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Penguin Pollution in $B\to J/ψV$ Decays and Impact on the Extraction of the $B_s-\bar B_s$ mixing phase

We formulate the most-general time-dependent decay distributions of $B_s\to J/ψ(\to l^+l^-) ϕ(\to K^+K^-)$ in which the direct CP violation is explicitly incorporated. We then investigate the $B \to J/ψV$ decays in the perturbative QCD approach where $V$ is a light vector meson. Apart from the leading-order factorizable contributions, we also take into account various QCD corrections and the hard-spectator diagrams. With the inclusion of these sizable corrections, our theoretical results for CP-averaged branching ratios, polarization fractions, CP-violating asymmetries, and relative phases are in good consistency with the available data. Based on the global agreement, we further explore the penguin contributions and point out that the $ϕ_s$ extracted from $B_s\to J/ψϕ$ can be shifted away by ${\cal O}(10^{-3})$.