Possible complex annihilation andB→KπdirectCPasymmetry

Abstract

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 ${m}_{b}$. In the limit of infinite heavy-quark mass, both the soft-collinear effective theory and the perturbative QCD approach based on the ${k}_{T}$ 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 $O(\ensuremath{\Lambda}/{m}_{b})$, with $\ensuremath{\Lambda}$ 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 ${B}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{\mp}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$ direct $CP$ asymmetry.