Equality of hemisphere soft functions for e+e−e+e−, DIS and pp collisions at O(αs2)

We present a simple observation about soft amplitudes and soft functions appearing in factorizable cross sections in ee, ep, and pp   collisions that has not clearly been made in previous literature, namely, that the hemisphere soft functions that appear in event shape distributions in e+e−→dijetse+e−→dijets, deep inelastic scattering (DIS), and in Drell–Yan (DY) processes are equal in perturbation theory up to O(αs2), even though individual amplitudes may have opposite sign imaginary parts due to changing complex pole prescriptions in eikonal propagators for incoming vs. outgoing lines. We also explore potential generalizations of this observation to soft functions for other observables or with more jets in the final state.