On the Determination of Nonleptonic Kaon Decays from $K\to\pi$ Matrix Elements

TL;DR

This paper derives complete $O(p^4)$ expressions for $K o ext{pi}$ and $K o 0$ matrix elements in Chiral Perturbation Theory, crucial for accurate lattice QCD calculations of nonleptonic kaon decays.

Contribution

It provides the full $O(p^4)$ corrections for these matrix elements in partially quenched QCD, including the role of the $ ext{eta}'$ meson and numerical estimates of chiral logs.

Findings

$O(p^4)$ corrections are significant for lattice QCD.

Chiral logarithms can be numerically large.

The role of the $ ext{eta}'$ meson is clarified.

Abstract

The coupling constants of the order $p^2$ low-energy weak effective lagrangian can be determined from the $K\to\pi$ and $K\to 0$ weak matrix elements, choosing degenerate quark masses for the first of these. However, for typical values of quark masses in Lattice QCD computations, next-to-leading $O(p^4)$ corrections are too large to be ignored, and will need to be included in future analyses. Here we provide the complete $O(p^4)$ expressions for these matrix elements obtained from Chiral Perturbation Theory, valid for partially quenched QCD with $N$ degenerate sea quarks. Quenched QCD corresponds to the special case N=0. We also discuss the role of the $\eta'$ meson in some detail, and we give numerical examples of the size of chiral logarithms.