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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 $J=T=1$ 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 π-π resonance is at 10μ², then a reasonable radius can be easily obtained. On the other hand, if the resonance is at 20μ² 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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Blankenbecler et al. (1962) studied this question.
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