  We perform a detailed study of the effective Lagrangian for the Goldstone
mode of a superfluid Fermi gas at zero temperature in the whole BCS-BEC
crossover. By using a derivative expansion of the response functions, we derive
the most general form of this Lagrangian at the next to leading order in the
momentum expansion in terms of four coefficient functions. This involves the
elimination of all the higher order time derivatives by careful use of the
leading order field equations. In the infinite scattering length limit where
conformal invariance is realized, we show that the effective Lagrangian must
contain an unnoticed invariant combination of higher spatial gradients of the
Goldstone mode, while explicit couplings to spatial gradients of the trapping
potential are absent. Across the whole crossover, we determine all the
coefficient functions at the one-loop level, taking into account the dependence
of the gap parameter on the chemical potential in the mean-field approximation.
These results are analytically expressed in terms of elliptic integrals of the
first and second kind. We discuss the form of these coefficients in the extreme
BCS and BEC regimes and around the unitary limit, and compare with recent work
by other authors.
