TMD factorization and the gluon distribution in high energy QCD

This paper is a part of a series of works where we in detail examine the concept of Transverse Momentum Dependent (TMD), or k_T, factorization, which is frequently encountered in the literature and is widely used in the phenomenological applications of QCD at very high energies. We address the question of what exactly factorization is, as it is meant in different contexts and formalisms, and we compare the formalisms to each other. We clarify some basic concepts regarding factorization and how it exactly is applied in high energy QCD, and we make important notes on some key and fundamental points that are often overlooked. We offer an extensive analysis of single inclusive particle production, and we analyze the TMD gluon distribution that plays a pivotal role in high energy QCD.