Spontaneous Breaking of Chiral Symmetry in a Vector-Gluon Model

Solutions of the self-consistent equation for fermion propagator in a vector-gluon model are fully examined. The equation is characterized by a set of parameters, i.e., the coupling constant g, the bare mass of the fermion m0 and the cutoff Λ. It is proved that with a suitable gauge chosen, the equation without cutoff has solutions only in the case m0 = 0. It is then shown that, if g2/4π< (16/33)2π, the number of the solution is infinity of continuum. This situation does not come from the freedom of fixing the mass scale since the gluon mass is chosen to be non-vanishing. When the cutoff is introduced, the equation has a unique solution for g2/4π< π/4. In this case, however, β turns out to be identically zero if we put m0 = 0, which means that any “superconducting” solutions do not exist for such a value of g irrespective of cutoff momentum; that is, there are no Nambu-Goldstone bosons. When g2 is large enough, such a “superconducting” solution does exist in the model in which the vector part of the inverse fermion propagator is identically set equal to unity. Furthermore the existence of many “superconducting” solutions is inferred in this model. It is also found that in the region g2/4π > 8π, the “normal-state” solution for the equation without cutoff, even if it existed, should necessarily have an unphysical singularity. This fact implies that the “normal-state” solution becomes unstable for a sufficiently large value of g2.