A two-loop relation between inclusive radiative and semileptonicB-decay spectra

Abstract

A shape-function independent relation is derived between the partialB → X u l −ν decay rate with a cut on P + = E X − | P X | ≤ ∆ and a weighted integral over the normalizedB → X s γ photon-energy spectrum. The leading-power contribution to the weight function is calculated at next-to-next-to-leading order in renormalization-group improved perturbation theory, including exact two-loop matching corrections at the scale µ i ∼ m b Λ QCD . The overall normalization of the weight function is obtained up to yet unknown corrections of order α 2 s (m b ). Power corrections from phase-space factors are included exactly, while the remaining subleading contributions are included at first order in Λ QCD /m b . At this level unavoidable hadronic uncertainties enter, which are estimated in a conservative way. The combined theoretical accuracy in the extraction of |V ub | is at the level of 5% if a value of ∆ near the charm threshold can be achieved experimentally.

Status of |Vcb| and |Vub| CKM matrix elements

“…The analyses from BaBar [102] and Belle [101] collaborations, as well as the HFAG averages [39], rely on at least four theoretical different QCD calculations of the inclusive partial decay rate: BLNP by Bosch, Lange, Neubert, and Paz [103,104,105]; GGOU by Gambino, Giordano, Ossola and Uraltsev [106]; ADFR by Aglietti, Di Lodovico, Ferrara, and Ricciardi [107,108,109]; DGE, the dressed gluon exponentiation, by Andersen and Gardi [110]. They can be roughly divided into approaches based on the estimation of the shape function (BLNP, GGOU) and on resummed perturbative QCD (ADFR, DGE).…”

Flavour, electroweak symmetry breaking and dark matter: state of the art and future prospects

“…While the status of B → X c ν calculations is excellent (see for instance ref. [223]), the B → X u ν transition is afflicted by a general breakdown of the OPE due to experimental cuts required to suppress the B → X c ν background that result in large uncertainties related to the B meson shape function [63,224] (more recently BaBar and Belle presented results that take into account approximatively 90% of the total available phase space [225]). Predictions for exclusive modes are controlled by the lattice QCD determination of the B → π and B → D ( * ) form factors (the former, in particular, requires an extrapolation at low-q 2 that is usually performed using a z-parametrization and a simultaneous fit of lattice and experimental results).…”

Recent Developments in Soft-Collinear Effective Theory

“…The analysis in Ref. 17 based on the two-loop result for W (∆, P + ) predicts Γ u (0.65 GeV) = (46.5 ± 4.1) |V ub | 2 ps −1 , where the quoted number also includes an estimate of power corrections. Situations with more complicated cuts have been studied in Ref.…”