Superconductivity from perturbative one-gluon exchange in high density quark matter

We study color superconductivity in QCD at an asymptotically large chemical potential. In this limit, pairing is dominated by perturbative one-gluon exchange. We derive the Eliashberg equation for the pairing gap and solve this equation numerically. Taking into account both magnetic and electric gluon exchanges, we find $\Delta \sim g^{-5}\exp (-c/g)$ with $c=3\mathrm{}{\pi }^2/\sqrt{2},$ verifying a recent result by Son. For chemical potentials that are of physical interest, $\mu <1\mathrm{}\hspace{0.5em}\mathrm{GeV},$ the calculation ceases to be reliable quantitatively, but our results suggest that the gap can be as large as 100 MeV.