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 ΛEFT, we derive cutoff-dependent bounds valid for arbitrary UV completions. Taking ΛEFT→∞ 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.
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BibTeX
@misc{lippstreu2026unitarity,
title = {{Unitarity and the Forward Direction in Theories with Long-Range Forces}},
author = {Luke Lippstreu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16896},
eprint = {2609.16896},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16896},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unitarity and the Forward Direction in Theories with Long-Range Forces
AU - Luke Lippstreu
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16896
UR - https://arxiv.org/abs/2609.16896
AB - 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_EFT, we derive cutoff-dependent bounds valid for arbitrary UV completions. Taking Λ_EFT→∞ 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.
ER -