Bethe-Salpeter Approach for Unitarized Chiral Perturbation Theory

The Bethe-Salpeter equation restores exact elastic unitarity in the $s-$ channel by summing up an infinite set of chiral loops. We use this equation to show how a chiral expansion can be undertaken in the two particle irreducible amplitude and the propagators accomplishing exact elastic unitarity at any step. Renormalizability of the amplitudes can be achieved by allowing for an infinite set of counter-terms as it is the case in ordinary Chiral Perturbation Theory. Crossing constraints can be imposed on the parameters to a given order. Within this framework, we calculate the leading and next-to-leading contributions to the elastic $\pi \pi$ scattering amplitudes, for all isospin channels, and to the vector and scalar pion form factors in several renormalization schemes. A satisfactory description of amplitudes and form factors is obtained. In this latter case, Watson's theorem is automatically satisfied. From such studies we obtain a quite accurate determination of some of the ChPT $SU(2)-$low energy parameters (${\bar l}_1 - {\bar l}_2 = -6.1\er{0.1}{0.3} $ and ${\bar l}_6= 19.14 \pm 0.19$). We also compare the two loop piece of our amplitudes to recent two--loop calculations.