Dynamics and stability of dynamical symmetry breaking without fundamental scalars

Starting with a theory that contains only fermionic and gauge field degrees of freedom, we find a closed-form expression for the effective potential describing the bound states in the fermion-antifermion channels, enabling one to study the stability of the system in a way that closely parallels the traditional treatment with fundamental scalars and to develop a stability criterion for candidate vacua in gauge-theory models which have nonzero ${〈\overline{\psi }\psi 〉}_0$ order parameters. To find this composite-object effective potential we introduce a complete set of sources coupled to $\overline{\psi }\psi$ which allows a closed-form evaluation of the Legendre transform to conjugate fields. We illustrate this within the context of a truncated Schwinger-Dyson-equation approximation to massless QED for which we are able to solve the Schwinger-Dyson equation for the fermion propagator and the concomitant hierarchy of Bethe-Salpeter equations. We find that for the coupling constant beyond a critical value the normal vacuum becomes unstable and a chiral-breaking solution of the Schwinger-Dyson equation is found. However, the chiral-breaking solution corresponds to a saddle point of the effective potential and hence does not define a stable vacuum.