Double penguin diagrams and the contribution of vector-meson-like states to the decaysB→K*γ,B→ργ

Using perturbative QCD, the contribution at the leading twist, leading ${\mathrm{\alpha }}_{\mathit{s}}$ level, of charm and up quark loops to the decays B→${\mathit{K}}^{\mathrm{*}}$γ and B→ργ is presented. In the case of B→ργ, the relative importance of these contributions depends upon the unknown CKM matrix elements ${\mathit{V}}_{\mathit{b}\mathit{u}}$ and ${\mathit{V}}_{\mathit{t}\mathit{d}}$. Assuming that the ratio r=${\mathit{V}}_{\mathit{b}\mathit{c}}$ ${\mathit{V}}_{\mathit{c}\mathit{d}}^{\mathrm{*}}$/${\mathit{V}}_{\mathit{b}\mathit{t}}$ ${\mathit{V}}_{\mathit{t}\mathit{d}}^{\mathrm{*}}$ is bounded between -2.25≤r≤-0.5, as suggested by the Particle Data Group, the error in extracting ‖${\mathit{V}}_{\mathit{t}\mathit{d}}$/${\mathit{V}}_{\mathit{t}\mathit{s}}$‖ by these decays is estimated. © 1996 The American Physical Society.