A binned Dalitz plot analysis of $B^\pm \to D K^\pm$ decays, with $D\to K_\text{S}^0\pi^+\pi^$ and $D\to K_\text{S}^0K^+K^$, is used to perform a measurement of the CPviolating observables $x_{\pm}$ and $y_{\pm}$, which are sensitive to the CabibboKobayashiMaskawa angle $\gamma$. The analysis is performed without assuming any $D$ decay model, through the use of information on the strongphase variation over the Dalitz plot from the CLEO collaboration. Using a sample of protonproton collision data collected with the LHCb experiment in 2015 and 2016, and corresponding to an integrated luminosity of 2.0$ \text{fb}^{1}$, the values of the CP violation parameters are found to be $x_ = ( 9.0 \pm 1.7 \pm 0.7 \pm 0.4) \times 10^{2}$, $y_ = ( 2.1 \pm 2.2 \pm 0.5 \pm 1.1) \times 10^{2}$, $x_+ = ( 7.7 \pm 1.9 \pm 0.7 \pm 0.4) \times 10^{2}$, and $y_+ = ( 1.0 \pm 1.9 \pm 0.4 \pm 0.9) \times 10^{2}$. The first uncertainty is statistical, the second is systematic, and the third is due to the uncertainty on the strongphase measurements. These values are used to obtain $\gamma = \left(87 ^{+11}_{12}\right)^\circ$, $r_B = 0.086^{+ 0.013}_{0.014}$, and $\delta_B = (101 \pm 11)^\circ$, where $r_B$ is the ratio between the suppressed and favoured $B$decay amplitudes and $\delta_B$ is the corresponding stronginteraction phase difference. This measurement is combined with the result obtained using 2011 and 2012 data collected with the \lhcb experiment, to give $\gamma = \left(80 ^{+10}_{ 9}\right)^\circ$, $r_B = 0.080 \pm 0.011$, and $\delta_B = (110 \pm 10)^\circ$.
Binning schemes for (left) $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^ $ decays and (right) $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^ $ decays. The diagonal line separates the positive and negative bins, where the positive bins are in the region in which $m^2_>m^2_+$ is satisfied. 
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Dalitz plots of long and downstream (left) $ B ^+ \rightarrow D K ^+ $ and (right) $ B ^ \rightarrow D K ^ $ candidates for (top) $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^ $ and (bottom) $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^ $ decays in which the reconstructed invariant mass of the $ B ^\pm$ candidate is in a region of $\pm 25 {\mathrm{ Me V /}c^2} $ around the $ B ^\pm$ mass. The narrow region is chosen to obtain high purity, as no background subtraction has been made. The Dalitz coordinates are calculated using the results of a kinematic fit in which the $ D $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}}$ masses are constrained to their known values. The blue lines show the kinematic boundaries of the decays. 
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Invariantmass distributions of (left) $ B ^\pm \rightarrow D K ^\pm $ and (right) $ B ^\pm \rightarrow D \pi ^\pm $ candidates, with $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^ $ , shown separately for the (top) long and (bottom) downstream $ K ^0_{\mathrm{ \scriptscriptstyle S}}$ categories. Fit results, including the signal component and background components due to misidentified companions, partially reconstructed decays and combinatorial background, are also shown. 
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Invariantmass distributions of (left) $ B ^\pm \rightarrow D K ^\pm $ and (right) $ B ^\pm \rightarrow D \pi ^\pm $ candidates, with $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^ $ , shown separately for the (top) long and (bottom) downstream $ K ^0_{\mathrm{ \scriptscriptstyle S}}$ categories. Fit results, including the signal component and background components due to misidentified companions, partially reconstructed decays and combinatorial background, are also shown. 
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Result of the simultaneous fit to $\optbar{ B}{} \rightarrow D ^{*\pm} \mu^\mp\optbarlowcase{\nu}{} _{ \mu}$ , $ D ^* ^\pm\rightarrow \optbar{ D}{} ^0(\rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^ )\pi ^\pm $ decays with downstream $ K ^0_{\mathrm{ \scriptscriptstyle S}}$ candidates, in 2016 data. The projections of the fit result are shown for (left) $m( K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^ )$ and (right) $\Delta m$. The (blue) total fit PDF is the sum of components describing (solid red) signal, (dashed black) combinatorial background and (dotted green) random soft pion background. 
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Example efficiency profiles of (left) $ B ^\pm \rightarrow D \pi ^\pm $ and (right) $\optbar{ B}{} \rightarrow D ^{*\pm} \mu^\mp\optbarlowcase{\nu}{} _{ \mu} X$ decays in the simulation. The top (bottom) plots are for $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^ $ ( $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^ $ ) decays. These plots refer to downstream $ K ^0_{\mathrm{ \scriptscriptstyle S}}$ candidates under 2016 data taking conditions. The normalisation is chosen so that the average over the Dalitz plot is unity. 
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Confidence levels at 68.2%, 95.5% and 99.7% probability for $(x_+,y_+)$ and $(x_{}, y_{})$ as measured in $B^\pm \rightarrow D K^\pm$ decays (statistical uncertainties only). The parameters $(x_+,y_+)$ relate to $ B ^+ $ decays and $(x_{}, y_{})$ refer to $ B ^ $ decays. The black dots show the central values obtained in the fit. 
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(Left) Comparison of total signal yields from the direct fit (points) to those calculated from the central values of $x_\pm$ and $y_\pm$ (solid line). The yields are given for the effective bin: $+i$ for $ B ^+ $ and $i$ for $ B ^ $ , and summed over $ B $ charge and $ K ^0_{\mathrm{ \scriptscriptstyle S}} $ decay category. (Right) Comparison of the difference between the $ B ^+ $ and $ B ^ $ yield obtained in the direct fit for each effective bin (points), the prediction from the central values of $x_\pm$ and $y_\pm$ (solid line), and the prediction assuming no $ C P$ violation (dotted line). 
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Rectangular binning schemes for (left) $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^ $ decays and (right) $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^ $ decays. The diagonal line separates the positive and negative bins, where the positive bins are in the region in which $m^2_>m^2_+$ is satisfied. 
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Twodimensional 68.3 %, 95.5 % and 99.7 % confidence regions for $( x_{\pm} , y_{\pm} )$ obtained in this measurement, as well as for the LHCb combination in Ref. [54], taking statistical and systematic uncertainties, as well as their correlations, into account. 
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Twodimensional 68.3 % and 95.5 % confidence regions for $( \gamma, r_B, \delta_B)$ for the $ x_{\pm}$ , $ y_{\pm}$ parameters obtained in the fit to 2015 and 2016 data, the fit to Run 1 data, and their combinations. 
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Animated gif made out of all figures. 
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Statistical correlation matrix for the fit to data. 
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Fit results for the total $ B ^\pm \rightarrow D K ^\pm $ yields in the signal region, where the invariant mass of the $ B $ candidate is in the interval 52495319 $ {\mathrm{ Me V /}c^2}$ , integrated over the Dalitz plots. 
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Summary of uncertainties for the parameters $ x_{\pm}$ and $ y_{\pm}$ . The various sources of systematic uncertainties are described in the main text. All entries are given in multiples of $10^{2}$. 
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Correlation matrix of the experimental and strongphase related systematic uncertainties. 
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Correlation matrix between Run 1 results (I) and the results presented in this paper (II), when fitting data while varying the inputs from the CLEO collaboration in a correlated way. 
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Total correlation matrix for the systematic uncertainties of the Run 1 results (I) and the results presented in this paper (II), including experimental and strong phase related systematic uncertainties. 
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Supplementary material full pdf 
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This ZIP file contains supplemetary material for the publication LHCbPAPER2018017. The files are: supplementary.pdf : An overview of the extra figures *.pdf, *.png, *.eps, *.C : The figures in variuous formats 
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Created on 14 January 2021.