Unitarity and the Forward Direction
Abstract: Most theorems of scattering theory assume that particles stop interacting at late times, an assumption violated by four-dimensional theories containing photons and gravitons. I will describe how theories with long-range forces have genuinely new analytic properties: no identity term in the S-matrix, a new distributional structure in the forward direction, and a modified optical theorem. As an application, I will turn to a puzzle in the S-matrix bootstrap: applied to long-range forces, standard techniques give bounds on couplings that depend on an arbitrary infrared scale. In a solvable model whose exact bound is known independently, accounting for these novel analytic properties, together with distorted-wave perturbation theory, removes the infrared scale and gives bounds that converge rapidly to the exact one. The same framework resums infinite families of Feynman diagrams, including diagrams involving multiple polylogarithms and elliptic integrals, into compact expressions. This suggests that a better treatment of long-range dynamics may also simplify multi-loop amplitudes.