Three-Photon-Annihilation Contributions to Positronium Energies at Order $m{\alpha }^7$

Positronium spectroscopy ($n=1$ hyperfine splitting, $n=2$ fine structure, and the $2S\text{−}1S$ 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 $m{\alpha }^7$. 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 $2.6216(11)m{\alpha }^7/(n\pi {)}^3=11.5/n^3\mathrm{kHz}$ for the spin-triplet $S$ state with principal quantum number $n$. The corresponding energy shift for true muonium (the ${\mu }^{+}{\mu }^{-}$ bound state) is $2.38/n^3\mathrm{MHz}$ with an additional $-5.33/n^3\mathrm{MHz}$ coming from electronic vacuum polarization.

Figure 1

The lowest order $3\gamma \mathrm{A}$ contribution. The positronium four momentum in the center-of-mass frame is $P=(2m,\overrightarrow{0})$, and $p$, $q$, and $P-p-q$ are the photon four momenta. Only one of the six permutations of annihilation photons is shown here.

Figure 2

The seven types of one-loop radiative corrections in the $3\gamma \mathrm{A}$ channel: (a) self-energy, (b) outer vertex, (c) inner vertex, (d) double vertex, (e) ladder, (f) light-by-light, and (g) vacuum polarization. Each of these diagrams represents the full set of permutations of the annihilation photons and additional contributions coming from the various places where the correction could act.