First measurement of the CKM angle $\ddot{I}{•}_3$ with $B^{\pm }\to D(K_{\mathrm{S}}^0\ddot{I}{€}^{+}\ddot{I}{€}^{-}\ddot{I}{€}^0)K^{\pm }$ decays

Abstract

We present the first model-independent measurement of the CKM unitarity triangle angle $\ddot{I}{•}_3$ using $B^{\pm }\to D(K_{\mathrm{S}}^0\ddot{I}{€}^{+}\ddot{I}{€}^{-}\ddot{I}{€}^0)K^{\pm }$ decays, where $D$ indicates either a $D^0$ or ${\overline{D}}^0$ meson. Measurements of the strong-phase difference of the $D\to K_{\mathrm{S}}^0\ddot{I}{€}^{+}\ddot{I}{€}^{-}\ddot{I}{€}^0$ amplitude obtained from CLEO-c data are used as input. This analysis is based on the full Belle data set of $772\times 10^6$ $B\overline{B}$ events collected at the $\hat{I}\yen (4S)$ resonance. We obtain $\ddot{I}{•}_3=(5.7{}_{-8.8}^{+10.2}\pm 3.5\pm 5.7{)}^{\circ }$ and the suppressed amplitude ratio $r_B=0.323\pm 0.147\pm 0.023\pm 0.051$. Here the first uncertainty is statistical, the second is the experimental systematic, and the third is due to the precision of the strong-phase parameters measured from CLEO-c data. The 95% confidence interval on $\ddot{I}{•}_3$ is $(-29.7,109.5{)}^{\circ }$, which is consistent with the current world average.