Effective Range Expansion with the Left-Hand Cut: Higher Order Improvements

TL;DR

This paper extends the effective-range expansion to include higher-order terms and relativistic effects, improving the modeling of low-energy scattering amplitudes with the left-hand cut for more accurate two-body scattering analysis.

Contribution

It introduces a higher-order, relativistic version of the effective-range expansion that systematically improves the description of near-threshold scattering with the left-hand cut.

Findings

The generalized ERE remains reliable with higher-order terms.

Relativistic effects enhance the robustness of the ERE.

The framework is applicable to various two-body scattering problems.

Abstract

A model-independent parameterization of the low-energy scattering amplitude that incorporates the left-hand cut from one-particle exchange, an extension of the conventional effective-range expansion (ERE), was recently proposed and successfully applied to the low-energy $DD^*$ system [Phys. Rev. Lett. 135, 011903 (2025)]. While the original formulation is based on a nonrelativistic approximation and is thus limited to a [1,1] approximant for self-consistency, we extend the framework by explicitly including the higher-order terms up to $\mathcal{O}(k^6)$. We systematically investigate the reliability and robustness of the generalized ERE by incorporating relativistic kinematic effects. In addition, we develop a relativistic version of the ERE that accounts for lhc contributions. These results affirm the generalized ERE as a robust and systematically improvable framework for near-threshold scattering processes, providing both analytical and numerical reliability for applications in two-body scattering problems with a particle exchange.