Charmless inclusive B decays and the extraction of V(ub)

Abstract

This work discusses charmless inclusive B decays and their application to the extraction of $|V_{ub}|$. Starting from first principles we relate the differential decay rate to the hadronic tensor in terms of optimal choice of kinematical variables. We review the traditional methods of calculating the hadronic tensor, expansion in $α_s$ and HQET, and discuss their shortcomings. In the kinematical region relevant for experiment ("shape function" region), the hadronic tensor can be factorized, at each order in $1/m_b$, as a product of calculable hard functions and a convolution of calculable jet functions with non perturbative shape functions. Using SCET, we calculate the leading order hard and jet function to first order in $α_s$. Large logarithms are resummed in RGE improved perturbation theory. Local OPE is used to relate moments of the renormalized shape function to HQET parameters, defined in the "shape function scheme". Beyond leading order in $1/m_b$, several subleading shape functions arise. We derive them at tree level, where they can be expressed as forward matrix elements of bi-local light-cone operators. Based on these theoretical calculations we present two applications. In the first, we present the "state-of-the-art " expressions for the triple differential $\bar B\to X_u l\barν$ decay rate and the $\bar B\to X_sγ$ photon spectrum. These expressions include all known contributions and smoothly interpolate between the "shape-function" and "OPE region". Based on these an event generator can be constructed, from which the theoretical prediction for any experimental cut can be extracted. In the second, a weight function is constructed that relates the $P_+$ spectrum in $\bar B\to X_u l\barν$ to the normalized $\bar B\to X_sγ$ photon spectrum.