Effective range expansion with the left-hand cut and its application to the $T_{cc}(3875)$

TL;DR

This paper introduces a new parameterization for low-energy scattering amplitudes that accounts for left-hand cuts, extending the effective range expansion's validity and improving analysis of systems like the $T_{cc}(3875)$.

Contribution

A novel parameterization that incorporates left-hand cuts, enhancing the effective range expansion's convergence and applicability to various physical systems.

Findings

Extends the convergence radius of the effective range expansion.

Enables extraction of coupling strengths between exchange particles and scattering particles.

Demonstrates effectiveness through application to $DD^*$ scattering with lattice QCD data.

Abstract

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.