Spontaneous symmetry breaking at two loop in 3-d massless scalar electrodynamics

In three-dimensional Maxwell-Chern-Simons massless scalar electrodynamics with φ6 coupling, the U(1) symmetry is spontaneously broken at two loop order regardless of the presence or absence of the Maxwell term. Dimensional transmutation takes place in pure Chern-Simons scalar electrodynamics. The beta function for the φ6 coupling is independent of gauge couplings.