Three-body scattering area of identical bosons in two dimensions

Abstract

We study the wave function ${\varphi }^{(3)}$ 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 ${\varphi }^{(3)}$ 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 ${\varphi }^{(3)}$ and the changes of the two-body and three-body potentials. We also study the effects of $D$ and ${\varphi }^{(3)}$ 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.

18 pages, 2 figures