Branching ratio and direct CP-violating rate asymmetry of the rare decays $B\to K^{\ast }\gamma$ and $B\to \rho \gamma$

We calculate CP-violating rate asymmetries in the rare radiative decays $B^{\pm }\to K^{\ast \pm }\gamma$ and $B^{\pm }\to {\rho }^{\pm }\gamma$. They arise because of the interference between leading-order penguin amplitudes and one-gluon corrections with absorptive phases, and provide unambiguous evidence for direct CP violation. Complementing earlier studies, we also investigate gluon exchange with the `spectator’ quark. The bound state effects in the exclusive matrix elements are taken into account by a covariant model, which yields a branching ratio $BR(B\rightarrow K^\ast \gamma) = (4-5)\times 10^{-5}$ in good agreement with the observed value. The bound state effects increase the CP asymmetry, which is of order $1\mathrm{\%}$ in the channel $B\to K^{\ast }\gamma$ and $15\mathrm{\%}$ for $B\to \rho \gamma$.