Theory of the Roton Lifetime in Superfluid Helium

Abstract

We have extended the theory of the roton lifetime to include two-roton resonances in the final-state interaction. When resonances are taken into account, the final density of states ${\rho }_2$ exhibits a complicated energy dependence and bears little resemblance to the noninteracting roton density ${\rho }_2^{\mathrm{}\left(0\right)\mathrm{}}$ used in the Landau-Khalatnikov theory. Using a point-interaction model for the roton coupling and including only direct roton-roton scattering, we obtain a single-roton linewidth which is consistent with the upper bound given by other authors; we then obtain a linewidth for the two-roton resonance which is smaller than the experimental value. In contrast to previous work, we have proceeded to include roton hydridization in the lifetime theory. The effect of hybridization is twofold: (a) The final density of states ${\rho }_2$ is markedly altered, and (b) the hybridization opens new scattering channels involving a single quasiparticle in the final state. Although the net effect of mechanism (a) on the roton lifetime seems to be small, the additional scattering (b) gives a non-negligible contribution. Furthermore, we discuss the importance of self-consistency and roton scattering in nonzero angular momentum channels. Our theory predicts an unusual pressure dependence for the roton linewidth which may be checked experimentally. We conclude that further experiments, such as light and neutron scattering, are needed to resolve the strength of the various contributions to the linewidth and thereby provide a truly quantitative analysis of the roton lifetime.