  The dynamic structure function $S(k,\omega)$ informs about the dispersion and
damping of excitations. We have recently (Phys. Rev. B {\bf 97}, 184520 (2018))
compared experimental results for $S(k,\omega)$ from high-precision neutron
scattering experiment and theoretical results using the ``dynamic many-body
theory'' (DMBT), showing excellent agreement over the whole experimentally
accessible pressure regime. This paper focuses on the specific aspect of the
propagation of low-energy phonons. We report calculations of the phonon
mean-free path and phonon life time in liquid \he4 as a function of wave length
and pressure. Historically, the question was of interest for experiments of
quantum evaporation. More recently, there is interest in the potential use of
$^4$He as a detector for low-energy dark matter (K. Schulz and Kathryn M.
Zurek, Phys. Rev. Lett. {\bf 117}, 121302 (2016)). While the mean free path of
long wave length phonons is large, phonons of intermediate energy can have a
short mean free path of the order of $\mu$m. Comparison of different levels of
theory indicate that reliable predictions of the phonon mean free path can be
made only by using the most advanced many--body method available, namely, DMBT.
