Vacuum Polarization and Positronium-Ground-State Splitting

The value of vacuum-polarization amplitude is investigated in the region of positronium bound-state poles, where the naive Feynman-graph expansion fails. The corresponding contribution to positronium-ground-state splitting is explicitly evaluated up to order ${\alpha }^6$. The previously given result for the ${\alpha }^6 \ln {\alpha }^{-1\mathrm{}}$ term coming from fourth-order vacuum-polarization corrections to the one-photon annihilation diagram is confirmed and put on a sounder basis. The ${\alpha }^6$ term is found to receive, besides the obvious contribution from ${\alpha }^2$ vacuum-polarization term, an additional ${\alpha }^6$-contribution which can be considered as generated by an infinite series of terms of higher nominal order in $\alpha$.