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Since the definition of the gluon to hadrons fragmentation amplitude$<H,X|g>$ involves gluon field $Q^{\mu a}(x)$ corresponding to single gluonincoming state $|g>$, the present definition of the gluon fragmentationfunction at high energy colliders [which involves non-abelian field tensor$F^{\mu \nu a}(x)=\partial^\mu Q^{\nu a}(x) -\partial^\nu Q^{\mua}(x)+gf^{abc}Q^{\mu b}(x)Q^{\nu c}(x)$ instead of $Q^{\mu a}(x)$ in theinitial state] is not consistent with the single gluon incoming state $|g>$. Inthis paper we derive the correct definition of the gluon fragmentation functionat high energy colliders from first principles which is obtained from thesingle gluon incoming state $|g>$ and is gauge invariant and is consistent withthe factorization theorem in QCD.

Correct Definition of the Gluon Fragmentation Function at High Energy Colliders