Light-by-light scattering in the Lamb shift and the bound electron g factor

We compute an $\mathcal{O}\left(\alpha^2(Z\alpha)^6\right)$ contribution to the hydrogen-atom Lamb shift arising from the light-by-light scattering. Analogous diagrams, with one atomic electric field insertion replaced by an external magnetic field, contribute to the gyromagnetic factor of the bound electron at $\mathcal{O}\left(\alpha^2(Z\alpha)^4\right)$. We also calculate the contribution to the gyromagnetic factor from the muon magnetic loop.