Relativistic Three-Body Theory with Applications to π − N Scattering

Linear relativistic three-body equations for the scattering of a particle from a bound state or correlated pair of the others are constructed by combining the quasiparticle or isobar idea with two- and three-body unitarity as suggested by Blankenbecler and Sugar. After a partial-wave decomposition, the equations turn out to be one-dimensional, and hence are easily solved numerically. Any exchange mechanism and any number of isobars or separable two-body interactions can be included in the equations without violating two- and three-body unitarity and Lorentz invariance. Higher integer-spin separable interactions or isobars are included, in very close analogy to the nonrelativistic case. Applying the equation to the $\ensuremath{\pi}\ensuremath{-}N$ system with pseudoscalar coupling, that is, with only nucleon exchange and no $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ interaction or $\ensuremath{\pi}\ensuremath{-}{N}^{*}$ intermediate state, gives a (3,3) resonance but no other interesting structure. That is just what one would expect from such a simple mechanism and encourages us to go on to richer input. Analyzing the answers as a function of nucleon mass shows that the static-model expansion converges very slowly.