arXiv subjects
Publications and source records attributed to Torben Hannah.
The amplitude for the anomalous process gamma pi -> pi pi 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 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.
From recent analysis of the $ππ$ scattering amplitude, it has been claimed that there exists a broad and light $σ$ 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 $\sqrt{s}=445-i235$ MeV. This gives further strong evidence for the existence of the $σ$ meson.
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 one-loop 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 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.