Abstract
The leading-order equations of the 1/N – expansion for a vector-matrix model with interaction gphi _a^*phi _bchi _{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.
Highlights
The solution of the equation for the phion propagator in the leading order of the 1/N – expansion 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 propagator behaves as free, which is consistent with the wide-spread opinion about the dominance of perturbation theory for this super-renormalizable model
In the strong-coupling region, the asymptotic behavior changes dramatically – the propagator in the deep Euclidean region tend to some constant limit
Summary
The solution of the equation for the phion propagator in the leading order of the 1/N – expansion shows a change of the asymptotic behavior in the deep Euclidean region in a vicinity of a certain critical value of the coupling constant. In the strong-coupling region, the asymptotic behavior changes dramatically – the propagator in the deep Euclidean region tend to some constant limit. In the present paper we obtain solutions of the nonlinear equation for the phion propagator in the linearized approximation, correctly describing both the asymptotic behavior and the behavior at small momenta, and investigate the asymptotic behavior of the vertex for zero transfer momentum. In the strong-coupling region, the solution for the vertex in coordinate space has the same shell structure as the propagator. An analogy is made between the phase transition in the model under consideration and the re-arrangement of the physical vacuum in the supercritical external field
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