Higher-order nuclear-polarizability corrections in atomic hydrogen

Nuclear-polarizability corrections that go beyond unretarded-dipole approximation are calculated analytically for hydrogenic (atomic) $S$ states. These retardation corrections are evaluated numerically for deuterium and contribute $-0.68$ kHz, for a total polarization correction of 18.58(7) kHz. Our results are in agreement with one previous numerical calculation, and the retardation corrections completely account for the difference between two previous calculations. The uncertainty in the deuterium polarizability correction is substantially reduced. At the level of 0.01 kHz for deuterium, only three primary nuclear observables contribute: the electric polarizability ${\alpha }_E,$ the paramagnetic susceptibility ${\beta }_M,$ and the third Zemach moment $〈r^3{〉}_{\mathrm{}\left(2\right)\mathrm{}}.$ Cartesian multipole decomposition of the virtual Compton amplitude and its concomitant gauge sum rules are used in the analysis.