  We show that the Grossman-Nir (GN) bound, $\text{Br}(K_L\to \pi^0\nu
\bar\nu)\leq 4.3 \, \text{Br}(K^+\to \pi^+\nu \bar\nu)$, can be violated in the
presence of light new physics with flavor violating couplings. We construct
three sample models in which the GN bound can be violated by orders of
magnitude, while satisfying all other experimental bounds. In the three models
the enhanced branching ratio $\text{Br}(K_L\to \pi^0+{\rm inv})$ is due to
$K_L\to \pi^0\phi_1$, $K_L\to \pi^0\phi_1\phi_1$, $K_L\to \pi^0\psi_1\bar
\psi_1$ transitions, respectively, where $\phi_1 (\psi_1)$ is a light scalar
(fermion) that escapes the detector. In the three models $\text{Br}(K^+\to
\pi^++{\rm inv})$ remains very close to the SM value, while $\text{Br}(K_L\to
\pi^0+{\rm inv})$ can saturate the present KOTO bound. Besides invisible
particles in the final state (which may account for dark matter) the models
require additional light mediators around the GeV-scale.

High Energy Physics - Phenomenology