We extend the application of lattice QCD to the two-photon-mediated, order ${\alpha}^{2}$ rare decay ${\pi}^{0}\to {e}^{+}{e}^{-}$. By combining Minkowski- and Euclidean-space methods we are able to calculate the complex amplitude describing this decay directly from the underlying theories (QCD and QED) which predict this decay. The leading connected and disconnected diagrams are considered; a continuum limit is evaluated and the systematic errors are estimated. We find $\mathrm{Re}\mathcal{A}=18.60(1.19)(1.05)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$, $\mathrm{Im}\mathcal{A}=32.59(1.50)(1.65)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$, a more accurate value for the ratio $(\mathrm{Re}\mathcal{A}/\mathrm{Im}\mathcal{A})=0.571(10)(4)$, and a result for the partial width $\mathrm{\Gamma}({\pi}^{0}\to \gamma \gamma )=6.60(0.61)(0.67)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$. Here the first errors are statistical and the second systematic. This calculation is the first step in determining the more challenging, two-photon-mediated decay amplitude that contributes to the rare decay $K\to {\mu}^{+}{\mu}^{-}$.

- Received 24 August 2022
- Accepted 11 April 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.191901

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP^{3}.

Published by the American Physical Society

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