Validity of the effective potential method to study PT-symmetric field theories

Abstract

We calculate the one loop effective potential for the class (-(iϕ)α) of PT-symmetric and non-Hermitian field theories in 0+1 space-time dimensions. To test the method, we showed that for the massless Hermitian ϕ4 theory, the method gives the exact power law behavior known from the literature. We show that this order of calculations goes beyond the truncation of the Schwinger-Dyson equations at the two-point green functions applied to the PT-symmetric (-ϕ4) theory in the literature. We found that the effective potential calculation represents good approximations of the vacuum energies of the class (-(iϕ)α) compared to the numerical results. For the vacuum condensate, the method gives also accurate results for the absolute values but gives both positive as well as negative imaginary condensates for even α which again agrees with the prediction of the Schwinger-Dyson equations. Unlike other methods, the effective potential can be directly extended to higher dimensions as it offers a way to implement the PT-symmetric boundary conditions as well as there exist well known methods to regularize the theory at higher dimensions.