Renormalization of the Two-Photon Vacuum Polarization and the Self Energy Vacuum Polarization for a Tightly Bound Electron

The renormalization method of Bogoljubov-Parasuik-Hepp-Zimmermann (BPHZ) is used in order to derive the renormalized energy shift due to the gauge invariant Källén-Sabry diagram of the two-photon vacuum polarization (VPVP) as well as the self energy vacuum polarization S(VP)E beyond the Uehling approximation. It is outlined, that no outer renormalization is required for the two-photon vacuum polarization and that only the inner renormalization has to be accomplished. It is shown that the so-called nongauge invariant sourious term is absent for a wide class of vacuum polarization (VP) diagrams if one applies the widely used sherical expansion of bound and free-electron propagator. This simplifies significantly calculations in bound state quantum electrodynamis. As one result of our paper the use of the BPHZ-approach in bound state QED is established.