  We consider the definition of a transverse-momentum-dependent parton
distribution function (TMDPDF) and its evolution properties given a
factorization theorem of Drell- Yan lepton pair production with a non-vanishing
transverse momentum. We discuss rapidity divergences that exist in both limits
of QCD: the soft and collinear. We argue that only when a specific combination
of the soft and collinear matrix elements is formed, rapidity divergences
cancel. We also argue that the soft matrix element, calculated in perturbation
theory, contains only rapidity divergences and there is no genuine
long-distance infra-red divergences.
