Universal relations for identical bosons from three-body physics

Systems consisting of identical bosons with a large scattering length satisfy universal relations determined by 2-body physics that are similar to those for fermions with two spin states. They require the momentum distribution to have a large-momentum 1/k(4) tail and the radio-frequency transition rate to have a high-frequency 1/ω(3/2) tail, both of which are proportional to the 2-body contact. Identical bosons also satisfy additional universal relations that are determined by 3-body physics and involve the 3-body contact, which measures the probability of 3 particles being very close together. The coefficients of the 3-body contact in the 1/k(5) tail of the momentum distribution and in the 1/ω(2) tail of the radio-frequency transition rate are log-periodic functions of k and ω that depend on the Efimov parameter.