An effective field theory of QCD at high density

We derive a (Wilsonian) effective field theory of QCD at high density by integrating out the states in the Dirac sea when the chemical potential $\mu\gg\Lambda_{\rm QCD}$. The quark-gluon coupling is effectively (1+1)-dimensional and the theory contains four-quark operators which become relevant as we approach to the Fermi sea. The gluons that couple to quarks in the high density limit get mass $M=g_s\mu/(\sqrt{2}\pi)$ while gluons perpendicular to the velocity of quarks remain massless but their couplings to quarks are supressed by $1/\mu$. At energy scale less than the gluon mass, the four-quark operator in the color anti-triplet channel with opposite momenta is most attractive and leads to color superconductivity.