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

We derive the low-energy expansion of $(Zalpha) ^{2}$ and $(Zalpha) ^{4}$ 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.