QCD factorization for the four-body leptonic $B$-meson decays

Employing the QCD factorization formalism we compute $B_{u}^{-} \to \gamma^{\ast} \, \ell \, \bar \nu_{\ell}$ form factors with an off-shell photon state possessing the virtuality of order $m_b \, \Lambda_{\rm QCD}$ and $m_b^2$, respectively, at next-to-leading order in QCD. Perturbative resummation for the enhanced logarithms of $m_b / \Lambda_{\rm QCD}$ in the resulting factorization formulae is subsequently accomplished at next-to-leading logarithmic accuracy with the renormalization-group technique in momentum space. In particular, we derive the soft-collinear factorization formulae for a variety of the subleading power corrections to the exclusive radiative $B_{u}^{-} \to \gamma^{\ast} \, W^{\ast}$ form factors with a hard-collinear photon at ${\cal O}(\alpha_s^{0})$. We further construct the complete set of the angular observables governing the full differential distribution of the four-body leptonic decays $B_{u}^{-} \to \ell^{\prime} \, \bar \ell^{\prime} \, \ell \, \bar \nu_{\ell}$ with $\ell, \, \ell^{\prime} \in \{ e, \, \mu \}$ and then perform an exploratory phenomenological analysis for a number of decay observables accessible at the LHCb and Belle II experiments, with an emphasis on the systematic uncertainties arising from the shape parameters of the leading-twist $B$-meson light-cone distribution amplitude in heavy quark effective theory.