Relativistic vortex electrons: paraxial versus non-paraxial regimes

A plane-wave approximation in particle physics implies that a width of a massive wave packet $\sigma_{\perp}$ is much larger than its Compton wavelength $\lambda_c = \hbar/mc$. For Gaussian beams or for packets with the non-singular phases (say, the Airy beams), corrections to this approximation are attenuated as $\lambda_c^2/\sigma_{\perp}^2 \ll 1$ and usually negligible. Here we show that this situation drastically changes for particles with the phase vortices associated with an orbital angular momentum $\ell\hbar$. For highly twisted beams with $|\ell| \gg 1$, the non-paraxial corrections get $|\ell|$ times enhanced and $|\ell|$ can already be as large as $10^3$. We describe the relativistic wave packets, both for vortex bosons and fermions, which transform correctly under the Lorentz boosts, are localized in a 3D space, and represent a non-paraxial generalization of the massive Laguerre-Gaussian beams. We compare such states with their paraxial counterpart paying specific attention to the relativistic effects and to the differences from the twisted photons. In particular, a Gouy phase is found to be Lorentz invariant and it generally depends on time rather than on a distance $z$. By calculating the electron packet's mean invariant mass, magnetic moment, etc., we demonstrate that the non-paraxial corrections can already reach the relative values of $10^{-3}$. These states and the non-paraxial effects can be relevant for the proper description of the spin-orbit phenomena in relativistic vortex beams, of scattering of the focused packets by atomic targets, of collision processes in particle and nuclear physics, and so forth.