BEC-BCS Crossover in the Epsilon Expansion
  The epsilon expansion (expansion around four spacial dimensions) developed by Nishida and Son for a cold fermi gas with infinite scattering length is extended to finite scattering length to study the BEC-BCS crossover. A resummation of higher order logarithms and a suitable extension of fermion coupling in d-dimensions are developed in order to apply the theory in the BCS regime. The ratio between the chemical potential and the Fermi energy, mu/eF, is computed to next-to-leading order in the epsilon expansion as a function of eta=1/(a kF), where a is the scattering length and kF is the Fermi momentum in a non-interacting system. Near the unitarity limit eta->0, we found mu/eF=0.475-0.707 eta-0.5 eta^2. Near the BEC limit eta->infinity, mu/eF=0.062/eta - eta^2, while near the BCS limit eta->-infinity, mu/eF=1+0.707/eta. Overall good agreement with Quantum Monte Carlo results is found. 
Jiunn-Wei Chen and Eiji Nakano