title: Three-Photon-Annihilation Contributions to Positronium Energies at Order mα^{7}.
abstract: 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.
authors.affiliation: Franklin & Marshall College, Lancaster, Pennsylvania 17604, USA.
authors.forename: Gregory S
authors.initials: GS
authors.lastname: Adkins
authors.affiliation: Franklin & Marshall College, Lancaster, Pennsylvania 17604, USA.
authors.forename: Minji
authors.initials: M
authors.lastname: Kim
authors.affiliation: Franklin & Marshall College, Lancaster, Pennsylvania 17604, USA.
authors.forename: Christian
authors.initials: C
authors.lastname: Parsons
authors.affiliation: Martin A. Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02453, USA.
authors.forename: Richard N
authors.initials: RN
authors.lastname: Fell