Pion-Pion Scattering Below 850 MeV. A Unique Solution

Abstract

Forward dispersion relations are used as the basis of an analysis of the information on pion-pion scattering available up to 850 MeV. These are shown to result in a model-independent relation between the scattering lengths ${\ensuremath{\alpha}}_{0}$ and ${\ensuremath{\alpha}}_{2}$, valid for all possible behaviors of ${{\ensuremath{\delta}}_{0}}^{0}$ in the $\ensuremath{\rho}$-region, and taking the value $\frac{{a}_{0}}{{a}_{2}}=\ensuremath{-}3.2\ifmmode\pm\else\textpm\fi{}1.0$ from recent low-energy work, the values ${a}_{0}=0.16\ifmmode\pm\else\textpm\fi{}0.04$, ${a}_{2}=\ensuremath{-}0.05\ifmmode\pm\else\textpm\fi{}0.01$ uniquely result. The ambiguity in ${{\ensuremath{\delta}}_{0}}^{0}$ in the $\ensuremath{\rho}$ region is also studied, by four different methods, and uniquely resolved, allowing a detailed picture of all the phases up to 850 MeV to be given.