Search for $K_{S,L}$ oscillations and invisible decays into the dark sector at NA64

The decays $K_{S,L}$ have never been experimentally tested. In the Standard Model their branching ratios for the decay into two neutrinos are predicted to be extremely small, $Br(K_{S,L}\to \nu \bar{\nu })\lesssim {10}^{-16}$. We consider several natural extensions of the SM, such as two-Higgs-doublet (2HDM), 2HDM and light scalar, and dark mirror sector models, that allow to enhance the $Br(K_{S,L})$ up to a measurable level. We briefly discuss the possible search for $K_{S,L}$ decays and $K_{S,L}$ oscillations into the dark sector at the NA64 experiment at CERN with the sensitivity to $Br(K_{S,L})\lesssim {10}^{-7}-{10}^{-5}$.