Pion-Pion Interaction and Pion-Nucleon S~attering

Pion-pion interaction is analyzed by using the dispersion relation for pion-nucleon scattering obtained by keeping the momentum transfer between an initial pion and a final nucleon constant. In order to take into account the singularity of two-pion threshold in the dispersion relation, the dispersion relation may be regarded as an integral equation for pion-nucleon scattering amplitude with the kernel of a pion-pion scattering amplitude. When this solution is compared with experiments on pion-nucleon scattering, it is found that the unknown quantity in the dispersion relation is only a pion-pion scattering amplitude. Therefore, if pion-pion amplitude is expressed in terms of an unknown parameter such as. scattering length, then this value can be determined via dispersion relation. Thus we are led to the following conclusion: In the isotopic spin state 1=0 (S-wave) of pion-pion system pion-pion interaction is attractive and the scattering length is of the order of one pion Compton wavelength, while in the isotopic spin state 1=1 (P-wave) of pion-pion state a definite conclusion could not be obtained. Finally the possibility of explaining the momentum dependence of pion-nucleon phase shift a13 related to the pion-pion interaction is briefly discussed