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It has by now been established that standard QCD factorization usingtransverse momentum dependent parton distribution functions fails inhadro-production of nearly back-to-back hadrons with high transverse momentum.The essential problem is that gauge invariant transverse momentum dependentparton distribution functions cannot be defined with process-independent Wilsonline operators, thus implying a breakdown of universality. This has lednaturally to proposals that a correct approach is to instead use a type of"generalized" transverse momentum dependent factorization in which the basicfactorized structure is assumed to remain valid, but with transverse momentumdependent parton distribution functions that contain non-standard, processdependent Wilson line structures. In other words, to recover a factorizationformula, it has become common to assume that it is sufficient to simply modifythe Wilson lines in the parton correlation functions for each separate hadron.In this paper, we will illustrate by direct counter-example that this is notpossible in a non-Abelian gauge theory. Since a proof of generalized transversemomentum dependent factorization should apply generally to any hardhadro-production process, a single counter-example suffices to show that ageneral proof does not exist. Therefore, to make the counter-argument clear andexplicit, we illustrate with a specific calculation for a double spin asymmetryin a spectator model with a non-Abelian gauge field. The observed breakdown ofgeneralized transverse momentum dependent factorization challenges the notionthat the role of parton transverse momentum in such processes can be describedusing separate correlation functions for each external hadron.

No Generalized TMD-Factorization in the Hadro-Production of High Transverse Momentum Hadrons