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 − → μ − τ − ν ¯ γ in differential and integrated form as a function of the photon energy cut-off $$ {E}_{\gamma}^{\textrm{cut}} $$ E γ 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 $$ {\left.{E}_{\gamma}^{\textrm{cut}}\right|}_{\mu } $$ E γ cut μ > 0 . 18(3) GeV and is subdominant for the tau channel even when fully inclusive.