Exact behavior of the density and spin susceptibilities of a Fermi liquid for large wave vectors: Derivation from diagrammatic many-body theory

The exact asymptotic behavior of the density and spin response functions of a uniform Fermi system for large wave vectors is derived from diagrammatic many-body theory. Various exact relationships between the self-energy, the particle-hole interaction, and the vertex corrections are derived for the first time and applied to this problem. The final results agree with those provided by the kinetic equation approach. Different behavior of the polarization potentials and local-field factors for interactions that decrease faster or slower than 1/$q^2$ for large wave vector q, is pointed out.