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

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.