Measurements of $C P$ observables in $B^\pm \rightarrow D K^\pm$ and $B^\pm \rightarrow D \pi^\pm$ decays are presented where the $D$ meson is reconstructed in the final states $K^\pm\pi^\mp$, $\pi^\pm K^\mp$, $K^+K^$, $\pi^+\pi^$, $K^\pm\pi^\mp \pi^+ \pi^$, $\pi^\pm K^\mp \pi^+ \pi^$ and $\pi^+ \pi^ \pi^+ \pi^$. This analysis uses a sample of charged $B$ mesons from $pp$ collisions collected by the LHCb experiment in 2011 and 2012, corresponding to an integrated luminosity of 3.0 fb$^{1}$. Various \CPviolating effects are reported and together these measurements provide important input for the determination of the unitarity triangle angle $\gamma$. The analysis of the fourpion $D$ decay mode is the first of its kind.
Invariant mass distributions of selected $ B ^\pm \rightarrow [ K ^\pm \pi ^\mp ]_{D}h^{\pm}$ candidates, separated by charge, with $ B ^ {\rm(} B ^+ {\rm)}$ candidates on the left (right). The top plots contain the $ B ^\pm \rightarrow DK^{\pm}$ candidate sample, as defined by a PID requirement on the bachelor particle. The remaining candidates are placed in the bottom row, reconstructed with a pion hypothesis for the bachelor. The red (thick, open) and green (hatchedarea) curves represent the $ B ^\pm \rightarrow D K ^\pm $ and $ B ^\pm \rightarrow D \pi ^\pm $ signals. The shaded part indicates partially reconstructed decays, the dotted line, where visible, shows the combinatorial component, and the total PDF is drawn as a thin blue line. 
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Invariant mass distributions of selected $ B ^\pm \rightarrow [\pi ^\pm K ^\mp ]_{D}h^{\pm}$ decays, separated by charge. The dashed pink line left of the signal peak shows partially reconstructed $ B _{s}^{0} \rightarrow [ K ^+ \pi ^ ]_D K ^ \pi ^+ $ decays, where the bachelor pion is missed. The favoured mode crossfeed is also included in the fit, but is too small to be seen. See the caption of Fig. 1 for other definitions. 
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Invariant mass distributions of selected $ B ^\pm \rightarrow [\pi ^+ \pi ^ ]_{D}h^{\pm}$ candidates, separated by charge. The dashed black line represents the residual contribution from charmless decays. This component is present in the $D$ final states considered, but is most visible in this case. See the caption of Fig. 1 for other definitions. 
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Invariant mass distributions of selected $ B ^\pm \rightarrow [ K ^+ K ^ ]_{D}h^{\pm}$ candidates, separated by charge. The dashed cyan line represents partially reconstructed $\Lambda ^0_ b \rightarrow [p^+ K ^ \pi ^+ ]_\Lambda ^+_ c h^$ decays, where the pion is missed and the proton is misidentified as a kaon. See the caption of Fig. 1 for other definitions. 
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Invariant mass distributions of selected $ B ^\pm \rightarrow [ K ^\pm \pi ^\mp \pi ^+ \pi ^ ]_{D}h^{\pm}$ candidates, separated by charge. See the caption of Fig. 1 for the definitions. 
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Invariant mass distributions of selected $ B ^\pm \rightarrow [\pi ^\pm K ^\mp \pi ^+ \pi ^ ]_{D}h^{\pm}$ candidates, separated by charge. The dashed pink line left of the signal peak shows partially reconstructed $ B _{s}^{0} \rightarrow [ K ^+ \pi ^ \pi ^+ \pi ^ ]_D K ^ \pi ^+ $ decays, where the bachelor pion is missed. See the caption of Fig. 1 for other definitions. 
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Invariant mass distributions of selected $ B ^\pm \rightarrow [\pi ^+ \pi ^ \pi ^+ \pi ^ ]_{D}h^{\pm}$ candidates, separated by charge. The dashed black line represents the residual contribution from charmless decays. See the caption of Fig. 1 for other definitions. 
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Comparison of selected results with the SM expectation (shaded) based on existing knowledge of the underlying parameters and $ D $ decay measurements, as described in the text. The first five rows show $ B ^ \rightarrow D K ^ $ observables, twobody $D$ decay on the left and fourbody on the right. The last three rows show $ B ^ \rightarrow D\pi ^ $ observables. 
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Animated gif made out of all figures. 
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Signal yields as measured in the $ B ^ \rightarrow [h^+h^]_{D}h^$ and $ B ^ \rightarrow [h^+h^\pi ^+ \pi ^ ]_{D}h^$ invariant mass fits, together with their statistical uncertainties. 
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Systematic uncertainties for the $ B ^ \rightarrow [h^+h^]_{D}h^$ $ C P$ observables quoted as a percentage of the statistical uncertainty on the observable. PID refers to the PID calibration procedure. Bkg refers to the choice of background shapes and yields in the fit. Sim refers to the use of finite samples of simulated events to determine efficiency ratios. Asym refers to the fixed pion and kaon detection asymmetries, and the assumption of no $ C P$ violation in $ B ^ \rightarrow D ^0 \pi ^ $ decays. 
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Statistical correlation matrix from for the 2body fit. 
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Statistical correlation matrix from for the 4body fit. 
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Correlation matrix for the systematic uncertainties in the 2body analysis. 
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Correlation matrix for the systematic uncertainties in the 4body analysis. 
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Created on 10 August 2020.