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

Effective field theory approach to lepton number violating decays  $K^\pm\rightarrow \pi^\mp l^{\pm}_\alpha l^{\pm}_\beta$: long-distance  contribution