$CP$ violation in charmed hadron decays into neutral kaons

We find a new $CP$ violating effect in charmed hadron decays into neutral kaons, which is induced by the interference between the Cabibbo-favored and doubly Cabibbo-suppressed amplitudes with the $K^0-{\bar{K}}^0$ mixing. It is estimated to be of order of $\mathcal{O}({10}^{-3})$, much larger than the direct $CP$ asymmetry, but missed in the literature. To reveal this new $CP$ violation effect, we propose a new observable, the difference of the $CP$ asymmetries in the $D^{+}\to {\pi }^{+}{K}_S^0$ and $D_s^{+}\to K^{+}{K}_S^0$ modes. Once the new effect is determined by experiments, the direct $CP$ asymmetry then can be extracted and used to search for new physics.