Coupling of phonons to a helium atom adsorbed on graphite

We compute the self-energy for a ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atom adsorbed on graphite to second order in the phonon coupling. The phonon contributions amount to several degrees Kelvin. The imaginary part corresponds to a lifetime of some ${10}^{-11\mathrm{}}$ s.