Universal Relations for Identical Bosons from 3-Body Physics

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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/k⁴ tail and the radio-frequency transition rate to have a high-frequency 1/omega³/² 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/k⁵ tail of the momentum distribution and in the 1/omega² tail of the radio-frequency transition rate are log-periodic functions of k and omega that depend on the Efimov parameter.