Factorization and Shape-Function Effects in Inclusive B-Meson Decays

TL;DR

The paper develops a three-scale factorization of inclusive B -> X_u l nu decays in the shape-function region using SCET and HQET, yielding a convolution of hard, jet, and shape functions with next-to-leading-log resummation. It provides complete one-loop matching for the hard and jet functions and a thorough one-loop treatment of the renormalized shape function, including its anomalous dimension and analytic evolution. A key advance is the introduction of a physical, subtraction-based shape-function mass m_b^{SF} and a running kinetic parameter μ_π^2, along with a scheme-independent analysis of shape-function moments and asymptotics that reveal a negative radiative tail. The work derives shape-function–independent relations between spectra, analyzes various experimental cuts (notably on P_+), and presents model shape functions with numerical predictions showing that P_+ cuts can retain ~80% of events while keeping shape-function uncertainties manageable. Overall, the framework enables more precise determinations of |V_{ub}| and provides robust guidance for phenomenology and future refinements.

Abstract

Using methods of effective field theory, factorized expressions for arbitrary B -> X_u l nu decay distributions in the shape-function region of large hadronic energy and moderate hadronic invariant mass are derived. Large logarithms are resummed at next-to-leading order in renormalization-group improved perturbation theory. The operator product expansion is employed to relate moments of the renormalized shape function with HQET parameters such as m_b, Lambda(bar) and lambda_1 defined in a new physical subtraction scheme. An analytic expression for the asymptotic behavior of the shape function is obtained, which reveals that it is not positive definite. Explicit expressions are presented for the charged-lepton energy spectrum, the hadronic invariant mass distribution, and the spectrum in the hadronic light-cone momentum P_+ = E_H - P_H. A new method for a precision measurement of |V_{ub}| is proposed, which combines good theoretical control with high efficiency and a powerful discrimination against charm background.

Figures (9)

• Figure 1: One-loop diagrams contributing to the current correlator in SCET. The effective current operators are denoted by crossed circles, and hard-collinear propagators are drawn as dashed lines. Mirror graphs obtained by exchanging the two currents are not shown.
• Figure 2: Radiative corrections to the shape function. The bilocal HQET operator is denoted by the black square. A mirror copy of the first graph is not shown.
• Figure 3: Hadronic phase space for the light-cone variables $P_-$ and $P_+$ (left), and theory phase space for $m_b=4.8$ GeV (right). The scatter points indicate the distribution of events as predicted by the model of DeFazio:1999sv. In each plot the solid line separates the regions where $s_H<M_D^2$ (dark gray) and $s_H>M_D^2$ (light gray), whereas the dashed line corresponds to $P_+=M_D^2/M_B$. The dotted line in the first plot shows the contour where $q^2=(M_B-M_D)^2$.
• Figure 4: Phase-space constraints (left) and weight functions (right) for combined cuts on the hadronic and leptonic invariant mass: $(s_0,\,q_0^2)=(M_D^2,\,0)$ (solid), $(M_D^2,\,6\,\hbox{GeV}^2)$ (dashed), and $((1.7\,\hbox{GeV})^2,\,8\,\hbox{GeV}^2)$ (dotted).
• Figure 5: Weight function $w(\Delta,P_+)$ entering the rate relation (\ref{['raterel']}) for $\Delta=M_D^2/M_B$ and three different choices of the intermediate scale, namely $\mu_i=1.5$ GeV (solid), 2.0 GeV (dashed), and 1.0 GeV (dotted). The weight function is formally independent of $\mu_i$.