High order perturbative corrections to the determination of |Vub| from the P+ spectrum in B -> Xu l nu
We investigate the behaviour of the perturbative relation between the photon energy spectrum in B -> Xs gamma and the hadronic P+ spectrum in semileptonic B -> Xu l nu decay at high orders in perturbation theory in the "large-beta_0" limit, in which only terms of order alpha_s^n beta_0^(n-1) are retained. The leading renormalon in the weight function W(Delta,P_gamma) relating the two spectra is confirmed to be at u=1/2, corresponding to nonperturbative corrections at O(Lambda_QCD/m_b). We show that the P_gamma dependent pieces of the weight function have no infrared renormalons in this limit, and so the factorial growth in perturbation theory arises solely from the constant terms. We find no numerical enhancement of leading logarithms, suggesting that fixed-order perturbation theory is more appropriate than a leading-log resummation for the extraction of |Vub|. The importance of various terms in the expansion of the weight function is studied using a model for the B -> Xs gamma photon spectrum. Our analysis suggests that higher order perturbative corrections do not introduce a significant uncertainty in the extraction of |Vub|.