Parapositronium decay into three photons and implications for the neutral pion

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 $\pi^0\to3\gamma$ amplitude.