The validity range of the widely used traditional effective range expansion can be severely limited by the presence of a left-hand cut near the two-particle threshold. Such a left-hand cut emerges in two-particle scattering processes involving either a light particle exchange in the $t$-channel or a particle exchange with a mass slightly heavier than the mass difference of the two particles in the $u$-channel, which occurs in a wide range of physical systems.
We propose a new parameterization for the low-energy scattering amplitude that incorporates these left-hand cuts arising from particle exchange diagrams. This parameterization extends the convergence radius of the effective range expansion beyond the branch point of the left-hand cut and is applicable to a broad range of systems.
The parameterization enables the extraction of coupling strengths between the exchange particle and the scattering particles, and reveals amplitude zeros resulting from the interplay between short- and long-range interactions. We demonstrate the effectiveness of this new parameterization through its application to $DD^*$ scattering with meson masses obtained in a lattice QCD calculation.