Charmless Hardronic Decays $B_u \to V V $: Angular Distributions, Direct CP Violation and Determination of the Unitary Triangle

Abstract

Two-body charmless nonleptonic decays of $B_u \to V V$ are studied within the generalized factorization approach using a recent calculation of the effective Wilson coefficients $c^{eff}_i$, which are not only renormalization-scale and -scheme independent but also gauge invariant and infrared finite. After making a universal ansatz for the nonfactorizable contributions, we parametrize these effects in terms of $N_c^{eff}(LL)$ and $N_c^{eff}(LR)$, the effective numbers of colors arising from $(V-A)(V-A)$ and $(V-A)(V+A)$ four-quark operators, respectively. Three different schemes for these contributions are considered: (i) the naive factorization, (ii) the large-N_c improved factorization, and (iii) our preferred choice: $(N_c^{eff}(LL), N_c^{eff}(RR))=(2,5)$. We present the full angular distribution of all charmless $B_u \to V V$ decays in both transversity and helicity frames.Direct CP violation in these normalized angular correlation coefficients is not negligible in $B^-_u \to K^{*-} ρ^0, K^{*-} ω$, and direct CP violation in the partial rate difference for $B^-_u \to K^{*-} ω, K^{*-} ρ$ and $ρ^- ω$ can be as large as 45%, 25%, -10%, respectively. Due to the sizable QCD penguin contributions in $ρ^- ω$, the determination of the unitary triangle $α$ via this decay mode is more promising than via $ρ^- ρ^0$. It is also encouraging to determine the unitary triangle $γ$ through $B^-_u \to K^{*-} ρ$ because of $N_c$-insensitivity and the not-so-small tree contribution. The impacts of a negative $ρ$ on the branching ratios and CP violation are studied. We also comment on the theoretical uncertainties and their possible impacts.