Double Penguins and the Contribution of Vector Meson--like States to the Decays $B \to K^* \gamma, \, B \to \rho \gamma$

Abstract

Using perturbative QCD, the contribution at the leading twist, leading $\alpha_s$ level, of charm and up quark loops to the decays $B \rightarrow K^* \gamma$ and $B \rightarrow \rho \gamma$ is presented. In the case of $B \rightarrow \rho \gamma$, the relative importance of these contributions depend upon the unknown CKM matrix elements $V_{bu}$ and $V_{td}$. Assuming that the ratio $r = V_{bc}V^*_{cd}/V_{bt}V^*_{td}$ is bounded between $-2.25 \le r \le -.5$ as is suggested by the Particle Data Group, the error in extracting $ |V_{td}/V_{ts}| $ by these decays is estimated.