Search for time-dependent $CP$ violation in $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decays

A measurement of time-dependent $CP$ violation in $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decays using a $pp$ collision data sample collected by the LHCb experiment in 2012 and from 2015 to 2018, corresponding to an integrated luminosity of 7.7${\mathrm{f}\mathrm{b}}^{-1}$, is presented. The initial flavour of each $D^0$ candidate is determined from the charge of the pion produced in the $D^{\ast }(2010{)}^{+}\to D^0{\pi }^{+}$ decay. The decay $D^0\to K^{-}{\pi }^{+}{\pi }^0$ is used as a control channel to validate the measurement procedure. The gradient of the time-dependent $CP$ asymmetry, $\mathrm{\Delta }Y$, in $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decays is measured to be \begin{equation*} ΔY = (-1.3 \pm 6.3 \pm 2.4) \times 10^{-4}, \end{equation*} where the first uncertainty is statistical and the second is systematic, which is compatible with $CP$ conservation.