Inverse amplitude method and chiral perturbation theory to two loops

The inverse amplitude method is analyzed to two-loop order in the chiral expansion in the case of $\pi \pi$ scattering and the pion form factors. The analysis is mainly restricted to the elastic approximation but the possible extension to the inelastic case is also discussed in some detail. It is shown how the two-loop approach improves the inverse amplitude method applied to one-loop order in the chiral expansion. For both $\pi \pi$ scattering and the pion form factors, it is in fact found that the inverse amplitude method to two-loop order agrees remarkably well with the experimental data up to energies where inelasticities become essential. At somewhat lower energies, the two-loop approach compares well with the one-loop approximation, and in the threshold region they both agree with chiral perturbation theory. This suggests that the inverse amplitude method is indeed a rather systematic way of improving chiral perturbation theory order by order in the chiral expansion.