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

TL;DR

This paper analyzes the radiative leptonic decays of $B^-$ and $D^-$ mesons using a factorization approach, emphasizing the significance of the soft photon region for accurate branching ratio predictions.

Contribution

It proves the validity of factorization at 1-loop level for these decays and highlights the importance of including the soft photon region in calculations.

Findings

Soft photon region significantly affects decay branching ratios.

Branching ratio for $B o ho e u_e$ is approximately three times larger when including soft photons.

Numerical results show the soft photon contribution is crucial for precise decay predictions.

Abstract

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 $O(\Lambda_{\rm QCD}\left/m_Q\right.)$. 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 $B\to \gamma e\nu_e$ is $5.21\times 10^{-6}$, 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 $D\to \gamma e\nu_e$, the result of the branching ratio is $1.92\times 10^{-5}$.