Four different phases of the two-flavor quark system, i.e., the normal quark matter and color superconducting phases with and without the charge neutrality condition, are investigated in the framework of the SU͑2͒ Nambu-Jona-Lasinio model. It is found that the color superconducting phase without charge neutrality has the lowest thermodynamic potential, and the charge neutral systems have higher thermodynamic potentials. However, the BCS pairing always lowers the system's thermodynamic potential, i.e., for both the charged and charge neutral systems, the superconducting phases always have lower thermodynamic potentials. Compared with the BCS gap ⌬ 0 for the color superconducting phase without charge neutrality, the BCS gap ⌬ in the charge neutral color superconducting phase has been largely reduced. When the thermodynamic potential for the charge neutral color superconducting phase equals that in the neutral normal quark matter, the diquark condensate disappears.

A more general expression for the quark propagator including both chiral and diquark condensates has been derived by using energy projectors. This makes it possible to study the phase transition from hadron phase to color superconductivity phase in the moderate baryon density region by using Feynman diagrammatic method or Green-function method. : 12.38.Aw; 11.10.Wx; 21.65.+f While the standard BCS (Bardeen Cooper and Schrieffer) theory predicted the existence of colorsuperconducting phase at high baryon density twenty years ago [1] [2], QCD (Quantum Chromodynamics) phase transitions along the baryon density direction attracts much attention after the appearance of [3] and [4]. The authors of the two papers found that due to the nonperturbative effects, the color-superconducting gap can be of the order of 100MeV, which is two orders larger than early perturbative estimates [5]. PACSThe color superconductivity with two massless flavors and the color-flavor-locking (CFL) phase with three degenerate massless quarks [6] have been widely discussed from first principle QCD calculations. For review of the rich QCD phase structure at high baryon density, see [7] and references therein. Usually, when the Green-function or Feynman diagrammatic method is used at the asymptotic densities [8] [9], current quark mass or chiral condensate is not necessary to be considered because it can be neglected comparing with the very high Fermi surface. At less-than-asymptotic densities, the importance of nonzero quark mass corrections to the pure CFL phase and to the meson excitation was discussed in [10], [11], [12] -[18], and the structure of the mass term was recently investigated in effective theories in [19]. The anti-particles were decoupled from the system, and the quark mass corrections were treated perturbatively by expanding the quark mass order by order [18] [19].However, for physical applications, we are more interested in moderate baryon density region which may be related to the neutron stars and even in very optimistic cases -to heavy-ion collisions. In this region, the usual way is to work out the gap equations from the thermodynamic potential [20]-[27] by using the variational methods. To work out the phase structure from hadron gas to color superconductivity, one should deal with the chiral condensate and diquark condensate simultaneously. Since the chiral condensate contributes a dynamic quark mass, it becomes unreasonable to treat the quark mass term perturbatively like in [18].In this paper, we are trying to evaluate the quark propagators including chiral condensate and diquark condensate simultaneously.At moderate baryon density, the four-fermion interaction models are usually used. With only scalar, pseudoscalar mesons and scalar diquark involved, the Lagrangian density has the formwhere q C = Cq T ,q C = q T C are charge-conjugated spinors, C = iγ 2 γ 0 is the charge conjugation matrix (the superscript T denotes the transposition operation), m 0 is the current quark mass, the quark field q ≡ q iα is a flavor do...