Effective potential for scalar QED in (2 + 1) dimensions

We calculate the renormalized one loop approximation to the effective potential for scalar electrodynamics in (2 + 1) dimensions. We study its gauge dependence and show that it satisfies the renormalization group equation since it is independent on any renormalization scale. This relies on the fact that we do not need to improve the effective potential in (2 + 1) dimensions.