Cross ratios between Dalitz plot amplitudes in three-body $D^0$ decays

A recent study of $D^0 \to π^0 K^+ K^-$ and $D^0 \to K_S π^+π^-$ describes a flavor-symmetric approach to calculate relative amplitudes and phases, for characteristic interferences between $D$ decays to a light pseudoscalar $P$ and a light vector $V$, on Dalitz plots for $D \to PPP$ decays. The flavor-symmetric approach used an earlier fit to $D \to P V$ decay rates and was found to agree fairly well with experiments for $D^0 \to π^0 π^+ π^-$ but not as well for $D^0 \to π^0 K^+ K^-$ and $D^0 \to K_S π^+π^-$. The present work extends this investigation to include $D^0 \to K^- π^+ π^0$. We use an SU(3) flavor symmetry relationship between ratios of Cabibbo-favored (CF) $D \to P V$ amplitudes in $D^0 \to K^- π^+ π^0$ and ratios of singly- Cabibbo-suppressed (SCS) $D \to P V$ amplitudes in $D^0 \to π^0 K^+ K^-$ and $D^0 \to π^0 π^+ π^-$. We observe that experimental values for Dalitz plot cross ratios obey this relationship up to discrepancies noted previously. The need for an updated Dalitz plot analysis of $D^0 \to K^- π^+ π^0$ is emphasized.

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