Next-to-next-to-leading order corrections to (B)over-bar → Xul(v)over-bar in the shape-function region

The inclusive decay (B) over bar -> X(u)l (v) over bar is of much interest because of its potential to constrain the CKM element vertical bar V-ub vertical bar. Experimental cuts required to suppress charm background restrict measurements of this decay to the shape-function region, where the hadronic final state carries a large energy but only a moderate invariant mass. In this kinematic region, the differential decay distributions satisfy a factorization formula of the form H center dot J circle times S, where S is the nonperturbative shape function, and the object H center dot J is a perturbatively calculable hard-scattering kernel. In this paper, we present the calculation of the hard function H at next-to-next-to-leading order (NNLO) in perturbation theory. Combined with the known NNLO result for the jet function J, this completes the perturbative part of the NNLO calculation for this process.