An analysis of the fragmentation function of gluon at next-to-leading order approximation

TL;DR

This paper studies the gluon fragmentation function at next-to-leading order using a Laplace transform technique to evolve it for heavy-quarkonium states, providing numerical solutions and comparisons with existing results.

Contribution

It introduces a method to numerically evolve the gluon fragmentation function at NLO for heavy-quarkonium states using Laplace transforms, extending previous leading-order analyses.

Findings

Numerical solutions for gluon fragmentation functions at NLO for heavy-quarkonium states.

Comparison with existing literature confirms the accuracy of the evolved fragmentation functions.

Provides initial scale values from recent literature for different quarkonium states.

Abstract

We are investigating the behavior of the fragmentation function of a gluon, denoted as $ D_{g}(x,\mu^2)$, where $\mu$ represents the observable scale. This function is derived from the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations. Our objective is to evolve the fragmentation function of a gluon for heavy-quark-antiquark bound states with large transverse momentum using a Laplace transform technique. This method enables us to calculate numerical solutions for two and four quarkonium states based on the known initial fragmentation function of gluons. We examine both leading-order (LO) and higher-order approximations for the fragmentation function of a gluon, $[g{\rightarrow}T_{nQ}]$, by integrating the evolved fragmentation function of the gluon at the initial scale. In our computations, we utilize the initial scales for $T^{2c}_{g}$ from Braaten-Yuan [E.Braaten and T.C.Yuan, Phys. Rev. Lett. {\bf71}, 1673 (1993)] and for $T^{4c}_{g}$ and $T^{4b}_{g}$ from Celiberto-Gatto-papa [ F. G. Celiberto, G. Gatto and A. Papa, Eur. Phys. J. C {\bf84}, 1071 (2024)] and Celiberto-Gatto [ F. G. Celiberto and G. Gatto, Phys. Rev. D {\bf111}, 034037 (2025)] respectively. Through comparing our predictions with existing literature results, we can accurately determine the evolution of the fragmentation function of a gluon at scale $\mu$.