  In large-momentum effective theory (LaMET), the transverse-momentum-dependent
(TMD) light-front wave functions and soft functions can be extracted from the
simulation of a four-quark form factor and equal-time correlation functions. In
this work, using expansion by regions we provide a one-loop proof of TMD
factorization of the form factor. For the one-loop validation, we also present
a detailed calculation of ${\cal O}(\alpha_s)$ perturbative corrections to
these quantities, in which we adopt a modern technique for the calculation of
TMD form factor based the integration by part and differential equation. The
one-loop hard functions are then extracted. Using lattice data from Lattice
Parton Collaboration on quasi-TMDWFs, we estimate the effects from the one-loop
matching kernel and find that the perturbative corrections depend on the
operator to define the form factor, but are less sensitive to the transverse
separation. These results will be helpful to precisely extract the soft
functions and TMD wave functions from the first-principle in future.
