Search for time-dependent $CP$ violation in $D^0 → π^+ π^- π^0$ decays

A measurement of time-dependent $CP$ violation in D⁰ → π⁺ π⁻ π⁰ 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\,fb⁻¹, is presented. The initial flavour of each D⁰ candidate is determined from the charge of the pion produced in the D^*(2010)⁺ → D⁰ π⁺ decay. The decay D⁰ → K⁻ π⁺ π⁰ is used as a control channel to validate the measurement procedure. The gradient of the time-dependent $CP$ asymmetry, Δ Y, in D⁰ → π⁺ π⁻ π⁰ decays is measured to be {equation*} Δ Y = (-1.3 ± 6.3 ± 2.4) × 10⁻⁴, {equation*} where the first uncertainty is statistical and the second is systematic, which is compatible with $CP$ conservation.