Original study of the $g\phi^2(i\phi)^{\epsilon}$ theory. Analysis of all orders in $\epsilon$ and resummations

TL;DR

This study investigates the $g\,\phi^2(i\phi)^\epsilon$ theory by calculating Green's functions at various orders in epsilon, finding it non-interacting at finite orders, and demonstrating the limitations of resummation techniques for this model.

Contribution

The paper provides a detailed analysis of the all-orders behavior of the $g\phi^2(i\phi)^\epsilon$ theory and critically assesses the effectiveness of epsilon-expansion and resummation methods.

Findings

Theory is non-interacting at finite epsilon orders for all dimensions d ≥ 2.

Epsilon expansion fails as a systematic renormalization method for the theory.

Resummation approaches yield trivial or poor results, questioning their applicability.

Abstract

In a recent work the Green's functions of the $\mathcal{PT}$-symmetric scalar theory $g \phi^{2}(i\phi)^\epsilon$ were calculated at the first order of the logarithmic expansion, i.e. at first order in $\epsilon$, and it was proposed to use this expansion in powers of $\epsilon$ to implement a systematic renormalization of the theory. Using techniques that we recently developed for the analysis of an ordinary (hermitian) scalar theory, in the present work we calculate the 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 non-interacting for any dimension $d \geq 2$. We then conclude that by no means this expansion can be used for a systematic renormalization of the theory. We are then lead to consider resummations, and we start with the leading contributions. Unfortunately, the results are quite poor. Specifying to the physically relevant $i g \phi^3$ model, we show that this resummation simply gives the trivial lowest order results of the weak-coupling expansion. We successively resum subleading diagrams, but again the results are rather poor. All this casts serious doubts on the possibility of studying the theory $g \phi^{2}(i\phi)^\epsilon$ with the help of such an expansion. We finally add that the findings presented in this work were obtained by us some time ago (December 2019), and we are delighted to see that these results, that we communicated to C.M. Bender in December 2019, are confirmed in a recent preprint (e-Print:2103.07577) of C.M. Bender and collaborators.