$W$ radiative decay to heavy-light mesons in HQET factorization through ${\cal O}(\alpha_s)$

Analogous to NRQCD factorization for heavy quarkonium exclusive production, in this work we propose to employ the heavy-quark-effective-theory (HQET) factorization, which has been predominantly applied to account for exclusive $B$ decays, to study the exclusive production of the heavy-flavored mesons. We take $W\to B(D_s)+\gamma$ as a prototype process. The validity of the HQET factorization rests upon the presumed scale hierarchy: $m_W\sim m_b\gg \Lambda_{\rm QCD}$. Through an explicit analysis at next-to-leading order in $\alpha_s$ yet at leading order in $1/m_b$, we verify that the decay form factors can indeed be expressed as the convolution between perturbatively calculable hard-scattering kernel and the $B$ meson light-cone distribution amplitude (LCDA) defined in HQET. It is observed that the factorization scale dependence becomes reduced after incorporating the NLO perturbative correction. An interesting future investigation is to identify and resum large collinear logarithms of $m_W/m_b$ that arise ubiquitously in the fixed-order expressions of the hard-scattering kernel in HQET factorization.