  In this paper, we study superconductivity of nonrelativistic fermions at
finite-density coupled to a transverse $U(1)$ gauge field, with the effective
interaction including the Landau-damping. This model, first studied by
Holstein, Norton, and Pincus [Phys. Rev B, {\bf 8}, 2649 (1973)] has been known
as an example of a non-Fermi liquid, {\i.e.} a metallic state in which the
decay rate of a quasiparticle is large compared to the characteristic
quasiparticle energy; other examples of the non-Fermi liquid includes the 2d
electron gas in a magnetic field at $\nu=1/2$ and the normal state of optimally
doped cuprate superconductors. Our study thus addresses the question of whether
or not non-Fermi liquids, like Fermi liquids, are unstable towards the
formation of superconductivity.The results are (i) the non-Fermi liquid is
stable against superconductivity below a critical gauge coupling, (ii) above
this critical coupling, the ground state is an unconventional superconductor
with angular momentum $\ell \ge 2$. Our results are obtained from a solution of
the Dyson-Nambu equation. Note that in this problem there is a quantum critical
point between a non-Fermi liquid state and the superconducting state, as the
critical coupling is nonzero. This is in contrast to a weakly coupled metal,
which exhibits superconductivity for infinitesimally weak interaction
regardless of its sign.
