Study of the non-Hermitian PT-symmetric g phi(2)(i phi)(epsilon) theory: Analysis of all orders in epsilon and resummations

In a recent work Bender et al. [Phys. Rev. D 98, 125003 (2018)] proposed to study non-Hermitian PT-symmetric quantum field theories within the framework of the logarithmic expansion. For a single component scalar field theory, the PT-symmetric interaction term is written as g phi(2)(phi)(epsilon) (generic values of epsilon). The logarithmic expansion is obtained by considering e as a small parameter, and expanding (i phi)(epsilon) in powers of epsilon. Approximations to Green's functions G(n) are obtained by truncating the interaction at finite orders in epsilon, and the first order contribution was calculated by Bender et al. The value of epsilon that corresponds to the original theory is then replaced in the final results. It was also suggested that this expansion could be used to implement a systematic renormalization of the theory order by order in epsilon. Using techniques that we developed recently for the analysis of a Hermitian scalar theory, in the present work we calculate Green's functions at O(epsilon(2)), pushing also the analysis to higher orders. We find that, at each finite order in epsilon, the theory is noninteracting for any dimension d >= 2. Therefore, differently from the weak-coupling and the (h) over bar expansions, a systematic renormalization of the theory order by order in epsilon cannot be realized. Actually, as long as we limit ourselves to finite orders in epsilon, it is not even possible to define the theory. We are then lead to consider resummations, and we start by resumming the ultraviolet leading diagrams. Unfortunately, the results are quite poor, as these resummations simply give the lowest order results of the weak-coupling expansion. We also consider the resummation of subleading terms, but the outcome does not improve much: we get nothing more than the O(g(2)) weak-coupling results.