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Measurement of the CKM angle $\gamma$ using $B^\pm \to D K^\pm$ with $D \to K^0_{\rm S} \pi^+\pi^-, K^0_{\rm S} K^+ K^-$ decays

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Abstract

A binned Dalitz plot analysis of $B^\pm \to D K^\pm$ decays, with $D \to K_S \pi^+\pi^-$ and $D \to K_S K^+ K^-$, is performed to measure the \CP-violating observables $x_{\pm}$ and $y_{\pm}$, which are sensitive to the Cabibbo-Kobayashi-Maskawa angle $\gamma$. The analysis exploits a sample of proton-proton collision data corresponding to 3.0\invfb collected by the LHCb experiment. Measurements from CLEO-c of the variation of the strong-interaction phase of the $D$ decay over the Dalitz plot are used as inputs. The values of the parameters are found to be $x_+ = (-7.7 \pm 2.4 \pm 1.0 \pm 0.4)\times 10^{-2}$, $x_- = (2.5 \pm 2.5 \pm 1.0 \pm 0.5) \times 10^{-2}$, $y_+ = (-2.2 \pm 2.5 \pm 0.4 \pm 1.0)\times 10^{-2}$, and $y_- = (7.5 \pm 2.9 \pm 0.5 \pm 1.4) \times 10^{-2}$. The first, second, and third uncertainties are the statistical, the experimental systematic, and that associated with the precision of the strong-phase parameters. These are the most precise measurements of these observables and correspond to $\gamma = (62^{\,+15}_{\,-14})^\circ$, with a second solution at $\gamma \to \gamma + 180^\circ$, and $r_B = 0.080^{+ 0.019}_{-0.021}$, where $r_B$ is the ratio between the suppressed and favoured $B$ decay amplitudes.

Figures and captions

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

KsPiPi[..].pdf [217 KiB]
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Invariant mass distributions of (left) $ B ^\pm \rightarrow D K ^\pm $ and (right) $ B ^\pm \rightarrow D \pi ^\pm $ candidates, with $ D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^- $ , divided between the (top) long and (bottom) downstream $ K ^0_{\rm\scriptscriptstyle S}$ categories. Fit results, including the signal and background components, are superimposed.

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Invariant mass distributions of (left) $ B ^\pm \rightarrow D K ^\pm $ and (right) $ B ^\pm \rightarrow D \pi ^\pm $ candidates, with $ D \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ K ^- $ , divided between the (top) long and (bottom) downstream $ K ^0_{\rm\scriptscriptstyle S}$ categories. Fit results, including the signal and background components, are superimposed.

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canvas_d2kskk_LL_merge_nopull.pdf
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canvas_d2kskk_DD_merge_nopull.pdf

Dalitz plots of $ B ^\pm \rightarrow D K ^\pm $ candidates in the signal region for $ D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^- $ decays from (left) $ B ^+ $ and (right) $ B ^- $ decays. Both long and downstream $ K ^0_{\rm\scriptscriptstyle S}$ candidates are included. The blue line indicates the kinematic boundary.

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Dalitz_d2kspipi_B2DK_KSboth_plus_graph.pdf
Dalitz[..].pdf [422 KiB]
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Dalitz_d2kspipi_B2DK_KSboth_minus_graph.pdf

Dalitz plots of $ B ^\pm \rightarrow D K ^\pm $ candidates in the signal region for $ D \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ K ^- $ decays from (left) $ B ^+ $ and (right) $ B ^- $ decays. Both long and downstream $ K ^0_{\rm\scriptscriptstyle S}$ candidates are included. The blue line indicates the kinematic boundary.

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Dalitz_d2kskk_B2DK_KSboth_minus_graph.pdf

Result of the simultaneous fit to $\optbar{B}{} \!\rightarrow D ^{*\pm} \mu^\mp\optbarlowcase{\nu}{} _{\!\!\mu}$ , $ D ^* \pm\rightarrow \optbar{D}{} ^0(\rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^- )\pi ^\pm $ decays with downstream $ K ^0_{\rm\scriptscriptstyle S}$ candidates, in 2012 data. A two-dimensional fit is performed in (left) $m( K ^0_{\rm\scriptscriptstyle S} h ^+ h ^- )$ and (right) $\Delta m$. The (blue) total fit PDF is constructed from (solid red) signal, (dashed black) combinatorial background and (dotted green) random soft pion background.

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Example efficiency profiles of (left) $ B ^\pm \rightarrow D \pi ^\pm $ and (right) $\optbar{B}{} \!\rightarrow D ^{*\pm} \mu^\mp\optbarlowcase{\nu}{} _{\!\!\mu}$ decays in simulation. These plots refer to downstream $ K ^0_{\rm\scriptscriptstyle S}$ candidates under 2012 data taking conditions.

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Fractional yield ratios of $ B ^\pm \rightarrow D \pi ^\pm $ to $\optbar{B}{} \!\rightarrow D ^{*\pm} \mu^\mp\optbarlowcase{\nu}{} _{\!\!\mu}$ decays, $f_i$, for each effective Dalitz plot bin. The vertical dashed line separates the ratios for $ K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^- $ (bins $-8$ to 8) and $ K ^0_{\rm\scriptscriptstyle S} K ^+ K ^- $ (bins $-2$ to 2). The left (right) plot shows the values of $f_i$ before (after) correcting for the efficiency differences.

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Confidence levels at 39.3%, 86.5% and 98.9% probability for $(x_+,y_+)$ and $(x_{-}, y_{-})$ as measured in $B^\pm \rightarrow D K^\pm$ decays (statistical uncertainties only). The parameters $(x_+,y_+)$ relate to $ B ^+ $ decays and $(x_{-}, y_{-})$ refer to $ B ^- $ decays. The stars represent the best fit central values.

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Combinations of signal yields (data points) in effective bins compared with prediction of $(x_\pm, y_\pm)$ fit (solid histogram) for $ D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^- $ and $ D \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ K ^- $ decays. The dotted histograms give the prediction for $x_\pm=y_\pm=0$. The left plot shows the sum of $B^+$ and $B^-$ yields and the right plot shows the difference of $B^+$ and $B^-$ yields.

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The rectangular binning schemes for the two decays. On the left (right) is plotted the scheme for the $ K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^- $ ( $ K ^0_{\rm\scriptscriptstyle S} K ^+ K ^- $ ) decay.

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The three-dimensional confidence volumes, corresponding to 19.9%, 73.9% and 97.1% confidence levels, are projected onto the $(\gamma, r_B)$ and $(\gamma, \delta_B)$ planes. The confidence levels are given by solid, dashed and dotted contours. The diamonds mark the central values.

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Animated gif made out of all figures.

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Tables and captions

Yields and statistical uncertainties in the signal region from the invariant mass fits, scaled from the full fit mass range, for candidates passing the $ B ^\pm \rightarrow D ( K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^- ) h ^\pm$ selection. Values are shown separately for candidates formed using long and downstream $ K ^0_{\rm\scriptscriptstyle S}$ decays. The signal region is between 5247 $ {\mathrm{\,MeV\!/}c^2}$ and 5317 $ {\mathrm{\,MeV\!/}c^2}$ and the full fit range is between 5080 $ {\mathrm{\,MeV\!/}c^2}$ and 5800 $ {\mathrm{\,MeV\!/}c^2}$ .

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Table_1.pdf

Yields and statistical uncertainties in the signal region from the invariant mass fits, scaled from the full fit mass range, for candidates passing the $ B ^\pm \rightarrow D ( K ^0_{\rm\scriptscriptstyle S} K ^+ K ^- ) h ^\pm$ selection. Values are shown separately for candidates formed using long and downstream $ K ^0_{\rm\scriptscriptstyle S}$ decays. The signal region is between 5247 $ {\mathrm{\,MeV\!/}c^2}$ and 5317 $ {\mathrm{\,MeV\!/}c^2}$ and the full fit range is between 5080 $ {\mathrm{\,MeV\!/}c^2}$ and 5800 $ {\mathrm{\,MeV\!/}c^2}$ .

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Results for $x_{\pm}$ and $y_{\pm}$ from fits of both the $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^-$ and $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^-$ samples, and from fits of the $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^-$ sample only. The first, second, and third uncertainties are the statistical, the experimental systematic, and the error associated with the precision of the strong-phase parameters, respectively.

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Summary of statistical, experimental, and strong-phase, uncertainties on $x_\pm$ and $y_\pm$ in the case where both $ D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^- $ and $ D \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ K ^- $ decays are included in the fit. All entries are given in multiples of $10^{-2}$.

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Correlation matrix between the $ x_{\pm}$ , $ y_{\pm}$ parameters for the full data set.

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Correlation matrix between the $ x_{\pm}$ , $ y_{\pm}$ parameters for $ K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^- $ decays only.

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Created on 09 December 2018.Citation count from INSPIRE on 18 December 2018.