  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.
