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

TL;DR

This work addresses near-threshold two-body scattering by incorporating the left-hand cut (lhc) from one-particle exchange into a generalized, model-independent effective-range expansion (ERE). It systematically extends the framework to higher orders in $k^2$ up to $ ext{O}(k^6)$ and develops a kinematically relativistic version that accounts for lhc effects, using an $N/D$ decomposition to separate left- and right-hand cuts and introducing a flexible $[m,n,ℓ]$ truncation. The authors validate the approach by comparing to Lippmann-Schwinger solutions for a Yukawa potential across a wide range of couplings, showing accurate reproduction of phase shifts and amplitudes, including near the lhc branch points and pole structures. They also provide closed-form expressions for the relativistic lhc contributions, enabling a consistent treatment in relativistic kinematics. The resulting framework is a robust, systematically improvable tool for near-threshold scattering in two-body systems with particle exchange, with broad applicability to hadronic and related quantum systems.

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.

Figures (7)

• Figure 1: (a) $t$-channel and (b) $u$-channel scattering processes mediated by one-particle exchange. Here, $g_{1,2}$ and $g$ denote the coupling constants, and $p_i$ represent the momenta of the scattering particles.
• Figure 2: Comparison between the generalized ERE of the [0,1] approximant in Eq. \ref{['eq:ere:mn']} (dashed curves) and the LSE solution of Eq. \ref{['eq:LSE']} (solid curves). Upper panel: $k\cot\delta$; lower panel: the scattering amplitude. The parameters of the $[0,1]$ approximant are determined only from the real parts of the phase shift $k\cot{\delta}$ of the LSE solutions. It is noteworthy that the imaginary parts of $k\cot\delta$ are also well reproduced by Eq. \ref{['eq:ere:mn']}. For comparison, both the real and imaginary parts of the amplitude derived from the generalized ERE and the LSE are displayed. In the cases $g=-20$ and $=-10$, the Yukawa potentials are repulsive and no pole appears. For $g=5$, the attraction is weak and leads to a virtual state. The vertical dotted lines marks the branch point of the lhc.
• Figure 3: Comparison between the generalized ERE of the [0,1] approximant in Eq. \ref{['eq:ere:mn']} (dashed curves) and the LSE solution of Eq. \ref{['eq:LSE']} (solid curves). Upper panel: $k\cot\delta$; lower panel: the scattering amplitude. The parameters of the $[0,1]$ approximant are determined only from the real parts of the phase shift $k\cot{\delta}$ of the LSE solutions for $g=40$, $20$ and $13$ in order. Note the imaginary parts of $k\cot\delta$ are properly reproduced by Eq. \ref{['eq:ere:mn']}. Both the real and imaginary parts of the amplitude derived from the generalized ERE and the LSE are also displayed. For $g=40$, the potential supports a very deep bound state ($E_\text{bind}\approx$1 GeV); $g=20$ gives rise to a bound state near the lhc branch point ($E_\text{bind}$=71 MeV); while for $g=13$, a very shallow bound state is formed ($E_\text{bind}$=0.1 MeV). The vertical dotted line indicates the branch point of the lhc.
• Figure 4: Comparison between the generalized ERE of the [0,1,1] and [1,1,1] approximants in Eq. \ref{['eq:ere:mnl']} (dashed curves) and the LSE solution of Eq. \ref{['eq:LSE']} (solid curves) for $g=20$. The parameters are determined by a direct fit to the LSE amplitude. The vertical dotted line marks the branch point of the lhc.
• Figure 5: Comparison between $u$-channel lhc contributions calculated up to $\mathcal{O}(k^2)$, $\mathcal{O}(k^4)$, $\mathcal{O}(k^6)$ and the full Yukawa potential \ref{['eq:YOPE_u']}. The black curves represent the real (solid) and imaginary (dashed) parts of the Yukawa potential from Eq. \ref{['eq:YOPE_u']} under the on-shell condition, and the remaining curves correspond to the real (solid) and imaginary (dashed) parts obtained from Eqs. \ref{['eq:L']}, \ref{['eq:L4']}, \ref{['eq:L6']}, in order.