Pion condensation in the Gross-Neveu model with nonzero baryon and isospin chemical potentials
aa r X i v : . [ h e p - ph ] F e b HU-EP-09/03
Pion condensation in the Gross-Neveu model with nonzero baryon and isospinchemical potentials
D. Ebert and K. G. Klimenko Institute of Physics, Humboldt-University Berlin, 10115 Berlin, Germany Institute for High Energy Physics, 142281 Protvino, Moscow Region, Russia
The properties of the two-flavored Gross-Neveu model in the (1+1)-dimensional spacetime areinvestigated in the presence of the isospin µ I as well as quark number µ chemical potentials bothat zero and nonzero temperatures. The consideration is performed in the limit N c → ∞ , i.e. inthe case with an infinite number of colored quarks. In the plane of parameters µ I , µ a rather richphase structure is found, which contains phases with and without pion condensation. Moreover, wehave found a great variety of one-quark excitations of these phases, including gapless and gappedquasiparticles. PACS numbers: 11.30.Qc, 12.39.-x, 12.38.MhKeywords: Gross – Neveu model; pion condensation
I. INTRODUCTION
During the last decade great attention was paid to the investigation of the QCD phase diagram in terms of baryonicas well as isotopic (isospin) chemical potentials. First of all, this interest is motivated by experiments on heavy-ioncollisions, where we have to deal with dense baryonic matter which has an evident isospin asymmetry, i.e. differentneutron and proton contents of initial ions. Moreover, the dense hadronic/quark matter inside compact stars isalso isotopically asymmetric. Generally speaking, one of the important QCD applications is just to describe thedense and hot baryonic matter. However, in the above mentioned realistic situations the density is rather small, andweak coupling QCD analysis is not applicable. So, different nonperturbative methods or effective theories such aschiral effective Lagrangians and especially Nambu – Jona-Lasinio (NJL) type models [1] are usually employed for theconsideration of the properties of dense and hot baryonic matter under heavy-ion experimental and/or compact starconditions, i.e. in the presence of such external conditions as temperature and chemical potentials, magnetic field,finite size effects etc (see, e.g., the papers [2, 3, 4, 5, 6, 7, 8, 9, 10] and references therein). In particular, the colorsuperconductivity [4, 5] as well as charged pion condensation [11, 12, 13, 14, 15, 16] phenomena of dense quark matterwere investigated in the framework of these QCD-like effective models.It is necessary to note that an effective description of QCD in terms of NJL models, i.e. through an employmentof four-fermionic theories in (3+1)-dimensional spacetime, is usually valid only at rather low energies and densities.Besides, at present time there is the consensus that another class of theories, the set of (1+1)-dimensional Gross-Neveu (GN) type models [17, 18], can also be used for a good qualitative consideration of the QCD properties withoutany restrictions on the energy/density values, which is in an encouranging contrast with NJL models. Indeed, theGN type models are renormalizable, the asymptotic freedom and spontaneous chiral symmetry breaking are anotherproperties inherent both for QCD and GN theories etc. In addition, the µ − T phase diagram is qualitatively the samein QCD and GN model [19, 20, 21, 22, 23] (here µ is the quark number chemical potential and T is the temperature).The GN type models are also very suitable for the description of physics in such quasi one-dimensional condensedmatter systems as polyacetylene [24]. Due to the relative simplicity of GN models in the leading order of the large N c -expansion ( N c is the number of colored quarks), their use is very convenient for the application of nonperturbativemethods in quantum field theory [25]. Moreover, it is necessary to note that just in the leading order of the large N c -expansion the well known no-go theorem by Mermin-Wagner-Coleman [26] apparently forbidding the spontaneousbreaking of continuous symmetries in the considered (1+1)-dimensional models is not valid [21, 22, 23].Thus, such phenomena of dense QCD as color superconductivity, where the color group is broken spontaneously,and charged pion condensation, where the spontaneous breaking of the continuous isospin symmetry takes place,might be simulated in terms of simpler (1+1)-dimensional GN type models (see, e.g., [22] and [27], correspondingly).In our previous paper [27] the phase diagram of the (1+1)-dimensional GN model with two massless quark flavorswas investigated under the constraint that quark matter occupies a finite space volume (see also the relevant papers[28]). In particular, there we have studied in the large N c -limit the charged pion condensation phenomenon in coldquark matter with zero baryonic density, i.e. at µ = 0, but nonzero isotopic density, i.e. with nonzero isospinchemical potential µ I . In contrast, in the present paper in the leading order of the 1 /N c -expansion we consider thephase portrait of the above mentioned GN model in a more general case, where both isospin- and quark numberchemical potentials are nonzero, i.e. µ I = 0 and µ = 0. Moreover, we deal with nonzero temperature and withspacetime of the usual topology, R × R . We hope that our investigations will shed some new light on the physicsof dense and hot isotopically asymmetric quark matter which can be observed in heavy-ion collision experiments ormight exist in compact stars, where baryon density is obviously nonzero (i.e. µ = 0). Our consideration is based,for simplicity, on the approach with homogeneous condensates (an extension to inhomogeneous condensates was alsoconsidered in [23, 29]).The paper is organized as follows. In Section II the thermodynamic potential of the two-flavored Gross-Neveumodel is obtained in the presence of quark number as well as isotopic chemical potentials. In Section III and IV thephase structure of the model is investigated at zero and nonzero temperatures, correspondingly. It turns out that atzero temperature, µ I = 0 and rather small values of µ the gapped pion condensed phase occurs. However, at largervalues of µ the dense quark matter phase with chirally broken phase and gapless quasiparticles is realized. II. THE MODEL AND ITS THERMODYNAMIC POTENTIAL
We consider a (1+1)-dimensional model which describes dense quark matter with two massless quark flavors ( u and d quarks). Its Lagrangian has the form L = ¯ q h γ ν i ∂ ν + µγ + µ I τ γ i q + GN c h (¯ qq ) + (¯ q i γ ~τ q ) i , (1)where the quark field q ( x ) ≡ q iα ( x ) is a flavor doublet ( i = 1 , i = u, d ) and color N c -plet ( α = 1 , ..., N c ) aswell as a two-component Dirac spinor (the summation in (1) over flavor, color, and spinor indices is implied); τ k ( k = 1 , ,
3) are Pauli matrices; the quark number chemical potential µ in (1) is responsible for the nonzero baryonicdensity of quark matter, whereas the isospin chemical potential µ I is taken into account in order to study propertiesof quark matter at nonzero isospin densities (in this case the densities of u and d quarks are different). Evidently,the model (1) is a generalization of the (1+1)-dimensional Gross-Neveu model [17] with a single massless quark color N c -plet to the case of two quark flavors and additional chemical potentials. As a result, in the considered case wehave a modified chiral symmetry group. Indeed, at µ I = 0 apart from the color SU( N c ) symmetry, the Lagrangian(1) is invariant under transformations from the chiral SU L (2) × SU R (2) group. However, at µ I = 0 this symmetry isreduced to U I L (1) × U I R (1), where I = τ / L, R mean that the corresponding group acts only on left, right handed spinors, respectively). Evidently,this symmetry can also be presented as U I (1) × U AI (1), where U I (1) is the isospin subgroup and U AI (1) is theaxial isospin subgroup. Quarks are transformed under these subgroups as q → exp(i ατ ) q and q → exp(i αγ τ ) q ,respectively.The linearized version of the Lagrangian (1), which contains composite bosonic fields σ ( x ) and π a ( x ) ( a = 1 , , L = ¯ q h γ ν i ∂ ν + µγ + µ I τ γ − σ − i γ π a τ a i q − N c G h σσ + π a π a i . (2)From the Lagrangian (2) one gets the following constraint equations for the bosonic fields σ ( x ) = − GN c (¯ qq ); π a ( x ) = − GN c (¯ q i γ τ a q ) . (3)Obviously, the Lagrangian (2) is equivalent to the Lagrangian (1) when using the constraint equations (3). Fur-thermore, it is clear from (3) that the bosonic fields transform under the isospin U I (1) and axial isospin U AI (1)subgroups in the following manner: U I (1) : σ → σ ; π → π ; π → cos(2 α ) π + sin(2 α ) π ; π → cos(2 α ) π − sin(2 α ) π ,U AI (1) : π → π ; π → π ; σ → cos(2 α ) σ + sin(2 α ) π ; π → cos(2 α ) π − sin(2 α ) σ. (4)Due to the transformation rules (4) such quantity as the thermodynamic potential (TDP) depends effectively onlyon the two combinations ( σ + π ) and ( π + π ) of the bosonic fields, which are invariants with respect to the U I (1) × U AI (1) group. In this case, without loss of generality, one can put π = π = 0, and study the TDP asa function of only two variables, M ≡ σ and ∆ ≡ π . Throughout the paper we suppose for simplicity that thecondensates M and ∆ are homogeneous quantities which do not depend on the space coordinate. In order to avoidthe no-go theorem [26], which forbids the spontaneous breaking of continuous symmetries in the considered case ofone space direction, all our considerations are performed in the leading order of the 1 /N c -expansion, i.e. in the limit N c → ∞ , where the TDP of the model looks like (see, e.g., [27])Ω µ,ν ( M, ∆) = M + ∆ G + i Z d p (2 π ) ln nh ( p + µ ) − ( E +∆ ) ih ( p + µ ) − ( E − ∆ ) io , (5)where E ± ∆ = p ( E ± ) + ∆ , E ± = E ± ν , ν = µ I / E = p p + M . It is clear that the TDP Ω µ,ν ( M, ∆) issymmetric under the transformations µ → − µ and/or ν → − ν . So it is sufficient to consider only the region µ ≥ ν ≥
0. Taking into account this constraint as well as integrating in (5) over p , we obtain for the TDP of the systemat zero temperature the following expression:Ω µ,ν ( M, ∆) = M + ∆ G − Z ∞−∞ dp π n E +∆ + E − ∆ + ( µ − E +∆ ) θ ( µ − E +∆ ) + ( µ − E − ∆ ) θ ( µ − E − ∆ ) o , (6)where θ ( x ) is the Heaviside step function. Since we are going to study the phase diagram of the initial GN model,the system of gap equations is needed:0 = ∂ Ω µ,ν ( M, ∆) ∂M ≡ M G − M Z ∞−∞ dp πE n θ ( E +∆ − µ ) E + E +∆ + θ ( E − ∆ − µ ) E − E − ∆ o , ∂ Ω µ,ν ( M, ∆) ∂ ∆ ≡ ∆2 G − ∆ Z ∞−∞ dp π n θ ( E +∆ − µ ) E +∆ + θ ( E − ∆ − µ ) E − ∆ o . (7)Evidently, the coordinates M and ∆ of the global minimum point of the TDP (6) supply us with two order parameters(gaps), which are proportional to the ground state expectation values h ¯ qq i and h ¯ q i γ τ q i , respectively. If only the gap M is nonzero, then in the ground state of the model the axial isospin symmetry U AI (1) (at µ I = 0) is spontaneouslybroken down. However, if only the gap ∆ = 0, then the ground state describes the phase with charged pion conden-sation, where the isospin U I (1) symmetry is spontaneously broken. Note that in this phase the space parity is alsospontaneously broken. III. PHASE PORTRAIT AT ZERO TEMPERATURE
First of all, let us consider the phase portrait of the initial GN model (1) using as a starting point the TDP (6).Since it is an ultraviolet divergent quantity, one should renormalize it, using a special dependence of the bare couplingconstant G ≡ G (Λ) on the cutoff parameter Λ (Λ restricts the integration region in the divergent integrals, | p | < Λ).In our previous paper [27] the following prescription for the bare coupling constant G (Λ) was used,12 G (Λ) = 1 π Z Λ − Λ dp p M + p = 2 π ln Λ + p M + Λ M ! , (8)where M is the dynamically generated quark mass in the vacuum, i.e. at µ = 0 and µ I = 0 (see below). Then,introducing the quantity Ω µ,ν ( M, ∆; Λ),Ω µ,ν ( M, ∆; Λ) = M + ∆ G (Λ) − Z Λ − Λ dp π n E +∆ + E − ∆ + ( µ − E +∆ ) θ ( µ − E +∆ ) + ( µ − E − ∆ ) θ ( µ − E − ∆ ) o + Λ π , (9)it is possible to obtain the renormalized (finite) expression for the TDP:Ω µ,ν ( M, ∆) = lim Λ →∞ Ω µ,ν ( M, ∆; Λ) . (10)(The renormalized expression for the gap equations is obtained in the limit Λ → ∞ , if the replacements G → G (Λ)and | p | < Λ are done in (7), or by a direct differentiation of the expression (10).) In particular, at µ = 0 and µ I = 0we have from (10): Ω µ,ν ( M, ∆) (cid:12)(cid:12)(cid:12) µ =0 ,ν =0 ≡ V ( p M + ∆ ) = M + ∆ π (cid:20) ln (cid:18) M + ∆ M (cid:19) − (cid:21) . (11)Since for a strongly interacting system the space parity is expected to be a conserved quantity in the vacuum, we put∆ equal to zero in (11). As a result, the global minimum of the TDP (11) (usually, this quantity is called effectivepotential in the vacuum) lies in the point M = M , which means that in the vacuum the dynamically generated quarkmass is just the parameter M introduced in (8). However, in the general case, i.e. at nonzero values of the chemicalpotentials, this quantity depends certainly on µ, µ I and obeys the gap equations (7).Numerical investigations show that local minima of the TDP (10) can occur only on the M - or ∆-axes (othersolutions of the gap system (7) correspond to saddle points of the TDP (10)). So it is enough to restrict theconsideration of the TDP (10) to the regions M = 0 , ∆ = 0 (∆-axis) or M = 0 , ∆ = 0 ( M -axis). Moreover, since theTDP is an even function with respect to the transformations M → − M or ∆ → − ∆, we will suppose further that ˜ µ ˜ ν
131 2 ˜ µ = 1 / √ µ = 1 / √ t ˜ µ A s FIG. 1: The (˜ µ, ˜ ν ) phase portrait of the model at T = 0and ˜ ν >
0. Here ˜ µ = µM , ˜ ν = µ I M , and M is the quarkmass in the vacuum. Number 1 denotes the symmetricphase with massless quarks, number 2 – the normal quarkmatter phase with massive quarks, and 3 denotes the pioncondensed phase. FIG. 2: The ( t, ˜ µ ) phase portrait of the model at µ I =0 . M ( t = TM , ˜ µ = µM ). The solid and dotted lines arethe critical curves of the 1st and 2nd order phase transi-tions, correspondingly. They are divided by the tricriticalpoint A. Other notions are the same as in Fig. 1. M, ∆ ≥
0. As a result we haveΩ µ,ν ( M = 0 , ∆) = V (∆) − ν π + θ ( µ − ∆) π " ∆ ln µ + p µ − ∆ ∆ ! − µ p µ − ∆ , (12)Ω µ,ν ( M, ∆ = 0) = V ( M ) + θ ( µ + ν − M )2 π " M ln µ + ν + p ( µ + ν ) − M M ! − ( µ + ν ) p ( µ + ν ) − M + θ ( | µ − ν | − M )2 π " M ln | µ − ν | + p ( µ − ν ) − M M ! − | µ − ν | p ( µ − ν ) − M i (cid:16) − θ ( M − ν ) θ ( ν − µ ) (cid:17) , (13)where the function V ( x ) is presented in (11). Compairing the global minima of the TDPs (12)-(13), it is possibleto obtain the global minimum point of the genuine TDP (10) of the model. Clearly, its form (and, as a result, thephase of the model) depends on the values of µ and ν . So, the total information about the behaviour of the globalminimum point vs µ , ν can be presented by the phase portrait given in Fig. 1. (Note, it is valid only for ν > ν = 0 case the phase structure is described below). There, in the corresponding (˜ ν, ˜ µ )-plane (excludingthe ˜ µ -axis), where ˜ µ = µM , ˜ ν = νM = µ I M , you can see three respective phase regions denoted by the numbers 1,2, 3. For the values of (˜ ν, ˜ µ ) from the region 1, 2, and 3 the global minimum point of the TDP (10) has the form( M = 0 , ∆ = 0), ( M = 0 , ∆ = 0), and ( M = 0 , ∆ = 0), correspondingly. As a result, in the region 1 the chirally U I (1) × U AI (1)-symmetric phase with massless quarks is arranged. In the region 2, where the order parameter M is nonzero, this symmetry is spontaneously broken down to the isospin U I (1) subgroup (in this region the orderparameter M is a smooth function vs µ and ν ). We call this phase the normal quark matter phase, since here quarksdynamically acquire a mass which is equal to the order parameter M , and space parity is not broken. Finally, theregion 3 corresponds to the charged pion condensed phase, because of the nonzero order parameter ∆. Furthermore,for all points of this region ∆ ≡ M . Note that on the phase boundaries phase transitions of the first order occur.It is worth to remark that in early papers (see e.g. [19, 20]), the phase structure of the GN model was studied in theexceptional case of ν = 0 , µ = 0 with homogeneous condensates. As it was shown there, at µ > M / √ µ < M / √ M . Let us denote this particular normal phase as phase 4, which is not depicted in Fig. 1. There are If an inhomogeneous ansatz for condensates is taken into account, then in some GN phase diagrams an additional crystalline phasemight appear at ν = 0 and intermediate values of µ [23, 29]. The existence of the crystalline phase in the general case with ν = 0 is notinvestigated up to now. We would like to stress once more that in this figure the phase 3 does not occupy the part of the ˜ µ -axis (˜ ν = 0, 0 < ˜ µ < / √ two essential differences between phases 2 and 4. The first one manifests itself in their dynamical properties such asquasiparticle excitations. Recall that in the most general case the energy spectrum of the u -, d -, ¯ u -, ¯ d -quasiparticles(quarks) has the following form (see, e.g., [13]): E u ( p ) = E − ∆ − µ, E d ( p ) = E +∆ − µ, E ¯ u ( p ) = E +∆ + µ, E ¯ d ( p ) = E − ∆ + µ. (14)It is easily seen from (14) that in the phase 4, where ν = 0, ∆ = 0, M = M and µ < M / √
2, there is a gap in thequasiparticle energy spectrum, i.e. in the phase 4 all quasiparticles are gapped excitations. In contrast, in the phase2 ( ν = 0) the situation is quite different and here in the energy spectrum of the u -quasiparticles the gap is absent. Itmeans that for each point of the region 2 in Fig. 1 there exists a space momentum p ⋆ such that E u ( p ⋆ ) = 0. (Forexample, the point (˜ ν = 0 . , ˜ µ ≈ .
92) is in the phase 2 (near its boundary), where ∆ = 0 and M ≈ . M . In thiscase p ⋆ ≈ . M etc.) So, in analogy with gapped and gapless color superconductivity [4], the phase 2 may be calledgapless normal quark matter phase.The second difference between phases 2 and 4 is a thermodynamic one and related with the density of quarks inboth phases. It is clear that in the phase 4, i.e. at ν = 0, M = M and µ < M / √
2, the TDP from (13) does notdepend on the quark chemical potential µ . As a result, we see that in this phase the quark density n q ≡ − ∂ Ω /∂µ isequal to zero. However, in the phase 2 the density of quarks is not zero. So the consideration of dense quark matterin the framework of the GN model is more adequate in terms of both nonzero isotopic and quark number chemicalpotentials, ν = 0 and µ = 0, than in the simpler case with ν = 0, µ = 0.It is seen from Fig. 1 that the symmetric phase 1, where M = 0 and ∆ = 0, is a disconnected manifold consisting oftwo different parts. (Note that in the ground state of this phase the quark number density is not zero.) The first partis arranged above the phase 2 and corresponds to µ > ν , whereas in the second one, which is below the phase 2, wehave µ < ν . It turns out that the dynamical properties of these two regions are different. Indeed, for the part of thephase 1 which is above the phase 2 both u - and d -quasiparticles are gapless. However, in the rest of the phase 1, whichcorresponds to µ < ν , only u -quasiparticles are gapless, as it is clear from the corresponding dispersion relations ofthese quasiparticles, E u ( p ) = | p − ν | − µ and E d ( p ) = p + ν − µ , in the phase 1. Hence, such dynamical propertiesof dense quark matter as transport phenomena (e.g., conductivities etc) can occur in a qualitatively different way inthese subphases of the symmetric phase 1.Finally, note that all quasiparticle excitations of the pion condensed phase 3 are gapped ones. Indeed, it is clearfrom (14) that at ∆ = M , M = 0 and µ ≤ M / √ IV. PHASE PORTRAIT AT NONZERO TEMPERATURES
Now let us stydy the influence of the temperature T on the phase structure of the considered GN model with twochemical potentials µ and ν ≡ µ I /
2. To get the corresponding thermodynamic potential Ω µ,ν, T ( M, ∆), one can simplystart from the expression for the TDP at zero temperature (5) and perform the following standard replacements: Z ∞−∞ dp π (cid:0) · · · (cid:1) → iT ∞ X n = −∞ (cid:0) · · · (cid:1) , p → p n ≡ iω n ≡ iπT (2 n + 1) , n = 0 , ± , ± , ..., (15)i.e. the p -integration should be replaced in favour of the summation over an infinite set of Matsubara frequencies ω n . Summing in the obtained expression over Matsubara frequencies (the corresponding technique is presented, e.g.,in [30]), one can find for the TDP:Ω µ,µ I , T ( M, ∆) = M + ∆ G − Z ∞−∞ dp π n E +∆ + E − ∆ + T ln (cid:2) e − β ( E +∆ − µ ) (cid:3) + T ln (cid:2) e − β ( E +∆ + µ ) (cid:3) + T ln (cid:2) e − β ( E − ∆ − µ ) (cid:3) + T ln (cid:2) e − β ( E − ∆ + µ ) (cid:3)o , (16)where β = 1 /T . As in the case with zero temperature, the renormalized expression for Ω µ,ν, T ( M, ∆) can be obtainedby the replacement G → G (Λ) (see formula (8)) which, along with the cutting of the integration region in (16), | p | < Λ, leads in the limit Λ → ∞ to the finite expression for the TDP. Numerical investigations show that allpossible local minima of the obtained TDP lie on the M - and ∆-axis. So it is sufficient to deal with correspondingrestrictions of the TDP on these axes, i.e. with the following functions,Ω µ,µ I , T ( M = 0 , ∆) = V (∆) − ν π − Tπ Z ∞ dp ln n h e − β ( E− µ ) i h e − β ( E + µ ) i o , (17)Ω µ,µ I , T ( M, ∆ = 0) = V ( M ) − Tπ Z ∞ dp ln n h e − β ( E + ν − µ ) i h e − β ( E + ν + µ ) i o − Tπ Z ∞ dp ln n h e − β ( E − ν − µ ) i h e − β ( E − ν + µ ) i o , (18) t ˜ µ
13 2 A s B s C s t ˜ µ
13 2 A s B s C s FIG. 3: The ( t, ˜ µ ) phase portrait of the model at µ I =1 . M ( t = TM , ˜ µ = µM ). The points A, B, C are tri-critical points. Other notions are the same as in Figs. 1,2. FIG. 4: The ( t, ˜ µ ) phase portrait of the model at µ I =2 M ( t = TM , ˜ µ = µM ). Other notions are the same asin the previous figures. where the effective potential V ( x ) is given in (11), E = p p + M , and E = p p + ∆ . Comparing the globalminima of the functions (17) and (18), it is possible to establish the global minimum point of the renormalized TDP(16). Then, the dependence of the global minimum point vs T, µ, ν defines the phase structure of the model.Using this prescription in our numerical investigations of the TDPs (17)-(18), we have found the three ( µ, T )-phaseportraits of the initial GN model, depicted in Figs. 2, 3, 4, for three qualitatively different values of the isospin chemicalpotentials, µ I = 0 . M , µ I = 1 . M , and µ I = 2 M , respectively. There, the solid and dotted lines correspond tocurves of first- and second order phase transitions. Moreover, there are several tricritical points, A, B, C, in thesephase diagrams. ( A point of the phase diagram is called a tricritical one, if an arbitrary small vicinity of it containsboth first- and second order phase transition points.) V. SUMMARY AND CONCLUSIONS
Recent investigations of the phase diagram of isotopically asymmetric dense quark matter in terms of NJL modelsshow that their pion condensation content is not yet fully understood. Indeed, the number of the charged pioncondensation phases of the phase diagram depends strictly on the parametrization set of the NJL model. It meansthat for different values of the coupling constant, cutoff parameter etc just the same NJL model predicts differentnumbers of pion condensation phases of quark matter both with or without electric neutrality constraint (see, e.g.,[13, 15]). So to obtain a more objective information about the pion condensation phenomenon of dense quark matterit is important to invoke alternative approaches. One of them, which qualitatively quite successfully imitates theQCD properties (see also the Introduction), is based on the consideration of this phenomenon in terms of the leadingorder of the large N c -technique in the framework of (1+1)-dimensional GN models.In the present paper we have studied the phase structure of the GN model (1) in terms of temperature, quarknumber ( µ )- as well as isospin ( µ I ) chemical potentials in the limit N c → ∞ . Firstly, we have found that at T = 0the charged pion condensation phase of the GN model is realized inside the chemical potential region µ I >
0, and µ is not greater than M / √ µ I > µ we have found the normalquark matter phase 2 (see Fig. 1) in which chiral symmetry is broken spontaneously. Moreover, some of its one-quarkexcitations are gapless quasiparticles, and the quark number density in the ground state is not zero. (In comparision,in the early investigations of the GN model at µ I = 0 and µ = 0 the chirally broken phase was also found, but withzero quark number density and gapped quasiparticles [19].) The region corresponding to this phase in the ( µ I , µ )-plane is not restricted. In contrast, in the chirally symmetric NJL model, i.e. with zero current quark mass, the( µ I , µ )-phase diagram does not contain the normal quark matter phase with such properties (i.e. with nonzero quarknumber density, gapless quasiparticles, and broken chiral invariance), at least at µ I > µ > ν , all quasiparticles are gapless. However, if µ < ν , then the gap is absent in the spectrum of only u -quasiparticles.We hope that our investigation of the GN phase diagram will shed some new light on the phase structure of QCDat nonzero baryonic and isotopic densities. Thus, even in the most simple approach to the GN phase diagram wehave found a variety of phases with rather rich dynamical contents. Obviously, a more realistic imitation of the QCDphase diagram requires to include also a nonzero bare quark mass, i.e. to study massive GN models, as well as totake into account the possibility of inhomogeneous condensates. [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. D , 345 (1961).[2] D. Ebert and M. K. Volkov, Yad. Fiz. , 1265 (1982); D. Ebert and H. Reinhardt, Nucl. Phys. B , 188 (1986).[3] M. Asakawa and K. Yazaki, Nucl. Phys. A504 , 668 (1989); P. Zhuang, J. H¨ufner and S. P. Klevansky, Nucl. Phys.
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