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arXiv:2601.20509

Three-body scattering area of identical bosons in two dimensions

Junjie Liang, Hongye Yu, Shina Tan

cond-mat.quant-gasphysics.atom-phquant-phSubmitted Jan 28, 2026• Announced Jan 29, 2026• 0 citations
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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.

Comments
18 pages, 2 figures

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We study the wave function φ(3)φ^{(3)}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 aaa. We derive asymptotic expansions of φ(3)φ^{(3)}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)\ln^3(B/a)ln3(B/a) at large hyperradius B=(s12+s22+s32)/2B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}B=(s12​+s22​+s32​)/2​. At order B−2ln⁡−3(B/a)B^{-2}\ln^{-3}(B/a)B−2ln−3(B/a), we identify a three-body parameter DDD 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, DDD 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 DDD in terms of φ(3)φ^{(3)}φ(3) and the changes of the two-body and three-body potentials. We also study the effects of DDD and φ(3)φ^{(3)}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.
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