Existence of a phase transition in the(φ4)3quantum field theory

We show that a properly renormalized ${\left({\phi }^4\right)}_3$ field theory undergoes a phase transition from a vacuum with broken symmetry to a vacuum with manifest symmetry as the ${\phi }^4$ coupling strength is increased with the mass parameter held fixed. The same behavior is exhibited by the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ model with ${\left({\overrightarrow{\phi }}^2\right)}^2$ coupling in three space-time dimensions.