Twisted photons are not plane waves, but superpositions of plane waves with a defined projection hm of the orbital angular momentum onto the propagation axis (m is integer and may attain values m≫ 1). Here, we describe in detail the possibility to produce high-energy twisted photons by backward Compton scattering of twisted laser photons on ultra-relativistic electrons with a Lorentz-factor γ=E/(mec2)≫ 1. When a twisted laser photon with the energy hω~1 eV performs a collision with an electron and scatters backward, the final twisted photon conserves the angular momentum m, but its energy hω' is increased considerably: ω'/ω=4γ2/(1+x), where x=4Ehω/(mec2)2. The S matrix formalism for the description of scattering processes is particularly simple for plane waves with definite 4-momenta. However, in the considered case, this formalism must be enhanced because the quantum state of twisted particles cannot be reduced to plane waves. This implies that the usual notion of a cross section is inapplicable, and we introduce and calculate an averaged cross section for a quantitative description of the process. The energetic upconversion of twisted photons may be of interest for experiments with the excitation and disintegration of atoms and nuclei, and for studying the photo-effect and pair production off nuclei in previously unexplored regimes.



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