  We study the renormalization of the Fermi surface coupled to a massless boson
near three spatial dimensions. For this, we set up a Wilsonian RG with
independent decimation procedures for bosons and fermions, where the
four-fermion interaction "Landau parameters" run already at tree level. Our
explicit one loop analysis resolves previously found obstacles in the
renormalization of finite density field theory, including logarithmic
divergences in non-local interactions and the appearance of multilogarithms.
The key aspects of the RG are the above tree level running, and a UV-IR mixing
between virtual bosons and fermions at the quantum level, which is responsible
for the renormalization of the Fermi velocity. We apply this approach to the
renormalization of $2 k_F$ singularities, and a companion paper considers the
RG for Fermi surface instabilities. We end with some comments on the
renormalization of finite density field theory with the inclusion of Landau
damping of the boson.
