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The optical conductivity of a metal near a quantum critical point (QCP) isexpected to depend on frequency not only via the scattering time but also viathe effective mass, which acquires a singular frequency dependence near a QCP.We check this assertion by computing diagrammatically the optical conductivity,$\sigma' (\Omega)$, near both nematic and spin-density wave (SDW) quantumcritical points (QCPs) in 2D. If renormalization of current vertices is nottaken into account, $\sigma' (\Omega)$ is expressed via the quasiparticleresidue $Z$ (equal to the ratio of bare and renormalized masses in ourapproximation) and transport scattering rate $\gamma_{\text{tr}}$ as $\sigma'(\Omega)\propto Z^2 \gamma_{\text{tr}}/\Omega^2$. For a nematic QCP($\gamma_{\text{tr}}\propto\Omega^{4/3}$ and $Z\propto\Omega^{1/3}$), thisformula suggests that $\sigma'(\Omega)$ would tend to a constant at $\Omega \to0$. We explicitly demonstrate that the actual behavior of $\sigma' (\Omega)$ isdifferent due to strong renormalization of the current vertices, which cancelsout a factor of $Z^2$. As a result, $\sigma' (\Omega)$ diverges as$1/\Omega^{2/3}$, as earlier works conjectured. In the SDW case, we considertwo contributions to the conductivity: from hot spots and from"lukewarm"regions of the Fermi surface. The hot-spot contribution is not affected byvertex renormalization, but it is subleading to the lukewarm one. For thelatter, we argue that a factor of $Z^2$ is again cancelled by vertexcorrections. As a result, $\sigma' (\Omega)$ at a SDW QCP scales as $1/\Omega$down to the lowest frequencies.

Optical conductivity of a two-dimensional metal near a quantum-critical point: the status of the "extended Drude formula"