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Measurement of the CKM angle $\gamma$ from a combination of LHCb results

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Abstract

A combination of measurements sensitive to the CKM angle $\gamma$ from LHCb is performed. The inputs are from analyses of time-integrated $B^{+}\rightarrow DK^+$, $B^{0} \rightarrow D K^{*0}$, $B^{0} \rightarrow D K^+ \pi^-$ and $B^{+} \rightarrow D K^+\pi^+\pi^-$ tree-level decays. In addition, results from a time-dependent analysis of $B_{s}^{0} \rightarrow D_{s}^{\mp}K^{\pm}$ decays are included. The combination yields $\gamma = (72.2^{+6.8}_{-7.3})^\circ$, where the uncertainty includes systematic effects. The 95.5 confidence level interval is determined to be $\gamma \in [55.9,85.2]^\circ$. A second combination is investigated, also including measurements from $B^{+} \rightarrow D \pi^+$ and $B^{+} \rightarrow D \pi^+\pi^-\pi^+$ decays, which yields compatible results.

Figures and captions

$ 1-{\rm CL}$ curves for the $ DK$ combination obtained with the {\sc Plugin} method. The $1\sigma$ and $2\sigma$ levels are indicated by the horizontal dotted lines.

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

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$ 1-{\rm CL}$ curves for the $ Dh$ combination obtained with the {\sc Plugin} method. The $1\sigma$ and $2\sigma$ levels are indicated by the horizontal dotted lines. \vspace{2cm}

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

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Dependence of the coverage for the one-dimensional {\sc Plugin} method (blue circles) and the profile likelihood method (red squares) on $ r_B^{DK}$ for the $ DK$ combination (left) and on $ r_B^{D\pi}$ for the $ Dh$ combination (right). The solid horizontal line shows the nominal coverage at $1\sigma$ of 68.3%.

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$ 1-{\rm CL}$ plots, using the profile likelihood method, for $ DK$ combinations split by the initial $B$ meson flavour (left) and split by analysis method (right). Left: (orange) $ B ^0_ s $ initial state, (yellow) $ B ^0$ initial states, (blue) $ B ^+$ initial states and (green) the full combination. Right: (yellow) GGSZ methods, (orange) GLW/ADS methods, (blue) other methods and (green) the full combination.

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

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

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Posterior probability density from the Bayesian interpretation for the $ DK$ combination.

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Two-dimensional posterior probability regions from the Bayesian interpretation for the $ DK$ combination. Light and dark regions show the 68.3% and 95.5% credible intervals respectively.

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Posterior probability density from the Bayesian interpretation for the $ Dh$ combination. The inset for $ r_B^{D\pi}$ shows the same distribution on a logarithmic scale.

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Two-dimensional posterior probability regions from the Bayesian interpretation for the $ Dh$ combination. Light and dark regions show the 68.3% and 95.5% credible intervals respectively.

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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 combinations.

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List of the auxiliary inputs used in the combinations.

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Confidence intervals and central values for the parameters of interest in the frequentist $ DK$ combination.

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Confidence intervals and central values for the parameters of interest in the frequentist $ Dh$ combination.

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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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Credible intervals and most probable values for the hadronic parameters determined from the $ DK$ Bayesian combination.

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Credible intervals and most probable values for the hadronic parameters determined from the $ Dh$ Bayesian combination.

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$ D ^0$ decay time acceptance coefficients (see Ref. [37]) for each analysis.

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Correlation matrix of the statistical uncertainties for the $ B ^+ \rightarrow D h^+$ , $ D ^0 \rightarrow h^+h^-$ observables [44].

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Correlation matrix of the systematic uncertainties for the $ B ^+ \rightarrow D h^+$ , $ D ^0 \rightarrow h^+h^-$ observables [44].

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Correlation matrix of the statistical uncertainties for the $ B ^+ \rightarrow D h^+$ , $ D ^0 \rightarrow h^\pm\pi^\mp\pi^+\pi^-$ observables [44].

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Correlation matrix of the systematic uncertainties for the $ B ^+ \rightarrow D h^+$ , $ D ^0 \rightarrow h^\pm\pi^\mp\pi^+\pi^-$ observables [44].

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Correlation matrix of the statistical uncertainties for the $ B ^+ \rightarrow D h^+$ , $ D ^0 \rightarrow h^\pm h^\mp \pi ^0 $ observables [52].

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Correlation matrix of the systematic uncertainties for the $ B ^+ \rightarrow D h^+$ , $ D ^0 \rightarrow h^\pm h^\mp \pi ^0 $ observables [52].

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Correlation matrix of the statistical uncertainties for the $ B ^+ \rightarrow D K ^+ $ , $ D ^0 \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} h^+h^-$ observables [46].

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Correlation matrix of the systematic uncertainties for the $ B ^+ \rightarrow D K ^+ $ , $ D ^0 \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} h^+h^-$ observables [46].

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Correlation matrix of the statistical uncertainties for the $ B ^0 \rightarrow D K^{*0}$ , $ D\rightarrow K^+\pi^-$ observables [49].

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Correlation matrix of the systematic uncertainties for the $ B ^0 \rightarrow D K^{*0}$ , $ D\rightarrow K^+\pi^-$ observables [49].

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Correlation matrix of the statistical uncertainties for $ B ^0 \rightarrow D ^0 K\pi$ , $ D\rightarrow h^+h^-$ observables [50].

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Correlation matrix of the systematic uncertainties for $ B ^0 \rightarrow D ^0 K\pi$ , $ D\rightarrow h^+h^-$ observables [50].

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Correlation matrix of the statistical uncertainties for the $ B ^0 \rightarrow D K^{*0}$ , $ D\rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi^+\pi^-$ observables [51].

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Correlation matrix of the statistical uncertainties for the $ B ^0_ s \rightarrow D_s^\mp K^\pm$ observables [52].

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Correlation matrix of the systematic uncertainties for the $ B ^0_ s \rightarrow D_s^\mp K^\pm$ observables [52].

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Correlations of the HFAG charm parameters (CHARM 2015, "Fit 3", $ C P$ violation allowed) [22].

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Correlations of the $ D ^0 \rightarrow K^\pm\pi^\mp\pi^+\pi^-$ and $ D ^0 \rightarrow K^\pm\pi^\mp \pi ^0 $ parameters from CLEO and LHCb [58].

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Fit parameter correlations for the $ DK$ combination. The fit results are given in Table 3

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Fit parameter correlations for the $ Dh$ combination solution 1. The fit results are given in Table 4

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Fit parameter correlations for the $ Dh$ combination solution 2. The fit results are given in Table 4

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Created on 02 May 2024.