Observation of $D^{+}\to K^{+}{\eta }^{(\mathrm{\prime })}$ and Search for CP Violation in $D^{+}\to {\pi }^{+}{\eta }^{(\mathrm{\prime })}$ Decays

We report the first observation of the doubly Cabibbo-suppressed decays $D^+ \rightarrow K^+ \eta^{(\prime)}$usinga791fb$^{-1}$ data sample collected with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. The ratio of the branching fractions of doubly Cabibbo-suppressed relative to singly Cabibbo-suppressed $D^{+}\to {\pi }^{+}{\eta }^{(\mathrm{\prime })}$ decays are $\mathcal{B}(D^{+}\to K^{+}\eta )\mathrm{/}\mathcal{B}(D^{+}\to {\pi }^{+}\eta )$ = (3.06 $\pm$ 0.43 $\pm$ 0.14)% and $$\mathcal{B}$(D^+ \rightarrow K^+ \eta’)/$\mathcal{B}$(D^+ \rightarrow \pi^+ \eta’)$=(3.77$\pm$0.39$\pm$ 0.10)%. From these, we find that the relative final-state phase difference between the tree and annihilation amplitudes in $D^{+}$ decays, ${\delta }_{TA}$, is (72 $\pm$ 9)${}^{\circ }$ or (288 $\pm$ 9)${}^{\circ }$. We also report the most precise measurements of CP asymmetries to date: $A_{CP}^{D^{+}\to {\pi }^{+}\eta }$ = (+1.74 $\pm$ 1.13 $\pm$ 0.19)% and $A_{CP}^{D^{+}\to {\pi }^{+}{\eta }^{\mathrm{\prime }}}$ = ($-$0.12 $\pm$ 1.12 $\pm$ 0.17)%.