  We present a theory of the influence of band renormalization and excitonic
electron-electron interaction effects on the optical conductivity
$\sigma(\omega)$ of doped bilayer graphene. Using the Keldysh formalism, we
derive a kinetic equation from which we extract numerical and approximate
analytic results for $\sigma(\omega)$. Our calculations reveal a previously
unrecognized mechanism which couples the Drude and interband response and
renormalizes the plasmon frequency, and suggest that screening must play an
essential role in explaining the weakly renormalized conductivity seen in
recent experiments.
