Searching for low mass dark matter via phonon creation in superfluid 4He

We consider the scattering of dark matter particles from superfluid liquid $^{4}\\mathrm{He}$, which has been proposed as a target for their direct detection. Focusing on dark matter masses below $\\ensuremath{\\sim}1\\text{ }\\text{ }\\mathrm{MeV}$, we demonstrate from sum-rule arguments the importance of the production of single phonons with energies $\\ensuremath{\\omega}\\ensuremath{\\lesssim}1\\text{ }\\text{ }\\mathrm{meV}$. We show further that the anomalous dispersion of phonons in liquid $^{4}\\mathrm{He}$ at low pressures [i.e., ${d}^{2}\\ensuremath{\\omega}(q)/d{q}^{2}g0$, where $q$ and $\\ensuremath{\\omega}(q)$ are the phonon momentum and energy] has the important consequence that a single phonon will decay over a relatively short distance into a shower of lower-energy phonons centered on the direction of the original phonon. Thus, the experimental challenge in this regime is to detect a shower of low-energy phonons, not just a single phonon. Additional information from the distribution of phonons in such a shower could enhance the determination of the dark matter mass.