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| Motivated by the possible large annihilation contributions implied by recent CDF and LHCb measurements on nonleptonic annihilation $B$-meson decays, and the refined experimental measurements on hadronic $B$-meson decays, we study the strength of annihilation contributions within QCD factorization (QCDF) in this paper. With the available measurements of two-body $B_{u,d,s}$ ${ o}$ ${pi}{pi}$, ${pi}K$, $KK$ decays, a comprehensive fit on the phenomenological parameters $X_{A}^{i,f}$ (or ${
ho}_A^{i,f}$ and ${phi}_A^{i,f}$) which are used to parameterize the endpoint singularity in annihilation amplitudes is performed with the statistical $chi^2$ approach. It is found that (1) flavour symmetry breaking effects are hardly to be distinguished between $X_{A,s}^i$ and $X_{A,d}^i$ due to the large experimental errors and theoretical uncertainties, where $X_{A,s}^i$ and $X_{A,d}^i$ are related to the nonfactorization annihilation contributions in $B_{s}$ and $B_{u,d}$ decay, respectively. So $X_{A,s}^i$ ${simeq}$ $X_{A,d}^i$ is a good approximation by now. (2) In principle, parameter $X_{A}^f$ who is related to the factorization annihilation contributions and independent of the initial state can be regarded as the same variable for $B_{u,d,s}$ decays. (3) Numerically, two solutions are found, one is $(
ho_A^i, phi_A^i[^{circ}])$ $=$ $(3.00^{+1.25}_{-0.96},-104^{+36}_{-26})$ and $(
ho_A^f, phi_A^f[^{circ}])$ $=$ $(1.18^{+0.24}_{-0.20},-42^{+13}_{-5})$, the other is $(
ho_A^i, phi_A^i[^{circ}])$ $=$ $(2.85^{+1.48}_{-0.79},-100^{+33}_{-28})$ and $(
ho_A^f, phi_A^f[^{circ}])$ $=$ $(2.83^{+0.21}_{-0.21},165^{+3}_{-4})$. Obviously, $(
ho_{A}^{i},phi_{A}^{i})$ $
eq$ $(
ho_{A}^{f},phi_{A}^{f})$, i.e., nonfactorization annihilation parameter $X_{A}^{i}$ is generally unequal to factorization annihilation parameter $X_{A}^{f}$, which differs from the traditional treatment. With the fitted parameters, all results for observables of $B_{u,d,s}$ $ o$ ${pi}{pi}$, ${pi}K$, $KK$ decays are in good agreement with experimental data. |
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| Keywords:B-meson, annihilation topology, QCD factorization |
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