Weak annihilation in the rare radiative $B\to \rho\gamma$ decay

Abstract

The ampitude of the $B\to\rho\gamma$ decay induced by the flavour-changing neutral currents contains the penguin contribution and the weak-annihilation contribution generated by the 4-quark operators in the effective Hamiltonian. The penguin contribution is known quite well. We analyse the weak-annihilation which is suppressed by the heavy-quark mass compared to the penguin contribution. In the factorization approximation, the weak annihilation amplitude is represented in terms of the leptonic decay constants and the meson-photon matrix elements of the weak currents. The latter contain the $B\gamma$, $\rho\gamma$ transition form factors and contact terms determined by the equations of motion. We calculate the $B\gamma$ and $\rho\gamma$ form factors within the relativistic dispersion approach and obtain numerical estimates for the weak annihilation amplitude.