  This is a sequel to our recent work [1] in which we calculated the lepton
number violating (LNV) $K^\pm$ decays due to contact dimension-9 (dim-9)
quark-lepton effective interactions that are induced at a high energy scale. In
this work we investigate the long-distance contribution to the decays arising
from the exchange of a neutrino. These decays can probe LNV interactions
involving the second generation of fermions that are not reachable in nuclear
neutrinoless double-$\beta$ decays. Our study is completely formulated in the
framework of effective field theories (EFTs), from the standard model effective
field theory (SMEFT) through the low energy effective field theory (LEFT) to
chiral perturbation theory. We work to the first nontrivial orders in each
effective field theory, collect along the way the matching conditions and
renormalization group effects, and express the decay branching ratios in terms
of the Wilson coefficients associated with the dim-5 and dim-7 operators in
SMEFT. Our result is general in that it does not depend on dynamical details of
physics at a high scale that induce the effective interactions in SMEFT and in
that it does not appeal to any hadronic models. We find that the long-distance
contribution overwhelmingly dominates over the contact or short-distance one.
Assuming the new physics scale to be around a TeV, the branching ratios are
predicted to be below the current experimental upper bounds by several orders
of magnitude.
