Estimates of the Weak Annihilation Contributions to the Decays $B\to \rho +\gamma$ and $B\to \omega +\gamma$

The dominant long-distance contributions to the exclusive radiative decays $B \to \rho (\omega) + \gamma$ involve photon emission from the light quarks at large distances, which cannot be treated perturbatively. We point out that this emission can be described in a theoretically consistent way as the magnetic excitation of quarks in the QCD vacuum and estimate the corresponding parity-conserving and parity-violating amplitudes using the light-cone QCD sum rule approach. These are then combined with the corresponding short-distance contribution from the magnetic moment operator in the same approach, derived earlier, to estimate the decay rates $\mathrm{\Gamma }(B\to \rho (\omega )+\gamma )$. The implications of this result for the determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements in radiative $B$ decays are worked out.