Vacuum Insertion Approximation and the ΔI=1/2ΔI=1/2 rule: A lattice QCD test of the naïve 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 naïve factorization hypothesis in K→ππK→ππ 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 ΔF=2ΔF=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  , DsDs, BsBs 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.