Path-integral formulation of high-energy scattering in quantum field theories

Abstract

We give a formulation of high-energy scattering via path integrals. This nonperturbative formulation allows the approximate treatment of both fixed-angle scattering as well as small-angle scattering. The only approximation made is replacing the summation of the paths of an external particle by the contribution of the classical path---a straight line for small-angle scattering and two straight lines joined at the origin at an angle for fixed-angle scattering. In this way, the scattering amplitude is factorized into a product of two amplitudes: the amplitude for the interaction between the external particles and the amplitude for vacuum-to-vacuum transition in the presence of the external fields produced classically by the external particles. We show that all of the exactly calculable factors (the eikonal formula, the Sudakov form factor, and the energy-dependent factor of multiphoton exchange for fixed-angle scattering) belong to the first amplitude and are easily produced by a semiclassical treatment. The second amplitude is fully quantum mechanical, and no justified approximation has been found. In the case of small-angle scattering, we deduce, with this formulation, the principle of the equivalence of phase space. For the case of fixed-angle scattering, we find that there are three time scales: ${\ensuremath{\omega}}^{\ensuremath{-}1}$, $\ensuremath{\lambda}$, and $\ensuremath{\omega}$, where $\ensuremath{\omega}$ is the incident c.m. energy and $\ensuremath{\lambda}$ is the photon mass.