Updated bounds on $CP$ asymmetries in $B^0\to {\eta }^{\prime }{K}_S$ and $B^0\to {\pi }^0{K}_S$

New rate measurements of $B^0$ decays into ${\pi }^0{\pi }^0$, ${\pi }^0\eta$, ${\pi }^0{\eta }^{\prime }$, $\eta \eta$, $\eta {\eta }^{\prime }$, ${\eta }^{\prime }{\eta }^{\prime }$ and $K^{+}{K}^{-}$ are used in conjunction with flavor $SU(3)$ to constrain the coefficients $S$ and $C$ of $\sin \Delta mt$ and $\cos \Delta mt$ in the time-dependent $CP$ asymmetries of $B^0\to {\eta }^{\prime }{K}_S$ and $B^0\to {\pi }^0{K}_S$. Experimental values of $S_{{\eta }^{\prime }K}$ are now seen to be closer to the Standard Model expectations, fully consistent with the new improved bounds.

Figure 1

Regions in the $(S_{{\eta }^{\prime }K},C_{{\eta }^{\prime }K})$ plane satisfying limits on the ratio $|A_C^{\prime }/A_P^{\prime }|$ and bounds (13) (region enclosed by the solid curve) or (14) (region enclosed by the dashed curve). The small plotted point denotes $(S_{{\eta }^{\prime }K},C_{{\eta }^{\prime }K})=(\sin 2\beta ,0)$. The points with experimental errors denote values from BABAR [9] (plain point) and Belle [11] (small square).

Figure 2

Top: points in the $S_{\pi K^{-}}{C}_{\pi K}$ plane satisfying $\pm 1\sigma$ limits (19) on the ratio $R_{\pi /K}$, with the small $B^0\to K^{+}{K}^{-}$ contribution ignored. The small plotted point denotes the pure-penguin value $S_{\pi K}=\sin 2\beta$, $C_{\pi K}=0$. Points with experimental errors denote values from BABAR [12] (plain point) and Belle [10] (small square). The dashed arc denotes the boundary of allowed values: $S_{\pi K}^2+C_{\pi K}^2\le 1$. Bottom: small $B^0\to K^{+}{K}^{-}$ contribution included.