Search for K_S,L oscillations and invisible decays into the dark sector at NA64

The decays K_S,L → invisible have never been experimentally tested. In the Standard Model (SM), their branching ratios for the decay into two neutrinos are predicted to be extremely small, Br (K_S,L → νν̄) ≲ 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 one to enhance the Br (K_S,L → invisible) up to a measurable level. We briefly discuss the possible search for K_S,L → invisible decays and K_S,L oscillations into the dark sector at the NA64 experiment at CERN with the sensitivity to Br (K_S,L → invisible) ≲ 10⁻⁷−10⁻⁵.