Effective Three-Boson Interactions using a Separable Potential

Abstract

Effective field theories (EFTs) are widely used to study many-body systems by describing two-body interactions using zero-ranged contact potentials. However, when extended to three-body processes, these contact interactions lead to divergences due to the absence of an intrinsic length scale. In EFT, this is typically resolved by introducing a zero-ranged three-body interaction, which can be renormalized to make the low-energy physics independent of the short-distance physics. However, when the two-body potential has a finite range, such as in separable potentials, there is no need for such renormalization. In this work, we derive the integral equation for the three-body scattering amplitude for separable potentials, and solve it in the strongly-interacting regime. With our model, we retrieve the known analytic form of the scattering amplitude for inelastic scattering processes and formulate a new scaling law for elastic three-body scattering processes.

Figures (6)

• Figure 1: A schematic overview of the scattering amplitudes is shown for the separable model of Eq. \ref{['AC']} in panel (a), and for the EFT model of Eq. \ref{['As_EFT']} in panel (b). Panel (c) displays the corresponding series for the full diatom propagator (thick solid line) given in Eq. \ref{['eq:DiatomPropagator']}. In all panels thin single lines denote single particle propagators, while the double-lines in (b) and (c) denote the free diatom propagator. (Figures (b) and (c) were adapted from Braaten2006.)
• Figure 2: Representation of the wavenumbers of the Efimov states as a function of the scattering length $a$ for a cutoff of $\Lambda=1$. The deepest trimers with $\kappa_*^{(0)}$ are displayed here in dashed blue, while the first excited state $\kappa_*^{(1)}$ is plotted here in dotted red. Both these curves end at the black solid line, which represents the dimer binding energy.
• Figure 3: Relationship between the Efimov trimer wavenumbers $\kappa_*^{(0)}$ and the cutoff $\Lambda$. All the parameters are expressed in units of the inverse van der Waals length vandekraats2024.
• Figure 4: The scattering amplitude $\mathcal{A}_s(p,k,E\rightarrow 0)$ form the separable model presented as a function of the momenta $p/\Lambda$ and $k/\Lambda$. Here, $p$ represents the momentum of the incoming diatom and particle, while $k$ represents the same for the outgoing particle and diatom. This is plotted in the low-energy limit $E \rightarrow 0$. The blue-green color bar denotes regions where the scattering amplitude is negative, while the red-yellow regions denote regions of positive scattering amplitudes.
• Figure 5: The scattering amplitudes $\mathcal{A}_s(p,k\rightarrow 0,E\rightarrow 0)$ presented as a function of the momentum $p$ in units of the cutoff on a log-log scale. Here, $p$ represents the momentum of the incoming diatom and particle, while $k$ represents the same for the outgoing particle and diatom. This is plotted in the low-energy limit $E \rightarrow 0$, and for inelastic scattering where the outgoing particles have small momentum $k \rightarrow 0$. Regions where $A_s(p,k\to 0, E\to 0)<0$ are not shown for simplicity.