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We present $O(\alpha_s^2)$ QCD corrections to the fully-differential decayrate of a $b$-quark into inclusive semileptonic charmless final states. Ourcalculation provides genuine two-loop QCD corrections, beyond theBrodsky-Lepage-Mackenzie (BLM) approximation, to any infra-red safe partonicobservable that can be probed in $b \to X_u e \bar \nu$ decays. Kinematic cutsthat closely match those used in experiments can be fully accounted for. Toillustrate these points, we compute the non-BLM corrections to moments of thehadronic invariant mass and the hadronic energy with cuts on the lepton energyand the hadronic invariant mass. Our results remove one of the sources oftheoretical uncertainty that affect the extraction of the CKM matrix element$|V_{ub}|$ from charmless inclusive B-decays.

On the $O(\alpha_s^2)$ corrections to $b \to X_u e \bar \nu$ inclusive decays