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. 
dkpp_b[..].pdf [211 KiB] HiDef png [937 KiB] Thumbnail [295 KiB] *.C file 

dkpp_k[..].pdf [198 KiB] HiDef png [644 KiB] Thumbnail [214 KiB] *.C file 

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. 
d2kspi[..].pdf [11 KiB] HiDef png [329 KiB] Thumbnail [264 KiB] *.C file 

d2kspi[..].pdf [9 KiB] HiDef png [250 KiB] Thumbnail [201 KiB] *.C file 

d2kspi[..].pdf [11 KiB] HiDef png [331 KiB] Thumbnail [264 KiB] *.C file 

d2kspi[..].pdf [9 KiB] HiDef png [267 KiB] Thumbnail [219 KiB] *.C file 

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. 
d2kskk[..].pdf [11 KiB] HiDef png [307 KiB] Thumbnail [246 KiB] *.C file 

d2kskk[..].pdf [9 KiB] HiDef png [250 KiB] Thumbnail [201 KiB] *.C file 

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. 
Dalitz[..].pdf [34 KiB] HiDef png [201 KiB] Thumbnail [179 KiB] *.C file 

Dalitz[..].pdf [34 KiB] HiDef png [212 KiB] Thumbnail [190 KiB] *.C file 

Dalitz[..].pdf [17 KiB] HiDef png [174 KiB] Thumbnail [143 KiB] *.C file 

Dalitz[..].pdf [17 KiB] HiDef png [175 KiB] Thumbnail [145 KiB] *.C file 

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. 
likescan.pdf [8 KiB] HiDef png [239 KiB] Thumbnail [175 KiB] *.C file 

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. 
BplusB[..].pdf [6 KiB] HiDef png [211 KiB] Thumbnail [216 KiB] *.C file 

BplusB[..].pdf [15 KiB] HiDef png [223 KiB] Thumbnail [231 KiB] *.C file 

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. 
fr_con[..].pdf [4 KiB] HiDef png [166 KiB] Thumbnail [159 KiB] *.C file 

fd_con[..].pdf [5 KiB] HiDef png [161 KiB] Thumbnail [163 KiB] *.C file 

Animated gif made out of all figures. 
PAPER2012027.gif Thumbnail 
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}$ . 
Table_1.pdf [39 KiB] HiDef png [56 KiB] Thumbnail [30 KiB] tex code 

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}$ . 
Table_2.pdf [39 KiB] HiDef png [57 KiB] Thumbnail [29 KiB] tex code 

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). 
Table_3.pdf [52 KiB] HiDef png [80 KiB] Thumbnail [46 KiB] tex code 

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 $10^{2}$. 
Table_4.pdf [50 KiB] HiDef png [123 KiB] Thumbnail [60 KiB] tex code 

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 $10^{2}$. 
Table_5.pdf [50 KiB] HiDef png [122 KiB] Thumbnail [60 KiB] tex code 
Created on 05 November 2019.