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

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 $$K\rightarrow (\pi \pi )_{I=0,2}$$, $$A_0$$ and $$A_2$$. In particular, we concentrate on the simpler $$K\rightarrow \pi $$ 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 $$K \rightarrow (\pi \pi )_{I=0,2}$$ isospin amplitudes can then be derived, which is in good agreement with the experimental value.