Isoscalar Nucleon Structure

The three-pion contribution to the isoscalar nucleon structure is examined in dispersion theory. We assume that the amplitudes (photon → three pions) and (three pions → nucleon pair) are characterized by the pions interacting in pairs. This interaction is taken to be the low energy $J=T=1\mathrm{}$ resonance, which consequently dominates the picture. The effects of a three-pion interaction are included so as to satisfy an extended unitarity condition. The lowest mass singularities, including the complex ones, are discussed, interpreted, and included. If the $\pi -\pi$ resonance is at $10{\mu }^2$, then a reasonable radius can be easily obtained. On the other hand, if the resonance is at $20{\mu }^2$ or higher, then a strong intrinsic three-pion resonance or a bound state seems to be needed for agreement with experiment.