Relativistic three-body bound states and the reduction from four to three dimensions

Abstract

Beginning with an effective field theory based upon meson exchange, the Bethe-Salpeter equation for the three-particle propagator (six-point function) is obtained. Using the one-boson-exchange form of the kernel, this equation is then analyzed using time-ordered perturbation theory, and a three-dimensional equation for the propagator is developed. The propagator consists of a pre-factor in which the relative energies are fixed by the initial state of the particles, an intermediate part in which only global propagation of the particles occurs, and a post-factor in which relative energies are fixed by the final state of the particles. The pre- and post-factors are necessary in order to account for the transition from states where particles are off their mass shell to states described by the global propagator with all of the particle energies on shell. The pole structure of the intermediate part of the propagator is used to determine the equation for the three-body bound state: a Schr{ö}dinger-like relativistic equation with a single, global Green's function. The role of the pre- and post-factors in the relativistic dynamics is to incorporate the poles of the breakup channels in the initial and final states. The derivation of this equation by integrating over the relative times rather than via a constraint on relative momenta allows the inclusion of retardation and dynamical boost corrections without introducing unphysical singularities.