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

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 $3.0{\mathrm{fb}}^{-1}$ 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.

Figure 1

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

Figure 2

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

Figure 3

Results of fits to $D{K}^{+}{\pi }^{-}$ candidates in the (a,b) $D\to K^{+}{\pi }^{-}$, (c,d) $D\to K^{+}{K}^{-}$, and (e,f) $D\to {\pi }^{+}{\pi }^{-}$ samples. The data and the fit results in each NN output bin have been weighted according to $\mathcal{S}/(\mathcal{S}+\mathcal{B})$ 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.

Figure 4

Results of the fit to $D{K}^{+}{\pi }^{-}$, $D\to K^{+}{\pi }^{-}$ candidates shown separately in the five bins of the neural network output variable. The bins are shown, from (a)–(e), in order of increasing $\mathcal{S}/\mathcal{B}$. 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.

Figure 5

Results of the fit to $D{K}^{+}{\pi }^{-}$, $D\to K^{+}{K}^{-}$ candidates shown separately in the five bins of the neural network output variable. The bins are shown, from (a)–(e), in order of increasing $\mathcal{S}/\mathcal{B}$. 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.

Figure 6

Results of the fit to $D{K}^{+}{\pi }^{-}$, $D\to {\pi }^{+}{\pi }^{-}$ candidates shown separately in the five bins of the neural network output variable. The bins are shown, from (a)–(e), in order of increasing $\mathcal{S}/\mathcal{B}$. 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.

Figure 7

Dalitz plots for candidates in the $B$ candidate mass signal region in the $D\to K^{+}{K}^{-}$ 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 ${\overline{B}}_s^0\to D^{*0}{K}^{+}{\pi }^{-}$ decays is expected at low $m(D{K}^{+})$ (i.e. along the top right diagonal).

Figure 8

Projections of the $D\to K^{+}{K}^{-}$ 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 $\mathcal{S}/(\mathcal{S}+\mathcal{B})$ and combined. The components are described in the legend.

Figure 9

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

Figure 10

Results of likelihood scans for (a) $\gamma$, (b) $r_B$, and (c) ${\delta }_B$.

Figure 11

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% C.L.

Figure 12

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