Anatomy of Non-Leptonic Two-Body Decays of Charmed Mesons into Final States with $\eta'$

We show that $D\rightarrow P\eta^\prime$ decay amplitudes, where $P=K,\pi,\eta$, cannot be simply related to their $D\rightarrow P\eta$ 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 $D\to P\eta^\prime$ decays. Using this framework, we perform a global analysis of $D\rightarrow P\eta^\prime$ 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 $D\rightarrow P\eta'$ modes. Our predictions deviate from the current data in case of the branching ratios $\mathcal{B}(D_s^+\rightarrow K^+\eta^\prime)$ and $\mathcal{B}(D^+\rightarrow K^+\eta')$. 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 $D\rightarrow P\eta^\prime$ decays.