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

Both Cabibbo allowed and doubly forbidden transitions contribute coherently to $D\rightarrow K_{S,L}+π's$ decays. This leads to several intriguing and even quantitatively significant consequences, among them: (i) A difference between $Γ(D^+\rightarrow K_S π^+)$ and $Γ(D^+\rightarrow K_L π^+)$ and between $Γ(D^0\rightarrow K_S π^0)$ and $Γ(D^0\rightarrow K_L π^0)$ of roughly 10\% ; similarly $Γ(D^+\rightarrow [K_Sπ^0]_{K^*} π^+) \neq \frac{1}{4}Γ(D^+\rightarrow [K^-π^+]_{K^*} π^+)$, and more generally $Γ(D\rightarrow \bar K^0+π's) \neq 2Γ(D\rightarrow K_S+π's)$. (ii) A change in the relative phase between the isospin 3/2 and 1/2 amplitudes as extracted from the observed branching ratios for $D^+\rightarrow K_Sπ^+$, $D^0\rightarrow K_Sπ^0 ,\, K^-π^+$. (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 $D^+\rightarrow K_Sπ^+$ vs. $D^-\rightarrow K_Sπ^-$, $D^0\rightarrow K_Sπ^0$ vs. $\bar D^0\rightarrow K_Sπ^0$, $D^+\rightarrow [K_Sπ^0]_{K^*}π^+$ vs. $D^-\rightarrow [K_Sπ^0]_{K^*}π^-$, 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.