content_image/386982.jpg content_image/386988.jpg content_image/386984.jpg content_image/386985.jpg content_image/386981.jpg content_image/386979.jpg content_image/386980.jpg content_image/386976.jpg content_image/386975.jpg content_image/386974.jpg content_image/386983.jpg content_image/386978.jpg content_image/386973.jpg content_image/386986.jpg content_image/386987.jpg content_image/386972.jpg
elements 

here operator V is unitary, and V − 1 = V † and ϕ$_{a}$ ( a = 4 , · · · , 7) and ϕ$_{8}$ are five Nambu-Goldstone diquarks, and we have not consider the topologically nontrivial case and therefore V ( x ) can be expanded uniformly according to the powers of ϕ ’s. In fact, V ( x ) is alway topologically trivial for a configuration of ϕ ’s that has a finite energy because of the trivial homotopy group π$_{2}$ ( SU (3) /SU (2)). Introducing a new quark field χ , which is connected with the original quark field q in Eq. (46) through a nonlinear transformation, 

and the charge-conjugate fields transform as 

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, 

<img src='content_image/386982.jpg'>

In the Nambu-Gor’kov space of the new spinors 

the nonlinear realization of the original Lagrangian den- sity Eq. (46) takes the form of 

with 

Here the explicit form of the free propagator for the new quark field is 

and 

Comparing with the free propagator in the original La- grangian density, the free propagator in the non-linear re- alization framework naturally takes into account the con- tribution from the Nambu-Goldstone currents or phase fluctuations, i.e., 

<img src='content_image/386988.jpg'>

<img src='content_image/386984.jpg'>

<img src='content_image/386985.jpg'>

<img src='content_image/386981.jpg'>

<img src='content_image/386979.jpg'>

<img src='content_image/386980.jpg'>

<img src='content_image/386976.jpg'>

<img src='content_image/386975.jpg'>

<img src='content_image/386974.jpg'>

which is the N$_{c}$N$_{f}$ × N$_{c}$N$_{f}$ -dimensional Maurer-Cartan one-form introduced in Ref. [148]. The linear order of the Nambu-Goldstone currents V µ and V µ C has the explicit form of 

The advantage of the non-linear realization framework Eq. (58) is that it can naturally take into account the contribution from the phase fluctuations or Nambu- Goldstone currents. The task left is to find the correct ground state by exploring the stability against the fluc- tuations of the magnitude and the phases of the order parameters. The free energy Ω( V$_{µ}$, Φ , µ, µ$_{8}$, µ$_{e}$ ) can be evaluated directly and it takes the form of 

<img src='content_image/386983.jpg'>

In the 2SC phase, the color symmetry SU (3)$_{c}$ is spon- taneously broken to SU (2)$_{c}$ and diquark field obtains a nonzero expectation value. Without loss of generality, one can always assume that diquark condenses in the anti-blue direction, i.e., only red and green quarks par- ticipate the Cooper pairing, while blue quarks remains as free particles. The ground state of the 2SC phase is characterized by 〈 Δ $^{3}$〉 ≡ Δ, and 〈 Δ $^{1}$〉 = 0, 〈 Δ $^{2}$〉 = 0. Considering the fluctuation of the order parameter, the 

diquark condensate can be parameterized as 

<img src='content_image/386978.jpg'>

where Φ $^{ρ}$( x ) = (0 , 0 , Δ + H ( x )) is the diquark field in the nonlinear realization framework, ϕ$_{a}$, ϕ$_{8}$ are Nambu- Goldstone bosons, and H is the Higgs field. ρ 

Expanding the diquark field Φ around the ground state: Φ ρ = (0 , 0 , Δ), the free-energy of the system takes the following expression as 

There are three contributions to the free-energy, the mean-field approximation free-energy part Ω$_{M}$ has the form of 

<img src='content_image/386973.jpg'>

<img src='content_image/386986.jpg'>

<img src='content_image/386987.jpg'>

<img src='content_image/386972.jpg'>

