Particle Theory Approach to the Two-Pion and Three-Pion Systems

The relation between the two-pion and three-pion resonances is discussed in terms of a model in which the motion of the pions is described by a partially relativistic Schrödinger-type wave equation, and the interaction between them is represented by a static potential. An attractive square-well potential that is almost strong enough to bind the di-pion gives a satisfactory account of the observed $P$-wave pion-pion scattering. If this potential is assumed to be additive between all pairs in the three-pion $I=0\mathrm{}$, $J=1^{-}$ state, it produces far too much binding to agree with observation. As an alternative to additivity, the interaction may be assumed to saturate, so that the total potential never exceeds that between any pair of pions. It is found that the model provides a qualitatively consistent explanation both of pion-pion scattering and the three-pion system if the pair interactions saturate or nearly saturate.