  One of the reasons for the tremendous success of a plane-wave approximation
in particle physics is that the non-paraxial corrections to such observables as
energy, magnetic moment, scattering cross section, and so on are attenuated as
$\lambda_c^2/\sigma_{\perp}^2 \ll 1$ where $\sigma_{\perp}$ is a beam width and
$\lambda_c = \hbar/mc$ is a Compton wavelength. This amounts to less than
$10^{-14}$ for modern electron accelerators and less than $10^{-6}$ for
electron microscopes. Here we show that these corrections are $|\ell|$ times
enhanced for vortex particles with high orbital angular momenta $|\ell|\hbar$,
which can already be as large as $10^3\hbar$. We put forward 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 Laguerre-Gaussian beams. We demonstrate that
it is $\sqrt{|\ell|}\, \lambda_c \gg \lambda_c$ that defines a paraxial scale
for such packets, in contrast to those with a non-singular phase (say, the Airy
beams). With current technology, the non-paraxial corrections can reach the
relative values of $10^{-3}$, yield a proportional increase of an invariant
mass of the electron packet, describe a spin-orbit coupling as well as the
quantum coherence phenomena in particle and atomic collisions.
