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Low-energy effective field theory for finite-temperature relativistic superfluids

We derive the low-energy effective action governing the infrared dynamics of relativistic superfluids at finite temperature. We organize our derivation in an effective field theory fashion-purely in terms of infrared degrees of freedom and symmetries. Our degrees of freedom are the superfluid phase ψ, and the comoving coordinates for the volume elements of the normal fluid component. The presence of two sound modes follows straightforwardly from Taylor-expanding the action at second order in small perturbations. We match our description to more conventional hydrodynamical ones, thus linking the functional form of our Lagrangian to the equation of state, which we assume as an input. We re-derive in our language some standard properties of relativistic superfluids in the high-temperature and low-temperature limits. As an illustration of the efficiency of our methods, we compute the cross-section for a sound wave (of either type) scattering off a superfluid vortex at temperatures right beneath the critical one.