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We consider the definition of a transverse-momentum-dependent partondistribution function (TMDPDF) and its evolution properties given afactorization theorem of Drell- Yan lepton pair production with a non-vanishingtransverse momentum. We discuss rapidity divergences that exist in both limitsof QCD: the soft and collinear. We argue that only when a specific combinationof the soft and collinear matrix elements is formed, rapidity divergencescancel. We also argue that the soft matrix element, calculated in perturbationtheory, contains only rapidity divergences and there is no genuinelong-distance infra-red divergences.

On Rapidity Divergences in the Soft and Collinear Limits of QCD