Probing the $\Delta U=0$ Rule in Three Body Charm Decays

Abstract

CP violation in charm decay was observed in the decays $D^0\to P^{\pm }{P}^{\mp }$ of a $D^0$ meson to two pseudoscalars. When interpreted within the SM, the results imply that the ratio of the relevant rescattering amplitudes has a magnitude and phase that are both of $O(1)$. We discuss ways to probe similar ratios in $D^0\to V^{\pm }{P}^{\mp }$ decays, where $V$ is a vector that decays to two pseudoscalars, from the Dalitz-plot analysis of time-integrated three-body decays. Compared to two-body decays, three-body decays have the advantage that the complete system can be solved without the need for time-dependent CP violation measurements or use of correlated $D^0$--${\overline{D}}^0$ production. We discuss the decays $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and $D^0\to K^{+}{K}^{-}{\pi }^0$ as examples by considering a toy model of only two overlapping charged resonances, treating the underlying pseudo two-body decays in full generality.

Comments: 32 pages. Additional references and clarifications. Conclusions unchanged. Matches published version