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.

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.

Invariant-mass 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.

Invariant-mass 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.

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.

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.

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.

(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).

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.

Two-dimensional 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.

Two-dimensional 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.

Animated gif made out of all figures.

Statistical correlation matrix for the fit to data.

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 5249--5319 $ {\mathrm{ Me V /}c^2}$ , integrated over the Dalitz plots.

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 $10^{-2}$.

Correlation matrix of the experimental and strong-phase related systematic uncertainties.

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.

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.

Supplementary material full pdf

This ZIP file contains supplemetary material for the publication LHCb-PAPER-2018-017. The files are: supplementary.pdf : An overview of the extra figures *.pdf, *.png, *.eps, *.C : The figures in variuous formats