< ψ α $_{Li}$ψ β Lj > = − < ψ α $_{Ri}$ψ β Rj > . (42) < ψ α $_{Li}$ψ β Lj ¯ ψ k Rα ¯ ψ l Rβ > ∼ < ψ α $_{Li}$ψ β Lj >< ¯ ψ k Rα ¯ ψ l Rβ > Δ $^{2}$ϵ$_{ijm}$ ϵ klm (43) < udsuds > ∼ < ΛΛ > . (44) Δ αβ ij = Δ ∑ I ϵ$_{ijI}$ϵ $^{αβI}$, (40) Δ αβ ij = Δ ϵ$_{ij}$ϵ $^{αβb}$, (39) Δ αβ ij = Δ( δ α i δ β j − δ α j δ β i ) (41) 11 that the order parameter Even though the antisymmetry in the attractive chan- nel signifies that only quarks with different colors can form Cooper pairs, color superconductivity has very rich phase structure because of its flavor, spin and other de- grees of freedom. In the following, I list some of the known color superconducting phases. describes the color-flavor locked (CFL) phase proposed in Ref. [86]. Many other different treatments [87–89] agreed that a condensate of the form (40) is the dominant con- densate in three-flavor QCD. In the CFL phase, all quark colors and flavors partici- pate in the pairing. The color gauge group is completely broken, and all eight gluons become massive [86, 90], which ensures that there are no infrared divergences as- sociated with gluon propagators. Electromagnetism is no longer a separate symmetry, but corresponds to gauging one of the flavor generators. A rotated electromagnetism (“ ˜ Q ”) remains unbroken. Two global symmetries, the chiral symmetry and the baryon number, are broken in the CFL phase, too. In zero-density QCD, the spontaneous breaking of chi- ral symmetry is due to the condensation of left-handed quarks with right-handed quarks. Here, at high baryon density, the chiral symmetry breaking occurs due to a rather different mechanism: locking of the flavor rota- tions to color. In the CFL phase, there is only pairing of left-handed quarks with left-handed quarks, and right- handed quarks with right-handed quarks, i.e., Renormalization group arguments [67, 83, 84] suggest that possible quark pairs always condense in the s − wave. This means that the spin wave function of the pair is anti-symmetric. Since the diquark condenses in the color antitriplet ¯ 3$_{c}$ channel, the color wave function of the pair is also anti-symmetric. The Pauli principle requires that the total wave function of the Cooper pair has to be anti- symmetric under the exchange of the two quarks forming the pair. Thus the flavor wave function has to be anti- symmetric, too. This determines the structure of the order parameter Firstly we consider a system with only massless u and d quarks, assuming that the strange quark is much heavier than the up and down quarks. The color superconducting phase with only two flavors is normally called the 2SC phase. A. Different color superconducting phases The 2SC phase Where L, R indicate left- and right-handed, respectively, α, β are color indices and i, j are flavor indices. A gauge invariant form [91, 92] where color indices α, β ∈ ( r, g, b ) and flavor indices i, j ∈ ( u, d ). From the order parameter Eq. (39), we can see that the condensate picks a color direction (here the blue direction, which is arbitrarily selected). The ground state is invariant under an SU (2)$_{c}$ subgroup of the color rotations that mixes the red and green colors, but the blue quarks are singled out as different. Thus the color SU (3)$_{c}$ is broken down to its subgroup SU (2)$_{c}$, and five of the gluons obtain masses, which indicates the Meissner effect [85]. captures the chiral symmetry breaking. The spectrum of excitations in the CFL phase contains an octet of Gold- stone bosons associated with the chiral symmetry break- ing. This looks remarkably like those at low density. In the excitation spectrum of the CFL phase, there is an- other singlet U (1) Goldstone boson related to the baryon number symmetry breaking, which can be described us- ing the order parameter In the 2SC phase, the Cooper pairs are ud − du sin- glets and the global flavor symmetry SU (2)$_{L}$ ⊗ SU (2)$_{R}$ is intact, i.e., the chiral symmetry is not broken. There is also an unbroken global symmetry which plays the role of U (1)$_{B}$. Thus no global symmetry are broken in the 2SC phase. In QCD with three degenerate light flavors, the spectrum in the CFL phase looks similar to that in the hyper- nuclear phase at low-density. It is suggested that the low density hyper-nuclear phase and the high density quark phase might be continuously connected [93]. In the case when the chemical potential is much larger than the strange quark mass, we can assume m$_{u}$ = m$_{d}$ = m$_{s}$ = 0, and there are three degenerate massless flavors in the system. The spin-0 order parameter should be color and flavor anti-symmetric, which has the form of In the case of only one-flavor quark system, due to the antisymmetry in the color space, the Pauli principle re- quires that the Cooper pair has to occur in a symmetric spin channel. Therefore, in the simplest case, the Cooper where color indices α, β ∈ ( r, g, b ) and flavor indices i, j ∈ ( u, d, s ). Writing ∑ I ϵ$_{ijI}$ϵ αβI = δ α i δ β j − δ α j δ β i , we can see Spin-1 color superconductivity The CFL phase