Time-dependent study of $K_{S} \to \pi^{+} \pi^{-}$ decays for flavour physics measurements

Nowadays High Energy Physics experiments can accumulate unprecedented statistics of heavy flavour decays that allows to apply new methods, based on the study of very rare phenomena, which used to be just desperate. In this paper we propose a new method to measure composition of $K^0$-$\overline{K}^0$, produced in a decay of heavy hadrons. This composition contains important information, in particular about weak and strong phases between amplitudes of the produced $K^0$ and $\overline{K}^0$. We consider possibility to measure these parameters with time-dependent $K^0 \to \pi^+ \pi^-$ analysis. Due to $CP$-violation in kaon mixing time-dependent decay rates of $K^0$ and $\overline{K}^0$ differ, and the initial amplitudes revealed in the $CP$-violating decay pattern. In particular we consider cases of charmed hadrons decays: $D^+ \to K^0 \pi^+$, $D_s^+ \to K^0 K^+$, $\Lambda_c \to p K^0$ and with some assumptions $D^0 \to K^0 \pi^0$. This can be used to test the sum rule for charmed mesons and to obtain input for the full constraint of the two body amplitudes of $D$-mesons.