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We show that as a result of causality-constrained coordinate spaceanalyticity, the Drell-Yan-shape transverse-momentum dependent soft factor inthe exponential regulator allows Euclidean-type parametric representationswithout cuts, to all orders in perturbation theory. Moreover, it is identicalto another soft factor defined with a single time-ordering that has a naturalinterpretation as a space-like form factor. Furthermore, this relationcontinues to hold for a larger class of TMD soft factors that interpolatebetween three different rapidity regulators: the off-light-cone regulator, thefinite light-front length regulator, and the exponential regulator.

Revisiting the time-ordering issue of TMD soft factors: causality, coordinate space analyticity and new equalities