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QCD-Based Interpretation of the Lepton Spectrum in Inclusive $\bar B\to X_u\,\ell\,\barν$ Decays

Matthias Neubert

TL;DR

Problem: accurately describing the endpoint region of the inclusive lepton spectrum in B-bar to X_u l nu decays requires nonperturbative QCD input. Approach: introduce a shape function S(y) that captures the nonperturbative fall-off near the endpoint and relate its moments to forward scattering matrix elements of local HQET operators, allowing a controlled all-orders 1/m_b expansion. Key results: the endpoint width is governed by the kinetic-energy parameter lambda1, while M0 is suppressed by a cancellation; the formal all-orders definition connects moments to A_n coefficients; a simple ACM-model illustrates the construction and highlights limitations due to missing gluon-field effects. Significance: this framework provides a path to more reliable extraction of V_ub and heavy-quark parameters, and can be extended to B->X_c decays and to nonperturbative determinations via lattice or sum rules.

Abstract

We present a QCD-based approach to the endpoint region of the lepton spectrum in $\bar B\to X_u\,\ell\,\barν$ decays. We introduce a genuinely nonperturbative form factor, the shape function, which describes the fall-off of the spectrum close to the endpoint. The moments of this function are related to forward scattering matrix elements of local, higher-dimension operators. We find that nonperturbative effects are dominant over a finite region in the lepton energy spectrum, the width of which is related to the kinetic energy of the $b$-quark inside the $B$ meson. Applications of our method to the extraction of fundamental standard model parameters, among them $V_{ub}$, are discussed in detail.

QCD-Based Interpretation of the Lepton Spectrum in Inclusive $\bar B\to X_u\,\ell\,\barν$ Decays

TL;DR

Problem: accurately describing the endpoint region of the inclusive lepton spectrum in B-bar to X_u l nu decays requires nonperturbative QCD input. Approach: introduce a shape function S(y) that captures the nonperturbative fall-off near the endpoint and relate its moments to forward scattering matrix elements of local HQET operators, allowing a controlled all-orders 1/m_b expansion. Key results: the endpoint width is governed by the kinetic-energy parameter lambda1, while M0 is suppressed by a cancellation; the formal all-orders definition connects moments to A_n coefficients; a simple ACM-model illustrates the construction and highlights limitations due to missing gluon-field effects. Significance: this framework provides a path to more reliable extraction of V_ub and heavy-quark parameters, and can be extended to B->X_c decays and to nonperturbative determinations via lattice or sum rules.

Abstract

We present a QCD-based approach to the endpoint region of the lepton spectrum in decays. We introduce a genuinely nonperturbative form factor, the shape function, which describes the fall-off of the spectrum close to the endpoint. The moments of this function are related to forward scattering matrix elements of local, higher-dimension operators. We find that nonperturbative effects are dominant over a finite region in the lepton energy spectrum, the width of which is related to the kinetic energy of the -quark inside the meson. Applications of our method to the extraction of fundamental standard model parameters, among them , are discussed in detail.

Paper Structure

This paper contains 6 sections, 30 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Resummation of the singular terms
  3. The shape function
  4. Formal definition of the shape function
  5. A model calculation
  6. Summary and conclusions

Figures (2)

  • Figure 1: An example of a resummed lepton energy spectrum according to (\ref{['resum']}). On the vertical axis, we show ${\rm d} \Gamma/{\rm d}y$ in units of $2\Gamma_b$.
  • Figure 3: The shape function of the ACM model (a) and the corresponding lepton spectrum (b) for $-\lambda_1=0.05~{\rm GeV^2}$ (dashed), 0.1 GeV$^2$ (solid), and 0.2 GeV$^2$ (dotted). The corresponding values of the Fermi momentum are $p_{\rm F}\simeq 180$ MeV, 260 MeV, and 365 MeV, respectively. In (b), the parton model spectrum is shown as a grey line.