Color superconductivity from the chiral quark-meson model

We study the two-flavor color superconductivity of low-temperature quark matter in the vicinity of chiral phase transition in the quark-meson model where the interactions between quarks are generated by pion and sigma exchanges. Starting from the Nambu-Gor'kov propagator in real-time formulation we obtain finite temperature (real axis) Eliashberg-type equations for the quark self-energies (gap functions) in terms of the in-medium spectral function of mesons. Exact numerical solutions of the coupled nonlinear integral equations for the real and imaginary parts of the gap function are obtained in the zero temperature limit using a model input spectral function. We find that these components of the gap display a complicated structure with the real part being strongly suppressed above $2\Delta_0$, where $\Delta_0$ is its on-shell value. We find $\Delta_0\simeq 40$ MeV close to the chiral phase transition.