document: studying qcd at finite baryon density is the traditional subject of nuclear physics . 
   the behaviour of qcd at finite baryon density and low temperature is central for astrophysics to understand the structure of compact stars , and conditions near the core of collapsing stars ( supernovae , hypernovae ) . 
   it is known that sufficiently cold and dense baryonic matter is in the color superconducting phase . 
   this was proposed several decades ago by frautschi @xcite and barrois @xcite by noticing that one - gluon exchange between two quarks is attractive in the color antitriplet channel . from bcs theory @xcite , we know that if there is a weak attractive interaction in a cold fermi sea , the system is unstable with respect to the formation of particle - particle cooper - pair condensate in the momentum space 
   . studies on color superconducting phase in 1980 s can be found in ref . 
   the topic of color superconductivity stirred a lot of interest in recent years @xcite . for reviews on recent progress of color superconductivity 
   see , for example , ref . 
   @xcite . 
   the color superconducting phase may exist in the central region of compact stars . to form bulk matter inside compact stars , the charge neutrality condition as well as @xmath0 equilibrium are required @xcite . 
   this induces mismatch between the fermi surfaces of the pairing quarks . 
   it is clear that the cooper pairing will be eventually destroyed with the increase of mismatch .    without the constraint from the charge neutrality condition , 
   the system may exhibit a first order phase transition from the color superconducting phase to the normal phase when the mismatch increases @xcite . 
   it was also found that the system can experience a spatial non - uniform loff ( larkin - ovchinnikov - fudde - ferrell ) state @xcite in a certain window of moderate mismatch . 
   it is still not fully understood how the cooper pairing will be eventually destroyed by increasing mismatch in a charge neutral system . 
   the charge neutrality condition plays an essential role in determining the ground state of the neutral system . if the charge neutrality condition is satisfied globally , and also if the surface tension is small , the mixed phase will be favored @xcite . 
   it is difficult to precisely calculate the surface tension in the mixed phase , thus in the following , we would like to focus on the homogeneous phase when the charge neutrality condition is required locally . 
   it was found that homogeneous neutral cold - dense quark matter can be in the gapless 2sc ( g2sc ) phase @xcite or gapless cfl ( gcfl ) phase @xcite , depending on the flavor structure of the system . 
   the gapless state resembles the unstable sarma state @xcite . 
   however , under a natural charge neutrality condition , i.e. , only neutral matter can exist , the gapless phase is indeed a thermal stable state as shown in @xcite . 
   the existence of thermal stable gapless color superconducting phases was confirmed in refs . 
   @xcite and generalized to finite temperatures in refs . 
   recent results based on more careful numerical calculations show that the g2sc and gcfl phases can exist at moderate baryon density in the color superconducting phase diagram @xcite . 
   one of the most important properties of an ordinary superconductor is the meissner effect , i.e. , the superconductor expels the magnetic field @xcite . in ideal color superconducting phases , e.g. , in the 2sc and cfl phases , the gauge bosons connected with the broken generators obtain masses , which indicates the meissner screening effect @xcite . 
   the meissner effect can be understood using the standard anderson - higgs mechanism . 
   unexpectedly , it was found that in the g2sc phase , the meissner screening masses for five gluons corresponding to broken generators of @xmath1 become imaginary , which indicates a type of chromomagnetic instability in the g2sc phase @xcite . 
   the calculations in the gcfl phase show the same type of chromomagnetic instability @xcite . remembering the discovery of superfluidity density instability @xcite in the gapless interior - gap state @xcite 
   , it seems that the instability is a inherent property of gapless phases . 
   ( there are several exceptions : 1 ) it is shown that there is no chromomangetic instability near the critical temperature @xcite ; 2 ) it is also found that the gapless phase in strong coupling region is free of any instabilities @xcite . ) 
   the chromomagnetic instability in the gapless phase still remains as a puzzle . by observing 
   that , the 8-th gluon s chromomagnetic instability is related to the instability with respect to a virtual net momentum of diquark pair , giannakis and ren suggested that a loff state might be the true ground state @xcite . 
   their further calculations show that there is no chromomagnetic instability in a narrow loff window when the local stability condition is satisfied @xcite . 
   latter on , it was found in ref . 
   @xcite that a charge neutral loff state can not cure the instability of off - diagonal 4 - 7th gluons , while a gluon condensate state @xcite can do the job . in ref . 
   @xcite we further pointed out that , when charge neutrality condition is required , there exists another narrow unstable loff window , not only off - diagonal gluons but the diagonal 8-th gluon can not avoid the magnetic instability .    in a minimal model of gapless color supercondutor , hong showed in ref . 
   @xcite that the mismatch can induce a spontaneous nambu - goldstone current generation . 
   the nambu - goldstone current generation state in u(1 ) case resembles the one - plane wave loff state or diagonal gauge boson s condensate . 
   we extended the nambu - goldstone current generation picture to the 2sc case in the nonlinear realization framework in ref . 
   we show that five pseudo nambu - goldstone currents can be spontaneously generated by increasing the mismatch between the fermi surfaces of the pairing quarks . 
   the nambu - goldstone currents generation state covers the gluon phase as well as the one - plane wave loff state . 
   this article is organized as follows . in sec . 
   [ sec - gnjl ] , we describe the framework of the gauged su(2 ) nambu  jona - lasinio ( gnjl ) model in @xmath0-equilibrium . 
   we review chromomagnetic instabilities in sec . 
   [ sec - mag - ins ] . 
   we discuss neutral baryon current instability and the loff state in sec . 
   [ sec - bc ] . 
   then sec . 
   [ sec - ng - current ] gives a general nambu - goldstone currents generation description in the non - linearization framework . at the end , we give the discussion and summary in sec . 
   [ sec - sum ] . 
   we take the gauged form of the extended nambu  jona - lasinio model @xcite , the lagrangian density has the form of @xmath2   \nonumber \\   & + & g_d[(i { \bar q}^c   \varepsilon   \epsilon^{b } \gamma_5 q )     ( i { \bar q } \varepsilon \epsilon^{b } \gamma_5 q^c ) ] , \label{lg}\end{aligned}\ ] ] with @xmath3 . here 
   @xmath4 are gluon fields and @xmath5 with @xmath6 are generators of @xmath7 gauge groups 
   . please note that we regard all the gauge fields as external fields , which are weakly interacting with the system . 
   the property of the color superconducting phase characterized by the diquark gap parameter is determined by the unkown nonperturbative gluon fields , which has been simply replaced by the four - fermion interaction in the njl model . however , the external gluon fields do not contribute to the properties of the system . 
   therefore , we do not have the contribution to the lagrangian density from gauge field part @xmath8 as introduced in ref .  @xcite . 
   .  [ sec - ng - current ] , by using the non - linear realization in the gnjl model , we will derive one nambu - goldstone currents state , which is equivalent to the so - called gluon - condensate state . )    in the lagrangian density eq . 
   ( [ lg ] ) , @xmath9 , @xmath10 are charge - conjugate spinors , @xmath11 is the charge conjugation matrix ( the superscript @xmath12 denotes the transposition operation ) . 
   the quark field @xmath13 with @xmath14 and @xmath15 is a flavor doublet and color triplet , as well as a four - component dirac spinor , @xmath16 are pauli matrices in the flavor space , where @xmath17 is antisymmetric , and @xmath18 , @xmath19 are totally antisymmetric tensors in the flavor and color spaces . 
   @xmath20 is the matrix of chemical potentials in the color and flavor space . in @xmath0-equilibrium 
   , the matrix of chemical potentials in the color - flavor space @xmath21 is given in terms of the quark chemical potential @xmath22 , the chemical potential for the electrical charge @xmath23 and the color chemical potential @xmath24 , @xmath25 @xmath26 and @xmath27 are the quark - antiquark coupling constant and the diquark coupling constant , respectively . in the following , we only focus on the color superconducting phase , where @xmath28 and @xmath29 . 
   after bosonization , one obtains the linearized version of the model for the 2-flavor superconducting phase , @xmath30 with the bosonic fields @xmath31 in the nambu - gorkov space , @xmath32 the inverse of the quark propagator is defined as @xmath33^{-1 } = \left(\begin{array}{cc } \left[g_0^{+}(p)\right]^{-1 } & \delta^- \\ \delta^+ & \left[g_0^{-}(p)\right]^{-1 } \end{array}\right ) , \label{prop}\ ] ] with the off - diagonal elements @xmath34 and the free quark propagators @xmath35 taking the form of @xmath36^{-1 } =   \gamma^0 ( p_0 \pm \hat{\mu } ) - \vec{\gamma } \cdot \vec{p}.   \label{freep}\ ] ] the 4-momenta are denoted by capital letters , e.g. , @xmath37 . 
   we have assumed the quarks are massless in dense quark matter , and the external gluon fields do not contribute to the quark self - energy . 
   the explicit form of the functions @xmath38 and @xmath39 reads @xmath40 [ g_i ] with @xmath41 and @xmath42 [ xi_i ] where @xmath43 is an alternative set of energy projectors , and the following notation was used : @xmath44 from the dispersion relation of the quasiparticles eq . 
   ( [ g - disp ] ) , we can read that , when @xmath45 , there will be excitations of gapless modes in the system . 
   the thermodynamic potential corresponding to the solution of the gapless state @xmath45 is a local maximum . however , under certain constraint , e.g. , the charge neutrality condition , the gapless 2sc phase can be a thermal stable state @xcite .      in this section , 
   we review the chromomagnetic instabilities driven by mismatch in 2sc and g2sc phases as shown in ref . 
   the polarization tensor in momentum space has the following general structure : @xmath46 . \label{pipscfng}\ ] ] the trace here runs over the dirac indices and the vertices @xmath47 with @xmath48 . 
   gluons @xmath52 of the unbroken @xmath53 subgroup couple only to the red and green quarks . 
   the general expression for the polarization tensor @xmath54 with @xmath55 is diagonal . after performing the traces over the color , flavor and nambu - gorkov indices , the expression has the form of @xmath56 , \label{pi11}\end{aligned}\ ] ]    by making use of the definition in eq . 
   ( [ def - debye ] ) and eq . 
   ( [ def - meissner ] ) . 
   we arrive at the following result for the threefold degenerate debye mass square : @xmath57 with @xmath58 . 
   the meissner mass square reads @xmath59    the debye screening mass in eq . 
   ( [ m_d_1 ] ) vanishes in the gapped phase ( i.e. , @xmath60 ) . as in the case of the ideal 2sc phase 
   , this reflects the fact that there are no gapless quasiparticles charged with respect to the unbroken su(2)@xmath61 gauge group . in the gapless 2sc phase , 
   such quasiparticles exist and the value of the debye screening mass is proportional to the density of states at the corresponding `` effective '' fermi surfaces . 
   the 8-th gluon can probe the cooper - paired red and green quarks , as well as the unpaired blue quarks . 
   after the traces over the color , the flavor and the nambu - gorkov indices are performed , the polarization tensor for the 8th gluon can be expressed as    @xmath63 , \label{pi88 } \\ 
   \pi_{88,b}^{\mu\nu}(p ) & = &   \frac{g^2t}{4}\sum_n\int \frac{d^3 { \mathbf k}}{(2\pi)^3 } \mbox{tr}_{\rm d }   \left [ \gamma^{\mu } g_{3}^+(k ) \gamma^{\nu } g_{3}^+(k ' )   + \gamma^{\mu } g_{3}^-(k ) \gamma^{\nu}g_{3}^-(k ' ) \right.\nonumber\\     & + & \left . \gamma^{\mu } g_{4}^+(k )   \gamma^{\nu}g_{4}^+(k ' )   + \gamma^{\mu } g_{4}^-(k ) \gamma^{\nu}g_{4}^-(k ' ) \right ] . 
   \label{pi88b}\end{aligned}\ ] ]          after performing the traces over the color , the flavor and the nambu - gorkov indices , the diagonal components of the polarization tensor @xmath68 with @xmath69 have the form @xmath70.\end{aligned}\ ] ] note that @xmath71 . 
   apart from the diagonal elements , there are also nonzero off - diagonal elements , @xmath72 with @xmath73.\end{aligned}\ ] ] the physical gluon fields in the 2sc / g2sc phase are the following linear combinations : @xmath74 and @xmath75 . 
   these new fields , @xmath76 and @xmath77 describe two pairs of massive vector particles with well defined electomagnetic charges , @xmath78 . 
   the components of the polarization tensor in the new basis read        in the static limit , all four eigenvalues of the polarization tensor are degenerate . by making use of the definition in eq . 
   ( [ def - debye ] ) , we derive the following result for the corresponding debye masses : @xmath81 \!\ ! . 
   \label{m_d_4}\ ] ] here we assumed that @xmath24 is vanishing which is a good approximation in neutral two - flavor quark matter . 
   the fourfold degenerate meissner screening mass of the gluons with @xmath67 reads @xmath82 \!\ ! . 
   \label{m_m_4}\ ] ] both results in eqs . 
   ( [ m_d_4 ] ) and ( [ m_m_4 ] ) interpolate between the known results in the normal phase ( i.e. , @xmath83 ) and in the ideal 2sc phase ( i.e. , @xmath84 ) . the instability of off - diagonal gluons appears in the whole region of gapless 2sc phases ( with @xmath66 ) and even in some gapped 2sc phases ( with @xmath85 ) . 
   it is noticed that the meissner screening mass square for the off - diagonal gluons decreases monotonously to zero when the mismatch increases from zero to @xmath86 , then goes to negative value with further increase of the mismatch in the gapped 2sc phase . 
   however , the behavior of diagonal 8-th gluon s meissner mass square is quite different . 
   it keeps as a constant in the gapped 2sc phase . in the gapless 2sc phase , 
   all these five gluons meissner mass square are negative . 
   it is not understood why gapless color superconducting phases exhibit chromomagnetic instability . 
   it sounds 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 the instability by requiring the electrical neutrality on the system . in order to understand what is really going ` wrong ' with the homogeneous g2sc phase , we want to know whether there exists other instabilities except the chromomagnetic instability . for that purpose , we probe the g2sc phase using different external sources , e.g. , scalar and vector diquarks , mesons , vector current , and so on . here 
   we report the most interesting result regarding the response of the g2sc phase to an external vector current @xmath87 , the time - component and spatial - components of this current correspond to the baryon number density and baryon current , respectively .    from the linear response theory , 
   the induced current and the external vector current is related by the response function @xmath88 , @xmath89 .\ ] ] the trace here runs over the nambu - gorkov , flavor , color and dirac indices . 
   the explicit form of vertices is @xmath90 . 
   the explicit expression of the vector current response function takes the form of @xmath91 with @xmath92 , \\ 
   % \label{piv }   \pi_{v , b}^{\mu\nu}(p ) & = &   \frac{t}{2}\sum_n\int \frac{d^3 { \mathbf k}}{(2\pi)^3 } \mbox{tr}_{\rm d }   \left [ \right . \nonumber\\   & & \left . 
   \gamma^{\mu } g_{3}^+(k ) \gamma^{\nu } g_{3}^+(k ' )   + \gamma^{\mu } g_{3}^-(k ) \gamma^{\nu}g_{3}^-(k ' ) \right.\nonumber\\     & + & \left . 
   \gamma^{\mu } g_{4}^+(k )   \gamma^{\nu}g_{4}^+(k ' )   + \gamma^{\mu } g_{4}^-(k ) \gamma^{\nu}g_{4}^-(k ' ) \right ] ,   % \label{pivb}\end{aligned}\ ] ] here the trace is over the dirac space .    comparing the explicit expression of @xmath93 with that of the 8-th gluon s self - energy @xmath94 , 
   i.e. , eq . 
   ( [ pi88 ] ) , it can be clearly seen that , @xmath93 and @xmath94 almost share the same expression , except the coefficients . 
   this can be easily understood , because the color charge and color current carried by the 8-th gluon is proportional to the baryon number and baryon current , respectively . in the static long - wavelength 
   ( @xmath95 and @xmath96 ) limit , the time - component and spatial component of @xmath93 give the baryon number susceptibility @xmath97 and baryon current susceptibility @xmath98 , respectively , @xmath99    in the g2sc phase , @xmath100 as well as @xmath98 become negative . 
   this means that , except the chromomagnetic instability corresponding to broken generators of @xmath1 , and the instability of a net momentum for diquark pair , the g2sc phase is also unstable with respect to an external color neutral baryon current @xmath101 . 
   the 8-th gluon s magnetic instability , the diquark momentum instability and the color neutral baryon current in the g2sc phase can be understood in one common physical picture . 
   the g2sc phase exhibits a paramagnetic response to an external baryon current . 
   naturally , the color current carried by the 8-th gluon , which differs from the baryon current by a color charge , also experiences the instability in the g2sc phase . 
   the paramagnetic instability of the baryon current indicates that the quark can spontaneously obtain a momentum , because diquark carries twice of the quark momentum , it is not hard to understand why the g2sc phase is also unstable with respect to the response of a net diquark momentum 
   .    it is noticed that , the instability of @xmath101 will be induced by mismatch in all the asymmetric fermi pairing systems , including superfluid systems , where @xmath101 can be interpreted as particle current .      the paramagnetic response to an external vector 
   current naturally suggests that a vector current can be spontaneously generated in the system . 
   the generated vector current behaves as a vector potential , which modifies the quark self - energy with a spatial vector condensate @xmath102 , and breaks the rotational symmetry of the system . 
   it can also be understood that the quasiparticles in the gapless phase spontaneously obtain a superfluid velocity , and the ground state is in an anisotropic state . 
   the quark propagator @xmath35 in eq . 
   ( [ freep ] ) is modified as @xmath103^{-1 } =   \gamma^0 ( p_0 \pm \hat{\mu } ) - \vec{\gamma } \cdot \vec{p } \mp \vec{\gamma } \cdot \vec{\sigma}_v ,   \label{freep - m}\ ] ] with a subscript @xmath104 indicating the modified quark propagator . 
   correspondingly , the inverse of the quark propagator @xmath105^{-1}$ ] in eq . 
   ( [ prop ] ) is modified as @xmath106^{-1 } = \left(\begin{array}{cc } \left[g_{0,v}^{+}(p)\right]^{-1 } & \delta^- \\ \delta^+ & \left[g_{0,v}^{-}(p)\right]^{-1 } \end{array}\right ) . 
   \label{prop - m}\ ] ] it is noticed that the expression of the modified inverse quark propagator @xmath107^{-1}$ ] takes the same form as the inverse quark propagator in the one - plane wave loff state shown in ref . 
   the net momentum @xmath108 of the diquark pair in the loff state @xcite is replaced here by a spatial vector condensate @xmath109 . 
   the spatial vector condensate @xmath102 breaks rotational symmetry of the system . 
   this means that the fermi surfaces of the pairing quarks are not spherical any more . 
   it has to be pointed out , the baryon current offers one doppler - shift superfluid velocity for the quarks . a spontaneously generated nambu - goldstone current in the minimal gapless model @xcite or a condensate of 8-th gluon s spatial component can do the same job . 
   all these states mimic the one - plane wave loff state . in the following , 
   we just call all these states as the single - plane wave loff state .    in order to determine the deformed structure of the fermi surfaces 
   , one should self - consistently minimize the free energy @xmath110 . 
   the explicit form of the free energy can be evaluated directly using the standard method , in the framework of nambu  jona - lasinio model @xcite , it takes the form of @xmath111^{-1 } ) + \frac{\delta^2}{4 g_d},\end{aligned}\ ] ] where @xmath12 is the temperature , and @xmath27 is the coupling constant in the diquark channel . 
   when there is no charge neutrality condition , the ground state is determined by the thermal stability condition , i.e. , the local stability condition . 
   the ground state is in the 2sc phase when @xmath112 with @xmath113 , in the loff phase when @xmath114 correspondingly @xmath115 , and then in the normal phase with @xmath116 when the mismatch is larger than @xmath117 . here 
   @xmath118 indicate the diquark gap in the case of @xmath119 and @xmath120 , respectively . 
   when charge neutrality condition is required , the ground state of charge neutral quark matter should be determined by solving the gap equations as well as the charge neutrality condition , i.e. , @xmath121 by changing @xmath122 or coupling strength @xmath27 , the solution of the charge neutral loff state can stay everywhere in the full loff window , including the window not protected by the local stability condition , as shown explicitly in ref . 
   @xcite .    from the lesson of charge neutral g2sc phase , 
   we learn that even though the neutral state is a thermal stable state , i.e. , the thermodynamic potential is a global minimum along the neutrality line , it can not guarantee the dynamical stability of the system . 
   the stability of the neutral system should be further determined by the dynamical stability condition , i.e. , the positivity of the meissner mass square . 
   the polarization tensor for the gluons with color @xmath123 should be evaluated using the modified quark propagator @xmath124 in eq . 
   ( [ prop - m ] ) , i.e. , @xmath125 , \label{piab}\ ] ] with @xmath126 and the explicit form of the vertices @xmath127 has the form @xmath128 . in the loff state 
   , the meissner tensor can be decomposed into transverse and longitudinal component . 
   the transverse and longitudinal meissner mass square for the off - diagonal 4 - 7 gluons and the diagonal 8-th gluon have been performed explicitly in the one - plane wave loff state in ref . 
   @xcite . 
   \1 ) the stable loff ( s - loff ) window in the region of @xmath129 , which is free of any magnetic instability . 
   please note that this s - loff window is a little bit wider than the window @xmath130 protected by the local stability condition . 
   \2 ) the stable window for diagonal gluon characterized by ds - loff window in the region of @xmath131 , which is free of the diagonal 8-th gluon s magnetic instability but not free of the off - diagonal gluons magnetic instability ;    \3 ) the unstable loff ( us - loff ) window in the region of @xmath132 , with @xmath133 . in this us - loff window , all the magnetic instabilities exist . 
   please note that , it is the longitudinal meissner mass square for the 8-th gluon is negative in this us - loff window , the transverse meissner mass square of 8-th gluon is always zero in the full loff window , which is guaranteed by the momentum equation .    us - loff is a very interesting window , it indicates that the loff state even can not cure the 8-th gluon s magnetic instability . in the charge neutral 2-flavor system 
   , it seems that the diagonal gluon s magnetic instability can not be cured in the gluon phase , because there is no direct relation between the diagonal gluon s instability and the off - diagonal gluons instability . 
   ( of course , it has to be carefully checked , whether all the instabilities in this us - loff window can be cured by off - diagonal gluons condensate in the charge neutral 2-flavor system . ) 
   it is also noticed that in this us - loff window , the mismatch is close to the diquark gap , i.e. , @xmath134 . 
   therefore it is interesting to check whether this us - loff window can be stabilized by a spin-1 condensate @xcite as proposed in ref . 
   @xcite .    in the charge 
   neutral 2sc phase , though it is unlikely , we might have a lucky chance to cure the diagonal instability by the condensation of off - diagonal gluons . 
   it is expected that this instability will show up in some constrained abelian asymmetric superfluidity system , e.g. , in the fixed number density case @xcite . 
   it will be a new challenge for us to really solve this problem . 
   we have seen that , except chromomagnetic instability corresponding to broken generators of @xmath1 , the g2sc phase is also unstable with respect to the external neutral baryon current . it is noticed that all the instabilities are induced by increasing the mismatch between the fermi surfaces of the cooper pairing . in order to understand the instability driven by mismatch , in the following , we give some general analysis .    a superconductor will be eventually destroyed and goes to the normal fermi liquid state , so one natural question is : how an ideal bcs superconductor will be destroyed by increasing mismatch ? to answer how a superconductor will be destroyed , one has to firstly understand what is a superconductor . 
   the superconducting phase is characterized by the order parameter @xmath135 , which is a complex scalar field and has the form of e.g. , for electrical superconductor , @xmath136 , with @xmath137 the amplitude and @xmath138 the phase of the gap order parameter or the pseudo nambu - goldstone boson . 
   there are two ways to destroy a superconductor . 
   one way is by driving the amplitude of the order parameter to zero . 
   this way is bcs - like , because it mimics the behavior of a conventional superconductor at finite temperature , the gap amplitude monotonously drops to zero with the increase of temperature ; another way is non - bcs like , but berezinskii - kosterlitz - thouless ( bkt)-like @xcite , even if the amplitude of the order parameter is large and finite , superconductivity will be lost with the destruction of phase coherence , e.g. the phase transition from the @xmath144wave superconductor to the pseudogap state in high temperature superconductors @xcite . 
   stimulating by the role of the phase fluctuation in the unconventional superconducting phase in condensed matter , we follow ref . 
   @xcite to formulate the 2sc phase in the nonlinear realization framework in order to naturally take into account the contribution from the phase fluctuation or pseudo nambu - goldstone current .    in the 2sc phase , the color symmetry @xmath145 breaks to @xmath146 . 
   the generators of the residual @xmath53 symmetry h are @xmath147 with @xmath148 and the broken generators @xmath149 with @xmath150 . 
   more precisely , the last broken generator is a combination of @xmath151 and the generator @xmath152 of the global @xmath153 symmetry of baryon number conservation , @xmath154 of generators of the global @xmath155 and local @xmath1 symmetry .    the coset space @xmath156 is parameterized by the group elements @xmath157\,\,,\ ] ] here @xmath158 and @xmath159 are five nambu - goldstone diquarks , and we have neglected the singular phase , which should include the information of the topological defects @xcite 
   . operator @xmath160 is unitary , @xmath161 . 
   introducing a new quark field @xmath162 , which is connected with the original quark field @xmath163 in eq . 
   ( [ lagr-2sc ] ) in a nonlinear transformation form , @xmath164 and the charge - conjugate fields transform as @xmath165 in high-@xmath166 superconductor , this technique is called charge - spin separation , see ref . 
   the advantage of transforming the quark fields is that this preserves the simple structure of the terms coupling the quark fields to the diquark sources , @xmath167 in mean - field approximation , the diquark source terms are proportional to @xmath168    introducing the new nambu - gorkov spinors @xmath169 the nonlinear realization of the original lagrangian density eq.([lagr-2sc ] ) takes the form of @xmath170 where @xmath171^{-1 } & \phi^- \\               \phi^+ & [ g^-_{0,nl}]^{-1 }         \end{array } \right)\,\ , .\ ] ] here the explicit form of the free propagator for the new quark field is @xmath172^{-1 } & = & i\ , \fsl{d } + { \hat \mu } \ , \gamma_0 + \gamma_\mu \ , v^\mu ,   % [ g^+_{0,nl}]^{-1 } & = & i\ , \gamma^\mu \partial_\mu + { \hat \mu } \ , 
   \gamma_0 + \gamma_\mu \ , 
   v^\mu , \end{aligned}\ ] ] and @xmath173^{-1 } & = & i\ , \fsl{d}^t - { \hat \mu } \ , 
   \gamma_0 + \gamma_{\mu } \ , v_c^\mu .\end{aligned}\ ] ] comparing with the free propagator in the original lagrangian density , the free propagator in the non - linear realization framework naturally takes into account the contribution from the nambu - goldstone currents or phase fluctuations , i.e. , @xmath174 which is the @xmath175-dimensional maurer - cartan one - form introduced in ref . 
   the linear order of the nambu - goldstone currents @xmath176 and @xmath177 has the explicit form of @xmath178    the lagrangian density eq . 
   ( [ lagr - nl ] ) for the new quark fields looks like an extension of the theory in ref . 
   @xcite for high-@xmath166 superconductor to non - abelian system , except that here we neglected the singular phase contribution from the topologic defects . the advantage of the non - linear realization framework eq . 
   ( [ lagr - nl ] ) is that it can naturally take into account the contribution from the phase fluctuations or nambu - goldstone currents . 
   the task left is to correctly solve the ground state by considering the phase fluctuations . 
   the free energy @xmath179 can be evaluated directly and it takes the form of @xmath180^{-1 } ) + \frac{\phi^2}{4 g_d}. \label{free - energy - ng}\end{aligned}\ ] ] to evaluate the ground state of @xmath179 as a function of mismatch is tedious and still under progress . in the following 
   we just give a brief discussion on the nambu - goldstone current generation state @xcite , one - plane wave loff state @xcite , as well as the gluon phase @xcite . 
   if we expand the thermodynamic potential @xmath179 of the non - linear realization form in terms of the nambu - goldstone currents , we will naturally have the nambu - goldstone currents generation in the system with the increase of mismatch , i.e. , @xmath181 and/or @xmath182 at large @xmath183 . 
   this is an extended version of the nambu - goldstone current generation state proposed in a minimal gapless model in ref . 
   @xcite . from eq . 
   ( [ lagr - nl ] ) , we can see that @xmath184 contributes to the baryon current . 
   @xmath182 indicates a baryon current generation or 8-th gluon condensate in the system , it is just the one - plane wave loff state . 
   this has been discussed in sec . 
   [ sec - usloff ] . 
   the other four nambu - goldstone currents generation @xmath181 indicates other color current generation in the system , and is equivalent to the gluon phase described in ref . 
   @xcite . 
   we do not argue whether the system will exprience a gluon condensate phase or nambu - goldstone currents generation state . 
   we simply think they are equivalent . 
   in fact , the gauge fields and the nambu - goldstone currents share a gauge covariant form as shown in the free propogator . however , we prefer to using nambu - goldstone currents generation than the gluon condensate in the gnjl model . as mentioned in sec . 
   [ sec - gnjl ] , in the gnjl model , all the information from unkown nonperturbative gluons are hidden in the diquark gap parameter @xmath185 . 
   the gauge fields in the lagrangian density are just external fields , they only play the role of probing the system , but do not contribute to the property of the color superconducting phase . therefore , there is no gluon free - energy in the gnjl model , it is not clear how to derive the gluon condensate in this model . in order to investigate the problem in a fully self - consistent way 
   , one has to use the ambitious framework by using the dyson - schwinger equations ( dse ) @xcite including diquark degree of freedom @xcite or in the framework of effective theory of high - density quark matter as in ref . 
   @xcite .        except the chromomagnetic instability 
   , the g2sc phase also exhibits a paramagnetic response to the perturbation of an external baryon current . 
   this suggests a baryon current can be spontaneously generated in the g2sc phase , and the quasiparticles spontaneously obtain a superfluid velocity . 
   the spontaneously generated baryon current breaks the rotational symmetry of the system , and it resembles the one - plane wave loff state . 
   we further describe the 2sc phase in the nonlinear realization framework , and show that each instability indicates the spontaneous generation of the corresponding pseudo nambu - goldstone current . 
   we show this nambu - goldstone currents generation state can naturally cover the gluon phase as well as the one - plane wave loff state . 
   we also 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-th gluon can not avoid the magnetic instability 
   . the diagonal gluon s magnetic instability in this us - loff window can not be cured by off - diagonal gluon condensate in color superconducting phase . 
   more interestingly , this us - loff window will also show up in some constrained abelian asymmetric superfluid system . 
   the us - loff window brings us a new challenge . 
   we need new thoughts on understanding how a bcs supercondutor will be eventually destroyed by increasing the mismatch , we also need to develop new methods to really resolve the instability problem 
   . some methods developed in unconventional superconductor field , e.g. , high-@xmath166 superconductor , might be helpful . 
   till now , the results on instabilities are based on mean - field ( mf ) approximation . 
   the bcs theory at mf can describe strongly coherent or rigid superconducting state very well . 
   however , as we pointed out in @xcite , with the increase of mismatch , the low degrees of freedom in the system have been changed . 
   for example , the gapless quasi - particle excitations in the gapless phase , and the small meissner mass square of the off - diagonal gluons around @xmath186 . 
   this indicates that these quasi - quarks and gluons become low degrees of freedom in the system , their fluctuations become more important . in order to correctly describe the system , 
   the low degrees of freedom should be taken into account properly . 
   the work toward this direction is still in progress . 
   the author thanks m. alford , f.a . 
   bais , k. fukushima , e. gubankova , m. hashimoto , t. hatsuda , l.y . 
   hong , w.v . 
   liu , m. mannarelli , t. matsuura , y. nambu , k. rajagopal , h.c . 
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   the work is supported by the japan society for the promotion of science fellowship program . 
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summary: we review recent progress of understanding and resolving instabilities driven by mismatch between the fermi surfaces of the pairing quarks in 2-flavor color superconductor . with the increase of mismatch , 
   the 2sc phase exhibits chromomagnetic instability as well as color neutral baryon current instability . 
   we describe the 2sc phase in the nonlinear realization framework , and show that each instability indicates the spontaneous generation of the corresponding pseudo nambu - golstone current . 
   the nambu - goldstone currents generation state covers the gluon phase as well as the one - plane wave 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-th gluon can not avoid the magnetic instability . in this us - loff window , the diagonal magnetic instability can not be cured by off - diagonal gluon condensate in the color superconducting phase .