Quark-meson diquark model and color superconductivity in dense quark matter

We consider the two- and three-flavor QMD models as renormalizable low-energy models for QCD at finite quark chemical potentials with quarks, mesons, and diquarks as effective degrees of freedom. Using the on-shell scheme the parameters in the scalar sector can be fixed and expressed in terms of observed meson masses and decay constants. The remaining parameters can be varied. In the QMD models, all the symmetries are global, including the $SU(N_c)$ symmetry. The breaking of the global symmetries gives rise to a number of Goldstone bosons depending on the symmetry-breaking pattern, i.e. whether the system is in the 2SC phase or the color-flavor-locked (CFL) phase. This is in contrast to perturbative QCD, where some of the gauge bosons become massive via the Higgs mechanism. We classify the Goldstone bosons and show that their type and number are in accordance with general counting rules. The thermodynamic potential $\Omega$ is calculated in the mean-field approximation, where we include quark loops, while mesons and diquarks are treated at tree level. As important applications, we study the properties of the pion-condensed phase at finite isospin chemical potential, and the 2SC and CFL phases at finite baryon chemical potential. We present a few numerical results focusing on the speed of sound, gaps, and condensates. It is shown that the BCS gaps approaches a constant for large isospin and baryon chemical potentials and that the speed of sound approaches the conformal value from above in the same limit.