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We study the transverse momentum dependent (TMD) parton distributions in thenewly proposed quasi-parton distribution function framework in Euclidean space.A soft factor subtraction is found to be essential to make the TMDs calculableon lattice. We show that the quasi-TMDs with the associated soft factorsubtraction can be applied in hard QCD scattering processes such as Drell-Yanlepton pair production in hadronic collisions. This allows future latticecalculations to provide information on the non-perturbative inputs and energyevolutions for the TMDs. Extension to the generalized parton distributions andquantum phase space Wigner distributions will lead to a complete nucleontomography on lattice.

Soft Factor Subtraction and Transverse Momentum Dependent Parton Distributions on Lattice