  Transverse momentum dependent parton distributions (TMDPDFs) which appear in
factorized cross sections involve infinite Wilson lines with edges on or close
to the light-cone. Since these TMDPDFs are not directly calculable with a
Euclidean path integral in lattice QCD, we study the construction of
quasi-TMDPDFs with finite-length spacelike Wilson lines that are amenable to
such calculations. We define an infrared consistency test to determine which
quasi-TMDPDF definitions are related to the TMDPDF, by carrying out a one-loop
study of infrared logarithms of transverse position $b_T\sim \Lambda_{\rm
QCD}^{-1}$, which must agree between them. This agreement is a necessary
condition for the two quantities to be related by perturbative matching.
TMDPDFs necessarily involve combining a hadron matrix element, which nominally
depends on a single light-cone direction, with soft matrix elements that
necessarily depend on two light-cone directions. We show at one loop that the
simplest definitions of the quasi hadron matrix element, the quasi soft matrix
element, and the resulting quasi-TMDPDF all fail the infrared consistency test.
Ratios of impact parameter quasi-TMDPDFs still provide nontrivial information
about the TMDPDFs, and are more robust since the soft matrix elements cancel.
We show at one loop that such quasi ratios can be matched to ratios of the
corresponding TMDPDFs. We also introduce a modified "bent" quasi soft matrix
element which yields a quasi-TMDPDF that passes the consistency test with the
TMDPDF at one loop, and discuss potential issues at higher orders.
