Parapositronium decay into three photons and implications for the neutral pion

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 $1\mathrm{/}{m}_Z^6$ 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 ${\pi }^0\to 3\gamma$ amplitude.