SU(3) analysis of four-quark operators: K → ππ 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 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 = $$ \frac{1}{2} $$ 1 2 enhancement, confirming old suggestions about its underlying dynamical origin.

Violations of quark-hadron duality in low-energy determinations of αs

“…In fact, the relatively large oscillations of the spectral function at s 0 m 2 τ have a very minor numerical role in the integrals A ω J (s 0 ). Additionally, as it is well-known in the QCD literature [27,29,60,[85][86][87][88], taking weight functions that vanish at s 0 (pinched weights), one is then further minimizing the numerical impact of the unwanted DV effects. Updated ALEPH spectral functions for the V , A and V + A channels [39].…”

The Euclidean Adler Function and its Interplay with $Δα^{\mathrm{had}}_{\mathrm{QED}}$ and $α_s$

“…This is the case of the two-point function of a left-handed and a right-handed currents (the V V − AA correlator), which is identically zero to all perturbative orders in α s but receives non-zero contributions from D ≥ 6 vacuum condensates that are order parameters of the chiral symmetry breaking. The sizes of the leading power corrections to this correlator are well known, since they can be directly extracted from the τ decay data [100,101]. Since there is no reason to neglect them, neither in the vector correlator nor in the axial one, it is then a must to incorporate power corrections for a complete description of the OPE-based Adler function.…”

Violations of Quark-Hadron Duality in Low-Energy Determinations of $α_s$

“…In fact, the relatively large oscillations of the spectral function at s 0 m 2 τ have a very minor numerical role in the integrals A ω J (s 0 ). Additionally, as it is well-known in the QCD literature [22,24,53,[75][76][77][78], taking weight functions that vanish at s 0 (pinched weights), one is then further minimizing the numerical impact of the unwanted DV effects.…”