CPviolation in inclusive and exclusive charmlessBdecays

Abstract

Absorptive parts in penguin amplitudes lead to $\mathrm{CP}$-violating partial rate asymmetries in $B$-meson decays. A careful study of the inclusive process reveals a subtlety in the implementation of this mechanism. Because of (1) the smallness of $\frac{V_{\mathrm{ub}}}{V_{\mathrm{cb}}}$, (2) the fact that ${\alpha }_s\left(m_b\right)$ is not too small, and (3) the kinematic suppression of $c\overline{c}$ absorptive contribution, $O\left({\alpha }_s^2\right)$ effects have to be kept consistently. We find a strong correlation between the $\mathrm{CP}$ rate asymmetries of the $b\to \mathrm{su}\overline{u}$ mode and the purely loop-induced $b\to \mathrm{sd}\overline{d} \mathrm{and} \mathrm{ss}\overline{s}$ modes, which is a consequence of the constraints of $\mathrm{CPT}$ invariance. The semi-inclusive charmless $b\to \mathrm{su}\overline{u}$ decay rate asymmetry is therefore negligibly small. With the (semi-)inclusive result as a guide, we estimate the rates and $\mathrm{CP}$ asymmetries for several exclusive modes. Keeping only the perturbative, penguin-induced absorptive part, one finds that the modes due to $b\to \mathrm{su}\overline{u}$ (e.g., $B^{-}\to K^{-}{\pi }^0$) could have $\mathrm{CP}$ rate asymmetries opposite in sign with respect to pure penguin modes (such as $B^{-}\to K^{-}\phi$), contrary to naive expectations. Charmless $b\to d$ transitions are also discussed. We emphasize that exclusive $b\to d$ modes (such as $B^{-}\to K^{-}{K}_S$) may be more promising for $\mathrm{CP}$ studies within the standard model. Many uncertainties and problems related to penguin processes are discussed and assessed in detail.