Structure-dependent QED in \( {B}^{-}\to {\ell}^{-}\overline{\nu}\left(\gamma \right) \)

Based on explicitly gauge invariant interpolating operators we compute complete next-leading order QED-corrections for leptonic decays. These are sizeable since the helicity-suppression in V-A interactions allows for structure-dependent collinear logs. We have explicitly checked that these logs are absent for helicity-unsuppressed Yukawa-type transitions. Based on B → γ form factors we present the rates for $B^{-}\to \left({\mu }^{-},{\tau }^{-}\right)\overline{\nu }\left(\gamma \right)$ \( {B}^{-}\to \left({\mu}^{-},{\tau}^{-}\right)\overline{\nu}\left(\gamma \right) \) in differential and integrated form as a function of the photon energy cut-off $E_{\gamma }^{\mathrm{cut}}$ \( {E}_{\gamma}^{\textrm{cut}} \) . The effect of the virtual structure-dependent corrections are approximately +5% and +3% for the μ- and τ-channel respectively. The structure dependence of the real radiation exceeds that of the virtual one for ${\left(E_{\gamma }^{\mathrm{cut}}\right|}_{\mu }$ \( {\left.{E}_{\gamma}^{\textrm{cut}}\right|}_{\mu } \) > 0.18(3) GeV and is subdominant for the tau channel even when fully inclusive.