Phase transitions in vector-gluon models: A solution to the U(3) problem

Abstract

We have examined the realizations of chiral symmetry in the Abelian vector-gluon model, in the absence of gluon self-energy. The chiral symmetry is broken spontaneously in the vacuum. A vacuum stability condition on the gluon coupling, $g$, and the gluon-fermion mass ratio $\frac{\ensuremath{\mu}}{m}$ of the form ${{f}_{\ensuremath{\pi}}}^{2}({g}^{2},\frac{{\ensuremath{\mu}}^{2}}{{m}^{2}})>0$ is obtained as a necessary condition for a Nambu-Goldstone realization with bound-state Goldstone bosons. Here ${f}_{\ensuremath{\pi}}$ is the decay constant of the bound-state Goldstone boson. It is shown that for those channels that do not communicate with pairs of vector gluons this condition is satisfied in the weak-coupling limit, ${g}^{2}\ensuremath{\rightarrow}0$. For axialvector currents that satisfy anomalous Ward identities and communicate with pairs of gluons we show that the renormalized integral equations for ${{f}_{\ensuremath{\pi}}}^{2}$ do not possess solutions. The failure to find a Goldstone solution for these channels is associated with the fact that these axialvector currents have an anomalous dimension greater than three. Instead of Goldstone bosons one finds that axial-vector-current conservation breaks down and the symmetry is broken explicitly. Hence, there is no Goldstone boson associated with "axial baryonic charge." This also answers in the negative the old question of whether electrodynamics can support a Goldstone mode. We also calculate ${g}_{A}\ensuremath{-}1$ to order ${g}^{2}$ and discuss phase transitions between the Wigner-Weyl mode, the Goldstone mode, and the mode in which the symmetry is explicitly broken. We also speculate on the implications of this work for gauge theories of the strong interactions.