The Ginzburg-Landau Free Energy Functional of Color Superconductivity at Weak Coupling

We derive the Ginzburg-Landau free energy functional of color superconductivity in terms of the thermal diagrams of QCD in its perturbative region. The zero mode of the quadratic term coefficient yields the same transition temperature, including the pre-exponential factor, as the one obtained previously from the Fredholm determinant of the two quark scattering amplitude. All coefficients of the free energy can be made identical to those of a BCS model by setting the Fermi velocity of the latter equal to the speed of light. We also calculate the induced symmetric color condensate near $T_c$ and find that it scales as the cubic power of the dominant antisymmetric color component. We show that in the presence of an inhomogeneity and a nonzero gauge potential, while the color-flavor locked condensate dominates in the bulk, the unlocked condensate, the octet, emerges as a result of a simultaneous color-flavor rotation in the core region of a vortex filament or at the junction of super and normal phases.