Theory of charmless inclusiveBdecays and the extraction ofVub

Abstract

We present "state-of-the-art" theoretical expressions for the triple differentialB → X u l −ν decay rate and for theB → X s γ photon spectrum, which incorporate all known contributions and smoothly interpolate between the "shape-function region" of large hadronic energy and small invariant mass, and the "OPE region" in which all hadronic kinematical variables scale with M B . The differential rates are given in a form which has no explicit reference to the mass of the b quark, avoiding the associated uncertainties. Dependence on m b enters indirectly through the properties of the leading shape function, which can be determined by fitting theB → X s γ photon spectrum. This eliminates the dominant theoretical uncertainties from predictions forB → X u l −ν decay distributions, allowing for a precise determination of |V ub |. In the shape-function region, short-distance and long-distance contributions are factorized at next-to-leading order in renormalization-group improved perturbation theory. Higher-order power corrections include effects from subleading shape functions where they are known. When integrated over sufficiently large portions in phase space, our results reduce to standard OPE expressions up to yet unknown O(α 2 s ) terms. Predictions are presented for partial B → X u l −ν decay rates with various experimental cuts. An elaborate error analysis is performed that contains all significant theoretical uncertainties, including weak annihilation effects. We suggest that the latter can be eliminated by imposing a cut on high lepton invariant mass.

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

“…The function H u is a linear combination of the hard functions entering the structures F i in (3), which in the notation of [7] is given by…”

“…Experiments typically reject semileptonic B-decay events with very low lepton energy. The effect of such an additional cut can be determined from [7]. Alternatively, it is possible to modify the weight function so as to account for a lepton cut, however at the expense of a significant increase in complexity.…”

“…One approach for measuring |V ub | is to first extract the shape function from theB → X s γ photon spectrum, and then to use this information for predictions of event distributions inB → X u l −ν . A comprehensive description of this program has been presented in [7]. Equivalently, it is possible to eliminate the shape function inB → X u l −ν decay rates in favor of theB → X s γ photon-energy spectrum.…”

“…Expressions for the structure functions F i valid at next-to-leading order (NLO) in renormalization-group (RG) improved perturbation theory and including first-and secondorder power corrections can be found in [7] (see also [13,14,15]). Symbolically, they are written as…”

“…The calculation of the weight function starts with the theoretical expressions for the P + spectra inB → X u l −ν andB → X s γ decays, which are given as [7] …”