Type: Article
Publication Date: 1999-10-08
Citations: 194
DOI: https://doi.org/10.1103/physrevd.60.094014
Two-body charmless nonleptonic decays of ${B}_{u}$ and ${B}_{d}$ mesons are studied within the framework of generalized factorization in which the effective Wilson coefficients ${c}_{i}^{\mathrm{eff}}$ are renormalization-scale and -scheme independent while factorization is applied to the tree-level hadronic matrix elements. Contrary to previous studies, our ${c}_{i}^{\mathrm{eff}}$ do not suffer from gauge and infrared problems. Nonfactorizable effects are parametrized in terms of ${N}_{c}^{\mathrm{eff}}(\mathrm{LL})$ and ${N}_{c}^{\mathrm{eff}}(\mathrm{LR}),$ the effective numbers of colors arising from $(V\ensuremath{-}A)(V\ensuremath{-}A)$ and $(V\ensuremath{-}A)(V+A)$ four-quark operators, respectively. Tree and penguin transitions are classified into six different classes. The data of ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\varphi}{K}^{\ensuremath{-}}$ clearly indicate that ${N}_{c}^{\mathrm{eff}}(\mathrm{LR})\ensuremath{\ne}{N}_{c}^{\mathrm{eff}}(\mathrm{LL}):$ The first measurement of the $\stackrel{\ensuremath{\rightarrow}}{b}u$ mode ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{\ensuremath{\pi}}^{\ensuremath{-}}$ and the experimental information on the tree-dominated mode ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\pi}}^{\ensuremath{-}}$ all imply that ${N}_{c}^{\mathrm{eff}}(\mathrm{LL})$ is less than 3, whereas the CLEO measurement of ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\varphi}{K}^{\ensuremath{-}}$ shows ${N}_{c}^{\mathrm{eff}}(\mathrm{LR})>3.$ For given input parameters, the prediction of $\mathcal{B}(\stackrel{\ensuremath{\rightarrow}}{B}{\ensuremath{\eta}}^{\ensuremath{'}}K)$ is largely improved by setting ${N}_{c}^{\mathrm{eff}}(\mathrm{LL})\ensuremath{\sim}2$ and ${N}_{c}^{\mathrm{eff}}(\mathrm{LR})>{N}_{c}^{\mathrm{eff}}(\mathrm{LL});$ in particular, the charm content of the ${\ensuremath{\eta}}^{\ensuremath{'}}$ contributes in the right direction. The decay rate of $\stackrel{\ensuremath{\rightarrow}}{B}\ensuremath{\varphi}{K}^{*}$ is very sensitive to the form-factor ratio ${A}_{2}{/A}_{1};$ the absence of $\stackrel{\ensuremath{\rightarrow}}{B}\ensuremath{\varphi}K$ events does not necessarily invalidate the factorization approach. If the branching ratio of ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\omega}{K}^{\ensuremath{-}}$ is experimentally found to be significantly larger than that of ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{K}^{\ensuremath{-}},$ we argue that inelastic final-state rescattering may account for the disparity between $\ensuremath{\omega}{K}^{\ensuremath{-}}$ and ${\ensuremath{\rho}}^{0}{K}^{\ensuremath{-}}.$ By contrast, if $\mathcal{B}{(B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{K}^{\ensuremath{-}})\ensuremath{\sim}\mathcal{B}{(B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\omega}{K}^{\ensuremath{-}})$ is observed, then W annihilation and/or spacelike penguin amplitudes could play a prominent role. The decay modes ${B}_{d}^{0}\ensuremath{\rightarrow}\ensuremath{\varphi}{\ensuremath{\pi}}^{0},\ensuremath{\varphi}\ensuremath{\eta},\ensuremath{\varphi}{\ensuremath{\eta}}^{\ensuremath{'}},\ensuremath{\varphi}{\ensuremath{\rho}}^{0},\ensuremath{\varphi}\ensuremath{\omega}{,B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\varphi}{\ensuremath{\pi}}^{\ensuremath{-}},\ensuremath{\varphi}{\ensuremath{\rho}}^{\ensuremath{-}}$ involving a vector meson \ensuremath{\varphi} are dominated by electroweak penguin amplitudes. We show that a unitarity angle \ensuremath{\gamma} larger than 90\ifmmode^\circ\else\textdegree\fi{} is helpful for explaining the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}},$ $\ensuremath{\pi}K,$ and ${\ensuremath{\eta}}^{\ensuremath{'}}K$ data. The relative magnitudes of tree, QCD penguin, and electroweak penguin amplitudes are tabulated for all charmless $\stackrel{\ensuremath{\rightarrow}}{B}PP,VP,VV$ decays. Our favored predictions for branching ratios are those for ${N}_{c}^{\mathrm{eff}}(\mathrm{LL})\ensuremath{\approx}2$ and ${N}_{c}^{\mathrm{eff}}(\mathrm{LR})\ensuremath{\sim}5.$