For identical bosons with a large scattering length, the dependence of the three-body recombination rate on the collision energy is determined in the zero-range limit by universal functions of a single scaling variable. There are six scaling functions for angular momentum zero and one scaling function for each higher partial wave. We calculate these universal functions by solving the Skorniakov-Ter-Martirosian equation. The results for the three-body recombination as a function of the collision energy are in good agreement with previous results from solving the three-body Schr\"odinger equation for
4He atoms. The universal scaling functions can be used to calculate the three-body recombination rate at nonzero temperature. We obtain an excellent fit to the data from Grimm and co-workers for
133Cs atoms with a large positive scattering length.
For atoms with large scattering length, the dependence of the three-body recombination rate on the collision energy is determined by the scattering length and the Efimov three-body parameters and can be expressed in terms of universal functions of a single scaling variable. We use published results on the three-body recombination rate for
4He atoms to constrain the universal functions. We then use those universal functions to calculate the three-body recombination rate for other atoms with large scattering length at nonzero temperature. The constraints from the
4He results are strong if the scattering length is near the minimum of the three-body recombination rate at threshold. We apply our results to
133Cs atoms with a large positive scattering length, and compare them with experimental results from the Innsbruck group.
We analyze the zero energy collision of three identical bosons in the same internal state with total orbital angular momentum
L=2, assuming short range interactions. By solving the Schrödinger equation asymptotically, we derive two expansions of the wave function when three bosons are far apart or a pair of bosons and the third boson are far apart. The scattering hypervolume
D is defined for this collision. Unlike the scattering hypervolume defined by one of us in 2008, whose dimension is length to the fourth power, the dimension of
D studied in the present paper is length to the eighth power. We then derive the expression of
D when the interaction potentials are weak, using the Born's expansion. We also calculate the energy shift of such three bosons with three different momenta
ℏk1,
ℏk2 and
ℏk3 in a large periodic box. The obtained energy shift depends on
D(0)/Ω2 and
D/Ω2, where
D(0) is the three-body scattering hypervolume defined for the three-body
L=0 collision and
Ω is the volume of the periodic box. We also calculate the contribution of
D to the three-body T-matrix element for low-energy collisions. We then calculate the shift of the energy and the three-body recombination rate due to
D(0) and
D in the dilute homogeneous Bose gas. The contribution to the three-body recombination rate constant from
D is proportional to
T2 if the temperature
T is much larger than the quantum degeneracy temperature but still much lower than the temperature scale at which the thermal de Broglie wave length becomes comparable to the physical range of interaction.