Practical Theory of Three-Particle States. I. Nonrelativistic

We use Faddeev's mathematical work on nonrelativistic three-particle systems to obtain a practical theory of three-particle states. Each of the two-particle subsystems is assumed to be dominated by a finite number of bound states and resonances. We then obtain two sets of equations. One expresses processes with three final particles in terms of unstable-particle scattering. It generalizes and justifies the isobar model. The other is a set of coupled Lippmann-Schwinger equations for the scattering of bound states and unstable particles. The potentials in these scattering equations for composite particles are nonlocal and energy-dependent, and become complex above the three-particle threshold. They are expressed in terms of the wave functions of the two-particle bound states and resonances—there are no new arbitrary constants. The solution satisfies three-particle unitarity. Unstable particles are not approximated by stable ones. The theory applies especially to "overlapping" final-state interactions. The equations are given in detail for the $3N$ and $N\pi \pi$ systems in the static limit. In the latter case, the nucleon is treated as an $N\pi$ bound state. The $N\pi \pi$ equations are solved approximately in closed form, giving analogs of the Chew-Low effective-range formula, for the coupled $N\pi$ and $N^{*}\pi$ systems. This predicts large inelasticity in the $P_{11}$ state, in agreement with experiment. Detailed solutions will be given in subsequent papers. If all elementary particles are treated as bound states of themselves, our approach gives the basis of a complete dynamical theory of strong interactions.