  We study Euclidean formulations of the transverse-momentum-dependent (TMD)
soft function, which is a cross section for soft gluon radiations involving
color charges moving in two conjugate lightcone directions in quantum
chromodynamics. We show it is related to a special form factor of a pair of
color sources traveling with nearly-lightlike velocities, which can be matched
to TMD physical observables in semi-inclusive deep-inelastic scattering and
Drell-Yan process in the framework of large momentum effective theory. It can
also be extracted by combining a large-momentum form factor of light meson and
its leading TMD wave function. These formulations are useful for initiating
nonperturbative calculations of this useful quantity.
