  Recent and ongoing experimental work on the positronium spectrum motivates
new efforts to calculate positronium energy levels at the level of three loop
corrections. We have obtained results for one set of such corrections involving
light-by-light scattering of the photons produced in a two-photon virtual
annihilation process. Our result is an energy shift $1.58377(8) m
\alpha^7/\pi^3$ for the n=1 singlet state, correcting the ground state
hyperfine splitting by -6.95 kHz. We also obtained a new and more precise
result for the light-by-light scattering correction to the real decay of
parapositronium into two photons.

High Energy Physics - Phenomenology