Measurements of the $CP$-even fractions of $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and $D^0\to K^{+}{K}^{-}{\pi }^0$ at BESIII

The $CP$-even fractions ($F_{+}$) of the decays $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and $D^0\to K^{+}{K}^{-}{\pi }^0$ are measured with a quantum-correlated $\psi (3770)\to D\bar{D}$ data sample collected by the BESIII experiment corresponding to an integrated luminosity of 7.93 ${\mathrm{f}\mathrm{b}}^{-1}$. The results are $F_{+}^{{\pi }^{+}{\pi }^{-}{\pi }^0}=0.9406\pm 0.0036\pm 0.0021$ and $F_{+}^{K^{+}{K}^{-}{\pi }^0}=0.631\pm 0.014\pm 0.011$, where the first uncertainties are statistical and the second systematic. These measurements are consistent with the previous determinations, and the uncertainties for $F_{+}^{{\pi }^{+}{\pi }^{-}{\pi }^0}$ and $F_{+}^{K^{+}{K}^{-}{\pi }^0}$ are reduced by factors of 3.9 and 2.6, respectively. The reported results provide important inputs for the precise measurement of the angle $\gamma$ of the Cabibbo-Kobayashi-Maskawa matrix and indirect $CP$ violation in charm mixing.