  We study the recently proposed effective field theory for the phonon of an
arbitrary non-relativistic superfluid. After computing the one-loop phonon
self-energy, we obtain the low temperature T contributions to the phonon
dispersion law at low momentum, and see that the real part of those can be
parametrized as a thermal correction to the phonon velocity. Because the
phonons are the quanta of the sound waves, at low momentum their velocity
should agree with the speed of sound. We find that our results match at order
T^4ln(T) with those predicted by Andreev and Khalatnikov for the speed of
sound, derived from the superfluid hydrodynamical equations and the phonon
kinetic theory. We get also higher order corrections of order T^4, which are
not reproduced pushing naively the kinetic theory computation. Finally, as an
application, we consider the cold Fermi gas in the unitarity limit, and find a
universal expression for the low T relative correction to the speed of sound
for these systems.
