Parapositronium decay into three photons and implications for the neutral pion

TL;DR

This work completes the Standard Model prediction for $p ext{-}Ps \to 3\gamma$ by incorporating $Z$-boson loop contributions, demonstrating that the leading $Z$-mediated amplitude begins at $\mathcal{O}(1/m_Z^6)$ despite an additional $m_e$-scale. The total decay rate, including both $W$ and $Z$ diagrams, is $\Gamma \simeq 1.1\times 10^{-80}$ eV, corresponding to a branching ratio of $\mathrm{BR} \simeq 2.1\times 10^{-75}$, which is about 47 orders of magnitude smaller than earlier estimates that did not account for the exact suppression. The calculation relies on a two-region expansion to handle the two mass scales, a tensor decomposition of the amplitude with four form factors, and careful renormalization of the $C$-violating channel, while identifying several diagrams that vanish or do not contribute. The results refine expectations for parity- and CP-violating processes in low-energy QED+EW, and provide guidance for assessing the analogous $\pi^0\to3\gamma$ amplitude, where hadronic effects and extra scales may yield further insights.

Abstract

We complete the determination of the parapositronium decay into three photons by evaluating amplitudes mediated by the $Z$ boson. We show that, contrary to the expectation that the extra mass scale $m_e$ may bring an enhancement to the overall scaling, the amplitude turns out to start at $1/m_Z^6$ order, similarly to the $W$ boson mediated amplitude. The decay rate with both $W$ and $Z$ boson contributions is found to be $1.1\cdot 10^{-80}$ eV, about 47 orders of magnitude smaller than previously estimated. We also discuss its implication for the $π^0\to3γ$ amplitude.

Figures (2)

• Figure 1: Contributions of the $Z$ boson (thick wavy lines) to the three-photon decay amplitude of p-Ps. Each diagram has six permutations of the photons. For diagrams (b,c,e), there are also diagrams with the $Z$ boson appearing at locations symmetric to ones shown. Diagrams (c,d,e) are divergent and have corresponding counterterm diagrams (f,g,h).
• Figure 2: Diagrams that do not contribute to the amplitude for reasons explained in the text. Diagrams (a,b,c) contain a virtual fermion loop. In diagrams (d-h), the virtual loop has a $W$ boson. Both $Z$ and $W$ bosons are drawn with a thicker wavy line, while the photons are depicted with a regular wavy line.