We consider the scattering of dark matter particles from superfluid liquid $^4$He, which has been proposed as a target for their direct detection. Focusing on dark matter masses below ~1 MeV, we demonstrate from sum-rule arguments the importance of the production of single phonons with energies $omega lesssim 1$ meV. We show further that the anomalous dispersion of phonons in liquid $^4$He at low pressures [i.e., $d^2omega(q)/dq^2>0$, where $q$ and $omega(q)$ 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.
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