We measure the branching fractions and $CP$ asymmetries for the singly Cabibbo-suppressed decays $D^{ 0}\to\pi^{ +}\pi^{ -}\eta$, $D^{ 0}\to K^{ +}K^{ -}\eta$, and $D^{ 0}\to\phi\eta$, using 980 fb$^{ -1}$ of data from the Belle experiment at the KEKB $e^+e^-$ collider. We obtain \begin{ eqnarray} \mathcal{ B}(D^{ 0}\to\pi^{ +}\pi^{ -}\eta) & = & [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^{ -}\eta) & = & [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}\to\phi\eta) & = & [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 $D^{ 0}\to K^{ -}\pi^{ +}\eta$. The color-suppressed decay $D^{ 0}\to\phi\eta$ is observed for the first time, with very high significance. The results for the $CP$ asymmetries are \begin{ eqnarray} A_{ CP}(D^{ 0}\to\pi^{ +}\pi^{ -}\eta) & = & [0.9\pm 1.2\,({ \rm stat})\pm 0.5\,({ \rm syst})]\%\,, \nonumber \\ A_{ CP}(D^{ 0}\to K^{ +}K^{ -}\eta) & = & [-1.4\pm 3.3\,({ \rm stat})\pm 1.1\,({ \rm syst})]\%\,, \nonumber \\ A_{ CP}(D^{ 0}\to\phi\eta)&= & [-1.9\pm 4.4\,({ \rm stat})\pm 0.6\,({ \rm syst})]\%\,. \nonumber \end{ eqnarray} The results for $D^{ 0}\to\pi^{ +}\pi^{ -}\eta$ are a significant improvement over previous results. The branching fraction and $A_{ CP}$ results for $D^{ 0}\to K^{ +}K^{ -}\eta$, and the $A_{ CP}$ result for $D^{ 0}\to\phi\eta$, are the first such measurements. No evidence for $CP$ violation is found in any of these decays.
Categories: hep-ex