Non-Trivial Phase Structure of $\phi ^{4}$-Theory at Finite Temperature

The effective potential for the local composite operator $\phi^{2}(x)$ in $\lambda \phi^{4}$-theory is investigated at finite temperature in an approach based on path-integral linearisation of the $\phi^4$-interaction. At zero temperature, the perturbative vacuum is unstable, because a non-trivial phase with a scalar condensate $\langle \phi ^{2} \rangle _{0}$ has lower effective action. Due to field renormalisation, $\langle \lambda \phi ^{2} \rangle _{0}$ is renormalisation group invariant and leads to the correct scale anomaly. At a critical temperature $T_{c}$ the non-perturbative phase becomes meta-stable implying a first order phase-transition to the perturbative phase. The ratio $\langle\lambda \phi^{2} \rangle _{0} / T_{c}^{2} \approx 62$ turns out to be a universal constant.