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 that highlights the necessary consistency between the exact Schwinger-Dyson equation for the fermion propagator and the exact Bethe-Salpeter equation for fermion-antifermion bound states. The approach is tailored to the case when a global chiral symmetry is dynamically broken. Criteria are provided for maintaining the consistency when the exact equations are modified by approximations. In particular, for gauge theories in which partial conservation of the axial vector 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.