Abstract
The three-pion contribution to the isoscalar nucleon structure is examined in dispersion theory. We assume that the amplitudes (photon \ensuremath{\rightarrow} three pions) and (three pions \ensuremath{\rightarrow} nucleon pair) are characterized by the pions interacting in pairs. This interaction is taken to be the low energy $J=T=1$ 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 $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ resonance is at $10{\ensuremath{\mu}}^{2}$, then a reasonable radius can be easily obtained. On the other hand, if the resonance is at $20{\ensuremath{\mu}}^{2}$ or higher, then a strong intrinsic three-pion resonance or a bound state seems to be needed for agreement with experiment.
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