Dynamical Chiral Symmetry Breaking, Goldstone's Theorem and the Consistency of the Schwinger--Dyson and Bethe--Salpeter Equations

A proof of Goldstone's theorem is given for the case in which global chiral symmetry is dynamically broken. The proof highlights a needed consistency between the exact Schwinger--Dyson equation for the fermion propagator and the exact Bethe--Salpeter equation for fermion--antifermion bound states. A criterion, based on the Cornwall, Jackiw and Tomboulis effective action for composite operators, is provided for maintaining the consistency when the equations are modified by approximations. For gauge theories in which partial conservation of the axial current (PCAC) should hold, a constraint on the approximations to the fermion--gauge boson vertex function is discussed, and a vertex model is given which satisfies both the PCAC constraint and the vector Ward--Takahashi identity.