New Analysis of the ∆ I = 1 / 2 Rule in Kaon Decays and the B̂ K Parameter

We present a new analysis of the ∆I = 1/2 rule in K → ππ decays and the B̂K parameter. We use the 1/Nc expansion within the effective chiral lagrangian for pseudoscalar mesons and compute the hadronic matrix elements at leading and nextto-leading order in the chiral and the 1/Nc expansions. Numerically, our calculation reproduces the dominant ∆I = 1/2 K → ππ amplitude. Our result depends only moderately on the choice of the cutoff scale in the chiral loops. The ∆I = 3/2 amplitude emerges sufficiently suppressed but shows a significant dependence on the cutoff. The B̂K parameter turns out to be smaller than the value previously obtained in the 1/Nc approach. It also shows a significant dependence on the choice of the cutoff scale. Our results indicate that corrections from higher order terms and/or higher resonances are large for the ∆I = 3/2 K → ππ amplitude and the (|∆S| = 2) K0 − K̄0 transition amplitude. PACS numbers: 12.39.Fe, 13.25.Es E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]