Energy-loss function in the two-pair approximation for the electron liquid

The imaginary part of the proper polarizability, Im Pi, arising due to excitations of two electron-hole pairs, is studied in detail for electron systems of arbitrary dimensionality, and taking into account arbitrary degeneracy of the electron bands. This allows an application to semiconductors with degenerate valleys, and to ferromagnetic metals. The results obtained not only confirm expressions already known for paramagnetic systems in the high-frequency region, but are also rigorously shown to be valid for all frequencies outside the particle-hole continuum. For a sufficiently high momentum transfer a cutoff frequency (below which Im Pi=0) is established for not only two-pair but also any n-pair processes. In contrast, there is no upper cutoff for n>1. The energy-loss function, including the discussed two-pair contributions, is calculated, The effects of screening are investigated. Numerical results, illustrating various aspects and properties of this function, especially showing finite-width plasmon peaks, are obtained for a two-dimensional electron gas.