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Update of the LHCb combination of the CKM angle $\gamma$

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Abstract

A combination of tree-level measurements of the CKM angle $\gamma$ at LHCb is performed. The results are obtained from time-integrated measurements of $ B ^+ \rightarrow D K ^+ $ , $ B ^+ \rightarrow D^* K ^+ $ , $ B ^+ \rightarrow D K^{*+}$ , $ B ^0 \rightarrow D K^{*0}$ , $ B ^0 \rightarrow D K ^+ \pi ^- $ and $ B ^+ \rightarrow D K ^+ \pi ^+ \pi ^- $ decays and time-dependent analyses of $ B ^0_ s \rightarrow D_s^\mp K^\pm$ and $ B ^0 \rightarrow D ^\mp \pi ^\pm $ decays. This combination includes new and updated results compared to the previous LHCb combination, giving a best fit value of $\gamma = 74.0 ^\circ$ with confidence intervals, set using a frequentist procedure, of $\gamma \in [68.2,79.0] ^\circ$ at 68.3% confidence level (CL) and $\gamma \in [61.6,83.7] ^\circ$ at 95.5% CL, where all values are modulo $180^\circ$. This gives the result \begin{align*} \gamma = ( 74.0 ^{+5.0}_{-5.8} )^\circ , \end{align*} where the uncertainty includes both statistical and systematic contributions.

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$ 1-{\rm CL}$ curves for the combination, with central values (solid vertical lines) and $1\sigma$ uncertainties (dashed vertical lines) labelled. The $1\sigma$ and $2\sigma$ levels are indicated by the horizontal dotted lines.

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Profile likelihood contours from the combination. The contours show the two-dimensional $1\sigma$ and $2\sigma$ confidence regions, corresponding to 68.3% and 95.5% CL, respectively.

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Profile likelihood contours from the combination. The contours show the two-dimensional $1\sigma$ and $2\sigma$ confidence regions, corresponding to 68.3% and 95.5% CL, respectively.

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Profile likelihood contours of $\gamma$ vs $ r_B^{DK}$ (left) and $\gamma$ vs $\delta_B^{DK}$ (right) for the GLW/ADS analysis of $ B ^+ \rightarrow D K ^+ $ , $ D\rightarrow h^+h^-$ (blue) alongisde the Run 1 (brown) and Run 2 (pink) GGSZ analyses of $ B ^+ \rightarrow D K ^+ $ , $ D\rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} h^+h^-$ .

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$ 1-{\rm CL}$ plots, using the profile likelihood method, for combinations split by the initial $B$ meson flavour: (orange) $ B ^0_ s $ initial states, (yellow) $ B ^0$ initial states, (blue) $ B ^+$ initial states and (green) the full combination.

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Profile likelihood contours of $\gamma$ vs. $ r_B^{DK}$ (left) and $\gamma$ vs. $\delta_B^{DK}$ (right) for various sub-combinations: (blue) $ B ^+ \rightarrow D K ^+ $ , $D \rightarrow h3\pi$/$hh^{\prime}\pi ^0 $, (pink) $ B ^+ \rightarrow D K ^+ $ , $D\rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} hh$, (light brown) $ B ^+ \rightarrow D K ^+ $ , $D\rightarrow KK$/$K\pi$/$\pi\pi$, (orange) all $ B ^+$ modes and (green) the full combination. Dark and light confidence regions correspond to 68.3% and 95.5% CL respectively.

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Profile likelihood contours of $\gamma$ vs. $ r_B^{DK^{*0}}$ (left) and $\gamma$ vs. $\delta_B^{DK^{*0}}$ (right) for various sub-combinations: (brown) $ B ^0 \rightarrow D K^{*0}$ , $D\rightarrow KK$/$K\pi$/$\pi\pi$, (pink) $ B ^0 \rightarrow D K^{*0}$ , $D\rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi\pi$, (purple) all $ B ^0$ modes and (green) the full combination. Dark and light confidence regions correspond to 68.3% and 95.5% CL respectively.

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The central value and $1\sigma$ uncertainty of the LHCb $\gamma$ combination (black points with error bars) as a function of time. The blue line (band) shows the central value (68.3% CL) for the latest result (BEAUTY 2018).

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Animated gif made out of all figures.

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Tables and captions

List of the LHCb measurements used in the combination, where TD is time-dependent and the method acronyms refer to the authors of Refs. [10,11,12,13,4,5,6,7,8,9].

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List of the auxiliary inputs used in the combination.

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Confidence intervals and central values for the parameters of interest. Note that $ r_B^{D^\mp\pi^\pm}$ is constrained as described in Sec. 3.

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Measured coverage $\alpha$ of the confidence intervals for $\gamma$ , determined at the best fit points, for both the one-dimensional {\sc Plugin} and profile likelihood methods. The nominal coverage is denoted as $\eta$.

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Statistical correlation matrix for the $ B ^+ \rightarrow D K ^+ $ , $ D\rightarrow h^+h^-$ observables [14].

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Systematic correlation matrix for the $ B ^+ \rightarrow D K ^+ $ , $ D\rightarrow h^+h^-$ observables [14].

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Statistical correlation matrix for the $ B ^+ \rightarrow D^* K ^+ $ , $ D\rightarrow h^+h^-$ observables [14].

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Systematic correlation matrix for the $ B ^+ \rightarrow D^* K ^+ $ , $ D\rightarrow h^+h^-$ observables [14].

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Statistical correlation matrix for the $ B ^+ \rightarrow D K^{*+}$ , $ D\rightarrow h^+h^-$ and $ D\rightarrow h^+\pi^-\pi^+\pi^-$ observables [20].

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Systematic correlation matrix for the $ B ^+ \rightarrow D K^{*+}$ , $ D\rightarrow h^+h^-$ and $ D\rightarrow h^+\pi^-\pi^+\pi^-$ observables [20].

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Statistical correlation matrix for the $ B ^+ \rightarrow D K ^+ $ , $ D\rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} h^+h^-$ observables [18].

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Systematic correlation matrix for the $ B ^+ \rightarrow D K ^+ $ , $ D\rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} h^+h^-$ observables [18].

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Statistical correlation matrix for the $ B ^0_ s \rightarrow D_s^\mp K^\pm$ observables [25].

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Systematic correlation matrix for the $ B ^0_ s \rightarrow D_s^\mp K^\pm$ observables [25].

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Statistical correlation matrix for the $ B ^0 \rightarrow D ^\mp \pi ^\pm $ observables [26].

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Systematic correlation matrix for the $ B ^0 \rightarrow D ^\mp \pi ^\pm $ observables [26].

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Created on 05 November 2019.