  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.
