Pion-Pion Scattering Below 850 MeV. A Unique Solution

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 ${\alpha }_0$ and ${\alpha }_2$, valid for all possible behaviors of ${{\delta }_0}^0$ in the $\rho$-region, and taking the value $\frac{a_0}{a_2}=-3.2\mathrm{}\pm 1.0$ from recent low-energy work, the values $a_0=0.16\mathrm{}\pm 0.04$, $a_2=-0.05\mathrm{}\pm 0.01$ uniquely result. The ambiguity in ${{\delta }_0}^0$ in the $\rho$ 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.