i.e., a superconductor expels the magnetic field, which was discovered by Meissner and Ochsenfeld in 1933 [66]. From theoretical point of view, the Messiner effect can be explained using the linear response theory. The in- duced current j ind i is related to the magnetic field A$_{j}$ by j ind i = Π$_{ij}$ A $^{j}$, where the response function Π$_{ij}$ is the pho- ton polarization tensor. The response function has two components, diamagnetic and paramagnetic part [137]. In the static and long-wavelength limit, for the normal metal, the paramagnetic component cancels exactly the diamagnetic component. While in the superconducting phase, the paramagnetic component is quenched by the energy gap and producing a net diamagnetic response. Thus the ordinary superconductor is a perfect diamag- net. In color superconducting phases, the gluon self-energy (the response function to an external color field), has been investigated in the ideal 2SC phase [85] and in the CFL phase [90]. The results show that the gauge bosons connected with the broken generators obtain masses in these phases, which indicate the Meissner screening effect in these phases. Refs. [144, 145], or even have an inhomogeneous phase structure, like a crystal [99–101] or a vortex [146] struc- ture. It is very interesting to know the chromomagnetic property in the g2SC phase. We studied the g2SC phase in the framework of the SU(2) NJL model, and the NJL model lacks gluons. As reflection of this, it possesses the global instead of gauged color symmetry. In addi- tion, there appear five Nambu-Goldstone (NG) bosons in the ground state of the model when the color symme- try is broken. In QCD, there is no room for such NG bosons. However, the NJL model can be thought of as the low energy theory of QCD in which the gluons, as independent degrees of freedom, are integrated out. The gluons could be reintroduced back by gauging the color symmetry in the Lagrangian density of the NJL model, providing a semirigorous framework for studying the ef- fect of the Cooper pairing on the physical properties of gluons. The existence of the g2SC phase can be regarded as a physical and model independent result under the restric- tion of local charge neutrality condition, the order param- eter for this phase is Δ < δµ . In Ref. [120], we calculated the gluon self-energy in the g2SC phase. It is found that, in this phase, the symmetry broken gauge bosons have imaginary Meissner screening masses, which is induced by the dominant paramagnetic contribution to the gluon self-energy. In condensed matter, this phenomenon is called the paramagnetic Meissner effect(PME) [138], and has been observed in some high temperature supercon- ductors and small superconductors. Unavoidably, the imaginary Meissner screening mass indicates a chromomagnetic instability of the g2SC phase. There are many proposals on resolving the chro- momagnetic instability [139–143]: One is through a gluon condensate to stabilize the system, which may not change the structure of the g2SC phase. It is also possible that the instability drives a new stable ground state, which may have a rotational symmetry breaking like in In the 2SC phase, the color symmetry G = SU (3)$_{c}$ breaks to H = SU (2)$_{c}$. The generators of the residual SU (2)$_{c}$ symmetry H are { S a = T $^{a}$} with a = 1 , 2 , 3 and the broken generators { X b = T b $^{+3}$} with b = 1 , · · · , 5. (More precisely, the last broken generator is a combina- tion of T$_{8}$ and the generator 1 of the global U (1) symme- try of baryon number conservation, B ≡ ( 1 + √$_{3}$$_{T}$$_{8}$$_{)}$$_{/}$$_{3}$ of generators of the global U (1)$_{B}$ and local SU (3)$_{c}$ sym- metry. ) The coset space G/H is parameterized by the group In order to understand the chromomagnetic instability in the gapless phases, we extend the gauged NJL model beyond mean-field approximation [142]. The supercon- ducting state is characterized by the order parameter Δ( x ), which is a complex scalar field and has the form of Δ( x ) = | Δ( x ) | e iϕ ( x $^{)}$, with | Δ | the amplitude and ϕ the phase of the gap order parameter. For a homogeneous condensate, Δ( x ) is a spatial constant. The fluctuations of the phase give rise to the pseudo Nambu-Goldstone bo- son(s), while that of the amplitude to the Higgs field, fol- lowing the terminology of the electroweak theory. Stim- ulated by the role of the phase fluctuation in the un- conventional superconducting phase [147] in condensed matter, we formulate the 2SC phase in the nonlinear re- alization framework [148] in order to naturally take into account the contribution from the phase fluctuation or pseudo Nambu-Goldstone current(s). FIG. 18: Squared values of the gluon Debye (upper panel) and Meissner (lower panel) screening masses, devided by m 2 g = 4 α s ¯ µ $^{2}$/ 3 π , as functions of the dimensionless parameter Δ /δµ . The red solid line denotes the results for the gluons with A = 1 , 2 , 3, the green long-dashed line denotes the results for the gluons with A = 4 , 5 , 6 , 7, and the blue short-dashed line denotes the results for the gluon with A = ˜ 8. The figure is taken from Ref. [120]. D. Sarma instability and Higgs instability 15 1 2 3 4 GLYPH<CapDelta> GLYPH<Slash1> ΔΜ GLYPH<Minus> 0.6 GLYPH<Minus> 0.4 GLYPH<Minus> 0.2 0.2 0.4 m$_{M}$$^{$^{2}$GLYPH<Slash1>}$m$_{g}$ 2