Invisible $K_L$ decays in the SM extensions

In the Standard Model (SM) the branching ratio for the decay $K_L \to \nu \bar{\nu}$ into two neutrinos is helicity suppressed and predicted to be very small $\leq O(10^{-17})$. We consider two natural extensions of the SM, such as two-Higgs-doublet model (2HDM) and the $\nu$MSM with additional singlet scalar, those main features is that they can lead to an enhanced $Br(K_L \to invisible)$. In the 2HDM the smallness of the neutrino mass is explained due to the smallness of the second Higgs doublet vacuum expectation value. Moreover, the $\nu$MSM extension with additional singlet field can explain the $(g - 2)$ muon anomaly. The considered models demonstrate that the $K_L \to invisible$ decay is a clean probe of new physics scales well above 100 TeV, that is complementary to rare $K \to \pi+invisible$ decay, and provide a strong motivation for its sensitive search in a near future low-energy experiment.