The pion-pion scattering amplitude
  We obtain reliable $\pi\pi$ scattering amplitudes consistent with experimental data, both at low and high energies, and fulfilling appropriate analyticity properties. We do this by first fitting experimental low energy ($s^{1/2}\leq1.42 {\rm GeV}$) phase shifts and inelasticities with expressions that incorporate analyticity and unitarity. In particular, for the S wave with isospin~0, we discuss in detail several sets of experimental data. This provides low energy partial wave amplitudes that summarize the known experimental information. Then, we impose Regge behaviour as follows from factorization and experimental data for the imaginary parts of the scattering amplitudes at higher energy, and check fulfillment of dispersion relations up to 0.925 GeV. This allows us to improve our fits. The ensuing $\pi\pi$ scattering amplitudes are then shown to verify dispersion relations up to 1.42 GeV, as well as $s - t - u$ crossing sum rules and other consistency conditions. The improved parametrizations therefore provide a reliable representation of pion-pion amplitudes with which one can test chiral perturbation theory calculations, pionium decays, or use as input for CP-violating $K$ decays. In this respect, we find $[a_0^{(0)}-a_0^{(2)}]^2=(0.077\pm0.008) M^{-1}_\pi$ and $\delta_0^{(0)}(m^2_K)-\delta_0^{(2)}(m^2_K)=52.9\pm1.6^{\rm o}$. 
J. R. Pelaez and F. J. Yndurain