Spectator Scattering and Annihilation Contributions as a Solution to the πK and ππ Puzzles within QCD Factorization Approach

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The large branching ratios for pure annihilation Bₛ⁰ → π+ π- and Bd⁰ → K+ K- decays reported by CDF and LHCb collaborations recently and the so-called πK and ππ puzzles indicate that spectator scattering and annihilation contributions are important to the penguin-dominated, color-suppressed tree dominated, and pure annihilation nonleptonic B decays. Combining the available experimental data for Bᵤ,d → ππ, πK and K K decays, we do a global fit on the spectator scattering and annihilation parameters Xₕ(ρₕ, ϕₕ), Xₐⁱ(ρₐⁱ,ϕₐⁱ) and Xₐᶠ(ρₐᶠ,ϕₐᶠ), which are used to parameterize the endpoint singularity in amplitudes of spectator scattering, nonfactorizable and factorizable annihilation topologies within the QCD factorization framework, in three scenarios for different purpose. Numerically, in scenario II, we get (ρₐⁱ,ϕₐⁱ[∘])=(2.88⁺¹.⁵²₋₁.₃₀,-103⁺³³₋₄₀) and (ρₐᶠ,ϕₐᶠ[∘])=(1.21⁺⁰.²²₋₀.₂₅,-40⁺¹²₋₈) at the 68% confidence level, which are mainly demanded by resolving πK puzzle and confirm the presupposition that Xₐⁱ≠ Xₐᶠ. In addition, correspondingly, the B-meson wave function parameter λB is also fitted to be 0.18⁺⁰.¹¹₋₀.₀₈ MeV, which plays an important role for resolving both πK and ππ puzzles. With the fitted parameters, the QCDF results for observables of Bᵤ,d → ππ, πK and K K decays are in good agreement with experimental measurements. Much more experimental and theoretical efforts are expected to understand the underlying QCD dynamics of spectator scattering and annihilation contributions.