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This is the first paper of our systematic efforts on lepton number violating(LNV) hadronic decays in the effective field theory approach. These decaysprovide information complementary to popular nuclear neutrinolessdouble-$\beta$ ($0\nu\beta\beta$) decay in that they can probe LNV interactionsinvolving heavier quarks and charged leptons. We may call them hadronic$0\nu\beta\beta$ decays in short, though $\beta$ refers to all charged leptons.In this work we investigate the decays $K^\pm\rightarrow\pi^\mp l^{\pm}l^{\pm}$that arise from short-distance or contact interactions involving four quarkfields and two charged lepton fields, which have canonical dimension nine(dim-9) at leading order in low energy effective field theory (LEFT). We make acomplete analysis on the basis of all dim-9 operators that violate leptonnumber by two units, and compute their one-loop QCD renormalization effects. Wematch these effective interactions in LEFT to those in chiral perturbationtheory ($\chi$PT) for pseudoscalar mesons, and determine the resulting hadroniclow energy constants (LECs) by chiral symmetry and lattice results in theliterature. The obtained decay rate is general in that all physics at and abovethe electroweak scale is completely parameterized by the relevant Wilsoncoefficients in LEFT and hadronic LECs in $\chi$PT. Assuming the standard modeleffective field theory (SMEFT) is the appropriate effective field theorybetween some new physics scale and the electroweak scale, we match our LEFTresults to SMEFT whose leading effective interactions arise from LNV dim-7operators. This connection to SMEFT simplifies significantly the interactionstructures entering in the kaon decays, and we employ the current experimentalbounds to set constraints on the relevant Wilson coefficients in SMEFT.

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