Measurement of $CP$ Violation Parameters with a Dalitz Plot Analysis of $B^{\pm }\to D_{{\pi }^{+}{\pi }^{-}{\pi }^0}{K}^{\pm }$

We report the results of a $CP$ violation analysis of the decay $B^{\pm }\to D_{{\pi }^{+}{\pi }^{-}{\pi }^0}{K}^{\pm }$, where $D_{{\pi }^{+}{\pi }^{-}{\pi }^0}$ indicates a neutral $D$ meson detected in the final state ${\pi }^{+}{\pi }^{-}{\pi }^0$, excluding $K_S^0{\pi }^0$. The analysis makes use of $324\times {10}^6$ $e^{+}{e}^{-}\to B\overline{B}$ events recorded by the BABAR experiment at the PEP-II $e^{+}{e}^{-}$ storage ring. Analyzing the ${\pi }^{+}{\pi }^{-}{\pi }^0$ Dalitz plot distribution and the $B^{\pm }\to D_{{\pi }^{+}{\pi }^{-}{\pi }^0}{K}^{\pm }$ branching fraction and decay rate asymmetry, we find the following one-standard-deviation constraints on the amplitude ratio and on the weak and strong phases: $0.06<r_B<0.78$, $-30°<\gamma <76°$, $-27°<\delta <78°$. We also measure the magnitudes and phases of the components of the $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decay amplitude.

Figure 1

(a) The two-dimensional ($s_{+}$, $s_{-}$) distribution of the $D^{*+}\to D^0{\pi }^{+}$ data. Charge conjugation is implied. (b) One-, two-, and three-standard-deviation contours of $\mathcal{L}$ as functions of ${\theta }_{\pm }$ vs ${\rho }_{\pm }$. The solid (dashed) curves correspond to $B^{+}$ ($B^{-}$) results. The no-interference point (${\rho }_{\pm }=x_0$, ${\theta }_{\pm }=180°$) is marked with an ×. (c) Projection of the three-dimensional confidence level $1-\alpha$ onto $r_B$ and $\gamma$. (d) $1-\alpha$ vs $\gamma$.