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

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 $τ$-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 $τ$ decay. Second, we fit our $SU(3)$ dynamical parameters to the most recent lattice data on $K\toππ$ matrix elements. The comparison of this numerical fit with results from previous analytical approaches provides an interesting anatomy of the $ΔI = \frac{1}{2}$ enhancement, confirming old suggestions about its underlying dynamical origin.