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now, no full calculation has been done for the bulk vis- cosity. The frequency dependence of the spectral density has been analyzed in Refs. [63] and [32] and the limita- tion of the ansatz Eq.(35) has been discussed. From Eq. (19), we see that the bulk viscosity is dominated by C$_{v}$ at T$_{c}$ . If C$_{v}$ diverges at T$_{c}$ , the bulk viscosity should also be divergent at the critical point and behave as t − $^{α}$. How- ever, the detailed analysis in the Ising model in Ref. [64] shows a very different divergent behavior ζ ∼ t − zν + $^{α}$, with z ≃ 3 the dynamic critical exponent and ν ≃ 0 . 630 the critical exponent in the Ising system. More careful calculation on the bulk viscosity is needed in the future. 

## III. COLOR SUPERCONDUCTING PHASES 

It is known that the deconfined cold dense quark mat- ter is in color superconducting phase. 

Let us start with the system of free fermion gas. Fermions obey the Pauli exclusion principle, which means no two identical fermions can occupy the same quantum state. The energy distribution for fermions (with mass m ) has the form of 

here E$_{p}$ = √ p 2 + m $^{2}$, µ is the chemical potential and T is the temperature. At zero temperature, f ( E$_{p}$ ) = θ ( µ − E$_{p}$ ). The ground state of the free fermion gas is a filled Fermi sea, i.e., all states with the momenta less than the Fermi momentum p$_{F}$ = √ µ 2 − m 2 are occu- pied, and the states with the momenta greater than the Fermi momentum p$_{F}$ are empty. Adding or removing a single fermion costs no free energy at the Fermi surface. 

For the degenerate Fermi gas, the only relevant fermion degrees of freedom are those near the Fermi surface. Con- sidering two fermions near the Fermi surface, if there is a net attraction between them, it turns out that they can form a bound state, i.e., Cooper pair [65]. The binding energy of the Cooper pair Δ( K ) ( K the total momen- tum of the pair), is very sensitive to K , being a maxium where K = 0. There is an infinite degeneracy among pairs of fermions with equal and opposite momenta at the Fermi surface. Because Cooper pairs are composite bosons, they will occupy the same lowest energy quantum state at zero temperature and produce a Bose-Einstein condensation. Thus the ground state of the Fermi sys- tem with a weak attractive interaction is a complicated coherent state of particle-particle Cooper pairs near the Fermi surface [17]. Exciting a quasiparticle and a hole which interact with the condensate requires at least the energy of 2Δ. 

In QED case in condensed matter, the interaction be- tween two electrons by exchanging a photon is repul- sive. The attractive interaction to form electron-electron Cooper pairs is by exchanging a phonon, which is a col- lective excitation of the positive ion background. The 

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Cooper pairing of the electrons breaks the electromag- netic gauge symmetry, and the photon obtains an effec- tive mass. This indicates the Meissner effect [66], i.e., a superconductor expels the magnetic fields. 

In QCD case at asymptotically high baryon density, the dominant interaction between two quarks is due to the one-gluon exchange. This naturally provides an at- tractive interaction between two quarks. The scattering amplitude for single-gluon exchange in an SU ( N$_{c}$ ) gauge theory is proportional to 

Where T$_{a}$ is the generator of the gauge group, and i, j and k, l are the fundamental colors of the two quarks in the incoming and outgoing channels, respectively. Under the exchange of the color indices of either the incoming or the outgoing quarks, the first term is antisymmetric, while the second term is symmetric. For N$_{c}$ = 3, Eq. (38) rep- resents that the tensor product of two fundamental colors decomposes into an (antisymmetric) color antitriplet and a (symmetric) color sextet, 

In Eq. (38), the minus sign in front of the antisymmet- ric contribution indicates that the interaction in this an- titriplet channel is attractive, while the interaction in the symmetric sextet channel is repulsive. 

For cold dense quark matter, the attractive interac- tion in the color antitriplet channel induces the conden- sate of the quark-quark Cooper pairs, and the ground state is called the “color superconductivity”. Since the diquark cannot be color singlet, the diquark conden- sate breaks the local color SU (3)$_{c}$ symmetry, and the gauge bosons connected with the broken generators ob- tain masses. Comparing with the Higgs mechanism of dynamical gauge symmetry breaking in the Standard Model, here the diquark Cooper pair can be regarded as a composite Higgs particle. The calculation of the energy gap and the critical temperature from the first principles has been derived systematically in Refs. [67–74]. 

In reality, we are more interested in cold dense quark matter at moderate baryon density regime, i.e., µ$_{q}$ ∼ 500 MeV , which may exist in the interior of neutron stars. It is likely that cold dense quark droplet might be created in the laboratory through heavy ion collisions in GSI- SPS energy scale. At these densities, an extrapolation of the asymptotic arguments becomes unreliable, we have to rely on effective models. Calculations in the framework of pointlike four-fermion interactions based on the instanton vertex [19, 75–77], as well as in the Nambu–Jona-Lasinio (NJL) model [78–82] show that color superconductivity does occur at moderate densities, and the magnitude of diquark gap is around 100 MeV. 

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