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We present a simple observation about soft amplitudes and soft functionsappearing in factorizable cross sections in $ee$, $ep$, and $pp$ collisionsthat has not clearly been made in previous literature, namely, that thehemisphere soft functions that appear in event shape distributions in$e^+e^-\to$ dijets, deep inelastic scattering (DIS), and in Drell-Yan (DY)processes are equal in perturbation theory up to $\mathcal{O}(\alpha_s^2)$,even though individual amplitudes may have opposite sign imaginary parts due tochanging complex pole prescriptions in eikonal propagators for incoming vs.outgoing lines. We also explore potential generalizations of this observationto soft functions for other observables or with more jets in the final state.

Equality of hemisphere soft functions for $e^+e^-$, DIS and $pp$ collisions at $\mathcal{O}(\alpha_s^2)$