SU(3) analysis of four-quark operators: $K\to \pi \pi$ and vacuum matrix elements

Abstract

Hadronic matrix elements of local four-quark operators play a central role in non-leptonic kaon decays, while vacuum matrix elements involving the same kind of operators appear in inclusive dispersion relations, such as those relevant in $\tau$-decay analyses. Using an $SU(3{)}_L\otimes SU(3{)}_R$ decomposition of the operators, we derive generic relations between these matrix elements, extending well-known results that link observables in the two different sectors. Two relevant phenomenological applications are presented. First, we determine the electroweak-penguin contribution to the kaon CP-violating ratio ${\epsilon }^{\prime }/\epsilon$, using the measured hadronic spectral functions in $\tau$ decay. Second, we fit our $SU(3)$ dynamical parameters to the most recent lattice data on $K\to \pi \pi$ matrix elements. The comparison of this numerical fit with results from previous analytical approaches provides an interesting anatomy of the $\Delta I=\frac{1}{2}$ enhancement, confirming old suggestions about its underlying dynamical origin.

46 pages, 7 figures. Published version

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