Polarizations of two vector mesons in B decays

Inspired by the small longitudinal polarizations (LPs) of $B\to K^* ϕ$ decays observed by BELLE and BABAR, we revise the theoretical uncertainties of perturbative QCD approach for determining hard scales of B decays, we find that the LPs of $B\to K^* ϕ$ could approach to 60% while the branching ratios (BRs) could be around $9\times 10^{-6}$. In addition, we also study the BRs and polarization fractions of $B\to ρ(ω) ρ(ω)$ and $B\to ρ(ω) K^*$ decays. For those tree dominant and color-allowed processes in $B\to ρ(ω) ρ(ω)$ decays, we get that the BRs of $(ρ^{+} ρ^{-}, ρ^{0} ρ^{+}, ωρ^{+})$ are $(23.06, 11.99, 14.78)\times 10^{-6}$ while their LPs are close to unity. Interestingly, due to significant tree contributions, we find that the BR(LP) of $ρ^{-} K^{*+}$ could be around $10.13 \times 10^{-6}(60$; and due to the tree and electroweak penguin, the BR(LP) of $ωK^{*+}$ could be around $5.67 \times 10^{-6}(61$.

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