On the phase structure of vector-matrix scalar model in four dimensions

The leading-order equations of the $$1/N$$ – expansion for a vector-matrix model with interaction $$g\phi _a^*\phi _b\chi _{ab}$$ in four dimensions are investigated. This investigation shows a change of the asymptotic behavior in the deep Euclidean region in a vicinity of a certain critical value of the coupling constant. For small values of the coupling the phion propagator behaves as free. In the strong-coupling region the asymptotic behavior drastically changes – the propagator in the deep Euclidean region tend to some constant limit. The phion propagator in the coordinate space has a characteristic shell structure. At the critical value of coupling that separates the weak and strong coupling regions, the asymptotic behavior of the phion propagator is a medium among the free behavior and the constant-type behavior in strong-coupling region. The equation for a vertex with zero transfer is also investigated. The asymptotic behavior of the solutions shows the finiteness of the charge renormalization constant. In the strong-coupling region, the solution for the vertex has the same shell structure in coordinate space as the phion propagator. An analogy between the phase transition in this model and the re-arrangement of the physical vacuum in the supercritical external field due to the “fall-on-the-center” phenomenon is discussed