Sum Rules of Charm $CP$ Asymmetries beyond the ${\mathrm{SU}(3)}_F$ Limit

We find new sum rules between direct $CP$ asymmetries in $D$ meson decays with coefficients that can be determined from a global fit to branching ratio data. Our sum rules eliminate the penguin topologies $P$ and $PA$, which cannot be determined from branching ratios. In this way, we can make predictions about direct $CP$ asymmetries in the standard model without ad hoc assumptions on the sizes of penguin diagrams. We consistently include first-order ${\mathrm{SU}(3)}_F$ breaking in the topological amplitudes extracted from the branching ratios. By confronting our sum rules with future precise data from LHCb and Belle II, one will identify or constrain new-physics contributions to $P$ or $PA$. The first sum rule correlates the $CP$ asymmetries $a_{CP}^{\text{dir}}$ in $D^0\to K^{+}{K}^{-}$, $D^0\to {\pi }^{+}{\pi }^{-}$, and $D^0\to {\pi }^0{\pi }^0$. We study the region of the $a_{CP}^{\text{dir}}(D^0\to {\pi }^{+}{\pi }^{-})-a_{CP}^{\text{dir}}(D^0\to {\pi }^0{\pi }^0)$ plane allowed by current data and find that our sum rule excludes more than half of the allowed region at 95% C.L. Our second sum rule correlates the direct $CP$ asymmetries in $D^{+}\to {\overline{K}}^0{K}^{+}$, $D_s^{+}\to K^0{\pi }^{+}$, and $D_s^{+}\to K^{+}{\pi }^0$.

Figure 1

SM predictions for $CP$ asymmetries obtained from our global fit. In Figures (a), (b), (e) the dashed (solid) red lines delimit the experimental 68% (95%) C.L. regions. The other dashed (solid) lines are the 68% (95%) C.L. regions of the respective fit scenarios explained in the captions of the subfigures. The generic error of order $\sim 30\%$ from ${\mathrm{SU}(3)}_F$ breaking in $P$ and $P+PA$ is not shown. The experimental error ellipses are obtained by scaling the errors quoted in Table 3 by factors of $\sqrt{2.28}$ and $\sqrt{5.99}$ in order to obtain the two-dimensional 68% and 95% C.L. regions from the corresponding one-dimensional ranges. The experimental error ellipse shown in (b) is calculated from the corresponding one for $\mathrm{\Delta }{a}_{CP}^{\text{dir}}$ and $\mathrm{\Sigma }{a}_{CP}^{\text{dir}}$. (a) Fit including $1/N_c$ counting in the topological amplitudes with current (future) branching ratio data in blue (green). The black (magenta) line delimits the 95% C.L. region found in a fit without using $1/N_c$ counting in the topological amplitudes for current (future) branching ratio data. Note that the black and magenta curves lie on top of each other. (b) Color coding as in (a). The dash-dotted line denotes the $\mathrm{SU}(3{)}_F$-limit sum rule of Eq. (3). (c) Prediction from current branching ratio data. The dashed, solid, and dash-dotted lines correspond to the $1\sigma ,2\sigma$, and $3\sigma$ intervals, respectively. The blue bars show our fit results for current branching ratio data. The corresponding result without $1/N_c$ counting [i.e., only using $\mathrm{SU}(3{)}_F$] is shown in black. (d) Same as in (c), but for our future scenario with smaller errors of the branching ratios. The green (magenta) bars correspond to the analysis with (without) $1/N_c$ counting. (e) Future scenario assuming $a_{CP}^{\mathrm{dir}}(D_s^{+}\to K^{+}{\pi }^0)=-1\%$ and branching ratios with current (future) errors including $1/N_c$ counting in blue (green). The corresponding allowed area found in a fit without $1/N_c$ counting is delimited by the black (magenta) line. Note that the black, magenta, and red curves lie on top of each other. The dash-dotted line denotes the $\mathrm{SU}(3{)}_F$-limit sum rule of Eq. (4).