Correct Definition of the Gluon Fragmentation Function at High Energy Colliders

Since the definition of the gluon to hadrons fragmentation amplitude $<H,X|g>$ involves gluon field $Q^{μa}(x)$ corresponding to single gluon incoming state $|g>$, the present definition of the gluon fragmentation function at high energy colliders [which involves non-abelian field tensor $F^{μνa}(x)=\partial^μQ^{νa}(x) -\partial^νQ^{μa}(x)+gf^{abc}Q^{μb}(x)Q^{νc}(x)$ instead of $Q^{μa}(x)$ in the initial state] is not consistent with the single gluon incoming state $|g>$. In this paper we derive the correct definition of the gluon fragmentation function at high energy colliders from first principles which is obtained from the single gluon incoming state $|g>$ and is gauge invariant and is consistent with the factorization theorem in QCD.

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