  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
$\varepsilon'/\varepsilon$, 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.

High Energy Physics - Phenomenology;High Energy Physics - Lattice