On a $CP$ anisotropy measurement in the Dalitz plot

We describe a novel use of the Dalitz plot to probe $CP$ symmetry in three-body modes of $B$ and $D$ mesons. It is based on an observable inspired by astronomers’ practice, namely the significance in the difference between corresponding Dalitz plot bins. It provides a model-independent mapping of local $CP$ asymmetries. We illustrate the method for probing $CP$ symmetry in the two complementary cases of $B$ and $D$ decays: in the former sizable or even large effects can be expected, yet have to be differentiated against leading standard model contributions, while in the latter one cannot count on sizable effects, yet has to deal with much less standard model background.

Figure 1

Dalitz plot distribution for $B^{+}\to K^{+}{\pi }^{+}{\pi }^{-}$ (top) and $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ (bottom) in model “${\rho }^0$.”

Figure 2

Asymmetry in the Dalitz plot bins for model “${\rho }^0$” as defined by Table .

Figure 3

Top: Significance ${}^{\mathrm{Dp}}S_{CP}$ plot for two $CP$ conserving 300 K signal $+200\mathrm{K}$ background samples for $CP$ symmetric decays. Bottom: Gaussian fit for the ${}^{\mathrm{Dp}}S_{CP}$ distribution; $P1$, $P2$, and $P3$ denote the fit values for the central value, width, and normalization parameter, respectively.

Figure 4

Top: Significance ${}^{\mathrm{Dp}}S_{CP}$ plot for $B^{\pm }\to K^{\pm }{\pi }^{\mp }{\pi }^{\pm }$ for model “${\rho }^0$.” Bottom: ${}^{\mathrm{Dp}}S_{CP}$ for the bins in the top figure that pass the statistical cut, fit to a centered Gaussian with unit width. $P1$ is the normalization parameter.

Figure 5

$B^{\pm }\to K^{\pm }{\pi }^{\mp }{\pi }^{\pm }$ Dalitz plot for model “${\rho }^0$” divided into regions.

Figure 6

Distribution of Fig. 4 divided in the regions shown in Fig. 5. $P1$ is the normalization parameter.

Figure 7

Dalitz plot distributions for $B^{+}\to K^{+}{\pi }^{+}{\pi }^{-}$ (top) and $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ (bottom) in model “$f_0$.”

Figure 8

Top: Asymmetry in the Dalitz plot bins for model “$f_0$.” Bottom: Plot of the significance ${}^{\mathrm{Dp}}S_{CP}$ for $B^{\pm }\to K^{\pm }{\pi }^{\mp }{\pi }^{\pm }$.

Figure 9

Top row: ${}^{\mathrm{Dp}}S_{CP}$ for the bins in Fig. 8 (bottom) that pass the statistical cut, fit to a centered Gaussian with unit width for model “$f_0$.” $P1$ is the normalization parameter. Bottom two rows: Distribution of top row divided into the regions shown in Fig. 5. $P1$ is the normalization parameter.

Figure 10

Dalitz plot distributions for $B^{+}\to K^{+}{\pi }^{+}{\pi }^{-}$ (top) and $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ (bottom) in a model “${\rho }^0$” with a 20° phase difference.

Figure 11

Dalitz plot distributions for $B^{+}\to K^{+}{\pi }^{+}{\pi }^{-}$ (top) and $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ (bottom) in a model “${\rho }^0$” with a 10° phase difference.

Figure 12

Plot of $\Delta (i)$ (top) and ${}^{\mathrm{Dp}}S_{CP}$ (bottom) for $B^{+}\to K^{+}{\pi }^{+}{\pi }^{-}$ and $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ in a model “${\rho }^0$” with a 20° phase difference.

Figure 13

Plot of $\Delta (i)$ (top) and ${}^{\mathrm{Dp}}S_{CP}$ (bottom) for $B^{+}\to K^{+}{\pi }^{+}{\pi }^{-}$ and $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ in a model “${\rho }^0$” with a 10° phase difference.

Figure 14

Top row: Distribution of ${}^{\mathrm{Dp}}S_{CP}$ that pass the statistical cut, fit to a centered Gaussian with unit width; $P1$ is the normalization parameter. Bottom two rows: Distribution of ${}^{\mathrm{Dp}}S_{CP}$ divided into the regions shown in Fig. 5 in a model “${\rho }^0$” with a 20° phase difference. $P1$ is the normalization parameter.

Figure 15

Top row: Distribution of ${}^{\mathrm{Dp}}S_{CP}$ that pass the statistical cut, fit to a centered Gaussian with unit width; $P1$ is the normalization parameter. Bottom two rows: Distribution of ${}^{\mathrm{Dp}}S_{CP}$ divided into the regions shown in Fig. 5 in a model “${\rho }^0$” with a 10° phase difference. $P1$ is the normalization parameter.

Figure 16

Dalitz plot for $D^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$ in a model with a 1% (3.6°) phase difference in the ${\rho }^0$ amplitude with subdomains I and II; distributions of the significance ${}^{\mathrm{Dp}}S_{CP}$ for the whole plot and the two subdomains I and II; $P1$ is the normalization parameter.