Remarks on CP Asymmetries in $D^0/\bar {D^0} \ra K_S \pi^+ \pi^-$

Abstract

We consider the interference of resonant amplitudes leading to the final state $K_S \pi^+ \pi^-$ in $D^0/\bar {D^0}$ decays. Each of these amplitudes consists of both Cabibbo allowed and doubly Cabibbo suppressed transitions. The role of strong phase arising out of Breit-Wigner resonant propagators is emphasised. Invoking the $\Delta S = \Delta Q$ rule, $K_S$ in the final state is identified as the mass eigenstate of superposed weak eigenstates $K^0$ and $\bar K^0$. A nonzero CP asymmetry appears to be possible in three body decays.