The inverse amplitude method is analysed to two-loop order in the chiral expansion in the case of ππ 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 oneloop order in the chiral expansion. For both ππ scattering and the pion form factors, it is in fact found that the inverse amplitude method to two-loop order agrees remarkable 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.

The amplitude for the anomalous process γπ → ππ is evaluated to two loops in the chiral expansion by means of a dispersive method. The two new coupling constants that enter at this order are estimated via sum rules derived from a non-perturbative chiral approach. With these coupling constants fixed, the numerical results are given and compared with the available experimental information.

The inverse-amplitude method is applied to the one-loop chiral expansion of the pion, kaon, and K l3 form factors. Since these form factors are determined by the same chiral low-energy constants, it is possible to obtain finite predictions for the inverse-amplitude method. It is shown that this method clearly improves one-loop chiral perturbation theory, and a very good agreement between the inverse-amplitude method and the experimental information is obtained. This suggests that the inverse-amplitude method is a rather systematic way of improving chiral perturbation theory. ͓S0556-2821͑96͒02619-7͔

From recent analysis of the scattering amplitude, it has been claimed that there exists a broad and light meson. However, if this meson really exists, it must also appear in other observables such as the pion scalar form factor. With the use of unitarity and dispersion relations together with chiral perturbation theory, this form factor is analyzed in the complex energy plane. The result agrees well with the empirical information in the elastic region and reveals a resonance pole at ͱsϭ445Ϫi235 MeV. This gives further strong evidence for the existence of the meson.

The inverse amplitude method has previously been successfully applied to ππ scattering in order to extend the range of applicability of chiral perturbation theory. However, in order to take the chiral zeros into account systematically, the previous derivation of the inverse amplitude method has to be modified. It is shown how this can be done to both one and two loops in the chiral expansion. In the physical region, the inclusion of these chiral zeros has very little significance, whereas they become essential in the sub-threshold region. Finally, the crossing properties of the inverse amplitude method are investigated in the sub-threshold region. PACS number(s): 13.75. Lb, 11.30.Rd, 11.55.Fv, 11.80.Et Chiral perturbation theory (ChPT) [1,2] has become a very successful methodology for low-energy hadron physics. Within this methodology one obtains a systematic expansion in powers of external momenta and light quark masses, or equivalently in the number of loops. However, unitarity is only satisfied perturbatively in the chiral expansion, which gives a severe restriction on the applicability of ChPT. Therefore, in order to extend the validity of ChPT, several methods have been proposed to combine exact unitarity and the chiral expansion. One such method is the inverse amplitude method (IAM), which has been analyzed to both one-loop [3] and two-loop [4] order in the chiral expansion. This has shown that the IAM is indeed a successful and systematic method to extend the range of applicability of ChPT.However, the IAM has to be somewhat modified when there are zeros in the amplitudes [5]. This is the case for ππ scattering where chiral dynamics demand that the S waves have zeros below threshold, whereas P and higher partial waves have fixed kinematical zeros at threshold. The previous derivation of the IAM [3,4] has been based upon the assumption that these zeros occur at the same energy as for the lowest order ChPT result. This assumption is in fact true for the P and higher partial waves, whereas the same is not necessarily the case for the S waves. Thus, the derivation of the IAM should be somewhat modified in the case of the S waves in order to systematically account for the occurrence of the chiral zeros. In this brief report it is shown how this is possible to both one and two-loop order in the chiral expansion. Since the main interest will be in the sub-threshold behavior, the following derivation of this generalized IAM will be restricted to the elastic approximation.The elastic unitarity relation for the ππ partial waves t is given bywhere s is the square of the c.m. energy, I and l are isospin and angular momentum indices, and σ is the phase-space factor. In ChPT the elastic ππ partial waves are now known to two loops in the chiral expansion [6,7]. These partial waves are given by the following expansionwhere t (0) is the lowest order result, t (1) is the one-loop correction, and t (2) the additional two-loop correction. They satisfy the elastic unitarity relation (1) perturbativelySince perturbative un...