Aspects of high density effective theory in QCD

We study an effective theory of QCD at high density in detail, including the finite temperature effects and the leading order correction in $1/\mu$ expansion. We investigate the Cooper pair gap equation and find that the color-flavor locking phase is energetically preferred at high density. We also find the color-superconducting phase transition occurs in dense quark matter when the chemical potential is larger than $250\pm 100~{\rm MeV}$ and the temperature is lower than 0.57 times the Cooper pair gap in the leading order in the hard-dense-loop approximation. The quark-neutrino four-Fermi coupling and the quark-axion coupling receive significant corrections in dense quark matter.