Dissecting the Δ I= 1 / 2 rule at large Nc

Abstract

We study the scaling of kaon decay amplitudes with the number of colours, N, in a theory with four degenerate flavours, N= 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) , A and A. In particular, we concentrate on the simpler K→ π amplitudes, A, mediated by these two operators. A diagrammatic analysis of the large-N scaling of these observables is presented, which demonstrates the anticorrelation of the leading O(1 / N) and O(Nf/Nc2) corrections in both amplitudes. Using our new N= 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) isospin amplitudes can then be derived, which is in good agreement with the experimental value.