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

Abstract

The effective potential for the local composite operator $ϕ^{2}(x)$ in $λϕ^{4}$-theory is investigated at finite temperature in an approach based on path-integral linearisation of the $ϕ^4 $-interaction. At zero temperature, the perturbative vacuum is unstable, because a non-trivial phase with a scalar condensate $\langle ϕ^{2} \rangle _{0}$ has lower effective action. Due to field renormalisation, $\langle λϕ^{2} \rangle _{0}$ 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 $\langleλϕ^{2} \rangle _{0} / T_{c}^{2} \approx 62$ turns out to be a universal constant.