Pion–pion scattering at low energy

We present technical details of the evaluation of the elastic ππ scattering amplitude to two loops in chiral perturbation theory. In particular, we elaborate on the renormalization procedure at the two–loop order and on the evaluation of the relevant Feynman diagrams that can all be expressed in terms of elementary functions. For the sake of clarity, we discuss these matters both in the N–component φ4 theory (in its symmetric phase) and in chiral perturbation theory. Estimates for the relevant low–energy constants of O(p6) are presented. Threshold parameters and phase shifts are then calculated for two sets of O(p4) coupling constants and compared with experiment. We comment on the extraction of threshold parameters from phase shift data.