Three-body scattering area of identical bosons in two dimensions

TL;DR

This paper develops a comprehensive 2D three-body framework for identical bosons with finite two-body scattering length $a$, introducing the three-body scattering area $D$ that governs the large-distance asymptotics of the zero-energy three-body wave function. By constructing 21- and 111-expansions and defining two-body special functions, the authors obtain explicit asymptotic forms for $\phi^{(3)}$ and show that $D$ appears at order $1/(B^2 \ln^3 \widetilde{B})$ in the 111-expansion, with $ ext{Im}\,D$ signaling three-body decay channels when two-body bound states exist. They also derive how small perturbations in two- and three-body potentials shift $D$ and connect these few-body parameters to many-body physics, yielding finite-volume energy shifts, the thermodynamic-energy density shift, and the temperature-dependent three-body recombination rate $L_3$ that scales with $|\mathrm{Im} D|$ and logarithmic factors. The results establish a nonuniversal 2D three-body parameter that links microscopic interactions to macroscopic observables in ultracold Bose gases, including the dilute Bose gas energy corrections and three-body loss processes.

Abstract

We study the wave function $φ^{(3)}$ of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length $a$. We derive asymptotic expansions of $φ^{(3)}$ in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as $\ln^3(B/a)$ at large hyperradius $B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}$. At order $B^{-2}\ln^{-3}(B/a)$, we identify a three-body parameter $D$ with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, $D$ acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of $D$ in terms of $φ^{(3)}$ and the changes of the two-body and three-body potentials. We also study the effects of $D$ and $φ^{(3)}$ on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.

Figures (2)

• Figure 1: Real (red solid line) and imaginary (blue dashed line) parts of $\Omega_2$ as functions of $\widetilde{R}$. Insets show the real and the imaginary parts of $\Omega_2$ at large values of $\log_{10}\widetilde{R}$ or $\widetilde{R}$, and they are consistent with Eq. \ref{['Omega2 expansion']} and Eq. \ref{['ImOmega2']} respectively.
• Figure 2: The real part (red solid line) and the imaginary part (blue dashed line) of $G_2(\widetilde{B},\alpha)$ as functions of $\widetilde{B}$ for hyper-angles $\alpha = 0$, $\pi/12$, $\pi/6$, $\pi/4$, $\pi/3$, $5\pi/12$. The insets show the real and the imaginary parts of $G_2(\widetilde{B},\alpha)$ at large $\widetilde{B}$.