  We develop an effective field theory to describe the superfluid pairing in
strongly interacting fermions with arbitrary short-range attractions, by
extending Kaplan's idea of coupling fermions to a fictitious boson state in
Nucl. Phys. B \textbf{494}, 471 (1997). This boson field is assigned with
unconventional kinetic term to recover the exact scattering phase shift
obtained either from scattering data or model calculations. The theory works
even if the explicit form of the interaction potential has not been constructed
from scattering data. The contact boson-fermion coupling allows us to go beyond
mean-field to include Gaussian pair fluctuations, yielding reliable predictions
on equations of state. As an application, we use our theory to address the
non-univerisal ground-state energy of strongly paired fermions, due to the
non-trivial momentum dependence of the phase shift characterized, for example,
by effective range. We find a good agreement between our predictions and recent
quantum Monte Carlo simulations on the effective-range dependence in both three
and two spatial dimensions. We propose that in cold-atom experiments, the
non-universal dependence in thermodynamics can be probed using dark-state
optical control of Feshbach resonances.
