  We study the quantum theory of a Fermi surface coupled to a gapless boson
scalar in $D=4-\epsilon$ spacetime dimensions as a simple model for non-Fermi
liquids (NFL) near a quantum phase transition. Our analysis takes into account
the full backreaction from Landau damping of the boson, and obtains an RG flow
that proceeds through three distinct stages. Above the scale of Landau damping
the Fermi velocity flows to zero, while the coupling evolves according to its
classical dimension. Once damping becomes important, its backreaction leads to
a crossover regime where dynamic and static damping effects compete and the
fermion self-energy does not respect scaling. Below this crossover and having
tuned the boson to criticality, the theory flows to a $z=3$ scalar interacting
with a NFL. By increasing the number of bosonic flavors, the phase diagram near
the quantum critical point interpolates between a superconducting dome fully
covering the NFL behavior, and a phase where NFL effects become important
first, before the onset of superconductivity. A generic prediction of the
theory is that the Fermi velocity and quasiparticle residue vanish with a
power-law $\omega^\epsilon$ as the fixed point is approached. These features
may be useful for understanding some of the phenomenology of high $T_c$
materials in a systematic $\epsilon$--expansion.
