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A Combined Fit on the Annihilation Corrections in $B_{u,d,s}$ $\to$ $PP$ Decays Within QCDF
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} -> pi pi, pi K, K K decays, a comprehensive fit on the phenomenological parameters X_A^{i,f} (or rho_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) flavor 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} decays, respectively. So X_{A,s}^i = X_{A,d}^i is a good approximation by now. (2) In principle, parameter X_{A}^f which 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 (rho_A^i, phi_A^i) = (2.98^+1.12_-0.86,-105^+34_-24) and (rho_A^f, phi_A^f) = (1.18^+0.20_-0.23,-40^+11_-8), the other is (rho_A^i, phi_A^i) = (2.97^+1.19_-0.90,-105^+32_-24) and (rho_A^f, phi_A^f) = (2.80^+0.25_-0.21,165^+4_-3). Obviously, nonfactorization annihilation parameter X_A^i is generally unequal to factorization annihilation parameter X_A^f, which differ from the traditional treatment. With the fitted parameters, all results for observables of B_{u,d,s} ->pi pi, pi K, K K decays are in good agreement with experimental data.
