Double Dalitz plot analysis of the decay $B^0\to D{K}^{+}{\pi }^{-}$, $D\to K_S^0{\pi }^{+}{\pi }^{-}$

It is shown that it is possible to perform a model-independent extraction of the Cabibbo-Kobayashi-Maskawa unitarity triangle angle $\gamma$ using only the decays $B^0\to D{K}^{+}{\pi }^{-}$ with $D\to K_S^0{\pi }^{+}{\pi }^{-}$ and $B^0\to D{K}^{+}{\pi }^{-}$ with flavor-specific $D$ decays. The proposed method can also utilize the $B^0\to D{K}^{+}{\pi }^{-}$ data with $CP$-eigenstate decays of the $D$ meson in a model-independent way.

Figure 1

Generated Dalitz plots for (a) $B^0\to {\overline{D}}^0{K}^{+}{\pi }^{-}$ and (b) $B^0\to D^0{K}^{+}{\pi }^{-}$ decays using amplitude model C.

Figure 2

(a) Binning of $B^0\to D{K}^{+}{\pi }^{-}$ Dalitz plot and (b) values of ${\varkappa }_i$, ${\sigma }_i$ parameters for optimal binning (crosses are calculated values, scattered points are the values obtained from the fit to toy MC samples).

Figure 3

(a) Binning of $D\to K_S^0{\pi }^{+}{\pi }^{-}$ Dalitz plot and (b) values of $c_i$, $s_i$ parameters for optimal binning (crosses are calculated values, scattered points are the values obtained from the fit to toy MC samples).

Figure 4

Accuracy of $\gamma$ measurement for the binned double Dalitz fit of $B^0\to D{K}^{+}{\pi }^{-}$, $D\to K_S^0{\pi }^{+}{\pi }^{-}$ (option 6) using model C. (a) Residual distribution for the optimal binning, (b) residual distribution for the rectangular binning with $Q=0.60$, and (c) dependence of the accuracy on the value of $\gamma$ and comparison with the combined fit with $B^0\to D_{CP}{K}^{+}{\pi }^{-}$ decays (option 8).