Binning schemes for (left) $ D \rightarrow K ^0_{\mathrm{S}} \pi ^+ \pi ^- $ decays and (right) $ D \rightarrow K ^0_{\mathrm{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 plot for $ D $ decays of (left) $ B ^+ \rightarrow D K ^+ $ and (right) $ B ^- \rightarrow D K ^- $ candidates in the signal region, in the (top) $ D \rightarrow K ^0_{\mathrm{S}} \pi ^+ \pi ^- $ and (bottom) $ D \rightarrow K ^0_{\mathrm{S}} K ^+ K ^- $ channels. The horizontal and vertical axes are interchanged between the $ B ^+ $ and $ B ^- $ decay plots to aid visualisation of the $ C P$ asymmetries between the two distributions.

Invariant mass distributions for the (left) $ B ^\pm \rightarrow D K ^\pm $ channel and (right) $ B ^\pm \rightarrow D \pi ^\pm $ channel with $ D \rightarrow K ^0_{\mathrm{S}} \pi ^+ \pi ^- $ . The top (bottom) plots show data where the $ K ^0_{\mathrm{S}}$ candidate is long (downstream). Square brackets in the legend denote a particle that has not been reconstructed.

Invariant mass distributions for the (left) $ B ^\pm \rightarrow D K ^\pm $ channel and (right) $ B ^\pm \rightarrow D \pi ^\pm $ channel with $ D \rightarrow K ^0_{\mathrm{S}} K ^+ K ^- $ . The top (bottom) plots show data where the $ K ^0_{\mathrm{S}}$ candidate is long (downstream). Square brackets in the legend denote a particle that has not been reconstructed.

Confidence levels at 68.2 % and 95.5 % probability for (left, blue) $( x_+ ^{ D K ^\pm }, y_+ ^{ D K ^\pm })$, (left, red) $( x_- ^{ D K ^\pm }, y_- ^{ D K ^\pm })$, and (right, green) $(x_\xi^{ D \pi ^\pm },y_\xi^{ D \pi ^\pm }) $ as measured in $ B ^\pm \rightarrow D K ^\pm $ and $ B ^\pm \rightarrow D \pi ^\pm $ decays with a profile likelihood scan. The black dots show the central values

The bin-by-bin asymmetries $(N^-_{-i}-N^+_{+i})/(N^-_{-i}+N^+_{+i})$ for each Dalitz-plot bin number for (left) $ B ^\pm \rightarrow D K ^\pm $ decays and (right) $ B ^\pm \rightarrow D \pi ^\pm $ decays. The prediction from the central values of the $ C P$ -violation observables is shown with a solid line and the asymmetries obtained in fits with independent bin yields are shown with the error bars. The predicted asymmetries in a fit that does not allow for $ C P$ violation are shown with a dotted line. The vertical dashed line separates the $ K ^0_{\mathrm{S}} \pi ^+ \pi ^- $ and $ K ^0_{\mathrm{S}} K ^+ K ^- $ bins on the horizontal axis.

Confidence limits for the CKM angle $\gamma$ obtained using the method described in Ref. [44].

The 68 % and 95 % confidence regions for combinations of the physics parameters $(\gamma , r_ B ^{ D K } , \delta_ B ^{ D K } , r_ B ^{ D \pi} , \delta_ B ^{ D \pi} )$ obtained using the methods described in Ref. [44].

Animated gif made out of all figures.

The signal and background yields in the region $m_B\in[5249, 5309]\text{ Me V /}c^2 $ as obtained in the fit. For the $ B ^\pm \rightarrow D K ^\pm $ candidates, the yield of the partially reconstructed background includes the contributions from $ B ^0_ s $ decays and misidentified partially reconstructed backgrounds.

Overview of all sources of uncertainty, $\sigma$, on $ x_\pm^{ D K }$ , $ y_\pm^{ D K }$ , $ x_\xi^{ D \pi}$ , and $ y_\xi^{ D \pi}$ . All uncertainties are quoted $\times 10^{-2}$.

Statistical uncertainties, $\sigma$, and correlation matrix for $ x_\pm^{ D K }$ , $ y_\pm^{ D K }$ , $ x_\xi^{ D \pi}$ , and $ y_\xi^{ D \pi}$ .

Total LHCb -related systematic uncertainties, $\sigma$, for $ x_\pm^{ D K }$ , $ y_\pm^{ D K }$ , $ x_\xi^{ D \pi}$ , and $ y_\xi^{ D \pi}$ , and the corresponding correlation matrix.

Systematic uncertainties, $\sigma$, for $ x_\pm^{ D K }$ , $ y_\pm^{ D K }$ , $ x_\xi^{ D \pi}$ , and $ y_\xi^{ D \pi}$ due to strong-phase inputs, the corresponding correlation matrix.

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