Considering disordered electron systems we suggest a scheme that allows us to include an electron-electron interaction into a supermatrix $\sigma$-model. The method is based on replacing the initial model of interacting electrons by a fully supersymmetric model. Although this replacement is not exact, it is a good approximation for a weak short range interaction and arbitrary disorder. The replacement makes the averaging over disorder and further manipulations straightforward and we come to a supermatrix $\sigma$-model containing an interaction term. The structure of the model is rather similar to the replica one, although the interaction term has a different form. We study the model making perturbation theory and renormalization group calculations. We check the renormalizability of the model in the first loop approximation and in the first order in the interaction. In this limit we reproduce the renormalization group equations known from earlier works. We hope that the new supermatrix $\sigma$-model may become a new tool for nonperturbative calculations for disordered systems with interaction.

• Received 21 September 2004

DOI:https://doi.org/10.1103/PhysRevB.71.134203