QED effects in inclusive semi-leptonic $B$ decays

We perform a detailed analysis of QED corrections to the total decay width and 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 ${\cal O}(\alpha)$ partonic terms and leading-logarithmic QED effects up to ${\cal O}(\Lambda^3_{\rm QCD}/m_b^3)$. In the case of the branching ratio and the electron energy moments we perform a detailed numerical comparison of our results against those obtained with the Monte Carlo (MC) tool PHOTOS. While the comparison indicates good overall agreement, our computation contains QED effects not included in PHOTOS and should therefore provide a better description of photon radiation to $B \to X_c e \nu$ as measured by the $B$-factories. The calculations performed here represent the first steps in the construction of a fully differential higher-order QED MC generator for inclusive semi-leptonic $B$ decays.