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

TL;DR

This paper proves Goldstone's theorem for dynamically broken chiral symmetry, emphasizing the importance of consistency between Schwinger--Dyson and Bethe--Salpeter equations, and discusses approximation criteria in gauge theories.

Contribution

It provides a proof of Goldstone's theorem emphasizing the consistency of key equations and offers criteria for maintaining this consistency under approximations.

Findings

Established a criterion for consistent approximations in chiral symmetry breaking.

Presented a vertex model satisfying PCAC and Ward--Takahashi identities.

Highlighted the necessity of equation consistency for Goldstone's theorem validity.

Abstract

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.