On soft contributions to the $B^-\to \gamma^*$ form factors

The photoleptonic decay $B^-\to \gamma \ell^-\bar{\nu}$ is the simplest low-energy process that probes the substructure of the $B$ meson, making it an excellent candidate to determine the parameters of $B$-meson light-cone distribution amplitudes from experimental data. More recently, the decay $B^-\to \gamma^*(\to \ell'^-\ell'^+)\ell^-\bar\nu$ has received attention as an alternative probe of these parameters. Both decays are described through a common set of hadronic form factors, which if computed within the framework of QCD factorization give rise to the sensitivity to the $B$-meson light-cone distribution amplitudes. Nevertheless, in this case the form factors still receive so-called soft contributions that can only be estimated but not rigorously computed. In this work, we provide results for the QCD factorization expressions for the form factors, including terms at next-to-leading power in the $b$ quark mass and in the photon energy. We further use these results to estimate the soft contributions within a light-cone sum rule setup. Finally, using numerical results obtained from a benchmark model of the light-cone distribution amplitudes, we show that the soft contributions are under significantly better theoretical control at a mildly spacelike photon virtuality.