Dissecting the Δ 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^± , mediate Δ S=1 transitions, such as the two isospin amplitudes of non-leptonic kaon decays for K→ (ππ )_I=0,2 , A_0 and A_2 . In particular, we concentrate on the simpler K→π amplitudes, A^± , 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 𝒪(1/N_c) and 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 Δ 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^± , can be determined. A NLO estimate of the K → (ππ )_I=0,2 isospin amplitudes can then be derived, which is in good agreement with the experimental value.