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 $O(\Lambda _{\rm QCD}\left/m_Q\right.)$ contribution is found to be very important, the correction to the decay amplitudes of $B^-\to \gamma e\bar{\nu}$ is about $20\% - 30\%$. For $D$ mesons, the $O(\Lambda _{\rm QCD}\left/m_Q\right.)$ contributions are more important.