Structure-dependent QED in $B^{-}\to {\mathrm{\ell }}^{-}\bar{\nu }(\gamma )$

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 $B\to \gamma$ form factors we present the rates for $B^{-}\to ({\mu }^{-},{\tau }^{-})\bar{\nu }(\gamma )$ in differential and integrated form as a function of the photon energy cut-off $E_{\gamma }^{\text{cut}}$. The effect of the virtual structure-dependent corrections are approximately $+5\mathrm{\%}$ and $+3\mathrm{\%}$ for the $\mu$- and $\tau$-channel respectively. The structure dependence of the real radiation exceeds that of the virtual one for $E_{\gamma }^{\text{cut}}{\mathrm{\mid }}_{\mu }>0.18(3)$ GeV and is subdominant for the tau channel even when fully inclusive.