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A model-independent Dalitz plot analysis of $B^\pm \to D K^\pm$ with $D \to K^0_{\rm S} h^+h^-$ ($h=\pi, K$) decays and constraints on the CKM angle $\gamma$

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Abstract

A binned Dalitz plot analysis of $B^\pm \to D K^\pm$ decays, with $D \to K^0_{\rm S} \pi^+\pi^-$ and $D \to K^0_{\rm S} K^+ K^-$, is performed to measure the CP-violating observables $x_{\pm}$ and $y_{\pm}$ which are sensitive to the CKM angle $\gamma$. The analysis exploits 1.0 $\rm fb^{-1}$ of data collected by the LHCb experiment. The study makes no model-based assumption on the variation of the strong phase of the D decay amplitude over the Dalitz plot, but uses measurements of this quantity from CLEO-c as input. The values of the parameters are found to be $x_- = (0.0 \pm 4.3 \pm 1.5 \pm 0.6) \times 10^{-2}$, $y_- = (2.7 \pm 5.2 \pm 0.8 \pm 2.3) \times 10^{-2}$, $x_+ = (-10.3 \pm 4.5 \pm 1.8 \pm 1.4)\times 10^{-2}$ and $y_+ = (-0.9 \pm 3.7 \pm 0.8 \pm 3.0)\times 10^{-2}$. The first, second, and third uncertainties are the statistical, the experimental systematic, and the error associated with the precision of the strong-phase parameters measured at CLEO-c, respectively. These results correspond to $\gamma = (44^{\,+43}_{\,-38})^\circ$, with a second solution at $\gamma \to \gamma + 180^\circ$, and $r_B = 0.07 \pm 0.04$, where $r_B$ is the ratio between the suppressed and favoured B decay amplitudes.

Figures and captions

Binning choices for (a) $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^-$ and (b) $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^-$. The diagonal line separates the positive and negative bins.

dkpp_b[..].pdf [211 KiB]
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dkpp_blur_manual.pdf
dkpp_k[..].pdf [198 KiB]
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dkpp_kskk_babar_manual.pdf

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

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d2kspipi_pass_both_LL_merge.pdf
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d2kspipi_fail_both_LL_merge.pdf
d2kspi[..].pdf [11 KiB]
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d2kspipi_pass_both_DD_merge.pdf
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d2kspipi_fail_both_DD_merge.pdf

Invariant mass distributions of (a) $B^\pm \rightarrow D K^\pm$ and (b) $B^\pm \rightarrow D \pi^\pm$ candidates, with $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^-$, shown with both $ K ^0_{\rm\scriptscriptstyle S}$ categories combined. Fit results, including the signal and background components, are superimposed.

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d2kskk_pass_both_comb_merge.pdf
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d2kskk_fail_both_comb_merge.pdf

Dalitz plots of $B^\pm \rightarrow D K^\pm$ candidates in the signal region for (a,b) $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^-$ and (c,d) $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+ K^-$ decays, divided between (a,c) $B^+$ and (b,d) $B^-$. The boundaries of the kinematically-allowed regions are also shown.

Dalitz[..].pdf [34 KiB]
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Dalitz_d2kspipi_B2DK_KSboth_plus_graph.pdf
Dalitz[..].pdf [34 KiB]
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Dalitz_d2kspipi_B2DK_KSboth_minus_graph.pdf
Dalitz[..].pdf [17 KiB]
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Dalitz_d2kskk_B2DK_KSboth_plus_graph.pdf
Dalitz[..].pdf [17 KiB]
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Dalitz_d2kskk_B2DK_KSboth_minus_graph.pdf

One (solid), two (dashed) and three (dotted) standard deviation confidence levels for $(x_+,y_+)$ (blue) and $(x_{-}, y_{-})$ (red) as measured in $B^\pm \rightarrow D K^\pm$ decays (statistical only). The points represent the best fit central values.

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likescan.pdf

Signal yield in effective bins compared with prediction of $(x_\pm, y_\pm)$ fit (black histogram) for $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^-$ and $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^-$. Figure (a) shows the sum of $B^+$ and $B^-$ yields. Figure (b) shows the difference of $B^+$ and $B^-$ yields. Also shown (dashed line and grey shading) is the expectation and uncertainty for the zero $ C\!P$ -violation hypothesis.

BplusB[..].pdf [6 KiB]
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BplusBminusdkFitSepSummed_bothModes.pdf
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BplusBminusdkFitSepDifference_bothModes.pdf

Two-dimensional projections of confidence regions onto the $(\gamma, r_B)$ and $(\gamma, \delta_B)$ planes showing the one (solid) and two (dashed) standard deviations with all uncertainties included. For the ($\gamma,r_B$) projection the three (dotted) standard deviation contour is also shown. The points mark the central values.

fr_con[..].pdf [4 KiB]
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fd_contours.pdf

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 fit, scaled from the full fit mass range, for candidates passing the $B^\pm \rightarrow D h^\pm$, $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^-$ selection. Values are shown separately for candidates containing 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 5110 $ {\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 fit, scaled from the full fit mass range, for candidates passing the $B^\pm \rightarrow D h^\pm$, $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^-$ selection. Values are shown separately for candidates containing 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 5110 $ {\mathrm{\,MeV\!/}c^2}$ and 5800 $ {\mathrm{\,MeV\!/}c^2}$ .

Table_2.pdf [39 KiB]
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Table_2.pdf

Results for $x_{\pm}$ and $y_{\pm}$ from the fits to the data in the case when both $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^-$ and $D \rightarrow K ^0_{\rm\scriptscriptstyle S} K^+K^-$ are considered and when only the $D \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi^+\pi^-$ final state is included. 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. The correlation coefficients are calculated including all sources of uncertainty (the values in parentheses correspond to the case where only the statistical uncertainties are considered).

Table_3.pdf [52 KiB]
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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 $10^{-2}$.

Table_4.pdf [50 KiB]
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Table_4.pdf

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

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