  Effects of the quantum interference in collisions of particles have a twofold
nature: they arise because of the auto-correlation of a complex scattering
amplitude and due to spatial coherence of the incoming wave packets. Both these
effects are neglected in a conventional scattering theory dealing with the
delocalized plane waves, although they sometimes must be taken into account in
particle and atomic physics. Here, we study the role of a transverse coherence
length of the packets, putting special emphasis on the case in which one of the
particles is twisted, that is, it carries an orbital angular momentum
$\ell\hbar$. In $ee, ep$, and $pp$ collisions the interference results in
corrections to the plane-wave cross sections, usually negligible at the
energies $\sqrt{s} \gg 1$ GeV but noticeable for smaller ones, especially if
there is a twisted hadron with $|\ell| > 10^3$ in initial state. Beyond the
perturbative QCD, these corrections become only moderately attenuated allowing
one to probe a phase of the hadronic amplitude as a function of $s$ and $t$. In
this regime, the coherence effects can compete with the loop corrections in QED
and facilitate testing the phenomenological models of the strong interaction at
intermediate and low energies.
