Search for $CP$ violation in neutral $D$ meson Cabibbo-suppressed three-body decays

Using $385{\mathrm{fb}}^{-1}$ of $e^{+}{e}^{-}$ collision data collected at center-of-mass energies around 10.6 GeV, we search for time-integrated $CP$ violation in the Cabibbo-suppressed decays $D^0/{\overline{D}}^0\to {\pi }^{-}{\pi }^{+}{\pi }^0$ and $D^0/{\overline{D}}^0\to K^{-}{K}^{+}{\pi }^0$ with both model-independent and model-dependent methods. Measurements of the asymmetries in amplitudes of flavor states and $CP$ eigenstates provide constraints on theories beyond the standard model, some of which predict $CP$ violation in amplitudes at the 1% level or higher. We find no evidence of $CP$ violation and hence no conflict with the standard model.

Figure 1

Normalized residuals in Dalitz plot elements, defined in Eq. (1), for (a) $D\to {\pi }^{-}{\pi }^{+}{\pi }^0$ and (b) $D\to K^{-}{K}^{+}{\pi }^0$.

Figure 2

Normalized residuals for the first three Legendre polynomial moments of the ${\pi }^{-}{\pi }^{+}$ (row 1), ${\pi }^{+}{\pi }^0$ (row 2), $K^{-}{K}^{+}$ (row 3), and $K^{+}{\pi }^0$ (row 4) subsystems. The confidence level for no $CP$ violation (dashed line) is obtained from the first eight moments. The error bars represent $\pm 1\sigma$.

Figure 3

Phase-space-integrated $CP$ asymmetry as a function of the cosine of the polar angle of the reconstructed $D$ candidate CM momentum for (a) $D\to {\pi }^{-}{\pi }^{+}{\pi }^0$ and (b) $D\to K^{-}{K}^{+}{\pi }^0$ decays. The dashed lines represent the central values, and the shaded regions the $1\sigma$ intervals.