We complete and improve the fits to experimental $\pi \pi$ scattering amplitudes, both at low and high energies, that we performed in the previous papers of this series. We then verify that the corresponding amplitudes satisfy analyticity requirements, in the form of partial wave analyticity at low energies, forward dispersion relations (FDR) at all energies, and Roy equations below$\overline{K}K$ threshold; the first by construction, the last two, inside experimental errors. Then we repeat the fits including as constraints FDR and Roy equations. The ensuing central values of the various scattering amplitudes verify very accurately FDR and, especially, Roy equations, and change very little from what we found by just fitting data, with the exception of the D2 wave phase shift, for which one parameter moves by $1.5\sigma$. These improved parametrizations therefore provide a reliable representation of pion-pion amplitudes with which one can test various physical relations. We also present a list of low energy parameters and other observables. In particular, we find $a_0^{(0)}=0.223\pm 0.009M_{\pi }^{-1}$, $a_0^{(2)}=-0.0444\pm 0.0045M_{\pi }^{-1}$, and ${\delta }_0^{(0)}(m_K^2)-{\delta }_0^{(2)}(m_K^2)=50.9\pm {1.2}^{\mathrm{o}}$.

• Received 2 November 2007

DOI:https://doi.org/10.1103/PhysRevD.77.054015