Dissecting the rule at large

Andrea Donini; Pilar Hernández; Carlos Pena; Fernando Romero-López

doi:10.1140/epjc/s10052-020-8192-3

EPJ

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

Andrea Donini¹, Pilar Hernández¹, Carlos Pena² and Fernando Romero-López¹^*

¹ IFIC (CSIC-UVEG), Edificio Institutos Investigación, Apt. 22085, 46071, Valencia, Spain
² Departamento de Física Teórica and Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, 28049, Madrid, Spain

Received: 30 March 2020
Accepted: 27 June 2020
Published online: 18 July 2020

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 $K\rightarrow (\pi \pi )_{I=0,2}$ , $$A_0$$

and

. 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.

15th European Conference on Atoms Molecules and Photons (ECAMP)
June 30 to July 4, 2025
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EPJ

Dissecting the rule at large

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