Goldstone Bosons as Bound States in the Quark-Gluon Model

We examine the implications of a Nambu-Goldstone realization of chiral symmetry in the quark-gluon model. The context of this examination is that of a renormalizable, finite theory, so eigenvalue conditions are assumed to be satisfied. We discuss the solutions to the Schwinger-Dyson gap equation for the fermion self-energy $\Sigma \left(p^2\right)$ that exhibit spontaneous breaking of the vacuum symmetry. In the leading-order Bethe-Salpeter approximation the boundary conditions to the homogeneous, linear integral equations stipulate the vacuum symmetry. It is shown how the Goldstone bosons emerge as bound states, as suggested by Nambu and Jona-Lasinio. We also examine the Goldstone alternative in the Bethe-Salpeter equation for fermion-fermion scattering. Explicit symmetry breaking is introduced by additional Abelian vector gluons coupling to hypercharge and isospin besides baryon number. The eigenvalue condition for the fine-structure constant is consequently model-dependent but takes a simple form. We also consider the influence of explicit symmetry breaking on the ground-state mesons and indicate how the solutions to the eigenvalue problem regulate the structure of symmetry breaking.