Note on Tests of the Factorization Hypothesis and the Determination of Meson Decay Constants

We discuss various tests of the factorization hypothesis making use of the close relationship between semi-leptonic and factorized nonleptonic decay amplitudes. It is pointed out that factorization leads to truely model-independent predictions for the ratio of nonleptonic to semi-leptonic decay rates, if in the nonleptonic decay a spin one meson of arbitrary mass or a pion take the place of the lepton pair. Where the decay constants of those mesons are known, these predictions represent ideal tests of the factorization hypothesis. In other cases they may be used to extract the decay constants. Currently available data on the decays $\bar B^0 \to D^+\pi^-,\, D^{*+}\pi^-,\, D^+\varrho^-,\, D^{*+}\varrho^-$ are shown to be in excellent agreement with the factorization results. A weighted average of the four independent values for the QCD coefficient $a_1$ extracted from the data gives $a_1=1.15\pm 0.06$ suggesting that it may be equal to the Wilson coefficient $c_1(\mu)$ evaluated at the scale $\mu = m_b$.