Three-body effective range and other properties for ultracold bosonic atoms near a three-body threshold

The two-body scattering length and the effective range are important parameters for ultracold atoms. We introduce an analogous concept of \emph{three-body} effective range for three bosonic atoms near a three-body threshold. This will be useful for ultracold atoms near low-energy three-body resonances. It may also be relevant for certain nuclear systems near three-body resonances. For three bosons with large two-body scattering length, which display the Efimov effect, we derive \emph{analytical} results for the three-body effective range and the two-body and three-body contacts at the three-body threshold.