Determination of Pion-Pion Scattering Amplitudes Satisfying Dispersion Relations and Unitarity

A method is developed for determining the partial-wave scattering amplitude in terms of the unitarity condition and the known branch cuts and poles of the inverse amplitude. The method is applied to the problem of pion-pion scattering and an implicit solution to the pion-pion partial-wave amplitude is derived for any angular momentum state and for both elastic and inelastic scattering. With the aid of this solution the low-energy resonance behavior of the pion-pion scattering system is studied by neglecting all inelastic processes and concentrating on $S$ and $P$ waves. It is found that a $P$-wave resonance with a position and width required by nucleon electromagnetic structure can be determined in terms of two parameters. An iteration procedure is described that is applicable when the $P$ wave dominates the equations and this procedure determines the contribution of the unphysical cut. The first iteration of the unphysical cut is numerically integrated on the IBM 709, and the results show that the shift of the resonance position due to the unphysical branch cut can be neglected.