Measurement of the Cabibbo-Kobayashi-Maskawa Angle $\gamma$ in $B^{\mp }\to D^{(*)}{K}^{\mp }$ Decays with a Dalitz Analysis of $D\to K_S^0{\pi }^{-}{\pi }^{+}$

We report on a measurement of the Cabibbo-Kobayashi-Maskawa $CP$-violating phase $\gamma$ through a Dalitz analysis of neutral $D$ decays to $K_S^0{\pi }^{-}{\pi }^{+}$ in the processes $B^{\mp }\to D^{(*)}{K}^{\mp }$, $D^{*}\to D{\pi }^0$, $D\gamma$. Using a sample of $227\times {10}^6$ $B\overline{B}$ pairs collected by the BABAR detector, we measure the amplitude ratios $r_B=0.12\pm 0.08\pm 0.03\pm 0.04$ and $r_B^{*}=0.17\pm 0.10\pm 0.03\pm 0.03$, the relative strong phases ${\delta }_B=(104\pm {45}_{-21-24}^{+17+16})°$ and ${\delta }_B^{*}=(-64\pm {41}_{-12}^{+14}\pm 15)°$ between the amplitudes $\mathcal{A}(B^{-}\to {\overline{D}}^{(*)0}{K}^{-})$ and $\mathcal{A}(B^{-}\to D^{(*)0}{K}^{-})$, and $\gamma =(70\pm {31}_{-10-11}^{+12+14})°$. The first error is statistical, the second is the experimental systematic uncertainty, and the third reflects the Dalitz model uncertainty. The results for the strong and weak phases have a twofold ambiguity.

Figure 1

Distributions of $m_{\mathrm{ES}}$ for (a) $B^{-}\to {\tilde{D}}^0{K}^{-}$, (b) $B^{-}\to {\tilde{D}}^{*0}({\tilde{D}}^0{\pi }^0)K^{-}$, and (c) $B^{-}\to {\tilde{D}}^{*0}({\tilde{D}}^0\gamma )K^{-}$. The curves represent the fit projections for signal plus background (solid lines) and background (dotted lines). The peaking structure of the background is due to remaining $B^{-}\to {\tilde{D}}^{(*)0}{\pi }^{-}$ events.

Figure 2

(a) The $D^0\to K_S^0{\pi }^{-}{\pi }^{+}$ Dalitz distribution from $D^{*+}\to D^0{\pi }^{+}$ events, and projections on (b) $m_{-}^2$, (c) $m_{+}^2$, and (d) $m_{{\pi }^{+}{\pi }^{-}}^2$. The curves are the fit projections.

Figure 3

Contours at 39.3% (dark) and 86.5% (light) confidence level (statistical only) in the ${\mathsf{z}}^{(*)}$ planes for (a) ${\tilde{D}}^0{K}^{-}$ and (b) ${\tilde{D}}^{*0}{K}^{-}$, separately for $B^{-}$ (thick and solid lines) and $B^{+}$ (thin and dotted lines). Projections in the $r_B^{(*)}\mathrm{\text{−}}\gamma$ planes of the five-dimensional one-standard (dark) and two-standard (light) deviation regions, for (c) ${\tilde{D}}^0{K}^{-}$ and (d) ${\tilde{D}}^{*0}{K}^{-}$.