Effective range expansion with a long-range force

TL;DR

The paper introduces a new parameterization for low-energy scattering amplitudes that incorporates long-range forces and left-hand cuts, improving the analysis of near-threshold resonances across various physical systems.

Contribution

It extends the effective range expansion by including nonanalytic terms from long-range forces, applicable to diverse systems with Yukawa interactions.

Findings

New parameterization accounts for left-hand cuts in scattering amplitudes.

Applicable to particle, hadronic, nuclear, cold atom, and quantum gas systems.

Enables extraction of exchanged particle couplings and analysis of amplitude zeros.

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 mass small or 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 parameterization for the low-energy scattering amplitude that accounts for the left-hand cut. The parameterization is like a Pad\'e approximation but with nonanalytic terms from the left-hand cut and can be regarded as an extension of the effective range expansion. This parameterization 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 parameterization 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.