Improved measurement of the CKM angle $\gamma$ in $B^{\mp }\to D^{(*)}{K}^{(*)\mp }$ decays with a Dalitz plot analysis of $D$ decays to $K_S^0{\pi }^{+}{\pi }^{-}$ and $K_S^0{K}^{+}{K}^{-}$

We report on an improved measurement of the Cabibbo-Kobayashi-Maskawa $CP$-violating phase $\gamma$ through a Dalitz plot analysis of neutral $D$ meson decays to $K_S^0{\pi }^{+}{\pi }^{-}$ and $K_S^0{K}^{+}{K}^{-}$ produced in the processes $B^{\mp }\to D{K}^{\mp }$, $B^{\mp }\to D^{*}{K}^{\mp }$ with $D^{*}\to D{\pi }^0$, $D\gamma$, and $B^{\mp }\to D{K}^{*\mp }$ with $K^{*\mp }\to K_S^0{\pi }^{\mp }$. Using a sample of $383\times {10}^6$ $B\overline{B}$ pairs collected by the BABAR detector, we measure $\gamma =(76\pm 22\pm 5\pm 5)°$ (mod 180°), where the first error is statistical, the second is the experimental systematic uncertainty, and the third reflects the uncertainty on the description of the Dalitz plot distributions. The corresponding 2-standard-deviation region is $29°<\gamma <122°$. This result has a significance of direct $CP$ violation ($\gamma \ne 0$) of 3.0 standard deviations.

Figure 1

Main Feynman diagrams contributing to the $B^{-}\to {\tilde{D}}^0{K}^{-}$ decay. 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 color suppressed.

Figure 2

Distributions of $m_{\mathrm{ES}}$ in $\Delta E$ signal region for (a), (e) $B^{-}\to {\tilde{D}}^0{K}^{-}$, (b), (f) $B^{-}\to {\tilde{D}}^{*0}[{\tilde{D}}^0{\pi }^0]K^{-}$, (c), (g) $B^{-}\to {\tilde{D}}^{*0}[{\tilde{D}}^0\gamma ]K^{-}$, and (d) $B^{-}\to {\tilde{D}}^0{K}^{*-}$, with (a)–(d) ${\tilde{D}}^0\to K_S^0{\pi }^{+}{\pi }^{-}$ and (e)–(g) ${\tilde{D}}^0\to K_S^0{K}^{+}{K}^{-}$. The curves superimposed represent the projections of the fit described in Sec. 2c: signal plus background (solid black lines), the continuum plus $B\overline{B}$ background contributions (dotted red lines), and the sum of the continuum, $B\overline{B}$, and $K/\pi$ misidentification background components (dashed blue lines).

Figure 3

Distributions of $m_{\mathrm{ES}}$ in $\Delta E$ signal region for (a), (e) $B^{-}\to {\tilde{D}}^0{\pi }^{-}$, (b), (f) $B^{-}\to {\tilde{D}}^{*0}[{\tilde{D}}^0{\pi }^0]{\pi }^{-}$, (c), (g) $B^{-}\to {\tilde{D}}^{*0}[{\tilde{D}}^0\gamma ]{\pi }^{-}$, and (d) $B^{-}\to {\tilde{D}}^0{a}_1^{-}$, with (a)–(d) ${\tilde{D}}^0\to K_S^0{\pi }^{+}{\pi }^{-}$ and (e)–(g) ${\tilde{D}}^0\to K_S^0{K}^{+}{K}^{-}$. The curves superimposed represent the projections of the fit described in Sec. 2c: signal plus background (solid black lines), the continuum plus $B\overline{B}$ background contributions (dotted red lines), and the sum of the continuum, $B\overline{B}$, and $K/\pi$ misidentification background components (dashed blue lines).

Figure 4

$D^0$ mass distributions after all selection criteria, for (a) $D^{*+}\to D^0{\pi }^{+}$, $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$ and (b) $D^{*+}\to D^0{\pi }^{+}$, $D^0\to K_S^0{K}^{+}{K}^{-}$. The curves superimposed represent the result from the $m_D$ fit (solid blue lines) and the linear background contribution (dotted red lines).

Figure 5

Dalitz plot distributions for (a) $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$ and (b) $D^0\to K_S^0{K}^{+}{K}^{-}$ from $D^{*+}\to D^0{\pi }^{+}$ events after all selection criteria, in the $D^0$ mass signal region. The contours (solid red lines) represent the kinematical limits of the $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$ and $D^0\to K_S^0{K}^{+}{K}^{-}$ decays.

Figure 6

$D^0\to K_S^0{h}^{+}{h}^{-}$ Dalitz plot projections from $D^{*+}\to D^0{\pi }^{+}$ events on (a), (d) $m_{-}^2$, (b), (e) $m_{+}^2$, and (c), (f) $m_0^2$, for (a)–(c) $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$ and (d)–(f) $D^0\to K_S^0{K}^{+}{K}^{-}$. The curves are the reference model fit projections.

Figure 7

${\tilde{D}}^0\to K_S^0{\pi }^{+}{\pi }^{-}$ Dalitz plot distributions for (a) $B^{-}\to {\tilde{D}}^0{K}^{-}$, (b) $B^{+}\to {\tilde{D}}^0{K}^{+}$, (c) $B^{-}\to {\tilde{D}}^{*0}[{\tilde{D}}^0{\pi }^0]K^{-}$, (d) $B^{+}\to {\tilde{D}}^{*0}[{\tilde{D}}^0{\pi }^0]K^{+}$, (e) $B^{-}\to {\tilde{D}}^{*0}[{\tilde{D}}^0\gamma ]K^{-}$, (f) $B^{+}\to {\tilde{D}}^{*0}[{\tilde{D}}^0\gamma ]K^{+}$, (g) $B^{-}\to {\tilde{D}}^0{K}^{*-}$, and (h) $B^{+}\to {\tilde{D}}^0{K}^{*+}$, for the $\Delta E$ signal region. The requirements $m_{\mathrm{ES}}>5.272\mathrm{GeV}/c^2$ and $\mathcal{F}>-0.1$ have been applied to reduce the background contamination, mainly from continuum events. The contours (solid red lines) represent the kinematical limits of the ${\tilde{D}}^0\to K_S^0{\pi }^{+}{\pi }^{-}$ decay.

Figure 8

${\tilde{D}}^0\to K_S^0{K}^{+}{K}^{-}$ Dalitz plot distributions for (a) $B^{-}\to {\tilde{D}}^0{K}^{-}$, (b) $B^{+}\to {\tilde{D}}^0{K}^{+}$, (c) $B^{-}\to {\tilde{D}}^{*0}[{\tilde{D}}^0{\pi }^0]K^{-}$, (d) $B^{+}\to {\tilde{D}}^{*0}[{\tilde{D}}^0{\pi }^0]K^{+}$, (e) $B^{-}\to {\tilde{D}}^{*0}[{\tilde{D}}^0\gamma ]K^{-}$, and (f) $B^{+}\to {\tilde{D}}^{*0}[{\tilde{D}}^0\gamma ]K^{+}$, for the $\Delta E$ signal region. The requirements $m_{\mathrm{ES}}\mathrm{GeV}/c^2$ and $\mathcal{F}>-0.1$ have been applied to reduce the background contamination, mainly from continuum events. The contours (solid red lines) represent the kinematical limits of the ${\tilde{D}}^0\to K_S^0{K}^{+}{K}^{-}$ decay.

Figure 9

Contours at 39.3% (dark) and 86.5% (light) 2-dimensional confidence level (CL) in the (a) $(x_{\mp },y_{\mp })$, (b) $(x_{\mp }^{*},y_{\mp }^{*})$, and (c) $(x_{s\mp },y_{s\mp })$ planes, corresponding to 1- and 2-standard-deviation regions (statistical only), for $B^{-}$ (thick and solid lines) and $B^{+}$ (thin and dotted lines) decays.

Figure 10

Contours at 39.3% (dark) and 86.5% (light) 2-dimensional CL in the (a), (c) $(x_{\mp },y_{\mp })$ and (b), (d) $(x_{\mp }^{*},y_{\mp }^{*})$ planes, corresponding to 1- and 2-standard-deviation regions (statistical only), for $B^{-}\to {\tilde{D}}^{(*)0}{K}^{-}$ (thick and solid lines) and $B^{+}\to {\tilde{D}}^{(*)0}{K}^{+}$ (thin and dotted lines) decays, for (a), (b) ${\tilde{D}}^0\to K_S^0{\pi }^{+}{\pi }^{-}$ and (c), (d) ${\tilde{D}}^0\to K_S^0{K}^{+}{K}^{-}$ only decay modes.

Figure 11

Contours at 39.3% (dark) and 86.5% (light) 2-dimensional CL in the (a) $(x_{\mp ,\pi },y_{\mp ,\pi })$ and (b) $(x_{\mp ,\pi }^{*},y_{\mp ,\pi }^{*})$ planes, corresponding to 1- and 2-standard-deviation regions (statistical only), for $B^{-}\to D^{(*)0}{\pi }^{-}$ (thick and solid lines) and $B^{+}\to {\overline{D}}^{(*)0}{\pi }^{+}$ (thin and dotted lines) control sample decays. Note the differences in scale when comparing to Figs. 9 and 10.

Figure 12

$\alpha =1-\mathrm{CL}$ as a function of $\gamma$ for $B^{-}\to {\tilde{D}}^0{K}^{-}$, $B^{-}\to {\tilde{D}}^{*0}{K}^{-}$, and $B^{-}\to {\tilde{D}}^0{K}^{*-}$ decays separately, and their combination, including statistical and systematic uncertainties and their correlations. The dashed (upper) and dotted (lower) horizontal lines correspond to the 1- and 2-standard-deviation intervals, respectively.

Figure 13

$\alpha =1-\mathrm{CL}$ as a function of (upper panel) $r_B$, $r_B^{*}$, and $\kappa r_s$, and (lower panel) ${\delta }_B$, ${\delta }_B^{*}$, and ${\delta }_s$, for $B^{-}\to {\tilde{D}}^0{K}^{-}$, $B^{-}\to {\tilde{D}}^{*0}{K}^{-}$, and $B^{-}\to {\tilde{D}}^0{K}^{*-}$ decays, including statistical and systematic uncertainties and their correlations. The dashed (upper) and dotted (lower) horizontal lines correspond to the 1- and 2-standard-deviation intervals, respectively.