First measurement of $T$-odd moments in $D^0\to K_S^0{\pi }^{+}{\pi }^{-}{\pi }^0$ decays

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

Figure 1

The signal-enhanced logarithmic distributions of (a) $\mathrm{\Delta }M$ and (b) $M_{D^0}$ for $D^0$ with $C_T>0$, (c) $\mathrm{\Delta }M$ and (d) $M_{D^0}$ for $D^0$ with $C_T<0$, (e) $\mathrm{\Delta }M$ and (f) $M_{D^0}$ for ${\overline{D}}^0$ with $-{\overline{C}}_T>0$ and (g) $\mathrm{\Delta }M$ and (h) $M_{D^0}$ for ${\overline{D}}^0$ with $-{\overline{C}}_T<0$; the $\mathrm{\Delta }M$ distributions have a selection criteria on $M_{D^0}$ in the signal region and vice versa. The black points with error bars are the data points and the solid blue curve is the projection of the total signal and background components. The dotted magenta and dashed red curves indicate combinatorial and random ${\pi }_{\mathrm{slow}}^{+}$ backgrounds, respectively. The normalized residuals (pulls) and the ${\chi }^2/\mathrm{DoF}$, where DoF is the number of degrees of freedom, are shown above each plot.