Three-scale factorization theorem and effective field theory: Analysis of nonleptonic heavy meson decays

We develop a perturbative QCD factorization theorem which is compatible with effective field theory. The factorization involves three scales: an infrared cutoff of order ${\Lambda }_{\mathrm{QCD}},$ a hard scale of the order of the $B$ meson mass, and an ultraviolet cutoff of the order of the $W$ boson mass. Our approach is renormalization-group invariant and moderates the scale-dependent problem in effective field theory. Employing this formalism with nonfactorizable contributions included, we clarify the controversy over the BSW parameters $a_2{/a}_1$ for charm and bottom decays.