Anatomy of Non-Leptonic Two-Body Decays of Charmed Mesons into Final States with ${\eta }^{\mathrm{\prime }}$

We show that $D\to P{\eta }^{\mathrm{\prime }}$ decay amplitudes, where $P=K,\pi ,\eta$, cannot be simply related to their $D\to P\eta$ counterparts with a single ${\eta }_0$–${\eta }_8$ mixing angle. We propose a novel, consistent treatment of ${\eta }_0$–${\eta }_8$ mixing for application in $D\to P\eta$ and $D\to P{\eta }^{\mathrm{\prime }}$ decays. Using this framework, we perform a global analysis of $D\to P{\eta }^{\mathrm{\prime }}$ decays employing SU(3)${}_F$ symmetry including linear SU(3)${}_F$ breaking. We find that the assumption of 30% SU(3)${}_F$ breaking is in slight tension ($2.5\sigma$) with the data when compared to a fit that allows for 50% SU(3)${}_F$ breaking, the latter giving a perfect description of the data. In order to allow for further scrutinization of SU(3)${}_F$-breaking effects in the future, we give branching ratio predictions for all $D\to P{\eta }^{\mathrm{\prime }}$ modes. Our predictions deviate from the current data in case of the branching ratios $\mathcal{B}(D_s^{+}\to K^{+}{\eta }^{\mathrm{\prime }})$ and $\mathcal{B}(D^{+}\to K^{+}{\eta }^{\mathrm{\prime }})$. Future more precise measurements of these channels are therefore highly important in order to clarify the quality of the SU(3)${}_F$ expansion in nonleptonic $D\to P{\eta }^{\mathrm{\prime }}$ decays.