Asymptotic behavior and critical coupling in the scalar Yukawa model from Schwinger–Dyson equations

Abstract

A sequence of n-particle approximations for the system of Schwinger–Dyson equations is investigated in the model of a complex scalar field ϕ and a real scalar field χ with the interaction gϕ*ϕχ. In the first non-trivial two-particle approximation, the system is reduced to a system of two nonlinear integral equations for propagators. The study of this system shows that for equal masses a critical coupling constant exists, which separates the weak- and strong-coupling regions with the different asymptotic behavior for deep Euclidean momenta. In the weak-coupling region (), the propagators are asymptotically free, which corresponds to the wide-spread opinion about the dominance of perturbation theory for this model. At the critical point, the asymptotics of propagators are ∼1/p. In the strong-coupling region (), the propagators are asymptotically constant, which corresponds to the ultra-local limit. For unequal masses, the critical point transforms into a segment of values, in which there are no solutions with a self-consistent ultraviolet behavior without Landau singularities.