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

TL;DR

This paper uses SU(3) symmetry to relate four-quark operator matrix elements in kaon decays and vacuum processes, deriving new relations and applying them to phenomenology and lattice data analysis.

Contribution

It extends known SU(3) relations between hadronic and vacuum matrix elements and applies these to analyze CP violation and lattice results in kaon physics.

Findings

Determined electroweak penguin contribution to ε'/ε using τ decay data.

Fitted SU(3) parameters to lattice K→ππ matrix elements.

Confirmed the dynamical origin of the ΔI=1/2 enhancement.

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 $\varepsilon'/\varepsilon$, 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.