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

Abstract

Using perturbative QCD, the contribution at the leading twist, leading ${\alpha }_s$ level, of charm and up quark loops to the decays $B\to K^{\ast }\gamma$ and $B\to \rho \gamma$ is presented. In the case of $B\to \rho \gamma$, the relative importance of these contributions depend upon the unknown CKM matrix elements $V_{bu}$ and $V_{td}$. Assuming that the ratio $r=V_{bc}{V}_{cd}^{\ast }/V_{bt}{V}_{td}^{\ast }$ 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.

Comments: 4 double column pages, revtex, 3 figures attached, 2 figures added indicating less optimism concerning the accuracy at which $|V_{td}/V_{ts}|$ can be extracted: to appear in PRD