Pion-Nucleon Scattering and Pion-Pion Interaction

Applying Cini-Fubini approximation to the Mandelstam representation for pion-nucleon scattering amplitude, we treat explicitly the effect of pion-pion interaction to be added to the dispersion relation given by Chew, Goldberger, Low and Nambu. On account of scanty information on the contribution to tl:e dispersion .integral from high-mass states, we ought to introduce constant parameters through a subtraction. In the dispersion relations thus obtained, the integral over the branch cut extending 4t-t2 to oo are calculated by a method similar to that of Frazer and Fulco, but using one more subtraction in order to avoid the divergence regarding the left-hand cut. From the comparison of the dispersion relations with experimental data of pion-nucleon scattering, we investigate the property of 1=0 (S-wave) pion-pion interaction, and with the result that the interaction is attractive and the magnitude of scattering length is 0.05-0.2 1-1-1. This is still one order smaller as compared with the recent experimental data. In connection with this, we briefly discuss about the necessity of another new parameter.