Heavy-Quark Fragmentation

TL;DR

The paper develops a unified, renormalon-based framework for heavy-quark fragmentation at large x, showing that in the joint large-N and mass limit the fragmentation function depends only on NΛ/m and can be expressed as a perturbative Sudakov factor convolved with a non-perturbative shape function of m(1−x). Using Dressed Gluon Exponentiation in the large-β0 limit, the authors derive a process-independent fragmentation function with a well-defined renormalon structure, enabling a PV-Borel resummed perturbative piece and a multiplicative non-perturbative correction that exponentiates in NΛ/m. They apply this to B-meson production in e+e− annihilation, perform factorization into fragmentation, jet, and evolution components, and compare with LEP data via moment-space fits, extracting non-perturbative parameters consistent with an O(1) shift and a small higher-order term. The work provides a theoretically controlled link between perturbative fragmentation and hadronization, offering improved descriptions near the endpoint and a universal approach applicable across observables sensitive to large-x dynamics.

Abstract

We study perturbative and non-perturbative aspects of heavy-quark fragmentation into hadrons, emphasizing the large-x region, where x is the energy fraction of the detected hadron. We first prove that when the moment index N and the quark mass m get large simultaneously with the ratio (N Lambda/m) fixed, the fragmentation function depends on this ratio alone. This opens up the way to formulate the non-perturbative contribution to the fragmentation function at large N as a shape function of m(1-x) which is convoluted with the Sudakov-resummed perturbative result. We implement this resummation and the parametrization of the corresponding shape function using Dressed Gluon Exponentiation. The Sudakov exponent is calculated in a process independent way from a generalized splitting function which describes the emission probability of an off-shell gluon off a heavy quark. Non-perturbative corrections are parametrized based on the renormalon structure of the Sudakov exponent. They appear in moment space as an exponential factor, with a leading contribution scaling as (N Lambda/m) and corrections of order (N Lambda/m)^3 and higher. Finally, we analyze in detail the case of B-meson production in e+e- collisions, confronting the theoretical predictions with LEP experimental data by fitting them in moment space.

Figures (10)

• Figure 1: The definition of the fragmentation function. The dashed line represents a path-ordered exponential.
• Figure 2: Single gluon emission off a heavy quark involved in a generic hard process ${\cal M}_0$.
• Figure 3: The diagram contributing to $\tilde{D}(N,m^2)$ in the light-cone axial gauge $A\cdot y=0$ and in the large $N_f$ limit.
• Figure 4: $QQg$ phase-space boundaries in the $x$--$\bar{x}$ plane, with varying gluon virtualities $\epsilon=(1-\sqrt{\rho})^2\,i/10$, for $i=0$ through $9$, with $\rho=0.04$.
• Figure 5: Factorization of the process $\gamma^*\longrightarrow Q+X$ at large $N$.