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.
Highlights
The non-Hermitian PT -symmetric scalar quantum field theory gφ2ðiφÞε was recently studied within the framework of the logarithmic expansion [1], that is an expansion in powers of ε, and it was suggested to use this expansion to implement a sensible definition of the theory, together with its systematic renormalization order by order in ε
In the present work we study the non-Hermitian PT symmetric gφ2ðiφÞε quantum field theory in d dimensions, and apply the logarithmic expansion to the calculation of Green’s functions Gn
The first order of this expansion was considered in[1], where it was suggested that such an expansion could be used to implement a systematic renormalization of the theory, order by order in ε
Summary
The non-Hermitian PT -symmetric scalar quantum field theory gφ2ðiφÞε was recently studied within the framework of the logarithmic expansion [1], that is an expansion in powers of ε, and it was suggested to use this expansion to implement a sensible definition of the theory, together with its systematic renormalization order by order in ε. In the case of a quantum field theory, the search for Stokes wedges for the integration over infinitely many field variables is an insurmountable task [1], and the restriction to real values of φ is of great help for the calculation of Green’s functions. Such a restriction is implemented by considering small values of ε. The goal of the present work is to study Green’s functions of the nonHermitian PT -symmetric gφ2ðiφÞε theory at higher orders in ε To perform this analysis, we will use techniques that we developed in a previous paper of ours [14], where we studied ordinary, i.e. Hermitian, scalar theories.
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