Implications of the recent measurement of pure annihilation $B_s \to \pi^+ \pi^-$ decays in QCD factorization

The CDF 3.7 sigma evidence of pure annihilation $B_s \to \pi^+ \pi^-$ decays, if confirmed, would imply a large annihilation scenario in the QCD factorization approach. This is somewhat unexpected as the large annihilation scenario was disfavored in previous studies. In this paper we reinvestigate the role of annihilation topology in QCD factorization. We find that it is not easy to reach the CDF central value of $B_s \to \pi^+ \pi^-$ decays when other decay channels are considered. Our analysis also reveals that, for well-measured charmless B decays into two final pseudoscalar mesons, the QCD factorization predictions with large annihilation parameters show good agreement with the experimental data except $B_s \to K^+ K^-$ and $B_d \to K^0 \bar{K}^0$ decays. Though other possibilities can not be excluded, this may indicate that the SU(3) flavor symmetry breaking should be taken into account for the annihilation topology. In addition, there are different annihilation topologies, so that somewhat different annihilation parameters may be chosen for different final states and different annihilation topologies. If so, the predictive power of the QCD factorization method may be rather limited for many decay channels.