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

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.