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.

Figures and captions

$ 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.	Fig1a.pdf [14 KiB] HiDef png [186 KiB] Thumbnail [147 KiB] *.C file
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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.	Fig2a.pdf [14 KiB] HiDef png [268 KiB] Thumbnail [163 KiB] *.C file
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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.	Fig3a.pdf [14 KiB] HiDef png [288 KiB] Thumbnail [174 KiB] *.C file
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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^-$ .	Fig4a.pdf [18 KiB] HiDef png [367 KiB] Thumbnail [213 KiB] *.C file
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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.	Fig5a.pdf [16 KiB] HiDef png [297 KiB] Thumbnail [174 KiB] *.C file
	Fig5b.pdf [12 KiB] HiDef png [107 KiB] Thumbnail [102 KiB] *.C file
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.	Fig6a.pdf [19 KiB] HiDef png [366 KiB] Thumbnail [197 KiB] *.C file
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	Fig6c.pdf [13 KiB] HiDef png [128 KiB] Thumbnail [118 KiB] *.C file
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.	Fig7a.pdf [19 KiB] HiDef png [424 KiB] Thumbnail [225 KiB] *.C file
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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).	Fig8.pdf [13 KiB] HiDef png [137 KiB] Thumbnail [107 KiB] *.C file
Animated gif made out of all figures.	CONF-2018-002.gif Thumbnail

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].	Table_1.pdf [113 KiB] HiDef png [151 KiB] Thumbnail [40 KiB] tex code
List of the auxiliary inputs used in the combination.	Table_2.pdf [97 KiB] HiDef png [124 KiB] Thumbnail [59 KiB] tex code
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.	Table_3.pdf [74 KiB] HiDef png [167 KiB] Thumbnail [83 KiB] tex code
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 $\eta$.	Table_4.pdf [51 KiB] HiDef png [44 KiB] Thumbnail [21 KiB] tex code
Statistical correlation matrix for the $ B ^+ \rightarrow D K ^+ $ , $ D\rightarrow h^+h^-$ observables [14].	Table_5.pdf [54 KiB] HiDef png [50 KiB] Thumbnail [12 KiB] tex code
Systematic correlation matrix for the $ B ^+ \rightarrow D K ^+ $ , $ D\rightarrow h^+h^-$ observables [14].	Table_6.pdf [55 KiB] HiDef png [65 KiB] Thumbnail [16 KiB] tex code
Statistical correlation matrix for the $ B ^+ \rightarrow D^* K ^+ $ , $ D\rightarrow h^+h^-$ observables [14].	Table_7.pdf [61 KiB] HiDef png [50 KiB] Thumbnail [24 KiB] tex code
Systematic correlation matrix for the $ B ^+ \rightarrow D^* K ^+ $ , $ D\rightarrow h^+h^-$ observables [14].	Table_8.pdf [62 KiB] HiDef png [58 KiB] Thumbnail [27 KiB] tex code
Statistical correlation matrix for the $ B ^+ \rightarrow D K^{*+}$ , $ D\rightarrow h^+h^-$ and $ D\rightarrow h^+\pi^-\pi^+\pi^-$ observables [20].	Table_9.pdf [76 KiB] HiDef png [38 KiB] Thumbnail [15 KiB] tex code
Systematic correlation matrix for the $ B ^+ \rightarrow D K^{*+}$ , $ D\rightarrow h^+h^-$ and $ D\rightarrow h^+\pi^-\pi^+\pi^-$ observables [20].	Table_10.pdf [76 KiB] HiDef png [49 KiB] Thumbnail [19 KiB] tex code
Statistical correlation matrix for the $ B ^+ \rightarrow D K ^+ $ , $ D\rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} h^+h^-$ observables [18].	Table_11.pdf [52 KiB] HiDef png [46 KiB] Thumbnail [22 KiB] tex code
Systematic correlation matrix for the $ B ^+ \rightarrow D K ^+ $ , $ D\rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} h^+h^-$ observables [18].	Table_12.pdf [52 KiB] HiDef png [47 KiB] Thumbnail [22 KiB] tex code
Statistical correlation matrix for the $ B ^0_ s \rightarrow D_s^\mp K^\pm$ observables [25].	Table_13.pdf [52 KiB] HiDef png [49 KiB] Thumbnail [25 KiB] tex code
Systematic correlation matrix for the $ B ^0_ s \rightarrow D_s^\mp K^\pm$ observables [25].	Table_14.pdf [52 KiB] HiDef png [50 KiB] Thumbnail [26 KiB] tex code
Statistical correlation matrix for the $ B ^0 \rightarrow D ^\mp \pi ^\pm $ observables [26].	Table_15.pdf [44 KiB] HiDef png [52 KiB] Thumbnail [19 KiB] tex code
Systematic correlation matrix for the $ B ^0 \rightarrow D ^\mp \pi ^\pm $ observables [26].	Table_16.pdf [51 KiB] HiDef png [38 KiB] Thumbnail [16 KiB] tex code