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On the Quark-Quark Superconducting Pairing Induced by Instantons in QCD.pdf

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On the Quark-Quark Superconducting Pairing Induced by Instantons in QCD

a r X i v : h e p - p h / 0 0 0 1 2 3 4 v 2 3 1 J a n 2 0 0 0 ON THE QUARK-QUARK SUPERCONDUCTING PAIRING INDUCED BY INSTANTONS IN QCD N. V. Hieu, H. N. Long, N. H. Son and N. A. Viet Institute of Physics, P. O. Box 429, Boho, Hanoi 10000, Vietnam. Abstract The superconducting pairing of quarks induced by instantons in QCD is studied by means of the functional integral method. The integral equation determining the critical temperature of the superconducting phase transition is established. It is shown that the Bose condensate of diquarks consists of color antitriplet (3?) scalar diquarks. The superconducting pairing of quarks in QCDwas investigated since more than twenty years[1?5]. At the present time a strong interest to this problem is motivated by recent works of Alford, Ra- jagopal and Wilczeck[6], Scha?fer and Wilczek[7], Rapp, Scha?fer, Shuryak and Velkovsky[8] and other authors[9?16]. In this work the functional integral method[15] is applied to the study of the super- conducting pairing of quarks induced by instantons[8,17,18], and the integral equation determining the critical temperature of phase transition is established. We work in the imaginary time formalism and use the notations x = (x, τ) , ∫ dx = ∫ β 0 dτ ∫ dx, β = 1 kT , k and T being the Boltzmann constant and the temperature. Denote ψA (x) = ψA (x, τ) the quark field, A being the set consisting of the index α of the Dirac spinor, the color and flavor indices a and i, A = (αai), a = 1, 2, ... , Nc, i = 1, 2, ... , Nf . Consider the system of massless quarks (and antiquarks) with the parity conserving and chirally symmetric interaction Lagrangian of the form Lint = 1 2 ψ A (x)ψ B (x)UCDBA ψD (x)ψC (x) , (1) UCDBA = ?U CD AB = ?U DC BA = U DC AB . (2) The partition function of the system equals Z = ∫ [Dψ] [ Dψ ] exp { ? ∫ dxψ A (x)DBAψB (x) } exp { 1 2 ∫ dxψ A (x)ψ B (x)UCDBA ψD (x)ψC (x) } (3) with DBA = [ γ4 ( ? ?τ ? μ ) + γ? ]β α δbaδ j i , (4) μ being the chemical potential. Introducing the