 )) has only one zero at p2 0 = 0. Therefore, only one NG-boson, which is an SUc (2)-singlet, exists in this sector (as in the ?1 , ?2 -sectors, each zero of det(Π(p0 )) in the p2 0 -plane means the mass squared of the ground-state excitation, this time in the mixed σ -?3 sector). 5 As it was mentioned above, in the 2SC phase of the model, where ? = 0, the color SUc (3) symmetry of the ground state is spontaneously broken down to the SUc (2) group. Hence, in accordance with the Goldstone theorem, it is generally expected that ?ve NG-bosons would appear in the theory. However, we have proved that in the ?1 , ?2 , ?3 sectors of the model only three massless bosons exist, which can be identi?ed with NG-bosons. Therefore, there seems to be a de?ciency of two NG-bosons in the framework of the model under consideration. In spite of this fact, there are, however, no contradictions with the NC theorem, since two of the three found NG-bosons have quadratic dispersion laws for |p| → 0. In addition, the model contains also light massive diquarks. Usually, in the framework of the NJL model, the 2SC phase is studied in the region of coupling constants (2). In this case, at ? = 0 we have a chirally noninvariant phase even if mo → 0, and the transition to the 2SC phase occurs at some ?nite value of the chemical potential. Formally, however, one can consider the region ω = {(G1 , G2 ) : G2 > π 2 /(4Λ2 ), π 2 (G2 ? G1 ) > 2G1 G2 Λ2 }, (29)

where (even at ? = 0) the 2SC phase is realized [31]. As can easily be seen from (27), in the case of ? = 0 we have H (0) = 0 and m1 = m2 = 0, and the total number of gapless excitations (NG-bosons) in the ?1 - and ?2 -sectors of the model equals four. They form two massless SUc (2)-antidoublets. Apart from this, there is one NG-boson, which is an SUc (2)-singlet, in the mixed σ -?3 sector. Naturally, all these excitations have a linear dispersion law, as it should be in the relativistically invariant case (see Appendix). However, at an arbitrary small ?, particles from one of these antidoublets acquire nonzero masses, and the dispersion laws for the two massless particles from the remaining antidoublet are changed essentially from the linear to quadratic laws. Note ?nally, that in the above consideration we used the real/imaginary part parametrization for diquark ?elds: √ √ √ ?1 (x) = (?1 (x) + i?2 (x))/ 2, ?2 (x) = (? ?1 (x) + i? ?2 (x))/ 2, ?3 (x) ? ? = (φ(x) + iψ (x))/ 2, (30) where ?ρ (x) are the unshifted diquark ?elds from Lagrangian (3) and ? = ?3 . However, one can use also an alternative ”polar coordinate” parametrization: ? ? ? ? ? 1 ? ξ5 ? iξ4 0 ? (x) t 1 1 λ ξ ( x ) a a ? + o(ξi , η ), ? = √ ? √ ξ7 ? iξ6 ? ?2 (x) ? = exp ?i 0 √ √ ?√ (31) 2 3 2? 2 2 √ 2? + η + i 3 ξ8 ? (x) a 2? + η (x)

where λa are Gell-Mann matrices and the summation over a = 4, .., 8 is implied (the form of an exponential multiplier in (31) corresponds to the fact that ?ρ (x) is an SUc (3)-antitriplet). Comparing (30) and (31) at small ξi , η , we see that 2 ?1 = ξ5 , ?2 = ?ξ4 , ? ?1 = ξ7 , ? ?2 = ?ξ6 , φ = η, ψ = √ ξ8 . 3

(32)

Further, one should insert (31) into (5) and expand the resulting expression into a series of meson- and ξa , η ?elds. It is easily seen that in the second order of the new variables the e?ective action is a sum similar to (17). In particular, it means that the diquark ?elds ξ4 , ξ5 are decoupled from other ?elds, and their e?ective action looks like (20) with the replacement (32) used. So, some elements of the inverse propagator matrix for ξ4 , ξ5 ?elds di?er by a sign from Γkl (x ? y ) (21), but the determinant of the whole momentum space matrix is not changed. Hence, the mass spectrum, the dispersion laws etc., in the ξ4 , ξ5 -?elds are the same, as in the case with old ?elds ?1 , ?2 . Similar conclusions are valid for the rest of new variables, i. e. such physical characteristics of the model, as particle masses, dispersion laws etc., do not depend on the choice of ?eld parametrizations, (30) or (31). In particular, the number of abnormal NG-bosons, which equals three in the model under consideration, is a parametrization invariant quantity6 .

5 6

A detailed investigation of the meson/diquark spectrum including the σ-?3 mixing is in preparation. In papers [4, 5] an abnormal number of NG-bosons in the same SU (2) × U (1) toy model was discovered. But in [4] the parametrization (30) was used for the scalar doublet, whereas in [5] a ”polar coordinate” parametrization, similar to (31), was used. Evidently, their results coincide, i.e. the abnormal number of NG-bosons, particle masses, formulae for dispersion laws are the same.

10

IV. COLOR–ASYMMETRY AND MASSES OF SCALAR DIQUARKS

As it was noted in [12], the color superconducting phase is automatically color-neutral in QCD, but this does not hold for the NJL model under consideration. Massive scalar diquarks and, as a consequence, the abnormal number of NG-bosons in the model discussed in previous sections testify for this. As mentioned in the Introduction, for the number of NG-bosons to equal the number of broken generators of the underlying symmetry group (in our case SUc (3)), the ground-state expectation value for the commutator of any two broken symmetry generators Qa and Qb must be zero: [Qa , Qb ] = 0 [5]. This criterion is not ful?lled for the NJL model considered here, namely, the ground state expectation value of the color charge operator Q8 = q ?γ 0 λ8 q does not vanish, and, therefore, the usual counting rule for NG-bosons is not applicable. Let us look at the function H (p0 ) at p0 = 0. One can rewrite the expression (27) in a di?erent form: H (0) = 1 2 d3 q 1 (2π )3 ?2 ?2θ(? ? E ) + E+ E? ? + ? E? E? . (33)

This formula can be expressed in terms of quark densities. For this purpose, we need the expression of the thermodynamical potential of the system, which in the mean-?eld approximation is equal to the value of Ve? in its global minimum point, supplied by the gap equations, i. e.

?δ ?(?) = Ve? (σo , πo , ?δ o , ?o ),

(34)

where Ve? was given in (8). Further, one can divide the thermodynamical potential into several terms:

3

?(?) =

c=1

?c (?) +

|?|2 (M ? mo )2 + , 4G1 4G2

(35)

where ?c (?) is the contribution from the color-c quark, de?ned as follows: ?1 (?) = ?2 (?) = ?2 ?3 (?) = ?2 d3 q + ? E? + E? , (2π )3 d3 q |E + | + |E ? | . (2π )3

(36)

The contribution from color-1 and color-2 quarks, ?1(2) (?) are equal due to the remaining SUc (2) color symmetry after the global SUc (3) symmetry group is spontaneously broken in the color superconducting phase. The third color quark, which does not condensate, gives ?3 (?). Then, one can calculate the densities nc of color-1, -2, and -3 quarks separately without introducing additional chemical potentials for each color simply by di?erentiating ?c (?) over ?, i. e. nc = ? d3 q (2π )3 ? ?c (?) , ?? c = 1, 2, 3; d3 q θ(? ? E ). (2π )3 (37)

n1 = n2 = 2 Eq. (33) can thus be transformed to

E+ E? ? + ? E? E?

,

n3 = 4

(38)

1 H (0) = (n1 + n2 ? 2n3 ) = 8?2

√ 3 Q8 . 8?2

(39)

From eq. (39) and the fact that the diquark mass m1 = H (0)/H ′ (0) is found to be nonzero (see Figs. 3 and 4), one concludes that the 2SC-ground state in the NJL model under consideration has ?nite color charge Q8 . Evidently, this fact is related to the inequality of color quark densities n1,2 > n3 for paired (1,2) and unpaired (3) quarks, i. e. to color-asymmetry. Moreover, as discussed in Appendix, the nonzero value of H (0) is responsible for the abnormal dispersion law p0 ? p 2 /H (0) for two NG-bosons. To restore the local color neutrality in the NJL model, one can, e. g., include an additional term, ?8 Q8 , and impose the color neutrality condition: Q8 = ? ? ?(?, ?8 ) = 0, ??8 (40)

or consider the superconducting matter as composed of colored domains, with the global color charge being equal zero. The latter, however, goes beyond the mean-?eld approximation exploited here and should be considered as an

11 approximation for the case of large colored domains. Anyway, the true ground state should be checked to give the absolute minimum of the thermodynamical potential. The inclusion of an additional chemical potential ?8 “by hand”, however, does not contradict QCD. As discussed in [12], a non-vanishing expectation value of the gluon ?eld A8 0 , corresponding to the eighth generator in the SUc (3) group, gives rise to a non-vanishing value of ?8 , which then cancels certain tadpole contributions and restores the color neutrality. This cannot occur in the NJL model automatically because there are no gluons in it.

V. SUMMARY AND DISCUSSIONS

Recently, it has been shown that in QCD with a nonzero strangeness chemical potential the number of NG-bosons can be less than the number of broken symmetry generators [4, 5], so that the usual counting rule for NG-bosons evidently does not hold. In the present paper, we have presented another relativistic model providing us with an abnormal number of NG-bosons. This is an NJL–type model with two light-quark ?avors, where a single quark chemical potential is taken into account. In the color superconducting phase of this model, the color SUc (3) group is spontaneously broken down to SUc (2), i. e. the number of broken symmetry generators equals ?ve. Despite of this fact, there appeared only three NG-bosons in the 2SC phase; two of them form an antidoublet with respect to the unbroken SUc (2) subgroup and have quadratic dispersion laws. The remaining one is an SUc (2)-singlet, so there are no contradictions with the NC theorem [3]. Moreover, there exists an SUc (2)-antidoublet of light diquarks ?1 , ?2 with masses m1 , the behavior of which at ?nal chemical potential ? is shown in Figs. 3 and 4 for the parameter sets A and B, respectively. Most interestingly, we found that the diquark masses m1 = H (0)/H ′ (0) are proportional to the ground state average √ of the color charge operator Q8 = 1/ 3(n1 + n2 ? 2n3 ). Here, the quark densities n1 , n2 of paired quarks with color1,2 are equal due to the remaining SUc (2) symmetry and larger than the density n3 of the unpaired color-3 quark. Thus, the ground state of the considered NJL-type model is evidently color-asymmetric. It is just the appearance of this nonvanishing expectation value Q8 which simultaneously leads to a violation of the criterion for the equality of the number of NG-bosons and broken symmetry generators [5], and to the appearance of the quadratic dispersion law for the NG-diquarks (compare eqs. (38), (A6)). In the investigations of color superconductivity in the two-?avored QCD, ?ve scalar NG-bosons are argued to exist. They are mixing with gluons and required as longitudinal components for ?ve massive gluons (color Meissner e?ect) [10]. The fact that we ?nd an abnormal number of three NG-bosons does not directly contradict QCD. Namely, in the color superconducting phase of QCD a nonvanishing condensate of the eighth gluon ?eld component A8 0 appears, which induces a color chemical potential ?8 and cancels certain tadpole contributions, responsible for the nonvanishing color charge of the ground state [12]. This mechanism just leads to color neutrality. Obviously, there arises the question, how can one reconcile the considered NJL approach with QCD? Insofar as NJL-type models do not contain gluon ?elds, the required contribution from the condensed eighth gluon ?eld does not follow automatically. Therefore, the tadpole contribution becomes unbalanced by gluon contributions. It can be be cancelled “by hand”, e. g. with the help of the additional color chemical potential ?8 simulating the omitted gluon contribution. In such enlarged NJL models the condition of color neutrality of the ground state, i. e. Q8 = 0, can be imposed as an additional physical requirement [11]. According to our results, one gets in this case two additional NG-bosons as well as a change of quadratic dispersion laws into normal linear ones. Thus, the abnormal number of three NG-bosons might, in principle, become converted into the normal number ?ve found in QCD. Nevertheless, what to do now with these ?ve NG-bosons arising in the color neutral 2SC-phase of an enlarged NJL-type model? Clearly, in the framework of the standard NJL model one cannot apply the color Meissner e?ect, in order to absorb these NG-bosons into longitudinal degrees of freedom of massive gluons. A possible way out of this problem could be an extension of the NJL mode via the inclusion of perturbative gluons, ?uctuating around the original low-energy gluonic ?elds (realized e. g. by instantons or other non-perturbative background ?elds, which were already integrated out to yield the e?ective four-quark interactions) [32]. Obviously, the color Meissner e?ect could then be realized for such perturbative gluons. In addition, let us remark that the above discussions refer to the case of locally neutral quark matter, where color neutrality should indeed be required. On the other hand, global color neutrality does not demand local color neutrality: for example, one can assume the ground state as being composed of colored domains. Anyway, to decide which of the considered ground states is in favor, the absolute minimum of the thermodynamic potential should be calculated in each case. (Note that the presence of domains means a violation of translational invariance and is not consistent with the simple mean-?eld approach.) Finally, notice that in the framework of the considered NJL-type of model there arises a mixing between the σ meson and ?3 -diquark in the 2SC phase. The behavior of σ -, ?3 - as well as π -meson masses vs. ? in the 2SC phase is now under consideration.

12

Acknowledgments

This work has been supported in part by DFG-project 436 RUS 113/477/0-2, RFBR grant 02-02-16194, the Heisenberg–Landau Program, and the “Dynasty” Foundation.

APPENDIX A: THE Γ-MATRIX ELEMENTS AND DISPERSION LAWS

Using the e?ective action (18), let us introduce auxiliary quantities Γ?? ? (z ) = ? Γ??? (z ) = ? δ (z ) δ 2 S1 ? iTrsc a(z )?1 γ 5 d(?z )?1 γ 5 , = 1 ? 1 δ ? (y )δ ? (x) 4G2 δ (z ) δ 2 S1 ? iTrsc d(z )?1 γ 5 a(?z )?1 γ 5 , = δ ??1 (y )δ ?1 (x) 4G2

(2) (2)

(A1)

where δ (z ) is the Dirac δ -function, z = x ? y and a(z ), d(z ) are operators (13) and (16), respectively. Since (2) (2) δ 2 S1 /(δ ?1 δ ?1 ) = δ 2 S1 /(δ ??1 δ ??1 ) = 0 and δ 1 = √ δ?1 2 δ δ + δ ? δ ?? , δ i = √ δ?2 2 δ δ ? δ ? δ ?? ,

the matrix elements of the inverse propagator matrix (21) have the form: Γ11 (z ) = Γ22 (z ) = 1 (Γ?? ? (z ) + Γ??? (z )) , 2 Γ12 (z ) = ?Γ21 (z ) = i (Γ?? ? (z ) ? Γ??? (z )) . 2 (A2)

For an arbitrary function F (z ), it is possible to de?ne the Fourier-transformed one, F (p), by the relation F (p) = d4 zF (z )eipz , i. e. F (z ) = d4 p F (p)e?ipz . (2π )4 (A3)

Now, using (A3), it is possible to get from (A1) 1 ? iTrsc 4G2 1 Γ??? (p) = ? iTrsc 4G2 Γ?? ? (p) = d4 q a(q + p)?1 γ 5 d(q )?1 γ 5 , (2π )4 d4 q d(q + p)?1 γ 5 a(q )?1 γ 5 , (2π )4

(A4)

?(q ) can be easily derived from (13), (16). It follows from (A4) that where the Fourier-transformed expressions a ?(q ), d Γ?? ? (?p) = Γ??? (p). So, taking into account (A2), one can easily obtain the relations (22). After tedious calculations we arrive at the expression Γ?? ? (p) = ? 4i (2π )4 4ip0 (2π )4 p0 + q0 + ? + Eq p0 + q0 + ? ? Eq d4 q ? + 2 ] [(p0 + q0 + ?)2 ? Ep D ( q ) D+ (q0 ) ? 0 +q

Eq p 2 + (q0 + ? + Eq )(p · q ) Eq p 2 ? (q0 + ? ? Eq )(p · q ) d4 q . + 2 ] [(p0 + q0 + ?)2 ? Ep Eq D? (q0 ) Eq D+ (q0 ) +q

(A5)

Here, the notations that were introduced after (10) are used. Moreover, Eq ≡ E = q 2 + M 2 . Note, that expression (A5) is obtained under the assumption that ? = 0, i. e. it is valid only in the 2SC phase. In this case, the gap equation (10) was used to eliminate the coupling constant G2 from relation (A4) in favour of other model parameters. Quasiparticle dispersion laws are de?ned by (23) or, equivalently, by the two equations Γ?? ? (±p) = 0. One can easily see that at p 2 = 0 the ?rst integral in RHS of (A5) coincides with (24) having an evident zero at p0 = 0. This is just the dispersion law of a gapless particle (NG-boson) at p 2 = 0. Let us ?nd the dispersion law for the NG-boson at small nonzero values of p 2 , i. e. try to solve the equation Γ?? ? (p) = 0 at p → 0. Expanding expression (A5) into a Taylor-series of (p0 , p) at the point (0, 0) and taking into account only leading terms, we have, at ? = 0, the following equation connecting the energy and momentum of the massless particle (in section III, it was proved that such a particle, NG-boson, exists in the ?1 -sector of the model): Γ?? ? (p) ≡ 4ip0 H (0) ? ? 4ip 2 (2π )4 4ip 2 (2π )4 d4 q 2] D? (q0 )[(q0 + ?)2 ? Eq 1+ 1? 2q 2 3Eq (q0 + ? ? Eq ) ? (A6)

d4 q 2] D+ (q0 )[(q0 + ?)2 ? Eq

2q 2 3Eq (q0 + ? + Eq )

+ · · · = 0,

13 where the quantity H (0) (which is nonzero at ? = 0) is given in (27). Obviously, the quadratic dispersion law, p0 ? p 2 , for a massless particle follows from (A6). However, in case of relativistic invariance of the system (i. e. at ? = 0 and for coupling constants from the region ω (29)), or for a color-neutral ground state, when H (0) = 0, the dispersion law ′ ′ for a NG-boson changes. Indeed, in this case the term 4ip0 H (0) from (A6) should be replaced by 4ip2 0 H (0) (H (0) is presented in (27)), and one arrives at a linear dispersion law for NG-bosons.

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- Quark description of the Nambu-Goldstone bosons in the color-flavor locked phase
- Goldstone bosons in presence of charge density
- Masses of the pseudo-Nambu-Goldstone bosons in two flavor color superconducting phase
- Goldstone bosons in the color-flavor locked phase
- Mesons and diquarks in the color neutral 2SC phase of dense cold quark matter
- On asymmetric Chern-Simons number diffusion
- Photons and Gravitons as Goldstone Bosons, and the Cosmological Constant
- Longitudinal gluons and Nambu-Goldstone bosons in a two-flavor color superconductor
- Linearizing N = 1 nonlinear supersymmetry with higher derivative terms of a Nambu-Goldstone
- Color Neutral Ground State of 2SC Quark Matter
- 关于进一步加强道路机动车辆生产一致性监督管理和注册登记工作的通知
- Calculation of Particle Production by Nambu Goldstone Bosons with Application to Inflation
- Longitudinal gluons and Nambu-Goldstone bosons in a two-flavor color superconductor
- Quark description of the Nambu-Goldstone bosons in the color-flavor locked phase
- Cosmology with Ultra-light Pseudo-Nambu-Goldstone Bosons

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