Microscopic approach to the theory of roton interactions in superfluid helium

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Abstract
A microscopic theory of quasiparticle interactions in liquid helium is derived in terms of the density-fluctuation variables. The collective-variable approach follows closely the formalism of Sunakawa and co-workers, who have previously calculated in detail the three-phonon interaction. In this paper, the direct scattering of quasiparticles is calculated from first principles. The amplitude for the scattering of two excitations is derived in the general form to leading order in perturbation theory. It is then used to estimate the effective roton-roton coupling in terms of the analysis based on the collective-variable approach. The microscopic theory yields an attractive coupling of two rotons, but the coupling strength is an order of magnitude larger than the phonomenological estimates based on experiments. A renormalization procedure is outlined which reduces the strength of the roton-roton interaction, which may bring the coupling in closer agreement with previous estimates. Thus the renormalization of vertex functions and quasiparticle propagators is of great importance in relating microscopic theories to phonomenological parameters obtained from experiments.