Superconducting quark matter in SU (2) colour group

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The effects of superconductivity of the quark matter in SU(2) colour group are studied. We analyze the QCD-inspired model of interaction of the quarks with the four quark contact interaction which represents relativistic extension of the BCS model. We construct, using the Gor'kov's propagator approach to description of the superconductivity, explicit expressions for the quark propagator, for the anomalous Green's functions, discuss their transformation properties, transformation properties of the order parameters, and find dispersion laws for quasiparticle modes in the one flavour quark matter. There exist two types of solutions of the superconductivity equations corresponding to condensation of the quark Cooper pairs in the triplet spin state. We show that the gap function depends on directions of the quasiparticle momentum and in the massless limit the energy gap vanishes for some directions of the momentum. We estimate the order parameters and critical temperaturs of the phase transition from the normal to the superconducting state, and give expressions for the fermion number density and spin density in the two states of the quark matter. We discuss also analogy between the superconducting quark matter and the nuclear matter in the case of SU(2) colour group. Two quarks in colourless state in SU(2) colour group carry fermion number and should therefore be identified with baryons, so the Cooper pairs in the quark matter can be considered as a kind of baryons present in the quark matter. The superconducting quark matter can be considered as a mixture of quarks and baryons. Such a type of the baryonic matter is shown to obey saturation condition — the well known property of nuclear matter in the real world. The possibility of extension of the model considered to the SU(3) colour group is discussed.

The authors discussed some of the problems and were supported much with the constant interest, before his untimely death, by I.Yu. Kobzarev. We are grateful to M.M. Giannini for many discussions of the superconducting quark matter. We like to express gratitude to the referee for careful reading the manuscript and valuable remarks. One of the authors (M.I.K.) is indebted to D.A. Kirzhnitz for pointing out earlier papers on the quark matter superconductivity. He is also grateful to the directional board of I.N.F.N. for grants.