Rare decays of the positronium ion and molecule, ${\mathrm{Ps}}^{-}$ $\to$ $e^{-}\gamma$ and ${\mathrm{Ps}}_2$ $\to$ $e^{+}{e}^{-}\gamma$, $\gamma \gamma$, $e^{+}{e}^{-}$

Decay rates of the positronium molecule ${\text{Ps}}_2$ into two photons and into an electron-positron pair are determined. Previous studies find that these rates are very different, $\mathrm{\Gamma }({\text{Ps}}_2\to e^{+}{e}^{-})/\mathrm{\Gamma }({\text{Ps}}_2\to \gamma \gamma )\simeq 250$. This is puzzling since both processes have two-body final states and are of the same order in the fine-structure constant. We propose a simple calculational method and test it with the well-established decay of the positronium ion into an electron and a photon. We then employ it to correct predictions for both these ${\text{Ps}}_2$ decays. We find that previous studies overestimated the $e^{+}{e}^{-}$ channel and underestimated the $\gamma \gamma$ channel by factors of about 5.44 and 3.93, respectively. Our results give $\mathrm{\Gamma }({\text{Ps}}_2\to e^{+}{e}^{-})/\mathrm{\Gamma }({\text{Ps}}_2\to \gamma \gamma )\simeq 11.7$.

Figure 1

Examples of contributions to the single-photon decay of the positronium ion, ${\text{Ps}}^{-}\to e^{-}\gamma$. Arrows indicate the spin projections on the $z$ axis: plain arrows denote spins 1/2 and a double arrow denotes spin 1 of the photon. Four more diagrams are obtained by switching the roles of the two electrons $e^{-}$. In all plots, time flows horizontally from left to right.

Figure 2

Three types of contributions to the two-photon decay ${\text{Ps}}_2\to \gamma \gamma$. Diagrams of type C turn out not to contribute to ${\text{Ps}}_2$ decays. One example of external particles' spins is shown: a spin-up fermion cannot emit a spin-down photon.

Figure 3

Four groups of contributions to the annihilation ${\text{Ps}}_2\to e^{+}{e}^{-}$. In group I, a single virtual photon produces the final state; such diagrams do not contribute to the decay of the molecule ${\text{Ps}}_2$, which has an angular momentum of zero.