  We show that colliding vortex beams instead of (approximate) plane waves can
lead to a direct measurement of how the overall phase of the plane wave
scattering amplitude changes with the scattering angle. Since vortex beams are
coherent superpositions of plane waves with different momenta, their scattering
amplitude receives contributions from plane wave amplitudes with distinct
kinematics. These contributions interfere, leading to the measurement of their
phase difference. Although interference exists for any generic wave packet
collision, we show that using vortex beams dramatically enhances sensitivity to
the phase in comparison with non-vortex beams. Since the overall phase is
inaccessible in a plane wave collision, this measurement would be of great
importance for a number of topics in hadronic physics, for example, meson
production in the resonance region, physics of nucleon resonances, and small
angle elastic hadron scattering.
