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.
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