  We show how the relativistic effective field theory for the superfluid phase
of helium-4 can replace the standard methods used to compute the production
rates of low momentum excitations due to the interaction with an external
probe. This is done by studying the scattering problem of a light dark matter
particle in the superfluid, and comparing to some existing results. We show
that the rate of emission of two phonons, the Goldstone modes of the effective
theory, gets strongly suppressed for sub-MeV dark matter particles due to a
fine cancellation between two different tree-level diagrams in the limit of
small exchanged momenta. This phenomenon is found to be a consequence of the
particular choice of the potential felt by the dark matter particle in helium.
The predicted rates can vary by orders of magnitude if this potential is
changed. We prove that the dominant contribution to the total emission rate is
provided by the phonons. Finally, we analyze the angular distributions for the
emissions of one and two phonons, and discuss how they can be used to measure
the mass of the hypothetical dark matter particle hitting the helium target.
