In this talk we consider differential cross sections in QCD depending on several disparate physical scales. Ratios of these scales can lead to large logarithms, spoiling the convergence of perturbative expansions. We use the soft-collinear effective theory (SCET) of QCD to factorize the differential cross sections into well behaved parts. For each part, renormalization group techniques allow to identify and resum the dominant contributions to all perturbative orders.
As an explicit and phenomenologically important example, we consider the transverse momentum spectra of heavy, color neutral finale states produced at hadron colliders. This leads us to the identification of process independent transverse momentum dependent parton distribution functions (TPDFs) in terms of matrix elements of gauge invariant operators. These are generalizations of (collinear) PDFs describing the initial state radiation in processes sensitive to the transverse momentum.
We discuss the perturbative calculation of their partonic versions to next-to-next-to-leading order (NNLO). To treat rapidity divergences, an analytic regulator is introduced on top of dimensional regularization. We demonstrate the cancellation of the rapidity divergences in physical observables at NNLO, which explicitly verifies that a definition of TPDFs is applicable beyond the first non-trivial order. From our results, we extract coefficient functions relevant for a next-to-next-to-next-to-leading logarithmic transverse momentum resummation in a large class of processes at hadron colliders.