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\overline{D}$ data sample collected by the BESIII experiment corresponding to an integrated luminosity of $ 7.93\text{}\text{}{\mathrm{fb}}^{-1}$. The results are $ {F}_{+}^{{\pi }^{+}{\pi }^{-}{\pi }^{0}}=0.9406\pm 0.0055\pm 0.0033$ and $ {F}_{+}^{{K}^{+}{K}^{-}{\pi }^{0}}=\phantom{\rule{0ex}{0ex}}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 2.6 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.