  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.
