Shape Function Effects in B -> X_c l ν_l

TL;DR

The work addresses the endpoint region of the inclusive decay B → X_c ℓ ν_ℓ, where m_c^2 ∼ m_b Λ_QCD and the B-meson light-cone distribution influences the lepton spectrum. It develops a light-cone OPE with a modified twist expansion that keeps Δ_+ exact, introducing new operators (e.g., O_1(ω), O_2(ω)) and a rich set of shape functions to describe subleading effects across the entire phase space. The authors derive the leading and subleading lepton-energy spectrum, express it in terms of leading and subleading shape functions (F_0, K_0, F_1, L_1, G_2, etc.), and show consistency with local O(Λ^2) results and the massless limit, while clarifying discrepancies with earlier Bauer analyses. This framework enables improved moment analyses and potential extraction of B-meson shape functions, with prospects for incorporating radiative corrections via SCET.

Abstract

Owing to the fact that m_c^2 ~ m_b Λ_QCD, the endpoint region of the charged lepton energy spectrum in the inclusive decay B -> X_c l ν_l is affected by the Fermi motion of the initial-state b quark bound in the B meson. This effect is described in QCD by shape functions. Including the mass of the final-state quark, we find that a different set of operators as employed in Ref. hep-ph/0205150 is needed for a consistent matching, when incorporating the subleading contributions in B -> X_q l ν_l for both q = u and q = c. In addition, we modify the usual twist expansion in such a way that it yields a description of the lepton energy spectrum which is not just valid in the endpoint region, but over the entire phase space.

Figures (1)

• Figure 1: Feynman diagrams for the zero- and one-gluon matrix elements (to leading order in $\alpha_s$).