A binned Dalitz plot analysis of $B^\pm \to D K^\pm$ decays, with $D \to K^0_{\rm S} \pi^+\pi^$ and $D \to K^0_{\rm S} K^+ K^$, is performed to measure the CPviolating observables $x_{\pm}$ and $y_{\pm}$ which are sensitive to the CKM angle $\gamma$. The analysis exploits 1.0 $\rm fb^{1}$ of data collected by the LHCb experiment. The study makes no modelbased assumption on the variation of the strong phase of the D decay amplitude over the Dalitz plot, but uses measurements of this quantity from CLEOc as input. The values of the parameters are found to be $x_ = (0.0 \pm 4.3 \pm 1.5 \pm 0.6) \times 10^{2}$, $y_ = (2.7 \pm 5.2 \pm 0.8 \pm 2.3) \times 10^{2}$, $x_+ = (10.3 \pm 4.5 \pm 1.8 \pm 1.4)\times 10^{2}$ and $y_+ = (0.9 \pm 3.7 \pm 0.8 \pm 3.0)\times 10^{2}$. The first, second, and third uncertainties are the statistical, the experimental systematic, and the error associated with the precision of the strongphase parameters measured at CLEOc, respectively. These results correspond to $\gamma = (44^{ +43}_{ 38})^\circ$, with a second solution at $\gamma \to \gamma + 180^\circ$, and $r_B = 0.07 \pm 0.04$, where $r_B$ is the ratio between the suppressed and favoured B decay amplitudes.
Binning choices for (a) $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^$ and (b) $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^$. The diagonal line separates the positive and negative bins. 
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Invariant mass distributions of (a,c) $B^\pm \rightarrow D K^\pm$ and (b,d) $B^\pm \rightarrow D \pi^\pm$ candidates, with $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^$, divided between the (a,b) long and (c,d) downstream $ K ^0_{\rm\scriptscriptstyle S}$ categories. Fit results, including the signal and background components, are superimposed. 
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Invariant mass distributions of (a) $B^\pm \rightarrow D K^\pm$ and (b) $B^\pm \rightarrow D \pi^\pm$ candidates, with $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^$, shown with both $ K ^0_{\rm\scriptscriptstyle S}$ categories combined. Fit results, including the signal and background components, are superimposed. 
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Dalitz plots of $B^\pm \rightarrow D K^\pm$ candidates in the signal region for (a,b) $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^$ and (c,d) $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+ K^$ decays, divided between (a,c) $B^+$ and (b,d) $B^$. The boundaries of the kinematicallyallowed regions are also shown. 
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One (solid), two (dashed) and three (dotted) standard deviation confidence levels for $(x_+,y_+)$ (blue) and $(x_{}, y_{})$ (red) as measured in $B^\pm \rightarrow D K^\pm$ decays (statistical only). The points represent the best fit central values. 
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Signal yield in effective bins compared with prediction of $(x_\pm, y_\pm)$ fit (black histogram) for $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^$ and $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^$. Figure (a) shows the sum of $B^+$ and $B^$ yields. Figure (b) shows the difference of $B^+$ and $B^$ yields. Also shown (dashed line and grey shading) is the expectation and uncertainty for the zero $ C P$ violation hypothesis. 
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Twodimensional projections of confidence regions onto the $(\gamma, r_B)$ and $(\gamma, \delta_B)$ planes showing the one (solid) and two (dashed) standard deviations with all uncertainties included. For the ($\gamma,r_B$) projection the three (dotted) standard deviation contour is also shown. The points mark the central values. 
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Animated gif made out of all figures. 
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Yields and statistical uncertainties in the signal region from the invariant mass fit, scaled from the full fit mass range, for candidates passing the $B^\pm \rightarrow D h^\pm$, $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^$ selection. Values are shown separately for candidates containing long and downstream $ K ^0_{\rm\scriptscriptstyle S}$ decays. The signal region is between 5247 $ {\mathrm{ Me V /}c^2}$ and 5317 $ {\mathrm{ Me V /}c^2}$ and the full fit range is between 5110 $ {\mathrm{ Me V /}c^2}$ and 5800 $ {\mathrm{ Me V /}c^2}$ . 
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Yields and statistical uncertainties in the signal region from the invariant mass fit, scaled from the full fit mass range, for candidates passing the $B^\pm \rightarrow D h^\pm$, $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^$ selection. Values are shown separately for candidates containing long and downstream $ K ^0_{\rm\scriptscriptstyle S}$ decays. The signal region is between 5247 $ {\mathrm{ Me V /}c^2}$ and 5317 $ {\mathrm{ Me V /}c^2}$ and the full fit range is between 5110 $ {\mathrm{ Me V /}c^2}$ and 5800 $ {\mathrm{ Me V /}c^2}$ . 
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Results for $x_{\pm}$ and $y_{\pm}$ from the fits to the data in the case when both $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^$ and $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^$ are considered and when only the $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^$ final state is included. The first, second, and third uncertainties are the statistical, the experimental systematic, and the error associated with the precision of the strongphase parameters, respectively. The correlation coefficients are calculated including all sources of uncertainty (the values in parentheses correspond to the case where only the statistical uncertainties are considered). 
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Summary of statistical, experimental and strongphase uncertainties on $x_\pm$ and $y_\pm$ in the case where both $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^$ and $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^$ decays are included in the fit. All entries are given in multiples of $10^{2}$. 
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Summary of statistical, experimental and strongphase uncertainties on $x_\pm$ and $y_\pm$ in the case where only $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^$ decays are included in the fit. All entries are given in multiples of $10^{2}$. 
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Created on 14 January 2021.