A comparative analysis of a model of complex scalar field
φ and real scalar field
χ with interaction
for the real and purely imaginary values of coupling
g in perturbative and non-perturbative regions is provided. In contrast to the usual Hermitian version (real
g), which is asymptotically free and energetically unstable, the non-Hermitian
PT-symmetric theory (imaginary
g) is energetically stable and not asymptotically free. The non-perturbative approach based on Schwinger-Dyson equations reveals new interesting feature of the non-Hermitian model. While in the Hermitian version of theory the phion propagator has the non-physical non-isolated singularity in the Euclidean region of momenta, the non-Hermitian theory substantially free of this drawback, as the singularity moves in the pseudo-Euclidean region.