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 τ-decay analyses. Using an SU(3)L ⊗ 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 ε′/ε, 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 → ππ matrix elements. The comparison of this numerical fit with results from previous analytical approaches provides an interesting anatomy of the ∆I = frac{1}{2} enhancement, confirming old suggestions about its underlying dynamical origin.
Highlights
Using an SU(3)L ⊗ 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
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
One runs perturbatively the EFT to energies as small as possible, so that all large short-distance logarithms can be reabsorbed into the computed Wilson coefficients, but the hadronic matrix elements of their associated operators must still be determined with non-perturbative methods
Summary
The covariant derivative DμU = ∂μU − irμU + iU μ includes auxiliary external left ( μ) and right (rμ) matrix-valued vector sources coupled to the quarks, which allow us to derive the low-energy realization of the QCD currents [36]. By taking functional derivatives with respect to the appropriate external sources in both terms of the equality, one finds the explicit low-energy expressions of the QCD quark currents. This dictionary will be exploited below to derive some useful relations among four-quark operators. The subsections compile them together, using a much simpler approach purely based on symmetry arguments
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