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We study transverse momentum dependent factorization and resummation atsub-leading power in Drell-Yan and semi-inclusive deep inelastic scattering. Inthese processes the sub-leading power contributions to the cross section enteras a kinematic power correction to the leptonic tensor, and the kinematic,intrinsic, and dynamic sub-leading contributions to the hadronic tensor. Byconsistently treating the power counting of the interactions, we demonstraterenormalization group consistency. We calculate the anomalous dimensions of thekinematic and intrinsic sub-leading correlation functions at one loop and findthat the evolution equations give rise to anomalous dimension matrices whichmix leading and sub-leading power distribution functions. Additionally wecalculate the hard and soft functions associated with each of thesecontributions. We find that these hard and soft contributions differ from thoseat the leading power. Finally, we calculate the rapidity anomalous dimensionfor the dynamic sub-leading distributions and find that it is the same as theleading power anomalous dimension. We then comment on the implications for thesoft function associated with this contribution. Using this information, weestablish the factorization formalism at sub-leading power for these processesat the one-loop level.

Transverse-momentum-dependent factorization at next-to-leading power