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

TL;DR

This work develops a low-energy EFT for finite-temperature relativistic superfluids using four scalar degrees of freedom (the Goldstone ψ and comoving coordinates φ^I) organized by a derivative expansion with strict symmetry constraints. The resulting action, ${\cal L}=F(b,X,y)$ with $b=\,sqrt{-J_\mu J^\mu}$, $X=\partial_\mu\psi\partial^\mu\psi$, and $y= u^\mu\partial_\mu\psi$, encapsulates the two-fluid dynamics and yields two propagating sound modes that reduce to known hydrodynamic results in the appropriate limits. The authors map the field-theory variables to standard thermodynamic quantities (μ=y, s=b, T=-F_b) and derive the stress tensor and current, showing consistency with established two-fluid formulations and providing a clear path to perturbative calculations. They illustrate the framework by deriving high- and low-temperature limits and by computing a relativistic sound-vortex scattering cross section, demonstrating both the method’s power and its connection to familiar fluid dynamics. The approach offers a clean, systematic avenue for exploring perturbations and potential extensions to anomalies via Wess-Zumino terms.

Abstract

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.