Final state interactions in K→ ππ decays: Δ I= 1 / 2 rule vs. ε^′/ ε

Dispersive effects from strong ππ rescattering in the final state interaction (FSI) of weak K→ ππ decays are revisited with the goal to have a global view on their relative importance for the Δ I= 1 / 2 rule and the ratio ε^′/ ε in the standard model (SM). We point out that this goal cannot be reached within a pure effective (meson) field approach like chiral perturbation theory in which the dominant current–current operators governing the Δ I= 1 / 2 rule and the dominant density–density (four-quark) operators governing ε^′/ ε cannot be disentangled from each other. But in the context of a dual QCD approach, which includes both long-distance dynamics and the UV completion, that is, QCD at short-distance scales, such a distinction is possible. We find then that beyond the strict large N limit, N being the number of colours, FSIs are likely to be important for the Δ I= 1 / 2 rule but much less relevant for ε^′/ ε. The latter finding diminishes significantly hopes that improved calculations of ε^′/ ε would bring its SM prediction to agree with the experimental data, opening thereby an arena for important new physics contributions to this ratio.