ar X iv : h ep - p h / 01 12 02 2 v 1 3 D ec 2 00 1 What can we learn from B → ρ ( ω ) K decays ?

We show that in the PQCD approach the B → ρ(ω)K decays display not only the dynamical penguin enhancement, the mechanism responsible for the large branching ratios (BRs) of B → φK decays, but also the importance of annihilation contributions. We find that the CP asymmetries (CPAs) of B → ρK, B → ρ(ω)K (class I) are all over 50%. PACS number(s): 12.38.BX, 12.38.QK, 13.25.HW The study of charmless B meson decays has an enormous progress, since many modes, such as B → ππ and B → πK, were measured by CLEO [1], as well as by BABAR [2] and BELLE [3]. Via the search of B decays, we can not only test the origin of CP violation in standard model (SM), which is the consequence of the Cabibbo-Kobayashi-Maskawa (CKM ) quark-mixing matrix [4], but also verify various QCD approaches, proposed to deal with hadronic effects in exclusive decays. Recently, the BRs of the modes involving a vector (V) meson, such as B → ρ(ω)π measured by CLEO [5] and B → φK by the B factories, B → φK = (5.5 −1.8 ± 0.6)× 10 (CLEO [6]), = (7.7 −1.4 ± 0.8)× 10 (BABAR [7]), = (11.2 −2.0 ± 1.4)× 10 (BELLE [8]) have been also observed. Although the present results of B → φK among the B factories do not match well each other, more accurate measurements will be obtained in the near future. Nevertheless, the theoretical predictions are not consistent either; for instance, the modified perturbative QCD (MPQCD) predicts (10.2 −2.1) × 10 [9, 10], but QCD factorization (QCDF) [11] gives (4.3 −1.4)× 10 [12]. Therefore, an interesting question is raised: besides B → φK decays, can we find other processes, that distinguish which QCD approach is proper and correct? or what processes can support either of them? Following the speculation of B → V P decays, P being a pseudoscalar meson, we find that it is important to investigate B → V K decays (V ′ = ρ and ω), because many properties, such as CPAs of over 50% and essential annihilation contributions, are remarkable. Although the V ′ meson carries spin degrees of freedom, only the longitudinal component contributes. Thus, in order to understand the properties of B → V K, it is useful to combine the analysis of B → πK and B → φK decays, all of which are described by the b → s transition. Note that the pion can be regarded as the longitudinal component of a ρ meson. The effective Hamiltonian for decays with the b → s transition is given by