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Positronium spectroscopy (n=1 hyperfine splitting, n=2 fine structure, andthe 1S-2S interval) has reached a precision of order 1 MHz. Vigorous ongoingefforts to improve the experimental results motivate the calculation of thepositronium energy levels at order $m \alpha^7$. In this work we present theresult for a complete class of such contributions--those involving virtualannihilation of positronium to three photons in an intermediate state. We findan energy shift of $2.6216(11) m \alpha^7/(n \pi)^3$ for the spin-triplet Sstate with principal quantum number n.

Three-photon-annihilation contributions to positronium energies at order $m \alpha^7$