  We derive the low energy effective action for the collective modes in systems
of fermions interacting via a short-range s-wave attraction, featuring unequal
chemical potentials for the two fermionic species (asymmetric systems). As a
consequence of the attractive interaction, fermions form a condensate that
spontaneously breaks the U(1) symmetry associated with total number
conservation. Therefore at sufficiently small temperatures and asymmetries, the
system is a superfluid. We reproduce previous results for the stability
conditions of the system as a function of the four-fermion coupling and
asymmetry. We obtain these results analyzing the coefficients of the low energy
effective Lagrangian of the modes describing fluctuations in the magnitude
(Higgs mode) and in the phase (Goldstone mode) of the difermion condensate. We
find that for certain values of parameters, the mass of the Higgs mode
decreases with increasing mismatch between the chemical potentials of the two
populations, if we keep the scattering length and the gap parameter constant.
Furthermore, we find that the energy cost for creating a position dependent
fluctuation of the condensate is constant in the gapped region and increases in
the gapless region. These two features may lead to experimentally detectable
effects. As an example, we argue that if the superfluid is put in rotation, the
square of the radius of the outer core of a vortex should sharply increase on
increasing the asymmetry, when we pass through the relevant region in the
gapless superfluid phase. Finally, by gauging the global U(1) symmetry, we
relate the coefficients of the effective Lagrangian of the Goldstone mode with
the screening masses of the gauge field.

Superconductivity;Statistical Mechanics;High Energy Physics - Phenomenology