Interference Between Cabibbo Allowed and Doubly Forbidden Transitions in $D\ra K_{S,L} + \pi '$s Decays

Abstract

Both Cabibbo allowed and doubly forbidden transitions contribute coherently to $D\rightarrow K_{S,L}+\pi 's$ decays. This leads to several intriguing and even quantitatively significant consequences, among them: (i) A difference between $\Gamma (D^+\rightarrow K_S \pi ^+)$ and $\Gamma (D^+\rightarrow K_L \pi ^+)$ and between $\Gamma (D^0\rightarrow K_S \pi ^0)$ and $\Gamma (D^0\rightarrow K_L \pi ^0)$ of roughly 10\% ; similarly $\Gamma (D^+\rightarrow [K_S\pi ^0]_{K^*} \pi ^+) \neq \frac{1}{4}\Gamma (D^+\rightarrow [K^-\pi ^+]_{K^*} \pi ^+)$, and more generally $\Gamma (D\rightarrow \bar K^0+\pi 's) \neq 2\Gamma (D\rightarrow K_S+\pi 's)$. (ii) A change in the relative phase between the isospin 3/2 and 1/2 amplitudes as extracted from the observed branching ratios for $D^+\rightarrow K_S\pi ^+$, $D^0\rightarrow K_S\pi ^0 ,\, K^-\pi ^+$. (iii) If New Physics intervenes to provide the required {\em weak} phase, then CP asymmetries of up to a few per cent can arise in $D^+\rightarrow K_S\pi ^+$ vs. $D^-\rightarrow K_S\pi ^-$, $D^0\rightarrow K_S\pi ^0$ vs. $\bar D^0\rightarrow K_S\pi ^0$, $D^+\rightarrow [K_S\pi ^0]_{K^*}\pi ^+$ vs. $D^-\rightarrow [K_S\pi ^0]_{K^*}\pi ^-$, etc.; an asymmetry of the same size, but opposite in sign occurs when the $K_S$ is replaced by a $K_L$ in the final state.