  Twisted photons are not plane waves, but superpositions of plane waves with a
defined projection hbar m 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 gamma=E/(m_e c^2) >> 1. When a twisted laser photon with the
energy hbar omega ~ 1 eV performs a collision with an electron and scatters
backward, the final twisted photon conserves the angular momentum m, but its
energy hbar omega' is increased considerably: omega'/omega=4 gamma^2/(1+x),
where x=4E hbar omega/(m_e c^2)^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.
