Measurement of the $CP$ asymmetry in $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decays at Belle II

We measure the time- and phase-space-integrated $CP$ asymmetry $A_{CP}$ in $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decays reconstructed in $e^{+}{e}^{-}\to c\bar{c}$ events collected by the Belle II experiment from 2019 to 2022. This sample corresponds to an integrated luminosity of 428 fb${}^{-1}$. We require $D^0$ mesons to be produced in $D^{\ast +}\to D^0{\pi }^{+}$ decays to determine their flavor at production. Control samples of $D^0\to K^{-}{\pi }^{+}$ decays are used to correct for reconstruction-induced asymmetries. The result, $A_{CP}(D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0)=(0.29\pm 0.27\pm 0.13)\mathrm{\%}$, where the first uncertainty is statistical and the second systematic, is the most precise result to date and is consistent with $CP$ conservation.