Perturbative Heavy Quark Fragmentation Function through O(α_s^2): Gluon Initiated Contribution

TL;DR

The paper addresses resummation of quasi-collinear logarithms in heavy-quark differential distributions by refining the Perturbative Fragmentation Function (PFF) formalism and solving the DGLAP evolution for NNLL accuracy. It develops a process-independent derivation of the gluon-initiated initial condition $D^{\rm ini}_g$ at ${\cal O}(\alpha_s^2)$, complementing existing fermion-initiated results to yield the full NNLO initial condition. The authors provide an explicit analytic expression for $d^{(2)}_g(z,\mu_0/m)$ with a detailed color decomposition and polylogarithmic structure up to rank 3, including mass logarithms and threshold-like features at $z=1/2$. This work enables NNLL resummation of large quasi-collinear logs once the three-loop time-like splitting functions are available, extending the PFF formalism to precise predictions of heavy-quark spectra in $e^+e^-$, hadron colliders, DIS, and top decays.

Abstract

We derive the gluon initiated contribution to the initial condition for the perturbative fragmentation function of a heavy quark through order O(α_s^2) in the MS-bar scheme. This result is needed for the resummation with next-to-next-to-leading logarithmic accuracy of quasi-collinear logarithms ln^k(m^2) in heavy quark differential distributions by solving the complete DGLAP equation. Together with the previously evaluated fermion initiated components, this result completes the derivation of the initial condition for the perturbative fragmentation function at next-to-next-to-leading order.

Figures (1)

• Figure 1: Examples of diagrams for the gluon decay process $g \to Q + X$ at ${\cal O}(\alpha_s^2)$ that contribute to the function $W$. The dashed vertical line indicates the intermediate state that has to be considered.