Optical conductivity of a two-dimensional metal at the onset of spin-density-wave order

We consider the optical conductivity of a clean two-dimensional metal at $T=0$ near a spin-density-wave instability. Critical fluctuations destroy fermionic coherence at ``hot spots'' of the Fermi surface but a large part of the Fermi surface is neither ``hot'' or ``cold'' but rather ``lukewarm,'' in a sense that quasiparticles there are strongly renormalized compared to the non-interacting case. We discuss the self-energy of lukewarm fermions and their contribution to the optical conductivity, $\sigma(\Omega)$, due to scattering off composite bosons made of two critical magnetic fluctuations. Recent study [S.A. Hartnoll et al., Phys. Rev. B {\bf 84}, 125115 (2011)] found that composite scattering leads to a singular fermionic self-energy of lukewarm fermions at the quantum critical point. We show that, at the lowest frequencies, the most singular, $\ln^3\Omega/\Omega^{1/3}$ contribution to the conductivity is canceled between the self-energy, vertex-correction, and Aslamazov-Larkin diagrams. However, the cancellation does not extent beyond logarithmic accuracy, and the remaining conductivity still diverges as $1/\Omega^{1/3}$. At larger $\Omega$, $\sigma (\Omega)$ scales in a marginal FL way, as $1/\Omega$.