  Non-Fermi liquids arise when strong interactions destroy stable fermionic
quasiparticles. The simplest models featuring this phenomenon involve a Fermi
surface coupled to fluctuating gapless bosonic order parameter fields, broadly
referred to as "Hertz-Millis" models. We revisit a controlled approach to
Hertz-Millis models that combines an expansion in the inverse number ($N$) of
fermion species with an expansion in the deviation of the boson dynamical
critical exponent $z$ from 2. The structure of the expansion is found to be
qualitatively different in the quantum critical regime $\Omega \ll q$ and in
the transport regime $\Omega \gg q$. In particular, correlation functions in
the transport regime involve infinitely many diagrams at each order in
perturbation theory. We provide an explicit and tractable recipe to classify
and resum these diagrams. For the simplest Hertz-Millis models, we show that
this recipe is consistent with non-perturbative anomaly arguments and correctly
captures the fixed point optical conductivity as well as leading corrections
from irrelevant operators. We comment on potential applications of this
expansion to transport in more complicated Hertz-Millis models as well as
certain beyond-Landau metallic quantum critical points.
