Particle distribution tail and related energy formula

We present a simple derivation of the energy formula found by Tan, relative to the single channel hamiltonian relevant for ultracold Fermi gases. This derivation is generalized to particles with different masses, to arbitrary mixtures, and to two-dimensional space. We show how, in a field theoretical approach, the $1/k^4$ tail in the momentum distribution and the energy formula arise in a natural way. As a specific example, we consider quantitative calculations of the energy, from different formulas within the ladder diagrams approximation in the normal state. The comparison of the results provides an indication on the quality of the approximation.

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