Analytic proof that the Quark Model complies with the PCAC theorems

Abstract

The Weinberg theorem, the Adler self consistency zero, the Goldberger and Treiman relation and the Gell-Mann Oakes and Renner relation are proved analytically in full detail for Quark Models. These proofs are independent of the particular quark-quark interaction, and they are displayed with Feynman diagrams in a compact notation. I assume the ladder truncation, which is natural in the Quark Model, and also detail the diagrams that must be included in each relation. Off mass shell and finite size effects are included in the quark-antiquark pion Bethe Salpeter vertices. The axial and vector Ward identities, for the quark propagator and for the ladder, exactly cancel any model dependence.