  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., $d^2\omega(q)/dq^2>0$, 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.
