ππScattering Amplitude in the Low-Energy Region

Recent unambiguous experimental $\pi \pi$ phase shifts are shown to be consistent at least up to 850 MeV with crossing and analyticity requirements, using the rigorously provable integral equations of Roy. We compute the $s$- and $p$-wave phase shifts in the experimentally unknown region from threshold to 500 MeV. We find the scattering lengths $a_0^0=0.15\mathrm{}\pm 0.07$ and $a_0^2=-0.053\mathrm{}\pm 0.028$. We also calculate the $s$-, $p$-, and $d$-wave amplitudes in the subthreshold region and show that the current-algebra predictions of Weinberg are strongly supported by the data. We consider some phenomenological models which are in agreement with the data and discuss why their predictions are similar to Weinberg's.