Perturbative Heavy Quark Fragmentation Function through O(alpha_s^2)

TL;DR

The work derives the initial condition for the heavy-quark perturbative fragmentation function $D^{\rm ini}$ through ${O}(\alpha_s^2)$ in the ${\overline{\mathrm{MS}}}$ scheme, enabling heavy-quark energy spectra to be obtained from massless calculations and allowing resummation of collinear logs $\ln(Q^2/m^2)$ via the DGLAP evolution at NNLL. The authors employ a process-independent framework, with a modified KLCC approach, to express $D^{\rm ini}$ in terms of universal collinear kernels $\Gamma$ and bare massive fragmentation functions $\widetilde{D}$, and compute the contributions from real and virtual collinear radiation up to NNLO. They provide explicit analytic results for the NNLO fermion-initiated coefficients $d^{(2)}_a(z,\mu_0/m)$, including complex polylogarithmic structures and color-decomposition terms, and perform consistency checks against soft-gluon resummation and large-$\beta_0$ limits. The findings open pathways for improved predictions of heavy-quark spectra in collider and QED contexts (e.g., muon decay) and set the stage for NNLL accuracy in heavy-quark fragmentation analyses.

Abstract

We derive the initial condition for the perturbative fragmentation function of a heavy quark through order O(alpha_s^2) in the MS-bar scheme. This initial condition is useful for computing heavy quark (or lepton, in case of QED) energy distributions from calculations with massless partons. In addition, the initial condition at O(alpha_s^2) can be used to resum collinear logarithms ln(Q^2/m^2) in heavy quark energy spectrum with next-to-next-to-leading logarithmic accuracy by solving the DGLAP equation.

Figures (2)

• Figure 1: Diagrams that contribute to perturbative fragmentation of a heavy quark $Q \to Q + X$ at ${\cal O}(\alpha_s)$. The dashed vertical line indicates the intermediate state that has to be considered.
• Figure 2: Examples of diagrams that contribute to perturbative fragmentation of a heavy quark $Q \to Q + X$ at ${\cal O}(\alpha_s^2)$. The dashed vertical line indicates the intermediate state that has to be considered.