Subleading Shape Functions in Inclusive B Decays

TL;DR

The paper develops a two-step SCET+HQET framework to quantify subleading power corrections in inclusive B decays within the shape-function region. It introduces subleading shape functions $s,t,u,v$ and, via four-quark operators, defines tilde variants that absorb certain effects, establishing moment relations tied to HQET parameters $lambda_1,lambda_2$. The authors provide explicit tree-level expressions for decay spectra (e.g., $ar{B} o X_u l u$ and $ar{B} o X_s\u03gamma$) in terms of these functions, analyze the difference from prior literature, and perform a numerical study showing power corrections are small over most of phase space, with potential enhancements near endpoints. The work also discusses isospin-breaking implications from four-quark operators and offers practical formulae and models for phenomenological use in extracting $|V_{ub}|$ and testing SM predictions.

Abstract

The contributions of subleading shape functions to inclusive decay distributions of B mesons are derived from a systematic two-step matching of QCD current correlators onto soft-collinear and heavy-quark effective theory. At tree-level, the results can be expressed in terms of forward matrix elements of bi-local light-cone operators. Four-quark operators, which arise at O(g^2), are included. Their effects can be absorbed entirely into a redefinition of other shape functions. Our results are in disagreement with some previous studies of subleading shape-function effects in the literature. A numerical analysis of B->X_u+l+nu decay distributions suggests that power corrections are small, with the possible exception of the endpoint region of the charged-lepton energy spectrum.

Figures (2)

• Figure 1: Representative examples of time-ordered products contributing to the current correlator $T_{ij}$ in SCET. Double lines show heavy-quark fields, dashed ones hard-collinear fields, and wavy lines denote the external currents. Lagrangian insertions and higher-order effective current operators are exemplified by soft gluons.
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