NNLO corrections to $\bar{B}\to X_{u}\ell \bar{\nu}_{\ell}$ and the determination of |V ub |

Abstract

We study the impact of next-to-next-to-leading order (NNLO) QCD corrections on partial decay rates in $\bar{B}\to X_{u}\ell \bar{\nu}_{\ell}$ decays, at leading-order in the 1/m b expansion for shape-function kinematics. These corrections are implemented within a modified form of the BLNP framework, which allows for arbitrary variations of the jet scale μ i ∼1.5 GeV. Our analysis includes a detailed comparison between resummed and fixed-order perturbation theory, and between the complete NNLO results and those obtained in the large-β 0 approximation. For the default choice μ i =1.5 GeV used in current extractions of |V ub | within the BLNP framework, the NNLO corrections induce significant downward shifts in the central values of partial decay rates with cuts on the hadronic variable P +, the hadronic invariant mass, and the lepton energy. At the same time, perturbative uncertainties are reduced, especially those at the jet scale, which are the dominant ones at next-to-leading order (NLO). For higher values of μ i and in fixed-order perturbation theory, the shifts between NLO and NNLO are more moderate. We combine our new results with known power-suppressed terms in order to illustrate the implications of our analysis on the determination of |V ub | from inclusive decays.