Current status of TMD structure functions and their connections to the collinear twist-3 functions

The origin of the single transverse-spin asymmetry(SSA) has been a mystery in high energy hadron physics since the first observation in the late 1970s. Transverse-momentum-dependent(TMD) factorization and the collinear twist-3 factorization are known as successful perturbative QCD frameworks in describing the data of the SSA. Although these frameworks basically have different applicable regions, there is the intermediate region where both frameworks are available. The relationship between the frameworks is important for the unified description of the origin of the SSA in the full kinematic region. There are two major sources of the SSA, Sivers effect and Collins effect. Sivers effect has drawn attention also in the context of the proton spin structure because it describes the correlation between the spin and the orbital motion of partons inside the proton. In this talk, I will review the current status of our understanding about the nonperturbative TMD functions and the relationship to the collinear twist-3 functions.