Dissecting the $\Delta I= 1/2$ rule at large $N_c$

We study the scaling of kaon decay amplitudes with the number of colours, $N_c$, in a theory with four degenerate flavours, $N_f=4$. In this scenario, two current-current operators, $Q^\pm$, mediate $\Delta S=1$ transitions, such as the two isospin amplitudes of non-leptonic kaon decays for $K\to (\pi\pi)_{I=0,2}$, $A_0$ and $A_2$. In particular, we concentrate on the simpler $K\to\pi$ amplitudes, $A^\pm$, mediated by these two operators. A diagrammatic analysis of the large-$N_c$ scaling of these observables is presented, which demonstrates the anticorrelation of the leading ${\mathcal O}(1/N_c)$ and ${\mathcal O}(N_f/N_c^2)$ corrections in both amplitudes. Using our new $N_f=4$ and previous quenched data, we confirm this expectation and show that these corrections are $naturally$ large and may be at the origin of the $\Delta I=1/2$ rule. The evidence for the latter is indirect, based on the matching of the amplitudes to their prediction in Chiral Perturbation Theory, from which the LO low-energy couplings of the chiral weak Hamiltonian, $g^\pm$, can be determined. A NLO estimate of the $K \to (\pi\pi)_{I=0,2}$ isospin amplitudes can then be derived, which is in good agreement with the experimental value.