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

The renormalization method of Bogoljubov-Parasiuk-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 b e accomplished.
It is shown that the so-called nongauge invariant spurious term
is absent for a wide class of vacuum polarization (VP)
diagrams if one applies the widely used spherical 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.