Virtual annihilation contribution to orthopositronium decay rate

Positronium, the bound state of an electron and positron, is an excellent laboratory to test our understanding of Quantum Electrodynamics of bound states. Although in the majority of cases the agreement between theory and experiment is very good, the case of orthopositronium (o-Ps) decay into three photons is outstanding, since the theoretical predictions differ by about 6 standard deviations from the most accurate experimental result [1] (see, however, an alternative result in [2]). Provided that the experiment [1] is correct, the theory can only be rescued if the second order correction to the o-Ps lifetime turns out to be ∼ 250(α/π)ΓLO. It is however difficult to imagine how such a large number could appear in the perturbative calculations, even if the bound state is involved. This point of view is supported by a recent complete calculation of the O(α2) correction to the parapositronium (p-Ps) decay rate into two photons [3]. It has shown that the “natural scale” of the gauge-invariant contributions is [several units]×(α/π)2ΓLO. For this reason, it was conjectured in [3] that the O(α2) correction to the orthopositronium decay rate o-Ps → 3γ most likely is of the same order of magnitude. At first sight, the result of Ref. [4],