  It has by now been established that standard QCD factorization using
transverse momentum dependent parton distribution functions fails in
hadro-production of nearly back-to-back hadrons with high transverse momentum.
The essential problem is that gauge invariant transverse momentum dependent
parton distribution functions cannot be defined with process-independent Wilson
line operators, thus implying a breakdown of universality. This has led
naturally to proposals that a correct approach is to instead use a type of
"generalized" transverse momentum dependent factorization in which the basic
factorized structure is assumed to remain valid, but with transverse momentum
dependent parton distribution functions that contain non-standard, process
dependent Wilson line structures. In other words, to recover a factorization
formula, it has become common to assume that it is sufficient to simply modify
the Wilson lines in the parton correlation functions for each separate hadron.
In this paper, we will illustrate by direct counter-example that this is not
possible in a non-Abelian gauge theory. Since a proof of generalized transverse
momentum dependent factorization should apply generally to any hard
hadro-production process, a single counter-example suffices to show that a
general proof does not exist. Therefore, to make the counter-argument clear and
explicit, we illustrate with a specific calculation for a double spin asymmetry
in a spectator model with a non-Abelian gauge field. The observed breakdown of
generalized transverse momentum dependent factorization challenges the notion
that the role of parton transverse momentum in such processes can be described
using separate correlation functions for each external hadron.
