Observation of direct $CP$ violation in the measurement of the Cabibbo-Kobayashi-Maskawa angle $\gamma$ with $B^{\mathbf{\pm }}\to D^{\mathbf{(}*\mathbf{)}}{K}^{\mathbf{(}*\mathbf{)}\mathbf{\pm }}$ decays

We report the determination of the Cabibbo-Kobayashi-Maskawa $CP$-violating angle $\gamma$ through the combination of various measurements involving $B^{\pm }\to D{K}^{\pm }$, $B^{\pm }\to D^{*}{K}^{\pm }$, and $B^{\pm }\to D{K}^{*\pm }$ decays performed by the BABAR experiment at the PEP-II $e^{+}{e}^{-}$ collider at SLAC National Accelerator Laboratory. Using up to 474 million $B\overline{B}$ pairs, we obtain $\gamma =({69}_{-16}^{+17})°$ modulo 180°. The total uncertainty is dominated by the statistical component, with the experimental and amplitude-model systematic uncertainties amounting to $\pm 4°$. The corresponding two-standard-deviation region is $41°<\gamma <102°$. This result is inconsistent with $\gamma =0$ with a significance of 5.9 standard deviations.

Figure 1

Dominant Feynman diagrams for the decays $B^{-}\to D^0{K}^{-}$ (left) and $B^{-}\to {\overline{D}}^0{K}^{-}$ (right). The left diagram proceeds via $b\to c\overline{u}s$ transition, while the right diagram proceeds via $b\to u\overline{c}s$ transition and is both CKM- and color-suppressed.

Figure 2

Experimental likelihoods as functions of the ADS charge-specific ratios $R_{\pm }$ (a,b), $R_{\pm }^{*}$ [$D{\pi }^0$] (c,d), $R_{\pm }^{*}$ [$D\gamma$] (e,f), and $R_{\pm }^{K\pi {\pi }^0}$ (g,h), from Refs. [23, 24], including systematic uncertainties. The P.D.F.s are normalized so that their maximum values are equal to 1. These distributions are well parametrized by sums of two asymmetric Gaussian functions with mean values as given in Table 3.

Figure 3

Two-dimensional $-2\Delta \ln \mathcal{L}=s^2$ contours (up to three standard deviations, i.e., $s=1$, 2, 3) in the ${\overline{\mathsf{z}}}_{\pm }$ (left column), ${\overline{\mathsf{z}}}_{\pm }^{*}$ (center column), and ${\overline{\mathsf{z}}}_{s\pm }$ (right column) planes, for the GGSZ measurement only (top row), the GGSZ and GLW combination (middle row), and the GGSZ, GLW, and ADS combination (bottom row). The solid (blue) and dashed (red) lines correspond to $B^{-}$ and $B^{+}$ decays.

Figure 4

$1-\mathrm{C}.\mathrm{L}.$ distributions for the combination of the GGSZ, GLW, and ADS methods as a function of $\gamma$ (top), $r_B^{(*)}$, and $\kappa r_s$ (middle), and ${\delta }_B^{(*)}$, ${\delta }_s$ (bottom), including statistical and systematic uncertainties, for $B^{\pm }\to D{K}^{\pm }$, $B^{\pm }\to D^{*}{K}^{\pm }$, and $B^{\pm }\to D{K}^{*\pm }$ decays. The combination of all the $B$-decay channels is also shown for $\gamma$. The dashed (dotted) horizontal line corresponds to the one- (two-) standard-deviation C.L.

Figure 5

Comparison of $1-\mathrm{C}.\mathrm{L}.$ as a function of $r_B$ (top), $r_B^{*}$ (middle), and $\gamma$ (bottom) for all $B$-decay channels combined with the GGSZ method only, the combination with the GLW measurements, and the global combination, including statistical and systematic uncertainties. The horizontal lines represent the one-, two-, three-, and four-standard-deviation C.L.