Spontaneous Symmetry Breaking at Two Loop in 3-d Massless Scalar Electrodynamics

In three dimensional Maxwell-Chern-Simons massless scalar electrodynamics with $ϕ^6$ 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 $ϕ^6$ coupling is independent of gauge couplings.

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