Anomalous contributions to the positronium hyperfine splitting and the radiation gauge

By recasting the Bethe-Salpeter equation in a more general form, the natural ensuing bound-state perturbation theory accounts for the anomalous ${\alpha }^6\ln {\alpha }^{-1\mathrm{}}$ contribution of the fourth-order vacuum polarization to the ground state of the positronium hyperfine structure as a binding correction of a two-photon diagram. Viewed as such, this contribution is not anomalous. Another two-photon diagram, in which such a binding correction makes it a candidate for an ${\alpha }^6\ln {\alpha }^{-1\mathrm{}}$ contribution, is found. By using the Bethe-Salpeter equation recast in the more general form and the perturbation theory associated with it, the formal connection with the external field approximation is given.