Soft factor subtraction and transverse momentum dependent parton distributions on the lattice

We study the transverse momentum dependent (TMD) parton distributions in the newly proposed quasiparton distribution function framework in Euclidean space. In this framework, the parton distributions can be extracted from lattice observables in a systematic expansion of 1/P-z where P-z is the hadron momentum. A soft factor subtraction is found to be essential to make the TMDs calculable on the lattice. We show that the quasi-TMDs with the associated soft factor subtraction can be applied in hard QCD scattering processes such as Drell-Yan lepton pair production in hadronic collisions. This allows future lattice calculations to provide information on the nonperturbative inputs and energy evolutions for the TMDs. Extension to the generalized parton distributions and quantum phase space Wigner distributions will lead to a complete nucleon tomography on the lattice.