Confronting hadronic tau decays with non-leptonic kaon decays

TL;DR

This paper links hadronic tau decays and non-leptonic kaon decays through the Operator Product Expansion, using spectral functions and lattice data to refine vacuum condensate values and related CP violation matrix elements.

Contribution

It establishes a novel connection between tau decay spectral functions and kaon decay matrix elements, providing new constraints on vacuum condensates and decay constants.

Findings

Determined the matrix element  for  decays using spectral functions.

Provided updated values for the D=8 condensate.

Derived a new short-distance constraint on vacuum condensates.

Abstract

In the chiral limit, the $D=6$ contribution to the Operator Product Expansion (OPE) of the $\mathrm{VV-AA}$ 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 $\langle(\pi\pi)_{I=2}|\mathcal{Q}_{8}|K\rangle$, 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.