  Collisions of particles prepared in non--plane-wave states with a non-trivial
phase structure, such as vortex states carrying an adjustable orbital angular
momentum (OAM), open novel opportunities in atomic, nuclear, and high-energy
physics unavailable for traditional scattering experiments. Recently, it was
argued that photoinduced processes such as $\gamma d \to pn$ and $\gamma p \to
\Delta^+$ initiated by a high-energy vortex photon should display a remarkable
threshold shift and a sizable cross section enhancement as the impact parameter
$b$ of the target hadron with respect to the vortex photon axis goes to zero.
In this work, we theoretically explore whether this effect exists within the
quantum-field-theoretic treatment of the scattering process. We do not rely on
the semiclassical assumption of pointlike, non-spreading target particle and,
instead, consider the toy process of heavy particle pair production in
collision of two light particles prepared as a Laguerre-Gaussian and a compact
Gaussian wave packets, paying special attention to the threshold behavior of
the cross section. We do observe threshold smearing due to non-monochromaticity
of the wave packets, but we do not confirm the near-threshold enhancement.
Instead we find an OAM-related dip at $b\to 0$ as compared with the two
Gaussian wave packet collision.
