Cubic interactions in superfluid $^4$He

High-resolution neutron resonance spin-echo measurements of superfluid $^4$He show that the roton energy does not have the same temperature dependence as the inverse lifetime, in contrast to the Landau-Khalatnikov theory. We present a diagrammatic analysis that attributes this effect to the interaction of rotons with thermally excited phonons via both four- and three-particle processes, the latter being allowed by the broken gauge symmetry of the Bose condensate.