  We study the dynamics of a quantum critical boson coupled to a Fermi surface
in intermediate energy regimes where the Landau damping of the boson can be
parametrically controlled, either via large Fermi velocity or by large N
techniques. We focus on developing a systematic approach to studying the BCS
instability, including careful treatment of the enhanced log^2 and log^3
singularities which appear already at 1-loop. We also treat possible
instabilities to charge density wave (CDW) formation, and compare the scales
Lambda_{BCS} and Lambda_{CDW} of the onset of the instabilities in different
parametric regimes. We address the question of whether the dressing of the
fermions into a non-Fermi liquid via interactions with the order parameter
field can happen at energies > Lambda_{BCS}, Lambda_{CDW}.
