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We studied the Dynamical Symmetry Breaking (DSB) mechanism in asupersymmetric Chern-Simons theory in $\left(2+1\right)$ dimensions coupled to$N$ matter superfields in the superfield formalism. For this purpose, wedeveloped 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-fixingparameter that is introduced in the quantization process. The possibledependence of the effective potential on the gauge parameter have been studiedin the context of quantum field theory. We developed the formalism of theNielsen identities in the superfield language, which is the appropriateformalism to study DSB when the effective potential is gauge dependent. We alsodiscuss how to calculate the effective superpotential via the RenormalizationGroup Equation (RGE) from the knowledge of the renormalization group functionsof the theory, i.e., $\beta$ functions and anomalous dimensions $\gamma$. Weperform a detailed calculation of these functions at two loops, finding thatthese do not depend on $\alpha$, and therefore, by using the RGE, we calculatethe effective superpotencial $K_{\mathrm{eff}}$, showing that it is alsoindependent of $\alpha$. Then we discuss the improvement of the calculation of$K_{\mathrm{eff}}$ by summing up leading logarithms, and we compare thisimprovement with the one obtained in the non supersymmetric version of themodel. Finally, we study the DSB finding that it is operational for allreasonable values of the free parameters, while the improvement obtained fromthe RGE in general only produces a small quantitative correction in theresults, instead of the more dramatic qualitative change found in nonsupersymmetric models.

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