Probing phase of a scattering amplitude beyond the plane-wave approximation

Within a plane-wave approach, a number of scattering events in a collision is insensitive to a general phase of a transition amplitude, although this phase is extremely important for a number of problems, especially in hadronic physics. In reality the particles are better described as wave packets, and here we show that the observables grow dependent upon this phase if one lays aside the simplified plane-wave model. We discuss two methods for probing how the Coulomb- and hadronic phases change with the transferred momentum t, either by colliding two beams at a non-vanishing impact-parameter or by employing such novel states as the vortex particles carrying orbital angular momentum or the Airy beams. For electron-electron collision, the phase contribution to the cross-section can reach values higher than 10−4-10−3 for well-focused beams with energies of hundreds of keV.