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

Abstract

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.

arXiv comment12 pages, 7 figures. Minor changes

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Cited by in corpus (7)

• FLAG Review 2021
• FLAG Review 2024
• Direct CP violation and the $\Delta I=1/2$ rule in $K\to \pi \pi$ decay from the Standard Model
• The Large $N_c$ limit of QCD on the lattice
• SU(3) analysis of four-quark operators: $K\to \pi \pi$ and vacuum matrix elements
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