Three-body scattering area for particles with infinite or zero scattering length in two dimensions

We derive the asymptotic expansions of the wave function of three particles having equal mass with finite-range interactions and infinite or zero two-dimensional scattering length colliding at zero energy and zero orbital angular momentum, from which a three-body parameter $D$ is defined. The dimension of $D$ is length squared, and we call $D$ three-body scattering area. We find that the ground state energy per particle of a zero-temperature dilute Bose gas with these interactions is approximately $\frac{\hbar^2 D }{6m}\rho^2$, where $\rho$ is the number density of the bosons, $m$ is the mass of each boson, and $\hbar$ is Planck's constant over $2\pi$. Such a Bose gas is stable at $D\geq 0$ in the thermodynamic limit, and metastable at $D<0$ in the harmonic trap if the number of bosons is less than $N_{cr}\approx 3.6413 \sqrt{\frac{\hbar}{m\omega |D|}}$, where $\omega$ is the angular frequency of the harmonic trap. If the two-body interaction supports bound states, $D$ typically acquires a negative imaginary part, and we find the relation between this imaginary part and the amplitudes of the pair-boson production processes. We derive a formula for the three-body recombination rate constant of the many-boson system in terms of the imaginary part of $D$.