Three-body scattering area of identical bosons in two dimensions

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.

💡 Research Summary

This paper presents a comprehensive theoretical analysis of three identical bosons scattering in two dimensions (2D) at zero collision energy, zero total momentum, and zero orbital angular momentum, interacting via short‑range potentials characterized by a finite two‑body scattering length a. The authors develop two complementary asymptotic expansions of the three‑body wave function ϕ^(3)(r₁,r₂,r₃): the “111‑expansion,” valid when all three pairwise distances s₁, s₂, s₃ become large simultaneously, and the “21‑expansion,” applicable when one particle is far from a tightly bound pair (distance R≫s≫range). By introducing the hyperradius B=√(s₁²+s₂²+s₃²)/√2, the 111‑expansion is written as ϕ^(3)=∑{i≥0}T^{(-i)}(r₁,r₂,r₃) with each term scaling as B^{-i} multiplied by possible logarithmic factors. The leading term T^{(0)} grows only as a power of ln B, reflecting the logarithmic nature of 2D two‑body scattering (ϕ(s)=ln(s/a)). The 21‑expansion similarly reads ϕ^(3)=∑{j≥0}S^{(-j)}(R,s), where S^{(0)}(R,s)=F₀(R) ϕ(s) and F₀(R) behaves as R⁰ lnⁿR.

A central result emerges at order B^{-2} ln^{-3}(B/a) in the 111‑expansion: a new dimensionful parameter D with dimensions of length squared appears. The authors name D the “three‑body scattering area.” Unlike the three‑body hypervolume defined in three dimensions or the three‑body scattering area previously studied for a→∞ or a→0, D exists only for finite a and encodes genuine three‑body physics beyond two‑body parameters a and the effective range r_s. When the two‑body interaction is attractive and supports bound dimers, D acquires a negative imaginary part. The paper derives a precise relation between Im D and the probability amplitudes for producing a two‑body bound state in a three‑body collision, namely Im D = −(π/2) |A_{2b}|², where A_{2b} is the dimer‑production amplitude. This establishes a direct link between the three‑body parameter and observable three‑body recombination loss rates.

The authors also treat the response of D to infinitesimal changes in the interaction potentials. By applying first‑order perturbation theory to the Schrödinger equation, they express the shift ΔD in terms of integrals over the unperturbed three‑body wave function ϕ^(3) and the variations of the two‑ and three‑body potentials, δU₂ and δU₃. This formalism shows that D can be tuned experimentally, for example via magnetic Feshbach resonances or optical lattice modulations, and that its variation can be predicted from the known wave function.

Having defined D, the paper explores its consequences for several many‑body and few‑body observables:

1. Three‑body ground‑state energy in a periodic box. For a large square of side L with periodic boundary conditions, the leading finite‑size correction to the three‑boson ground‑state energy is E₃(L)= (6πħ²/mL²)