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

Abstract

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 $μ_f$ and of the non-perturbative parameters $ξ^{NF}_1 (m_b)$ and $ξ^{NF}_2 (m_b)$ which are supposed to measure non-factorizable contributions to hadronic matrix elements with $ξ^{NF}_i (μ_f) = 0$. We emphasize that both $μ_f$ and $ξ^{NF}_i (m_b)$ are renormalization scheme dependent and we demonstrate analytically and numerically that for any chosen scale $μ_f=\ord(m_b)$ it is possible to find a renormalization scheme for which $ξ^{NF}_1 (μ_f) = ξ^{NF}_2 (μ_f)=0$. 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 $μ$ and renormalization scheme independent suffer from gauge dependences and infrared dependences.