Vacuum Insertion Approximation and the $\Delta I=1/2$ rule:a lattice QCD test of the na\"ive factorization hypothesis for $K$, $D$, $B$ and static mesons
Motivated by a recent paper by the RBC-UKQCD Collaboration, which observes large violations of the na\"ive factorization hypothesis in $K \to \pi \pi$ decays, we study in this paper the accuracy of the Vacuum Insertion Approximation (VIA) for the matrix elements of the complete basis of four fermion $\Delta F=2$ operators. We perform a comparison between the matrix elements in QCD, evaluated on the lattice, and the VIA predictions. We also investigate the dependence on the external meson masses by computing matrix elements for $K$, $D_s$, $B_s$ and static mesons. In commonly used renormalization schemes, we find large violations of the VIA in particular for one of the two relevant Wick contractions in the kaon sector. These deviations, however, decrease significantly as the meson mass increases and the VIA predictions turn out to be rather well verified for B-meson matrix elements and, even better, in the infinite mass limit.