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Constraints on the unitarity triangle angle $\gamma$ from Dalitz plot analysis of $B^0 \to D K^+ \pi^-$ decays

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Abstract

The first study is presented of CP violation with an amplitude analysis of the Dalitz plot of $B^0 \to D K^+ \pi^-$ decays, with $D \to K^+ \pi^-$, $K^+ K^-$ and $\pi^+ \pi^-$. The analysis is based on a data sample corresponding to $3.0\,{\rm fb}^{-1}$ of $pp$ collisions collected with the LHCb detector. No significant CP violation effect is seen, and constraints are placed on the angle $\gamma$ of the unitarity triangle formed from elements of the Cabibbo-Kobayashi-Maskawa quark mixing matrix. Hadronic parameters associated with the $B^0 \to D K^*(892)^0$ decay are determined for the first time. These measurements can be used to improve the sensitivity to $\gamma$ of existing and future studies of the $B^0 \to D K^*(892)^0$ decay.

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Feynman diagrams for the contributions to $ B ^0 \rightarrow D K ^+ \pi ^- $ from (a) $ B ^0 \rightarrow D_2^*(2460)^- K ^+ $, (b) $ B ^0 \rightarrow \overline{ D }{} {}^0 K ^* (892)^0$ and (c) $ B ^0 \rightarrow D ^0 K ^* (892)^0$ decays.

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Illustration of the method to determine $\gamma$ from Dalitz plot analysis of $ B ^0 \rightarrow D K ^+ \pi ^- $ decays [12,13]: (left) the $V_{cb}$ amplitude for $ B ^0 \rightarrow \overline{ D }{} {}^0 K ^{*0} $ compared to that for $ B ^0 \rightarrow D_2^{*-} K ^+ $ decay; (right) the effect of the $V_{ub}$ amplitude that contributes to $ B ^0 \rightarrow D_{ C\!P } K ^{*0} $ and $\overline{ B }{} {}^0 \rightarrow D_{ C\!P }\overline{ K }{} {}^{*0} $ decays provides sensitivity to $\gamma$. The notation $D_{ C\!P }$ represents a neutral $ D $ meson reconstructed in a $ C\!P$ eigenstate, while $D^{*-}_{2\, C\!P }$ denotes the decay chain $D_2^{*-}\rightarrow D _{ C\!P }\pi ^- $, where the charge of the pion tags the flavour of the neutral $ D $ meson, independently of the mode in which it is reconstructed, so there is no contribution from the $V_{ub}$ amplitude.

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Results of fits to $D K ^+ \pi ^- $ candidates in the (a,b) $D\rightarrow K ^+ \pi ^- $, (c,d) $D\rightarrow K ^+ K ^- $ and (e,f) $D\rightarrow \pi ^+ \pi ^- $ samples. The data and the fit results in each NN output bin have been weighted according to ${\cal S}/({\cal S}+{\cal B})$ as described in the text. The left and right plots are identical but with (left) linear and (right) logarithmic $y$-axis scales. The components are as described in the legend.

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Results of the fit to $ D K ^+ \pi ^- $, $ D \rightarrow K ^+ \pi ^- $ candidates shown separately in the five bins of the neural network output variable. The bins are shown, from (a)--(e), in order of increasing ${\cal S}/{\cal B}$. The components are as indicated in the legend. The vertical dotted lines in (a) show the signal window used for the fit to the Dalitz plot.

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Results of the fit to $ D K ^+ \pi ^- $, $ D \rightarrow K ^+ K ^- $ candidates shown separately in the five bins of the neural network output variable. The bins are shown, from (a)--(e), in order of increasing ${\cal S}/{\cal B}$. The components are as indicated in the legend. The vertical dotted lines in (a) show the signal window used for the fit to the Dalitz plot.

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Results of the fit to $ D K ^+ \pi ^- $, $ D \rightarrow \pi ^+ \pi ^- $ candidates shown separately in the five bins of the neural network output variable. The bins are shown, from (a)--(e), in order of increasing ${\cal S}/{\cal B}$. The components are as indicated in the legend. The vertical dotted lines in (a) show the signal window used for the fit to the Dalitz plot.

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Dalitz plots for candidates in the $ B $ candidate mass signal region in the $ D \rightarrow K ^+ K ^- $ and $\pi ^+ \pi ^- $ samples for (a) $\overline{ B }{} {}^0$ and (b) $ B ^0$ candidates. Only candidates in the three purest NN bins are included. Background has not been subtracted, and therefore some contribution from $\overline{ B }{} {}^0_ s \rightarrow D ^{*0} K ^+ \pi ^- $ decays is expected at low $m(D K ^+ )$ ( i.e. along the top right diagonal).

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Projections of the $D\rightarrow K ^+ K ^- $ and $\pi ^+ \pi ^- $ samples and the fit result onto (a,b) $m( D \pi ^\mp )$, (c,d) $m( K ^\pm \pi ^\mp )$ and (e,f) $m( D K ^\pm )$ for (a,c,e) $\overline{ B }{} {}^0$ and (b,d,f) $ B ^0$ candidates. The data and the fit results in each NN output bin have been weighted according to ${\cal S}/({\cal S}+{\cal B})$ and combined. The components are described in the legend.

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Contours at $68\,\%$ CL for the (blue) $(x_+,y_+)$ and (red) $(x_-,y_-)$ parameters associated with the $ B ^0 \rightarrow D K ^* (892)^0$ decay, with statistical uncertainties only. The central values are marked by a circle and a cross, respectively.

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Results of likelihood scans for (a) $\gamma$, (b) $r_B$ and (c) $\delta_B$.

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Confidence level contours for (a) $\gamma$ and $r_B$, (b) $\gamma$ and $\delta_B$ and (c) $r_B$ and $\delta_B$. The shaded regions are allowed at $68\,\%$ CL.

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Distributions of (a) $\kappa$, (b) $\bar{R}_B$ and (c) $\Delta \bar{\delta}_B$, obtained as described in the text.

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Animated gif made out of all figures.

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Tables and captions

Yields in the signal window of the fit components in the five NN output bins for the $ D \rightarrow K ^+ \pi ^- $ sample. The last column indicates whether or not each component is explicitly modelled in the Dalitz plot fit.

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Yields in the signal window of the fit components in the five NN output bins for the $ D \rightarrow K ^+ K ^- $ sample. The last column indicates whether or not each component is explicitly modelled in the Dalitz plot fit.

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Yields in the signal window of the fit components in the five NN output bins for the $ D \rightarrow \pi ^+ \pi ^- $ sample. The last column indicates whether or not each component is explicitly modelled in the Dalitz plot fit.

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Results for the unconstrained parameters obtained from the fits to the three data samples. Entries where no number is given are fixed to zero. Fractions marked * are not varied in the fit, and give the difference between unity and the sum of the other fractions.

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Experimental systematic uncertainties.

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Model uncertainties.

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Results for the complex coefficients $c_j$ from the fit to data. Uncertainties are statistical only. All reported quantities are unconstrained in the fit, except that the $D^{*}_{2}(2460)^{-}$ component is fixed as a reference amplitude, and the magnitude of the $D^{*}_{s1}(2700)^{+}$ component is constrained. The $ K ^+ \pi ^- $ S-wave is the coherent sum of the $ K ^*_0 (1430)^{0}$ and the nonresonant $K\pi$ S-wave component [52].

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Correlation matrices associated to the (left) statistical and (right) systematic uncertainties of the $ C\!P$ violation parameters associated with the $ B ^0 \rightarrow D K ^* (892)^0$ decay.

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