$CP$ asymmetries in three-body final states in charged $D$ decays and $CPT$ invariance

The study of regional $CP$ asymmetry in Dalitz plots of charm (& beauty) decays gives us more information about the underlying dynamics than the ratio between total rates. In this paper we explore the consequences of the constraint from $CPT$ symmetry with emphasis on three-body $D$ decays. We show simulations of $D^{\pm }\to {\pi }^{\pm }{K}^{+}{K}^{-}$ and discuss correlations with measured $D^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$. There are important comments about analysis of recent LHCb data in $CP$ asymmetries for $B^{\pm }$ decays to three-body final states.

Figure 1

A simulation of the Dalitz plot of the decay $D\to K^{-}{K}^{+}{\pi }^{+}$. The decay model is taken from CLEO-c (see text for details) and is used as the starting point of our studies.

Figure 2

A simulation of the Dalitz plot of the decays $D^{+}\to K^{-}{K}^{+}{\pi }^{+}$ and $D^{-}\to K^{+}{K}^{-}{\pi }^{-}$. No CPV is seeded: the same set of coefficients $c_j$ is used for the simulation of the $D^{+}$ and the $D^{-}$ samples. Values of ${\mathcal{S}}_{CP}^i$ are, in this case, distributed according to a unit Gaussian centered at zero. No excess of one charge over the other is observed in any region of the Dalitz plot, apart from statistical fluctuations.

Figure 3

A simulation of $CP$ violation in the decay $D\to K^{-}{K}^{+}{\pi }^{+}$. A 3° difference in the $K^{*}{K}^{+}$ and $\varphi {\pi }^{+}$ relative phase between $D^{+}$ and $D^{-}$ is introduced. The difference in relative phase cause the $CP$ asymmetry to change sign across the Dalitz plot, according to the phase variation of the interfering Breit-Wigners functions.

Figure 4

A simulation of the Dalitz plot of the decay $D^{+}\to K^{-}{K}^{+}{\pi }^{+}$. The same set of coefficients $c_j$ are used for both $D^{+}$ and $D^{-}$. A rescattering term in the decay amplitude is introduced (see text for details), being different for $D^{+}$ and $D^{-}$. The distribution in the left panel is fitted to a Gaussian with free mean and width.

Figure 5

A simulation of $CP$ violation in the decay $D^{+}\to K^{-}{K}^{+}{\pi }^{+}$. A 3° difference in the $K^{*}{K}^{+}$ and $\varphi {\pi }^{+}$ relative phase between $D^{+}$ and $D^{-}$ is introduced, in addition to the difference in the rescattering term.

Figure 6

Simulation the decays $D^{+}\to K^{-}{K}^{+}{\pi }^{+}$ (left) and $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ (right). $CP$ violation is seeded inspired in a ND scenario in which there is an asymmetry between the coupling between the $D$ meson and the light scalars.