Perturbative nature of color superconductivity

Color superconductivity is a possible phase of high density QCD. We present a systematic derivation of the transition temperature $T_C$ from the QCD Lagrangian through study of the di-quark proper vertex. With this approach, we confirm the dependence of $T_C$ on the coupling $g,$ namely $T_C\sim \mu g^{-5}{e}^{-\kappa /g},$ previously obtained from the one-gluon exchange approximation in the superconducting phase. The diagrammatic approach we employ allows us to examine the perturbative expansion of the vertex and the propagators. We find an additional $O\left(1\right)\mathrm{}$ contribution to the prefactor of the exponential from the one-loop quark self energy and that the other one-loop radiative contributions and the two gluon exchange vertex contribution are subleading.