Pseudopotential theory of interacting roton pairs in superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$

A configuration-space pseudopotential, which is closely related to that used by Aldrich and Pines to describe the effective interaction between background particles in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, is constructed and used to calculate the roton-roton scattering amplitude. From that amplitude we obtain a theory that is completely congruent with the roton-liquid theory of Bedell, Pines, and Fomin. We calculate two-roton bound states, roton-liquid parameters, and roton lifetimes, as well as information about the hybridization of the two-roton bound state with excitations of higher and lower energy. Excellent agreement between theory and experiment is obtained for the $l=2\mathrm{}$ bound state at zero pair momentum, the roton lifetime, the roton contribution to the normal-fluid viscosity and the normal-fluid density, and the temperature variation of the roton energy. The effective roton-roton coupling parameters at large pair momentum are found to be an order of magnitude larger than those for small or vanishing pair momentum. At SVP we find that a substantial number of two-roton bound states of varying symmetry exist for pair momentum up to ∼ 3 ${\mathrm{\AA }}^{-1\mathrm{}}$; at standard pressure, however the roton-roton interaction for momenta ∼ 1 ${\mathrm{\AA }}^{-1\mathrm{}}$ is found to become repulsive, so that both the $l=2\mathrm{}$ bound state at zero pair momentum and bound states at intermediate momenta are predicted to disappear under pressure.