  Transverse momentum dependent (TMD) parton correlators describing the
partonic structure of hadrons contain gauge links, required by color gauge
invariance. The required gauge links enter in the matrix elements that contain
the parton fields and depend on the color flow in the hard process. The
correlators are expanded in terms of transverse momentum dependent parton
distribution functions, referred to as TMD PDFs, or in short TMDs. In this
paper, we introduce gluon TMDs of definite rank, by making an expansion of the
TMD gluon correlator with the help of irreducible tensors built from the
transverse momenta. The process dependence is isolated in gauge link dependent
gluonic pole factors multiplying the TMDs. It is important to account for the
different 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 the
leading tree level result for a hard process in terms of process dependent
gluon correlators which are expressed in a finite set of universal TMDs. We
tabulate the gluonic pole factors for various gauge links, among them those
that are relevant for 2 to 2 processes.

High Energy Physics - Phenomenology;High Energy Physics - Theory;Nuclear Theory