Density-correlation function for liquid helium nearTλin the symmetric planar-spin model

The density-correlation function $S(k,\omega )$ for liquid helium near $T_{\lambda }$ is calculated in the symmetric planar-spin model in three dimensions, both above and below the transition. The approximation used is a self-consistent generalization of the lowest-order $\epsilon$ expansion, which resembles the mode-coupling expressions of Kawasaki and others. The spectrum is evaluated numerically for different values of $k\xi$, and predictions are made for the exponents and amplitudes of various singular dynamic quantities. The results are consistent with earlier ones based on an extrapolation of the second-order $\epsilon$ expansion. Comparison with experiment is possible without any adjustable parameters, and it is found that the results are consistent with existing measurements of the thermal conductivity, but inconsistent with the observed amplitude of second-sound damping. The results also disagree with light-scattering experiments at and below $T_{\lambda }$. These disagreements are at present unexplained.