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Effective-Range Expansion with a Long-Range Force

Meng-Lin Du ^1,*, Feng-Kun Guo ^2,3,4,†, and Bing Wu ^1,‡

¹School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China
²Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
³School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
⁴Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China

^*Contact author: du.ml@uestc.edu.cn
^†Contact author: fkguo@itp.ac.cn
^‡Contact author: wu.bing@uestc.edu.cn

Phys. Rev. Lett. 135, 011903 – Published 2 July, 2025

DOI: https://doi.org/10.1103/cwdt-dj6z

Abstract

The validity range of the time-honored effective range expansion can be very limited due to the presence of a left-hand cut close to the two-particle threshold. Such a left-hand cut arises in the two-particle interaction involving a light particle exchange with a small mass or a mass slightly heavier than the mass difference of the two particles, identified as a long-range force, a scenario encountered in a broad range of systems. This can hinder a precise extraction of low-energy scattering observables and resonance poles. To address this issue, we propose a new parametrization for the low-energy scattering amplitude that accounts for the left-hand cut. The parametrization is like a Padé approximation but with nonanalytic terms from the left-hand cut and can be regarded as an extension of the effective range expansion. This parametrization is versatile and applicable to a broad range of systems with Yukawa-type interactions, including particle, hadronic, nuclear, cold atom, and quantum gas systems. In particular, it should be invaluable in understanding various near-threshold hadron resonances. As byproducts, we also show that the parametrization can be used to extract the couplings of the exchanged particle to the scattering particles, and derive expressions for amplitude zeros caused by the interplay between the short- and long-range interactions.

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Phys. Rev. Lett. 135, 011903– Published 2 July, 2025

Vol. 135, Iss. 1 — 2 July 2025

Received 2 September 2024
Revised 13 June 2025
Accepted 17 June 2025

DOI: https://doi.org/10.1103/cwdt-dj6z

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Effective-Range Expansion with a Long-Range Force

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