Retarded Interactions in Fermi Systems. II. Quasiparticle Lifetimes

Abstract

Properties of the inverse lifetime $\frac{1}{\tau }$ of quasiparticles in an infinite system of nucleons ("nuclear matter") interacting via the exchange of a neutral scalar boson are studied in the dielectric formulation of the many-body problem. It is shown that the polarization of the nuclear medium plays an essential role in determining $\frac{1}{\tau }$. The presence of the medium mediates the interaction between two nucleons in it through the introduction of an effective "dielectric constant" for nuclear matter. The interaction becomes "dressed" in a fashion analogous to, but different in its details from, the screening of the Coulomb interaction between electrons in metals. Calculations of $\frac{1}{\tau }$ are made for nuclear matter both in second-order perturbation theory, and using the fully dressed interaction calculated in the random-phase approximation (RPA). The RPA estimate is lower by about a factor of 3 over a wide range of momenta and is in good agreement with empirical optical-model estimates of $\frac{1}{\tau }$. The second-order perturbation estimate is by contrast entirely inadequate, emphasizing the importance of taking the polarization effects into account for scattering processes at actual nuclear densities. The use of a dielectric constant for the medium also introduces another feature into the properties of $\frac{1}{\tau }$. The behavior of $\frac{1}{\tau }$ near the Fermi surface now becomes sensitive to whether or not the interacting ground state is unstable with respect to a permanent density fluctuation. If it is, then $\frac{1}{\tau }$ behaves like ($p-p_F$) instead of ${(p-p_F)}^2$ near the Fermi surface ($p_F$ is the Fermi momentum). This is not a new physical result, but rather a reflection of the inadequacy of the RPA if the system becomes unstable. It is shown that as the momentum increases the effect of such an instability, if present, decreases, and becomes entirely negligible when $p\gtrsim \sqrt{2}{p}_F$.