  In this work we investigate the implication of $K\to \pi \nu \bar{\nu}$ from
the recent KOTO and NA62 measurements for generic neutrino interactions and the
new physics scale in effective field theories. The interactions between quarks
and left-handed Standard Model (SM) neutrinos are first described by the low
energy effective field theory (LEFT) below the electroweak scale. We match them
to the chiral perturbation theory ($\chi$PT) at the chiral symmetry breaking
scale to calculate the branching fractions of Kaon semi-invisible decays and
match them up to the SM effective field theory (SMEFT) to constrain new physics
above the electroweak scale. In the framework of effective field theories, we
prove that the Grossman-Nir bound is valid for both dim-6 and dim-7 LEFT
operators, and the dim-6 vector and scalar operators dominantly contribute to
Kaon semi-invisible decays based on LEFT and chiral power counting rules. They
are induced by multiple dim-6 lepton-number-conserving operators and one dim-7
lepton-number-violating operator in the SMEFT, respectively. In the
lepton-number-conserving $s\to d$ transition, the $K\to \pi \nu \bar{\nu}$
decays provide the most sensitive probe for the operators with $\tau\tau$
component and point to a corresponding new physics scale of $\Lambda_{\rm NP}
\in[47~\text{TeV},~72~\text{TeV}]$ associated with a single effective
coefficient. The lepton-number-violating operator can also explain the observed
$K\to\pi\nu\bar{\nu}$ discrepancy with the SM prediction within a narrow range
$\Lambda_{\rm NP}\in [19.4~\text{TeV},~21.5~\text{TeV}]$, which is consistent
with constraints from Kaon invisible decays.
