  We examine the effective theory of critical dynamics near superfluid phase
transitions in the framework of the Keldysh-Schwinger formalism. We focus on
the sector capturing the dynamics of the complex order parameter and the
conserved current corresponding to the broken global symmetry. After
constructing the theory up to quadratic order in the $a$-fields, we compare the
resulting stochastic system with Model F as well as with holography. We
highlight the role of a time independent gauge symmetry of the effective theory
also known as ``chemical shift". Finally, we consider the limiting behaviour at
energies much lower than the gap of the amplitude mode by integrating out the
high energy degrees of freedom to reproduce the effective theory of
superfluids.
