First measurement of $boldsymbol{T}$-odd moments in $boldsymbol{D^{0} rightarrow K_{S}^{0} pi^{+} pi^{-} pi^{0}}$ decays

We report the first measurement of the $T$-odd moments in the decay $D^{0} rightarrow K_{S}^{0} pi^{+} pi^{-} pi^{0}$ from a data sample corresponding to an integrated luminosity of $966,{rm fb}^{-1}$ collected by the Belle experiment at the KEKB asymmetric-energy $e^+ e^-$ collider. From these moments we determine the $CP$-violation-sensitive asymmetry $a_{CP}^{Ttext{-odd}} = left[-0.28 pm 1.38 ~(rm{stat.}) ^{+0.23}_{-0.76} ~(rm{syst.})right] times 10^{-3}$, which is consistent with no $CP$ violation. In addition, we perform $a_{CP}^{Ttext{-odd}}$ measurements in different regions of the $D^{0} rightarrow K_{S}^{0} pi^{+} pi^{-} pi^{0}$ phase space; these are also consistent with no $CP$ violation.