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

We study the scaling of kaon decay amplitudes with the number of colours, $$N_c$$Nc, in a theory with four degenerate flavours, $$N_f=4$$Nf=4. In this scenario, two current-current operators, $$Q^\pm $$Q±, mediate $$\Delta S=1$$ΔS=1 transitions, such as the two isospin amplitudes of non-leptonic kaon decays for $$K\rightarrow (\pi \pi )_{I=0,2}$$K→(ππ)I=0,2, $$A_0$$A0 and $$A_2$$A2. In particular, we concentrate on the simpler $$K\rightarrow \pi $$K→π amplitudes, $$A^\pm $$A±, mediated by these two operators. A diagrammatic analysis of the large-$$N_c$$Nc scaling of these observables is presented, which demonstrates the anticorrelation of the leading $${{\mathcal {O}}}(1/N_c)$$O(1/Nc) and $${{\mathcal {O}}}(N_f/N_c^2)$$O(Nf/Nc2) corrections in both amplitudes. Using our new $$N_f=4$$Nf=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 $$\Delta I=1/2$$ΔI=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, $$g^\pm $$g±, can be determined. A NLO estimate of the $$K \rightarrow (\pi \pi )_{I=0,2}$$K→(ππ)I=0,2 isospin amplitudes can then be derived, which is in good agreement with the experimental value.