Factorization of Radiative Leptonic Decays of $B^{-}$ and $D^{-}$ Mesons

In this work, we study the factorization of the radiative leptonic decays of $B^{-}$ and $D^{-}$ mesons, the contributions of the order $O(\Lambda _{\rm QCD}\left/m_Q\right.)$ are taken into account. The factorization is proved to be valid explicitly at the order $O(\alpha _s\Lambda _{\rm QCD}\left/m_Q\right.)$. The hard kernel is obtained. The numerical results are calculated using the wave-function obtained in relativistic potential model. The $O({\mathrm{\Lambda }}_{\mathrm{Q}\mathrm{C}\mathrm{D}}/m_Q)$ contribution is found to be very important, the correction to the decay amplitudes of $B^{-}\to \gamma e\bar{\nu }$ is about $20\mathrm{\%}-30\mathrm{\%}$. For $D$ mesons, the $O(\Lambda _{\rm QCD}\left/m_Q\right.)$ contributions are more important.