Asymptotic behavior in a model with Yukawa interaction from Schwinger-Dyson equations

A system of Schwinger-Dyson equations for pseudoscalar four-dimensional Yukawa model in the two-particle approximation is investigated. The simplest iterative solution of the system corresponds to the mean-field approximation (or, equivalently, to the leading order of 1/N-expansion) and includes a non-physical Landau pole in deep-Euclidean region for the pseudoscalar propagator $\Delta$. It is argued, however, that a full solution may be free from non-physical singularities and has the self-consistent asymptotic behavior $p^2_e\Delta\simeq C\,\log^{-4/5}\frac{p^2_e}{M^2}$. An approximate solution confirms the positivity of $C$ and the absence of Landau pole.