We discuss the application of the MPSTV non-perturbative method cite{NPM} to the operators relevant to kaon decays. This enables us to reappraise the long-standing question of the $Delta I=1/2$ rule, which involves power-divergent subtractions that cannot be evaluated in perturbation theory. We also study the mixing with dimension-six operators and discuss its implications to the chiral behaviour of the $B_K$ parameter.
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