  A general problem of $2\rightarrow N_f$ scattering is addressed with all the
states being wave packets with arbitrary phases. Depending on these phases, one
deals with coherent states in $(3+1)$ D, vortex particles with orbital angular
momentum, the Airy beams, and their generalizations. A method is developed in
which a number of events represents a functional of the Wigner functions of
such states. Using width of a packet $\sigma_p/\langle p\rangle$ as a small
parameter, the Wigner functions, the number of events, and a cross section are
represented as power series in this parameter, the first non-vanishing
corrections to their plane-wave expressions are derived, and generalizations
for beams are made. Although in this regime the Wigner functions turn out to be
everywhere positive, the cross section develops new specifically quantum
features, inaccessible in the plane-wave approximation. Among them is
dependence on an impact parameter between the beams, on phases of the incoming
states, and on a phase of the scattering amplitude. A model-independent
analysis of these effects is made. Two ways of measuring how a Coulomb phase
and a hadronic one change with a transferred momentum $t$ are discussed.
