2023 Impact factor 4.2
Particles and Fields
Eur. Phys. J. C 10, 271-292
DOI 10.1007/s100529900084

New analysis of the $\Delta I = 1/2$ rule in kaon decays
and the $\hat{B}_K$ parameter

T. Hambye1 - G.O. Köhler2 - P.H. Soldan2

1 INFN - Laboratori Nazionali di Frascati, P.O. Box 13, I-00044 Frascati, Italy (hambye@lnf.infn.it)
2 Institut für Physik, Universität Dortmund, D-44221 Dortmund, Germany (koehler, soldan@doom.physik.uni-dortmund.de)

Received: 22 February 1999 / Published online: 12 August 1999

We present a new analysis of the $\Delta I = 1/2$ rule in $K\rightarrow
\pi\pi$ decays and the $\hat{B}_K$ parameter. We use the $1/N_{\mathrm
{c}}$ expansion in the effective chiral lagrangian for the pseudoscalar mesons and compute the hadronic matrix elements at leading and next-to-leading order in the chiral and the $1/N_{\mathrm
{c}}$ expansions. Numerically, our calculation reproduces the dominant $\Delta I = 1/2$ $K\rightarrow
\pi\pi$ amplitude. Our result depends only moderately on the choice of the cutoff scale in the chiral loops. The $\Delta I = 3/2$ amplitude emerges sufficiently suppressed but shows a significant dependence on the cutoff. The $\hat{B}_K$ parameter turns out to be smaller than the value previously obtained in the $1/N_{\mathrm
{c}}$ 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 $\Delta I = 3/2$ $K\rightarrow
\pi\pi$amplitude and the $(\vert\Delta S\vert=2)$ K0-$\bar{K}^0$ transition amplitude.

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