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Motivated by recent optical measurements on a number of strongly correlatedelectron systems, we revisit the dependence of the conductivity of a Fermiliquid, \sigma(\Omega,T), on the frequency \Omega and temperature T. Using theKubo formalism and taking full account of vertex corrections, we show that theFermi liquid form Re\sigma^{-1}(\Omega,T)\propto \Omega^2+4\pi^2T^2 holds undervery general conditions, namely in any dimensionality above one, for a Fermisurface of an arbitrary shape (but away from nesting and van Hovesingularities), and to any order in the electron-electron interaction. We alsoshow that the scaling form of Re\sigma^{-1}(\Omega,T) is determined by theanalytic properties of the conductivity along the Matsubara axis. If a systemcontains not only itinerant electrons but also localized degrees of freedomwhich scatter electrons elastically, e.g., magnetic moments or resonant levels,the scaling form changes to Re\sigma^{-1}(\Omega,T)\propto \Omega^2+b\pi^2T^2,with 1\leq b<\infty. For purely elastic scattering, b =1. Our analysis impliesthat the value of b\approx 1, reported for URu_2Si_2 and some rare-earth baseddoped Mott insulators, indicates that the optical conductivity in thesematerials is controlled by an elastic scattering mechanism, whereas the valuesof b\approx 2.3 and b\approx 5.6, reported for underdoped cuprates andorganics, correspondingly, imply that both elastic and inelastic mechanismscontribute to the optical conductivity.

First-Matsubara-frequency rule in a Fermi liquid. Part II: Optical conductivity and comparison to experiment