Light-by-light-scattering contributions to the Lamb shift in light muonic atoms

We consider one-loop light-by-light-scattering contributions to the Lamb shift of the $1s, 2s, 2p$ states in light muonic hydrogen like atoms at $Z\leq10$. The contributions are of the order $\alpha^5m_\mu$ (with diverse dependence on the nuclear charge $Z$). Those include the contributions of the so-called Wichmann-Kroll potential ($\alpha(Z\alpha)^4m_\mu$), the virtual Delbr\"uck scattering ($\alpha^2(Z\alpha)^3m_\mu$), etc. The results are obtained in a nonrelativistic approximation. For the calculation of the virtual-Delbr\"uck-scattering contribution, we have constructed an effective potential in the coordinate space which may be applied to other calculations in muonic atoms.