Factorization of Radiative Leptonic Decays of $B^-$ and $D^-$ Mesons Including the Soft Photon Region

In this work, we study the radiative leptonic decays of $B^-$ and $D^-$ mesons using factorization approach. Factorization is proved to be valid explicitly at 1-loop level at any order of $O(Lambda_{rm QCD}left/m_Qright.)$. We consider the contribution in the soft photon region that $E_{gamma} sim left. Lambda^2_{rm QCD} /right. m_Q$. The numerical results shows that, the soft photon region is very important for both the $B$ and $D$ mesons. The branching ratios of $Bto gamma e u_e$ is $5.21times 10^{-6}$, which is about $3$ times of the result obtained by only considering the hard photon region $E_{gamma}sim m_Q$. And for the case of $Dto gamma e u_e$, the result of the branching ratio is $1.92times 10^{-5}$.