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In a recent work the Green's functions of the $\mathcal{PT}$-symmetric scalartheory $g \phi^{2}(i\phi)^\epsilon$ were calculated at the first order of thelogarithmic expansion, i.e. at first order in $\epsilon$, and it was proposedto use this expansion in powers of $\epsilon$ to implement a systematicrenormalization of the theory. Using techniques that we recently developed forthe analysis of an ordinary (hermitian) scalar theory, in the present work wecalculate the Green's functions at $O(\epsilon^2)$, pushing also the analysisto higher orders. We find that, at each finite order in $\epsilon$, the theoryis non-interacting for any dimension $d \geq 2$. We then conclude that by nomeans this expansion can be used for a systematic renormalization of thetheory. We are then lead to consider resummations, and we start with theleading contributions. Unfortunately, the results are quite poor. Specifying tothe physically relevant $i g \phi^3$ model, we show that this resummationsimply gives the trivial lowest order results of the weak-coupling expansion.We successively resum subleading diagrams, but again the results are ratherpoor. 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 addthat 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 wecommunicated to C.M. Bender in December 2019, are confirmed in a recentpreprint (e-Print:2103.07577) of C.M. Bender and collaborators.

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