  We provide a thermodynamic derivation of the only-phase Popov action
functional, which is often adopted to study the low-energy effective
hydrodynamics of a generic nonrelativistic superfluid. It is shown that the
crucial assumption is the use of the saddle point approximation after
neglecting the quantum-pressure term. As an application, we analyze charged
superfluid (superconductors) coupled to the electromagnetic field at zero
temperature. Our only-phase and minimally-coupled theory predicts the decay of
the electrostatic field inside a superconductor with a characteristic length
much smaller than the London penetration depth of the static magnetic field.
This result is confirmed also by a relativistic only-phase Popov action we
obtain from the Klein-Gordon Lagrangian.
