Final State Interactions in $K\to \pi \pi$ Decays: $\mathrm{\Delta }I=1\mathrm{/}2$ Rule vs. ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$

Dispersive effects from strong $\pi \pi$ rescattering in the final state (FSI) of weak $K\to \pi \pi$ decays are revisited with the goal to have a global view on their {\it relative} importance for the $\mathrm{\Delta }I=1\mathrm{/}2$ rule and the ratio ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ 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 $\mathrm{\Delta }I=1\mathrm{/}2$ rule and the dominant density-density (four-quark) operators governing ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ 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, FSI are likely to be important for the $\mathrm{\Delta }I=1\mathrm{/}2$ rule but much less relevant for ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$. The latter finding diminishes significantly hopes that improved calculations of ${\epsilon }^{\mathrm{\prime }}\mathrm{/}\epsilon$ would bring its SM prediction to agree with the experimental data, opening thereby an arena for important new physics contributions to this ratio.