A study of Schwinger-Dyson Equations for Yukawa and Wess-Zumino Models

TL;DR

This paper investigates Schwinger-Dyson equations in Yukawa and Wess-Zumino models, revealing a critical coupling for dynamical mass generation in Yukawa models and no mass in Wess-Zumino models, consistent with non-renormalization theorems.

Contribution

It provides a detailed analysis of dynamical mass generation and wavefunction renormalization in these models, extending beyond simple approximations and discussing supersymmetry preservation.

Findings

Fermions acquire dynamical mass above a critical coupling in Yukawa models.

No dynamical mass for fermions in Wess-Zumino models due to cancellations.

Wavefunction renormalization is analytically and numerically evaluated near critical coupling.

Abstract

We study Schwinger-Dyson equation for fermions in Yukawa and Wess-Zumino models, in terms of dynamical mass generation and the wavefunction renormalization function. In the Yukawa model with $\gamma_5$-type interaction between scalars and fermions, we find a critical coupling in the quenched approximation above which fermions acquire dynamical mass. This is shown to be true beyond the bare 3-point vertex approximation. In the Wess-Zumino model, there is a neat cancellation of terms leading to no dynamical mass for fermions. We comment on the conditions under which these results are general beyond the rainbow approximation and also on the ones under which supersymmetry is preserved and the scalars as well do not acquire mass. The results are in accordance with the non-renormalization theorem at least to order $\alpha$ in perturbation theory. In both the models, we also evaluate the wavefunction renormalization function, analytically in the neighbourhood of the critical coupling and numerically, away from it.