Positronium energy levels at order $m{\alpha }^7$: Product contributions in the two-photon-annihilation channel

Ongoing improvements in the measurement of positronium transition intervals motivate the calculation of the $O\left(m{\alpha }^7\right)$ corrections to these intervals. In this work we focus on corrections to the spin-singlet parapositronium energies involving virtual annihilation to two photons in an intermediate state. We have evaluated all contributions to the positronium $S$-state energy levels that can be written as the product of a one-loop correction on one side of the annihilation event and another one-loop correction on the other side. These effects contribute $\mathrm{\Delta }E=-0.561971\left(25\right)m{\alpha }^7/{\pi }^3$ to the parapositronium ground-state energy.

Figure 1

Representatives of the four classes of two-photon-annihilation corrections that contribute to the parapositronium energy levels at $O\left(m{\alpha }^7\right)$. They are (a) light-by-light corrections, (b) terms involving vacuum polarization corrections to the annihilation photons, (c) terms containing the product of two one-loop corrections, one on each side of the virtual annihilation, and (d) terms with a two-loop correction on one side of the virtual annihilation.

Figure 2

The LO $2\gamma \mathrm{A}$ contributions. Both (a) uncrossed and (b) crossed photons are included to account for Bose symmetry. The wave functions bracketing these kernels on the left and right are implicit.

Figure 3

The four types of one-loop radiative corrections in the $2\gamma \mathrm{A}$ channel that contribute at NLO: (a) vacuum polarization, (b) self-energy, (c) vertex, and (d) ladder. Additional contributions having crossed annihilation photons and with the radiative corrections acting on different parts of the graphs give contributions equal to those shown and are included by means of appropriate symmetry factors.

Figure 4

The seven types of NNLO terms in the $2\gamma \mathrm{A}$ channel having a one-loop correction on each side of the annihilation event. The contributions are (a) self-energy–self-energy, (b) self-energy–vertex, (c) self-energy–ladder, (d) vertex-vertex (type A), (e) vertex-vertex (type B), (f) vertex-ladder, and (g) ladder-ladder. Each of these diagrams represents two configurations of annihilation photons (uncrossed and crossed); additional factors have been included to account for the various places in the diagram where the corrections could act.