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Inspired by the recent measurements of the $\eta_c$ meson production at LHC,we investigate the relativistic correction effect for the fragmentationfunction of the gluon into $\eta_c$, which constitutes the crucialnonperturbative elements to understand $\eta_c$ production at high $p_T$.Employing three distinct methods, we calculate the leading relativisticcorrection to the $g\to\eta_c$ fragmentation function in the NRQCDfactorization framework, as well as verify the existing NLO result for the$c\to \eta_c$ fragmentation function. We also study the evolution behavior ofthese fragmentation functions with the aid of DGLAP equation.

Relativistic correction to gluon fragmentation function into pseudoscalar quarkonium