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Model-independent measurement of the CKM angle $\gamma$ using $B^0 \to D K^{\ast 0}$ decays with $D \to K_{S}^{0} \pi^{+} \pi^{-}$ and $K_{S}^{0} K^{+} K^{-}$

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Abstract

A binned Dalitz plot analysis of the decays $B^0 \to D K^{\ast 0}$, with $D \to K_{S}^{0} \pi^{+} \pi^{-}$ and $K_{S}^{0} K^{+} K^{-}$, is performed to measure the observables $x_\pm$ and $y_\pm$, which are related to the CKM angle $\gamma$ and the hadronic parameters of the decays. The $D$ decay strong phase variation over the Dalitz plot is taken from measurements performed at the CLEO-c experiment, making the analysis independent of the $D$ decay model. With a sample of proton-proton collision data, corresponding to an integrated luminosity of $3.0\,\rm{fb}^{-1}$, collected by the LHCb experiment, the values of the $CP$ violation parameters are found to be $x_+ = 0.05 \pm 0.35 \pm 0.02$, $x_-=-0.31\pm 0.20 \pm 0.04$, $y_+=-0.81\pm 0.28\pm 0.06$ and $y_-=0.31\pm 0.21 \pm 0.05$, where the first uncertainties are statistical and the second systematic. These observables correspond to values $\gamma$ = $(71 \pm 20)^\circ$, $r_{B^0} = 0.56\pm 0.17$ and $\delta_{B^0} = (204\,^{+21}_{-20})^\circ$. The parameters $r_{B^0}$ and $\delta_{B^0}$ are the magnitude ratio and strong phase difference between the suppressed and favoured $B^0$ decay amplitudes, and have been measured in a region of $\pm 50$ MeV/$c^2$ around the $K^{\ast}(892)^{0}$ mass and with the magnitude of the cosine of the $K^{\ast}(892)^{0}$ helicity angle larger than 0.4.

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Feynman diagrams of the (left) $ B ^0 \rightarrow D ^0 K ^{*0} $ and (right) $ B ^0 \rightarrow \overline{ D }{} {}^0 K ^{*0} $ amplitudes, which interfere in the $ B ^0 \rightarrow D K ^{*0} $ decay.

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Binning schemes for (left) $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ and (right) $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ . The diagonal line separates the positive and negative bin numbers, where the positive bins are in the region $m^2_- \geq m^2_+$.

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Invariant mass distributions of $ B ^0 \rightarrow D K ^{*0} $ candidates with (top) $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ and (bottom) $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ . The fit results, including the signal and background components, are superimposed.

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Dalitz plots of candidates in the signal region for $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ decays from (left) $\overline{ B }{} {}^0 \rightarrow D\overline{ K }{} {}^{*0} $ and (right) $ B ^0 \rightarrow D K ^{*0} $ decays. The solid blue line indicates the kinematic boundary.

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Dalitz plots of candidates in the signal region for $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ decays from (left) $\overline{ B }{} {}^0 \rightarrow D\overline{ K }{} {}^{*0} $ and (right) $ B ^0 \rightarrow D K ^{*0} $ decays. The solid blue line indicates the kinematic boundary.

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Result of the simultaneous fit to $ B ^0 \rightarrow D ^{*-} \mu^+\nu_\mu$ , $ D ^{*-} \rightarrow \overline{ D }{} {}^0 (\rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- )\pi ^- $ decays with downstream $ K ^0_{\mathrm{ \scriptscriptstyle S}}$ candidates, in 2012 data. A two-dimensional fit is performed in (left) $m( K ^0_{\mathrm{ \scriptscriptstyle S}} h ^+ h ^- )$ and (right) $\Delta m$. The (blue) total fit PDF and the signal and background components are superimposed.

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Example efficiency profiles of (left) $ B ^0 \rightarrow D K ^{*0} $ and (right) $ B ^0 \rightarrow D ^{*-} \mu^+\nu_\mu$ decays in the simulation. The top (bottom) plots are for $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ ( $ D \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ ) decays.

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Confidence levels at (solid) 68.3% and (dotted) 95.5% for (red, light) $(x_+,y_+)$ and (blue, dark) $(x_{-}, y_{-})$ as measured in $ B ^0 \rightarrow D K ^{*0} $ decays (statistical uncertainties only). The parameters $(x_+,y_+)$ relate to $ B ^0$ decays and $(x_{-}, y_{-})$ refer to $\overline{ B }{} {}^0$ decays. The points represent the best fit values.

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The three-dimensional confidence volumes projected onto the $(\gamma, r_{ B ^0 })$ and $(\gamma, \delta_{ B ^0 })$ planes. The confidence levels correspond to 68.3% and 95.5% confidence levels when projected onto one dimension and are denoted by solid and dotted contours, respectively. The diamonds mark the central values.

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Animated gif made out of all figures.

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Tables and captions

Functional forms of the $DK\pi$ invariant mass distribution, $m$, in partially reconstructed decays of $ B ^0_ s \rightarrow ( D ^{*0} \rightarrow D ^0 \{\pi ^0 ,\gamma\})\overline{ K }{} {}^{*0} $, where either the $\pi ^0$ or $\gamma$ is not reconstructed. The $ D ^{*0}$ helicity state is given by $\lambda$. The quantities $a_X$ and $b_X$ are the minimum and maximum kinematic boundaries of the reconstructed $DK\pi$ invariant mass, where $X$ is the particle that is missed.

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Results of the simultaneous fit to the invariant mass distribution of $ B ^0 \rightarrow D K ^{*0} $ decays, with the $ D $ meson decaying to $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ .

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Summary of the systematic uncertainties for the parameters $ x_{\pm} , y_{\pm} $ . The various sources of systematic uncertainties are described in the main text.

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Total correlation matrix, including statistical and systematic uncertainties, between the $ x_{\pm}$ , $ y_{\pm}$ parameters used in the extraction of $\gamma$.

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Supplementary Material [file]

Supplementary material full pdf

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This ZIP file contains supplemetary material for the publication LHCb-PAPER-2016-006. The files are: supplementary.pdf : PDF containing the extra figures and some text *.pdf, *.png, *.eps, *.C : The figures in various formats

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Created on 09 December 2018.Citation count from INSPIRE on 09 December 2018.