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.

• Received 27 November 2001

DOI:https://doi.org/10.1103/PhysRevD.65.076008