Asymptotic behavior and critical coupling in the scalar Yukawa model from Schwinger-Dyson equations
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 $g^2_c$ exists, which separates the weak- and strong-coupling regions with the different asymptotic behavior for deep Euclidean momenta. In the weak-coupling region ($g^2 g^2_c$), 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.