  We show that as a result of causality-constrained coordinate space
analyticity, the Drell-Yan-shape transverse-momentum dependent soft factor in
the exponential regulator allows below-threshold (Euclidean) parametric
representations without cuts, to all orders in perturbation theory. Moreover,
it is identical to another soft factor with natural interpretation as a
space-like form factor and this relation continues to hold for a larger class
of TMD soft factors that interpolate between three different rapidity
regulators: the off-light-cone regulator, the finite light-front length
regulator and the exponential regulator.
