Shape function effects in B -> X_s gamma and B -> X_u l nu decays

TL;DR

The paper addresses inclusive B decays in the endpoint region where conventional OPE breaks down due to large radiative corrections. It develops a soft-collinear effective theory framework to perform multi-scale matching from QCD to SCET, then to bilocal shape-function operators, and finally to local HQET operators, including order alpha_s coefficients and next-to-leading-log resummation. A key finding is that the shape function and its moments do not obey the same renormalization-group evolution as local operator moments, implying that moments of the shape function are not simply related to local HQET matrix elements; perturbative effects are also embedded in the shape function at multiple scales, challenging previous convolution prescriptions. The results provide a more reliable way to incorporate radiative corrections with the shape function, refine predictions for photon- and lepton-energy spectra, and impact precise extractions of CKM parameters such as Vub.

Abstract

We calculate the decay distributions for inclusive B -> X_s gamma and B -> X_u l nu decays in the endpoint region, where radiative corrections are large. The computation is done using effective field theory methods. The matching coefficients are computed to O(alpha_s), and the anomalous dimensions to next-to-leading order. The final expressions for the differential decay spectra include the complete O(alpha_s) corrections, and sum the leading and next-to-leading Sudakov series. We present results for regions of phase space where the shape function can be expanded in local operators, and give the matching coefficients of the resulting enhanced non-perturbative effects to order alpha_s. We show that moments of the shape function {\it are not} given by moments of local operators once perturbative effects are included, explain why the shape function and its moments satisfy different renormalization group equations, and contrast this with the situation for deep inelastic scattering. We show that there are large perturbative corrections in the usual definition of the shape function. This renders incorrect previous prescriptions for combining radiative corrections with the shape function.

Figures (7)

• Figure 1: Tree graph for the product of two currents. The dashed line depicts a collinear particle.
• Figure 2: Tree level matrix element of the shape function operator. The double line between the two currents depicts a Wilson line in the $n$ direction.
• Figure 3: One loop correction to the current product. Graphs (a), (b) and (e) also have mirror image graphs where the gluon attaches to the other side.
• Figure 4: One loop correction to the shape function operator.
• Figure 5: One loop corrections to the local operator.