Measurements of the $CP$-even fractions of $D^0\toπ^{+}π^{-}π^{0}$ and $D^0\to K^{+}K^{-}π^{0}$ at BESIII
The $CP$-even fractions ($F_{+}$) of the decays $D^0\toπ^{+}π^{-}π^{0}$ and $D^0\to K^{+}K^{-}π^{0}$ are measured with a quantum-correlated $ψ(3770)\to D\bar{D}$ data sample collected by the BESIII experiment corresponding to an integrated luminosity of 7.93 $\mathrm{fb}^{-1}$. The results are $F_{+}^{π^{+}π^{-}π^{0}}=0.9406\pm0.0036\pm0.0021$ and $F_{+}^{K^{+}K^{-}π^{0}}=0.631\pm0.014\pm0.011$, where the first uncertainties are statistical and the second systematic. These measurements are consistent with the previous determinations, and the uncertainties for $F_{+}^{π^{+}π^{-}π^{0}}$ and $F_{+}^{K^{+}K^{-}π^{0}}$ are reduced by factors of 3.9 and 2.6, respectively. The reported results provide important inputs for the precise measurement of the angle $γ$ of the Cabibbo-Kobayashi-Maskawa matrix and indirect $CP$ violation in charm mixing.