Three-Photon-Annihilation Contributions to Positronium Energies at Order mα^{7}.

Positronium spectroscopy (n=1 hyperfine splitting, n=2 fine structure, and the 2S-1S interval) has reached a precision of order 1 MHz. Vigorous ongoing efforts to improve the experimental results motivate the calculation of the positronium energy levels at order mα^{7}. In this Letter, we present the result for a complete class of such contributions-those involving virtual annihilation of positronium to three photons in an intermediate state. We find an energy shift of 2.6216(11)mα^{7}/(nπ)^{3}=11.5/n^{3}  kHz for the spin-triplet S state with principal quantum number n. The corresponding energy shift for true muonium (the μ^{+}μ^{-} bound state) is 2.38/n^{3}  MHz with an additional -5.33/n^{3}  MHz coming from electronic vacuum polarization.