Two-body and Three-body Contacts for Identical Bosons near Unitarity

In a recent experiment with ultracold trapped $^{85}$Rb atoms, Makotyn et al. have studied a quantum-degenerate Bose gas in the unitary limit where its scattering length is infinitely large. We show that the observed momentum distributions are compatible with a universal relation that expresses the high-momentum tail in terms of the 2-body contact $C_2$ and the 3-body contact $C_3$. We determine the contact densities for the unitary Bose gas with number density $n$ to be ${\cal C}_2 \approx 20~n^{4/3}$ and ${\cal C}_3 \approx 2~n^{5/3}$. We also show that the observed atom loss rate is compatible with that from 3-atom inelastic collisions, which gives a contribution proportional to $C_3$, but the loss rate is not compatible with that from 2-atom inelastic collisions, which gives a contribution proportional to $C_2$. We point out that the contacts $C_2$ and $C_3$ could be measured independently by using the virial theorem near and at unitarity, respectively.