Possible complex annihilation and B→Kπ direct CP asymmetry

We point out that a strong phase from the penguin annihilation channel in B meson decays is sensitive to the treatment of subleading terms with physical b quark mass mb. In the limit of infinite heavy-quark mass, both the soft-collinear effective theory and the perturbative QCD approach based on the kT factorization theorem agree that the strong phase is suppressed. For finite heavy-quark mass, the two approaches predict different results, which are related to the treatment of subleading terms. This is illustrated by taking a toy model for soft-collinear effective theory, in which a small quantity, suppressed by O(Λ/mb), with Λ being a hadronic scale, is kept in the denominators of internal particle propagators. This model can generate a sizable strong phase to accommodate the data of the B0→K π± direct CP asymmetry.