A two-loop relation between inclusive radiative and semileptonic B-decay spectra

A shape-function independent relation is derived between the partial →Xu l− decay rate with a cut on P+ = EX−|X| ≤ Δ and a weighted integral over the normalized →Xsγ photon-energy spectrum. The leading-power contribution to the weight function is calculated at next-to-next-to-leading order in renormalization-group improved perturbation theory, including exact two-loop matching corrections at the scale μi ~ (mbΛQCD)1/2. The overall normalization of the weight function is obtained up to yet unknown corrections of order αs2(mb). Power corrections from phase-space factors are included exactly, while the remaining subleading contributions are included at first order in ΛQCD/mb. At this level unavoidable hadronic uncertainties enter, which are estimated in a conservative way. The combined theoretical accuracy in the extraction of |Vub| is at the level of 5% if a value of Δ near the charm threshold can be achieved experimentally.