  We examine in depth a recent proposal to utilize superfluid helium for direct
detection of sub-MeV mass dark matter. For sub-keV recoil energies, nuclear
scattering events in liquid helium primarily deposit energy into long-lived
phonon and roton quasiparticle excitations. If the energy thresholds of the
detector can be reduced to the meV scale, then dark matter as light as ~MeV can
be reached with ordinary nuclear recoils. If, on the other hand, two or more
quasiparticle excitations are directly produced in the dark matter interaction,
the kinematics of the scattering allows sensitivity to dark matter as light as
~keV at the same energy resolution. We present in detail the theoretical
framework for describing excitations in superfluid helium, using it to
calculate the rate for the leading dark matter scattering interaction, where an
off-shell phonon splits into two or more higher-momentum excitations. We
validate our analytic results against the measured and simulated dynamic
response of superfluid helium. Finally, we apply this formalism to the case of
a kinetically mixed hidden photon in the superfluid, both with and without an
external electric field to catalyze the processes.
