arXiv2021
We measure the branching fractions and $CP$ asymmetries for the singly Cabibbo-suppressed decays $D^{0}\toπ^{+}π^{-}η$, $D^{0}\to K^{+}K^{-}η$, and $D^{0}\toϕη$, using 980 fb$^{-1}$ of data from the Belle experiment at the KEKB $e^+e^-$ collider. We obtain \begin{eqnarray} \mathcal{B}(D^{0}\toπ^{+}π^{-}η) & = & [1.22\pm 0.02\,({\rm stat})\pm 0.02\,({\rm syst})\pm 0.03\,(\mathcal{B}_{\rm ref})]\times 10^{-3}\,, \nonumber \\ \mathcal{B}(D^{0}\to K^{+}K^{-}η) & = & [1.80\,^{+0.07}_{-0.06}\,({\rm stat})\pm 0.04\,({\rm syst})\pm 0.05\,(\mathcal{B}_{\rm ref})]\times 10^{-4}\,, \nonumber \\ \mathcal{B}(D^{0}\toϕη) & = & [1.84\pm 0.09\,({\rm stat})\pm 0.06\,({\rm syst})\pm 0.05\,(\mathcal{B}_{\rm ref})]\times 10^{-4}\,, \nonumber \end{eqnarray} where the third uncertainty ($\mathcal{B}_{\rm ref}$) is from the uncertainty in the branching fraction of the reference mode $D^{0}\to K^{-}π^{+}η$. The color-suppressed decay $D^{0}\toϕη$ is observed for the first time, with very high significance. The results for the $CP$ asymmetries are \begin{eqnarray} A_{CP}(D^{0}\toπ^{+}π^{-}η) & = & [0.9\pm 1.2\,({\rm stat})\pm 0.5\,({\rm syst})]\%\,, \nonumber \\ A_{CP}(D^{0}\to K^{+}K^{-}η) & = & [-1.4\pm 3.3\,({\rm stat})\pm 1.1\,({\rm syst})]\%\,, \nonumber \\ A_{CP}(D^{0}\toϕη)&= & [-1.9\pm 4.4\,({\rm stat})\pm 0.6\,({\rm syst})]\%\,. \nonumber \end{eqnarray} The results for $D^{0}\toπ^{+}π^{-}η$ are a significant improvement over previous results. The branching fraction and $A_{CP}$ results for $D^{0}\to K^{+}K^{-}η$, and the $A_{CP}$ result for $D^{0}\toϕη$, are the first such measurements. No evidence for $CP$ violation is found in any of these decays.