Lifetime Effects in Condensed Helium-3

The condensation of a Fermion system by forming $d$-type bound pairs is discussed with the help of time-dependent correlation functions both at absolute zero and finite temperatures, for the purpose of applying this study to the case of liquid helium-3. We use essentially Gor'kov's method, suitably generalized to take into account the anisotropy of the bound pairs and also the finite lifetime of the quasi-particles which make up the pairs. The treatment proposed here goes one step further than the Hartree approximation in the sense that the finite decay rate of the quasi-particles is introduced by means of a model spectral density for the renormalized propagator (Green's function). This model features a single broad peak instead of the infinitely sharp peak which characterizes the Hartree approximation. Considerable care is taken to relate this microscopic model to the available experimental data on the scattering probability in liquid helium-3. It is concluded that the effect of scattering on the condensation can be adequately described by a cutoff $\Lambda$ of the order of 1°K, limiting the domain in momentum space of the quasi-particles which participate effectively in the condensation process. This entails a reduction of the transition temperature estimated previously on the basis of the Hartree approximation, down to a value of the order of 0.02-0.03°K.