Correct Definition of the Gluon Fragmentation Function at High Energy Colliders

TL;DR

This paper derives a gauge-invariant, consistent definition of the gluon fragmentation function at high energy colliders directly from first principles, correcting previous inconsistencies related to the gluon field representation.

Contribution

The paper provides the first-principles derivation of a correct, gauge-invariant gluon fragmentation function consistent with QCD factorization, addressing prior theoretical inconsistencies.

Findings

Derived a gauge-invariant gluon fragmentation function from first principles.

Resolved inconsistencies in the previous definitions involving non-abelian field tensors.

Ensured the new definition aligns with the QCD factorization theorem.

Abstract

Since the definition of the gluon to hadrons fragmentation amplitude $<H,X|g>$ involves gluon field $Q^{\mu a}(x)$ 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 $F^{\mu \nu a}(x)=\partial^\mu Q^{\nu a}(x) -\partial^\nu Q^{\mu a}(x)+gf^{abc}Q^{\mu b}(x)Q^{\nu c}(x)$ instead of $Q^{\mu a}(x)$ 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.