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

The large branching ratios for pure annihilation $\bar{B}_s^0$ $\to$ $π^+ π^-$ and $\bar{B}_d^0$ $\to$ $K^+ K^-$ 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 $B_{u,d}$ ${\to}$ $ππ$, $πK$ and $K \bar{K}$ decays, we do a global fit on the spectator scattering and annihilation parameters $X_H(ρ_H$, $ϕ_H)$, $X_A^i(ρ_A^{i},ϕ_A^{i})$ and $X_A^f(ρ_A^{f},ϕ_A^{f})$, 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 $(ρ_A^{i},ϕ_A^{i}[^{\circ}])=(2.88^{+1.52}_{-1.30},-103^{+33}_{-40})$ and $(ρ_A^{f},ϕ_A^{f}[^{\circ}])=(1.21^{+0.22}_{-0.25},-40^{+12}_{-8})$ at the $68\%$ confidence level, which are mainly demanded by resolving $πK$ puzzle and confirm the presupposition that $X_A^i\neq X_A^f$. In addition, correspondingly, the $B$-meson wave function parameter $λ_B$ 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 $B_{u,d}$ $\to$ $ππ$, $πK$ and $K \bar{K}$ 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.

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