QED effects in inclusive semi-leptonic $B$ decays

We analyse in detail the QED corrections to the total decay width and the moments of the electron energy spectrum of the inclusive semi-leptonic $B\to X_{c}e\nu$ decay. Our calculation includes short-distance electroweak corrections, the complete $\mathcal{O}(\alpha )$ partonic terms and leading-logarithmic QED effects up to $\mathcal{O}({\mathrm{\Lambda }}_{\mathrm{Q}\mathrm{C}\mathrm{D}}^3\mathrm{/}{m}_b^3)$. A comprehensive numerical comparison of our results against those obtained with the Monte Carlo (MC) tool PHOTOS is presented. While the comparison indicates good overall agreement, our computation contains QED effects not included in PHOTOS and should therefore better describe photon radiation to $B\to X_{c}e\nu$ as measured by the $B$-factories. Our calculations represent the first steps in the construction of a fully differential higher-order QED MC generator for inclusive semi-leptonic $B$ decays.