Effect ofπ−πInteraction on High-Energyπ−pScattering

The object of this paper is to examine the possibility of explaining the second maximum in ($\pi -p$) scattering in the framework of the Tamm-Dancoff approximation for ($\pi -p$) interaction, by postulating a resonant $\pi -\pi$ interaction in the isotopic spin state $I=0\mathrm{}$ of the $\pi -\pi$ system. For this purpose the Tamm-Dancoff method in pseudosclar theory, as formulated by Dyson and others, is first extended so as to include the effects of a $\pi -\pi$ interaction. This is achieved by introducing a Green's function whose form is determined on the assumption of a zero-range $\pi -\pi$ interaction in the state $I=0\mathrm{}$ of the $\pi -\pi$ system. The solution of the "modified" integral equation is greatly simplified by assuming the $\pi -p$ interaction to be weak compared with the $\pi -\pi$ interaction (the reasons for which are given). The total inelastic cross section is then derived by using the unitary property of the $S$ matrix.