  We develop a formalism to study the Resonant Inelastic X-ray Scattering
(RIXS) response in metals based on the diagrammatic expansion for its cross
section. The standard approach to the solution of the RIXS problem relies on
two key approximations: short-range potentials and non-interacting conduction
electrons. However, these approximations are inaccurate for charged particles
in metals, where the long-range Coulomb interaction and dynamic screening
effects are very important. In this work we study how to extract important
information about collective excitations in the Coulomb plasma, plasmons and
electron-hole pairs, from RIXS data. We find that single- and multi-plasmon
excitations can easily be distinguished by positions of the corresponding
peaks, singularities, and their intensities. We also discuss the hybrid
processes, where plasmon emission is accompanied by excitation of electron-hole
pairs, and study how they manifest themselves.
