Resummation of Nonperturbative Corrections to the Lepton Spectrum in Inclusive $B\to X\,\ell\,\barν$ Decays

TL;DR

The paper addresses the endpoint region of the lepton spectrum in inclusive B decays where a naive operator product expansion fails due to a second large scale. It extends the shape-function resummation to finite final-state quark mass by introducing a universal light-cone structure function $f(k_+)$, whose moments relate to HQET matrix elements, and shows that the lepton spectrum is a convolution of the free-quark rate with $f(k_+)$. This framework applies to both $B o X_u ext{ℓ}ar u$ and $B o X_c ext{ℓ}ar u$ when $ ho=m_q^2/m_b^2$ is $ ext{O}(rac{ extLambda}{m_b})$, with nonperturbative corrections exhibited mainly near the endpoint. A simple one-parameter model illustrates the smoothing of the endpoint and suggests practical strategies to extract $f(k_+)$ (or its moments) from data, while noting that radiative corrections must be incorporated for precision phenomenology.

Abstract

We apply the operator product expansion to resum the leading nonperturbative corrections to the endpoint region of the lepton spectrum in inclusive semileptonic $B\to X_q\,\ell\,\barν$ decays, taking into account a finite quark mass $m_q$ in the final state. We show that both for $b\to c$ and $b\to u$ transitions, it is consistent to describe these effects by a convolution of the parton model spectrum with a fundamental light-cone structure function. The moments of this function are proportional to forward matrix elements of higher-dimension operators. The prospects for an extraction of the structure function from a measurement of the lepton spectrum are discussed.

Figures (5)

• Figure 1: Shape of the lepton spectrum predicted in the free-quark decay model. We use $\rho=(m_c/m_b)^2 =0.08$.
• Figure 2: Nonperturbative corrections to the lepton spectrum obtained using a naive operator product expansion. The solid line corresponds to $\lambda_1=-0.3$ GeV$^2$, the dashed one to $\lambda_1=-0.1$ GeV$^2$.
• Figure 3: Model ansatz (\ref{['ftoy']}) for the structure function $f(k_+)$, evaluated for $\bar{\Lambda}=0.57$ GeV.
• Figure 4: (a) Charged-lepton spectrum $(1/\Gamma_b)\,\hbox{d}\Gamma/ \hbox{d}E_\ell$ in $B\to X_c\,\ell\,\bar{\nu}$ decays. The solid line is obtained from the convolution in (\ref{['convol']}) using the ansatz (\ref{['ftoy']}) for the structure function. The dashed line shows the prediction of the free-quark decay model. (b) The shape function $S(y,\rho)$, which is obtained from the difference of the two curves in (a).
• Figure 5: Same as Fig. \ref{['fig:4']}, but for $B\to X_u\,\ell\,\bar{\nu}$ decays.