Energy-Level Shift for Atoms with Point Nuclei Due to Higher Order Vacuum Polarization

The energy-level shifts for hydrogen and mesonic hydrogen due to higher order vacuum polarization effects have been calculated using Schwinger's result for the Lagrangian density which takes into account the effect of vacuum polarization so as to include Feynman diagrams involved in photon-photon scattering and similar higher order diagrams. The calculations are carried out by defining a field-dependent dielectric constant. The values of the level shifts for hydrogen and mesonic hydrogen are estimated, respectively, to be about ${10}^{-4\mathrm{}}$ times and ${10}^{-2\mathrm{}}$ times that obtained for the lowest order diagram. Results for $Z=2\mathrm{} \mathrm{and} 3$ are also given.