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Ultraviolet and soft X--ray photon--photon elastic scattering in an electron gas

We have considered the processes which lead to elastic scattering between two far ultraviolet or X--ray photons while they propagate inside a solid, modeled as a simple electron gas. The new ingredient, with respect to the standard theory of photon--photon scattering in vacuum, is the presence of low--energy, nonrelativistic electron--hole excitations. Owing to the existence of two--photon vertices, the scattering processes in the metal are predominantly of second order, as opposed to fourth order for the vacuum case. The main processes in second order are dominated by exchange of virtual plasmons between the two photons. For two photons of similar energy $\hbar Ω$, this gives rise to a cross section rising like $Ω^2$ up to maximum of around $10^{-32}$~cm$^2$, and then decreasing like $Ω^{-6}$. The maximal cross section is found for the photon wavevector $k \sim k_{F}$, the Fermi surface size, which typically means a photon energy $\hbar Ω$ in the keV range. Possible experiments aimed at checking the existence of these rare but seemingly measurable elastic photon--photon scattering processes are discussed, using in particular intense synchrotron sources.