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

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, $[g\to T_{nQ}]$, by integrating the evolved fragmentation function of the gluon at the initial scale. In our computations, we utilize the initial scales for $T_g^{2c}$ from Braaten-Yuan [E.Braaten and T.C.Yuan, Phys. Rev. Lett. {\bf71}, 1673 (1993)] and for $T_g^{4c}$ and $T_g^{4b}$ 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 $\mu$.