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 $O(Λ_{\rm QCD}\left/m_Q\right.)$. We consider the contribution in the soft photon region that $E_γ \sim \left. Λ^2_{\rm QCD} /\right. m_Q$. The numerical results shows that, the soft photon region is very important for both the $B$ and $D$ mesons. The branching ratios of $B\to γeν_e$ is $5.21\times 10^{-6}$, which is about $3$ times of the result obtained by only considering the hard photon region $E_γ\sim m_Q$. And for the case of $D\to γeν_e$, the result of the branching ratio is $1.92\times 10^{-5}$.

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