Transverse momentum dependent TMD factorization at next to leading power and TMD observables

The study of the three-dimensional (3-D)-longitudinal and intrinsic transverse momentum-and spin structure of the nucleon emerged from studies of power-suppressed/next to leading power (NLP) contributions to semi-inclusive deep inelastic scattering (SIDIS) observables. In this respect the NLP cosφ azimuthal modulation--so called Cahn effect-- was important for the development of the TMD field, since one of the first topics of transverse parton momentum in DIS is related to this observable. In fact, the first observed transverse single spin asymmetry (SSA) in SIDIS was a sizable power suppressed target SSA for pion production from the HERMES collaboration. Generally, although typically suppressed by Λ_QCD/Q where Q indicates the hard scale, NLP TMD observables are not small. New measurements of the Cahn-like effect from the COMPASS collaboration have re-established the importance of this fundamental observable to unfolding the 3-D partonic structure of the nucleon. In this talk I will present our latest work on establishing TMD factorization at NLP. We emphasize the role of renormalization group consistency as a necessary condition establishing TMD factorization at NLP. We also discuss the criteria of matching large and small transverse momentum of the cross section in relation to establishing TMD factorization at NLP. Establishing TMD factorization at NLP power is crucial for performing 3D imaging of hadrons in present and future DIS experiments and therefore for on global analyses for both transversity studies, as well as SSA studies going forward in the EIC era.