Scattering lengths and medium and high energyππscattering

Crucial tests of chiral perturbation theory and determinations of chiral coupling constants rely on fine details of dispersion relation analysis of $\pi \pi$ scattering data. Whereas a recent Roy equation analysis of low energy $S$ and $P$ waves with $a_0^0$ in the range predicted by standard one-loop chiral perturbation theory, yielded sharp determinations of certain $S$- and $P$-wave threshold parameters only, we now extend the discussion to medium and high energy. This is shown to contribute to the $D$--wave scattering lengths $a_2^0$ and $a_2^2$ significantly, roughly 10% and 30% of their central experimental values, remaining consistent with chiral predictions. Results of our Roy equation fits to the $F$-wave scattering length $a_3^1$, a quantity for which two-loop chiral perturbation theory holds important predictions, are presented.