Second generation of “Miranda procedure” for $CP$ violation in Dalitz studies of $B$ (and $D$ and $\tau$) decays

The “Miranda procedure” proposed for analyzing Dalitz plots for $CP$ asymmetries in charged $B$ and $D$ decays in a model-independent manner is extended and refined in this paper. The complexity of Cabibbo-Kobayashi-Maskawa $CP$ phenomenology through order ${\lambda }^6$ is needed in searches for new dynamics (ND). Detailed analyses of three-body final states offer great advantages: (i) They give us more powerful tools for deciding whether an observed $CP$ asymmetry represents the manifestation of ND and its features. (ii) Many advantages can already be obtained by the Miranda procedure without construction of a detailed Dalitz plot description. (iii) One studies $CP$ asymmetries independent of production asymmetries. We illustrate the power of a second generation Miranda procedure with examples with time integrated rates for $B_d/{\overline{B}}_d$ decays to final states $K_S{\pi }^{+}{\pi }^{-}$ as trial runs, with comments on $B^{\pm }\to K^{\pm }{\pi }^{+}{\pi }^{-}/K^{\pm }{K}^{+}{K}^{-}$.

Figure 1

Distribution of $A_{CP}^{\mathrm{bin}}$ across the $B_d\to K_S{\pi }^{+}{\pi }^{-}$ Dalitz plot. In this example, a single source of $CP$ violation—constant excess of $B_d$ over ${\overline{B}}_d$ restricted to the low $K_S{\pi }^{+}/{\pi }^{+}{\pi }^{-}$ mass region—was simulated. Bins have different sizes in order to contain the same number of events.

Figure 2

Distribution of $A_{CP}^{\mathrm{bin}}$ for a single, constant source of $CP$ violation. The distribution was fitted to two Gaussians. The one centered at 0 represents bins where $CP$ is conserved, whereas the second Gaussian represents the $CP$-violating bins.

Figure 3

Distribution of $A_{CP}^{\mathrm{bin}}$ across the $B_d\to K_S{\pi }^{+}{\pi }^{-}$ Dalitz plot for the case of $CP$ violation seeded as a difference in the magnitude of the $\rho K_S$ mode between $B_d$ and ${\overline{B}}_d$. The excess of $B_d$ over ${\overline{B}}_d$ events is concentrated along the $\rho$ band. Note that the values of $A_{CP}^{\mathrm{bin}}$ for these bins are always positive.

Figure 4

Distribution of $A_{CP}^{\mathrm{bin}}$ for the case of $CP$ violation through a difference in the magnitude of the $\rho K_S$ mode. The Gaussian in red (centered at 0.12) is an empirical representation of $A_{CP}^{\mathrm{bin}}$ for the $CP$-violating bins.

Figure 5

Distribution of $A_{CP}^{\mathrm{bin}}$ across the $B_d\to K_S{\pi }^{+}{\pi }^{-}$ Dalitz plot for the third example. When the $CP$ violation is seeded as a relative phase difference, the values of $A_{CP}^{\mathrm{bin}}$ change sign.

Figure 6

Enlarged view of the $CP$-violating bins of Fig. 5.

Figure 7

Distribution of $A_{CP}^{\mathrm{bin}}$ for the third example. In addition to the Gaussian representing the $CP$-conserving bins (green curve, centered at zero), two other Gaussians were used to empirically represent the $A_{CP}^{\mathrm{bin}}$ distribution of the $CP$-violating bins (red and blue curves).