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Spontaneous current generation in the gapless 2SC phase∗ Mei Huang† Physics Department, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Itisfoundthat,exceptchromomagneticinstability,thegapless2SCphasealsoexhibitsaparam- agneticresponsetotheperturbationofanexternalcolorneutralbaryoncurrent. Thespontaneously generated baryon current driven by the mismatch is equivalent to the one-plane wave LOFF state. Wedescribethe2SCphaseinthenonlinearrealizationframework,andshowthateachinstabilityin- dicatesthespontaneousgenerationofthecorrespondingpseudoNambu-Golstonecurrent. Weshow thisNambu-Goldstonecurrentsgenerationstatecoversthegluonphaseaswellastheone-planewave LOFF state. We further point out that, when charge neutrality condition is required, there exists a narrow unstable LOFF (Us-LOFF) window, where not only off-diagonal gluons but the diagonal 8-thgluon cannotavoid themagneticinstability. Wediscussthatthediagonal magneticinstability inthisUs-LOFFwindowcannotbecuredbyoff-diagonalgluoncondensateincolorsuperconducting 6 phase,anditwillalsoshowupinsomeconstrainedAbelianasymmetricsuperfluid/superconducting 0 system. 0 2 PACSnumbers: 12.38.-t,12.38.Aw,26.60.+c n a J I. INTRODUCTION vor structure of the system. The gapless state resembles 6 theunstableSarmastate[15,16]. However,underanat- 2 uralchargeneutralitycondition,i.e.,onlyneutralmatter Sufficiently cold and dense baryonic matter is a color 2 superconductor, such a state of matter may exist in the can exist, the gapless phase is indeed a thermal stable state as shown in [13, 14]. The existence of thermal sta- v centralregionofcompactstars. Forthisreason,thetopic ble gapless color superconducting phases was confirmed 5 of color superconductivity stirred a lot of interest in re- 3 cent years [1, 2, 3, 4]. (For reviews on color supercon- in Refs. [17] and generalized to finite temperatures in 2 Refs. [18]. Recent results based on more careful numer- ductivity see, for example, Ref. [5].) 4 ical calculations show that the g2SC and gCFL phases To form bulk matter inside compact stars, the charge 0 can exist at moderate baryon density in the color super- 5 neutralityconditionaswellasβ equilibriumarerequired conducting phase diagram [19] . 0 [6, 7, 8]. This induces mismatch between the Fermi sur- / faces of the pairing quarks. It is clear that the Cooper One of the most important properties of an ordinary h pairing will be eventually destroyed with the increase of superconductor is the Meissner effect, i.e., the supercon- p - mismatch. However, there are no certain answer on the ductor expels the magnetic field [20]. In ideal color su- p truegroundstateofneutralcold-densequarkmatterwith perconducting phases, e.g., in the 2SC and CFL phases, e a moderate mismatch. the gauge bosons connected with the broken generators h obtain masses, which indicates the Meissner screening : Withouttheconstraintfromthechargeneutralitycon- v effect [21]. The Meissner effect can be understood us- dition, the systemmayexhibit afirstorderphasetransi- i ingthestandardAnderson-Higgsmechanism. Unexpect- X tionfromthe colorsuperconducting phaseto the normal edly, it was found that in the g2SC phase, the Meissner phasewhenthemismatchincreases[9]. Itwasalsofound r a that the system can experience a spatial non-uniform screening masses for five gluons correspondingto broken generatorsof SU(3) become imaginary,which indicates LOFF (Larkin-Ovchinnikov-Fudde-Ferrell)state [10, 11] c a type of chromomagnetic instability in the g2SC phase in a certain window of moderate mismatch. [22, 23]. The calculations in the gCFL phase show the The chargeneutrality condition plays anessentialrole same type of chromomagnetic instability [24]. Remem- indeterminingthegroundstateoftheneutralsystem. If bering the discovery of superfluidity density instability the charge neutrality condition is satisfied globally, and [25] in the gapless interior-gap state [26], it seems that also if the surface tension is small, the mixed phase will the instability is a inherent property of gapless phases, be favored [12]. It is difficult to precisely calculate the thoughtheresultinRef. [27]showsthatthereisnochro- surfacetensioninthemixedphase,thusinthefollowing, momangetic instability near the critical temperature. we would like to focus on the homogeneous phase when the charge neutrality condition is required locally. The chromomagnetic instability in the gapless phase It was found that homogeneous neutral cold-dense still remains as a puzzle. By observing that, the 8-th quarkmattercanbeinthegapless2SC(g2SC)phase[13] gluon’s chromomagnetic instability is related to the in- or gaplessCFL (gCFL) phase [14], depending onthe fla- stability with respect to a virtual net momentum of diquark pair, Giannakis and Ren suggested that a LOFF state might be the true ground state [28]. Their further calculationsshowthatthereisnochromomagneticinsta- ∗TKYNT-05-8 bility inanarrowLOFFwindowwhenthe localstability †[email protected] condition is satisfied [29, 30]. Latter on, it was pointed 2 outinRef. [31]thatachargeneutralLOFFstatecannot color triplet, as well as a four-component Dirac spinor, cure the instability of off-diagonal 4-7th gluons, while a τ = (τ1,τ2,τ3) are Pauli matrices in the flavor space, gluoncondensatestate[32]candothejob. Wewillpoint whereτ2 isantisymmetric,and(ε)ik εik,(ǫb)αβ ǫαβb ≡ ≡ out that, when charge neutrality condition is required, are totally antisymmetric tensors in the flavor and color there exists anothernarrowunstable LOFF window,not spaces. µîsthematrixofchemicalpotentialsinthecolor onlyoff-diagonalgluonsbutthe diagonal8-thgluoncan- and flavor space. In β-equilibrium, the matrix of chemi- not avoid the magnetic instability. calpotentialsinthecolor-flavorspaceµˆ isgiveninterms This paper is organized as follows. In Sec. II, we of the quark chemical potential µ, the chemical poten- describe the framework of the gauged SU(2) Nambu– tial for the electrical charge µe and the color chemical Jona-Lasinio(NJL) model in β-equilibrium. In Sec. III, potential µ , 8 we probe the g2SC phase using different external fields, 2 andwe find that, exceptchromomagneticinstability, the µαijβ =(µδij −µeQij)δαβ + √3µ8δij(T8)αβ. (2) g2SC phase is also unstable with respect to a color neutral baryon current. Then in Sec. IV, we propose that GS and GD are the quark-antiquark coupling constant a baryon current can be spontaneously generated in the and the diquark coupling constant, respectively. In the ground state due to the mismatch, and this baryon cur- following, we only focus on the color superconducting rentgenerationstateresemblestheone-planewaveLOFF phase, where <q¯q >=0 and <q¯γ5τq >=0. state. We further analyze the instabilities of the LOFF After bosonization, one obtains the linearized version state under the requirement of charge neutrality condi- of the model for the 2-flavor superconducting phase, tionandpointoutthatthereexistsanarrowLOFFwin- ∆∗b∆b dow,wherenotonlyoff-diagonalgluonsbutthediagonal = q¯(iD/ +µˆγ0)q 2SC L − 4G 8-th gluon, cannot avoid the magnetic instability. Sec. D 1 1 V gives a general Nambu-Goldstone currents generation ∆∗b(iq¯Cεǫbγ q) ∆b(iq¯εǫbγ qC) (3) 5 5 description in the non-linearization framework. At the − 2 − 2 end, we give the discussion and summary in Sec. VI. with the bosonic fields ∆b iq¯Cεǫbγ q, ∆∗b iq¯εǫbγ qC. (4) 5 5 ∼ ∼ II. THE GAUGED SU(2) In the Nambu-Gor’kov space, NAMBU–JONA-LASINIO (GNJL) MODEL q Ψ= , (5) qC We take the gauged form of the extended Nambu– (cid:18) (cid:19) Jona-Lasinio model [33], the Lagrangiandensity has the the inverse of the quark propagator is defined as form of [ (P)]−1 = G+0(P) −1 ∆− , (6) = q¯(iD/ +µˆγ0)q+GS[(q¯q)2+(q¯iγ5τq)2] S (cid:2) ∆+(cid:3) G−0(P) −1 ! L + GD[(iq¯Cεǫbγ5q)(iq¯εǫbγ5qC)], (1) with the off-diagonal elements (cid:2) (cid:3) gwliutohnDfiµeld≡s∂aµnd−pihgoAtoaµnTfiae−ldi,eTAaµaQn.dHQeraereAtaµheagnedneArµataorres and the∆fre−e≡qu−airǫkbεpγr5o∆pa,gator∆s+G≡±(−Pi)ǫbtεaγk5i∆ng∗,the fo(r7m) 0 ofSU(3)candU(1)EMgaugegroups,respectively. Please of notethatweregardallthegaugefieldsasexternalfields, whichareweaklyinteractingwiththesystem. Theprop- G±0(P) −1 =γ0(p0±µˆ)−~γ·p~. (8) erty ofthe colorsuperconducting phase characterizedby The 4-mome(cid:2)nta are(cid:3)denoted by capital letters, e.g., P = the diquark gapparameter is determined by the unkown (p ,p~). We have assumed the quarks are massless in 0 nonperturbative gluon fields, which has been simply re- dense quark matter, andthe externalgluonfields do not placedbythefour-fermioninteractionintheNJLmodel. contribute to the quark self-energy. While, the external gluon fields do not contribute to the The explicit form of the functions G± and Ξ± reads I IJ properties of the system. Therefore, we do not have the contribution to the Lagrangian density from gauge field G± = (k0−Ed±g)γ0Λ˜+k + (k0+Ed∓g)γ0Λ˜−k , npoanrt-lLingeaarsrienatlrizoadtuiocendininth[3e2g].N(JILnmSeocd.eVl,,wbeywuilslindgertihvee 1 (k0∓δµ)2−E∆±2 (k0∓δµ)2−E∆∓2 oneNambu-Goldstonecurrentsstate,whichisequivalent G± = (k0−Eu±r)γ0Λ˜+k + (k0+Eu∓r)γ0Λ˜−k , to the so-called gluon-condensate state.) 2 (k δµ)2 E±2 (k δµ)2 E∓2 In the Lagrangian density Eq. (1), qC = Cq¯T, q¯C = 0± − ∆ 0± − ∆ 1 1 qTC are charge-conjugate spinors, C = iγ2γ0 is the G± = γ0Λ˜++ γ0Λ˜−, 3 k +E± k k E∓ k charge conjugation matrix (the superscript T denotes 0 bu 0− bu wthiethtrian=spous,idtioanndopαera=tiorn,)g.,bTihseaqfluaavrkorfideoldubqle≡t aqniαd G±4 = k +1E±γ0Λ˜+k + k 1E∓γ0Λ˜−k, 0 bd 0− bd 3 with E± E µ and in Ref. [23], iα ≡ k± iα i∆γ5Λ˜+ i∆γ5Λ˜− Ξ± = − k + − k , Πµν(P) = Π˜µν(P)+Πµν(P), (14) 12 (k δµ)2 E±2 (k δµ)2 E∓2! V V V,b 0± − ∆ 0± − ∆ T d3k Ξ±21 = (k0∓−iδ∆µγ)25Λ−˜+kE∆±2 + (k0∓−iδ∆µγ)25Λ˜−−kE∆∓2!, Π˜µVν(P) = γ2µXGn+1(ZK)(γ2πνG)3+1T(rKD′[)+γµG−1(K)γνG−1(K′) + γµG+(K)γνG+(K′)+γµG−(K)γνG−(K′) where 2 2 2 2 γµΞ−(K)γνΞ+(K′) γµΞ+(K)γνΞ−(K′) Λ˜±k = 12 1±γ0γ·kE−m (11) −− γµΞ1−221(K)γνΞ2+112(K′)−−γµΞ1+221(K)γνΞ2−112(K′) , (cid:18) k (cid:19) T d3k Πµν(P) = Tr [ (cid:3) is analternative setofenergy projectors,and the follow- V,b 2 (2π)3 D n Z ing notation was used: γµXG+(K)γνG+(K′)+γµG−(K)γνG−(K′) 3 3 3 3 Ek± ≡ Ek±µ¯, + γµG+4(K)γνG+4(K′)+γµG−4(K)γνG−4(K′) , E± (E±)2+∆2, (cid:3) ∆,k ≡ k here the trace is over the Dirac space. qµ +µ µ +µ µ µ µ¯ ur dg = ug dr =µ e + 8, ComparingtheexplicitexpressionofΠµν(P)withthat ≡ 2 2 − 6 3 V of the 8-th gluon’s self-energy Πµν(P), i.e., Eq. (55) δµ µdg−µur = µdr−µug = µe. in Ref. [23] , it can be clearly se8e8n that, Πµν(P) and ≡ 2 2 2 Πµν(P)almostsharethe sameexpression,exceVpttheco- 88 efficients. This can be easily understood, because the color charge and color current carried by the 8-th gluon III. PARAMAGNETIC RESPONSE TO A is proportional to the baryon number and baryon cur- COLOR NEUTRAL BARYON CURRENT rent, respectively. In the static long-wavelength (p = 0 0 and p~ 0 ) limit, the time-component and spatial com- It is not understood why gapless color superconduct- ponent→of Πµν(P) give the baryon number susceptibility V ing phasesexhibit chromomagneticinstability. It sounds ξ and baryon current susceptibility ξ , respectively, n c quite strange especially in the g2SC phase, where it is the electrical neutrality not the color neutrality playing the essential role. It is a puzzle why the gluons can feel ξ limΠ˜00(0,p~) m2 , (15) the instability by requiring the electrical neutrality on n ≡ −p~→0 V ∝ 8,D the system. In order to understand what is really go- ξ 1 lim g + pipj Πij(0,~p) m2 . (16) ing ‘wrong’ with the homogeneous g2SC phase, we want c ≡ −2p~→0 ij p2 V ∝ 8,M (cid:18) (cid:19) to know whether there exists other instabilities except the chromomagnetic instability. For that purpose, we probe the g2SC phase using different external sources, Intheg2SCphase,m2 aswellasξ becomenegative. 8,M c e.g., scalar and vector diquarks, mesons, vector current, This means that, except the chromomagnetic instability andsoon. Inthispaper,weonlyreportthemostinterest- corresponding to broken generators of SU(3) , and the c ingresultregardingtheresponseoftheg2SCphasetoan instabilityofanetmomentumfordiquarkpair,theg2SC externalvectorcurrentVµ =ψ¯γµψ,thetime-component phase is also unstable with respect to an external color andspatial-componentsofthiscurrentcorrespondtothe neutral baryon current ψ¯~γψ. baryonnumber density andbaryoncurrent,respectively. The 8-thgluon’s magneticinstability,the diquarkmo- From the linear response theory, the induced current mentum instability and the color neutral baryoncurrent andtheexternalvectorcurrentisrelatedbytheresponse in the g2SC phase can be understood in one common function Πµν(P), V physical picture. The g2SC phase exhibits a paramagnetic response to an external baryon current. Naturally, 1 T Πµν(P)= Tr Γˆµ (K)Γˆν (K P) . (13) the color currentcarriedby the 8-th gluon, which differs V 2 V VS VS − from the baryon current by a color charge, also experi- XK h i ences the instability in the g2SC phase. The paramag- ThetracehererunsovertheNambu-Gorkov,flavor,color netic instability of the baryon current indicates that the and Dirac indices. The explicit form of vertices is Γˆµ quark can spontaneously obtain a momentum, because diag(γµ, γµ). V ≡ diquark carries twice of the quark momentum, it is not − The explicit expression of the vector current response hard to understand why the g2SC phase is also unstable functioncanbeevaluateddirectlybyusingthenotations withrespecttotheresponseofanetdiquarkmomentum. 4 IV. SPONTANEOUS BARYON CURRENT is in the 2SC phase when δµ < 0.706∆0 with ∆ ∆0, ≃ GENERATION AND THE LOFF STATE in the LOFF phase when 0.706∆ < δµ < 0.754∆ cor- 0 0 respondingly 0 < ∆/∆ < 0.242, and then in the nor- 0 A. Spontaneous baryon current generation mal phase with ∆=0 when the mismatch is larger than 0.754∆ . Here∆,∆ indicatethediquarkgapinthecase 0 0 of δµ=0 and δµ=0, respectively. The paramagnetic response to an external vector cur- 6 rentnaturallysuggeststhatavectorcurrentcanbespon- taneouslygeneratedinthesystem. Thegeneratedvector currentbehavesasavectorpotential,whichmodifiesthe B. Unstable neutral LOFF window quarkself-energywithaspatialvectorcondensate~γ Σ~ , V · andbreakstherotationalsymmetryofthesystem. Itcan NowcometothechargeneutralLOFFstate,andinves- also be understood that the quasiparticles in the gapless tigate whether the LOFF state can resolve all the mag- phasespontaneouslyobtainasuperfluidvelocity,andthe netic instabilities. ground state is in an anisotropic state. The quark prop- When charge neutrality condition is required, the agator G±(P) in Eq. (8) is modified as ground state of charge neutral quark matter should be 0 determined by solving the gap equations as well as the −1 G± (P) =γ0(p µˆ) ~γ ~p ~γ Σ~ , (17) charge neutrality condition, i.e., 0,V 0± − · ∓ · V h i ∂Γ ∂Γ ∂Γ ∂Γ with a subscript V indicating the modified quark propa- =0, =0, =0, =0. (20) gator. Correspondingly, the inverse of the quark propa- ∂ΣV ∂∆ ∂µe ∂µ8 gator [ (P)]−1 in Eq. (6) is modified as S Bychanging∆ orcouplingstrengthG ,thesolutionof 0 D −1 thechargeneutralLOFFstatecanstayeverywhereinthe G+ (P) ∆− [ (P)]−1 = 0,V . (18) full LOFF window, including the window not protected SV  h ∆+ i G− (P) −1  by the local stability condition, as shown explicitly in 0,V Ref. [31].    h i  Fromthelessonofchargeneutralg2SCphase,welearn It is noticed that the expression of the modified inverse quark propagator [S (P)]−1 takes the same form as the that even though the neutral state is a thermal sta- V ble state, i.e., the thermodynamic potential is a global inverse quark propagator in the one-plane wave LOFF minimum along the neutrality line, it cannot guarantee state shown in Ref. [29]. The net momentum ~q of the the dynamical stability of the system. The stability of diquark pair in the LOFF state [29] is replaced here by the neutral system should be further determined by the a spatial vector condensate Σ~ . The spatial vector con- V dynamical stability condition, i.e., the positivity of the densate~γ Σ~ breaksrotationalsymmetryofthesystem. · V Meissner mass square. ThismeansthattheFermisurfacesofthepairingquarks The polarization tensor for the gluons with color A= are not spherical any more. 4,5,6,7,8 should be evaluated using the modified quark It has to be pointed out, the baryon current offers propagator in Eq. (18), i.e., V one Doppler-shift superfluid velocity for the quarks. A S spontaneously generated Nambu-Goldstone current in 1 T the minimal gapless model [41] or a condensate of 8-th ΠµAνB(P)= 2 V Tr ΓˆµASV(K)ΓˆνBSV(K−P) , gluon’sspatialcomponentcandothesamejob. Allthese XK h i (21) states mimic the one-plane waveLOFFstate. In the fol- with A,B = 4,5,6,7,8 and the explicit form of the ver- lowing, we just call all these states as the single-plane tices Γˆµ has the form Γˆµ diag(gγµT , gγµTT). In wave LOFF state. A A ≡ A − A the LOFFstate, the Meissnertensorcanbe decomposed In order to determine the deformed structure of the into transverse and longitudinal component. The trans- Fermisurfaces,oneshouldself-consistentlyminimize the verse and longitudinal Meissner mass square for the off- free energy Γ(Σ ,∆,µ,µ ,µ ). The explicit form of the V e 8 diagonal 4-7 gluons and the diagonal 8-th gluon have free energy can be evaluated directly using the standard been performed explicitly in the one-plane wave LOFF method,intheframeworkofNambu–Jona-Lasiniomodel state in Ref. [29]. [8, 13], it takes the form of According to the dynamical stability condition, i.e., T d3p~ ∆2 the positivity of the transverse as well as longitudinal Γ= Trln([ (P)]−1)+ , (19) −2 (2π)3 SV 4G Meissner mass square, we can devide the LOFF state D n Z X into three LOFF windows [34]: whereT isthe temperature,andG isthecouplingcon- 1) The stable LOFF (S-LOFF) window in the region D stant in the diquark channel. of0<∆/∆ <0.39,whichis freeofanymagnetic insta- 0 When there is no charge neutrality condition, the bility. Please note that this S-LOFF window is a little groundstateisdeterminedbythethermalstabilitycondi- bit wider than the window 0<∆/∆ <0.242 protected 0 tion, i.e., the local stability condition. The ground state by the local stability condition. 5 2)The stable windowfor diagonalgluoncharacterized 1) The superconducting phase is charaterized by the by Ds-LOFF window in the region of 0.39 < ∆/∆ < nonzerovacuumexpectationvalue,i.e.,<∆>=0,which 0 6 0.83, which is free of the diagonal 8-th gluon’s magnetic means the amplitude of the gap is finite, and the phase instability but not free of the off-diagonal gluons’ mag- coherence is also established. netic instability; 2)Iftheamplitudeisstillfinite,whilethephasecoher- 3) The unstable LOFF (Us-LOFF) window in the re- enceislost,thisphaseisinaphasedecoherentpseudogap gion of 0.83< ∆/∆ <r , with r ∆(δµ =∆)/∆ state characterized by ∆ = 0, but < ∆ >= 0 because 0 c c 0 1. In this Us-LOFF window, all th≡e magnetic instabil≃i- of <eiϕ(x) >=0. | | 6 ties exist. Please note that, it is the longitudinal Meiss- 3) The normal state is characterized by ∆ =0. | | ner mass square for the 8-th gluon is negative in this There are two ways to destroy a superconductor. One Us-LOFF window, the transverse Meissner mass square way is by driving the amplitude of the order parameter of 8-th gluon is always zero in the full LOFF window, to zero. This way is BCS-like, because it mimics the which is guaranteed by the momentum equation. behavior of a conventionalsuperconductor at finite tem- Us-LOFFisaveryinterestingwindow,itindicatesthat perature,the gapamplitude monotonouslydropsto zero the LOFF state even cannot cure the 8-th gluon’s mag- with the increase of temperature; Another way is non- netic instability. Inthechargeneutral2-flavorsystem,it BCSlike,butBerezinskii-Kosterlitz-Thouless(BKT)-like seemsthatthediagonalgluon’smagneticinstabilitycan- [36],eveniftheamplitudeoftheorderparameterislarge not be cured in the gluon phase, because there is no di- and finite, superconductivity will be lost with the de- rectrelationbetweenthediagonalgluon’sinstabilityand struction of phase coherence, e.g. the phase transition the off-diagonal gluons’ instability. (Of course, it has to fromthe d wavesuperconductortothe pseudogapstate − be carefully checked, whether all the instabilities in this in high temperature superconductors [37]. Us-LOFF window can be cured by off-diagonal gluons’ Stimulating by the role of the phase fluctuation in condensate in the charge neutral 2-flavor system.) It is the unconventional superconducting phase in condensed alsonoticedthatinthisUs-LOFFwindow,themismatch matter, we follow Ref. [38] to formulate the 2SC phase is close to the diquark gap, i.e., δµ ∆. Therefore it is in the nonlinear realization framework in order to natu- ≃ interesting to check whether this Us-LOFF window can rally take into account the contribution from the phase be stabilized by a spin-1 condensate [35] as proposed in fluctuation or pseudo Nambu-Goldstone current. Ref. [41]. In the 2SC phase, the color symmetry G = SU(3) c In the charge neutral 2SC phase, though it is un- breaks to H = SU(2) . The generators of the residual c likely, we might have a lucky chance to cure the diago- SU(2) symmetry H are Sa = Ta with a =1,2,3 and c nalinstabilitybythecondensationofoff-diagonalgluons. the broken generators X{b = Tb+3} with b = 1, ,5. { } ··· However, this instability still exists in some constrained Moreprecisely,thelastbrokengeneratorisacombination Abelian asymmetric superfluidity system, and it’s a new ofT andthegenerator1oftheglobalU(1)symmetryof 8 challenge for us to really solve this problem. baryon number conservation, B (1+√3T )/3 of gen- 8 ≡ erators of the global U(1) and local SU(3) symmetry. B c The coset space G/H is parameterized by the group V. SPONTANEOUS NAMBU-GOLDSTONE elements CURRENTS GENERATION 7 1 (x) exp i ϕ (x)T + ϕ (x)B , (22) a a 8 We have seen that, except chromomagnetic instability V ≡ " √3 !# a=4 corresponding to broken generatorsof SU(3) , the g2SC X c phaseisalsounstablewithrespecttotheexternalneutral here ϕ (a = 4, ,7) and ϕ are five Nambu-Goldstone a 8 baryoncurrent. It is noticed that all the instabilities are diquarks, and·w··e have neglected the singular phase, induced by increasing the mismatch between the Fermi which should include the information of the topological surfaces of the Cooper pairing. In order to understand defects [39, 40]. Operator is unitary, −1 = †. the instability driven by mismatch, in the following, we Introducing a new quarVk field χ, whVich is cVonnected give some general analysis. with the original quark field q in Eq. (3) in a nonlinear A superconductor will be eventually destroyed and transformation form, goes to the normal Fermi liquid state, so one natural questionis: howanidealBCSsuperconductorwillbede- q = χ , q¯=χ¯ † , (23) stroyedbyincreasingmismatch? Toanswerhowasuper- V V conductorwillbedestroyed,onehastofirstlyunderstand and the charge-conjugate fields transform as whatis a superconductor. The superconducting phase is characterized by the order parameter ∆(x), which is a q = ∗χ , q¯ =χ¯ T . (24) C C C C V V complex scalar field and has the form of e.g., for electrical superconductor, ∆(x) = ∆eiϕ(x), with ∆ the In high-T superconductor, this technique is called c | | | | amplitude and ϕ the phase of the gap order parameter charge-spin separation, see Ref. [39]. The advantage of or the pseudo Nambu-Goldstone boson. transforming the quark fields is that this preserves the 6 simplestructureofthetermscouplingthequarkfieldsto from the topologic defects. The advantage of the non- the diquark sources, linear realization framework Eq. (28) is that it can naturally take into account the contribution from the phase q¯C∆+q χ¯CΦ+χ , q¯∆−qC χ¯Φ−χC . (25) fluctuations or Nambu-Goldstone currents. ≡ ≡ The task left is to correctly solve the ground state In mean-field approximation, the diquark source terms by considering the phase fluctuations. The free energy are proportionalto Γ(V ,∆,µ,µ ,µ ) can be evaluated directly and it takes µ 8 e Φ+ χ χ¯ , Φ− χχ¯ . (26) the form of C C ∼h i ∼h i Introducing the new Nambu-Gor’kov spinors T d3p~ Φ2 Γ= Trln([ (P)]−1)+ . (35) −2 (2π)3 Snl 4G D X χ , X¯ (χ¯, χ¯ ), (27) Xn Z ≡ χC ≡ C (cid:18) (cid:19) To evaluate the ground state of Γ(V ,∆,µ,µ ,µ ) as a µ 8 e the nonlinear realization of the original Lagrangianden- function of mismatch is tedious and still under progress. sity Eq.(3) takes the form of In the following we just give a brief discussion on the Nambu-Goldstone current generation state [41], one- Φ+Φ− X¯ −1X , (28) plane wave LOFF state [28, 29], as well as the gluon Lnl ≡ Snl − 4GD phase [32]. If we expand the thermodynamic potential where Γ(V ,∆,µ,µ ,µ ) of the non-linear realization form µ 8 e [G+ ]−1 Φ− in terms of the Nambu-Goldstone currents, we will −1 0,nl . (29) Snl ≡ Φ+ [G− ]−1 naturally have the Nambu-Goldstone currents genera- (cid:18) 0,nl (cid:19) tion in the system with the increase of mismatch, i.e., Here the explicit formof the free propagatorfor the new < 7a=4▽~ϕa >=6 0 and/or < ▽~ϕ8 >=6 0 at large δµ. quark field is This is an extended version of the Nambu-Goldstone P current generation state proposed in a minimal gapless [G+ ]−1 = iD/ +µˆγ +γ Vµ, (30) model in Ref. [41, 42]. From Eq. (28), we can see that 0,nl 0 µ ▽~ϕ contributes to the baryon current. < ▽~ϕ >= 0 8 8 6 and indicates a baryon current generation or 8-th gluon condensate in the system, it is just the one-plane wave [G−0,nl]−1 = iD/T −µˆγ0+γµVCµ. (31) LOFF state. This has been discussed in Sec. IV. The other four Nambu-Goldstone currents generation Comparing with the free propagator in the original La- < 7 ▽~ϕ >=0 indicates other color current genera- a=4 a 6 grangiandensity,thefreepropagatorinthenon-linearre- tion in the system, and is equivalent to the gluon phase alizationframeworknaturallytakesintoaccountthecon- desPcribed in Ref. [32]. tribution from the Nambu-Goldstone currents or phase We do not argue whether the system will exprience fluctuations, i.e., a gluon condensate phase or Nambu-Goldstone currents Vµ † (i∂µ) , generationstate. Wesimplythinktheyareequivalent. In ≡ V V fact, the gaugefields andthe Nambu-Goldstonecurrents Vµ T (i∂µ) ∗, (32) C ≡ V V share a gauge covariant form as shown in the free pro- pogator. However, we prefer to using Nambu-Goldstone which is the N N N N -dimensional Maurer-Cartan c f × c f currents generation than the gluon condensate in the one-formintroducedinRef. [38]. The linearorderofthe gNJL model. As mentioned in Sec. II, in the gNJL Nambu-Goldstone currents Vµ and Vµ has the explicit C model,alltheinformationfromunkownnonperturbative form of gluons are hidden in the diquark gap parameter ∆. The gauge fields in the Lagrangian density are just external 7 1 Vµ (∂µϕ ) T (∂µϕ ) B , (33) fields, they only play the role of probing the system, but a a 8 ≃ − − √3 do not contribute to the property of the color supercon- a=4 X ducting phase. Therefore, there is no gluon free-energy 7 1 Vµ (∂µϕ ) TT + (∂µϕ ) BT . (34) in the gNJL model, it is not clear how to derive the C ≃ a a √3 8 gluon condensate in this model. In order to investigate a=4 X the problem in a fully self-consistentway, one has to use The Lagrangian density Eq. (28) for the new quark the ambitious framework by using the Dyson-Schwinger fieldslookslikeanextensionofthetheoryinRef. [39]for equations(DSE)[43]includingdiquarkdegreeoffreedom high-T superconductor to Non-Abelian system, except [44]orintheframeworkofeffectivetheoryofhigh-density c that here we neglected the singular phase contribution quark matter as in Ref. [45]. 7 VI. CONCLUSION AND DISCUSSION need new thoughts on understanding how a BCS super- condutor will be eventually destroyed by increasing the In this paper, we show that, except the chromomag- mismatch,wealsoneedtodevelopnewmethodstoreally neticinstability,theg2SCphasealsoexhibitsaparamag- resolvetheinstabilityproblem. Somemethodsdeveloped netic response to the perturbation of an external baryon in unconventionalsuperconductor field, e.g., High-Tc su- current. This suggests a baryon current can be spon- perconductor, might be helpful. The work toward this taneously generated in the g2SC phase, and the quasi- direction is in progressing. particlesspontaneouslyobtainasuperfluidvelocity. The In this paper, we did not discuss the magnetic insta- spontaneouslygeneratedbaryoncurrentbreaksthe rota- bilty in the gCFL phase. After the first version of this tionalsymmetryofthesystem,anditisequivalenttothe paper appeared, the author was informed by D. T. Son one-plane wave LOFF state. thatthe baryoncurrentgenerationwasalsofound inthe We further describe the 2SC phase in the nonlinear gCFL phase [46]. For more discussion on solving the realizationframework,andshowthateachinstabilityin- magnetic instabilty in the gCFL phase, please refer to dicates the spontaneous generation of the corresponding Ref. [47]. pseudoNambu-Goldstonecurrent. WeshowthisNambu- Goldstone currents generation state can naturally cover the gluon phase as well as the one-plane wave LOFF Acknowledgments state. We also point out that, when charge neutrality condition is required, there exists a narrow unstable LOFF The author thanks M. Alford, F.A. Bais, K. 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