We report results from a study of $B^\pm \rightarrow DK^\pm$ decays followed by $D$ decaying to $CP$~eigenstates, where $D$ indicates a $D^0$ or $\bar{ D}^{ 0}$ meson. These decays are sensitive to the Cabibbo-Kobayashi-Maskawa unitarity-triangle angle $\phi_{ 3}$. The results are based on a combined analysis of the final data set of $772 \times 10^6~B\bar{ B}$ pairs collected by the Belle experiment and a data set of $198 \times 10^6~B\bar{ B}$ pairs collected by the Belle~II experiment, both in electron-positron collisions at the $\Upsilon(4S)$ resonance. We measure the $CP$ asymmetries to be $\mathcal{ A}_{ CP +} =~(+12.5 \pm 5.8 \pm 1.4)\% $ and $\mathcal{ A}_{ CP -} =~(-16.7 \pm 5.7 \pm 0.6)\%$, and the ratios of branching fractions to be $\mathcal{ R}_{ CP+}=~1.164 \pm 0.081 \pm 0.036 $ and $\mathcal{ R}_{ CP-} =~1.151 \pm 0.074 \pm 0.019$. The first contribution to the uncertainties is statistical, and the second is systematic. The asymmetries $\mathcal{ A}_{ CP +}$ and $\mathcal{ A}_{ CP -}$ have similar magnitudes and opposite signs; their difference corresponds to 3.5~standard deviations. From these values we calculate 68.3\% confidence intervals of ($8.5^{ \circ}<\phi_{ 3}<16.5^{ \circ}$) or ($84.5^{ \circ}<\phi_{ 3}<95.5^{ \circ}$) or ($163.3^{ \circ}<\phi_{ 3}<171.5^{ \circ}$) and $0.321<r_{ B}<0.465$.
Categories: hep-ex