Flavor dependence of annihilation parameters in QCD factorization

For $B_{d,s} \to \pi^\mp K^\pm$ and $K^{(\ast)} K^{(\ast)}$ decays, the flavor symmetry breaking effects may be particularly small since the final state interactions should be the same between the corresponding $B_d$ and $B_s$ decays due to the charge conjugation symmetry of the final states. This is consistent with the newly measured direct CP asymmetry of $B_s \to \pi^+ K^-$. These decays are thus supposed to be important in testing the Standard Model and in probing new physics effects. However, the observation of pure annihilation decay $B_s \to \pi^+ \pi^-$ appears to imply a large annihilation scenario with $\rho_A \sim 3$, in contrast to the case of $\rho_A \sim 1$ in $B_{u,d}$ decays in the framework of QCD factorization. This seems to indicate unexpectedly large flavor symmetry breaking effects between the annihilation amplitudes of $B_s$ and $B_{u,d}$ decays. This apparent contradiction could be resolved by noticing that there is a priori no reason to justify the common practice of assuming the universality of annihilation parameters for different Dirac structures of effective operators. We then argue that, for $B_{d,s} \to \pi^\mp K^\pm$ decays, the flavor symmetry breaking effects of annihilation amplitudes have all been included in the initial state decay constants and are thus small. But the flavor symmetry breaking effects in $B_{d,s} \to K^{(\ast)} K^{(\ast)}$ decays are likely to be much larger, as part of the annihilation topologies of $B_s \to K K$ decay could be related to $B_s \to \pi^+ \pi^-$ decay. Therefore when new physics effects are searched for in these decay channels, care must be taken to consider the potentially large flavor symmetry breaking effects in more details.