Full-Phase-Space Twist Expansion in Semileptonic and Radiative B-Meson Decays

TL;DR

This work develops a full-phase-space twist expansion for inclusive B decays by performing a QCD light-cone operator expansion and matching directly to light-cone operators. It provides a smooth interpolation between local OPE and twist-dominated descriptions, valid across the full hadronic phase space except near resonances, and yields complete tree-level expressions for the triple differential rate in B → X_u ℓν and the photon energy spectrum in B → X_s γ. By introducing a hadronic-variable framework and exploiting reparametrization invariance, the authors derive a comprehensive set of hadronic shape functions and show how subleading twist corrections can be organized and constrained. They also present practical models for shape-functions, derive diverse spectra and partial rates under realistic kinematic cuts, and demonstrate reduced shape-function sensitivity for certain observables, enabling potentially more precise determinations of |V_ub|. The framework lays the groundwork for including radiative corrections and renormalization effects, and highlights hadronic-energy–based observables as promising tools for CKM analyses.

Abstract

We study the Lambda_QCD/M_B corrections from subleading shape functions in inclusive B-meson decays. We propose a natural and smooth interpolation from the endpoint region to the full phase space, and derive expressions for the triple differential decay rate in B -> X_u l nu and the photon energy spectrum in B -> X_s gamma. Our results are valid to order Lambda_QCD/M_B for hadronic invariant masses of order Lambda_QCD M_B and to order Lambda_QCD^2/M_B^2 for larger hadronic masses. They allow a systematic investigation of the transition between the separate regimes of the local and nonlocal expansions, and can be used to study decay distributions in any kinematic variables. We consider several examples of interest and point out that a combined measurement of hadronic energy and invariant mass provides an alternative to the extraction of |V_ub| which is largely independent of shape function effects and in principle allows a higher accuracy than the combined measurement of leptonic and hadronic invariant masses. We perform the expansion directly in QCD light-cone operators, and give a discussion of the general basis of light-cone operators. Reparametrization invariance under the change of the light-cone direction reduces the number of independent shape functions. We show that differing previous results for the lepton energy spectrum obtained from different choices of light-cone coordinates are in agreement.

Figures (20)

• Figure 1: (color online) Phase space regions in the $P_\pm$ plane as discussed in the text. The solid line shows $s_H/M_B^2 = \Lambda$ and the dashed one $s_H = M_D^2$. The light (orange) filled region is $P_+/P_- < \sqrt{\Lambda}$, the light (green) and dark (violet) hatched regions are $P_+/M_B > \sqrt{\Lambda}$ and $P_-/M_B < \sqrt{\Lambda}$, and the dark (violet) filled region is the resonance region with $P_-/M_B < \Lambda$. We take $\Lambda = 0.1$ in all cases.
• Figure 2: Tree-level Feynman diagrams for the zero- and one-gluon matrix element.
• Figure 3: Four-quark diagrams. On the right we only show the routing of residual momenta.
• Figure 4: (color online) Shape functions. Model 1, 2, and 3 are dark (violet), medium (orange), and light (green). The left plot shows $F_0(\omega)$. The right one shows $G_5(\omega)$ (solid), $H_5(\omega)$ (dashed), and $R_4(\omega)$ (dotted).
• Figure 5: (color online) $E_\gamma$ spectrum. The solid lines show the full result, the dashed lines include only the contributions proportional to $F_0(\omega)$, and the dotted ones show the subleading twist result.