What is the $i\varepsilon$ for the S-matrix?

Understand

Can the S-matrix be complexified in a way consistent with causality? Since the 1960's, the affirmative answer to this question has been well-understood for $2 \to 2$ scattering of the lightest particle in theories with a mass gap at low momentum transfer, where the S-matrix is analytic everywhere except at normal-threshold branch cuts.

• We ask whether an analogous picture extends to realistic theories, such as the Standard Model, that include massless fields, UV/IR divergences, and unstable particles.
• Especially in the presence of light states running in the loops, the traditional $i\varepsilon$ prescription for approaching physical regions might break down, because causality requirements for the individual Feynman diagrams can be mutually incompatible.
• We demonstrate that such analyticity problems are not in contradiction with unitarity.