Final state interactions in [Formula: see text] decays: [Formula: see text] rule vs. [Formula: see text]

Dispersive effects from strong [Formula: see text] rescattering in the final state interaction (FSI) of weak [Formula: see text] decays are revisited with the goal to have a global view on their relative importance for the [Formula: see text] rule and the ratio [Formula: see text] 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 [Formula: see text] rule and the dominant density–density (four-quark) operators governing [Formula: see text] 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 [Formula: see text] rule but much less relevant for [Formula: see text] . The latter finding diminishes significantly hopes that improved calculations of [Formula: see text] would bring its SM prediction to agree with the experimental data, opening thereby an arena for important new physics contributions to this ratio.