Chirally symmetric quark description of low energy \pi-\pi scattering

Abstract

Weinberg's theorem for \pi-\pi scattering, including the Adler zero at threshold in the chiral limit, is analytically proved for microscopic quark models that preserve chiral symmetry. Implementing Ward-Takahashi identities, the isospin 0 and 2 scattering lengths are derived in exact agreement with Weinberg's low energy results. Our proof applies to alternative quark formulations including the Hamiltonian and Euclidean space Dyson-Schwinger approaches. Finally, the threshold \pi-\pi scattering amplitudes are calculated using the Dyson-Schwinger equations in the rainbow-ladder truncation, confirming the formal derivation.