Directional detection of light dark matter from three-phonon events in superfluid He-4

We present the analysis of a new signature for light dark matter detection with superfluid He-4: 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 theoretical tools to deal with such an observable, and compute the associated differential distributions.