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Effective field theory and dispersion law of the phonons of a non-relativistic superfluid

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.