Meissner screening masses in gluonic phase
aa r X i v : . [ h e p - ph ] D ec UWO-TH-07/16
Meissner screening masses in gluonic phase
Michio Hashimoto ∗ and Junji Jia † Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada (Dated: October 27, 2018)A numerical analysis for the Meissner mass in the simplest gluonic phase (the minimal cylindricalgluonic phase II) is performed in the framework of the gauged Nambu-Jona-Lasinio model with coldtwo-flavor quark matter. We derive Meissner mass formulae without using the numerical secondderivative. It is revealed that the gapless mode yields a characterized contribution to the Meissnermass. We also find that there are large and positive contributions from the tree gluon potentialterm to the transverse modes of gluons. It is shown that the simplest gluonic phase resolves thechromomagnetic instability in a rather wide region.
PACS numbers: 12.38.-t, 11.15.Ex, 11.30.Qc
I. INTRODUCTION
Quark matter at sufficiently high density and low tem-perature is expected to be in a color superconductingstate driven by the BCS mechanism [1, 2, 3]. This isanalogous to the electron Cooper paring in a supercon-ducting metal. However, quarks, unlike electrons, havecolor and flavor degrees of freedom as well as spin, sothat the phase structure is quite rich. In nature, decon-fined quark matter might exist in the interior of neutronstars [4]. Thus the dynamics of the color superconduc-tivity has been intensively studied [5].Bulk matter in compact stars should be in equilibriumunder the weak interaction ( β -equilibrium), and be elec-trically and color neutral. The electric and color neu-trality conditions play a crucial role in the dynamics ofthe quark pairing [6, 7, 8, 9]. In addition, the strangequark mass cannot be neglected in moderately densequark matter as in the compact stars. Then a mismatch δµ between the Fermi momenta of the pairing quarks isinduced.As the mismatch δµ increases, the conventional colorsuperconducting state tends to be destroyed. Before thecomplete destruction, however, the Meissner mass of glu-ons turns to be imaginary in the gapped (2SC) and gap-less (g2SC) two-flavor color superconducting phases [10]:In the g2SC phase with the diquark gap ∆ < δµ the 8th gluon has an imaginary Meissner mass, whilethe Meissner masses for the 4-7th gluons are imagi-nary also in the 2SC phase δµ < ∆ < √ δµ . Thischromomagnetic instability implies that there should ex-ist a more stable vacuum other than the 2SC/g2SCphase. Later a chromomagnetic instability was found ∗ Electronic address: [email protected] † Electronic address: [email protected] also in the three-flavor gapless color-flavor locked (gCFL)phase [11, 12, 13]. One of the central issues in this fieldis to establish the genuine ground state for realistic val-ues of δµ . Besides the gluonic phase [14, 15], a numberof other candidates for the true vacuum have been pro-posed [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].Connected with the chromomagnetic instability, it wasrevealed that there appear tachyonic plasmons in the 4-7th and 8th gluonic channels [27]. It clearly shows thatthe physical vectorial excitations carry the instabilitiesand thus supports the scenario with gluon condensates(gluonic phase).It is also known that the physical diquark excitation(the diquark Higgs mode) suffers from the Sarma insta-bility [28] in the g2SC region, which corresponds to thenegative mass squared of the diquark Higgs at zero mo-mentum. Furthermore, it was found that the diquarkHiggs mode has a negative velocity squared v < δµ ≈ ∆ / √
2. Outside thescaling region around δµ ≈ ∆ / √
2, the self-consistentanalysis by solving the gap equations and the neutral-ity conditions was recently performed in Ref. [32]: It isshown that the gluonic phase is actually realized in awide region of the parameter space and it is energeti-cally more favorable than the normal, 2SC/g2SC, andthe single plane wave Larkin-Ovchinnikov-Fulde-Ferrell(LOFF) [16, 33, 34] phases. It is also found that thevalues of ∆ and δµ in the gluonic phase are significantlydifferent from those in the 2SC/g2SC phase. It is notice-able that the values of the gluon condensate are large,say, O (100–250MeV) in the almost whole region where itexists. On the other hand, the values of the color chemi-cal potentials are relatively small. For the earlier works inother approaches, see Refs. [35, 36]. The extension to themodel with nonzero temperature is studied in Ref. [37].In this paper, we examine whether or not the gluonicphase resolves the chromomagnetic instability. We deriveformulae for the Meissner mass without using the numer-ical second derivative. It turns out that the gapless modegives a special contribution to the Meissner mass. In thenumerical analysis, we consider the gluonic phase withthe simplest ansatz which is called the minimal cylin-drical gluonic phase II [15, 32]. As a benchmark, wealso analyze the single plane wave LOFF and 2SC/g2SCphases including the non-HDL corrections.We find that in the gluonic phase the tree gluon po-tential term yields positive and large contributions to theMeissner masses of the transverse modes of gluons. Ac-tually, in the minimal cylindrical gluonic phase II, thechromomagnetic instability is resolved in the weak andintermediate coupling region, 65 . < ∆ < µ = 400MeV and Λ = 653 . µ andΛ denote the quark chemical potential and the cutoffin the (gauged) Nambu-Jona-Lasinio (NJL) model, re-spectively. We here introduced the 2SC gap parameter∆ defined at δµ = 0, which essentially corresponds tothe four-diquark coupling constant in the (gauged) NJLmodel. In the intermediate and strong coupling region130MeV < ∆ < ≃ . < ∆ < . < ∆ < < . II. MODEL
We study the gauged Nambu-Jona-Lasinio (NJL)model with two light quarks. We neglect the currentquark masses and the ( ¯ ψψ ) -interaction channel. TheLagrangian density is given by L = ¯ ψ ( i / D + µ γ ) ψ + G ∆ (cid:20) ( ¯ ψ C iεǫ α γ ψ )( ¯ ψiεǫ α γ ψ C ) (cid:21) − F aµν F a µν , (1)with D µ ≡ ∂ µ − igA aµ T a , F aµν ≡ ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν , (2)where ε and ǫ α are the totally antisymmetric tensors inthe flavor and color spaces, respectively. We also intro-duced gluon fields A aµ , the QCD coupling constant g , thegenerators T a of SU (3), and the structure constants f abc .The quark field ψ is a flavor doublet and color triplet.The charge-conjugate spinor is defined by ψ C ≡ C ¯ ψ T with C = iγ γ . We do not introduce the photon field.On the other hand, the whole theory contains free elec-trons, although we do not show them explicitly in Eq. (1).In β -equilibrium, the chemical potential matrix µ for upand down quarks is µ = µ − µ e Q em , (3)with ≡ c ⊗ f , and Q em ≡ c ⊗ diag(2 / , − / f ,where µ and µ e are the quark and electron chemical po-tentials, respectively. (The baryon chemical potential µ B is given by µ B ≡ µ .) The subscripts c and f mean thatthe corresponding matrices act on the color and flavorspaces, respectively. Hereafter, we abbreviate the unitmatrices, , c and f , if it is self-evident. By introduc-ing the diquark field Φ α ∼ i ¯ ψ C εǫ α γ ψ , we can rewritethe Lagrangian density (1) as L = ¯ ψ ( i / D + µ γ ) ψ − | Φ α | G ∆ −
12 Φ α [ i ¯ ψεǫ α γ ψ C ] −
12 [ i ¯ ψ C εǫ α γ ψ ]Φ ∗ α − F aµν F a µν . (4)In the 2SC/g2SC phase, we can choose the anti-bluedirection without loss of generality, h Φ r i = 0 , h Φ g i = 0 , h Φ b i = ∆ , (5)where the diquark condensate ∆ is real. In this basis, byimposing the color neutrality condition, the color chem-ical potential µ is induced [38]. We can interpret µ asthe vacuum expectation value (VEV) of the time compo-nent of the 8th gluon.Let us define the Nambu-Gor’kov spinor,Ψ ≡ (cid:18) ψψ C (cid:19) . (6)The propagator inverse of Ψ in the 2SC/g2SC phase isgiven by S − ( P ) = (cid:18) [ G +0 ] − ∆ − ∆ + [ G − ] − (cid:19) , (7)with[ G +0 ] − ( P ) ≡ ( p + ¯ µ − δµτ − µ b ) γ − ~γ · ~p, (8)[ G − ] − ( P ) ≡ ( p − ¯ µ + δµτ + µ b ) γ − ~γ · ~p, (9)and∆ − ≡ − iεǫ b γ ∆ , ∆ + ≡ γ (∆ − ) † γ = − iεǫ b γ ∆ , (10)where P µ ≡ ( p , ~p ) is the energy-momentum four vector.We also defined τ ≡ diag(1 , − f , b ≡ diag(0 , , c ,and ¯ µ ≡ µ − δµ µ , δµ ≡ µ e . (11)The 2SC/g2SC phase is not the genuine ground state inthe region δµ > ∆ / √
2, because it suffers from the chro-momagnetic instability. A candidate to resolve the chro-momagnetic instability is the gluonic phase with gluoncondensates [14].Let us introduce the gluon condensates h A aµ i 6 = 0.When the space-component gluon condensates h ~A a i 6 = 0are incorporated into the theory, the time-componentVEVs of the gluon fields other than the 8th one are gen-erally induced as well. We may interpret them as thecolor chemical potentials [39]: µ ˘ a = g h A ˘ a i , (˘ a = 1 , , · · · , , µ = √ g h A i . (12)The propagator inverse S − g of Ψ including the gluon con-densates is written as S − g ( P ) = (cid:18) [ G +0 ,g ] − ∆ − ∆ + [ G − ,g ] − (cid:19) , (13) In an appropriate basis, the color chemical potentials can bereduced only into µ and µ , because the color chemical potentialmatrix µ c ≡ µ a T a ( a = 1 , , · · · ,
8) is hermite and traceless. Inthis case, however, the basis for the diquark field changes fromEq. (5). with[ G +0 ,g ] − ( P ) ≡ ( p + µ ) γ − ~γ · ~p + g h / A a i T a , (14)[ G − ,g ] − ( P ) ≡ ( p − µ ) γ − ~γ · ~p − g h / A a i ( T a ) T . (15)In the fermion one-loop approximation, the bare effec-tive potential including both gluon and diquark conden-sates is given by V bareeff = ∆ G ∆ + 14 F aµν F a µν − µ e π − Z d Pi (2 π ) Tr ln S − g , (16)where we added the free electron contribution. Since thebare potential has a divergence, a counter term is re-quired. We take into account only differences of the freeenergies with and without the chemical potentials. Wethus define the renormalized effective potential by V R eff ≡ V bareeff − V c . t . , (17)with the counter term, V c . t . = − Z d Pi (2 π ) Tr ln S − g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ = µ e = µ a =0 , ∆=0 , h ~A a i6 =0 . (18)In this prescription, even if we use the regularizationscheme with the sharp cutoff Λ for the loop integral, wecan remove artificial mass terms of gluons like Λ ~A a .In general, we can reduce the 32 homogeneous gluoncondensates to 25 ones [15]. However the general case isquite complicated and hence it is difficult to find the self-consistent solutions of the gap equations and the colorneutrality conditions for the 25 VEVs at present.In this paper, we consider the minimal ansatz for thecylindrical gluonic phase II [15, 32], µ ≡ g h A i , µ ≡ √ g h A i , B ≡ g h A z i . (19)In order to make the physical meaning of the gluon con-densates clearer, it is convenient to use the unitary gaugein which all gauge dependent degrees of freedom are re-moved. We shall fix the gauge as follows [15]:Φ r ≡ , Φ g ≡ , ImΦ b ≡ , (20)and A z ≡ , A z ≡ , A z ≡ . (21)As a benchmark, we also consider the single plane-wave2SC-LOFF phase [16, 33, 34], h Φ r i = h Φ g i = 0 , h Φ b i = ∆ e − i~q · ~x . (22) s o l u t i on s o f t he g l uon i c pha s e ( M e V ) ∆ (MeV)B δµ ∆ µ µ FIG. 1: The dynamical solutions for the minimal cylindricalgluonic phase II. The bold solid, bold dashed, thin solid, thindashed and thin dot-dashed curves represent the values of B ,∆, δµ , µ and µ , respectively. The values Λ = 653 . µ = 400 MeV were used. We also took α s = 1. Introducing the quark field ψ ′ = e i~q · ~x ψ with the color-singlet phase, we can erase the x -dependent phase of theLOFF order parameter and instead, the x -independentcolor-singlet term ¯ ψ~γ · ~qψ is induced in the kinetic term forquarks. Although the vector ~q is also gauge equivalent tothe condensate h ~A i , there is subtlety with respect to thetree gluon kinetic term: Notice that the tree gluon po-tential does not give any contribution to the free energy,while it is relevant to the Meissner masses for the trans-verse modes of the 4-7th gluons, because T does notcommute to T − and thereby the corresponding Meiss-ner masses are of the order of q ( ≡ | ~q | ). The point is thatthe values of q are large in a wide range of the parameterregion where the LOFF phase exists, as we will see below.In order to avoid confusion, we may use the color-singlettransformation or we simply do not incorporate the treegluon potential term into the LOFF phase in any case .The dynamics of the minimal cylindrical gluonicphase II is analyzed in Ref. [32] by solving the gap equa-tions and the neutrality conditions in a self-consistentway. We explicitly show the results for the gluonic, sin-gle plane wave LOFF and 2SC/g2SC phases in Figs.1–3, respectively. In the analysis, we took realistic values µ = 400MeV and Λ = 653 . G ∆ to the 2SC gapparameter ∆ defined at δµ = 0 and varied the valuesof ∆ from the weak coupling regime (∆ ∼
60 MeV) tothe strong coupling one (∆ ∼
200 MeV). For the gluonicphase, it is required to specify the value of α s [ ≡ g / (4 π )],although the results for the minimal cylindrical gluonicphase II are not sensitive to the choice of the values of -20 0 20 40 60 80 100 120 140 60 70 80 90 100 110 120 130 140 s o l u t i on s o f t he L O FF pha s e ( M e V ) ∆ (MeV)q δµ ∆ µ FIG. 2: The dynamical solutions for the single plane waveLOFF phase. The bold solid, bold dashed, thin solid andthin dot-dashed curves represent the values of q ( ≡ | ~q | ), ∆, δµ and µ , respectively. The values Λ = 653 . µ = 400 MeV were used. α s [32]. We here took α s = 1.While the neutral normal phase without the diquarkcondensate always exists, the neutral gluonic, LOFF,g2SC and 2SC phases do only in the regions,65.4MeV < ∆ < , (gluonic) (23)64.9MeV < ∆ < , (LOFF) (24)92.2MeV < ∆ < , (g2SC) (25)and ∆ > , (2SC) (26)respectively.The analysis for the free energies has been done inRef. [32]: The normal phase is realized in the weak cou-pling regime with ∆ < < ∆ < < ∆ < > B is large in the almost whole regionwhere the gluonic phase exists. (See Fig. 1.) This featureis crucial for the Meissner masses in the gluonic phase,as we will see in the next section. s o l u t i on s o f t he S C / g2 S C pha s e ( M e V ) ∆ (MeV) δµ∆ µ FIG. 3: The dynamical solutions for the 2SC/g2SC phase.The solid, dashed and dot-dashed curves represent the valuesof ∆, δµ and µ , respectively. At ∆ = 134 . . µ = 400 MeV were used. III. MEISSNER MASSES IN GLUONIC PHASEA. Formulae
Let us derive formulae for the numerical calculation ofthe Meissner screening mass.The squared Meissner mass can be expressed throughthe second derivative of the effective potential: ∂ V R eff ∂A aµ ∂A bν = Π µν tree + g Z d Pi (2 π ) Tr (cid:20) S g Γ µa S g Γ νb (cid:21) − g Z d Pi (2 π ) Tr (cid:20) S g Γ µa S g Γ νb (cid:21) c . t . , (27) where we defined the tree contributionΠ µν tree ≡ g f a ab f a a a A a µ A a ν + g f a aa f a ba g µν A a λ A a λ + g f a aa f a a b A a µ A a ν , (28)and the vertexΓ µa ≡ g − ∂S − g ∂A aµ = (cid:18) γ µ T a − γ µ ( T a ) T (cid:19) . (29)In Eq. (27), “c.t.” denotes the counter term and we used0 = ∂∂X ( S g S − g ) = ∂S g ∂X S − g + S g ∂S − g ∂X , (30)for X = A µ , ∆ , δµ , and linearity of S − g with respect toall variables, i.e., ∂ S − g ∂X∂Y = 0. We also abbreviated thebracket h· · ·i for the gluon condensates.For the numerical calculation, it is useful to diagonalize S − g and/or S g . Although the propagator inverse S − g inEq. (13) is a 48 ×
48 matrix in the flavor, color, spinor andNambu-Gor’kov spaces, we can block-diagonalize S − g inthe flavor and chirality spaces. This technique reducesour labour.Let us transform the propagator inverse S − g as follows; S − g ( P ) = (cid:18) γ iεγ (cid:19) ˜ S − g ( P ) (cid:18) − iεγ γ (cid:19) , (31)with˜ S − g ( P ) = p + H g , H g ≡ − δµτ + (cid:18) ¯¯ µ + µ c − γ ~γ · ~p − γ ~γ · ~A ǫ b ∆ − ǫ b ∆ − ¯¯ µ − µ Tc + γ ~γ · ~p − γ ~γ · ~A T (cid:19) , (32)where¯¯ µ ≡ µ − δµ , µ c ≡ gA a T a , ~A ≡ g ~A a T a . (33)We here decomposed ˜ S − g into the diagonal p -part andthe “Hamiltonian” H g . (One can check easily hermitic-ity of H g .) Notice that the flavor dependence of ˜ S − g exists only in the first term of H g and therefore ˜ S − g is flavor-diagonal. Since the inverse of Eq. (31) yields theexpression for the propagator, S g ( P ) = (cid:18) iεγ γ (cid:19) ˜ S g ( P ) (cid:18) γ iεγ (cid:19) , (34)the second derivative then reads ∂ V R eff ∂A aµ ∂A bν = Π µν tree + g Z d Pi (2 π ) Tr (cid:20) ˜ S g ˜Γ µa ˜ S g ˜Γ νb (cid:21) − g Z d Pi (2 π ) Tr (cid:20) ˜ S g ˜Γ µa ˜ S g ˜Γ νb (cid:21) c . t . , (35)with ˜Γ µa ≡ (cid:18) γ iεγ (cid:19) Γ µa (cid:18) iεγ γ (cid:19) , (36)= (cid:18) γ γ µ T a − γ µ γ ( T a ) T (cid:19) . (37)Since the current quark masses are ignored, the theory ischiral invariant. Thus we can decompose the vertex andalso the propagator into the right and left-handed parts. One can easily show that the trace over the spinor spacein Eq. (35) is the sum of the two parts.In virtue of hermiticity, we can diagonalize H g and ˜ S g by a unitary matrix U , H g = U H diag U † , H diag = diag( E τ , E τ , · · · , E τn ) , (38)˜ S g = U diag (cid:18) p + E τ , p + E τ , · · · , p + E τn (cid:19) U † , (39)where τ = ± for τ = ± E τ , , ··· ,n denote the en-ergy eigenvalues. It is not difficult to find numerically the energy eigenvalues and the unitary matrix by using astandard method. Noting that the integrand of Eq. (35)contains the p -dependence only in ˜ S g with the expres-sion (39), we can explicitly perform the integral over p and thereby obtain ∂ V R eff ∂A aµ ∂A bν = Π µν tree − g X τ = ± X E τi = E τj Z d p (2 π ) θ ( E τi ) − θ ( E τj ) E τi − E τj ( U † ˜Γ µa U ) ij ( U † ˜Γ νb U ) ji − g X τ = ± X E τi = E τj Z d p (2 π ) δ ( E τi )( U † ˜Γ µa U ) ij ( U † ˜Γ νb U ) ji − (counter term) . (40)We can derive similar formulae for other second deriva-tives, ∂ V R eff ( ∂ ∆) , ∂ V R eff ( ∂µ e ) , ∂ V R eff ∂ ∆ ∂A aµ , etc.. (41)It is also straightforward to extend the formulae to theversion with a finite temperature.The formula (40) has an advantage over the numericalderivative of the effective potential: The numerical sec-ond derivative requires a quite precise calculation for thefree energy and thereby it takes a long time. On the otherhand, we can reach a sufficiently accurate result via (40)in a reasonable time. Furthermore, in the expression of(40), it is clear that the contributions of the gapped andgapless modes to the Meissner masses are quite differ-ent. (Compare the second and third terms in Eq. (40).)This might help us to understand why the existence ofthe gapless modes yields a sudden change of the squaredMeissner mass for the 8th gluon at the border of the 2SCand g2SC phases [10]. B. Numerical analysis
Before the numerical calculation, we describe severalfeatures of the Meissner masses in the minimal cylindricalgluonic phase II.In this phase, the symmetry breaking structure is [15][ SU (3) c ] local × U (1) em × SO (3) rot∆ ,B −→ ˜˜ U (1) em × SO (2) rot , (42)where the unbroken ˜˜ U (1) em is connected with the newelectric charge ˜˜ Q em = Q em − √ T − T . The rotationalsymmetry breaking leads to different Meissner masses forthe transverse ( µ, ν = x, y ) and longitudinal ( µ, ν = z )modes. It is thus convenient to define( M ) Tab ≡ ∂ V R eff ∂A ax ∂A bx = ∂ V R eff ∂A ay ∂A by , (43)( M ) Lab ≡ ∂ V R eff ∂A az ∂A bz . (44)Since we took the unitary gauge (21), the squared Meiss- -0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 60 80 100 120 140 160 180 ( M ) T / M ∆ (MeV)g2SCLOFF gluonic 2SC FIG. 4: The squared Meissner masses of the transverse modeof the 4th gluon in the unit of M g [ ≡ α s µ / (3 π )]. Thebold solid, dashed and dot-dashed curves are for the minimalcylindrical gluonic phase II, the single plane-wave LOFF andthe 2SC/g2SC phases, respectively. The values Λ = 653 . µ = 400 MeV and α s = 1 were used. ner masses for the physical degrees of freedom are( M ) T,L = ( M ) T,L , ( M ) T = ( M ) T , (45)and ( M ) T,L , ( M ) T,L , ( M ) T , ( M ) T,L , (46)where the relations (45) hold owing to the unbroken˜˜ U (1) em symmetry. For the transverse modes of the 3rdand 8th gluons, it turns out that there exists a large mix-ing term ( M ) T , so that we define the diagonal mass-squared terms for them,( M ) T , diag , ( M ) T , diag . (47)Do there exist two Nambu-Goldstone (NG) bosonsconnected with the symmetry breaking SO (3) rot → SO (2) rot ? This is nontrivial because Goldstone’s theo-rem for relativistically invariant theories is not necessar-ily valid in noninvariant systems [40, 41]. We find thatthe answer is formally “yes” in this case, as we will seebelow.The point is that the rotational symmetry is sponta-neously broken only by the condensate h A z i 6 = 0 in theminimal cylindrical gluonic phase II. Therefore, beforetaking the z -direction, the effective potential should de-pend on the SO (3) rot invariant B ≡ X i = x,y,z h A i i , (48) -0.4-0.2 0 0.2 0.4 60 80 100 120 140 160 180 ( M ) T / M ∆ (MeV)g2SCgluonic 2SC FIG. 5: The squared Meissner masses of the transverse modeof the 8th gluon in the unit of M g [ ≡ α s µ / (3 π )]. The boldsolid and dot-dashed curves are for the minimal cylindricalgluonic phase II and the 2SC/g2SC phases, respectively. Thevalues Λ = 653 . µ = 400MeV and α s = 1 were used.In the g2SC phase, the squared Meissner mass monotonouslydecreases and goes to minus infinity at ∆ = 134 . i.e., V R eff = V R eff ( B ) . (49)We then find ∂ V R eff ∂ h A i i ∂ h A j i = 2 δ ij ∂V R eff ∂ B + 4 h A i ih A j i ∂ V R eff ( ∂ B ) . (50)Since we take the direction h A x,y i = 0, h A z i 6 = 0 and thegap equation for h A z i yields ∂V R eff ∂ B = 0, we formally findthat the squared Meissner mass for the transverse modeof the 6th gluon is vanishing,( M ) T = 0 . (51)It implies that A x,y correspond to the two NG bosons.A crucial difference between the gluonic and 2SC/g2SCphases is the existence of the tree gluon potential term.Although the effect is negligible for the free energy in theminimal cylindrical gluonic phase II, it is quite importantfor the Meissner masses. Neglecting the suppressed terms ∼ O ( µ ) , O ( µ µ ) , O ( µ ), we obtain the tree terms of thesquared Meissner masses:( M ) T , ≃ ( M ) T , ≃ ( M ) T = B , (52a)( M ) T ≃ B , ( M ) T = 34 B , (52b)( M ) T = − √ B . (52c) -0.3-0.2-0.1 0 0.1 0.2 60 80 100 120 140 160 180 ( M ) L66 / M ∆ (MeV)g2SCLOFFgluonic 2SC FIG. 6: The squared Meissner masses of the longitudinalmode of the 6th gluon in the unit of M g [ ≡ α s µ / (3 π )].The bold solid, dashed and dot-dashed curves are for theminimal cylindrical gluonic phase II, the single plane-waveLOFF and the 2SC/g2SC phases, respectively. The valuesΛ = 653 . µ = 400MeV and α s = 1 were used. Thus the transverse modes except for ( M ) T and( M ) T , diag have the positive and large contributions ofthe order of B . (For the values of B , see Fig. 1.) Thisis one of the reasons why the Meissner masses tend to bepositive compared with those in the 2SC/g2SC phase.We also note some features of the Meissner masses forthe single-plane wave 2SC-LOFF phase. The A − µ glu-ons should be massless and the relations( M ) T,L = ( M ) T,L = ( M ) T,L = ( M ) T,L (53)hold because of the unbroken SU (2) c gauge symmetry.In addition, similarly to (51), we formally obtain( M ) T = 0 . (54)Let us now turn to the numerical analysis of the Meiss-ner masses.We depict the results in Figs.4–8 in the unit of M g [ ≡ α s µ / (3 π )]. In the analysis, we used µ = 400MeV, Λ =653 . α s = 1.For the minimal cylindrical gluonic phase II, thesquared Meissner masses ( M ) T = ( M ) T are positivein the region 65.4MeV < ∆ < < ∆ < M ) T becomesnegative in the small region around ∆ ∼ M ) T , ( M ) T and ( M ) T ,the instability in ( M ) T is converted into ( M ) T , diag , -0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 60 80 100 120 140 160 180 ( M ) L88 / M ∆ (MeV)g2SCLOFF gluonic2SC FIG. 7: The squared Meissner masses of the longitudinalmode of the 8th gluon in the unit of M g [ ≡ α s µ / (3 π )].The bold solid, dashed and dot-dashed curves are for theminimal cylindrical gluonic phase II, the single plane-waveLOFF and the 2SC/g2SC phases, respectively. The valuesΛ = 653 . µ = 400MeV and α s = 1 were used. As inFig.5, the curve for the g2SC phase below the figure is cut off. because we define the diagonalized squared masses as( M ) T , diag < ( M ) T , diag .) It is quite noticeable thatthere exist spikes and valleys around ∆ ∼ ∼ α s ? Although the dynamical solutions of ∆, B , δµ , µ and µ in the minimal cylindrical gluonic phase IIare almost independent of α s [32], the Meissner massesfor the transverse modes can be sensitive. Note thatthe one-loop contributions are proportional to α s andthus the influence of the tree contributions (52) is rel-atively stronger (weaker) as the values of α s decrease(increase). For example, ( M ) T becomes negative at∆ = 140 , , α s = 0 . , . , .
15, respec-tively. For the other transverse modes in Eq. (52), thereshould appear similar sensitivities. On the other hand,for the longitudinal modes, the ratio ( M ) Lab /M g is in-sensitive to α s , because the tree contributions are sup-pressed.For the single plane wave LOFF phase, like in the min-imal cylindrical gluonic phase II, the transverse modes ofthe 4-7th gluons are the most problematic. However thechromomagnetic instability occurs in the earlier region,80MeV < ∆ < ∼ ( M ) T , L11 / M ∆ (MeV)(M ) L11 (M ) T11 ( M ) L33 / M , ( M ) T / M ∆ (MeV)(M ) L33 (M ) T77 -0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 60 80 100 120 140 160 ( M ) T , , / M ∆ (MeV)(M ) T33 (M ) T38 (M ) T88 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 60 80 100 120 140 160 ( M ) T , d i ag / M , ( M ) T , d i ag / M ∆ (MeV)(M ) T33,diag (M ) T88,diag
FIG. 8: The squared Meissner masses for several gluonicmodes in the minimal cylindrical gluonic phase II in theunit of M g [ ≡ α s µ / (3 π )]. The values Λ = 653 . µ = 400MeV and α s = 1 were used. In the right bottomfigure, we showed the diagonalized mass squared terms forthe transverse modes of the 3rd and 8th gluons. these results are consistent with the analysis based onthe HDL approximation [33, 34].The 2SC/g2SC phase has the chromomagnetic in-stability numerically in the region 92.2MeV < ∆ < A − µ gluons acquire the non-HDL contribu-tions like ∆ log Λ / ∆ in the sharp-cutoff regularizationscheme [12, 44]. In order to settle this problem, a moresophisticated regularization scheme is required. It willbe studied elsewhere. IV. SUMMARY AND DISCUSSIONS
We analyzed the Meissner screening masses in thesimplest gluonic phase (the minimal cylindrical glu-onic phase II) as well as the single plane wave LOFFand 2SC/g2SC phases. We derived the formulae forthe Meissner masses without any help of the numericalderivative. It was found that in the formulae the gap-less mode makes the contribution characterized by theDirac’s δ -function. We showed that the simplest gluonic phase removes the chromomagnetic instability in the re-gion 65.4MeV < ∆ < < ∆ < . µ = 400MeV and α s = 1. The2SC phase does not have the chromomagnetic instabilityin the strong coupling regime ∆ > < ∆ < < ∆ < ∼ ∼ ∼ ∼ B takes its maximum around ∆ ∼ B ≃ δµ is satisfied around ∆ ∼ µ = √ / g h A i and K = g h A y i = g h A z i [15, 32] ishopeful, for example. An important point is that the freeenergy for the GCSL phase is slightly lower than that forthe minimal cylindrical gluonic phase II [32]. Inhomoge-neous gluonic phases are also interesting [27, 45].Independently of the chromomagnetic instability, theSarma instability for the diquark Higgs mode should beremoved as well. This problem will be considered else-where.0 Acknowledgments
The authors thank V. A. Miransky for fruitful discus-sions. J.J. acknowledges useful discussions with Razvan Nistor. The numerical calculations were carried out onAltix3700 BX2 at YITP in Kyoto University. The workwas supported by the Natural Sciences and EngineeringResearch Council of Canada. [1] D. Bailin and A. Love, Phys. Rept. , 325 (1984).[2] M. Iwasaki and T. Iwado, Phys. Lett. B , 163 (1995).[3] M. G. Alford, K. Rajagopal and F. Wilczek, Phys. Lett.B , 247 (1998); R. Rapp, T. Sch¨afer, E. V. Shuryakand M. Velkovsky, Phys. Rev. Lett. , 53 (1998).[4] D. Ivanenko and D. F. Kurdgelaidze, Astrofiz. , 479(1965); Lett. Nuovo Cim. , 13 (1969); N. Itoh, Prog.Theor. Phys. , 291 (1970); F. Iachello, W. D. Langerand A. Lande, Nucl. Phys. A , 612 (1974).[5] For a recent comprehensive review, see, e.g., M. G. Al-ford, K. Rajagopal, T. Schaefer and A. Schmitt, hep-ph/0709.4635.[6] K. Iida and G. Baym, Phys. Rev. D , 074018 (2001)[Erratum-ibid. D , 059903 (2002)].[7] M. Alford and K. Rajagopal, JHEP , 031 (2002).[8] A. W. Steiner, S. Reddy and M. Prakash, Phys. Rev. D , 094007 (2002).[9] M. Huang, P. f. Zhuang and W. q. Chao, Phys. Rev. D , 065015 (2003).[10] M. Huang and I. A. Shovkovy, Phys. Rev. D ,051501(R) (2004); ibid , 094030 (2004).[11] R. Casalbuoni, R. Gatto, M. Mannarelli, G. Nardulli andM. Ruggieri, Phys. Lett. B , 362 (2005) [Erratum-ibid. B , 297 (2005)].[12] M. Alford and Q. Wang, J. Phys. G , 719 (2005).[13] K. Fukushima, Phys. Rev. D , 074002 (2005).[14] E. V. Gorbar, M. Hashimoto and V. A. Miransky, Phys.Lett. B , 305 (2006).[15] E. V. Gorbar, M. Hashimoto and V. A. Miransky, Phys.Rev. D , 085012 (2007).[16] M. G. Alford, J. A. Bowers and K. Rajagopal, Phys. Rev.D , 074016 (2001). For a review, see, e.g., R. Casal-buoni and G. Nardulli, Rev. Mod. Phys. , 263 (2004).[17] J. A. Bowers and K. Rajagopal, Phys. Rev. D , 065002(2002).[18] S. Reddy and G. Rupak, Phys. Rev. C , 025201 (2005).[19] M. Huang, Phys. Rev. D , 045007 (2006).[20] D. K. Hong, hep-ph/0506097.[21] A. Kryjevski, hep-ph/0508180.[22] T. Sch¨afer, Phys. Rev. Lett. , 012305 (2006).[23] R. Casalbuoni, R. Gatto, N. Ippolito, G. Nardulli andM. Ruggieri, Phys. Lett. B , 89 (2005) [Erratum-ibid.B , 565 (2006)]. [24] K. Rajagopal and R. Sharma, Phys. Rev. D , 094019(2006).[25] M. Mannarelli, K. Rajagopal and R. Sharma,hep-ph/0702021.[26] R. Gatto and M. Ruggieri, Phys. Rev. D , 114004(2007).[27] E. V. Gorbar, M. Hashimoto, V. A. Miransky andI. A. Shovkovy, Phys. Rev. D , 111502(R) (2006).[28] G. Sarma, J. Phys. Chem. Solids , 1029 (1963).[29] M. Hashimoto, Phys. Lett. B , 93 (2006).[30] K. Iida and K. Fukushima, Phys. Rev. D , 074020(2006).[31] I. Giannakis, D. Hou, M. Huang and H. c. Ren, Phys.Rev. D , 011501(R) (2007); ibid , 014015 (2007).[32] M. Hashimoto and V. A. Miransky, Prog. Theor. Phys. , 303 (2007).[33] I. Giannakis and H. C. Ren, Phys. Lett. B , 137(2005); Nucl. Phys. B , 255 (2005); I. Giannakis,D. f. Hou and H. C. Ren, Phys. Lett. B , 16 (2005).[34] E. V. Gorbar, M. Hashimoto and V. A. Miransky, Phys.Rev. Lett. , 022005 (2006).[35] K. Fukushima, Phys. Rev. D , 094016 (2006).[36] O. Kiriyama, D. H. Rischke and I. A. Shovkovy, Phys.Lett. B , 331 (2006).[37] O. Kiriyama, hep-ph/0709.1083.[38] A. Gerhold and A. Rebhan, Phys. Rev. D , 011502(R)(2003); D. D. Dietrich and D. H. Rischke, Prog. Part.Nucl. Phys. , 305 (2004).[39] M. Buballa and I. A. Shovkovy, Phys. Rev. D , 097501(2005).[40] H. B. Nielsen and S. Chadha, Nucl. Phys. B , 445(1976).[41] V. A. Miransky and I. A. Shovkovy, Phys. Rev. Lett. ,111601 (2002); T. Schafer, D. T. Son, M. A. Stephanov,D. Toublan and J. J. M. Verbaarschot, Phys. Lett. B , 67 (2001).[42] O. Kiriyama, Phys. Rev. D , 074019 (2006); ibid ,114011 (2006).[43] L. He, M. Jin and P. Zhuang, Phys. Rev. D , 036003(2007).[44] D. H. Rischke, Phys. Rev. D62