Effective Superpotential and the Renormalization Group Equation in a Supersymmetric Chern-Simons-Matter Model in the Superfield Formalism

TL;DR

This paper investigates dynamical symmetry breaking in a supersymmetric Chern-Simons-matter model using superfield formalism, developing a gauge-independent effective superpotential calculation via the renormalization group equation, and analyzing the impact of quantum corrections.

Contribution

It introduces a method to compute the effective superpotential in supersymmetric Chern-Simons theories, ensuring gauge independence and applying RGE techniques for improved calculations.

Findings

DSB occurs for all reasonable parameters

Effective superpotential is gauge independent at two loops

RGE improvement yields small quantitative corrections

Abstract

We studied the Dynamical Symmetry Breaking (DSB) mechanism in a supersymmetric Chern-Simons theory in $\left(2+1\right)$ dimensions coupled to $N$ matter superfields in the superfield formalism. For this purpose, we developed a mechanism to calculate the effective superpotencial $K_{\mathrm{eff}}\left(\sigma_{\mathrm{cl}},\alpha\right)$, where $\sigma_{\mathrm{cl}}$ is a background superfield, and $\alpha$ a gauge-fixing parameter that is introduced in the quantization process. The possible dependence of the effective potential on the gauge parameter have been studied in the context of quantum field theory. We developed the formalism of the Nielsen identities in the superfield language, which is the appropriate formalism to study DSB when the effective potential is gauge dependent. We also discuss how to calculate the effective superpotential via the Renormalization Group Equation (RGE) from the knowledge of the renormalization group functions of the theory, i.e., $\beta$ functions and anomalous dimensions $\gamma$. We perform a detailed calculation of these functions at two loops, finding that these do not depend on $\alpha$, and therefore, by using the RGE, we calculate the effective superpotencial $K_{\mathrm{eff}}$, showing that it is also independent of $\alpha$. Then we discuss the improvement of the calculation of $K_{\mathrm{eff}}$ by summing up leading logarithms, and we compare this improvement with the one obtained in the non supersymmetric version of the model. Finally, we study the DSB finding that it is operational for all reasonable values of the free parameters, while the improvement obtained from the RGE in general only produces a small quantitative correction in the results, instead of the more dramatic qualitative change found in non supersymmetric models.