Shape function effects in $\overrightarrow{B}{X}_s\gamma$ and $\overrightarrow{B}{X}_u\mathcal{l}\overline{\nu }$ decays

We calculate the decay distributions for inclusive $\overrightarrow{B}{X}_s\gamma$ and $\overrightarrow{B}{X}_u\mathcal{l}\overline{\nu }$ decays in the endpoint region, where radiative corrections are large. The computation is done using effective field theory methods. The matching coefficients are computed to $\mathcal{O}\left({\alpha }_s\right)$ and the anomalous dimensions to next-to-leading order. The final expressions for the differential decay spectra include the complete $\mathcal{O}\left({\alpha }_s\right)$ corrections and sum the leading and next-to-leading Sudakov series. We present results for regions of phase space where the shape function can be expanded in local operators and give the matching coefficients of the resulting enhanced non-perturbative effects to order ${\alpha }_s.$ We show that moments of the shape function are not given by moments of local operators once perturbative effects are included, explain why the shape function and its moments satisfy different renormalization group equations, and contrast this with the situation for deep inelastic scattering. We show that there are large perturbative corrections in the usual definition of the shape function. This renders incorrect previous prescriptions for combining radiative corrections with the shape function.