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

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.