  We present a field-theoretic renormalization group analysis of Abanov and
Chubukov's model of the spin density wave transition in two dimensional metals.
We identify the independent field scale and coupling constant renormalizations
in a local field theory, and argue that the damping constant of spin density
wave fluctuations tracks the renormalization of the local couplings. The
divergences at two-loop order overdetermine the renormalization constants, and
are shown to be consistent with our renormalization scheme. We describe the
physical consequences of our renormalization group equations, including the
breakdown of Fermi liquid behavior near the "hot spots" on the Fermi surface.
In particular, we find that the dynamical critical exponent z receives
corrections to its mean-field value z = 2. At higher orders in the loop
expansion, we find infrared singularities similar to those found by S.-S. Lee
for the problem of a Fermi surface coupled to a gauge field. A treatment of
these singularities implies that an expansion in 1/N, (where N is the number of
fermion flavors) fails for the present problem. We also discuss the
renormalization of the pairing vertex, and find an enhancement which scales as
logarithm-squared of the energy scale. A similar enhancement is also found for
a modulated bond order which is locally an Ising-nematic order.
