Perturbative corrections to the determination of Vub from the P+ spectrum in B->X_u l nu

We investigate the relation between the E_gamma spectrum in B->X_s gamma decay and the P+ spectrum in semileptonic B->X_u l nu decay (P+ is the hadronic energy minus the absolute value of the hadronic three-momentum), which provides in principle the theoretically simplest determination of Vub from any of the "shape function regions" of B->X_u l nu spectra. We calculate analytically the P+ spectrum to order alpha_s^2 beta_0, and study its relation to the B->X_s gamma photon spectrum to eliminate the leading dependence on nonperturbative effects. We compare the result of fixed order perturbation theory to the next-to-leading log renormalization group improved calculation, and argue that fixed order perturbation theory is likely to be a more appropriate expansion. Implications for the perturbative uncertainties in the determination of Vub from the P+ spectrum are discussed.