Effective Potential for $\mathcal{P}\mathcal{T}$-Symmetric Quantum Field Theories

Abstract

Recently, a class of $\mathcal{P}\mathcal{T}$-invariant scalar quantum field theories described by the non-Hermitian Lagrangian $\mathcal{L}$=$\frac{1}{2}$(∂ϕ)²+gϕ²(iϕ)^ε was studied. It was found that there are two regions of ε. For ε<0 the $\mathcal{P}\mathcal{T}$-invariance of the Lagrangian is spontaneously broken, and as a consequence, all but the lowest-lying energy levels are complex. For ε≥0 the $\mathcal{P}\mathcal{T}$-invariance of the Lagrangian is unbroken, and the entire energy spectrum is real and positive. The subtle transition at ε=0 is not well understood. In this paper we initiate an investigation of this transition by carrying out a detailed numerical study of the effective potential V _eff (ϕ_c) in zero-dimensional spacetime. Although this numerical work reveals some differences between the ε<0 and the ε>0 regimes, we cannot yet see convincing evidence of the transition at ε=0 in the structure of the effective potential for $\mathcal{P}\mathcal{T}$-symmetric quantum field theories.