Directional detection of light dark matter from three-phonon events in
  superfluid $^4$He
  We present the analysis of a new signature for light dark matter detection
with superfluid $^4$He: the emission of three phonons. We show that, in a
region of mass below the MeV, the kinematics of this process can offer a way to
reconstruct the dark matter interaction vertex, while providing background
rejection via coincidence requirements and directionality. We develop all the
necessary technology to deal with such an observable, and compute the
associated differential distributions.
