Generalized Factorization in Non-leptonic Two-Body B-Decays: a Critical Look

We reanalyze critically the generalized factorization hypothesis in non-leptonic two-body B-decays discussed recently by several authors. In particular we address the determination of the factorization scale $\mu_f$ and of the non-perturbative parameters $\xi^{NF}_1 (m_b)$ and $\xi^{NF}_2 (m_b)$ which are supposed to measure non-factorizable contributions to hadronic matrix elements with $\xi^{NF}_i (\mu_f) = 0$. We emphasize that both $\mu_f$ and $\xi^{NF}_i (m_b)$ are renormalization scheme dependent and we demonstrate analytically and numerically that for any chosen scale $\mu_f=\ord(m_b)$ it is possible to find a renormalization scheme for which $\xi^{NF}_1 (\mu_f) = \xi^{NF}_2 (\mu_f)=0$. The existing data indicate that such "factorization schemes" differ from the commonly used schemes NDR and HV. Similarly we point out that the recent extractions of the effective number of colours $N^{eff}$ from two-body non-leptonic B-decays while $\mu$ and renormalization scheme independent suffer from gauge dependences and infrared dependences.