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Transverse momentum dependent (TMD) parton correlators describing thepartonic structure of hadrons contain gauge links, required by color gaugeinvariance. The required gauge links enter in the matrix elements that containthe parton fields and depend on the color flow in the hard process. Thecorrelators are expanded in terms of transverse momentum dependent partondistribution functions, referred to as TMD PDFs, or in short TMDs. In thispaper, we introduce gluon TMDs of definite rank, by making an expansion of theTMD gluon correlator with the help of irreducible tensors built from thetransverse momenta. The process dependence is isolated in gauge link dependentgluonic pole factors multiplying the TMDs. It is important to account for thedifferent possibilities in the color structure within the matrix elements,leading to multiple TMDs at a given rank. In this way we are able to write theleading tree level result for a hard process in terms of process dependentgluon correlators which are expressed in a finite set of universal TMDs. Wetabulate the gluonic pole factors for various gauge links, among them thosethat are relevant for 2 to 2 processes.

Generalized Universality of Definite Rank Gluon Transverse Momentum Dependent Correlators