$(Z\alpha {)}^4$ order of the polarization operator in Coulomb field at low energy

We derive the low-energy expansion of $(Z\alpha {)}^2$ and $(Z\alpha {)}^4$ terms of the polarization operator in the Coulomb field. Physical applications such as the low-energy Delbr"{u}ck scattering and “magnetic loop” contribution to the $g$ factor of the bound electron are considered.