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

TL;DR

This paper investigates the asymptotic behavior and critical coupling in a scalar Yukawa model using Schwinger-Dyson equations, revealing a phase transition at a critical coupling that separates different asymptotic regimes of the propagators.

Contribution

It introduces a two-particle approximation analysis of the scalar Yukawa model, identifying a critical coupling and characterizing the asymptotic behaviors of propagators in different regimes.

Findings

Existence of a critical coupling constant $g_c^2$ separating weak and strong coupling behaviors.

In the weak-coupling region, propagators are asymptotically free with $1/p$ behavior.

In the strong-coupling region, propagators become asymptotically constant, indicating an ultra-local limit.

Abstract

A sequence of $n$-particle approximations for the system of Schwinger-Dyson equations is investigated in the model of a complex scalar field $\phi$ and a real scalar field $\chi$ with the interaction $g\phi^*\phi\chi$. 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 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 $\sim 1/p$. In the strong 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.