Unitarity and the Forward Direction in Theories with Long-Range Forces

Abstract

Integrating scattering amplitudes over the forward direction, and consequently the unitarity constraints one extracts from such integrals, can appear ambiguous in theories with long-range forces. Standard techniques for bounding EFT couplings then typically produce bounds that depend on an arbitrary infrared scale. We study a non-relativistic model in which the standard techniques produce such an ambiguous bound, but which is simple enough that the exact bound can also be derived non-perturbatively and shown to involve no infrared scale. We then show how to derive bounds perturbatively, with no infrared scale entering at any stage. This requires two ingredients: accounting for the modified distributional structure of long-range amplitudes, and using distorted-wave perturbation theory (DWPT), which treats the Coulomb dynamics exactly. Together they give amplitudes with well-defined partial-wave projections and no spurious infrared divergences. Interpreting the model as an EFT with an ultraviolet cutoff $Λ_{\rm EFT}$, we derive cutoff-dependent bounds valid for arbitrary UV completions. Taking $Λ_{\rm EFT}\to\infty$ at each order yields bounds that rapidly converge to the exact bound, reaching $0.002\%$ accuracy at fourth order in the expansion. Finally, we demonstrate that every order of the DWPT expansion resums infinitely many Feynman diagrams of short-range perturbation theory, which individually evaluate to multiple polylogarithms and complete elliptic integrals, into a compact expression. At the orders we compute, no integration is even required, suggesting that DWPT may offer a simpler representation of scattering amplitudes than standard perturbation theory.