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. 
Fig1left.pdf [684 KiB] HiDef png [1 MiB] Thumbnail [135 KiB] *.C file 

Fig1right.pdf [757 KiB] HiDef png [2 MiB] Thumbnail [138 KiB] *.C file 

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. 
Fig2to[..].pdf [364 KiB] HiDef png [288 KiB] Thumbnail [102 KiB] *.C file 

Fig2to[..].pdf [364 KiB] HiDef png [295 KiB] Thumbnail [103 KiB] *.C file 

Fig2bo[..].pdf [178 KiB] HiDef png [180 KiB] Thumbnail [62 KiB] *.C file 

Fig2bo[..].pdf [179 KiB] HiDef png [182 KiB] Thumbnail [64 KiB] *.C file 

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. 
Fig3top.pdf [37 KiB] HiDef png [242 KiB] Thumbnail [64 KiB] *.C file 

Fig3bottom.pdf [35 KiB] HiDef png [244 KiB] Thumbnail [65 KiB] *.C file 

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. 
Fig4top.pdf [30 KiB] HiDef png [244 KiB] Thumbnail [64 KiB] *.C file 

Fig4bottom.pdf [35 KiB] HiDef png [240 KiB] Thumbnail [62 KiB] *.C file 

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. 
Fig5a.pdf [24 KiB] HiDef png [284 KiB] Thumbnail [86 KiB] *.C file 

Fig5b.pdf [30 KiB] HiDef png [226 KiB] Thumbnail [69 KiB] *.C file 

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. 
Fig6a.pdf [480 KiB] HiDef png [1 MiB] Thumbnail [159 KiB] *.C file 

Fig6b.pdf [475 KiB] HiDef png [1 MiB] Thumbnail [157 KiB] *.C file 

Fig6c.pdf [674 KiB] HiDef png [1 MiB] Thumbnail [190 KiB] *.C file 

Fig6d.pdf [670 KiB] HiDef png [1 MiB] Thumbnail [191 KiB] *.C file 

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. 
Fig7.pdf [30 KiB] HiDef png [204 KiB] Thumbnail [55 KiB] *.C file 

(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). 
Fig8left.pdf [15 KiB] HiDef png [158 KiB] Thumbnail [68 KiB] *.C file 

Fig8right.pdf [15 KiB] HiDef png [169 KiB] Thumbnail [73 KiB] *.C file 

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. 
Fig9left.pdf [679 KiB] HiDef png [1 MiB] Thumbnail [121 KiB] *.C file 

Fig9right.pdf [690 KiB] HiDef png [2 MiB] Thumbnail [149 KiB] *.C file 

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. 
Fig10.pdf [20 KiB] HiDef png [359 KiB] Thumbnail [68 KiB] *.C file 

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. 
Fig11left.pdf [15 KiB] HiDef png [377 KiB] Thumbnail [67 KiB] *.C file 

Fig11right.pdf [15 KiB] HiDef png [354 KiB] Thumbnail [66 KiB] *.C file 

Animated gif made out of all figures. 
PAPER2018017.gif Thumbnail 
Statistical correlation matrix for the fit to data. 
Table_1.pdf [37 KiB] HiDef png [45 KiB] Thumbnail [7 KiB] tex code 

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. 
Table_2.pdf [50 KiB] HiDef png [39 KiB] Thumbnail [7 KiB] tex code 

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 $10^{2}$. 
Table_3.pdf [56 KiB] HiDef png [99 KiB] Thumbnail [15 KiB] tex code 

Correlation matrix of the experimental and strongphase related systematic uncertainties. 
Table_4.pdf [38 KiB] HiDef png [33 KiB] Thumbnail [6 KiB] tex code 

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. 
Table_5.pdf [43 KiB] HiDef png [78 KiB] Thumbnail [13 KiB] tex code 

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. 
Table_6.pdf [44 KiB] HiDef png [81 KiB] Thumbnail [14 KiB] tex code 
Supplementary material full pdf 
supple[..].pdf [188 KiB] 

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 
SupMat[..].pdf [15 KiB] HiDef png [276 KiB] Thumbnail [56 KiB] *C file 

SupMat[..].pdf [15 KiB] HiDef png [335 KiB] Thumbnail [64 KiB] *C file 

SupMatFig2.pdf [39 KiB] HiDef png [379 KiB] Thumbnail [84 KiB] *C file 
Created on 15 April 2019.Citation count from INSPIRE on 24 April 2019.