Confronting hadronic tau decays with non-leptonic kaon decays

Abstract

In the chiral limit, the $D=6$ contribution to the Operator Product Expansion (OPE) of the $\mathrm{VV}-\mathrm{AA}$ correlator of quark currents only depends on two vacuum condensates, which can be related to hadronic matrix elements associated to CP violation in non-leptonic kaon decays. We use those relations to determine $⟨(\pi \pi {)}_{I=2}|{\mathcal{Q}}_8|K⟩$, using the updated ALEPH spectral functions. Alternatively, we use those relations in the opposite direction. Taking the values of the matrix elements from the lattice to obtain the $D=6$ vacuum elements provides a new short-distance constraint which allows for an inclusive determination of $f_{\pi }$ and an updated value for the $D=8$ condensate.

Comments: 6 pages, 3 figures, talk given at 21th International Conference in Quantum Chromodynamics (QCD 18), 2 - 6 July 2018, Montpellier - FR