Correct Definition of the Gluon Distribution Function at High Energy Colliders

TL;DR

This paper derives a gauge-invariant, consistent definition of the gluon distribution function at high energy colliders from first principles, addressing inconsistencies in previous definitions related to the number operator interpretation.

Contribution

It provides the first-principles derivation of a gluon distribution function that aligns with the number operator interpretation and the factorization theorem in QCD.

Findings

Corrected the definition of the gluon distribution function.

Ensured gauge invariance of the gluon distribution function.

Aligned the definition with the factorization theorem.

Abstract

Unlike QED, since F_{\mu \nu}^a(x)F^{\mu \nu a}(0) in QCD contains cubic and quartic powers of the gluon field the present definition of the gluon distribution function at high energy colliders is not consistent with the number operator interpretation of the gluon. In this paper we derive the correct definition of the gluon distribution function at high energy colliders from first principles which is consistent with the number operator interpretation of the gluon and is gauge invariant and is consistent with the factorization theorem in QCD.