Analytic solution to the problem of three-particle collisions in a model with eikonal Hamiltonian

The three-particle eikonal model of the process $3\to3$ is investigated in the case when the relative motions of the particles of each pair correspond to high energies. No other restrictions (such as in the fixed-center approximation) are imposed on the motion of the particles. It is shown that in this case the three-particle problem admits an analytic solution. An explicit expression is found for the off-shell amplitude of the $3\to3$ process; it is obtained by exact summation of the multiple scattering series in a model with eikonal Hamiltonian. On the mass shell, this series terminates (there are no terms with multiplicity higher than three). A formula is obtained that describes the mutual canceling of the terms of higher multiplicity in the series.