Effective potential for PT-symmetric quantum field theories

Recently, a class of PT-invariant scalar quantum field theories described b y the non-Hermitian Langrangian L = 1/2(partial derivative phi)(2) + g phi( 2)(i phi)(epsilon) was studied. It was found that there are two regions of epsilon. For epsilon < 0 the PT-invariant of the Langrangian is spontaneous ly broken, and as a consequence, all but the lowest-lying energy levels are complex. For epsilon greater than or equal to 0 the PT-invariance of the L angrangian is unbroken, and the entire energy spectrum is real and positive . The subtle transition at epsilon = 0 is not well understood. In this pape r we initiate an investigation of this transition by carrying out a detaile d numerical study of the effective potential V-eff(phi(o)) in zero-dimensio nal spacetime. Although this numerical work reveals some differences betwee n the epsilon < 0 and the epsilon > 0 regimes, we cannot yet see convincing evidence of the transition at epsilon = 0 in the structure of the effectiv e potential for PT-symmetric quantum filed theories.